“Everything flows?”: elastic effects on startup flows of yieldstress fluids
 1.5k Downloads
 11 Citations
Abstract
It is now 30 years since Barnes and Walters published a provocative paper in which they asserted that the yield stress is an experimental artifact. We now know that the situation is far more complicated than understood at the time, and that the mechanics of the solid material prior to yielding must be considered carefully. In this paper, we examine the response of a wellstudied “simple” yieldstress material, namely a Carbopol gel that exhibits no thixotropy, and demonstrate the significance of the preyielding behavior through a number of elementary measurements.
Keywords
Rheology experiments Yield stress Viscoelasticity Flow curves KelvinVoigt Maxwell fluidIntroduction
In 1985, Howard Barnes and Ken Walters published a provocative paper entitled “The yield stress myth?” (Barnes and Walters 1985), in which they asserted that the yield stress is an experimental artifact, and notably that all fluids will show viscous (indeed, Newtonian) behavior at sufficiently small stresses. They stated that “the yield stress hypothesis, which has hitherto been a useful empiricism, is no longer necessary, and … fluids which flow at high stresses will flow at all lower stresses, i.e., the viscosity, although large, is always finite and there is no yield stress.”^{1} This assertion by two very prominent rheologists caused a flurry of discussion and publication, with substantial parsing of the meaning of the words “yield stress;” i.e., is the yield stress a material property or a useful approximation for materials that exhibit a large reduction in viscosity over a narrow shear stress range? Barnes and Walters supported their assertion with data obtained using a constantstress rheometer that showed a Newtonian regime at stresses lower than the apparent yield stress, and Barnes subsequently showed similar data on a number of different materials (Barnes 1999; Roberts and Barnes 2001), including Carbopol.
The concept of a yieldstress fluid was popularized by Bingham, who included such fluids in the context of yielding in many classes of materials in his 1922 book Fluidity and Plasticity (Bingham 1922). Barnes (1999) has written a comprehensive review of the history of the study of yielding, in which he places the common yieldstress fluids currently being studied in the context of phenomena like creep in metals and plastics. Modern interest in yieldstress fluids largely dates from work by Oldroyd (1947) and Prager (Hohenemser and Prager 1932; Prager 1961) that put the description of such materials into an invariant continuum formulation that can be applied to flows in complex geometries. Both Oldroyd and Prager assumed that there is a transition between a solid and a fluid at a critical value of a stress invariant, typically taken to be a yield surface defined by the von Mises criterion (Prager 1961). Prager assumed that no deformation was possible on the “solid” side of the yield surface. Oldroyd assumed that the material is an incompressible elastic (Hookean) solid before yielding, with a stress proportional to the strain, and a viscous material thereafter, with a stress that is linear in the rate of deformation. Most subsequent investigators have assumed that the solid has an infinite modulus, in which case no deformation is possible prior to yielding, and the assumption of linearity after yielding has been generalized to include powerlaw behavior and even viscoelasticity. The Oldroyd–Prager formulation, with a discontinuous transition between solid and liquid, is at the heart of the yieldstress controversy initiated by Barnes and Walters.
Møller et al. (2009a) showed that the apparent Newtonian viscosity observed by Barnes at stresses below the apparent yield stress was not a true viscosity but was in fact an experimental artifact whose value depends on the waiting time prior to measurement (i.e., the elapsed time between initiating the deformation and recording the measurement), increasing with a powerlaw dependence on the waiting time; the exponents were between ½ and 1, depending on the material. What is in fact being observed is a response of the unyielded material that gives a ratio of stress to shear rate that is independent of the imposed stress, hence appearing to be a constant viscosity.

Is the yield stress a flow to noflow transition? i.e., as per Barnes and Walters (1985), is there viscous flow below the yield stress?

Can the yield stress be inferred by extrapolation of the flow curve to zero shear rate?

Can the yield stress be inferred from startup experiments?

Are nonlinear oscillatory shear measurements a better way to infer the yield stress?
Materials
Rheology experiments
Extrapolation to zero shear rate: flow curves
Flow to noflow transition
Startup: experiments at a constant shear rate
Oscillatory measurements
The effect of viscoelasticity: preyielding mechanics
Of course, a material cannot be roughly a Maxwell fluid in one class of deformations and roughly a Kelvin–Voigt solid in another unless there is a hidden variable in a more general formulation that interpolates between the behaviors. This is the case in the kinematic hardening model used by Dimitriou et al. (2013), for example, in which the “back stress” evolves dynamically and affects the mechanics. The back stress can be viewed as a “lambda parameter” (e.g., Denn and Bonn (2011)) in simple shear flow and causes the location of the yield surface to adjust, depending on the deformation state, as in the general framework of the evolution of the yield stress surface for elastoviscoplastic solids that was developed by Naghdi and Srinivasa (1992). The kinematic hardening model can be shown to be roughly Maxwellian for small deformations at a constant shear rate and to be Maxwellian for the difference τ − τ _{ y } close to yielding, so it reflects the behavior seen in Fig. 5. Dimitriou et al. (2013) have shown via a numerical simulation at constant stress that the model predicts behavior qualitatively like that shown in Fig. 4.
Concluding remarks
This short article is intended to highlight the significance of the description of the preyielded material in considering the mechanics of yieldstress fluids. For the simple yieldstress fluid considered here, the transition appears to be based on a critical strain, with the possibility of dissipative deformations in a viscoelastic solid that make the critical stress under transient conditions deformation dependent. It is clear experimentally that the appearance of a Newtonian fluid regime at stresses below the yield stress is an artifact that would be observed with the simplest viscoelastic solid representation, namely a Kelvin–Voigt solid. We have not addressed the likely failure of the Oldroyd–Prager formalism following yielding, but there is convincing evidence that a viscoelastic fluid description is necessary for materials like the Carbopol studied here. Indeed, Fraggedakis et al. (2016) have employed both kinematic hardening and a viscoelastic model by Saramito (2007) to describe the kinematics and settling dynamics of a spherical particle through a Carbopol gel.
Finally, we note that for the ideal (nonthixotropic) yieldstress fluid studied here, the transition in a plot of total stress versus strain in finiteamplitude oscillatory shear gives a value of the yield stress that is consistent with the yield stress obtained by extrapolation of the flow curve and the value obtained in a startup experiment, with the added information of the yield strain. This method has the advantage of eliminating artifacts associated with startup flows or extrapolation of the flow curve. Any of these methods properly used, however, can give a reliable value of the yield stress.
Footnotes
 1.
Barnes has described the paper as having been presented at the Fourth International Congress on Rheology in 1984 in a number of publications, but the paper does not appear in the Congress proceedings.
References
 Balmforth NJ, Frigaard IA, Ovarlez G (2014) Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu Rev Fluid Mech 46:121–46CrossRefGoogle Scholar
 Barnes HA (1999) The yield stress—a review or “ πα ν τ α ρ ε ι”—everything flows J NonNewt Fluid Mech 81:133–178CrossRefGoogle Scholar
 Barnes HA, Nguyen Q (2001) Rotating vane rheometry—a review. J NonNewtonian Fluid Mech 98:1–14CrossRefGoogle Scholar
 Barnes HA, Walters K (1985) The yield stress myth. Rheol Acta 24:323–326CrossRefGoogle Scholar
 Bingham EC (1922) Fluidity and plasticity. McGrawHill, New YorkGoogle Scholar
 Bonn D, Denn MM (2009) Yield stress fluids slowly yield to analysis. Science 324:1401–1402CrossRefGoogle Scholar
 Bonn D, Paredes J, Denn MM, Berthier L, Divoux T, Manneville S (2015) Yield stress materials in soft condensed matter. arXiv preprint. arXiv:1502.05281
 Bonnecaze RT, Brady JF (1992) Yield stresses in electrorheological fluids. J Rheol 36:73–115CrossRefGoogle Scholar
 Coussot P (2014) Yield stress fluid flows: a review of experimental data. J NonNewtonian Fluid Mech 211:31–49CrossRefGoogle Scholar
 Christopoulou C, Petekidis G, Erwin B, Cloitre M, Vlassopoulos D (2009) Ageing and yield behaviour in model soft colloidal glasses. Phil Trans Roy Soc A367:5051–5071CrossRefGoogle Scholar
 Denn MM, Bonn D (2011) Issues in the flow of yieldstress fluids. Rheol Acta 50:307–315CrossRefGoogle Scholar
 Dimitriou CJ, Ewoldt RH, McKinley GH (2013) Describing and prescribing the constitutive response of yield stress fluids using large amplitude oscillatory shear stress (LAOStress). J Rheology 57:27–70CrossRefGoogle Scholar
 Dinkgreve M, Paredes J, Denn MM, Bonn D (2016) On different ways of measuring “the” yield stress. J NonNewtonian Fluid Mech 238:233–241Google Scholar
 Fraggedakis D, Dimakopoulos Y, Tsamopoulos J (2016) Yielding the yield stress analysis: a study focused on the settling of a single spherical particle in a yield stress fluid. Soft Matter 24:5378–5401CrossRefGoogle Scholar
 Hohenemser K, Prager W (1932) Über die Ansätze der Mechanik isotroper Kontinua. Z Ang Math Mech 12:216–226CrossRefGoogle Scholar
 Liddell PV, Boger DV (1996) Yield stress measurements with the vane. J NonNewtonian Fluid Mech 63:235–261CrossRefGoogle Scholar
 Møller PCF, Fall A, Bonn D (2009a) Origin of apparent viscosity in yield stress fluids below yielding. EPL 87:38004CrossRefGoogle Scholar
 Møller P, Fall A, Chikkadi V, Derks D, Bonn D (2009b) An attempt to categorize yield stress fluid behavior. Phil Trans Roy Soc A367:5139–5155CrossRefGoogle Scholar
 Naghdi P, Srinivasa AR (1992) On the dynamical theory of rigidviscoplastic materials. Quart J Mech Applied Math 45:747–773CrossRefGoogle Scholar
 Nguyen QD, Boger DV (1983) Yield stress measurements concentrated suspensions. J Rheol 27:321–349CrossRefGoogle Scholar
 Oldroyd JG (1947) A rational formulation of the equations of plastic flow for a Bingham solid. Proc Camb Phil Soc 43:100–105CrossRefGoogle Scholar
 Ovarlez G, CohenAddad S, Krishnan K, Goyon J, Coussot P (2013) On the existence of a simple yield stress behavior. J NonNewtonian Fluid Mech 193:68–79CrossRefGoogle Scholar
 Prager W (1961) Introduction to mechanics of continua. Ginn & Co., BostonGoogle Scholar
 Roberts GP, Barnes HA (2001) New measurements of the flowcurves for Carbopol dispersions without slip artefacts. Rheol Acta 40:499–503CrossRefGoogle Scholar
 Saramito P (2007) A new constitutive equation for elastoviscoplastic fluid flows. J NonNewtonian Fluid Mech 145:1–14CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.