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In this paper we consider the equilibrium problem in the relaxed linear modelof micromorphic elastic materials. The basic kinematical fields of thisextended continuum model are the displacement $u\in \mathbb{R}^3$ and thenon-symmetric micro-distortion density tensor $P\in \mathbb{R}^{3\times 3}$. Inthis relaxed theory a symmetric force-stress tensor arises despite the presenceof microstructure and the curvature contribution depends solely on themicro-dislocation tensor ${\rm Curl}\, P$. However, the relaxed model is ableto fully describe rotations of the microstructure and to predict non-polarsize-effects. In contrast to classical linear micromorphic models, we allow theusual elasticity tensors to become positive-semidefinite. We prove that,nevertheless, the equilibrium problem has a unique weak solution in a suitableHilbert space. The mathematical framework also settles the question of whichboundary conditions to take for the micro-distortion. Similarities anddifferences between linear micromorphic elasticity and dislocation gauge theoryare discussed and pointed out.

The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations