Simulation of Apparent Elastic Property in the Two-Dimensional Modelof Aluminum Foam Sandwich Panels

Sawei Qiu1, Xinna Zhang2, Qingxian Hao1, Renjun Dou1, Yan Ju1 and Yuebo Hu1,+

1Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology,Kunming 650500, Yunnan, China2Faculty of Automation and Electric Power, Kunming Metallurgy College, Kunming 650033, China

Based on its structural features, Aluminum Foam Sandwich (AFS) panels were properly simplified to two-dimensional random modelsthrough C++ and ANSYS software in this paper. The apparent elastic property of AFS panels was studied through simulation under thecircumstances of different relative density, pore size range or geometric imperfections. The obtained results showed that the microstructuraldeformation in AFS was caused during uniaxial compression, which leaded to the increase of apparent elastic modulus with the increase ofrelative density. It was also found that the apparent elastic modulus of AFS panels with non-homogeneous structure was the highest and theapparent elastic modulus of AFS panels with geometric imperfections was the lowest. Moreover, the pore size range almost had no effect on theslope of the elastic stage. However, the apparent elastic modulus showed great sensitivity to the geometric imperfections. In addition, theexponential relationship between the relative densities and the relative elastic modulus was fitted. The simulation results in this work were inagreement with those found in previous theoretical and experimental results reported in other literature, which confirmed the feasibility andrationality of two-dimensional random models. [doi:10.2320/matertrans.M2015008]

(Received January 6, 2015; Accepted February 16, 2015; Published April 25, 2015)

Keywords: aluminum foam sandwich panels, two-dimensional random model, simulation, apparent elastic modulus

1. Introduction

The ideal of sandwich structure with two compositeskins and a low density core was originally proposed byFairbairn.1,2) Aluminum Foam Sandwich (AFS) panels, dueto its excellent properties such as light weight, high strength-to-weight and stiffness-to-weight ratios and good energy-absorbing capacity, and so on, have been developed into anovel structural material which has broad application inaerospace, shipbuilding, electronics, automobile sectors,construction fields and other fields.3,4)

So far, many studies have been conducted on theproperties of AFS panels. Harte et al.5) presented a studyabout AFS panels tested in cyclic four point bend, anddetermined S­N fatigue curves for the failure modes of facefatigue, core shear and core indentation. Kesler et al.6)

analyzed the effect of the beam depth on the limit loadunder the three points bending test of AFS panels. Tagarielliet al.7) investigated the plastic collapse modes for clampedsandwich beams for the case of AFS panels and pointed outthe effect of clamped boundary conditions was to drive thedeformation mechanism toward plastic stretching of the facesheets. However, as far as we know, the study on theproperties of AFS panels by numerical simulating is few. Inthis paper, based on the structural features of AFS panels,two-dimensional random models were created by C++ andANSYS software. The apparent elastic property of AFSpanel was studied through simulation under the circum-stances of different relative density, pore size range orgeometric imperfections. A comparative analysis of thesimulation results in this work and the theoretical andexperimental results reported in previous literature wascarried out.

2. Modeling and Simulating

Compared with three-dimensional models, two-dimen-sional random model used in this paper to study the elasticbehavior of AFS panels had the advantages of the simplegeometry shapes and the easy dividing of meshes in ANSYSsoftware, and it had been confirmed that two-dimensionalrandom model had a good application in understanding someimportant features of real foams.8) The finite element modelsused to study the apparent elastic modulus of AFS panelswere generated through the method of combining the C++and ANSYS software, the process of modeling could beconstructed as follows. Firstly, porosity, diameter pore sizeand minimum wall thickness were set as input variables;center coordinate (x, y) and radius r were set as outputvariables. Secondly, a Cartesian coordinate system waschosen, n hollow circular being created in a two-dimensionalspace A0. Among them, each coordinate of circle center (x, y)and radius r were randomly generated through C++program. Finally, two-dimensional random model containedn hollow circular automatic was generated in the ANSYSsoftware through reading data. The models of AFS panelswere based on a highly porous aluminum foam core (50mmwidth, 25mm thickness) and two aluminum sheets (50mmwidth, 1mm thickness) bonded to the core on either side. Therandom model with non-homogeneous structure is shown inFig. 1(a): the size of radius varies from 0.5 to 2mm. Therandom model with relative homogeneous structure is shownin Fig. 1(b): the size of radius varies from 1 to 2mm. Therandom model with geometric imperfections is shown inFig. 1(c): the size of radius varies from 1 to 2mm.

The following material data referring to two-dimensionalrandom models were used: Es = 69Gpa, Vs = 0.31,EAl = 71Gpa and VAl = 0.34, where Es is Young’s modulusof aluminium foam core, Vs is Poisson’s ratio of aluminium+Corresponding author, E-mail: huyb@kmust.edu.cn

Materials Transactions, Vol. 56, No. 5 (2015) pp. 687 to 690©2015 The Japan Institute of Metals and Materials

foam core, EAl is Young’s modulus of aluminium sheets, VAlis Poisson’s ratio of aluminium sheets.9) Due to the unevenstructure of aluminum foam, it would present local plasticdeformation under a small load. Therefore, the matrixmaterials of aluminum foam could be set as multi-linearelastic model which could describe the nonlinear stress-strainrelationship. The stress-strain data were shown in Table 1.10)

After modeling, setting material properties and generatingmesh, the boundary constrains and loading conditions wereapplied (shown in Fig. 2). Finally, finite element solution wasobtained. The finite element analysis process is composed ofthe following basic algebraic equation.11)

½K�½D� ¼ fPg ð1Þ½¾� ¼ ½A�T½U� ð2Þ½·� ¼ ½D�½¾� ð3Þ

where [K] is whole stiffness matrix of model, {P} is the nodeload matrix, [A] and [D] are the geometry matrix and theelastic coefficient matrix of model, respectively.

In this simulation, to measure the apparent elastic modulus,a series of quasi-static uniaxial compression tested were doneat different relative densities (µ = 0.583, 0.538, 0.495, 0.452and 0.398). The simulation tests were carried out on theANSYS software with same testing conditions and specimendimensions as previously shown. Three specimens weresimulated at the same density values, and the average resultswere reported. The calculation results showed all of themodels with relative density (µ = 0.398) were in the state ofelastic deformation when the loading pressure under 1100 Pa.So, the loading pressure used in analyzing the apparentelastic modulus of the model samples selected to be 1100 Pa.In addition, the thickness of AFS panel sample was not largeand the direction of the force was always perpendicular to theloading surface. Y direction caused the maximum amount ofcompression (The Y-component of displacement diagram isshown in Fig. 3) was less than 10¹7m and the maximumdeflection angle caused by the pressure was 10¹7/(50 ©10¹3) = 0.2 © 10¹5 rad. The angle of deflection was so smallthat it was negligible. So, we might consider that the pressurewas always along the height direction of specimen and the

models were also consistent with the plane strain condition.Therefore, the two-dimensional planar strain models used insimulation were proved to suitable with the practical 3Dstructure.

3. Results and Analysis

On the horizontal line of the aluminum plate defined a pathand divided it into 12 parts, with a total of 13 data points.Amount of compression along the loading direction could beobtained by the post-processing module in ANSYS software.After each compression UY of specimens obtained, accordingto the formula (4) and (5),11) the arithmetic mean UYE and theapparent elastic modulus could be obtained.

¾ ¼ UYE=h ð4ÞE ¼ ·=¾ ð5Þ

where E is the apparent elastic modulus of AFS panels, h isthe height of the specimen model, · is the stress, ¾ is thestrain. The above obtained apparent elastic modulus E ofdifferent models are shown in Table 2, Table 3 and Table 4.

Drawing from the above mentioned results in the samecoordinate could gain the relationships between relativedensities and elastic modulus as shown in Fig. 4.

Many physical properties of the metal foams could beestimated by the exponential rule which was composed bythe relative density and a constant of the performance.Therefore, assuming that relative elastic modulus and relativedensity of AFS panel conformed to the following formula:

E

Es

¼ Cµ

µs

� �n

ð6Þ

(a) (b) (c)

Fig. 1 Part of Two-dimensional random models.

Table 1 The stress-strain data of aluminum foam core.

Stress (Mpa) 69 170 185 213 230

Strain (mm) 0.001 0.005 0.01 0.023 0.035

Fig. 2 Model loading and constraints diagram.

Fig. 3 Y-component of displacement diagram.

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where E is the apparent elastic modulus of aluminum foam(Mpa), Es is the elastic modulus of the matrix material (Mpa),µ is the density of aluminum foam (g/cm3), µs is the densityof the matrix material, C is a proportional constant. E/Es

is defined as Ex, µ/µs is defined as µx. Taking logarithmon both sides of the equation at the same time can beobtained:

lgEx ¼ n lg µx þ n lgC ð7ÞThe above equation shows that lgEx is of linear relation-

ship with lg µx, where n is the slope and n lgC is the intercept.The simulation data are fitted to Origin software, line L1, L2

and L3 are obtained shown in Fig. 4. Then the value of n andlgC could be calculated. The elastic modulus expression ofeach models as follows, respectively.

The random model with non-homogeneous structure:

Ex ¼ 0:908ðµxÞ2:33 ð8ÞThe random model with relative homogeneous structure:

Ex ¼ 0:989ðµxÞ2:81 ð9ÞThe random model with geometric imperfections:

Ex ¼ 0:928ðµxÞ3:03 ð10ÞIn flatwise compression, load was mainly taken up by the

core due to the collective microscopic elastic bending of thecell wall, as reported in the Ref. 12). As seen from Fig. 4,AFS panels were in elastic phase under the test pressurewhere the apparent elastic modulus kept a linear relation withthe relative density. At same time, it was found from this

figure the apparent elastic modulus of the AFS panel withnon-homogeneous structure was the highest and that one ofthe AFS panel with geometric imperfections was the lowest.The reason may lie in, compared with other models, therandom model with non-homogeneous structure has wideraperture size distribution and more cells at the same porosity,which lead to more cell edges and a framework that canwithstand larger pressure. So, under the same pressure, thecompression of the AFS panels with non-homogeneousstructure in the minus Y direction was the smallest. Thisresult revealed the micro-geometry structural characteristicsof the different cells. The deformation mechanism wasmainly the bending of cell walls and faces which means thatthe topology of the cells caused the cell edges to bend whenthe structure was loaded.13­15) A conclusion was drawn thatrelative density and diameter pore size had a great influenceon the apparent elastic modulus of AFS panels, and relativedensity kept an approximately linear relation with apparentelastic modulus. In addition, the slopes of fitting line L1 andL2 were basically identical, which showed that pore sizerange almost had no effect on the slope of the elastic portion.The result was in agreement with the earlier report.16) Thiscould be due to the round holes in material were subjectedto a uniform deformation within the gauge length of thespecimen when the structure was loaded. The other smallslope observed in line L3 indicated the apparent elasticmodulus of AFS panel had a great sensitivity to the geometricimperfections in aluminum foam cores. This could be due tothe existence of geometric imperfections in aluminum foamcores change the size distribution and morphology character-istics of the cell edges (as shown in Fig. 1). In addition, itwas well known that the foam structure with a comparativelyregular cell edges could withstand a higher stress values incontrast to those with irregular cell edges. Hence, the samplewith the geometric imperfections showed a relatively poorapparent elastic modulus. Another conclusion was drawn thatthe introduction of more geometric imperfections during theinternal structure resulting in a reduction of the apparentelastic modulus of AFS panels. This obtained results alsoreported in Ref. 17).

Table 3 The apparent elastic modulus of the random model with relativehomogeneous structure.

µx 0.583 0.538 0.495 0.452 0.398

14810 10663 9746 6820 4730

E (Mpa) 14427 10990 9832 7172 4941

14254 10598 9714 7029 4744

Mean Value 14494 10750 9764 7007 4805

Table 4 The apparent elastic modulus of the random model with geometricimperfections.

µx 0.583 0.538 0.495 0.452 0.398

9901 8864 7285 5021 3184

E (Mpa) 10079 9034 6852 5059 3277

10312 8625 6793 5132 3209

Mean Value 10097 8837 6976 5070 3223

Fig. 4 The relationship between relative density and apparent elasticmodulus.

Table 2 The apparent elastic modulus of the random model with non-homogeneous structure.

µx 0.583 0.538 0.495 0.452 0.398

14734 13905 10122 7825 6501

E (Mpa) 14873 13682 9909 7816 6434

14714 13289 10034 8171 6615

Mean Value 14773 13625 10021 7970 6516

Simulation of Apparent Elastic Property in the Two-Dimensional Model of Aluminum Foam Sandwich Panels 689

Moreover, the obtained expressions of elastic moduluswhich were basically consistent with others research results.S. K. Maiti et al.18) pointed out that the elastic modulus ofmetal foam satisfied the relationship Ex = k(µx)2.0. K. Carolinet al.19) noted that the elastic modulus of metal foam metEx = 1.47(µx)3.0. Dai20) deduced the expression of elasticmodulus from the experiments Ex = 0.948(µx)2.78. Comparedwith the above simulation results, n and C of the expressionwere slightly different. However, this was a kind of AFSpanel models where n and C were affected by substratematerial and the structure of foams, which fully confirmedthat the selected models and the simulation results werereliable.

4. Conclusion

In this paper, three kinds of two-dimensional randommodels of AFS panels were presented and the apparent elasticmodulus of AFS panels was analyzed by ANSYS software.The results indicated that the apparent elastic modulus ofAFS panels had a close relationship with porosity, pore sizerange and geometric imperfections. In addition, the expo-nential relational expressions between relative densities andrelative elastic modulus were obtained. The study on theelastic behavior of AFS panels offered a reliable method thatwas time-saving, labor-saving and low-cost, and it was ofgreat significance for the structural properties research ofAFS panels.

REFERENCES

1) X. Li, G. Y. Li, C. H. Wang and M. You: Appl. Compos. Mater. 19(2012) 315­331.

2) D. Zenkert: An Introduction to Sandwich Construction, (EngineeringMaterials Advisory Services Ltd., UK, 1995).

3) G. Zhou and M. D. Hill: J. Sandw. Struct. Mater. 11 (2009) 329­356.4) J. Banhart and H. W. Seeliger: Adv. Eng. Mater. 14 (2012) 1082­1087.5) A. M. Harte, N. A. Fleck and M. F. Ashby: Int. J. Fatigue 23 (2001)

499­507.6) O. Kesler and L. J. Gibson: Mater. Sci. Eng. A 326 (2002) 228­234.7) V. L. Tagarielli and N. A. Fleck: J. Appl. Mech. 72 (2005) 408­417.8) K. R. Mangipudi and P. R. Onck: Int. J. Solids Struct. 49 (2012) 2823­

2829.9) S. B. Han and Z. H. Lv: Acta Armamentarii 31 (2010) 1468­1474.10) S. Y. Xing and S. H. Wang: Aluminum Alloy and Titanium Alloy, (China

Machine Press, Beijing, 1987) pp. 55­93.11) P. Wei: Doctoral Thesis, Huazhong University of Science and

Technology, (2006) pp. 69­80.12) S. J. Kumar and K. Sabeel Ahmed: J. Reinf. Plast. Comp. 32 (2013)

1370­1379.13) C. Swetha and R. Kumar: Mater. Des. 32 (2011) 4152­4163.14) M. Hassan and W. J. Cantwell: J. Compos. Mater. 46 (2011) 1191­

1199.15) C. Tekoglu and P. R. Onck: J. Mech. Phys. Solids 56 (2008) 3541­

3564.16) Y. H. Yu, Z. Yang and Y. Liang: Comput. Simul. 27 (2010) 317­320.17) Y. X. Gan, C. Chen and Y. P. Shen: Int. J. Solids Struct. 42 (2005)

6628­6642.18) S. K. Maiti, L. J. Gibson and M. E. Ashby: Acta Met. 32 (1984) 1963­

1975.19) C. Körner, M. Thies and R. F. Singer: Adv. Eng. Mater. 4 (2002) 765­

769.20) C. L. Dai: Master Thesis, Yanshan University, (2012) pp. 42­55.

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