Navigating the Pipkin Map: route planning and the edge of linearity

The Pipkin Map is a foundational tool of the rheologist to separately consider effects of timescale and forcing strength[1-3]. This talk will describe how to use the Pipkin Map to understand test conditions on an instrument, relate those test conditions to processing-relevant time scales and forcing strengths, and understand the limits of the linear viscoelastic regime across a wide range of timescales (Deborah number). The forcing strength can be represented as either a strain, strain rate, or stress magnitude. This talk will outline options for dimensionless groups for this axis including but not limited to the Weissenberg number, Plasticity number, Capillary number, and Mnemosyne number. All these dimensionless numbers are subsets of the broader concept of a dimensionless quantity comparing a characteristic strength of deformation to a critical forcing strength for material nonlinearity to appear. In practice, the linear limit is often defined by a subjective threshold of nonlinear emergence. A more rigorous definition, independent of this subjective threshold, will be described, which is most clearly revealed from oscillatory deformation known as medium-amplitude oscillatory shear (MAOS) that independently controls the forcing timescale and amplitude [4-5]. Recent work with stress-controlled MAOS has revealed that stress is a more fundamental measure of nonlinearity strength across a wide range of Deborah number, challenging prior perspectives which have generally fixated on using strain or strain rate to understand the limit of the linear viscoelastic regime [5].

REFERENCES:

[1] Pipkin, A. C., Lectures on Viscoelasticity Theory. Springer (1972) https://doi.org/10.1007/978-1-4612-1078-8

[2] Corman, R. E., and R. H. Ewoldt, “Mapping linear viscoelasticity for design and tactile intuition,” Applied Rheology (2019). https://doi.org/10.1515/arh-2019-0013

[3] Ewoldt, R.H. and G. H. McKinley, “Mapping thixo-elasto-visco-plastic behavior,” Rheologica Acta (2017). http://dx.doi.org/10.1007/s00397-017-1001-8

[4] Ewoldt, R. H. and N. A. Bharadwaj, “Low-dimensional intrinsic material functions for nonlinear viscoelasticity,” Rheologica Acta (2013). http://dx.doi.org/10.1007/s00397-013-0686-6

[5] Ramwali, N., M. T. Hossain, A. Shetty, and R. H. Ewoldt, “Stress-controlled medium-amplitude oscillatory shear (MAOStress) of PVA-Borax,” Journal of Rheology (2024) https://doi.org/10.1122/8.0000843