The Dislocation-Particle Analogy and Plasma-Crystal Models

Abstract

Dislocation theory has been interacting quite fruitfully with various fields of pure and applied mathematics, mainly tensor calculus, differential geometry and the theory of generalized functions. And last, but not least, some ideas born within dislocation theory might prove useful for solving some fundamental problems of physics in the general theory of relativity and particle physics.S.I.Ben Abraham, Fundamental Aspects of Dislocation Theory, USDC, 1970Now many years ago, metallurgists were puzzled by the fact that a simple plastic deformation model for metals, in which planes of atoms were imagined to slide over one another, predicted shear strengths which were about 1000 times higher than those which were actually measured. The problem was of course that they were making the same mistake that inexperienced hotel staff might make. Faced with moving a very large carpet by a small distance across an assembly room, naïve staff might try to drag it bodily across the floor; probably an impossible task given the considerable weight and coefficient of friction. The task can in fact be accomplished quite easily by exploiting the first, and most elementary analogy to be considered here: that is, the carpet-ruck of Orowan, a phenomenon which has itself been explored in quite surprising detail with regard to its creation 1 , statics and inertial dynamics 2,3 . It is found that friction permits sufficiently large rucks to remain when the initial compression is removed. The analogy breaks down somewhat with regard to movement because a simple elastic treatment does not involve consideration of an extensive field, like that of a dislocation. The shape of a static or dynamic ruck (figure 1) is determined by the boundary conditions at the two contact lines which define where the carpet loses and regains contact with the floor. Without adhesion, the curvature vanishes at either line, due to the absence of localized torques at the contact lines. For small amplitude rucks, the excess length, e, (contained in the ruck, and analogous to a Burgers vector) is much smaller than the freely suspended length of the ruck and the projected length, L, is about equal to the freely suspended length. The ruck height, H, is then given by, H ≅ √(eL)