For macroscopically anisotropic media in which the variations in the phase stiffness tensor are small, formal solutions to the boundary-value problem have been developed in the form of perturbation series (Dederichs and Zeller, 1973; Gubernatis and Krumhansl, 1975 ; Willis, 1981). Due to the nature of the integral operator, one must contend with conditionally convergent integrals. One approach to this problem is to carry out a “renormalization” procedure which amounts to identifying physically what the conditionally convergent terms ought to contribute and replacing them by convergent terms that make this contribution (McCoy, 1979). For the special case of macroscopically isotropic media, the first few terms of this perturbation expansion have been explicitly given in terms of certain statistical correlation functions for both three-dimensional media (Beran and Molyneux, 1966 ; Milton and Phan-Thien, 1982) and two-dimensional media (Silnutzer, 1972 ; Milton, 1982). A drawback of all of these classical perturbation expansions is that they are only valid for media in which the moduli of the phases are nearly the same, albeit applicable for arbitrary volume fractions. In this paper we develop new, exact perturbation expansions for the effective stiffness tensor of macroscopically anisotropic composite media consisting of two isotropic phases by introducing an integral equation for the so-called “cavity” strain field. The expansions are not formal but rather the nth-order tensor coefficients are given explicitly in terms of integrals over products of certain tensor fields and a determinant involving n-point statistical correlation functions that render the integrals absolutely convergent in the infinite-volume limit. Thus, no renormalization analysis is required because the procedure used to solve the integral equation systematically leads to absolutely convergent integrals. Another useful feature of the expansions is that they converge rapidly for a class of dispersions for all volume fractions, even when the phase moduli differ significantly.

针对宏观各向异性介质，其中相位刚度张量的变化很小，边值问题的形式解已经通过摄动级数的形式得到了发展（Dederichs和Zeller，1973年；Gubernatis和Krumhansl，1975年；Willis，1981年）。由于积分算子的性质，必须处理条件收敛积分。解决这个问题的一种方法是进行“重整化”程序，即识别物理上条件收敛的项应该做出的贡献，并用能够产生这种贡献的收敛项替换它们（McCoy，1979年）。对于宏观各向同性介质的特殊情况，这个摄动展开的前几项已经明确给出了三维介质（Beran和Molyneux，1966年；Milton和Phan-Thien，1982年）和二维介质（Silnutzer，1972年；Milton，1982年）的某些统计相关函数的表达式。所有这些经典摄动展开的缺点在于它们仅适用于相位模量几乎相同的介质，尽管适用于任意的体积分数。在本文中，我们通过引入所谓的“腔体”应变场的积分方程，为由两个各向同性相组成的宏观各向异性复合介质的有效刚度张量开发了新的精确摄动展开。这些展开不是形式上的，而是第n阶张量系数用一些张量场和涉及n点统计相关函数的行列式的积分显式给出，这些统计相关函数使得积分在无限体积极限下绝对收敛。因此，不需要重整化分析，因为用于解决积分方程的程序系统地导致绝对收敛积分。展开的另一个有用特性是，它们在所有体积分数的某些分散类别中都快速收敛，即使相位模量差异很大。