Unit Cell Analysis of the Superelastic Behavior of Open-Cell Tetrakaidecahedral Shape Memory ...

12 Pages·2015·3.31 MB·English

Hindawi Publishing Corporation
Smart Materials Research
Volume 2014, Article ID 870649, 11 pages
http://dx.doi.org/10.1155/2014/870649
Research Article
Unit Cell Analysis of the Superelastic Behavior of
Open-Cell Tetrakaidecahedral Shape Memory Alloy Foam
under Quasi-Static Loading
GuillaumeMaîtrejean,1,2PatrickTerriault,1DiegoDevísCapilla,1andVladimirBrailovski1
1DepartmentofMechanicalEngineering,E´coledetechnologiesupe´rieure,Montre´al,QC,CanadaH3C1K3
2LaboratoireRhe´ologieetProce´de´s,363ruedelaChimie-BaˆtimentB,DomaineUniversitaire,BP53,38041GrenobleCedex9,France
CorrespondenceshouldbeaddressedtoGuillaumeMaˆıtrejean;[email protected]
Received16January2014;Revised2April2014;Accepted18April2014;Published8July2014
AcademicEditor:OutiSo¨derberg
Copyright©2014GuillaumeMaˆıtrejeanetal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttribution
License,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperly
cited.
Cellularsolidmaterialsand,morespecifically,foamsareincreasinglycommoninmanyindustrialapplicationsduetotheirattractive
characteristics.Thetetrakaidecahedralfoammicrostructure,whichcanbeobservedinmanytypesoffoams,isstudiedinthepresent
workinassociationwithshapememoryalloys(SMA)material.SMAfoamsareofparticularinterestastheyassociateboththe
shapememoryeffectandthesuperelasticitywiththecharacteristicsoffoam.AUnitCellFiniteElementMethodapproachisused,
anapproachthatallowsaccuratepredictingofthemacroscaleresponseofthefoamwithahighlyreducednumericaleffort.The
tetrakaidecahedralfoam’sresponses,bothintheelasticandinthesuperelasticstages,arethenextractedandcomparedwithresults
fromtheliterature.ThetetrakaidecahedralgeometryisfoundtobeofparticularinterestwhenassociatedwithSMAasittakesmore
advantageofthesuperelasticpropertyofthematerialthanfoamswithrandomlydistributedporosity.
1.Introduction Tetrakaidecahedralcellularsolidsareofinteresttomany
researchers,sincetheywereprovenasfarbackas1887byLord
Shape memory alloy materials are of major interest in a
Kelvin[8]thattheyarethemostefficientmeanstofillspace
broadrangeofapplicationsfortheiruncommonmechanical
withinagivenvolumewithasingleshape;thatis,theyarethe
properties—shape memory effect and superelasticity [1, 2].
structure that provide the minimum surface area per given
Overthelastdecade,numerousstudieshavefocusedonthe
volume [7]. The tetrakaidecahedron, also called the Kelvin
porous form of SMA, which combines the intrinsic assets
cellortheregulartruncatedoctahedron,ischaracterizedby
of both SMA and foam materials. From an experimental
14faces,eightregularhexagonsandsixsquares,andcanbe
pointofview,SMAfoamshavebeenprocessedusingdifferent
generatedbytruncatingthecornersofacube[9].Asdepicted
manufacturingtechniquesandanumberofresearchpapers
in Figure1, a number of foams exhibit geometry which
havebeenpublishedinvariousresearchfields,especiallyfor
canbeaccuratelymodeledusingatetrakaidecahedron-based
biomedicalapplications(see,e.g.,[3,4]or,morerecently,[5]).
model.Eventhoughsuchfoamsnormallypossessaveryhigh
Fromanumericalpointofview,intensiveresearchhasbeen
conductedonboththedeterminationofaccurateconstitutive porosity,itisnowpossibletoobtain,usingmodernadditive
modelsofdenseSMA(areviewofsomemodelscanbefound manufacturingmethods,tetrakaidecahedralgeometry-based
in [6]) and the SMA foam modelling. Several studies have foamswithporosityranginginverylargelimits,thatis,from
shown the importance of the foam’s geometrical character- 1to99%ofporosity[10,11].
istics,thatis,thesize,shape,orientation,anddistributionof Forthesemultiplereasons,thetetrakaidecahedralgeom-
theporemicrostructureonitsmacroscalebehavior[7]. etryoffoammodelshasbeenwidelystudiedintheliterature
2 SmartMaterialsResearch
(a) (b)
Figure1:Tetrakaidecahedral-likemetallicfoam,from[12](a)andfrom[13](b).
fromtheanalytical[9,14,15],experimental[12],andnumer- Table1:MaterialparametersfordenseSMAconstitutivemodel.
ical[16–18]perspectives.
Materialparameters Values
In this study, we focus on the mechanical responses of
tetrakaidecahedralopen-cellSMAfoamwithadensityratio 𝐸𝐴(GPa) 50
𝜌/𝜌𝑑rangingfrom0.01to1,where𝜌denotesthedensityand ] 0.3
thesubscriptd isthefullydensevalue.Notethattheterms 𝜎𝐴𝑀(MPa) 150
𝑠
porosity,densityratio,andporevolumefractionaresynonyms 𝜎𝐴𝑀(MPa) 200
𝑓
andwillalternativelyandindistinctlybeusedinthiswork. 𝜎𝑀𝐴(MPa) 100
𝑠
In particular, we aim to study the influence of the 𝜎𝑀𝐴(MPa) 50
𝑓
superelastic effect associated with open-cell tetrakaideca-
hedral foam. For this purpose, we define the model of 𝜀𝐿 0.01
tetrakaidecahedral foam from both a geometrical and a
numericalpointofviewinSection2.Thetetrakaidecahedral
ThefollowingconstantsareusedtomodeltheSMAbehavior
foamresponsesintheelasticandinthesuperelasticstageare
andtheyaregraphicallyrepresentedin[21]:
extracted in Section3 and compared with results from the
literature.WedrawourconclusionsinSection4. (i)𝜎𝐴𝑀isthestartingstressforthephasetransformation
𝑠
fromAustenitetoMartensite;
2.ComputationalSetup (ii)𝜎𝐴𝑀 is the final stress for the phase transformation
𝑓
fromAustenitetoMartensite;
2.1. SMA ConstitutiveModel. Allthecomputationsareper-
formedusingthecommercialfiniteelementsoftwareAnsys. (iii)𝜎𝑀𝐴isthestartingstressforthephasetransformation
𝑠
The SMA model has already been implemented [19]. The fromMartensitetoAustenite;
modelisbrieflysummarizedbelow; theinterestedreaderis (iv)𝜎𝑀𝐴 is the final stress for the phase transformation
invited to refer to [20] and the references therein for more 𝑓
fromMartensitetoAustenite;
information.
The material model considers the nonlinear behavior (v)𝜀𝐿isthemaximumsuperelasticstrain;
occurring during the reversible phase transformation from (vi)𝐸𝐴istheYoung’smodulusofAustenite;
Austenite to Martensite or from Martensite to Austenite.
For the sake of clarity, we adopt the following convention (vii)𝐸𝑀istheYoung’smodulusofMartensite;
concerning exponents or subscripts: 𝐴,𝑀, 𝐴𝑀, and 𝑀𝐴 (viii)𝐸𝐴𝑀isthephasetransformationmodulusofaustenite
will refer to Austenite, Martensite, Austenite to Martensite toMartensite;
and Martensite to Austenite phase transformations, respec- (ix)𝐸𝑀𝐴isthephasetransformationmodulusofMarten-
tively. The Auricchio’s model is able to model the tension-
sitetoAustenite.
compressionasymmetrybehaviorbut,asafirstapproxima-
tion, we do not account for such asymmetry in the present Throughout this paper, the porous SMA material will be
study.Itisworthpointingout,however,thattheasymmetry definedbytheparameterssummarizedinTable1.Theasso-
behaviorhasagreatinfluenceonthemechanicalresponseof ciatedstress-strainresponseforfullydensefoamisdepicted
themodelandthereadermustkeepinmindtheassumption inFigure2.
ofasymmetricmechanicalbehavior. Note that to reduce the number of iterations required
Itisassumedthatthematerialbehaviordoesnotshowany for a complete phase transformation, 𝜀𝐿 is set to 0.01,
permanentstrainandthatthematerialisperfectlyisotropic. which is approximately 1/10 of the Ti-Ni superelastic range.
SmartMaterialsResearch 3
the macroscale behaviour of the foam, provided that the
250
specific boundary conditions are applied (addressed in the
next section). This approach, the so-called Unit Cell Finite
200 Element Method (UCFEM), has already been widely used
[27, 28] and makes it possible to circumvent the huge
Pa) 150 numericalweightexhibitedbyafullfoammodel.
M The tetrakaidecahedral cell, that is, the UCFEM model
ess ( associated with tetrakaidecahedral foam, is obtained by
Str 100 joining several tetrakaidecahedra together and performing
appropriatesectioning.Thefinalcell,depictedinFigure4,is
verysimilartotheoneobtainedbyZhuetal.[9]andSullivan
50
et al. [15], who performed reference works on tetrakaidec-
ahedral cells via an analytical approach. Note that this cell
0 possessessixsymmetryplanesand,consequently,thestress
0 0.005 0.01 0.015 and the strain fields can be extended for the whole porous
Strain SMA.Thestressisthentheratioofthetotalappliedforceto
theareaoftheunitcellboundary.Thestrainalongtheloading
Figure2:Stress-straincurveobtainedfor0.015maximumstrain.
pathistheratioofthedisplacementintheloadingdirection
oftheunitcelltotheinitiallengthoftheunitcell.
However, contrary to Zhu et al. [9] and Sullivan et al.
This scaling down does not affect the validity of the results
[15],whomodeledthecellusingbeamelements(whichlimit
obtained.
the validity of the model for porosity less than 0.05), the
NoteabouttheModel.TheAuricchiomodelasimplemented presentfiniteelementmodelismeshedusingbrickelements
in Ansys software may be considered somewhat weak, as (20 nodes) exhibiting a quadratic displacement behavior. A
moresophisticatedmodelshavebeenpublishedbydifferent widerrangeofporosityvariationcanthenbeaddressed.
research groups [23, 24] since its appearance, even by the
Auricchio’s own research group [25]. However, this model
2.2.2. Boundary Conditions. The boundary conditions are
presentstwofundamentalassetswhichmakeitverysuitable
derivedfromtheconsiderationofthesymmetryconditions
forthepurposeofthepresentwork.
onthesurfacesoftheboxenclosingtheunitcell.Following
the approach of [29], the reflection symmetries of the unit
(i)Firstly, the model focuses on the superelastic be-
cell require that the shear stress vanishes at all points of
haviour of the SMA using only a few constitutive
theenclosingboxandthatthenormaldisplacementsonthe
parameters,andithasbeensuccessfullyvalidatedin
surfaces are constant. A local coordinate system (𝑥⃗,𝑦,⃗ 𝑧⃗) is
anumberofstudies.
then defined by the 𝑥, 𝑦, 𝑧 axes of the edges of the unit
(ii)Secondly, its direct implementation in the commer-
cell box (Figure4), corresponding to the global coordinate
cial software Ansys precludes the need to apply the system(𝑥𝑔,𝑦𝑔,𝑧𝑔)witha𝜋/4rotationaroundthe𝑧-axis.
USER Material Subroutine (USERMAT), thereby Thedisplacements(𝑢,V,𝑤)inthislocalcoordinatesystem
allowing the reader to easily verify the results ob- aregivenalong(𝑥⃗,𝑦,⃗ 𝑧⃗),respectively.Inthisframework,the
tainedinthiswork.
displacementboundaryconditionscanbeexpressedby
Insum,inlightoftheseassets,thismodelhasbeensuccess-
𝑢(0,𝑦,𝑧)=V(𝑥,0,𝑧)=𝑤(𝑥,𝑦,0)=0,
fullyappliedin[21]aswellasin[26]andtheuseofamore
sophisticatedmodelinthisworkisnotjustified.
𝑢(2𝑎,𝑦,𝑧)=𝑢, V(𝑥,2𝑎,𝑧)=V,
(1)
2.2.UnitCellFiniteElementMethodApproach 𝑤(𝑥,𝑦,−√2𝑎)=𝑤
2.2.1.GeometricalAspects. Thegeometricalmodeladdressed
andtheshearstressconditionsby
in this work is based on an equisided tetrakaidecahedron
which is defined by the side length 𝑎. The open-cell geom-
𝜎 (0,𝑦,𝑧)=𝜎 (𝑥,0,𝑧)=𝜎 (2𝑎,𝑦,𝑧)
etry is obtained from a fully dense tetrakaidecahedron by 𝑥𝑦 𝑥𝑦 𝑥𝑦
removingasimilar,butsmaller,tetrakaidecahedronvolume =𝜎 (𝑥,2𝑎,𝑧)=0,
𝑥𝑦
fromtheinnerspaceandbyopeningeachofthe14faces.The
final tetrakaidecahedral geometry is depicted in Figure3 in 𝜎 (0,𝑦,𝑧)=𝜎 (𝑥,𝑦,0)=𝜎 (2𝑎,𝑦,𝑧)
theglobalcoordinatesystem(𝑋𝑔⃗ ,𝑌𝑔⃗,𝑍𝑔⃗ )andtheassociated 𝑥𝑧 𝑥𝑧 𝑥𝑧 (2)
tetrakaidecahedralfoamisobtainedbyrepeatingthisgeom- =𝜎 (𝑥,𝑦,−√2𝑎)=0,
𝑥𝑧
etrytofillspace.
Due to the periodicity and symmetry exhibited by this 𝜎𝑦𝑧(𝑥,0,𝑧)=𝜎𝑦𝑧(𝑥,𝑦,0)=𝜎𝑦𝑧(𝑥,2𝑎,𝑧)
foam, there is no need to reproduce a complete foam
=𝜎 (𝑥,𝑦,−√2𝑎)=0,
structure, since only a small part of it is needed to capture 𝑦𝑧
4 SmartMaterialsResearch
a
2√2a
Z
Z g
g
Yg Yg
X
g
Xg 3a
(a) (b)
Figure3:Equisidedtetrakaidecahedron.
z
z
x
y
w=w=c
u=0
√2a
(cid:3)=(cid:3)
Z (cid:3)=0
g
y
u=u
2a
Y
2a X g w=0
g
x
Figure4:UCFEMmodelusedforstressandstraincalculations.
Figure 5: Schematic boundary conditions where (𝑢,V,𝑤) are the
displacements, denotesthemeanvalue,and𝑐isahomogeneous
displacementoftheupperface.
where denotesthemeanvalue,whichimposesthatallthe
materialpointsontherespectiveunitcellboundarieshavethe
samenormaldisplacement.
Theimplementationofthedisplacementboundarycon-
ditions stated in (1) in Ansys software is straightforward,
usingcoupleddegreesoffreedom.Theshearstressboundary Y
conditions given by (2), however, cannot be explicitly set
X
in Ansys, but are implicitly defined using the kinematic Z
boundary condition-defined (1). Figure5 is a schematic
illustrationoftheseboundaryconditions.
Finally, the finite element model of the Unit Cell
is meshed using 20-nodes, mixed tetrahedral hexahedral,
quadraticelementsasdepictedFigure6.
Thisfiniteelementmodelpresentsgeometricsingularities
thatmayreducethestresscalculationreliability.Toremove
or,atleast,tolimittheserisks,itisgenerallyadvisedtoround
the edges and fillet the corners. However, that approach
would lead to a more complex geometry, higher number
of elements, and thus higher computational cost. For these
reasons,themaximumstressesobtainedwithourmodelare
definitelyoverestimated.
2.2.3. Simplifications. When experiencing high strains, tet-
rakaidecahedralfoamissubjectedtohighlynonlineareffects Figure6:Meshofthefiniteelementmodelforthe𝜌/𝜌𝑑 =0.1unit
such as cracks or nonlinear buckling that may eventually cellporosity.
SmartMaterialsResearch 5
Y Y Y
Z X Z X Z X
MX
MN MX
MX
MN MN
4 3 6 1 3 7 9 4 7 7 3 9 2 4 6 8 5 2 7 9 4 2 5 1 7 3 8 4 5 6
9 2 1 3 0 9 6 8 7 0 7 8 0 1 2 3 0 6 8 2 3 9 7 7 6 6 5 5 7 4
43.92 160.7 277.5 394. 511.1 627.8 744. 861.4 978.2 1095. 15.57 138.9 262.4 385.8 509.2 632.6 756. 879.4 1002. 1126. 1.983 54.57 107.1 159.7 212.3 264.9 317.5 370.1 422. 475.3
𝜌 𝜌 𝜌
=0.3 =0.5 =0.9
𝜌 𝜌 𝜌
d d d
(a) (b) (c)
Figure7:IllustrationofmicroscaleresponsewiththeVon-Misesstressofaunitcellincompression(𝜀=0.015)duringtheloadingstagefor
differentporosities.
result either in a collapse or a destruction of the cell under that martensitic transformation will never be completed in
compressive or tensile loading, respectively. However, the porousSMAs.
present work aims at studying the superelastic response of
tetrakaidecahedral foam and, for this reason, only linear- 3.2. Macroscale Response. Stress-strain curves from both
elasticandsuperelasticbehavioursareaddressedhere. tension and compression tests are depicted in Figure9 for
variousporosities.Wecanobservethelargerangeofstresses
exhibited by the samples of different porosity (more than
3.Results:SuperelasticBehaviour
three orders of magnitude). It has been shown for random
foam[21,22]that,forhighporosityfraction,afullmartensitic
Themechanicalresponseoftetrakaidecahedralfoamcanbe
transformation can never be reached, even at very large
observedeitheratamicro-oramacroscale,thatis,bylooking
strain. To confirm this conclusion, the maximum applied
attheresponseatthecellscaleorattheoverallsamplescale.
Mostoftheexperimentalresultsthatcanbefoundintheliter- macroscalestrainissetto0.03,whichisthreetimesthe𝜀𝐿(see
aturearenaturallylimitedtothemacroscalegivendifficulties Table1).Asexpected,from𝜌/𝜌𝑑 > 0.3tohigherporosities,
𝜎𝐴𝑀 and 𝜎𝑀𝐴 stresses are never reached. Furthermore in
obtainingthisinformationatthemicroscale.Fortunately,we 𝑓 𝑠
canillustratebothscalesbyusingnumericalapproaches,and this work, no difference is assumed between compression
weaddressbothperspectivesinthefollowingsubsections. andtensionintheAnsysSMAconstitutivemodel.Thevery
closeresponsesbetweenthecompressionandtensiontestsof
Figure9indicatethatthebehaviorofallthefoamstestedis
3.1.MicroscaleResponse. Figure7illustratesthestressstatein
notaffectedbytheloaddirection,and,therefore,onlytensile
aunitcellforamaximumcompressivestrainof𝜀=0.015.
resultscouldbeaddressedhereafter.
Asonecanexpect,weobservehighstressheterogeneity
Figure10depictstheevolutionoftheconstitutivemodel
with stress concentration due to the sharp edges of the parameters ratio of, respectively, 𝐸𝐴, 𝐸𝐴𝑀, 𝐸𝑀, 𝐸𝑀𝐴, and
geometry. 𝜎𝐴𝑀 and 𝜎𝑀𝐴 as functions of porosity. By “ratio” of any
𝑠 𝑓
In order to see how the phase transformation evolves
parameter we denote the apparent value corresponding
during loading, we depict, in Figures 8(a) and 8(b), the
to a given porosity divided by the fully dense value. We
volume fraction of material having an equivalent stress 𝜎
eq observethattheratiosarealmostidenticalforcriticalstresses
greaterthan𝜎𝑠𝐴𝑀and𝜎𝑓𝐴𝑀asfunctionsofappliedstrain.In (Figure10(a)),aswellasforthemoduli(Figure10(b)),even
bothfigures,wecanobservethatforallporosityfoams,the if𝐸𝑀showsaslightoffset.
higher the density, the higher the volume fractions of the The figures are scaled logarithmically, meaning that the
material undergoing phase transformation and being fully slopeoftheplotscorrespondstotheexponent𝑛ofapower
transformed, respectively, in Figure8(a) and in Figure8(b) lawfunctionequation𝑦 = 𝑐𝑥𝑛.Asobservedexperimentally
(whichcorrespondstoavolumefractionof1).Theseobser- in[30]andnumericallyin[16],theexponent𝑛isnotconstant
vationssupportthosemadeforrandomSMAfoams[21,22] over the whole porosity fraction range. We observe three
6 SmartMaterialsResearch
1 1
0.8 0.8
M M
A𝜎s A𝜎f
> >
q q
𝜎e0.6 𝜎e0.6
n n
o o
cti cti
a a
e fr e fr
m m
u 0.4 u 0.4
ol ol
v v
ge ge
a a
Aver 0.2 Aver 0.2
0 0
0 0.005 0.01 0.015 0 0.005 0.01 0.015
Strain Strain
Porosity:
Porosity: 0% 70%
0% 70% 10% 90%
10% 90% 30% 99%
30% 99% 50%
50%
(a) (b)
Figure8:Volumefractionofthemodelsexhibitinganequivalentstress𝜎 greaterthanthephasetransformationstress𝜎𝐴𝑀(a)andgreater
eq 𝑠
thanthephasetransformationstress𝜎𝐴𝑀(b)asafunctionoftheappliedstrain.
𝑓
different regions, corresponding to high porosity (𝜌/𝜌𝑑 < (3)Finally,forhighporosities(Figure11(c)),weobserve
0.25),mediumporosity(0.25<𝜌/𝜌𝑑 <0.8),andlowporosity two different trends. On one hand, the fractions of
(𝜌/𝜌𝑑 >0.8)fractions. fullytransformedmaterialarealmostequalforboth
geometries. On the other hand, the volume fraction
of material undergoing 𝐴 → 𝑀 transformation is
4.Discussion
farmoresignificantforthetetrakaidecahedralfoam,
butonlywhenstrainexceeds0.005.
4.1. Superelastic Response. We compare, in Figure11, the
volumefractionsofthematerialundergoingphasetransfor-
Note that these conclusions are deemed valid despite the
mation (from Figure8) with the ones obtained for random
slight differences in the threshold phase transformation
foam of same porosity [21]. Significant differences between strainsusedintheseworks: 𝜀𝑀𝐴 = 0.003and𝜀𝑀𝐴 = 0.013
theregularandrandomfoams’susceptibilitytophasetrans- 𝑠 𝑓
formationcanbeobservedasfollows. (thiswork)and𝜀𝑠𝑀𝐴 =0.0025and𝜀𝑓𝑀𝐴 =0.0125[21].
The differences revealed in the results indicate that
(1)Forlowporosities(Figure11(a)),thevolumefraction tetrakaidecahedral foam will exhibit better superelastic
of material for which the phase transformation is behaviourthanrandomfoam,inaccordancewiththefollow-
initiated (𝜎 > 𝜎𝐴𝑀) for tetrakaidecahedral foam ingobservations.
eq 𝑠
is far less notable than for random foam. The same
(i)For low-porosity (Figure11(a)), the periodic distri-
trendisalsoobservedforthevolumefractionoffully
transformedmaterial(𝜎 >𝜎𝐴𝑀). bution and the pore shape regularity of tetrakaidec-
eq 𝑓 ahedral foams lead to a decrease of the material
volume fraction that is fully transformed. This is an
(2)For moderate porosities (Figure11(b)), the vol-
interestingfeature,sincealesserproportionofsucha
ume fraction of material, under transformation or
materialwillultimatelyreachtheyieldstress,generate
fully transformed, follows a similar trend until a
irreversiblestrains,andleadtofracture.
macroscale strain of 0.0075. For larger strains, the
phasetransformationinthetetrakaidecahedralfoam (ii)For high-porosity (Figure11(c)), the periodic pore
islessthanintherandomfoam. structure tends to drastically increase the amount
SmartMaterialsResearch 7
600 300
400 200
Pa) 200 Pa) 100
M M
Stress ( −2000 Stress ( −1000
−400 −200
−600 −300
−0.03 −0.02 −0.01 0 0.01 0.02 0.03 −0.03 −0.02 −0.01 0 0.01 0.02 0.03
Strain Strain
Traction Compression Traction Compression
(a) 𝜌/𝜌𝑑=0.9 (b) 𝜌/𝜌𝑑=0.7
30
100
20
50 10
a) a)
P P
ess (M 0 ess (M 0
Str Str −10
−50
−20
−100
−30
−150
−0.03 −0.02 −0.01 0 0.01 0.02 0.03 −0.03 −0.02 −0.01 0 0.01 0.02 0.03
Strain Strain
Traction Compression Traction Compression
(c) 𝜌/𝜌𝑑=0.5 (d) 𝜌/𝜌𝑑=0.3
3
2
1
a)
P
M
0
ess (
Str −1
−2
−3
−4
−0.03 −0.02 −0.01 0 0.01 0.02 0.03
Strain
Traction Compression
(e) 𝜌/𝜌𝑑=0.1
Figure9:Macroscalestress-strainresponsesforseveralporosities.
8 SmartMaterialsResearch
100
100
10−1 10−1
d
Ed10−2 𝜎/𝜎 10−2
/
E
10−3
10−3
10−4
10−2 10−1 100 10−2 10−1 100
𝜌/𝜌
𝜌/𝜌 d
d
EA EM 𝜎AsM
EAM EMA 𝜎MfA
(a) Elasticmoduli (b) Criticalstresses
Figure10:Evolutionoftheconstitutivemodelparametersasafunctionofporosityfraction:(a)𝐸𝐴,𝐸𝐴𝑀,𝐸𝑀,and𝐸𝑀𝐴;(b)𝜎𝑠𝐴𝑀and𝜎𝑓𝑀𝐴.
of material undergoing phase transformation, while Finally,theresultsfrom[21],whichdepicttheevolution
keepingthecompletelytransformedmaterialfraction of the elastic moduli for SMA foams with random pores
unchanged. With a larger proportion of the mate- (randomshape,size,anddistribution),produceatrendclose
rial participating in the phase transformation, the tothatofGibsonandAshby,withaslightoffsettothepresent
macroscopic strain is better distributed throughout results.
theentiresample.
4.3. Stress Concentration Factor. Following the definition of
4.2. Apparent Elastic Modulus. In the global coordinate the stress concentration factor introduced by Panico and
system in which the tetrakaidecahedral geometry has been Brinson [22] in the apparent elastic regime as being the
defined,themodelisisotropicalongtheprincipalaxes.The ratioofthemaximumaxialstresscomponenttotheapplied
apparent elastic modulus ratio of the unit cell, that is, the macroscopicstress,Figure13depictsthestressconcentration
ratio of the apparent elastic modulus of the unit cell and factorfortetrakaidecahedroncellandrandomfoams[22].
ofthefullydensematerial,isdepictedinFigure12forboth
As expected, the concentration factor obtained in the
tractionandcompressiontestsatdifferentporosityfractions
present work follows the same trend as the factor from
andcomparedtoresultsfromtheliterature.
random foams: the higher the porosity, the higher the
Wefirstobservethatthecompressionandtensionresults
concentrationfactor.However,forthelow-porosityfraction,
are perfectly superimposed, meaning that no difference
thestressconcentrationfactorofthetetrakaidecahedralcell
emerges between these two loadings as far as it concerns
is lower (Figure13 is a log-log plot) than the factors from
apparentelasticmodulus.
[22],whichindicatesthattheperiodicfoamstudiedherewill
Next,whencomparingwiththewell-knownGibsonand
exhibit an increased fatigue strength than Panico’s random
Ashby cellular solid equation of the evolution of the elastic
modulus, that is, 𝐸/𝐸𝑑 = 𝐶(𝜌/𝜌𝑑)2 (with 𝐶 = 1 generally foams.
With the geometrical models used in this work, a large
used),weobservethattheevolutionisparalleltothatofthe
present work from very high (𝜌/𝜌𝑑 = 0.01) to moderately amount of sharp edges are obtained. Local stress concen-
low (𝜌/𝜌𝑑 = 0.7) porosities. For lower porosities, the trations will then be inevitably generated, but it is not
tetrakaidecahedral elastic modulus ratio converges to the obvious that a significant influence would be observed at
fully dense value fraction, 𝐸/𝐸𝑑 = 1. The analytical results the macroscale level. To address this concern, other similar
from [9] for tetrakaidecahedral foams of high porosity, simulations should be carried out by rounding the sharp
based on beam theory, are in very good agreement with edges or by building the tetrakaidecahedral unit cell with
our tetrakaidecahedral cells modeled using circular beam circularextrusions.
elements.Theelasticmoduliofrandomporescalculatedby
RobertsandGarboczi[16]areinperfectagreementwiththe 4.4. Linear Buckling. The analysis of linear buckling in tet-
GibsonandAhsby’spredictions,and,thus,theirevolutionis rakaidecahedralfoamcellshasbeenthoroughlyinvestigated
paralleltothatofthetetrakaidecahedralfoam. in[14],basedonthebeamtheorydevelopedbyTimoshenko
SmartMaterialsResearch 9
0.7 0.9
0.8
0.6
0.7
n 0.5 n
o o 0.6
me fracti 0.4 me fracti 0.5
Volu 0.3 Volu 0.4
0.3
0.2
0.2
0.1
0.1
0 0
0 0.005 0.01 0.015 0 0.005 0.01 0.015
Strain Strain
𝜎eq>𝜎AsM, Maˆıtrejean et al. 𝜎eq>𝜎AsM, present work 𝜎eq=𝜎AsM, Maˆıtrejean et al. 𝜎eq=𝜎AsM, present work
𝜎eq>𝜎AfM, Maˆıtrejean et al. 𝜎eq>𝜎AfM, present work 𝜎eq=𝜎AfM, Maˆıtrejean et al. 𝜎eq=𝜎AfM, present work
(a) PVF=0.3 (b) PVF=0.5
0.5
0.4
n
o
cti
a 0.3
r
e f
m
u
Vol 0.2
0.1
0
0 0.005 0.01 0.015
Strain
𝜎eq>𝜎AsM, Maˆıtrejean et al. 𝜎eq>𝜎AsM, present work
𝜎eq>𝜎AfM, Maˆıtrejean et al. 𝜎eq>𝜎AfM, present work
(c) PVF=0.7
Figure11:Averagevolumefractionofphasesinthemodelasafunctionofstrainforthreedifferentporevolumefractions:(a)PVF = 0.3,
(b)PVF = 0.5,and(c)PVF = 0.7inhighly-randomporefoam,from[21],andintetrakaidecahedralfoam.Thevolumefractionthatis
experiencingphasetransformationishighlighted.
andGere[31].Itisassumedthattheedgesofthegeometryare InFigure14,bycomparingthepredictedonsetbucklingstress
longthinbeams;thatis,thattherelativedensityofthefoamis from (3) with the phase transformation stress 𝜎𝐴𝑀 of the
𝑠
low.Zhuetal.[14]showedthat,forpolymericfoamexhibiting
material, we can see that this geometry is not subjected
viscoplastic behaviour, buckling was unlikely to happen. A
to linear buckling for all the density fractions, as the yield
more general approach by Gibson and Ashby [7], based on
stress reached is at least one order of magnitude lower.
experimentalelasticcollapse,givestheonsetbucklingstress
Furthermore, in light of these results, it is more than likely
asfollows:
that the tetrakaidecahedral foam will experience plastic
𝜌 2 𝜌 1/2 2
𝜎∗ ≈0.03𝐸( ) (1+( ) ) . (3) deformation and cracking before being subjected to any
el 𝜌 𝜌 nonlinearbucklingeffect.
𝑑 𝑑
10 SmartMaterialsResearch
100
10−1
102
10−2 Pa)
d M
E
E/ (𝜎
10−3
100
10−4
10−2 10−1 100
10−2 10−1 100 𝜌/𝜌
d
𝜌/𝜌
d 𝜎∗ Ashby and Gibson
el
Present work-tension Zhu et al.
𝜎AMTetrakaidecahedron
Present work-compression Roberts et al. s
Gibson and Ashby Maîtrejean et al.
Figure14:Evolutionofboththeonsetbucklingstressandtheyield
Figure 12: Comparison of the evolution of the apparent elastic stress𝜎𝑠𝐴𝑀asafunctionofthedensityratio.
modulusbetweenthepresentwork(compressionandtension)and
resultsfrom[7,9,16,21].
as well as in the superelastic regime for the whole range of
porosity.
104
Furthermore, it was observed that, due to the regular
spatialdistributionandshapeoftheporesinthetetrakaidec-
ahedral foam, we can expect random foam to be stiffer
or 103
ct thanperiodicfoamwithmoderately-to-lowporosity.Onthe
a
n f contrary,forhighporosities,wewouldexpectperiodicfoam
o
ati tobestifferthanrandomfoam.
ntr 102 Finally,wehaveseen,bothintheelasticregimethrough
e
nc the stress factor and in the inelastic regime, by comparing
o
ress c 101 ttehteravkoaluidmeceafhreadctriaolngeoofmphetarsyeptrraonvsidfoersmanedombvaioteursiasly,ntheargtetthice
St
effectwhencoupledwithsuperelasticity.Forexample,when
comparing regular and random foams in the 0.5 to 0.99
100 porosity range, both structures show similar quantity of
10−2 10−1 100 the completely transformed material, while the fraction of
𝜌/𝜌0 material undergoing martensitic transformation is much
largerintheregularthanintherandomfoams,makingthe
Present work
Panico and Brinson loadsharingmuchmoreefficient.
Figure13:Stressconcentrationfactorasafunctionofporosityfor
bothtetrakaidecahedralandrandomfoamfrom[22]. ConflictofInterests
The authors declare that there is no conflict of interests
regardingthepublicationofthispaper.
5.Conclusion
This work was designed to gain a deeper insight of the References
mechanical response of superelastic SMA foam based on
open-celltetrakaidecahedralgeometry.UtilizingtheUCFEM [1] V.Brailovski,S.Prokoshkin,P.Terriault,andF.Trochu,Shape
Memory Alloys: Fundamentals, Modeling and Applications,
approach, a finite element model has been defined to accu-
E´coledeTechnologieSupe´rieure,2003.
ratelystudythecompleteporosityfraction,fromfullydense
to its highest-porosity limit, that is, to as much as 99% [2] D.C.Lagoudas,ShapeMemoryAlloys:ModelingandEngineer-
of empty space. The numerical responses were obtained by ingApplications,Springer,NewYork,NY,USA,2008.
using a simple but, from a numerical point of view, very [3] Y. Zhao, M. Taya, Y. Kang, and A. Kawasaki, “Compression
effectiveSMAconstitutivemodelintheAnsysenvironment. behaviorofporousNiTishapememoryalloy,”ActaMaterialia,
Numerous results were then extracted in the elastic regime vol.53,no.2,pp.337–343,2005.