Comparison of FEM results with prior results

In Fig. 9, we compare FEM data for the closed-cell foams with the prior results discussed in section 2. The results for the random Voronoi tessellation are seen to be around 10 % greater than Simone and Gibson's results for the tetrakaidecahedral model. It is not clear if this implies that the disordered tessellation is stiffer than a regular tessellation. Indeed, Grenestedt [6] found that a tessellation of a randomly perturbed BCC lattice was in fact 10 % weaker than the tetrakaidecahedral model. Therefore, it is possible that differences between the models are explicable in terms of systematic discretization errors (which we estimate to be around 10 %). The modulus of the Gaussian random field model is considerably below the estimate for the tetrakaidecahedral model. This can be attributed to the highly irregular cell shapes and curved faces of the Gaussian model. At low densities, Christensen's result significantly overestimates data for both random models, indicating that the assumption of "straight- through" faces is not justified for random foams.

Figure 9: Comparison of the FEM data (symbols) with theory (lines). The data shown is for the closed-cell Voronoi tessellation ($\circ$) and the closed-cell Gaussian random field model ($\Box$). The theories are due to Christensen [17] (- $\cdot$ -) and Simone and Gibson [5] ($\cdots$). We also show the Hashin-Shtrikman bound for all isotropic materials (-- --) and the 3-point bound on the modulus of the Gaussian random field model (---).

As expected, all of the data fall below the Hashin-Shtrikman bound for isotropic materials. For the Gaussian random field model it is possible to evaluate the 3-point statistical correlation function [27], and calculate the more restrictive 3-point upper bound [20]. Figure 9 shows that the 3-point bound still does not provide a good estimate of the Young's modulus and computation is necessary.

To apply the semi-empirical formula of Gibson and Ashby given in Eq. (4), we need to estimate the fraction of mass contained in the cell edges $\phi$. To determine $\phi$ for the Voronoi tessellation we have deleted all the faces from the model and recorded the remaining density (see Table 1). For / _s < 0.3, we find < ½, indicating that the prediction of Eq. (4) is in fact greater than the Hashin-Shtrikman bound. For the closed cell Gaussian random field model, an analytic result for $\phi$ can be derived as follows. If, instead of a union set, we form the intersection set of two level-cut random-fields, we obtain an open-cell model comprised of the cell-edges of the closed-cell model. Denoting the reduced density of the open- and closed-cells Gaussian models as p_op and P_cl , it can be shown that $p_{\rm cl}=\sqrt p_{\rm op}(2-\sqrt p_{\rm op})$ [29]. The fraction of mass in the edges is then

$$\phi=\frac{p_{\rm op}}{p_{\rm cl}}=\frac{(1-\sqrt{1-p_{\rm cl}})^2}{p_{\rm cl}}.$$ (12)

Now for p_cl < 8/9, < ½, indicating that the semi-empirical formula will exceed the Hashin-Shtrikman upper bound at lower densities. Therefore the semi-empirical formula does not provide an accurate estimate of the elastic modulus for either model.

The Poisson's ratio of the closed cell foams are compared with predictions in Fig. 10. The FEM data for the closed cell Voronoi tessellation and Gaussian random field models increase from 0.2 (the solid value) to about 0.24 and 0.28, respectively, as density decreases. The results lie between the predictions $\nu=0.2$ = 0.2 and $\nu=0.33$ = 0.33. In a related study [30], we have shown that the Poisson's ratio becomes independent of the solid-Poisson's ratio at low densities indicating that the predictions are not correct for the models studied here.

Figure 10: Comparison of the FEM data for closed-cell foams (symbols) with theory (lines). The data shown is for the closed-cell Voronoi tessellation ($\circ$) and the closed-cell Gaussian random field model ($\Box$). The lines correspond to Christensen's theory [17] (- $\cdot$ -) and the empirical result $\nu _s$_s = 0.33 of Gibson and Ashby [1] (- - -).