## 3 Nonlinear Kinematic Hardening Model  
### Simplified Nonlinear Kinematic Hardening (SNKH) Model  
The nonlinear kinematic hardening model has frequently been used for soil in dynamic analyses using ABAQUS [5; 6; 7; 16; 17]. In previous studies, a simplified nonlinear kinematic hardening (SNKH) model was applied with the implied von-Mises criterion and a back stress (\(\alpha\)), which control the translation of yield stress. The yield criterion is defined as:  
\[\mathrm{F=f\left(\sigma-\alpha\right)-\sigma_{o}=0} \tag{2}\]  
where \(\sigma_{o}\) is the initial yield stress at zero plastic strain. The increment of back stress is described as follows [5; 18]:  
\[\dot{\alpha}=C_{1}\frac{1}{\sigma_{o}}(\sigma-\alpha)\tilde{\varepsilon}^{ pl}-\gamma_{1}\alpha\tilde{\varepsilon}^{pl} \tag{3}\]  
where \(\tilde{\varepsilon}^{pl}\) is the increment of equivalent plastic strain. The three parameters required for the SNKH model are the initial yield stress (\(\sigma_{o}\)), initial kinematic hardening modulus (\(C_{1}\) is equal to [PERSON]'s modulus E), and rate of decrement of kinematic hardening modulus (\(\gamma_{1}\)). The model parameters could be estimated based on the assumption of equaling the stress at large plastic strain to the maximum yield stress (\(\sigma_{y}\)) of soil, which can be estimated from the soil strength parameters (i.e., friction angle \(\phi\), and cohesion, \(C\)) [5; 7]. The \(\sigma_{o}\) value can be estimated as a fraction (\(\lambda\)) of the yield stress (\(\sigma_{o}=\lambda\cdot\sigma_{y}\)), and the \(\gamma_{1}\) parameter could be calculated as:  
\[\gamma_{1}=C_{1}/\left(\sigma_{y}-\sigma_{o}\right) \tag{4}\]  
Figure 2: (**a**) Cylindrical column, and (**b**) torsional shear strain.  
By calibrating the \(\lambda\) parameter, which typically ranges from 0.1 to 0.3 for soil, the model parameters could be estimated to fit to the \(\mathrm{G}/\mathrm{G}_{\mathrm{max}}\) curve. The mean-average-error (MAE) was used to estimate the difference between the model and target curves, as:  
\[\mathrm{MAE}=\sum\lvert A_{m}-A_{t}\rvert \tag{5}\]  
where \(A_{m}\) and \(A_{t}\) are the \(\mathrm{G}/\mathrm{G}_{\mathrm{max}}\) values obtained from the SNKH model and target curve, respectively.  
Figure 1a-c show the \(\mathrm{G}/\mathrm{G}_{\mathrm{max}}\) and \(\mathrm{D}\) curves with SNKH model compared to the target curves with the model parameters indicated in Table 1. The \(\lambda\) parameter for the three soils varied (0.15 to 0.3). Note that the main point of the calibrating method was modifying the back-stress curve by changing the \(\sigma_{o}\) and \(\gamma_{1}\) parameters, without changing the ultimate shear strength, which is represented by summation of the \(\sigma_{o}\) and \(\mathrm{C}_{1}/\gamma_{1}\) values [5]. In this study, the \(\lambda\) parameter was chosen as the value that produced the lowest MAE, as shown in Figure 1d.  
The degradation of shear modulus and damping properties simulated in ABAQUS by the SNKH model partially matched the target curves. However, there were several error points: (1) the threshold shear strain (\(\epsilon_{th}^{SNKH}\)), the strain level that separates material behavior from elastic to nonlinear plastic that can be estimated as shear strain at \(\mathrm{G}/\mathrm{G}_{\mathrm{max}}\) of 0.99 [19; 20] in the model was much higher than those in experiments (\(\epsilon_{th}^{RCT}\)); (2) the \(\mathrm{G}/\mathrm{G}_{\mathrm{max}}\) curve at small strain range was not well simulated; and (3) the hysteretic damping in the SNKH model at low high strain level was much smaller than in experiment, as shown in Figure 1a-c. Similar errors could also be seen in the results reported by [PERSON] and [PERSON] [6] and [PERSON] and [PERSON] [7].