Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Abstract

Abstract. This article gives a short description and a slight refinement of recent work [MSZ15], [SZ12] on the derivation of gradient plasticity models from discrete dislocations models. We focus on an array of parallel edge dislocations. This reduces the problem to a two-dimensional setting. As in the work Garroni, Leoni & Ponsiglione [GLP10] we show that in the regime where the number of dislocation N ε is of the order log 1 ε (where ε is the ratio of the lattice spacing and the macroscopic dimensions of the body) the contributions of the self-energy of the dislocations and their interaction energy balance. Upon suitable rescaling one obtains a continuum limit which contains an elastic energy term and a term which depends on the homogenized dislocation density. The main novelty is that our model allows for microscopic energies which are not quadratic and reflect the invariance under rotations. A key mathematical ingredient is a rigidity estimate in the presence of dislocations which combines the nonlinear Korn inequality of IntroductionClassical theories of plasticity are scale independent. Nonetheless experiments show a notable size dependence of plastic behaviour in the micron and submicron range.

Strain-Gradient Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

“…Remark 4.5. In [31], the authors construct a recovery sequence β j which fulfills det β j > 0. This construction could also be used in our case.…”

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

“…The existing mathematical results in this direction are limited to the special case of straight dislocations in a cylindrical domain, which reduces to a two dimensional model where dislocations are point singularities [20,23,[32][33][34]. The situation of line singularities is much richer as one has to take into account the possibility of complex dislocation structures.…”

Dislocation microstructures and strain-gradient plasticity with one active slip plane

“…Correspondingly, the elastic energy behaves as M 2 . Therefore if M ∼ ln 1 ε the total line-tension energy and the total elastic energy are of the same order, see [37,38,39] for mathematically rigorous treatments of this heuristics.…”

“…Strain-gradient plasticity models can be rigorously derived from discrete models, or regularized semidiscrete models, using Γ-convergence with a choice of the scaling of the energy which balances the contributions of the elastic field and of the dislocation core energies. This was performed for the first time for point dislocations in the plane by Garroni, Leoni and Ponsiglione [37] in a geometrically linear setting with a core regularization approach, and by Müller, Scardia and Zeppieri with a geometrically nonlinear formulation [38,39]. Both results rely on a well-separation assumption, which permits to locally estimate the self-energy of each individual dislocation.…”