## 4 Semi-Automated Procedure to Estimate the NKH Model Parameters for Soils and Rocks
### Semi-Automated Procedure to Estimate the NKH Model Parameters  
In the previous section, the NKH model parameters, including the initial yield stress \(\sigma_{o}\) and three pairs of kinematic hardening parameters (i.e., C\({}_{1}\)-\(\gamma_{1}\), C\({}_{2}\)-\(\gamma_{2}\), and C\({}_{3}\)-\(\gamma_{3}\)), were found for the three soils by manually using the trial-and-error method, which consumed a lot of time and effort. Therefore, in this section, an algorithm is proposed with the main purpose of rapidly finding the NKH model parameters to match the target G/Gmax and damping curves obtained from experimental results based on the basic properties of soil, such as the Young's modulus (E) or shear modulus (G), density (\(\rho\)), and Poisson's ratio (\(\
u\)). Figure 7 shows the flowchart of the semi-automated procedure estimating the NKH model parameters. The outputs of the process are the seven parameters of the NKH model  
Figure 5: Algorithm for estimating back stress and plastic strain in MATLAB. previously mentioned from the input, including E or G, \(\rho\), and \(\
u\), and the target G/Gmax and damping curves. The procedure could be divided into two parts: (1) finding the initial parameters from the G/Gmax curve; and (2) updating the parameters with consideration of both the G/Gmax and damping curves with a weighting factor (\(w\)). Figure 6: Comparison of (**a**) shear stress estimation, (**b**) shear stress–shear strain hysteretic loop, and (**c**) G/Gmax and damping curves between ABAQUS and MATLAB typically for silica sand (**left**) and Toyoura sand (**right**). There are three steps to estimating the initial parameter from the \(\mathrm{G/G_{max}}\) curve. The first parameter, initial yield stress, \(\sigma_{o}\), was estimated from G and the threshold shear strain (\(\varepsilon_{th}^{RCT}\)), as shown in Figure 7, following the method mentioned above. At a shear strain larger than \(\varepsilon_{th}^{RCT}\), the G of the material was reduced compared to \(\mathrm{G_{max}}\), and plastic strain and back stress are produced. In the SNKH and NKH models, the yield stress at a strain level is the summation of the initial yield stress and the back stress. Therefore, the back-stress target curve with the plastic shear strain could be estimated for a target \(\mathrm{G/G_{max}}\) curve in the second step, as shown in Figure 7. The target \(\mathrm{G/G_{max}}\) curve was discretized to approximately 200 data points in the logarithm scale for the shear strain space. At the shear strain \(\varepsilon_{k}\) with increment shear strain \(\Delta\varepsilon_{k}\), the shear stress, \(\sigma\), could be estimated by the shear modulus G calculated from the \(\mathrm{G/G_{max}}\) value. Plastic shear stress could be defined from the shear stress and the elastic shear stress, which is equal to \(\sigma\)/\(\mathrm{G_{max}}\). Back stress could also be calculated from the \(\sigma\) and \(\sigma_{o}\) values. Finally, by applying the equation of evolution of back stress with plastic strain (i.e., the positive phase of Equation (6)), the initial parameters (i.e., \(\mathrm{C_{1}}\), \(\gamma_{1}\), \(\mathrm{C_{2}}\), \(\gamma_{2}\), \(\mathrm{C_{3}}\), and \(\gamma_{3}\)) could be predicted. Figure 8 compares the \(\mathrm{G/G_{max}}\) and damping curves with the initial parameters with the target curves for the different materials. The results show slight differences between the \(\mathrm{G/G_{max}}\) curve obtained by the model with the target in the shear strain range (0.0005% to 0.01%), especially for the Toyoura sand in Figure 8c. Therefore, these initial parameters needed to be upgraded. In the second part of the procedure, the model parameters were modified to better fit the \(\mathrm{G/G_{max}}\) and/or damping curves. The \(\mathrm{G/G_{max}}\) and damping curves were divided into two or three segments by manually assigning the strain level at a slope-changing point, as shown in Figure 3b.