Volterra kernels, Oldroyd models, and interconversion in superposition rheometry

Abstract

The purpose of this paper is to demonstrate how the general problem of interconversion between parallel and orthogonal superposition protocols can be treated using the kernels in a Fréchet series expansion about the base viscometric flow. Such series differ from Fréchet series expanded about the rest history which are encountered in the theory of Green–Rivlin materials and the simple fluid theory of Coleman and Noll, in that nonlinear response of the material is captured at first order. Unlike first and second-order functional derivatives evaluated at the rest history, the derivatives we discuss require more than one kernel in their integral representation. However, all the kernels are inter-related. The strategy in the paper involves identifying the kernels which specify the components in the first and second order Fréchet derivatives for the nonlinear constitutive models to be used in the interconversion. Interconversion between parallel and orthogonal protocols can then be effected by establishing the relationships between the kernels. Step-strain perturbations are treated by allowing differentiation in the sense of distributions. The theory is illustrated throughout by evaluating the kernels and superposition moduli associated with the incompressible corotational Maxwell and Oldroyd models. • Treats interconversion between protocols in superposition rheometry. • Explores non-monotonicity and changes of sign in superposition moduli. • Treats oscillatory and step-strain perturbations. • Analyses Fréchet series expanded about viscometric flow history. • Explains second-order consistency of model parameterization.