Hypoplastic constitutive models are able to describe history dependence using a single nonlinear tensorial function with aset of parameters. A hypoplastic model including a structure tensor for consolidation history was introduced in our previous paper (Wang and Wu in Acta Geotechnica, 2020, https://doi.org/10.1007/s11440-020-01000-z ). The present paper focuses mainly on the model validation with experiments. This model is as simple as the modiﬁed Cam Clay model but with betterperformance. The model requires ﬁve parameters, which are easy to calibrate from standard laboratory tests. In particular, the model is capable of capturing the unloading behavior without introducing loading criteria. Numerical simulations of element tests and comparison with experiments show that the proposed model is able to reproduce the salient features ofnormally consolidated and overconsolidated clays. Keywords Critical state /C1Hypoplastic model /C1Overconsolidation ratio /C1Structure tensor /C1Validation 1 Introduction Overconsolidation is one of the most important factors that signiﬁcantly inﬂuence the mechanical behavior of over-consolidated (OC) clays. Over the past decades, signiﬁcant progress has been made in the development of constitutive models for predicting overconsolidation behavior of clays [1,4,8,9,12,13,20,25,47–49]. One of the main con- siderations for the evaluation of the models is that themodel parameters can be easily obtained from conventional laboratory tests. For this purpose, ﬁrstly, the model parameters should possess clear physical meaning, andsecondly, the model parameters should be independent of the initial states, such as the initial conﬁning pressure and the initial overconsolidation ratio (OCR). Many elasto-plastic models can properly describe the mechanical behavior of OC clays using a set of model parameters. However, model capacity is frequently gained at the sac-riﬁce of simplicity.As an alternative to the prevailing plasticity theory, the hypoplastic models have gained much attention in the recent years [ 5–7,15,26–28,30,34–37,39,45]. Beneﬁted from the nonlinear tensorial function, hypoplastic modelsare able to describe the overconsolidation behavior without resource to the concept in elastoplastic theory such as yield surface, plastic potential, differentiation between elastic and plastic behaviors, and loading criterion [ 38,41–44]. Therefore, compared to the elastoplastic models,hypoplastic models are usually characterized by simpler formulations and fewer model parameters. Consequently, the numerical implementation of hypoplastic models isquite straightforward. Hypoplastic models were originally developed for granular materials, and most applications of hypoplasticmodels focused on the simulation of sands. There are a number of hypoplastic models for clays. For instances, a series of hypoplastic models for ﬁne-grained soils wereproposed by Mas ˇı´n[16,18,19]. These models open a new avenue for modeling of OC clays. Based on these models, some extensions for OC clays were made by Shi et al. [ 33] and Wang et al. [ 39]. Whereas the aforementioned models provide a fairly decent prediction for OC clays, their for- mulations are rather intricate and further extensions basedon these models will inevitably increase the complexity of &Wei Wu wei.wu@boku.ac.at 1Institut fu ¨r Geotechnik, Universita ¨tf u¨r Bodenkultur Wien, Feistmantelstrasse 4, 1180 Vienna, Austria 123Acta Geotechnica (2021) 16:31–41 https://doi.org/10.1007/s11440-020-01105-5 (0123456789().,-volV) (0123456789(). ,- volV) the model and the number of the parameters. A hypoplastic model possessing simple formulations and fewer parame- ters is therefore pressing to beneﬁt the development of more sophisticated models for clays. This paper follows the line of our previous work [ 38], in which a general hypoplastic framework for OC soils was outlined. A structure tensor was included in a hypoplasticmodel to account for the consolidation history of soils. The theory and performance of the model were however pre- sented based on a basic hypoplastic model for granular materials [ 44]. Hence, the model adopts a set of material parameters for sand, which might be unsuitable for clay. Inthe present paper, the parameters of the basic hypoplastic model are calibrated for clays. This leads to a reduction of the constitutive parameters, which are easy to calibratefrom conventional laboratory experiments. Moreover, the Matsuoka-Nakai failure criterion is included in the new model. The inﬂuence of model parameters on the modelprediction is analyzed by performing a series of elementary tests. In the end, the model is validated by comparing the numerical predictions with experiments on different OCclays. 2 Hypoplastic constitutive model for OC clays 2.1 General framework Let us consider the following hypoplastic constitutive model proposed by Wu and Kolymbas [ 40]: /C23T¼L:DþNkDk ð1Þ where LandNare fourth- and second-order tensorial functions, respectively. The colon : denotes an inner pro- duct between two tensors. Dis the strain rate (stretching) tensor, and kDk¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ trðD2Þq stands for the norm of the strain rate tensor. /C23Tis the Jaumann stress rate deﬁned as follows: /C23T¼_TþTW /C0WT ð2Þ where Tand and _Tare the Cauchy stress tensor and the material time derivative of the Cauchy stress, respectively;Wis the spin tensor. In order to describe the overconsolidation behavior using hypoplasticity, Wang and Wu [ 38] introduced a structure tensor to account for history dependence in the following way: /C23T¼LðTþSÞ:DþNðT/C0SÞkDk ð3Þ where Sis a structure tensor representing theoverconsolidation. The following simple expression is used for the structure tensor: S¼aln1 R/C18/C19 T ð4Þ where ais a model parameter, which controls the inﬂuence of the consolidation history; and Ris a stress ratio repre- senting the degree of overconsolidation with a smaller value of Rcorresponding to a larger OCR. It reads R¼pþ e peð5Þ in which pþ eandpeare the preconsolidation pressure in the modiﬁed Cam Clay (MCC) model and the Hvorslev equivalent pressure, respectively. Their formulations aregiven as follows: p þ e¼ph 1þðq MpÞ2i and pe¼exphN/C0lnð1þeÞ k/C3i ð6Þ where pandqare the mean stress and deviatoric stress, respectively; the inclination Mcorresponds to the stress ratio at the critical state; eis the current void ratio; the index k/C3is the slope of the normal compression line in the double logarithmic ln ð1þeÞ/C0lnpplane; Ndenotes the logarithmic value of the speciﬁc volume at the reference stress pr¼1 kPa. Note that the parameter Nis different from the tensorial function Nin Eq. ( 1). In addition to the structure tensor, an additional multi- plier fuis introduced to improve the model performance for isochoric strain rate, giving /C23T¼LðTþSÞ:DþfuNðT/C0SÞkDk ð7Þ in which the multiplier fuis deﬁned as fu¼kB:Dk kBkkDkð8Þ where B¼/C0L/C01:Ndenotes the ﬂow rule of the model. One may refer to [ 38] for the mechanism of the function fu on the improvement of undrained triaxial tests. 2.2 Hypoplastic equation With the framework incorporating consolidation history, we proceed to consider the following speciﬁc constitutivemodel, which is rearranged from a basic hypoplastic model proposed by Wu et al. [ 44]: /C23T¼f sh ðtrTÞDþfvðtrDÞTþa2trðTDÞ trTTþaðTþT/C3ÞkDki ð9Þ32 Acta Geotechnica (2021) 16:31–41 123 where T/C3is the deviatoric part of T;fsis the stiffness factor andfvis a multiplier inﬂuencing the volumetric response of the model; the material constant acorresponds to the limit stress at the critical state. The deﬁnition of aincluding the Matsuoka-Nakai failure criterion will be given in Section 2.4. Combining Eqs. ( 7) and ( 9), we obtain the following simple hypoplastic model: /C23T¼fsh ðtr/C20TÞDþfvðtrDÞ/C20Tþa2trð/C20TDÞ tr/C20T/C20T þfuað^Tþ^T/C3ÞkDkið10Þ where /C20T¼TþSand ^T¼T/C0Sare adopted for simplicity. The constitutive model ( 9) was originally developed for sand. For modeling the OC behavior of clays, the stiffness factor fsand the multiplier fvshould be calibrated to allow this model to describe the stiffness and volumetric response of OC clays. These multipliers can be related to somewidely used parameters in soil mechanics. In the following, we proceed to determine f sand fvfor OC clays. The complete formulation of constitutive model ( 10) is given in Appendix A. 2.3 Calibration of fvand fs Bearing in mind that normal consolidation with ORC = 1 leads to R¼1, and thus, the structure tensor vanishes with S=0. This implies fsandfvcan be determined by using the so-called consistency assumption [2,16], i.e., isotropically compressing a soil sample from its normal consolidationstate. To this end, we turn our attention to the model ( 9) and write out the decomposed formulations as follows: /C0_p¼pf s"/C16 1þa2 3þfv/C17 Dv þa2^qDqþaﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D2 v 3þ3 2D2 qs # ð11aÞ _q¼3pfs" 3 2Dqþ/C0a2 3þfv/C1 Dv^qþa2^qDq^q þ2^qaﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D2 v 3þ3 2D2 qs # ð11bÞ where _pand _qare the rates of the mean stress pand deviatoric stress q, respectively; ^q¼q=trTis the normal- ized deviatoric stress; DvandDqstand for the volumetric and deviatoric strain rates, respectively.First, we consider the stiffness factor fsfor a normally consolidated clay. Following the work by Butterﬁeld [ 3], the rate of the mean stress _pcan be expressed as follows: _p¼/C0p_e k/C3ð1þeÞð12Þ where _edenotes the rate of the current void ratio. Note that the constitutive model ( 10) should predict the same _pfor an isotropic compression tests. To this end, let us recall the _pexpressed in Eq. ( 11a). Initially, we have Dq¼0 and Dv¼_e=ð1þeÞ\0, thus Eq. ( 11a) becomes _p¼pfs_e ð1þeÞ1þ1 3a2þfv/C0ﬃﬃﬃ 3p 3a/C18/C19 ð13Þ Comparison between Eqs. ( 12) and ( 13) yields a general form for fs: fs¼/C01 k/C3ð1þ1 3a2þfv/C0ﬃﬃﬃ 3p 3aÞ/C01 ð14Þ Next, we conﬁne our attention to the multiplier fv. Analo- gously, Eq. ( 11a) is able to predict the initial bulk modulus of a soil in the following way: /C0_p¼pfs/C18 1þ1 3a2þfv/C0ﬃﬃﬃ 3p 3a/C19 Dv,KiDv ð15Þ where Ki¼pfsð1þa2=3þfv/C0ﬃﬃﬃ 3p =3aÞrepresents the initial bulk modulus of a sample in an isotropic compres- sion test. On the other hand, in an undrained test with constant volume, i.e., Dv/C170, Eq. ( 11b) is recast to _qi¼_qj^q¼0¼9 2fspDq,3GiDq ð16Þ in which Gi¼1:5pfsdenotes the initial shear stiffness at an isotropic stress state. As assumed in previous works [ 10,11,16,39], the ratio vi¼Ki=Gican be considered as a material constant, which has a similar physical meaning to Poisson’s ratio. As a consequence, a simple formulation for the multiplier fvis obtained as follows: fv¼3 2vi/C01 3ð3þa2/C0ﬃﬃﬃ 3p aÞ ð17Þ Accordingly, the stiffness factor fsis obtained by substi- tuting Eq. ( 17) into Eq. ( 14): fs¼/C02 3vik/C3 ð18Þ It is worth noting that the formulations of fsandfvfor clays contain both the deformation parameters ( viandk/C3) and the strength parameter ( aor/c) through simple formulations.Acta Geotechnica (2021) 16:31–41 33 123 2.4 Failure criterion Equation ( 1) describes a critical state condition for con- tinuing deformation when the directional stiffness van- ishes, then we have /C23T¼0, which corresponds to B¼/C0L/C01:N¼D kDkð19Þ Taking the norm of both sides gives rise to the failure criterion at the critical state. f¼kBk/C01¼0 ð20Þ At the critical state, the model ( 10) degrades to ( 9) with S= 0andfu¼1, indicating the same failure criterion for the two models. To obtain the failure criterion, we can writeout the explicit expression of Bfor model ( 9): B¼a trT" T/C3þTðtrTÞ2/C0a2kT/C3k2 ð1þfvÞðtrTÞ2þa2kTk2# ð21Þ With the above equation, the failure condition ( 20)i s equivalent to f¼akT/C3 ck/C0trTc¼0 ð22Þ where the TcandT/C3 cdenote the stress tensor at the critical state and its deviatoric part, respectively. Obviously, the parameter ain Eq. ( 22) is related to the critical state value of the normalized deviatoric stress kT/C3 ck=trTc. With a constant value for a, Eq. ( 22) has a conical shape in the principal stress space, which resembles that of the Drucker- Prager failure surface [ 36,44]. In order to describe the soil behavior under true triaxial condition, the following for-mula is adopted for a: a¼ﬃﬃﬃ 3p gð3/C0sin/cÞ 2ﬃﬃﬃ 2p sin/cð23Þ where /cis the critical state friction angle of the material; the factor gis adopted to incorporate the Matsuoka-Nakai failure criterion according to Yao et al. [ 46]. It reads g¼2I1 3ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðI1I2/C0I3ÞðI2 1/C03I2Þ=ð11I2/C09I3Þp /C0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ðI2 1/C03I2Þp ð24Þ in which I1,I2, and I3are stress invariants. It should be noted that a weighted factor can be applied on gto smoothly change the failure criterion from the Drucker- Prager to the Matsuoka-Nakai criterion. More details can be found in the reference [ 46].3 Model parameters 3.1 Summary of model parameters The proposed hypoplastic model contains ﬁve parameters, i.e.,/c,k/C3,N,vi, and a. The ﬁrst three have the same physical meaning as those used in the MCC model: /cis the critical state friction angle; k/C3is the slope of the iso- tropic normal compression line in the double logarithmicplane ln ð1þeÞ/C0lnp;Nis the value of ln ð1þeÞat the isotropic normal compression line for p r¼1 kPa; and the parameter viis the ratio of the bulk modulus in the isotropic compression and the shear modulus in the undrained shear test on the isotropic consolidated sample; it has a similar physical meaning to Poisson’s ratio. Finally, the additionalparameter acontrols the magnitude of the structure tensor. Overall, the parameters k /C3and Ncan be determined from an isotropic consolidation test. The parameters /cand vican be determined either from a triaxial compression tests with constant mean stress or a conventional triaxial compression test. Therefore, only the determination of the parameter arequires curve ﬁtting, and the other parameters can be measured in a straightforward way. 3.2 Effects of model parameters on model predictions The model parameters k/C3,N, and /care the same with those used in the MCC model [ 31,32]. In the following, therefore, a parametric study considering only aandviis performed. Several triaxial compression tests are modeled to show the inﬂuences of aandvion the model prediction. For this purpose, the parameters /c¼20/C14,k/C3¼0:05, and N¼1:0 are ﬁxed, while aandvivary in a certain range. The conﬁning pressure is assumed to be 100 kPa. Three different OCRs (5, 2.5, and 1.2) are considered in drained and undrained conditions. Figure 1a and c show the stress–strain relation of drained and undrained tests. It can be seen that a greater a leads to a larger initial stiffness. Consequently, steeperstress–strain curves are obtained in the simulations. For heavy overconsolidation, i.e., OCR = 5, the increase of a will also gives rise to an enhancement of the peak shearstrength. Meanwhile, more dilatant volumetric responses are observed at the onset of drained tests, as shown in Fig. 1b. The normalized effective stress paths under undrained condition are shown in Fig. 1d. Obviously, the effective stress paths move to the right with the increase of a, indicating the decrease of the excess pore pressure. Nevertheless, adoes not inﬂuence the model prediction at critical state. The above analysis shows that amainly inﬂuences the initial stiffness of OC clays, and it can be34 Acta Geotechnica (2021) 16:31–41 123 also used to adjust the peak shear strength for heavy overconsolidation in drained simulations. The same numerical tests are performed to show the effects of parameter vion the model prediction. The results are plotted on the planes e1/C0qandp=p/C3 e/C0q=p/C3 e. Figure 2a and c shows the stress-strain relation of drained and undrained tests. As indicted by Eq. ( 18), the parameter vi inﬂuences the stiffness multiplier fs. Increasing of vigives rise to the decrease of fsand eventually leads to softer stiffness responses. Figure 2b and d shows the normalized effective stress paths under drained and undrained condi-tions. For both conditions, the effective stress paths move to the left with the increase of v i. The change of effective stress path, however, becomes less signiﬁcant with theincrease of the OCR. The simulations reveal that the parameter v imainly affects the stiffness response of OC clays.4 Validation with experimental results In this section, the model predictions are compared with experimental results of different ﬁne-grained soils, namely Fujinomori clay [ 21,22], Lower Cromer till [ 29], and Boston blue clay [ 29]. Different tests are simulated to validate the proposed model. The model parameters for different soils are summarized in Table 1. The ﬁrst three parameters are collected from the literature, while the parameters viandaare obtained through optimization methods [ 50,51] to gain the best match with experimental data. 4.1 Fujinomori clay First, let us inspect the prediction of drained and undrainedtriaxial tests on normally consolidated Fujinomori clay.The test results were used to calibrate several hypoplastic models [ 11,17] and elastoplastic models [ 14,23,47]. In the simulations, the initial state p 0¼196 kPa and e0¼ 0:915 is considered. The parameters /c,k/C3, and Nlisted in Table 1are similar to those used in the literature [ 17].(a) (b) (c) (d) Fig. 1 Inﬂuence of the parameter aon model prediction in drained and undrained triaxial compression testsActa Geotechnica (2021) 16:31–41 35 123 Details for the properties of Fujinomori clay can be found in [21]. Figure 3shows the comparison between numerical and experimental results of drained triaxial tests. These results are shown in terms of the normalized deviatoric stress and volumetric strain against the axial strain. The numericalpredictions reproduce well the observed soil behavior, except that a more contractive volume change is obtained at the vicinity of shearing. A further concern of the proposed model lies in the capacity of modeling undrained loading. In the simulation of undrained compression tests, the same initial state asthat used in the drained compression test is considered. The comparison between predictions and experiments is shownin Figure 4. It can be seen that the proposed model predicts correctly the deviatoric stress, the stress path, as well as the variation of the axial and radial effective stresses withrespect to the deviatoric strain. The next simulation is concerned with the prediction of overconsolidation behavior. To this end, a set of drainedtriaxial tests with constant mean stress on OC Fujinomori clay are performed. The initial OCRs are between 1 and 8. The specimen with initial OCR = 8 was tested at a meaneffective stress of p 0= 98 kPa, while other specimens were tested at mean effective stress of 196 kPa. The initial void ratios for OCR = 2, 4, and 8 are e0¼0.863, 0.833, and 0.857, respectively. The predicted stress–strain and volu- metric change curves using the proposed model are shown in Figure 5. An excellent agreement in the stress ratio curves is achieved in the compression tests. More obvious difference between predictions and experimental results is observed in the volumetric strain. Speciﬁcally, the modelunderestimates the dilatancy of the Fujinomori clay at OCR = 2 and 4. At these OCRs, a larger contractive volume changes are predicted by the model. Nevertheless, the mainfeatures of the variation of stress ratio and volumetric strain can be reasonably captured by the proposed model.(a) (b) (c) (d) Fig. 2 Inﬂuence of the parameter vion model prediction in drained and undrained triaxial compression tests Table 1 Model parameters for simulation of different clays Soils /c=/C14k/C3Nv i a Fujinomory clay 33.7 0.0445 0.88 0.6 0.4 Lower Cromer till 30.0 0.047 0.62 (0.61) 0.8 1.0Boston blue clay 33.5 0.025 0.815 1.0 0.336 Acta Geotechnica (2021) 16:31–41 123 4.2 Lower Cromer till The next simulation will focus on illustrating model capabilities to describe the overconsolidation behavior in standard drained triaxial compression tests. The soil, Lower Cromer till, is classiﬁed as a low plasticity sandysilty–clay with a clay fraction of approximately 17 % [ 29]. Several drained triaxial compression tests are performed using the parameters listed in Table 1. Except v ianda, all the other parameters can be obtained from experiments given in [ 29]. First, normal compression tests (both hydrostatic and K0 compression conditions) with loading and unloading are simulated. Note that two different values for the material constant Nare used in the isotropic compression test and the oedometer test. Figure 6shows the predictions together with experimental results. The behavior of Lower Cromer Till in both hydrostatic and K0conditions is well described by a linear normal compression line in the e/C0lnpplane. The predicted normal compression lines agree well with the experiments because the stiffness factor fsof the pro- posed model ensures to predict the predeﬁned normal compression line. In particular, an excellent agreement inthe prediction of unloading lines is achieved in the simulations. Note that an unloading index, e.g., j/C3, is used in most of the existing hypoplastic clay models [ 11,16,19]. It can be expected that the inclusion of an unloading index will facilitate the numerical modeling for one cycle ofunloading, while it has little contribution in capturing the cyclic behavior with a large number of loading cycles [ 38]. In contrast, the numerical result indicates that, although theunloading index is not included in this model, a reasonable(a) (b) Fig. 3 Comparison between experimental and numerical drained triaxial tests on normally consolidated Fujinomory clay. Experimentaldata from Nakai and Mihara [ 21](a) (b) (c) Fig. 4 Comparison between experimental and numerical undrained triaxial tests on normally consolidated Fujinomory clay. Experimental data from Nakai and Mihara [ 21]Acta Geotechnica (2021) 16:31–41 37 123 unloading behavior can be predicted by this model. This property is signiﬁcantly different from most elastoplastic models developed in the framework of MCC, in which the elastic behaviors are solely controlled by the unloading index. In the proposed hypoplastic model, however, there is not a clear boundary between the elastic and the plasticdeformation. Therefore, it is not necessary to include the unloading index for a hypoplastic clay model. We now turn our attention to the prediction of over- consolidation behavior in drained compression tests. Fig-ure7compares the model predictions with experimental results from a series of drained triaxial compression tests on isotropically consolidated specimens with differentOCRs. As the OCR increases, the behavior changes from contractive to nearly neutral to dilative at higher OCRs (e.g., OCR = 10). Generally, the shear stress–strain response and volumetric behavior are very well described by the proposed model although the model slightly over-estimates the measured dilation and contraction at heavily consolidated condition (OCR = 10) and normally consoli- dated condition (OCR = 1), respectively. Overall, theoverconsolidation behavior predicted by the model is in excellent agreement with experimental data. 4.3 Boston blue clay Finally, we consider the capability of the proposed model in describing undrained response of OC clays. Several simulations of undrained triaxial compression tests on(a) (b) Fig. 5 Comparison between experimental and numerical drained triaxial tests with constant pon OC Fujinomory clay. Experimental data from Nakai et al. [ 22] Fig. 6 Comparison between experiments and predicted isotropic and oedometer compression tests. Experimental data from Pestana et al. [29](a) (b) Fig. 7 Comparison between experimental and numerical drained constant ptriaxial tests on OC Lower Cromer Till. Experimental data from Pestana et al. [ 29]38 Acta Geotechnica (2021) 16:31–41 123 Boston blue clay are carried out. The specimens were isotropically consolidated to reach different initial over- consolidation ratios. The initial void ratio used in the simulations are e0= 0.918, 0.93, 0.94, and 0.95 for OCR = 1, 2, 4, and 8, respectively. The model parameters are listed in Table 1. Figure 8compares the predictions of stress paths for undrained triaxial compression tests with experimental results. The experiment at OCR = 8 is not provided in the literature [ 29], and thus it is not considered in the simu- lation of strain–stress simulation. Generally, the model gives excellent predictions of the deviatoric stresses andstress paths for all OCRs. A slight difference in the stress path is observed for OCR = 2. This less good result may be attributed to the model that underestimates the stiffnessresponse at higher strains, thereby leading to a higher shear-induced pore pressures.5 Conclusions Consolidation history is one of the most important factors that affect the mechanical behaviors of OC clays. However, hypoplastic constitutive models that describe, in a quanti- tative manner, the stress–strain and deformation behaviorsof clays are relatively rare compared with the well-devel- oped frameworks for sands. A general framework of hypoplasticity including a structure tensor for historydependence is proposed in our recent paper [ 38]. In the present paper, calibration and validation of a basic hypoplastic model for sand are carried out to allow thismodel to describe the overconsolidation behavior of clays. In comparison with the MCC model, our model pos- sesses simpler formulations and better performance par-ticularly for OC clays. Whereas the MCC model captures properly the salient features of clay with normal consoli- dation, it shows large elastic region, sudden transition from elastic into plastic region, as well as poor predictions of peak failure stresses and dilatancy for clays with heavyoverconsolidation. Purely elastic behavior within the yield surface makes the MCC model inadequate for OC clays. Our model remedies these shortcomings. Numerical pre-dictions show that it is able to properly describe the main behaviors of OC clays, such as strain softening and volu- metric expansion under the drained condition, as well asthe evolution of undrained shear strength and excess pore pressure during the undrained shearing process. The model requires only ﬁve material parameters. All the parameters except ahave clear physical meaning and can be easily determined from conventional laboratory tests. Although a swelling index is not adopted in ourmodel, a reasonable unloading behavior can be reproduced. This feature makes our model simpler than the existing hypoplastic models for clays [ 11,16,19]. Therefore, it is an ideal basic version to build more sophisticated models for predicting behaviors of clays under, e.g., partially sat- urated and thermal conditions. Moreover, numericalimplementation of the model is also quite straightforward even with the Matsuoka-Nakai failure criterion, and hence it is particularly suitable for practical applications. Acknowledgements The study was ﬁnancially supported by the H2020 Marie Skłodowska-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). The ﬁrst author wishes to thank theOtto Pregl Foundation for ﬁnancial support in Austria. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this(a) (b) Fig. 8 Comparison between experimental and numerical undrained triaxial tests on OC Boston Blue Clay. Experimental data from Pestana et al. [ 29]Acta Geotechnica (2021) 16:31–41 39 123 article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/ . Funding Open access funding provided by University of Natural Resources and Life Sciences Vienna (BOKU). Appendix The proposed model is formulated as follows: /C23T¼fsh ðtr/C20TÞDþfvðtrDÞ/C20Tþa2trð/C20TDÞ tr/C20T/C20T þfuað^Tþ^T/C3ÞkDkið25Þ where /C20T¼TþSand ^T¼T/C0Swith Sdenoting the structure tensor for overconsolidation. It is expressed asfollows: S¼aln/C16 1 R/C17 T ð26Þ in which Rstands for the degree of overconsolidation: R¼/C16 pþq2 M2p/C17 exphlnð1þeÞ/C0N k/C3i ð0\R61Þ ð27Þ The stiffness factor fsand the multipliers fvand fuare deﬁned as follows: fs¼/C02 3vik/C3;fv¼3 2vi/C01 3ð3þa2/C0ﬃﬃﬃ 3p aÞ;fu¼kB:Dk kBkkDk ð28Þ where Bdenotes the ﬂow rule of the model. In the above equations, aandMcorrespond to the limit stress at critical state, giving a¼ﬃﬃﬃ 3p gð3/C0sin/cÞ 2ﬃﬃﬃ 2p sin/c;andM¼6sin/c gð3/C0sin/cÞð29Þ where the factor gis adopted to incorporate the Matsuoka- Nakai failure criterion, to give g¼2I1 3ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðI1I2/C0I3ÞðI2 1/C03I2Þ=ð11I2/C09I3Þp /C0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ðI2 1/C03I2Þp ð30Þ in which I1,I2, and I3are stress invariants: I1¼trT;I2¼0:5½T:T/C0ðI1Þ2/C138;I3¼detT ð31Þ detTis the determinant of T.There are ﬁve parameters for the proposed model: /c,vi, k/C3,N, and a./cis the critical state friction angle; viis the ratio of the bulk modulus in the isotropic compression and the shear modulus in the undrained shear test on the iso- tropic consolidated sample; k/C3is the slope of the isotropic normal compression line in the double logarithmic plane lnð1þeÞ/C0lnp;Nis the value of ln ð1þeÞat the isotropic normal compression line for pr¼1 kPa; and acontrols the magnitude of the structure tensor.