Plasticity and geophysical flows: A review

Abstract

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.

Experimental investigation of the spreading of viscoplastic fluids on inclined planes

“…Geophysical flows offer many examples of finite volumes of plastic materials spreading on a slope. Typical examples include snow avalanches and debris flows [11,12] as well as volcanic lava [13].…”

Slumps of viscoplastic fluids on slopes

“…The dynamical regime under consideration corresponds to relatively thin flows ( 1), so that the pressure is hydrostatic to leading order, and to situations with negligible inertia ( 2 Re 1) so the motion is dynamically controlled by a balance of downslope acceleration and streamwise pressure gradients with the divergence of viscous stresses. (This balance yields the velocity scale, U, introduced above.)…”

“…Many materials such as muds, volcanic lava, concrete and various food stuffs exhibit mechanical features typical of viscoplastic fluids (see [1] for a recent comprehensive review). These materials only flow once a yield stress has been exceeded and exhibit a non-linear dependence of shear stress upon the rate of strain.…”

“…For instance, concentrated particulate suspensions show evidence of yielding as the configuration of the particles changes upon the action of shear and the inter-particle contacts are rearranged, but currently there is not a widely accepted mathematical model to link this micro-mechanics to the bulk motion. The concept of a yield stress, therefore, is often interpreted as an extrapolation of measured shear stresses to the case of vanishing rates of strain [1].…”

“…Free surface flows of viscoplastic materials have wide application to geophysical and industrial situations [1][2][3] and in recent years considerable progress has been made in developing consistent mathematical models of their motion when the flows are relatively shallow and predominantly parallel to the underlying boundary [4]. Viscoplasticity affects these flows in important ways: first the shear stresses developed within the flow diminish as the free surface is approached and so fluid close to the surface must exhibit 'unyielded' behaviour.…”