Advantages of formulating evolution equations for elastic-viscoplastic materials in terms of the velocity gradient instead of the spin tensor

Abstract

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.