The effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations

Abstract

This paper concerns the determination of the effective elastic moduli of an isotropic composite material containing randomly distributed spheres of the same size at non-dilute concentrations φ. By employing the solutions[18, 19] for the elasticity problems of two interacting spheres in the presence of four different applied strains at infinity and a method developed by Batchelor[15, 27] and by Jeffrey[16, 17] for computing bulk quantities which involve conditionally convergent integrals, we evaluate the effective moduli of the composite exactly to order φ2 and thereby extend the Einstein formula.In the particular case of an incompressible matrix, the expression for the bulk modulus κ, when rearranged, leads to an extension of Taylor's result[20] for the expansion viscosity of an incompressible fluid containing air bubbles.The present calculations have a wider significance than just to elasticity in that they give a better understanding of the method of normalization[17] for converting a conditionally convergent integral into one that is absolutely convergent. Specifically, when the applied strain is isotropic, two sources of indeterminancy are uncovered. The first arises from the unusual property of S(1)ij, the additional dipole of one sphere due to the interaction with a second which is required for the evaluation of the bulk moduli to O(φ2), whose trace for R ⪢ 1 is O(R−6), when R is the distance between the two sphere centers, whereas, all its other elements are O(R−3). This suggests that there may exist a method for calculating the effective bulk modulus which does not require a normalization to lead to an absolutely convergent integral and which gives, apparently, a different result. Secondly, the exact method of normalization is not unambiguous in that two possible ways of normalizing are shown to exist. However, when higher-order particle interactions (especially the three-particle interactions) are taken into account, this indeterminancy is resolved and a unique type of expansion applies.