Variational approach to determine the properties of dislocations at finite deformation

Abstract

A generalised version of the continuum theory of curved dislocations describing the spatial and temporal evolution of the fields: statistically stored dislocation density, geometrically necessary dislocation density, and curvature has recently been proposed. The dynamics of the system are derived from a scalar functional of the relevant fields that cannot increase during the evolution of the system. However, the framework was established only for small deformations. The aim of the present paper is to discuss the fundamentals of the elasticity theory of finite deformation in cases where individual dislocations are present in the system. The equilibrium equations are derived within a variational formalism. It is shown that introducing dislocations into the finite deformation framework is a nontrivial task. Moreover, if the deformation is large, the force acting on a dislocation segment is not the well-known Peach-Koehler force. In a forthcoming paper, the results obtained will be applied to the generalisation of the dislocation continuum theory of curved dislocations.

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