# Multiscale Modeling and Optimization of Aluminum Foam Material Properties Under Dynamic Load
## 2 Materials and Methods
### Specim Preparation  
As part of the characterization process, each of the 11 test specimens underwent detailed weighing and dimensional analysis. To ensure accuracy and account for possible irregularities along the specimen length, measurements were taken at three distinct points along each sample: one near each end and one at the midpoint. These measurements included width, height, and length. The three values obtained for each dimension were then averaged to determine a representative value for each specimen.  
Following this procedure, the mass of each specimen was measured using a high-precision KERN balance. With both the average volume and the precise mass available, the density of each specimen was calculated using the standard formula, the results are presented in Table 1.  
Two samples, from classes B and D, were selected for computed tomography (CT) analysis. The CT scan results provided a detailed report quantifying the solid volume within the total analyzed volume, as illustrated in Figure 2. This ratio effectively represents the air fraction within the material, offering a percentage-based assessment of the air content relative to the total volume [34; 35; 36].  
Having only two analyses, for two samples, we developed a linear interpolation formula, as seen in Equation (1) [37; 38], to estimate the air fraction volume for the remaining nine test samples. This approach allowed us to approximate the air content in the untested samples based on the data obtained from the analyzed ones.  
\begin{table}
\begin{tabular}{c c c} \hline \multirow{2}{*}{**Density Class**} & \multicolumn{2}{c}{**Range of Density**} & \multirow{2}{*}{**Number of Specimens**} \\  & \(\left[\frac{\mathbf{g}}{\mathbf{cm}^{3}}\right]\) & & \\ \hline A & 0.87–0.96 & 3 \\ B & 0.82–0.83 & 2 \\ C & 0.68–0.71 & 2 \\ D & 0.54–0.62 & 4 \\ \hline \end{tabular}
\end{table}
Table 1: Distribution of specimens based on density classes.  
Figure 1: Aluminum foam specimens.  
\[\phi(\rho)=\phi_{i}+\left(\phi_{i+1}-\phi_{i}\right)\frac{\rho-\rho_{i}}{\rho_{i+1} -\rho_{i}} \tag{1}\]  
\(\phi(\rho)\) - Estimated air fraction volume for a sample with density \(\rho\); \(\phi_{i}\) and \(\phi_{i+1}\) - Known air fraction volumes for two reference samples (sample \(i\) and sample \(i+1\)); \(\rho_{i}\) & \(\rho_{i+1}\) - Densities of those two reference samples; \(\rho\) - Density of the unknown sample, for which you are estimating the air fraction.  
The linear interpolation method used to estimate the air fraction for intermediate densities is justified based on standard numerical analysis principles. Specifically, the interpolation follows the classical linear approximation, Equation (1), which is mathematically equivalent to  
\[p(x)=f_{(x_{0})}+\frac{f_{(x_{1})}-f_{(x_{0})}}{x_{1}-x_{0}}.(x-x_{0}) \tag{2}\]  
According to interpolation theory, if the function \(f_{(\rho)}\), (in this case, the air fraction as a function of density) is smooth and has a continuous second derivative, the interpolation error is bound by  
\[|R_{T}|\leq\frac{(\rho_{i+1}-\rho_{i})^{2}}{8}\max_{\rho_{i}\leq\rho\leq\rho_ {i+1}}|f^{\prime\prime}(\rho)| \tag{3}\]  
This ensures that the approximation remains valid and reasonably accurate within the range of known data points. Moreover, similar linear interpolation approaches for estimating foam morphology properties have been applied and validated in the literature, including works such as [39,40].