Mechanical properties and non-homogeneous deformation of … · 2012. 1. 31. · Materials Science and Engineering A289 (2000) 276–288 Mechanical properties and non-homogeneous

Materials Science and Engineering A289 (2000) 276–288

Mechanical properties and non-homogeneous deformation ofopen-cell nickel foams: application of the mechanics of cellular

solids and of porous materials

X. Badiche a, S. Forest a,*, T. Guibert a, Y. Bienvenu a, J.-D. Bartout a, P. Ienny b,M. Croset c, H. Bernet d

a Ecole des Mines de Paris/CNRS, Centre des Materiaux/UMR 7633, BP 87 91003 E6ry, Franceb Ecole des Mines d’Ales, Laboratoire de Metrologie et Mecanique, 6 A6. de Cla6ieres, 30319 Ales, France

c NiTECH, ZI du Clos Marquet, rue Michel Rondet, 42400 Saint-Chamond, Franced Suez Industrie, 68 rue du Fg Saint-Honore, 75008 Paris, France

Received 8 November 1999; received in revised form 9 February 2000

Abstract

The mechanical properties of open-cell nickel foams are investigated for the range of densities used in industrial applicationsfor energy storage. The obtained Young’s modulus, compression yield stress and tensile fracture stress are compared to thepredictions of models based on periodic, Penrose and Voronoi beam networks. It is found that Gibson and Ashby’s model [L.J.Gibson, M.F. Ashby, Cellular Solids, Cambridge University Press, Cambridge, 1998] provides the proper scaling laws with respectto relative density for almost all investigated properties. The strong anisotropy of the observed overall responses can also beaccounted for. The two-dimensional strain field during the tension of a nickel foam strip has been measured using aphotomechanical technique. Non-homogeneous deformation patterns are shown to arise. The same technique is used to obtain thestrain field around a circular hole in a nickel foam strip. The observed deformation fields are compared to the results of a finiteelement analysis using anisotropic compressible continuum plasticity. © 2000 Elsevier Science S.A. All rights reserved.

Keywords: Cellular solid; Porous material; Nickel foam; Strain field measurement; Anisotropic compressible plasticity; Penrose lattice

www.elsevier.com/locate/msea

1. Introduction

Metal foams are used in many industrial applicationsranging from thermal insulation, electromagneticshielding, to energy absorption during car crash [1].Nickel foams are well-suited for battery applicationsand are involved for instance in portable computersand mobile phones. This requires outstanding mechani-cal and electrical properties. The use of light and highlyconductive foams leads to a considerable increase of theenergy density, whereas a high tensile strength is neces-sary for smooth processing of the foam during thebattery production steps like ‘pasting’, calendering andcoiling. The nickel foam studied in this work is pro-duced in the form of sheets or coils.

The design of an optimal foam for wanted mechani-cal and electrical properties requires a sound knowledgeof the relation between microstructural parameters likedensity or number of cells or struts per unit volume andthe effective properties. That is why linear propertieslike elasticity moduli (and electrical conductivity inappendix) and non-linear properties like yield and frac-ture strengths are investigated in this work for relativedensities of industrial interest ranging from 0.01 to0.05. These properties turn out to be strongly an-isotropic and this anisotropy is mainly due to materialprocessing. The deformation of heterogeneous highlyporous materials often is associated with the develop-ment of non-homogeneous deformation patterns likestrain localization, in the form of bands during thecompression of aluminium foams for instance [2,3].Although the deformation of nickel foam strips intension does not lead to significant localization phe-

* Corresponding author. Tel.: +33-1-60763051; fax: +33-1-60763150.

E-mail address: [email protected] (S. Forest)

0921-5093/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved.PII: S0921 -5093 (00 )00898 -4

X. Badiche et al. / Materials Science and Engineering A289 (2000) 276–288 277

nomena before initiation of a final crack and fracture,photomechanical measurement methods are used in this

work to display the two-dimensional strain field duringthe test. Similar strain field measurement techniques,together with X-ray tomography, have been used foraluminium foams in [4]. The present results are given inSection 3 and reveal the existence of slightly inhomoge-neous deformation patterns appearing at the very be-ginning of the test. A description of nickel foamprocessing, microstruture and of the experimental tech-niques used in this work is given in Section 2 and theexperimental results are presented in Section 3. Theelectrical properties of the material are of the utmostimportance for industrial applications. Since this workmainly focuses on mechanical properties of metalfoams, the results on electrical conductivity measure-ments are given in appendix. It is shown that theelectrical properties can be interpreted using the sametype of models as for the mechanical properties.

Two classes of material models are available to simu-late and predict the response of metal foams undervarious loading conditions. The micromechanical ap-proach relies on the knowledge of the behaviour of theconstituent and of the cellular structure to derive theeffective properties. Simple and efficient models can befound in [5] and apply for a wide range of foams. Theyare based on idealized cellular structures like grids orhoneycombs in the two dimensional case. In the three-dimensional case, the idealized unit cell proposed in [5]can be used for instance to predict elastic propertiesand the plateau stress in compression. A straightfor-ward extension exists accounting for anisotropic be-haviour (Fig. 1 and [6]). As can be seen from Fig. 2, thecell distribution is actually non periodic, which stronglyaffects the effective non-linear properties for instance.For that purpose, random models for cellular materialshave been introduced in [7], in the form of two-dimen-sional Voronoi honeycombs. Since the representativeunit cell of the nickel foam rather is a pentagonaldodecahedron which cannot lead to periodic space tes-selation [8,9], the case of quasiperiodic beam networksis introduced in this work and represents an intermedi-ate model between periodic and random lattices. Theexample of a two-dimensional Penrose beam networkand its elastic deformation in tension is given in Fig. 3.

The second class of constitutive models for metalfoams is the phenomenological approach that leads tofull three-dimensional models that can be implementedin finite element codes for structural applications. Themain advantage of this approach is that it can accountfor non-homogenous deformation of structures made ofmetal foam. The mechanics of porous materials hasbeen developed to describe the processing and constitu-tive behaviour of powder metallurgy materials [10] butalso ductile fracture of materials (see, e.g. [11]). Al-though micromechanical arguments are available toconstruct three-dimensional compressible plasticity

Fig. 1. Anisotropic idealized unit cell for the mechanical analysis ofmetal foams (after [5]).

Fig. 2. Cellular structure of a nickel foam.

Fig. 3. Quasiperiodic Timoshenko beam network and its deformedstate in tension for linear elasticity.

X. Badiche et al. / Materials Science and Engineering A289 (2000) 276–288278

criteria [12], the parameters appearing in these modelscan be identified phenomenologically from macroscopictests. There is a need for simple and reliable engineeringconstitutive equations for metal foams. That is why thephenomenological approach has been used for alu-minium foams in [13,14]. This is done in this work fornickel foams and the prediction capability is illustratedin the case of a nickel foam strip with a hole in tension.The two classes of models are confronted with experi-mental results in Section 4. One original aspect of thiswork is the use of an anisotropic plasticity criterion forwhich an identification procedure is applied, based onthe measurement of length, width and thickness varia-tions during uniaxial straining. The parameters of suchmodels can also be identified using multiaxial tests likeaxisymmetric compression in [14]. But multiaxial homo-geneous loading conditions are difficult to design forfoam sheets.

2. Experimental procedures

The main features of nickel foam processing and ofthe obtained microstructure are described in this partand this is followed by a presentation of the differentmechanical tests performed in this work.

2.1. Material processing and cellular structure

The production of nickel foams can be decomposedinto five main steps [15]. A polyurethane foam havingthe thickness and the porosity of the final product isselected and covered with an electrically conductivecoating. Nickel is then electroplated on the coatedsurface, before burning out the polyurethane foam. Afinal thermal treatment is necessary to obtain thewanted mechanical properties. The firm NiTECH hasdeveloped an original process for the second step,namely cathodic magnetron nickel sputtering, whichallows the treatment of polyurethane of different thick-nesses, from 1.2 to 6 mm and of different porosities.Impurities like carbon and sulphur are present in nickelafter material processing, in contents lower than 100and 50 ppm, respectively. The successive processingsteps being continuous, the nickel foam sheets aresubmitted to a tensile force in the rolling directioncalled RD. The transverse direction will be denoted TDin the sequel.

The obtained cell structure is given on Fig. 2 wheremainly dodecahedral cells with open pentagonal facescan be seen. The mean cell diameter is about 500 mmbut can be varied from 400 mm to 1.2 mm depending onthe final application. The structure of nickel foams isthe quasi exact replication of that of the originalpolyurethane foam. In particular, the section of a struttypically is a so-called ‘Plateau border’ which corre-

sponds to a structure controlled by surface tension andinterfacial effects during the foaming of thepolyurethane substrate and the explosion of the cellwalls to get an open-cell foam structure. Since thepolymer is eventually burnt out, the nickel struts arehollow. Typical length and diameter of a strut are 150and 70 mm. The walls of the struts are about 10 mmthick. The roughness of the surface of the struts is dueto the electroplating process. The final heat treatmentslead to nickel grain sizes from 1 to 10 mm. The investi-gated properties will be shown to depend mainly on therelative density of the foam, defined as the ratio of themass density of the cellular material and that of purenickel:

r�=rfoam

rNi

, with rNi=8900 kg m−3 (1)

An other important parameter is the number n ofstruts in the thickness of the nickel foam sheet. Thisnumber must be large enough for the material to beconsidered homogeneous on a macroscopic scale. If it istoo small, strain will tend to localize in some weakersection and homogeneous deformation of the foam willbe impossible. This number is defined as the ratiobetween the thickness e and the mean strut length a. Inthis work, values of n from 6 to 18 have been studied.Other parameters like specific surface areas and tortu-osity have been studied in [16] for very similar nickelfoams.

2.2. Heterogenity and anisotropy

Several aspects of the material processing contributeto introduce a variability of the cell structure in thematerial. First of all, the thickness te of the depositednickel layer at the surface of the foam is greater thanthe thickness ti in the mid-section of the foam. Theelectroplating technique is intrinsically associated witha thickness deposition ratio (TDR= te/ti) differentfrom one. The obtained values of TDR range from 1.2to 1.6. This may play an important role in the interpre-tation of the compression tests discussed in Section 3.2.

The foaming process of polyurethane gives rise to anelongation of the cells in the direction of gravity. Twocutting techniques are used to obtain the skeleton of thenickel foam. The peeling of parallelepipedicpolyurethane foam blocks leads to a periodic variationof the structure of the peeled foam, since the elongatedpolyurethane cells are successively cut along their longand short axes. In contrast, the slit cutting of a longblock of polyurethane gives a more uniform isotropicstructure. Many results given in this work have beenobtained for the latter product called LS (for loop-slit-ting). There is however another source of anisotropyfor the final product and even for LS nickel foams,namely the tensile force applied to the foam during the


Fig. 4. Tensile behaviour of a nickel foam (LS foam with r�=0.028);the results of the simulation using the model developed in Sections4.2 and 4.3 are also presented (stress in MPa).

the extensometer itself may lead to some bias in themeasurement of mechanical response of particularlythin foams. That is why a strain field measurementmethod has been used to investigate these differentaspects of foam deformation.

Optical methods are well-suited for non-destructivetesting because they are non-intrusive and do not affectthe surface of the specimens [17]. The used opticalextensometer provides a bidimensional field of in-planedisplacements. When an optically rough surface is illu-minated by light, the surface acquires a granular ap-pearance. The porous surface of metal foams isparticularly adapted for such observations. An imagingsystem (CCD camera) is used to record the successivestates of a zone (or even the whole specimen) of the flatspecimen under loading. Natural characteristic patternsarising from the surface roughness can be identified andfollowed during deformation thanks to digital imageprocessing. The determination of displacements is basedon digital image correlation (see [17]). Three compo-nents of the strain field are then deduced from thetwo-dimensional displacement field.

The idea of applying strain field measurement meth-ods to cellular solids goes back to Chen and Lakes [18]who resorted to a holographic method. A similarspeckle measurement method has been used in [3,4] forthe study of closed-cell aluminium foams. However inthe latter case, the cell size was not negligible whencompared to the specimen size so that the discretedeformation of individual cells could be observed. Incontrast, the size of the zone of the nickel foam speci-men investigated here is large enough for the materialto be regarded as statistically homogeneous and treatedas a continuous medium. The continuous deformationfields observed in tension are reported in the next part.

The technique has been applied to tensile tests onnickel foam strips, extension tests (plane tension ofwide strips leading to almost vanishing lateral deforma-tion) and to the tension of a strip with a circular hole.

3. Results

The observed tensile and compressive properties ofnickel foams are reported below, as well as the newinsights brought by the use of photomechanical strainfield measurement methods.

3.1. Tensile tests

Fig. 4 shows the material response to tensile loadingconditions in both directions RD and TD. The finalcrack leading to fracture usually starts from one lateralboundary of the sample far from the grips. No signifi-cant dependence on material response and ductility ofthe way of cutting the samples out of the coils or sheets

processing. This results in cells that are more elongatedin average in direction RD that in the transverse direc-tion. Considering the idealized unit cell of Fig. 1, thisanisotropy can be expressed by the aspect ratio:

R=Dd

(2)

where D is measured in direction RD. A typical valueof 1.5 is deduced from the analysis of elastic properties(see Section 3.1) and electrical properties (see ap-pendix). The resulting strong anisotropy of the overallmaterial properties will be shown in the next sections.

2.3. Mechanical testing methods

An electro-mechanical machine with a load cell of 50kg is used to perform tensile tests on nickel foamsamples. The specimens are 200 mm long and 20 mmwide nickel foam strips. The thickness varies from 1.2to 6 mm depending on the density of the studied foam.The specimen’s ends are compressed in the tensile ma-chine grips and deformation is measured using an ex-tensometer clipped on the specimen. The directions RDand TD are carefully distinguished in the tests.

Nickel foams can be compressed in the directionnormal to the sheet plane. In order to perform com-pression tests, disks of 20-mm diameter have been cutout of the nickel foam sheet. Due to the limited thick-ness of the specimen, only the relative displacement ofthe test machine platens can be measured. It has notbeen possible to significantly reduce friction betweenthe specimen and the machine platens.

2.4. Photomechanical strain field measurement method

The use of an extensometer clipped on the samplegives no indication on the homogenity of deformationalong the specimen. Morever the weight and rigidity of


has been noticed. The linear and non-linear tensileproperties are successively reported.

The elastic part of the curve is hard to estimate sothat unloading conditions have been prescribed at thebeginning of the test in order to measure Young’smodulus in both directions. The unloading curves ofFig. 5 exhibit a slight hysteresis loop but elastic moduli

Fig. 8. Failure of a nickel strut: cracks develop at the boundarybetween the wall and the edges of the Plateau border that exhibit amass excess due to the electroplating process.

Fig. 5. Unloading condition at the beginning of a tensile test for thedetermination of Young’s modulus in each direction (LS foam r�=0.028).

can be unambiguously determined and display a stronganisotropy. The relative Young’s modulus in each di-rection is given as a function of relative density on Fig.6 and is defined by

E�=Efoam

ENi

with ENi=214 GPa (3)

ENi for bulk isotropic nickel. The Poisson ratio couldnot be measured using this technique and the problemof transverse deformation is tackled in Section 3.3 usingphotomechanics. It must be noted that the elastic prop-erties vary with deformation. For a LS foam (withr�=0.028), Young’s modulus in direction RD in-creases from 300 MPa at 0.3% strain to 360 MPa at5%. In direction TD Young’s modulus goes from 150Mpa at 0.3% to 200 MPa at 10%. The Young’s mod-ulus is found to increase very sligthly with n, thenumber of struts through the thickness.

The elastoplastic response of nickel foams shown onFig. 4 displays a very low initial yield stress, a non-lin-ear regime followed by almost linear hardening. Thestress is computed using the prescribed force and theinitial section of the specimen, whereas the reportedstrain is the ratio between the specimen elongation andthe initial gauge length. The nickel foams exhibit astrongly anisotropic elastoplastic behaviour: the ductil-ity in direction TD is almost twice that in direction RD.The maximum stress reached before fracture is plottedas a function of relative density in Fig. 7. An exampleof a fracture process in a strut can be seen on Fig. 8,where the role played by the Plateau border shapeclearly appears. At the level of the nickel strut, fractureis intergranular as shown on Fig. 9. The embrittlementof grain boundaries can likely be attributed to thepresence of sulphur in the material.

Fig. 6. Relative Young’s modulus as a function of relative density.

Fig. 7. Relative fracture stress in tension as a function of relativedensity.


3.2. Compression tests

Some compression curves are presented on Fig. 10.Several features of these curves are common to manycellular solids under compression: a plateau is reachedafter an almost linear regime and followed by a pro-nounced hardening stage corresponding to nickel den-sification. The short plateau at the very beginning ofthe curves may be attributed to defects in the flatness ofthe specimens mainly due to the deformation inducedby the machining. In general, the curves display a peakfollowed by apparent softening or a short plateau. Asimilar behaviour can be observed on polymer foams[6]. The peak tends to disappear for foams with rathersmall n and therefore large cells. Interrupted tests be-fore, at and after the peak followed by SEM observa-tions have shown that the hinge deformationmechanism proposed in [5] applies here and starts at

the peak. Observations on homogenity of deformationwithin the thickness during the test were not possiblebut interrupted tests indicate that the slightly less densecore of the sheet (due to the existence of a TDRdifferent from 1) fails earlier.

3.3. Strain distribution during a tensile test

The photomechanical technique enables one to com-pute the components o11,o22 and o12 of the strain fieldduring the deformation of a nickel foam strip in tensionalong direction 2 (which will be RD or TD in thefollowing tests). In particular, averaged values of o11

can be computed over a central part of the sample inorder to have access to transverse deformation. Theratio −o11/o22 of transverse versus longitudinal defor-mation is found to be not far from 0.5 when thetensile direction 2 coincides with RD, whereas a verydifferent result, namely about 0.25, is obtained for theother direction. This confirms the very strong an-isotropy of the behaviour of nickel foams. Precise mea-surements in the elastic regime for the determination ofPoisson ratios for each direction are still to be per-formed.

Deformation maps of a LS nickel foam in tensionalong RD are presented on Fig. 11 for a global defor-mation of about o22=6.5%. The strain field turns out tobe relatively inhomogoneous and this has been ob-served on the 12 specimens of various densities thathave been investigated using the photomechanical tech-nique. A fluctuation Do22 equal to about 1.8% developsat the very beginning of deformation and remainsalmost constant during further deformation. Someband-like deformation patterns are generally observedbut the crack leading to final fracture does not system-atically appear within them. A similar heterogenity intransverse deformation exists. The effects of the grips atthe sample ends where the transverse deformation isconstrained to be nearly zero, can be seen on Fig. 11(b)since the whole gauge length has been analysed. Notethat the obtained heterogenity cannot be explained onlyby some variations of the specimen thickness. Theinitial thickness of the sample was measured and foundto vary from 1.95 to 2.05 mm. A finite element analysis,similar to the simulations to be reported in the discus-sion, has been carried out including random variationsof specimen thickness: maximal fluctuations Do22 ofabout 0.1% have been obtained. On the other hand, theobserved heterogeneous deformation patterns cannotbe related to edge effects mentioned in [19]. Differentcutting techniques of the specimens have been used anddo not affect significantly the deformation of the sam-ples considered here. However, the observed heterogen-ities remain small and stable enough, so that they donot significantly affect the overall properties determinedin the previous sections.

Fig. 9. View of a crack tip in a nickel wall: intergranular crackpropagation.

Fig. 10. Compression tests on nickel foams with different relativedensities and numbers of cells within the thickness.


Fig. 11. Strain field measurements on a nickel foam in tension: axialstrain o22 (a) and transverse strain o11 (b); distances are given in mm,the tensile direction is the vertical one and coincides with RD;r�=0.025.Fig. 14. Strain localization (above) and deformation band propaga-tion (below) in an elastoplastic quasi-periodic beam network intension along the vertical direction; equivalent plastic deformation inthe beams; the deformation within the band is about 10 time largerthan in the surrounding material and saturates at 2% before the nextband starts.

3.4. Deformation maps of a foam sheet with a circularhole in tension

Metal foams are used in industrial structures withspecific geometries that may lead to non-homogeneousdeformation during straining of the component. As anexample of simple structure, we consider a 124-mmlong and 50-mm large nickel foam strip with a circularhole, 25.5 mm in diameter, at its center. When submit-ted to tensile loading conditions at its ends, a non-ho-mogeneous strain field develops that can be recordedusing the photomechanical technique. The two straincomponents along the ligaments of the specimen (fromthe notch to the free boundary) are given on Fig. 12and Fig. 13 together with the finite element simulationsprovided at the end of next part. The results are givenfor tension in RD for 0.3% overall deformation. Thetests have been carried out in direction TD too, but theresults are not provided for conciseness. Note thatsignificantly higher strains are reached at the notch fortension in direction TD than in direction RD.

4. Discussion and constitutive modelling

The micromechanical and phenomenological ap-proaches of the mechanics of cellular solids are succes-sively applied to interpret the previous experimentalresults and to predict the overall and structural non-lin-ear mechanical properties of nickel foams.

4.1. Linear and non-linear beam models

4.1.1. Young’s modulusPredictions of Young’s modulus for cellular solids

can be derived from the models based on beam net-works presented in the introduction. They are reportedin Table 1. The predicted relative Young modulus isfound to be proportional to r�

m where the exponent mstrongly depends on the retained unit cell [20,21]. Alinear relationship is found if the struts are regarded asbars carrying only normal forces, whereas powers 2–3are expected if the bending of beams is introduced.Note that the Penrose lattice proposed in this workleads to a power not far from that obtained in the caseof random networks. However the chosen quasi-peri-odic lattice does not respect the geometric properties ofthe structure of the present foam, in particular thenumber of edges ending at a given node, but a morerealistic network should be three-dimensional.

The experimental results are compatible with Gibsonand Ashby’s model [5] for both directions RD and TDas can be seen on Fig. 6. The use of the anisotropic unitcell of Fig. 1 [6,22] leads to the following expression ofthe ratio between the predicted longitudinal and trans-verse Young’s moduli:


Fig. 12. Axial deformation along a horizontal line (ligament) startingfrom the notch; tensile direction 2 coincides with RD; distance fromthe hole in mm (overall deformation of 0.3%)

of the non-linear mechanical response of the con-stituent. The determination of the elastoplastic be-haviour of a nickel strut is a challenging problem thathas been tackled in [23]. For that purpose, the tensilebehaviour of nickel films of 400 mm thickness whichhave been submitted to the same heat treatments as thenickel foams, has been determined. The measured elas-tic properties significantly differ from the usual onesmentioned in textbooks due to the texture of the filmsand to the fact that a nickel film is a multicrystallinematerial with sometimes only one grain through thethickness. These results and the constitutive modellingof elastoplastic pure nickel have been reported in [23].They have been used to compute the non-linear re-sponse of periodic and random beam networks and tocompare it with experimental results. A good agreementhas been obtained at least under uniaxial loading condi-tions. In contrast, we report here the very specificnon-linear response of the quasi-periodic lattice. Thetensile behaviour of the Penrose network of Timo-shenko beams of Fig. 3 is considered again but anelastoplastic behaviour with non-linear isotropic hard-ening is attributed to each beam. The deformationmaps of Fig. 14 show that deformation immediatelytends to localize within deformation bands. Once aband has formed, deformation further proceeds bylocalization band propagation spreading over the entirespecimen. The inclination of the bands of about p/5(36°) with respect to the tensile direction reflects theunderlying quasi-periodic structure of the network.Note that this angle is close to the orientation of shearbands in an isotropic incompressible elastoplasticmedium under plane stress conditions (namely 35.26°).In contrast, periodic structures like grids and honey-combs have not been found to be prone to localizationphenomena, at least under tensile conditions. Shearbands in Voronoi honeycombs under compression havebeen simulated in [7] but the tensile case remains to beconsidered. This propensity of quasi-periodic lattices tostrain localization may perhaps explain to some extentthe observed systematic strain heterogenity in tension(see Section 3.3) but the more realistic 3D extensionremains to be done.

4.1.3. Fracture stress in tensionThe maximum stress reached before fracture is plot-

ted as a function of relative density in Fig. 7. A strong

Fig. 13. Lateral deformation along the ligament starting from thenotch; tensile direction 2 coincides with RD; distance from the hole inmm (overall deformation of 0.3%)

E�RD

E�TD =

2R2

1+R−3 (4)

Experimentally, E�RD/E�

TD#3.45, which implies R#1.5. Note that a similar simple beam model exists topredict the electrical conductivity of the foam (seeAppendix) and is compatible with the same value of theaspect ratio.

4.1.2. Hardening beha6iour and localization phenomenaThe extension of the beam network models to ac-

count for the non-linear regime requires the knowledge

Table 1Prediction of relative Young’s modulus using periodic and non-periodic beam networks

Hexagonal Bernet [23]Warren and Kraynik [26]Gibson and Ashby [5] Penrose networkVoronoi honeycomb [7]honeycomb

3.2r�2E� 2.6r�

3.44

:r�2 0.56r�

2.43r�6

3

2r�

3


Fig. 15. Plateau stress of nickel foams under compression.

f(s4 )

=�3

2C(as11

2 +bs222 +cs33

2 +2(ds122 +es23

2 + fs312 ))

+F(us11+6s22+ws33)2�1/2

−R (5)

where s4 is the deviator of the stress tensor s4 . Thisrepresents the anisotropic generalization of thequadratic criterion used in [14] for which parametersa– f and u, 6, w are set to 1. A similar anisotropicextension of the Drucker–Prager criterion and themore general ones proposed in [13] is also possible. Thecomponents of these tensors are given with respect tothe frame associated with the orthotropy axes of thefoam. The ratio C/F accounts for the respective influ-ence of the deviatoric and spherical parts of the stresson plastic yielding, whereas parameters a– f and u–waccount for the anisotropic response. Plastic flow fol-lows from assumed normality rule:

o;4 p=p; (f(s4 (6)

where p; is the plastic multiplier. An associated flow ruleis also assumed in [13,14]. The expression of the plasticmultiplier is the following in the isotropic case:

p; =� 23C

o;4 devp :o;4 dev

p +1

9F(tro;4 p)2�1/2

(7)

where o4; devp denotes the deviatoric part of the plastic

deformation tensor. In the general case, a viscoplasticframework is used according to which:

p; =#f(s4 )K

$N

(8)

where MacCauley brackets and viscosity parameters K,N have been introduced. Large values of N and smallvalues of K lead to the rate-independent elastoplasticcase considered in present work.

The non-linear work-hardening behaviour of thefoam is described by the following hardening rule:

R=R0+Q(1−e−Bp)+Hp (9)

which involves 4 parameters, R0 being the initialthreshold. The coefficients (Q, B) and H, respectivelyaccount for the non-linear initial and linear final partsof the tensile curves. This model is implemented in thefinite element code Z-set described in [24]. Fully implicitglobal resolution and local integration schemes areused.

4.3. Parameter identification

The identification of the numerous parameters ap-pearing in the anisotropic function and in the evolutionequations requires sufficient experimental information.

correlation seems to exist involving a power 1.7–1.75 ofthe relative density. In contrast, no correlation couldbe found between deformation before fracture and rela-tive density. A model accounting for the found rela-tionship between fracture stress and density remainsto be developed, since no corresponding result hasbeen found in literature yet.

4.1.4. Plateau stress in compressionWhen plotted as a function of relative density (Fig.

15), the stress reached at the plateau or at the peak isfound to follow the power law (m=1.5). This relation-ship is compatible with the hinge model proposed in[5]. The dispersion of the results is however morepronounced than for the properties previously investi-gated.

4.2. General constituti6e framework based on themechanics of porous media

The main drawback of the previous approach is thatit can generally not provide a fully three-dimensionalconstitutive model to be used for structural calcula-tions. That is why we resort here to a classical phe-nomenological constitutive framework based oncontinuum plasticity theory. The validity of such acontinuum approach will be tested in the case of thenickel foam strip with a hole in tension. Metal foamsbeing porous compressible materials, a compressibleplasticity model originally proposed in [12] is retainedand is similar to the one used in the work [10] dealingwith powder metallurgy. An additional feature of themodel is the introduction of plastic anisotropy as ob-served experimentally. Note that the model can beformulated within the framework of small deformationssince the ductility in tension of the nickel foam issufficiently low. The general form of the anisotropicelliptical plasticity criterion reads:


Table 2Material parameters of the anisotropic compressible plasticity model for a LS foam with r�=0.028

aC b c F u 6 w R0 (MPa) Q (MPa) B H (Mpa)

2.071.0 2.00.42 0.0018 1.0 20.6 23.3 0.45 0.37 3.3 8.17

When no multiaxial test is available, it is possible toidentify some parameters from the knowledge of thestress-strain tensile curve and from the apparent Pois-son ratio [13]. In the present anisotropic case, we haveto consider at least two tensile tests in two differentdirections (here RD and TD) and to measure bothlateral deformation components during the unaxial test.That is why the variations of the width and thickness ofthe tensile specimens have been recorded and used inthe identification procedure. The criterion (Eq. (5))leads to the following ratio of the yield stress in direc-tion 1 and 2:�sY

1

s2Y

�2

=C(a+4b+c)+6F62

C(4a+b+c)+6Fu2 (10)

The ratio of lateral versus longitudinal plastic defor-mation for tension tests in direction 1 is a consequenceof application of normality rule (Eq. (6)):

o22p

o11p =

C(−2a−2b+c)+6Fu6C(4a+b+c)+6Fu2 (11)

o33p

o11p =

C(−2a+b−2c)+6FuwC(4a+b+c)+6Fu2 (12)

Two similar relations hold for tension in direction 2.The criterion (Eq. (5)) can be multiplied by any con-stant so that the product Ca is arbitrary. In Eqs.(10)–(12), F, u, 6, w are not independent and the actualunknowns are Fu2, 6/u, w/u, so that we can fix forinstance u=1. Five unknowns remain b, c, F, 6, w thatcan be unambiguously determined using the previousfive equations. However, due to experimental scatter,we do not directly solve these equations but ratherresort to an optimization procedure based on a Leven-berg–Marquardt algorithm and a quadratic cost func-tion to determine a set of parameters describingexperimental data with sufficient accuracy. Theparameters R0, Q, B and H appearing in the nonlinearhardening rule R(p) are also included in this optimiza-tion loop in order to reproduce at best the whole tensilecurves.

In contrast, in the available experimental data, thereis not the necessary information to determine theparameters d– f characterizing shear anisotropy. Ac-cordingly, shear isotropy d=e= f=1 is assumed in thefollowing. The importance of this assumption can beestimated in the structural analysis of next section.

The anisotropic elastic behaviour must also be takeninto account. Some components of the orthotropic

elasticity matrix can be identified. The remaining con-stants and in particular the shear moduli are estimatedaccording to Voigt’s approximation of homogenizationtheory [21].

The coefficients of the constitutive equations can beassumed to be constant since foam density does notvary significantly in the considered tensile tests. Theidentification of the material parameters has been per-formed for a LS foam (with r�=0.028) using thefollowing experimental basis: tensile tests in directionRD and TD. The information about the transversedeformation in both tests (obtained by photomechani-cal analysis) thickness reduction has been taken intoaccount in the identification procedure. The results ofthe parameter identification are shown on Fig. 4 andthe parameters are given in Table 2. Note that theidentification concerns only the tensile domain of thestress space, compression results have not been in-cluded, which would probably leads to a more sophisti-cated model [13].

The predictive capabilities of the model have beentested using extension tests for which wide specimenswith a short gauge length are used in order to imposealmost vanishing transverse deformation (o11#0) atleast in a large part of the specimen. This gives rise toslightly higher stresses as shown on Fig. 16 that arecorrectly reproduced by the model using the parametersidentified from tensile tests.

Note that the parameters obtained for the hardeningrule Eq. (9) are not those of the behaviour of purenickel. Such an approach may be possible but in thiscase C cannot be taken equal to 1 as done here. In the

Fig. 16. Simulation of the extension (plane tension) of nickel foams indirections RD and TD; confrontation with experiment.


Fig. 17. F.E. simulation of the tensile deformation of a nickel foamplate with a hole: o22 strain field (in %); direction coincides with TD.

has been performed under plane stress conditions andthe obtained deformed state is shown on Fig. 17.Deformation is not maximum on the ligament butrather in an inclined band starting from the equatorand that tends to develop. This is due to the stronglyanisotropic material behaviour. In experiment, finalcracking is actually found to occur in such an inclinedband (slightly less than 45° from the tensile axis).Quantitative comparison between experiment and simu-lation is illustrated here by the strain components ob-tained along the ligament of the sample. Only onefourth of the specimen is represented in the simulationfor symmetry reasons, whereas experimental informa-tion about the two ligaments are reported. The mea-sured components o22 and o11 along the ligament arecompared with the finite element results on Figs. 12 and13 for RD tensile test. The corresponding results forTD tensile test are available but they are not given herefor conciseness. The model properly describes the meanvariation of the strain field in both cases, althoughimportant fluctuations are observed experimentally thatcannot be accounted for by the deterministic contin-uum plasticity approach. The origin of the fluctuationsremains unclear but they are in accordance with thedevelopment of heterogenity during a tensile test on ahomogeneous sample reported in Section 3.3. Thestrongly anisotropic response is correctly described bythe numerical analysis. Note that shear component s12

of the stress remains significantly smaller than thecomponents s11 and s22 in the computation, whichexplains why the crude hypothesis of isotropic elasticand plastic shear properties does not alter theprediction.

Structural tests like compression of samples with ahole have already been performed on aluminium foams(see [3,25]) but strain field measurements combined withfinite element analyses are well-suited tools to interpretthem.

5. Conclusions

Non-homogeneous deformation seems to be an im-portant feature of the mechanical behaviour of metalfoams. Strain localization phenomena have been re-peatedly observed in the compression of closed-cellaluminium foams [2–4]. In the tensile deformation ofnickel foams, slightly non-homogeneous deformationpatterns have been detected using photomechanicaltechniques. The ultimate strain at fracture being amajor request of the users of the nickel foam product,a precise knowledge of the heterogeneous deformationpatterns that may arise during a characterization test isan important information. Beam models for open-cellfoams have been shown to provide good estimates ofthe overall properties of nickel foams or at least the

literature of the mechanics of porous media, relation-ships between parameters C, F and the relative densityhave been derived considering for instance ideal ar-rangements of spheres or cylinders [12] or cubic net-works. One of them [12] reads for instance:

C=� 3−2(1−r�)

1/4

3(1− (1−r�)1/3)

�2

, F=1

4(log(1−r�))2 (13)

These formula are usually applied to higher densitiesthan for metal foams but it is interesting to extrapolatethem to the densities considered in this work. Forinstance, formula (Eq. (13)) leads in our case to a ratioC/F of 4.2 for r�=0.03. This is in relatively goodagreement with the value 5.6 of the correspondingindicative ratio C(a+b+c)2/(F(u+6+w)2) that canbe defined for the present anisotropic model, using theparameters of Table 2.

4.4. Application to the deformation of a foam platewith a hole

To check the validity of the previous continuumplasticity model, it is applied to the simulation of thedeformation of a foam plate with a circular hole. Theresults are compared to experimental photomechanicalmeasurements. A finite element simulation of the tests


Fig. 18. Relative electric resistivity as a function of relative density.

during the preparation of the manuscript. The authorsthank the referees for drawing their attention on refer-ences [4] (that appeared during the reviewing proce-dure) and [14]. The second author thanks ProfessorN.A. Fleck for providing the manuscript [14] prior topublication.

Appendix A. Electrical conductivity of nickel foams

An electrical current of given intensity has beenpassed through (L=120-mm) long and (l=20-mm)wide nickel foam strips and the necessary tension hasbeen measured in order to obtain the resistance r andassociated resistivity V of nickel foam in a givendirection:

V=rleL

(14)

The obtained relative resistivity V�=V/VNi withVNi=6.9×10−8 Vm for pure nickel, is plotted in Fig.18 as a function of relative density of the foam and forboth directions RD and TD. The observed relationshipcan be accounted for using a simple model of the foambased on a quadratic network which is a three-dimen-sional anisotropic extension of the square lattice. Thelength (direction RD) and width (direction TD) of thequadratic unit cell are denoted D and d respectively,similarily to Fig. 1. The predicted resistivity in bothdirections is:

V�RD=

1+2/Rr�

andVRD

VTD =1R

(15)

In the isotropic case, this model gives the resultV�

RD=3/r�, which is not far from more sophisticatedestimations like in [27,28]. If the previous model ap-plies, the ratio of the observed longitudinal and trans-verse resistivities implies R#1.5, which is consistentwith the result obtained for anisotropic elasticity.

The resistivity of nickel foams is found to be almostindependent of n.

References

[1] Innovation 128, Les materiaux en mousse et leurs applicationsindustrielles, Techtendances, Etudes Technologiques, Paris, 1998.

[2] Y. Chastel, E. Hudry, S. Forest, C. Peytour, Mechanical be-haviour of aluminium foam for various deformation paths, in: J.Banhart, M.F. Ashby, N.A. Fleck. (eds.), Experiment and Mod-elling, Metal Foam and Porous Metal Structures, Verlag MITPublishing, 1999, pp. 263–268. Journees d’Automne de laSF2M, Paris, 1998.

[3] K.Y.G. McCullough, N.A. Fleck, M.F. Ashby, Acta Mater. 47(1999) 2323–2330.

[4] A.F. Bastawros, H. Bart-smith, A.G. Evans, J. Mech. Phys.Solids 48 (2000) 301–322.

correct dependency of these properties on relative den-sity for the investigated range. However, in the non-lin-ear regime, the available models are much lessnumerous. The proposed quasi-periodic beam modelfor instance is not adequate for nickel foams, since it ismuch more prone to localization than the actual mate-rial, at least for the two-dimensional version of themodel investigated here. Furthermore, the prediction oftwo-dimensional strain fields in structures made ofmetal foam requires a full three-dimensional elastoplas-tic model. Compressible continuum plasticity has beenshown to be a well-suited tool for this purpose, al-though material parameters must be identified for eachgiven density. The proposed purely phenomenologicalapproach represents a pragmatic way for computingindustrial components involving metal foams using con-ventional finite element codes for which similar com-pressible plasticity models are already available.Improvements of the beam or continuum models arenecessary to describe the observed compressionbehaviour.

A combination of the micromechanical and phe-nomenological approaches with a view to deriving rela-tions between the parameters entering compressibleplasticity criteria and material relative density surelyrepresents the most promising framework for the gen-eral modelling of metal foam behaviour. The modellingof crack propagation then requires the introduction ofdamage mechanisms into this framework in a waysimilar to the treatment of ductile fracture of metals[11]. This further work is under progress.

Acknowledgements

R. Descamps and Ch. Bourgin, during their trainingperiod (Engineering degree at Ecole des Mines de Paris,speciality Engineering Materials) have contributed tothe first stages of this study. The authors thank Profes-sor A. Pineau for stimulating discussions and his help


[5] L.J. Gibson, M.F. Ashby, Cellular Solids, 1st edition, Cam-bridge University Press, Cambridge, 1998.

[6] A.T. Huber, L.J. Gibson, J. Mater. Sci. 23 (1988) 3031–3040.[7] M.J. Silva, L.J. Gibson, Int. J. Mech. Sci. 39 (1997) 549–563.[8] B. Grunbaum, G.C. Shephard, Tilings and Patterns, W.H. Free-

man, San Francisco, CA, 1986.[9] C. Janot, Quasycristals a Primer, Clarendon Press, Oxford, 1992.

[10] J. Besson, M. Abouaf, Mater. Sci. Eng. A 109 (1989) 37–43.

[11] A. Pineau, Effect of inhomogeneities in the modelling of me-chanical behaviour and damage of metallic materials, 7th Inter-national Conference on Mechanical Behaviour of Materials,ESIS, The Hague, The Netherlands, 1995, pp. 1–22.

[12] R.J. Green, Int. J. Mech. Sci. 14 (1972) 215–224.[13] R.E. Miller, Int. J. Mech. Sci. 42 (2000) 729–754.[14] V.S. Deshpande, N.A. Fleck, J. Mech. Phys. Solids (2000) in

press.[15] M. Croset, W. Pruyn, Nickel Foam for High Performances NiCd

and NiMH Batteries, CIBF, Beijing, 1997.[16] A. Montillet, J. Comiti, J. Legrand, J. Mater. Sci. 27 (1992)

4460–4464.

[17] F. Laraba-Abbes, P. Ienny, R. Piques, Kautsch. Gummi Kun-stst. 52 (1998) 209–214.

[18] C.P. Chen, R.S. Lakes, J. Mater. Sci. 26 (1991) 5397–5402.[19] R. Brezny, D.J. Green, J. Mater. Sci. 25 (1990) 4571–4578.[20] J.L. Grenestedt, Int. J. Solids Struct. 36 (1999) 1471–1501.[21] S. Torquato, L.V. Gibiansky, M.J. Silva, L.J. Gibson, Int. J.

Mech. Sci. 40 (1998) 71–82.[22] S.V. Kanakkanatt, J. Cell. Plast. 9 (1973) 50–53.[23] Ch. Bourgin, Th. Guibert, J.-D. Bartout, Y. Bienvenu, S. Forest,

H. Bernet, M. Croset, Caracterisation et modelisation du com-portement mecanique des mousses de nickel, Colloque Nationalde Metallurgie des Poudres, SF2M, Commission de Metallurgiedes Poudres et Frittage, Grenoble, 6–8 April, SF2M ed., 1998,pp. 41–48.

[24] J. Besson, R. Foerch, Comp. Methods Appl. Mech. Eng. 142(1997) 165–187.

[25] A. Paul, T. Seshacharyulu, U. Ramamurty, Scripta Mater. 40(1999) 809–814.

[26] W.E. Warren, A.M. Kraynik, J. Appl. Mech. 55 (1988) 341–346.[27] A.K. Datye, R. Lemlich, Int. J. Multiph. Flow 9 (1983) 627–

636.[28] R. Lemlich, J. Colloid Interface Sci. 64 (1978) 107–110.

.

Mechanical properties and non-homogeneous deformation of … · 2012. 1. 31. · Materials Science and Engineering A289 (2000) 276–288 Mechanical properties and non-homogeneous

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