Bounds on Elastic Moduli of Composite Media

Abstract

Two variational principles are formulated which characterize the displacement and the stress in a time harmonic deformation of a nonuniform isotropic elastic material containing inclusions of different materials. These principles are used to obtain upper and lower bounds on the static effective elastic moduli of the composite material in terms of trial displacements and stresses. Admissible trial displacements and trial stresses are constructed and used to obtain specific bounds on the effective moduli. These bounds, which involve the statistics of the distribution of inclusions, are the main results. At low volume concentrations, for each modulus the upper and lower bounds coincide, yielding explicit expressions for the moduli. For spherical and circular cylindrical inclusions, these low concentration results have been derived before. In the case of identical equally spaced spherical inclusions, bounds valid for all concentrations are presented.