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

Abstract

We consider target-space homogenization for energies defined on partitions parametrized\ud by a discrete lattice B ⊂ RN. For a smallσ > 0, the variable is a piecewise constant\ud function taking values in σB, and the energy depends on the jumps and their orientation. In\ud the limit as σ → 0 we obtain a homogenized functional defined on functions of bounded\ud variation. This result is relevant in the study of dislocation structures in plastically deformed\ud crystals. We review recent literature on the topic and propose our limiting effective energy\ud as a continuum model for strain-gradient plasticity

Derivation of strain-gradient plasticity from a generalized Peierls-Nabarro model

“…The transition from scaled line-tension functionals to a functional with a continuous distribution of dislocations was studied in [CGM17]. Here two effects are present.…”

“…We present here a joint convergence result, in which the two terms are derived in one step from the Peierls-Nabarro model in the limit of small lattice spacing. The study of a single limiting process is crucial to obtain the limiting behavior of both the local and the nonlocal term; if the various homogenization and relaxation steps are taken separately then the nonlocal (interaction) term disappears [CGM17].…”