Granular Matter

Granular materials are poorly understood at a fundamental level. While there is a number of continuum models, usually based on elastoplastic type of constitutive relations, it is still very difficult to understand some basic features of granular materials based on continuum picture alone. For this reason, a significant part of the research is carried out using molecular dynamics /discrete element simulations. Such simulations, as well as experiments with photoelastic particles carried out at Duke University in the group of Robert P. Behringer, have uncovered the existence of force networks, mesoscopic features that are large compared to the particle size, but small compared to the size of computational and experimental domains.

Recent group activities in the field of granular matter have focused on the analysis of force networks using a variety of methods and approaches. A significant part of this research is carried out by using the tools based on computational topology. These tools allow for extracting a significant new information about static and dynamic properties of force networks, both in two and three spatial dimensions.

Earlier works on granular materials include consideration of a variety of granular systems, flow types and geometries, including dynamics of impact on granular matter, flow out of silo, flow of energy through excited granular systems, flow of cohesive materials, and others. The relevant recent publications can be found below, and older works are listed at the publication list or here.

We acknowledge past and current support by NSF, DARPA, NASA, and DTRA.

External Collaborators:

Martin Brinkmann, U. Saarbrucken (Research Gate)
Robert P. Behringer, Duke U. (Website)
Manuel Carlevaro, IFlySB (Research Gate)
Abe Clark, Yale U. (Email)
Miro Kramar, INRIA (Email)
Konstantin Mischaikow, Rutgers U. (Website)
Luis Pugnaloni, UTN (Website)
Ralf Seemann, U. Saarbrucken (Website)

Recent Publications

Kondic, Lou; Kramár, Miroslav; Kovalčinová, Lenka; and Mischaikow, Konstanin, Evolution of force networks in dense granular matter close to jamming, EPJ Web Conf., 140, 15014, (2017)

When dense granular systems are exposed to external forcing, they evolve on the time scale that is typically related to the externally imposed one (shear or compression rate, for example). This evolution could be characterized by observing temporal evolution of contact networks. However, it is not immediately clear whether the force networks, defined on contact networks by considering force interactions between the particles, evolve on a similar time scale. To analyze the evolution of these networks, we carry out discrete element simulations of a system of soft frictional disks exposed to compression that leads to jamming. By using the tools of computational topology, we show that close to jamming transition, the force networks evolve on the time scale which is much faster than the externally imposed one. The presentation will discuss the factors that determine this fast time scale.

Clark, Abram H; Kondic, Lou; and Behringer, Robert P, Steady flow dynamics during granular impact, Physical Review E, 93, 050901, (2016)

We study experimentally and computationally the dynamics of granular flow during impacts where intruders strike a collection of disks from above. In the regime where granular force dynamics are much more rapid than the intruder motion, we find that the particle flow near the intruder is proportional to the instantaneous intruder speed; it is essentially constant when normalized by that speed. The granular flow is nearly divergence free and remains in balance with the intruder, despite the latters rapid deceleration. Simulations indicate that this observation is insensitive to grain properties, which can be explained by the separation of time scales between intergrain force dynamics and intruder dynamics. Assuming there is a comparable separation of time scales, we expect that our results are applicable to a broad class of dynamic or transient granular flows. Our results suggest that descriptions of static-in-time granular flows might be extended or modified to describe these dynamic flows. Additionally, we find that accurate grain-grain interactions are not necessary to correctly capture the granular flow in this regime.

Kovalcinova, Lenka; Goullet, Arnaud; and Kondic, Lou, Scaling properties of force networks for compressed particulate systems, Physical Review E, 93, 042903, (2016)

We consider, computationally and experimentally, the scaling properties of force networks in the systems of circular particles exposed to compression in two spatial dimensions. The simulations consider polydisperse and monodisperse particles, both frictional and frictionless, and in experiments we use monodisperse and bidisperse frictional particles. While for some of the considered systems we observe consistent scaling exponents describing the behavior of the force networks, we find that this behavior is not universal. In particular,we find that frictionless systems, independently of whether they partially crystallize under compression or not, show scaling properties that are significantly different compared to the frictional disordered ones. The findings of nonuniversality are confirmed by explicitly computing fractal dimension for the considered systems. The results of the physical experiments are consistent with the results obtained in simulations of frictional disordered systems.

Kondic, L.; Kramár, M.; Pugnaloni, Luis A.; Carlevaro, C. Manuel; and Mischaikow, K., Structure of force networks in tapped particulate systems of disks and pentagons. II. Persistence analysis, Physical Review E, 93, 062903, (2016)

In the companion paper [Pugnaloni et al., Phys. Rev. E 93, 062902 (2016)], we use classical measures based on force probability density functions (PDFs), as well as Betti numbers (quantifying the number of components, related to force chains, and loops), to describe the force networks in tapped systems of disks and pentagons. In the present work, we focus on the use of persistence analysis, which allows us to describe these networks in much more detail. This approach allows us not only to describe but also to quantify the differences between the force networks in different realizations of a system, in different parts of the considered domain, or in different systems. We show that persistence analysis clearly distinguishes the systems that are very difficult or impossible to differentiate using other means. One important finding is that the differences in force networks between disks and pentagons are most apparent when loops are considered: the quantities describing properties of the loops may differ significantly even if other measures (properties of components, Betti numbers, force PDFs, or the stress tensor) do not distinguish clearly or at all the investigated systems.

Pugnaloni, Luis A.; Carlevaro, C. Manuel; Kramár, M.; Mischaikow, K.; and Kondic, L., Structure of force networks in tapped particulate systems of disks and pentagons. I. Clusters and loops, Physical Review E, 93, 062902, (2016)

The force network of a granular assembly, defined by the contact network and the corresponding contact forces, carries valuable information about the state of the packing. Simple analysis of these networks based on the distribution of force strengths is rather insensitive to the changes in preparation protocols or to the types of particles. In this and the companion paper [Kondic et al., Phys. Rev. E 93, 062903 (2016)], we consider two-dimensional simulations of tapped systems built from frictional disks and pentagons, and study the structure of the force networks of granular packings by considering network's topology as force thresholds are varied. We show that the number of clusters and loops observed in the force networks as a function of the force threshold are markedly different for disks and pentagons if the tangential contact forces are considered, whereas they are surprisingly similar for the network defined by the normal forces. In particular, the results indicate that, overall, the force network is more heterogeneous for disks than for pentagons. Such differences in network properties are expected to lead to different macroscale response of the considered systems, despite the fact that averaged measures (such as force probability density function) do not show any obvious differences. Additionally, we show that the states obtained by tapping with different intensities that display similar packing fraction are difficult to distinguish based on simple topological invariants.

Clark, Abram H; Petersen, Alec J; Kondic, Lou; and Behringer, Robert P, Nonlinear force propagation during granular impact, Physical Review Letters, 114, 144502, (2015)

We experimentally study nonlinear force propagation into granular material during impact from an intruder, and we explain our observations in terms of the nonlinear grain-scale force relation. Using highspeed video and photoelastic particles, we determine the speed and spatial structure of the force response just after impact.We show that these quantities depend on a dimensionless parameter, M' = tc v0/d, where v0 is the intruder speed at impact, d is the particle diameter, and tc is the collision time for a pair of grains impacting at relative speed v0. The experiments access a large range of M by using particles of three different materials. When M' ≪ 1, force propagation is chainlike with a speed, vf, satisfying vf ∝ d=vc. For larger M', the force response becomes spatially dense and the force propagation speed departs from vf ∝ d=tc, corresponding to collective stiffening of a strongly compressed packing of grain

P. Mort; J.N. Michaels; R.P. Behringer; C.S. Campbell; L. Kondic; M. Kheiripour Langroudi; M. Shattuck; J. Tang; G.I. Tardos; and C. Wassgren, Dense granular flow - A collaborative study, Powder Technology, 284, 571 - 584, (2015)

The International Fine Powder Research Institute (IFPRI) has funded an extensive program in dry powder and granular flows, including a focused study on dense flows of interest to a range of industrial handling and process unit operations, especially dense flows at relatively high shear rates. The dense flow program included experimental studies of granular rheology in 3D axial Couette and 2D hopper geometries, wherein the effect of force chains and jamming interactions were investigated as relevant to flow, stress and packing dynamics. The program cumulated in a collaborative study funded by the NSF, wherein a group of academic collaborators was invited to model experimental systems used in IFPRI-sponsored projects. This paper provides a summary of the IFPRI program, details of the collaborative modeling study, and perspective on what is needed to progress the work further.

Kovalcinova, L; Goullet, A; and Kondic, L, Percolation and jamming transitions in particulate systems with and without cohesion, Physical Review E, 92, 032204, (2015)

We consider percolation and jamming transitions for particulate systems exposed to compression. For the systems built of particles interacting by purely repulsive forces in addition to friction and viscous damping, it is found that these transitions are influenced by a number of effects, and in particular by the compression rate. In a quasistatic limit, we find that for the considered type of interaction between the particles, percolation and jamming transitions coincide. For cohesive systems, however, or for any system exposed to even slow dynamics, the differences between the considered transitions are found and quantified.

Kramár, Miroslav; Goullet, Arnaud; Kondic, Lou; and Mischaikow, Konstantin, Evolution of force networks in dense particulate media, Physical Review E, 90, 052203, (2014)

We utilize the tools of persistent homology to analyze features of force networks in dense granular matter, modeled as a collection of circular, inelastic frictional particles. The proposed approach describes these networks in a precise and tractable manner, allowing us to identify features that are difficult or impossible to characterize by other means. In contrast to other techniques that consider each force threshold level separately, persistent homology allows us to consider all threshold levels at once to describe the force network in a complete and insightful manner. We consider continuously compressed system of particles characterized by varied polydispersity and friction in two spatial dimensions.We find significant differences between the force networks in these systems, suggesting that their mechanical response may differ considerably as well.

Animations: 1 2

Kondic, Lou, Simulations of two dimensional hopper flow, Granular Matter, 16, 235--242, (2014)

We report on discrete element simulations of the flow out of a hopper. The geometry of the setup, as well as the material parameters, are taken directly from the exper- iments reported by Tang and Behringer (Chaos 21:041107, 2011; IFRPI-NSF collaboratory: 2D hopper data summary, 2010 ). The simulations consider flow of elastic, frictional, disk-like particles as the slope of the hopper walls and the sizeoftheopeningarevaried,sothatwecanconsiderboththe regime where jamming happens frequently (small openings) and where it occurs very rarely, or not at all (large openings), as well as the influence of the slope of hopper walls. We dis- cuss the distribution of jamming events, the mass flux out of the hopper, and the influence of material parameters on these quantities. In addition, we consider velocity and pres- sure fields in the hopper, as well as their fluctuations.

Miroslav Kramár; Arnaud Goullet; Lou Kondic; and Konstantin Mischaikow, Quantifying force networks in particulate systems, Physica D: Nonlinear Phenomena, 283, 37 - 55, (2014)

We present mathematical models based on persistent homology for analyzing force distributions in particulate systems. We define three distinct chain complexes of these distributions: >digital, position, and interaction, motivated by different types of data that may be available from experiments and simulations, e.g. digital images, location of the particles, and the forces between the particles, respectively. We describe how algebraic topology, in particular, homology allows one to obtain algebraic representations of the geometry captured by these complexes. For each complex we define an associated force network from which persistent homology is computed. Using numerical data obtained from discrete element simulations of a system of particles undergoing slow compression, we demonstrate how persistent homology can be used to compare the force distributions in different systems, and discuss the differences between the properties of digital, position, and interaction force networks. To conclude, we formulate well-defined measures quantifying differences between force networks corresponding to the different states of a system, and therefore allow to analyze in precise terms dynamical properties of force networks.


Kramar, M; Goullet, A; Kondic, L; and Mischaikow, K, Persistence of force networks in compressed granular media, Physical Review E, 87, 042207, (2013)

We utilize the tools of persistent homology to analyze features of force networks in dense granular matter, modeled as a collection of circular, inelastic frictional particles. The proposed approach describes these networks in a precise and tractable manner, allowing us to identify features that are difficult or impossible to characterize by other means. In contrast to other techniques that consider each force threshold level separately, persistent homology allows us to consider all threshold levels at once to describe the force network in a complete and insightful manner. We consider continuously compressed system of particles characterized by varied polydispersity and friction in two spatial dimensions. We find significant differences between the force networks in these systems, suggesting that their mechanical response may differ considerably as well.

Kondic, L; Fang, X; Losert, W; O'Hern, CS; and Behringer, RP, Microstructure evolution during impact on granular matter, Physical Review E, 85, 011305, (2012)

We study the impact of an intruder on a dense granular material. The process of impact and interaction between the intruder and the granular particles is modeled using discrete element simulations in two spatial dimensions. In the first part of the paper we discuss how the intruder’s dynamics depends on (1) the intruder's properties, including its size, shape and composition, (2) the properties of the grains, including friction, polydispersity, structural order, and elasticity, and (3) the properties of the system, including its size and gravitational field. It is found that polydispersity and related structural order, and frictional properties of the granular particles, play a crucial role in determining impact dynamics. In the second part of the paper we consider the response of the granular system itself. We discuss the force networks that develop, including their topological evolution. The influence of friction and structural order on force propagation, including the transition from hyperbolic-like to elastic-like behavior is discussed, as well as the affine and nonaffine components of the grain dynamics. Several broad observations include the following: tangential forces between granular particles are found to play a crucial role in determining impact dynamics; both force networks and particle dynamics are correlated with the dynamics of the intruder itself.

Animations: 1 2

Kondic, L; Goullet, A; O'Hern, CS; Kramar, M; Mischaikow, K; and Behringer, RP, Topology of force networks in compressed granular media, EPL (Europhysics Letters), 97, 54001, (2012)

Using numerical simulations, we investigate the evolution of the structure of force networks in slowly compressed model granular materials in two spatial dimensions. We quantify the global properties of the force networks using the zeroth Betti number B0, which is a topological invariant. We find that B0 can distinguish among force networks in systems with frictionless vs. frictional disks and varying size distributions. In particular, we show that 1) the force networks in systems composed of frictionless, monodisperse disks differ significantly from those in systems with frictional, polydisperse disks and we isolate the effect (friction, polydispersity) leading to the differences; 2) the structural properties of force networks change as the system passes through the jamming transition; and 3) the force network continues to evolve as the system is compressed above jamming, e.g., the size of connected clusters with forces larger than a given threshold decreases significantly with increasing packing fraction.

Clark, Abram H; Kondic, Lou; and Behringer, Robert P, Particle scale dynamics in granular impact, Physical Review Letters, 109, 238302, (2012)

We perform an experimental study of granular impact, where intruders strike 2D beds of photoelastic disks from above. High-speed video captures the intruder dynamics and the local granular force response, allowing investigation of grain-scale mechanisms in this process. We observe rich acoustic behavior at the leading edge of the intruder, strongly fluctuating in space and time, and we show that this acoustic activity controls the intruder deceleration, including large force fluctuations at short time scales. The average intruder dynamics match previous studies using empirical force laws, suggesting a new microscopic picture, where acoustic energy is carried away and dissipated.