A Geometric Field Theory of Dislocation Mechanics

Abstract

In this paper a geometric field theory of dislocation dynamics and finite plasticity in single crystals is formulated. Starting from the multiplicative decomposition of the deformation gradient into elastic and plastic parts, we use Cartan's moving frames to describe the distorted lattice structure via differential $1$-forms. In this theory the primary fields are the dislocation fields, defined as a collection of differential $2$-forms. The defect content of the lattice structure is then determined by the superposition of the dislocation fields. All these differential forms constitute the internal variables of the system. The evolution equations for the internal variables are derived starting from the kinematics of the dislocation 2-forms, which is expressed using the notions of flow and of Lie derivative. This is then coupled with the rate of change of the lattice structure through Orowan's equation. The governing equations are derived using a two-potential approach to a variational principle of the Lagrange-d'Alembert type. As in the nonlinear setting the lattice structure evolves in time, the dynamics of dislocations on slip systems is formulated by enforcing some constraints in the variational principle. Using the Lagrange multipliers associated with these constraints, one obtains the forces that the lattice exerts on the dislocation fields in order to keep them gliding on some given crystallographic planes. Moreover, the geometric formulation allows one to investigate the integrability -- and hence the existence -- of glide surfaces, and how the glide motion is affected by it. Lastly, a linear theory for small dislocation densities is derived, allowing one to identify the nonlinear effects that do not appear in the linearized setting.