On the Mechanics of Crystalline Solids with a Continuous Distribution of Dislocations

TL;DR

This paper develops a differential geometric framework to describe the dynamics and equilibrium states of crystalline solids with a continuous distribution of dislocations, leading to nonlinear PDEs for static configurations.

Contribution

It introduces a novel geometric approach to model dislocation dynamics and equilibrium in crystalline solids, connecting Lie group theory with elasticity.

Findings

Formulated new laws for dislocation dynamics using differential geometry.

Derived nonlinear elliptic PDEs for static equilibrium configurations.

Solved the equations in simplified cases of interest.

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.