The parameters of constitutive models of soils are often obtained by comparing experimental results and numerical simulations using finite elements at the element test level [

31]. In this study, experimental data from an undrained cyclic triaxial test are used to calibrate the parameter

$C$, and two methods for calibrating the parameters

$\lambda $,

${\gamma}_{o}$, and

${\sigma}_{0}$ are presented. First, the curve of the undrained Young’s modulus versus axial strain is drawn using the results obtained from the cyclic triaxial test and numerical simulations of the test to obtain the

$\lambda $ parameter. Second, data from a cyclic shear test and Ishibashi and Zhang’s equations [

32] are used.

The values of the parameters such as the undrained shear strength (

${S}_{u}$), maximum shear modulus (

${G}_{max}$) or shear wave velocity (

${V}_{s}$), and curve of the undrained Young’s modulus versus axial strain (

${E}_{u}$ Vs.

${\epsilon}_{1}$) are required to calibrate the parameters. A series of undrained cyclic triaxial and monotonic tests of kaolin clay were conducted by Wichtmann [

33] in both stress and strain control modes and under different level stress amplitudes.

In this study, the results of monotonic and cyclic tests under the stress control condition with an effective mean pressure of 200 kPa are used to calibrate the constitutive model parameters. The process mentioned above, by which the parameters of the model are calibrated, is fully described below.

#### 2.2.1. Determination of $C$

We use the results of the undrained cyclic triaxial tests at different stress amplitudes (C1–C8 and C32–C36) [

33] to determine the nonlinear undrained Young’s modulus. For instance,

Figure 2 shows the deviatoric stress (

$q$) and undrained Young’s modulus (

${E}_{u}$) versus the axial strain (

${\epsilon}_{1}$) curve with stress amplitudes of 60 kPa and 70 kPa for undrained cyclic triaxial tests of C7 and C8, respectively. Each loading and unloading loop (

Figure 2a,c) is used to calculate the undrained Young’s modulus value at the desired strain amplitude, using the equation

${E}_{u}=\frac{{q}^{ampl}}{{\epsilon}_{1}^{ampl}}$ (

Figure 2b,d). Therefore, each point in

Figure 2b,d is the value of the undrained Young’s modulus of the corresponding cycle in

Figure 2a,c. Thus,

Figure 2 depicts the stiffness degradation of soil under cyclic loading and unloading. As shown in the diagrams, by increasing the number of cycles and axial strain amplitudes, the undrained Young modulus or the shear modulus decrease, and the amount of damping caused by the soil plasticity increases. Although by increasing number of cycles the amount of axial strain is also increased, the increase in axial strain on both sides of the horizontal coordinate axis is unequal, such that there is more on the positive side (compression).

Figure 3 shows the normalized undrained Young’s modulus (

$\raisebox{1ex}{${E}_{u}$}\!\left/ \!\raisebox{-1ex}{${f}_{E,e}$}\right.$) versus strain amplitude (

${\epsilon}_{1}^{ampl}$) based on the results of numerical analysis, Equation (11), and the experimental results of the several undrained cyclic triaxial tests, where these are performed under different stress amplitudes (C1–C8 and C32–C36) [

33] to calibrate the parameters

$C$ and

$\lambda $ of the constitutive model. The calibration of parameter

$\lambda $ using the data in

Figure 3 is described in

Section 2.2.2. Each experiment test [

33] is based on a number of specified cycles, and numerical simulations were performed based on the same number of cycles. The parameters of the model have been calibrated in order to best fit the numerical model points with the experimental results. In this figure, the second and fifth cycles of curves of deviatoric stress (

$q$) versus axial strain (

${\epsilon}_{1}$) of the undrained cyclic triaxial tests are used to obtain the undrained Young’s modulus values (

${E}_{u}$); then, these values are divided by the function of the void ratio (Equation (10)) and plotted versus the corresponding strain amplitude

${\epsilon}_{1}^{ampl}$.

The equation of the void ratio (used in

Figure 3) obtained on the basis of experimental data is presented as follows [

33] (Equation (10)):

where

$e$ is the void ratio of soil.

As shown in

Figure 3, by increasing the strain amplitude, the values of the undrained Young’s modulus decrease, and this decreasing trend is proportional to Equation (11) [

34]:

where

${\epsilon}_{1,r}^{ampl}=3\ast {10}^{-4}$, corresponding to

$\frac{{E}_{u}}{{E}_{{u}_{max}}}={f}_{{E}_{u},ampl}=0.5$ and

$\alpha =0.9$ (for kaolin clay) [

33]. In addition, by subjecting the experimental data to extrapolation operations, e.g.,

${E}_{{u}_{max}}\approx 30\ast {f}_{{E}_{u},e}$, Equations (9)–(11) can be used to express parameter

$C$ as follows (Equation (12)):

Using the results obtained from the undrained cyclic triaxial test at very small strains (${\epsilon}_{1}^{ampl}<{10}^{-6}$), the value of the maximum undrained Young’s modulus can be calculated.

Then, according to the equation ${G}_{max}=\frac{{E}_{{u}_{max}}}{2\left(1+{\upsilon}_{max}\right)}$, the value of the maximum shear modulus can be obtained, where ${E}_{{u}_{max}}$ and ${\upsilon}_{max}$ are the maximum undrained Young’s modulus and Poisson ratio, respectively, and ${\upsilon}_{max}=0.49$ in the undrained case.

#### 2.2.2. Determination of $\lambda $, ${\sigma}_{0}$, and ${\gamma}_{o}$

The undrained shear strength (

${S}_{u}$) is obtained from the results of a monotonic consolidated undrained triaxial test.

Figure 4 shows the stress–strain relationship in the monotonic undrained triaxial test [

33]. According to this figure and using the following equation, the maximum value of deviatoric stress and undrained shear strength can be calculated as follows (Equation (13)).

where

${S}_{u}$ is the undrained shear strength,

${\sigma}_{1}$ and

${\sigma}_{3}$ are the maximum and minimum mean stress, respectively, and the deviatoric stress is

$q={\sigma}_{1}-{\sigma}_{3}$.

The effective maximum yield stress considering the triaxial test condition is calculated as follows (the detailed expression can be found in

Appendix B):

The effective maximum yield stress considering the pure shear test is calculated as follows:

The next parameter for calibration is

$\lambda $. The first method of calibrating

$\lambda $ in the kinematic hardening model is based on the undrained cyclic triaxial test data. The deviatoric stress (

$q$) versus axial strain (

${\epsilon}_{1}$) under the stress control condition at different stress amplitudes are plotted based on the simulation results of the undrained cyclic triaxial test, and similar to the experimental method, the undrained elastic modulus (

${E}_{u}$) of the second and fifth cycles of each test is obtained. Then, the curve of the undrained Young’s modulus versus axial strain is plotted for both the experimental data and simulation results to calibrate

$\lambda $. The simulation results of the undrained cyclic triaxial test and their comparison with the experimental results are presented in

Figure 3. Here,

$\lambda $ = 0.14 is found to provide a reasonable fit to the experimental results, where the simulation is performed using Abaqus software under the stress control condition.

If undrained cyclic triaxial test data are not available, the results of the cyclic shear test under the strain control condition and the Ishibashi and Zhang experimental equations using the plasticity index of soil are used to calibrate the parameter λ (second method). In this method, the curve data of shear modulus versus shear strain (

$G-\gamma $ curve) is required to calibrate this parameter. Thus, the hysteresis loops of the cyclic shear test simulation in different strain amplitudes should first be plotted. For instance, the hysteresis loops of three different strain levels are shown in

Figure 5. As can be seen, the value of the damping caused by the soil plasticity increases, and the shear modulus decreases if the strain levels are increased. Then, the shear modulus corresponding to the stabilized cycle is obtained for each strain amplitude to enable the plotting of the

$G-\gamma $ points to calibrate

$\lambda $ (see

Figure 6). For example, the shear stress–strain loops of three strain amplitudes in

Figure 5 correspond to the designated points in

Figure 6. These two figures illustrate the validation of the kinematic hardening model against the published

$G-\gamma $ curve of Ishibashi and Zhang [

32]. After calibration using this method,

$\lambda $ = 0.12 is obtained.

After determining the parameters ${S}_{u}$, ${\sigma}_{y}$, and $\lambda $, the initial size of yield surface (${\sigma}_{0}$) is obtained from Equation (7) (${\sigma}_{0}=\lambda {\sigma}_{y}$).

The last parameter to be determined is the rate decrease of the kinematic hardening modulus (

${\gamma}_{o}$). According to

Figure 1, the amount of effective maximum yield stress is determined by Equation (16):

Finally, from Equations (14)–(16), parameter

${\gamma}_{o}$ is calculated according to Equations (17) and (18):

(Triaxial test condition)

(Simple pure shear test condition).

Generally, the triaxial test is an acceptable method for assessing the cyclic behavior of soils, and usually this test is used to calibrate the parameters of different constitutive models. Applying relatively simple boundary conditions and creating proper meshes in simulations of this test afford logical and reliable results. Thus, the results of the cyclic triaxial test using nonlinear kinematic hardening and Mohr–Coulomb models on a clay sample are compared with the experimental model results to validate and investigate the kinematic hardening constitutive model. This test is modeled in the 2D axisymmetric case to achieve acceptable results. Application of a 200 kPa confining pressure is simulated as a uniformly distributed load applied normally inward to all external elements around the specimen; then, the cyclic load ($q=60\text{}\mathrm{kPa}\text{}\text{}70\text{}\mathrm{kPa}$) is applied to the top of the sample.

In the Mohr–Coulomb model, since the test is carried out in an undrained condition (i.e., an undrained cyclic triaxial test), the internal friction angle of soil (

$\phi $) is zero and the cohesion value (

$C$) equals the undrained shear strength of soil (

${S}_{u}$).

Figure 7 shows the comparison of deviatoric stress versus axial strain in numerical simulations (kinematic hardening model and Mohr–Coulomb) with the experimental results in stress control mode. As shown in the diagrams, the behavior of undrained soils using the kinematic hardening model is in relatively good agreement with the results obtained from the experimental test, while the Mohr–Coulomb model fails to create the hysteresis loops, and the results thus differ from the experimental results. In addition, the material behavior in the Mohr–Coulomb model is in the elastic range.

It is worth noting that this numerical simulation for the Mohr–Coulomb model is performed in two modes without considering and taking into account the damping ratios, and the results of the two cases are similar, which indicates that the linear elastic–perfectly plastic Mohr–Coulomb model cannot accurately predict the soil behavior under cyclic loading. In contrast, the kinematic model is capable of creating hysteresis loops and considers the soil stiffness degradation as the number of loading cycles increases. As shown in

Figure 7, the undrained Young’s modulus ratio of the numerical model to the experimental model (

$\frac{{E}_{u\left(\mathrm{NemericalModel}\right)}}{{E}_{u\left(\mathrm{ExperomentalModel}\right)}}$) increases with an increasing number of loading cycles, so that in the last cycle, it reaches its maximum value of approximately 1.4. Thus, a numerical model generally overestimates the soil stiffness degradation. The experimental curves in

Figure 7 are related to tests C7 and C8.

Finally, after calibrating and using the corresponding equations, the values of the different hardening parameters are calculated for all five soil layers (see

Figure 8). Thus, the values of the constitutive model parameters used in all of the analyses can be calibrated according to the second method (

Table 1).