Exact results concerning multifield moduli of two-phase composites

TL;DR

This paper derives new exact relations for the elastic moduli of two-phase composites, independent of microstructure, providing a novel mathematical framework that generalizes previous results and allows expressing any modulus as a linear combination of others.

Contribution

The authors present a new derivation of Chen's expansions, generalize them, and establish microstructure-independent relations between component and composite moduli.

Findings

Exact microstructure-independent relations between component and composite moduli.

Any composite modulus can be expressed as a linear combination of other moduli with fixed coefficients.

The relations are equivalent to known compatibility conditions but are presented in a more advantageous form.

Abstract

Chen has recently shown how the response matrix of a two-phase composite can be written as linear combinations of products of the component matrices. We elaborate on Chen's expansions by deriving them in a different way, which a. shows them in a different light, and b. permits us to generalize them further. As an application of our results we find exact microstructure-independent relations between the moduli of the two components and those of any composite. The body of these relations is equivalent to the compatibility relations of Milgrom and Shtrikman (1989), but they are cast in a rather different form, which has certain advantages. As an example, we show how any modulus of an arbitrary two-phase composite can be written in closed form as a linear combination of any other $n$ of its moduli, with coefficients that depend only on the component moduli, but not on the volume fractions, or the microstructure.