simulation of yielding and stressâ€“stain behavior of ... · simulation of yielding and...

Computers and Geotechnics 38 (2011) 341–353

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Computers and Geotechnics

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Simulation of yielding and stress–stain behavior of shanghai soft clay

Maosong Huang a,⇑, Yanhua Liu a,b, Daichao Sheng c

a Department of Geotechnical Engineering, Tongji University, Shanghai 200092, Chinab School of Highway, Chang’an University, Xi’an 710064, Chinac School of Engineering, University of Newcastle, NSW 2308, Australia

a r t i c l e i n f o

Article history:Received 16 March 2010Received in revised form 30 November 2010Accepted 20 December 2010Available online 22 January 2011

Keywords:Soft clayStructureAnisotropyYieldingBounding surface model

0266-352X/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compgeo.2010.12.005

⇑ Corresponding author. Address: Department oTongji University, 1239 Siping Road, Shanghai 20065983980.

E-mail address: [email protected] (M. Huan

a b s t r a c t

In this paper, a simple bounding surface plasticity model is used to reproduce the yielding and stress–strain behavior of the structured soft clay found at Shanghai of China. A series of undrained triaxial testsand drained stress probe tests under isotropic and anisotropic consolidation modes were performed onundisturbed samples of Shanghai soft clay to study the yielding characteristics. The degradation of theclay structure is modeled with an internal variable that allows the size of the bounding surface to decaywith accumulated plastic strain. An anisotropic tensor and rotational hardening law are introduced toreflect the initial anisotropy and the evolution of anisotropy. Combined with the isotropic hardening rule,the rotational hardening rule and the degradation law are incorporated into the bounding surface formu-lation with an associated flow rule. Validity of the model is verified by the undrained isotropic and aniso-tropic triaxial test and drained stress probe test results for Shanghai soft clay. The effects of stressanisotropy and loss of structure are well captured by the model.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The Modified Cam-clay (MCC) model, which is based on thecritical state theory, is one of the most widely used constitutivemodels for clay [23]. It was originally formulated for remoldedclays under isotropic consolidation condition. Although the MCCmodel is widely used to represent the behavior of clayey soils, itsprediction ability is not considered adequate for natural clay. Thisis because of the complicated properties such as anisotropy, struc-ture and strain rate. Structure and anisotropy are the essential nat-ure of naturally deposited soft clay, which have considerableinfluence on the strength and stress–strain response of naturalclays. Sometimes loading causes degradation of the initial struc-ture, and this is particularly true in soft clays (e.g. Leda clay [21];Bothkennar clay [27]). Neglecting the anisotropy of soil behaviormay lead to highly inaccurate predictions of soil response underloading [33]. From engineering point of view, the last two decadesor so have seen an increased trend of construction activities on softsoils and hence a quantitative model that can accurately predictthe behavior of the soil is highly desirable.

There are various approaches for the constitutive modelling ofnatural clays (e.g. [22,6]). To model the destructuration of natural

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g).

clays, it is logical to start with a model that has had some successin predicting the behavior of remoulded material and then add to itsome measure of structure and destructuration [24,1,5,20,28].After all, a structured soil can eventually become something likea remolded soil given sufficient loading and destructuration.Meanwhile, numerous constitutive models that account for plasticanisotropy of natural clays have been proposed, in which S-CLAY1model proposed by Wheeler et al. [31] is a relatively simple elasto-plastic anisotropic model. Most existing models in the literature(e.g. [26,3,14]) account for either structure or anisotropy, but onlyfew models consider both properties of natural clays (e.g. [16,2]).The two properties can be related, but not always equivalent. Sev-eral researchers (e.g. [15,24,11]) developed constitutive models fornatural soils within the framework of kinematic hardening, whichconsider simultaneously the anisotropic and structural effect onthe mechanical behavior of soils. Those models can in generalachieve good results but often at a cost of complexity. They oftenrequire special techniques to ensure that the current stress pointsare located on the inner yield surface at every integration step inthe finite element implementation [35]. An alternative approachto avoid this complexity is to remove the kinematic hardeningyield surface, only preserving the bounding surface [7,8]. Thus,the kinematic hardening yield surface is degenerated to a loadingstress point, and the plastic modulus at the current stress pointcan be defined by a simple interpolation rule using values at thebounding surface. By means of vanishing pure elastic region, theclassical kinematic hardening bounding surface model can besimplified into the single bounding surface model [9].

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α

Reference surface

pc rpc

K0 line

CSL

q

Bounding surface

p

NCL

Fig. 1. Reference surface and bounding surface of anisotropic model for structuredclays.

342 M. Huang et al. / Computers and Geotechnics 38 (2011) 341–353

The main purpose of this paper is to investigate the yieldingcharacteristics of Shanghai soft clay and to demonstrate that a sim-plified bounding surface model is sufficient to simulate both theanisotropic and structural properties of natural clays. Experimentaldata from undrained triaxial tests and drained stress probe tests onShanghai soft clay are presented to support the proposed model.

2. Model description

The constitutive model, which is presented in this section, isdeveloped in p–q stress space, with p being the effective meanstress and q the deviatoric stress. Attention is restricted to rate-independent behavior and full saturation. Thus, the basic elasto-plastic assumption is the additive decomposition of total strainrate _eij, into elastic and plastic parts, _ee

ij and _epij

_eij ¼ _eeij þ _ep

ij ð1Þ

The response associated with the elastic part is expressed interms of the bulk and shear modulus, K and G, which are assumedto depend on the current mean stress p

K ¼ pj�; G ¼ 3ð1� 2mÞ

ð1þ mÞ K ð2Þ

where m is a constant Possion’s ratio; j⁄ = j/(1 + e0); e0 is initial voidratio and j is the slope of the swelling line in e-lnp plane.

The corresponding elastic incremental constitutive relation isgiven by

_rij ¼ Deijkl

_eekl ð3Þ

where Deijkl is elastic matrix, being in general a function of K and G.

The plastic part _epij is developed within the framework of the

critical state theory and the bounding surface plasticity. The for-mulation of the proposed model is given in detail in the following.

2.1. Bounding surface

Based on the experimental observations, an anisotropic refer-ence surface, which is an inclined ellipse on the p–q plane, is usedto model the intrinsic behavior of the reconstituted soils. Theanisotropic reference surface can describe the effect of initialanisotropy caused by one-dimensional deposition and K0-consoli-dation process. A structure surface or bounding surface whichhas the same elliptical shape as the reference surface is adoptedto describe the effect of the initial structure and control the processof destructuration. For simplicity, the loss of structure is assumedto affect merely the size of yielding surface. A scalar variable rcalled structural parameter is defined:

r ¼ �pc=pc ð4Þ

where �pc is the structural yielding stress; pc is the initial consolida-tion stress. Namely, r defines the ratio between the sizes of thestructure surface and reference surface. The curves of the referencesurface and structure surface are shown in Fig. 1. The value of r isalways larger than or equal to 1, due to its physical meaning. Forr = 1.0, the soil is completely destructured and the size of the struc-ture surface is related only to pc, corresponding to the size of thereference surface. We consider that the yield surface for describingthe behavior of the destructured or remolded soil, which is calledthe reference surface, is an anisotropic elliptical form. Accordingto Ling et al. [19], the mathematical equation of the reference sur-face is defined by

f ¼ ðp� pcÞ pþ R� 2R� pc

� �þ ðR� 1Þ2 q2

a

v ¼ 0 ð5Þ

The structure surface, which can be thought of as a boundingsurface, controls the process of destructuration. For simplicity,we consider that the structure surface has the same elliptical shapeas the reference surface. The mathematical equation of structuresurface is given by

F ¼ ðp� �pcÞ pþ R� 2R� �pc

� �þ ðR� 1Þ2 q2

a

v ¼ 0 ð6Þ

where

a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3aijaij=2

qð7Þ

qa ¼ffiffiffiffiffiffiffi3Ja

p; Ja ¼

12

saijs

aij

� �12

ð8Þ

saij ¼ sij � rkkaij=3 ð9Þ

v ¼ ðM � aÞ½2aðR� 1Þ2 þM � aþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4aðR� 1Þ2M þ ðM � aÞ2

q�=2

ð10Þ

where aij is the anisotropic tensor defining the anisotropy of clays;a defines the inclination of yield surface in p–q stress space, whichis the second invariant of anisotropic tensor; rij is stress tensor; sij isdeviatoric stress tensor; sa

ij is the reduced deviatoric stress tensor;qa is the reduced equivalent shear stress; Ja is the reduced secondstress invariant; R is the shape parameter, which controls the ratioof the two major axes of the yield surface; M is the slope of criticalstate line in triaxial space, which is defined by the reduced Lode an-gle ha as follows:

ha ¼13

sin�1 3ffiffiffi3p

2Sa

Ja

� �3" #

ð11Þ

Sa ¼13

saijs

ajksa

ik

� �13

ð12Þ

M ¼ Mc2m4

ð1þm4Þ � ð1�m4Þ sin 3ha

� �1=4

ð13Þ

where Sa is the reduced third stress invariant; m is a materialparameter defined as m = Me/Mc in which Mc and Me are the criticalstate stress ratios for triaxial compression and triaxial extension inp–q stress space. According to Sheng et al. [25], the yield surface isconvex provided m P 0.6, which coincides with the Mohr–Coulombhexagon at all vertices in the deviatoric plane.

As defined in Liang and Ma [18] and Ling et al. [19], the initialanisotropic tensor a0

ij are expressed through a constant A0 withthe initial stress states

M. Huang et al. / Computers and Geotechnics 38 (2011) 341–353 343

a0ij ¼ A0

s0ij

pc; s0

ij ¼ r0ij � pcdij ð14Þ

For the initial stress ratio K0 ¼ r03=r0

1, the components of thetensor are given as follows:

a011 ¼ 2k0

; a022 ¼ a0

33 ¼ �k0; a0

12 ¼ a023 ¼ a0

13 ¼ 0 ð15Þ

where

k0 ¼ A01� K0

1þ 2K0ð16Þ

For isotropic consolidated specimens K0 = 1.0, thus k0 = 0. Forthe K0-consolidated specimen, A0 = 0.65–1.0 (Liang and Ma [18]).

2.2. Hardening rules

The isotropic, rotational/anisotropic hardening rules, anddestructuration law are used to control the size, rotation and theprocess of destructuration of the bounding surface.

2.2.1. Isotropic hardeningIn line with the Cam-clay model, a volumetric hardening rule is

adopted. The internal variable pc is used to reflect the effect of pre-consolidation, which is independent of the bonding of soils andcontrols the size of the yield surface. pc is controlled only by theplastic volumetric strain rate _ep

v , given by

_pc ¼ pc _epv=ðk

� � j�Þ ð17Þ

where k� ¼ k=ð1þ e0Þwith k being the slope of the normal compres-sion line in the e-lnp space.

2.2.2. Rotational/anisotropic hardeningThe rotational rate of the bounding surface is controlled by the

evolution of the anisotropic tensor aij. We adopt a similar form tothe anisotropic/rotational law proposed by Wheeler et al. [31].

The proposed form of modified hardening law is

_aij ¼ lq3sij

4p� aij

� �� h _ep

vi þ bsij

3p� aij

� �� _ep

s

�� ð18Þ

where the parameter b controls the relative effectiveness of plasticshear strains _ep

s and plastic volumetric strains _epv in determining the

overall current target value for aij; and the soil constant l controlsthe absolute rate at which aij approaches its current target value[31] . An extra parameter q is introduced in order to control thechange rate of aij as the stress ratio g = q/p approaches the criticalstate value M. This suggests the following expression for q

q ¼ 1� gM

�� D Eð19Þ

Fig. 2. Mapping rule in bounding surface model.

2.2.3. Destructuration lawThe scalar variable r represents the progressive degradation of

soils, which controls the ratio between the sizes of structure sur-face and reference surface. According to Rouainia and Muir Wood[24], the scalar variable r is assumed to be a monotonicallydecreasing function of the plastic strain. The following exponentialdestructuration law is adopted

r ¼ 1þ ðr0 � 1Þ exp�kdep

d

k� � j�

� �ð20Þ

where r0 denotes the initial structure and kd is a parameter whichdescribes the rate of destructuration with strain. This equationtakes the main effect of damage by both plastic volumetric andplastic deviatoric strains into account through the plastic destruc-turation strain ep

d, which has the following form

epd ¼

Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� BÞ � ðdep

vÞ2 þ B � ðdeps Þ2

qð21Þ

where depv is plastic volumetric strain increment; dep

s is plastic shearstrain increment; B is a non-dimensional scaling parameter whichcontrols the relative contributions to damage of volumetric and dis-tortional plastic strain increments dep

v and deps . The form of Eq. (21)

suggests that for B = 1 the destructuration is entirely distortional,while for B = 0 the destructuration is entirely volumetric.

2.3. Mapping rule

In the proposed model, the projection center is fixed at the ori-gin of p–q stress space. And following the linear radial mappingrule (Dafalias and Herrmann [9]), for any actual stress point rij,there is a unique image stress point �rij on the bounding surfacecorresponding to the current stress point. The following relation-ships are used in relating current stress states to those at thebounding surface:

�rij ¼ brij; b ¼ d0

d0 � dð22Þ

where d and d0 denote respectively, the distance and the ultimatedistance between current stress point and image stress point.Fig. 2 shows the mapping rule in the proposed model.

2.4. Bounding plastic modulus

Experimental evidence from Shanghai soft clay and Korhonenand Lojander [17], suggests that the associated flow rule is a rea-sonable approximation for natural clay when combined with an in-clined yield curve. Thus the bounding surface function also servesas the plastic potential function. The plastic strain rate is deter-mined as

_epij ¼ h _/i

@F@�rmn

ð23Þ

where _/ is the plastic loading index, is defined as follows

_/ ¼ 1Hp

@F@�rij

_rij ¼1�Hp

@F@�rij

_�rij ð24Þ

Substituting the hardening rules and plastic loading index intothe consistency conditions, the bounding plastic modulus is ob-tained as

�Hp ¼ �@F@qn

_qn ¼ �@F@pc

_pc þ@F@aij

_aij þ@F@r

_r� �

ð25Þ

where qn denote a set of internal variables of material. The deriva-tives @F=@�rmn can be evaluated in terms of @F=@�I, @F=@�Ja and @F=@�ha.The details are presented in Appendix A of this paper. For details of


the derivatives of F with respect to hardening variables pc, aij and r,the readers is also referred to Appendix A.

The plastic modulus is related to the bounding plastic modulusthrough the following relationships [18,13]

Hp ¼ Hð�Hp; d;rij; qnÞ

¼ �Hp þ 1Pa@F@p

� �2

þ @F@q

� �2" #

d0

d0 � d

� �w

� 1

" #ð26Þ

Table 1Index properties of Shanghai soft clay.

Index property Value

Water content, w(%) 51.8Liquid limit, wL(%) 44.17Plastic limit, wP(%) 22.4Plastic index, IP 21.77Liquid index, IL 1.35Specific gravity, Gs 2.74Sensitivity, St 4.86Initial void ratio, e0 1.402Over consolidation ratio, OCR 1.0Coefficient of lateral pressure at rest, K0 0.6

w ¼ w0 exp �neps

� ð27Þ

where eps ¼

R_ep

s , which denotes the cumulative plastic deviatoricstrain. The function w is introduced to reflect the effect of strainhistory on the plastic modulus. As ep

s increases continuously withloading, the plastic modulus Hp decreases. Pa is the atmospherepressure, 1 and w0 and n are the model parameters. For d = 0, the ac-tual stress point rij will coincide with the image stress point �rij, sothe actual plastic modulus Hp will be equal to the bounding surfaceplastic modulus �Hp. It is emphasized that when the cumulative plas-tic deviator strain ep

s acquires very large values, the function w willapproach to zero, which may also result in Hp ¼ �Hp. Nevertheless,that does not mean the actual stress rij has to equal the image stress�rij, and there is nothing wrong with such an eventuality.

2.5. Model parameters

The proposed model requires 13 material parameters as well asthe initial stress state parameters (e0, pc, r0, A0). The parameters arerelated to critical state soil mechanics (k, j, Mc, Me, m), shape ofbounding surface (R), loss of structure (kd, B), evolution of anisot-ropy (l, b), and interpolation of plastic modulus (w0, n, 1).

The procedure for determining soil parameter values and initialvalues of the state variables for the proposed model is relativelystraightforward. The initial state parameter r0 defines the degreeof initial structure, which can be determined by comparing struc-tural yielding stress and pre-consolidation stress in a one-dimen-sional compression test on structural soft clays. Increasing r0

increases the initial degree of structure so that a higher structuralstiffness is reached. In case of r0 = 1.0, the sample of natural soil haslittle or nothing any initial structure, i.e., the structureless state isreceived. The initial anisotropic state parameter A0 is a model cal-ibration constant, and typically in the range of 0.65–1.0 [18].

k and j are determined from isotropic consolidation tests. Theymay also be obtained from the compression index Cc and swellingindex Cs of one-dimensional consolidation tests, wherek ¼ Cc=2:303 and j = Cs/2.303. Mc and Me are determined fromthe slope of the critical state line or indirectly from the angle ofinternal friction /. m may be specified as a constant.

Shape parameter R is a material parameter. It geometricallycontrols the extent of the tensile section in the stress space diago-nal (R P 2.0), which can be viewed as a parameter controlling theshape of the yield function. Larger values of R imply a flatter shapeof the yield function. Undrained stress path of the normal consol-idated soil may be used to obtain the value of shape parameter Ras mentioned by Ling et al. [19], where R = 2.0 is a typical value.

According to Rouainia and Muir Wood [24], the structuralparameter kd influences the rate of destructuration with strain. Ahigh value of kd can lead to very rapid loss of structure, whereasthis destructuration is much slower with a smaller value of kd.The structural parameter B controls the relative contributions todamage of volumetric and distortional plastic strain incrementsdep

v and deps , the range of the value of B is 0–1.0. The structural

parameters can be determined by comparing triaxial compressionand one-dimensional compression tests [24].

The anisotropic hardening parameter l controls the rate, atwhich aij tends towards its current target value. According to

Wheeler et al. [31], it is difficult to suggest a simple and directmethod for deriving the value of l for a given soil. They proposedto conduct model simulations with several different values of land then to compare these simulations with the observed behaviorto select the most appropriate value for the parameter l. The mostsuitable experimental tests would be ones involving significantrotation of the yield curve. In practice, performing suitable labora-tory tests and then undertaking model simulations with differentvalues of l for each deposit may not be feasible. Zentar et al.[34] suggested that the value of l for a particular soil will normallylie in the range 10/k to 15/k. With l in this range, the modelpredicts that an anisotropic natural soil must be subjected to anisotropic stress approximately three times larger than the yieldstress if the anisotropy of plastic behavior is to be erased (thismatches reported behavior for a number of clays). The modelparameter b defines the relative effectiveness of plastic shearstrains _ep

s and plastic volumetric strains _epv in rotating the yield

surface, which can be determined by the normally consolidatedK0 stress ratio and the critical stress ratio. Wheeler et al. [31]suggested the value of b/M between 0.5 and 1.0 for normally orlightly overconsolidated natural soft clays.

The parameters for stiffness interpolation w0, n and 1 are ob-tained by best fitting the experimental results. The parametersfor stiffness interpolation w0, n and 1 are obtained by best fittingthe experimental results.

3. Simulations of experiments on Shanghai soft clay

3.1. Summary of experiments

A programme of tests on samples of Shanghai soft clay wasundertaken to investigate the validity of the proposed model. Thetesting programme consisted of oedometer tests and triaxial tests.The important aims were to determine the initial shape and size ofthe structure surface, and to supply essential parameters for pro-posed model.

For the present study, undisturbed samples were taken atdepths of 10 m, with in situ horizontal consolidation stressr0hc = 41 kPa and vertical consolidation stress r0vc = 68.6 kPa. Theinitial mean effective stress pc was determined to be 50.3 kPa.Some physical properties of Shanghai soft clay at the depth ofinterest are presented in Table 1.

3.1.1. Consolidation characteristicsOne-dimensional consolidation characteristics from 24 h

oedometer tests are investigated in the present study. Fig. 3 showsthe results of the oedometer tests on undisturbed samples. Basedon the results, the consolidation yield stress (r0y) was determinedto be 110.5 kPa. The initial structural parameter, r0 = 1.61, is deter-mined approximately from r0y/r0vC . The compression index andswelling index (Cc and Cs) of one-dimensional consolidation testsare 0.489 and 0.107 respectively. k and j are obtained from Cc

and Cs where k ¼ Cc=2:303 and j = Cs/2.303.

1 10 100 1000 10000

σ'v(kPa)

0.4

0.6

0.8

1

1.2

1.4

1.6

eσ'vc=68.6 kPa

σ'y=110.5 kPae0=1.402

Fig. 3. Void ratio e-log r0v relationships in oedometer tests.

Table 2Test conditions for undrained triaxial tests on Shanghai soft clay.

Test number Horizontal and vertical reconsolidation stress r0r/r0a (kPa)

CIU-1 50/50CIU-2 100/100CIU-3 150/150CIU-4 200/200CIU-5 300/300CAU-1 41/68.6, p0 = 50CAU-2 81.8/136.4, p0 = 100CAU-3 245/408.3, p0 = 300

0 50 100 150 200 250 300 350

Mean effective stress: kPa

0

50

100

150

200

250

Dev

iato

r stre

ss: k

Pa

CIU-1CIU-2CIU-3CIU-4CIU-5CAU-1CAU-2CAU-3

M=1.277c'=0ϕ'=31.8o

CSL

K0 line

Fig. 4. Stress paths in undrained triaxial tests.

0 5 10 15 20 25Axial strain: %

0

20

40

60

80

100

120

140

Dev

iato

r stre

ss: k

Pa

CIU-1CIU-2CIU-3CIU-4

Fig. 5. Stress–strain curve of CIU tests.

0 5 10 15 20 25Axial strain: %

0

20

40

60

80

100

Dev

iato

r stre

ss: k

Pa

CAU-1CAU-2

Fig. 6. Stress–strain curve of CAU tests.


3.1.2. Yielding characteristicsIn this section, fundamental deformation and yielding charac-

teristics of Shanghai soft clay, such as strain softening, yield or lim-it surface, etc., are discussed based on the results of triaxial tests onthe Shanghai soft clay. The size of specimen was used: 39.1 mm indiameter and 80 mm long.

3.1.2.1. Consolidated undrained tests. Undrained triaxial tests underisotropic and anisotropic (K0 = 0.6) consolidation modes were per-formed on natural undisturbed samples. The initial horizontalreconsolidation stress r0r and vertical reconsolidation stress r0rare given in Table 2.

Figs. 4–6 show the stress paths and stress–strain relationshipsof the CIU (isotropically consolidated undrained) and CAU (aniso-tropically consolidated undrained) tests with a constant axialstrain rate.

In Fig. 4, with the progress of strain, it is observed that stresspaths reach their peak strength and finally approach a narrow zonein the stress space. This phenomenon shows that the critical stateconcept could be applied to natural clay at large strains. The slopeof critical state line, M, was determined to be 1.277, which corre-sponds to effective angle of internal friction /0 = 31.8.

Figs. 5 and 6 present stress–strain data from CIU and CAU testsof Shanghai soft clay. Strain softening is observed on the conditionthat mean effective stress is under yielding stress r0y, i.e., the stressdecreases with an increase of strain after the stress has reached itspeak. With the increase of mean effective stress, the relationship ofstress–strain presents gradually hardening character.

3.1.2.2. Consolidated drained tests. Several stress-controlled drainedtriaxial tests plus a group of undrained tests were carried out in or-der to investigate progressively the yielding characteristics ofShanghai soft clay. The principal features of the yielding test pro-gramme are summarized in Table 3 and Fig. 7.

All samples were reconsolidated to the in situ stress state alongpath that retraced their normal consolidated stress histories. PointA is p0, q = 50.3, 27.6 kPa. From point A the specimens were eithersheared undrained (SEU and SCU tests) or subjected to continuousdrained probing tests radiating from point A at a range of angles(x = tan�1Dq/Dp0)-type SCD and SED tests.

Defining yield is a useful approach to quantify deformationbehavior of clays within the context of elasto-plasticity. So theyielding phenomenon has often been discussed for natural soft clayby many researchers [32,12,27,4]. According to Smith et al. [27], ifan element of soil which located at a stable point in triaxial stress

Table 3Triaxial tests performed for yielding study on Shanghai soft clay.

Test number Comment

SCU Undrained test in compression, x = 90�SEU Undrained test in extension, x = �90�SCD0� Drained probing test, x = 0�SCD15� Drained probing test, x = 15�SCD29� Drained probing on K0-line from point A, x = 29�SCD50� Drained probing test, x = 50�SCD60� Drained probing test, x = 60�SCD72� Drained probing test, x = 72�SCD90� Drained probing test, x = 90�SED-15� Drained probing test, x = �15�SED-29� Drained probing test, x = �29�SED-56� Drained probing test, x = �56�

0 100 200 300 p'(kPa)

-100

0

100

200

q(kP

a)

SCD0o

SCD15o

SCD29o

SCD50oSCD60oSCD72oSCD90o

SED-15o

SED-29oSED-56o

A ω

SEU

SCU

Fig. 7. Standard consolidation stress paths.


space is loaded along a path such as that shown in Fig. 8, the stressspace within the initial bounding surface may be divided into threezones separated by yielding surface of different types termed asY1–Y3. The innermost zone bounded by the Y1 surface (i.e., Zone1), where strains are fully recoverable and particles remain lockedtogether, represents the ‘‘true’’ elastic region. In general, for softersoils, the size of Zone 1 is extremely small in stress space. Becauseof its limited size, it is difficult to map the Y1 surface. In Zone 2,which is enveloped by Y2 surface, soil behavior is characterizedby the rapid reduction of the tangent stiffness and hysteresis, withstiffness being highly dependent on the recent stress and strainhistory. The Zone 2 envelop can be mapped formally only by theperforming of a large number of drained stress cycles. In Zone 3,where particles start to move relative to one another, soil behavioris characterized by the hysteretic energy dissipation with irrecov-erable strains. Smith et al. [27] indicated that the Y3 surface con-

Y1

Y2

Zone 3

Zone 1

Zone 2

p

Bounding surfaceq

Y3

Fig. 8. Definition of yield surface [27].

ceptually coincides with the conventional yield surface, and thelarge-scale changes in soil structure are delayed until the stresspath reaches the Y3 surface, i.e., the proportion of plastic strain in-creases progressively as the initial structure surface is approached.Therefore, the bounding surface (structure surface) in Fig. 1, whichis related to Eq. (6), coincides with the Y3 surface. In view of singlestructural yield surface model proposed in this study, it is notedthat the determinations of Y1 and Y2 surface are beyond the scopeof this paper.

It is rather complicated to define the yield points in the stressspace. There are a number of approaches one can take to definethe stress state at yielding. The yield points determined by variousplots of the deviator stress q against the shear strain es and themean effective stress p0 against the volumetric strain ev, etc., how-ever, are not generally identical [30]. In addition, although theyield points should be determined traditionally by the onset ofthe development of plastic strain, it is rather difficult to dividethe strain precisely into elastic and plastic components. Takingthe yielding characters of structure into account in the present pa-per, therefore, the yield points are identified as the points wherethe total strain develops extensively. In practice, as shown inFig. 9, a yield is defined herein at the foot of a perpendicular, atwhich the intersection of rectilinear extrapolations of the pre-yieldand post-yield portions of the stress–strain curve verticals to curve.For each stress path, such points were identified both in the p0–evcurve and in the q–es curve. Then the average stress was taken asa yield point. Data from the drained probing experiments (SCDand SED) and undrained triaxial tests (SCU and SEU) on the Shang-hai soft clay are presented in Appendix B where graphs of p0 plottedagainst ev and q plotted against es represent a considerable amountof data.

Fig. 10 shows the yield points obtained from SCD, SED, SCU andSEU tests. The predicted limit state surface (structure surface) fromEq. (6) is also shown in Fig. 10. In drawing the yield curve, the valueof M has been taken as 1.277, as estimated from undrained shear-ing in triaxial compression. The value of r0 = 1.61, giving the initialstructure, was determined by one-dimensional consolidation test.And pc = 50.3 kPa for the initial stress state has been used. Theparameter A0 = 0.844 was determined from the experimental yieldpoints, which corresponds to a = 0.46, giving the inclination of theyield curve. Inspection of Fig. 10 shows that the yield curve expres-sion of Eq. (6) is a reasonable fit to the experimental data. Whenexamined in detail, being similar to many other clays [32,29], theshape of limit state surface which is approximately elliptical, isnot symmetrical with respect to the p0-axis. However, in the caseof Shanghai soft clay, the difference is that the limit state surfaceis not symmetrical as well with respect to the K0-line, just belowthe K0-line. This indicates that the clay is initially anisotropic andthat an anisotropic quasi-initial yield surface, not an isotropicone like that used in the original Cam-clay model, is necessaryfor the construction of an elasto-plastic constitutive model. At

Yield point

Strain

Stre

ss

Fig. 9. Definition of the yield point.

0 20 40 60 80 100 120

-60

-40

-20

0

20

40

60

80

P'(kPa)

q(kPa)K0 line

CSL

(p'0,q0)

α0 rpc

NCL

Fig. 10. Yielding surface and plastic flow direction of Shanghai soft clay.

-150 -100 -50 0 50 100 150

-40

-30

-20

-10

0

10

20

30

40

average value -0.13

deviation from normality (clockwise)

ω

Fig. 11. Relationship of plastic flow direction and stress path angle.


0

20

40

60

80

100

Dev

iato

r stre

ss: k

Pa

Experiment (CIU-1)Experiment (CIU-2)Model of this paperMCC model

CSL

Axial strain: %

0

20

40

60

80

100

Dev

iato

r stre

ss: k

PaExperiment (CIU-1)Experiment (CIU-2)Model of this paperMCC model

0 20 40 60 80 100 120

0 5 10 15 20 25

0 5 10 15 20 25Axial strain: %

0

20

40

60

80

100

Pore

pre

ssur

e: k

Pa

Experiment (CIU-1)Experiment (CIU-2)Model of this paperMCC model

(a)

(b)

(c)


the same time, the change in anisotropy can play an important rolein yielding of clays.

To verify the applicability of associated flow rule, the directionsof the plastic strain increment vectors at the appropriate yieldpoints were plotted in Fig. 10. The immediate impression is thatthese plastic strain increment vectors are roughly normal to theyield locus. Closer examination shows that the deviation from nor-mality does vary between ±20� with an average value of about�0.13� as seen in Fig. 11. This indicates the proposition of normal-ity is acceptable for Shanghai soft clay.

Fig. 12. Simulation of undrained triaxial compression tests on isotropicallyconsolidated clay. (a) Stress path; (b) stress–strain curve; (c) pore pressure-straincurve.

3.2. Model simulations

Triaxial loading tests performed both on isotropically and aniso-tropically (K0 = 0.6) compressed samples of Shanghai soft clay aresimulated with the proposed model and MCC model. The calibra-tion of material parameters was based on the results of isotropicallyconsolidated specimens, so that the behaviors of anisotropicallyconsolidated specimens were predicted. Table 4 shows the valueof model parameters for Shanghai soft clay. Two groups of

Table 4Model parameters for Shanghai soft clay.

Traditional Struct

k j Mc Me m R kd

0.212 0.046 1.277 0.9 0.2 2.0 0.65

undrained triaxial tests and a series of drained stress probe testshave been simulated using previous material parameters. Thesevalues e0 = 1.402, pc = 50.3 kPa, r0 = 1.61, A0 = 0.844 for the initialstress state have been used for all simulations. In addition, to verifythe effect of degradation of structure, the K0-consolidation test re-

ural Anisotropic Stiffness interpolation

B l b w0 n 1

0.5 50.0 0.7 10.0 1.5 3.0


0

20

40

60

80

100

Dev

iato

r stre

ss: k

Pa

Experiment (CAU-1)Experiment (CAU-2)Model of this paperMCC model

CSL

K0 line

Axial strain: %

0

20

40

60

80

100

Pore

pre

ssur

e: k

Pa


0 20 40 60 80 100 120

0 5 10 15 20 25

0 5 10 15 20 25Axial strain: %

0

20

40

60

80

100

Dev

iato

r stre

ss: k

Pa


(a)

(b)

(c)

Fig. 13. Simulation of undrained triaxial compression tests on anisotropicallyconsolidated clay. (a) Stress path; (b) stress–strain curve; (c) pore pressure-straincurve.

εv(%)

0

40

80

120

160

200

p'(k

Pa)

ExperimentModel of this paper

p'0

SCD0o

0 2 4 6 8 10 12 14 16

0 1 2 3 4

εs(%)

0

20

40

60

80

100

q(kP

a)ExperimentModel of this paper

q0

SCD0o

(a)

(b)

Fig. 14. Stress–strain curve of test SCD0� (a) p0–ev; (b) q–es.


sult for Shanghai soft clay is simulated respectively by the struc-tural model of this paper and non-structural model which the struc-tural mechanism is switched off.

Fig. 12a–c presents the undrained compression behavior ofShanghai soft clay with two different isotropic consolidation pres-sures of 50 and 100 kPa. Fig. 13a–c shows the comparison betweenthe results of two models and the experimental data for two un-drained compression tests on anisotropically reconsolidated speci-mens. The solid and dashed lines show the predicted results by theproposed model and MCC model, respectively. The open and closedpoints are the experimental results for the isotropic tests and aniso-tropic tests, respectively. Several conclusions were obtained asfollows:

(a) The general trend is well captured by the proposed model interms of stress path, deviatoric stress and excess pore pres-sure versus strain response. The predicted effective stresspaths converge towards ultimate remoulded undrainedstrengths on the critical state line. In tests on the isotropicconsolidation samples, the peaks of the stress–strain curvesare obtained after approximately 2% of axial strain. And thepeak shear stress occurs after approximately 1–2% axialstrain in the process of undrained compression tests onanisotropic consolidation specimens. In addition, the charac-ters of high stiffness and strain softening for structured clayare well reflected by the proposed model.

(b) In general, the results predicted by the MCC model were lesssatisfactory in tests on both the isotropic samples and theanisotropic samples, because the behaviors of anisotropyand structure are not estimated effectively. Under the rela-tively higher consolidation stress (p0 = 100 kPa) which is onthe verge of yield stress, however, the MCC model performsslightly better than under the lower consolidation stress(p0 = 50 kPa) because of the damage of structure. At the sametime, the prediction for isotropic tests is somewhat betterthan anisotropic tests.

(c) Though underpredicting the yield stress in both consolida-tion modes, the MCC model gives a relatively better predic-tion for pore pressure, especially under the low meaneffective stress.

Figs. 14–25 in Appendix B show the simulated results on rosetteof drained stress paths by the proposed model. It can be seen thatthe general quality of the simulations is good. Comparing the pre-dicted results between compression paths (SCD and SCU series)and extension paths (SED and SEU series), the former is better.

Fig. 26 shows the comparison results between K0-consolidationtest and structural model proposed by this paper. As shown in

εv(%)

0

40

80

120

160

200p'

(kPa

)


p'0

SCD15o

0 2 4 6 8 10 12 14 16

0 1 2 3 4 5 6 7 8εs(%)

0

20

40

60

80

100

q(kP

a)


q0

SCD15o

(a)

(b)


εv(%)

0

40

80

120

160

200

p'(k

Pa)


p'0

SCD29o

0 2 4 6 8 10 12 14 16

0 2 4 6 8 10 12

εs(%)

0

20

40

60

80

100

120

q(kP

a)


q0

SCD29o

(a)

(b)


εv(%)

0

40

80

120

160

200

p'(k

Pa)


p'0

SCD50o

0 2 4 6 8 10 12 14 16

0 2 4 6 8 10 12 14 16 18 20 22 24

εs(%)

0

40

80

120

160

200

q(kP

a)ExperimentModel of this paper

q0

SCD50o

(a)

(b)


εv(%)

0

10

20

30

40

50

60

70

80

p'(k

Pa)


p'0

SCD60o

0 1 2 3 4 5 6 7 8

0 2 4 6 8 10 12 14 16 18 20 22 24

εs(%)

0

20

40

60

80

100

120

q(kP

a)


q0

SCD60o

(a)

(b)



εv(%)

0

10

20

30

40

50

60

70

80

p'(k

Pa)


p'0

SCD72o

0 1 2 3 4 5 6 7 8

0 2 4 6 8 10 12 14 16 18 20 22 24

εs(%)

0

20

40

60

80

100

q(kP

a)


q0

SCD72o

(a)

(b)


εv(%)

0

10

20

30

40

50

60

70

80

p'(k

Pa)


p'0

SCD90o

0 1 2 3 4

0 2 4 6 8 10 12 14 16 18 20 22 24

εs(%)

0

20

40

60

80

q(kP

a)


q0

SCD90o

(a)

(b)


εv(%)

0

40

80

120

160

200

p'(k

Pa)


p'0

SED-15o

0 2 4 6 8 10 12 14 16

-8 -7 -6 -5 -4 -3 -2 -1 0 1

-60

-40

-20

0

20

40


εs(%)

q(kP

a)q0SED-15o

(a)

(b)

Fig. 21. Stress–strain curve of test SED-15� (a) p0–ev; (b) q–es.

εv(%)

0

40

80

120

160

200

p'(k

Pa)


p'0

SED-29o

0 2 4 6 8 10 12 14 16

-8 -7 -6 -5 -4 -3 -2 -1 0 1

-60

-40

-20

0

20

40


εs(%)

q(kP

a)

q0SED-29o

(a)

(b)



Fig. 26, the compression curve of K0-consolidation test can be wellinterpreted by two parts in the e-lgp plot, i.e., pre-yield state andpost-yield state. In the pre-yield state which refers to that the ap-plied stress level is less than the consolidation yield stress, the

εv(%)

0

40

80

120

160

200p'

(kPa

)


p'0

SED-56o

0 2 4 6 8 10 12 14 16

-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2

-200

-160

-120

-80

-40

0

40


εs(%)q(

kPa)

q0

SED-56o

(a)

(b)


0 2 4 6 8 10 12 14 16 18 20 22 24

εs(%)

0

20

40

60

80

q(kP

a)


q0

SCU

Fig. 24. q–es Curve of test SCU.

-24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

-80

-60

-40

-20

0

20

40


εs(%)

q(kP

a)

q0

SEU

Fig. 25. q–es Curve of test SEU.

1 10 100 1000

σ 'v(kPa)

0.4

0.6

0.8

1

1.2

1.4

1.6

e

ExperimentStructural modelNon-structural model

e0=1.402

Fig. 26. Simulation of K0-consolidation test on Shanghai clay.


mechanical behavior of soil is hardly unchanged because of theresistance of initial structure. When the applied load is beyondthe consolidation yield stress, the compressive behavior of soil

enters into post-yield state, in which a small increment of forcecan lead to greater change of void ratio in that the original struc-ture of soil is mostly destroyed. Though a little difference, thestructural effect was captured on the whole by the proposed mod-el. When the structural mechanism is switched, the predictedcurve presents straight line corresponding to the remouldedsample.

4. Conclusions

A simple model, based on the critical state concept and bound-ing surface plasticity, has been formulated to describe structureand plastic anisotropy of natural soft clay. The model consideredisotropic, rotational hardening and degradation of structure usinga total of 13 material parameters as well as the initial stress states.The anisotropic reference surface used here is proposed by Linget al. [19], which introduced the shape parameter of distorted el-lipse suggested originally by Dafalias [10]. Based on the anisotropicreference surface, a structural inner variable is introduced to de-scribe the structure of soft clay. With the process of destructur-ation, the structural parameter which is a monotonicallydecreased function, controls the contraction of structure/boundingsurface to the reference surface. When the structure of clays is fulldestroyed, the structure/bounding surface is the same with the ref-erence surface. The proposed form of bounding surface equationhas been validated by a substantial programme of stress probetests on Shanghai soft clay. As compared to the kinematic harden-ing model recently developed by Rouainia and Muir Wood [24], thepresent model has the advantage of being much simpler as a resultof removing the kinematic hardening yield surface. The compari-sons with undrained triaxial and drained triaxial stress path testresults of Shanghai soft clay under isotropic and anisotropic con-solidation modes, revealed the predictive capability of the pro-posed model.

Acknowledgements

This research is jointly supported by the National Natural Sci-ence Foundation of China through Grant No. 50778132 and the Na-tional Science Fund for Distinguished Young Scholars of Chinathrough Grant No. 50825803.

Appendix A

The purpose of this appendix is to provide detailed expressionsfor the normal to the bounding surface @F=@�rmn and derivatives of


bounding surface function F with respect to the hardeningvariables.

(1) Normal to the bounding surface

The normal to the bounding surface is given by

Lmn ¼@F@�rmn

¼ @F@�I

@�I@�rmn

þ @F@�Ja

@�Ja@�rmn

þ @F@�ha

@�ha

@�rmnð28Þ

where

I ¼ �rijdij ð29Þ

Ja ¼12

saijs

aij

� �12

ð30Þ

ha ¼13

sin�1 3ffiffiffi3p

2Sa

Ja

� �3" #

ð31Þ

Sa ¼13

saijs

ajksa

ik

� �13

ð32Þ

The reduced second stress invariant Ja and third stress invariantSa are defined in terms of the reduced deviatoric stress tensor sa

ij asfollows:

saij ¼ sij � rkkaij=3 ð33Þ

sij ¼ rij � rkkdij=3 ð34Þ

The normal to the bounding surface Lmn can be determined:

@F@�I¼ 1

3@F@�p¼ 2

3p� 1

Rrpc

� �ð35Þ

@F@�Ja¼

ffiffiffi3p @F

@�qa¼ 2

ffiffiffi3pðR� 1Þ2�qa

v ð36Þ

@F@�ha¼ @F@v

@v@M

@M@�ha

ð37Þ

@F@v ¼ �

ðR� 1Þ2�q2a

v2 ð38Þ

@v@M¼ 1

22aðR� 1Þ2 þM � aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4aðR� 1Þ2M þ ðM � aÞ2

q� �

þM � a2

1þ 2aðR� 1Þ2 þ ðM � aÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4aðR� 1Þ2M þ ðM � aÞ2

q264

375 ð39Þ

@M@�ha¼ 3M5ð1�m4Þ

8m4M4c

cos 3�ha ð40Þ

@�I@�rmn

¼ dmn ð41Þ

@�Ja@�rmn

¼ 12Ja

Samn �

13

dmnaijSaij

� �ð42Þ

@�ha

@�rmn¼ tan 3h

1Sa

@�Sa

@�rmn� 1

Ja

@�Ja@�rmn

� �ð43Þ

@�Sa

@�rmn¼ 1

3S2a

SamkSa

nk �13

2J2admn þ aijS

ajkSa

kidmn

�� ð44Þ

(2) Derivative of bounding surface function with respect to thehardening variables

Derivative of F with respect to pc:

@F@pc¼ �2

Rr½pþ rðR� 2Þpc� ð45Þ

Derivative of F with respect to r:

@F@r¼ �2

Rppc � 2rp2

c 1� 2R

� �ð46Þ

Derivative of F with respect to aij:

@F@aij¼ @F@v

@v@a

@a@aijþ @v@M

@M@�ha

@�ha

@aij

� �þ @F@�qa

@�qa

@aijð47Þ

where

@F@v ¼ �

ðR� 1Þ2�q2a

v2 ð48Þ

@v@a¼ �1

22aðR� 1Þ2 þM � aþ


q� �

þM � a2

2ðR� 1Þ2 � 1þ 2ðR� 1Þ2M � ðM � aÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4aðR� 1Þ2M þ ðM � aÞ2

q264

375 ð49Þ

@a@aij¼ 3aij

2að50Þ

@v@M¼ 1

2½2aðR� 1Þ2 þM � aþ


q�

þM � a2

1þ 2aðR� 1Þ2 þ ðM � aÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4aðR� 1Þ2M þ ðM � aÞ2

q264

375 ð51Þ

@M@�ha¼ 3M5ð1�m4Þ

8m4M4c

cos 3�ha ð52Þ

@�ha

@aij¼ @

�ha

@Ja

@Ja@amn

þ @�ha

@Sa

@Sa

@amnð53Þ

@�ha

@Ja¼ �3

ffiffiffi3p

2 cos 3hS3a

J4a

ð54Þ

@�ha

@Sa¼ 3

ffiffiffi3p

2 cos 3hS2a

J3a

ð55Þ

@Ja@amn

¼ � 12Ja

pSamn ð56Þ

@Sa

@amn¼ � 1

3S2a

pSamkSa

nk ð57Þ

@F@�qa

@�qa

@aij¼ @F@�Ja

@�Ja@aij

ð58Þ

@F@�Ja¼

ffiffiffi3p @F

@�qa¼ 2

ffiffiffi3pðR� 1Þ2�qa

v ð59Þ

@�Ja@aij¼ � 1

2�Ja�p�Sa

ij ð60Þ


Appendix B

Stress–strain curves of drained stress probing tests on Shanghaisoft clay (Figs. 14–25).

References

[1] Asaoka A, Nakano M, Noda T. Superloading yield surface concept for highlystructured soil behavior. Soils Found 2000;40(2):99–110.

[2] Nakano M, Nakai K, Noda T, Asaoka A. Simulation of shear and one-dimensional compression behavior of naturally deposited clays by super/subloading yield surface Cam-clay model. Soils Found 2005;45(1):141–51.

[3] Baudet B, Stallebrass S. A constitutive model for structured clays.Geotechnique 2004;54(4):269–78.

[4] Callisto L, Calabresi G. Mechanical behavior of a natural soft clay. Geotechnique1998;48(4):495–513.

[5] Callisto L, Gajo A, Muir Wood D. Simulation of triaxial and true triaxial tests onnatural and reconstituted Pisa clay. Geotechnique 2002;52(9):649–66.

[6] Crouch RS, Wolf JP, Dafalias YF. Unified critical bounding surface plasticitymodel for soil. J Eng Mech, ASCE 1994;120(11):2251–70.

[7] Dafalias YF. On cyclic and anisotropic plasticity. Thesis presented to theUniversity of California, at Berkeley, Calif., in partial fulfilment of therequirements for the degree of Doctor of Philosophy; 1975.

[8] Dafalias YF, Popov EP. Cyclic loading for materials with a vanishing elasticregion. Nucl Eng Des 1977;41:293–302.

[9] Dafalias YF, Herrmann LR. Bounding surface formulation of soil plasticity. In:Pande GN, Zienkiewicz OC, editors. Soil mechanics—transient and cyclicloads. New York: John Wiley & Sons; 1982. p. 253–82.

[10] Dafalias YF. An anisotropy critical state soil plasticity model. Mech ResCommun 1986;13:341–7.

[11] Gajo A, Muir Wood D. A new approach to anisotropic, bounding surfaceplasticity: general formulation and simulations of natural and reconstitutedclay behavior. Int J Numer Anal Methods Geomech 2001;23(3):207–41.

[12] Graham J, Noonan ML, Lew KV. Yield states and stress–strain relationships in anatural plastic clay. Can Geotech J 1983;20:502–16.

[13] Huang M, Wei X. An anisotropic bounding surface model for natural clays. In:Yin H, Li XS, Yeung AT, Desai CS, editors. Proceedings of international work-shop on constitutive modelling-development, implementation, evaluation andapplication, 12–13 January 2007, Hong Kong. Hong Kong: AdvancedTechnovation Limited; 2007. p. 317–26.

[14] Karstunen M, Koskinen M. Plastic anisotropy of soft reconstituted clays. CanGeotech J 2008;45:314–28.

[15] Kavvadas M, Amorosi A. A constitutive model for structured soils.Geotechnique 2000;50(3):263–73.

[16] Kobayashi I, Soga K, Iizuka A, et al. Numerical interpretation of a shape of yieldsurface obtained from stress probe tests. Soils Found 2003;43(3):95–103.

[17] Korhonen KH, Lojander M. Yielding of Perno clay. In: Proceedings of the 2ndinternational conference on constitutive laws for engineering materials. NY:Tucson, Ariz. Elsevier; 1987. p. 1249–55.

[18] Liang RY, Ma FG. Anisotropic plasticity model for undrained cyclic behavior ofclays. I: theory. J Geotech Eng, ASCE 1992;118(2):246–65.

[19] Ling HI, Yue D, Kaliakin VN, Themelis NJ. Anisotropic elastoplastic boundingsurface model for cohesive soils. J Eng Mech, ASCE 2002;129(7):748–58.

[20] Liu MD, Carter JP. A structured Cam clay model. Can Geotech J2002;39:1313–32.

[21] Mitchell RJ. On the yielding and mechanical strength of Leda clay. Can GeotechJ 1970;7:297–312.

[22] Ohta H, Wroth CP. Anisotropy and stress reorientation in clay under load. In:Proceeding of 2nd international conference on numerical methods ingeomechanics, Blacksburg 1; 1976. p. 319–28.

[23] Roscoe KH, Burland JB. On the generalized stress–strain behavior of ‘wet’ clay.In: Herman J, Leckie FA, editors. Engineering plasticity. Cambridge,UK: Cambridge University Press; 1968. p. 535–609.

[24] Rouainia M, Muir Wood D. A kinematic hardening constitutive model fornatural clays with loss of structure. Geotechnique 2000;50(2):153–64.

[25] Sheng D, Sloan SW, Yu HS. Aspects of finite element implementation of criticalstate models. Comput Mech 2000;26:185–96.

[26] Sivakumar V, Doran IG, Graham J, et al. The effect of anisotropic elasticity onthe yielding characteristics of overconsolidated natural clay. Can Geotech J2001;38:125–37.

[27] Smith PR, Jardine RJ, Hight DW. The yielding of Bothkennar clay. Geotechnique1992;42(2):257–74.

[28] Taiebat M, Dafalias YF, Peek R. A destructuration theory and its application tosaniclay model. Int J Numer Anal Methods Geomech 2010;34(10):1009–40.

[29] Tavenas F, Leroueil S. Effects of stresses and time on yield of clays. In:Proceedings of 9th ICSMFE, Tokyo; 1977. p. 319–26.

[30] Tavenas F, Leroueil S. Laboratory and its stress–strain-time behavior of softclays: A STATE-OF-THE-ART. In: Proceedings of the symposium on geotech.Engineering of soft soils, Mexico City, 1987.

[31] Wheeler SJ, Naatanen A, Karstunen M, Lojander M. An anisotropic elasto-plastic model for soft clays. Can Geotech J 2003;40(2):403–18.

[32] Wong PK, Mitchell RJ. Yielding and plastic flow of sensitive cemented clay.Geotechnique 1975;25(4):763–82.

[33] Zdravkovic L, Potts DM, Hight DW. The effect of strength anisotropy on thebehavior of embankments on soft clay. Geotechnique 2002;52(6):447–57.

[34] Zentar R, Karstunen M, Wiltafsky C, Schweiger HF, Koskinen M. Comparison oftwo approaches for modeling anisotropy of soft clays. In: Proceedings of the8th international symposium on numerical models in geomechanics (NUMOGVIII), Rome; 2002b. p. 115–21.

[35] Zhao J, Sheng D, Rouainia M, Sloan SW. Explicit stress integration for complexsoil models. Int J Numer Anal Methods Geomech 2005;29(12):1209–29.

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