Stress-free states of continuum dislocation fields: Rotations, grain boundaries, and the Nye dislocation density tensor

Abstract

We derive general relations between grain boundaries, rotational deformations, and stress-free states for the mesoscale continuum Nye dislocation density tensor. Dislocations generally are associated with long-range stress fields. We provide the general form for dislocation density fields whose stress fields vanish. We explain that a grain boundary (a dislocation wall satisfying Frank's formula) has vanishing stress in the continuum limit. We show that the general stress-free state can be written explicitly as a (perhaps continuous) superposition of flat Frank walls. We show that the stress-free states are also naturally interpreted as configurations generated by a general spatially-dependent rotational deformation. Finally, we propose a least-squares definition for the spatially-dependent rotation field of a general (stressful) dislocation density field.

New Perspectives in Plasticity Theory: Dislocation Nucleation, Waves, and Partial Continuity of Plastic Strain Rate

“…A series of works by Limkumnerd and Sethna [8][9][10] that have since appeared is based on essentially the same theory as recognized by the authors [10]. Motivated by their numerical results, these authors suggest that the theory admits singularities in the Nye tensor field even when the elastic response is linear, an interesting claim that would be well worth establishing rigorously.…”

“…This leads to "generation" of Burgers vector and line direction in 9 2 with crystallographic sense. In turn, the relation (31) leads to transport of the excess density in roughly the sense that one would expect from the Peach-Koehler relation [3].…”

“…This was demonstrated in modeling of the torsion of ice single crystals [48], where the predominant movement of screw dislocations developed in torsion was on basal planes. The interplay of internal stresses associated with 9 1 and the applied stress may indeed provide a driving force with (31) allowing for transport out of the slip plane. Such role is critical in the present work to representing a mechanism of transport of excess dislocation density along the specimen axis.…”

A statistical analysis of the elastic distortion and dislocation density fields in deformed crystals

“…Later, Kröner (1981) incorporated the elastic strain into the definition and established the dislocation density tensor as a central quantity in his continuum theory of dislocations. The importance of the dislocation density tensor has become apparent in plasticity theories both as a primary field in continuum dislocation dynamics (Acharya, 2001;Limkumnerd and Sethna, 2007) and as a definition for geometrically necessary dislocation (GND) densities in strain gradient plasticity theories (Gao et al, 1999;Huang et al, 2000;Gao and Huang, 2003). The inclusion of the dislocation density tensor in plasticity models has motivated experimental measurements through the electron backscattering diffraction (EBSD) technique and x-ray microbeam diffraction.…”

Shocks and slip systems: Predictions from a mesoscale theory of continuum dislocation dynamics

“…The fact that our walls form from relatively homogeneous initial conditions suggests that a theory evolving to locally minimize the energy does form walls. On the other hand, our theory forms walls to minimize the energy, but they are sharp only because of the particular dynamical evolution law by which they are formed; a continuum blur of walls, within our theory, would also minimize the energy (Limkumnerd and Sethna, 2007).…”