# Experimental and Numerical Evaluation of Calcium-Silicate-Based Mineral Foam for Blast Mitigation
## 3 Numerical Modeling
### Material Modeling  
Experimental findings indicate that both the clamping plate and the rigid steel frame endure elastic deformation throughout the process [37]. Therefore, these components are modeled using a linear elastic material model (*MAT_1). This material model parameters are as follows: density of 7850 kg/m\({}^{3}\), [PERSON]'s modulus of 210 GPa and Poisson's ratio of 0.3.  
Due to the intense blast loading, the aluminum plates deform significantly. The modified Johnson-Cook material model (*MAT_107) is used to capture the strain rate effect, incorporating strain hardening and thermal softening. Equation (1) describes the corresponding yield stress, dependent on strain, strain rate and temperature:  
\[\sigma_{y}=\Big{[}A+B\epsilon^{n}+\sum_{i=1}^{2}Q_{i}(1-\exp(-L_{i \epsilon}))\Big{]}\bigg{(}1+Cln\bigg{(}\frac{\dot{\epsilon}}{\dot{\epsilon}_{0 }}\bigg{)}\bigg{)}\bigg{(}1-\bigg{(}\frac{T-T_{a}}{T_{a}-T_{f}}\bigg{)}^{m} \bigg{)}, \tag{2}\]  
The material parameters \(A\), \(B\), \(C\), \(m\), \(n\), \(Q_{1}\), \(L_{1}\), \(Q_{2}\), and \(L_{2}\) govern the modified Johnson-Cook model. These parameters account for the plastic strain rate \(\dot{\epsilon}\), quasi-static threshold strain rate \(\dot{\epsilon}_{0}\), ambient temperature \(T_{a}\), and fusion temperature \(T_{f}\). The specific values used for the aluminum plates are detailed in Table 2, derived from the literature [43].  
[PERSON] et al. [9; 11] and [PERSON] et al. [42] showed that the deformability of the front skin influences the stress distribution within the crushable core, thereby influencing the energy absorption of the SC. Consequently, to explore how the front skin's deformability influences the effectiveness of the calcium-silicate-based SC, a rigid constitutive law, i.e., *MAT_20, is also considered while maintaining constant front skin mass. The parameter values used in the adopted material model are as follows: density of 2710 kg/m\({}^{3}\), [PERSON]'s modulus of 69 GPa and Poisson's ratio of 0.33.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\rho\) (kg/m\({}^{-3}\)) & \(E\) (GPa) & \(\
u\) (\(-\)) & \(C_{p}\) (J/(kg\(\cdot\)K)\({}^{-1}\)) & \(\dot{\epsilon}\) (K\({}^{-1}\)) & \(\dot{\epsilon}_{0}\) (ms\({}^{-1}\)) & \(T_{a}\) (K) & \(m\) (\(-\)) \\ \hline
2710 & 69 & 0.33 & 899 & \(1.2\times 10^{-5}\) & 1 & 293 & 1 \\ \hline \(A\) (GPa) & \(B\) (GPa) & \(C\) (\(-\)) & \(Q_{1}\) (GPa) & \(L_{1}\) (\(-\)) & \(Q_{2}\) (\(-\)) & \(L_{2}\) (\(-\)) & \(n\) (\(-\)) \\ \hline
0.065 & 0.014 & 0.014 & 0.025 & 3324 & 0.019 & 533 & 0.36 \\ \hline \end{tabular}
\end{table}
Table 2: Modified Johnson-Cook parameters used for the aluminum plates [42].  
Figure 14: Convergence study of the mesh size on the aluminum plates and the mineral foam.  
The Fu Chang Foam (*MAT_083) and the Crushable Foam (*MAT_63) material models are used for modeling the mineral foam in this study. Fu Chang Foam is a one-dimensional model assuming zero Poisson's ratio, allowing uni-axial or tri-axial compression strain with strain rate effects considered for uni-axial compression. The input parameters for *MAT_083 are derived from [37] and presented in Table 3. Contrary to the Fu Chang Foam material model, the Crushable Foam material model does not consider strain rate effects. However, from the simulations in [37], it is known that in these test-specific conditions, the foam is crushed at a mean strain rate of about 800 s\({}^{-1}\). Therefore, a single stress- strain curve at this strain rate is input in the Crushable Foam material model.