Disk Packings Research Articles

Dynamical arrest of topological defects in 2D hyperuniform disk packings

We investigate collective motions of points in 2D systems, orchestrated by Lloyd algorithm. The algorithm iteratively updates a system by minimising the total quantizer energy of the Voronoi landscape of the system. As a result of a tradeoff between energy minimisation and geometric frustration, we find that optimised systems exhibit a defective landscape along the process, where strands of 5- and 7-coordinated dislocations are embedded in the hexatic phase. In particular, dipole defects, each of which is the simplest possible pair of a pentagon and a heptagon, come into the picture of dynamical arrest, as the system freezes down to a disordered hyperuniform state. Moreover, we explore the packing fractions of 2D disk packings associated to the obtained hyperuniform systems by considering the maximum inscribed disks in their Voronoi cells.

Stress propagation in locally loaded packings of disks and pentagons

The mechanical strength and flow of granular materials can depend strongly on the shapes of individual grains. We report quantitative results obtained from photoelasticimetry experiments on locally loaded, quasi-two-dimensional granular packings of either disks or pentagons exhibiting stick-slip dynamics. Packings of pentagons resist the intruder at significantly lower packing fractions than packings of disks, transmitting stresses from the intruder to the boundaries over a smaller spatial extent. Moreover, packings of pentagons feature significantly fewer back-bending force chains than packings of disks. Data obtained on the forward spatial extent of stresses and back-bending force chains collapse when the packing fraction is rescaled according to the packing fraction of steady state open channel formation, though data on intruder forces and dynamics do not collapse. We comment on the influence of system size on these findings and highlight connections with the dynamics of the disks and pentagons during slip events.

Rotational diffusion and rotational correlations in frictional amorphous disk packings under shear.

We show here that rotations of round particles in amorphous disk packing reveal various nontrivial microscopic features when the packing is close to rigidification. We analyze experimental measurements on disk packing subjected to simple shear deformation with various inter-particle friction coefficients and across a range of volume fractions where the system is known to stiffen. The analysis of measurements indicates that shear induces diffusive microrotation, that can be both enhanced and suppressed depending upon the volume fraction as well as the inter-particle friction. Rotations also display persistent anticorrelated motion. Spatial correlations in microrotation are observed to be directly correlated with system pressure. These observations point towards the broader mechanical relevance of collective dynamics in the rotational degree of freedom of particles.

Strain dependent vorticity in sheared granular media

Displacement fields in sheared particle packings often display vortex-like structures that reveal essential features about the mechanical state of the collection of particles. There are several metrics to quantify these flow field features, yet extracting such quantitative metrics from flow field or particle tracking data involves making numerous choices on the time and length scales over which to average. Here we employ a much used experimental data set on sheared disk packings to explore how such arbitrary data mining choices affect the obtained results. We focus on calculating the strain dependent vorticity, as this metric is a differential method hence potentially sensitive to the way it is computed. We find that the total surface area with an absolute vorticity above a certain threshold approaches a plateau value as shear progresses. This plateau value exhibits a non-monotonic dependence on packing fraction. We also show which range of choices yields results that can support an analysis method independent, physical interpretation of the flow field data.

Sheared Amorphous Packings Display Two Separate Particle Transport Mechanisms.

Shearing granular materials induces nonaffine displacements. Such nonaffine displacements have been studied extensively, and are known to correlate with plasticity and other mechanical features of amorphous packings. A well known example is shear transformation zones as captured by the local deviation from affine deformation, D_{min}^{2}, and their relevance to failure and stress fluctuations. We analyze sheared frictional athermal disc packings and show that there exists at least one additional mesoscopic transport mechanism that superimposes itself on top of local diffusive motion. We evidence this second transport mechanism in a homogeneous system via a diffusion tensor analysis and show that the trace of the diffusion tensor equals the classic D_{min}^{2} when this second mesoscopic transport is corrected for. The new transport mechanism is consistently observed over a wide range of volume fractions and even for particles with different friction coefficients and is consistently observed also upon shear reversal, hinting at its relevance for memory effects.

Packing Disks by Flipping and Flowing

We provide a new type of proof for the Koebe–Andreev–Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph G, one can remove any flippable edge $$e^-$$ of this graph and then continuously flow the disks in the plane, so that at the end of the flow, one obtains a new disk packing whose contact graph is the graph resulting from flipping the edge $$e^-$$ in G. This flow is parameterized by a single inversive distance.

Elasticity of jammed packings of sticky disks

Numerous soft materials jam into an amorphous solid at high packing fraction. This non-equilibrium phase transition is best understood in the context of a model system in which particles repel elastically when they overlap. Recently, however, it was shown that introducing any finite amount of attraction between particles changes the universality class of the transition. The properties of this new ``sticky jamming'' class remain almost entirely unexplored. We use molecular dynamics simulations and scaling analysis to determine the shear modulus, bulk modulus, and coordination of marginal solids close to the sticky jamming point. In each case, the behavior of the system departs sharply and qualitatively from the purely repulsive case.

Governing equations for stress distribution in rhombic disk packings

The governing equations of stress tensor fields are significant to understand the mechanical behaviors of static granular matter. A multiscale method based on the algebra of translation operators is proposed to derive the governing equations of stress transmission in rhombic lattice packings of disks with friction. Except for the two well-known equations, i.e. the symmetry of stress tensor and the stress equilibrium equation, we also obtain a stress-structure equation as the constitutive relation of the lattice packings. A key variable called friction-geometry tensor is introduced for the expression of the constitutive equation and the Janssen coefficient is given from the eigenvalues of this tensor. DEM (Discrete Element Method) simulations are performed to calculate the numerical values of Janssen coefficient and the results are in good agreement well with theoretical predictions.

Rotational Transport via Spontaneous Symmetry Breaking in Vibrated Disk Packings.

It is shown that vibrated packings of frictional disks self-organize cooperatively onto a rotational-transport state where the long-time angular velocity ω[over ¯]_{i} of each disk i is nonzero. Steady rotation is mediated by the spontaneous breaking of local reflection symmetry, arising when the cages in which disks are constrained by their neighbors acquire quenched disorder at large packing densities. Experiments and numerical simulation of this unexpected phenomenon show excellent agreement with each other, revealing two rotational phases as a function of excitation intensity, respectively, the low-drive (LDR) and the moderate-drive (MDR) regimes. In the LDR, interdisk contacts are persistent and rotation happens due to frictional sliding. In the MDR, disks bounce against each other, still forming a solid phase. In the LDR, simple energy-dissipation arguments are provided, that support the observed dependence of the typical rotational velocity on excitation strength.

Calculating loads and life-time reduction of wind turbine gearbox and generator bearings due to shaft misalignment

During the last two decades, wind turbine industries have faced high failure rates, downtimes and costly repairs. Gearbox and generator have contributed to this, especially, because their high speed shaft bearings have often failed. In this article, an analytical method is proposed to calculate the reaction loads of flexible connecting couplings installed between wind turbine gearbox and generator. Raction loads are determined from joint kinematics and metal disk pack deformations as well as axial and angular shaft misalignment. The calculations are executed for both flexible connecting couplings and a universal joint shaft and applied to the gearbox high speed shaft. The performance of flexible connecting couplings and universal joint shaft is compared with respect to the bearing loads and life-time of the gearbox high speed shaft. It is shown that the early, unplanned bearing failures of gearbox and generator high speed shaft can often be attributed to the flexible connecting couplings installed between gearbox and generator.

Fast and efficient particle packing algorithms based on triangular mesh

The point of departure for the particle-based discrete element simulations is the generation of disk or sphere packing of interest domain. In this paper, based on a triangular mesh, two geometric packing algorithms are proposed for 2D problems. New disk(s) are obtained element-wise by solving system of equations, which can be easily derived according to the local topological relationships. In this way, the adjacent disks are guaranteed to be in contact with each other exactly. Related issues are discussed in detail, such as boundary treatment for good definition of the boundaries, elimination of incorrect overlaps, random perturbation for avoiding over-homogeneous packing. The main advantages of these two algorithms are the high efficiency and applicability for arbitrary domains. Furthermore, with a refined mesh, a refined packing can be easily obtained, which is a potential alternative for multiscale simulations. The proposed methods are verified by several examples.

A geometric separation method for non-uniform disk packing with prescribed filling ratio and size distribution

This paper proposes an approach to obtain a set of particles with a prescribed filling ratio and size distribution, using only geometric procedures. An essential step in the analysis of discontinuous media using the discrete element method is to obtain an initial set of particles that represents the granular model. Various particle packing approaches are currently employed to achieve a numerical model similar to a real discrete medium. In this context, this paper proposes an alternative particle packing method using the geometric separation procedure. This strategy aims to generate granular models that present particles with desired sizes, without overlapping and that fill a prescribed area of the domain. The method consists of five macro-steps: (a) definition of the initial set of particles; (b) geometric separation; (c) particle reallocation; (d) particle removing, and (e) particle inserting. This work focus on the bi-dimensional approach, where disks represent the particles. An initial set of particles is generated using input data related to the size distribution and filling rate, obtaining random positions for them. The other steps are used to eliminate overlaps between the particles. The packing algorithm provides good performance and approximation to the input parameters filling ration and grain size distribution. Granular models generated from real soil data are presented for validation of the proposed strategy, and practical examples from the literature are reproduced to illustrate its applicability.

VOROPACK-D: Real-time disk packing algorithm using Voronoi diagram

The disk packing problem (DPP) is to find an arrangement of circular disks within the smallest possible container without any overlap. We discuss a DPP for polysized disks in a circular container. DPP is known NP-hard and reported algorithms are slow for finding good solutions even with the problem instances of small to moderate sizes. Here we introduce a heuristic algorithm which finds sufficiently good solutions in realtime for small to moderate-sized problem instances and in pseudo-realtime for large problem instances. The proposed algorithm, VOROPACK-D, takes advantage of the spatial reasoning property of Voronoi diagram and finds an approximate solution of DPP in O(nlog n) time with O(n) memory by making incremental placement of n disks in the order of non-increasing disk size, thus called a big-disk-first method. The location of a placement is determined using the Voronoi diagram of already-placed disks. If needed, we further enhance a big-disk-first realtime packing solution using the Shrink-and-Shake algorithm by taking an additional O(Mn2) time for each shrinkage where M ≪ n is the number of protruding disks for each shrinkage. Experimental results show that the proposed algorithm is faster than other reported ones by several orders of magnitude, particularly for large problem instances. Theoretical observations are verified and validated by a thorough experiment. This study suggests that Voronoi diagram might be useful for solving other hard optimization problems related with empty spaces. VOROPACK-D is freely available at Voronoi Diagram Research Center (http://voronoi.hanyang.ac.kr/software/voropack-d).

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.

Comparison of shear and compression jammed packings of frictional disks

We compare the structural and mechanical properties of mechanically stable (MS) packings of frictional disks in two spatial dimensions (2D) generated with isotropic compression and simple shear protocols from discrete element modeling (DEM) simulations. We find that the average contact number and packing fraction at jamming onset are similar (with relative deviations $< 0.5\%$) for MS packings generated via compression and shear. In contrast, the average stress anisotropy $\langle {\hat \Sigma}_{xy} \rangle = 0$ for MS packings generated via isotropic compression, whereas $\langle {\hat \Sigma}_{xy} \rangle >0$ for MS packings generated via simple shear. To investigate the difference in the stress state of MS packings, we develop packing-generation protocols to first unjam the MS packings, remove the frictional contacts, and then rejam them. Using these protocols, we are able to obtain rejammed packings with nearly identical particle positions and stress anisotropy distributions compared to the original jammed packings. However, we find that when we directly compare the original jammed packings and rejammed ones, there are finite stress anisotropy deviations $\Delta {\hat \Sigma}_{xy}$. The deviations are smaller than the stress anisotropy fluctuations obtained by enumerating the force solutions within the null space of the contact networks generated via the DEM simulations. These results emphasize that even though the compression and shear jamming protocols generate packings with the same contact networks, there can be residual differences in the normal and tangential forces at each contact, and thus differences in the stress anisotropy.

Large-scale unconstrained optimization using separable cubic modeling and matrix-free subspace minimization

We present a new algorithm for solving large-scale unconstrained optimization problems that uses cubic models, matrix-free subspace minimization, and secant-type parameters for defining the cubic terms. We also propose and analyze a specialized trust-region strategy to minimize the cubic model on a properly chosen low-dimensional subspace, which is built at each iteration using the Lanczos process. For the convergence analysis we present, as a general framework, a model trust-region subspace algorithm with variable metric and we establish asymptotic as well as complexity convergence results. Preliminary numerical results, on some test functions and also on the well-known disk packing problem, are presented to illustrate the performance of the proposed scheme when solving large-scale problems.

Critical scaling for yield is independent of distance to isostaticity

Using discrete element simulations, we demonstrate that critical behavior for yielding in soft disk and sphere packings is independent of distance to isostaticity over a wide range of dimensionless pressures. Jammed states are explored via quasistatic shear at fixed pressure, and the statistics of the dimensionless shear stress $\mu$ of these states obey a scaling description with diverging length scale $\xi \propto |\mu-\mu_c|^{-\nu}$. The critical scaling functions and values of the scaling exponents are nearly independent of distance to isostaticity despite the large range of pressures studied. Our results demonstrate that yielding of jammed systems represents a distinct nonequilibrium critical transition from the isostatic critical transition which has been demonstrated by previous studies. Our results may also be useful in deriving nonlocal rheological descriptions of granular materials, foams, emulsions, and other soft particulate materials.

Jamming and irreversibility

We investigate irreversibility in soft frictionless disk packings on approach to the unjamming transition. Using simulations of shear reversal tests, we study the relationship between plastic work and irreversible rearrangements of the contact network. Infinitesimal strains are reversible, while any finite strain generates plastic work and contact changes in a sufficiently large packing. The number of irreversible contact changes grows with strain, and the stress–strain curve displays a crossover from linear to increasingly nonlinear response when the fraction of irreversible contact changes approaches unity.

The Isostatic Conjecture

We show that a jammed packing of disks with generic radii, in a generic container, is such that the minimal number of contacts occurs and there is only one dimension of equilibrium stresses, which have been observed with numerical Monte Carlo simulations. We also point out some connections to packings with different radii and results in the theory of circle packings whose graph forms a triangulation of a given topological surface.

Rigidity for sticky discs.

We study the combinatorial and rigidity properties of disc packings with generic radii. We show that a packing of n discs in the plane with generic radii cannot have more than 2n - 3 pairs of discs in contact. The allowed motions of a packing preserve the disjointness of the disc interiors and tangency between pairs already in contact (modelling a collection of sticky discs). We show that if a packing has generic radii, then the allowed motions are all rigid body motions if and only if the packing has exactly 2n - 3 contacts. Our approach is to study the space of packings with a fixed contact graph. The main technical step is to show that this space is a smooth manifold, which is done via a connection to the Cauchy-Alexandrov stress lemma. Our methods also apply to jamming problems, in which contacts are allowed to break during a motion. We give a simple proof of a finite variant of a recent result of Connelly et al. (Connelly et al. 2018 (http://arxiv.org/abs/1702.08442)) on the number of contacts in a jammed packing of discs with generic radii.