Spin tensors associated with corotational rates and corotational integrals in continua

Abstract

In many cases of constitutive modeling of continua undergoing large deformations, use of corotational rates and integrals is inevitable to avoid the effects of rigid body rotations. Making corotational rates associated with specific spin tensors is a matter of interest, which can help for a better physical interpretation of the deformation. In this paper, for a given kinematic tensor function, say G , a tensor valued function F as well as a spin tensor Ω 0 is obtained in such a way that the corotational rate of F associated with the spin tensor Ω 0, becomes equal to G . In other words, F is the corotational integral of G associated with the spin tensor Ω 0. Here, G is decomposed additively into the principally diagonal and the principally off-diagonal parts. The symmetric tensors with the same principally diagonal parts have the same tensor valued corotational integrals, but associated with different spin tensors, which depends on the principally off-diagonal part. For the validity of the results, the Eulerian logarithmic strain tensor is shown to be the tensor valued corotational integral of the strain rate tensor, associated with the logarithmic spin. This result has separately been derived by Reinhardt & Dubey and Xiao et al. In addition, a specific corotational rate called Γ-rate and its associated spin tensor are introduced.