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 (>10 6 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.