Graded and Perfect Disorder in Random Media Elasticity

Abstract

The basic problem considered concerns the calculation of effective elastic moduli of a random elastic medium such as a polycrystalline aggregate or a disordered composite, given some statistical information about its microstructure. A rigorous formal general solution is obtained as a Neumann-type series of multiple integrals which contain the statistical information in terms of n-point correlation functions of increasing order n. The integrals can be calculated provided the medium is highly disordered. To describe the disorder in a quantitative fashion, the concept of graded and perfect (=maximum) disorder is introduced. Rigorous bounds of nth order are derived for the effective elastic moduli of random media which are classified as disordered of grade n. The effective elastic moduli of the perfectly disordered medium concide in rigor with the well-known self-consistent moduli.