Relaxed micromorphic model of transient wave propagation in anisotropic band-gap metastructures

Abstract

In this paper, we show that the transient waveforms arising from several localised pulses in a micro-structured material can be reproduced by a corresponding generalised continuum of the relaxed micromorphic type. Specifically, we compare the dynamic response of a bounded micro-structured material to that of bounded continua with special kinematic properties: (i) the relaxed micromorphic continuum and (ii) an equivalent Cauchy linear elastic continuum. We show that, while the Cauchy theory is able to describe the overall behaviour of the metastructure only at low frequencies, the relaxed micromorphic model goes far beyond by giving a correct description of the pulse propagation in the frequency band-gap and at frequencies intersecting the optical branches. In addition, we observe a computational time reduction associated with the use of the relaxed micromorphic continuum, compared to the sensible computational time needed to perform a transient computation in a micro-structured domain.

Nonstandard micro-inertia terms in the relaxed micromorphic model: well-posedness for dynamics

“…This insight has opened a way to obtain new generalized continuum models allowing to describe the behavior of metastructures in the simplified framework of continuum mechanics with homogenized material behavior. Moreover, due to the a priori relation between macro and micro parameters of the considered samples obtained in [4], the constitutive parameters of our model can be identified on real meta-materials [5,13] even for anisotropic materials, opening the way to the efficient design and realization of meta-structures.…”

Existence results for non‐homogeneous boundary conditions in the relaxed micromorphic model

“…This choice has allowed us to reduce the number of constitutive coefficients involved in the model. In contrast to the classical micromorphic theory, considering a wave ansatz in the partial differential equations arising in the relaxed micromorphic model, some band gaps occur in the diagrams describing the dispersion curves; see previous studies 7‐14 . This means that there exists an interval of wave frequencies for which no wave propagation may occur 12,13,15,16 .…”

“…In contrast to the classical micromorphic theory, considering a wave ansatz in the partial differential equations arising in the relaxed micromorphic model, some band gaps occur in the diagrams describing the dispersion curves; see previous studies. [7][8][9][10][11][12][13][14] This means that there exists an interval of wave frequencies for which no wave propagation may occur. 12,13,15,16 Such an interesting phenomena is obtained in experiments when metamaterials are designed and cannot be captured by the classical micromorphic approach.…”

Analysis of planes within reduced micromorphic model

“…Furthermore, the coupling between the micro-strain and the macro-strain is eliminated in this model. The relaxed micromorphic model has been used to model the wave propagation in metamaterials 24 – 27 .…”