M. Ainsworth and J. T. Oden, A posteriori error estimators for second order elliptic systems part 2. An optimal order process for calculating self-equilibrating fluxes, Computers & Mathematics with Applications, vol.26, issue.9, pp.75-87, 1993.
DOI : 10.1016/0898-1221(93)90007-I
URL : http://doi.org/10.1016/0898-1221(93)90227-m

A. M. Barrientos, N. G. Gatica, and P. E. Stephan, A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a-posteriori error estimate, Numerische Mathematik, vol.91, issue.2, pp.197-222, 2002.
DOI : 10.1007/s002110100337

F. Bassi, L. Botti, A. Colombo, D. A. Di-pietro, and P. Tesini, On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations, Journal of Computational Physics, vol.231, issue.1, pp.45-65, 2012.
DOI : 10.1016/j.jcp.2011.08.018
URL : https://hal.archives-ouvertes.fr/hal-00562219

L. Beirão-da-veiga, F. Brezzi, and L. D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal, vol.2, issue.51, pp.794-812, 2013.

L. Beirão-da-veiga, C. Lovadina, and D. Mora, A Virtual Element Method for elastic and inelastic problems on polytope meshes, Computer Methods in Applied Mechanics and Engineering, vol.295, pp.327-346, 2015.
DOI : 10.1016/j.cma.2015.07.013

C. Bi and Y. Lin, Discontinuous Galerkin method for monotone nonlinear elliptic problems, Int. J. Numer. Anal. Model, vol.9, pp.999-1024, 2012.

S. O. Biabanaki, A. R. Khoei, and P. Wriggers, Polygonal finite element methods for contact-impact problems on non-conformal meshes, Computer Methods in Applied Mechanics and Engineering, vol.269, pp.198-221, 2014.
DOI : 10.1016/j.cma.2013.10.025

S. C. Brenner, Korn's inequalities for piecewise $H^1$ vector fields, Mathematics of Computation, vol.73, issue.247, pp.1067-1087, 2004.
DOI : 10.1090/S0025-5718-03-01579-5
URL : http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.153.8201&rep=rep1&type=pdf

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, 2010.
DOI : 10.1007/978-0-387-70914-7

M. Cervera, M. Chiumenti, and R. Codina, Mixed stabilized finite element methods in nonlinear solid mechanics: Part II: Strain localization, Comput. Methods in Appl. Mech. and Engrg, vol.199, pp.37-402571, 2010.

H. Chi, L. Beirão, G. H. Veiga, and . Paulino, Some basic formulations of the virtual element method (VEM) for finite deformations, Computer Methods in Applied Mechanics and Engineering, vol.318, pp.148-192, 2017.
DOI : 10.1016/j.cma.2016.12.020

H. Chi, C. Talischi, O. Lopez-pamies, and G. H. Paulino, Polygonal finite elements for finite elasticity, International Journal for Numerical Methods in Engineering, vol.306, issue.1496, pp.305-328, 2015.
DOI : 10.1098/rsta.1982.0095
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.691.1565

K. Deimling, Nonlinear functional analysis, 1985.
DOI : 10.1007/978-3-662-00547-7

M. Destrade and R. W. Ogden, On the third- and fourth-order constants of incompressible isotropic elasticity, The Journal of the Acoustical Society of America, vol.128, issue.6, pp.3334-3343, 2010.
DOI : 10.1121/1.3505102

D. A. Di-pietro and J. Droniou, A Hybrid High-Order method for Leray???Lions elliptic equations on general meshes, Mathematics of Computation, vol.86, issue.307, pp.2159-2191, 2017.
DOI : 10.1090/mcom/3180
URL : https://hal.archives-ouvertes.fr/hal-01183484

D. A. Di-pietro and J. Droniou, Ws,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray???Lions problems, Mathematical Models and Methods in Applied Sciences, vol.9, issue.05, pp.879-908, 2017.
DOI : 10.1007/PL00005470

D. A. Di-pietro, J. Droniou, and A. Ern, A Discontinuous-Skeletal Method for Advection-Diffusion-Reaction on General Meshes, SIAM Journal on Numerical Analysis, vol.53, issue.5, pp.2135-2157, 2015.
DOI : 10.1137/140993971
URL : https://hal.archives-ouvertes.fr/hal-01079342

D. A. Di-pietro, J. Droniou, and G. Manzini, Discontinuous skeletal gradient discretisation methods on polytopal meshes, 2017, p.9683

D. A. Di-pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods, of Mathématiques & Applications
DOI : 10.1007/978-3-642-22980-0

D. A. Di-pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes, Computer Methods in Applied Mechanics and Engineering, vol.283, pp.1-21, 2015.
DOI : 10.1016/j.cma.2014.09.009
URL : https://hal.archives-ouvertes.fr/hal-00979435

D. A. Di-pietro, B. Kapidani, R. Specogna, and F. Trevisan, An Arbitrary-Order Discontinuous Skeletal Method for Solving Electrostatics on General Polyhedral Meshes, IEEE Transactions on Magnetics, vol.53, issue.6, pp.1-4, 2017.
DOI : 10.1109/TMAG.2017.2666546
URL : https://hal.archives-ouvertes.fr/hal-01399505

D. A. Di-pietro and S. Lemaire, An extension of the Crouzeix???Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow, Mathematics of Computation, vol.84, issue.291, p.2015
DOI : 10.1090/S0025-5718-2014-02861-5
URL : https://hal.archives-ouvertes.fr/hal-00753660

D. A. Di-pietro and S. Nicaise, A locking-free discontinuous Galerkin method for linear elasticity in locally nearly incompressible heterogeneous media, Applied Numerical Mathematics, vol.63, pp.105-116, 2013.
DOI : 10.1016/j.apnum.2012.09.009
URL : https://hal.archives-ouvertes.fr/hal-00685020

D. A. Di-pietro and R. Tittarelli, Lectures from the fall 2016 thematic quarter at Institut Henri Poincaré, chapter An introduction to Hybrid High-Order methods Accepted for publication, 2017.

J. Droniou, Finite volume schemes for fully non-linear elliptic equations in divergence form, ESAIM: Mathematical Modelling and Numerical Analysis, vol.40, issue.6, pp.1069-1100, 2006.
DOI : 10.1051/m2an:2007001
URL : https://hal.archives-ouvertes.fr/hal-00009614

J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R. Herbin, The gradient discretisation method, 2016.
DOI : 10.1007/978-3-319-57397-7_24
URL : https://hal.archives-ouvertes.fr/hal-01382358

J. Droniou and B. P. Lamichhane, Gradient schemes for linear and non-linear elasticity equations, Numerische Mathematik, vol.29, issue.4, pp.251-277, 2015.
DOI : 10.1002/nme.1620290802

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics. Grundlehren der mathematischen Wissenschaften 219, 1976.
DOI : 10.1115/1.3424078

L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions. Studies in advanced mathematics, Boca Raton, 1992.

G. N. Gatica, A. Márquez, and W. Rudolph, A priori and a posteriori error analyses of augmented twofold saddle point formulations for nonlinear elasticity problems, Computer Methods in Applied Mechanics and Engineering, vol.264, pp.23-48, 2013.
DOI : 10.1016/j.cma.2013.05.010

G. N. Gatica and E. P. Stephan, A mixed-FEM formulation for nonlinear incompressible elasticity in the plane, Numerical Methods for Partial Differential Equations, vol.43, issue.1, pp.105-128, 2002.
DOI : 10.1002/num.1046

G. N. Gatica and W. L. Wendland, Coupling of Mixed Finite Elements and Boundary Elements for A Hyperelastic Interface Problem, SIAM Journal on Numerical Analysis, vol.34, issue.6, pp.2335-2356, 1997.
DOI : 10.1137/S0036142995291317

R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, Finite Volumes for Complex Applications V, pp.659-692, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00429843

D. S. Hughes and J. L. Kelly, Second-Order Elastic Deformation of Solids, Physical Review, vol.9, issue.5, pp.1145-1149, 1953.
DOI : 10.1063/1.1710417

L. D. Landau and E. M. Lifshitz, Theory of elasticity, 1959.

S. E. Leon, D. W. Spring, and G. H. Paulino, Reduction in mesh bias for dynamic fracture using adaptive splitting of polygonal finite elements, International Journal for Numerical Methods in Engineering, vol.23, issue.01, pp.555-576, 2014.
DOI : 10.1142/S0218202512500492

G. J. Minty, ON A "MONOTONICITY" METHOD FOR THE SOLUTION OF NONLINEAR EQUATIONS IN BANACH SPACES, Proceedings of the National Academy of Sciences of the United States of America, pp.1038-1041, 1963.
DOI : 10.1073/pnas.50.6.1038

J. Ne?-cas, Introduction to the theory of nonlinear elliptic equations, 1986.

S. Nicaise, K. Witowski, and B. I. Wohlmuth, An a posteriori error estimator for the Lame equation based on equilibrated fluxes, IMA Journal of Numerical Analysis, vol.28, issue.2, p.331, 2008.
DOI : 10.1093/imanum/drm008

C. Ortner and E. Süli, Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems, SIAM Journal on Numerical Analysis, vol.45, issue.4, pp.1370-1397, 2007.
DOI : 10.1137/06067119X
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.154.5882

M. Pitteri and G. Zanotto, Continuum models for phase transitions and twinning in crystals, 2003.
DOI : 10.1201/9781420036145

S. Soon, B. Cockburn, and H. K. Stolarski, A hybridizable discontinuous Galerkin method for linear elasticity, International Journal for Numerical Methods in Engineering, vol.16, issue.1, pp.1058-1092, 2009.
DOI : 10.1007/978-3-642-59721-3_1

D. W. Spring, S. E. Leon, and G. H. Paulino, Unstructured polygonal meshes with adaptive refinement for the numerical simulation of dynamic cohesive fracture, International Journal of Fracture, vol.33, issue.1, pp.33-57, 2014.
DOI : 10.1002/nme.1620330703

C. Talischi, G. H. Paulino, A. Pereira, and I. F. Menezes, Polymesher: a general-purpose mesh generator for polygonal elements written in Matlab. Structural and Multidisciplinary Optimization, pp.309-328, 2012.

C. Wang, J. Wang, R. Wang, and R. Zhang, A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation, Journal of Computational and Applied Mathematics, vol.307, issue.C, pp.346-366, 2016.
DOI : 10.1016/j.cam.2015.12.015

J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Mathematics of Computation, vol.83, issue.289, pp.2101-2126, 2014.
DOI : 10.1090/S0025-5718-2014-02852-4
URL : http://arxiv.org/abs/1202.3655

P. Wriggers, W. T. Rust, and B. D. Reddy, A virtual element method for contact, Computational Mechanics, vol.79, issue.16, pp.1039-1050, 2016.
DOI : 10.1017/CBO9781139171731