## 4 Validation and Application
### Application: High-Velocity Impact of Projectile on Ceramic Plate  
The SPH method with the present contact algorithm is now applied to the simulation of a practical problem originating from the reference [29]. The problem involves an alumina ceramic plate normally impacted by a blunt tungsten heavy alloy projectile with a high velocity. The ceramic plate has a square shape. Its side length is 101.6 mm and its thickness is 12.7 mm. The projectile is cylindrical. Its diameter is 6.14 mm and its length is 20.86 mm. It has an initial velocity of 903.9 m/s. Experiment results for the problem were reported by [PERSON] et al. in the reference [29]. A more systematic study conducted by [PERSON] and his co-workers [30] also provided experiment results for reference. The experimental results showed that extremely large deformation and material failure occur for both the projectile and the ceramic plate, so it is challenging for numerical simulation. Figure 14 shows the SPH model established for the problem. The projectile is discretized by particles with a spacing of about 0.6 mm. The ceramic plate is discretized by uniform  
Figure 13: Comparison of global z-direction velocity histories of the blocks calculated by SPH and FEM. particles with a spacing of 1.0 mm. There is a total of about 140 thousand particles in the model. For the simulation of this problem, the time step is set to 2 \(\times\) 10\({}^{-8}\) s, the artificial viscosity constants \(C_{L}\) and \(C_{Q}\) are 0.5 and 1.0, respectively, and surface reconstruction is performed in every time step. On the constitutive model, the JH-2 model [31] is implemented in our SPH program to model the dynamic behavior of the ceramic material. With the JH-2 model, the yield strength of the material is expressed in a normalized form as  
\[\sigma_{\rm y}^{*}=\sigma_{\rm i}^{*}-D(\sigma_{\rm i}^{*}-\sigma_{\rm f}^{*}) \tag{30}\]  
where \(D\) is the damage variable between 0 and 1, \(\sigma_{\rm i}^{*}\) is the normalized intact strength, and \(\sigma_{\rm f}^{*}\) is the normalized fracture strength. \(\sigma_{\rm y}^{*}\), \(\sigma_{\rm i}^{*}\) and \(\sigma_{\rm f}^{*}\) are normalized by the stress at Hugoniot elastic limit (HEL), \(\sigma_{\rm HEL}\). The expressions for \(\sigma_{\rm i}^{*}\) and \(\sigma_{\rm f}^{*}\) are  
\[\sigma_{\rm i}^{*}=a_{1}(P^{*}+P_{\rm T}^{*})^{n_{\rm I}}\Big{(}1+c_{1}\ln \dot{\varepsilon}^{*}\Big{)} \tag{31}\]  
and  
\[\sigma_{\rm f}^{*}=b_{1}P^{*m_{1}}\Big{(}1+c_{1}\ln\dot{\varepsilon}^{*}\Big{)}, \tag{32}\]  
respectively. In Equations (31) and (32), \(a_{1}\), \(n_{1}\), \(c_{1}\), \(b_{1}\) and \(m_{1}\) are material constants, \(P^{*}=P/P_{\rm HEL}\) is the normalized pressure, \(P\) and \(P_{\rm HEL}\) are the actual pressure and the pressure at the HEL, respectively, \(P_{\rm T}^{*}=P_{\rm T}/P_{\rm HEL}\) is the normalized maximum tensile hydrostatic pressure (tensile strength), \(P_{\rm T}\) is the maximum tensile hydrostatic pressure that material can undergo, and \(\dot{\varepsilon}^{*}=\dot{\varepsilon}/\dot{\varepsilon}_{0}\) where \(\dot{\varepsilon}\) and \(\dot{\varepsilon}_{0}\) are the actual strain rate and the reference strain rate, respectively. The damage \(D\) is defined as  
\[D=\sum\Delta\varepsilon_{\rm P}/\varepsilon_{\rm f}, \tag{33}\]  
where \(\Delta\varepsilon_{\rm P}\) is the plastic strain increment in a time step, and \(\varepsilon_{\rm f}\) is the failure strain which is defined as  
\[\varepsilon_{\rm f}=D_{1}(P^{*}+P_{\rm T}^{*})^{D_{2}}, \tag{34}\]  
where \(D_{1}\) and \(D_{2}\) are damage parameters. In the JH-2 model, the pressure for intact material (\(D=0\)) is determined as  
\[P=\left\{ \begin{array}{ll}K_{1}\mu+K_{2}\mu^{2}+K_{3}\mu^{3},&\mu\geq 0\\ K_{1}\mu,&\mu<0\end{array} \right.. \tag{35}\]  
For damaged material (\(D>0\)), the pressure of material under compression (\(\mu\geq 0\)) is modified as  
\[P=K_{1}\mu+K_{2}\mu^{2}+K_{3}\mu^{3}+\Delta P, \tag{36}\]  
Figure 14: SPH model for high velocity impact of ceramic plate impacted by projectile.