The First Pressure Derivative of the Shear Modulus of Porous Materials

Abstract

Summary A general theory for the calculation of the second order effective elastic moduli of porous materials in which the porosity is in the form of isolated cavities is presented. The particular case of spherical cavities distributed randomly within an isotropic matrix in such a manner that the material is macroscopically isotropic is then considered in detail and an expression for the first pressure derivative of the effective shear modulus of such a material is obtained correct to first order in the porosity. 1. Introduction In a paper by Walton (1973), hereafter referred to as Paper I, the first pressure derivative of the effective bulk modulus of a porous material was calculated. The particular porous medium considered was that of a homogeneous isotropic matrix containing a dilute distribution of spherical cavities, not necessarily of the same size but such that the total porosity c (that is, the ratio of cavity volume to total volume) is so small that terms of order c2 may be neglected in comparison with unity. Furthermore, the distribution was assumed to be random and such that the material is macroscopically homogeneous and isotropic. The aim of the present paper is to extend the method used in Paper I to the calculation of the first pressure derivative of the effective shear modulus of such a material. 2. Second order effective moduli The method is based on considerations of the overall constitutive law and, in the spirit of Hill (1963), the problem of the calculation of the effective elastic moduli of porous materials may be formulated as follows. The model to be considered is that of a large volume V of some porous material subjected to a uniform strain in its outer boundary. The matrix material is assumed both perfectly elastic and homogeneous, although not necessarily isotropic. The porosity, on the other hand, is assumed to be in the form of isolated cavities distributed throughout the matrix in such a manner that the material is macroscopically homogeneous, although not necessarily isotropic. Finally, there is no restriction at this stage on the size of the porosity c. With the 9-vectors S and D denoting the nominal stress and displacement gradient respectively and with superscripts (m) and (c) referring respectively to the solid matrix and the cavities, the constitutive law for the matrix material may be written, correct to second order in D(m),