## 3 Perforation Model Validation
### Perforation Test and FE Model  
With an ogival-shaped nose, hard projectiles were used for penetration tests in which no apparent erosion occurs. The total projectile mass was 428 g, the shank diameter was 25.3 mm and the caliber-radius-head (CRH) of the ogival nose was 3.0, as shown in Figure 4.  
Figure 4: Projectile dimension [14]. Figure 3: Petaling diagram [17]. Simulations of the SC walls perforation tests were conducted with 200-mm thick concrete panels. Figure 5 depicts the target dimension together with its reinforced mesh. The projectile impact location point is denoted by '\(\times\)'. For the investigated concrete, the unconfined compressive strength of cylinder sample (\(f_{c}^{\prime}\)) was 41 MPa. For the numerical study, LS-DYNA, the extensively applied explicit solver, was adopted for impact simulations. Since the projectile material was of high strength and great hardness, the projectile was modeled as MAT_RIGID in the simulation. Successfully applied for concrete penetration simulations [30], the Holmquist-Johnson-Cook (HJC) model [31] was chosen to model the concrete material. Originally presented by [PERSON], [PERSON] and [PERSON] [32], the HJC concrete model was developed for the purpose of impact computations where the material experiences large strains, high strain rates and high pressures. Coupled with isotropic damage, the HJC concrete model is an elastic-viscoplastic model [33] where the deviatoric response is determined by the following constitutive law:  
\[\sigma_{\text{Y}}^{*}=\Big{[}A(1-D)+Bp^{*N}\Big{]}\big{[}1+C\ln\epsilon^{*} \big{]} \tag{12}\]  
in which \(\sigma_{\text{Y}}^{*}\)=\(\sigma_{\text{Y}}/f_{c}^{\prime}\) is the normalized equivalent stress, and \(p^{*}\)=\(p/f_{c}^{\prime}\) is the normalized pressure. The equation of state in HJC is characterized by three stages: plate elastic (for \(p<p_{c}\)) \(p=K\mu\); pore collapse (for \(p_{c}\leq p\leq p_{1}\)) \(p=p_{c}+(p_{1}-p_{c})(\mu-\mu_{1})/(\mu_{1}-\mu_{c})\); compaction (for \(p>p_{1}\)) \(p=k_{1}\overline{\mu}+k_{2}\overline{\mu}^{2}+k_{3}\overline{\mu}^{3}\) with \(\overline{\mu}=(\mu-\mu_{1})/(1+\mu_{1});k_{1},k_{2}\), and \(k_{3}\) are constants. The HJC model includes a scalar damage formulation, where the damage evolution is accumulated from both the equivalent plastic strain increment \(\Delta\varepsilon_{\text{eq}}^{\text{P}}\) and the equivalent plastic volumetric strain increment \(\Delta\mu_{\text{eq}}^{\text{P}}\). The damage evolution is expressed as:  
\[\Delta D=\frac{\Delta\varepsilon_{\text{eq}}^{\text{P}}+\Delta\mu_{\text{eq} }^{\text{P}}}{\varepsilon_{\text{p}}^{\text{f}}+\mu_{\text{p}}^{\text{f}}} \tag{13}\]  
where, \(\varepsilon_{\text{p}}^{\text{f}}+\mu_{\text{p}}^{\text{f}}=D_{1}(p^{*}+T^{* })^{D_{2}}\), \(\varepsilon_{\text{p}}^{\text{f}}\) and \(\mu_{\text{p}}^{\text{f}}\) are plastic strain and plastic volumetric strain corresponding to fracture, \(T^{*}\) is the normalized tensile strength, and \(D_{1}\) and \(D_{2}\) are damage constants. The steel plate was modeled by the Johnson-Cook (JC) model [34] for its wide adoption in the metal impact engineering domain. JC is a strain rate and temperature-dependent (adiabatic assumption) visco-plastic material model [35; 36]. This model is suitable for problems in which strain rates vary over a large range. The JC model expresses the flow stress with the form:  
\[\sigma_{\text{Y}}=\Big{[}A+B\varepsilon_{\text{p}}^{N}\Big{]}\big{[}1+C\ln \epsilon^{*}\big{]} \tag{14}\]  
Figure 5: Geometric dimension of SC walls target. where \(\sigma_{\rm Y}\) is the effective stress, \(\varepsilon_{\rm P}\) is the effective plastic strain, \(\epsilon^{\ast}\) is the normalized effective plastic strain rate (typically normalized to a strain rate of 1.0 s\({}^{-1}\)), \(N\) is the work hardening exponent, and \(A\), \(B\), \(C\) are constants determined by calibration. With reference to [37], the steel rebar was modeled using MAT_PLASTIC_KINEMATIC.