Reflection and transmission of elastic waves in non-local band-gap metamaterials: A comprehensive study via the relaxed micromorphic model

Abstract

In this paper we propose to study wave propagation, transmission and reflection in band-gap mechanical metamaterials via the relaxed micromorphic model. To do so, guided by a suitable variational procedure, we start deriving the jump duality conditions to be imposed at surfaces of discontinuity of the material properties in non-dissipative, linear-elastic, isotropic, relaxed micromorphic media. Jump conditions to be imposed at surfaces of discontinuity embedded in Cauchy and Mindlin continua are also presented as a result of the application of a similar variational procedure. The introduced theoretical framework subsequently allows the transparent set-up of different types of micro-macro connections granting the description of both i) internal connexions at material discontinuity surfaces embedded in the considered continua and, as a particular case, ii) possible connections between different (Cauchy, Mindlin or relaxed micromorphic) continua. The established theoretical framework is general enough to be used for the description of a wealth of different physical situations and can be used as reference for further studies involving the need of suitably connecting different continua in view of (meta-)structural design. In the second part of the paper, we focus our attention on the case of an interface between a classical Cauchy continuum on one side and a relaxed micromorphic one on the other side in order to perform explicit numerical simulations of wave reflection and transmission. This particular choice is descriptive of a specific physical situation in which a classical material is connected to a phononic crystal. The reflective properties of this particular interface are numerically investigated for different types of possible micro-macro connections, so explicitly showing the effect of different boundary conditions on the phenomena of reflection and transmission. Finally, the case of the connection between a Cauchy continuum and a Mindlin one is also presented as a numerical study, so showing that band-gap description is not possible for such continua, in strong contrast with the relaxed micromorphic case.

Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics

“…This is consistent with the classical notation of equation (48). Now, if we consider a quadratic energy in ε -β, recalling equation (46) and (47), we can write it as:…”

Real wave propagation in the isotropic-relaxed micromorphic model

“…In subsequent works [15][16][17][18], the model has shown its wider applicability compared with the classical Mindlin-Eringen micromorphic model in diverse areas [10][11][12]19].…”

“…We suppose that the space dependences of all introduced kinematic fields are limited to a direction defined by a unit vectorξ ∈ R 3 , which is the direction of propagation of the wave and which is assumed given. Hence, we look for solutions of (2.15) in the form 18) where…”

On the role of micro-inertia in enriched continuum mechanics

“…The associated equations of motion in strong form, obtained by a classical least action principle, take the form [22][23][24][25] …”

“…-Micro-inertia terms involving time derivatives of the extra kinematic degrees of freedom η P ,t 2 allow us to describe and control optic branches in the dispersion relations of classical and relaxed micromorphic continuum models [1,2,[19][20][21][22][23][24][25][26]. -The relaxed micromorphic model with micro-inertia of the type η P ,t 2 is able to describe the onset of the first band-gaps in mechanical metamaterials [20][21][22][23][24]. -The relaxed micromorphic model with both micro-inertia terms η P ,t 2 andη ∇u ,t 2 allows us to account for the first and also for the second band-gap which occurs for higher frequencies.…”