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.
Highlights
The spin tensor is the velocity gradient required to generate rigid body rotations
The deformation, i.e. rotation and shape change, of a small homogeneous volume element is described by the velocity gradient tensor L
The angular velocity ω and the spin tensor W estimate a rotation of the line segments contained by the considered small volume in the considered continuum
Summary
The spin tensor is the velocity gradient required to generate rigid body rotations. The Liutex component of the vorticity vector defines a corresponding spin tensor that can be applied in continuum elasticity and plasticity theories, and that does not generate rotations during simple shear deformation. The deformation, i.e. rotation and shape change, of a small homogeneous volume element is described by the velocity gradient tensor L This rigid body rotation can be described by the angular velocity vector ω. Along with the corresponding angular velocity vector ω, it is defined for cases that are not pure rigid body rotation, by the skew part of any velocity gradient tensor L. The angular velocity ω and the spin tensor W estimate a rotation of the line segments contained by the considered small volume in the considered continuum
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