On objective corotational rates and their defining spin tensor

In lhis paper, we prove a general result on objective corotational rates and their defining spin tensors: let f~* be a spin tensor that is associated with the rotation and deformation of a deforming material body in an arbitrary manner indicated b 3 fl* - 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 # defined by the spin ~*, i.e., the tensor field 6-* = dr+#f~* ~*#, is objective for every time-differentiable ol2jective Eulerian symmetric tensor field ~ if and on b if the spin tensor ~* assumes the form .Q* - W +/ʹ(B, D). where Iʹ(B,D) is an antisymmetric tensor-valued isotropic fimction. 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 w)rticity 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. 1998 Elsevier Science Ltd. All rights reserved