Bending Mechanical Property and Multi-objective Optimization of Gradient Foam Aluminum Sandwich Panel Based On Stochastic Pore Model

Bending Mechanical Property and Multi-objective Optimization of Gradient Foam Aluminum Sandwich Panel Based On Stochastic Pore Model | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Bending Mechanical Property and Multi-objective Optimization of Gradient Foam Aluminum Sandwich Panel Based On Stochastic Pore Model Kun Yang, yuan yao, Yunjie Sha, Yibo Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4799516/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Foam aluminum, as a new type of filling composite structure, has good energy absorption characteristics, it has good energy absorption characteristics. In this paper, the mechanical response and energy absorption characteristics of gradient foam aluminum sandwich panel under bending conditions are simulated and analyzed. The two-dimensional random distribution model of gradient foam aluminum was built by Digimat, and the feasibility and validity of the model were verified by comparing the compression simulation and quasi-static compression experiment results. Simulation of three-point bending of gradient foam aluminum sandwich panel under impact loading based on stochastic pore model, the bending mechanical property of gradient aluminum foam sandwich panel under impact loading were studied from the aspects of deformation mode and energy absorption performance. The results show that under the same impact displacement, the energy absorption effect of the gradient foam aluminum sandwich panel are obviously better than that of the homogeneous foam aluminum sandwich panel of the same quality. In order to further improve the crashworthiness of gradient foam aluminum sandwich panel, so that the pressure can be evenly distributed to other parts of the sandwich panel, the design point was selected by Latin hypercube sampling, and the response surface was constructed by simulation model. The optimal parameter matching design of gradient foam aluminum sandwich panel under bending condition was obtained by Moga algorithm.It provides some reference for the design and application of gradient foam aluminum sandwich panel in bending. Gradient foam aluminum Bending Mechanical Property Stochastic Pore Model Response surface Analysis Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 1 Introduction Foam aluminum material is a type of lightweight high-strength material that boasts excellent heat insulation, corrosion resistance, as well as sound absorption and sound insulation capabilities. Additionally, foam aluminum exhibits good plasticity and workability[ 1 – 3 ]. The complex internal structure and variable density determine the superior energy absorption capacity of foam aluminum, in order to achieve more effective energy absorption effect, the idea of functional gradient foam aluminum instead of homogeneous foam aluminum has become the next goal. As the core and other structural components, the laminated structure not only improves the mechanical property of the original structure, but also avoids the disadvantage of low strength of foam aluminum itself[ 4 ]. Because the homogeneous foam aluminum sandwich structure is subjected to bending impact load, the force transmission and diffusion capacity is poor, therefore, in order to disperse the pressure to other parts of the structure, the density gradient foam aluminum can be used as the core to reduce the cross-sectional deformation and improve the load-carrying capacity of the structure[ 5 ] .A lot of experimental and numerical studies have been carried out on the bending mechanical property of foam aluminum sandwich structure. Sipei Cai et al.[ 6 ] pointed out that under the impact of explosion, compared with non gradient foam aluminum, gradient foam aluminum core layer can reduce the deformation of the backplane to protect the contents.Dong Wang et al.[ 7 ]divided the three-point bending mechanical response of foam aluminum sandwich panel into three stages: elastic deformation stage, yield stage and complete failure stage, the yield stage of the structure lasts the longest, and the main failure mode of the sandwich structure is core shear. Ashutosh et al.[ 8 ] compared the bending property of foam aluminum sandwich structure and pure aluminum foam, it is found that the bending performance, load-bearing capacity and energy-absorbing performance of sandwich structure are significantly improved compared with pure foam aluminum; Furthermore, the bending deformation behavior of the carbon fiber reinforced gradient foam aluminum sandwich structure was analyzed, and the results showed that the sandwich structure's bending load-bearing capacity, bending stiffness, and energy absorption capacity were significantly improved with the increase in the thickness of the foam aluminum core material. Xuding Song et al.[ 9 ] studied the failure mode of foam aluminum sandwich panel by using three-point bending fatigue test. The results show that both the panel thickness and the core density affect the service life of the sandwich panel, and that there is an optimal match between the two for the best bending performance of the sandwich structure, the main failure mode of sandwich structure is core shear. In addition, the mechanical behavior of single foam aluminum sandwich structure under dynamic impact load has been studied by theoretical analysis, experiment and numerical simulation. T. M. McCormack et al.[ 10 ] obtained the failure load diagram of foam aluminum sandwich panel under dynamic three-point bending load by means of test and simulation. Xuhui Deng et al.[ 11 ] found that in the same loading conditions, the double-layer foam aluminum sandwich panel has stronger impact resistance compared to the single-layer foam aluminum sandwich panel. Xu Hang Hang et al.[ 12 ] conducted experimental and numerical studies on the mechanical property and deformation patterns of dented aluminum honeycomb filled with foam aluminum. The results indicated that the composite structure with a designed functional gradient outperformed the uniform composite structure in terms of both the initial peak stress and the stress value during the plateau phase.These results fully show that the bending performance, load-bearing capacity and energy-absorbing performance of foam aluminum sandwich structure are significantly improved compared with pure foam aluminum, gradient foam aluminum sandwich structure can further enhance the bending performance of the structure. However, there are few studies on the bending behavior of gradient aluminum foam sandwich panel under impact loading, And most of the previous research is the use of experimental analysis, there is a lack of analysis on the collapse process of the meso-pore structure of the gradient foam aluminum core. In order to simulate and analyze the collapse process and mechanical property of foam aluminum, Many scholars focus on the modeling methods of foam aluminum. The cells of Kelvin model are composed of 8 regular hexagons and 6 regular quadrangles. The cells of Kelvin model can fill up the whole space according to periodic regular arrangement. It is considered to be the closest structural model to the foam metal[ 13 ]. Dou et al.[ 14 ] established two-dimensional stochastic models with different relative densities (20%, 30%, and 40%) in combination with C + + and ANSYS/LS-DYNA, the influence of micro-inertial effect on strain rate effect under different relative densities was investigated by using the model of random distribution of circular cell. COSTANZA et al.[ 15 ] pointed out that the morphology of cell structure was studied by X-ray computed tomography (CT), which was expressed by the distribution of equivalent diameter and roundness value of cells in sample slices. This technique has been shown to be particularly suitable for in-house studies of foam metals, along with basic microscopy. At present, there are three kinds of simulation models of porous foam aluminum: simple cell model, random model and three-dimensional CT reconstruction model. The simple cell model can simulate the pore structure of foam aluminum to some extent, but it can not simulate the complex and changeable pore structure. Random model is formed by random arrangement of simple cell pores in a certain plane or space using Bourg's operation. The pore structure can be changed by adjusting cell pore size parameters and number, and the whole random model can be constructed. Voronoi model uses space segmentation method to divide the space into a specified number of seamless units by defining the distance of cutting points. However, because of its modeling method, the edge of the bubble appears sharp edge, easy to produce stress concentration. Three-dimensional reconstruction modeling can almost realize 1∶1 lossless modeling of material structure, but it is costly and time-consuming. In this paper, the bending simulation models of homogeneous and gradient foam aluminum sandwich panel with random distribution under transverse impact are established by Digimat, the bending deformation modes and energy absorption property of different types of sandwich panel under impact loading are analyzed. Taking energy absorption and peak stress as optimization objectives, taking the thickness and gradient coefficient of sandwich panel as design variables, multi-objective optimization design of gradient foam aluminum sandwich panel were carried out. Latin hypercube sampling is used to select design points, and the selected design points are brought into the model for simulation to construct response surface, in order to improve the energy absorption performance, the optimum size parameters of the gradient foam aluminum sandwich panel under impact condition were obtained by Moga algorithm. 2 Experiment and simulation analysis of foam aluminum compression based on Stochastic Pore Model 2.1 The preparation of aluminum foam materials The foam aluminum material used in this paper is a closed-cell foam aluminum, produced by Shanghai Ciqi Foam Aluminum New Material Technology Co., Ltd., through melting and casting. The average pore diameter isφ2mm-3mm. In order to, study the effect of density on the axial compressive property of foam aluminum. According to, the porosity range of foam aluminum provided by the manufacturer. Finally, the densities of foam aluminum are selected as follows 0.54g/cm 3 (Porosity 80%), 0.81g/cm 3 (Porosity 70%) and 1.08g/cm 3 (Porosity 60%).,three density models were used for experimental research. The national standard of metal material performance test samples and the "size effect" of porous materials are comprehensively considered. The final compression size of the specimen is set to 30 mm×30 mm×30 mm. Processed experimental specimen is shown in Fig. 1. Foam aluminum of quasi static compression mechanics experiments using the tesla CMT − 5305 microcomputer control electronic universal testing machine. The downward movement speed of the indenter is set at 2mm/min, and the target compression displacement is 70% of the specimen, that is, 21mm. Three dimensional specimens were carried out three compression, take the average density of specimen experimental results of the final result. 2.2 Analysis of experimental results of foam aluminum axial compression Three kinds of foam aluminum density engineering stress strain curve of the specimens is shown in Fig. 2 . It can be seen that the stress-strain curves of three densities of closed-cell foam aluminum can be divided into three stages: linear elastic stage, plastic platform stage and densification stage. Figure 3 shows the macro-deformation results for each stage represented by the curves during the axial compression of foam aluminum of three densities. Figure 4 shows the final compression results of three densities of homogeneous foam aluminum. The results show that the stress-strain curves of three kinds of foam aluminum with different densities show three stages of linearity, plasticity and high strength. In Fig. 3 , the macroscopic deformation states of foam aluminum with three different densities under axial compression are displayed, where s represents the magnitude of the displacement of the indenter downward. By observing the compressive stress-strain curves of foam aluminum with three densities, it can be found that the deformation mode of foam aluminum is three stages regardless of the density. (1) in the elastic stage, the compressive deformation mode of foam aluminum is linear elastic deformation, and the stress increases linearly with the strain in a small range of strain, when the first peak load appears, it marks the end of the elastic deformation stage of foam aluminum. (2) In the platform stage, the foam aluminum matrix selected in this experiment is made of pure aluminum, which is relatively brittle. The plateau stress of 60% porosity foam aluminum is in a continuously increasing state compared with that of low density foam aluminum, and it is easy to rupture during compression. A small amount of debris was observed when the material was compressed to a certain extent, and several small pieces of the high-density foam aluminum were also observed in the final compression results. Low-density foam aluminum exhibit excellent foam structure characteristics, and the pore structure of foam aluminum is compressed to produce deformation during the platform stage, the force and deformation of the entire structure are relatively balanced, and as the strain increases, the stress does not exhibit a significant increase but shows a very gentle trend, ultimately resulting in a compacted whole upon compression. (3) In the densification stage, with the increasing of the displacement, the holes in the structure were completely crushed, and the cell wall structure between the holes began to contact. The internal space of the structure is gradually squeezed out, so that the foam aluminum structure can no longer appear large strain, and the stress begins to bear by the compacted foam aluminum matrix, so the stress will appear a larger increase. By comparing the axial compression Stress–strain curve of foam aluminum with three porosity, the following conclusions can be drawn:(1) In the elastic stage, the slope of the elastic segment increases with the increase of the density of foam aluminum, the faster the peak stress of high-density foam aluminum is reached. (2) At the platform stage, the stress of 60% porosity foam aluminum fluctuates obviously because of its rupture, the platform stress is around 15 Mpa, and the compression platform stress of two kinds of low-density foam aluminum is relatively stable, about 4 Mpa and 2.4 Mpa, respectively. (3) Although the density of 60% porosity foam aluminum with is only 10% different from that of 70% porosity foam aluminum, but there is a big difference between the two from the stress-strain curve and the compression deformation process, the property of 60% porosity foam aluminum is more similar to the mechanical property of solid metal, and do not show obvious pore structure. 2.3 Finite element model of foam aluminum axial compression based on Stochastic Pore Model The pore structure of the three-dimensional random model is closer to the real foam aluminum, and the simulation results are in good agreement with the experimental results. However, the requirements for computers by this model are excessively high. When it constructs models with excessively high porosity, the grid size is too small, the total number of elements is too large, the computation time is excessively long, and convergence is difficult to achieve. In addition, at the boundary of the hole wall, the thickness direction is only a layer of grid, which affects the calculation accuracy. At this point, the shell unit effect is better, can greatly reduce the number of units.However, the random distribution leads to the non-uniform thickness at the boundary of the hole wall, so it is not practical to use the same thickness shell element. Therefore, this paper uses a two-dimensional stochastic model, which can better simulate the complex internal structure of foam aluminum, but also reduce the impact of computing power. Based on the Digimat-FE module, referring to the experimental foam aluminum size, in the 30 × 30 two-dimensional model created by random hole placement, shape selection circle, the percentage of hole is set to 60%, 70%, 80%, and the basic diameter of holes is set to 3mm. The quasi-static compression finite element model of foam aluminum was constructed by LS-DYNA, which is mainly divided into three parts: the bottom restraint plate, the top loading plate and the foam aluminum body, as shown in Fig. 5. Because of the transverse expansion of foam aluminum during compression, the size of the top and bottom plates is slightly larger than that of foam aluminum, set at 50mm × 3mm. The size of the foam aluminum model is 30mm × 30mm, and the cell grid is divided by multi-node triangle. Due to the narrowest width of 0.3mm in the foam aluminum core, to ensure the accuracy of the simulation, the planar model was set as a single-layer mesh model with a thickness of 0.3mm in the direction of the normal to the surface. This facilitates the application of the contact algorithm in LS-DYNA. Material model Considering that pure aluminum serves as the experimental specimen in this work. As a result, the perfect elastic-plastic materials are specified as the matrix material characteristics of foam metal, which are input as pure aluminum attribute data. Table 1 and Table 2 displays the model parameters. Table 1 Performance parameters of aluminum matrix material Parameter Value Density ρ/(kg•m -3 ) 2700 Modulus of elasticity E /MPa 69000 Poisson's ratio µ 0.3 Yield strength σ s /MPa 76 Table 2 Material parameters of low carbon steel Parameter Value Density \(\rho\) / \({\text{(kg}} \cdot {{\text{m}}^{ - 3}})\) 7860 Elastic modulus E/MPa 206000 Tangent modulus T/MPa 180 Poisson's ratio \(\mu\) 0.28 yield strength \({\sigma _{\text{s}}}\) /MPa 235 Contact conditions Setting up the model appropriately and accurately requires a thorough analysis of the particular experimental contact possibility during the preparation phase of the simulation. Thus, a method for automatic single-side contact defines the foam aluminum model. Therefore, an automatic single-side contact is the definition of the foam aluminum model. Simultaneously, the foam model and rigid plate are specified as the automatic node-to-surface contact on both ends. with a friction coefficient of 0.2 set. Boundary conditions In order to, save time cost effectively, and to set the loading rate is 100 mm/s. The simulation should copy all the conditions in the experiment as much as possible. According to the experimental method, the upper platen is set to be compressed along the Y-axis, the loading displacement is -21mm, and the axial compression loading time of the finite element model is set to 0.21s. 2.4 Simulation results of two-dimensional finite element model for axial compression of foam aluminum The simulation results of the model are compared with the experimental results, ε is strain, and the comparison results are shown below. From Fig. 5–8, it can be found that the internal holes of foam aluminum are uniformly deformed during compression. In the process of collapse, the strain increases continuously. However, the stress is reduced due to cell collapse and gas release in the cell. With the intensification of compression, the hole wall is bent and wrinkled. the foam aluminum matrix presents a tendency of sinking and collapsing. Contrast experimental results, the compression deformation of low-density foam aluminum in each stage is very similar. However, high-density foam aluminum is brittle, fragmentation occurred in the experiment. Therefore, the maximum principal strain failure criterion is added to it separately, the critical value is 0.08. The overall change trend of simulation and experimental results tends to be consistent. Figure 9 compares the stress-strain curves of each density foam aluminum sample under axial compression. Because the simulation is a two-dimensional model. There will be some differences between the actual stress-strain curve of foam aluminum and the experimental compression. However, the axial compression simulation of foam aluminum is basically consistent with the experimental data. It is proved that the accuracy of the model can meet the research requirements. It lays a foundation for the simulation research of the following foam aluminum sandwich panel model. 3 Simulation analysis of impact bending of gradient foam aluminum sandwich panel based on stochastic pore model 3.1 Equal density gradient foam aluminum The mechanical property of foam aluminum were improved by changing the density distribution. Foam aluminum with equal density can be divided into several density layers with the same thickness according to the different density, and the density difference between two adjacent density layers is the same[16]. Figure 11 shows equal density gradient foam aluminum. L represents the foam aluminum core layer with a density of 1.08g/cm 3 (60% porosity). Q represents the foam aluminum core layer with a density of 0.81g/cm 3 (70% porosity). B represents the foam aluminum core layer with a density of 0.54g/cm 3 (80% porosity). Three densities of foam aluminum are combined in a certain order to form a gradient foam aluminum structure. The gradient coefficient of relative density distribution along the height of the specimen is defined as: $$\:{y}^{i}=\frac{{\rho\:}_{y}^{i+1}-{\rho\:}_{y}^{i}}{\varDelta\:ℎ/ℎ}$$ 1 In this formula \(\:\:\text{ℎ}\) for total height of specimen, \(\:\varDelta\:\text{ℎ}\) for height of each gradient layer, \(\:\text{i}\) for number of gradient layers of specimen, \(\:{\rho\:}_{\gamma\:}^{i}\) and \(\:{\rho\:}_{\gamma\:}^{i+1}\) for the relative densities of i layer and i + 1 layer respectively. According to the difference between positive and negative values of the calculated gradient coefficient \(\:{\gamma\:}\) , When \(\:{\gamma\:}\) >0, the gradient foam aluminum is defined as positive gradient foam aluminum. When \(\:\gamma\:\) <0, the gradient foam aluminum is defined as negative gradient foam aluminum. When \(\:{\gamma\:}\) =0, it is uniform foam aluminum. The change speed of the density of gradient foam aluminum along the direction \(\:\:\text{ℎ}\) can be expressed by the absolute value of gradient coefficient \(\:\left|{\gamma\:}\right|\) . The larger \(\:\left|{\gamma\:}\right|\) is, the faster the density of gradient foam aluminum changes along the direction \(\:\:\text{ℎ}\) , the greater the density difference between the two ends. The gradient coefficient of L-Q-B is -0.81, that of Q-Q-Q is 0, and that of B-Q-L is 0.81. 3.2 Establishment of finite element model Set the section size of sandwich panel to 30 * 60mm and the upper and lower panel are 1.2 mm thick. The span of three-point bending of foam aluminum sandwich panel is Ls = 6d, the length of sandwich panel is L = Ls + d, d is the length of cross-section of sandwich panel. Therefore, the span Ls = 180mm and the length L = 240mm for three-point bending of foam aluminum sandwich panel are obtained. The diameters of the two supporting cylindrical heads and the force loading ram are both 25mm. The side length of foam aluminum core is 27.6mm. Figure 12 is the schematic diagram of three-point bending of foam aluminum sandwich panel. The three-point bending finite element model of homogeneous and gradient sandwich panel is shown in Fig. 12. 3.3 Setting of material model and simulation parameters The effect of strain rate on Q235 low carbon material should be considered in impact simulation. The Johnson-Cook model is selected to describe the influence on the steel plate. The equation considers the influence of strain, strain rate, temperature and other factors. Its parameter settings and foam aluminum material model parameters are shown in Table 1 and Table 2 . The materials used for the three-point bending sandwich panel in this chapter are Q235, and considering the effect of strain rate, the constitutive relationship under impact is adopted from the Johnson-Cook constitutive model. The complete expression of the Johnson-Cook constitutive equation is: In the given context, T* represents the dimensionless temperature, T is the environmental temperature of the sample, Tr is the room temperature during the experiment, and Tm is the melting point of the material. This simulation experiment primarily considers the effects of material property such as strain hardening and strain rate sensitivity on the energy absorption performance, without involving temperature factors. Therefore, the model's environmental temperature is set to be equal to the laboratory temperature. T = Tr , m = 0. The simplified form of the Johnson-Cook constitutive equation is expressed as: In the formula: σ represents the dynamic yield strength of Q235, A is the static yield strength, 235 MPa; B is the hardening constant, 400 MPa; n is the hardening exponent, 0.36; C is the strain rate hardening parameter, 0.0391. The sandwich panel and the foam aluminum core are made of quadrilateral meshes with the size of 0.3 mm. Simultaneously, the planar model is configured as a single-layer mesh model with a thickness of 0.3 mm along the normal direction of the surface, facilitating the application of contact algorithms within LS-DYNA. Both the support heads and the force application head are considered rigid bodies during the impact compression, without taking into account their own deformation. As the core layer of composite structure, aluminum foams exhibit good property of energy absorption, load buffering and shock wave dissipation, therefore, it is widely used in the fields of national defense, aerospace, automotive engineering and anti-explosion engineering. Foam aluminum sandwich panel is mainly the role of energy absorption, buffer, in the low-speed impact situation, it is about 5–35 m/s, so in the simulation process, the supporting head is fixed, and the pressure head strikes down along the y-axis at a speed of 35m/s, and the loading displacement is 35mm, In order to speed up the calculation, a proper mass scaling method is used to increase the incremental step size of the steady-state time. The Workbench shows the stress-strain cloud at the corresponding point in time. The process of the internal structure of closed-cell foam aluminum changing with the load movement can be observed by combining the cloud diagram. 3.4 Analysis of simulation results of three-point bending of gradient foam aluminum sandwich panel The simulation results of three-point bending of foam aluminum sandwich panel under impact loading are shown in Fig. 13 –17. S is the displacement distance of the head. When the mean foam aluminum core sandwich panel is subjected to a bending impact load, the impact force cannot be effectively diffused within the structure, leading to excessive concentration of deformation and consequently phenomena such as fracture of the external plate. As shown in Figs. 14 (a), 15(a), and 16(a), when the homogenous foam aluminum core sandwich panel is subjected to a compressive displacement of s = 17.5 mm under the impact load of the press head, a local indentation appears at the center of the contact end, and the foam aluminum structure near the impact end is damaged. However, at the position near the boundary on the other end, the deformation of the rear surface and the core is minimal, indicating that the structure is in the stage of local deformation at this time. Upon the continued application of the impact load to s = 35 mm, noticeable indentation deformation occurs at the contact end, with significant overall inelastic deformation observed at both the front and rear surfaces. The deformation of single-layer foam aluminum initiates from the thin walls of the cell cavities, whereas the gradient foam aluminum collapses layer by layer. Moreover, the energy absorption efficiency of the structure is positively correlated with the relative density of the foam aluminum. If gradient foam aluminum sandwich panel are employed, it can further enhance the structure's crashworthiness and overall energy absorption efficiency, allowing the impact force to be effectively distributed to other regions, thus avoiding premature failure. As depicted in Fig. 17(a), upon the impact press head making contact with the front end plate of the gradient foam aluminum sandwich panel, localized deformation occurs in the area directly under the impact of the press head, with pronounced structural damage to the B-type cell structure of the core layer. At this stage, the rear end plate and the core layers Q and L maintain their overall structure essentially intact, with no failure areas observed. As the impact load continues to be applied, the deflection of the rear end plate gradually reaches its maximum value. Throughout the entire impact response process of the B-Q-L type gradient structure, the degree of local compression in the core layer B is higher than that in the homogeneous sandwich panel structures with 60% and 70% porosity rates. Moreover, the sandwich panel enters the overall deformation response at a later stage, significantly reducing the displacement distance at the midpoint of the rear end plate. Compared to the 70% porosity foam aluminum sandwich panel, the B-Q-L type gradient aluminum foam sandwich panel, due to the lower inherent strength of its core layer B, exhibits more significant deformation during the impact process. Through its own compression and deformation, it dissipates a portion of the impact kinetic energy, which results in fewer shear failures occurring in the core layers Q and L. Unlike the 70% porosity foam aluminum sandwich panel and the B-Q-L type gradient aluminum foam sandwich panel, the L-Q-B type gradient foam aluminum sandwich panel features an L-layer at the contact end of the core. The L-layer is characterized by its high density, high bending stiffness, and strong resistance to bending deformation. As shown in Fig. 18 (a), under the action of the impact load from the loading press head, the supporting role of the L-layer results in a low degree of local deformation of the front end plate, and the deformation at the entire impact end is also relatively minor. Due to the L-layer of the L-Q-B type gradient foam aluminum sandwich panel experiencing minimal local compression, the L-Q-B type gradient foam aluminum sandwich panel enters the overall deformation stage earlier. As the overall bending deformation of the sandwich panel increases, shear failure occurs in the core B-layer at the boundary, leading to the formation of shear cracks. From the photographs of the core layer failure of the specimens, it can be observed that when the impact distance reaches a maximum of 35 mm, the low-density core layer of the foam aluminum exhibits the greatest bending deformation. At this point, the maximum displacement of the center point of the far end face for the five models are as follows: L(60) = 24.966 mm; L(70) = 23.011 mm; L(80) = 18.227 mm; L(L-Q-B) = 22.589 mm; L(B-Q-L) = 20.118 mm. Table 3 presents the peak load, maximum energy absorption, and the final displacement of the center of the rear end face for the five models in this simulation. The comparison results of the peak load and maximum energy absorption are shown in Figs. 18 and 19 , respectively. Table 3 Simulation results Peak load (N) Maximum energy absorption (J) Final displacement of the center of the rear end face (mm) 60% porosity 16181.1 335.87 24.966 70% porosity 11423.3 244.625 23.011 80% porosity 6912.9 204.064 18.227 L-Q-B 16898.2 291.09 22.589 B-Q-L 12340.5 286.24 20.118 Upon examining the impact bending load-displacement curves and energy absorption curves of the foam aluminum sandwich panel, it was found that under equivalent impact conditions, the energy absorbed by the positive and negative gradient foam aluminum sandwich panel structures increased by 17% and 19%, respectively, compared to the homogeneous foam aluminum sandwich panel of the same mass (i.e., the structure with 70% porosity). Additionally, the maximum displacement at the center of the far end face was reduced by 12.57% and 1.8%, respectively. 4 Multi-Objective Optimization of Gradient Foam Aluminum Sandwich Panel Structures Based on Stochastic Pore Models 4.1 Experimental design methods The multi-objective optimization design process can be succinctly summarized into three parts: the experimental design for selecting design points, the construction of surrogate models, and the optimization solving. The flowchart of the multi-objective optimization process based on the response surface method in this paper is shown in Fig. 20 . 4.2 The mathematical model of multi-objective optimization In identical impact conditions, the energy absorption effect of the negatively gradient foam aluminum sandwich panel is significantly better than that of the homogeneous foam aluminum sandwich panel structure of equal mass. However, the drawback is that the displacement of the structure's rear end plate is slightly larger. Therefore, by applying the same optimization process, the gradient foam aluminum sandwich panel structure under impact load is subjected to multi-objective optimization design. Table 4 Design Variable Parameters Design variables Name Initial size value Range of change P2 Wall thickness of sandwich panel /mm 1.2 1ཞ1.4 P5 Gradient coefficient -0.81 -0.81ཞ0.81 Optimization mathematical model for the bending performance of the foam aluminum sandwich panel structure under impact load is depicted in formula (5). $$\:\left\{\begin{array}{c}min\left\{PL\left(D,T\right),-EA\left(D,T\right),S\left(D,T\right)\right\}\\\:s,t.1mm\le\:D\le\:1.4mm\\\:-0.81\le\:T\le\:0.81\end{array}\right.$$ 5 In the formula, D represents the thickness of the sandwich panel wall, with units in millimeters (mm); T represents the gradient coefficient, which is a dimensionless quantity; PCS represents the peak compressive stress, with units in megapascals (MPa); S represents the maximum displacement of the sandwich panel's far end face, with units in millimeters (mm); EA represents the total energy absorption, with units in (mJ) 4.3 Test design The multi-objective optimization design process under impact bending conditions is primarily divided into three parts: experimental design, construction of surrogate models, and optimization solving. The distinction lies in the three optimization objectives: peak load, maximum displacement of the far end face, and total energy absorption. The sample points and response values for the multi-objective optimization design of gradient foam aluminum sandwich panel under impact bending conditions are shown in Table 5 Table 5 Sample points and response values of multi-objective optimization design Name P4-Thickness of the upper wall of the sandwich panel (mm) P6-Gradient coefficient P1-Peak load (N) P2-Maximum displacement of distal face(mm) P3-Maximum internal energy (mJ) P5-Thickness of the lower wall of the sandwich panel (mm) 1 1 -0.18 11323.8 19.866 254247 1 2 1.1 -0.54 13952.4 21.632 275418 1.1 3 1.1 0.18 10943.5 21.066 238014 1.1 4 1.2 -0.72 16298.1 22.589 293052 1.2 5 1.2 0 11423.3 23.011 244625 1.2 6 1.2 0.72 12581.4 20.118 280923 1.2 7 1.3 0.36 14201.5 22.332 304524 1.3 8 1.3 -0.36 17504.3 24.036 329281 1.3 (P4 = P5) 4.4 Proxy model Based on the sample points and response values in Table 5 , the response surface approximation model is established, and the 3D response surface diagram is shown in Fig. 21 . The response surface quality is tested with the Goodness of fit and the results are shown in table 8. The response surface approximation model fitting accuracy is shown in Fig. 22 . Table 6 Quality accuracy test of response surface P1-Peak load P3-Maximum internal energy P2-Maximum displacement of distal face Decision coefficient 0.99912 0.99896 0.99902 Relative maximum absolute error 3.1446% 3.4063% 3.3073 Relative average absolute error 2.7863% 3.0249% 2.9309 4.5 Optimization of structural parameters Based on the Moga, the objective is to minimize the peak load P3, the distal displacement P4 and the internal energy P5. After 519 evaluations, the optimal structural dimensions are obtained, as shown in Table 7 . Table 7 Optimal structure size of impact bending of gradient foam aluminum sandwich panel Structure parameters Wall thickness of sandwich panel P2/mm Gradient coefficient Peak load/N Maximum internal energy/J Maximum displacement of distal face /mm Initial value 1.2 -0.81 16898.2 291.09 22.589 Optimization value 1.0015 0.60791 10774 303.94 17.79 The results show that the peak load, the maximum displacement and the maximum internal energy of foam aluminum sandwich panel is reduced by 36.24%, 21.24% and 4.41% respectively. In order to verify the reliability of the optimization results, the optimal solution is taken as the verification point and replaced by the finite element simulation to compare the difference between the optimization results and the simulation results. The values of P3, P4 and P5 are 10133.54 mpa, 18.38.278.26J respectively. It can be seen that the error of the optimized result is 5.94%, 3.39% and 8.44% respectively. The precision meets the application requirements. 5 Conclusion A finite element model of impact bending of foam aluminum sandwich panel with various porosity was established. The simulation results show that the energy absorption effect of foam gradient aluminum sandwich panel is better than that of homogeneous foam aluminum sandwich panel with the same impact displacement. The energy absorbed by positive and negative gradient foam aluminum sandwich panel is increased by 17% and 19% respectively compared with that of homogeneous aluminum foam sandwich panel with 70% porosity. At the same time, the maximum displacement of the center of the rear face is shortened by 12.57% and 1.8% respectively. The multi-objective mathematical model of gradient structure is constructed. Furthermore, the response surface optimization was conducted using Ansys Workbench to obtain the Par Workbench to obtain the Pareto optimal solutions. The reliability of the optimization result is proved by reintroducing the optimal scheme parameters into the model. Under the bending impact condition, the peak load decreases by 36.24%, the maximum displacement of far end surface decreases by 21.24%, and the maximum internal energy increases by 4.41%. Compared with the gradient structure before optimization, the optimized functional gradient foam aluminum sandwich panel has more superior mechanical property. The optimized structure can improve the energy absorption performance and reduce the peak load and the maximum displacement at the far end. Declarations Authors’ contributions All authors contributed to the study conception and design. Yiwen Chen did all the tests and simulations and wrote the paper about it. Kun Yang instructed Yiwen Chen to revise the paper. Material preparation, data collection and analysis were performed by Yiwen Chen, Kun Yang and Yushan Zhang. The first draft of the manuscript was written by Yiwen Chen and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. Availability of data and materials: The data used to support the findings of this study are available from the corresponding author upon request. References Brooks H, Smith. Material Characterization and Computational Simulation of Steel Foam for Use in Structural Applications[D]. Amherst: University of Massachusetts, 2014. Tianjian Lu, Deping He, Changqing Chen, Changying Zhao, Daining Fang, Xiaolin Wang. Multifunctional properties and applications of ultra-lightweight porous metallic materials [J]. Advances in Mechanics, 2006, 36(4): 517-535. Mingxia Diao, Chunhuan Guo, Huabing Gao, Haixin Li, Tao Dong, Mingying Xiao, Zhenlin Yang, Fengchun Jiang. Research progress on foam metal composite materials [J]. Materials Engineering, 2022, 50(12): 60-70. Yunfei Cao, Yanjun Zhao, Liuyang Li, Liwen Pan, Peng Tang. Research progress on foam aluminum filled tubes [J]. Precision Forming Engineering, 2023, 15(02): 19-28. Kun Yang, Yunjie Sha, Tao Yu. Research on Three-point Bending Mechanical Performance of Square Tube Structure Filled with Foam Aluminum[J]. Mechanical, 2021, 27(6):442-450. Sipei Cai, Jun Liu, Pan Zhang, et al. Experimental study on failure mechanisms of sandwichpanels with multilayered aluminum foam/UHMWPE laminate core under combined blastand fragments loading[J]. Thin-Walled Structures, 2021, 159(107227): 1-15. Xuding Song, Dong Wang, Chuanhe Jing. Research on bending fatigue life and failure modes of foam aluminum sandwich panels [J]. Hot Working Technology, 2022, 51(10): 56-60. Pandey Ashutosh, Muchhala Dilip, Kumar Rajeev, et al. Flexural deformation behavior ofcarbon fiber reinforced aluminium hybrid foam sandwich structure[J]. Composites Part B:engineering, 2020, 183(107729): 1-11. Dong Wang, Chang Yan, Xuding Song. Experimental study and simulation of three-point bending of foam aluminum sandwich panels [J]. Equipment Manufacturing Technology, 2018, 285(09): 94-95+110. McCormack T.M., Miller R., Kesler O., et al. Failure of sandwich beams with metallic foamcores[J]. International Journal of Solids and Structures, 2001, 38(49): 1-20. Xuhui Deng, Yabin Li. Numerical study on impact resistance performance of double-layer foam aluminum sandwich panels [J]. Journal of Railway Science and Engineering, 2019, 16(10): 2603-2611. Hang H X ,Chen H L ,Gang X Z , et al.Mechanical properties of aluminum foam filled re-entrant honeycomb with uniform and gradient designs[J].International Journal of Mechanical Sciences,2023,244 DOU R J,QIU S W,JU Y,et al.Simulation of compression behavior and strain-rate effect for aluminum foam sandwich panels Computational Materials Science[J],2016,112:205-209. ZHANG Z C,FENG H M,XU T,et al.Composite Structures[J],2022,283:115090. COSTANZA G,GIUDICE F,SILI A,et al.Correlation Modeling between Morphology and Compression Behavior of Closed-Cell Al Foams Based on X-ray Computed Tomography Observations Metals[J],2021,11(9):1370. Bingbing Zhang. Anisotropic Crushing Mechanical Behavior of Gradient Foam Aluminum [D]. Harbin: Journal of Harbin Institute of Technology, 2018. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4799516","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":349925244,"identity":"ea54c5c9-dbfa-44d8-bf63-3284da1c81cd","order_by":0,"name":"Kun Yang","email":"","orcid":"","institution":"Liaoning University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Kun","middleName":"","lastName":"Yang","suffix":""},{"id":349925245,"identity":"84c9c04c-6913-4dff-ae09-49722ee68dcb","order_by":1,"name":"yuan yao","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVRIiWNgGAWjYBACNmbmw4//VPyr5+dvPkCcFj52tjQDnjMHEiRnHEsgToscP4+BBG/bgQSDAzkGxDqMx8BAgu1OnsGBMx9vvGGwk9NtIKiFreCBAc+zYsnDvZst5zAkG5sdIKiFeYNBggQzY9+Bs9ukeRgOJG4jrIXBQOKAATNjw4GcZ8RqYTGQbEg4nDjhQA4bsVrY0owZDqQZAwPZ2HKOARF+ke8/fPgx4z8bOWBUPrzxpsJOjqAWFCDBQ2TUIGshVccoGAWjYBSMCAAA5GFBp8QFyBwAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0009-5436-994X","institution":"Liaoning University of Technology","correspondingAuthor":true,"prefix":"","firstName":"yuan","middleName":"","lastName":"yao","suffix":""},{"id":349925246,"identity":"65e94608-821b-4a51-b6bc-e102b88bc288","order_by":2,"name":"Yunjie Sha","email":"","orcid":"","institution":"Liaoning University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Yunjie","middleName":"","lastName":"Sha","suffix":""},{"id":349925247,"identity":"26cfb9b9-bf44-4b4f-9c5f-44570503f795","order_by":3,"name":"Yibo Wang","email":"","orcid":"","institution":"Liaoning University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Yibo","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2024-07-25 06:39:03","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4799516/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4799516/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":66146388,"identity":"6e0b8b08-d0d0-477f-9daa-26011430593d","added_by":"auto","created_at":"2024-10-08 07:17:08","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":334846,"visible":true,"origin":"","legend":"\u003cp\u003eFinished aluminum foam specimen\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/203971281f9601d3862e3ff2.png"},{"id":66146387,"identity":"9460c32c-08fb-4536-8032-28aa99a2d6a0","added_by":"auto","created_at":"2024-10-08 07:17:08","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":27415,"visible":true,"origin":"","legend":"\u003cp\u003eStress-strain curves of foam aluminum specimens with three densities\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/ebbc2f1de7c701baaa457636.png"},{"id":66146395,"identity":"fca8314e-7b5f-413b-8148-9a8370ffee1b","added_by":"auto","created_at":"2024-10-08 07:17:08","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":356726,"visible":true,"origin":"","legend":"\u003cp\u003eAxial compression process of foam aluminum with three densities\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/8f815f3d2f15e03788310551.png"},{"id":66148042,"identity":"4889a112-f5d5-44e6-85b4-68f95194e294","added_by":"auto","created_at":"2024-10-08 07:25:08","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":116413,"visible":true,"origin":"","legend":"\u003cp\u003eFinal compression results of foam aluminum with three densities\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/90f514d5d0e0a2a4c2dd877d.png"},{"id":66146391,"identity":"533971b0-2c90-4a5d-bd32-6cac3af0ed48","added_by":"auto","created_at":"2024-10-08 07:17:08","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":160806,"visible":true,"origin":"","legend":"\u003cp\u003eTwo-dimensional finite element quasi-static compression model of homogeneous 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comparison of simulation data\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/f0127f17e2957433a61977ca.png"},{"id":66146403,"identity":"953b8284-4fce-45f6-8544-8d5b63bab75d","added_by":"auto","created_at":"2024-10-08 07:17:08","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":32889,"visible":true,"origin":"","legend":"\u003cp\u003eThe density distribution diagram of equal density gradient foam aluminum\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/0f98b4472aa5ad663cd6c3b6.png"},{"id":66148049,"identity":"57042c11-d05b-426c-a58a-ca4bb995fdc8","added_by":"auto","created_at":"2024-10-08 07:25:08","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":82862,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of three-point bending of foam aluminum sandwich panel\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/bbc0d337ed65ad60970ac70c.png"},{"id":66146398,"identity":"61a59f57-6e20-4e1c-85ac-681990dd0b44","added_by":"auto","created_at":"2024-10-08 07:17:08","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":187852,"visible":true,"origin":"","legend":"\u003cp\u003eTwo dimensional finite element model for three-point bending of homogeneous and gradient foam aluminum sandwich panel\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/6cf76c1aa1142cf3c1b71859.png"},{"id":66148803,"identity":"45113ccd-ba51-422c-bbd4-39feda62dd89","added_by":"auto","created_at":"2024-10-08 07:33:08","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":278557,"visible":true,"origin":"","legend":"\u003cp\u003e60% porosity foam aluminum sandwich panel impact bending simulation results\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/ca216fcef78d7d4160a611ff.png"},{"id":66146407,"identity":"b3cbb703-cd67-4087-b498-91c1a0341fcc","added_by":"auto","created_at":"2024-10-08 07:17:09","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":301217,"visible":true,"origin":"","legend":"\u003cp\u003e70 % porosity foam aluminum sandwich panel impact bending simulation results\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/413f9c0f01743895ab6cdeb2.png"},{"id":66148806,"identity":"8ed968d3-ad5b-48ae-8cbf-5c1b237d7e13","added_by":"auto","created_at":"2024-10-08 07:33:08","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":340196,"visible":true,"origin":"","legend":"\u003cp\u003e80 % porosity foam aluminum sandwich panel impact bending simulation results\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/d4813824f0e00afe6278a841.png"},{"id":66148055,"identity":"08d0f4bf-b72b-44b1-92c2-d176304fc168","added_by":"auto","created_at":"2024-10-08 07:25:09","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":289512,"visible":true,"origin":"","legend":"\u003cp\u003eL-Q-B gradient aluminum foam sandwich panel impact bending simulation results\u003c/p\u003e","description":"","filename":"16.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/a30b15d334d51b5f25da5e3a.png"},{"id":66148801,"identity":"a61d1f7b-6205-4263-a837-5adb7b45e298","added_by":"auto","created_at":"2024-10-08 07:33:08","extension":"png","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":325333,"visible":true,"origin":"","legend":"\u003cp\u003eB-Q-L gradient aluminum foam sandwich panel impact bending simulation results\u003c/p\u003e","description":"","filename":"17.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/59d44429c3c71f7e720e359b.png"},{"id":66150189,"identity":"a0163ef1-37ff-4795-83aa-9ce8d1909745","added_by":"auto","created_at":"2024-10-08 07:41:08","extension":"png","order_by":18,"title":"Figure 18","display":"","copyAsset":false,"role":"figure","size":93030,"visible":true,"origin":"","legend":"\u003cp\u003eImpact bending load displacement curve of foam aluminum sandwich panel\u003c/p\u003e","description":"","filename":"18.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/60800216c484e069f53b81e3.png"},{"id":66148044,"identity":"c02dd89f-c0d3-431e-bf3f-215c2141873b","added_by":"auto","created_at":"2024-10-08 07:25:08","extension":"png","order_by":19,"title":"Figure 19","display":"","copyAsset":false,"role":"figure","size":58534,"visible":true,"origin":"","legend":"\u003cp\u003eImpact bending energy absorption curve of foam aluminum sandwich panel\u003c/p\u003e","description":"","filename":"19.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/f79964caf835e2c6917a35f9.png"},{"id":66150188,"identity":"61022d69-9854-455a-9ad8-ac986d9056d8","added_by":"auto","created_at":"2024-10-08 07:41:08","extension":"png","order_by":20,"title":"Figure 20","display":"","copyAsset":false,"role":"figure","size":36758,"visible":true,"origin":"","legend":"\u003cp\u003eMulti-objective optimization flow chart\u003c/p\u003e","description":"","filename":"20.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/6d6a3e42d1e6fe021f18ca3c.png"},{"id":66148047,"identity":"9f799258-e653-4911-97b4-5c8969d52ef1","added_by":"auto","created_at":"2024-10-08 07:25:08","extension":"png","order_by":21,"title":"Figure 21","display":"","copyAsset":false,"role":"figure","size":209995,"visible":true,"origin":"","legend":"\u003cp\u003e3D response surface\u003c/p\u003e","description":"","filename":"21.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/dcf6fd412fbb3c1a988ee54f.png"},{"id":66148804,"identity":"8a348e6e-cc78-40a8-bbff-e49eea28c3bb","added_by":"auto","created_at":"2024-10-08 07:33:08","extension":"png","order_by":22,"title":"Figure 22","display":"","copyAsset":false,"role":"figure","size":86164,"visible":true,"origin":"","legend":"\u003cp\u003eResponse surface fitting accuracy\u003c/p\u003e","description":"","filename":"22.png","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/62d2a98b6ebca61ed5746fc3.png"},{"id":88519214,"identity":"2dd5435f-c6b2-491a-bb56-f06c90a3dd26","added_by":"auto","created_at":"2025-08-07 09:30:12","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4897716,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4799516/v1/a7f69975-0099-4de8-9d2a-3c24c2a44ba4.pdf"}],"financialInterests":"","formattedTitle":"Bending Mechanical Property and Multi-objective Optimization of Gradient Foam Aluminum Sandwich Panel Based On Stochastic Pore Model","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eFoam aluminum material is a type of lightweight high-strength material that boasts excellent heat insulation, corrosion resistance, as well as sound absorption and sound insulation capabilities. Additionally, foam aluminum exhibits good plasticity and workability[\u003cspan additionalcitationids=\"CR2\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. The complex internal structure and variable density determine the superior energy absorption capacity of foam aluminum, in order to achieve more effective energy absorption effect, the idea of functional gradient foam aluminum instead of homogeneous foam aluminum has become the next goal. As the core and other structural components, the laminated structure not only improves the mechanical property of the original structure, but also avoids the disadvantage of low strength of foam aluminum itself[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Because the homogeneous foam aluminum sandwich structure is subjected to bending impact load, the force transmission and diffusion capacity is poor, therefore, in order to disperse the pressure to other parts of the structure, the density gradient foam aluminum can be used as the core to reduce the cross-sectional deformation and improve the load-carrying capacity of the structure[\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e5\u003c/span\u003e] .A lot of experimental and numerical studies have been carried out on the bending mechanical property of foam aluminum sandwich structure. Sipei Cai et al.[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e6\u003c/span\u003e] pointed out that under the impact of explosion, compared with non gradient foam aluminum, gradient foam aluminum core layer can reduce the deformation of the backplane to protect the contents.Dong Wang et al.[\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e7\u003c/span\u003e]divided the three-point bending mechanical response of foam aluminum sandwich panel into three stages: elastic deformation stage, yield stage and complete failure stage, the yield stage of the structure lasts the longest, and the main failure mode of the sandwich structure is core shear. Ashutosh et al.[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e8\u003c/span\u003e] compared the bending property of foam aluminum sandwich structure and pure aluminum foam, it is found that the bending performance, load-bearing capacity and energy-absorbing performance of sandwich structure are significantly improved compared with pure foam aluminum; Furthermore, the bending deformation behavior of the carbon fiber reinforced gradient foam aluminum sandwich structure was analyzed, and the results showed that the sandwich structure's bending load-bearing capacity, bending stiffness, and energy absorption capacity were significantly improved with the increase in the thickness of the foam aluminum core material. Xuding Song et al.[\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e9\u003c/span\u003e] studied the failure mode of foam aluminum sandwich panel by using three-point bending fatigue test. The results show that both the panel thickness and the core density affect the service life of the sandwich panel, and that there is an optimal match between the two for the best bending performance of the sandwich structure, the main failure mode of sandwich structure is core shear. In addition, the mechanical behavior of single foam aluminum sandwich structure under dynamic impact load has been studied by theoretical analysis, experiment and numerical simulation. T. M. McCormack et al.[\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e10\u003c/span\u003e] obtained the failure load diagram of foam aluminum sandwich panel under dynamic three-point bending load by means of test and simulation. Xuhui Deng et al.[\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e11\u003c/span\u003e] found that in the same loading conditions, the double-layer foam aluminum sandwich panel has stronger impact resistance compared to the single-layer foam aluminum sandwich panel. Xu Hang Hang et al.[\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e12\u003c/span\u003e] conducted experimental and numerical studies on the mechanical property and deformation patterns of dented aluminum honeycomb filled with foam aluminum. The results indicated that the composite structure with a designed functional gradient outperformed the uniform composite structure in terms of both the initial peak stress and the stress value during the plateau phase.These results fully show that the bending performance, load-bearing capacity and energy-absorbing performance of foam aluminum sandwich structure are significantly improved compared with pure foam aluminum, gradient foam aluminum sandwich structure can further enhance the bending performance of the structure. However, there are few studies on the bending behavior of gradient aluminum foam sandwich panel under impact loading, And most of the previous research is the use of experimental analysis, there is a lack of analysis on the collapse process of the meso-pore structure of the gradient foam aluminum core.\u003c/p\u003e \u003cp\u003eIn order to simulate and analyze the collapse process and mechanical property of foam aluminum, Many scholars focus on the modeling methods of foam aluminum. The cells of Kelvin model are composed of 8 regular hexagons and 6 regular quadrangles. The cells of Kelvin model can fill up the whole space according to periodic regular arrangement. It is considered to be the closest structural model to the foam metal[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. Dou et al.[\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e14\u003c/span\u003e] established two-dimensional stochastic models with different relative densities (20%, 30%, and 40%) in combination with C\u0026thinsp;+\u0026thinsp;+\u0026thinsp;and ANSYS/LS-DYNA, the influence of micro-inertial effect on strain rate effect under different relative densities was investigated by using the model of random distribution of circular cell. COSTANZA et al.[\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e15\u003c/span\u003e] pointed out that the morphology of cell structure was studied by X-ray computed tomography (CT), which was expressed by the distribution of equivalent diameter and roundness value of cells in sample slices. This technique has been shown to be particularly suitable for in-house studies of foam metals, along with basic microscopy. At present, there are three kinds of simulation models of porous foam aluminum: simple cell model, random model and three-dimensional CT reconstruction model. The simple cell model can simulate the pore structure of foam aluminum to some extent, but it can not simulate the complex and changeable pore structure. Random model is formed by random arrangement of simple cell pores in a certain plane or space using Bourg's operation. The pore structure can be changed by adjusting cell pore size parameters and number, and the whole random model can be constructed. Voronoi model uses space segmentation method to divide the space into a specified number of seamless units by defining the distance of cutting points. However, because of its modeling method, the edge of the bubble appears sharp edge, easy to produce stress concentration. Three-dimensional reconstruction modeling can almost realize 1∶1 lossless modeling of material structure, but it is costly and time-consuming. In this paper, the bending simulation models of homogeneous and gradient foam aluminum sandwich panel with random distribution under transverse impact are established by Digimat, the bending deformation modes and energy absorption property of different types of sandwich panel under impact loading are analyzed. Taking energy absorption and peak stress as optimization objectives, taking the thickness and gradient coefficient of sandwich panel as design variables, multi-objective optimization design of gradient foam aluminum sandwich panel were carried out. Latin hypercube sampling is used to select design points, and the selected design points are brought into the model for simulation to construct response surface, in order to improve the energy absorption performance, the optimum size parameters of the gradient foam aluminum sandwich panel under impact condition were obtained by Moga algorithm.\u003c/p\u003e"},{"header":"2 Experiment and simulation analysis of foam aluminum compression based on Stochastic Pore Model","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 The preparation of aluminum foam materials\u003c/h2\u003e \u003cp\u003eThe foam aluminum material used in this paper is a closed-cell foam aluminum, produced by Shanghai Ciqi Foam Aluminum New Material Technology Co., Ltd., through melting and casting. The average pore diameter isφ2mm-3mm. In order to, study the effect of density on the axial compressive property of foam aluminum. According to, the porosity range of foam aluminum provided by the manufacturer. Finally, the densities of foam aluminum are selected as follows 0.54g/cm\u003csup\u003e3\u003c/sup\u003e (Porosity 80%), 0.81g/cm\u003csup\u003e3\u003c/sup\u003e (Porosity 70%) and 1.08g/cm\u003csup\u003e3\u003c/sup\u003e (Porosity 60%).,three density models were used for experimental research. The national standard of metal material performance test samples and the \"size effect\" of porous materials are comprehensively considered. The final compression size of the specimen is set to 30 mm×30 mm×30 mm. Processed experimental specimen is shown in Fig.\u0026nbsp;1.\u003c/p\u003e \u003cp\u003eFoam aluminum of quasi static compression mechanics experiments using the tesla CMT − 5305 microcomputer control electronic universal testing machine. The downward movement speed of the indenter is set at 2mm/min, and the target compression displacement is 70% of the specimen, that is, 21mm. Three dimensional specimens were carried out three compression, take the average density of specimen experimental results of the final result.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Analysis of experimental results of foam aluminum axial compression\u003c/h2\u003e \u003cp\u003eThree kinds of foam aluminum density engineering stress strain curve of the specimens is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIt can be seen that the stress-strain curves of three densities of closed-cell foam aluminum can be divided into three stages: linear elastic stage, plastic platform stage and densification stage. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows the macro-deformation results for each stage represented by the curves during the axial compression of foam aluminum of three densities. Figure\u0026nbsp;4 shows the final compression results of three densities of homogeneous foam aluminum. The results show that the stress-strain curves of three kinds of foam aluminum with different densities show three stages of linearity, plasticity and high strength. In Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the macroscopic deformation states of foam aluminum with three different densities under axial compression are displayed, where s represents the magnitude of the displacement of the indenter downward.\u003c/p\u003e \u003cp\u003eBy observing the compressive stress-strain curves of foam aluminum with three densities, it can be found that the deformation mode of foam aluminum is three stages regardless of the density. (1) in the elastic stage, the compressive deformation mode of foam aluminum is linear elastic deformation, and the stress increases linearly with the strain in a small range of strain, when the first peak load appears, it marks the end of the elastic deformation stage of foam aluminum. (2) In the platform stage, the foam aluminum matrix selected in this experiment is made of pure aluminum, which is relatively brittle. The plateau stress of 60% porosity foam aluminum is in a continuously increasing state compared with that of low density foam aluminum, and it is easy to rupture during compression. A small amount of debris was observed when the material was compressed to a certain extent, and several small pieces of the high-density foam aluminum were also observed in the final compression results. Low-density foam aluminum exhibit excellent foam structure characteristics, and the pore structure of foam aluminum is compressed to produce deformation during the platform stage, the force and deformation of the entire structure are relatively balanced, and as the strain increases, the stress does not exhibit a significant increase but shows a very gentle trend, ultimately resulting in a compacted whole upon compression. (3) In the densification stage, with the increasing of the displacement, the holes in the structure were completely crushed, and the cell wall structure between the holes began to contact. The internal space of the structure is gradually squeezed out, so that the foam aluminum structure can no longer appear large strain, and the stress begins to bear by the compacted foam aluminum matrix, so the stress will appear a larger increase. By comparing the axial compression Stress–strain curve of foam aluminum with three porosity, the following conclusions can be drawn:(1) In the elastic stage, the slope of the elastic segment increases with the increase of the density of foam aluminum, the faster the peak stress of high-density foam aluminum is reached. (2) At the platform stage, the stress of 60% porosity foam aluminum fluctuates obviously because of its rupture, the platform stress is around 15 Mpa, and the compression platform stress of two kinds of low-density foam aluminum is relatively stable, about 4 Mpa and 2.4 Mpa, respectively. (3) Although the density of 60% porosity foam aluminum with is only 10% different from that of 70% porosity foam aluminum, but there is a big difference between the two from the stress-strain curve and the compression deformation process, the property of 60% porosity foam aluminum is more similar to the mechanical property of solid metal, and do not show obvious pore structure.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Finite element model of foam aluminum axial compression based on Stochastic Pore Model\u003c/h2\u003e \u003cp\u003eThe pore structure of the three-dimensional random model is closer to the real foam aluminum, and the simulation results are in good agreement with the experimental results. However, the requirements for computers by this model are excessively high. When it constructs models with excessively high porosity, the grid size is too small, the total number of elements is too large, the computation time is excessively long, and convergence is difficult to achieve. In addition, at the boundary of the hole wall, the thickness direction is only a layer of grid, which affects the calculation accuracy. At this point, the shell unit effect is better, can greatly reduce the number of units.However, the random distribution leads to the non-uniform thickness at the boundary of the hole wall, so it is not practical to use the same thickness shell element. Therefore, this paper uses a two-dimensional stochastic model, which can better simulate the complex internal structure of foam aluminum, but also reduce the impact of computing power.\u003c/p\u003e \u003cp\u003eBased on the Digimat-FE module, referring to the experimental foam aluminum size, in the 30 × 30 two-dimensional model created by random hole placement, shape selection circle, the percentage of hole is set to 60%, 70%, 80%, and the basic diameter of holes is set to 3mm. The quasi-static compression finite element model of foam aluminum was constructed by LS-DYNA, which is mainly divided into three parts: the bottom restraint plate, the top loading plate and the foam aluminum body, as shown in Fig.\u0026nbsp;5. Because of the transverse expansion of foam aluminum during compression, the size of the top and bottom plates is slightly larger than that of foam aluminum, set at 50mm × 3mm. The size of the foam aluminum model is 30mm × 30mm, and the cell grid is divided by multi-node triangle. Due to the narrowest width of 0.3mm in the foam aluminum core, to ensure the accuracy of the simulation, the planar model was set as a single-layer mesh model with a thickness of 0.3mm in the direction of the normal to the surface. This facilitates the application of the contact algorithm in LS-DYNA.\u003c/p\u003e \u003cp\u003eMaterial model\u003c/p\u003e \u003cp\u003eConsidering that pure aluminum serves as the experimental specimen in this work. As a result, the perfect elastic-plastic materials are specified as the matrix material characteristics of foam metal, which are input as pure aluminum attribute data. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e displays the model parameters.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePerformance parameters of aluminum matrix material\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"2\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDensity ρ/(kg•m\u003csup\u003e-3\u003c/sup\u003e)\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2700\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eModulus of elasticity \u003cem\u003eE\u003c/em\u003e/MPa\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e69000\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePoisson's ratio µ\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eYield strength σ\u003csub\u003e\u003cb\u003es\u003c/b\u003e\u003c/sub\u003e/MPa\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e76\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e \u003cp\u003e\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMaterial parameters of low carbon steel\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"2\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameter\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDensity\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\rho\\)\u003c/span\u003e\u003c/span\u003e/\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{(kg}} \\cdot {{\\text{m}}^{ - 3}})\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e7860\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eElastic modulus E/MPa\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e206000\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTangent modulus T/MPa\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e180\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePoisson's ratio \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mu\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.28\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eyield strength \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma _{\\text{s}}}\\)\u003c/span\u003e\u003c/span\u003e/MPa\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e235\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e \u003cp\u003e\u003c/p\u003e \u003cp\u003eContact conditions\u003c/p\u003e \u003cp\u003eSetting up the model appropriately and accurately requires a thorough analysis of the particular experimental contact possibility during the preparation phase of the simulation. Thus, a method for automatic single-side contact defines the foam aluminum model. Therefore, an automatic single-side contact is the definition of the foam aluminum model. Simultaneously, the foam model and rigid plate are specified as the automatic node-to-surface contact on both ends. with a friction coefficient of 0.2 set.\u003c/p\u003e \u003cp\u003eBoundary conditions\u003c/p\u003e \u003cp\u003eIn order to, save time cost effectively, and to set the loading rate is 100 mm/s. The simulation should copy all the conditions in the experiment as much as possible. According to the experimental method, the upper platen is set to be compressed along the Y-axis, the loading displacement is -21mm, and the axial compression loading time of the finite element model is set to 0.21s.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Simulation results of two-dimensional finite element model for axial compression of foam aluminum\u003c/h2\u003e \u003cp\u003eThe simulation results of the model are compared with the experimental results, \u003cem\u003eε\u003c/em\u003e is strain, and the comparison results are shown below.\u003c/p\u003e\u003cp\u003eFrom Fig.\u0026nbsp;5–8, it can be found that the internal holes of foam aluminum are uniformly deformed during compression. In the process of collapse, the strain increases continuously. However, the stress is reduced due to cell collapse and gas release in the cell. With the intensification of compression, the hole wall is bent and wrinkled. the foam aluminum matrix presents a tendency of sinking and collapsing. Contrast experimental results, the compression deformation of low-density foam aluminum in each stage is very similar. However, high-density foam aluminum is brittle, fragmentation occurred in the experiment. Therefore, the maximum principal strain failure criterion is added to it separately, the critical value is 0.08. The overall change trend of simulation and experimental results tends to be consistent.\u003c/p\u003e \u003cp\u003eFigure 9 compares the stress-strain curves of each density foam aluminum sample under axial compression. Because the simulation is a two-dimensional model. There will be some differences between the actual stress-strain curve of foam aluminum and the experimental compression. However, the axial compression simulation of foam aluminum is basically consistent with the experimental data. It is proved that the accuracy of the model can meet the research requirements. It lays a foundation for the simulation research of the following foam aluminum sandwich panel model.\u003c/p\u003e"},{"header":"3 Simulation analysis of impact bending of gradient foam aluminum sandwich panel based on stochastic pore model","content":"\u003ch2\u003e3.1 Equal density gradient foam aluminum\u003c/h2\u003e\u003cp\u003eThe mechanical property of foam aluminum were improved by changing the density distribution. Foam aluminum with equal density can be divided into several density layers with the same thickness according to the different density, and the density difference between two adjacent density layers is the same[16]. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e11\u003c/span\u003e shows equal density gradient foam aluminum. L represents the foam aluminum core layer with a density of 1.08g/cm\u003csup\u003e3\u003c/sup\u003e(60% porosity). Q represents the foam aluminum core layer with a density of 0.81g/cm\u003csup\u003e3\u003c/sup\u003e(70% porosity). B represents the foam aluminum core layer with a density of 0.54g/cm\u003csup\u003e3\u003c/sup\u003e(80% porosity). Three densities of foam aluminum are combined in a certain order to form a gradient foam aluminum structure.\u003c/p\u003e\u003cp\u003eThe gradient coefficient of relative density distribution along the height of the specimen is defined as:\u003c/p\u003e\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{y}^{i}=\\frac{{\\rho\\:}_{y}^{i+1}-{\\rho\\:}_{y}^{i}}{\\varDelta\\:ℎ/ℎ}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cp\u003eIn this formula\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\text{ℎ}\\)\u003c/span\u003e\u003c/span\u003e for total height of specimen, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:\\text{ℎ}\\)\u003c/span\u003e\u003c/span\u003e for height of each gradient layer, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{i}\\)\u003c/span\u003e\u003c/span\u003e for number of gradient layers of specimen, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\rho\\:}_{\\gamma\\:}^{i}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\rho\\:}_{\\gamma\\:}^{i+1}\\)\u003c/span\u003e\u003c/span\u003e for the relative densities of \u003cem\u003ei\u003c/em\u003e layer and \u003cem\u003ei +\u003c/em\u003e 1 layer respectively.\u003c/p\u003e\u003cp\u003eAccording to the difference between positive and negative values of the calculated gradient coefficient \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\gamma\\:}\\)\u003c/span\u003e\u003c/span\u003e, When \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\gamma\\:}\\)\u003c/span\u003e\u003c/span\u003e \u0026gt;0, the gradient foam aluminum is defined as positive gradient foam aluminum. When \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\gamma\\:\\)\u003c/span\u003e\u003c/span\u003e\u0026lt;0, the gradient foam aluminum is defined as negative gradient foam aluminum. When \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\gamma\\:}\\)\u003c/span\u003e\u003c/span\u003e=0, it is uniform foam aluminum. The change speed of the density of gradient foam aluminum along the direction\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\text{ℎ}\\)\u003c/span\u003e\u003c/span\u003e can be expressed by the absolute value of gradient coefficient \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left|{\\gamma\\:}\\right|\\)\u003c/span\u003e\u003c/span\u003e. The larger \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left|{\\gamma\\:}\\right|\\)\u003c/span\u003e\u003c/span\u003e is, the faster the density of gradient foam aluminum changes along the direction\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\text{ℎ}\\)\u003c/span\u003e\u003c/span\u003e, the greater the density difference between the two ends. The gradient coefficient of L-Q-B is -0.81, that of Q-Q-Q is 0, and that of B-Q-L is 0.81.\u003c/p\u003e\u003ch2\u003e3.2 Establishment of finite element model\u003c/h2\u003e\u003cp\u003eSet the section size of sandwich panel to 30 * 60mm and the upper and lower panel are 1.2 mm thick. The span of three-point bending of foam aluminum sandwich panel is Ls = 6d, the length of sandwich panel is L = Ls + d, d is the length of cross-section of sandwich panel. Therefore, the span Ls = 180mm and the length L = 240mm for three-point bending of foam aluminum sandwich panel are obtained. The diameters of the two supporting cylindrical heads and the force loading ram are both 25mm. The side length of foam aluminum core is 27.6mm. Figure\u0026nbsp;12 is the schematic diagram of three-point bending of foam aluminum sandwich panel.\u003c/p\u003e\u003cp\u003eThe three-point bending finite element model of homogeneous and gradient sandwich panel is shown in Fig.\u0026nbsp;12.\u003c/p\u003e\u003ch2\u003e3.3 Setting of material model and simulation parameters\u003c/h2\u003e\u003cp\u003eThe effect of strain rate on Q235 low carbon material should be considered in impact simulation.\u003c/p\u003e\u003cp\u003eThe Johnson-Cook model is selected to describe the influence on the steel plate. The equation considers the influence of strain, strain rate, temperature and other factors. Its parameter settings and foam aluminum material model parameters are shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eThe materials used for the three-point bending sandwich panel in this chapter are Q235, and considering the effect of strain rate, the constitutive relationship under impact is adopted from the Johnson-Cook constitutive model. The complete expression of the Johnson-Cook constitutive equation is:\u003c/p\u003e\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"475\" height=\"29\"\u003e\u003c/p\u003e\u003cp\u003eIn the given context, \u003cem\u003eT*\u003c/em\u003e represents the dimensionless temperature, \u003cem\u003eT\u003c/em\u003e is the environmental temperature of the sample, \u003cem\u003eTr\u003c/em\u003e is the room temperature during the experiment, and \u003cem\u003eTm\u003c/em\u003e is the melting point of the material.\u003c/p\u003e\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"420\" height=\"51\"\u003e\u003c/p\u003e\u003cp\u003eThis simulation experiment primarily considers the effects of material property such as strain hardening and strain rate sensitivity on the energy absorption performance, without involving temperature factors. Therefore, the model's environmental temperature is set to be equal to the laboratory temperature. \u003cem\u003eT = Tr\u003c/em\u003e, m = 0.\u003c/p\u003e\u003cp\u003eThe simplified form of the Johnson-Cook constitutive equation is expressed as:\u003c/p\u003e\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"450\" height=\"38\"\u003e\u003c/p\u003e\u003cp\u003eIn the formula: \u003cem\u003eσ\u003c/em\u003e represents the dynamic yield strength of Q235, \u003cem\u003eA\u003c/em\u003e is the static yield strength, 235 MPa; \u003cem\u003eB\u003c/em\u003e is the hardening constant, 400 MPa; \u003cem\u003en\u003c/em\u003e is the hardening exponent, 0.36; \u003cem\u003eC\u003c/em\u003e is the strain rate hardening parameter, 0.0391.\u003c/p\u003e\u003cp\u003eThe sandwich panel and the foam aluminum core are made of quadrilateral meshes with the size of 0.3 mm. Simultaneously, the planar model is configured as a single-layer mesh model with a thickness of 0.3 mm along the normal direction of the surface, facilitating the application of contact algorithms within LS-DYNA. Both the support heads and the force application head are considered rigid bodies during the impact compression, without taking into account their own deformation.\u003c/p\u003e\u003cp\u003eAs the core layer of composite structure, aluminum foams exhibit good property of energy absorption, load buffering and shock wave dissipation, therefore, it is widely used in the fields of national defense, aerospace, automotive engineering and anti-explosion engineering. Foam aluminum sandwich panel is mainly the role of energy absorption, buffer, in the low-speed impact situation, it is about 5–35 m/s, so in the simulation process, the supporting head is fixed, and the pressure head strikes down along the y-axis at a speed of 35m/s, and the loading displacement is 35mm, In order to speed up the calculation, a proper mass scaling method is used to increase the incremental step size of the steady-state time. The Workbench shows the stress-strain cloud at the corresponding point in time. The process of the internal structure of closed-cell foam aluminum changing with the load movement can be observed by combining the cloud diagram.\u003c/p\u003e\u003ch2\u003e3.4 Analysis of simulation results of three-point bending of gradient foam aluminum sandwich panel\u003c/h2\u003e\u003cp\u003eThe simulation results of three-point bending of foam aluminum sandwich panel under impact loading are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e13\u003c/span\u003e–17. S is the displacement distance of the head.\u003c/p\u003e\u003cp\u003eWhen the mean foam aluminum core sandwich panel is subjected to a bending impact load, the impact force cannot be effectively diffused within the structure, leading to excessive concentration of deformation and consequently phenomena such as fracture of the external plate. As shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e14\u003c/span\u003e(a), 15(a), and 16(a), when the homogenous foam aluminum core sandwich panel is subjected to a compressive displacement of s = 17.5 mm under the impact load of the press head, a local indentation appears at the center of the contact end, and the foam aluminum structure near the impact end is damaged. However, at the position near the boundary on the other end, the deformation of the rear surface and the core is minimal, indicating that the structure is in the stage of local deformation at this time. Upon the continued application of the impact load to s = 35 mm, noticeable indentation deformation occurs at the contact end, with significant overall inelastic deformation observed at both the front and rear surfaces. The deformation of single-layer foam aluminum initiates from the thin walls of the cell cavities, whereas the gradient foam aluminum collapses layer by layer. Moreover, the energy absorption efficiency of the structure is positively correlated with the relative density of the foam aluminum. If gradient foam aluminum sandwich panel are employed, it can further enhance the structure's crashworthiness and overall energy absorption efficiency, allowing the impact force to be effectively distributed to other regions, thus avoiding premature failure. As depicted in Fig.\u0026nbsp;17(a), upon the impact press head making contact with the front end plate of the gradient foam aluminum sandwich panel, localized deformation occurs in the area directly under the impact of the press head, with pronounced structural damage to the B-type cell structure of the core layer. At this stage, the rear end plate and the core layers Q and L maintain their overall structure essentially intact, with no failure areas observed. As the impact load continues to be applied, the deflection of the rear end plate gradually reaches its maximum value. Throughout the entire impact response process of the B-Q-L type gradient structure, the degree of local compression in the core layer B is higher than that in the homogeneous sandwich panel structures with 60% and 70% porosity rates. Moreover, the sandwich panel enters the overall deformation response at a later stage, significantly reducing the displacement distance at the midpoint of the rear end plate. Compared to the 70% porosity foam aluminum sandwich panel, the B-Q-L type gradient aluminum foam sandwich panel, due to the lower inherent strength of its core layer B, exhibits more significant deformation during the impact process. Through its own compression and deformation, it dissipates a portion of the impact kinetic energy, which results in fewer shear failures occurring in the core layers Q and L. Unlike the 70% porosity foam aluminum sandwich panel and the B-Q-L type gradient aluminum foam sandwich panel, the L-Q-B type gradient foam aluminum sandwich panel features an L-layer at the contact end of the core. The L-layer is characterized by its high density, high bending stiffness, and strong resistance to bending deformation. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e18\u003c/span\u003e(a), under the action of the impact load from the loading press head, the supporting role of the L-layer results in a low degree of local deformation of the front end plate, and the deformation at the entire impact end is also relatively minor. Due to the L-layer of the L-Q-B type gradient foam aluminum sandwich panel experiencing minimal local compression, the L-Q-B type gradient foam aluminum sandwich panel enters the overall deformation stage earlier.\u003c/p\u003e\u003cp\u003eAs the overall bending deformation of the sandwich panel increases, shear failure occurs in the core B-layer at the boundary, leading to the formation of shear cracks. From the photographs of the core layer failure of the specimens, it can be observed that when the impact distance reaches a maximum of 35 mm, the low-density core layer of the foam aluminum exhibits the greatest bending deformation. At this point, the maximum displacement of the center point of the far end face for the five models are as follows: L(60) = 24.966 mm; L(70) = 23.011 mm; L(80) = 18.227 mm; L(L-Q-B) = 22.589 mm; L(B-Q-L) = 20.118 mm.\u003c/p\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e presents the peak load, maximum energy absorption, and the final displacement of the center of the rear end face for the five models in this simulation. The comparison results of the peak load and maximum energy absorption are shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e18\u003c/span\u003e and \u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e19\u003c/span\u003e, respectively.\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSimulation results\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePeak load (N)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMaximum energy absorption (J)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eFinal displacement of the center of the rear end face (mm)\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e60% porosity\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e16181.1\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e335.87\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e24.966\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e70% porosity\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e11423.3\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e244.625\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e23.011\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e80% porosity\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e6912.9\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e204.064\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e18.227\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eL-Q-B\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e16898.2\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e291.09\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e22.589\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB-Q-L\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e12340.5\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e286.24\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e20.118\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003eUpon examining the impact bending load-displacement curves and energy absorption curves of the foam aluminum sandwich panel, it was found that under equivalent impact conditions, the energy absorbed by the positive and negative gradient foam aluminum sandwich panel structures increased by 17% and 19%, respectively, compared to the homogeneous foam aluminum sandwich panel of the same mass (i.e., the structure with 70% porosity). Additionally, the maximum displacement at the center of the far end face was reduced by 12.57% and 1.8%, respectively.\u003c/p\u003e"},{"header":"4 Multi-Objective Optimization of Gradient Foam Aluminum Sandwich Panel Structures Based on Stochastic Pore Models","content":"\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Experimental design methods\u003c/h2\u003e \u003cp\u003eThe multi-objective optimization design process can be succinctly summarized into three parts: the experimental design for selecting design points, the construction of surrogate models, and the optimization solving. The flowchart of the multi-objective optimization process based on the response surface method in this paper is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e20\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e4.2 The mathematical model of multi-objective optimization\u003c/h2\u003e \u003cp\u003eIn identical impact conditions, the energy absorption effect of the negatively gradient foam aluminum sandwich panel is significantly better than that of the homogeneous foam aluminum sandwich panel structure of equal mass. However, the drawback is that the displacement of the structure's rear end plate is slightly larger. Therefore, by applying the same optimization process, the gradient foam aluminum sandwich panel structure under impact load is subjected to multi-objective optimization design.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eDesign Variable Parameters\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDesign variables\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eName\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eInitial size value\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRange of change\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eP2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWall thickness of sandwich panel /mm\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1ཞ1.4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eP5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGradient coefficient\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-0.81\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.81ཞ0.81\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eOptimization mathematical model for the bending performance of the foam aluminum sandwich panel structure under impact load is depicted in formula (5).\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\left\\{\\begin{array}{c}min\\left\\{PL\\left(D,T\\right),-EA\\left(D,T\\right),S\\left(D,T\\right)\\right\\}\\\\\\:s,t.1mm\\le\\:D\\le\\:1.4mm\\\\\\:-0.81\\le\\:T\\le\\:0.81\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the formula, \u003cem\u003eD\u003c/em\u003e represents the thickness of the sandwich panel wall, with units in millimeters (mm); \u003cem\u003eT\u003c/em\u003e represents the gradient coefficient, which is a dimensionless quantity; \u003cem\u003ePCS\u003c/em\u003e represents the peak compressive stress, with units in megapascals (MPa); \u003cem\u003eS\u003c/em\u003e represents the maximum displacement of the sandwich panel's far end face, with units in millimeters (mm); \u003cem\u003eEA\u003c/em\u003e represents the total energy absorption, with units in (mJ)\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Test design\u003c/h2\u003e \u003cp\u003eThe multi-objective optimization design process under impact bending conditions is primarily divided into three parts: experimental design, construction of surrogate models, and optimization solving. The distinction lies in the three optimization objectives: peak load, maximum displacement of the far end face, and total energy absorption. The sample points and response values for the multi-objective optimization design of gradient foam aluminum sandwich panel under impact bending conditions are shown in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSample points and response values of multi-objective optimization design\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eName\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eP4-Thickness of the upper wall of the sandwich panel (mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eP6-Gradient coefficient\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eP1-Peak load (N)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eP2-Maximum displacement of distal face(mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eP3-Maximum internal energy (mJ)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eP5-Thickness of the lower wall of the sandwich panel (mm)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-0.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e11323.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e19.866\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e254247\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-0.54\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e13952.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e21.632\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e275418\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10943.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e21.066\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e238014\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-0.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e16298.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e22.589\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e293052\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e11423.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e23.011\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e244625\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e12581.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e20.118\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e280923\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e14201.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e22.332\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e304524\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-0.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e17504.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e24.036\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e329281\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"7\"\u003e(P4\u0026thinsp;=\u0026thinsp;P5)\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Proxy model\u003c/h2\u003e \u003cp\u003eBased on the sample points and response values in Table \u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, the response surface approximation model is established, and the 3D response surface diagram is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e21\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe response surface quality is tested with the Goodness of fit and the results are shown in table 8. The response surface approximation model fitting accuracy is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e22\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eQuality accuracy test of response surface\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eP1-Peak load\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eP3-Maximum internal energy\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eP2-Maximum displacement of distal face\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDecision coefficient\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.99912\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.99896\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.99902\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRelative maximum absolute error\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.1446%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.4063%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.3073\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRelative average absolute error\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.7863%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.0249%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.9309\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e4.5 Optimization of structural parameters\u003c/h2\u003e \u003cp\u003eBased on the Moga, the objective is to minimize the peak load P3, the distal displacement P4 and the internal energy P5. After 519 evaluations, the optimal structural dimensions are obtained, as shown in Table \u003cspan refid=\"Tab7\" class=\"InternalRef\"\u003e7\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab7\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eOptimal structure size of impact bending of gradient foam aluminum sandwich panel\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStructure parameters\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWall thickness of sandwich panel\u003c/p\u003e \u003cp\u003eP2/mm\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eGradient coefficient\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePeak load/N\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMaximum internal energy/J\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMaximum displacement of distal face /mm\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eInitial value\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-0.81\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e16898.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e291.09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e22.589\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOptimization value\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.0015\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.60791\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e10774\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e303.94\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e17.79\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe results show that the peak load, the maximum displacement and the maximum internal energy of foam aluminum sandwich panel is reduced by 36.24%, 21.24% and 4.41% respectively. In order to verify the reliability of the optimization results, the optimal solution is taken as the verification point and replaced by the finite element simulation to compare the difference between the optimization results and the simulation results. The values of P3, P4 and P5 are 10133.54 mpa, 18.38.278.26J respectively. It can be seen that the error of the optimized result is 5.94%, 3.39% and 8.44% respectively. The precision meets the application requirements.\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Conclusion","content":"\u003cp\u003eA finite element model of impact bending of foam aluminum sandwich panel with various porosity was established. The simulation results show that the energy absorption effect of foam gradient aluminum sandwich panel is better than that of homogeneous foam aluminum sandwich panel with the same impact displacement. The energy absorbed by positive and negative gradient foam aluminum sandwich panel is increased by 17% and 19% respectively compared with that of homogeneous aluminum foam sandwich panel with 70% porosity. At the same time, the maximum displacement of the center of the rear face is shortened by 12.57% and 1.8% respectively. The multi-objective mathematical model of gradient structure is constructed. Furthermore, the response surface optimization was conducted using Ansys Workbench to obtain the Par Workbench to obtain the Pareto optimal solutions. The reliability of the optimization result is proved by reintroducing the optimal scheme parameters into the model. Under the bending impact condition, the peak load decreases by 36.24%, the maximum displacement of far end surface decreases by 21.24%, and the maximum internal energy increases by 4.41%. Compared with the gradient structure before optimization, the optimized functional gradient foam aluminum sandwich panel has more superior mechanical property. The optimized structure can improve the energy absorption performance and reduce the peak load and the maximum displacement at the far end.\u003c/p\u003e"},{"header":"Declarations","content":" \u003ch2\u003e \u003cb\u003eAuthors\u0026rsquo; contributions\u003c/b\u003e \u003c/h2\u003e \u003cp\u003eAll authors contributed to the study conception and design. Yiwen Chen did all the tests and simulations and wrote the paper about it. Kun Yang instructed Yiwen Chen to revise the paper. Material preparation, data collection and analysis were performed by Yiwen Chen, Kun Yang and Yushan Zhang. The first draft of the manuscript was written by Yiwen Chen and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.\u003c/p\u003e \u003ch2\u003eAvailability of data and materials:\u003c/h2\u003e \u003cp\u003eThe data used to support the findings of this study are available from the corresponding author upon request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eBrooks H, Smith. Material Characterization and Computational Simulation of Steel Foam for Use in Structural Applications[D]. Amherst: University of Massachusetts, 2014.\u003c/li\u003e\n\u003cli\u003eTianjian Lu, Deping He, Changqing Chen, Changying Zhao, Daining Fang, Xiaolin Wang. Multifunctional properties and applications of ultra-lightweight porous metallic materials [J]. Advances in Mechanics, 2006, 36(4): 517-535.\u003c/li\u003e\n\u003cli\u003eMingxia Diao, Chunhuan Guo, Huabing Gao, Haixin Li, Tao Dong, Mingying Xiao, Zhenlin Yang, Fengchun Jiang. Research progress on foam metal composite materials [J]. Materials Engineering, 2022, 50(12): 60-70.\u003c/li\u003e\n\u003cli\u003eYunfei Cao, Yanjun Zhao, Liuyang Li, Liwen Pan, Peng Tang. Research progress on foam aluminum filled tubes [J]. Precision Forming Engineering, 2023, 15(02): 19-28.\u003c/li\u003e\n\u003cli\u003eKun Yang, Yunjie Sha, Tao Yu. Research on Three-point Bending Mechanical Performance of Square Tube Structure Filled with Foam Aluminum[J]. Mechanical, 2021, 27(6):442-450.\u003c/li\u003e\n\u003cli\u003eSipei Cai, Jun Liu, Pan Zhang, et al. Experimental study on failure mechanisms of sandwichpanels with multilayered aluminum foam/UHMWPE laminate core under combined blastand fragments loading[J]. Thin-Walled Structures, 2021, 159(107227): 1-15.\u003c/li\u003e\n\u003cli\u003eXuding Song, Dong Wang, Chuanhe Jing. Research on bending fatigue life and failure modes of foam aluminum sandwich panels [J]. Hot Working Technology, 2022, 51(10): 56-60.\u003c/li\u003e\n\u003cli\u003ePandey Ashutosh, Muchhala Dilip, Kumar Rajeev, et al. Flexural deformation behavior ofcarbon fiber reinforced aluminium hybrid foam sandwich structure[J]. Composites Part B:engineering, 2020, 183(107729): 1-11.\u003c/li\u003e\n\u003cli\u003eDong Wang, Chang Yan, Xuding Song. Experimental study and simulation of three-point bending of foam aluminum sandwich panels [J]. Equipment Manufacturing Technology, 2018, 285(09): 94-95+110.\u003c/li\u003e\n\u003cli\u003eMcCormack T.M., Miller R., Kesler O., et al. Failure of sandwich beams with metallic foamcores[J]. International Journal of Solids and Structures, 2001, 38(49): 1-20.\u003c/li\u003e\n\u003cli\u003eXuhui Deng, Yabin Li. Numerical study on impact resistance performance of double-layer foam aluminum sandwich panels [J]. Journal of Railway Science and Engineering, 2019, 16(10): 2603-2611.\u003c/li\u003e\n\u003cli\u003eHang H X ,Chen H L ,Gang X Z , et al.Mechanical properties of aluminum foam filled re-entrant honeycomb with uniform and gradient designs[J].International Journal of Mechanical Sciences,2023,244\u003c/li\u003e\n\u003cli\u003eDOU R J,QIU S W,JU Y,et al.Simulation of compression behavior and strain-rate effect for aluminum foam sandwich panels Computational Materials Science[J],2016,112:205-209.\u003c/li\u003e\n\u003cli\u003eZHANG Z C,FENG H M,XU T,et al.Composite Structures[J],2022,283:115090.\u003c/li\u003e\n\u003cli\u003eCOSTANZA G,GIUDICE F,SILI A,et al.Correlation Modeling between Morphology and Compression Behavior of Closed-Cell Al Foams Based on X-ray Computed Tomography Observations Metals[J],2021,11(9):1370.\u003c/li\u003e\n\u003cli\u003eBingbing Zhang. Anisotropic Crushing Mechanical Behavior of Gradient Foam Aluminum [D]. Harbin: Journal of Harbin Institute of Technology, 2018.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Gradient foam aluminum, Bending Mechanical Property, Stochastic Pore Model, Response surface Analysis","lastPublishedDoi":"10.21203/rs.3.rs-4799516/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4799516/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFoam aluminum, as a new type of filling composite structure, has good energy absorption characteristics, it has good energy absorption characteristics. In this paper, the mechanical response and energy absorption characteristics of gradient foam aluminum sandwich panel under bending conditions are simulated and analyzed. The two-dimensional random distribution model of gradient foam aluminum was built by Digimat, and the feasibility and validity of the model were verified by comparing the compression simulation and quasi-static compression experiment results. Simulation of three-point bending of gradient foam aluminum sandwich panel under impact loading based on stochastic pore model, the bending mechanical property of gradient aluminum foam sandwich panel under impact loading were studied from the aspects of deformation mode and energy absorption performance. The results show that under the same impact displacement, the energy absorption effect of the gradient foam aluminum sandwich panel are obviously better than that of the homogeneous foam aluminum sandwich panel of the same quality. In order to further improve the crashworthiness of gradient foam aluminum sandwich panel, so that the pressure can be evenly distributed to other parts of the sandwich panel, the design point was selected by Latin hypercube sampling, and the response surface was constructed by simulation model. The optimal parameter matching design of gradient foam aluminum sandwich panel under bending condition was obtained by Moga algorithm.It provides some reference for the design and application of gradient foam aluminum sandwich panel in bending.\u003c/p\u003e","manuscriptTitle":"Bending Mechanical Property and Multi-objective Optimization of Gradient Foam Aluminum Sandwich Panel Based On Stochastic Pore Model","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-10-08 07:17:03","doi":"10.21203/rs.3.rs-4799516/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"327efc90-287d-459d-8701-57deb0b996e1","owner":[],"postedDate":"October 8th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-08-07T09:22:01+00:00","versionOfRecord":[],"versionCreatedAt":"2024-10-08 07:17:03","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4799516","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4799516","identity":"rs-4799516","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}