Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics

Arash Yavari, Alain Goriely

Research output: Contribution to journalArticlepeer-review

127 Scopus citations


We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold-where the body is stress free-is a Weitzenböck manifold, that is, a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan's moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields, assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance. © 2012 Springer-Verlag.
Original languageEnglish (US)
Pages (from-to)59-118
Number of pages60
JournalArchive for Rational Mechanics and Analysis
Issue number1
StatePublished - Mar 9 2012
Externally publishedYes


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