Analytical and Numerical Investigation of Energy Absorption in Graded Aluminum Open Cell Foam under Low Velocity Impact Loading
Subject Areas : Mechanics of SolidsS Davari 1 , Seyed Ali Galehdari 2 * , Amir Atrian 3
1 - Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
2 - Faculty Member of Mechanical Engineering Department, Islamic Azad University, Najafabad Branch,Iran
3 - Department of Mechanical Engineering, Islamic Azad University, Najafabad Branch, Najafabad, Esfahan, Iran
Keywords: Open cell foam, Optimization, Specific energy absorption, Low velocity impact, Graded structure,
Abstract :
Given the significance of energy absorption in various industries, light shock absorbers such as structures made of metal foam have been considered. In this study, analytical equation of plateau stress is presented for an open cell foam based on the Gibson-Ashby model, which follows elastic perfectly plastic behavior. For comparison of acquired analytical equations, the problem for a cell and then for three cells that make up an aluminum open cell foam is simulated in ABAQUS/CAE. Using the stress strain diagram, plateau stress and densification strain equations, the specific energy absorbed of the open cell metal foam is extracted. The capacity of absorb energy for an aluminum open cell foam with three cell is obtained once using analytical equations and again by using numerical simulation in ABAQUS/CAE. Numerical results retain an acceptable accordance with analytical equations with less than 3% occurred error for absorbed energy. To ensure the accuracy of numerical simulation, the results of simulating are compared with the results of the simulation of the same foam in a reference whose accuracy is verified by the experiment. Based on the results, the effective cross-sectional area of the foam with Gibson-Ashby cell does not follow the cross-sectional that is used for the calculation of plateau stress in adsorbent structures. Then tow equations are extracted to calculate the effective cross-sectional area and the transfer force. Applying sequential quadratic programming method (SQP) and genetic algorithm (GA), to design a graded metal foam with high specific Energy absorption.
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[20] B. H. M. Ashby , 2000. , "Metal Foams: A Design Guide," Boston: Hutterworth, pp. 1-50, 2000.
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Journal of Solid Mechanics Vol. 15, No. 4 (2023) pp. 469-487 DOI: 10.60664/jsm.2024.3051777 |
Research Paper Analytical And Numerical Investigation of Energy Absorption in Graded Aluminum Open Cell Foam Under Low Velocity Impact Loading |
S. Davari1,2, S.A. Galehdari1,2 1, A. Atrian | |
1 Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran 2 Modern Manufacturing Technologies Research Center, Najafabad Branch, Islamic Azad University, Najafabad, Iran | |
Received 25 May 2023; accepted 27 August 2023 | |
| ABSTRACT |
| Given the significance of energy absorption in various industries, light shock absorbers such as structures made of metal foam have been considered. In this study, analytical equation of plateau stress is presented for an open cell foam based on the Gibson-Ashby model, which follows elastic perfectly plastic behavior. For comparison of acquired analytical equations, the problem for a cell and then for three cells that make up an aluminum open cell foam is simulated in ABAQUS/CAE. Using the stress strain diagram, plateau stress and densification strain equations, the specific energy absorbed of the open cell metal foam is extracted. The capacity of absorb energy for an aluminum open cell foam with three cell is obtained once using analytical equations and again by using numerical simulation in ABAQUS/CAE. Numerical results retain an acceptable accordance with analytical equations with less than 3% occurred error for absorbed energy. To ensure the accuracy of numerical simulation, the results of simulating are compared with the results of the simulation of the same foam in a reference whose accuracy is verified by the experiment. Based on the results, the effective cross-sectional area of the foam with Gibson-Ashby cell does not follow the cross-sectional that is used for the calculation of plateau stress in adsorbent structures. The reason for this is reasonably stated. Then, in order to determine the effective cross-section in this foam, and also with regard to the importance of calculating the force transmitted to the protected structure in impact loading, tow equations are extracted to calculate the effective cross-sectional area and the transfer force. In order to create a suitable impact absorber, grade affection is used to design this foam. Applying sequential quadratic programming method (SQP) and genetic algorithm (GA), to design a graded metal foam with high specific Energy absorption. © 2023 IAU, Arak Branch.All rights reserved. |
| Keywords: Open cell foam; Specific energy absorption; Low velocity impact; Graded structure; Optimization. |
1 INTRODUCTION
D
UE to the importance of energy absorption in various industries, light-weight impact absorbents including honeycomb structures [1] and metal foams have gained increased attention [2]. Metal foam is a porous metal structure with vastly different characteristics compared to a continuous metal part. These metal foams offer physical, mechanical and electrical properties which are different from their bulk metal which has resulted in several applications as impact, vibration and sound absorbents. One of the important applications of metal foams is in automobile and aerospace industries due to their high energy absorption capacity in compression stresses. A metal foam is defined as a metal structure with uniformly distributed gas-filled pores. If these pores are not connected to each other, the structure will be known as a closed cell metal foam while structures with connected pores are known as open cell metal foams [3]. Materials with cellular structure are abundantly used in nature. The examples of these natural porous structures include prismatic or honeycomb cells in wood or cork, the cellular structure in the inner part of plant stems and cellular structures in the inner bones of human skull.
In the previous decade, various structures with high energy absorption capacity have been investigated. Galehdari et.al. [1] carried out an analytical, numerical and empirical investigation of graded honeycomb structures at low speeds. While optimizing the mass and geometry of honeycomb structures, they observed that graded structure and plane loading result in decrease in the force transferred to the protected structure and reduced damage. Sawei et.al. [4] investigated different modeling methods for simulation of metal foams and introduced six models with different advantages and disadvantages. Stone [5] investigated material test methods in different relevant standards and stated that these methods are not sufficient for test of metal foams. He created twenty-one finite element models of open-cell and close-cell metal foams and carried out more than seventy numerical simulations. His finite element models were made from beam elements while using Gibson-Ashby and Kelvin cells. He then repeated a single cell in order to reach the desired foam density. Kremer [6] investigated the effects of metal foams in car seats for reducing damage to the passengers and used head injury critria (HIC) for his tests. Gibson [2] also investigated various natural foam structures including wood, cork, bone, plant stems and other similar structures. Pinnoji et.al. [7] replaced the thermoplastic material in motorcycle helmets with a metal foam and investigated the dynamic behavior of the helmet. They also separately investigated head injury criterion in a helmet with metal foam and one with ABS foam and concluded that using metal foam leads to better results. Moreira et.al. [8] investigated the numerical behavior of open cell aluminum foam under impact scenarios. They investigated the behavior of aluminum foam with different spherical cavity dimensions and presented displacement force diagrams for each cavity dimension. Among the dimensions of 4, 5 and 6 mm holes, the energy absorption property of the 5 mm hole was better than other samples. Li et.al. [9] investigated the effect of radial supports on the behavior of open-cell aluminum foam under quasi-static and dynamic compression. The results showed that this type of support condition increases the stiffness strain in the open cell aluminum foam and increases the absorbed energy per unit volume of the open cell aluminum foam. Ramirez et.al. [10] investigated the elasto-plastic behavior of open cell aluminum foam which was created by aluminum-silicon-magnesium alloy and by infiltration method and by vacuum pressure under compressive loading in the experiment. They created the geometric model by 3D CT scan method and used it in finite element analysis. And they concluded that plane stress and energy absorption increase with decreasing the size of the cavity and increasing the density. Anwar al-Hassan [11] presented a model for closed-cell aluminum foam modeling, whose finite element analysis results were in good agreement with the test results. The model presented in this reference provided a better prediction for the mechanical properties of the foam than the previous models. Goga [12] presented a new model to predict the behavior of foams. The results of this model were consistent with the results of experiments. In this model, the definition of a spring and damper was used to create a foam mathematical model. Bin et.al. [13] studied the repeatability and predictability of void size and relative density of an open-cell aluminum foam with spherical voids using the space-holder method and evaluated the mechanical properties of the foam. Kaoua et.al. [14] used a numerical simulation and a finite element modeling. In this way, the foam structure was created by a regular network of the Kelvin model. For the elements, they used the beam model and defined four types of sections for the beam. The numerical analysis results were in agreement with the experimental results. Branca et.al. [15] used Gresen's model to simulate failure in metal foams. Two aspects were used in Gresen's model and hypoelastic and hyperelastic formulations were compared. Lopatnikov et al. [16] investigated an aluminum metal foam in four different ballistic velocity regimes and presented the time history of the stress wave propagated in the foam along with the impact energy and deformation of the foam and finally obtained a relationship between optimal energy absorption and foam density.
Mukai et al. [17] investigated an open-cell magnesium foam in a dynamic test with a strain rate of 103s-1 and evaluated its energy absorption. Also, this test was performed for foams with different densities and the results were compared with foams of other types.
Zhihua et al. [18] evaluated the static and dynamic behavior of open-cell aluminum foam with different cell sizes but with the same structure. The results showed that the dynamic response of foams is sensitive to the strain rate and depends on the cell size. Davari et.al. [19] designed an aluminum open cell foam as a shock absorber for use in helicopters in emergency landing condition. By design a graded foam and performing optimization operations, they achieved an optimal structure for energy absorption that compliance the criteria of aviation standards. Base on the reviewed articles, the equation of the absorbed energy for Gibson- Ashby open cell foam has not been derived yet. In this paper this equation has been derived. This equation can be used to measure the capacity and optimization of energy absorption of open cell foams.
2 MECHANICS OF METAL FOAM
2.1 Stress – strain graph of energy absorbents
When a foam structure is subjected to compression loading, it first enters an elastic region where the stress increases until reaching its yield stress. Then, the foam enters a plastic region where its cells undergo plastic collapse one by one. At this stage, the stress – strain graph of the foam is an almost horizontal line which shows significant increase in strain without significant increase in stress. This stress is known as the plateau stress. With increased in load, material is then compressed to its ultimate level, reaching a strain known as densification strain and shown with εD. After this stage, stress will increase with a sharp slop and material loses its capacity to absorb energy. Figure 1 shows an ideal stress – strain graph for energy absorbent materials.
2.2 Foam behavior under compression loadings
When investigating the behavior of foams under compression loading, it is necessary to investigate various behaviors including linear elastic behavior, non-linear elastic behavior and plastic behavior. During the linear elastic phase, materials don’t show suitable energy absorption capabilities. However, non-linear elastic behavior and plastic behavior result in significant strain in foams which is suitable for energy absorption. Among these two, non-linear elastic behavior is often seen in hyper elastic materials such as elastomers which are not the subject of the current study. However, plastic behavior is often seen in metals including aluminum foams under impact loading which is explained in the following sections.
When investigating the mechanical behavior of materials, it is first necessary to have a suitable parametric model which offers a mathematical explanation of mechanical behaviors. To this end, Gibson and Ashby introduced a geometrical model for studying the behavior of foams. They offered two models for open and close-cell foams and proved their accuracy using empirical experiments [3]. This model for open-cell foams is shown in figure 2.
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Fig.1 Ideal stress – strain graph of energy absorbent materials [19]. |
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Fig.2 Gibson – Ashby model for open-cell foams [3]. |
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Fig.3 Foam densification due to plastic behavior [3]. |
According to figure 3, compression loading results in bending of the foundations and creates plastic hinge in junctions. Continuous loading results in densification of the foam. The formation of plastic hinge is shown in figure 3.
If σpl* is the plateau stress resulting from plastic behavior of the foam, σys is the yield stress of base metal, ρ* is the density of the foam and ρs is the density of the base metal, then their relation can be defined as shown in equation (1) [3].
(1) |
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3 THE ENERGY ABSORPTION EQUATION OF OPEN-CELL FOAM
If σpl* is the plateau stress, it can be calculated with the help of equation (1) as shown in equation (2) [3].
(2) |
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On the other hand, densification strain can be calculated using equation (3) [3].
(3) |
|
Therefore, a constant value for plateau stress in stress – strain graph is based on the assumption of elastic perfectly plastic behavior and necessary for Gibson – Ashby model. The shaded area in figure 1 is the amount of absorbed energy and is calculated using equation (4).
(4) |
|
|
Fig.4 The naming of cell dimensions in a Gibson – Ashby cell. |
Where V is the volume of the foam. So we have an energy function that its independent variables are: density of foam (which is dependent on cell dimensions), density of the base metal and yield stress of the base metal. This function will then calculate the amount of energy absorbed by the foam. However, if the goal is to provide an energy absorption equation for open-cell foams based on cell dimensions, these dimensions can be named as shown in figure 4 with bases in other directions shown by b and c.
The volume occupied by these parts is:
(5) |
|
While the mass of the basses is calculated using:
(6) |
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Therefore, foam density can be written as:
(7) |
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Finally, energy absorption of a single cell is calculated using equation (8).
(8) |
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For example, using equation (8), it is possible to predict that a cell with dimensions of a=b=c=0.04(m) and t=0.002(m) and yield stress of 1.53×108(Pa) is capable of absorbing 8.35 joules of energy.
4 NUMERICAL SIMULATION IN ABAQUS ENVIRONMENT
ABAQUS software environment was used to compare the results obtained from analytical equations with the results of numerical simulations. To this end, a single Gibson – Ashby cell with dimensions of a=b=c=0.04(m) and t=0.002(m) was modeled in ABAQUS environment and investigated under impact loading. Geometrical modeling of a Gibson – Ashby cell along with an impacting plane above the cell and a support plane below the cell was carried out in Assembly environment of CATIA software. Then, the geometrical model (including three parts of cell, moving plane and stationary plane) was imported into ABAQUS environment. Model dimensions were presented in the metric system. After introducing material properties to the software, a 5(Kg) weight was modeled on top of striker plane. After assembly of the parts, an explicit dynamic step with time step of 0.02(s) was defined in the Step environment. The contact between cell base and striker and support planes was defined without friction penalty. In the loading section and support conditions, an initial velocity of 1.82 (m/s) was applied to the weight on the striker plane. The cells used solid element meshing while shell element was used for meshing of the rigid planes. The lower rigid plane was fully fixed and support conditions at the end of cell bases was considered to be symmetrical. This is because this cell is in fact part of a continuous foam structure. Finite element model of the cell and rigid planes is shown in figure 5.
In order to introduce material characteristics to the software, simple tensile strength test results for an Al-6061 sample presented in reference [20] were used. Given the fact that Gibson – Ashby model assumes elastic perfectly plastic stress – strain behavior in materials, the strain for the yield stress is equal to zero which is written in the first row while the ultimate strain is added in the next row while specifying that the stress associated with this strain should be the yield stress of material. Table 1 shows the plastic properties defined in the software.
The simulation results can be compared to the results of analytical equations only when the energy of the impact in the software is fully absorbed by the cell. In other words, after the end of simulation, the striker weight should have a kinetic energy of zero. According to equation (9), the kinetic energy of the striker weight in simulation can be calculated:
(9) |
|
After the end of analysis, the kinetic energy graph of the striker weight is created using the History output section of ABAQUS software and presented in figure 6. This graph shows that all kinetic energy is absorbed by the cell.
Furthermore, figure 7 shows the position of the striker weight at the final time step which shows full absorption of kinetic energy. On the other hand, deformation of the cell also indicates that the cell has used all its energy absorption capacity. The animated time history output of the software also shows that kinetic energy reaches zero only due to bending capacity of cell parts without compressing two vertical supports after bending of horizontal supports. In other words, observing cell deformation steps under impact loading shows that impact stops before bended bases can reach each other.
Table 1
Defined plastic properties of the material
Strain | Stress (Pa) |
0 | 1.53×108 |
0.181478 | 1.53×108 |
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Fig.5 Finite element model of a single cell along with striker and support planes. |
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Fig.6 Changes in kinetic energy during simulation. |
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Fig.7 Deformed cell after impact loading. |
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Fig.8 Comparison of force transferred to the support in three different meshing. |
Therefore, numerical simulation results are fully compatible with the results obtained from analytical equations.
In order to ensure that the results are independent of the finite element meshing, analysis is repeated three times with different meshing sizes. After each simulation, the force applied to the support is extracted from the software. The support force in the second simulation is different from the first simulation while the results of second and third simulations are similar. This shows that results are convergent after the second simulation which is shown in figure 8.
Since selecting proper element type can affect the analysis results in finite element simulations, the results obtained using solid element and beam element in cell modeling were compared. Figure 9 shows the results of simulation using beam element.
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Fig.9 Finite element model of Gibson – Ashby cell using beam element. |
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Fig.10 The transferred force for two different elements in impact loading. |
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Fig.11 Maximum displacement of striker in impact loading. |
After preparing the model, its numerical solution was determined using the same method used for the model with solid element. The results show that the force transferred to the support plain in this model is similar to the model using solid element as shown in figure 10.
However, it’s important to note that due to the ability of solid elements in creating self-contact between cell parts and densification, it is necessary to use solid element for modeling Gibson – Ashby cells.
In order to compare the results of impact loading for Gibson – Ashby cell with the results of quasi-static loading, the same model was investigated under quasi-static loading. To this end, maximum displacement of striking plane in impact loading in software simulation results was identified. The maximum displacement can be seen in figure 11.
The maximum striker displacement in impact loading (where initial kinetic energy is fully absorbed) was then defined as mandatory displacement in the amplitude of ABAQUS software. It’s worth noting that in this case, it’s not necessary to define striking weight or initial velocity for the moving place because plane movement is only controlled by the mandatory displacement. In order to solve this problem, the force transferred to the base is determined and compared to the transferred force under impact loading which is shown in figure 12.
Based on the results, the force transferred to the support plain in the impact loading is higher than the transferred force under quasi-static loading.
5 CONFIDENCE AND ACCURACY OF NUMERICAL SIMULATION RESULTS
The results presented in reference [5] were used to ensure the accuracy of numerical results. This reference conducts an extensive investigation of Gibson – Ashby and Kelvin models in simulation of metal foams in ANSYS software and compares the results of empirical tests with numerical simulations. An important part of this reference is that it includes the results of modeling and simulation for a five-cell Gibson – Ashby foam under different loadings.
The geometrical characteristics of this modeled foam and loadings are also provided. Figure 13 shows the open-cell foam investigated in this work [5].
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Fig.12 Comparison between transferred force under quasi-static and dynamic loadings. |
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Fig.13 Five-cell foam modelled with beam elements [5]. |
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Fig.14 Finite element model of five-cell foam in ABAQUS software. |
© 2023 IAU, Arak Branch. All rights reserved. 
[1] Corresponding author. Tel.: +98 086 33412563.
E-mail address: m-najafizadeh@arak.iau.ir (M.M.Najafizadeh)