A corotational time derivative for surface tensors, constitutive relations and a new measure of bending strain

Abstract

In the construction of constitutive relations of the rate-type for a bulk continuun, it is important to utilize a time derivative of tensor quantities (e.g. stress, rate-of-strain) which is both objective (invariant to rigid body motions of the reference frame) and which commutes with the raising and lowering of indices by the metric tensor of the space. In addition for second-order tensors, the vanishing of this tensor rate should imply that the corresponding invariants of the tensor be stationary. Such a corotational derivative was first constructed for three-dimensional space tensors by Jaumann. In this paper an analogous operator is developed for surface tensors in an evolving metric space. This operator can then be used to construct constitutive relations for viscoelastic surface fluids. It also gives rise to a new measure of bending strain which is needed in the constitutive relation for the moment tensor (e.g. when the surface material resists bending) This new measure of bending strain is compared to more traditional measures used in elastic shell theory , and for a number of reasons it is found to be preferable.