Justification of an effective field method in elasto-statics of heterogeneous solids

A heuristic scheme for predicting the overall properties of composite media has been proposed, explored and widely used in the last 25 years by Kanaun and Levin. The numerous results and technical details are collected in the impressive volume Kanaun and Levin (Petroza-vodsk: Izdatel'stvo Petrozavodskogo Universiteta, 1993 (in Russian)). The method is based on a certain old, natural and appealing physical reasoning, which well explains its name effective field method (EFM). The predictions of the methods agree very well with many experimental data. However, no attempt has been made to justify the method, putting it into the frame of a more rigorous theory of heterogeneous media. The present study represents such an attempt. The main conclusion is that the EFM approach, at least in elasto-statics when the effective moduli of the composite are looked for, is in essence a variational procedure of Hashin-Shtrikman type. Hence Kanaun-Levin's results provide, for the microstructures treated by the authors, rigorous variational prescriptions. The latter represent bounds on the effective moduli of the composites, if the matrix is weaker or stiffer than the inclusions. Moreover, it is shown that in the special, though quite general case of a matrix containing several kinds of ellipsoidal inclusions, distributed with "ellipsoidal" symmetry, an obvious generalization of the EFM method yields results that coincide with the variational estimates, recently obtained by Ponte Castaneda and Willis (J. Mech. Phys. Solids 43 (1995) 1919). (C) 2001 Elsevier Science Ltd. All rights reserved.