The growth of a curved twin boundary, involving a twinning step in 2 dimensions under an applied stress, was studied. The twinning deformation was described as being an anti-plane shear deformation with discontinuous strains. The material was assumed to be hyper-elastic, with a non-convex multi-well stored-energy function. Twin-boundary evolution was governed by an orientation-dependent kinetic relationship whose form was derived from a dislocation model for the twin boundary. Since steady-state evolution was precluded by the model, a study was made of quasi-steady growth. This allowed for transient effects and for twin-boundary shape-changes which were slow when compared with the average growth rate. Twin-boundary evolution was governed by a non-linear integro-differential equation. A particular analytical solution was found, and was shown to be globally asymptotically stable. It described the long-term behavior of twinning steps having an arbitrary initial shape.

Quasi-Steady Growth of Twins under Stress. H.Tsai, P.Rosakis: Journal of the Mechanics and Physics of Solids, 2001, 49[2], 289-312