Overlapping Ellipsoidal Pores

A common method of analyzing the effect of pore shape on elastic properties is to study ellipsoidal pores. In analytic formulae, it is possible to treat the limiting cases of needles and platelets, although the difficulty of resolving these fine structures prohibits these limits from being treated with the finite element method. However, the percolation properties of these limiting cases can be computationally studied [26]. To gauge the effect of deviations from spherical shaped pores we considered isotropically oriented overlapping oblate ellipsoidal pores bounded by the surface (x/a)²+(y/b )²+(z/c) ²=1with a=b=1 µm and c=0.25 µm (see Fig. 1c). For this case, the pore phase becomes connected at porosity $\phi=0.2$ = 0.2 [26]. Statistical errors were found to be acceptable for a computational cube of size T=10 µm. Using M=96 pixels the discretization errors were 3 % for $\phi=0.5$ = 0.5 and 2 % for $\phi=0.3$ = 0.3. As for the case of spherical pores, these errors were considered sufficiently small, so that the added computational burden of the extrapolation technique could be again avoided.

Again the Young's modulus was found to be independent of the solid Poisson's ratio to a very good approximation. The results, shown in Fig. 2, can be accurately described by Eq. (2) with n=2.25 and $\phi _0$₀ = 0.798. The Poisson's ratio (Fig. 5) can be roughly fit using the Eq. (3) with $\nu _0$₀ = 0.166 and $\phi _0$₀ = 0.604. A better fit is obtained using Eq. (4) with m=1.91, $\nu _0$ ₀ = 0.161 and $\phi _0$₀ = 0.959. A flow diagram similar to that seen before is obtained.

Figure 5: Poisson's ratio of the overlapping ellipsoidal pore model as a function of solid Poisson's ratio and porosity. The solid lines are an empirical fit to Eq. (4) and the dashed lines correspond to the linear fit to Eq. (3) with $\nu _0=0.166$₀ = 0.166 and $\phi _0=0.604$₀ = 0.604. The intercepts of the lines at zero porosity correspond to the solid Poisson's ratio.

The CPU time and memory required for these computations are an important "experimental" detail. The memory requirement for a given model was 230 x M³ bytes, where M was the edge length in pixels of a cubic unit cell. So for the largest computations carried out, M = 128, the memory requirement was about 500 Mbytes. The amount of CPU time consumed was approximately 3000 hours, divided among different modern workstations.