Cellular solids based on Gaussian random fields

Very different types of models can be generated using the level-cut Gaussian random field (GRF) scheme. One starts with a GRF field y(r) which assigns a (spatially correlated) random number to each point in space. A two-phase solid-pore model [Berk, 1987, Roberts & Knackstedt, 1996] can be defined by letting the region in space where −β < y(r) < be solid, while the remainder [ | y(r )  β ] corresponds to the pore-space. Open-cell solids can be obtained from the model by forming the intersection sets of two statistically independent level cut GRF models [Roberts, 1997]. An example is shown in Figure 4(d). Details for generating the models have been previously described [Roberts & Garboczi, 1999].

The model (Figure 4d) shows a highly irregular structure, with curved 'struts' of variable thickness. The morphology is reminiscent of the nickel and copper cellular solids in Figure 2.4 of Gibson and Ashby [Gibson, 1988] and the sponges shown in Figure 2.5 (ibid.) and Figure 31 of Weaire and Fortes [Weaire & Fortes, 1994]. The small angle scattering intensities of the model have also been shown to be consistent with experimental data for organic aerogels [Roberts, 1997]. At low densities, the Young's modulus can be described (to within 12 %) by

$$\frac EE_s = 4.20 \left(\frac{\rho}{\rho_s} \right)^{3.15} \rm {for}\;\; 0.05 < \frac\rho\rho_s < 0.20.$$ (12)

Equation (9) describes the higher density data ( ρ / ρ_s > 0.2) with m=2.15 and p₀=0.029 to within 4 % (and the low density data to within 12 %). The data and fitting formulae are shown in Figure 6. We are confident that the model exhibits power-law behaviour in the low-density limit, such that the correct data fall on a straight line on a log-log graph such as in Figure 6. There should be a cross over from a slope of m ≈ 2 to a steeper power-law decay at low densities. However, the data show a small, but persistent, curvature as the density decreases below 0.2. This may be due to a non-zero percolation threshold (thus altering the assumed form for the power law, which takes the percolation threshold to be zero) or resolution effects. Note that the underlying two level cut GRF remains connected at all finite densities. This follows from the fact that there must always be a thin surface separating the regions in space where y>0 and y<0. The struts of the intersection model correspond to the lines where surfaces from each of two independent two level cut models intersect. Hence they will exist at all densities, and the model does not have a finite percolation threshold. Therefore, assuming the power law is correct, we conclude that the curvature of the data is likely due to finite size effects. These finite size effects obviously have not been totally eliminated by our error-reduction procedure.

Figure 6: The Young's modulus of the non-foam-like cellular models;. high coordination number node-bond model ($\Box$), low coordination number node-bond ($\circ$) and Gaussian random field model ($\triangle$). The lines are the empirical fits to the data given in the text.