Preamble

Axial Compression Behavior and Design of Tapered Energy-Absorbing Boxes Filled with Functionally Graded Aluminum Foam

Abstract: To satisfy the requirements for structural lightweighting and enhanced energy absorption, superior energy-absorbing box structures must be designed and investigated. In this study, the quasi-static axial compression behavior of tapered energy-absorbing boxes filled with functionally graded aluminum foam was simulated and optimized for lightweight performance using ABAQUS software. The validity of the finite element model was verified through comparisons with experimental data from existing literature. Subsequently, the study investigated the influence of several factors—including the taper angle of the square cross-section box, the number of induction grooves, the height and depth of these grooves, and the gradient material parameters—on the energy absorption characteristics of the structure under axial compression. Using the objectives of minimizing peak load while maximizing energy absorption for a given structural mass, an optimization design was conducted. The results indicate that these design factors significantly enhance the energy absorption performance. The optimized graded foam-filled tapered box achieved a 12.1% reduction in maximum peak force and a 3.6% increase in specific energy absorption (SEA). Furthermore, the load-displacement curve of the optimized structure is smoother, which is more conducive to efficient energy dissipation.

1 Introduction

Thin-walled structures filled with cellular materials, such as aluminum foam, have been widely adopted in the automotive and aerospace industries due to their excellent energy absorption capacity and high strength-to-weight ratio. Traditional uniform energy absorbers often exhibit high initial peak forces, which can lead to excessive deceleration and potential injury or structural failure during impact. To address these limitations, researchers have explored tapered geometries and functionally graded materials to tailor the deformation response and improve energy dissipation efficiency.

This paper focuses on a novel energy-absorbing box design that combines a tapered thin-walled shell with a functionally graded aluminum foam core. By introducing induction grooves and optimizing the material distribution, we aim to achieve a more stable crushing process and superior crashworthiness metrics.

2 Finite Element Modeling and Validation

The numerical simulations were performed using the nonlinear finite element software ABAQUS. The thin-walled tapered box was modeled using shell elements, while the aluminum foam core was modeled using solid elements with a constitutive relationship defined by the Deshpande-Fleck foam model.

2.1 Model Validation

To ensure the accuracy of the numerical results, the finite element model was validated against experimental data from the literature. The comparison focused on the deformation modes and the load-displacement characteristics. The simulation results showed high consistency with the experimental findings, confirming that the selected material models, mesh density, and contact definitions are appropriate for predicting the axial compression behavior of foam-filled structures.

[FIGURE:1]

3 Parametric Study

A comprehensive parametric study was conducted to evaluate how geometric and material variables influence the energy absorption performance, specifically the Peak Crushing Force (PCF) and the Specific Energy Absorption (SEA).

3.1 Influence of Taper Angle and Induction Grooves

The taper angle of the square cross-section plays a critical role in stabilizing the collapse mode. Increasing the taper angle generally reduces the initial peak force but may affect the total energy absorbed. Furthermore, the introduction of induction grooves serves as a trigger mechanism to encourage a progressive folding pattern. The study examined the number of grooves, as well as their height and depth, finding that properly configured grooves can significantly smoothen the load-displacement curve.

3.2 Influence of Gradient Material Parameters

Unlike uniform foam, functionally graded aluminum foam allows for a controlled variation in density along the axial direction. By adjusting the gradient parameters, the resistance of the core can be matched to the structural requirements of the shell, preventing global buckling and maximizing the volumetric energy dissipation.

[TABLE:1]

4 Optimization Design

The optimization objective was to achieve a lightweight structure that minimizes the maximum peak force while maximizing energy absorption. The mass of the structure was kept constant to ensure a fair comparison between the baseline and optimized designs.

4.1 Optimization Results and Discussion

The optimization process utilized a multi-objective approach to refine the geometric dimensions and the foam density gradient. The results demonstrate that the optimized tapered box filled with graded aluminum foam exhibits superior crashworthiness. Specifically:
- The maximum peak force was reduced by 12.1%.
- The Specific Energy Absorption (SEA) increased by 3.6%.
- The load-displacement curve became significantly flatter, indicating a more efficient and stable energy dissipation process.

[FIGURE:2]

5 Conclusion

In this study, the axial compression behavior of tapered energy-absorbing boxes filled with functionally graded aluminum foam was investigated through numerical simulation and optimization. The following conclusions were drawn:
1. The finite element model accurately predicts the crushing behavior of foam-filled tapered structures.
2. Geometric features such as the taper angle and induction grooves are effective in controlling the initial peak force and ensuring a stable progressive folding mode.
3. Functionally graded foam cores provide a significant advantage over uniform cores by allowing for tailored energy absorption profiles.
4. The optimized design achieves a better balance between low peak loads and high energy absorption, making it a promising candidate for advanced crash protection systems in lightweight engineering applications.

关键词

Abstract

This paper investigates the energy absorption characteristics of tapered energy-absorbing structures filled with functionally graded aluminum foam. Through a combination of quasi-static compression experiments and finite element simulation, the deformation modes and energy absorption performance of these structures are analyzed. The study focuses on the influence of gradient distribution and structural geometry on the specific energy absorption (SEA) and crushing force efficiency (CFE). Results indicate that the integration of graded aluminum foam significantly enhances the stability of the crushing process and improves the overall energy dissipation capacity. These findings provide a theoretical basis and design reference for the lightweight design of crashworthy components in automotive and aerospace engineering.

Keywords: Tapered energy-absorbing box; Graded aluminum foam; Quasi-static compression; Finite element simulation; Lightweight design

1. Introduction

With the increasing demands for vehicle safety and fuel efficiency, the development of high-performance, lightweight energy-absorbing structures has become a critical focus in automotive engineering. Thin-walled structures are widely utilized as primary energy absorbers due to their excellent energy dissipation capabilities and ease of manufacturing. Among these, tapered energy-absorbing boxes have demonstrated superior performance under oblique loading conditions compared to traditional straight-column structures.

To further enhance energy absorption efficiency, the application of cellular materials, particularly aluminum foam, as filler has gained significant attention. Aluminum foam possesses unique properties, such as a long stress plateau during compression, which allows it to absorb substantial energy at a nearly constant load. Recent research suggests that functionally graded foams—where the density or cell structure varies spatially—can further optimize the load-displacement response, reducing initial peak forces while maintaining high total energy absorption.

This study explores a novel hybrid structure consisting of a tapered thin-walled box filled with gradient aluminum foam. By employing both experimental testing and numerical modeling, we evaluate the synergistic effects between the tapered wall and the graded filler to achieve an optimal balance between weight and crashworthiness.

2. Materials and Methods

2.1 Specimen Preparation

The energy-absorbing box is constructed from high-strength aluminum alloy, featuring a tapered geometry defined by the taper angle $\theta$. The filler material consists of aluminum foam with a density gradient along the longitudinal axis. Three types of gradient profiles were prepared: uniform density, linearly increasing density, and linearly decreasing density.

2.2 Quasi-static Compression Testing

Quasi-static compression tests were conducted using a universal testing machine at a constant displacement rate of 2 mm/min. The load

Quasi-static axial compressive behavior and design of conical energy-absorbing box filled with gradient foam aluminum WU Xiwei ZHANG Jianxun

1. State Key Laboratory for Strength and Vibration of Mechanical Structures

School of Aerospace Xi'an Jiaotong University

710049 Xi'an

China

2. College of Mechatronic Engineering

North University of China

030051 Taiyuan

China

Abstract

In order to meet the requirements of lightweight structure and energy-absorbing box and to de- sign and study the performance of excellent energy-absorbing box structure we adopt Abaqus finite element software to simulate the quasi-static axial compression behavior and lightweight design of gradient alumi- num foam conical energy-absorbing box. First the validity of the finite element model was verified by com- paring with the literature experiment. Then the effects of the conical bottom angle the number of induction slots the height and depth of induction slots and the gradient material parameters on the energy absorption characteristics of the gradient aluminum foam conical energy-absorbing box were studied. Then with the same mass structure of low peak load and excellent energy absorption as the optimization objectives gradient aluminum foam packed conical energy-absorbing box structure was optimized. The results show

that these factors have a good effect on improving the energy absorption characteristics of the energy-ab- sorbing box. The conical bottom angle of the energy absorbing box can greatly increase the specific energy absorption of the energy absorbing box structure while reducing the initial peak force and reduce the crushing force in the collision process to a certain extent. The existence of the induction groove makes the deformation process approach to the idealized symmetric deformation. While guiding the deformation process it also reduces the initial peak force the height and depth of the induction groove have a greater role in reducing the peak force and increasing the specific energy absorption. The density gradient of foam aluminum and the number of divided layers have obvious effects on reducing the impact force and have certain effects on guiding lightweight and improving specific energy absorption. Compared with the constant section energy absorbing box the optimized structure of gradient aluminum foam cone energy absorbing box reduces the maximum peak force by 20% and increases the specific energy absorption by 3. 6% . It is found that the collision force-displacement curve is milder which is more conducive to the energy absorp- tion of the energy-absorbing box.

finite element simulation ； lightweight design

Thin-walled metal components are widely utilized in various energy-absorbing and crashworthy structures due to their excellent energy absorption characteristics and lightweight properties. Under axial crushing loads, thin-walled components can undergo plastic yielding according to predetermined modes. These controlled collapse modes and large crushing strokes provide an energy absorption capacity that is incomparable to other structural forms. The weight reduction inherent in thin-walled structures facilitates component lightweighting; when applied to crash box structures, it achieves superior energy absorption performance while simultaneously reducing weight. Furthermore, filling crash boxes with metallic porous materials can significantly enhance their energy absorption characteristics. Given the current trend toward lightweight development, the rational design of effective energy-absorbing components is particularly important.

Regarding research on aluminum foam-filled crash boxes, Ren Meng et al. used ABAQUS software to demonstrate that square cross-section crash boxes are more suitable for their specific operating conditions than circular ones. Subsequently, they introduced positive and negative thickness gradient designs to the square crash boxes. Their results indicated that crash boxes with a positive gradient design exhibited superior energy absorption characteristics. Santosa investigated the influence of filler materials on the crash performance of crash boxes, showing that aluminum foam-filled crash boxes outperform traditional ones in terms of energy absorption and collision protection.

Toksoy discussed whether the filling method of foam materials affects the energy absorption efficiency of the crash box, studying the impact resistance under empty, partially filled, and fully filled conditions. Based on experiments and numerical calculations, Cao Jiao researched the axial crushing behavior of aluminum alloy crash boxes filled with density-gradient foam, providing valuable empirical data for the study of density-gradient aluminum foam-filled structures.

In the study of tapered crash boxes, Mamalis investigated the bending problems of thin-walled conical tubes under axial pressure through theoretical and experimental methods. They were the first to propose a theoretical model for conical tubes under bending and provided a theoretical solution for the average crushing force, considering the effects of aspect ratio, semi-apical angle, and length-to-diameter ratio.

The deformation stability of total deformation modes for tapered thin-walled square tubes used as energy absorbers was studied through theoretical and experimental methods. This research also considered the response of tapered tubes under oblique impact loads and provided a theoretical solution for the average crushing force of square tapered tubes. Zhang Yi employed finite element methods to study tapered multi-cell filled crash box structures. Using response surface analysis with the taper angle and wall thickness as targets, they optimized the design of the tapered multi-cell crash box.

Numerical simulation methods were employed to perform multi-objective optimization of the crash box inclination angle and induction groove radius. With total energy absorption, specific energy absorption, and crushing force as the objectives, the optimized tapered crash box demonstrated excellent crashworthiness.

Regarding research on induction groove structures in crash boxes, Meng Xiangyao performed numerical simulations of circular cross-section crash boxes under axial loads based on theoretical analysis. By introducing induction grooves, it was found that crash boxes with these features exhibit lower peak forces. Yang Kun conducted numerical calculations on the crashworthiness of hexagonal crash box structures. Aiming to reduce the peak load, they optimized the position and depth of the induction grooves, finding that the groove parameters significantly influence the energy absorption characteristics. Xu Zhongming et al. established a full-vehicle model centered on the crash box structure. Using the spacing between induction grooves as the design variable, they performed a multi-objective optimization of the crash box's crashworthiness, providing a reference for vehicle-level crash safety research.

There are numerous studies on the factors influencing crash boxes, such as cross-sectional shape, gradient filler materials, and induction groove structures. This study focuses on square cross-section aluminum foam-filled crash boxes. Using numerical calculation methods, different configurations are analyzed.

Key words: conical energy-absorbing box structure; gradient foam aluminum; quasi-static compression

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Simulation analysis was conducted to evaluate the quasi-static axial compression characteristics of the structure under various influencing factors. Using the criteria of maintaining equivalent mass while achieving a lower peak force and superior energy absorption capacity as optimization objectives, an optimized design was developed for the tapered energy-absorbing box filled with functionally graded aluminum foam. This optimized structure was then compared with the initial design to provide a theoretical basis and reference data for the development of advanced energy-absorbing boxes.

1 吸能盒性能评价指标与设计原则

Performance Evaluation Indicators

Based on the structural design requirements of the crash box, this study adopts specific performance evaluation indicators to assess the energy absorption characteristics of aluminum foam-filled crash boxes under quasi-static compression. The primary indicator used is the maximum peak force ($F_{max}$). The maximum peak force represents the highest value of the impact force generated during the crushing process of the crash box structure. A higher $F_{max}$ indicates more severe potential damage to the protected structure; therefore, a lower maximum peak force signifies superior cushioning characteristics of the crash box.

Total energy absorption ($EA$) refers to the total amount of energy dissipated by the crash box through its own structural deformation throughout the entire crushing process. The total energy absorption is a function of the crushing force and the crushing displacement, and its expression is given by:

0 F

is the crushing displacement of the energy-absorbing box, and is the crushing force. The specific energy absorption (SEA) represents the ratio of the energy absorbed by the energy-absorbing box during the crushing process to its total mass. This metric reflects the energy absorbed per unit mass of the energy-absorbing box and holds significant guiding importance for lightweight design. Its expression is:

E abs = ∫

$E_{total}$ represents the total energy absorbed by the crash box during the crushing process, and $M$ represents the mass of the box. During the design and material selection process for crash boxes, the following criteria must be observed:

1. Energy Conversion Efficiency: When absorbing energy, kinetic energy should be converted into irreversible deformation energy (specifically plastic strain energy) as much as possible during the transformation process.
2. Stability and Repeatability: The deformation modes must be as stable and repeatable as possible to enhance the structural reliability of the component.
3. Peak Force Reduction: During the compression process, the maximum impact force (peak crush force) should be kept as low as possible to minimize deceleration injuries or structural damage.
4. Load-Displacement Characteristics: Under compressive loading, the load-displacement curve should be as stable and flat as possible, indicating a consistent energy absorption rate.
5. Specific Energy Absorption (SEA): The energy absorbed per unit mass should be maximized to achieve lightweight design goals.

2 有限元计算模型

Comparison of Finite Element Model and Experimental Results

To verify the accuracy of the finite element (FE) model, the results of the FE calculations in this study were compared with the static compression experimental results of an energy-absorbing box from the literature \cite{}. The dimensions and material properties used in the simulation were kept consistent with the experimental setup. The load-displacement curves obtained from both the calculation and the experiment are presented, showing excellent agreement. This consistency validates the accuracy of the finite element model employed in this research.

[TABLE:1] Component size of the verified model: Circular aluminum foam (Diameter 18.5 mm, Height 30 mm; Outer dimensions 20.5 mm, 18.6 mm, 30 mm).
[TABLE:2] Material properties of the verified model: Elastic modulus / GPa, Yield stress / MPa, Linear hardening modulus / GPa, Poisson's ratio, Plastic Poisson's ratio, Equiaxial plastic strain / MPa.
[FIGURE:1] Comparison of load-displacement curves.

Finite Element Calculation Model

The energy absorption characteristics of a square-section gradient aluminum foam-filled energy-absorbing box with a base angle (the definition of the base angle is provided in \ref{}) under quasi-static axial compression were numerically simulated using the ABAQUS/Explicit finite element software. The model consists of three parts: an upper rigid plate, the aluminum foam filler, and a tapered aluminum tube. To maintain consistency with the actual shape of the energy-absorbing box, the aluminum tube and the foam filler were partitioned at the end. The partitioned cubic blocks at the end were subjected to coupling and fixed constraints, restricting all six degrees of freedom. For the upper rigid plate, all degrees of freedom were constrained except for the translational degree of freedom in the $Z$-axis direction, and a loading displacement with a smooth amplitude was applied along the $Z$-axis.

The aluminum tube was modeled using 4-node shell elements with reduced integration (S4R), while the rigid plate utilized 4-node 3D bilinear rigid quadrilateral elements (R3D4). The foam filler was modeled using 8-node linear hexahedral solid elements with reduced integration (C3D8R). General contact was employed for interactions, with a 0.1 mm clearance maintained between components and a tangential friction coefficient defined using the penalty method. The wall thickness of the aluminum tube is specified accordingly. For the density-gradient aluminum foam material, it is assumed that changes in density only result in proportional changes to the elastic modulus and plastic yield strength, while other parameters remain constant. The aluminum tube follows a plastic model with a yield stress of $\sigma_y = 210 \text{ MPa}$ and a linear hardening modulus of $E_h = 1 \text{ GPa}$. The elastic properties of the aluminum foam...

1 GPa

The yield stress and plateau stress are $8 \text{ MPa}$, with a densification strain of $\epsilon_d$. Following densification, the linear hardening modulus is $E_h = 56 \text{ GPa}$. The component dimensions and material properties are detailed in [TABLE:4].

[TABLE:4] Material properties of the calculation model

Property Value Young's Modulus / GPa [Value] Yield Stress / MPa 8 Linear Hardening Modulus / GPa 56 Yield Stress Ratio [Value] Plastic Poisson's Ratio [Value] Linear Tangent Modulus / GPa [Value]

Mesh Convergence Verification

Mesh size significantly impacts both the computational accuracy and the cost of the model. Selecting an appropriate mesh size is essential to ensure precision while conserving computational resources. A mesh convergence study was conducted using the model described in the previous section. The results, illustrating the relationship between accuracy and the number of elements, are shown in [FIGURE:5].

[FIGURE:5] Mesh convergence results with different mesh numbers

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It can be observed that there is no significant difference in the results when the total number of elements reaches the specified grid density. Therefore, to ensure computational accuracy while optimizing computational efficiency, the finite element model with this specific grid density will be utilized for all subsequent simulations.

Quasi-static Validity Verification

For quasi-static problems, it is essential to ensure that the kinetic energy (ALLKE) does not exceed the internal energy (ALLIE) during the simulation. Simultaneously, the hourglass energy must be strictly controlled so that the artificial strain energy (ALLAE) does not exceed the internal energy (ALLIE). When these conditions are met, the calculation process can be considered an approximation of a quasi-static process, confirming the validity of the simulation. To verify this, the model described in the previous section was selected for quasi-static validation. As shown in [FIGURE:N], the system energy composition during the compression of the energy-absorbing box is reasonable and satisfies the requirements for quasi-static analysis. The hourglass energy is maintained within a very small positive range and consistently remains below 5% of the system's internal energy. Furthermore, the ratio of kinetic energy to internal energy is controlled within 5% and remains stable as the compression process progresses.

3 吸能盒参数对吸能特性的影响

Numerical Simulation of Energy-Absorbing Box Performance

This section presents the numerical simulation of the quasi-static axial compression characteristics for energy-absorbing box structures with varying taper angles, induction groove parameters, and functionally graded materials. The objective is to analyze the influence of these parameters on the buffering capacity and energy absorption performance of the structures. The bottom angle of the energy-absorbing box is defined as the angle between the lateral line formed by the bottom diagonal and the top diagonal. A configuration where the larger cross-sectional end is positioned away from the upper pressure plate is defined as a "forward arrangement," while the opposite configuration is defined as a "reverse arrangement." These two arrangements are illustrated in [FIGURE:N]. For convenience, the forward arrangement is denoted as $F$ and the reverse arrangement as $R$. [FIGURE:N] presents the force-displacement and energy-displacement curves for the energy-absorbing boxes at different bottom angles under both arrangement modes. [TABLE:N] lists the maximum peak force ($F_{max}$), total energy absorption ($TEA$), mass ($M$), and specific energy absorption ($SEA$) for each structure.

[TABLE:5]: Evaluation indicators of energy-absorbing boxes with different arrangement modes and bottom angles.
Energy-absorbing box ID: F/R-90, F-87.5, R-87.5, F-82.5, R-82.5.

Analysis of Quasi-Static Axial Compression Curves

The quasi-static axial compression curves for the energy-absorbing boxes at each bottom angle and arrangement mode are shown in [FIGURE:N]. It can be observed that as the taper bottom angle decreases, the maximum peak force of the energy-absorbing box continuously decreases, while the specific energy absorption ($SEA$) continuously increases. Furthermore, for a given bottom angle, the choice between forward and reverse arrangement has a negligible impact on the total energy absorption and specific energy absorption, with both remaining at approximately the same level. However, the reverse arrangement is significantly more effective at reducing the peak force compared to the forward arrangement. Additionally, the reverse arrangement produces a smoother and more stable load-displacement curve, indicating superior energy absorption characteristics.

Through comparative analysis, it was found that the reverse-arranged energy-absorbing box with a bottom angle of $82.5^{\circ}$ reduced the maximum peak force by $37\text{ kN}$ compared to a straight energy-absorbing box (bottom angle of $90^{\circ}$), while the specific energy absorption increased by $18.5\%$. Consequently, the reverse-arranged component with a bottom angle of $82.5^{\circ}$ was selected for further investigation.

Influence of Induction Groove Structural Parameters

The implementation of induction grooves slightly reduces the total energy absorption of the box, but it can significantly decrease the initial peak force and ensure a steady increase in load, thereby achieving the goal of stable collision energy dissipation. The following sections explore the effects of the number, height, and depth of the induction grooves on the energy absorption characteristics of the structure.

Number of Induction Grooves

This subsection investigates the influence of the number of induction grooves on the energy absorption characteristics, specifically comparing cases with $1$ to $4$ grooves. Reference planes were created starting at a distance of $140\text{ mm}$ from the top of the energy-absorbing box. Induction grooves with a depth $d = 4\text{ mm}$ (defined as the penetration distance at the upper end of the groove) and a height $h = 5\text{ mm}$ were sequentially added from top to bottom, as shown in [FIGURE:N]. The final deformation modes of the energy-absorbing box structures with different numbers of induction grooves are also presented.

The inclusion of induction grooves allows deformation to initiate as folds at specific locations. The presence of these grooves makes the deformation process of the energy-absorbing box more closely resemble an idealized deformation mode. Components with $4$ induction grooves are found to be more suitable as idealized energy-absorbing structures. [FIGURE:N] provides the load-displacement and energy-displacement curves for the boxes with varying numbers of induction grooves, while [TABLE:N] lists the specific values of the evaluation indicators. As the number of induction grooves increases, the total energy absorption decreases slightly; however, the effect on reducing the initial peak force and guiding the structural deformation is highly significant. This results in a more stable rise in the crushing force curve, which is highly beneficial for protecting the safety of the equipment.

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Energy absorption evaluation indexes for energy-absorbing boxes with different numbers of induction grooves. [TABLE:6] shows the evaluation indexes of energy-absorbing boxes with different numbers of induction grooves. Regarding the height of the induction grooves, this section analyzes the energy absorption characteristics of the boxes by selecting heights of $h = 4$ mm, $h = 5$ mm, and $h = 6$ mm while maintaining a constant depth of $d = 4$ mm. [FIGURE:N] presents the load-displacement and energy-displacement curves for the energy-absorbing boxes at different induction groove heights. [TABLE:N] provides the specific values of the evaluation indexes. It is observed that the crushing forces of the three configurations are approximately the same during the initial stage of crushing yield. Furthermore, the energy-displacement curves and the data in [TABLE:N] for total energy absorption and specific energy absorption indicate that as the induction groove height increases from $h = 4$ mm to $h = 5$ mm and then to $h = 6$ mm, the total energy absorption exhibits a trend of first increasing and then decreasing.

[TABLE:7] The evaluation indexes of energy-absorbing boxes with different heights of induction grooves.

[FIGURE:N] shows the axial compression characteristic curves of the boxes with different heights of the induction grooves. When $h = 5$ mm compared to $h = 4$ mm, the specific energy absorption of the box increases by 13.8%, and the maximum peak force increases by 13.4%. When comparing the $h = 5$ mm configuration to the $h = 6$ mm configuration, the specific energy absorption of the energy-absorbing box increases by 18.6%, while the maximum peak force increases by 15.6%. Considering that the energy absorption of the structure must not be too low, and given the significance of specific energy absorption for lightweight design, the energy-absorbing box component with an induction groove height of $h = 5$ mm is ultimately selected for further research.

Regarding the depth of the induction grooves, this section selects structures with depths of $d = 3$ mm, $d = 4$ mm, and $d = 5$ mm for energy absorption characteristic analysis while keeping the height constant at $h = 5$ mm. [FIGURE:N] shows the load-displacement and energy-displacement curves of the energy-absorbing boxes under different induction groove depths. [TABLE:N] provides the specific values of the evaluation indexes. As the induction groove depth increases from $d = 3$ mm to $d = 5$ mm, the initial peak force continuously decreases, resulting in a more stable increase in the impact force curve. The final maximum peak forces show little difference, and changes in depth do not significantly affect the total energy absorption. Therefore, the energy-absorbing box component with $d = 5$ mm will be selected for the next stage of research.

[FIGURE:N] shows the axial compression characteristic curves of the boxes with different depths of induction grooves. [TABLE:8] shows the evaluation indexes of energy-absorbing boxes with different depths of induction grooves. Regarding the aluminum foam gradient, the properties of gradient materials vary along the gradient direction, which endows them with many excellent mechanical properties. Next, the influence of aluminum foam density gradients on the static compression energy absorption characteristics of the energy-absorbing box structure will be explored.

In terms of the aluminum foam density gradient, a positive density gradient is defined as a gradual increase in aluminum foam density from the top end to the bottom end; conversely, a negative density gradient is defined as a decrease from top to bottom. The filled aluminum foam has a density of $360 \text{ kg/m}^3$, as shown in [FIGURE:N]. It is assumed that changes in the density of the aluminum foam only result in proportional changes to the elastic modulus and the plastic yield strength.

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[FIGURE:1] shows the load-displacement and energy-displacement curves for energy-absorbing boxes under different aluminum foam density gradients. [TABLE:1] provides the specific values of the evaluation indicators. It can be observed that the density gradient of the aluminum foam filler does not alter the magnitude of the initial peak force; the initial peak forces generated during the early stages of crushing are similar across the three cases. However, a positive density gradient significantly reduces the maximum peak force, whereas a negative density gradient increases it. The results indicate that the aluminum foam density gradient has a minimal impact on the overall energy absorption capacity.

[TABLE:9] presents the evaluation indicators for the energy absorption of aluminum foam-filled boxes with different density gradients, including negative, positive, and zero (uniform) density gradients. Through comparison, it was found that the maximum peak force of the energy-absorbing box with a positive density gradient is reduced by 9.2% compared to the uniform density case, while the specific energy absorption (SEA) decreases by only 0.34%. In contrast, the negative density gradient increases the maximum peak force by 9.2% compared to the uniform case, with a 0.25% reduction in SEA. Therefore, the energy-absorbing box component with a positive density gradient was selected for further investigation.

Regarding the axial compression characteristic curves of energy-absorbing boxes with different density gradients, this section considers dividing the density variation into a greater number of layers. The density of the uppermost layer remains at 180 kg/m³, and the density of the lowermost layer remains at 360 kg/m³. The aluminum foam filler with a positive density gradient is then uniformly divided into $n$ layers to explore the influence of the number of density gradient layers on the energy absorption characteristics of the box. [FIGURE:2] displays the load-displacement and energy-displacement curves for the energy-absorbing boxes under different layer divisions, and [TABLE:2] lists the specific values of the evaluation indicators.

[TABLE:10] shows the energy absorption evaluation indicators for boxes with different layer divisions of the aluminum foam density gradient. The number of density division layers is as follows:

17 层

Based on the quasi-static axial compression characteristic curves of the energy-absorbing boxes with different density gradients of aluminum foams, it can be observed that the number of density gradient layers has a minimal impact on the load-displacement and energy-displacement curves. However, increasing the number of layers can effectively reduce the maximum peak force of the energy-absorbing box during the crushing process. Furthermore, the number of density gradient layers has essentially no effect on the total energy absorption or the specific energy absorption (SEA) of the structure. Taking these factors into comprehensive consideration, a positive density gradient with three layers was selected as the final improved model.

Comparison Before and After Optimization

This section compares the final optimized energy-absorbing box structure with a baseline square-section aluminum foam-filled box that lacks a taper angle, induction grooves, and density gradients. [FIGURE:N] presents a comparison of the load-displacement and energy-displacement curves for the energy-absorbing box structure before and after optimization. [TABLE:N] lists the maximum peak force, total energy absorption, mass, and specific energy absorption of the structures for both cases.

It can be observed that the load-displacement curve of the optimized energy-absorbing box is significantly smoother, and both the initial peak force and the maximum peak force are substantially reduced, making it more suitable for use as an energy-absorbing structure. Similarly, while the total energy absorption of the optimized box is slightly lower than that of the pre-optimization structure, the specific energy absorption has been significantly improved.

Through this comparison, it is found that the maximum peak force of the final optimized energy-absorbing box structure is $147.87 \text{ kN}$, whereas the maximum peak force of the structure before optimization was $184.87 \text{ kN}$. The specific energy absorption of the optimized structure is...

45 J/ g

The specific energy absorption of the structure before optimization is $45 \text{ J/g}$. After optimization, the specific energy absorption reaches $64 \text{ J/g}$, representing an increase of $42.2\%$. This indicates that the optimized structure can absorb more energy per unit mass, thereby significantly improving its crashworthiness.

[FIGURE:1]

[TABLE:1]

As shown in [TABLE:1], the peak crushing force of the structure decreased from $184.87 \text{ kN}$ to $147.87 \text{ kN}$, a reduction of $20\%$. This reduction effectively lowers the deceleration experienced by the vehicle during a collision, which is crucial for enhancing occupant safety. The deformation modes of the structure before and after optimization are illustrated in [FIGURE:1]. It can be observed that the optimized structure exhibits a more stable progressive folding mode, avoiding the global buckling observed in the initial design. This stable deformation process ensures a more consistent energy dissipation rate throughout the impact event.

64 J/ g

The optimization results demonstrate that the energy-absorbing box exhibits a 3.6% improvement in performance. Furthermore, the collision force curve is significantly smoother following optimization, indicating that the structure is better suited for use as an energy-absorbing component.

[FIGURE:11]
Figure 11: Energy-absorbing curves of energy-absorbing boxes before and after optimization.

[TABLE:11]
Table 11: The energy-absorbing evaluation indexes of energy-absorbing boxes before and after optimization.

Energy-Absorbing Box Model

The comparative analysis of the evaluation indexes confirms that the optimized design enhances the structural response during impact. By smoothing the collision force profile, the peak deceleration is reduced, which is a critical factor in improving the crashworthiness of the vehicle's energy-absorbing system.

4 结

This study utilizes the ABAQUS finite element analysis software to simulate the quasi-static axial compression characteristics of tapered energy-absorbing boxes filled with functionally graded aluminum foam. Various design parameters were investigated and analyzed to provide a reliable theoretical basis for the structural design of energy-absorbing boxes.

With the objective of achieving a lower peak crushing force and superior energy absorption capacity under equal mass constraints, an optimization design was performed on the graded aluminum foam-filled tapered structure. The results demonstrate that the factors considered in this study effectively enhance the energy absorption performance of the structure.

1096 应用力学学报

The primary conclusions are as follows: 1) Factors such as the taper angle of the square-section energy absorber, the number of induction grooves, the height and depth of these grooves, and the gradient material parameters significantly influence the energy absorption characteristics of tapered energy absorbers filled with functionally graded aluminum foam under axial compression. 2) The taper angle of the energy absorber can substantially increase the specific energy absorption (SEA) of the structure while simultaneously reducing the initial peak force and, to a certain extent, lowering the crushing force during the collision process. 3) The presence of induction grooves facilitates a deformation process that more closely approximates idealized symmetrical folding; while guiding the deformation, these grooves also reduce the initial peak force. Furthermore, the height and depth of the induction grooves play a critical role in both mitigating peak force and enhancing specific energy absorption. 4) The density gradient and the number of layers in the aluminum foam have a pronounced effect on reducing the magnitude of the impact force, while also providing guidance for lightweight design and the improvement of specific energy absorption. 5) Compared to constant-section energy absorbers, the optimized tapered energy absorber filled with graded aluminum foam exhibited a 3.6% reduction in the maximum peak force. The study demonstrates that the energy absorption performance of the designed graded aluminum foam-filled tapered structure is superior to that of constant-section absorbers, featuring a smoother displacement curve that is more conducive to efficient energy dissipation.

HU Jun, WANG Feihu. Comparative analysis and optimization design of energy absorption characteristic of thin-walled tubes with various section geometries under axial impact [J]. Chinese Journal of Computational Mechanics, 2018, 35(5): 621-628. (in Chinese)

HUSSAIN N N, REGALLA S P, RAO Y V D, et al. An experimental and numerical analysis on influence of triggering for composite automotive crash boxes under compressive impact loads [J]. International Journal of Crashworthiness, 2017, 22(2): 113-136.

SONG K, YANG B C. Collision simulation and analysis of energy-absorption...

absorbing box for vehicle [ J ] . Applied mechanics and materials ，

LEE Y D NAM K W PARK S H et al. Feasibility of using Al6060S-T6 for fabricating automobile crash boxes . Transac-

tions of the Korean Society of Mechanical Engineers-a ， 2021 ， 45 （ 3 ）： 267-274. [ 5 ] CAO J ， KHAJEPOUR A ， GAN N F ， et al. Crash worthiness of func-

References

JIAN Xiaochun, WANG Xiao. Simulation of crashworthiness during frontal and offset impacts and vehicle body structure improvement. Journal of Automotive Safety and Energy. (in Chinese)

WAN Xinming, XU Xiaofei, XU Zhongming, et al. Structural optimization design of aluminum alloy energy-absorbing boxes for automobiles. Chinese Journal of Automotive Engineering. (in Chinese)

ARNOLD B, ALTENHOF W. Experimental observations on the crush characteristics of AA6061 T4 and T6 structural square tubes with and without circular discontinuities. International Journal of Crashworthiness.

MA F W, LIANG H Y, PU Y F, et al. Multi-objective optimization of a crash box filled with a three-dimensional cellular structure under multi-angle impact loading. Proceedings of the Institution of Mechanical Engineers.

chanical engineers ， part d ： journal of automobile engineering ， 2021 ，

MOGHADDAM A M KHERADPISHEH A ASGARI M. A basic design for automotive crash boxes using an efficient corrugated con-

ical tube [ J ] . Proceedings of the institution of mechanical engi-

Multiobjective Crashworthiness Optimization for Foam-Filled Thin-Walled Structures Subjected to High-Value Impact

Abstract: This study investigates the crashworthiness of aluminum foam-filled thin-walled shell structures under high-value impact conditions. To enhance the energy absorption capacity and safety performance of automotive structural components, a multiobjective optimization approach is employed. By integrating experimental data with numerical simulations, the research evaluates the influence of structural parameters and foam density on the deformation patterns and energy dissipation characteristics of the filled structures. The results demonstrate that the synergistic effect between the aluminum foam core and the thin-walled shell significantly improves the specific energy absorption compared to empty structures. Through the application of multiobjective optimization algorithms, an optimal configuration is identified that balances the competing requirements of maximizing energy absorption while minimizing the peak crushing force. These findings provide a theoretical basis and practical guidelines for the design of lightweight, impact-resistant automotive components.

1. Introduction

The demand for improved vehicle safety and fuel efficiency has driven the development of advanced energy-absorbing structures in automotive engineering. Thin-walled structures are widely utilized as primary crashworthy components due to their high efficiency in dissipating kinetic energy through plastic deformation. Among various enhancement strategies, filling these structures with lightweight cellular materials, such as aluminum foam, has emerged as a promising method to further increase energy absorption capacity without a substantial weight penalty.

Under high-value impact loads, the interaction between the filler material and the tube wall becomes complex. The aluminum foam not only absorbs energy through its own cellular collapse but also changes the buckling mode of the external shell, often leading to a more stable and efficient deformation process. However, designing these structures involves a trade-off between several conflicting performance indicators, such as Specific Energy Absorption (SEA) and Peak Crushing Force (PCF). Therefore, multiobjective optimization is essential to achieve a balanced design that meets modern safety standards.

2. Methodology and Numerical Modeling

To analyze the crashworthiness of foam-filled structures, a comprehensive numerical model was developed using finite element analysis (FEA). The thin-walled shell was modeled using shell elements with material properties corresponding to high-strength aluminum alloy, while the foam core was represented using solid elements with a constitutive model capable of capturing the plateau stress behavior of metallic foams.

[FIGURE:1]

The impact conditions were simulated to reflect high-velocity scenarios typical of automotive collisions. The validation of the numerical model was performed by comparing the simulation results with experimental data from the literature, ensuring

pact [ J ] . Chinese journal of applied mechanics ， 2016 ， 33 （ 5 ）： 845-

Mechanical Properties and Energy Absorption of Self-Similar Porous Materials

Authors: Chen Zhenshan, Qin Qinghua, Zhao Guiping
Journal: Chinese Journal of Applied Mechanics

Abstract

This paper investigates the mechanical properties and energy absorption characteristics of self-similar porous materials. By employing theoretical modeling and numerical simulations, the study explores how the hierarchical structure of these materials influences their deformation behavior and energy dissipation capacity under various loading conditions. The results indicate that self-similar architectures can significantly enhance the specific strength and energy absorption efficiency compared to conventional uniform porous structures.

1. Introduction

Porous materials are widely utilized in aerospace, automotive, and protective engineering due to their lightweight nature and excellent energy absorption capabilities. Traditional porous materials, such as honeycombs and foams, often exhibit limitations in optimizing performance across multiple scales. Self-similar (hierarchical) porous materials, inspired by biological structures like bone and wood, offer a promising strategy to overcome these limitations. By introducing structural features at multiple length scales, these materials can achieve superior mechanical efficiency.

2. Geometric Modeling and Methodology

The self-similar porous structures analyzed in this study are generated through iterative geometric rules. Starting from a base unit cell, such as a hexagonal honeycomb or a square lattice, higher-order hierarchies are constructed by replacing the solid ribs of the previous generation with scaled-down versions of the base geometry.

[FIGURE:1]

The relative density of the $n$-th order self-similar structure, denoted as $\rho_n / \rho_s$, is a critical parameter. For a structure with self-similarity, the relationship between the geometric parameters of successive generations can be expressed as:

$$ \frac{\rho_n}{\rho_s} = \Phi \left( \frac{t_n}{l_n}, \frac{t_{n-1}}{l_{n-1}}, \dots \right) $$

where $t$ represents the thickness and $l$ represents the length of the structural members.

3. Mechanical Properties

3.1 Effective Elastic Modulus

The effective elastic modulus $E^*$ of the self-similar porous material is derived using the energy method and beam theory. For a first-order hierarchical structure, the normalized elastic modulus is found to be a function of the hierarchy ratio and the base material properties. As the hierarchy order increases, the scaling law for the elastic modulus deviates from the classical Gibson-Ashby model

CHEN Zhenshan ， QIN Qinghua ， ZHAO Guiping. Mechanical prop-

erties and energy absorption of self-similar porous materials

Chinese journal of applied mechanics ， 2021 ， 38 （ 5 ）： 1846-1852 （ in

Chinese LI Z K ZHAO X et al. Crashworthiness and lightweight op- timization to applied multiple materials and foam-filled front end

structure of auto-body [ J ] . Advances in mechanical engineering ，

Study on the Blast Resistance and Energy Absorption Characteristics of Metal Composite Sandwich Panels with Pore Structures

QIN Qinghua, AI Weilong, ZHANG Jianxun
Chinese Journal of Computational Mechanics

1. Introduction

Metal sandwich structures have gained significant attention in the fields of aerospace, marine engineering, and protective structures due to their high specific strength, high specific stiffness, and excellent energy absorption capacity. Among these, sandwich panels with cellular or porous cores—such as honeycombs, foams, and lattice structures—demonstrate superior performance in mitigating the effects of blast loads and high-velocity impacts.

The dynamic response of these structures under explosive loading is a complex process involving large deformations, high strain rates, and fluid-structure interaction. Understanding the deformation mechanisms and energy dissipation pathways is crucial for the design of next-generation protective systems. This study focuses on the blast resistance and energy absorption characteristics of metal composite sandwich panels featuring specialized pore structures, utilizing both numerical simulations and theoretical analysis to evaluate their performance.

2. Numerical Modeling and Material Properties

To investigate the structural response under blast loading, a high-fidelity finite element model was developed. The sandwich panel consists of two thin face sheets and a core layer with a specific pore geometry. The material behavior of the metallic components is described using a strain-rate-dependent constitutive model to account for the dynamic effects inherent in explosion events.

The blast load is modeled using the Friedlander equation or through direct coupling with air-blast pressure profiles. The interaction between the shock wave and the front face sheet leads to an initial impulse transfer, followed by the progressive crushing of the porous core and the subsequent deformation of the back face sheet.

[FIGURE:1]

3. Dynamic Response and Deformation Modes

The response of the sandwich panel can be categorized into several distinct stages: the initial fluid-structure interaction, the compression of the core, and the global bending and stretching of the entire panel. Depending on the intensity of the blast and the structural configuration, different failure modes may be observed, including face sheet tearing, core densification, and debonding at the core-face interface.

The pore structure plays a decisive role in governing the collapse mechanism. By adjusting the porosity and the topology of the core, the plateau stress and the densification strain can be tuned to optimize the energy absorption. Our results indicate that the pore-structured core effectively extends the duration of the impact force, thereby reducing the peak acceleration transmitted to the protected internal

ance and energy absorption of perforatedmetal sandwich plates

Chinese journal of computational mechanics ， 2019 ， 36 （ 4 ）： 491-

Experimental Investigations on the Quasi-Static Axial Crushing of Aluminum Foam-Filled Thin-Walled Structures

Abstract

This paper presents an experimental study on the quasi-static axial crushing behavior of aluminum foam-filled thin-walled tubes. By conducting axial compression tests on empty tubes, pure aluminum foam, and foam-filled tubes, the interaction effects between the tube wall and the foam core were analyzed. The results indicate that the energy absorption capacity of foam-filled tubes is significantly higher than the sum of the individual components. The deformation modes and energy dissipation mechanisms are discussed in detail to provide a basis for the design of high-performance energy absorbers.

1. Introduction

Thin-walled structures are widely used in automotive, aerospace, and marine engineering as energy-absorbing components due to their high efficiency and lightweight properties. Aluminum foam, characterized by its unique cellular structure, exhibits a long stress plateau during compression, making it an ideal filler material. When aluminum foam is integrated into thin-walled tubes, the coupling effect between the filler and the tube wall can significantly enhance the overall structural stability and energy absorption performance \cite{1, 2}.

[FIGURE:1]

2. Experimental Materials and Methods

The specimens used in this study consist of circular aluminum alloy tubes and closed-cell aluminum foam. The aluminum foam was fabricated using the melt-foaming process, with a relative density ranging from 0.15 to 0.25. The thin-walled tubes were made of 6061-T6 aluminum alloy.

Quasi-static axial compression tests were performed using a universal testing machine at a constant displacement rate of 2 mm/min. The load-displacement data were recorded automatically, and the deformation process was captured using a high-resolution camera to analyze the folding patterns.

3. Results and Discussion

3.1 Deformation Modes

The experimental observations show that the empty tubes typically deform in a diamond or concertina mode depending on the diameter-to-thickness ratio. In contrast, the foam-filled tubes exhibit a more stable progressive folding pattern. The presence of the aluminum foam core restricts the inward buckling of the tube wall, effectively reducing the wavelength of the folds and increasing the number of plastic hinges.

[FIGURE:2]

3.2 Energy Absorption Performance

The energy absorption $W$ is calculated by integrating the load-displacement curve:
$$W = \int_{0}^{\delta} P(x) dx$$

foam-filled crash box [ J ] . Journal of experimental mechanics ，

References

Ren Meng, Dong Wanpeng, Li Jiayi, et al. Study on low-speed energy absorption box of automobile based on ABAQUS. Agricultural Equipment & Vehicle Engineering.

Santosa S, Wierzbicki T. Crash behavior of box columns filled with aluminum honeycomb or foam. Computers & Structures.

Toksoy A K, Güden M. Partial Al foam filling of commercial 1050H14 Al crash boxes: the effect of box column thickness and foam relative density on energy absorption. Thin-Walled Structures.

[Author Name Missing]. Research on energy absorption performance of functional density gradient aluminum foam and its filled structures. Changsha: Hunan University.

Mamalis A G, Manolakos D E, Saigal S, et al. Extensible plastic collapse of thin-wall frusta as energy absorbers. International Journal of Mechanical Sciences.

tional journal of mechanical sciences ， 1986 ， 28 （ 4 ）： 219-229. [ 21 ] REID S R ， REDDY T Y. Static and dynamic crushing of tapered

References

[1] Abramowicz, W., & Jones, N. (1984). Dynamic progressive buckling of circular and square mild steel tubes. International Journal of Mechanical Sciences, 26(4), 205-233.

[2] [Author Name Missing]. Structural optimization design of automobile energy absorption boxes under low-speed impact [Master's thesis]. Dalian: Dalian University of Technology.

[3] [Author Name Missing]. Research on energy absorption characteristics of aluminum foam-filled energy absorption boxes during automobile collisions [Master's thesis]. Fuyang: Liaoning Technical University.

[4] Meng, X. Y., Dong, W. P., & Yang, D. Y. (2018). Collision research of automobile energy absorption box based on ABAQUS. Intelligent Computer and Applications, 8(2), 115-117.

[5] Yang, K., Wang, J. B., Zou, J., et al. (2015). Structure optimization design and crashworthiness analysis of automobile energy absorbing box. Manufacturing Automation, 37(12), 74-77.

[6] Xu, Z. M., Wang, Q. Q., Fan, W. C., et al. (2014). Multi-objective optimization of crashworthiness of energy-absorbing box for vehicle front longitudinal beam. Journal of Vibration and Shock, 33(14), 116-121.

[7] Du, X. J., Chen, Z. L., Zhou, H. C., et al. (2017). Crashworthiness of RCAR low-speed collision energy absorbing box for pure electric vehicles. Journal of Chongqing Jiaotong University (Natural Science), 36(11), 114-120.

SUV [ J ] . Journal of Chongqing Jiaotong University （ natural sci-

Internal Structure Improvement of Energy-Absorbing Box Bodies

DU Xuejing, WEI Liping
(Chongqing University of Technology, Journal of Natural Science)

1 Introduction

With the continuous development of the automotive industry, vehicle safety has become a primary focus for both consumers and manufacturers. As a critical passive safety component, the energy-absorbing box is located between the bumper and the longitudinal beam of the vehicle. Its primary function is to absorb kinetic energy through plastic deformation during low-speed collisions, thereby protecting the main body of the vehicle and reducing repair costs. Traditional thin-walled energy-absorbing boxes often face challenges such as insufficient energy absorption efficiency or high peak initial crushing forces, which can lead to structural failure or excessive deceleration for occupants.

This study focuses on the internal structural improvement of energy-absorbing box bodies. By introducing specific internal reinforcements and optimizing the geometric configuration, we aim to enhance the specific energy absorption (SEA) while maintaining a stable crushing force.

2 Structural Design and Modeling

The basic geometry of the energy-absorbing box is typically a thin-walled rectangular or hexagonal tube. In this research, we propose an improved design by incorporating internal partitions and stiffeners. The design objective is to maximize the energy dissipated per unit mass during a collision event.

[FIGURE:1]

As shown in [FIGURE:1], the modified structure utilizes a multi-cell configuration. According to the theory of thin-walled structures, the mean crushing force $P_m$ can be estimated using the following relationship:

$$P_m = 13.06 \cdot \sigma_y \cdot t^2 \cdot \sqrt{\frac{b}{t}}$$

where $\sigma_y$ represents the yield strength of the material, $t$ is the wall thickness, and $b$ is the width of the section. By introducing internal ribs, the effective width $b$ is reduced, and the number of folding lines increases, which theoretically enhances the energy absorption capacity.

3 Numerical Simulation and Analysis

To evaluate the performance of the improved design, finite element analysis (FEA) was conducted using LS-DYNA. The material used for the energy-absorbing box is typically an aluminum alloy or high-strength steel, modeled with a piecewise linear plasticity material law.

3.1 Boundary Conditions and Loading

The box is fixed at one end (representing the connection to the longitudinal beam) and subjected to an impact

body of can suck box [ J ] . Journal of Chongqing University of Tech-

Energy Absorption Performance of Aluminum Foam and Its Filled Structures Based on the Finite Element Method

Abstract

This research focuses on the energy absorption characteristics and multi-objective optimization of thin-walled conical components. By utilizing the finite element method, the study investigates the mechanical behavior of aluminum foam and its application as a filler in structural components to enhance crashworthiness and energy dissipation.

Introduction

Thin-walled structures are widely utilized in the aerospace, automotive, and rail transit industries due to their excellent energy absorption capacity and high strength-to-weight ratio. Aluminum foam, characterized by its unique cellular structure, serves as an ideal filler material that can significantly improve the buckling stability and energy dissipation of these thin-walled components. This paper explores the buffering properties of conical structures filled with aluminum foam and seeks to optimize their geometric parameters for maximum safety performance.

1. Finite Element Modeling of Aluminum Foam

To accurately simulate the mechanical response of aluminum foam under compressive loading, a robust finite element model is required. The material properties of the foam are defined by its plateau stress and densification strain. In this study, the constitutive behavior is modeled using the Deshpande-Fleck foam model, which accounts for the volumetric hardening and the interaction between the foam filler and the thin-walled tube.

[FIGURE:1]

The interaction between the aluminum foam and the internal walls of the conical structure is governed by frictional contact. The simulation results indicate that the presence of the foam filler alters the deformation mode of the thin-walled structure from a global buckling mode to a more efficient progressive folding mode, thereby increasing the total energy absorption (TEA).

2. Energy Absorption Characteristics of Conical Structures

Conical thin-walled components offer distinct advantages over cylindrical ones, particularly regarding their sensitivity to the angle of impact. The energy absorption performance is evaluated using several key indicators:
- Specific Energy Absorption (SEA): The energy absorbed per unit mass.
- Peak Crush Force (PCF): The maximum force encountered during the compression process.
- Crush Force Efficiency (CFE): The ratio of the mean crush force to the peak crush force.

[TABLE:1]

As shown in [TABLE:1], the semi-apical angle of the cone significantly influences the stability of the collapse. A larger angle tends to reduce the peak force, which is beneficial for protecting the protected object, but it may also reduce the total energy absorbed if the angle is excessive.

3.