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 Ĝ 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. Copyright © 2006 John Wiley & Sons, Ltd.