.. Well-known-facts and S. , 78 3.2 Playing with homogenization structures, p.83

S. Bloch-transform and .. Definition-of-ellipticity-constants, 98 3.5.3 Application to nonlinear problems, p.102

. Strong-ellipticity and ]. Discrete-homogenization-[-glo11b, 124 4.5.1 Loss of strong ellipticity in the periodic discrete case, Perturbation result for stochastic polymer networks . . . . . . . . . . . . . . . . 130

R. 1. Alicandro and M. Cicalese, A General Integral Representation Result for Continuum Limits of Discrete Energies with Superlinear Growth, SIAM Journal on Mathematical Analysis, vol.36, issue.1, pp.1-37, 2004.
DOI : 10.1137/S0036141003426471

J. Babadjian and M. Barchiesi, A variational approach to the local character of G-closure: the convex case, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.26, issue.2, pp.351-373, 2009.
DOI : 10.1016/j.anihpc.2007.08.002

URL : https://hal.archives-ouvertes.fr/hal-00265694

G. Bal, Central Limits and Homogenization in Random Media, Multiscale Modeling & Simulation, vol.7, issue.2, pp.677-702, 2008.
DOI : 10.1137/070709311

J. M. Ball, Some Open Problems in Elasticity, Geometry, mechanics, and dynamics, pp.3-59, 2002.
DOI : 10.1007/0-387-21791-6_1

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Archive for Rational Mechanics and Analysis, vol.8, issue.4, pp.337-403, 1977.
DOI : 10.1007/BF00279992

M. Barchiesi, Loss of polyconvexity by homogenization: a new example, Calculus of Variations and Partial Differential Equations, vol.19, issue.3, pp.215-230, 2007.
DOI : 10.1007/s00526-006-0085-2

M. Bellieud and G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects, Ann. Scuola Norm. Sup. Pisa Cl. Sci, vol.26, issue.43, pp.407-436, 1998.
URL : https://hal.archives-ouvertes.fr/hal-01283228

A. Bensoussan, J. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, of Studies in Mathematics and its Applications, 1978.

P. Binev, A. Cohen, W. Dahmen, R. Devore, G. Petrova et al., Convergence Rates for Greedy Algorithms in Reduced Basis Methods, SIAM Journal on Mathematical Analysis, vol.43, issue.3
DOI : 10.1137/100795772

URL : https://hal.archives-ouvertes.fr/hal-00767082

X. Blanc and C. L. Bris, Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings, Networks and Heterogeneous Media, vol.5, issue.1, pp.1-29, 2010.
DOI : 10.3934/nhm.2010.5.1

URL : https://hal.archives-ouvertes.fr/inria-00387214

X. Blanc, C. L. Bris, and P. L. Lions, Une variante de la th??orie de l'homog??n??isation stochastique des op??rateurs elliptiques, Comptes Rendus Mathematique, vol.343, issue.11-12, pp.717-724, 2006.
DOI : 10.1016/j.crma.2006.09.034

S. Boyaval, Reduced-Basis Approach for Homogenization beyond the Periodic Setting, Multiscale Modeling & Simulation, vol.7, issue.1, pp.466-494, 2008.
DOI : 10.1137/070688791

URL : https://hal.archives-ouvertes.fr/inria-00132763

A. Braides, Homogenization of some almost periodic functionals, Rend. Accad. Naz. Sci. XL, vol.103, pp.261-281, 1985.

A. Braides, Loss of polyconvexity by Homogenization, Archive for Rational Mechanics and Analysis, vol.19, issue.2, pp.183-190, 1994.
DOI : 10.1007/BF00377660

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, of Oxford Lecture Series in Mathematics and Its Applications, 1998.

M. Briane, Homogenization??of Non-Uniformly Bounded Operators:??Critical Barrier for Nonlocal Effects, Archive for Rational Mechanics and Analysis, vol.164, issue.1, pp.73-101, 2002.
DOI : 10.1007/s002050200196

A. Buffa, Y. Maday, A. T. Patera, C. Prud-'homme, and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis
URL : https://hal.archives-ouvertes.fr/hal-00659314

L. A. Caffarelli and P. E. Souganidis, Rates of convergence for the homogenization of??fully nonlinear uniformly elliptic pde in??random media, Inventiones mathematicae, vol.23, issue.2, pp.301-360, 2010.
DOI : 10.1007/s00222-009-0230-6

Y. Chen, On strong ellipticity and the Legendre-Hadamard condition, Archive for Rational Mechanics and Analysis, vol.50, issue.2, pp.165-175, 1991.
DOI : 10.1007/BF00380415

C. Choquet and A. Sili, Homogenization of a model of displacement with unbounded viscosity, Networks and Heterogeneous Media, vol.4, issue.4, pp.649-666, 2009.
DOI : 10.3934/nhm.2009.4.649

P. G. Ciarlet, Mathematical elasticity. Volume I: three-dimensional elasticity, of Studies in Mathematics and its Applications, 1988.
URL : https://hal.archives-ouvertes.fr/hal-01077424

A. Cohen, R. Devore, and C. Schwab, Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs, Foundations of Computational Mathematics, vol.60, issue.6, pp.615-646, 2010.
DOI : 10.1007/s10208-010-9072-2

A. Cohen, R. Devore, and C. Schwab, ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S, Analysis and Applications, vol.09, issue.01, pp.11-47, 2011.
DOI : 10.1142/S0219530511001728

J. G. Conlon and A. Naddaf, On Homogenization Of Elliptic Equations With Random Coefficients, Electronic Journal of Probability, vol.5, issue.0, pp.1-58, 2000.
DOI : 10.1214/EJP.v5-65

G. , D. Maso, and L. Modica, Integral functionals determined by their minima, Rend. Semin. Mat. Univ, vol.76, pp.255-267, 1986.

G. , D. Maso, and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math, vol.368, pp.28-42, 1986.

A. Dalibard, Homogenization of Non-linear Scalar Conservation Laws, Archive for Rational Mechanics and Analysis, vol.5, issue.1, pp.117-164, 2009.
DOI : 10.1007/s00205-008-0123-7

D. Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital, vol.8, issue.4, pp.391-411, 1973.

A. De-masi, P. A. Ferrari, S. Goldstein, and W. D. Wick, An invariance principle for reversible Markov processes. Applications to random motions in random environments, Journal of Statistical Physics, vol.38, issue.1, pp.787-855, 1988.
DOI : 10.1007/BF01041608

B. Delaunay, Sur la sphère vide, Bull. Acad. Sci. URSS, VII, issue.6, pp.793-800, 1934.

R. L. Dobrushin and S. B. Shlosman, Completely Analytical Gibbs Fields, Statistical physics and dynamical systems, pp.371-403, 1984.
DOI : 10.1007/978-1-4899-6653-7_21

R. L. Dobrushin and S. B. Shlosman, Constructive Criterion for the Uniqueness of Gibbs Field, Statistical physics and dynamical systems, pp.347-370, 1984.
DOI : 10.1007/978-1-4899-6653-7_20

R. Figari, E. Orlandi, and G. Papanicolaou, Mean Field and Gaussian Approximation for Partial Differential Equations with Random Coefficients, SIAM Journal on Applied Mathematics, vol.42, issue.5, pp.1069-1077, 1982.
DOI : 10.1137/0142074

P. J. Flory, Statistical mechanics of chain molecules, 1969.

S. Fortune, VORONOI DIAGRAMS and DELAUNAY TRIANGULATIONS, Computing in Euclidean geometry, pp.225-265, 1995.
DOI : 10.1142/9789812831699_0007

G. Friesecke, R. D. James, and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Communications on Pure and Applied Mathematics, vol.120, issue.3, pp.1461-1506, 2002.
DOI : 10.1002/cpa.10048

G. Geymonat, S. Müller, and N. Triantafyllidis, Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity, Archive for Rational Mechanics and Analysis, vol.52, issue.3, pp.231-290, 1993.
DOI : 10.1007/BF00380256

A. Gloria, A direct approach to numerical homogenization in finite elasticity, Netw. Heterog. Media, vol.1, pp.109-141, 2006.
URL : https://hal.archives-ouvertes.fr/inria-00070383

M. Grüter and K. Widman, The Green function for uniformly elliptic equations, Manuscripta Mathematica, vol.26, issue.3, pp.303-342, 1982.
DOI : 10.1007/BF01166225

N. Hansen, The CMA evolution strategy: a comparing review Advances on estimation of distribution algorithms, pp.75-102, 2006.

T. Y. Hou, Numerical Approximations to Multiscale Solutions in Partial Differential Equations, Frontier in Numerical Analysis, pp.241-302, 2003.
DOI : 10.1007/978-3-642-55692-0_6

T. Iwaniec, L. V. Kovalev, and J. Onninen, Diffeomorphic Approximation of Sobolev Homeomorphisms, Archive for Rational Mechanics and Analysis, vol.7, issue.2, pp.1047-1067, 2011.
DOI : 10.1007/s00205-011-0404-4

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, 1994.
DOI : 10.1007/978-3-642-84659-5

C. Kipnis and S. R. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Communications in Mathematical Physics, vol.28, issue.1, pp.1-19, 1986.
DOI : 10.1007/BF01210789

K. Klenke and . Wahrscheinlichkeitstheorie, English version appeared as: Probability theory. A comprehensive course, 2006.

S. M. Kozlov, AVERAGING OF RANDOM OPERATORS, Mathematics of the USSR-Sbornik, vol.37, issue.2, pp.188-202, 1979.
DOI : 10.1070/SM1980v037n02ABEH001948

S. M. Kozlov, AVERAGING OF DIFFERENCE SCHEMES, Mathematics of the USSR-Sbornik, vol.57, issue.2, pp.351-369, 1987.
DOI : 10.1070/SM1987v057n02ABEH003072

J. Kristensen, On the non-locality of quasiconvexity, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.16, issue.1, pp.1-13, 1999.
DOI : 10.1016/S0294-1449(99)80006-7

W. Kuhn and F. Grün, Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe, Kolloid-Zeitschrift, vol.101, issue.3, pp.248-271, 1942.
DOI : 10.1007/BF01793684

R. Künnemann, The diffusion limit for reversible jump processes onZ d with ergodic random bond conductivities, Communications in Mathematical Physics, vol.80, issue.1, pp.27-68, 1983.
DOI : 10.1007/BF01209386

L. Tallec, Numerical methods for nonlinear three-dimensional elasticity, Handbook of numerical analysis, pp.465-622, 1994.
DOI : 10.1016/S1570-8659(05)80018-3

M. Ledoux, Logarithmic Sobolev Inequalities for Unbounded Spin Systems Revisited, Séminaire de Probabilités, pp.167-194, 2001.
DOI : 10.1007/978-3-540-44671-2_13

W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa, vol.17, issue.3, pp.43-77, 1963.

K. A. Lurie and A. V. Cherkaev, Synopsis, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol.264, issue.1-2, pp.71-87, 1984.
DOI : 10.1063/1.1728579

Y. Maday, A. T. Patera, and G. Turinici, Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations, Comptes Rendus Mathematique, vol.335, issue.3, pp.289-294, 2002.
DOI : 10.1016/S1631-073X(02)02466-4

URL : https://hal.archives-ouvertes.fr/hal-00798389

P. Marcellini, Periodic solutions and homogenization of non linear variational problems, Annali di Matematica Pura ed Applicata, Series 4, vol.22, issue.1, pp.139-152, 1978.
DOI : 10.1007/BF02417888

P. Mathieu, Quenched Invariance Principles for Random Walks with??Random Conductances, Journal of Statistical Physics, vol.129, issue.2, pp.1025-1046, 2008.
DOI : 10.1007/s10955-007-9465-z

R. Meester and R. Roy, Continuum percolation, Cambridge Tracts in Mathematics, vol.119, 1996.
DOI : 10.1017/CBO9780511895357

K. Messaoudi and G. Michaille, Stochastic homogenization of nonconvex integral functionals, ESAIM: Mathematical Modelling and Numerical Analysis, vol.28, issue.3, pp.329-356, 1994.
DOI : 10.1051/m2an/1994280303291

N. Meunier, O. Pantz, and A. Raoult, Eslastic limit of square lattices with three point interactions, 2011.

J. Mourrat, Variance decay for functionals of the environment viewed by the particle, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.47, issue.1, pp.294-327, 2011.
DOI : 10.1214/10-AIHP375

URL : https://hal.archives-ouvertes.fr/hal-01271688

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rat. Mech. Anal, vol.99, pp.189-212, 1987.

S. Müller, Variational models for microstructure and phase transitions, Lecture Notes in Math, vol.19, pp.85-210, 1996.
DOI : 10.1007/BFb0092670

S. Müller and S. Neukamm, On the Commutability of Homogenization and Linearization in Finite Elasticity, Archive for Rational Mechanics and Analysis, vol.14, issue.6, pp.465-500, 2011.
DOI : 10.1007/s00205-011-0438-7

F. Murat, H-convergence. Séminaire d'Analyse fonctionnelle et numérique, 1978.

G. Nguetseng, Homogenization structures and applications. I, Z. Anal. Anwendungen, vol.22, issue.1, pp.73-107, 2003.
DOI : 10.4171/zaa/1133

G. Nguetseng, Homogenization Structures and Applications II, Zeitschrift f??r Analysis und ihre Anwendungen, vol.23, issue.3, pp.483-508, 2004.
DOI : 10.4171/ZAA/1208

J. Nolen and G. Papanicolaou, Fine scale uncertainty in parameter estimation for elliptic equations, Inverse Problems, vol.25, issue.11, 2010.
DOI : 10.1088/0266-5611/25/11/115021

R. W. Ogden, Large deformation isotropic elasticity: on the correlation of theory and experiment for incompressible rubberlike solids, Proc. R. Soc. Lond. A, pp.565-584, 1972.

H. Owhadi, Approximation of the effective conductivity of ergodic media by periodization, Probability Theory and Related Fields, vol.125, issue.2, pp.225-258, 2003.
DOI : 10.1007/s00440-002-0240-4

URL : https://hal.archives-ouvertes.fr/hal-00138275

G. C. Papanicolaou, Diffusions and random walks in random media, Lecture Notes in Math, vol.18, pp.391-399, 1983.
DOI : 10.1007/BF01646091

G. C. Papanicolaou and S. R. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random fields, pp.835-873, 1979.

M. D. Penrose, Random Parking, Sequential Adsorption,??and the Jamming Limit, Communications in Mathematical Physics, vol.218, issue.1, pp.153-176, 2001.
DOI : 10.1007/s002200100387

B. Perthame and A. E. Tzavaras, Kinetic Formulation for Systems of Two??Conservation Laws and Elastodynamics, Archive for Rational Mechanics and Analysis, vol.155, issue.1, pp.1-48, 2000.
DOI : 10.1007/s002050000109

U. Raitums, On the Local Representation of G-Closure, Archive for Rational Mechanics and Analysis, vol.158, issue.3, pp.213-234, 2001.
DOI : 10.1007/PL00004244

R. H. Rubinstein, M. , and C. , Polymer physics, 2003.

D. Ruelle, Statistical mechanics. Rigorous results, 1999.
URL : https://hal.archives-ouvertes.fr/hal-00126389

T. Schreiber, M. D. Penrose, and J. E. Yukich, Gaussian Limits for Multidimensional Random Sequential Packing at Saturation, Communications in Mathematical Physics, vol.74, issue.4, pp.167-183, 2007.
DOI : 10.1007/s00220-007-0218-2

V. Sidoravicius and A. Sznitman, Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields, pp.219-244, 2004.

V. Sverák, New examples of quasiconvex functions, Archive for Rational Mechanics and Analysis, vol.120, issue.4, pp.293-300, 1992.
DOI : 10.1007/BF01837111

V. Sverák, On the problem of two wells, pp.183-189, 1993.

L. Tartar, Estimations fines de coefficients homogénéisés, Reseach Notes in Mathematics, vol.125, pp.168-187, 1985.
DOI : 10.1007/978-1-4612-2032-9_2

M. Vidrascu, Solution of non-linear elasticity problems using the continu software, Inria Report Research, 2001.
URL : https://hal.archives-ouvertes.fr/inria-00072500

M. Vogelius, A homogenization result for planar, polygonal networks, ESAIM: Mathematical Modelling and Numerical Analysis, vol.25, issue.4, pp.483-514, 1991.
DOI : 10.1051/m2an/1991250404831

W. Woess, Random walks on infinite graphs and groups, volume 138 of Cambridge Tracts in Mathematics, 2000.

X. Yue and W. E. , The local microscale problem in the multiscale modeling of strongly heterogeneous media: Effects of boundary conditions and cell size, Journal of Computational Physics, vol.222, issue.2, pp.556-572, 2007.
DOI : 10.1016/j.jcp.2006.07.034

V. V. Yurinski?-i, Averaging of symmetric diffusion in random medium, Siberian Mathematical Journal, vol.34, issue.No. 4, pp.167-180, 1986.
DOI : 10.1007/BF00969174