Overlapping Spherical Pores

The overlapping spherical pore (or swiss cheese) model [14,21] is generated by interchanging the roles of the solid and pore phase of the overlapping solid sphere model (see Fig. 1b). The morphology corresponds to isolated spherical pores at low porosity, with the pores becoming macroscopically interconnected at $\phi\approx0.3$ 0.3. The solid phase remains connected up to $\phi\approx0.97$ 0.97. This type of morphology may arise in ceramics generated with a particulate filler [1], or where bubbles form in a molten state [25]. We consider solid Poisson's ratios in the range -0.1 _s 0.4.

We determined that statistical errors were acceptable for a computational cube of size T=12 µm with pores of radii r=1 µm. Using M=80 pixels, the discretization errors were less than 3 % for $\phi=0.5$ = 0.5 and 2 % for $\phi=0.3$ = 0.3. Therefore, it was not considered necessary to generate samples at different discretizations (M) and extrapolate the results. As for solid spheres, the Young's modulus was independent of the solid Poisson's ratio to a very good approximation. The Young's modulus can be described by Eq. (2) with n=1.65 and $\phi _0$₀ = 0.818 (Fig. 2). Poisson's ratio of the porous material is shown in Fig. 4 and is simply described by the linear relation given in Eq. (3) with $\nu _0$₀ = 0.221 and $\phi _0$₀ = 0.840. Again, a flow diagram is observed.

Figure 4: Poisson's ratio of the overlapping spherical pore model as a function of solid Poisson's ratio and porosity. The lines are an empirical fit to the relation = _s + / ₀ x (₀ - _s ) with $\nu _0$₀ = 0.221 and $\phi _0$₀ = 0.840. The intercepts of the lines at zero porosity correspond to the solid Poisson's ratio.