Numerical implementation of a 3D continuum theory of dislocation dynamics and application to micro-bending

Abstract

Crystal plasticity is governed by the motion of lattice dislocations. Although continuum theories of static dislocation assemblies date back to the 1950s, the line-like character of these defects posed serious problems for the development of a continuum theory of plasticity which is based on the averaged dynamics of dislocation systems. Only recently the geometrical problem of performing meaningful averages over systems of moving, oriented lines has been solved. Such averaging leads to the definition of a dislocation density tensor of second order along with its evolution equation. This tensor can be envisaged as the analogue of the classical dislocation density tensor in an extended space which includes the line orientation as an independent variable. In the current work, we discuss the numerical implementation of a continuum theory of dislocation evolution that is based on this dislocation density measure and apply this to some simple benchmark problems as well as to plane-strain micro-bending.

Keywords:

• dislocations dynamics
• continuum theory of dislocation
• dislocation density
• micro-bending

Acknowledgement

Financial support of the European Commission under NEST-2005-PATH-COM-043386 is gratefully acknowledged.

Notes

Notes

1. We speak of a kinematically closed equation as we assume the dislocation velocity v at this point to be a given quantity. In general, v is a function of the local stress, which in turn depends on the dislocation arrangement. Hence, a mathematically closed theory requires additional relationships between the dislocation state, as expressed by α, and the dislocation velocity v. If v is a function of stress and line direction only, these relationships may be derived from Kröner's theory of eigenstresses 1.

2. It is worth noting that the definition of the second order dislocation density tensor does not necessarily require a metric or volume element. However, an invariant definition requires the use of advanced mathematical concepts such as e.g. differential forms. We refrain from introducing these concepts in the current paper and refer the interested reader to 16 and especially 18 for a more thorough treatment.

3. In our case, we pragmatically consider a numerical scheme to be well-behaving within the ‘working regime’ if a positive half-wave can be propagated for more than 10 000 steps with a velocity v = 1.0, the mesh width h = 1.0 and a time step Δt = 0.1, such that the peak of the wave can clearly be identified and no oscillations occur.