Relative Helicity and Jog Densities in Continuum Descriptions of Dislocations
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Relative Helicity and Jog Densities in Continuum Descriptions of Dislocations Thomas Hochrainer Universität Bremen, IW3, Am Biologischen Garten 2, 28359 Bremen, Germany.
ABSTRACT Dislocations are line like crystal defects mediating plasticity in single crystals. In the current contribution we review classical continuum concepts of dislocation theory from a topological view point. Subsequently, we introduce a new measure for the density of jogs mutually impaired on each other by dislocations on different slip systems. This jog density is closely related to a topological measure of the interlinkage of the dislocations on the involved slip systems, known as relative helicity in other branches of physics. INTRODUCTION Crystal plasticity is mediated by the motion of line like crystal defects called dislocations. At low temperatures dislocations are typically confined to move in crystallographically defined planes (slip planes). Upon moving, dislocations slip crystal parts separated by their swept surface against each other by their Burgers vector b . Though single dislocations and their interactions are well understood and ensembles of dislocations may be simulated with discrete dislocation dynamics, the most conspicuous feature of metal plasticity, i.e. strain hardening, may still not be predicted from a continuum theory based on dislocations. Much progress has been made recently towards a continuum theory of plasticity through the advent of continuum dislocation dynamics (CDD) [1]. But CDD has thus far mostly been developed for slip on a single slip system, while strain hardening is essentially linked to simultaneous slip on several slip systems. One process connected to multiple slip, which is assumed to play a rôle in strain hardening, is the emergence of jogs on dislocation lines upon crossing each other, e.g. [2,3]. Especially jogs perpendicular to the slip plane are less mobile than their parent dislocations and may produce point defects when moving, trigger dislocation sources or cross slip [2]. The question we address in the current contribution is what we can infer on the appearance of jogs from well-known dislocation measures used in continuum dislocation theory. Motivated from a continuum version of Gauss’ linking number named the relative helicity of two vector fields [4,5], we argue that a contraction of slip system specific dislocation density vectors and plastic distortion tensors can be interpreted as a topologically necessary jog density. We briefly discuss what consequences we expect from considering the jog density in constitutive laws of crystal plasticity. TOPOLOGICAL PRELIMINARIES Topologically speaking, dislocations are oriented closed curves in crystals. A natural measure characterizing the entanglement of two dislocations is the Gauss’ linking number, which determines the number and sense of windings of two closed curves around each other. It was
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