# Analysis of Liquefaction in Tailings Deposits by Fem Modeling of Underained Cyclic Triaxial
## 4 UBC3D-PLM Constitutive Soil Model  
The UBC3D-PLM constitutive model was used to simulate the cycling stage of the CTX tests to obtain the response of the excess pore pressure and deviatoric stress. This constitutive model is based on the classical elastoplasticity theory developed to simulate liquefaction [33; 44]. The model has two yield surfaces to model the cyclic behavior of the soil, ensuring a smooth transition before liquefaction occurs. The primary yield surface is based on an isotropic hardening rule through a Mohr-Coulomb failure function in a three-dimensional principal stress space. The secondary surface contemplates a simplified kinematic hardening that generates smaller plastic strains than are generated on the primary surface. When the stress state is within the space defined by the yield surfaces, elastic behavior is assumed (see Figure 4a). However, once the stress path touches the yield surface, plasticity begins to occur (see Figure 4b). The transition between the first and second load is determined by the maximum value of the mobilized friction angle (\(\phi_{mob}\)). When the primary yield surface reaches the maximum stress state (governed by the maximum friction angle), the secondary yield surface is activated; that is, it starts to control the plastic behavior of the soil to ensure a smooth transition to the liquefaction state. The secondary yield surface is deactivated; that is, stops controlling the behavior when the mobilized stress state exceeds the previous maximum level defined by the peak friction angle. At this point, a stiffness that varies as a function of the input parameter \(f_{post}\) is used (see Figure 4c). The response of the elastic component is determined by the shear modulus \(\mathrm{K}^{e}_{\mathrm{C}}\) and the volumetric modulus [45]. These variables are defined using the following equations:  
\[K^{e}_{\mathrm{G}}=k^{e}_{\mathrm{G}}P_{A}\left(\frac{p^{\prime}}{P_{A}} \right)^{ne} \tag{1}\]  
\[K^{e}_{\mathrm{B}}=k^{e}_{\mathrm{B}}P_{A}\left(\frac{p^{\prime}}{P_{A}} \right)^{me} \tag{2}\]  
where \(k^{e}_{\mathrm{G}}\) and \(k^{e}_{\mathrm{B}}\) are the values of the reference shear and volumetric modulus, respectively, that are input by the user; \(P_{A}\) is the atmospheric pressure; \(p^{\prime}\) is the mean effective stress; and \(ne\) and \(me\) are exponents that define the dependency of the mean effective stress. The plastic shear modulus \(K^{p}_{\mathrm{G}}\) during primary loading is identical to that entered as an input parameter and is used to control the first yield surface. During secondary loading, the shear modulus \(K^{p}_{\mathrm{G}}\) increases as a function of the number of cycles to capture the effect of soil densification according to Equation (3), which was calibrated based on a cyclic single  
Figure 4: (**a**) Primary and secondary yield surfaces; (**b**) the transition between the primary and secondary surfaces; (**c**) deactivation of the secondary surface and post-liquefaction behavior. Modified from reference [46]. shear test. Subsequently, to include the post-liquefaction behavior of the soil, the shear modulus \(K_{G}^{p}\) is calculated using Equation (4):  
\[K_{G_{secondary}}^{p}=k_{G}^{p}\Big{(}4+\frac{n_{rep}}{2}\Big{)}f_{hard}*f_{dens} \tag{3}\]  
\[K_{G_{post-liquefaction}}^{p}=K_{G}^{p}*f_{post} \tag{4}\]  
where \(n_{rep}\) is the number of shear stress reversals, \(f_{hard}\) is a factor that corrects the densification rule for loose soil, \(f_{dens}\) is a user-entered multiplier to adjust this densification rule, and \(f_{post}\) is the input parameter used as the post-liquefaction factor.