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

The “no-slip” is a fundamental assumption and generally-accepted boundary condition in rheology, tribology, and fluid mechanics with strong experimental support. The violations of this condition, however, are widely recognized in many situations, especially in the flow of non-Newtonian fluids. Wall slip could lead to large errors and flow instabilities, such as sharkskin formation and spurt flow, and hence complicates the analysis of fluid systems and introduces serious practical difficulties. In this article, we discuss slip at fluid-solid interface in an attempt to highlight the main issues related to this diverse complex phenomenon and its implications.

Slip at Fluid-Solid Interface

Bedload, the transport of sediment remaining in contact with the stream bed, has mainly been studied from the perspective of the correlation between fluid driving forces and the responding sediment flux. Yet grain–grain interactions are important and bedload should also be considered as a granular phenomenon. We review progress made recently in the study of granular flows, especially on segregation and rheology, that better illuminates the nature of bedload. Granular flows may exhibit gas-like or fluid-like flow, or quasi-solid deformation. All three conditions might be duplicated in bedload. Understanding of intense bedload transport occurring continuously in a layer several grains deep – typical of sand beds – might greatly benefit from results in granular physics, as illustrated by grain-inspired bedload results. However, processes restricted to the surface of the bed, when particles move intermittently and the bed becomes structured, while characteristic in gravel-bed channels, are not well addressed in granular physics. Mutual study of these phenomena may benefit both physics and fluvial geomorphology. We intend, therefore, to contribute to an enhanced dialogue between granular physics and bedload science communities. Copyright © 2010 John Wiley & Sons, Ltd.

Bedload: a granular phenomenon

We focus our attention on granular suspensions made up of noncolloidal spherical particles within a Newtonian fluid. The main objective of this paper is to provide a general framework for the formulation of the bulk stress tensor. The bulk stress within granular suspensions is mainly generated at the particle level by strong interactions between particles, such as friction, collision, and lubricated contact. The existence of a few local mechanisms is a major cause of behavior complexity at the macroscopic scale. A direct consequence is that the constitutive equation is only known for some flow conditions and given types of mixture. Here we have used a microstructural approach, which consists of considering the mixture as an effective continuum at the macroscopic level and inferring the bulk stress tensor from averaging of local interactions and local stresses. The bulk stress tensor may be split into elementary contributions pertaining to particle interactions. A complementary equation standing for the bu...

A theoretical framework for granular suspensions in a steady simple shear flow

In the first part of this essay we investigate the thermomechanical behaviour of carbon black-filled rubber under dynamic loading conditions. The loadings consist of static predeformations which are superimposed by small sinusoidal oscillations. The frequency was varied between 0.1 Hz and 100 Hz, the deformation amplitude between 0.006 and 0.06, the temperature between 253 K and 373 K and the storage and dissipation moduli measured. The data show that the frequency dependence of the moduli is of the powerlaw type. They also depend on the deformation amplitude and the temperature. If the temperature is constant, increasing amplitudes lead to decreasing moduli. As discussed by Payne (1965), this behaviour can be interpreted in terms of a thixotropic change. If, on the other hand, the deformation amplitude is kept constant, increasing temperature levels lead to decreasing moduli. In the second part of this work we develop a constitutive theory to represent the material behaviour observed. We consider the dissipation principle of thermodynamics and formulate the model for three-dimensional finite deformations. To describe thermal expansion effects, we decompose the deformation gradient into a thermal and a mechanical part as proposed by Lu and Pister (1975). We prescribe the thermal part by a constitutive function and assume the mechanical part to be the driving force for the stress tensor. As motivated in earlier works we split the total stress into an equilibrium stress and a rate-dependent overstress. We represent the equilibrium stress using a modified Mooney-Rivlin strain energy function and the overstress by a series of Maxwell elements whose springs are of Neo-Hookean type. Both the kinematic tensors and the associated stress measures are defined by using the concept of dual variables proposed by Haupt and Tsakmakis (1989). In order to represent the thixotropic effects, the viscosities depend on the temperature and the deformation history. The history dependence is implied by an internal variable which is a measure for the deformation amplitude and has a relaxation property as discussed by Payne (1965). To determine the material parameters, we linearise the constitutive equations with respect to the static predeformation and take the analytical solution for harmonic strain-controlled loadings into account. Numerical simulations demonstrate that the constitutive theory describes all phenomena experimentally observed with a fairly good approximation. The model is compatible with the second law of thermodynamics in the form of the Clausius Duhem inequality.

Thixotropic behaviour of rubber under dynamic loading histories: Experiments and theory

A simple constitutive framework is used to review some phenomenological models for solid-fluid mixtures. Several successful descriptions of viscoplasticity assume that stress is proportional to a normalized deformation rate. The modulus in this constitutive equation evolves with time for thixotropic materials. This approach is used in most models based on scalar measures of structure. Such measures are determined by an evolution equation that is insensitive to rate reversals during shear flow. This behavior appears to be characteristic of inelastic materials such as an aqueous solution of bentonite. However, the same solution with guargum does not exhibit this response due to elasticity induced by the polymer. Models for granular media extend this constitutive framework by including compressibility effects and a scalar measure of particle interactions. The effect of fluid viscosity on particle interactions is incorporated using a mixture theory approach. Pipe flow of viscoplastic materials and shear flow of granular media are analyzed using boundary conditions that allow slip at solid surfaces.

A. Berker

Papers

Phenomenological models of viscoplastic, thixotropic, and granular materials