Strain versus Stress in a Model Granular Material: A Devil's Staircase

Abstract

The series of equilibrium states reached by disordered packings of rigid, frictionless discs in two dimensions, under gradually varying stress, are studied by numerical simulations. Statistical properties of trajectories in configuration space are found to be independent of specific assumptions ruling granular dynamics, and determined by geometry only. A monotonic increase in some macroscopic loading parameter causes a discrete sequence of rearrangements. For a biaxial compression, we show that, due to the statistical importance of such events of large magnitudes, the dependence of the resulting strain on stress direction is a Lévy flight in the thermodynamic limit. The mechanical properties of granular media are currently an active field of research, both in the condensed matter physics and in the mechanics and engineering communities [1, 2,3].Granular packings close to mechanical equilibrium are traditionnally modelled, in the framework of continuum mechanics, with elastoplastic constitutive laws [3,4], i.e., incremental stress-strain relations. Such laws, despite their practical success, were never clearly related to grain-level mechanics. Moreover, cohesionless granular systems seem to be quite different from ordinary solids. Many experimental, theoretical and numerical studies [2,5,6,7,8] have recently been devoted to the peculiar features of stress transmission in granular systems at equilibrium, with correlations over length scales significantly larger than the grain size.Observations of displacement fields and strains, as the system moves from one equilibrium to another, are scarcer. Systems of rigid grains are expected to deform because of rearrangements of the packing, rather than contact elasticity. How such rearrangements average to produce a macroscopic strain, related to stress variations, remains rather mysterious. The rather singular, unilateral form of the local interaction in such systems led some authors [9] to expect quite unusual macroscopic properties, for which the very concept of strain, so familiar in mechanics of solids, would be irrelevant.Direct grain-level approaches are, in principle, possible by numerical simulations. However, one has then to define a complete mechanical model to enable a calculation of particle trajectories. In practice, dynamical parameters ruling energy dissipation are often chosen according to computational convenience as much as physical accuracy. It would be desirable to assess the influence of such choices on the results.The present numerical study addresses those problems, as follows.Disordered, dense assemblies of rigid, circular, frictionless discs, are prepared by isotropic compaction. The force law reduces to the condition that contacts transmit repulsive normal forces of unknown magnitude. Then, the direction of the load is gradually altered, thus simulating the biaxial test of fig.

Unjamming of granular packings due to local perturbations: Stability and decay of displacements

“…Here we deal with rigid (undeformable) grains with help of the simulation technique of contact dynamics [10,11] therefore elastic deformations are excluded. As opposed to the weak perturbations, in our case, plastic deformation of the packing is initiated [3,4]. Plastic rearrangements caused by local perturbations have been studied in recent experiments [5] where interesting power law decay of the rearrangement field was found.…”

“…When the external load on a static assembly of grains is changed at a certain point the load may become incompatible with the inner structure of the packing and the solid state looses its stability. How exactly this happens on grain-scale and what are the key features of the transition between statics and flow are intriguing and unresolved problems [3,4,5,6]. Moreover, it is of essential importance in many applications to be able to predict, initiate or prevent such transitions.…”

Mechanical fluctuations suppress the threshold of soft-glassy solids: The secular drift scenario

“…Consequently, for infinitesimal deformations around a reference state, a granular packing should possess a true elastic response and displays no ageing [31]. Note however, in the limit of very small if not zero friction the establishment of a linear elastic response under finite shear is questionable [32]. Moreover, for real granular materials, the actual pressures at contact are generically high and contacts may creep plastically.…”

Marginal Stability Constrains Force and Pair Distributions at Random Close Packing

“…Despite these successes, the hypothesis of linear marginal stability yields an incomplete insight on the non-linear processes occurring in amorphous materials, which are critical to understand plasticity, thermal activation or granular flows [7]. When interactions are short-range one key source of non-linearity is the creation or destruction of contacts between particles [8,9]. Combe and Roux have observed numerically [8] that such rearrangements occur intermittently, in bursts or avalanches whose size is power-law distributed, a kind of dynamics referred to as crackling noise [10].…”

“…When interactions are short-range one key source of non-linearity is the creation or destruction of contacts between particles [8,9]. Combe and Roux have observed numerically [8] that such rearrangements occur intermittently, in bursts or avalanches whose size is power-law distributed, a kind of dynamics referred to as crackling noise [10].Interestingly some glassy systems with long-range interactions display such dynamics, in particular Coulomb glasses [11] and mean-field spin glasses [12]. In both cases the requirement of stability toward discrete excitations (flipping two spins or moving one electron) leads to bounds on important physical quantities: Efros and Shklovskii showed that the density of states in a Coulomb glass must vanish at the Fermi energy [13], implying the presence of the so-called Coulomb gap.…”