los

a b,0 Yonof Adv

Article history:

Received in revised form 11 August 2008

The mechanical properties of the cell wall, such as the elastic modulus, 0.2% offset yield

both experimental measurements and nite element analyses. A 6 6 12 cm3 ingot of

streams of research have been developed subsequently:rst, the analysis of the signicant effects of geometricparameters related to the cell structure on the perfor-mance of the foam material (Miyoshi et al., 1999, 2000;Simone and Gibson, 1998a; Grenestedt, 1998; Grenestedt

The wavy imperfection and cell shape variation decreasethe stiffness of foam materials (Grenestedt, 1998;Grenestedt and Tanaka, 1999). The effects of cell wallmicrostructures, which are formed differently dependingon the production processes and the composition of thecell wall raw materials, on the compressive response ofclosed-cell foams were also presented (Markaki and Clyne,2001). Recently, the effects of intrinsic structural defects of

0167-6636/$ - see front matter 2008 Elsevier Ltd. All rights reserved.

* Corresponding author. Tel.: +82 62 530 1688; fax: +82 62 530 1689.E-mail address: i_jeon@chonnam.ac.kr (I. Jeon).

Mechanics of Materials 41 (2009) 6073

Contents lists available at ScienceDirect

Mechanics of

elsedoi:10.1016/j.mechmat.2008.08.0021. Introduction

As a functional light weight material, closed-cell Alfoam is widely used in transportation applications, militaryequipment, machine tools and so on. Therefore, themechanical performances of the foam material, includingits effective elastic modulus, compressive strength, and en-ergy absorption capacity, have become an important sub-ject of research. Much research has been done onperformance improvement of the foam material. Two main

and Tanaka, 1999; Markaki and Clyne, 2001; Jeon andAsahina, 2005), and second, the analysis on the effects ofthe cell wall material properties on the performance ofthe foam material (Jeon et al., submitted for publication).

With respect to the effects of the geometric parameters,decreased cell size and increased thickness of the cell wallincrease the compressive strength of the aluminum foammaterial (Miyoshi et al.,1999, 2000). Furthermore, cell facecurvatures and corrugations decrease the elastic modulusand plastic collapse stress (Simone and Gibson, 1998).the cell wall base material, which is sampled from melted Al-1.5 wt.%Ca alloy before foam-ing, is prepared, and its mechanical properties are initially measured to set the limit valuesof the mechanical properties of the Al cell wall. Two 5 5 5 mm3 Al foam specimens ofcompletely different structures are fabricated, and directly modeled for the nite elementanalysis by using a microfocus X-ray CT system, 3D reconstruction program, 3D scanneddata processing software, and commercial mesh generation program. Subsequently, uniax-ial compression tests are carried out on the specimens, and the numerical simulations ofthese tests are performed using the nite element models. For the simulations, variousmechanical properties for the cell wall selected from the measured properties of the basematerial are used. Then, the Al cell wall mechanical properties are precisely determined bycomparing the computed forcedisplacement curves with the measured ones. Finally, theeffects of each mechanical property on the compressive behavior of the foam material areanalyzed.

2008 Elsevier Ltd. All rights reserved.Received 3 March 2008 stress and power-law hardening exponent of the closed-cell Al foam are determined usingCell wall mechanical properties of c

Insu Jeon a,*, Kiyotaka Katou b, Tsutomu Sonoda School of Mechanical Systems Engineering, Chonnam National University, 30bMaterials Research Institute for Sustainable Development, National InstituteNagoya 463-8560, Japan

a r t i c l e i n f o a b s t r a c t

journal homepage: www.ed-cell Al foam

Tadashi Asahina b, Ki-Ju Kang a

gbong-dong, Buk-gu, Gwangju 500-757, Republic of Koreaanced Industrial Science and Technology (AIST), Shimoshidami, Moriyama-ku,

Materials

vier .com/locate /mechmat

the closed-cell Al foam on the decrease of the effectivemodulus as well as the plateau stress of the foam materialhave been reported (Jeon and Asahina, 2005).

With respect to the effects of the material properties ofthe cell wall, research results have not been extensive. Jeonet al. (submitted for publication), however, demonstratedthe effects of cell wall plastic properties, such as the 0.2%yield stress and power-law hardening exponent, on the en-ergy absorption capacity of the closed-cell Al foam. Theyalso compared the stressstrain curve measured from thea uniaxial compression test on a 10 10 10 mm3closed-cell Al foam with the stressstrain curves obtainedfrom numerical simulations of the test. Although the plas-tic properties of the cell wall contribute signicantly to themechanical performance of the foam material, research onthe measurement of the precise values of these propertieshave been limited.

An approximation method was applied to measure theyield stress of the Al cell wall, which takes the yield stressvalues to be one-third of the cell wall hardness (Sugimuraet al., 1997; Simone and Gibson, 1998b; Andrews et al.,1999). Another measurement method using nanoindenta-tion has been recently introduced to measure cell wallmechanical properties (Kim and Tunvir, 2006; Hasanet al., 2008). This method, however, has some difculties

Fig. 1. (a) A 6 6 12 cm3 base material ingot, (b) three cylindrical specimens of 25 mm height and 12.7 mm diameter, and (c) a specimen betweencompression platens.

ain cu

Table 1Measured elastic moduli and elastic limit stresses of the base material

E (GPa) Elastic limit stress, re (MPa)

Comp. test case #1 66.09 12Comp. test case #2 56.24 9.9Comp. test case #3 62.8 11.6Average 61.7 11.16

I. Jeon et al. /Mechanics of Materials 41 (2009) 6073 61Fig. 2. True stress-true str rves of the base material.

Fig. 3. (a) Specimen A (5 5 5 mm3), (b) its tomographic images and (c) reconstructed model.

Fig. 4. (a) Specimen B (5 5 5 mm3), (b) its tomographic images and (c) reconstructed model.

62 I. Jeon et al. /Mechanics of Materials 41 (2009) 6073

in obtaining actual cell wall mechanical properties: it usu-ally uses the indentation loaddisplacement curves, whichinclude considerable variations arising from the inhomo-geneous microstructure of the cell wall cross section, andcan not account for the effect of cell wall oxidation in themeasurement of the mechanical properties.

In this research, we suggest a new approach to deter-mine the mechanical properties of the cell wall of theclosed-cell Al foam. After measuring the mechanical prop-erties of the cell wall base material, taken from melted Al-1.5 wt.%Ca alloy before foaming, we perform a direct niteelement modeling (Jeon et al., submitted for publication) oftwo actual 5 5 5 mm3 foam specimens. Uniaxial com-pression tests using the specimens and numerical simula-

tions of the tests using the nite element models arecarried out. In the simulations, the mechanical propertiesfor the cell wall selected from the measured properties ofthe base material are used. Then, the cell wall mechanicalproperties and their effects on the compressive behavior offoam materials are precisely determined by comparing thecomputed forcedisplacement curves with the measuredones.

2. Mechanical properties of the cell wall base material

A 6 6 12 cm3 cell wall base material ingot, takenfrom melted Al-1.5 wt.%Ca alloy before foaming, was pre-pared to measure the mechanical properties of the base

surfa

I. Jeon et al. /Mechanics of Materials 41 (2009) 6073 63Fig. 5. (a) Modied and smoothened outer cells and inner voids, (b) NURBS3D nite element model of Specimen A.ces of the outer cells and inner voids, (c) fabricated geometric solid and (d)

material, in order to establish the limit values of themechanical properties of the Al cell wall (see Fig. 1a).From the ingot, three cylindrical specimens of 25 mmheight and 12.7 mm diameter were fabricated by usingthe electrical discharge machine for the uniaxial com-pression test (see Fig. 1b). The compression tests werecarried out by using a universal testing machine AG-I ofSHIMADZU Corp. with a 100 kN load cell. The specimenswere compressed between two circular steel compressionplatens of 200 mm in diameter. The surfaces of the plat-ens were lubricated to reduce the friction effect of thecontact surfaces between the specimens and the platenson the compression test results (ASTM E9-89A, 2000).

Three strain gauges were attached on each specimensurface to measure the precise strain values (see Fig. 1c).A displacement rate of 0.5 mm/min was applied to thetop surface of the specimen. The strain was accurately

determined only up to 3.5% because of the measurementlimitation of the strain gauge. The three, measured, truestress-true strain curves are shown in Fig. 2.

From the curves in Fig. 2, the elastic modulus E andelastic limit stress re were obtained and tabulated togetherwith their average values, E = 61.7 GPa and re = 11.16 MPa,in Table 1. The nonlinear part of the curves can be tted byusing the RambergOsgood model as follows:

ep

ee a r

re

nfor r > re 1

where r, the Mises stress; ep, the equivalent plastic strain;ee = re/E, the reference strain component; a, a non-dimen-sional constant set to 1 for computation; n, the power-law hardening exponent.

The properly tted curve using the RambergOsgoodmodel for the measured curves of the base material is plot-

surfa

64 I. Jeon et al. /Mechanics of Materials 41 (2009) 6073Fig. 6. (a) Modied and smoothened outer cells and inner voids, (b) NURBS3D nite element model of Specimen B.ces of the outer cells and inner voids, (c) fabricated geometric solid and (d)

ted together in Fig. 2. For this curve tting, the power-lawhardening exponent, n, was determined as 3 obtained fromthe comparison of the measured curves with the calculatedcurves after substituting E = 61.7 GPa and re = 11.16 MPa,and various n values into Eq. (1).

3. Preparation of two small foam specimens

Two 5 5 5 mm3 specimens A and B of the closed-cell Al foam named ALPORAS of completely differentstructures were prepared for the uniaxial compressiontests and nite element modeling (see Figs. 3a and 4a).Before preparing these specimens, several 5 5 5 mm3foam specimens were fabricated by using the electricaldischarge machine to ensure both surface atness andclearness. Then, of these specimens, two specimens havingno structural defects that would degrade the compressiveperformance of foam materials were carefully selected bysurface inspection: these two nal specimens were thespecimens A and B. The relative density of Specimen Awas q/qs = 0.0785 and that of Specimen B was

q/qs = 0.0837. Here, q is the density of the foam material,and qs = 2.7 g/Cm3 is the density of the Al cell wall.

4. Finite element modeling of the two foam specimens

A thorough introduction of the direct nite elementmodeling procedure for a closed-cell foam was presented(Jeon et al., submitted for publication). Following the pro-cedure, two 5 5 5 mm3 foam specimens A and B weremodeled independently. The microfocus X-ray CT systemof Shimadzu Corp. was used to scan the outer and innerstructures of the specimens. The scanned grayscale tomo-graphic images of the specimens A and B are shown in Figs.3b and 4b, respectively. For each specimen, 257 imageswere taken along the specimen height of 5 mm. Theseimages include the structures of the cell wall and the airregion within the foam materials.

Using these images, each foam specimen was recon-structed using a 3D reconstruction program TRI/3D-BONof Ratoc System Engineering Co. Ltd. The prole of the

I. Jeon et al. /Mechanics of Materials 41 (2009) 6073 65Fig. 7. (a) Specimen A between two platens and (b) their nite elementmodels.Fig. 8. (a) Specimen B between two platens and (b) their nite elementmodels.

cellular structure and the air region of a specimen in thetomographic images were binarized using the 3D recon-struction program and the isolated small particles fromthe cellular structure in the binarized images were re-moved. The 3D cellular structures were then reconstructedusing the marching cubes algorithm (see Figs. 3c and 4c)(Sone et al., 2004; Jeon et al., submitted for publication).

For geometric modeling of each foam specimen, everycell in each reconstructed model was disjointed, and allthe disjointed cells were modied and smoothened oneby one to fabricate their nonuniform rationale B-spline(NURBS) surfaces. Figs. 5a and 6a show the modied and

smoothened outer cells and inner voids of the specimensA and B, respectively. The cube-shaped frames in Figs. 5aand 6a represent the disjointed status of each cell, andthe bold solid lines in the disjointed cells indicate theboundary of each cell. For simplicity of modeling, all innervoids having maximum diameters that were less than0.1 mm were disregarded. Then, the NURBS surfaces werefabricated using each cell. Figs. 5b and 6b show the fabri-cated NURBS surfaces for the outer cells and inner voidsof specimen A and B, respectively. Two geometric solidsthat have the same structure of the two foam specimenswere fabricated using Boolean operations between two

of the

66 I. Jeon et al. /Mechanics of Materials 41 (2009) 6073Fig. 9. Variation of true stress-true strain curves obtained from curvesexponent, (b) elastic limit stress and (c) elastic modulus.base material according to the change of (a) the power-law hardening

5 5 5 mm3 geometric solid cubes and the outer and in-ner NURBS surfaces of each specimen (Jeon et al., submit-ted for publication). The completed geometric solids forthe closed-cell foam specimen A and B are shown in Figs.5c and 6c, respectively. The 3D scanned data processingsoftware RapidFormTM of INUS Tech. Inc. was used for allgeometric modeling (Shin et al., 2007; Jeon et al., submit-ted for publication).

Subsequently, compatible 3D nite element mesheswere generated directly in each fabricated geometric solidafter iterative calculations using the commercial meshgeneration software PATRAN of MSC Software Corp. (seeFigs. 5d and 6d). In this process the volume errors, VmVaVa ,where Vm is the volume of the meshed model and Va isthe actual volume of the specimen, were 5.5% for SpecimenA and 4.94% for Specimen B, respectively.

Using the nite element models, two Al foam specimensbetween two platens were completely modeled (see Figs. 7and 8). Quadratic tetrahedral elements with 10 nodes wereused for the specimens and platens. The total number ofelements and nodes used for specimen A were 79311 and145725, and for specimen B were 75994 and 137585,respectively. For simulating the compression test process,a package code ABAQUS of Dassault Systems was used.

ment was loaded at over 20% strain on the top surface ofeach upper platen. The frictionless contact condition forthe contact surfaces between the specimens and the plat-ens was applied, which represents the lubricated surfacecondition of the platens.

As for the material properties of the platens,E = 214 GPa and v = 0.3 of Tool Steel SKS3 were applied.For the Al cell wall, various values of the power-law hard-ening exponents, n = 3, 5, 6, 8.5, 9 and elastic limit stresses,re = 11.16, 25, 28, 30, 31, 32, 34 MPa, were selected; thesevarious values were the measured mechanical propertiesof the cell wall base material and selected values basedon the measured values. For the cell wall elastic modulus,E = 61.7 GPa of the base material, E = 68 GPa of the pub-lished values of ALPORAS (Sugimura et al., 1997) andE = 73.1 GPa of Aluminum 2024-T6 were selected for thenumerical tests. Fig. 9ac show the calculated true stress-true strain curves after substituting the various selectedvalues of n, re and E in Eq. (1) together with the measuredcurves of the base material.

5. Uniaxial compression tests of the two foamspecimens

(conti

I. Jeon et al. /Mechanics of Materials 41 (2009) 6073 67The incremental plasticity theory of isotropic-hardeningmaterials was selected for the elasticplastic behavior ofthe specimens, which uses the objective stress rate for -nite deformation of the Al cell wall. Moreover, thepower-law hardening rule in Eq. (1) was used to calculatethe plastic hardening index in the incremental plasticitytheory (McMeeking, 1977; Jeon and Im, 2001; Lee et al.,2007; Jeon et al., submitted for publication). As the bound-ary condition for the simulations, x- and y-direction de-grees of freedom were constrained at several nodesaround the center on the lower surface of each specimento prevent rigid body motions, and z-direction displace-

Fig. 9 nued)A universal testing machine AG-I of SHIMADZU Corp.with a 5 kN load cell was used for the uniaxial compressiontests of the two 5 5 5 mm3 foam specimens. The spec-imens were compressed between two circular platens of100 mm in diameter, which was sufciently larger thanthe specimens (see Figs. 7a and 8a). The surfaces of theplatens were lubricated to reduce the friction effect ofthe contact surfaces between the foam specimens andthe platens (ASTM E9-89A, 2000).

A displacement rate of 0.15 mm/min was applied tothe top surface of the specimens up to 60% strain to

obtain the entire compressive behavior of the foam spec-imens. Because of the difculty in attaching the straingauge to the nonuniform surfaces of the foam specimens,the machine compliance, which represents the relationbetween the displacement of the testing machine andthe applied loading, was used to determine precise spec-imen displacements (Jeon and Asahina, 2005).

6. Results and discussion

The forcedisplacement curves of the specimens A andB obtained from the experimental measurements and the

numerical simulations are plotted in Figs. 10 and 11,respectively. To dene the changes in the obtained forcedisplacement curves, three evaluation parameters wereintroduced: the peak force in the curves, fpeak, the displace-ment at the peak force, dpeak, and the elastic stiffness in theelastic interval of the curves, f/d.

Figs.10a and 11a show the dependence of the forcedis-placement curve on the power-law hardening exponents ofthe cell wall. Five different computations for each speci-men were carried out: the same measured values ofE = 61.7 GPa and re = 11.16 MPa of the cell wall base mate-rial were used, but ve different selected values of n: n = 3,

stress

68 I. Jeon et al. /Mechanics of Materials 41 (2009) 6073Fig. 10. Effects of (a) the power-law hardening exponent, (b) elastic limit and (c) elastic modulus on the forcedisplacement curve of Specimen A.

5, 6, 8.5, 9 were used. From the two gures, it is found thatfpeak and dpeak of the computed curves rapidly decreasewith the increase in the power-law hardening exponent.The % errors of the evaluation parameters,Exp: oneNum: one

Exp: one 100, which were obtained from Figs. 10aand 11a, are tabulated in Table 2. In Table 2, the % errorsof dpeak obtained from the computation using n = 8.5 are6.8692% and 8.42798% for the specimens A and B, respec-tively. These % error values are close to the volumetric er-rors of the specimens. Moreover, these small % error valuesmean that the peak force positions in the computed forcedisplacement curves are horizontally very close to those ofthe experimentally measured curves. For these reasons, thepower-law hardening exponent of the Al cell wall wasdetermined as n = 8.5 in this research.

Figs. 10b and 11b show the effects of the elastic limitstress of the cell wall on the forcedisplacement curve.For these computations, the same values of E = 61.7 GPaand n = 8.5 but seven different selected values of re:re = 11.16, 25, 28.5, 30, 31, 32, 34 MPa were used for eachspecimen. In the two gures, dpeak is insensitive to thechange of the re but fpeak and f/d dramatically increase withthe increase of re. Table 3 shows the % errors of the evalu-ation parameters according to the change of re. The closestvalues of each % error to each volume error of the speci-mens A and B were obtained from computation using

are shown compared with the measured curves in Figs.10c and 11c. In these gures, it is hard to nd signicantdifferences among the computed curves. This means thatthe effect of the elastic moduli of the cell wall in the rangeof a general Al alloy, E = 61.7 73.1 GPa, on the forcedis-placement curve is very small. Table 4 shows the % errorsof the evaluation parameters obtained from the curves. InTable 4, all of the % error values stay within 10% and closeto the volumetric errors of the specimens. Therefore, weselected the elastic modulus of the Al cell wall to beE = 68 GPa, which was also measured by Sugimura et al.(1997).

From these results, we determined the mechanicalproperties of the Al cell wall as E = 68 GPa, re = 31 MPaand n = 8.5. Finally, the 0.2% offset yield stress, rY, wasdetermined as 35.5 MPa by using the stress correspondingto the intersection of the calculated true stress-true straincurve, which was obtained after substituting the deter-mined properties in Eq. (1), and the line parallel to the elas-tic part of the curve offset by a 0.2% strain. In particular,this rY is remarkably smaller than the published values,rY = 120170 MPa (Sugimura et al., 1997; Simone andGibson, 1998b; Andrews et al., 1999), which were obtainedfrom the approximation method. However, the mechanicalproperties determined in this research were veriedthrough the computation for the stressstrain behavior of

3

(cont

I. Jeon et al. /Mechanics of Materials 41 (2009) 6073 69re = 31 MPa. In this case, the congurations of the com-puted forcedisplacement curves of each specimen closelyresemble the measured ones. Therefore, re = 31 MPa isdetermined for the value of the elastic limit stress of theAl cell wall.

The computed forcedisplacement curves of each spec-imen using the same values of n = 8.5 and re = 31 MPa, andthree different elastic moduli, E = 61.7, 68, and 73.1 GPa

Fig. 10a 10 10 10 mm closed-cell Al foam specimen undercompression and were successfully used to analyze theplastic collapse mechanism of the foam material (see Jeonet al., submitted for publication).

Further, we demonstrated that the power-law harden-ing exponent of the cell wall had a signicant effect onthe magnitude of the peak force and the displacement atthe peak force in the forcedisplacement curve, and that

inued)

the elastic limit stress of the cell wall had an important ef-fect on the magnitude of the peak force as well as the elas-tic stiffness. However, the elastic limit stress did not affectthe displacement at the peak force. Moreover, we showedthat the elastic modulus in the range of a general Al alloy,E = 61.7 73.1 GPa did not have any signicant effects onthe forcedisplacement curve.

Using the determined mechanical properties of the cellwall, the compressive behavior of the two foam specimenswas numerically investigated. Fig. 12ac and Fig. 13ac

show the deformation shapes of two specimens with thedistributions of the equivalent plastic strain eld greaterthan 0.2%. Note that the equivalent plastic strain less than0.2% are regarded as elastic strain for clear evaluation ofthe actual plastic strain in the specimen.

Figs. 12a and 13a represent the initial state of the spec-imen A and B. In this stage, no equivalent plastic strainelds were detected in the specimens. Figs. 12b and 13bshow the plastically buckled shapes of the cell wall in thetwo specimens (see the regions within the closed curves)

stress

70 I. Jeon et al. /Mechanics of Materials 41 (2009) 6073Fig. 11. Effects of (a) the power-law hardening exponent, (b) elastic limit and (c) elastic modulus on the forcedisplacement curve of Specimen B.

Fig. 11 (continued)

Table 2% Errors of the evaluation parameters according to the changes in the power-law hardening exponent, n

E (GPa) re (MPa) n dpeak error(%) fpeak error (%) Stiffness (f/d) error(%)

Spec. A Spec.B Spec.A Spec. B Spec. A Spec. B

Cal. #1 61.7 11.16 3 97.97169 94.44165 33.73116 18.29626 46.26194 45.73207Cal. #2 61.7 11.16 5 31.29902 25.9021 33.04793 42.87535 61.50028 60.37315Cal. #3 61.7 11.16 6 26.23276 15.65987 44.07231 51.92538 64.59853 64.85735Cal. #4 61.7 11.16 8.5 6.8692 8.42798 55.82837 62.49337 68.53542 72.39752Cal. #5 61.7 11.16 9 3.39854 9.21198 57.17856 63.6846 68.60759 73.20666

Table 3% Errors of the evaluation parameters according to the changes in the elastic limit stress, re

E (GPa) re (MPa) n dpeak error(%) fpeak error(%) Stiffness (f/d) error(%)

Spec. A Spec. B Spec. A Spec. B Spec. A Spec. B

Cal. #1 61.7 11.16 8.5 6.8692 8.42798 55.82837 62.49337 68.53542 72.39752Cal. #2 61.7 25 8.5 6.49058 8.73957 11.05809 23.88477 36.29876 27.43667Cal. #3 61.7 28 8.5 6.64383 0.40493 22.11082 Cal. #4 61.7 30 8.5 5.24655 8.77475 4.07765 10.73255 13.1409 14.40955Cal. #5 61.7 31 8.5 5.24655 8.78983 7.08129 8.14667 5.40935 7.87728Cal. #6 61.7 32 8.5 6.52664 9.41803 10.04247 5.5482 0.54771 7.63744Cal. #7 61.7 34 8.5 9.59895 0.42946 1.33613

Table 4% Errors of the evaluation parameters according to the changes in the elastic modulus, E

E (GPa) re (MPa) n dpeak error (%) fpeak error(%) Stiffness(f/d) error(%)

Spec. A Spec. B Spec. A Spec. B Spec. A Spec. B

Cal. #1 61.7 31 8.5 5.24655 8.78983 7.08129 8.14667 5.40935 7.87728Cal. #2 68a 31 8.5 8.86144 8.70942 8.51777 7.01161 1.19502 8.66801Cal. #3 73.1 31 8.5 9.86208 9.05619 9.57739 6.14852 0.2301 6.65899

a See (Sugimura et al., 1997).

I. Jeon et al. /Mechanics of Materials 41 (2009) 6073 71

72 I. Jeon et al. /Mechanics of Materials 41 (2009) 6073at the compressive displacements of 0.14 mm for specimenA and 0.155 mm for specimen B, respectively, which arethe deformation state before the peak force in Figs. 10cand 11c. In these gures, localized plastic strain regionsaround the buckled cell wall were found. Finally, Figs.12c and 13c show the plastically collapsed cell walls ofeach specimen (see the regions within the closed curves).These gures were taken at the compressive displacementsof 0.30 mm for specimen A and 0.34 mm for specimen B,respectively, which are the deformation state after thepeak force in Figs. 10c and 11c. At this stage, the fullydeveloped plastic zones were also observed in thespecimens.

7. Conclusions

A new approach was attempted to determine themechanical properties of the cell wall of the closed-cellAl foam, such as the elastic modulus, 0.2% offset yieldstress, and power-law hardening exponent. To set the limitvalues for the mechanical properties of the Al cell wall, theproperties of the cell wall base material was initially mea-sured using a 6 6 6 cm3 base material ingot. Two

Fig. 12. The computed deformation shape of the Specimen A with its equivalent pand (c) the cell wall collapse.5 5 5 mm3 Al foam specimens of completely differentstructures were prepared and used for uniaxial compres-sion tests as well as for direct nite element modeling.Numerical simulations were carried out by using the twomodels and various mechanical properties selected fromthe measured properties of the base material. The Al cellwall mechanical properties were precisely determined bycomparing the computed forcedisplacement curves andthe measured ones.

Based on the results of this research, the mechanicalproperties were determined as E = 68 GPa, rY = 35.5 MPaand n = 8.5. Further, the effects of each mechanical prop-erty on the forcedisplacement curve of the foam materialwere elucidated. An increase in the power-law hardeningexponent of the cell wall rapidly decreased both the mag-nitude of the peak force and the displacement at the peakforce in the forcedisplacement curves. An increase in theelastic limit stress of the cell wall dramatically increasedthe magnitude of the peak force and the elastic stiffnessbut did not change the displacement at the peak force.Also, the elastic modulus of the cell wall in the range of ageneral Al alloy, E = 61.7 73.1 GPa, did not give any sig-nicant effects on the forcedisplacement behavior of thefoam material.

lastic strain eld at the state of (a) the initiation, (b) the cell wall buckling

I. Jeon et al. /Mechanics of MaAcknowledgement

I. Jeon thanks Dr. Tetsuji Miyoshi of Shinko Wire Co.LTD. for supplying the base material ingot of the ALPORAS.This study was supported by National Research Lab pro-gram of the Korea Science & Engineering Foundation(R0A-2006-000-10249-0).

References

Andrews, E., Sanders, W., Gibson, L.J., 1999. Compressive and tensilebehavior of aluminum foams. Mater. Sci. Eng. A 270, 113124.

E 9-89A, 2000. Standard test methods of compression testing of metallicmaterials at room temperature. ASTM, Philadelphia.

Grenestedt, J.L., 1998. Inuence of wavy imperfections in cell walls onelastic stiffness of cellular solids. J. Mech. Phys. Solids 46,2950.

Grenestedt, J.L., Tanaka,K., 1999. Inuenceof cell shapevariationsonelasticstiffness of closed cell cellular solids. Scripta Mater. 40, 7177.

Hasan, M.A., Kim, A., Lee, H.J., 2008. Measuring the cell wall mechanicalproperties of Al-alloy foams using the nanoindentation method.Compos. Struct. 83, 180188.

Jeon, I., Asahina, T., 2005. The effect of structural defects on the compressivebehavior of closed-cell Al foam. Acta Mater. 53, 34153423.

Jeon, I., Asahina, T., Kang, K.J., Im, S., Lu, T.J., submitted for publication.Finite element simulation of the plastic collapse of closed-cellaluminum foams with X-ray computed tomography.

Jeon, I., Im, S., 2001. The role of higher order eigenelds in elasticplasticcracks. J. Mech. Phys. Solids 49, 27892818.

Fig. 13. The computed deformation shape of the Specimen B with its equivalent pand (c) the cell wall collapse.terials 41 (2009) 6073 73Kim, A., Tunvir, K., 2006. Study of Al-alloy foam compressive behaviorbased on instrumented sharp indentation technology. J. Mech. Sci.Technol. 20, 819827.

Lee, Y.H., Lee, B.K., Jeon, I., Kang, K.J., 2007. Wire-woven Bulk Kagome(WBK) truss cores. Acta Mater. 55, 60846094.

Markaki, A.E., Clyne, T.W., 2001. The effect of cell wall microstructure onthe deformation and fracture of aluminum-based foams. Acta Mater.49, 16771686.

McMeeking, R.M., 1977. Finite deformation analysis of crack-tip openingin elasticplastic materials and implications for fracture. J. Mech.Phys. Solids 25, 357381.

Miyoshi, T., Itoh, M., Akiyama, S., Kitahara, A., 2000. ALPORAS aluminumfoam: Production process, properties, and applications. Adv. Eng.Mater. 2, 179183.

Miyoshi, T., Itoh, M., Mukai, T., Kanahashi, H., Kohzu, H., Tanabe, S.,Higashi, K., 1999. Enhancement of energy absorption in a closed-cellaluminum by the modication of cellular structures. Scripta Mater.41, 10551060.

Shin, D.S., Lee, K., Kim, D., 2007. Biomechanical study of lumbar spinewith dynamic stabilization device using nite element method.Comput. Aided Des. 39, 559567.

Simone, A.E., Gibson, L.J., 1998a. The effects of cell face curvature andcorrugations on the stiffness and strength of metallic foams. ActaMater. 46, 39293935.

Simone, A.E., Gibson, L.J., 1998b. Aluminum foams produced by liquid-state processes. Acta Mater. 46, 31093123.

Sone, T., Tamada, T., Jo, Y., Miyoshi, H., Fukunaga, M., 2004. Analysis ofthree-dimensional microarchitecture and degree of mineralization inbone metastases from prostate cancer using synchrotronmicrocomputed tomography. Bone 35, 432438.

Sugimura, Y., Meyer, J., He, M.Y., Bart-Smith, H., Grenstedt, J., Evans, A.G.,1997. On the mechanical performance of closed cell Al alloy foams.Acta Mater. 45, 52455259.

lastic strain eld at the state of (a) the initiation, (b) the cell wall buckling

Cell wall mechanical properties of closed-cell Al foamIntroductionMechanical properties of the cell wall base materialPreparation of two small foam specimensFinite element modeling of the two foam specimensUniaxial compression tests of the two foam specimensResults and discussionConclusionsAcknowledgementReferences