the coupled theory of mixtures in geomechanics with

The Coupled Theory of Mixtures in Geomechanics with Applications

George Z. Voyiadjis and Chung R. Song

The Coupled Theory ofMixtures in Geomechanicswith Applications

ABC

Dr. George Z. VoyiadjisDepartment of Civiland Environmental EngineeringLouisiana State UniversityCEBA Building, Room 3508-BBaton Rouge, LA 70803-6405U.S.A.E-mail: [email protected]

Dr. Chung R. Song218 CarrierDepartment of Civil EngineeringUniversity of MississippiUniversity, MS 38677U.S.A.E-mail: [email protected]

Library of Congress Control Number: 2005936354

ISBN-10 3-540-25130-8 Springer Berlin Heidelberg New YorkISBN-13 978-3-540-25130-9 Springer Berlin Heidelberg New York

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Preface

Geomaterials consist of a mixture of solid particles and void space that maybe filled with fluid and gas. The solid particles may be different in sizes,shapes, and behavior; and the pore liquid may have various physical andchemical properties. Hence, physical, chemical or electrical interaction be-tween the solid particles and pore fluid or gas may take place. Therefore,the geomaterials in general must be considered a mixture or a multiphasematerial whose state is described by physical quantities in each phase. Thestresses carried by the solid skeleton are typically termed “effective stress”while the stresses carried by the pore liquid are termed “pore pressure.” Thesummation of the effective stress and pore pressure is termed “total stress”(Terzaghi, 1943). For a free drainage condition or completely undrained con-dition, the pore pressure change is zero or depends only on the initial stresscondition; it does not depend on the skeleton response to external forces.Therefore, a single phase description of soil behavior is adequate. For anintermediate condition, however, some flow (pore pressure leak) may takeplace while the force is applied and the skeleton is under deformation. Dueto the leak of pore pressure, the pore pressure changes with time, and theeffective stress changes and the skeleton deforms with time accordingly. Thesolution of this intermediate condition, therefore, requires a multi-phase con-tinuum formulations that may address the interaction of solid skeleton andpore liquid interaction.

The pore pressure leak(flow) is related to the hydraulic conductivity, itimplies that the hydraulic conductivity affects the behavior of geomaterials.Conversely, when the behavior of a soil is known, the hydraulic characteristicsof the soil can be known from the hydro-mechanical analysis – the so-called“coupled theory of mixtures”.

Biot (1955, 1978) was one of the first scientists to develop a coupled the-ory for an elastic porous medium. However, experiments have shown thatthe stress-strain-strength behavior of the soil skeleton is strongly non-linear,anisotropic, and elasto-plastic. An extension of Biot’s theory into the non-linear, anisotropic range is therefore necessary in order to analyze the tran-sient response of soil deposits. This extension has acquired considerable im-portance in recent years because of the increased concern with the dynamicbehavior of saturated soil deposits and associated liquefaction of saturated

VIII Preface

sand deposits under seismic loading conditions. Such an extension of Biot’sformulation was proposed by Prevost (1980).

Prevost (1980)’s theory of mixture was further extended for the updatedLagrangian reference frame by Kiousis and Voyiadjis (1985), and Voyiadjisand Abu-Farsakh (1996), thus the application to the large strain behaviorbecomes possible.

This book addresses the coupled theory of mixtures for geo-materials. Italso presents the formulation and the numerical procedures using the coupledtheory of mixtures for geo-materials for the solution of a variety of problemsin geo-mechanics, including the applicability and use of the cone penetrom-eter (for evaluation of soil properties), soft soil tunneling, Implementation ofNano-mechanics to geo-materials, Application of Geo-acoustics, and so forth.

Baton Rouge George Z. VoyiadjisOxford Chung R. SongDecember 29, 2005

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Historical Review of the Theory of Mixtures

for Geo-materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 The Early Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 The Classical Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 The Modern Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Flow in Geo-materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Nature of Geo-materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Flow of Water in Geo-Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Transient or Non-steady Flow . . . . . . . . . . . . . . . . . . . . . . 13

3 Coupled Theory of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 Traditional Theory of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Terzaghi’s Consolidation Theory . . . . . . . . . . . . . . . . . . . 153.1.2 Biot’s Consolidation Theory . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Modern Theory of Mixtures for Finite Strain . . . . . . . . . . . . . . . 173.2.1 Balance of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Coupling of Mass Balance Equation and Darcy’s Law . 20

4 Coupling Yield Criteria and Micro-mechanicswith the Theory of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1 Soil Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Anisotropic Soil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 Anisotropic Modified Cam Clay Model . . . . . . . . . . . . . . 334.3 Elasto-Plastic Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . 364.4 Micro-Mechanical Considerations/Bridging Different Length

Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4.1 RVE Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4.2 Characteristic Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4.3 Bridging Different Length Scales . . . . . . . . . . . . . . . . . . . 45

4.5 Micro-mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.5.1 Back Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.5.2 Rotation of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

X Contents

4.5.3 Grain Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.5.4 Rate Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5.5 Damage of Solid Grains . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.5.6 Summary of Micro-mechanisms . . . . . . . . . . . . . . . . . . . . 74

4.6 Equation of Equilibrium of the Externaland Internal Forces in an Updated Lagrangian ReferenceFrame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Finite Element Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.1 Updated Lagrangian Reference Sheme . . . . . . . . . . . . . . . . . . . . . 835.2 Finite Element Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3 Remeshing and Return to Yield Surface . . . . . . . . . . . . . . . . . . . 89

6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.1 Piezocone Penetration Test (PCPT) . . . . . . . . . . . . . . . . . . . . . . 91

6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.1.2 Current Practice of Determining Hydraulic Properties

from the Piezocone Penetrometer . . . . . . . . . . . . . . . . . . . 936.1.3 New Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.1.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2 (Shield) Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2.2 Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.2.3 Finite Element Numerical Simulation . . . . . . . . . . . . . . . 1226.2.4 Remeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.2.5 Mapping Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.2.6 Modeling of Interface Friction . . . . . . . . . . . . . . . . . . . . . . 1346.2.7 Case Study of N-2 Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.3 Estimation of Hydraulic Conductivityusing Acoustic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.3.2 Basics of Wave Propagation in Saturated Media . . . . . . 1516.3.3 Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.1 Nano-mechanics for Geotechnical Engineering . . . . . . . . . . . . . . 161

7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.1.2 Brief History of Nano-mechanics . . . . . . . . . . . . . . . . . . . 1627.1.3 Nano-mechanics as a General Platform

for Studying Detailed Behavior of Geo-materials . . . . . . 1637.1.4 Nano-mechanics as a Tool to Study Macro-level

Material Properties Through Continuumization . . . . . . 1677.1.5 Space Science Application . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.2 Coupled Behavior of Micro-Mechanisms . . . . . . . . . . . . . . . . . . . 172

Contents XI

7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1727.2.2 Simplification of Equations Incorporating

the Physical Behavior of Soils . . . . . . . . . . . . . . . . . . . . . . 1737.2.3 Rate Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.2.4 (Strain) Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1797.2.5 Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.2.6 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827.2.7 Anisotropy and Plastic Spin . . . . . . . . . . . . . . . . . . . . . . . 1847.2.8 Experimental Verification of Coupled Pore Pressure

Around a Cone Tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887.2.9 Back Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.2.10 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Appendix: Fortran Codes of CS-S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

1 Introduction

Geomaterials consist of assemblages of particles with different sizes andshapes that form a skeleton (porous matrix) whose voids are filled with wateror other liquids and air or gas. The word ‘soil’ therefore implies a mixtureof assorted mineral grains with various fluids and must for that reason beconsidered a multi phase material.

The study of this type of material is of great importance in applied civilengineering, particularly in geotechnical engineering. The analysis of the re-sponse of multi phase materials (i.e. soils or porous media) has a multitudeof important applications: analyzing the settlement of underlying soil de-posits, determining the dissipation of excess pore water pressures resultingfrom foundation loading, designing foundations for vibrating machines, ana-lyzing the vulnerability of offshore structures under wave loading, studyingthe propagation of earthquake impulses in geologic materials, and measuringthe catastrophic loss of strength of a soil associated with the increase of porewater pressure (the phenomenon known as “liquefaction”). The importanceof understanding these phenomena has been recognized since the early 1940’s,when the colossal devastation caused by major earthquakes heightened theneed to better understand the behavior of soils in such events in order tomitigate higher losses that could be avoided by adequate engineering design.

During the last three decades, significant progress has been made in un-derstanding the behavior of a porous matrix interacting with one or morefluids. In the area of geotechnical engineering, three approaches may be iden-tified as a result of this process: (1) field observations prior, during, and afterearthquakes; (2) laboratory experimentation; and (3) theoretical/numericalstudies. While many researchers have focused on what could be observed fromlaboratory experiments and field evidence, others have emphasized their workin the development of appropriate theoretical procedures to describe theseobservations. Theoretical studies have yielded three different philosophies:decoupled methods (or the total stress approach), indirectly coupled meth-ods (the quasi-effective stress approach), and fully coupled methods (or theeffective stress approach).

This book focuses on the fully coupled methods of analysis, which arebased on the theory of mixtures. The premise of these theories is that amixture may be viewed as a superposition of “n” individual continua, each

2 1 Introduction

following its own motion. It is also assumed that at any time t, the entire spacein the mixture is occupied simultaneously by several different particles, onefrom each constituent. In addition, each component of a mixture is regardedas an open system for which local balance relations are postulated. In thiscondition, their production densities differ from zero because of the fact thatthe interaction between the individual constituents must be incorporated.Furthermore, because it is natural to expect that the mixture as a whole willact as one-component single body, restrictive conditions must be postulatedin terms of conservation principles.

In the light of these theories, the behavior of porous media cannot bedescribed directly because in general it is not possible to know a priori whichspatial position will be occupied by which particular constituent. For thisreason, substitute continuum models must be used to replace the particulatestructure, where a single material component is assumed to occupy the totalregion. The use of volume fraction theories constitutes the necessary connec-tion between global bulk average quantities and the local effective quantities.Within this framework, the problem generally leads to geometrical and phys-ical nonlinear relations. In order to obtain a simple and practical theory,further simplifications must be introduced. For this reason, a mixture com-posed of only two constituents, i.e. the porous matrix and a saturating fluid,frequently is considered. The different components are assumed to share acommon temperature with a vanishing temperature gradient. In addition,the material under consideration frequently assumed to be isotropic, and it isassumed that the compressibility of the solid particles is much smaller thanthe compressibility of the body as a whole. The deviator stress in the mediafluid is assumed to be negligible in comparison to that in the solids skeleton.Finally, the development of the constitutive equations is restricted to the caseof small strains initially: it is extended to the case of large strains later.

From this analysis, the differential equations that govern the motion of asoil mixture become coupled with the global mass balance equation, an opera-tion that leads to a set of coupled differential equations. To obtain a solution,the set of differential equations may be approximately be a weighted residualmethod such as Galerkin’s method, and the approximated integral may bediscretized using finite element techniques and robust temporal integrationmethods. This approach results in a full coupling of the pore pressure gener-ation/dissipation with the deformation of the soil skeleton. In this sense, theability that the selected constitutive model may have to predict permanentvolume changes during loading becomes a major factor in the global perfor-mance of the algorithm. The historical review in Sect. 1.1 is obtained fromthe reference by Arduino (1996) with the author’s permission. This section ismostly verbatim from this reference with minor updates on some additionalresearch works.

1.1 Historical Review of the Theory of Mixtures for Geo-materials 3

1.1 Historical Review of the Theory of Mixturesfor Geo-materials

Three major periods may be identified in the historical development of theporous media theory (De Boer, 1996): The early era (18th and 19th centuries),the classical era (1910–1960), and the modern era (1960-present). In the earlyera, the concept of volume fractions was stated, some fundamental laws werediscovered, and the mixture theory was founded. In the classical era, scientistsfirst attempted to clarify the mechanical interaction of liquids, gases, and rigidporous solids to explore the behavior of deformable saturated porous solids.In the modern era, researchers developed and continue to study theories ofimmiscible mixtures.

1.1.1 The Early Era

By 1794, R. Woltman had made the first contributions to the theory of porousmedia. Motivated by a competition at the Imperial Academy of Sciencesin St. Petersburg, Woltman developed a sophisticated earth pressure the-ory, introducing a concept for the calculation of failure conditions in soilmechanics – the angle of internal friction. Woltman had already separatedsoils into four types: sand, lime, clay, and compost-earth. At that time hepointed out that friction is a common property in all kinds of earth and thatthis friction effect differentiates soils from fluids. He stated that only by totalsaturation with water is the friction effect lost. Woltman called this state ofthe mixture “quicksand” or “mud”. In connection with his discussion of themechanical behavior of “mud”, Woltman spoke of a mixture and (surpris-ingly) introduced the concept of volume fractions. He was probably the firstscientist to formulate this concept.

After the development of the concept of volume fractions by Woltman, nosubstantial contributions to the theory of heterogeneously composed bodieswere published for a long time. Around the mid-nineteenth century, importantfindings in the theory of mixtures and in the theory of porous bodies weremade principally by Delesse, Fick, and Darcy (De Boer, 1996).

Delessian Law

With the concept of volume fractions in hand, scientists were able to treatheterogeneously composed bodies with existing continuum mechanical mod-els. However, it was also necessary to develop a corresponding concept forthe surface element on a saturated porous body. In 1848, Delesse made adecisive contribution. As a mining engineer, he was interested in determiningthe ratios of single minerals in valuable ores without destroying samples, andafter some theoretical considerations, he concluded that the surface rationsare equal to the volume fractions. Although this concept was, and must be,

4 1 Introduction

considered as a statistical necessity, it was extensively used in the followingdevelopments of the porous media theory.

Fick’s Law

The first attempts to develop a phenomenological theory of mixtures weremade by Fick (1855), who studied the problem of diffusion. Fick, who was aphysician, was inspired by hydro-diffusion through membranes. He remarkedthat this problem is not only important for organic life, but also for other im-portant physical processes. Following the development of the Fourier equationof heat propagation, Fick arrived at the differential equation of the diffusionstream which in the case of a constant cross-section takes the form;

∂y

∂t= −k

∂2y

∂x2(1.1)

where y is concentration, t is time, k is a constant which depends on thenature of the constituents, and x is a measure of height. Today, this relationis known as Fick’s second law of diffusion.

Darcy’s Law

Darcy (1856) observed in tests with natural sand both the proportionality ofthe total volume of running water through the sand and the loss of pressure.Although his investigations were purely of experimental nature, his resultsare essential for a continuum mechanical treatment of the motion of a liquidin a porous solid. In Darcy’s work, the interaction of different constituents ina multi-phase continuum was studied for the first time. Today, Darcy’s law istheoretically well founded with thermodynamic principles. However, Darcy’slaw becomes invalid for liquids at high velocities, and for gases at very lowand at very high velocities.

Although Delesse, Fick and Darcy discovered their fundamental laws moreor less heuristically rather than by development from the fundamental rela-tions of mechanics and thermodynamics, they established fundamental state-ments that have been extensively applied in the theories that have followed,such as the mixture and the porous media theories. At the end of the nine-teenth century, the development of a new branch of mechanics began that isof essential importance for the creation of a consistent porous media theory:the theory of mixtures. As previously stated, it was Fick who discovered thedifferential equation for the diffusion problem of liquids with different con-centrations. However, Maxwell was the first scientist who, starting from thebasic principles of mechanics, developed the hydrodynamic equations for gasmixtures (Stefan, 1871). His investigations finally led to the creation of thekinetic gas theory.

A decisive step to the continuum mechanical theory of mixtures was ac-complished by Stefan (1871):


If the true processes in a mixture should be calculated, it is not suffi-cient anymore, to consider the mixture as a uniform body as commonmechanics does; equations must be set up which should contain thecondition of equilibrium and the laws of motion for every individualconstituent in the mixture.

He then introduced the main assumption concerning the interaction forcesbetween constituents: “In a mixture, each particle of a gas, if it is in motion,suffers from each of the other gases a resistance which is proportional to thedensity of this gas and the relative velocity of both.”

Stefan subsequently formulated the equations of equilibrium for a mixtureof two gases and the balance of mass excluding any mass exchange. Follow-ing the derivation of the differential equations, Stefan discussed solutions forspecial initial conditions cases. In the eighth section of his valuable paper, hetreated the problem of the diffusion of a gas through a porous diaphragm.This study can be considered as the first time that the mixture theory re-stricted by the volume fractions concept was applied to a binary model withinthe framework of continuum mechanics.

1.1.2 The Classical Era

From the second to the fourth decades of the current century, decisive progresswas made towards creating a consistent porous media theory. There were twonotable steps in this progression. First, scientists discovered important me-chanical effects in a liquid saturated rigid porous solid. Second, in the 1920’sand 1930’s, the scientists first attempted to investigate saturated deformableporous solids. The discovery of fundamental mechanical effects in saturatedporous solids and the formulation of the first porous media theories are mainlydue to two distinguished professors at the Technische Hocheschule of Vienna:Paul Fillunger and Karl von Terzaghi.

From 1913 to 1934, these two professors described for the first time impor-tant physical phenomena in rigid liquid-saturated porous media, specificallywith reference to the effects of uplift, friction, capillarity, and effective stress.It was Fillunger (De Boer, 1996) who pioneered the porous media theory ofliquid-saturated porous solids, investigating the problem of uplift and frictionforces acting on heavy weight masonry dams. At the same time, von Terzaghiaddressed the theoretical problem of capillarity and formally established theconcept of effective stresses. The effective stress concept had already beenpresented at the beginning of the century, but definitive evidence was pro-vided by von Terzaghi and later by Fillunger, (Bjerrum et al., 1960; De Boerand Ehlers, 1990a,b; De Boer, 1996).

Although much success had been gained with the model of a saturatedrigid porous solid, the theory of porous media remained incomplete becausethe description of the deformation and the determination of the stress statein saturated deformable porous bodies were excluded from Terzaghi’s and

6 1 Introduction

Fillunger’s research. The first author to deal with the important problemsof fluid-filled deformable porous solids was von Terzaghi. He recognized thata water-filled and deformable soil body, despite being of great relevance forthe foundation of buildings, had still not been scientifically treated. In afamous paper presented to the Academy of Sciences in Vienna in June 1923,von Terzaghi showed the derivation of his consolidation theory. This theory,which brought him immediate fame, was later published in an internationallyacclaimed expanded study that is considered the first substantial book in soilmechanics (Terzaghi, 1943).

Subsequently, Fillunger also studied in detail the consolidation problemand published his results in a pamphlet called “Erdbaumechanik?” whereKarl Terzaghi’s work was strongly criticized (De Boer, 1996). He proceededfrom a two-phase system which he described with sound mechanical ax-ioms and principles. Certain statements of inexorable criticism and personaldefamatory attacks against von Terzaghi led to a personal controversy be-tween von Terzaghi and Fillunger known as the “Viennese Affair” whichended with the tragic suicide of Fillunger and his wife. (For more details seeBjerrum et al., 1960; De Boer, 1996).

Despite this tragic episode, the discovery of fundamental mechanical ef-fects in liquid-filled porous solids by these two Viennese professors in the firsthalf of this century represents a brilliant achievement in engineering. At thattime, thermodynamics, in the modern sense, was not yet developed and theconstitutive theory with the procedure to gain restrictions from the entropyinequality was completely unknown. Therefore it is not surprising that in thesequel of the discoveries, many errors and incorrect proofs appeared.

After the fundamental contributions of von Terzaghi and Fillunger, thetheory of porous media was further developed, in particular, by Maurice Biotwho followed, at the beginning of his career, von Terzaghi’s direction. In theearly 40’s Biot (1941) generalized von Terzaghi’s theory of consolidation byextending it to the three dimensional case and by establishing equations validfor any arbitrary load varied with time. In the following years, Biot gener-alized his theory to include properties of anisotropy, variable permeability,linear visco-elasticity, and the propagation of elastic waves in a fluid saturatedporous solid (Biot, 1941, 1955, 1956b). The three-dimensional theory of wavepropagation developed by Biot is rather intuitive (but consistent) and showsthe existence of two dilational and one rotational body waves. A chronolog-ical review of Biot’s work is discussed in his publications on the mechanicsof deformation and acoustic propagation in porous media (Biot, 1956a,c,d).The main disadvantage of Biot’s model, however, is that the correspondingtheory is not developed from the fundamental axioms and principles of me-chanics and thermodynamics. Thus, some derivations are complicated andobscure. Finally, Biot (1965) developed, within the framework of quasi-staticand isothermal deformations, a theory of finite deformations of porous media.


1.1.3 The Modern Era

Since the beginning of the 1960’s the study of porous media has advanced inseveral directions:

A. In the geotechnical field, the necessity of obtaining practical solutions ledto the development of simplified theories and empirical relations. Scien-tists developed experimental and sometimes incomplete formulations toacquire a better understanding of the soil-fluid interaction problem. Thiswork is important, for it sheds light on the physical behavior of saturatedsoils. With von Terzaghi’s consolidation theory in hand, the problem ofthe static (or quasi-static) analysis of saturated porous media found apractical solution that was applied to many “real life” problems. However,the colossal devastation caused by earthquakes required the study of thedynamic problem as well. Prior to 1975, dynamic analyses in geotechni-cal engineering were based on total stresses because of the deficiency inpractical models that could predict pore water pressures. A widely usedmethod was the Equivalent Linear Method (ELM), which provided anapproximate solution based on elastic soil stiffness and damping that arecompatible with induced strains in the soil. In this method, results fromlaboratory tests relate the damping ratio and the shear modulus to cyclicshear strain levels. A linear solution is based on initial values of the shearmodulus and the damping ratio. From the time variation of the shearstrain, an equivalent strain magnitude is estimated and used to obtainnew values of the shear modulus and damping ratio. A new solution iscalculated, and the procedure is repeated until convergence is achieved.The method became popular after Seed and co-workers and was imple-mented in computer applications such as SHAKE (Schanabel et al. 1972),QUAD (Idriss et al., 1983), and FLUSH (Lysmer et al., 1975). However,none of these models could predict either the increase of pore water pres-sure or its effect in the effective stresses.

In 1975, (Martin et al., 1975) presented a model that clarified the den-sification mechanism that occurs during liquefaction. They stated that fora saturated sand, if drainage cannot occur during the time span of theloading sequence, the tendency for volume reduction during each cycleof loading results in a corresponding progressive increase in pore-waterpressure. In this model, it was assumed that the plastic volumetric strainthat occurs during one cycle of uniform shear strain in an undrained sim-ple shear test is equal to the total volumetric strain in a drained simpleshear test. The plastic volumetric strains during undrained cyclic load-ing are absorbed by the elastic rebound in the soil skeleton due to thedecrease in effective stress and a constant volume is retained, which is abasic assumption in critical state theory as applied to undrained loading.Because of its ability to estimate pore pressures, the model was imple-mented in non-linear dynamic effective stress analyses which originated

8 1 Introduction

the programs DESRA (Lee and Finn, 1978) and TARA (Finn et al. 1986).In 1976 Bazant and Krizek developed an endochronic constitutive law forthe study of the liquefaction of sand. The development of the endochronictheory is attributed to Valanis (1971) and is based on an intrinsic timeparameter which is an independent scalar variable that depends on bothtime and deformation increments. Bazant and Krizek used endochronicvariables in order to express the densification of loose sand that developsduring cyclic shearing. Lee and Finn (1978) investigated the possibility ofexpressing the pore water pressures and the volumetric strains in termsof endochronic variables. They presented an efficient response function torepresent the pore pressure data by simplifying the family of curves usedin previous methods to a single curve.

B. The work of Biot also received great attention and was extensively used.The literature is replete with publications pertaining to the analyticalsolution of the general governing equations of motion for two-phase me-dia based on the work of Biot. Deresiewicz (1960, 1962) solved Biot’sgoverning equations of motion for an elastic half-space under harmonictime variations using displacement potentials. Derski (1978) used veloc-ity terms to express the relative motion of different phases. Burridge andVargas (1979) obtained the time domain fundamental solution (Green’sfunction) for an infinite space; they also studied the disturbance in aporo-elastic infinite space due to application of an instantaneous pointbody force. Simon et al. (1984) presented an analytical one-dimensionalsolution for the transient response of an infinite domain by using Laplacetransformations. Gazetas and Petrakis (1981) evaluated the complianceof a poroelastic half-space for swaying and rocking motions of an infi-nitely long, rigid and pervious strip that permitted complete drainage atthe contact surface. Finally, Halpern and Christiano (1986) evaluated thecompliances of three-dimensional square footings considering pervious aswell as impervious cases.

C. At the same time, the development of new continuum theories of mix-tures opened once again the dilemma about the validity of many of theapproaches that had been used in the past. In 1960, Truesdell and Toupinpresented a treatise on the classical field theories in which they developedin detail the properties of motion and the fundamental physical principlesof balance. They also listed mathematical principles that help to formu-late definite constitutive models. The treatise contains a wide reference toprevious work, and constitutes a rational work based on methods of mod-ern continuum mechanics. In 1965, Green and Naghdi (1965b, 1967, 1968)developed a dynamic theory for the relative flow of two continua based onan energy equation and an entropy production inequality for the entirecontinuum. This approach takes into account the use of invariance condi-tions under superposed rigid body motions, theoretical approach that ismore in line with the work of Green and Rivlin (1964), which examines a


new type of continuum theory. The new theory, known as multi-polar con-tinuum mechanics, is based on some concepts developed by Truesdell andToupin, who introduced generalized forces, body and surface forces, andgeneralized stresses. Green and Rivlin also discussed Truesdell’s theoryand related it to their new approach. The theories of mixtures were actu-ally studied by many researchers, who developed their variations to gainfurther understanding of the behavior of mixtures. The work of Bowen(1976), which contributed to the formulation of rational mixture theoriesand the understanding of wave propagation through porous media, wasstudied, by Garg et al. (1974) and Derski (1978), among others. Aifantis(1980) and co-workers also contributed useful works, including studies ofporous media. Atkin and Craine (1976) outlined the historical develop-ment of mixture theories and demonstrated the applicability of mixturetheory to the case of chemically inert mixtures of ideal gases. It seems thatMorland (1972) was the first scientist to use the volume fraction conceptin connection with modern mixture theories to describe the behavior ofporous media. Drumheller (1978) presented a theoretical treatment ofporous solid using a mixture theory in which the volume fraction conceptwas introduced. The key point in his derivation was in the formulationof pore collapse relations to express the rate of change of the volumefractions. Bowen (1982) summarized all findings of the mixture theoryand introduced the volume fractions concept for the saturated condition.Bowen substituted the free Helmholtz energy function per unit mass in theClausius-Duhem inequality by a free energy review concerning these andother thoeories for structured mixtures (e.g. porous media). Katsube andCarroll (1987) modified the mixture theory of Green and Naghdi by intro-ducing a micro-mechanical constitutive theory. Ehlers (1989) presented amacroscopic description of fluid-saturated porous media via mixture the-ories extended by the volume fractions concept where elasto-plasticitywas taken into account by means of a multiplicative decomposition of thedeformation gradient. Following this line of research, Ehlers (1993) wasable to describe the behavior of compressible, incompressible, and hybridporous materials.

D. With the advances in modern computational science and the developmentof rigorous numerical techniques such as the finite element method andnumerical implementations of the consolidation theory, Biot’s equationsand mixture theories found wide applications. A variational formulationof the dynamics of fluid-saturated porous solids was the basis of a nu-merical method that Ghaboussi and Dikmen (1978) developed for thepurpose of discretizing a partial media into finite elements. Shandu andWilson (1969) first applied the finite element method to study fluid flow insaturated porous media. With the introduction of the FEM as a sound nu-merical technique, it became possible to extend the mixture theory to en-compass elasto-plastic non-linear constitutive models and obtain reliable

10 1 Introduction

solutions of the field displacements and pressures. Prevost (1980) pre-sented a general analytical procedure that accounts for non-linear effects.In his work, Prevost focused on the integration of the discretized fieldequations based on the mixture theories of Green and Naghdi (1965b).Later, he worked on several numerical applications to study the consol-idation of inelastic porous media (Prevost, 1981) and on the non-lineartransient phenomena and wave propagation effects in saturated porousmedia (Prevost, 1982, 1985; Lacy and Prevost, 1987). Because of theincreasing necessity of non-linear applications, Zienkiewicz, and other re-searchers published a series of papers that elucidated various numerical so-lutions for pore–fluid interaction analysis. Zienkiewicz and Shiomi (1984)classified different methods of analysis in a comprehensive paper on nu-merical solutions of the Biot formulation. These numerical solutions werefurther studied and used in several numerical applications related to theundrained, consolidating, and dynamic behavior of saturated soils (Simonet al., 1986; Zienkiewicz et al., 1990; Simon et al., 1986). A continuumtheory for saturated porous media that is applicable for soils exhibitinglarge strains was formulated later by Kiousis and Voyiadjis (1985) usinga Lagrangian reference frame.

E. In their later references, the porous solid was considered as being com-pletely saturated with the fluid. In reality, problems of soil mechanicsoften involve partially saturated soils, where the pores are filled with thefluid and air. The mechanics of a solid-fluid-gas medium was consideredby Raats and Klute (1968) and De Boer and Ehlers (1986) for establishingfield equations based on mixture theories. Schrefler et al. (1989) followedBiot’s equations to establish equivalent relations for partially saturatedsoils (Zienkiewicz et al., 1990). Nevertheless, no complete and totally ac-cepted theory for partially saturated media is yet available. The modernversion of the porous media theory is based on the mixture theory devel-oped in the 1960’s. The research in porous media theories is now mainlyfocused on three directions: first, the implementation and validation ofthe developed porous media models into numerical algorithms; second,the incorporation of different material behavior into the developed math-ematical models; and third, the investigation of special phenomena whichappear in saturated and empty porous solids based on available theoreti-cal solutions and experimental results.

This book presents the fundamental pore water flow in geo-materials inChap. 2, the coupled theory of mixtures in Chap. 3, the yield criteria inChap. 4, the finite element formulations in Chap. 5, applications in Chap. 6,advanced topics in Chap. 7, and Fortran Code in Appendix.

2 Flow in Geo-materials

2.1 Nature of Geo-materials

Soils are Mother Nature’s products and their composition is quite complex.Typically soils are composed of three phases: solid, water, and air. Whensoils are under the water table, they are inherently saturated. Under thewater table, air voids in the soils are completely filled with water, and thesoils are essentially two phase materials. Above the water table, there aremixtures of air and water in the voids and the soils are three phase materi-als. This multi-phase nature of soils develops the coupled behavior of theirconstituents. A simple example of coupled behavior is the stress distributionof a composite beam such as a reinforced concrete beam. The reinforcingsteel carries greater stress than the concrete due to its higher modulus. Insoils, the phases as well as moduli of their various constituents are different.Therefore, additional coupled behavior from different phases is expected inaddition to that from the different moduli. The coupled behavior from thedifferent phases involves the flow characteristics of pore liquids (e.g. air andwater). Traditional coupled theories of mixtures (Biot 1956b, Prevost 1980)address the effects of pore liquid flow; however, more recent coupled theo-ries of mixtures address the additional coupling with rate dependency andmicro-mechanical mechanisms such as grain rotations, grain interactions, anddamages Voyiadjis and Song (2005a).

This chapter, however, focuses primarily on the flow of water in soils andfundamental coupling concepts.

When the soils are not saturated (above the water table), the air reservessome volume in the pore spaces and the soils are three phase materials asshown in Fig. 2.1. For three phase conditions (unsaturated soils), coupledbehavior operates in a tri-valent relationship involving solids, water and air.When soils are saturated (under the water table), the volume of air is zero andthe soils are essentially two phase materials (e.g. solid and water) as shown inFig. 2.1. For two phase conditions (saturated soils), coupled behavior existsonly between the two constituents. The classical effective stress equation forsolids and water (two phase materials) is expressed by Terzaghi (1943) asfollows:

σ′ = σ − u (2.1)

12 2 Flow in Geo-materials

Fig. 2.1. (a) Soil element In natural state; (b) three phases of soil element (Das,2005)

where, σ′ is the effective stress, σ is the total stress and u is the pore (water)pressure. An expanded version of (2.1), that includes the air pressure, isderived by Bishop et al. (1960) as shown in (2.2):

σ′ = σ − ua + χ(ua − uw) (2.2)

where, ua is the pore air pressure, χ is an experimental parameter and uw

is the pore water pressure. Equations (2.1) and (2.2) deal with the effectivestress, pore water pressure, and pore air pressure as three separate quanti-ties. Therefore, these two equations represent weakly-coupled relations amongthose three quantities. The concept of weakly or uncoupled behavior is gen-erally correct when there is no flow in the soils (static condition). When thereis some kind of flow, it affects the pore water pressure and ultimately the ef-fective stress is affected. Flow of water in soils is controlled by many factors,but the most important factor governing soil flow property is the hydraulicconductivity (permeability) of soils. Ultimately, hydraulic conductivity comesinto play in (2.1) and (2.2) for dynamic (or pseudo static) condition. Incor-porating the hydraulic conductivity into the effective stress equations is thekey concept of the coupled theory of mixtures.

2.2 Flow of Water in Geo-Materials 13

2.2 Flow of Water in Geo-Materials

Flow of water in the geo-materials is divided into steady flow and transient(or non-steady) flow. Steady flow is the condition in which flow rate is con-stant with time. Therefore one can apply (2.1) and (2.2) without sacrificingtheoretical integrity. Transient or non-steady flow is the condition in whichflow rate is not constant with time. In this condition, continuous changes inpore pressure are present, and using (2.1) and (2.2) may result in a severetheoretical compromise. (Both for steady flow and transient flow, Darcy’s lawis assumed to be valid. For some cases, fluid flow may be caused by chemicalpotential difference or electrical potential difference; nevertheless, only theDarcy’s law is considered in this book.)

2.2.1 Steady Flow

Steady flow in the soil cause the seepage pressure in the soils and affects thepore water pressure. Steady flow can increase or decrease the measured porewater pressure depending on the direction of the seepage as shown in (2.3).

σ′′ = σ′ ± us = σ′ ± izγw (2.3)

where, σ′′ is the effective stress considering seepage, us is the pore waterpressure caused by seepage, i is the hydraulic gradient, z is the depth andγw is the unit weight of water. The ± sign is positive for downward flowand negative for upward flow. The seepage pressure us affects the pore waterpressure; however, it does not change with time in the steady flow while itchanges with time in the transient flow.

2.2.2 Transient or Non-steady Flow

Transient flow occurs when the flow condition is forced to change by externalenergy. Common examples of transient flow are the vibration-induced porewater pressure and its associated flow, stress (deformation)-induced flow,consolidation- induced flow, and many others. During transient flow, the flowrate will depend on the hydraulic conductivity and time, as described byDarcy’s law. Therefore one may expect the incorporation of the hydraulicconductivity and time in (2.3). One can anticipate that the effective stressalso will be affected by the pore water pressure as expressed in (2.4).

∆u = B[∆s1 + A(∆s1 − ∆σ3)] (2.4)

where, ∆u is the generated (excess) pore water pressure, A and B are porepressure parameters, ∆σ1 is the change in the deviator stress, and ∆σ3 is thechange in the confining stress. From (2.3) and (2.4) we can see that the porewater pressure and the effective stresses are inter-related. Consequently, wemay say that the effective stress, the pore water pressure, and the hydraulicconductivity are all coupled in the transient flow; thus these relations showthe fundamental idea of the coupled theory of mixtures.

3 Coupled Theory of Mixtures

3.1 Traditional Theory of Mixtures

The consolidation is a well known time dependent volume contraction be-havior of saturated clayey soils. The consolidation is triggered by externalloading, void spaces are compressed, pore water pressure is increased, andthe pore water pressure starts to dissipate (flow out). In turn, the pore waterpressure is decreased and the effective stress is increased. This consolidationis the classical example of the coupled behavior of soils though it was notrecognized as one of them in the past. This chapter addresses Terzaghi’s andBiot’s consolidation theories in terms of the coupled theory of mixtures fortwo phase (saturated) materials.

3.1.1 Terzaghi’s Consolidation Theory

Terzaghi’s one-dimensional consolidation theory is a classical consolidationtheory as expressed in the following equation;

∂u

∂t= cv

∂2u

∂t2(3.1)

where, u is the excess pore water pressure, t is the time, and cv is the consoli-dation coefficient. Terzaghi (1943) obtained (3.1) intuitively from the analogyof the consolidation and heat diffusion phenomena. Equation (3.1) is read-ily found in thermodynamics or partial differential equation text books. Acomparison of the consolidation and heat diffusion equation is expressed asfollows by Terzaghi (1943).

From (3.1) and Table 3.1 we can see that (3.1) is exactly the same asthe heat diffusion equation; therefore, the assumptions in the heat diffusionequation may be transferred into (3.1). One of assumptions is that uncoupledbehavior exists between the effective stress and excess pore water pressure.This assumption is implicitly shown in (3.1) through the constant cv. cv isa constant in the heat diffusion equation, and therefore, as we have seen, itis also a constant in the consolidation equation. One of the solutions of (3.1)may be expressed (Craig, 1979) as follows:

16 3 Coupled Theory of Mixtures

Table 3.1. Analogy of consolidation and heat diffusion

Conclusion Symbol Heat Diffusion

Excess pore water pressure u TemperatureTime t TimeCoefficient of consolidation cv Diffusivity

u =m=∞∑

m=0

2ui

M

(sin

Mz

d

)exp(−M2Tv

)(3.2)

where, u is the excess pore pressure at certain time t, ui is the initial ex-cess pressure, M = (π/2)(2m + 1), z is the depth, d is the length of longestdrainage path, and Tv is the time factor that is expressed in Tv = (cvt/d2).In reality, because of the coupling of the effective stress and the excess porewater pressure, cv is not a constant. Terzaghi (1943) assumed it to be aconstant for simplicity. This simplification made the calculation easier, butit also lost some details as a trade-off. One example of the consequences ofsuch a significant loss of detail is the fact that Terzaghi’s consolidation isnot able to predict the ‘Mandel-Cryer’ effect, which can be predicted onlyby coupled multi-dimensional consolidation theory. Biot’s more recent con-solidation theory has the advantage of considering the coupled behavior ofthe effective stress and the excess pore pressure, but Biot’s equations are notconvenient to use.

3.1.2 Biot’s Consolidation Theory

Biot (1955) assumed that the soil is a linearly elastic isotropic solid expe-riencing small strain (Miga et al., 1998). The equations incorporating two-dimensional plane strain mechanical equilibrium in Cartesian coordinates fora homogeneous linear elastic media is described by the following equations;

G∇2u +G

1 − 2ν

∂ε

∂x− α

∂p

∂x= 0 (3.3)

G∇2v +G

1 − 2ν

∂ε

∂y− α

∂p

∂y= 0 (3.4)

where, G is shear modulus, ν is Poisson ratio, α is the ratio of water volumeextracted to the volume change of the soil, u, v are the x, y displacementsin the Cartesian plane, p is the pore water pressure, and ε is the volumetricstrain. Biot (1955) added a constitutive relationship relating volumetric strainand fluid pressure in order to complete the continuum model. The constitutiveequation describing this relationship is shown as follows:

∇ · k∇p − α∂ε

∂t− 1

S

∂p

∂t= 0 (3.5)

3.2 Modern Theory of Mixtures for Finite Strain 17

Fig. 3.1. Comparison of Terzaghi’s one dimensional consolidation and Biot’s twodimensional consolidation

where, k is the coefficient of hydraulic conductivity and 1/S is the amount ofwater that can be forced into the soil under pressure while the volume of thesoil is kept constant. The first two terms in this equation provide the couplingrelationships among volumetric strain, pore fluid pressure, and hydraulic con-ductivity. Generalization of (3.3) through (3.5) using a displacement tensoru and pressure p will produce the following relations;

G∇ · ∇u +G

1 − 2ν∇(∇ · u) − α∇p = 0 (3.6)

α∂

∂t(∇ · u) +

1S

∂p

∂t−∇ · k∇p = 0 (3.7)

Equations (3.6) and (3.7) are Biot’s coupled consolidation equations. Biot’sconsolidation equations, of course, are more difficult to use than Terzaghi’sconsolidation equations. Biot’s consolidation equations, however, are morerational and are able to predict so called ‘Mandel-Cryer’ effect (Lambe andWhitman, 1979), the fact that the excess pore water pressure can be biggerthan the applied stress at the initial stage of the consolidation. The Mandel-Cryer effect was observed in the laboratory and in the field, and it could bepredicted by Biot’s consolidation equations. Figure 3.1 shows the compari-son of Terzaghi’s one-dimensional consolidation and Biot’s two-dimensionalcoupled consolidation to exemplify the Mandel Cryer effects predicted byAbu-Farsakh (1997).

3.2 Modern Theory of Mixtures for Finite Strain

Biot (1955, 1978) first developed the foundation of modern coupled the-ory of mixtures for an elastic porous medium. However, experiments have


shown that the stress-strain-strength behavior of the soil skeleton may bestrongly non-linear, anisotropic, and elastoplatic characteristics that Biot’stheory does not adequately account for. An extension of Biot’s theory into thenon-linear, anisotropic range is, therefore, necessary in order to analyze thetransient response of soil deposits. This extension has acquired considerableimportance in recent years due to the increased concern with the dynamicbehavior of saturated soil deposits and the associated liquefaction of satu-rated sand deposits under seismic loading conditions. Such an extension ofBiot’s formulation was proposed by Prevost (1980). Prevost (1980)’s theoryof mixture was subsequently coupled with Terzaghi (1943)’s effective stresstheory for finite strain by Kiousis and Voyiadjis (1985) and Voyiadjis andAbu-Farsakh (1997), and therefore a coupled theory of mixtures for finitestrain condition is obtained. This book further extends the work of Voyi-adjis and Abu-Farsakh (1997) for the anisotropic stress condition with themicro-mechanical consideration such as grain rotations.

The main goal of this book is to illuminate the coupled behavior of soils.This method uses the hydro-mechanical analysis by the coupled theory ofmixtures at finite strains. To perform this analysis, the formulation of thecoupled field equations for soils using the theory of mixtures in an updatedLagrangian frame based on the principle of virtual work and implemented ina finite element program is used. When it is saturated, soil consists of twodeformable media, the solid grains and water as in shown in Fig. 3.1. Basedon this figure, the following quantities are defined;

γw =Ww

Vw; γs =

Ws

Vs(3.8)

ρw =γw

g; ρs =

γs

g(3.9)

nw =Vw

VT(3.10)

ρw =Ww

gVT= nwρw (3.11)

ρs =ws

gVT= nsρs = (1 − nw)ρs (3.12)

where, Ww and Ws are the weight of the water and solid phase respectively;Vw and Vs are the volume of the water and solids respectively; γw and γs

are the unit weights of the water and the solid phase respectively; g is theacceleration due to gravity; ρw and ρs are the intrinsic mass densities of thewater and solid phase respectively; nw is the porosity of the soil; ρw and ρs

are the apparent mass densities of the water and the solid phase respectively.

3.2.1 Balance of Mass

The amount of mass (mα) of the constituent (α) in the continuum occupyingthe spatial volume V at time t is given by;


mα =∫

v

ραdV (α = s or w) (3.13)

where, ρα is the macroscopic mass density of the α-constituent. If the soilmixture is assumed to be chemically inert, then the law of conservation ofmass requires that the rate of change of the mass of each constituent in thecontinuum to be equal to zero and hence the material derivate of mα be zero.Therefore the balance of mass for each constituent α implies that

D

Dt

∫

v

mαdV =D

Dt

∫

v

ραdV =∫

v

(∂ρα

∂t+ vα

i

∂ρα

∂zi

)dV = 0 (3.14)

where vα is the velocity of the α constituent. Applying the divergence theoremto (3.14), it becomes

∂

∂t

∫

v

ραdV +∂

∂t

∫

A

ραvα · ndA =∂

∂t

∫

v

ραdV +∫

A

div(ραvα)dV = 0 (3.15)

Making use of the following relation,

Div(ραvα) = ραdiv(vα) + ∇ρα · vα (3.16)

Equation (3.14) then becomes∫

v

[(∂ρα

∂t+ ∇ρα · vα

)+ ραdiv(vα)

]dV = 0 (3.17)

Because (3.17) is valid for any arbitrary volume V , the balance of massfor each constituent (α) can be written as follows:

D

Dtρα + ραdiv(vα) = 0 (3.18)

orρα + ραdiv(vα) = 0 (3.19)

Similarly, the balance of mass for the mixture leads to

ρ + ρ div(v) = 0 (3.20)

In the Lagrangian configuration, the balance of mass becomes

ραJα = ρao ; ρJ = ρo (3.21)

where ρao is the apparent mass density of the α-constituent in its reference

configuration and ρ is the mass density of the mixture expressed as follows:

ρ =∑

αρα (3.22)


Using the relation between the apparent mass densities and the intrinsicmass densities of the water and solid grains, the material derivatives thenbecomes

ρα = nαρα + nαρα (3.23)

Equation (3.19) may then be reduced to

nα + nαdiv(vα) = −nα ρα

ρα(3.24)

If one assumes the water phase is incompressible (ρw), one obtains

nw + nwdiv(vw) = 0 (3.25)

If the solid grains are taken incompressible (ρs) then

ns + nsdiv(vs) = 0 (3.26)

butnw + ns = 1 (3.27)

Therefore, one obtains∇ns = −∇nw (3.28)

∂nw

∂t+

∂ns

∂t= 0 (3.29)

ns − vs∇ns + nw − vw∇nw = 0 (3.30)

Substituting ns and ns in (3.26) and subtracting (3.25) from the resultingexpression, the balance of mass can be written in terms of the soil porositynw, solid velocity vs, and water velocity vw as follows;

div(vs) − div(vw) =1

nw[div(vs) + (vw − vs)grad(nw)] (3.31)

Equation (3.31) is the mass balance equation for saturated porous mediaderived by Prevost (1980).

3.2.2 Coupling of Mass Balance Equation and Darcy’s Law

The general form of Darcy’s law for the flow of water through porous mediumis given by

(vw − vs) = − 1γw

Kws(grad(Pw) − ρwb) (3.32)

where b is the body force vector, γw is the unit weight of water, and Kws

is the permeability tensor in (m/sec). Taking the divergence of both sides of(3.32) and substitute the result into (3.31) one obtains

div(vs)−nwdiv

[1γw

Kws(grad(Pw) − ρwb)]+(vw−vs)grad(nw) = 0 (3.33)


Substitute (3.32) into (3.33) one obtains

div(vs) − div

[nw

γwKws(grad(Pw) − ρwb

]= 0 (3.34)

Making use of the following relation,

div(vs) = dskk = Xs

k,aXsK,BεAB (3.35)

(3.34) can be rewritten as follows:

dskk − ∂

∂Za[Kws

ab (grad(ρw) − ρwbb)] = 0 (3.36)

Making use of the following relations

Kwsab = Xs

a,AXsb,BKWS

AB (3.37)bb = Xs

b,BBB (3.38)∂Pw

∂Zb= Xs

C,b

∂Pw

∂Xc(3.39)

∂

∂Za= Xs

D,a

∂

∂Xd(3.40)

one obtains the coupled equation in an updated Lagrangian formulation asfollows;

sCs−1ij εij − JsCs−1

ij Cs−1ij Xs

D,a

∂

∂Xd

[nw

γwKWS

AB Xsa,A

(∂Pw

∂XB− ρwBB

)]= 0

(3.41)where, Cs

ij = Xsk,iX

sk,j .

If one assumes incompressibility of the solid grains, the porosity, nwcanbe updated from n configuration to at n+1 configuration using the Jacobianof solid grains, Js as follows:

1 − nwn+1

1 − nwn

=1Js

(3.42)

As discussed previously, the drainage condition in the soil is somewherebetween the fully drained and the fully undrained condition. This conditionis called the partially drained condition or the transient flow condition. Forthe transient flow condition, it may be presumed that the pore pressure isa function of the hydraulic conductivity and other parameters (stress-strainparameters). (This present study presents a relatively concise derivation ofthe coupled theory of mixtures: for a full derivation, see Prevost (1980)).

4 Coupling Yield Criteria andMicro-mechanics with the Theory of Mixtures

The behavior of geo-materials follows not only the theory of mixtures, butit also follows yield criteria. The yield behavior of various components (e.g.water, solid) of geo-materials differs. However, applying a complex paradigmof mechanics that deals with individual soil components is too complicatedand time consuming at present. For this reason, the authors have restrictedthis chapter to macroscopic yield behavior, as shown in the coupling of theoryof mixtures and yield criteria of geo-materials.

4.1 Soil Models

There are many models for analyzing yield behavior of soils. One of the oldestsoil models is Mohr-Coulomb yield criteria. In Mohr-Coulomb yield criteria,the soil behavior is expressed in terms of the friction angle and cohesion.And it is assumed that soils fail when the shear stress is greater than theshear strength of the soils. The shape of Mohr-Coulomb criteria in π-plane isshown in Fig. 4.1. It is a distorted hexagon because the compressive strengthis greater than the extensional strength in soils. Mohr-Coulomb criteria aresimilar to Tresca criteria: in both sets of criteria, the materials fail at thehighest shear stress (and the shape of yield criteria in π-plane is hexagon).Tresca criteria do not differentiate extension part from compression part asshown in Fig. 4.2 because Tresca criteria were initially developed for yieldanlaysis of metals, and metals usually show identical tensile and compressivestrength. Therefore, the shape of the Tresca yield criteria in π-plane is anundistorted hexagon.

Considering the fact that soils exhibit different strength in extension andcompression, Mohr–Coulomb criteria are more widely used for soils. Anotherdifference between Mohr-Coulomb criteria and Tresca criteria is that Mohr-Coulomb can incorporate J1 (mean stress) dependency while Tresca cannot.Soils are essentially particulate materials; therefore, the J1 dependency isessential.

Another widely used fundamental yield criterion is a von Mises yield cri-teria (Fig. 4.3). Von Mises criteria are also originally used for yield behavioranalysis of metals. Von Mises criteria assume that materials fail when secondstress invarient (J2) reaches its maximum state. Therefore, von Mises criteria

24 4 Coupling Yield Criteria and Micro-mechanics

Mohr-Coulomb

(a) (b)

Fig. 4.1. Mohr-Coulomb yield criteria (a) in stress space, (b) in π plane (Chenand Mizuno, 1990)

(a) (b)

Fig. 4.2. Tresca yield criteria (a) in stress space, (b) in π-plane (Chen and Mizuno,1990)

are very similar to Tresca, however these criteria yield a different shape inπ-plane. The shape of Von Mises criteria in the π-plane is a circle with itscenter at the origin. Von Mises criteria do not incorporate the J1 dependency,either. For this reason von Mises is not used widely for the yield behavioranalysis of soils. For metals, the true yield behavior is known to be somewherein between von Mises and Tresca yield criteria. However, soils are different:readers should remember they are particulate materials, and models that areonly J2-dependent have limitations on their applicability.

4.1 Soil Models 25

Fig. 4.3. Von Mises yield criteria in the stress space (Chen and Mizuno, 1990)

The Drucker–Prager model, introduced relatively recently, compares withMohr–Coulomb, Tresca, and von–Mises. The uniqueness of Drucker–Prager isthe incorporation of J1 dependency in von–Mises model as shown in Fig. 4.4.The Drucker-Prager yield criteria is a circle in π-plane. The size (diameter)of the circle, however, varies depending on the mean principal stress (J1).When one takes away the J1 dependency from Drucker–Prager yield criteria,it is similar to von-Mises. Because the J1 dependency of Drucker–Prager yieldcriteria is a more realistic representation of the particulate characteristics ofsoils, it is one of the widely used soil models. Drucker–Prager assumes thesame compressive and tensile behavior of the material and therefore its yieldcriteria are essentially the equation of a circle; the mathematics for Drucker–Prager is quite favorable. Compared to Drucker–Prager’s single equation ofyield surface and its associated mathematical simplicity, the Mohr–Coulombapproach, however, is composed of six lines (equations) and the mathemat-ics dealing with the manipulation of the equations is very complicated. Forexample, finding the normal direction in Mohr–Coulomb at one of the breakpoints in Fig. 4.1, clearly involves exceptional mathematical difficulties.

(Students and beginning engineers: note that the Mohr–Coulomb is thesame Mohr–Coulomb in most soil mechanics textbooks. When one plotsMohr–Coulomb in the mean stress and deviatoric stress axis, one obtainsa straight line that has slope φ [internal friction angle] and ordinate c [cohe-sion]).


Fig. 4.4. Drucker–Prager yield criteria in stress space (Chen and Mizuno, 1990)

The above descriptions are discussions of classical soil models that areused for rigid plastic or elastic-perfect plastic behavior. The elastic-perfectplastic models assume that, after yielding, the material stays in yield (per-fectly plastic) condition. However, field soils may not show this behavior.Field soils may show recovery of their strength, elastic rebound, and (some-times) increased or decreased load carrying capacity even after yielding. Thisbehavior is called hardening. Even during the loading process, a soil hardens(changes its strength); therefore, the incorporation of hardening process willbe beneficial to formulating a realistic simulation of soil behavior.

One of the pioneering soil models that can incorporate some of the abovecharacteristics is the Cam Clay model, formulated by Roscoe, Schofield andWroth (1957). One of the features of Cam Clay model is that it can incorpo-rate the volume change (consolidation) phenomenon in the yield behavior ofsoils as shown in Fig. 4.5. By adding consolidation in the yield criteria, CamClay model yield criteria can simulate changes of J1(mean normal stress) dur-ing shear. Therefore, the Cam Clay model yield criteria can expand duringshear if J1 increases as shown in Fig. 4.6(a). If J1 decreases, the yield criteriado not shrink, but in this model, such a condition is considered an over-consolidated state. The Cam Clay model attracted great interest and showedgood agreement with a vast range of test results, as shown in Fig. 4.6(b):

The Cam Clay model assumes that the dissipated energy is carried outonly through the plastic shear strain only. Later, this assumption changed,

4.1 Soil Models 27

Fig. 4.5. Cam Clay model in p, q and e (Chen and Mizuno, 1990)

(a) (b)

Fig. 4.6. Cam Clay model (a) in stress space (Chen and Mizuno, 1990) (b) in p-qspace (Wood, 1990)


Yield Surface

PPo/2

Po

CSL

q

Fig. 4.7. Yield Locus of the modified Cam Clay model in p-q space

and it was assumed that the dissipated energy is carried out through boththe plastic shear strain and plastic volumetric strain. This modified modelis called the modified Cam Clay model. The advantage of the modified CamClay model is that it is more rational in fundamentals and mathematicallymore convenient than the original Cam Clay model: it uses only one equationfor the yield surface while original Cam Clay model uses two equations in pvs. q space. This advantage solved the important mathematical problem ofnormality of plastic strain to the yield surface.

The original Cam Clay model places the break point (point A and C inFig. 4.6) between the Roscoe surface and the Hvoslev surface. This breakpoint makes it mathematically difficult to find the normal vector to the yieldsurface as mentioned previously. In the modified Cam Clay model, the wholeyield surface is a smooth ellipsoidal shape as shown in Figs. 4.7 and 4.8.Therefore, finding the normality of plastic strain with respect to the yieldsurface is not difficult at all.

Although the modified Cam Clay model has the above-described advan-tages, it does not agree well with experimental data for over-consolidatedsoils. Typically, the modified Cam Clay model tends to over-estimate thestrength of soils in over-consolidated state (Chen and Mizuno, 1990).

4.1 Soil Models 29

Space diagonal

Fig. 4.8. Yield criteria of modified Cam Clay Model in stress space

DiMaggio and Sandler (1971) and Baladi and Rohani (1979) proposed amathematically advantageous yield criteria that also have good agreementwith experimental data. This model is composed of two lines, one line forthe over-consolidated state and another line for the normally consolidatedstate, as shown in Fig. 4.9. These two lines meet smoothly at the connectionpoint (point B). At the connection point, the slopes of two lines are identical

Fig. 4.9. Cap Model (Desai and Siriwardane, 1984)


(the two lines are tangential to each other at point B). Therefore, the nor-mality rule of plastic strains with respect to the yield surface does not causemathematical problems. (For readers who wonder why the authors imposethe normality rule, please note that the normality rule does not have to beimposed. However, employing the normality rule will simplify the formulationprocedure.)

Also, by taking flatter lines at the over-consolidated state, the above Capmodel achieves a better agreement with experimental data than does themodified Cam Clay model. These models are some traditional models, thedetails of which can be found in Chen and Mizuno (1990), Desai and Siriwar-dane (1984) and Wood (1990). Readers should also note that although thereare many soil models other than these traditional ones, this book restrictsitself to a thorough discussion of the widely regarded and widely used soilmodel – the modified Cam Clay model.

4.2 Anisotropic Soil Model

The above traditional models are isotropic models. Therefore, their applica-tion is of inherently limited use in addressing the anisotropic nature of soils.It is the same situation for hardening; even though soils may have anisotropichardening characteristics, the previous models incorporate isotropic harden-ing only. Typically, the anisotropy of soils is divided into two kinds: inher-ent anisotropy and (stress)-induced anisotropy. Inherent anisotropy is causedby the natural texture of soils as shown in Fig. 4.10(a); (stress)-induced

(a) (b)

Fig. 4.10. Anisotropic textures of clays (a) in natural deposits (b) in the shearplane (Mitchell, 1993)

4.2 Anisotropic Soil Model 31

anisotropy is caused by the re-orientation of the soil particles during shear asshown in Fig. 4.10(b). The shape of the modified Cam Clay model in π-planeis a circle with center at the origin. The overall shape in principal stress spaceis an egg-shaped ellipse, as shown in Fig. 4.8. With hardening, the size of thecircle increases in π-plane, or the size of the egg-shape ellipse increases inthe principal stress space. In order to incorporate the anisotropic character-istics of soils, the soil models need to have some additional features to thetraditional modified Cam Clay model (that is, if one intends to use the modi-fied Cam Clay model). Dafalias (1987), for example, proposed an anisotropicmodified Cam Clay model that can incorporate the inherent anisotropy andthe stress induced anisotropy. Dafalias incorporated the back stress conceptin the modified Cam Clay model. Back stress may be interpreted as the in-ternally embedded stress that changes the structure of the soil. When thestructure of a soil changes, it becomes an essentially different soil. For thisdifferent soil, the yield surface is different in size and location, as shown inFig. 4.11. In the π-plane, this behavior is represented as the translation ofthe yield criteria; in the principal stress space, this behavior is representedas the rotation of the yield criteria.

Rotation in the principal stress space or translation in the π-plane rep-resents the different material behavior depending on the three-dimensionalstress conditions. In Fig. 4.11(b), the two dotted circles mean that the mate-rial is stronger to compression in σ1 and σ3 directions (if we say compressionis positive). For σ2 direction, the soil has almost zero strength or some ten-sile strength. Figure 4.11 is a conceptual diagram to show that the requiredfeatures for the anisotropic modified Cam Clay model. The following twosections address the essentials of (isotropic) modified Cam Clay model andformulate an anisotropic modified Cam Clay model.

Modified Cam Clay Model (Isotropic)

From the equilibrium of work done and dissipated energy, (4.1) is obtained(Burland, 1965; Schofield and Wroth, 1968) as follows:

pdεpv + qdεp

s = p(dεpv)2 + M2(dεp

s)2)1/2 (4.1)

where, the variables are defined as follows:

p = mean principal stress = (σ1 + σ2 + σ3)/3 = (σ1 + 2σ3)/3 for triaxialcondition

q = deviatoric stress = (σ1 − σ2)/2εp

v = plastic-volumetric strainεp

s = plastic-shear strainM = slope of critical state line in p vs. q space.


1

3

2

Ko-linebackstress

1 = 2 = 3

(a)

2

1

3

(b)

Fig. 4.11. Concepts of anisotropic Cam Clay model (a) in stress space (afterBanerjee and Yousif (1986)) (b) in π-plane

From (4.1), one can derive the well known yield criterion as follows:

f = p2 − pop + (1/M2)q2 = 0 (4.2)

In (4.2), po is the p at the hydrostatic condition (q = 0). Equation (4.2) canbe rewritten as follows:


(p − po

2

)2(

po

2

)2 +q2

(M po

2

)2 = 1 (4.3)

Equation (4.3) represents the equation of ellipse with one axis length po andanother axis length Mpo as shown in Fig. 4.7.

When the stress condition is inside the ellipse, the material undergoes elas-tic behavior. When the stress condition is at the ellipse and load increment ispositive, the material is in plastic deformation. The increase of effective stressdue to consolidation during loading causes the increase of the mean principalstress po in (4.2). Increased mean principal stress causes the expansion ofyield surface in Fig. 4.7. On unloading, the stress point lies inside the yieldsurface, and elasticity is incorporated.

4.2.1 Anisotropic Modified Cam Clay Model

Equation (4.2) shows the evolution of the yield surface with an isotropichardening behavior; po in (4.2) is the isotropic hardening parameter. However,the real soil is subjected to the anisotropic stress condition, and thereforethe Anisotropic Modified Cam Clay Model (hereafter called AMCCM) isdiscussed here following the work of Dafalias (1987). The shape of AMCCMin the principal stress space is egg shaped as shown in Fig. 4.7. The crosssection of the oval shape can be a circle, hexagon, or similar shape (notspecifically known). However, one can guess that shape will be somewherebetween the anisotropic von-Mises and Tresca type. When the yield locus inthe principal stress space undergoes kinematic hardening, the yield surfacewill move around the π-plane. This behavior will appear as the rotation ofthe oval shape yield locus in the principal stress space. In the p vs. q plane,where p is the space diagonal and q is the deviatoric stress, the shape ofthe yield locus is the inclined cut of the three dimensional yield locus thatappears in Fig. 4.12; that is, the yield locus of AMCCM is a tilted ellipse.Thus the kinematic hardening will result in the up and down rotation of theelliptical yield locus in the p vs. q space for which the origin does not change.

We see, then, that the shape of the yield locus will be the distorted ellipsewhen a soil is subjected to anisotropic hardening. For ease of understanding,the AMCCM is first derived for the triaxial stress condition (σ2 = σ3). In(4.1), one can introduce the anisotropic hardening parameter α as shown in(4.4):

pdεpv + qdεp

s = p(

dεpv

)2 + M2(dεp

s

)2)1/2 + 2α dεpv dεp

s

1/2

(4.4)

In (4.4), α represents the so-called ‘back stress.’ One may use different ex-pression for the inclusion of back stress in (4.4), such as α or 5α instead of2α; however, the results will be the same. When one uses 2α, the integrationbecomes easier. Rearranging the terms in (4.4) yields the following relation:


CSL

q

P

B

A

C

PoPcO

q

qo= Po

M

Fig. 4.12. Kinematic hardening of AMCCM in p vs. q space (p is the mean principalstress, q is the deviatoric stress) (after Dafalias (1987))

dεpv

dεps

=2αp2 − 2pq

q2 − p2M2(4.5)

Let η = q/p (η is different from M the slope of critical state line. η is thestress ratio at any condition, such as in elastic condition. At the critical stateη and M become the same.) and making use of it in (4.5), one obtains thefollowing relation:

dεps

dεpv

=2η − 2α

M2 − η2(4.6)

From the normality rule, one can obtain (4.7):

dεps

dεpv

= −dp

dq(4.7)

Let the variable Ψ be defined as follows:

Ψ =dεp

v

dεps

=M2 − η2

2η − 2α(4.8)

Making use of (4.7) in (4.8), one obtains

dq

dp= −Ψ = −M2 − η2

2η − 2α(4.9)


From the relationship q = ηp and (4.9), one obtains (4.10)

dq = pdη + ηdp = −Ψdp . (4.10)

Rearranging the terms in (4.10) and making use of (4.7) into (4.10), oneobtains the following expression:

−dp

p=

dη

η + M2−η2

2η−2α

(4.11)

Integrating (4.11) for p (pi to p) and η( ηi to η) (where subscript i impliesinitial value), (4.12) is obtained as follows:

p

po=

M2 − α2

M2 + η2 − 2αη(4.12)

Rearranging (4.12), one obtains

f = p2 − ppo +1

M2(q2 − 2αpq + α2ppo) = 0 (4.13)

Recalling the similar equation for the (isotropic) modified Cam Clay Modelyields the following equation

f = p2 − pop +1

M2q2 = 0 (4.14)

The same equations may be expressed as follows:

(p − po

2

)2(

po

2

)2 +q2

(M po

2

)2 = 1 (equation of ellipse) (4.15)

One understands that the (4.13) is also the equation of ellipse with capabilityof rotating with α in p vs. q space. Equation (4.13) is identical to the oneobtained by Dafalias (1987). Therefore (4.13) can have yield surface transla-tion in the π-plane. For α in (4.13), Dafalias (1987) proposed the followingevolution equation that is similar to that of the Armstrong and Frederick(1966):

α = 〈 ˙λ〉

1 + eo

λ − κ

∣∣∣∣∂f

∂p

∣∣∣∣c

po(q − xαp)

(4.16)

where, 〈〉 is the Macauley bracket, 〈 ˙λ〉 is the loading index, λ is the compres-sion index from the e vs. ln p curve, κ is the recompression index from thee vs. ln p curve, eo is the initial void ratio, and c and x are constants. Theexpression 〈 ˙λ〉(∂f/∂p) represents the plastic strain rate dεp. Making use of(4.6), (4.17) through (4.19) are obtained as shown below:


dεpv =

λ − κ

1 + eo

(dp

p+

2η − 2α

M2 + η2 − 2αηdη

)(4.17)

dεps =

λ − κ

1 + eo

(dp

p+

2η − 2α


)(2η − 2α

M2 − η2

)(4.18)

dεv = dεev + dεp

v (4.19a)

=λ

1 + eo

dp

p+(1 − κ

λ

) 2η − 2α


(4.19b)

Equations (4.13) through (4.19) are for the triaxial stress condition. For thegeneralized stress condition, some changes are made for q and α such as q =(3/2)sijsij1/2 and α = (3/2)αijαij1/2. Equations (4.13) and (4.16) arenow generalized to (4.20) and (4.21) as follows:

f = p2 − ppo +3

2M2(sij − pαij)(sij − pαij) + (po − p)pαijαij = 0

(4.20)

αij = 〈 ˙λ〉

1 + eo

λ − κ

∣∣∣∣tr∂f

∂σmn

∣∣∣∣c

po(sij − xpαij)

(4.21)

Equations (4.20) and (4.21) now represents an egg-shaped, three-dimensionalyield surface in the principal stress space which can rotate anisotropically ac-cording to the magnitudes and directions of αij . Furthermore, note that theinitial values of αij represent the (initial) inherent anisotropy due to its initialtextures. Conversely, the evolution equation of the back stress αij representsthe induced anisotropy due to shearing. Therefore (4.20) and (4.21) are ex-pressions which can incorporate both the inherent anisotropy and inducedanisotropy.

4.3 Elasto-Plastic Constitutive Relations

Now one needs to incorporate the AMCCM into the constitutive relations toanalyze the anisotropic behavior of soils. Assuming the normality rule (flowrule), (4.22) is obtained,

dεpij = d ˙λ

∂f

∂σij= d ˙λBij (4.22)

where, f is the yield function, s is the current stress and εpij is the plastic

strain. For the case of small elastic strains, one may make use of the additivedecomposition of the incremental strain as follows:

dεeij = dεij − dεp

ij (4.23)

4.3 Elasto-Plastic Constitutive Relations 37

In order to obtain the constitutive relation, one uses the following relations

dσij = Cijkl(dεkl − dεpkl) (4.24a)

= Cijkldεkl − Cijkldεpkl (4.24b)

= Cijkldεkl − Cijkld˙λBkl (4.24c)

where Cijkl is the elastic stiffness matrix. Using equations (4.24), the incre-mental stress strain relation is expressed as follows:

dσij =

(Cijkl −

CijklBmnBTrsCmnrs + Cijpq

∂f∂αmn

dαmnBpq13dε−1

kl

BabCabcdBcd − ∂f∂εp

vBaa

)dεkl

(4.25a)

dσij =

(Cijkl −

CijklBmnBTrsCmnrs + Cijpq

∂f∂αmn

BpqAmnkl

BabCabcdBcd − ∂f∂εp

vBaa

)dεkl (4.25b)

Equations (4.25) express the well-known elasto-plastic stiffness equations sim-ilar to the form, [Dep]−1= [De]−1 + [Dp]−1 (where, [Dep] represents theelasto-plastic stiffness, [De] represents the elastic stiffness, and [Dp] representsthe plastic stiffness). The expression Amnkl in (4.25b) represents dαmn/dεkl,which essentially is back stress stiffness. From (4.25b), one sees that the backstress αijaffects the plastic stiffness. This behavior is also physically correctbecause the different back stress means the different location of the stresspoint in the yield surface and the elasto-plastic behavior of the materialshould be different.

In equations (4.25) Cijkl is the elastic stiffness matrix which is directlyobtained from experiments. Bij is the gradient of the yield function with re-spect to the stress, as defined in (4.26). The terms in (4.26) are also expressedas follows:

Bij =∂f

∂σij=

∂f

∂p

∂p

∂σij+

∂f

∂sij

∂sij

∂σij(4.26)

∂f

∂p= 2p − po +

32M2

−2αijsij + poαijαij (4.27)

∂p

∂σij=

13δij (4.28)

∂f

∂sij=

3M2

(sij − pαij) (4.29)

∂sij

∂σmn= δimδjn − 1

3δijδmn (4.30)

Substituting equations (4.27) through (4.30) into (4.26), one obtains the fol-lowing equation:


Bij =(2p − po)

3δij +

12M2

(poαkl − 2skl)αklδij

+3

M2(skl − pαkl)

(δikδjl −

13δijδkl

)(4.31)

In (4.25b), ∂f/∂αij is the rate of change of the yield function with respectto the back stress. This rate of change is given in (4.32) by making use of(4.20);

∂f

∂αij=

3M2

−p(sij − pαij) + (po − p)pαij (4.32a)

=3

M2(ppoαij − psij) (4.32b)

From Dafalias (1987) αij is defined by the relation given by (4.21). Integrating(4.21) with respect to time t, and expressing it in incremental form, oneobtains

dαij = 31 + eo

λ − κ|dεp

v|c

po(sij − xpαij) (4.33)

∂f/∂εpv is obtained from the chain rule such that

∂f

∂εpv

=∂f

∂po

∂po

∂εpv

=[

− 1 +3

2M2αijαij

1 + eo

λ − κppo

](4.34)

Substituting equations (4.31) through (4.34) into (4.25b), one obtains elasto-plastic constitutive relationships for the anisotropic modified Cam Claymodel. After this section, represents the usual plastic multiplier.

4.4 Micro-Mechanical Considerations/BridgingDifferent Length Scales

Many soil models implicitly assume the uniform stresses or strains for bothsoil particles (micro-structure) and soil mass (macro-structure). However, itis known that micro-structural quantities may be quite different from those ofmacro-structural quantities. It is also known that micro-behaviors may affectmacro-behaviors. The typical micro-behaviors that are known to affect macrobehavior are the rotation of particles, the interaction of particles, the ratedependency, and the damage. Details of individual mechanisms are discussedin the following sections.

Incorporating micro-behavior into macro-behavior of soil materials can beperformed either by modifying some terms in the constitutive relations or byobtaining the macro-behavior from the average response of micro-behavior.Some micro- mechanisms are incorporated by the first method, and someother micro-mechanisms are incorporated by the second method. The secondmethod is the more fundamental method for bridging two different scales,

4.4 Micro-Mechanical Considerations/Bridging Different Length Scales 39

and it is gaining greater attention in current research. This section describesdetails of modern RVE (Representative Volume Element) concepts and aver-aging schemes.

4.4.1 RVE Concept

The RVE concept is based on the premise that the representative macroscale material properties are obtained by the averaging of the micro scaleproperties for the representative volume. According to this premise, at amaterial point, a non-local tensor, A, is expressed as the weighted averageof its local counterpart A over a surrounding volume V at a small distance|s| ≤ l (l is an internal characteristic length, which is a material property)from x such that

A =1V

∫

V

h (s) A (x + s)dV (4.35)

where, h(s) is an empirical weighting function subject to the normalizingcondition

∫V

h (s) dV = V . The normalizing condition ensures that A = Awhen A(x) is a constant. Bazant and Chang (1984) examined a numberof weighting functions that are used in the integral equation of non-localtheories. For simplicity, it is typically assumed that the weighting function isthe identity tensor such that the non-local measure is simply the average ofthe local measure:

A =1V

∫

V

A (x + s)dV (4.36)

In order to develop the gradient dependent approximation of this aver-aging equation, the local tensor, A must be first approximated by a Taylorseries expansion at s = 0 such that

A (x + s) ≈ A + ∇A · s +12!∇(2)A · s ⊗ s + · · · (4.37)

where ∇(i) denotes the i-th order gradient operator evaluated at the macro-scale. For a general three-dimensional case, the surrounding volume can beassumed to be a sphere with a radius equal to the material characteristiclength such that V = 4

3πl3. Furthermore, truncating the Taylor series af-ter the second order gradient term, the following expression is used for thenonlocal tensor:

A = A +1 4πl5

2!15 V∇2A (4.38)

where ∇2 is the Laplacian operator and is defined as the trace of the secondgradient. By substituting the volume over which the local variable is averaged,we obtain the form used in this work for the non-local measure:

A = A + c∇2A (4.39)


Equations (4.35) through (4.39) are based on the constant length scale l andthe volume of RVE determined from the l. As we shall see, the use of thisconstant length scale in this manner calls for additional discussion (Voyiadjisand Abu al-Rub, 2005).

4.4.2 Characteristic Length

A constant length scale l is easy to use; however, this approach fails to capturethe full behavior of the variation in material strength with sizes as shown bythe experimental results of different size shear band formations in differentgrain sizes. Length scales also change with the degree of accumulated plas-tic strain in the material. An evolution law for the length scale parameteris needed to address the proper modification required for the full utiliza-tion of the current gradient plasticity theories in solving the size effect prob-lem. Gradient approaches typically retain terms in the constitutive equationswith specific order spatial gradients with coefficients that represent length-scale measures of the deformed microstructures associated with the non-localcontinuum. Aifantis (1984) was one of the first to study the gradient regular-ization in solid mechanics. An extensive review of the recent developments ingradient theories can be found in Voyiadjis et al. (2003). The full utilizationof the gradient-type theories hinges on the ability to determine the intrinsicmaterial length that scales with strain gradients.

Gracia (1994) approximated the evolution of the mean dislocation spacingLs in metals with the following equation,

Ls =δd

δ + dp1/m(4.40)

where d is the diameter of the grain size. It is taken as the mean diameter(D50) of the grains. δ is a constant coefficient on the order of 1.0 µm (formetals), p is the effective plastic strain, and m is a material constant. Inthis work it is assumed that the dislocation spacing is directly related to theintrinsic material length-scale. Thus one can write l in terms of the size dand the effective plastic strain p as follows,

l =hδD50

δ + D50p1/m(4.41-a)

where h is a constant. The intrinsic material length-scale can then be assumedto decrease from an initial value lo at yield to a final value of l → 0 atthe saturation point due to the plastic deformation (corresponding to theconventional plasticity limit) at a rate characterized by a constant coefficientk1, such that

l = lo exp(−k1p) (4.41-b)


This two-parameter function is thought to be the simplest form that givesenough freedom for the evolution equation of the material intrinsic length-scale. Equations (4.40) through (4.41) show the asymptotic variation of thelength scale l. The asymptotic variation of the length scale with the effectiveplastic strain is physically more appropriate than a constant value of l . Equa-tions (4.40) through (4.41) assume the same length scale Ls or l. However, aquestion may arise regarding the cogency of an isotropic expression for l, asdiscussed below.

The three dimensional Taylor expansion of the strain function γ is ex-pressed as follows:

γ(x, y, z) = γ(xo, yo, zo)

+∂f(xo, yo, zo)

∂x(∆x) +

∂f(xo, yo, zo)∂y

(∆y)

+∂f(xo, yo, zo)

∂z(∆z) +

12

∂2f(xo, yo, zo)∂x2

(∆x2)

+12

∂2f(xo, yo, zo)∂y2

(∆y2) +12

∂2f(xo, yo, zo)∂z2

(∆z2)

+∂2f(xo, yo, zo)

∂x∂y∆x∆y +

∂2f(xo, yo, zo)∂x∂z

∆y∆z

+∂2f(xo, yo, zo)

∂z∂x∆z∆x + · · · (4.42-a)

When one assumes symmetric gradient distributions and neglecting higherorder terms, the above expression reduces to

γ(x, y, z) = γ(xo, yo, zo) +12

∂2f(xo, yo, zo)∂x2

(∆x2)

+12

∂2f(xo, yo, zo)∂y2

(∆y2) +12

∂2f(xo, yo, zo)∂z2

(∆z2) (4.42-b)

The above expression maybe expressed into the familiar gradient expressionsuch that

γ(x, y, z) = γ(xo, yo, zo)

+[12(∆x)2(∇2γx) +

12(∆y)2(∇2γy) +

12(∆z)2(∇2γz)

](4.43)

where the terms 12 (∆x)2, 1

2 (∆y)2, 12 (∆z)2 may be considered a mathematical

based length scale and the whole of the second terms may be considered amathematical gradient. Equation (4.37) is the one dimensional Taylor expan-sion and (4.43) is the three dimensional Taylor expansion. Using the threedimensional gradients is a more realistic approach to the problems involvedin multi dimensional stress or strain field. Shear bands, for example, will


Large

Small

Fig. 4.13. Shear Band and Strain Gradient

have enormous strain gradients across the shear plane; however, there will beminimal strain gradients along the shear plane, as shown in Fig. 4.13.

Therefore, the use of different length scales for the three different direc-tions will be more appropriate for incorporating multi-dimensional problems.The fact that there are three different length scales implies that one may haverectangular or elliptical RVE’s instead of spherical or square block RVE’s.This implication will be especially valid for anisotropic cohesive soils in thenatural alluvial deposits. Incorporating the concept of physical gradients to(4.43), one obtains

γ(x, y, z) = γ(xo, yo, zo) + c1xl2x(∇2γx) + c1yl2y(∇2γy) + c1zl2z(∇2γz) (4.44)

where cx, cy, cz are material parameters and lx, ly, lz represent the lengthscales. The above equation may be expressed in a simplified form as follows:

γij = γ(xo, yo, zo)ij +3∑

k=1

ckl2k(∇2γk) (4.45)

By comparing equations (4.45) and (4.37), one observes that relation (4.37)is a kind of the isotropic versions of (4.45). An isotropic length scale may bejustified for an isotropic material. By applying (4.45) to isotropic quantitiessuch as the effective plastic strain, one obtains

p = p(xo, yo, zo) + c1l21(∇2p) (4.46)

Comparing equations (4.45), and (4.46) and using the typical strain gradi-ent expression in relation (4.47), one concludes that (4.47) is an undesirableexpression of (4.45).

γij = γ(xo, yo, zo)ij + c1l21∇2γij (4.47)


Equation (4.47) is only approximately correct. If one simplifies equation(4.45) for computational purposes, that is strictly for the computational pur-pose not for the realistic analysis.

Equation (4.45) may be further reduced to the following form by incor-porating c1 into the length scale itself:

γij = γ(xo, yo, zo)ij +3∑

k=1

(cklk)2(∇2γk)

= γ(xo, yo, zo)ij +3∑

k=1

(l′k)2(∇2γk) (4.48)

Note that in the above equations, lk and ∇2γk provide a mutually compen-sating behavior because ∇2γk is expressed as ∂2γk

∂l2k. A larger lk (this is really

∆lk) will produce a smaller ∂2γk

∂l2k. This is why it can reduce the mesh size

dependency of the numerical results. Anisotorpic nature of l′k, therefore, maynot be meaningful if one uses an average number. The use of single lengthscale may be regarded an approximate length scale.

l′k in soils represents the physical distance range in which the soil strainswill be affected. One may expect that l′k depends on many physical propertiesof the material. The mean diameter, the angularity, and the aspect ratio ofsoils are the major factors that may govern this length scale. One should ex-pect changes in the magnitude of the length scale with the increase/decreasein the magnitude of the above factors. In addition to the above inherent ma-terial properties, other physical conditons such as the the effective plasticstrain will affect the length scale because the material becomes more local-ized as the strains become higher. At elevated strains, the localization willbecome stabilized, and the length scale should be constant. In addition tothe physical material properties discussed above, secondary properties suchas the texture, the mineral composition, OCR (over consolidation ratio) andthe strength may be involved in the change of the length scale. However,the only four detrimental parameters that may be easily determined will beincorporated in the further formulation as follows:

l′k = f(D50, R,As, p) (4.49)

where, D50 is the mean grain diameter, R is the roughness, As is the aspectratio, and p is the effective plastic strain. Roughness of grains with reasonableaccuracy may be easily obtained by comparing the roughness and shape ofthe grains to the chart shown in Fig. 4.14 for standard roughness and shape ofgrains (Alshibli and Alsaleh, 2004). Alsaleh (2004) also shows how to calculatethe rougness from the microscopic image of the grains. The range of theroughness is zero to one. In natural soil deposit, the range of roughness ratiois 0.1 (very angular) to 0.85 (well rounded) (Alshibli and Alshaleh, 2004). The


Fig. 4.14. Visual Comparison Chart for Estimating Roundness and Sphericity(Alshibli and Alsaleh, 2004)

aspect ratio of the grain is obtained from the shape of the grain. A perfectsphere has an aspect ratio of unity, while a thin plate has an aspect ratio closeto zero. When one uses the mean grain diameter as the primary parameterfor the initial length scale lo, and assumes the effective plastic strain as theprimary factor in controlling the length scale, one can set up the followingexpression:

l = lo exp(−k1p)= a1D50[a2R + a3As] exp(−k1p) (4.50)

Equation (4.50) is similar in form to (4.41), but additional arameters a2

and a3 are added to better reflect the physical peoperties of soils. Equation(4.50) is merely an expression for one direction; similar expressions for otherdirections may be obtained accordingly.


RVE1

RVE2

Fig. 4.15. Multiple averaging with multiple RVE’s (Voyiadjis and Dorgan, 2003)

4.4.3 Bridging Different Length Scales

The RVE technique is aimed to obtain the macro-properties from the micro-properties. This RVE technique, however, is not limited to integrate onlyonce. Depending on the level of micro-mechanism, multiple RVE’s may usedas required. The concept is illustrated in Fig. 4.15.

The multiple RVE and multiple averaging scheme is widely used inVoyiadjis and Deliktas (2000), Voyiadjis and Dorgan (2003). Voyiadjis andDeliktas (2000) derived a multi-scale gradient theory based on consistentthermodynamic formulations for metals and metal matrix composites. In thisbook, similar equations are derived for the anisotropic modified Cam Claymodel.

The thermoelastic Helmholtz free energy is expressed in terms of theinternal state variables such that

Ψ = Ψ(εe, T, A

(p)(k),∇A

(p)(k),∇

2A(p)(k), ∇A

(p)(k), ∇2A

(p)(k)

)(4.51)

where εe is the elastic strain, T is the temperature, superscript p representsplasticity, and A

(p)(k) is a macroscale internal state variable such as for the

isotropic hardening or the kinematic hardening in plasticity. ∇ representsthe first order gradient, ∇2 represents the second order gradient, and A

(p)(k)

represents the microscale internal state variable in a representative volumeelement. At this point, readers may not feel comfortable with the inclusion ofgradient terms in (4.51). Details of gradient terms will be explained in laterchapters. However, readers may refer to (4.41) through (4.48) to see why thegradient terms may be included in the free energy equation.


When one assumes symmetrical distribution of the gradients, the oddorder gradient terms vanish, and (4.51) becomes

Ψ = Ψ(εe, T, A

(p)(k),∇

2A(p)(k), ∇2A

(p)(k)

)(4.52)

The link between macroscale internal variables and microscale internalvariables is obtained by averaging the microscale internal variables in therepresentative volume element (RVE),

A(p)(k) =

1VRVE

∫

VRVE

A(p)(k)dVRVE (4.53)

where VRVE is the volume of the RVE.The first and second order gradients of the macro and micro internal

variables are defined as follows:

∇2A(p)(k) =

∂2Aij

∂Xk∂Xk

(4.54)

∇2A(p)(k) =

1VRVE

∫

VRVE

∂2A(p)(k)

∂Xk∂XkdVRVE (4.55)

This integration maybe performed over a sub-volume of the RVE. One canexpress the time derivative of (4.52) in terms of its higher order state variablesas follows,

Ψ =∂Ψ∂εe

: εe +∂Ψ

∂A(p)(k)

: A(p)(k) +

∂Ψ

∂∇2A(p)(k)

: ∇2A(p)(k) +

∂Ψ

∂∇2A(p)(k)

.

∇2A(p)(k) (4.56)

where super dots represent the time derivative.By substituting (4.56) into the Clausius–Duhem inequality, one obtains

(σ − ρ

∂Ψ∂εe

): εe + σ : εp − ρ

∂Ψ

∂A(p)(k)

:

A(p)(k) − ρ

∂Ψ

∂∇2A(p)(k)

: ∇2A(p)(k) − ρ

∂Ψ

∂∇2A(p)(k)

.

: ∇2A(p)(k) ≥ 0 (4.57)

where the mesoscale gradient terms ∇2A(p)(k) are dependent on the macroscale

variables, A(p)(k). However, the random periodic boundary condition ensures

that there is no net flux of mesoscale gradients across the RVE boundary.Such a constraint would effectively prevent coupling between macroscale andmesoscale gradient terms.

The first term in (4.57) represents the thermoelastic law of thermody-namics, and the remaining terms represent the total dissipation process due


to plasticity. This relationship can be expressed as the sum of the plasticdissipation as follows,

Π = σ : εp − V(p)(k) : A

(p)(k) − W

(p)(k) : ∇2A

(p)(k) − X

(p)(k) : ∇2 ˙

A(p)(k) (4.58)

where V(r)(k) , W

(r)(k) , and X

(r)(k) are the thermodynamic force conjugates and are

expressed as follows:

V(p)(k) = ρ

∂Ψ

∂A(p)(k)

(4.59a)

W(p)(k) = ρ

∂Ψ

∂∇2A(p)(k)

(4.59b)

X(p)(k) = ρ

∂Ψ

∂∇2A(p)(k)

(4.59c)

One can now express the analytical form of the Helmholtz free energy as thequadratic form of its internal state variables as follows:

ρΨ =12εe : E(φ) : εe +

12a(p)(k)A

(p)(k) : A

(p)(k) +

12b(p)(k)∇

2A(p)(k) : ∇2A

(p)(k)

+12c(p)(k)∇2A

(p)(k) : ∇2A

(p)(k) (4.60)

Using (4.59) and (4.60), the following definitions can be obtained for thethermodynamic forces as follows;

V(p)(k) = a

(p)(k)A

(p)(k) (4.61a)

W(p)(k) = b

(p)(k)∇

2A(p)(k) (4.61b)

Z(p)(k) = c

(p)(k)∇

2A(p)(k) (4.61c)

The value of the thermodynamic forces can be obtained through the evolutionrelations of the internal state variables. However, it should be pointed outthat there are two classes of evolution equations that need to be developed,normally one at the macroscale and the other at the mesoscale level. Theformer can be obtained by assuming the physical existence of the dissipationpotential at the macroscale. The latter can be obtained by a micromechanicalor phenomenological approach.

One may also consider that the evolution equations of the internal statevariables A

(p)(k) can be obtained by integrating the evolution equations of the

local internal state variables at the mesoscale; that is, A(p)(k) over the domain

of the RVE. However, integration of the A(p)(k) is a cumbersome task because

at the mesoscale, A(p)(k) is a function of many different aspects of the material


inhomogeneities such as interaction of defects, size of defects, spacing betweenthem, and distribution of defects within the sub RVE. Therefore, in this workthe evolution equations of the mesoscale internal state variables are obtainedthrough the use of the generalized normality rule of thermodynamics. In thisregard the macroscale dissipation potential is defined only in terms of themacroscale flux variables as follows:

Θ = Θ(εp, A

(p)(k),∇

2A(p)(k),∇

2 ˙A

(p)(k)

)(4.62)

By using the Legendre–Fenchel transformation of the dissipation potential(Θ), one can obtain complementary laws in the form of the evolution laws offlux variables as function of the dual variables as follows:

Θ∗ = Θ∗(σ, B

(p)(k),∇

2B(p)(k),∇2B

(p)(k)

). (4.63)

Equation (4.63) is also expressed as follows,

Θ∗ = F

(σ,

N∑

i=1

po,i,

N∑

i=1

αi

)(4.64)

where po,i and αi are the isotropic and kinematic hardening forces respec-tively. The evolution equations of parameters in (4.64) are obtained as fol-lowing way:

B(p)(k) = CA

(p)(k) = C

∂Ψ

∂B(p)(k)

(4.65)

The evolution equations of po,i and αi in (4.64) are also obtained from thedissipation. When one assumes Perzyna type elastic/viscoplasticity (1963,1966) and the normality rule, one obtains the following evolution equations,

εpv = γ〈φ〉tr

∣∣∣∣∂F

∂σij

∣∣∣∣ = γ〈φ〉tr∣∣∣∣

∂f

∂σij

∣∣∣∣ (4.66a)

αij = C1γ〈φ〉∂F

∂αij= C1γ〈φ〉

∂f

∂αij(4.66b)

˙∇2

αij = C2γ〈φ〉∂F

˙∂∇2

αij

= C2γ〈φ〉∂f

˙∂∇2

αij

(4.66c)

˙∇2

po= C3γ〈φ〉∂F

˙∂∇2

po

= C3γ〈φ〉∂f

˙∂∇2

po

(4.66d)

where αij represent the back stress, γ(φ) represents the viscoplastic mul-tiplier, 〈〉 is the MaCauley bracket, and γ is the viscosity. When one usesthe anisotropic modified Cam Clay model expressed by Voyiadjis and Song(2000), the following expression is obtained for the yield criterion,


f = p2 − ppo +3

2M2(sij − pαij)(sij − pαij) + (po − p)pαij αij = 0 (4.67)

where p is the mean principal stress, po is the initial mean principal stress,M is the slope of the critical state line (CSL) in p vs. q space, sij is thedeviatoric stress, and αij is the rotation of the yield surface with respect tothe p axis (the dimensionless back stress factor). When one incorporates thegradient to the isotropic hardening factor po and the anisotropic hardeningfactor αij , the following expressions are obtained,

po = po − a1∇2po − b1

˙∇2

po (4.68a)

˙αij = αij − a2∇2αij − b2

˙∇2

αij (4.68b)

where po, po are global and local mean principal stresses for the isotropic hard-ening factor, and ˙αij and αij are global and local dimensionless back stressesfor the kinematic hardening factor, respectively. In (4.68a) and (4.68b), theparameters a1, b1, a2, b2, also take into account the dimensional discrepanciesbetween a quantity and the corresponding second order gradient of that quan-tity. One obtains the evolution equation for αij from equations (4.66b) (notethat αij = αij/p, p = mean principle stress). One also obtains the evolutionequation for ∇2 ˙αij from (4.66c). In this study, the evolution equation of ˙αij

is used to obtain the evolution equation of ∇2 ˙αij . The partial derivatives ofthe yield function with respect to αij and ∇2 ˙αij are as follows,

∂f

∂αij=

3M2

ppo(αij − a2∇2αij − b2∇2αij) − sij (4.69a)

∂f

ζ∇2αij

=3

ζM2[−b2psij − p(αij − a2∇2αij − b2∇2αij)

− (po − a1∇2po − b1∇2po)pb2(αij − a2∇2αij − b2∇2αij)] (4.69b)

where ζ accounts for the dimensional discrepancy problem b2 in (4.68b) ac-counts for both the dimensional discrepancy problem and the material char-acteristics.

One also obtains similarly the evolution equation for po from (4.68a) asfollows:

po = −1 + eo

λ − k

po(po + ∆p)∆p

εpv = Kεp

v (4.70)

In the same way as in (4.69b), the following equation is obtained,

∂f

ξ∇2po

=1ξ

pb1 +

32M2

pb1(αij − a2∇2αij − b2∇2αij)

× (αij − a2∇2αij − b2∇2αij))

(4.71)

where ξ is a parameter similar to ζ in (4.69b).


These derivations are based on normality rule assumptions. If one intro-duces other dissipation functions, the evolution equations of state variableswill be changed accordingly.

4.5 Micro-mechanisms

Micro-mechanisms of soils are reported by many researchers (Aifantis 1980,1984; Dafalias 1983, 1984, 1985; Zbib 1993, 1994; Voyiadjis and Song 2005a,2005b; Song and Voyiadjis 2005a; and many others). Traditional micro-mechanisms are studied for individual mechanisms. Recent study of Voyiadjisand Song (2005a) showed the coupled effects of micro-mechanisms in soils.Coupled effects of micro-mechanisms on soil behavior were quite differentfrom those expected from other continua such as steel or composite materi-als. Soils consist of solids and pore spaces. Saturated soils consist of solids andpore water. Some micro-mechanisms affect the solid phase and some othermicro-mechanisms affect the pore phase; therefore, some micro-mechanismsbehave so as to cancel each other’s effect, and some micro-mechanisms behaveso as to enhance each other’s effect. Details of mico-mechanisms are dealt inthe following sections.

Micro-mechanisms are essentially the products of the internal behavior ofsoils; therefore, many micro-mechanisms are related to the internal (embed-ded) stress that is also called back stress in continuum mechanics. Therefore,Sect. 4.5 starts with the discussions of back stress.

4.5.1 Back Stress

The meaning of the back stress is better expressed as “embedded stress” or“internal stress.” The internal stress is embedded in the grain and at thecontact points between the grains and so forth. The embedded stress insidethe grain is termed “short range back stress,” and the embedded stress atthe contact points between the grains (or outside the grain boundaries) istermed “long range back stress” (Lowe and Miller 1984, 1986; Moosbrug-ger and McDowell 1988; Lamar 1989). They are differentiated because theircharacteristics are different. For typical continua such as metals, a linearhardening rule is used for the evolution equation of short range back stress,and a non-linear hardening rule is used for the evolution equation of longrange back stress. The use of independent corotational rates for the shortrange and the long range back stresses is also suggested by Zbib and Aifantis(1988). Justification or verification for using multiple back stresses is moreusually found in discussion of metals than in discussions of geo-materials. Forthat reason, we will introduce at this point the concept of back stress as itrelates to metallurgy.

The ratcheting behavior of metals has been examined by many researchers(Abdel-Karim and Ohno 2000; Bari and Hassan 2000; Barbe et al. 2001;

4.5 Micro-mechanisms 51

Fig. 4.16. Variation of multiple back stresses (Lowe and Miller, 1986)

Moosbrugger et al. 2000; Yoshida 2000), and it has been explained by theconcept of short range back stress. Experimentally, Montheillet et al. (1984)reported the existence of multiple induced textures of metals (short rangeback stress) and independent evolution rules during shear. Mughrabi (1975)has reported the evidence of separate short range and long range internalstress fields in copper. Mughrabi measured the radii of curvature of free pri-mary dislocations pinned by irradiation-induced point defects using trans-mission electron microscopy. Long wavelength (∼6 µm) and short wavelength(∼1 µm) fluctuations in dislocation curvature were observed, indicating shortrange and long range internal stresses. There are many other sources of ev-idence for multiple back stresses; the interested reader may refer to Hirschand Mitchell (1966), and to Kressel and Brown (1968). The graphical repre-sentation of short range and long range back stresses as reported by Loweand Miller (1986) is illustrated in Fig. 4.16.

Figure 4.16 clearly shows the different nature of short range and longrange back stresses, and thus one can see that independent treatment ofmultiple back stresses is justified.

The combined short range and long range back stresses are expressed in(4.72) as follows,

α = α1 +N∑

i=2

αi (4.72)

where α is the composite back stress, α1 is a short range back stress, andαi (i = 2, 3, 4 . . .) is a long range back stress. The physical interpretation ofmultiple back stresses in soils is illustrated in Figs. 4.17 and 4.18. Figure 4.17shows the assembly of clay particles: clusters (aggregations) and conglomer-ation of clusters. The back stresses α1,α2, and α3 for these clay structuresare for different structural scales and conceptually understood from Fig. 4.17.Please note that Collins and McGown (1974) did not refer to the back stresses.


2 ( intra cluster inter granular

level)

3 (inter cluster level)

1 (intra granular level)

Fig. 4.17. Overall microfabrics in Tucson Silty Clay (freshwater alluvial deposit)(Collins and McGown, 1974)

1

2

3

(a) Fabric and particle orientationin Portsea Beach Sand (Lefeber

(b) Idealization of load carrying scheme of Sand (Petrakis andDobry, 1986)

Fig. 4.18. Structure of granular materials


However, one can see the different nature of back stresses. Blenkinsop (2000)introduced 33 different microstructures for geo-materials, and ambitious read-ers may want to include all these 33 microstructures with 33 back stresses.This book, however, deals with only 2 to three back stresses for simplicity.

Figure 4.18a shows the structure of sandy soils. It is observed that somesand grains have continuous contacts and contribute to the load carryingchain, and other sand grains do not have continuous contacts and do notcontribute to the load carrying chain. However, sandy soils are believed toconstitute hidden clustering structures as shown in Fig. 4.18b and are fre-quently modeled by a group of spherical grains through which the stresses arecarried out (Petrakis, 1986). In Fig. 4.18b, the rectangular chains show thestress transferring mechanism. When the rectangular chains are continuous,the load carrying chain is effective; otherwise, the load carrying chain endsat the particle and the grain does not contribute to the load carrying chain.These grains are orphans. The applied stresses are not carried out by orphansbut by the load carrying chains.

In Fig. 4.18b, one can see that many particles are involved in more thanone load carrying chain. That means these particles are subjected to morethan one back stress. The number of back stresses are many, but this presentstudy incorporate only two back stresses because of the lack of informationabout α3 and higher-order back stresses.

The graphical representation of the evolution of multiple back stressesis shown in Fig. 4.19. The threshold shear strain is taken as 10−2% for α1

because the typical threshold strain for the degradation of the shear modulusis experimentally reported to be approximately 10−2% for most clayey soils(Song, 1986).

Figure 4.19 shows the independent evolution of the two different backstresses by Yoshida (2000). The evolution equation of α1 is completely inde-pendent from the evolution of α2. Similarly, adjusting parameters in the α2

curve does not give the properties of the α1 curve. Therefore, independentevolution equations are to be used for a more rational consideration of theembedded stress.

One may assume that α1 is the back stress within the soil particles (shortrange), and α2 is the combination of the back stresses outside the soil parti-cles (long range) as shown in Fig. 4.19. Equation (4.72) is now expressed asfollows:

α = α1 + α2 (4.73)

The quantification of back stress should be incorporated with the fabric,inter-particle charges, and so forth. However, those relations are not availablepresently, and the authors applied the phenomenological approach using theexisting evolution equations of back stresses. When one uses Prager’s linearevolution equation (See Chaboche (1991) and Zbib and Aifantis (1988)) forα1, and Dafalias’ non-linear evolution equation for α2 (Dafalias, 1987), oneobtains the following evolution equations for the back stresses,


(overall)

1 (short range)

2 (long range)

Bac

k S

tres

s

Strain Range

Fig. 4.19. Schematics of evolution of short range back stress (α1), long range backstress (α2) and composite back stress (α)

α1 =23C1ε

p (4.74)

α2 = C2|εpv|(s − X2 pα2) (4.75)

where C1 is a material parameter for the linear kinematic hardening rule, εp

is the plastic strain rate, εpv is the volumetric strain rate, s is the deviatoric

stress, and p is the mean principal stress. C2 and X2 are similar to thosegiven by Dafalias (1987). Also one may find that (4.74) and (4.75) show thesame evolution relationship to the lines in Fig. 4.19.

Each corotation of the different back stresses will follow its own corota-tional rate. Therefore, one should use the following corotational equations,

αi = αi − W si αi + αi W s

i (no sum) (4.76)

where αi represents the corotational rate of the back stress, αi is the timerate of back stress, and W s

i is the modified spin for the back stress. Zbib andAifantis (1988) suggested the following expressions for the modified spin forthe short range back stress and the long range back stress:

W s1 = W s − ξ1(α1d − dα1) (short range) (4.77)

W s2 = W s − ξ2(α2d − dα2) (long range) (4.78)

In (4.77), W s1 is the modified spin for short range back stress, W s is the

spin, ξ1 is a material constant, and d is the plastic strain rate. In (4.78), W s2

is the modified spin for long range back stress, and ξ2 is a material constant.Equations (4.77) and (4.78) are essentially derived for metals. However, testresults for soils (Song, 1986) have shown the consistent axial strain devel-opment during torsional cyclic loading by the resonant column test. (This


strain is not caused by void ratio change.) The authors have assumed a simi-lar mechanism for soils as that used by Zbib and Aifantis (1988) in metals forthe double back stresses and corresponding plastic spin in soils. More detailsof the plastic spin will be introduced in the next section.

We will use the combination of back stresses α1 and α2 to incorporateback stresses into the yield function. When one uses the anisotropic modifiedCam Clay model as shown in (4.79), one may substitute α1 + α2 into α:

f = p2 = ppo+3

2M2(sij−pαij)(sij−pαij)+(po−p)pαijαij = 0 (4.79)

However, for the plastic spin, the back stress α1 may not have to be in-cluded in conjunction with the spin tensor because intra-granular rotationstiffness will be much higher than inter-granular rotation stiffness (and there-fore, intra-granular rotation may be neglected compared to inter-granularrotation.)

4.5.2 Rotation of Particles

Dramatically different behavior of the material at large strain was first re-ported by Truesdell (1955), who noted that an oscillatory stress solution isobtained when a standard linearly hypoelastic material is subject to a largedeformation as shown in Fig. 4.20.

Figure 4.20 shows the behavior of large strain models with or withoutconsidering the rotation of grains. The model that does not consider therotation of grains (W,α = 0) shows unrealistic oscillation of stresses at thehigher strain level. Voyiadjis and Kattan (1999) also reported the similarbehavior.

Fig. 4.20. Oscillatory behavior for large strains (after Zbib and Aifantis, 1988)


a) Plastic Spin

The plastic spin tensor modifies the constitutive equation that follows fromthe change in the substructure of the material. To clearly understand thephysical concept of plastic spin, the stress co-rotation concept is addressedas follows. The stress transformation between two coordinate systems can beeasily expressed as

T ∗ = QTQT (4.80)

where T ∗ is the stress tensor transformed in the new coordinate system, T isthe stress tensor in the original coordinate system, and Q is the transforma-tion matrix. However, when one uses a stress rate in the above constitutiveexpression, the coordinate transformation is not as easy as (4.80). The timederivative of (4.80) is given below,

T∗

= QTQT + Q ˙TQT

+ QTQT

(4.81)

where the upper dot represents the time rate. However, one cannot assumethat the first and third terms are zero in (4.81); therefore, one must introducethe spin tensor W to compensate for these non-zero terms as shown in (4.82),

T∗

= T −WT + TW (4.82)

where, W is called the “spin tensor”. The spin tensor W corrects the con-stitutive relations for the rigid body rotation. Note also that even with theincorporation of the spin tensor W , the stress-strain response of the ma-terial is unstable at high strains in applications such as the simple shearproblem (Voyiadjis and Kattan 1991; Dafalias 1983; Lee et al. 1983) appar-ently because of the absence of the micro-mechanical spin of the material inthe constitutive model. The micro-mechanical spin having been incorporated,it becomes clear that some internal stress change is incorporated. Thus thespin tensor must be modified to reflect such a micro-mechanical change. The“modified spin tensor” is commonly expressed as W ∗ or Ω and is given asfollows,

T∗

= T −W ∗T + TW ∗ = T − ΩT + TΩ (4.83)W ∗ = Ω = W −W P (4.84)

where, W p is the plastic spin. One observes that the plastic spin is the termthat incorporates the spin tensor W and reflects micro-mechanical changes ofthe material. The micro-mechanical changes may be a single-source phenom-enon or may be a multi-source phenomenon. From intuition and the resultsof Anandarajah (1995), one may deduce that the plastic spin may be relatedto the plastic strain (or stress) and back stress. Dafalias (1983) quantifiedthis concept in the following equation,


W p = η(αT − σT ) = η(ασ − σα) (4.85)

where η is a constant, T is a tensor of physical quantities such as stresses orstrains, and α is a constant. Equation (4.85) may be expressed as follows,

W p = ξ(αd ′′ − d ′′α) (4.86)

where d ′′ represents the plastic strain rate and ξ is a function of η and theflow rule. (We will introduce more details of η and ξ later.) Equation (4.86) isthe expression of plastic spin, and it can be incorporated into the conventionalconstitutive relationships.

b) The Microplane Model

The microplane model was proposed by Zienkiewicz and Pande (1977),Bazant (1984), Bazant and Kim (1986), and Prat and Bazant (1989). Thismodel is quite different from the relative spin or plastic spin model. While therelative spin or plastic spin model is a plasticity-based model, the microplanemodel is a microscopic material characteristics-based model that is a kind ofthe classical slip theory of plasticity.

The microplane method was first proposed by Taylor (1938), who sug-gested that the stress-strain relation be specified independently on planes ofvarious orientations in the material. He also assumed that either the stresseson that plane (now called the microplane) are the resolved components ofthe macroscopic stress tensor (static constraint), or the strains on the planeare the resolved components of the macroscopic strain tensor (kinematic con-straint). The responses on the planes of various orientations are then relatedto the macroscopic response simply by super position or, as has been done inrecent works (Bazant, 1984; Carol et al., 1990), by means of the principle ofvirtual work. In the initial application to metals, beginning with Batdorf andBudiansky (1949), only the static constraint was considered. So it was theearly applications to soils (Zienkiewicz and Pande, 1977; Pande and Sharma,1980, 1983; Pande and Xiong, 1982) that successfully described some basicaspects of soil behavior other than strain softening. It appeared, however,that the microplane system under a static constraint becomes unstable whenstrain softening takes place (Bazant and Oh, 1983, 1985; Bazant and Gam-barova, 1984). To cope with these problems, Prat and Bazant (1989) improvedthe microplane model for dynamic constraint.

In the application of this model to clays, the microplanes may be imag-ined to represent the slip on the contact planes between clay platelets orthe planes normal to the platelets on which slip is manifested by normalstrain as shown in Fig. 4.21. Although the correlation to the microstruc-tural mechanism of inelastic deformations is largely intuitive, the microplanemodel has the advantage of being able to distinguish among the intensities ofinelastic strains at various orientations and describe how they are mutually


Clay Particles

1 2

Fig. 4.21. Microplane in Cohesive Soil: 1 = micro-plane as slip plane between clayplatelets; 2 = micro-plane as normal plane to clay platelets (after Prat and Bazant,1990)

constrained. Therefore, the microplane strains εN and εT in Fig. 4.22 maybe imagined to represent the sum of the inelastic relative displacements onall the weak planes contained within a unit volume of the material plus theassociated elastic deformations of all the particles. The equilibrium equationfor constitutive law was set up by equating the macro level strain energyto the summation of the micro level strain energy caused by εN and εT asfollows:

∆V oσijδεij =∫

Ω

∆Vo(σ′NδεN + σ′

T δεT )ΨNdΩ (4.87)

where, ∆Vo = unit volume, σεij = macroscopic stress tensor, δij = incre-ment of macroscopic strain tensor, σ′

N = normal stress in microplane, σ′

T =tangential stress in microplane ΨN = αNvN , and dΩ = sin θdθdφ, with θ andφ = angular spherical coordinates.

The prior advantage of the microplane model is that the constitutive lawis written in terms of the current stresses and strains (not in terms of theirincrements), which allows the model to be explicit with all the numericaladvantages shown by Carol et al. (1990). However, it is also pointed outby Prat and Bazant (1989) that it is impossible for the effective microplane


n

NT

Fig. 4.22. Strain Components on microplane (after Prat and Bazant, 1990)

stresses σ′N and σ′

T to equilibrate the effective macrostress σ′ij exactly since

the microplanes are constrained kinematically.

c) Cosserat “Micropolar” Continuum

Cosserat “Micropolar” continuum is a classical but one of the best continuummechanics to account for the particle rotation (independently to the displace-ment). The Cosserat theory was first developed by the Cosserat brothers in1859; 50 years later, researchers started revisiting and republishing Cosseratwork until the 1960s, when Gunther marked the rebirth of the micromechanicsin his papers. This section discusses the essentials of Cosserat “Micropolar”continuum.

The Cosserat theory can separate the grain rotation from its translation,adding three other degrees of freedom to any point in the 3D continua. Inclassical continuum mechanics, one might have two different strain tensors,the Green–Lagrangian strain tensor and the Eulerian strain tensor. Either oneof these tensors can be decomposed into a symmetric part (stretch tensor)and an antisymmetric part (spin tensor). Granular materials undergo highrotational and translational deformations at failure. However, the classicalstrain tensor fails to describe the real kinematics, such as micro-rotation in


granular materials, and other alternative tensors need to be used instead(Vardoulakis and Sulem, 1995; Oda and Iwishita, 1999).

In geomechanics, interests in the Cosserat continuum began to show upin mid 1980’s, when links were made between Cosserat kinematics and strainlocalization phenomenon. In 1979, Kanatani used the micropolar theory tostudy the flow of granular materials; in this study, the grains were assumed tobe homogenous rigid spheres with the same size. In his formulations, the ve-locity (vi) and the rotation (ωij) of the grains are considered two independentvariables that describe the deformation of the continua. Making use of theseindependent kinematics in the macro conservation laws of mass, linear mo-mentum, and angular momentum, Kanatani was able to propose equations ofmotion for the granular materials. However, the assumption that grains haveuniform contact distribution because of the spherical shape of the particlesis not correct: the non-uniformity of the shape must be accounted for.

Muhlhaus and Vardoulakis (1987) used the Cosserat kinematics for 2Dspace to investigate the thickness of shear bands in granular materials. Theirwork shows one of the strongest links between the Cosserat continuum andstrain localization in granular materials. In their approach, the continuumhas an overall rotation (ωij) which is different from that of the grain or theCosserat rotation (ωc

ij). The deviation in the rotation would actually causenon-symmetry in the strain and stress tensors, and as a result, those tensorswould be different from the classical ones. Assuming infinitesimal deformationin the pre-banding regime, the following kinematics are proposed for planestrain case:

εij = e ij + ωij − ωcij (4.88)

κi = w c3,i (4.89)

where

e ij =u i,j + uj,i

2;ωij =

u i,j − uj,i

2(4.90)

And

ωcij = −e ij3w

c3 (4.91)

where eijk is the Ricci permutation tensor. The curvature or the rotationgradient ki is a measure of the relative rotation of a single grain with respectto the neighboring grain. If the rotation of the continuum coincides withthe grain rotation, then (4.103) collapses into a classical strain tensor. TheCosserat continuum explicitly includes the rotation of grains, and thereforeits equilibrium equations include the moment equilibrium as follows (refer toFig. 4.23):

σ11,1 + σ12,2 + f1 = 0 (4.92)σ21,1 + σ22,2 + f2 = 0 (4.93)

m1,1 + m2,2 + (σ21 − σ12) + Φ = 0. (4.94)


Fig. 4.23. Equilibrium condition for Cosserat continuum

Also the equation of virtual work is expressed as the combination of thestretch and rotation as follows,

δW (i) =∫

V

(σijδγij + miδki) dV (4.95)

where δ indicate the variation in the related quantity.In summary, the beauty of Cosserat continuum is that it considers grain

rotations explicitly while many traditional continuum mechanics do not in-corporate them. (For more details of Cosserat continuum, see Vardoulakisand Sulem (1995) and Oda and Iwishita (1999)).


d) Particulate Mechanics

This approach may be said to be the advanced method since it deals withthe mechanics of individual soil particles. By its nature, this method needsextended computation time. At the current state of knowledge, its applica-tion is primarily used for sandy soils. Particulate mechanics assumes thatthe behavior of sand aggregate is very similar to that of polycrystals becausethe individual grain packing within the sand could be considered in first ap-proximation to behave like randomly oriented crystals (Voyiadjis et al., 1995,1992). However, the main difference between sand particles and randomlyordered crystals is that the properties of these packed sand grains are pres-sure dependent, and the amount of slip affecting each of these packed grains,in contrast to the polycrystalline aggregate, depends on the mean stress.For example, a simple cubic array of equal spheres is a pressure dependentmonocrystal with three sliding planes, with each plane containing two slidingdirections (Voyiadjis and Foroozesh, 1990). Moreover, sand may experiencedilation under shear that does not occur in polycrystalline aggregates. Fi-nally, unlike metals, soils exhibit nonlinear inelastic stress strain behavioreven at very small strains.

The definition of “yielding” in granular media is critical because yieldingin soils is likely to be a controversial topic. This yielding is a result of thenonlinear force-deformation behavior (Mindlin and Deresiewicz, 1953) at theinterparticle contacts. This nonlinear force-deformation causes granular me-dia to exhibit nonlinear inelastic stress-strain behavior at very small strainlevels. Therefore, strictly speaking, cohesionless aggregates, unlike metals, donot have a clear “linear elastic region” defined by an initial yield surface.

There are two distinct deformation mechanisms which operate duringloading of a granular medium. At very small strains (γ < 10−2%) there areno particles sliding and all macroscopic nonlinearity is the result of nonlinear-ities at the inter-granular contacts and of the redistribution of contact forces(one aspect of the material fabric) during loading. The normal componentof the deformation at the contact is nonlinear elastic, while the tangentialcomponent is nonlinear inelastic as a result of the slip at the edges of thecontact annulus between two spheres (Mindlin and Deresiewicz, 1953). Insoils, hysteric behavior is observed during low level shear strain cycling inthe resonant column device, but no permanent volumetric changes or porepressure buildup accumulates.

At larger strains (γ > 10−2%), there is sliding between particles whichmove and rearrange themselves. Therefore, the geometric aspects of the fabricchange as well. This change of geometric fabric manifests itself by irreversiblevolumetric changes if the loading takes place under drained conditions. Thestrain level at which this occurs has been experimentally determined (Dobry,1985) to be on the order of 10−2% for the level of mean stress used in soiltesting (40–270 kPa) and it is called the threshold strain, γt. This slidingof particles is directly analogous to the “slip” in metals. The macroscopic


strain caused by grain slipping at the contact annulus is an order of magni-tude smaller than the macroscopic strain caused by grain sliding and can beconsidered to be a second order effect. Therefore, one possible definition ofyielding, in a manner directly analogous to yielding in metals, is the point instress space at which the geometric fabric of the material changes irreversibly;that is when the first grain slides. Since the sliding of the first particle is verydifficult to monitor in the laboratory, yielding could be defined as the locusof all points in stress space at which the value of the octahedral shear strainis equal to, or less than, 10−2%.

Following the above logic, the yield is defined as the locus of all points instress space [τzθ(σzz − σrr)/2] that have the same value of total (elastic plusplastic) octahedral shear strain, γc

oct = γeoct + γp

oct. The value of γcoct should

be as close to the threshold value as possible, given the restrictions posed bythe experimental device, so that γp

oct will be close to zero. While a criterionof an octahedral shear strain of 10−2% does not necessarily imply that onlyone sphere has slipped, it is assumed that a small percentage of particles haveslipped and that the yield loci obtained using this approach are homothetic tothe true yield surface. Numerical simulations (Petrakis et al., 1991b) supportthis last hypothesis. In the experiments performed by Petrakis and Dobry(Petrakis et al., 1991a; Dobry et al., 1991), the “yield” criterion was set to3×10−2%. This value caused plastic strains to accumulate during the probingportion of the tests. The strain for yield is typically set at 1× 10−2% or less.

With the proper constitutive model, this method is expected to includemost of the micro-mechanical behavior and hardening characteristics of theyield surfaces of a granular medium under various levels of pre-strain. Asmentioned previously, this model is one of the more advanced. However, itsmajor difficulties are its associated computation costs and the fact that itsapplication to clayey soils is not yet widely appreciated.

4.5.3 Grain Interaction

As discussed in Sect. 4.5.2d), there is no substantial inter-grain motion whenthe shear strain is low. However, there may be substantial inter-grain mo-tion when the shear strain is high. At finite strain, the severe grain interac-tion manifests itself and causes the stress transfer to the surrounding areas.Figure 4.24 shows the illustration of the grain interaction at the shear band(plane) by Di Prisco and Aifantis (1999).

During shear, the shear strain in the shear band is greater than that inthe surrounding area. In the case of large shear straining, the particles shouldbe rearranged. Some particles may roll over neighboring particles, and someother particles may drop into the hollow space, thereby creating a changein volume. During volume expansion (dilatancy), the particles in the shearband push out the neighboring particles and increase the contact pressure; theultimate result is higher shear strength in the shear band. When expansionis partly allowed, this behavior causes the stress to transfer from the shear


SHEAR BAND

Fig. 4.24. Illustration of grain interaction (Di Prisco and Aifantis, 1999)

band to the neighboring area. When the neighboring particles are actuallypushed back, the stress in the shear band is redistributed and reduced.

Note, however, that this mechanism should not be confused with thatproposed by Vardoulakis and Aifantis (1989). According to Vardoulakis andAifantis (1989), as volume increases, grain contact decreases, causing grainsto move apart from each other. If, on the other hand, the volume decreases,then new contact is generated and the two grains move closer together. Dila-tancy loosens grain-interlocking and reduces the stress or strain level in thatregion. For the volume contraction region, the quality of grain-interlockingis enhanced. This phenomenon alters stress or strain levels that are causedby the stress or strain gradients (Zbib 1994; Zbib and Aifantis 1988, 1989;Vardoulakis and Aifantis 1991).

Although these two different mechanisms seem to conflict, they are essen-tially the same. Figure 4.25 illustrates the general stress-strain behavior ofdense sand and loose sand specimens in drained condition. In Fig. 4.25, thedense specimen develops higher shear resistance at point 2 than at point 1.One the other hand, the void ratio at point 2 is greater than it is at point1. This result shows that there is a dilatancy from point 1 to point 2, andthe specimen shows higher strength. This is the mechanism of first particleinteraction illustrated by Fig. 4.25 . The void ratio at point 2 is the maximumvoid (the loosest condition) that the specimen may experience during shear.Therefore, the specimen behaves similar to the loose specimen.

The loose specimen, one the other hand, shows lower shear resistance atpoint 5 than at point 4. Because the void ratio at point 5 is greater thanthat at point 4, it is evident that the loose specimen is subject to void ratioreduction (contraction) during shear and gains strength as the number ofcontacts among the particles increases.

This process is not different from the mechanism of second particle inter-action illustrated by Vardoulakis and Aifantis (1989). Therefore, one must becautious when considering the grain interactions in the constitutive relationsbecause the positive and negative effects of grain interactions will be different


(b)

1

2

43

5

Fig. 4.25. Triaxial tests on “loose” and “dense” specimens of a typical sand: (a)stress-strain curves; (b) void ratio changes during shear (after Hirschfeld, 1963 andHoltz and Kovacs, 1981)


depending on the initial condition of the soils. The criterion by which soilsmay be judged “dense” or “loose” is easily determined from Fig. 4.25b. Whenthe initial void ratio of the specimen is higher than the critical void ratio, thenthe soil is loose; if the contrary is the case, the soil is dense. The critical voidratio is defined as the ultimate void ratio at which continuous deformationoccurs with no change in principal stress difference. (For further explanationof the critical void ratio, see Casagrande (1936)).

Mathematically, the gradient concept is expressed as follows,

σij = σij − a1∇2σij (4.96)

where σij represents the stress considering stress transfer (a state that is alsocalled “homogenized stress”), and σij represents the stress before consideringstress transfer (a state that is also called “localized stress”). Equation (4.96)is the gradient equation for stresses; however, the similar equation may beused for other physical quantities. The above discussions may be summarizedas follows: “A stress or strain gradient around the shear plane will cause achange in volume and, eventually, a redistribution of stresses or strains.” Thenext section will discuss the formulation of this gradient concept.

So far the gradient is described in terms of physical meaning. Gradienttheory; however, is known for contributing to the “well posedness” of nu-merical solutions. This effect may be illustrated from a first order Taylorexpansion (Chapra and Canale, 1988)

f(x) ∼= f(x) + f ′(x)(x − x) (4.97)

This relationship can be employed to estimate the relative error of f(x) as in

e[f(x)] =f(x) − f ′(x)

f(x)∼=

f ′(x)f(x)

(x − x) (4.98)

The relative error of x is given by

e(x) =x − x

x(4.99)

A condition number is defined as the ratio of these relative errors as follows:

Conditional Number =e[f(x)]e(x)

=xf ′(x)f(x)

(4.100)

The condition number provides a measure of convergence or divergence ofa numerical error. When the condition number is greater than 1, the errordiverges; when the condition number is less than 1, the error converges. Whenthe condition number is greater than one, it is called “ill conditioned” or “illposed.” In the following example, one can see that the numerical solution canbe ill posed in the shear plane while it is well posed in other areas.


Assume a function f(x) = tan x for x = π/2+0.1(π/2); then the conditionnumber is computed as

Condition number =x(1/ cos2 x)

tan xFor x = π/2 + 0.1(π/2),

the condition number =1.7279(4086)

−6.314= −11.2 (4.101)

Thus, the condition is ill conditioned. Equation (4.101) shows that the majorcause of the ill conditioning appears to be the derivative. For the areas ofsharp strain change (or sharp stress change), one can expect a high derivativeand an ill posed condition. For the same solution at the regular area (wherewe do not have high stress change), the derivative is small and the conditionis well posed.

When one includes the gradient it actually enhances the well posednessof a numerical solution as in the following second Taylor expansions:

f(x) ∼= f(x) + f ′x(x − x) +f ′′ ˜(x)

2(x − x)2. (4.102)

Of course a second order Taylor expansion is more accurate than a first orderTaylor expansion, and the error should be smaller in the second order Taylorexpansion. A smaller error will make a smaller condition number and improvethe well posedness of a numerical solution. Note also that the third term onthe right side of the equation is nothing but the second order gradient term.Therefore, the addition of higher order gradient term actually improves thestability of a numerical solution. The same example shows the effects of thegradient term on the condition number. For (4.102) the condition number isexpressed as follows:

Condition Number =e[f(x)]e(x)

=x[f(x) + 1

2f ′′(x − x)]f(x)

= 0.0968 for x = π/2 + 0.2(π/2). (4.103)

Voila! Equation (4.103) becomes a very well posed numerical condition justby the addition of the second order gradient term. An even higher ordergradient may be added, but one can see from (4.103) that adding the secondorder gradient yields sufficiently stable results. It is therefore clear that thegradient theory has numerical justification as well as physical justification.

In addition, note that the expression in (4.103) suggests that a constanta1 in (4.96) may be the same as (x−x)2

2 . However, that conclusion is notexactly true (The mathematical gradient and the physical gradient may notbe the same.). The gradient constant in real life may be the whole amountof (x−x)2

2 and may be smaller or larger. For that reason, researchers use aconstant that is a material parameter to generalize it.


Fig. 4.26. Effects of strain rate on normalized undrained shear (Kulhawy andMayne, 1990)

4.5.4 Rate Dependency

The viscous property of soils affects the rate dependency of materials. In lo-calized shearing zones such as the shear band, the strain rate is much higherthan it is outside the shear band. Therefore, materials within the shear bandresponds differently from the same materials outside the shear band. Tradi-tionally, researchers have believed that a higher strain rate results in a higherstrength of materials as shown in Fig. 4.26.

The trend shown in Fig. 4.26 is also accepted in other material scienceareas. This trend is easy to understand; one may feel very little resistancewhen one is gently moving one’s hand in water, however, one will feel strongresistance when one is moving one’s hand quickly. Generally, the shearingresistance of a fluid is approximately proportional to the velocity of shearingmechanism. This is one of the reasons why a man can dive into the waterwhile an airplane cannot. The rate dependency in soils, however, may notfollow the general trend explained above. It is addressed below.

Viscosity Dependent Rate Dependency

The following discussion first describes the traditional viscosity related ratedependency; second, it describes non-traditional rate dependency.

The additive decomposition of total strain rate into elastic, visco-plasticand damage parts gives the following relation,


εij = εeij + εvp

ij + εdij (4.104)

where εij is the total strain rate, εeij is the elastic strain rate, εvp

ij is the visco-plastic strain rate, and εd

ij is the damage strain rate. The yield function fneeds to be defined in terms of the effective stresses and strains. For modelingthe rate dependency of soils, this study uses a Perzyna (1963) type visco-plasticity as follows,

εvpij = γ〈Φ(F )〉 ∂fd

∂σij(4.105)

where · is the viscosity, 〈〉 is the McCauly Brackets, fd is a dynamic yieldfunction incorporated with rate dependency, and Φ(F ) is a visco-plastic mul-tiplier that is a function of F , which is defined as follows,

F =fd − ks

ks(4.106)

where ks is a hardening factor in static loading. Oka (1981) obtained fromexperimental results the expression of Φ(F ) as follows,

Φ(F ) = C exp(

mpo

po

)(4.107)

where C and m are the material constants, po is an initial principal stressfor the static condition, and po is an initial principal stress for the dynamiccondition. Equation (4.105) with Oka’s (1981)’s elaborated approach presentsa workable solution for the rate dependency of soils. Later, Oka et al. (1999)developed a more sophisticated rate dependency. However, researchers arefamiliar with the flow rule that has expressions as follows:

εpij = dλ

∂f

∂σij(4.108)

When one uses visco-plasticity, the yield function always expands or shrinksdepending on the strain rate as shown in Fig. 4.27; therefore, researchers callthis condition the dynamic yield surface. Since it is “dynamic”, it does notsatisfy the consistency condition (df = 0) for the general flow rule from whichthe flow scalar dλ is determined. This was the major technical inconveniencein using Perzyna (1963) type visco-plasticity for continuum mechanics. Re-cent research results obtained by a Louisiana State University research groupand Perzyna overcame the above difficulties and presented an easier way ofincorporating visco-plasticity in the form of (4.22).

For a modified Cam-Clay type with isotropic and kinematic hardening,the static yield function is defined as follows:

fs = p2−ppo +3

2M2(sij −pαij)(sij −pαij)+(po−p)pαijαij = 0. (4.109)


Fs

Fd

Ko line

Fig. 4.27. Dynamic yield surface for rate dependent modified Cam Clay model

By mathematical arrangement of (4.109), the isotropic hardening parameterpo is separated as follows:

fs =1

M2 − 1.5αmnαmn

(pM2 +

1.5p

sijsij − 2sijαij

)= po (4.110)

The extension of (4.110) to include the rate-dependent plasticity (viscoplas-ticity) implies that the stress state is no longer constrained to remain on theyield surface but can have fs ≥ 0. Here, the homogenization effect by ratedependency for anisotropic modified Cam Clay model is derived in line witha similar formulation for the von Mises material by Voyiadjis et al. (2003).We define the overstress as

σv = 〈fs − R〉 (4.111)

where σv is the viscous stress in the effective configuration (or the overstress;that is, the difference between the dynamic stress and its static counterpart),R is the isotropic hardening function, and 〈〉 denotes the MacAuley bracketsdefined by 〈x〉 = (x + |x|)/2. σv is the common notion of visco-plasticity(Perzyna, 1966), which implies that an inelastic process can only take placeif σv is positive. In that case, fs ≥ 0. Therefore, we define the dynamic yieldsurface, f , as follows,

f = fs − R(p) − σv(p, p) ≤ 0 (4.112)

where p and p are the effective accumulative visco-plastic strain and its rate.The effective rate of the accumulative visco-plastic strain, p, is defined by


p =

√23ε

vpij ε

vpij (4.113)

where εvp

is the visco-plastic strain rate in the effective configuration. Onecan write the evolution equation for the visco-plastic strain in the effectiveconfiguration, ε

vp, as follows:

εvp

ij = λvp ∂f

∂σij(4.114)

It can be easily shown that p expressed by (4.113) is related to λvp by

λvp =

√32

p√Bii

(4.115)

In classical visco-plastic models of the Perzyna-type (Perzyna, 1963, 1966),which are considered as penalty regularization of rate-dependent plasticity(visco-plasticity), the consistency parameter λvp in the effective configurationcan be replaced by an increasing function of the overstress, as in the followingexample,

λvp =1

ηvp

⟨σv

σγp + po

⟩ml

(4.116)

where m1 is the viscoplastic rate sensitivity parameter and ηvp is the viscosityor fluidity parameter, which is referred to as the relaxation time accordingto the notation given by Perzyna (1988).

By making use of (4.111) and (4.116), one can write an expression for theoverstress function σv as follows:

σv =

[ηvp

√32

p√Bii

]l/ml

[σyp + po] (4.117)

Substituting σv into (4.112) gives the following expression for the dynamicyield surface f in the effective configuration:

f = fs − po

[1 + ηvp

√32

p√Bii

)1/ml]≡ 0 (4.118)

This function is a generalization of the anisotropic modified Cam Clay modelfor rate-dependent materials. The rate-independent condition can be sim-ply recovered by imposing ηvp = 0 (no viscosity effect), so that one has theplasticity case f = fs ≤ 0. In the elastic domain, both fs and f are equiv-alent since, in that case, p = 0 . Therefore, the admissible stress states areconstrained to remain on or within the elastic domain so that one obtains asimilar formulation to the rate-independent plasticity f ≤ 0. However, duringthe unloading process in rate dependent behavior, f < 0 and for a particular


strain-rate does not imply that the material is in the elastic domain; it mayalso be in a visco-plastic state with a smaller strain-rate. Moreover, from therelation in (4.26), it can be seen that as the viscosity parameter ηvp goes tozero (rate-independent case), the consistency parameter λvp remains finiteand positive (though indeterminate) because σv also goes to zero. The ex-tended criterion given by (4.118) will play a crucial rule in the dynamic finiteelement formulation described hereafter. It also allows a generalization of thestandard Kuhn-Tucker loading/unloading conditions:

f ≤ 0, λvp ≥ 0, λvpf = 0 (4.119)

Thus, f still satisfies the constraint equation (f = 0 ). For this reason, the dy-namic yield surface can expand and shrink not only by softening or hardeningeffects, but also due to softening/hardening rate effects.

A Gradient Dependent Formulation for a Rate DependentAnisotropic Modified Cam Clay Model

When one incorporates gradients in the dynamic yield surface equation ex-pressed in (4.120), the following relations are obtained,

f = fs − R(p,∇2p

)− σv

(p,∇2, p, p,∇2p

)≤ 0 (4.120)

fs =1

M2 − 1.5αmnαmn

(pM2 +

1.5p

sij sij − 2sijαij

)= po (4.121)

where upper “∼” represents a gradient implemented non-local quantity. Inthe above equation, sij is determined from the strain gradient. When oneincorporates the gradient into the isotropic hardening factor po and theanisotropic hardening factor αij , the following expressions are obtained,

˙po = po − a1∇2po (4.122)

˙αij = αij − a2∇2αij (4.123)

where ˙po, po are global and local mean principal stresses for the isotropichardening effect and ˙aij , aij are global and local dimensionless back stressesfor the kinematic hardening effect, respectively. In (4.122) and (4.123), theparameters a1 and a2 are length scales that account for both material proper-ties and dimensional consistencies in the equations. One obtains the followingevolution equation for αij , εvp

v and βij assuming negligible micro level gradi-ents (note that aij = αij/p, p = mean principle stress),

εvpv = λvptr

∣∣∣∣∂F

∂σij

∣∣∣∣ = λvptr

∣∣∣∣∂f

∂σij

∣∣∣∣ (4.124a)

αij = C1λvp ∂F

∂αij= C1λ

vp ∂f

∂αij(4.124b)

βij = C4λvp ∂F

∂βij

= C4λvp ∂f

∂βij

(4.124c)


where αij represent the back stress for plasticity, βij represent the back stressfor damage, and λ′vp is the viscoplastic multiplier.

Effects of Ground Water Flow on Rate Dependencyof Saturated Soils

So far, we have discussed the traditional viscosity related rate dependency.Saturated soils, however, are composed of two different phases: solids andpore water. Pore water flows in soils during the dynamic condition. (Pleasenote that the “dynamic” in this section does not denote “vibration”; it justmeans a non-steady state.) When pore water flows in the soil, it changes theexcess pore pressure and, ultimately, changes the effective stress. Pore waterflow is a time-dependent behavior; therefore, it causes another kind of ratedependency.

For the present, let us disregard the viscosity-related rate dependency. Weknow that the slower loading rate will generate smaller excess pore pressure(because there is ample time for pore pressure dissipation), larger effectivestress, and ultimately greater shear strength. This result is a phenomenonthat is exactly opposite to the viscosity- related rate dependency. There-fore, one should note that the rate dependency caused by the porewater flow is opposite to that caused by the viscosity . For the correctevaluation of rate dependency of saturated soils, one should use a coupledrate dependency equation that is nothing other than the coupling of (4.118)and (4.125). Detailed derivation of (4.125) was addressed in Sect. 3.2

(vw − vs) = − 1γw

Kws(grad(Pw) − ρwb) (4.125)

4.5.5 Damage of Solid Grains

Damage of solid grains may be caused by the micro-failure of contact points orby micro-cracks. Clearly, such damage causes the reduction of modulus andaffects the behavior of soils. Typically, sandy soils show more pronouncedchange in behavior from damage (perhaps the primary loading carrying ca-pacity of sands comes from the stresses at the grain contacts.), and mostresearch for soil damage is focused on sandy soils.

Damage is incorporated with the gradient-implemented anisotropic modi-fied Cam Clay model with visco-plasticity by a rather simple approach. Dam-age strain rate εd

ij is expressed similarly to the plastic strain as follows:

εdij = λd ∂g

∂σij(4.126)

The Damage potential g may be defined as follows:

g = (Y − Γ) : P : (Y − Γ) − 1 ≤ 0 (4.127)


The term Y is the damage conjugate force and Γ is similar to the back stressαij . Γ is the gradient dependent kinematic hardening of damage. Using theabove damage function and from Voyiadjis and Deliktas (2000), the damagemultiplier for damage λd is expressed as follows;

λd =∂g∂σ : σ

∂g

∂φ

∂g

∂Y+ ∂g

∂K C d + ∂g∂Γ : Bd : Ad

. (4.128)

Equation (4.128) is a thermodynamics-based damage formulation. How-ever, note that another popular approach for the damage for geo-materialscalled the “disturbed state” concept by Desai and Zhang (1998) and Desaiet al. (1996). A simpler version of “disturbed state” damage formulation byKatti et al.(1999, 2000) is adopted in this study. The damage function g isassumed to be the same as the yield function f , and we further assumed theadditive decomposition of dλ = dλp + dλd. Katti and Yazdani (2001) alsofound that the damage of sand particles depends primarily on the confiningpressure. They suggested the following function for the scalar form of dλd:

dλd

dλ= (1 −A(p)) (4.129)

A = 1 − e−A1∗pA2 (4.130)

where A1 and A2 are material parameters. Combining the above equationswith dλ = dλvp + dλd for viscoplastic material, one obtains the followingequation:

dλ =dλvp

A(p)(4.131)

The above equation assumes isotropic damage and does not differentiate be-tween the damage of micro-structure and macro-structure. This equation isvery easy to use. There is perhaps some doubt that the analysis of damage forsand can be applied to clays; however, we will assume that the fundamentalmechanism of damage for sands and clays is similar.

4.5.6 Summary of Micro-mechanisms

When a material is subjected to large deformation, substructure changes maytake place. For this reason, substructure change needs to be taken into ac-count for a more accurate analysis of the behavior of geo-materials. Thissubstructure change is minimal in small strain problems such as linear elasticproblems. However, it becomes very important for finite strains. The sub-structure change is caused by the external work energy, and thus it shouldbe taken into account for the correct evaluation of the behavior of materials.

For geo-materials, this behavior is more prominent because the bondingforces between the particles are relatively weak compared to other materials


such as steel. Also, the strain range for some geo technical problems such aspost-failure problems, cone penetration tests, and so forth, are several tensto several hundreds percent in magnitude. Thus, severe substructure changescan take place.

Another main characteristic of geo-materials is the anisotropy. As a con-sequence of the nature of the deposition of granular materials, inherentanisotropy exists in geo-materials. During deformation, the material under-goes an induced anisotropy because of the substructure change of the mate-rial. This change means that the initial anisotrophy of geo-materials evolveswith the strain. Thus a valid assessment of the anisotropy of the geo-materialsshould also incorporate substructure changes.

Comprehensive understanding of this substructure change and anisotropyis not yet well known. However, intensive experimental and theoretical studieshave been performed by many researchers (see Masad et al. 1998; Anandara-jah et al. 1996; Anandarajah 1995; Anandarajah and Kuganenthira 1994,1995). Masad et al. (1998) showed a clear relationship between the micro-structures and induced anisotropy, and presented an internal plastic energydissipation formulation to account for fabric re-arrangement. Anandarajahand Kuganenthira (1995, 1994) presented experimental results that indicatethe change of anisotropy during shear is accompanied by the microstructuralchange. This microstructural change may be illustrated by the rotation orrealignment of the soil particles. The microstructural change is inevitablyaccompanied by the new arrangement of the inter-particle attraction andrepulsion forces as pointed by Anandarajah (1995). This change of inter-particle forces is caused by the so called “embedded stress,” and quite oftenthis stress is referred to as the back stress. Thus, one may conclude that thechange of anisotropy is related to the back stress.

The embedded residual stress energy is part of the applied energy, andthus the constitutive equation must consider these terms for the correct equi-librium conditions. Application of this concept to constitutive relationshiphas been performed by many researchers (for example, Lee et al. 1983; Pratand Bazant 1990). This book presents a theoretical basis for the considera-tion of the substructure change and anisotropy. It also presents applicationsof this model by employing the anisotropic modified Cam Clay model andplastic spin. To deal with substructure change and its related anisotropy, onemust incorporate the substructure level (micro-mechanical behavior of thegeo-materials).

There are several approaches for incorporating the micro-mechanical be-havior of soils into the macro-mechanical modeling of soils, such as the micor-plane approach (Prat and Bazant 1990), double slip approach (Zbib 1993),modified spin tensor (Dafalias 1998; Lee et al. 1983), micro-mechanical mod-els (Dobry et al. 1991a, 1991b), and so forth. Even though the atomic or mole-cular level approach is the ultimate goal for implementing the micro-behavior


n+1 configuration

o configuration

Zo, Zn, Zn+1

Yo, Yn, Yn+1

Xo, Xn, Xn+1

Vn

Vn+1

n configuration

Vo

Fig. 4.28. Updated Lagrangian reference frame

of the material, the previously mentioned methods are computationally fea-sible at this time (computation time, numerical error accumulation).

4.6 Equation of Equilibrium of the Externaland Internal Forces in an Updated LagrangianReference Frame

To avoid the large numerical errors caused by the large distortion and ro-tation, this book uses an updated Lagrangian reference frame. (Beginninggeotechnical engineers should note that the difference between the engineer-ing stress and true stress is significant when the deformation is large. Toovercome this numerical dilemma, one can calculate the stresses for a smallincremental deformation and repeat the calculation to the final deformationlevel. This kind of calculation is called an incremental scheme, and whenone uses it together with the traditional coordinate system, this approach iscalled the updated Lagrangian frame.) A schematic diagram of the updatedLagrangian reference frame is shown in Fig. 4.28. As shown in Fig. 4.28, theincremental scheme is used and the configuration is updated at every calcu-lation step. In this chapter, the basic coupled equilibrium equations for finitestrain with plastic spin is derived. The equilibrium equations addressing othermicro-mechanisms are discussed in individual section in Chap. 6.

Following Bathe (1996), the principle of virtual work in an updated La-grangian reference frame is obtained by (4.132),

4.6 Equation of Equilibrium of the External and Internal Forces 77

n+1R =∫

nV

n+1n SABδ

(n+1n εAB

)dnV (4.132)

where nV is the volume of the element at the nth configuration, n+1nSAB is

the second Piola–Kirchoff stress from nth to n + 1th configuration, δn+1n εAB is

the increment of Green–Lagrangian strain from nth to (n+1)th configuration,and n+1R is the external force at the (n + 1)th configuration.

Equation (4.132) can be now expressed as follows,

n+1R =∫

nV

(nσAB + ∆nSAB)δ(neAB + nηAB)dnV

=∫

nV

(nσABδ(neAB + nηAB)dnV

+∫

nV

∆nSABδ(neAB + nηAB)dnV (4.133)

where ∆nSAB is the increment of the second Piola–Kirchoff stress at the nth

configuration, neAB is the linear strain at the nth configuration, and nηAB

is the non-linear strain at the nth configuration. In (4.133), ∆nSAB can beexpressed as

∆nSAB =

t+∆t∫

t

SABdt (4.134)

where SAB is the time rate of second Piola–Kirchoff stress. From Voyiadjisand Abu-Farsakh (1997), Voyiadjis (1988), and Voyiadjis and Kattan (1989),SAB can be expressed as follows, considering the effective stress and the porewater pressure, and assuming the plastic spin to be zero:

SAB = D∗ABCD εCD + JsXs

A,aXsB,bPwδab (4.135)

The superscript “s” is used to distinguish the stress and strains of soils fromthose of water. In (4.135), D∗

ABCD is expressed as follows:

D∗ABCD =

[D

abcd− σ′s

cbδad − σ′sacδbd + σ′s

abδcd + Pwδabδbd

−2Pwδacδbd

]J

sXs

A,aXsB,bX

sC,cX

sD,d (4.136)

The symbol “ ′ ” is used to describe the effective stress. In (4.135) and(4.136), D∗

ABCD is the modified elasto-plastic modulus, XsA,a = ∂n+1Xs

A/

∂nXsa, εCD, is the strain rate, Pw is the pore water pressure, Pw is the pore

pressure rate, Dabcd is the elasto-plastic modulus, and J is the Jacobian. Inthis chapter, kinematic hardening is coupled with anisotropic modified CamClay model, and plastic spin is incorporated. Accounting for the plastic spin,(4.135) is modified such that


SAB = [Dabcd − σ′scbδad − σ′s

acδbd + σ′sabδcd + Pwδabδbd − 2Pwδacδbd]J

sXs

A,a

XsB,bX

sC,cX

sD,dεCD + [σ′s

mbWs′′

ma + σ′sajW

s′′

jb ]JsXs

A,aXsB,b

+JsXsA,aXs

B,bPwδab (4.137)

Equation (4.137) is expressed in a simpler form such that

SAB = D∗ABCD εCD + D∗∗

ABCD εCD + JsXsA,aXs

B,bPwδab (4.138)

where D∗ABCD is defined by (4.135).

D∗∗ABCD is explained as follows. The relation between the plastic spin

tensor and the backstress tensor as given by Dafalias (1983, 1985), Lee et al.(1983), Paulun and Pecherski (1985), Voyiadjis and Kattan (1989,1990,1991)is shown below,

W s′′= ξ(αds′′

− ds′′α) (4.139)

where, ξ is a function of the plastic strain. Paulun and Pecherski (1987)expressed ξ as follows,

ξ =

√32

3ε2eq

1 + 3ε2eq

εeq (4.140)

where, εeq and εeq are the equivalent plastic strain and its rate, respectively.The strain rate ds represents the strain rate of solid grains; it can be decom-posed as shown below,

dsmn = ds′

mn + ds′′

mn (4.141)

where, dsmn is the total strain rate, ds′

mn is the elastic strain rate, and ds′′

mn

is the plastic strain rate. dsmn is the same as εs

mn; however, it is denotedas ds

mn for consistency with most notations implying strain rates for finitedeformations. Lower case subscripts indicate the spatial coordinate systemwhile upper case subscripts indicate the material coordinate system. Theplastic component of the strain rate can be expressed as follows,

ds′′

ij =〈L〉H

nij (4.142)

where L is σ′sijnij (the back stress is already reflected in σ

′sij ), H is the

hardening modulus, and nij is the normal to the yield surface. For the elasticbehavior, one obtains

σ′skl = Es

klmndsmn (4.143)

where E′sklmn is the modulus of elasticity corresponding to the soil skeleton,

as expressed in

E′sklmn = Λsδklδmn + Gs(δkmδln + δknδlm) (4.144)

where Λs and Gs are Lame’s constants for the soil skeleton. Using equations(4.141), (4.142), and (4.143), one obtains (4.145):


σ′skl = E

′sklmn

(ds

mn − 〈L〉H

nmn

)(4.145)

Taking the inner product of the corotational stress tensor with the normalto the yield surface σ

′skl nkl, one obtains

σ′sklnkl = Es

klmndsmnnkl −

EsklmnLn

mnnkl

H(4.146)

However, σ′skl nkl = L, thus (4.144) is rewritten into the following form:

L(H + E′sklmnnklnmn) = HE

′sklmnds

mnnkl (4.147)

One can now solve for L:

L =HE

′sklmndmnnkl

H + E′sabcdnabncd

(4.148a)

= cE′sklmnds

mnnkl (4.148b)

In (4.148b), c is given as follows:

c =H

H + E′sabcdnabncd

(4.149)

Substituting (4.148b) into (4.142), one obtains

ds′′

ij =cE

′sklmnds

mnnkl

Hnij (4.150)

Equation (4.150) may be rewritten in a simpler form as follows,

ds′′

ij = Mijmndsmn (4.151)

where M = cE′sklmnnklnij/H .

Substituting (4.151) into (4.149) gives the following relation,

W s′′= ξ(αamds′′

mb − ds′′

anαnb) (4.152a)= ξ(αamMmbcdd

scd − Mancdd

scdαnb) (4.152b)

= ξ(αamMmbcd − Mancdαnb)dscd (4.152c)

= ξNabcddscd (4.153)

where Nabcd is defined as (αamMmbcd−Mancdαnb). The expression [σ′smbW

s′′

ma+σ

′sajW

s′′

jb ] in (4.137) is obtained as follows:[σ

′smbW

s′′

ma + σ′sajW

s′′

jb

]=[σ

′smbξNmacdd

scd + σ

′sajξNjbcdd

scd

](4.154a)

= ξ[σ′smbNmacd + σ

′sajNjbcd]ds

cd (4.154b)


By making use of the relation dscd = XC,cXD,dε

sCD and multiplying both

sides of the equation by J XA,aXB,b, expression equation (4.154b) may berewritten in the following form,

[σ′smbW

s′′

ma + σ′sajW

s′′

jb ]JXA,aXB,b

= ξ[σ′smbNmacd + σ

′sajNjbcd](JXA,aXB,bXC,cXD,d)εs

CD (4.155a)= D∗∗

ABCD εsCD (4.155b)

where D∗∗ABCD is defined as

D∗∗ABCD = ξ[σ

′smbNmacd + σ

′sajNjbcd](JXA,aXB,bXC,cXD,d) (4.156)

From (4.156) and (4.137), one obtains (4.157),

SAB = D∗∗∗ABCD εs

AB + JsXsA,aXs

B,bPwδab (4.157)

where D∗∗∗ABCD is defined as D∗

ABCD + D∗∗ABCD. Substituting (4.157) into

(4.134), one obtains the new expression for the second Piola–Kirchoff stress:

∆nSAB = D∗∗∗ABCD

t+∆t∫

t

εsABdt +

t+∆t∫

t

JsXsA,aXs

B,bPwδabdt (4.158a)

= D∗∗∗ABCD∆εs

CD + JsXsA,aXs

B,b∆Pwδab (4.158b)

Substituting (4.158b) into (4.134) and neglecting the ∆η · η term (because itis very small), one obtains (4.1.61):

n+1R =∫

nV

(nσ′sAB + nPwδAB)δneABdnV

+∫

nV

(nσ′sAB + nPwδAB)δnηABdnV

+∫

nV

D∗∗∗ABCD(∆neAB + ∆nηAB)δneABdnV

+∫

nV

D∗∗∗ABCD(∆neAB)δnηABdnV

+∫

nV

JsXsA,aXs

B,bwδab∆Pw(neAB + nηAB)dnV (4.1.61 = 4.6.28)

Equation (4.6.28) is the equation of equilibrium of the external and internalforces in an updated Lagrangian reference frame expressed in terms of theeffective stress and pore water pressure.


So far, the relationship between the pore water pressure and the hydraulicconductivity has not been shown. The Prevost (1980) coupled theory of mix-tures that was derived in Sect. 3.2 is used here as follows,

JsCsij εij − JCs−1

ij Cs−1ij Xs

D,a

∂

∂Xp×[nw

ρwKws

ABXsa,A

(∂Pw

∂XB− ρwBB

)]= 0

(4.159)

where Csij = Xs

K,IXsK,I , εij , is the strain rate tensor, Xs

a,A = ∂n+1Xa/∂nXA,

Bb = bb/Xsb,B , and J is the Jacobian. Using (4.6.28) and (4.159), one obtains

the coupling of the stress, the deformation, the pore water pressure, and thehydraulic conductivity. In a matrix form, the coupling of (4.6.28) and (4.159)is expressed as shown below:

[nK − nΩ−nΩt

nΨ δt

] [∆U∆W

]=[

nΦnΠ

](4.160)

In equations (4.160), nK is the stiffness matrix, nΩ is the coupling matrix,nΨ is the flow vector, ∆U is the incremental nodal displacement, ∆W is theincremental pore water pressure, and t is the incremental time. More detailsof (4.160) are given below:

nK = nKL + nKnL + nKTnL + nK

s (4.161)

nKL =∫

nv

nBTLD∗∗∗

nBLdnV (4.162)

nKnL =∫

nv

BTLD∗∗∗

nBNLdnV (4.163)

nKs =

∫

nv

nB∗TNL

nσbB∗NLdnV (4.164)

B∗NL is the geometric nonlinear strain displacement matrix ∆U = incre-

mental nodal displacement

nΩ =∫

nV

J sX sA,aX

sB,b(nB

TL + nB

Tnl)Nabd

nV (4.165)

N = mN ,mT = 1, 1, 0 (4.166)

∆W = incremental nodal pore water pressure


nΦ = n+1R −∫

nV

nBTL

nσdnV (4.167)

nΨ =∫

nV

J s(nw/ρw)C s−1ij C s−1

ij nKwsABN ,AN,BdnV (4.168)

nΠ = δtG − δtΨW n +∫

nS

qnPwdnV (4.169)

G = −∫

nV

J snwC s−1ij Cs−1

ij nKwsABN ,AN ,BdnV (4.170)

Thus, by solving (4.160), one can predict the behavior of the soil us-ing an anisotropic modified Cam Clay model that is coupled with the largestrain elasto-plastic constitutive equations in an updated Lagrangian refer-ence frame that is again coupled with the theory of mixtures. Full derivationsof (4.160) are shown in Sect. 5.1.

5 Finite Element Formulations

In solving finite strain problems, typical numerical issues are related to thedifference between the initial configuration and the final configuration. Theformulations that are based on the initial configuration are no longer valid.Typical example is the difference between the “engineering stress” and “truestress”. For infinitesimal strain problems, the engineering stress and truestress are reasonably the same. However, for finite strain problems, these twostresses may be quite different.

To overcome these difficulties, two reference frames are used. One is an up-dated Lagrangian reference frame and another is an Eulerian reference frame.In geotechnical engineering, both formulations are used. In this book, an up-dated Lagrangian reference frame is used for the convenience of formulationas discussed in Sect. 4.6.

5.1 Updated Lagrangian Reference Sheme

The finite element discretization is used here for the displacement u and thepore water pressure Pw as follows,

u = h · U (4.1)P w = N · W (4.2)

where h is the displacement shape function, N is the pore water pressureshape functions, U is the nodal displacement, and W is the nodal pore waterpressure. The linear and nonlinear strains may be expressed as follows:

e = BL · U (5.3a)

η =12BNL · U (5.3b)

The linear and nonlinear strain variations are given as follows:

δe = BL · δU (5.4a)

δη =12BNL · δU (5.4b)

The pore water pressure gradient is given such that

84 5 Finite Element Formulations

∂Pw

∂XB= N ,BW (5.4c)

In (5.4), BL and BNL are the linear and nonlinear strain-displacement ma-trices. Substitution of (4.2) and (5.4c) into (4.159), yields

δUT (nKL +n KNL +n KTNL +n Ks)∆U − δUT

nΩ∆W = δUTnΦ (5.5)

The above equation is valid for any δUT ; therefore one obtains

(nKL +nKNL +n KTNL +n Ks)∆U −n Ω∆W =n Φ (5.6)

Making use of the following expression,

nK = (nKL +n KNL +n KTNL +n Ks)

Equation (5.6) is expressed as follows:

nK∆U −n Ω∆W =n Φ (5.7)

The components of nK are expressed as follows,

nKL =∫

nV

nBTLD∗∗∗nBLdnV (linear stiffness matrix) (5.8)

nKNL =∫

nV

nBTLD∗∗∗nBNLdnV (non-linear stiffness matrix) (5.9)

nKs =∫

nV

nCNLdnV (non-linear geometric stiffness matrix) (5.10)

where CNL is defined as;

nCNL =n B∗TNL

nσnBNL (5.11)

nΩ in (5.6) [See also (4.165)] is expressed as

nΩ =∫

nV

JsXsA,aXs

B,b(nBTL + nBT

NL)NabdnV (coupling matrix) (5.12)

whereN = mN (5.13)

and mT = 1 1 0 for two dimensions, and mT = 1 1 1 0 0 0 for threedimensions such that

σ = σ′ + mPw (5.14)

nΦ in (5.7) is expressed as follows:

nΦ = n+1R −∫

nV

nBTL

n σ dnV (5.15)

5.1 Updated Lagrangian Reference Sheme 85

From (3.41) and using Galerkin’s weighted residual method requires that∫

nV

[JsCs

ij−1εij − JsCs

ij−1Cs

ij−1

× ∂

∂XA

nw

γwKWS

AB

(∂Pw

∂XB− ρwBB

)]PwdnV = 0 (5.16)

The weak form of above equation is obtained by applying Green’s theory(Zienkiewicz and Pande 1977) as follows,∫

nV

[JsCs−1

ij εijPwdnV −∫

nV

nw

γwJsCs−1

RS Cs−1

RS KWSAB

(∂Pw

∂XB− ρwBB

)]

× ∂Pw

∂XAdnV −

∫

nS

qnPwdnA = 0 (5.17)

wherePw = Nw (5.18)

is the weighted residual (virtual pore pressure). The pressure gradient is givenas follows:

∂Pw

∂XA= N,Aw (5.19)

qn is the seepage velocity normal to the boundary surface. Substituting (5.1.4)and (5.19) into (5.17), one obtains

−nΩT u + nΨW = G −∫

nS

qnPwdnA (5.20)

nΨ =∫

nV

nw

γwJsCs

RS−1Cs

RS−1KWS

AB N,AN,BBBdnV (5.21)

Equation (5.20) can be solved by various processes of time stepping as follows,

un+1 = un + δtnun+β (5.22)

whereun+β = (1 − β)un + βun+1 (5.23)

such that a particular value of β corresponds to a particular integration rule.For example, β = 0 corresponds to a forward difference integration, β = 1/2corresponds to a linear variation and to the trapezoidal integration, and β =1 corresponds to a backward difference integration. Considering the stabilityof the numerical time integration scheme, it reveals that for stability, β ≥ 1/2is required (Prevost, 1981). Here the backward difference scheme is adaptedwith β = 1. Therefore (5.22) becomes

un+1 = un + δtnun+1 (5.24)


Equation (5.20) maybe expressed as follows:

−nΩT δtun+1 + nΨδtnWn+1 = δtnG − δt

∫

nS

qnPwdnA (5.25)

Utilizing the following relationships,

∆U = δtnun+1 (5.26)W n+1 = W n + ∆W (5.27)

Equation (5.25) can be rewitten as

−nΩT ∆U + nΨδt∆W = nΠ (5.28)

where nΠ is defined as

nΠ = δtnG − δtnΨW n −∫

nS

qnPwdnA (5.29)

Assembling (5.7) and (5.28), one obtains the following matrix form of coupledequations for the two-phase media:

[nK −nΩ−nΩt δtnΨ

] [∆U∆W

]=[

nΦ

nΠ

](5.30)

5.2 Finite Element Implementation

The proposed coupled system of equations derived earlier is implementedinto the finite element program CS-Soil. This program is used to solve alarge strain non-linear behavior of soils. In this process, element matricesare derived for 8-noded isoparametric plane strain element Q8P4. The useof the isoparametric elements has the advantage of describing the curvedboundaries in the deformed configuraions. The shape functions N for the8-noded isoparametric element are given as follows:

h =

[h1 0 h2 0 h3 0 h4 0 h5 0 h6 0 h7 0 h8 0

0 h1 0 h2 0 h3 0 h4 0 h5 0 h6 0 h7 0 h8

](5.31)

The element displacements are related to the nodal displacement using theshape functions h as follows:

ui =8∑

k=1

hku(k)i (5.32)

The linear and nonlinear incremental strains are related to the displacementderivatives as follows:

5.2 Finite Element Implementation 87

neAB =12(nuA,B + nuB,A) (5.33)

nηAB =12(nuK,A − nuK,B) (5.34)

Because the shape functions h are expressed in terms of local coordinates rand s, a chain rule is applied in order to refer the displacement derivatives interms of the global coordinates. The chain rule implies the following,

∂h

∂r

∂h

∂s

= [J ]

∂h

∂nX1

∂h

∂nX2

(5.35)

where

[J ] =

[J11 J12

J21 J22

]=

∂nX1

∂r

∂nX2

∂r

∂nX1

∂s

∂nX2

∂s

(5.36)

The inverse of (5.35) gives the following relations:

∂h

∂nX1

∂h

∂nX2

= [J ]−1

∂h

∂r

∂h

∂s

(5.37)

The displacement derivatives with respect to the global coordinates are givenas follows,

∂ui

∂nXj=

8∑

k=1

∂hk

∂nXjU (k)(i = 1, 2; j = 1, 2) (5.38)

where∂hk

∂nXj= J−1

j1

∂hk

∂r+ J−1

j2

∂hk

∂s(5.39)

By using the previous expressions and the nodal displacements, the defor-mation gradient matrix F for the increment displacement can be obtained asfollows:

F =

∂ui

∂nXj

=

∂u1

∂nX1

∂u1

∂nX2

∂u2

∂nX1

∂u2

∂nX2

(5.40)

Once the deformation gradient matrix is obtained, the linear and nonlinearincremental strains can be computed. Using the definition of (5.33), the linearstrain displacement matrix BL is expressed as follows:


[BL] =

nh1,1 0 nh2,1 0 nh3,1 . . . nh7,1 0 nh8,1 0

0 nh1,2 0 nh2,2 0 . . . 0 nh7,2 0 nh82

nh1,2nh1,1

nh2,2nh2,1

nh3,2 . . . nh7,2nh7,1

nh82nh8,1

(5.41)

In accordance with (5.41), the nonlinear strain-displacement matrix BNL maybe expressed as follows,

BNL = G · Q · H (5.42)

where

[G] =

nh1,1 0 nh1,1 0 nh2,1 0 . . . nh8,1 0 nh8,1 0

0 nh1,2 0 nh1,2 0 nh2,2 . . . 0 nh8,2 0 nh82

nh1,2nh1,1

nh1,2nh1,1

nh2,2nh2,1 . . . nh8,2

nh8,1nh82

nh8,1

(5.43)

[Q]T =

nu11 0 0 0 nu2

1 0 0 0 . . . nu81 0 0 0

0 nu11 0 0 0 nu2

1 0 0 0 . . . nu81 0 0

0 0 nu12 0 0 0 nu2

2 0 0 0 . . . nu82 0

0 0 0 nu12 0 0 0 nu2

2 0 0 0 . . . nu82

(5.44)

[H] =

nh1,1 0 nh2,1 0 . . . nh7,1 0 nh8,1 0nh1,2 0 nh2,2 0 . . . nh7,2 0 nh82 0

0 nh1,1 0 nh2,1 . . . 0 nh7,1 0 nh8,1

0 nh1,2 0 nh2,2 . . . 0 nh7,2 0 nh8,2

(5.45)

The nonlinear matrix nCNL is given as follows:

[nCNL] = [nB∗NL]T [nσ][nB∗

NL] (5.46)

The geometric nonlinear strain displacement matrix [B∗NL] may be arranged

in the following form,

[B∗NL] =

nh1,1 0 nh2,1 0 . . . nh7,1 0 nh8,1 0nh1,2 0 nh2,2 0 . . . nh7,2 0 nh82 0

0 nh1,1 0 nh2,1 . . . 0 nh7,1 0 nh8,1

0 nh1,2 0 nh2,2 . . . 0 nh7,2 0 nh8,2

(5.47)

and the stress [nσ] is given by

5.3 Remeshing and Return to Yield Surface 89

[nσ] =

nσ11nσ21 0 0

nσ21nσ22 0 0

0 0 nσ11nσ21

0 0 nσ21nσ22

(5.48)

5.3 Remeshing and Return to Yield Surface

In order to solve the system of nonlinear equations that arises from themathematical formulation of the coupled equations, a finite strain problem isprocessed incrementally and iterations are performed within each incremen-tal penetration. The full Newton-Raphson iterative method is used in orderto obtain convergency. An abridged algorithm procedure is shown in this sec-tion. Full details of the algorithm procedure can be referred to Abu-Farsakh(1997) and Kiousis (1985).

Incremental Loop

The total penetration length is divided into smaller increments. At the be-ginning of each penetration increment, the incremental displacement ∆Uappl

is computed by dividing the total penetration length by the number of incre-ments. The Newton-Raphson iterative procedure is carried out within eachincrement in order to solve for the incremental load Rinc and excess porepressures ∆Pw.

Iterative Loop

The applied iterative incremental load, Riter, for the first iteration is givenby:

Riter = Rinc

The Newton–Rapson iteration loop is then carried out as described by thefollowing steps:

(1) Convert the iterative applied loads, Riter, to account for the skew bound-aries, such that the degrees of freedom at the skew boundary nodes arenormal and tangential to the skew boundary.

(2) Loop over the whole elements.(3) Assemble the global stiffness matrix.(4) Use a linear solver to solve the nonlinear equations for the iterative

incremental scheme.(5) Rotate back the iterative incremental displacements and loads at the

skew boundaries to the original coordinate system.


(6) Add the iterative incremental displacement to the previously computedquantities.

(7) Compute the Lagrangian iterative incremental strains ∆εi with respectto the previous configuration from the iterative incremental displace-ments ∆Ui.

(8) Compute the iterative incremental stresses ∆σi using the sub-incrementation technique and applying certain corrections due to cross-ing the yield surface and the return to the yield surface [See Abu-Farsakh(1997) and Kiousis (1985) for details].

(9) Update the constitutive matrix D and calculate the equilibrating forcesfor the element stresses (Requil).

(10) Calculate the out-of-balance (the corrected) load vector from the accu-mulated applied load vector Rappl and the equilibrium load vector Reqt

as follows:Rcor = Rappl − Requil

(11) Check the convergence of the solution using an appropriate convergencecriterion. In this book, the displacement criterion is adapted as follows:

√∑uiter uiter ≤ tol

√∑uinc uinc

where tol is the tolerence.If convergence does not meet, repeat the iterative steps 1 through 11. Ifconvergence meets the criteria, then proceed to step 12.

(12) Update the nodal coordinates by adding the incremental nodal displace-ments.

(13) Move to the next load increment until the total load is applied.

6 Applications

6.1 Piezocone Penetration Test (PCPT)

6.1.1 Introduction

A cone penetrometer is a device that is pushed into the ground at a constantrate to measure ground responses continuously. Typical measurements aremade for tip resistance, friction resistance, and pore pressure response. Fromthese measured quantities, the properties of ground are determined directlyor indirectly. Crude types of penetrometers were introduced as early as theRoman era; at that time, the number of slaves required to push the rod intothe ground was counted and used to quantify the strength of the ground.With the advent of modern science, this method was diversified and severaldifferent penetrometers were developed, such as the cone penetrometer, thestandard penetrometer, the Swedish penetrometer, and others. Among thesepenetrometers, the cone penetrometer becames one of the widely used pen-etrometers because of its superior performance (repeatability, convenience,economy, and so forth).

A modern cone penetrometer, the mechanical cone penetrometer, wasfirst introduced in the early 20th century in Europe. The mechanical conepenetrometer pushed the cone into the ground by a mechanical driving sys-tem (chain + gear system), and measured the end resistance by probingrings. Later, a cone penetrometer that could simultaneously measure bothend bearing and friction resistance appeared, and it was called the mantlecone penetrometer.

Holland, which contributed much to the development of the modern conepenetrometer (including its unofficial but internationally known standardname, “Dutch Cone”) perhaps used the cone penetrometer most widely inearlier days. The early “Dutch Cone” was not equipped with modern sensorsand automatic driving systems, but it presented more consistent and reliableresults with less cost compared to its strong cousin, the SPT (Standard Pen-etration Test); thus, it continued to evolve with continued use. Traditionalmechanical cone penetrometers use the double rod system – inner rods andouter rods. Measurement was carried out by pushing the inner rods for re-sistance measurements, and subsequently by pushing the outer rods for the

92 6 Applications

advancement of the whole system. Thus, the test procedure was not contin-uous; it was a “stop and go” process.

With the advent of the modern sensors and electric technology, electriccone penetrometers were introduced in the 1970’s (Torstensson, 1975), a de-velopment that resulted in the greatly increased productivity and overallperformance. The electric cone penetrometer used load cells and a motorizeddriving system instead of probing rings and a manual driving system; thus,readings could be recorded electronically without stopping the penetrationprocess. The penetration speed also could be controlled more accurately.

The piezocone is a type of electric cone penetrometer which, by virtureof having pore pressure monitoring capability, can enhance the assessment ofengineering parameters, especially the hydraulic properties of soils. Measure-ment of the pore water pressure generated while advancing a cone tip into theground (and its subsequent dissipation when the penetration stops) was firstmade in Sweden in the early 1970’s (Wissa et al., 1975; Torstensson 1975).Early types of piezocone penetrometers did not have the ability to measureboth cone resistance (end bearing and/or friction) and pore pressure simulta-neously. Subsequent developments in transducer technology during the early1980’s involved the incorporation of piezometric elements into the standardelectric cone penetrometers. This development made it possible to measurepore pressure, cone resistance, and skin friction simultaneously. Tumay et al.(1981) is known as the first group who utilized the simultaneous measure-ment of cone resistance and pore pressure (Zuidberg et al., 1982). Later,many researchers contributed the valuable application of the simultaneousmeasurement of cone resistance and pore pressure (Baligh et al., 1981; Cam-panella and Robertson, 1981; Muromachi, 1981; de Ruiter, 1981; Zuidberget al., 1982; Smiths, 1982; Lunne et al., 1997) , and opened the era of thefully equipped piezocone penetrometer.

At present, many other sensors can be attatched to the cone probesimultaneously in order to obtain the various soil properties. Accelerome-ters can be attatched to detect the seismic response of soils (Campanella,1994). Electric resistivity sensors, thermal sensors or infrared sensors canbe attatched to detect ground contamination. Microphones may also be at-tatched to detect the sonic response of the ground, while the radioactivesensors can be attatched to detect the radioactive materials in the ground(Muromachi, 1981; Lunne et al., 1997). Recent developments include attach-ing a video camera and high intensity light to physically see the undergroundconditions (Envi, 1996). Virtually any kind of sensor can be incorporated withthe modern cone penetrometers. For geotechnical purposes, the most widelyused combination is two load cells for measuring the end bearing and theside friction, one piezometer for pore pressure response, and one inclinationsensor for the inclination check. All of these sensors are electronic sensors,and an industrial-computer based electronic readout is adopted, so most ofthe measured response is recorded and analyzed on a real-time basis.

6.1 Piezocone Penetration Test (PCPT) 93

Fig. 6.1. Typical cone penetrometer tips (Courtesy of Dr. M. Tumay)

Recently, there have been several efforts to increase the efficiency of thepiezocone penetration test (PCPT) even further by mechanical improvement.Tumay et al. (1998) and Tumay and Kurup (1999) devised a spiral rod pen-etrometer system for non-inturrupted continuous advancing of the cone tip, adevelopment that eliminates time for connecting push rods. This same tech-nique was used for underwater ground exploration at the sea bottom. Envi.Corp. (1996) developed the wireless piezocone penetrometer, which does notrequire the hassle of wiring during PCPT. Mayne and Rix (1996) developedthe combined system of pressure meter and cone penetrometer for the simul-taneous testing of the pressure meter and cone penetrometer. Typical conepenetrometer tips are shown in Fig. 6.1. Measuring sensors are built intothe cone body. The white rings shown in Fig. 6.1 are the porous elementsthat protect piezometers. To improve the mobility of the cone penetrom-eter system, the system is typically mounted on all-wheel driven trucks orcrawler-equipped vehicles. For off-shore application, the systems are mountedon the jack-up pontoons or barges. These vehicles or barges have their ownpower supply and data logging system for real time data processing. Fig-ure 6.2 shows a typical truck-mounted driving system and data-collectingsystem that are currently used in LTRC (Louisiana Transportation ResearchCenter).

6.1.2 Current Practice of Determining Hydraulic Propertiesfrom the Piezocone Penetrometer

Determination or estimation of ground properties from the piezocone pene-tration tests is made by using empirical equations or theoretical analyses.

94 6 Applications

Fig. 6.2. Inside view of a truck-mounted driving and data collection system (Cour-tesy of Dr. M. Tumay, 1999)

Stress strain parameters are typically obtained from cone tip resistanceand friction resistance. Hydraulic properties are typically obtained from thepore pressure dissipation response.

Measurements of pore water pressures generated while advancing a probeinto the ground and their subsequent dissipation were first made in the early1970’s. The determination of the coefficient of consolidation and hydraulicconductivity from PCPT utilizes the pore pressure dissipation test data ob-tained during the stopping period of the Piezocone (Acar et al., 1982; Tumayand Acar, 1985). However, the above method has two major explicit draw-backs which make the required time for dissipation test very long, thereforemaking the continuous hydraulic conductivity profile virtually impossible.Typically, it takes several hours (including idling time) for one dissipationtest. Without the dissipation test, the whole procedure for one piezoconepenetration test takes 1.5 to 2 hours for a 30 m penetration. For this reason,the dissipation test greatly decreases the efficiency of the piezocone penetra-tion test. Moreover, because of the prolonged waiting time for the dissipationtest, only a limited number of dissipation tests can be carried out. As a result,continuous hydraulic conductivity profiles cannot be obtained even thoughother profiles (cone resistance, friction resistance, etc.) are practically con-tinuous.

This chapter presents a new method of determining the hydraulic con-ductivity of soils that utilizes the coupled theory of mixtures but does notrequire dissipation tests; hence, not only does it not require additional testtime, but it potentially provides a continuous hydraulic conductivity profile.


It has been argued that the piezocone obtained hydraulic property is forthe disturbed condition, not for the intact condition (Lunne et al., 1985).It is believed that the change of soil structures may affect the hydraulicconductivity. Previous research (Baligh and Levadoux, 1986; Robertson et al.,1992), however, had shown reasonable agreement between the laboratory testresults and the piezocone dissipation test results. In general, it is believedthat the hydraulic properties obtained from the piezocone penetration testare enough close to the field values.

Another aspect of this chapter is the development of analytical tools forpiezocone penetration tests. Along with the significant evolution of mechan-ical aspects of PCPT, there were also great achievements in the analysismethod for the PCPT results. However, many of these works concentratedon the interpretation of stress-strain properties of soils. From the point ofutilization of the penetration pore pressure response, most of the efforts werefocused on the correction of end bearing and side friction, or on soil classifica-tion. Relatively fewer efforts were focused on the direct interpretation of pen-etration pore pressure response itself. This chapter focuses on the transientpore pressure response during the penetration, thereby extracting valuableinformation that has been overlooked in past research.

6.1.3 New Approaches

Two new approaches for estimating hydraulic properties are presented in thissection. The first method utilizes only one pore pressure data and it is termedthe one-point method (OPM). The second method utilizes two pore pressuredata and it is termed the two-point method (TPM). The one-point methodis based on the idea that a measured pore pressure from PCPT is directly afunction of hydraulic conductivity as follows,

u = f(σ, k, . . . ) (6.1)

where, u is a pore pressure, σ is a stress tensor, and k is a hydraulic condutiv-ity. Equation (6.1) is nothing but a conceptual expression of Biot’s coupledof theory of mixtures, and (6.2) is a working equation of this concept:

[nK −n Ω−nΩt

nΨδt

] [∆U∆W

]=[

nΦ

nΠ

](6.2)

The two-point method utilizes the pore pressure measurement results attwo different locations and time so that the pore pressure dissipation effectis computed. The hydraulic conductivity is computed from the pore pressuredissipation response.

a) The One-Point Method (OPM)

The deformation and consolidation of clayey soils around the advancingcone tip is essentially a large strain problem with partial drainage condi-

96 6 Applications

tions. Advanced consolidation theories incorporating the above conditionwere developed by Gibson et al. (1981); Schiffman (1980); Prevost (1980), andVoyiadjis and Abu-Farsakh (1997). While other formulations are based on anEulerian reference frame, Voyiadjis and Abu-Farsakh (1997) adopted the up-dated Lagrangian reference frame to reduce numerical errors. The study inthis book adopts the Voyiadjis and Abu-Farsakh (1997) updated Lagrangianreference frame approach.

Traditional methods for estimating the hydraulic conductivity of soilsfrom piezocone penetration tests are based on pore pressure dissipation testdata (Torstensson, 1977; Baligh et al., 1980; Jamiolkowski et al., 1985; Cam-panella et al., 1985; Carter et al., 1979; Houlsby and Teh, 1988). The es-sentials of these conventional methods are suggested by Torstensson (1977).These methods are similar to solving the heat diffusion equation. The porepressure dissipation test assumes a certain dissipation pattern (like a cylin-drical or spherical heat sink in a homogeneous media in some ideal boundaryconditions). Thus, the conventional method inherently has some limitationsdepending on the real soil conditions. The limitation of the conventionalmethod is discussed below.

The partial differential equations, boundary conditions and initial condi-tions for the conventional method are expressed respectively:

Partial Differential Equation:

∂σ′/∂t = (∂σ/∂t) − (∂u/∂t) = ch[(∂2u/∂r2 + (1/R)(∂u/∂r)] + cz[(∂u/∂z2](6.3)

Boundary Conditions:

u = 0 at r = ∞, z = ∞, and z = −∞u′ = k at r = R, z = 0 (where k is a constant)

Initial Condition:

F = g (stress strain functions such as those obtained from the cavityexpansion theory)

In (6.3), σ′ is the effective stress, σ is the total stress, u is the excess porepressure, t is the elapsed time, ch is the horizontal consolidation coefficient,cz is the vertical consolidation coefficient, R is the radius of the cone, r is theradial axis, and z is the vertical axis of the cylindrical coordinate system. Thesolution of (6.3) is not straightforward, but it can be solved using a specialnumerical technique such as the Crank – Nicholson technique. Torstensson(1975, 1977) simplified (6.3) by assuming negligible vertical drainage (assumethe effect of cz and related term is minor) and a constant total stress (as-sume (∂σ/∂t) =0). Torstensson (1975, 1977) presented a convenient graphical


solution that is similar to that of Terzaghi’s (1943) one-dimensional consoli-dation solution. However, inaccurate results can also result if field conditionsare not close to the assumptions given above. Gupta and Davidson (1986)used (6.3) instead of Torstensson’s simplified version (1975, 1977) with theassumed boundary conditions (given that (6.3) cannot be solved with bound-ary conditions such as u = 0 at r = ∞.) Although Gupta and Davidson (1986)obtained better results, they did not eliminate the fundamental drawbacks of(6.3) and its simplified solution by Torstensson (1975, 1977). The advantageof the conventional method is that because it does not require the coupledtheory of mixtures, it is convenient to interprete. Torstensson (1977) also didnot fully consider the different dissipation mechanisms at the cone tip, shaftand face. Levadoux and Baligh (1986) and Teh and Houlsby (1991) furtherimproved the Torstensson (1977) method by adopting different initial porepressure magnitudes and dissipation conditions at the cone tip, shaft, andface.

Essentials of Pore Pressure Response

The pore pressure response from the piezocone penetration test is not knownin detail yet. Intuitively, however, we may expect that the response will followthe curves shown in Fig. 6.3.

This figure shows a conceptual and a slightly exaggerated excess porepressure response of a soil element that is located at the projected center-lineof the piezocone penetrometer’s travel route. Initially, the piezocone tip islocated far above this soil element, and there is no excess pore pressure. As

Fig. 6.3. Illustration of pore pressure response for a soil element during piezoconepenetration (Song, 1999)

98 6 Applications

time passes, the penetrating cone tip comes closer to the soil element, andthe induced stress by the penetrating cone tip gradually builds up over thissoil element. This increase in the induced stress results in a build up of excesspore water pressure (an increase that may be either linear or non-linear). Asthe penetrating cone tip passes through this soil element, severe disturbanceoccurs and high excess pore pressure occurs. When the penetrating cone tipstops at this soil element, there is an immediate drop in the excess pore pres-sure as a result of the reduced axial force. In the mean time, the interactionof pore pressure between the near field and far field takes place (near field:location radially close to the cone tip; far field: location radially far from thecone tip), resulting in the small increase or decrease in the pore pressure. Asobserved previously by Voyiadjis and Abu-Farsakh (1997), the pore pressureduring penetration is at a maximum at the cone face (typically known as au1 position). Thus, if one has a porous element at the cone shoulder location(typically known as a u2 position), the measured pore pressure at during pen-etration will be smaller than the pore pressure at the cone face. Therefore,there will be a tendency toward a small increase of pore pressure at u2 po-sition because of pore water inflow from the u1 position (high pore pressureposition). (This explanation applies to the normally consolidated soils. Thepore pressure response of an overconsolidated soil is not the same as that of anormally consolidated soil, and it is beyond the scope of this book.) We mayconclude from the above discussion that the drainage condition during thecone penetration test is that of a partially drained condition and not that ofthe fully drained condition or fully undrained condition.

Figure 6.3, also shows the difference between the assumed dissipationcurve and the real dissipation curve. Comparing these two curves shows thatthe magnitude and initial timing of the excess pore pressure are different.Therefore, it is clear that the dissipation test is usually analyzed with anincorrect initial time and magnitude of initial pore pressure. As the drainagecondition deviates more and more from the assumption of the fully undrainedcondition, the reliability of the assumed dissipation curve progressively de-creases. Senneset et al. (1988) indirectly pointed out this aspect by question-ing the validity of the magnitude of the initial pore pressure when Bq is lessthan 0.4 where Bq is the ratio of excess pore pressure to net cone resistance.The value Bq = 0.4 corresponds roughly to clayey silt. A small value of Bq

implies a higher hydraulic conductivity. For soils with small Bq values, onecan expect a substantial drop of ∆uo (initial pore pressure) as a result of porepressure dissipation during penetration and pore water pressure interactionbetween the far field and the near field. Silty soils that have negative Bq

values and high hydraulic conductivity exhibit negative excess pore pressure.Moreover, the validity of ∆uo for these soils is questionable. In other words,for materials of relatively high permeability, the penetration process of CPTis a partially drained condition and unlikely the fully undrained condition


to be valid. This deviance from the fully undrained condition may cause asubstantial difference between the peaks of the two curves in Fig. 6.3.

Elsworth (1993) showed that Bq varies with the coefficient of consolida-tion when Bq is less than 0.5. Elsworth (1993) also showed that Bq is almostconstant when the coefficient of consolidation is less than a certain num-ber. These results show that practically speaking, the undrained conditionis obtained only when the coefficient of consolidation is less than a certainnumber. This phenomenon may again be caused by the fact that for higherpermeable soils (Bq < 0.5), the validity of initial excess pore pressure is in-accurate because of the fact that in reality, the drainage condition is notthat of a fully undrained condition. The results of Senneset et al. (1988) andElsworth (1993) showed that the conventional method for determination ofthe hydraulic conductivity is desirable when the hydraulic conductivity is lowenough and the interaction of pore pressure between the far field and nearfield is minimal.

Another aspect of the piezocone-induced excess pore water pressureregime is that the measured penetration pore pressure is the result of si-multaneous generation and dissipation. Thus, the measured pore pressuresat the piezometer tip are the sum of the early-generated and dissipated porepressures and the newly generated pore pressures. This casts some doubt onthe the accuracy of the measurement of the magnitude of the initial pore pres-sure (Kurup and Tumay, 1997). Kurup and Tumay (1997) have shown thatthe dissipation of pore pressure during piezocone penetration is unavoidableand that the phenomenon results in the interference of the spatial distrib-ution of pore pressures. Therefore, the pore pressure dissipation curve mayresult in a more gentle dissipation slope than a theoretically predicted one.Typically, the curvature represents the pore pressure dissipation rate. There-fore, the computed hydraulic conductivity or coefficient of consolidation maynot be correct. The error induced from the above difficulties may be negligi-ble, or it may be significant – depending on the soil condition. Therefore, animproved method for estimating the hydraulic conductivity from piezoconedata is needed.

New approaches to estimating the hydraulic conductivity from CPT-induced steady state excess pore pressure were proposed by Elsworth (1993,1998), Manassero (1994) and House et al. (2001). Elsworth (1993, 1998) es-timated the hydraulic properties of soils with a linear elastic soil model in-corporating the dislocation method. The linear elastic soil model may shownot be a perfect fit for an elastic-plastic model for the given strain. There-fore, higher or lower excess pore pressures may be obtained from the analy-sis. Typically, the prediction of the hydraulic conductivity based on the lin-ear elastic soil model underpredicts the hydraulic conductivity of the soil.Elsworth (1993)’s comparison with experimental data also showed a lowerpredicted hydraulic conductivity than that obtained from the experimen-tal data. Manassero (1994) correlated Bq and the hydraulic conductivity bysemi-empiricism. Manassero’s correlation (1994) is obtained from a linear cor-relation between the given hydraulic conductivity data and the Bq coefficient.

100 6 Applications

Manassero’s method may therefore be used when one has existing data basefor the hydraulic conductivity and the Bq parameter. House et al. (2001)experimentally incorporated the variable penetration speed of the piezoconepenetrometer and accompanying excess pore pressure response and corre-lated them to the hydraulic conductivity. From the view of cost, the linearelastic model and the semi-empirical approach are quite efficient. However,considering the non-linear behavior of soils at the vicinity of the cone tip andthe insufficient data base available for the empirical relationship between thehydraulic conductivity and the Bq or penetration speed, one sees the neces-sity of incorporating an elasto-plastic large strain approach in the analysis ofthe piezocone penetration test.

Essentials of OPM

The main objective of the OPM is to develop a method for the determina-tion of the hydraulic conductivity that can overcome the drawbacks of theconventional dissipation test method and provide a more realistic theoreticalframework. The method proposed in this study is based on the analysis ofthe steady state pore pressure during the piezocone penetration test so thatthe full interaction between the piezocone and the soil is considered. As aresult, the problems related to the initial time and initial magnitude of thepore pressure discussed in the previous section are naturally resolved.

The formulation of the coupled field equations for soils, using the theory ofmixtures in an updated Lagrangian reference frame based on the principle ofvirtual work, is used in this work. The authors make use of an axi-symmetricfinite element program developed for simulating the behavior of soils withthe advancement of the piezocone tip. Finally, the results obtained using ofthe proposed method is compared with well-documented field test data aswell as experimental results obtained using the Louisiana State Universitycalibration chamber system.

The piezocone penetrometer typically penetrates into the ground withthe speed of 2 cm/sec, thereby inducing complete failure of soils around thecone tip. Researchers have shown that the strain at the vicinity of the conetip ranges between ten percent to more than a hundred percent (Levadouxand Baligh 1986; Kiousis et al. 1986; Voyiadjis and Abu-Farsakh 1997). Thusthe penetration of the piezocone is essentially a time-dependent large strainproblem. Considering the complexity of the piezocone penetration test andthe soil characterization, both the large strain theory and the visco-plasticityare desirable features for the analysis of the problem. However, in this sec-tion only the isotropic, elasto-plastic large strain approach with time depen-dent loading condition is adopted for the sake of simplicity, and viscosity-dependent time rate effects are disregarded. Also, the modified Cam claymodel is used in this study to describe the plastic behavior of soils. The influ-ence of soil compressibility on the estimation of the hydraulic conductivity isincorporated in the modified Cam clay model by using the compression indexof the soil.


The drainage condition around the penetrating cone tip is again neitherfully drained nor fully undrained. This condition is the partially drained con-dition or the transient flow condition. For the transient flow condition, it canbe presumed that the pore pressure is a function of one or several of variousfactors such as hydraulic conductivity, material stiffness, and so forth. Hence,the soil must be looked upon as a multiphase material whose state is to bedescribed by the stresses and displacements within each individual phase.In this work, Prevost (1980)’s theory of mixtures for two phase materials(saturated soil) is coupled with Terzaghi’s (1943) effective stress theory asshown by Kiousis et al. (1988) and Voyiadjis and Abu-Farsakh (1997). Voyi-adjis and Abu-Farsakh (1997) implemented the coupled theory of soil-watermixtures to the effective stress and pore water pressure and derived the cou-pled equations of mixture in an updated Lagrangian reference frame. In thischapter, the coupling of the stress, deformation, pore water pressure, andhydraulic conductivity is obtained by using the work outlined in Voyiadjisand Abu-Farsakh (1997). (The details of the coupling equation are shown inChap. 4.)

Numerical Simulation

From the work of Voyiadjis and Abu-Farsakh (1997), the coupled equationsin the updated Lagrangian reference frame are obtained as shown in (6.3).

The numerical simulation is conducted using an axi-symmetric finite ele-ment analysis code with the mesh shown in Fig. 6.4. The excess pore pressureat the cone face (typically known as u1) is used in this chapter. The piezoconepenetrometer is assumed to be infinitely stiff and therefore, tensile stressesare not allowed to develop along the centerline boundaries. This assumption isreasonable because the test is assumed to take place at a depth great enoughto assure that the initial stresses in the ground prevent the development oftensile stress. The continuous penetration of the piezocone is numericallysimulated by applying an incremental vertical penetration rate of the cone of2 cm/sec. This is the same rate as the piezocone penetration, and it allows thepartial pore pressure dissipation during the penetration. A simple constraintapproach at the nodal level of the interface is adapted in this study to accountfor the soil-penetrometer interface friction. An angle of friction, δ = 14, isassumed between the soil and the piezocone face. During the piezocone pene-tration, three interface states can be identified: the fixed state, the slip state,and the free state. Solution for the new nodal interface state is determinediteratively. Based on the previous interface state (fixed, slip or free) and theloading criterion, a new interface state is assumed and solved in order to ob-tain a trial solution. The trial solution is then used to check if the assumedtrial state is correct. If not, a new state is assumed that is more likely tobe correct (Voyiadjis et al., 1998). (The input parameters used in this studyare obtained from typical oedometer tests and triaxial compression tests asshown in Table 6.1.)

102 6 Applications

Fig. 6.4. Finite element mesh for the piezocone penetration analysis

At the beginning of penetration, all the nodes along the inclined conicalsurface and along the piezocone shaft are prevented from sliding along thesurface and are forced to move vertically with the cone boundary incrementalmovement until the sliding potential occurs. The sliding potential is reachedwhen the tangent frictional force (Ft) of the node along the boundary surfacereaches the allowable friction force (Ft > Fs) given by (Voyiadjis et al. 1998)as follows:

Table 6.1. Input parameters for the finite element analysis

Parameter Value Units

Compression index, λ 0.11 dimensionlessRecompression index, κ 0.024 dimensionlessInitial void ratio, eo 1.0 dimensionlessPoisson’s ratio, v 0.3 dimensionlessEarth pressure coefficient (Ko) 0.5 dimensionlessSlope of critical line, M 1.16 dimensionlessUnit Wt. of soil, γt 1.8 ton/m3

Depth (from ground surface) 20 mUnit Wt. of water, γw 1.0 ton/m3


Fs = Fka + Fk

n tan δ (6.4)

In (6.4), Fka is the soil-piezocone adhesion force at the load increment k, Fk

n isthe normal effective force at the load increment k, and δ is the angle of fric-tion between the soil and the piezocone surface. These nodes are afterwardsallowed to slide along the skew boundary surface and along the cone shaftsurface.

During the incremental penetration, the nodes along the boundary arecontinuously checked for appropriate boundary condition adjustments. Basedon the previous boundary conditions at load increment (k-1), a new bound-ary condition state is assumed for the load increment (k), depending on theload and displacement criteria. The validity of the assumed trial boundarycondition is tested before proceeding to the next loading increment. (Meshsensitivity of this problem was tested by Abu-Farsakh et al. (1998), whodemonstrated that the finite element mesh used in this present study is ade-quate. More details of the numerical simulation may also be found in Voyiadjiset al. (1998).)

Hydraulic Conductivity Estimation Procedure

In (6.2), ∆W is the incremental excess pore pressure matrix, and nΨ andnΠ are functions of the hydraulic conductivity matrix. Thus one can solvethe equations if the hydraulic conductivity is known. Computing the hy-draulic conductivity matrix is possible if the ∆W matrix is known, but the∆W matrix represents the distribution of the incremental excess pore pres-sures around the cone tip, a distribution that is not known. The accumulatedexcess pore pressure at the piezometric-element location is the only known(measured) quantity. Thus, a straightforward procedure is not possible.

It is possible, by assuming the hydraulic conductivity matrix, to proceedusing a trial and error method. With the assumed hydraulic conductivitymatrix, the excess pore pressure at the piezometric-element location can becomputed. This computed value is compared to the measured value. If thesetwo values are within 10% of each other, the assumed hydraulic conductivityis considered a good estimation of the hydraulic conductivity of the soil. Thisgeneral trial and error method is time consuming, and without a good initialhydraulic conductivity matrix, this process may take a substantially longtime.

Beside the numerical simulation of the piezocone penetration test for thesoil properties given in Table 6.1, additional numerical simulations were car-ried out to check the sensitivity of the material parameters. The effect of theslope M of the critical state line in the p-q space was investigated for therange of M from 1 to 1.2. The effect of the void ratio was investigated forthe range of initial void ratio from 0.8 to 1.16, where the initial void ratio(H) implies the void ratio at ln(p)= 0 in the e-ln p plot. The variability in

104 6 Applications

0

100

200

300

400

500

600

70070E

x. P

WP

, kP

a

M=1.2, H=1.16M=1.2, H=1.0M=1.2, H=0.8M=1.0, H=0.95M=1.0, H=0.8

20

30

10

40

50

0

60

M : slope of critical state lineH : Initial void ratio when ln(p)= 0

10 -1 8 10 -1 4 10 -1 0 10 -6 10 -2

Hydraulic conductivity, (m/sec)

Fig. 6.5. Numerically predicted results for excess pore pressure and hydraulicconductivity (σ′

v = 200 kPa)

the results with elastic stiffness is expected. However, this effect is not in-vestigated here since the experimental data are normalized with respect tothe undrained shear strength (which also can be said to be normalized withthe undrained modulus). Figure 6.5 shows the effect of the variability of thematerial parameters on the excess pore pressure and hydraulic conductiv-ity. These effects are negligible for the hydraulic conductivity higher than10−9 m/sec., a finding that concurs with the results of a study by Houseet al. (2001). Figure 6.5 also indicates that there are clear and predominantrelationships between the hydraulic conductivity and the piezocone-inducedexcess pore pressures considering that the effect of other material parameterssuch as stiffness is also included. These relationships are clearer when thehydraulic conductivity is in the range of 10−9 m/sec to 10−6 m/sec. The soilsin this range of hydraulic conductivity are clayey silt to fine sand. Typically,soils out of this range of hydraulic conductivity are sands or very plasticclays. These results indicate that the coupled theory of mixtures has strongpotential for making acceptable predictions of the hydraulic conductivity forthese soils.

Comparison with Test Results

Test results were collected from well-documented piezocone penetration tests.Three piezocone penetration tests were also conducted at the LSU calibrationchamber for a K-33 specimen (a mixture of 33% kaolinite and 67% sand) forevaluation of this approach. The average of the results of these three calibra-tion chamber tests is shown in Fig. 6.6. Test data from the literature were


0

10

20

30

40

50

60

-100 0 100 200 300 400

Excess Pore Pressure (kPa)

Pen

etra

tio

n D

epth

(cm

)u1

u2

u3

u4

u1

u2

u3

u4

Fig. 6.6. Test set up and results from the LSU calibration chamber specimen

normalized to an undrained shear strength 60 kPa, which is the shear strengthof the specimen tested in the LSU calibration chamber in this study. Threepenetration tests were carried out in the soil specimens. The hydraulic systemused for the cone penetration features dual-piston, double-acting hydraulicjacks on a collapsible frame. The frame is mounted on top of the upper lidof the chamber and allows for penetration of the sample in a single strokeof 640 mm or less. Such a single-stroke, continuous penetration is desirableespecially in saturated cohesive specimens where stress relaxation and porepressure dissipation can occur during a pause between strokes. The penetra-tion depth is measured using an electronic analog to the digital converterdepth decoding system. All tests are conducted at the standard penetrationrate of 2 cm/sec. A total of three penetration tests are performed. Tests 1and 2 are performed for the two different piezo-element configurations, theu1, u3, u4 configuration (see Fig. 6.6 for piezo-element configuration) and theu2, u3, u4 configuration, respectively. Test 3 has the same configuration asTest 2.

The main purpose of Test 3 is to check the repeatability of the tests. (Thediscussions of these u2, u3, u4 will be published in a forthcoming paper; inthis book, only the results of u1 are further discussed.)

To include the scale effects in comparing the experimental data with thetheoretical results, the following dimensionless excess pore water pressure

106 6 Applications

and dimensionless hydraulic conductivity will be used. ∆u/σ′v is used for the

vertical axis, and (kv )( r

rref) is used for the horizontal axis. ((k

v )( rrref

) is similarto the dimensionless term used by House et al. (2001).) ∆u is the excesspore water pressure, σ′

v is the effective vertical overburden, k is the hydraulicconductivity, v is the penetration speed of the piezocone penetrometer, ris the radius of the piezcone penetrometer’s head, and rref (1.784 cm) isthe radius of the reference piezocone penetrometer’s head. The rationale forthe normalization of the field data in Table 6.2 is based on the fact that theinduced excess pore pressure is proportional to the undrained shear strengthfrom the Cavity Expansion Theory (Vesic, 1972). Through this normalization,the dependency of the pore water pressure to the undrained shear strengthof the material is reduced.

To evaluate the effects of some factors (such as the penetration speedand effective overburden), the authors subjected performed several numer-ical analyses. Figure 6.7 shows the effect of the penetration speed on theexcess pore pressure response. This effect is not quite linear; however, onecan approximately assume a linear function, and in this study we assumethat the effects of penetration speed are linear. Figure 6.8 shows the effectsof the effective overburden on the excess pore pressure response. This effectis also not quite linear, but once again, one can reasonably assume a linearfunction, and we assume in this study that the effects of confining pressureare linear.

As discussed previously, test results from the literature shown in Table 6.2are plotted in Fig. 6.9 and compared with the theoretical results. Figure 6.9shows reasonably good agreement between the test data and the theoreticallypredicted results. This agreement is good, considering that these data are ob-tained from different locations for different soils with different properties suchas stiffness, and so forth. In Fig. 6.9, the test data covers a wide range of soilswith dimensionless hydraulic conductivities ranging from 10−7. to 10−4. Onecan see from Fig. 6.7 that the change in the excess pore pressure is very smallwhen the hydraulic conductivity is less than 10−7 or larger than 10−4. There-fore, the drainage condition is practically that of a fully undrained conditionfor the dimensionless hydraulic conductivity of lower than 10−7, and that of afully drained condition for the dimensionless hydraulic conductivity of higherthan 10−4. A simpler theoretical approach that uses the drained cavity ex-pansion or undrained cavity expansion may be applicable. However, for soilsbetween these boundaries, the partial drainage effect is not negligible. Also,such a simpler approach may not provide reliable results.

As we have noted above, the work of Senneset et al. (1988) and Elsworth(1993) showed that the partially drained condition is obtained when Bq isless than 0.4 or 0.5. The work presented here gives similar results by showingthat the excess pore pressure during the piezocone penetration test is affectedwhen the dimensionless hydraulic conductivity is lower than 10−10.


Table 6.2. Test data of cone penetration induced excess pore pressure at u1 posi-tion and the hydraulic conductivity (σ′

v = 200 kPa)

Su k, (m/sec) ∆u ∆unorm

No. Soil (kPa) Test Method (kPa) (kPa) OCR Reference

1 Stjørdal

Silty Clay,

83 (0.7 −5.2)×10−8

Oedometer

338 244= 338 ×60 ÷ 83

N.A. Senn set

et al. (1988)

2 Glava

StjørdalSilty Clay

90 (3.6–6.3) ×10−10

Oedometer

800 533 3–4 Sandven(1990)

3 BakklandetTrondheimSilty Clay

100 (0.2–1.1) ×10−9

Oedometer

800 480 1.7 Sandven(1990)

4 ValøyaTrondheimSilty Clay

125 (0.2–1.0) ×10−9

Oedometer

1250 600 2–3 Sandven(1990)

5 Halsen

StjørdalSilty Clay

83 (3–9) × 10−8

Oedometer

300–400 216–289 N.A. Sandven

(1990)

6 Norco SiltyClay

50–60 (0.30–5) ×10−10

Oedometer

500–550 545–600 1–1.5 Tumay andAcar (1985)

7 AmherstClayey Silt

71.8 (1–2) × 10−9

Lab. Perm.Test

450–500 376–417 1.3–3 Baligh andLevadoux(1986)

8 Pentre Clay 62.5 (2–8) × 10−9

Oedometer

600 576 1.2–1.8 Powell and

Quarterman

(1997)

9 BothkennerSilty Clay

40–75 (1.4–3) × 10−9

Oedometer830–870 664–696 1.0–1.5 Lunne et al.

(1997)

10 LSUCalibrationChamber

Artificial

60 8 × 10−9

Oedometer560–624 560–624 1 Kurup (1993),

K−50 (50%Kaolinite +50% Sand)

11 Soil 85 2.1 × 10−8

Lab. Perm.Test

350–370 250–260 1.5 This study,K−33 (33%Kaolinite +67% Sand)

12 Quiou Sand N.A. 3 × 10−4

Hazen’s Law10–15 10–15 1 Almeida

et al. (1991)

13 Glass Bead N.A. 9 × 10−3

Hazen’s Law0 0 1 Peterson

(1991)

108 6 Applications

Fig. 6.7. Effects of confirning pressure to excess pore pressure response of thepiezocone penetrometer

0

100

200

300

400

500

600

700

0 1 2 3 4

Penetration Speed (cm/sec)

Ex.

P.W

.P.

(kP

a)

k=5x10-12 m/seck=5x10-10 m/seck=5x10-9 m/seck=5x10-8 m/sec

Fig. 6.8. Effects of penetration speed to excess pore pressure response of thepiezocone penetrometer


Fig. 6.9. Comparison of actual test data and predicted results of the excess porepressure and hydraulic conductivity (for M = 1.2, H = 1.16)

Summary of OPM

A new theoretical interpretation and experimental verification of the conepenetration induced excess pore pressures is presented in this study. In addi-tion, it adopts the large strain coupled theory of mixtures formulation usingan updated Lagrangian reference frame. Using this theory and the presentednumerical simulation technique, the cone penetration-induced excess porepressure is predicted and compared with existing test data.

The test data agree well with the theoretically predicted results. There-fore, the potential exists for use of this method to interpret the continuouspore pressure measurements, and it may be possible to use this approachfor the real time analysis for the hydraulic conductivity of saturated soils.Two threshold dimensionless hydraulic conductivities are obtained as 10−7

and 10−4 for the undrained condition and the free drainage condition, respec-tively. The coupled theory of mixtures should be used to predict the behaviorof soils within the range of these threshold values.

b) Two Points Method (TPM)

OPM utilizes the coupled theory of mixtures to estimate the hydraulic con-ductivitis of soils. TPM, however, utilizes traditional (uncoupled) consolida-tion theory to estimate the hydraulic conductivities of soils.

This method utilizes the difference of pore water pressure at u2 and u3

locations (See Fig. 6.6 for u2 and u3 locations). This method is valid when

110 6 Applications

Fig. 6.10. Excess pore pressure distribution around the cone tip (after Whittleand Aubeny, 1991)

there is a measurable difference between the pore pressures measured at u2

and u3. The test data and analytical results show that there is a clear differ-ence between u2 and u3 locations. Whittle and Aubeny (1991) analyticallyshowed that there is a clear difference in pore pressure at u2 and u3 locations(See Fig. 6.10). Robertson et al. [1986] and Juran and Tumay [1989] showedthis behavior experimentally. Also, the typical data showed the clear differ-ence in pore pressures at u2 and u3 locations (See Fig. 6.11). Undoubtedly,there is a clear difference between the pore pressures at u2 and u3 locations.However, one can predict that the difference between the pore pressures atu2 and u3 locations for the fully undrained condition (very low hydraulic con-ductivity) and the fully drained condition (very high hydraulic conductivity)will be very small. The conceptual response of pore pressure difference thatone can expect is shown in Fig. 6.12.

From Fig. 6.12, one can expect that two hydraulic conductivities may beobtained from the one value of the pore pressure difference between the u2

and u3 locations. Figure 6.12 shows that the direct relationship between theamount of pore pressure difference at u2 and u3 locations and the hydraulicconductivity may not be an easy way for the quantification. A second methodis the consolidation approach. The pore pressure at u2 and u3 locations is


Fig. 6.11. Field measured pore pressure from PCPT at Pentre, U.K. (after Powelland Quarterman, 1997)

Por

e pr

essu

re d

iffe

renc

e, (

u 2 -

u 3

)

Hydraulic conductivity

Fig. 6.12. Conceptual relation between pore pressure difference and hydraulicconductivity

different because of both consolidation and the stress conditions. One cansee the possibility that the pore pressure at the u3 location is the dissipatedpore pressure of the u2 location. At the steady state penetration, one canreasonably assume that the shearing stresses at u2 and u3 locations are iden-tical. Then the pore pressure difference between u2 and u3 locations is due tothe normal stress difference and pore pressure dissipation. Thus, if one canseparate the shear stress-induced pore pressure and normal stress-induced

112 6 Applications

pore pressure, one can compute the amount of the pore pressure dissipationbetween the u2 and u3 locations. This computation would give the hydraulicproperty of the soil the u2 that is directly measured during the piezoconepenetration test.

In conclusion, the basic idea is that the pore pressure distribution at u2

and u3 represents the dissipation curve of the normal stress-induced excesspore pressure. Thus, by starting the virtual consolidation at the u2 locationfor the normal stress-induced pore pressure with the assumed hydraulic con-ductivity, one can obtain the pore pressure at u3 location. For the consolida-tion, u2 is not taken as the initial pore pressure without a proper justification.The simultaneous generation and dissipation for the pore water is taken intoaccount from the beginning of the PCPT throughout the virtual consolida-tion. Thus, both the implicit and the explicit drawbacks of the conventionalmethod are inherently removed.

Formulation of the Uncoupled Consolidation

For comparison with the result of coupled consolidation, approximate for-mulations for the uncoupled consolidation are derived. Using the notationof Fig. 6.6, one can show that the measured pore pressures at u2 and u3

locations are as follows:u2 = us + ∆u2 (6.5)

u3 = us + ∆u3 (6.6)

In (6.5) and (6.6), us is the hydrostatic pressure, ∆u2 is the excess porepressure at u2 location, and ∆u3 is the excess pore pressure at u3 location.The increment between u2 and u3 locations in (6.5) and (6.6) is expressed as(6.7);

u2 − u3 = (us + ∆u2) − (us + ∆u3) = (∆u2 − ∆u3) (6.7)

Also, by noting that the excess pore pressure is the function of both thenormal stress change and the shear stress change, one obtains the followingequation,

∆u = ∆un + ∆us

= ∆uoct + 3a∆τoct

= ∆uoct + 1.4142∆τoct (Assume A = 1, a = (√

2/3)A) (6.8)

where, ∆uoct is the excess pore pressure induced by octahedral normal stress,3a∆τoct is the excess pore pressure induced by octahedral shear stress, “a”is Henkel’s pore pressure coefficient, and “A” is Skempton’s pore pressurecoefficient.

Consequently, (6.7) reduces to (6.9):

∆u2 − ∆u3 = (∆uoct + 1.4142∆τoct)2 − (∆uoct + 1.4142∆τoct)3 (6.9)


As mentioned previously, when one assumes that the shear stress at u2 andu3 location is identical, (6.9) reduces (6.10),

∆u2 − ∆u3 = ∆uoct,2 − ∆uoct,3 (6.10)

where, ∆uoct,2 is the octahedral normal stress induced excess pore pressureat the u2 location and ∆uoct,3 is the octahedral normal stress induced excesspore pressure at u3. Also, one can see that ∆u2 is not exactly ∆uo because ofthe preceding dissipation: ∆u2 is smaller than ∆uo in nature (∆u2 < ∆uo).Thus, one can assume that ∆uo,oct = ∆uo − 3a∆τoct ⇒ ∆uo,oct < ∆uo. ⇒∆u2 ≈ ∆uo,oct (taking into account the preceding dissipation). Then thedegree of consolidation is calculated as follows:

U = (∆uo−∆ut)/∆uo = (∆u2−∆u3)/∆uo,oct ≈ (∆u2−∆u3)/∆u2 (6.11)

From (6.11) and Torstensson’s (1975, 1977) graphical solution, one can calcu-late the radial coefficient of consolidation cr or permeability k (k = crγw/M,

where k is the permeability, M is the constraint modulus, and γw, is the unitweight of water) from U. Because of the inherent assumption of (6.11), theresults are interpreted to have approximate values.

Comparison with Test Results

The collected field test results in Table 6.2 and Table 6.3 are plotted in Fig. 6.8together with the theoretically predicted results. Because the collected fieldtest results are for various soils, a normalization of the field test results isperformed for the reference undrained shear strength of 60 kPa. This normal-ization technique is based on the fact that the induced excess pore pressureis proportional to the undrained shear strength as shown in (6.12) (Vesic,1972).

∆u = su[0.817αf + 2 ln(Rp/r)] (6.12)

In (6.12) su is the undrained shear strength, αf is Henkel’s pore pressureparameter, Rp is the radius of the plastic zone, and r is the distance to thecenter of cavity.

To evaluate the relationships between the PCPT-induced pore pressuresand the permeability using the coupled theory of mixtures in an updatedLagrangian reference frame with an incremental elasto-plastic constitutivemodel, the analyses are carried out utilizing the finite element program CS-S(Coupled System – Soils) as described in OPM. This evaluation is performedfor various permeabilities and strength parameters.

In Fig. 6.13, the parameter (∆u2 −∆u3)/∆u2 is used as the referenceparameter for the quantification of the relationship between the excess porepressure and permeability. (Note that both ∆u2 and ∆u3 are normalized.)This parameter is used because the direct increment of pore pressure (∆u2−∆u3) is not a function of a single permeability value, as previously discussed.

114 6 Applications

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-12 -10 -8 -6 -4 -2 0

Log of Permeability (M/SEC.)

(u2-

u3)

/(u

2-u

s)

M=1.2, H=0.9

M=1.0, H=1.0

Fig. 6.13. Change of pore pressure ratio [(∆u2−∆u3)/∆u2] with hydraulic con-ductivity [In the figure, (u2–u3)/(u2–us) is the same as (∆u2–∆u3)/∆u2, rectanglesrepresent the field test data of Table 6.2, and solid dots represent the predicted val-ues of permeability for the sites in Table 6.2 from the uncoupled consolidation]

In Fig. 6.13, the two solid lines represent the change of (∆u2−∆u3)/∆u2

with hydraulic conuctivity (k) for different values of M and H, respectively(where M and H are the properties of Cam-Clay model). From the theoreti-cally predicted lines in Fig. 6.13, one can see that there is a clear relationshipbetween the (∆u2− ∆u3)/∆u2 and the hydraulic conductivity in the rangefrom 10−10 to 10−6 m/sec. Soils with hydraulic conductivity smaller than10−10 m/sec are clayey soils and the soils with very low hydraulic conductiv-ity higher than 10−6 m/sec are sandy soils.

Considering the typical permeability criteria for the clay liners for sanitaryland fill is 10−9 m/sec and that for the free drainage materials for verticaldrains and (or) horizontal drains is 10−5 m/sec, the curves in Fig. 6.13 showthe new possibility of estimating hydraulic conductivities for most field soils.

In Fig. 6.13, the test results are indicated with rectangles instead of points.The rectangles reflect the transient variational nature of the measured porepressures during the PCPT. The center of the rectangles is the same as thevalues in Table 6.2. In Table 6.3, ∆u2 and ∆u3 values are taken from themean ∆u2 and ∆u3 of the field-measured pore pressure values.

It seems that the agreement between the test results and the theoreti-cally predicted results is not very excellent. However, considering that therepresentative field permeability values are typically larger than that of thelaboratory values, as reported by Baligh and Levadoux (1980, 1986), Songet al. (1992) and Funeki (1976), one may recognize the tendency that the


Table 6.3. Cases of the cone penetration induced excess pore pressure and thepermeability

Normalized

Excess PorePressure for

Excess Pore Cohesion =Hydraulic Pressure, ∆u 60 kPa,conducti- (kPa) ∆u (kPa)

Cohesion vity, k,

Site (kPa) (m/sec) ∆u2 ∆u3 ∆u2 ∆u3 OCR Reference

Bakklandet 100 1.1 × 10−9 550 250 330 150 1.7 SandvenTrondheim [53]

(Norway)

Pentre 62.5 (2–8) × 10−9 400 170 384 163 1.2–1.8 Powell and(U.K) Quarterman

[46]

Bothkenner 40–75 (1.4–3) × 10−9 500 340 400 272 1.0–1.5 Lunne et al.(U.K) [1]

Glava 90 6.3 × 10−10 600 420 400 280 3–4 SandvenStjørdal [53](Norway)

LSU 60 7.4 × 10−9 330 200 330 200 1.5 This studyCalibration (K-33)Chamber

rectangles of the test results in Fig. 6.13 should move to the right. With thistendency having been recognized, one can obtain better agreement betweenthe two. The arrows in Fig. 6.13 represent this tendency.

The solid and thick dots in Fig. 6.13 represent analytical results by theuncoupled consolidation. The results from the uncoupled consolidation showa remarkably good agreement with the curves obtained from the coupledconsolidation. Considering the computational simplicity and implicit assump-tions, these results are unexpected. However, considering that the uncoupledconsolidation theory does not take into account the pore pressure interactionsaround the cone penetrometer, the results of the uncoupled consolidation maydeviate substantially in certain conditions: thus the uncoupled theory shouldbe used very cautiously.

6.1.4 Evaluation

The one-point method is applicable for soils with permeabilities in the range10−9 m/sec to 10−6 m/sec. The two-point method shows a wider range of ap-plication. The OPM has its own advantages in that it can be used without

116 6 Applications

Table 6.4. Computational Results

Hydraulic Conductivity, k (m/sec)

Coupled Uncoupled Lab. TestSites Consolidation∗ Consolidation (vertical)

Bakklandet Trondheim 2 × 10−8 2.3 × 10−8 .11 × 10−8

(Norway)Pentre 3 × 10−8 1.2 × 10−8 (.2 − .8) ×10−8

(U.K)Bothkenner 2 × 10−8 6 × 10−9 (1.4 − 3)×10−9

(U.K)Glava Stjørdal 1.5 × 10−8 1.5 × 10−8 .063 × 10−8

(Norway)LSU Calibration 1.3 × 10−8 1.7 × 10−8 .74 × 10−8

Chamber

∗ Hydraulic conductivity is back calculated from data in Table 6.3 using CS-S.

the modification of the existing piezocone penetrometer which has a piezo-element at the u1 position. However, the TPM has the advantage that it canbe applied for a wider range of hydraulic conductivities. Thus, one can seethese two methods are compensating each other’s disadvantages and consti-tute new methods when they are used together.

In conclusion, the discussions in this study show the ability of the coupledtheory of mixtures to predict the hydraulic conductivity of soils utilizing thepenetrating pore pressure of PCPT. This study shows that the agreementwith the test data is quite reasonable. Thus, one can see the possibility ofobtaining the continuous permeability profile, an objective that was not pos-sible in the past. Also, with the incorporation of the high-speed processor,the continuous permeability profile can be obtained with real time basis (“onthe fly”).

6.2 (Shield) Tunneling

6.2.1 Introduction

Rapid growth in urban development has called for continual upgrading andexpansion of the existing infrastructure to meet all the demands of urbanlife, not least of which is the demand for space. Space must be found fortransportation systems and various other kinds of networks, including com-munication systems, utilities, water supply, and sewage disposal pipelines.Space must also be found for storage of many kinds of materials and for facil-ities (shelters) that offer protection against natural and man-made disasters.

6.2 (Shield) Tunneling 117

Because underground structures can accommodate new demands for spaceas well as various other demands for new infrastructure, and because suchstructures have minimal impact on surface features, tunneling will play anincreasingly important role in meeting these demands.

The development of shield tunnel methods throughout the history hasbeen concentrated on how to stabilize and support the cutting face duringexcavation. An unstable cutting face during excavation can be supported indifferent ways, such as by mechanical means, by compressed air, by fluid,and by the excavated soil itself. Mechanical support is not suitable and isvery risky in soft ground, especially bellow groundwater level (Babenderede,1991). The compressed air shield was the first approach used in soft ground.The next approach was the slurry shield, in which the face is supported bya fluid (usually water and additives such as bentonite). In 1974, the EarthPressure Balance (EPB) shield system was introduced in Japan, in which theexcavated soil material itself supports the cutting face. The pressure appliedto the tunnel face counter-balances, in theory, the existing overburden andhydrostatic pressures. The EPB shield was first used in the USA in 1981 todrive a 3.7 m diameter, 915 m long tunnel for the San Francisco clean waterproject.

Despite these advances, construction of tunnels in soft grounds, especiallyin urban regions, continues to pose a unique challenge to engineers. DesignEngineers are facing an increasing challenge, with more responsibilities tobuild tunnels under different ground conditions and in congested town re-gions for various needs and purposes that meet both natural and legislatedenvironmental restrictions. In urban areas, consideration must be given toprotect pre-existing structures and underground conduits from damage dur-ing shield tunneling.

Recent advances in tunneling technology reduce construction time withconsequent decrease in cost. Unfortunately, theoretical advances have notkept pace with the recent advances in the tunneling technology. Construc-tion of tunnels in soft ground, especially in urban regions, poses a uniquechallenge to engineers, and careful consideration must be given to the mag-nitude and distribution of settlements. At present, there is no generally validmethod for predicting, prior to tunneling, the subsidence caused by tunneling.Up-to-date empirical procedures (e.g. Peck, 1969) have been widely used toasses potential ground deformation owing to tunneling, but empirical formu-las have limitations in their applicability to different tunnel geometries, dif-ferent construction techniques, and different soil conditions (Lee et al., 1992).It has been reported that deformation caused by tunneling (Ng et al., 1986)and the subsequent potential damage to adjacent and overlying services andstructures depends on various factors: ground and groundwater condition,tunnel depth and geometry, and construction procedures, the last being themost important factor. Recently, the prediction of ground deformation andstress patterns during shield tunneling have been carried out by numerical

118 6 Applications

analysis based on the finite element method. Due to simplicity and cost effec-tiveness, in many cases, researchers (e.g. Rowe et al., 1983; Rowe and Kack,1983; Ng et al., 1986; Finno and Clough, 1985; Rowe and Lee, 1993) haveadapted the two dimensional plane strain or axi-symmetrical approach for thetunnel transverse or longitudinal section. However, field results and theoret-ical analyses show that the general stress and displacement patterns aroundthe tunnel are three dimensional and very different from that of the planestrain transverse section (Lee and Rowe, 1990a, 1990b). During the shieldadvance, before the face of the tunnel shield reaches the section, the soil issubjected to small settlement or heave movement. As the tunnel shield passesthe section, rapid downward settlement of the soil occurs immediately afterthe tailpiece cleara the section invading the space (gap) between the tunnelboundary and the lining. At this stage, the lined tunnel section approachesthe plain strain condition. The distance required for ground displacement toreach the plane strain condition depends on the amount of plasticity devel-oped around the tunnel. Under idealized construction, when the tunnelingmachine is kept hard against the face, minimizing the stress changes and de-formation into the face, the tunneling advances under perfect alignment andcan be treated as a plane strain case. Under less conservative constructionprocedures, three-dimensional movement ahead of the tunnel face may besignificant. Construction difficulties such as steering and alignment problemscan cause over-excavation and remolding of adjacent soils. Usually, duringtunneling, a significant zone of plastic behavior is induced around the tun-nel. Several attempts have been made to use 3-D finite elements model (Leeand Rowe, 1989a,b, 1990a,b; Akagi, 1994) to simulate the tunneling process,especially for open face tunneling shields. Akagi (1994) used excavated ele-ments ahead of the shield face to simulate the advance of the shield in the3-D model. Finno and Clough (1985) introduced a 2-D model that is basedon the combination of both the transverse and longitudinal plane sections.The longitudinal section analysis is used to provide information on the natureof the pressure distribution in order to simulate the heaving process in thetransverse section.

In this chapter, a two-dimensional computational model is developed andused to simulate the continuous advance of the Earth Pressure Balance (EPB)shield during the tunneling process in cohesive soils. The model is basedon the plane strain “transverse-longitudinal” sections that are capable ofsimulating the continuous advances of the shield and incorporating the 3-Ddeformation of the soil around and ahead of the shield face. An elasto-plasticcoupled system of equations (as described in Chap. 4) is used here in order todescribe the time-dependent deformation of the saturated cohesive soil. TheRemeshing technique is used in the longitudinal section in order to rearrangethe finite element mesh ahead of the shield face do that the size and dimensionof the excavated elements match the geometric shape and size of the shieldadvance. The computational model is used to analyze the N-2 tunnel project


excavated in 1981 in San Francisco using the EPB shield tunnel machine.The results of this analysis are compared with the in-situ field measurementsof the N-2 tunnel project.

6.2.2 Method of Analysis

Empirical formulas have been widely used to assess potential ground move-ments owing to tunneling (e.g., Peck, 1969; Mair et al., 1993). These formulasare based on the assumption that the transverse surface settlement (S) profilefollows a normal probability (Gaussian distribution) curve as given below,

S = Smaxe e(−x2/2i) (6.13)

where Smax is the maximum settlement that occurs above the tunnel center-line; x is the distance from tunnel centerline; and i is the distance from thetunnel centerline to the point of inflection (as shown in Fig. 6.14). The totalhalf width of the settlement profile is given by 2.5 times i.

Nevertheless, empirical formulas have limitations in applicability to differ-ent tunnel geometries, ground conditions, and construction procedures (Leeet al., 1992). In addition, these formulas do not take into consideration theredistribution of the stress state and the development of excess pore pressureresulting from the advancing of the tunnel shield. For these reasons, an al-ternate method is used to calculate the stress distribution around the tunnelopening for the lining design. In recent years, several researchers have used nu-merical analysis based on the finite element technique in order to predict theground deformation and stress patterns caused by the tunneling process. Inmany cases, numerical modeling has been treated as a two-dimensional plane

Fig. 6.14. Gaussian distribution of surface and subsurface settlement profiles (afterMair et al., 1993)

120 6 Applications

strain or an axi-symmetric problem of the tunnel transverse or longitudinalsections. However, field studies and theoretical analysis show that the dis-placements around the tunnel are three-dimensional (Lee and Rowe, 1990a,b).Limited theoretical research has been conducted on the three-dimensional be-havior of the ground movement ahead the tunnel face, particularly for softsoils, where nonlinear elasto-plastic conditions exist (Clough and Leca, 1993).

For the sake of simplicity and cost effectiveness, most of the previousworkers (e.g. Rowe and Kack, 1983; Finno and Clough, 1985; and Ng et al.,1986) adapted the two-dimensional, finite element approach. In this approachthe tunnel process is modeled using a “transverse section” plane strain model.The transverse section cuts perpendicularly to the tunnel axis so that theunder formed tunnel appears as a circle in the finite element mesh.

There are two approaches to incorporating the effect of the three-dimensional movement into the two-dimensional “transverse section” analy-sis. The first approach, adapted by Finno and Clough (1985), makes use of asecond plane strain “longitudinal section” in order to represent a section cutparallel to the tunnel axis for the case of EPB shield tunneling. The “longitu-dinal section” is used to simulate the shield advance by applying incrementalrigid translation, causing the soil to displace away from the tunnel face. Theinduced out-of-plane stresses are used to estimate the pressure distribution,which is used to simulate the initial heaving process in the “transverse sec-tion.” The plane strain “transverse section” is then used to simulate theclosure of the tail gap and the long time-dependent deformations.

The second approach has been used by Rowe et al. (1983), Rowe and Kack(1983), Ng et al. (1986), Rowe and Lee, (1992), and Lee et al. (1992) suggestthe use of what is called the “gap parameter” to represent quantitatively thethree-dimensional ground loss resulting from three-dimensional movement foran open face shield tunneling. The gap parameter (GAP) (Fig. 6.15) can beexpressed (Rowe and Lee, 1992; Lee et al., 1992) as

GAP = Gp + U∗3D + ω (6.14)

where Gp = 2∆ + δ = the physical gap representing the geometric clearancebetween the outer skin of the shield and the lining, the gap being comprisedof the thickness of the tailpiece (∆) and the clearance required for erection ofthe lining (∂); U

∗3D represents the equivalent three-dimensional elasto-plastic

deformation at the tunnel face, which can be calculated from formulas givenby Lee et al. (1992) and Rowe and Lee (1992), and ω takes into account thequality of workmanship.

These movements can be approximately incorporated into the two-dimensional plane strain model by assuming a larger excavated tunnel di-ameter with an additional volume corresponding to the volume of groundloss ahead and over the shield. Excavation of tunnel is then simulated byremoving the excavated elements and replacing them with tractions aroundthe tunnel opening. Tractions are then removed incrementally from aroundthe tunnel periphery.


Fig. 6.15. Two-dimensional ground loss simulation (after Lee et al., 1992)

For the closed-face shield (such as the EPB) Yi et al. (1993) suggestapplying a peripheral pressure at the tunnel opening until the observed heaveis achieved in order to simulate the heaving caused by EPB. Tail gap closureis then simulated, allowing the soil to move inward until the closure of thetheoretical gap.

Lee and Rowe (1990a, 1990b, and 1991) used a three-dimensional finiteelement model to simulate the advance of an open-face tunneling shield. Thecontinuous advance of the tunnel face was simulated by a two-stage analy-sis. At the beginning, the axial pressure ahead of the tunnel and the radialpressure around the periphery is released gradually, allowing the soil to movefreely into the zone to be excavated until the total radical convergence of thesoil at the tunnel crown and invert corresponds to the tunnel clearance soilin front of the heading moves both radially and axially toward the face. Thevolume between the final cut surface and the original position of the soil rep-

122 6 Applications

resents the ground loss due to the three-dimensional movement (U∗3D). Once

U∗3D is known, the total gap can be calculated. A second three-dimensional

computation is then performed on the new dimensions.Pelli et al. (1991) used a three-dimensional finite elements analysis to

simulate tunneling process in rocks. The excavation process was modeledby eliminating the “excavated” elements from the stiffness matrix. Akagi(1994) introduced excavated elements ahead of the shield face to simulatethe advance of the shield in the three-dimensional model.

6.2.3 Finite Element Numerical Simulation

The continuous advance of the tunneling process is simulated here by us-ing the two-dimensional finite element analysis (Voyiadjis and Abu-Farsakh,1998; Abu-Farsakh and Voyiadjis, 1999) that is based on the combinationof both the plane strain “longitudinal and transverse” sections. The planestrain longitudinal section will be used to simulate the continuous advanceof the shield and to study the short-term soil deformations, stress redistri-bution, and excess pore pressure ahead and around the shield, as well asthe resulting initial heave or settlement caused from the shield advance. Theanalysis of the longitudinal section will provide important information – suchas the initial surface heave or settlement and the distribution of the excesspore pressure around the tunnel opening – that is needed in the transverseanalysis.

a) Two-dimensional Longitudinal Section

A 2-D Longitudinal section will be used to simulate the advancement of theEPB shield and the associated ground deformation and stresses redistributionresulting from the shield advancement. The advancement of the tunnelingmachine involves a change in geometry and removal of excavated soil, so astep-by-step incremental excavation procedure will be used in this simulation.The EPB shield machine can be operated in such a way that the rate ofexcavation of the soil is more less or equal to the rate of advancing the shield.This flexibility allows over-, under-, or perfect excavation simultaneously byadjusting the applied earth pressure at the shield face. In most cases, the EPBshield is operated so that the rate of excavation is less than the machine’s rateof advancement, a discrepancy that forces the soil away from the machine’sface in such a way as to cause small initial heave. This initial heave willreduce the amount of final settlement. The magnitude of the heave is directlyproportional to the applied earth pressure. However, the proposed model willbe capable of handling the three mentioned cases, and for both open- andclosed-face shields. For the over excavation case, during each incrementaladvance, part of the soil will move inside the shield, causing the incrementalexcavation to be higher than the incremental advance. In the case of under-


Fig. 6.16. Stages of two-dimensional longitudinal section (Abu-Farsakh, 1996)

excavation, during each incremental advance, part of the soil ahead of theface will be excavated and part will be displayed away from the shield face.In the case of perfect excavation, the soil is neither displaced away nor movedinside.

In the longitudinal plane strain analysis, the simulation of the advance-ment of the EPB shield tunneling will be accomplished in the following threestages as described in Fig. 6.16.

Stage 1: Removal of the part of the soil ahead of the shield that has tobe excavated, replacing it with the equivalent traction acting around thetunnel, and moving the face of the shield just ahead of the excavated part.The equivalent nodal tractions, Req, are calculated by integrating the totalstresses over the excavated elements as given below:

124 6 Applications

Req =∫

A

BTσdA (6.15)

Stage 2: Under excavation: Apply incremental rigid translations of thenodes representing the face and the body of the EPB shield until the averageearth pressure at the face reaches the specified controlled value (based on themachine operation). This will simulate the initial heaving process caused bydisplacing the soil away from the face.Over Excavation: Gradually release the nodes representing the face of theEPB shield until the average earth pressure at the face reaches the speci-fied controlled value. This process will simulate the initial settlement processcaused by inward movement of the soil into the shield chamber.Perfect Excavation: disregard this stage.

Stage 3: Remeshing: The finite element mesh will be rearranged in the por-tion ahead of the shield face so that the size and dimensions of the excavatedelements. For the next incremental step, match the geometric shape and sizeof the incremental shield advance (see Fig. 6.17).

Stages 1–3 will be repeated for other excavation increments until a planestrain condition is reached just behind the shield tail or until a specifiedexcavation distance is achieved. During the tunneling process, a region ofdisturbed (remolded) zone is created ahead of the shield face. The strengthand the Poison’s ratio for the elements immediately ahead of the shield faceare reduced in order to represent this disturbed zone.

Fig. 6.17. Remeshing of the finite element ahead of the tunnel boring machine(Voyiadjis and Abu-Farsakh, 1998)


Iso-parametric interface elements are used between the soil and the shieldbody in order to model the soil-shield interface friction. The Mohr-Coulombelestoplastic yield function as given by Matsu and San (1989) is used in thiswork in order to describe the incremental stress-strain behavior of the inter-face element. This constitutive model is capable of describing the restraineddilatancy of the soil at the interface. More details of the soil-shield interfacemodeling are discussed later in Sect. 6.2.6.

b) Two-dimensional Transverse Section

In order to incorporate the three-dimensional deformation into the planestrain transverse section analysis, one must know the magnitude and distri-bution of the radial heave/settlement pressure that needs to be applied inthe transverse section. Results of the longitudinal section analysis provideinformation about the amount of the initial surface heave/settlement (Spl),the distribution of the excess pore pressure, and the stress change. The ac-tual three-dimensional surface heave/settlement (S3D) is expected to be lessthan the plane strain value (S3D < Spl or S3D = A.Spl, where A < 1). Inthis analysis the distribution of the heave/settlement pressure is obtained byapplying a rigid translation of the nodes representing the shield body andface in a simple 3-D model as shown in Fig. 6.18. This pressure will be ap-plied incrementally in the transverse section analysis until the excess porepressure at the spring line reaches the value obtained from the longitudinalsection. In future work, the author will simulate the 3-D model in order toobtain a good correlation between the 3-D model and the plane strain lon-gitudinal section. In this way, enough information will be obtained from thelongitudinal section analysis to feed the transverse section analysis (i.e. thethree-dimensional initial heave/settlement S3D).

Simulation in transverse plane strain section will be accomplished in thefollowing five stages as described in Fig. 6.19 (Voyiadjis and Abu-Farsakh,1998, Abu-Farsakh and Voyiadjis, 1999).

Stage 1: Apply (or unload for initial settlement) incrementally the heave orsettlement pressure until the specified excess pore pressure at the spring lineis achieved. (For perfect excavation, disregard this stage.) In this stage theperipheral nodal displacement can be estimated.

Stage 2: Once the peripheral nodal displacement is known, repeat stage 1with a anew mesh using a different tunnel opening so that at the end of theheaving stage, the actual opening is reached.

Stage 3: Apply incremental unloading pressure around the tunnel peripheryuntil the tail gap is closed (in case of grouting, unload until the groutingpressure is achieved).

In this operation, nodes around the peripheral of the tunnel are allowed tomove inward. The tail gap is considered closed when the relative movement

126 6 Applications

Fig. 6.18. Three dimensional mesh

of the node at the crown and at the invert is equal to the theoretical size ofthe tail gap. It is better to choose the distribution of the unloading pressurein such a way as to insure the closure of the gap at the spring line and at boththe crown and the invert simultaneously. Reasonable results can be obtainedby using a uniform unloading pressure distribution.

Stage 4: Once the soil comes into contact with the lining, the soil-lininginteraction will be activated by changing the material properties of the innerelement (must have the same size of the lining).in order to represent theactual lining material. In addition, the weight of the erected lining and theremaining of the peripheral pressure will be applied incrementally to the soilelements around the tunnel.

Stage 5: The time-dependent long term deformation resulting from the dis-sipation of the developed excess pore pressure is simulated here.

6.2.4 Remeshing

The advancement of the tunneling machine during the excavation processinvolves removal of the excavated soil and change in geometry of the soil aheadand around the shield. Therefore, the finite element mesh is rearranged in theportion ahead and around the shield body so that the size and dimension ofthe excavated elements for the next incremental step match the geometricshape and size of the incremental shield advance as illustrated in Fig. 6.20.


Fig. 6.19. Two dimensional transverse section simulation

Adaptive methods for remeshing can be classified into three main types:the r-method, where the total number of nodes and elements are kept con-stant but the nodes are relocated (or moved) to new location with new co-ordinates; the h-method, where the elements of the initial mesh are refinedinto smaller elements or redefined into larger elements so as the error is dis-tributed uniformly over the finite elements of the specified domain; and thep-method, which calls for applying higher (or lower) order interpolation poly-nomial shape functions used for element interpolation while keeping the totalnumber of elements and nodes constant. A fourth type–the hp-method–mayalso be used, which is a combination of h and p method. Implementationof the p-method is more complicated than the h-method because extensivemodification of the analysis is required. However, the convergence of the p-method is better than the h-method.

The r-method is used in this work to rearrange the finite element mesh inthe portion ahead and around the shield body so that the geometric shapes

128 6 Applications

Fig. 6.20. Rearrangement of the finite element mesh ((Voyiadjis and Abu-Farsakh,1998; Abu-Farsakh and Voyiadjis, 1999)

and sizes of the finite elements in front of the shield match the next incremen-tal shield advance during the tunneling process. The idea is to relocate thenode coordinated of the mesh in order to serve the afore-mentioned goal. Sincethe number of nodes is unchanged during this adoption, the computational ef-ficiency of the r-method makes it the best among the other methods (Tezuka,1992). Nonetheless, though the r-method does not increase the number of el-ements, if the shape of the domain is complicated, some elements might behighly distorted and may lead to unacceptable results (Tezuka, 1992).

6.2.5 Mapping Variables

In general, the mapping process mainly consists of three essential steps (Leeand Bathe 1994):

(1) Element Identification (EI), to identify the element Eo that contains thenode n.

(2) Iso-Parametric inversion (IIN), to determine the iso-parametric local co-ordinates (ξn, ηn) of the node n in the element Eo.


(3) Data transfer (DTR), to transfer data (values and variables) from elementEo to the node n with local coordinates, (ξn, ηn).

a) Mapping of Nodal Variables (Displacementsand Pore Pressures)

Referring to Fig. 6.20, the old mesh ID denoted by (α), while the new meshis denoted by (αRM ) element. One needs first to identify the elements in theold mesh(α) that contain the node and thus determine the corresponding iso-parametric local coordinates of that node using the modifies Newton-Raphsoniterative method.

If Nk is the interpolation function corresponding to the node k in theelement, then the nodal coordinates (Xn, Yn) of the nodes n in the newelement are given by

Xn =m∑

k=1

Nk(ξn, ηn)Xk (6.15a)

Y n =m∑

k=1

Nk(ξn, ηn)Y k (6.15b)

where m is the number of nodes in the old element that contains node n,and (Xk, Yk) are the coordinates of node k. To obtain the iso-parametriclocal coordinates ξn, ηnof the node n, one needs to solve (6.15a). To ensurethat the node n lies inside the element Eo., one must satisfy the followingconstraints:

−1 < ξn < 1 (6.16a)−1 < ηn < 1 (6.16b)

In the case of the r-method adaptivity with no change of element node con-nectivity, step1 and 2 are done simultaneously. That is, the element identifi-cation (EI) and the iso-parametric inversion (IIN) are done at the same timefor each node as follows:

• Loop over the whole nodes of the new element in (αRM) mesh.• Loop over the whole elements in the old mesh (α).

For Each node, one needs to solve (6.16a) in order to obtain theiso-parametric local coordinates. The modified Newton-Raphson iterativemethod is used to solve (4.4) as follows:Let

f1(ξn, ηn) = Xn −m∑

k=1

Nk(ξn, ηn)Xk = 0 (6.17a)

f2(ξn, ηn) = Y n −m∑

k=1

Nk(ξn, ηn)Y k = 0 (6.17b)

130 6 Applications

Using the above equations, the ith iterative local coordinates are given asfollows;

ξni+1 = ξn

i − f1(ξni , ηn

i )∂f1∂ξ

∣∣∣(ξn

i+1,ηni )

(6.18a)

ηni+1 = ηn

i −f1(ξn

i+1, ηni )

∂f1∂ξ

∣∣∣(ξn

i+1,ηni )

(6.18b)

The iterations will continue until the square difference between two sub-sequent values is within acceptable tolerance (tol), such as

√(error)2 ≤ tol

• Once the iso-parametric local coordinates (ξn, ηn) are determined, onemust check the conditions of (6.17a). If they are satisfied, then the nodeis identified inside that element; otherwise, one moves to other elements inthe (α) domain.

In some cases, the node lies along the border of two or more elements(ξ or η ∼= 1) and is identified in more than one element. In this case thenodal variables are averaged smoothed. The method of obtaining the iso-parametric local coordinates ξn, ηnof node n is referred as the “InverseIso-parametric Mapping Technique”.

Once the iso-parametric local coordinates (ξn, ηn) are known, then thenodal displacements and pore pressure can be interpolated from the Eo-nodes to node n using the interpolation functions Nk (may be taken tobe different interpolation functions than those used in the finite elementanalysis) as follows:

Un =m∑

k=1

Nk(ξn, ηn)Uk (6.19a)

Pn =mp∑

k=1

Nk(ξn, ηn)P k (6.19b)

b) Mapping of Stresses and Strains at Gauss Points

Since the mapping of stresses at the Gaussian elements interpolation points(I.P’s) is done after each increment, the new I.P’s of the (αRM) element willremain inside the α element, the least square smoothing will be used for thispurpose. The first step before smoothing is to obtain the iso-parametric localcoordinates of the new I.P’s with respect to the old element “Iso-ParametricInversion”. The modified Newton- Raphson iterative method is also used inorder to obtain the local coordinates of the new I.P’s with respect to the αelements as follows:

Let


f1ξnp, ηnp = Xnp −

mip∑

k=1

Nk(ξnp, ηnp)Xk = 0 (6.20a)

f2(ξnp, ηnp) = Y np −mip∑

k=1

Nk(ξnp, ηnp)Y k = 0 (6.20b)

where mip is the number of integration points; (ξnp, ηnp) is the total coor-dinate of the new integration points (I.P), np, of the (αRM) elements withrespect to α elements; (ξnp, ηnp). And (Xk, Y k) are the global coordinates ofthe new integration point, np, and the previous integration point, k, respec-tively; and Nk is the interpolation function.

Then the ith iterative coordinates of the np integration points will begiven as follows (similar to the nodal local coordinates):

ξnpi+1 = ξnp

i − f1(ξnpi , ηnp

i )∂f1∂f2

∣∣∣(ξnp

i ,ηnpi )

(6.21a)

ηnpi+1 = ηnp

i −f1(ξ

npi+1, η

npi )

∂f1∂ξ

∣∣∣(ξnp

i+1,ηnpi )

(6.21b)

The iteration stops once a certain accuracy is achieved, (i.e√

(error)2 ≤ tol.).Once the iso-parametric local coordinates (ξnp, ηnp) are obtained for all

the new I.P’s in the αRM (with respect to the αRM elements), the leastsquare smoothing of stresses and strains can be carried out over the wholefinite elements domain, until “global smoothing” is attained. The problemis then to find the set of smoothed stresses from the unsmoothed stressesthat minimize the squares of errors using smoothing shape function (may betaken to be different interpolation functions than those in the finite elementanalysis).

c) Least Square Smoothing

The least square smoothing procedure may be carried out over the wholefinite element domain global smoothing, or performed separately over eachindividual element, a process called “local smoothing” (Hinton and Campbell,1974). The local smoothing of the finite elements is adapted in this work.

There are two types of local smoothing:

1) Local functional smoothing, in which the smoothed function g(ξnp, ηnp) isassumed to be the least squares to the unsmoothed function σ(ξnp, ηnp).

2) Local distance smoothing, in which the smoothed function g(ξnp, ηnp) isassumed an exact least squares fit to selected values of σ(ξnp, ηnp) (at theGaussian integration points).

132 6 Applications

d) Functional Smoothing

Let the smoothed function of stresses (for example) at any point, n, withinan element in the finite element formulation be given by the following ex-pression,

g(ξnp, ηnp) =nip∑

k=1

∼Nk

∼σk (6.22)

where∼

Nk are the smoothed shape functions, which may be taken as beingof different order from the interpolation shape functions used in the finiteelement analysis, and

∼σk are the smoothed IP stresses. If σ(ξ, η) are the

unsmoothed stresses at any point within the element, then the error betweenthe smoothed and unsmoothed stresses at any point within the element isgiven by

e(ξ, η) = σ(ξ, η) − g(ξ, η) (6.23)

The problem now becomes finding the smoothed stresses, σk, which minimizethe functional as follows;

E =∫∫

A

e(ξ, η)2 dx dy (6.24)

For the error E to be minimized,

∂E

∂σi= 0 for I = 1,nip (6.25)

To find the smoothed stresses, one needs to solve the following set of equa-tions,

[A]e ∼σ = Fe (6.26)

where, the elements-smoothing matrix will be

[A]e =

[∫∫A

N∼1 σ det JDξdη . . .

∫∫A

∼N1N

∼n det Jdξη

∫∫A

Nn1

∼N1σ det JDξdη. . . .

∫∫A

∼Nn

∼Nn det Jdξη

](6.27)

and the elements force vector will be

[F]e =

[∫∫A

∼N1σ det Jdξdη.

]

∫∫A

∼N1σ det Jdξdη.

(6.28)

e) Discrete Smoothing:

In discrete smoothing the problem is to minimize the following:


E =nip∑

k=1

e(ξ, η)2 (6.29)

In order to minimize the error E, the following expression is obtained, whichsatisfies the condition of (6.23),

[AD]e∼‘σ = FDe (6.30)

where the discrete element smoothing matrix will be as follows,

[AD]e =

nip∑k=1

∼N1(ξk, ηk)

∼N1(ξk, ηk) . . .

nip∑k=1

∼‘

N1(ξk, ηk)∼

N1(ξk, ηk)

nip∑k=1

∼N1(ξk, ηk)

∼N1(ξk, ηk) . . .

nip∑k=1

∼‘

N1(ξk, ηk)∼

N1(ξk, ηk)

(6.31)

and the discrete element force vector will be as follows:

[FD]e =

nip∑k=1

∼N1(ξk, ηk)σk

nip∑k=1

∼N1(ξk, ηk)σk

(6.32)

f) Determination of the Yield Stress at the New Gauss Points

In the elastoplastic analysis, one needs to transfer the model’s yield surfacefrom the old I.P’s of the α element to the new I.P’s of the αRM element. Oneway to do this transfer is by interpolating or by calculating the least squaresmoothing of the yield sureface from old I.P’s to the new I.P’s. This transfercan lead to values that are self-consistent (Lee and Bathe 1994); that is tosay, the effective stress σ may be greater than the yield stress σy). Lee andBathe (1994) suggested that the yield stress (σy) at the new I.P’s should beobtained from the mapped equivalent plastic strain and the strain hardeningfunction as

σy = f(εpeq) (6.33)

In the modified Cam Clay model, the evolution of the hardening parameterdPc is given as a function of the incremental volumetric plastic strain, dεp

v as

dPc =(1 + e)(λ − k)

Pcdεpv (6.34)

Taking the integration of both sides of (6.33) over the increment, one obtains

pc∫

pco

dPc

Pc=

(1 + eav)(λ − k)

epv∫

εpvo

dεpv (6.35)

134 6 Applications

Therefore,

ln(Pc) − ln(Pco) =(1 + eav)(λ − k)

(εpv − εp

vo) (6.36)

where Pco and εpvo are the reference hardening parameter and volumetric

plastic strain respectively. Because the hardening parameter is a function ofthe plastic volumetric strain εp

v, the author proposes to map the equivalentplastic strain by the least square smoothing from the old IP’s in the α elementto the new IP’s in the αRM element. The next step is to find the hardeningparameter Pc at the new IP’s. Because remeshing is done after each loadincrement, the new IP’s are expected to be very close to the old IP’s. Voyiadjisand Abu-Farsakh (1998) propose to relate the hardening parameter at thenew IP’s in the αRM element to the hardening parameter of the old closestIP’s in the α element by the formula

ln(Pc)new − ln(Pco)ln(Pc)old − ln(Pco)

=(1 + enew)(1 + eold)

εPvnew

εpvold

(6.37)

The equivalent plastic strain can be mapped by applying the same methodused to map stresses and strains. Another option may be used by directmapping of ln(Pc) by the least square smoothing technique similar to thestress and strain mapping.

In mapping the hardening parameter, one needs to check the consistencybetween the stress state and tehe new mapped yield surface in the new IP’ssuch that the stress state can not be outside the yield surface, and the fol-lowing condition must be satisfied,

Py

Pc≤ 1.0 (6.38)

where Py is the hardening parameter corresponding to the current state.In the case Py

Pc> 1.0, correction of the yield surface will be carried out by

returning the stress state to the yield surface.

6.2.6 Modeling of Interface Friction

The engineering literature contains a variety of interface elements formula-tions. Goodman et al. (1968) developed a simple rectangular, two-dimensionalelement with eight degrees of freedom. The element formulation is derivedon the basis of relative nodal displacements of the solid elements adjacent tothe interface element. The thickness of the element is assumed to be zero.Zienkiewicz et al., (1970) proposed the use of continuous solid elements asinterface elements with a simple nonlinear material property for shear andnormal stresses, assuming uni-form strain in the thickness direction. However,not many critical and systematic studies and implementations of the conceptare available in the published literature (Desai et al., 1984). Numerical diffi-culties may arise for zero-thick elements, from ill conditioning of the stiffness


Fig. 6.21. Thin-layer interface element

matrix due to very large off-diagonal or small diagonal terms (Ghaboussiet al., 1973). In recent years, a number of investigators have considered theuse of thin-layer interface elements (i.e Desai et al., 1984; Sharma and Desai,1992). In this present study, the 6-noded thin-layer iso-parametric element isused to model the soil-shield interface friction.

a) Interface Element

Consider a 6-noded nonlinear iso –parametric thin layer interface elementas shown in Fig. 6.21. This element is used to model the soil–shield inter-face friction. Using the local coordinates, (ξ, η), the shape functions, Ni, aredefined as follows:

N1 = −(ξ/4)(1 − ξ)(1 − η)N2 = (ξ/4)(1 + ξ)(1 − η)N3 = (ξ/4)(1 + ξ)(1 + η)N4 = −(ξ/4)(1 − ξ)(1 + η)N5 = (1/2)(1 − ξ2)(1 − η)N6 = (1/2)((1 − ξ2)(1 + η)

One defines the nodal displacement vector, d, as follows:

d = u1 v1 u2 v2 u3 v3 u4 v4 u6 v6

The displacement at any arbitrary point can be released to the nodal dis-placement vector using the shape function, Ni, as

136 6 Applications

u

v

= [N ]d (6.39)

where

[N] =[

N1 00 N1

N2 00 N2

N3 00 N3

N4 00 N4

N5 00 N5

N6 00 N6

](6.40)

b) The Elasto-Plastic Stress-Strain Relation

An elastoplastic constitutive model of the Coulomb yield function and itsassociated flow rule (elastic-perfectly plastic model) is used in this study asproposed by Matsui and Sam (1989) and as shown in Fig. 6.22. This modelcan represent the restrained dilatancy. The shear stress (τ) is given by

τ = Ca + σn · tan δ (6.41)

where Ca is the adhesion, σ′n is the effective normal to the surface stress,

and δ is the angle of friction between the soil and the shield body. The yieldfunction, f, can be defined (Matsui and San, 1989) as

F = τ2 − (Ca + σ′n · tan δ)2 = 0 (6.42)

Assuming the associated plastic flow rule, the plastic spatial strain rate,d p, is given by

dp = λ∂f

∂σ′ = 0 (6.43)

Fig. 6.22. Coulomb yield function for interfaces (Matsui and San, 1989)


The consistency condition implies

df =∂f

∂σ′ dσ′ = 0 (6.44)

Substitution of (6.44) into (6.42) leads to

df = 2τdτ − 2(Ca + σ′n · tan δ) tan δ · dσ′

n = 0 (6.45)

The total spatial strain rate, d, can be decomposed into elastic, de, andplastic, d p, spatial strains rate components as

d = de + dp (6.46)

de = d − dp (6.47)

The plastic spatial strain is then becomes

Dp = λ∂f

∂σ′ = λ

0−2S2τ

(6.48)

where S = (Ca + σ′n · tan δ) tan δ.

The co-rotational stress rate, σ′, is related to the spatial strain rate as

σ′= [De]de= [De](d − dp)

σ′s

σ′n

τ

= [De]

ds

dn

dγ

− λ

0−2S2τ

(6.49)

where the elastic stiffness matrix, [De], is given by

[De] =

C1 C2 0C2 C1 00 0 G

(6.50)

in which

C1 =E(1 − v)

(1 + v)(1 − 2v); C1 =

Ev

(1 + v)(1 − 2v)(6.51)

where E is the elastic (young’s) modulus, v is the poison’s ratio, and G isthe shear modulus. By substituting (6.47) into (6.43) one obtains the scalarparameter, λ, such that

λ =τGdγs − SEdεn

2τ2G + 2S2E(6.52)

Substituting (6.52) into (6.50) leads to

138 6 Applications

σ′s

σ′n

τs

= [Dep]

ds

dn

dγ

− 1τ2G + S2E

0 0 00 S2E −τSG0 −τSG τ2G

ds

dn

dγ

(6.53)

or

σ′s

σ′n

τs

= [Dep]

ds

dn

dγ

(6.54)

where[Dep] = [De] − [Dp] (6.55)

The plastic stiffness matrix is given by

[Dp] =

0 0 00 D11 D12

0 D21 D22

(6.56)

where

D11 =S2E2

S2E + τ2G;D22 =

τ2G2

S2E + τ2G; and D12 = D21 =

−τSEG

S2E + τ2G

The element stiffness matrix is then given by

[K]ξ,η =∫

A

[B]T [Dep][B]detJdξdη (6.57)

Transforming the stiffness matrix from local (ξ, η) to global (x,y) coordinatesaxis, one obtains

[K]xy = [T ][K]ξ,η[T ] (6.58)

where [T] is the transformation matrix containing the direction cosines asgiven below,

[T ] =

cn2 sn2 cn.snsn2 cn −cn.sn

−2cn.sn 2cn.sn cn2 − sn2

(6.59)

where cn and sn are the direction cosines between the local and global coor-dinates.

6.2.7 Case Study of N-2 Tunnel

a) Subsurface Condition

The proposed Computational model is used here in order to analyze the N-2tunnel project located in San Francisco and constructed in 1981 (Abu-Farsakhand Voyiadjis, 1999). The soil condition of the N-2 tunnel site consists of anaverage of 6.1 m of rubble fill underlain by 9.1 m of soft sediment, knownlocally as Recent Bay Mud as illustrated in Fig. 6.23.


Fig. 6.23. Subsurface profile

A stratum of colluviums and residual sandy clay is encountered below theBay Mud (compete description of the subsurface condition can be found inClough et al., 1983). A tunnel of 3.7 m diameter was advanced within theRecent Bay Mud stratum. The fill is classified as loose to medium dense.The recent Bay Mud consists mainly of silt and lean clay that is normallyconsolidated. The ground water table is located at 3 m below ground surface.

b) Description of the Earth Pressure Balance (EPB) Shield

The EPB tunneling machine, as shown in Fig. 6.24, usually is a cylindricalshape within which the excavation takes place and the liner is erected. Itadvances itself by jacking against the in-place linear segments. After theshield is pushed forward by a full stroke of propulsive jacks, the jacks arewithdrawn and the next ring of liner segment is erected within the “tail”of the shield. As the shield advances, soil is excavated through slots in arotating cutter head and deposited into a soil-retaining area located betweenthe cutter head and the bulkhead. Two rows of cutting bits are usually setbetween the open slots in the cutter head to cut the soil at the face of theshield. The soil is removed from the enclosed spoil retaining area via a screw

140 6 Applications

Fig. 6.24. Earth pressure balance (EPB) shield section (Abu-Farsakh, 1996)

auger which takes it through the bulkhead and deposits it onto a conveyerbelt. The rates of soil excavation and removal are controlled by the operationof the shield.

The fundamental idea of the earth pressure balance shield is to allowfor the control of soil removed from the spoil-retaining area as the shieldadvances. The total earth pressures are measured inside the spoil retainingarea as the shield advances. This pressure is applied to the tunnel face tocounter-balance, in theory, the existing earth and hydrostatic pressure. Ifthis pressure is too large, it produces an upheaval at the surface, and if it istoo small, it leads to excessive settlements at the surface.

c) Finite Elements Analysis

Figure 6.25 represents the finite element mesh used in the longitudinal sec-tion analysis, while Fig. 6.26 represent the finite element mesh used in thetransverse section analysis. The eight-nodded iso-parametric finite element,Q8P4, is used to represent the cohesion soil of the Recent Bay Mud. The Q8element is used to represent the cohesion-less soil of the fill and colluviums.The pre pressures are kept fixed at the bottom of the rubble fill and at thetop of the colluviums. Six-nodded, thin-layer iso-parametric interface slip el-ements (Q6P4) are arranged between the shield machine and the soil in orderto model the soil-shield interface friction.

Desai et al. (1984) carried out parametric study on the effect of the thick-ness (t) to length (l) ratio of the interface element. They reported that the


Fig. 6.25. Finite element mesh for the longitudinal section analysis

use of t/l ratio in the range of 0.01 to 0.1 gives satisfactory results for inter-face behavior. In this study, the ratio t/l is taken as 0.1. The modified CamClay is used in this study in order to describe the plastic behavior of nor-mally consolidated cohesive soil. (The modified Cam-Clay model is discussedin Chap. 4.)

A nonlinear hyperbola model is used to describe the cohesionless soilresponse (fill and colluviums). This model has been used by several inves-tigators. Detailed Derivations of the hyperbolic formulas for Et and νt canbe found in various publications (i.e. Clough and Duncan, 1972; Desai, 1971;and Chang and Duncan, 1971). The final equations are given below,

Et = Ei(1 − λt)2 (6.60)

vt =A1−B1log

(σ′3

Pa

)

[1− (σ′1−σ′

3)GEi(1−λt)

](6.61)

where the initial tangent modulus, Ei, is related to the confining pressure,σ′

3, through the following empirical formula (Janbu 1963),

Ei = K ′Pa

(σ′

3

Pa

)n

(6.62)

and

λt =Rf (1 − Sinφ)(σ′

1 − σ′3)

2cCosφ + 2σ′3 Sinφ

(6.63a)

142 6 Applications

Fig. 6.26. Finite element mesh for the transverse section analysis

Rf =(σ′

1 − σ′3)f

(σ′1 − σ′

3)ult(6.63b)

where c and φ the cohesion and international friction angle as in the Mohr-Coulomb shear strength; Rf is the failure ratio; (σ′

1 − σ′3) is the stress dif-

ference; Pa is the atmospheric pressure expressed in appropriate units; andthe soil parameter A1, B1,G, K and n can be determined from conventionallaboratory tests such as the tri-axial tests.

For unloading-reloading of the soil, the unloading-reloading modulus, Eur,has been found to be related to the confining pressure, σ′

3, in the same man-ner as shown in (6.62) for the initial tangent modulus (Chang and Duncan,1971),

Eur = K ′urPa

(σ′3Pa

)n

(6.64)

where the unloading-reloading modulus number, K′ur, can be determined from

cyclic unloading-reloading laboratory tests. An elasto-plastic constitutive fric-tional model of the Mohr-Coulomb type is used to describe the response ofthe interface elements as described earlier. The interface friction angle, δ, canbe taken as δ = (2/3)φ, in which φ is the angle of friction of the soil.

The soil parameters used in this analysis are taken from Finno and Clough(1985). Table 6.5 describes the soil parameters from the nonlinear hyper-bolic model used for the fill and colluviums stratas chosen from the testresults for similar materials. The modified Cam-clay soil parameters usedfor the recent Bay Mud presented in Table 6.6 are based on iso-tropicallyconsolidated undrained triaxial compression (CIU) tests. For the interfaceelements, the soil parameters are the following: The Young’s modulus E =14,700 kPa; Poisson’s ratio v = 0.3; adhesion Ca = 0; and shield-soil interface


Table 6.5. Soil Parameters for the Hyperbolic Model (after Finno and Clough,1985)

Parameter Fill Colluvium

v, Poisson’s ratio 0.3 0.35Rf , Failure ratio 0.9 0.9φ, friction angle 30 20

C, Cohesion, in kPa 14.36 47.88n, modulus exponent 0.5 0.4K, Primary loading modulus 400 945Kur, unloading-reloading modulus 600 1400Ko, Lateral earth pressure co-efficient 0.5 0.8γt, Total unit weight, in kN/M3 15.75 19.68

Table 6.6. Soil Parameters for Cam-Clay Model (after Finno and Clough, 1985)

Parameter Recent Bay Mud

λ, Slope of isotropic compression line 0.326k , Slop of isotropic unloading-reloading line 0.043M, Slope of the critical state line 1.2Void ratio at critical state and unit pressure 3.72Ratio of Shear modulus of effective overburden pressure 40Kv; kh = 5 kv, coefficient of permeability, in m/sec 10−8

γt, Total unit weight, in kN/m3 16.53

angle δ = 20. In the longitudinal analysis, the average applied face pressureis kept at 74.25 kPa. The EPB shield is successfully advanced a distanceequal to 1.5 times the shield length as illustrated in Fig. 6.25 by the dashedelement.

It has been accomplished by six subsequent excavation steps. At eachstep of excavation, a rigid translation is applied to the nodes representingthe shield face and body until the average applied earth pressure reaches thespecified value (74.25 kPa). The strength of the soil is reduced by 20% forthe elements just ahead of the shield face in order to represent the disturbed(remolded) zone created ahead of the tunnel face.

The distribution of the heaving pressure in the transverse section isdetermined from the analysis of a simple 3-D with rigid translation of theshield face and body. An elliptical pressure distribution of the ratio 1:5:1 forthe crown, spring line, and invert respectively is adapted in this analysis.The heaving process is completed when a certain criterion is reached. Thecriterion is taken by many workers to be based on the field measurements.This criterion can be taken as either when the observed (measured) field

144 6 Applications

heave is reached (Rowe et al., 1990a) or when the lateral observed displace-ment is achieved (Finno and Clough, 1985). These two versions of the onecriterion assume that one knows the field measurements before the analysis.The aim in this study is to predict the soil response before the event.

At the present state of study, the heaving process is considered completedwhen the predicted excess pore pressure at the spring line during heaving inthe transverse section reaches the predicted excess pore pressure from thelongitudinal analysis. This assumption seems reasonable in this study. Theauthors suggest that in the future, the initial heave/settlement be obtainedfrom the longitudinal plane strain analysis.

The distribution of the unloading pressure during the closure of the gap of15 cm is taken as 1:1:5:1 for the crown, spring line, and invert (respectively)in order to ensure that the closure of the gap at the spring line, invert, andcrown occurs simultaneously. If a uniform unloading pressure distributionis adapted, consideration must be given to the fact that closure of the gaparound the tunnel opening may not occur simultaneously for crown, springline, and invert. Proper considereation of this possibility can be achieved, forexample, by monitoring each of the peripheral nodes separately. Once anynode closes the corresponding gap, that node will be prevented from furthermovements. This monitoring will be continued until the all of the peripheralnodes close the gap.

d) Results

Deformation of the Longitudinal Section

The predicted longitudinal displacements ahead of the shield that are ob-tained from the longitudinal section analysis of the shield advance are com-pared with field data measured at 1.2 m and 5.5 m in front of the shield asshown in Fig. 6.27:

Most of the longitudinal displacement occurs in the Recent Bay Mudlayer. Figure 6.28 compares the predicted and observed surface heaving. Goodagreement can be seen between the predicted and observed longitudinal dis-placements at 1.2 m. However, the predicted displacements at 5.5 m and thepredicted surface heaving are higher than the measured ones. This result isexpected because the soil is restricted from moving laterally, and thereforethe plane train displacements are expected to be higher than those obtainedfrom the three-dimensional analysis.

A 3-D computational model needs to be developed in the future that willhelp to obtain a correlation between the actual 3-D displacements and thelongitudinal plane strain displacements. This correlation will help analysizingthe transverse section, especially with regard to the value of the initial surfaceheave/settlement.


Fig. 6.27. Longitudinal displacement ahead of the shield; (a) at 1.2 m ahead ofthe face; (b) at 5.5 m ahead of the face

Excess Pore Pressure and Stress Changesin the Longitudinal Section

The contour of the predicted excess pore pressure in the Recent Bay MudLayer resulting from the shield-advance in the longitudinal analysis is pre-sented in Fig. 6.29. The time required to advance the shield 7.5 m is estimatedto be 0.82 days based on the average daily advancement (9.1 m/day). Thistime allows partial dissipation of the soil. The predicted excess pore pressureat the end of the shield advancement is 32.5 kPa immediately in front of the

146 6 Applications

Fig. 6.28. Predicted and observed surface initial heaving at the longitudinal section

Fig. 6.29. Contours of Excess pore pressure (kPa)

shield face and decreases rapidly with increasing distance from the shieldface.

Figures 6.30 and 6.31 represent the contour of axial stress change andshear stress respectively. The maximum predicted axial stress change of


Fig. 6.30. Contours of axial stress change (kPa) at the longitudinal section

Fig. 6.31. Contour of normal stress (kPa) at the longitudinal section

21.8 kPa occurs around the bottom corner of the shield face. The predictedshear stress ranges from −16 kPa along the bottom of the shield body to−6 kPa along the top of the shield body. This variation is due to the factthat the effective normal stress is higher near the bottom of the shield.

148 6 Applications

Fig. 6.32. Lateral displacement at maximum heave; (a) at 2.05 m from centerline;(b) at 5.5 m from centerline

Fig. 6.33. Predicted and observed lateral displacement distribution at maximumheave


Deformations of the Transverse Section

The predicted lateral deformation at the time of maximum heave as comparedwith the lateral measurements at a distance of 2.05 m and 5, 5 m from thetunnel centerline is shown in Fig. 6.32. Good agreement id found between thepredicted and observed lateral deformation. Both the observed and predicteddeformations indicate that the maximum lateral deformation occurs slightlyabove the spring line and that most of the lateral deformation occurs in theRecent Bay Mud layer.

The predicted and observed lateral displacement distributions obtainedat the time of the maximum heave near the crown, spring line, and invert areshown in Fig. 6.33. A good agreement with the field data can be seen for thepredicted deformations that are less than 5 m from the centerline. Figure 6.33shows that the lateral displacement decreases rapidly with distance from thetunnel centerline.

The development of the surface initial heave and final settlement (af-ter consolidation) profiles are drawn in Fig. 6.34. The surface heaves up to

Fig. 6.34. Predicted and observed distribution of surface displacements

150 6 Applications

0.88 cm initially above the tunnel vertical center axis and ultimately settlesdown to 2.89 cm. The observed final settlements are quite close to the pre-dicted values.

6.3 Estimation of Hydraulic Conductivityusing Acoustic Technique

6.3.1 Introduction

Hydraulic conductivity of geo-materials is one of the major geotechnical pa-rameters used for designing and analyzing geotechnical structures. However,the procedure for determining the hydraulic conductivity (sometimes termedpermeability with some controversy) of these materials is complicated. Al-though considerable theoretical and experimental work has been done ondeveloping accurate methods for determining the hydraulic conductivity ofsoils, existing methods are still time consuming and expensive. Conventionaltests, both in situ and in the laboratory, tend to change the texture of soilsin the process of drilling and sampling.

This chapter will discuss the development of a new method for estimatingthe hydraulic conductivity of soils using an acoustic technique. Because thismethod does not disturb soil texture, it may deliver an accurate assessmentof the hydraulic conductivity.

The theoretical background of this study is based on the established factthat there are two compression waves for saturated soils: (1) the wave thathas in- phase motion of fluids and solids, and (2) the wave that has out ofphase motion of fluids and solids. The first wave, called the fast P-wave, hasmuch less attenuation and propagates faster. The second wave, called the slowP-wave, has significant attenuation compared to the first one and propagatesslower. The different behavior of the fast P-wave and the slow P-wave was firstpredicted by Biot (1956a, b, c, d) theoretically, and experimentally confirmedby others later (Berryman 1981).

The measurement of slow P-wave velocity involves complex procedures,including the capture of a weak signal associated with the high attenuation.Yamamoto, (2003), Xiang and Sabatier, (2003), Bouzidi and Schmitt (2002),and Batzle et al. (2001) have recently reported the measurement of the slowP-wave velocity by implementing a spread spectrum technology called PseudoRandom Noise, and the measurements of the slow P-wave are now consideredwell documented (Yamamoto, 2003; Xiang and Sabatier, 2003).

Recently Song and Kim (2005) reported an easier acoustic method ofestimating soil hydraulic conductivity, which is based on the attenuation offast P-wave with different frequencies. This chapter discusses essentials ofacoustic methods for soil permeability estimation.

6.3 Estimation of Hydraulic Conductivity using Acoustic Technique 151

6.3.2 Basics of Wave Propagation in Saturated Media

Seismic wave is one of the most conveniently used ground investigation toolsfor profiling geotechnical conditions. Cross Hole technique is used for anaccurate and direct measurement of seismic wave velocities, while SASW(Spectral Analysis of Surface Wave) technique is used for convenient andquick determination of seismic wave velocities. There are also other methodssuch as Down Hole technique, Up Hole technique, and many others. Seismicwave velocities obtained by such methods are used for the determination ofmodulus, for the evaluation of liquefaction potential, for the profiling ofground layers, and for many other purposes.

The essentials of seismic waves and acoustic waves are not very different.Acoustic wave (a compression wave) is an extension of seismic wave, butthe frequency range is that of audio wave. Typically the seismic frequency isbetween 1 and 100 Hz, but the acoustic frequency is between 20 and 20,000 Hz.Seismic wave is easy to feel but acoustic wave is easy to hear. Typically thefield-measured P-wave velocity from the seismic technique is the first P-wavevelocity because the magnitude of slow P-wave velocity is very weak and istypically hidden in the noise. However, the slow P-wave has much interactionwith pore properties that it can be used to detect the pore properties.

Theoretical background for the relationship between the slow P-wavevelocity and the permeability can be found in Biot (1956a,b,c,d). Biot(1956a,b,c,d) developed macroscopic equations for the propagation of elasticwaves in poro-elastic media. In addition to the existence of a fast compres-sional P-wave and a shear wave, Biot’s theory predicted the existence of athird wave mode referred to as the slow compressional P-wave. The essentialsof Biot (1956a,b,c,d) are shown in this chapter.

Assuming a conservative physical system which was statistically isotropic,Biot derived the following stress-strain relations containing four distinct elas-tic constants:

σx = 2Nex + Ae + Qε (6.65)σy = 2Ney + Ae + Qε (6.66)σz = 2Nez + Ae + Qε (6.67)τx = Nγx (6.68)τ y = Nγy (6.69)τ z = Nγz (6.70)s = Qe + Rε (6.71)

where, N is the shear modulus, A is the Lame’s constant, Q and R arecoupling moduli, e is the strain of solids, ε is the strain of the pore fluid, ands is the pore pressure.

152 6 Applications

The dynamic equilibrium relations assuming no dissipation are expressedas follows:

∂σx

∂x+

∂τ z

∂y+

∂τ y

∂z=

∂2

∂t2(ρ11ux + ρ12U x) (6.72)

∂s

∂x=

∂2

∂t2(ρ12ux + ρ22U x), etc (6.73)

where ρ11 is the volume fraction of solids, ρ12 is the coupled volume fraction ofsolids and pore fluid, and ρ22 is the volume fraction of pore fluid. The equa-tions for wave propagation are obtained by substituting expressions (6.65)through (6.71) into (6.72) and (6.73) and applying divergence operation:

∇2(Pe + Qε) =∂2

∂t2(ρ11e + ρ12ε) (6.74)

∇2(Qe + Rε) =∂2

∂t2(ρ12e + ρ22ε), with definitionP = A + 2N (6.75)

With additional mathematical arrangements, the above equations are ex-pressed thus:

∇2(σ11e + σ12ε) =1

V 2c

∂2

∂t2(γ11e + γ12ε) (6.76)

∇2(σ12e + σ22ε) =1

V 2c

∂2

∂t2(γ12e + γ22ε) (6.77)

Solutions of these equations are written in the form,

e = C1 exp[i(lx + αt)] (6.78)ε = C2 exp[i(lx + αt)] (6.79)

And the velocity V of these waves is

V =α

l

This velocity is determined by substituting expressions (6.78) and (6.79)into (6.76) and (6.77). Putting z = V 2

c /V 2 and by applying mathematicalarrangements and orthogonality relations, one obtains:

zi =γ11C

(i)21 + 2γ12C

(i)1 + γ22C

(i)22

σ11C(i)21 + 2σ12C

(i)1 + σ22C

(i)22

(6.80)

In the above equation, there are two z’s for two phase materials (i = 1, 2),and they are:

V 21 = V 2

c /z1 (6.81)V 2

2 = V 2c /z2 (6.82)


Equation (6.80) is obtained using orthogonality relations, an expression thatmeans that one solution represents the fast P-wave and another solutionrepresents the slow P-wave. This orthogonal relation also means that onesolution is for in-phase motion, and another one should be for out-of-phasemotion. Equations (6.65) though (6.82) are shown just to indicate the originof two different wave velocities. (For details of derivation, see Biot (1956a)).

When one adds dissipation to the above equations, the following equationsare obtained:

V1

Vc= 1 − 1

2

(f

fc

)2 (σ11σ22 − σ12)2

(γ12 + γ22)2ζ1ζ2

(ζ1 + ζ2 +

12ζ1ζ2

)(6.83)

Lc

x1=

12|ζ1ζ2|

σ11σ22 − σ212

γ12 + γ22

(f

fc

)2

(6.84)

V2

Vc=(

2f

fc

(σ11σ22 − σ212)

(γ12 + γ22)

)1/2

(6.85)

Lc

x2=(

12

f

fc

σ11σ22 − σ212

γ12 + γ22

)1/2

. (6.86)

Where, f is the frequency, fc is the characteristic frequency, ζ1 = z1 − 1, ζ2 =z2 − 1, Lc

xirepresent attenuation of the first P-wave and the second P-wave.

Above results are for the low frequency range where the input frequency isless than the characteristic frequency. The Biot characteristic frequency is thefrequency near which propagation and diffusion effects have approximatelyequal contributions so that the material shows the maximum attenuation.One can also derive similar relations for the high frequency range. When oneplots relations (6.73) through (6.86) for low and high frequency ranges, oneobtains Figs. 6.35 and 6.36.

Fig. 6.35. Sweep Curve for Velocity

154 6 Applications

Fig. 6.36. Sweep Curve for attenuation

Figures 6.35 and 6.36 do not appear exactly same for the fast P-wave andthe slow P wave; however, the overall trend is the same. Figures 6.35 and6.36 show that there is a major change at the characteristic frequency. Biot(1956a) relates the characteristic to the viscosity and permeability as follows:

ωc =ηφ

Kρ(6.87)

where, ωc is the characteristic frequency (angular frequency), η is the viscos-ity, φ is the porosity, K is the intrinsic permeability (Darcy unit), and ρ is thefluid density. When ωc is determined from the experiment, the permeabilityK can be estimated from (6.87). Modification of (6.87) for familiar quantitiesin soil mechanics presents (6.88),

fc =φ · g2π · k (6.88)

where g is the gravity, f is the frequency in hertz, and k is the hydraulicconductivity in m/sec unit.

Equations (6.87) and (6.88) assume Darcy flow. Similar equations areavailable for different flow conditions. For squirt flow, (6.89) is suggested asfollows by Batzle et al. (2001),

ωc =Kα3

η(6.89)

where, K is frame modulus and α is crack aspect ratio.Based on the Kozeny equation, Kelder (1998) also suggested the following

equation,

ko =V 2

pf · η · φ · ρb

Vps · f(w) · [Kf · (Kb + 4G3 )]

(6.90)

where ko is the hydraulic conductivity, Vpf is the fast P-wave velocity, Vps isthe slow P-wave velocity, ρb is the bulk density, Kf and Kb are respectivelythe fluid and bulk moduli, G is the shear modulus, and f(w) is a functionof frequency that approaches one for w equal to zero. Recently, Yamamoto


(2003) reported a “super-k regime” for a more reliable imaging of permeabil-ity profile of highly permeable carbonate formations.

The studies cited above show that a promising theoretical base is al-ready in place. The main problem yet to be worked out is how to catch thecharacteristic frequency. The experimental program that follows this sectionpresents a suitable methodology for solving this problem.

6.3.3 Applicability

The applicability of the above equations has some limitations. These equa-tions are based one the fact that the characteristic frequency or wave ve-locities are reliably obtained. Batzle et al. (2001) reported that for low per-meability materials (k < 200 mD), the characteristic frequency could notbe captured since the characteristic frequency is lower than the measurablerange. However, they reported that the estimation of permeability for highlypermeable materials such as sand would be easier to ascertain.

Converted into a conventional geotechnical hydraulic conductivity, theexpression k = 200 mD is somewhere near 10−6 m/s. This is a typical per-meability range for silt to sandy soils. Therefore, application of the proposedmethod will be feasible for some soils. The quick calculation of Biot’s char-acteristic frequency by (6.89) gives the following results:

Table 6.7. Approximate range of Biot’s characteristic frequency

Permeability (m/s) fc (kHz) Remarks

10−6 500 ultrasonic10−5 50 ultrasonic10−4 5 sonic10−3 0.5 sonic

Researchers (Batzle et al. 2001, Yamato, 2003) presented the measuredslow P-wave velocity up to 106 Hz or a higher frequency range. Of course theseresults are for rock specimens. The received signals for soils (from the sameintensity of source signal) will be weaker than these same signals for rocksbecause of a higher attenuation. Fortunately, the widespread availability ofspread spectrum signal processing technique and precision sensors make cap-turing this weak signal more effective than ever. Yamamoto (2003) and Xiangand Sabatier (2003) recently reported the implementation of spread spectrumtechnology called PSRN(Pseudo Random Noise). Yamamoto reported thathe could achieve S/N ratio higher than 40% by using PSRN encoded signal.Also using the singular value decomposition method developed by Bregmannet al. (1989a,b), Yamanoto (2003) calculated the attenuation of the slow P-wave reliably.

156 6 Applications

wave

Fig. 6.37. Equipment set up

The widely used acoustic experimental program is very similar to that ofconventional surface wave technique. Figure 6.37 shows equipment set ups.This procedure requires a digital oscilloscope with one source and one re-ceiver and triggering capability. Measurement technique is similar to thatfor the conventional seismic wave measurement. The oscilloscope records thesource signal and received signal. Two signals will be shown in the oscilloscopescreen, and the arrival time of slow P-wave is calculated.

So far, the measurement scheme is about the same as that for a typicalsurface reflection test. For this purpose, a frequency controlled source is usedinstead of random frequency shock wave because the velocity of slow P-waveis also a function of frequency.

Sweeping from low frequency to high frequency presents the variation ofslow P-wave velocity or attenuation as shown in Figs. 6.35 and 6.36. Fromthis sweep curve reveals a characteristic frequency. Once the characteristicfrequency is found, the characteristic slow P-wave velocity is found. Fromthis characteristic slow P-wave velocity or characteristic frequency, one canestimate the permeability with proper equations.

Recently Song and Kim (2005) reported the application of the similarmethod to sandy soils as shown in Fig. 6.38. Song and Kim (2005) usedthe fast P-wave to capture the characteristic frequency. Figure 6.38 clearlyshows that the characteristic frequency can be captured and may be used forthe hydraulic conductivity computation. The hydraulic conductivity as mea-sured in the laboratory by Song and Kim (2005) is 0.0000542 m/sec., whilethe predicted by the Biot equation is 0.000167 m/sec. This discrepancy, how-ever, is not substantial considering the nature of the hydraulic conductivity.Moreover, the predicted hydraulic conductivity by Biot equation may not

6.4 Evaluation 157

Saturated soil

frequency (Hz)

dam

ping

Fig. 6.38. Frequency response of damping characteristic of mortar sand (Song andKim, 2005)

necessarily be exactly the same as that measured in the laboratory as pointedout by Berryman (2005). As the more experimental data are collected, thebetter relationship between the static (laboratory) hydraulic conductivityand dynamic hydraulic conductivity may be established.

6.4 Evaluation

There are many methods to evaluate the hydraulic conductivity of soils. Typ-ical laboratory tests include falling head tests and constant head tests. Bothtests may be performed for disturbed and intact specimens. However, lab-oratory tests for intact specimens are very difficult to conduct; therefore,disturbed specimens are commonly used. Because of this unavoidable sam-ple disturbance, laboratory test results may not be very reliable even thoughthey are performed under well controlled experimental conditions.

Typical field tests include in situ falling head tests and in situ constanthead tests. Field tests are less prone to the sample disturbance; however,field tests are not performed in a highly controlled environment. For exam-ple, typical field tests are performed in drilled bore holes, and one does nothave much control over the wall condition of bore holes. The bore hole di-ameter may be larger than it should be, or the bore hole wall may be coatedwith drilling fluid. Many times engineers do not have any other option but touse the bore hole as is. Another disadvantage of the field hydraulic conduc-tivity tests is that it is time-consuming. Typically, such a test requires severaldays. Therefore the amount of test data is limited, and a detailed hydraulicconductivity profile can seldom be provided.

158 6 Applications

The BAT is used to minimize the sample disturbance problem. It operatesin a bore hole bottom. BAT uses a hypodermic needle to inject pressurizedwater into the ground and monitors the dissipation of this injected water withan associated piezometer. By analyzing the dissipation of injected pressurizedwater, it obtains the hydraulic conductivity of the soil at the bore-hole bot-tom. This method is one of the best methods invented so far; however, borehole bottoms are usually contaminated with the drilling fluid and sludge.When the bore hole bottom is not clean (sometimes, thorough cleaning ofbore hole bottom is impossible.), the BAT results may bring out the hy-draulic conductivity of the drilling sludge, not that of the soils.

Several methods to evaluate the hydraulic conductivity using the piezo-cone penetrometer have recently been developed. Piezocone is a devicethat penetrates into the ground and measures the pore pressure and pen-etration resistance. Rust et al. (1995) estimated the coefficient of con-solidation from the pore pressure dissipation curve obtained during theresting time (that is required to reconnect the driving rod) Robertsonet al. (2000) applied the same analytical method as the one used forthe analysis of conventional dissipation curve for piezocone holding tests.Song et al. (1999) tried to estimate the hydraulic conductivity of soilfrom the pore pressure difference between u2 (the measured pore pres-sure at the shoulder of the conventional piezocone penetrometer tip) andu3 (the measured pore pressure at the cone shaft that is approximately14 cm apart from u2 location of the piezocone penetration test). Compar-ison of the pore pressures at u2 and u3 (u3 is measured seven seconds af-ter u2), shows that the pore pressure at u3 is usually smaller than that atu2. The difference is caused by the dissipation of excess pore pressure inseven seconds, because the piezocone penetrometer moves with 2 cm/sec be-tween u2 and u3. Using this dissipation data, one can estimate the hydraulicconductivity of the soil.

House et al. (2001) tried to estimate the pore pressure responses of soils bychanging the penetration speed of the piezocone penetration test. They foundthat the soils with high hydraulic conductivity show no significant differencebetween measured pore pressures for two different penetration speed, and thesoils with a low hydraulic conductivity show significant difference betweenmeasured pore pressures for two different penetration speed. When one hasthe difference of those two pore pressures, one can estimate the hydraulicconductivity of the soil.

Voyiadjis and Song (2003) reported the calculation of the hydraulic con-ductivity based on back analysis of pore pressure measured during the pene-tration of piezocone. They used the coupled theory mixtures and soil plastic-ity to estimate the hydraulic conductivity. Using Voyiadjis and Song’s (2003)methods, one can obtain virtually continuous hydraulic conductivity pro-files of the ground without additional time-consuming experiments. Theserecent efforts have contributed significantly to the easy evaluation of hydraulic

6.4 Evaluation 159

conductivity. They also show how desperate the engineers are to find moreeffective and reliable ways to evaluate hydraulic conductivities.

Although the acoustic method is in line with these recent studies, it isa completely different method. This method does not use any penetrometerthat may disturb grounds; it uses non destructive wave techniques, which donot disturb the structure of soils.

Another benefit of this method is that it may provide the complete spec-trum of the permeability of the ground when it is combined with the inver-sion technique. Used in conjunction with the inversion technique, the acousticmethod can provide the tomographical view of hydraulic conductivities of ajob site. Continuous and three-dimensional hydraulic conductivity informa-tion will enhance the current practice of geotechnical engineering substan-tially (e.g. design and analysis of soft soil problems).

7 Advanced Topics

Traditionally, coupled theory for soils was mainly focused on micro-mechanisms. However, modern theories of molecular and quantum mechan-ics combined with formidable computation power extends the coupled theoryto the nano level. This chapter discusses some topics of Nano mechanics forgeo-materials (7.1). Also, the coupled behavior of micro-mechanisms is dif-ferent in saturated soils from the coupled behavior of other continua such ascomposite materials or metals. This chapter also discusses topics of coupledbehavior of micro-mechanisms (7.2).

7.1 Nano-mechanics for Geotechnical Engineering

7.1.1 Introduction

The properties of soils vary extensively. Naturally, the prediction of soil be-havior is definitely challenging even with modern geotechnical engineering.For accurate predictions of soil behavior, sophisticated constitutive relationsare currently used. However, the prediction of detailed behavior (e.g. fluid-clay interactions) is extremely difficult; these sophisticated constitutive rela-tions work with new input parameters that are not easy to determine. Someparameters, especially parameters for micro-mechanics, are known to exist,but testing methods are not well developed at present. Many researchers (in-cluding these authors) are striving to find these parameters. A new methodthat provides these material parameters rationally and predicts the behaviorof soils will provide a quantum leap for the whole geotechnical community. Inthis section the authors present a method for implementing a Nano-mechanicsapproach into geotechnical Engineering to overcome the associated difficultiesand to provide a new method for predicting and analyzing soil behaviors.

Nano-mechanics is an emerging technology that deals with Nano (10−9 m)level particle size, that is the size of a cluster of atoms and molecules. Thus,Nano-mechanics is essentially based on quantum and molecular mechanics,and naturally it deals with the equilibrium states of atomic and molecularforces. One advantage of using Nano-mechanics is that its input parame-ters are well established physical quantities, such as electron charge, dihedralangle, electron permittivity, and so on. Another advantage of using Nano-mechanics is that the detailed physical properties (or input parameters for

162 7 Advanced Topics

continuum mechanics) of materials including soils are obtained from the sta-tistical average behavior of atoms and molecules. The principal method inthis concept is that one starts from Nano-level material behavior and con-tinue to integrate to obtain the real (macro) scale material behavior withoutresorting to difficult input parameters. The authors have addressed manyapplications in nanocomposites (Voyiadjis et al., 2003; Meng and Voyiadjis,2003; Srinivasan et al., 2005).

The great challenge to this averaging and integrating process is the sub-stantial amount of computation time required. However, promising time-saving averaging and integrating techniques have recently been developed,and some researchers (including one of the co-authors of this book) success-fully predicted the behavior of materials such as carbon-nano-fibers withina reasonable computation time (Srinivasan et al., 2005). Soils of course aredifferent from carbon-nano-fibers in many ways; however, both materials aremade up with nothing but atoms and molecules, and the operating mechanicsare fundamentally the same. Eventually, the behavior of soils can be under-stood and predicted in great detail using Nano-mechanics. The applicationsof this method are many. One of the applications may be to the NASA spaceprogram, such as the prediction of soils behavior in the Moon and Mars.

7.1.2 Brief History of Nano-mechanics

Richard P. Feynman provoked the scientific community by predicting anemerging nanotechnology on December 29th, 1959 at the annual meetingof the American Physical Society at the California Institute of Technologyin his speech “There’s Plenty of Room at the Bottom.” Regis (1995) com-ments that “Some in the audience (Many of them were physicists, after all,and they’d earned all their hundreds of advanced degrees by reading books,studying books, going to the library for books, big damned fat square vol-umes with page after page of dense text)” suspected Feynman’s nano-conceptwhen he talked about writing a full volume of Encyclopedia Britannica ona pin head. Some of them took it as a typical dinner table joke.” However,nanotechnology emerged just as Feynman predicted, and is now a reality inthis modern scientific world.

The discovery of carbon nanotubes by Iijima in 1991 has inspired a num-ber of scientific investigators to explore their unique properties and potentialapplications. In 1992, the eminent scientist K. Eric Drexler testified at theSenate Committee on Commerce, Science, and Transportation, Subcommit-tee on Science, Technology, and Space chaired by the ex vice-president AlGore. He convinced the country’s leaders of the strategic potential of nan-otechnology (Regis, 1995; US Congress, 1992). Since then, nanotechnologyhas found its way to broader and more profound applications because of itsability to make things smaller, faster, or stronger. Consequently, new ma-terials with exceptional properties have emerged, allowing for the design ofmachines that will lead to new manufacturing paradigms. (CMP Cientifica,

7.1 Nano-mechanics for Geotechnical Engineering 163

2002). So far, nanotechnology has been successfully focused on the fabrica-tion of nano-materials. Subsequently, analytical works by nano-mechanics arerequired to better understand the underlying principles and mechanisms atatomic and molecular level.

Nano-mechanics is a combination of molecular mechanics and quantummechanics; because these are the fundamental building blocks of every me-chanics, the same technique can be used for geo-continuum mechanics. Inspite of advancement of nanotechnology and nano-mechanics in many sci-entific areas, it has had little influence and use in geotechnical engineering.This chapter addresses the application of nano-mechanics to the geotechnicalarea, its purpose being to improve the current state of practice in geotechnicalengineering.

7.1.3 Nano-mechanics as a General Platformfor Studying Detailed Behavior of Geo-materials

Because of nano-level length scales and also pico to femto seconds time scales,nano-mechanics provides extremely detailed material behavior. Modern con-tinuum mechanics (Cui et al. 1996; Jasiuk and Ostoja-Starzewski 2003; Kali-akin et al. 2000; Ling et al. 2002, 2003; Manzari 2004; Song and Voyiadjis2005a; Voyiadjis and Song 2005b, Voyiadjis et al. 1999) and discrete mechan-ics (Anandarajah, 2004, 1994; Peters, 2004) approaches are very capable ofproviding macro-scale material behavior but their ability to provide detailedmaterial behavior is limited. The contrasts of different scales are well de-picted in Fig. 7.1. Some of the advanced continuum mechanics are capable of

Fig. 7.1. Comparison of length scales and time scales of different mechanics(Cygan, 2001)


providing detailed material behavior to some extent; however, the determi-nation of input parameters is still a challenging task. To analyze the detailedbehavior of geo-materials such as clay-fluid interaction, nano-mechanics isthe tool of choice at present (Smith, 1998; Shroll and Smith, 1999).

Some of the merits of nano-mechanics are its extraordinarily well estab-lished theories and simple input parameters. One of the theories could be apotential energy equation shown in (7.1); some of the input parameters aretraditional physical and chemical constants, such as permittivity of a vacuumand angle bend force constant which are already well defined in Table 7.1.

ETotal = ECoul + EV DW + EBond Stretch + EAngle Bend + ETorsion (7.1)

where ECoul, the Coulombic energy, and EV DW , the van der Waals energy,representing so-called non-bonded energy components, and the final threeterms represent the explicit bonded energy components associated with bondstretching, angle bending, and torsion dihedral, respectively. Important fea-tures of energy components are summarized in Table 7.1.

Using (7.1) and from the optimized energy or force field, one can computematerial properties easily. One can compare the potential energy along acertain direction and equalize it to the strain energy in the same direction andcompute spring constant or modulus. This is the key concept of Anderson-Parrinello-Rahmen theory (Ray and Rahman, 1984; Ray, 1988; Parrinelloand Rahmen, 1981). A preliminary result by the authors using the aboveprocedure is presented in Fig. 7.2.

The major challenges to this process are the difficulties associated with thetheory of molecular mechanics or quantum mechanics and the great amountof computation time. Yet this challenge is very much attenutated nowadays,since the concepts for molecular mechanics or quantum mechanics are well es-tablished. Ever since the establishment of the time-independent Schrodingerequation and the Born-Oppenheimer approximation that effectively decou-ples nuclear and electronic motions, molecular interactions are easily handledand computed. This challenge is also partly overcome by the advent of ac-cessible molecular mechanics software such as LAMMPS, NAMD, BioCoRE,and Materials Studio.

With these accessible software programs and the basic knowledge ofPhysics and Chemistry, one can even perform quantum level analyses sat-isfactorily.

Dynamic conditions occur when one has interaction of two different sta-ble molecular systems. When the two materials are mixed (allowed to inter-act), molecular interaction takes place and finally achieves a new equilibriumwith new optimum (minimum) potential energy. The chemical procedureof obtaining the new equilibrium strictly follows molecular dynamics rules(Katti et al., 2004). Therefore, the interaction of two different materials isconveniently analyzed by molecular dynamics. When the chemical reactionis completed, one can use molecular mechanics to predict the physical prop-erties. A part of computer outputs in this respect obtained by the Song and


Table 7.1. Important features of potential energy components

Name Governing Equations Remark

Coulombic ECoul = e2

4πεo

∑i=j

qiqj

rijwhere qi

and qj represent the charge ofthe two interacting atoms (ions),e is the electron charge, and εo isthe permittivity (dielectric con-stant) of a vacuum.

The Coulombic energy isbased on the classical de-scription of charged parti-cle interactions and variesinversely with the distancesrij .

van der Waals(2nd term ofLennard-Jones12-6 potentialenergy)

Lennard-Jones Potential EnergyEV DW

=∑i=j

Do

[[Rorij

]12− 2[

Rorij

]6]

where Do and Ro representempirical parameters.

The second term is the vander Waals energy represent-ing the attractive molecularinteractions.

Bond Stretch EBond Stretch = k1(r − ro)2

where r is the separation dis-tance for the bonded atoms, ro

is the equilibrium bond distance,and k1 is an empirical force con-stant.

The bond stretch term canbe represented as a sim-ple quadratic (harmonic)expression.

Angle Bend EAngle Bend = k2(θ−θo)2 where

θ is the measured bond angle forthe configuration, θo is the equi-librium bond angle and k2is theangle bend force constant.

The energy equation for an-gle bend for a bonded sys-tem is typically expressedusing a harmonic potential.

Torsion ETorsion = k3(1 + cos 3ϕ) wherek3 is an empirical force constantand ϕ is the dihedral angle.

The torsional dihedral in-teractions expressed withthe dihedral angle ϕ is de-fined as the angle formed bythe terminal bonds of quar-tet of sequentially bondedatoms as viewed along theaxis of the intermediatebond.

Remarks Additional terms may be added to the total potential energyexpression of (7.1), such as an out-of-plane stretch term forsystems that have a planar equilibrium structure.


Fig. 7.2. A part of computer outputs showing calculated material properties frommolecular mechanics (Software: Material Studio)

Tentative Results

Olemiss Civil Engineering

Polymer molecules are placedaround CNT. (t=0 s.)

System aft er molecular dynamics simulation(15 ps.)

This simulation would help us understandinginterfacial mechanical properties.

Fig. 7.3. A part of computer outputs showing molecular dynamics (The left figureshows the initial condition when the polymer is introduced around the carbon nanotube. The right figure shows the condition after 15 pico seconds)

Al-Ostaz (2005) for a carbon nano-tube is shown in Fig. 7.3. It is believedthat the same concept may be used to evaluate the interaction between clayminerals (e.g. Yazoo clay in Mississippi) and pore water.

However, the process of applying nano-mechanics to geo-materials isnot without difficlties. Although equations and minimization techniques for


potential energies are known to be correct, each equation or technique has itsown intrinsic assumption(s) and approximation(s) which may not be quitereasonable for geo-materials (see Belytschko, 2005). A typical myth is “Mole-cular mechanics does not assume anything and solves the problem very accu-rately”. In reality, substantial research is required to evaluate and select theproper optimization technique(s) for geo-materials.

7.1.4 Nano-mechanics as a Tool to Study Macro-level MaterialProperties Through Continuumization

The previous section described the capability of nano-mechanics to evaluatedetailed properties of geo-materials. On the other hand, nano-mechanics isnot as capable as continuum mechanics in providing and predicting macro-scale material behavior because of its extensive computation time. Thereforeone needs a carrier that bridges nano-mechanics and continuum mechanicsand also provides nano-scale material behavior in detail as well as contin-uum scale macro-material behavior. With the carrier, one can make use ofnano-mechanics in conjunction with the well established modern continuummechanics described in Table 7.2. Essentially, nano-mechanics will supply thefundamental properties that continuum mechanics needs (e.g. continuum me-chanics parameters shown in Table 7.2, and continuum mechanics will workout the micro and macro scale behavior of geo-materials.

In this way, one can enjoy the full features of modern continuum mechanicswithout expending exhaustive efforts to evaluate input parameters.

One of the promising techniques in continuum mechanics called RVE(Representative Volume Element) casts helpful light on the feasibility of com-bining nano-mechanics and continuum mechanics so that the detailed mate-rial behavior as well as macro-scale material behavior is obtained withoutconsuming excessive computational costs.

The underlying idea of RVE concept is based on the concept that themacro scale behavior is nothing but an averaged lower scale behavior. Uponimplanting nano-mechanics in the lowest RVE of continuum mechanics, onemay obtain detailed properties with minimum input parameters.

Another technique called “Equivalent Beam or Truss Element Method”is also useful. These approaches represent the lattice structure of moleculesinto joints and beams in traditional numerical analyses. Elongation stiffnessand torsional stiffness are obtained from the equalization of a bond stretchpotential and a twisting potential in molecular mechanics to the elongationstrain energy and the torsional strain energy in traditional mechanics. Inthis way the molecular level simulations are effectively performed in popularcommercial software such as ANSYS. This method turned out to be quiteeffective for continua (Srinivasan et al., 2005).


Table 7.2. Micro-mechanical behavior in continuum mechanics

Micro-mechanicalBehavior Form of Governing Equations* Remark

Rotation ofparticles

W s′′ = ξ(αds′′ − ds′′α)where, W s′′,ξ, α, and ds′′ are theplastic spin tensor, constant, backstress, and deformation, respectively.

Dafalias (1998),Cosserat (1909), Songand Voyiadjis (1999 to2005), Voyiadjis andKattan (1990, 1991)

Interaction ofparticles

˙εvp = εvp−k∇2εvp

where, ˙εvp

εvpv , and k are the

homogenized strain rate, visco-plasticvolumetric strain rate and a constant,respectively.

Di Prisco and Aifantis(1999), Zbib (1994),Zbib and Aifantis(1988), Voyiadjis andSong (2005), Song andVoyiadjis (2005a)

Damage ofparticles

εdij = λd ∂g

∂σij, where, εd

ij ,

λd, g and σ are the damage strain,damage multiplier, damage potential,and stress, respectively.

Voyiadjis and Song(2005a)

Viscosity ofpore fluid

f = fs − po[1 + ηvp

√32

˙p√Bii

)1/m1 ] ≡ 0

where, f, fs,po,ηvp, ˙p, Bii, and m1 arethe dynamic yield surface, static yieldsurface, mean principal stress,viscosity, time rate of the mean stress,the differentiation of fwith stress, anda constant, respectively,

Perzyna (1963, 1966,1988), Song andVoyiadjis (2005a),Voyiadjis and Song(2005)

Flowcharacteristicsof pore fluid

−div [(nw/ρw) Kws(grad Pw − ρwb+ ρwaw)] + div vs = 0where, n, ρs, ρw awKwsvsPw and bare the porosity, mass density of thesoil, mass density of the water,acceleration of water, permeabilitytensor, solid velocity, pore waterpressure, and body force vector.

Coupled theory ofmixtures Biot (1955,1978), Prevost (1980,1982), Muraleetharanet al. (1994), Schrefleret al. (1990), Wei andMuraleetharan (2002)Song and Voyiadjis(1999 to 2005)Voyiadjis and Song(2000 to 2005)

• Equations may vary for different researchers. These equations are generallyaccepted forms.

Detailed concepts and comparisons of these two methods are shown inTable 7.3. Using this continuumization technique, researchers, including oneof the co-authors of this study (Srinivasan et al., 2005), have successfullypredicted the behavior of materials such as carbon nano-fibers based on ad-


Table 7.3. Features of averaging techniques

Method Governing Rules* Remark

RVE A′ = 1V

∫v

AdV

Where, A′ is the property atupper scale, V is the volume ofRVE, A is the property at lowerscale.

Similar to homogenization tech-nique in continuum mechanics.

Full capability of incorporatingmolecular interactions.

Belytschko and Xiao (2003)Huang and Jiang (2005)Liu et al. (2005)Belytschko (2005)Song and Al-Ostaz (2005)

Equivalentbeam ortrusselement

Ku = f where,u = [uxi, uyi, uzi, θxi, θyi, θzi,uxj , uyj , uzj , θxj , θyj , θzj ]

T

f = [fxi, fyi, fzi, mxi, myi, mzi,

fxj , fyj , fzj , mxj , myj , mzj ]T

K =

[KiiKij

KijKjj

]

This is the continuumexpression of molecularmechanics.

Input parameters for stiffness,such as Young’s modulus, wasobtained by comparing thepotential energy for stretchingto the equivalent strain energyas follows:

EBondStretch = k1(r − ro)2

= Eaxialstretch = 12

EAL

(∆L)2

This method is not veryeffective for simulating dynamicprocedure such as interaction ofminerals and pore fluid.Li and Chou (2003)Odegard et al. (2001)Ostoja-Starzewski, (2002)Srinivasan et al. (2005)Wang et al. (2005)

• Equations may vary for different researchers. These equations are generally ac-cepted forms.


vanced cotinuumization algorithms (Belytschko and Xiao 2003; Belytschko2005; Li and Chou 2003; Odegard et al. 2001; Ostoja-Starzewski, 2002).

Soils, however, have both continuum properties and particulate proper-ties. For example, sands are continua inside the grain boundary; they arediscrete media outside the grain boundary.

Therefore, nano-mechanics provides fundamental properties to discreteelement, and the continuumization process between discrete element and con-tinuum (Cosserat brothers 1909; Anandarajah 2004; Tordesillas et al. 2004;Peters, 2004) will provide macro-behavior.

The Nano-mechanics approach for soils is quite challenging and must bedifferent from that for other continua. The rational procedure shall be asfollows:

1) Implement nano-mechanics for grains and grains-liquid interaction.2) Obtain the material properties for discrete particles and surrounding liq-

uid3) Use a continuumization technique to expand the discrete properties to

continuum properties

(Maiti et al., 2004) presented a different method for averaging called “DPD(Dissipative Particle Dynamics)” that is quite frequently used in Chemistryto analyze the interaction of two different fluids. The DPD approach utilizesthe positions and momenta of “fluid droplets” illustrated in Fig. 7.4 ratherthan individual atoms for dynamic analysis. Therefore, DPD technique solvesmolecular dynamics in terms of RVE, and it may be used to compute the equi-librium condition for long length and time scales. Maiti et al., 2004 revisedDPD theory to incorporate the interactions between carbon nano-tubes andfluid, and they obtained the equilibrium morphology that can be used as aninput to traditional finite element codes. This method is partly similar tothe RVE concept in continuum mechanics; but it is also different because itis still at molecular level; it is commonly called “coarse grained” moleculardynamics. The revised DPD (Maiti et al., 2004) therefore is equivalent to amolecular level RVE, and it requires no extra averaging scheme. As a result,DPD technique seems very cost effective to study interactions between clayparticles and the pore fluid. The material properties from nano-mechanicswill be transferred to the continuum properties by RVE and accompanyingintegration technique.

The link between nano-scale molecular mechanics and macro-scale con-tinuum mechanics, therefore, may be summarized as follows:

• Clayey Soils: Molecular mechanics with DPD (or similar mechanism) –Lowest RVE for continuum mechanics

• Sandy Soils: Molecular mechanics to evaluate grain properties – DEM toevaluate grain interactions – Lowest RVE for continuum mechanics

The linking procedure is not easy; nevertheless, the preliminary research hasshown quite convincing possibilities.


Fig. 7.4. Fluid droplets in DPD (Maiti et al., 2004)

This method also has some challenges. Traditional continuumization tech-niques are primarily good for the continua but not for particulate media. DPDtechnique is promising for clayey soils; however, further extensive research isnecessary before it becomes fully feasible.

7.1.5 Space Science Application

Nano-mechanics predicts material properties based on the chemical, physicaland electrical interactions of the constituents. Its advantage in space explo-ration is evident: whereas performing physical or chemical experiments wouldbe extremely difficult, a nano-mechanics approach can reasonably predict theproperties and behavior of new materials in the outer space (insofar as oneknows the chemical composition of the materials). One of the great dreamsof human beings is to create habitats on the Moon or Mars. The behaviorof materials will be different on the Moon from their behavior on the Earth.The properties of muscovite and montmorillonite in wet condition are verydifferent from each other, but they are similar in dry conditions. The surfaceof the Moon and Mars are completely dry, and properties of materials therewill be very different from their properties on Earth. It is predicted thatthe behavior of these materials in outer space is very different from that onEarth; geotechnical engineering based in nano-mechanics cast the new possi-bility in this aspect for it can predict the material behaviors in such extremeconditions.

One example of a material that calls for nano-mechanics is beta quartzformed at high temperature and pressure – a material that does not exist innormal conditions at thte earth’s surface. Nano-mechnics makes the analysisof these special materials possible without much difficulty.


Nano-mechanics for geo-materials is an area of science that is still growingand which stands in need of much more research. However, it is quite sure thatmodern nano-mechanics has a strong potential for providing a new paradigmfor geotechnical engineering.

7.2 Coupled Behavior of Micro-Mechanisms

7.2.1 Introduction

The macro-behavior of geo-materials is the result of the complex interac-tion of the coupled micro-behaviors of many constituent elements. Micro-mechanical behavior includes the rotation of grains, grain interactions suchas grain interlocking, the viscosity of the pore fluid, and the integrity of grains(Voyiadjis and Song, 2002).

The rotation of grains affects the equilibrium equations because part ofthe applied energy is used for the rotation of the grains. There are many waysto consider the rotation of grains; this study accounts for it by employing theconcept of plastic spin.

Grain interaction affects the stress-strain redistribution and well-posedness of numerical methods. When soils with lower than the critical voidratio are subject to shear, the dilation (volume expansion) occurs primarilyalong the shear band; the soil particles along the shear band are more highlystressed than those in the surrounding area; and the higher stress along theshear plane is transferred to the surrounding area followed by the volumeexpansion. This mechanism reduces the stress in the shear band, and thusstrain is reduced accordingly. The reduced strain also contributes to the well-posedness of the numerical methods as discussed in Sect. 4.5.3. Due to thestress (or strain) redistribution (or homogenization), this phenomenon is alsotermed (almost officially) the homogenization mechanism.

The viscous property of the soil affects the rate dependency of materials.In localized shearing zones such as the shear band region, the strain rate ismuch higher than that outside the shear band. Therefore, the material withinthe shear band responds more strongly than the material outside the shearband. This phenomenon tends to expand the shear band to the neighboringarea. Rate dependency, therefore, brings out another kind of localization orhomogenization mechanism.

The integrity of grains may be degraded during the loading and unload-ing process because of particle crushing, micro-cracks or wear-out of sharpedges. The degradation of the friction angle due to particle crushing is welldocumented by de Beer (1963), Lee (1965), and Lee and Seed (1967). Thisphenomenon will cause the reduction of the modulus. Desai and Zhang (1998)and Desai et al. (1997) discussed the reduction of modulus by “disturbedstate”. Ultimately all damage mechanisms will cause the reduction of themacroscopic stiffness of the soils. For the localized shearing zone such as the

7.2 Coupled Behavior of Micro-Mechanisms 173

shear band, the reduction in the stiffness of the material will cause the soft-ening of the material with localized high strains. This mechanism enhancesthe strain localization.

From the above discussion, one can see that the micro-mechanical behav-ior is coupled. Some mechanisms enhance localization while some mechanismsenhance homogenization.

Saturated soils are essentially a two-phase material (e.g. solid grains andpore water). Interaction between the two components of two-phase materialproduces effective stresses and pore pressure. The effective stress and porepressure compete with each other and reduce each other’s magnitude. There-fore the effects of the above micro-mechanical mechanisms are expected tobe even more complicated in soils. Saturated soils are also pervious, whichin addition causes the response of the soils to be affected by the hydraulicconductivity.

This section investigates the coupled effects of the above mentioned micro-mechanical mechanisms on the behavior of saturated soils around a penetrat-ing object. Several factors – the rotation of particles, the interaction of par-ticles, the rate dependency, the soundness of the particles, and the couplingof particles with pore fluid – are all incorporated in the anisotropic modifiedCam Clay model through the plastic spin, the gradient theory, the visco-plasticity, the damage theory, and the coupled theory mixtures, respectively.

Material parameters are not easy to define for the above micro-mechanicalbehavior; therefore, a parametric study is performed to estimate the propermaterial properties.

7.2.2 Simplification of Equations Incorporatingthe Physical Behavior of Soils

The rate-dependent anisotropic modified Cam Clay model with strain gradi-ent and plastic spin are incorporated with the coupled theory of mixtures aspresented in Sect. 4.5. The formulations presented in the above chapter arefurther simplified, reflecting the physical behavior of soils. In this way, manyparameters introduced in above sections are simplified for practical purposes.Viscosity-dependent rate dependency is reduced to (7.2) as shown by Horsky(2002);

f = fs − po

[1 +( ˙p

η

)m]≡ 0 (7.2)

where fs is the static yield surface, η is the viscosity parameter that has sec−1

unit, and m is a dimensionless parameter. In this work m is used instead of1/m1 to reduce numerical instabilities. Equation (7.2) also assumes a constantBii.

A non-local accumulative effective visco-plastic strain rate is introducedas follows;

˙εvp = εvp−k∇2εvp (7.3)


Fig. 7.5. Finite element mesh for the piezocone penetration analysis

Considering that the effective visco-plastic strain gradient is related tothe volume change of the material (Di Prisco and Aifantis, 1999), (7.3) isfurther simplified as follows;

˙εvp = εvp−Kεvvp (7.4)

where, εvpv is the visco-plastic volumetric strain rate. Equation (7.4) naturally

accommodates strain reduction for dilative behavior and strain increase forthe contractive behavior depending on the sign of K.

Numerical Simulation of the Cone Penetration Test

To investigate the performance of the proposed formulations, numerical sim-ulations of the cone penetration tests are performed and compared with theexperimental results. For the numerical simulations, the cone penetrationtests are modeled as shown in Fig. 7.5. The detailed procedure of numericalsimulation is also shown in Sect. 6.1.


Fig. 7.6. Gradation of K-33 specimen

Model Calibration

To calibrate the fundamental model parameters, anisotropic triaxial test re-sults for artificially manufactured K-33 soil (33% Kaolinite, 67% Sand) car-ried out by Kim (1999) are used. The fundamental properties of K-33 soilare shown in Table 7.3 and Fig. 7.6. Determination of material properties orparametric studies for rate dependency, gradient, and damage are explainedin each relevant section.

7.2.3 Rate Dependency

Rate dependency of the cone penetration test was analyzed for 7 different pen-etration velocities, e.g. 0.02 cm/s, 0.2 cm/s, 2 cm/s, 5 cm/s, 10 cm/s, 20 cm/s,and 200 cm/s. To determine the material properties η and m of (7.2), a seriesof parametric study was carried out. Trial η was changed from 0.1 to 100,000(η is infinity for rate independent materials with the assumption of positivem). Trial m was changed from 0.1 to 1. The responses of the cone penetra-tion test turned out to be most reasonable to these parameters when η isapproximately 10 and m is 1. These numbers are used for the analyses. Thepore pressure responses of the cone penetration tests for these 7 velocitiesare shown in Fig. 7.7a.

Figure 7.7a shows the magnitude of the excess pore pressure is low atthe high permeability, and it is high at the low permeability. This trendagrees with the expected behavior. Figure 7.7a also shows the two-stage ratedependency. The rate dependency is minimal when the penetration velocity islower than 10 cm/s. The rate dependency is substantial when the penetrationvelocity is higher than 10 cm/s. The rate dependency is subdued again when


Table 7.4. Basic material properties

Parameter Value Units

Compression index, λ 0.11 dimensionlessRecom pression index, κ 0.024 dimensionlessInitial void ratio, eo 1.0 dimensionlessPoisson’s ratio, ν 0.3 dimensionlessSlope of critical line, M 1.16 dimensionlessUnit Wt. of soil, γt 1.8 ton/m3

Depth (from ground surface) 20 mUnit Wt. of water, γw 1.0 ton/m3

Atterberg Limit LL=20, PL=14 %Specific Gravity, Gs 2.67 dimensionlessWater Content, w 18.56 %Undrained Shear Strength, su 80 kPa

Fig. 7.7a. Excess pore pressure response of cone penetration test for differentpenetration velocities and different hydraulic conductivities

the penetration velocity is higher than 20 cm/s. Analysis for the penetrationvelocity higher than 200 cm/s. was not performed in this analysis because theadditional dynamic effects need to be considered for the analysis of such afast penetration. This variation of excess pore pressure with the penetrationvelocity is somewhat unexpected because continuously increasing excess porepressure with increasing penetration velocity was expected. To analyze thisstrange behavior, Figs. 7.7b and 7.7c are plotted below.

Figure 7.7b shows the magnitude of the effective stress at the cone tipis higher for high permeability soils (silty soils), while it is lower for the lowpermeability soils (fat clays). This is because one has high excess pore pres-sure in low conductivity soils. This trend is a generally expectable behavior.


Fig. 7.7b. Effective stress response at the cone tip for different penetration veloc-ities and different hydraulic conductivities

675

680

685

690

695

700

705

710

715

0.01 0.1 1 10 100 1000

Penetration Velocity (cm/S)

To

tal V

ert

ical S

tress (

kP

a)

k=10-15 m/s

k=19-9 m/s

k=10-8 m/s

k=10-7 m/s

Fig. 7.7c. Total stress response at the cone tip for different penetration velocitiesand different hydraulic conductivities

Figure 7.7b also shows that the variation of the effective stress at the conetip decreases with increasing penetration velocity. But it does not show sub-stantial difference for different hydraulic conductivities when the penetrationvelocity exceeds 2 cm/s. In other words, the magnitude of the effective stressat the cone tip changes with the permeability only when the penetrationvelocities are lower than 2 cm/s. The reasons for this behavior are not con-firmed here due the absence of existing experimental data to study suchphenomena.

Therefore Fig. 7.7c is plotted for a better understanding of this behavior.


Fig. 7.8. Influence of rate of penetration on cone resistance in a varved clay(reprinted form Roy et al., 1982)

Figure 7.7c shows the total stress at the cone tip for different penetrationvelocities and different hydraulic conductivities. It shows essentially rate in-dependent total vertical stress at the low penetration velocity (when velocityis less than 0.2 cm/s), reduced total vertical stress at the intermediate pene-tration velocity (when velocity is less than 10 cm/s and larger than 2 cm/s),sharp increase of total vertical stress at the critical penetration velocity (whenvelocity is between 10 cm/s and 20 cm), and subdued response at the higherpenetration velocity (when velocity is higher than 20 cm/s). This behavior isquite interesting and unprecedented. However, comparison of Fig. 7.7c withRoy et al. (1982) and Bemben and Myers (1974) shows that the soil behaviorexhibited in Fig. 7.7c is the expected behavior for saturated soils. A copyof a figure from Roy et al. (1982) is presented in Fig. 7.8 for completeness.Figure 7.8 shows variation of the normalized cone resistance qc with respectto the preconsolidation pressure.

To compare Fig. 7.7c and Fig. 7.8, one needs to understand that the totalvertical stress in Fig. 7.7c multiplied by the cross sectional area of the conepenetrometer is approximately the same as the qc in Fig. 7.8. Therefore,Figs. 7.7c and 7.8 should have linear relationships, and the trend should bethe same. These two figures show exactly the same trend. Bemben and Myers(1974) attributed the shape of the curve to a combination of viscosity andpore pressure drainage effects. They further elaborately predicted the mech-anism of this behavior such as “up to about 0.05 mm/s drained conditionsapply and above 5 cm/s undrained conditions apply”. Based on the resultsobtained from Bemben and Myers (1974) and Roy et al. (1982), Lunne et al.(1997) speculated that the background mechanisms are as follows:


For very slow rates of penetration qc is predominantly of a drained nature.As the rate of penetration increases, qc decreases due to the decrease ineffective stress and reduction in strength. As the penetration rate increases,the viscous forces offset the strength reduction and the curve will pass througha minimum. Then viscous forces will tend to dominate the process and qc willincrease.

This mechanism was the hypothesis predicted by Lunne et al. (1997) whenBemben and Myers (1974) and Roy et al. (1982) reported the test results. Atthis time, however, one can confirm that this mechanism is a fact, and themechanisms that are displayed in Figs. 7.7a to 7.7c are generally true.

7.2.4 (Strain) Gradient

Gradient dependency of the cone penetration test was simulated with twodifferent gradient conditions. Gradient parameter K of (7.4) was determinedfrom a series of parametric studies. Parameter K was increased from 0 with 0.1increments. K equals zero implies zero gradient effect. Bigger K means a morepronounced effect of the gradient. From the parametric study, it turned outthat K cannot be larger than 0.3. At that point, the singularity condition wasobserved for a certain element. In (7.4) one can observe that a very big valueof K will change the sign of the strain. This implies that a very big value of Kwill change dilative behavior to contractive behavior or contractive behaviorto dilative behavior. These results do not represent the correct behavior ofthe gradient. Therefore, two gradient conditions (e.g. K = 0.1 and K = 0.2)are used for further study. Gradient causes the homogenization of the stressand strain, and therefore, one can expect the reduced maximum pore pressureand maximum effective vertical stress. Figure 7.9a shows these behaviors.

Figure 7.9a demonstrates the noticeable effects of the gradient on theexcess pore pressure. One observes that the flat curve of the excess porepressure is extended up to a penetration velocity of 20 cm/s when the gradientis incorporated. The effects of the gradient coefficient K are not especiallylarge; however, bigger values of K produce smaller excess pore pressure.

Figure 7.9b shows the effects of gradient on the effective vertical stress.The overall trend is similar to Fig. 7.9b. The effects of gradients on themagnitude of the effective vertical stress are not pronounced. The authorsbelieve that slightly reduced excess pore pressure in Fig. 7.9a causes slightlyhigh effective stress in Fig. 7.9b. However, that effect is minimized when thepenetration velocity is higher than 2cm/s. Therefore, the total vertical stressis not significantly affected by the gradient.

Figure 7.9c shows the variation of total stress at the cone tip: the responseis similar to Fig. 7.9c. However, the flat part of the total stress is extendedagain to a penetration velocity of 20 cm/s. when the gradient is considered.


530

540

550

560

570

580

590

0.01 0.1 1 10 100 1000

Penetration Velocity (cm/s)

Exc

ess

Po

re P

ress

ure

(kP

a) k=10-15 m/s. K=0k=10-15 m/s. K=0.1k=10-15 m/s. K=0.2k=10-9 m/s. K=0k=10-9 m/s. K=0.1k=10-9 m/s. K=0.2

Fig. 7.9a. Excess pore pressure response at the cone tip for different penetrationvelocities, hydraulic conductivities, and gradient constants

100

110

120

130

140

150

160

0.01 0.1 1 10 100 1000


Eff

ectiv

eV

ertic

al S

tres

s (k

Pa)

k=10-15 m/s. K=0k=10-15 m/s. K=0.1k=10-15 m/s. K=0.2k=10-9 m/s. K=0k=10-9 m/s. K=0.2k=-9,kd02k=10-9 m/s. K=0.1k=10-9 m/s. K=0.2

Fig. 7.9b. Effective stress response at the cone tip for different penetration veloc-ities, hydraulic conductivities, and gradient constants

7.2.5 Damage

Damage dependency of the cone penetration test was simulated with twodifferent damage magnitudes. Damage parameters A1 and A2 of (4.77) weredetermined from a series of parametric studies. Considering that the skeleton


675

680

685

690

695

700

705

710

715

0.01 0.1 1 10 100 1000


Tota

l Ver

tical

Str

ess

(kP

a)

k=10-15 m/s. K=0k=10-15 m/s. K=0.1k=10-15 m/s. K=0.2k=10-9 m/s. K=0k=10-9 m/s. K=0.2k=-9,kd02k=10-9 m/s. K=0.1k= 10-9 m/s. K=0.2

Fig. 7.9c. Total stress response at the cone tip for different penetration velocities,hydraulic conductivities, and gradient constants

stiffness of soils is much smaller than that of the soil particle, a smaller mag-nitude of damage is assumed. One or two percent of damage is assumed froma 200 kPa to a 1000 kPa confining pressure. These assumptions correspond toA1 = 24.04, A2 = −0.23535 and A1 = 50, A2 = −0.35366, respectively.

Figure 7.10a shows the variation of excess pore pressures with differentdamage magnitudes. For higher damage, lower excess pore pressures are ob-served. However, sharp changes of excess pore pressures at the critical pene-tration velocity are not observed for the damaged condition in Fig. 7.10a.

This result is believed to be effected by the reduced stiffness of particlescaused by damage at high penetration velocity.

Figure 7.10b shows the variation of the effective stress at the cone tipwith different damage magnitudes. It shows higher effective stresses at higherdamage magnitudes. This phenomenon is somewhat unexpected since dam-age causes a reduction in the physical quantity. The authors speculate thisphenomenon is due to the interaction of grain damage and excess pore pres-sure. Excess pore pressure in Fig. 7.10a is decreased for higher damage. Thedecrease in excess pore pressure causes the increase in the effective stressalthough damage might decrease the effective stress itself. However, at thepresent the authors do not have experimental data to support this hypothesis.

Total stress variation at the cone tip is shown in Fig. 7.10c. The overallshape is similar to Fig. 7.9c. The curves with damage show smaller totalstresses.


450

470

490

510

530

550

570

590

0.01 0.1 1 10 100 1000

Penetration Velocity(cm/s)

Exc

ess

Po

re P

ress

ure

(kP

a)

D=0%D=0.1%D=0.2%

Fig. 7.10a. Excess pore pressure response at the cone tip for different penetra-tion velocities, different hydraulic conductivities, and different damages (gradientconstant = 0.2)

100

120

140

160

180

200

220

0.01 0.1 1 10 100 1000


Effec

tive

Str

ess

(kP

a)

D=0%D=0.1%D=0.2%

Fig. 7.10b. Effective stress response at the cone tip for different penetration ve-locities, hydraulic conductivities, and damages (gradient constant = 0.2)

7.2.6 Permeability

Effects of permeability on the cone penetration tests are reported by Voyiadjisand Song (2003). Figure 7.11 is replotted from Voyiadjis and Song (2003).Figure 7.11 shows that the Piezocone penetration pore pressure is sensitive tothe permeability only within a certain range of the permeability. However, thepenetration pore pressure is not very sensitive to the permeability at very highor very low permeability ranges. From Figs. 7.7b to 7.10b one can also see thatthe effective stress at the cone tip shows little change for different hydraulic


640

650

660

670

680

690

700

710

0.01 0.1 1 10 100 1000Penetration Velocity (cm/s)

Tota

l Str

ess

(kPa)

D=0%D=0.1%D=0.2%

Fig. 7.10c. Total stress response at the cone tip for different penetration velocities,hydraulic conductivities, and damages (gradient constant = 0.2)

Fig. 7.11. Variation of excess pore pressure with hydraulic conductivity (Voyiadjisand Song, 2003)

conductivities, except at very slow penetration velocities. These results showthat the permeability affects the excess pore pressure response rather thanthe effective stress response. These responses are more pronounced within acertain range of permeability (10−9 m/sec ≤ k ≤ 10−6 m/sec).


7.2.7 Anisotropy and Plastic Spin

In this section a comparison is made to see the effect of anisotropic andisotropic models. Figure 7.12 shows the excess pore pressure contours ob-tained from the finite element analysis. Figures 7.12a and 7.12b show theresults of anisotropic modified Cam Clay model (AMCCM) with the plasticspin and without the plastic spin, respectively. Figure 7.12c shows the resultsof isotropic modified Cam Clay model (IMCCM) with a mean principle stressthat is the same as the vertical effective stress (hydrostatic stress condition).Figure 7.12d shows the IMCCM with a mean principal stress that is the sameas Fig. 7.12a. The results of IMCCM with initial anisotropic stress conditionare shown in Fig. 7.12e. The initial confining condition is the same as inFig. 7.12a. Figure 7.12f shows the results of AMCCM with initial isotropicstress condition at which the mean principle stress is the same as the verticaleffective stress.

In Figs. 7.12a and 7.12b one can see that the spatial distributions ofthe excess pore pressures in both figures look alike whether the plastic spinis incorporated or not. However, different maximum excess pore pressuresare obtained in each case. This is due to the high concentration of micro-structural changes at the vicinity of the cone tip where the strains are ex-tremely large. This behavior is shown indirectly in Fig. 7.13 by the tensorcomponent (N2133) of the plastic spin. In the region of the cone face adjacentto the shoulder, one notes that most of the plastic spin activity occurs. Thisis also the region of maximum strain (Voyiadjis and Abu-Farsakh, 1997).

Comparison of Fig. 7.12a with Fig. 7.12c, Fig. 7.12d, and Fig. 7.12e showsthe difference between the AMCCM and the IMCCM constitutive relations.Figure 7.12a, Fig. 7.12d, and Fig. 7.12e essentially have the same mean prin-ciple stress. The difference is the initial confining condition and the appliedsoil model. From Fig. 7.12a and Fig. 7.12d, one can see that the pore pressureresponse of AMCCM is substantially higher than that of IMCCM (for thesame mean principle stress). However, Fig. 7.12c shows that when one uses aconfining pressure that is the same as the vertical effective stress in IMCCM,one obtains almost similar results with the AMCCM and the plastic spin.However, it should be noted that this agreement is mainly due to the highermean confining pressure. Figure 7.12e shows that even with an anisotropicinitial confining condition for IMCCM, the response is not much differentwhen the mean principle stress is the same. Figure 7.12f shows that the com-bination of high initial mean principle stress (same as vertical effective stress)with AMCCM results in the highest pore pressure response.

From this discussion, one can see that the pore pressure response of conepenetration tests is substantially affected by the applied model and appliedinitial confining condition. However, one should acknowledge that the combi-nation of anisotropic initial stress condition and AMCCM with plastic spinis the most realistic condition.


(a)

AM

CC

M w

ith

plas

tic

spin

(

b) A

MC

CM

wit

hout

pla

stic

spi

n

Fig

.7.1

2.

Exce

sspore

pre

ssure

conto

urs

ofA

MC

CM

and

IMC

CM


(c)

IMC

CM

for

mea

npr

inci

ple

(d)

IMC

CM

for

the

sam

e m

ean

s

tres

s sa

me

as v

erti

cal s

tres

s pr

inci

ple

stre

ss a

s (a

)Fig

.7.1

2(C

onti

nued

)


(e)

IMC

CM

for

sam

e in

itia

l(f

)A

MC

CM

wit

h in

itia

l a

niso

trop

ic s

tres

s co

ndit

ion

as (

a)

iso

trop

ic s

tres

s co

ndit

ion

as (

b)Fig

.7.1

2(C

onti

nued

)


Fig. 7.13. Distribution of plastic spin around a cone tip

Figure 7.14 shows the variation of the cone-penetrometer-induced porepressure with different permeability. It is noted that Fig. 7.14 is for the case ofground depth 20 m (saturated) and the stiffness similar to the LSU (LouisianaState University) calibration chamber test specimen (recompression indexκ = 0.024, Poisson’s ratio ν = 0.3). For the different soils such as stiffersoils, the back-bone curves in Fig. 7.14 may be shifted up especially at lowerpermeability range. From Fig. 7.14, one can know that the pore pressureresponse from the AMCCM (the one which considers both initial anisotropyand induced anisotropy) is closer to the experimental data than that fromIMCCM. This condition (the one which considers both initial anisotropy andinduced anisotropy) is closer to true field conditions; thus the results mustagree well with the field test results.

This behavior is quite rational considering the realistic behavior of soilsalways reflect the anisotropy and some degree of texture changes during shear.

7.2.8 Experimental Verification of Coupled Pore PressureAround a Cone Tip

This chapter experimentally investigates the distribution of excess pore pres-sures along the surface of a penetrating object during the penetration. Apenetrometer embodied with three pore pressure measuring units is used,and tests are performed in the calibration chamber facility at Louisiana StateUniversity. A large strain elasto-plastic coupled theory of mixtures that incor-porates the modified Cam-Clay model is used for analyzing the problem. Toincorporate the effects of micro-mechanical grain rotations at the close vicin-ity of the penetrating object, the back stress and plastic spin are included


0

100

200

300

400

500

600

700

800

1.0E 02 1.0E+00

Exc

ess

Po

re P

ress

ure

(kP

a)

-16 1.0E-14 1.0E-12 1.0E-10 1.0E-08 1.0E-06 1.0E-04 1.0E-

Hydraulic Conductivity (m/sec)

All Isotropic

Initial Aniso + Induced Aniso

Initial Iso + Induced Aniso

Size of rectangles shows therange of test dataSolid circles represent theLSU/CALCHAS test results

Fig. 7.14. FEM results of pore pressure prediction of PCPT and experimentalresults

in the numerical model. The results show that the maximum excess porepressure is observed at the tip (face) of the penetrating object while smallerexcess pore pressure is observed at the shaft of the penetrating object. Asimilar trend is expected for other researches (Torstensson, 1977; Houlsbyand Teh, 1988; Teh and Houlsby, 1991), and the same behavior is confirmedin a highly controlled test environment in this study. It is also shown that theinteraction and accompanying dissipation of the shear-induced pore pressureand compression-induced pore pressure play a major role for the excess porepressure response at the shaft of the penetrating object.

The Three-Piezo-element Miniature Penetrometer

The three-piezo-element miniature penetrometer used to obtain test resultswas specially designed and fabricated especially. A photograph of the pen-etrometer is shown in Fig. 7.15. It has a projected penetrometer area of 2 cm2

and an apex angle of 60. The maximum pore pressure capacity is 700 kPa.The penetrometer tip is detachable so that u1 and u2 porous elements are ex-changeable. The u1, u2, u3 numbering system follows the typical numberingsystem for the piezocone penetrometer. The u4 is numbered last because itis behind u3. The penetrometer can have two set-ups depending on the com-bination of porous elements. The possible configurations are pore pressuremeasurements at u1, u3, and u4 or at u2, u3, and u4.

The pore pressure transducers are Precision Measurement Miniature Pres-sure Transducer Model 150F full bridge electric resistor type strain gauges.


Penetrometer tip, u1

is attached here for u1, u3, u4 configuration

u2 (with porous tip removed ) Small dots are theleader holes for pore pressuretransmission

u3 (with porous tip removed)Circular dot is the sensing membraneof piezometer

u4 (with porous tip attached)

61

15 (

adju

stab

le)

0.61

Diameter = 1.61 cmArea = 2 cm2

For u1, u3, u4 configurationthe current penetrometer tip is removed and the newpenetrometer tip with u1 is attached.

Sleeve is exchange-able for adjusting u3 – u4 dis-tance

Fig. 7.15. The three-piezo-element miniature penetrometer (showing u2, u3, u4

configuration. Units are in cm.)

For the u1 or u2 location, conventional leader holes are used to transmit thewater pressure to the sensor (see Fig. 7.15). During the saturation process,a hypodermic needle is used to inject water into the leader hole. However,for the u3 and u4 locations, because the sensing membrane is directly placedjust next to the porous protective cover (see u3 in Fig. 7.15), the leader hole


is not needed. With this configuration, the saturation process for u3 andu4 piezometers is substantially simplified and the volumetric displacementduring the pore pressure measurement is minimized.

Measurements for friction resistance or end resistance are not performedfor this penetrometer. Only three pore pressure transducers are used be-cause of the limited space in the penetrometer body. For the end resistanceand friction data, independent penetration tests are carried out with anotherminiature penetromenter by Lim (1999). This calibration chamber test sys-tem and sample preparation procedure are essentially the same as those usedby Lim (1999) and Kim (1999). The essentials of the test system and samplepreparation procedure are outlined below. The details are given in the worksof Lim (1999) and Kim (1999).

The specimen is essentially a combination of sand, kaolin and deionizedwater at a water content of twice the liquid limit. This water content is foundto be adequate to minimize the segregation of soil grains and air entrapmentin the slurry. The grain size distribution of the kaolin and fine sand is shownin Fig. 7.6. Mixing is done in two large 40 gallon polyethylene tanks using aspecially designed, hand-held heavy duty agitator. A mixture of 33% kaolinand 67% Edgar fine sand by weight is used to prepare the K-33 specimens.The Atterberg limits of the soil mixture are shown in Table 7.4. Slurry isplaced very carefully inside the consolidometer (H×D = 1624 mm × 525 mm)with a large spoon.

A vertical consolidation pressure of 138 kPa is gradually applied to theslurry by an ENERPAC hydraulic pump. The vertical pressure is selected inorder to obtain an initial soil specimen of minimum strength to withstandits own weight. Pore pressures are monitored at two different elevations andat various radial distances. During the slurry consolidation, a Ko conditionis maintained due to the rigid wall structure of the slurry consolidometer.The drainage is allowed at the top and bottom of the specimen through thefilter papers, but the drainage is not allowed at the side wall. At the endof the first stage of slurry consolidation, the specimen enclosed in the mem-brane is transferred to the calibration chamber (H×D = 945 mm × 525 mm)where it is subjected to a second stage of consolidation at higher stresses.The calibration chamber has similar boundary conditions as Houlsby andTeh (1988): flexible walls and rigid top and bottom ends. The drainage con-dition of the calibration chamber is the same as the slurry consolidometer:drainage through the top and bottom plates, no drainage through the sidewall. The chamber consolidation is performed with the initial back pressure(uo) of 138 kPa while the effective stress is kept intact to ensure saturation.After checking the B parameter for saturation, the confining pressure is ad-justed for the desired pressure condition. In this work, the vertical pressure isincreased to σv

′ = 273 kPa and adjusted to σv′ = 182 kPa (vertical) to make

a lightly overconsolidated specimen. The horizontal pressure is increased toσh

′ = 112 kPa and reduced to σh′ = 75 kPa (horizontal). In this way a lightly


overconsolidated (OCR = 1.5) clayey specimen is obtained that is very sim-ilar to specimens obtained in typical field conditions. The vertical stress isapplied through a vertical loading jack. Horizontal stress is applied throughthe cell pressure.

Test Procedure Using Three-Piezo-ElementMiniature Penetrometer

Three penetration tests are carried out in the soil specimens. Dissipationtests are performed at the end of the penetration tests. The hydraulic systemused for the penetration consists of dual piston, double-acting hydraulic jackson a collapsible frame. The frame is mounted on the top lid of the chamberand allows for penetration of the sample up to 640 mm or less in a singlestroke. Such a single stroke continuous penetration is desirable, especiallyin saturated cohesive specimens where stress relaxation and pore pressuredissipation can occur during a pause between strokes. Data acquisition iscarried out both manually and automatically. The pore pressure response isrecorded with digital voltmeters, which are hooked up to the computer’s dataacquisition system. The readings of the digital voltmeters are also recorded bya video camera in case of data acquisition system failure. Data is taken everyone second. During the dissipation test, the sampling frequency is lowered toreduce the data file size. The test equipment set up is shown in Fig. 7.16.

Validity of the penetration rate is carefully investigated because of thescale difference of the three-piezo-element miniature penetrometer and otherpenetrating objects. This penetrometer has cross sectional area 2 cm2, andthis area is much smaller than that of the typical international referencecone penetrometer or piles. Regarding this scale difference, an adjustment ofpenetration speed for the miniature penetrometer was considered.

Comprehensive comparison of penetration speed effects for the cone pen-etrometer by Lunne et al. (1997) showed an inconsistent trend. Also, theresults of Roy et al. (1982) showed a non-significant difference in excess porepressure change for the penetration speed range 3 cm/sec – 240 cm/sec. Thetest results by Lim (1999) also showed a non-significant difference in ex-cess pore pressure response between a reference penetrometer and a minia-ture penetrometer with 1 cm2 cross sectional area as shown in Fig. 7.17.Voyiadjis and Song (2003) investigated the scale effect of the miniature conepenetrometer on the pore pressure response extensively, and they showedthat there is no significant scale effects for a penetration speed of around2 cm/sec. Consequently, no predominant effect of the strain rate is consid-ered, and 2 cm/sec penetration speed is adopted in this test. Therefore, theeffects of penetration speed are not considered in this study.


3 ChannelSignal Amplifier

3 Digital Voltmeters

Data Logger (PC)

Pen

etro

met

er

Fig. 7.16. Test set up

Results

The test results are shown in Fig. 7.18. The combined penetration results forthe u1, u2, u3, and u4 pore pressure measuring locations are indicated in thefigure. The pore pressure response for the u1 location shows a similar trendto a typical pore pressure response obtained from the penetration test. Thesteady state pore pressure for u1 is obtained for approximately two secondsafter the initiation of penetration. Also, u2 shows a similar overall responseto u1. The magnitude of the pore pressure for u2 is a little smaller than thatof u1 and is similar to the results determined by Robertson et al. (1992) andPowell and Quartermann (1988).

The results of the pore pressure for u3 and u4 in Fig. 7.18 show a sub-stantial initial fluctuation and an increase up to the steady state condition.This is an unexpected behavior. However, one should note that the distancesfrom the penetrometer tip to the u3 and u4 locations are substantially large.They are 7 cm and 22 cm, respectively. These results imply that the initialequilibrium condition and “on-the-penetration” equilibrium condition for u3

and u4 may be very different from those for u1 and u2. Therefore the dis-cussion of pore pressure response is provided with respect to the equilibriumcondition as follows:

Figure 7.18 shows that the steady state condition for u1 is obtained ina relatively short time (about 2 seconds). This response is similar to the


Fig. 7.17. Comparison of penetration results for the reference cone (10 cm2) andminiature cone (1 cm2) penetrometer (Lim, 1999)


0

10

20

30

40

50

60

-100 0 100 200 300 400

Excess Pore Pressure (kPa)

Pen

etra

tio

n D

epth

(cm

)

u1

u2

u3

u4

u1

u2

u3

u4

Fig. 7.18. Combined penetration test results for u1, u2, u3, and u4 locations

u1 response of other researchers (Kurup et al. 1994; Lunne et al. 1997). Therapid achievement of the steady state represents the rapid achievement of thenew (on-the-penetration) equilibrium condition. The steady state includesconstant equilibrium conditions such as stabilized pore water flow and stress-strain conditions. u1 is initially under the Ko condition without any flow.After initiation of penetration, the Ko condition and the zero-flow conditionare no longer valid. Once “on-the-penetration” new equilibrium conditionsare developed and stabilized for u1, they reach the steady state pore pressurecondition.

The pore pressure response for u2 is slightly different from that for u1. Theu2 response shows a slight initial hesitation and that is followed by a trendsimilar to that of u1. Regarding the initial hesitation, it is important to notethe pore pressure response for u3 because the initial hesitation is more clearlyobserved in the response for u3. In u3, this initial hesitation period is muchlonger than that for u2. The enlarged version of the pore pressure responsefor u3 is shown in Fig. 7.19. Note that u3 shows a two-stage, steady-statecondition.


0

5

10

15

20

25

30

35

40

45

50

-50 0 50 100 150 200

Excess Pore Pressure (kPa)P

enet

ratio

n D

epth

(cm

)

Entering into the passage of u1

location

Entering into old u2 location

Reduced Shear-Induced Porepressure (note: OCR=1.5)

Maximum Shear-Induced PorePressure

Ko Condition

Fig. 7.19. Details of pore pressure at u3

Initially u3 is subjected to the Ko stress condition, and with no flow con-dition. With the initiation of penetration, u3 undergoes shear motion. There-fore u3 shows the shear induced pore pressure, which is maximum at a 2 cmpenetration. As penetration exceeds 2 cm, the shear induced pore pressuredecreases. The reason for this decrease should be attributed to the overcon-solidation of the test specimen. The overconsolidation ratio (OCR) of the testspecimen is 1.5, which implies a lightly overconsolidated condition. However,it seems that this much OCR is enough to cause the dilative behavior of thespecimen at the cone shaft. During this dilative behavior, the pore pressurefor u3 achieves a “stage one” steady condition even though that is only for avery short time period. (Note that u4 shows a longer and clearer “stage one”steady state condition). The reduction of pore pressure by dilative behaviorstops when the u3 element enters into the old cone tip location (initial loca-tion of u2 and u1). The initial location of u2 and u1 has some residual pore


pressure that was generated by the cone tip penetration. Therefore, when u3

enters into the old cone tip location, the coupling of shear-generated nega-tive pore pressure and the residual from the compression-generated positivepore pressure is triggered. After some time, the coupling process achieves theequilibrium that is the “stage two” steady state condition.

Note that u2 is located at the penetrometer shaft; however, it is veryclose to u1. The authors could not capture the initial fluctuation of the porepressure for u2. However, the essentials of initial fluctuation for u2 should besame as that of u3.

Independently from this study, Elsworth (1993) predicted the time toreach the steady state penetration pore pressure based on the dislocationscheme. For the case of a penetration speed of 2 cm/sec and a coefficient ofconsolidation 28.3 × 10−3 m2/sec, the time required to reach 95% of steadystate penetration pore pressure is about 0.5 seconds for the penetrometertip. That number is 10 seconds for the penetrometer shaft at a location tentimes the radius of the penetrometer behind the face. One notes Elsworth(1993) that the minimum penetration depth for achieving the steady statecondition is quite short for the penetrometer tip. However, that distance isincreased substantially for the penetrometer shaft. Elsworth’s 1993 findingsdo not exactly agree with the experimental results in this study, but theyshow a conceptual agreement with our results.

From this discussion, one can see that for a better analysis of the pen-etration test, a substantial amount of penetration is required, especially forthe analysis of the friction on the penetrometer shaft. Figure 7.19 also showsthe steady state pore pressure distribution which is high at the penetrome-ter tip and low (gradual decrease) along the penetrometer shaft. This trendagrees well with Levadoux and Baligh’s experimental results (1986), and withWhittle and Aubeny’s (1991) and Elsworth’s (1993) theoretical results.

Figure 7.20 shows the dissipation test results carried out at the end of thepenetration for all four piezometer locations. Note that u1 shows a typicaldissipation curve. The computed permeability from the u1 dissipation curveis about 2 × 10−8 m/sec (t50 method of Robertson et al., 1992). Consideringthe fact that the measured permeability value in the laboratory (constanthead permeability tests for the triaxial specimen with a confining pressurehaving the same magnitude as the effective vertical overburden pressure of thespecimen) is 2.1×10−8 m/sec, one concludes that the computed permeabilityis in good agreement with the laboratory test results. However, u2 shows somedeviation from the standard back-bone shape of the dissipation curve. Notealso that u3 and u4 show even a larger deviation from the standard dissipationcurve.

The following observations are made with reference to Fig. 7.19 withrespect to the mechanism of the unstable pore pressure dissipation curve.For the u1 location, a faster pore pressure dissipation is observed, which mayindicate a spherical dissipation. The spherical dissipation is proportional to


Fig. 7.20. Dissipation curves of u1, u2, u3, and u4 locations

r3, while the cylindrical dissipation is proportional to r2 (where r is the radiusof the penetrometer).

For the u2, u3, and u4 locations, the dissipation condition may be as-sumed to be of a cylindrical drainage. However, the initial pore pressure isvery unstable and is far from the standard back-bone shape. Through the pre-viously discussed two stage equilibrium, one observes that small (negative)pore pressure is generated at the penetrometer shaft, and it approaches the“stage one” steady state condition until it enters a new equilibrium phase.After the penetration of the penetrometer stops, an interaction starts be-tween the penetrometer tip generated pore pressure and the penetrometershaft generated pore pressure (possibly negative pressure) in order to attainpore pressure equalization. The high pore pressure from the far field flowsinto the near field of the penetrometer shaft. Therefore, it shows an initialincrease in the pore pressure response for the u2, u3 and u4 as indicated inFig. 7.20. The above discussion explains the non-standard back bone shapeof the pore pressure dissipation curves for u1, u2, u3 and u4.


Therefore one can say that pore pressure dissipation curves for u2, u3, andu4 are adequate to use for the evaluation of the coefficient of consolidationor the permeability.

Comparison with Numerical Results

The calibration chamber test results are compared with numerical simula-tion results. Numerical simulation is carried out based on the Prevost (1980)coupled theory of mixtures and an anisotropic modified Cam Clay model(Song and Voyiadjis 2000). An updated Lagrangian reference frame is usedfor the large strain elasto-plasticity model. Numerical simulation results areshown in Fig. 7.21, which shows a high excess pore pressure at the face (u1

position) and smaller excess pore pressure at the penetrometer shoulder (u2

position). Figure 7.21 also shows that the magnitude of u1 is very close to theexperimental result. In addition, u2 shows smaller pore pressure than u1, aresult that also agrees fairly well with the experimental result. Therefore, onecan say that Fig. 7.21 and Fig. 7.18 agree fairly well for the pore pressuresof u1 and u2. The background theory for Fig. 7.21 is the coupled theory ofmixtures. The coupled theory of mixtures takes into account the interaction

Fig. 7.21. Analytical prediction of the excess pore pressure distribution around apenetrating object


of the cone shaft generated pore pressure and the cone face (tip) generatedpore pressure. The micro-level rotation of the soil grains is incorporated inthe numerical simulation through the plastic spin, which essentially producesmore accurate pore pressures. (Song and Voyiadjis 2000) showed that theeffect of the plastic spin is especially large at the u1 location.

The Possible interaction of the near field pore pressure and the far fieldpore pressure can be also predicted from Fig. 7.21. In Fig. 7.21, one canexpect the dissipation of the pore water pressure from the face (shown byarrows) to the cone shoulder direction.

As the penetration proceeds, u2 will go into the high excess pore pressureregion, an effect that accounts for the observed gradual increase. Fig. 7.21does not show the pore pressure response for large distances from the tipsuch as those at u3 and u4 because the numerical simulation was performedfor a penetration distance of 2.5 cm, and the required penetration length forsteady state is more than 10 cm and 35 cm for u3 and u4, respectively (seeFig. 7.18). Numerical simulation for this long penetration is not possible atthis time due to the accumulated numerical errors. Considering that the testcone used in this study is a miniature cone that has 2 cm2 cross sectionalarea compared to the reference cone that has 10 cm2 cross sectional area,one can see that the numerical simulation of the cone shaft behavior forthe reference cone requires even a larger penetration length. The numericalsimulation of several centimeters of penetration for the shaft friction will notpresent reliable results.

Evaluation

This study evaluates a new concept for the experimental investigation ofthe penetration induced excess pore pressure. Our analysis is based on thelarge strain coupled theory of mixtures formulation using the modified CamClay model and incorporating micro-mechanics through the plastic spin inan updated Lagrangian reference frame. To carry out our experiment, weused a three-piezo-element penetrometer that was designed and tested in theLouisiana State University Calibration Chamber.

We offer the following conclusions:

• The generated pore pressure at the tip of the penetrating object is mainlydue to the compressive stress, which is usually positive.

• The generated pore pressure at the shaft of the penetrating object is mainlydue to the shear stress.

• The measured pore pressure at the shaft of the penetrating object is a com-bination of shear induced pore pressure, compression induced pore pres-sure, and dissipation of early generated pore pressure.

• Sometimes, the shear induced pore pressure is negative, and the combinedpore pressure can be negative or very small.


• Due to this interaction of shear induced pore pressure and compressioninduced pore pressure at the shoulder of the penetrating object, the dissi-pation of pore pressure does not show the standard back-bone shape whenthe pore pressure measurement position is not at the tip.

• The effect of this interaction is ever more severe as the distance from thecone tip increases, and the steady state pore pressure magnitude is verydifferent from that of the tip or face of the penetrating object.

• The steady state for the shaft of the penetrating object is obtained in twostages because of the change in equilibrium conditions. In the case of thepenetrating object tip, the steady state is obtained in one stage because ofthe constant equilibrium conditions. For the pore pressure at the cone shaft,the pore water dissipation curve (especially u3, and u4) deviates largelyfrom the standard back-bone shape curve due to the different mechanismin achieving the steady state.

7.2.9 Back Stress

Multiple back stresses are discussed in Sect. 4.5.1, which investigates theeffects of multiple back stresses. Figure 7.22 shows the comparison of excesspore pressure around the cone tip for the case of single back-stress and forthe case of dual back-stress. As shown in Fig. 7.22, the difference between thetwo back-stress models is not substantial in close proximity to the penetratingcone tip. However, the difference is observed at a relatively far distance fromthe cone tip (see the area encircled by the dotted line.). This behavior iscontrary to what was expected. It is typically known that the activity ofthe back-stress is pronounced in the high strain region (close vicinity of thecone tip). However, recalling the basic concept of the short range back-stress,one can explain the above behavior. The material at close proximity to thecone tip undergoes very large strains, and effect of short range back-stressis smudged. However, the behavior of the material at a far distance fromthe cone tip usually undergoes small strains; the behavior will be affectedprimarily by the short range back-stress. This behavior is predicted by thecareful observation of Fig. 7.22. In Fig. 7.22, the short range back-stressis mobilized rapidly and its amount is larger than that of the long rangeback-stress at relatively low strains. The long range back-stress, however, ismobilized relatively slowly and the initial magnitude is smaller than that ofthe short range back-stress. Therefore its effect is not substantial at low strainrange. Figure 7.22 shows the excess pore pressure response of the penetratingcone tip predicted by the isotropic modified Cam Clay model (without anyback-stress). Comparing Figs. 7.22(b) and 7.16, one can see that the effect ofthe long range back-stress is mostly pronounced at close vicinity of the conetip.

It may be assumed that the effect of the short range back-stress for steelis substantial, especially for repeated cyclic loading (Montheillet et al.. 1984).


Con

e S

haft

(a) Short Range + Long RangeBack-Stress Condi-tion

(b) Long Range OnlyBack-Stress Condi-tion

____ Umax=357.3 kPa ____ Umax=357.1 kPa

Con

e S

haft

Fig. 7.22. Comparison of excess pore pressure contours for dual back-stress andsingle back-stress condition

One of the most widely known behaviors for steel which is related to the shortrange back-stress is “ratcheting behavior”.

Ratcheting behavior in metals is easily observed by the length change ofthe test specimen subjected to the cyclic torsional loading. Similar behaviorfor soil (sand) is reported by Song (1986). Measurement of the height changeof resonant column test specimens (ASTM C-109 Ottawa Sand) shows a


consistent change of the height of the test specimen (Song, 1986). This changeis distinctly different from the void ratio reduction which is typically causedby dynamic loading. This behavior is similar to the ratcheting behavior formetals, and it is believed to be caused by the short range back-stress. Theresearch for the ratcheting behavior of clay is not reported yet. However,the authors believe that the ratcheting behavior of the clay may not be assignificant as ratcheting behavior in sand because the magnitude of the shortrange back-stress relative to the long range back-stress is believed to dependon the stiffness of the grain relative to the stiffness of the skeleton of thestructure. Because the stiffness of the grain is larger than that of the materialskeleton, the internal stress will cause the relative movement of the grainsrather than cause the deformation of the grains. This mechanism addressesthat the soil grains dos not store much energy. In metals, the stiffness of themacroscopic skeleton structure is much greater that that of the sand or clay;however, the macroscopic stiffness is similar or lower. Therefore, relativelylarger short range back-stresses can be stored in the metal microstructure. Insands, the stiffness of the sand grains is larger than that of the soil skeleton.It follows that the probability for short range back-stress storage is relativelylower for sand than for steel. In clays, the stiffness of the clay particles ismuch larger than that of the clay skeleton, and the short range back-stressis hardly stored in the clay particles.

This result shows the relatively insignificant effect of the short range back-stress on the very large strain region. However, it does not mean that the shortrange back-stress in not important for geo-materials. As discussed previously,the short range back-stress is more important for materials that have higherstiffness, such as sands or over-consolidated clays.

It is noted that the effects of α3 are not taken into account here. It isbeyond the scope of this study, since the material parameters for α3 are notavailable at this moment.

Evaluation

In this section the effect of the micro-structural change for soils is investigatedthrough the dual back-stress approach – the so called short range back-stressand long range back-stress approach. The anisotropic modified Cam Claymodel is used, which incorporates the plastic spin as an internal state variable.This soil model is used to solve the problem of the cone penetrometer test thatis a typical example for large strain problems in soils. This study shows thatthe effect of the short range back-stress is not significant in the large strainregion. However, its effect is shown to be significant in the low strain region.This is primarily because the ultimate magnitude of the short range back-stress is much smaller than that of the long range back-stress. The magnitudeof the short range back-stress is greater than that of the long range back-stress at the earlier stage (low strain stage). This relationship holds for the


soil condition used in this study. However, the effect of the short range back-stress may be larger for some other soils such as sands or stiffer skeletonmaterials because the stiffness of the clay particles is much greater than thatof the clay skeleton, and it is much easier for the internal energy to causethe relative movement of the clay particles than to store the back-stress inthe clay particles. The authors also expect that this short range back-stressmay play a substantial role in soils that have higher skeleton stiffness, suchas sands.

7.2.10 Evaluation

In our discussion of the individual micro-mechanical mechanisms, we havenoted that each individual micro-mechanical mechanism has its own effect.The penetration rate effect, however, is not very strong for the typical pen-etration velocity range. The mechanism is such that when the penetrationis initiated, the response of the soil is governed by two different rate depen-dency mechanisms: the viscosity related rate dependency and the pore fluidflow dependent rate dependency. When the effect of the viscosity relatedrate dependency is dominant (as is the case for the relatively high penetra-tion speed), the penetration resistance is increased as the penetration speedis increased. When the effective stress is dominant (as is the case for therelatively low penetration speed), the penetration resistance is increased asthe penetration speed is decreased. In the typical penetration velocity range,the viscosity related rate dependency and the porous fluid flow related ratedependency are balanced, and no pronounced rate effects are observed.

The effects of gradients are expected to cause the reduction of the peakeffective stress and the peak excess pore pressure. In this section, the reduc-tion of the excess pore pressure due to the gradient appears to be biggerthan that of the effective stress. This behavior causes the overall reductionof the total stress at the cone tip. Negligible changes in the effective stressare believed to be caused by the combination of the reduced effective stressby gradient mechanism and the increased effective stress by reduction of thepore pressure. The damage effect causes a substantial reduction in the ex-cess pore pressure. It is reasonable to anticipate that the effective stress atthe cone tip will be reduced. However, the effective stress at the cone tipis actually increased because of the substantial decrease of the excess porepressure.

The authors conclude that the effects of the individual micro-mechanicalmechanisms are inter-linked with each other. The saturated clayey soil is es-sentially a two-phase material composed of the pore fluid and soil grains.For the multi-phase materials, the coupled effects of these micro-mechanicalmechanisms are quite different from what is to be expected from the uncou-pled soil models. The primary reason is the interaction of the pore fluid andthe soil grains, which is essentially the interaction of the pore pressure and the


effective stress. Pore pressure and effective stress nullify each other; therefore,the effects of the micro-mechanical mechanism are not always consistent.

The summary of the coupled micro-mechanisms may be outlined as fol-lows:

• Homogenization mechanisms such as gradient, damage, rate dependencyimpact both the effective stress and pore pressure.

• Viscosity governs the rate dependency of the Piezocone tip resistance onlyat higher penetration speeds such as higher than 10 cm/sec.

• Pore fluid flow governs the rate dependency of the Piezocone tip resistanceonly at lower penetration speeds (i.e., lower than 0.2 cm/sec).

• Viscosity related rate dependency and pore pressure related rate depen-dency are balanced at the intermediate penetration speed range, and thecorresponding Piezocone tip resistance is not sensitive to the penetrationspeed.

• Gradient theory reduces the peak pore pressure. However, the effectivestress is not affected substantially.

• This behavior is believed to be due to the increase of effective stress bythe reduced pore pressure and the decrease of the effective stress by thegradient.

• Damage reduced the pore pressure around the Piezocone tip. The reducedpore pressure increases the effective stress even with damage of grains.

Appendix: Fortran Codes of CS-S

c PROGRAM CS-S.f

C----------------------------------------------------------------------

C THIS PROGRAM ANALYSE GENERAL FINITE DEFORMATION, FINITE STRAIN

C INELASTIC TIME INDEPENDENT PROBLEMS USING UPDATED LAGRANGIAN

C REFERENCE FRAME.

C----------------------------------------------------------------------

C Back up is cone8.f

C ORIGINALLY CODED BY S. M. Sivakumar

C Modifid by MURAD ABU-FARSAKH for CPT with modified Cam Clay model

C MODIFIED BY CHUNG R. SONG (1/26/98) for anisotrophy & plastic spin

C Modified by Chung R. Song (4/27/00) for multiple back stress

C Last modified by Chung R. Song (8/20/03)

C Last accessed by Chung R. Song (8/12/03)

C NOTE BY CHUNG R. SONG (4/27/00)

C FINITE STRAIN : consider PLASTIC SPIN.

C COUPLED THEORY OF MIXTURE : PREVOST (1980, 1981)

C TIME INDEPENDENT : ACTUALLY SEMI TIME DEPENDENT BY

CONSIDERING

C THE DISSIPATION OF PORE WATER DURING

C EACH PENETRATION STEP. -- NOT VISCO ELASTO-PLASTIC

C UPDATED LAGRANGIAN : INCREMENTAL APPROACH FOR LARGE STRAIN

PROBLEM

C SOIL MODEL : Anisotropic - MODIFIED CAM CLAY MODEL WITH GENERAL

HARDENING AND YIELD

C SURFACE CORRECTION

C This program also considers the micro-mechanical behavior of

soil by

c plastic spin.

C Rate Dependency 08/07/03

c Damage 08/08/03

c Gradient 08/12/03

C ----- INITIAL SETUP

IMPLICIT REAL*8(A-H,O-Z)

INCLUDE ’PARM.FOR’

CHARACTER*60

208 Appendix: Fortran Codes of CS-S

INFILE,OUTFILE1,OUTFILE2,OUTFILE3,OUTFILE4,OUTFILE5

COMMON/DEVICE/LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV

COMMON/PARS/PYI,ASMVL,ZERO

COMMON/NSIZE/NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL,

1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW

DIMENSION XYZ(3,MNODES),NCONN(NTPE,MEL),LTYP(MEL),MAT(MEL)

DIMENSION NQ(MNODES),NW(MNODES+1),NLST(NTPE)

C ----- OPEN INPUT & OUTPUT FILE

C WRITE(*,*)’ENTER INPUT FILE NAME>’

C READ(*,’(A)’)INFILE

INFILE=’Chung21.TXT’

C WRITE(*,*)’ENTER MAIN OUTPUT FILE NAME>’

C READ(*,’(A)’)OUTFILE1

OUTFILE1=’SONG2’

C WRITE(*,*)’ENTER EQUIL OUTPUT FILE NAME>’



C WRITE(*,*)’ENTER STRESS OUTPUT FILE NAME>’



C WRITE(*,*)’ENTER CAM-PAR OUTPUT FILE NAME>’

c READ(*,’(A)’)OUTFILE4



OPEN(1,FILE=INFILE,FORM=’FORMATTED’,STATUS=’UNKNOWN’)

OPEN(2,FILE=OUTFILE1,FORM=’FORMATTED’,STATUS=’UNKNOWN’)




OPEN(10,FILE=’CONSOLV’,FORM=’UNFORMATTED’,STATUS=’UNKNOWN’)

c OPEN(8,FILE=OUTFILE5,FORM=’FORMATTED’,STATUS=’UNKNOWN’)

LINP=1

LOUT1=2

LOUT2=3

LOUT3=4

LOUT4=5

c LOUT5=8

LSOLV=10

C ----- CALCULATION SEQUENCY

C

C INPUT NODE COORDINATE & ELEMENT DATA

CALL INPUT(XYZ,NCONN,MAT,LTYP,NLST)

Appendix: Fortran Codes of CS-S 209

C CALCULATE NO OF D.O.F FOR EACH NODE

CALL MAKENZ(NEL,NNODES,NCONN,LTYP,NQ,INXL)

C GENERATE GLOPAL NUMBERS FOR ALL D.O.F

CALL CALDOF(NNODES,NNOD1,NDF,NW,NQ)

C PRINT OUT ARRAYS

CALL GPOUT(LOUT1,NEL,NNODES,NDF,NCONN,MAT,LTYP,NLST)

C CALL THE MAIN SUMROUTINE

CALL MINT(XYZ,NCONN,MAT,LTYP,NW)

C

C --- MAJOR SUBROUTINES ARE ATTATCHED BY ITS ORDER RIGHT AFTER

THE CONTROL PROGRAM.

C --- MINOR SUBROUTINES ARE ATTATCHED BY ITS ORDER AFTER THE

MAJOR SUBROUTINES

STOP

END

C ----- END OF CONTROL PROGRAM

C ----- MAJOR SUBROUTINE -----

C**********************************************************************

SUBROUTINE INPUT(XYZ,NCONN,MAT,LTYP,NLST)

C**********************************************************************

C THIS SUBPROGRAM GETS ALL THE VALUES NEEDED AS INPUT FOR THE

PROGRAM



CHARACTER*80 TITLE

CHARACTER*80 HEADER1




COMMON/DEVICE/LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV

COMMON/PARS/PYI,ASMVL,ZERO

COMMON/NSIZE/NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL,


COMMON/ELINF/LINFO(50,15)

COMMON/SKBC/ISPB(20),DIRCOS(20,3)

DIMENSION DUMMY(6),NLST(NTPE),M(20)

DIMENSION XYZ(3,MNODES),NCONN(NTPE,MEL),MAT(MEL),

1 LTYP(MEL),TEMP(3)


C ----- INITIALIZE ARRAYS

CALL ZEROR2(XYZ,3,MNODES)

CALL ZEROR2(DIRCOS,20,3)

CALL ZEROI2(NCONN,NTPE,MEL)

CALL ZEROI1(LTYP,MEL)

CALL ZEROI1(MAT,MEL)

CALL ZEROI1(NLST,NTPE)

CALL ZEROI1(ISPB,20)

C ----- SET SOME CONSTANTS

PYI=4.0D0*ATAN(1.0D0)

ASMVL=1.0D-20

ZERO=0.0D0

C ASMVL=ASSIGNED MINIMUM VALUE

C ----- READ THE TITLE

READ(LINP,101)TITLE

WRITE(LOUT1,101)TITLE

101 FORMAT(A80)

C ----- READ PL. STR-AXI. SYM INDEX NPLAX & THE LINEARITY

INDIX LINR

READ(LINP,101)HEADER1

READ(LINP,*)NPLAX,LINR,NDIM

WRITE(LOUT1,*)’LINEARITY=’,LINR

C ----- SET SOME VARIABLES

C INXL - INDEX TO NO. OF D.O.F OF FIRST NODE OF ELEMENT

C NSP - ONE DIMENSIONAL INTEGRATION NUMBER OF SAMPLING POINTS

C NS - SIZE OF D-MATRIX

C NPT - MAXM NUMBER OF DISPLACEMENT NODES ALONG ELEMENT EDGE

C MXEN,MXLD - SIZE OF ARRAYS IN COMMON BLOCKS PRSLD,PRLDI

C MXLD - MAXIMUM NUMBER OF ELEMENT EDGES WITH PRESSURE LOADING

C MXEN - MAXIMUM NUMBER OF DISPLACEMENT NODES ALONG AN EDGE X 2

C MXFXT - MAXIMUM NUMBER OF FIXITIES (SIZE OF ARRAYS MF,

NFIX,DXYT)

C -----

INXL=20

NSP=5

IF(NDIM.EQ.2)NS=4

IF(NDIM.EQ.3)NS=6

NPT=LV

MXEN=10

MXLD=100

MXFXT=200


C ----- READ AND GENERATE THE NODAL COORDINATES

I=0

READ(LINP,*)NNODES,NSKEW

WRITE(LOUT1,*)’NUMBER OF NODES =’,NNODES


NNOD1=NNODES+1

310 READ(LINP , *) K,(DUMMY( IDIR ) , IDIR = 1 ,NDIM),INCR

DO IDIR=1,NDIM

XYZ(IDIR,K)= DUMMY(IDIR)

ENDDO

I=I+1

C ----- Interpolation of the nodal coordinate

IF (INCR.NE.0)THEN

N=(K-K1)/INCR

DX=(XYZ(1,K)-XYZ(1,K1))/N

DY=(XYZ(2,K)-XYZ(2,K1))/N

DZ=(XYZ(3,K)-XYZ(3,K1))/N

K2=K-INCR

DO J=K1,K2,INCR

N1=(J-K1)/INCR

XYZ(1,J)=XYZ(1,K1)+N1*DX

XYZ(2,J)=XYZ(2,K1)+N1*DY

XYZ(3,J)=XYZ(3,K1)+N1*DZ

I=I+1

ENDDO

I=I-1

ENDIF

K1=K

IF(I.LT.NNODES) GO TO 310

C

WRITE(LOUT1, 20)

DO K1 = 1, NNODES

WRITE(LOUT1, 10)K1,(XYZ(IDIR,K1),IDIR=1,NDIM)

ENDDO

C ---- READ SKEW BOUNDARY NODES & ITS DIRECTION COSINES


IF(NSKEW.GT.0) THEN

DO K=1,NSKEW

READ(LINP,*)INODE

C ,(TEMP(IDIR),IDIR=1,2)

TEMP(1)=5.640D0

TEMP(2)=9.780D0

C --- Above values are for 60 degree cone tip.

ISPB(K)=INODE

TEMP3=SQRT(TEMP(1)**2+TEMP(2)**2)


DIRCOS(K,1)=TEMP(2)/TEMP3

DIRCOS(K,2)=TEMP(1)/TEMP3

IF(NDIM.EQ.3)DIRCOS(K,3)=TEMP(3)

ENDDO

END IF

C ----- READ, WRITE AND GENERATE THE ELEMENTS


I = 0

READ(LINP,*)NEL,NELNOD

WRITE(LOUT1,*)’NUMBER OF ELEMENTS =’,NEL

410 READ(LINP, *)K,ITYP,IMAT,INCR,(NLST(IK),IK=1,NELNOD)

C

NDN=LINFO(1,ITYP)

LTYP(K)=ITYP

MAT(K)=IMAT

C

DO IK=1,NDN

NUM=NLST(IK)

NCONN(IK,K)=NUM

ENDDO

C

I = I + 1

IF(INCR.EQ.0) THEN

K1 = K

C

ELSE

K2 = (K - K1)/INCR

DO NODE = 1, NDN

M( NODE ) = (NCONN(NODE,K)-NCONN(NODE,K1))/K2

ENDDO

C

DO IELEM = K1+INCR, K-INCR, INCR

LTYP(IELEM)=LTYP(K)

MAT(IELEM)=MAT(K)

I = I + 1

IELEM1 = IELEM - INCR

DO NODE = 1,NDN

NCONN(NODE,IELEM) = NCONN(NODE,IELEM1) + M( NODE )

ENDDO

ENDDO

END IF

IF(I.LT.NEL) GO TO 410

C

DO I=1,NEL


LT=LTYP(I)

NDN=LINFO(1,LT)

C WRITE(LOUT1,*)I,(NCONN(J,I),J=1,NDN)

ENDDO

RETURN

10 FORMAT(I5,4X,3F12.4)

20 FORMAT(/,12X,’COORDINATES OF THE NODES’/,45(1H-),/,3X,’NODE’,

1 11X,’X’,11X,’Y’,11X,’Z’,/,45(1H-))

END

C**********************************************************************

BLOCK DATA

C**********************************************************************


C----------------------------------------------------------------------

C DATA PRESENTED BY LIN (FIRST INDEX)

C 1 - TOTAL NUMBER OF NODES (DISPLACEMENT + POREPRESSURE)......NDPT

C 2 - TOTAL NUMBER OF VERTEX NODES..............................NVN

C 3 - TOTAL NUMBER OF ELEMENT EDGES............................NEDG

C 4 - TOTAL NUMER OF ELEMENT FACES (3D)........................NFAC

C 5 - TOTAL NUMBER OF DISPLACEMENT NODES........................NDN

C 6 - TOTAL NUMBER OF POREPRESSURE NODES........................NPN

C 7 - NO. OF DISPLACEMENT NODES PER EDGE (EXCLUDING END NODES).NDSD

C 8 - NO. OF POREPRESSURE NODES PER EDGE (EXCLUDING END NODES).NPSD

C 9 - NUMBER OF INNER DISPLACEMENT NODES.......................NIND

C 10 - NUMBER OF INNER POREPRESSURE NODES.......................NINP

C 11 - NUMBER OF INTEGRATION POINTS (=GAUSS POINT)...............NGP

C 12 - INDEX TO WEIGHTS AND INTEGRATION POINT COORDINATES.......INDX

C 13 - INDEX TO VERTEX NODES OF ELEMENTS (ARRAY NFC).............INX

C 14 - INDEX TO NODES ALONG EDGE (ARRAYS NP1, NP2).............INDED

C 15 - NUMBER OF LOCAL OR AREA COORDINATES........................NL

C 16 - TOTAL NUMBER OF DEGREES OF FREEDOM (D.O.F.) IN ELEMENT...MDFE

C 17 - CENTROID INTEGRATION POINT NUMBER........................NCGP

C 21 - ONWARDS THE NUMBER OF D.O.F. OF EACH NODE OF ELEMENT.....NDFN

C

C ELEMENT TYPES (SECOND INDEX)

C 1 - 3-NODED BAR ....................(2-D) **

C 2 - 6-NODED LST TRIANGLE............(2-D)

C 3 - 6-NODED LST TRIANGLE............(2-D CONSOLIDATION)

C 4 - 8-NODED QUADRILATERAL...........(2-D)

C 5 - 8-NODED QUADRILATERAL...........(2-D CONSOLIDATION)

C 6 - 15-NODED CUST TRIANGLE..........(2-D)

C 7 - 22-NODED CUST TRIANGLE..........(2-D CONSOLIDATION)


C 8 - 20-NODED BRICK..................(3-D)

C 9 - 20-NODED BRICK..................(3-D CONSOLIDATION)

C 10 - 10-NODED TETRA-HEDRA............(3-D) **

C 11 - 10-NODED TETRA-HEDRA............(3-D CONSOLIDATION) **

C

C ** ELEMENT TYPES NOT IMPLEMENTED IN THIS VERSION

C**********************************************************************

COMMON /ELINF/ LIN(50,15)

COMMON /DATL / SL(4,100)

COMMON /DATW / W(100)

COMMON /SAMP / POSSP(5),WEIGP(5)

DATA LIN(1,1),LIN(2,1),LIN(3,1),LIN(4,1),LIN(5,1),LIN(6,1),

1 LIN(7,1),LIN(8,1),LIN(9,1),LIN(10,1),LIN(11,1),LIN(12,1),

2 LIN(13,1),LIN(14,1),LIN(15,1),LIN(16,1),LIN(17,1),

3 LIN(21,1),LIN(22,1),LIN(23,1)/

3 3,2,1,1,3,0,1,0,0,0,5,0,0,0,1,6,3,2,2,2/




3 LIN(21,2),LIN(22,2),LIN(23,2),LIN(24,2),LIN(25,2),LIN(26,2)/

4 6,3,3,1,6,0,1,0,0,0,7,5,0,0,3,12,7,2,2,2,2,2,2/




3 LIN(22,3),LIN(23,3),LIN(24,3),LIN(25,3),LIN(26,3)/

4 6,3,3,1,6,3,1,0,0,0,7,5,0,0,3,15,7,3,3,3,2,2,2/





4 LIN(26,4),LIN(27,4),LIN(28,4)/

4 8,4,4,1,8,0,1,0,0,0,4,12,4,3,2,16,9,2,2,2,2,2,2,2,2/





4 LIN(26,5),LIN(27,5),LIN(28,5)/

4 8,4,4,1,8,4,1,0,0,0,4,12,4,3,2,20,9,3,3,3,3,2,2,2,2/




3 LIN(21,6),LIN(22,6),LIN(23,6),



5 15,3,3,1,15,0,3,0,3,0,16,21,0,0,3,30,16,

6 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2/


DATA LIN(1,7),LIN(2,7),LIN(3,7),LIN(4,7),LIN(5,7),







7 22,3,3,1,15,10,3,2,3,1,16,21,0,0,3,40,16,3,3,3,2,2,2,2,2,2,2,2,2,

8 2,2,2,1,1,1,1,1,1,1/





4 LIN(26,8),LIN(27,8),LIN(28,8),



6 20,8,12,6,20,0,1,0,0,0,8,37,4,3,3,60,27,3,3,3,3,3,3,3,3,3,3,3,

7 3,3,3,3,3,3,3,3,3/







6 LIN(38,9),LIN(39,9),LIN(40,9)/

7 20,8,12,6,20,8,1,0,0,0,8,37,4,3,3,68,27,4,4,4,4,4,4,4,4,

8 3,3,3,3,3,3,3,3,3,3,3,3/






5 LIN(29,10),LIN(30,10)/

5 10,4,6,4,10,0,1,0,0,0,4,64,28,15,4,30,0,3,3,3,3,3,3,3,3,3,3/






5 LIN(29,11),LIN(30,11)/

5 10,4,6,4,10,4,1,0,0,0,4,64,28,15,4,34,0,4,4,4,4,3,3,3,3,3,3/

C----------------------------------------------------------------------

C AREA COORDINATES - LINEAR STRAIN TRIANGLE - ELEMENT TYPE 2,3

C----------------------------------------------------------------------

DATA SL(1,6),SL(2,6),SL(3,6),SL(1,7),SL(2,7),SL(3,7),SL(1,8),

1 SL(2,8),SL(3,8),SL(1,9),SL(2,9),SL(3,9),SL(1,10),SL(2,10),

1 SL(3,10),SL(1,11),SL(2,11),SL(3,11),SL(1,12),SL(2,12),SL(3,12)/

1 .797426985353087245,.101286507323456343,.101286507323456343


1,.101286507323456343,.797426985353087245,.101286507323456343

1,.101286507323456343,.101286507323456343,.797426985353087245

1,.597158717897698279E-01,.470142064105115082,.470142064105115082

1,.470142064105115082,.597158717897698279E-01,.470142064105115082

1,.470142064105115082,.470142064105115082,.597158717897698279E-01

1,.333333333333333329,.333333333333333329,.333333333333333329/

C----------------------------------------------------------------------

C LOCAL COORDINATES - LINEAR STRAIN QUADRILATERAL - ELEM TYPE 4, 5

C----------------------------------------------------------------------

DATA SL(1,13),SL(2,13),SL(1,14),SL(2,14),SL(1,15),SL(2,15),

1 SL(1,16),SL(2,16),SL(1,17),SL(2,17),SL(1,18),SL(2,18),

1 SL(1,19),SL(2,19),SL(1,20),SL(2,20),SL(1,21),SL(2,21)/

1 -0.577350269189626,-0.577350269189626,

1 0.577350269189626,-0.577350269189626,

1 0.577350269189626, 0.577350269189626,

1 -0.577350269189626, 0.577350269189626,

1 0.0D0,0.0D0,

1 0.0D0,0.0D0,

1 0.0D0,0.0D0,

1 0.0D0,0.0D0,

1 0.0D0,0.0D0/

C----------------------------------------------------------------------

C AREA COORDINATES - CUBIC STRAIN TRIANGLE - ELEMENT TYPE 6,7

C----------------------------------------------------------------------


1 SL(1,24),SL(2,24),SL(3,24),SL(1,25),SL(2,25),SL(3,25),

1 SL(1,26),SL(2,26),SL(3,26),SL(1,27),SL(2,27),SL(3,27),

1 SL(1,28),SL(2,28),SL(3,28),SL(1,29),SL(2,29),SL(3,29)/

1 0.898905543365938,0.050547228317031,0.050547228317031,

1 0.050547228317031,0.898905543365938,0.050547228317031,

1 0.050547228317031,0.050547228317031,0.898905543365938,

1 0.658861384496478,0.170569307751761,0.170569307751761,

1 0.170569307751761,0.658861384496478,0.170569307751761,

1 0.170569307751761,0.170569307751761,0.658861384496478,

1 0.081414823414554,0.459292588292723,0.459292588292723,

1 0.459292588292723,0.081414823414554,0.459292588292723/


1 SL(1,32),SL(2,32),SL(3,32),SL(1,33),SL(2,33),SL(3,33),

1 SL(1,34),SL(2,34),SL(3,34),SL(1,35),SL(2,35),SL(3,35),

1 SL(1,36),SL(2,36),SL(3,36),SL(1,37),SL(2,37),SL(3,37)/

1 0.459292588292723,0.459292588292723,0.081414823414554,

1 0.008394777409958,0.728492392955404,0.263112829634638,

1 0.008394777409958,0.263112829634638,0.728492392955404,

1 0.263112829634638,0.008394777409958,0.728492392955404,

1 0.728492392955404,0.008394777409958,0.263112829634638,

1 0.728492392955404,0.263112829634638,0.008394777409958,

1 0.263112829634638,0.728492392955404,0.008394777409958,

1 0.333333333333333,0.333333333333333,0.333333333333333/


C----------------------------------------------------------------------

C LOCAL COORDINATES - 20-NODED BRICK - ELEM TYPE 8, 9

C----------------------------------------------------------------------


1 SL(1,40),SL(2,40),SL(3,40),SL(1,41),SL(2,41),SL(3,41),

1 SL(1,42),SL(2,42),SL(3,42),SL(1,43),SL(2,43),SL(3,43),

1 SL(1,44),SL(2,44),SL(3,44),SL(1,45),SL(2,45),SL(3,45)/

1 -0.577350269189626,-0.577350269189626, 0.577350269189626,

1 0.577350269189626,-0.577350269189626, 0.577350269189626,

1 0.577350269189626, 0.577350269189626, 0.577350269189626,

1 -0.577350269189626, 0.577350269189626, 0.577350269189626,

1 -0.577350269189626,-0.577350269189626,-0.577350269189626,

1 0.577350269189626,-0.577350269189626,-0.577350269189626,

1 0.577350269189626, 0.577350269189626,-0.577350269189626,

1 -0.577350269189626, 0.577350269189626,-0.577350269189626/

C 1 SL(1,46),SL(2,46),SL(3,46),SL(1,47),SL(2,47),SL(3,47),

C 1 SL(1,48),SL(2,48),SL(3,48),SL(1,49),SL(2,49),SL(3,49),

C 1 SL(1,50),SL(2,50),SL(3,50),SL(1,51),SL(2,51),SL(3,51),

C 1 SL(1,52),SL(2,52),SL(3,52),SL(1,53),SL(2,53),SL(3,53),

C 1 SL(1,54),SL(2,54),SL(3,54),SL(1,55),SL(2,55),SL(3,55),

C 1 SL(1,56),SL(2,56),SL(3,56),SL(1,57),SL(2,57),SL(3,57),

C 1 SL(1,58),SL(2,58),SL(3,58),SL(1,59),SL(2,59),SL(3,59),

C 1 SL(1,60),SL(2,60),SL(3,60),SL(1,61),SL(2,61),SL(3,61),

C 1 SL(1,62),SL(2,62),SL(3,62),SL(1,63),SL(2,63),SL(3,63),

C 1 SL(1,64),SL(2,64),SL(3,64)/

C 1 0.0D0,0.0D0/

C----------------------------------------------------------------------

C WEIGHTS - LINEAR STRAIN TRIANGLE - ELEMENT TYPE 2,3

C----------------------------------------------------------------------

DATA W(6),W(7),W(8),W(9),W(10),W(11),W(12)/

1 .062969590272413570,.062969590272413570,.062969590272413570,

1 .066197076394253089,.066197076394253089,.066197076394253089,

1 .112499999999999996/

C----------------------------------------------------------------------

C WEIGHTS - LINEAR STRAIN QUADRILATERAL - ELEMENT TYPE 4,5

C----------------------------------------------------------------------

DATA W(13),W(14),W(15),W(16),W(17),W(18),W(19),W(20),W(21)/

1 1.00000000000000,1.00000000000000,

1 1.00000000000000,1.00000000000000,

1 0.0D0,0.0D0,

1 0.0D0,0.0D0,

1 0.0D0/

C----------------------------------------------------------------------

C WEIGHTS - CUBIC STRAIN TRIANGLE - ELEMENT TYPE 6,7

C----------------------------------------------------------------------

DATA W(22),W(23),W(24),W(25),W(26),W(27),W(28),W(29),

1 W(30),W(31),W(32),W(33),W(34),W(35),W(36),W(37)/


1 .016229248811599,.016229248811599,.016229248811599,

1 .051608685267359,.051608685267359,.051608685267359,

1 .047545817133642,.047545817133642,.047545817133642,

1 .013615157087217,.013615157087217,.013615157087217,

1 .013615157087217,.013615157087217,.013615157087217,

1 .072157803838893/

C----------------------------------------------------------------------

C WEIGHTS - 20-NODED BRICK - ELEMENT TYPE 8,9

C----------------------------------------------------------------------

DATA W(38),W(39),W(40),W(41),W(42),W(43),W(44),W(45)/

1 1.000000000000000,1.000000000000000,

1 1.000000000000000,1.000000000000000,

1 1.000000000000000,1.000000000000000,

1 1.000000000000000,1.000000000000000/

C----------------------------------------------------------------------

C ONE-DIMENSIONAL INTEGRATION

C----------------------------------------------------------------------

DATA POSSP(1),POSSP(2),POSSP(3),POSSP(4),POSSP(5)/

1 -0.906179845938664,-0.538469310105683,0.0D0,

1 0.538469310105683,0.906179845938664/

DATA WEIGP(1),WEIGP(2),WEIGP(3),WEIGP(4),WEIGP(5)/

1 0.236926885056189,0.478628670499366,0.568888888888889,

1 0.478628670499366,0.236926885056189/

END

C**********************************************************************

SUBROUTINE MAKENZ(NEL,NN,NCONN,LTYP,NQ,INXL)

C**********************************************************************

C SETS UP THE NQ ARRAY WHICH CONTAINS THE NUMBER

C OF DEGREES OF FREEDOM ASSOCIATED WITH EACH NODE

C FOR ELEMENTS IN THIS ASSEMBLY.

C 1 CALLED BY INSITU.



DIMENSION NCONN(NTPE,MEL),LTYP(MEL),NQ(MNODES)

COMMON /ELINF/ LINFO(50,15)

C----------------------------------------------------------------------

C INXL - INDEX TO NO. OF DEGREES OF FREEDOM OF FIRST NODE OF

ELEMENT

C (SEE BLOCK DATA ROUTINES BDATA1, MAIN2)

C----------------------------------------------------------------------

DO 8 J=1,NN

8 NQ(J)=0

C

DO 20 J=1,NEL

IF(LTYP(J).LT.0) GOTO 20

LT=LTYP(J)


NDPT=LINFO(1,LT)

C

DO 10 I=1,NDPT

NDFN=LINFO(I+INXL,LT)

NOD=NCONN(I,J)

IF(NDFN.GT.NQ(NOD)) NQ(NOD)=NDFN

10 CONTINUE

20 CONTINUE

C

RETURN

END

C**********************************************************************

SUBROUTINE CALDOF(NN,NN1,NDF,NW,NQ)

C**********************************************************************

C ROUTINE TO CALCULATE GLOBAL NUMBER FOR D.O.F.


DIMENSION NW(MNODES+1),NQ(MNODES)

C

NC=1

NW(1)=1

C

DO 10 I=1,NN

NC=NC+NQ(I)

10 NW(I+1)=NC

C

NDF=NW(NN1)-1

C

RETURN

END

C

C**********************************************************************

SUBROUTINE GPOUT(LOUT1,NEL,NN,NDF,NCONN,

1 MAT,LTYP,NLST)

C**********************************************************************

C ROUTINE TO PRINTOUT ARRAYS SET-UP IN GEOMETRY PART OF PROGRAM



DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL),NLST(NTPE)

COMMON /ELINF / LINFO(50,15)

C

WRITE(LOUT1,902)


C

DO 20 JU=1,NEL

IF(NEL.EQ.0)GOTO 20

MPR=JU

LT=LTYP(MPR)

NDPT=LINFO(1,LT)

C

DO 10 IN=1,NDPT

NP=NCONN(IN,MPR)

10 NLST(IN)=NP

C

WRITE(LOUT1,906)JU,LT,MAT(MPR),(NLST(IN),IN=1,NDPT)

20 CONTINUE

C

C WRITE(LOUT1,908)(NQ(IN),IN=1,NN)

C

C WRITE(LOUT1,910)(NW(IN),IN=1,NN1)

C

WRITE(LOUT1,911)NN

WRITE(LOUT1,912)NDF

C

RETURN

902 FORMAT(//10X,30H ELEMENT MATERIAL TYPE AND,

1 15H NODE NUMBERS//1X,7HELEMENT,1X,4HTYPE,2X,3HMAT,

2 19H 1 2 3 4,

3 55H 5 6 7 8 9 10 11 12 13 14 15,

4 35H 16 17 18 19 20 21 22/)

906 FORMAT(I5,2I6,22I5)

911 FORMAT(//25H TOTAL NUMBER OF NODES =,I8)

912 FORMAT(/40H TOTAL DEGREES OF FREEDOM IN SOLUTION =,I8)

END

C**********************************************************************

SUBROUTINE MINT(XYZ,NCONN,MAT,LTYP,NW)

C**********************************************************************

C This is the main calculation subroutine



COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV

COMMON /PARS / PYI,ASMVL,ZERO

COMMON/NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL,


COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF

COMMON /PRSLD / PRESLD(10,100),LEDG(100),NDE1(100),NDE2(100),NLED

COMMON /PRLDI / PRSLDI(10,100),LEDI(100),NDI1(100),NDI2(100),ILOD


C

DIMENSION NW(MNODES+1),NP1(NPL),NP2(NPL)

DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL)

DIMENSION STRESS(NVRS,NIP,MEL),XYZ(3,MNODES)

DIMENSION PR(NPR,NMT),NTY(NMT),PCONI(MDOF),PEQT(MDOF)

c write(*,*)pr(1,1)

C ----- CALL SUBROUTINE SETUP NP

CALL SETNP(NP1,NP2,NPL)

C ----- CALCULATE INITIAL INSITU STRESSES

LINK1=1

CALL INITIAL(LINK1,XYZ,STRESS,PCONI,PEQT,NCONN,MAT,LTYP,NW,

1 NP1,NP2,PR,NTY,INCS,INCF)

C ----- DEFINE INITIAL STRESS STATUS FOR THE MODEL

CALL INMODST(STRESS,MAT,PR,NTY,LTYP,NEL,NS,NDIM)

C ----- Major calculation

CALL TOTSOL(XYZ,STRESS,PCONI,PEQT,NCONN,MAT,LTYP,NW,PR,NTY,

1 NP1,NP2,INCS,INCF)

C

RETURN

END

C----- FROM HERE, MINOR SUBROUTINES BEGIN

C

C

C----- MINOR SUBROUTINES FOR SUBROUTINE INPUT(...)

C

SUBROUTINE ZEROR2(V,L1,L2)

C**********************************************************************

C ROUTINE TO INITIALISE A 2-DIMENSIONAL REAL ARRAY

C**********************************************************************

c 1 CALLED BY INPUT

C 2 CALLED BY TOTSOL



DIMENSION V(L1,L2)

C

DO 10 I=1,L1

DO 10 J=1,L2

10 V(I,J)=0.0D0

RETURN

END


SUBROUTINE ZEROI2(N,L1,L2)

C**********************************************************************

C ROUTINE TO INITIALISE A 2-DIMENSIONAL INTEGER ARRAY

C**********************************************************************

C 1 CALLED BY INPUT

C 2 CALLED BY INMODST


DIMENSION N(L1,L2)

C

DO 10 J=1,L2

DO 10 I=1,L1

10 N(I,J)=0

RETURN

END

SUBROUTINE ZEROI1(N,LN)

C**********************************************************************

C ROUTINE TO INITIALISE A 1-DIMENSIONAL INTEGER ARRAY

C**********************************************************************

C 1 CALLED BY INPUT

DIMENSION N(LN)

C

DO 10 I=1,LN

10 N(I)=0

RETURN

END

C----- MINOR SUBROUTINES FOR SUBROUTINE MINT(...)

C

SUBROUTINE SETNP(NP1,NP2,NPL)

C**********************************************************************

C SET UP ARRAYS NP1 AND NP2 WHICH GIVE THE INDEX TO ARRAY

C NCONN FOR NODES AT EITHER END OF EACH ELEMENT EDGE

C**********************************************************************

C 1 CALLED BY MINT SETNP(NP1,NP2,NPL)

DIMENSION NPL1(21),NPL2(21),NP1(NPL),NP2(NPL)

C----------------------------------------------------------------------

C INDEXES OF ARRAYS NPL1,NPL2,NP1,NP2

C INDEX ELEMENT TYPE

C 1 - 3 1, 2, 3, 6, 7

C 4 - 7 4, 5

C 4 - 15 8, 9

C 16 - 21 10,11

C----------------------------------------------------------------------


DATA NPL1(1),NPL1(2),NPL1(3),NPL1(4),NPL1(5),NPL1(6),NPL1(7),

1 NPL1(8),NPL1(9),NPL1(10),NPL1(11),NPL1(12),NPL1(13),NPL1(14),

2 NPL1(15),NPL1(16),NPL1(17),NPL1(18),NPL1(19),NPL1(20),NPL1(21)/

3 1,2,3,1,2,3,4,5,6,7,8,1,2,3,4,1,2,3,1,2,3/

DATA NPL2(1),NPL2(2),NPL2(3),NPL2(4),NPL2(5),NPL2(6),NPL2(7),

1 NPL2(8),NPL2(9),NPL2(10),NPL2(11),NPL2(12),NPL2(13),NPL2(14),

2 NPL2(15),NPL2(16),NPL2(17),NPL2(18),NPL2(19),NPL2(20),NPL2(21)/

3 2,3,1,2,3,4,1,6,7,8,5,5,6,7,8,2,3,1,4,4,4/

C

DO 10 I=1,NPL

NP1(I)=NPL1(I)

10 NP2(I)=NPL2(I)

C

RETURN

END

SUBROUTINE INITIAL(LINK1,XYZ,STRESS,PCONI,PEQT,NCONN,MAT,LTYP,NW,

+ NP1,NP2,PR,NTY,INCS,INCF)

C**********************************************************************

C MAIN CONTROLLING ROUTINE - INSITU STRESSES

C**********************************************************************

C 1 Called by MINT

INITIAL(LINK1,XYZ,STRESS,PCONI,PEQT,NCONN,MAT,LTYP,NW,NP1,NP2,PR,NTY)




DIMENSION XYZ(3,MNODES),STRESS(NVRS,NIP,MEL),P(MDOF),

1 PT(MDOF),PCOR(MDOF),PEQT(MDOF),XYFT(MDOF),PCONI(MDOF)

DIMENSION NW(MNODES+1),IDFX(MDOF),NP1(NPL),NP2(NPL)


DIMENSION NTY(NMT),PR(NPR,NMT)












COMMON /ANLYS/ TTIME,DTIMEI,TGRAV,DGRAVI,FRACT,FRACLD,

+ ICOR ,IDCHK,IOUT ,INCT ,IWL ,

+ NOIB ,JS ,JINCB,NLOD ,NLDS

C -----

LINK2=1

TTIME=ZERO

TGRAV=ZERO


READ(LINP,*)IDCHK

WRITE(LOUT1,922)IDCHK

IF(IDCHK.EQ.0)WRITE(LOUT1,930)



C -----IF ONLY TO TEST GEOMETRY DATA STOP HERE

IF(IDCHK.EQ.1)STOP

IF(LINK1.EQ.LINK2) GO TO 1

WRITE(LOUT1,904)LINK1,LINK2

STOP

C ----- ROUTINE TO READ CONTROL OPTIONS AND MATERIAL PROPERTIES

1 CALL RDPROP(NPLAX,NDIM,NMAT,NOIB,INCS,INCF,INCT,

1 ICOR,PR,NTY)

C ----- READ & SETUP IN-SITU STRESSES AND CHECK FOR EQUILIBRIUM

IF(INCS.EQ.1)CALL INSITU(XYZ,STRESS,P,PT,PCOR,PEQT,

+ XYFT,PCONI,NCONN,MAT,LTYP,NW,IDFX,NP1,NP2,PR,NTY)

RETURN

101 FORMAT(A80)

904 FORMAT(//10X,32HERROR ---- LINK CODE MISMATCH,2I5)

922 FORMAT(/1X,20HDATA CHECK OPTION =,I5/)

930 FORMAT(1X,32HCOMPLETE ANALYSIS IS CARRIED OUT/)

935 FORMAT(1X,30HONLY GEOMETRY DATA ARE CHECKED/)

940 FORMAT(1X,42HGEOMETRY DATA AND IN-SITU STRESSES CHECKED/)

END

SUBROUTINE INMODST(STRESS,MAT,PR,NTY,LTYP,NEL,NS,NDIM)

C**********************************************************************

C This subroutine define initial stress status for the model.

C**********************************************************************

C 1 CALLED BY MINT




COMMON/ELINF /LINFO(50,15)

COMMON /MODEL /PQMOD(NIP,MEL,2),MCODE(NIP,MEL)

DIMENSION STRESS(NVRS,NIP,MEL),MAT(MEL),PR(NPR,NMT),NTY(NMT),

1 LTYP(MEL),TEMP(6)

C ----- Initializing

CALL ZEROI2(MCODE,NIP,MEL)

CALL ZEROR3(PQMOD,NIP,MEL,2)

CALL ZEROR1(TEMP,6)

C

DO MR=1,NEL

KM=MAT(MR)

KGO=NTY(KM)

IF(KGO.NE.3.AND.KGO.NE.4)GO TO 100

PRM=PR(4,KM)

LT=LTYP(MR)

NGP=LINFO(11,LT)

DO IP=1,NGP

PC=STRESS(NS+3,IP,MR)

PQMOD(IP,MR,1)=PC

P=(STRESS(1,IP,MR)+STRESS(2,IP,MR)+STRESS(3,IP,MR))/3.0D0

DO IS=1,NS

TEMP(IS)=STRESS(IS,IP,MR)

ENDDO

QE=Q(TEMP,NS,NDIM)

PQMOD(IP,MR,2)=QE/P

C

IF(KGO.EQ.3) THEN

PCS=PC/2.0D0

PY=P+QE*QE/(P*PRM*PRM)

ELSEIF(KGO.EQ.4) THEN

PCS=PC/EXP(1.0d0)

PY=P*EXP(QE/(PRM*P))

ENDIF

C

IF(P.GE.PCS) THEN !NC

IF(PY.GE.0.9950D0*PC)THEN

MCODE(IP,MR)=2 !Iitial point at Pc

ELSE

MCODE(IP,MR)=1 !Roscoe surface & inside

ENDIF

ELSEIF(P.LT.PCS) THEN !OC

IF(PY.GE.0.9950D0*PC)THEN

MCODE(IP,MR)=4 !OC+hardening(expansion)

ELSE

MCODE(IP,MR)=3 !OC, Hvoslev surface & inside

ENDIF


ENDIF

C

ENDDO

100 CONTINUE

ENDDO

RETURN

END

SUBROUTINE TOTSOL(XYZ,STRESS,PCONI,PEQT,NCONN,MAT,LTYP,NW,PR,NTY,

1 NP1,NP2,INCS,INCF)

C**********************************************************************

C This is a main controlling routine.

C**********************************************************************

C 1 Called by MINT.








COMMON /LOADS / FB(2,15)

COMMON /OUT / IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF




COMMON /PTLOAD/PT1(MDOF),INDPT

COMMON/NSIZE / NN,NEL,NDF,NNOD1,NS,NPT,NSP,INXL,




+ NOIB ,JS ,JINCB ,NLOD ,NLDS

COMMON/BSTRESS/ PREP,PREQ,PREETA,PREALPH !back stress

variable

C

DIMENSION NQ(MNODES),NW(MNODES+1),IDFX(MDOF)



DIMENSION PR(NPR,NMT),NTY(NMT),NP1(NPL),NP2(NPL)

DIMENSION

PINC(MDOF),PCOR(MDOF),PREV(MDOF),PT(MDOF),PEQT(MDOF),

1 PCONI(MDOF),PITER(MDOF)

DIMENSION DINC(MDOF),DITER(MDOF),DIPR(MDOF),DA(MDOF)

DIMENSION PDISLD(3,LV),PRES(3,LV)

DIMENSION STRAIN(NVRN,NIP,MEL)

DIMENSION


XYFIB(MDOF),PIB(MDOF),PEXIB(MDOF),PEXI(MDOF),XYFT(MDOF)

DIMENSION JEL(MEL),FXYZ(3),DXYT1(4,200)

DIMENSION RINCC(2500),DTM(2500),IOPT(2500)

DIMENSION VARC(9,NIP,MEL),MCS(MEL),MNGP(MEL),NELCM(MEL)

DIMENSION LCS(NIP,MEL),LNGP(NIP,MEL)

DIMENSION NCHAIN(100,2),NUMD(MDOF,2),DD(4,200),DD1(4,200)

DIMENSION PNOD(MDOF),DP(MDOF),DP1(MDOF)

DIMENSION FRICT(MDOF),FRICT1(MDOF),FRICTPR(MDOF),XMUFR(2)

C ----- MAXIMUM NUMBER OF INCREMENTS IN A INCREMENT BLOCK

INCZ=2500

ILINR=0

IBCAL=0

FRMAX0=0.0D0

FRMAX3=0.0D0

TOTPEN=0.0D0

NDIM1=NDIM+1

IF(IDCHK.EQ.0)GOTO 10 !=0 for me

WRITE(LOUT1,907)

STOP

C ----- INITIALIZE SOME ARRAYS

10 CALL ZEROR3(STRAIN,NVRN,NIP,MEL)

CALL ZEROR2(DD,4,200)

CALL ZEROI2(NUMD,MDOF,2)

CALL ZEROR1(XMUFR,2)

DO IDF=1,MDOF

XYFT(IDF)=0.0D0

PEXI(IDF)=0.0D0

PCOR(IDF)=0.0D0

DP(IDF)=0.0D0

DP1(IDF)=0.0D0

PNOD(IDF)=0.0D0

FRICT(IDF)=0.0D0

FRICT1(IDF)=0.0D0

FRICTPR(IDF)=0.0D0

DA(IDF)=0.0D0

ENDDO

C ----- START OF INCREMENT BLOCK LOOP

INDPT=0

CALL ZEROR1(PT1,MDOF) ! Initializing

DO 250 JINCB=1,NOIB ! Start of big Do loop, NOIB=1 for me.

INCR1=0


WRITE(LOUT1,908) JINCB




C ----- INITIALISE LOAD VECTORS

DO IDF=1,MDOF

XYFIB(IDF)=0.0D0

PIB(IDF)=0.0D0

PEXIB(IDF)=0.0D0

PINC(IDF)=0.0D0

PITER(IDF)=0.0D0

DINC(IDF)=0.0D0

DIPR(IDF)=0.0D0

ENDDO

CALL ZEROR2(DXYT1,4,200) ! Initializing

CALL ZEROR2(PRSLDI,10,100) ! Initializing

C

ILOD=0

CALL ZEROI1(JEL,MEL) ! Initializing

CALL ZEROI1(IOPT,2500) ! Initializing

CALL ZEROR1(DTM,2500) ! Initializing

CALL ZEROR1(RINCC,2500) ! Initializing

FRACT=0

C ----- READ INCREMENT CONTROL OPTIONS


READ(LINP,*)IBNO,NLOD,ILDF,NFXEL,NFXNOD,IOUTS,

1IOCD,DTIME,ITMF,DGRAV

INC1=INCS

INC2=INCF

IDUMMY=1

IF(IBNO.EQ.2)INC2=2

C

WRITE(*,*)IBNO,INC1,INC2,IDUMMY,NLOD,ILDF,NFXEL,NFXNOD,IOUTS,

C 2 IOCD,DTIME,ITMF !,DGRAV

WRITE(LOUT1,912)IBNO,INC1,INC2,IDUMMY,NLOD,ILDF,NFXEL,NFXNOD,IOUTS,

2IOCD !,! DTIME,ITMF !,DGRAV

NOINC=INC2+1-INC1

IF(NOINC.LE.INCZ)GOTO 70 !NOINC=10 INCZ=2500 -- MAXIMUM VALUE

OF NOINC

WRITE(LOUT1,950)NOINC

STOP

70 IF(IBNO.EQ.JINCB) GO TO 76 !IBNO=1 JINCB=1

WRITE(LOUT1,913) IBNO,JINCB


STOP

101 FORMAT(A80)

C ----- CALCULATE BODY FORCE LOAD VECTOR

C ----- FOR SELF-WEIGHT LOADING AND GRAVITY LOADING

76 CALL SEL1(LOUT1,NDIM,NEL,XYZ,PIB,NCONN,MAT,

1 LTYP,NW,PR,DGRAV)

C ----- READ LOAD FACTORS, TIME FACTORS AND OUTPUT OPTIONS

CALL FACTOR(LINP,LOUT1,NOINC,ILDF,IOCD,ITMF,IOUTS,

1RINCC,DTM,IOPT,DTIME)

IF(NLOD.EQ.0)GO TO 95 !NLOD=0

IF(NLOD.GT.0)GO TO 82

C ----- Skip from here to 95

C ----- PRESSURE LOADING ALONG ELEMENT EDGE

WRITE(LOUT1,1000)

NLDS=IABS(NLOD)

IF(NDIM.EQ.2)GOTO 78

WRITE(LOUT1,955)

955 FORMAT(/1X,’NO OPTION TO CALCULATE NODAL LOADS’,1X,

1 ’FROM PRESSURE LOADING IN 3-D PROBLEM (ROUTINE TOTSOL)’)

STOP

C

78 DO 80 KLOD=1,NLDS ! NLDS = No. of loaded element. (guess)

READ(LINP,*)LNE,ND1,ND2,((PDISLD(ID,IV),ID=1,NDIM),IV=1,NPT)

WRITE(LOUT1,1002)LNE,ND1,ND2,((PDISLD(ID,IV),ID=1,NDIM),IV=1,NPT)

C

CALL ZEROR2(FB,2,15)

DO 100 IV=1,NPT

DO 100 ID=1,NDIM

IDR=NDIM+1-ID

100 PRES(ID,IV)=PDISLD(IDR,IV)

C

DO 110 IV=1,NPT

DO 110 ID=1,NDIM

110 PDISLD(ID,IV)=PRES(ID,IV)

C

CALL EDGLD(LOUT1,NDIM,NCONN,LTYP,LNE,ND1,ND2,NP1,NP2,

1PDISLD,PRES,KLOD,NPT,0,MXLD)

C

CALL DISTLD(LOUT1,XYZ,PIB,NCONN,LTYP,NW,NP1,

1 NP2,PRES,LNE,ND1,ND2,1,1,1.0D0)

80 CONTINUE

GO TO 95

C ----- READ INCREMENTAL POINT LOADS


82 WRITE(LOUT1,916)

C

DO 90 JJ=1,NLOD

READ(LINP,*)KK,(FXYZ(ID),ID=1,NDIM)

WRITE(LOUT1,940)KK,(FXYZ(ID),ID=1,NDIM)

C ----- NO PROVISION FOR PORE PRESSURE TERMS IN ’APPLIED’ NODAL

LOADS

FTT=ZERO

KJ=KK

N1=NW(KJ)-1

IDF=NW(KJ+1)-NW(KJ)

IF(IDF.EQ.1)GO TO 84

C

DO 83 ID=1,NDIM

83 XYFIB(N1+ID)=FXYZ(ID)

IF(IDF.EQ.NDIM1)XYFIB(N1+NDIM1)=FTT

GO TO 90

84 XYFIB(N1+1)=FTT

90 CONTINUE

C

95 IF(NFXEL.EQ.0.AND.NFXNOD.EQ.0) GO TO 137 !NFXEL=8, NFXNOD=0

C ----- READ CHANGE TO NODAL FIXITIES

WRITE(LOUT1,931)

C

IF(NDIM.EQ.2)THEN


CALL FIXX2(LINP,LOUT1,NDIM,NCONN,LTYP,NP1,NP2,NFXEL,

1 NFXNOD)

ENDIF

IF(NDIM.EQ.3)THEN


CALL FIXX3(LINP,LOUT1,NDIM,NCONN,LTYP,NFXEL,NFXNOD)

ENDIF

137 CONTINUE

C ----- FIND NQ

CALL MAKENZ(NEL,NN,NCONN,LTYP,NQ,INXL)

C ----- READ DATA FOR PENETRATION PROBLEM


READ(LINP,*)NPEN,IPEN

IF(NPEN.GE.1)THEN !

IF(IPEN.EQ.1)CALL INPENT(LINP,NPEN,XREF,YREF,DYREF,DXPEN,

!IPEN=1

1 XCOS,XSIN,XMUFR,NCHAIN,KSLID0,KSLID3)


IF(KSLID3.EQ.0)THEN

NSKEW1=NSKEW

NSKEW=0

ENDIF

ENDIF

C ----- Initializing incremental variables for back stress by

Song 2/22/98

PREP=1

PREQ=0

PREETA=0

PREALPH=0

C ----- START OF INCREMENT LOOP

INCT=0 ! ADDED BY SONG 9/2/98

DO 200 JS=INC1,INC2 !*** INCREMENTAL LOOP STARTS***

INCT=INCT+1

INCR1=INCR1+1

IF(JS.EQ.INCT)GO TO 138

WRITE(LOUT1,933)JS,INCT

STOP

138 JC=JS+1-INC1

FRACLD=RINCC(JC) !FRACLD=RINCC=FSTD=1/No. of Inc

C FRACLD=Fragment !Song

FRACT=FRACT+FRACLD !AccumulatED total displ for unit pen. leng.

DTIMEI1=DTM(JC)

TTIME=TTIME+DTIMEI1

DTIMEI=DTIMEI1

DGRAVI=FRACLD*DGRAV !DGRAV = increment in gravity field =0

TGRAV=TGRAV+DGRAVI

IOUT=IOPT(JC)

C ----- ASSIGN THE APPROPRIATE POINTERS TO THE DOF AND STIFFNESS

MATRIX

CALL LOCINIT(LTYP,NCONN,NQ,NW,NDF,NDIM)

C

IWL=0

IF(JINCB.EQ.NOIB.AND.JS.EQ.INC2)IWL=1

C----------------------------------------------------------------------

C BOUNDARY CONDITIONS (LOADS AND DISPLACEMENTS) ARE PRINTED

C EVERY IBC INCREMENTS

C IBC = 0 NOT PRINTED IN ANY INCREMENT

C IBC = 1 PRINTED IN EACH INCREMENT

C IBC = N PRINTED IN EVERY NTH INCREMENT

C----------------------------------------------------------------------

IOUTP=0

IF(IBC.EQ.0)GOTO 130

NJS=IBC*(JS/IBC)


IF(NJS.EQ.JS)IOUTP=1

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

IF(INCT.LE.5.OR.INCT.EQ.8)IOUTP=1

IF(INCT.EQ.10.OR.INCT.EQ.15)IOUTP=1




IF(INCT.EQ.150)IOUTP=1

IF(IBNO.GE.2)IOUTP=1

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

WRITE(*,*)’INCR=’,INCT

IF(IOUTP.EQ.1)THEN

WRITE(LOUT1,919)DTIMEI,TTIME




ENDIF

C

130 DO 140 IM=1,NDF

C XYFT(IM)=XYFT(IM)+XYFIB(IM)*FRACLD+FRICT(IM)

C 140 PINC(IM)=FRACLD*PIB(IM)+FRACLD*XYFIB(IM)+DP(IM)+FRICT(IM)

XYFT(IM)=XYFT(IM)+XYFIB(IM)*FRACLD+DP1(IM)+FRICT(IM)+FRICT1(IM)

140 PINC(IM)=FRACLD*PIB(IM)+FRACLD*XYFIB(IM)+DP(IM)+DP1(IM)+FRICT1(IM)

CALL ZEROR1(FRICT,MDOF)

CALL ZEROR1(FRICT1,MDOF)

C

DO 145 IM=1,NDF

145 PEXI(IM)=(1.0D0-FRACT)*PEXIB(IM)+PNOD(IM)

C ----- UPDATE LIST OF PRESSURE LOADING ALONG ELEMENT EDGES

IF(NLOD.GE.0)GO TO 162

C

DO 160 ISD=1,NLDS

LNE=LEDI(ISD)

ND1=NDI1(ISD)

ND2=NDI2(ISD)

ICT=0

C *** N2D = 2 FOR TWO DIMENSIONAL PROBLEMS

N2D=2

DO 150 IK=1,NPT

DO 150 IJ=1,N2D

ICT=ICT+1

150 PRES(IJ,IK)=FRACLD*PRSLDI(ICT,ISD)

CALL LODLST(LOUT1,NDIM,LNE,ND1,ND2,PRES,NPT,0,MXLD)

160 CONTINUE

162 CONTINUE

C ----- INITIALISE INCREMENTAL DISPLACEMENTS


ICONV=0

ITER=1

MAXITER=10

TOL=0.080D0

C

DO I10=1,NDF

PITER(I10)=PINC(I10)+PCOR(I10)

c PITER(I10)=PINC(I10)

ENDDO

DO IDF=1,MDOF

DINC(IDF) =0.0D0

DIPR(IDF) =0.0D0

DITER(IDF)=0.0D0

PREV(IDF) =0.0D0

PINC(IDF) =0.0D0

ENDDO

C----------------------------------------------------------------------

c IF(INCR1.EQ.1.AND.LINR.GE.1)THEN

c ILINR=LINR

c LINR =0

c ELSEIF(INCR1.GT.1.AND.ILINR.GE.1)THEN

c LINR =ILINR

c ILINR=0

c ENDIF

c write(2,*)’linr=’,linr

C----------------------------------------------------------------------

C ----- START OF ITERATION LOOP ******ITERATION STARTS******

DO WHILE (ICONV.EQ.0.AND.ITER.LE.MAXITER)

C ----- CONVERT PITER TO COUNT FOR SKEW BOUNDARIES

IF(NSKEW.GT.0)CALL ROTBC(PITER,NW,NDIM,NSKEW,1)

C ----- SOLVE EQUATIONS USING SKY LINE

CALL SKSOLV(XYZ,DA,DITER,DIPR,DD,STRESS,PITER,NQ,NW,LTYP,

1 NTY,MAT,NCONN,PR,IOUTP,ITER) !js REMOVED TENTATIVELY

IF(ITER.EQ.1) THEN

DO IJ=1,NF

DO ID=1,NDIM

DXYT1(ID,IJ)=DXYT(ID,IJ)

DXYT(ID,IJ)=0.0D0

DD1(ID,IJ)=DD(ID,IJ)

DD(ID,IJ)=0.0D0

ENDDO

ENDDO

ENDIF

C ----- ROTATE BACK DISPL & LOADS AT SKEW BOUNDARIES TO ORIGINAL


COORD.

IF(NSKEW.GT.0) THEN

CALL ROTBC(DITER,NW,NDIM,NSKEW,-1)

CALL ROTBC(PITER,NW,NDIM,NSKEW,-1)

END IF

C ----- FIND THE OUT-OF-BALANCE LOAD PCOR

CALL EQUIBLOD(XYZ,NCONN,MAT,LTYP,NQ,NW,NP1,NP2,PR,NTY,DITER,

1 DIPR,STRESS,STRAIN,PEXI,IDFX,PITER,PT,PCOR,PEQT,XYFT,PCONI,

2 LCS,LNGP,NELCM,MCS,MNGP,NCAM,ITER,IOUTP,JS)

C ----- ROTATE BACK PCOR AT SKEW BOUNDARIES TO ORIGINAL COORD.

IF(NSKEW.GT.0)CALL ROTBC(PCOR,NW,NDIM,NSKEW,-1)

C

DO I10=1,NDF

DINC(I10)=DINC(I10)+DITER(I10)

ENDDO

C ----- CHECK CONVERGENCE

CALL CONVCH(NDF,NN,NDIM,TOL,ICONV,2,NW,DITER,DINC,PREV,PCOR,

1 ITER,RNRM1)

C

DO I10=1,NDF

PINC(I10)=PINC(I10)+PITER(I10)

PITER(I10)=PCOR(I10)

PREV(I10)=PCOR(I10)

DIPR(I10)=DITER(I10)

ENDDO

CALL ZEROR1(DITER,MDOF)

c DTIMEI=0.0001D0

DTIMEI=DTIMEI1/100.0D0

WRITE(*,*)" Iter. No.",ITER

ITER=ITER+1

ENDDO !***** END OF ITERATION LOOP (iteration until converge)

C

NITER=ITER-1

IF(IOUTP.EQ.1)WRITE(LOUT1,*)’ICONV=’,ICONV,

1 ’NO. OF ITERATIONS=’,NITER

C

DO IJ=1,NF

DO ID=1,NDIM

DXYT(ID,IJ)=DXYT1(ID,IJ)

DXYT1(ID,IJ)=0.0D0


DD(ID,IJ)=DD1(ID,IJ)

DD1(ID,IJ)=0.0D0

ENDDO

ENDDO

C ----- UPDATE AND OUTPUT CALCULATIONS

CALL UPOUT(XYZ,DINC,DA,STRESS,STRAIN,PINC,PT,PCOR,PEQT,NCONN,

1 MAT,NTY,PR,LTYP,NW,NQ,IDFX,VARC,YREF,IOUTP)

C ----- ADJUST B.C FOR THE PENETRATION PROBLEM

IF(NPEN.GE.1)THEN

YREF=YREF-DYREF*FRACLD !FRACLD=FSTD=RINCC=1/NOINC

TOTPEN=TOTPEN+DYREF*FRACLD

IF(KSLID3.NE.0) THEN

c

CALL BCADJST(NPEN,NDIM,NSKEW,XREF,YREF,DXPEN,XSIN,XCOS,

1 NCHAIN,XYZ,NW,PEQT,PT,FRICT1,FRICTPR,XMUFR,PNOD,DD,DP,DP1,

2 INCT,NUMD,IOUTP,TOTPEN,IBCAL,FRMAX0,FRMAX3)

c

ENDIF

IF(KSLID0.EQ.0.OR.KSLID3.EQ.0)THEN

CALL CHSLIDE(NPEN,NSKEW,NSKEW1,DXPEN,KSLID0,KSLID3,XSIN,XCOS,

1 XMUFR,NCHAIN,NW,PEQT,PT,FRICT,FRICTPR,INCT,FRMAX0,FRMAX3)

ENDIF

ENDIF

200 CONTINUE !*** END OF INCREMENTAL LOOP (for incremental penet.)

C ----- ZERO ALL NON-ZERO PRESCRIBED VALUES

IF(NF.EQ.0)GOTO 240

C

DO JJ=1,MXFXT

DO II=1,4

DD(II,JJ)=0.0d0

DXYT(II,JJ)=0.0d0

ENDDO

ENDDO

C

DO JJ=1,NDF

FRICT1(JJ)=0.0D0

DP1(JJ)=0.0D0

PNOD(JJ)=PNOD(JJ)+DP(JJ)

DP(JJ)=0.0D0

ENDDO

240 CONTINUE

C

250 CONTINUE !*** END OF INCREMENTAL LOOP (for the incremental

blocknumber)


907 FORMAT(/1X,24HANALYSIS NOT CARRIED OUT/)

908 FORMAT(//120(1H=)//

1 1X,43HSTART OF LOAD INCREMENT BLOCK NUMBER ,I5/1X,48(1H-))

912 FORMAT(/

11X,23HINCR BLOCK NUMBER.....=,I5,4X,23HSTARTING INCR NUMBER..=,I8/

21X,23HFINISHING INCR NUMBER.=,I5,4X,23HNO. OF ELEMENT

CHANGES=,I8/

31X,23HNUMBER OF LOADS.......=,I5,4X,23HLOAD RATIO OPTION.....=,I8/

41X,23HNUM OF ELEM FIXITIES..=,I5,4X,23HNUM OF NODE FIXITIES..=,I5/

51X,23HSTD OUTPUT CODE.......=,I8/ !i8

61X,23HOUTPUT OPTION.........=,I8/)

C 74X,23HTIME INCREMENT........=,F10.1/

C 71X,23HTIME RATIO OPTION.....=,I5/)

C 84X,23HINCR IN GRAVITY FIELD.=,F10.1/) !temp kill

913 FORMAT(//1X,26HERROR IN INCR BLOCK NUMBER,2I6)

914 FORMAT(//28H LIST OF ELEMENT ALTERATIONS/1X,27(1H-)/)

916 FORMAT(//,3X,32H LIST OF INCREMENTAL NODAL LOADS/3X,33(1H-),/,

1 3X,4HNODE,6X,1HX,9X,1HY,9X,1HZ,/,3X,33(1H-),/)

920 FORMAT(1X,10I8)

931 FORMAT(/1X,29HPRESCRIBED BOUNDARY CONDITONS/1X,29(1H-)/)

933 FORMAT(//1X,’ERROR IN INCREMENT NUMBER’,2I6,2X,’(ROUTINE

TOTSOL)’)

940 FORMAT(1X,I5,3F10.3)

950 FORMAT(/1X,46HINCREASE SIZE OF ARRAYS RINCC, DTM AND IOPT TO,

1 I5,2X,’ALSO SET INCZ IN ROUTINE TOTSOL’)

1000 FORMAT(39H SPECIFIED NODAL VALUES OF SHEAR/NORMAL,

1 36H STRESSES AND EQUIVALENT NODAL LOADS/1X,74(1H-)/5H0ELEM,

2 1X,4HNDE1,2X,4HNDE2,2X,4HSHR1,8X,4HNOR1,8X,4HSHR2,8X,4HNOR2,

3 8X,4HSHR3,8X,4HNOR3,8X,4HSHR4,8X,4HNOR4,8X,4HSHR5,8X,4HNOR5/

1 1X,16H(LOAD DIRECTION),2X,3H(X),9X,3H(Y),9X,3H(X),9X,3H(Y),

2 9X,3H(X),9X,3H(Y),9X,3H(X),9X,3H(Y),9X,3H(X),9X,3H(Y)/)

1002 FORMAT(1X,3I4,10E12.4)

C 915 FORMAT(//120(1H=)//

C 1 1X,32HSTART OF LOAD INCREMENT NUMBER ,I5,

C 2 4X,22HINCREMENT BLOCK NUMBER,I5,4X,13HLOAD RATIO =,F5.2/

C 3 1X,90(1H-))

C 917 FORMAT(/22H INCR GRAVITY LEVEL =,E12.4,

C 1 24H TOTAL GRAVITY LEVEL =,E12.4)

919 FORMAT(/80(1H=),/18H TIME INCREMENT =,G12.4,/,

1 14H TOTAL TIME =,G12.4)

RETURN

END

C----- MINOR OF MINOR SUBROUTINES

C

C

SUBROUTINE


RDPROP(NPLAX,NDIM,NMAT,NOIB,INCS,INCF,INCT,ICOR,PR,NTY)

C**********************************************************************

C READ CONTROL OPTIONS AND MATERIAL PROPERTIES

C**********************************************************************

C 1 CALLED BY INITIAL




DIMENSION PR(NPR,NMT),NTY(NMT)



COMMON/MATPROP/C,X

C----------------------------------------------------------------------

C ICOR - OPTION TO APPLY OUT-OF-BALANCE LOADS AS CORRECTING

C LOADS IN THE NEXT INCREMENT

C ICOR = 0 - CORRECTING LOADS ARE NOT APPLIED

C ICOR = 1 - CORRECTING LOADS ARE APPLIED

C----------------------------------------------------------------------

ICOR=0

C


READ(LINP,*)NMAT,NOIB,INCS,INCF

WRITE(LOUT1,922)NMAT,NOIB,INCS,INCF

NOINC=INCF-INCS+1

IF(NOINC.GT.0)GOTO 5

WRITE(LOUT1,925)NOINC,INCS,INCF

STOP

C

5 CONTINUE

C----------------------------------------------------------------------

C INCT - COUNTER OF INCREMENT NUMBER

C----------------------------------------------------------------------

INCT=INCS-1

IF(NDIM.NE.3)GOTO 8

WRITE(LOUT1,928)

GOTO 10

8 IF(NPLAX.EQ.0)WRITE(LOUT1,930)

IF(NPLAX.EQ.1)WRITE(LOUT1,931)

10 CONTINUE

C ----- READ OUTPUT REDUCING OPTIONS. THIS OPERATES ON RE


READ(LINP,*)IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF

WRITE(LOUT1,945)IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF

C ----- READ MATERIAL PROPERTIES


CALL ZEROR2(PR,NPR,NMT)

WRITE(LOUT1,932)


DO 20 I=1,NMAT

READ(LINP,*)II,NTY(II),(PR(JJ,II),JJ=1,NPR)

WRITE(LOUT1,936)II,NTY(II),(PR(JJ,II),JJ=1,NPR)

20 CONTINUE

READ(LINP,101)HEADER81 !back stress parameter

READ(LINP,*)pr(6,2),pr(7,2) !C,X

read(LINP,101)HEADER82 !VISCOSITY HEADER

read(linp,*)pr(1,2),pr(2,2) !VISCOSITY DATA

read(linp,101)header83 !Damage header

read(linp,*)pr(3,2),pr(4,2) !A1 & A2

read(linp,101)header84 !Gradient Header

read(linp,*)pr(5,2) !gradcon

c write(*,*)pr(5,2)

RETURN

101 FORMAT(A80)

922 FORMAT(/

1 10X,46HNUMBER OF MATERIALS..........................=,I5/

2 10X,46HNUMBER OF INCREMENT BLOCKS...................=,I5/

3 10X,46HSTARTING INCR NUMBER OF ANALYSIS.............=,I5/

4 10X,46HFINISHING INCR NUMBER OF ANALYSIS............=,I5/

5 /120(1H*)/)

C 6 10X,46HNUMBER OF PRIMARY ELEMENT CHANGES............=,I5/

C 7 10X,46HOPTION TO UPDATE COORDINATES.................=,I5/

C 8 10X,46HOPTION TO STOP/RESTART ANALYSIS..............=,I5/

C 9 /120(1H*)/)

925 FORMAT(/1X,29HERROR IN NO. OF INCREMENTS =,I5,

1 4X,7HINCS =,I5,4X,7HINCF =,I5,2X,16H(ROUTINE RDPROP))

928 FORMAT(//1X,22H3-DIMENSIONAL ANALYSIS)

930 FORMAT(//1X,21HPLANE STRAIN ANALYSIS)

931 FORMAT(//1X,22HAXI-SYMMETRIC ANALYSIS)

932 FORMAT(//24H MATERIAL PROPERTY TABLE

1 /1X,23(1H-)

2 //2X,8HMAT TYPE,7X,1H1,11X,1H2,11X,1H3,11X,1H4,11X,1H5,

3 11X,1H6,11X,1H7,11X,1H8,11X,1H9,11X,2H10/)

936 FORMAT(1X,2I5,(10E12.4/))

945 FORMAT(//120(1H*)/

1 10X,46HOPTION TO PRINT BOUNDARY CONDITIONS..........=,I5/

2 10X,46HOPTION TO PRINT REACTIONS....................=,I5/

3 10X,46HSTARTING VERTEX NODE NUMBER FOR OUTPUT.......=,I5/

4 10X,46HFINISHING VERTEX NODE NUMBER FOR OUTPUT......=,I5/

5 10X,46HSTARTING MIDSIDE NODE NUMBER FOR OUTPUT......=,I5/

6 10X,46HFINISHING MIDSIDE NODE NUMBER FOR OUTPUT.....=,I5/

7 10X,46HSTARTING ELEMENT NUMBER FOR OUTPUT...........=,I5/


8 10X,46HFINISHING ELEMENT NUMBER FOR OUTPUT..........=,I5/

9 /120(1H*)/)

END

SUBROUTINE LODLST(LOUT1,NDIM,LNE,ND1,ND2,PRES,NPT,ILST,MXLD)

C**********************************************************************

C ROUTINE TO STORE CUMULATIVE LIST OF APPLIED

C PRESSURE LOADING ALONG ELEMENT EDGES

C**********************************************************************

C 1 Called by EDGLD



DIMENSION PRES(3,LV)


C ----- SKIP IF NEW LIST

IF(NLED.EQ.0.OR.ILST.EQ.1)GO TO 22

C ----- SEARCH FOR LNE IN EXISTING LIST

DO 20 J=1,NLED

IF(LNE.NE.LEDG(J))GO TO 20

N1=NDE1(J)

N2=NDE2(J)

IF(N1.EQ.ND1.AND.N2.EQ.ND2)GO TO 25

20 CONTINUE

C ----- ADD NEW EDGE TO THE LIST

22 NLED=NLED+1

IF(NLED.LE.MXLD)GO TO 23

WRITE(LOUT1,900)

900 FORMAT(/27H INCREASE SIZE OF ARRAYS IN,

1 51H COMMON BLOCK PRSLD ALSO SET MXLD IN ROUTINE MAXVAL/

2 25X,16H(ROUTINE LODLST))

STOP

23 JE=NLED

GO TO 30

C ----- UPDATE EXISTING LIST

25 JE=J

GO TO 35

C

30 LEDG(JE)=LNE

NDE1(JE)=ND1

NDE2(JE)=ND2

C

35 IC=0


DO 40 IPT=1,NPT

DO 40 IK=1,NDIM

IC=IC+1

40 PRESLD(IC,JE)=PRESLD(IC,JE)+PRES(IK,IPT)

RETURN

END

SUBROUTINE ZEROR1(A,LA)

C**********************************************************************


C**********************************************************************




DIMENSION A(LA)

C

DO 10 I=1,LA

10 A(I)=0.0D0

RETURN

END

SUBROUTINE ZEROR3(V,L1,L2,L3)

C**********************************************************************


C**********************************************************************




DIMENSION V(L1,L2,L3)

C

DO 10 I=1,L1

DO 10 J=1,L2

DO 10 K=1,L3

10 V(I,J,K)=0.0D0

RETURN

END

SUBROUTINE DETJCB(LOUT1,NDIM,DJACB,NDN,ELCOD,DS,IP,MUS,KSTGE)

C**********************************************************************

C CALCULATES DETERMINANT OF JACOBIAN MATRIX *

C**********************************************************************

C 1 CALLED BY SELF





DIMENSION ELCOD(3,NDMX),DS(3,20),XJAC(3,3)

C ----- NXJ - SIZE OF ARRAY XJAC

C NXJ=3

CALL ZEROR2(XJAC,3,3)

C

DO 10 ID=1,NDIM

DO 10 JD=1,NDIM

DO 10 IN=1,NDN

10 XJAC(ID,JD)=XJAC(ID,JD)+DS(ID,IN)*ELCOD(JD,IN)

C

IF(NDIM.NE.2)GOTO 20

DJACB=XJAC(1,1)*XJAC(2,2)-XJAC(1,2)*XJAC(2,1)

GOTO 50

C

20 DJACB=XJAC(1,1)*(XJAC(2,2)*XJAC(3,3)-XJAC(2,3)*XJAC(3,2))

DJACB=DJACB-XJAC(1,2)*(XJAC(2,1)*XJAC(3,3)-XJAC(2,3)*XJAC(3,1))

DJACB=DJACB+XJAC(1,3)*(XJAC(2,1)*XJAC(3,2)-XJAC(2,2)*XJAC(3,1))

C

50 IF(DJACB.GT.ZERO)GO TO 60

WRITE(LOUT1,900)DJACB,MUS,IP

900 FORMAT(1X,10H JACOBIAN,E16.5,3X,11HIS NEGATIVE,2X,

1 7HELEMENT,I5,2X,10HINT. POINT,I5,2X,16H(ROUTINE DETJCB))

C

WRITE(LOUT1,910)KSTGE

910 FORMAT(/1X,’CODE TO INDICATE STAGE OF ANALYSIS =’,I5//

1 4X,’CODE’,20X,’STAGE OF THE ANALYSIS’,//

1 6X,’1 - CALLED BY INSITU/EQLOD/SELF CALCULATION OF’,

2 1X,’INSITU SELF WEIGHT LOADS’,/6X,’2 - CALLED BY’,

3 1X,’TOTSOL/CHANGE/SELF LOADS DUE TO ELEMENT CHANGES’,/

4 6X,’3 - CALLED BY TOTSOL/SEL1/SELF INCREMENTAL SELF’,

5 1X,’WEIGHT LOADS/6X,25H4 - CALLED BY UPOUT/EQLOD’,

6 ’/SELF SELF WEIGHT LOADS FOR EQUILIBRIUM CHECK’)

STOP

60 RETURN

END

SUBROUTINE DETMIN(LOUT1,NDIM,XJACM,XJACI,DJACB,JL,IP,ISTGE)

C**********************************************************************

C CALCULATES DETERMINANT AND INVERSE OF A SQUARE 3X3 MATRIX

C**********************************************************************

C 1 Called by FORMB2

C 2 Called by ELMSTIF

C 3 Called by EQUIBLOD

C 4 Called by DMCAM

C 5 Called by UPOUT





DIMENSION XJACM(3,3),XJACI(3,3)

C


DJACB=XJACM(1,1)*XJACM(2,2)-XJACM(1,2)*XJACM(2,1)

IF(DJACB.GT.ZERO)GOTO 15

GOTO 60

C

15 XJACI(1,1)= XJACM(2,2)/DJACB

XJACI(2,2)= XJACM(1,1)/DJACB

XJACI(1,2)=-XJACM(1,2)/DJACB

XJACI(2,1)=-XJACM(2,1)/DJACB

RETURN

C

20 XJACI(1,1)= (XJACM(2,2)*XJACM(3,3)-XJACM(2,3)*XJACM(3,2))

XJACI(1,2)=-(XJACM(1,2)*XJACM(3,3)-XJACM(1,3)*XJACM(3,2))

XJACI(1,3)= (XJACM(1,2)*XJACM(2,3)-XJACM(1,3)*XJACM(2,2))

C




C




C

DJACB=XJACM(1,1)*XJACI(1,1)+XJACM(1,2)*XJACI(2,1)+

1 XJACM(1,3)*XJACI(3,1)

IF(DJACB.GT.ZERO)GOTO 32

GOTO 60

C

32 DJACBI=1.0D0/DJACB

C

DO 35 ID=1,NDIM

DO 35 JD=1,NDIM

35 XJACI(ID,JD)=XJACI(ID,JD)*DJACBI

RETURN

60 WRITE(LOUT1,900)DJACB,JL,IP

900 FORMAT(/1X,9HJACOBIAN ,E16.5,3X,10HOF ELEMENT,I6,3X,

1 17HINTEGRATION POINT,I5,3X,29HIS NEGATIVE (ROUTINE DETMIN))

WRITE(LOUT1,910)ISTGE

910 FORMAT(/1X,36HCODE TO INDICATE STAGE OF ANALYSIS =,I5//

1 4X,4HCODE,20X,21HSTAGE OF THE ANALYSIS//

2 6X,49H1 - CALLED BY RDSTRS/EQLIB/FORMB2 LOAD EQUIVALENT,

3 19H TO INSITU STRESSES/6X,33H2 - CALLED BY CHANGE/EQLIB/FORMB2,


4 32H CALCULATION OF IMPLIED LOADINGS/6X,

5 35H3 - CALLED BY SKSOLV/ELMSTIF/FORMB2,

6 32H CALCULATION OF STIFFNESS MATRIX/

7 6X,38H4 - CALLED BY UPOUT/FORMB2 CALCULATION,

8 1X,24HOF STRAINS. OUTPUT STAGE)

STOP

END

SUBROUTINE DETMIN1(LOUT1,NDIM,XJACM,XJACI,DJACB,JL,IP,ISTGE)

C**********************************************************************

C CALCULATES DETERMINANT AND INVERSE OF A SQUARE 3X3 MATRIX

C**********************************************************************




DIMENSION XJACM(3,3),XJACI(3,3)

20 XJACI(1,1)= (XJACM(2,2)*XJACM(3,3)-XJACM(2,3)*XJACM(3,2))



C




C




C

DJACB=XJACM(1,1)*XJACI(1,1)+XJACM(2,1)*XJACI(1,2)+

1 XJACM(3,1)*XJACI(1,3)

C

32 DJACBI=1.0D0/DJACB

C

DO 35 ID=1,NDIM

DO 35 JD=1,NDIM

35 XJACI(ID,JD)=XJACI(ID,JD)*DJACBI

RETURN

END

SUBROUTINE DISTLD(LOUT1,XYZ,RHS,NCONN,LTYP,

1 NW,NP1,NP2,PRES,LNE,ND1,ND2,IPRINT,IST,FC)

C**********************************************************************

C ROUTINE TO CALCULATE EQUIVALENT NODAL LOADS FOR SPECIFIED *


C PRESSURE LOADING ALONG ELEMENT EDGES USING 5 POINT (NSP) *

C INTEGRATION RULE. INTEGRATES POLYNOMIAL OF ORDER NINE OR LESS *

C EXACTLY. ARRAYS ILOC,PRES,PEQLD,ELCD,SHF,DERIV ARE *

C TO CATER FOR A MAXIMUM OF FIVE NODES (NPT) ALONG AN ELEMENT

EDGE *

C (ALL 2-D ELEMENTS UP TO ORDER FIVE). *

C**********************************************************************

C 1 Called by EQLOD







COMMON /SAMP / POSSP(5),WEIGP(5)

COMMON /LOADS / FB(2,15)

DIMENSION NCONN(NTPE,MEL),LTYP(MEL),

1 NW(MNODES+1),NP1(NPL),NP2(NPL)

DIMENSION RHS(MDOF),XYZ(3,MNODES),PRES(3,LV)

DIMENSION ILOC(5),PSP(2),DSP(2),PEQLD(3,5),ELCD(2,5)

DIMENSION SHF(5),DERIV(5),PCOM(3)

NP=5

TPI=2.0D0*PYI

C NE=MREL(LNE)

NE=LNE

LI1=ND1

C LI1=NREL(ND1)

LT=LTYP(NE)

IF(IST.EQ.1)GOTO 5

LT=IABS(LT)

5 IF(LT.GT.0)GOTO 10

WRITE(LOUT1,900)LNE

900 FORMAT(/1X,44H**** ERROR : YOU HAVE PUT A PRESSURE LOAD ON,

1 8H ELEMENT,I5,2X,28HWHICH IS NOT PRESENT IN MESH,

2 17H (ROUTINE DISTLD)/)

RETURN

10 NVN=LINFO(2,LT)

NEDG=LINFO(3,LT)

NDSD=LINFO(7,LT)

NTSD=NDSD+2

INDED=LINFO(14,LT)

C

DO 20 K1=1,NEDG

J1=NP1(K1+INDED)

J2=NP2(K1+INDED)

I1=NCONN(J1,NE)


IF(LI1.EQ.I1)GOTO 25

20 CONTINUE

WRITE(LOUT1,903)LNE,ND1,ND2

903 FORMAT(/21H **** ERROR : ELEMENT,I5,

1 2X,22H DOES NOT HAVE NODES :,2I5,

2 3X,16H(ROUTINE DISTLD))

RETURN

C ----- STORE LOCATIONS OF NODE (IN NCONN) IN ARRAY ILOC

25 LC1=NVN+(J1-1)*NDSD

ILOC(1)=J1

ILOC(NTSD)=J2

IF(NDSD.EQ.0)GOTO 31

C

DO 30 JP=1,NDSD

30 ILOC(JP+1)=LC1+JP

C ----- SET UP LOCAL ARRAY FOR CO-ORDINATES IN ELCD

31 DO 32 KC=1,NTSD

ILC=ILOC(KC)

NDE=NCONN(ILC,NE)

C

DO 32 ID=1,NDIM

32 ELCD(ID,KC)=XYZ(ID,NDE)

C INITIALISE PEQLD

CALL ZEROR2(PEQLD,3,5)

C ----- LOOP FOR NUMERICAL INTEGRATION

DO 60 ISP=1,NSP

XI=POSSP(ISP)

C ----- EVALUATE SHAPE FUNCTION FOR SAMPLING POINT

CALL SFR1(LOUT1,XI,SHF,DERIV,NTSD,LNE,LT)

C ----- CALCULATE COMPONENTS OF THE EQUIVALENT NODAL LOADS -

PEQLD

DO 40 IDOF=1,NDIM

PSP(IDOF)=ZERO

DSP(IDOF)=ZERO

C

DO 40 IEDG=1,NTSD

PSP(IDOF)=PSP(IDOF)+PRES(IDOF,IEDG)*SHF(IEDG)

40 DSP(IDOF)=DSP(IDOF)+ELCD(IDOF,IEDG)*DERIV(IEDG)

C

DV=WEIGP(ISP)

IF(NPLAX.EQ.0)GOTO 48

RAD=0.0D0

C


DO 45 IEDG=1,NTSD

45 RAD=RAD+ELCD(1,IEDG)*SHF(IEDG)

DV=DV*TPI*RAD

48 PCOM(1)=DSP(1)*PSP(2)-DSP(2)*PSP(1)

PCOM(2)=DSP(1)*PSP(1)+DSP(2)*PSP(2)

C

DO 50 IEDG=1,NTSD

DO 50 ID=1,NDIM

50 PEQLD(ID,IEDG)=PEQLD(ID,IEDG)+PCOM(ID)*SHF(IEDG)*DV

C

60 CONTINUE

IF(IPRINT.EQ.1)WRITE(LOUT1,905)LNE,ND1,ND2,

1 ((PEQLD(ID,IP),ID=1,2),IP=1,NTSD)

905 FORMAT(1X,3I4,10E12.4/)

C ----- SLOT LOADS INTO ARRAY RHS

DO 80 IJ=1,NTSD

JL=ILOC(IJ)

IF(JL.GT.15)WRITE(LOUT1,*)’WARNING..JL > 15 (S. DISTLD)’

NDE=NCONN(JL,NE)

N1=NW(NDE)-1

C

DO 80 ID=1,NDIM

FB(ID,JL)=FB(ID,JL)+PEQLD(ID,IJ)

80 RHS(N1+ID)=RHS(N1+ID)+PEQLD(ID,IJ)*FC

RETURN

END

SUBROUTINE EDGLD(LOUT1,NDIM,NCONN,LTYP,

1 LNE,ND1,ND2,NP1,NP2,PDISLD,PRES,KLOD,NPT,KINS,MXLD)

C**********************************************************************

C ROUTINE TO ALIGN NODES ALONG LOADED EDGE IN THE ANTI-CLOCKWISE *

C ORDER AND TO STORE THE INFORMATION *

C THE PRESSURES AT THE BEGINNING OF AN INCREMENT BLOCK ARE STORED *

C IN A TEMPORARY ARRAY COMMON BLOCK PRLDI *

C THE RATIOS OF THESE LOADING ARE ADDED TO THE CUMULATIVE LIST *

C (COMMON BLOCK PRSLD) *

C OF PRESSURE LOADS AT THE BEGINNING OF EACH INCREMENT *

C**********************************************************************

C Called by INSITU





DIMENSION NCONN(NTPE,MEL),LTYP(MEL),NP1(NPL),NP2(NPL)

DIMENSION PDISLD(3,LV),PRES(3,LV)

C


CALL ZEROR2(PRES,3,LV)

NE=LNE

LI1=ND1

LI2=ND2

LT=LTYP(NE)

IF(LT.GT.0)GOTO 15

WRITE(LOUT1,901)NE

901 FORMAT(1X,7HELEMENT,I6,2X,27HNOT PRESENT IN CURRENT MESH,

1 1X,16H(ROUTINE EDGLD))

RETURN

15 NEDG=LINFO(3,LT)

NDSD=LINFO(7,LT)

NTSD=NDSD+2

INDED=LINFO(14,LT)

C

DO 20 K1=1,NEDG

J1=NP1(K1+INDED)

J2=NP2(K1+INDED)

I1=NCONN(J1,NE)

I2=NCONN(J2,NE)

IF(LI1.EQ.I1.AND.LI2.EQ.I2)GO TO 25


20 CONTINUE

WRITE(LOUT1,903)KLOD,LNE,ND1,ND2

903 FORMAT(/13H **** ERROR :,I5,17H TH LOAD. ELEMENT,I5,

1 2X,25H DOES NOT CONTAIN NODES :,2I5,

2 2X,15H(ROUTINE EDGLD))

STOP

C ----- ALIGN NODES IN SEQUENCE

21 LIT=LI1

LI1=LI2

LI2=LIT

NT=ND1

ND1=ND2

ND2=NT

C ----- PRES - CONTAINS THE PRESSURE COMPONENTS ALIGNED IN SEQUENCE

DO 24 J=1,NTSD

JBACK=NTSD+1-J

DO 24 I=1,2

24 PRES(I,J)=PDISLD(I,JBACK)

GO TO 35

25 DO 30 J=1,NTSD

DO 30 I=1,2

30 PRES(I,J)=PDISLD(I,J)

C ----- UPDATE OR READ IN A NEW LIST


35 IF(KINS.EQ.0)GO TO 40

C ----- PRESSURE LOADS IN EQUILIBRIUM WITH IN-SITU STRESSES

C NEW LIST - READ DIRECTLY INTO COMMON PRSLD

C ----- Call LODLST ! Store cumulative list of applied pressure

C loading along element edges.

CALL LODLST(LOUT1,NDIM,LNE,ND1,ND2,PRES,NPT,1,MXLD)

GO TO 55

C ----- PRESSURE LOADS FOR NEW INCREMENT BLOCK READ INTO

COMMON PRSLDI

40 ILOD=KLOD

LEDI(ILOD)=LNE

NDI1(ILOD)=ND1

NDI2(ILOD)=ND2

IC=0

DO 50 IV=1,NTSD

DO 50 IJ=1,2

IC=IC+1

50 PRSLDI(IC,ILOD)=PRES(IJ,IV)

55 CONTINUE

RETURN

END

SUBROUTINE EQLBM(LOUT2,NDIM,IEQL,NN,NDF,NW,NQ,IDFX,P,PT,PCOR,

1 PEQT,IEQOP,ICOR,IRAC,IOUTP)

C**********************************************************************

C CARRIES OUT AN EQUILIBRIUM CHECK

C CALCULATE AND PRINTOUT UNBALANCED NODAL LOADS

C**********************************************************************

C 1 CALLED BY EQLOD




DIMENSION NW(MNODES+1),NQ(MNODES),IDFX(MDOF)

DIMENSION P(MDOF),PT(MDOF),PCOR(MDOF),PEQT(MDOF)

DIMENSION PAR(6),RMAX(6),TER(3)

C ----- MP - ARRAY SIZE OF PAR, RMAX

MP=6

NDIM1=NDIM+1

NDIM2=2*NDIM

IF(IRAC.EQ.1)CALL REACT(LOUT2,NDIM,NN,NW,NQ,IDFX,PEQT,PT,IOUTP)

C --- Above subroutine calculate the reaction at the restrained


points.

C ----- INCLUDE ALL PORE-PRESSURE TERMS IN THE LIST OF FIXED D.O.F.

C ALL EXCESS PORE PRESSURE D.O.F. ARE CONSIDERED TO BE FIXED

DO 2 NI=1,NN

NQL=NQ(NI)

IF(NQL.NE.1.AND.NQL.NE.NDIM1)GO TO 2

ILC=NW(NI)+NQL-1

IDFX(ILC)=1

2 CONTINUE

C ----- CALCULATE OUT-OF-BALANCE LOADS FOR ALL FREE D.O.F.

DO 5 IK=1,NDF

IF(IDFX(IK).EQ.1) GO TO 3

PCOR(IK)=PT(IK)-PEQT(IK)

GO TO 5

3 PCOR(IK)=ZERO

5 CONTINUE

IF(IEQL.NE.1.OR.IOUTP.NE.1) RETURN

C ----- OUTPUT EQUILIBRIUM, OUT-OF-BALANCE AND APPLIED NODAL LOADS

IF(IEQOP.EQ.0)GOTO 25

IF(NDIM.EQ.3) GOTO 22

WRITE(LOUT2,900)

WRITE(LOUT2,904)

GOTO 23

22 WRITE(LOUT2,930)

WRITE(LOUT2,934)

23 CONTINUE

DO 20 JR=1,NN

J=JR

NQL=NQ(J)

IF(NQL.LE.1)GOTO 20

c IF(IEQOP.EQ.1.AND.JR.GT.NDZ)GOTO 20

N1=NW(J)

N2=N1+NDIM-1

IF(NDIM.EQ.2)WRITE(LOUT2,901)JR,(P(JJ),JJ=N1,N2),

1 (PT(JJ),JJ=N1,N2),(PEQT(JJ),JJ=N1,N2),(PCOR(JJ),JJ=N1,N2)

IF(NDIM.EQ.3)WRITE(LOUT2,931)JR,(P(JJ),JJ=N1,N2),

1 (PT(JJ),JJ=N1,N2),(PEQT(JJ),JJ=N1,N2),(PCOR(JJ),JJ=N1,N2)

20 CONTINUE

25 CALL ZEROR1(RMAX,6)

C ----- CALCULATE MAXIMUM OF APPLIED AND OUT-OF-BALANCE

C ----- LOADS IN ALL DIRECTIONS

DO 50 IK=1,NN

NQL=NQ(IK)

IF(NQL.LE.1)GOTO 50


N1=NW(IK)

N2=N1+NDIM-1

IC=0

C

DO 35 KN=N1,N2

IC=IC+1

PAR(IC)=PT(KN)

35 PAR(IC+NDIM)=PCOR(KN)

C

DO 40 IC=1,NDIM2

RV=PAR(IC)

IF(ABS(RV).LT.ASMVL)GOTO 40

IF(ABS(RV).GT.RMAX(IC))RMAX(IC)=ABS(RV)

40 CONTINUE

50 CONTINUE

C ----- OUTPUT MAXIMUM OF (1) APPLIED LOADS (2) OUT-OF-BALANCE

LOADS

C ----- IN ALL DIRECTIONS

WRITE(LOUT2,902)

C

IWARN=0

PMAXT=RMAX(1)

DO 55 ID=2,NDIM

55 IF(RMAX(ID).GT.PMAXT)PMAXT=RMAX(ID)

IF(PMAXT.LT.ASMVL) GOTO 132

DO 130 ID=1,NDIM

130 TER(ID)=100.0D0*RMAX(ID+NDIM)/PMAXT

GOTO 125

132 IWARN=1

DO 135 ID=1,NDIM

135 TER(ID)=ZERO

C

125 IF(NDIM.EQ.3) GOTO 122

WRITE(LOUT2,903)

WRITE(LOUT2,905)

WRITE(LOUT2,907)(RMAX(JQ),JQ=1,NDIM2),(TER(ID),ID=1,NDIM)

GOTO 123

122 WRITE(LOUT2,933)

WRITE(LOUT2,935)

WRITE(LOUT2,937)(RMAX(JQ),JQ=1,NDIM2),(TER(ID),ID=1,NDIM)

123 CONTINUE

IF(IWARN.EQ.1)WRITE(LOUT2,910)

C ----- ZERO PCOR IF NO CORRECTING LOADS ARE TO BE APPLIED IN NEXT

INCR

IF(ICOR.NE.0)RETURN

C


DO 140 IK=1,NDF

140 PCOR(IK)=ZERO

RETURN

900 FORMAT(//59X,19HLOADS EQUIVALENT TO/9X,

1 24HINCREMENTAL APPLIED LOAD,3X,18HTOTAL APPLIED LOAD,

1 6X,16HELEMENT STRESSES,7X,19HOUT-OF-BALANCE LOAD/

2 9X,24(1H-),3X,18(1H-),6X,16(1H-),7X,19(1H-))

901 FORMAT(1X,I5,2X,8E12.4)

902 FORMAT(//1X,17HEQUILIBRIUM CHECK/1X,17(1H-))

903 FORMAT(/3X,20HMAXIMUM APPLIED LOAD,2X,

1 24HMAXM OUT-OF-BALANCE LOAD,3X,

2 31HPERCENTAGE ERROR IN EQUILIBRIUM/

3 3X,20(1H-),2X,24(1H-),3X,31(1H-)/)

904 FORMAT(/1X,5H NODE,8X,1HX,12X,1HY,11X,1HX,11X,1HY,11X,1HX,

1 11X,1HY,11X,1HX,11X,1HY//)

905 FORMAT(8X,1HX,11X,1HY,11X,1HX,10X,1HY,15X,1HX,15X,1HY/)

907 FORMAT(1X,4E12.4,4X,F10.4,5X,F10.4)

910 FORMAT(/40H WARNING **** NO APPLIED LOADING - CHECK,

1 1X,49HWHETHER ALL BOUNDARY CONDITIONS ARE DISPLACEMENTS,

2 2X,15H(ROUTINE EQLBM))

930 FORMAT(/15X,24HINCREMENTAL APPLIED LOAD,5X,

1 38HLOADS EQUIVALENT TO TOTAL APPLIED LOAD,

1 8X,16HELEMENT STRESSES,20X,19HOUT-OF-BALANCE LOAD/

2 15X,24(1H-),5X,38(1H-),8X,16(1H-),20X,19(1H-))

931 FORMAT(1X,I5,2X,12E12.4)

933 FORMAT(/10X,20HMAXIMUM APPLIED LOAD,13X,

1 24HMAXM OUT-OF-BALANCE LOAD,11X,

2 31HPERCENTAGE ERROR IN EQUILIBRIUM/

3 10X,20(1H-),13X,24(1H-),11X,31(1H-)/)

934 FORMAT(/2X,5H

NODE,8X,1HX,11X,1HY,11X,1HZ,11X,1HX,11X,1HY,11X,1HZ,

1 11X,1HX,11X,1HY,11X,1HZ,11X,1HX,11X,1HY,11X,1HZ//)

935 FORMAT(8X,1HX,11X,1HY,11X,1HZ,11X,1HX,11X,1HY,11X,1HZ,

1 13X,1HX,11X,1HY,11X,1HZ/)

937 FORMAT(1X,6E12.4,3F12.4)

END

SUBROUTINE EQLIB(JJ,LT,NGP,INDX,NDN,NAC,XYZ,STRESS,F,NCONN,ISTGE)

C**********************************************************************

C ROUTINE TO CALCULATE FORCES EQUILIBRATING

C ELEMENTAL STRESSES

C**********************************************************************

C 1 CALLED BY RDSTRS









COMMON /JACB / XJACI(3,3),DJACB

DIMENSION XYZ(3,MNODES),STRESS(NVRS,NIP,MEL),ELCOD(3,NDMX),

1 DS(3,20),SHFN(20),CARTD(3,NDMX),B(6,NB),

2 F(3,NDMX),SLL(4),NCONN(NTPE,MEL)

CR=1.0D0

IF(NPLAX.EQ.1)CR=2.0D0*PYI

CALL ZEROR2(F,3,NDMX) !Initializing

DO 20 KN=1,NDN

NDE=NCONN(KN,JJ)

DO 20 ID=1,NDIM

20 ELCOD(ID,KN)=XYZ(ID,NDE)

DO 60 IP=1,NGP

IPA=IP+INDX

C

DO 30 IL=1,NAC

30 SLL(IL)=SL(IL,IPA)

C ----- Call FORMB2 - Forms B matrix

CALL FORMB2(JJ,R,RI,NDN,NAC,ELCOD,DS,SHFN,CARTD,B,SLL,LT,IP,ISTGE)

F9=CR*DJACB*W(IPA)

IF(NPLAX.EQ.1)F9=F9*R

C

U=STRESS(NS+1,IP,JJ)

SIGXT=STRESS(1,IP,JJ)+U

SIGYT=STRESS(2,IP,JJ)+U

SIGZT=STRESS(3,IP,JJ)+U

TXY=STRESS(4,IP,JJ)


C

TYZ=STRESS(5,IP,JJ)

TZX=STRESS(6,IP,JJ)

C

DO 50 IN=1,NDN

F(1,IN)=F(1,IN)+(CARTD(1,IN)*SIGXT+CARTD(2,IN)*TXY

1 +CARTD(3,IN)*TZX)*F9

F(2,IN)=F(2,IN)+(CARTD(2,IN)*SIGYT+CARTD(1,IN)*TXY

1 +CARTD(3,IN)*TYZ)*F9

F(3,IN)=F(3,IN)+(CARTD(3,IN)*SIGZT+CARTD(2,IN)*TYZ



50 CONTINUE

GOTO 60

C

35 DO 40 IN=1,NDN

F(1,IN)=F(1,IN)+(CARTD(1,IN)*SIGXT+SHFN(IN)*SIGZT*RI

1 +CARTD(2,IN)*TXY)*F9

40 F(2,IN)=F(2,IN)+(CARTD(2,IN)*SIGYT+CARTD(1,IN)*TXY)*F9

60 CONTINUE

RETURN

END

SUBROUTINE EQLOD(NCONN,MAT,LTYP,NW,NQ,IDFX,NP1,NP2,XYZ, P,PT,

1 PCOR,PEQT,XYFT,PCONI,PR,IEQOP,ICOR,TGRAV,IRAC,KSTGE,IEQL,IOUTP)

C----------------------------------------------------------------------

C ROUTINE TO CALCULATE EQUIVALENT NODAL LOADS FOR

C APPLIED LOADING TO CARRY OUT AN EQUILIBRIUM CHECK

C**********************************************************************

C 1 Called by INSITU



DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL),

1 NW(MNODES+1),NQ(MNODES),IDFX(MDOF),NP1(NPL),NP2(NPL) DIMENSION

XYZ(3,MNODES),P(MDOF),PT(MDOF),PCOR(MDOF),PEQT(MDOF),

1 XYFT(MDOF),PCONI(MDOF),F(3,NDMX),PR(NPR,NMT),PRES(3,LV)





COMMON /PTLOAD/PT1(MDOF),INDPT



C

CALL ZEROR1(PT,MDOF)

C ----- (1) PRESSURE LOADING ALONG ELEMENT EDGE

IF(NLED.EQ.0.AND.TGRAV.LT.ASMVL)GO TO 62

IF(NLED.EQ.0.OR.INDPT.NE.0)GO TO 32

C

DO 30 KE=1,NLED

LNE=LEDG(KE)

NE=LNE

LT=LTYP(NE)


IF(LT.GT.0)GOTO 10

IF(KSTGE.EQ.4)GOTO 30

WRITE(LOUT1,900)LNE

900 FORMAT(/1X,45H *** ERROR : IN SITU PRESSURE LOAD APPLIED TO,1X,

1 7HELEMENT,I5,2X,28HWHICH IS NOT PRESENT IN MESH,1X,

2 15H(ROUTINE EQLOD)/)

GOTO 30

10 ND1=NDE1(KE)

ND2=NDE2(KE)

C *** N2D = 2 FOR TWO DIMENSIONAL PROBLEMS

N2D=2

ICT=0

DO 20 IV=1,NPT

DO 20 ID=1,N2D

ICT=ICT+1

20 PRES(ID,IV)=PRESLD(ICT,KE)

C

C ----- Call DISTLD

CALL DISTLD(LOUT1,XYZ,PT1,NCONN,LTYP,NW,NP1,

1 NP2,PRES,LNE,ND1,ND2,0,1,1.0D0)

30 CONTINUE

INDPT=1

32 DO IDF=1,NDF

PT(IDF)=PT1(IDF)

ENDDO

C ----- (2) SELF WEIGHT LOADING

IF(TGRAV.LT.ASMVL) GO TO 62

DO 60 KL=1,NEL

LT=LTYP(KL)

IF(LT.LT.0)GO TO 60

JK=KL

NDN=LINFO(5,LT)

INDX=LINFO(12,LT)

NAC=LINFO(15,LT)

KM=MAT(KL)

C ----- FIND IF ELEMENT HAS BEEN ADDED IN THIS INCREMENT BLOCK

C ----- THEN USE LOAD RATIO FRACT ON GRAVITY LOADING

C DO 40 IM=1,NEL

C MUS=JEL(IM)

C IF(MUS.EQ.0)GO TO 42

C MPR=MREL(MUS)

C IF(KL.EQ.MPR)GO TO 44

C 40 CONTINUE

FA=1.0D0


GO TO 45

C 44 FA=FRACT

45 DENS=PR(8,KM)*TGRAV*FA

C

C ----- Call Self

CALL

SELF(LOUT1,KL,NDN,NAC,XYZ,F,NCONN,MAT,LT,INDX,DENS,JK,KSTGE)

C

DO 55 KK=1,NDN

NCOR=NCONN(KK,KL)

KKK=NW(NCOR)-1

C

DO 55 ID=1,NDIM

55 PT(KKK+ID)=PT(KKK+ID)+F(ID,KK)

60 CONTINUE

62 CONTINUE

C ----- ADD CONTRIBUTIONS FROM POINT LOADS

DO 70 J=1,NDF

70 PT(J)=PT(J)+XYFT(J)+PCONI(J)

C ----- CONVERT PT TO COUNT FOR SKEW BOUNDARIES

IF(NSKEW.GT.0) CALL ROTBC(PT,NW,NDIM,NSKEW,1) ! Nskew=8 for me

C --- This subroutine turn back the cal. displ. to original coord.

system

C --- for skew boundary nodes.

C ----- FIND DOF WHICH ARE RESTRAINED

CALL RESTRN(NDIM,NW,IDFX)

C ----- EQUILIBRIUM CHECK

CALL EQLBM(LOUT2,NDIM,IEQL,NN,NDF,NW,NQ,IDFX,

1 P,PT,PCOR,PEQT,IEQOP,ICOR,IRAC,IOUTP)

RETURN

END

C**********************************************************************

SUBROUTINE FIXX2(LINP,LOUT1,NDIM,NCONN,LTYP,NP1,NP2,NFX,NFXNOD)

C----------------------------------------------------------------------

C ROUTINE TO MAINTAIN A LIST OF NODAL FIXITIES. INTERPRETS

C FIXITIES ALONG ELEMENT EDGES INTO NODAL FIXITIES

C----------------------------------------------------------------------

C 1 Called by INSITU

C 2 Called by TOTSOL


c NFX=NFXEL IN THE INPUT HEADER 11



DIMENSION NCONN(NTPE,MEL),LTYP(MEL)

DIMENSION NP1(NPL),NP2(NPL),IND(5),FV(5),V(LV)

! LV=3 from PARM.FOR



C

NFZ=200

NDIM1=NDIM+1

IF(NFX.EQ.0)RETURN

CALL ZEROI1(IND,5)

CALL ZEROR1(FV,5)

WRITE(LOUT1,900)

C----------------------------------------------------------------------

C LOOP ON ALL FIXED EDGES I.E. EDGES WITH PRESCRIBED

C DISPLACEMENT/EXCESS PORE PRESSURES

C----------------------------------------------------------------------

C Read in fixity data

C WRITE(LOUT1,101)NFX

C 101 FORMAT(//I5)

IF(NFX.EQ.0) GO TO 201

DO 200 JX=1,NFX !NFX=NFXEL

READ(LINP,*)ML,ND1,ND2,IVAR,IFX,V ! V is an array of size (3).

WRITE(LOUT1,902)JX,ML,ND1,ND2,IVAR,IFX,V

NE=ML

LI1=ND1

LI2=ND2

LT=LTYP(NE)

LT=IABS(LT)

NVN=LINFO(2,LT)

NEDG=LINFO(3,LT)

NDSD=LINFO(7,LT)

IF(IVAR.EQ.NDIM1)NDSD=LINFO(8,LT)

NTSD=NDSD+2

INDED=LINFO(14,LT)

C

DO 20 K1=1,NEDG

J1=NP1(K1+INDED)

J2=NP2(K1+INDED)

I1=NCONN(J1,NE)

I2=NCONN(J2,NE)




20 CONTINUE

WRITE(LOUT1,903)JX,ML,ND1,ND2

GOTO 200

C----------------------------------------------------------------------

C ALIGN END NODES OF EDGE IN CORRECT SEQUENCE. (ANTICLOCKWSIE

C OREDER ABOUT ELEMENT CENTRE)

C----------------------------------------------------------------------

21 LIT=LI1

LI1=LI2

LI2=LIT

NT=ND1

ND1=ND2

ND2=NT

C

DO 24 J=1,NTSD

JBACK=NTSD+1-J

24 FV(J)=V(JBACK)

GO TO 35

C

25 DO 30 J=1,NTSD

30 FV(J)=V(J)

C----------------------------------------------------------------------

C IND - LIST OF NODES ALONG EDGE. START WITH END NODES

C----------------------------------------------------------------------

35 IND(1)=LI1

IND(NTSD)=LI2

IF(NTSD.EQ.2)GO TO 42

LC1=NVN+(K1-1)*NDSD

IF(IVAR.EQ.NDIM1)LC1=LINFO(5,LT)+(K1-1)*NDSD

C----------------------------------------------------------------------

C INTERMEDIATE NODES (IF NTSD=2 NO INTERMEDIATE NODES)

C----------------------------------------------------------------------

DO 40 JP=1,NDSD

ILC=LC1+JP

40 IND(JP+1)=NCONN(ILC,NE)

C----------------------------------------------------------------------

C LOOP ON ALL NODES ALONG EDGE

C----------------------------------------------------------------------

42 DO 100 KND=1,NTSD

I=IND(KND)

IF(NF.EQ.0)GO TO 58

C

DO 50 J=1,NF

IF(I.EQ.MF(J))GO TO 55

50 CONTINUE

C

GO TO 58


C----------------------------------------------------------------------

C UPDATE EXISTING VALUES

C----------------------------------------------------------------------

55 JF=J

GO TO 60

C

58 NF=NF+1

IF(NF.LE.NFZ)GO TO 59

WRITE(LOUT1,904)

STOP

59 JF=NF

60 MF(JF)=I

NFIX(IVAR,JF)=IFX

DXYT(IVAR,JF)=FV(KND)

100 CONTINUE

200 CONTINUE

201 IF(NFXNOD.EQ.0) RETURN ! NFXNOD=0 for me.

WRITE(LOUT1,905)

C ----- This subroutine is not used now. 1/26/98

DO 202 INODE=1,NFXNOD

READ(LINP,*)NODE,IVAR,IFX,VALUE

WRITE(LOUT1,906)NODE,IVAR,IFX,VALUE

DO J=1,NF

KNODE=MF(J)

IF(KNODE.EQ.NODE) THEN

NFIX(IVAR,J)=IFX

DXYT(IVAR,J)=VALUE

GO TO 202

ENDIF

ENDDO

NF=NF+1

MF(NF)=NODE

NFIX(IVAR,NF)=IFX

DXYT(IVAR,NF)=VALUE

202 CONTINUE

RETURN

900 FORMAT(/1X,4HSIDE,4X,7HELEMENT,3X,5HNODE1,3X,5HNODE2,

1 3X,3HDOF,4X,11HFIXITY CODE,8X,4HVAL1,6X,4HVAL2,6X,4HVAL3,

2 6X,4HVAL4,6X,4HVAL5/)

902 FORMAT(1X,I3,4X,I5,5X,I4,4X,I4,5X,I2,12X,I3,3X,5F9.4)

903 FORMAT(/13H **** ERROR :,I5,19H TH FIXITY. ELEMENT,

1 I5,25H DOES NOT CONTAIN NODES :,2I5,2X,14H(ROUTINE FIXX))

904 FORMAT(/42H INCREASE SIZE OF ARRAYS MF, NFIX AND DXYT/

1 1X,34HIN COMMON BLOCK FIX (ROUTINE FIXX))


905 FORMAT(/3X,4HNODE,3X,3HDOF,6X,11HFIXITY CODE,6X,3HVAL/)

906 FORMAT(1X,I4,5X,I2,10X,I3,5X,F12.6)

END

C**********************************************************************

SUBROUTINE FIXX3(LINP,LOUT1,NDIM,NCONN,LTYP,NFX,NFXNOD)

C**********************************************************************

C ROUTINE TO MAINTAIN A LIST OF NODAL FIXITIES. INTERPRETS

C FIXITIES ALONG (3-D) ELEMENT FACE INTO NODAL FIXITIES.

C AT PRESENT TO CATER FOR THE 3-D BRICK ELEMENTS ONLY.

C----------------------------------------------------------------------




DIMENSION IND(8),FV(8)

DIMENSION KX(48),NDU(8),NDP(8),NXC(4),NXM(4),KNL(8)



C----------------------------------------------------------------------

C ARRAY KX(48) GIVES THE INDEX TO ARRAY NCONN FOR THE FOUR

C CORNER NODES OF EACH FACE OF THE ELEMENT FOLLOWED BY THE

C MIDSIDE NODES.

C----------------------------------------------------------------------

DATA KX(1),KX(2),KX(3),KX(4),KX(5),KX(6),KX(7),KX(8),KX(9),

1 KX(10),KX(11),KX(12),KX(13),KX(14),KX(15),KX(16),KX(17),




1 KX(42),KX(43),KX(44),KX(45),KX(46),KX(47),KX(48)/

1 1,2,3,4,9,10,11,12,6,5,8,7,13,16,15,14,1,5,6,2,17,13,18,9,

1 2,6,7,3,18,14,19,10,4,3,7,8,11,19,15,20,5,1,4,8,17,12,20,16/

C

DO 5 IU=1,8

KNL(IU)=0

NDU(IU)=0

NDP(IU)=0

5 CONTINUE

C

NFZ=200

NDIM1=NDIM+1

IF(NFX.EQ.0) GO TO 201

WRITE(LOUT1,900)

C----------------------------------------------------------------------

C IF NEW 3-D ELEMENT TYPES ARE ADDED THEN NC, NFCD

C AND LVL (=NFCD) SHOULD BE OBTAINED FROM ARRAY LINFO


C IN ORDER TO MAKE THE ROUTINE GENERAL.

C NC - NUMBER OF VERTEX NODES ON ELEMENT FACE

C NFCD- TOTAL NUMBER OF DISPLACEMENT NODES ON FACE

C----------------------------------------------------------------------

NC=4

NFCD=8

C----------------------------------------------------------------------

C LOOP ON ALL FACES WITH FIXITIES I.E FACES WITH PRESCRIBED

C DISPLACEMENT/EXCESS PORE PRESSURES.

C----------------------------------------------------------------------

LVL=NFCD

DO 200 JX=1,NFX

READ(LINP,*)ML,(NDU(J),J=1,NC),IVAR,IFX,(FV(K),K=1,LVL)

WRITE(LOUT1,910)JX,ML,(NDU(J),J=1,NC),IVAR,IFX,(FV(K),K=1,LVL)

NE=ML

C

DO 30 IN=1,NC

ND=NDU(IN)

30 NDP(IN)=ND

C

LT=LTYP(NE)

LT=IABS(LT)

NFAC=LINFO(4,LT)

C----------------------------------------------------------------------

C LOOP ON ALL FACES OF ELEMENT TO IDENTIFY THE FACES OF THE

C ELEMENT WITH PRESCRIBED VALUES

C----------------------------------------------------------------------

DO 90 IFAC=1,NFAC

ISX=NFCD*(IFAC-1)

C GET INDEXES OF NODES TO NCONN

DO 40 IN=1,NC

NXC(IN)=KX(ISX+IN)

C IF NOT PORE-PRESSURE D.O.F, ADDITIONAL NODES ALONG

C EDGE ARE PRESENT

IF(IVAR.NE.NDIM1)NXM(IN)=KX(ISX+NC+IN)

40 CONTINUE

C GET VERTEX NODES OF FACE FROM NCONN

DO 50 IN=1,NC

IP=NXC(IN)

50 KNL(IN)=NCONN(IP,NE)

C----------------------------------------------------------------------

C LOOP ON ALL STARTING NODES

C TRY TO MATCH THE NODES SPECIFIED BY THE USER

C WITH THE NODES OF EACH FACE. EACH NODE IN TURN

C IS CONSIDERED AS A STARTING NODE.

C----------------------------------------------------------------------

DO 80 IS=1,NC

ISV=IS


C TRY MATCHING THE NODES

DO 60 IN=1,NC

IF(NDP(IN).NE.KNL(IN))GOTO 65

60 CONTINUE

GOTO 95

C START WITH NEXT NODE. THE SEQUENCES OF THE NODES ARE STILL THE

SAME

65 CALL ALTER(LOUT1,KNL,NC)

80 CONTINUE

90 CONTINUE

C FACE NOT FOUND

WRITE(LOUT1,930)JX,ML,(NDU(J),J=1,NC)

C

GOTO 200

C

95 IF(ISV.EQ.1)GOTO 105

IS1=ISV-1

C SORT THE INDEXES TO MATCH WITH NODE SEQUENCE KNL

DO 100 IM=1,IS1

CALL ALTER(LOUT1,NXC,NC)

IF(IVAR.NE.NDIM1)CALL ALTER(LOUT1,NXM,NC)

100 CONTINUE

C IF PORE-PRESSURE FIXITY

105 CONTINUE

IF(IVAR.NE.NDIM1)GOTO 125

C

DO 120 IL=1,NC

IP=NXC(IL)

120 IND(IL)=NCONN(IP,NE)

NSDN=NC

GOTO 132

C IF DISPLACEMENT FIXITY

125 DO 130 IL=1,NC

IM=NXC(IL)

IN=NXM(IL)

IND(2*IL-1)=NCONN(IM,NE)

130 IND(2*IL)=NCONN(IN,NE)

NSDN=NFCD

132 CONTINUE

C----------------------------------------------------------------------

C LOOP ON ALL NODES ALONG FACE

C----------------------------------------------------------------------

142 DO 180 KND=1,NSDN

I=IND(KND)

IF(NF.EQ.0)GOTO 158

C

DO 150 J=1,NF

IF(I.EQ.MF(J))GOTO 155


150 CONTINUE

C

GOTO 158

C----------------------------------------------------------------------

C UPDATE EXISTING VALUES

C----------------------------------------------------------------------

155 JF=J

GOTO 160

C

158 NF=NF+1

IF(NF.LE.NFZ)GOTO 159

WRITE(LOUT1,940)

STOP

159 JF=NF

160 MF(JF)=I

NFIX(IVAR,JF)=IFX

DXYT(IVAR,JF)=FV(KND)

180 CONTINUE

200 CONTINUE

C

201 IF(NFXNOD.EQ.0) RETURN

WRITE(LOUT1,905)

DO 202 INODE=1,NFXNOD

READ(LINP,*)NODE,IVAR,IFX,VALUE

WRITE(LOUT1,906)NODE,IVAR,IFX,VALUE

DO J=1,NF

KNODE=MF(J)


NFIX(IVAR,J)=IFX

DXYT(IVAR,J)=VALUE

GO TO 202

ENDIF

ENDDO

NF=NF+1

MF(NF)=NODE

NFIX(IVAR,NF)=IFX

DXYT(IVAR,NF)=VALUE

202 CONTINUE

RETURN

C

900 FORMAT(/19X,16H......NODES.....,8X,6HFIXITY//

1 1X,4HFACE,4X,7HELEMENT,3X,16H1 2 3 4,

2 3X,3HDOF,3X,4HCODE,7X,4HVAL1,5X,4HVAL2,5X,4HVAL3,

3 5X,4HVAL4,5X,4HVAL5,5X,4HVAL6,5X,4HVAL7,5X,4HVAL8//)

910 FORMAT(1X,I3,4X,I5,3X,I4,1X,I4,1X,I4,1X,I4,4X,I2,3X,I3,3X,8F8.4)

930 FORMAT(/1X,20H***** ERROR : FIXITY,I4,2X,8HIN LIST.,3X,

1 7HELEMENT,I5,2X,29HDOES NOT HAVE FACE WITH NODES,4I5)


940 FORMAT(/42H INCREASE SIZE OF ARRAYS MF, NFIX AND DXYT/

1 1X,35HIN COMMON BLOCK FIX (ROUTINE FIXX3))

905 FORMAT(/1X,4HNODE,3X,3HDOF,3X,11HFIXITY CODE,3X,3HVAL)

906 FORMAT(1X,I4,5X,I2,10X,I3,5X,F10.3)

END

C**********************************************************************

SUBROUTINE ALTER(LOUT1,IM,N)

C**********************************************************************

C ROUTINE TO SHIFT ARRAY FORWARD BY ONE PLACE

C----------------------------------------------------------------------


DIMENSION IM(N)

C

IF(N.LE.1)GOTO 100

NM1=N-1

IMT=IM(1)

C

DO 10 K=1,NM1

10 IM(K)=IM(K+1)

IM(N)=IMT

RETURN

100 WRITE(LOUT1,900)N

900 FORMAT(/1X,45HERROR * ARRAY CONTAINS LESS THAN OR EQUAL TO,I5,2X,

1 40HMEMBERS (ROUTINE ALTER) CALLED BY FIXX3)

RETURN

END

SUBROUTINE FORMB2(J,R,RI,NDN,NAC,ELCOD,DS,SHFN,

1 CARTD,B,SLL,LT,IP,ISTGE)

C**********************************************************************

C FORMS B MATRIX FROM AREA/LOCAL COORDS SLL(NAC)

C IN ELEMENT J FOR INTEGRATION POINT IP

C**********************************************************************

C 1 CALLED BY EQLIB



DIMENSION ELCOD(3,NDMX),DS(3,20),SHFN(20),

1 CARTD(3,NDMX),B(6,NB),SLL(4),XJACM(3,3)







C ----- INITIALISE SHAPE FUNCTION AND DERIVATIVES (LOCAL COORDS)

CALL ZEROR2(DS,3,20)

CALL ZEROR1(SHFN,20)

CALL ZEROR2(B,6,NB)

C

CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,2,J)

CALL ZEROR2(XJACM,3,3)

C

NDN2=2*NDN

C

DO 15 IDIM=1,NDIM

DO 15 JDIM=1,NDIM

SUM=ZERO

C

DO 12 IN=1,NDN

12 SUM=SUM+DS(IDIM,IN)*ELCOD(JDIM,IN)

15 XJACM(IDIM,JDIM)=SUM

C ----- Call DETMIN

CALL DETMIN(LOUT1,NDIM,XJACM,XJACI,DJACB,J,IP,ISTGE)

C ----- CALCULATE RADIUS FOR AXI-SYM B MATRIX

R=ZERO

RI=ZERO


C

DO 25 IN=1,NDN

25 R=R+ELCOD(1,IN)*SHFN(IN)

RI=-1.0D0/R

C

28 DO 35 IN=1,NDN

DO 35 ID=1,NDIM

SUM=ZERO

C

DO 30 JD=1,NDIM

30 SUM=SUM-DS(JD,IN)*XJACI(ID,JD)

35 CARTD(ID,IN)=SUM

C

IF(NDIM.NE.2)GOTO 52

C ----- 2 - D ELEMENT

DO 50 IN=1,NDN

B(1,IN)=CARTD(1,IN)

B(2,NDN+IN)=CARTD(2,IN)


B(3,IN)=SHFN(IN)*RI

45 B(4,NDN+IN)=B(1,IN)

50 B(4,IN)=B(2,NDN+IN)


C

52 IF(NDIM.NE.3)GOTO 62

C ----- 3 - D ELEMENT

DO 60 IN=1,NDN

B(1,IN)=CARTD(1,IN)


B(3,NDN2+IN)=CARTD(3,IN)

B(4,IN)=CARTD(2,IN)




B(6,IN)=CARTD(3,IN)


60 CONTINUE

C

62 CONTINUE

RETURN

END

SUBROUTINE INSITU(XYZ,STRESS,P,PT,PCOR,PEQT,XYFT,PCONI,

3 NCONN,MAT,LTYP,NW,IDFX,NP1,NP2,PR,NTY)

C================================================================

=======

C SETUP INSITU STRESSES AND CHECK FOR EQUILIBRIUM

C================================================================

=======

C 1 CALLED BY RDPROP










COMMON /ANLYS/ TTIME,DTIMEI,TGRAVI,DGRAVI,FRACT,FRACLD,


+ NOIB ,JS ,JINCB,NLOD ,NLDS


DIMENSION XYZ(3,MNODES),STRESS(NVRS,NIP,MEL),P(MDOF),

1 PT(MDOF),PCOR(MDOF),PEQT(MDOF),XYFT(MDOF),PCONI(MDOF)

DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL),

1 NW(MNODES+1),NQ(MNODES),IDFX(MDOF),NP1(NPL),NP2(NPL)

DIMENSION PR(NPR,NMT),PDISLD(3,LV),PRES(3,LV),NTY(NMT),TEMP(3)

C ----- CODE TO INDICATE STAGE OF THE ANALYSIS

KSTGE=1

C ----- INITIALISE PRESSURE LOADS

NDIM1=NDIM+1

CALL ZEROR1(PCONI,MDOF)

CALL ZEROR1(TEMP,3)

CALL ZEROR2(PRESLD,10,100)

CALL ZEROI1(LEDG,100)

CALL ZEROI1(NDE1,100)

CALL ZEROI1(NDE2,100)

CALL ZEROI1(MF,200)

CALL ZEROI2(NFIX,4,200)

CALL ZEROR2(DXYT,4,200)

C ----- SET UP IN-SITU STRESS SYSTEM


READ(LINP,*)KT,NI

WRITE(LOUT1,926)KT,NI

IF(NI.EQ.0)NI=1 ! If NI = 0 use a value of 1 to avoid array

size of 0.

IF(NI.LE.100)GOTO 40

WRITE(LOUT1,910)NI

101 FORMAT(A80)

910 FORMAT(/1X,’INCREASE SIZE OF ARRAYS NLI AND NHI TO’,I6,2X,

1 ’IN ROUTINE INSITU’)

STOP

40 CONTINUE

C ----- Read in the initial stress data

CALL RDSTRS(KT,XYZ,STRESS,PEQT,NCONN,MAT,LTYP,NW,PR,NTY,NI)

C INITIALISE FIXED LOADS, TOTAL POINT LOADS AND TOTAL

DISPLACEMENTS

C NF - NUMBER OF FIXITIES

c ----- Initializing

NF=0

CALL ZEROR1(PCOR,MDOF)

CALL ZEROR1(XYFT,MDOF)

CALL ZEROR1(P,MDOF)


C ----- READ LOADS IN EQUILIBRIUM WITH IN-SITU STRESSES

NLED=0

TGRAVI=ZERO

IF(KT.EQ.0)GO TO 62

C


READ(LINP,*)NLODI,NLDNOD,NFXEL,NFXNOD,TGRAVI

WRITE(LOUT1,952)NLODI,NLDNOD,NFXEL,NFXNOD,TGRAVI

C

IF(NLODI.EQ.0)GO TO 52

WRITE(LOUT1,960)

C


DO 50 KL=1,NLODI

READ(LINP,*)LNE,ND1,ND2,((PDISLD(ID,IV),ID=1,2),IV=1,NPT)

WRITE(LOUT1,964)LNE,ND1,ND2,((PDISLD(ID,IV),ID=1,2),IV=1,NPT)

C

DO 100 IV=1,NPT

DO 100 ID=1,NDIM

IDR=NDIM+1-ID

100 PRES(ID,IV)=PDISLD(IDR,IV)

C

DO 110 IV=1,NPT

DO 110 ID=1,NDIM

110 PDISLD(ID,IV)=PRES(ID,IV)

C

C ----- Call EDGLD ! Allign nodes along the loaded edge

CALL EDGLD(LOUT1,NDIM,NCONN,LTYP,LNE,ND1,ND2,NP1,NP2,PDISLD,

1 PRES,KL,NPT,1,MXLD)

50 CONTINUE

52 IF(NLDNOD.EQ.0) GO TO 53

C ----- Read point load data

WRITE(LOUT1,965)


DO KL=1,NLDNOD

READ(LINP,*)NODE,(TEMP(IDIR),IDIR=1,NDIM)

WRITE(LOUT1,966)NODE,(TEMP(IDIR),IDIR=1,NDIM)

DO IDIR=1,NDIM

N1=NW(NODE)-1

NID=N1+IDIR

PCONI(NID)=TEMP(IDIR)

ENDDO

ENDDO

C

53 IF(NFXEL.EQ.0.AND.NFXNOD.EQ.0)GO TO 62

C ----- IN-SITU BOUNDARY CONDITIONS


WRITE(LOUT1,930)

C READ(LINP,101)HEADER14

IF(NDIM.EQ.2) THEN


CALL FIXX2(LINP,LOUT1,NDIM,NCONN,LTYP,NP1,NP2,NFXEL,

1 NFXNOD) !Interpret fixities along element edges into nodal

fixities.

ENDIF

IF(NDIM.EQ.3) THEN


CALL FIXX3(LINP,LOUT1,NDIM,NCONN,LTYP,NFXEL,NFXNOD)

ENDIF

C ----- Call Make NZ

CALL MAKENZ(NEL,NN,NCONN,LTYP,NQ,INXL)

TTGRV=1.0D0

C ----- Call EQLOD ! Convert the applied load to equal nodal force.

CALL EQLOD(NCONN,MAT,LTYP,NW,NQ,IDFX,NP1,NP2,XYZ,P,

1PT,PCOR,PEQT,XYFT,PCONI,PR,2,0,TTGRV,IRAC,KSTGE,1,1)

C

62 RETURN

907 FORMAT(//1X,38HLIST OF REMOVED ELEMENTS TO FORM,

1 14H PRIMARY MESH/1X,52(1H-)/)

920 FORMAT(20I6/)

926 FORMAT(//10X,30HIN-SITU STRESS OPTION........=,I10

1 /10X,30HNUMBER OF IN-SITU NODES......=,I10/)

930 FORMAT(/1X,27HIN-SITU BOUNDARY CONDITIONS/1X,27(1H-)/)

952 FORMAT(/

1 10X,46HNUMBER OF EDGES WITH PRESSURE LOAD...........=,I5/

2 10X,46HNUMBER OF POINT LOAD NODES...................=,I5/

3 10X,46HNUMBER OF EDGES RESTRAINED...................=,I5/

4 10X,46HNUMBER OF INDIVIDUAL RESTRAINED NODES........=,I5/

5 10X,46HIN-SITU GRAVITY ACCELERATION FIELD...........=,F8.1,2X,

6 1HG//)

960 FORMAT(/1X,38HSPECIFIED NODAL VALUES OF SHEAR/NORMAL,

1 19H STRESSES (IN-SITU)/1X,57(1H-)/1X,4HELEM,

2 1X,4HNDE1,2X,4HNDE2,2X,4HSHR1,8X,4HNOR1,8X,4HSHR2,8X,4HNOR2,

3 8X,4HSHR3,8X,4HNOR3,8X,4HSHR4,8X,4HNOR4,8X,4HSHR5,8X,4HNOR5/)

964 FORMAT(1X,3I4,10E12.4)

965 FORMAT(/5X,’SPECIFIED POINT LOAD VALUES’,/5X,27(1H-),/1X,’NODE’,

1 5X,’X’,6X,’Y’,6X,’Z’/)

966 FORMAT(1X,I4,3E12.4)

END


SUBROUTINE INSTRS(LOUT1,NEL,NDIM,NS,XYZ,STRESS,NCONN,MAT,LTYP,NTY)

C**********************************************************************

C ROUTINE TO PRINT OUT INITIAL IN-SITU STRESSES

C**********************************************************************

C 1 Called by RDSTRS



DIMENSION XYZ(3,MNODES),STRESS(NVRS,NIP,MEL)


DIMENSION ELCOD(3,NDMX),DS(3,20),SHFN(20),TEMP(6),

1 CIP(3),SLL(4),NTY(NMT)




C

CALL ZEROR1(TEMP,6) ! Initializing

NS1=NS+1

WRITE(LOUT1,900)

900 FORMAT(/1X,34HINTEGRATION POINT IN-SITU STRESSES/

1 1X,34(1H-)/)

IF(NDIM.EQ.2)WRITE(LOUT1,901)


C

DO 60 MR=1,NEL

IF(MR.EQ.0)GO TO 60

J=MR

LT=LTYP(J)

IF(LTYP(J).LT.0)GO TO 60

NDN=LINFO(5,LT)

NGP=LINFO(11,LT)

INDX=LINFO(12,LT)

NAC=LINFO(15,LT)

KM=MAT(J)

KGO=NTY(KM)

GO TO(11,11,12,60,60),KGO

WRITE(LOUT1,910)MR,KGO

GOTO 60

11 ICAM=0

GO TO 14

12 ICAM=1

14 CONTINUE

C

DO 18 KN=1,NDN

NDE=NCONN(KN,J)

DO 18 ID=1,NDIM



C

DO 40 IP=1,NGP

IPA=IP+INDX

C

DO 25 IL=1,NAC


C ----- Call shape function

CALL SHAPE (LOUT1,SLL,NAC,DS,SHFN,LT,1,MR)

C

DO 35 ID=1,NDIM

SUM=ZERO

DO 30 I=1,NDN

30 SUM=SUM+SHFN(I)*ELCOD(ID,I)

35 CIP(ID)=SUM

C

DO IS=1,NS


ENDDO

IF(ICAM.NE.1)GO TO 38

EI=STRESS(NS+2,IP,J)

PCI=STRESS(NS+3,IP,J)

PE=(STRESS(1,IP,J)+STRESS(2,IP,J)+STRESS(3,IP,J))/3.0D0

QE=Q(TEMP,NS,NDIM)

IF(NDIM.EQ.2)WRITE(LOUT1,903)J,IP,(CIP(ID),ID=1,NDIM),

1 (STRESS(IK,IP,J),IK=1,NS1),PE,QE,PCI,EI

IF(NDIM.EQ.3)WRITE(LOUT1,933)J,IP,(CIP(ID),ID=1,NDIM),

1 (STRESS(IK,IP,J),IK=1,NS1),PE,QE,PCI,EI

GO TO 40

38 WRITE(LOUT1,903)J,IP,(CIP(ID),ID=1,NDIM),

1 (STRESS(IK,IP,J),IK=1,NS1)

40 CONTINUE

60 CONTINUE

RETURN

901 FORMAT(1X,7H ELM-IP,4X,1HX,6X,1HY,9X,2HSX,10X,

1 2HSY,10X,2HSZ,10X,3HTXY,9X,1HU,10X,2HPE,

2 11X,1HQ,10X,2HPC,7X,4HVOID/)

903 FORMAT(1X,I3,I2,2F8.4,8E12.4,F7.4)

910 FORMAT(1X,7HELEMENT,I5,2X,27HIS OF UNKNOWN MATERIAL TYPE,I5,

1 2X,16H(ROUTINE INSTRS))

931 FORMAT(1X,7H ELM-IP,4X,1HX,7X,1HY,7X,1HZ,8X,2HSX,10X,

1 2HSY,10X,2HSZ,9X,3HTXY,9X,3HTYZ,9X,3HTZX,11X,1HU,10X,2HPE,

2 11X,1HQ,10X,2HPC,6X,4HVOID/)

933 FORMAT(1X,I3,I2,3F8.4,10E12.4,F7.4)

END


SUBROUTINE RDSTRS(KT,XYZ,STRESS,PEQT,NCONN,MAT,LTYP,NW,PR,NTY,NI)

C**********************************************************************

C SET UP IN-SITU STRESSES

C**********************************************************************



DIMENSION XYZ(3,MNODES),STRESS(NVRS,NIP,MEL),PEQT(MDOF)

DIMENSION ELCOD(3,NDMX),DS(3,20),SHFN(20),

1 FI(3,NDMX)


DIMENSION NW(MNODES+1)

DIMENSION YI(100),VAR(NVRS,100),NLI(100),NHI(100)

DIMENSION CIP(3),SLL(4),PR(NPR,NMT),NTY(NMT)







C ----- ISTGE - CODE TO INDICATE STAGE OF THE ANALYSIS

ISTGE=1

C ----- INITIALISE STRESS - INTEGRATION POINT VARIABLES

CALL ZEROR3(STRESS,NVRS,NIP,MEL)

C ----- INITIALISE PEQT - CONTRIBUTION OF FORCES DUE TO ELEMENT

IN-SITU

C STRESSES

CALL ZEROR1(PEQT,MDOF)

IF(KT.EQ.0) WRITE(LOUT1,904)

IF(KT-1) 200,8,82

8 IF(NDIM.EQ.2)WRITE(LOUT1,906)


C ----- Read in Node Y coordinate and initial stress data


101 FORMAT(A80)

DO 10 J=1,NI

NDAT=NS+3 ! NS=4 for 2 D problems

READ(LINP,*)YI(J),(VAR(JJ,J),JJ=1,NDAT)

IF(NDIM.EQ.2)WRITE(LOUT1,910) J,YI(J),(VAR(JJ,J),JJ=1,NDAT)

IF(NDIM.EQ.3)WRITE(LOUT1,930) J,YI(J),(VAR(JJ,J),JJ=1,NDAT)

10 CONTINUE

C

MI=NI-1


DO 20 IN=1,MI

N1=IN

N2=IN+1

Y1=YI(N1)

Y2=YI(N2)

C

IF(Y1.LT.Y2) THEN

NMIN=N1

NMAX=N2

ELSE

NMAX=N1

NMIN=N2

END IF

NLI(IN)=NMIN

NHI(IN)=NMAX

20 CONTINUE

C ----- LOOP ON ALL GEOMETRY MESH ELEMENTS

DO 80 J=1,NEL

LT=LTYP(J)

IF(LT.LT.0)GOTO 80

CC LT=IABS(LT)

JUS=J

GO TO(80,22,22,22,22,22,22,22,22,22,22,80,80,80,80),LT

WRITE(LOUT1,915)JUS,LT

GOTO 80

22 KM=MAT(J)

NGP=LINFO(11,LT)

NDN=LINFO(5,LT)

INDX=LINFO(12,LT)

NAC=LINFO(15,LT)

C

DO 30 KN=1,NDN

NDE=NCONN(KN,J)

DO 30 ID=1,NDIM

ELCOD(ID,KN) = XYZ(ID,NDE)

30 CONTINUE

C ----- LOOP ON ALL INTEGRATION POINTS

DO 60 IP=1,NGP

C ----- CALCULATE INTEGRATION POINT COORDINATES

IPA=IP+INDX

DO 35 IL=1,NAC


C ----- Call Shape functin

CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,1,JUS)


C

DO 40 ID=1,NDIM

SUM=ZERO

DO 38 I=1,NDN

38 SUM=SUM+SHFN(I)*ELCOD(ID,I)

40 CIP(ID)=SUM

YY=CIP(2)

C ----- SEARCH FOR RELEVANT IN-SITU LAYER

DO 45 JJJ=1,MI

NSM=NLI(JJJ)

NLA=NHI(JJJ)

YMIN=YI(NSM)

YMAX=YI(NLA)

C

IF(YY.LT.YMIN.OR.YY.GT.YMAX)GO TO 45

GO TO 48

C

45 CONTINUE

WRITE(LOUT1,950)JUS,IP

GO TO 60

C ----- DIRECT INTERPOLATION FROM IN-SITU MESH NODES

48 DY=YI(JJJ)-YI(JJJ+1)

YR=(YY-YMIN)/DY

C

DO 50 I=1,NVRS

50 STRESS(I,IP,J)=VAR(I,NSM)+(VAR(I,JJJ)-VAR(I,JJJ+1))*YR

KGO=NTY(KM)

GO TO(60,60,52,60,60),KGO

C ----- CALCULATE MEAN EFFECTIVE STRESS P’

52 P=(STRESS(1,IP,J)+STRESS(2,IP,J)+STRESS(3,IP,J))/3.0D0

C CALCULATE PC’ (PC) AND CRITICAL STATE VALUE OF P’(PU)

PC=STRESS(NS+3,IP,J)

IF(KGO.NE.3)GO TO 54

PU=0.50D0*PC

GO TO 55

54 PU=PC/EXP(1.0d0)

C ----- CALCULATE VOID RATIO

C 55 STRESS(NS+2,IP,J)=PR(3,KM)-PR(1,KM)*ALOG(P)-

C 1(PR(2,KM)-PR(1,KM))*ALOG(PU)

55 STRESS(NS+2,IP,J)=PR(3,KM)-PR(1,KM)*LOG(P)-

1(PR(2,KM)-PR(1,KM))*LOG(PU)

60 CONTINUE

80 CONTINUE


GOTO 92

C ----- DIRECT SPECIFICATION OF IN-SITU STRESSESS

82 IF(KT.NE.2)GO TO 92 ! KT=1 for me -- skip

WRITE(LOUT1,955)

C *** READ FOR ALL INTEGRATION POINTS

DO 90 IM=1,NEL

READ(LINP,*)MUS

C IL=MREL(MUS)

IL=MUS

LT=LTYP(IL)

NGP=LINFO(11,LT)

C

DO 85 IP=1,NGP

READ(LINP,*)(STRESS(JJJ,IP,IL),JJJ=1,NVRS)

85 WRITE(LOUT1,960)(STRESS(JJJ,IP,IL),JJJ=1,NVRS)

90 CONTINUE

C ----- CALCULATE EQUILIBRIUM LOADS FOR INSITU STRESSES

C ----- ASSEMBLE ELEMENT CONTRIBUTION (FI) INTO PEQT

92 CR=1.0D0


DO 100 J=1,NEL

LT=LTYP(J)

IF(LT.LE.0)GO TO 100

NDN=LINFO(5,LT)

NGP=LINFO(11,LT)

INDX=LINFO(12,LT)

NAC=LINFO(15,LT)

C ----- Call EQLIB (Calculate equilibrium loads for in situ

stresses)

CALL EQLIB(J,LT,NGP,INDX,NDN,NAC,XYZ,STRESS,FI,NCONN,ISTGE)

C ----- SLOT EQUILIBRIUM LOADS INTO PEQT

DO IK=1,NDN

NCOR=NCONN(IK,J)

N1=NW(NCOR)-1

C

DO 95 ID=1,NDIM

NID=N1+ID

95 PEQT(NID)=PEQT(NID)+FI(ID,IK)

END DO


100 CONTINUE

C ----- COUNT FOR SKEW BOUNDARIES

C ----- Turn back the displacement to the original coordinate.

(Global to local)

IF(NSKEW.GT.0) CALL ROTBC(PEQT,NW,NDIM,NSKEW,1) !NSKEW>0 for

my case

C ----- PRINT OUT INITIAL IN-SITU STRESSES

C CALL INSTRS(LOUT1,NEL,NDIM,NS,XYZ,STRESS,NCONN,MAT,LTYP,NTY)

C

200 CONTINUE

RETURN

904 FORMAT(//1X,36HIN-SITU STRESSES ALL SET TO ZERO/1X,36(1H-))

906 FORMAT(//1X,19HIN-SITU MESH DATA/1X,19(1H-)/

1 /3X,4HNODE,8X,1HY,10X,2HSX,10X,2HSY,10X,2HSZ,

2 9X,3HTXY,10X,1HU,22X,2HPC/)

910 FORMAT(1X,I5,10F12.3)

915 FORMAT(1X,7HELEMENT,I5,2X,18HIS OF UNKNOWN TYPE,I5)

926 FORMAT(//1X,19HIN-SITU MESH DATA/1X,19(1H-)/

1 /3X,4HNODE,8X,1HY,10X,2HSX,10X,2HSY,10X,2HSZ,

2 9X,3HTXY,10X,3HTYZ,9X,3HTZX,9X,1HU,22X,2HPC/)

930 FORMAT(1X,I5,12F12.3)

950 FORMAT(1X,46HWARNING --- POINT OUTSIDE IN-SITU STRESS SPACE,

1 2X,9HELEMENT =,I5,2X,4HIP =,I5,2X,16H(ROUTINE RDSTRS))

CC951 FORMAT(2I4,7E14.4)

955 FORMAT(//1X,40HDIRECT SPECIFICATION OF IN-SITU STRESSES

1 /1X,39(1H-))

960 FORMAT(1X,10E12.5)

C 985 FORMAT(/1X,37HEQUILIBRIUM LOADS FOR INSITU STRESSES/

C 1 1X,37(1H-)//(10E12.4))

END

SUBROUTINE REACT(LOUT2,NDIM,NN,NW,NQ,IDFX,PEQT,PT,IOUTP)

C**********************************************************************

C CALCULATES REACTION TO EARTH AT RESTRAINED NODES

C**********************************************************************

C 1 CALLED BY EQLBM



DIMENSION PEQT(MDOF),PT(MDOF),NW(MNODES+1),NQ(MNODES),IDFX(MDOF)

DIMENSION R(500),NDENO(500),NDIR(500)

C ----- NCT - SIZE OF ARRAYS R, NDENO AND NDIR

NCT=500


C ----- ICT - COUNTER OF TOTAL NO. OF REACTIONS

ICT=0

C

DO 25 JR=1,NN

IF(JR.EQ.0)GOTO 25

J=JR

NQL=NQ(J)

C ----- SKIP IF NODE HAS PORE PRESSURE D.O.F. ONLY

IF(NQL.LE.1)GOTO 25

N1=NW(J)

N2=N1+NDIM-1

IDF=0

C

DO 20 KN=N1,N2

IDF=IDF+1

IF(IDFX(KN).NE.1)GOTO 20

ICT=ICT+1

IF(ICT.GT.NCT)GOTO 30

R(ICT)=-(PEQT(KN)-PT(KN))

NDENO(ICT)=JR

NDIR(ICT)=IDF

20 CONTINUE

25 CONTINUE

C

IF(IOUTP.NE.1) RETURN

WRITE(LOUT2,901)

WRITE(LOUT2,903)(NDENO(JCT),NDIR(JCT),R(JCT),JCT=1,ICT)

RETURN

30 WRITE(LOUT2,906)

STOP

901 FORMAT(//1X,18H LIST OF REACTIONS/2X,17(1H-)/

1 2X,3(4HNODE,4X,9HDIRECTION,7X,8HREACTION,11X)/)

903 FORMAT(3(1X,I5,5X,I4,5X,E14.4,10X))

906 FORMAT(/1X,35HINCREASE ARRAY SIZE OF R,NDENO,NDIR,

1 1X,16HIN ROUTINE REACT)

END

SUBROUTINE RESTRN(NDIM,NW,IDFX)

C**********************************************************************

C ROUTINE TO IDENTIFY ALL DISPLACEMENT BOUNDARY CONDITIONS WHICH

C ARE SPECIFIED. (SET IDFX = 1 FOR ALL DOF WHICH ARE RESTRAINED)

C**********************************************************************

C 1 CALLED BY EQLOD




DIMENSION NW(MNODES+1),IDFX(MDOF)


C ----- LOOP ON ALL NODES WITH ONE OR MORE FIXITIES

CALL ZEROI1(IDFX,MDOF)

C DO 10 J=1,NDF

C 10 IDFX(J)=0

C

IF(NF.EQ.0)RETURN

DO 40 JN=1,NF

NDE=MF(JN)

NFS=NW(NDE)-1

C ----- BY-PASS IF NODE HAS ONLY PORE-PRESSURE DOF

JP=NW(NDE+1)-NW(NDE)

IF(JP.EQ.1)GO TO 40

C

DO 20 JF=1,NDIM

NCDE=NFIX(JF,JN)

IF(NCDE.EQ.0)GO TO 20

IDFX(NFS+JF)=1

20 CONTINUE

40 CONTINUE

RETURN

END

SUBROUTINE SELF(LOUT1,I,NDN,NAC,XYZ,F,NCONN,

1 MAT,LT,INDX,DENS,MUS,KSTGE)

C**********************************************************************

C CALCULATES SELF WEIGHT LOADS

C**********************************************************************

C 1 CALLED BY EQLOD

C 2 Called by SEL1



DIMENSION NCONN(NTPE,MEL),MAT(MEL)

DIMENSION XYZ(3,MNODES),ELCOD(3,NDMX),DS(3,20),

1 SHFN(20),F(3,NDMX),SLL(4),GCOM(3)








C

TPI=2.0D0*PYI

NGP=LINFO(11,LT)

K=MAT(I)

C ----- INITIALISE ARRAY F

CALL ZEROR2(F,3,NDMX)

C

IF(DENS.LE.ASMVL)GO TO 100

GCOM(1)= ZERO

GCOM(2)=-DENS

GCOM(3)= ZERO

C ----- SET UP LOCAL ARRAY FOR CO-ORDINATES

DO 10 KC=1,NDN

NDE=NCONN(KC,I)

C

DO 10 ID=1,NDIM

10 ELCOD(ID,KC)=XYZ(ID,NDE)

C ----- LOOP FOR NUMERICAL INTEGRATION

DO 60 IP=1,NGP

IPA=IP+INDX

C

DO 35 IL=1,NAC


C ----- EVALUATE SHAPE FUNCTION FOR INTEGRATION POINT

C ----- Call shape & detjcb

CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,2,MUS)

CALL DETJCB(LOUT1,NDIM,DJACB,NDN,ELCOD,DS,IP,MUS,KSTGE)

!Determinant of J.

DV=DJACB*W(IPA)

IF(NPLAX.EQ.0)GO TO 45

C

RAD=ZERO

C

DO 40 IN=1,NDN

40 RAD=RAD+ELCOD(1,IN)*SHFN(IN)

DV=DV*TPI*RAD

C

45 DO 50 IN=1,NDN

DO 50 ID=1,NDIM

50 F(ID,IN)=F(ID,IN)+GCOM(ID)*SHFN(IN)*DV

60 CONTINUE

100 CONTINUE

RETURN

END


SUBROUTINE SFR1(LOUT1,S,SHF,DERIV,NSD,LNE,LT)

C**********************************************************************

C SHAPE FUNCTIONS AND DERIVATIVES FOR ONE-DIMENSIONAL

C GAUSSIAN INTEGRATION ALONG ELEMENT EDGE

C**********************************************************************

C 1 Called by DISTLD


DIMENSION SHF(5),DERIV(5)

C ----- INITIALISE

CALL ZEROR1(SHF,5)

CALL ZEROR1(DERIV,5)

C

GO TO(80,21,31,41,51),NSD

WRITE(LOUT1,900)LNE,LT

900 FORMAT(1X,7HELEMENT,I5,2X,7HOF TYPE,I5,2X,

1 22HUNKNOWN (ROUTINE SFR1))

STOP

C ----- 2 NODES ALONG EDGE

21 CONTINUE

WRITE(LOUT1,910)LT

910 FORMAT(/1X,12HELEMENT TYPE,I5,2X,

1 30HNOT IMPLEMENTED (ROUTINE SFR1))

GO TO 80


31 CONTINUE

SHF(1)=0.50D0*S*(S-1.0D0)

SHF(2)=(1.0D0-S)*(1.0D0+S)

SHF(3)=0.50D0*S*(S+1.0D0)

DERIV(1)=S-0.50D0

DERIV(2)=-2.0D0*S

DERIV(3)=S+0.50D0

GO TO 80


41 CONTINUE

WRITE(LOUT1,910)LT

GO TO 80


51 S0=S

S1=S+0.50D0

S2=S-0.50D0

S3=S+1.0D0

S4=S-1.0D0


C1=2.0D0/3.0D0

C2=8.0D0/3.0D0

C3=4.0D0

SHF(1)= C1*S0*S1*S2*S4

SHF(2)=-C2*S0*S2*S3*S4

SHF(3)= C3*S1*S2*S3*S4

SHF(4)=-C2*S0*S1*S3*S4

SHF(5)= C1*S0*S1*S2*S3

DERIV(1)= C1*(S2*S4*(S1+S0)+S0*S1*(S2+S4))

DERIV(2)=-C2*(S2*S4*(S3+S0)+S0*S3*(S2+S4))

DERIV(3)= C3*(S3*S4*(S1+S2)+S1*S2*(S3+S4))

DERIV(4)=-C2*(S3*S4*(S1+S0)+S1*S0*(S3+S4))

DERIV(5)= C1*(S2*S3*(S1+S0)+S1*S0*(S2+S3))

80 CONTINUE

RETURN

END

SUBROUTINE SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,ICODE,MUS)

C================================================================

=======

C SHAPE FUNCTIONS AND DERIVATIVES FOR DIFFERENT ELEMENT TYPES

C================================================================

=======


C 2 CALLED BY INSTRS

C 3 CALLED BY SELF



DIMENSION SLL(4),SHFN(20),DS(3,20)

C

AC1=SLL(1)

AC2=SLL(2)

IF(NAC.LT.3)GOTO 10

AC3=SLL(3)

IF(NAC.LT.4)GOTO 10

AC4=SLL(4)

C

10 GOTO(11,13,13,14,14,15,15,17,17,18,18),LT

WRITE(LOUT1,910)MUS,LT

STOP

C ----- SHAPE FUNCTIONS AND DERIVATIVES FOR BAR ELEMENT

11 CONTINUE


GOTO 80

C ----- SHAPE FUNCTIONS AND DERIVATIVES FOR LST


13 SHFN(1)=AC1*(2.0D0*AC1-1.0D0)

SHFN(2)=AC2*(2.0D0*AC2-1.0D0)

SHFN(3)=AC3*(2.0D0*AC3-1.0D0)

SHFN(4)=4.0D0*AC1*AC2



IF(ICODE.EQ.1)GOTO 80 !icode=1 means initial stress condition

C otherwise icode=2

C

DS(1,1)=4.0D0*AC1-1.0D0

DS(1,2)=0.0D0

DS(1,3)=-(4.0D0*AC3-1.0D0)

DS(1,4)=4.0D0*AC2

DS(1,5)=-4.0D0*AC2

DS(1,6)=4.0D0*(AC3-AC1)

C

DS(2,1)=0.0D0

DS(2,2)=4.0D0*AC2-1.0D0

DS(2,3)=-(4.0D0*AC3-1.0D0)

DS(2,4)=4.0D0*AC1

DS(2,5)=4.0D0*(AC3-AC2)

DS(2,6)=-4.0D0*AC1

GO TO 80

C ----- SHAPE FUNCTIONS AND DERIVATIVES FOR QUADRILATERALS

14 SHFN(1)=-0.250D0*(AC1-1.0D0)*(AC2-1.0D0)*(1.0D0+AC1+AC2)

SHFN(2)=0.250D0*(AC1+1.0D0)*(AC2-1.0D0)*(1.0D0-AC1+AC2)

SHFN(3)=0.250D0*(AC1+1.0D0)*(AC2+1.0D0)*(AC1+AC2-1.0D0)

SHFN(4)=-0.250D0*(AC1-1.0D0)*(AC2+1.0D0)*(AC2-AC1-1.0D0)

SHFN(5)=0.50D0*(AC1**2-1.0D0)*(AC2-1.0D0)

SHFN(6)=-0.50D0*(AC1+1.0D0)*(AC2**2-1.0D0)

SHFN(7)=-0.50D0*(AC1**2-1.0D0)*(AC2+1.0D0)

SHFN(8)=0.50D0*(AC1-1.0D0)*(AC2**2-1.0D0)

IF(ICODE.EQ.1)GOTO 80

C derivatives for ac1

DS(1,1)=-0.250D0*(AC2-1.0D0)*(2.0D0*AC1+AC2)

DS(1,2)=0.250D0*(AC2-1.0D0)*(-2.0D0*AC1+AC2)

DS(1,3)=0.250D0*(AC2+1.0D0)*(2.0D0*AC1+AC2)

DS(1,4)=-0.250D0*(AC2+1.0D0)*(-2.0D0*AC1+AC2)

DS(1,5)=AC1*(AC2-1.0D0)

DS(1,6)=-0.50D0*(AC2**2-1.0D0)

DS(1,7)=-AC1*(AC2+1.0D0)

DS(1,8)=0.50D0*(AC2**2-1.0D0)

C derivatives for ac2

DS(2,1)=-0.250D0*(AC1-1.0D0)*(2.0D0*AC2+AC1)

DS(2,2)=0.250D0*(AC1+1.0D0)*(2.0D0*AC2-AC1)

DS(2,3)=0.250D0*(AC1+1.0D0)*(2.0D0*AC2+AC1)

DS(2,4)=-0.250D0*(AC1-1.0D0)*(2.0D0*AC2-AC1)


DS(2,5)=0.50D0*(AC1**2-1.0D0)

DS(2,6)=-AC2*(AC1+1.0D0)

DS(2,7)=-0.50D0*(AC1**2-1.0D0)

DS(2,8)=AC2*(AC1-1.0D0)

c

GOTO 80

C ----- SHAPE FUNCTIONS AND DERIVATIVES FOR CUBIC STRAIN TRIANGLE

15 CONTINUE

C1=32.0D0/3.0D0

C2=64.0D0

C3=128.0D0/3.0D0

C4=128.0D0

T11=AC1-0.250D0

T12=AC1-0.50D0

T13=AC1-0.750D0

T21=AC2-0.250D0

T22=AC2-0.50D0

T23=AC2-0.750D0

T31=AC3-0.250D0

T32=AC3-0.50D0

T33=AC3-0.750D0

C ----- SHAPE FUNCTIONS

SHFN(1) =C1*AC1*T11*T12*T13

SHFN(2) =C1*AC2*T21*T22*T23

SHFN(3) =C1*AC3*T31*T32*T33

SHFN(4) =C3*AC1*AC2*T11*T12






SHFN(10)=C3*AC1*AC3*T31*T32



SHFN(13)=C4*AC1*AC2*AC3*T11




C

DS(1,1)=C1*(T12*T13*(T11+AC1)+AC1*T11*(T13+T12))

DS(1,2)= 0.0D0

DS(1,3)=-C1*(T32*T33*(AC3+T31)+AC3*T31*(T32+T33))

DS(1,4)= C3*AC2*(T11*T12+AC1*(T11+T12))

DS(1,5)= C2*AC2*T21*(AC1+T11)

DS(1,6)= C3*AC2*T21*T22


DS(1,7)=-C3*AC2*T21*T22

DS(1,8)=-C2*AC2*T21*(AC3+T31)

DS(1,9)=-C3*AC2*(T31*T32+AC3*(T31+T32))

DS(1,10)=-C3*(AC1*AC3*(T31+T32)-T31*T32*(AC3-AC1))

DS(1,11)= C2*(AC1*AC3*(T31-T11)+T31*T11*(AC3-AC1))

DS(1,12)= C3*(AC1*AC3*(T11+T12)+T11*T12*(AC3-AC1))

DS(1,13)= C4*AC2*(AC1*AC3+T11*(AC3-AC1))

DS(1,14)= C4*AC2*T21*(AC3-AC1)

DS(1,15)=-C4*AC2*(AC1*AC3+T31*(AC1-AC3))

C

DS(2,1) = 0.0D0

DS(2,2) = C1*(T22*T23*(AC2+T21)+AC2*T21*(T22+T23))

DS(2,3) =-C1*(T32*T33*(AC3+T31)+AC3*T31*(T32+T33))

DS(2,4) = C3*AC1*T11*T12

DS(2,5) = C2*AC1*T11*(AC2+T21)

DS(2,6) = C3*AC1*(T21*T22+AC2*(T21+T22))

DS(2,7) = C3*(AC2*AC3*(T21+T22)+T21*T22*(AC3-AC2))

DS(2,8) = C2*(AC2*AC3*(T31-T21)+T21*T31*(AC3-AC2))

DS(2,9) =-C3*(AC2*AC3*(T31+T32)+T31*T32*(AC2-AC3))

DS(2,10)=-C3*AC1*(T31*T32+AC3*(T31+T32))

DS(2,11)=-C2*AC1*T11*(AC3+T31)

DS(2,12)=-C3*AC1*T11*T12

DS(2,13)= C4*AC1*T11*(AC3-AC2)

DS(2,14)= C4*AC1*(AC2*AC3+T21*(AC3-AC2))

DS(2,15)=-C4*AC1*(AC2*AC3+T31*(AC2-AC3))

GO TO 80

C ----- SHAPE FUNCTIONS AND DERIVATIVES FOR BRICK ELEMENT

C ----- SHAPE FUNCTIONS

17 SHFN(1)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC2)*(1.0D0+AC3)

1 *(-AC1-AC2+AC3-2.0D0)

SHFN(2)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC2)*(1.0D0+AC3)

2 *(AC1-AC2+AC3-2.0D0)

SHFN(3)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC2)*(1.0D0+AC3)

3 *(AC1+AC2+AC3-2.0D0)

SHFN(4)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC2)*(1.0D0+AC3)

4 *(-AC1+AC2+AC3-2.0D0)

SHFN(5)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC2)*(1.0D0-AC3)

5 *(-AC1-AC2-AC3-2.0D0)

SHFN(6)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC2)*(1.0D0-AC3)

6 *(AC1-AC2-AC3-2.0D0)

SHFN(7)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC2)*(1.0D0-AC3)

7 *(AC1+AC2-AC3-2.0D0)

SHFN(8)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC2)*(1.0D0-AC3)

8 *(-AC1+AC2-AC3-2.0D0)

SHFN(9) =0.250D0*(1.0D0-AC1**2)*(1.0D0-AC2)*(1.0D0+AC3)

SHFN(10)=0.250D0*(1.0D0+AC1)*(1.0D0-AC2**2)*(1.0D0+AC3)

SHFN(11)=0.250D0*(1.0D0-AC1**2)*(1.0D0+AC2)*(1.0D0+AC3)


SHFN(12)=0.250D0*(1.0D0-AC1)*(1.0D0-AC2**2)*(1.0D0+AC3)

SHFN(13)=0.250D0*(1.0D0-AC1**2)*(1.0D0-AC2)*(1.0D0-AC3)

SHFN(14)=0.250D0*(1.0D0+AC1)*(1.0D0-AC2**2)*(1.0D0-AC3)

SHFN(15)=0.250D0*(1.0D0-AC1**2)*(1.0D0+AC2)*(1.0D0-AC3)

SHFN(16)=0.250D0*(1.0D0-AC1)*(1.0D0-AC2**2)*(1.0D0-AC3)

SHFN(17)=0.250D0*(1.0D0-AC1)*(1.0D0-AC2)*(1.0D0-AC3**2)

SHFN(18)=0.250D0*(1.0D0+AC1)*(1.0D0-AC2)*(1.0D0-AC3**2)

SHFN(19)=0.250D0*(1.0D0+AC1)*(1.0D0+AC2)*(1.0D0-AC3**2)

SHFN(20)=0.250D0*(1.0D0-AC1)*(1.0D0+AC2)*(1.0D0-AC3**2)


C DERIVATINES

DS(1,1)=0.1250D0*(1.0D0-AC2)*(1.0D0+AC3)*(2.0D0*AC1+AC2-AC3+1.0D0)

DS(1,2)=0.1250D0*(1.0D0-AC2)*(1.0D0+AC3)*(2.0D0*AC1-AC2+AC3-1.0D0)

DS(1,3)=0.1250D0*(1.0D0+AC2)*(1.0D0+AC3)*(2.0D0*AC1+AC2+AC3-1.0D0)

DS(1,4)=0.1250D0*(1.0D0+AC2)*(1.0D0+AC3)*(2.0D0*AC1-AC2-AC3+1.0D0)

DS(1,5)=0.1250D0*(1.0D0-AC2)*(1.0D0-AC3)*(2.0D0*AC1+AC2+AC3+1.0D0)

DS(1,6)=0.1250D0*(1.0D0-AC2)*(1.0D0-AC3)*(2.0D0*AC1-AC2-AC3-1.0D0)

DS(1,7)=0.1250D0*(1.0D0+AC2)*(1.0D0-AC3)*(2.0D0*AC1+AC2-AC3-1.0D0)

DS(1,8)=0.1250D0*(1.0D0+AC2)*(1.0D0-AC3)*(2.0D0*AC1-AC2+AC3+1.0D0)

DS(1,9) =-0.50D0*AC1*(1.0D0-AC2)*(1.0D0+AC3)

DS(1,10)= 0.250D0*(1.0D0-AC2**2)*(1.0D0+AC3)

DS(1,11)=-0.50D0*AC1*(1.0D0+AC2)*(1.0D0+AC3)

DS(1,12)=-0.250D0*(1.0D0-AC2**2)*(1.0D0+AC3)

DS(1,13)=-0.50D0*AC1*(1.0D0-AC2)*(1.0D0-AC3)

DS(1,14)= 0.250D0*(1.0D0-AC2**2)*(1.0D0-AC3)

DS(1,15)=-0.50D0*AC1*(1.0D0+AC2)*(1.0D0-AC3)

DS(1,16)=-0.250D0*(1.0D0-AC2**2)*(1.0D0-AC3)

DS(1,17)=-0.250D0*(1.0D0-AC2)*(1.0D0-AC3**2)

DS(1,18)= 0.250D0*(1.0D0-AC2)*(1.0D0-AC3**2)

DS(1,19)= 0.250D0*(1.0D0+AC2)*(1.0D0-AC3**2)

DS(1,20)=-0.250D0*(1.0D0+AC2)*(1.0D0-AC3**2)

C

DS(2,1)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC3)*(AC1+2.0D0*AC2-AC3+1.0D0)

DS(2,2)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC3)*

1 (-AC1+2.0D0*AC2-AC3+1.0D0)

DS(2,3)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC3)*(AC1+2.0D0*AC2+AC3-1.0D0)

DS(2,4)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC3)*

1 (-AC1+2.0D0*AC2+AC3-1.0D0)

DS(2,5)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC3)*(AC1+2.0D0*AC2+AC3+1.0D0)

DS(2,6)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC3)*

1 (-AC1+2.0D0*AC2+AC3+1.0D0)

DS(2,7)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC3)*(AC1+2.0D0*AC2-AC3-1.0D0)

DS(2,8)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC3)*

1 (-AC1+2.0D0*AC2-AC3-1.0D0)

DS(2,9) =-0.250D0*(1.0D0-AC1**2)*(1.0D0+AC3)

DS(2,10)=-0.50D0*(1.0D0+AC1)*AC2*(1.0D0+AC3)

DS(2,11)= 0.250D0*(1.0D0-AC1**2)*(1.0D0+AC3)

DS(2,12)=-0.50D0*(1.0D0-AC1)*AC2*(1.0D0+AC3)


DS(2,13)=-0.250D0*(1.0D0-AC1**2)*(1.0D0-AC3)

DS(2,14)=-0.50D0*(1.0D0+AC1)*AC2*(1.0D0-AC3)

DS(2,15)= 0.250D0*(1.0D0-AC1**2)*(1.0D0-AC3)

DS(2,16)=-0.50D0*(1.0D0-AC1)*AC2*(1.0D0-AC3)

DS(2,17)=-0.250D0*(1.0D0-AC1)*(1.0D0-AC3**2)

DS(2,18)=-0.250D0*(1.0D0+AC1)*(1.0D0-AC3**2)

DS(2,19)= 0.250D0*(1.0D0+AC1)*(1.0D0-AC3**2)

DS(2,20)= 0.250D0*(1.0D0-AC1)*(1.0D0-AC3**2)

C

DS(3,1)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC2)*

1 (-AC1-AC2+2.0D0*AC3-1.0D0)

DS(3,2)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC2)*(AC1-AC2+2.0D0*AC3-1.0D0)

DS(3,3)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC2)*(AC1+AC2+2.0D0*AC3-1.0D0)

DS(3,4)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC2)*

1 (-AC1+AC2+2.0D0*AC3-1.0D0)

DS(3,5)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC2)*(AC1+AC2+2.0D0*AC3+1.0D0)

DS(3,6)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC2)*

1 (-AC1+AC2+2.0D0*AC3+1.0D0)

DS(3,7)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC2)*

1 (-AC1-AC2+2.0D0*AC3+1.0D0)

DS(3,8)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC2)*(AC1-AC2+2.0D0*AC3+1.0D0)

DS(3,9) = 0.250D0*(1.0D0-AC1**2)*(1.0D0-AC2)

DS(3,10)= 0.250D0*(1.0D0+AC1)*(1.0D0-AC2**2)

DS(3,11)= 0.250D0*(1.0D0-AC1**2)*(1.0D0+AC2)

DS(3,12)= 0.250D0*(1.0D0-AC1)*(1.0D0-AC2**2)

DS(3,13)=-0.250D0*(1.0D0-AC1**2)*(1.0D0-AC2)

DS(3,14)=-0.250D0*(1.0D0+AC1)*(1.0D0-AC2**2)

DS(3,15)=-0.250D0*(1.0D0-AC1**2)*(1.0D0+AC2)

DS(3,16)=-0.250D0*(1.0D0-AC1)*(1.0D0-AC2**2)

DS(3,17)=-0.50D0*(1.0D0-AC1)*(1.0D0-AC2)*AC3

DS(3,18)=-0.50D0*(1.0D0+AC1)*(1.0D0-AC2)*AC3

DS(3,19)=-0.50D0*(1.0D0+AC1)*(1.0D0+AC2)*AC3

DS(3,20)=-0.50D0*(1.0D0-AC1)*(1.0D0+AC2)*AC3

GOTO 80

C ----- SHAPE FUNCTIONS AND DERIVATIVES FOR TETRA-HEDRA

18 CONTINUE


910 FORMAT(/1X,7HELEMENT,I5,2X,14HIS OF TYPE ***,I5,2X,

1 31HNOT IMPLEMENTED (ROUTINE SHAPE))

80 CONTINUE

RETURN

END


SUBROUTINE FACTOR(LINP,LOUT1,NOINC,ILDF,IOCD,ITMF,IOUTS,

1 RINCC,DTM,IOPT,DTIME)

C**********************************************************************

C LOAD RATIOS, TIME RATIOS (CONSOLIDATION ANALYSIS) AND OUTPUT

C OPTIONS FOR ALL INCREMENTS IN THE BLOCK

C**********************************************************************



DIMENSION RINCC(2500),DTM(2500),IOPT(2500)


C ----- READ LOAD RATIOS FOR INCREMENTS

FSTD=1.0D0/FLOAT(NOINC) !FSTD=factor of increment for unit pen.

length

IF(ILDF.EQ.0)GO TO 98 !ILDF=0 and skip --Song

WRITE(LOUT1,948)

READ(LINP,*)(RINCC(IN),IN=1,NOINC)

WRITE(LOUT1,954)(RINCC(IN),IN=1,NOINC)

GO TO 122

98 DO 100 IK=1,NOINC

100 RINCC(IK)=FSTD

C ----- READ OUTPUT OPTIONS

122 IF(IOCD.EQ.0)GO TO 127 !IOCD=0

WRITE(LOUT1,960)

READ(LINP,*)(IOPT(IN),IN=1,NOINC)

WRITE(LOUT1,964)(IOPT(IN),IN=1,NOINC)

GO TO 131

C


130 IOPT(IK)=IOUTS

C ----- READ TIME RATIOS FOR INCREMENTS

131 IF(DTIME.LT.ASMVL.OR.ITMF.EQ.0)GO TO 132 !ITMF=0

WRITE(LOUT1,965)

READ(LINP,*)(DTM(IN),IN=1,NOINC)

WRITE(LOUT1,968)(DTM(IN),IN=1,NOINC)

GO TO 136

C


135 DTM(IK)=FSTD*DTIME

136 CONTINUE

RETURN

948 FORMAT(/1X,34HLIST OF LOAD RATIOS FOR INCREMENTS/1X,34(1H-)/)

954 FORMAT(1X,10F8.1)

960 FORMAT(/1X,35HLIST OF OUTPUT CODES FOR INCREMENTS/1X,35(1H-)/)

964 FORMAT(1X,10I6)


965 FORMAT(/1X,33HLIST OF TIME STEPS FOR INCREMENTS/1X,33(1H-)/)

968 FORMAT(1X,8G10.1)

END

C**********************************************************************

SUBROUTINE SKSOLV(XYZ,DA,DITER,DIPR,DD,STRESS,PITER,NQ,NW,

1 LTYP,NTY,MAT,NCONN,PR,IOUTP,ITER)

C**********************************************************************

C

C----------------------------------------------------------------------



DIMENSION XYZ(3,MNODES),DA(MDOF),DITER(MDOF),DAPPL(MDOF),

1 DIPR(MDOF),DSOLVD(MDOF),STRESS(NVRS,NIP,MEL),PITER(MDOF),

2 P(MDOF),GSTIF(MKSIZ)

DIMENSION NQ(MNODES),NW(MNODES+1)


DIMENSION PR(NPR,NMT),NTY(NMT),DD(4,200)

COMMON/GLBSTF/LOCDOF(MDOF),JDIAG(MDOF),NKSIZ,NEQTNS







C INITIALIZE THE GLOBAL STIFFNESS MATRIX

DO I=1,MKSIZ

GSTIF(I)=0.0D0

ENDDO

C INITIALIZE DAPPL,P

CALL ZEROR1(DAPPL,MDOF)

CALL ZEROR1(P,MDOF)

CALL ZEROR1(PDIS,MDOF)

CALL ZEROR1(DITER,MDOF)

NDIM1=NDIM+1

C

IF(ITER.EQ.1) THEN

DO INODE=1,NF

NODE=MF(INODE)

IDF=NW(NODE)-1

KDF=NQ(NODE)

IF(KDF.EQ.1)GO TO 10


DO ID=1,KDF

IF(ID.EQ.NDIM1)GO TO 10

DAPPL(IDF+ID)=DXYT(ID,INODE)*FRACLD+DD(ID,INODE)

DITER(IDF+ID)=DAPPL(IDF+ID)

ENDDO

GO TO 20

10 NTTI=NFIX(NDIM1,INODE)

IF(NTTI.EQ.2) THEN

DAPPL(IDF+KDF)=DXYT(NDIM1,INODE)-DA(IDF+KDF)

DITER(IDF+KDF)=DAPPL(IDF+KDF)

NFIX(NDIM1,INODE)=1

DXYT(NDIM1,INODE)=0.0D0

ENDIF

20 CONTINUE

ENDDO

ENDIF

c THE LOAD MATRIX IS REARRANGED SUCH THAT BCS ARE TAKEN AWAY.

DO INODE=1,NNODES

KDF=NQ(INODE)

NWDF=NW(INODE)-1

DO IDF=1,KDF

LOC=NWDF+IDF

LOCGLB=LOCDOF(LOC)

IF(LOCGLB.GT.0)P(LOCGLB)=PITER(LOC)

enddo

enddo

C

CALL GLASSEM(GSTIF,XYZ,DA,DAPPL,DIPR,STRESS,P,NQ,NW,LTYP,

1 NTY,MAT,NCONN,PR,DTIMEI,ITER,JS)

C

DO I=1,NEQTNS

DSOLVD(I)=P(I)

ENDDO

C

CALL SOLSYM(GSTIF,DSOLVD)

c

DO I=1,NDF

c DITER(I)=0.0D0

LOC=LOCDOF(I)

IF(LOC.GT.0) THEN

DITER(I)=DSOLVD(LOC)


ENDIF

ENDDO

C

RETURN

END

C**********************************************************************

SUBROUTINE GLASSEM(GSTIF,XYZ,DA,DAPPL,DIPR,STRESS,P,NQ,NW,

1 LTYP,NTY,MAT,NCONN,PR,DTIMEI,ITER,JS)

C**********************************************************************

C THIS SUBROUTINE ASSEMBLES THE GLOBAL STIFFNESS MATRIX AND

CALCULATE

C THE EQUIVALENT LOAD FOR THE APPLIED DISPLACEMENT

C----------------------------------------------------------------------



DIMENSION XYZ(3,MNODES),DA(MDOF),DAPPL(MDOF),DIPR(MDOF),

1 STRESS(NVRS,NIP,MEL),P(MDOF),GSTIF(MKSIZ),ESTIF(MDFE,MDFE),

2 EDINC(NB)

DIMENSION NQ(MNODES),NW(MNODES+1),LOCELM(MDFE)







C LOOP OVER EACH ELREMENT

DO IELEM=1,NEL

ITYPE=LTYP(IELEM)

NELN=LINFO(1,ITYPE)

NEDOF=LINFO(16,ITYPE)

C GET THE ELEMENT DISPL FROM GLOBAL DISPL

DO JDN=1,NELN

NIN=NDIM*(JDN-1)

JN=ABS(NCONN(JDN,IELEM))

JL=NW(JN)-1

DO ID=1,NDIM

EDINC(NIN+ID)=DIPR(JL+ID)

ENDDO

ENDDO

C GET THE CORRESPONDANCE BETWEEN ELEMENT AND GLOBAL DOFS

CALL DOFL2G(LOCELM,NW,NQ,NCONN,NELN,IELEM)

C CALCULATE ELEMENT STIFNESS MATRIX


CALL ELMSTIF(IELEM,ESTIF,DTIMEI,ITYPE,XYZ,DA,EDINC,STRESS,P,

2 NCONN,MAT,NW,PR,NTY,IOUTP,ITER,JS)

C

C IF(IELEM.EQ.1) THEN

C WRITE(2,*)’ESTIF’

C DO I1=1,20

C WRITE(2,*)(ESTIF(I1,J1),J1=1,20)

C ENDDO

C ENDIF

C

DO I=1,NEDOF

IGDF=LOCELM(I)

IDOF=LOCDOF(LOCELM(I))

IF(IDOF.GT.0) THEN

LOCD=JDIAG(IDOF)

DO J=1,NEDOF

JGDF=LOCELM(J)

JDOF=LOCDOF(LOCELM(J))

IF(IDOF.GE.JDOF.AND.JDOF.GT.0) THEN

LOCA=LOCD+IDOF-JDOF

GSTIF(LOCA)=GSTIF(LOCA)+ESTIF(I,J)

C WRITE(*,*)"STIFF",GSTIF(LOCA)

ENDIF

C IF(JDOF.GT.0) THEN

C P(JDOF)=P(JDOF)+ESTIF(I,J)*DAPPL(IGDF)

C PDIS(JGDF)=PDIS(JGDF)+ESTIF(I,J)*DAPPL(IGDF)

C ENDIF

ENDDO

ELSE IF(IDOF.EQ.0) THEN

DO J=1,NEDOF

JDOF=LOCDOF(LOCELM(J))

IF(JDOF.GT.0) THEN

P(JDOF)=P(JDOF)-ESTIF(I,J)*DAPPL(IGDF)

ENDIF

ENDDO

ENDIF

ENDDO

C

ENDDO

RETURN

END


C**********************************************************************

SUBROUTINE SOLSYM(A,R)

C**********************************************************************

C THIS SUBROUTINE SOLVES FOR THE DISPLACEMENTS BY SOLVING THE

C SYMMETRIC LINEAR SET OF EQUATIONS

C----------------------------------------------------------------------




DIMENSION A(MKSIZ),R(MDOF)

c perform l*d*l factorization of the stiffness matrix

do n=1,NEQTNS

kn = jdiag( n )

kl = kn + 1

ku = jdiag(n+1) - 1

kh = ku - kl

if (kh.gt.0)then

k = n - kh

ic = 0

klt = ku

do j = 1, kh

ic = ic + 1

klt = klt - 1

ki = jdiag( k )

nd = jdiag( k + 1 ) - ki - 1

if (nd.gt.0)then

kk = min0(ic,nd)

c = 0.

do l = 1, kk

c = c + a(ki + l)*a(klt + l)

enddo

a( klt ) = a( klt ) - c

endif

k = k + 1

enddo

endif

if(kh.ge.0)then

k = n

b = 0.d0

do kk = kl, ku

k = k - 1

ki = jdiag( k )

c = a( kk )/ a( ki )


b = b + c*a( kk )

a( kk ) = c

enddo

a( kn ) = a( kn ) - b

endif

C if (a(kn).le.0)then

if (a(kn).EQ.0)then

write( *, 2000) n, a( kn )

stop

endif

enddo

c reduce the right-hand-side load vector

do n = 1, NEQTNS

kl = jdiag( n ) + 1

ku = jdiag( n + 1) - 1

if(ku-kl.ge.0)then

k = n

c = 0.d0

do kk = kl, ku

k = k - 1

c = c + a( kk )*r( k )

enddo

r( n ) = r( n ) - c

endif

enddo

C

c back-substitute

C

do n = 1, NEQTNS

k = jdiag( n )

r( n ) = r( n )/ a( k )

enddo

if (NEQTNS.eq.1) return

n = NEQTNS

do l = 2, NEQTNS

kl = jdiag( n ) + 1

ku = jdiag( n + 1 ) - 1

if (ku-kl.ge.0)then

k = n

do kk = kl, ku

k = k - 1

r( k ) = r( k ) - a( kk )*r( n )

enddo

endif

n = n - 1


enddo

return

2000 format(//1x,’stop - stiffness matrix not positive definite ’//

1 1x,’ ZERO PIVOT FOR EQUATION ’,i4//1x,’pivot = ’,e20.12)

end

C**********************************************************************

SUBROUTINE LOCINIT(LTYP,NCONN,NQ,NW,NDF,NDIM)

C**********************************************************************

C THS SUBROUTINE REASSIGN THE APPROPRIATE POINTERS TO THE DOF MATRIX

C AND STIFFNESS MATRIX FOR STORAGE ALLOCATION AND DEPENDNG ON THE WAY

C THE LINEAR SYSTEM IS SOLVED.

C----------------------------------------------------------------------







C ELIMINATE THE DOFS THAT ARE CONSTRAINED BY ASSIGNING ZERO TO

LOCDOF

C CORRESPOND TO THAT DOF

M=0

NDIM1=NDIM+1

CALL ZEROI1(LOCDOF,MDOF)

CALL ZEROI1(JDIAG,MDOF)

C

C WRITE(2,*)’NODE NFIX LOCDOF ’

DO INODE=1,NF

NODE=MF(INODE)

IDF=NW(NODE)-1

KDF=NQ(NODE)

IF(KDF.EQ.1) THEN

LOCDOF(IDF+KDF)=NFIX(NDIM1,INODE)

ELSE

DO ID=1,KDF

LOCDOF(IDF+ID)=NFIX(ID,INODE)

ENDDO

ENDIF

ENDDO

C WRITE(2,*)’ LOCDOF 1’

C WRITE(2,*)(LOCDOF(IDF),IDF=1,NDF)


C

DO IDOF=1,NDF

IF(LOCDOF(IDOF).EQ.0) THEN

M=M+1

LOCDOF(IDOF)=M

ELSE IF(LOCDOF(IDOF).GT.0) THEN

LOCDOF(IDOF)=0

ENDIF

ENDDO

NEQTNS=M

C WRITE(2,*)’ LOCDOF 2’

C WRITE(2,*)(LOCDOF(IDF),IDF=1,NDF)

CALL DIAG(LTYP,NCONN,NQ,NW)

RETURN

END

C**********************************************************************

SUBROUTINE DOFL2G(LOCGLB,NW,NQ,NCONN,NELN,IELEM)

C**********************************************************************

C THIS SUBROUTINE FINDS THE CORRESPONDENCE BETWEEN ELEMENT AND GLOBAL

C DOFS

C----------------------------------------------------------------------




DIMENSION NCONN(NTPE,MEL),LOCGLB(MDFE)

C

IESTRT=0

NWQ=0

DO IELN=1,NELN

NODE=NCONN(IELN,IELEM)

NODDOF=NQ(NODE)

IESTRT=IESTRT+NWQ

IGSTRT=NW(NODE)-1

DO IDOF=1,NODDOF

IELOC=IESTRT+IDOF

LOCGLB(IELOC)=IGSTRT+IDOF

ENDDO

NWQ=NODDOF

ENDDO

RETURN

END


C**********************************************************************

SUBROUTINE DIAG(LTYP,NCONN,NQ,NW)

C**********************************************************************

C THIS SUBROUTINE FINDS THE DIAGONAL LOCATION OF THE ELEMENT IN THE

C STIFFNESS MATRIX (JDIAG)

C----------------------------------------------------------------------









C FIND THE COLUMN HEIGHTS

DO IELEM=1,NEL

ITYPE=LTYP(IELEM)

NELN=LINFO(1,ITYPE)

MAXDOF=0

MINDOF=1000000

DO IELN=1,NELN


NODDOF=NQ(NODE)

DO IDOF=1,NODDOF

K=NW(NODE)-1+IDOF

IF(LOCDOF(K).GT.0) THEN

C IF(LOCDOF(K).GT.MAXDOF)MAXDOF=LOCDOF(K)

C IF(LOCDOF(K).LT.MINDOF)MINDOF=LOCDOF(K)

MAXDOF=MAX0(MAXDOF,LOCDOF(K))

MINDOF=MIN0(MINDOF,LOCDOF(K))

ENDIF

ENDDO

ENDDO

C

DO IELN=1,NELN


NODDOF=NQ(NODE)

DO IDOF=1,NODDOF

K=NW(NODE)-1+IDOF

IF(LOCDOF(K).GT.0) THEN

ID=LOCDOF(K)

MHT=ID-MINDOF+1

IF(MHT.GT.JDIAG(ID)) JDIAG(ID)=MHT

ENDIF


ENDDO

ENDDO

ENDDO

C

MHT=1

ID=0

DO K=1,NEQTNS+1

ID=ID+MHT

MHT=JDIAG(K)

JDIAG(K)=ID

ENDDO

NKSIZ=JDIAG(NEQTNS+1)-JDIAG(1)

IF(NKSIZ.GT.MKSIZ)THEN

WRITE(2,*)’NKSIZ=’,NKSIZ

WRITE(2,*)’NKSIZ GREATER THAN MKSIZ’

STOP

ENDIF

RETURN

END

SUBROUTINE SEL1(LOUT1,NDIM,NEL,XYZ,P,NCONN,MAT,

1 LTYP,NW,PR,DGRAV)

C ------------------------------------------------------------------




DIMENSION XYZ(3,MNODES),P(MDOF),F(3,NDMX)

DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL),NW(MNODES+1)

DIMENSION PR(NPR,NMT)



C -----CODE TO INDICATE STAGE OF THE ANALYSIS

KSTGE=3

C ----- ITERATE FOR ALL ELEMENTS

DO 50 J=1,NEL

JK=J

C ----- BY-PASS ADDITION IF ELEMENT NOT IN CURRENT MESH

LT=LTYP(J)


IF(LT.LT.0)GO TO 50

GOTO(50,22,22,22,22,22,22,22,22,22,22),LT

WRITE(LOUT1,900)JK,LT

900 FORMAT(1X,7HELEMENT,I5,2X,18HIS OF UNKNOWN TYPE,I5,

1 14H(ROUTINE SEL1))

22 INDX=LINFO(12,LT)

NDN=LINFO(5,LT)

NAC=LINFO(15,LT)

K=MAT(J)

DENS=DGRAV*PR(8,K)

IF(DENS.LE.ASMVL)GO TO 50

C ----- Call Self

CALL

SELF(LOUT1,J,NDN,NAC,XYZ,F,NCONN,MAT,LT,INDX,DENS,JK,KSTGE)

C

DO 30 JJ=1,NDN

JN=NCONN(JJ,J)

JL=NW(JN)-1

C

DO 30 ID=1,NDIM

30 P(JL+ID)=P(JL+ID)+F(ID,JJ)

50 CONTINUE

RETURN

END

C**********************************************************************

SUBROUTINE ELMSTIF(K,STIFF,DTIME,LT,XYZ,DA,EDISPINC,STRESS,P,

2 NCONN,MAT,NW,PR,NTY,IOUTP,ITER,JS)

C**********************************************************************

C CALCULATION AND ASSEMBLY OF STIFFNESS MATRIX

C**********************************************************************

C This subroutine calculate the stiffness of each element.

C It is combined with global stiffness in GLobalASSEMbly. Song 3/31/99

C Called by Glassem

C K=ILEM=element number



DIMENSION PERM(3)

DIMENSION SG(KES),XYZ(3,MNODES),DA(MDOF),DAB(3,3),

1 STRESS(NVRS,NIP,MEL),P(MDOF),D(6,6),ELCOD(3,NDMX),DS(3,20),

2 EDISPINC(NB),SHFN(20),CARTD(3,NDMX),BL(6,NB),

3 BNL(6,NB),BNLS(9,NB),BL1(9,NB),DB(6,NB),

3 EKSTIF(NB,NB),ELCODP(3,NPMX),XJACM(3,3),

4 E(3,NPMX),RN(NB),AA(NPMX),EFLOW(NPMX,NPMX),

5 ECOUPT(NB,NPMX),SPK(9,9),S(6),F(3,3),FINV(3,3)


dimension ETE(NPMX,NPMX),RLT(NB,NPMX)

DIMENSION NCONN(NTPE,MEL),MAT(MEL),NW(MNODES+1),

1 NWL(NPMX),SLL(4),PR(NPR,NMT),NTY(NMT)

DIMENSION STIFF(MDFE,MDFE)

DIMENSION GDT(NPMX),BOD(3),PE(3,NPMX)

DIMENSION SALPHA(4),A(6) !A added just for matching








COMMON /NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL,


C----------------------------------------------------------------------

CR=1.0D0


C----------INITIALISE EKSTIF,ECOUPT AND EFLOW

CALL ZEROR2(EKSTIF,NB,NB)

CALL ZEROR2(ECOUPT,NB,NPMX)

CALL ZEROR2(RLT,NB,NPMX)

CALL ZEROR2(EFLOW,NPMX,NPMX)

CALL ZEROR2(ETE,NPMX,NPMX)

CALL ZEROR1(GDT,NPMX)

CALL ZEROR1(BOD,3)

C WRITE(*,*)"ELMSTIF"

C

NDN=LINFO(5,LT)

NPN=LINFO(6,LT)

NGP=LINFO(11,LT)

INDX=LINFO(12,LT)

NAC=LINFO(15,LT)

NDV=NDIM*NDN

NDPT=LINFO(1,LT)

GOTO(1,1,2,1,2,1,2,1,2,1,2),LT

WRITE(LOUT1,910)K,LT

910 FORMAT(1X,7HELEMENT,I5,2X,18HIS OF UNKNOWN TYPE,I5,

1 2X,17H(ROUTINE ELMSTIF))

STOP

C

1 ICPL=0

IBLK=1

NPN=0

GOTO 14

2 ICPL=1

IBLK=0


C----------------------------------------------------------------------

C SETUP LOCAL ARRAY OF NW AS NWL GIVING THE INDEX TO

C PORE-PRESSURE VARIABLES

C----------------------------------------------------------------------

IPP=0

C----------INXL - INDEX TO NODAL D.O.F. (SEE BLOCK DATA)

DO 12 IV=1,NDPT

IQ=LINFO(IV+INXL,LT)

IF(IQ.EQ.NDIM)GO TO 12

IPP=IPP+1

NDE=NCONN(IV,K)

NDE=IABS(NDE)

C----------COORDINATES OF POREPRESSURE NODES OF ELEMENT

DO 10 ID=1,NDIM

10 ELCODP(ID,IPP)=XYZ(ID,NDE)

NWL(IPP)=NW(NDE)+IQ-1

12 CONTINUE

C

14 KM=MAT(K)

C----------------------------------------------------------------------

C LOCAL ARRAY OF COORDINATES OF DISPLACEMENT NODES OF

ELEMENT

C----------------------------------------------------------------------

DO 20 KN=1,NDN

NDE=NCONN(KN,K)

NDE=IABS(NDE)

C

DO 20 ID=1,NDIM


C

IF(NTY(KM)-2)26,28,28

C----------CONSTANT ELASTICITY D MATRIX

26 CALL DCON(K,IBLK,MAT,PR,D,NDIM,BK)

C----------------------------------------------------------------------

C ITERATE FOR ALL INTEGRATION POINTS

C----------------------------------------------------------------------

28 DO 80 IP=1,NGP !NGP = No. of iteration point = No. of Gauss Point

IPA=IP+INDX

ICOD=MCODE(IP,K) !icod=1,2,3,4,5 depend on the stress point

C

DO 30 IL=1,NAC


ISTGE=3

C----------------------------------------------------------------------

C INITIALISE SHAPE FUNCTION AND DERIVATIVES (LOCAL COORDS)

C----------------------------------------------------------------------




C----------------------------------------------------------------------

C CALCULATE SHAPE FUNCTIONS AND DERIVATIVES W.R.T LOCAL COOR.

C----------------------------------------------------------------------

CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,2,K)


C----------------------------------------------------------------------

C CALCULATE [JAC], [JAC]-1 & DETJAC

C----------------------------------------------------------------------

DO 15 IDIM=1,NDIM

DO 15 JDIM=1,NDIM

SUM=ZERO

C

DO 112 IN=1,NDN



C

CALL DETMIN(LOUT1,NDIM,XJACM,XJACI,DJACB,K,IP,ISTGE)

C----------------------------------------------------------------------

C CALCULATE RADIUS FOR AXI-SYM B MATRIX

C----------------------------------------------------------------------

R=ZERO

RI=ZERO

IF(NDIM.EQ.3.OR.NPLAX.EQ.0)GOTO 38

C

DO 25 IN=1,NDN

25 R=R+ELCOD(1,IN)*SHFN(IN)

RI=-1.0D0/R

C----------------------------------------------------------------------

C CALCULATE CARTESIAN DERIVATIVES OF SHAPE FUNCTIONS

C----------------------------------------------------------------------

38 DO 35 IN=1,NDN

DO 35 ID=1,NDIM

SUM=ZERO

C

DO 130 JD=1,NDIM


35 CARTD(ID,IN)=SUM

C----------------------------------------------------------------------

C CALCULATE THE LINEAR STRAIN-DISPL MATRIX [BL]

C----------------------------------------------------------------------

CALL BLNR(NDIM,NDN,NPLAX,RI,SHFN,CARTD,BL)

F9=CR*DJACB*W(IPA)

IF(NDIM.EQ.2.AND.NPLAX.EQ.1)F9=F9*R

C

IF(LINR.GT.0) THEN


C----------------------------------------------------------------------

C CALCULATE THE GEOMETRIC NONLINEAR MATRIX B\_NLS

C----------------------------------------------------------------------

CALL BNLNRS(SHFN,CARTD,RI,BNLS,NDIM,NDN,NPLAX)

C----------------------------------------------------------------------

C RETRIEVE THE SECOND PIOLA-KIRCHHOF STRESS VECTOR

C----------------------------------------------------------------------

DO IS=1,NS

S(IS)=STRESS(IS,IP,K)

ENDDO

IF(NDIM.NE.3.AND.NPLAX.NE.1) THEN

S(3)=S(4)

S(4)=0.0D0

END IF

C----------------------------------------------------------------------

C CONVERT S VECTOR TO SPK MATRIX

C----------------------------------------------------------------------

CALL CONVERT(S,SPK,NDIM,NPLAX)

C

IF(LINR.GT.1) THEN

C----------------------------------------------------------------------

C CALCULATE THE NONLINEAR STRAIN-DISPL MATRIX BNL

C----------------------------------------------------------------------

CALL BNLNR(EDISPINC,SHFN,CARTD,RI,NDIM,NDN,BNL,NS,NPLAX)

C

CALL ADDBMAT(BL,BNL)

END IF

C----------------------------------------------------------------------

C COMPUTE [BNLS]T[S][BNLS] AND ADD TO THE EKSTIF

C----------------------------------------------------------------------

NDD=NDIM*NDIM

IF(NDIM.EQ.2.AND.NPLAX.EQ.1) NDD=NDD+1

CALL BTDB(BNLS,SPK,EKSTIF,F9,NDD)

C

END IF

C----------------------------------------------------------------------

C CALCULATE THE LINEAR DISPL-DISPL MATRIX BL1

C----------------------------------------------------------------------

CALL BLNR1(CARTD,BL1,NDIM,NDN)

C----------------------------------------------------------------------

C CALCULATE THE DEFORMATION GRADIENT MATRIX [F],[FINV] & DETF

C----------------------------------------------------------------------

CALL DEFGRAD(BL1,F,FINV,DETF,EDISPINC,NDIM,LINR)

C----------------------------------------------------------------------

C CALCULATE PORE PRESSURE SHAPE FUNCTIONS & DERIVATIVES

C----------------------------------------------------------------------

IF(ICPL.EQ.1)CALL JPC(K,NDIM,NPN,NAC,NPLAX,

1 DS,CARTD,BL,E,RN,AA,SLL,LT)


C

KGO=NTY(KM)

C----------------------------------------------------------------------

C COMPUTE [D] MATRIX

C----------------------------------------------------------------------

GO TO(39,32,33),KGO

32 CALL DLIN(K,IBLK,NDIM,NDN,ELCOD,SHFN,MAT,D,PR,BK)

GO TO 39

33 CALL DMCAM(IP,K,IBLK,NDIM,NS,STRESS,MAT,D,PR,BK,ITER,JS,DAB,

+SALPHA,A,H)

GO TO 39

39 CONTINUE

C----------------------------------------------------------------------

C CALCULATE EKSTIF MATRIX

C----------------------------------------------------------------------

CALL LSTIFA(EKSTIF,BL,D,DB,F9,NS)

C----------------------------------------------------------------------

C BYPASS IF NOT COUPLED CONSOLIDATION

C----------------------------------------------------------------------

IF(ICPL.EQ.0)GO TO 80

C----------------------------------------------------------------------

C FORM PERM*E

C----------------------------------------------------------------------

PERM(1)=PR(9,KM)

PERM(2)=PR(10,KM)

PERM(3)=PERM(1)

GAMMAW=PR(7,KM)

XGAM=GAMMAW

C

KGO=NTY(KM)

GO TO(60,60,52,60,60),KGO

C----------------------------------------------------------------------

C CALCULATE VOID RATIO

C----------------------------------------------------------------------

52 EI=STRESS(NS+2,IP,K)

POR=EI/(1.0D0+EI)

c POR=POR*DETF

STRESS(NS+2,IP,K)=POR/(1.0D0-POR)

XGAM=GAMMAW/POR

C XGAM=GAMMAW

60 CONTINUE

C----------------------------------------------------------------------

C CALCULATE ELEMENT FLOW MATRIX

C----------------------------------------------------------------------

c WRITE(*,*)"2222"

CALL ELFLOW(NDIM,NPN,FINV,DETF,E,F9,EFLOW,PERM,XGAM,DTIME)


C

DO 40 JJ=1,NPN

DO 40 IM=1,NDIM

PE(IM,JJ)=PERM(IM)*E(IM,JJ)

40 CONTINUE

C----------------------------------------------------------------------

c FORM ET*PERM*E

C----------------------------------------------------------------------

DO 41 II=1,NPN

DO 41 JJ=1,NPN

DO 41 KK=1,NDIM

41 ETE(II,JJ)=ETE(II,JJ)+E(KK,II)*PE(KK,JJ)*DTIME*F9/GAMMAW

C----------------------------------------------------------------------

C FORM LT

C----------------------------------------------------------------------

DO 42 II=1,NDV

DO 42 JJ=1,NPN

42 RLT(II,JJ)=RLT(II,JJ)+RN(II)*AA(JJ)*F9

C----------------------------------------------------------------------

C FORM G.DT

DO 50 II=1,NPN

DO 50 KK=1,NDIM

50 GDT(II)=GDT(II)+PE(KK,II)*BOD(KK)*DTIME*F9*POR

C----------------------------------------------------------------------

C CALCULATE THE ELEMENT COUPLING MATRIX [ECOUP]T

C----------------------------------------------------------------------

CALL ELCPT(AA,BL,FINV,DETF,ECOUPT,NPN,NDV,NS,F9)

C----------------------------------------------------------------------

C END OF INTEGRATION POINT LOOP

C----------------------------------------------------------------------

80 CONTINUE

C TRANSFORM THE EKSTIF & ECOUPT MATRIX TO COUNT FOR SKEW

BOUNDARIES

IF(NSKEW.GT.0) CALL

STFTRN(K,NDIM,NSKEW,EKSTIF,ECOUPT,NCONN,LT)

C

C WRITE(LOUT1,*)’EKSTIF MATRIX OF’,K

C DO I1=1,NB

C WRITE(LOUT1,1104)(EKSTIF(I1,J1),J1=1,20)

C ENDDO

C WRITE(LOUT1,*)’EFLOW MATRIX’

C DO I=1,NPN

C WRITE(LOUT1,1103)(EFLOW(I,J),J=1,NPN)

C ENDDO

C WRITE(LOUT1,*)’ECOUPT MATRIX’


C DO I3=1,NB

C WRITE(LOUT1,1103)(ECOUPT(I3,J3),J3=1,NPN)

C ENDDO

C----------------------------------------------------------------------

C FORM STIFFNESS MATRIX SG FROM EKSTIF, ECOUPT AND EFLOW

C----------------------------------------------------------------------

CALL ASSEMPLE(SG,DA,GDT,P,EKSTIF,EFLOW,ECOUPT,NWL,

1 NPN,NDIM,NDN,LT,ICPL,ITER)

II=0

DO J1=1,20

DO I1=1,J1

II=II+1

STIFF(I1,J1)=SG(II)

STIFF(J1,I1)=STIFF(I1,J1)

ENDDO

ENDDO

c WRITE(*,*)"4444"

C

C

c IF(IOUTP.EQ.1.AND.K.EQ.5) then

c WRITE(2,*)’EKSTIF MATRIX’

c DO I1=1,10

c WRITE(2,1104)(EKSTIF(I1,J1),J1=1,10)

c ENDDO

c

C WRITE(2,*)’EFLOW MATRIX’

C DO I3=1,NPN

C WRITE(2,1103)(EFLOW(I3,J3),J3=1,NPN),(ETE(I3,J3),J3=1,NPN)

C ENDDO

C WRITE(2,*)’ECOUPT MATRIX’

C DO I3=1,16

C WRITE(2,1103)(ECOUPT(I3,J2),J2=1,NPN),(RLT(I3,J3),J3=1,NPN)

C ENDDO

c ENDIF

1101 FORMAT(6E10.3)

1103 FORMAT(8E10.3,/)

1104 FORMAT(20E10.3,/)

C WRITE(LOUT1,*)’ELEMENT NO=’,K

C WRITE(LOUT1,*)’[D] MATRIX’

C DO I=1,NS

C WRITE(LOUT1,1101)(D(I,J),J=1,NS)

C ENDDO

C 1101 FORMAT(6E10.3,/)


C WRITE(LOUT1,*)’BLNR MATRIX’

C DO I=1,NS

C WRITE(LOUT1,1102)(BL(I,J),J=1,40)

C ENDDO

C 1102 FORMAT(40E10.3,/)

C WRITE(LOUT1,*)’FULL ELEMENT STIFFNESS MATRIX’

C DO I=1,68

C WRITE(LOUT1,1102)(STIFF(I,J),J=1,20)

C ENDDO

C 1102 FORMAT(20E10.3,/)

C

RETURN

END

C**********************************************************************

SUBROUTINE FORMP(J,NDIM,NPN,NAC,DS,SFP,CARTD,SLL,LT)

C**********************************************************************

C FORMS CARTD MATRIX FOR AREA COORDS SLL(NAC)

C IN TRIANGLE J FOR INTEGRATION POINT IP

C**********************************************************************



DIMENSION SLL(4)

DIMENSION DS(3,10),SFP(10),CARTD(3,NPMX)




C----------------------------------------------------------------------

C CALCULATE SHAPE FUNCTION AND DERIVATIVES (LOCAL COORDS)

C----------------------------------------------------------------------

CALL SHFNPP(LOUT1,SLL,NAC,DS,SFP,LT,1,J)

C

DO 35 IN=1,NPN

DO 35 ID=1,NDIM

SUM=ZERO

C

DO 30 JD=1,NDIM


35 CARTD(ID,IN)=SUM

RETURN

END

C**********************************************************************

SUBROUTINE JPC(J,NDIM,NPN,NAC,NPLAX,DS,CARTD,B,E,RN,AA,SLL,LT)

C----------------------------------------------------------------------

C CALCULATES SHAPE FUNCTIONS AND DERIVATIVES

C FOR EXCESS PORE PRESSURE VARIATION

C----------------------------------------------------------------------




DIMENSION DS(3,10),CARTD(3,NPMX),B(6,NB),

1 E(3,NPMX),RN(NB),AA(NPMX),SLL(4)


C

CALL FORMP(J,NDIM,NPN,NAC,DS,AA,CARTD,SLL,LT)

C----------------------------------------------------------------------

C FORM RN

C----------------------------------------------------------------------

NCOM=NDIM

IF(NPLAX.EQ.1.AND.NCOM.EQ.2)NCOM=NDIM+1

C

DO 30 IB=1,NB

SUM=ZERO

C

DO 20 ID=1,NCOM

20 SUM=SUM+B(ID,IB)

30 RN(IB)=SUM

C----------------------------------------------------------------------

C FORM E

C----------------------------------------------------------------------

DO 50 IN=1,NPN

DO 50 ID=1,NDIM

50 E(ID,IN)=CARTD(ID,IN)

RETURN

END

C**********************************************************************

SUBROUTINE ASSEMPLE(SG,DA,GDT,P,EKSTIF,EFLOW,ECOUPT,NWL,NPN,

1 NDIM,NDN,LT,ICPL,ITER)

C**********************************************************************

C FORM ELEMENT STIFFNESS MATRIX SG FROM EKSTIF, ECOUPT AND

EFLOW

C**********************************************************************



DIMENSION KP(29),KD(94),NXP(15),NXD(15)

DIMENSION SG(KES),DA(MDOF),P(MDOF),EKSTIF(NB,NB),

1 EFLOW(NPMX,NPMX),ECOUPT(NB,NPMX),NWL(NPMX),GDT(NPMX)

COMMON /PARS /PYI,ASMVL,ZERO


C----------------------------------------------------------------------

C INDEX TO ROWS/COLUMNS OF SG FOR ROWS/COLUMNS OF ETE

C INDEX TO COLUMNS OF SG FOE COLUMNS OF ECOUPT (FOR

CONSOLIDATION)

C----------------------------------------------------------------------

C----------ELEMENT TYPE 3 - LST----------------------------------------

DATA KP(1),KP(2),KP(3)/

1 3,6,9/


C----------ELEMENT TYPE 5 - QUADRILATERAL------------------------------

DATA KP(4),KP(5),KP(6),KP(7)/

1 3,6,9,12/

C----------ELEMENT TYPE 7 - CUST---------------------------------------

DATA KP(8),KP(9),KP(10),KP(11),KP(12),KP(13),KP(14),KP(15),

1 KP(16),KP(17)/

2 3,6,9,34,35,36,37,38,39,40/

C----------ELEMENT TYPE 9 - BRICK--------------------------------------

DATA KP(18),KP(19),KP(20),KP(21),KP(22),KP(23),KP(24),KP(25)/

2 4,8,12,16,20,24,28,32/

C----------ELEMENT TYPE 11 - TETRA-HEDRA-------------------------------

DATA KP(26),KP(27),KP(28),KP(29)/

1 4,8,12,16/

C----------------------------------------------------------------------

C INDEX TO FIRST DISPLACEMENT VARIABLE OF EACH NODE IN SG

C INDEX TO ROWS/COLUMNS OF SG FROM ROWS/COLUMNS OF SS

C INDEX TO ROWS OF SG FOR ROWS OF ECOUPT (FOR CONSOLIDATION

ELEMENT)

C----------------------------------------------------------------------

C----------ELEMENT TYPE 1(2), 2(6), 4(8), 6(15)------------------------

DATA KD(1),KD(2),KD(3),KD(4),KD(5),KD(6),KD(7),KD(8),KD(9),KD(10),

2 KD(11),KD(12),KD(13),KD(14),KD(15)/

3 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29/

C----------ELEMENT TYPE 8(20), 10(10)----------------------------------

DATA KD(16),KD(17),KD(18),KD(19),KD(20),KD(21),KD(22),KD(23),

1 KD(24),KD(25),KD(26),KD(27),KD(28),KD(29),KD(30),KD(31),

2 KD(32),KD(33),KD(34),KD(35)/

3 1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58/

C----------ELEMENT TYPE 3(6)-------------------------------------------

DATA KD(36),KD(37),KD(38),KD(39),KD(40),KD(41)/

1 1,4,7,10,12,14/

C----------ELEMENT TYPE 5(8)-------------------------------------------

DATA KD(42),KD(43),KD(44),KD(45),KD(46),KD(47),KD(48),KD(49)/

1 1,4,7,10,13,15,17,19/

C----------ELEMENT TYPE 7(15)------------------------------------------


1 KD(58),KD(59),KD(60),KD(61),KD(62),KD(63),KD(64)/

2 1,4,7,10,12,14,16,18,20,22,24,26,28,30,32/

C----------ELEMENT TYPE 9(20)------------------------------------------


1 KD(73),KD(74),KD(75),KD(76),KD(77),KD(78),KD(79),KD(80),

2 KD(81),KD(82),KD(83),KD(84)/

3 1,5,9,13,17,21,25,29,33,36,39,42,45,48,51,54,57,60,63,66/

C----------ELEMENT TYPE 11(10)-----------------------------------------

DATA KD(85),KD(86),KD(87),KD(88),KD(89),

1 KD(90),KD(91),KD(92),KD(93),KD(94)/

2 1,5,9,13,17,20,23,26,29,32/

C----------------------------------------------------------------------


C NXP AND NXD GIVE STARTING INDEX TO ARRAYS KP AND KD

C RESPECTIVELY FOR DIFFERENT ELEMENT TYPES

C----------------------------------------------------------------------

DATA NXP(1),NXP(2),NXP(3),NXP(4),NXP(5),NXP(6),NXP(7),

1 NXP(8),NXP(9),NXP(10),NXP(11)/

2 0,0,0,0,3,0,7,0,17,0,25/

DATA NXD(1),NXD(2),NXD(3),NXD(4),NXD(5),NXD(6),NXD(7),

1 NXD(8),NXD(9),NXD(10),NXD(11)/

2 0,0,35,0,41,0,49,15,64,15,84/

C----------------------------------------------------------------------

C---------- SIZE OF ARRAYS KP AND KD-----------------------------------

NKP=29

NKD=94

C----------------------------------------------------------------------

INXD=NXD(LT)

C----------BYPASS IF NOT COUPLED CONSOLIDATION

IF(ICPL.EQ.0)GOTO 96

C----------------------------------------------------------------------

C COUPLED CONSOLIDATION

C----------------------------------------------------------------------

INXP=NXP(LT)

C----------CALCULATE RIGHT HAND SIDE FOR PORE PRESSURES

DO 94 II=1,NPN

N1=NWL(II)

SUM=ZERO

C

DO 92 JJ=1,NPN

N2=NWL(JJ)

IF(ITER.EQ.1) THEN

SUM=SUM+EFLOW(II,JJ)*DA(N2)

ELSE

SUM=0.0D0

ENDIF

92 CONTINUE

LOCGLB=LOCDOF(N1)

94 P(LOCGLB)=P(LOCGLB)+SUM+GDT(II)

C----------------------------------------------------------------------

C FORM SG FROM EKSTIF

C----------------------------------------------------------------------

96 DO 150 J=1,NDN

NJ=KD(J+INXD)-1

C

DO 150 JD=1,NDIM

NJA=NJ+JD

JA=JD+(J-1)*NDIM

NCN=NJA*(NJA-1)/2

C

DO 150 I=1,NDN


NI=KD(I+INXD)-1

C

DO 140 ID=1,NDIM

NIA=NI+ID

IA=ID+(I-1)*NDIM

IF(NIA.GT.NJA)GOTO 140

LOC=NCN+NIA

SG(LOC)=EKSTIF(IA,JA)

140 CONTINUE

150 CONTINUE

C

IF(ICPL.EQ.0)GOTO 200

C----------------------------------------------------------------------

C SLOT ECOUPT

C----------------------------------------------------------------------

DO 160 JA=1,NPN

NJA=KP(JA+INXP)

NCN=NJA*(NJA-1)/2

C

DO 160 I=1,NDN

NI=KD(I+INXD)-1

C

DO 160 ID=1,NDIM

NIA=NI+ID

IA=ID+(I-1)*NDIM

LOC=NIA+NCN

IF(NIA.GT.NJA)LOC=NIA*(NIA-1)/2+NJA

160 SG(LOC)=ECOUPT(IA,JA)

C----------------------------------------------------------------------

C SLOT EFLOW

C----------------------------------------------------------------------

DO 180 JE=1,NPN

NJ=KP(JE+INXP)

NCN=NJ*(NJ-1)/2

C

DO 180 IE=1,JE

NI=KP(IE+INXP)

180 SG(NI+NCN)=-EFLOW(IE,JE)

200 CONTINUE

RETURN

END

C**********************************************************************

SUBROUTINE LSTIFA(EKSTIF,B,D,DB,F9,NS)

C**********************************************************************

C ROUTINE TO CALCULATE D*B AND BT*D*B

C FOR EACH INTEGRATION POINT

C**********************************************************************




DIMENSION EKSTIF(NB,NB),D(6,6),DB(NVRN,NB),B(6,NB)

C----------------------------------------------------------------------

C FORM D*B

C----------------------------------------------------------------------

CALL ZEROR2(DB,NVRN,NB)

C

DO 20 JJ=1,NB

DO 20 II=1,NS

DO 20 KK=1,NS

20 DB(II,JJ)=DB(II,JJ)+D(II,KK)*B(KK,JJ)

C----------------------------------------------------------------------

C FORM BT*D*B

C----------------------------------------------------------------------

DO 30 JJ=1,NB

DO 30 II=1,JJ

DO 30 KK=1,NS

30 EKSTIF(II,JJ)=EKSTIF(II,JJ)+DB(KK,JJ)*B(KK,II)*F9

DO 40 JJ=2,NB

JJM1=JJ-1

DO 40 II=1,JJM1

40 EKSTIF(JJ,II)=EKSTIF(II,JJ)

RETURN

END

C**********************************************************************

SUBROUTINE SHFNPP(LOUT1,SLL,NAC,DS,SFP,LT,IFL,MUS)

C**********************************************************************

C SHAPE FUNCTIONS AND DERIVATIVES FOR PORE PRESSURE VARIATION

C**********************************************************************



DIMENSION SFP(10),DS(3,10),SLL(4)

C

RL1=SLL(1)

RL2=SLL(2)

IF(NAC.LT.3)GOTO 10

RL3=SLL(3)

IF(NAC.LT.4)GOTO 10

RL4=SLL(4)

C

10 GOTO(80,80,13,80,25,80,37,80,49,80,71),LT


900 FORMAT(/1X,7HELEMENT,I5,2X,22HIS OF UNKNOWN TYPE ***,I5,2X,

1 16H(ROUTINE SHFNPP))

STOP

C----------------------------------------------------------------------


C LINEAR STRAIN TRIANGLE

C----------------------------------------------------------------------

13 IF(IFL.EQ.0)GO TO 23

DS(1,1)=1.0D0

DS(1,2)=0.0D0

DS(1,3)=-1.0D0

DS(2,1)=0.0D0

DS(2,2)=1.0D0

DS(2,3)=-1.0D0

C

23 SFP(1)=RL1

SFP(2)=RL2

SFP(3)=RL3

RETURN

C----------------------------------------------------------------------

C QUADRILATERAL ELEMENT

C----------------------------------------------------------------------


DS(1,1)=0.250D0*(RL2-1.0D0)

DS(1,2)=-0.250D0*(RL2-1.0D0)

DS(1,3)=0.250D0*(RL2+1.0D0)

DS(1,4)=-0.250D0*(RL2+1.0D0)

DS(2,1)=0.250D0*(RL1-1.0D0)

DS(2,2)=-0.250D0*(RL1+1.0D0)

DS(2,3)=0.250D0*(RL1+1.0D0)

DS(2,4)=-0.250D0*(RL1-1.0D0)

C

35 SFP(1)=0.250D0*(RL1-1.0D0)*(RL2-1.0D0)

SFP(2)=-0.250D0*(RL1+1.0D0)*(RL2-1.0D0)

SFP(3)=0.250D0*(RL1+1.0D0)*(RL2+1.0D0)

SFP(4)=-0.250D0*(RL1-1.0D0)*(RL2+1.0D0)

RETURN

C----------------------------------------------------------------------

C CUBIC VARIATION IN PORE-PRESSURE

C----------------------------------------------------------------------

37 C1=9.0D0/2.0D0

C2=27.0D0/2.0D0

C3=27.0D0

T11=RL1-1.0D0/3.0D0

T12=RL1-2.0D0/3.0D0

T21=RL2-1.0D0/3.0D0

T22=RL2-2.0D0/3.0D0

T31=RL3-1.0D0/3.0D0

T32=RL3-2.0D0/3.0D0

IF(IFL.EQ.0)GO TO 40

C

DS(1,1)=C1*(T11*T12+RL1*(T11+T12))

DS(1,2)=0.0D0


DS(1,3)=-C1*(T31*T32+RL3*(T31+T32))

DS(1,4)=C2*RL2*(RL1+T11)

DS(1,5)=C2*RL2*T21

DS(1,6)=-C2*RL2*T21

DS(1,7)=-C2*RL2*(RL3+T31)

DS(1,8)=C2*RL3*T31-C2*RL1*(RL3+T31)

DS(1,9)=C2*RL3*(RL1+T11)-C2*RL1*T11

DS(1,10)=C3*RL2*RL3-C3*RL2*RL1

C

DS(2,1)=0.0D0

DS(2,2)=C1*(T21*T22+RL2*(T21+T22))

DS(2,3)=-C1*(T31*T32+RL3*(T31+T32))

DS(2,4)=C2*RL1*T11

DS(2,5)=C2*RL1*(RL2+T21)

DS(2,6)=C2*RL3*(RL2+T21)-C2*RL2*T21

DS(2,7)=C2*RL3*T31-C2*RL2*(RL3+T31)

DS(2,8)=-C2*RL1*(RL3+T31)

DS(2,9)=-C2*RL1*T11

DS(2,10)=C3*RL1*RL3-C3*RL1*RL2

C

40 SFP(1) =C1*RL1*T11*T12

SFP(2) =C1*RL2*T21*T22

SFP(3) =C1*RL3*T31*T32

SFP(4) =C2*RL1*RL2*T11






SFP(10)=C3*RL1*RL2*RL3

RETURN

C----------------------------------------------------------------------

C PORE-PRESSURE SHAPE FUNCTIONS AND DERIVATIVES FOR BRICK ELEMENT

C----------------------------------------------------------------------


C DERIVATIVES

DS(1,1)=-0.1250D0*(1.0D0-RL2)*(1.0D0+RL3)

DS(1,2)= 0.1250D0*(1.0D0-RL2)*(1.0D0+RL3)

DS(1,3)= 0.1250D0*(1.0D0+RL2)*(1.0D0+RL3)

DS(1,4)=-0.1250D0*(1.0D0+RL2)*(1.0D0+RL3)

DS(1,5)=-0.1250D0*(1.0D0-RL2)*(1.0D0-RL3)

DS(1,6)= 0.1250D0*(1.0D0-RL2)*(1.0D0-RL3)

DS(1,7)= 0.1250D0*(1.0D0+RL2)*(1.0D0-RL3)

DS(1,8)=-0.1250D0*(1.0D0+RL2)*(1.0D0-RL3)

C

DS(2,1)=-0.1250D0*(1.0D0-RL1)*(1.0D0+RL3)

DS(2,2)=-0.1250D0*(1.0D0+RL1)*(1.0D0+RL3)

DS(2,3)= 0.1250D0*(1.0D0+RL1)*(1.0D0+RL3)


DS(2,4)= 0.1250D0*(1.0D0-RL1)*(1.0D0+RL3)

DS(2,5)=-0.1250D0*(1.0D0-RL1)*(1.0D0-RL3)

DS(2,6)=-0.1250D0*(1.0D0+RL1)*(1.0D0-RL3)

DS(2,7)= 0.1250D0*(1.0D0+RL1)*(1.0D0-RL3)

DS(2,8)= 0.1250D0*(1.0D0-RL1)*(1.0D0-RL3)

C

DS(3,1)= 0.1250D0*(1.0D0-RL1)*(1.0D0-RL2)

DS(3,2)= 0.1250D0*(1.0D0+RL1)*(1.0D0-RL2)

DS(3,3)= 0.1250D0*(1.0D0+RL1)*(1.0D0+RL2)

DS(3,4)= 0.1250D0*(1.0D0-RL1)*(1.0D0+RL2)

DS(3,5)=-0.1250D0*(1.0D0-RL1)*(1.0D0-RL2)

DS(3,6)=-0.1250D0*(1.0D0+RL1)*(1.0D0-RL2)

DS(3,7)=-0.1250D0*(1.0D0+RL1)*(1.0D0+RL2)

DS(3,8)=-0.1250D0*(1.0D0-RL1)*(1.0D0+RL2)

C SHAPE FUNCTIONS

45 SFP(1)=0.1250D0*(1.0D0-RL1)*(1.0D0-RL2)*(1.0D0+RL3)

SFP(2)=0.1250D0*(1.0D0+RL1)*(1.0D0-RL2)*(1.0D0+RL3)

SFP(3)=0.1250D0*(1.0D0+RL1)*(1.0D0+RL2)*(1.0D0+RL3)

SFP(4)=0.1250D0*(1.0D0-RL1)*(1.0D0+RL2)*(1.0D0+RL3)

SFP(5)=0.1250D0*(1.0D0-RL1)*(1.0D0-RL2)*(1.0D0-RL3)

SFP(6)=0.1250D0*(1.0D0+RL1)*(1.0D0-RL2)*(1.0D0-RL3)

SFP(7)=0.1250D0*(1.0D0+RL1)*(1.0D0+RL2)*(1.0D0-RL3)

SFP(8)=0.1250D0*(1.0D0-RL1)*(1.0D0+RL2)*(1.0D0-RL3)

RETURN

C----------------------------------------------------------------------

C TETRA-HEDRA ELEMENT

C----------------------------------------------------------------------

71 CONTINUE


910 FORMAT(/1X,7HELEMENT,I5,2X,14HIS OF TYPE ***,I5,2X,

1 31HNOT IMPLEMENTED (ROUTINE SHAPE))

80 RETURN

END

C**********************************************************************

SUBROUTINE ELFLOW(NDIM,NPN,FINV,DETF,E,F9,EFLOW,

1 PERM,GAMMAW,DTIME)

C**********************************************************************

C THIS SUBPROGRAM CALCULATES THE ELEMENT FLOW MATRIX

C**********************************************************************



DIMENSION FINV(3,3),E(3,NPMX),EFLOW(NPMX,NPMX),PERM(3)

DIMENSION PE(3,NPMX),TEMP(3,3)

C

CALL ZEROR2(TEMP,3,3)

CALL ZEROR2(PE,3,NPMX)


C CALCULATE THE FACTOR [CIJ]-1 * [CIJ]-1

DO I=1,NDIM

DO J=1,NDIM

TEM = 0.0D0

DO K=1,NDIM

TEM = TEM + FINV(I,K)*FINV(J,K)

END DO

TEMP(I,J)=TEM

END DO

END DO

C

TEMP1=0.0D0

DO I=1,NDIM

DO J=1,NDIM

TEMP1=TEMP1+TEMP(I,J)*TEMP(I,J)/NDIM

ENDDO

ENDDO

CNST=F9*DETF*TEMP1

C FORM PERM*E

C

DO 10 JJ=1,NPN

DO 10 IM=1,NDIM

10 PE(IM,JJ)=PERM(IM)*E(IM,JJ)

C--------------------------------------------------------------------

C FORM [EFLOW]

C--------------------------------------------------------------------

DO 20 II=1,NPN

DO 20 JJ=1,NPN

DO 20 KK=1,NDIM

20 EFLOW(II,JJ)=EFLOW(II,JJ)+E(KK,II)*PE(KK,JJ)*DTIME*CNST/GAMMAW

RETURN

END

C *********************************************************************

SUBROUTINE ELCPT(AA,BL,FINV,DETF,ECOUPT,NPN,NDV,NS,F9)

C *********************************************************************

C THIS SUBPROGRAM CALCULATES THE ELEMENT COUPLING MATRIX

C ---------------------------------------------------------------------



DIMENSION FINV(3,3),AA(NPMX),TEMP(3,3),BL(6,NB)

DIMENSION ECOUPT(NB,NPMX),TEMP1(NVRN),RN(NB)

C

CALL ZEROR1(TEMP1,NVRN)


CALL ZEROR2(TEMP,3,3)

CALL ZEROR1(RN,NB)

C CALCULATE [FINV]*[FINV]*KRONICOR DELTA =[TEMP]

DO 10 I=1,3

DO 10 J=1,3

TEM=0.0D0

DO 10 K=1,3

TEM=TEM+FINV(I,K)*FINV(J,K)

10 TEMP(I,J)=TEM

C CONVERT [TEMP] INTO VECTOR, MULTIPLY BY BL, PUT IN RN

TEMP1(1)=TEMP(1,1)

TEMP1(2)=TEMP(2,2)

TEMP1(3)=TEMP(3,3)

TEMP1(4)=TEMP(1,2)

C

DO 20 IB=1,NB

SUM=0.0D0

DO 30 ID=1,NS

30 SUM=SUM+BL(ID,IB)*TEMP1(ID)

20 RN(IB)=SUM

C CALCULATE THE ELEMENT COUPLING MATRIX

C [ECOUP]T=RN*AA*DETF*F9

DO 60 II=1,NDV

DO 60 JJ=1,NPN

60 ECOUPT(II,JJ)=ECOUPT(II,JJ)+RN(II)*AA(JJ)*F9*DETF

RETURN

END

C *********************************************************************

SUBROUTINE BLNR1(CARTD,BL1,NDIM,NDN)

C *********************************************************************

C THIS SUBPROGRAM CALCULATES THE B\_L* MATRIX

C ---------------------------------------------------------------------



DIMENSION CARTD(3,NDMX),BL1(9,NB)

C INITIALIZE B\_L MATRIX TO ZERO

CALL ZEROR2(BL1,9,NB)

C ASSIGN THE APPROPRIATE VALUES TO THE APPROPRIATE ELEMENTS OF

B\_L

C

IF(NDIM.NE.2) GO TO 10

C


C 2-DIMENSIONAL ELEMENT

DO IELN=1,NDN

IDOF1=2*IELN-1

IDOF2=IDOF1+1

BL1(1,IDOF1)=CARTD(1,IELN)




ENDDO

10 IF(NDIM.NE.3) GO TO 50


DO IELN=1,NDN

IDOF1=3*IELN-2

IDOF2=IDOF1+1

IDOF3=IDOF2+1










ENDDO

50 CONTINUE

RETURN

END

C *********************************************************************

SUBROUTINE BLNR(NDIM,NDN,NPLAX,RI,SHFN,CARTD,BL)

C *********************************************************************

C THIS SUBPROGRAM IN CALCULATES THE B\_L MATRIX

C ---------------------------------------------------------------------



DIMENSION SHFN(20),BL(6,NB),CARTD(3,NDMX)

C INITIALIZE B\_L MATRIX TO ZERO

CALL ZEROR2(BL,6,NB)

C


C----------------------------------------------------------------------


C----------------------------------------------------------------------

DO IELN=1,NDN


IDOF1=2*IELN-1

IDOF2=IDOF1+1

BL(1,IDOF1)=CARTD(1,IELN)


IF(NPLAX.NE.1) GO TO 20

BL(3,IDOF1)=SHFN(IELN)*RI

20 BL(4,IDOF1)=CARTD(2,IELN)


ENDDO


C----------------------------------------------------------------------


C----------------------------------------------------------------------

DO IELN=1,NDN

IDOF1=IELN*3-2

IDOF2=IDOF1+1

IDOF3=IDOF2+1










ENDDO

50 CONTINUE

RETURN

END

C *********************************************************************

SUBROUTINE BNLNRS(SHFN,CARTD,RI,BNLS,NDIM,NDN,NPLAX)

C *********************************************************************

C THIS SUBPROGRAM CALCULATES GEOMETRIC NONLIN STRESS-DISP

MATRIX B\_NLS.

C ---------------------------------------------------------------------



DIMENSION SHFN(20),CARTD(3,NDMX),

. BNLS(9,NB)

C INITIALIZE B\_NL MATRIX

CALL ZEROR2(BNLS,9,NB)


C ASSIGN THE APPROPRIATE VALUES TO THE SPECIFIC ELEMENTS OF B\_NLS


C


DO IELN=1,NDN

IDOF1=2*IELN-1

IDOF2=IDOF1+1

BNLS(1,IDOF1)=CARTD(1,IELN)



BNLS(3,IDOF1)=SHFN(IELN)*RI



GO TO 30

20 BNLS(3,IDOF2)=CARTD(1,IELN)


30 CONTINUE

ENDDO

C



DO IELN=1,NDN

IDOF1=IELN*3-2

IDOF2=IDOF1+1

IDOF3=IDOF2+1










END DO

50 CONTINUE

RETURN

END

C *********************************************************************

SUBROUTINE CONVERT(S,SPK,NDIM,NPLAX)

C *********************************************************************

C THIS SUBPROGRAM CONVERTS THE S VECTOR CONSISTING OF 4 COMPONENTS

C INTO SPK MATRIX OF 5X5

C --------------------------------------------------------------------



DIMENSION S(6),SPK(9,9)


C INITIALIZE SPK TO ZERO

DO I=1,9

DO J=1,9

SPK(I,J)=0.0D0

ENDDO

ENDDO

C ASSIGN THE ELEMENTS OF S APPROPRIATELY IN SPK



SPK(1,1)=S(1)

SPK(2,2)=S(2)


SPK(3,3)=S(3)

SPK(4,4)=S(1)

SPK(5,5)=S(2)

SPK(4,5)=S(4)

SPK(5,4)=S(4)

SPK(1,2)=S(4)

SPK(2,1)=S(4)

GO TO 10

20 SPK(3,3)=S(1)

SPK(4,4)=S(2)

SPK(1,2)=S(3)

SPK(2,1)=S(3)

SPK(3,4)=S(3)

SPK(4,3)=S(3)


DO K=1,NDIM

K1=3*K-2

K2=K1+1

K3=K1+2

SPK(K1,K1)=S(1)

SPK(K2,K2)=S(2)

SPK(K3,K3)=S(3)

SPK(K1,K2)=S(4)

SPK(K2,K1)=S(4)

SPK(K2,K3)=S(5)

SPK(K3,K2)=S(5)

SPK(K1,K3)=S(6)

SPK(K3,K1)=S(6)

ENDDO

40 CONTINUE

RETURN

END


C *********************************************************************

SUBROUTINE BTDB(B,D,ESTIF,CNSTIP,NS)

C *********************************************************************

C THIS SUBPROGRAM CALCULATE THE MATRIX PRODUCT B\^T D B GIVEN B AND D

C MATRICES.

C ---------------------------------------------------------------------



DIMENSION B(9,NB),D(9,9),ESTIF(NB,NB),TEMP(9,NB)

DO I=1,NS

DO J=1,NB

TEM=0.0D0

DO K=1,NS

TEM=TEM+D(I,K)*B(K,J)

ENDDO

TEMP(I,J)=TEM

ENDDO

ENDDO

DO I=1,NB

DO J=1,NB

TEM=0.0D0

DO K=1,NS

TEM=TEM+B(K,I)*TEMP(K,J)

ENDDO

ESTIF(I,J)=ESTIF(I,J)+TEM*CNSTIP

ENDDO

ENDDO

RETURN

END

C *********************************************************************

SUBROUTINE BNLNR(EDISP,SHFN,CARTD,RI,NDIM,NDN,BNL,NS,NPLAX)

C *********************************************************************

C THIS SUBPROGRAM CALCULATES THE NONLIN STRESS-DISP. MATRIX B\_NL

C ---------------------------------------------------------------------



DIMENSION SHFN(20),CARTD(3,NDMX),EDISP(NB),BNL(6,NB)

DIMENSION BB(NVRN+1,NB),C(NVRN+1)

NS1=NS+1

C INITIALIZE MATRICES

CALL ZEROR2(BB,NVRN+1,NB)

CALL ZEROR2(BNL,6,NB)



C----------------------------------------------------------------------


C----------------------------------------------------------------------

C FIND THE INTERMEDIATE MATRIX BB

DO IELN=1,NDN

IDOF1=IELN*2-1

IDOF2=IDOF1+1

BB(1,IDOF1)=CARTD(1,IELN)



BB(3,IDOF1)=SHFN(IELN)*RI

20 BB(4,IDOF1)=CARTD(2,IELN)


ENDDO

C FIND [C]=[BB][U]

DO 1 I=1,NS1

C(I)=0.0D0

DO 1 J=1,NB

1 C(I)=C(I)+BB(I,J)*EDISP(J)

C CALCULATE BNL MARIX

DO IELN=1,NDN

IDOF1=IELN*2-1

IDOF2=IDOF1+1

BNL(1,IDOF1)=C(1)*CARTD(1,IELN)





BNL(3,IDOF1)=C(3)*SHFN(IELN)*RI

15 BNL(4,IDOF1)=C(4)*CARTD(1,IELN)+C(1)*CARTD(2,IELN)

BNL(4,IDOF2)=C(2)*CARTD(1,IELN)+C(5)*CARTD(2,IELN)

ENDDO

C


C----------------------------------------------------------------------


C----------------------------------------------------------------------

C FIND THE INTERMEDIATE MATRIX BB

DO IELN=1,NDN

IDOF1=IELN*3-2

IDOF2=IDOF1+1

IDOF3=IDOF2+1





ENDDO

100 CONTINUE

RETURN

END

C**********************************************************************

SUBROUTINE DEFGRAD(BL1,F,FINV,DETF,EDISPINC,NDIM,LINR)

C**********************************************************************

C THIS SUBPROGRAM CALCULATES THE DEFORMATION GRADIENT MATRIX [F],

C [FINV] AND DET[F]

C----------------------------------------------------------------------



DIMENSION EDISPINC(NB),BL1(9,NB)

DIMENSION UDRV(9),F(3,3),FINV(3,3),ff(3,3)

NDD=NDIM*NDIM

CALL ZEROR1(UDRV,9)

CALL ZEROR2(F,3,3)

CALL ZEROR2(Ff,3,3)

CALL ZEROR2(FINV,3,3)

C----------------------------------------------------------------------

C FIND THE DERIVATIVES OF DISPLACEMENTS U1,1 U1,2 U2,1 U2,2 FOR 2-D

C AND U1,1 U1,2 U1,3 U2,1 U2,2 U2,3 U3,1 U3,2 U3,3 FOR 3-D

C----------------------------------------------------------------------

DO I=1,NDD !ndd=ndim*ndim

TEMP=0.0D0

DO J=1,NB !NB=60 from parm.for

TEMP=TEMP+BL1(I,J)*EDISPINC(J)

!BL1=linear strain-displ matrix

!edispinc=element incremental displacement=EDINC.

!temp=accumulated strain???? Wrong logic?

ENDDO

UDRV(I)=TEMP

ENDDO

C FIND THE DEFORMATION GRADIENT F

DO I=1,3

DO J=1,3

F(I,J)=0.0D0

FINV(I,J)=0.0D0

IF(I.EQ.J)THEN

F(I,J)=1.0D0

FINV(I,J)=1.0D0


ENDIF

ENDDO

ENDDO

IF(LINR.GT.0) THEN



F(1,1)=1.0D0+UDRV(1)

F(2,2)=1.0D0+UDRV(4)

F(3,3)=1.0D0

F(1,2)=UDRV(2)

F(2,1)=UDRV(3)

GO TO 101


10 F(1,1)=1.0D0+UDRV(1)

F(1,2)=UDRV(2)

F(1,3)=UDRV(3)

F(2,1)=UDRV(4)

F(2,2)=1.0D0+UDRV(5)

F(2,3)=UDRV(6)

F(3,1)=UDRV(7)

F(3,2)=UDRV(8)

F(3,3)=1.0D0+UDRV(9)

101 CONTINUE

END IF

C FIND THE INVERSE OF THE DEFORMATION GRADIENT FINV



DETF=F(1,1)*F(2,2)-F(1,2)*F(2,1) !DET(F) IS J

FINV(1,1)=F(2,2)/DETF

FINV(2,2)=F(1,1)/DETF

FINV(3,3)=1.0D0

FINV(1,2)=-F(1,2)/DETF

FINV(2,1)=-F(2,1)/DETF

GO TO 100


15 FINV(1,1)= (F(2,2)*F(3,3)-F(2,3)*F(3,2))

FINV(1,2)=-(F(1,2)*F(3,3)-F(1,3)*F(3,2))

FINV(1,3)= (F(1,2)*F(2,3)-F(1,3)*F(2,2))

C

FINV(2,1)=-(F(2,1)*F(3,3)-F(2,3)*F(3,1))

FINV(2,2)= (F(1,1)*F(3,3)-F(1,3)*F(3,1))

FINV(2,3)=-(F(1,1)*F(2,3)-F(1,3)*F(2,1))


C

FINV(3,1)= (F(2,1)*F(3,2)-F(2,2)*F(3,1))

FINV(3,2)=-(F(1,1)*F(3,2)-F(1,2)*F(3,1))

FINV(3,3)= (F(1,1)*F(2,2)-F(2,1)*F(1,2))

C

DETF=F(1,1)*FINV(1,1)+F(1,2)*FINV(2,1)+

1 F(1,3)*FINV(3,1)

C

DO 35 ID=1,NDIM

DO 35 JD=1,NDIM

35 FINV(ID,JD)=FINV(ID,JD)/DETF

100 CONTINUE

c

DETF=Finv(1,1)*Finv(2,2)-Finv(1,2)*Finv(2,1) !DET(F) IS J

c DETF=1.0d0 !DET(F) IS J

RETURN

END

C *********************************************************************

SUBROUTINE ADDBMAT(BL,BNL)

C *********************************************************************



DIMENSION BL(6,NB),BNL(6,NB)

DO I=1,6

DO J=1,NB

BL(I,J)=BL(I,J)+BNL(I,J)

ENDDO

ENDDO

RETURN

END

C**********************************************************************

SUBROUTINE STFTRN(IELEM,NDIM,NSKEW,EKSTIF,ECOUPT,NCONN,LT)

C**********************************************************************

C THIS SUBRPROGRAM TRANFORMS THE STIFFNESS MATRIX & COUPLING MATRIX

C TO ACCOUNT FOR THE ROTATION OF DOFS IN THE SKEW BOUNDARIES

C----------------------------------------------------------------------




COMMON/SKBC /ISPB(20),DIRCOS(20,3)

DIMENSION EKSTIF(NB,NB),ECOUPT(NB,NPMX),ROTMAT(3,3)

DIMENSION NCONN(NTPE,MEL),TEMP(3),TEMP1(3)


C

NDPT =LINFO(1,LT)

NEDOF=LINFO(16,LT)

NPN=LINFO(6,LT)

C

DO 100 I=1,NDPT

NODE=ABS(NCONN(I,IELEM))

DO ISKEW=1,NSKEW

INODE=ISPB(ISKEW)

IF(INODE.EQ.NODE) THEN

K1=NDIM*(I-1)

CALL ROTM(DIRCOS,ROTMAT,NDIM,ISKEW)

DO J = 1,NEDOF

DO K = 1,NDIM

TEMP( K ) = 0.0D0

DO IDOF = 1,NDIM

ID = K1 + IDOF

TEMP(K) = TEMP(K)+EKSTIF(J,ID)*ROTMAT(IDOF,K)

ENDDO

ENDDO

DO K = 1,NDIM

ID = K1 + K

EKSTIF(J,ID) = TEMP( K )

ENDDO

ENDDO

C

DO J = 1,NEDOF

DO K = 1,NDIM

TEMP(K) = 0.0D0

DO IDOF = 1,NDIM

ID = K1 + IDOF

TEMP(K) = TEMP(K)+EKSTIF(ID,J)*ROTMAT(IDOF,K)

ENDDO

ENDDO

DO K = 1, NDIM

ID = K1 + K

EKSTIF(ID,J) =TEMP(K)

ENDDO

ENDDO

C

DO J = 1,NPN

DO K = 1,NDIM

TEMP1(K)= 0.0D0

DO IDOF = 1,NDIM


ID = K1 + IDOF

TEMP1(K)= TEMP1(K)+ECOUPT(ID,J)*ROTMAT(IDOF,K)

ENDDO

ENDDO

DO K = 1, NDIM

ID = K1 + K

ECOUPT(ID,J)=TEMP1(K)

ENDDO

ENDDO

C

ENDIF

ENDDO

100 CONTINUE

RETURN

END

SUBROUTINE ROTBC(R,NW,NDIM,NSKEW,NUM)

C*********************************************************************

C THIS SUBPROGRAM ROTATE BACK THE OBTAINED CURRENT DISPLACEMENTS TO

C THE ORIGINAL COORD. SYSTEM AT THE SKEW BOUNDARY.

C----------------------------------------------------------------------




COMMON/SKBC /ISPB(20),DIRCOS(20,3)

DIMENSION R(MDOF),NW(MNODES+1)

DIMENSION TEMP(6),ROTMAT(3,3)

C

DO ISKEW=1,NSKEW

INODE=ISPB(ISKEW)

ICODE=NW(INODE)-1

C

C ----- Call ROTM ! Cal Rotation Matrix

CALL ROTM(DIRCOS,ROTMAT,NDIM,ISKEW)

DO I=1,NDIM

TEMP1=0.0D0

DO J=1,NDIM

LOC=ICODE+J

IF(NUM.GT.0) TEMP1=TEMP1+R(LOC)*ROTMAT(J,I)

IF(NUM.LT.0) TEMP1=TEMP1+R(LOC)*ROTMAT(I,J)


ENDDO

TEMP(I)=TEMP1

ENDDO

C

DO I=1,NDIM

LOC=ICODE+I

R(LOC)=TEMP(I)

ENDDO

ENDDO

RETURN

END

SUBROUTINE ROTM(DIRCOS,ROTMAT,NDIM,ISKEW)

C**********************************************************************

C THIS SUBPROGRAM FINDS THE ROTATION MATRIX NEEDED FOR ROTATING

C THE DOFS W.R.T. THAT NODE

C----------------------------------------------------------------------

C 1 CALLED BY ROTBC

C 2 CALLED BY ROTBC BY EQLOD



DIMENSION DIRCOS(20,3),ROTMAT(3,3)

IF(NDIM.EQ.3) GO TO 10

C ASSIGN COS(TH\_X) & COS(TH\_Y) FOR TWO DIMENSION

ROTMAT(1,1)= DIRCOS(ISKEW,1)

ROTMAT(2,1)=-DIRCOS(ISKEW,2)



10 RETURN

END

C**********************************************************************

SUBROUTINE DCON(I,IET,MAT,PR,D,NDIM,BK)

C**********************************************************************

C CALCULATES STRESS-STRAIN MATRIX FOR ANISOTROPIC ELASTICITY

C**********************************************************************



DIMENSION MAT(MEL),D(6,6),PR(NPR,NMT)


C

KM=MAT(I)

AN=PR(1,KM)/PR(2,KM)

A=PR(2,KM)/((1.0D0+PR(3,KM))*(1.0D0-PR(3,KM)-2.0D0*AN*PR(4,KM)*

1 PR(4,KM)))

D(1,1)=A*AN*(1.0D0-AN*PR(4,KM)*PR(4,KM))

D(1,2)=A*AN*PR(4,KM)*(1.0D0+PR(3,KM))

D(1,3)=A*AN*(PR(3,KM)+AN*PR(4,KM)*PR(4,KM))

D(2,1)=D(1,2)

D(2,2)=A*(1.0D0-PR(3,KM)*PR(3,KM))

D(2,3)=D(1,2)

D(3,1)=D(1,3)

D(3,2)=D(2,3)

D(3,3)=D(1,1)

D(4,4)=PR(5,KM)

BK=(D(2,2)+2.0D0*D(2,1))/3.0D0

IF(NDIM.EQ.2)GOTO 5

D(5,5)=PR(5,KM)

D(6,6)=PR(5,KM)

5 IF(IET.EQ.0) GO TO 20

C

DO 10 J=1,3

DO 10 JJ=1,3

10 D(JJ,J)=D(JJ,J)+PR(7,KM)*BK

20 RETURN

END

C**********************************************************************

SUBROUTINE DLIN(I,IET,NDIM,NDN,ELCOD,SHFN,MAT,D,PR,BK)

C**********************************************************************

C CALCULATES STRESS-STRAIN MATRIX FOR LINEAR ELASTIC

C BEHAVIOUR WHEN ELASTIC PROPERTIES VARY LINEARLY WITH DEPTH

C**********************************************************************



DIMENSION ELCOD(3,NDMX),SHFN(20),D(6,6)

DIMENSION MAT(MEL),PR(NPR,NMT)


C

KM=MAT(I)

CC IPA=IP+INDX

YY=ZERO

DO 5 IN=1,NDN

5 YY=YY+SHFN(IN)*ELCOD(2,IN)

E=PR(1,KM)+PR(3,KM)*(PR(2,KM)-YY)

G=E/(2.0D0*(1.0D0+PR(4,KM)))

A=E/((1.0D0+PR(4,KM))*(1.0D0-2.0D0*PR(4,KM)))


BK=E/(3.0D0*(1.0D0-2.0D0*PR(4,KM)))

D(1,1)=A*(1.0D0-PR(4,KM))

D(1,2)=A*PR(4,KM)

D(1,3)=D(1,2)

D(2,1)=D(1,2)

D(2,2)=D(1,1)

D(2,3)=D(1,3)

D(3,1)=D(1,3)

D(3,2)=D(2,3)

D(3,3)=D(1,1)

D(4,4)=G

IF(NDIM.EQ.2)GOTO 8

D(5,5)=G

D(6,6)=G

8 IF(IET.EQ.0)RETURN

DO 10 J=1,3

DO 10 JJ=1,3

10 D(JJ,J)=D(JJ,J)+PR(7,KM)*BK

RETURN

END

C**********************************************************************

SUBROUTINE DMCAM(IP,I,IET,NDIM,NS,STRESS,MAT,D,PR,BK,ITER,JS,DAB,

+SALPHA,A,H)

C**********************************************************************

C CALCULATES STRESS-STRAIN MATRIX [Dep] FOR MODIFIED CAM-CLAY

C**********************************************************************

C IP=Gauss point, I=element number, IET=iteration number

C ITER=iteration number, JS=increment number



C I=ELEMENT NUMBER


COMMON/BSTRESS/ PREP,PREQ,PREETA,PREALPH !back stress

variable

COMMON/NORMFACT/XI

DIMENSION STRESS(NVRS,NIP,MEL),D(6,6),MAT(MEL)

DIMENSION S(6),A(6),B(6),PR(NPR,NMT)

DIMENSION DUM4(4),DUM5(4),HPA(4,4),DEPSI(4),DAB(3,3),DABI(3,3)

DIMENSION SALPHA(4),SALPH22(4),SALPH11(4),BB21(4),BB31(4)

DIMENSION DUM1(4),DELALPH(4)

dimension epsiequi(2,170,4),epsiaccu(2,170,4,3,3),graddab(3,3)

C I=ELEMENT NUMBER

C

KM=MAT(I)

PRM=PR(4,KM)

ICOD=MCODE(IP,I)


SX=STRESS(1,IP,I) !new sigma x

SY=STRESS(2,IP,I) !new sigma y and so and so

SZ=STRESS(3,IP,I)

TXY=STRESS(4,IP,I)

E=STRESS(NS+2,IP,I) !Void ratio

PC=ABS(STRESS(NS+3,IP,I))

c Correction of stress for rate dependency ------- 08/13/03

call rate(epsiequi,epsiaccu,i,ip)

c call rate1(dabequi,dab)

viscos=pr(1,2)

pm1=pr(2,2) !pm1=m1 just for making real number

SX=STRESS(1,IP,I)*(1+(epsiequi(1,i,ip)/viscos)**pm1) !

SY=STRESS(2,IP,I)*(1+(epsiequi(1,i,ip)/viscos)**pm1)

SZ=STRESS(3,IP,I)*(1+(epsiequi(1,i,ip)/viscos)**pm1)

delsx=STRESS(1,IP,I)*((epsiequi(1,i,ip)/viscos)**pm1)

delsy=STRESS(2,IP,I)*((epsiequi(1,i,ip)/viscos)**pm1)

delsz=STRESS(3,IP,I)*((epsiequi(1,i,ip)/viscos)**pm1)

TXY=STRESS(4,IP,I)+0.5*(delsx-delsy) !! correct?

c Correction of py for rate dependency ----------- 08/07/03

c viscos=pr(1,2)

c pm1=pr(2,2) !pm1=m1 just for making real number

c pc=(1+(epsiequi(1,i,ip)/viscos)**pm1)*pc !i=ien in sub-rate

c write(*,*)pc

c pause

P=(SX+SY+SZ)/3.0D0

Q2=SX*(SX-SY)+SY*(SY-SZ)+SZ*(SZ-SX)+3.0D0*TXY*TXY

!general exp.


C

TYZ=STRESS(5,IP,I)

TZX=STRESS(6,IP,I)

Q2=Q2+3.0D0*TYZ*TYZ+3.0D0*TZX*TZX

10 Q=SQRT(Q2)

ETA=Q/P

c write(*,*)pr(4,km)

PY=P+Q*Q/(P*PR(4,KM)*PR(4,KM))

c Correction of py for rate dependency ----------- 08/07/03

c viscos=pr(1,2)

c pm1=pr(2,2) !pm1=m1 just for making real number

c py=(1+(epsiequi(1,i,ip)/viscos)**pm1)*py !i=ien in sub-rate

c if(epsiequi(1,i,ip).gt.0.0)write(*,*)"999",py,pc

BK=(1.0D0+E)*P/PR(1,KM) !K constrained modulus

C----------------------------------------------------------------------

C CALCULATE ELASTIC STRESS-STRAIN MATRIX

C----------------------------------------------------------------------

G=PR(5,KM) !G=Poisson’s ratio


IF(G.LT.1.0D0) G=BK*1.50D0*(1.0D0-2.0D0*PR(5,KM))/(1.0D0+PR(5,KM))

AL=(3.0D0*BK+4.0D0*G)/3.0D0

DL=(3.0D0*BK-2.0D0*G)/3.0D0

C

CALL ZEROR2(D,6,6)

D(1,1)=AL

D(2,1)=DL

D(3,1)=DL

D(1,2)=DL

D(2,2)=AL

D(3,2)=DL

D(1,3)=DL

D(2,3)=DL

D(3,3)=AL

D(4,4)=G


D(5,5)=G

D(6,6)=G

C

12 IF(PY.LT.0.9950d0*PC) GO TO 50 !PY=Po !bypass

C----------------------------------------------------------------------

C CALCULATE PLASTIC STRESS-STRAIN MATRIX IF CURRENT

C POINT ON YIELD LOCUS AND SET PC NEGATIVE

C----------------------------------------------------------------------

PCS=.50D0*PC

PB=P/PCS

S(1)=SX-P

S(2)=SY-P

S(3)=SZ-P

S(4)=2.0D0*TXY


S(5)=2.0D0*TYZ

S(6)=2.0D0*TZX

C -----------------------------

16 CONTINUE

c Adjustment of Dab for gradient

gradcon=pr(5,2)

volepsi=dab(1,1)+dab(2,2)-dab(1,1)*dab(2,2)

c write(*,*)"vol",volepsi,dab(1,1)

do ii=1,3

do jj=1,3

c if(volepsi.ge.0)goto 4 !consider gradient when volume expand

c subtraction for volume exp. addition for volume contr.


graddab(ii,jj)=dab(ii,jj)-gradcon*volepsi

4 enddo

enddo

c Turn back gradient dab to dab for further calculation

do ii=1,3

do jj=1,3

dab(ii,jj)=graddab(ii,jj)

enddo

enddo

C Correction to avoid tension failure

C DO II=1,3

C IF(S(II).LE.0.0)S(II)=0.0

C ENDDO

IF(STRESS(1,IP,I).LE.10.0)GOTO 200



c Calculate the back stress.

17 CALL BKSTRS2(P,Q,ETA,PRM,SALPH22,PR,ITER,JS,IP,DELALPH,I,S)

c i=ien=element number watch out!!!!!

CALL BKSTRS1(P,Q,ETA,PRM,PR,ITER,JS,IP,I,SALPH11,DAB,

1epsiaccu)

c write(*,*) "song"

c write(*,*) epsiaccu(1,1,1,1,1)

SALPHA(1)=SALPH22(1)+SALPH11(1)




c write(*,*)"sapha",salpha(1),SALPH11(1),SALPH22(1)

GOTO 210

C -----------------------------

C---BB=(df/dp)(dp/dsig) without Kronecker dij 3x3 tensor

C--- C=(df/dq)(dq/dsig) without Sij

C--- A=Bij

C--- S=Sij

200 do ii=1,4

salpha(ii)=0 ! 11,22,33,13

delalph(ii)=0

enddo

C---Calculate Bij

210 BB11=(2.0d0*P-PC)*PRM**2.0d0

BB11=BB11/3 ! really x delta ij


BB12=0D0

DO II=1,4

BB12=BB12+1.5D0*(PC*SALPHA(II)*SALPHA(II)-2D0*SALPHA(II)

1*S(II))

c !above term is scalar quantity.

ENDDO

BB12=BB12/3

BB21(1)=(2D0/3D0)*S(1)-(1D0/3D0)*S(2)-(1D0/3D0)*S(3)

BB21(1)=3*BB21(1)

BB21(2)=-(1D0/3D0)*S(1)+(2D0/3D0)*S(2)-(1D0/3D0)*S(3)

BB21(2)=3*BB21(2)

BB21(3)=-(1D0/3D0)*S(1)-(1D0/3D0)*S(2)+(2D0/3D0)*S(3)

BB21(3)=3*BB21(3)

BB21(4)=(2D0/3D0)*S(4)

BB21(4)=3*BB21(4)

BB31(1)=3D0*P*((2D0/3D0)*SALPHA(1)-(1D0/3D0)*SALPHA(2)-

6(1D0/3D0)*SALPHA(3))

BB31(2)=3D0*P*(-(1D0/3D0)*SALPHA(1)+(2D0/3D0)*SALPHA(2)-

1(1D0/3D0)*SALPHA(3))

BB31(3)=3D0*P*(-(1D0/3D0)*SALPHA(1)-(1D0/3D0)*SALPHA(2)+

1(2D0/3D0)*SALPHA(3))

BB31(4)=3D0*P*(2D0/3D0)*SALPHA(4)

DO II=1,3

A(II)=BB11+BB12+BB21(II)-BB31(II)

ENDDO

A(4)=0+0+BB21(4)-BB31(4)

BII=A(1)+A(2)+A(3) !Bii tr(df/dsigma)

c WRITE(*,*)"111"

C Complete Bij !A=Bij matrix

18 DO 20 J=1,3

B(J)=0.0D0

DO 20 JJ=1,3

20 B(J)=B(J)+D(J,JJ)*A(JJ) ! B=[C]ijkl[B]kl = (De)(df/dsig)

B(4)=D(4,4)*A(4)


B(5)=D(5,5)*A(5)

B(6)=D(6,6)*A(6)

c WRITE(*,*)"222"

C---df/dev

25 XI=(1.0D0+E)/(PR(2,KM)-PR(1,KM)) !XI=(1+e)/(lambda-kappa)

ALPHA=0D0 !ALPHA ij Alpha ij

DO II=1,4

ALPHA=ALPHA+SALPHA(II)*SALPHA(II) !ALPHA=scalar, aijxaij sum


ENDDO

AA1=XI*P*PC*(-2D0*PRM**2D0+3D0*ALPHA) !SONG df/d epsilon vp

AA2=BII

AA=AA1*AA2 !AA=[Hp] currently negative

H=AA !To transfer H to the next routine

C WRITE(*,*)"AA1,AA2",AA1,AA2,P,PC,ALPHA

C---Compute the additional change of stiffness matrix by back stress.

C by Chung R. Song 3/08/99

DUM2=0

DO II=1,4

DUM1(II)=3D0*(P*PC*SALPHA(II)-P*S(II)) !df/dalpha

DELALPH(II)=DELALPH(II)

DUM2=DUM2+DUM1(II)*DELALPH(II) ! (df/dalpha)(dalpha)

SCALAR !magic

ENDDO

C WRITE(*,*)"333"

C----- Inversion of depsilon

C DAB=d edpsilon, DABI=inverse of d epsilon

C DEPS=DAB This is the vectorial form Dab Song

C DAB is depsi !3/23/99 Song

C WRITE(*,*)"DAB",DAB(1,1),DAB(1,2)

C WRITE(*,*)"DAB",DAB(2,1),DAB(2,2),ITER

DJACOB1=DAB(1,1)*(DAB(2,2)*DAB(3,3)-DAB(3,2)*DAB(2,3))



DJACOB=DJACOB1-DJACOB2+DJACOB3

C DJACOB=DAB(1,1)*DAB(2,2)-DAB(1,2)*DAB(2,1) !radial strain = 0

IF(DJACOB.EQ.0)GOTO 27 !skip singular condition

CALL DETMIN1(LOUT1,NDIM,DAB,DABI,DJACB,JL,IP,ISTGE)

!special

c IF(ITER.EQ.1)GOTO 27

C CALL DETMIN1(LOUT1,NDIM,DAB,DABI,DJACB,JL,IP,ISTGE)

C DABI(1,1)=DAB(2,2)/DJACOB


C DABI(1,2)=-DAB(2,1)/DJACOB


GOTO 75

27 DO 70 JJJ1=1,4

70 DEPSI(JJJ1)=0D0

GOTO 77


C WRITE(*,*)"444"

C-----Change the depsilon\_1 tensor to depsilon\_1 vector

75 DEPSI(1)=DABI(1,1)

DEPSI(2)=DABI(2,2)

DEPSI(3)=0.0 !DABI(3,3)

DEPSI(4)=DABI(1,2) !Axi-symmetric

C-----Substitute depsilon inverse, into [Hpa] equation

77 CONTINUE

DO JS1=1,4

DUM5(JS1)=0.0D0 !clear

ENDDO

C WRITE(*,*)"555"

DO 90 JS1=1,4 !2-D case, Axisymmetric

DUM4(JS1)=DUM2*A(JS1) ! (df/dalpha)(dalpha)[B] vectorized tensor

DO 91 JS2=1,4 !2-D case

91 DUM5(JS1)=DUM5(JS1)+D(JS1,JS2)*DUM4(JS2)

!2nd oderized 4th order,sum tensor

DO 90 JS2=1,4

90 HPA(JS1,JS2)=(1D0/3D0)*DUM5(JS1)*DEPSI(JS2)

![HPA]=(1/3)[C](df/dalpha)(dalpha)[B][depsil]

IF(ETA.LT.PRM.AND.AA.LT.0.0D0)AA=0000000000.0 !AA=infinitive

C IF(ETA.GT.PRM.AND.AA.GT.0.0D0)AA=0.0D0

IF(ICOD.EQ.5)AA=0.0D0

AB=0.0D0 !clear AB

DO 30 J=1,NS

30 AB=AB+A(J)*B(J) !AB=[He] (df/dsig)(De)(df/dsig)

C

c WRITE(*,*)"666"

BETA=-AA+AB ![Hp]+[He]

BETAA=10.0D17

IF(BETA.GT.BETAA)BETA=BETAA !prevent overflow

DO 40 J=1,NS

DO 40 JJ=1,NS

C WRITE(*,*)"H, D",BETA

Call Damage(pr,npr,nmt,pc,ap) !Damage consideration

40 D(JJ,J)=D(JJ,J)-(B(JJ)*B(J)+HPA(JJ,J))/(BETA*ap)

! [Cijkl]-[C]ijkl[B]kl[C]ijkl[B]kl+[HPP]/[Hp]+[He]

C 40 WRITE(*,*)"Dep",D(JJ,J)


c

50 CONTINUE

C

IF(IET.EQ.0) GOTO 80

C

DO 60 J=1,3

DO 60 JJ=1,3

60 D(JJ,J)=D(JJ,J)+PR(7,KM)*BK !Bulk Modulus

80 CONTINUE

c WRITE(*,*)"777"

c rate dependency

c rate depencency --------------------------

call rate(epsiequi,epsiaccu,i,ip)

c write(*,*)epsiequi(1,1,1)

RETURN

END

C *********************************************

Subroutine damage(pr,npr,nmt,pc,ap)

c This subroutine cnsiders the damage.

c Called from DMCAM

c Chung R. Song 08/09/2003

c *********************************************


dimension PR(NPR,NMT)

c write(*,*)"999"

A1=pr(3,2)

A2=pr(4,2)

dum=A1*pc**A2

c write(*,*)dum

c write(*,*)"9999",pc

Ap=1-2.71828**(-dum)

if(ap.LE.0.95)ap=0.95

c write(*,*)"99999"

c beta=beta*Ap !!!! Cancelled. ap is used directly in dmcam

if(ap.le.0.5)write(*,*)"Too much damage!!!!"

return

end

C *********************************************

subroutine rate(epsiequi,epsiaccu,i,ip)

c This subroutine change the yield function based on Song

and Voyiadjis (2003)

c Called from DMCAM


c *********************************************






DIMENSION epsiequi(2,170,4),epsiaccu(2,170,4,3,3)

ien=i

EPSIEQUI(1,ien,ip)=epsiaccu(2,ien,ip,1,1)

1*epsiaccu(2,ien,ip,1,1)

2+epsiaccu(2,ien,ip,2,2)*epsiaccu(2,ien,ip,2,2)

3+epsiaccu(2,ien,ip,3,3)*epsiaccu(2,ien,ip,3,3)

EPSIEQUI(1,ien,ip)=epsiequi(1,ien,ip)*sqrt(0.666667)

c write(*,*) ttime

epsiequi(1,ien,ip)=epsiequi(1,ien,ip)/ttime !ttime=elapsed time

c write(*,*) epsiequi(1,ien,ip)

c Adjustment of pc is made in the beginning part of DMCAM

return

end

C *********************************************

subroutine rate1(dabequi,dab)

c This subroutine change the yield function based on Song and

Voyiadjis (2003)

c Called from DMCAM


c*********************************************





DIMENSION dab(3,3)

ien=i

dabEQUI=dab(1,1)*dab(1,1)+dab(2,2)*dab(2,2)+dab(3,3)*dab(3,3)

dabEQUI=dabequi*sqrt(0.666667)

c write(*,*) ttime

dabequi=dabequi/ttime

c Adjustment of po is made by adjusting sigmaxx, sigmayy, sigmazz.

return

end

C**********************************************************************

subroutine bkstrs2(p,q,eta,prm,salph22,pr,iter,js,ip,delalph

1,ien,s)

c SUBROUTINE BKSTRS2(P,Q,ETA,PRM,SALPH22,PR,ITER,JS,

c 1IP,DELALPH,IEN,S)

c CALL BKSTRS2(P,Q,ETA,PRM,SALPH22,PR,ITER,JS,IP,DELALPH,I,S)

C**********************************************************************


C 1. CALLED FROM EQUIBLOD, DMCAM

C 2. This subroutine calculate the back stress and supply it

to yield f.

C 3. Chung R. Song, 2/25/99.

C 4. Modified for multiple back stress

C****** Long range back stress, ALPHA 2 **********


COMMON/BSTRESS/ PREP,PREQ,PREETA,PREALPH !Back stress variable

COMMON/MATPROP/C,X

DIMENSION PREP2(170,2,4,8),PREQ2(170,2,4,8),PREETA2(170,2,4,8)

DIMENSION DALPHA(170,2,4,8,4),SSALPHA(170,4,4),DELS(4),

1SALPH22(4),PR(10,10) !PR added just for matching

DIMENSION PRE\_S(170,2,4,8),S(6) !6 just for matching with Dmcam

DIMENSION DELALPH(4)

C elem no, Incr no, gauss pt, iteration, direction

c (IEN,JS,IP,ITER,J)

C Js was reduced to 2 by magic. (repeated use)

c WRITE(*,*)"BKSTRS",IEN

C---Clear

DO J=1,4

SALPH22(J)=0.0

DELS(J)=0.0

ENDDO

C---Iteration for direction ij (11,22,33,12)

DO J=1,4 !Beginning of loop

c--- Prevent numerical noise

C--- Do not remove this part. This part makes the program more stable.

IF(P.LE.0.0000000001)P=0

IF(Q.LE.0.0000000001)Q=0

c IF(S(J).LE.0.0000000001)S(J)=0

IF(ETA.LE.0.0000000001)ETA=0

IF(SALPH22(J).LE.0.0000000001)SALPH22(J)=0

DO I=1,2

DO II=1,8

IF(PREP2(IEN,I,IP,II).LE.0.0000000001)PREP2(IEN,I,IP,II)=0.0

IF(PREQ2(IEN,I,IP,II).LE.0.0000000001)PREQ2(IEN,I,IP,II)=0.0

IF(PRE\_S(IEN,I,IP,II).LE.0.0000000001)PRE\_S(IEN,I,IP,II)=0.0

IF(PREETA2(IEN,I,IP,II).LE.0.0000000001)PREETA2(IEN,I,IP,II)=

10.0

IF(DALPHA(IEN,I,IP,II,J).LE.0.0000000001)

1DALPHA(IEN,I,IP,II,J)=0.0

ENDDO

ENDDO


IF(JS.EQ.1.AND.ITER.EQ.1) THEN !Very Beginning

DELP=0.0D0 !P-PREP

DELQ=0.0D0 !Q-PREQ

DELS(J)=0.0D0

DELETA=0.0D0 !ETA-PREETA

DELALPH(I)=0.D0 !SALPHA-PREALPH

ENDIF

IF(JS.EQ.1.AND.ITER.EQ.2) THEN ! Beginning

DELP=P-PREP2(IEN,JS,IP,ITER-1)

DELQ=Q-PREQ2(IEN,JS,IP,ITER-1)

DELS(J)=S(J)-PRE\_S(IEN,JS,IP,ITER-1)

DELETA=ETA-PREETA2(IEN,JS,IP,ITER-1)

DELALPH(I)=DALPHA(IEN,JS,IP,1,J)

ENDIF

IF(JS.EQ.1.AND.ITER.GE.3) THEN ! Beginning

DELP=P-PREP2(IEN,JS,IP,ITER-1)

DELQ=Q-PREQ2(IEN,JS,IP,ITER-1)

DELS(J)=S(J)-PRE\_S(IEN,JS,IP,ITER-1)

DELETA=ETA-PREETA2(IEN,JS,IP,ITER-1)

DELALPH(I)=DALPHA(IEN,1,IP,ITER-1,J)-DALPHA(IEN,1,IP,ITER-2,J)

ENDIF

IF(JS.EQ.2.AND.ITER.EQ.1) THEN

DELP=P-PREP2(IEN,1,IP,2) !JS=increm. penet, IP=Gauss point No.

DELQ=Q-PREQ2(IEN,1,IP,2)

DELS(J)=S(J)-PRE\_S(IEN,1,IP,2)

DELETA=ETA-PREETA2(IEN,1,IP,2)

DELALPH(I)=DALPHA(IEN,1,IP,2,J) !Previous value SALPHA-

PREALP2(JS-1,IP)

ENDIF

IF(JS.GT.2.AND.ITER.EQ.1) THEN !Intermediate Beginning

DELP=P-PREP2(IEN,2,IP,2) !JS=increm. penet, IP=Gauss point No.




DELALPH(I)=DALPHA(IEN,2,IP,2,J) !Previous value SALPHA-

PREALP2(JS-1,IP)

ENDIF

IF(JS.GT.1.AND.ITER.EQ.2) THEN !Intermediate 2

DELP=P-PREP2(IEN,2,IP,1)




DELALPH(I)=DALPHA(IEN,2,IP,1,J)-DALPHA(IEN,1,IP,2,J)


ENDIF

IF(JS.GT.1.AND.ITER.GE.3) THEN !Intermediate 3 and others

DELP=P-PREP2(IEN,2,IP,ITER-1)

DELQ=Q-PREQ2(IEN,2,IP,ITER-1)

DELS(J)=S(J)-PRE\_S(IEN,2,IP,ITER-1)

DELETA=ETA-PREETA2(IEN,2,IP,ITER-1)

DELALPH(I)=DALPHA(IEN,2,IP,ITER-1,J)-DALPHA(IEN,2,IP,ITER-2,J)

ENDIF

C=PR(6,2)

X=PR(7,2)

ALP=ALP !SSALPHA(IEN,IP,3)-( 1)

! Use alpha1-alpha3 from previous calculation

c WRITE(*,*)"P,PRM,ETA,ALP",P,PRM,ETA,ALP

SALPHA1=DELP/P+(2D0*ETA-

2D0*ALP)*DELETA/(PRM**2D0+ETA**2D0-

12D0*ALP*ETA)

SALPHA2=(C/P)*(DELS(J)-X*DELP*SALPH22(J))

C WRITE(*,*)"DELP",ITER,IEN,IP,J,ETA

SALPH22(J)=SALPH22(J)+3*(SALPHA1*SALPHA2)

C WRITE(*,*)"FDDF",J,JS,SALPHA1,SALPHA2,SALPHA(J),DELS(J)

C WRITE(*,*)"DFD", DELP,P,ETA,ALP,DELETA,PRM

C IF(ABS(SALPHA(J)).GT.PRM)THEN

C SALPHA(J)=PRM

C ENDIF

C SALPHA(J)=0 !MAKE ISOTROPIC

C ---------- Save current P, Q, ETA, SDELS, SALPHA for next

iteration

IF(JS.EQ.1)THEN !JS=incremental number

PREP2(IEN,1,IP,ITER)=P

PREQ2(IEN,1,IP,ITER)=Q

PRE\_S(IEN,1,IP,ITER)=S(J)

PREETA2(IEN,1,IP,ITER)=ETA

DALPHA(IEN,1,IP,ITER,J)=SALPH22(J)

SSALPHA(IEN,IP,J)=SSALPHA(IEN,IP,J)+DALPHA(IEN,1,IP,ITER,J)

1 -DALPHA(IEN,1,IP,ITER-1,J)

SALPH22(J)=SSALPHA(IEN,IP,J)

ALP=SALPH22(1)-SALPH22(3)

ENDIF


IF(JS.EQ.2)THEN










ENDIF

IF(JS.GE.3)THEN

PREP2(IEN,1,IP,ITER)=PREP2(IEN,2,IP,ITER)


PREQ2(IEN,1,IP,ITER)=PREQ2(IEN,2,IP,ITER)


PRE\_S(IEN,1,IP,ITER)=PRE\_S(IEN,2,IP,ITER)


PREETA2(IEN,1,IP,ITER)=PREETA2(IEN,2,IP,ITER)


DALPHA(IEN,1,IP,ITER,J)=DALPHA(IEN,2,IP,ITER,J)






ENDIF

C ----------

ENDDO !End of big do loop

RETURN

END

C**********************************************************************

SUBROUTINE BKSTRS1(P,Q,ETA,PRM,PR,ITER,JS,IP,IEN,SALPH11

1,DAB,epsiaccu)

C**********************************************************************

C CALLED FROM Dmcam

C This subroutine compute the short range back stress by Prager’s

C linear hardening rule. 5/30/00

C alpha1(dot)=(2/3)C1*epsilonp(dot)---------



DIMENSION EPSIACCU(2,170,4,3,3),DAB(3,3) !elem,gaus,i,j

DIMENSION ALP1DOT(170,4,3,3),SALPH11(4),PR(10,10) !PR added for

matching

C elem no, Incr no, gauss pt, iteration, direction

c (IEN,JS,IP,ITER,J)

c accumulation for incremental steps only (JS)

c for each element, for each Gauss point, for each direction i,j !!!

c ----- Clear ALP1DOT

do i=1,3

do j=1,3

ALP1DOT(IEN,IP,I,J)=0.0

enddo

enddo

c DAB(i,j) is the strain increment.

inew=1

iold=2

c ----- Initializing EPSIACCU

if(js.EQ.1)EPSIACCU(iold,ien,ip,i,j)=0.0d0

do i=1,3

do j=1,3

C do ien=1,150 !already in iterating loop

c do ip=1,8 !already in iterating loop

c ien=element no. ip=gauss no.

EPSIACCU(inew,ien,ip,I,J)=EPSIACCU(iold,ien,ip,I,J)+DAB(i,j)

!EPSIACCU=Accumulated Plastic Strain

C ----- STABILIZE alpha1 after THRESHOLD strain limit

if (epsiaccu(inew,ien,ip,i,j).GT.0.001)GOTO 100

C1=.0 !temperal

ALP1DOT(ien,ip,I,J)=(2/3.0)*C1*EPSIACCU(inew,ien,ip,I,J)

!ip added 5/10/00

c ----- Vectorize ALP1dot(ien,ip,i,j), indecies in salpha( ) is

same as J

c ----- in previous Bkstrs subroutine

SALPH11(1)=ALP1DOT(IEN,IP,1,1)



SALPH11(4)=ALP1DOT(IEN,IP,1,2) !REST = 0

c ENDDO

c ENDDO


ENDDO

ENDDO

do i=1,3

do j=1,3

EPSIACCU(iold,ien,ip,i,j)=EPSIACCU(inew,ien,ip,i,j)

enddo

enddo

100 RETURN

END

C**********************************************************************

SUBROUTINE STRSCOR(IP,I,NDIM,NS,STRESS,VAR,MAT,PR,FYLD,DSIG,PC,

1ITER,JS,DAB,SALPHA,ICOD,H,A)

C**********************************************************************

C CALLED FROM EQUIBLOD

C This subroutine correct the stress which cross the yield

surface. 2/25/99.

C Corrected again at 5/1/99 by C.R. Song



DIMENSION STRESS(NVRS,NIP,MEL),D(6,6),MAT(MEL)

DIMENSION DSIG(6),VAR(6),S(6),A(6),B(6),PR(NPR,NMT)

DIMENSION DAB(3,3),DABI(3,3),HPA(4,4),DEPSI(4),DUM4(4),DUM5(4)

DIMENSION SALPHA(4),BB21(4),BB31(4),DUM1(4),DELALPH(4)

C Subroutine modified by Song for default KGO=3 4/30/99

C

C WRITE(*,*)"STRSCOR"

KM=MAT(I)

PRM=PR(4,KM)

SX=VAR(1)

SY=VAR(2)

SZ=VAR(3)

TXY=VAR(4)

C WRITE(*,*)"VAR",VAR(1),VAR(2),VAR(3)

E=STRESS(NS+2,IP,I)

P=(SX+SY+SZ)/3.0D0

Q2=SX*(SX-SY)+SY*(SY-SZ)+SZ*(SZ-SX)+3.0D0*TXY*TXY


C

TYZ=VAR(5)

TZX=VAR(6)

Q2=Q2+3.0D0*TYZ*TYZ+3.0D0*TZX*TZX

10 Q=SQRT(Q2)

ETA=Q/P

PY=P+Q*Q/(P*PR(4,KM)*PR(4,KM))

PCS=PC/2.0D0


PB=P/PCS

BK=(1.0D0+E)*P/PR(1,KM)

C--- CALCULATE ELASTIC STRESS-STRAIN MATRIX

G=PR(5,KM)

IF(G.LT.1.0D0)G=BK*1.50D0*(1.0D0-2.0D0*PR(5,KM))/(1.0D0+PR(5,KM))

AL=(3.0D0*BK+4.0D0*G)/3.0D0

DL=(3.0D0*BK-2.0D0*G)/3.0D0

C

CALL ZEROR2(D,6,6)

D(1,1)=AL

D(2,1)=DL

D(3,1)=DL

D(1,2)=DL

D(2,2)=AL

D(3,2)=DL

D(1,3)=DL

D(2,3)=DL

D(3,3)=AL

D(4,4)=G


D(5,5)=G

D(6,6)=G

C--- CALCULATE A=DF/DSIG YIELD FUNCTION DERIVATIVES

12 S(1)=SX-P

S(2)=SY-P

S(3)=SZ-P

S(4)=2.0D0*TXY


S(5)=2.0D0*TYZ

S(6)=2.0D0*TZX

16 CONTINUE

C Correction to avoid tension failure by C.R. Song

C DO II=1,3

C IF(S(II).LE.0.0)S(II)=0.0

C ENDDO

IF(VAR(1).LE.10.0)GOTO 200



C--- Calculate the back stress.

C 17 CALL BKSTRS2(P,Q,ETA,PRM,SALPHA,PR,ITER,JS,IP,DELALPH,I,S)

GOTO 210


C WRITE(*,*)"BSTRSED IN STRSCOR"

C -----------------------------

c 16 BB=-2.0D0*(1.0D0-PB)/(3.0D0*PCS)

c C=3.0D0/(PCS*PCS*PR(4,KM)*PR(4,KM))

C BB=PRM**2.0D0*(2.0d0*P-PC)/3.0D0 !TEMP. BY SONG

C C=3.0D0 !TEMP. BY SONG

C SJJ2 IS THE SECOND STRESS DETERMINANT J2 !SONG

C WRITE(*,*)"ALPHA",SALPHA

C SJJ2=(Q**2D0)/(3.0D0) !SONG

C---BB=(df/dp)(dp/dsig) without Kronecker dij

C--- C=(df/dq)(dq/dsig) without Sij

C--- A=Bij

C--- S=Sij

200 do ii=1,4

salpha(ii)=0

delalph(ii)=0

enddo

C---Caculate Bij

210 BB11=(2.0d0*P-PC)*PRM**2.0d0

BB11=BB11/3

BB12=0D0

DO II=1,4

BB12=BB12+1.5D0*(PC*SALPHA(II)*SALPHA(II)-2D0*SALPHA(II)

1*S(II))

ENDDO

BB12=BB12/3

BB21(1)=(2D0/3D0)*S(1)-(1D0/3D0)*S(2)-(1D0/3D0)*S(3)

BB21(1)=3*BB21(1)

BB21(2)=-(1D0/3D0)*S(1)+(2D0/3D0)*S(2)-(1D0/3D0)*S(3)

BB21(2)=3*BB21(2)

BB21(3)=-(1D0/3D0)*S(1)-(1D0/3D0)*S(2)+(2D0/3D0)*S(3)

BB21(3)=3*BB21(3)

BB21(4)=(2D0/3D0)*S(4)

BB21(4)=3*BB21(4)

BB31(1)=3D0*P*((2D0/3D0)*SALPHA(1)-(1D0/3D0)*SALPHA(2)-

6(1D0/3D0)*SALPHA(3))

BB31(2)=3D0*P*(-(1D0/3D0)*SALPHA(1)+(2D0/3D0)*SALPHA(2)-

1(1D0/3D0)*SALPHA(3))

BB31(3)=3D0*P*(-(1D0/3D0)*SALPHA(1)-(1D0/3D0)*SALPHA(2)+

1(2D0/3D0)*SALPHA(3))

BB31(4)=3D0*P*(2D0/3D0)*SALPHA(4)

DO II=1,3


A(II)=BB11+BB12+BB21(II)-BB31(II)

ENDDO

A(4)=0+0+BB21(4)-BB31(4)

BII=A(1)+A(2)+A(3) !Bii

C Complete Bij !A=Bij matrix

18 DO 20 J=1,3

B(J)=0.0D0

DO 20 JJ=1,3

20 B(J)=B(J)+D(J,JJ)*A(JJ) ! B=[C]ijkl[B]kl = (De)(df/dsig)

B(4)=D(4,4)*A(4)


B(5)=D(5,5)*A(5)

B(6)=D(6,6)*A(6)

C---df/dev

25 XI=(1.0D0+E)/(PR(2,KM)-PR(1,KM)) !XI=(1+e)/(lambda-kappa)

ALPHA=0D0 !ALPHA ij Alpha ij

DO II=1,4

ALPHA=ALPHA+SALPHA(II)*SALPHA(II) !ALPHA=scalar

ENDDO

AA1=XI*P*PC*(-2D0*PRM**2D0+3D0*ALPHA) !SONG df/d epsilon vp

AA2=BII

AA=AA1*AA2 !AA=[Hp]

H=AA !To transfer H to the next routine

C WRITE(*,*)"H in STRscor",H,AA1,AA2

C--------- Compute the additional change of stiffness matrix

by back stress.

C--------- by Chung R. Song 3/08/99

DUM2=0

DO II=1,4

DUM1(II)=3D0*(P*PC*SALPHA(II)-P*S(II)) !df/dalpha

DELALPH(II)=DELALPH(II)

DUM2=DUM2+DUM1(II)*DELALPH(II) ! (df/dalpha)(dalpha)

SCALAR !magic

ENDDO








C DJACOB=DAB(1,1)*DAB(2,2)-DAB(1,2)*DAB(2,1)

C IF(DJACOB.EQ.0)GOTO 27 !skip singular condition

C

C IF(ITER.EQ.1)GOTO 27

C CALL DETMIN(LOUT1,NDIM,DAB,DABI,DJACB,JL,IP,ISTGE)





C GOTO 75

C 27 DO 70 JJJ1=1,4

C 70 DEPSI(JJJ1)=0D0

C GOTO 77










DJACOB=DJACOB1-DJACOB2+DJACOB3

C DJACOB=DAB(1,1)*DAB(2,2)-DAB(1,2)*DAB(2,1) !radial strain = 0

IF(DJACOB.EQ.0)GOTO 27 !skip singular condition

CALL DETMIN1(LOUT1,NDIM,DAB,DABI,DJACB,JL,IP,ISTGE)

!special

c IF(ITER.EQ.1)GOTO 27

C CALL DETMIN1(LOUT1,NDIM,DAB,DABI,DJACB,JL,IP,ISTGE)





GOTO 75

27 DO 70 JJJ1=1,4

70 DEPSI(JJJ1)=0D0

GOTO 77

C WRITE(*,*)"444"

C-----Change the depsilon\_1 tensor to depsilon\_1 vector

75 DEPSI(1)=DABI(1,1)


DEPSI(2)=DABI(2,2)

DEPSI(3)=0.0 !DABI(3,3)

DEPSI(4)=DABI(1,2) !Axi-symmetric

C-----Substitute depsilon inverse, into [Hp] equation

77 CONTINUE

DO JS1=1,4 !clear

DUM4(JS1)=0.0D0

ENDDO

DO 90 JS1=1,4 !2-D case, Axisymmetric

DUM4(JS1)=DUM2*A(JS1) ! (df/dalpha)(dalpha)[B] vectorized

tensor

DO 91 JS2=1,4 !2-D case

91 DUM5(JS1)=DUM5(JS1)+D(JS1,JS2)*DUM4(JS2)

!2nd oderized 4th order tensor

DO 90 JS2=1,4

90 HPA(JS1,JS2)=(1D0/3D0)*DUM5(JS1)*DEPSI(JS2)

![HPA]=(1/3)[C](df/dalpha)(dalpha)[B][depsil]

IF(ETA.LT.PRM.AND.AA.LT.0.0D0)AA=0000000000.0 !AA=infinitive

C IF(ETA.GT.PRM.AND.AA.GT.0.0D0)AA=0.0D0

IF(ICOD.EQ.5)AA=0.0D0

AB=0.0D0

C CALCULATE ALPHA PARAMETER

C---------------------------------------------------------------------

DO 30 J=1,NS

30 AB=AB+A(J)*B(J)

BETA=AA+AB

ALPHA=FYLD/BETA

C---------------------------------------------------------------------

C CALCULATE DSIG = ALPHA*[D]*A

C---------------------------------------------------------------------

DO 40 J=1,NS

40 DSIG(J)=ALPHA*B(J)

RETURN

END

C**********************************************************************

SUBROUTINE EQUIBLOD(XYZ,NCONN,MAT,LTYP,NQ,NW,NP1,NP2,PR,NTY,DI,

1 DIPR,STRESS,STRAIN,PEXI,IDFX,P,PT,PCOR,PEQT,XYFT,PCONI,

+ LCS,LNGP,NELCM,MCS,MNGP,NCAM,ITER,IOUTP,JS)


C----------------------------------------------------------------------

C THIS SUBROUTINE COMPUTES THE EQUILIBRATING GLOBAL LOAD VECTOR

C----------------------------------------------------------------------



DIMENSION STRESS(NVRS,NIP,MEL),XYZ(3,MNODES),STRAIN(NVRN,NIP,MEL)

DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL),NW(MNODES+1),

1 NQ(MNODES),NP1(NPL),NP2(NPL)

DIMENSION BL(6,NB),BNL(6,NB),BL1(9,NB),FT(3,NDMX)

DIMENSION SLL(4),NWL(NPMX),AA(NPMX)


DIMENSION D(6,6),ELCOD(3,NDMX),DS(3,20),SHFN(20),CARTD(3,NDMX)

DIMENSION MCS(MEL),MNGP(MEL),ST(6),ST1(6)

DIMENSION DI(MDOF),DIPR(MDOF),EDINC(NB),EDINCP(NB),ED(2),

1 XJACM(3,3)

DIMENSION P(MDOF),PCOR(MDOF),PT(MDOF),XYFT(MDOF),PCONI(MDOF)

DIMENSION PEXI(MDOF),PEQT(MDOF),IDFX(MDOF)

DIMENSION LCS(NIP,MEL),LNGP(NIP,MEL),NELCM(MEL)



COMMON /COUNT / NCS,NNGP










C----------------------------------------------------------------------

LED=2

IEL=0

NCAM=0

NDIM1=NDIM+1

CR=1.0D0


C INITIALIZE THE EQUILIBRIATING LOAD VECTOR

CALL ZEROR1(PEQT,MDOF)

IF(ITER.EQ.1) CALL ZEROR1(PT,MDOF)

C----------------------------------------------------------------------

C INITIALISE

C----------------------------------------------------------------------

DO 18 IM=1,NEL

MCS(IM)=0


MNGP(IM)=0

NELCM(IM)=0

DO 18 IP=1,NIP

LCS(IP,IM)=0

18 LNGP(IP,IM)=0

C----------------------------------------------------------------------

IEL=0

NCAM=0

C GET THE ELEMENT DISPLACEMENTS FROM THE GLOBAL DISPLACEMENT VECTOR

DO 200 MR=1,NEL

ICAM=0

C

LT=LTYP(MR)

IF(LT.LT.0)GOTO 200

NDN=LINFO(5,LT)

NGP=LINFO(11,LT)

INDX=LINFO(12,LT)

NPN=LINFO(6,LT)

NDPT=LINFO(1,LT)

NAC=LINFO(15,LT)

C CALCULATE ELEMENT INCREMENTAL DISPLACEMENT EDINC

CALL ZEROR1(EDINCP,NB)

CALL ZEROR1(EDINC,NB)

DO JDN=1,NDN

NIN=NDIM*(JDN-1)

JN=ABS(NCONN(JDN,MR))

JL=NW(JN)-1

DO ID=1,NDIM

EDINCP(NIN+ID)=DIPR(JL+ID)

EDINC(NIN+ID)=DI(JL+ID)

ENDDO

ENDDO

C----------------------------------------------------------------------

C SETUP LOCAL NODAL COORDINATES OF ELEMENT

C----------------------------------------------------------------------

DO 20 KN=1,NDN

NDE=NCONN(KN,MR)

DO 20 ID=1,NDIM


C

GOTO(25,25,23,25,23,25,23,25,23,25,23),LT

C----------------------------------------------------------------------

C SETUP LOCAL ARRAY OF NW AS NWL GIVING THE INDEX TO

C PORE-PRESSURE VARIABLES

C----------------------------------------------------------------------

23 IPP=0

DO 24 IV=1,NDPT

IQ=LINFO(IV+INXL,LT)


IF(IQ.NE.NDIM1.AND.IQ.NE.1)GOTO 24

IPP=IPP+1

NDE=NCONN(IV,MR)

NWL(IPP)=NW(NDE)+IQ-1

24 CONTINUE

25 KM=MAT(MR)

KGO=NTY(KM) !Explanation of KGO

IF(NTY(KM)-2)27,28,28

27 CALL DCON(MR,0,MAT,PR,D,NDIM,BK)

28 IEL=IEL+1

C INITIALIZE THE EQUILIBRIATING LOAD VECTOR FT

CALL ZEROR2(FT,3,NDMX)

C----------------------------------------------------------------------

C LOOP ON INTEGRATION POINTS

C----------------------------------------------------------------------

DO 125 IP=1,NGP

IPA=IP+INDX

C

DO 35 IL=1,NAC


C----------------------------------------------------------------------


C----------------------------------------------------------------------




CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,2,MR)



DO 15 IDIM=1,NDIM

DO 15 JDIM=1,NDIM

SUM=ZERO

DO 112 IN=1,NDN



c

C WRITE(*,*)"EQUIBLOD"

CALL DETMIN(LOUT1,NDIM,XJACM,XJACI,DJACB,MR,IP,ISTGE)

C----------------------------------------------------------------------

C CALCULATE RADIUS FOR AXI-SYM B MATRIX

C----------------------------------------------------------------------

R=ZERO

RI=ZERO


C

DO IN=1,NDN

R=R+ELCOD(1,IN)*SHFN(IN)


ENDDO

RI=-1.0D0/R

C CALCULATE CARTESIAN DERIVATIVES OF SHAPE FUNCTIONS

38 DO IN1=1,NDN

DO ID=1,NDIM

SUM=ZERO

C

DO 130 JD=1,NDIM

130 SUM=SUM-DS(JD,IN1)*XJACI(ID,JD)

CARTD(ID,IN1)=SUM

ENDDO

ENDDO

C CALCULATE THE LINEAR STRAIN-DISPL MATRIX [BL]

CALL BLNR(NDIM,NDN,NPLAX,RI,SHFN,CARTD,BL)

C

IF(LINR.GT.1) THEN

C CALCULATE THE NONLINEAR STRAIN-DISPL MATRIX BNL

CALL BNLNR(EDINCP,SHFN,CARTD,RI,NDIM,NDN,BNL,NS,NPLAX)

END IF

C CALCULATE THE LINEAR DISPL-DISPL MATRIX BL1

CALL BLNR1(CARTD,BL1,NDIM,NDN)

C

CALL ZEROR1(ST,NS)

C

DO 44 II=1,NDN

IN=NCONN(II,MR)

N1=NW(IN)

N2=N1+1

ST(1)=ST(1)+CARTD(1,II)*DI(N1)


ST(3)=ST(3)+SHFN(II)*DI(N1)*RI

ST(4)=ST(4)+CARTD(1,II)*DI(N2)+CARTD(2,II)*DI(N1)


N3=N1+2




44 CONTINUE

C

DO IS=1,NS

ST1(IS)=STRAIN(IS,IP,MR)

ENDDO

ED(1)=EDS(ST1,NS,NDIM)

C

GOTO(70,70,66,70,66,70,66,70,66,70,66),LT


66 CALL SHFNPP(LINP,SLL,NAC,DS,AA,LT,0,MR)

SUM=0.0D0

DO 68 IC=1,NPN

IVR=NWL(IC)

68 SUM=SUM+AA(IC)*DI(IVR)

V=ST(1)+ST(2)+ST(3)

UI=SUM

GOTO 72

70 V=ST(1)+ST(2)+ST(3)

UI=PR(7,KM)*V*BK

72 CONTINUE

C

CALL STRSTN(IP,MR,NDN,KGO,ELCOD,SHFN,MAT,PR,D,BK,

1 EDINC,BL,BNL,BL1,STRESS,UI,ST,ITER,JS)

C

DO 45 IS=1,NS

45 STRAIN(IS,IP,MR)=STRAIN(IS,IP,MR)+ST(IS)

DO IS=1,NS

ST1(IS)=STRAIN(IS,IP,MR)

ENDDO

ED(2)=EDS(ST1,NS,NDIM)

C

IF(KGO.NE.3.AND.KGO.NE.4)GOTO 85

C----------------------------------------------------------------------

C CALCULATE EXTRA VARIABLES FOR CAM-CLAY ONLY

C----------------------------------------------------------------------

85 CALL STRSEQ(MR,IP,IPA,NDIM,NDN,NS,STRESS,SHFN,CARTD,FT,DJACB,

1 R,RI,CR,NPLAX)

C END OF INTEGRATION POINTS LOOP

125 CONTINUE

C----------------------------------------------------------------------

C ASSEMBLE EQUILIBRATING NODAL FORCES INTO GLOBAL ARRAY - PEQT

C----------------------------------------------------------------------

DO 150 IK=1,NDN

II=NCONN(IK,MR)

N1=NW(II)-1

C

DO 150 ID=1,NDIM

150 PEQT(N1+ID)=PEQT(N1+ID)+FT(ID,IK)

C END OF ELEMENTS LOOP

200 CONTINUE


C----------------------------------------------------------------------

C CALCULATE OUT-OF-BALANCE NODAL LOADS

C----------------------------------------------------------------------

DO 230 IM=1,NDF

230 PEQT(IM)=PEQT(IM)+PEXI(IM)

C COUNT FOR SKEW BOUNDARIES

IF(NSKEW.GT.0) CALL ROTBC(PEQT,NW,NDIM,NSKEW,1)

C

KSTGE=4

TTGRV=1.0D0

C

IF(ITER.EQ.1) THEN

CALL EQLOD(NCONN,MAT,LTYP,NW,NQ,IDFX,NP1,NP2,XYZ,P,

1 PT,PCOR,PEQT,XYFT,PCONI,PR,0,1,TTGRV,0,KSTGE,0,IOUTP)

ELSE

CALL RESTRN(NDIM,NW,IDFX)

DO 5 IK=1,NDF

IF(IDFX(IK).EQ.1) THEN

PCOR(IK)=ZERO

ELSE

PCOR(IK)=PT(IK)-PEQT(IK)

END IF

5 CONTINUE

ENDIF

C

RETURN

END

C *********************************************************************

SUBROUTINE CONVCH(NDF,NNODES,NDIM,TOL,ICONV,ICRIT,NW,UITER,UINC,

1 REPRE,RE,ITER,RNRM1)

C----------------------------------------------------------------------

C THIS SUBPROGRAM FINDS OUT IF THE CONVERGENCE HAS OCCURED BY USING

C THE PREVIOUS AND CURRENT UNBALANCED LOADS OR DISPLACEMENTS OR BY

C ENERGY CRITERIA

C ---------------------------------------------------------------------



DIMENSION

UITER(MDOF),UINC(MDOF),REPRE(MDOF),RE(MDOF),NW(MNODES+1)

ICONV=0

C INTERNAL ENERGY CRITERION:

ICONV1=0


IF(ICRIT.EQ.1)THEN

ERGPREV=ERG

ERG=0.0D0

C

DO I=1,NNODES

N1=NW(I)-1

DO JJ=1,NDIM

ERG=ERG+UITER(N1+JJ)*(RE(N1+JJ)-REPRE(N1+JJ))

ENDDO

ENDDO

IF(ITER.EQ.1)THEN

ERG1=ERG

RETURN

ELSE

IF(ABS(ERG).LT.ABS(TOL*ERG1))ICONV=1

IF(ABS(ERG).GT.ABS(ERGPREV))ICONV1=1

ENDIF

C DISPLACEMENT CRITERION:

ELSE IF(ICRIT.EQ.2)THEN

TOTNRM=0.0D0

CURNRM=0.0D0

C

DO I=1,NNODES

N1=NW(I)-1

DO JJ=1,NDIM

TOTNRM=TOTNRM+UINC(N1+JJ)*UINC(N1+JJ)

CURNRM=CURNRM+UITER(N1+JJ)*UITER(N1+JJ)

ENDDO

ENDDO

C

TOTNRM=SQRT(TOTNRM)

CURNRM=SQRT(CURNRM)

IF(CURNRM.LT.TOL*TOTNRM)ICONV=1

C IF UNBALANCED FORCE CRITERION:

ELSE IF(ICRIT.EQ.3)THEN

DO I=1,NDF

RTMPNRM=0.0D0

RTMPNRM=RTMPNRM+RE(I)*RE(I)

ENDDO

RNRM=SQRT(RTMPNRM)


IF(ITER.EQ.1)THEN

RNRM1=RNRM

RETURN

ELSE

IF(RNRM.LT.TOL*RNRM1)ICONV=1

ENDIF

ENDIF

RETURN

END

C**********************************************************************

SUBROUTINE UPOUT(XYZ,DI,DA,STRESS,STRAIN,P,PT,PCOR,PEQT,NCONN,MAT,

1 NTY,PR,LTYP,NW,NQ,IDFX,VARC,YREF,IOUTP)

C----------------------------------------------------------------------

C UPDATE AND OUTPUT ROUTINE

C----------------------------------------------------------------------



DIMENSION NQ(MNODES),NW(MNODES+1),IDFX(MDOF)

DIMENSION

NCONN(NTPE,MEL),MAT(MEL),NTY(NMT),LTYP(MEL),PR(NPR,NMT)


DIMENSION P(MDOF),PCOR(MDOF),PT(MDOF),PEQT(MDOF)

DIMENSION DI(MDOF),DA(MDOF)

DIMENSION ELCOD(3,NDMX),DS(3,20),SHFN(20)

DIMENSION CIP(3),SLL(4)

DIMENSION VARC(9,NIP,MEL),STRAIN(NVRN,NIP,MEL)

DIMENSION NELPR(MEL),NELUS(MEL)

DIMENSION SPA(3)

DIMENSION XJACM(3,3)





COMMON /COUNT / NCS,NNGP









C

ISTGE=4

LED=2

NS1=NS+1

NDIM1=NDIM+1

C----------------------------------------------------------------------

C BREAK OUTPUT CODE

C----------------------------------------------------------------------

IOUT4=IOUT/1000

IOUT3=(IOUT-1000*IOUT4)/100

IOUT2=(IOUT-1000*IOUT4-100*IOUT3)/10

IOUT1=(IOUT-1000*IOUT4-100*IOUT3-10*IOUT2)

IF(IOUT1.LT.1)GOTO 66

LT1=LTYP(1)

LT1=IABS(LT1)

IF(IOUTP.NE.1) GO TO 66

C GOTO(1,1,2,1,2,1,2,3,4,1,2),LT1

C 1 WRITE(LOUT1,902)

C GOTO 66


C GOTO 66


C GOTO 66


66 CONTINUE

C----------------------------------------------------------------------

C UPDATE ABSOLUTE DISPLACEMENTS

C----------------------------------------------------------------------

CR=1.0D0


DO 5 KD=1,NDF

5 DA(KD)=DA(KD)+DI(KD)

C

DO 10 JR=1,NN

IF(JR.EQ.0)GOTO 10

J=JR

NQL=NQ(JR)

IF(NQL.EQ.0) GOTO 10

N1=NW(JR)

IF(IOUT1.EQ.0)GOTO 10

IF(IOUT1.EQ.1.AND.JR.GT.NN)GOTO 10

IF(JR.LT.NN)GOTO 6

IF(JR.LT.NMOS.OR.JR.GT.NMOF)GOTO 10

GOTO 8

6 CONTINUE

IF(JR.LT.NVOS.OR.JR.GT.NVOF)GOTO 10

8 CONTINUE


C

N2=N1+NQL-1


C IF(NDIM.EQ.3) GOTO 9

C IF(NQL.EQ.3)WRITE(LOUT1,900)JR,(DI(JJ),JJ=N1,N2),(DA(JJ),JJ=N1,N2)


C IF(NQL.EQ.1)WRITE(LOUT1,911)JR,DI(N1),DA(N1)

C GOTO 10

C 9 CONTINUE



10 CONTINUE

IF(NDIM.EQ.3) GOTO 12

IF(IOUT2.EQ.2)WRITE(LOUT3,904)


GOTO 14

12 CONTINUE



14 CONTINUE

C----------------------------------------------------------------------

C INITIALISE

C----------------------------------------------------------------------

DO 18 IM=1,NEL

NELPR(IM)=0

NELUS(IM)=0

18 CONTINUE

C----------------------------------------------------------------------

C UPDATE NODAL CO-ORDINATES

C----------------------------------------------------------------------

ND=NN

DO 220 J=1,NN

N1=NW(J)-1

DO 220 ID=1,NDIM

220 XYZ(ID,J)=XYZ(ID,J)+DI(N1+ID)

C----------------------------------------------------------------------

C OUTPUT NODAL COORDINATES AND PWP

C----------------------------------------------------------------------

IF(IOUTP.EQ.1)THEN



DO I33=1,NN

NWL=NW(I33)-1

NQL=NQ(I33)

ND1=NDIM+1

NDL=NWL+ND1

XNODE=XYZ(1,I33)

YNODE=XYZ(2,I33)


YLIM=YREF-0.040D0

c IF(XNODE.LT.0.040D0.AND.YNODE.GE.YLIM)THEN

XNORM=XNODE/0.005640D0

YNORM=(YNODE)/0.005640D0

IF(NQL.EQ.ND1) THEN

c IF(NDIM.EQ.2)WRITE(LOUT1,303)I33,XNORM,YNORM,

IF(NDIM.EQ.2)WRITE(LOUT1,303)I33,XNODE,YNODE,

1 DA(NDL)

IF(NDIM.EQ.3)WRITE(LOUT1,333)I33,(XYZ(IDIM,I33),IDIM=1,NDIM),

1 DA(NDL)

c ELSE

c IF(NDIM.EQ.2)WRITE(LOUT1,313)I33,(XYZ(IDIM,I33),IDIM=1,NDIM)

c IF(NDIM.EQ.3)WRITE(LOUT1,323)I33,(XYZ(IDIM,I33),IDIM=1,NDIM)

ENDIF

c ENDIF

ENDDO

ENDIF

C----------------------------------------------------------------------

C CALCULATE AND OUTPUT STRESSES AT THE NODES

C----------------------------------------------------------------------

IF(IOUTP.EQ.1)CALL NODSTRS(NEL,NN,NS,LTYP,XYZ,NCONN,STRESS,

1 YREF)

C----------------------------------------------------------------------

C CALCULATE AND OUTPUT STRAINS AT THE NODES

C----------------------------------------------------------------------

IF(IOUTP.EQ.1)CALL NODSTRN(NEL,NN,NS,LTYP,XYZ,NCONN,STRAIN,

1 YREF)

C----------------------------------------------------------------------

C LOOP OVER THE ELEMENTS

C----------------------------------------------------------------------

IEL=0

DO 200 MR=1,NEL

KMAT=MAT(MR)

KGO=NTY(KMAT)

J=MR

C IF(J.EQ.0)GOTO 200

LT=LTYP(J)

NDN=LINFO(5,LT)

NGP=LINFO(11,LT)

INDX=LINFO(12,LT)

NPN=LINFO(6,LT)

NDPT=LINFO(1,LT)

NAC=LINFO(15,LT)

C----------------------------------------------------------------------

C SETUP LOCAL NODAL COORDINATES OF ELEMENT

C----------------------------------------------------------------------

DO 20 KN=1,NDN


NDE=NCONN(KN,J)

DO 20 ID=1,NDIM


C

25 IF(IOUT2.NE.2.OR.IOUTP.NE.1)GOTO 26

IF(MR.LT.NELOS.OR.MR.GT.NELOF)GOTO 26

WRITE(LOUT3,908)MR



26 CONTINUE

IEL=IEL+1

NELUS(IEL)=MR

NELPR(IEL)=J

C----------------------------------------------------------------------

C LOOP ON INTEGRATION POINTS

C----------------------------------------------------------------------

DO 125 IP=1,NGP

IPA=IP+INDX

C

DO 35 IL=1,NAC


C----------------------------------------------------------------------


C----------------------------------------------------------------------




CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,2,MR)



DO 15 IDIM=1,NDIM

DO 15 JDIM=1,NDIM

SUM=ZERO

DO 112 IN=1,NDN



C

CALL DETMIN(LOUT1,NDIM,XJACM,XJACI,DJACB,MR,IP,ISTGE)

C----------------------------------------------------------------------

C OUTPUT ABSOLUTE STRESSES

C----------------------------------------------------------------------

CALL PRINC(STRESS(1,IP,J),STRESS(2,IP,J),STRESS(4,IP,J),SPA)



IKM=IP

GOTO 122

120 IF(IOUT2.NE.1.OR.IP.NE.NGP)GOTO 175


IKM=MR

C

122 DO 124 ID=1,NDIM

SUM=ZERO

C

DO 123 IN=1,NDN

123 SUM=SUM+SHFN(IN)*ELCOD(ID,IN)

124 CIP(ID)=SUM


IF(MR.LT.NELOS.OR.MR.GT.NELOF)GOTO 175

IF(NDIM.EQ.2)WRITE(LOUT3,916)IKM,(CIP(ID),ID=1,NDIM),

1 (STRESS(IK,IP,J),IK=1,NS1),(SPA(JL),JL=1,3)

IF(NDIM.EQ.3)WRITE(LOUT3,946)IKM,(CIP(ID),ID=1,NDIM),

1 (STRESS(IK,IP,J),IK=1,NS1)

C

175 IF(KGO.NE.3.AND.KGO.NE.4)GOTO 125

C----------------------------------------------------------------------

C CALCULATE EXTRA VARIABLES FOR CAM-CLAY AND UPDATE MCODE

C----------------------------------------------------------------------

CALL UPDCAM(STRESS,IP,MR,KMAT,NS,NDIM,PR,KGO,VARC)

125 CONTINUE

C

200 CONTINUE

C----------------------------------------------------------------------

C OUTPUT ADDITIONAL PARAMETERS FOR CAM-CLAYS

C----------------------------------------------------------------------

CALL CAMOUT(LOUT4,LTYP,MAT,NTY,IOUT3,NEL,VARC,IOUTP)

C 225 CONTINUE

KSTGE=4

CALL EQLBM(LOUT2,NDIM,1,NN,NDF,NW,NQ,IDFX,P,PT,PCOR,PEQT,

1 IOUT4,1,IRAC,IOUTP)

C----------------------------------------------------------------------

C WRITE RESULTS ON SAVE FILE

C----------------------------------------------------------------------

C IF(ISR.EQ.0)GOTO 250

C IF(ISR.EQ.2)GOTO 240

C IF(ISR.EQ.1.AND.IWL.EQ.1)GOTO 240

C GOTO 250

C 240 WRITE(IW2)

TTIME,TGRAV,XYZ,STRESS,STRAIN,DA,XYFT,PCOR,PCONI,LTYP,NMOD

C WRITE(IW2) NF,MF,NFIX,DXYT

C WRITE(IW2) NLED,LEDG,NDE1,NDE2,PRESLD

C

C 250 CONTINUE


RETURN

301 FORMAT(//,14X,’UPDATED NODE COORDINATES’,/,4X,48(’-’),/,

1 5X,’NODE’,12X,’X’,12X,’Y’,12X,’PWP’,/,4X,48(’-’))

303 FORMAT(4X,I4,4X,2F12.6,4X,F10.3)

313 FORMAT(4X,I4,4X,2F12.6)

331 FORMAT(//,20X,’UPDATED NODE COORDINATES’,/,4X,58(’-’),

1 /,5X,’NODE’,10X,’X’,11X,’Y’,11X,’Z’,12X,’PWP’,/,4X,58(’-’))

333 FORMAT(4X,I4,4X,4F12.4)

323 FORMAT(4X,I4,4X,3F12.4)

900 FORMAT(1X,I5,6E12.4)

901 FORMAT(//46H NODAL DISPLACEMENTS AND EXCESS PORE PRESSURES/

1 1X,45(1H-)//21X,11HINCREMENTAL,26X,8HABSOLUTE//

12X,4HNODE,7X,2HDX,10X,2HDY,10X,2HDU,10X,2HDX,10X,2HDY,10X,2HDU/)

902 FORMAT(//20H NODAL DISPLACEMENTS/1X,19(1H-)//

1 18X,11HINCREMENTAL,33X,8HABSOLUTE//

1 2X,4HNODE,7X,2HDX,13X,2HDY,28X,2HDX,13X,2HDY/)

904 FORMAT(//40H ABSOLUTE STRESSES AT INTEGRATION POINTS/1X,39

(1H-)/)

906 FORMAT(//30H STRESSES AT ELEMENT CENTROIDS/1X,29(1H-)//8H

ELEMENT,

1 3X,1HX,13X,1HY,11X,2HSX,11X,2HSY,11X,2HSZ,10X,3HTXY,12X,1HU,

1 10X,5HSIG-1,8X,5HSIG-2,7X,5HTH-XY)

908 FORMAT(/15H ELEMENT NUMBER,I5/1X,19(1H-))

910 FORMAT(1X,I5,2E12.4,12X,2E12.4)

911 FORMAT(1X,I5,30X,E12.4,30X,E12.4)

914 FORMAT(2X,2HIP,7X,1HX,9X,1HY,10X,2HSX,10X,2HSY,10X,2HSZ,

1 9X,3HTXY,9X,1HU,9X,5HSIG-1,7X,5HSIG-2,7X,5HTH-XY/)

916 FORMAT(1X,I3,2F10.4,7E12.4,F10.1)

933 FORMAT(//20H NODAL DISPLACEMENTS/1X,19(1H-)//

1 18X,11HINCREMENTAL,51X,8HABSOLUTE//

1 2X,4HNODE,7X,2HDX,13X,2HDY,13X,2HDZ,28X,2HDX,13X,2HDY,13X,2HDZ/)

934 FORMAT(//46H NODAL DISPLACEMENTS AND EXCESS PORE PRESSURES/

1 1X,45(1H-)//21X,11HINCREMENTAL,37X,8HABSOLUTE//

1 4X,4HNODE,5X,2HDX,10X,2HDY,10X,2HDZ,10X,2HDU,

1 10X,2HDX,10X,2HDY,10X,2HDZ,10X,2HDU/)

936 FORMAT(//30H STRESSES AT ELEMENT CENTROIDS/1X,29(1H-)//8H

ELEMENT,

1 3X,1HX,13X,1HY,12X,1HZ,11X,2HSX,11X,2HSY,11X,2HSZ,11X,3HTXY,

1 11X,3HTYZ,10X,3HTZX,11X,1HU/)

940 FORMAT(1X,I5,8E12.4)

941 FORMAT(1X,I5,3E12.4,12X,3E12.4)

944 FORMAT(2X,2HIP,7X,1HX,9X,1HY,9X,1HZ,8X,2HSX,

1 10X,2HSY,10X,2HSZ,10X,3HTXY,9X,3HTYZ,10X,3HTZX,9X,1HU/)

946 FORMAT(1X,I3,3F10.4,7E12.4)

END


C**********************************************************************

SUBROUTINE STRSTN(IP,MR,NDN,KGO,ELCOD,SHFN,MAT,PR,D,

1 BK,EDINC,BL,BNL,BL1,STRESS,UI,DEPS,ITER,JS)

C**********************************************************************






DIMENSION EDINC(NB),F(3,3),FINV(3,3),EDOT(3,3),TEMP2(3,3),

1 DAB(3,3),SPK(3,3),SIG(3,3),SIGP(6),DEPS(6),TEMP1(6),TEMP3(6),

2 STRESS(NVRS,NIP,MEL),BL(6,NB),BNL(6,NB),BL1(9,NB)

DIMENSION ELCOD(3,NDMX),SHFN(20),MAT(MEL),PR(NPR,NMT),D(6,6),

1 DSIG(6),SIGDOT(3,3),SPKDOT(3,3),DSIGCOR(6),VAR(6)

DIMENSION DD(6,6),DDD(6,6),A(6),SALPHA(4),D2(3,3,3,3),IJ1(6)

C WRITE(*,*)"STRSTN"

C

C IK(M,N)=(M/N)*(N/M)

C

KM=MAT(MR)

PRM=PR(4,KM)

ICOD=MCODE(IP,MR)

C INITIALIZE ARRAYS

DO I1=1,6

DSIGCOR(I1)=0.0D0

TEMP1(I1)=0.0D0

TEMP3(I1)=0.0D0

VAR(I1)=0.0D0

SIGP(I1)=0.0D0

DSIG(I1)=0.0D0

ENDDO

DO I1=1,3

DO J1=1,3

TEMP2(I1,J1)=0.0D0

DAB(I1,J1)=0.0D0

EDOT(I1,J1)=0.0D0

SIG(I1,J1)=0.0D0

SIGDOT(I1,J1)=0.0D0

SPKDOT(I1,J1)=0.0D0

ENDDO

ENDDO

C

IF(LINR.GT.1)CALL ADDBMAT(BL,BNL)


C

DO II1=1,NS

TEMP=0.0D0

DO II2=1,NB

TEMP=TEMP+BL(II1,II2)*EDINC(II2)

C write(2,*)’TEMP=’,TEMP

ENDDO

TEMP1(II1)=TEMP

ENDDO

C----------------------------------------------------------------------

C CONVERT IT INTO MATRIX TEMP1----> EDOT(I,J)

C----------------------------------------------------------------------

DO II1=1,3

DO II2=1,3

EDOT(II1,II2)=0.0D0

ENDDO

ENDDO

EDOT(1,1)=TEMP1(1)

EDOT(2,2)=TEMP1(2)

EDOT(3,3)=TEMP1(3)

EDOT(1,2)=TEMP1(4)

EDOT(2,1)=EDOT(1,2)


EDOT(2,3)=TEMP1(5)

EDOT(3,2)=EDOT(2,3)

EDOT(1,3)=TEMP1(6)

EDOT(3,1)=EDOT(1,3)

101 CONTINUE

C----------------------------------------------------------------------

C CALCULATE THE DEFORMATION GRADIENT MATRIX [F],[FINV] & DETF

C----------------------------------------------------------------------

CALL DEFGRAD(BL1,F,FINV,DETF,EDINC,NDIM,LINR)

C----------------------------------------------------------------------

C TRANSFORM THE STRAIN [EDOT] TO [DAB] BY: [DAB]=[FINV]\^T

[EDOT][FINV]

C----------------------------------------------------------------------

DO II=1,3

DO IJ=1,3

TEMP=0.0D0

DO K=1,3

TEMP=TEMP+EDOT(II,K)*FINV(K,IJ)

ENDDO

TEMP2(II,IJ)=TEMP

ENDDO

ENDDO

DO II=1,3

DO IJ=1,3


TEMP=0.0D0

DO K=1,3

TEMP=TEMP+FINV(K,II)*TEMP2(K,IJ)

ENDDO

DAB(II,IJ)=TEMP

ENDDO

ENDDO

C WRITE(*,*)"STRSTN"

C WRITE(*,*)"INI",DAB(1,1),DAB(1,2)

C WRITE(*,*)"INI",DAB(2,1),DAB(2,2)

C CORRECTION FOR ANTI-TENSION FAILURE

c DO IJ=1,3

c IF(STRESS(IJ,IP,MR).LE.10.0)STRESS(IJ,IP,MR)=10.0

c ENDDO

C----------------------------------------------------------------------

C CALCULATE 2-ND PIOLA-KIRCHHOFF STRESS TENSOR SPK(3,3)

C----------------------------------------------------------------------

CALL ZEROR2(SPK,3,3)

SPK(1,1)=STRESS(1,IP,MR)





IF(NDIM.EQ.3)THEN





ENDIF

C----------------------------------------------------------------------

C TRANSFORM SPK INTO SIG (CAUCHY STRESS)

C----------------------------------------------------------------------

DO II=1,3

DO IJ=1,3

TEMP=0.0D0

DO K=1,3

TEMP=TEMP+SPK(II,K)*F(K,IJ)

ENDDO

TEMP2(II,IJ)=TEMP

ENDDO

ENDDO

DO II=1,3

DO IJ=1,3

TEMP=0.0D0

DO K=1,3


TEMP=TEMP+F(K,II)*TEMP2(K,IJ)

ENDDO

SIG(II,IJ)=TEMP/DETF

ENDDO

ENDDO

C----------------------------------------------------------------------

C CHANGE THE TENSORS TO VECTORS (SIG AND DAB)

C----------------------------------------------------------------------

DEPS(1)=DAB(1,1)

SIGP(1)=SIG(1,1)

DEPS(2)=DAB(2,2)

SIGP(2)=SIG(2,2)

DEPS(3)=DAB(3,3)

SIGP(3)=SIG(3,3)

DEPS(4)=DAB(1,2)

SIGP(4)=SIG(1,2)

DEPS(5)=DAB(2,3)

SIGP(5)=SIG(2,3)

DEPS(6)=DAB(1,3)

SIGP(6)=SIG(1,3)

C

STRESS(1,IP,MR)=SIG(1,1)




IF(NDIM.EQ.3)THEN



ENDIF

C----------------------------------------------------------------------

C CALCULATE F(P,PC,Q)

C----------------------------------------------------------------------

IF(KGO.EQ.3.OR.KGO.EQ.4) THEN

P1=(STRESS(1,IP,MR)+STRESS(2,IP,MR)+STRESS(3,IP,MR))/3.0D0

PC1=STRESS(NS+3,IP,MR)

Q1=Q(SIGP,NS,NDIM)

ETA1=Q1/P1

C

ENDIF

C----------------------------------------------------------------------

C CALL THE CONSTITUTIVE MODEL TO FIND [D]

C----------------------------------------------------------------------

C WRITE(*,*)"ip=",IP,’MR=’,mr

C PAUSE

izero=0

GOTO(1,2,3),KGO


2 CALL DLIN(IP,izero,NDIM,NDN,ELCOD,SHFN,MAT,D,PR,BK)

GOTO 1

3 CALL DMCAM(IP,MR,izero,NDIM,NS,STRESS,MAT,D,PR,BK,ITER,JS,DAB,

+SALPHA,A,H)

C WRITE(*,*)"HH",salpha(1)

GOTO 1

C----------------------------------------------------------------------

C COMPUTE EFFECTIVE COROTATIONAL/CAUCHY STRESS RATE TENSOR

C----------------------------------------------------------------------

1 CONTINUE

DO 60 II=1,NS

DSIG(II)=0.0D0

DO 60 JJ=1,NS

c WRITE(*,*)"DDD",DSIG(II),D(II,JJ),DEPS(JJ)

60 DSIG(II)=DSIG(II)+D(II,JJ)*DEPS(JJ)

C----------------------------------------------------------------------

C CONVERT DSIG VECTOR TO SIGDOT TENSOR

C----------------------------------------------------------------------

SIGDOT(1,1)=DSIG(1)

SIGDOT(2,2)=DSIG(2)

SIGDOT(3,3)=DSIG(3)

SIGDOT(1,2)=DSIG(4)

SIGDOT(2,1)=DSIG(4)

SIGDOT(2,3)=DSIG(5)

SIGDOT(3,2)=DSIG(5)

SIGDOT(1,3)=DSIG(6)

SIGDOT(3,1)=DSIG(6)

C----------------------------------------------------------------------

C FIND OUT EFFICTIVE SPK STRESS INCREMENT AS:

C [SPKDOT]=DETF [FINV][FINV]\[SIGDOT]-[DAB][SIG]-[DBC][SIG]+DCC[SIG]

C----------------------------------------------------------------------

DCC=DAB(1,1)+DAB(2,2)+DAB(3,3)

DO I=1,3

DO J=1,3

TEMP=0.0D0

TEM=0.0D0

DO K=1,3

TEMP=TEMP+DAB(I,K)*SIG(K,J)

TEM=TEM+SIG(I,K)*DAB(J,K)

ENDDO

c WRITE(*,*)"SIGDOT",DCC,SIG(I,J),TEMP,TEM,SIGDOT(I,J)

C WRITE(*,*)"LINT",LINR

IF(LINR.GT.1)THEN


C SPKDOT(I,J)=DCC*SIG(I,J)-TEMP-TEM+SIGDOT(I,J) !alive 5/30/99 by

Song

SPKDOT(I,J)=SIGDOT(I,J)

ELSE

SPKDOT(I,J)=SIGDOT(I,J)

ENDIF

ENDDO

ENDDO

DO I=1,3

DO J=1,3

TEMP=0.0D0

DO K=1,3

TEMP=TEMP+SPKDOT(I,K)*FINV(K,J) ! Sik Xkj

ENDDO

TEMP2(I,J)=TEMP

ENDDO

ENDDO

DO I=1,3

DO J=1,3

TEMP=0.0D0

DO K=1,3

TEMP=TEMP+FINV(K,I)*TEMP2(K,J) ! Xki Sik Xkj

ENDDO

SPKDOT(I,J)=TEMP*DETF ! Xki Sik Xkj J

ENDDO

ENDDO

C----------------------------------------------------------------------

C UPDATE PORE WATER PRESSURE U

C----------------------------------------------------------------------

TEMP=0.0D0

DO I=1,NDIM

TEMP=TEMP+F(I,I)*F(I,I)

ENDDO

STRESS(NS+1,IP,MR)=STRESS(NS+1,IP,MR)*TEMP/(NDIM*DETF)

PWP=STRESS(NS+1,IP,MR) !Change of variable for D* routine

C----------------------------------------------------------------------

C UPDATE EFFECTIVE STRESSES STRESS

C----------------------------------------------------------------------

VAR(1)=STRESS(1,IP,MR)+SPKDOT(1,1)




IF(NDIM.EQ.3)THEN



ENDIF


c correction for anti-tension failure

c DO IJ=1,3

c IF(STRESS(IJ,IP,MR).LE.10.0)STRESS(IJ,IP,MR)=10.0

c IF(VAR(IJ).LE.10.0)VAR(IJ)=10.0

c ENDDO

C

IF(KGO.EQ.3.OR.KGO.EQ.4) THEN

C----------------------------------------------------------------------

C CORRECT THE YIELD SURFACE

C----------------------------------------------------------------------

P2=(VAR(1)+VAR(2)+VAR(3))/3.0D0

Q2=Q(VAR,NS,NDIM)

IF(KGO.EQ.3) THEN

PY=P2+Q2*Q2/(P2*PRM*PRM)

ELSEIF(KGO.EQ.4)THEN

PY=P2*EXP(Q2/(PRM*P2))

ENDIF

IF(PY.LT.PC1.AND.ICOD.NE.4)GOTO 10

C----------------------------------------------------------------------

C UPDATE THE HARDENING PARAMETER PC

C----------------------------------------------------------------------

ETA2=Q2/P2

DP12=P2-P1

DQ12=Q2-Q1

P12=(P1+P2)/2.0D0

ETA12=(ETA1+ETA2)/2.0D0

DETA=DQ12/P12-DP12*ETA12/P12

DPC=PC1*(DP12/P12+(2.0D0*ETA12*DETA)/(ETA12*ETA12+PRM*PRM))

IF(ICOD.EQ.4.AND.DPC.GT.0.0D0)DPC=0.0D0

IF(ICOD.NE.4.AND.DPC.LT.0.0D0)DPC=0.0D0

IF(ICOD.EQ.5)DPC=0.0D0

PC2=PC1+DPC

write(2,*)’P2=’,p2,’ Q2=’,q2

c WRITE(2,*)’ETA2=’,eta2

c write(2,*)’DPC=’,dpc,’ PC2=’,pc2

C-------------------------------------------------------------------

C CORRECT STRESSES DUE TO DRIFTING OF YIELD SURFACE

C-------------------------------------------------------------------

FYB=FPQ(P2,PC2,Q2,PRM,KGO)

c WRITE(2,*)’FYB2=’,FYB

20 CALL STRSCOR(IP,MR,NDIM,NS,STRESS,VAR,MAT,PR,FYB,DSIGCOR,PC2,

1 ITER,JS,DAB,SALPHA,ICOD,H,A)

VAR(1)=VAR(1)-DSIGCOR(1)





IF(NDIM.EQ.3)THEN



ENDIF

C

P2=(VAR(1)+VAR(2)+VAR(3))/3.0D0

Q2=Q(VAR,NS,NDIM)

C FYC=FPQ(P2,PC2,Q2,PRM,KGO)

IF(KGO.EQ.3) THEN




ENDIF

STRESS(NS+3,IP,MR)=PY

c WRITE(2,*)’FYC=’,FYC

ENDIF

C--------------------------------------------------------------------

C UPDATE STRESSES AND PORE PRESSURE

C--------------------------------------------------------------------

10 STRESS(1,IP,MR)=VAR(1)

STRESS(2,IP,MR)=VAR(2)



IF(NDIM.EQ.3)THEN



ENDIF

STRESS(NS+1,IP,MR)=STRESS(NS+1,IP,MR)+UI

C Update the constitutive element stiffness matrix D*

call Dstar(D,DD,SIG,FINV,DETF,NDIM,NS,LINR,PWP,D2)

C Update the constitutive element stiffness matrix for plastic

spin D***=D*+D**

C

C WRITE(*,*)"LKK",SALPHA(1)

CALL Dstar2(D2,A,SALPHA,DDD,SIG,FINV,DETF,NDIM,NS,LINR,H,IJ1,

1P2,IP,MR)

C Combination of D* and D**, D*** = D* + D**


C WRITE(*,*),D(1,1)

do i=1,6

do j=1,6

D(i,j)=DD(i,j)+DDD(i,j)

enddo

enddo

C WRITE(*,*)"3",D(1,1)

C Save the modified constitutive matrix for use in ELMSTIF routine

C write(pp,11)(DD(I,J),J=1,NS),I=1,NS)

RETURN

END

C *********************************************************************

SUBROUTINE Dstar(D1,DD,SIG,FINV,DETF,NDIM,NS,LINR,PWP,D2)

C ---------------------------------------------------------------------

C COMPTUTE [DD]=(D* TENSOR)=JF-1F-1F-1(D-SIG*IK-SIG*IK....)

C ---------------------------------------------------------------------

C This subroutine was initially called from nowhere.

C Now it is corrected & called from ELMSTIF Song, 4/16/99



INTEGER*2 A,B

DIMENSION D1(6,6),DD(6,6),DD1(6,6),SIG(3,3),FINV(3,3)

DIMENSION D2(3,3,3,3),D3(3,3,3,3),IJ(6),MN(33),IK(3,3)

DIMENSION De(6,6)

C D1=Dij, uncorrected D 6x6

C D2=DABCD, uncorrected D 3x3x3x3

C D3=D*ABCD

C DD=Dij, corrected D 6x6

C DD1=Dij, uncorrected D 6x6

C Definition of Kronecker delta

DO M=1,3

DO N=1,3

IF(M.EQ.N) THEN

IK(M,N)=1

ELSE

IK(M,N)=0

ENDIF

ENDDO

ENDDO


C IK(M,N)=(M/N)*(N/M)

LOUT1=2 ! ?

C Give the numbers to index IJ

DO I=1,6

IJ(I)=0

ENDDO

IJ(1)=1

IJ(2)=5

IJ(3)=9

IJ(4)=2

IF(NDIM.EQ.3) THEN

IJ(5)=6

IJ(6)=3

ENDIF

C Give the numbers to index MN

DO I=1,33

MN(I)=0

ENDDO

MN(11)=1

MN(22)=2

MN(33)=3

MN(21)=4

MN(12)=4

MN(23)=5

MN(32)=5

MN(13)=6

MN(31)=6

C Give the numbers to M and M1 / Convert 9x9 D tensor

to 3x3x3x3 D tensor.

C logic confirmed by Song, 4/14/99

C--- Set D14, D24,D34 7/16/99

C This is for the non-coaxiality of principal direction & x,y,z

D1(1,4)=D1(1,1)/10.0

D1(2,4)=D1(1,1)/10.0

D1(3,4)=D1(1,1)/10.0

D1(4,1)=D1(1,4)

D1(4,2)=D1(2,4)

D1(4,3)=D1(3,4)

DO I=1,3

DO J=1,3

C M=(I-1)*3+J

M=J+10*I ! 11,12,13,21,22,23,31,32,33

M1=MN(M) ! 1,4,6,4,2,5,6,5,3


DO K=1,3

DO L=1,3

C N=(K-1)*3+L

N=L+10*K ! 11,12,13,21,22,23,31,32,33

N1=MN(N) ! 1,4,6,4,2,5,6,5,3

D2(I,J,K,L)=D1(M1,N1) ! 11111,1112,1113,1121,1122,1123,

1131,1132,1133,...

C ! 11,14,16,14,12,15,16,15,13,...

c WRITE(*,*)"111",I,J,K,L,M1,N1

c WRITE(*,*)"222",D2(I,J,K,L),D1(M1,N1)

c PAUSE

C D2 is also transferred and used in Dstar2 instead of [E]klmn

ENDDO

ENDDO

ENDDO

ENDDO

C Compute D* (Dstar) [D3] represents D*

IF(LINR.GT.1)THEN

DO A=1,3

DO B=1,3

DO C=1,3

DO D=1,3

TEMP=0.0D0

DO I=1,3

DO J=1,3

DO K=1,3

DO L=1,3

TEMP1=D2(I,J,K,L)-SIG(K,J)*IK(I,L)-SIG(I,K)*IK(J,L)

+ +SIG(I,J)*IK(K,L)-PWP*IK(I,J)*IK(K,L)+2.0D0*PWP*IK(I,K)*IK(J,L)

TEMP2=FINV(I,A)*FINV(J,B)*FINV(K,C)*FINV(L,D)

IF(LINR.LT.2)TEMP2=1

TEMP=TEMP+TEMP1*TEMP2

ENDDO

ENDDO

ENDDO

ENDDO

D3(A,B,C,D)=TEMP*DETF

ENDDO

ENDDO

ENDDO

ENDDO

ENDIF

C Return back 3x3x3x3 D3 matrix to 9x9 DD matrix

C Logic confirmed by Song. 4/14/99

DO I=1,NS


M=IJ(I) ! 1,5,9,2,6,3

II=(M-1)/3+1 ! 1,2,3,1,2,1

JJ=M-(II-1)*3 ! 1,2,3,2,3,3

DO J=1,NS

N=IJ(J) ! 1,5,9,2,6,3

KK=(N-1)/3+1 ! 1,2,3,1,2,1

LL=N-(KK-1)*3 ! 1,2,3,2,3,3

IF(LINR.GT.1)THEN

DD(I,J)=D3(II,JJ,KK,LL) !1111,1122,1133,1112,1123,1113,2211,

2222,....

ELSE

DD(I,J)=D2(II,JJ,KK,LL)

ENDIF

DD1(I,J)=D2(II,JJ,KK,LL)

ENDDO

ENDDO

C Convert DD to D

DO I=1,NS

DO J=1,NS

D1(I,J)=DD(I,J)

ENDDO

ENDDO

RETURN

END

C *********************************************************************

SUBROUTINE Dstar2(D2,A,SALPHA,DDD,SIG,FINV,DETF,NDIM,NS,LINR,H,IJ,

1P2,IP,MR)

C ---------------------------------------------------------------------

C COMPTUTE [D**]=n[sigmbNmacd + sigajNjbcd](JXA,aXB,b...)

C ---------------------------------------------------------------------

C This subroutine compute the D** matrix which considers plastic

spin.

C Song, 4/16/99



COMMON/PLSPIN/PSPIN(27,500)

DIMENSION DD1(6,6),TM(3,3,3,3),TN(3,3,3,3),B(3,3),Eta(3,3)

DIMENSION Alpha(3,3),D4(3,3,3,3),DDD(6,6),IK(3,3),A(6),IJ(6)

DIMENSION D2(3,3,3,3),SIG(3,3),FINV(3,3),SALPHA(4),Dumm(3,3)


c WRITE(*,*)"DSTAR2"

C D2=DABCD, uncorrected D 3x3x3x3

C A=Bij from DMCAM

C D3=D*ABCD

C DDD=Dij, corrected D 6x6

C DDD=Dij, uncorrected D 6x6

C Definition of Kronecker delta !This routine is O.K. checked

4/26/99

DO M=1,3

DO N=1,3

IF(M.EQ.N) THEN

IK(M,N)=1

ELSE

IK(M,N)=0

ENDIF

ENDDO

ENDDO

LOUT1=2 ! ?

C---- Hardening Modulus H (roundf/roundepsilon v)Bii

C Hardening Modulus H was taken from H at DMCAM

C---- Elastic Stiffness E

C E was taken from D2 of previous step n-1 because this is

incremental scheme.

C D2 is 3x3x3x3 matrix from initial

C---- (Eta)kl = [B]ij/norm [B]ij

C [B]ij is taken from [A]n of DMCAM

C Change the vector [B] to tensor [B]

C Clear Bij first

do i=1,3

do j=1,3

B(i,j)=0

enddo

enddo

B(1,1)=A(1)

B(2,2)=A(2)

B(3,3)=A(3)

B(1,2)=A(4)

B(1,3)=A(5)

B(2,3)=A(6) !Other numbers are zero

Bnorm1 = B(1,1)**2+B(2,2)**2

Bnorm2= B(3,3)**2+2*B(1,2)**2

Bnorm3=2*B(1,3)**2


Bnorm4=2*B(2,3)**2 !Scalar this time

Bnorm=SQRT(Bnorm1+Bnorm2+Bnorm3+Bnorm4)

c WRITE(*,*)"BNORM",Bnorm

do i=1,3 !Cal. Eta

do j=1,3

Eta(i,j)=B(i,j)/Bnorm

enddo

enddo

C---- parameter c !Scalar this time

C c=H/(H+Eabcd nab ncd)

C Calculate E nab ncd first

dum=0 ! Initialize dum

Do k=1,3

Do l=1,3

Dumm(k,l)=0 !Clear

enddo

enddo

Do k=1,3

Do l=1,3

Do i=1,3

Do j=1,3

Dumm(k,l)=Dumm(k,l)+D2(i,j,k,l)*Eta(i,j)

enddo

enddo

enddo

enddo

Do kk=1,3

Do ll=1,3

dum=dum+Dumm(kk,ll)*Eta(kk,ll)

enddo

enddo

c=H/(H+dum)

C---- [M] tensor

C [M]=(c E n n)/H

C Clear

Do i=1,3

Do j=1,3

Do k=1,3

Do l=1,3

TM(i,j,k,l)=0

enddo

enddo


enddo

enddo

c WRITE(*,*)"H in Dstar2",H

Do k=1,3

Do l=1,3

Dumm(k,l)=0 !Clear

enddo

enddo

Do m=1,3

Do n=1,3

Do k=1,3

Do l=1,3

Dumm(m,n)=Dumm(m,n)+D2(k,l,m,n)*Eta(k,l)

c WRITE(*,*)"K",k,l,m,n,D2(k,l,m,n)

c pause

enddo

enddo

enddo

enddo

Do i=1,3

Do j=1,3

Do m=1,3

Do n=1,3

TM(m,n,i,j)=c*Dumm(m,n)*Eta(i,j)/(-H)

c WRITE(*,*)"m,n,i,j",m,n,i,j

c WRITE(*,*)"M",TM(m,n,i,j),Dumm(m,n),Eta(i,j)

c pause

enddo

enddo

enddo

enddo

C---- Back stress alpha

C Back stress was taken from DMCAM & BKSTRS

C Alpha was taken as Salpha times identity matrix.

do i=1,3

do j=1,3

Alpha(i,k)=0

enddo

enddo

Alpha(1,1)=P2*SALPHA(1)






c WRITE(*,*)"111",SALPHA(1),ALPHA(2,2),ALPHA(3,3),ALPHA(1,2)

C---- [N] matrix

C [N]=alpha M - M alpha

do ia=1,3

do ib=1,3

do ic=1,3

do id=1,3

TM1=0

TM2=0

do m=1,3

TM1=TM1+alpha(ia,m)*TM(m,ib,ic,id)

enddo

do m=1,3

TM2=TM2+TM(ia,m,ic,id)*alpha(m,ib)

enddo

TN(ia,ib,ic,id)=TM1-TM2

ielem=MR

if(TN(ia,ib,ic,id).EQ.0.0D0)GOTO 10

c write(*,*),TN(ia,ib,ic,id),ia,ib,ic,id

c pause

PSPIN(IP,ielem)=TN(ia,ib,ic,id)

10 CONTINUE

c write(6,100),TM1,TM2,TN(ia,ib,ic,id)

c pause

enddo

enddo

enddo

enddo

c 100 FORMAT(/,3x,F15.13,3x,F15.13,3x,F15.13)

C---- Etadot ! Scalar function

C Eta was assumed 0.001 to 0.1 as constant

Etadot=0.001

C---- D** ! Final Product

C D**=Eta*[sig N +sig N](J XA,a XB,b XC,c XD,d)

do iaa=1,3

do ibb=1,3

do icc=1,3

do idd=1,3

do ia=1,3


do ib=1,3

do ic=1,3

do id=1,3

Dum1=0

do m=1,3

Dum1=Dum1+SIG(m,ib)*TN(m,ia,ic,id)+SIG(ia,m)*TN(m,ib,ic,id)

enddo

Dum2=Dum2+Dum1*FINV(iaa,ia)*FINV(ibb,ib)*FINV(icc,ic)*

1FINV(idd,id)

enddo

enddo

enddo

enddo

D4(iaa,ibb,icc,idd)=Etadot*Dum2*DETF

enddo

enddo

enddo

enddo

C Give the numbers to index IJ

DO I=1,6

IJ(I)=0

ENDDO

IJ(1)=1

IJ(2)=5

IJ(3)=9

IJ(4)=2

IF(NDIM.EQ.3) THEN

IJ(5)=6

IJ(6)=3

ENDIF

DO I=1,NS

M=IJ(I) ! 1,5,9,2,6,3

II=(M-1)/3+1 ! 1,2,3,1,2,1

JJ=M-(II-1)*3 ! 1,2,3,2,3,3

DO J=1,NS

N=IJ(J) ! 1,5,9,2,6,3

KK=(N-1)/3+1 ! 1,2,3,1,2,1

LL=N-(KK-1)*3 ! 1,2,3,2,3,3

DDD(I,J)=D4(II,JJ,KK,LL) !1111,1122,1133,1112,1123,1113,2211,

2222,....


C WRITE(*,*)"DDD",DDD(1,1)

ENDDO

ENDDO

RETURN

END

C**********************************************************************

SUBROUTINE UPDCAM(STRESS,IP,MR,KM,NS,NDIM,PR,KGO,VARC)

C**********************************************************************

C CALCULATE EXTRA STRESS PARAMETERS FOR CAM-CLAYS

C**********************************************************************




DIMENSION STRESS(NVRS,NIP,MEL),PR(NPR,NMT)

DIMENSION VARC(9,NIP,MEL),TEMP(6)

C

CALL ZEROR1(TEMP,6)

PRM=PR(4,KM)

U=STRESS(NS+1,IP,MR)

C----------------------------------------------------------------------

C CHECK IF MCODE STATUS FOR CAM MODEL NEEDS TO BE CHANGED

C----------------------------------------------------------------------

DO IS=1,NS


ENDDO

PC1=PQMOD(IP,MR,1)

ETA1=PQMOD(IP,MR,2)

P2=(STRESS(1,IP,MR)+STRESS(2,IP,MR)+STRESS(3,IP,MR))/3.0D0

Q2=Q(TEMP,NS,NDIM)

PC2=STRESS(NS+3,IP,MR)

ETA2=Q2/P2

ICOD=MCODE(IP,MR)

IF(KGO.EQ.3) THEN

PCS=PC2/2.0D0



PCS=PC2/EXP(1.0d0)


ENDIF


C Mcode is the index parameter for elastic, plastic, hardening

condition.

C Later it is changed to ICOD in DMCAM

IF(ICOD.EQ.2) THEN !Initial Pc

IF(PY.LT.0.9950D0*PC2)THEN !Pc2=new Pc

IF(P2.LT.PCS)MCODE(IP,MR)=3 !OC - Hvoslev

IF(P2.GE.PCS)MCODE(IP,MR)=1 !Roscoe Same as initial cond.

ELSEIF(PY.GE.0.9950D0*PC2)THEN !Initial point Pc

IF(ETA2.GE.PRM.AND.PC2.GT.PC1) THEN !OC above and need

expansion YC

C PC2=PC1+(PRM-ETA1)*(PY-PC1)/(ETA2-ETA1)

MCODE(IP,MR)=5

C GO TO 100

ELSEIF(P2.LT.PCS.AND.PC2.LE.PC1)THEN !OC elastic

MCODE(IP,MR)=4 !Condition stress adjusted to much

ENDIF

ENDIF

GO TO 200

ELSEIF(ICOD.EQ.4) THEN !

IF(ETA2.LT.PRM.AND.PC2.LT.PC1) THEN !below CSL OC or NC

C PC2=PC1-(ETA1-PRM)*(PC1-PY)/(ETA1-ETA2)

MCODE(IP,MR)=5

C GO TO 100

ELSEIF(PC2.GE.PC1) THEN !Need hardening

IF(P2.LT.0.9950D0*PY.AND.P2.LT.PCS)MCODE(IP,MR)=3 !+OC

IF(P2.LT.0.9950D0*PY.AND.P2.GE.PCS)MCODE(IP,MR)=1 !+NC

ENDIF

GO TO 200

ELSEIF(ICOD.EQ.1) THEN

IF(PY.LT.0.9950D0*PC2.AND.P2.LT.PCS)MCODE(IP,MR)=3

IF(PY.GE.0.9950D0*PC2)THEN

IF(ETA2.GT.PRM.AND.PC2.GT.PC1) THEN

C PC2=PC1+(PRM-ETA1)*(PY-PC1)/(ETA2-ETA1)

MCODE(IP,MR)=5

C GO TO 100

ELSEIF(ETA2.LT.PRM.AND.P2.GE.PCS)THEN

MCODE(IP,MR)=2

ELSEIF(P2.LT.PCS.AND.PC2.LE.PC1)THEN

MCODE(IP,MR)=4

ENDIF

ENDIF

GO TO 200

ELSEIF(ICOD.EQ.3) THEN

IF(PY.LT.0.9950D0*PC2.AND.P2.GE.PCS)MCODE(IP,MR)=1

IF(PY.GE.0.9950D0*PC2.AND.P2.GE.PCS)MCODE(IP,MR)=2


IF(PY.GE.0.9950D0*PC2.AND.P2.LT.PCS)MCODE(IP,MR)=4

GO TO 200

C ELSEIF(ICOD.EQ.5) THEN

C IF(PY.LT.0.990D0*PC2.AND.P2.GE.PCS)MCODE(IP,MR)=1

C IF(PY.LT.0.990D0*PC2.AND.P2.LT.PCS)MCODE(IP,MR)=3

ENDIF

GO TO 200

C 100 STRESS(NS+3,IP,MR)=PC

C MCODE(IP,MR)=5

C FYD=FPQ(P2,PC,Q2,PRM,KGO)

C CALL STRSCOR(IP,MR,NDIM,NS,STRESS,MAT,PR,FYD,DSIGCOR,KGO)

C STRESS(1,IP,MR)=STRESS(1,IP,MR)-DSIGCOR(1)




C IF(NDIM.EQ.3)THEN



C ENDIF

C 200 CONTINUE

C

C DO IS=1,NS

C TEMP(IS)=STRESS(IS,IP,MR)

C ENDDO

C P2=(STRESS(1,IP,MR)+STRESS(2,IP,MR)+STRESS(3,IP,MR))/3.0D0

C Q2=Q(TEMP,NS,NDIM)

C ETA2=Q2/P2

C PC=STRESS(NS+3,IP,MR)

C FYD=FPQ(P2,PC,Q2,PRM,KGO

C

200 PQMOD(IP,MR,1)=PC2

PQMOD(IP,MR,2)=ETA2

c ICOD2=MCODE(IP,MR)

c WRITE(2,*)’ICOD1=’,ICOD,’ICOD2=’,ICOD2

C DO IS=1,NS

C TEMP(IS)=STRESS(IS,IP,MR)

C ENDDO

C QT=Q(TEMP,NS,NDIM)

C PE=(STRESS(1,IP,MR)+STRESS(2,IP,MR)+STRESS(3,IP,MR))/3.0D0

C EE=STRESS(NS+2,IP,MR)

C PC=STRESS(NS+3,IP,MR)

C

C IF(KGO.EQ.3) THEN


C PY=PE+QT*QT/(PE*PRM*PRM)

C PCS=PC/2.0D0

C ELSEIF(KGO.EQ.4) THEN

C PY=PE*EXP(QT/(PRM*PE))

C PCS=PC/EXP(1.0d0)

c ENDIF

C

C FYC=FPQ(PE,PC,QT,PRM,KGO)

C

IF(PCS.LT.0.0D0) THEN

WRITE(*,*)’WARNING: PC LESS THAN ZERO FOR ELM-IP’,MR,IP

STOP

ENDIF

C EE=PR(3,KM)-PR(1,KM)*ALOG(P2)-(PR(2,KM)-PR(1,KM))*ALOG(PCS)

IF(P2.GT.0.0D0)EE=PR(3,KM)-PR(1,KM)*LOG(P2)-

1(PR(2,KM)-PR(1,KM))*LOG(PCS)

C

VARC(1,IP,MR)=P2

VARC(2,IP,MR)=Q2

VARC(3,IP,MR)=P2+U

VARC(4,IP,MR)=PC2

VARC(5,IP,MR)=ETA2

VARC(6,IP,MR)=Q2/(P2*PRM)

VARC(7,IP,MR)=PY/PC2

VARC(8,IP,MR)=EE

STRESS(NS+2,IP,MR)=EE

C

CALL ANGTH(STRESS,IP,MR,THETA)

VARC(9,IP,MR)=THETA

C

IF(NDIM.EQ.3)VARC(9,IP,MR)=0.0D0

RETURN

END

C**********************************************************************

SUBROUTINE ANGTH(STRESS,IP,J,THETA)

C**********************************************************************

C ROUTINE TO CALCULATE ANGLE IN PI PLANE

C**********************************************************************



DIMENSION STRESS(NVRS,NIP,MEL)


C

ALAR=1.0D+25

SX=STRESS(1,IP,J)


SY=STRESS(2,IP,J)

SZ=STRESS(3,IP,J)

TXY=STRESS(4,IP,J)

C

PIBY4=0.250D0*PYI

SD=0.50D0*(SX-SY)

SM=0.50D0*(SX+SY)

RAD=SQRT(SD*SD+TXY*TXY)

SIG1=SM+RAD

SIG3=SM-RAD

DY=SY-SM

IF(ABS(TXY).LT.ASMVL.AND.ABS(DY).LT.ASMVL)GOTO 8

THXY2=ATAN2(TXY,DY)

GOTO 9

8 THXY2=0.50D0*PYI

9 THXY=0.50D0*THXY2

THXYD=THXY*180.0D0/PYI

IF(ABS(THXY).LT.PIBY4)GOTO 10

PSIGX=SIG1

PSIGY=SIG3

GOTO 15

10 PSIGX=SIG3

PSIGY=SIG1

15 PSIGZ=SZ

C

SIGX=(PSIGZ-PSIGY)/SQRT(2.0D0)

SIGY=(2.0D0*PSIGX-PSIGY-PSIGZ)/SQRT(6.0D0)

CC RADO=SQRT(SIGX*SIGX+SIGY*SIGY)

IF(ABS(SIGX).LT.ASMVL.AND.ABS(SIGY).LT.ASMVL)GOTO 20

C

THETA=ATAN2(SIGY,SIGX)

IF(THETA.LT.ZERO)THETA=2.0D0*PYI+THETA

THETA=THETA*180.0D0/PYI

GOTO 25

C

20 THETA=ALAR

25 CONTINUE

C

RETURN

END

C**********************************************************************

SUBROUTINE STRSEQ(JJ,IP,IPA,NDIM,NDN,NS,STRESS,SHFN,CARTD,F,DJACB,

1 R,RI,CR,NPLAX)

C**********************************************************************

C ROUTINE TO CALCULATE FORCES EQUILIBRATING

C ELEMENTAL STRESSES (INTEGRATION POINT CONTRIBUTION)

C**********************************************************************




DIMENSION STRESS(NVRS,NIP,MEL),SHFN(20),CARTD(3,NDMX)

DIMENSION F(3,NDMX)


C

F9=CR*DJACB*W(IPA)

IF(NPLAX.EQ.1)F9=F9*R

C

U=STRESS(NS+1,IP,JJ)

C

SIGXT=STRESS(1,IP,JJ)+U

SIGYT=STRESS(2,IP,JJ)+U

SIGZT=STRESS(3,IP,JJ)+U

TXY=STRESS(4,IP,JJ)


C

TYZ=STRESS(5,IP,JJ)

TZX=STRESS(6,IP,JJ)

C

DO 30 IN=1,NDN

F(1,IN)=F(1,IN)+(CARTD(1,IN)*SIGXT+CARTD(2,IN)*TXY


F(2,IN)=F(2,IN)+(CARTD(2,IN)*SIGYT+CARTD(1,IN)*TXY

1 +CARTD(3,IN)*TYZ)*F9

F(3,IN)=F(3,IN)+(CARTD(3,IN)*SIGZT+CARTD(2,IN)*TYZ


30 CONTINUE

GOTO 60

C

35 DO 40 IN=1,NDN

F(1,IN)=F(1,IN)+(CARTD(1,IN)*SIGXT+SHFN(IN)*SIGZT*RI

1 +CARTD(2,IN)*TXY)*F9

40 F(2,IN)=F(2,IN)+(CARTD(2,IN)*SIGYT+CARTD(1,IN)*TXY)*F9

60 CONTINUE

RETURN

END

C**********************************************************************

SUBROUTINE PRINC(C,D,E,B)

C**********************************************************************

C CALCULATES PRINCIPAL STRESSES AND THEIR DIRECTIONS

C**********************************************************************


DIMENSION B(3)


C

AP=C+D


AD=C-D

S=SQRT(.250D0*AD*AD+E*E)

B(1)=.50D0*AP+S

B(2)=.50D0*AP-S

B(3)=90.0D0

IF(ABS(AD).LT.ASMVL) GO TO 2

B(3)=28.64790D0*ATAN(2.0D0*E/AD)

2 RETURN

END

C**********************************************************************

SUBROUTINE NODSTRS(NEL,NNODES,NS,LTYP,XYZ,NCONN,STRESS,YREF)

C**********************************************************************



DIMENSION NCONN(NTPE,MEL),LTYP(MEL),XYZ(3,MNODES),SHAPE(4)

DIMENSION STRESS(NVRS,NIP,MEL),SIG(MNODES,6),ICOUNT(MNODES)

DIMENSION R(8),S(8),PRST(3),SPIN(537)


COMMON/PLSPIN/PSPIN(NIP,MEL)

DATA R(1),R(2),R(3),R(4),R(5),R(6),R(7),R(8)/

1 -1.73205080D0,1.73205080D0,1.73205080D0,-1.73205080D0,

2 0.0D0,1.73205080D0,0.0D0,-1.73205080D0/

DATA S(1),S(2),S(3),S(4),S(5),S(6),S(7),S(8)/

1 -1.73205080D0,-1.73205080D0,1.73205080D0,1.73205080D0,

2 -1.73205080D0,0.0D0,1.73205080D0,0.0D0/

C----------------------------------------------------------------------

C INITIALIZE

C----------------------------------------------------------------------

CALL ZEROI1(ICOUNT,MNODES)

CALL ZEROR2(SIG,MNODES,6)

CALL ZEROR1(SHAPE,4)

C----------------------------------------------------------------------


C----------------------------------------------------------------------

DO IELM=1,NEL

LT=LTYP(IELM)

NDN=LINFO(5,LT)

NGP=LINFO(11,LT)

C----------------------------------------------------------------------

C RETURN IF THE ELEMENT DOES NOT HAVE 8-NODES,4 INTEGRATION

POINT

C----------------------------------------------------------------------


IF(NDN.NE.8.OR.NGP.NE.4)THEN

WRITE(*,*)’CANNOT CALCULATE NODAL STRESSES FOR ELM=’,IELM,NDN,NGP

RETURN

ENDIF

C----------------------------------------------------------------------

C LOOP OVER THE ELEMENT NODES

C----------------------------------------------------------------------

DO INOD=1,NDN

NODE=NCONN(INOD,IELM)

ICOUNT(NODE)=ICOUNT(NODE)+1

C----------------------------------------------------------------------

C SHAPE FUNCTIONS FOR 8-NODES,4 INTEGRATION POINT CASE

C----------------------------------------------------------------------

SHAPE(1)=0.250D0*(1.0D0-R(INOD))*(1.0D0-S(INOD))

SHAPE(2)=0.250D0*(1.0D0+R(INOD))*(1.0D0-S(INOD))

SHAPE(3)=0.250D0*(1.0D0+R(INOD))*(1.0D0+S(INOD))

SHAPE(4)=0.250D0*(1.0D0-R(INOD))*(1.0D0+S(INOD))

C----------------------------------------------------------------------

C LOOP OVER THE INTEGRATION POINTS

C----------------------------------------------------------------------

DO IP=1,NGP

DO IS=1,NS

SIG(NODE,IS)=SIG(NODE,IS)+STRESS(IS,IP,IELM)*SHAPE(IP)

ENDDO

SPIN(NODE)=SPIN(NODE)+PSPIN(IP,IELM)*SHAPE(IP) !other spin = 0

ENDDO

ENDDO

ENDDO

C

WRITE(2,10)

C----------------------------------------------------------------------

C FIND AVERAGE STRSSS AT NODES

C----------------------------------------------------------------------

DO INOD=1,NNODES

A=FLOAT(ICOUNT(INOD))

DO IS=1,NS

SIG(INOD,IS)=SIG(INOD,IS)/A

ENDDO

C----------------------------------------------------------------------

C CALCULATE PRINCIPLE STRESSES

C----------------------------------------------------------------------

CALL PRINC(SIG(INOD,1),SIG(INOD,2),SIG(INOD,4),PRST)

C----------------------------------------------------------------------


C OUTPUT NODAL STRESSES

C----------------------------------------------------------------------

XNODE=XYZ(1,INOD)

YNODE=XYZ(2,INOD)

YLIM=YREF-0.040D0




c WRITE(2,100)INOD,XNORM,YNORM,(SIG(INOD,J),J=1,NS),

WRITE(2,100)INOD,XNODE,YNODE,(SIG(INOD,J),J=1,NS),

1 (PRST(K),K=1,3),SPIN(INOD)

c ENDIF

ENDDO

10 FORMAT(//,3X,’NODE’,3X,’NODE COORDINATES’,19X,’NODE STRESSES’,

1 18x,’PRINCIPAL STRESSES’,/,1X,120(’-’),/,11X,’X’,8X,’Y’,

2 12X,’SX’,10X, ’SY’,10X,’SZ’,9X,’SXY’,9X,’S1’,10X,’S2’,

3 10X,’TH’,10X,’SPIN’/,1X,120(’-’))

100 FORMAT(1X,I4,1X,2F9.5,2X,6E12.4,F9.2,2X,E12.4)

RETURN

END

C**********************************************************************

SUBROUTINE NODSTRN(NEL,NNODES,NS,LTYP,XYZ,NCONN,STRAIN,YREF)

C**********************************************************************

C 1This subroutine update the nodal strain

C 2. Called by DMCAM



DIMENSION NCONN(NTPE,MEL),LTYP(MEL),XYZ(3,MNODES),SHAPE(4)

DIMENSION STRAIN(NVRN,NIP,MEL),SIG(MNODES,6),ICOUNT(MNODES)

DIMENSION R(8),S(8),PRST(3)


DATA R(1),R(2),R(3),R(4),R(5),R(6),R(7),R(8)/

1 -1.73205080D0,1.73205080D0,1.73205080D0,-1.73205080D0,

2 0.0D0,1.73205080D0,0.0D0,-1.73205080D0/

DATA S(1),S(2),S(3),S(4),S(5),S(6),S(7),S(8)/

1 -1.73205080D0,-1.73205080D0,1.73205080D0,1.73205080D0,

2 -1.73205080D0,0.0D0,1.73205080D0,0.0D0/


C----------------------------------------------------------------------

C INITIALIZE

C----------------------------------------------------------------------

CALL ZEROI1(ICOUNT,MNODES)

CALL ZEROR2(SIG,MNODES,6)

CALL ZEROR1(SHAPE,4)

C----------------------------------------------------------------------


C----------------------------------------------------------------------

DO IELM=1,NEL

LT=LTYP(IELM)

NDN=LINFO(5,LT)

NGP=LINFO(11,LT)

C----------------------------------------------------------------------

C RETURN IF THE ELEMENT DOES NOT HAVE 8-NODES,4 INTEGRATION

POINT

C----------------------------------------------------------------------

IF(NDN.NE.8.OR.NGP.NE.4)THEN

WRITE(*,*)’CANNOT CALCULATE NODAL STRAINS FOR

ELM=’,IELM,NDN,NGP

RETURN

ENDIF

C----------------------------------------------------------------------

C LOOP OVER THE ELEMENT NODES

C----------------------------------------------------------------------

DO INOD=1,NDN

NODE=NCONN(INOD,IELM)

ICOUNT(NODE)=ICOUNT(NODE)+1

C----------------------------------------------------------------------

C SHAPE FUNCTIONS FOR 8-NODES,4 INTEGRATION POINT CASE

C----------------------------------------------------------------------

SHAPE(1)=0.250D0*(1.0D0-R(INOD))*(1.0D0-S(INOD))

SHAPE(2)=0.250D0*(1.0D0+R(INOD))*(1.0D0-S(INOD))

SHAPE(3)=0.250D0*(1.0D0+R(INOD))*(1.0D0+S(INOD))

SHAPE(4)=0.250D0*(1.0D0-R(INOD))*(1.0D0+S(INOD))

C----------------------------------------------------------------------

C LOOP OVER THE INTEGRATION POINTS

C----------------------------------------------------------------------

DO IP=1,NGP

DO IS=1,NS !NS=size of D matrix=4 for 2-D,=6 for 3-D

SIG(NODE,IS)=SIG(NODE,IS)+STRAIN(IS,IP,IELM)*SHAPE(IP)

!Eventhough sig is used here that is really strain.

!Just to save the memory.


ENDDO

ENDDO

ENDDO

ENDDO

C

WRITE(2,10)

C----------------------------------------------------------------------

C FIND AVERAGE STRSSS AT NODES

C----------------------------------------------------------------------

DO INOD=1,NNODES

A=FLOAT(ICOUNT(INOD))

DO IS=1,NS

SIG(INOD,IS)=SIG(INOD,IS)/A

ENDDO

C----------------------------------------------------------------------

C CALCULATE PRINCIPLE STRAINS

C----------------------------------------------------------------------

CALL PRINC(SIG(INOD,1),SIG(INOD,2),SIG(INOD,4),PRST)

C----------------------------------------------------------------------

C OUTPUT NODAL STRAINS

C----------------------------------------------------------------------

XNODE=XYZ(1,INOD)

YNODE=XYZ(2,INOD)

YLIM=YREF-0.040D0




c WRITE(2,100)INOD,XNORM,YNORM,(SIG(INOD,J),J=1,NS),

WRITE(2,100)INOD,XNODE,YNODE,(SIG(INOD,J),J=1,NS),

1 (PRST(K),K=1,3)

c ENDIF

ENDDO

10 FORMAT(//,3X,’NODE’,3X,’NODE COORDINATES’,19X,’NODE STRAINS’,

1 18x,’PRINCIPAL STRAINS’,/,1X,106(’-’),/,11X,’X’,8X,’Y’,

1 12X,’STX’,9X, ’STY’,9X,’STH’,9X,’STXY’,9X,’S1’,10X,’S2’,

1 10X,’TH’,/,1X,106(’-’))

100 FORMAT(1X,I4,1X,2F9.5,2X,6E12.4,F9.2)

RETURN

END

C**********************************************************************

FUNCTION FPQ(P,PC,Q,PM,KGO)

C**********************************************************************

C THIS FUNCTION CALCULATES F(P,PC,Q)


C----------------------------------------------------------------------


IF(KGO.EQ.3) THEN

FPQ=(P*P)/(PC*PC)-(P/PC)+(Q*Q)/(PC*PC*PM*PM)


FPQ=LOG(P)-LOG(PC)+Q/(P*PM)

C FPQ=ALOG(P)-ALOG(PC)+Q/(P*PM)

ENDIF

RETURN

END

C**********************************************************************

SUBROUTINE CAMOUT(LOUT4,LTYP,MAT,NTY,IOUT3,NEL,VARC,IOUTP)

C**********************************************************************

C *** OUTPUT ADDITIONAL PARAMETERS CAM-CLAYS

C**********************************************************************






DIMENSION MAT(MEL),NTY(NMT),LTYP(MEL),VARC(9,NIP,MEL)

C

IF(IOUT3.EQ.0.OR.IOUTP.NE.1)GOTO 25





C

DO 20 J=1,NEL

KM=MAT(J)

KGO=NTY(KM)

LT=LTYP(J)

NGP=LINFO(11,LT)

IF(KGO.NE.3.AND.KGO.NE.4)GO TO 20


IF(J.LT.NELOS.OR.J.GT.NELOF)GOTO 20


IF(IOUT3.EQ.2)WRITE(LOUT4,904)J

C

DO 10 IGP=1,NGP

WRITE(LOUT4,905)IGP,(VARC(IK,IGP,J),IK=1,9),MCODE(IGP,J)

10 CONTINUE

GOTO 20

12 WRITE(LOUT4,905)J,(VARC(IK,NGP,J),IK=1,9),

1 (MCODE(IP,J),IP=1,NGP)


20 CONTINUE

WRITE(LOUT4,935)

C

25 CONTINUE

RETURN

901 FORMAT(2X,6HELM-IP,6X,2HPE,11X,1HQ,11X,2HPT,11X,

1 2HPC,9X,3HETA,5X,5HETA/M,6X,2HYR,4X,6HE-VOID,3X,

2 4HTH-3,2X,3HCDE)

902 FORMAT(2X,6HELM-IP,6X,2HPE,11X,1HQ,11X,2HPT,11X,

1 2HPC,9X,3HETA,5X,5HETA/M,6X,2HYR,4X,6HE-VOID,3X,

2 4HTH-3,2X,14H 1 2 3 4 5 6 7)

904 FORMAT(I4)

905 FORMAT(2X,I4,4E13.5,2F9.3,3X,F6.3,F8.4,2X,F7.1,

1 2X,8I2/5X,9I2)

911 FORMAT(/33H CAM CLAY PARAMETERS AT CENTROIDS/

1 1X,32(1H-)/)

912 FORMAT(/42H CAM CLAY PARAMETERS AT INTEGRATION POINTS/

1 1X,41(1H-)/)

935 FORMAT(//)

END

C**********************************************************************

FUNCTION Q(A,N,NDIM)

C**********************************************************************


DIMENSION A(N)

Q2=0.50D0*((A(1)-A(2))*(A(1)-A(2))+(A(2)-A(3))*(A(2)-A(3))

1 +(A(3)-A(1))*(A(3)-A(1)))+3.0D0*A(4)*A(4)


Q2=Q2+3.0D0*A(5)*A(5)+3.0D0*A(6)*A(6)

10 Q=SQRT(Q2)

RETURN

END

C**********************************************************************

FUNCTION EDS(A,N,NDIM)

C**********************************************************************


DIMENSION A(N)

EDS2=0.50D0*((A(1)-A(2))*(A(1)-A(2))+(A(2)-A(3))*(A(2)-A(3))

1 +(A(3)-A(1))*(A(3)-A(1)))+.750D0*A(4)*A(4)


EDS2=EDS2+0.750D0*A(5)*A(5)+0.750D0*A(6)*A(6)

10 EDS=2.0D0*SQRT(EDS2)/3.0D0

RETURN

END


SUBROUTINE INPENT(LINP,NPEN,XREF,YREF,DYREF,DXPEN,

1 XCOS,XSIN,XMUFR,NCHAIN,KSLID0,KSLID3)

C**********************************************************************

C 1 Called by TOTSOL



DIMENSION NCHAIN(100,2),XMUFR(2)

C ----- INITIALIZE NCHAIN MATRIX

CALL ZEROI2(NCHAIN,100,2)

C ----- READ INPUT DATA FOR PENETRATION


101 FORMAT(A80)

READ(LINP,*)XREF,YREF,DXPEN,XDIM,YDIM,KSLID0,KSLID3,

1 (XMUFR(I),I=1,2)

TEMP=SQRT(XDIM**2+YDIM**2)

XCOS=YDIM/TEMP

XSIN=XDIM/TEMP

C DYREF=DXPEN/XSIN !DYREF=incremental distance at cone face

DYREF=DXPEN/XCOS !by Song to control it by vertical displ.

C !Above manipulation should not make cal. diff.


DO I=1,NPEN

READ(LINP,*)(NCHAIN(I,J),J=1,2)

ENDDO

C

RETURN

END

SUBROUTINE CHSLIDE(NPEN,NSKEW,NSKEW1,DXPEN,KSLID0,KSLID3,XSIN,

1 XCOS,XMUFR,NCHAIN,NW,PEQT,PT,FRICT,FRICTPR,INCR,FRMAX0,FRMAX3)

C**********************************************************************




DIMENSION NCHAIN(100,2),NW(MNODES+1),PEQT(MDOF),

1 PT(MDOF),XMUFR(2),FRICT(MDOF),FRICTPR(MDOF)

C----------------------------------------------------------------------

NODSLID0=0

NODSLID3=0

ICOUNT0=0

ICOUNT3=0


TOTFR0=0.0D0

TOTFR3=0.0D0

TOTALFR0=0.0D0

TOTALFR3=0.0D0

FVNOD0=0.0D0

FMNOD0=0.0D0

FVNOD3=0.0D0

FMNOD3=0.0D0

ICV0=0

ICM0=0

ICV3=0

ICM3=0

C----------------------------------------------------------------------

DO 100 I=1,NPEN

NODE =NCHAIN(I,1)

INDEX=NCHAIN(I,2)

IVM=NODE-2*(NODE/2)

N1=NW(NODE)-1

IF(INDEX.NE.0.AND.INDEX.NE.3) GO TO 100

C

DO II=1,NF

INODE=MF(II)

IF(INODE.EQ.NODE) GO TO 1

ENDDO

1 CONTINUE

C----------------------------------------------------------------------

C CHECK NODES WITH INDEX=0

C----------------------------------------------------------------------

IF(INDEX.EQ.0.AND.KSLID0.EQ.0) THEN

ICOUNT0=ICOUNT0+1

RNORM = -PEQT(N1+1)

FRICT0=-PT(N1+2)+PEQT(N1+2)

TOTFR0=TOTFR0+FRICT0

ALLFR0=RNORM*XMUFR(1)

IF(IVM.EQ.1)THEN

FVNOD0=FVNOD0+ALLFR0

ICV0=ICV0+1

ELSEIF(IVM.EQ.0)THEN

ICM0=ICM0+1

FMNOD0=FMNOD0+ALLFR0

ENDIF

TOTALFR0=TOTALFR0+ALLFR0

IF(FRICT0.LT.ALLFR0) THEN

NODSLID0=NODSLID0+1

ENDIF


C----------------------------------------------------------------------

C CHECK NODES WITH INDEX=3

C----------------------------------------------------------------------

ELSEIF(INDEX.EQ.3.AND.KSLID3.EQ.0)THEN

ICOUNT3=ICOUNT3+1

RXX= -PEQT(N1+1)

RYY= -PEQT(N1+2)

RNORM =RXX*XCOS-RYY*XSIN

FRICT3=-RXX*XSIN-RYY*XCOS

TOTFR3=TOTFR3+FRICT3

ALLFR3=RNORM*XMUFR(2)

IF(IVM.EQ.1)THEN

ICV3=ICV3+1

FVNOD3=FVNOD3+ALLFR3

ELSEIF(IVM.EQ.0)THEN

ICM3=ICM3+1

FMNOD3=FMNOD3+ALLFR3

ENDIF

TOTALFR3=TOTALFR3+ALLFR3

IF(FRICT3.LT.ALLFR3) THEN

NODSLID3=NODSLID3+1

ENDIF

ENDIF

100 CONTINUE

C----------------------------------------------------------------------

C CHECK IF SLIDING OCCURS AT NODES WITH INDEX=0

C----------------------------------------------------------------------

IF(NODSLID0.GE.(ICOUNT0))GO TO 2

IF(TOTFR0.LT.TOTALFR0)GO TO 2

GO TO 20

2 CONTINUE

AVRM0=FMNOD0/ICM0

AVRV0=FVNOD0/ICV0

FRMAX0=TOTALFR0/ICOUNT0

KSLID0=1

DO 200 I=1,NPEN

NODE =NCHAIN(I,1)

INDEX=NCHAIN(I,2)

INDFOR=NCHAIN(I+1,2)

N1=NW(NODE)-1

IF(INDEX.NE.0)GO TO 200


C

DO II=1,NF

INODE=MF(II)


ENDDO

3 CONTINUE

IF(INDFOR.EQ.3.AND.KSLID3.EQ.0)GO TO 200

NFIX(2,II)=0

DXYT(1,II)=0.0D0

DXYT(2,II)=0.0D0

FRICT(N1+2)=-PT(N1+2)+PEQT(N1+2)

FRICTPR(N1+2)=-PT(N1+2)+PEQT(N1+2)

200 CONTINUE

20 CONTINUE

C----------------------------------------------------------------------

C CHECK IF SLIDING OCCURS AT NODES WITH INDEX=3

C----------------------------------------------------------------------

IF(NODSLID3.GE.ICOUNT3)GO TO 4

IF(TOTFR3.LT.TOTALFR3)GO TO 4

GO TO 30

4 CONTINUE

AVRM3=FMNOD3/ICM3

AVRV3=FVNOD3/ICV3

FRMAX3=TOTALFR3/ICOUNT3

KSLID3=1

NSKEW=NSKEW1

DO 300 I=1,NPEN

NODE =NCHAIN(I,1)

INDEX=NCHAIN(I,2)

INDPR=NCHAIN(I-1,2)

N1=NW(NODE)-1

IF(INDEX.NE.3)GO TO 300

C

DO II=1,NF

INODE=MF(II)


ENDDO

5 CONTINUE

IF(INDPR.EQ.0.AND.KSLID0.NE.0)THEN

DO IIP=1,NF

INODEP=MF(IIP)

IF(INODEP.EQ.(NODE-1)) GO TO 11

ENDDO

11 N11=NW(NODE-1)-1

NFIX(2,IIP)=0

DXYT(1,IIP)=0.0D0

DXYT(2,IIP)=0.0D0


FRICT(N11+2)=-PT(N11+2)+PEQT(N11+2)

FRICTPR(N11+2)=-PT(N11+2)+PEQT(N11+2)

ENDIF

NFIX(1,II)=1

NFIX(2,II)=0

DXYT(1,II)=DXPEN

DXYT(2,II)=0.0D0

RXX= -PEQT(N1+1)

RYY= -PEQT(N1+2)

RYPAR=-RXX*XSIN-RYY*XCOS

FRICT(N1+1)=RYPAR*XSIN

FRICT(N1+2)=RYPAR*XCOS

FRICTPR(N1+1)=RYPAR*XSIN

FRICTPR(N1+2)=RYPAR*XCOS

300 CONTINUE

30 CONTINUE

RETURN

END

C**********************************************************************

SUBROUTINE BCADJST(NPEN,NDIM,NSKEW,XREF,YREF,DXPEN,XSIN, 1

XCOS,NCHAIN,XYZ,NW,PEQT,PT,FRICT,FRICTPR,XMUFR,PNOD,DD,DP,DP1,

2 INCR,NUMD,IOUTP,TOTPEN,IBCAL,FRMAX0,FRMAX3)

C**********************************************************************




COMMON /SKBC /ISPB(20),DIRCOS(20,3)

DIMENSION XYZ(3,MNODES),NCHAIN(100,2),NW(MNODES+1),PEQT(MDOF),

1 PT(MDOF),DD(4,200),DP(MDOF),DP1(MDOF),NUMD(MDOF,2),PNOD(MDOF)

DIMENSION QC(MDOF),FRICT(MDOF),FRICTPR(MDOF),XMUFR(2)

C

IBCAL=IBCAL+1

FRMAX0=-2.0d-03

FRMAX3=-0.8082640d-03

c XMUFR(1)=0.250D0

c XMUFR(2)=0.250D0

C

CALL ZEROR1(QC,MDOF)

NUM=40

NUMB=40

I1=0

C

DO 100 I=1,NPEN

NODE =NCHAIN(I,1)

INDEX=NCHAIN(I,2)


XNODE=XYZ(1,NODE)

YNODE=XYZ(2,NODE)

N1=NW(NODE)-1

C

DO II=1,NF

INODE=MF(II)


ENDDO

1 CONTINUE

C----------------------------------------------------------------------

C CHECK NODES WITH INDEX=0 IF TENSION OCCURS (0)

C----------------------------------------------------------------------

IF(INDEX.EQ.0) THEN

QC(N1+2)=PT(N1+2)-PEQT(N1+2)

REACTX=PT(N1+1)-PEQT(N1+1)

INDFOR1=NCHAIN(I+1,2)


IF(IBCAL.EQ.1.AND.INDFOR1.EQ.3)THEN

DP1(N1+2)=-PEQT(N1+2)/10.0D0

NUMD(N1+2,1)=10

c WRITE(2,*)’NODE=’,NODE,’DP1=’,DP1(N1+2)

ENDIF

IF(REACTX.GT.0.0D0.AND.INDFOR1.NE.0.AND.INDFOR2.NE.0) THEN

C IF(REACTX.GT.0.0D0) THEN

C CHANGE INDEX FROM 0 TO 4 AND FREE THE NODE

NCHAIN(I,2)=4

NFIX(1,II)=0

NFIX(2,II)=0

DXYT(1,II)=0.0D0

DXYT(2,II)=0.0D0

DD(1,II)=0.0D0

C APPLY LOAD=REACT & THEN UNLOAD IT INCREMENTALLY

NUMBER=INT(ABS(REACTX)/1.0D-04)+1

DP(N1+1) =REACTX/NUMBER

PNOD(N1+1)=REACTX

NUMD(N1+1,1)=NUMBER

c WRITE(2,*)’NODE=’,NODE,’NUM=’,NUMBER,’PNOD=’,PNOD(N1+1),

c 1 ’DP=’,DP(N1+1)

C CHECK FOR FRICTION

ELSEIF(REACTX.LT.0.0D0)THEN

C

ALLFRICT=REACTX*XMUFR(1)

FPND=ALLFRICT+PNOD(N1+2)

IF(FPND.GT.0.0D0)GO TO 10

APPLFR=FRICTPR(N1+2)


IF(ALLFRICT.LT.APPLFR.AND.APPLFR.GT.FRMAX0)THEN

FRDIF=ALLFRICT-FRICTPR(N1+2)

IF(FRDIF.GE.(-2.0D-05)) THEN

FRICT(N1+2)=FRDIF

ELSE

FRICT(N1+2)=-2.0D-05

ENDIF

FRICTPR(N1+2)=FRICTPR(N1+2)+FRICT(N1+2)

c WRITE(2,*)’NODE=’,NODE,’FRICT=’,FRICT(N1+2),’FRICTPR=’,

c 1 FRICTPR(N1+2),’RNORM=’,REACTX

C

ENDIF

ENDIF

GO TO 10

C----------------------------------------------------------------------

C CHECK NODES WITH INDEX=4 IF XNODE < XREF (4)

C----------------------------------------------------------------------

ELSEIF(INDEX.EQ.4) THEN



QC(N1+2)=PT(N1+2)-PEQT(N1+2)

DXNODE=XNODE-XREF

INDPR=NCHAIN(I-1,2)

IF(DXNODE.LT.0.0D0) THEN

C CONSTRAIN THE NODE IN THE X-DIRECTION & CHANGE INDEX FROM 4 TO 0

NFIX(1,II)=1

NFIX(2,II)=0

DXYT(1,II)=0.0D0

DXYT(2,II)=0.0D0

DP(N1+1)=0.0D0

PNOD(N1+1)=0.0D0

NUMD(N1+1,1)=0

NUMBER2=INT(ABS(DXNODE)/(1.0D-05))+1

NUMD(N1+1,2)=NUMBER2

DD(1,II)=-DXNODE/NUMBER2

NCHAIN(I,2)=0

WRITE(2,*)’NODE’,NODE,’N=’,NUMBER2,’dd=’,(dd(ifr,ii),IFR=1,2)

GOTO 10

C

ELSEIF(INDFOR1.EQ.0.AND.INDFOR2.EQ.0)THEN

C CONSTRAIN THE NODE IN THE X-DIRECTION & CHANGE INDEX FROM 4 TO 0

NFIX(1,II)=1

NFIX(2,II)=0

DXYT(1,II)=0.0D0

DXYT(2,II)=0.0D0


DP(N1+1)=0.0D0

PNOD(N1+1)=0.0D0

NUMD(N1+1,1)=0

NUMBER2=INT(ABS(DXNODE)/(1.0D-05))+1


DD(1,II)=-DXNODE/NUMBER2


NCHAIN(I,2)=0

ENDIF

GO TO 10

C----------------------------------------------------------------------

C CHECK NODES WITH INDEX=3 (SKEW BOUNDARY) (3)

C----------------------------------------------------------------------

ELSE IF(INDEX.EQ.3) THEN

INDPR=NCHAIN(I-1,2)

RXX = PT(N1+1)-PEQT(N1+1)

RYY = PT(N1+2)-PEQT(N1+2)

QC(N1+2)=-(-PEQT(N1+1)*XSIN+PEQT(N1+2)*XCOS)

IF(XNODE.GT.XREF.AND.INDPR.NE.3) THEN

C FREE THE SKEW NODE AND CHANGE INDEX FROM 3 TO 4

NFIX(1,II)=0

NFIX(2,II)=0

DXYT(1,II)=0.0D0

DXYT(2,II)=0.0D0

DD(1,II)=0.0D0

DD(2,II)=0.0D0

NCHAIN(I,2)=4

C REMOVE THE NODE FROM SKEW BOUNDARY AND SHIFT THE ISPB ARRAY

CALL CHSKEW(NSKEW,NDIM,NODE,XCOS,XSIN,-1)

C ADD LOADS = REACTIONS AND THEN UNLOAD THEM

RXX = PT(N1+1)-PEQT(N1+1)

RYY = PT(N1+2)-PEQT(N1+2)

ABRXX=ABS(RXX*XCOS)

NUMBER=INT(ABRXX/1.0D-04)+1

DP(N1+1) = RXX*XCOS/NUMBER

DP(N1+2) =-RXX*XSIN/NUMBER

PNOD(N1+1)=RXX*XCOS

PNOD(N1+2)=-RXX*XSIN

NUMD(N1+1,1)=NUMBER

NUMD(N1+2,1)=NUMBER

NUMD(N1+1,2)=0


NUMD(N1+2,2)=0


c 1 PNOD(N1+2),’DP=’,DP(N1+1),DP(N1+2)

ELSE

C CHECK FOR FRICTION

RXX1=RXX+PNOD(N1+1)*XCOS

IF(RXX1.LT.0.0D0)THEN

ALLFRICT=RXX1*XMUFR(2)

APPLFR=FRICTPR(N1+1)*XSIN+FRICTPR(N1+2)*XCOS

IF(ALLFRICT.LT.APPLFR.AND.APPLFR.GT.FRMAX3)THEN

c IF(ALLFRICT.LT.APPLFR)THEN

IF(XNODE.LT.0.0010D0)GOTO 55

FRDIF=ALLFRICT-APPLFR

IF(FRDIF.GE.(-2.0D-05)) THEN

RYPAR=FRDIF

ELSE

RYPAR=-2.0D-05

ENDIF

FRICT(N1+1)=RYPAR*XSIN

FRICT(N1+2)=RYPAR*XCOS



c WRITE(2,*)’NODE=’,NODE,’FRICT=’,(FRICT(N1+IFR),IFR=1,2),

c 1 ’FRICTPR=’,applfr,’FRMAX3=’,frmax3

c write(2,*)’pt=’,(pt(n1+ifr),ifr=1,2),’peqt=’,

c 1 (peqt(n1+ifr),ifr=1,2),’rxx=’,rxx,’ryy=’,ryy

ENDIF

55 CONTINUE

ENDIF

C

C

IF(XNODE.LT.0.0D0) THEN

C APPLY DISPL TO MOVE THE NODE BACK ALONG THE POSITIVE SKEW

BOUNDARY

DD(1,II)=0.0D0

XNODE3=-XNODE/XSIN

NUMBER2=INT(ABS(XNODE3)/(1.0D-05))

IF(NUMBER2.LT.1)NUMBER2=1

DD(2,II)=XNODE3/NUMBER2


NUMD(N1+1,2)=0


ENDIF

ENDIF

GO TO 10


C----------------------------------------------------------------------

C CHECK THE NODE WITH INDEX = 2 (2)

C----------------------------------------------------------------------

ELSE IF(INDEX.EQ.2) THEN

DYNODE=YNODE-(YREF+XNODE*XCOS/XSIN)

INDPR=NCHAIN(I-1,2)

IF(XNODE.LT.0.0D0.AND.YNODE.LT.YREF) THEN

C RECONSTRAIN THE NODE & CHANGE INDEX FROM 2 TO 1

I1=I1+1

NCHAIN(I,2)=1

NFIX(1,II) =1

NUMBER2=INT(ABS(XNODE)/(1.0D-05))


DD(1,II) =-XNODE/NUMBER2


NUMD(N1+2,2)=0


PNOD(N1+1)=0.0D0

DP(N1+1)=0.0D0

GO TO 20

ELSEIF(XNODE.GT.0.0D0.AND.DYNODE.GE.0.0D0) THEN

C CHANGE INDEX FROM 2 TO 3 AND ADD THE NODE TO SKEW BOUNDARY

NCHAIN(I,2)=3

CALL CHSKEW(NSKEW,NDIM,NODE,XCOS,XSIN,1)

C CHANGE THE BOUNDARY CONDITION OF THE NODE

NFIX(1,II)=1

NFIX(2,II)=0

DXYT(1,II)=DXPEN

DXYT(2,II)=0.0D0

C

XNOD2=DYNODE*XSIN

NUMBER2=INT(ABS(XNOD2)/(1.0D-05))+1

DD(1,II) = DYNODE*XSIN/NUMBER2

DD(2,II) = 0.0D0

C PNOD(N1+2)=PNOD(N1+1)*XSIN

C DP(N1+2) =DP(N1+1)*XSIN

C PNOD(N1+1)=0.0D0

C DP(N1+1) =0.0D0

C NUMD(N1+2,1)=NUMD(N1+1,1)

C NUMD(N1+1,1)=0


NUMD(N1+2,2)=0



C

GO TO 10

ELSEIF(XNODE.LE.0.0D0.AND.YNODE.GE.YREF) THEN

C CHANGE THE INDEX FROM 2 TO 3 AND ADD THE NODE TO SKEW BOUNDARY

NCHAIN(I,2)=3

NFIX(1,II)=1

NFIX(2,II)=0

DXYT(1,II)= DXPEN

DXYT(2,II)= 0.0D0

A=YNODE-YREF

B=-XNODE*XSIN/XCOS

IF(A.GE.B) THEN

XNOD2=DYNODE*XSIN

NUMBER2=INT(ABS(XNOD2)/(1.0D-05))+1

DD(1,II)=DYNODE*XSIN/NUMBER2

DD(2,II)=0.0D0


NUMD(N1+2,2)=0


ELSEIF(A.LT.B) THEN

NUMBER2=INT(ABS(B)/(1.0D-05))


DD(1,II)=(-XNODE*XCOS+(YNODE-YREF)*XSIN)/NUMBER2

DD(2,II)=(-XNODE*XSIN-(YNODE-YREF)*XCOS)/NUMBER2




ENDIF


C PNOD(N1+2)=PNOD(N1+1)*XSIN

C DP(N1+2) =DP(N1+1)*XSIN

C PNOD(N1+1)=0.0D0

C DP(N1+1) =0.0D0

C NUMD(N1+2,1)=NUMD(N1+1,1)

C NUMD(N1+1,1)=0

C

ENDIF

GO TO 10

C----------------------------------------------------------------------

C CHECK THE FIRST NODE WITH INDEX=1 IF TENSION OCCURS (1)

C----------------------------------------------------------------------


ELSE IF(INDEX.EQ.1.AND.I1.EQ.0) THEN

I1=I1+1

INDPR=NCHAIN(I-1,2)

DYNODE=YNODE-YREF

REACTX=PT(N1+1)-PEQT(N1+1)

C

IF(YNODE.GE.YREF) THEN

C CHANGE INDEX FROM 1 TO 3 AND ADD THE NODE TO SKEW BOUNDARY

I1=I1-1

NCHAIN(I,2)=3


C CHANGE THE BOUNDARY CONDITION OF THE NODE

NFIX(1,II)=1

NFIX(2,II)=0

DXYT(1,II)=DXPEN

DXYT(2,II)=0.0D0

C

NUMBER2=INT(ABS(DYNODE*XSIN)/(1.0D-05))+1

DD(1,II) = DYNODE*XSIN/NUMBER2

DD(2,II) = 0.0D0

PNOD(N1+1)=REACTX


DP(N1+1)=REACTX/NUMBER

NUMD(N1+1,1)=NUMBER


NUMD(N1+2,2)= 0


c 1 ’DP=’,DP(N1+1)


C

ELSEIF(YNODE.LT.YREF.AND.REACTX.GT.0.0D0.AND.INDPR.NE.2) THEN

C CHANGE INDEX FROM 1 TO 2 AND FREE THE NODE

NCHAIN(I,2)=2

NFIX(1,II)=0

NFIX(2,II)=0

DXYT(1,II)=0.0D0

DXYT(2,II)=0.0D0

C APPLY LOAD=REACT AND THEN UNLOAD IT INCREMENTALLY


DP(N1+1) =REACTX/NUMBER

PNOD(N1+1)=REACTX

NUMD(N1+1,1)=NUMBER


c 1 ’DP=’,DP(N1+1)

ENDIF

GO TO 10

ENDIF


10 CONTINUE

100 CONTINUE

20 CONTINUE

c IF(IOUTP.EQ.1) THEN

c DO I33=1,20

c NODE =NCHAIN(I33,1)

c INDEX=NCHAIN(I33,2)

c XNODE=XYZ(1,NODE)

c YNODE=XYZ(2,NODE)

c ND1=NW(NODE)

c ENDDO

c ENDIF

DO INODE=1,NF

NODE=MF(INODE)

IDF=NW(NODE)-1

DO ID=1,NDIM

NUM1=NUMD(IDF+ID,1)

NUM2=NUMD(IDF+ID,2)

IF(NUM1.GT.0) THEN

PNOD(IDF+ID)=PNOD(IDF+ID)*(NUM1-1)/NUM1

NUMD(IDF+ID,1)=NUM1-1

ELSEIF(NUM1.EQ.0) THEN

PNOD(IDF+ID)=0.0D0

DP(IDF+ID)=0.0D0

DP1(IDF+ID)=0.0D0

ENDIF

IF(NUM2.GT.0) THEN

NUMD(IDF+ID,2)=NUM2-1

ELSEIF(NUM2.EQ.0) THEN

DD(ID,INODE)=0.0D0

ENDIF

ENDDO

c

ENDDO

C

IF(IOUTP.EQ.1) THEN

WRITE(2,30)INCR,YREF,TOTPEN

WRITE(2,40)

DO I=1,NPEN

NODE =NCHAIN(I,1)

N1=NW(NODE)-1

WRITE(2,50)(NCHAIN(I,J),J=1,2),(XYZ(J,NODE),J=1,2),QC(N1+2)


ENDDO

ENDIF

30 FORMAT(/,2X,’INCR=’,I5,4X,’YREF=’,F12.6,4X,’TOTPEN=’,F12.6)

40 FORMAT(/,6X,’NCHAIN’,9X,’COORDINATES’,15X,’QC’,/,55(’-’))

50 FORMAT(2X,2I5,2F12.6,4X,E15.7)

c

C WRITE(2,*)’ NODE+ DXYT+ NFIX’

C DO II=1,20

C WRITE(2,100)MF(II),(DXYT(J,II),J=1,2),(NFIX(J,II),J=1,2)

C ENDDO

C 100 FORMAT(1X,I5,2F12.6,2I5)

C 200 FORMAT(1X,I3,4I4,2X,6E12.4)

RETURN

END

C**********************************************************************

SUBROUTINE CHSKEW(NSKEW,NDIM,NODE,XCOS,XSIN,IND)

C**********************************************************************


COMMON /SKBC /ISPB(20),DIRCOS(20,3)

IF(IND.EQ.1) THEN

C ADD THE NODE FROM SKEW BOUNDARY

DO ISKEW=1,NSKEW

KNODE=ISPB(ISKEW)

IF(KNODE.GT.NODE) THEN

DO KSKEW=ISKEW,NSKEW

KK=NSKEW-KSKEW+1

ISPB(ISKEW+KK)=ISPB(ISKEW+KK-1)

DIRCOS(ISKEW+KK,1)=DIRCOS(ISKEW+KK-1,1)

DIRCOS(ISKEW+KK,2)=DIRCOS(ISKEW+KK-1,2)

ENDDO

ISPB(ISKEW)=NODE

DIRCOS(ISKEW,1)=XCOS

DIRCOS(ISKEW,2)=XSIN

GO TO 25

ENDIF

ENDDO

ISPB(NSKEW+1)=NODE

DIRCOS(NSKEW+1,1)=XCOS

DIRCOS(NSKEW+1,2)=XSIN

25 NSKEW=NSKEW+1

ELSEIF(IND.EQ.-1) THEN

C REMOVE THE NODE FROM SKEW BOUNDARY AND SHIFT THE ISPB ARRAY

DO ISKEW=1,NSKEW

KNODE=ISPB(ISKEW)



DO IK=ISKEW,NSKEW-1

ISPB(IK)=ISPB(IK+1)

DO ID=1,NDIM

DIRCOS(IK,ID)=DIRCOS(IK+1,ID)

ENDDO

ENDDO

ISPB(NSKEW)=0

DO ID=1,NDIM

DIRCOS(NSKEW,ID)=0.0D0

ENDDO

NSKEW=NSKEW-1

GO TO 30

ENDIF

ENDDO

30 CONTINUE

ENDIF

RETURN

END

C******************************************************************

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Index

Acoustic propagation 6Acoustic technique 149, 151, 153, 155Anderson-Parrinello-Rahmen theory

164Anisotropy 6, 30, 31, 35, 75, 184, 188Angle of internal friction 3Aspect ratio 44Atom 161Attenuation 150–156

Back stress 21–38, 48–56, 72–78, 168,188–203

BAT 158Binary model 5BioCoRE 164Body force 8, 20, 168(Non) bonded energy 164Born-Oppenheimer approximation

164

Cam clay model 26-55, 70–83, 100,114, 133–143, 184–207

Capillarity 5Cap model 29, 30Carbon-nano-fiber 162Cartesian coordinate 16Characteristic frequency 153–155Characteristic length 39, 40Clausius-Duham inequality 46Clay 3, 15, 30, 41, 63, 74, 170Clay-fluid interaction 164Clay liner 114Coefficient of consolidation 197, 199Cohesion 24, 26, 125, 142Cohesionless 62, 141Compressibility 2, 21 100Compression index 35, 100, 102, 176Compression wave 150, 151

Condition number 66, 67Cone face 98, 101, 184, 200Cone penetrometer 91–106, 158, 168,

192, 203Cone resistance 92, 94, 98, 178Cone shoulder 98, 200

Conservation principle 2Consolidation 6–17, 26–43, 94–115,

158, 178, 191Consolidation coefficient 15, 96

Constitutive equation 2, 16, 40, 56,75, 82

Constitutive law 8, 58, 63, 64Constitutive model 2, 8, 10, 113,

125, 136, 184Constitutive relation 16, 36–38, 56,

57, 75, 161Constitutive theory 6, 9

Continuum 4–19, 40, 50, 59–69,162–170

Continuum models 2, 16Cosserat (continuum), 59–61Coulombic energy 164Coupled consolidation 112, 115

Coupled theory 10–21, 81, 94–116,158–200

Coupled theory of mixtures 10, 12,13–21, 81, 94, 97, 104, 109, 113,116, 173, 188, 199, 200

Coupling modulus 152Coupling matrix 84Crank-Nicholson technique 96Critical void ratio 65, 66, 172Cross hole technique 151Cyclic shear strain 7

Damage 11, 38, 68–74, 117, 168–205Damping (ratio), 7, 157

436 Index

Darcy’s law 13Densification 7, 8Deviator stress 2, 1Diffusion 4, 15, 16, 96, 97, 153Diffusivity 16Dihedral angle 161Dilatancy 63Dilational wave 6Discrete mechanics 163Displacement tensor 17Dissipation potential 47, 48Dissipation test 94, 96, 98, 100, 192,

197Disturbed state 74Divergence 19, 20, 66, 152DPD (Dissipative Particle Dynamics),

170Drucker-Prager 25, 26Dutch cone 91Dynamic yield function 69

Earthquake 1, 7Effective stress 1, 5, 7–18, 33, 69–80,

96, 102, 133, 173–205Elasticity 33, 78Electric cone penetrometer 92Electron charge 161Electron permittivity 161Embedded stress 31, 40, 43, 75Endochronic (theory) 8Entropy 6, 8EPB shield 117Equivalent beam or truss element

method 167Equivalent Linear Method (ELM) 7Equivalent plastic strain 78, 133, 134Excess pore water pressure 1, 13–17,

98, 99, 105–106

Failure 3, 59, 63, 65, 100, 142, 143,192

Fast P-wave 150, 151, 153Finite deformation 6FEM (Finite Element Method) 2,

9–18, 72, 83, 86, 90–103, 113–132,170–174

Finite strain 17–21, 63, 74, 76, 83, 89Flow rule 36, 57, 69, 136Fluid droplet 170

Fourier equation 4Friction angle 23, 26, 142, 143, 172Friction resistance 91, 94, 191Fully drained 21, 98, 101, 106, 111Fully undrained 21, 98, 101, 106, 111

Gauss point 133, 213Gradient 2, 9, 13, 37, 39–49, 60–87,

173–207Grain interaction 11, 63, 64, 170, 172Grain rotation 11, 18, 59, 61, 188

Hardening 26, 30–34, 45–54, 64–78134, 135

Heat diffusion 15, 16, 96, 97Henkel’s pore pressure coefficient 112Helmholtz free energy 46, 47Hvoslev surface 28Hydraulic conductivity 12–21, 94–116,

149–183Hydraulic gradient 13Hydrodynamic 4Hydro-mechanical 18

Ill conditioned 66Ill posedness 66, 67Immiscible mixture 3Incremental scheme 76, 89Induced anisotropy 30, 31, 36, 75, 188Inelastic 10, 57, 58, 62, 70Inherent anisotropy 30, 31, 36, 76In-phase motion 150Interaction 3–11, 38, 38, 48, 63,

64, 98–200, 115, 151, 161–173,181–204

Interface element 125, 134, 135, 142Internal friction angle 26Internal state variable 45, 47, 48, 203Intrinsic permeability 154Isoparametric (element) 86, 129, 130Isothermal 6

Jacobian 21, 77, 81

Kinetic theory 4

Lagrangian reference frame 10, 76,80–93, 96–113, 199, 200

Lame’s constant 152

Index 437

Laplacian operator 39Large strain 2, 10, 55, 82, 86, 95, 100,

109, 119, 188, 198–207Length scale 38–45, 73, 163Lennard-Jones 12–6 potential energy

165Lime 3Linear stiffness matrix 84Liquefaction 1, 7, 8, 18, 151Liquefaction potential 151Long range back stress 50, 51, 54,

201–203

MaCauley bracket 35, 48, 70Macroscale 39, 45–48, 167Macroscopic yield 35Mandel-Cryer effect 16, 17Mantle cone 91Material Studio 164Mathematical gradient 41, 67Mechanical cone 91Mesh size dependency 43Mesoscale 46–48Micro-mechanical 9–18, 38–56, 63, 75,

168, 173, 188, 204–207Microplane model 57, 58Micropolar (continuum) 59, 60Microscale 45, 46Minimization 166Mixture 1–23, 81, 94–104, 113 115

168 173 188–207Modified spin tensor 56Mohr-Coulomb 23–25, 125, 142Molecular mechanics 163Molecule 161Mud 3Multi phase 1 4 11 204Multi-polar continuum 9NAMD 164Nanocomposite 162Nano-mechanics 161–167, 170, 171Nanotechnology 162(Non) bonded energy 164Nonlinear 2, 10, 62, 81–90, 120, 134,

135, 141, 142Non-linear stiffness matrix 84

Octahedral normal stress 112Octahedral shear stress 112

OPM (One Point Method) 95, 100Orthogonality relation 152Oscilloscope 156Out-of-phase motion 150Overstress 70, 71

Partial differential equation 15, 96Partially drained 21, 98, 101, 106Partially saturated 10Permeability 6–20, 98–116, 143–199Permeability tensor 20, 168Physical gradient 42, 67Piezocone penetrometer test 91–96,

100, 104, 115, 158Piola-Kirchoff stress 77, 80π-plane 23–25, 31–35, 58, 120, 165Poisson’s ratio 16, 166Pore water pressure 1, 7–17, 77–109,

168–200Poro-elastic 8, 151Porosity 18, 20, 154, 168Porous media 1–10, 20Preconsolidation pressure 178PSRN (Pseudo Random Noise), 155

Quantum mechanics 163Quasi-static 6, 7Quicksand 3

Radius of plastic zone 113Rate dependency 11, 38, 68–74, 172,

175Recompression index 35, 102, 188Remeshing 89, 118, 124–127, 134Resonant column test 54Rigid plastic 26Rigid porous solid 5Roscoe surface 28Rotational wave 6Roughness 43RVE 39, 40, 42–46, 167, 170

Sand 3–8, 18, 52, 53, 62, 64, 73, 74,104, 139, 155, 170, 191, 203, 204

SASW 151Seepage 13, 85Senate Committee on Commerce

Science and Transportation 162Shear band 40–42, 63, 64, 68, 172

438 Index

Shear modulus 7, 16, 53, 137, 143,152, 166

Shear wave 151Shield tunnel 116, 117Short range back stress 50, 51, 54,

201–203Schrodinger equation 164Simple shear 7Singular value decomposition method

155Skeleton 1, 7, 18, 78, 203, 204Skempton’s pore pressure coefficient

112Sliding potential 102Slip theory 57Slow P-wave 150, 151, 156Slurry consolidation 191Small strain 2, 16, 62, 74, 201S/N ratio 155Source 155Spin tensor 56Spread spectrum 150SPT 91Steady flow 13Stiffness 7, 37, 55, 81, 84, 89–106, 122,

134–138, 167–204Stress invariant 23Stress tensor 59Swedish penetrometer 91

Taylor expansion 41, 66, 67Thermodynamics 4, 15, 46, 48, 74Three phase material 11Threshold strain 62

Total stress 1, 7, 12, 96, 178–183, 204TPM (Two Point Method) 95, 109Transient flow 13, 21, 101Tresca 23–25, 33Trigger 155Two phase material 11, 101, 152, 173,

204

Uncoupled consolidation 112, 114, 115Updated Lagrangian 18, 21, 76,

80–113, 199–200

van der Walls energy 164Varved clay 178Virtual consolidation 112Virtual work 18, 57, 61, 76, 100Visco-elasticity 33Visco-plasticity 48, 69, 70Viscosicity 48, 68–73, 100, 154, 168,

172, 173, 178, 204Void 1, 11, 15. 35, 55, 66, 76, 102–104,

143, 157, 172, 203, 266Volume fraction 2, 3, 5, 9, 152Volumetric strain 7, 16–31, 54, 134,

168, 174von Mises 23–25, 33, 70

Wave propagation 6, 9, 10, 151, 152Well posedness 66, 67

Yazoo clay 166Yield criteria 10, 23–40, 42–46, 58, 80Yield surface 26–36, 49, 63, 64, 69–72,

78, 79, 89, 90, 133, 134, 168, 173

the coupled theory of mixtures in geomechanics with

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