On objective corotational rates and their defining spin tensors

Abstract

In this paper, we prove a general result on objective corotational rates and their defining spin tensors: let Ω* be a spin tensor that is associated with the rotation and deformation of a deforming material body in an arbitrary manner indicated by Ω* = Y(B, D, W), where B and D and W are the left Cauchy Green tensor and the stretching tensor and the vorticity tensor, respectively. Then the corotational rate of σ defined by the spin Ω*, i.e., the tensor field σ* = σ+σΩ*—Ω*σ, is objective for every time-differentiable objective Eulerian symmetric tensor field a if and only if the spin tensor Ω* assumes the form Ω* = W + Y(B, D). where Y(B, D) is an antisymmetric tensor-valued isotropic function. Furthermore, by virtue of certain necessary or reasonable requirements, it is found that a single antisymmetric function of two positive real variables can be introduced to characterize a general class of spin tensors defining objective corotational rates. Accordingly, a general explicit basis-free expression for the latter is established in terms of the left Cauchy-Green tensor B, the vorticity tensor W and the stretching tensor D as well as the introduced antisymmetric function. By choosing several particular forms of the latter, it is shown that all commonly-used spin tensors are incorporated into this general expression in a natural way.