## 2 Constitutive Model and Calibration
### Calibration of Parameters and Model Validation
#### 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 (\(\varepsilon_{1}\)) 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 \(E_{u}=\frac{q^{ampl}}{\varepsilon_{1}^{ampl}}\) (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 (\(\frac{E_{u}}{f_{E_{e}}}\)) versus strain amplitude (\(\varepsilon_{1}^{ampl}\)) 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  
Figure 2: Deviatoric stress and undrained Young’s module vs. axial strain: (**a,b**) test C7 with \(p_{0}=200\) kPa, \(q^{ampl}=60\) kPa, \(e=1.202\); (**c,d**) test C8 with \(p_{0}=200\) kPa, \(q^{ampl}=70\) kPa, \(e=1.202\)[33]. 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 (\(\varepsilon_{1}\)) 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 \(\varepsilon_{1}^{ampl}\). The Equation of the void ratio (used in Figure 3) obtained on the basis of experimental data is presented as follows [33] (Equation (10)):  
\[f_{E_{u},e}=\frac{(4.4-e)^{2}}{(1+e)} \tag{10}\]  
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]:  
\[\frac{E_{u}}{E_{u_{max}}}=f_{E_{u},ampl}=\frac{1}{1+\left(\frac{\varepsilon_{1 }^{ampl}}{\varepsilon_{1,r}^{ampl}}\right)^{\alpha}} \tag{11}\]  
where \(\varepsilon_{1,r}^{ampl}=3*10^{-4}\), 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., \(E_{u_{max}}\approx 30*f_{E_{u},e}\), Equations (9)-(11) can be used to express parameter \(C\) as follows (Equation (12)):  
\[C=E_{u_{max}}=30*f_{E_{u},ampl}*f_{E_{u},e}. \tag{12}\]  
Using the results obtained from the undrained cyclic triaxial test at very small strains (\(\varepsilon_{1}^{ampl}<10^{-6}\)), the value of the maximum undrained Young's modulus can be calculated.