An anholonomic crystal reference system was determined from the given elastic strain (metric) and dislocation density (Cartan torsion). It was shown that the incompatible inverse elastic deformation satisfied a system of linear first-order differential equations. It was found that the integrability condition required that the curvature of the crystal connection should vanish. The equivalent system of equations and integrability conditions for the inverse elastic conditions were also obtained. Formulae were derived for the inverse elastic deformation, and the inverse elastic rotation, in the form of tensor series. The dual problem of determining the plastic deformation and the plastic rotation from the plastic strain and the dislocation density was solved in an analogous manner.

On the Determination of the Crystal Reference System in the Non-Linear Continuum Theory of Dislocations. K.C.Le, H.Strumpf: Proceedings of the Royal Society A, 1996, 452[1945], 359-71