M. F. Azeez and A. F. Vakakis, Numerical and experimental analysis of the nonlinear dynamics due to impacts of a continuous overhung rotor, Proceedings of DETC'97, ASME Design Engineering technical Conferences, 1997.

M. F. Azeez and A. F. Vakakis, PROPER ORTHOGONAL DECOMPOSITION (POD) OF A CLASS OF VIBROIMPACT OSCILLATIONS, Journal of Sound and Vibration, vol.240, issue.5, 1998.
DOI : 10.1006/jsvi.2000.3264

A. Chaterjee, An introduction to the proper orthogonal decomposition, Current Sciences, pp.808-817, 2000.

G. Hariton, I. Socolsky, and E. A. , Neo-Hookean fibber reinforced composites in finite elasticity, J. Mech. Phys. Solids, vol.54, issue.3, pp.533-559, 2006.

A. Dür, On the Optimality of the Discrete Karhunen--Lo??ve Expansion, SIAM Journal on Control and Optimization, vol.36, issue.6, pp.1937-1939, 1998.
DOI : 10.1137/S0363012997315750

G. Dvorak, Y. A. Bahei-el-din, and A. M. Wafa, The modeling of inelastic composite materials with the transformation field analysis, Modelling and Simulation in Materials Science and Engineering, vol.2, issue.3A, pp.571-586, 1994.
DOI : 10.1088/0965-0393/2/3A/011

B. F. Feeny and R. Kappagantu, ON THE PHYSICAL INTERPRETATION OF PROPER ORTHOGONAL MODES IN VIBRATIONS, Journal of Sound and Vibration, vol.211, issue.4, pp.189-192, 1998.
DOI : 10.1006/jsvi.1997.1386

F. Feyel and J. Chaboche, FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials, Computer Methods in Applied Mechanics and Engineering, vol.183, issue.3-4, pp.309-330, 2000.
DOI : 10.1016/S0045-7825(99)00224-8

F. Feyel, A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua, Computer Methods in Applied Mechanics and Engineering, vol.192, issue.28-30, pp.3233-3244, 2003.
DOI : 10.1016/S0045-7825(03)00348-7

W. Feng and R. M. Christensen, Nonlinear deformation of elastomeric foams, International Journal of Non-Linear Mechanics, vol.17, issue.5-6, pp.335-367, 1982.
DOI : 10.1016/0020-7462(82)90005-1

J. Fish, K. Shek, M. Pandheeradi, and M. S. Shepard, Computational plasticity for composite structures based on mathematical homogenization: Theory and practice, Computer Methods in Applied Mechanics and Engineering, vol.148, issue.1-2, pp.53-73, 1997.
DOI : 10.1016/S0045-7825(97)00030-3

A. N. Gent and A. G. Thomas, The deformation of foamed elastic materials, Journal of Applied Polymer Science, vol.1, issue.1, pp.107-113, 1959.
DOI : 10.1002/app.1959.070010117

S. Ghosh, K. Lee, and S. Moorthy, Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and Voronoi cell finite element model, Computer Methods in Applied Mechanics and Engineering, vol.132, issue.1-2, pp.63-116, 1996.
DOI : 10.1016/0045-7825(95)00974-4

S. Ghosh, K. Lee, and P. Raghavan, A multi-level computational model for multi-scale damage analysis in composite and porous materials, International Journal of Solids and Structures, vol.38, issue.14, pp.2335-2385, 2001.
DOI : 10.1016/S0020-7683(00)00167-0

J. M. Guedes and N. Kikuchi, Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods, Computer Methods in Applied Mechanics and Engineering, vol.83, issue.2, pp.143-198, 1990.
DOI : 10.1016/0045-7825(90)90148-F

H. Harman, Modern factor analysis, 1960.

Z. Hashin, Large isotropic elastic deformation of composites and porous media, International Journal of Solids and Structures, vol.21, issue.7, pp.711-720, 1985.
DOI : 10.1016/0020-7683(85)90074-5

R. Hill, On constitutive macro-variables for heterogeneous solids at finite stains, Proc. R. Soc. London Ser, pp.131-147, 1972.

R. Hill and J. R. Rice, Elastic Potentials and the Structure of Inelastic Constitutive Laws, SIAM Journal on Applied Mathematics, vol.25, issue.3, pp.448-461, 1973.
DOI : 10.1137/0125045

H. Hotelling, Analysis of complex statistical variables in principal components, J. Exp. Psy, vol.24, p.417, 1953.

G. A. Holzapfel, Nonlinear Solid Mechanics, a Continuum Approach for Engineering, 2001.

J. D. Humphrey, Cardiovascular Solid Mechanics: Cells, Tissues, and Organs, Applied Mechanics Reviews, vol.55, issue.5, 2002.
DOI : 10.1115/1.1497492

M. L. Joyner, Comparison of two techniques for implementing the proper orthogonal decomposition method in damage detection problems, Mathematical and Computer Modelling, vol.40, issue.5-6, pp.553-571, 2004.
DOI : 10.1016/j.mcm.2003.10.050

K. Karhunen, Zur spektraltheorie stochastischer prozesse, Ann. Acad. Sci. Fennicae, vol.37, 1946.

V. Kousnetzova, W. A. Brekelmans, and F. T. Baaijens, An approach to micro-macro modeling of heterogeneous materials, Computational Mechanics, vol.27, issue.1, pp.37-48, 2001.
DOI : 10.1007/s004660000212

V. G. Kousnetzova, M. G. Geers, and W. A. Brekelmans, Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy, Computer Methods in Applied Mechanics and Engineering, vol.193, issue.48-51, pp.5525-5550, 2004.
DOI : 10.1016/j.cma.2003.12.073

P. Ladevèze, O. Loiseau, and D. Dureisseix, A micro???macro and parallel computational strategy for highly heterogeneous structures, International Journal for Numerical Methods in Engineering, vol.46, issue.12, pp.121-138, 2001.
DOI : 10.1002/nme.274

S. Lall, P. Krysl, and J. E. Marsden, Structure-preserving model reduction for mechanical systems, Physica D: Nonlinear Phenomena, vol.184, issue.1-4, pp.304-318, 2003.
DOI : 10.1016/S0167-2789(03)00227-6

Y. C. Liang, H. P. Lee, S. P. Lim, W. Z. Lin, and K. H. Lee, PROPER ORTHOGONAL DECOMPOSITION AND ITS APPLICATIONS???PART I: THEORY, Journal of Sound and Vibration, vol.252, issue.3, pp.527-544, 2002.
DOI : 10.1006/jsvi.2001.4041

T. Lieu and M. Lesoinne, Parameters adaptation of reduced order models for threedimensional flutter analysis, pp.2004-0888

T. Lieu, C. Farhat, and M. Lesoinne, Reduced-order fluid/structure modelling of a complete aircraft configuration, Comput. Methods Appl. Mech. Engrg in press

O. Lopez-pamies, Second-Order Estimates for the Macroscopic Response and Loss of Ellipticity in Porous Rubbers at Large Deformations, Journal of Elasticity, vol.21, issue.9, pp.247-287, 2005.
DOI : 10.1007/s10659-005-1405-z
URL : https://hal.archives-ouvertes.fr/hal-00111435

E. N. Lorenz, Empirical orthogonal eigenfunctions and statistical weather prediction, 1956.

J. L. Lumley, The structure of inhomogeneous turbulent flows Atmospheric turbulence and radio wave propagation, pp.166-178, 1967.

J. C. Michel, H. Moulinec, and P. Suquet, Effective properties of composite materials with periodic microstructure: a computational approach, Computer Methods in Applied Mechanics and Engineering, vol.172, issue.1-4, pp.109-143, 1999.
DOI : 10.1016/S0045-7825(98)00227-8

J. C. Michel and P. Suquet, Nonuniform transformation field analysis, International Journal of Solids and Structures, vol.40, issue.25, pp.6937-6955, 2003.
DOI : 10.1016/S0020-7683(03)00346-9
URL : https://hal.archives-ouvertes.fr/hal-00088331

J. C. Michel and P. Suquet, Computational analysis of nonlinear composite structures using the nonuniform transformation fields analysis Comput, Methods Appl. Mech. Engrg, vol.193, pp.48-515477, 2004.

C. Miehe, . Schröder, and J. Schotte, Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials, Computer Methods in Applied Mechanics and Engineering, vol.171, issue.3-4, pp.387-418, 1999.
DOI : 10.1016/S0045-7825(98)00218-7

C. Miehe and A. Koch, Computational micro-to-macro transitions of discretized microstructures undergoing small strains, Archive of Applied Mechanics (Ingenieur Archiv), vol.72, issue.4-5, pp.300-317, 2002.
DOI : 10.1007/s00419-002-0212-2

C. Miehe, Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy, Computer Methods in Applied Mechanics and Engineering, vol.192, issue.5-6, pp.559-591, 2003.
DOI : 10.1016/S0045-7825(02)00564-9

C. Miehe, J. Schotte, and M. Lambrecht, Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals, Journal of the Mechanics and Physics of Solids, vol.50, issue.10, pp.2123-2167, 2002.
DOI : 10.1016/S0022-5096(02)00016-9

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

S. Nemat-nasser and M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland Series in Applied Mathematics and Mechanics, 1993.
DOI : 10.1115/1.2788912

A. Newman and P. S. Krishnaprasad, Nonlinear model reduction for RTCVD, Proceedings of the 32nd Conference in Information Sciences ans Systems, 1998.

R. W. Ogden, On the overall moduli of non-linear elastic composite materials, Journal of the Mechanics and Physics of Solids, vol.22, issue.6, pp.541-553, 1974.
DOI : 10.1016/0022-5096(74)90033-7

R. W. Ogden, Extremum principles in non-linear elasticity and their application to composites???I, International Journal of Solids and Structures, vol.14, issue.4, pp.265-282, 1978.
DOI : 10.1016/0020-7683(78)90037-9

P. Castañeda and P. , The overall constitutive behavior of nonlinearly elastic composites, Proc. R. Soc. London Ser, pp.147-171, 1989.

P. Castañeda, P. Suquet, and P. Nonlinear-composites, Nonlinear Composites, Adv. Appl. Mech, vol.34, pp.171-302, 1998.
DOI : 10.1016/S0065-2156(08)70321-1

P. Castañeda and P. , Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I???theory, Journal of the Mechanics and Physics of Solids, vol.50, issue.4, pp.737-757, 2002.
DOI : 10.1016/S0022-5096(01)00099-0

E. Schmidt, Zur Theorie der linearen und nichtlinearen Integralgleichungen. III. Teil, Mathematische Annalen, vol.65, issue.3, pp.433-476, 1907.
DOI : 10.1007/BF01456418

R. Schmidt and M. Glauser, Improvements in low dimensional tools for flowstructure interaction problems: using global POD, pp.2004-0889

N. Sheng, M. C. Boyce, D. M. Parks, G. C. Rutledge, J. I. Abes et al., Multiscale micromechanical modeling of polymer/clay nanocomposites and the effective clay particle, Polymer, vol.45, issue.2, pp.487-506, 2004.
DOI : 10.1016/j.polymer.2003.10.100

R. J. Smit, . Brekelmans, and H. E. Meijer, Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling, Computer Methods in Applied Mechanics and Engineering, vol.155, issue.1-2, pp.181-192, 1998.
DOI : 10.1016/S0045-7825(97)00139-4

P. M. Suquet, Local and global aspects in the mathematical theory of plasticity, Plasticity Today: Modelling, Methods and Applications, pp.279-310, 1985.

P. M. Suquet, Introduction, Homogenization Techniques for composites Materials, pp.193-278, 1987.
DOI : 10.1007/3-540-17616-0_15
URL : https://hal.archives-ouvertes.fr/hal-00404025

J. Tatyor, Dynamics of large scale structures in turbulent shear layers, 2001.

K. Terada and N. Kikuchi, Nonlinear homogenization method for practical applications, Computational Methods in Micromechanics Plasticity Today: Modeling, Methods and Applications, pp.279-310, 1985.

K. Terada and . Kikuchi, A class of general algorithms for multi-scale analyses of heterogeneous media, Computer Methods in Applied Mechanics and Engineering, vol.190, issue.40-41, pp.5427-5464, 2001.
DOI : 10.1016/S0045-7825(01)00179-7

J. R. Willis, Variational and Related Methods for the Overall Properties of Composites, Advances in Applied Mechanics, pp.1-78, 1981.
DOI : 10.1016/S0065-2156(08)70330-2