Medium amplitude parallel superposition (MAPS) rheology. Part 1: Mathematical framework and theoretical examples

Journal Article · Fri May 01 00:00:00 EDT 2020 · Journal of Rheology

DOI:https://doi.org/10.1122/1.5132693· OSTI ID:1603728

Lennon, Kyle R.  ^[1]; McKinley, Gareth H.  ^[2]; Swan, James W. ^[1]

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1. Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142
2. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

A new mathematical representation for nonlinear viscoelasticity is presented based on the application of the Volterra series expansion to the general nonlinear relationship between shear stress and shear strain history. This theoretical and experimental framework, which we call medium amplitude parallel superposition (MAPS) rheology, reveals a new material property, the third-order complex modulus, which describes completely the weakly nonlinear response of a viscoelastic material in an arbitrary simple shear flow. In this first part, we discuss several theoretical aspects of this mathematical formulation and new material property. For example, we show how MAPS measurements can be performed in strain- or stress-controlled contexts and provide relationships between the weakly nonlinear response functions measured in each case. We show that the MAPS response function is a superset of the response functions that have been previously reported in medium amplitude oscillatory shear and parallel superposition rheology experiments. We also show how to exploit inherent symmetries of the MAPS response function to reduce it to a minimal domain for straightforward measurement and visualization. We compute this material property for a few constitutive models to illustrate the potential richness of the datasets generated by MAPS experiments. Finally, we discuss the MAPS framework in the context of some other nonlinear, time-dependent rheological probes and explain how the MAPS methodology has a distinct advantage over these others because it generates data embedded in a very high-dimensional space without driving fluid mechanical instabilities, and is agnostic to the flow protocol.

• Sponsoring Organization: USDOE
• Grant/Contract Number: SC0020347
• OSTI ID: 1603728
• Journal Information: Journal of Rheology, Journal Name: Journal of Rheology Vol. 64 Journal Issue: 3; ISSN 0148-6055
• Publisher: Society of Rheology
• Country of Publication: United States
• Language: English