## 4 Numerical Work
### Constitutive Relations  
The loading conditions during blast events may involve large plastic strains, high strain rates and thermal softening due to self-heating in the affected materials. To account for these effects, a thermoelastic-thermoviscoplastic constitutive model like the one proposed by e.g., [PERSON] et al. [35] should be applied for the aluminium skins. However, studies by [PERSON] et al. [30] on perforation of 5000-series aluminium plates and [PERSON] et al. [36] on low-velocity impact of Docol 600 DL steel plates showed that thermal softening effects are limited at moderate plastic strains and strain rates. Therefore, thermal effects in the aluminium skins will be neglected in this study. For the foam cores, the compression tests at elevated strain rates indicate a strain-rate sensitivity for all densities studied. A similar conclusion was given by [PERSON] and [PERSON] [37] and [PERSON] and [PERSON] [38] for polystyrene foams. Thus, an extension of the Deshpande-Fleck [39] crushable foam model was proposed and implemented as a user-material subroutine in LS-DYNA to account for rate sensitivity in the foam cores. The main assumptions in the applied models are an isotropic material and small elastic strains, while the plastic strains and rotations may be finite. Thus, a hypoelastic formulation is adopted through an additive decomposition of the rate-of-deformation tensor  
\[\mathbf{D}=\mathbf{D}^{\mathrm{e}}+\mathbf{D}^{\mathrm{P}} \tag{2}\]  
where \(\mathbf{D}^{\mathrm{e}}\) and \(\mathbf{D}^{\mathrm{P}}\) are the elastic and plastic parts, respectively. The elastic part is defined as  
\[\mathbf{D}^{\mathrm{e}}=\frac{1+\
u}{E}\sigma^{\mathrm{VJ}}-\frac{\
u}{E} \mathrm{tr}\big{(}\sigma^{\mathrm{VJ}}\big{)}\mathbf{I} \tag{3}\]  
where \(\
u\) and \(E\) are elastic constants, \(\sigma^{\mathrm{VJ}}\) is the Jaumann rate of the Cauchy stress tensor \(\sigma\) and \(\mathbf{I}\) is the second-order unit tensor. The plastic rate-of-deformation tensor \(\mathbf{D}^{\mathrm{P}}\) is defined by the associated flow rule, viz. \[\mathbf{D}^{\mathrm{P}}=\dot{p}\frac{\partial f}{\partial\sigma} \tag{4}\]  
Figure 13: Post-test images of back skins (**top**), core (**middle**) and front skins (**bottom**) for series 1 at load level P15. where \(\dot{p}\) is the equivalent plastic strain rate. The yield function is adopted in the form  
\[f(\mathbf{\sigma},R)=\sigma_{\text{eq}}(\mathbf{\sigma})-\sigma_{\text{Y}}(R) \tag{5}\]  
where \(\sigma_{\text{eq}}\) is the equivalent plastic strain and \(R=R(p)\) is the isotropic hardening variable which depends on the equivalent plastic strain \(p\), and \(\sigma_{\text{Y}}\) is the flow stress of the material. For the pressure insensitive aluminium skins, the rate-dependent \(J_{2}\) flow theory is adopted. The von Mises equivalent stress is then given by  
\[\sigma_{\text{eq}}(\mathbf{\sigma})=\sigma_{\text{eq}}\big{(}\mathbf{\sigma}^{{}^{ \prime}}\big{)}=\sqrt{3J_{2}}=\sqrt{\frac{3}{2}\mathbf{\sigma}^{{}^{\prime}}:\mathbf{ \sigma}^{{}^{\prime}}} \tag{6}\]  
where \(\mathbf{\sigma}^{{}^{\prime}}=\sigma-\frac{1}{3}\text{tr}(\mathbf{\sigma})\textbf{I}\) is the stress deviator. The plastic part of the rate-of-deformation tensor in Equation (4) for the von Mises material is then obtained as  
\[\textbf{D}^{\text{P}}=\dot{p}\frac{\partial f}{\partial\mathbf{\sigma}}=\frac{3}{ 2}\dot{p}\frac{\mathbf{\sigma}^{{}^{\prime}}}{\sigma_{\text{eq}}} \tag{7}\]  
The flow stress is defined as  
\[\sigma_{\text{Y}}=\sigma_{0}+R(p)=\sigma_{0}+\left[\sum_{i=1}^{2}Q_{i}(1-\exp (-C_{i}p))\right] \tag{8}\]  
where \(\sigma_{0}\) is the yield stress and \((Q_{i},C_{i})\) are material parameters governing strain hardening based on the extended Voce hardening rule. The equivalent plastic strain rate \(\dot{p}\) is given as  
\[\dot{p}=\left\{ \begin{array}{ll}0,&f\leq 0\\ \dot{p}_{0}\Bigg{[}\left(\frac{\sigma_{\text{eq}}(\mathbf{\sigma})}{\sigma_{\text{ Y}}(R)}\right)^{\frac{1}{\text{MoC}}}-1\Bigg{)}&f>0\end{array} \right.