A unifying perspective: the relaxed linear micromorphic continuum

TL;DR

This paper introduces a relaxed linear micromorphic continuum model that captures microstructure rotation and size-effects in heterogeneous materials, with improved mathematical properties and potential applications in dislocation dynamics and seismic modeling.

Contribution

It presents a novel relaxed micromorphic model with symmetric stresses and a micro-dislocation tensor, unifying and simplifying existing models while ensuring well-posedness.

Findings

Supports well-posedness for static and dynamic cases

Relates to dislocation dynamics and seismic processes

Handles non-polar size-effects in heterogeneous materials

Abstract

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict non-polar size-effects. It is intended for the homogenized description of highly heterogeneous, but non polar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. 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. It unifies and simplifies the understanding of the linear micromorphic models.