Strain Rates and Material Spins

Heng Xiao — 2007-04-13T15:49:18-00:00

Journal of Elasticity, Vol. 52, No. 1. (16 July 1998), pp. 1-41.

On objective corotational rates and their defining spin tensors

H Xiao — 2007-04-06T18:21:07-00:00

International Journal of Solids and Structures, Vol. 35, No. 30. (October 1998), pp. 4001-4014.

In this paper, we prove a general result on objective corotational rates and their defining spin tensors: let [Omega]* be a spin tensor that is associated with the rotation and deformation of a deforming material body in an arbitrary manner indicated by [Omega]* = 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 [sigma] defined by the spin [Omega]*, i.e., the tensor field [sigma]* = [sigma]+[sigma][Omega]*--[Omega]*[sigma], is objective for every time-differentiable objective Eulerian symmetric tensor field a if and only if the spin tensor [Omega]* assumes the form [Omega]* = 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.

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Strain Rates and Material Spins

On objective corotational rates and their defining spin tensors