The yield stress—a review or ‘παντα ρει’—everything flows?

Abstract

An account is given of the development of the idea of a yield stress for solids, soft solids and structured liquids from the beginning of this century to the present time. Originally, it was accepted that the yield stress of a solid was essentially the point at which, when the applied stress was increased, the deforming solid first began to show liquid-like behaviour, i.e. continual deformation. In the same way, the yield stress of a structured liquid was originally seen as the point at which, when decreasing the applied stress, solid-like behaviour was first noticed, i.e. no continual deformation. However as time went on, and experimental capabilities increased, it became clear, first for solids and lately for soft solids and structured liquids, that although there is usually a small range of stress over which the mechanical properties change dramatically (an apparent yield stress), these materials nevertheless show slow but continual steady deformation when stressed for a long time below this level, having shown an initial linear elastic response to the applied stress. At the lowest stresses, this creep behaviour for solids, soft solids and structured liquids can be described by a Newtonian-plateau viscosity. As the stress is increased the flow behaviour usually changes into a power-law dependence of steady-state shear rate on shear stress. For structured liquids and soft solids, this behaviour generally gives way to Newtonian behaviour at the highest stresses. For structured liquids this transition from very high (creep) viscosity (>10⁶ Pa.s) to mobile liquid (<0.1 Pa.s) can often take place over a single order of magnitude of stress. This extreme behaviour, when viewed on a linear basis, gave every reason for believing that the material had a yield stress, and in many cases the flow curve seemed to be adequately described by Bingham’s simple straight-line-with-intercept equation. However, if viewed on a logarithmic basis, the equally simple Newtonian/power-law/Newtonian description is clearly seen. (One evident implication of these statements is that παντα ρει—everything flows!) Although we have shown that, as a physical property describing a critical stress below which no flow takes place, yield stresses do not exist, we can, without any hesitation, say that the concept of a yield stress has proved—and, used correctly, is still proving—very useful in a whole range of applications, once the yield stress has been properly defined. This proper definition is as a mathematical curve-fitting constant, used along with other parameters to produce an equation to describe the flow curve of a material over a limited range of shear rates. This equation can then be used to predict the behaviour of that material in different geometries. However, it should only be used over the same range of shear rates that the original characterisation and curve fitting were undertaken. Here we show how best to deal with such situations, and we emphasise that the simplest-possible adequate ‘yield-stress’ equation should be used.

Introduction

The subject of yield stresses has, in the last decade or so, generated a fair amount of debate in the rheological literature (see below for details). Following a previous paper by this author in 1985 [1], some writers using the expression tried to clarify what they meant by yield stress, rather than simply introducing it without cautious definition. However, there is still enough unenlightened and uncritical use of the expression by some authors-who seem unaware of the controversy that has gone on-to justify a review of the subject, and possibly to round off discussion in the area.

As an example of the continued importance of yield stresses and allied ideas, we note that in the last dozen years or so the phrases yield stress or yield point have been cited nearly 2500 times in the general scientific and engineering literature (All citation statistics quoted here are derived from the Institute for Scientific Information Citation Databases, 1985 to date). As to the particular continued use of the concept in Rheology, we note that in a recent, extra-large volume of Rheology Abstracts [40 (3), 1997], over 500 papers were reviewed, and more than 1 in 15 of them made reference to either yield stress or plasticity.

A typical dictionary definition of the verb ‘to yield’ would be ‘to give way under pressure’ and this implies an abrupt and profound change in behaviour to a less resistant state. As a qualitative description of a general change in the mechanical response of a material, it is completely satisfactory and unobjectionable, since many materials show a radical transformation in flow (or deformation) behaviour over a relatively narrow range of stress, and have obviously ‘yielded’, for example as seen in Fig. 1 where the flow properties of a 6% suspension of iron oxide in mineral oil (redrawn from [2]), change from a very viscous liquid (∼10⁵ Pa s) at low stresses (∼0.7 Pa) to a quite mobile liquid (0.5 Pa s) at a slightly higher stress (∼3 Pa).

Historically this kind of behaviour was graphically portrayed in linear plots of applied stress against shear rate, which is not surprising, given the limited shear-rate ranges available—typically 1–100 s⁻¹. However, plotted as, say, the logarithm of viscosity against the logarithm of applied stress, the phenomenon is much better illustrated. Within the limited range of measurable shear rates, it looked as though the viscosity of such a ‘yield-stress’ liquid was increasing asymptotically as the applied shear stress was decreased. Indeed, for all practical purposes it looked as though there was a definite stress where the viscosity did become infinite, see for example Fig. 2 which is derived from Fig. 3.8 of Han’s book [3]. Looking at it the other way around, below this critical stress the material in question appeared to be an infinite-viscosity solid, and above the critical stress, a (very shear-thinning) liquid!.

The controversy begins here, since many people would then go on to strictly define a solid as a material that does not flow, i.e. continue to deform under stress. Hence, for them, below the yield stress there is no flow! This came about because those involved truly could not measure any sensible flow below the yield stress, that is to say, within the experimental means at their disposal at the time. The position taken here is that such experimental limitations have now been removed, and the picture has changed profoundly. However, without any further hesitation, we can say that the concept of a yield stress has proved and, used correctly, is still proving very useful in a whole range of applications, once it is properly defined, delimited and circumscribed.

Obviously, the idea of a yield stress was well imbedded in the domain of metals long before it was brought over into the liquids area by Schwedoff and Bingham early in this century. Indeed, the Bingham model was introduced in one of his books in 1922 in a chapter headed ‘The Plasticity of Solids’. (The original publication of the Bingham equation was in 1916 [4].)

However, the definition of the yield stress of a solid (for instance a metal) is itself far from being simple and easily described, and before we embark on a study of the subject for liquids, it is well worthwhile devoting some space to this topic. For instance, looking at some popular definitions offered nowadays, we have:Penguin Dictionary of Physics [5]:

• “yield point—a point on a graph of stress versus strain… at which the strain becomes dependent on time and the material begins to flow.

• yield stress—the minimum stress for creep to take place. Below this value any deformation produced by an external force will be purely elastic.

• yield value—the minimum value of stress that must be applied to a material in order that it shall flow”.

McGraw-Hill Dictionary of Science and Technology, 4th Edition [6]:

• “yield point—the lowest stress at which strain increases without increase in stress.

• yield strength—the stress at which a material exhibits a specified deviation from [linear] proportionality between stress and strain”.

Chambers 21st Century Dictionary [7]:

• “yield stress—the level of stress at which substantial deformation suddenly takes place.”

Chambers Dictionary of Science and Technology [8]:

• “yield stress—the stress at which a substantial amount of plastic deformation takes place under constant load. This sudden yielding is characteristic of iron and annealed steels. In other metals, deformation begins gradually…”

Van Nostrand’s Science Encyclopaedia, 5th edition [9]:

• “yield stress—the minimum stress at which a… material will deform without significant increase in load… some materials do not have a well-defined yield point and in others it is not a well-defined value.”

Clearly the definition is quite varied even within this small sample of everyday science publications: for instance, at one point it is ‘departure from linearity’; then ‘increase in deformation without increase in stress’ and then ‘the onset of creep or flow’, etc. Secondly, it is hedged about by all kinds of non-quantitative qualifiers, such as ‘specified deviation’, ‘substantial deformation’, ‘substantial amount’, ‘without significant increase in load’, ‘not… well-defined’, etc. All of this is in contrast, for instance, to the Young’s modulus of the same material, which is a well-specified quantity, which can be assigned precisely.

However we could say that a simplistic definition of the yield stress of a solid is essentially the point at which, when increasing the applied stress, the solid first shows liquid-like behaviour, i.e. continual deformation. If this is the case, then we can say that conversely, a simple definition of the yield stress of a liquid is the point at which, when decreasing the applied stress, solid-like behaviour is first seen, i.e. no continual deformation.

The liquids that appear to have a yield stress are legion. Among them are Bingham’s many original examples such as clay, oil paint, toothpaste, drilling mud, molten chocolate, etc., then later, materials as diverse as creams of all sorts, ketchups and other culinary sauces, molten filled rubbers and printing inks, etc., were found to show similar behaviour. Now such disparate systems as ceramic pastes, electro-viscous fluids, thixotropic paints, heavy-duty washing liquids, surface-scouring liquids, mayonnaise, yoghurts, purees, liquid pesticides, bio-mass broths, blood, water-coal mixtures, molten liquid-crystalline polymers, plastic explosives, foams, rocket propellant pastes, etc., can be added to the extensive list.

In the following sections, we will trace the history of the subject and then look at the present position with regard to these kinds of materials in order to see where there is use and abuse of the yield-stress concept.