Rigid Cluster Decomposition Reveals Criticality in Frictional Jamming

Abstract

We study the nature of the frictional jamming transition within the framework of rigidity percolation theory. Slowly sheared frictional packings are decomposed into rigid clusters and floppy regions with a generalization of the pebble game including frictional contacts. We discover a second-order transition controlled by the emergence of a system-spanning rigid cluster accompanied by a critical cluster size distribution. Rigid clusters also correlate with common measures of rigidity. We contrast this result with frictionless jamming, where the rigid cluster size distribution is noncritical.The interplay of constraints, forces, and driving gives rise to the jamming transition in granular media. It is now wellestablished that the frictionless jamming transition has characteristics of both second-and first-order transitions. Both the average coordination number and the largest rigid cluster size jump at the transition, yet there exists a diverging lengthscale [1][2][3][4]. Frictional jamming is more puzzling: The hysteresis observed in the stress-strain rate curves of stresscontrolled flow simulations [5][6][7][8] and experiments [9] has lead to an interpretation as a first-order transition. Yet, signs of second-order criticality appear when treating the fraction of contacts at the Coulomb threshold as an additional parameter [10][11][12].To elucidate the frictional jamming transition from a microscopic viewpoint, we extend concepts and tools from rigidity percolation, i.e., the onset of mechanical rigidity in disordered spring networks [13][14][15][16], to frictional packings. The former is driven by the emergence of a system-spanning rigid cluster that can be mapped out (in 2d) using the pebble game [17], an improved constraint counting method that goes beyond meanfield by identifying redundant constraints. We, for the first time, implement a generalized pebble game for 2d frictional systems and use it to identify rigid clusters in very slowly sheared packings. As we show below, this allows us to identify a second-order rigidity transition and to link stresses and nonaffine motions to the microscopic structure of frictionally jammed packings.Generalized isostaticity: To establish context, we first review the application of Maxwell constraint counting to jamming [18]. For N particles in d dimensions and a mean number of contacts per particle z, interparticle forces yield dzN/2 constraints. Since each particle has 1 2 d(d + 1) translational and rotational degrees of freedom, there are 1 2 (N − 1)d(d + 1) total degrees of freedom (subtracting out global degrees of freedom). When these match the force constraints, we arrive at the isostatic criterion, or dzN/2 = 1 2 (N − 1)d(d + 1). In the limit N → ∞, z iso = d + 1 for frictional granular materials. For frictionless packings, we ignore rotations and obtain the familiar z iso = 2d.Despite being mean field, i.e. neglecting spatial correla- tions, isostaticity works seemingly well to locate the jamming transition in static frictionless systems [1]. For frictional syste...

Unifying Suspension and Granular flows near Jamming

“…For example, the relative velocity between contacting particles differs from the amplitude of non-affine velocities. This additional complexity is plausibly caused by the intermittent and non-extensive localization of strain observed in several studies [49][50][51][52]. A similar localization is observed in the plasticity of soft particles [53], whose connection to hard frictional particles is, however, unclear.…”

Partial jamming and non-locality in dense granular flows

“…More generally, correlated motion of particles [20][21][22], and avalanche-like rearrangements have been observed in various glassy materials like sand [23], granular materials [15,24,25], cohesive grains [26], foams [27], suspensions [8,28], colloids [29,30], and Lennard-Jones glass [31,32]. These observations suggest that clusters of particles may exist in many glassy materials.…”

Forecasting failure locations in 2-dimensional disordered lattices

“…We do a total of 14 experiments, which we test each network at least once in compression and once in tension; for the networks with z 0 = 2.60 and z 0 = 3.00, we do an additional tensile test on a second set of fully intact samples. To obtain a sample that is as close as possible to the isostatic value z iso = 3.00 of an infinite friction packing [55], we prune a network that initially has a value of z 0 = 3.60 by progressively removing its contacts with the smallest force values.…”