Why the Cox–Merz rule and Gleissle mirror relation work: A quantitative analysis using the Wagner integral framework with a fractional Maxwell kernel

In this work, we mathematically derive the conditions for which empirical rheometric relations such as the Cox–Merz rule and Gleissle mirror relationship are satisfied. We consider the Wagner integral constitutive framework, which is a special limiting case of the Kaye–Bernstein Kearsley Zapas (K-BKZ) constitutive equation to derive analytical expressions for the complex viscosity, the steady shear viscosity, and the transient stress coefficient in the start-up of steady shear. We use a fractional Maxwell liquid model as the linear relaxation modulus or memory kernel within a non-linear integral constitutive framework. This formulation is especially well-suited for describing complex fluids that exhibit a broad relaxation spectrum and can be readily reduced to the canonical Maxwell model for describing viscoelastic liquids that exhibit a single dominant relaxation time. To incorporate the nonlinearities that always become important in real complex fluids at large strain amplitudes, we consider both an exponential damping function as well as a more general damping function. By evaluating analytical expressions for small amplitude oscillatory shear, steady shear, and the start-up of steady shear using these different damping functions, we show that neither the Cox–Merz rule nor the Gleissle mirror relation can be satisfied for materials with a single relaxation mode or narrow relaxation spectrum. We then evaluate the same expressions using asymptotic analysis and direct numerical integration for more representative complex fluids having a wide range of relaxation times and nonlinear responses characterized by damping functions of exponential or Soskey–Winter form. We show that for materials with broad relaxation spectra and sufficiently strong strain-dependent damping the empirical Cox–Merz rule and the Gleissle mirror relations are satisfied either exactly, or to within a constant numerical factor of order unity. By contrast, these relationships are not satisfied in other classes of complex viscoelastic materials that exhibit only weak strain-dependent damping or strain softening.