Comparison of FEM results with existing theory

The Young's modulus and Poisson's ratio of the open-cell foams are shown in Figures 8 and 9 along side four relevant theories. For low densities (0.04 < ρ / ρ_s < 0.15) data for the high-coordination number node-bond model (Figure 4c) is reasonably well described by Christensen's results for isotropic foams with straight-through struts, indicating that longitudinal compression dominates the deformation. The power law dependence (n=1.3) is higher than predicted (n=1), because, being random, the model has no completely straight-through struts. This does not lead to a significant bending (indicated by a quadratic decay) because there are sufficient struts emanating from each node to 'lock' the relative node positions, and reduce the bending component of deformation. In contrast, data for the low-coordination number node-bond model ( 0.03 < ρ / ρ_s < 0.30 ) are well described by the semi-empirical result given by Gibson and Asbhy [Gibson, 1988] (n=2). This confirms the predominance of the beam-bending mechanism for deformation for this model. The Young's modulus of the open-cell Voronoi tessellation also follows the conventional quadratic decay with density, but as noted above (Figure 5), the bulk modulus actually scales linearly with density.

Figure 8: Comparison of the FEM data for open-cell foams (symbols) with theory. The data shown is for the high- ($\triangle$) and low- ($\Box$) coordination number node-bond models, the open-cell Voronoi tesselation ($\circ$) and the open cell Gaussian random field model ($\diamond$). The theories are due to Christensen [Christensen, 1986] ($\cdots$), Gibson and Ashby [Gibson, 1988] (---), Warren and Kraynik [Warren & Kraynik, 1988] (-- --) and Zhu et al  [Zhi et al.] (- $\cdot$ -).

Figure 9: Comparison of the FEM data for open-cell foams (symbols) with theory (lines). The same nomenclature as in Figure 8 is used.

The modulus of the open-cell GRF model is considerably lower than any of the predictions. In this case, the 'struts' of the network are themselves bent, therefore allowing greater deformation. Moreover, the struts have varying thickness. Since the stiffness of the struts is limited by their thinnest sections, the mass in the thickest regions contributes little to the overall stiffness. This has the effect of reducing the moduli at a given density, compared to models having struts with a uniform cross-sectional area.

Results for the Poisson's ratio of the open-cell foams are shown in Figure 9. Data for the high and low co-ordination number node-bond models and open-cell GRF model are approximately constant, showing a slight increase from the solid value of ν = 0.2 (the solid value) to ν ≈ 0.22 0.24 at low densities. We have recently shown [Garboczi & Roberts, 2000] that the Poisson's ratio of a wide range of three-dimensional porous materials converges nearly linearly from the Poisson ratio of the solid material (ν_s ) at high densities to a microstructure-dependent fixed point at low densities. Therefore, if the fixed point is close to ν_s , as it is for the three models under discussion with ν_s =0.2, ν will be nearly independent of density as observed in the data. The actual low density limits are close to 0.25, which is predicted by numerous independent theories; see (equation 6) for example.

In contrast, the Poisson's ratio of the open-cell Voronoi tessellation shows a sharp increase towards ν  ≈ 0.5 with decreasing density. This is associated with the foam being much stiffer under uniform compression (n ≈ 1 for the bulk modulus) than under shear or uniaxial compression (n ≈ 2). A Poisson's ratio of 0.5 is the highest attainable by an isotropic material and physically means that if a cubic sample is uni-axially compressed, the decrease in volume in the direction of compression is exactly balanced by the lateral expansion of the material in the perpendicular directions. This 'incompressible' behaviour is observed in solid rubber and in liquids.

The unusual density dependence of both the bulk modulus and Poisson's ratio is actually very well described by the Warren-Kraynik result [(equation 7)] for isotropically averaged tetrahedral joints, and qualitatively similar to the results for the tetrakaidecahedral foam [(equation 4)]. It is easy to understand the behaviour of the simple models. If the unit-cell of the tetrakaidecahedral foam model is placed under uniform compression the strut(s) indicated by an arrow in Figure 1(d) only undergo axial (not bending) deformation, and hence the bulk modulus varies linearly with density. Alternately, if equal forces are applied along each axis of the four struts of a tetrahedral element, the central node is not displaced and the struts are only axially compressed. This 'node-locking' will not occur under shearing or uniaxial compression, therefore allowing bending of the struts in each case.

In general, four cells touch at each node of the Voronoi tessellation [Stoyan et al., 1995], so that there are four struts associated with each node in the open-cell model. Therefore, approximating the behaviour of the model by tetrahedral elements would seem appropriate [Warren & Kraynik, 1988]. However one would expect the disorder in the random Voronoi tessellations to upset the local force balance at the nodes, and allow significant bending to occur. Our results show that this is not so. This indicates that the exact force balance that occurs in a perfect tetrahedral joint under uniform compressions is fairly robust to large angular deviations (from 109º) between the struts. Although we have not quantified the distribution of angles for the foam struts, the high Poisson's ratio (ν ≈ 0.4 at ρ / ρ_s= 0.05) at ) of the polydisperse random tessellations considered above (§ 3(a) and Figure 7b) support this idea.

Since experiments indicate that real cellular solids have lower Poisson's ratio's (≈ 0.33), the open-cell Voronoi tessellation may be missing salient features. One possibility is that the strut edges of real foams are not straight. However, if the struts are bent both the Young's and bulk modulus will decrease (possibly leaving Poisson's ratio ν = 1/2 − E/ 6K unchanged). Indeed, Grenestedt (1998) has shown that the ratio E/K is constant for a simple model with curved beams. A second alternative is suggested by the low-coordination number node-bond model, which has ν ≈ 0.25. In this model, some nodes may be attached to less than four struts, and even when the number is four or more, they may not be geometrically well balanced. In a real foam, this would correspond to missing bonds, perhaps due to imperfections in the solidification process. To test the magnitude of this effect we deleted bonds at random from a BCC open-cell Voronoi tessellation with 27 cells. For a 2 % reduction in mass the bulk modulus decreased by 22 %, while the Young's modulus only decreased by 6 %. A 15 % reduction in mass was required before the Poisson's ratio decreased to ν ≈ 0.33 (in accord with experiment). The same results were found for deletion of bonds from an isotropic model with 64 cells. Therefore, defective bonds provide a plausible explanation for the fact that ν ≈ 0.33 for real foams.

Note also that the bulk modulus of the open cell Voronoi tessellation (which has a co-ordination number of $\bar z \approx 4$) and the high co-ordination number ( $\bar z \approx 12.5$) node-bond foam are about the same at ρ / ρ_s ≈ 0.075 (the numerical value is K / E_s ≈ 0.009). This shows that the geometrical arrangement of struts can be more important than coordination number in determining the bulk modulus. Our results indicate that there is strong positive correlation between the coordination number and the Young's modulus. The foams with a coordination number of around 4-5 have similar Young's moduli, but the magnitude is higher (by a factor of three at ρ / ρ_s ≈ 0.05) for the high coordination number foam.