For years, engineers have used elastic and plastic models to describe the properties of granular solids, such as sand piles and grains in silos1,2,3. However, there are theoretical4,5,6 and experimental7,8,9,10,11,12,13,14 results that challenge this approach. Specifically, it has been claimed4,5,6 that stress in granular solids propagates in a manner described by wave-like (hyperbolic) equations, rather than the elliptic equations of static elasticity. Here we report numerical simulations of the response of a two-dimensional granular slab to an external load, revealing that both approaches are valid—albeit on different length scales. For small systems that can be considered mesoscopic on the scale of the grains, a hyperbolic-like, strongly anisotropic response is expected. However, in large systems (those typically considered by engineers), the response is closer to that predicted by traditional isotropic elasticity models. Static friction, often ignored in simple models, plays a key role: it increases the elastic range and renders the response more isotropic, even beyond this range.
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Nedderman, R. M. Statics and Kinematics of Granular Materials (Cambridge Univ. Press, Cambridge, 1992)
Savage, S. B. in Physics of Dry Granular Media (eds Herrmann, H. J., Hovi, J. P. & Luding, S.) 25–95 (NATO ASI Series, Kluwer, Dordrecht, 1998)
Jackson, R. in Theory of Dispersed Multiphase Flow (ed. Meyer, R. E.) 291–337 (Academic, New York, 1983)
Wittmer, J. P., Claudin, P., Cates, M. E. & Bouchaud, J.-P. An explanation for the central stress minimum in sand piles. Nature 382, 336–338 (1996)
Bouchaud, J.-P., Claudin, P., Levine, D. & Otto, M. Force chain splitting in granular materials: a mechanism for large-scale pseudo-elastic behaviour. Eur. Phys. J. E 4, 451–457 (2001)
Tkachenko, A. V. & Witten, T. A. Stress propagation through frictionless granular material. Phys. Rev. E 60, 687–696 (1999)
Drescher, A. & de Josselin de Jong, G. Photoelastic verification of a mechanical model for the flow of a granular material. J. Mech. Phys. Solids 20, 337–351 (1972)
Geng, J. et al. Footprints in sand: the response of a granular material to local perturbations. Phys. Rev. Lett. 87, 035506 (2001)
Geng, J., Reydellet, G., Clément, E. & Behringer, R. P. Green's function measurements of force transmission in 2D granular materials. Physica D 182, 274–303 (2003)
Mueggenburg, N. W., Jaeger, H. M. & Nagel, S. R. Stress transmission through three-dimensional ordered granular arrays. Phys. Rev. E 66, 031304 (2002)
Moukarzel, C. F., Pacheco-Martinez, H., Ruiz-Suarez, J. C. & Vidales, A. M. Static response in disk packings. Granular Matter 6, 61–66 (2004)
da Silva, M. & Rajchenbach, J. Stress transmission through a model system of cohesionless elastic grains. Nature 406, 708–710 (2000)
Smid, J. & Novosad, J. Pressure distribution under heaped bulk solids. In Proc. 1981 Powtech Conf., Institution of Chemical Engineers Symp. 63, D3/V/1–12 (1981).
Brockbank, R., Huntley, J. M. & Ball, R. C. Contact force distribution beneath a three-dimensional granular pile. J. Phys. II 7, 1521–1532 (1997)
Vanel, L., Howell, D., Clark, D., Behringer, R. P. & Clément, E. Memories in sand: experimental tests of construction history on stress distributions under sandpiles. Phys. Rev. E 60, R5040–R5043 (1999)
Serero, D., Reydellet, G., Claudin, P., Clément, E. & Levine, D. Stress response function of a granular layer: quantitative comparison between experiments and isotropic elasticity. Eur. Phys. J. E 6, 169–179 (2001)
Goldenberg, C. & Goldhirsch, I. Force chains, microelasticity, and macroelasticity. Phys. Rev. Lett. 89, 084302 (2002)
Reydellet, G. & Clément, E. Green's function probe of a static granular piling. Phys. Rev. Lett. 86, 3308–3311 (2001)
Cundall, P. A. & Strack, O. D. L. A discrete numerical model for granular assemblies. Geotechnique 29, 47–65 (1979)
Otto, M., Bouchaud, J.-P., Claudin, P. & Socolar, J. E. S. Anisotropy in granular media: classical elasticity and directed-force chain network. Phys. Rev. E 67, 031302 (2003)
Duffy, J. & Mindlin, R. D. Stress–strain relations and vibrations of a granular medium. J. Appl. Mech. 24, 585–593 (1957)
Bathurst, R. J. & Rothenburg, L. Micromechanical aspects of isotropic granular assemblies with linear contact interactions. J. Appl. Mech. 55, 17–23 (1988)
Chang, C. S. & Ma, L. Elastic material constants for isotropic granular solids with particle rotation. Int. J. Solids Struct. 29, 1001–1018 (1992)
Gay, C. & da Silveira, R. Anisotropic elastic theory of preloaded granular media. Europhys. Lett. 68, 51–57 (2004)
da Silveira, R., Vidalenc, G. & Gay, C. Stress propagation in two dimensional frictional granular matter. Preprint at 〈http://arXiv.org/cond-mat/0208214〉 (2002).
Johnson, K. L. Contact Mechanics 220 (Cambridge Univ. Press, Cambridge, 1985)
Tournat, V. Probing weak forces in granular media through nonlinear dynamic dilatancy: clapping contacts and polarization anisotropy. Phys. Rev. Lett. 92, 085502 (2004)
We thank A. P. F. Atman, R. P. Behringer, P. Claudin, E. Clément, J. Geng, N. Mueggenburg, M. van Hecke, W. van Saarloos and T. A. Witten for discussions. This work was supported by the Israel Science Foundation (ISF) and the US-Israel Binational Science Foundation (BSF).
The authors declare that they have no competing financial interests.
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Goldenberg, C., Goldhirsch, I. Friction enhances elasticity in granular solids. Nature 435, 188–191 (2005). https://doi.org/10.1038/nature03497
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