Geometrical theory of dislocations in bodies with microstructure (Q556190)

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.