The relaxed linear micromorphic continuum: Existence, uniqueness and continuous dependence in dynamics

Abstract

We study well-posedness for the relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. In contrast to classical micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. Another interesting feature concerns the prescription of boundary values for the micro-distortion field: only tangential traces may be determined which are weaker than the usual strong anchoring boundary condition. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes.

The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results

“…Many results are available by now (e.g. [9,63,71,[106][107][108][109][110][111]) indicating that it is physically needed or mathematically consistent to consider macroscopic continuum models where contact interactions expend work on high order virtual displacement gradients on dividing surfaces. These interactions are exactly those which are called s-forces, following Green & Rivlin [43,44,46] or [28,33].…”

Real wave propagation in the isotropic-relaxed micromorphic model

“…The model is well posed in the statical and dynamic case even for zero Cosserat couple modulus μ c = 0; see [21,22]. In that case, it is non-redundant in the sense of [23].…”

On the role of micro-inertia in enriched continuum mechanics

“…-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].…”