Riemann-Cartan Geometry of Nonlinear Dislocation Mechanics

作者：Yavari, Arash^*; Goriely, Alain

来源：Archive for Rational Mechanics and Analysis, 2012, 205(1): 59-118.

DOI：10.1007/s00205-012-0500-0

• CONTINUOUSLY DISTRIBUTED DISLOCATIONS
• VERSETZUNGEN UND EIGENSPANNUNGEN
• PHYSICAL FIELD-THEORY
• CRYSTAL PLASTICITY
• GENERAL-RELATIVITY
• AFFINE MANIFOLD
• SIMPLE BODIES
• SPACE-TIME
• TORSION
• DEFECTS

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 Weitzenbock 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.

• 出版日期2012-7