Isostatic Phase Transition and Instability in Stiff Granular Materials

Cristian F. Moukarzel

doi:10.1103/PhysRevLett.81.1634

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Isostatic Phase Transition and Instability in Stiff Granular Materials

Cristian F. Moukarzel^*

Instituto de Física, Universidade Federal Fluminense, 24210-340 Niterói RJ, Brazil

^*Email address: cristian@if.uff.br

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Phys. Rev. Lett. 81, 1634 – Published 24 August, 1998

DOI: https://doi.org/10.1103/PhysRevLett.81.1634

Abstract

Structural rigidity concepts are used to understand the origin of instabilities in granular aggregates. It is first demonstrated that the contact network of a noncohesive granular aggregate becomes exactly isostatic when $I = k ε / f_{l} ≫ 1$ , where $k$ is the stiffness, $ε$ is the typical interparticle gap, and $f_{L}$ is the typical stress induced by loads. Thus random packings of stiff particles are typically isostatic. Furthermore isostaticity is responsible for the anomalously large susceptibility to perturbation observed in granular aggregates. The load-stress response function of granular piles is critical (power-law distributed) in the isostatic limit, which means that slight overloads will produce internal rearrangements.

References (17)

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Exceptions are packings with periodic boundary conditions because they are not bounded, and idealized regular close-packings, because they have no length mismatches and therefore no self-stress even on overconstrained graphs.
S. Ouaguenouni and J.-N. Roux, Europhys. Lett. 35, 449 (1995); Europhys. Lett. 39, 117 (1997); see also Ref. [[4]].
C. Moukarzel and P. M. Duxbury, Phys. Rev. Lett. 75, 4055 (1995); see also C. Moukarzel, P. M. Duxbury, and P. L. Leath, 78, 1480 (1997).
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F. Radjai, M. Jean, J. J. Moreau, and S. Roux, Phys. Rev. Lett. 77, 3110 (1996).

[c1] P. Dantu, in Proceedings of the 4th International Conference on Soil Mechanics and Foundation Engineering (Butterworths, London, 1957); Ann. Ponts Chaussees 4, 144 (1967); M. Ammi, D. Bideau, and J. P. Troadec, J. Phys. D 20, 424 (1987); B. Miller, C. O'Hern, and R. Behringer, Phys. Rev. Lett. 77, 3110 (1996).

[c2] C. Liu et al., Science 269, 513 (1995).

[c3] H. M. Jaeger and S. R. Nagel, Rev. Mod. Phys. 68, 1259 (1996).

[c4] Friction, Arching, Contact Dynamics, D. E. Wolf and P. Grassberger (World Scientific, Singapore, 1997).

[c5] S. N. Coppersmith et al., Phys. Rev. E 53, 4673 (1996).

[c6] P. Claudin, J.-P. Bouchaud, M. E. Cates, and J. P. Wittmer, Phys. Rev. E 57, 4441 (1998).

[c7] P. Claudin and J. P. Bouchaud, Phys. Rev. Lett. 78, 231 (1997).

[c8] M. Nicodemi, Phys. Rev. Lett. 80, 1340 (1998).

[c9] Henry Crapo, Struct. Topology 1, 26–45 (1979); T. S. Tay and W. Whiteley, 11, 21–69 (1985).

[c10] E. Guyon, S. Roux, A. Hansen, D. Bideau, J.-P. Troadec, and H. Crapo, Rep. Prog. Phys. 53, 373 (1990).

[c11] Exceptions are packings with periodic boundary conditions because they are not bounded, and idealized regular close-packings, because they have no length mismatches and therefore no self-stress even on overconstrained graphs.

[c12] S. Ouaguenouni and J.-N. Roux, Europhys. Lett. 35, 449 (1995); Europhys. Lett. 39, 117 (1997); see also Ref. [[4]].

[c13] C. Moukarzel and P. M. Duxbury, Phys. Rev. Lett. 75, 4055 (1995); see also C. Moukarzel, P. M. Duxbury, and P. L. Leath, 78, 1480 (1997).

[c14] A. Coniglio, J. Phys. A 15, 3829 (1982).

[c15] C. Moukarzel (to be published); see also cond-mat/9807004.

[c16] S. Roux, D. Stauffer, and H. J. Herrmann, J. Phys. (Paris) 48, 341 (1987); J. Hemmingsson, H. J. Herrmann, and S. Roux, J. Phys. I (France) 7, 291 (1997).

[c17] F. Radjai, M. Jean, J. J. Moreau, and S. Roux, Phys. Rev. Lett. 77, 3110 (1996).

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Isostatic Phase Transition and Instability in Stiff Granular Materials

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