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Quasi-static rheology

The term “quasi-static” refers to slow deformations that can be thought of as a succession of statically stable configurations of grains. Many intermediate states are, however, unstable or metastable. For this reason, the concept of quasi-static deformation in granular materials should be considered as an approximation. Macroscopically, we may consider a deformation path to be quasi-static when the relaxation time under load (return time to an equilibrium state after a perturbation, which depends on particle mass) is small compared to the inverse of deformation rate. The ratio of the latter to the relaxation time is what has come to be known as inertial number.

Sheared granular material with bi-periodic boundary conditions.

The quasi-static behavior (or rheology) of granular materials was mainly developed in soil mechanics. The Mohr-Coulomb plastic behavior is and remains at the heart of soil mechanics, which has been primarily interested in predicting soil failure. In the 10th Rankine Lecture in 1970, Roscoe emphasized the need to understand the stress-strain behavior of soils well before failure under complex loading conditions as encountered in engineering practice (Roscoe 1970). Fully aware of the need for a fundamental approach, he indicated the route towards a fundamental understanding of soils by working “with soils in their simplest possible states (e.g., well graded sands and saturated remolded clays) so that their properties can be defined by the minimum possible number of parameters,” preparing “soil samples in initially uniform states,” developing test equipment and “nondestructive methods of checking the uniformity of the behavior of the soils at all stages of these tests,” and developing “scanning electron microscopy methods of studying the change of soil fabric during mechanical deformation.” This is the route followed during the coming decades.

A consistent model of soil plasticity was actually achieved through the critical-state soil mechanics of the Cambridge School (Roscoe et al. 1958; Schofield and Wroth 1968; Wood 1990). By accounting for both frictional and volume-change behaviors of soils and, more importantly, by recognizing a family of memoryless states reached after sufficiently long shearing, it provided for the first time a general framework for the quasi-static behavior of granular soils and clays. The critical state theory is the core of nearly all constitutive models that were developed later to account for complex loadings paths. Critical states are asymptotic states approached for large enough strains, applied monotonically and quasi-statically. Constitutive laws predicting this gradual approach are traditionally elastoplastic in nature. The elastic properties of granular materials have been clarified, over the past 25 years, thanks to improved experimental techniques apt to measure very small strain intervals (Shibuya et al. 1992; Hicher 1996). Elastic waves (Goddard 1990) are actually the reflection of the quasi-static elastic behavior of small-amplitude perturbations about an equilibrium state of a solid granular assembly (Thomann and Hryciw 1990; Shibuya et al. 1992; Geoffroy et al. 2003). Due to nonlinear contact elasticity, their velocity depends on the stress level. Although grain-scale disorder induces incoherent propagation, larger wavelengths propagate coherently (Liu and Nagel 1992; Liu 1994; Liu and Nagel 1994; Jia et al. 1999).

Micromechanical approaches to the plastic behaviors (Christoffersen et al. 1981; Bathurst and Rothenburg 1988; Chang and Misra 1990) and elastic (Walton 1987; La Ragione and Jenkins 2007) of granular materials have been an active area from the outset of modern research on granular materials. The onset of instabilities in granular materials as a function of the loading program has also been a subject of extensive investigation under homogeneous boundary conditions (Vardoulakis 1979; Lade 1994, 2002; Vardoulakis and Sulem 1995; Nova 1994; Darve and Laouafa 1999; Nicot and Darve 2007; Chang et al. 2011). Directional loading reveals that material instability can occur in a diffuse form or be localized in a shear band. Imaging techniques have been used to analyze shear bands in which plastic deformations are fully developed (Desrues et al. 1985, 1996).

See my review paper for further details following this link: Modeling Granular Materials: Century-Long Research across Scales.

The numerical simulations based on the so-called Discrete-Element Method even in two dimensions and with simple disk-like particles interacting through friction contacts yield quite realistic behaviors of collections of such particles. See, for instance, my paper on Contact Dynamics Study of 2D Granular Media: Critical States and Relevant Internal Variables. Later simulations in 2D and 3D focus on the influence of material parameters (particle properties and their interactions) on the quasi-static behavior. Most of the focus has, however, been on the steady states (critical state). See for example the collective paper Particle shape dependence in 2D granular media and the paper on Packings of irregular polyhedral particles: strength, structure, and effects of angularity.