classification of constitutive models for soils (icold bulletin 122)

Classification of constitutive models for soils.

This classification is meant to help the practising engineer to identify those models which are of relevance in a

particular situation and readily correlate new models with existing ones.

Nonlinear or piecewise elastic models. This type of model is simpliest of all. Elastic moduli (Young’

modulus, E, and Poisson’s ratio,, or bulk, B, and shear moduli, G) are assumed to be nonlinear functions of

stress. These models involve many (up to 9) constants with little physical significance but deriving from

standard triaxial compression tests.

The most well known model in this category is the Duncan-Chang hyperbolic model [89] which has been used

in many static analyses of embankment dams, mainly earth and earth zoned dams. Some dam designers have

gained sufficient experience in the use of this model and appear to get good correlation with the field

performance of homogeneous earth and earth zones dams.

There are, however, serious objections to the use of these models for static analyses of embankment dams,

especially, rockfill dams with clay cores, asphalt concrete diaphragms and concrete facings.

The main drawback of a nonlinear elastic analysis is that all strains are assumed to be elastic and thus the

strain increment direction is dependent upon the direction of the strain increment. However, this assumption is

not valid for many geotechnical problems, particularly, for embankment dams where rotations of the principal

stress direction and changes in the load direction occur during impounding and emptying of reservoir. Thus,

these models are not parth-dependent and can not simulate the important dilatant effect in compacted soils and

their nonlinear behaviour during unloading-reloading.

Therefore, these models can be recommended only for the use in the static analysis of earth dams.

Elastic perfectly plastic models. The mathematic theory of elasto-plasticity is well established and has been a

fertile ground for the development of various soil models. Various combinations of yield functions, flow and

hardening rules give rise to different models modelling the complex behaviour of soils in both monotonic and

cyclic loading. Among them the most simplified models are elastic perfectly plastic models with Mohr-

Coulomb, von Mises, and Drucker-Prager yield criteria. The latter two criteria were used geotechnical

analyses as a simplification of Tresca and Mohr-Coulomb yield criteria. It is, however, well established that

these approximations give poor results and offer no real advantage in numerical analyses of embankment

dams. The dilatancy of cohesionless soils controls the collapse loads of embankment dams. However, it

predicted by the Mohr-Coulomb yield criteria with associated flow rule is unrealistically high. Only a partial

remedy for this situation lies in adopting a non-associated flow rule which makes this simple simple plasticity

formulation is applicable only to one important class of analysis of undrained saturated soils in total stresses.

Hardening elasto-plastic models (HEP models).

Isotropic, kinematis and mixed (isotropic-kinematic) hardening rules with hardening due to plastic volumetric

and plastic shear strains give rise to these models of varying complexity. The Critical State (CS) model [90],

the Infinite Number of Surface (INS) model due to Mroz and Norris [91], models due to Lade [92], Nova [93],

Prevost [94], Dafalias and Herrman [95] and Zaretsky [96] are a few examples in this category.

At the present stage of models development only HEP models are the ones to be able to represent soil

behaviour satisfactorily. The list of these models with the range of their applicability and types of loadings

and model formulation are given in Table 7. The well-established CS model (modified Cam Clay model) is

the basis of many (about 30) models proposed for monotonic and cyclic loads and therefore, has been placed

in the first position in Table 7. Other models have been arranged in alphabetic order. In the type of

formulation of models the type of flow rule (associated or non-associated) has been indicated as this has an

important bearing on the computer implementation of the models. The type of hardening rule (isotropic or

kinematic) has also indicated.

Many models such as No 2,5,7,8 and 10 in Table 7 are modifications of the Critical State (CS) model.

CS model was originally developed under Rosco‘s leadership at Cambridge (UK) in the 1960’s. CS model

developed for normally consolidated (N-C) and lightly overconsolidated (O-C ratio 2) clays is a relatively

simple model with 5 material parameters which is capable to predict, at least qualitatively, a great number of

fundamental aspects of soil behaviour.

The SC model is known to have the following principal features:

Table 7. Principal hardening elasto-plastic (HEP) models of soils.

No Model Types of soil Type of loading Type of formulation

1 Critical state (CS)

model (modified Cam

Clay model)

Normally consolidated,

lightly overconsolidated

clays

Monotonic Isotropic hardening,

associated flow rule

2 Bound surface plasticity

(BSP) model

Clays, sands Monotonic,

cyclic, transient

Kinematic hardening,


3 Densification model Clays, sands Cyclic, transient Empirical isotropic

hardening, associated flow

rule

4

Ghauboussi’s model Sands Monotonic,

cyclic, transient


non-associated flow rule

5 Infinite number of

surface (INS) model

Clays Monotonic,

cyclic, transient



6 Lade’s model Sands Monotonic Isotropic-kinematic

hardening, associated-


7 Multi-laminated (ML)

models

Clays Monotonic Isotropic-kinematic



8 Nova’s model All soils Monotonic,

cyclic, transient

Isotropic-kinematic



9 Prevost’s model All soils Monotonic,

cyclic, transient



10 RS model Clays Monotonic,

cyclic, transient



11 Zaretsky’s model All soils Monotonic,

cyclic, transient

Isotropic hardening,


-For normally consolidated clays in drained triaxial tests, the CS model predicts shear stress-shear strain

response, volumetric strains and failure stress close to the observed experimental results.

- For normally consolidated clays in drained triaxial tests, although failure stress, stress path and pore

pressures are reasonably well predicted, the predicted shear strain for a given shear stress is much lower than

the observed experimental shear strain.

-The value of K0 (the ratio of horizontal to vertical stress) is overestimated in all cases.

Prediction of the correct level of shear strain in monotonic loading is at least as important as the prediction of

the failure stress if the model is to be extended for application to cyclic and transient loads. Further, the

importance of initial stress can not overemphasized in any analysis of embankment dams and soil model have

an important role in predicting the critical the initial stresses through the prediction of K0.

Table 8. Comparison of principal hardening elasto-plastic models for normally consolidated clays and sands.

No Model % error in predicted a at 0.9 qf Predicted a at 0.9 qf

1 Critical state (CS) model -200% 1%

2 Boundary surface plasticity (BSP) model 0% 10%

3 Ghaboussi’s model 0% 1.5%

4 Infinite number of surface (INS) model -72% 2%

5 Nova’s model -85% 3%

6 Prevost’s model 0% n/a

7 RS model -166% 1.2%

8 Zaretsky’s model 0% 3%

In view of the above, Table 8 shows the % error in predicted axial strain (a) at 90% of failure stress qf with

reference to experimental results in undrained triaxial tests for principal hardening elasto-plastic (HEP)

models for normally consolidated (N-C) clays and sands. It is noted from Table 8 that most critical state based

models greatly underestimate the strains with an exception of the BSP model of Dafalias and Herrman [95].

All models listed in Table 8 show that the axial strain (a) attained at 90% of the failure stress (qf) is in the

range of 1-3% while for the BSP and Zaretsky’s models it is about 8-10% a value more in line with

experimental results. Table 9 lists data similar to Table 8 for over-consolidated (O-C) clays (undrained triaxial

tests).

Table 9. Comparison of principal hardening elasto-plastic models for over-consolidated clays

No Model %error in prediction of

peak shear strength

%error in prediction of strain

at 90% of peak shear strength

%error in prediction of

pore pressure at failure

1 CS model +2% -360% 0%

2 BSP model 1% 0% 0%

3 INS model +6% -300% 0%

4 Prevost’s model 0% 0% n/a

5 RS model +8% n/a -8%

6 Zaretsky model <2% 0% 0%

The picture here is similar to that of N-C clays. The INS model seriously underestimates the axial strains.

Prevost’s initial model [94] was a total stress model and thus was not capable of predicting pore pressures. For

his last effective stress model [97] no results of pore pressure during undrained triaxial tests are given.

Except the INS, Lade’s and Zaretsky’s models none of the model listed in Table 7 appear to predict realistic

values of K0. The quality of K0 predictions for the models of Ghaboussi [98], Nova and Prevost is not known.

Thus, the models which are extensions of the CS model suffer from the drawbacks that their response is too

stiff and K0 values are overestimated. The first drawback implies that these models cannot be successfully

used for cyclic and transient loads. The second drawback prohibits the use of the model to predict the initial

stress conditions, which is an important role of the model in embankment dam analysis.

Table 10 gives the qualitative comparison of the principal hardening elasto-plastic (HEP) models based on the

chosen criteria for evaluation.

Table 10. Qualitative comparison of principal hardening elasto-plastic models

Criterion CS BPS Densific

ation

Ghabo

ussi

INS Lade ML Nova Prev

ost

RS Zaret

sky

Normally consolidated

soils: Prediction of

strains in undrained

triaxial tests

G G P G P G P P G P G

Overconsolidated soils:

Prediction of strains in

undrained triaxial tests

P G P G P G P n/a G P G

K0 prediction P P n/a n/a G G P n/a n/a P G

P-poor, G-good, n/a - not available

The first modified Cam Clay Model (CCM) with the ellipse yield surface proposed by Roscoe and Burland

[99] makes compatible associated plasticity with a frictional envelope and zero dilatation for ultimate

conditions. Another feature of soil behaviour successfully predicted by the CCM is the different volumetric

response of soil depending on its stress history. The model performs best for roughly radial effective stress

paths leading to subcritical yielding. In the supercritical region, deformation predictions are highly suspect.

The model also predicts a unique state boundary surface and a unique void ratio-critical stress relationship

which is in accordance with experimental results. The consolidation/swelling behaviour and the yelding of

soils at preconsolidation pressure are also well predicted.

When predictions are compared quantitatively with experimental results, it is found that the simple models

like CS models are not capable to reproduce exactly the real behaviour of soils. However, this basic

formulation often gives sufficiently accurate predictions particularly in absence of stress reversals or rotations.

Even at present, modified CCM remains the most widely used CS model in numerical geotechnical analyses.

Its simplicity and small number (5) of parameters required outweighs frequently the possible better predictive

advantages of more elaborate models.

The great majority of computational applications of CS models are static geotechnical problems (mainly,

embankments and embankment dams on soft clays and silts), dynamic analyses are still few. The basic Cam

Clay formulation is particularly suited for the stress path followed by the soil under an embankment dam. This

explains the popularity of this model in embankments and embankment dams analyses and good predictions

which have been generally obtained for this type of geotechnical problems.

A great number of modifications have been proposed to the basic formulation of CS model to achieve a better

agreement between predicted and observed soil behaviour and to model cyclic loading effects.

Zienkiewicz and Naylor [100] used a non-associated flow rate with dilatancy increasing linearly from zero at

the critical state state point to some fixed value at p=0.

The Cap models due to Di Maggio and Sandler [101] can be considered, basically, as CS model with modified

supercritical (softening) yield surface and associated plasticity. The Cap yield surface moves according to the

changes in plastic volumetric strains but the failure surface is fixed and, therefore, no softening behaviour is

predicted. The simplicity and flexibility of the model has meant that it has been used in a number of numerical

geotechnical analyses.

Although modifications to the supercritical yield surface surfaces are necessary to predict realistic failure

stresses in that region, they are not generally implemented. Most of the computational applications of CS

models in the embankment dams analyses refer to soil materials in the subcritical (hardening) region and

supercritical side are seldom employed.

The basic CS formulation did not impose any condition on the shear elastic component, inside the yield locus.

The CS model is not conservative and energy may, therefore, be extracted from certain cyclic cycles. The fact

may not be too important for monotonic loading but may become significant if the loading involves many

stress reversals.

For problems involving cyclic loading it is particularly important to adopt more complex soil behaviour inside

the yield locus since irrecoverable, cumulative soil behaviour should be accurately modelled if realistic

predictions are to be made. To achieve this, many different models have been proposed with various forms of

plasticity (bounding, two-surface, multi-surface, general plasticity, etc.).

The formulations used in numerical geotechnical analyses can be divided into two main groups:

(a) Models in which cyclic loading effects are described by means of a separate formulation which is added to

a suitable static model (Van Eekelen and Potts, [102]; Zienkiewicz et al., [103]).

(b) Complete models in which cyclic loading effects are the consequence of the overall formulation of the full

model (Dafalias and Herrman, [95]; Prevost, [97]; Zienkiewicz, Leung, Pastor, [104]).

The second type of models can describe more accurately the real soil behaviour including hysteresis effect but

they become costly if a large number of cycles must be considered as in embankment dam seismic analysis.

Van Eekelen and Potts [102] used the pore pressure due to cyclic loading as the fatigue parameters of a

separate formulation. Its increase per cycle depends on the normalized stress amplitude of the cycle. The static

part of the model is a form of the CS formulation called Drammen clay with yielding.

Zienkiewicz et al. [103] used the volumetric strains as the fatigue variable and its increase depends on the total

length of the deviatoric strain path. This Densification model coupled to a CS formulation is used for the

analysis of a layer of saturated sand subjected to a horizontal earthquake shock. The same problem has been

used [105] to compare the results of Densification model with two other constitutive laws of type (b). The

pore pressure build-up predictions of the three models show significant differences, the faster increase

corresponds to Densification model.

A further development has been the introduction of a generalized Bounding Surface Plasticity (BSP) model in

the basic CS formulation (Zienkiewicz et al., [104]). It leads to a model with relatively few parameters with a

good predictive capacity in analysis of dynamic problems. The availability of realistic boundary value

problems using CS models requires the use of the finite element methods. The implementation of CS models

of embankment dams in F.E. codes has been made in many geotechnical computer packages such as

ABAQUS (Manchester Unic. Comp. Centre), CRISP (Cambridge Univ.), FEASAS (TRRL), FINETAN/CS

(Swansea Univ.), ICFEP (Imperial College), ROSALIE (Labor. Central Ponts et Chaussees) and others.

Unfortunately, in the literature, little information is given on the details of model implementation and only

few comparisons have been published about the relative merits and efficiency of the various implementations

The soil materials modelled appear to be mostly limited to saturated clays and silts. This is related to the most

characteristic feature of the CS models which is the yield in the subcritical (or ‘wet’) region. Hence, the use of

CS models is especially rewarding for soil materials like soft clays and silts. In contrast, stiff over-

consolidated clays do not appear to be satisfactorily modelled with CS formulations.

The dilatancy effects of dense sand and gravel require further elaboration of the model incorporating, for

instance, hardening due to shear strains. The ‘double’ hardening models do not appear to have been widely

used in numerical geotechnical analyses. Although some interesting proposals for extending CS to cover rock-

fill behaviour exist, the computational applications have been scarce. (Liapichev, [106]).

Elasto-viscoplastic models.

Post-constructional settlements of earth and, especially, rockfill dams are controlled by creep or time-

dependent plastic strains of soils and, especially, rockfill. Due to the wet clay cores, delayed consolidation is

observed and the post-constructional performance of these dams is controlled more by creep than by primary

consolidation. Neither the assessment of cracking in the clay cores nor the safe design of the concrete

upstream facing can be carried out without taking creep into account.

The various approaches to numerical modelling of creep differ by the rheological constitutive relations and

numerical techniques applied. As regard the constitutive relations, four groups can be distinguished:

1) linear viscoelastic; 2) nonlinear viscoelastic; 3) elastic-viscoplastic; 4) rate-type viscoplastic relations.

Linear viscoplastic constitutive relations derived from rheological models allow of obtaining closed form

analytical solutions for a limited class of geotechnical problems. The strain is separated into elastic and

viscous (creep) components and first a linear elastic solution of the problem is obtained. Then either an

incremental procedure or integral transformation can be applied. The use of the incremental procedure is

efficient when together with creep, the nonlinear behaviour of soils, the change of the dam shape due to the

construction sequence or some complicated loading is also considered. The integral transformation procedure

is recommended for the linear problems with very long periods of time.

The simplest way to take creep into account is to compute the stresses and strains for each loading increment

by the variable-stiffness method and then to determine the creep increment corresponding to this stress state

and the elapsed time. The equivalent nodal forces introducing these creep increments into the solution are

computed by the initial strain method. Using these techniques, nonlinear consolidation problems taking creep

effects into account were solved by Dolezalova [107] and Liapichev [108] for stress-strain analyses of rockfill

dams with concrete facing, using rheological constitutive relations suggested by Feda [109]. These relations

between the uniaxial creep rate, volumetric creep and distortional creep rate were found out from the results of

odometer and ring shear tests of different soil materials. When solving 2D, 3D rheological problems by

numerical methods, the determination of the value and direction of all creep components is necessary. In the

above linear and incrementally linear solutions, these problems were solved by transforming the relations to

invariant form and supposing the coincidence of the directions of stress and strain tensors in a given time step.

A successful elasto-viscoplastic approach suggested by Cormeau [110] has been used and analyzed by a

number of authors during the last decade. In Cormeau’s approach time-dependent elastic and viscoplastic

strains and plastic flow at failure are supposed. The pure plastic strains corresponding to failure are computed

by the imaginary viscosity procedure, which helps to obtain a stable solution. The initial strain and stress

methods are used to handle the associated and nonassociated flow rules with equal ease. Nevertheless, the

important a priori criteria of numerical stability are only derived for the perfect plastic case and applied for the

associated theory. The potential surfaces that are smooth in the deviatoric plane (von Mises, Drucker-Prager)

differ considerably from those with corners (Morh-Coulomb, Tresca). The latter surfaces are less convenient

as they require considerably smaller time steps and much computer effort to get a stable solution.

As for the soft clays the approaches using the Cam Clay model and its modifications are physically well based

and most often accepted. The achievement in this field is a nonassociated, two-surface plasticity model

adopted by Hsieh and Kavazanijian [111] to describe the rheological behaviour of soft clays. A porosity-

dependent permeability and both the deviatoric and volumetric components of creep are considered in this

model. The elliptical and horizontal plastic potential surfaces have time-independent and dependent portions

when expanding. The initial stress method is used for incorporating the creep component into the solution.

Difficulties with deriving plastic potential surfaces for different soils and the complexity of the nonassociated

theory led to the development of new elasto-plastic and viscoplastic constitutive models which depart

completely from the plastic potential theory. As a successful approach, the Kolymbas [112] relations can be

mentioned, which have a remarkable prediction capability with a small number of parameters. The time-

dependent behaviour is taken into account in these equations by an additional term containing second time

derivatives of the strain. These constitutive relations are incrementally nonlinear and hence they require the

solution of a system of nonlinear equations at each loading and time step.

6. References

[89] J.M. Duncan, Y.Y. Chang. Nonlinear analysis of stress and strain in soils. Journal of Soil Mechanics &

Found. Div., ASCE, Vol. 96, No SM5, September 1970, pp. 1629-1653.

[90] K.H. Roscoe, A.N. Schofield. Mechanical behaviour of an idealized ‘wet clay’. Proc., 2-nd European

Conf. on Soil Mechanics, Wiesbaden, 1963, Vol. 1, pp. 47-54.

[91] Z. Mroz , V.A. Norris and O.C. Zienkiewicz. An anosotropic hardening model for soils and its

application to cyclic loading. Int. J. Num. & Anal. Methods in Geomechanics, Vol.2, 1978, pp.203-221.

[92] P. V. Lade. Elastoplastic stress-strain theory for cohesionless soil with curved yield surfaces. Int. Journal

of Solids and Structures, Vol. 13, 1977, pp. 1019-1035.

[93] R. Nova and T. Hueckel. A unified approach to the modelling of liquefaction and cyclic mobility of

sands. Soils and Foundation, Vol. 21, 1981, pp. 13-28.

[94] J.H. Prevost. Plasticity theory for soil stress-strain behaviour. Journ. Eng. Mechanics Div., ASCE, Vol.

104, 1978, pp. 1174-1194.

[95] J.F. Dafalias and L.R. Herrman. Bounding surface formation of soil plasticity. Soil Mechanics- Transient

and Cyclic Loads (Eds. G.N. Pande and O.C. Zienkiewicz), Wiley, Chichester, 1982, pp. 253-282.

[96] Yu. K. Zaretsky. Soil viscoplasticity and design of structures. Balkema, Rotterdam, 1996, 512 pp.

[97] J.H. Prevost. Anisotropic undrained stress-strain behavior of clays. Jour. Geotechn. Eng. Div., ASCE,

Vol. 104, No GT8, August, 1978, pp. 1075-1090.

[98] J. Ghaboussi and K.J. Kim. Analysis of construction pore pressure in embankment dams. Numerical

Models in Engineering Practice, Balkema, Rotterdam, Netherlands, 1986.

[99] K.H. Roscoe, J.B. Burland. On the generalized stress-strain behaviour of ‘wet clay’. Engineering

Plasticity , ed. by J. Heyman & F. Leckie, Cambridge University Press, 1968, pp. 535-609.

[100] O.C. Zienkiewicz and O.J. Naylor. Finite element studies of soils and porous media. Lect. Finite

Elements in Continuum Mechanics (Eds. J.T. Odens, E.R. de Arantes), UAH Press, 1973, pp. 459-493.

[101] F.L. Di Maggio, I.S. Sandler and G.Y. Ballandi. Generalized Cap model for geologic materials. Journ.

Geotechn. Eng. Div., ASCE, Vol. 102, 1976, pp. 683-697.

[102] H.A. Van Eekelen and D.M. Potts. The behaviour of Drammen clay under cyclic loading.

Geotechnique, Vol. 28, 1978, pp. 173-196.

[103] O.C. Zienkiewicz, C.T. Chang and E. Hinton. Nonlinear seismic response and liquefaction. Int. Journ.

Numer. & Anal. Methods in Geomechanics, Vol. 2, 1978, pp. 381-404.

[104] O.C. Zienkiewicz, K. Leung, M. Pastor. Simple model for transient soil loading in earthquake analysis.

I. Basic model and its application. Int. J. Num. & Anal. Meth. in Geomech., Vol. 9, 1985, p. 453-476.

[105] O.C. Zienkiewicz, K. Leung, E. Hinton and C.T. Chang. Liquefaction and permanent deformation

under dynamic conditions - numerical solutions and constitutive relations. (Eds. O.C. Zienkiewicz &

G.N. Pande). Soil Mechanics- Transient and Cyclic Loads, Wiley, Chichester, 1982, Chapter 5, pp. 71-

104.

[106] Yu. P. Liapichev. Extensions of the modified Cam Clay model for modelling of compacted soil and

rockfill materials of embankment dams (in Russian). Structural Mechanics of Engineering Structures and

Works. Interuniversities’ Transaction, Moscow, Vol. 4, 1994, pp. 86-100.

[107] M. Dolezalova, A. Horeni and V. Zemanova. Experience with numerical modelling of dams. Proc., Int.

Conf. Num. Methods in Geomech., Innsbruck 1988, Balkema, Rotterdam, Netherlands, pp. 1279-1290.

[108] Yu. P. Liapichev. Stress-strain state analysis of concrete faced rockfill dam taking into account creep of

rockfill (in Russian). Studies of 3-D Structures, Proc., Univ. of Russia, Moscow, 1996, pp.73-80.

[109] J. Feda. Creep of Soils Related Phenomena. Developments in Geotechnical Engineering, Vol. 68,

Elsevier, Amsterdam, 1992, 403 pp.

[110] I. C. Cormeau. Numerical stability in quasi-static elasto-viscoplasticity. Int. Jour. Numer. Methods in

Eng. Vol. 9, 1975, pp. 109-127.

[111] H.S. Hsieh and E. Kavazanjian. A non-associative Cam Clay plasticity model for the stress-strain-time

behavior of soft clays. Geotech. Eng. Research, Report No. GT4, 1987, Stanford Univ., Stanford, Calif.

[112] D. Kolymbas. A constitutive law of the rate type for soils and other granular materials. Proc., 1-st

Chechoslovak Conf. on Numerical Methods in Geomechanics, High Tatras, 1987.

classification of constitutive models for soils (icold bulletin 122)

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