Test case: Overlapping sphere model

Figure 2: Cross-sections of (a) the overlapping sphere model, and (b) the best reconstruction (model I₁₀ c=0). The volume fraction is p=20% and the images are 96 x 96 pixels.

Before going ahead and reconstructing the W-Ag composite we first tested the procedure for the over-lapping sphere model for which the statistical properties can be analytically evaluated [44]. A 3-D realization of this model is also easy to generate, so that the reconstruction can be carefully tested. The overlapping sphere model [Fig 2(a)] has previously been reconstructed using the models derived from the level cut GRF's [34]. Model I₁₀ (c=0) was found to be the best reconstruction [Fig 2(b)]. To gauge the accuracy of the procedure for the elastic properties of the tungsten-silver composite (see the following section) we have computed the elastic moduli of the overlapping sphere model and its reconstruction at volume fraction p=20% (of the phase outside the spheres). The moduli of each phase at each temperature is set to the corresponding value for silver and tungsten. The results, shown in Fig. 3, indicate that the procedure performs very well. When the silver has non-zero elastic moduli, for temperatures below the melting point of silver, the reconstruction provides an extremely good prediction of E (error <1%). At temperatures above the melting point of silver, the silver is taken to have zero shear and bulk moduli, and the reconstructed model is 9% stiffer than overlapping spheres. Since the moduli depend most strongly on microstructure at high contrast (contrast = the ratio of the Young's moduli between the two phases) the latter error is likely to be more indicative of the ability of the model to reproduce the microstructure of overlapping spheres. Nevertheless the reconstruction provides a reasonable model. Similar agreement was seen for the Poisson ratios (not shown).

Figure 3: Young's modulus of the overlapping sphere model (OS) compared with data obtained from the best reconstruction (Recon.) [model I ₁₀ (c=0)] (Finite element method). The Beran, Molyneux, Milton and Phan-Thien (BMMP) bounds are seen to be more restrictive than the Hashin and Shtrikman (HS) bounds for both models.

An advantage of the method is that the BMMP bounds [Eqn. (18)] can be evaluated (since p⁽³⁾_rec can be computed). The microstructure parameters for the reconstruction [model I ₁₀ (c=0)] were estimated as $\zeta_1=0.43$₁ = 0.43, $\eta_1=0.35$ ₁ = 0.35 [34]. The bounds (see Fig. 3) are significantly more restrictive than those of Hashin and Shtrikman, and are seen to bound the finite element moduli of both the overlapping sphere and reconstructed models. Above the melting point the lower bounds are identically zero, as we have taken the elastic properties of silver to be zero at this point. Actually, above the melting point of silver, it would be more reasonable to suppose that the silver has a non-zero bulk modulus, with a zero shear modulus. This is only important when comparing with experimental results, however, and not in this model-model comparison.