Hostname: page-component-7bb8b95d7b-dtkg6 Total loading time: 0 Render date: 2024-10-05T09:46:36.919Z Has data issue: false hasContentIssue false
  • English
  • Français

Frictional–collisional constitutive relations for granular materials, with application to plane shearing

Published online by Cambridge University Press:  21 April 2006

P. C. Johnson  and
R. Jackson
P. C. Johnson
Affiliation:
Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
R. Jackson
Affiliation:
Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Within a granular material stress is transmitted by forces exerted at points of mutual contact between particles. When the particles are close together and deformation of the assembly is slow, contacts are sustained for long times, and these forces consist of normal reactions and the associated tangential forces due to friction. When the particles are widely spaced and deformation is rapid, on the other hand, contacts are brief and may be regarded as collisions, during which momentum is transferred. While constitutive relations are available which model both these situations, in many cases the average contact times lie between the two extremes. The purpose of the present work is to propose constitutive relations and boundary conditions for this intermediate case and to solve the corresponding equations of motion for plane shear of a cohesionless granular material between infinite horizontal plates. It is shown that, in general, not all the material between the plates participates in shearing, and the solutions for the shearing material are coupled to a yield condition for the non-shearing material to give a complete solution of the problem.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 176 , March 1987 , pp. 67 - 93
Copyright
© 1987 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ackerman, N. L. & Shen, H. 1982 Stresses in rapidly sheared fluid–solid mixtures. J. Engng Mech. Div. ASCE, 108, 95–113.Google Scholar
Bagnold, R. A. 1954 Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. A 225, 49–63.Google Scholar
Bridgwater, J. 1972 Stress–velocity relationships for particulate solids. ASME paper no. 72-MH-21.
Carr, J. F. & Walker, D. M. 1967 An annular shear cell for granular materials. Powder Tech. 1, 369–373.Google Scholar
Coulomb, C. A. 1776 Essai sur une application des règles de maximis et minimis à quelques problèmes de statique, relatifs à l'architecture. Acad. R. Sci. Mém. Math. Phys. par Divers Savants 7, 343–382.Google Scholar
de Jong, G. de J. 1959 Statics and kinematics in the failable zone of granular material. thesis, University of Delft.
Drucker, D. C. & Prager, W. 1952 Soil mechanics and plastic analysis or limit design. Q. Appl. Maths. 10, 157–165.Google Scholar
Haff, P. K. 1983 Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430.Google Scholar
Hanes, D. M. & Inman, D. L. 1985 Observations of rapidly flowing granular-fluid materials. J. Fluid Mech. 150, 357–380.Google Scholar
Hui, K., Haff, P. K., Ungar, J. E. & Jackson, R. 1984 Boundary conditions for high-shear grain flows. J. Fluid Mech. 145, 223–233.Google Scholar
Jenike, A. W. & Shield, R. T. 1959 On the plastic flow of Coulomb solids beyond failure. Trans. ASME E: J. Appl. Mech. 26, 599–602Google Scholar
Jenkins, J. T. & Richman, M. W. 1986 Boundary conditions for plane flows of smooth, nearly elastic, circular disks. J. Fluid Mech. 171, 53–69.Google Scholar
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202.Google Scholar
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. 140, 223–256.Google Scholar
Reynolds, O. 1885 On the dilatancy of media composed of rigid particles in contact. Phil. Mag. 20, 469–481.Google Scholar
Roscoe, K. H. 1970 The influence of strains in soil mechanics. Géotechnique 20, 129–170.Google Scholar
Roscoe, K. H., Schofield, A. N. & Wroth, C. P. 1958 On the yielding of soils. Géotechnique 8, 22–53.Google Scholar
Savage, S. B. 1982 Granular flows down rough inclines–review and extension. In Proc. of U.S.-Japan Seminar on New Models and Constitutive Relations in the Mechanics of Granular Materials (ed. J. T. Jenkins & M. Satake). Elsevier.
Savage, S. B. & Jeffrey, D. J. 1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 225–272.Google Scholar
Savage, S. B. & Sayed, M. 1984 Stresses developed by dry cohesionless granular materials sheared in an annular shear cell. J. Fluid Mech. 142, 391–430.Google Scholar
Scarlett, B. & Todd, A. C. 1969 The critical porosity of free flowing solids. Trans. ASME B: J. Engng Ind. 91, 477–488Google Scholar
Schofield, A. N. & Wroth, C. P. 1968 Critical State Soil Mechanics. McGraw-Hill.
Walton, O. R. & Braun, R. L. 1985 Viscosity, granular-temperature and stress calculations for shearing assemblies of inelastic frictional disks. Quarterly Technical report for U.S. DOE, contract number W-7405-Eng-48, 25 June.