MODELLING OF CLOSED-CELL FOAMS INCORPORATING CELL...

11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V)

6th European Conference on Computational Fluid Dynamics (ECFD VI) E. Oñate, J. Oliver and A. Huerta (Eds)

MODELLING OF CLOSED-CELL FOAMS INCORPORATING CELL SIZE AND CELL WALL THICKNESS VARIATIONS

YOUMING CHEN, RAJ DAS* AND MARK BATTLEY

Center for Advanced Composite Materials Department of Mechanical Engineering, University of Auckland

Auckland, New Zealand. *e-mail: [email protected]

Key Words: Micromechanical modelling, Closed-cell foams, Laguerre tessellation, Cell size variation, Cell wall thickness variation.

Abstract. This paper concerns with the micromechanical modelling of closed-cell polymeric foams (M130) using Laguerre tessellation models incorporated with realistic foam cell size and cell wall thickness distributions. It is found that when the cell size and cell wall thickness are assumed to be uniform in the models, the Kelvin, Weaire-Phelan and Laguerre models overpredict the stiffness of the foam. However, the Young’s modulus and shear modulus predicted by the Laguerre models incorporating measured foam cell size and cell wall thickness distributions agree well with the experimental data. This emphasizes the fact that the integration of realistic cell wall and cell size variations is vital for foam modelling. Subsequently the effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams were investigated using Laguerre models. It is found that the Young’s modulus and shear modulus decrease with increasing cell size and cell wall thickness variations. The degree of stiffness variation of closed-cell foams resulting from the cell size dispersion and cell wall thickness dispersion are comparable. There is little interaction between the cell size variation and cell wall thickness variation as far as their effects on foam moduli are concerned. 1 INTRODUCTION

Foam materials are increasingly being used in automotive, aerospace, aircraft, marine, construction and packaging industries, partly due to their unique characteristics, such as light weight, impact absorbing, thermal insulation, flotation, acoustic isolation and noise abatement, and partly owing to the great progresses made in foam manufacturing and processing over the last decades. With the enormous usage of foams, extensive attention has been paid to their mechanical behaviour, especially when foams play a role in load-bearing in structures, such as sandwich panels and impact protective components. It is well recognized that foam mechanical properties depend on the properties of base material (from which the foam is made), relative density (ratio of the foam density to the density of base material) and foam microstructures. Micromechanical modelling can predict the macroscopic properties of heterogeneous materials based on the properties of constituent materials and microstructures,

Youming Chen, Raj Das and Mark Battley.

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and is thus well-suited to the study of property-microstructure relationship of foams [1-4]. In micromechanical modelling of foams, one challenge is to approximate foam

microstructures, which is fairly irregular and random, but still complies with a few rules. Matzke [5] observed 600 liquid bubbles using a microscope. It was found that the number of cell faces per cell ranges from 11 to 17 with an average of 13.7. They concluded in such structures more than two-thirds of cell faces are pentagonal and 99.6% are quadrilateral, pentagonal and hexagonal. Voronoi tessellations can be constructed with topology, to some extent, close to the above description. Köll and Hallström [6] developed Voronoi foam models with seed points arranged by random sequential adsorption (RSA) algorithm and random close packing (RCP) algorithm. The Voronoi models based on RSA algorithm have 14.9 cell faces per cell on average and have relatively homogenous cell size. The Voronoi models based on RCP algorithm have 14.2 faces per cell and have a considerably wide range of cell size. However, it is virtually impossible to generate Voronoi tessellation models with a prescribed cell size distribution. Laguerre tessellation, a type of weighted Voronoi tessellation, is capable of accomplishing so. In Laguerre tessellations, each seed point has a weight, which plays a role in determining the size of the cell that encloses the seed point. In addition, Laguerre tessellations produce a microstructural configuration that is in good agreement with the observation in [5], with average number of cell faces per cell ranging from 14.11 to 13.04 and the average number of edges per face from 5.14 to 5.09 [7]. Therefore, Laguerre tessellations are fairly effective in approximating foam microstructural geometry and recently has been applied to foam modelling [8, 9].

The variability in cell size and cell wall thickness is common and remarkable in real foams. The influence of the variation in cell wall thickness on the stiffness of two dimensional cellular solids and open-cell foams was investigated in [10, 11]. It was found that both Young’s modulus and shear modulus substantially decease with increasing dispersion of cell wall thickness. Grenestedt and Bassinet [12] studied the effect of variation of cell wall thickness on the stiffness of Kelvin closed-cell foams and found bulk modulus and shear modulus are reduced by roughly 19% when the thickest walls are 19 times thicker than the thinnest walls. Redenbach and Shklyar [9] investigated the effect of variation of cell size on the elastic constants of closed-cell foams using Laguerre tessellation models with cell sizes following a gamma distribution. The stiffness of the foam is found to reduce slightly with increasing variation of cell size. However, the cell wall thickness distributions in [10-12] are uniform distribution, different from realistic cell wall thickness distribution, and only 112 different thicknesses were assigned in [12], which may not be able to capture the full range of cell wall thickness. Since both cell size variation and cell wall thickness variation have an effect on foam properties, they must be integrated into numerical models while performing foam analysis. Fischer and Lim [13] incorporated cell sizes in the range of ±30% of the measured average diameter into finite element models by statistically perturbing Kelvin models. However, the integrated cell sizes are limited to 60% of the measured cell sizes, and the Kelvin models are severely distorted after perturbation. Foam modelling integrating realistic cell size or cell wall thickness distributions has been limited. In addition, studies on the combined effect and interactions of cell size and cell wall thickness variation have not been reported so far.

Therefore, the present study will focus on micromechanical modelling of a closed-cell foam using Laguerre tessellation models integrating realistic cell size and cell wall thickness


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distributions. The influences of the variations of cell size and cell wall thickness on the elastic constants of closed-cell foams including their combined effects will be investigated. The present study is structured as follows. Firstly, the characterisations of the studied foam are carried out, including the measurements of the macroscopic Young’s modulus and shear modulus of the foam, the Young’s modulus of cell wall material, and cell size and cell wall thickness distributions. Secondly, the construction of Laguerre models is described. Then the stiffness predicted by Laguerre models incorporating measured cell size and cell wall thickness distributions is compared with experimental data. A parametric study concerning the effects of cell size and cell wall thickness variations on foam stiffness is followed.

2 EXPERIMENTS

2.1 Stiffness and density measurements The foam studied herein is M130 from Gurit that is widely used in marine industry

nowadays. Foam M130 is a closed-cell foam and made from styrene-acrylonitrile (SAN). Five specimens of dimension 20 mm × 80 mm × 80 mm were first weighed. The recalculated density was 148 ± 3.7 kg/m3, slightly higher than its nominal density. The density of SAN is around 1070 kg/m3 [14, 15]; hence the relative density of the foam was 13.83 ± 0.3%. Compression tests were subsequently performed with the five specimens, following the standard ISO 844. Single block shear tests following the standard ASTM C273-07 were also conducted. According to the requirement of the standard for the minimum dimensions of specimens, the specimens chosen were 20 mm × 50 mm × 240 mm. The measured Young’s modulus and shear modulus were 119 ± 2.45 MPa and 42.1 ± 5.97 MPa, respectively.

2.2 Nanoindentation tests It has been pointed out that the properties of cell wall material may differ from that of the

bulk material from which the foam is made, due to polymer chain alignment during foaming processes and chemical changes [1, 16]. Hence, nanoindentation tests were employed in this study to determine the Young’s modulus of cell wall material of the foam. Firstly, a few cubes of the foam with size of 10 mm were cut out and mounted into thermo-set epoxy resin cylinders. Then the mounted specimens were grinded and polished on one face using 1 µm diamond dust suspension to create a flat surface for indentations. Next, a large number of indentations were made on thick junctions of cell walls at constant loading rates of 30, 200 and 400 µN/s, and the loads were increased to up to 300, 600 and 1200 µN, respectively, using Hysitron TI-950 TriboIndenter. The loads were subsequently held at their respective maximum values for 10 s and then removed. Figure 1 shows parts of the load-depth curves of the tests. The indents produced in the tests were around 3 µm, whereas the thicknesses of the cell wall junctions where these indents were made are larger than 50 µm, which ensures that edge effect is negligible according to the criteria in [17].

Generally, in nanoindentation tests Young’s modulus is calculated using the Oliver-Pharr method [18]. However, the deformation response of cell walls is time-dependent, whereby indentation depth increases when the loads are held at maximum values, as shown in Figure 1. It has been reported that the Oliver-Pharr method overestimates Young’s modulus when the tested material is time-dependent [16, 19-21]. Instead, the Lu and Wang method [19] is


4

employed here (that overcomes this aspect) and the calculated Young’s modulus is 3.58 ± 0.34 GPa. The Young’s modulus of SAN was reported to be in the range from 3.2 to 3.44 GPa [14, 15, 22, 23]. In the following simulations, the average value of 3.58 GPa will be used.

Figure 1: Load-depth curves of nanoindentation tests at loading rates of 30, 200 and 400 µN/s

2.3 Cell size and cell wall thickness measurements Microstructures of the foam were imaged using the Olympus BX60m microscope. In the present study, the diameter of the incircle of a cell is taken as cell size, and cell wall thickness is measured near the middle area of cell walls where thickness is relatively uniform. From these micrographs, the sizes of 473 cells and the thicknesses of 281 cell walls were measured. Figure 2 presents parts of the measurements. The measured cell diameter (size) and cell wall thickness approximately follow lognormal distribution. Cell size has an average of 256 µm and standard deviation of 91.8 µm, as shown in Figure 3a. Cell wall thickness has an average of 9.2 µm and standard deviation of 9.3 µm, as shown in Figure 3b. In comparison to cell size, cell wall thickness shows a larger dispersion.

(a) (b) Figure 2: Micrographs of foam cells: (a) Cell size measurements, and (b) cell wall thickness measurements

0

200

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600

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1200

1400

0 100 200 300 400 500

Load

(µN

)

Depth (nm)

300µN

600µN

1200µN


5

(a)

(b)

Figure 3: (a) Measured cell size distribution and its probability distribution fit, and (b) measured cell wall thickness distribution and its probability distribution fit

3 LAGUERRE MODEL CONSTRUCTION

3.1 Random close packing

Random close packing of spheres is a useful approach for numerous material modeling scenarios, such as powder compaction [24, 25] and polycrystalline structure simulation [7]. Packing density, the ratio of the total volume of spheres to the volume of the container in which the spheres are packed, describes the degree of packing, i.e. how closely these spheres are packed. Provided that the centres of a bunch of random closely packed spheres are taken as the seed points of a Laguerre tessellation and the radii of these spheres are chosen as the weights of the seed points, the constructed Laguerre tessellation will have a cell size distribution close to the diameter distribution of these spheres. Therefore, the first, essential step in constructing Laguerre tessellation is random close packing. The algorithm adopted in the present study is based on the collective rearrangement algorithm proposed in [24]. In order to obtain models with different sizes, random close packing with 1000, 1500 and 2000

60 100 140 180 220 260 300 340 380 420 460 500 5400%

2%

4%

6%

8%

10%

12%

Cell diameter (µm)

Pro

babi

lity

MeasurementLognormal distribution fit

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

2%

4%

6%

8%

10%

12%

14%

Cell wall thickness (µm)

Pro

babi

lity

MeasurementLognormal distribution fit


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spheres was conducted. With the measured foam cell size distribution, the packing densities achieved were 0.5908, 0.5974 and 0.6088, respectively. Figure 4a shows random close packing of 1000 spheres.

3.2 Laguerre tessellations Let S = {P1, P2, … , Pn} be a set of points in space R. For any point Q in the space, dV(Q,

Pi) denotes the Euclidean distance between Q and Pi. Then the region V(Pi) is called Voronoi cell for seed point Pi, and is defined by

V(Pi) = {Q|Q�R, dV(Q,Pi) < dV(Q, Pj), j≠i} (1)

Each seed point in S dominates a Voronoi cell. These cells partition space into an array of convex, space-filling polyhedrons, forming a Voronoi tessellation.

Laguerre tessellation is a type of weighted Voronoi tessellation. Assuming we assign a weight ri to seed point Pi, then there is a corresponding weight set R = {r1, r2 ,…, rn}. Let us redefine the distance between Q and Pi by

dL(Q, Pi)= [dV(Q, Pi)]2- ri2 (2)

Similarly, the region defined by

V(Pi) = {Q|Q�R, dL(P,Pi) < dL(P, Pj), j≠i} (3)

is called Laguerre cell for seed point Pi. Each seed point in S dominates a Laguerre cell. These cells partition space into an array of convex, space-filling polyhedrons, forming a Laguerre tessellation. The algorithm of generating Laguerre tessellations in the present study is based on that presented in [7]. With random close packing of spheres, Laguerre tessellations with prescribed cell size distributions were created, as shown in Figure 4b.

(a) (b)

Figure 4: (a) Randomly close packed spheres, and (b) Laguerre tessellation based on random closed packing

3.3 Finite element model Geometric models were constructed in the software FreeCAD which is python scriptable.

To avoid the presence of small elements, short edges and small faces under a threshold were


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removed. Then cubes were cut out from the geometrical models and meshed in the pre-processor Hypermesh with shell elements, as shown in Figure 5.

(a) (b)

Figure 5: (a) Geometric model, and (b) Finite element mesh

Firstly, the measured cell wall thicknesses were randomly assigned to cell faces (around 5100 faces) in a Laguerre model, and then the relative density of the model was calculated. Then the thicknesses of cell faces in the model were scaled by a factor so that the foam model had the prescribed relative density. Symmetry boundary conditions were applied on three orthogonal faces of the cubic models. A small displacement was imposed on one face, and the reaction forces were obtained and used to calculate the Young’s modulus. With respect to evaluation of shear modulus, biaxial load tests were performed instead of pure shear tests which require larger model size. A tensile strain in y direction and a compressive strain in z direction were applied in the biaxial load tests. Shear modulus is calculated as follows.

/ /2 ( )y z z y

x y z

F L F LG

Lτγ ε ε

−= =

− (4)

where Fy and Fz are reaction forces along y and z directions. Lx, Ly and Lz are dimensions of models along x, y and z directions. εy and εz are strains along y and z directions.

4 RESULTS AND DISCUSSION

4.1 Mesh and model size sensitivity

Mesh size and model size are important for the accuracy of prediction and computational expense. Uniaxial compression tests were first performed on a Laguerre model (based on random close packing of 1000 spheres) with the element sizes of 0.03 0.0225 and 0.015 mm. The predicted Young’s moduli were 148.50 MPa, 149.08 MPa and 149.64 MPa, respectively for the three meshes. Considering the accuracy of results and computational expense, an element size of 0.0225 mm was chosen for further study. To study the sensitivity of model size, Laguerre models based on random close packing of 1000, 1500 and 2000 spheres were considered. For each model size, four Laguerre models were generated. All the cell faces in


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the models were assigned with uniform thickness (around 15.4 µm). Results show that the Laguerre models based on random close packing of 1500 spheres can yield relatively accurate results.

4.2 Validation of simulations The Young’s modulus and shear modulus predicted by the Laguerre models integrated

with the measured cell size and cell wall thickness distributions are 117.81 and 45.21 MPa. a, within 1% and 7% of the experimental data (119 and 42.1 MPa), respectively. Weair-Phelan model, Kelvin model and Laguerre model with uniform cell size and cell wall thickness were also applied to calculate the stiffness of the foam. Table 1 lists the results of these models. Apart from Laguerre models with measured cell size and cell wall thickness, all the models considerably overestimate the stiffness of the foam, which suggests that the integration of real cell size and cell wall thickness distribution into numerical models is critical for foam micromechanical modelling. Comparing the three types of Laguerre models in Table 1, it is found that the non-uniformity of cell wall thickness is the main cause for the stiffness of the foam to deviate from ideal foams (i.e. foams of uniform cell size and wall thickness).

Table 1. Prediction of foam moduli by different models

E (MPa) G (MPa) Laguerre model + uniform cell size and wall thickness 168.43 63.80

Laguerre model + measured cell size + uniform wall thickness 156.50 60.15 Laguerre model + measured cell size and wall thickness 117.81 45.21

Weaire-Phelan model 189.75 54.46 Kelvin model 170.36 64.90

Experimental data 119.00 42.10

4.3 Effect of cell size variation on foam stiffness To address this, five lognormal cell size distributions that have the same average but

different standard deviations were studied. The lognormal distribution parameters are µ1 and σ1, with σ1 reflecting the dispersion of cell size. For each distribution, four Laugerre models were generated. To isolate the effect of cell size, the relative density of each model was kept the same (13.83%), and the cell walls in each model had uniform thickness. Figure 6 shows the variation of normalised Young’s modulus and shear modulus with dispersion parameter σ1. The simulation results, as presented in Figure 6, show that Young’s modulus and shear modulus decrease nearly linearly with cell size variation (σ1). The decreasing slope of normalized Young’s modulus over cell size variation is around 0.25, which is close to that of the normalized shear modulus over cell size variation (0.21).

4.4 Effect of cell wall thickness variation on foam stiffness To study this, four lognormal cell wall thickness distributions that have the same mean but

different standard deviation were investigated. The lognormal distribution parameters are µ2 and σ2, where σ2 reflects the dispersion of cell wall thickness. To consider the effect of cell wall thickness variation only and eliminate the effect of cell size variation, only Laguerre models with uniform cell size were employed here. As mentioned above, all the cell wall


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thicknesses were scaled by a factor so that the relative densities of all the models were constant at 13.83%. The calculated Young’s modulus and shear modulus were found to decrease with increasing cell wall thickness variation, as shown in Figure 7. The normalized Young’s modulus is close to the normalized shear modulus, which means the Poisson’s ratio remains the same. The normalized Young’s modulus and shear modulus are fitted well by the following expression.

22

22( ) e

σ

ϕ σ−

≈ (5)

Comparing Figure 7 and Figure 6, one can notice that the effect of cell size variation on moduli reduction is comparable to that of cell wall thickness variation on moduli reduction when cell size and cell wall thickness have the same degree of variability. However, in real foams, cell wall thickness may have larger dispersion than cell size, such as the foam studied, and thus may primarily contribute to the reduction of foam moduli from ideal foams.

Figure 6: Variation of normalized moduli with variability in cell size

Figure 7: Variation of normalized moduli with variability in cell wall thickness.

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

E(σ 1

)/E(σ

1=0)

, G(σ

1)/G

(σ1=

0)

Cell size variation σ1

Young's modulusShear modulus

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

E(σ

2)/E

(σ2=

0), G

(σ2)

/G(σ

2=0)

Cell wall thickness variation σ2

Young's modulus

Shear modulus

exp(-σ₂^2/2)


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4.5 Combined effect of variations of cell size and cell wall thickness In real foams, both cell size and cell wall thickness are non-uniform. To investigate their

combined effect, Laguerre models with simultaneous non-uniform cell size and non-uniform wall thickness were employed. Figure 8 shows the variations of Young’s modulus and shear modulus with variability in cell wall thickness for models with different cell size dispersion (σ1). It is noteworthy that the moduli-cell wall thickness variation curves are essentially parallel, which implies there is little interaction between the cell size variation and the cell wall thickness variation as far as their effects on foam moduli are concerned, and thus the combined effect of them can be calculated by simply multiplying the individual effects.

(a)

(b)

Figure 8: Variation of (a) Young’s modulus and (b) shear modulus with cell wall thickness variations for models with differing cell size variations

5 CONCLUSIONS

In this paper, Laguerre tessellations based on random close packing of spheres were employed to model closed-cell M130 foam. The cell size and cell wall thickness distributions

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Youn

g's

mod

ulus

(MP

a)


Cell size variation σ₁=0.0000Cell size variation σ₁=0.1783Cell size variation σ₁=0.3475

35

40

45

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Shea

r mod

ulus

(MPa

)


Cell size variation σ₁=0.0000




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of the foam were measured from images captured under a microscope and then integrated into the Laguerre tessellation based numerical models. The Young’s modulus of cell wall material was determined from nanoindentation tests. The foam stiffness predicted by the Laguerre models was validated against experimental data. The effects of cell size and cell wall thickness variations, and their combined effect on the moduli of foams were investigated. Based on the present study, the following conclusions can be drawn:

- Laguerre models incorporated with realistic cell size and cell wall thickness distributions can yield fairly accurate prediction of stiffness of closed-cell foams. By contrast, Kelvin model, Weaire-Phelan model and Laguerre models with uniform cell size and cell wall thickness overpredict the stiffness of the foam.

- For M130 foams, the variation of cell wall thickness is larger than that of cell size and thus is the main cause for stiffness reduction from ideal foams.

- Both Young’s modulus and shear modulus decrease nearly linearly with cell size variation. The decreasing slopes of normalized Young’s modulus and shear modulus over cell size variation are 0.25 and 0.21, respectively.

- Both Young’s modulus and shear modulus decrease with cell wall thickness variation. For a specified level of variation, the effect of cell size variation on stiffness reduction is comparable to that of cell wall thickness variation.

- Little interaction between the effect of cell size variation and the effect of cell wall thickness variation on moduli reduction of foams is observed.

REFERENCES [1] Gibson, L.J. and Ashby, M.F. Cellular solids : structure and properties. 2nd. Cambridge ;

New York: Cambridge University Press. (1997). [2] Zhu, H., Knott, J. and Mills, N. Analysis of the elastic properties of open-cell foams with

tetrakaidecahedral cells. Journal of the Mechanics and Physics of Solids (1997) 45 (3):319-343.

[3] Zhu, H., Mills, N. and Knott, J. Analysis of the high strain compression of open-cell foams. Journal of the Mechanics and Physics of Solids (1997) 45 (11):1875-1904.

[4] Mills, N. and Zhu, H. The high strain compression of closed-cell polymer foams. Journal of the Mechanics and Physics of Solids (1999) 47 (3):669-695.

[5] Matzke, E.B. The three-dimensional shape of bubbles in foam-an analysis of the rôle of surface forces in three-dimensional cell shape determination. American Journal of Botany (1946):58-80.

[6] Köll, J. and Hallström, S. Morphology effects on constitutive properties of foams. (2011).

[7] Fan, Z., Wu, Y., Zhao, X. and Lu, Y. Simulation of polycrystalline structure with Voronoi diagram in Laguerre geometry based on random closed packing of spheres. Computational Materials Science (2004) 29 (3):301-308.

[8] Kanaun, S. and Tkachenko, O. Mechanical properties of open cell foams: Simulations by Laguerre tesselation procedure. International Journal of Fracture (2006) 140 (1-4):305-312.

[9] Redenbach, C., Shklyar, I. and Andrä, H. Laguerre tessellations for elastic stiffness simulations of closed foams with strongly varying cell sizes. International Journal of


12

Engineering Science (2012) 50 (1):70-78. [10] Li, K., Gao, X.L. and Subhash, G. Effects of cell shape and cell wall thickness variations

on the elastic properties of two-dimensional cellular solids. International Journal of Solids and Structures (2005) 42 (5–6):1777-1795.

[11] Li, K., Gao, X.L. and Subhash, G. Effects of cell shape and strut cross-sectional area variations on the elastic properties of three-dimensional open-cell foams. Journal of the Mechanics and Physics of Solids (2006) 54 (4):783-806.

[12] Grenestedt, J.L. and Bassinet, F. Influence of cell wall thickness variations on elastic stiffness of closed-cell cellular solids. International Journal of Mechanical Sciences (2000) 42 (7):1327-1338.

[13] Fischer, F., Lim, G., Handge, U. and Altstädt, V. Numerical simulation of mechanical properties of cellular materials using computed tomography analysis. Journal of cellular plastics (2009) 45 (5):441-460.

[14] Stretz, H. and Paul, D. Properties and morphology of nanocomposites based on styrenic polymers. Part I: Styrene-acrylonitrile copolymers. Polymer (2006) 47 (24):8123-8136.

[15] Kutz, M. Mechanical engineers' handbook. John Wiley & Sons Inc. (2006). [16] Daphalapurkar, N., Hanan, J., Phelps, N., Bale, H. and Lu, H. Tomography and

simulation of microstructure evolution of a closed-cell polymer foam in compression. Mechanics of Advanced Materials and Structures (2008) 15 (8):594-611.

[17] Jakes, J.E. and Stone, D.S. The edge effect in nanoindentation. Philosophical Magazine (2011) 91 (7-9):1387-1399.

[18] Oliver, W.C. and Pharr, G.M. Improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal of Materials Research (1992) 7 (6):1564-1583.

[19] Lu, H., Wang, B., Ma, J., Huang, G. and Viswanathan, H. Measurement of creep compliance of solid polymers by nanoindentation. Mechanics of time-dependent materials (2003) 7 (3-4):189-207.

[20] Fischer-Cripps, A. A simple phenomenological approach to nanoindentation creep. Materials Science and Engineering: A (2004) 385 (1):74-82.

[21] Tweedie, C.A. and Van Vliet, K.J. Contact creep compliance of viscoelastic materials via nanoindentation. Journal of Materials Research (2006) 21 (6):1576-1589.

[22] Majumdar, B., Keskkula, H. and Paul, D. Mechanical properties and morphology of nylon-6/acrylonitrile-butadiene-styrene blends compatibilized with imidized acrylic polymers. Polymer (1994) 35 (25):5453-5467.

[23] Nicodemo, L. and Nicolais, L. Mechanical properties of metal/polymer composites. Journal of materials science letters (1983) 2 (5):201-203.

[24] He, D. and Ekere, N. Computer simulation of powder compaction of spherical particles. Journal of materials science letters (1998) 17 (20):1723-1725.

[25] Liu, L. and Yuan, Y. Dynamic simulation of powder compact by random packing of monosized and polydisperse particles. Journal of materials science letters (2000) 19 (10):841-843.

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