# Single-point parallel disk correction for asymptotically nonlinear oscillatory shear

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## Abstract

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* ∼ *γ*^{α}, where typically *α* = 3 has been assumed in the past. The stress corrections are a multiplicative amplification by a factor \(f\left (\alpha \right )=\frac {\alpha +3}{4}\), 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.

### Keywords

Parallel disk rheometry Single point correction Asymptotically nonlinear rheology Large amplitude oscillatory shear Uncertainty propagation in MAOS MAOS## Notes

### Acknowledgments

This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-FG02-07ER46471, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign.

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