A strong contrast homogenization formulation for multi-phase anisotropic materials

Homogenization relations, linking a material's properties at the mesoscale to those at the macroscale, are fundamental tools for design and analysis of microstructure. Recent advances in this field have successfully applied spectral techniques to Kroner-type perturbation expansions for polycrystalline and composite materials to provide efficient inverse relations for materials design. These expansions have been termed 'weak-contrast' expansions due to the conditionally convergent integrals, and the reliance upon only small perturbations from the reference property. In 1955, Brown suggested a different expansion for electrical conductivity that resulted in absolutely convergent integrals. Torquato subsequently applied the method to elasticity, with good results even for high-contrast materials; thus it is commonly referred to as a 'strong contrast' expansion. The methodology has been applied to elasticity for two phases of isotropic material, generally assuming macroscopic isotropy (with noted exceptions), thus resulting in a rather elegant form of the solution. More recently, a multi-phase form of the solution was developed for conductivity. This paper builds upon this result to apply the method to elasticity of polycrystalline materials with both local and global anisotropy. New spectral formulations are subsequently developed for both the weak and strong contrast solutions. These form the basis for efficient microstructure analysis using these frameworks, and subsequently for inverse design applications. The process is taken through to demonstration of a property closure, which acts as the basis for materials design; the closure delineates the envelope of all physically realizable property combinations for the chosen properties, based on the particular homogenization relation being used. (C) 2008 Elsevier Ltd. All rights reserved.