First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model

TL;DR

This paper provides the first experimental estimation of key parameters in the relaxed micromorphic model for real band-gap metamaterials, revealing the non-locality scale related to the microstructure of phononic crystals.

Contribution

It introduces a novel procedure to identify relaxed micromorphic model parameters from experimental wave transmission data in metamaterials.

Findings

Estimated 5 of 6 elastic parameters for the model.

Determined the micro-inertia parameter.

Estimated the non-locality length scale as about one-third of the hole diameter.

Abstract

In this paper we propose the first estimate of some elastic parameters of the relaxed micromorphic model on the basis of real experiments of transmission of longitudinal plane waves across an interface separating a classical Cauchy material (steel plate) and a phononic crystal (steel plate with fluid-filled holes). A procedure is set up in order to identify the parameters of our model by superimposing the experimentally-based profile of the reflection coefficient (plotted as function of the frequency of the traveling waves) with the analogous profile obtained via simulations based upon the relaxed micromorphic model. We end up with the determination of 5 out of 6 constitutive parameters which are featured by the relaxed micromorphic model in the isotropic case, plus the determination of the micro-inertia parameter. The sixth elastic parameter, namely the Cosserat couple modulus $\mu_{c}$, still remains undetermined, since experimental data concerning the transmission properties of the considered interface for transverse incident waves are not yet available. A fundamental result of the present paper is the estimate of the non-locality intrinsically associated to the underlying microstructure of the metamaterial. As a matter of fact, we appraise that the characteristic length $L_{c}$ measuring the non-locality of the considered phononic crystal is of the order of $1/3$ of the diameter of the considered fluid-filled holes.