Assessing continuum postulates in simulations of granular flow

Continuum mechanics relies on the fundamental notion of a mesoscopic volume"element" in which properties averaged over discrete particles obey deterministic relationships. Recent work on granular materials suggests a continuum law may be inapplicable, revealing inhomogeneities at the particle level, such as force chains and slow cage breaking. Here, we analyze large-scale three-dimensional Discrete-Element Method (DEM) simulations of different granular flows and show that an approximate"granular element" defined at the scale of observed dynamical correlations (roughly three to five particle diameters) has a reasonable continuum interpretation. By viewing all the simulations as an ensemble of granular elements which deform and move with the flow, we can track material evolution at a local level. Our results confirm some of the hypotheses of classical plasticity theory while contradicting others and suggest a subtle physical picture of granular failure, combining liquid-like dependence on deformation rate and solid-like dependence on strain. Our computational methods and results can be used to guide the development of more realistic continuum models, based on observed local relationships betweenaverage variables.

[1]  H. Jaeger,et al.  Physics of the Granular State , 1992, Science.

[2]  Lev S Tsimring,et al.  Continuum theory of partially fluidized granular flows. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  C Goldenberg,et al.  Force chains, microelasticity, and macroelasticity. , 2002, Physical review letters.

[4]  F. Stillinger,et al.  Jamming in hard sphere and disk packings , 2004 .

[5]  C. Rycroft,et al.  Analysis of granular flow in a pebble-bed nuclear reactor. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Silke Henkes,et al.  Entropy and temperature of a static granular assembly: an ab initio approach. , 2007, Physical review letters.

[7]  J. Jenkins,et al.  A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles , 1983, Journal of Fluid Mechanics.

[8]  A. Schofield,et al.  Critical State Soil Mechanics , 1968 .

[9]  Vladimir Lorman,et al.  Local mechanism for crystal-quasicrystal transformations , 1996 .

[10]  C. Rycroft,et al.  Dynamics of random packings in granular flow. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  César Treviño,et al.  Velocity field measurements in granular gravity flow in a near 2D silo , 1998 .

[12]  G. Midi,et al.  On dense granular flows , 2003, The European physical journal. E, Soft matter.

[13]  S.F.Edwards,et al.  Statistical Mechanics of Vibration-Induced Compaction of Powders , 1997, cond-mat/9707276.

[14]  G. Grest,et al.  Granular flow down an inclined plane: Bagnold scaling and rheology. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  L S Tsimring,et al.  Continuum description of avalanches in granular media. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Andrea J. Liu,et al.  Jamming at zero temperature and zero applied stress: the epitome of disorder. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Anaël Lemaître,et al.  Rearrangements and dilatancy for sheared dense materials. , 2002, Physical review letters.

[18]  Jean-Noël Roux,et al.  Rheophysics of dense granular materials: discrete simulation of plane shear flows. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  T. Majmudar,et al.  Contact force measurements and stress-induced anisotropy in granular materials , 2005, Nature.

[20]  C. H. Liu,et al.  Force Fluctuations in Bead Packs , 1995, Science.

[21]  M. Bazant,et al.  Velocity profile of granular flows inside silos and hoppers , 2005, cond-mat/0501568.

[22]  H. Jaeger,et al.  Force distributions in three-dimensional granular assemblies: effects of packing order and interparticle friction. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Martin Ostoja-Starzewski,et al.  Scale effects in plasticity of random media: status and challenges , 2005 .

[24]  Gary S Grest,et al.  Confined granular packings: structure, stress, and forces. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  J. Langer,et al.  Dynamics of viscoplastic deformation in amorphous solids , 1997, cond-mat/9712114.

[26]  K. K. Rao.  Statics and kinematics of granular materials , 1995 .

[27]  Martin van Hecke,et al.  Kinematics: Wide shear zones in granular bulk flow , 2003, Nature.

[28]  Gary S Grest,et al.  Plug flow and the breakdown of Bagnold scaling in cohesive granular flows. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  P. W. Rowe.  The stress-dilatancy relation for static equilibrium of an assembly of particles in contact , 1962, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[30]  J. Kurchan,et al.  Testing the thermodynamic approach to granular matter with a numerical model of a decisive experiment , 2002, Nature.

[31]  Heinrich M. Jaeger,et al.  FORCE DISTRIBUTION IN A GRANULAR MEDIUM , 1998 .

[32]  D. C. Drucker,et al.  Soil mechanics and plastic analysis or limit design , 1952 .

[33]  R. M. Nedderman,et al.  A kinematic model for the flow of granular materials , 1979 .

[34]  I. Goldhirsch,et al.  The classical granular temperature and slightly beyond , 2007, cond-mat/0702545.

[35]  C Goldenberg,et al.  On the microscopic foundations of elasticity , 2002, The European physical journal. E, Soft matter.

[36]  Gary S Grest,et al.  Statistics of the contact network in frictional and frictionless granular packings. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Martin van Hecke,et al.  Stresses in smooth flows of dense granular media , 2006 .

[38]  Z. Hashin.  Analysis of Composite Materials—A Survey , 1983 .

[39]  I. Goldhirsch,et al.  Small and large scale granular statics , 2003, cond-mat/0308603.

[40]  Sam F. Edwards,et al.  The equations of stress in a granular material , 1998 .

[41]  A. Mehta,et al.  Challenges in Granular Physics , 2003 .

[42]  Arshad Kudrolli,et al.  Diffusion and mixing in gravity-driven dense granular flows. , 2004, Physical review letters.

[43]  P. Cundall,et al.  A discrete numerical model for granular assemblies , 1979 .

[44]  S. Nemat-Nasser,et al.  A Micromechanical Description of Granular Material Behavior , 1981 .

[45]  Martin Z. Bazant,et al.  The stochastic flow rule: a multi-scale model for granular plasticity , 2006, cond-mat/0611391.

[46]  A.J.M. Spencer,et al.  A theory of the kinematics of ideal soils under plane strain conditions , 1964 .

[47]  Olivier Pouliquen,et al.  A constitutive law for dense granular flows , 2006, Nature.

[48]  Pierre A. Gremaud,et al.  On the Computation of Steady Hopper Flows: I. Stress Determination for Coulomb Materials , 2001 .

[49]  J. Rice,et al.  CONDITIONS FOR THE LOCALIZATION OF DEFORMATION IN PRESSURE-SENSITIVE DILATANT MATERIALS , 1975 .

[50]  H. Jaeger,et al.  Granular solids, liquids, and gases , 1996 .

[51]  Lallit Anand,et al.  Granular materials: constitutive equations and strain localization , 2000 .

[52]  Measurements of particle dynamics in slow, dense granular Couette flow. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  Ken Kamrin,et al.  Stochastic flow rule for granular materials. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  R. M. Nedderman,et al.  Analytical Methods in Bin-load Analysis. , 1992 .

[55]  David G. Schaeffer,et al.  On the computation of steady hopper flows III: Model comparisons , 2006, J. Comput. Phys..

[56]  P. G. de Gennes,et al.  Granular Matter: A Tentative View , 1999 .

[57]  Jaehyuk Choi.  Transport-limited aggregation and dense granular flow , 2005 .

[58]  Numerical tests of constitutive laws for dense granular flows. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[59]  G Combe,et al.  Scale separation in granular packings: stress plateaus and fluctuations. , 2006, Physical review letters.

[60]  Martin Z. Bazant.  The Spot Model for random-packing dynamics , 2005 .

[61]  P. Gremaud,et al.  On the computation of steady Hopper flows II: von Mises materials in various geometries , 2004 .

[62]  Olivier Pouliquen,et al.  SCALING LAWS IN GRANULAR FLOWS DOWN ROUGH INCLINED PLANES , 1999 .