Determnation of elasticity constants for microheterogeneous media

Abstract

The foundations of the theory of stochastically heterogeneous solids were laid a long time ago by Voigt [1J, who developed a method for determining the macroscopic parameters of polycrystalline materials by averaging the appropriate crystallite parameters with respect to orientations. Lifshits and Rozentzveig [2] showed that it was necessary to consider the correlation properties of the field in computations of macroscopic parameters. They calculated the first corrections for the averaged elastic constants of polycrystallites for the case of cubic and hexagonal crystallites. Assuming a low degree of heterogeneity, these authors used an approximation which corresponds to the Born approximation in the theory of scattering [3]. This method and its modifications were subsequently used by several authors for the computation of macroscopic parameters of polycrystallites [4– 6] and of other microheterogeneous materials [8]. Moreover, the assumption of a low degree of heterogeneity of the properties is very restrictive. It precludes use of the method in the case of macroscopically isotropic polycrystallites formed from essentially anisotropic crystallite stochastically glass reinforced plastics, and similar microheterogeneous materials. This rises the problem of developing procedures that could be applied in cases of a high degree of heterogeneity. This problem presents serious analytical difficulties, however. It is sufficient to point out that even computation of the second approximation (i.e., the one following the Born approximation) has not yet been completed. Analogous problems in the classical and quantum theories of scattering are also, as a rule, considered only in the Born approximation. More complicated methods (e.g., Feyman's method) make possible only partial summation of infinite sequences in which the result is obtained. A method analogous to that of a selfconsistent field in quantum mechanics [9,10] is promising; however, this method is approximate and the magnitude of its error has not yet been estimated. The possibility of accurate determination of mascroscopic parameters for certain classes of microheterogeneous media was demostrated in [11], in which a detailed analysis was presented of parameters forming a second order tensor and characterizing the distribution in the medium of a certain scalar value obeying an equation similar to the steady-state heat-conduction equation. Accurate formulas for macroscopic coefficients of thermal conductivity (diffusion) were derived for the case of a strongly anisotropic medium and for that of a medium with a high degree of transverse isotropy. We made a comparison with various approximate methods and evaluated their degree of error. This article describes an accurate method of computing macroscopic elastic constants for polycrystalline media with a high degree of anisotropy; for the case of polycrystals with a cubic structure [12] the error margin and range of application of approximate methods are estimated.