Benefits and shortcomings of the continuous theory of dislocations

Abstract

Out of the vast field of microstructural mechanical behaviour of solids, we choose the area of elastoplasticity of crystalline solids. It is emphasized that elastoplastic deformation proceeds through defects in the ordered crystalline structure. Most important, at least in our investigation, are the defects dislocations that produce plasticity by motion at all temperatures and, in addition, point defects that become active at a higher temperature. It is shown that for two reasons, the elastoplasticity of crystalline solids does not fit well into the scheme of continuum mechanics: (i) The conventional tensor of dislocation density counts only excess dislocations of one sign, whereas the observed hardening and softening is due to the dislocations of two signs. (ii) The motion of the typical defects in the crystalline structure destroys the particles that constitute the body whose particles, therefore, do not persist during the elastoplastic motion. For this reason, the elastoplastic crystalline solid is not a differentiable material manifold. During the elastoplastic deformation, an irregular, often densifying dislocation network develops that can be seen in the electromicroscope and therefore is characteristic for the internal mechanical state. The network can be described by the infinite set on n-point correlation functions of dislocations. It is proposed that solutions are classified as of first, second, third, etc. order according to the highest order of correlation function which is included. The first-order theory is the so-called mean field theory, a well-known concept within the statistical physics. The two-point autocorrelation function gives the often used total length of dislocations in a unit volume, also a state quantity. The present state of the theory, in particular of the dynamics, is still rather underdeveloped.