A unifying perspective: the relaxed linear micromorphic continuum

Abstract

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy forcestresses 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.

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

“…In this paper we do not present the historic development of enriched continua, referring the reader to [52,69] for this purpose. Furthermore, we restrict our attention to the linearized framework noting that the first 3D body x Ω ϕ(Ω) Figure 1: Enriched kinematics for a micromorphic continuum.…”

Real wave propagation in the isotropic-relaxed micromorphic model

“…Well-posedness results for the statical and dynamic cases have been provided in [14], making decisive use of recently established new coercive inequalities, generalizing Korn's inequality to incompatible tensor fields [24][25][26][27][28]. Decisive for the relaxed micromorphic formulation is the definition of the elastic energy in terms of suitable strain tensors.…”

Untitled

“…Its extension to higher gradient theories like those proposed in Placidi 2016;Neff et al 2014] is possible although there are technical details that need to be carefully analyzed.…”