Geometrical theory of dislocations in bodies with microstructure
type: Item
instance of: scholarly article
title: Geometrical theory of dislocations in bodies with microstructure
zbMATH Open document ID: 1081.53026
author: Ioan Bucataru
author: Marcelo Epstein
published in: Journal of Geometry and Physics
publication date: 2005-06-13T00:00:00Z
review text: A materially simple body \(B\) can be modeled as a base manifold of a fiber-bundle. The fiber bundle formalism is used to define material connections and other geometrical quantities. In the case of bodies with internal structure (microstructure) the body itself can be modeled as a fiber-bundle \(M\to B\), with a fiber \(F\) carrying the microstructure information. The authors develop geometrical formalism for the description of properties of bodies whose mechanical behavior is local. The energy density \(w\) is assumed to be a function of the first order jet of the configuration. The authors introduce the Lie groupoid \(J^1(M,B)\) of first order jets of local fiber-preserving diffeomorphisms and a Lie group \(G\) such that \(J^1_x(M,B)\) for \(x\in B\) is a principal fiber-bundle over \(B\) with the structural group \(G.\) They derive necessary and sufficient conditions for the fiber \(G\)-structure to be integrable. Next, \(B\) is assumed to be \(3\)-dimensional coverable by one chart and the bundle \(M\to B\) to be trivializable. Under this assumption the authors show that the body \(M\to B\) is materially uniform if and only if the subgroupoid \(J^1_w (M,B)\) of \(J^1(M,B)\) (that consists of jets of local fiber and energy density preserving diffeomorphisms) is a Lie groupoid. The body bundle is shown to be homogeneous if and only if the principal fiber bundle \(J^1_{x_0, w}(M,B)\to B\) is integrable.
Mathematics Subject Classification ID: 53C10
Mathematics Subject Classification ID: 55R10
Mathematics Subject Classification ID: 74A30
Mathematics Subject Classification ID: 74N15
zbMATH DE Number: 2175244
zbMATH Keywords: theory of dislocations
zbMATH Keywords: microstructure
zbMATH Keywords: G-structure
zbMATH Keywords: uniformity
zbMATH Keywords: homogeneity
MaRDI profile type: Publication
full work available at URL: j.geomphys.2004.01.006
OpenAlex ID: W1991786147
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DOI: 10.1016/J.GEOMPHYS.2004.01.006
description: scientific article; zbMATH DE number 2175244