The elastic properties of two-phase (solid-pore) porous materials depend on the geometrical nature of the pore space and solid phase, as well as the value of porosity [1,2,3,4]. Relevant aspects of porous materials may include pore shape and size as well as the size and type of the interconnections between solid regions. These features, which generally lack precise definition, comprise the microstructure of the material. In order to predict properties, or properly interpret experimental property-porosity relationships, it is necessary to have an accurate method of relating elastic properties to porosity and microstructure. In this paper we use the finite element method to derive simple formulae that relate Young's modulus and Poisson's ratio to porosity and microstructure, for three different models of microstructure.
There have been several different approaches to deriving property-porosity relations for porous materials. Formulae derived using the micro-mechanics method [5,6,7] are essentially various methods of approximately extending exact results for small fractions of spherical or ellipsoidal pores to higher porosities. This includes the differential  and self consistent methods [9,10,11,12] as well as the commonly used semi-empirical correction to the dilute result made by Coble and Kingery  to explain the properties of porous alumina. A drawback of this approach is that the microstructure corresponding to a particular formula is not precisely known; hence agreement or disagreement with data can neither confirm nor reject a particular model. A second problem is that these types of models provide no predictions for the case where the microstructure is comprised of incompletely sintered grains, which is a common morphology in porous ceramics. A second class of results [3,13] have been termed minimum solid area (MSA) models. In this approach purely geometrical reasoning is used to predict the elastic moduli based on the weakest points within the structure. Again, the microstructure that corresponds to the MSA predictions is not exactly known. A number of semi-empirical relations have also been proposed , which generally provide a reasonable means of describing data, extrapolating results and comparing data among materials. However, lacking a rigorous connection with microstructure, these results do not offer either predictive or interpretive power. Theoretical bounds [5,14] exist for the elastic properties, but the vanishing of the lower bound for porous materials lessens their predictive power. There are numerous other approaches, such as the generalized method of cells [6,4], which are not considered in this paper.
Another approach is to computationally solve the equations of elasticity for digital models of microstructure [15,16]. In principle this can be done exactly. However, large statistical variations and insufficient resolution, have limited the accuracy of results obtained to date. Only recently have computers been able to handle the large three-dimensional models and number of computations needed to obtain reasonable results. As input to the method, we employ three different microstructural models that broadly cover the types of morphology observed in porous ceramics. The results, which can be expressed simply by two (or sometimes three) parameter relations, correspond to a particular microstructure and explicitly show how the properties depend on the nature of the porosity. Therefore, the results can be used as a predictive tool for cases where the microstructure of the ceramic is similar to one of the models, and as an interpretive tool if the microstructure is unknown. The numerically exact FEM results are compared with various well-known micro-mechanics and MSA results to determine how close an approximation a particular formula provides for each model. In the FEM, we can freely vary the properties of the solid phase, allowing us to determine the dependence of Young's modulus and Poisson's ratio on the solid Poisson's ratio as well as on the porosity. This question has attracted recent interest in the ceramics literature [17,18,19].