Mechanical response of packings of nonspherical particles: A case study of two-dimensional packings of circulo-lines.

We investigate the mechanical response of jammed packings of circulo-lines in two spatial dimensions, interacting via purely repulsive, linear spring forces, as a function of pressure P during athermal, quasistatic isotropic compression. The surface of a circulo-line is defined as the collection of points that is equidistant to a line; circulo-lines are composed of a rectangular central shaft with two semicircular end caps. Prior work has shown that the ensemble-averaged shear modulus for jammed disk packings scales as a power law, 〈G(P)〉∼P^{ß}, with ß∼0.5, over a wide range of pressure. For packings of circulo-lines, we also find robust power-law scaling of 〈G(P)〉 over the same range of pressure for aspect ratios R≳1.2. However, the power-law scaling exponent ß∼0.8-0.9 is much larger than that for jammed disk packings. To understand the origin of this behavior, we decompose 〈G〉 into separate contributions from geometrical families, G_{f}, and from changes in the interparticle contact network, G_{r}, such that 〈G〉=〈G_{f}〉+〈G_{r}〉. We show that the shear modulus for low-pressure geometrical families for jammed packings of circulo-lines can both increase and decrease with pressure, whereas the shear modulus for low-pressure geometrical families for jammed disk packings only decreases with pressure. For this reason, the geometrical family contribution 〈G_{f}〉 is much larger for jammed packings of circulo-lines than for jammed disk packings at finite pressure, causing the increase in the power-law scaling exponent for 〈G(P)〉.

Contact network changes in ordered and disordered disk packings.

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.

Pressure Dependent Shear Response of Jammed Packings of Frictionless Spherical Particles.

The mechanical response of packings of purely repulsive, spherical particles to athermal, quasistatic simple shear near jamming onset is highly nonlinear. Previous studies have shown that, at small pressure p, the ensemble-averaged static shear modulus ⟨G-G_{0}⟩ scales with p^{α}, where α≈1, but above a characteristic pressure p^{**}, ⟨G-G_{0}⟩∼p^{ß}, where ß≈0.5. However, we find that the shear modulus G^{i} for an individual packing typically decreases linearly with p along a geometrical family where the contact network does not change. We resolve this discrepancy by showing that, while the shear modulus does decrease linearly within geometrical families, ⟨G⟩ also depends on a contribution from discontinuous jumps in ⟨G⟩ that occur at the transitions between geometrical families. For p>p^{**}, geometrical-family and rearrangement contributions to ⟨G⟩ are of opposite signs and remain comparable for all system sizes. ⟨G⟩ can be described by a scaling function that smoothly transitions between two power-law exponents α and ß. We also demonstrate the phenomenon of compression unjamming, where a jammed packing unjams via isotropic compression.

Jammed packings of 3D superellipsoids with tunable packing fraction, coordination number, and ordering.

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 zJ < ziso, where ziso = 2df and df = 5 or 6 depending on whether the particles are axi-symmetric 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 A and reduced aspect ratio ß of the particles.

Hypostatic jammed packings of frictionless nonspherical particles.

We perform computational studies of static packings of a variety of nonspherical particles including circulo-lines, circulo-polygons, ellipses, asymmetric dimers, dumbbells, and others to determine which shapes form packings with fewer contacts than degrees of freedom (hypostatic packings) and which have equal numbers of contacts and degrees of freedom (isostatic packings), and to understand why hypostatic packings of nonspherical particles can be mechanically stable despite having fewer contacts than that predicted from naive constraint counting. To generate highly accurate force- and torque-balanced packings of circulo-lines and cir-polygons, we developed an interparticle potential that gives continuous forces and torques as a function of the particle coordinates. We show that the packing fraction and coordination number at jamming onset obey a masterlike form for all of the nonspherical particle packings we studied when plotted versus the particle asphericity A, which is proportional to the ratio of the squared perimeter to the area of the particle. Further, the eigenvalue spectra of the dynamical matrix for packings of different particle shapes collapse when plotted at the same A. For hypostatic packings of nonspherical particles, we verify that the number of "quartic" modes along which the potential energy increases as the fourth power of the perturbation amplitude matches the number of missing contacts relative to the isostatic value. We show that the fourth derivatives of the total potential energy in the directions of the quartic modes remain nonzero as the pressure of the packings is decreased to zero. In addition, we calculate the principal curvatures of the inequality constraints for each contact in circulo-line packings and identify specific types of contacts with inequality constraints that possess convex curvature. These contacts can constrain multiple degrees of freedom and allow hypostatic packings of nonspherical particles to be mechanically stable.