# Calibration of Finn Model and UBSCAN Model for Simplified Liquefaction Analysis Procedures
## 2 Constitutive Models
### UBCSAND Model  
The UBCSAND model [17; 18] is a simple elastoplastic stress/strain model for simulating the liquefaction phenomenon of sand with a relative density less than 80%. The elastic and plastic responses of the model are briefly described in this section. The elastic response is assumed to be isotropic and governed by shear modulus, \(G^{\prime}\), and bulk modulus, \(B^{\prime}\), as follows:  
\[G^{\prime}=K_{\rm G}{}^{e}\times P_{a}\times(\sigma^{\prime}/P_{a})^{me}, \tag{3}\]  
\[B^{\prime}=K_{B}{}^{e}\times P_{a}\times(\sigma^{\prime}/P_{a})^{me}, \tag{4}\]  
where \(K_{\rm G}{}^{e}\) is the elastic shear modulus number which can be related to the SPT N-value, \(P_{a}\) is the atmospheric pressure, \(\sigma^{\prime}\) is the mean effective stress in the plane of loading, \(ne\) is the exponent variable that relates the elastic shear modulus to the mean effective stress, \(K_{\rm B}{}^{e}\) is the elastic bulk modulus number (which depends on Poisson's ratio), and \(me\) is the elastic exponent variable that relates the bulk modulus to the mean effective stress.  
Plastic responses are divided into the plastic shear strain, \(\gamma^{p}\), and the plastic volumetric strain, \(\varepsilon_{p}{}^{p}\), controlled by the yield surface and the flow rule. The yield surface is represented by a radial line from the origin in \(\sigma^{\prime}-\tau\) space (mean effective stress-shear stress space, Figure 1). When the shear stress ratio (\(\eta=\tau/\sigma^{\prime}\)) changes, the plastic shear strain increment, \(d\gamma^{p}\), occurs as follows:  
\[d\gamma^{p}=(\sigma^{\prime}/G^{p})\times d\eta, \tag{5}\]  
\[G^{p}=G_{i}{}^{p}\times(1-(\eta/\eta_{f}\times R_{f}))^{2}, \tag{6}\]  
\[G_{i}{}^{p}=K_{\rm G}{}^{p}\times P_{a}\times(\sigma^{\prime}/P_{a})^{mp}, \tag{7}\]  
where \(G^{p}\) is the plastic shear modulus, \(d\eta\) is the stress ratio increment, \(G_{i}{}^{p}\) is the plastic shear modulus at a low shear stress ratio level, \(\eta_{f}\) (\(=sin\varphi_{\rm peak}\)) is the failure stress ratio, \(\varphi_{peak}\) is the peak friction angle, \(R_{f}\) is the failure ratio, \(K_{\rm G}{}^{p}\) is the plastic shear modulus number, and \(np\) is an exponent variable that relates the plastic shear modulus to the mean effective stress.  
Figure 1: Yield surface, flow rule, and directions of plastic potential increments of the UBCSAND model.  
A non-associated flow rule is adopted to connect the plastic volumetric strain increment, \(d\varepsilon_{v}p\), with the plastic shear strain increment, \(d\gamma^{p}\), as follows:  
\[d\varepsilon_{v}p=(sin\varphi_{cv}-\tau/\sigma^{\prime})\times d\gamma^{p}, \tag{8}\]  
where \(\varphi_{cv}\) is the constant volume friction angle or phase transformation angle. The direction of the plastic potential increment (i.e., the direction of the arrow on the yield surface) is shown in Figure 1. When the shear stress ratio \(\eta\) (\(\tau\) /\(\sigma^{\prime}\)) is equal to \(sin\varphi_{cv}\), there is zero plastic volumetric strain increment, and the material is in a constant-volume condition. When \(\eta\) is higher than \(sin\varphi_{cv}\), \(d\varepsilon_{v}p\) is negative, and the material is in a dilation condition. When \(\eta\) is lower than \(sin\varphi_{cv}\), \(d\varepsilon_{v}p\) is positive, and the material is in a contraction condition. The detailed information of the UBCSAND model is described in the literature [17; 18; 19].  
The UBCSAND model was coded with FLAC [16] and validated by many studies [17; 18; 20; 21; 22; 23; 24] via application to centrifuge tests, laboratory tests, and earthquake measurements. [PERSON] and [PERSON] [19] provide a set of calibrated model input parameters against the liquefaction triggering curve of the NCEER procedure [10].