Abstract
A complete continuum mechanical theory for granular media, including explicit expressions for the energy current and the entropy production, is derived and explained. Its underlying notion is: granular media are elastic when at rest, but turn transiently elastic when the grains are agitated—such as by tapping or shearing. The theory includes the true temperature as a variable, and employs in addition a granular temperature to quantify the extent of agitation. A free energy expression is provided that contains the full jamming phase diagram, in the space spanned by pressure, shear stress, density and granular temperature. We refer to the theory as GSH, for granular solid hydrodynamics. In the static limit, it reduces to granular elasticity, shown previously to yield realistic static stress distributions. For steady-state deformations, it is equivalent to hypoplasticity, a state-of-the-art engineering model.
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References
Jaeger H.M., Nagel S.R., Behringer R.P.: Granular solids, liquids, and gases. Rev. Mod. Phys. 68(4), 1259 (1996)
Liu A.J., Nagel S.R.: Jamming is not just cool any more. Nature 396, 21 (1998)
de Gennes P.G.: Granular matter: a tentative view. Rev. Mod. Phys. 71(2), 347 (1999)
Kadanoff L.P.: Built upon sand: theoretical ideas inspired by the flow of granular materials. Rev. Mod. Phys. 71(1), 435 (1999)
Haff P.K.: Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401 (1983)
Jenkins J.T., Savage S.B.: A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 187 (1983)
Campbell C.S.: Rapid granular flows. Annu. Rev. Fluid Mech. 22, 57 (1990)
Herrmann, H.J.,Hovi, J.-P., Luding, S. (eds.): Physics of DryGranular Media. Kluwer, Dordrecht (1998)
Goldhirsch I.: Scales and kinetics of granular flows. Chaos 9, 659 (1999)
Mehte A.: Granular Physics. Cambridge University Press, Cambridge (2007)
Silbert L.E., Ertas D., Grest G.S., Halsey T.C., Levine D., Plimpton S.J.: Granular flow down an inclined plane: bagnold scaling and rheology. Phys. Rev. E 64, 051302 (2001)
MiDi G.D.R.: On dense granular flows. Eur. Phys. J. E 14, 341 (2004)
Jop P., Forterre Y., Pouliquen O.: A constitutive law for dense granular flows. Nature 441, 727 (2006)
Alonso-Marroquin F., Herrmann H.J.: Calculation of the incremental stress-strain relation of a polygonal packing. Phys. Rev. E 66, 021301 (2002)
Alonso-Marroquin F., Herrmann H.J.: Ratcheting of granular materials. Phys. Rev. Lett. 92, 054301 (2004)
Garcia-Rojo R., Alonso-Marroquin F., Herrmann H.J.: Characterization of the material response in granular ratcheting. Phys. Rev. E 72, 041302 (2005)
Nedderman R.M.: Statics and Kinematics of Granular Materials. Cambridge University Press, Cambridge (1992)
Schofield A., Wroth P.: Critical State Soil Mechanics. McGraw-Hill, London (1968)
Huang, W.X. (eds): Engineering Properties of Soil, 1st edn. Hydroelectricity Publishing, Beijing (1983)
Kolymbas D.: Introduction to Hypoplasticity. Balkema, Rotterdam (2000)
Kolymbas, D., Wu, W., Kolymbas, D.: In: Kolymbas, D. (ed.) Constitutive Modelling of Granular Materials. Springer, Berlin (2000), and references therein
Truesdell C., Noll W.: The Nonlinear Field Theories of Mechanics, Handbuch der Physik III/c. Springer, Berlin (1965)
Truesdell C.: Continuum Mechanics, vol 1. and 2. Gordon and Breach, New York (1965)
Liu I.-S.: Continuum Mchanics. Springer, Berlin (2002)
Landau L.D., Lifshitz E.M.: Fluid Mechanics. Butterworth-Heinemann, Oxford (1987)
Landau L.D., Lifshitz E.M.: Theory of Elasticity. Butterworth-Heinemann, Oxford (1986)
Khalatnikov I.M.: Introduction to the Theory of Superfuidity. Benjamin, New York (1965)
de Groot S.R., Masur P.: Non-Equilibrium Thermodynamics. Dover, New York (1984)
Forster D.: Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions. Benjamin, New York (1975)
de Gennes P.G., Prost J.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1993)
Martin P.C., Parodi O., Pershan P.S.: Unified hydrodynamic theory for crystals, liquid crystals, and normal fluids. Phys. Rev. A 6, 2401 (1972)
Lubensky T.C.: Hydrodynamics of cholesteric liquid crystals. Phys. Rev. A 6, 452 (1972)
Liu M.: Hydrodynamic theory near the nematic Smectic-A transition. Phys. Rev. A 19, 2090 (1979)
Liu M.: Hydrodynamic theory of biaxial nematics. Phys. Rev. A 24, 2720 (1981)
Liu M.: Maxwell equations in nematic liquid crystals. Phys. Rev. E 50, 2925 (1994)
Pleiner, H., Brand, H.R.: In: Buka, A., Kramer, L. (eds.) Pattern Formation in Liquid Crystals. Springer, New York (1996)
Graham R.: Hydrodynamics of 3He in Anisotropic A Phase. Phys. Rev. Lett. 33, 1431 (1974)
Graham R., Pleiner H.: Spin hydrodynamics of 3He in the anisotropic A phase. Phys. Rev. Lett. 34, 792 (1975)
Liu M.: Hydrodynamics of 3He near the A-transition. Phys. Rev. Lett. 35, 1577 (1975)
Liu M., Cross M.C.: Broken spin-orbit symmetry in superfluid 3He and the B-phase dynamics. Phys. Rev. Lett. 41, 250 (1978)
Liu M., Cross M.C.: Gauge Wheel of Superfluid 3He. Phys. Rev. Lett. 43, 296 (1979)
Liu M.: Relative broken symmetry and the dynamics of the A 1-phase. Phys. Rev. Lett. 43, 1740 (1979)
Liu M.: Rotating superconductors and the frame-independent London equations. Phys. Rev. Lett. 81, 3223 (1998)
Jiang Y.M., Liu M.: Rotating superconductors and the London moment: thermodynamics versus microscopics. Phys. Rev. B 6, 184506 (2001)
Liu M.: Superconducting hydrodynamics and the higgs analogy. J. Low Temp. Phys. 126, 911 (2002)
Henjes K., Liu M.: Hydrodynamics of polarizable liquids. Ann. Phys. 223, 243 (1993)
Liu M.: Hydrodynamic theory of electromagnetic fields in continuous media. Phys. Rev. Lett. 70, 3580 (1993)
Liu M.: Mario Liu replies. Phys. Rev. Lett. 74, 1884 (1995)
Jiang Y.M., Liu M.: Dynamics of dispersive and nonlinear media. Phys. Rev. Lett. 77, 1043 (1996)
Shliomis M.I.: Magnetic fluids. Sov. Phys. Usp. 17, 153 (1974)
Rosensweig R.E.: Ferrohydrodynamics. Dover, New York (1997)
Liu M.: Fluiddynamics of colloidal magnetic and electric liquid. Phys. Rev. Lett. 74, 4535 (1995)
Liu M.: Off-Equilibrium, static fields in dielectric ferrofluids. Phys. Rev. Lett. 80, 2937 (1998)
Liu M.: Electromagnetic fields in ferrofluids. Phys. Rev. E 59, 3669 (1999)
Müller H.W., Liu M.: Structure of ferro-fluiddynamics. Phys. Rev. E 64, 061405 (2001)
Müller H.W., Liu M.: Shear excited sound in magnetic fluid. Phys. Rev. Lett. 89, 67201 (2002)
Müller O., Hahn D., Liu M.: Non-Newtonian behaviour in ferrofluids and magnetization relaxation. J. Phys. Condens. Matter 18, 2623 (2006)
Mahle S., Ilg P., Liu M.: Hydrodynamic theory of polydisperse chain-forming ferrofluids. Phys. Rev. E 77, 016305 (2008)
Temmen H., Pleiner H., Liu M., Brand H.R.: Convective nonlinearity in non-newtonian fluids. Phys. Rev. Lett. 84, 3228 (2000)
Temmen H., Pleiner H., Liu M., Brand H.R.: Temmen et al. reply. Phys. Rev. Lett. 86, 745 (2001)
Pleiner H., Liu M., Brand H.R.: Nonlinear fluid dynamics description of non-newtonian fluids. Rheol. Acta 43, 502 (2004)
Müller, O.: Die Hydrodynamische Theorie Polymerer Fluide. PhD Thesis, University of Tübingen (2006)
Jiang Y.M., Liu M.: Granular elasticity without the Coulomb condition. Phys. Rev. Lett. 91, 144301 (2003)
Jiang Y.M., Liu M.: Energy instability unjams sand and suspension. Phys. Rev. Lett. 93, 148001 (2004)
Jiang Y.M., Liu M.: A brief review of “granular elasticity”. Eur. Phys. J. E 22, 255 (2007)
Krimer D.O., Pfitzner M., Bräuer K., Jiang Y., Liu M.: Granular elasticity: general considerations and the stress dip in sand piles. Phys. Rev. E 74, 061310 (2006)
Bräuer K., Pfitzner M., Krimer D.O., Mayer M., Jiang Y., Liu M.: Granular elasticity: stress distributions in silos and under point loads. Phys. Rev. E 74, 061311 (2006)
Jiang Y.M., Liu M.: From elasticity to hypoplasticity: dynamics of granular solids. Phys. Rev. Lett. 99, 105501 (2007)
Ono I.K., O’Hern C.S., Durian D.J., Langer S.A., Liu A.J., Nagel S.R.: Effective temperatures of a driven system near jamming. Phys. Rev. Lett. 89, 095703 (2002)
Kostädt P., Liu M.: Three ignored densities, frame-independent thermodynamics, and broken Galilean symmetry. Phys. Rev. E 58, 5535 (1998)
Liu M.: Comment on “weakly and strongly consistent formu- lation of irreversible processes”. Phys. Rev. Lett. 100, 098901 (2008)
Onoda G.Y., Liniger E.G.: Random loose packings of uniform spheres and the dilatancy onset. Phys. Rev. Lett. 64, 2727 (1990)
Hardin B.O., Richart F.E.: Elastic wave velocities in granular soils. J. Soil Mech. Found. Div. ASCE 89(SM1), 33–65 (1963)
Kuwano R., Jardine R.J.: On the applicability of cross-anisotropic elasticity to granular materials at very small strains. Geotechnique 52, 727 (2002)
Jiang Y.M., Liu M.: Incremental stress-strain relation from granular elasticity: comparison to experiments. Phys. Rev. E 77, 021306 (2008)
Bocquet L., Errami J., Lubensky T.C.: Hydrodynamic Model for a dynamical jammed-to-flowing transition in gravity driven granular media. Phys. Rev. Lett. 89, 184301 (2002)
Losert W., Bocquet L., Lubensky T.C., Gollub J.P.: Particle dynamics in sheared granular matter. Phys. Rev. Lett. 85, 1428 (2000)
Bocquet L., Losert W., Schalk D., Lubensky T.C., Gollub J.P.: Granular shear flow dynamics and forces: experiment and continuum theory. Phys. Rev. E 65, 011307 (2002)
Goldhirsch I.: Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267 (2003)
Johnson P.A., Jia X.: Nonlinear dynamics, granular media and dynamic earthquake triggering. Nature 437/6, 871 (2005)
Edwards S.F., Oakeshott R.B.S.: Theory of powders. Physica A 157, 1080 (1989)
Edwards S.F., Grinev D.V.: Statistical mechanics of granular materials: stress propagation and distribution of contact forces. Granul. Matter 4, 147 (2003)
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Jiang, Y., Liu, M. Granular solid hydrodynamics. Granular Matter 11, 139–156 (2009). https://doi.org/10.1007/s10035-009-0137-3
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DOI: https://doi.org/10.1007/s10035-009-0137-3