## 4 Strain Energy  
Under the effect of the external applied load, Al foams undergo plastic deformations. The greatest part of the applied energy is heat dissipated, with a small amount is the storage in the material in the form of strain energy, and just a little amount is present in the form of residual stress. The strain energy stored in the pseudo-elastic stage is almost equal to the work conducted by an external force (work-energy theorem). Until the pseudo-elastic limit is reached, it can also be approximated by computing the area under the curve of the stress vs. strain graph. According to [PERSON]'s law, for minor deformations of elastic materials, the strain is directly proportional to the applied stress. [PERSON]'s law only applies to minor deformations in an elastic material (up to the proportional limit) under the hypothesis of linear elasticity. Under plastic deformation, the strain energy density is the amount of strain energy per unit volume that is uniformly distributed inside a material. It can be determined by:  
\[Uv=\frac{Total\;Strain\;Energy}{Volume\;of\;the\;Material}\]  
From the analysis of Figures 3-6 the plateau stress, the maximum strain, and the maximum energy per unit of volume have been extrapolated and reported in Figures 7 and 8 and summarized in Tables 6 and 7. \begin{table}
\begin{tabular}{c c c} \hline  & **Before Compression** & **After Compression** \\ \hline Weight (g) & 7.6 & 6.0 \\ Diameter (mm) & 16 & 21 \\ Height (mm) & 45 & 9 \\ \hline \end{tabular}
\end{table} 
Table 4: Properties of sample 3 (0.6% TiH\({}_{2}\), 2.8% SiC) before and after compression. \begin{table}
\begin{tabular}{c c c} \hline  & **Before Compression** & **After Compression** \\ \hline Weight (g) & 5.8 & 3.9 \\ Diameter (mm) & 16 & 21 \\ Height (mm) & 28 & 7 \\ \hline \end{tabular}
\end{table} 
Table 2: Properties of sample 1 (0.2% TiH\({}_{2}\), 2.8% SiC) before and after compression. \begin{table}
\begin{tabular}{c c c} \hline  & **Before Compression** & **After Compression** \\ \hline Weight (g) & 5.8 & 3.9 \\ Diameter (mm) & 16 & 21 \\ Height (mm) & 28 & 7 \\ \hline \end{tabular}
\end{table} 
Table 3: Properties of sample 2 (0.4% TiH\({}_{2}\), 2.8% SiC) before and after compression. \begin{table}
\begin{tabular}{c c c} \hline  & **Before Compression** & **After Compression** \\ \hline Weight (g) & 7.6 & 6.0 \\ Diameter (mm) & 16 & 21 \\ Height (mm) & 45 & 9 \\ \hline \end{tabular}
\end{table} 
Table 5: Properties of sample 4 (0.2% TiH\({}_{2}\), 6% SiC) before and after compression. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multirow{2}{*}{**Plateau Stress (MPa)**} & \multirow{2}{*}{**Maximum Strain (\%)**} & \multicolumn{2}{c}{**Maximum Energy per Unit Volume (m/mm\({}^{3}\))**} \\  & & & \\ \hline
1 & 9.2 & 79.5 & 2.9 \\
2 & 7.6 & 78.0 & 5.9 \\
3 & 7.5 & 82.9 & 5.0 \\
4 & 10.3 & 81.4 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Compression test data for each sample. Figure 8: Energy per unit volume for different compositions of SiC. \begin{table}
\begin{tabular}{c c} \hline \hline \multirow{2}{*}{**Sample**} & \multicolumn{2}{c}{**Energy per Unit Volume (m/mm\({}^{3}\)) at 80\%**} \\  & **Strain** \\ \hline
1 & 2.9 \\
2 & 5.9 \\
3 & 3.9 \\
4 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Energy per unit volume at 80% of strain for each sample. Figure 7: Energy per unit volume for different compositions of TiH\({}_{2}\). The ability of closed-cell aluminum foams to absorb energy improves as their relative density and strain rate increase [14]. This improvement in energy absorption at high strain rates is promising for the application of closed-cell aluminum foams in high-impact energy absorption scenarios [14]. By limiting the computation of absorbed energy up to 80% of the strain, the following results can be evidenced (Table 7). These results evidence that Sample 2 (SiC = 2.8%, TiH\({}_{2}\) = 0.4% and Al = 96.8%) shows higher strain energy when compared to other samples.