Spin and vorticity with vanishing rigid-body rotation during shear in continuum mechanics

Abstract

A long-standing challenge in continuum mechanics has been how to separate shear deformation and corresponding shape changes from the rotation of a continuum. This can be obtained by a new decomposition of the spin tensor, i.e. of the skew part of the velocity gradient, into two parts, where one of them vanishes during shear flows. The same decomposition applies to the vorticity vector. In both cases, the two spin components are interpreted as generating plastic shear deformation and rigid body rotation. In continuum plasticity theories, the suggested rotational part of the spin tensor can be applied to avoid spurious behavior of the objective Lie derivatives of second order tensors, e.g. of the stress tensor. It provides a history-independent spin corresponding to a time-averaged angular velocity of the rotating line segments in a small homogeneous volume. In fluid mechanics, the new spin component can be used to quantify vortexes in shear flows and turbulent structures, and it provides a sound interpretation and generalization of the Δ and swirling-strength criteria for visualization of vortices.