Session T29: Focus Session: Jamming: Marginal Solids I

From Crystals to Point J: how changing the order affects disordered systems

The theory of crystalline solids is well established as the basis for our understanding of periodically ordered materials. While less developed, much progress has been made in understanding solids that lack periodic order. Specifically, the jamming transition of idealized soft spheres is a critical point that corresponds to the opposite limit of the fully disordered solid--the epitome of disorder. We seek to bridge the gap between these two extreme limits--the completely disordered solid and the perfect crystal--to understand how partially ordered systems behave. Can they always be considered as perturbations away from these two limits, or are they fundamentally different? We find that systems with intermediate bond orientational order exist but that most systems display either very high or very low order. We study mechanically stable configurations that are very ordered but whose contact number is not far from the marginal value. Despite their ordered structure, these states show the same excess low-frequency modes, elastic properties and scalings typically associated with systems near the jamming transition. This suggests that the signatures of the jamming transition are more robust than previously thought and sheds light on the physical mechanism that makes jamming unique.   

The equilibration of temperature-like variables in jammed granular subsystems

Although jammed granular systems are athermal, several thermodynamic-like descriptions have been proposed which make quantitative predictions about the distribution of volume and stress within a system and provide a corresponding temperature-like variable. We perform experiments with an apparatus designed to generate a large number of independent, jammed, two-dimensional configurations. Each configuration consists of a single layer of photoelastic disks supported by a gentle layer of air. New configurations are generated by alternately dilating and re-compacting the system through a series of boundary displacements. Within each configuration, a bath of particles surrounds a smaller subsystem of particles with a different inter-particle friction coefficient than the bath. The use of photoelastic particles permits us to find all particle positions as well as the vector forces at each inter-particle contact. By comparing the temperature-like quantities in both systems, we find compactivity (conjugate to the volume) does not equilibrate between the systems, while the angoricity (conjugate to the stress) does. Both independent components of the angoricity are linearly dependent on the hydrostatic pressure, in agreement with predictions of the stress ensemble.   

Shape effect on dynamical properties of granular materials

We have investigated the effect of shape on dynamical and rheological properties of granular materials through Couette shear and cyclic isotropic compression experiments. We track the evolution of our systems by measuring the mean velocity local density, orientational order, and local stress. This set of experiments which were performed on systems of bidisperse disks and identical ellipses at exactly same conditions, reveals striking differences between the dynamics of disks and ellipses. In particular we observe a very slow relaxation in various dynamical quantities for systems of ellipses. We also demonstrate that the strain history of the system (i.e. shear vs. compression) highly impacts the aging process.   

Fracture mechanics and crack propagation in fragile matter

Using simulations and theory, we investigate fracture processes and the formation of cracks in near-isostatic networks derived from jammed packings in both the quasi-static limit and with molecular dynamics. We study how localized cracks in networks with high coordination number become randomly distributed and isolated bond breakages near the isostatic point and suggest that this may be related to the scaling of the size of the process zone with characteristic lengths from jamming.   

Evolution of Triangle Decomposition During Jamming

We use simulations of soft 2D bidisperse disks to determine the properties of jammed packings and investigate the statistical mechanics of these systems. We have created a novel method for the classification of structural subunits of a packing and use the subunits to calculate relevant physical quantities. The classification scheme is based on a 20 type decomposition of the Delaunay triangles extracted from the centers of the particles. The distribution of triangle types evolve as systems are jammed by compression or as they are sheared. We analyze the statistics of the triangle types and identify specific transition events during compression, jamming, and shear.   

Network and Dynamical System Analysis of a Granular Stick-Slip Experiment

We describe analysis of stick-slip behavior in a granular material under shear from a slider that is pulled across the granular surface. We extend previous statistical analysis, focusing on size distributions of failure events by applying nonlinear time series analysis, including surrogate data, and complex network methods. Local dimension measures suggest a robust evolution law of stick-slip dynamics needs at least 4 to 6 degrees of freedom. Surrogate methods indicate that individual stick-slip events may contain more complex nonlinear determinism periodic dynamics, although models with periodic dynamics are adequate for some cases. Within each stick-slip ``cycle'', we found evidence of nonlinear determinism but no long term memory across cycles. Representing the observed time series as a complex network, however, revealed that despite no evidence for long term dynamical correlations between distinct stick-slip events there is consistency in the structure of individual subnetworks associated with the onset of each slip event, possibly reflecting a single driving mechanism of failure, i.e. dynamics of force chains. When the data is representated as a complex network, it appears to present a new stratification of system dynamics with a previously unreported ranking, or genus,   

On the local construction of jamming graphs

We extend the concept of minimal rigidity to particulate systems, or nonbonded networks, in two-dimensions with the introduction of the jamming graph. The jamming graph is a planar Laman graph with each vertex satisfying the Hilbert local stability requirement. In other words, the jamming graph contains both property of global and local mechanical stability at the onset of rigidity for the model system of frictionless, repulsive soft spheres. We demonstrate how such graphs can be constructed using purely local moves interestingly enough. To make comparisons with the model system, we first associate springs with the edges of the graph and then associate shapes with each vertex and determine various mechanical properties as the spring density, or particle packing fraction, is increased. The jamming graph not only provides for a rigorous starting point for the onset of rigidity, the local rules used to construct it can be easily modified to account for friction and/or particle shapes beyond spheres so that a more general framework for the onset of rigidity in particulate systems may ultimately be established.   

The crossover from random close to random loose packings of frictional disks

Mechanically stable packings of frictionless disks with contact interactions form through fast quenches at random close packing (RCP). However, for frictional particles with static friction coefficient $\mu $ greater than $\mu $*, the packing density slides toward random loose packing (RLP) at large friction. We elucidate the crossover from random close to random loose packing through simulations of bidisperse disks using the geometric asperity (GA)[1] and Cundall-Strack (CS) friction models. We demonstrate that a change takes place in the structure of allowed mechanically stable packings in configuration space: From uncorrelated points at zero friction to linear and other low-dimensional structures at small friction to higher dimensional structures at large friction. Further, we use the GA model to study dynamical mechanical properties without ad hoc assumptions for sliding contacts, and we find that low-frequency vibrational modes with significant rotational content display a strong peak below $\mu $*. Their rotational content drastically changes from co-rotating contacting particles for low friction to counter-rotating, gear-like, for $\mu $ greater than $\mu $* and the groups of particles with gear-like dynamical contributions percolate at $\mu $*. Finally, the very existence of the low-frequency vibrational peak gives rise to a change in the scaling of the static shear modulus with pressure compared to the frictionless behavior. \\[4pt] [1] S. Papanikolaou, C. S. O' Hern and M. D. Shattuck, arxiv:1207.6010 (2012)   

Vibrational modes of jammed and unjammed packings

We showed previously that granular packings composed of frictionless particles with repulsive contact interactions are strongly nonharmonic. Weakly vibrated packings possess well-defined average positions that differ from those of the unvibrated packing and other nearby static packings, and when excited along a single vibrational mode from the dynamical matrix energy quickly leaks to other modes during vibration due to contact breaking. We now measure the displacement correlation matrix for weakly vibrated systems and the velocity autocorrelation function averaged over fluctuations to extract the associated density of vibrational modes. We find that there is an increase in the number of low-frequency eigenmodes of the displacement matrix compared to that for the dynamical matrix in linear response, and these modes provide a more accurate description of the dynamics. The new set of modes from the displacement correlation matrix persists over several orders of magnitude in the input energy of the vibrations. Futhermore, the new vibrational modes are insensitive to pressure, i.e. packings prepared above and below jamming yield the same set of vibrational modes. We also perform vibration experiments as a function of amplitude and frequency, and compare our findings.   

Energy Transfers in Coupled Ordered Granular Chains with No Precompression

We study the dynamics of coupled one-dimensional granular chains mounted on elastic foundations. No dissipative effects, such as plasticity or dry friction effects are taken into account in our analysis. Assuming no pre-compression between beads, the dynamics of the system under consideration is strongly nonlinear and, in an acoustic analogy they can be viewed as `sonic vacua'. Sources of strong nonlinearity in these systems are nonlinearizable Hertzian interactions between adjacent beads in compression, and also possible separations between beads in the absence of compressive forces leading to bead collisions. We find that demonstrate that in weakly coupled granular chains there can occur strong energy exchanges in the form of nonlinear beat phenomena of spatially periodic traveling waves, stationary breathers or propagating breathers. We employ analytical techniques to study these dynamical phenomena.   

Rearrangements in 2D packings

Using computer simulations of frictionless, harmonic, packings, we have investigated the effects of global shear deformations on a local scale. We have focused on the making and breaking of contacts between particles, as a change in the contact network signals a departure from linear response. We show the deformation at which the first contact change happens can be predicted, using simple scaling arguments, from the initial pressure and the number of particles. In addition, we show the also probability of creating versus breaking a contact can be understood. Finally, we are able to show the locality of the rearrangements in the packing.   

Lattice model of correlated forces in granular solids near jamming

We have devised a lattice model to study force correlations in granular solids as the isostatic limit is approached. We apply biased Monte Carlo simulations to the Tighe ``wheel move'' model to progressively starve the system of force-bearing bonds. Increasingly long-ranged correlations are visible as point J is approached, not in the structure of the network of force-bearing bonds, but in the spatial extent of perturbations of the forces consistent with a given starved network. The correlation length so defined diverges as the isostatic point is approached, as a power law $\xi = \delta Z^{-4.78}$. This divergence is much stronger than for the length scale of ``soft modes'' observed in jammed systems approaching point J from above. We can relate the correlated regions we observe to a certain definition of percolation clusters. The probability distribution of cluster sizes, and the bulk and surface fractal dimensions of the clusters, all scale analogously to classical percolation, but with distinctly different scaling exponents.   

Tuning with tension: Controlling elasticity in nearly isostatic spring networks

We show that the shear stiffness of random spring networks can be controlled by exploiting their strong susceptibility to tensile loading. Unstressed networks below the isostatic point are floppy and cannot sustain shear. But floppiness can be ``pulled out'' with tension, rendering the loaded system rigid. Using scaling arguments and computer simulations, we determine the dependence of stretched networks' shear modulus on tension and show how this effect can be leveraged to generate ``smart networks'' with tunable stiffness.