On shear-rate dependent relaxation spectra in superposition rheometry: A basis for quantitative comparison/interconversion of orthogonal and parallel superposition moduli

In 1971, in a seminal paper, Yamamoto derived integral relationships between dynamic moduli and ratedependent relaxation spectra, H(tau, gamma(2)), in parallel superposition of oscillatory shear on steady shear flow, where both the flows and deformation gradients exist in the same plane. These integral relationships are more complicated than their counterparts for orthogonal superposition (where the oscillatory and unidirectional flow fields occur in orthogonal planes), since they involve not only the spectrum, but also its derivative with respect to unidirectional shear-rate. Herein we derive (i) expressions for determining rate-dependent relaxation spectra directly from parallel superposition rheometry data and (ii) expressions to convert from parallel to orthogonal dynamic moduli in a stable manner. These results facilitate the physical interpretation of parallel superposition dynamic moduli, and direct model-based comparison of parallel and orthogonal superposition moduli in the study of weak nonlinear response.