Dislocation group dynamics IV. General solutions of the continuum approximation

Abstract

The general motion of a dislocation distribution is considered for the case where the individual dislocations follow a linear stress—velocity law. A function of a complex variable is introduced and it is shown that the dislocation motion corresponds to the flow of this function along characteristics in the complex plane. For screw dislocations the stress field of the distribution and the distribution itself are different aspects of the same function and are determined together. It is shown that this complex representation is fully determined for an applied stress field which is a constant, a linear, or a quadratic function of position and the extention of this method to other stress fields is discussed.