P. Introduction-de-la and .. , 19 II 21 II.A. Lois de comportement des constituants 21 II.B. Formulation du problème hétérogène local (versions initiale et incrémentale), E. Ecriture du problème auxiliaire thermoélastique, p.28

.. Microstructures-À-fibres-longues, I. A. Microstructure-réalisant-la-borne-inférieure-d-'hashin-shtrikman, and .. , 36 III.A.1. Microstructure et chargement 36 III.A.2. Solution exacte de référence : principe de correspondance 37 III.A.3 Estimation de la réponse macroscopique : taux de fibres de 0,45 39 III.A.5 41 III.B. Microstructure périodique : mise en place des outils et premiers résultats 42 III.B.2. Solution exacte de référence : simulations en champs-complets Approche incrémentale : méthode de détermination du pas de calcul optimal et critère de convergence de la réponse asymptotique, 53 III.C. Microstructure périodique : porosité et effets des contrastes.. 57 III.C.2.a) Propriétés de la fibre 58 III.C.2.b) Propriétés visqueuses de la matrice, p.59

I. C. Bilan and .. , 60 III.D. Estimation du second moment 60 III.D.1. Effets du contraste et du taux de fibres sur la projection déviatorique du second moment???, 61 III.D.2. Projection déviatorique du second moment pour le cas d'étude, p.63

I. Généralisation-de-l-'approche-incrémentale and .. , 69 IV.A. Lois de comportement des constituants 69 IV.B. Formulation du problème hétérogène local (version incrémentale) 70 IV.C. Hypothèses de travail (nouvelle version), ., p.77

.. Application-de-l-'approche-incrémentale-pour-différentes-lois-viscoélastiques-locales, V. A. Modèle-de-maxwell-À-variable-interne-non-déviatorique, and .. , 78 V.A.1. Cas d'un coefficient de Poisson constant 79 V.A.2. Réponses macroscopique et par phase 84 V.A.5. Pour aller un peu plus loin : effet des temps caractéristiques, Bilan, p.86

.. Modèle-À-deux-variables-internes, 94 V.C.2.a) Temps de relaxation proches 95 V.C.2.b) Temps de relaxation éloignés

V. B. Microstructures-À-inclusions-ellipsoïdales,-prolates-et-oblates and .. , 109 VI.B.2. Effet du taux d'inclusions sur la réponse asymptotique homogénéisée, p.112

V. C. Composites-À-copeaux-de-bois-lamellés and .. , 112 VI.C.1. Définition de la cellule élémentaire et propriétés élastiques effectives 113 VI.C.2, p.116

P. Introduction-de-la and .. , 119 VII Approche incrémentale avec couplage thermoviscoélastique 121 VII.B. Formulation du problème hétérogène thermomécanique couplé (versions initiale et discrétisée) 122 VII, C. Hypothèses de travail

.. Température-uniforme-imposée-en-tout-point, V. B. Fibres-Élastiques, and .. , 131 VIII.A. Comportements des phases et chargements retenus 134 VIII.B.1. Réponse homogénéisée sous chargement purement mécanique seul 134 VIII.B.2. Réponses homogénéisées pour différents chargements thermomécaniques, 138 VIII.C. Fibres thermoélastiques 140 VIII.C.1. Réponses homogénéisées pour différents chargements thermomécaniques ....... 141 VIII.C.2. Effet de la vitesse du chargement thermique sur les réponses homogénéisées .... 142 VIII.C.3. Contraintes asymptotiques homogénéisées et moyennes par phase, p.144

M. Le-chargement, 146 IX.A. Comportement des phases et intégration de l'équation de la chaleur dans la procédure de résolution, p.148

.. Microstructure-périodique, Comparaison entre l'estimation de la projection déviatorique du second moment dans la matrice avec l'approximation de Luciano et BarberoLB) et la solution de référence (EF) en fonction du contraste et pour différents taux de fibres, Figure, vol.23, p.62, 1994.

J. Aboudi, Mechanics of Composite Materials, A Unified Micromechanical Approach, 1991.

S. R. Arwade, P. L. Clouston, and R. Winans, Measurement and Stochastic Computational Modeling of the Elastic Properties of Parallel Strand Lumber, Journal of Engineering Mechanics, vol.135, issue.9, pp.897-905, 2009.
DOI : 10.1061/(ASCE)EM.1943-7889.0000020

J. Auriault, Effective macroscopic description for heat conduction in periodic composites, International Journal of Heat and Mass Transfer, vol.26, issue.6, pp.861-869, 1983.
DOI : 10.1016/S0017-9310(83)80110-0

C. Badulescu, N. Lahellec, and P. Suquet, Field statistics in linear viscoelastic composites and polycrystals, European Journal of Mechanics - A/Solids, vol.49, pp.329-344, 2015.
DOI : 10.1016/j.euromechsol.2014.07.012

URL : https://hal.archives-ouvertes.fr/hal-01108248

E. Barbero and R. Luciano, Micromechanical formulas for the relaxation tensor of linear viscoelastic composites with transversely isotropic fibers, International Journal of Solids and Structures, vol.32, issue.13, pp.1859-1872, 1995.
DOI : 10.1016/0020-7683(94)00233-M

Y. Benveniste, A new approach to the application of Mori-Tanaka's theory in composite materials, Mechanics of Materials, vol.6, issue.2, pp.147-157, 1987.
DOI : 10.1016/0167-6636(87)90005-6

S. Berbenni, F. Dinzart, and H. Sabar, A new internal variables homogenization scheme for linear viscoelastic materials based on an exact Eshelby interaction law, Mechanics of Materials, vol.81, pp.110-124, 2015.
DOI : 10.1016/j.mechmat.2014.11.003

M. Berveiller and A. Zaoui, An extension of the self-consistent scheme to plastically-flowing polycrystals, Journal of the Mechanics and Physics of Solids, vol.26, issue.5-6, p.325, 1979.
DOI : 10.1016/0022-5096(78)90003-0

M. Bornet, T. Bretheau, and P. Gilormini, Homogénéisation en mécanique des matériaux 1 ? Matériaux aléatoires élastiques et milieu périodiques, 2001.

M. Bornet, T. Bretheau, and P. Gilormini, Homogénéisation en mécanique des matériaux 2 ? Comportements non-linéaires et problèmes ouverts, 2001.

L. Brassart, L. Stainier, I. Doghri, and L. Delannay, A variational formulation for the incremental homogenization of elasto-plastic composites, Journal of the Mechanics and Physics of Solids, vol.59, issue.12, pp.2455-2475, 2011.
DOI : 10.1016/j.jmps.2011.09.004

URL : https://hal.archives-ouvertes.fr/hal-01004965

L. Brassart, L. Stainier, I. Doghri, and L. Delannay, Homogenization of elasto-(visco) plastic composites based on an incremental variational principle, International Journal of Plasticity, vol.36, pp.86-112, 2012.
DOI : 10.1016/j.ijplas.2012.03.010

URL : https://hal.archives-ouvertes.fr/hal-01006931

R. Brenner and R. Masson, Improved affine estimates for nonlinear viscoelastic composites, European Journal of Mechanics - A/Solids, vol.24, issue.6, 2005.
DOI : 10.1016/j.euromechsol.2005.06.004

M. Brinson, I. C. Lin, and W. S. , Comparison of micromechanics methods for effective properties of multiphase viscoelastic composites, Composite Structures, vol.41, issue.3-4, pp.353-367, 1998.
DOI : 10.1016/S0263-8223(98)00019-1

J. Chaboche, S. Kruch, J. Maire, and T. Pottier, Towards a micromechanics based inelastic and damage modeling of composites, International Journal of Plasticity, vol.17, issue.4, pp.411-439, 2001.
DOI : 10.1016/S0749-6419(00)00056-5

F. Chinesta, A. Leygue, M. Beringhier, L. T. Ngyen, J. C. Grandidier et al., Towards a framework for non???linear thermal models in shell domains, International Journal of Numerical Methods for Heat & Fluid Flow, vol.23, issue.1, pp.55-73, 2013.
DOI : 10.1108/09615531311289105

URL : https://hal.archives-ouvertes.fr/hal-01007391

R. M. Christensen, Viscoelastic properties of heterogeneous media, Journal of the Mechanics and Physics of Solids, vol.17, issue.1, pp.23-41, 1969.
DOI : 10.1016/0022-5096(69)90011-8

R. M. Christensen and K. H. Lo, Solutions for effective shear properties in three phase sphere and cylinder models, Journal of the Mechanics and Physics of Solids, vol.27, issue.4, pp.315-330, 1979.
DOI : 10.1016/0022-5096(79)90032-2

R. M. Christensen, A critical evaluation for a class of micro-mechanics models, Journal of the Mechanics and Physics of Solids, vol.38, issue.3, pp.379-404, 1990.
DOI : 10.1016/0022-5096(90)90005-O

C. Czarnota, K. Kowalczyk-gajewska, A. Salahouelhadj, M. Martiny, and S. Mercier, Modeling of the cyclic behaviour of elastic-viscoplastic composites by the additive tangent Mori-Tanaka approach and its validation by finite element calculations, Int. J. of Solids and Structures, pp.56-57, 2015.

G. Debotton and L. Tevet-deree, The Response of a Fiber-Reinforced Composite with a Viscoelastic Matrix Phase, Journal of Composite Materials, vol.10, issue.14, pp.1255-1277, 2004.
DOI : 10.1177/0021998304042732

J. Dirrenberger, S. Forest, and D. Jeulin, Towards gigantic RVE sizes for 3D stochastic fibrous networks, International Journal of Solids and Structures, vol.51, issue.2, pp.359-376, 2014.
DOI : 10.1016/j.ijsolstr.2013.10.011

URL : https://hal.archives-ouvertes.fr/hal-00931281

I. Doghri, L. Adam, and N. Bilger, Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method, International Journal of Plasticity, vol.26, issue.2, pp.219-238, 2010.
DOI : 10.1016/j.ijplas.2009.06.003

G. Dvorak, Transformation Field Analysis of Inelastic Composite Materials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.437, issue.1900, pp.311-327, 1992.
DOI : 10.1098/rspa.1992.0063

G. Dvorak and Y. Benveniste, On Transformation Strains and Uniform Fields in Multiphase Elastic Media, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.437, issue.1900, pp.291-310, 1992.
DOI : 10.1098/rspa.1992.0062

J. D. Eshelby, The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.241, issue.1226, pp.376-396, 1957.
DOI : 10.1098/rspa.1957.0133

G. A. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity Archive for rational mechanics and analysis, pp.307-334, 1986.

E. Ghossein and M. Lévesque, A fully automated numerical tool for a comprehensive validation of homogenization models and its application to spherical particles reinforced composites, International Journal of Solids and Structures, vol.49, issue.11-12, pp.1387-1398, 2012.
DOI : 10.1016/j.ijsolstr.2012.02.021

E. Ghossein and M. Lévesque, Random generation of periodic hard ellipsoids based on molecular dynamics: A computationally-efficient algorithm, Journal of Computational Physics, vol.253, pp.471-490, 2013.
DOI : 10.1016/j.jcp.2013.07.004

E. Ghossein and M. Lévesque, A comprehensive validation of analytical homogenization models: The case of ellipsoidal particles reinforced composites, Mechanics of Materials, vol.75, pp.135-150, 2014.
DOI : 10.1016/j.mechmat.2014.03.014

E. Ghossein and M. Lévesque, Homogenization models for predicting local field statistics in ellipsoidal particles reinforced composites: Comparisons and validations, International Journal of Solids and Structures, vol.58, pp.91-105, 2015.
DOI : 10.1016/j.ijsolstr.2014.12.021

M. Gigliotti, J. C. Grandidier, and M. C. Lafarie-frenot, Assessment of chemo-mechanical couplings in polymer matrix materials exposed to thermo-oxidative environments at high temperatures and under tensile loadings, Mechanics of Materials, vol.43, issue.8, pp.431-443, 2011.
DOI : 10.1016/j.mechmat.2011.04.007

S. Giordano, Explicit nonlinear homogenization for magneto-electro-elastic laminated materials, Mechanics Research Communications, vol.55, 2014.
DOI : 10.1016/j.mechrescom.2013.10.008

URL : https://hal.archives-ouvertes.fr/hal-00913578

S. Giordano, M. Goueygou, N. Tiercelin, A. Talbi, P. Pernod et al., Magneto-electro-elastic effective properties of multilayered artificial multiferroics with arbitrary lamination direction, International Journal of Engineering Science, vol.78, pp.134-153, 2014.
DOI : 10.1016/j.ijengsci.2014.02.011

URL : https://hal.archives-ouvertes.fr/hal-00957770

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, J. de Mécanique, vol.14, issue.1, pp.39-63, 1975.

J. Halpin and J. Kardos, Halpin-Tsai Equations -Review, Polym. Eng. Sci, vol.16, issue.5, pp.344-352, 1976.

Z. Hashin, The Elastic Moduli of Heterogeneous Materials, Journal of Applied Mechanics, vol.29, issue.1, pp.143-150, 1962.
DOI : 10.1115/1.3636446

Z. Hashin and S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials, Journal of the Mechanics and Physics of Solids, vol.11, issue.2, p.127, 1963.
DOI : 10.1016/0022-5096(63)90060-7

Z. Hashin and W. Rosen, The Elastic Moduli of Fiber-Reinforced Materials, Journal of Applied Mechanics, vol.31, issue.2, pp.223-232, 1964.
DOI : 10.1115/1.3629590

Z. Hashin, On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry, Journal of the Mechanics and Physics of Solids, vol.13, issue.3, pp.119-134, 1965.
DOI : 10.1016/0022-5096(65)90015-3

Z. Hashin, Complex moduli of viscoelastic composites???I. General theory and application to particulate composites, International Journal of Solids and Structures, vol.6, issue.5, pp.539-552, 1970.
DOI : 10.1016/0020-7683(70)90029-6

Z. Hashin, Complex moduli of viscoelastic composites???II. Fiber reinforced materials, International Journal of Solids and Structures, vol.6, issue.6, pp.797-807, 1970.
DOI : 10.1016/0020-7683(70)90018-1

Z. Hashin, Analysis of Composite Materials???A Survey, Journal of Applied Mechanics, vol.50, issue.3, pp.481-505, 1983.
DOI : 10.1115/1.3167081

R. Hill, Continuum micro-mechanics of elastoplastic polycrystals, Journal of the Mechanics and Physics of Solids, vol.13, issue.2, pp.89-101, 1965.
DOI : 10.1016/0022-5096(65)90023-2

R. Hill, The essential structure of constitutive laws for metal composites and polycrystals, Journal of the Mechanics and Physics of Solids, vol.15, issue.2, 1967.
DOI : 10.1016/0022-5096(67)90018-X

M. Hori and S. Nemat-nasser, Double-inclusion model and overall moduli of multi-phase composites, Mechanics of Materials, vol.14, issue.3, pp.189-206, 1993.
DOI : 10.1016/0167-6636(93)90066-Z

J. W. Hutchinson, Bounds and Self-Consistent Estimates for Creep of Polycrystalline Materials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.348, issue.1652, 1976.
DOI : 10.1098/rspa.1976.0027

M. Idiart, K. Danas, P. Castañeda, and P. , Second-order theory for nonlinear composites and application to isotropic constituents, Comptes Rendus M??canique, vol.334, issue.10, pp.575-581, 2006.
DOI : 10.1016/j.crme.2006.06.006

URL : https://hal.archives-ouvertes.fr/hal-00111455

D. Gueribiz, F. Jacquemin, and S. Fréour, A moisture diffusion coupled model for composite materials, European Journal of Mechanics - A/Solids, vol.42, pp.81-89, 2013.
DOI : 10.1016/j.euromechsol.2013.04.008

URL : https://hal.archives-ouvertes.fr/hal-01004849

T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin, Determination of the size of the representative volume element for random composites: statistical and numerical approach, International Journal of Solids and Structures, vol.40, issue.13-14, pp.13-14, 2003.
DOI : 10.1016/S0020-7683(03)00143-4

T. Kanit, F. N-'guyen, S. Forest, D. Jeulin, M. Reed et al., Apparent and effective physical properties of heterogeneous materials: Representativity of samples of two materials from food industry, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.33-36, pp.33-36, 2006.
DOI : 10.1016/j.cma.2005.07.022

URL : https://hal.archives-ouvertes.fr/hal-00139164

P. Kanoute, D. P. Boso, J. L. Chaboche, and B. A. Schrefler, Multiscale Methods for Composites: A Review, Archives of Computational Methods in Engineering, vol.69, issue.1, pp.31-75, 2009.
DOI : 10.1007/s11831-008-9028-8

G. Karami and M. Garnich, Effective moduli and failure considerations for composites with periodic fiber waviness, Composite Structures, vol.67, issue.4, pp.461-475, 2005.
DOI : 10.1016/j.compstruct.2004.02.005

K. Khan, R. Barello, A. Muliana, and M. Levesque, Coupled heat conduction and thermal stress analyses in particulate composites, Mechanics of Materials, vol.43, issue.10, pp.608-625, 2011.
DOI : 10.1016/j.mechmat.2011.06.013

K. Khan and A. Muliana, Fully coupled heat conduction and deformation analyses of nonlinear viscoelastic composites, Composite Structures, vol.94, issue.6, pp.2025-2037, 2012.
DOI : 10.1016/j.compstruct.2012.01.010

K. Kowalczyk-gajewska and H. Petryk, Sequential linearization method for viscous/elastic heterogeneous materials, European Journal of Mechanics - A/Solids, vol.30, issue.5, pp.650-664, 2011.
DOI : 10.1016/j.euromechsol.2011.04.002

URL : https://hal.archives-ouvertes.fr/hal-00769679

N. Lahellec and P. Suquet, Composites non linéaires à deux potentiels : estimations affine et du second-ordre, pp.49-56, 2003.

N. Lahellec and P. Suquet, Effective behavior of linear viscoelastic composites: A time-integration approach, International Journal of Solids and Structures, vol.44, issue.2, pp.507-529, 2007.
DOI : 10.1016/j.ijsolstr.2006.04.038

URL : https://hal.archives-ouvertes.fr/hal-00091303

N. Lahellec and P. Suquet, On the effective behavior of nonlinear inelastic composites: II, Journal of the Mechanics and Physics of Solids, vol.55, issue.9, 2007.
DOI : 10.1016/j.jmps.2007.02.004

URL : https://hal.archives-ouvertes.fr/hal-00214215

N. Lahellec and P. Suquet, On the effective behavior of nonlinear inelastic composites: II, Journal of the Mechanics and Physics of Solids, vol.55, issue.9, pp.1964-1992, 2007.
DOI : 10.1016/j.jmps.2007.02.004

URL : https://hal.archives-ouvertes.fr/hal-00214215

N. Lahellec and P. Suquet, Effective response and field statistics in elasto-plastic and elasto-viscoplastic composites under radial and non-radial loadings, International Journal of Plasticity, vol.42, pp.1-30, 2013.
DOI : 10.1016/j.ijplas.2012.09.005

URL : https://hal.archives-ouvertes.fr/hal-00753819

R. Largenton, J. C. Michel, and P. Suquet, Extension of the Nonuniform Transformation Field Analysis to linear viscoelastic composites in the presence of aging and swelling, Mechanics of Materials, vol.73, pp.76-100, 2014.
DOI : 10.1016/j.mechmat.2014.02.004

URL : https://hal.archives-ouvertes.fr/hal-00978984

A. Launay, Y. Marco, M. H. Maitournam, and I. Raoult, Modelling the influence of temperature and relative humidity on the time-dependent mechanical behaviour of a short glass fibre reinforced polyamide, Mechanics of Materials, vol.56, pp.1-10, 2013.
DOI : 10.1016/j.mechmat.2012.08.008

URL : https://hal.archives-ouvertes.fr/hal-00778275

V. Levin, Thermal expansion coefficients of heterogeneous materials, Mekh. Tverd. Tela, vol.2, pp.83-94, 1967.

G. Lielens, P. Pirotte, A. Couniot, F. Dupret, and R. Keunings, Prediction of thermo-mechanical properties for compression moulded composites, Composites Part A: Applied Science and Manufacturing, vol.29, issue.1-2, pp.63-70, 1998.
DOI : 10.1016/S1359-835X(97)00039-0

J. Llorca, M. Elices, and Y. Termonia, Elastic properties of sphere-reinforced composites with a mesophase, Acta Materialia, vol.48, issue.18-19, pp.4589-4597, 2000.
DOI : 10.1016/S1359-6454(00)00245-7

R. Luciano and E. J. Barbero, Formulas for the stiffness of composites with periodic microstructure, International Journal of Solids and Structures, vol.31, issue.21, 1994.
DOI : 10.1016/0020-7683(94)90060-4

S. Malekmohammadi, B. Tressou, C. Nadot-martin, F. Ellyin, and R. Vaziri, Analytical micromechanics equations for elastic and viscoelastic properties of strand-based composites, Journal of Composite Materials, vol.29, issue.15, pp.1857-1874, 2014.
DOI : 10.1177/0021998313490977

J. Mandel, Contribution théorique à l'étude de l'écrouissage et des lois de l'écoulement plaastique th Int, Congr. Appl. Mech, pp.11-502, 1964.

R. Masson and A. Zaoui, Self-consistent estimates for the rate-dependentelastoplastic behaviour of polycrystalline materials, Journal of the Mechanics and Physics of Solids, vol.47, issue.7, pp.1543-1568, 1999.
DOI : 10.1016/S0022-5096(98)00106-9

R. Masson, M. Bornert, P. Suquet, and A. Zaoui, An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals, Journal of the Mechanics and Physics of Solids, vol.48, issue.6-7, pp.1203-1227, 2000.
DOI : 10.1016/S0022-5096(99)00071-X

URL : https://hal.archives-ouvertes.fr/hal-00114467

R. Masson, R. Brenner, and O. Castelnau, Incremental homogenization approach for ageing viscoelastic polycrystals, Comptes Rendus M??canique, vol.340, issue.4-5, pp.378-386, 2012.
DOI : 10.1016/j.crme.2012.02.021

URL : https://hal.archives-ouvertes.fr/hal-01205563

J. 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

B. Miled, I. Doghri, L. Brassart, and L. Delannay, Micromechanical modeling of coupled viscoelastic???viscoplastic composites based on an incrementally affine formulation, International Journal of Solids and Structures, vol.50, issue.10, pp.1755-1769, 2013.
DOI : 10.1016/j.ijsolstr.2013.02.004

A. Molinari, S. Ahzi, and R. Kouddane, On the self-consistent modeling of elastic-plastic behavior of polycrystals, Mechanics of Materials, vol.26, issue.1, pp.43-62, 1997.
DOI : 10.1016/S0167-6636(97)00017-3

A. Molinari, Averaging Models for Heterogeneous Viscoplastic and Elastic Viscoplastic Materials, Journal of Engineering Materials and Technology, vol.124, issue.1, pp.62-70, 2002.
DOI : 10.1115/1.1421052

T. Mori and K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metallurgica, vol.21, issue.5, pp.571-574, 1973.
DOI : 10.1016/0001-6160(73)90064-3

H. Moulinec and P. Suquet, A fast numerical method for computing the linear and nonlinear properties of composites, C. R. Acad. Sc. Paris, II, pp.318-1417, 1994.

H. Moulinec and P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Computer Methods in Applied Mechanics and Engineering, vol.157, issue.1-2, pp.69-94, 1998.
DOI : 10.1016/S0045-7825(97)00218-1

URL : https://hal.archives-ouvertes.fr/hal-01282728

H. Moulinec and P. Suquet, Comparison of FFT-based methods for computing the response of composites with highly contrasted mechanical properties, Physica B: Condensed Matter, vol.338, issue.1-4, pp.58-60, 2003.
DOI : 10.1016/S0921-4526(03)00459-9

A. Muliana and K. Khan, A time-integration algorithm for thermo-rheologically complex polymers, Computational Materials Science, vol.41, issue.4, pp.576-88, 2008.
DOI : 10.1016/j.commatsci.2007.05.021

A. Muliana and K. Li, Time-dependent response of active composites with thermal, electrical, and mechanical coupling effect, International Journal of Engineering Science, vol.48, issue.11, pp.1481-1497, 2008.
DOI : 10.1016/j.ijengsci.2010.08.014

T. Mura, Micromechanics of Defects in Solids, 1987.

A. Noor and R. Shah, Effective thermoelastic and thermal properties of unidirectional fiber-reinforced composites and their sensitivity coefficients, Composite Structures, vol.26, issue.1-2, pp.7-23, 1993.
DOI : 10.1016/0263-8223(93)90040-W

S. T. Nguyen, S. Castagnet, and J. C. Grandidier, Nonlinear viscoelastic contribution to the cyclic accommodation of high density polyethylene in tension: Experiments and modeling, International Journal of Fatigue, vol.55, pp.166-177, 2013.
DOI : 10.1016/j.ijfatigue.2013.06.013

L. Olivier, Prévision du vieillissement thermo-oxydant de composites à matrice organique dédiés à l'aéronautique : Prise en compte des couplages multiphysiques, Thèse de doctorat, 2008.

M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates, Computer Methods in Applied Mechanics and Engineering, vol.171, issue.3-4, 1999.
DOI : 10.1016/S0045-7825(98)00219-9

URL : https://hal.archives-ouvertes.fr/hal-01007379

I. Özdemir, . Brekelmans, . Wam, and M. Geers, Computational homogenization for heat conduction in heterogeneous solids, International Journal for Numerical Methods in Engineering, vol.47, issue.2, pp.185-204, 2008.
DOI : 10.1002/nme.2068

C. Pelissou, J. Baccou, Y. Monerie, and F. Perales, Determination of the size of the representative volume element for random quasi-brittle composites, International Journal of Solids and Structures, vol.46, issue.14-15, pp.2842-2855, 2009.
DOI : 10.1016/j.ijsolstr.2009.03.015

L. D. Perez-fernandez, J. Bravo-castillero, R. Rodriuez-ramos, and F. Sabina, On the constitutive relations and energy potentials of linear thermo-magneto-electro-elasticity. Mech, Research Communications, vol.36, pp.343-350, 2008.

M. Pindera, H. Khatam, A. S. Drago, and Y. Bansal, Micromechanics of spatially uniform heterogeneous media: A critical review and emerging approaches, Composites Part B: Engineering, vol.40, issue.5, pp.349-378, 2009.
DOI : 10.1016/j.compositesb.2009.03.007

P. Castaneda and P. , The effective mechanical properties of nonlinear isotropic composites, Journal of the Mechanics and Physics of Solids, vol.39, issue.1, 1992.
DOI : 10.1016/0022-5096(91)90030-R

P. Castaneda and P. , New variational principles in plasticity and their application to composite materials, Journal of the Mechanics and Physics of Solids, vol.40, issue.8, pp.1757-1788, 1992.
DOI : 10.1016/0022-5096(92)90050-C

P. Ponte-castañeda, 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

G. Rambert, Modélisation des couplages thermo-diffuso-mécaniques dans les matériaux polymères. Application numérique à la décompression explosive, Thèse de doctorat, 2002.

G. Rambert and J. C. Grandidier, An approach to the coupled behaviour of polymers subjected to a thermo-mechanical loading in a gaseous environment, European Journal of Mechanics - A/Solids, vol.24, issue.1, pp.151-168, 2005.
DOI : 10.1016/j.euromechsol.2004.10.005

G. Rambert, G. Jugla, J. C. Grandidier, and L. Cangemi, A modelling of the direct couplings between heat transfer, mass transport, chemical reactions and mechanical behaviour. Numerical implementation to explosive decompression, Composites Part A: Applied Science and Manufacturing, vol.37, issue.4, pp.571-584, 2006.
DOI : 10.1016/j.compositesa.2005.05.021

G. Rambert, J. C. Grandidier, and E. C. Aifantis, On the direct interactions between heat transfer, mass transport and chemical processes within gradient elasticity, European Journal of Mechanics - A/Solids, vol.26, issue.1, pp.68-87, 2007.
DOI : 10.1016/j.euromechsol.2005.12.002

A. Rekik and R. Brenner, Optimization of the collocation inversion method for the linear viscoelastic homogenization, Mechanics Research Communications, vol.38, issue.4, pp.305-308, 2011.
DOI : 10.1016/j.mechrescom.2011.04.003

URL : https://hal.archives-ouvertes.fr/hal-00653303

J. Ricaud and R. Masson, Effective properties of linear viscoelastic heterogeneous media: Internal variables formulation and extension to ageing behaviours, International Journal of Solids and Structures, vol.46, issue.7-8, pp.1599-1606, 2009.
DOI : 10.1016/j.ijsolstr.2008.12.007

B. W. Rosen and Z. Hashin, Effective thermal expansion coefficients and specific heats of composite materials, International Journal of Engineering Science, vol.8, issue.2, pp.157-173, 1970.
DOI : 10.1016/0020-7225(70)90066-2

Y. Rougier, C. Stolz, and A. Zaoui, Self-consistent modelling of elastic?viscoplastic polycrystals, C.R, 1994.
URL : https://hal.archives-ouvertes.fr/hal-00092035

S. Roussette, J. Michel, and P. Suquet, Nonuniform transformation field analysis of elastic???viscoplastic composites, Composites Science and Technology, vol.69, issue.1, pp.22-27, 2009.
DOI : 10.1016/j.compscitech.2007.10.032

URL : https://hal.archives-ouvertes.fr/hal-00499000

M. Salmi, F. Auslender, M. Bornert, and M. Fogli, Various estimates of Representative Volume Element sizes based on a statistical analysis of the apparent behavior of random linear composites, Comptes Rendus M??canique, vol.340, issue.4-5, 2012.
DOI : 10.1016/j.crme.2012.02.007

URL : https://hal.archives-ouvertes.fr/hal-01157361

R. Schapery, Approximate methods of transform inversion for viscoelastic stress analysis, Congr. Appl. Mech. ASME 4th, vol.2, pp.1075-1085, 1962.

P. Suquet, E. Sanchez-palencia, Z. Et, and A. , Elements of homogeneization for inelastic solid mechanics, Editeurs : Homogenization Techniques for Composite, Media, pp.193-278, 1987.

P. Suquet, Overall Properties of Nonlinear Composites, CR Acad. SCi. Paris IIb, vol.320, pp.563-571, 1995.
DOI : 10.1007/978-94-009-1756-9_19

P. Suquet, Effective Properties of Nonlinear Composites, Continuum Micromechanics. CISM Lecture Notes N0 377, pp.197-264, 1997.
DOI : 10.1007/978-3-7091-2662-2_4

URL : https://hal.archives-ouvertes.fr/hal-00111291

C. Sun, P. Saffari, R. Ranade, K. Sadeghipour, and G. Baran, Finite element analysis of elastic property bounds of a composite with randomly distributed particles, Composites Part A: Applied Science and Manufacturing, vol.38, issue.1, pp.80-86, 2007.
DOI : 10.1016/j.compositesa.2006.01.010

C. L. Tucker and E. Liang, Stiffness predictions for unidirectional short-fiber composites: Review and evaluation, Composites Science and Technology, vol.59, issue.5, pp.655-671, 1999.
DOI : 10.1016/S0266-3538(98)00120-1

M. Vincent, C. Nadot-martin, Y. Nadot, and A. Dragon, Fatigue from defect under multiaxial loading: Defect Stress Gradient (DSG) approach using ellipsoidal Equivalent Inclusion Method, International Journal of Fatigue, vol.59, pp.176-187, 2014.
DOI : 10.1016/j.ijfatigue.2013.08.027

J. Willis, Variational and Related Methods for the Overall Properties of Composites, Adv. Appl. Mech, vol.21, pp.1-78, 1981.
DOI : 10.1016/S0065-2156(08)70330-2