On the representation formula for well-ordered elastic composites: a convergence of measure approach

The aim of this paper is to derive an integral representation formula for the effective elasticity tensor for a two-component well-ordered composite of elastic materials without using a third reference medium and without assuming the completeness of the eigenspace of the operator G defined in (2.16) in (J. Mech. Phys. Solids 1984; 32(1):41-62). As shown in (J. Mech. Phys. Solids 1984; 32(1):41-62) and (Math. Meth. Appl. Sci. 2006; 29(6):655-664), this integral representation formula implies a relation which links the effective elastic moduli to the N-point correlation functions of the microstructure. Such relation not only facilitates a powerful scheme for systematic incorporation of microstructural information into bounds on the effective elastic moduli but also provides a theoretical foundation for de -homogenization. The analysis presented in this paper can be generalized to an n-component composite of elastic materials. The relations developed here can be applied to the de-homogenization for a special class of linear viscoelastic composites. The results will be presented in another paper.