Book chapter

Inserted: 28 apr 2015
Last Updated: 28 apr 2015

Journal: Analysis and Computation of Microstructure in Finite Plasticity. Lecture Notes in Applied and Computational Mechanics.
Volume: 78
Pages: 175--204
Year: 2015
Links: Analysis and Computation of Microstructure in Finite Plasticity - Chapter 7

Abstract:

This article gives a short description and a slight refinement of recent works by Müller, Scardia, and Zeppieri (MSZ15) and Scardia and Zeppieri (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 by Garroni, Leoni, and Ponsiglione (GLP10) we show that in the regime where the number of dislocations $N_{\varepsilon}$ is of the order $\log\frac{1}{\varepsilon}$ (where $\varepsilon$ 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 Friesecke, James, and Müller (FJM02) and the linear Bourgain and Brezis estimate (BB07) for vector fields with controlled divergence. The main technical improvement of this article compared to (MSZ15) is the removal of the upper bound $W(F)\leq C {\rm dist}^2(F,SO(2))$ on the stored energy function.