Summary
Characterization of material behavior can be divided into two parts, the analysis of deformation and the underlying physics, though these are intimately related. A significant advance in the analysis of deformation was made when the polar decomposition theorem was introduced, making it possible to separate large deformations into a stretch and a rotation. Consequences of the theorem affect the way rate processes should be characterized. In particular, rate of material rotation is different from vorticity, and the stress rate for finite strains is different from the usual stress rate of Zaremba, Jaumann, and Noll. It is convenient to define a strain rate that is different from the stretching that is the symmetric part of the velocity gradient. These concepts are described in detail in a 1979 paper. Various criticisms of that paper have appeared in the Journal of Applied Mechanics, which are discussed herein. To illustrate the distinction it is shown that the rate of rotation in a classical vortex does not vanish, though the vorticity is zero. It is also shown that the rate of material rotation recently computed by Nemat-Nasser, which involves an eigenvalue expansion, is equivalent to the one given in the 1979 paper, which makes use of matrix inversion, and it is asseverated that the matrix inversion approach is computationally more efficient.
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Dienes, J.K. A discussion of material rotation and stress rate. Acta Mechanica 65, 1–11 (1987). https://doi.org/10.1007/BF01176868
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DOI: https://doi.org/10.1007/BF01176868