Effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams

Accepted Manuscript Effects of cell size and cell wall thickness variations on the stiffness of closedcell foams Youming Chen, Raj Das, Mark Battley PII: DOI: Reference:

S0020-7683(14)00369-2 http://dx.doi.org/10.1016/j.ijsolstr.2014.09.022 SAS 8518

To appear in:

International Journal of Solids and Structures

Received Date: Revised Date:

2 July 2014 21 September 2014

Please cite this article as: Chen, Y., Das, R., Battley, M., Effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams, International Journal of Solids and Structures (2014), doi: http://dx.doi.org/ 10.1016/j.ijsolstr.2014.09.022

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Effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams Youming Chen∗, Raj Das and Mark Battley

Centre for Advanced Composite Materials, Department of Mechanical Engineering, University of Auckland, Auckland, New Zealand

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. The cell size and cell wall thickness distributions of the foam were measured from microscope images. The Young’s modulus of cell wall material of the foam was characterized by nanoindentation tests. 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. Based on the simulation results, expressions incorporating cell size and cell wall thickness variations were formulated for predicting the stiffness of closed-cell foams. Lastly, a simple spring system model was proposed to explain the effects of cell size and cell wall thickness variations on the stiffness of cellular structures. Keywords: Micromechanical modelling; Closed-cell foams; Laguerre tessellation; Cell size variation; Cell wall thickness variation.

∗

Corresponding author at: CACM, Tamaki campus, University of Auckland, Auckland, NZ. Email address: [email protected] Tel: +64 (0) 9923 4774

1. Introduction Foam materials are increasingly being used in automotive, aerospace, marine, aircraft, 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 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 cores in sandwich panels and impact absorbers. It is well recognized that foam mechanical properties depend on the properties of the base material (from which the foam is made), relative density (ratio of the foam density to the density of base material) and microstructural geometry. In terms of investigation on the property-microstructure of foams, experimental study has limitation in that because foams with prescribed microstructures are hardly available, whereas micromechanical modelling can predict the macroscopic properties of heterogeneous materials based on the properties of constituent materials and their microstructures and thus is well-suited and widely used for the task. For example, using scaling law and classic beam and shell theory, Gibson and Ashby [1] (based on cubic unit cell) and Mills and Zhu [2-4] (based on Kelvin unit cell) analysed the response of foams to different types of loads and related foam mechanical properties to foam relative density. Voronoi tessellations, to some extent, resemble the microstructures of real foams and thus have been often utilised in conjunction with the finite element method [5-8]. With the development of X-ray computer tomography (CT) techniques, foam finite element models based on the reconstruction of real foams using CT techniques have also been reported in [914]. In micromechanical modelling of foams, one big challenge is to approximate foam microstructures which are fairly irregular and random, but still complies with a few rules. In 1946, Matzke [15] observed 600 liquid bubbles using a microscope and 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 pentagons and 99.6% are quadrilateral, pentagonal and hexagonal. With seed points arranged by random sequential adsorption (RSA) algorithm and random close packing (RCP) algorithm, Voronoi tessellations can be constructed with microstructural topology close to Matzke’s observation [16, 17]. However, Voronoi tessellations cannot be produced with cells of size following a prescribed 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. Provided that the centres of a set of random close packed spheres are taken as the seed points of a Laguerre tessellation and the radii of these spheres are chosen as the weights, then the constructed Laguerre tessellation will have a cell size distribution close to the diameter distribution of these spheres. In addition, Laguerre tessellations constructed in this manner have microstructures that agree well with Matzke’s observation, with the 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 [18], Therefore, Laguerre tessellations are fairly effective in approximating foam microstructural geometry and recently have begun to be applied to foam modelling [19-21]. 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 numerically investigated in [6, 7]. It was found that both Young’s modulus and shear modulus substantially decease with increasing dispersion of cell wall thickness. Grenestedt and Bassinet [22] studied the effect of variation of cell wall thickness on the stiffness of Kelvin closed-cell foams, and found the bulk modulus and shear modulus are reduced by roughly 19% when the thickest cell walls are 19 times thicker than the thinnest cell wall. However, the cell wall thickness distribution applied in [6, 7, 22] is uniform distribution, different from realistic cell wall thickness distribution, and only 112 different thicknesses were assigned in [22], which may not be able to capture the full range of cell wall thickness. Redenbach and Shklyar [20] carried out research on 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, and found that foam stiffness reduces slightly with increasing variation of cell size. Studies on the combined effect and interactions of cell size and cell wall thickness variation have not been reported so far. Since both cell size and wall thickness variations have an effect on foam properties, they must be integrated into numerical models while performing microstructural foams analysis. Fischer and Lim [23] incorporated a variation of cell size in the range of ±30% of measured mean diameter into finite element models by statistically distortion of regular Kelvin model. On one hand, the integrated cell size variation is limited; on the other hand, the cells in models are severely distorted. Foam microstructural modelling integrated with full measured cell size distribution or cell wall thickness distribution has been limited.

Therefore, the present study will focus on micromechanical modelling of a closed-cell foam using Laguerre tessellation models integrating with realistic cell size and cell wall thickness distributions. The simulation results will be validated against experimental data. Then the influences of variations of cell size and cell wall thickness including their combined effect on the elastic constants of closed-cell foams will be investigated. This paper 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. Lastly, a spring system model is proposed to explain the reduction of stiffness resulting from cell size and cell wall thickness variations.

2. Experiments The foam studied herein is M130 from Gurit which is widely used in marine industry nowadays. Foam M130 is a closed-cell foam made of styrene-acrylonitrile (SAN). This section concerns with the characterization of the foam, including uniaxial compressive and single block shear tests, nanoindentation test, and cell size and cell wall thickness measurements. 2.1 Stiffness and density measurements Five specimens of dimension 20 mm × 80 mm × 80 mm were first weighed so as to calculate the actual density of the foam. The recalculated density is 148 ± 3.7 kg/m3, slightly larger than its nominal density. The density of SAN is around 1070 kg/m3 [24, 25]; hence the relative density of the foam was 13.83 ± 0.3%. Uniaxial 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 conducted to determine the response of the foam under shear. According to the requirement of the standard for the minimum dimensions of specimens, the specimens chosen for shear tests 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 walls of a polymeric foam 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 by the addition of foaming agents [1, 26]. But direct measurement of foam cell walls is difficult because of the small size of cell walls. Nanoindentation tests, originally developed for the characterization of thin films, have been applied to a wide range of materials with small dimensions including foam cell walls [26-28]. In the present study, nanoindentation tests were employed to determine the Young’s modulus of cell wall material of the foam. Firstly, a few cubes of the foam with dimension around 10 mm were cut out and mounted into thermo-set epoxy resin cylinders. After the resin cylinders were fully hardened, one face of the mounted specimens were ground and polished using 1 µm diamond dust suspension to create a flat surface for indentations. Next, a large number of indentations were made on the thick junctions of cell walls at constant loading rates of 30, 200 and 400 µN/s, and the loads were increased 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 the load-depth curves of the nanoindentation tests. Theoretically, indentations should be conducted on a semi-infinite field. However, the largest indent produced in the tests is around 3.5 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 [29]. Theoretically speaking, the material properties of cell walls may differ from that of cell wall junctions because cell wall materials are stretched more than materials in junctions during the process of foaming. But the difference between them might be minor since they are made of the same base material. Since there is no published research exploring this type of heterogeneity in polymeric closed-cell foams and it is fairly difficult to find cell wall thicker than 50 µm to perform nanoindentation tests, we assume cell walls and junctions have the same material properties in the present study.

1400 1200µN

1200

Load (µN)

1000 800 600µN

600 400 300µN 200 0 0

100

200

300

400

500

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

Generally, Young’s modulus is calculated in nanoindentation tests using the Oliver-Pharr method [30], in which Young’s modulus is related to the initial slopes of unloading curves. The Young’s modulus of cell wall material computed in this manner is 5.04 ± 0.19 GPa for the foam studied. 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 a time-dependent polymeric material [26, 31-33]. Lu and Wang [31] derived the creep compliances of time-dependent materials by fitting the initial elastic part of loading curves into a compliance function based on viscoelastic contact analysis. The Young’s relaxation moduli obtained in this approach agree well with the macroscopic tensile test data of the materials. Therefore, the Lu and Wang method was employed herein to calculate the Young’s modulus of cell wall material. It was found that the indentation marks left after the removal of the loads were hardly identifiable in micrographs with the loading rate of 30 µN/s (see Figure 2a); hence the corresponding maximum depth 174 nm was taken as an approximate limit of elasticity. For the convenience of fitting, a compliance function of two terms was chosen. Fitting the linear part of a load-depth curve into Eq. (17) in [31], we obtained a creep compliance function as follows.

J (t ) ≈ 0.0206 + 0.7728(1 − et /3.884 )

(1)

where J(t) is creep compliance in shear in 1/GPa and t is time in second. Figure 3 shows the load-depth curve and its fit. Converting Eq. (1) into Young’s modulus, the Young’s relaxation modulus obtained is 3.40 GPa. The same analyses were performed for the other load-depth curves, and the Young’s modulus calculated 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 [24, 25, 34, 35]. In the following simulations, the average value of 3.58 MPa will be used.

(a)

(b)

Figure 2: Indentation patterns (a) loading rate = 30 µN (b) loading rate = 400 µN. 350 Experiment 300

Fit

Load (µm)

250 200 150 100 50 0 0

20

40

60

80

100

120

140

160

180

200

Depth (nm)

Figure 3: A load-depth curve obtained from the nanoindentation test and its fit.

2.3 Cell size and cell wall thickness measurements

Generally, foam cells are approximately irregular polyhedrons. The most reasonable indicator of cell size is cell volume which was applied in [36]. However, measuring cell volume requires not only 3D reconstruction of foam samples, but also sophisticated image processing to extract the polyhedral profile of each cell. Cell length along one direction is relatively easy to measure and has also been used to estimate the size of foam cells [23, 37]. By contrast, 2D micrographs of foams are easy to acquire and measure with. But, because cells have different sizes at different altitudes, with this method some cell sizes measured may deviate from actual values. Nevertheless, overall cell size measured with 2D micrographs has been reported to be

close to that measured with 3D reconstructed foam models [23]. Hence, microstructures of the foam were imaged using the microscope Olympus BX60m. The diameters of the incircles of cells were taken as cell sizes. The sizes of 473 cells were measured and Figure 4a shows parts of the measurements. The measured cell diameter approximately follows a lognormal distribution with an average of 256 µm and standard deviation of 91.8 µm, as shown in Figure 5a.

(a)

(b)

Figure 4: (a) Cell size measurements, and (b) cell wall thickness measurements. 12%

Probability

10%

Measurement Lognormal distribution fit

8% 6% 4% 2% 0% 60

100 140 180 220 260 300 340 380 420 460 500 540

Cell diameter (µm) (a)

14% 12%

Measurement Lognormal distribution fit

Probability

10% 8% 6% 4% 2% 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Cell wall thickness (µm) (b)

Figure 5: (a) Measured cell size distribution and its probability distribution fit, and (b) measured cell wall thickness distribution and its probability distribution fit.

Cell wall thickness is difficult to measure accurately due to its small size and non-uniformity in a given cell wall (thinner in middle area than around edges). Moreover, some cell walls may be inclined to the cutting surface and some cuttings may be right through plateau borders, as a result, some measured wall thicknesses may be slightly larger than the actual thicknesses of cell walls. To reduce the errors induced by the non-uniformity of cell wall thickness, measurements were all made near the middle area of cell walls where thickness is relatively uniform. To reduce the error induced by cell wall inclination, we selected cell walls that seem vertical to cutting surface while measuring. In the present study, the thicknesses of 281 cell walls were measured, and Figure 4b shows a few cell wall thickness measurements on a micrograph. The measured cell wall thickness approximately follows a lognormal distribution with an average of 9.2 µm and standard deviation of 9.3 µm, as shown in Figure 5b. In comparison to the cell size, the cell wall thickness shows a larger dispersion.

3. Methodology In this section, the construction of Laguerre foam models incorporating prescribed cell size and cell wall thickness distributions is described. Firstly, random close packing is discussed and the algorithm of packing is introduced. Then the definition of Laguerre tessellations and the algorithm for constructing Laguerre tessellations are presented. Lastly, the creation of finite element models for foam micromechanical analysis is followed.

3.1 Random close packing

Indeed, random close packing of spheres is a useful approach for numerous material modelling scenarios, such as powder compaction [38, 39] and polycrystalline structure simulation [18]. As discussed early, random close packing of spheres is also essential for the construction of Laguerre tessellations. Packing density, the ratio of the total volume of the 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. For random close packing of spheres of uniform size, the highest packing density that can be achieved is 0.6366 ± 0.005 [39, 40]. Packing density increases when the spheres have non-uniform sizes. For the computer simulation of random close packing, a variety of algorithms have been proposed. They generally fall into two categories: the sequential generation method and the collective rearrangement algorithms [41]. In the present study the collective rearrangement algorithm is adopted because it is relatively easy to implement. A certain number of spheres are packed in a cubic space. In each loop, spheres are relocated in a random order based on their positions relative to their surrounding spheres. If the summation of total overlap of the spheres over the last 50 loops is greater than that over the 50 loops before the last 50 loops, all the spheres are shrunk by a factor of 0.995. When the total overlap is less than one third of the radius of the smallest sphere, the simulation is terminated. For the purpose of demonstration, random close packing of 2D disks was performed and the packed disks are shown in Figure 6a. In order to obtain models with different size, random close packing with 1000, 1500 and 2000 spheres was conducted. With the cell size distribution that we measured with the M130 foam, the packing densities that we achieved are 0.5908, 0.5974 and 0.6088, respectively. Figure 6b shows the 1000 spheres that are randomly closely packed.

(a)

(b)

Figure 6: Random close packing of (a) disks and (b) 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|QϵR, dV(Q,Pi) < dV(Q, Pj), j≠i}

(2)

Physically speaking, this region consists of points which are closer to seed point Pi than other seed points in S. 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. Suppose 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

(3)

Similarly, the region defined by V(Pi) = {Q|QϵR, dL(P,Pi) < dL(P, Pj), j≠i}

(4)

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.

If the centres of a certain number of random closely packed spheres are taken as the seed points and their radii are chosen as the weights of a Luguerre tessellation, then the constructed 3D Laugerre tessellation will have cell size distribution close to that of the packed spheres. Figure 7a shows 2D Laguerre tessellation constructed in this manner. It is seen that each Laguerre cell surrounds a disk closely and thus the Laguerre tessellation has cell size distribution close to the distribution of disk diameter. The algorithm of generating 3D Laguerre tessellations in the present study is based on that presented in [18]. The main procedures are as follows: 1) We map a set of random closely packed spheres C = {C1, C2, …, Cn} onto a set of 4D points P*={ P1*, P2 *,……, Pn*}. Let (xi, yi, zi) be the coordinates of the centre and ri be the radius of sphere Ci. Then the coordinates of point Pi* is (xi, yi, zi, xi2+yi2+zi2-ri2). 2) We calculate the convex hull of set P* using the program Qhull and collect the lower facets of the convex hull to obtain the lower hull. 3) Each facet fi is composed of four 4D points. Let us suppose these points be Pr*, Ps*, Pt* and Pu*, then the coordinates (Xi, Yi, Zi) of a vertex of a Laguerre cell can be calculated by Eq.(5).  2 xr 2x  s  2 xt   2 xu

2 yr

2 zr

2 ys 2 yt

2 zs 2 zt

2 yu

2 zu

−1  X i   xr 2 + yr 2 + z r 2 − rr 2    −1  Yi   xs 2 + ys 2 + z s 2 − rs 2  = −1  Zi   xt 2 + yt 2 + zt 2 − rt 2      −1 Wi   xu 2 + yu 2 + zu 2 − ru 2 

(5)

4) We collect all the vertices that are calculated by facets containing the same point Pi*. These vertices indeed form a Laguerre cell that encloses the sphere Ci. Calculating the convex of these vertices, we can obtain the topological information of this Laguerre cell. 5) We repeat procedure (4) for all the points in P* and then store the information of vertices, lines, faces and cells in a hierarchical order. 6) With the topologic information obtained in (5), the geometry of Voronoi tessellations is generated in the software FreeCAD which is python scriptable, starting with creating point, then connecting the points with lines and finally generating the covering cell faces with shells. With this algorithm Laguerre tessellations with prescribed cell size distributions were generated. Figure 7 shows 2D and 3D Laguerre tessellations based on random close packing.

(a)

(b)

Figure 7: (a) 2D Laguerre tessellation and packed disks, and (b) 3D Laguerre tessellation.

3.3 Finite element model

To avoid the presence of very small and distorted elements, short edges and small faces under a threshold were removed. Specifically, in the present study when an edge was shorter than 1/25th of the longest edge in a model, it was shrunk to its midpoint. When the area of a face was smaller than 1/500th of that of the largest face, it was shrunk to it central points. During this process only around 1.6% of the faces were removed, which accounted for only around 0.002% of the total face area. Therefore, the removal of small edges and faces did not effectively change the geometry of models and thus did not affect the stiffness of models. Cubes were cut out from the geometric models and meshed with shell elements in the preprocessor Hypermesh, as shown in Figure 8. Struts in closed-cell foams can be modelled as beam elements reinforcing along face edges (refer to [8]). However, the effect of struts on the elastic behaviour of closed-cell foams is not considered here. The reason for that is, on one hand, it fairly difficult to measure the fraction of material in struts in a real closed-cell foam. On the other hand, struts affect the elastic modulus of closed-cell foams slightly when the fraction of material in struts is low (for the studied foam of relative density 13.83%, struts affect elastic moduli slightly when the fraction of material in struts is lower than 80%, refer to [42]). The measured Young’s modulus of 3.58 GPa and a Poisson’s ratio of 0.35 were assigned to all cell walls. The cell wall thicknesses that follow a prescribed distribution were first assigned to cell faces (around 5100 faces), and subsequently the relative density of the model was calculated. Then the thicknesses of the cell faces in the model were scaled by a

factor so that the model has a prescribed relative density. Hence, actually, it was the shape of the cell wall thickness distribution that is incorporated into the model. Symmetry boundary conditions were applied on the cubic models. A small displacement was imposed on one face and the resultant reaction forces were obtained to calculate the Young’s modulus. With respect to shear modulus, biaxial load tests were performed instead of pure shear tests (refer to [6, 7]).

(a)

(b)

Figure 8: (a) Geometric model, and (b) Finite element mesh.

4. Results and Discussion Mesh size and model size are first considered in this section because they may affect the accuracy of numerical prediction and computational cost. Next, the stiffness predicted by the Laguerre models integrated with measured cell size and cell wall thickness distributions is compared with experimental data. Lastly, a parametric study concerning the effects of cell size and cell wall thickness variations on foam stiffness is followed. 4.1 Mesh and model size sensitivity

To study mesh sensitivity, compressive tests were 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, 149.08 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. Model size is critical in micromechanical modelling. Insufficient model size may underestimate the stiffness of materials and lead to

large scatter of results, whereas models with excessive size give rise to high computational cost. In the present study, Laguerre models based on random close packing of 1000, 1500 and 2000 spheres were considered. For each model size, four Laguerre models were generated. Because there are no Laguerre cells created for the spheres packed at the outermost layer, the number of Laguerres cells constructed was actually less than the packed spheres. The Young’s moduli predicted by these models are shown in Figure 9. Considering the accuracy of results and computational expense, the Laguerre models based on random close packing of 1500 spheres are employed in the further study.

Young's modulus (MPa)

165 160 155 150 145 140 135 130 900

1100

1300

1500

1700

1900

2100

Number of packed spheres

Figure 9: Young’s moduli predicted by Laguerre models with different sizes.

4.2 Validation of simulations

With the measured cell sizes but uniform cell wall thickness, the Young’s modulus and shear modulus predicted by Laguerre models are 156.50 and 60.15 MPa, respectively, which are greater than experimental data (see Table 1). With the measured cell sizes and cell wall thicknesses integrated, Laguerre models predict Young’s modulus and shear modulus within the 1% and 7% of experimental data, being 117.81 and 45.21 MPa, respectively (see Table 1), respectively. Weair-Phelan model, Kelvin model and Laguerre model with uniform cell size and cell wall thickness were employed as well. Table 1 lists the results of these models. It was found that all these models (with uniform cell size and cell wall thickness) 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 vital for foam micromechanical modelling. Comparing the three Laguerre models in Table 1, it could be concluded that the non-uniformity of cell wall thickness is the main cause for the stiffness of the foam to deviate from the ideal foam (i.e. foam of uniform cell size and wall thickness).

Table 1. Predictions of foam moduli by different models

Laguerre model + uniform cell size + uniform wall thickness Laguerre model + measured cell sizes + uniform wall thickness Laguerre model + measured cell sizes + measured wall thicknesses Weaire-Phelan model Kelvin model Experimental data

E (MPa) 168.43 156.50 117.81 189.75 170.36 119.00

G (MPa) 63.80 60.15 45.21 54.46 64.90 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, as listed in Table 2, were studied. The lognormal distribution parameters are µ 1 and σ1, with σ1 reflecting the dispersion of cell size. For each distribution, four Laugerre models were generated. The topological statistics of the generated Laguerre models, including the number of faces per cell and edges per face, and equivalent cell diameter (the diameter of the sphere of equal volume to a cell), is shown in Figure 10, and the average of them is listed in Table 2. 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. Simulation results are normalized by the Young’s modulus and shear modulus of the Laguerre model of uniform cell wall thickness and cell size (168.43 MPa and 63.80 MPa, respectively) , and shown in Figure 11. It can be seen that the predicted Young’s modulus and shear modulus decrease nearly linearly with cell size variation (σ1) and exhibit more scatter as cell size variation increases. The decreasing slope of normalized Young’s modulus over cell size variation (σ1) is 25.7 %, which is close to that of the normalized shear modulus over cell size variation (21.1%).

40% σ₁=0.3475

35%

Probability

σ₁=0.2648 30%

σ₁=0.1783

25%

σ₁=0.0897 σ₁=0.0000

20% 15% 10% 5% 0% 6

8

10

12

14

16

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Faces per cell (a)

22

24

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45% σ₁=0.3475

40%

σ₁=0.2648 35% σ₁=0.1783 30%

Probability

σ₁=0.0897

25%

σ₁=0.0000

20% 15% 10% 5% 0% 3

4

5

6

7

8

9

10

11

Edges per face (b) 60% σ₁=0.3475 50%

σ₁=0.2648 σ₁=0.1783

Probability

40%

σ₁=0.0897 σ₁=0.0000

30% 20% 10% 0% 140 180 220 260 300 340 380 420 460 500 540 580

Equivalent diameter (µm) (c) Figure 10: Topological statistics of the Laguerre models: (a) faces per cell (b) edges per face (c) equivalent cell diameter.

E(σ1)/E(σ1=0), G(σ1)/G(σ1=0)

1.00

Normalized Young's modulus

0.98

Normalized shear modulus

0.96 0.94 0.92 0.90 0.88 0.86 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Cell size variation σ1 Figure 11: Variation of normalized moduli with cell size variation.

In addition, the relationship between foam moduli and relative density was investigated for foams with different cell size variations. Laguerre models with cell size variation σ1=0.3475 and with uniform cell size (σ1=0) were employed, and six relative densities were considered. Results show that Young’s modulus and shear modulus increase virtually linearly with relative density for the models with σ1=0.3475 and these with σ1=0 (see Figure 12). Since the modulirelative density curves pass through the origin, the fractional reduction of moduli resulting from cell size variation is independent of relative density, that is, the effect of cell size variation on moduli is independent of the relative density of foams. Noteworthy, the moduli predicted by the Laguerre models with uniform cell size are fairly close to those predicted by the Kelvin model, despite the fact that Laguerre models have cells (see Figure 13) different from the Kelvin cell. Table 2. Studied cell size distributions and topological statistics of models

1 2 3 4 5

Average diameter (µm) 256 256 256 256 256

Standard deviation of diameter (µ m) 92 69 46 23 0

µ1

σ1

5.4853 5.5101 5.5293 5.5412 5.5452

0.3475 0.2648 0.1783 0.0897 0.0000

Average faces per cell 13.95 14.14 14.16 14.24 14.25

Average edges per face 5.13 5.14 5.14 5.14 5.14

Average equivalent diameter (µ m) 330 324 308 303 305

* µ 1, σ1 are the lognormal distribution parameters of cell size, with σ1 reflecting the dispersion of cell size

180

Young's modulus (MPa)

160 140 120 100 80 Laguerre model σ₁=0.3475

60

Laguerre model σ₁=0.0000

40

Kelvin model

20 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Relative density (a) 70

Shear modulus (MPa)

60 50 40 30 Laguerre model σ₁=0.3475

20

Laguerre model σ₁=0.0000

10

Kelvin model

0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Relative density (b) Figure 12: (a) Variation of predicted Young’s modulus with relative density, and (b) variation of predicted shear modulus with relative density.

Figure 13: Cells in the Laguerre models of uniform cell size.

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 (listed in Table 3) were investigated. The lognormal distribution parameters are µ 2 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 thicknesses were scaled by a factor so that the relative densities of all the models were constant at 13.83%. Simulation results are normalized by the Young’s modulus and shear modulus of the Laguerre model of uniform cell wall thickness and cell size and shown in Figure 14. It can be seen that the calculated Young’s modulus and shear modulus decrease with increasing cell wall thickness variation. The normalized Young’s modulus is almost the same 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

ϕ (σ ) ≈ e

−σ 2 2 2

(6)

Comparing Figure 14 and Figure 11, it is seen that moduli reductions resulting from cell size variation and from cell wall thickness variation are comparable 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 studied foam, and thus might be the main cause for the reduction of foam moduli from ideal foams. Table 3. Studied cell wall thickness distributions

1 2 3 4

Average cell wall thickness (µ m) 9.2655 9.2655 9.2655 9.2655

Standard deviation of cell wall thickness (µ m) 0 3 6 9

µ2

σ2

2.226 2.176 2.051 1.8945

0.0000 0.3157 0.5918 0.8147

*µ 2, σ2 are the lognormal distribution parameters of cell wall thickness, with σ2 reflecting the dispersion of cell wall thickness.

As with cell size variation, the relationship between moduli and relative density was studied for foams of different cell wall thickness variations. Laguerre models with uniform cell wall thickness (σ2=0) and with cell wall thickness variation σ2=0.8147 were used and six relative densities were calculated. It was found that both the Young’s modulus and shear modulus increase virtually linearly with relative density for these two kinds of models, as shown in Figure 15. Similarly, this indicates that the effect of cell wall thickness variation on the fractional reduction in moduli is independent of relative density.

E(σ2)/E(σ2=0), G(σ2)/G(σ2=0)

1.00 0.95 0.90 0.85 0.80 0.75

Normalized Young's modulus

0.70

Normalized shear modulus

0.65

Exp(-σ^2/2)

0.60 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Cell wall thickness variation σ2 Figure 14: Variation of normalized moduli with cell wall thickness variation. 180 Laguerre model σ₂=0.0000

Young's modulus (MPa)

160

Laguerre model σ₂=0.8147

140 120 100 80 60 40 20 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Relative density (a) 70 Laguerre model σ₂=0.0000

Shear modulus (MPa)

60

Laguerre model σ₂=0.8147

50 40 30 20 10 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Relative density (b) Figure 15: Variation of (a) predicted Young’s modulus and (b) predicted shear modulus with relative density.

4.5 Combined effect of cell size and cell wall thickness variations

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 16 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. 170 Cell size variation σ₁=0.0000 160

Young's modulus (MPa)

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

150 140 130 120 110 100 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Cell wall thickness variation σ2

(a) 65

Shear modulus (MPa)

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

60

Cell size variation σ₁=0.3475 55 50 45 40 35 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Cell wall thickness variation σ2

(b) Figure 16: Variation of (a) predicted Young’s modulus and (b) predicted shear modulus with cell wall thickness variation for models with different cell size variation.

Since the effects of cell size and cell wall thickness variations on moduli are independent of the relative density, and there is little interaction between these two effects, foam stiffness can be approximated by Eq.(7) and (8), which incorporate the effects of cell size and cell wall thickness variations.

E ≈ C1 ⋅ R ⋅ λ (σ 1 ) ⋅ ϕ (σ 2 ) E*

(7)

G ≈ C2 ⋅ R ⋅ λ (σ 1 ) ⋅ ϕ (σ 2 ) E*

(8)

where R is the relative density of a foam, E and G are the macroscopic Young’s modulus and shear modulus of the foam, respectively. E* is the Young’s modulus of cell wall material, respectively. C1 and C2 are constants of proportionality. By fitting the Young’s modulus and shear modulus predicted by Laguerre models with uniform cell size and cell wall thickness in Figure 12, we obtain C1 = 0.3361and C2 = 0.1311. λ(σ1) and φ(σ2) are correction factors which are functions of cell size variation and cell wall thickness variation parameters, respectively. Fitting curves in Figure 11 and Figure 14, we obtained the following expressions for the correction factors, λ and φ. 1 4

(9)

−σ 2 2 2

(10)

λ (σ 1 ) ≈ 1 − σ 1 ϕ (σ 2 ) ≈ e

Figure 17 shows the Young’s modulus and shear modulus calculated from Eq. (7) and (8), and the simulation results in Figure 16. It can be seen that the surfaces of Young’s modulus and shear modulus defined by Eq. (7) and (8) match well with the simulation results.

(a)

(b) Figure 17: (a) Young’s modulus calculated from Eq.(7) and the simulation results in Figure 16a and (b) Shear modulus calculated from Eq.(8) and the simulation results in Figure 16b.

5. Theoretical analysis In this section, the effect of cell wall thickness variation on the stiffness of unit cells of cellular solids is first examined by 2D simplified models. Then a spring system is employed to study the effect of cell wall thickness variation on the stiffness of general cellular solids. Mathematical proof for the effect of cell wall thickness variation on the reduction of stiffness of cellular solids is presented. Lastly, two specific distributions, lognormal and uniform distribution, of cell wall thickness are investigated. 5.1 Analysis of unit cell of cellular solids

Let us first consider a unit cell of 2D hexagonal honeycombs. There are four representative cell wall thickness distributions, as shown in Figure 18a-d. In case (a) all the cell walls have the same thickness. In cases (b)-(d) cell walls have non-uniform thicknesses. In all of the four cases, the structures have the same total mass and thus have the same relative densities. To calculate the stiffnesses of these structures, we simplified the structure as a spring system, as shown in Figure 18e, with each spring representing one load-carrying beam. The two beams at the same side (right or left side) are in series, and these are in parallel with beams at the other side. Hence, the total stiffness of the spring system can be computed by Eq. (11). When honeycombs are subjected to in-plane compression, cell wall bending is the dominant deformation mechanism. Therefore, the stiffness of the springs in Figure 18e is nearly equal to the bending stiffness of their corresponding cell walls, which is proportional to the cube of wall thickness. For each case, the bending stiffness of each cell wall and the compressive stiffness of the structure were calculated, as shown in Figure 18a-d. C is a combination of constants (the Young’s modulus of cell wall material, cell wall lengths and the angle between cell walls). The structure in case (c) has the highest compressive stiffness, 1.75 times the stiffness of the structure in case (a). The compressive stiffnesses of the structures in cases (b) and (d) are lower than that of structure in case (a).

ε

ε E1 = Ct t

t

1

3

2

4

t

E1 = C (1.5t )3

3

E2 = Ct 3

1.5t

E3 = Ct 3 t

(a)

E4 = Ct 3 1 1 Ea = Ct 3 + Ct 3 2 2 = Ct 3

2

t

3

0.5t

4

(c)

E3 = C (0.5t )3 t

E4 = Ct 3 27 3 1 3 Eb = Ct + Ct 35 9 278 3 Ct = 315

1.5t

E1 = C (1.5t )3

3

1.5t

1.5t

E2 = C (0.5t )3 E3 = C (1.5t )3

2

E2 = Ct 3

ε E1 = C (0.5t )

1

4

0.5t

(b)

ε 0.5t

3

1

E4 = C (1.5t )3 1.5t 1 27 Ec = Ct 3 + Ct 3 16 16 28 3 = Ct 16

0.5t

1

3

2

4

E2 = C (0.5t ) 3 E3 = C (1.5t ) 3 E4 = C (0.5t ) 3

0.5t

27 3 27 3 Ct + Ct 224 224 27 3 Ct = 112

Ed = (d)

E1

E3

E2

E4 (e)

Figure 18: (a)-(d) 2D unit hexagonal cellular structures with different wall thickness distribution, and (e) spring system.

E=

1 1 + 1 1 1 1 + + E1 E2 E3 E4

(11)

Similarly, the structures in Figure 18 can be used to represent cells of open-cell foams. However, the relative density of open-cell foams depends on the square of cell wall thickness, and the bending stiffness of beam-like cell walls scales with the power of 4 of cell wall thickness. In order to keep relative density the same in each case, the sum of the square of cell wall thicknesses needs to be the same. Hence, four cell wall thickness distributions, as shown in Figure 19, were considered. Likewise, the bending stiffness of cell walls and the compressive stiffness of the structures were calculated, as shown in Figure 19. D is a combination of constants (the Young’s modulus of cell wall material, cell wall lengths and the angle between cell walls). It is noticeable that the structure in case (c) has higher compressive

stiffness than that of the structure in case (a), and the compressive stiffnesses of the structures in cases (b) and (d) are lower than that of the structure in case (a).

ε

ε t

3

1

t

E2 = Dt 4

t

E3 = Dt 4

4

t

1 4 1 4 Dt + Dt 2 2 4 = Dt

Ea =

(a)

ε 0.5t

0.5t

7 t 2

E4 = Dt 4

2

1

3

2

4

(c)

7 t 2

7 t 2

1 Dt 4 16 1 E2 = Dt 4 16 49 4 E3 = Dt 16 49 4 E4 = Dt 16 1 49 Ec = Dt 4 + Dt 4 32 32 25 4 Dt = 16

49 4 Dt 16 E2 = Dt 4 0.5t 1 3 E3 = Dt 4 16 E4 = Dt 4 4 49 4 1 t Eb = Dt + Dt 4 65 17 ≈ 0.81Dt 4

E1 =

E1 = Dt 4

t

1

2

(b)

ε

E1 =

49 4 t 16 7 1 t E2 = D t 4 3 2 16 49 E3 = D t 4 16 1 4 0.5t E4 = D t 4 16 1 1 Ed = Dt 4 + Dt 4 16 16 1 4 = Dt 8 E1 = D

7 t 2

0.5t

1

2

(d)

Figure 19: 2D unit hexagonal cellular structures with different wall thickness distributions.

However, for closed-cell foams the main deformation mechanism is not cell wall bending, but cell wall stretching or compression. Therefore, we used a 2D rectangular cellular structure to represent a cell of closed-cell foams, as shown in Figure 20a-d. The relative density of closedcell foams scales linearly with cell wall thickness; hence the sum of thickness of cell walls was kept the same for all the cases. The compressive stiffness of each cell wall and the compressive stiffness of the structures are calculated, as shown in Figure 20a-d. K is a combination of constants (the Young’s modulus of cell wall material, cell wall lengths and cell wall width). It is to be noted that the structures in cases (a) and (c) have the same stiffness, greater than that of the structures in cases (b) and (d).

ε

ε E1 = Kt

t

t

1

3

2

4

E2 = Kt

t

E1 = K (1.5t )

1.5t

3 0.5t

1

E3 = Kt E4 = Kt

t

1 1 Kt + Kt 2 2 = Kt

0.5t

4

2

t

Ea =

(a)

(b)

E1 = K (1.5t )

E1 = K (0.5t ) 3

1

1.5t

E2 = K (0.5t )

1.5t

1

1.5t

3

2

4

(c)

1.5t

E4 = K (1.5t ) 1 3 Ec = Kt + Kt 4 4 = Kt

E2 = K (0.5t ) E3 = K (1.5t )

E3 = K (1.5t )

0.5t

E4 = Kt 3 1 Eb = Kt + Kt 5 3 14 = Kt 15

ε

ε

0.5t

E2 = Kt E3 = K (0.5t )

0.5t 2

4

(d)

0.5t

E4 = K (0.5t )

3 3 Ed = Kt + Kt 8 8 3 = Kt 4

Figure 20: 2D unit rectangular cellular structures with different cell wall thickness distributions.

5.2 Analysis of an assembly of cells

We have previously discussed the stiffness of cellular solids by one single cell. It can be seen that increasing cell wall thickness variation may increase or decrease the compressive stiffness of the structures, depending on the distribution of cell wall thickness. Real honeycombs and foams are an assembly of cells. To represent this, we used a spring system of n rows and m columns to analyse the compressive stiffness of an assembly of cells, as shown in Figure 21, with each spring representing a cell wall. The compressive stiffness of the spring system can be described by

E11

Em1

E21

… … …… E22

E12

Em2

. ..

. .. E1n-1

E2n-1

E1n

E2n

..

..

..

. .. .

Emn-1

… … ……

Emn

Figure 21: 2D spring system representing an assembly of cells m

E = ∑ Ei

(12)

i =1 n

1 1 =∑ Ei j =1 Eij

(13)

where Eij is the stiffness of the spring at i column and j row, Ei is the total stiffness of the springs at i column, and E is the total compressive stiffness of the spring system. To further simplify the analysis, all the cell walls are assumed to be of equal size and thus the stiffness of cell walls depended only on cell wall thickness. As discussed before, the stiffness of representative springs is proportional to the power of 3, 4 and 1 of cell wall thickness for honeycombs, open-cell foams and closed-cell foams, respectively. Without any loss of generality, we define a unified cell wall thickness T, instead of the actual cell wall thickness t. The relation between them is given by

tij  Tij =  tij2 t  ij

(honeycombs) (open-cell foams)

(14)

(closed-cell foams)

Then the stiffnesses of springs can be calculated in terms of T as follows CTij3  Eij =  DTij2  KT  ij

(honeycombs) (open-cell foams)

(15)

(closed-cell foams)

Taking the cellular solids of uniform cell wall thickness as a reference case, the Young’s modulus of the reference case (E0) is

 CT03 m n   DT 2 E0 = m 0 n   KT0 m n 

(honeycombs) (open-cell foams)

(16)

(closed-cell foams)

where T0 is the average of cell wall thicknesses, described by T0 =

1 m n ∑∑ Tij mn i =1 j =1

(17)

Next we apply the concepts of power mean and power mean inequality. If p is a non-zero real number, the power mean with p of a set of positive real numbers x1, x2…, xn is defined as: 1

1 n M p ( x1 ,..., xn ) = ( ∑ xip ) p n i =1

(18)

Power mean inequality: Suppose r > s, then Mr ≥ Ms, with equality holds if and only if x1 = x2 = … = xn. Substituting Eq. (15) into Eq. (12) and rearranging E in the form of power mean m  C [ M −3 (Ti1 , Ti 2 ... Tin )] 3 ∑ m n  C ∑ (∑ Tij−3 )−1 = i =1 n  i =1 j =1  m  m n D ∑ [ M −2 (Ti1 , Ti 2 ... Tin )] 2  −2 −1 E =  D ∑ (∑ Tij ) = i =1 n  i =1 j =1 m  K ∑ M −1 (Ti1 , Ti 2 ... Tin )  m n  K ( T −1 ) −1 = i =1 ∑ ij  ∑ n i =1 j =1  

Rearranging E0 in the form of power mean,

(honeycombs)

(open-cell foams)

(closed-cell foams)

(19)

3 m   1  3 C  ∑ M1 (Ti1, Ti 2 ... Tin )    m C  M1 (T11, T12 ... Tij ... Tmn ) =m  m i =1 (honeycombs)  n n  2 1 m   2 D M ( T , T ... T )  ∑ 1 i1 i 2 in   D  M1 (T11, T12 ... Tij ... Tmn )  = m  m i =1  (open-cell foams) E0 = m  n n  m  K M1 (Ti1 , Ti 2 ... Tin ) ∑  K  M1 (T11 , T12 ... Tij ... Tmn )    i =1 m (closed-cell foams) = n n   

(20)

According to power mean inequality, we obtain E ≤ E0 for closed-cell foams, and the equality holds if and only if Ti1 = Ti2 =…= Tin, i ϵ [1, m], such as for the structure in case (d) in Figure 20. But for honeycombs and open-cell foams, it cannot be ascertained whether E is less than E0 by comparing Eq. (19) and (20). Let us suppose n is large enough and cell wall thickness of the cellular solid is macroscopically homogenous, then Eq. (21) is satisfied for any i ϵ [1, m]. T0 =

(21)

1 m n 1 n T = Tij ∑∑ ij n ∑ mn i =1 j =1 j =1

Rearranging E0 in the form of power mean again, m  3 C [ M1 (Ti1 , Ti 2 ... Tin )] ∑ 3  CT  m 0 = i =1  n n E0 =  m 2  D∑ [ M 1 (Ti1 , Ti 2 ... Tin ) ]  DT02 = i =1 m  n n

(honeycombs )

(22)

(open-cell foams)

Comparing Eq. (19) and Eq. (22), we obtain E ≤ E0 for honeycombs and open-cell foams according to the power mean inequality. However, there may be cases in which Eq. (21) does not hold, such as for the structures in cases (b) and (c) in Figure 18 and Figure 19. Firstly, let us consider a specific case in which the cell wall thickness is uniform within each column, such as case (c), that is

∀ i ϵ [1, m]

Ti1 = Ti2 =…= Tin

(23)

Then

T0 =

1 m ∑ Ti = M1 (T1 , T2 , ..., Tm ) m i =1

(24)

where Ti= Tij Rewriting E and E0 in terms of Ti, m  3 C  ∑ Ti m  i =1 = C [ M 3 (T1 ,T2 , ..., Tm )]3  n n E= m D T 2 i  ∑ m i =1 =C [M 2 (T1 ,T2 , ..., Tm )]2   n n

 m 3 C n [ M 1 (T1 ,T2 , ..., Tm )] E0 =   D m [M (T ,T , ..., T ) ]2 m  n 1 1 2

(honeycombs )

(25) (open-cell foams)

(honeycombs) (26) (open-cell foams)

Applying power mean inequality, we find E ≥ E0 for honeycombs and open-cell foams. However, in this case cell wall thickness distribution is regular, rather than random and macroscopically homogenous. The cellular structures are thus not isotropic. For example, Figure 22 shows a honeycomb with wall thickness satisfying the requirement for this case. The compressive stiffness of this structure in y direction is larger than that of a honeycomb of the same relative density but with uniform cell wall thickness, while the compressive stiffness in x direction is smaller than that.

y x Figure 22. Honeycomb structure with uniform cell wall thickness in each column

Therefore, it is mathematically possible for E to be greater than E0, depending on how the cell wall thickness is distributed. However, if the cell wall thicknesses are completely random statistically, the probability of E being greater than E0 is found to be nearly zero when n and m are sufficiently large. To demonstrate this, numerical simulations were conducted using Matlab with n = m. For each n, 100000 sets of Tij was randomly assigned with a value in the range from 0 to 100. The Young’s moduli, E in Eq.(19) and E0 in Eq. (20). were calculated and compared. The number of cases for which E > E0 was recorded to calculate the probability of E > E0. Figure 23 shows the variation of the probability of E > E0 with n. For n > 7 the probability of E > E0 is zero for honeycombs and open-cell foams. In real cellular structures n is much greater than 8. Therefore, we can conclude that in real cellular structures the variation of cell wall thickness reduces the stiffness. For the spring system in Figure 21, the greater the degree of homogeneity in the stiffness of the springs is, the greater the total stiffness of the system will be. By analogy, when cell wall size in cellular structures becomes more non-uniform, the bending or stretching stiffness of cell walls becomes more non-uniform, and the macroscopic stiffness of the cellular structures is thus reduced. 30% Honeycombs Open-cell foams

Probabity (E>E0)

25% 20% 15% 10% 5% 0% 2

3

4

5

6

7

8

9

10

n Figure 23. Variation of the probability of E > E0 with n

5.3 Case study

Next, let us consider two specific distributions: lognormal distribution and uniform distribution. It is known that if Xi follows a lognormal distribution with parameters µ and σ, then the q-th moment of Xi exists and it holds that

2

qσ qµ + 1 m q 2 Xi = e ∑ m i =1

2

(27)

If Xi follows a uniform distribution in the range [a, b], then its q-th moment holds that b − a  2   ln b − ln a 1 m q  b−a ∑ Xi =  1 m i =1   ab  ( a + b)  2 2  2a b

q =1 q = −1 (28) q = −2 q = −3

Let us suppose that cell wall thickness in the cellular structures is macroscopically homogenous, substituting Eq. (27) into Eq. (19) and Eq. (20), we can obtain   mC 2  −3 µ + 9 σ 2  ne  mD E≈ 4σ 2  ne −2 µ + 2   mK  −µ +σ 2  ne 2 σ  µ+ 3  mC (e 2 )  n  σ2 µ+  2  mD(e 2 ) E0 ≈  n  σ2  µ+  mKe 2  n 

(honeycombs)

(open-cell foams)

(29)

(closed-cell foams)

2

(honeycombs )

(open-cell foams)

(30)

(closed-cell foams)

Then the stiffness ratio is given by e−6σ E  −3σ 2 = e E0  2 −σ  e 2

(honeycombs ) (open-cell foams) (closed-cell foams)

(31)

Likewise, substituting Eq. (28) into Eq. (19) and Eq. (20), we can obtain the stiffness ratio   (1 − α 2 )2 E  = 1 − α 2 E0  1+ α  2α / ln( ) 1−α 

(honeycombs) (open-cell foams)

(32)

(closed-cell foams)

where α is the ratio of (a+b)/2 to (b-a)/2. In real cellular structures, each column of cells is connected with each other, rather than isolated from each other. For cellular structures of uniform cell wall thicknesss, the cell walls normal to loading direction carry little load, while in cellular structures having non-uniform cell wall thickness, they prevent cells from rotation, thereby increasing the stiffness of the structures. Hence, the actual stiffness ratio for real cellular structures is greater than the predictions by Eqs. (31) and (32). Let us suppose the actual stiffness ratio has the modified forms given in Eqs. (33) and (34). Taking β3=1/ 2 , Eq. (33) fits well with the variation of moduli ratio with cell wall thickness dispersion, as shown in Figure 14. The Young’s modulus ratios obtained in [6, 7, 22] with cell wall thickness following uniform distributions are shown in Figure 24. Taking γ1= 1/ 4 , γ2= 1/ 3 and γ3= 1/ 2 , Eq. (34) fits well with the data in the literatures.  e−6( β1σ ) E  −3( β2σ )2 = e E0  − ( β3σ )2  e 2

  1 − (γ 1α )2  2   E  2 =  1 − (γ 2α ) E0  2γ 3α   ln(1 + γ 3α ) − ln(1 − γ 3α )

(honeycombs )

(33)

(open-cell foams) (closed-cell foams)

(honeycombs) (open-cell foams) (closed-cell foams)

(34)

1.00 0.95 0.90

E/E0

0.85 0.80 0.75

Closed-cell foams (Grenestedt, 2000)

0.70

Open-cell foams (Li, 2006) 2D cellular solids (Li,2005)

0.65

Closed-cell foam fit

0.60

Open-cell foam fit

0.55

2D cellular solid fit

0.50 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cell wall thickness variaiton α Figure 24: Stiffness ratio data in the literatures [6, 7, 22] and their fits by Eq. (34).

6. Conclusions In this paper, Laguerre tessellations based on random close packing of spheres were applied to model the closed-cell M130 foam. The cell size and cell wall thickness distributions of the foam were measured from microscopy images and then integrated into the Laguerre tessellation based models. The Young’s modulus assigned to the cell walls in the numerical models was determined from nanoindenation tests. The foam stiffness predicted by the Laguerre models was validated against experimental data. The effect of cell size and cell wall thickness variations, and their combined effect on the moduli of foams were investigated. Lastly, a spring system model was proposed to demonstrate how the cell size and cell wall thickness variation affect the macroscopic stiffness of foams. Based on the present study, the following conclusions can be drawn: (i) 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 over predict the stiffness of the foam. For M130 foam, cell wall thickness has a larger dispersion than cell size does and thus is the main cause for stiffness reduction from ideal foams. (ii) 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. The effect of cell size variation on moduli is

independent of relative density. The moduli predicted by Laguerre models with uniform cell size and cell wall thickness are close to those predicted by the Kelvin model, although the cells in the Laguerre models differ from the Kelvin cells. (iii) Both Young’s modulus and shear modulus decrease with cell wall thickness variation. The effect of cell wall thickness variation on moduli is independent of relative density. For a given level of variation, the effect of cell size variation on the stiffness reduction is comparable to that of the cell wall thickness variation. (iv) Little interaction between the effect of cell size variation and the effect of cell wall thickness variation on moduli reduction of foams is observed. Expressions for predicting the stiffness of closed-cell foams which corporate cell size and cell wall thickness variation were formulated based on the simulation results. (v) Cellular structures behaviour can be modelled as a spring system, with each cell wall represented by a spring. It can be proven that increasing the variation of stiffnesses of the springs decreases the total stiffness of the spring system. In cellular structures increasing cell size and cell wall thickness variation leads to increase in the dispersion of bending or stretching stiffness of cell walls, and as a result the macroscopic stiffness of the cellular structures is reduced.

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