This paper presents a hypoplastic constitutive model for overconsolidated clays. The model is developed based on a basic model proposed recently for sand. New density and sti ness factors are introduced to account for history dependence. The model is capable of predicting a desirable asymptotic state boundary surface for overconsolidated clays. Validation with experimental results shows this model can properly describe the main features of overconsolidated clays. Keywords: Hypoplastic model, Critical state, Overconsolidation, Response envelope, Asymptotic state boundary surface 1. Introduction Constitutive modeling of overconsolidated (OC) clays has a long history, pioneered by the Modi ed Cam-clay (MCC) model based on the critical state concept [32, 33]. The MCC model is capable of describing the behavior of normally consolidated and lightly OC clays. The capacity of the model for OC clays has been largely enhanced by intro- ducing various concepts, such as bounding surface [6] and subloading surface [12]. These theories can be integrated with various dilatancy relations to form constitutive models for overconsolidated soils [7, 47,48,49]. An inspection of the relevant literature reveals, however, that most of the existing models for OC clays are based on elastoplasticity. Hypoplastic models are developed as an alternative approach to the prevailing elasto- plastic models in describing the overconsolidation behavior. Following the pioneer work by Kolymbas and Wu [17, 18,40,41], di erent versions of hypoplastic models were de- veloped for sand [4, 9,37] . These models served as the basis for the model development of clay. The rst attempt to extend hypoplastic model for clay was made by Niemunis [26,27], who combined the critical state concept with hypoplasticity to describe the behavior of OC clays. Meanwhile, Herle and Kolymbas [14] modi ed the model by von Wol erdor [37] (VW model) for soils with low friction angle. Based on these works, Ma s n [21] proposed the rst hypoplastic model for clay (DM model), which employs a density factor to describe the consolidation history. Corresponding author. E-mail: wei.wu@boku.ac.at Preprint submitted to Computers and Geotechnics March 4, 2021 However, later Ma s n himself found that the DM model gives rise to an unrealistic asymptotic state boundary surface (ASBS) when a certain combination of parameters is used [23]. The reason for this de ciency is that the adopted density factor is not only dependent on the consolidation history but also related to other material parameters. To address this issue, Ma s n [22, 23] proposed a new form of hypoplastic models with explicit ASBS. Nevertheless, the original form is still valuable for further developments and enhancements. For instance, Wu et al. [45] reported a basic hypoplastic model for sand. This model employs the basic form of previous hypoplastic models but additionally includes a new tensorial term, which provides an opportunity to re ne the modelling of the dilatant behaviour and asymptotic state. In this work, we will extend the basic model [45] for OC clays. In the following, we rst inspect the limitation of the basic model. Next, a detail process for developing the model for OC clays is presented. A new density factor is introduced to account for consolidation history. The employment of the density factor allows the model to predict reasonable ASBSs for OC clays. The characteristics of the model are then investigated by a parametric study. At the end, model performance is demonstrated by simulating several element tests, and comparing the prediction with experiments on di erent OC clays. 2. Theory of hypoplastic constitutive model 2.1. General framework To gain perspective, we start with the rate-form constitutive equation by Wu and Kolymbas [41], which has the following general form: T=L:D+NkDk (1) whereLandNare fourth- and second-order tensors, respectively. Dis the strain rate (stretching) tensor. kDk =p tr(D2) stands for the norm of the strain rate tensor. Tis the Jaumann stress rate de ned by: T=_T+TWWT (2) where _Tis the material time-derivative of the Cauchy stress tensor and Wthe spin tensor. Bear in mind that the failure criterion and ow rule of the model (1) is the interac- tional outcome of LandN. This implies the intensity and direction of ow are mutually dependent. To remedy this shortcoming, Niemunis [26] proposed a simple rearrangement of the constitutive framework (1) by introducing a tensor function B: B=Ym kmk;andN=L:B (3) where the tensor Bincludes both the ow direction and intensity of the ow by the functionYand a tensor m, respectively. 3 With the above rearrangement, a generalized form of hypoplastic models is obtained as follows: T=fsL: DfdYm kmkkDk (4) wherefsandfdare the sti ness and density factors, accounting for the e ects of stress level and density, respectively [21, 26]. 2.2. A basic hypoplastic model To develop a new constitutive model for OC clays, we start by considering the fol- lowing basic model proposed by Wu et al. [45]: T=C1(trT)D+C2(trD)T+C3tr(TD) trTT+C4(T+T)kDk (5) in whichCi(i= 1;2;3;4) are material constants. Tis the deviatoric stress tensor de ned as T=TI=3trTwithIbeing the unit tensor. The material constants can be related to some well established parameters in soil mechanics, such as the initial tangential modulus Ei, the initial Poisson's ratio i, the critical friction angle c, and dilatancy angle . Moreover, for critical state with = 0, the model (5) can be further rearranged as the following form: T=fsh D+c2(trD)^T+a2tr(^TD)^T+a(^T+^T)kDki (6) where ^Tis the normalized stress de ned as ^T=T=tr(T); the multiplier c2permits a re ned modelling of the dilatancy behavior. The constant aaccounts for the critical state through: a=p 3(3sin c) 2p 2sin c(7) Note that the equation (6) is a special case of equation (5) with = 0. This basic model has several merits such as simplicity and robustness in numerical simulation. These advantages make it an interesting stand-alone model for many applications[30, 31,36,38,46] and further developments [19, 20,39]. 2.3. Limitation of the basic model for OC clays The performance of the basic model is well documented in the literature [45]. The model is capable of modeling the basic behavior of normally consolidated clay, such as failure and contractive deformation. However, there exist some shortcomings for modeling OC clays. The most important ones are summarized as follows: The consolidation history is not incorporated in this model, thus it cannot describe strain softening and volume dilatancy of OC clays. This model does not include the swelling index . Therefore, the directional sti ness for loading and unloading is not adjustable. 4 This model has an constant sti ness factor fs. It is adequate to describe the initial shear modulus of normally consolidated soil but it will underestimate signi cantly the initial shear moduli of OC clays. Note that the VW model and DM model have similar forms with model (6) except that the second term tr( D)^Tis not included. These two models address some of the above mentioned shortcomings. However, the VW model is seriously awed because the underlying basic equation does not perform properly [45]. The DM model also su ers from some shortcomings [23]. 3. Proposed model for OC clays To address the aforementioned shortcomings of the basic model (6), we proceed to extend it by considering some salient features of OC clays. 3.1. Model framework We rst consider the linear behavior of the model (6) by manipulating the linear terms. To doing so, the model (6) can be expressed as follows: T=c1L1+c2L2+c3L3+N (8) whereLi(i= 1;2;3) are the linear tensorial function in equation (6) with ci(i= 1;2;3) being the material constants. The e ects of the linear terms can be best appreciated by considering the response envelope. For a rosette of strain rates with a constant magnitude, the response envelope at a given initial stress state T0is the surface spanned by all stresses T=T0+T [10]. Figure 1shows some response envelopes depicted on the Rendulic plane with T22=T33. the response envelope of the model (6) represents an ellipse. Obviously, the multiplier c1 scales the magnitude of D, and thus can expand or shrink the response envelopes. The multipliers c2andc3have similar e ects, that is increase c2andc3gives rise to longer major axis while does not change the minor axis of the ellipse. Note that the distance from the initial stress T0to any point on the response envelope represents the sti ness along certain strain rate. As shown in Figure 1(a), the vectors j~ oajandj~objrepresent, respectively, the directional sti ness of loading and unloading at a isotropic stress state. j~ ocjandj~odjdenote the directional sti ness with isochoric strain. For clays, the ratio j~ oaj=j~objis speci ed by the loading and swelling indexes, i.e., and . It is known that a desirable ratio of j~ oajtoj~ ocjcan be achieved by implementing two scale multipliers with L1andL3[14]. Analogously, with the help of the term c2L2, an adjustable ratio of loading to unloading sti ness is available by introducing the swelling index. Another remark concerns with the enhancement of the initial shear sti ness, Gi, for OC clays. This subject has been investigated by several experiments [1, 2]. The ex- perimental results indicate that the dependence of Gion stress level and OCR can be described by exponential functions [11]. However, hypoplastic models usually underes- timateGiof OC clays, which is an intrinsic de ciency of many hypoplastic models. As 5 0 0.5 1 1.5 2 2.5 32 T22/trT00.511.523 T11/trTc1 = 1.0 c1 = 0.5 a bc do(a) e ect of c1 0 0.5 1 1.5 2 2.5 32 T22/trT00.511.523 T11/trTc2 = 1.1 c2 = 0.1 (b) e ect of c2 0 0.5 1 1.5 2 2.5 32 T22/trT00.511.523 T11/trTc3 = 1.0 c3 = 0.2 (c) e ect of c3 Figure 1: The e ects of the tensorial function Lon response envelopes illustrated in Figure 1, the initial shear sti ness can be enhanced by multiplying the whole model. Following this way, we introduce an additional multiplier into the model (4): T=mRfsL: DfdYm kmkkDk (9) wheremRrepresents the sti ness enhancement of OC clays. In the following, the el- ements for the model (9) are derived. The complete formulations are given in the appendix. 3.2. Tensor function L The linear tensorial function of model (6) is considered for the new model. It can be written as: L=c1I+c2^T 1+c3^T ^T (10) 6 whereIand1are the fourth-order and the second-order unit tensor, respectively. Next, we proceed to determine the constants c1,c2,c3by assuming that the proposed model has the same form with the model (6) at an isotropic stress state. Therefore, the model (9) can be expressed with the following decomposed forms: _p=1 3fs" c1+c2+1 3c3 Dv+c3^qDq+ar 1 3D2 v+3 2D2 q# (11a) _q=fs" 3 2c1Dq+ c2Dv+1 3c3Dv+c3^qDq ^q+ 2^qar 1 3D2 v+3 2D2 q# (11b) where _pand _qare the rates of the mean stress and the deviatoric stress, respectively. ^q=q=tr(T) is the normalized deviatoric stress. DvandDqare the volumetric strain and deviatoric strain, respectively. For an isotropic stress state, we have initially ^ q= 0, Dq= 0, and D v<0, and thus equation (11a) becomes: _pi=1 3fs c1+c2+1 3c3p 3 3a Dv,K+ iDv (12) whereK+ i=fs(c1+c2+c3=3ap 3=3)=3 represents the initial bulk modulus of the soil for loading. Since the modi cation of the model does not in uence the formulation of the model (6) at an isotropic stress state, and therefore we have: c3=a2+ 3(1c1) (13) For unloading in an isotropic compression test, we have D v>0. The rate of mean stress can be gained as follows: _pi=1 3fs c1+c2+1 3c3+p 3 3a Dv,K iDv (14) in whichK i=fs(c1+c2+c3=3 +ap 3=3)=3 represents the initial bulk modulus of the soil for unloading. On the other hand, in a drained test with constant volume change, i.e., D v= 0 (isochoric shear), the equation (11b) is recast to: _qi= _qj^q=0=3 2fsc1Dq,3GiDq (15) whereinGi= 0:5fsc1is the initial tangential shear modulus for isochoric shear. Comparison between the initial tangential bulk modulus in loading and unloading conditions yields K+ i=K i==. In addition, the ratio of K+ itoGican be considered as a material constant vi. Combining the equations (12 to15) and making use of the 7 above conditions, we obtain the formulations for the multipliers c1,c2: c1=2p 3a( 1) 9vi; c2=1 3(p 3a a23) (16) with being the ratio ( +)=(). The third constant c3is then obtained from equation (13). 3.3. Flow rule mand limit stress condition Y The hypoplastic ow rule mindicates the ow direction, which should obey the conditions m^Tat a critical state, and ~m=I=p 3 at an isotropic stress state [26]. The tensorial function Bof the model (6) ful ls these requirements. In this paper, a similar form including Matsuoka-Nakai failure criterion is adopted: m=^TF2a2k^Tk2 F2(1 +c2) +a2k^Tk2^T (17) whereFis an interpolation function to account for the Matsuoka-Nakai failure criterion [21,37]. The failure criterion is obtained by incorporating the Matsuoka-Nakai failure criterion into the non-linearity Y. In this work, a similar Yas that used in [21] is adopted. It reads: Y=( 1)(I1I2=I39) + 8tan2 c 8 tan2 c(18) 3.4. The density factor fd The factor fdis used to consider the consolidation history. Bearing in mind that fd= 1 shall be guaranteed for normal consolidation (OCR = 1), we proceed to introduce the following simple expression for the density factor: fd=R (0<R61) (19) where is a model parameter; Ris the stress ratio representing the overconsolidation [47]. A smaller value of Rcorresponds to a larger OCR to give: R=p+ e p e(20) in which the p+ eandp eare the pre-consolidation pressure in MCC model and the Hvorslev equivalent pressure respectively. Their de nitions are shown in Figure 2with the for- mulations given as follows: p+ e=ph 1 + (q M()p)2i ; p e= exphNln(1 +e) i (21) 8 in which the inclination Mcorresponds to the stress ratio at the critical state. It is interpolated with the function Fto de ne the critical state line at di erent Lode angles , i.e.,M= 6Fsin c=(3sin c);Nstands for the logarithmic value of the speci c volume at the reference stress pr= 1 kPa; the index is the slope of the normal compression line in the double logarithmic ln(1 + e)lnpplane. pe* pe*pe* pe+pe+*λ *κ ln p p p qln (1+e) Isotr. normal compression line CSLN current stateIsotr. unloading line R = /q=Mp1 1 m (b)(a) Figure 2: The de nition of (a) Hvorslev equivalent pressure p e, and (b) the stress ratio R 3.5. The sti ness factor fs The factor fscan be obtained from the pre-de ned isotropic normal compression line. After Butter eld [5], the normal compression line is expressed as: ln(1 +e) =Nln(p=pr) (22) Time di erentiation of equation (22) gives rise to the rate form: _p=p_e (1 +e)(23) 9 Comparing equations (23) and (12), and making use of Dv= _e=(1+e), the expression forfsis obtained: fs=p 3trT 2a(1 1 ) (24) It can be seen that the sti ness factor fsincludes both the strength parameter aand the deformation parameters ( ;) through a simple formulation. 3.6. The multiplier mR In this paper, we propose a simple formulation for mR: mR= exp[ (1R)] (25) where is a constant controlling the in uence of consolidation history on the initial shear modulus. Obviously, we have mR= 1 for normal consolidation with R= 1; for extreme case with R!0, the multiplier mR= exp( ). If needed, can be considered as a material parameter. In the present simulation, however, a xed value = 1 is used. 4. Model performance In this section, we discuss the basic properties of the proposed model. First, a para- metric study is performed to facilitate the selection of proper parameters in numerical simulation. Next, the performance of the proposed model is presented. Additionally, the in uence of parameters on the shape and size of the ASBS is investigated. 4.1. Model parameters The proposed hypoplastic model contains six constitutive parameters, i.e., c,N,, ,vi, and . The rst ve have the same physical interpretation as those used in the MCC model: cis the critical state friction angle; Nis the value of ln(1 + e) at the isotropic normal compression line for p= 1 kPa;is the slope of the isotropic normal compression line on the ln(1 + e)lnpplane;denotes the slope of unloading line in the same plane, and viis ratio between the initial bulk and shear moduli. Since the rst ve parameters of our model have the same physical meaning as those in the DM model. In the following, therefore, a series of triaxial tests are simulated to show the e ects of . In the simulations, two di erent conditions with OCR = 5 and 1.2 are considered. The con ning pressure is assumed to be 100 kPa. The parameters used in the simulations are listed in Table 1. Table 1: Parameters for model performance Parameter c=N vi Model test 30 0.1 0.01 1.0 0.2 0.5,1.0,1.5 Figure 3shows the stress { strain relationship and e ective stress paths of the model. A close inspection of the results reveals that mainly in uences the peak friction angle 10 -100-50 0 50 100 150 200 250 300 -0.3 -0.2 -0.1 0 0.1 0.2 0.3Drained testq [kPa] Axial strain, ε1 [-]α = 0.5 α = 1.0 α = 1.5(a) -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1Drained test ASBSq/p* e p/p* eα = 0.5 α = 1.0 α = 1.5 (b) -200-150-100-50 0 50 100 150 200 250 300 -0.3 -0.2 -0.1 0 0.1 0.2 0.3Undrained testq [kPa] Axial strain, ε1 [-]α = 0.5 α = 1.0 α = 1.5 (c) -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1Undrained testq/p* e p/p* eα = 0.5 α = 1.0 α = 1.5 ASBS (d) Figure 3: In uence of the parameter on the stress-strain and stress path curves in drained and undrained triaxial compression and extension tests of OC clays. As shown in Figure 3(a, c), The parameter governs the rate to achieve the critical state; a smaller leads to a faster convergence to the critical state. In addition, in uences the e ective stress paths in triaxial tests. As shown in Figure 3(b, d), the normalized stress paths expand out with the increase of in triaxial compression tests. However, this in uence is less signi cant in triaxial extension tests. Note that in the DM model is a variable related to ,, and c. In a recent model by Ma s n [23], a xed value = 2 is adopted for numerical simulations. 4.2. In uence of parameters on the ASBS Asymptotic behaviors are among the most signi cant mechanical properties of clays. For instance, one of the most important asymptotic states of clay is the well - known critical state behavior, which describes the asymptotic behavior of clay with constant volume during shearing [22, 23,24]. The asymptotic states of a hypoplastic model can be achieved after a suciently long proportional stretching, i.e. stretching with a constant direction of the strain rate. This leads to the ASBS that bounds all admissible asymptotic states in the stress - void ratio space. The size of ASBS should be controlled by the critical state friction angle c. As shown in Figure 4, the ASBS of the proposed model expands with the increase of c, 11 This expansion does not leads the ASBS to the tensile stress region. It is found that the ASBS of our model is close to the elliptic shape of the yield surface of the MCC model [32] in the range p > p e=2. The elliptic shape, however, becomes narrow in the range p<p e=2, coinciding with the nonlinear Hvorslev surface on the dry side. This property ensures the model predicts properly the peak strength of heavily OC clays [3, 34]. −1−0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1α = 1.0, κ*/λ*=0.1152025 30 35 40 90 90 40 35 30252015q/p* e p/p* e Figure 4: Asymptotic sate boundary surface for c= 15;20;25;30;35;40 Ma s n and Herle [24] reported that the ratio =has a signi cant in uence on the shape of the ASBS predicted by the DM model. As shown in Figure 5(a), for high ratios of =, the DM model predicts unrealistic shape of the ASBS. However, this shortcoming does not exist in the proposed model. As shown in Figure 5(b), the proposed model predicts very similar ASBS with varying ratios of =, even with high values of=the shape of ASBS is reasonable. For the DM model, the dependence of ASBS on the ratio of =is due to the connection between =and the parameter . As is not an independent parameter in the DM model, high ratio of =will give rise to small value of . In the proposed model, however, the parameter is given as an independent material constant, and thus the ratio of =does not in uence . As shown in Figure 6, increase of gives rise to expansion in the wet side ( p=pe>0:5), and shrinkage in the dry side ( p=pe<0:5). This is consistent with the results presented in Figure 3, that the increase of leads to increased peak friction angle of OC clays, and this in uence becomes more obvious with low value . However, the shape of ASBS remains almost unchanged with increasing from 1.0 to 3.0. 5. Validation with experimental data In this section, model predictions are compared with experimental results on both remoulded and undisturbed OC clays. Di erent types of triaxial tests, including isotrop- ically consolidated undrained triaxial tests (CIU), isotropically consolidated drained tri- axial tests with constant mean pressure (CIDP) and constant con ning pressure (CID), are simulated to validate this model. Parameters used in the simulations are summarised 12 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2Mašín modelq/p* e p/p* eκ*/λ*=0.1 κ*/λ*=0.2 κ*/λ*=0.25 κ*/λ*=0.3(a) −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2proposed modelq/p* e p/p* eκ*/λ*=0.1 κ*/λ*=0.2 κ*/λ*=0.25 κ*/λ*=0.3 (b) Figure 5: In uence of ratios =on the shape of ASBS predicted by (a) Ma s n's model [21] and (b) proposed model −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2q/p* e p/p* eα = 0.5 α = 1.0 α = 2.0 α = 3.0 Figure 6: The in uence of on the shape of the ASBS of the proposed model in Table 2. The rst ve parameters are collected from the literature, while the param- etersviand are obtained through optimization methods [16, 50,51] to gain the best 13 match with experimental results. Table 2: Parameters for simulation of di erent clays Soils c=N v i Wenzhou clay [8] 29.8 0.0684 0.025 0.95 1.0 0.89 Fujinomori clay [25] 33.7 0.0445 0.0108 0.86 0.6 0.5 Weald clay [29] 23.5 0.059 0.028 0.8 1.5 1.0 Bangkok sti clay 1 [13] 31.5 0.032 0.0072 0.89 1.2 0.5 Bangkok sti clay 2 [13] 30.2 0.0505 0.02 0.892 0.6 2.0 5.1. CIU test, Remoulded Wenzhou clay The Wenzhou clays are mainly marine deposits from Wenzhou, eastern coastal China. The deposit usually reaches a depth of 150 m and exhibits evident heavily OC behavior. A series of triaxial compression tests on remoulded clay was carried out by Gu et al. [8] using a GDS device. The samples contain clay fraction of 63.8% and a ne fraction of 93.4%. The natural water content can reach to wL= 67:7% with plasticity index Ip= 37:2 and initial void ratio, e0, ranging from 1.55 - 1.59 [8]. 0 50 100 150 200 OCR = 8 OCR = 4 OCR = 2 OCR = 1 250 0 0.05 0.1 0.15 0.2 0.25q [kPa] Axial strain, ε1 [−]model prediction experiment (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 1.2q/pc p/pc model prediction experiment (b) Figure 7: Comparison between experimental and numerical undrained triaxial tests on OC Wenzhou clay. Experimental data from Gu et al. [8] 14 Four samples were initially subjected to pre-consolidation pressure ( pc)of 100, 200, 400, 800 kPa. After the completion of pre-consolidation, the e ective con ning pressure was reduced to p0= 100 kPa in order to produce OC samples with speci c OCR values. The test results are presented in Figure 7together with model prediction. As shown in Figure 7(a), the samples with di erent OCRs exhibit various shear strengths, this history dependence can be predicted by the proposed model. The peak strength can also be captured quite accurately. However, the stress { strain curves agree not very well at the bends of the curves, as it underestimates the initial sti ness and also the degradation rate of the sti ness. Figure 7(b) shows that the predicted e ective stress paths agree well with experimental results for heavy consolidation, while less well prediction is observed for normal consolidation. 5.2. CIDP test, Remoulded Fujinomori clay Experimental data on reconstituted Fujinomori clay by Nakai and Hinoko [25] is a classical set of experiments used in the literature to evaluate di erent constitutive models within the framework of elastoplasticity [47 ]. The test was carried out with constant mean e ective stress. Both drained triaxial compression and extension tests were performed on OC Fujinomori clay with OCR from 1 to 8; The sample with OCR = 8 was tested at mean e ective stress of p0= 98 kPa, while other samples were tested at mean e ective stress of 196 kPa. The predicted stress - strain and volumetric change curves using the proposed hy- poplastic model are shown in Figure 8. Good agreement in the stress ratio curves is achieved in the compression tests, while the model underestimates the peak strength in the extension tests. A more signi cant di erence is observed in the results of volumetric strain, especially in the extension tests. Nevertheless, the main features of the variation of stress ratio and volumetric strain can be reasonably captured by the proposed model. 5.3. CIU and CID test, Remoulded Weald clay A series of tests have been carried out by Parry [29] on reconstituted Weald clay with both normal consolidated ( pc= 206:8 kPa, OCR = 1) and heavy consolidated (pc= 827:4 kPa, OCR = 12) states. The experiments are simulated and the model prediction are compared with experimental data. Parameters for this simulation are collected from the literature [23] by Ma s n, who compared his two hypoplastic models for clay using this set of data. Predictions of the drained and undrained triaxial compression and extension tests, together with experimental results, are shown in Figure 9. It can be seen that the model predictions agree very well with the experimental results. Especially, excellent agreements are achieved in the stress - strain curves of the drained tests. On the other hand, this model underestimates the shear strength in undrained triaxial extension test for OCR = 12. 5.4. CID and CIU test, undisturbed Bangkok sti clay In the previous evaluation, the model is validated with di erent types of reconsti- tuted clays. This model is able to capture most properties of reconstituted OC clay. To 15 -1.5-1-0.5 0 0.5 1 1.5 2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15q/q0 [-] Axial strain εa [-]model prediction OCR = 1 OCR = 2 OCR = 4 OCR = 8(a) -0.06-0.05-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 -0.15 -0.1 -0.05 0 0.05 0.1 0.15-εv [-] Axial strain εa [-]OCR = 1 OCR = 2OCR = 4 OCR = 8 (b) Figure 8: Comparison between experimental and numerical drained constant mean pressure triaxial compression and extension tests on OC Fujinomori clay, data from Nakai and Hinoko [25] broaden the validation of the proposed model, the nal set of modeling will focus on an undisturbed clay, Bangkok sti clay, from Bong Ngoo Hao site and Rangsit site. Accord- ing to the original test results [13], although taken from di erent sites, these two soils exhibit similar overconsolidation behavior. In this study, selected triaxial compression test are simulated using the proposed model. The reported peak friction angle and cohesion for samples from Bong Ngoo Hao site are p= 26andc= 30 kPa, respectively. This friction angle is apparently higher than the critical friction angle. Considering the high cohesion value and taking into account that the failure envelope for sti , OC clays is curvilinear, a critical friction angle c= 31:5is adopted for the Bong Ngoo Hao sti clay. Results from drained and undrained triaxial tests on these samples show that a slightly lower critical friction angle was obtained for samples from the Rangsit site. These results are adopted in our simulation. The parameters andare calibrated from oedometer tests presented in [35], and the other parameters are listed in Table 1. Comparison between numerical and experimental results of the drained compression tests on undisturbed Bangkok clay from Bong Ngoo Hao site is presented in Figure 16 −150−100−50 0 50 100 150 200 250 300 −0.1 −0.05 0 0.05 0.1 0.15 0.2q [kPa] Axial strain εa [−]model prediction experimentOCR = 12 OCR = 1OCR = 12OCR = 1(a) −0.06−0.05−0.04−0.03−0.02−0.01 0 0.01OCR = 12 OCR = 1OCR = 1OCR = 12 0.02 0.03 −0.1 −0.05 0 0.05 0.1 0.15 0.2−εv [−] Axial strain εa [−]model prediction experiment (b) −150−100−50 0 50OCR =12 OCR =12OCR =1 OCR =1 100 150 200 −0.1 −0.05 0 0.05 0.1 0.15 0.2q [kPa] Axial strain εa [−]model prediction experiment (c) −150−100−50 0 50 100 150 200 0 50 100 150 200 250q [kPa] p [kPa]model prediction experiment OCR =1 OCR =12(d) Figure 9: Comparison between experimental and numerical drained (a,b) and undrained (c,d) triaxial tests on remoulded Weald clay. Experimental data from Parry and Aust [29] 10. The adopted pre-consolidation pressure for the simulations is 700 kPa with various con ning pressures of 35, 105, and 414 kPa. Inspection of the results suggests that the undisturbed clay samples show evident strain softening behavior. Particularly, for the sample under con ning pressure 414 kPa, a sudden failure at an axial strain of 0.06 is observed. This may be most likely due to the natural structure of intact samples. The proposed model underestimates the peak strength for the samples with con ning pressures of 35 and 105 kPa, while overpredicts the peak strength for the samples with con ning pressure 414 kPa. However, an excellent agreement with volume change is observed in the simulations. The model is further validated with experimental results obtained from undrained triaxial tests on samples taken from the Rangsit site. Seven di erent con ning pressures, i.e., 14, 70, 140, 207, 276, 345, 414 kPa, with the same pre-consolidation pressure of 700 kPa are used in this simulation. Comparison between predictions and experimental results is shown in Figure 11. Note that the results for con ning pressures of 140, 276, and 345 kPa are not shown in the p=q"1curves because these testing results showed signi cant deviation from the general experimental trend. The corresponding e ective stress paths are repented in the pqcurves with solid symbols. An excellent agreement with experimental results is achieved. This again suggests that the proposed model can 17 0 100 200 300 400 500 600 700 800 900 0 0.02 0.04 0.06 0.08 0.1 0.12 0.1435 kPa105 kPa414 kPaq [kPa] Axial strain εa [−]model prediction experiment(a) −0.03−0.02−0.01 0 0.01 0.02 0.03 0.04 0 0.02 0.04 0.06 0.08 0.1 0.12 0.1435 kPa105 kPa414 kPa−εv [−] Axial strain εa [−]model prediction experiment (b) Figure 10: Comparison between experimental and numerical drained triaxial compression tests on undisturbed Bangkok sti clay at Bong Ngoo Hao site. Experimental data from Hassan [13] correctly predict the behaviors of OC clays. 6. Conclusions In this paper, we present extend a basic hypoplastic constitutive model for OC clays. New density and sti ness factors are introduced to account for consolidation history. The model is capable of predicting a desirable ASBS. The shape of the ASBS is close to the elliptic shape of the yield surface of the MCC model on the wet side, while similar to the nonlinear Hvorslev surface on the dry side. This feature ensures that the model can properly predict the peak and critical strengths of OC clays. This model is validated by experiments on both reconstituted and undisturbed OC clays. The validation shows that the proposed model can capture the most common features of OC clays. Acknowledgments This work was funded by the H2020 Marie Sk lodowska-Curie Actions RISE 2017 HERCULES (778360) and FRAMED (734485); the Erasmus+ KA2 project Re-built (2018-1-RO01-KA203-049214); and the Nazarbayev University Research Fund (SOE2017001). 18 0 0.5 1 1.5 2 2.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14q/p Axial strain, ε1 [−]model prediction 14 kPa 70 kPa 207 kPa 414 kPa(a) 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 450q [kPa] p [kPa]model prediction 14 kPa 70 kPa 140 kPa 207 kPa 276 kPa 345 kPa 414 kPa (b) Figure 11: Comparison between experimental and numerical undrained triaxial compression tests on undisturbed Bangkok sti clay at Rangsit site. Experimental data from Hassan [13] The rst author wishes to thank the Otto Pregl Foundation for nancial support. All numerical simulations were performed using the Triax element test driver. The provider Dr. David Ma s n is acknowledged. Appendix The concrete model formulation is: T=mRfsL: DfdYm kmkkDk (.1) The tensor Lis: L=c1I+c2^T 1+c3^T ^T (.2) where the normalized stress ^T=T=trT.Iand1are the fourth-order and the second- order unit tensor, respectively. The constants c1,c2, andc3are: c1=2p 3a( 1) 9vi; c2=p 3a a23 3; c3=a2+ 33c1 (.3) 19 The hypoplastic ow rule mis m=^TF2a2k^Tk2 F2(1 +c2) +a2k^Tk2^T (.4) with the interpolation factor Fgiven as F=" 3 8k^Tk2+11:5k^Tk2 1 +p 1:5k^Tkcos3#1=2 r 3 8k^Tk (.5) where the Lode angle is de ned as: cos3=p 6tr^T^T^T k^Tk3=2(.6) The limit stress condition Yis given as: Y=( 1)(I1I2=I39) + 8tan2 c 8 tan2 c(.7) whereI1,I2, andI3are stress invariants expressed as: I1= trT; I 2= [T :T(I1)2]=2; I 3= det T (.8) The multipliers fs,fd, andmRare: fs=p 3trT 2a1 1  ; fd=1 R ; mR= exp 11 R (.9) with the stress ratio Rbeing de ned as: R=p+ e p e(.10) wherep+ eis the pre-consoldiation pressure in the MCC model, and p eis the Hvorslev equivalent pressure. p+ e=ph 1 + (q Mp)2i ; p e= exphNIn(1 +e) i (.11) In the above equations, ,a, andMare material-related constants: =+ ; a=p 3(3sin c) 2p 2sin c;andM=3p 3F a(.12) Finally, there are six parameters for the proposed model: c,,,N,vi, and . 20