* Corresponding author. Tel.: #1-617-253-2104; fax:
#1-617-253-1962.E-mail address: [email protected] (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