A Generalized Strain Measure in a Single-Integral Constitutive Equation

Abstract

The Bird-Carreau single-integral constitutive equation is considered with an unspecified and unrestricted form of non-linear memory function, and with an arbitrary strain measure, I, in place of the normally used Finger and Cauchy strain tensors. It is shown that the form of the shear component of the strain measure, I12, can be derived from experimental results obtained in the start-up of steady shear flow; and that normal stresses can be predicted from shear stresses. In contrast to the Finger and Cauchy tensors, the form of I12 obtained from experiment is decidedly non-linear in the amount of shear, and this results in markedly improved normal stress predictions. It is concluded that some of the previous failures of the Bird-Carreau type of equation have been due to a poor choice of strain measure, and not (as has been suggested) to the use of an inappropriate form of memory function. The new strain measure might usefully be applied to other flows.