F. Alonso-marroqu?n, Micromechanical investigation of soil deformation: incremental response and granular ratcheting, 2004.

F. Alonso-marroquin and H. Herrmann, The incremental response of soils. An investigation using a discrete-element model, Journal of Engineering Mathematics, vol.53, issue.9, pp.11-34, 2005.
DOI : 10.1201/9780203023631.ch15

S. Antony, R. Momoh, and M. Kuhn, Micromechanical modelling of oval particulates subjected to bi-axial compression, Computational Materials Science, vol.29, issue.4, pp.494-498, 2004.
DOI : 10.1016/j.commatsci.2003.12.007

J. Bardet, Numerical simulations of the incremental responses of idealized granular materials, International Journal of Plasticity, vol.10, issue.8, pp.879-908, 1994.
DOI : 10.1016/0749-6419(94)90019-1

D. Bigoni, Bifurcation and Instability of Non-Associative Elastoplastic Solids, 2000.
DOI : 10.1007/978-3-7091-2562-5_1

D. Bigoni and T. Hueckel, Uniqueness and localization???I. Associative and non-associative elastoplasticity, International Journal of Solids and Structures, vol.28, issue.2, pp.197-213, 1991.
DOI : 10.1016/0020-7683(91)90205-T

R. R. Bonilla, Numerical simulations of undrained granular media, 2004.

F. Calvetti, G. Viggiani, and C. Tamagnini, A numerical investigation of the incremental behavior of granular soils, Rivista italiana di geotecnica, vol.37, issue.3, pp.11-29, 2003.

C. S. Campbell, A problem related to the stability of force chains, Granular Matter, vol.5, issue.3, pp.129-134, 2003.
DOI : 10.1007/s10035-003-0138-6

R. Chambon and D. Caillerie, Existence and uniqueness theorems for boundaryvalue problems involving incrementally non linear models, International Journal of Solids and Structures, vol.36, issue.33, pp.5089-5099, 1999.
DOI : 10.1016/S0020-7683(98)00212-1

R. Chambon, D. Caillerie, and G. Viggiani, Loss of uniqueness and bifurcation vs instability: some remarks. Revue française de génie civil, pp.5-6517, 2004.

B. Chareyre, Discrete element modeling of composites soil-geosynthetics structures: application to trench anchorages at the top of slopes. Theses, 2003.
URL : https://hal.archives-ouvertes.fr/tel-00486807

P. A. Cundall and O. D. Strack, A discrete numerical model for granular assemblies, G??otechnique, vol.29, issue.1, pp.47-65, 1979.
DOI : 10.1680/geot.1979.29.1.47

F. Darve and F. Laouafa, Instabilities in granular materials and application to landslides, Mechanics of Cohesive-frictional Materials, vol.3, issue.8, pp.627-652, 2000.
DOI : 10.1515/JMBM.1994.5.2.193

F. Darve, G. Servant, F. Laouafa, and H. Khoa, Failure in geomaterials: continuous and discrete analyses, Computer Methods in Applied Mechanics and Engineering, vol.193, issue.27-29, pp.27-293057, 2004.
DOI : 10.1016/j.cma.2003.11.011
URL : https://hal.archives-ouvertes.fr/ineris-00962876

J. Desrues, Tracking strain localization in geomaterials using computerized tomography. X-ray CT for Geomaterials, pp.15-41, 2004.

J. Desrues and G. Viggiani, Strain localization in sand: an overview of the experimental results obtained in Grenoble using stereophotogrammetry, International Journal for Numerical and Analytical Methods in Geomechanics, vol.28, issue.4, pp.279-321, 2004.
DOI : 10.1002/nag.338

D. Prisco, C. Imposimato, and S. , Experimental analysis and theoretical interpretation of triaxial load controlled loose sand specimen collapses, Mechanics of Cohesive-frictional Materials, pp.93-120, 1997.
DOI : 10.1115/1.3167042

D. C. Drucker and . Document, A definition of stable inelastic material, 1957.
DOI : 10.21236/AD0143756

B. Gardiner and A. Tordesillas, Micromechanics of shear bands, International Journal of Solids and Structures, vol.41, issue.21, pp.415885-5901, 2004.
DOI : 10.1016/j.ijsolstr.2004.05.051

G. Gudehus, A comparison of some constitutive laws for soils under radially symmetric loading and unloading, Canadian Geotechnical Journal, vol.20, pp.502-516, 1979.

N. Hadda, F. Nicot, and F. Darve, Aspects micromécaniques de la rupture dans les milieux granulaires, Thèse de doctorat Mécanique, 2013.

N. Hadda, F. Nicot, R. Wan, and F. Darve, Microstructural self-organization in granular materials during failure, Comptes Rendus M??canique, vol.343, issue.2, pp.143-154, 2015.
DOI : 10.1016/j.crme.2014.09.009

R. Hill, A general theory of uniqueness and stability in elastic-plastic solids, Journal of the Mechanics and Physics of Solids, vol.6, issue.3, pp.236-249, 1958.
DOI : 10.1016/0022-5096(58)90029-2

G. W. Hunt, A. Tordesillas, S. C. Green, and J. Shi, Force-chain buckling in granular media: a structural mechanics perspective, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.52, issue.3, pp.368249-262, 1910.
DOI : 10.1063/1.2746816

M. Iordache and K. Willam, Localized failure analysis in elastoplastic Cosserat continua, Computer Methods in Applied Mechanics and Engineering, vol.151, issue.3-4, pp.559-586, 1998.
DOI : 10.1016/S0045-7825(97)00166-7

K. Iwashita and M. Oda, Micro-deformation mechanism of shear banding process based on modified distinct element method, Powder Technology, vol.109, issue.1-3, pp.192-205, 2000.
DOI : 10.1016/S0032-5910(99)00236-3

W. T. Koïter, On the thermodynamic background of elastic stability theory. problems of hydro-dynamics and continuum mechanics, In SIAM, pp.423-433, 1969.

M. R. Kuhn, Micro-mechanics of fabric and failure in granular materials, Mechanics of Materials, vol.42, issue.9, pp.827-840, 2010.
DOI : 10.1016/j.mechmat.2010.07.004

M. R. Kuhn and K. Bagi, Contact rolling and deformation in granular media, International Journal of Solids and Structures, vol.41, issue.21, pp.415793-5820, 2004.
DOI : 10.1016/j.ijsolstr.2004.05.066

M. R. Kuhn and K. Bagi, On the relative motions of two rigid bodies at a compliant contact: Application to granular media, Mechanics Research Communications, vol.32, issue.4, pp.463-480, 2005.
DOI : 10.1016/j.mechrescom.2005.01.006

J. Lanier and M. Jean, Experiments and numerical simulations with 2D disks assembly, Powder Technology, vol.109, issue.1-3, pp.206-221, 2000.
DOI : 10.1016/S0032-5910(99)00237-5

A. Lyapunov, Problème général de la stabilité des mouvements, Annales de la faculté des sciences de Toulouse (France), pp.203-474, 1907.

A. Mahboubi, A. Ghaouti, and B. Cambou, La simulation num??rique discr??te du comportement des mat??riaux granulaires, Revue Fran??aise de G??otechnique, issue.76, 1996.
DOI : 10.1051/geotech/1996076045

J. Mandel, Cours de mécanique des milieux continus, 1966.

K. Mehlhorn and D. Michail, Implementing minimum cycle basis algorithms, Journal of Experimental Algorithmics (JEA), vol.11, pp.2-5, 2007.
DOI : 10.1145/1187436.1216582
URL : http://edoc.mpg.de/get.epl?fid=22484&did=279141&ver=0

M. Neilsen and H. Schreyer, Bifurcations in elastic-plastic materials, International Journal of Solids and Structures, vol.30, issue.4, pp.521-544, 1993.
DOI : 10.1016/0020-7683(93)90185-A

T. Ng, Shear strength of assemblies of ellipsoidal particles, G??otechnique, vol.54, issue.10, pp.659-669, 2004.
DOI : 10.1680/geot.2004.54.10.659

F. Nicot and F. Darve, A micro-mechanical investigation of bifurcation in granular materials, International Journal of Solids and Structures, vol.44, issue.20, pp.6630-6652, 2007.
DOI : 10.1016/j.ijsolstr.2007.03.002

F. Nicot, F. Darve, D. Vu-khoa, and H. , Bifurcation and second-order work in geomaterials. International journal for numerical and analytical methods in geomechanics, pp.311007-1032, 2007.

F. Nicot, N. Hadda, F. Bourrier, L. Sibille, and F. Darve, Failure mechanisms in granular media: a discrete element analysis, Granular Matter, vol.11, issue.5, pp.255-260, 2011.
DOI : 10.1007/s10035-009-0146-2

F. Nicot, L. Sibille, and F. Darve, Bifurcation in granular materials: An attempt for a unified framework, International Journal of Solids and Structures, vol.46, issue.22-23, pp.3938-3947, 2009.
DOI : 10.1016/j.ijsolstr.2009.07.008
URL : https://hal.archives-ouvertes.fr/hal-01005291

F. Insa-lyon, L. Sibille, and F. Darve, Failure in rate-independent granular materials as a bifurcation toward a dynamic regime, International Journal of Plasticity, issue.0, pp.29136-154, 2012.

R. Nova, CONTROLLABILITY OF THE INCREMENTAL RESPONSE OF SOIL SPECIMENS SUBJECTED TO ARBITRARY LOADING PROGRAMMES, Journal of the Mechanical Behavior of Materials, vol.100, issue.2, pp.193-202, 1994.
DOI : 10.1061/(ASCE)0733-9399(1988)114:12(2173)

M. Oda, Microstructure in shear band observed by microfocus X-ray computed tomography, G??otechnique, vol.54, issue.8, pp.539-542, 2004.
DOI : 10.1680/geot.2004.54.8.539

M. Oda and K. Iwashita, Study on couple stress and shear band development in granular media based on numerical simulation analyses, International Journal of Engineering Science, vol.38, issue.15, pp.1713-1740, 2000.
DOI : 10.1016/S0020-7225(99)00132-9

V. Osinov and W. Wu, Instability and ill-posedness in the deformation of plastic solids: some correlations through simple examples, Trends in Applications of Mathematics to Mechanics, pp.361-370, 2005.

O. Sullivan and C. , Particulate discrete element modelling, 2011.

J. Peters, M. Muthuswamy, J. Wibowo, and A. Tordesillas, Characterization of force chains in granular material, Physical Review E, vol.46, issue.4, p.41307, 2005.
DOI : 10.1103/PhysRevE.58.2079

H. Petryk, Theory of Bifurcation and Instability in Time-Independent Plasticity, 1993.
DOI : 10.1007/978-3-7091-2712-4_3

F. Prunier, F. Laouafa, and F. Darve, 3D bifurcation analysis in geomaterials, European Journal of Environmental and Civil Engineering, vol.8, issue.2, pp.135-147, 2009.
DOI : 10.1002/nag.591
URL : https://hal.archives-ouvertes.fr/ineris-00963176

F. Prunier, F. Nicot, F. Darve, F. Laouafa, and S. Lignon, Three-Dimensional Multiscale Bifurcation Analysis of Granular Media, Journal of Engineering Mechanics, vol.135, issue.6, pp.493-509, 2009.
DOI : 10.1061/(ASCE)EM.1943-7889.0000003
URL : https://hal.archives-ouvertes.fr/ineris-00963153

S. Pucilowski and A. Tordesillas, Building blocks for the self-assembly of granular materials in three-dimensions, 2010.

F. Radjai, S. Roux, and J. J. Moreau, Contact forces in a granular packing, Chaos: An Interdisciplinary Journal of Nonlinear Science, vol.6, issue.3, pp.544-550, 1999.
DOI : 10.1103/PhysRevB.57.R3201
URL : https://hal.archives-ouvertes.fr/hal-01407368

F. Radjai, D. E. Wolf, M. Jean, and J. Moreau, Bimodal Character of Stress Transmission in Granular Packings, Physical Review Letters, vol.54, issue.1, pp.61-64, 1998.
DOI : 10.1103/PhysRevE.54.861
URL : https://hal.archives-ouvertes.fr/hal-01407369

B. Raniecki, Uniqueness criteria in solids with non-associated plastic-flow laws at finite deformations, 1979.

B. Raniecki and O. Bruhns, Bounds to bifurcation stresses in solids with non-associated plastic flow law at finite strain, Journal of the Mechanics and Physics of Solids, vol.29, issue.2, pp.153-172, 1981.
DOI : 10.1016/0022-5096(81)90021-1

J. R. Rice, The localization of plastic deformation, Theoretical and Applied Mechanics. Citeseer, 1976.

P. Royis and T. Doanh, Theoretical analysis of strain response envelopes using incrementally non-linear constitutive equations. International journal for numerical and analytical methods in geomechanics, pp.97-132, 1998.

J. W. Rudnicki and J. Rice, Conditions for the localization of deformation in pressure-sensitive dilatant materials, Journal of the Mechanics and Physics of Solids, vol.23, issue.6, pp.371-394, 1975.
DOI : 10.1016/0022-5096(75)90001-0

G. Servant, F. Darve, J. Desrues, G. , and I. , Diffuse modes of failure in geomaterials. Deformation characteristics of geomaterials, pp.181-198, 2005.

L. Sibille, Mod??lisation discr??te de la rupture dans les milieux granulaires, Revue Europ??enne de G??nie Civil, vol.10, issue.8, 2006.
DOI : 10.1080/17747120.2006.9692899

L. Sibille, Modélisations discrètes de la rupture dans les milieux granu- laires, 2006.

L. Sibille, F. Nicot, F. Donzé, and F. Darve, Analysis of failure occurrence from direct simulations, European Journal of Environmental and Civil Engineering, vol.3, issue.2, pp.187-201, 2009.
DOI : 10.1002/nag.591
URL : https://hal.archives-ouvertes.fr/hal-01005248

L. Sibille, F. Nicot, F. Donzé, and F. Darve, Material instability in granular assemblies from fundamentally different models. International journal for numerical and analytical methods in geomechanics, pp.31457-481, 2007.

A. G. Smart and J. M. Ottino, Evolving loop structure in gradually tilted two-dimensional granular packings, Physical Review E, vol.29, issue.4, p.41307, 2008.
DOI : 10.1103/PhysRevLett.89.204302

V. ?milauer, E. Catalano, B. Chareyre, S. Dorofeenko, J. Duriez et al., Yade Reference Documentation, ?milauer, editor, Yade Documentation . The Yade Project, 2010.

K. Thompson and J. Gray, Granular materials: fundamentals and application. Granular materials: fundamentals and application, 2004.

C. Thornton, Numerical simulations of deviatoric shear deformation of granular media, G??otechnique, vol.50, issue.1, pp.43-53, 2000.
DOI : 10.1680/geot.2000.50.1.43

A. Tordesillas, Force chain buckling, unjamming transitions and shear banding in dense granular assemblies, Philosophical Magazine, vol.94, issue.32, pp.4987-5016, 2007.
DOI : 10.1103/PhysRevLett.94.015701

A. Tordesillas, Q. Lin, J. Zhang, R. Behringer, and J. Shi, Structural stability and jamming of self-organized cluster conformations in dense granular materials, Journal of the Mechanics and Physics of Solids, vol.59, issue.2, pp.265-296, 2011.
DOI : 10.1016/j.jmps.2010.10.007

A. Tordesillas and M. Muthuswamy, On the modeling of confined buckling of force chains, Journal of the Mechanics and Physics of Solids, vol.57, issue.4, pp.706-727, 2009.
DOI : 10.1016/j.jmps.2009.01.005

A. Tordesillas, J. Shi, and T. Tshaikiwsky, Stress-dilatancy and force chain evolution, International Journal for Numerical and Analytical Methods in Geomechanics, vol.76, issue.6, pp.264-292, 2011.
DOI : 10.1103/PhysRevE.76.011301

A. Tordesillas, D. M. Walker, L. , and Q. , Force cycles and force chains, Physical Review E, vol.81, issue.1, p.11302, 2010.
DOI : 10.1142/S0219525901000140
URL : https://hal.archives-ouvertes.fr/hal-00654281

A. Tordesillas, S. D. Walsh, and M. Muthuswamy, The Effect of Local Kinematics on the Local and Global Deformations of Granular Systems, Mathematics and Mechanics of Solids, vol.116, issue.1, pp.3-41, 2010.
DOI : 10.1680/geot.2005.55.4.333

A. Tordesillas, J. Zhang, and R. Behringer, Buckling force chains in dense granular assemblies: physical and numerical experiments. Geomechanics and Geoengineering, An International Journal, vol.4, issue.1, pp.3-16, 2009.

K. Valanis, On the Uniqueness of Solution of the Initial Value Problem in Softening Materials, Journal of Applied Mechanics, vol.52, issue.3, pp.649-653, 1985.
DOI : 10.1115/1.3169115

K. Valanis, Banding and stability in plastic materials, Acta Mechanica, vol.23, issue.1-2, pp.113-141, 1989.
DOI : 10.1007/BF01181483