Dynamic Rotation and Stretch Tensors from a Dynamic Polar Decomposition

TL;DR

This paper introduces a new dynamic polar decomposition that provides a uniquely consistent rotation tensor and an objective stretch tensor, improving the analysis of continuum deformation over time.

Contribution

It derives a generalized polar decomposition yielding a dynamically consistent rotation tensor and an objective stretch tensor, addressing limitations of the classical polar decomposition.

Findings

The dynamic rotation tensor satisfies the superposition principle.

The dynamic stretch tensor is objective and shares principal strains with the classic tensor.

The method provides objective measures of deformation angles.

Abstract

The local rigid-body component of continuum deformation is typically characterized by the rotation tensor, obtained from the polar decomposition of the deformation gradient. Beyond its well-known merits, the polar rotation tensor also has a lesser known dynamical inconsistency: it does not satisfy the fundamental superposition principle of rigid-body rotations over adjacent time intervals. As a consequence, the polar rotation diverts from the observed mean material rotation of fibers in fluids, and introduces a purely kinematic memory effect into computed material rotation. Here we derive a generalized polar decomposition for linear processes that yields a unique, dynamically consistent rotation component, the dynamic rotation tensor, for the deformation gradient. The left dynamic stretch tensor is objective, and shares the principal strain values and axes with its classic polar counterpart. Unlike its classic polar counterpart, however, the dynamic stretch tensor evolves in time without spin. The dynamic rotation tensor further decomposes into a spatially constant mean rotation tensor and a dynamically consistent relative rotation tensor that is objective for planar deformations. We also obtain simple expressions for dynamic analogues of Cauchy's mean rotation angle that characterize a deforming body objectively.