Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-09T18:59:08.298Z Has data issue: false hasContentIssue false
  • English
  • Français

The stress tensor for simple shear flows of a granular material

Published online by Cambridge University Press:  26 April 2006

Charles S. Campbell
Charles S. Campbell
Affiliation:
University of Southern California, Department of Mechanical Engineering, Los Angeles, CA 90089–1453, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The complete stress tensor has been measured using a computer simulation of an assemblage of rough, inelastic spheres in an imposed simple shear flow. Only five components of the stress tensor were found to be significantly different from zero. These represent the disperssive normal stresses τxx, τyy and τzz and the in-the-shear-plane shear stresses τxy and τyx; furthermore, the two off-diagonal stresses, τxy and τyx, were found to be equal so that the resultant stress tensor is symmetric. Two modes of microscopic momentum transport produce the final macroscopic stress tensor: the streaming or kinetic mode by which particles carry the momentum of their motion as they move through the bulk material, and the collisional mode by which momentum is transported by interparticle collisions. The contribution of each to the final result is examined separately. The friction coefficient, the ratio of shear to normal force, is shown to decrease at dense packings for both the collisional and streaming modes. Also observed were normal stress differences, both in and out of the shear plane, reflecting anisotropies in the granular temperature.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 203 , June 1989 , pp. 449 - 473
Copyright
© 1989 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

Ackermann, N. L. & Shen, H., 1979 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 particles in a Newtonian fluid under shear. Proc. R. Soc. Lond. A 225, 49–63.Google Scholar
Campbell, C. S.: 1982 Shear flows of granular materials. Ph.D. thesis and Rep. E-200.7. Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA.
Campbell, C. S.: 1986a The effect of microstructure development on the collisional stress tensor in a granular flow. Acta Mech. 63, 61–72.Google Scholar
Campbell, C. S.: 1986b Computer simulation of rapid granular flows. Proc. 10th Natl Cong. of Appl. Mech., Austin Texas, June 1986.
Campbell, C. S.: 1987 Boundary interactions for two-dimensional granular flows: asymmetric stresses and couple stresses. Proc. US-Japan Seminar on the Micromechanics of Granular Material, Sendai-Zao, Japan, October 26–30, 1987.Google Scholar
Campbell, C. S. & Brennen, C. E., 1985a Computer simulation of granular shear flows. J. Fluid Mech. 151, 167–188.Google Scholar
Campbell, C. S. & Brennen, C. E., 1985b Chute flows of granular material: some computer simulations. Trans. ASME E: J. Appl. Mech. 52, 172–178.Google Scholar
Campbell, C. S. & Gong, A., 1986 The stress tensor in a two-dimensional granular shear flow. J. Fluid Mech. 164, 107–125.Google Scholar
Campbell, C. S. & Gong, A., 1987 Boundary conditions for two-dimensional granular flows. Proc. Sino-US Intl Symp. on Multiphase Flows, Hangzhou, China, August, 1987, vol I, pp. 278–283Google Scholar
Campbell, C. S. & Wang, D. G., 1986 The effective conductivity of shearing particle flows. Proc. 8th Intl Heat Trans. Conf. vol. 4, pp. 2567–2572.Google Scholar
Chapman, S. & Cowling, T. G., 1970 The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press.
Cundall, P. A.: 1974 A computer model for rock-mass behavior using interactive graphics for input and output of geometrical data. US Army Corps. of Engrs (Missouri River Div.) Tech. Rep. MRD-2074.Google Scholar
Haff, P. K.: 1983 Grain flow as a fluid mechanical phenomenon. J. Fluid Mech. 134, 401–430.Google Scholar
Haff, P. K. & Werner, B. T., 1986 Computer simulation of the mechanical sorting of grains. Powder Technol. 48, 239–245.Google Scholar
Hanes, D. M.: 1983 Studies on the mechanics of rapidly flowing granular-fluid materials. Ph.D. thesis, University of California, San Diego.
Hanes, D. M. & Inman, D. L., 1985 Observations of rapidly flowing granular-fluid flow. J. Fluid Mech. 150, 357–380.Google Scholar
Heyes, D. M.: 1986 The nature of extreme shear thinning in simple liquids. Molec. Phys. 57, 1265–1282.Google Scholar
Hopkins, M. A.: 1985 Collisional stresses in a rapidly deforming granular flow. MS thesis, Clarkson University, Potsdam, NY, USA.
Hopkins, M. A. & Shen, H. H., 1987 A Monte Carlo simulation of a rapid simple shear flow of granular materials. Proc. US-Japan Seminar on the Micromechanics of Granular Material, Sendai-Zao, Japan, October 26–30, 1987.Google Scholar
Jenkins, J. T. & Richmond, M. W., 1985a Grad's 13-moment system for a dense gas of inelasic spheres. Arch. Rat. Mech. Anal. 87, 355–377.Google Scholar
Jenkins, J. T. & Richmond, M. W., 1985b Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks, Phys. Fluids 28, 3485–3494.Google Scholar
Jenkins, J. T. & Richmond, 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 particles. J. Fluid Mech. 130, 187–202.Google Scholar
Kanatani, K.: 1979a A micropolar continuum theory of granular materials. Intl J. Engng Sci. 17, 419–432.Google Scholar
Kanatani, K.: 1979b A continuum theory for the flow of granular materials. In Theoretical and Applied Mechanics (ed. Japan Natl Comm. for Theor. and App. Mech.), vol. 27, pp. 571–578.
Kanatani, K.: 1980 A continuum theory for the flow of granular materials (II), In Theoretical and Applied Mechanics (ed. Japan Natl Comm. for Theor. and App. Mech.), vol. 28, pp. 485–497.
Lees, A. W. & Edwards, S. F., 1972 The computer study of transport processes under extreme conditions. J. Phys. C: Solid State Phys. 5, 1921–1929.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
Lun, C. K. K. & Savage, S. B. 1987 A simple kinetic theory for granular flow of rough, inelastic, spherical particles. Trans. ASME E: J. Appl. Mech. 63, 47–53.Google Scholar
McTigue, D. F.: 1978 A model for stresses in shear flows of granular material. Proc. US–Japan Seminar on Continuum-Mechanical and Statistical Approaches in the Mechanics of Granular Materials, pp. 266–271.Google Scholar
Nakagawa, M.: 1987 Kinetic theoretical approach for rapidly deforming granular material, Proc. US-Japan Seminar on the Micromechanics of Granular Material, Sendai-Zao, Japan, October 26–30, 1987.Google Scholar
Ogawa, S. & Oshima, N., 1977 A thermomechanical theory of soil-like materials, In Theoretical and Applied Mechanics (ed. Japan Natl Comm. for Theor. and App. Mech.), vol. 25, pp. 229–244.
Oshima, N.: 1978 Continuum model of fluidized granular media. Proc. US-Japan Seminar on Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, pp. 189–202.Google Scholar
Oshima, N.: 1980 Dynamics of fluidized granular material. In Theoretical and Applied Mechanics (ed. Japan Natl Comm. for Theor. and App. Mech.), vol. 28, pp. 475–484.
Savage, S. B.: 1979 Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92, 53–96.Google Scholar
Savage, S. B. & Jeffrey, D. J., 1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255–272.Google Scholar
Savage, S. B. & Sayed, M., 1984 Stresses developed by dry cohesionless granular materials in an annular shear cell. J. Fluid Mech. 142, 391–430.Google Scholar
Walton, O. R. & Braun, R. L., 1986a Viscosity and temperature calculations for shearing assemblies of inelastic, frictional disks. J. Rheol. 30, 949–980.Google Scholar
Walton, O. R. & Braun, R. L., 1986b Stresses calculations for assemblies of inelastic spheres in uniform shear. Acta Mech. 63, 73–86.Google Scholar
Werner, B. T. & Haff, P. K., 1985 The collisional interaction of a small number of confined, inelastic grains. Proc. Intl Symp.-Workshop on Particulate and Multiphase Processes, 16th Annual Meeting of the Fine Particle Society, Miami Beach, Florida, USA, April 22–26.Google Scholar
Werner, B. T. & Haff, P. K., 1986 A simulation study of the low energy ejecta resulting from single impacts in eolian saltation. Proc. ASCE Conf., Minneapolis, Minnesota, USA.Google Scholar