## 4 Numerical Work
### Constitutive Relations
\tag{9}\]  
Here, \(c_{\text{MJC}}\) is a material parameter governing the strain-rate sensitivity and \(\dot{p}_{0}\) is a user-defined reference strain rate. In the viscoplastic domain (\(f>0\)), the equivalent stress is found from Equation (9) as  
\[\sigma_{\text{eq}}=\sigma_{\text{Y}}(R)\Bigg{[}1+\frac{\dot{p}}{\dot{p}_{0}} \Bigg{]}^{\text{MJC}}=\Bigg{[}\sigma_{0}+\sum_{i=1}^{2}Q_{i}(1-\exp(-C_{i}p) )\Bigg{]}\Bigg{[}1+\frac{\dot{p}}{\dot{p}_{0}}\Bigg{]}^{\text{MJC}} \tag{10}\]  
This constitutive relation was implemented in LS-DYNA by [PERSON] et al. (2018) as material model 107 (\"MAT_MODIFIED_JOHNSON_COOK). For the pressure-sensitive foam cores, the yield function should include a hydrostatic stress term in addition to the deviatoric stress term to account for the volumetric change when the cells of the material collapse under compression. The continuum-based isotropic constitutive relation for crushable foams proposed by [PERSON] and [PERSON] (2017) was adopted for this purpose. In this model, the equivalent stress \(\sigma_{\text{eq}}\) is given by  
\[\sigma_{\text{eq}}^{2}(\mathbf{\sigma})=\sigma_{\text{eq}}^{2}\Big{(}\mathbf{\sigma}^ {{}^{\prime}},\sigma_{\text{H}}\Big{)}=\frac{1}{\Bigg{[}1+\Big{(}\frac{\dot{ \sigma}}{\dot{\sigma}}\Big{)}^{2}\Bigg{]}}\Big{[}\sigma_{\text{vM}}^{2}+\alpha ^{2}\sigma_{\text{H}}^{2}\Big{]} \tag{11}\]  
where \(\sigma_{\text{vM}}\) is the von Mises equivalent stress defined by Equation (6) and \(\sigma_{\text{H}}=\frac{1}{3}\text{tr}(\mathbf{\sigma})\) is the mean hydrostatic stress. The parameter \(\alpha\) governs the shape of the yield surface and is defined as  
\[\alpha^{2}=\frac{3G}{K}=\frac{9(1-2v^{\text{P}})}{2(1+v^{\text{P}})} \tag{12}\]where \(K\) and \(G\) are the bulk- and shear modulus, respectively, and \(v^{\rm P}\) is the plastic coefficient of contraction. It is seen from Equation (12) that when \(v^{\rm P}=0.5\), as for pressure insensitive materials, \(\alpha^{2}=0\), and the equivalent stress \(\sigma_{\rm eq}(\sigma)\) in Equation (11) reduces to the von Mises equivalent stress \(\sigma_{\rm vM}\). The plastic rate-of-deformation tensor for the pressure-sensitive material is decomposed into a deviatoric and a hydrostatic part, viz. \[\mathbf{D}^{\rm P}=\dot{p}\frac{\partial f}{\partial\sigma}=\dot{p}\frac{ \partial f}{\partial\sigma_{\rm vM}}\frac{\partial\sigma_{\rm vM}}{\partial \sigma}+\dot{p}\frac{\partial f}{\partial\sigma_{\rm H}}\frac{\partial\sigma_ {\rm H}}{\partial\sigma}=\dot{\varepsilon}_{\rm vM}\mathbf{n}+\frac{1}{3} \dot{\varepsilon}_{\rm v}\mathbf{I} \tag{13}\]  
where the von Mises equivalent plastic strain rate, \(\dot{\varepsilon}_{\rm vM}\), and the volumetric plastic strain rate, \(\dot{\varepsilon}_{\rm v}\), are defined as  
\[\dot{\varepsilon}_{\rm vM}=\frac{\dot{p}}{1+\left(\frac{\alpha}{3}\right)^{2 }}\frac{\sigma_{\rm vM}}{\sigma_{\rm eq}},\ \dot{\varepsilon}_{\rm v}=\frac{\alpha^{2}\dot{p}}{1+\left(\frac{\alpha}{3} \right)^{2}}\frac{\sigma_{\rm H}}{\sigma_{\rm eq}} \tag{14}\]  
By combining the equations above, it is possible to express the equivalent plastic strain rate explicitly in terms of \(\dot{\varepsilon}_{\rm vM}\) and \(\dot{\varepsilon}_{\rm v}\), i.e.,  
\[\dot{p}=\sqrt{\left[1+\left(\frac{\alpha}{3}\right)^{2}\right]\left(\dot{ \varepsilon}_{\rm vM}^{2}+\frac{1}{\alpha^{2}}\ \dot{\varepsilon}_{\rm v}\right)} \tag{15}\]  
Based on the work of [PERSON] et al. [40] and [PERSON] et al. [41], the flow stress is taken as  
\[\sigma_{\rm Y}(R)=\sigma_{\rm P}+R(p)=\sigma_{\rm P}+\gamma\frac{p}{ \varepsilon_{\rm D}}+\alpha_{2}\ln\left(\frac{1}{1-\left(\frac{p}{\varepsilon _{\rm D}}\right)^{\beta}}\right) \tag{16}\]  
where \(\sigma_{\rm P}\) is the plateau stress, \(\varepsilon_{\rm D}\) is the true densification strain, and \(\gamma\), \(\alpha_{2}\) and \(\beta\) are material parameters governing the work hardening. This constitutive relation was implemented in LS-DYNA by [PERSON] et al. [41] as material model 154 (\"MAT_DESHPANDE_FLECK_FOAM). [PERSON] et al.