## 2 Numerical Model Validation
### Material Constitutive Model and Parameters
Thus, the interface shear stiffness can be expressed as the following:  
\[\frac{K_{\text{si}}\cdot H}{P}=\chi(\sigma_{n})\cdot\ln(\tan\varphi_{i})+\eta (\sigma_{n}) \tag{1}\]  
where \(K_{\text{si}}\) = the interface shear stiffness, \(H\) = the wall height, \(P\) = the normal atmospheric pressure (i.e., 101.3 kPa), \(\varphi_{\text{i}}\) = the interface friction angle, \(\sigma_{\text{n}}\) = the interface vertical stress, and \(\chi(\sigma_{\text{n}})\) and \(\eta(\sigma_{\text{n}})\) = coefficients in relation to \(\sigma_{\text{n}}\). The values of \(\chi(\sigma_{\text{n}})\) and \(\eta(\sigma_{\text{n}})\), corresponding to different vertical stresses, are plotted in Figure 3b and can be expressed as the following:  
\[\left\{ \begin{array}{l}\chi(\sigma_{\text{n}})=a_{1}\big{(}\frac{\sigma_{ \text{n}}}{P}\big{)}^{2}+b_{1}\big{(}\frac{\sigma_{\text{n}}}{P}\big{)}+c_{1} \\ \eta(\sigma_{\text{n}})=a_{2}\big{(}\frac{\sigma_{\text{n}}}{P}\big{)}^{2}+b_{ 2}\big{(}\frac{\sigma_{\text{n}}}{P}\big{)}+c_{2}\end{array} \right. \tag{2}\]  
where \(a_{1,2}\), \(b_{1,2}\), and \(c_{1,2}\) = fitting parameters, and their values are given in Figure 3b. In GRS segmental walls, the interface vertical stress \(\sigma_{\text{n}}\) at the base of the facing is a function of the wall height \(H\): \(\sigma_{\text{n}}\) = \(H\)\(\cdot\)\(\gamma_{\text{fb}}\), where \(\gamma_{\text{fb}}\) = the unit weight of the facing block. The value of \(\gamma_{\text{fb}}\) can be considered constant because the plain concrete block used as the facing block usually has a density of 2200 kg/m\({}^{3}\)[12; 43; 44]. Therefore, if \(H\) and \(\varphi_{\text{i}}\) are known, the concrete-concrete interface shear stiffness \(K_{\text{si}}\) can be computed using Equations (1) and (2). Figure 4 shows the values of \(K_{\text{si}}\) corresponding to \(H\) in the range of 3 to 10 m and \(\varphi_{\text{i}}\) in the range of 5\({}^{\circ}\) to 45\({}^{\circ}\), calculated using Equations (1) and (2). This shear stiffness calculation model can provide concrete-concrete interface parameters for the numerical simulation of segmental retaining walls. Figure 3: Relationships among the shear stiffness \(K_{\text{si}}\), the friction angle \(\varphi_{\text{i}}\), and the normal stress \(\sigma_{\text{n}}\) of concrete–concrete interface: (**a**) \(K_{\text{si}}\) versus \(\varphi_{\text{i}}\); and (**b**) fitting functions \(\chi(\sigma_{\text{n}})\) and \(\eta(\sigma_{\text{n}})\). For the facing block-leveling pad interface in this wall, the friction angle was taken to be 36\({}^{\circ}\) according to the results of an interface shear test performed by [PERSON] et al. [8]. The shear stiffness of this interface can be set to 23 MPa, according to Figure 4. A small cohesion of 1 kPa instead of 0 was adopted for this interface to avoid numerical instability. For the interfaces between the facing blocks, the cohesion increased from 0 to 58 kPa as a result of the shear keys between the facing blocks [8]. Also because of the shear keys, the shear stiffness of these interfaces was set to 40 MPa, as recommended by [PERSON] and [PERSON] [17,18] and [PERSON] et al. [41], instead of being determined from Figure 4. A typical normal stiffness of 1000 MPa was applied to these two interfaces. For the concrete-soil interfaces, such as the facing block-backfill and the leveling pad-foundation soil interfaces, the values of the friction angle and cohesion were the same as those used by [PERSON] et al. [41]. The leveling pad-foundation soil interface had a normal stiffness of 1000 MPa and a shear stiffness of 10 MPa. The value of the shear stiffness was derived from the shear stress-shear displacement curves of a concrete-sand interface reported by [PERSON] et al. [32]; one of the curves was based on the normal stress that was close to 60 kPa, which is the normal stress value at the base of the leveling pad for this wall.