Renormalized Energy and Forces on Dislocations

Abstract

In this work we discuss, from a variational viewpoint, the equilibrium problem for a finite number of Volterra dislocations in a plane domain. For a given set of singularities at fixed locations, we characterize elastic equilibrium as the limit of the minimizers of a family of energy functionals, obtained by a finite-core regularization of the elastic-energy functional. We give a sharpasymptotic estimate of the minimum energy as the core radius tends to zero, which allows one to eliminate this internal length scale from the problem. The energy content of a set of dislocations is fully characterized by the regular part of the asymptotic expansion, the so-called renormalized energy, which contains all information regarding self- and mutual interactions between the defects. Thus our result may be considered as the analogue for dislocations of the classical result of Bethuel, Brezis and Hélein for Ginzburg--Landau vortices. We view the renormalized energy as the basic tool for the study of the discrete-to-continuum limit in plasticity of crystals, i.e., the passage from models of isolated defects to theories of continuous distributions of dislocations. The renormalized energy is a function of the defect positions only: we prove that its derivative with respect to the position of a given dislocation is the resultant of the Eshelby stress on that dislocation, which can be identified in turn with the classical Peach--Köhler force.