## 2 Constitutive Model and Calibration
### Calibration of Parameters and Model Validation
#### 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)). \[S_{u}=\frac{\left(\sigma_{1}-\sigma_{3}\right)_{max}}{2}=\frac{q_{max}}{2} \tag{13}\]  
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):  
\[\sigma_{y}=\frac{2\sqrt{3}}{3}\;S_{u}. \tag{14}\]  
The effective maximum yield stress considering the pure shear test is calculated as follows:  
\[\sigma_{y}=\sqrt{3}\;S_{u}. \tag{15}\]  
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 (\(\varepsilon_{1}\)) 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 \(\lambda\) (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. Figure 4: Stress–strain relationship in the monotonic triaxial test (\(q\) and \(S_{u}\) ) [33]. 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 [PERSON] and [PERSON] [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) (\(\sigma_{0}=\lambda\sigma_{y}\)). Figure 5: Hysteresis loops of a cyclic simple shear test at different strain levels: **(a)** engineering shear strain vs. time, **(b)**\(\gamma=0.001\), **(c)**\(\gamma=0.007\), **(d)**\(\gamma=0.01\). Figure 6: Comparison of an Finite Element-computed \(G-\gamma\) curve with the experimental curve of [PERSON] and [PERSON] (1993).