Isostatic Phase Transition and Instability in Stiff Granular Materials

Abstract

Structural rigidity concepts are used to understand the origin of instabilities in granular aggregates. It is first demonstrated that the contact network of a noncohesive granular aggregate becomes exactly isostatic when I = kǫ/f l >> 1, where k is stiffness, ǫ is the typical interparticle gap and fL is the typical stress induced by loads. Thus random packings of stiff particles are typically isostatic. Furthermore isostaticity is responsible for the anomalously large susceptibility to perturbation observed in granular aggregates. The load-stress response function of granular piles is critical (power-law distributed) in the isostatic limit, which means that slight overloads will produce internal rearrangements.Photoelastic visualization experiments [1][2][3] show clearly defined stress-concentration paths in non-cohesive granular materials under applied load. These often suffer sudden rearrangement on a global scale when the load conditions are slightly changed, evidencing a degree of susceptibility to perturbation not usually present in elastic materials. It is rather possible that this intrinsic instability be responsible for much of the interesting phenomenology of granular materials [3,4]. Recently a number of phenomenological models [2,5-8] have been put forward, which succeed to reproduce several aspects of stress propagation in granular systems, and the issue of instability has been addressed by noting that the loadstress response function may take negative values [6]. It is the purpose of this letter to show that structural rigidity concepts help us understanding the origin of instability in granular materials, linking it to the topological properties of the system's contact network.Structural rigidity [9] studies the conditions that a network of points connected by rotatable bars (representing central forces) has to fulfill in order to sustain applied loads. A network with too few bars is flexible, while if it has the minimum number required to be rigid it is isostatic. Networks with bars in excess of minimal rigidity are overconstrained, and are in general self-stressed. Concepts from structural rigidity were first introduced in the study of granular media by Guyon et al [10], who stressed that granular systems are not entirely equivalent to linear elastic networks since in the former only compressive interparticle forces are possible. We next show that this constraint has far-reaching consequences for the static behavior of stiff granular aggregates.Consider a d-dimensional frictionless granular pile in equilibrium under the action of external forces F i (gravitational, etc) on its particles. Imagine building an equivalent linear-elastic central-force network (the contact network ), in which two sites are connected by a bond if and only if there is a nonzero compression force between the two corresponding particles. Because of linearity, stresses f ij on the bonds of this equivalent system can be decomposed as f ij = f self ij + f load ij where f self ij are self-stresses, and f load ij are load-depen...

Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

“…Numerical simulations and theoretical work suggest that at the jamming transition the system becomes exactly isostatic [29,31,27,34,61,33,60]. But no rigorous proof of this statement exists.…”

“…In practice, it is widely believed that the isostatic condition is necessary for a jammed disordered packing following the Alexander conjecture [59,60,61] which was tested in several works [27,29,30,33] It is well known that mechanical equilibrium imposes an average coordination number larger or equal than a minimum coordination where the number of force variables equals the number of force and torque balance equations [59,61,60]. The so-called isostatic condition.…”

The jamming scenario—an introduction and outlook

“…The answer is related to the fact that at the jamming threshold, packings of frictionless discs and spheres are isostatic, i.e., they are marginal solids that can just maintain their stability [5,[12][13][14][15][16][17]. The origin of this is the following.…”

Landscape analysis of constraint satisfaction problems

“…The study of problems with disorder raises doubts on the fact that isostaticity of a frictionless particle system is a necessary (apart from sufficient [14,54], although see [66]) condition for criticality. This may be checked just by testing the J-point procedure with a packing of non-spherical (e.g.…”