The mathematics of oscillatory recovery rheology with applications to experiments, the Cox-Merz rules, and the nonlinear modeling of common amplitude sweep behaviors

Oscillatory shear tests are frequently used to determine viscoelastic properties of complex fluids. Both the amplitude and frequency of the input signal can be independently varied, allowing rheologists to probe a wide range of material responses. Historically, most oscillatory tests have focused on the measurement and application of the total strain. However, the total strain is a composite parameter consisting of recoverable and unrecoverable components. Use of only the total strain therefore provides an incomplete description of the rheology. In this work, we provide a mathematical derivation for the determination of the recoverable and unrecoverable components in steady-state linear viscoelastic oscillatory flows via a simple experimental procedure. The relationship between the total strain and its components is fully explored and challenged in the context of how rheologists define moduli and common rheological models. These relations are demonstrated via experimental measurements on model viscoelastic and viscoplastic materials: wormlike micelles and Carbopol 980. Additionally, we show how the derived mathematics fully details the conditions where the Cox-Merz rules are valid in terms of recovery rheology. Finally, we demonstrate how a thorough understanding of the strain components can be used to create a simple nonlinear model that reproduces all common amplitude sweep behaviors.