Orientation distribution function in non-affine rotations

The orientation distribution function (ODF) for a class of directional elements, represented by unit vectors or orthonormal triads, which are embedded in a deforming continuum, is obtained by a novel approach in closed analytical form as a function of appropriate deformation and rotation measures, for orientational processes which can be characterized as non-affine rotations. The main conceptual innovation is the introduction of a fictitious motion defined locally in the neighborhood of a material point, such that a directional element which follows non-affine rotation in relation to the real motion, follows affine rotation in relation to the fictitious motion. A key point of the analysis is the time integration of the velocity gradient of the fictitious motion under steady-state conditions for the real velocity gradient, in order to obtain the corresponding local fictitious deformation gradient, from which the ODF for non-affine rotations is calculated by standard methods applicable to affine rotations. Many examples illustrate the foregoing theoretical findings. The ODFs associated with sequences of different deformation processes, which were too difficult to yield a solution in the past, are now readily obtained by the above approach for both affine and non-affine rotations, while previously reported results are retrieved as particular cases with a modicum of analytical calculations. (C) 2001 Elsevier Science Ltd. All rights reserved.