Two-parameter integral representation formula for the effective elastic moduli of two-phase composites

Abstract

In this article, the integral representation formula (IRF) for the family of composites described in (1) is derived and the relation between moments of positive measures and the microstructure are established explicitly. This family represents composites consisting of one phase with fixed isotropic elasticity tensor and the other phase with varying elasticity tensor, e.g. viscoelastic materials whose complex-valued moduli change with frequency. Every choice of complex bulk modulus and shear modulus for constituent in Ω₂ corresponds to a unique choice of parameters ξ₁ and ξ₂, the contrast between the bulk moduli and shear moduli, respectively. All choices with Im(ξ₁) · Im(ξ₂) > 0 have well-defined effective elasticity tensor C*(ξ₁, ξ₂). For each component of C* that can be obtained by ε ₀ : C* :  ε ₀ with some real symmetric matrix ε ₀, the Borel measure in the IRF is independent of the choice of (ξ₁, ξ₂); it is determined by the microstructure and the reference medium C ⁽¹⁾. This implies that information of microstructure of the composites and the contrast of constituent properties are separated. Hence it potentially provides a way for retrieving microstructural information from measurement of C* of different (ξ₁, ξ₂) and C ⁽¹⁾. The relation between the moments of the Borel measure and statistical information of the microstructure is established by first deriving the Taylor series expansion of C* near the homogeneous case ξ₁ = ξ₂ = 0 and then analytically extending it by using the IRF, which is valid for every choice of (ξ₁, ξ₂) ∈ U ² or (−U)².

Keywords:

• two-parameter integral representation formula
• elasticity
• analytic continuation

AMS Subject Classification:

• 32A25
• 35J25
• 78M40
• 74A40

Acknowledgements

This work is partially supported by NSF-DMS Math. Bio Grant No. 0920852. The author would like to thank Robert Lipton for stimulating discussions.