Bounds on Effective Dynamic Properties of Elastic Composites

TL;DR

This paper develops rigorous, computable bounds for the dynamic elastic properties of heterogeneous composites, applicable to any geometry or boundary condition, extending static bounds to frequency-dependent cases.

Contribution

It introduces explicit, strict bounds for the frequency-dependent elastic properties of composites, generalizing Hashin-Shtrikman bounds to dynamic conditions without restrictions on homogeneity or isotropy.

Findings

Derived explicit bounds for dynamic elastic moduli.

Applicable to any geometry and boundary conditions.

Extends static bounds to frequency-dependent properties.

Abstract

We present general, computable, improvable, and rigorous bounds for the total energy of a finite heterogeneous volume element or a periodically distributed unit cell of an elastic composite of any known distribution of inhomogeneities of any geometry and elasticity, undergoing a harmonic motion at a fixed frequency or supporting a single-frequency Bloch-form elastic wave of a given wave-vector. These bounds are rigorously valid for \emph{any consistent boundary conditions} that produce in the finite sample or in the unit cell, either a common average strain or a common average momentum. No other restrictions are imposed. We do not assume statistical homogeneity or isotropy. Our approach is based on the Hashin-Shtrikman (1962) bounds in elastostatics, which have been shown to provide strict bounds for the overall elastic moduli commonly defined (or actually measured) using uniform boundary tractions and/or linear boundary displacements; i.e., boundary data corresponding to the overall uniform stress and/or uniform strain conditions. Here we present strict bounds for the dynamic frequency-dependent constitutive parameters of the composite and give explicit expressions for a direct calculation of these bounds.