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1 Computers and Geotechnics 8 (11) 41 5 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: 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 9, China b School of Highway, Chang an University, Xi an 7164, China c School of Engineering, University of Newcastle, NSW 8, Australia article info abstract Article history: Received 16 March 1 Received in revised form November 1 Accepted December 1 Available online January 11 Keywords: Soft clay Structure Anisotropy Yielding Bounding surface model 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 tests and drained stress probe tests under isotropic and anisotropic consolidation modes were performed on undisturbed samples of Shanghai soft clay to study the yielding characteristics. The degradation of the clay structure is modeled with an internal variable that allows the size of the bounding surface to decay with accumulated plastic strain. An anisotropic tensor and rotational hardening law are introduced to reflect 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 formulation with an associated flow rule. Validity of the model is verified by the undrained isotropic and anisotropic triaxial test and drained stress probe test results for Shanghai soft clay. The effects of stress anisotropy and loss of structure are well captured by the model. Ó 11 Elsevier Ltd. All rights reserved. 1. Introduction The Modified Cam-clay (MCC) model, which is based on the critical state theory, is one of the most widely used constitutive models for clay []. It was originally formulated for remolded clays under isotropic consolidation condition. Although the MCC model is widely used to represent the behavior of clayey soils, its prediction ability is not considered adequate for natural clay. This is because of the complicated properties such as anisotropy, structure and strain rate. Structure and anisotropy are the essential nature of naturally deposited soft clay, which have considerable influence on the strength and stress strain response of natural clays. Sometimes loading causes degradation of the initial structure, and this is particularly true in soft clays (e.g. Leda clay [1]; Bothkennar clay [7]). Neglecting the anisotropy of soil behavior may lead to highly inaccurate predictions of soil response under loading []. From engineering point of view, the last two decades or so have seen an increased trend of construction activities on soft soils and hence a quantitative model that can accurately predict the behavior of the soil is highly desirable. There are various approaches for the constitutive modelling of natural clays (e.g. [,6]). To model the destructuration of natural Corresponding author. Address: Department of Geotechnical Engineering, Tongji University, 19 Siping Road, Shanghai 9, China. Tel./fax: address: mshuang@tongji.edu.cn (M. Huang). clays, it is logical to start with a model that has had some success in predicting the behavior of remoulded material and then add to it some measure of structure and destructuration [4,1,5,,8]. After all, a structured soil can eventually become something like a remolded soil given sufficient loading and destructuration. Meanwhile, numerous constitutive models that account for plastic anisotropy of natural clays have been proposed, in which S-CLAY1 model proposed by Wheeler et al. [1] is a relatively simple elastoplastic anisotropic model. Most existing models in the literature (e.g. [6,,14]) account for either structure or anisotropy, but only few models consider both properties of natural clays (e.g. [16,]). The two properties can be related, but not always equivalent. Several researchers (e.g. [15,4,11]) developed constitutive models for natural soils within the framework of kinematic hardening, which consider simultaneously the anisotropic and structural effect on the mechanical behavior of soils. Those models can in general achieve good results but often at a cost of complexity. They often require special techniques to ensure that the current stress points are located on the inner yield surface at every integration step in the finite element implementation [5]. An alternative approach to avoid this complexity is to remove the kinematic hardening yield surface, only preserving the bounding surface [7,8]. Thus, the kinematic hardening yield surface is degenerated to a loading stress point, and the plastic modulus at the current stress point can be defined by a simple interpolation rule using values at the bounding surface. By means of vanishing pure elastic region, the classical kinematic hardening bounding surface model can be simplified into the single bounding surface model [9]. 66-5X/$ - see front matter Ó 11 Elsevier Ltd. All rights reserved. doi:1.116/j.compgeo.1.1.5

2 4 M. Huang et al. / Computers and Geotechnics 8 (11) 41 5 The main purpose of this paper is to investigate the yielding characteristics of Shanghai soft clay and to demonstrate that a simplified bounding surface model is sufficient to simulate both the anisotropic and structural properties of natural clays. al data from undrained triaxial tests and drained stress probe tests on Shanghai soft clay are presented to support the proposed model.. Model description The constitutive model, which is presented in this section, is developed in p q stress space, with p being the effective mean stress and q the deviatoric stress. Attention is restricted to rateindependent behavior and full saturation. Thus, the basic elastoplastic assumption is the additive decomposition of total strain rate _e ij, into elastic and plastic parts, _e e ij and _e p ij _e ij ¼ _e e ij þ _e p ij ð1þ The response associated with the elastic part is expressed in terms of the bulk and shear modulus, K and G, which are assumed to depend on the current mean stress p K ¼ p j ; ð1 mþ G ¼ ð1 þ mþ K where m is a constant Possion s ratio; j = j/(1 + e ); e is initial void ratio and j is the slope of the swelling line in e-lnp plane. The corresponding elastic incremental constitutive relation is given by _r ij ¼ D e ijkl _ e e kl where D e ijkl is elastic matrix, being in general a function of K and G. The plastic part _e p ij is developed within the framework of the critical state theory and the bounding surface plasticity. The formulation of the proposed model is given in detail in the following..1. Bounding surface Based on the experimental observations, an anisotropic reference surface, which is an inclined ellipse on the p q plane, is used to model the intrinsic behavior of the reconstituted soils. The anisotropic reference surface can describe the effect of initial anisotropy caused by one-dimensional deposition and K -consolidation process. A structure surface or bounding surface which has the same elliptical shape as the reference surface is adopted to describe the effect of the initial structure and control the process of destructuration. For simplicity, the loss of structure is assumed to affect merely the size of yielding surface. A scalar variable r called structural parameter is defined: r ¼ p c =p c where p c is the structural yielding stress; p c is the initial consolidation stress. Namely, r defines the ratio between the sizes of the structure surface and reference surface. The curves of the reference surface and structure surface are shown in Fig. 1. The value of r is always larger than or equal to 1, due to its physical meaning. For r = 1., the soil is completely destructured and the size of the structure surface is related only to p c, corresponding to the size of the reference surface. We consider that the yield surface for describing the behavior of the destructured or remolded soil, which is called the reference surface, is an anisotropic elliptical form. According to Ling et al. [19], the mathematical equation of the reference surface is defined by f ¼ðp p c Þ p þ R R p c þðr 1Þ q a ¼ ð5þ v ðþ ðþ ð4þ The structure surface, which can be thought of as a bounding surface, controls the process of destructuration. For simplicity, we consider that the structure surface has the same elliptical shape as the reference surface. The mathematical equation of structure surface is given by F ¼ðp p c Þ p þ R R p c where q a ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ij a ij = q a ¼ ffiffiffiffiffiffiffi p J a; Ja ¼ 1 sa ij sa ij s a ij ¼ s ij r kk a ij = þðr 1Þ q a v ¼ 1 v ¼ðM aþ½aðr 1Þ þ M a þ ð6þ ð7þ ð8þ ð9þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4aðR 1Þ M þðm aþ Š= ð1þ where a ij is the anisotropic tensor defining the anisotropy of clays; a defines the inclination of yield surface in p q stress space, which is the second invariant of anisotropic tensor; r ij is stress tensor; s ij is deviatoric stress tensor; s a ij is the reduced deviatoric stress tensor; q a is the reduced equivalent shear stress; J a is the reduced second stress invariant; R is the shape parameter, which controls the ratio of the two major axes of the yield surface; M is the slope of critical state line in triaxial space, which is defined by the reduced Lode angle h a as follows: " h a ¼ 1 p ffiffiffi # S a sin 1 ð11þ J a S a ¼ q 1 1 sa ij sa jk sa ik Bounding surface Reference surface α ð1þ m 4 1=4 M ¼ M c ð1þ ð1 þ m 4 Þ ð1 m 4 Þ sin h a where S a is the reduced third stress invariant; m is a material parameter defined as m = M e /M c in which M c and M e are the critical state stress ratios for triaxial compression and triaxial extension in p q stress space. According to Sheng et al. [5], the yield surface is convex provided m P.6, which coincides with the Mohr Coulomb hexagon at all vertices in the deviatoric plane. As defined in Liang and Ma [18] and Ling et al. [19], the initial anisotropic tensor a ij are expressed through a constant A with the initial stress states p c CSL rp c K line NCL Fig. 1. Reference surface and bounding surface of anisotropic model for structured clays. p

3 a ij ¼ A s ij ; s ij p ¼ r ij p cd ij ð14þ c For the initial stress ratio K ¼ r =r 1, the components of the tensor are given as follows: a 11 ¼ k ; a ¼ a ¼ k ; a 1 ¼ a ¼ a 1 ¼ where M. Huang et al. / Computers and Geotechnics 8 (11) ð15þ e p d ¼ Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 BÞðde p vþ þ B ðde p s Þ ð1þ where de p v is plastic volumetric strain increment; de p s is plastic shear strain increment; B is a non-dimensional scaling parameter which controls the relative contributions to damage of volumetric and distortional plastic strain increments de p v and de p s. The form of Eq. (1) suggests that for B = 1 the destructuration is entirely distortional, while for B = the destructuration is entirely volumetric. k ¼ A 1 K 1 þ K ð16þ For isotropic consolidated specimens K = 1., thus k =. For the K -consolidated specimen, A = (Liang and Ma [18])... Hardening rules The isotropic, rotational/anisotropic hardening rules, and destructuration law are used to control the size, rotation and the process of destructuration of the bounding surface...1. Isotropic hardening In line with the Cam-clay model, a volumetric hardening rule is adopted. The internal variable p c is used to reflect the effect of preconsolidation, which is independent of the bonding of soils and controls the size of the yield surface. p c is controlled only by the plastic volumetric strain rate _e p v, given by _p c ¼ p c _e p v =ðk j Þ ð17þ where k ¼ k=ð1 þ e Þ with k being the slope of the normal compression line in the e-lnp space.... Rotational/anisotropic hardening The rotational rate of the bounding surface is controlled by the evolution of the anisotropic tensor a ij. We adopt a similar form to the anisotropic/rotational law proposed by Wheeler et al. [1]. The proposed form of modified hardening law is _a ij ¼ lq s ij 4p a ij h_e p v iþb s ij p a ij _e p s ð18þ where the parameter b controls the relative effectiveness of plastic shear strains _e p s and plastic volumetric strains _e p v in determining the overall current target value for a ij ; and the soil constant l controls the absolute rate at which a ij approaches its current target value [1]. An extra parameter q is introduced in order to control the change rate of a ij as the stress ratio g = q/p approaches the critical state value M. This suggests the following expression for q D q ¼ 1 g E ð19þ M.. Mapping rule In the proposed model, the projection center is fixed at the origin of p q stress space. And following the linear radial mapping rule (Dafalias and Herrmann [9]), for any actual stress point r ij, there is a unique image stress point r ij on the bounding surface corresponding to the current stress point. The following relationships are used in relating current stress states to those at the bounding surface: r ij ¼ br ij ; b ¼ d d d ðþ where d and d denote respectively, the distance and the ultimate distance between current stress point and image stress point. Fig. shows the mapping rule in the proposed model..4. Bounding plastic modulus al evidence from Shanghai soft clay and Korhonen and Lojander [17], suggests that the associated flow rule is a reasonable approximation for natural clay when combined with an inclined yield curve. Thus the bounding surface function also serves as the plastic potential function. The plastic strain rate is determined as _e p ij ¼h_ r mn where _ / is the plastic loading index, is defined as follows _/ ¼ 1 H r ij _r ij ¼ 1 r ij _r ij ðþ ð4þ Substituting the hardening rules and plastic loading index into the consistency conditions, the bounding plastic modulus is obtained as H p n _q n ¼ _p c þ ij þ _r ð5þ where q n denote a set of internal variables of material. The derivatives =@ r mn can be evaluated in terms of =@ I, = and =@ h a. The details are presented in Appendix A of this paper. For details of... Destructuration law The scalar variable r represents the progressive degradation of soils, which controls the ratio between the sizes of structure surface and reference surface. According to Rouainia and Muir Wood [4], the scalar variable r is assumed to be a monotonically decreasing function of the plastic strain. The following exponential destructuration law is adopted r ¼ 1 þðr 1Þ exp k de p d k j ðþ where r denotes the initial structure and k d is a parameter which describes the rate of destructuration with strain. This equation takes the main effect of damage by both plastic volumetric and plastic deviatoric strains into account through the plastic destructuration strain e p d, which has the following form Fig.. Mapping rule in bounding surface model.

4 44 M. Huang et al. / Computers and Geotechnics 8 (11) 41 5 the derivatives of F with respect to hardening variables p c, a ij and r, the readers is also referred to Appendix A. The plastic modulus is related to the bounding plastic modulus through the following relationships [18,1] H p ¼ ^Hð H p ; d; r ij ; q n Þ " ¼ H p þ 1P a þ #" # w d d w ¼ w exp ne p s ð6þ ð7þ where e p s ¼ R _e p s, which denotes the cumulative plastic deviatoric strain. The function w is introduced to reflect the effect of strain history on the plastic modulus. As e p s increases continuously with loading, the plastic modulus H p decreases. P a is the atmosphere pressure, 1 and w and n are the model parameters. For d =, the actual stress point r ij will coincide with the image stress point r ij,so the actual plastic modulus H p will be equal to the bounding surface plastic modulus H p. It is emphasized that when the cumulative plastic deviator strain e p s acquires very large values, the function w will approach to zero, which may also result in H p ¼ H p. Nevertheless, that does not mean the actual stress r ij has to equal the image stress r ij, and there is nothing wrong with such an eventuality..5. Model parameters The proposed model requires 1 material parameters as well as the initial stress state parameters (e, p c, r, A ). The parameters are related to critical state soil mechanics (k, j, M c, M e, m), shape of bounding surface (R), loss of structure (k d, B), evolution of anisotropy (l, b), and interpolation of plastic modulus (w, n, 1). The procedure for determining soil parameter values and initial values of the state variables for the proposed model is relatively straightforward. The initial state parameter r defines the degree of initial structure, which can be determined by comparing structural yielding stress and pre-consolidation stress in a one-dimensional compression test on structural soft clays. Increasing r increases the initial degree of structure so that a higher structural stiffness is reached. In case of r = 1., the sample of natural soil has little or nothing any initial structure, i.e., the structureless state is received. The initial anisotropic state parameter A is a model calibration constant, and typically in the range of [18]. k and j are determined from isotropic consolidation tests. They may also be obtained from the compression index C c and swelling index C s of one-dimensional consolidation tests, where k ¼ C c =: and j = C s /.. M c and M e are determined from the slope of the critical state line or indirectly from the angle of internal friction /. m may be specified as a constant. Shape parameter R is a material parameter. It geometrically controls the extent of the tensile section in the stress space diagonal (R P.), which can be viewed as a parameter controlling the shape of the yield function. Larger values of R imply a flatter shape of the yield function. Undrained stress path of the normal consolidated soil may be used to obtain the value of shape parameter R as mentioned by Ling et al. [19], where R =. is a typical value. According to Rouainia and Muir Wood [4], the structural parameter k d influences the rate of destructuration with strain. A high value of k d can lead to very rapid loss of structure, whereas this destructuration is much slower with a smaller value of k d. The structural parameter B controls the relative contributions to damage of volumetric and distortional plastic strain increments de p v and de p s, the range of the value of B is 1.. The structural parameters can be determined by comparing triaxial compression and one-dimensional compression tests [4]. The anisotropic hardening parameter l controls the rate, at which a ij tends towards its current target value. According to Wheeler et al. [1], it is difficult to suggest a simple and direct method for deriving the value of l for a given soil. They proposed to conduct model simulations with several different values of l and then to compare these simulations with the observed behavior to select the most appropriate value for the parameter l. The most suitable experimental tests would be ones involving significant rotation of the yield curve. In practice, performing suitable laboratory tests and then undertaking model simulations with different values of l for each deposit may not be feasible. Zentar et al. [4] suggested that the value of l for a particular soil will normally lie in the range 1/k to 15/k. With l in this range, the model predicts that an anisotropic natural soil must be subjected to an isotropic stress approximately three times larger than the yield stress if the anisotropy of plastic behavior is to be erased (this matches reported behavior for a number of clays). The model parameter b defines the relative effectiveness of plastic shear strains _e p s and plastic volumetric strains _e p v in rotating the yield surface, which can be determined by the normally consolidated K stress ratio and the critical stress ratio. Wheeler et al. [1] suggested the value of b/m between.5 and 1. for normally or lightly overconsolidated natural soft clays. The parameters for stiffness interpolation w, n and 1 are obtained by best fitting the experimental results. The parameters for stiffness interpolation w, n and 1 are obtained by best fitting the experimental results.. Simulations of experiments on Shanghai soft clay.1. Summary of experiments A programme of tests on samples of Shanghai soft clay was undertaken to investigate the validity of the proposed model. The testing programme consisted of oedometer tests and triaxial tests. The important aims were to determine the initial shape and size of the structure surface, and to supply essential parameters for proposed model. For the present study, undisturbed samples were taken at depths of 1 m, with in situ horizontal consolidation stress r = 41 kpa and vertical consolidation stress hc r vc = 68.6 kpa. The initial mean effective stress p c was determined to be 5. kpa. Some physical properties of Shanghai soft clay at the depth of interest are presented in Table Consolidation characteristics One-dimensional consolidation characteristics from 4 h oedometer tests are investigated in the present study. Fig. shows the results of the oedometer tests on undisturbed samples. Based on the results, the consolidation yield stress (r y ) was determined to be 11.5 kpa. The initial structural parameter, r = 1.61, is determined approximately from r y /r vc. The compression index and swelling index (C c and C s ) of one-dimensional consolidation tests are.489 and.17 respectively. k and j are obtained from C c and C s where k ¼ C c =: and j = C s /.. Table 1 Index properties of Shanghai soft clay. Index property Value Water content, w 51.8 Liquid limit, w L Plastic limit, w P.4 Plastic index, I P 1.77 Liquid index, I L 1.5 Specific gravity, G s.74 Sensitivity, S t 4.86 Initial void ratio, e 1.4 Over consolidation ratio, OCR 1. Coefficient of lateral pressure at rest, K.6

5 M. Huang et al. / Computers and Geotechnics 8 (11) e e =1.4 σ' vc =68.6 kpa σ' y =11.5 kpa CIU-1 CIU- CIU- CIU-4 CIU-5 CAU-1 CAU- CAU- CSL K line M=1.77 c ' = ϕ ' =1.8 o σ' v (kpa) Fig.. Void ratio e-log r v relationships in oedometer tests. Table Test conditions for undrained triaxial tests on Shanghai soft clay. Test number Horizontal and vertical reconsolidation stress r r /r a (kpa) CIU-1 5/5 CIU- 1/1 CIU- 15/15 CIU-4 / CIU-5 / CAU-1 41/68.6, p =5 CAU- 81.8/16.4, p = 1 CAU- 45/48., p =.1.. Yielding characteristics In this section, fundamental deformation and yielding characteristics of Shanghai soft clay, such as strain softening, yield or limit surface, etc., are discussed based on the results of triaxial tests on the Shanghai soft clay. The size of specimen was used: 9.1 mm in diameter and 8 mm long Consolidated undrained tests. Undrained triaxial tests under isotropic and anisotropic (K =.6) consolidation modes were performed on natural undisturbed samples. The initial horizontal reconsolidation stress r r and vertical reconsolidation stress r r are given in Table. Figs. 4 6 show the stress paths and stress strain relationships of the CIU (isotropically consolidated undrained) and CAU (anisotropically consolidated undrained) tests with a constant axial strain rate. In Fig. 4, with the progress of strain, it is observed that stress paths reach their peak strength and finally approach a narrow zone in the stress space. This phenomenon shows that the critical state concept could be applied to natural clay at large strains. The slope of critical state line, M, was determined to be 1.77, which corresponds to effective angle of internal friction / = 1.8. Figs. 5 and 6 present stress strain data from CIU and CAU tests of Shanghai soft clay. Strain softening is observed on the condition that mean effective stress is under yielding stress r y, i.e., the stress decreases with an increase of strain after the stress has reached its peak. With the increase of mean effective stress, the relationship of stress strain presents gradually hardening character Consolidated drained tests. Several stress-controlled drained triaxial tests plus a group of undrained tests were carried out in order to investigate progressively the yielding characteristics of Shanghai soft clay. The principal features of the yielding test programme are summarized in Table and Fig Mean effective stress: kpa Fig. 4. Stress paths in undrained triaxial tests. CIU-1 CIU- CIU- CIU Axial strain: % Fig. 5. Stress strain curve of CIU tests. CAU-1 CAU Axial strain: % Fig. 6. Stress strain curve of CAU tests. All samples were reconsolidated to the in situ stress state along path that retraced their normal consolidated stress histories. Point Aisp, q = 5., 7.6 kpa. From point A the specimens were either sheared undrained (SEU and SCU tests) or subjected to continuous drained probing tests radiating from point A at a range of angles (x = tan 1 Dq/Dp )-type SCD and SED tests. Defining yield is a useful approach to quantify deformation behavior of clays within the context of elasto-plasticity. So the yielding phenomenon has often been discussed for natural soft clay by many researchers [,1,7,4]. According to Smith et al. [7], if an element of soil which located at a stable point in triaxial stress

6 46 M. Huang et al. / Computers and Geotechnics 8 (11) 41 5 Table Triaxial tests performed for yielding study on Shanghai soft clay. Test number Comment SCU Undrained test in compression, x = 9 SEU Undrained test in extension, x = 9 SCD Drained probing test, x = SCD15 Drained probing test, x = 15 SCD9 Drained probing on K -line from point A, x =9 SCD5 Drained probing test, x = 5 SCD6 Drained probing test, x = 6 SCD7 Drained probing test, x = 7 SCD9 Drained probing test, x = 9 SED-15 Drained probing test, x = 15 SED-9 Drained probing test, x = 9 SED-56 Drained probing test, x = SCD9 o SCU A SCD7 o SCD6 o SCD5 o SCD9 o SCD15 o SCD o 1 SED-15 o SEU ω SED-56 o Fig. 7. Standard consolidation stress paths. SED-9 o q Bounding surface space is loaded along a path such as that shown in Fig. 8, the stress space within the initial bounding surface may be divided into three zones separated by yielding surface of different types termed as Y 1 Y. The innermost zone bounded by the Y 1 surface (i.e., Zone 1), where strains are fully recoverable and particles remain locked together, represents the true elastic region. In general, for softer soils, the size of Zone 1 is extremely small in stress space. Because of its limited size, it is difficult to map the Y 1 surface. In Zone, which is enveloped by Y surface, soil behavior is characterized by the rapid reduction of the tangent stiffness and hysteresis, with stiffness being highly dependent on the recent stress and strain history. The Zone envelop can be mapped formally only by the performing of a large number of drained stress cycles. In Zone, where particles start to move relative to one another, soil behavior is characterized by the hysteretic energy dissipation with irrecoverable strains. Smith et al. [7] indicated that the Y surface conceptually coincides with the conventional yield surface, and the large-scale changes in soil structure are delayed until the stress path reaches the Y surface, i.e., the proportion of plastic strain increases progressively as the initial structure surface is approached. Therefore, the bounding surface (structure surface) in Fig. 1, which is related to Eq. (6), coincides with the Y surface. In view of single structural yield surface model proposed in this study, it is noted that the determinations of Y 1 and Y surface are beyond the scope of this paper. It is rather complicated to define the yield points in the stress space. There are a number of approaches one can take to define the stress state at yielding. The yield points determined by various plots of the deviator stress q against the shear strain e s and the mean effective stress p against the volumetric strain ev, etc., however, are not generally identical []. In addition, although the yield points should be determined traditionally by the onset of the development of plastic strain, it is rather difficult to divide the strain precisely into elastic and plastic components. Taking the yielding characters of structure into account in the present paper, therefore, the yield points are identified as the points where the total strain develops extensively. In practice, as shown in Fig. 9, a yield is defined herein at the foot of a perpendicular, at which the intersection of rectilinear extrapolations of the pre-yield and post-yield portions of the stress strain curve verticals to curve. For each stress path, such points were identified both in the p ev curve and in the q e s curve. Then the average stress was taken as a yield point. Data from the drained probing experiments (SCD and SED) and undrained triaxial tests (SCU and SEU) on the Shanghai soft clay are presented in Appendix B where graphs of p plotted against ev and q plotted against e s represent a considerable amount of data. Fig. 1 shows the yield points obtained from SCD, SED, SCU and SEU tests. The predicted limit state surface (structure surface) from Eq. (6) is also shown in Fig. 1. In drawing the yield curve, the value of M has been taken as 1.77, as estimated from undrained shearing in triaxial compression. The value of r = 1.61, giving the initial structure, was determined by one-dimensional consolidation test. And p c = 5. kpa for the initial stress state has been used. The parameter A =.844 was determined from the experimental yield points, which corresponds to a =.46, giving the inclination of the yield curve. Inspection of Fig. 1 shows that the yield curve expression of Eq. (6) is a reasonable fit to the experimental data. When examined in detail, being similar to many other clays [,9], the shape of limit state surface which is approximately elliptical, is not symmetrical with respect to the p -axis. However, in the case of Shanghai soft clay, the difference is that the limit state surface is not symmetrical as well with respect to the K -line, just below the K -line. This indicates that the clay is initially anisotropic and that an anisotropic quasi-initial yield surface, not an isotropic one like that used in the original Cam-clay model, is necessary for the construction of an elasto-plastic constitutive model. At Zone Y Y Stress Zone 1 Y 1 Yield point Zone p Strain Fig. 8. Definition of yield surface [7]. Fig. 9. Definition of the yield point.

7 M. Huang et al. / Computers and Geotechnics 8 (11) CSL K line 6 NCL 4 (p ',q ) α P'(kPa) rp c Fig. 1. Yielding surface and plastic flow direction of Shanghai soft clay Mean effective stress: kpa 1 CSL (CIU-1) (CIU-) MCC model 1 average value ω the same time, the change in anisotropy can play an important role in yielding of clays. To verify the applicability of associated flow rule, the directions of the plastic strain increment vectors at the appropriate yield points were plotted in Fig. 1. The immediate impression is that these plastic strain increment vectors are roughly normal to the yield locus. Closer examination shows that the deviation from normality does vary between ± with an average value of about.1 as seen in Fig. 11. This indicates the proposition of normality is acceptable for Shanghai soft clay... Model simulations deviation from normality (clockwise) Fig. 11. Relationship of plastic flow direction and stress path angle. Triaxial loading tests performed both on isotropically and anisotropically (K =.6) compressed samples of Shanghai soft clay are simulated with the proposed model and MCC model. The calibration of material parameters was based on the results of isotropically consolidated specimens, so that the behaviors of anisotropically consolidated specimens were predicted. Table 4 shows the value of model parameters for Shanghai soft clay. Two groups of (c) Pore pressure: kpa (CIU-1) (CIU-) MCC model Axial strain: % (CIU-1) (CIU-) MCC model Axial strain: % Fig. 1. Simulation of undrained triaxial compression tests on isotropically consolidated clay. Stress path; stress strain curve; (c) pore pressure-strain curve. undrained triaxial tests and a series of drained stress probe tests have been simulated using previous material parameters. These values e = 1.4, p c = 5. kpa, r = 1.61, A =.844 for the initial stress state have been used for all simulations. In addition, to verify the effect of degradation of structure, the K -consolidation test re- Table 4 Model parameters for Shanghai soft clay. Traditional Structural Anisotropic Stiffness interpolation k j M c M e m R k d B l b w n

8 48 M. Huang et al. / Computers and Geotechnics 8 (11) CSL SCD o K line (CAU-1) (CAU-) MCC model Mean effective stress: kpa p ' SCD o Axial strain: % (CAU-1) (CAU-) MCC model 6 4 q 1 4 Fig. 14. Stress strain curve of test SCD p ev; q e s. (c) Pore pressure: kpa Axial strain: % (CAU-1) (CAU-) MCC model Fig. 1. Simulation of undrained triaxial compression tests on anisotropically consolidated clay. Stress path; stress strain curve; (c) pore pressure-strain curve. sult for Shanghai soft clay is simulated respectively by the structural model of this paper and non-structural model which the structural mechanism is switched off. Fig. 1a c presents the undrained compression behavior of Shanghai soft clay with two different isotropic consolidation pressures of 5 and 1 kpa. Fig. 1a c shows the comparison between the results of two models and the experimental data for two undrained compression tests on anisotropically reconsolidated specimens. The solid and dashed lines show the predicted results by the proposed model and MCC model, respectively. The open and closed points are the experimental results for the isotropic tests and anisotropic tests, respectively. Several conclusions were obtained as follows: The general trend is well captured by the proposed model in terms of stress path, deviatoric stress and excess pore pressure versus strain response. The predicted effective stress paths converge towards ultimate remoulded undrained strengths on the critical state line. In tests on the isotropic consolidation samples, the peaks of the stress strain curves are obtained after approximately % of axial strain. And the peak shear stress occurs after approximately 1 % axial strain in the process of undrained compression tests on anisotropic consolidation specimens. In addition, the characters of high stiffness and strain softening for structured clay are well reflected by the proposed model. In general, the results predicted by the MCC model were less satisfactory in tests on both the isotropic samples and the anisotropic samples, because the behaviors of anisotropy and structure are not estimated effectively. Under the relatively higher consolidation stress (p = 1 kpa) which is on the verge of yield stress, however, the MCC model performs slightly better than under the lower consolidation stress (p = 5 kpa) because of the damage of structure. At the same time, the prediction for isotropic tests is somewhat better than anisotropic tests. (c) Though underpredicting the yield stress in both consolidation modes, the MCC model gives a relatively better prediction for pore pressure, especially under the low mean effective stress. Figs in Appendix B show the simulated results on rosette of drained stress paths by the proposed model. It can be seen that the general quality of the simulations is good. Comparing the predicted results between compression paths (SCD and SCU series) and extension paths (SED and SEU series), the former is better. Fig. 6 shows the comparison results between K -consolidation test and structural model proposed by this paper. As shown in

9 M. Huang et al. / Computers and Geotechnics 8 (11) SCD15 o SCD5 o p ' p ' SCD15 o SCD5 o q q Fig. 15. Stress strain curve of test SCD15 p ev; q e s. Fig. 17. Stress strain curve of test SCD5 p ev; q e s. 16 SCD9 o SCD6 o p ' 4 p ' SCD9 o 1 SCD6 o q q Fig. 16. Stress strain curve of test SCD9 p ev; q e s. Fig. 18. Stress strain curve of test SCD6 p ev; q e s.

10 5 M. Huang et al. / Computers and Geotechnics 8 (11) SCD7 o 16 SED-15 o 5 4 p ' p ' SCD7 o SED-15 o 4 q q Fig. 1. Stress strain curve of test SED-15 p ev; q e s. Fig. 19. Stress strain curve of test SCD7 p ev; q e s. SED-9 o p ' SCD9 o p ' SED-9 o 4 q 8 SCD9 o q Fig.. Stress strain curve of test SCD9 p ev; q e s. Fig. 6, the compression curve of K -consolidation test can be well interpreted by two parts in the e-lgp plot, i.e., pre-yield state and post-yield state. In the pre-yield state which refers to that the applied stress level is less than the consolidation yield stress, the -4-6 Fig.. Stress strain curve of test SED-9 p ev; q e s.

11 M. Huang et al. / Computers and Geotechnics 8 (11) SED-56 o e = e 1. 1 p ' SED-56 o 4 q Fig.. Stress strain curve of test SED-56 p ev; q e s. enters into post-yield state, in which a small increment of force can lead to greater change of void ratio in that the original structure of soil is mostly destroyed. Though a little difference, the structural effect was captured on the whole by the proposed model. When the structural mechanism is switched, the predicted curve presents straight line corresponding to the remoulded sample. 4. Conclusions.8.6 Structural model Non-structural model σ' v (kpa) Fig. 6. Simulation of K -consolidation test on Shanghai clay q SEU Fig. 4. q e s Curve of test SCU. mechanical behavior of soil is hardly unchanged because of the resistance of initial structure. When the applied load is beyond the consolidation yield stress, the compressive behavior of soil SCU Fig. 5. q e s Curve of test SEU. 4 q A simple model, based on the critical state concept and bounding surface plasticity, has been formulated to describe structure and plastic anisotropy of natural soft clay. The model considered isotropic, rotational hardening and degradation of structure using a total of 1 material parameters as well as the initial stress states. The anisotropic reference surface used here is proposed by Ling et al. [19], which introduced the shape parameter of distorted ellipse suggested originally by Dafalias [1]. Based on the anisotropic reference surface, a structural inner variable is introduced to describe the structure of soft clay. With the process of destructuration, the structural parameter which is a monotonically decreased function, controls the contraction of structure/bounding surface to the reference surface. When the structure of clays is full destroyed, the structure/bounding surface is the same with the reference surface. The proposed form of bounding surface equation has been validated by a substantial programme of stress probe tests on Shanghai soft clay. As compared to the kinematic hardening model recently developed by Rouainia and Muir Wood [4], the present model has the advantage of being much simpler as a result of removing the kinematic hardening yield surface. The comparisons with undrained triaxial and drained triaxial stress path test results of Shanghai soft clay under isotropic and anisotropic consolidation modes, revealed the predictive capability of the proposed model. Acknowledgements This research is jointly supported by the National Natural Science Foundation of China through Grant No and the National Science Fund for Distinguished Young Scholars of China through Grant No Appendix A The purpose of this appendix is to provide detailed expressions for the normal to the bounding surface =@ r mn and derivatives of

12 5 M. Huang et al. / Computers and Geotechnics 8 (11) 41 5 bounding surface function F with respect to the hardening variables. (1) Normal to the bounding surface The normal to the bounding surface is given by L mn ¼ h r r mn where I ¼ r ij d ij J a I r mn r h a ð8þ ð9þ 1 1 sa ij sa ij ðþ " h a ¼ 1 p ffiffiffi # S a sin 1 J a S a ¼ 1 1 sa ij sa jk sa ik ð1þ ðþ The reduced second stress invariant J a and third stress invariant S a are defined in terms of the reduced deviatoric stress tensor s a ij as follows: s a ij ¼ s ij r kk a ij = s ij ¼ r ij r kk d ij = ðþ ð4þ The normal to the bounding surface L mn can be I ¼ ¼ p 1 R rp c ð5þ ¼ p ffiffiffi a pffiffiffi ðr 1Þ q a v ð6þ () Derivative of bounding surface function with respect to the hardening variables Derivative of F with respect to p c c ¼ R r½p þ rðr Þp cš Derivative of F with respect to ¼ R pp c rp c 1 R Derivative of F with respect to a ij : ¼ ðr 1Þ q a h ij h ij ¼ 1 aðr 1Þ þ M a þ þ M ¼ a ij a ij q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4aðR 1Þ M þðm aþ ð45þ ð46þ ð47þ ð48þ 6 ðr 1Þ 1 þ ðr 1Þ M ðm aþ 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 ð49þ 4aðR 1Þ M þðm ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½aðR 1Þ þ M a þ 4aðR 1Þ M þðm aþ Š þ M a 6 1 þ aðr 1Þ þðm aþ 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 ð51þ 4aðR 1Þ M þðm ¼ @ h ¼ M5 ð1 m 4 Þ cos h h a 8m 4 M 4 a c ¼ ðr 1Þ q a ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðr 1Þ þ M a þ 4aðR 1Þ M þðm aþ þ M a 6 1 þ aðr 1Þ þðm aþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9þ 4aðR 1Þ M þðm h ij h mn h h a ¼ pffiffiffi S a cos h J 4 h a ¼ pffiffiffi S a cos h J mn ð5þ ¼ M5 ð1 m 4 Þ cos h h a 8m 4 M 4 a c a ¼ 1 a mn mn J a r mn ¼ d mn ¼ 1 r a mn mn J 1 a d mna ij S a h r mn ¼ tan h 1 S S r mn 1 J J r S a ¼ 1 r a mn S mk Sa nk 1 J a d mn þ a ij S a jk Sa ki d mn a ð41þ ð4þ ð4þ a ¼ 1 a mn S mk Sa nk ij ij ¼ p ffiffiffi a pffiffiffi ðr 1Þ q a v ð57þ ð58þ ð59þ ¼ ij p S a ij ð6þ J a

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