. Agoras, Incremental variational procedure for elasto-viscoplastic composites and application to polymer-and metal-matrix composites reinforced by spheroidal elastic particles, International Journal of Solids and Structures, pp.668-686, 2016.

A. Kassem-;-al-kassem, G. Allain, S. Bouaziz, and O. , Microstructure based modeling for the mechanical behavior of ferrite-pearlite steels suitable to capture isotropic and kinematic hardening, Publikationsserver der RWTH Aachen University, vol.496, pp.329-336, 2008.

A. , F. Armstrong, P. Frederick, and C. , A mathematical representation of the multiaxial bauschinger effect, 1966.

. Barthélémy, The eshelby inclusion problem in ageing linear viscoelasticity, International Journal of Solids and Structures, pp.530-542, 2016.

[. Berbenni, 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.
URL : https://hal.archives-ouvertes.fr/hal-01513855

[. Berbenni, Micromechanical modeling of the elastic-viscoplastic behavior of polycrystalline steels having different microstructures, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00128379

, Eng. A, vol.372, pp.128-164

Z. Berveiller, M. Berveiller, and A. Zaoui, An extension of the self-consistent scheme to plastically flowing polycrystals, J. Mech. Phys. Solids, vol.26, pp.325-344, 1979.

[. Boudet, An incremental variational formulation for the prediction of the effective work-hardening behavior and field statistics of elasto-(visco)plastic composites, International Journal of Solids and Structures, vol.83, pp.90-113, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01310906

[. Bruno, An investigation on microscopic and macroscopic stability phenomena of composite solids with periodic microstructure, International Journal of Solids and Structures, vol.47, pp.2806-2824, 2010.

J. Chaboche, Constitutive Modeling of Ratcheting Effects, Part I: Experimental Facts and Properties of the Classical Models, ASME J. Eng. Mater. Technol, vol.111, issue.4, pp.384-392, 1989.

L. Chaboche, J. Chaboche, and J. Lemaitre, Mechanics of Solid Materials, 1990.

. Cheng, A homogenization model for porous ductile solids under cyclic loads comprising a matrix with isotropic and linear kinematic hardening, International Journal of Solids and Structures, vol.121, pp.174-190, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01537602

R. M. Christensen, Viscoelastic properties of heterogeneous media, J. Mech. Phys. Solids, vol.17, pp.23-41, 1969.

[. Czarnota, Modeling of the cyclic behavior of elastic-viscoplastic composites by the additive tangent Mori-Tanaka approach and validation by finite element calculations, International Journal of Solids and Structures, vol.56, pp.96-117, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01513622

[. Doghri, Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method, Int. J. Plast, vol.26, pp.219-238, 2010.

J. D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. Roy. Soc. Lond. A, vol.241, pp.376-396, 1957.

A. Frederick, C. Frederick, and P. Armstrong, A mathematical representation of the multiaxial bauschinger effect, Materials at High Temperatures, vol.24, issue.1, pp.1-26, 2007.

J. Gomez and C. Basaran, Damage mechanics constitutive model for Pb/Sn solder joints incorporating nonlinear kinematic hardening and rate dependent effects using a return mapping integration algorithm, Mechanics of Materials, vol.38, issue.7, pp.585-598, 2006.

[. Guo, A cyclic visco-plastic constitutive model for timedependent ratchetting of particle-reinforced metal matrix composites, Int. J. Plasti, vol.10, pp.101-125, 2013.

Z. Hashin, The inelastic inclusion problem, Int. J. Engng Sci, vol.7, pp.11-36, 1969.

Z. Hashin, Analysis of Composite Materials -A Survey, J. Appl. Mech, vol.50, pp.481-505, 1983.

. Hashin, Z. Shtrikman-;-hashin, and S. Shtrikman, A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids, vol.11, pp.127-140, 1962.

R. Hill, Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, vol.13, pp.89-101, 1965.

R. Hill, A self-consistent mechanics of composite materials, J. Mech. Phys. Solids, vol.13, pp.213-222, 1965.

J. W. Hutchinson, Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. Roy. Soc. Lond. A, vol.348, pp.101-127, 1976.

M. I. Idiart and N. Lahellec, Estimates for the overall linear properties of pointwise heterogeneous solids with application to elasto-viscoplasticity, Journal of the Mechanics and Physics of Solids, vol.97, pp.317-332, 2016.

[. Kouddane, Self-consistent modelling of heterogeneous viscoelastic and elastoviscoplastic materials, pp.129-141, 1993.

. Kowalczyk-gajewska, K. Petryk-;-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.
URL : https://hal.archives-ouvertes.fr/hal-00769679

. Kroner, E. Koch-;-kroner, and H. Koch, Effective properties of disordered materials, Solid Mechanics Archives, vol.1, pp.183-238, 1976.

. Lahellec, N. Suquet-;-lahellec, and P. Suquet, On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles, J. Mech. Phys. Solids, vol.55, pp.1932-1963, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00214209

. Lahellec, N. Suquet-;-lahellec, and P. Suquet, Effective response and field statistics in elastoplastic and elasto-viscoplastic composites under radial and non-radial loadings, Int. J. Plasticity, vol.42, pp.1-30, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00753819

M. Laws, N. Laws, and R. Mclaughlin, Self-consistent estimates for viscoelastic creep compliance of composite materials, Proc. R. Soc. London, vol.359, pp.251-273, 1978.

[. Lubarda, V. Benson-;-lubarda, and D. Benson, On the numerical algorithm for isotropickinematic hardening with the Armstrong-Frederick evolution of the back stress, Computer methods in applied mechanics and engineering, vol.191, issue.33, pp.3583-3596, 2002.

[. Lucchetta, A double incremental variational procedure for elastoplastic composites with combined isotropic and linear kinematic hardening, International Journal of Solids and Structures, 2018.

[. Luciano, R. Willis-;-luciano, and J. R. Willis, Non-local constitutive response of a random laminate subjected to configuration-dependent body force, J. Mech. Phys. Solids, vol.49, pp.431-444, 2001.

B. Mareau, C. Mareau, and S. Berbenni, An affine formulation for the self-consistent modelling of elasto-viscoplastic heteregeneous materials based on the translated field method, Int. J. Plast, vol.64, pp.134-150, 2015.

. Marfia, S. Sacco-;-marfia, and E. Sacco, Multiscale technique for nonlinear analysis of elastoplastic and viscoplastic composites, Composites Part B, vol.136, pp.241-253, 2018.

Z. Masson, R. Masson, and A. Zaoui, Self-consistent estimates for the rate-dependent elastoplastic behavior of polycrystalline materials, J. Mech. Phys. Solids, vol.47, pp.1543-1568, 1999.

[. Mercier, Validation of an interaction law for the eshelby inclusion problem in elasto-viscoplasticity, Int. J. Solids Structures, vol.42, pp.1923-1941, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00455522

S. Mercier and A. Molinari, Homogenization of elastic-viscoplastic heterogeneous materials: Self-consistent and Mori-Tanaka schemes, Int. J. Plast, vol.25, issue.6, pp.1024-1048, 2009.

G. W. Milton, The Theory of Composites, 2002.

A. Molinari-;-molinari, Averaging models for heterogeneous viscoplastic and elastic viscoplastic materials, J. Engng. Mat. Tech, vol.124, pp.62-70, 2002.

[. Molinari, A self consistent approach of the large deformation polycrystal viscoplasticity, Acta Metall, vol.35, pp.2983-2994, 1987.

[. Molinari, Validation of the tangent formulation for the solution of the non-linear eshelby inclusion problem, Int. J. Plasticity, vol.20, pp.291-307, 2004.

. Molinari, A. Toth-;-molinari, and L. S. Toth, Tuning a self-consistent viscoplastic model by finite elements results -I modeling, Acta Metall. Mater, vol.42, pp.2453-2458, 1994.

T. Mori, T. Mori, and K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall, vol.21, pp.571-574, 1973.

[. Mortazavi, Modeling of two-phase random composite materials by finite element, Mori-Tanaka and strong contrast methods, Composites Part B, vol.45, pp.1117-1125, 2013.

. Paquin, Integral formulation and self-consistent modelling of elasto-viscoplastic behavior of heterogeneous materials, Arch. of Appl. Mech, vol.69, pp.14-35, 1999.

P. Perzyna, Internal state variable description of dynamic fracture of ductile solids, Int. J. Solids Structures, vol.22, pp.797-818, 1986.

[. Pichler, Generalized self-consistent scheme for upscaling of viscoelastic properties of highly-filled matrix-inclusion composites -Application in the context of multiscale modeling of bituminous mixtures, Composites Part B, vol.43, pp.457-464, 2012.

O. Pierard and I. Doghri, An enhanced affine formulation and the corresponding numerical algorithms for the mean-field homogenization of elasto-viscoplastic composites, Int. J. Plast, vol.22, pp.131-157, 2006.

[. Pierard, Micromechanics of elasto-plastic materials reinforced with ellipsoidal inclusions, Int. J. Solids Struct, vol.44, pp.6945-6962, 2007.

, New variational principles in plasticity and their application to composite materials, J. Mech. Phys. Solids, vol.40, pp.1757-1788, 1992.

P. Castañeda, P. P. Suquet, and P. , Advances in Applied Mechanics, vol.34, pp.171-302, 1997.

W. Prager, Recent developments in the mathematical theory of plasticity, Journal of Applied Physics, vol.20, pp.235-241, 1949.

C. Qu, J. Qu, M. Cherkaoui, J. Ricaud, and R. Masson, Effective properties of linear viscoelastic heterogeneous media: Internal variables formulation and extension to ageing behaviours, Int. J. Solids Struct, vol.46, pp.1599-1606, 2006.

[. Rougier, Représentation spectrale en viscoélasticité linéaire des matériaux hétérogènes, C. R. Acad. Sci, vol.316, pp.1517-1522, 1993.

[. Rougier, Self-consistent modelling of elastic-viscoplastic polycrystals, C. R. Acad. Sci. Paris, vol.318, pp.145-151, 1994.
URL : https://hal.archives-ouvertes.fr/hal-00092035

[. Sabar, A new class of micro-macro models for elastic-viscoplastic heterogeneous materials, Int. J. Solids Structures, vol.39, pp.3257-3276, 2002.
URL : https://hal.archives-ouvertes.fr/hal-00128381

. Sabiston, Micromechanics based elasto-visco-plastic response of long fibre composites using functionally graded interphases at quasi-static and moderate strain rates, Composites Part B, vol.100, pp.31-43, 2016.

P. Suquet, Elements of homogeneization for inelastic solid mechanics, pp.193-278, 1985.

G. J. Weng, A unified, self-consistent theory for the plastic-creep deformation of metals, J. Appl. Mech, vol.49, pp.728-734, 1982.

[. Wu, An incremental-secant mean-field homogenization method with second statistical moments for elasto-visco-plastic composite materials, Mechanics of Materials, vol.114, pp.180-200, 2017.

, Evaluation of the normalized slope of the overall strain-stress curve in uniaxial tension after reaching the threshold ? Y in both phases as a function of kinematic hardening modulus h according to the Kroner MT model (relation (C.11)). The elastic parameters of the phases are given in Table 1. The volume fraction of inclusions is c = 25%

, Case of hard inclusions. Material parameters for the inclusion and matrix phases

, Case of soft inclusions. Material parameters for the inclusion and matrix phases. Viscoplasticity is described by the Perzyna-type law