Discussion and Conclusions

We have derived empirical theories for the dependence of the Young's modulus on porosity for three distinct models of porous ceramics, based on careful finite element computations. An advantage of these results over many conventional theories is that they correspond to a priori known microstructures. The dilute result (extended by Coble-Kingery to all porosities), the differential method, and the self consistent methods all have a "built-in" microstructure, but apart from the dilute case, it is not clear what that microstructure is. Therefore, agreement or disagreement with a particular analytic theory neither confirms nor rejects a particular physical model. For the minimum solid area models, the microstructure is exactly known, but the approximation involved in making the Young's modulus directly proportional to the contact area leads to a similar conclusion. Indeed, we found that the MSA models did not provide quantitative agreement with the moduli of the random microstructures studied. We found that the differential method (Eqs. 12 and 13) gave results in reasonable agreement with computed data for the cases of overlapping spherical and ellipsoidal pores, probably due to the similarities between the assumptions of the model and the definitions of the microstructure. Results for the granular model of overlapping solid spheres were not well modeled by any of the analytic theories, demonstrating the importance of finite element techniques in this case of great physical interest.

We have also generated data that shows the dependence of Poisson's ratio on porosity and the solid Poisson's ratio. It is difficult to study this question experimentally because of the inability to vary the Poisson's ratio of the solid independently, and the well known difficulties of accurately measuring the Poisson's ratio at moderate to high porosities [18]. At sufficiently high porosities we find that the Poisson's ratio converges to a fixed non-zero value $\nu _0$ ₀ irrespective of the solid Poisson's ratio. For overlapping solid spheres $\nu_0=0.14$ ₀ = 0.14, spherical pores $\nu_0=0.22$ ₀ = 0.22 and oblate ellipsoidal pores $\nu_0=0.16$₀ = 0.16. This behavior is exact in two dimensions [23,24] and is exhibited by many of the analytic theories in three dimensions. At present the available experimental data cannot confirm this qualitative behavior [18]. We have shown that the Poisson's ratio does not vanish at high porosities as has been recently argued [19].

It is not simple to attribute our results to features of the solid-pore morphology such as the size, shape, distribution and connectivity of pores or solid grains, since these features have no obvious definition for complex bi-continuous random microstructures. A few general observations can be made, and interpreted in terms of interrelated geometrical and mechanical features of the models. For a given porosity, the sintered grain structure of the overlapping solid sphere model is relatively weak. The small solid contacts between spheres and the highly interconnected porosity (which becomes macroscopically connected at $\phi$ = 0.03) lead to a weak structure. We also assume that the valleys which occur between grains will provide sites of large stress concentrations, and consequently, large deformations. In contrast, spherical pores provide high (near optimal) stiffness at a given porosity. The dispersed nature of the porosity (which is macroscopically disconnected for $\phi <0.3$ < 0.3) corresponds to a well connected solid matrix. Ellipsoidal pores tend to weaken a structure more than spherical pores due to a combination of a less well connected solid phase (the pores become macroscopically connected at $\phi=0.2$ = 0.2), and greater stresses and deformations near the high curvature regions of the ellipsoid.

We have compared our FEM results with several sets of previously published experimental data. In cases where the microstructure of the porous ceramics roughly matched that of the models, the agreement was very good. Since the FEM results correspond to a known microstructure, it was possible to explain deviations in terms of specific microstructural features. Thus, comparison of experimental data with the three computational results provides a useful interpretive tool. Note that a given elastic modulus does not correspond to a particular microstructure. Therefore, it is important to corroborate microstructural interpretations obtained from the elastic moduli with information about the particular material (such as a micrograph). In the future it would be useful to extend this work to higher porosities and to other relevant models (such as non-overlapping porous spheres). It is also possible to use statistical microstructural information obtained from two-dimensional micrographs to generate models [29] that actually mimic physical microstructures.

Acknowledgments A.R. thanks the Fulbright Foundation and Australian Research Council for financial support. We also thank the Partnership for High-Performance Concrete Technology program of the National Institute of Standards and Technology for partial support of this work.