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Mechanics of Materials

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Research paper

Crushing and densification of rapid prototyping polylactide foam:Meso-structural effect and a statistical constitutive model

Wang Shilonga, Zheng Zhijuna,⁎, Zhu Changfenga, Ding Yuanyuanb, Yu Jilina

a CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026,ChinabMechanics and Materials Science Research Center, Ningbo University, Ningbo 315211, China

A R T I C L E I N F O

Keywords:Polylactide foam3D printingCompression testMeso-structural effectStatistical constitutive model

A B S T R A C T

The deformation characteristics and mechanical behavior of closed-cell polylactide (PLA) foams under quasi-static uniaxial compression are investigated. The PLA foam specimens are constructed numerically with 3DVoronoi technique and prepared with fused deposition modeling (FDM). The effects of meso‑structure, layerdeposition strategy and relative density on the crushing and densification behaviors of PLA foams are analyzed.The experimental results indicate that PLA foams with regular meso‑structures are more likely to induce localcollapse during compression, and those with random meso‑structures show good performances, such as highspecific energy absorption, low stress fluctuation and stable deformation process. The experimental resultsfurther reveal that random PLA foams with the deposition layers perpendicular to the loading direction havehigher strength and smaller lateral expansion than those with other orientations. The quasi-static compressionstress-strain curves of random PLA foams exhibit three typical stages, namely linear elastic stage, plastic collapsestage and densification stage. A stress drop after reaching the initial peak stress is observed. To characterize thesefeatures, we propose a statistical constitutive model which takes into account the meso‑structural characteristicsand deformation mechanism at the cell scale. This constitutive model contains six parameters and fits the ex-perimental results very well. In particular, it can capture the peak stress at the onset of plastic collapse and thesubsequent stress drop of cellular materials. Finally, the dependence of the six parameters in the constitutivemodel on the relative density is analyzed and the corresponding quantitative statistical relationships are de-termined.

1. Introduction

Cellular materials are extensively used as impact energy absorbersdue to their considerable capacity in energy absorption (Merrett et al.,2013; Ding et al., 2016). The typical stress–strain response of cellularstructure under uniaxial compression indicates that external energy isdissipated in a smooth manner with a nearly constant stress level for awide strain range (Gibson and Ashby, 1997). Different failure me-chanisms and compressive strengths are demonstrated on the designedcellular materials with micro-truss or micro-shell structures (Jens et al.,2014) and the meso‑structural dependence of cellular materials is evi-dently marked. Actually, the meso‑structure design of cellular structurelargely determines the deformation behavior in the bending-dominatedform or in the stretching-dominated form (Deshpande et al., 2001;Evans et al., 2001). In addition, the relative density also has remarkableeffect on the crushing behavior of cellular materials. The linear

relationship between the initial peak stress and the relative density foropen-cell foams was documented (Jang and Kyriakides, 2009;Gaitanaros and Kyriakides, 2015). Therefore, the presence of a varietyof specific meso‑structures provides a powerful design capability forcellular structures to meet the specific needs of the corresponding ap-plication areas.

Traditional methods used to produce cellular materials, especiallythe metallic foams, include melt solidification, solid-state sintering andmelt deposition, as detailed in Ref. (Degischer and Kriszt, 2002).Though these methods have been improved greatly in recent years, howto effectively control the size, shape and pores distribution of cells, andhow to achieve better reproducibility of cellular specimens, is still anurgent problem. Recently, rapid prototyping/3D printing becomes analternative method to produce cellular structures. It has a great ad-vantage over the traditional methods on unparalleled flexibility andunrestricted complexity of meso‑structures (Yan and Gu, 1996). Among

https://doi.org/10.1016/j.mechmat.2018.09.003Received 22 February 2017; Received in revised form 3 September 2018; Accepted 6 September 2018

⁎ Corresponding author.E-mail address: zjzheng@ustc.edu.cn (Z. Zheng).

Mechanics of Materials 127 (2018) 65–76

Available online 15 September 20180167-6636/ © 2018 Elsevier Ltd. All rights reserved.

T

rapid prototyping processes, selective laser sintering is used to fabricatemetallic cellular structures (Chantarapanich et al., 2014) or nylon mi-crostructures (Andreassen et al., 2014), while fused deposition mod-eling (FDM) is applied to fabricate polymeric foams (Maiti et al., 2016).The dimensions of cellular structures fabricated with 3D printingtechnique can range from submicron to centimeter level with variousarchitectures and base materials. However, the current research oncellular structures produced by 3D printing technique is mainly con-fined to open-cell foams or periodic lattice structures. Such structuresdeform mainly in the form of struts bending. In contrast, the dominateddeformation mechanism is the collapse and stretching of cell walls forclosed-cell foams (Gibson and Ashby, 1997), which are more suitablefor energy absorption and crashworthiness applications.

Stochastic cellular materials are highly heterogeneous at meso‑scalein geometry and porosity distribution, with coupling effect among cells.Therefore, the stress–strain response of cellular material under quasi-static compression exhibits high nonlinearity because of the compli-cated cell structures. Several empirical models of stress–strain response(Rusch, 1969; Hanssen et al., 2002; Liu and Subhash, 2004; Avalleet al., 2007; Zheng et al., 2014; Li et al., 2012) were proposed and themechanical characteristics of cellular materials for particular applica-tions were discussed. Rusch (Rusch, 1969) suggested a simple approx-imation in terms of two power laws to quantitatively describe the me-chanical behaviors of polymeric foams. The Rusch model is not accurateenough to describe the densification stage of cellular materials, andsome more accurate constitutive models were presented to considerhardening behavior (Hanssen et al., 2002; Liu and Subhash, 2004;Avalle et al., 2007; Zheng et al., 2014) and strain-rate effect (Li et al.,2012). These phenomenological constitutive models can depict themechanical behavior of various cellular materials under large de-formation, but they are not able to account for the heterogeneous fea-ture and the statistical behavior of cells during crushing. Essentially, themacroscopic mechanical response is the statistical average of the micro-mechanical behavior of numerous cells or representative elements(Hu et al., 2007). For such trans-scale problem, the statistical mechanicswell bridges the gap between the mesoscopic description and themacroscopic behavior of materials (Bai et al., 2005). The statisticalanalysis of tensile and compression behaviors of aluminum foams wascarried out by Blazy et al. (2004) and McCullough et al. (1999) throughintroducing the Weibull distribution to characterize the heterogeneityof aluminum foams. Later, a stochastic constitutive model for dis-ordered cellular materials was developed by Schraad andHarlow (2006) by assembling the geometric stiffness of foam cells witha proper probability density function. However, this model can not welldescribe the feature that the load-carrying capacity of cellular materialdrops after the onset of collapse, and also it does not provide an explicitform of stress–strain relation. Another constitutive relation was con-structed for open-cell aluminum foams by considering the mesoscopic

characteristics of cells (Hu et al., 2007) and yet it is also not capable ofcapturing the feature of stress drop.

In this paper, an FDM manufacturing technique of rapid prototypingis employed to fabricate closed-cell foams using polylactide (PLA) as thebase material. The mechanical behavior of PLA foams under quasi-staticuni-axial compression is investigated experimentally and theoretically.In Section 2, the preparation conditions of specimens and the experi-mental scheme are elaborated, together with a brief introduction of theprinciple of 3D printing technique and the construction of printablemodel. In Section 3, the effects of meso‑structure, layer depositionstrategy (i.e. the orientation of layer deposition) and relative density onthe quasi-static compressive behaviors of PLA foams are explored. InSection 4, a six-parameter statistical constitutive model is proposed byconsidering the micro-mechanical mechanisms, and the material para-meters in this model are identified and analyzed.

2. Material and methods

2.1. Voronoi structures

The geometric model of closed-cell foam is constructed by usingVoronoi technique (Okabe et al., 1992), as the growth mechanism ofcells in the foaming process of many types of foams, e.g. closed-cell Alfoam, is similar to the principle of Voronoi diagram. Firstly, N nucleiare randomly seeded in a desired region one by one, which follows theprinciple that the distance between any possibly added nucleus and allthe existing nuclei must be larger than the minimum distance (Zhu andWindle, 2002). Then, the nuclei are translated to the surroundingneighboring volumes to ensure periodic boundary conditions. Finally, aVoronoi structure, in which each cell consists of several polygonalsurfaces that are mid-vertical planes of two adjacent nuclei, is gener-ated and the geometric model of closed-cell foam with the desired sizeis reserved by cutting the large sample, as shown in Fig. 1(a). In ad-dition, it is noted that to generate a regular structure, nuclei should beseeded regularly.

Geometrically, cell walls in a Voronoi structure should be operatedto have a “thickness” that allows thermoplastic material being stackedby FDM manufacturing technique to materialize Voronoi specimens. Astrategy to achieve such aim is that all nodes on one cell are moved to aspecific distance by assigning the nucleus of current cell as reference.Since each surface in a Voronoi structure is shared by two adjacentcells, new surfaces are mirrors about the original surface essentially.The gap region between the two new surfaces represents a primitivecell-wall thickness of cellular structure. The schematic diagram of cell-wall thickening process is presented in Fig. 2.

It is noticed that cells inside configuration are self-closed, but theexposed cells which intersect with boundaries are somewhat broken.Therefore, the gaps between the original and the new cell surfaces are

Fig. 1. Cell-based Voronoi structure (a), a model with thickened cell walls by topological operation (b) and the corresponding produced PLA foam (c).

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filled up with additional surfaces to assure the requirement of closedpaths in 3D printing technique (Yan and Gu, 1996). After these topo-logical operations, the Voronoi structure is closed both outside andinside, see Fig. 1(b). And the space, which is turned into an inter-connected cavity, between any two adjacent surfaces is solidified with aspecific thermoplastic material via 3D printing technique.

2.2. Process of 3D printing

By reassembling the topologies of cells, a 3D Voronoi structure withclosed paths is generated. However, unlike solid models that aremodeled with engineering modeling software, the so-called solidVoronoi structure can not be directly exported as a STL file which isapplied to rapid prototyping of 3D System Inc. originally (Yan andGu, 1996). Actually, the STL file follows a simple rule that the exteriorsurfaces of a model are tessellated into a set of triangles, of which thenormal vector points to material-free. Therefore, each surface in aVoronoi structure is discretized with triangles and then these triangularfacets are recorded in an ASCII format file in forms of anticlockwise sortof vertex coordinates with the corresponding normal vector pointingoutward. Then, the solidifying paths (outline of the solid parts) of theVoronoi structure are mathematically sliced by FDM software Quick-Slice™ with parallel horizontal planes, as depicted in Fig. 3(b). Duringthe process of fabricating, a thermoplastic material is melted intoflowing status and the extruded filament from nozzle deposits along theexact outline of solid part. Then the solidified part inside the outline isbuilt with the liquid filament weaving alternately through the move-ment of the extrusion head. The next layer is bonded to the previouslayer through thermal heating and this procedure is repeated layer bylayer in the Z-direction until the model is fully constructed. A schematicof the 3D printing process and the specimen orientations are shown inFig. 3(a), where the X, Y and Z directions are associated with the printerwhile the coordinates 1, 2 and 3 are associated with samples.

2.3. Fabricating conditions

PLA and Acrylonitrile Butadiene Styrene (ABS) are two commonthermoplastic materials for FDM technique. Since the strength andtoughness of PLA are higher than those of ABS (Tymrak et al., 2014),we choose PLA as the base material of Voronoi structures. Actually, aVoronoi structure has similar cell morphology of PLA foam foamingwith supercritical CO2 as blowing agent (Corre et al., 2011), whichindicates that a Voronoi structure can well characterize the micro-architectural features of PLA foams. The density of PLA material used inthis study is ρs= 1240 kg/m3 and the 3D printer of FDM is supplied byShanghai Best Industry Co. Ltd in China. The dimensions of the pre-pared samples are 40× 40×60 mm3 with an equivalent cell diameterof about 7.5mm. The degree of cell irregularity of random configura-tion is set to be 0.4 (Zheng et al., 2005). The regular configurations

were constructed with corresponding specific nuclei arrangements. Thebuilding temperature of nozzle is 210 °C and the temperature of heatedbed is 60 °C. The slice thickness along printing direction layer-wisely is0.1 mm and the printing speed is set as 1.2 mm3/s. A fabricated spe-cimen of PLA foam is demonstrated in Fig. 1(c).

The relation between the relative density and the cell-wall thicknessof the fabricated random PLA foams is shown in Fig. 4. It is found thatthere is a deviation of the measured relative density of the fabricatedPLA foams to the pre-set relative density. The possible reasons could besummarized as numerical errors in facet approximation, non-uniformcell-wall thickness due to residual material scraps, topological re-construction, and the fully filled small size cells due to the limitedprinting accuracy. Besides, an exception is noticeable for PLA foam withcell-wall thickness of 0.6mm whose relative density is a little lowerthan the expected value. This may be explained as the partial loss of cellwalls during printing for the limited printer precision. The cell-wallthickness for this case is small and close to the size of the nozzle dia-meter (0.4 mm). Hence, for the sake of success and reliably, the cell-wall thickness of Voronoi structures should be larger than 0.4mm. Inthis study, only samples with the cell-wall thickness no less than0.8 mm are tested and the relative density used in the following sectionsis the measured one consistently.

2.4. Uniaxial compression

Quasi-static compression tests were carried out on Material TestingSystem MTS 809, which is located in the Engineering and MaterialTesting Center of USTC. All specimens were compressed along thelongitudinal axis 3 with layer deposition orientation D3 (see Fig. 3(b)).During testing, a strain rate of 0.001/s was imposed on the moving headthat was controlled precisely by hydraulic pressure. The experiencedtime, load and displacement along the longitudinal direction were de-tected and recorded constantly.

To assess the mechanical performance of cellular material as energyabsorbers, the specific energy absorption capacity (SEA) (Kim, 2002)and the undulation of load-carrying capacity (ULC) (Wang et al., 2011)are chosen as two evaluating indicators. The SEA is a dimensional in-dicator and is defined as the ratio of total energy absorbed up to thedensification strain to the mass of foam, written as

∫=ρ ρ

σSEA 1 (ɛ)dɛ,s 0

ɛD

(1)

where ρ is the relative density of foam and ρs the density of base ma-terial. The densification strain εD is determined as the strain corre-sponding to the maximal value of the energy absorption efficiencyfunction η(ε), which is defined as (Tan et al., 2005)

∫=ησ

σ(ɛ) 1(ɛ)

(ɛ)dɛ.0

ɛ

(2)

Fig. 2. Thickening cell walls of closed-cell foam to obtain the regions of materialization.

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The energy absorption efficiency characterizes the variation of ef-fective specific energy absorption with strain during uniaxial com-pression of cellular structures. The densification strain, which char-acterizes the transition from the plateau stage to the compaction stage,is identified by the strain corresponding to the maximum energy ab-sorption efficiency.

The dimensionless indicator ULC is defined as (Wang et al., 2011)

∫∫

=−σ σ

σULC

(ɛ) dɛ

(ɛ)dɛ,0

ɛavg

0ɛ

D

D(3)

where σavg is the mean stress averaged over the effective strain range(from zero to the densification strain εD). The undulation of load-car-rying capacity characterizes the stability of compression stress of cel-lular structures during crushing.

3. Results and discussion

3.1. Effect of meso‑structures

The presence of meso‑structures provides possibility for cellularstructures to be optimized to possess better mechanical and physicalperformances (Evans et al., 2001). For a given base material, the in-fluence of meso‑structures on the macroscopic behaviors of PLA foamsis investigated by constructing five structures with different config-urations. The unit cells of the five configurations are Random Cell (RC),Tetrakaidecahedron (Tet), Hexagonal Prism (HP), Rectangular Prism(RP) and Clipped Rectangular Prism (CRP). Actually, a CRP config-uration is formed by cutting the eight corners of rectangular prism cellswithin an RP configuration and then eight surfaces is additionallyadded to the remained part of the operated RP cell while the removedpart forms the octahedron. The unit cells and the corresponding PLAfoams with different meso‑structures are shown in Table 1.

The average relative density for configurations RC, Tet, HP, RP andCRP with cell-wall thickness 0.8mm are measured as 0.237, 0.207,0.174, 0.188 and 0.213, respectively. It is noted that configuration HPhas the lowest relative density while configuration RC appears to havethe highest one among the five configurations. An explanation for thisdiscrepancy is that the regular convex polytope could occupy morespace than that with irregular ones for the same equivalent cell size.

Fig. 3. Schematic diagram of the 3D printer by FDM technique (a) and the samples fabricated in three different layer deposition orientations (b).

Fig. 4. Variations of the relative density of random foams with the pre-set cell-wall thickness.

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Besides, for given cell-wall thickness and dimension of foams, the re-lative density is negatively correlated with cell size. The uniaxial re-sponses of the five configurations of PLA foams are investigated underquasi-static compression, as shown in Fig. 5. In the early stage of elasticresponse, the stress–strain curves of the five configurations are linearand nearly overlap with each other. However, dramatic fluctuations ofthe stress–strain response can be found in the plateau stage for con-figurations HP, RP and CRP, while configurations RC and Tet have re-latively stable performance in this stage. The high fluctuations in theearly plateau stage for configurations HP, RP and CRP are related to theplastic collapse of cell walls parallel to the loading direction, whichbehaves like the crooked plates with Mode II deformation(Calladine and English, 1984).

The deformation processes of the five configurations under quasi-static loading are demonstrated in Fig. 6. For configuration RC, thecrushing band initially occurs in the region where the strength of cellwall is relatively weak. As the heterogeneous strain concentrationcaused by the anisotropy of cells, the spread of crushing band may notbe continuous for insufficient perturbation from the existing cell-wallbuckling (Bastawros et al., 2000), and new shear bands may occurelsewhere, as demonstrated in Fig. 6(a). Compared to configuration RC,configuration Tet deforms more regularly, i.e. the collapse of cells in-itiates from one end of the specimen and then propagates to the otherend in a progressive manner. As cells within foam are almost com-pacted, the foam becomes a solid block that consists of stacked cellswith cracks along the adjoining bond of filaments. Different to config-urations RC and Tet, configurations HP, RP and CRP present a globalbuckling mode. This could be attributed to the inclined crushing bandacross the entire section. The initial failure of imperfect sites (such ascorrugation, non-uniform thickness or incomplete bond adjoining) de-teriorates the collapse strength of those local regions, which aggravates

the imbalance in force distribution. The cells, which are squeezed tobulge outward, are not completely compacted but with the cell wallsbeing torn.

A quantitative evaluation of energy absorption characteristic of PLAfoams with these five configurations is carried out. By applying Eqs. (1)and (3), the values of SEA and ULC of the five configurations are cal-culated and compared in a histogram, as shown in Fig. 7. It is found thatthe values of SEA for configurations HP, CRP and RP are higher thanthose for configurations RC and Tet; while the values of ULC exhibit asimilar behavior. The tube-like arrangement of meso‑structures inconfigurations HP, CRP and RP makes the cell walls form relatively fullplastic folds. Additionally, the collapsed cells are further stretchedduring compression. However, the external energy is dissipated in anoscillating manner because abrupt instability happens in the layer-by-layer collapse of cells. For configurations RC and Tet, the oblique cellwalls in the loading direction may easily induce the collapse of weakcells and the cracks along the adjoining bond. Moreover, it should benoticed that configurations RC and Tet present macroscopically a moreuniform deformation pattern compared to the other three configura-tions. For practical applications of cellular materials as energy absor-bers, the desirable mechanical behaviors and designable structuralproperties should be taken into consideration to maximize the effec-tiveness of cellular materials under specific needs.

3.2. Effect of layer deposition strategy

It has been found that cracks of cell walls grow perpendicular to thebuilding direction Z (i.e. between two successive layers), as observed inSection 3.1. Herein, three groups of random PLA foam samples werefabricated with an identical Voronoi structure but different depositionstrategies, denoted as D1, D2 and D3 as illustrated in Fig. 3(b), but all ofthem were compressed along the longitudinal axis (direction 3). Twospecimens with the same deposition strategy were prepared and tested.Slight difference in the relative density was found. The relative den-sities of the three groups of PLA foams are 0.240 and 0.230 for D1,0.233 and 0.227 for D2, and 0.239 and 0.233 for D3.

For foams D1 and D2, the stretching force acting on the cell wallsgive rise to crack along the bonds at a nominal strain of about 5%, asillustrated in Fig. 8. Then, the cracks develop into crushing bands alonga relatively weak region. It is also noticed that the initial crushing bandsof foam D1 occur at the middle region approximately, which seeminglypropagates with the direction oblique to the loading direction. Lateralvolume expansion occurs obviously during compression, as demon-strated in Fig. 8 for D1 and D2 at a nominal strain of about 40%, whichimplies that the plastic Poisson's ratio effect cannot be neglected forsuch PLA foams prepared with a certain printing direction. It is worthnoticing that the plastic Poisson's ratio for foam D1 can exceed 0.5, i.e.the upper limit of dense solids. This feature for cellular material is re-lated to the fracture of structures. A similar feature is captured in foam

Table 1The unit cells and the corresponding samples of foams.

Fig. 5. Quasi-static nominal stress–strain curves of PLA foams with differentmeso‑structures.

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D2, although the location of initial failure and the apparent shapeformed with the interactions of shear bands are different because foamD2 is not completely equivalent with foam D1 due to the randomness ofcell geometry. Different from foams D1 and D2, foam D3 initially de-forms with a random crushing band in the weakest site, and then newcrushing bands appear with the enhancement of the existing band in-teraction (Zheng et al., 2014), as illustrated in Fig. 8. The difference in

deformation patterns of PLA foams with different deposition strategiescan be attributed to the anisotropic property of cell walls that is in-troduced during the manufacturing process via FDM technique. Sincethe stress to form plastic collapse is higher than that to break thebinding layer apart, cell walls are ruptured more easily along the bondadjoining filaments when the layer deposition orientation is parallel tothe loading direction. Accordingly, the plateau stress of foam D3 is

Fig. 6. Deformation patterns of PLA foams with five configurations: (a) RC, (b) Tet, (c) HP, (d) RP and (e) CRP.

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nearly twice as high as those of foams D1 and D2, see Fig. 9.The quasi-static stress–strain curves of PLA foams under uniaxial

compression present a feature of three distinct stages, similar to that ofmetallic foams (Banhart and Baumeister, 1998), namely a linear elasticstage, a long plastic plateau stage and a rapid compaction stage, asshown in Fig. 9. During loading, the weakest part of a PLA foam col-lapses firstly (Bastawros et al., 2000) and it results in a rapid decrease of

load-carrying strength. As the load continues, the PLA foam deforms ata relatively stable stress level until all cells are nearly compacted.Subsequently, a rapid increase of stress within a small strain rangehighlights the compaction behavior. It is noticed that the initial

Fig. 7. Comparison of SEA and ULC of the five configurations.

Fig. 8. The deformation processes and the failure modes of PLA foams with different layer deposition strategies: (a) D1, (b) D2 and (c) D3.

Fig. 9. The stress–strain curves of PLA foams fabricated along different de-position directions.

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crushing stresses for foams D1 and D2 are slightly less than those offoam D3. It transpires that the compressive strength of PLA foamsfabricated in the vertical direction (D1 and D2) is relatively weakerthan that fabricated in the horizontal direction. In other words, theproperties of PLA foams remarkably depend on the layer depositionorientation with respect to the loading direction. Such phenomenonwas also observed in steel cellular lattices that were manufacturedusing the selective laser melting technique (Alsalla et al., 2016).

Hence, the specimens used in the following investigations areprinted along the longitudinal direction uniformly, i.e. the layer ma-terial deposition along direction 3 (foams D3).

3.3. Effect of relative density

The effect of relative density on the mechanical behavior of cellularmaterials is carried out by fabricating random PLA foams with 3Dprinting technique for a given Voronoi structure. Evidently, the greatadvantage of 3D printing technique over traditional methods to preparecellular foams is that the same meso‑structure can be reached for dif-ferent specimens. A random Voronoi structure with different cell-wallthicknesses is used for this purpose and PLA foams with different re-lative densities are prepared.

A group of PLA foams with cell-wall thickness ranging from 0.8 mmto 1.4mm with an interval of 0.2mm have been fabricated and theaverage relative densities of them are 0.244, 0.275, 0.305, 0.339 and0.374, respectively. The quasi-static compression stress–strain curves ofthe PLA foams are plotted in Fig. 10. Three typical stages can be foundfor these PLA foams and the relatively smooth plateau stages indicate astable deformation process. A stress drop appears after the first peakstress and the decrease has little dependence on the relative density.This feature of stress drop has been found in many types of cellularmaterials, e.g. closed-cell aluminum foam (Mu et al., 2010). As thecollapsed cells are nearly compacted, the stress is then slightly re-covered to the plateau stress level. Additionally, the hardening behaviorof PLA foams is slightly enhanced with the increase of relative densitybecause cellular materials with thicker cell walls have smaller voids toaccommodate plastic folds (Wang et al., 2017), so the probability of cellwall collapse decreases.

The quantitative relationship between the strength and the relativedensity of PLA foams is investigated by calculating the initial crushstress σ0 and the plateau stress σpl. The plateau stress σpl is defined asthe average stress over a strain range from the yield strain εy to thedensification strain εD, which is expressed as

∫=−

σ σ1ɛ ɛ

(ɛ)dɛ.plD y ɛ

ɛ

y

D

(4)

The yield strain εy is taken as 0.02, as used in Ref. (Liu et al., 2009).The variations of the initial crush stress σ0 and the plateau stress σplwith the relative density are shown in Fig. 11. It appears that both theinitial crush stress σ0 and the plateau stress σpl follow the power-lawrelations with the relative density ρ, written as

⎧⎨⎩

==

σ α ρσ α ρ ,

n

n0 1

pl 2

1

2 (5)

where the fitting parameters are determined as α1= 60.6 ± 6.7MPa,n1= 1.67 ± 0.09, α2= 59.8 ± 4.7MPa and n2= 1.82 ± 0.07.Moreover, the difference between the initial crush stress σ0 and theplateau stress σpl increases with the increase of relative density becausethe deformation resistance of cell walls is enhanced.

4. A statistical constitutive model for random PLA foams

The three-stage characteristics of deformation of PLA foams hasbeen captured by many phenomenological constitutive models in theliterature (Rusch, 1969; Hanssen et al., 2002; Liu and Subhash, 2004;Avalle et al., 2007; Zheng et al., 2014), but no constitutive model cancapture the feature of stress drop, as demonstrated by a typical phe-nomenological model in Fig. 12. Physically, cells in a real foam materialare totally heterogeneous on the meso‑scale due to random distribu-tions of shape, size and imperfections. Moreover, the meso‑instabilitywithin cells during crushing greatly affects the mechanical response ofcellular foams and is responsible for the stress drop. A few micro-sta-tistical models (Hu et al., 2007; Schraad and Harlow, 2006) have beenproposed by taking the random meso‑structure into consideration.

Fig. 10. Quasi-static stress–strain curves of random PLA foams with the sameVoronoi structure but different relative densities.

Fig. 11. Variations of the initial crush stress σ0 and the plateau stress σpl withthe relative density.

Fig. 12. A quasi-static stress–strain curve of PLA foam and two fitting models.

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However, these statistical models are still unable to capture the peakstress at the onset of collapse and the subsequent stress drop. In thisstudy, we aim to propose a new statistical constitutive model that cancapture the feature of stress drop as well as the three-stage character-istics of deformation.

4.1. Derivation of a new statistical constitutive model

For a single cell in a foam structure, its shape and size are randomand its mechanical state is greatly affected by the surrounding cells.Thus, an exact description of the crushing and densification behaviorsof any cell is obviously impossible. However, as expected, each cellexperiences a deformation process of elastic, plastic collapse andcompacting stages, as illustrated in Fig. 13(a). Therefore, we consider avirtually representative cell with an assumed stress–strain response,which implicitly includes the complicated interaction among cells.

It is supposed that the stress of the virtually representative cell dropsat the onset of collapse corresponding to a strain of ε0, before whichthere is a linear elastic stage with σ= Eε, where E is the elastic mod-ulus. Then, the load-carrying capacity of the cell drops to σ= B, whichdominates the mechanical behavior during plastic buckling. Actually,the strain hardening behavior plays a certain role in the buckling-dominated stage, which may be considered as an average response ofnumerous cells during plastic buckling, as shown in plateau regime inFig. 13(a). Finally, as the cell is nearly compacted, the contact of cellwalls contributes mostly to a strain hardening response. To characterizethe strain-hardening behavior, we assume a strain-hardening law whichfollows σ ∼ Cε/(εm− ε)2, where C is the strain-hardening parameter(C << E) and εm the maximal strain that the virtually representativecell can reach. This strain-hardening law can be deduced from theHugoniot relation of shock wave speed versus impact velocity (Barneset al., 2014; Sun et al., 2016; Zaretsky and Ben-dor, 1995), and gives aconsiderable value only when the strain is relatively large. Therefore, aunified form of the stress–strain response of the virtually representativecell can be given by

= − + − +−

σ E B C(ɛ; ɛ ) ɛH(ɛ ɛ) H(ɛ ɛ ) ɛ(ɛ ɛ)

,cell 0 0 0m

2 (6)

where H(ε) is the Heaviside step function (Weisstein, 2002).The material parameters E, B, C, ε0 and εm are related to the relative

density of the virtually representative cell, which is taken to be iden-tical to the relative density of the foam considered. Besides, the collapsestrain ε0 may be very sensitive to the defects or the strain rate of cell.Thus, we assume the collapse strain of cell follows a probability dis-tribution f(ε0), as illustrated in Fig. 13(b). It is noted that∫ =∞ f (ɛ )dɛ 10 0 0 and the value of f(ε0) when ε0> 1 is small enough tobe ignored. Although a nominal strain should be less than unity, wehave extended the upper limit of the integral to infinity, which does notbring appreciable error.

The macroscopic response should be a statistical ensemble of allcells, given by

∫=∞

σ σ f(ɛ) (ɛ; ɛ ) (ɛ )dɛ .0 cell 0 0 0 (7)

Substituting Eq. (6) into Eq. (7), we have

∫ ∫= + +−

∞σ E f B f C(ɛ) ɛ (ɛ )dɛ (ɛ )dɛ ɛ

(ɛ ɛ).

0

ɛ0 0 ɛ 0 0

m2 (8)

For heterogeneous foam cells, it is assumed that the collapse strainε0 follows the Weibull distribution, written as (Weibull, 1951)

= −−f kλ

λ λ(ɛ ) ·(ɛ / ) exp[ (ɛ / ) ],k k0 0

10 (9)

where k and λ are the shape parameter and the scale parameter, re-spectively.

Then, by substituting Eq. (9) into Eq. (8), a six-parameter statisticalconstitutive model of cellular material is obtained as

= − + +−

−σ E B B C(ɛ) ( ɛ )e ɛ(ɛ ɛ)

.λ(ɛ/ )

m2

k

(10)

It can be seen from Fig. 12 that this statistical constitutive model isable to capture the main features of the stress-strain curve of a PLAfoam under compression. Moreover, this model is elegant in form andcontinuously differentiable. This model contains six parameters E, B, C,λ, k and εm, which may be related to the relative density of foam.

By applying the statistical constitutive relation Eq. (10), the com-pressive stress–strain curves of PLA foams with different relative den-sities presented in Fig. 10 are fitted, as shown in Fig. 14. The newstatistical constitutive relation exhibits considerable ability to accu-rately represent the entire process of PLA foams during crushing.

Fig. 13. The collective stress–strain response of a virtually representative cell (a) and the distribution function of collapse strain (b).

Fig. 14. Characterization of the statistical model for the compressive stress-strain responses of PLA foams.

S. Wang et al. Mechanics of Materials 127 (2018) 65–76

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4.2. Parameters identification of the statistical constitutive model

The material parameters in our proposed statistical constitutivemodel are determined from tests and are plotted in Fig. 15. The resultsshow that parameters E, B and C depend on the relative density bypower law, see Fig. 15(a) and (b). The data fitting gives E=1653ρ1.43 MPa, B=60.3 ρ1.94 MPa and C=1.91 ρ1.06 MPa. For the elasticmodulus E, the proportional constant is related to the geometric to-pology and the property of base material, and the exponent refers todeformation mechanism (Roberts and Garboczi, 2002). For closed-cellPLA foams fabricated with sacking layers of thermal filament, cell-wallstretching is the predominant deformation mechanism, which results inthe exponent falling within a range between 1 and 2 (Gibson andAshby, 1997). For the strain-hardening parameter C, similar power-lawexpressions have been proposed for expanded polypropylene foam withan exponent of 2.43 and expanded polystyrene foam with an exponentof 1.39 (Avalle et al., 2007), or 3D closed-cell foam with an exponent of1.50 (Yang et al., 2017). The exponent difference of different cellularmaterials is mainly due to different deformation mechanisms caused bymeso‑structures. Here, the exponents in the power-law fitting expres-sions for parameters E, B and C are close to 1.5, 2 and 1, respectively.For simplicity and convergence, we suggest the following fitting ex-pressions

⎧

⎨⎩

===

E aρB bρC cρ

,

3/2

2

(11)

with the fitting parameters determined as a=1796MPa, b=64.6MPaand c=1.78MPa, as illustrated in Fig. 15(a) and (b).

The effects of relative density on the distribution parameters k and λthat are used to characterize the meso‑mechanical behaviors of cells areinvestigated, as depicted in Fig. 15(c). It transpired that the scaleparameter λ is almost independent of the relative density, and thus wetake a constant fitting with

=λ λ ,0 (12)

where the fitting parameter λ0 is determined to be 0.0405 for therandom PLA foam used. It is found that the shape parameter k decreaseswith the increase of relative density. We first use a power-law expres-sion to fit the relation of the shape parameter k with the relative densityand obtain k=0.50ρ−0.98. Taking the exponent −1 instead of −0.98,we have a fitting expression

=k k ρ/0 (13)

with the fitting parameter k0 determined to be 0.48.For a given base material, the variation of maximal strain with the

relative density of PLA foams is shown in Fig. 15(a). It transpires thatthe maximal strain intensively depends on the feature of the meso‑s-tructure. A linear dependence of the maximal strain εm on the relativedensity is approximately exhibited. In fact, with the relative densityapproaching to zero, the maximal strain εm can reach unity. Thus, wechoose the following expression,

= − βρɛ 1 ,m (14)

to fit the experimental data, where the fitting parameter β is determinedto be 0.44 for the random PLA foam used. If one takes the assumptionthat the plastic Poisson's ratio of cellular material is negligible duringcrushing, the maximum strain that a cellular material to be compactedinto dense solid should be equal to 1− ρ. However, the value of the

Fig. 15. Variations of the six material parameters with the relative density: (a) the elastic modulus E and the maximal strain εm; (b) the collapse stress B and thestrain-hardening parameter C; (c) the shape parameter k and the scale parameter λ of the Weibull distribution.

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critical strain εm determined from Eq. (14) is greater than 1− ρ, whichindicates that the effect of plastic Poisson's ratio on mechanical beha-viors of PLA foams should be considered. Moreover, it should be no-ticed that the plastic Poisson's ratio of PLA foams under uniaxial com-pression may be larger than 0.5, which is very different from that of adense solid. This exception is attributed to the effect of meso‑structuraldamage, as discussed in Section 3.2.

Therefore, for the random PLA foam used in this study, we obtain acompressive stress–strain relation, which involves the relative densityand can be written as

= − − + +− −

[ ]σ aρ bρ λ bρcρβρ

(ɛ) ( ɛ )exp (ɛ/ )ɛ

(1 ɛ),k ρ3/2 2

0/ 2

20

(15)

where a, b, c, k0, λ0 and β are constant. In this study, a=1796MPa,b=64.6MPa, c=1.78MPa, k0= 0.48, λ0= 0.0405 and β=0.44.

The investigation of the relation between the meso‑characteristics ofrandom cells and the macroscopic behavior of PLA foams provides anidea to study the trans-scale problem raised in cellular materials.

5. Conclusions

Closed-cell foams are constructed based on the 3D Voronoi tech-nique and a batch of polylactide (PLA) foams is fabricated by usingrapid prototyping Fused Deposition Modeling (FDM) technique. Themechanical behavior and deformation mechanism of PLA foams areinvestigated and analyzed. Additionally, a new statistical constitutivemodel is proposed, which demonstrates a versatile capability to re-present the stress–strain relations of random foams.

(a) Five configurations with cells in types of Random Cell (RC),Tetrakaidecahedron (Tet), Hexagonal Prism (HP), RectangularPrism (RP) and Clipped Rectangular Prism (CRP) are fabricated tostudy the effect of meso‑structure on the quasi-static compressivebehaviors of PLA foams. During compression, configuration Tetdeforms regularly with the crushing band spreading from the weakend to the other end layer by layer, but configuration RC presentsrandom collapse bands that are oblique to the loading direction. Forconfigurations HP, RP and CRP, the localized cell failure forms aninclined crushing band across the entire section, and then thestrength of these collapsed regions is further deteriorated, leadingto a global instability. Quantitative evaluation shows that config-urations HP, CRP and RP have higher specific energy absorptioncapacity and larger undulation of load-carrying capacity than thoseof configurations RC and Tet.

(b) The PLA foams with the orientation of layer deposition perpendi-cular to the loading direction have higher strength and less lateralexpansion than those fabricated with the layer deposition orienta-tion parallel to the loading direction. This can be attributed to theanisotropic property of cell walls introduced during the fused de-position modeling process.

(c) By changing the cell-wall thickness, PLA foams with the sameVoronoi configuration but different thicknesses are fabricated tostudy the dependence of mechanical behaviors on the relativedensity under quasi-static compression. The results show that boththe initial crush stress σ0 and the plateau stress σpl increase in apower-law manner with the increase of relative density.

(d) Finally, a statistical constitutive model with six material parametersis proposed by introducing the Weibull distribution of collapsestrain of foam cells. This model is versatile enough to capture theinitial stress drop and the three-stage deformation of cellular ma-terials. The quantitative investigation of material parameters in themodel show that the shape parameter k decreases in a power-lawrelation with the increase of relative density, and the scale para-meter λ is nearly independent of the relative density. Besides,parameters E, B and C present power-law dependences on the re-lative density, and the maximal strain εm can be expressed by a

linear function of the relative density. As a result, a quantitativestatistical relation for cellular materials is determined as a functionof relative density.

Acknowledgments

The research reported herein is supported by the National NaturalScience Foundation of China (Projects nos. 11372308 and 11372307)and the Fundamental Research Funds for the Central Universities(Grant nos. WK2480000003 and WK2090050023), which are gratefullyacknowledged. The second author would like to thank Prof. Yilong Baifor valuable communication on the statistical constitutive model.

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