geomechanics notes

7/30/2019 Geomechanics Notes

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THEME 1. INTRODUCTION TO GEOMATERIALS 4

1.1. Saturated 4

1.1.1. Low density. Natural geomaterials 4

1.1.2. Low density. Artificial geomaterials. Hydraulic fill 41.1.3. High density. Natural materials 4

1.2. Unsaturated 4

1.2.1. Natural low density geomaterials 4

1.2.2. Artificial geomaterials 5

1.3. Soft rocks 5

1.4. Specials soils 5

THEME 2. HYDRO MECHANICAL COUPLING IN GEOMATERIALS 6

2.1. Formulation for saturated soils 6

2.1.1. Saturated soils: main assumptions 6

2.1.2. Biot / hydro mechanical formulation: equations involved (flow and deformation

coupling) 6

2.1.3. u p formulation (Formulation based on displacement and pressures) 8

2.1.4. FEM Spatial discretization 10

2.1.5. Time discretization 10

2.1.6. Mechanical behaviour 11

2.1.7. Undrained strength 14

2.2. Formulation for Thermo hydro mechanical (T H M) problems in porous media 15

2.2.1. Basic formulation 15

2.2.2. The total mass flux of a species in a phase (e.g. flux of air present in gas phase) 16

2.2.3. Momentum balance for the medium. Unknown u 18

2.2.4. Energy balance for the medium 18

2.2.5. Species mass balance (reactive transport). Unknown c 19

2.2.6. Boundry conditions 19

2.3. Constitutive equations for T H M problems in porous media 20

2.3.1. Hydraulic problem 202.3.2. Thermal problem. 232.3.3. Mechanical problem 24

2.4. Introduction to numerical methods in geotechnical analysis 24

2.4.1. Finite elements in geotechnics: General aspects 24

2.4.2. The boundaries 25

2.4.3. Initial stresses 25

2.4.4. Effective/total stresses. Drained, undrained, consolidation 25

2.4.5. Excavation and construction 26

2.4.6. Constitutive laws 26


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THEME 3. GEOMECHANICAL BEHAVIOUR OF CLAYS AND SANDS 27

3.1. Behaviour of clays 27

3.1.1. Experimental behaviour of clays in the triaxial test 27

3.1.2. An attempt to simulate experimental behaviour of clays: the Cam Clay model 28

3.1.3. The Cam Clay model: predictions 32

3.2. Behaviour of sands 37

3.2.1. Experimental behaviour of sands 37

3.2.2. Critical state in sands: difficulties and achievements 38

3.2.3. Liquefaction (static and cyclic) 38

THEME 4. UNSATURATED SOILS 39

4.1. Unsaturated soils. Reference material 39

4.1.1. Introduction 39

4.1.2. Experimental behaviour of a reference material 40

4.1.3. Barcelona Basic model 43

4.1.4. Response of the model with different trajectories 47

4.2. Expansive unsaturated soils 48

4.2.1. Variation of humidity and suction in expansive unsaturated soils 48

4.2.2. Parameters of expansiveness 49

4.2.3. Geotechnical problems due to expansive soils 50

4.2.4. Foundations 50

4.3. Unsaturated soils. Rockfill 51

4.3.1. Some observation of rockfill behaviour in the fiel 51

4.3.2. Mechanisms of particle breakage 52

4.3.3. Results of an experimental investigation 53

4.3.4. A model for rockfill compressibility 53

4.3.5. Unsaturated soils vs unsaturated rockfill 54

THEME 5. HARD SOILS AND SOFT ROCKS 56

5.1. Behaviour of bonded soils 56

5.1.1. Introduction. Reconstituted soils and natural soils 56

5.1.2. Structure development 57

5.1.3. Effects of structure. The Limit State Surface concept 60

5.1.4. Destructuration 60

5.2. Shear strength of clays 61

5.2.1. Residual strength 61

5.2.2. Shear strength of stiff clays and weak argillaceous rocks 62

5.2.3. Operational strength in brittle materials 64

THEME 6. VERY SMALL STRAINS IN SOILS HIGH STIFFNESS 65

6.1. Introduction 65


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6.2. Soil behaviour far from failure 65

6.4. Synthetic Example: excavation far from failure using different models 68

THEME 7. ANISOTROPY AND PRINCIPAL STRESS ROTATION 70

7.1. Introduction and definitions 70

7.1.1. Introduction Error! Bookmark not defined.

7.2. Rotation of principal stresses 71

7.3. Laboratory equipment to examine anisotropy 71

7.4. Anisotropic behaviour 71

7.4.1. Reconstituted materials 71

7.4.2. Natural materials 72

7.5. Case history 72


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THEME 1. Introduction to geomaterials

1.1.Saturated1.1.1. Low density. Natural geomaterialsNormally Consolidated (NC) clays and silts:

Deltaic medium Low permeability Contracting behaviour Undrained strength (Basic parameters: undrained shear strength , consolidation ratio)Low-density sands:

Liquefaction (it also occurs in silts) Shear modulus has a great influence in how waves propagates in the ground1.1.2. Low density. Artificial geomaterials. Hydraulic fill Liquefaction

1.1.3. High density. Natural materialsOver Consolidated (OC) clays and silts

High plasticity Marked brittleness? Peak strength Strength residual (rotura progresiva) Low permeability Expanding behaviour Drained strength Case of Guadalquivir blue clay (WL=64%, IP=37%). Asnalcllar Failure. Lessons learned

- The difficulty to interpret, in practice, the behaviour of hard clayey soils/soft clay rockshaving: high plasticity, low permeability, marked brittleness, low residual friction

- The risk of some construction procedures of tailings dams founded on brittle clays- The relevance of correctly estimating at the design stage of pore water pressures.

Standard hypothesis (stationary flow) goes against safety.

Dense sands

1.2.Unsaturated1.2.1. Natural low density geomaterialsClays, silts and low density sands


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1.2.2. Artificial geomaterialsClays, silts, sands

Compacted rockfill

The behavior of rockfill is dominated by particle breakage Particle breakage explains the qualitative differences observed between the behavior of

sands (at low and moderate levels) and rockfill

Particle breakage in rockfill depends on:- Strength of individual particles- Grain size distribution- Stress level- Relative humidity prevailing at the rockfill voids

Valle del Gualdaquivir

Sedimentation planes:- Quasi horizontal stratification- Slickensides detected at some places- High continuity (>40m)

1.3.Soft rocks Clayed materials with carbonates and sulphates In case of unaltered samples, main strength is high in compression test. If it is altered by an

increment of humidity, rock strength decreases quickly

Low permeability in cases of Argillite (interstitial water pressure dissipation requires along term, due to embankments)

Brittle behaviour under undrained conditions (marked peak strength) due to cementation A lot of expansiveness cases. Weathering of soft rocks when they are exposed to the

environment.

1.4.Specials soils

Volcanic soils: tuff


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THEME 2. Hydro mechanical coupling in geomaterials

2.1.Formulation for saturated soils2.1.1. Saturated soils: main assumptionsOnly two phases: solid particles + liquid (No gas phase)

Two hypotheses of classical soil mechanics

water incompressible Solid particles not deformable: voids change size and/or shape (water may escape or may

not)

Hypotheses of soil mechanics:

Continuum media. This hypothesis is not reasonable in large solid blocks (very lowporosity) + discontinuities (higher porosity)

2.1.2. Biot / hydro mechanical formulation: equations involved (flow and deformationcoupling)

Biot general formulation is for dynamic problems and saturated soils

Sign convention: stresses in compression are negatives, but pore water pressure in

compression is positive

Small strains are always assumed in both formulations (but generalisation is possible)

Preliminary considerations:

The coordinate system moves with the solid phase (material coordinates), so convectiveaccelerations in terms of relative velocity applies only to the fluid phase.

Displacement of the solid matrix: Displacement of the fluid, relative to the particles: Fluid velocity, relative to the solid particles and in Darcys sense (average velocity of the

whole section):

Actual fluid velocity: , where is the porosity Absolute velocity of the fluid: Density of the mixture: Densities are assumed constant (because of small deformations)

Strains


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Effective stresses (Biot)

: Total stresses

: Water pressureGeneralization of effective stresses (Biot y Willis):

: Bulk modulus of the whole porous medium: Bulk modulus of the solid particlesMechanical constitutive equation

Relates stresses and strains. Generally nonlinear (incremental form):

: Tangential stiffness matrix (involves material parameters, like , in elasticity)

:Strains that depend only on effective stress changes (In nonlinear models depends

on state variables (history variables) : Deformations due to other effects (no stresses involved like chemical processes)It cannot work with a function because it would have two or more point for a samestrain, this is why it is used the incremental form.

Momentum balance equations

Conservation for fluid phase

Water density assumed constant mass conservation volume conservationFlow entering per unit volume (Gauss therorem):


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When the generalisation of the effective stress law is considered

Volumetric strain of the soil skeleton in a : Volumetric fluid deformation in a (compressible fluid): Volumetric deformation of solid grains (Solid particles deformable):

Generalized Darcys Law

In classical Soil Mechanics:

It can be written as:

When fluid accelerations are involved, it is generalized as:

2.1.3. u p formulation (Formulation based on displacement and pressures)Looking for a simplified version: selected

and

as main unknowns

Terms including can be eliminated if we assume This is reasonable for geotechnical earthquake engineering problems, because high

frequencies are not involved


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Coupled H M formulation, so it is not possible to solve the equations separately

Very general: most of the Classical Soil Mechanics situations are included in the formulation

Undrained, dynamic case

No water flow, that is: Keep equations , , , . In equation , neglect permeability terms

If water is assumed incompressible, then no volume change

Consolidation

Keep equations , , . Delete terms including in equations and

It is a typical case after an earthquake

Drained, static case

Delete terms including time derivatives: equations , , and

Uncoupled: H and M problems can be solved separately (loads are not modify the water

pressure in drained conditions)


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2.1.4. FEM Spatial discretizationDisplacements, is interpolated by using quadratic shape functions, we need secondderivatives.

Pore water pressure, is interpolated by using linear shape functions, we only need firstderivatives. Mass matrix Stiffness matrix Coupling matrix Compressibilitymatrix

Permeability

matrix

Vector forces Vector of flows Compressibility Matrix is not taken into account in Classical Mechanics.2.1.5. Time discretizationA Finite Difference scheme is used for time derivatives, based on original work by Newmark by

Finite Differences ( ). Our main unknown are: and

Models:

Elasticity: linear system of equations Elastoplasctic models: nonlinear system of equations


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2.1.6. Mechanical behaviourStresses and strains

Total stresses Effective stresses Strains

Lambe

Cambridge

Cambridge

(plane strain) Elasticity + Isotropy (elastic deformations)

Recoverable

Bulk and shear modulus are uncoupled

No failure

If , which is common in Soil Mechanics, using the Cambridge representation

Plasticity

Objective: avoid the disadvantages of the theory of Elasticity:

Characterize the ultimate and failure states Model non-recoverable deformations Model in a rigorous manner brittle or quasi-brittle behaviour


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Different type of plasticity behaviour:

Perfect rigid-plastic Perfect elastic-plastic Hardening (yield limit depend on the strain) Softening (yield limit depend on the strain)Yield surface

To calculate the plastic strains: generalize the one-dimensional experimental results Fixed yield surface (perfect plasticity): Expanding yield surface (hardening plasticity): Contracting yield surface (softening plasticity): The stress state must always be either inside or on the yield surface (never outside):

- Inside,

: only elastic deformations

- On, : elastic and plastic deformations In Soil Mechanics, the stresses are represented with the variables , Therefore, the yield surface is of the form

: hardening parameter that controls the expansion or contraction of the yield

surface

Plastic potential

To calculate the plastic deformations, a plastic potential function is postulated from whichthe deformations can be obtained:

: Parameter that controls the size and position of the plastic potential surfaceFlow rule

: Controls the magnitude of the plastic strains : Controls the direction of the plastic strains The yield surface,, and the plastic potential, , are in general different functions If , then the plastic model is said to be associated


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The plastic strain components are related: there is coupling between them, given by theflow rule

The plastic strains depend on the stress state, NOT on the applied stress incrementsHardening law

A function describing the change of size and/or position of the yield surface must beprovided:

Plastic deformations (consistency condition)

Once the elasto-plastic regime has been reached (i.e. one there are plastic strainsdeveloping), the stress state point must be on the yield surface. Therefore, the following

condition must be satisfied:

If the stress state changes, but continues in the elasto-plastic regime, the stress point will

always remain on the yield surface and therefore

* + * +

* + * +

*+ * +

The elastoplastic stiffness tensor is non-symmetric, except when

(associate

plasticity) In general


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: Elastoplastic stiffness tensor: Plastic modulus Perfect plasticity: Hardening plasticity Softening plasticity MohrCoulomb ModelElastic perfect plastic 2.1.7. Undrained strengthIn undrained conditions it was very difficult to predict the pore water pressure generated

during loading (couple system). Now with the H M formulations and FEM it is possible.

Alternative: Instead of using Mohr Coulomb and effective stresses, use and total stresses.Now the problem is how to evaluate , but it is simpler than evaluating pore pressure.Mohr Coulomb criterion in total stresses

The effective stress paths are difficult to know We could work with total stresses, because and failure is reached when the radius of Mohrs circle in effectives (and in totals) , tangent to the strength line Any Mohrs circle in total stresses is tangent to the line

Therefore it is as if the undrained strength were And in this way, working with total stresses, we can forget about the water pressure But note that is not a true parameter...


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2.2.Formulation for Thermo hydro mechanical (T H M) problems in porous media2.2.1. Basic formulationThe three phases are:

Solid phase (s): mineral

Liquid phase (l): water + air dissolved Gas phase (g): mixture of dry air and water vapourThe three species are

Solid (-): the mineral is coincident with solid phase Water (w): as liquid or evaporated in the gas phase Air (a): dry air, as gas or dissolved in the liquid phaseUnsaturated soil: Porosity and degree of saturation

Total volume: Gas phase:

Liquid phase:

Solid phase:

: Degree of saturation of liquid and gas phases

: PorosityUnsaturated soil: Mass in phases

Gas phase:Mass of air: Mass of water:

Liquid phase:Mass of water:

Mass of air:

Solid phase: Mass of solid: Unsaturated soil: Mass in porous medium

Gas phase:Mass of air: Mass of water:

Liquid phase:Mass of water:

Mass of air:

Solid phase: Mass of solid:

: Mass fraction (mass of a component with respect to the total mass of the phase)


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2.2.2. The total mass flux of a species in a phase (e.g. flux of air present in gas phase)

The advective flux caused by fluid motion (phase) w.r.t. solid

The advective flux caused by solid motion w.r.t. fixed framework The non-advective flux w.r.t. phase (i.e. diffusive/dispersive) The following balance equation will be applied

Mass balance:

Material derivative:

Mass balance of solid

This equation expresses the variation of porosity caused by volumetric deformation and solid

density variation.

Mass balance of water. Unknown

: External supply of water


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Flux referred to a fixed framework

: Flux referred to the solid skeleton

The use of the material derivative and after substitution by solid balance leads to:

Porosity appears in this equation as:

A coefficient in storage terms

In a term involving its variation caused by different processes Hidden in variables that depend on porosity (e.g. intrinsic permeability)If solid density changes are neglected in this equation:

Finally, volumetric strain can be viewed as source term in water conservation equation.

Velocities associated to fluxes of mass

Mass averaged velocity

Mass balance of air. Unknown

: External supply of air


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2.2.3. Momentum balance for the medium. Unknown u

2.2.4. Energy balance for the mediumFirst thermodynamics principle can be in two different ways:

In terms of internal energy

In terms of enthalpy

: Amount of heat supplied per unit mass: Pressure: Internal energy: Is the volume per unit mass: EnthalpyInternal energy balance for the medium. Unknown T

The equation for internal energy balance for the porous medium is established taking into

account the internal energy in each phase ( ): : Energy flux due to conduction through the porous medium: Advective fluxes of energy caused by mass motions: Internal/external energy supplyA non - advective mass flux causes an advective heat flux because a species inside a phase

moves and transports energy.


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The internal energy in each phase can be calculated as a function of internal energy of each

component

Following this decomposition, the advective fluxes of energy in each phase are calculated as:

2.2.5. Species mass balance (reactive transport). Unknown c

2.2.6. Boundry conditions

The first term is the mass inflow or outflow that takes places when a flow rate of gas () is

prescribed

The second term is the mass inflow or outflow that takes place when gas phase pressure() is prescribed at a node. The coefficient is a leakage coefficient, i.e., a parameterthat allows a boundary condition of the Cauchy Type.

The third term is the mass inflow or outflow that takes place when vapor mass fraction isprescribed at the boundary. This term naturally comes from nonadvective flux (Ficks law).

Mass fraction and density prescribed values are only when inflow takes place. For outflow

the values in the medium are considered.


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2.3.Constitutive equations for T H M problems in porous media2.3.1. Hydraulic problemLiquid phase density. As a first approximation, the density of liquid phase can be calculated as: : Water compressibility ( ): Volumetric expansion coefficient ( )Gas phase density.

If ideal gases law is used to calculate vapor and air density, it follows that:

: Molecular masses for vapour ( ): Molecular masses for dry-air ( ): Ideal gas constant ( ): TemperaturePartial pressure Principle:

If is given, air pressure can be calculated as where it is assumed that vaporpressure is also known.

Vapor pressure is primarily a function of temperature. In a formulation that includes also

presence of air and capillary effects, vapor pressure can be determined from the following

function:

Retention curve. This law relates capillary pressure and degree of saturation:


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: Essentially controls the shape of the retention curve: Is a measure of the capillary pressure required to start the saturation of the soil

Distribution of pores plays a major role in the shape of retention curve. can be expressed asfunction of pore structure

Since capillary pressure can be scaled with surface tension it appears that also does. This canbe shown if Laplace law is recalled: : Surface tension: Curvature radius of the meniscus

Hydraulic constitutive law.

,

Darcys law

Advective fluxes of liquid gas are calculated using Darcys law in the generalized form:

: Intrinsic permeability(should be calculated): Relative permeability(should be calculated)

: Viscosity (should be calculated)Intrinsic permeability

This parameter depends primarily on the porous medium structure (it does not depend on

characteristics of fluid):

: Intrinsic permeability at the reference porosity

: Reference porosity


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This dependency required for modeling the hydraulic behavior of the clay barriers because the

soil undergoes variations of porosity which imply a change in permeability that can reach a

factor of 5.

Relative permeability

This law has the advantage that avoids the determination of relative permeability

experimentally, which is very difficult.

: Effective saturation (defined for retention curve): Parameter responsible for the shape of the retention curveLiquid viscosity Gas viscosity

Ficks law. Moleculardiffusion. Molecular diffusion of vapor in the gas phase is a process-modeled using Ficks law. Dissolved

air is also modeled with the same law. It is written in the following way:

: Molecular diffusion depending if diffusion takes place in the liquid or in the gas

phase

Ficks law. Mechanical Dispersion.

|

|

||

: Mechanical dispersion tensor


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: Specific heat of the phase: Mechanical dispersion heat flux2.3.2. Thermal problem.

A simple way to calculate enthalpy in porous media in which air and capillary effects should be

taken into account is:

: Latent heat for phase change: Specific heatsDensities for solid, liquid, vapor and air are calculated as described in constitutive equationsfor hydraulic problem. Degree of saturation is calculated using retention curve.

Non isothermal unsaturated soils approach

Soil desaturation:

Liquid pressure decrease or air pressure increase: two phase flow with nearly immisciblefluids. (Moderate T)

Vapor pressure increase: gas pressure is dominated by vapor pressure.

(relatively

high T)

Highlights:

Pressures are state variables, instead of degree of saturation. Gas pressure is equal to airplus vapor pressure.

Surface tension is a function of temperature. Capillary effects vanish as T increases. Balance of enthalpy tends to dominate saturation as capillary effects reduce.Enthalpy of solid phase

Commonly a constant value of the specific heat of the solid phase is used. It is, however

reasonable to consider also a linear variation with temperature:

: Specific heat at Enthalpy of water

The phase change diagram for pure water displays that depending on the pressure of water

and the enthalpy per unit mass three regions are distinguished.


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It is interesting to highlight that from 100 C to 180 C, vapor pressure changes by one order of

magnitude.

(1) Single phase region (liquid water) - (2) Two phase region (liquid water +vapor) (3) Single

phase region (heated vapor)

Thermal constitutive law(Fouriers law). Thermal conductivity is used in Fouriers law to compute conductive heat flux, i.e.:

: Thermal conductivity of porous mediumThe dry and saturated thermal conductivities can be calculated or directly determined

experimentally.

2.3.3. Mechanical problem

2.4.Introduction to numerical methods in geotechnical analysis2.4.1. Finite elements in geotechnics: General aspectsWhat aspects may be different in Geo-Mechanics when using F.E.?

Usually domain very large. Only a part of the geometry can be considered There are initial stresses in the soil/rock Water may be a fundamental issue. In undrained analysis, a total stress calculation may be

performed. In fully drained analyses water pressure does not change with loading. And H

M couple formulation may be required

The geometry can change. Ex.: excavation + construction Some especial elements may be required: structural elements (i.e. concrete wall), interface

elements (rock joint, contact soil-concrete), anchor elements, etc.

Constitutive laws should be appropriate for soils/rocks


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2.4.2. The boundaries Check if there is any symmetry Boundaries should be always far away from the zone where changes occur Check the results when changing boundaries Mechanical problems

- Close to failure, a collapse mechanism develops: local geometry- General drained analysis- General undrained analysis (no volume). Larger geometry involved

Elasticity: boundary control displacement pattern Plasticity: displacements are controlled by the mechanism of failure (more loca)

2.4.3. Initial stresses Soils & rocks have always initial stresses. They are very important in excavation problems

- Vertical stresses controlled by selfweight- Horizontal stresses defined from coefficient- In excavations problems: stresses acting on the excavation boundaries are the same as

the initial stresses, but with opposite sign.

Defining initial stresses:- Horizontal ground: they can be defined directly from specific weight and coefficient- Non horizontal ground: Stresses in equilibrium should be computed. Define the

geometry, mesh and compute equilibrium using an elastic model (

,

). Keep the

stresses and deleted displacement and deformations (Note that )2.4.4. Effective/total stresses. Drained, undrained, consolidationUndrained scenario

Use total stresses. Water not involved in the analysis Appropriate parameters (modulus, undrained strength) No volume change (i.e. Posissonss ratio

if elasticity is used)

Drained scenario

Use effective stresses Water pressure obtained form a flow analysis. They do not change due to loading Appropriate parameters (modulus, Mohr coulomb, strength)When using Finite Elements:

If a Solid Mechanics code is available, then the classical approach is useful


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If a soil Mechanics code solving H M formulation is available, then all scenarios may becomputed. Advise: check all scenarios to verify the analyses. In undrained conditions,

water pressures are particularly difficult to predict properly.

2.4.5. Excavation and constructionExcavation and construction implies change of geometry. A good code should be able to cope

with this.

Excavation and construction requires several steps in general. Displacements are very sensitive

to that, even when using linear models.

Excavation

Apply on the excavated boundary the initial stresses on the boundary but with opposite sign.

Construction

Apply the weight of the new elements. Different strategies... stiffness of the new elements

may be low the first step they exist, and they get the appropriate stiffness in the next step.

2.4.6. Constitutive lawsElasticity: always useful as a preliminary analysis

Elasto-plasticity: modern analyses require elasto-plastic models, but... be careful with

parameters, assumptions, etc.

Elastic + Mohr Coulomb (perfect plasticity, hardening, softening) Cam clay Models with two yield surfaces Etc.


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THEME 3. Geomechanical behaviour of clays and sands

3.1.Behaviour of clays3.1.1. Experimental behaviour of clays in the triaxial testStage I: isotropic consolidation

Stage II (deviator), CD test


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Stage II (Deviator), CU test

3.1.2. An attempt to simulate experimental behaviour of clays: the Cam Clay modelSome previous remarks

The model is based on experience gathered from the conventional triaxial tests, using theCambridge variables:

It is a good model to reproduce laboratory tests, not so good to reproduce the real (insitu) behaviour.

Yield surface:

With the change of variables the equation of the yield surface becomes:


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Volumetric deformations: a) elastic

Assume an isotropic and elastic behaviour inside the yield surface, and that volumetric andshear deformations are uncouple:

and are not constant

All points inside a yield line including in the limit points belonging to the line itself, are

represented by the same unloading-reloading line (url)

Elastic volume change:

: Elastic volume change: Elastic volumetric deformation


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Volumetric deformation: a) plastic

: Total volumetric deformationShear deformation: b) elastic

Shear deformation: b) plastic

If the relationship is known, then the shear deformation would be knowbecause is already known

The plastic deformation vector ( ) be measured in triaxial tests when the stresspath crosses the yield surface.

Sand: non-associated plasticity

Plastic potential: Cam Clay is an associated plasticity model, therefore


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As a consequence of the flow rule:

Hardening law: these equations define the change of size of the yield surface as a

function of the accumulated plastic deformations:

The change of size of the yield surface

only depends on the plastic volumetric

deformation

Elasto-plastic stiffness matrix

The relationship has meaning only in the elasto-plastic regime

The stiffness matrix is symmetric because The determinant of the matrix is 0 because the plastic volume and shear deformations are

related

*+

Deformations due to a stress path

Deduce as a function of We know that Calculate the elastic deformations Check whether after application of a stress increment the new stress point is in the elastic

or in the elasto-plastic regime:


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Critical state theory

The critical state is defined as the combination of stresses for which the shear plasticdeformation increases indefinitely without changes of the mean effective stress, the shear

stress, or the volume:

This happens when: 3.1.3. The Cam Clay model: predictionsConventional drained triaxial test (CD)

Plastic shear deformation

Hardening law


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Normally consolidate clay: CD test

Slightly overconsolidated clay: CD test


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Very overconsolidated clay: CD Test

Conventional undrained triaxial test (CU)

Effective shear deformation

and have opposite signs Combining this relationship with the yield function, we obtain differential equation of the

undrained effective stress path:

This equation has meaning only when plastic volumetric deformations are involving


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If That signals the end of the stresspath

If the stress path stars in the elastic zone: The undrained effective stress path does not depend on the total stress path: it is unique.

For differential total stress paths, different porewater pressure are obtained. Porewater pressure: parameter a:

a) Elastic Zone b) Soil undergoing plastic deformation

Normally consolidated clay: CU test


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Slightly overconsolidated clay: CU test

Very overconsolidated clay: CU Test


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3.2.Behaviour of sands3.2.1. Experimental behaviour of sandsSands are characterized by their relative density (or by their void ratio), difficult to measure:

{ Dilatancy of sands: it is an increment of volume when it is applied shear loads (elastic materials

do not have this behaviour)

Sand: CD Tests

Sand: CU tests


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3.2.2. Critical state in sands: difficulties and achievementsIs a critical state theory possible for sand?

There is no unique iso-ncl curve for sands: it depends on the initial state of the sand(loose, dense)

The (, ) plane cannot be explored from compression tests in a manner similar to clays Sands have memory of its initial state, even if it has been highly compressed However, a final state or critical state may be defined. Some authors claim that CSL seem

to be a zone rather than a line (steady state and critical state)

State parameter

Sand behaviour depends on where is the current state with respect to the critical state line

captures density and captures the final reference state. The difference takes intoaccount the stress level The further away from the final Critical State, the faster dilation or contraction happens Models for sands can be defined by using and a set of parameters.3.2.3. Liquefaction (static and cyclic) Liquefaction means zero shear strength (null effective strength

liquid) i.e. due to

increment of water pressure

Failure means usually large strains Loose sands: liquefaction implies directly large deformations and failure Dense sands: liquefaction is a transient situation. Due to dilation, water pressure

decreases.

Under cyclic loading, this process is repeated and finally large deformation may occur

(Cyclic mobility).


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THEME 4. Unsaturated soils

4.1.Unsaturated soils. Reference material4.1.1. IntroductionApplications

Stability of slopes Displacements and instability due to natural humidity changes of soils

- Swelling and collapse deformations of pavements- Foundations on collapsible or expansible soils

Displacements or failure of compacted soils and compacted structures (dams) Isolation barriers:

- Storage of industrial waste- Radioactive waste

Immiscible liquids: petroleum reservesMaterials

How does water affect different kinds of materials?

: relative humidity controls rupture velocity of particles : capillary pressure modifies intergranular pressure in sands, silts and

clays. Energy changes of water lead to swelling, retraction.

Variables of work

Variables of stress

Total stress: Air pressure: Water pressure: Suction: Net mean stress: Ratios of amount of water

Natural water content: Degree of saturation: Ratio of water:


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4.1.2. Experimental behaviour of a reference materialVolumetric strain

Suction increases compaction stress

Higher suction values tend to make the soil structure more rigid and therefore lesscompressible

Compaction effects in the loading collapse shape


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Unsaturated clayey soils swell/collapse depending on applied vertical stress.

This behaviour could be represented in a curved surface on the plane (, , ) If a soil is in elastic state, then it swells when it is saturated If a soil is in plastic state, then it collapses when it is saturated

Volumetric strains change of sign when unsaturated soils are saturated (firstly swelling andfinally collapse)

Collapse always ends up on a unique consolidation line

Different trajectories of stress-suction tends to unique saturated normal consolidation line(NCL) when it is saturated


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Maximum potential collapse

Collapse reach the maximum at a given vertical stress and then it decreases with higherstress

We should be saturate the sample when the collapse is minimum

Shear strain and strength

Triaxial response with constant suction

CSL are not parallel. These lines depend on suction

Suction increases the size of the elastic surface


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4.1.3. Barcelona Basic modelCollapse and swelling depending on stress level

All LC points have the same pore ratio

, so it does not matter to load or to wet

LC is the line of preconsolidation stress with different suctions

Predictions of the BBM with trajectories of stress with constant suction and wetting with

constant net stress.

If we wet when the sample is in the elastic part, then it will swell up If we wet when the sample is in the plastic part, then it will collapse If we wet, being in the elastic part and then in plastic part, it will swell up and collapse


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Loading-unloading and drying-wetting

It is the same to load and then to wet, that to wet and then to load. It does not depend onthe stress trajectory

It is not the same to load and then to dry, that to dry and then to load. It does depend onthe stress trajectory


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Compressibility ratio

Barcelona Basic Model (BBM)

Compressibility equations

Unsaturated Saturated Elastic Elastic suction changes

Loading wetting trajectories: Yield curve (LC) (plane p - s)

: Virgin compressibility ratio in saturate states: Elastic compressibility ratio for net stress changes (independent of suction): Reference stress: Parameter which fix a minim compressibility value for high values of suction: Velocity of stiffness variation w.r.t. suction: Poisson coefficient

: Slope of the critic state lane

: Parameter which controls the increment of cohesion: Parameter which defines the non associatively of plastic potential


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Yield surface

Yield surface related to increment of suction Suction Increase (SI): is a yield surface needed to reproduce plastic strain during the phase

of drying

: historic maxim suction of the soil

CompressibilityUnsaturated

Reversible drying and wetting Yield: Hardening law

LC and SI hardening surfaces depend on volumetric plastic strains (coupled hardening of LCand SI surfaces)

Volume changes

P loadElastic

Elasto plastic


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plastic S load

Elastic Elasto plastic plastic

Total plastic strains

4.1.4. Response of the model with different trajectoriesYield surface at triaxial space ()

Yield surface LC


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Hardening law

Flow law

Elastic strains

Shear tests with constant main net stress and different suctions

4.2.Expansive unsaturated soils4.2.1. Variation of humidity and suction in expansive unsaturated soilsHumidity in superficial layers of the soil, it is controlled by:

Position of groundwater level Humidity transfer between ground and atmosphereWater flows from low suctions to high suctions

Distribution of water pressure is hydrostatic (without water fluxes)


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Sin embargo, la infiltracin procedente de lluvias y la evapotranspiracin modifican

profundamente esta situacin ideal, sobre todo en climas ridos y semiridos. Las succiones

medidas son en general muy superiores a las que se deducen del equilibrio esttico.

Active zone is the zone where it is expected to be deformed due to humidity changes so this

zone is not suitable to building a structure. This zone is determined by sounding line during dry

and wet time.

4.2.2. Parameters of expansivenessAtterberg limits:

Activity: : Inactive Normal ActiveMineralogy

Classification of expansive unsaturated soils according to colloids, PI and retraction limit

Index Expansiveness

(% volume

change)

Expansion

degree

Content of

colloids PI Retraction limit

>28 >35 30 Very high

20 31 25 41 7 12 20 30 High

13 23 15 28 10 16 10 20 Normal

35 >4 >1.5 High

50 60 25 35 1.5 4 0.5 1.5 Marginal


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4.2.3. Geotechnical problems due to expansive soilsDry and semidry climate: strong suction changes (including trees and vegetation)

Expanding and contracting: horizontal and vertical movements

Characteristics cracks: tensile stresses are normal to the cracks.

4.2.4. FoundationsAnalysis 1D of swelling

vertical strain of layer

: thickness of layer

: thickness of active layer

Swelling due to a stress equal to geostatic pressure (saturation)


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Filosofa Procedimiento Observaciones

Impedir movimientos

Aislar la estructura

Pilotes de cimentacin y vigas de

atado

Barreras contra cambios de

humedad

Eficaz en casos de potencial

hinchamiento alto

Eliminar movimientosdiferenciales

Losas reforzadas de cimentacin Eficaz en casos de potencialde hinchamiento medio-alto

Resistir los movimientos

diferenciales

Zapatas continuas perimetrales.

Tensiones de cimentacin

incrementadas

Riesgo si suelo muy

expansivo

4.3.Unsaturated soils. Rockfill4.3.1. Some observation of rockfill behaviour in the fielPartial saturation or suction in a rockfill

Rockfill structures collapse when they are totally or partially wetted

Capillary attraction forces between two spherical particles bonded by a water meniscus Computed capillary stresses for a simple cubic arrangement

Behaviour of rockfill

The behaviour of rockfill is dominated by particle breakage Particle breakage explains the quantitative differences observed between the behaviour of

sands (at low moderate stress levels) and rockfill

Particle breakage depends on:- Strength of individual particles- Grain size distribution- Stress level- Relative humidity prevailing at the rockfill voids


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4.3.2. Mechanisms of particle breakageFracture mechanics. Basic principles

Stress intensity factor

: Depends on geometry

: Loading modes: I tensile, II shear normal to crack tip, III shear parallel to crack tip

Classical criteria of the linear elastic fracture theory

: crack propagates : fracture toughnessSubcritical crack propagation

: cracks may also grow at some speed * + * + : Relative humidity: measure of the chemical potential or suction of water. It

controls propagation speed of particle breakage: Molar volume of water: Stress intensity factor in the crack: Absolute temperature: Gas constant: Activation energy for unstressed material

: Material constant

: Total suction


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4.3.3. Results of an experimental investigationOedometer tests of rockfill

Conclusions

A unique NCL exists for a given (or ) If increases the material stiffens An elastic domain may be defined Yield stress (preconsolidation stress) decreases as (Relative humidity) increases The effect of wetting depends on applied confining stress

Very low swelling strains are measured for low stress levelCollapse strains occur beyond a certain confining stress value. Collapse is more relevant

than swelling

Collapsed states end in saturated virgin line Saturation of rock particles seems sufficient to produce the collapse deformation as much

as the full flooding of the rock specimen

Time dependent strains are relatively low for low confining stresses and dry states Beyond a certain threshold stress value the time-dependent strains are strongly affected

by water

4.3.4. A model for rockfill compressibilityMechanisms of plastic deformation

Particle rearrangements: instantaneous and independent of water action

Clastic yielding: onset of particle crushing: , Instantaneous. Independent of water action. It is due to particle breakage:

Time dependent. Dependent of water action (RH): They are negligible for a very dry state


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Elastic strains

Stress induced: independent of water action

Water content induced: independent of stress level. (Swelling/shrinkage)

Elastoplastic strains

For : only instantaneous

For : instantaneous and delayed

Rockfill behaviour is independent of water action:

Low confining stresses Very dry states

4.3.5. Unsaturated soils vs unsaturated rockfillRockfill Unsaturated soil

Collapse is associated with particles breakage

a subsequence rearrangement of structure

Particle toughness is a fundamental property

The effect of suction is to control particle

breakage velocity

Threshold toughness to initiate fracture

propagation is included in the model through

a parameter, for no time delayeddeformation exist (no collapse)

Total suction () controls water inducedeffects

Time delayed deformations (and hence

collapse) is inhibited for dry states

Collapse is associated with particle

rearrangement

Particle strength does not affect the overall

behaviour

The effect of suction is to prestress soils

structure

There is no equivalent parameter

Matric suction (s) controls water induced

effects

There is no equivalent concept


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Yield stress for the very dry state is

conveniently chosen as a hardening

parameter

Elastoplastic strains (instantaneous and

delayed) are linearly related to confiningstress for the relevant range of stresses in

practice

Yield stress of the saturated soil is

conveniently chosen as the hardening

parameter

Elastoplastic strains are linearly related to log

(Confining stress)

4.3.6. Conclusions1) Particle breakage introduces a size effect on the constitutive behaviour of rockfill2) Mechanisms of rockfill deformation also include sliding and rotation of particles. Scaling

laws applicable to particle strength are unlikely to apply to the behaviour of a granular mix

3) Scale effects have been investigated through an elastoplastic constitutive model forrockfill. The model uses subcritical crack propagation phenomena (in rock particles) as a

convenient background

4) The variation of material parameters with a grain size parameter of samples with ascaled gradation has been stablished

5) The delayed compressibility index and the parameter describing the rate of change ofcompressibility index with RH (), decreases as decreases

6) The remaining model parameters are essentially independent of gradation7) Inconclusive results were found for the clastic yield stress, 8) The work presented provides a methodology to derive rockfill constitutive parameters in

practice


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THEME 5. Hard soils and soft rocks

5.1.Behaviour of bonded soils5.1.1. Introduction. Reconstituted soils and natural soilsThe geotechnical cycle

Diagenesis is changes to sediment or sediment rocks during and after rock formation

(lithification)

Reconstituted soils vs Natural soils

The behaviour of reconstituted soils depends on soil fabric (arrangement particles, aggregates

etc...)

The behaviour of natural soils depends on soil structure

Structure is the combination of fabric and bonding Fabric refers to particle size and arrangement porosity Bonding: is a general terms that usually refers to particle connections set up by geological

processes


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Sudden change of porosity How do we know if a soil is destructing

5.1.2. Structure developmentSedimentation compression curves for normally consolidated clays

Liquid Index removes plasticity

effects

Normalizing for plasticity with IL,

data for slurried clays are similar

In situ states of natural clays plot

above slurried

Natural soils have more open

structure (due to bonding) than

reconstituted soils with the same

applied load.

Separation of natural from

reconstituted depends on sensitivity

Low St rapid deposition inactive water

High St slow deposition in stillwater


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Intrinsic and sedimentation compression lines

We use Void Index instead of using Liquid Index because the latter is more difficult to obtain

: Void Ratio of a reconstituted soil when it is applied a load of 100KPaSedimentation compression lines: effect of sensitivity


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Natural soils are above reconstituted soils, it is due to bonding and sensitivity usually goes

between It does not make sense to speak about Over consolidation Rate (OCR). We speak about Yield

Stress Rate (YSR)

: Stress when natural soils behaviour begin to changeExceptions

Intensely fissured (scaly) clays: in this case, the subsequent loading curve is not able to reach

ICL and it remains on the left side of ICL.

There are kind of clays whose subsequent loading curve does not converge to intrinsic

behavior:

Coastal and alluvial soils: rapid decomposition in changing environment Layered macro fabric Structure of low sensitivity Little post yield convergence Stable structure: fabric no removed even at large strains

The conceptual framework based on Skempton and Burland provides a unifying perspective in

which the effects of gravitational compaction and structure can be readily assessed. It brings

together clays and clayrocks of different composition, plasticity and burial depth

Moreover, we have to take into account:

Materials with different sensitivities Time and rate of cementation/bonding (structure development) Intensely fissured clays and clayrocks


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Stable fabrics different from the intrinsic onesTwo remarks:

Generally, the sedimentation consolidation curve is practically parallel to intrinsicconsolidation curve

The subsequent loading curve rarely (if ever) go beyond the sedimentation consolidationline

5.1.3. Effects of structure. The Limit State Surface concept

5.1.4. Destructuration

By compression

Swelling sensitivity

: Swelling slope


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5.2.Shear strength of clays5.2.1. Residual strengthRing shear test

Residual strength is measured with high precision by the ring shear test

Large relative displacements Drained strength (porous stone) Fixing a vertical stress, the bottom of the ring is turned around It is measured the torque

There will be materials with residual strength which will not be important and there will be

others with low strength residual which will be important to know it.

With low clay fraction, the normally consolidated and residuals friction coefficient are similar

With low clay fraction there is not enough clay to get oriented surfaces in one direction so it is

not possible to cause sliding problems.

Fine fraction (silt and clay) is not a suitable measure to evaluate the clay fraction, it is better to

use Plasticity Index. Residual strength is directly proportional to PI


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Residual strength is not sensitive to the alteration of the sample. Initial state is not important

with respect to final state.

5.2.2. Shear strength of stiff clays and weak argillaceous rocks

A failure plane is create and the strength begins to decrease in the 2nd graphic

Argillaceous hard soils and weak rocks generally fail in a brittle manner.

Brittleness means that the strength decreases before peak strength and it does not mean that

it breaks.

Brittleness is higher when residual strength is low.


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Conceptual scheme for the drained strength of argillaceous hard soils weak rocks

Peak, postrupture and residual strength parameters

Strength of discontinuities

In fissures and joints with no (or little) relative displacement, strength is similar to post-rupture

strength. Bonding is destroyed with no opportunity for reaching residual strength.

In discontinuities with large relative displacements, shear strength is probably close (or equal

to) residual strength


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Brittle behaviour

Under conventional geotechnical conditions, argillaceous hard soils weak rocks tend tofail in a brittle manner

Often the reduction of strength occurs in two stages: a sharp one after peak and a moregentle one up to residual

- The first reduction is generally associated with the loss of bonding. Sometimes (but notalways) cohesion reduces to zero. Friction angle reductions are at most moderate

- The second reduction is associated with the reorientation of clay particles. Frictionangle reduces now to small value (residual strength)

Post-rupture strength is normally identified with the strength after the first stage ofstrength reduction:

- Post-rupture strength is often similar to intrinsic (reconstituted strength), the reasonsare unclear and it may be just fortuitous

- The strength of discontinuities that have undergone tension but not (or very limiteddisplacement) correspond closely to post-rupture strength

5.2.3. Operational strength in brittle materialsWhat is the operational strength for stiff clays?

It is not correct to use peak strength for the analysis of sedimentary clay slope stability (1sttime slide)

fully softened strength (reconstituted material) fits 1st time slides in sedimentary slopes The presence of fissures probably instrumental in causing strength reduction The progressive loss of cohesion appear to explain the delayed failure of cut in stiff clays Peak strength parameters apply in slides in (unfissured) boulder clay.Enhanced understanding

Concept of residual strength (depends on clay plasticity and previous relativedisplacement). Applicable after a slip has occurred involving large displacements.

Pore pressure equilibration controls the time of failure in stiff clays The idea of a cohesion progressively reducing with time discarded The primary difference between boulder clays and sedimentary clays is not the presence

or absence of fissures. Boulder clays are ductile and plastic, sedimentary clays are brittle.

Discontinuities (fissures, joints, faults) reduce initial mass strength. A full understanding oftheir role in strength development and degradation still pending.

Operational strength is an average and must depend on:

Initial mass strength (dependent on presence of fissures/discontinuities) Degree of brittleness Rate of strength degradation Geometry and loading history


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THEME 6. Very small strains in soils high stiffness

6.1.IntroductionFar from failure, behaviour is expected to be

elastic, but is not constant: can be very large (tg90 ): it is not used for very small strains: it is used for local strains

Cam Clay model

Non linear elasticity reversible but not constant is difficult to measure in practice

- is fixed or- is assume constant (but then check if is reasonable correct or not, )

In many practical situations, strains are below 1% Predictions of displacements are very sensitive to and nonlinear law6.2.Soil behaviour far from failureShear modulus is the key parameter

Measurements of from dynamics tests showed very high values if compared with obtainedin triaxial tests.


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Dynamic measurements give the soil response for very small strains. They can be useful for

monotonic loading also.

Models simulating the decay of G

New version of the Hardin Drnevich

: reference shear strain ( ), thisparameter gives the decaying speed of: is a monotonic shear strain historyparameter

Typical values of:

*

+

If soil is assumed to be frictional only, is expected to depend on mean effective stresswith

If soil is considered a set of spheres in contact, then Hertz theory applies and a value of is found At very small strains we may expect close to (around ). At large strains,

should be closer to 1, as slippage and rearrangement occur.

When the sample is very confined, then is high


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6.3.Measurement of soil properties at very low strainsIn situ tests

Measurement of shear wave velocity:

As very small strains are involved, is in fact

Cone penetration test + seismic device: CPTU + Vs

Laboratory tests

Bender elements: for horizontally propagating waves

Resonant column test: frequency of resonance is related to shear Modulus (very smallstrains)


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6.4.Synthetic Example: excavation far from failure using different modelsMain assumptions

Initially horizontal ground surface, homogeneous soil

Construction of diaphragm walls an soil excavation No water, drained conditions Initial stresses: conditions ( )Models considered

Linear elasticity- The soil never fails but it is useful because it is too easy to calculate- Tension stresses may appear in excavation problems- Only two parameters involved:

,

. Alternatively:

,

or

,

Elastic Mohr Coulomb- Elasto - plastic model. Perfect plasticity. Yield surface = strength criterion- Elastic Mohr Coulomb model involves five input parameters: , for soil elasticity,, for soil plasticity and as angle of dilatancy

- This model is represents a first order approximation of soil or rock behavior- It is recommended to use this model for a firs analysis of the problem considered- Cons 1: elastic part is constant ( constant)

Cases of close to failure approximates well

Cases of far from failure approximates to bad

Hardening Soil Model (HS)- Elasto-plastic model. 2 yield surfaces- The HS model is an advanced model for the simulation of soil behaviour- Soil stiffness is described much more accurately by using three different input

stiffnesses: the triaxial loading stiffness,

. The triaxial unloading stiffness

and

the odometer loading stiffness - Hyperbolic relationship between vertical strain and deviatoric stress triaxial loading


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- The elastic zone is small- is not constant, increases with the confinement ( )- Failure Parameters: , , ,- Basic parameters for soil stiffness:

,

,

,

Hardening Soil Model with Small Strains (HSsmall)-

Elasto-platic model. 2 yield surface with nonlinear elastic & high stiffness at smallstrains

- The HSsmall is a modification of the HS model that accounts for the increased stiffnessof soils at small strains

- At low strain levels most soils exhibit a higher stiffness than at engineering strainlevels, and this stiffness varies non-linearly with strain

- Add and as additional parameters to the previous HS model


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THEME 7. Anisotropy and principal stress rotation

7.1.Introduction and definitions7.1.1. ElasticityGeneral case. Anisotropic elastic solid: 21 parameters

In matrix format, the stress strain relation showing the 36 (66) independentcomponents of stiffness can be represented as:

(

)

[

]

1 plane of symmetry: 13 parameters

If xy is the plane of symmetry:

(

)

(

)

3 planes of symmetry. Orthotropic elastic solid: 9 parameters

[

] (

)

1 axis of symmetry: 5 parameters

( )

(

)


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Isotropic: 2 parameters

[

] (

)

7.2.Rotation of principal stresses

Direct simple shear

Plane strain Uncontrolled rotation of principal stresses Uniform strains only possible in undrained tests

7.3.Laboratory equipment to examine anisotropy

7.4.Anisotropic behaviour7.4.1. Reconstituted materialsApplications

Stability of slopes


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7.4.2. Natural materialsApplications

Stability of slopes

7.5.Case history

geomechanics notes

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