Engineering Mechanics

Proceedings Vol. 15 (2009)

ENGINEERING MECHANICS 2009

Copyright © 2009 Institute of Theoretical and Applied Mechanics, Academy of Sciences of the Czech Republic, v.v.i., Prague

 

Z. Fiala
pages 227 - 240, full text

The paper raises a question whether the logarithmic time derivative, expressed in specific coordinate system, is the Zaremba-Jaumann time derivative, and if not why. In fact, it has been already proved that the Z-J derivative represents the geometrically consistent linearization of tensor fields in terms of the covariant derivative in the space of right Cauchy-Green deformation tensors C. This is the space of symmetric, positive-definite 3×3 matrices of real numbers Sym+ (3, R) ∼ = + GL (3, R)/SO(3, R), which has a natural geometry of a Riemannian (globally) symmetric space of constant curvature, with the covariant derivative based on its Riemannian metric. Since in this geometry matrix exponentials stand for geodesics (that is a generalization of straight lines), the logarithmic strain log(C) can be interpreted simply as the change of coordinates in Sym+ (3, R), called the normal coordinates. There are some indications that the Z-J time derivative expressed in this normal coordinate system might be the logarithmic time derivative.

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