The variation inelastic modulus throughout the compression of foam materials
Yongle Sun1, B. Amirrasouli1, S.B. Razavi1, Q.M. Li1,3, T. Lowe2, P.J. Withers2
1School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Sackville Street, Manchester M13 9PL, UK
2Henry Moseley X-ray Imaging Facility, School of Materials, The University of Manchester, Manchester M13 9PL, UK
3State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China
Abstract:We present a comprehensiveexperimentalstudy ofthe variation in apparent unloading elastic modulus of polymer (largely elastic), aluminium (largely plastic) and fibre-reinforced cement (quasi-brittle) closed-cell foams throughoutuniaxial compression. The results show a characteristic “zero-yield-stress” responseand thereaftera rapid increase inunloading modulusduring the supposedly “elastic” regime of the compressive stress-strain curve. The unloading modulus then falls with strain due to the localised cell-wall yielding or failure in the pre-collapse stage and the progressive cell crushing inthe plateau stage, before rising sharply during the densification stage which isassociated with global cell crushing and foam compaction. A finite element model based on the actual 3D cell structure of the aluminium foam imaged by X-ray computed tomography (CT) predicts an approximately linear fall of elastic modulus from zero strain until a band of collapsed cells forms. It shows that the subsequentgradual decrease in modulus is caused by theprogressive cell collapse. The elastic modulus rises sharply after the densification initiation strain has been reached. However,the elastic modulus is still well below that of the constituent material even when the “fully” dense state is approached. This work highlights the fact that the unloading elastic modulus varies throughout compressionand challenges the idea that a constant elastic modulus canbe applied in a homogenised foam model. It is suggestedthat the most representative value of elastic modulus may be obtained by extrapolating the measured unloading modulus to zero strain.
Keywords:Cellular materials;Foam crushing; Young’s modulus; Unloading tests; Image-based modelling
1. Introduction
Foam materials, made of polymers, metalsand otherengineering ornaturalmaterials, are beingused increasingly in many engineering fields for energy absorption, thermal insulation, acoustic damping and sandwich cores. In contrast to condensed solid materials,foams normally have anapparent elastic modulus that is quite different from the initial loadingslope of the uniaxial compressive stress-strain curve,even when the deformation is small and the strains are well within the commonlyrecognised “elastic regime”[1-7].Non-linearity in the stress-strain curve and elastic hysteresis may also occur during unloading-reloading loops, making it difficult to idealise and quantify the elastic properties of foams.
Consequently, the international standard [8]for compression testing of porous and cellular metals is cautious in giving guidelinesfor the measurement of elastic modulus. It defines an elastic gradient as the slope of a straight line connecting two points in the unloading-reloading curve andstates that “the elastic gradient represents a porosity-dependent rigidity, not a modulus of the material, and generally changes during the course of compression”. The complex nature of the elastic behaviour has caused much confusion and uncertainty when a homogenised foam model is to be defined, and therefore, this issue should be clarified.
Despite the lack of a recognised standard method, Ashby et al. [7] recommended that a constant unloading slope can be used to characterise the elastic modulus of metal foams. Furthermore, they suggested the following equations for the correlation between the elastic modulus and the structural parameters of a cellular material
Open-cell foam: (1)
Closed-cell foam: (2)
whereE and are the elastic modulus of the foam and the constituentmaterial, respectively, is the relative density, and andare material constantsthatrange from 0.1 to 4.0 and 0.1 to 1.0, respectively. Although theirrecommended equations have been accepted by many researchers for different types of foams [2, 3, 5, 6, 9, 10], the unloading slope has been determined at different compressive strains to obtain the elastic modulus and the material constants in Eqs. (1) and (2). For instance, Jeon and Asahina[11]used the unloading slope at a strain of 0.06for aluminium Alporas foamwhile Andrew et al. [2]determined it when the stress reaches75% of the expected plastic collapse stress. These previous studies assumed that the elastic modulus isinvariant at small strains, but variations in modulus with strain have been observed even well before the collapse (or yield) strain, see Fig. 4 in Ref. [5]. Therefore, a more reliable method should be used to accurately determine the elastic modulusfor cellular solidsbased on a better understanding of its variation with strain.
Ideally, one would like to evaluate the elastic modulus at a small strain (approaching zero),but this can present experimental difficulties[11]. An unexpectedly low stiffnesshas been observed at small strains for aluminium foamsand this is usually attributed to the premature local yielding of cell walls occurring well before macroscopic yielding, as shown by surface strain measurements[5] and realistic finite element simulations[12]. However, it is not yet clear whether other factors (e.g. weakened load transfer through uneven surface cells due to mechanical cutting) also make contributions. Furthermore, since uniaxial compression actually involves complex modes ofcelldeformation at the meso-scale, it is reasonable to ask whether the discrepancy in stiffness is an intrinsic characteristic of aluminium foams or an experimental observation depending ontest conditions. It is also worthwhile to examine othertypes offoam (e.g. polymer and cement foams) to evaluate how generic this behaviour is and to identify theunderlying mechanisms.
When compression proceeds into the so-called plateau stage over which the stress isalmostconstant over a large range of strain, the unloading modulus has been found to decreasewith strain until the densification stage begins. Thishas beenobserved for aluminium [4, 10] and polymer[13]foams. McCullough et al. [4] attributed this to the change of cell geometry with deformation. Flores-Johnson et al.[13]suggested that the decrease in the elastic modulus arises from heterogeneousdeformation(i.e. progressive cell crushing),proposinga formula for modulus degradation in terms of the nominal strain.For trabecular bone,Moore and Gibson [14, 15] observed the accumulation of micro-damage (i.e. micro-cracks) under uniaxial compression and developed an analytical model to predict the reduction of the resulting modulus.These studies haveexplained some featuresof the modulusvariationduring the plateau stage; however, a systematic study relating changes in cell structure to variations in elastic modulus for both ductile and brittle foams is still outstanding.
Eventually, with increasing compression, all cells become crushed, accompanied by a significant increase in density and a rapid rise ofelastic modulus during the densification stage[10, 13]. The elastic properties of foams at very large deformation have attracted littleattention in the past, partly because this state lies beyond the conventional engineering foam design window. However, it is of fundamental importance to know how closely, and how quickly, the elastic modulusapproaches tothat of the constituent materialas the foam is compressed towardsthe density of the constituent material. This is needed whenthe foam is used under extreme loading conditions, such as intensive blast loading or highspeed impact, which can introduce very large deformations.
This paper presents a systematic study of apolymer (largely elastic), analuminium (largely plastic) and a fibre-reinforced cement (quasi-brittle) foam,overthe whole course of compression. To this end uniaxial compression tests have been undertaken involving intermittent unloading-reloading cyclesto sufficiently high nominalstrains (~80-95%) to approach “fully” dense states.Alongside this, a 3D finite element (FE) simulation based on a realistic cell geometry extracted from computed tomography (CT) imaging wasundertaken to help understand the experimentally observed variation in elastic modulusup to the plateau stage. Then an analytical modelwas usedto elucidate the factors that influence the variation in modulus during the progressive collapse of the foam.Finally, an empirical equation is proposed to describe the elastic modulus at the densification stage.
2. Materials and methods
2.1. Foam samples
The materials studied here are closed-cell foams made of aluminium, polymer and fibre-reinforced cement, as shown in Fig. 1a. The aluminium foam has the trade name ofAlporas(Shinko Wire Company, Ltd) andisproduced by a batch casting process during whichthe pure molten aluminium is thickened by 1.5 wt.-% Ca and then foamed by adding blowing agent of 1.6 wt.-% TiH2at . Thesamples (~Ø29.8×30.2 mm) were prepared by electrical discharge wiremachining to minimize membrane damage. The polymer foam (Rohacell® 51 WF) isproduced by the thermal expansion of a thermoset polymer made from methacrylonitrile (C4H5N) and methacrylic acid (C4H6O2) using alcohol as a foaming agent. Thesamples(~Ø24.8×26.7 mm) were prepared by mechanical cutting. The fibre-reinforced cement foam was produced in the laboratory by mixing cement (Ordinary Portland), fly ash (CEMEX 450-S), fibres (fibrillated polypropylene, 8 mm in length and 50 µm in diameter), foam agent (EABASSOC) and water.It waspoured intoa cylindrical mould and then cut into shorter samples (~Ø53.4×21.3 mm) after curing.
Five samples for each type of foam were prepared and then measured usingan electronic calliper and an accurate weighing scale (see Table 1).The relative density (RD) is defined as the ratio of the sample density to the density of the cell-wall material. The average cell size is about 2.8 mm and0.4 mmfor the commercial aluminium and polymer foams, respectively[3, 9], and is 1.0 mm for the fibre-reinforced cement foam, estimated by cross-sectional image analysis. Therefore, the shortest dimensions of the aluminium, polymer and cement foam samples represent11, 62 and 21 cellson average, respectively, whichare sufficient to obtain representative bulk properties[6].
2.2. Uniaxial compression testswith intermittent unloading-reloading
A series of uniaxial compression tests were undertaken on a standardInstron 200 kNservo-hydraulic testing machine ata nominal strain-rate of 1×10-3 s-1. The samples were compressed to a sufficiently high strain (~80-95%) in order to obtain the whole compressive response. To determine the unloading elastic modulus the samples were intermittently unloaded (to zero force) and reloaded during each test.
To measure the machine compliance the loading platens werecompressed without any sample and the mean slope of the load-displacement curvewas obtained as . However, at large loads (>50 kN) the machine begins to exhibitelastic hysteresis, the effect of which on the unloading modulus measurement was corrected. For each test, the sample displacement at a given compressive loadwasdetermined by subtracting the machine displacement from the cross-head displacement at the same load, as suggested by Jeon and Asahina[11] to accurately determine the elastic modulus. The results, after the correction of the machine compliance,have an accuracy similar to that based on a laser displacement sensor [3]. Then the nominal (engineering) strain and stresswereobtained as the ratio of the derived sample displacementto the original sample height and the loadto the original cross-sectional area, respectively. Asvery large compressive deformationis encountered at the plateau and densification stages, thelogarithmic (or true)strain, which is used in many constitutive formulations and commercial FE codes (e.g. ABAQUS) to modelelasto-plastic material behaviour, is adoptedfor the determination of modulus based on the unloading stress-strain curve. It is worth noting that in practice it is difficult to obtain the true stress since foams are compressible and the Poisson’s ratio varies over the different compression stages. The Poisson’s ratio is reported to be close to zero during the plateau stage forthe aluminium [16]and polymer [9]foams similar to those tested here. In other words, there will be negligibledifference between the nominal stress and the true stress over the large strain range covered by the plateau stage. By contrast, over the densification stage the foams are naturally much less compressible as evidenced by the increased cross-section and density after crushing of the foams, see Fig. 1b. To obtain consistent results, the nominal stress was used for the whole compression process. Theunloading modulusis calculated using the following equation
(3)
whereσ is the nominal stress and is the elastic component of thelogarithmic strain defined aswhere is the nominal strainwith positive sign in compression. A similar calculation was also performed by Bastawros et al. [5]. Note that hereinafter adding a bar above a variable indicates a definition based on logarithmic strain. In fact, the “true” unloading modulus can be derived from the nominal one, i.e.
(4)
since during unloading we have
(5)
where is the elastic component of nominal strain. From Eq. (4) it is also seen that the “true” unloading modulus is smaller than the nominal one, but for small strainstheir difference is negligible.
In the experiments and simulation (described later), the apparent unloading elastic modulus was determined as the mean slope of the curve between the 30-70% of the stress at which the unloading started.Conventionally the nominal strain is used to describe the compression of foams because the compression limit is clearly defined when . To follow this conventionthe elastic modulusstudied hereisregarded as a function of nominal strain while the large deformation effect in constitutive equation is taken into account by Eq. (3).The nominal elastic strain was also measuredas the recoverable nominal strain upon complete unloading,
2.3. Defining three compression stages
One unique feature of the compressive stress-strain curve of a foam is the presence of a plateau stageduring which the stress is almost constant, as shown schematically in Fig. 2. In this study, we define the regime before the initial peak stress as the pre-collapse stage (with the peak stress denoted by ).Terms such as“elastic regime” and “linear regime” for the pre-collapse stage are not really appropriate as inelastic deformation and stress-strain nonlinearity can occur. The initial loading slopeis defined as the mean slope of the curve at the stresses up to aboutwhich has been chosen to be small enough to represent the initial compression and avoid the subsequent nonlinearity effect, but large enough to obtain valid test data.The unloading modulus at the collapse initiation strain,defined as the strain corresponding to the peak stress ,is denoted by .
The densification stage is characterised by a rapid risein stress at the end of the plateau stage. An energy absorption efficiency methodis adopted to determine the densification initiation strain, which gives a unique and consistent measurefor the onset of the densification[17]. The energy absorption efficiency[18] is defined by
(6)
The densification initiation strain is obtained as the strain corresponding to the maximum energy absorption efficiency, i.e.
(7)
A typical curve is shown in the inset of Fig. 2 wherein the determination of is depicted.
2.4. Image-based finite element model for thealuminium foam
In order to examine the extent to which the observed unloading modulus variation is an inherent property of the foam ora consequence of the practical difficulties associated with loading a foam sample, a realistic 3D image-based model of the aluminium foam was created. This was necessary because cell morphology and topology are crucial geometrical factors in determining the compressive properties of foams[19].Three-dimensional modelling based on CT images (so-calledimage-based modelling [20]) has been used previously by a number of researchers to study the deformation of open-cell [21, 22]and closed-cell[12, 23] foams. Herea CT imageof the aluminium foamwasused toobtainthe cell structure for the image-based FE modellingso as to investigate the elastic response and local yielding prior to cell crushing.
A cylindrical Alporas foam sample (Ø15.0×14.4 mm), small enough to achieve high spatial resolution, was imaged using a Nikon Metris CT system housed in a customised bay at the Manchester Henry Moseley X-ray Imaging Facility (MXIF, Manchester, UK) and then reconstructed into a 3D CT image using Nikon Metris CT-Pro software. The accelerating voltage and current of the X-rays were set as 75 kV and 125 μA, respectively, and the original voxel size was 10.9 µm and later down-sampled to 50.0 µm so as to reduce the element number of the FE mesh created.The 3D CT imagewas cropped to the desired size (Ø11.0×13.5 mm) and some cells on theboundaries were cleaned up (some surface damages were apparent in the image arising fromsample preparation) in order to facilitate FE model construction (e.g. meshing and boundary condition definition).Tinypores (<400 µm)were filtered out and the greyscale-based segmentation process was applied. The relative density after meshingis0.109, close to the average relative density (namely 0.100) of the aluminium foam samples tested here.The FE mesh, as shown in Fig. 3, was generated using the commercial code ScanIP (Simpleware Ltd, UK), and comprises493150 quadratic tetrahedral elementsto ensure thenumerical precision of thegeometrical discretisation and the simulations of the bending and buckling of thin cell walls.
The cell-wall material propertiesare assumed to be homogeneous and isotropic, and are modeled by Hooke’s law and von Mises plasticity. The consitutive relationship for the cell-wall material can be determined from the following uniaxial stress-strain equations
(8)
(8)
where and are the uniaxial true stress and strain, respectively, is the Young’s modulus, is the yield strength and n is the hardening exponent of the cell-wall material. For the Alporas foam,, and [24].