Continuum theory of evolving dislocation fields

Abstract

Continuum theory of moving dislocations is used to set up a non-local constitutive law for crystal plasticity in the form of partial differential equations for evolving dislocation fields. The concept of single-valued dislocation fields that enables us to keep track of the curvature of the continuously distributed gliding dislocations with line tension is utilized. The theory is formulated in the Eulerian as well as in the so-called dislocation-Lagrangian forms. The general theory is then specialized to a form appropriate to formulate and solve plane-strain problems of continuum mechanics. The key equation of the specialized theory is identified as a transport equation of diffusion–convection type. The numerical instabilities resulting from the dominating convection are eliminated by resorting to the dislocation-Lagrangian approach. Several examples illustrate the application of the theory.

Acknowledgements

The authors are grateful for financial support by grants DFG We 2351/8-3 (RS, CS, EW) and VZ-MŠMT 6840770003 (JK).

Notes

† A non-vanishing elastic distortion that fulfils the relation is a compatible one, it can be derived from a potential (displacement) and as such it is not a field of a dislocation encircled by the curve C.

† A more detailed relation distinguishing, for example, between the strength of coplanar and non-coplanar dislocation interactions could be set up as well.

† To be consistent with other parts of the text, we keep the meaning of ρ[m ⁻²] as a volume density, although the dislocations in this example are distributed in a plane. As a consequence, the ‘length’ computed in equation (49), being reduced by a unit length measured in the m direction, is a dimensionless quantity.

† Due to the non-linearity in equation (73) it is not exactly the dimensionless Péclet number itself.