On the Limit and Applicability of Dynamic Homogenization

TL;DR

This paper investigates the accuracy of dynamic homogenization in modeling wave reflection in layered composites, demonstrating its effectiveness at low frequencies and exploring negative effective properties in multi-phase composites.

Contribution

It quantifies the approximation involved in dynamic homogenization and analyzes reflection at interfaces, establishing conditions for its validity and exploring negative properties in multi-phase composites.

Findings

Homogenization yields negligible reflection at low frequencies.

Second branch of 3-phase composite exhibits negative group velocity.

Reflection analysis supports the negative nature of certain composite branches.

Abstract

Recent years have seen considerable research success in the field of dynamic homogenization which seeks to define frequency dependent effective properties for heterogeneous composites for the purpose of studying wave propagation. There is an approximation involved in replacing a heterogeneous composite with its homogenized equivalent. In this paper we propose a quantification to this approximation. We study the problem of reflection at the interface of a layered periodic composite and its dynamic homogenized equivalent. It is shown that if the homogenized parameters are to appropriately represent the layered composite in a finite setting and at a given frequency, then reflection at this special interface must be close to zero at that frequency. We show that a comprehensive homogenization scheme proposed in an earlier paper results in negligible reflection in the low frequency regime, thereby suggesting its applicability in a finite composite setting. In this paper we explicitly study a 2-phase composite and a 3-phase composite which exhibits negative effective properties over its second branch. We show that based upon the reflected energy profile of the two cases, there exist good arguments for considering the second branch of a 3-phase composite a true negative branch with negative group velocity. The results open intriguing questions regarding the effects of replacing a semi-infinite periodic composite with its Bloch-wave (infinite domain) dynamic properties on such phenomenon as negative refraction.