Static size-effects meet the dynamic scattering properties of finite-sized mechanical metamaterials: a relaxed micromorphic study with parameter identification via two-stage static-dynamic optimization

TL;DR

This study explores the relationship between static size-effects and dynamic wave scattering in mechanical metamaterials using a relaxed micromorphic model with a two-stage parameter identification process.

Contribution

It introduces a novel two-stage static-dynamic optimization method to identify material parameters of the relaxed micromorphic model for metamaterials.

Findings

Static size-effects correlate with dynamic scattering patterns.

The two-stage optimization accurately fits static and dynamic behaviors.

The relaxed micromorphic model captures complex wave phenomena.

Abstract

Mechanical metamaterials exhibit size-effects when a few unit-cells are subjected to static loading because no clear micro-macro scale separation holds and the characteristic length of the deformation becomes comparable to the unit-cell size. These size-effects typically manifest themselves as a strengthening of the response in a form summarized as "smaller is stiffer". Moreover, the dynamical behavior of mechanical metamaterials is very remarkable, featuring unique phenomena such as dispersive behavior and band-gaps where elastic waves cannot propagate over specific frequency ranges. In these frequency ranges, the wavelength becomes gradually comparable to the unit-cell size, giving rise to microstructure related phenomena which become particularly visible in the reflection/transmission patterns where an incident wave hits the metamaterial's interfaces. This raises the question of whether the static size-effects and dynamic reflection/transmission patterns are correlated. In this work, we investigate the interaction of the static size-effects and the dynamic scattering response of mechanical metamaterials by employing the relaxed micromorphic model. We introduce a two-stage optimization procedure to identify the material parameters. In the first stage, the static material parameters are identified by exploiting the static size-effects through a least squares fitting procedure based on the total energy. The dynamic parameters are determined in the second stage by fitting the dispersion curves of the relaxed micromorphic model to those of the fully discretized microstructure. At this second stage, we assess the results obtained by fitting the dispersion curves in one and in two propagation directions, for both the relaxed micromorphic model (RMM) with curvature and its reduced counterpart (RRMM) without curvature. The full abstract is presented in the paper.