Multiscale Modeling and Optimization of Aluminum Foam Material Properties Under Dynamic Load

Abstract

Aluminum foam materials have gained significant attention over the past decade, particularly in the automotive industry, due to their excellent stiffness-to-weight ratio and superior energy absorption capabilities. In this study, a multiscale numerical material model was developed to accurately and efficiently simulate the vibrational behavior of aluminum foams. The foam specimens were categorized into four density classes based on their measured mass and calculated volume. Two specimens were selected to conduct CT (computerized tomography) scans and quantify the volume of air in their density class. Based on the CT measurements, a representative volume element (RVE) was built using ANSYS Material Designer (MD). The newly obtained material was employed in conducting normal mode numerical simulations. The resonance frequencies and response amplitudes were compared with physical experiments and showed correlation within 3%. These findings underscore the efficacy of using CT scans in foam to develop material models and accurately predict structural behavior. By conducting comprehensive investigations and numerical simulations, we established a correlation between physical tests and simulation results, highlighting the reliability of the developed models.

1. Introduction

Due to their lightweight structure and multifunctional characteristics, cellular structures offer excellent mechanical energy absorption, thermal insulation, and acoustic attenuation, making them ideal for various engineering applications [1,2,3]. Porous materials have emerged as essential components for advanced lightweight structural designs, functioning both as standalone elements and as reinforcements within composite materials combined with stiffer phases [4]. The properties of porous materials are strongly influenced by the morphological characteristics of the cells within their microstructure [5,6]. Among the various types of porous materials, those with a metallic matrix, commonly referred to as metallic foams, exhibit superior mechanical and physical performance. Their versatility enables applications ranging from everyday consumer products to advanced aerospace systems [7,8,9,10]. Due to the relatively low cost of raw materials and the availability of well-established manufacturing methods, metallic foams based on aluminum alloys are the most widely used [11,12]. According to the literature, the continuous enhancement of their mechanical and functional properties has enabled metallic foams to increasingly replace conventional materials in sectors such as automotive, construction, and railway engineering [13,14,15,16]. The wide spectrum of metallic foam applications requires a thorough understanding of their mechanical behavior. Despite ongoing advancements in manufacturing techniques, metallic foams still exhibit imperfections and microstructural inconsistencies [17,18]. Consequently, continued investigation into both their production processes and mechanical response remains essential. Among their properties, static mechanical behavior, particularly under compression and bending, has been the most extensively studied [19,20,21].

In addition to static loading conditions, dynamic mechanical behavior, particularly under vibration, has become increasingly important in understanding the full performance spectrum of metallic foams. Given their inherent damping capacity and complex microstructure, MFs offer promising characteristics for vibration attenuation, which is critical in various structural applications such as automotive panels, aerospace components, and mechanical assemblies. Despite their potential, studies on the vibrational performance of metallic foams remain limited, often constrained by manufacturing inconsistencies and complex geometrical modeling. To address this gap, this study introduces experimental vibration testing to evaluate the frequency response and damping properties of aluminum-based closed-cell metallic foams. The aim is to provide experimental insight that can be used to inform and validate numerical models [22,23,24].

Over the years, computational homogenization techniques have been developed to address the shortcomings of traditional scaling laws, particularly their reliance on fit parameters. Unlike analytical models based on simplistic geometries, these approaches utilize finite element simulations to capture the complex mechanical behavior of foam structures with more realistic cell morphologies. Simulations using idealized polyhedral cells have shown that elastic properties, under small strains, are influenced by wall stretching mechanisms. When irregularities are introduced into the structural layout—such as variations in wall thickness or the removal of certain cell faces—the deformation behavior shifts toward face bending, often resulting in a measurable reduction in stiffness. Despite the improvements in realism, most of these numerical studies have been limited in scale, typically modeling only a few hundred cells, which constrains their ability to represent the full heterogeneity of actual foam materials [25,26,27].

Accurate modeling of closed-cell aluminum alloy foams requires a realistic representation of both the foam’s microstructure and the mechanical behavior of its cell walls. Recent approaches have combined nanoindentation experiments with numerical simulations to extract more reliable material properties for use in finite element models. Building on this, an improved representative model has been proposed, featuring a mix of spherical and cruciform-shaped cells with varying wall thicknesses, better capturing the heterogeneity observed in real foams. Unlike traditional models, which tend to overpredict foam strength, this new approach offers significantly better agreement with experimental results and can replicate various deformation patterns seen in metallic foams [28,29,30].

Modeling the mechanical behavior of closed-cell cellular materials remains a significant challenge due to the intricate microstructure and extremely thin cell walls characteristic of these systems. Traditional finite element approaches, particularly those relying on continuum elements, often fail to capture the complexity of low-density foams, especially when the wall thickness approaches the resolution limit of meshing techniques. Recent developments have leveraged X-ray tomography to generate detailed three-dimensional representations of the foam structure, offering a pathway to more accurate numerical modeling. By extracting direct thickness measurements from tomographic images and applying them to shell-based finite element models, it becomes possible to simulate larger volumes of material with improved fidelity. This modeling approach enables both global mechanical responses and local stress–strain distributions to be analyzed, offering insights into structural deformation patterns that correlate well with experimental compression data [31,32,33].

This study begins by analyzing CT scan data to accurately extract the volume air fraction of closed-cell foam samples. This volumetric information is then used to determine the corresponding material properties, which serve as input parameters for numerical simulations. A computational model is developed to investigate the dynamic response of the material under vibrational loading. The simulated behavior is then compared against results from experimental vibration tests carried out on the same foam specimens. This integrated approach enables us to assess the accuracy of the model and to explore how the foam’s microstructure influences its macroscopic mechanical performance. The paper is organized as follows: The introduction reviews relevant theoretical background on the mechanical behavior of closed-cell foams. Section 2 outlines the method used to model the foam structure. Section 2 also details the computational model employed to simulate the material’s dynamic response. In the third section, simulation results are compared with experimental data, and key mechanical characteristics such as stiffness and damping behavior are discussed in relation to the foam’s relative density and internal structure.

2. Materials and Methods

2.1. Materials

Aluminum foams composed of AlMg₁Si_0.5 + 0.4 wt.% Ti were produced using the powder metallurgy method as follows: Aluminum alloy powder 6061 was dry mixed with a foaming agent (0.4 wt.% TiH₂). The powder mixture was cold isostatically pressed at 200 MPa and hot extruded at 400 °C with an extrusion ratio of 16:1. The obtained precursor had a porosity of less than 3% and cross section of 20 mm × 5 mm.

To achieve a homogeneous foam structure, the foaming precursor was first cut into 20 mm long pieces. These precursor segments, each measuring 20 mm × 20 mm × 5 mm, were then arranged inside a mold, nearly filling it completely (approximately 121 pieces in total), similar to checkerboard structure.

This specific arrangement was chosen to minimize anisotropy in the X-Y plane, a common issue in aluminum foam plates where the thickness is significantly smaller than the length and width.

Foaming was carried out inside a steel mold with internal dimensions of 300 mm × 300 mm × 7 mm. The mold was placed in an electric furnace and heated to 680 °C, where it remained for 10 min to facilitate foam formation. Once the foaming process was complete, the mold was cooled to room temperature to preserve the foam plate. The final foam specimens, measuring 296 mm × 296 mm × 7 mm, retained a surface skin after the foaming process.

The metallic foam plate was initially cut using a Water jet cutting was performed using a Streamline SL-VI 30 HP pump unit Streamline SL-VI OEM 30HP Water Jet Cutting (WJC), KMT Waterjet Systems Inc., Baxter Springs, KS, USA machine to produce parallelepiped-shaped specimens. These specimens were then machined to achieve a uniform width of 10 mm. Their mass was measured using a KERN PCB 350-3 precision balance (KERN & Sohn GmbH, Balingen-Frommern, Balingen, Germany), and their density was subsequently calculated. The final specimen dimensions were approximately 130 mm × 10 mm × 7 mm (Figure 1).

2.2. Specimen 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. Distribution of specimens based on density classes.

Density Class	Range of Density $\left[\frac{\mathbf{g}}{{\mathbf{c}\mathbf{m}}^3}\right]$	Number of Specimens
A	0.87–0.96	3
B	0.82–0.83	2
C	0.68–0.71	2
D	0.54–0.62	4

.

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.

$$\varphi (\mathit{\rho })={\varphi }_i+({\varphi }_{i+1}-{\varphi }_i){\displaystyle \frac{\mathit{\rho }-{\mathit{\rho }}_i}{{\mathit{\rho }}_{i+1}-{\mathit{\rho }}_i}}$$

(1)

$\varphi (\mathit{\rho })$	- Estimated air fraction volume for a sample with density ρ;
${\varphi }_i$$\mathrm{and} {\varphi }_{i+1}$	- Known air fraction volumes for two reference samples (sample i and sample i + 1);
${\mathit{\rho }}_{i }$$\& {\mathit{\rho }}_{i+1}$	- Densities of those two reference samples;
$\mathit{\rho }$	- 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

$$p(x)=f_{(x_0)}+{\displaystyle \frac{f_{(x_1)}-f_{(x_0)}}{x_1-x_0}}·(x-x_0)$$

(2)

According to interpolation theory, if the function $f_{(\mathit{\rho }),}$ (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

$$\left|R_T\right|\le {\displaystyle \frac{{({\mathit{\rho }}_{i+1}-{\mathit{\rho }}_i)}^2}{8}}\underset{{\mathit{\rho }}_i\le \mathit{\rho }\le {\mathit{\rho }}_{i+1}}{\max }\left|f^{″}(\mathit{\rho })\right|$$

(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].

2.3. Methods

2.3.1. Vibration Test

An essential phase of this study focused on assessing the dynamic behavior of the tested material by identifying its resonance frequencies within the 10–2000 Hz range. This analysis provides valuable insights into the mechanical response under dynamic conditions, including the measurement of resonance frequencies, amplitude variations, and the quality factor. The results obtained will contribute to the calculation of the material’s modal damping coefficients, essential for understanding its vibrational properties.

To conduct these tests, an LDS V406 (Brüel & Kjær, Nærum, Denmark) 196 N electromechanical actuator was used. As illustrated in Figure 3a, this device efficiently converts electrical energy into mechanical energy, generating a consistent sinusoidal vibration profile.

Designed for high-performance applications, the LDS V406 actuator operates within a frequency range of 5 to 9000 Hz, with a nominal peak force of 98 N for sinusoidal waves. However, when combined with an amplifier and cooled via an external fan, its output force increases to 196 N, making it highly effective for precision testing.

Furthermore, the actuator features a 38 mm armature diameter and a lightweight moving armature, ensuring stable support for test loads without compromising force capacity. To enhance stability and flexibility, two fiber-laminated flexures are bonded to the armature, providing axial support and improving lateral and rotational degrees of freedom.

By leveraging these advanced testing techniques, the study aims to deepen the understanding of the material’s mechanical performance under dynamic conditions, paving the way for improved material design and application in engineering fields.

For the vibration measurements, a Polytec CLV-3D Compact 3-D, Polytec GmbH, Waldbronn, Germany system, laser vibrometer was used, as shown in Figure 3b. This system operates based on the Doppler effect, detecting frequency shifts in the laser beam reflected from a moving object. The CLV-3D system includes three laser beams, each inclined at 12° relative to the surface (when using the 160 mm focal length lens) and spaced 120° apart in the output plane. These beams converge at a single measurement point located precisely 160 mm from the front of the sensor head. To ensure accurate measurements, the sensor must be positioned at this exact distance from the test specimen.

Because laser vibrometers require a reflective surface for optimal signal return, small reflective tape markers were placed on the specimens. The system captures vibration data in all three spatial directions (x, y, and z) at the focal point. Figure 4 shows the full setup used during the test procedure.

Although the CLV-3D system does not include an automatic calibration routine based on the size or position of the reflective patch, correct placement at the focal point was manually verified. During testing, a 1 g sine excitation was applied via the shaker, and the system’s response was cross-checked to ensure signal consistency and measurement reliability. While this procedure does not constitute a formal calibration, it confirmed that the reflective patch was properly positioned and that the recorded vibration data were accurate.

To assess the energy dissipation characteristics of the materials, the structural damping coefficient ξ was also evaluated as part of the vibration test procedure. The structural damping coefficient was calculated using the half-power bandwidth method, based on classical vibration theory [41]. The damping was derived from the resonance behavior of each specimen, which was excited under a broadband sine sweep profile ranging from 10 Hz to 2000 Hz, with a constant acceleration of 1 g and a linear sweep rate of 1 Hz/s. The quality factor Q, which quantifies the sharpness of resonance peaks, was calculated as the ratio between the center (resonant) frequency and the half-power bandwidth, as shown in Figure 5 and Equation (4). The half-power bandwidth method involves identifying the frequencies $f_1$ and $f_2$, which correspond to the points on the frequency response curve where the amplitude drops to ${\displaystyle \frac{A_{max}}{\sqrt{2}}}$, i.e., approximately 70.7% of the maximum amplitude. These two frequencies define the bandwidth around the resonance frequency and are used to calculate the Q factor, thereby enabling the estimation of the damping characteristics. A higher Q factor indicates a sharper, less damped resonance, while a lower Q factor reflects more significant damping [42,43]. Subsequently, the structural damping coefficient ξ was derived using Equation (5), which establishes the inverse relationship between damping and the Q factor. This approach allows for a quantitative estimation of the internal damping behavior specific to each material sample, taking into account both material properties and vibrational response.

$$Q={\displaystyle \frac{f_{\mathrm{n}}}{f_2-f_1}}$$

(4)

$$\xi ={\displaystyle \frac{1}{2Q}}$$

(5)

2.3.2. Analytic Calculations

To evaluate the mechanical properties of the aluminum specimens—specifically, mass M, density ρ, and elastic modulus E—the analytical approach was based on equations derived within the framework of the Bernoulli–Euler beam theory.

This approach enabled the theoretical determination of the natural/resonance frequency of each specimen,

Table 2. Table of analytical equations.

Description	Equation
Volume	$V=L·W·H$	(6)
Total mass	$M_{Total}={\mathit{\rho }}_{AL}·V$	(7)
Foam mass	$M_{Foam }=M_{Total}·\left(1-F_{Air}\right)$	(8)
Foam density	${\mathit{\rho }}_{Foam }={\mathit{\rho }}_{AL}·\left(1-F_{Air}\right)+(F_{Air}·{\mathit{\rho }}_{air})$	(9)
Foam elastic modulus	$E_{Foam }=E_{AL}·{({\displaystyle \frac{{\mathit{\rho }}_{Foam }}{{\mathit{\rho }}_{AL}}})}^2$	(10)
Moment of inertia	$I_z={\displaystyle \frac{W·H^3}{12}}$	(11)
Resonance frequency	$f_n={\displaystyle \frac{{1.875}^2}{2·\pi }}·\sqrt{{\displaystyle \frac{E_{Foam }·I_z}{{\mathit{\rho }}_{Foam }·W·H·L^4}}}$	(12)

, which was then compared with the experimental results obtained from vibration testing.

As illustrated in Figure 6, a portion of each specimen was clamped into a vibration shaker during testing. This fixed section does not participate in the free vibration of the beam and thus must be excluded from the effective length L when the beam theory is applied.

2.3.3. Material Modeling Through RVE and TPMS

In the virtual validation stage, the Material Designer module from the ANSYS suite was employed to homogenize the material and obtain its effective properties. These properties were subsequently used in a numerical vibration simulation, whose results were compared with both experimental data and the analytical predictions (Figure 7).

In the case of the investigated material, two distinct methods were employed:

A. The Random Particle Method—as shown in Figure 8, where the first step involved selecting the constituent materials (aluminum and air), and the second step included defining parameters such as

• Particle volume fraction (obtained from CT data);
• Particle diameter distribution;
• Mean particle diameter;
• Standard deviation of particle diameter;
• Particle wall thickness;
• Size ratio (defined as the ratio between the particle diameter and the side length of the representative cube).

B. TPMS Method: The second modeling approach employed the TPMS (Triply Periodic Minimal Surface) function. As with the previous method, the final model included the same two constituent materials: aluminum and air. However, in this case, the modeling parameters focused on the gyroid surface type and the solid volume fraction. This fraction represents the proportion of solid material within the representative cube—specifically, in a 10 mm × 10 mm × 10 mm cube, 29% of the volume consists of solid material, while the remaining 71% is void (air). The size parameter previously discussed was also applied in this configuration (Figure 9).

2.3.4. Numerical Analysis

The numerical simulation was carried out using Altair SimLab 2024 software, incorporating the material properties obtained during the material validation stage. The model was meshed using hexahedral elements, and a modal frequency response analysis was performed under a 1 g acceleration, within the 10–2000 Hz range. A logarithmic distribution over 500 intervals was selected for the frequency range. The previously calculated damping coefficients were also incorporated to improve the accuracy of the simulation (Figure 10).

3. Results

3.1. Volume Air Fraction Calculation

The first dataset was obtained through interpolation, based on CT scan measurements of specimens B2 and D4, to estimate the air volume fraction across all samples relative to their densities (Figure 11). The labels (e.g., A1–D4) represent density classes (A–D), with the number indicating the specimen ID within each class.

The observed pattern clearly demonstrates a direct inverse relationship between material density and volume air fraction. Specimens classified under group A, which exhibit the highest densities, correspond to the lowest volume air fractions. In contrast, group D specimens representing the lowest density range show the highest air volume fractions. This trend confirms the expected behavior, where increased porosity (and thus air content) results in lower overall material density. The consistent gradient across the density classes reinforces the reliability of the interpolation method used and supports the logical correlation between density and internal structure.

3.2. Test Results

This section presents the results of the sine vibration tests, categorized according to the predefined density classes A, B, C, and D. The analysis focuses on identifying the dynamic response characteristics of each specimen under control vibrational input. As shown in Figure 12, Figure 13, Figure 14 and Figure 15, the modal frequency responses exhibit distinct groupings corresponding to each density class.

Overall, most specimens demonstrated consistent and repeatable vibrational behavior, particularly in the first and second resonant modes. This consistency suggests reliable material properties within each density class and validates the classification method used.

However, three specimens deviated significantly from the expected pattern, exhibiting notably lower acceleration amplitudes in both the first and second modes. These anomalies may indicate variations in internal structure, potential defects, or experimental inconsistencies and warrant further investigation. The reduced response could be due to increased internal damping or localized heterogeneities affecting the modal behavior.

Table 3. Overview of results.

Measurement	1st Resonance Frequency [Hz]	1st Amplitude [g]	2nd Resonance Frequency [Hz]	2nd Amplitude [g]
A	268.6	81.9	1550.9	284.3
	253.0	86.6	1547.0	254.3
	253.0	73.9	1547.0	212.0
B	270.5	82.9	1620.8	286.5
	275.3	80.9	1591.7	323.8
C	233.6	79.9	1478.0	226.3
	201.5	27.6	1265.0	131.7
D	199.6	29.7	1262.1	169.2
	258.9	105.8	1437.1	272.1
	272.5	71.7	1479.9	255.5
	157.8	12.4	1212.6	141.5

provides a consolidated overview of the vibration test results, listing the first and second resonance frequencies along with their corresponding peak acceleration amplitudes for all eleven specimens. This tabular summary enables a clear comparison across samples and density classes. While most specimens exhibit consistent resonance behavior, three tests stand out due to notably reduced acceleration amplitudes in both modes. Despite maintaining resonance frequencies above 200 Hz—comparable to the rest—the significant drop in acceleration suggests either measurement anomalies or material inconsistencies, such as internal damping variations or structural defects. This observation reinforces the importance of considering both frequency and amplitude when evaluating dynamic response characteristics.

After completing the experimental testing, a finite element analysis (FEA) was performed. The damping coefficients calculated for each specimen based on the test results were then incorporated into the FEA to ensure accurate correlation between the experimental and numerical data.

The damping ratios obtained using the half-power bandwidth method are summarized in Figure 16.

3.3. Material Modeling Results

In the material modeling process, the elastic modulus is extracted from the material reconstructed based on available CT scan data. The analysis focuses on specimens for which CT data is available.

3.3.1. Random Particle Model

Figure 17a illustrates the selection of geometric parameters of our RVE model. In this case, the particle volume fraction was assigned based on the air fraction previously extracted from the CT scan for specimen B2. The particle size distribution was set to follow a logarithmic trend, as this better captures the inherently random and unpredictable nature of aluminum foams.

In addition to the volume fraction, two other key parameters were defined: the mean particle diameter and its standard deviation. Since the model assumes both phases as aluminum for simplification, we accounted for the presence of air by geometrically representing the particles as hollow spheres. This allowed us to preserve the influence of porosity, while the particles themselves act as thin interfacial shells with negligible material stiffness.

Finally, the dimension ratio was introduced, representing the ratio between the particle diameter and the edge length of the cubic RVE domain. For example, with a particle diameter of 10 μm and a size ratio of 4, the resulting RVE cube measures 40 μm along each axis.

After all input parameters were defined, the micromodel was generated and automatically discretized. Using the Ansys Material Designer (2025 R1) tool, the effective material properties were computed based on the defined input geometry and volume fractions. These results, shown in Figure 17b, represent the homogenized elastic behavior of the porous microstructure, as predicted by the Random Particle Method.

3.3.2. TPMS Model

This paragraph presents the second modeling strategy, which involves the use of Triply Periodic Minimal Surfaces (TPMSs). In this approach, the same two material constituents were employed as in the previous method. However, unlike the earlier model where both phases were explicitly defined, here only the solid phase was introduced as an input parameter, with its contribution specified as a percentage of the total volume (Figure 18a). The porous geometry was implicitly defined through the TPMS function, enabling a more organic and continuous representation of the cellular structure.

After all necessary parameters were configured, the model was discretized using the automated meshing algorithm. Subsequently, a new material card was generated, containing the computed mechanical properties, as shown in Figure 18b.

3.4. Analytic Results

In the analytical calculations,

Table 4. Table of solved equations for Specimen B2.

Description	Equation
Volume	$V=L·W·H=9072 {\mathrm{m}\mathrm{m}}^3$	(13)
Total mass	$M_{Total}={\mathit{\rho }}_s·V=24 \mathrm{g}$	(14)
Foam mass	$M_{Foam }=M_{Total}·\left(1-F_{Air}\right)=7.47 \mathrm{g}$	(15)
Foam density	${\mathit{\rho }}_{Foam }=824 {\displaystyle \raisebox{1ex}{$\mathrm{k}\mathrm{g}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^3$}\right.}$	(16)
Foam elastic modulus	$E_{Foam }=E_{AL}·{\left({\displaystyle \frac{{\mathit{\rho }}_{Foam }}{{\mathit{\rho }}_{AL}}}\right)}^2=6615 \mathrm{M}\mathrm{P}\mathrm{a}$	(17)
Moment of inertia	$I_z={\displaystyle \frac{W·H^3}{12}}{=283.5 \mathrm{m}\mathrm{m}}^4$	(18)
Resonance frequency	$f_n={\displaystyle \frac{{1.875}^2}{2·\pi }}·\sqrt{{\displaystyle \frac{E_{Foam }·I_z}{{\mathit{\rho }}_{Foam }·A·L^4}}}=190 \mathrm{H}\mathrm{z}$	(19)

, 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 ${\displaystyle \frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^3}}$) 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 (ϕ) 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 × 6 × 6 mm³ 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 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. The resulting value showed excellent agreement with the experimental data, demonstrating that the analytical model, despite its simplifications, can reliably predict the dynamic behavior of the porous material.

This analytical section primarily provides a brief overview of the Euler–Bernoulli beam theory as applied to cellular materials. While this method allows for estimating the fundamental resonance frequency, it does not capture the full dynamic behavior—specifically, the acceleration response observed in the two vibration modes recorded during experimental testing. The Euler–Bernoulli theory has inherent limitations when applied to porous and heterogeneous materials such as metallic foams, particularly under dynamic loading. These limitations stem from its assumptions of material homogeneity, neglect of shear deformation and rotary inertia, and inability to account for local microstructural effects. As a result, the theory may yield inaccurate predictions of dynamic behavior—especially resonance frequencies—since it does not incorporate scale-dependent properties, density variations, or the complex internal geometries characteristic of cellular materials.

3.5. FEA Results

The material properties obtained through material modeling techniques were considered in the finite element analysis and then compared with the experimental data.

In Figure 19, the comparison between the experimental data on Specimen B2 and the numerical simulations made using the material properties generated through RVE and TPMS techniques can be observed. The results showed a good correlation for the acceleration response, as both methods had less than 1% error. On the other hand, we observe a 24.77% error in the frequency for the TPMS method and a 37.09% error for the Random Particle Method. These could be caused by the geometrical accuracy of the material models. The TPMS method typically has a better approximation of the regularity and connectivity of cellular material, leading to more realistic stiffness and mass distribution; on the other hand, the random particle model often introduces irregularities, disconnected pores, or non-uniform stress distribution, which can lead to an overestimation of stiffness and higher resonance frequencies, hence the 37% error.

Following the analysis of the data presented in the previous subsection, the density values were determined analytically by measuring and weighing each specimen. It was concluded that the key parameter requiring optimization was the elastic modulus, since density was already aligned with the experimental values.

To achieve this, the experimental simulation setup was preserved unchanged, except for updating the density input to match the real measured value of the specimen. A parameter optimization study was conducted using OptiSLang by ANSYS (2025 R1), a robust tool for automated design exploration and optimization. The objective of the optimization process was to minimize the error between the simulated and experimental resonance frequency, specifically targeting a deviation of less than 1% from the experimental value.

In the optimization workflow, the elastic modulus of the material was treated as the design variable, while the density was held constant. The target response was the first resonance frequency, with the optimization goal set to match the experimentally measured value of 275 Hz, within a 1% tolerance.

As shown in Figure 20, Design 11 emerged as the optimal configuration. In this scenario, the elastic modulus was iteratively adjusted until the simulation reached a resonance frequency sufficiently close to the experimental reference. The best match was a simulated frequency of 272 Hz, which resulted in a relative error of around 0.83%.

Following the same methodology used for Specimen B2, the full process was repeated from the beginning for Specimen A1. The only modification consisted in replacing the geometry based on the CT scan of Specimen A1. All other steps, material modeling, simulation setup, and frequency extraction, remained identical. At the final stage, elastic modulus was identified by correlating the resonance frequency from the numerical simulation with the experimental result, as shown in Figure 21, ensuring consistency in methodology and validation.

4. Discussion

The results presented provide a comprehensive understanding of the methodology’s effectiveness and limitations in characterizing the dynamic behavior of metallic foams. By integrating experimental data with advanced multiscale modeling and optimization techniques, several important observations can be drawn regarding the adaptability, accuracy, and practical applications of the approach. The following discussion elaborates on these aspects in detail:

One of the main strengths of this methodology lies in its adaptability: with just a single CT scan and one dynamic test, it becomes feasible to estimate material behavior across a range of foam densities. This significantly reduces the number of physical tests required, offering a faster and more cost-effective pathway for characterizing cellular materials in dynamic environments.

Furthermore, the multiscale modeling approach applied to Specimen B2—incorporating both TPMS-based structures and stochastic RVE models—enabled a valuable comparison between idealized and more randomized microstructural configurations. Specifically, the TPMS-based model predicted a resonance frequency of 206 Hz with an amplitude of 82 g, while the random particle RVE predicted 377 Hz and 82 g. These predictions were compared with the experimental values of 275 Hz and 81 g.

Although the error margins between experimental and numerical results varied (24.77% for the TPMS model and 37.09% for the random particle RVE), the outcomes offered meaningful insights into the limitations and applicability of each modeling technique.

These discrepancies arise not only from geometrical accuracy, but also from several key factors:

▪

Differences in microstructural scale (micrometer-scale resolution for the random particle model versus millimeter-scale for the TPMS model);

▪

Variations in material property assignment due to scale-dependent behavior captured using ANSYS Material Designer;

▪

The inherent stochasticity of the random particle model versus the idealized periodicity of the TPMS structure;

▪

The influence of localized microstructural features, such as cell wall thickness distribution and connectivity, which affect the effective elastic modulus.

Moreover, the optimization routine implemented—whether through a parameter sweep or by using tools such as OptiSLang—demonstrated that inverse identification of mechanical properties is not only possible but can be performed with reasonable precision when experimental constraints are present. This proves especially useful for industries where in situ testing is limited or impossible, such as aerospace or automotive, where accurate material behavior must be predicted with minimal destructive testing.

The optimization method implemented via OptiSLang, used to identify the elastic modulus based on resonance frequency and density for Specimen A1, was applied consistently across all tested specimens.

The results showed strong agreement, with errors under 5% in both resonance frequency and amplitude for the first vibration mode, confirming the robustness and repeatability of the approach; specifically, the optimized numerical model predicted a resonance frequency of 266 Hz and amplitude of 88 g, while the experimental values were 269 Hz and 81 g, respectively.

This method also proves to be highly practical in scenarios where experimental frequency response data is available but direct correlation with simulation is hindered by unknown or difficult-to-model damping coefficients. In such cases, by using the resonance frequency and the amplitude of the frequency response as input data, it becomes possible to indirectly identify the damping coefficient. This makes the method especially valuable in practical applications, where full material characterization is limited, enabling a reliable estimation of mechanical properties even under constrained experimental conditions.

The observed discrepancies in frequency predictions between the TPMS and stochastic RVE models primarily stem from inherent differences in their microstructural representations, which influence stiffness and mass distributions. TPMS models exhibit idealized, smooth, and periodic geometries that distribute material uniformly, thereby simplifying dynamic behavior but potentially overlooking local heterogeneities. In contrast, stochastic RVE models reflect the random particle distributions and irregularities found in real metallic foams, leading to local variations in stiffness and mass that significantly affect resonance characteristics. As a result, the two approaches capture different aspects of foam complexity, yielding frequency deviations of approximately 24% for the TPMS model and 37% for the stochastic model relative to experimental data. However, both models show strong agreement in amplitude responses, suggesting their capability to replicate key dynamic features despite frequency shifts.

To address these limitations, an optimization framework was developed based on the initial models and calibrated using representative specimens. This framework enables the prediction of dynamic responses—including resonance frequency, amplitude, damping, and air volume fraction—for foams of varying densities, without requiring additional experimental testing. While it does not predict exact local distributions, it accurately captures global behavior, making the models suitable for industrial applications. Metallic foams characterized in this way are widely employed in the automotive and aerospace industries for lightweight structural components, vibration damping, and energy absorption, where overall dynamic trends are more critical than precise frequency matching.

The work also highlights the importance of considering geometric constraints in modal analysis. Specifically, accounting for the fixed portion of the sample during testing (e.g., 10 mm embedded in the vibration shaker) is crucial for accurate frequency prediction using the Euler-Bernoulli beam theory.

In conclusion, the methodology presented can serve as a solid foundation for rapid material characterization of metallic foams or similar cellular materials subjected to vibration loading.

5. Conclusions

In this study, a comprehensive approach was developed and validated for evaluating the dynamic behavior of metallic foams, using a combination of experimental, numerical, and data-driven techniques. The integration of computed tomography (CT) scans with modal vibration testing allowed for an in-depth understanding of both the internal microstructure and the global dynamic response of foam specimens. By subjecting the samples to sine load excitation, key modal parameters such as the resonance frequency were extracted with high accuracy. These values were then used to back-calculate the effective elastic modulus through a parameter-based inverse optimization method, correlating density and resonance frequency.

Based on the analysis, the following conclusions provide a comprehensive overview of the methodology’s strengths, limitations, and practical implications:

▪

Efficient characterization strategy

The methodology successfully reduces experimental effort by requiring only a single CT scan and one dynamic test to estimate material behavior across different foam densities—a clear practical advantage for early-stage assessments or industrial constraints.

▪

Mixed performance of modeling approaches

The TPMS and stochastic RVE models provided useful comparative insights but showed notable limitations in frequency prediction accuracy. TPMS had ~25% errors, and the RVE model ~37%, indicating these models may be more suitable for qualitative trend analysis than precise prediction.

▪

Limited added value from stochastic RVE modeling:

While intended to reflect more realistic microstructures, the stochastic RVE approach did not significantly improve dynamic response accuracy and added computational complexity, raising questions about its practical benefit in this context.

▪

Amplitude predictions were consistent:

Despite discrepancies in frequency, both modeling approaches captured amplitude responses well, suggesting their potential for predicting certain dynamic characteristics even if frequency tuning remains imprecise.

▪

Optimization workflow demonstrated strong potential:

The inverse identification process, particularly using OptiSLang, yielded resonance frequency and amplitude predictions within 5% of experimental values, confirming its robustness and potential as a reliable calibration tool.

▪

Useful damping estimation technique:

The ability to indirectly estimate damping coefficients from available resonance data adds practical value, especially in cases where direct measurements are not feasible.

▪

Applicable for trend-level industrial use:

While not suitable for high-precision modeling of dynamic behavior, the proposed framework captures overall trends effectively, making it a valuable tool for applications where broad dynamic behavior is more critical than exact local accuracy—such as in preliminary design for automotive or aerospace components.