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Journal articles on the topic 'Granular Dynamics'

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Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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1

Hayakawa, Hisao, and Daniel C. Hong. "Dynamics of Granular Compaction." International Journal of Bifurcation and Chaos 07, no. 05 (May 1997): 1159–65. http://dx.doi.org/10.1142/s0218127497000960.

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We investigate the way the disordered granular materials organize themselves in a vibrating bed, the intensity of which is given by the dimensionless parameter Γ. Based on the recognition that an assembly of mono-disperse and cohesionless granular materials is a collection of spinless hard sphere Fermions, we first demonstrate that the time averaged steady state density profile for weak excitation with Γ ≈ 1 is given by the Fermi distribution. This is consistent with the observed experimental data and the results of Molecular dynamics. We then present a dynamic model to study the dynamics of granular compaction, namely the dynamic evolution of the initial state ultimately relaxing toward this steady state. Our preliminary investigation reveals that the relaxation is exponential, which is not inconsistent with the available experimental data for low Γ.

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2

Sánchez, Rodrigo. "Granular dynamics and gravity." Soft Matter 16, no. 40 (2020): 9253–61. http://dx.doi.org/10.1039/d0sm01203c.

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3

Herrmann, Hans J., S. Luding, and R. Cafiero. "Dynamics of granular systems." Physica A: Statistical Mechanics and its Applications 295, no. 1-2 (June 2001): 93–100. http://dx.doi.org/10.1016/s0378-4371(01)00059-0.

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4

LOGUINOVA, NADEJDA, and YURI VLASOV. "OSCILLATIONS IN GRANULAR DYNAMICS." Advances in Complex Systems 10, no. 03 (September 2007): 287–99. http://dx.doi.org/10.1142/s0219525907001203.

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A new effect of granular dynamics in a bounded domain is reported. Oscillations arise when the system evolves from a given (non-equilibrium) initial state. The oscillations obtained are of importance for vibrated granular systems since they reveal some kind of fundamental frequencies and they lead to resonant frequencies under vibration.

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5

Makse, Hernán A., Robin C. Ball, H. Eugene Stanley, and Stephen Warr. "Dynamics of granular stratification." Physical Review E 58, no. 3 (September 1, 1998): 3357–67. http://dx.doi.org/10.1103/physreve.58.3357.

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6

Mehta, A., G. C. Barker, and J. M. Luck. "Heterogeneities in granular dynamics." Proceedings of the National Academy of Sciences 105, no. 24 (June 9, 2008): 8244–49. http://dx.doi.org/10.1073/pnas.0711733105.

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7

Johnson, Paul A., and Xiaoping Jia. "Nonlinear dynamics, granular media and dynamic earthquake triggering." Nature 437, no. 7060 (October 2005): 871–74. http://dx.doi.org/10.1038/nature04015.

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8

Murdoch, Naomi, Patrick Michel, Derek C. Richardson, Kerstin Nordstrom, Christian R. Berardi, Simon F. Green, and Wolfgang Losert. "Numerical simulations of granular dynamics II: Particle dynamics in a shaken granular material." Icarus 219, no. 1 (May 2012): 321–35. http://dx.doi.org/10.1016/j.icarus.2012.03.006.

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9

Goh, Y. K., and R. L. Jacobs. "Coarsening dynamics of granular heaplets in tapped granular layers." New Journal of Physics 4 (October 28, 2002): 81. http://dx.doi.org/10.1088/1367-2630/4/1/381.

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10

Armanini, Aronne, Luigi Fraccarollo, and Michele Larcher. "Liquid–granular channel flow dynamics." Powder Technology 182, no. 2 (February 2008): 218–27. http://dx.doi.org/10.1016/j.powtec.2007.08.012.

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11

Radjai, Farhang. "Multicontact dynamics of granular systems." Computer Physics Communications 121-122 (September 1999): 294–98. http://dx.doi.org/10.1016/s0010-4655(99)00337-9.

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12

Hayakawa, Hisao, Hiraku Nishimori, and Sin'ichi Sasa. "Dynamics of Granular Matter*1." Japanese Journal of Applied Physics 34, Part 1, No. 2A (February 15, 1995): 397–408. http://dx.doi.org/10.1143/jjap.34.397.

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13

Herminghaus *, S. "Dynamics of wet granular matter." Advances in Physics 54, no. 3 (May 2005): 221–61. http://dx.doi.org/10.1080/00018730500167855.

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14

Nie, X., E. Ben-Naim, and S. Y. Chen. "Dynamics of vibrated granular monolayers." Europhysics Letters (EPL) 51, no. 6 (September 15, 2000): 679–84. http://dx.doi.org/10.1209/epl/i2000-00392-7.

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15

Mehta, Anita, and G. C. Barker. "Glassy dynamics in granular compaction." Journal of Physics: Condensed Matter 12, no. 29 (July 6, 2000): 6619–28. http://dx.doi.org/10.1088/0953-8984/12/29/333.

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16

Chen, Kuo-Ching, and Jeng-Yin Lan. "Micromorphic modeling of granular dynamics." International Journal of Solids and Structures 46, no. 6 (March 2009): 1554–63. http://dx.doi.org/10.1016/j.ijsolstr.2008.11.022.

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17

Sinkovits, Robert S., and Surajit Sen. "Nonlinear Dynamics in Granular Columns." Physical Review Letters 74, no. 14 (April 3, 1995): 2686–89. http://dx.doi.org/10.1103/physrevlett.74.2686.

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18

Liao, Chun-Chung. "Density-induced granular migration dynamics in sheared slurry granular materials." Powder Technology 338 (October 2018): 931–36. http://dx.doi.org/10.1016/j.powtec.2018.07.070.

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19

Tingting, Li, Luo Chao, and Shao Rui. "The structure and dynamics of granular complex networks deriving from financial time series." International Journal of Modern Physics C 31, no. 06 (May 7, 2020): 2050087. http://dx.doi.org/10.1142/s0129183120500874.

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High noise and strong volatility are the typical characteristics of financial time series. Combined with pseudo-randomness, nonsteady and self-similarity exhibiting in different time scales, it is a challenging issue for the pattern analysis of financial time series. Different from the existing works, in this paper, financial time series are converted into granular complex networks, based on which the structure and dynamics of network models are revealed. By using variable-length division, an extended polar fuzzy information granule (FIGs) method is used to construct granular complex networks from financial time series. Considering the temporal characteristics of sequential data, static networks and temporal networks are studied, respectively. As to the static network model, some features of topological structures of granular complex networks, such as distribution, clustering and betweenness centrality are discussed. Besides, by using the Markov chain model, the transfer processes among different granules are investigated, where the fluctuation pattern of data in the coming step can be evaluated from the transfer probability of two consecutive granules. Shanghai composite index and foreign exchange data as two examples in real life are applied to carry out the related discussion.

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20

Liu, Y. Q., Y. H. Kong, R. Zhang, X. Zhang, F. S. Wong, J. H. Tay, J. R. Zhu, W. J. Jiang, and W. T. Liu. "Microbial population dynamics of granular aerobic sequencing batch reactors during start-up and steady state periods." Water Science and Technology 62, no. 6 (September 1, 2010): 1281–87. http://dx.doi.org/10.2166/wst.2010.408.

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This study investigates microbial population dynamics in granular sequencing batch reactors (GSBR). The experimental results of DGGE fingerprint of sludge demonstrated that the microbial community structure of sludge shifted significantly during granulation period and nutrient removal improvement period. After reactor performance and physical characteristics of sludge reached steady state, microbial population of sludge became relatively stable. The high similarity of microbial community structure between co-existed flocculated sludge and granular sludge in GSBR at different operation phases indicated that similar microbial consortium could exist in compact aggregated form or in amorphous flocculated form. Therefore, strong selection pressure was still required to wash out flocs to maintain the stability of reactor operation. In addition, it was found that substrate type had considerable impact on microbial species selection and enrichment in granular sludge. The clone library of granular sludge showed that microbial species in divisions of α-Proteobacteria, β-Proteobacteria, γ-Proteobacteria and Bacteroidetes existed within acetate-fed granule communities and Thauera spp. from β-Proteobacteria accounted for 49% of the total clones in the whole clone library. It is thus speculated that Thauera spp. are important for the formation of acetate-fed granules under the conditions used in this study, maintaining the integrity of granules or substrate degradation.

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21

Peti-Peterdi, János, Attila Fintha, Amanda L. Fuson, Albert Tousson, and Robert H. Chow. "Real-time imaging of renin release in vitro." American Journal of Physiology-Renal Physiology 287, no. 2 (August 2004): F329—F335. http://dx.doi.org/10.1152/ajprenal.00420.2003.

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Renin release from juxtaglomerular granular cells is considered the rate-limiting step in activation of the renin-angiotensin system that helps to maintain body salt and water balance. Available assays to measure renin release are complex, indirect, and work with significant internal errors. To directly visualize and study the dynamics of both the release and tissue activity of renin, we isolated and perfused afferent arterioles with attached glomeruli dissected from rabbit kidneys and used multiphoton fluorescence imaging. Acidotropic fluorophores, such as quinacrine and LysoTrackers, clearly and selectively labeled renin granules. Immunohistochemistry of mouse kidney with a specific renin antibody and quinacrine staining colocalized renin granules and quinacrine fluorescence. A low-salt diet for 1 wk caused an approximately fivefold increase in the number of both individual granules and renin-positive granular cells. Time-lapse imaging showed no signs of granule trafficking or any movement, only the dimming and disappearance of fluorescence from individual renin granules within 1 s in response to 100 μM isoproterenol. There appeared to be a quantal release of the granular contents; i.e., an all-or-none phenomenon. Using As4.1 cells, a granular cell line, we observed further classic signs of granule exocytosis, the emptying of granule content associated with a flash of quinacrine fluorescence. Using a fluorescence resonance energy transfer-based, 5-(2-aminoethylamino)naphthalene-1-sulfonic acid (EDANS)-conjugated renin substrate in the bath, an increase in EDANS fluorescence (renin activity) was observed around granular cells in response to isoproterenol. Fluorescence microscopy is an excellent tool for the further study of the mechanism, regulation, and dynamics of renin release.

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22

DOPPLER, DELPHINE, PHILIPPE GONDRET, THOMAS LOISELEUX, SAM MEYER, and MARC RABAUD. "Relaxation dynamics of water-immersed granular avalanches." Journal of Fluid Mechanics 577 (April 19, 2007): 161–81. http://dx.doi.org/10.1017/s0022112007004697.

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We study water-immersed granular avalanches in a long rectangular cell of small thickness. By video means, both the angle of the granular pile and the velocity profiles of the grains across the depth are recorded as a function of time. These measurements give access to the instantaneous granular flux. By inclining the pile at initial angles larger than the maximum angle of stability, avalanches are triggered and last for a long time, up to several hours for small grains, during which both the slope angle and the granular flux relax slowly. We show that the relaxation is quasi-steady so that there is no inertia: the relaxation at a given time is controlled only by the slope angle at that time. This allows us to adapt a frictional model developed recently for dry or water-immersed grains flowing in stationary conditions. This model succeeds well in reproducing our unsteady avalanche flows, namely the flowing layer thickness, the granular flux and the temporal relaxation of the slope. When a water counter-flow is applied along the pile, the granular avalanches are slowed down and behave as if granular friction were increased by an amount proportional to the water flow. All these findings are also reproduced well with the same friction model by taking into account the additional fluid force.

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23

Sanchez-Castillo, Francisco X., Jamshed Anwar, and David M. Heyes. "Molecular Dynamics Simulations of Granular Compaction." Chemistry of Materials 15, no. 18 (September 2003): 3417–30. http://dx.doi.org/10.1021/cm030176a.

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24

Ristow, Gerald H. "Simulating granular flow with molecular dynamics." Journal de Physique I 2, no. 5 (May 1992): 649–62. http://dx.doi.org/10.1051/jp1:1992159.

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25

Guillard, François, Pouya Golshan, Luming Shen, Julio R. Valdès, and Itai Einav. "Compaction dynamics of crunchy granular material." EPJ Web of Conferences 140 (2017): 07012. http://dx.doi.org/10.1051/epjconf/201714007012.

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26

Schulz, B. M., and M. Schulz. "The dynamics of wet granular matter." Journal of Non-Crystalline Solids 352, no. 42-49 (November 2006): 4877–79. http://dx.doi.org/10.1016/j.jnoncrysol.2006.03.125.

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27

TenCate, James A., Eric Smith, and Robert A. Guyer. "Universal Slow Dynamics in Granular Solids." Physical Review Letters 85, no. 5 (July 31, 2000): 1020–23. http://dx.doi.org/10.1103/physrevlett.85.1020.

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28

Losert, W., L. Bocquet, T. C. Lubensky, and J. P. Gollub. "Particle Dynamics in Sheared Granular Matter." Physical Review Letters 85, no. 7 (August 14, 2000): 1428–31. http://dx.doi.org/10.1103/physrevlett.85.1428.

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29

Scherer, Michael A., Thomas Mahr, Andreas Engel, and Ingo Rehberg. "Granular dynamics in a swirled annulus." Physical Review E 58, no. 5 (November 1, 1998): 6061–72. http://dx.doi.org/10.1103/physreve.58.6061.

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30

HERRMANN, HANS J. "MOLECULAR DYNAMICS SIMULATIONS OF GRANULAR MATERIALS." International Journal of Modern Physics C 04, no. 02 (April 1993): 309–16. http://dx.doi.org/10.1142/s012918319300032x.

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When sand or other granular materials are shaken, poured or sheared many intriguing phenomena can be observed. We will model the granular medium by a packing of elastic spheres and simulate it via Molecular Dynamics. Dissipation of energy and shear friction at collisions are included. The onset of fluidization can be determined and is in good agreement with experiments. On a vibrating plate we observe the formation of convection cells due to walls or amplitude modulations. Density and velocity profiles on conveyor belts are measured and the influence of an obstacle discussed. We mention various types of rheology for flow down an inclined chute or through a pipe and outflowing containers.

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31

Barrat, A., E. Trizac, and M. H. Ernst. "Granular gases: dynamics and collective effects." Journal of Physics: Condensed Matter 17, no. 24 (June 3, 2005): S2429—S2437. http://dx.doi.org/10.1088/0953-8984/17/24/004.

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32

Colombo, R. M., G. Guerra, and F. Monti. "Modelling the dynamics of granular matter." IMA Journal of Applied Mathematics 77, no. 2 (March 16, 2011): 140–56. http://dx.doi.org/10.1093/imamat/hxr007.

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33

Tarasov, V. P., and S. V. Krivenko. "Gas dynamics of a granular bed." Steel in Translation 44, no. 5 (May 2014): 359–62. http://dx.doi.org/10.3103/s0967091214050143.

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34

Krabbenhoft, K., J. Huang, M. Vicente da Silva, and A. V. Lyamin. "Granular contact dynamics with particle elasticity." Granular Matter 14, no. 5 (July 26, 2012): 607–19. http://dx.doi.org/10.1007/s10035-012-0360-1.

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35

HONG, DANIEL C. "DYNAMIC MODEL FOR GRANULAR ASSEMBLY." International Journal of Modern Physics B 07, no. 09n10 (April 20, 1993): 1929–47. http://dx.doi.org/10.1142/s0217979293002699.

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In this paper, we investigate in detail the complex dynamics and flow patterns of granular materials based on two simple yet quite realistic models: the diffusing void model and the nonlinear dynamic model. We first show how the diffusing void model describes some of the unusual and unique features of granular flows in a confined geometry such as the deformation of the free surface, the formation of dead zones, the flow around obstacles, the shock front with its companion void regions, and the front profile of the propagating density waves. We then provide theoretical framework for the diffusing void model by deriving it from the continuity equation and the microscopic force balance equation. This approach shows how the nonlinear term arises naturally, leading to the nonlinear dynamic equation, whose numerical solutions do exhibit the features shown by the diffusing void model and the experiment. When nonlocal interaction between grains is taken into account, the mass term appears in the dynamic equation in the thermodynamic limit, leading to exponential decay of the granular pile. This might account fot the different behaviors for large and small granular piles. We also present exact results of the stress distribution of a hexagonally packed granular pile in two dimensions and show that the load acting on each grain at the bottom layer is identical. We also present the results of molecular dynamics simulations which show that the speed of the outgoing grains in a hopper is independent of the depth. We then outline a few outstanding problems that are technologically important, yet can be handled by the models proposed and developed in this paper.

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36

Cheng, Chao, Aghil Abed Zadeh, and Lou Kondic. "Correlating the force network evolution and dynamics in slider experiments." EPJ Web of Conferences 249 (2021): 02007. http://dx.doi.org/10.1051/epjconf/202124902007.

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The experiments involving a slider moving on top of granular media consisting of photoelastic particles in two dimensions have uncovered elaborate dynamics that may vary from continuous motion to crackling, periodic motion, and stick-slip type of behavior. We establish that there is a clear correlation between the slider dynamics and the response of the force network that spontaneously develop in the granular system. This correlation is established by application of the persistence homology that allows for formulation of objective measures for quantification of time-dependent force networks. We find that correlation between the slider dynamics and the force network properties is particularly strong in the dynamical regime characterized by well-defined stick-slip type of dynamics.

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37

Mykulyak, S. V., V. V. Kulich, and S. I. Skurativskyi. "On the similarity of shear deformation of a granular massif and a fragmented medium in the seismically active area." Geofizicheskiy Zhurnal 43, no. 3 (July 28, 2021): 161–69. http://dx.doi.org/10.24028/gzh.v43i3.236386.

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In recent research, the dynamics of the medium located in the seismic region at the boundary of tectonic plates is considered as the behavior of a complex open system that is in a state of self-organized criticality. Such an approach results from the very laws of earthquake generation and the complex structure of these areas. The network of faults and cracks makes seismic zones significantly heterogeneous and fragmented. Therefore, discrete models are increasingly used to model the dynamics of these media. The basis for comparing the model and the full-scale object serves the statistical regularities of their dynamic deformation. Relying on this concept, in the paper it is modeled the shear dynamics of a granular massif composed of identical cubic granules and is compared system’s statistical characteristics with the similar characteristics obtained for the earthquake generation zone. Shear deformation is carried out by means of the box consisting of two parts — movable and immovable ones. The movable part possesses the cover which receives kinetic energy from the granular massif in the process of shear deformation. For numerical simulations of the shear dynamics, the discrete element method is applied. The numerical calculations result in the distribution of cover’s kinetic energy jumps simulating the perturbations transmitted from the granular system to an external medium. It turned out that the distribution for these perturbations is the power dependence with an exponent that is inherent in earthquakes (Gutenberg-Richter law). Before and after large perturbations it is observed the swarms of smaller perturbations which are the analogues of foreshocks and aftershocks. The distributions of element’s velocity fluctuations and the correlation of velocity fluctuations are calculated as well. It is revealed the similarity of distributions for velocity fluctuations in the model massif and in the seismically active region of California, which includes the San Andreas fault. Moreover, the similarity of corresponding correlation functions is shown. They both are the functions of the stretched exponent. The obtained result indicates that shear processes in granular massifs and natural seismic processes in the San Andreas Fault are statistically similar.

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38

Oono, Yoshitsugu. "Cell-Dynamics Modeling of Vibrating Powder." International Journal of Modern Physics B 07, no. 09n10 (April 20, 1993): 1859–64. http://dx.doi.org/10.1142/s0217979293002638.

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The instabilities in vibrating granular materials may be interpreted as segregation processes of vacuum and powder particles. Through an attempt of mesocale modeling of the instabilities within the cell-dynamics scheme, a notable distinction between the ordinary segregation processes such as binary alloy spinodal decomposition and the granular processes is highlighted. ‘Vibrating particles in a horizontal Hele-Shaw cell’ experiment is proposed.

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39

Bi, Dapeng, and Bulbul Chakraborty. "Rheology of granular materials: dynamics in a stress landscape." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367, no. 1909 (December 28, 2009): 5073–90. http://dx.doi.org/10.1098/rsta.2009.0193.

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We present a framework for analysing the rheology of dense driven granular materials, based on a recent proposal of a stress-based ensemble. In this ensemble, fluctuations in a granular system near jamming are controlled by a temperature-like parameter, the angoricity, which is conjugate to the stress of the system. In this paper, we develop a model for slowly driven granular materials based on the stress ensemble and the idea of a landscape in stress space. The idea of an activated process driven by the angoricity has been shown by Behringer et al . (Behringer et al. 2008 Phys. Rev. Lett. 101 , 268301) to describe the logarithmic strengthening of granular materials. Just as in the soft glassy rheology (SGR) picture, our model represents the evolution of a small patch of granular material (a mesoscopic region) in a stress-based trap landscape. The angoricity plays the role of the fluctuation temperature in the SGR. We determine (i) the constitutive equation, (ii) the yield stress, and (iii) the distribution of stress dissipated during granular shearing experiments, and compare these predictions with the experiments of Hartley & Behringer (Hartley & Behringer 2003 Nature 421 , 928–931.).

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40

Liao, Chun-Chung, Shu-San Hsiau, and Yu-Ming Hu. "Density-driven sinking dynamics of a granular ring in sheared granular flows." Advanced Powder Technology 28, no. 10 (October 2017): 2597–604. http://dx.doi.org/10.1016/j.apt.2017.07.011.

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41

Behringer, R. P. "The Scientist in the Sandbox: Time-Dependence, Fractals and Waves." International Journal of Bifurcation and Chaos 07, no. 05 (May 1997): 963–78. http://dx.doi.org/10.1142/s0218127497000789.

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Granular materials exhibit a rich variety of dynamical behavior, much of which is poorly understood. Fractal-like stress chains, convection, a variety of wave dynamics, including waves which resemble capillary waves, 1/f noise, and fractional Brownian motion provide examples. Although granular materials consist of collections of interacting particles, there are important differences between the dynamics of a collection of grains and the dynamics of a collection of molecules. In particular, the ergodic hypothesis is generally invalid for granular materials, so that ordinary statistical physics does not apply. Fluctuations on laboratory scales in such quantities as the stress can be very large — as much as an order of magnitude greater than the mean. Below is a brief review of some of the theoretical approaches to granular flow followed by a discussion of several recent experiments. These experiments focus on complex structures and fluctuations in the flow of granular materials in a hopper or in simple shear flow. The experimental work at Duke has been carried out in collaboration with a number of investigators, including G. W. Baxter, R. Leone, H. K. Pak, E. Van Doorn, C. O'Hern, and B. Miller. Elsewhere in this issue, Pak et al. discuss experiments to characterize and better understand the convective flows which occur when granular materials are shaken with accelerations exceeding that of gravity.

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42

Arsenović, D., S. B. Vrhovac, Z. M. Jakšić, Lj Budinski-Petković, and A. Belić. "Simulation Study of Granular Compaction Dynamics under Vertical Tapping." Materials Science Forum 555 (September 2007): 107–12. http://dx.doi.org/10.4028/www.scientific.net/msf.555.107.

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We study by numerical simulation the compaction dynamics of frictional hard disks in two dimensions, subjected to vertical shaking. Shaking is modeled by a series of vertical expansions of the disk packing, followed by dynamical recompression of the assembly under the action of gravity. The second phase of the shake cycle is based on an efficient event−driven molecular−dynamics algorithm. We analyze the compaction dynamics for various values of friction coefficient and coefficient of normal restitution. We find that the time evolution of the density is described by ρ(t)=ρ∞ − ρEα[−(t/τ)α], where Eα denotes the Mittag−Leffler function of order 0<α<1. The parameter τ is found to decay with tapping intensity Γ according to a power law τ ∝ Γ−γ , where parameter γ is almost independent of the material properties of grains. Also, an expression for the grain mobility during compaction process has been obtained.

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43

Wang, Dengming, and Youhe Zhou. "Particle dynamics in dense shear granular flow." Acta Mechanica Sinica 26, no. 1 (December 1, 2009): 91–100. http://dx.doi.org/10.1007/s10409-009-0322-y.

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44

Pastor, J. M., D. Maza, I. Zuriguel, A. Garcimartín, and J. F. Boudet. "Time resolved particle dynamics in granular convection." Physica D: Nonlinear Phenomena 232, no. 2 (August 2007): 128–35. http://dx.doi.org/10.1016/j.physd.2007.06.005.

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45

Eckern, Ulrich, and Albert Schmid. "Quantum vortex dynamics in granular superconducting films." Physical Review B 39, no. 10 (April 1, 1989): 6441–54. http://dx.doi.org/10.1103/physrevb.39.6441.

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46

Petri, Alberto, Andrea Baldassarri, Fergal Dalton, Giorgio Pontuale, and Stefano Zapperi. "Stick‐slip dynamics of a granular medium." Journal of the Acoustical Society of America 123, no. 5 (May 2008): 3269. http://dx.doi.org/10.1121/1.2933595.

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47

Pöschel, Thorsten, and Volkhard Buchholtz. "Molecular Dynamics of Arbitrarily Shaped Granular Particles." Journal de Physique I 5, no. 11 (November 1995): 1431–55. http://dx.doi.org/10.1051/jp1:1995208.

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48

Bobaru, Florin, J. S. Chen, and Joseph A. Turner. "Advances in the Dynamics of Granular Materials." Mechanics of Materials 41, no. 6 (June 2009): 635–36. http://dx.doi.org/10.1016/j.mechmat.2009.02.007.

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49

Maynar, P., M. I. García de Soria, and J. Javier Brey. "Homogeneous dynamics in a vibrated granular monolayer." Journal of Statistical Mechanics: Theory and Experiment 2019, no. 9 (September 16, 2019): 093205. http://dx.doi.org/10.1088/1742-5468/ab3410.

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Porter, Mason A., Panayotis G. Kevrekidis, and Chiara Daraio. "Granular crystals: Nonlinear dynamics meets materials engineering." Physics Today 68, no. 11 (November 2015): 44–50. http://dx.doi.org/10.1063/pt.3.2981.

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