The limiting speeds of dislocations
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I. INTRODUCTION IN the early literature on moving dislocations, there is an often quoted article by Frank,[1] which intends to show that the elastic displacement field of a screw dislocation with a constant velocity satisfies the equation for elastic shear wave propagation if the displacement field of the dislocation is transformed by means of a Lorentz transformation. This causes it to undergo a contraction in the direction of its motion and limits its velocity to that of elastic shear waves. The behavior is often referred to as “relativistic” by analogy with the theory of electromagnetic waves. However, both this solution and its interpretation have some serious flaws if they are not to contradict standard physical mechanics. Stationary dislocations can be described in terms of an inelastic localized core region plus a nonlocalized elastic field surrounding the core. This elastic field derives from the fact that the solid medium is multiply-connected and in a state of self-equilibrated strain. However, if a crystal dislocation moves, it transports a finite amount of displacement (the Burgers displacement). This inelastic process produces plastic deformation that can be described in terms of a shear angle. Part of the solid (crystal) is displaced while the rest is not. A typical schema is that the top half of the solid slides over the bottom half. Thus, the greater the relative displacement, the greater the shear angle; or, in terms of rates, the greater the dislocation velocity, the greater the angular rotation rate. Angular rotation is, of course, accompanied by angular momentum because solids (crystals) have mass densities. But, the Frank theory and variations on it that have followed over many years do not conserve angular momentum. Therefore, this body of theory is fundamentally flawed. II. FLAWS OF THE CONVENTIONAL THEORY Frank’s theory discusses a straight screw dislocation that is moving at a constant velocity. No overall equation of JOHN J. GILMAN, Research Professor, is with the Department of Materials Science and Engineering, University of California at Los Angeles, Los Angeles, CA 90095. This article is based on a presentation given in the symposium entitled “Dynamic Behavior of Materials—Part II,” held during the 1998 Fall TMS/ ASM Meeting and Materials Week, October 11–15, 1998, in Rosemont, Illinois, under the auspices of the TMS Mechanical Metallurgy and the ASM Flow and Fracture Committees. METALLURGICAL AND MATERIALS TRANSACTIONS A
motion is given, and the necessity for giving boundary conditions is avoided by specifying that the dislocation lies in an infinite medium. However, under these conditions, a straight screw dislocation would have infinite strain energy. Additional serious problems are associated with the fact, pointed out previously, that a moving dislocation produces both angular momentum and plastic deformation in addition to its linear momentum. The elastic displacement field used by Frank is valid for a stationary dislocation, but as soon as either an edge or a screw dislocation m
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