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