An activation mechanism for cyclic degradation of clays in bounding surface plasticity

During undrained cyclic loading, clayey soils experience substantial stiffness and strength degradation when subjected to shear amplitudes exceeding a critical threshold. This paper presents an enhanced bounding surface rate‐independent plasticity model, an evolution of the previous SANICLAY model, tailored to capture this specific behavior during cyclic loading. A distinguishing feature of the proposed model is the introduction of an activation mechanism. This mechanism triggers degradation modeling based on the applied cyclic shear amplitude. To measure this amplitude, the activation mechanism incorporates a novel state variable that serves as a proxy for the applied cyclic stress. The effectiveness of the proposed model is demonstrated by comparing it to experimental data from various materials subjected to cyclic shearing under undrained conditions. The study encompasses a broad range of constant strain or stress amplitudes. Compared to the reference model, the proposed model exhibits improved predictive accuracy for the stress‐strain response of clays at small amplitudes of cyclic loading and large number of cycles. Furthermore, it accounts for strength degradation due to cyclic loading.

rearrangement, which strengthens the soil microstructure until reaching a "stable state" capable of accommodating further cyclic loading without substantial damage.Conversely, major micro-structural modifications arise from the application of CSRs greater than the threshold.These modifications are subsequently manifested in the degradation of the undrained shear strength ( u ), as evident in the outcomes of monotonic tests conducted on cyclically loaded specimens. 1,6uring the simulation of transient loading conditions, such as earthquakes, the performance and stability of structures founded on saturated clayey deposits should be evaluated with constitutive models capable of capturing the salient features of the undrained response of clays under any applied cyclic shear amplitude (either stress or strain) and number of cycles, while still leading to satisfactory predictions under monotonic loading conditions.For modeling the anisotropic response of clays, a distorted and rotated yield surface along with rotational hardening was incorporated into a Critical State (CS) theory-compatible constitutive model with an associated flow rule. 7Non-associated flow rule was further implemented to capture the softening response exhibited by  0 -consolidated specimens during undrained compression. 8Eliminating the rotational hardening and changing the flow rule to associative render the model identical to the well-known Modified Cam-Clay (MCC).For this model, the name SANICLAY was chosen as the acronym for Simple ANIsotropic CLAY.0][11] Among all, Taiebat et al. 12 introduced isotropic and frictional destructuration into the SANICLAY framework for capturing the extreme strain softening of natural clays, which yields stiffness and strength degradation during undrained monotonic shearing. 13n parallel, the bounding surface (BS) concept was developed by Dafalias et al. 14 for capturing the strain-amplitude dependency of the stiffness at small strains.This type of modeling was achieved by enclosing the yield surface of the classical elastoplastic framework with a BS and by expressing the plastic modulus  p as a function of the distance between the stress point and the BS.6][17][18][19][20][21][22][23][24] Likewise, this concept has been implemented into a SANICLAY model by Taiebat et al. 25 and subsequently employed by the SANICLAY models by Seidalinov and Taiebat 26 and Shi et al. 27 for the cyclic loading conditions.In these models, improved predictions of the cyclic stiffness degradation were obtained with the inclusion of a damage mechanism producing additional plastic deviatoric strains with cycles.A SANICLAY model with BS, damage, and destructuration was implemented into the OpenSEES program and used for nonlinear site response analysis 28 and multidirectional cyclic shearing. 29However, with the exception of the work by Palmieri et al., 30,31 which represents a precursor of this work, the currently existing SANICLAY model extensions for cyclic loading, do not account for the presence of a stress and/or strain threshold and, therefore, always predict cyclic stiffness degradation regardless of the applied cyclic shear amplitude.To overcome this limitation, different sets of model parameters for the same material could be used depending on the cyclic shear amplitude.However, this solution, although practical for simulating nearly constant amplitude loading, would not be applicable to boundary value problems that impose variable amplitudes of cyclic shearing.
The aim of this research is to develop a SANICLAY model extension, named SANICLAY-BS, that effectively captures clayey soil degradation under varying cyclic loading amplitudes and cycle numbers.In contrast to existing SANICLAY models for cyclic loading, SANICLAY-BS employs destructuration, rather than damage, as the degradation mechanism.
The key feature of this proposed model is the inclusion of an activation mechanism, which governs the application of destructuration based on the cyclic shear level.By introducing a threshold for a newly introduced state variable, the model can replicate both the existence of a cyclic stress threshold in stress-controlled cyclic tests and the presence of a degradation strain threshold in strain-controlled cyclic tests.

Basic framework
In this section, the salient constitutive equations of the proposed SANICLAY-BS model are presented using triaxial stress quantities, namely  = ( a + 2 r )∕3 and  =  a −  r , along with strain quantities  v =  a + 2 r and  q = 2( a −  r )∕3.These definitions employ subscripts a and r to denote the axial and radial directions, respectively, and subscripts v and q to denote the volumetric and deviatoric components, respectively.The additive decomposition of the strain rate is assumed, denoted as ε = εe + εp , with superscripts e and p indicating the elastic and plastic components, respectively, while the superimposed dot implies rate.The complete multiaxial formulation is presented in Section 2.4. Figure 1 illustrates the characteristic surfaces of the proposed model.This model adheres to the original SANICLAY model formulation by Dafalias et al., 8 incorporating isotropic destructuration as per Taiebat et al., 12 and BS concept by Dafalias. 15In this framework, a primary player of the BS concept is the distance  between the stress state (, ) and its corresponding image stress on the BS ( p, q), which is related to the non-dimensional distance ratio : where the stress distances  and  are defined by mapping the current stress onto BS by adopting a specific projection center ( c ,  c ) as depicted in Figure 1.Following Seidalinov and Taiebat, 26 Equation (2) shows  p at the actual stress (, ) is related to the bounding plastic modulus Kp at the corresponding image stress ( p, q), by means of the distance ratio  as follows: where  0 and  are the state variables describing the size and orientation (anisotropy) of the BS, respectively, ℎ is a positive shape hardening parameter, and  ≥ 1 is an indirect measure of the size of the elastic nucleus that is a surface homologous to the BS with projection center as center of homology (Figure 1).Here Kp is defined by the consistency condition Ḟ = 0, and  p ≥ Kp .For defining , a radial mapping rule is employed which adopts a projection center adjusting its position at each stress reversal event.Stress reversal takes place when the loading index  ≤ 0. In cases with no stress reversal, the projection center only tracks along with the BS, ensuring that its location remains within the BS.This mechanism guarantees the uniqueness of the image stress and has been adopted by Seidalinov and Taiebat 26 and Shi et al. 27 By setting a large number to , the size of the elastic nucleus becomes so large that the model behaves as if no BS formulation exists.Hence, the model does not show any plastic deformation during the cycles of loading that fall within the BS.The default setting of  = 1 26 reduces the nucleus to a point, activating the BS for elastoplastic behavior under positive loading index .This setting affects plastic modulus interpolation, enabling a smooth transition from near-elastic to regular SANICLAY elastoplastic response as stress approaches the BS ( → 1 and  p → Kp ).

State variable 𝒃 sr
Building upon the BS concept, the proposed model introduces a discrete memory state variable  sr as the value of  updated at the last stress reversal (see Figure 2B).In this definition, the instance of stress reversal is specifically chosen to represent the applied cyclic stress level.As such, this variable preserves the applied cyclic stress level information during loading.

(A) (B)
F I G U R E 2 Definition of the state variable  sr (A) and its evolution (B) during CTXU test.
The evolution of  sr during three loading cycles of an undrained cyclic triaxial (CTXU) shearing with a constant CSR is illustrated in Figure 2B.Initial conditions are assumed isotropic stress state with an overconsolidation ratio (OCR) of 1, resulting in an initial  sr value 1. Cyclic shearing is applied following a two-way symmetric pattern, with the deviatoric stress  varying between ± cyc , where  cyc represents the single amplitude of the deviatoric stress.As depicted in the figure, the value of  sr updates at each stress reversal, leading to two different values of  sr within a single cycle.These values of  sr depend on the applied shearing directions, as the image stress is mapped to different portions of the BS (e.g., compression or extension in triaxial cyclic shearing), thereby affecting the value of  and, correspondingly,  sr .In addition, the evolution of  sr is influenced by changes in the image stress ratio q∕ p, which asymptotically aligns with the Critical State (CS) stress ratio, represented by  c and  e , depending on the shearing direction.When the image stress ratio aligns with the CS stress ratio for a specific shearing direction, the value of  sr tends toward a constant, that depends on initial conditions and the level of constant cyclic shear stress applied.This stabilization of  sr is a consequence of the interaction between two specific constitutive ingredients: (i) the application of volumetric hardening laws, and (ii) the updating of the projection center at stress reversal, in conjunction with the stress amplitude-controlled loading conditions.Further elaboration on this aspect will be provided in Section 2.5.
In the proposed model, the state variable  sr functions as a proxy for the cyclic shear stress ratio.Its integration with a specific threshold parameter is used for capturing the effects of cyclic degradation at different amplitudes of cyclic shearing and is essential for incorporating the concept of a cyclic stress threshold into the model.

The degradation mechanism
As a degradation mechanism for cyclic loading, the proposed model combines the isotropic destructuration of Taiebat et al. 12 with a novel activation mechanism capable of triggering destructuration based on the amplitude of cyclic loading.
In contrast to the existing BS SANICLAY models for cyclic loading, 26,27 which adopt a damage mechanism for simulating cyclic stiffness degradation, here, the destructuration is preferred to additionally allow for the prediction of pore water pressure accumulation and strength degradation with cycles.A comparison of the performances of damage and destructuration during the simulation of undrained cyclic shear loading can be found in Palmieri. 32he destructuration law employed follows the original formulation by Taiebat et al., 12 which quantifies the size of the structured BS by means of: where  0,d and  i are state variables representing the size of the destructured BS and the isotropic structuration factor, respectively (Figure 1).To link the destructuration rate to the plastic strain rate, the destructuration plastic strain rate is introduced as: where a is a model constant that controls the relative contributions of deviatoric and volumetric plastic strain increments to εp d .Compared to the original formulation, a lowercase symbol is adopted here for this model constant.While the value of a can be adjusted between 0 and 1, a default value of 0.5 is recommended for this constant. 12he evolution equation for the state variable  i , which governs the size of the structured BS, is proposed as follows: where  represents the current void ratio,  and  are the slopes of the compression and rebound lines in the  − ln  plane, respectively, and  i is a model constant that controls the destructuration rate.The term  represents the activation mechanism, which can only assume values of 0 or 1, following: where  sr represents the state variable introduced in Section 2.2, and  thr sr is a model constant representing the threshold.The Heaviside step function () ensures that  = 0 for non-positive arguments ( ≤ 0) and  = 1 for positive arguments ( > 0).Combining the activation mechanism with the destructuration law prevents destructuration ( = 0) when  sr >  thr sr , while it leaves the occurrence of destructuration unaltered ( = 1) when  sr ≤  thr sr .The term  undergoes dynamic activation and deactivation in scenarios extending beyond constant CSR loading.This activation or deactivation follows each loading reversal and is contingent on the updated value of  sr in relation to the model parameter  thr sr .Additionally, if a cyclic shearing phase is followed by a different monotonic shearing or a consolidation phase, the state of  is governed by the outcome of the last reversal.

Multiaxial formulation
Table 1 presents the proposed model multiaxial formulation, while listing all model constants and state variables.The basic notation includes second-order tensors denoted with boldface symbols, inner product with the symbol ∶, and norm operator with ‖ ⋅ ‖.The  = tr is the mean effective stress where  represents the effective stress tensor and tr⋅ the trace operator;  =  −  is the deviatoric stress tensor with  being the second-order identity tensor.Similarly, volumetric and deviatoric components of the strain tensor  are indicated as  v = tr and  =  − ( v ∕3), respectively.The tensorial state variable  describes the orientation of the BS (anisotropy).In addition, the CS stress ratio is Lode angle dependent, that is, ().Similar to , the image stress tensor  and projection center stress tensor  c are also decomposed in hydrostatic, that is, p = tr() and  c = tr( c ), and deviatoric, that is, s =  −  and  c =  c −  c  components.The generalization of the triaxial formulation to the multiaxial space is obtained considering that for any deviatoric tensor  and its triaxial counterpart  =  1 −  3 , with  1 and  2 =  3 being the normal components of , the relation (3∕2) ∶  =  2 holds true.For the deviatoric stress tensor s, this consideration results into (3∕2) ∶  =  2 , where s 1 − s 3 =  1 −  3 =  in triaxial conditions.For the deviatoric strain tensor, it results in (2∕3) ∶  =  2 q , with e 1 − e 3 =  1 −  3 = (3∕2) q in triaxial conditions.
Table 2 highlights the changes from the reference model 26 to the proposed SANICLAY-BS model equations.
The model has been numerically integrated based on the second-order (or modified) Euler method, also known as the second-order accurate explicit method.This integration method is enhanced by an automatic error control mechanism where the errors between the first-order (forward Euler) and the second-order (modified Euler) estimates of stress and internal variables are computed.If the maximum of the computed errors exceeds a specified tolerance, substepping of the input strain increment is employed.This approach was proposed in the original work of Sloan et al. 33 Details of the application of this integration technique to the SANICLAY class of models are presented in Seidalinov, 34 which served as the basis for the numerical implementation followed in the present study.The resulting constitutive driver is compatible with the OpenSees finite element program.; a Some constants have indicated default values.

2.5
The stable state as q∕ p =  c .Recalling that both isotropic and rotational hardening laws exclusively depend on εp v , when q∕ p =  c , it follows that εp v = 0, which leads to ṗ = 0, resulting in cyclic loading at an almost constant .In such conditions, the image stress ratio q∕ p varies between  c and  e upon subsequent stress reversals.When the image stress ratio takes on values indicated as q∕ p = , the stress state is consistently mapped onto a BS that does not exhibit any hardening.This leads to a constant  value and, consequently, a constant  p value for repeated cycles.The hysteresis loops in Figure 3B primarily evolve within the first 10 cycles due to hardening induced by εp v , while negligible degradation is observed during the subsequent 90 cycles.
It is noteworthy that the interruption of stiffness degradation upon reaching q∕ p =  has been identified as a limitation in various studies, prompting the development of damage mechanisms and hybrid flow rules. 26,27In this model, however, this phenomenon is strategically utilized to simulate the behavior of clay under CSRs below the threshold.Following Qian et al., 5 this condition is referred to as the "stable state", characterized by a BS without hardening, that is, a stable BS.The stable state can be interpreted as an asymptotic state achievable after initial hardening caused by volumetric plastic strains.In line with the interpretation by Qian et al., 5 this initial hardening mimics the strengthening effect of particle rearrangement, which leads the material to its stable state, capable of enduring further cyclic loading without incurring plastic damage.
Similar to the simulated response observed under two-way cyclic loading, during one-way cyclic loading (simulations not detailed in this paper), the stress path in the - space freezes vertically when the q∕ p reaches  (stable state), but with one-way cyclic strain accumulation, consistent with the experimental results presented in ref. 3. It is worth mentioning the slight stress path loop observed in the experimental data, indicating a minor variation between loading and reverseloading phases, which is less pronounced in the simulations.This emphasizes the close agreement between observations and experimental findings on stress path freezing and strain accumulation, with the loop serving as a detailed point of further exploration in future studies.

MODEL CALIBRATION
Tables 3 and 4 present the constants and state variables of the SANICLAY-BS model, respectively.In comparison to the basic SANICLAY model by Dafalias, 8 the proposed model introduces three additional constants (ℎ,  i ,  thr sr ) and one state variable ( i ).The constant ℎ is employed to capture the dependency of stiffness on the shear strain level, without considering the influence of the number of cycles.The parameter  thr sr is used to emulate the presence of a cyclic stress threshold, and its value depends on the material state.The parameter  i and state variable  i regulate the rate of destructuration.Default values are recommended by ref. 26 for the constants (= 1) and (= 0.5), and therefore they are not listed in Tables 3.
For the initialization and calibration of the parameters in the proposed model, it is advisable to adopt a systematic approach that builds upon the hierarchical evolution of its features.To this end, one can commence the calibration by determining the MCC parameters based on data from monotonic shearing tests, then proceed to calibrate , , and  based on the simulation of monotonic shear tests, and finally address the fitting of the remaining parameters through simulation of cyclic shearing tests.More specifically, concerning the new model parameters, the following order of calibration has proven effective: (i) Determine parameter ℎ from a shear modulus reduction curve to relate cyclic shear strain and secant stiffness.(ii) Specify the threshold  thr sr for a soil state based on simulations without destructuration ( i = 0) to define the cyclic stress threshold.(iii) Determine parameters  i and the initial state variable  i from a cyclic strength curve to understand soil behavior under cyclic loading.
The application of these steps is illustrated using experimental data from the natural Cloverdale clay subjected to CTXU tests, as documented in the work by Zergoun and Vaid. 4,6The database includes cyclic tests conducted at a CSR ranging between 0.36 and 0.79, with  u = 56 kPa.Before applying cyclic shearing in these tests, the specimens underwent isotropic normal consolidation to mean effective stress of 200 kPa.These steps are applied following the calibration of the basic SANICLAY model constants, which were adopted from Seidalinov and Taiebat, 26 except for the parameter .This parameter was originally calibrated using monotonic tests with larger shear strains, leading to an underestimation of the shear modulus at small strains in cyclic tests, a common issue for Cam-Clay based models with hypo-elastic laws.To address this,  has been reduced compared to the original calibration to better capture the increased stiffness observed at small strains.strain amplitude.These values are derived from experiments and simulations of undrained cyclic simple shear (CSS) tests conducted with different values of ℎ for N = 1, without destructuration ( i = 0).The figure demonstrates that ℎ can be calibrated to accurately capture stiffness non-linearity, with higher values leading to a stiffer soil response.It is important to note that, with the subsequent activation of destructuration, model predictions can vary at larger strains when  sr ≤  thr sr .Therefore, when calibrating ℎ, particular attention should be given to smaller strains.Figure 5 illustrates the calibration of  thr sr using simulation results conducted without destructuration ( i = 0).In this figure, the variable  sr is plotted at the end of each cycle for CSR values of 0.54 and 0.57.These specific CSR values were chosen to encompass the threshold CSR of 0.55 observed in the experiments.The figure shows that during loading, the variable  sr gradually stabilizes at a constant value, indicating the presence of a stable BS.As part of the calibration process,  thr sr could be set between the constant values approached by  sr during the two simulations.Since the value of  thr sr is based on simulation results, this step should be carried out after calibrating the basic SANICLAY model and BS constants.In the absence of laboratory tests for determining the cyclic stress threshold and subsequent calibration of  thr sr , an alternative approach is to consider the degradation strain threshold proposed by Vucetic 36 and Díaz-Rodríguez and Santamarina. 1 Additionally, semi-empirical procedures, as discussed in Palmieri, 32 offer another viable option.Finally, Figure 6 illustrates the effect of  i and the pre-shear value of  i on the cyclic strength curve derived from simulations in comparison with experiments.Following Zergoun and Vaid, in this figure, the cycle at failure  f is defined as the point where the stress path reaches the monotonic failure envelope due to pore water pressure accumulation.This specific type of clay failure during cyclic loading has been observed in various other clayey materials (e.g., refs.37-40) and can be predicted by the proposed model, which adopts destructuration (and not damage) as the degradation mechanism.The figure demonstrates that by varying  i and  i , it is possible to control the location and curvature of the cyclic strength curve, thus capturing the destructuration rate under different CSRs.
It must be noted that manual calibration, often relying on trial and error, can be demanding due to the complex interplay of various model parameters.Recent advancements in model parameter optimization techniques, such as ExCalibre, 41 GA-cal, 42 and ACT, 43,44 have demonstrated efficiency in calibrating advanced constitutive model parameters, offering a superior alternative to manual methods.These approaches present a promising avenue for future research in the context of the proposed model, suggesting potential areas for future study.

Stress amplitude-controlled cyclic tests
This section assesses the performance of the proposed model during CTXU tests and includes a comparison with the model developed by Seidalinov and Taiebat, 26 hereafter referred to as the reference model.The evaluation is based on experimental data from natural Cloverdale clay 4,6 and reconstituted Ariake clay. 38igures 7 and 8 present the predictions of the proposed model and the reference model compared to data from Cloverdale clay.For the proposed model, parameters and initial values of the state variables were determined as explained in the previous section and summarized in Tables 3 and 4, respectively.For the reference model, calibration and initialization following Seidalinov and Taiebat 26 were performed.Figure 7 presents the stress-strain response at various levels of CSR.It is clear that the proposed model outperforms the reference model, particularly at the lowest CSR (= 0.47).In this case, the reference model predicts significant degradation, while the proposed model accurately replicates the minor degradation observed in the experimental stress-strain loops.Additionally, at CSR = 0.79, the proposed model provides a better match to the experimentally observed hysteretic response.This improved performance can be attributed to the absence of the damage mechanism, which is typically associated with high values of ℎ and leads to low damping predictions.
Figure 8 assesses the abilities of the two models to replicate the evolution of the accumulated axial strain  acc a with N under varying CSRs. Figure 8A illustrates that the reference model predicts accelerating strains under all CSRs and can significantly overestimate the strains for CSRs below the threshold.On the other hand, Figure 8B demonstrates that the proposed model can capture both attenuating and accelerating strains depending on the applied CSR.This suggests that the proposed model provides a more flexible and accurate representation of strain accumulation under different loading conditions compared to the reference model.In the subsequent evaluation of model performance, the study employs experimental data obtained from tests conducted on Ariake clay, as presented in the work by Yasuhara et al. 38 In these experiments, reconstituted clay specimens were isotropically normally consolidated to a stress of 200 kPa before subjecting them to various CSRs, expressed here as  cyc ∕2 in for ease of comparison with the original reference.For the proposed model, the constants and pre-shear values of the state variables utilized are summarized in Tables 3 and 4.These values were determined through the procedure outlined in Section 3 and are documented in Palmieri. 32For simulations with the reference model, the study adopts values provided by Seidalinov and Taiebat. 26It is important to note that while Ariake clay is reconstituted, the proposed model initializes the state variable  i with a value greater than 1.This differs from the initialization approach used by Seidalinov and Taiebat, where  i = 1 characterizes a reconstituted state as structureless, in accordance with the original concept by Gens and Nova. 457][48] Given this insight, it is acknowledged that a truly structureless soil may not exist.Therefore, in characterizing reconstituted soils, the adoption of a value  i > 1 is deemed appropriate.
Figure 9 illustrates the influence of CSR on the evolution of cyclic axial strain  a,cyc for N = 1000.Similar to Figure 8, this figure demonstrates that only the proposed model can accurately capture both attenuating and accelerating strains, depending on the applied CSR.Furthermore, both models predict a symmetrical evolution of  a,cyc in compression and extension.This contrasts with the experimental response, which exhibits a more rapid development of  a,cyc in extension.To account for this specific feature, the proposed model could be further refined by introducing a Lode angle-dependent BS with a lower value of  in extension than compression.This refinement has not be explored in the present study.

Strain amplitude-controlled cyclic tests
This section evaluates the performance of the proposed model in strain amplitude-controlled undrained CSS tests, where the shear strain amplitude is kept constant in subsequent loading cycles.The evaluations are conducted against experimental data ranges for soils with OCR 1-15 and various plasticity. 36,49,50Simulations representative of the model's performance adopt model constants and initial values of state variables for Cloverdale clay, as detailed in Tables 3 and 4.
The initial stress state in the simulation is defined by an axial stress of 80 kPa, under K 0 conditions, with an OCR of 2.5.
The response of the reference model was similarly analyzed by Palmieri et al. 51 Two levels of cyclic shear strain amplitude,  cyc , specifically 0.1% and 1%, are examined.The corresponding shear stressstrain responses are illustrated in Figure 10 for the first 100 simulation loading cycles.At a  cyc of 0.1%, the stress-strain loops exhibit minor degradation, primarily within the first 10 cycles.In contrast, at a  cyc of 1%, marked non-linearity is Comparison of representative simulation results with experimental ranges for  s ∕ max , damping ratio curves, and linear and degradation strain thresholds in strain amplitude-controlled undrained CSS at various cyclic shear strain amplitudes and number of loading cycles N. observed, distinctly revealing the hysteresis loops, which continuously degrade throughout the loading cycles.The varying levels of cyclic shear strain amplitude mobilize different cyclic stress amplitudes, influencing the corresponding values of  sr and its impact on activation.This explains the limited and non-progressive degradation observed at  cyc = 0.1%, as opposed to the more pronounced and persistent degradation at  cyc = 1%.
Figure 11 presents normalized secant shear modulus ( s ∕ max ) and damping ratio () as functions of cyclic shear strain amplitude ( cyc ) ranging between 0.0001% and 1%, obtained from simulations with varying values of the model constant ℎ.The  s for each  cyc level is measured based on the extreme shear strain levels of the N th stress-strain loop, with  max taken as the  s value at  cyc = 0.0001% and N = 1.Normalized modulus reduction curves are presented for N = 1, 5, and 10 cycles to illustrate modulus degradation.Damping values are calculated using  =  d ∕(4 s ), where  d represents the dissipative energy within a cycle (equivalent to the area of the stress-strain loop), and  s is the elastic energy stored at maximum strain (equal to the area of a triangle with vertices at the origin of the stress-strain space, the point of maximum shear strain, and its projection on the shear strain axis).Damping ratio curves are exclusively plotted for N = 5 as they are less affected by N compared to the  s ∕ max curves.Model simulations are compared with characteristic ranges of  s ∕ max and  curves for clays with a Plasticity Index (PI) between 0 and 200. 49The figure also includes experimental ranges for the variation of the linear strain threshold  l in clays with a PI between 0% -60% 36 and the degradation strain threshold  d in clays with a PI between 10% -55%. 50The cyclic strain thresholds  l and  d define the onset of stiffness reduction (i.e.,  s ∕ max < 1) and cyclic stiffness degradation, respectively.The model simulations are presented using three levels of constant ℎ and the remaining model constants for Cloverdale clay.Inspection of the figure shows a reasonable match between the ranges from the experimental database, and those resulting from the model simulations.
Figure 12 presents a comparison of the degradation index, denoted as  N ∕ 1 , 52,53 between the proposed model and the experimental data range.In this figure,  1 and  N represent the secant shear moduli at the 1 st and N th cycle, respectively, quantifying the degradation behavior.The simulations primarily utilize Cloverdale clay parameters with  i = 0.21.To demonstrate the effect of the destructuration rate parameter, the figure also includes results from simulations with an increased  i value of 0.8.
Under  cyc = 0.1%, degradation is minimal and primarily occurs within the initial 10 cycles.It can also be observed that at this level of  cyc the effect of  i is negligible, suggesting that destructuration remains inactive.In contrast, under  cyc = 1%, active destructuration leads to a reduction in the degradation index.This reduction continues until  i approaches F I G U R E 1 2 Comparison of representative simulation results with the experimental range 49 for degradation index variation in strain amplitude-controlled undrained CSS at cyclic shear strain amplitudes of 0.1% and 1%.
1, and the rate of reduction is controlled by  i .Notably, destructuration is only activated under  cyc = 1% and does not occur at  cyc = 0.1%.Consequently, the model predicts a degradation strain threshold  d falling within the range of 0.1%-1%, which aligns with the corresponding experimental range in Figure 11.Furthermore, the figure demonstrates that the proposed model predictions of  N ∕ 1 are in reasonable agreement with the experimental data range, particularly within the specified experimental ranges.This suggests that the model effectively captures the degradation behavior under different levels of cyclic shear strain.

Post-cyclic monotonic tests
An important implication arises when modeling cyclic stiffness degradation through destructuration, rather than damage.Specifically, employing destructuration enables the coupling between strength and stiffness degradations.Consequently, if the final stress reversal results in  sr >  thr sr (i.e., for CSRs below the threshold), strength degradation is inhibited.The behavior described here aligns with the response of clays observed in post-cyclic undrained monotonic tests, as documented in previous studies. 1,6he capabilities of the proposed model in capturing this specific aspect of clay behavior are illustrated in Figure 13, based on the experimental work on Cloverdale clay by Zergoun. 6Within the figure, the stress-strain responses observed during undrained monotonic triaxial shearing following CTXU tests at different CSRs and N are plotted, along with the results from undrained monotonic triaxial tests conducted on specimens that were not previously cyclically loaded (pre-cyclic).The pre-shear (cyclic and monotonic) condition of the specimens corresponds to the isotropic state of 200 kPa, the same as in the experiments in Figures 7 and 8.No drainage was allowed between the cyclic and post-cyclic monotonic phases.Model simulations are conducted with constants and initial values of the state variables reported in Tables 3 and 4, respectively.The comparison between experiments and simulation results shows that the proposed model is capable of capturing the degradation of the monotonic undrained shear strength, which develops only for CSRs larger than the threshold.

CONCLUSIONS
The SANICLAY-BS model is introduced in this paper as an enhanced version of the model by Seidalinov and Taiebat 26 for simulating the cyclic degradation of clays.In contrast to the previous model, the SANICLAY-BS model adopts destructuration as the degradation mechanism while abandoning the use of damage.This simplification enables the SANICLAY-BS model to effectively simulate continuous volumetric plastic strains and strength degradation during cyclic loading, allowing it to predict phenomena such as pore water pressure accumulation and the degradation of the monotonic strength during cyclic loading.One of the distinguishing features of the proposed model is the incorporation of an activation mechanism that triggers destructuration.This mechanism relies on a newly introduced state variable, serving as a surrogate for the applied cyclic stress level.This approach offers advantages over using a specific CSR value, ensuring broader applicability across various loading conditions.By establishing a threshold for this state variable, the model accommodates the cyclic stress threshold phenomenon observed in clay behavior during undrained cyclic triaxial shearing at different CSRs.Specifically, when cyclic stress levels fall below the threshold, the activation mechanism inhibits destructuration.Under these conditions, the model approaches a state known as the "stable state" within the BS, leading to the rapid cessation of cyclic degradation without exhibiting any hardening behavior.In contrast, when cyclic stress levels surpass the threshold, the activation of destructuration initiates continuous degradation until ultimate failure.The proposed model maintains hierarchical compatibility with the SANICLAY and MCC models.Compared to the SANICLAY model, it necessitates the inclusion of three additional constants and one state variable, all of which can be readily calibrated using a straightforward procedure.
The model's validation involves a comparison between the predicted and experimental responses for both natural Cloverdale clay and reconstituted Ariake clay.The experimental datasets encompass results from symmetrical two-way undrained cyclic triaxial tests at varying CSRs and post-cyclic undrained monotonic triaxial test results, considering an isotropic initial state.Notably, the simulations consistently exhibit a reasonable degree of agreement with the corresponding experimental results.Moreover, the model's performance is assessed through undrained CSS tests carried out at varying strain amplitudes, utilizing data from diverse clayey soils under a  0 -consolidated state.Results emphasize the model's effectiveness in this specific loading scenario and demonstrate a remarkable ability to predict the presence of a degradation strain threshold.This adaptability empowers the model to deliver accurate simulations for different amplitudes of cyclic shear stress or strain.Compared to other constitutive models with similar capabilities, the proposed model stands out for its ability to adjust its degradation mechanism's predictions based on the amplitude of cyclic loading.This adaptability addresses a significant challenge in developing constitutive models for cyclic loading, making the model highly valuable for applications involving more loading cycles.
The presented validation is limited to certain initial state and loading conditions.It would be of interest to assess the model response for other initial states and loading conditions and explore whether a constant threshold parameter ( thr sr ) can accurately capture the experimental results in those loading scenarios.Future research should explore different initial states and more complex loading conditions, including but not limited to different static shear stresses, changes in cyclic shearing amplitude, and various drainage conditions.

F I G U R E 1
Characteristic surfaces of the proposed SANICLAY-BS model in the triaxial stress space.

Figure 3 3
Figure3illustrates the stress path and stress-strain loops predicted by the proposed model with inactive destructuration during simulations of CTXU test characterized by a two-way symmetrical cyclic loading pattern up to 100 loading cycles ( = 100).In Figure3A, it is observed that at  = 10, the image stress ratio has reached the CS stress ratio, denoted

Figure 4 4 F I G U R E 5
focuses on the model's ability to represent shear stiffness at various strain levels.The figure presents the modulus reduction curve  s vs  cyc , where  s represents the secant shear modulus, and  cyc represents the cyclic shear Calibration of ℎ based on the modulus reduction curve for Cloverdale clay.4Calibration of  thr sr for Cloverdale clay based on results of simulations with CSR = 0.54 and 0.57 in the absence of degradation.CSR, cyclic stress ratio.

6
Calibration of the (A) destructuration constant  i and (B) initial value of state variable  i based on a cyclic strength curve for Cloverdale clay.4

7 F I G U R E 8
Stress-strain response in (A), (D) the experiments for Cloverdale clay 4 in comparison with those simulated by (B), (E) the reference model and (C),(F) the proposed model.Note that simulations adopt different values of elastic constant .Prediction of the cyclic strain accumulation during CTXU test at various CSRs in comparison with experiments for Cloverdale clay 4 : (A) reference model, and (B) proposed model.

9
Predicted evolution of the cyclic axial strain during CTXU test at various CSRs in comparison with experiments for Ariake clay 38 : (A) reference model, and (B) proposed model.Stress-strain response simulation of the proposed model under strain amplitude-controlled undrained CSS at cyclic shear strain amplitudes of 0.1% and 1%.Note: Axes have different limits.

3
Performances of the proposed model in capturing the reduction of the post-cyclic undrained shear strength due to various CSRs and  for Cloverdale clay: (A) experiments, 6 and (B) simulations. 35 Constitutive equations of the proposed model in the multiaxial stress-strain space.
TA B L E 1 Description of the proposed model constants and their calibrated values for Cloverdale and Ariake clays.Description of the proposed model state variables and initial values for Cloverdale and Ariake clays.
TA B L E 3