Theory of elastic constants of heterogeneous media

The theory of elastic equilibrium of heterogeneous, macrohomogeneous bodies is cast in the form of a linear integral equation with the following properties: (1) The equation contains a constant elastic tensor which can be chosen freely to optimize any approximation method. (2) The singularity of the kernel is such that the integral must be interpreted as the principal value, thereby providing a clear separation between structure-dependent and structure-independent contributions. (3) The mean elastic tensor equals the above-mentioned constant tensor plus the space average of the dependent variable. (4) A term proportional to this average appears separately in the equation. (5) The integral equation facilitates the rederivation of several well-known approximations and opens new possibilities, of which some are discussed, in particular, a self-consistent imbedding method and an approximation based on correlation functions.