## 4 Numerical Work
### Material Calibration
#### 4.2.2 Foam Cores  
The strain hardening parameters for the foam cores were calibrated against the uniaxial compression tests conducted by [PERSON] and [PERSON] [6], see Section 2.2.1. As the foams in this study are assumed to have a negligible plastic contraction, the yield surface parameter \(\alpha\) is found from Equation (12) to be 2.12 for all foams studied. The negligible plastic contraction also allowed for the hardening parameters to be directly calibrated from the experimental results without inverse modelling. For each test performed in the normal direction (ND), a set of hardening parameters (i.e., \(\sigma_{\text{p}},\gamma,\alpha_{2}\) and \(\beta\) in Equation (16)) was calibrated to the experimental Cauchy stress-logarithmic plastic strain curves using a least-squares method. The acquired set of parameters for all foam densities was then used for fitting constants in the density-dependent hardening model in Equation (17). As \(\varepsilon_{\text{D}}\) is directly calculated based on the relative densities of the foams and the base material, according to Equation (18), no calibration of this parameter is needed. For all applications of this relation for \(\varepsilon_{\text{D}}\), a density of 1050 kg/m\({}^{3}\) for polystyrene \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\sigma_{0}\) & \(Q_{1}\) & \(C_{1}\) & \(Q_{2}\) & \(C_{2}\) \\
**[MPa]** & **[MPa]** & **[MPa]** & **[MPa]** & **[MPa]** \\ \hline
80.0 & 25.7 & 1000.0 & 7.6 & 21.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material hardening parameters for aluminium alloy AA1050-H14. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(E\) & \(\
u\) & \(\rho\) & \(c_{\text{MD}}\) & \(\dot{\vec{p}}_{0}\) \\
**[GPa]** & **[–]** & **[kg/m\({}^{3}\)]** & **[–]** & **[s\({}^{-1}\)]** \\ \hline
70.0 & 0.30 & 2700 & 0.014 & \(5\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Material constants for aluminium alloy AA1050-H14 (from [34]). was assumed [2]. The obtained constants for the density-dependent hardening model are presented in Table 6. As the density-dependent model only applies to the plastic hardening, elastic constants and thus contact behaviour in the numerical models are given by a single input based on the average density of the modelled material. The density and Young's modulus are taken from Reyes and Borvik [6], and are given in Table 2, while the elastic contraction is taken as \(v^{\mathrm{e}}=0.01\)[43]. To verify the density-dependent model and the model constants, a numerical model of the foam sample consisting of a 50 mm cube discretised by 5 mm cubic 8-node elements with reduced integration was established in LS-DYNA. When calculating the hardening parameters, the average density of each foam type was used as input in the numerical model. Nodal displacements were applied to one side of the model to represent the compressive load, while nodes on the opposite side were constrained in all degrees of freedom. The remaining four sides of the cube where unconstrained from movement. The numerical stress-strain curves are compared to representative experimental results in Figure 14a. Here, it is seen that the density-dependent model describes the foam hardening behaviour with reasonable accuracy for the three studied densities. The viscoplastic extension of the Deshpande-Fleck foam model was calibrated and verified by numerical simulations of the low-velocity impact tests described in Section 2.2.2. The same foam sample as described above was used, while the impactor was modelled as a rigid body with a total mass of 5.339 kg. In these simulations, an initial velocity of 5 m/s was given to the impactor. The contact force between the impactor and sample was obtained by a force transducer contact (\"CONTACT_FORCE_TRANSDUCER_PENALTY) on the top plane of the sample. The strain-rate sensitivity of the foams was obtained by inverse modelling of the low-velocity impact tests.