Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-19T17:25:09.629Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistical Mechanics of Stress Transmission in Static Arrays of Rigid Grains

Published online by Cambridge University Press:  01 February 2011

Dmitri V. Grinev  and
Sam F. Edwards
Dmitri V. Grinev
Affiliation:
Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 OHE, U.K.
Sam F. Edwards
Affiliation:
Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 OHE, U.K.
Get access

Abstract

We develop the statistical-mechanical theory that delivers the fundamental equations of stress equilibrium for static arrays of rigid grains. The random geometry of static granular packing composed of rigid cohesionless particles can be visualised as a network of intergranular contacts. The contact network and external loading determine the network of intergranular forces. In general, the contact network can have an arbitrary coordination number varying within the system. It follows then that the network of intergranular forces is indeterminate i.e. the number of unknown forces is larger than the number of Newton's equations of mechanical equilibrium. Thus, in order for the network of intergranular forces to be determined, the number of equations must equal the number of unknowns. We argue that this determines the contact network with a certain fixed coordination number. The complete system of equations for the stress tensor is derived from the equations of intergranular force and torque balance, given the geometric specification of the packing. The granular material fabric gives rise to corrections to the Euler-Cauchy equation that become significant at mesoscopic lengthscales. The stress-geometry equation establishes the relation between various components of the stress tensor, and depends on the topology of the granular array.

Type
Research Article
Information
MRS Online Proceedings Library (OPL) , Volume 627: Symposium BB – The Granular State , 2000 , BB6.1
Copyright
Copyright © Materials Research Society 2000

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

REFERENCES

1. Feda, J., Mechanics of Particulate Materials: The Principles (Elsevier, 1982).Google Scholar
2. Wood, D. M., Soil Behaviour and Critical State Soil Mechanics (Cambridge University Press, 1990).Google Scholar
3. Nedderman, R. M., Statics and Kinematics of Granular Materials (Cambridge University Press, 1992).Google Scholar
4. Edwards, S. F., Oakeshott, R. B. S., Physica D38, 88 (1989).Google Scholar
5. Bideau, D., Hansen, A. (eds.), Disorder and Granular Media (Elsevier, 1993).Google Scholar
Herrmann, H. J., Hovi, J.-P. and Luding, S. (eds.), Physics of Dry Granular Media, (Kluwer Academic, 1998).Google Scholar
6. Focus issue: Granular materials, Chaos 9, 3 (1999).Google Scholar
7. Brockbank, R., Huntley, J. M. and Ball, R. C., J. de Physique II (France) 7, 1521 (1997).Google Scholar
8. Liu, C. H. et al., Science 269, 513 (1995);Google Scholar
Liu, C., Nagel, S. R., Phys. Rev. B48, 15646 (1993);Google Scholar
Jia, X., Caroli, C., Velicky, B., Phys. Rev. Lett. 82, 1863 (1999).Google Scholar
9. Wittmer, J. P., Claudin, P., Cates, M. E., J. de Physique I (France), 7, 39 (1997).Google Scholar
10. Edwards, S. F. and Mounfield, C. C., Physica A226, 1 (1996).Google Scholar
11. Miller, B., O'Hern, C. and Behringer, R. P., Phys. Rev. Lett. 77, 3110 (1996).Google Scholar
12. Thornton, C., KONA: Powder and Particle, 15, 81, (1997).Google Scholar
13. Radjai, F. et al., Phys. Rev. Lett. 80, 61 (1998).Google Scholar
14. Ball, R. C. and Grinev, D. V., http://xxx.lanl.gov, cond-mat/9810124;Google Scholar
Edwards, S. F. and Grinev, D.V., Phys. Rev. Lett. 82, 5397 (1999); D.V. Grinev and S. F. Edwards, preprint, to be submitted to Phys. Rev. E.Google Scholar
15. Mueth, D. M., Jaeger, H. M. and Nagel, S. R., Phys. Rev. E 57, 3164 (1998).Google Scholar
16. Cates, M. E., Wittmer, J. P., Claudin, P., and Bouchaud, J-P., Phys. Rev. Lett. 81, 1841 (1998).Google Scholar