General relationships between grain boundaries, rotational deformations and stress-free states were derived for the mesoscale continuum Nye dislocation density tensor. Dislocations were generally associated with long-range stress fields. The general form for dislocation density fields whose stress-fields vanished was derived. It was explained that a grain boundary (a dislocation wall satisfying Frank’s formula) had a vanishing stress in the continuum limit. It was shown that the general stress-free state could be written explicitly as a (perhaps continuous) superposition of flat Frank walls. It was shown that the stress-free states were also naturally interpreted as configurations generated by a general spatially dependent rotational deformation. Finally, a least-squares definition was proposed for the spatially dependent rotation field of a general (stressful) dislocation density field.

Stress-Free States of Continuum Dislocation Fields - Rotations, Grain Boundaries, and the Nye Dislocation Density Tensor. S.Limkumnerd, J.P.Sethna: Physical Review B, 2007, 75[22], 224121 (9pp)