## 2 Experimental and Numerical Methods
### Numerical Calculation Model and Effective Verification
#### 2.3.3 Material Model  
(a) Q235 steel  
The water tank and steel plate are made of Q235 steel. The Cowper-Symonds constitutive model is adopted, and its dynamic yield strength \(\sigma_{\text{d}}\) is:  
\[\sigma_{d}=\bigg{(}\sigma_{0}+\frac{EE_{h}}{E-E_{h}}\varepsilon_{p}\bigg{)}[1+ \bigg{(}\frac{\dot{e}}{D}\bigg{)}^{1/n}\bigg{]} \tag{1}\]  
where \(\sigma_{0}\) is the static yield strength, \(E\) is the elastic modulus, \(E_{h}\) is the strain hardening modulus, \(\dot{e}\) is the equivalent strain rate, and \(D\) and \(n\) are the strain rate parameters. Table 2 lists the material parameters.  
#### 2.3.3 Material Model  
The material of the projectile is 45# steel, and Johnson-Cook constitutive model is selected:  
\[\sigma=\big{(}A+B\varepsilon_{p}{}^{n}\big{)}\bigg{(}1+C\ln\frac{\dot{\varepsilon }_{p}}{\dot{\varepsilon}_{po}}\bigg{)}\bigg{[}1-\bigg{(}\frac{T-T_{0}}{T_{m}-T_ {0}}\bigg{)}^{m}\bigg{]} \tag{2}\]  
where \(\sigma\) Is the dynamic yield strength of steel, \(\varepsilon_{p}\) is the plastic strain, \(A\) is the static yield limit, \(B\) is the strain hardening modulus, \(n\) is the strain hardening index \(C\) is the strain rate coefficient, \(\varepsilon_{p0}\) is the critical strain rate, \(m\) is the thermal softening index, \(T\) is the temperature, \(T_{m}\) is the melting point of the material, and \(T_{0}\) is the room temperature.  
The J-C failure model is adopted to describe the failure of the materials:  
\[\varepsilon_{f}=\Bigg{\{}D_{1}+D_{2}\exp\bigg{[}D_{3}\frac{\sigma_{h}}{ \sigma_{eff}}\bigg{]}\Bigg{\}}\Big{[}1+D_{4}\ln\dot{\varepsilon}^{*}\Big{]}(1 +D_{5}T^{*}) \tag{3}\]  
where \(D_{1}\)-\(D_{5}\) are material parameters, \(\sigma_{eff}\) is the von Mises stress, \(\sigma_{h}\) is the hydrostatic pressure of the material under the triaxial stress, \(T^{*}=(T-T_{r})/(T_{m}-T_{r})\) is the dimensionless temperature, \(T_{r}\) is the room temperature, and \(T_{m}\) is the melting point of the material.  
The material parameters of the projectile are presented in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  
The Gruneisen equation of state (EOS) is adopted for water:  
\[P=\frac{\rho_{0}C^{2}\mu\big{[}1+\big{(}1-\frac{\gamma_{0}}{2}\big{)}\mu-\frac{ \alpha}{2}\mu^{2}\big{]}}{\bigg{[}1-(S_{1}-1)\mu-S_{2}\frac{\mu^{2}}{\mu+1}-S_{ 3}\frac{\mu^{3}}{(\mu+1)^{2}}\bigg{]}^{2}}+(\gamma_{0}+\mathrm{a}\mu)E \tag{4}\]  
where \(\rho_{0}\) is the density, \(C\) is the speed of sound, \(\gamma_{0}\) is the Gruneisen parameter, \(\mu=\rho/\rho_{0}-1\), a is volume correction, and \(S_{1}\), \(S_{2}\) and \(S_{3}\) are curve fitting parameters.  
(d) Air  
When air is an ideal gas without viscosity, its EOS is a linear polynomial:  
\[P=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+\Big{(}C_{4}+C_{5}\mu+C_{6}\mu^{2} \Big{)}E \tag{5}\]  
where \(C_{0}\)-\(C_{6}\) are the parameters and \(E\) is the internal energy. Let \(C_{0}=C_{1}=C_{2}=C_{3}=C_{6}=0\), \(C_{4}=C_{5}=\gamma-1\), so that it has ideal gas characteristics, where \(\gamma\) is the adiabatic exponent. The material parameters of water and air are shown in Table 4.