Energy, configurational forces and characteristic lengths associated with the continuum description of geometrically necessary dislocations

The objectives of this study are: (i) to define the portion of work on plastic deformation of crystals associated with the existence of geometrically necessary (GN) dislocations – on the basis of the mechanics of dislocations, (ii) to express this microstructural work in terms of the continuum representation of GN dislocations – the Nye’s dislocation density tensor – and thus establish a firm connection between the mechanics of dislocations and a continuum theory, and (iii) to explain the nature of resulting characteristic lengths. (i) Analysis of elementary energy sinks in dislocation mechanics leads to the conclusion that (a) the interaction energies amongst GN dislocation segments, and (b) the interaction energies between the GN segments and the boundaries must be identified with the microstructural work. (ii) After establishing a connection between the discrete and the continuum descriptions of the state of dislocations in a solid, the interaction energies – both, amongst the GN segments and between the GN segments and the boundaries – are expressed as functions of Nye’s dislocation densities. Configurational forces are defined as work-conjugates to Nye’s densities. The resulting microstructural work density and the microstructural constitutive relations are nonlocal. The microstructural constitutive law takes the form of a convolution integral over the domains with non-vanishing Nye’s densities. This is a direct consequence of the long-range interactions in dislocation mechanics. (iii) Continuum characteristic lengths are intimately connected with the nonlocal nature of the microstructural work density. They represent the dimensions of domains of integration in the convolution integrals associated with the nonlocal configurational forces. No characteristic length can be described as a material length. Dislocation mechanics lengths are absorbed into continuum fields. The continuum lengths are identified as problem-dependent (dimensions of obstacles to dislocation motion) and solution-dependent (the widths of boundary layers associated with each slip system).