Comparison with Experiment

We now use the FEM results to analyze experimental measurements of the elastic properties of porous ceramic materials. The dependence of the elastic moduli on porosity has been the subject of many studies  [13,17,18]. Data for porous alumina from numerous studies  [35] are shown in Fig. 10. The Coble-Kingery [1] material is markedly stiffer than other materials, and is in very good agreement with the FEM results for the overlapping spherical pore model. The pores in the alumina matrix were actually created by the incorporation of a particulate filler [1], which corresponds well with the definition of the model microstructure. The remaining data closely follow the overlapping solid sphere FEM result for $\phi<0.25$ < 0.25, indicating that the solid alumina phase has the sintered granular morphology exhibited by the model microstructure (Fig.1a). However, Knudsen notes that several of the samples summarized were also created using particulate fillers. At higher porosities the solid sphere result underestimates the data. One reason for this might be that the model contains isolated solid spheres which artificially reduce the actual porosity. This was checked and found not to be the case for the porosities studied. Therefore, the solid connections in these samples of porous alumina are likely stiffer than those found in the solid sphere model at porosities $\phi>0.25$ > 0.25. Overlapping spheres can create very sharp "valleys" between a pair of overlapping solid spheres (see Fig. 1a), which would be rounded off in the sintering process, presumably strengthening the solid-solid connection.

Figure 10: Data for alumina (E_s=410 GPa) compiled by Knudsen [35] ($\circ$). The Coble-Kingery [1] (E_s=386 GPa) data are also shown ($\Box$). The lines correspond to the FEM theories computed in this paper: overlapping spherical pores (---), overlapping oblate ellipsoidal pores ($\cdots$) and overlapping solid spheres (- - -).

Hunter et al. [36,37,38 ,39] have studied the Young's modulus of several different oxides. In all cases, the porous material was created by sintering a powder of the pure oxide. The results for the Young's modulus are reproduced in Fig. 11. For low porosities ( $\phi < 0.1$ < 0.1) all of the data followed the FEM results for overlapping spherical pores. For Gd₂O₃ the FEM result continues to provide excellent agreement up to the maximum porosity measured ($\phi=0.4$ = 0.4) indicating that the microstructure is similar to that of the model (overlapping pores). In contrast, the data for the other three oxides decreases towards the result for overlapping solid spheres indicating a more granular character.

Figure 11: Data for various oxides measured by Hunter et al  [36,37,38,39] compared with the FEM theories for overlapping spherical pores (---), and overlapping solid spheres (- - -). Sm₂O₃  [36] E_s =145 GPa ($\diamond$); Lu₂O₃ [37] E_s=193 GPa ($\Delta$); Gd₂O₃  [38] E_s= 150 GPa ($\circ$); HfO₂  [39] E_s =246 GPa ($\Box$).

The data of Walsh et al. [25] for porous glass is compared with the FEM results for overlapping spherical pores in Fig. 12. The agreement is good for small to moderate porosities ($\phi <0.3$ < 0.3), but the FEM results underestimate the data at higher porosities. Walsh et al. point out that the pores in the glass are actually not interconnected (unlike the overlapping pores of the model). This would account for the increased stiffness. It is interesting that the FEM results begin to deviate from the experimental data at the threshold where the pores become macroscopically connected ($\phi$ = 0.3). Data for sintered MgAl₂ O₂ [40] powder is shown in Fig. 13, and is well modeled by the FEM results for overlapping solid spheres. Micrographs of the ceramic indicate a granular structure similar to that of the model microstructure (although the grains appear more like polyhedra, not spheres).

Figure 12: Data for porous glass [25] (K_s=46 GPa, $\nu _s$_s = 0.23). The line corresponds to the FEM theory for overlapping spherical pores (---).

Figure 13: Data for MgAl₂O₄  [40]. We used the value E_s=41.2 x 10⁶ psi (284 GPa) indicated on Fig. 3(A) of the reference, rather than the reported value of E_s =43.4 x 10⁶ psi which appears to be a misprint. The line corresponds to the FEM theory for overlapping solid spheres.