Evolution equations for tensors that characterize elastic-viscoplastic materials
are often formulated in terms of a Jaumann derivative based on the spin
tensor. Typically, numerical integration algorithms for such equations split
the integration operation by first calculating the response due to rate of
deformation, followed by a finite rotation. Invariance under superposed rigid body
motions of algorithms, incremental objectivity and strong objectivity are
discussed. Specific examples of steady-state simple shear at constant rate and
steady-state isochoric extension relative to a rotating coordinate system
are used to analyze the robustness and accuracy of different algorithms.
The results suggest that it is preferable to reformulate evolution equations
in terms of the velocity gradient instead of the spin tensor, since strongly
objective integration algorithms can be developed using the relative deformation
gradient. Moreover, this relative deformation gradient can be calculated
independently of the time dependence of the velocity gradient during a typical time
step.
Dedicated to Charles and Marie-Louise
Steele, who advanced the field of mechanics with their wise
editorial leadership