Line-Tension Model for Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy

Abstract

Abstract. In this paper we rigorously derive a line-tension model for plasticity as the Γ-limit of a nonlinear mesoscopic dislocation energy, without resorting to the introduction of an ad hoc cut-off radius. The Γ-limit we obtain as the length of the Burgers vector tends to zero has the same form as the Γ-limit obtained by starting from a linear, semi-discrete dislocation energy. The nonlinearity, however, creates several mathematical difficulties, which we tackled by proving suitable versions of the Rigidity Estimate in non-simply-connected domains and by performing a rigorous two-scale linearisation of the energy around an equilibrium configuration.

Geometric rigidity for incompatible fields and an application to strain-gradient plasticity

“…Unlike the case of fixed dislocations studied in [20], where the dislocation density µ was constant (up to an ε-scaling) and the energy (1.2) depended only on the strain β, in the present case the distribution of dislocations µ = N i=1 εb i δ x i is a variable of the problem, and therefore the dislocation energy (1.6) depends on both β and µ. Notice that, due to the quadratic growth of the energy density, also in this nonlinear setting the ε-regularization of the dislocation energy is needed, as in the linear case [11].…”

“…Considering a more general, nonlinear dislocation energy is therefore desirable. This general principle triggered the analysis done in [20], where the authors considered a nonlinear dislocation energy of the form…”

“…In [20] the authors considered the case of a finite number N of fixed edge dislocations located at points x 1 , . .…”

“…In (1.3), the strain β and the dislocation density which is encoded in the measure µ = N i=1 εb i δ x i are coupled via the relation Curl β = µ. In [20] it was shown that the energies (1.3) give rise in the limit to the line-tension plasticity model described by…”

“…Coming back to our model, in the present paper we treat the case of "infinitely many" dislocations in the nonlinear setting, although under more restrictive coerciveness assumptions on the energy density W than in [20]. More precisely, the dislocation energy in our model is given by 6) where the nonlinear energy density W behaves essentially as dist 2 (β, SO (2)) (see Section 2 for more details), and the strain β satisfies at any dislocation point x i the incompatibility condition…”