## 2 Constitutive Model and Calibration
### Calibration of Parameters and Model Validation
#### 2.2.2 Determination of \(\lambda\), \(\sigma_{0}\), and \(\gamma_{o}\)
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):  
\[\sigma_{y}=\frac{C}{\gamma_{o}}+\sigma_{0}. \tag{16}\]  
Finally, from Equations (14)-(16), parameter \(\gamma_{o}\) is calculated according to Equations (17) and (18):  
\[\sigma_{y}=\frac{2\sqrt{3}}{3}\;S_{u}. \tag{14}\]0  
(Triaxial test condition)  
\[\sigma_{y}=\frac{2\sqrt{3}}{3}\;S_{u}. \tag{14}\]1  
(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 (\(q=60\) kPa & 70 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 (\(\varphi\)) 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. \[\sigma_{y}=\frac{2\sqrt{3}}{3}\;S_{u}. \tag{14}\]2 _Cont._  
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 (\(\frac{F_{u}(\text{N}_{\text{N}}\text{N}_{\text{N}})}{F_{u}(\text{N}_{\text{N} }\text{N}_{\text{N}})}\)) 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).