Mathematical Modelling of Granular Materials

  • Mehrdad MassoudiEmail author


In this chapter, we provide a brief overview of the important issues in modelling of granular materials. A continuum mechanics approach is taken where it is assumed that the material behaves similar to a compressible non-linear fluid where the effects of density gradients are incorporated in the stress tensor. We discuss and solve the heat transfer in granular materials flowing down an inclined plane with radiation effects at the free surface. For a fully developed flow, the equations simplify to a system of three non-linear ordinary differential equations. The equations are made dimensionless and a parametric study is performed where the effects of various dimensionless numbers representing the effects of heat conduction, viscous dissipation, radiation, etc. are presented.


Free Surface Constitutive Relation Granular Material Kinetic Theory Viscous Dissipation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Adams MJ, Briscoe BJ, Kamjab M (1993) The deformation and flow of highly concentrated dispersions. Adv Colloid Interface Sci 44:143–182CrossRefGoogle Scholar
  2. 2.
    Ahmadi GA (1982) Generalized continuum theory for granular materials. Int J Non-Linear Mech 17:21–33zbMATHCrossRefGoogle Scholar
  3. 3.
    Antman, SS (1995) Nonlinear problems of elasticity. Springer-Verlag, New YorkzbMATHGoogle Scholar
  4. 4.
    Antony SJ, Hoyle W, Ding Y (2004) Granular materials: Fundamentals and applications. The R Soc of Chemistry, Cambridge, UKGoogle Scholar
  5. 5.
    Astarita T, Ocone R (1999) Unsteady compressible granular materials. Ind Eng Chem Res 38:1177–1182CrossRefGoogle Scholar
  6. 6.
    Baek S, Rajagopal KR, Srinivasa AR (2001) Measurements related to the flow of granular materials in a torsional rheometer. Part Sci Technol 19:175–186CrossRefGoogle Scholar
  7. 7.
    Bagnold RA (1954) Experiments on a gravity free dispersion of large solid spheres in a Newtonian fluid under shear. Proc R Soc Lond 225:49CrossRefGoogle Scholar
  8. 8.
    Batra RC (2006) Elements of continuum mechanics. American Institute of Aeronautics and Astronautics (AIAA) Inc., Reston, VAzbMATHCrossRefGoogle Scholar
  9. 9.
    Bingham EC (1922) Fluidity and plasticity. McGraw-Hill, New YorkGoogle Scholar
  10. 10.
    Bingham EC, Green H (1919) Testing materials II. American Assoc 19:610Google Scholar
  11. 11.
    Blinowski A (1978) On the dynamic flow of granular media. Arch Mech 30:27–34MathSciNetzbMATHGoogle Scholar
  12. 12.
    Boyle EJ, Massoudi M (1990) A theory for granular materials exhibiting normal stress effects based on Enskog’s dense gas theory. Int J Eng Sci 28:1261–1275zbMATHCrossRefGoogle Scholar
  13. 13.
    Brown RL, Richards JC (1970) Principles of powder mechanics. Pergamon Press, LondonGoogle Scholar
  14. 14.
    Caswell B (2006) Non-Newtonian flow at lowest order, the role of the Reiner-Rivlin stress. J Non-Newt Fluid Mech 133:1–13zbMATHCrossRefGoogle Scholar
  15. 15.
    Chapman S, Cowling TG (1990) The mathematical theory of non-uniform gases. Cambridge University Press, Cambridge, MAGoogle Scholar
  16. 16.
    Collins IF, Houlsby GT (1997) Applications of thermomechanical principles to the modeling of geotechnical materials. Proc R Soc Lond A 453:1975–2001zbMATHCrossRefGoogle Scholar
  17. 17.
    Collins IF (2005) Elastic/plastic models for soils and sands. Int J Mech Sci 47:493–508zbMATHCrossRefGoogle Scholar
  18. 18.
    Coussot P (2005) Rheometry of pastes, suspensions, and granular materials. Wiley, Hoboken, NJCrossRefGoogle Scholar
  19. 19.
    Cowin SC (1974) A theory for the flow of granular material. Powder Tech 9:61–69CrossRefGoogle Scholar
  20. 20.
    Cowin SC (1974) Constitutive relations that imply a generalized Mohr-Coulomb criterion. Acta Mech 20:41–46zbMATHCrossRefGoogle Scholar
  21. 21.
    Craig K, Buckholtz RH, Domoto G (1987) The effects of shear surface boundaries on stresses for the rapid shear dry powders. ASME J Tribol 109:232CrossRefGoogle Scholar
  22. 22.
    de Gennes PG (1998) Reflections on the mechanics of granular matter. Physica A 261: 267–293CrossRefGoogle Scholar
  23. 23.
    Duran J (2000) Sands, powders and grains. Springer, New YorkzbMATHCrossRefGoogle Scholar
  24. 24.
    Elaskar SA, Godoy LA (1998) Constitutive relations for compressible granular materials using non-Newtonian fluid mechanics. Int J Mech Sci 40:1001–1018zbMATHCrossRefGoogle Scholar
  25. 25.
    Fan LS, Zhu C (1998) Principles of gas–solid flows. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  26. 26.
    Fang C, Wang Y, Hutter K (2006) A thermo-mechanical continuum theory with internal length for cohesionless granular materials. Part I. A class of constitutive models. Continuum Mech Thermodyn 17:545–576MathSciNetzbMATHGoogle Scholar
  27. 27.
    Fuchs HU (1996) The dynamics of heat. Springer-Verlag Inc., New YorkzbMATHGoogle Scholar
  28. 28.
    Gidaspow D (1994) Multiphase flow and fluidization. Academic Press, San DiegozbMATHGoogle Scholar
  29. 29.
    Goddard JD (1984) Dissipative materials as models of thixotropy and plasticity. J Non-Newt Fluid Mech 14:141zbMATHCrossRefGoogle Scholar
  30. 30.
    Goldhirsch I (2003) Rapid granular flows. Annu Rev Fluid Mech 35:267–293MathSciNetCrossRefGoogle Scholar
  31. 31.
    Gudhe R, Rajagopal KR, Massoudi M (1994) Heat transfer and flow of granular materials down an inclined plane. Acta Mech 103:63–78MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Gutt GM, Haff PK (1991) Boundary conditions on continuum theories of granular material flow. Int J Multiphase Flow 17:621zbMATHCrossRefGoogle Scholar
  33. 33.
    Haff PK (1983) Grain flow as a fluid-mechanical phenomenon. J Fluid Mech 134:401–430zbMATHCrossRefGoogle Scholar
  34. 34.
    Haff PK (1986) A physical picture of kinetic granular fluids. J Rheol 30:931CrossRefGoogle Scholar
  35. 35.
    Hanes DM, Inman DL (1985) Observations of rapidly flowing granular–fluid materials. J Fluid Mech 150:357CrossRefGoogle Scholar
  36. 36.
    Hermann HJ (1999) Statistical models for granular materials. Physica A, 263:51–62MathSciNetCrossRefGoogle Scholar
  37. 37.
    Hermann HJ, Luding S (1998) Modeling granular media on the computer. Continuum Mech Thermodyn 10:189–231CrossRefGoogle Scholar
  38. 38.
    Hui K, Haff PK, Unger JE et al (1984) Boundary conditions for high-shear grain flows. J Fluid Mech 145:223–233zbMATHCrossRefGoogle Scholar
  39. 39.
    Hutter K (1983) Theoretical glaciology. Kluwer, Hingham, MACrossRefGoogle Scholar
  40. 40.
    Hutter K, Szidarovszky F, Yakowitz S (1986) Plane steady shear flow of a cohesionless granular material down an inclined plane: A model for flow avalanches Part I. Acta Mech 63:87–112zbMATHCrossRefGoogle Scholar
  41. 41.
    Hutter K, Szidarovszky F, Yakowitz S (1986) Plane steady shear flow of a cohesionless granular material down an inclined plane: A model for flow avalanches Part II: Numerical Results. Acta Mech 63:239–261Google Scholar
  42. 42.
    Hutter K, Rajagopal KR (1994) On the flows of granular materials. Continuum Mech. Thermodyn 6:81–139MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Jaeger HM, Nagel SR, Behringer RP (1996) Granular solids, liquids, and gases. Rev Modern Phys 68:1259CrossRefGoogle Scholar
  44. 44.
    Jackson R (2000) The dynamics of fluidized particles. Cambridge University Press, Cambridge, MAzbMATHGoogle Scholar
  45. 45.
    Jeffrey DJ (1973) Conduction through a random suspension of spheres. Proc R Soc Lond A 335:355–367CrossRefGoogle Scholar
  46. 46.
    Johnson PC, Jackson R (1987) Frictional–collisional constitutive relations for granular materials with application to plane shearing. J Fluid Mech 176:67–93CrossRefGoogle Scholar
  47. 47.
    Jyotsna R, Kesava Rao K (1997) A frictional-kinetic model for the flow of granular materials through a wedge-shaped hopper. J Fluid Mech 346:239–270zbMATHCrossRefGoogle Scholar
  48. 48.
    Kalman H, Tardos GI (2005) Elements of particle technology in the chemical industry. Particulate Sci Tech 23:1–19CrossRefGoogle Scholar
  49. 49.
    Kanatani KI (1979) A micropolar continuum theory for the flow of granular materials. Int J Eng Sci 17:419–432zbMATHCrossRefGoogle Scholar
  50. 50.
    Kaviany M (1995) Principles of heat transfer in porous media. 2nd edn. Springer, New YorkzbMATHCrossRefGoogle Scholar
  51. 51.
    Klausner Y (1991) Fundamentals of continuum mechanics of soils. Springer, New YorkCrossRefGoogle Scholar
  52. 52.
    Liu IS (2002) Continuum mechanics. Springer, BerlinzbMATHGoogle Scholar
  53. 53.
    Lu SY, Kim S (1990) Effective thermal conductivity of composites containing spheroidal inclusions. AIChE J 36:927–938CrossRefGoogle Scholar
  54. 54.
    Lugt HJ, Schot JW (1974) A review of slip flow in continuum physics. In: Lugt HJ (ed) Proc Second Symp Fluid–Solid Surface Interactions. Naval Research and Development Center, Bethesda, MDGoogle Scholar
  55. 55.
    Marcus RD, Leung LS, Klinzing GE et al (1990) Pneumatic conveying of solids. Chapman and Hall, LondonCrossRefGoogle Scholar
  56. 56.
    Massoudi M (2001) On the flow of granular materials with variable material properties. Int J Non-Linear Mech 36:25MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Massoudi M (2004) Constitutive modelling of flowing granular materials: A continuum approach. In: Antony SJ, Hoyle W, Ding Y (eds) Granular materials: Fundamentals and applications. The Royal Society of Chemistry, Cambridge, UK, 63–107Google Scholar
  58. 58.
    Massoudi M (2005) An anisotropic constitutive relation for the stress tensor of a rod-like (fibrous-type) granular material. Math Probl Eng 679–702Google Scholar
  59. 59.
    Massoudi M (2006) On the heat flux vector for flowing granular materials, Part I: Effective thermal conductivity and background. Math Methods Appl Sci 29:1585–1598MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Massoudi M (2006) On the heat flux vector for flowing granular materials, Part II: Derivation and special cases. Math Methods Appl Sci 29: 1599–1613MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Massoudi M, Ahmadi GA (1994) Rapid flow of granular materials with density and fluctuation energy gradients. Int J Non-Linear Mech 29:487–492zbMATHCrossRefGoogle Scholar
  62. 62.
    Massoudi M, Boyle EJ (2001) A continuum-kinetic theory approach to the flow of granular materials: The effects of volume fraction gradient. Int J Non-Linear Mech 36:637–648zbMATHCrossRefGoogle Scholar
  63. 63.
    Massoudi M, Mehrabadi MM (2001) A continuum model for granular materials: Considering dilatancy, and the Mohr-Coulomb criterion. Acta Mech 152:121–138zbMATHCrossRefGoogle Scholar
  64. 64.
    Massoudi M, Phuoc TX (2006) Boundary value problems in heat transfer studies of granular materials modeled as compressible non-linear fluids. Math Probl in Eng Article ID 56046 pp. 1–31. doi:10.1155/MPE/2006/56046CrossRefGoogle Scholar
  65. 65.
    Massoudi M, Phuoc TX (2007) Conduction and dissipation in the shearing flow of granular materials modeled as non-Newtonian fluids. Powder Technol 175:146–162CrossRefGoogle Scholar
  66. 66.
    Maugin GA (1999) The thermomechanics of nonlinear irreversible behaviors. World Scientific Publishing Co, River Edge, NJzbMATHGoogle Scholar
  67. 67.
    McQuarrie DA (1976) Statistical mechanics. Harper & Row Publishers, New YorkGoogle Scholar
  68. 68.
    McTigue DF (1982) A non-linear constitutive model for granular materials: Applications to gravity flow. J Appl Mech 49:291–296zbMATHCrossRefGoogle Scholar
  69. 69.
    Mehrabadi MM, Nemat-Nasser S, Oda M (1982) On statistical description of stress and fabric in granular materials. Int J Numer Anal Methods Geomech 6:95–108MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Mehrabadi MM, Loret B, Nemat-Nasser S (1993) Incremental constitutive relations for granular materials based on micromechanics. Proc R Soc Lond A 441:433–463zbMATHCrossRefGoogle Scholar
  71. 71.
    Mehta A (ed) (1994) Granular matter. Springer, New YorkGoogle Scholar
  72. 72.
    Műller I (1967) On the entropy inequality. Arch Rat Mech and Anal 26:118–141CrossRefGoogle Scholar
  73. 73.
    Na TY (1979) Computational methods in engineering boundary value problems. Academic Press, New YorkzbMATHGoogle Scholar
  74. 74.
    Nedderman RM (1992) Statics and kinematics of granular materials. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  75. 75.
    Nemat-Nasser S, Mehrabadi MM (1984) Micromechanically based rate constitutive description for granular materials. In: Desai CS, Gallagher RH (eds) Mechanics of engineering materials. Wiley, 451–463Google Scholar
  76. 76.
    Nozad I, Carbonell RG, Whitaker S (1985) Heat conduction in multiphase systems-I: Theory and experiment for two-phase systems. Chem Eng Sci 40:847–855Google Scholar
  77. 77.
    Oda M (1972) The mechanism of fabric change during compressional deformation of sand. Soils Foundations 12:1–18CrossRefGoogle Scholar
  78. 78.
    Ogawa S (1978) Multitemperature theory of granular materials. In: Cowin SC, Satake M (eds) Proc U.S.–Japan Seminar on Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials. Sendai, Japan, 208–217Google Scholar
  79. 79.
    Ogawa S, Umemura A, Oshima N (1980) On the equations of fully fluidized granular materials. J Appl Math Phys (ZAMP) 31:483–493zbMATHCrossRefGoogle Scholar
  80. 80.
    Plasynski SI, Peters WC, Passman SL (1992) The department of energy solids transport, multiphase flow program. Proc NSF–DOE Joint Workshop on Flow of Particulates and Fluids, Gaithersburg, September 16–18Google Scholar
  81. 81.
    Prager W (1989) Introduction to mechanics of continua. Dover Publications, Inc, Mineola, NYGoogle Scholar
  82. 82.
    Prakash JR, Kesava Rao K (1988) Steady compressible flow of granular materials through a wedge-shaped hopper: The smooth wall, radial gravity problem. Chem Eng Sci 43:479–494CrossRefGoogle Scholar
  83. 83.
    Prakash JR, Kesava Rao K (1991) Steady compressible flow of cohesionless granular materials through a wedge-shaped bunker. J Fluid Mech 225:21–80MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Rajagopal KR (2003) On implicit constitutive theories. Appl Math 48:579–319MathSciNetCrossRefGoogle Scholar
  85. 85.
    Rajagopal KR (2006) On implicit constitutive theories for fluids. J Fluid Mech 550:243–249MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Rajagopal KR, Massoudi M (1990) A method for measuring material moduli of granular materials: Flow in an orthogonal rheometer. Topical Report, DOE/PETC/TR-90/3Google Scholar
  87. 87.
    Rajagopal KR, Troy WC, Massoudi M (1992) Existence of solutions to the equations governing the flow of granular materials. Eur J Mech B/Fluids 11:265–276MathSciNetzbMATHGoogle Scholar
  88. 88.
    Rajagopal KR, Massoudi M, Wineman AS (1994) Flow of granular materials between rotating disks. Mech Res Comm 21:629–634zbMATHCrossRefGoogle Scholar
  89. 89.
    Rajagopal KR, Tao L (1995) Mechanics of mixtures. World Scientific Publishing. River Edge, NJzbMATHGoogle Scholar
  90. 90.
    Rajagopal KR, Gupta G, Yalamanchili RC (2000) A rheometer for measuring the properties of granular materials. Part Sci Tech 18:39–55CrossRefGoogle Scholar
  91. 91.
    Rajagopal KR, Srinivasa AR (2004) On thermomechanical restrictions of continua. Proc R Soc Lond A 460:631–651MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Ranade VV (2002) Computational flow modeling for chemical reactor engineering. Academic Press, San DiegoGoogle Scholar
  93. 93.
    Rao K, Kesava, Nott PR (2008) An introduction to granular flow. Cambridge University Press, New YorkCrossRefGoogle Scholar
  94. 94.
    Reiner M (1945) A mathematical theory of dilatancy. Am J Math 67:350–362MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Reiner M (1948) Elasticity beyond the elastic limit. Am J Math 70:433–466MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Reiner M (1958) Rheology. In: Flugge S (ed) Handbuch Der Physik, Vol. VI. Springer, BerlinGoogle Scholar
  97. 97.
    Reynolds O (1885) On the dilatancy of media composed of rigid particles in contact with experimental illustrations. Phil Mag Series 5(20):469–481CrossRefGoogle Scholar
  98. 98.
    Reynolds O (1886) Experiments showing dilatancy, a property of granular material, possibly connected with gravitation. Proc R Inst GB 11:354–363Google Scholar
  99. 99.
    Rivlin RS (1948) The hydrodynamics of non-Newtonian fluids. I Proc R Soc Lond 193:260–281MathSciNetzbMATHCrossRefGoogle Scholar
  100. 100.
    Saldanha da Gama RM (2004) On the conduction/radiation heat transfer problem in a body with wavelength-dependent properties. Appl Math Model 28:795–816zbMATHCrossRefGoogle Scholar
  101. 101.
    Schotte W (1960) Thermal conductivity of packed beds. AIChE J 6:63–67CrossRefGoogle Scholar
  102. 102.
    Savage SB (1979) Gravity flow of cohesionless granular materials in chutes and channels. J Fluid Mech 92:53–96zbMATHCrossRefGoogle Scholar
  103. 103.
    Savage SB (1984) The mechanics of rapid granular flows. Adv Appl Mech 24:289–366zbMATHCrossRefGoogle Scholar
  104. 104.
    Savage SB, Sayed M (1984) Stress developed by dry cohesionless granular materials in an annular shear cell. J Fluid Mech 142:391–430CrossRefGoogle Scholar
  105. 105.
    Schaeffer DG (1987) Instability in the evolution equations describing incompressible granular flow. J Diff Eq 66:19–50MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    Soo SL (1990)Multiphase fluid dynamics. Science Press, BrookfieldGoogle Scholar
  107. 107.
    Spencer AJM (1982)Deformation of ideal granular materials. In: Mechanics of solids. Pergamon Press, Oxford and New York 607–652Google Scholar
  108. 108.
    Stepanoff AJ (1969) Gravity flow of bulk solids and transportation of solids in suspension. Wiley, New YorkGoogle Scholar
  109. 109.
    Straughan B, Greve R, Ehrentraut H et al (eds) (2001) Continuum mechanics and applications in geophysics and the environment. Springer, BerlinzbMATHGoogle Scholar
  110. 110.
    Szeri AZ (1998) Fluid film lubrication. Cambridge University Press, Cambridge, MAzbMATHCrossRefGoogle Scholar
  111. 111.
    Szeri AZ, Rajagopal KR (1985) Flow of a non-Newtonian fluid between heated parallel plates. Int J Non-Linear Mech 20:91MathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    Tardos GI (1997) A fluid mechanistic approach to slow, frictional flow of powders. Powder Technol 92:61–74CrossRefGoogle Scholar
  113. 113.
    Tardos GI, McNamara S, Talu I (2003) Slow and intermediate flow of a frictional bulk powder in the Couette geometry. Powder Technol 131:3–39CrossRefGoogle Scholar
  114. 114.
    Torquato S (1987) Thermal conductivity of disordered heterogeneous media from the microstructure. Rev Chem Eng 4:151CrossRefGoogle Scholar
  115. 115.
    Truesdell C (1976) The meaning of viscometry in fluid mechanics. Annu Rev Fluid Mech 6:111–146CrossRefGoogle Scholar
  116. 116.
    Truesdell C, Muncaster RG (1980) Fundamentals of Maxwell’s kinetic theory of a simple monatomic gas. Academic Press, New YorkGoogle Scholar
  117. 117.
    Truesdell C, Noll W (1992) The non-linear field theories of mechanics. Springer, New YorkzbMATHGoogle Scholar
  118. 118.
    Tsotsas E, Martin H (1987) Thermal conductivity of packed beds: A review. Chem Eng Process 22:19–37CrossRefGoogle Scholar
  119. 119.
    Walton OR, Braun RL (1986) Stress calculations for assemblies of inelastic spheres in uniform shear. Acta Mech 63(1–4):73–86CrossRefGoogle Scholar
  120. 120.
    Walton OR, Braun RL (1986) Viscosity, granular-temperature, and stress calculations for shearing assemblies of inelastic, frictional disks. J Rheol 30:949–980CrossRefGoogle Scholar
  121. 121.
    Zhang X, Jeffrey RG, Mai YW (2006) A micromechanical-based Cosserat-type model for dense particulate solids. Z Angew Math Phys (ZAMP) 57:682–707MathSciNetzbMATHCrossRefGoogle Scholar
  122. 122.
    Zhu H, Kim YD, De Kee D (2005) Non-Newtonian fluids with a yield stress. J Non-Newt Fluid Mech 129:177–181zbMATHCrossRefGoogle Scholar
  123. 123.
    Ziegler H (1983) An introduction to thermomechanics. 2nd revised edn. North-Holland, AmsterdamzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.U.S. Department of EnergyNational Energy Technology Laboratory (NETL)PittsburghUSA

Personalised recommendations