Abstract
The partial differential equation (PDE) is often the tool of choice for describing the behavior of complicated fluid or solid mechanical systems. Sometimes, however, mechanical systems are composed of physically distinct elements that are not so large in number that a continuum description of the entire system is feasible or possible, or, although large in number, cannot be linked to macroscopic behavior through presently known constitutive laws. If interaction forces between individual elements are known or can be estimated and modeled, then the behavior of these elements or “particles” can be studied by solving the Newtonian equations of motion for each particle in the group simultaneously. This method (Cundall and Strack, 1979; Walton, 1983), variously called the distinct element method (DEM) or the particle dynamics method (PDM), is one example of several approaches that seek to follow the motion of selected “elementary” mechanical entities composing a larger system, and constitutes one approach to what we call here “discrete mechanics.” The purpose of this chapter is not to review the field of discrete mechanics, but to address in an eclectic way issues and problems arising in the implementation and application of these methods. For reviews see the articles by Campbell (1990) and by Cundall and Hart (1989).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Allen, M.P. and Tildesley, D.J., Computer Simulation of Liquids, Clarendon Press, Oxford, 1987, p. 385.
Anderson, R.S. and Haff, P.K. Simulation of eolian saltation, Science, 241, 820–823, 1988.
Barbosa, R. and Ghaboussi, J., Discrete finite element method, in Proc. 1st U.S. Conf. on Discrete Element Methods, Mustoe, G.G.W., Henriksen, M., and Huttelmaier, H.-P., Eds., CSM Press, Golden, Colorado, 1989.
Baxter, G.W. and Behringer, R.P., Cellular automata models of granular flow, Phys. Rev. A, 42, 1017–1020, 1990.
Baxter, G.W. and R.P. Behringer, Cellular automata models for the flow of granular materials, Physica D, 51, 465–471, 1991.
Campbell, C.S., Rapid granular flows, Annu. Rev. Fluid Mech., 22, 57–92, 1990.
Campbell, C.S. and Brennen, C.E., Computer simulation of granular shear flows, J. Fluid Mech., 151, 167–188, 1985.
Campbell, C.S. and Gong, A., The stress-tensor in a two-dimensional granular flow, J. Fluid Mech., 164, 107–125, 1986.
Cundall, P.A. and Hart, R.D., Numerical modeling of discontinua, in Proc. 1st U.S. Conf. on Discrete Element Methods, Mustoe, G.G.W., Henriksen, M., and Huttelmaier, H.-P., Eds., CSM Press, Golden, Colorado, 1989.
Cundall, P.A. and Strack, O.D.L., A discrete numerical model for granular assemblies, Geotechnique, 29, 47–65, 1979.
Forrest, S.B. and Haff, P.K., Mechanics of wind ripple stratigraphy, Science, 255, 1240–1243, 1992.
Frisch, U., Hasslacher, B., and Pomeau, Y., Lattice-gas automata for the Navier-Stokes equation, Phys. Rev. Lett., 56, 1505–1508, 1986.
Goddard, J.D., Nonlinear elasticity and pressure dependent wave speeds in granular media, Proc. R. Soc. Lond. A, 430, 105–131, 1990.
Goldsmith, W., Impact: The Theory and Physical Behavior of Colliding Solids, Edward Arnold, London, 1960, p. 379.
Gutt, G.M., The Physics of Granular Systems, Ph.D. thesis, California Institute of Technology, Pasadena, California, 1989, p. 185.
Gutt, G.M. and Haff, P.K., An automata model of granular materials, in Proc. 5th Distributed Memory Computing Conference, Vol. 1, IEEE Computer Society Press, Los Alamitos, California, 1990, pp. 522–529.
Haff, P.K., Grain flow as a fluid mechanical phenomena, J. Fluid Mech., 134, 401–430, 1983.
Haff, P.K., A physical picture of kinetic granular fluids, J. Rheol., 30, 931–948, 1986.
Haff, P.K. and Anderson R.S., Grain-scale simulations of loose sedimentary beds: the example of grain-bed impacts in aeolian saltation, Sedimentology, 1993, in press.
Haff, P.K. and Werner, B.T., Collisional interaction of a small number of confined inelastic grains, in Colloidal and Interfacial Phenomena, Vol. 3., Particulate and Multiphase Processes, Ariman, T. and Veziroglu, T.N., Eds., Hemisphere, Washington, D.C., 1987, pp. 483–501.
Hakuno, M., Tarumi, Y., and Meguro, K., A DEM simulation of concrete fracture, fault rupture and sand liquefaction, in Proc. 1st U.S. Conf. on Discrete Element Methods, Mustoe, G.G.W., Henriksen, M., and Huttelmaier, H.-P., Eds., CSM Press, Golden, Colorado, 1989.
Hanes, D.M. and Inman, D.L., Observations of rapidly flowing granular-fluid materials, J. Fluid Mech., 150, 357–380, 1985.
Hopkins, M.A. and Louge, M.Y., Inelastic microstructure in rapid granular flows of smooth disks, Phys. Fluids A, 3, 47–57, 1990.
Hopkins, M.A. and Shen, H., Constitutive relations for a planar, simple shear flow of rough disks, Int. J. Eng. Sci., 24, 1717–1730, 1986.
Hui, K. and Haff, P.K., Kinetic grain flow in a vertical channel, Int. J. Multiphase Flow, 12, 289–298, 1986.
Hui, K., Haff, P.K., Ungar, J.E., and Jackson, R., Boundary conditions for high-shear grain flows, J. Fluid Mech., 145, 223–233, 1984.
Jenkins, J.T. and Savage, S.B., A theory for the rapid flow of identical, smooth, nearly elastic particles, J. Fluid Mech., 130, 187–202, 1983.
Jiang, Z. and Haff, P.K., Multiparticle simulation methods applied to the micro-mechanics of bed load transport, Water Resources Res., 29, 399–412, 1993.
Johnson, K.L., Contact Mechanics, Cambridge University Press, Cambridge, 1985, p. 452.
Johnson, P.C. and Jackson, R., Frictional-collisional constitutive relations for granular materials, with application to plane shearing, J. Fluid Mech., 176, 67–93, 1987.
Johnson, P.C., Nott, P., and Jackson, R., Frictional-collisional equations of motion for particulate flows and their application to chutes, J. Fluid Mech., 210, 501–535, 1990.
Liepmann, H.W. and Roshko, A., Elements of Gasdynamics, John Wiley & Sons, New York, 1957, p. 439.
Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, 4th ed., Dover, New York, 1944, p. 643.
Lun, C.K.K., Savage, S.B., Jeffrey, D.J., and Chepurniy, N., 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, 1984.
Margolis, N., Tommaso, T. and Vichniac, G., Cellular-automata supercomputers for fluid-dynamics modeling. Phys. Rev. Lett., 56, 1694–1696, 1986.
Richman, M.W. and Chou, C.S., Boundary effects on granular shear flows of smooth disks, Z. angew. Math. Phys. (J. Appl. Phys. Math.), 39, 885–901, 1988.
Rosato, A.D., Vreeland, T., Jr., and Prinz, F.B., Manufacture of powder compacts, Int. Materials Rev., 36, 45–61, 1991.
Savage, S.B., The mechanics of rapid granular flows, Adv. Appl. Mech., 24, 289–366, 1984.
Ungar, J.E. and Haff, P.K., Stready-state saltation in air, Sedimentology, 34, 289–299, 1987.
Walton, O.R., Particle-dynamics calculations of shear flows, in Mechanics of Granular Materials: New Models and Constitutive Relations, Jenkins, J.T. and Satake, M., Eds., Elsevier, Amsterdam, 1983, pp. 327–338.
Walton, O.R. and Braun, R.L., Stress calculations for assemblies of inelastic spheres in uniform shear, Acta Mech., 63, 73–86, 1986a.
Walton, O.R. and Braun, R.L., Viscosity, granular temperature and stress calculations for shearing assemblies of inelastic, frictional disks, J. Rheol., 30, 949–980, 1986b.
Walton, O.R., Kim, H., and Rosato, A.D., Microstructure and stress differences in shearing flows, in Mechanics Computing in 1990s and Beyond. Proc. Engineering Mechanics Div., ASCE, Columbus, Ohio, May 20–22, 1991, pp. 1249–1253.
Williams, J.R. and Pentland, A.P., Superquadrics and modal dynamics for discrete elements in concurrent design, in Proc. 1st U.S. Conf. on Discrete Element Methods, Mustoe, G.G.W., Henriksen, M., and Huttelmaier, H.-P., Eds., CSM Press, Golden, Colorado, 1989.
Williams, J., Chen, A., and Petrie, D., Ice island interaction with conical production structures, in Proc. 1st U.S. Conf. on Discrete Element Methods, Mustoe, G.G.W., Henriksen, M., and Huttelmaier, H.-P., Eds., CSM Press, Golden, Colorado, 1989.
Worgan, K.J. and Mustoe, G.G.W., Application of the discrete element method to modelling the subsurface penetration of a uniform ice cover, in Proc. 1st U.S. Conf. on Discrete Element Methods, Mustoe, G.G.W., Henriksen, M., and Huttelmaier, H.-P., Eds., CSM Press, Golden, Colorado, 1989.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Haff, P.K. (1994). Discrete Mechanics. In: Mehta, A. (eds) Granular Matter. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4290-1_5
Download citation
DOI: https://doi.org/10.1007/978-1-4612-4290-1_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-8725-4
Online ISBN: 978-1-4612-4290-1
eBook Packages: Springer Book Archive