Mechanics of Dislocation Fields

Claude Fressengeas

• Publisher: Wiley-ISTE
• Year: 2017
• Print ISBN: 9781848213753
• eBook ISBN: 9781118578186
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Materials Science
• Index: Technology & Engineering

Accompanying the present trend of engineering systems aimed at size reduction and design at microscopic/nanoscopic length scales, Mechanics of Dislocation Fields describes the self-organization of dislocation ensembles at small length scales and its consequences on the overall mechanical behavior of crystalline bodies.

The account of the fundamental interactions between the dislocations and other microscopic crystal defects is based on the use of smooth field quantities and powerful tools from the mathematical theory of partial differential equations. The resulting theory is able to describe the emergence of dislocation microstructures and their evolution along complex loading paths. Scale transitions are performed between the properties of the dislocation ensembles and the mechanical behavior of the body.

Several variants of this overall scheme are examined which focus on dislocation cores, electromechanical interactions of dislocations with electric charges in dielectric materials, the intermittency and scale-invariance of dislocation activity, grain-to-grain interactions in polycrystals, size effects on mechanical behavior and path dependence of strain hardening.

1
Continuous Dislocation Modeling

1.1. Introduction

Dislocations are conventionally viewed in the Volterra construct as discontinuities of the elastic displacement across bounded surfaces terminating at the dislocation line [VOL 07], as sketched in Figure 1.1. However, it is rather well known in the world of physics modeling that defects viewed at sufficiently small scales do not involve discontinuities and singularities, but instead appropriately localized smooth fields. The Peierls-Nabarro model is such a paradigmatic model in the field of dislocation mechanics [NAB 47, PEI 40]. If only from the point of view of mathematical analysis and numerical computation, smooth localized representations of dislocations are essential to build regular sets of partial differential equations leading to their dynamics through the solution of well-posed boundary value problems. As line defects, dislocations imply a duality between the terminating curve of discontinuity of the elastic displacement field on a bounded surface and the smooth incompatibility of its “gradient” field, i.e. the elastic distortion field. Following this idea, the plan developed in this chapter is that geometrically defined incompatibility fields allow smooth modeling of the line defects supporting the singularity.

Figure 1.1. Cross-section view of a dislocation line terminating the elastic displacement discontinuity across a bounded surface S. S and F are the starting and finishing points of an oriented closed Burgers circuit. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip

1.2. Lattice incompatibility

The continuous description of a crystalline body B containing lattice defects such as dislocations and grain boundaries, and possibly submitted to loads on its external boundaries, consists of a set of points, perhaps coincident with atom sites, of vectors connecting these points and of continuous mappings and transformations relating these objects. The fixed set of atoms composing B lies, at any given time t, in a region of space called a configuration B(t). Each spatial point P of the initial configuration B₀ = B(0) corresponding to the as-received state of the body B at time t = 0 is associated with a “material particle”...

Table of contents

• Cover
• Table of Contents
• Title
• Copyright
• Acknowledgements
• Introduction
• 1 Continuous Dislocation Modeling
• 2 Elasto-static Field Equations
• 3 Dislocation Transport
• 4 Constitutive Relations
• 5 Elasto-plastic Field Equations
• 6 Case Studies
• 7 Review and Conclusions
• Appendix: Complements
• Bibliography
• Index
• End User License Agreement