Complete band gaps including non-local effects occur only in the relaxed micromorphic model

TL;DR

The paper demonstrates that complete band gaps with non-local effects are uniquely achievable by the relaxed micromorphic model among linear, isotropic, elastic continuum models, through a comparative analysis of different micromorphic models.

Contribution

It establishes that only the relaxed micromorphic model can describe complete band gaps with non-local effects at a phenomenological level.

Findings

Standard micromorphic model cannot produce partial band gaps.

Div-model shows size effects but no band gaps.

Relaxed micromorphic model uniquely supports complete band gaps.

Abstract

In this paper we substantiate the claim implicitly made in previous works that the relaxed micromorphic model is the only linear, isotropic, reversibly elastic, nonlocal generalized continuum model able to describe complete band-gaps on a phenomenological level. To this end, we recapitulate the response of the standard Mindlin-Eringen micromorphic model with the full micro-distortion gradient of P, the relaxed micromorphic model depending only on the Curl P of the micro-distortion P, and a variant of the standard micromorphic model in which the curvature depends only on the divergence Div P of the micro distortion. The Div-model has size-effects but the dispersion analysis for plane waves shows the incapability of that model to even produce a partial band gap. Combining the curvature to depend quadratically on Div P and Curl P shows that such a model is similar to the standard Mindlin-Eringen model which can eventually show only a partial band gap.