Estimation of the Elastic Moduli of Porous Materials using Analytical Methods, Numerical Methods, and Image Analysis

Abstract: The effective bulk modulus and effective shear modulus of porous materials having various types of pore shapes are investigated, using both analytical and numerical methods. These solutions, and the scaling laws that are derived with the aid of these solutions, are then used to make predictions of the effective elastic moduli of some sandstones and ceramics, based on two-dimensional images of the pore space.The complex variable method is used to find the hydrostatic and shear compliances of a large family of pores that have N-fold rotational symmetry, and which have at most four terms in their conformal mapping function. This solution is validated using boundary element (BEM) calculations, and is also used to test two scaling laws that estimate the compliances based on the area and perimeter of the pore.The boundary perturbation method is used to study the effect of small-scale roughness on the compressibility and shear compliance of a nominally circular pore. The solution is carried out to fourth order in the roughness parameter for the case of hydrostatic loading, and to second order for shear loading. These solutions allow one to judge the scale of roughness that can safely be ignored when obtaining images of the pores.Predictions are then made of the elastic moduli of some porous materials – two sandstones and a ceramic. Starting with scanning electron micrographs, image analysis software is used to isolate and extract each pore from the host material. The bulk and shear compliances are estimated using both BEM and the two scaling laws. Areally-weighted mean values of these compliances are calculated for each material, and the differential effective medium scheme is used to obtain expressions for the moduli as functions of porosity. These predictions agree well with the experimental values found in the literature.