Discussion and Conclusions

We have used the finite element method to calculate the elastic properties of a wide variety of realistic porous media. Our data confirm that several interesting and related results for two-dimensional materials (Day et al. , 1992),(Cherkaev et al. , 1992), are very nearly true in three-dimensions. These results demonstrate how useful the study of two-dimensional elasticity problems can be to understanding more difficult three-dimensional problems.

We have confirmed that the Young's modulus of three-dimensional porous materials is nearly independent of the solid matrix Poisson's ratio in the physically realistic range, 0 < $0< \nu_s < 0.5$_s < 0.5. Our conclusion supports (Christensen, 1993) analysis of the Young's modulus based on exact dilute results and effective medium theory. This simplification allowed us to generate simple structure-property relations for the models studied. The results are summarised in table 2. From the empirical formula for Young's modulus and Poisson's ratio, the bulk and shear moduli can be simply calculated. Because the structure-property relations corresponds to a known microstructure, the relations can be used to interpret experimental data, and aid in the optimisation of porous materials.

We have shown that Poisson's ratio of porous materials generally becomes independent of the solid Poisson's ratio near the percolation threshold. This implies that in this limit, the shape of the solid matrix dominates lateral expansion under uni-axial compression, rather than the material properties of the matrix. Similar to the case of 2-D Thorpe & Jasiuk, 1992), this Poisson's ratio flow behaviour seems ubiquitous, but with the value of the fixed point varying with microstructure. Of the seven models studied the relatively stiff closed-cell model provided the only exception to this behaviour. The high stiffness may explain why the matrix material retains an influence on the effective Poisson's ratio. Interestingly, the limiting value of the Poisson's ratio was generally close to $\nu_1=0.2$₁ = 0.2, irrespective of microstructure. This value is predicted by effective medium theory, as was shown in this and other papers (Garboczi & Day, 1995), and as seen in figure 13. The numerical data (extrapolated to the apparent percolation threshold) varies in the range $\nu_1=0.14$₁ = 0.14-0.28.

In addition to the fixed point behaviour, the effective Poisson's ratio was generally independent of solid fraction if _s = $\nu_s=\nu_1$₁ (closed-cell models provided the exception). This was clearly evident as a 'critical' point on plots of $\nu$ vs. $\nu _s$_s for different solid fractions. When graphed against $\nu _s$_s, the porous Poisson's ratio, for all values of p considered, intersected the = $\nu=\nu_s$_s line at the same point, = _s = $\nu=\nu_s=\nu_1$₁. We would conjecture that this behaviour shows that the dilute limit, in which $\nu _1$₁ can be calculated exactly for various shape pores, must play some role in the critical behaviour. This is because _s = ₁ = $\nu_s=\nu_1=\nu$ being an invariant for all values of p implies that the fixed point must be equal to the critical point. Although most of the theories indicate a special significance for the value $\nu=0.2$ = 0.2, we do not know of a physical explanation for this behaviour, other than the fact that this is the value that also comes from the dilute spherical pore analytical limit.

By comparing our data with various predictions we were able to establish, on a model by model basis, the accuracy of theoretical results at a particular solid fraction. Since the accuracy of theoretical results is not generally known a priori, this represents a practical advance. We showed that the differential method gave a reasonable prediction for materials with overlapping spheroidal pores at moderate to high solid fractions (p $p\geq0.6$ 0.6). We also found that a recent rigorous result due to Torquato gave reasonable predictions at p $p\geq 0.7$ 0.7 for all models tested with the exception of the overlapping solid spheres model. This model has highly interconnected pores (the pore space is macroscopically connected for p < 0.97), which may have been a source of greater error in the expansion. We have calculated the microstructure parameters $\zeta$ and $\eta$ for four models based on Gaussian random fields. These may also be used to evaluate Torquato's expansion for non-porous composites. Generally, porous materials represent a worst-case scenario for predictive methods, leading us to conjecture that the expansion will be quite accurate for composite materials with moderate contrast between the properties of each phase. The accuracy of the expansion at moderate and high solid fractions, using only third order information, provides impetus to undertake the difficult task (Helte, 1995) of including higher order microstructural information.