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 (

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

Modelling the semisolid processing of metallic alloys

The slump test, originally used to determine the “workability” of fresh concrete, has since been used in many industrial fields (e.g., mining and food industries). It offers a quick and easy way to measure the yield stress of suspensions or pasty materials. The model used for estimating the yield stress from the measured conical slump was first written by Murata [Mater. Struct. 98, 117–129 (1984)], corrected by Schowalter and Christensen [J. Rheol., 42, 865–870 (1988)] and adapted for a cylindrical geometry by Pashias [J. Rheol. 40, 1179–1189 (1996)]. However, a discrepancy between experimental and predicted slumps still appears in the case of conical slumps [Clayton et al., Int. J. Miner. Process. 70, 3–21 (2003)] and for high-yield stress materials. In the present paper, we extend the theoretical analysis of this simple practical test by including different flow regimes according to the ratio between the radius (R) and the height (H) of the slumped cone. We propose analytical solutions of the flow for t...

“Fifty-cent rheometer” for yield stress measurements: From slump to spreading flow

Yield stress fluids are encountered in a wide range of applications: toothpastes, cements, mortars, foams, muds, mayonnaise, etc. The fundamental character of these fluids is that they are able to flow (i.e., deform indefinitely) only if they are submitted to a stress above some critical value. Otherwise they deform in a finite way like solids. The flow characteristics of such materials are difficult to predict as they involve permanent or transient solid and liquid regions that are generally hard to locate a priori. Here we review the present state of the art as it appears from experimental data for flows of simple (non-thixotropic) yield stress fluids under various conditions, viz., uniform flows in straight channels or rheometrical geometries, complex stationary flows in channels of varying cross-section such as extrusion, expansion, flow through a porous medium, transient flows such as flows around obstacles, spreading, spin-coating, squeeze flow, and elongation. The effects of surface tension, confinement, and secondary flows are also reviewed. We focus especially on experimental work identifying internal flow characteristics that can be compared with numerical predictions. It is shown in particular that: (i) deformations in the solid regime can play a critical role in transient flows; (ii) the yield character is not apparent in the flow field when the boundary conditions impose large deformations; (iii) the yield character is lost in secondary flows.

Yield stress fluid flows: A review of experimental data

The objective of this review is to examine how the concept of plasticity is used in geophysical fluid dynamics. Rapid mass movements such as snow avalanches or debris flows involve slurries of solid particles (ice, boulder, clay, etc.) within an interstitial fluid (air, water). The bulk behavior of these materials has often been modeled as plastic materials, i.e., a plastic material yields and starts to flow once its stress state has significantly departed from equilibrium. Two plastic theories are of common use in fluid dynamics: Coulomb plasticity and viscoplasticity. These theories have little in common, since ideal Coulomb materials are two-phase materials for which pore pressure and friction play the key role in the bulk dynamics, whereas viscoplastic materials (e.g., Bingham fluids) typically behave as single-phase fluids on the macroscopic scale and exhibit a viscous behavior after yielding. Determining the rheological behavior of geophysical materials remains difficult because they encompass coarse, irregular particles over a very wide range of size. Consequently, the true nature of plastic behavior for geophysical flows is still vigorously debated. In this review, we first set out the continuum-mechanics principles used for describing plastic behavior. The notion of yield surface rather than yield stress is emphasized in order to better understand how tensorial constitutive equations can be derived from experimental data. The notion of single-phase or two-phase behaviors on the macroscopic scale is then examined using a microstructural analysis on idealized suspensions of spheres within a Newtonian fluid; for these suspensions, the single-phase approximation is valid only at very high or low Stokes numbers. Within this framework, the bulk stress tensor can also be constructed, which makes it possible to give a physical interpretation to yield stress. Most of the time, depending on the bulk properties (especially, particle size) and flow features, bulk behavior is either Coulomb-like or viscoplastic in simple-shear experiments. The consequences of the rheological properties on the flow features are also examined. Some remarkable properties of the governing equations describing thin layers flowing down inclined surfaces are discussed. Finally, the question of parameter fitting is tackled: since rheological properties cannot be measured directly in most cases, they must be evaluated from field data. As an example, we show that the Coulomb model successfully captures the main traits of avalanche motion, but statistical analysis demonstrates that the probability distribution of the friction coefficient is not universal.

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Plasticity and geophysical flows: A review

Steady, two‐dimensional flows of Bingham fluids are analyzed by means of a modified constitutive relation that applies everywhere in the flow field, in both yielded and practically unyielded regions. The conservation equations and the constitutive relation are solved simultaneously by Galerkin finite element and Newton iteration. This combination eliminates the necessity for tracking yield surfaces in the flow field. The analysis is applied to a one‐dimensional channel flow, a two‐dimensional boundary layer flow, and a two‐dimensional extrusion flow. The finite element predictions compare well with available analytic solutions for limiting cases.

Flows of Materials with Yield

Steady axisymmetric thin film flows on a rotating cone of Newtonian and non‐Newtonian liquids are analyzed by means of the axisymmetric boundary layer with swirl equations, Galerkin finite‐element discretization with free surface parameterization, and Newton iteration, which permits the simultaneous evaluation of the boundary layer thickness with the primary unknowns which are the three velocity components. The significance of the inlet boundary conditions on the accuracy of the finite‐element predictions is discussed in detail. It is shown that natural inlet boundary conditions compatible with the local dynamics are superior to essential boundary conditions which may give rise to artificial finite‐element solutions. The analysis accounts for inertia, viscous, and surface tension effects. Both centrifugal and Coriolis accelerations are included. Results are presented and compared for Newtonian, shear‐thinning and shear‐thickening, Bingham plastic, and second‐order viscoelastic liquids at different operating conditions. In the limiting case of a rotating disk, in spin coating, the effects of the Coriolis acceleration, which have been omitted in earlier analyses are investigated. Closed form asymptotic solutions agree with the finite‐element predictions at high rotation. At low rotation, the asymptotic solution breaks down and so the finite‐element solution becomes essential.

Tasos C. Papanastasiou

Papers

Flows of Materials with Yield

Rotating Thin Films in Bell Sprayers and Spin Coating