Analogy Between the Groundstate of an Antiferromagnet and the Mode of Quasi-Static Deformation of a Packing of Spheres

Abstract

We report on local and global processes during the mechanical deformation of packings of spheres. We demonstrate that solid friction imposes a counter-rotation interaction to two adjacent grains during the deformation in order to minimize losses. This is equivalent to introduce an anti-ferromagnetic spin-like interaction in the structure; the search for the deformation process is quite similar to that one of finding the groundstate of an anti-ferromagnetic disordered structure with special boundary conditions. We demonstrate experimentally that boundaries may produce leading frustration processes; in this case, they govern most of the deformation mode and cancel mostly the influence of bulk mechanical properties (at least in the case when global “antiferromagnetic” order of the bulk is allowed by the absence of frustration). Frustration due to triangular local structure is described and existence of “anti-ferromagnetic-like” domains is proved; these domains are induced by the adequation between the boundary conditions and the deformation process and the domain wall is a stress-localization band as named in classical soil mechanics.

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References

  1. 1.

    a) S.B. Savage S.B. & Y. Nohguchi, Acta Mechanica 75, 153–174, (1988); b) G.W. Baxter, R.P. Behringer, T. Fagert. and G.A. Johnson. A., Phys. Rev. Lett. 62, 2825, (1989); BAXTER G.W. and BEHRINGER R.P., Phys. Rev. A42, 1017, (1990)

    Article  Google Scholar 

  2. 2.

    A.N. Schofield & C.P. Wroth Critical State of Soil Mechanics, (Mac Graw Hill, London, 1968)

    Google Scholar 

  3. 3.

    I.F. Collins, J. Mech. Phys. Solid 38, 1–25, (1990); ROWE P.W., Proc. Roy. Soc. Lndn A269, 500–527, (1962)

    Article  Google Scholar 

  4. 4.

    P. Evesque and C. Stefani, J. Phys. II France 1, 1337–47, (1991)

    CAS  Article  Google Scholar 

  5. 5.

    C. Laroche, S. Douady & S. Fauve, J. Phys. France, 50, 699–706, (1989); P. Evesque & J. Rajchenbach, Phys. Rev. Lett. 62, 44, (1988); S.B. Savage, J. Fluid Mech., 194, 457, (1988); O. Zik. & J. Stavans, Europhys. Lett.

    Article  Google Scholar 

  6. 6.

    P. Evesque, J. Phys. France, 51, 697–700, (1990); E. Szmatula, J.-P.Denis, P. Evesque, C. R. Acad. Sci. Paris, 310, serie II, 1175–1178, (1990)

    CAS  Article  Google Scholar 

  7. 7.

    P. Evesque, P. Alfonsi, C. Stefani, B. Barbé, C.R. Acad. Sc. Paris, 311, série II, 393–398, (1990)

    CAS  Google Scholar 

  8. 8.

    P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381, (1987); and Phys. Rev. A38, 364, (1988)

    CAS  Article  Google Scholar 

  9. 9.

    P. Evesque & J. Rajchenbach, C.R. Acad. Sc. (Paris) 307, Série II, 223, (1988); H.M. Jagger, CH. Liu, S. Nagel, Phys. Rev. Lett. 62, 40, (1989); G.A. Held, D.H. Solin, D.T. Kean, W.J. Horn, G. Grinstein, Phys. Rev. Lett. 65, 1120, (1990)

    Google Scholar 

  10. 10.

    P. Evesque, J. Phys. France 51, 2515, (1990); Europh. Lett 14, 427, (1991); Phys. Rev. A43, 2720–40, (1991)

    Article  Google Scholar 

  11. 11.

    R.A. Bagnold, Proc. Roy. Soc. Lndn Ser. A225, 49 (1954) and ib. A235, 219 (1966)

    Google Scholar 

  12. 12.

    J. Biarez & P.-Y. Hicher, in “Powders and Grains”, ed. J. Biarez & R. Gourvés, (Balkema, Rotterdam, 1989), pp. 1–13; and in “Lois de Comportement des Sols Remanics”, Cours de D.E. A. de Mécanique des Sols et Structures, Ecole Centrale Paris, 1989)

  13. 13.

    P.A. Cundall, O.D.L. Strack, in Mechanics of Granular Materials and Constitutive Relations, ed. J.T. Jenkins & M. Satake, (Elsevier, Amsterdam, 1983), pp. 137–149; C.S. Campbell and L.N. Brennen., J. Fluid Mech. 151, 167, (1985); Y.M. Bashir & J.D. Goddard, J. Rheol. 35, 849–85, (1991) Cundall

  14. 14.

    P. Evesque submitted to publication; P. Evesque & D. Sornette., to appear in the Proceedings of the International Conf. on Mechanics, Physics and Structure of Materials “A Celebration of Aristotle’s 23 Centuries, in J. of the Mechanical Behavior of Materials

  15. 15.

    K. Binder & A.P. Young, Rev. Mod. Phys. 58, 801–976, (1986); M. Mézard, G. Parisi & M.A. Virasoro, Spin Glass Theory and Beyond. World Scientist Lecture Notes in Physics, vol.9, (1987)

    Article  Google Scholar 

  16. 16.

    A.J. Bray, M.A. Moore, Phys. Rev. Lett. 58, 57, (1987); ZHANG Y.C., Phys. Rev. Lett. 59, 2125-8, (1987)

    CAS  Article  Google Scholar 

  17. 17.

    P. Dantu, Géotechnique 18, 50–55, (1968); Annales des Ponts et Chaussées IV, 1–10, (1967)

    Article  Google Scholar 

  18. 18.

    accepted for publication in C.R. Acad. Sciences Paris, France.

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Acknowledgments

We thank Prof. J. BIAREZ for kind advices and encouragements and Dr D. SORNETTE for very stimulating and encouraging discussions. This work has been supported by GRECO-géomatériaux. P.E. wants to thank E.S.A. for partial financial support.

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Evesque, P. Analogy Between the Groundstate of an Antiferromagnet and the Mode of Quasi-Static Deformation of a Packing of Spheres. MRS Online Proceedings Library 291, 97–102 (1992). https://doi.org/10.1557/PROC-291-97

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