H. M. Jaeger, S. R. Nagel, and R. P. Behringer, The Physics of Granular Materials, Physics Today, vol.49, issue.4, pp.32-38, 1996.
DOI : 10.1063/1.881494

A. Van-blaaderen, R. Ruel, and P. Wiltzius, Template-directed colloidal crystallization, Nature, vol.385, issue.6614, pp.321-324, 1997.
DOI : 10.1038/385321a0

P. G. De-gennes, Granular matter: a tentative view, Reviews of Modern Physics, vol.71, issue.2, pp.374-382, 1999.
DOI : 10.1103/RevModPhys.71.S374

O. Dazel and V. Tournat, Nonlinear Biot waves in porous media with application to unconsolidated granular media, The Journal of the Acoustical Society of America, vol.127, issue.2, pp.692-702, 2010.
DOI : 10.1121/1.3277190

H. Hertz, On the contact of elastic solids, J. Reine Angew. Math, vol.92, p.1882

C. Coste and B. Gilles, On the validity of Hertz contact law for granular material acoustics, The European Physical Journal B, vol.7, issue.1, pp.155-168, 1998.
DOI : 10.1007/s100510050598

R. D. Mindlin, Compliance of Elastic Bodies in Contact, ASME J. Appl. Mech, vol.16, pp.259-268, 1949.
DOI : 10.1007/978-1-4613-8865-4_24

R. D. Mindlin and H. Deresiewicz, Elastic Spheres in Contact Under Varying Oblique Forces, ASME J. Appl. Mech, vol.75, pp.327-342, 1953.
DOI : 10.1007/978-1-4613-8865-4_35

J. Duffy and R. D. Mindlin, Stress-Strain Relations and Vibrations of a Granular Medium, ASME Trans. J. Appl. Mech, vol.24, pp.585-593, 1958.
DOI : 10.1007/978-1-4613-8865-4_49

L. Landau and E. Lifchitz, Theory of Elasticity, 1967.

K. L. Johnson, Contact Mechanics, 1985.

N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1976.

L. Brillouin, Wave Propagation in Periodic Structures, 1953.

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chang et al., Ultrasound Tunneling through 3D Phononic Crystals, Physical Review Letters, vol.88, issue.10, p.104301, 2002.
DOI : 10.1103/PhysRevLett.88.104301

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chang et al., Focusing of Sound in a 3D Phononic Crystal, Physical Review Letters, vol.93, issue.2, p.24301, 2004.
DOI : 10.1103/PhysRevLett.93.024301

J. O. Vasseur, P. A. Deymier, B. Djafari-rouhani, Y. Pennec, and A. Hladky-hennion, Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates, Physical Review B, vol.77, issue.8, p.85415, 2008.
DOI : 10.1103/PhysRevB.77.085415

URL : https://hal.archives-ouvertes.fr/hal-00357331

M. T. Dove, Introduction to Lattice Dynamics, 1993.

L. J. Raubenheimer and G. Gilat, Accurate Numerical Method of Calculating Frequency Distribution Functions in Solids. II. Extension to hcp Crystals, Physical Review, vol.157, issue.3, pp.586-599, 1967.
DOI : 10.1103/PhysRev.157.586

B. Gilles and C. Coste, Low-Frequency Behavior of Beads Constrained on a Lattice, Physical Review Letters, vol.90, issue.17, p.174302, 2003.
DOI : 10.1103/PhysRevLett.90.174302

URL : https://hal.archives-ouvertes.fr/hal-01406751

C. Coste and B. Gilles, Sound propagation in a constrained lattice of beads: High-frequency behavior and dispersion relation, Physical Review E, vol.77, issue.2, p.21302, 2008.
DOI : 10.1103/PhysRevE.77.021302

URL : https://hal.archives-ouvertes.fr/hal-01406704

F. Radjai, M. Jean, J. Moreau, and S. Roux, Force Distributions in Dense Two-Dimensional Granular Systems, Physical Review Letters, vol.77, issue.2, pp.274-277, 1996.
DOI : 10.1103/PhysRevLett.77.274

URL : https://hal.archives-ouvertes.fr/hal-00759668

J. F. Peters, J. Muthuswamy, J. Wibowo, and A. Tordesillas, Characterization of force chains in granular material, Physical Review E, vol.72, issue.4, p.41307, 2005.
DOI : 10.1103/PhysRevE.72.041307

T. S. Majmudar and R. P. Behringer, Contact force measurements and stress-induced anisotropy in granular materials, Nature, vol.20, issue.1079, pp.1079-1082, 2005.
DOI : 10.1103/PhysRevLett.92.054302

G. P. Srivastava, The Physics of Phonons. Taylor and Francis Group, 1990.

F. Radjai, S. Roux, and J. Moreau, Contact forces in a granular packing, Chaos: An Interdisciplinary Journal of Nonlinear Science, vol.9, issue.3, pp.544-550, 1999.
DOI : 10.1063/1.166428

URL : https://hal.archives-ouvertes.fr/hal-01407368

O. Mouraille, Sound Propagation in Dry Granular Materials : Discrete Element Simulations, Theory, and Experiments, 2008.

G. Theocharis, M. Kavousanakis, P. G. Kevrekidis, C. Daraio, M. A. Porter et al., Localized breathing modes in granular crystals with defects, Physical Review E, vol.80, issue.6, p.66601, 2009.
DOI : 10.1103/PhysRevE.80.066601

X. Jia, C. Caroli, and B. Velicky, Ultrasound Propagation in Externally Stressed Granular Media, Physical Review Letters, vol.82, issue.9, pp.1863-1866, 1999.
DOI : 10.1103/PhysRevLett.82.1863

URL : https://hal.archives-ouvertes.fr/hal-00351103

E. Somfai, J. Roux, J. H. Snoeijer, M. Van-hecke, and W. Van-saarloos, Elastic wave propagation in confined granular systems, Physical Review E, vol.72, issue.2, p.21301, 2005.
DOI : 10.1103/PhysRevE.72.021301

URL : https://hal.archives-ouvertes.fr/hal-00354215

V. Tournat, C. Inserra, and V. Gusev, Non-cascade frequency-mixing processes for elastic waves in unconsolidated granular materials, Ultrasonics, vol.48, issue.6-7, pp.492-497, 2008.
DOI : 10.1016/j.ultras.2008.03.014

URL : https://hal.archives-ouvertes.fr/hal-00341160

C. Liu and S. R. Nagel, Sound in sand, Physical Review Letters, vol.68, issue.15, pp.2301-2304, 1992.
DOI : 10.1103/PhysRevLett.68.2301

H. Lamb, On the vibration of an elatic sphere, Proc. Lond, pp.61-68

A. E. Love, A Treatrise on the Mathematical Theory of Elasticity, 1944.

Z. Ye, On the low frequency elastic response of a spherical particle. Chin, J. Phys, vol.38, pp.103-110, 2000.

A. Hladky-hennion, F. Cohen-tenoudji, A. Devos, and M. De-billy, On the existence of subresonance generated in a one-dimensional chain of identical spheres, The Journal of the Acoustical Society of America, vol.112, issue.3, pp.850-855, 2002.
DOI : 10.1121/1.1497369

URL : https://hal.archives-ouvertes.fr/hal-00147757

J. Anfosso, Propagation Acoustique dans des Milieux Granulaires de Billes de Verres et d'Acier, 2003.

M. De-billy, Experimental study of sound propagation in a chain of spherical beads, The Journal of the Acoustical Society of America, vol.108, issue.4, pp.1486-1495, 2000.
DOI : 10.1121/1.1289365

M. De-billy, Frequency analysis of the acoustic signal transmitted through a one-dimensional chain of metallic spheres, The Journal of the Acoustical Society of America, vol.110, issue.2, pp.710-716, 2001.
DOI : 10.1121/1.1385179

J. Anfosso and V. Gibiat, Elastic wave propagation in a three-dimensional periodic granular medium, Europhysics Letters (EPL), vol.67, issue.3, pp.376-382, 2004.
DOI : 10.1209/epl/i2004-10085-9

V. Gusev and V. Tournat, How acoustic waves are guided in buried subsurface channels in unconsolidated granular media, Physical Review E, vol.78, issue.3, p.36602, 2008.
DOI : 10.1103/PhysRevE.78.036602

URL : https://hal.archives-ouvertes.fr/hal-00341140

V. F. Nesterenko, Propagation of nonlinear compression pulses in granular media, Journal of Applied Mechanics and Technical Physics, vol.24, issue.5, 1983.
DOI : 10.1007/BF00905892

V. F. Nesterenko, Dynamics of Heterogeneous Materials, 2001.
DOI : 10.1007/978-1-4757-3524-6

V. Tournat, V. E. Gusev, V. Y. Zaitsev, and B. Castagnède, Acoustic second-harmonic generation with shear to longitudinal mode conversion in granular media, Europhysics Letters (EPL), vol.66, issue.6, pp.798-804, 2004.
DOI : 10.1209/epl/i2003-10264-2

V. Tournat, V. E. Gusev, and B. Castagnède, Self-demodulation of elastic waves in a one-dimensional granular chain, Physical Review E, vol.70, issue.5, p.56603, 2004.
DOI : 10.1103/PhysRevE.70.056603

S. Job, F. Santibanez, F. Tapia, and F. Melo, Nonlinear waves in dry and wet Hertzian granular chains, Ultrasonics, vol.48, issue.6-7, pp.506-514, 2008.
DOI : 10.1016/j.ultras.2008.03.006

S. Job, F. Santibanez, F. Tapia, and F. Melo, Wave localization in strongly nonlinear Hertzian chains with mass defect, Physical Review E, vol.80, issue.2, p.25602, 2009.
DOI : 10.1103/PhysRevE.80.025602

URL : https://hal.archives-ouvertes.fr/hal-00624740

C. Daraio, V. F. Nesterenko, E. B. Herbold, and S. Jin, Energy Trapping and Shock Disintegration in a Composite Granular Medium, Physical Review Letters, vol.96, issue.5, p.58002, 2006.
DOI : 10.1103/PhysRevLett.96.058002

V. F. Nesterenko, C. Daraio, E. B. Herbold, and S. Jin, Anomalous Wave Reflection at the Interface of Two Strongly Nonlinear Granular Media, Physical Review Letters, vol.95, issue.15, p.158702, 2005.
DOI : 10.1103/PhysRevLett.95.158702

I. D. Mayergoyz, Mathematical Models of Hysteresis, Physical Review Letters, vol.56, issue.15, pp.1518-1521, 1986.
DOI : 10.1103/PhysRevLett.56.1518

R. A. Guyer, K. R. Mccall, and G. N. Boitnott, Hysteresis, Discrete Memory, and Nonlinear Wave Propagation in Rock: A New Paradigm, Physical Review Letters, vol.74, issue.17, pp.3491-3494, 1995.
DOI : 10.1103/PhysRevLett.74.3491

URL : https://hal.archives-ouvertes.fr/hal-00301808

V. Gusev, Amplitude- and frequency-dependent nonlinearities in the presence of thermally-induced transitions in the Preisach model of acoustic hysteresis, Physical Review B, vol.72, issue.5, p.54104, 2005.
DOI : 10.1103/PhysRevB.72.054104

V. Aleshin and K. Van-den-abeele, Preisach analysis of the Hertz???Mindlin system, Journal of the Mechanics and Physics of Solids, vol.57, issue.4, pp.657-672, 2009.
DOI : 10.1016/j.jmps.2009.01.004

V. Gusev, Theory of non-collinear interactions of acoustic waves in an isotropic material with hysteretic quadratic nonlinearity, The Journal of the Acoustical Society of America, vol.111, issue.1, pp.80-94, 2002.
DOI : 10.1121/1.1382621

V. Tournat, V. E. Gusev, and B. Castagnède, Subharmonics and noise excitation in transmission of acoustic wave through unconsolidated granular medium, Physics Letters A, vol.326, issue.5-6, pp.340-348, 2004.
DOI : 10.1016/j.physleta.2004.04.042

L. Fillinger, V. Yu, V. Zaitsev, B. Gusev, and . Castagnède, Nonlinear relaxational absorption/transparency for acoustic waves due to thermoelastic effect, Acust. Acta Acust, vol.92, pp.24-34, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00133999

K. W. Winkler, Contact stiffness in granular porous materials: Comparison between theory and experiment, Geophysical Research Letters, vol.277, issue.11, pp.1071-1076, 1983.
DOI : 10.1029/GL010i011p01073

J. P. Castagna, M. L. Batzle, and R. L. Eastwood, Relationships between compressional???wave and shear???wave velocities in clastic silicate rocks, GEOPHYSICS, vol.50, issue.4, 1985.
DOI : 10.1190/1.1441933

W. F. Murphy, K. W. Winkler, and R. L. Kleinberg, Acoustic relaxation in sedimentary rocks: Dependence on grain contacts and fluid saturation, GEOPHYSICS, vol.51, issue.3, pp.757-766, 1986.
DOI : 10.1190/1.1442128

R. Bachrach, J. Dvorkin, and A. M. Nur, Seismic velocities and Poisson???s ratio of shallow unconsolidated sands, GEOPHYSICS, vol.65, issue.2, pp.556-564, 2000.
DOI : 10.1190/1.1444751

P. J. Digby, The Effective Elastic Moduli of Porous Granular Rocks, Journal of Applied Mechanics, vol.48, issue.4, pp.803-808, 1981.
DOI : 10.1115/1.3157738

K. Walton, The effective elastic moduli of a random packing of spheres, Journal of the Mechanics and Physics of Solids, vol.35, issue.2, pp.213-226, 1987.
DOI : 10.1016/0022-5096(87)90036-6

S. N. Domenico, ELASTIC PROPERTIES OF UNCONSOLIDATED POROUS SAND RESERVOIRS, GEOPHYSICS, vol.42, issue.7, p.1339, 1977.
DOI : 10.1190/1.1440797

J. T. Jenkins, P. A. Cundall, and I. Ishibashi, Micromechanical modeling of granular materials with the assistance of experiments and numerical simulations, Powder and Grains. Balkema, 1989.

H. A. Makse, N. Gland, D. L. Johnson, and L. M. Schwartz, Why Effective Medium Theory Fails in Granular Materials, Physical Review Letters, vol.83, issue.24, pp.5070-5073, 1999.
DOI : 10.1103/PhysRevLett.83.5070

I. Agnolin, J. Roux, P. Massaad, X. Jia, and P. Mills, Sound waves velocities in dry and lubricated granular packings : numerical simulations and experiments

L. , L. Ragione, and J. T. Jenkins, The initial response of an idealized granular material, Proc. R. Soc. A, pp.735-758, 2007.

M. A. Koenders, The incremental stiffness of an assembly of particles, Acta Mechanica, vol.29, issue.1, pp.31-49, 1987.
DOI : 10.1007/BF01174645

F. Trentadue, An equilibrium-based approach for the micromechanical modelling of a non-linear elastic granular material, Mechanics of Materials, vol.36, issue.4, pp.323-334, 2004.
DOI : 10.1016/S0167-6636(03)00061-9

J. T. Jenkins and M. A. Koenders, The incremental response of random aggregates of identical round particles, The European Physical Journal E - Soft Matter, vol.13, issue.2, pp.113-123, 2004.
DOI : 10.1140/epje/e2004-00048-9

J. T. Jenkins, D. Johnson, L. L. Ragione, and H. Makse, Fluctuations and the effective moduli of an isotropic, random aggregate of identical, frictionless spheres, Journal of the Mechanics and Physics of Solids, vol.53, issue.1, pp.197-225, 2005.
DOI : 10.1016/j.jmps.2004.06.002

I. Agnolin and J. Roux, Internal states of model isotropic granular packings. I. Assembling process, geometry, and contact networks, Physical Review E, vol.76, issue.6, p.61302, 2007.
DOI : 10.1103/PhysRevE.76.061302

URL : https://hal.archives-ouvertes.fr/hal-00148554

I. Agnolin and J. Roux, Internal states of model isotropic granular packings. II. Compression and pressure cycles, Physical Review E, vol.76, issue.6, p.61303, 2007.
DOI : 10.1103/PhysRevE.76.061303

URL : https://hal.archives-ouvertes.fr/hal-00148554

I. Agnolin and J. Roux, Internal states of model isotropic granular packings. III. Elastic properties, Physical Review E, vol.76, issue.6, p.61304, 2007.
DOI : 10.1103/PhysRevE.76.061304

URL : https://hal.archives-ouvertes.fr/hal-00148554

V. Magnanimo, L. La-ragione, J. T. Jenkins, P. Wang, and H. A. Makse, Characterizing the shear and bulk moduli of an idealized granular material, EPL (Europhysics Letters), vol.81, issue.3, p.34006, 2008.
DOI : 10.1209/0295-5075/81/34006

I. Agnolin and J. Roux, On the elastic moduli of three-dimensional assemblies of spheres: Characterization and modeling of fluctuations in the particle displacement and rotation, International Journal of Solids and Structures, vol.45, issue.3-4, pp.1101-1123, 2008.
DOI : 10.1016/j.ijsolstr.2007.07.016

URL : https://hal.archives-ouvertes.fr/hal-00353678

B. Velicky and C. Caroli, Pressure dependence of the sound velocity in a two-dimensional lattice of Hertz-Mindlin balls: Mean-field description, Physical Review E, vol.65, issue.2, p.21307, 2002.
DOI : 10.1103/PhysRevE.65.021307

L. M. Schwartz, D. L. Johnson, and S. Feng, Vibrational Modes in Granular Materials, Physical Review Letters, vol.52, issue.10, pp.831-834, 1984.
DOI : 10.1103/PhysRevLett.52.831

A. S. Suiker, R. De-borst, and C. S. Chang, Micro-mechanical modelling of granular material. Part 1: Derivation of a second-gradient micro-polar constitutive theory, Acta Mechanica, vol.42, issue.1-4, pp.161-180, 2001.
DOI : 10.1007/BF01261670

A. S. Suiker, R. De-borst, and C. S. Chang, Micro-mechanical modelling of granular material. Part 2: Plane wave propagation in infinite media, Acta Mechanica, vol.32, issue.1-4, pp.181-200, 2001.
DOI : 10.1007/BF01261671

A. S. Suiker and R. De-borst, Enhanced continua and discrete lattices for modelling granular assemblies, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.35, issue.1836, pp.2543-2580, 2005.
DOI : 10.1098/rsta.2005.1586

S. A. Lisina, A. I. Potapov, and V. F. Nesterenko, A nonlinear granular medium with particle rotation: A one-dimensional model, Acoustical Physics, vol.47, issue.5, pp.598-606, 2001.
DOI : 10.1134/1.1403551

I. S. Pavlov, A. I. Potapov, and G. A. Maugin, A 2D granular medium with rotating particles, International Journal of Solids and Structures, vol.43, issue.20, pp.6194-6207, 2006.
DOI : 10.1016/j.ijsolstr.2005.06.012

C. S. Chang and J. Gao, Non-linear dispersion of plane wave in granular media, International Journal of Non-Linear Mechanics, vol.30, issue.2, pp.111-128, 1995.
DOI : 10.1016/0020-7462(94)00039-D

A. S. Suiker, A. V. Metrikine, and R. De-borst, DYNAMIC BEHAVIOUR OF A LAYER OF DISCRETE PARTICLES, PART 1: ANALYSIS OF BODY WAVES AND EIGENMODES, Journal of Sound and Vibration, vol.240, issue.1, pp.1-18, 2001.
DOI : 10.1006/jsvi.2000.3202

A. S. Suiker, A. V. Metrikine, and R. De-borst, DYNAMIC BEHAVIOUR OF A LAYER OF DISCRETE PARTICLES, PART 2: RESPONSE TO A UNIFORMLY MOVING, HARMONICALLY VIBRATING LOAD, Journal of Sound and Vibration, vol.240, issue.1, pp.19-39, 2001.
DOI : 10.1006/jsvi.2000.3203

H. Mühlhaus and F. Oka, Dispersion and wave propagation in discrete and continuous models for granular materials, International Journal of Solids and Structures, vol.33, issue.19, pp.2841-2858, 1996.
DOI : 10.1016/0020-7683(95)00178-6

H. Arslan and S. Sture, Evaluation of a physical length scale for granular materials, Computational Materials Science, vol.42, issue.3, pp.525-530, 2008.
DOI : 10.1016/j.commatsci.2007.08.016

H. Arslan and S. Sture, Finite element simulation of localization in granular materials by micropolar continuum approach, Computers and Geotechnics, vol.35, issue.4, pp.548-562, 2008.
DOI : 10.1016/j.compgeo.2007.10.006

]. H. Bibliographie, N. Makse, D. L. Gland, L. M. Johnson, and . Schwartz, Why effective medium theory fails in granular materials, Phys. Rev. Lett, vol.83, issue.1, pp.5070-5073, 1999.

J. Jenkins, D. Johnson, L. L. Ragione, and H. Makse, Fluctuations and the effective moduli of an isotropic, random aggregate of identical, frictionless spheres, Journal of the Mechanics and Physics of Solids, vol.53, issue.1, pp.197-225, 2005.
DOI : 10.1016/j.jmps.2004.06.002

L. , L. Ragione, and J. T. Jenkins, The initial response of an idealized granular material, Proc. R. Soc. A, pp.735-758, 2006.

V. Magnanimo, L. La-ragione, J. T. Jenkins, P. Wang, and H. A. Makse, Characterizing the shear and bulk moduli of an idealized granular material, EPL (Europhysics Letters), vol.81, issue.3, p.34006, 2008.
DOI : 10.1209/0295-5075/81/34006

H. M. Jaeger, S. R. Nagel, and R. P. Behringer, The Physics of Granular Materials, Physics Today, vol.49, issue.4, pp.32-38, 1996.
DOI : 10.1063/1.881494

P. G. De-gennes, Granular matter: a tentative view, Reviews of Modern Physics, vol.71, issue.2, pp.374-382, 1999.
DOI : 10.1103/RevModPhys.71.S374

W. Wyart, On the rigidity of amorphous solids, Annales de Physique, vol.30, issue.3, pp.3-96, 2005.
DOI : 10.1051/anphys:2006003

W. G. Ellenbroek, E. Somfai, M. Van-hecke, and W. Van-saarloos, Critical Scaling in Linear Response of Frictionless Granular Packings near Jamming, Physical Review Letters, vol.97, issue.25, p.258001, 2006.
DOI : 10.1103/PhysRevLett.97.258001

C. Liu, S. R. Nagel, D. A. Schecter, S. M. Coppersmith, S. Majumbar et al., Force Fluctuations in Bead Packs, Science, vol.269, issue.5223, pp.513-515, 1995.
DOI : 10.1126/science.269.5223.513

J. Anfosso and V. Gibiat, Elastic wave propagation in a three-dimensional periodic granular medium, Europhysics Letters (EPL), vol.67, issue.3, pp.376-382, 2004.
DOI : 10.1209/epl/i2004-10085-9

D. L. Blair, N. W. Mueggenbur, A. H. Marshall, H. M. Jaeger, and S. R. Nagel, Force distributions in three-dimensional granular assemblies: Effects of packing order and interparticle friction, Physical Review E, vol.63, issue.4, p.41304, 2001.
DOI : 10.1103/PhysRevE.63.041304

Y. Nahmad-molinari and J. C. Ruiz-suarez, Epitaxial Growth of Granular Single Crystals, Physical Review Letters, vol.89, issue.26, p.264302, 2002.
DOI : 10.1103/PhysRevLett.89.264302

A. B. Yu, X. Z. An, R. P. Zou, R. Y. Yang, and K. Kendall, Self-Assembly of Particles for Densest Packing by Mechanical Vibration, Physical Review Letters, vol.97, issue.26, p.265501, 2006.
DOI : 10.1103/PhysRevLett.97.265501

O. Carvente and J. C. Ruiz-suarez, Crystallization of Confined Non-Brownian Spheres by Vibrational Annealing, Physical Review Letters, vol.95, issue.1, p.18001, 2005.
DOI : 10.1103/PhysRevLett.95.018001

Z. Cheng, W. B. Russel, and P. M. Chaikin, Controlled growth of hard-sphere colloidal crystals, Nature, vol.401, pp.893-895, 1999.

J. D. Joannopoulos, Photonics self-assembly lights up, Nature, vol.414, issue.6861, pp.257-258, 2001.
DOI : 10.1038/35104718

A. Van-blaaderen, R. Ruel, and P. Wiltzius, Template-directed colloidal crystallization, Nature, vol.385, issue.6614, pp.321-324, 1997.
DOI : 10.1038/385321a0

I. I. Tarhan and G. H. Watson, Photonic Band Structure of fcc Colloidal Crystals, Physical Review Letters, vol.76, issue.2, pp.315-318, 1996.
DOI : 10.1103/PhysRevLett.76.315

A. V. Akimov, Hypersonic Modulation of Light in Three-Dimensional Photonic and Phononic Band-Gap Materials, Physical Review Letters, vol.101, issue.3, p.33902, 2008.
DOI : 10.1103/PhysRevLett.101.033902

I. A. Kunin, Three-Dimensional Models, Elastic Media with Microstructure, 1983.

L. J. Raubenheimer and G. Gilat, Accurate Numerical Method of Calculating Frequency Distribution Functions in Solids. II. Extension to hcp Crystals, Physical Review, vol.157, issue.3, pp.586-599, 1967.
DOI : 10.1103/PhysRev.157.586

V. Tozzini and M. P. Tosi, Lattice vibrations and elastic constants of crystalline and near low-temperature melting, Physica B: Condensed Matter, vol.262, issue.3-4, pp.369-380, 1999.
DOI : 10.1016/S0921-4526(98)01049-7

K. L. Johnson, Contact Mechanics, 1985.

F. Gassmann, ELASTIC WAVES THROUGH A PACKING OF SPHERES, GEOPHYSICS, vol.16, issue.4, pp.673-685, 1951.
DOI : 10.1190/1.1437718

J. Duffy and R. D. Mindlin, Stress-Strain Relations and Vibrations of a Granular Medium, ASME Trans. J. Appl. Mech, vol.24, pp.585-593, 1958.
DOI : 10.1007/978-1-4613-8865-4_49

R. D. Stroll, Sediment Acoustics, 1989.

J. E. White, Underground Sound, 1983.

V. Gusev and V. Tournat, How acoustic waves are guided in buried subsurface channels in unconsolidated granular media, Physical Review E, vol.78, issue.3, p.36602, 2008.
DOI : 10.1103/PhysRevE.78.036602

URL : https://hal.archives-ouvertes.fr/hal-00341140

O. Mouraille and S. Luding, Sound wave propagation in weakly polydisperse granular materials, Ultrasonics, vol.48, issue.6-7, pp.498-505, 2008.
DOI : 10.1016/j.ultras.2008.03.009

L. M. Schwartz, D. L. Johnson, and S. Feng, Vibrational modes in granular crystals, Phys. Rev. Lett, vol.51, pp.831-834, 1984.

A. N. Bogdanov and A. T. Skvortsov, Nonlinear shear waves in granular medium, Sov. Phys. Acoust, vol.38, pp.224-226, 1992.

V. Tournat, V. E. Gusev, and B. Castagnede, Self-demodulation of elastic waves in a one-dimensional granular chain, Physical Review E, vol.70, issue.5, p.56603, 2004.
DOI : 10.1103/PhysRevE.70.056603

A. S. Suiker, A. V. Metrikine, and R. De-borst, Comparison of wave propagation characteristics of the Cosserat continuum model and corresponding discrete lattice models, International Journal of Solids and Structures, vol.38, issue.9, pp.1563-1583, 2001.
DOI : 10.1016/S0020-7683(00)00104-9

I. S. Pavlov, A. I. Potapov, and G. A. Maugin, A 2D granular medium with rotating particles, International Journal of Solids and Structures, vol.43, issue.20, pp.6194-6207, 2006.
DOI : 10.1016/j.ijsolstr.2005.06.012

C. Inserra, V. Tournat, and V. Gusev, A method of controlling wave propagation in initially spatially periodic media, Europhysics Letters (EPL), vol.78, issue.4, p.44001, 2007.
DOI : 10.1209/0295-5075/78/44001

C. Coste and B. Gilles, Sound propagation in a constrained lattice of beads: High-frequency behavior and dispersion relation, Physical Review E, vol.77, issue.2, p.21302, 2008.
DOI : 10.1103/PhysRevE.77.021302

URL : https://hal.archives-ouvertes.fr/hal-01406704

N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1976.

G. P. Srivastava, The Physics of Phonons. Taylor and Francis Group, 1990.

B. Velicky and C. Caroli, Pressure dependence of the sound velocity in a two-dimensional lattice of Hertz-Mindlin balls: Mean-field description, Physical Review E, vol.65, issue.2, p.21307, 2002.
DOI : 10.1103/PhysRevE.65.021307

V. F. Nesterenko, Dynamics of Heterogeneous Materials, 2001.
DOI : 10.1007/978-1-4757-3524-6

V. Tournat and V. Gusev, Acoustics of Unconsolidated ???Model??? Granular Media: An Overview of Recent Results and Several Open Problems, Acta Acustica united with Acustica, vol.96, issue.2, p.208, 2010.
DOI : 10.3813/AAA.918271

H. A. Makse, N. Gland, D. L. Johnson, and L. M. Schwartz, Why Effective Medium Theory Fails in Granular Materials, Physical Review Letters, vol.83, issue.24, pp.5070-5073, 1999.
DOI : 10.1103/PhysRevLett.83.5070

L. , L. Ragione, and J. T. Jenkins, The initial response of an idealized granular material, Proc. R. Soc. A, pp.735-758, 2007.

V. Magnanimo, L. La-ragione, J. T. Jenkins, P. Wang, and H. A. Makse, Characterizing the shear and bulk moduli of an idealized granular material, EPL (Europhysics Letters), vol.81, issue.3, p.34006, 2008.
DOI : 10.1209/0295-5075/81/34006

H. M. Jaeger, S. R. Nagel, and R. P. Behringer, The Physics of Granular Materials, Physics Today, vol.49, issue.4, pp.32-38, 1996.
DOI : 10.1063/1.881494

P. G. De-gennes, Granular matter: a tentative view, Reviews of Modern Physics, vol.71, issue.2, pp.374-382, 1999.
DOI : 10.1103/RevModPhys.71.S374

R. D. Mindlin and H. Deresiewicz, Elastic Spheres in Contact Under Varying Oblique Forces, ASME J. Appl. Mech, vol.75, pp.327-344, 1953.
DOI : 10.1007/978-1-4613-8865-4_35

J. Duffy and R. D. Mindlin, Stress-Strain Relations and Vibrations of a Granular Medium, ASME Trans. J. Appl. Mech, vol.24, pp.585-593, 1958.
DOI : 10.1007/978-1-4613-8865-4_49

V. Tournat, V. E. Gusev, and B. Castagnède, Self-demodulation of elastic waves in a one-dimensional granular chain, Physical Review E, vol.70, issue.5, p.56603, 2004.
DOI : 10.1103/PhysRevE.70.056603

N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1976.

M. T. Dove, Introduction to Lattice Dynamics, 1993.

G. P. Srivastava, The Physics of Phonons. Taylor and Francis Group, 1990.

L. M. Schwartz, D. L. Johnson, and S. Feng, Vibrational Modes in Granular Materials, Physical Review Letters, vol.52, issue.10, pp.831-834, 1984.
DOI : 10.1103/PhysRevLett.52.831

J. T. Jenkins and M. A. Koenders, The incremental response of random aggregates of identical round particles, The European Physical Journal E - Soft Matter, vol.13, issue.2, pp.113-123, 2004.
DOI : 10.1140/epje/e2004-00048-9

I. Agnolin and J. Roux, On the elastic moduli of three-dimensional assemblies of spheres: Characterization and modeling of fluctuations in the particle displacement and rotation, International Journal of Solids and Structures, vol.45, issue.3-4, pp.1101-1123, 2008.
DOI : 10.1016/j.ijsolstr.2007.07.016

URL : https://hal.archives-ouvertes.fr/hal-00353678

A. S. Suiker, A. V. Metrikine, and R. De-borst, Comparison of wave propagation characteristics of the Cosserat continuum model and corresponding discrete lattice models, International Journal of Solids and Structures, vol.38, issue.9, pp.1563-1583, 2001.
DOI : 10.1016/S0020-7683(00)00104-9

I. S. Pavlov, A. I. Potapov, and G. A. Maugin, A 2D granular medium with rotating particles, International Journal of Solids and Structures, vol.43, issue.20, pp.6194-6207, 2006.
DOI : 10.1016/j.ijsolstr.2005.06.012

B. Velicky and C. Caroli, Pressure dependence of the sound velocity in a two-dimensional lattice of Hertz-Mindlin balls: Mean-field description, Physical Review E, vol.65, issue.2, p.21307, 2002.
DOI : 10.1103/PhysRevE.65.021307

Z. Liu, C. T. Chan, P. Cheng, A. L. Goertzen, and J. H. Page, Elastic wave scattering by periodic structures of spherical objects: Theory and experiment, Physical Review B, vol.62, issue.4, pp.2446-2457, 2000.
DOI : 10.1103/PhysRevB.62.2446

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan et al., Ultrasound Tunneling through 3D Phononic Crystals, Physical Review Letters, vol.88, issue.10, p.104301, 2002.
DOI : 10.1103/PhysRevLett.88.104301

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan et al., Focusing of Sound in a 3D Phononic Crystal, Physical Review Letters, vol.93, issue.2, p.24301, 2004.
DOI : 10.1103/PhysRevLett.93.024301

O. Mouraille, W. A. Mulder, and S. Luding, Sound wave acceleration in granular materials, Journal of Statistical Mechanics: Theory and Experiment, vol.2006, issue.07, p.7023, 2006.
DOI : 10.1088/1742-5468/2006/07/P07023

V. A. Akimov, Hypersonic Modulation of Light in Three-Dimensional Photonic and Phononic Band-Gap Materials, Physical Review Letters, vol.101, issue.3, p.33902, 2008.
DOI : 10.1103/PhysRevLett.101.033902

A. C. Arsenault, From colour fingerprinting to the control of photoluminescence in elastic photonic crystals, Nature Materials, vol.1, issue.3, pp.179-184, 2006.
DOI : 10.1038/nmat1588

W. Cheng, J. Wang, U. Jonas, G. Fytas, and N. Stefanou, Observation and tuning of hypersonic bandgaps in colloidal crystals, Nature Materials, vol.69, issue.10, pp.830-836, 2006.
DOI : 10.1038/nmat1727

E. B. Herbold, J. Kim, V. F. Nesterenko, S. Y. Wang, and C. Daraio, Pulse propagation in a linear and nonlinear diatomic periodic chain: effects of acoustic frequency band-gap, Acta Mechanica, vol.82, issue.1-4, pp.85-103, 2009.
DOI : 10.1007/s00707-009-0163-6

C. Daraio, V. F. Nesterenko, E. B. Herbold, and S. Jin, Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals, Physical Review E, vol.73, issue.2, p.26610, 2006.
DOI : 10.1103/PhysRevE.73.026610

C. Coste and B. Gilles, Sound propagation in a constrained lattice of beads: High-frequency behavior and dispersion relation, Physical Review E, vol.77, issue.2, p.21302, 2008.
DOI : 10.1103/PhysRevE.77.021302

URL : https://hal.archives-ouvertes.fr/hal-01406704

Y. Nahmad-molinari and J. C. Ruiz-suarez, Epitaxial Growth of Granular Single Crystals, Physical Review Letters, vol.89, issue.26, p.264302, 2002.
DOI : 10.1103/PhysRevLett.89.264302

A. B. Yu, X. Z. An, R. P. Zou, R. Y. Yang, and K. Kendall, Self-Assembly of Particles for Densest Packing by Mechanical Vibration, Physical Review Letters, vol.97, issue.26, p.265501, 2006.
DOI : 10.1103/PhysRevLett.97.265501

O. Carvente and J. C. Ruiz-suarez, Crystallization of Confined Non-Brownian Spheres by Vibrational Annealing, Physical Review Letters, vol.95, issue.1, p.18001, 2005.
DOI : 10.1103/PhysRevLett.95.018001

Z. Cheng, W. B. Russel, and P. M. Chaikin, Control growth of hard-sphere colloidal crystals, Nature, vol.401, pp.893-895, 1999.

J. D. Joannopoulos, Self-assembly lights up, Nature, vol.414, issue.6861, pp.257-258, 2001.
DOI : 10.1038/35104718

A. Van-blaaderen, R. Ruel, and P. Wiltzius, Template-directed colloidal crystallization, Nature, vol.385, issue.6614, pp.321-324, 1997.
DOI : 10.1038/385321a0

A. Merkel, V. Tournat, and V. Gusev, Elastic waves in noncohesive frictionless granular crystals, Ultrasonics, vol.50, issue.2, pp.133-138, 2010.
DOI : 10.1016/j.ultras.2009.09.032

F. Gassmann, ELASTIC WAVES THROUGH A PACKING OF SPHERES, GEOPHYSICS, vol.16, issue.4, pp.673-685, 1951.
DOI : 10.1190/1.1437718

C. Inserra, V. Tournat, and V. Gusev, A method of controlling wave propagation in initially spatially periodic media, Europhysics Letters (EPL), vol.78, issue.4, p.44001, 2007.
DOI : 10.1209/0295-5075/78/44001

C. S. Chang and J. Gao, Non-linear dispersion of plane wave in granular media, International Journal of Non-Linear Mechanics, vol.30, issue.2, pp.111-128, 1995.
DOI : 10.1016/0020-7462(94)00039-D

L. J. Raubenheimer and G. Gilat, Accurate Numerical Method of Calculating Frequency Distribution Functions in Solids. II. Extension to hcp Crystals, Physical Review, vol.157, issue.3, pp.586-599, 1967.
DOI : 10.1103/PhysRev.157.586

J. P. Castagna, M. L. Batzle, and R. L. Eastwood, Relationships between compressional???wave and shear???wave velocities in clastic silicate rocks, GEOPHYSICS, vol.50, issue.4, 1985.
DOI : 10.1190/1.1441933

R. De-borst and L. J. Sluys, Localisation in a Cosserat continuum under static and dynamic loading conditions, Computer Methods in Applied Mechanics and Engineering, vol.90, issue.1-3, pp.805-827, 1991.
DOI : 10.1016/0045-7825(91)90185-9

]. E. Bibliographie, F. Cosserat, and . Cosserat, Théorie des Corps Déformables. Hermann et Fils Microcontinuum Field Theories. I. Foundations and Solids, 1909.

R. Lakes, Experimental methods for study of Cosserat elastic solids and other generalized elastic continua, Continuum Models for Materials with Microstructure

W. Nowacki, Theory of Asymmetric Elasticity, 1986.

E. Pasternak and A. Dyskin, Measuring of Cosserat Effects and??Reconstruction of Moduli Using??Dispersive??Waves, Mechanics of Generalized Continua : One Hundred After the Cosserats, 2010.
DOI : 10.1007/978-1-4419-5695-8_8

A. Kiris and E. Inan, On the identification of microstretch elastic moduli of materials by using vibration data of plates, International Journal of Engineering Science, vol.46, issue.6, pp.585-597, 2008.
DOI : 10.1016/j.ijengsci.2008.01.001

R. D. Gauthier, EXPERIMENTAL INVESTIGATIONS ON MICROPOLAR MEDIA, Mechanics of Micropolar Media, 1982.
DOI : 10.1142/9789812797247_0007

I. S. Pavlov, A. I. Potapov, and G. A. Maugin, A 2D granular medium with rotating particles, International Journal of Solids and Structures, vol.43, issue.20, pp.6194-6207, 2006.
DOI : 10.1016/j.ijsolstr.2005.06.012

J. Duffy and R. D. Mindlin, Stress-Strain Relations and Vibrations of a Granular Medium, ASME Trans. J. Appl. Mech, vol.24, pp.585-593, 1958.
DOI : 10.1007/978-1-4613-8865-4_49

L. M. Schwartz, D. L. Johnson, and S. Feng, Vibrational Modes in Granular Materials, Physical Review Letters, vol.52, issue.10, pp.831-834, 1984.
DOI : 10.1103/PhysRevLett.52.831

A. Merkel, V. Tournat, and V. Gusev, Dispersion of elastic waves in three-dimensional noncohesive granular phononic crystals: Properties of rotational modes, Physical Review E, vol.82, issue.3, p.31305, 2010.
DOI : 10.1103/PhysRevE.82.031305

I. Agnolin and J. Roux, On the elastic moduli of three-dimensional assemblies of spheres: Characterization and modeling of fluctuations in the particle displacement and rotation, International Journal of Solids and Structures, vol.45, issue.3-4, pp.1101-1123, 2008.
DOI : 10.1016/j.ijsolstr.2007.07.016

URL : https://hal.archives-ouvertes.fr/hal-00353678

O. Mouraille, Sound Propagation in Dry Granular Materials : Discrete Element Simulations, Theory, and Experiments, 2009.

M. De-billy, Frequency analysis of the acoustic signal transmitted through a one-dimensional chain of metallic spheres, The Journal of the Acoustical Society of America, vol.110, issue.2, pp.710-716, 2001.
DOI : 10.1121/1.1385179

J. Anfosso and V. Gibiat, Elastic wave propagation in a three-dimensional periodic granular medium, Europhysics Letters (EPL), vol.67, issue.3, pp.376-382, 2004.
DOI : 10.1209/epl/i2004-10085-9

C. Coste and B. Gilles, Sound propagation in a constrained lattice of beads: High-frequency behavior and dispersion relation, Physical Review E, vol.77, issue.2, p.21302, 2008.
DOI : 10.1103/PhysRevE.77.021302

URL : https://hal.archives-ouvertes.fr/hal-01406704

M. T. Dove, Introduction to Lattice Dynamics, 1993.

L. J. Raubenheimer and G. Gilat, Accurate Numerical Method of Calculating Frequency Distribution Functions in Solids. II. Extension to hcp Crystals, Physical Review, vol.157, issue.3, pp.586-599, 1967.
DOI : 10.1103/PhysRev.157.586

F. Auger and P. Flandrin, Improving the readability of time-frequency and time-scale representations by the reassignment method, IEEE Transactions on Signal Processing, vol.43, issue.5, pp.1068-1089, 1995.
DOI : 10.1109/78.382394

P. Flandrin, Time-Frequency / Time-Scale Analysis, 1999.

]. P. Bibliographie1 and . Westervelt, Parametric acoustic array Possible exploitation on non-linear acoustics in underwater transmitting applications, J. Acoust. Soc. Am. J. Sound Vib, vol.35, issue.2, pp.535-537435, 1963.

V. A. Zverev, How the idea of a parametric acoustic array was conceived, Acoust. Phys, vol.45, pp.611-618, 1999.

V. Yu, A. B. Zaitsev, V. E. Kolpakov, and . Nazarov, Detection of acoustic pulse in river sand : Experiment, Acoust. Phys, vol.45, pp.202-208, 1999.

V. Yu, A. B. Zaitsev, V. E. Kolpakov, and . Nazarov, Detection of acoustic pulse in river sand : Theory, Acoust. Phys, vol.45, pp.305-310, 1999.

V. Tournat, Effet Non-linéaires d'Auto-démodulation d'Amplitude dans les Milieux Granulaires : Théories et Expériences, 2004.

V. Tournat, V. Zaitsev, V. Gusev, V. Nazarov, P. Béquin et al., Probing Weak Forces in Granular Media through Nonlinear Dynamic Dilatancy: Clapping Contacts and Polarization Anisotropy, Physical Review Letters, vol.92, issue.8, p.85502, 2004.
DOI : 10.1103/PhysRevLett.92.085502

URL : https://hal.archives-ouvertes.fr/hal-00171056

V. Tournat, V. E. Gusev, and B. Castagnède, Self-demodulation of elastic waves in a one-dimensional granular chain, Physical Review E, vol.70, issue.5, p.56603, 2004.
DOI : 10.1103/PhysRevE.70.056603

V. Yu and . Zaitsev, A model of anomalous elastic nonlinearity of microinhomogeneous media, Acoust. Lett, vol.19, pp.171-174, 1996.

I. Yu, V. Y. Belyaeva, and . Zaitsev, Nonlinear elastic properties of microinhomogeneous hierarchically structured media, Acoust. phys, vol.43, pp.510-515, 1997.

C. Inserra, Caratérisation de la Compaction Granulaire par des Méthodes d'Acoustique Linéaires et Non Linéaires, 2007.

L. Fillinger, V. Yu, V. Zaitsev, B. Gusev, and . Castagnède, Nonlinear relaxational absorption/transparency for acoustic waves due to thermoelastic effect, Acust. Acta Acust, vol.92, pp.24-34, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00133999

[. L. Blair, N. W. Mueggenburg, A. H. Marshall, H. M. Jaeger, and S. R. Nagel, Force distributions in three-dimensional granular assemblies: Effects of packing order and interparticle friction, Physical Review E, vol.63, issue.4, p.41304, 2001.
DOI : 10.1103/PhysRevE.63.041304

J. C. Nahmad-molinari and . Ruiz-suarez, Epitaxial Growth of Granular Single Crystals, Physical Review Letters, vol.89, issue.26, p.264302, 2002.
DOI : 10.1103/PhysRevLett.89.264302

[. Carvente and J. C. Ruiz-suarez, Crystallization of Confined Non-Brownian Spheres by Vibrational Annealing, Physical Review Letters, vol.95, issue.1, p.18001, 2005.
DOI : 10.1103/PhysRevLett.95.018001

[. M. Schwartz, D. L. Johnson, and S. Feng, Vibrational Modes in Granular Materials, Physical Review Letters, vol.52, issue.10, pp.831-834, 1984.
DOI : 10.1103/PhysRevLett.52.831

[. Gassmann, ELASTIC WAVES THROUGH A PACKING OF SPHERES, GEOPHYSICS, vol.16, issue.4, pp.673-685, 1951.
DOI : 10.1190/1.1437718

[. A. Inserra, V. Tournat, and V. Gusev, A method of controlling wave propagation in initially spatially periodic media, Europhysics Letters (EPL), vol.78, issue.4, p.44001, 2007.
DOI : 10.1209/0295-5075/78/44001

[. A. Merkel, V. Tournat, and V. E. Gusev, Elastic waves in noncohesive frictionless granular crystals, Ultrasonics, vol.50, issue.2, pp.133-138, 2010.
DOI : 10.1016/j.ultras.2009.09.032

[. A. Merkel, V. Tournat, and V. E. Gusev, Dispersion of elastic waves in three-dimensional noncohesive granular phononic crystals: Properties of rotational modes, Physical Review E, vol.82, issue.3, p.31305, 2010.
DOI : 10.1103/PhysRevE.82.031305

V. P. Kulesh, I. N. Matveenko, and . Shardakov, Construction and Analysis of an Analytical Solution for the Surface Rayleigh Wave within the Framework of the Cosserat Continuum, Journal of Applied Mechanics and Technical Physics, vol.28, issue.6, pp.556-563, 2005.
DOI : 10.1007/s10808-005-0108-3

[. A. Van-blaaden, R. Ruel, and P. Wiltzius, Template-directed colloidal crystallization, Nature, vol.385, issue.6614, pp.321-324, 1997.
DOI : 10.1038/385321a0

[. D. Leahy, Geometric Stability and Elastic Response of a Supported Nanoparticle Film, Physical Review Letters, vol.105, issue.5, p.58301, 2010.
DOI : 10.1103/PhysRevLett.105.058301

]. W. Cheng, M. J. Campolongo, J. J. Cha, S. J. Tan, C. C. Umbach et al., Free-standing nanoparticle superlattice sheets controlled by DNA, Nature Materials, vol.96, issue.6, pp.519-525, 2009.
DOI : 10.1038/nmat2440

[. A. Cheng, Free-standing nanoparticle superlattice sheets controlled by DNA, Nature Materials, vol.96, issue.6, p.519, 2009.
DOI : 10.1038/nmat2440

[. A. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, Surface generation and detection of phonons by picosecond light pulses, Physical Review B, vol.34, issue.6, p.4129, 1986.
DOI : 10.1103/PhysRevB.34.4129