On the mechanics of crystalline solids with a continuous distribution of dislocations

Abstract

We formulate the laws governing the dynamics of a crystalline solid in which a continuous distribution of dislocations is present. Our formulation is based on new differential geometric concepts, which in particular relate to Lie groups. We then consider the static case, which describes crystalline bodies in equilibrium in free space. The mathematical problem in this case is the free minimization of an energy integral, and the associated Euler-Lagrange equations constitute a nonlinear elliptic system of partial differential equations. We solve the problem in the simplest cases of interest.