A three-dimensional continuum theory of dislocation systems: kinematics and mean-field formulation

Abstract

We propose a dislocation density measure which is able to account for the evolution of systems of three-dimensional curved dislocations. The definition and evolution equation of this measure arise as direct generalizations of the definition and kinematic evolution equation of the classical dislocation density tensor. The evolution of this measure allows us to determine the plastic distortion rate in a natural fashion and therefore yields a kinematically closed dislocation-based theory of plasticity. A self-consistent theory is built upon the measure which accounts for both the long-range interactions of dislocations and their short-range self-interaction which is incorporated via a line-tension approximation. A two-dimensional kinematic example illustrates the definitions and their relations to the classical theory.

Acknowledgements

We gratefully acknowledge partial financial support from the European Commission through the Marie Curie Research Training Network SIZEDEPEN under contract no. MRTN-CT-2003-504634.

Notes

†Once a kinematic evolution equation is established, the crucial physical problem in statistical modelling is to determine the velocity υ and establish its relation with the dislocation densities. In the cited case it turns out that, under the assumption of overdamped dislocation motion, the velocity can be written as a functional of the dislocation densities which encompasses several stress contributions accounting for different kinds of dislocation interactions.

†Here and in the following we adopt the Einstein summation convention for automatically contracting over pairs of upper and lower indices.

†El-Azab additionally classifies the dislocations by their velocity. If the dislocation velocity can be formulated as a function of the orientation and position in space of a dislocation, this classification is redundant (it is needed, on the other hand, if dislocations possess inertia). We will not consider the velocity-dependent classification within the present paper, as it complicates the presentation without helping to clarify the basic geometrical ideas.