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

TL;DR

This paper establishes the mathematical well-posedness of a relaxed linear micromorphic continuum model, addressing existence, uniqueness, and continuous dependence, with implications for dislocation dynamics, gradient plasticity, and seismic processes.

Contribution

It introduces a relaxed micromorphic model with weaker boundary conditions and non-uniform positive definiteness, extending the mathematical framework for continuum mechanics.

Findings

Proves well-posedness of the relaxed micromorphic model.

Establishes boundary conditions weaker than classical models.

Connects the model to physical phenomena like earthquakes and plasticity.

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.