Energetic Approach to Large Strain Gradient Crystal Plasticity

Jan Kratochvíl,Martin Kružík

doi:10.14311/1666

Abstract

We formulate a problem of the evolution of elasto-plastic materials subjected to external loads in the framework of large deformations and multiplicative plasticity. We focus on a spontaneous inhomogenization interpreted as a structuralization process. Our model includes gradients of the plastic strain and of hardening variables which provide a relevant length scale of the model. A simple computational experiment interpreted as a hint of a deformation substructure formation is included.

Highlights

The elastic-plastic behavior of crystalline materials poses a challenge for mathematical analysis on the microscopic, the mesoscopic, and the macroscopic scales
We study a rate-independent model arising in the crystal plasticity
A common and successful approach to the analysis of crystalline materials is by means of energy minimization; see e.g. Ortiz & Repetto [30]

Summary

Introduction

The elastic-plastic behavior of crystalline materials poses a challenge for mathematical analysis on the microscopic, the mesoscopic, and the macroscopic scales. The applicability of variational methods has been broadened to include rate-independent evolution These models are characterized by energy minimization of a functional including macroscopic quantities such as the macroscopic deformation gradient as well as a dissipation functional. To expose the essence of the mathematical structure of the energetic approach, we first analyze a proto-model called here a material with internal variables. It freely follows the exposition of Francfort & Mielke [11] and we recall it here to motivate the notion of the energetic solution. The subdifferential of f will be denoted ∂subf and its elements will be called subgradients of f at x0

Materials with internal variables

Energetic solution

Problem statement