In the presence of dislocations, the elastic deformation tensor
is not a gradient but
satisfies the condition
(with
the dislocation density
a tensor-valued measure concentrated in the dislocation
).
Then
with
.
This peculiarity is at the origin of the mathematical difficulties encountered by
dislocations at the mesoscopic scale, which are here modeled by integral
-currents
free to form complex geometries in the bulk. In this paper, we
first consider an energy-minimization problem among the couples
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.
Departamento de Matemática
Centro de Matemática, Aplicações Fundamentais e Investigação
Operacional
Universidade de Lisboa
Alameda da Universidade C6
1749-016 Lisboa
Portugal
Departamento de Matemática
Centro de Matemática, Aplicações Fundamentais e Investigação
Operacional
Universidade de Lisboa
Alameda da Universidade C6
1749-016 Lisboa
Portugal