Vol. 4, No. 2, 2016

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ISSN: 2325-3444 (e-only)
ISSN: 2326-7186 (print)
Constraint reaction and the Peach–Koehler force for dislocation networks

Riccardo Scala and Nicolas Van Goethem

Vol. 4 (2016), No. 2, 105–138
DOI: 10.2140/memocs.2016.4.105
Abstract

In the presence of dislocations, the elastic deformation tensor F is not a gradient but satisfies the condition CurlF = Λ T (with the dislocation density Λ a tensor-valued measure concentrated in the dislocation ). Then F Lp with 1 p < 2. This peculiarity is at the origin of the mathematical difficulties encountered by dislocations at the mesoscopic scale, which are here modeled by integral 1-currents free to form complex geometries in the bulk. In this paper, we first consider an energy-minimization problem among the couples (F,) of strains and dislocations, and then we exhibit a constraint reaction field arising at minimality due to the satisfaction of the condition on the deformation curl, hence providing explicit expressions of the Piola–Kirchhoff stress and Peach–Koehler force. Moreover, it is shown that the Peach–Koehler force is balanced by a defect-induced configurational force, a sort of line tension. The functional spaces needed to mathematically represent dislocations and strains are also analyzed and described in a preliminary part of the paper.

Keywords
dislocations, finite elasticity, variational problem, Peach–Koehler force, constraint reaction, integer-multiplicity currents
Mathematical Subject Classification 2010
Primary: 49Q15, 74B20, 74G65
Milestones
Received: 8 October 2015
Accepted: 17 July 2016
Published: 21 November 2016

Communicated by Francesco dell'Isola
Authors
Riccardo Scala
Departamento de Matemática
Centro de Matemática, Aplicações Fundamentais e Investigação Operacional
Universidade de Lisboa
Alameda da Universidade C6
1749-016 Lisboa
Portugal
Nicolas Van Goethem
Departamento de Matemática
Centro de Matemática, Aplicações Fundamentais e Investigação Operacional
Universidade de Lisboa
Alameda da Universidade C6
1749-016 Lisboa
Portugal