Rheologica Acta

, Volume 54, Issue 3, pp 223–233 | Cite as

Single-point parallel disk correction for asymptotically nonlinear oscillatory shear

Original Contribution


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, σγα, 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.


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



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.


  1. Beckwith TG, Marangoni RD, Lienhard JH (1993) Mechanical measurements, 5th ed., Addison-Wesley, New York, p 82Google Scholar
  2. Bharadwaj NA, Ewoldt RH (2014) The general low-frequency prediction for asymptotically nonlinear material functions in oscillatory shear. J Rheol 58:891–910CrossRefGoogle Scholar
  3. Bharadwaj NA, Ewoldt RH (2015) Constitutive models under medium-amplitude oscillatory shear (MAOS). J Rheol. In pressGoogle Scholar
  4. Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids, vol. 1, 2nd edn. Fluid mechanics. Wiley, New YorkGoogle Scholar
  5. Blackwell BC, Ewoldt RH (2014) A simple thixotropic-viscoelastic constitutive model produces unique signatures in large-amplitude oscillatory shear (LAOS). J Non-Newtonian Fluid Mech 208-209:27–41CrossRefGoogle Scholar
  6. Bozorgi Y, Underhill PT (2014) Large amplitude oscillatory shear rheology of dilute active suspensions. Rheol Acta 53:899–909Google Scholar
  7. Brunn P, Asoud H (2002) Analysis of shear rheometry of yield stress materials and apparent yield stress materials. Rheol Acta 41:524–531CrossRefGoogle Scholar
  8. Carvalho MS Padmanabhan M, Macosko CW (1994) Single-point correction for parallel disks rheometry. J Rheol 38:1925–1936CrossRefGoogle Scholar
  9. Cross MM (1965) Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic systems. J Colloid Sci 20:417–437CrossRefGoogle Scholar
  10. Cross MM, Kaye A (1987) Simple procedures for obtaining viscosity/shear rate data from a parallel disc viscometer. Polymer 28:435–440CrossRefGoogle Scholar
  11. Davis WM, Macosko CW (1978) Nonlinear Dynamic Mechanical Moduli for Polycarbonate and PMMA. J Rheol 22:53–71CrossRefGoogle Scholar
  12. Dealy JM, Wissbrun KF (1990) Melt rheology and its role in plastics processing : theory and applications. Van Nostrand Reinhold, New YorkCrossRefGoogle Scholar
  13. De Souza Mendes PR, Thompson RL, Alicke AA, Leite RT (2014) The quasilinear large-amplitude viscoelastic regime and its significance in the rheological characterization of soft matter. J Rheol 58:537–561CrossRefGoogle Scholar
  14. Dimitriou CJ, Ewoldt RH, McKinley GH (2013) Describing and prescribing the constitutive response of yield stress fluids using large amplitude oscillatory stress (LAOStress). J Rheol 57:27–70CrossRefGoogle Scholar
  15. Ding F, Giacomin JA, Bird BR, Kweon C-B (1999) Viscous dissipation with fluid inertia in oscillatory shear flow. J Non-Newtonian Fluid Mech 86:359–374CrossRefGoogle Scholar
  16. Ewoldt RH (2013) Defining nonlinear rheological material functions for oscillatory shear. J Rheol 57:177–195CrossRefGoogle Scholar
  17. Ewoldt RH, Bharadwaj NA (2013) Low-dimensional intrinsic material functions for nonlinear viscoelasticity. Rheol Acta 52:201–219CrossRefGoogle Scholar
  18. Ewoldt RH, Johnston MT, Caretta LM (2015) Experimental challenges of shear rheology: how to avoid bad data. In: Spagnolie S (ed) Complex fluids in biological systems, Biological and Medical Physics, Biomedical Engineering. Springer, Berlin, New York, pp 207–241 Springer, BerlinGoogle Scholar
  19. Ewoldt RH, Winter P, Maxey J, McKinley GH (2009) Large amplitude oscillatory shear of pseudoplastic and elastoviscoplastic materials. Rheol Acta 49:191–212CrossRefGoogle Scholar
  20. Fahimi Z, Broedersz CP, Kempen THS, Florea D, Peters GWM, Wyss HM (2014) A new approach for calculating the true stress response from large amplitude oscillatory shear (LAOS) measurements using parallel plates. Rheol Acta 53:75–83CrossRefGoogle Scholar
  21. Fan X-J, Bird RB (1984) A kinetic theory for polymer melts VI. Calculation of additional material functions. J Non-Newtonian Fluid Mech 15:341–373CrossRefGoogle Scholar
  22. Férec J, Heuzey MC, Ausias G, Carreau PJ (2008) Rheological behavior of fiber-filled polymers under large amplitude oscillatory shear flow. J Non-Newtonian Fluid Mech 151:89–100CrossRefGoogle Scholar
  23. Giacomin AJ, Bird RB, Johnson L M, Mix AW (2011) Large-amplitude oscillatory shear flow from the corotational Maxwell model. J Non-Newtonian Fluid Mech 166:1081–1099CrossRefGoogle Scholar
  24. Gurnon AK, Wagner NJ (2012) Large amplitude oscillatory shear (LAOS) measurements to obtain constitutive equation model parameters: Giesekus model of banding and nonbanding wormlike micelles. J Rheol 56:333–351CrossRefGoogle Scholar
  25. Helfand E, Pearson DS (1982) Calculation of the nonlinear stress of polymers in oscillatory shear fields. J Polym Sci Polym Phys Ed 20:1249–1258CrossRefGoogle Scholar
  26. Hyun K, Wilhelm M (2009) Establishing a new mechanical nonlinear coefficient Q from FT-rheology: first investigation of entangled linear and comb polymer model systems. Macromolecules 42:411–422CrossRefGoogle Scholar
  27. Hyun K, Wilhelm M, Klein CO, Cho KS, Nam JG, Ahn KH, Lee SJ, Ewoldt RH, McKinley GH (2011) A review of nonlinear oscillatory shear tests: Analysis and application of large amplitude oscillatory shear (LAOS). Prog Polym Sci 36:1697–1753CrossRefGoogle Scholar
  28. Läuger J, Stettin H (2010) Differences between stress and strain control in the non-linear behavior of complex fluids. Rheol Acta 49:909–930CrossRefGoogle Scholar
  29. Liu J, Yu W, Zhou W, Zhou C (2009) Control on the topological structure of polyolefin elastomer by reactive processing. Polymer 50:547–552CrossRefGoogle Scholar
  30. Macosko CW (1994) Rheology principles, measurements and applications. Wiley-VCH, New YorkGoogle Scholar
  31. Mattes KM, Vogt R, Friedrich C (2008) Analysis of the edge fracture process in oscillation for polystyrene melts. Rheol Acta 47:929–942CrossRefGoogle Scholar
  32. McMullan JM, Wagner NJ (2009) Directed self-assembly of suspensions by large amplitude oscillatory shear flow. J Rheol 53:575–588CrossRefGoogle Scholar
  33. Merger D, Wilhelm M (2014) Intrinsic nonlinearity from LAOStrain—experiments on various strain- and stress-controlled rheometers: a quantitative comparison. Rheol Acta 53:621–634CrossRefGoogle Scholar
  34. Nam JG, Hyun K, Ahn KH, Lee SJ (2008) Prediction of normal stresses under large amplitude oscillatory shear flow. J Non-Newtonian Fluid Mech 150:1–10CrossRefGoogle Scholar
  35. Ng TSK, McKinley GH, Ewoldt RH (2011) Large amplitude oscillatory shear flow of gluten dough: a model power-law gel. J Rheol 55:627–654CrossRefGoogle Scholar
  36. Onogi S, Masuda T, Matsumoto T (1970) Non-linear behavior of viscoelastic materials. I. Disperse systems of polystyrene solution and carbon black. J Rheol 14:275–294CrossRefGoogle Scholar
  37. Pearson DS, Rochefort WE (1982) Behavior of concentrated polystyrene solutions in large-amplitude oscillating shear fields. J Polym Sci Polym Phys Ed 20:83–98CrossRefGoogle Scholar
  38. Phan-Thien N, Newberry M, Tanner RI (2000) Non-linear oscillatory flow of a soft solid-like viscoelastic material. J Non-Newtonian Fluid Mech 92:67–80CrossRefGoogle Scholar
  39. Philippoff W (1966) Vibrational measurements with large amplitudes. Trans Soc Rheol 10:317–334CrossRefGoogle Scholar
  40. Ravindranath S, Wang S-Q (2008) Large amplitude oscillatory shear behavior of entangled polymer solutions: particle tracking velocimetric investigation. J Rheol 52:341–358CrossRefGoogle Scholar
  41. Ravindranath S, Wang S-Q, Olechnowicz M, Chavan VS, Quirk RP (2011) How polymeric solvents control shear inhomogeneity in large deformations of entangled polymer mixtures. Rheol Acta 50(2):97–105Google Scholar
  42. Schrag JL (1977) Deviation of velocity gradient profiles from the “gap loading” and “surface loading” limits in dynamic simple shear experiments. J Rheol 21:399–413CrossRefGoogle Scholar
  43. Shaw M, Liu Z (2006) Single-point determination of nonlinear rheological data from parallel-plate torsional flow. Appl Rheol 16:70–79Google Scholar
  44. Soskey PR, Winter HH (1984) Large step shear strain experiments with parallel-disk rotational rheometers. J Rheol 28:625–645CrossRefGoogle Scholar
  45. Stickel JJ, Knutsen JS, Liberatore MW (2013) Response of elastoviscoplastic materials to large amplitude oscillatory shear flow in the parallel-plate and cylindrical-Couette geometries. J Rheol 57:1569–1596Google Scholar
  46. Wagner MH, Rolon-Garrido VH, Hyun K, Wilhelm M (2011) Analysis of medium amplitude oscillatory shear data of entangled linear and model comb polymers. J Rheol 55:495–516CrossRefGoogle Scholar
  47. Yeow YL, Chandra D, Sardjono AA, et al. (2004) A general method for obtaining shear stress and normal stress functions from parallel disk rheometry data. Rheol Acta 44:270–277CrossRefGoogle Scholar
  48. Yoshimura AS, Prudhomme RK (1987) Response of an elastic Bingham fluid to oscillatory shear. Rheol Acta 26:428–436CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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