Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics

Type
Article

Authors
Yavari, Arash
Goriely, Alain

KAUST Grant Number
KUKC1-013-04

Online Publication Date
2012-03-09

Print Publication Date
2012-07

Date
2012-03-09

Abstract
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.

Citation
Yavari A, Goriely A (2012) Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics. Archive for Rational Mechanics and Analysis 205: 59–118. Available: http://dx.doi.org/10.1007/s00205-012-0500-0.

Acknowledgements
A. YAVARI benefited from discussions with ARKADAS OZAKIN and AMIT ACHARYA. This publication was based on work supported in part by Award No KUKC1-013-04, made by King Abdullah University of Science and Technology (KAUST). A. YAVARI was partially supported by AFOSR-Grant No. FA9550-10-1-0378 and NSF-Grant No. CMMI 1042559.

Publisher
Springer Nature

Journal
Archive for Rational Mechanics and Analysis

DOI
10.1007/s00205-012-0500-0

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