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Homogenization of elasto-plastic evolutions driven by the flow of dislocations

Paolo Bonicatto, Filip Rindler

Abstract

Starting from a prototypical model of elasto-plasticity in the small-strain and quasi-static setting, where the evolution of the plastic distortion is driven exclusively by the motion of discrete dislocations, this work performs a rigorous homogenization procedure to a model involving continuously-distributed dislocation fields. Our main result shows the existence of rate-independent evolutions driven by the motion of dislocation fields, obtained as limits of discrete dislocation evolutions. For all notions of solutions we employ the recent concepts of space-time integral and normal currents, which is richer than the classical approach using the Kröner dislocation density tensor. The key technical challenge is to find discrete dislocation evolutions approximating a given dislocation field evolution, which requires a careful recovery construction of space-time slip trajectories and associated displacements. These methods enable one to transfer the properties, most importantly the quasi-static stability, from the discrete to the field regime.

Homogenization of elasto-plastic evolutions driven by the flow of dislocations

Abstract

Starting from a prototypical model of elasto-plasticity in the small-strain and quasi-static setting, where the evolution of the plastic distortion is driven exclusively by the motion of discrete dislocations, this work performs a rigorous homogenization procedure to a model involving continuously-distributed dislocation fields. Our main result shows the existence of rate-independent evolutions driven by the motion of dislocation fields, obtained as limits of discrete dislocation evolutions. For all notions of solutions we employ the recent concepts of space-time integral and normal currents, which is richer than the classical approach using the Kröner dislocation density tensor. The key technical challenge is to find discrete dislocation evolutions approximating a given dislocation field evolution, which requires a careful recovery construction of space-time slip trajectories and associated displacements. These methods enable one to transfer the properties, most importantly the quasi-static stability, from the discrete to the field regime.
Paper Structure (32 sections, 37 theorems, 331 equations)

This paper contains 32 sections, 37 theorems, 331 equations.

Table of Contents

  1. Introduction
  2. Kinematics and energetics
  3. Dislocations and their dynamics
  4. Boundary conditions
  5. Relation to the existing literature
  6. Notation and preliminaries
  7. Linear and multilinear algebra
  8. Currents
  9. Integral currents
  10. Normal currents
  11. BV-theory of space-time currents
  12. Approximation of normal currents
  13. Spaces of Banach-space valued functions
  14. Dislocations and slip trajectories
  15. Dislocation systems
  16. ...and 17 more sections

Key Result

Lemma 2.1

Let $S = m \, \vec{S} \, \mathcal{H}^{1+k} \begin{picture}(10,8)\put(2,0){\line(0,1){7}}\put(1.8,0){\line(1,0){7}}\end{picture} R \in \mathrm{I}_{1+k}([\sigma,\tau] \times \overline{\Omega})$. Then, and

Theorems & Definitions (66)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • proof
  • ...and 56 more