Curvilinear coordinates for mode III crack plastic zone in a work hardening solid

For a problem in antiplane strain, a curvilinear coordinate system always exists in which one of the shear stress components (expressed in that coordinate system) is equal to zero and the other component is not. In an elastic solid, the radii of curvature RF and RT of the coordinate system trajectories must satisfy, ∂(1/RF)/∂F−∂(1/RT)/∂T=0, where F and T stand for finger and thumb directions. In a power law work hardening solid of exponent m the relationship that must be satisfied is ∂(1/RF)/∂F−m∂(1/RT)/∂T+(1−m)(1/RF)(1/RT)=0, when the shear stress exerted across a thumb trajectory is zero. The curvilinear coordinate system has yet to be found, which satisfies this equation for the problem of the mode III crack in general yielding in a power law work hardening solid. In this paper, curvilinear coordinates trajectory equations are obtained, which are valid only in the asymptotic region near a crack tip, in the asymptotic region near the center of the crack, and in the region distant from the crack itself.