## 3 The Numerical Model of Wedge-Ice-Water-Air Coupled Collision
### Material Model and Equation of State  
The process of the wedge breaking through ice into water involves four materials: wedge, ice, air, and water. The wedge and the ice material model will be introduced in the following sections. For air and water, it is necessary to define the material model and the corresponding fluid state equation to solve the governing equation.  
In LS-DYNA, the material properties of air and water can be simulated using the empty material (\"MAT_NULL) model, by simply specifying the fluid density. The equation of state for the fluid describes the relationship between volume deformation and pressure. In this study, the linear POLYNOMIAL equation of state is used for air, and the GRUNESIEN equation of state is used for water.  
The state equation for air medium in LS-DYNA is defined by the keyword *EOS-LINER-POLYNOMIAL. In the linear POLYNOMIAL equation of state, the relationship between pressure value and relative volume is expressed as follows:  
\[P=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+\left(C_{4}+C_{5}\mu+C_{6}\mu^{2} \right)E \tag{6}\]  
\[\mu=\frac{1}{V}-1 \tag{7}\]  
where \(C_{0}\)-\(C_{6}\) are the constant; \(V\) is the relative volume; and \(E\) is the specific internal energy of the fluid.  
In this study, the values of each constant are \(C_{0}=C_{1}=C_{2}=C_{3}=C_{6}=0\), \(C_{4}=C_{5}=0.4\), and \(E=2.5e5\).  
The state equation of water is defined by the keyword *EOS-GRUNESIEN in the LS-DYNA program. The GRUNESIEN equation of state results in different expressions between pressure and relative volume depending on the compression or expansion state of the material. The pressure expression in the compressed state is as follows:  
\[P=\frac{\rho_{0}C^{2}\mu\left[1+\left(1-\frac{\gamma_{0}}{2}\right)\mu-\frac{ \beta}{2}\mu^{2}\right]}{\left[1-\left(S_{1}-1\right)\mu-S_{2}\frac{\mu^{2}}{ \mu+1}-S_{3}\frac{\mu^{3}}{\left(\mu+1\right)^{3}}\right]^{2}}+\left(\gamma_{ 0}+\alpha\mu\right)E \tag{8}\]  
\[\mu=\frac{1}{V}-1 \tag{9}\]  
where \(\rho_{0}\) is the fluid density; \(C\) is the speed of sound in water; \(E\) is the specific internal energy of the fluid; and \(\alpha\), \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(\gamma_{0}\) are the input parameters related to pressure propagation. The constants for the water materials used in this study are shown in Table 1.  
In this study, the wedge is defined as a rigid body, and the elastic-plastic strain rate material model is selected for the ice layer. The interaction between the wedge and the ice layer, as well as the interaction between the ice cells, is realized using the erosion contact algorithm, where the ice cells are constantly damaged and cracks are formed by removing the failing cells. Once the grid cells fail and the contact surfaces are changed, the surface shapes can be re-established via erosion contact.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\rho_{0}\) & \(C\) & \(S1\) & \(S2\) & \(S3\) & \(\alpha\) & \(\gamma_{0}\) & \(E\) \\ \hline kg/m\({}^{3}\) & m/s & & & & & & \\ \hline
998 & 1480 & 1.92 & \(-\)0.096 & 0 & 0 & 0.35 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Water parameters.