2017 Volume 58 Issue 6 Pages 880-885
In this article, to study the local compression deformation of aluminum foam sandwich (AFS) panels by orthogonal method, two-dimensional random simulation models of AFS panels were constructed by C++ language combining with ANSYS software, and four parameters, including aluminum foam core's geometrical characteristic parameters porosity P and pore size R and experimental parameters loading force F and local compressing length L, were simultaneously considered. Furthermore, the dependences of the local compression deformation of AFS panels on the above-mentioned parameters were investigated by range analysis and variance analysis means and the related mechanisms were described. The results of range analysis confirmed that factors influencing local compressive deformation of AFS panels are in turn: porosity P, pore size R, local compressing length L, loading force F, in other words, P > R > L > F. Furthermore, variance analysis results showed, compared to loading conditions, geometrical parameters of samples were more influential on the local compressive deformation of AFS panels.
Aluminum foam sandwich (AFS) panels are a new-type of structure function materials with a special structure containing two composite skins and a low density core and have a broad application prospect in aerospace, marine, electronics, automotive, construction engineering, and other fields because of their excellent properties such as light weight, high strength-to-weight, good blast resistance, good energy-absorbing capacity, good electromagnetic shielding and low thermal conductivity and so on.1–4)
AFS panels with metallic bonding between face panels and core have excellent mechanical properties.5) However, due to the limitations of material fabrication technology, this kind of sandwich panels with above-mentioned connection properties are not yet easily obtained in academic study and practical application, many researchers turn to apply finite element methods by using digital simulation models6–8) to study the mechanical responses of this kind of materials. Moreover, with the unique advantages of simulation, finite element method has become the important tool for studying AFS panels.
A finite element analysis of sandwich panels with positive gradient metallic foam core under indentation response was performed by Zhao et al.7) It was found that the indentation force decreased as the increasing of gradient change under the same indentation depth. Lu et al.8) simulated the resistant behavior and energy absorbing performance of sandwich panels under blast loading, and the result showed that the sandwich panels with denser and thicker core had a better energy dissipation. Zhao et al.9) studied the dynamic response of sandwich shells with metallic foam cores by simulating and found that the loading intensity and geometric configuration had great impact upon its deformation and energy absorption. In addition, Tekoğlu and Onck10) studied the size effects in mechanical behavior of cellular solids by two-dimensional Voronoi model. Discrete analyses results showed, with decreasing sample size, the macroscopic shear stiffness increased, while the macroscopic uniaxial compressive and bending stiffness decreased. Furthermore, the results indicated that local compressing length affected the mechanical properties of cellular material. However, as far as we known, there were few reports about the simulation studies simultaneously considering different parameters effect on the mechanical properties of AFS panels with metallic bonding between face panels and core.
In this work, based on the inner features of AFS panels, two-dimensional random models were generated by combing C++ and ANSYS/LS-DYNA. Four parameters including structural parameters and loading conditions were considered, porosity P, pore size R, loading force F and local compressing length L. The local compressive deformation of AFS panel was simulated under different combination of parameter levels. The influence degree of different parameters on the local compressive deformation behavior of AFS panel and the optimal combination of above mentioned parameters which could produce the maximum deformation were studied by the orthogonal design method. Furthermore, the results in this work and those reported in previous studies were comprehensively analyzed.
Two-dimensional model is often used to analyze the deformation behavior of porous aluminum foam due to the simple geometry shape and the easy dividing of meshes, and it has been applied in a large number of simulation experiments.11,12) So, based on the inner structure features of aluminum foam core and used C++ language combining with ANSYS software, AFS panels were simplified into two-dimensional random models with a core containing many circles with different size in this work. The modeling processes of AFS panels have been illustrated in details in our previous work.13) The model of AFS panel is consist of two thin aluminum panels (70 mm length and 0.8 mm thickness) and a highly porous aluminum foam core (70 mm length and 12 mm thickness) as shown in Fig. 1. The joints between panels and core are metal-on-metal connections.
Two-dimensional random model of AFS panel.
Loading and bounded constraint.
The compressive deformation process of AFS panels.
In fact, the local compression deformation of AFS panels is affected by many factors, and the variation range of each factor is wide. So, it is difficult and expensive to study the optimum operating parameters of the AFS panels compressed with different material geometrical characteristic parameters and external experimental parameters using the comprehensive experiment method. For example, for a full factorial experiment in this work with 4 influential parameters (factors) and 4 levels, the number of necessary trials would be 44 = 256, the workload will be very heavy. Therefore, in this work, the orthogonal test method was used stead of full factorial experiment. The selected representative points are distributed uniformly within an orthogonal array, which be introduced to set up industrial experiments since Kackar14), and thus can adequately represent the overall situation15). In experiment design, to decrease the interaction of each other, the effect of each influencing factor was independent estimated. In addition, in this study, based on the previous researches on the mechanical properties of AFS panels, the influence factors and level values selected were summarized in Table 1.
| parameters | Factor A | Factor B | Factor C | Factor D |
|---|---|---|---|---|
| P (porosity, %) | Rmin～max (pore size, mm) | F (loading force, N) | L (local compressing length, mm) |
|
| Level 1 | 55 (1) | 0.5～1 (1) | 4200 (1) | 18 (1) |
| Level 2 | 58 (2) | 1～1.5 (2) | 4450 (2) | 20 (2) |
| Level 3 | 61 (3) | 1.5～2 (3) | 4700 (3) | 22 (3) |
| Level 4 | 64 (4) | 2～2.5 (4) | 4950 (4) | 24 (4) |
The orthogonal array (a subset of the full factorial array with 256 experiments) is described as La(bc), where L is the symbol of orthogonal experiment design, a is the number of experimental runs, b is the number of levels and c the number of factors. In this work, the factors were considered including aluminum foam core's porosity P (factor A) and pore size R (factor B) and experimental parameters loading force F (factor C) and local compressing length L (factor D). And the evaluation indexes include the average compression deformation and deformation-time. Consequently, the orthogonal matrix L16(45) was adopted and the 16 orthogonal experiments selected for this study, also taking the unknown uncertainties into consideration (1 additional null column). The additional factor is added into the mathematical description to account for possible interactions of the original 4 factors as well as other possible sources of errors and uncertainties by the follow variance analysis in this article. The tested factors and levels were given in Table 2.
| Experiment (No.) |
Factor A (%) |
Factor B (mm) |
Factor C (N) |
Factor D (mm) |
Null column |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 2 | 2 | 2 | 2 |
| 3 | 1 | 3 | 3 | 3 | 3 |
| 4 | 1 | 4 | 4 | 4 | 4 |
| 5 | 2 | 1 | 2 | 3 | 4 |
| 6 | 2 | 2 | 1 | 4 | 3 |
| 7 | 2 | 3 | 4 | 1 | 2 |
| 8 | 2 | 4 | 3 | 2 | 1 |
| 9 | 3 | 1 | 3 | 4 | 2 |
| 10 | 3 | 2 | 4 | 3 | 1 |
| 11 | 3 | 3 | 1 | 2 | 4 |
| 12 | 3 | 4 | 2 | 1 | 3 |
| 13 | 4 | 1 | 4 | 2 | 3 |
| 14 | 4 | 2 | 3 | 1 | 4 |
| 15 | 4 | 3 | 2 | 4 | 1 |
| 16 | 4 | 4 | 1 | 3 | 2 |
After collecting simulation data from the orthogonal experiment method, range analysis and variance analysis were performed in this work. The range analysis assumes that the influence of other factors on the result is balanced when the impact of a factor is analyzed. The variance analysis is used to estimate the relative significance of each parameter in terms of percentage contribution to the overall response.16) In addition, due to the used model of AFS panel in this work is random model which is more closed to actual sample, there are large differences in the structure properties of aluminum foam core between models. The simulation results also verified this fact shown in Fig. 4. So, to ensure accuracy data and make results more reasonable and reliable, three specimens were adopted to simulate local compression deformation of AFS panels at the same levels and then the obtained data were averaged. It deserves noting that the error bars in this plot denote the standard deviations of the data. It was calculated that the variability in local compression in negative Y direction all for porosity P, pore size R, loading force F and local compressing length L can be up to 10.69%. This fully shows a good randomness of the model used.
The average compression variation in negative Y direction corresponding to parameters at different levels.
Based on the generated models in ANSYS software and the orthogonal design array in Table 2, numerical simulations experiments were performed to study the dependence of the local compression deformation of AFS panels on the above-mentioned four factors and validate the material model and boundary conditions. In this work, the used face sheets of the AFS panel were LY8 aluminum plates, and the material properties of plate and aluminum foam core were shown in Table 3, respectively.17,18) The briefly numerical simulation and analysis processes have been illustrated in our previous work.13) In especial, in this work, unit type adopted plane162 which is defined by four nodes and used in the plane problem. Due to the property of inhomogeneous structure of aluminum foam core, the local plastic deformation will occur when under a little load. At the same time, based the nature of the studied metal materials, the bilinear isotropic hardening plasticity model was adopted for the matrix material of aluminum foam. In addition, the two-dimensional model was meshed by intellectual mesh division tool. This could be better adapted to geometric shape and improve the computational accuracy due to it can automatically adjusts the grid size based the geometric shape of aluminum foam core. The level of mesh was set to 2. Moreover, the solution option in the solver options dialog box was set up as large displacement static due to it was possible part of sample entered the plastic deformation when AFS panel as a whole is still in the elastic deformation stage. To speed up convergence, the function of line search in nonlinear options was opened. Furthermore, Auto-dimensional contact was chosen in ANSYS to reflect the actual contact situation. In the compress process, to prevent movement of model, the displacements of the lower panel in both directions X and Y axes were constrained. In addition, loading by the flat punch was acted on the appropriate boundary nodes of the upper panel. The model loaded and constrained is shown in Fig. 2. The local compressive deformation progress of AFS panels was shown in Fig. 3. A measurement path was defined at the loading location of panel, named PATH1. The compress variations in the negative Y direction were taken from the post-processing module in LS-DYNA, and the data of the local compressive deformation simulation of three models each level were listed in Table 4.
| Density (kg/m3) |
Young's modulus (GPa) |
Poisson's ratio |
σ0 (MPa) |
Tangent modulus (MPa) |
|
|---|---|---|---|---|---|
| LY8 | 2700 | 71 | 0.31 | 110 | 700 |
| Aluminum foam | 2700 | 69 | 0.3 | 76 | 690 |
| Experiment (No.) |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Compress variation (mm) |
5.919 | 6.103 | 6.475 | 6.141 | 5.944 | 6.009 | 6.9123 | 6.841 |
| 5.998 | 6.149 | 6.277 | 6.432 | 6.05 | 6.105 | 7.155 | 6.989 | |
| 5.930 | 6.135 | 6.67 | 6.286 | 5.61 | 6.065 | 7.023 | 7.092 | |
| Average (mm) | 5.949 | 6.129 | 6.474 | 6.286 | 5.868 | 6.060 | 7.0301 | 6.974 |
| Experiment (No.) |
9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
| Compress variation (mm) |
5.866 | 6.748 | 6.611 | 7.3475 | 7.161 | 7.386 | 7.208 | 7.303 |
| 6.077 | 6.831 | 6.737 | 7.2034 | 6.84 | 7.436 | 7.192 | 7.231 | |
| 6.101 | 6.781 | 7.208 | 7.29 | 6.989 | 7.464 | 6.969 | 7.091 | |
| Average (mm) | 6.015 | 6.787 | 6.852 | 7.2803 | 6.997 | 7.429 | 7.123 | 7.208 |
Under different combination of parameter levels, compress variation was the investigated index value in this orthogonal design experiment and set as Yi (i = 1, 2, 3, ..., n) which were the average values of three models each level listed in Table 4. Seen from these results, each kind of parameter generated a maximum deformation of AFS panels.
3.1 Range analysisIn the range analysis, Kjl denotes the sum of corresponding value (Yi) of the level “l” (l = 1, 2, ..., r) located at the jth column (j = 1, 2, ..., 5). Therefore, K11 = Y1 + Y2 + Y3 + Y4, K23 = Y3 + Y7 + Y11 + Y15, K54 = Y4 + Y5 + Y11 + Y14. According to the following formula12), the range value can be obtained.
| \[k_{jl} = K_{jl}/4\] | (3-1) |
| \[R_j = \max \{ k_{j1},k_{j2},k_{j3},k_{j4} \} - \min \{ k_{j1},k_{j2},k_{j3},k_{j4} \}\] | (3-2) |
| Experiment (No.) |
Factor A (%) |
Factor B (mm) |
Factor C (N) |
Factor D (mm) |
Null column | Yi |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 5.949 |
| 2 | 1 | 2 | 2 | 2 | 2 | 6.129 |
| 3 | 1 | 3 | 3 | 3 | 3 | 6.474 |
| 4 | 1 | 4 | 4 | 4 | 4 | 6.286 |
| 5 | 2 | 1 | 2 | 3 | 4 | 5.868 |
| 6 | 2 | 2 | 1 | 4 | 3 | 6.060 |
| 7 | 2 | 3 | 4 | 1 | 2 | 7.030 |
| 8 | 2 | 4 | 3 | 2 | 1 | 6.974 |
| 9 | 3 | 1 | 3 | 4 | 2 | 6.015 |
| 10 | 3 | 2 | 4 | 3 | 1 | 6.787 |
| 11 | 3 | 3 | 1 | 2 | 4 | 6.852 |
| 12 | 3 | 4 | 2 | 1 | 3 | 7.280 |
| 13 | 4 | 1 | 4 | 2 | 3 | 6.996 |
| 14 | 4 | 2 | 3 | 1 | 4 | 7.429 |
| 15 | 4 | 3 | 2 | 4 | 1 | 7.123 |
| 16 | 4 | 4 | 1 | 3 | 2 | 7.208 |
| Kj1 | 24.838 | 24.828 | 26.069 | 27.688 | 26.833 | |
| Kj2 | 25.932 | 26.405 | 26.400 | 26.951 | 26.382 | |
| Kj3 | 26.934 | 27.479 | 26.892 | 26.337 | 26.810 | |
| Kj4 | 28.756 | 27.748 | 27.099 | 25.484 | 26.435 | |
| kj1 | 6.210 | 6.207 | 6.517 | 6.922 | 6.708 | |
| kj2 | 6.483 | 6.601 | 6.600 | 6.738 | 6.596 | |
| kj3 | 6.734 | 6.870 | 6.723 | 6.584 | 6.703 | |
| kj4 | 7.189 | 6.937 | 6.775 | 6.371 | 6.609 | |
| Range value (Rj) |
0.907 | 0.730 | 0.258 | 0.551 | ||
| Priority | P4 | R4 | L1 | F4 |
As seen from Table 5, the range values corresponding to porosity P, pore size R, loading force F and local compressing length L are 0.907, 0.730, 0.258 and 0.551, respectively. So, the influence degree of the above mentioned parameters on the local compressive deformation behavior of AFS panels is in turn: porosity P, pore size R, local compressing length L, loading force F, in other words, P > R > L > F. Furthermore, from Table 5, it can also be found that the optimal combination is P4R4L1F4 which leads a maximum amount of compression.
Figure 4 showed the influence regularity and degree of each parameter on the local compressive deformation of AFS panels. As seen from Fig. 4, the influence of porosity on the compress variation of AFS panel was the most significant, and the amount of deformation of AFS panel was almost proportional to the porosity. Moreover, the second important influence factor on AFS panel was the pore size, and it was found that the compress variation increases with the increasing of the pore size and the growth trend reduces gradually. When the pore size approached to R1.5～2, cell had little influence on local compressive deformation behavior of AFS panels. These results indicated that compressive properties decreased with the increasing in hollow microsphere content and the size of microsphere. The similar result was reported in Refs. 19) and 20). The major reasons could lie in higher porosity and pore size in microsphere generated lower wall thickness which couldn't withstand higher stress values in contrast to those with higher wall thickness.21) The third-ranking parameter was local compressing length. However, it was interesting that the change trend of the compress variation of AFS panel induced by local compressing length was opposite to others studied variables. In other words, in a certain range, the higher local compressing length was, the less deformation obtained was. It can be seen from Fig. 4, loading force was the slightest influence factor on AFS panel among the above mentioned parameters, and the obtained compress variation traded in a tight range and almost was proportional to the loading force.
The results of range analysis indicated that porosity had a dominate position in above four parameters. The deformation-time curves of AFS panels with different porosity corresponding to the largest deformation were shown in Fig. 5 which was made up of the results coming from the experiments group 3, group 7, group 12 and group 14 designed in Table 5. A comprehensive analysis indicated that the higher porosity was, the greater corresponding compress variation was. Although the range size of cell (R2～2.5) of curve C was two grades larger than curve D (R1～1.5), curve D was still higher than curve C. This result confirmed that porosity had a significant impact on local compressive deformation behavior of AFS panel, which was identical with the earlier works on aluminum foam materials.22,23)
The deformation-time curves of AFS panels with different porosity corresponding to the largest deformation.
From Fig. 5, it could be observed that in period of 0～0.00013 s, curves B, C and D raise slowly, but each of them was faster than curve A. The major reason could lie in that, at the beginning, AFS panels displayed an elastic stage where porous structure subjects no severe stress concentration and just a slight cell wall bending and cell face stretching,24) as shown in Fig. 6 (a). In addition, aluminum foam with higher porosity would generate more hollow content, which resulted in a poor bearing capacity.21) Therefore, at the beginning of elastic stage, curve D raised at the fastest rate and curve A just was the opposite. In period of 0.00013～0.00045 s, curves A, B, C and D all shown a sharp upward tendency. However, the increase rate of curve A was slower than others. This phenomenon means that, with the continued loading force, AFS panels display plastic plateau stage due to the stress concentration and the cell walls reach to the yield limit that results in the internal pore structure overwhelmed and presents a wide range of structure collapses,25) as shown in Fig. 6 (b). Furthermore, compared with the high porosity of aluminum foam, the lower porosity of aluminum foam had more basis material and thicker cell wall. So, lower porosity of aluminum foam had the ability to resist larger loading and slower decrease of voids under compressed state.17) These results confirmed, at plastic plateau stage, the increase rate of curve A was slower than other curves which had close increase tendency. After 0.00045 s, curves A, C and D increased very gentle. This phenomenon means that AFS panels display a densification stage where the pore structure is collapsed providing more space due to the destroyed enclosed-hollow for the compressed material to occupy, and the contact between cell walls reach a certain degree resulting the material is difficult to be compressed further26,27), as shown in Fig. 6 (c).
(a) AFS panels in elastic stage, (b) AFS panels in plastic plateau stage and (c) AFS panels in densification stage.
As mentioned above, range analysis had neither distinguished data fluctuations that came from experiment conditions and experimental error, nor had provided a judgment standard to determine the influence degree of different parameters. The following variance analysis would complete this part of work. According to the formula (3-3) to (3-9),28) the statistical value F of each parameter was shown in Table 5.
| \[S_j^2 = -CT + \left( \sum_{l=1}^r K_{jl}^2 \right) \Biggm/ m\] | (3-3) |
| \[CT = \left( \sum_{i=1}^n Y_i \right)^2 \Biggm/ n\] | (3-4) |
| \[f_j = r - 1\] | (3-5) |
| \[\bar{S}_j^2 = S_j^2/f_j\] | (3-6) |
| \[f_e = n - f_1 - f_2 - f_3 - f_4\] | (3-7) |
| \[\bar{S}_e^2 = S_e^2/f_e\] | (3-8) |
| \[F = \bar{S}_j^2/\bar{S}_e^2 = (S_j^2/f_j)/(S_e^2/f_e)\] | (3-9) |
Seen from Table 6, the results of variance analysis indicated that the corresponding F value of porosity P, pore size R, loading force F and local compressing length L are 48.201, 30.555, 3.803 and 15.202, respectively. Under the significant level α = 0.05 and 0.025, when F > F1−α(r − 1, fe), the corresponding parameter of statistical value F would show a significant influence.16) By checking mathematical statistics tables, the critical values F0.95(3, 3) = 9.28 and F0.975(3, 3) = 15.44 were obtained. The following results were derived:
| \[ \begin{split} & F_{\rm P} > F_{0.975}(3,3),\ F_{\rm R} > F_{0.975}(3,3),\ F_{\rm F} < F_{0.95}(3,3), \\ & F_{0.975}(3,3) > F_{\rm L} > F_{0.95}(3,3). \end{split} \] |
| Factor A (%) |
Factor B (mm) |
Factor C (N) |
Factor D (mm) |
Error | |
|---|---|---|---|---|---|
| $S_j^2$ | 2.078($S_1^2$) | 1.317($S_2^2$) | 0.164($S_3^2$) | 0.655($S_4^2$) | 0.043($S_e^2$) |
| $f$ | 3(f1) | 3(f2) | 3(f3) | 3(f4) | 3(fe) |
| $\bar{S}_j^2$ | 0.693($\bar{S}_1^2$) | 0.439($\bar{S}_2^2$) | 0.055($\bar{S}_3^2$) | 0.218($\bar{S}_4^2$) | 0.014($\bar{S}_e^2$) |
| $F$ | 48.201 | 30.555 | 3.803 | 15.202 | |
| Significance | ** | ** | * |
“*” and “**” represent the significance of each parameter under the significant level α = 0.05 and 0.025, respectively.
Therefore, the result of variance analysis confirmed that level fluctuation of porosity and pore size had a significant impact on the deformation behavior of AFS panel and the level fluctuations of local compressing length was less significance. However, the local compressive deformation of AFS panel received very little effects from the level fluctuation of loading force. Obviously, porosity, pore size and local compressing length were needed to be noticed in the following structure design and application of porous composites. The result of variance analysis further demonstrated that the local compressive deformation behavior of AFS panels was more sensitive to geometrical parameters than loading conditions. The result was in good agreement with the above result which comes from the range analysis. In addition, the further related mechanism research of above four parameters effect on AFS panels in the process of compression will be completed in follow-up work.
In this paper, orthogonal simulation experiment has been conducted to investigate the influence degrees of structure parameters and loading conditions on the local compressive deformation behavior of AFS panels. Range analysis shows that the influence degree of four parameters on the local compressive deformation behaviors of AFS panel is in turn porosity P > pore size R > local compressing length L > loading force F. Furthermore, through variance analysis, it is further demonstrated that the local compressive deformation behavior of AFS panels is more sensitive to its geometrical parameters and secondary to its loading conditions. For AFS panels, this simulation results are of great significance to improve their overall application performance and have provided a certain theoretical guidance for experimental research, in especially, for the study on connection of AFS panels in the application of transportation equipment.
We would like to thank Associate Professor Ruijie Wang for providing valuable advices and recommendations with regards to the preparation of this paper.
