Single-point parallel disk correction for asymptotically nonlinear oscillatory shear

We derive exact single-point corrections for parallel disk measurements of all four asymptotically nonlinear measures under strain-controlled oscillatory shear. In this regime, sometimes called medium-amplitude oscillatory shear (MAOS), the derivatives appearing in the general stress correction are constant over the range of interest. This enables an exact single-point correction of all four shear stress components and material functions in the asymptotically nonlinear regime. This greatly simplifies the data processing and allows convenient measurements of true nonlinear material functions with parallel disk geometries. We use a strain amplitude expansion for the stress response, introducing a general non-integer strain amplitude scaling for the leading order nonlinearity, σ ∼ γα, where typically α = 3 has been assumed in the past. The stress corrections are a multiplicative amplification by a factor fα=α+34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f\left (\alpha \right )=\frac {\alpha +3}{4}$\end{document}, shown for the first time for all four asymptotically nonlinear coefficients. Experimental measurements are presented for the four asymptotically nonlinear signals on an entangled polymer melt of cis-1,4-polyisoprene, using both parallel disk and cone fixtures. The polymer melt follows a cubic (α = 3) strain amplitude scaling in the MAOS regime. The theoretical corrections indicate a 50 % amplification of the apparent signals measured with the parallel disk fixture. The corrected (amplified) signals match the measurements with the cone.