## 3 Nonlinear Kinematic Hardening Model
### Nonlinear Kinematic Hardening (NKH) Model  
In SNKH model, the \(\mathrm{G}/\mathrm{G}_{\mathrm{max}}\) and damping curves were not well modelled; therefore, in this study, the nonlinear kinematic hardening (NKH) model, which was basically developed from the SNKH model, was recommended. The NKH model has been a very common model used in mechanical engineering, and several researchers have proposed the procedure to estimate model parameters for metal materials based on experimental results [21; 22; 23; 24]. Using a similar yield surface function and the von-Mises failure criterion to the SNKH model, the NKH model was upgraded to three kinematic hardening moduli (i.e., \(\mathrm{C}_{1}\), \(\mathrm{C}_{2}\), and \(\mathrm{C}_{3}\)), and three rates of decrement of kinematic hardening modulus (i.e., \(\gamma_{1}\), \(\gamma_{2}\), and \(\gamma_{3}\), respectively). With the initial yielding stress, \(\sigma_{o}\), the model had seven input parameters, resulting in difficulties in finding appropriate parameters for ground materials. The back stress that included the three components, \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) with their evolution could be estimated from each pair of kinematic parameters (i.e., \(\mathrm{C}_{1}\)-\(\gamma_{1}\), \(\mathrm{C}_{2}\)-\(\gamma_{2}\), and \(\mathrm{C}_{3}\)-\(\gamma_{3}\), respectively) by Equation (3). Figure 3a shows the back-stress curves for silica sand in the SNKH and NKH models with the plastic shear strain using the equation proposed by [PERSON] and [PERSON] [21]. In the NKH model, the total back stress is the summation of three back-stress components. It can be seen that with the increasing \(\gamma^{\mathrm{p}}_{\mathrm{eq}}\) or shear strain, each back-stress component including \(\alpha_{1}\), \(\alpha_{1}\), and \(\alpha_{3}\), respectively, sequentially prevail in the total back-stress (\(\alpha\) in NKH model), and the total back-stress curve can be divided into three segments by the limitation of each back stress component. Therefore, each pair of \(\mathrm{C}_{i}\) and \(\gamma_{1}\) parameters controls the \(\mathrm{G}/\mathrm{G}_{\mathrm{max}}\) curve at each level of shear strain, as indicated in Figure 3b. A manual process was performed to estimate the seven input parameters for the NKH model. Firstly, the initial yield stress, \(\sigma_{o}\), was estimated from \(\mathrm{G}\) and the threshold shear strain (\(\epsilon_{th}^{RCT}\)), which is defined from the \(\mathrm{G}/\mathrm{G}_{\mathrm{max}}\) curve. Compared to those in the SNKH model, the \(\sigma_{o}\) value in the NKH model was slightly reduced, because the SNKH model produced a larger threshold strain. Due to the reduction in the \(\sigma_{o}\), three pairs of kinematic hardening parameters were firstly chosen to have slightly higher back stress at all strain range (Figure 3a). For each trial, the \(\mathrm{G}/\mathrm{G}_{\mathrm{max}}\) curve was obtained, and each pair of kinematic hardening parameters was modified following the instruction indicated  
in Figure 3b. By variation in its slope, the target G/G\({}_{\rm max}\) curves are divided into three parts, which are controlled by each pair of kinematic parameters, C\({}_{1}\)-\(\gamma_{1}\), C\({}_{2}\)-\(\gamma_{2}\), and C\({}_{3}\)-\(\gamma_{3}\), respectively. For each part of the G/G\({}_{\rm max}\) curve with its pair of kinematic hardening parameters, the reduction in C and increment in \(\gamma\) resulted in the increase in slope of the G/G\({}_{\rm max}\) curve, and _vice versa_, as shown in Figure 3b with blue and green arrows. This is because the reduction in C and/or increase in \(\gamma\) in each segment results in reduction in limitation of back stress; therefore, G reduces in that segment. The estimation process for kinematic hardening parameters should be performed in order from C\({}_{1}\)-\(\gamma_{1}\) to C\({}_{2}\)-\(\gamma_{2}\), and finally C\({}_{3}\)-\(\gamma_{3}\); and as a result, the G/G\({}_{\rm max}\) curve could gradually be fitted to the experimental curve.