On the role of micro-inertia in enriched continuum mechanics

Abstract

In this paper, the role of gradient micro-inertia termsη ∇u ,t 2 and free micro-inertia terms η P ,t 2 is investigated to unveil their respective effects on the dynamic behaviour of band-gap metamaterials. We show that the termη ∇u ,t 2 alone is only able to disclose relatively simplified dispersive behaviour. On the other hand, the term η P ,t 2 alone describes the full complex behaviour of bandgap metamaterials. A suitable mixing of the two micro-inertia terms allows us to describe a new feature of the relaxed-micromorphic model, i.e. the description of a second band-gap occurring for higher frequencies. We also show that a split of the gradient micro-inertiaη ∇u ,t 2 , in the sense of Cartan-Lie decomposition of matrices, allows us to flatten separately the longitudinal and transverse optic branches, thus giving us the possibility of a second band-gap. Finally, we investigate the effect of the gradient inertiaη ∇u ,t 2 on more classical enriched models such as the Mindlin-Eringen and the internal variable ones. We find that the addition of

A panorama of dispersion curves for the weighted isotropic relaxed micromorphic model

“…), the resulting dispersion curves are basically those of classical elasticity: two straight lines starting from the origin. The problem is simply that we do not give to the model the possibility to express its dynamical behavior since there is no micro-inertia that is able to trigger micro-vibrations (see also [22]).…”

“…The following identities will be useful in the following. Remembering the definitions of u (1,k) , P (1k) , P D , P S , u [1,k] − P [1k] given in (20)- (22), the scalar components of this vectorial system are therefore (1,2) − P (12) u (1,3) − P (13) [1,2] − P [12] u [1,3] − P [13] ⎞ ⎠ ⎞ ⎠ , [1,2] + P [12] 0 u [1,3] − P [13] ⎞ ⎠ ⎞ ⎠ , [1,3] + P [13] −u [2,3] + P [23] 0…”

“…[2-9, 11, 17, 35, 40, 42, 44]), but this is indeed not the case when considering pure elasticity in which the standard formulations commonly introduce the whole gradient ∇ P of the micro-distortion tensor P . As a matter of fact, the use of micromorphic models which only consider the Curl of the micro-distortion in a purely linear-elastic framework can shed light on the modeling of non-local metamaterials which exhibit band-gap behaviors [20,[22][23][24][25][26][27].…”

Anisotropic wave dispersion and band‐gaps in mechanical metamaterials via the relaxed micromorphic model

“…In this proceeding, we want to present a continuum model which is able to describe the exotic behavior of metamaterials in a very precise way. Among all possible choices present in the literature, the relaxed micromorphic model [1][2][3][4][5][6][7][8] is the only one which allows us to describe the presence of band-gaps in metamaterials.…”

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.…”