The contact percolation transition

Physical Review E, 2012

Typical quasistatic compression algorithms for generating jammed packings of purely repulsive, frictionless particles begin with dilute configurations and then apply successive compressions with the relaxation of the elastic energy allowed between each compression step. It is well known that during isotropic compression these systems undergo a first-order-like jamming transition at packing fraction φ J from an unjammed state with zero pressure and no force-bearing contacts to a jammed, rigid state with nonzero pressure, a percolating network of force-bearing contacts, and contact number z = 2d, where d is the spatial dimension. Using computer simulations of two-dimensional systems with monodisperse and bidisperse particle size distributions, we investigate the second-order-like contact percolation transition, which precedes the jamming transition with φ P < φ J and signals the formation of a system-spanning cluster of non-force-bearing contacts between particles. By measuring the number of nonfloppy modes of the dynamical matrix, the displacement field between successive compression steps, and the overlap between the adjacency matrix, which represents the network of contacting grains, at φ and φ J , we find that the contact percolation transition also signals the onset of a nontrivial mechanical response to applied stress. Our results show that cooperative particle motion occurs in unjammed systems significantly below the jamming transition for φ P < φ < φ J , not only for jammed systems with φ > φ J .

Physical review, 2018

We perform computational studies of repulsive, frictionless disks to investigate the development of stress anisotropy in mechanically stable (MS) packings at jamming onset. We focus on two protocols for generating MS packings at jamming onset: 1) isotropic compression and 2) applied simple or pure shear strain γ at fixed packing fraction φ. MS packings of frictionless disks occur as geometric families (i.e. quasi-parabolic segments with positive curvature) in the φ-γ plane. MS packings from protocol 1 populate parabolic segments with both signs of the slope, dφ/dγ > 0 and dφ/dγ < 0. In contrast, MS packings from protocol 2 populate segments with dφ/dγ < 0 only. For both simple and pure shear, we derive a relationship between the stress anisotropy and local dilatancy dφ/dγ obeyed by MS packings along geometrical families. We show that for MS packings prepared using isotropic compression, the stress anisotropy distribution is Gaussian centered at zero with a standard deviation that decreases with increasing system size. For shear jammed MS packings, the stress anisotropy distribution is a convolution of Weibull distributions that depend on strain, which has a nonzero average and standard deviation in the large-system limit. We also develop a framework to calculate the stress anisotropy distribution for packings generated via protocol 2 in terms of the stress anisotropy distribution for packings generated via protocol 1.

Physical Review Letters, 2008

We numerically study the jamming transition in particulate systems with attraction by investigating their mechanical response at zero temperature. We find three regimes of mechanical behavior separated by two critical transitions-connectivity and rigidity percolation. The transitions belong to different universality classes than their lattice counterparts, due to force balance constraints. We also find that these transitions are unchanged at low temperatures and resemble gelation transitions in experiments on colloidal and silica gels.

Physical Review E, 2013

We perform numerical simulations of repulsive, frictionless athermal disks in two and three spatial dimensions undergoing cyclic quasistatic simple shear to investigate particle-scale reversible motion. We identify three classes of steady-state dynamics as a function of packing fraction φ and maximum strain amplitude per cycle γ max. Point-reversible states, where particles do not collide and exactly retrace their intracycle trajectories, occur at low φ and γ max. Particles in loop-reversible states undergo numerous collisions and execute complex trajectories but return to their initial positions at the end of each cycle. For sufficiently large φ and γ max , systems display irreversible dynamics with nonzero self-diffusion. Loop-reversible dynamics enables the reliable preparation of configurations with specified structural and mechanical properties over a broad range of φ.

Soft Matter, 2019

We carry out numerical studies of static packings of frictionless superellipsoidal particles in three spatial dimensions. We consider more than 200 different particle shapes by varying the three shape parameters...

Soft Matter, 2020

We investigate the mechanical response of packings of purely repulsive, frictionless disks to quasistatic deformations. The deformations include simple shear strain at constant packing fraction and at constant pressure, "polydispersity" strain (in which we change the particle size distribution) at constant packing fraction and at constant pressure, and isotropic compression. For each deformation, we show that there are two classes of changes in the interparticle contact networks: jump changes and point changes. Jump changes occur when a contact network becomes mechanically unstable, particles "rearrange", and the potential energy (when the strain is applied at constant packing fraction) or enthalpy (when the strain is applied at constant pressure) and all derivatives are discontinuous. During point changes, a single contact is either added to or removed from the contact network. For repulsive linear spring interactions, second-and higher-order derivatives of the potential energy/enthalpy are discontinuous at a point change, while for Hertzian interactions, third-and higher-order derivatives of the potential energy/enthalpy are discontinuous. We illustrate the importance of point changes by studying the transition from a hexagonal crystal to a disordered crystal induced by applying polydispersity strain. During this transition, the system only undergoes point changes, with no jump changes. We emphasize that one must understand point changes, as well as jump changes, to predict the mechanical properties of jammed packings. c = dN -d + 1 (for systems with periodic boundary conditions), N is the number of (non-rattler ) grains, and d = 2, 3 is the spatial dimension. Ordered or compressed jammed packings can be hyperstatic with N c ≥ N iso c . Each jammed packing exists in a local energy minimum in configuration space, and therefore

Physical Review Letters, 2013

Discontinuous shear thickening (DST) observed in many dense athermal suspensions has proven difficult to understand and to reproduce by numerical simulation. By introducing a numerical scheme including both relevant hydrodynamic interactions and granularlike contacts, we show that contact friction is essential for having DST. Above a critical volume fraction, we observe the existence of two states: a low viscosity, contactless (hence, frictionless) state, and a high viscosity frictional shear jammed state. These two states are separated by a critical shear stress, associated with a critical shear rate where DST occurs. The shear jammed state is reminiscent of the jamming phase of granular matter. Continuous shear thickening is seen as a lower volume fraction vestige of the jamming transition.

Physical review, 2017

We focus on the response of mechanically stable (MS) packings of frictionless, bidisperse disks to thermal fluctuations, with the aim of quantifying how nonlinearities affect system properties at finite temperature. In contrast, numerous prior studies characterized the structural and mechanical properties of MS packings of frictionless spherical particles at zero temperature. Packings of disks with purely repulsive contact interactions possess two main types of nonlinearities, one from the form of the interaction potential (e.g. either linear or Hertzian spring interactions) and one from the breaking (or forming) of interparticle contacts. To identify the temperature regime at which the contact-breaking nonlinearities begin to contribute, we first calculated the minimum temperatures T cb required to break a single contact in the MS packing for both single and multiple eigenmode perturbations of the T = 0 MS packing. We find that the temperature required to break a single contact for equal velocity-amplitude perturbations involving all eigenmodes approaches the minimum value obtained for a perturbation in the direction connecting disk pairs with the smallest overlap. We then studied deviations in the constant volume specific heat CV and deviations of the average disk positions ∆r from their T = 0 values in the temperature regime T cb < T < Tr, where Tr is the temperature beyond which the system samples the basin of a new MS packing. We find that the deviation in the specific heat per particle ∆C 0 V /C 0 V relative to the zero temperature value C 0 V can grow rapidly above T cb , however, the deviation ∆C 0 V /C 0 V decreases as N −1 with increasing system size. To characterize the relative strength of contact-breaking versus form nonlinearities, we measured the ratio of the average position deviations ∆r ss /∆r ds for single-and double-sided linear and nonlinear spring interactions. We find that ∆r ss /∆r ds > 100 for linear spring interactions is independent of system size. This result emphasizes that contact-breaking nonlinearities are dominant over form nonlinearities in the low temperature range T cb < T < Tr for model jammed systems.

Soft Matter, 2019

We carry out numerical studies of static packings of frictionless superellipsoidal particles in three spatial dimensions. We consider more than 200 different particle shapes by varying the three shape parameters that define superellipsoids. We characterize the structural and mechanical properties of both disordered and ordered packings using two packing-generation protocols. We perform athermal quasi-static compression simulations starting from either random, dilute configurations (Protocol 1) or thermalized, dense configurations (Protocol 2), which allows us to tune the orientational order of the packings. In general, we find that superellipsoid packings are hypostatic, with coordination number z J < z iso , where z iso = 2d f and d f = 5 or 6 depending on whether the particles are axisymmetric or not. Over the full range of orientational order, we find that the number of quartic modes of the dynamical matrix for the packings always matches the number of missing contacts relative to the isostatic value. This result suggests that there are no mechanically redundant contacts for ordered, yet hypostatic packings of superellipsoidal particles. Additionally, we find that the packing fraction at jamming onset for disordered packings of superellipsoidal depends on at least two particle shape parameters, e.g. the asphericity and reduced aspect ratio β of the particles.

arXiv (Cornell University), 2023

Jammed packings of granular materials display complex mechanical response. For example, the ensemble-averaged shear modulus ⟨G⟩ increases as a power-law in pressure p for static packings of soft spherical particles that can rearrange during compression. We seek to design granular materials with shear moduli that can either increase or decrease with pressure without particle rearrangements even in the large-system limit. To do this, we construct tessellated granular metamaterials by joining multiple particle-filled cells together. We focus on cells that contain a small number of bidisperse disks in two dimensions. We first study the mechanical properties of individual disk-filled cells with three types of boundaries: periodic boundary conditions (PBC), fixed-length walls (FXW), and flexible walls (FLW). Hypostatic jammed packings are found for cells with FLW, but not in cells with PBC and FXW, and they are stabilized by quartic modes of the dynamical matrix. The shear modulus of a single cell depends linearly on p. We find that the slope of the shear modulus with pressure, λc < 0 for all packings in single cells with PBC where the number of particles per cell N ≥ 6. In contrast, single cells with FXW and FLW can possess λc > 0, as well as λc < 0, for N ≤ 16. We show that we can force the mechanical properties of multi-cell granular metamaterials to possess those of single cells by constraining the endpoints of the outer walls and enforcing an affine shear response. These studies demonstrate that tessellated granular metamaterials provide a novel platform for the design of soft materials with specified mechanical properties.