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

Abstract

AM, 0000-0003-1940-9853 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 the relaxed micromorphic model by superimposing the experimentally based profile of the reflection coefficient (plotted as function of the wave-frequency) with the analogous profile obtained via numerical simulations. We determine five out of six 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 μ c , still remains undetermined, since experiments on transverse incident waves are not yet available. A fundamental result of this paper is the estimate of the non-locality intrinsically associated with the underlying microstructure of the metamaterial. We show that the characteristic length L c measuring the non-locality of the phononic crystal is of the order of

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

“…Indeed, the relaxed micromorphic model has already been a source of inspiration for researchers working on granular materials [55]. 8 Moreover, the clear and transparent application of the relaxed micromorphic model in the isotropic case has recently been successfully achieved for the description of band-gap metamaterials (see [43,44]). …”

Real wave propagation in the isotropic-relaxed micromorphic model

“…The definition of the unknown vectors was made considering the coupling of the variables in the equations of motion; see [15][16][17][18][32][33][34][35][36]. More particularly, it has been shown in these previous works that three sets of equations can be isolated: one involving only longitudinal quantities, one involving only transverse quantities and one of three completely uncoupled equations.…”

“…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].…”

On the role of micro-inertia in enriched continuum mechanics

“…We have already shown in [20] that the relaxed micromorphic model with free micro-inertia can be successfully used to describe the dynamic behaviour of actual band-gap metamaterials. In that case, we showed that the model is perfectly able to capture experimental results related to the transmission coefficient at an interface between a classical Cauchy material and a specific band-gap metamaterial.…”

“…-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.…”