Integral Micromorphic Model Reproducing Dispersion in 1D Continuum

TL;DR

This paper introduces a new integral micromorphic model that accurately reproduces dispersion relations, including band gaps, in 1D metamaterials by nonlocal averaging and calibration using Fourier transforms.

Contribution

It presents a novel integral micromorphic continuum model that can exactly match dispersion curves of band-gap metamaterials through a calibration process involving Fourier analysis.

Findings

Successfully reproduces dispersion curves including band gaps.

Validated on a mass-spring chain with explicit dispersion relation.

Enables accurate modeling of wave propagation in metamaterials.

Abstract

The paper develops a new integral micromorphic elastic continuum model, which can describe dispersion properties of band-gap metamaterials, i.e., metamaterials that inhibit propagation of waves in a certain frequency range. The enrichment consists in nonlocal averaging of three terms in the expression for the potential energy density of the standard micromorphic continuum. After proper calibration, such a formulation can exactly reproduce two given branches of the dispersion curve (acoustic and optical), even in cases with a band gap. The calibration process exploits Fourier images of the unknown weight functions, which are analytically deduced from the dispersion relation of the material of interest. The weight functions are then reconstructed in the spatial domain by numerical evaluation of the inverse Fourier transform. The presented approach is validated on several examples, including a discrete mass-spring chain with two alternating masses, for which the dispersion relation has an explicit analytical form and the optical and acoustic branches are separated by a band gap.