# Multiscale Modeling and Optimization of Aluminum Foam Material Properties Under Dynamic Load
## 3 Results
### Analytic Results  
In the analytical calculations, Table 4, both the geometric dimensions of the specimen and its measured mass were considered to determine the mechanical properties of the aluminum foam. The first step was to calculate the total volume of the specimen based on its external dimensions. This allowed for the determination of the cross-sectional area and overall volume, which were later used to evaluate the apparent density of the material. The theoretical density of solid aluminum (2700 \(\frac{\text{kg}}{\text{m}^{3}}\)) was used as a reference value, which is critical in porous material analysis. Since the foam structure contains air voids, its overall density is significantly lower than that of solid material. To account for this, the air volume fraction (\(\phi\)) was introduced in the calculations. Using the known total mass of the specimen (experimentally measured) and the theoretical density of the solid phase, the air volume fraction was adjusted iteratively until the analytically calculated mass matched the real one. This correlation was achieved using Equation (5), which expresses the foam density as a function of the solid density and the air volume fraction. It is important to note that although the CT scan performed on a small 6 \(\times\) 6 \(\times\) 6 mm\({}^{3}\) volume indicated an air fraction of 0.718, this volume represented only about 2% of the entire specimen. Therefore, the global value used in the calculation, 0.695, was obtained through mass matching, providing a more accurate representation of the actual specimen. Once the air volume fraction was determined, the foam's apparent density was verified using Equation (13), which relates density to the measured mass and calculated volume. This step served as a consistency check to ensure that the density used in the subsequent mechanical models reflected the physical behavior of the material. With the density defined, the elastic modulus of the foam was estimated based on the Gibson-Ashby relationship. This model describes the dependence of the elastic modulus \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Equation** & \\ \hline Volume & \(V=L\cdot W\cdot H=9072\) mm\({}^{3}\) & (13) \\ Total mass & \(M_{Total}=\rho_{s}\cdot V=24\) g & (14) \\ Foam mass & \(M_{Foam}=M_{Total}\cdot(1-F_{Air})=7.47\) g & (15) \\ Foam density & \(\rho_{Foam}=824\)\(\frac{\text{kg}}{\text{m}^{3}}\) & (16) \\ Foam elastic modulus & \(E_{Foam}=E_{AL}\cdot(\frac{\rho_{Foam}}{\rho_{AL}})^{2}=6615\) MPa & (17) \\ Moment of inertia & \(I_{z}=\frac{W\cdot H^{3}}{12}=283.5\) mm\({}^{4}\) & (18) \\ Resonance frequency & \(f_{n}=\frac{1.8\text{g}^{2}}{2\cdot\pi}\cdot\sqrt{\frac{E_{Foam}\cdot L}{\rho _{Foam}\cdot A\cdot L^{4}}}=190\) Hz & (19) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Table of solved equations for Specimen B2. Figure 18: (**a**) Parameter selection for TPMS; (**b**) resulting material properties for density class B.  
on the foam's porosity and was applied using a solid-phase modulus of aluminum, 71 GPa. The resulting value for the foam's modulus aligned closely with the modulus calculated in the Material Designer software, with a deviation of less than 2%. Next, the moment of inertia _Iz_ was calculated from the specimen's width and height. This geometric parameter is essential for characterizing the bending stiffness of the beam and is directly used in the equation governing the resonance frequency. Finally, the resonance frequency was calculated using the Euler-Bernoulli beam theory for a cantilevered beam, Table 4. To better replicate the real experimental conditions, where approximately 10 mm of the specimen is fixed within the actuator clamp, this length was subtracted from the total length in the model. The adjusted length, along with the foam's density, elastic modulus, cross-sectional area, and moment of inertia, was used to compute the resonance frequency analytically.