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

2014/03/13 by Patrizio Neff, Neff, Patrizio, Ionel‐Dumitrel Ghiba +5 · 1 citation
Engineering · Materials Science · #Analysis of PDEs (math.AP) #Elasticity and Material Modeling #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlocal and gradient elasticity in micro/nano structures #Thermoelastic and Magnetoelastic Phenomena

paper · pdf · doi:10.48550/arxiv.1403.3442

openalex publication_date 2014/03/13 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the equilibrium problem in the relaxed linear model\nof micromorphic elastic materials. The basic kinematical fields of this\nextended continuum model are the displacement u\∈ \ℝ³ and the\nnon-symmetric micro-distortion density tensor P\∈ \ℝ^{3\× 3}. In\nthis relaxed theory a symmetric force-stress tensor arises despite the presence\nof microstructure and the curvature contribution depends solely on the\nmicro-dislocation tensor rm Curl , P. However, the relaxed model is able\nto fully describe rotations of the microstructure and to predict non-polar\nsize-effects. In contrast to classical linear micromorphic models, we allow the\nusual elasticity tensors to become positive-semidefinite. We prove that,\nnevertheless, the equilibrium problem has a unique weak solution in a suitable\nHilbert space. The mathematical framework also settles the question of which\nboundary conditions to take for the micro-distortion. Similarities and\ndifferences between linear micromorphic elasticity and dislocation gauge theory\nare discussed and pointed out.\n

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