# Cnoidal Wave-Induced Residual Liquefaction in Loosely Deposited Seabed under Coastal Shallow Water
## 2 Numerical Model and Constitutive Model  
In this study, the integrated numerical model FSSI-CAS 2D developed by [PERSON] et al. (2013) [19] and [PERSON] et al. (2013) [20] is utilized as the computational platform. FSSI-CAS 2D is an integrated numerical model especially for the problem of fluid-structure-seabed interaction (FSSI) in the field of offshore geotechnics, in which the VARANS equation is used to govern the hydrodynamics of seabed water, and the Biot 'u-p' equation is used to govern the dynamics of an offshore structure and its seabed foundation under external dynamic loading. A one-way coupling algorithm based on the time unmatched and mesh unmatched conception was first proposed by [PERSON] et al. (2013) [19] and [PERSON] et al. (2013) [20] to integrate the VARANS equation and the Biot 'u-p' equation, finally forming the model FSSI-CAS 2D. The validation of FSSI-CAS 2D for the FSSI problem has been widely verified by taking analytical solutions, a series of laboratory wave flume tests, and centrifuge geotechnical tests (for detailed information, please see [PERSON] et al. (2013)) [19]. This integrated model also has been successfully utilized to investigate the interaction between the offshore breakwater and various ocean waves, such as regular waves, breaking waves, and tsunami waves. Furthermore, FSSI-CAS 2D has been further extended to study the seismic dynamics and failure process of offshore breakwater built on a loosely deposited seabed foundation (see [PERSON] and [PERSON] (2015)) [21]. Overall, FSSI-CAS 2D is an excellent computational platform for FSSI problems in the field of offshore geotechnics. It has been widely recognized that the poroelastic soil model can be only applicable when the seab soil is in a very dense state. For loose seabed soil, elastoplastic soil constitutive models must be used due to its complicated and nonlinear mechanical behavior. In this study, the soil model Pastor-Zienkiewicz Mark III (PZIII) is utilized to describe the nonlinear behavior of the loosely deposited seabed under the action of the cnoidal wave. PZIII is already available in FSSI-CAS 2D, which was first incorporated into the numerical model by [PERSON] (1988) [22]. PZIII was established by [PERSON] and [PERSON] (1984) [23] based on the generalized plasticity theory, where there is no need to explicitly give the specific form of the functions describing the yield surface \(f\) and the plastic potential surface \(g\) in stress space. The unit normal vector of the yield and plastic potential surfaces in the \(p\)-\(q\) coordinates and the plastic modulus \(A\) are directly and explicitly given out. According to the plasticity theory, the elastoplastic matrix \(D_{\text{ep}}\) can be formulated as  
\[D_{ep}=D_{e}-\frac{D_{e}\cdot n_{g}\cdot n_{f}^{T}\cdot D_{e}}{A+n_{f}^{T}\cdot D _{e}\cdot n_{g}} \tag{1}\]  
where \(D_{e}\) is the 6 \(\times\) 6 elastic matrix, relating to elastic parameters of soil. \(n_{g}=\frac{\partial g}{\partial x_{ij}}\) and \(n_{f}=\frac{\partial f}{\partial x_{ij}}\) are the unit normal vector of the yield and plastic potential surfaces in stress space. \(A\) is the plastic modulus. In PZIII, \(n_{g}\) and \(n_{f}\) are directly given out in \(p\)-\(q\) coordinates as  
\[n_{g}=\left( \begin{array}{c}\frac{d_{g}}{\sqrt{1+d_{g}^{2}}}\\ \frac{1}{\sqrt{1+d_{g}^{2}}}\end{array} \right),n_{f}=\left( \begin{array}{c} \frac{d_{f}}{\sqrt{1+d_{f}^{2}}}\\ \frac{1}{\sqrt{1+d_{f}^{2}}}\end{array} \right) \tag{2}\]  
in which \(d_{g}=\left(1+\alpha_{g}\right)\left(M_{g}-\eta\right)\), and \(d_{f}=\left(1+\alpha_{f}\right)\left(M_{f}-\eta\right)\). \(M_{g}\) is the slope of the critical state line (CSL), and \(\eta=q/p\) is the stress ratio. \(\alpha_{g}\), \(\alpha_{f}\), \(M_{f}\) are parameters characterizing the deformation of soil. Among them, \(M_{f}\) is closely related to the density and \(M_{g}\). It is suggested that \(M_{f}/\ M_{g}=D_{r}\), where \(D_{r}\) is the relative density of soil.