* Corresponding author. Tel.: #1-617-253-2104; fax: #1-617-253-1962.E-mail address: wierz@mit.edu (T. Wierzbicki)International Journal of Impact Engineering 24 (2000) 509}534Experimental and numerical studies of foam-"lled sectionsSigit P. Santosa, Tomasz Wierzbicki*, Arve G. Hanssen', Magnus Langseth'Impact & Crashworthiness Laboratory, Massachusetts Institute of Technology, Room 5-218,77 Massachusetts Avenue, Cambridge, MA-02139, USA'Department of Structural Engineering, The Norwegian University ofScience and Technology,N-7034, Trondheim, NorwayReceived 30 September 1998; received in revised form 29 June 1999AbstractAcomprehensiveexperimental andnumerical studiesof thecrushbehaviorof aluminumfoam-"lledsections undergoing axial compressive loading is performed. Non-linear dynamic "nite element analyses arecarriedouttosimulatequasi-statictestconditions. Thepredictedcrushingforceandfoldformationarefound tobe in good agreement withthe experimental results. Based onthe numerical simulations, simpleclosed-form solution is developed to calculate the mean crushing force of the foam-"lled sections. It is foundthat the increase of mean crushing force of a "lled column has a linear dependence with the foam compressiveresistance and cross-sectional area of the column. The proposed solution is within 8% of the experimentaldata for wide range of column geometries, materials and foam strengths. 2000 Elsevier Science Ltd. Allrights reserved.Keywords: Thin-walled column; Aluminum foam; Axial crush1. IntroductionRecent developments of cost-e!ective processes for the productionof low-densitymetalliccellular material, suchas aluminumfoam, have clearedthe wayfor usingit inlight-weightstructuralmembers. Thisisduetotheuniquecharacteristicsofthecellularmaterialwhichcanundergolargestraindeformationwhilemaintainingitslowstresslevelbeforethedensi"cation,which occurs at the densi"cation strain in the range of 60}90%. One potential application of thistype of material is to reinforce thin-walled prismatic columns in space frame structures. It has been0734-743X/00/$- see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 7 3 4 - 7 4 3 X( 9 9 ) 0 0 0 3 6 - 6NomenclatureE Young's modulusH half-wave folding lengthP instantaneous crushing forceb column widthn strain hardening exponentt column thicknessC'foam strengthening constantEH elastic modulus of foam materialF", F'characteristic load of empty and "lled columnGH foam shear modulusPmean crushing force of empty columnP'mean crushing force for "lled columnP'mean crushing force for bonded-"lled columnAPincrease of the mean crushing forceo instantaneous shortening distancev Poisson's ratio0 rotation angle of superfolding elementc'densi"cation strainjH foam densityjQdensity of solid cell wall of foamo"plastic #ow stresso"ultimate strengtho`yield stresso'crushing strength of foamoH'plastic collapse stress of foamo'''tensile strength of adhesivetH'plastic shear strength of foamt'''shear strength of adhesiveshownthroughnumericalstudiesthatthecrushingcharacteristicsof athin-walledcolumnareimproved dramatically by "lling it with aluminum foam [1].A comprehensive experimental study on the e!ect of "lling thin-walled columns with aluminumfoamwasdonebyHanssenet al. [2,3]. Theyinvestigatedtheaxial crushingbehaviorof thefoam-"lled aluminum extrusion under quasi-static loading condition. They found that signi"cantincreases ofcrushingforcewereobtainedfromthedirectcompressivestrengthofthefoam andfrom the interaction between the foam and wall column. The interaction at the foam}wall interfacedecreases the folding length, and therefore increases the crushing force. For a typical column witha length to width ratio of 3, the non-"lled extrusion formed 5 lobes, while the foam-"lled sectionsformed as many as 9 lobes. Similar experimental results were obtained by Seitzberger et al. [4], ontheaxiallycompressedsteel tubes"lledwithaluminumfoam. Theexperimental resultsonthe510 S.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534foldingmodeof aluminumfoam"llingmethodagreedqualitativelywithearlierresultsonthelow-density polyurethane foam "lling reported by Thornton [5], Lampinen and Jeryan [6], andReidet al. [7]. However, Thorntonetal. [8]summarizedthee!ect of polyurethanefoambyconcluding that even though a considerable increase of collapse load was achieved, thickening ofthe column wall was still more weight e$cient than polyurethane foam "lling.Numerical investigations on the e!ect of aluminumfoam"lling of thin-walled prismatic columnsundergoing axial crushing were recently carried out by Santosaand Wierzbicki [9].In terms ofenergyabsorptionper unit total mass, aluminumfoam"llingwas foundtobepreferabletothickeningof thecolumnwall. Therefore, theenergyabsorptioncharacteristicsof thin-walledcolumns can be improved signi"cantly with aluminum foam "lling. This is due to the prevention ofthe inward fold formations of the thin-walled column by the presence of the foam "ller, leading tolarge plastic membrane deformation and, accordingly, increased energy dissipation. Santosa andWierzbicki have also expanded their numerical analysis to the case of torsion and bending withcross-sectional crushing [10,11]. Inboth cases, numerical analyses showed thataluminum foam"lling reduced the amount of sectional collapse, resulting in the increase of the energy absorption ofthe "lled columns.The objective of this paper is to validate the numerical prediction of the crushing behavior ofaluminum foam-"lled columns using available experimental data. Of interests in the study are theinstantaneous crushing force, the mean crushing force, and the deformation mode of the aluminumfoam-"lledcolumns. Thenumerical studywasconductedat theImpact andCrashworthinessLaboratory, MIT, whiletheexperimentalstudywasconductedattheNorwegianUniversityofScience and Technology. Both studies involve aluminum extrusion and HYDRO aluminum foam"ller. Furthermore, experimentalvalidationisalsoconductedforthecaseofsteelcolumnwithMEPURAaluminumfoam, inwhichthedatawasobtainedfromRef. [4]. Simpleclosed-formsolutions for the mean crushing load are constructed based on the analytical and numerical resultsand compared to the experiments.2. Theoretical predictionTheenergydissipationoffoam-"lled columnsundergoingacrushingprocessdependsonthemembrane and bending energy of the empty column, crushing energy of the foam, and the inter-actionbetween thesetwomechanisms. Theexistenceofthecouplingbetween thefoamandthecolumn in the deformed geometrical parameters posses a complex analytical problem. AbramowiczandWierzbicki [12]developedanapproximatesolutiontotheproblemof axial crushingoffoam-"lledcolumns. Theinteractionwasaccountedforusingthedependencyof thedissipatedenergy on the volumetric strain.Simpleclosed-formsolutionof thecrushingcharacteristicof thefoam-"lledcolumncanbeobtained by assuming that the contribution of the dissipated energy from the compressed foam isindependent from the deformed geometry of the column. This assumption e!ectively decouples thedeformation of the column and the "lling foam. This method was adopted by Reddy and Wall [13]to calculate the crushing response of axially compressed foam-"lled cylindrical tubes. The meancrushingforcePwasthencalculatedbyasimplesumofthecrushingresistanceoftheemptycolumn and the strengthening interaction of the foam "ller.S.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534 511Fig. 1. (a) Deformation pattern of box column, (b) superfolding element.Recently, SantosaandWierzbicki[9]developedaformulaforthecrushingresistanceof thefoam-"lledcolumnbyusingnumerical simulationresults. Similarly, Hanssenet al. [3] alsosuggestedanempirical formulafor the crushingresistance of the foam-"lledcolumns. Bothmethods used the following representation for the mean crushing force of the foam-"lled column:F'"F"#C'f (o", o', ), (1)where F', F" are respectively characteristic load of "lled and empty column, while o" and o' are theplastic #ow stress of column material and the crushing strength of the foam "ller, respectively, andis a geometrical parameter. C'represents the strengtheningconstant, andf (o",o',) is aninteraction function, which is determined from the dimensional analysis to capture the strengthen-ing mechanism. Furthermore, in the case of axial compression, the strengthening interaction can bedivided into two di!erent components, which are the direct uniaxial compressive strength of thefoamand the wall}foam strengthening mechanism. Each contributing term in Eq. (1) is discussed inthe following subsection.2.1. Thin-walled column crushing resistanceSimple closed-form solution for the crushing resistance of empty box columns undergoing axialcrushing adaptedhere is basedona concept of superfolding developedby Wierzbicki andAbramowicz[14]. Thecorrespondingtheoryfor aright angleelement waslater extendedtomulti-cornered, arbitrarily shaped column [15]. Here only the main points of the above theory willbe summarized. For a complete analysis, the reader should refer to the above references.Considerasquareboxcolumnwithcross-sectionof b;bandthicknesst undergoingaxialcrushing as shown in Fig. 1. As depicted in the "gure, the folding modes considered in the analysisis quasi-inextensional axial folding mode. The plastic deformation is localized at a portion of thecolumn and the plastic energy is dissipated through the formation of hinge lines and membraneaction zones. The localized plastic deformation zone is de"ned as the superfolding element, which isalarge"niteelementwithaprescribedknowledgeof thedeformationprocessandonlyafewdegrees of freedom. The superfolding element is characterized by the half-folding element H, which512 S.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534is obtained from the condition of minimum plastic work as [14]H"t`b``. (2)The instantaneous axial crushing force P(0) for small rotation is given byP(0)"P

0.6#0.5120 . (3)Notethat0correspondstotherotationangleof thesuperfoldingelementwithrespecttothevertical axis (see Fig. 1b) and the values varies from 0 to /2 for a complete formation of one fold.The mean crushing force is given byP"13.05o"t``b`, (4)where o" is the #ow stress of the column material. To take strain hardening e!ects into account, theenergy equivalent #ow stress can be calculated by using [16,17]o""

o`o"n#1,(5)where o`and o"are the yield and ultimate strength, and n is the strain hardening exponent of thethin-walled material.2.2. Mean crushing resistance of foam-xlled columnsDuring a complete axial deformation process, the column wall will be progressively crushed. Inthis respect, of interest in the crashworthiness analysis is not the actual load-shortening character-istics, but rather an average crushing resistance P. To asses the structural e$ciency of an energyabsorbingsystem, asinglecharacteristicof themeancrushingresistanceissu$cient. Asimpleclosed-formsolutionfor themeancrushingforceis givenbyEq. (4). Numerically, themeancrushing load can also be de"ned byP"1oB"P(o) do, (6)where P(o)istheinstantaneouscrushingloadcorresponding totheinstantaneousshortening o.The instantaneous crushing load data can be obtained from the experiments or from the numericalsimulations.By using the argument that the energy dissipations in the column and the foam were decoupled,Santosa and Wierzbicki [9] developed a strengthening function of the foam"ller, C'f (o",o',) termin Eq. (1). Based on the numerical simulations, they found that the lateral foam}wall strengtheninginteraction was ofthe sameorder asthe directuniaxial compressive strength ofthe foam. Theyconcludedthat theadditional strengthof thefoam-"lledcolumncouldthenbeapproximatedastwiceof theaxial strengthof thefoam"ller. Forasquareboxcolumnwithacrosssectionb;b, Santosa and Wierzbicki prediction forthe foam-"lled column mean crushing force canbeS.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534 513written asP'"P#2b`o', (7)where Pis the mean crushing force of empty column given in Eq. (4).Hanssen et al. [3] developed the foam strengthening mechanism as two separate terms, i.e. thedirect compressive and the wall}foam interaction terms. The wall}foam strengthening interactionwas obtained by dimensional analysis of the parameters involved in the crushing process, whichwas modeled as C`(o'o"bt, where t is the column thickness, and C`is interaction constant tobe determined from the experiments. The model was then curve-"tted with the experimental dataand gave an excellent agreement if the value of interaction constant set equal to 5. The empiricaladditive model of Hanssen et al. can then be written asP'"P#b`o'#5 b t(o'o", (8)3. Test programFig. 2 shows the test matrix designed for the present experimental investigation. The in#uence ofthree parameters on the energy absorbing behavior was studied, i.e. the density of the foam, thewall thickness of the extrusion and bonding between the foam and extrusion by applying adhesive.Atypical factorial designapproachwasselectedforthis investigation[18]. Threedi!erentdensities of foam were investigated in addition to empty extrusions. For each density of foam, thewall thickness was varied between two levels, namely thin wall and thick wall. In addition, eachwall thickness was tested with and without adhesive. This resulted in a total of four tests that had tobe carried out for each density of foam (Fig. 2). As a reference, tests were also carried out withoutfoam. This program consequently comprised 14 tests, see Fig. 2.3.1. Experimental setupThealuminumextrusionsweremachinedtoalengthof 245mm. Fourteenspecimenswithanaveragecross-sectional geometryof 80;80;1.88mmwereprepared. Sevenof theseweremachinedontheoutsidetoobtainawallthicknessof1.0mm. Theresultinggeometryoftheseseven test specimens was approximately 78;78;1.0 mm, (see Fig. 3). These specimens had sharpcorners, which were di!erent from the original extrusions with rounded edges.The aluminum foam was supplied as three sheets, each manufactured with a di!erent density. Sixspecimens with a geometry of 75;75;245 mm were cut from each sheet and further machined inorder to get an exact "t in the extrusions. This machining required water as a cooling agent, whichsoaked the test specimens. In order to obtain an accurate foam density, the specimens were allowedto dry in an oven at 1003C for 1 h. The subsequent weighing showed average densities of 0.20, 0.35and 0.48g/cm`. Only four foam specimens of each of the three densities were needed for the testprogram and those closest to the average value of each density were selected.Six test specimens required adhesive between foam and the walls of the extrusion. On the basisof previous experience, a two-component epoxy resin was chosen, namely Araldite 2011514 S.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534Fig. 2. Test program and symbol de"nition.(AW106/HV953U). Before the adhesive was applied, boththe inside of the extrusions andtheoutsideof thefoamspecimens werecarefullydegreasedusingacetone. Theadhesivewasappliedevenly tothe inside of the extrusions witha sti!brush. In addition, the adhesivewasappliedtotheoutsideof thefoamspecimensusingapaint roller. Afterthisprocess, thefoamspecimenswerepushedgentlyintotheextrusions. Theadhesivewasallowedtoset forthree days beforetesting. The components containingfoams of the lightest densityrequiredthe largest amount of adhesive; see the appendix for component details. For the densest foam j`,theweightof theadhesiveamountedto5%, forj`10%, andforj16%, comparedtototalcomponent weight.The tests were carried out in a Dartec 500 kN testing machine (accuracy $0.2% of 150 kN). Alltests werequasi-static, startingwitharateof 0.015mm/s tocapturethepeakforces at thebeginning. After this the rate was increased to 0.15 mm/s. The data logging system was running ata constant rate of 1 Hz. The specimens were placed between two 50 mm steel plates in the testingmachine. Fig. 3 gives an overview of the test specimen geometry and experimental setup used in thetests. Therelativedisplacementbetweenthetwosteel plateswasmeasuredusinganinductivedisplacement transducer, accuracy $1% of full scale (100 mm).S.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534 515Fig. 3. Test specimen geometry and experimental set-up.3.2. Material propertiesThe aluminum extrusions used in the tests were made up of alloy AA6060 temper T4. A typicalengineering tensile}stress strain curve of the material, obtained from a specimen taken parallel tothe direction of extrusion, is shown in Fig. 4. The aluminum foam was made up of alloy AlSi8Mg.Fig. 4 shows the engineering stress}strain curve of the foam in compression.4. Finite element modelingThe explicit dynamics non-linear "nite element code PAM CRASH 97 was used to numericallysimulate the axial crushing process of foam-"led columns. The "nite model was created by meshgenerator program HYPERMESH 2.1. The column wall was modeled with a Belytschko-Tsay-4-node thin shell element, while the foam core was modeled with an 8}node solid element. Since thefoam core can undergo a large strain deformation, solid element using selective reduced integrationrule was chosen to avoid volumetric locking. The selective reduced integration rule have 8 integra-tion points for the deviatoric strain part and 1 integration point for the volumetric strain.Intheaxialloadingcondition, thedeformationofthesquareboxcolumnhastwosymmetryplaneswithrespecttoitscrosssection, i.e. X0>and >0Zplanes(Fig. 5). Duetotheexpectedsymmetry of the deformation, only one quarter of the column was modeled to represent the axialcrushing problem. Notriggering imperfection was introduced tothemodel. Clampedboundaryconditions were applied at the bottom of the column, and the symmetry boundary conditions wereapplied on all free vertical edges. There are two sets of geometrical models considered in the presentwork. First, a square aluminum extrusion with cross section b"80 mm, length "245 mm, and516 S.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534Fig. 4. Typical stress}strain diagrams, wall of extrusion and aluminum foam.Fig. 5. Geometric modeling of a foam-"lled column.thickness t"1.88 mm "lled with HYDRO aluminum foam. Secondly, a square steel column withb"40 mm, "150 mm, and t"1.4 mm"lled with MEPURA aluminum foam. The results of thenumerical simulations of the "rst and the second models are compared with the experiments givenin Refs. [3,4], respectively.S.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534 517To simulate the displacement controlled experiment, velocity boundary conditions were appliedonthe topportionor the column. Inthe actual quasi-static experiments, the speedof thecross-beam is usually set to 0.01}1 mm/s [3]. This range of velocity is too slow for the numericalsimulation. This is due to the fact that the explicit time integration method is only conditionallystable, and therefore in general very small time increments have to be used. For the present work,theappliedvelocitywasarti"ciallyspeedupto2m/s. Thevelocitypro"lewasrampedduringthe "rst 50 ms to 2 m/s, and then this velocity was held constant. Quasi-static process still can beachievedwitharti"cial highvelocityprovidedthat theinertial e!ect isminimized, whichcanbe done by scaling the mass density. Two simulation responses need to be checked to verify thequasi-static process is held. First, the total kinetic energy has to be very small compared to the totalinternal energy over the period of the crushing process. Secondly, the crushing force-displacementresponse must be independent from the applied velocity.The strengtheninginteractionbetweenthe foamandthe columnwall was simulatedwitha surface-to-surface sliding contact in the case of unbonded "lling. During progressive formation ofplasticfolds, interpenetrationbetweentwofolds inthecolumnwall was preventedbyusingself-contactinterface. Thestrengtheninge!ectduetofrictionbetweenthewallcolumnandthefoam "llerwasstudied forvariousfriction coe$cients.Furthermore, internalsolidanti-collapsehas to be applied on the solid element. This internal contact can prevent numerical problem thatcan arise when solid elements are heavily compressed and distorted.In the bonding case, the adhesive was modeled with tied contact. Failure due to excessive tensionandshearforcesisallowedfromnodetonode. Onsetoffailurewasgovernedbythefollowingfailure criterion [19]:

oo'''

?#

tt'''

@)1, (9)where o'''and t'''aretensileandshearstrengthoftheadhesivematerial, respectively. Inthenumerical simulation, thevaluesfortensileandshearstrengthweretakentobeo'''"t'''"150MPa, while the exponential constants were taken to be a"b"2, as suggested bySeggewiss [20].5. Material modeling5.1. Thin-walled prismatic columnTheconstitutive behavior ofthethin shellelement forthecolumn material wasbased ontheelastic}plastic material model with Von Mises's isotropic plasticity algorithm. The transverse sheare!ect was considered by this material model. Plastic hardening was based on the polygonal curvede"nition, in which pairs of the plastic tangent modulus and the plastic stress were speci"ed.The aluminum extrusion AA 6060 T4 mechanical properties are: Young's modulusE"6.8210" N/mm`, initialyieldstress o`"80N/mm`, Poisson'sratio v"0.3, andthepowerlawexponentn"0.23. Thesteelcolumn pro"lewasmadeofmildsteelRSt37withmechanicalproperties: Young'smodulusE"210`N/mm`, initial yieldstresso`"251N/mm`, Poisson's518 S.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534Table 1Strain hardening data for AA 6060 T4 and Mild steel RSt37AA 6060 T4 RSt37Plastic plastic plasticstrain (%) stress (Mpa) stress (Mpa)0.0 80 2512.4 115 2644.9 139 2957.4 150 3169.9 158 32612.4 167 33414.9 171 33617.4 173 339ratiov"0.3, andthe power lawexponent n"0.12. The engineering stress}strainfor bothmaterials were given in Table 1. These stress}strain data were adopted from the experiments takenfrom Fig. 4 and Ref. [4].5.2. Aluminum foam coreThealuminumfoammechanical responseincompressionshowsatypicalbehaviorofhighlyporouscellularsolids: aninitialapproximatelylinearelasticregimeisfollowedbyanextendedplastic plateau, truncated by a densi"cation response at high strains during which the stress againincreases steeply (Fig. 6). Based on this characteristic, mechanical behavior of aluminum foam ischaracterized byelasticmodulusEH, plasticcollapsestress oH', shearmodulus GH, plasticshearstrength tH', and densi"cation strain c'. These parameters strongly depend on the aluminum foamdensity jH.PAM CRASH o!ers material modeling for metallic cellular solids capable of undergoing largestrain deformation such as aluminum foam. The mechanical properties of the aluminum foam aresmearedinthreeorthogonal directions(x, x`, x`), seeFig. 6. Inthecaseof aluminumfoammaterial, all three orthogonal directions have the same mechanical behavior, i.e. cubic symmetry.Currently, nointeractionbetweencomponentsof thestresstensorisincorporatedintheyieldcondition.Mechanical properties of aluminum foam for various relative densities are given as follows:EH"E'

jHj' `, (10)GH"38E'

jHj' `, (11)oH'"o"'

jHj' ``, (12)S.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534 519Fig. 6. Material constitutive modeling of aluminum foam.tH'"0.5o"'

jHj' ``, (13)c'"1!1.4jHj', (14)where E',o"',j'are the Young's modulus, plastic #ow stress, and mass density of solid cell wall ofthe foam material, respectively. The plastic collapse stress of the foam given in Eq. (12) is obtainedfrom [21], while the Young's and shear moduli are obtained from [22]. The plastic shear strength istakentobeahalf of theplasticcollapsestress. ThetangentmodulusE'tocapturethestrainhardening of the foam at the plastic stress plateau is assumed to be 2% of the elastic modulus EH foranyfoamdensities. ThereisherearoomforimprovementbymakingE'dependsonthefoamdensityj'andamount of plastic deformation. This canonlybe done once the mechanismresponsible for hardening of the foam is well understood.6. Quasi-static simulationThe explicit solutionmethodis a true dynamic procedure originally developedtomodelhigh-speedimpacteventsinwhichinertiaplaysadominantroleinthesolution. Therefore, ina quasi-static analysis, the goal is to model the process in the shortest time period in which inertialforces remain insigni"cant.Twowaysof achievingaquasi-staticprocessbyusingtheexplicit dynamicsprocedurearepresented in this section. The "rst is scaling down the mass of the material so that the inertial forces520 S.P. Santosa et al. / International Journal of Impact Engineering 24 (2000) 509}534Fig. 7. Quasi-static simulation: (a) energy, (b) crushing force response.will be minimum. However, scaling down the mass results in smaller time step, and therefore theanalysis will take a large number of time increments. The minimum stable time increment in theexplicit dynamic analysis can be expressed asAt" jE,where isthecharacteristicelementlength, EistheYoung'smodulus, andjisthematerialdensity. According to the above equation, scaling down the mass density by factor of f ` decreasesthe time step increment by factor of f. Therefore to limit the large number of time step, the loadingrate is accelerated. In the analysis, the material density is decreased by 1000 times of its originalvalue, whiletheloadingrateisappliedwithvelocityof