A Constitutive Model for Structured Soils Kavvadas GE500305

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A constitutive model for structured soils

M. KAVVADAS and A. AMOROSI{

The paper proposes a constitutive model for structured soils(MSS) which describes the engineering effects of structuredevelopment and degradation, such as: high intact stiffnessand strength, appreciable reduction of stiffness and strengthdue to de-structuring, and evolution of stress- and structure-induced anisotropy. A key feature of the model is thetreatment of pre-consolidation as a structure-inducing pro-cess and the unied description of all such processes via a`bond strength envelope' associated with the onset of appre-ciable de-structuring and distinguished from the onset ofplastic yielding. Other features include: a damage-type me-chanism to model volumetric and deviatoric structure degra-dation, the evolution of stress- and bond-induced anisotropyusing a fading memory scheme, adaptable predictive cap-abilities depending on the sophistication of the available testdata, modularity to extend its applicability in several soiltypes, and mathematical formulation in a general tensorialspace to facilitate its incorporation in nite element codes.The predictive capabilities of the model are evaluatedagainst the results of laboratory tests on the stiff overcon-solidated Vallericca clay: (a) isotropic and anisotropic con-solidation tests up to very high pressures; and (b) aniso-tropically consolidated triaxial shearing at both lowpressures (structured material response) and high pressures(de-structured material response).

KEYWORDS: constitutive relations; plasticity; fabric/structure ofsoils; numerical modelling and analysis; anisotropy; clays.

Cette communication propose un Modele constitutif pourSols Structures (MSS) qui decrit les effets techniques dudeveloppement et de la degradation de la structure, parexemple: rigidite intacte et resistance elevees, reductionsensible de la rigidite et de la resistance sous l'effet de ladestructuration et de l'evolution de l'anisotropie induite parla contrainte et la structure. Une des principales caracteris-tiques du modele est le traitement de pre-consolidation, entant que procede induisant la structure et la descriptionuniee de ces procedes a travers une Enveloppe a resistanced'adhesion, qui est en rapport avec le debut d'une destruc-turation sensible et qui se distingue du debut de la deforma-tion non elastique. Parmi les autres caracteristiques, onindiquera les suivantes: un mecanisme de type d'endom-magement, pour modeliser la degradation volumetrique et lastructure deviatrice, l'evolution de l'anisotropie induite parcontrainte et adhesion en utilisant un systeme a memoired'evanouissement; des dispositifs a prediction adaptables, enfonction de la sophistication des donnees d'essai disponibles,la modularite pour etendre son application a plusieurs typesde terrain; et la formulation mathematique dans un espaced'allongement general, pour faciliter son incorporation dansdes codes aux elements nis. On evalue les capacites deprevision du modele en fonction des resultats des tests delaboratoire sur de l'argile Vallericca sur-consolidee rigide:(i) tests de consolidation isotropiques et anisotropiques jus-qu'a des pressions tres elevees et (ii) cisaillement triaxialconsolide par anisotropie a basse pression (reponse struc-turee de la matiere) et hautes pressions (reponse destruc-turee de la matiere).

INTRODUCTION

The analysis of geotechnical problems requires constitutivemodels which can describe the deformation and strength ofnatural soils with reasonable accuracy. The development of suchmodels is usually based on experimental studies of reconstitutedsoils and the classical principles of soil mechanics whichinvolve the current state of the material (expressed by theeffective stresses and the void ratio) and its stress history(usually compounded in the maximum pre-consolidation pres-sure). The critical state theory and its extensions have providedthe theoretical basis for many such models in the last 30 years(e.g. Roscoe & Burland, 1968; Nova & Wood, 1979; Gens &Potts, 1982; Kavvadas, 1983; Al-Tabbaa & Wood, 1989; Whittle& Kavvadas, 1994). At the same time, it has become recognizedthat natural soils have components of stiffness and strengthwhich cannot be accounted for by classical soil mechanics andstem from the inuence of structure caused by cementation,ageing or even overconsolidation (e.g. Leroueil & Vaughan,1990; Burland, 1990; Gens & Nova, 1993; Clayton & Serratrice,1993; Muir Wood, 1995a; Burland et al., 1996). Experimentalevidence of the effects of structure is reported in a wide varietyof natural soils and weak rocks, including soft clays (e.g.Leroueil, 1977; Tavenas & Leroueil, 1990; Smith et al., 1992),

stiff clays and clay shales (e.g. Calabresi & Scarpelli, 1985;Rampello, 1989; Anagnostopoulos et al., 1991; Burland et al.,1996; Cotecchia, 1996), granular soils (e.g. Mitchell & Solymar,1984; Coop & Atkinson, 1993), residual soils (e.g. Vaughan,1985, 1988; Wesley, 1990) and weak weathered rocks (e.g. Eliot& Brown, 1985; Addis & Jones, 1990).

The importance of improving constitutive models to includeour present knowledge of the behaviour of structured soils hasbeen perceived by many researchers, who have generally fol-lowed three kinds of procedures

(a) renement of the `small strain' response, by incorporatingstiffness non-linearity in the `elastic' domain (e.g. Dafalias& Herrmann, 1980; Jardine et al., 1986, 1991; Gunn, 1993;Whittle & Kavvadas, 1994);

(b) renement of the material's memory of its stress history, byadding a number of `yield' or `history' surfaces whichrecord key characteristics of the stress path (e.g. Mroz etal., 1978, 1979; Prevost, 1978; Hashiguchi, 1985; Al-Tabbaa & Wood, 1989; Stallebrass & Taylor, 1997);

(c) description of the effects of material structure by a damage-type mechanism which permits the reduction of the size ofthe yield surface due to bond degradationwhile theprinciples of such damage-type models have been knownfor a long time (e.g. Nova, 1977; Wilde, 1977), theirsystematic application in soil modelling is fairly recent (e.g.Pastor et al., 1990; Kavvadas, 1995; Lagioia & Nova, 1995;Muir Wood, 1995b; Chazallon & Hicher, 1998).

This paper describes and evaluates a constitutive model forstructured soils (MSS) which combines and extends the aboveprocedures.

263

Kavvadas, M. & Amorosi, A. (2000). Geotechnique 50, No. 3, 263273

Manuscript received 16 December 1998; revised manuscript accepted 22October 1999.Discussion on this paper closes 26 September 2000; for further detailssee p. ii. National Technical University of Athens, Greece.{ Technical University of Bari, Italy; formerly University of Rome `LaSapienza'.

DESCRIPTION OF THE MSS MODEL

The proposed model is based on the theory of incrementalplasticity and the critical state concepts. Since the modeldescribes the response of the soil skeleton, all stresses areeffective stresses (the primes have been dropped for simplicity).The dots over symbols indicate an innitesimal increment ofthe corresponding quantity. Bold-face symbols indicate tensors.

Characteristic surfacesThe MSS model has two characteristic surfaces: an internal

plastic yield envelope (PYE) and an external bond strengthenvelope (BSE). Fig. 1 depicts these surfaces in a stress spacewhich consists of the isotropic mean stress ( p) and thedeviatoric hyper-plane (s), or, equivalently, the transformeddeviatoric hyper-plane fS1 S2 S3 S4 S5g (see Appendix 1).While the ( , s) space is a tensorial space, geometrical insightis preserved, since in the `triaxial' mode of deformation, thedeviatoric hyper-plane has only one non-zero component,S1 p(2=3)(1 3) p(2=3)q and thus the representationreduces to the common ( p, q) space. The internal surface(PYE) plays the role of the classical `yield surface', that is itdelimits elastic and plastic states. The term `plastic' was addedto the `yield surface' to point out the difference between plasticyielding and large-scale yielding (de-structuring) (e.g. Jardine etal., 1991). Since most soils behave elastically in a very limiteddomain, the model can accommodate an arbitrarily small PYEwithout adverse side-effects, since the size of the PYE ispractically independent of the maximum pre-consolidation pres-sure (as in most classical models). A small plastic yieldenvelope also helps in the realistic modelling of cyclic loading,since it allows the accumulation of permanent strains even forlow-intensity stress cycles, the development of appreciable ex-cess pore pressures during undrained loading (which mayeventually lead to liquefaction/cyclic mobility), and a progres-sive structure degradation due to a fatigue-type accumulation ofplastic strains.

The external surface (BSE) corresponds to material statesassociated with appreciable rates of structure degradation. Sinceplastic strains can develop inside the BSE, structure degradationinitiates before the material reaches the BSE, but, in this case,the rate of de-structuring is small. Experimental evidence (e.g.Vaughan, 1988; Smith et al., 1992; Lagioia & Nova, 1995)suggests that the onset of appreciable de-structuring is usuallyabrupt and easily identiable, thus facilitating the experimentaldetermination of the BSE by probing various directions in thestress space.

In structured soils, the size of the BSE is controlled by themagnitude of the bond strength, that is the pre-consolidationpressure of overconsolidated clays and the strength of thecementation/thixotropic bonds in cemented/aged clays. Since theBSE is not necessarily spherical, the model can describe bondswhich degrade more easily by shearing than by compression.Furthermore, since the BSE is not necessarily isotropic (i.e. it is

not centred on the isotropic axis), the model can account foranisotropic bond development due to preferred particle orienta-tions: for example, bonds may degrade more easily in extensionthan in compression, or by shearing along a specic plane.Finally, since the BSE is not necessarily circular in the devia-toric hyper-plane, the model can describe shear strength aniso-tropy by independently controlling the shear strength in variousmodes of deformation (triaxial, plane strain, simple shear, etc.),provided that such test data are available. In this way, the modelcan improve the predictions of the modied CamClay (MCC)family of models which over-predict the shear strength intriaxial extension and simple shear when the model parametersare selected by matching the shear strength in triaxial compres-sion. (This feature of the model was not used in the followingevaluation, as the calibration of the model parameters with testdata of Vallericca clay was limited in the triaxial plane.)

In overconsolidated clays without appreciable ageing or ce-mentation, the size and location of the BSE are controlled bythe stress history, in a way analogous to the classical criticalstate models which determine the size of the (unique) yieldsurface by the maximum pre-consolidation pressure. The pro-posed model generalizes this concept and records several othercharacteristics of the stress history in the hardening variables ofthe BSE (in addition to the maximum pre-consolidation pres-sure), namely the principal stress ratios and the directions of theprincipal stresses at the maximum pre-consolidation pressure.This is achieved via the degrees of freedom of the BSE (inaddition to its size along the isotropic axis), that is theeccentricities along the deviatoric axes and the location of itscentre in the stress space. In strongly cemented soils, thesecharacteristics are controlled by the magnitude and spatialdistribution of the bond strengths, since the effects of the stresshistory are masked by cementation. In weakly cemented soils,both structure-inducing effects (i.e. the stress history and thecementation bonds) have comparable magnitudes and the BSErepresents their combined effect. The MSS model allows stan-dard overconsolidation to be modelled in the same way as anyother process causing irreversible bonding at the inter-particlecontacts (such as ageing and cementation), a fact which is alsoappealing in conceptually unifying the effects of all theseprocesses.

The BSE is described by the function (the symbol `:'indicates a summation of the products)

F(, K , ) 1c2

(s sK ): (s sK ) ( K )2 2 0(1)

which reduces to the yield function of the MCC model

FMCC(, ) 1c2

s: s ( )2 2 0 (2)

for K and sK 0. The geometrical representation of theBSE in the stress space ( , s) is an ellipsoid centred atpoint K with coordinates K sK K I, where I is theisotropic unit tensor. The half-axes of the ellipsoid are equal to along the isotropic axis and equal to c along each of thedeviatoric axes. The size of the BSE along all deviatoric axesneed not be the same, that is the ellipsoid need not besymmetric about the isotropic axis: the half-axis along each ofthe ve deviatoric stress axes (Si) may be equal to ci where ciis the corresponding eccentricity. In this case, equation (1) canbe written as (see Appendix 1)

F X5i1

1

c2i(Si SKi)2 ( K )2 2 0

The introduction of more than one material constant ci permitsthe independent control of the shear strength in the variousshearing modes (triaxial, simple shear, plane strain, etc.). Thisfeature of the MSS model is very useful in modelling soils withappreciable shear strength anisotropy; in such soils, the shearstrengths in the various modes are not interdependent andFig. 1. Characteristic surfaces of the MSS model

264 KAVVADAS AND AMOROSI

certainly cannot be predicted by knowing the value of the shearstrength in triaxial compression.

For numerical simplicity, the PYE is assumed to be geome-trically similar to the BSE (scaled by a factor 1) and isdescribed by the function

f (, L, ) 1c2

(s sL): (s sL) ( L)2 ()2 0(3)

The PYE is centred at point L with coordinates L sL LI,has size along the isotropic axis equal to , size along each ofthe deviatoric axes equal to cor ciand is fully con-tained inside the BSE. Although the size of the PYE is scaledto the size of the BSE, this dependence is very weak, as thescaling factor is a very small number (of the order of 0001).

Hardening rulesThe MSS model requires the hardening variables (; K , L)

which control the size and position of the characteristic sur-faces. The evolution of the hardening variables during plasticdeformation is described by the hardening rules. Followingstandard plasticity, it is assumed that the material does notharden during elastic deformation (i.e. when the state is insidethe PYE). The proposed model possesses isotropic and kine-matic hardening rules. The isotropic hardening controls the sizeof the BSE, that is it describes the evolution of materialstructure, while the kinematic hardening governs the motion ofthe characteristic surfaces in the stress space, that is it describesthe evolution of the anisotropy.

Isotropic hardening. The change of the size of the BSE due tothe plastic strain increment ( _pv, _

pq) is

_ 1 e k

v exp(v, pv)

_pv

fq q exp(q, pq)g _pq

(4)

where _pv _p: I is the plastic volumetric strain increment, _pq p[(2=3)( _ep:ep)] is the modulus of the plastic deviatoric strain

increment [ _ep _p ( _pvI=3)], (pv, pq) are the accumulated plas-tic volumetric and deviatoric strains, e is the void ratio, (, k)are the intrinsic compressibility parameters during virgin com-pression and rebound, (v, v) are the volumetric structuredegradation parameters, and (q, q, q) are deviatoric structuredegradation parameters. Equation (4) reduces to the hardeningrule of the MCC model if all structure degradation parametersare zero. The isotropic hardening of the MSS model has twocomponents, as follows.

(a) A volumetric component, which depends on the plasticvolumetric strain increment _pv and models the intrinsicvolumetric hardening and the volumetric-strain-inducedstructure degradation. In non-structured soils (v v 0), the volumetric component is identical to theisotropic hardening of the MCC model, that is the intrinsicvirgin compression is linear in a (log pe) plot. Instructured soils, the parameters (v, v) dene the rate ofvolumetric structure degradation in an exponential damage-type form analogous to that proposed by Wilde (1977),Kavvadas (1995), Muir Wood (1995b) and Lagioia & Nova(1995). This form decays at large accumulated plasticstrains with a rate depending on the value of the positiveparameter v. Positive values of the parameter v tend toreduce the size of the BSE (and thus decrease the shearstrength) with the accumulation of plastic volumetricstrains. This collapse-type behaviour cannot be predictedby classical critical state models, where volume reduction isalways associated with an increased shear strength.

(b) A deviatoric component, depending on the modulus of theplastic deviatoric strain increment _pq, which uses an

exponential damage-type form similar to the volumetriccomponent and decays at large plastic shear strains with arate depending on the parameter q. A non-zero value of theparameter q gives permanent structure degradation (orhardening) but, in most applications, q 0. The deviatoriccomponent can be used to model shear-induced structuredegradation (q . 0), since shearing can cause a gradualreduction in the size of the BSE even for stress paths insidethe BSE (fatigue-type structure degradation).

Kinematic hardening. The kinematic hardening describes theevolution of material anisotropy during plastic deformation. Thisis achieved by the translation of the characteristic surfaces (BSEand PYE) in the stress space, that is by controlling the motion oftheir centres K and L.

During plastic deformation, the centre K of the BSE movesas follows.

For material states inside the BSE

_K _ K (5a)

that is, the centre K of the BSE moves along a radial pathpassing through the origin. In this respect, the proposed modelreduces to the MCC model if K I.

For material states on the BSE

_K _ K _

s

KsK

(5b)

where , are parameters. The second term of the aboveexpression causes the centre K of the BSE to deviate from theradial direction, that is to move in the deviatoric hyper-plane,thus altering the anisotropy.

The kinematic hardening rule introduces a primary anisotro-py tensor, bK sK=K which controls the offset of the centreK of the BSE from the isotropic axis and depicts the tangent ofthe angle of OK with the isotropic axis (Fig. 1). The model alsouses a secondary anisotropy tensor, bL sL=L, which controlsthe deviation of the centre L of the PYE from the isotropicaxis. It can be seen that the MCC model lacks both types ofanisotropy, since its (single) yield surface is always centred onthe isotropic axis. The primary anisotropy of the proposedmodel changes only during plastic deformation from materialstates on the BSE and thus it represents the bond strengthanisotropy. During plastic deformation inside the BSE, thecentre K moves along a radial path (equation (5a)), and thusthe primary anisotropy does not change. The primary anisotropyalso does not change during loading along stabilized radialstress paths, that is after sustained loading along a radial stresspath so that the material anisotropy has fully adjusted to thepreferred directions of this path. In fact, equation (5b) impliesthat when a radial stress path (with direction s= ) has beenstabilized (i.e. the centre K also moves on a radial path), theprimary anisotropy of the material is such that s= (sK=K ), that is the centre K moves along a radial path withslope (sK=K ) which forms an angle with respect to the stresspath (s= ) controlled by the material constant .

During plastic deformation, the centre L of the PYE movesas follows (see Fig. 1).

For material states on the BSE (i.e. when the two character-istic surfaces are in contact at a point corresponding to thecurrent stress state), the two surfaces remain in contact and theposition of L is dictated by the position of K

L

K

) L (1 ) K (6)

For material states inside the BSE, the motion of point L issuch that the PYE moves towards point M9, which is theconjugate of the current state (point M). The geometric similar-ity of the two surfaces permits the denition of conjugate points(M and M9) on the PYE and BSE, respectively, such that thenormal vectors at these points are parallel. The stress at the

A CONSTITUTIVE MODEL FOR STRUCTURED SOILS 265

conjugate point M9 is (M9) K ( L)= and the direc-tion vector is

MM9 (M9) 1

( L) ( K ) (7)

According to the above, the translation of the centre L is

_ L _ L _ (8)

where the factor _ is evaluated below. The rst term of thisrule, [( _=) L], is a homothetic transformation which preservesparallelity of the direction of the vector and as such it ensuresthat the characteristic surfaces do not intersect even for niteincrements of the material state; when the two surfaces comeinto contact, they contact at conjugate points which coincidewith the stress state. Similar translation rules have been pro-posed by Hashiguchi (1985), Al-Tabbaa & Wood (1989) andStallebrass & Taylor (1997). The factor _ is determined fromthe `consistency condition', that is a requirement that duringplastic deformation the stress point remains on the PYE( _f 0), which gives

_ (1=c2)(s sL): (_s ( _=)s ( L)( _ ( _=) )

2 [(1=c2)(s sL): (s sK ) ( L)( K ] (9)

Flow ruleThe ow rule determines the plastic strain increment _p and

generally has the form

_p _P _ 1H

(Q: _) (10)

The scalar _ and the plastic potential P give the magnitude andthe direction of the plastic strain increment, _ is the corre-sponding effective stress increment, H is a `plastic modulus' asdescribed in a following section, and Q @ f =@ is the gradi-ent of the PYE. The plastic gradient Q has the followingisotropic and deviatoric components (using equation (3))

Q Q: I @ f@ 2( L)

Q9 Q 13

QI @ f@s 1

3

@ f

@s: I

I 2

c2(s sL) (11)

The proposed model uses an associated ow rule, that isP Q.

ElasticityThe elastic component describes the behaviour inside the

PYE where deformation is by denition elastic. According tostandard plasticity, strain increments consist of elastic (i.e.reversible) and plastic (i.e. irreversible) components which canbe split into volumetric and deviatoric parts as follows

_ _e _p ) _v _ev _pv and _e _ee _ep (12)The elastic component of the strain increment is assumed to belinearly related to the corresponding effective stress incrementvia an elastic stiffness Ce

_ Ce: _e (13)In linear isotropic elasticity, the elastic stiffness depends on twomaterial constants, the bulk modulus K and the shear modulusG, and the stressstrain relationships are

_ K _ev _s 2G _ee (14)Critical state models usually employ poro-elasticity, which as-sumes that the elastic volume compressibility is linearly relatedto the logarithm of the mean effective stress. Such models havea pressure-dependent bulk modulus given by K (1 e)=k,where k is the CamClay compressibility parameter. In order toimprove the accuracy of the predictions (e.g. Wroth et al.,1979; Houlsby, 1981), the elastic shear modulus is also assumed

to be pressure dependent (i.e. G=K constant). This choicecan lead to theoretical and numerical difculties, especially incyclic loading, since the elasticity becomes non-conservative(Houlsby, 1985). A solution which preserves the pressure depen-dence of (K, G) and, at the same time, maintains the conserva-tive nature of elasticity, is hyper-elasticity. Hyper-elasticity hasthe additional advantage of introducing coupling between thevolumetric and shear components in the stressstrain relation-ships, a fact commonly observed in practice.

The MSS model uses either poro-elasticity (equation (14)) orhyper-elasticity (Houlsby, 1985)

_ pr exp ev

k

1

k 132k

(eq)2

_ev

2

k

(ee _ee)

(15a)

_s pr exp ev

k

2k

ee _ev 2 _ee

(15b)

Poro-elasticity requires the material constants k and G=K, whilehyper-elasticity requires material constants k k=(1 e), ,and a reference pressure pr. An additional advantage of hyper-elasticity (compared to isotropic poro-elasticity) is the ability topredict the development of shear-induced excess pore pressuresduring elastic undrained loading, and the related improvementof the predicted effective stress path.

Plastic modulus HFor material states on the BSE, the plastic modulus is

determined from the `consistency condition', which ensures thatthe stress point remains on the BSE (see Appendix 2)

H 2RT (16)It can be seen that the critical state (where H 0) is achievedat the top of the BSE, where R 0 (since P 0 and shear-induced degradation has ceased). For material states inside theBSE, the plastic modulus is determined from the requirementfor a continuous variation of its magnitude as the PYE ap-proaches the BSE. This requirement is satised if the plasticmodulus is obtained from the following interpolation rule (seeAppendix 2)

H H 0 jH 0jf[1 (=o)] 1g (17)where H 0 is the value of the plastic modulus at point M 0where vector

!OM intersects the BSE (Fig. 1), and is the

normalized length of MM 0 (M is the current state). Equation(17) interpolates between: H 1 (upon initiation of yielding)and: H H 0 (when the stress state reaches the BSE). Moredetails are included in Appendix 2.

Summary of model parametersThe MSS model requires the following eleven parameters.

Four of them are the parameters of the MCC model, while theremaining seven control the structure degradation and anisotro-pic characteristics of the proposed model.

(a) k: poro-elastic compressibility. The corresponding para-meter in hyper-elasticity is k k=(1 e).

(b) G=K or : elastic shear parameter in poro-elasticity orhyper-elasticity, respectively.

(c) : intrinsic compressibility.(d ) c (or ci): eccentricity of the BSE. Controls the shear

strength in the appropriate mode (if different ci values areused). In the simplest case, it is proportional to the Mparameter of the MCC model: c p(2=3)M .

(e) (v, v) and (q, q): volumetric and deviatoric structuredegradation parameters.

( f ) (, ): parameters controlling the evolution of materialanisotropy, that is the motion of the BSE in the deviatoricspace.


(g) : parameter controlling the variation of the elasto-plasticmodulus (H) in the early stages of structure degradation(i.e. before the BSE is engaged). It controls the stiffness ofthe stressstrain curve after the onset of plastic strains.

The MSS model may use the following optional parameters.

(a) : ratio of the sizes of the BSE and PYE. Controls the sizeof the elastic domain. Usually it is set to a small number(typically 0:0050:05).

(b) q: steady-state deviatoric structure degradation/hardeningparameter (usually q 0).

Each of the above parameters controls a specic aspect ofthe model (modulus, strength, structure, anisotropy, etc.) withoutappreciable interaction among parameters. In this way, thedetermination of their values for a specic soil using standardlaboratory tests is simplied. Furthermore, the model parametersare such that the sophistication of the predictions is adaptableto the available test data.

In addition to the above parameters, the implementation ofthe proposed model requires the determination of the initialstate of the material, which involves the following state vari-ables.

(a) : effective stress components(b) e: void ratio(c) : size of the BSE (controls the bond strength and

consequently the shear strength)(d ) K : position of the centre of the BSE (controls the primary

structure anisotropy)(e) L: position of the centre of the PYE (controls the

secondary anisotropy).

The MCC model requires only the rst three state variables,since it does not include strength anisotropy.

EVALUATION OF THE PROPOSED MODEL

The capabilities of the proposed model are investigated bycomparing its predictions with the results of a series of labora-tory tests on Vallericca clay, a natural Plio-Pleistocene marineclay from a site a few kilometres north of Rome (Italy).Vallericca clay is a stiff, overconsolidated, medium plasticityand activity material characterized by a calcium carbonate con-tent of about 30% and a remarkable absence of major macro-structures. Its average index properties are listed in Table 1.

Vallericca clay has been extensively studied in the lastdecade (e.g. Rampello et al., 1993); a considerable proportionof this research has been aimed at the investigation of the roleof structure in the mechanical behaviour of the material.Burland et al. (1996) identied the inuence of microstructuraleffects on the compressibility and shear strength of Vallericcaclay, comparing the results of oedometer and triaxial testscarried out on natural and reconstituted samples. Further experi-mental research on Vallericca clay, recently carried out byAmorosi (1996), conrmed that the mechanical behaviour of thesoil is signicantly affected by its natural structured state;depending on the direction of the stress path, de-structuring canoccur during both the consolidation and the shear stages of thetests, and is related to the cumulative volumetric and deviatoricplastic strains. These features are explicitly described by theproposed constitutive model, thus making meaningful the com-

parison of laboratory test results with model predictions. Theexperimental programme and the testing procedures are de-scribed in detail by Amorosi (1996); a brief summary is givenbelow, with the objective of providing the necessary informationfor the evaluation of the model parameters and the comparisonswith the model predictions.

Large block samples of the natural clay were used fortrimming 38 mm dia. and 78 mm high cylindrical specimens fortriaxial testing. The experimental programme was performed inthree computer-controlled stress path triaxial apparatus, capableof applying cell pressures up to 3, 10 and 14 MPa. It consistedof two series of shear tests on samples consolidated to relativelymedium and high pressures, the results of which are shown inFigs 27.

The rst series of tests, referred to as medium-pressure (MP)tests, was performed on samples anisotropically consolidatedalong the effective stress path shown by dotted lines in Fig. 5.This stress path was selected to ensure that the radial strain wasvery close to zero. All samples were compressed from an initialisotropic effective stress state ( pk 400 kPa) to a nal statehaving mean effective stress pmax 1770 kPa, and deviatoricstress qmax 1210 kPa (effective stress ratio 3= 1 0:53).The vertical effective stress at the nal state (vmax 2570 kPa)is slightly lower than the vertical stress (vy 2600 kPa) corre-sponding to the onset of appreciable rates of structure degrada-tion, that is to the intersection of the stress path with the BSE.After consolidation, performed in small steps to ensure minimal

Table 1. Average index properties of Vallericca clay

Property Value: %

Clay fraction (,2 m) 47Calcium carbonate content (CaCO3) 32Liquid limit 55Plasticity index 29Plastic limit 26Natural moisture content 264Specic gravity 278

Fig. 2. Vallericca clay. Comparison between model prediction andexperimental data in: (a) anisotropic compression, (b) isotropiccompression


excess pore water pressure, the samples were either sheared(OCR vmax=vo 1) or rebounded to OCR values of 17,24 and 4 and then sheared. Drained and undrained shearingwas carried out at axial strain rates equal to 11% and 58% perday, respectively. The comparison of the proposed model withthe results of the MP series of tests allows us to assess themodel capability in predicting material behaviour for deviatoricstress paths intersecting the initial BSE.

The second series of tests, referred to as high-pressure (HP)tests, was performed on samples anisotropically compressed insmall steps along the extension of the consolidation path of theMP tests, that is a path corresponding to a constant effectivestress ratio ( 3= 1 0:53); the consolidation path of the HPtests is shown by the dotted line in Fig. 7. At the nal state( pmax 4630 kPa, qmax 3160 kPa), the vertical effective stress(vmax 6750 kPa) is equal to about 26 times the value cor-responding to the intersection of the consolidation path with theBSE (vy 2600 kPa). After consolidation, the samples wereeither sheared directly (OCR 1) or rebounded to OCR valuesof 17 and 24 prior to shearing. The comparison of the

proposed model with the results of the HP series of tests allowsus to assess the model capability in reproducing the mechanicalbehaviour of Vallericca clay as observed after the de-structuringprocess induced during high-pressure anisotropic consolidation.

The numerical predictions of the observed response duringthe MP and HP series of tests were obtained using the set ofmodel parameters listed in Table 2. A very small elastic domainwas assumed ( 0:005) and behaviour in that region wasdescribed by the hyper-elastic option of the model. The elasticparameter k was determined from the slope of the initialportion of the unloadingreloading line during anisotropic con-solidation plotted in an ln pln(1 e) plane. The elastic para-meter was determined from the orientation of the initial partof the stress paths obtained from undrained triaxial compressiontests carried out on overconsolidated samples, as suggested byBorja et al. (1997).

The isotropic hardening parameters of the model were esti-mated by a trial and error procedure, as described below, takinginto account the different relative weights of the deviatoric andvolumetric hardening during drained and undrained conditions.

2000

1000

1000

500

0

2000

1000

0

0

50

q: kP

au

: kPa

q: kP

aVo

lum

etric

stra

in: %

ModelExperiment

0 5 10Axial strain: %

Axial strain: %0 5 10 0 5 10


OCR = 1, undrained OCR = 1, drained

Fig. 3. Medium-pressure undrained and drained compression tests on anisotropically con-solidated Vallericca clay. Normally consolidated samples (vo vmax 2570 kPa). Comparisonbetween model predictions and experimental data. Plots of deviatoric stress and excess porewater pressure versus axial strain

2000

1000

1000

500

0

0

q: kP

au

: kPa

ModelExperiment


1000

500

0

u: k

Pa


500

500

0

u: k

Pa


Axial strain: %0 5 10

2000

1000

0

q: kP

a


2000

1000

0q:

kP

a


OCR = 17 OCR = 24 OCR = 4

Fig. 4. Medium pressure undrained triaxial compression tests on anisotropically consolidated Vallericca clay (vmax 2570 kPa). Over-consolidated samples: OCR 17 (vo 1512 kPa), OCR 24 (vo 1071 kPa), and OCR 4 (vo 642 kPa). Comparison between modelpredictions and experimental data. Plots of deviatoric stress and excess pore water pressure versus axial strain


The deviatoric hardening parameters (q, q) were evaluated bymatching the observed behaviour along the undrained shearingstress paths, while the volumetric hardening parameters (v, v)were determined by a similar process using data from theanisotropic consolidation and the drained shearing stress paths.

Model2000

1500

1000

500

0

2000

1500

1000

500

00 500 1000 1500 2000

q: kP

aq:

kP

a

p : kPa

0 500 1000 1500 2000

OCR = 4

OCR = 24

OCR = 17

OCR = 1

OCR = 4

OCR = 24

OCR = 17

OCR = 1

Experiment

U D

Fig. 5. Medium-pressure triaxial compression tests on anisotropi-cally consolidated Vallericca clay (vmax 2570 kPa). Normally con-solidated samples OCR 1 (No vmax) (D drained, U un-drained); overconsolidated samples (u): OCR 17 (vo 1512 kPa),OCR 24 (vo 1071 kPa), and OCR 4 (vo 642 kPa). Com-parison between model predictions and experimental data. Effectivestress paths in pq plane

2000

3000

4000

1000

3000

1000

2000

0

0

2000

3000

4000

1000

3000

1000

2000

0

0

2000

3000

4000

1000

3000

1000

2000

0

0

q: kP

au

: kPa

ModelExperiment


u: k

Pa


u: k

Pa



q: kP

a


q: kP

a


OCR = 17OCR = 1 OCR = 24

Fig. 6. High-pressure undrained triaxial compression tests on anisotropically consolidated Vallericca clay (vmax 6750 kPa). Normallyconsolidated and overconsolidated samples: OCR 1 (vo 6750 kPa), OCR 17 (vo 3970 kPa), and OCR 24 (vo 2812 kPa).Comparison between model predictions and experimental data. Plots of deviatoric stress and excess pore water pressure versus axial strain

1000

2000

3000

4000

00 1000 2000 3000 4000 5000

q: kP

a

p : kPa

OCR = 24

OCR = 17

OCR = 1

1000

2000

3000

4000

00 1000 2000 3000 4000 5000

q: kP

a

OCR = 24

OCR = 17

OCR = 1

Model

Experiment

Fig. 7. High-pressure undrained triaxial compression tests onanisotropically consolidated Vallericca clay (vmax 6750 kPa).Normally consolidated and overconsolidated samples: OCR 1(vo 6750 kPa), OCR 17 (vo 3970 kPa), and OCR 24(vo 2812 kPa). Comparison between model predictions andexperimental data. Effective stress paths in pq plane. The stresspaths of the medium-pressure tests are also shown for comparison


The parameter (q) was set to zero, based on the hypothesis ofa negligible residual rate of structure degradation at largedeviatoric strain. The volumetric hardening parameter () wasevaluated with reference to the nal stages of the anisotropicconsolidation path in the HP series of tests; it can be seen thatthe slope of this curve in an eln p plot is not exactly equal to, as it would be according to classical critical state theory,because the material is not fully de-structured and its compres-sibility is also inuenced by the structure degradation para-meters. For the selected values of the structure degradationparameters, the model predicts a more rapid degradation due tovolumetric plastic strains than due to deviatoric plastic strains.However, since the magnitude of the deviatoric strains is largerthan that of the volumetric strains, structure degradation iscaused by a combination of both mechanisms.

The kinematic hardening parameters were selected assumingthat the motion of the BSE in the stress space was relativelyslow ( 0:1) and that, for continuous radial stress paths, thecentre of the BSE is located on the line of the stress path( 1).

The parameter c1 was estimated from the results of undrainedtriaxial compression tests carried out on the MP and HP nor-mally consolidated samples. It was further assumed that c2 c3 c1, since only tests in the triaxial plane were available andevidence regarding shear strength anisotropy off the triaxialplane could not be substantiated. However, as the model permitsindependent control of the shear strength in the various shearingmodes (by varying c1, c2, c3, c4 and c5), calibration off thetriaxial plane can be performed without affecting the calibrationin the triaxial plane, provided that such test data are available.

The material parameter () controlling the variation of theplastic modulus H inside the BSE was evaluated from theundrained triaxial compression tests carried out on overconsoli-dated samples.

For each of the MP and HP tests, the proposed model wasused to simulate the complete sequence of the consolidation,rebound and shearing stress paths followed by the specimen inthe laboratory, starting from an initial isotropic effective stressstate ( pk 400 kPa). In particular, the simulations were per-formed under stress-controlled conditions during the consolida-tion and rebound stages of the tests, followed by the strain-controlled shearing stage. The initial size of the BSE wasdetermined from the anisotropic consolidation tests shown inFig. 2, and specically from the state where an abrupt stiffnessloss was observed (vy 2600 kPa, hy 1378 kPa). This statewas considered to represent the intersection of the consolidationpath with the BSE. Using this information, the initial values ofthe state variables were determined as: K 1400 kPa,S1K 230 kPa. Fig. 2 also shows the observed and predictedcompression curves during isotropic consolidation plotted in thelog p versus specic volume (1 e) plane. The two types oftests are reproduced well by the proposed model. The ellipticalshape of the BSE appears to represent reasonably well thestructural characteristics of Vallericca clay, since the stress levelcorresponding to a major loss in stiffness is accurately predictedalong both the isotropic and the anisotropic consolidation paths.

Figures 3 and 4 compare the experimental and the predictedcurves of the deviatoric stress (q) and the excess pore pressure(u)/volumetric strain (v) observed in selected undrained anddrained MP tests. The corresponding effective stress paths areplotted in Fig. 5. The hyper-elastic formulation employed in themodel reproduces well the stiffness and the slope of the effec-

tive stress paths in the early stages of shearing, that is beforethe onset of plastic strains and the initiation of structuredegradation. Fig. 3 shows that the model can reproduce well thebrittle stressstrain behaviour and the post-peak strain softeningobserved in the normally consolidated undrained tests. Themodel is also successful in predicting the monotonic strainhardening observed in the drained test. The observed rates ofexcess pore pressure development in the undrained tests and thevolumetric compression in the drained test are also reasonablywell predicted. According to the proposed model, structuredegradation inuences drained and undrained shearing differ-ently: drained specimens are subjected to larger de-structuringthan undrained specimens, because of the additive deleteriouseffects of the volumetric and deviatoric strains in the drainedspecimen. Despite that, strain-softening is observed and pre-dicted only in the undrained tests; this is due to the appreciablevolumetric compression of the drained tests, which enhances thefrictional shearing resistance of the material and masks thedeleterious effects of de-structuring, causing a net enlargementof the BSE. In contrast, in the undrained tests, the size of theBSE decreases due to the prevailing effects of shear-inducedstructure degradation (since volumetric hardening is nil).

The MP overconsolidated samples sheared in undrained mode(Figs 4 and 5) exhibit a brittle stressstrain behaviour which isreproduced reasonably well by the proposed model. The samplesstrain harden until the effective stress paths are inside the BSE.When the paths reach the BSE, the rate of de-structuringbecomes appreciable and the material starts to strain soften. Asthe OCR increases, the dilatant behaviour of the soil is enhanced,while the rate of de-structuring decreases due to the largeraccumulated shear strains inside the BSE (Figs 4 and 5). Thistype of behaviour is correctly reproduced by the proposed model.

The capabilities of the proposed model are also evaluatedusing the results of the HP series of tests. Figs 6 and 7 comparethe experimental and predicted deviatoric stress and excess porepressure curves and the associated effective stress paths inspecimens anisotropically consolidated to very high pressure(well above the BSE) and then sheared undrained atOCR 1:0, 17 and 24. The experimental results show a brittlestressstrain behaviour, coupled with an increase of u in thenal part of the tests. Accordingly, the stress paths bend to theleft after peak strength, showing decreasing values of p and q.This feature can only partly be attributed to the initial structuredstate, as the high-pressure consolidation stage is likely to havecaused an appreciable amount of structure degradation. Theobserved behaviour is more likely to be related to the aniso-tropic consolidation stress path imposed prior to shear. In fact,similar softening responses were also observed on reconstitutedsamples of other clays sheared undrained after anisotropic com-pression and swelling (e.g. Gens, 1982; Rossato et al., 1992).

To reproduce such observations, the model was calibrated inorder to retain some deviatoric structure degradation even athigh pressure. Figs 6 and 7 indicate that the model satisfactorilyreproduces the stressstrain behaviour during the HP shearing,while it tends to overestimate the corresponding excess porewater pressure. It should be pointed out that the model wascalibrated mainly by using the consolidation and MP test resultsand was then employed in a comparison with the HP testresults. While this last comparison is not always satisfactory, itshould be realized that the stress levels of the MP and HP testsare very different and thus the model was used under `unfavour-able' circumstances.

Table 2. Values of the model parameters for Vallericca clay

Parameter Value Parameter Value Parameter Value

k 0013 v 5 x 1 103 v 50 01 0118 q 3 14c 085 q 05 0005


CONCLUSIONS

The paper describes and evaluates a critical-state incremen-tal-plasticity model for structured soils (MSS). The modelsimulates the engineering effects of processes causing structuredevelopment (pre-consolidation, ageing, cementation, etc.) andstructure degradation (remoulding by volumetric and/or deviato-ric straining), such as high stiffness and strength at the intactstates, appreciable reduction of stiffness and strength during de-structuring, and the evolution of stress-induced and structure-induced anisotropy. A novel feature of the model is the treat-ment of pre-consolidation as a structure-inducing process andthe unied description of all such processes via the BSE. Theproposed model distinguishes the concepts of `yielding' (i.e. theonset of irreversible deformation upon reaching the PYE) andthe onset of major de-structuring which occurs when the BSE isengaged. Thus, the model avoids the large elastic domain ofcritical state models and permits the development of irreversiblestrains even for small variations of the stress levels. Otherfeatures of the MSS model include

(a) a general-purpose damage-type mechanism which canmodel the structure degradation induced by volumetricand deviatoric strains

(b) stress- and bond-induced anisotropy as well as memory ofthe stress history, achieved by recording the offset of thetwo model surfaces from the isotropic axisthese charac-teristics are gradually erasable (fading memory) as thesurfaces move and the material state adapts to more recentstressing

(c) formulation in a tensorial space consisting of the isotropicaxis and the deviatoric hyper-planethis formulationensures the generality required for incorporation in niteelement codes without losing the geometrical insight of thetriaxial pq plane, and it facilitates the modelling of shearstrength anisotropy by decoupling the shear strengthparameters in the various shearing modes (triaxial, planestrain, simple shear, etc.), thus permitting independentcontrol of the shear strength in these modes

(d ) downward compatibility with the MCC model when allstructural and anisotropic features are turned offfurther-more, these features can be turned on and off according tothe type of the available test data, thus adapting the level ofpredictive sophistication to the available data.

The model is evaluated by comparing the predicted andobserved behaviour of the stiff overconsolidated Vallericca clay.The experimental data used to investigate the predictive capabil-ities of the model consist of drained and undrained triaxialcompression tests performed on natural samples after consolida-tion and swelling along anisotropic stress paths to reach differ-ent levels of maximum stress and overconsolidation ratio. Forsamples re-consolidated to stress levels below the BSE (MPtests), the model predictions are in good agreement with theobserved behaviour. These results are of particular interest inthe prediction of the behaviour of geotechnical structures, sincemost of these interact with natural soils subjected to low stresslevels. For samples re-consolidated to stress levels well abovethe BSE (HP tests), the model satisfactorily reproduces thestressstrain behaviour during undrained shearing. Comparisonof the observed and predicted effective stress paths of all testsindicates that the model can reproduce with a satisfactorydegree of accuracy the overall behaviour of Vallericca clay asobserved in a wide range of stresses and loading conditions.

APPENDIX 1. TRANSFORMED STRESS AND STRAIN SPACESThe MSS model is formulated in a general effective stress space

(Prevost, 1978; Kavvadas, 1983) consisting of the isotropic (mean) stressaxis ( x y z)=3 and the deviatoric hyper-plane fS1 S2S3 S4 S5g, where S1 (2y x z)=p6, S2 ( z x)=p2;S3 xyp2, S4 xzp2 and S5 yzp2. The corresponding strainmeasures consist of the volumetric strain v (x y z) and thedeviatoric vector fE1 E2 E3 E4 E5g where E1 (2 y x z)=p6,E2 (z x)=p2, E3 xy=p2, E4 xz=p2 and E5 yz=p2.

These transformed stress and strain measures are energy conjugate and,compared to the standard tensorial quantities (, ), have the advantagethat the size of the space required to represent any loading path is theabsolute minimum; for example, a triaxial test can be represented in thetwo-dimensional space ( , S1), a plane strain test in the three-dimensional space ( , S1, S2), etc.

APPENDIX 2. CALCULATION OF THE PLASTIC MODULUS HFor material states on the BSE, the plastic modulus is determined

from the `consistency condition', which requires that the material stateshould remain on the BSE, that is

_F 0) @F@

: _ @F@K

_ K @F@sK

_sK @F@

_ 0 (18)

However

@F

@: _ 1

Q

: _ 1

_H ;

@F

@K 2( K );

@F

@sK 2

c2(s sK );

_K _K ; _sK _

sK _ s

KsK

;@F

@ 2

It can be seen that, in such cases, the material state () coincides withthe contact point of the PYE and the BSE. Furthermore (using equation(10))

_pv _P and _pq p

[23( _ep: _ep)] (sign _) _p[2

3(P9: P9)]

where P and P9 are the volumetric and deviatoric components of theplastic potential tensor P, respectively.

Thus, equation (4) gives _ _R, where

R 1 e k

v exp(vpv)

P

(sign _)fq q exp(qpq)gp

[23(P9: P9)]

Substitution of the above into equation (18) gives the plastic modulus

H 2RT (19)where

T ( K ) 1c2

(s sK ): s s K sK

For material states inside the BSE, the plastic modulus H can bedetermined from the requirement for a continuous variation of its valueas the PYE approaches the BSE and eventually the two surfaces comeinto contact. At that nal stage, the material state will be located on theBSE and the plastic modulus will be determined from equation (19). It isnoted that the consistency condition has already been used in thedetermination of the translation of the PYE (equation (9)).

The requirement for a continuous variation of H is satised if theplastic modulus is obtained from the following interpolation rule

H H 0 jH 0jf[1 (=o)] 1g (20)where H 0 is the value of the plastic modulus at a state corresponding topoint M 0 where vector

!OM intersects the BSE (Fig. 1) and is computed

via equation (19). Point M 0 has coordinates ( > 1) and M is thecurrent stress state (coordinates: ). The parameter is computed fromthe relationship: [(BpB2 A )]=A, where

A 1c2

(s: s) 2; B 1c2

(s: sK ) K ; 1c2

(sK : sK ) 2K 2

The parameter is the normalized length of MM9, dened by therelationship

S Q2kQk :

and o is the value of the parameter upon initiation of yielding; that is,o is reset to the value of each time yielding is reinitiated. Thus,=o 1 upon initiation of yielding, =o , 1 at any later stage, and 0 when the material state lies on the BSE. Equation (20) isessentially an interpolation rule between the value H 1 uponinitiation of yielding, and the value: H H 0 when the stress statereaches the BSE. The material constant . 0 determines the rate ofvariation of H in the range (1, H 0).


ACKNOWLEDGEMENTS

The authors would like to acknowledge the assistance andsupport offered by Professor G. Calabresi, Professor S. Rampel-lo and Dr M. R. Coop during the experimental research onVallericca clay.

NOTATIONBSE bond strength envelope

c (or ci) eccentricity of the BSE and the PYEdot (over a symbol) innitesimal increment of this quantity

e void ratioe (superscript) elastic component of strain

G=K elastic shear parameter in poro-elasticityF function of the BSEf function of the PYE

H elasto-plastic modulusI unit second-order tensor

OCR overconsolidation ratiop (superscript) plastic component of strain

PYE plastic yield envelopeq scalar stress deviatorR auxiliary scalar quantity (dened in Appendix 2)s tensorial stress deviator

Si deviatoric stress componentsT auxiliary scalar quantity (dened in Appendix 2) size of the BSE elastic shear parameter in hyper-elasticity parameter controlling the variation of the elasto-

plastic modulus (H)u excess pore pressure strain tensorv volumetric strainq scalar deviatoric strain

v, v, q, q volumetric and deviatoric structure degradationparameters

q steady-state deviatoric structure degradation/hardening parameter

k poro-elastic compressibilityk hyper-elastic compressibility intrinsic compressibility

, p mean effective stressvmax maximum vertical pre-consolidation pressurevo vertical consolidation pressure effective stress tensor

K coordinates of the centre of the BSE in the stressspace

L coordinates of the centre of the PYE in the stressspace

ratio of the sizes of the BSE and PYE(, ) parameters controlling the evolution of material

anisotropy

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