. .. Fluctuation-normale-de-la-plasticité, , vol.54

. .. Cas-d'une-symétrie-de-maclage, 60 2.2.1 Macles de type I et de type II

. .. Applications, 70 2.4.1 Début de plasticité dans des micropiliers bicristallins. 70 2.4.2 Structures lamellaires de macles, p.79

. .. Conclusions,

, Modélisations micromécaniques, p.85

. , Plasticité intra-granulaire distribuée en bandes de glissement 86 3.1.1 Contrainte cinématique moyenne, vol.87

, 90 3.2.1 Principe de la méthode de superposition des bicristaux 90 3.2.2 Application auprobì eme d'Eshelby-Kröner, p.93

. .. , 96 3.3.1 Observations expérimentales dans l'alliage AZ31, p.101

. .. , 104 3.4.2 Modélisation auto-cohérente avec hypothèses spécifiques107 3.4.3 Résultats des simulations et discussion, p.111

. .. Conclusions,

, A l'aide d'un modèlè a champs moyens viscoplastique

. Bibliographie,

R. J. Asaro and D. M. Barnett, The non-uniform transformation strain problem for an anisotropic ellipsoidal inclusion, Journal of the Mechanics and Physics of Solids, vol.23, pp.77-83, 1975.

A. Acharya and J. L. Bassani, Lattice incompatibility and a gradient theory of crystal plasticity, Journal of the Mechanics and Physics of Solids, vol.48, pp.1565-1595, 2000.

A. Acharya, A model of crystal plasticity based on the theory of continuously distributed dislocations, Journal of the Mechanics and Physics of Solids, vol.49, pp.761-784, 2001.

A. Acharya, Driving forces and boundary conditions in continuum dislocation mechanics, Proceedings of the Royal Society London A, vol.459, pp.1343-1363, 2003.

A. Acharya, Constitutive analysis of finite deformation field dislocation mechanics, Journal of the Mechanics and Physics of Solids, vol.52, pp.301-316, 2004.

A. Acharya, Jump condition for GND evolution as a constraint on slip transmission at grain boundaries, Philosophical Magazine, vol.87, pp.1349-1359, 2007.

E. C. Aifantis, On the microstructural origin of certain inelastic models. Transactions of the ASME, Journal of Engineering Materials and Technology, vol.106, pp.326-330, 1984.

E. C. Aifantis, The physics of plastic deformation, International journal of plasticity, vol.3, pp.211-247, 1987.

D. Ando, J. Koike, and Y. Sutou, Relationship between deformation twinning and surface step formation in AZ31 magnesium alloys, Acta Materialia, vol.58, pp.4316-4324, 2010.

S. Allain, Caractérisation et modélisation thermomécaniques multi-´ echelles des mécanismes de déformation et d'´ ecrouissage d'aciers austénitiquesausténitiquesà haute teneur en manganèse : applicationàapplicationà l'effet TWIP, 2004.

S. Allain, Comportement mécanique des aciers : des mécanismes fondamentauxàfondamentauxà la déformation macroscopique. HabilitationàHabilitationà Diriger des Recherches, 2012.

K. E. Amouzou, Caractérisation et modélisation multi-´ echelles de l'anisotropie et de l'hétérogénéité de la déformation plastique du ?-titane en conditions de traction, 2015.

A. Arsenlis, D. M. Parks, R. Becker, and V. V. Bulatov, On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals, Journal of the Mechanics and Physics of Solids, vol.52, pp.1213-1246, 2004.

A. Acharya and A. Roy, Size effects and idealized dislocation microstructure at small scales : Predictions of a Phenomenological model of Mesoscopic Field dislocation Mechanics : Part i, Journal of the Mechanics and Physics of Solids, vol.54, pp.1687-1710, 2006.

K. E. Amouzou, T. Richeton, A. Roth, M. A. Lebyodkin, and T. A. Lebedkina, Micromechanical modeling of hardening mechanisms in commercially pure ?-titanium in tensile condition, International Journal of Plasticity, vol.80, pp.222-240, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01514731

R. J. Asaro, Crystal plasticity, Journal of Applied Mechanics, vol.50, pp.921-934, 1983.

M. F. Ashby, The deformation of plastically non-homogeneous materials. Philosophical Magazine, vol.21, pp.399-424, 1970.

S. Ahmad and R. W. Whitworth, Dislocation motion in ice : a study by synchroton x-ray topography, Philosophical Magazine A, vol.57, pp.749-766, 1988.

D. J. Bammann and E. C. Aifantis, On a proposal for a continuum with microstructure, Acta Mechanica, vol.45, pp.91-121, 1982.

M. R. Barnett, Twinning and the ductility of magnesium alloys Part II. contraction twins, Materials science and Engineering A, vol.464, pp.8-16, 2007.

O. Bouaziz, S. Allain, and C. Scott, Effect of grain and twin boundaries on the hardening mechanisms of twinning-induced plasticity steels, Scripta Materialia, vol.58, pp.484-487, 2008.

D. J. Bacon, D. M. Barnett, ;. Benmhenni, S. Bouvier, R. Brenner et al., Micromechanical modelling of monotonic loading of cp ?-ti : Correlation between macroscopic and microscopic behaviour, Materials Science and Engineering A, vol.23, pp.222-233, 1979.

C. J. Bayley, W. A. Brekelmans, and M. G. Geers, A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity, International Journal of Solids and Structures, vol.43, pp.7268-7286, 2006.

S. Berbenni, M. Berveiller, and T. Richeton, Intra-granular plastic slip heterogeneities : Discrete vs. Mean Field approaches, International Journal of Solids and Structures, vol.45, pp.4147-4172, 2008.

B. A. Bilby, R. Bullough, E. Smith, ;. Barkia, V. Doquet et al., Continuous distributions of dislocations : a new application of the methods of non-Riemannian geometry, Proceedings of the Royal Society London A, vol.231, pp.91-102, 1955.

B. Barkia, V. Doquet, E. Héripré, and I. Guillot, Characterization and analysis of deformation heterogeneities in commercial purity titanium, Materials Characterization, vol.108, pp.94-101, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01227164

F. Barbe, L. Decker, D. Jeulin, and G. Cailletaud, Intergranular and intragranular behavior of polycrystallie aggregates, International Journal of Plasticity, vol.17, pp.513-536, 2001.

M. Berveiller, Thèse d'etat : ContributionàContributionà l'´ etude du comportement plastique et des textures de déformation des polycristaux métalliques. Sciences et techniques de l'armement, vol.54, 1980.

T. R. Bieler, P. Eisenlohr, C. Zhang, H. J. Phukan, and M. A. Crimp, Grain boundaries and interfaces in slip transfer, Current Opinion in Solid State and Materials Science, vol.18, pp.212-226, 2014.

S. Berbenni, V. Favier, and M. Berveiller, Impact of the grain size distribution on the behaviour of heterogeneous materials, International Journal of Plasticity, vol.23, pp.114-142, 2007.

F. Barbe, S. Forest, and G. Cailletaud, Intergranular and intragranular behavior of polycrystallie aggregates. part 2 : Results, International Journal of Plasticity, vol.17, pp.513-536, 2001.

M. Berveiller, O. Fassi-fehri, A. Hihi, ;. Beausir, C. Fressengeas et al., The problem of 2 plastic and heterogeneous inclusions in an anisotropic medium, International Journal of. Engineering Science, vol.25, pp.5382-5395, 1987.

J. E. Bailey and P. B. Hirsch, The recrystallization process in some polycrystalline metals, Proceedings of the Royal Society of London A, vol.267, pp.11-30, 1962.

M. R. Barnett, Z. Keshavarz, A. G. Beer, and X. Ma, Non-Schmid behaviour during secondary twinning in a polycrystalline magnesium alloy, Acta Materialia, vol.56, pp.5-15, 2008.

H. Becker and W. Pantleon, Work-hardening stages and deformation mechanism maps during tensile deformation of commercially pure titanium, Computational Materials Science, vol.76, pp.52-59, 2013.

S. Berbenni, B. Paliwal, and M. Cherkaoui, A micromechanics-based model for shear-coupled grain boundary migration in bicrystals, International Journal of Plasticity, vol.44, pp.68-94, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01947296

L. Bracke, ;. L. Bao, C. Schuman, J. Lecomte, M. Philippe et al., Study of deformation mechanisms in titanium by interrupted rolling and channel die compression tests, Deformation behaviour of austenitic FeMn alloyas by twinning and martensitic transformations, vol.15, pp.113-127, 2007.

C. Blochwitz and W. Tirschler, Influence of texture on twin boundary cracks in fatigued austenitic stainless steel, Crystal Research and Technology, vol.339, pp.318-327, 2003.

C. Blochwitz, W. Tirschler, ;. Bertin, C. N. Tomé, I. J. Beyerlein et al., On the strength of dislocation interactions and their effect on latent hardening in pure magnesium, International Journal of Plasticity, vol.40, pp.72-92, 2005.

T. B. Britton and A. J. Wilkinson, High resolution electron backscatter diffraction measurements of elastic strain variations in the presence of larger lattice rotationss, Ultramicroscopy, vol.114, pp.82-95, 2012.

J. W. Christian, A. G. Crocker-;-f, and . Nabarro, Dislocations and lattice transformations, In Dislocations in Solids, issue.3, 1980.

O. Castelnau, P. Duval, M. Montagnat, and R. Brenner, Elastoviscoplastic micromechanical modeling of the transient creep of ice, Journal of Geophysical Research, vol.113, pp.11203-11204, 2008.
URL : https://hal.archives-ouvertes.fr/insu-00378329

N. M. Cordero, S. Forest, E. P. Busso, S. Berbenni, and M. Cherkaoui, Grain size effects on plastic strain and dislocation density tensor fields in metal polycrystals, Computational Materials Science, vol.52, pp.7-13, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00658644

P. Cermelli, M. E. Gurtin-;-n, A. Cordero, S. Gaubert, E. P. Forest et al., Analysis of particle induced dislocation structures using three-dimensional dislocation dynamics and strain gradient plasticity, Journal of the Mechanics and Physics of Solids, vol.49, pp.33-39, 1963.

J. Chevy, Viscoplasticité et Hétérogénéités de déformation du monocristal de glace : expériences et simulations, 2008.

A. T. Churchman, The slip modes of titanium and the effect of purity on their occurence during tensile deformation of single crystals, Proceedings of the Royal Society London A, vol.226, pp.216-226, 1954.

J. W. Christian and S. Mahajan, Deformation twinning, Progress in Materials Science, vol.39, pp.1-157, 1995.

T. Chauve, M. Montagnat, and P. Vacher, Strain field evolution during dynamic recrystallization nucleation ; A case study on ice, Acta Materialia, vol.101, pp.116-124, 2015.
URL : https://hal.archives-ouvertes.fr/hal-02014830

H. Conrad, Effect of interstitial solutes on the strength and ductility of titanium, Progress in Materials Science, vol.26, pp.123-403, 1981.

B. Clausen, C. N. Tomé, D. W. Brown, and S. R. Agnew, Reorientation and stress relaxation due to twinning : Modeling and experimental characterization for Mg, Acta Materialia, vol.56, pp.2456-2468, 2008.

]. W. Clark, R. H. Wagoner, T. C. Lee, Z. Y. Shen, I. M. Robertson et al., On the criteria for slip transmission across interfaces in polycrystals, Scripta Metallurgica et Materialia, vol.26, pp.203-206, 1992.

S. Caré and A. Zaoui, Cavitation at triple nodes in ?-zirconium polycrystals, Acta Materialia, vol.44, pp.1323-1336, 1996.

P. Duval, M. F. Ashby, and I. Anderman, Rate-controlling processes in the creep of polycrystalline ice, The Journal of Physical Chemistry, vol.87, pp.4066-4074, 1983.

M. Dupeux and R. Bonnet, Stresses, displacements and energy calculations for interfacial dislocations in anisotropic two-phase media, Acta Metallurgica, vol.28, pp.721-728, 1980.

V. Doquet and B. Barkia, A micromechanical model of the viscoplastic behaviour of titanium accounting for its anisotropic and strain-rate dependent viscosity, Mechanics of Time-Dependent Materials, vol.19, pp.153-166, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01116597

V. Doquet, B. Barkia, ;. S. Dancette, L. Delannay, K. Renard et al., Combined AFM, SEM and crystal plasticity analysis of grain boundary sliding in titanium at room temperature, Mechanics of Materials, vol.103, pp.2135-2145, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01391790

B. Devincre, Dislocation dynamics simulations of slip systems interactions and forest strengthening in ice single crystal, Philosophical Magazine, vol.93, pp.235-246, 2013.

B. Devincre, T. Hoc, and L. P. Kubin, Collinear interactions of dislocations and slip systems, Materials Science and Engineering A, pp.182-185, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00019070

B. Devincre, L. Kubin, and T. Hoc, Physical analyses of crystal plasticity by dd simulations, Scripta Materialia, vol.54, pp.741-746, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00019068

Y. Estrin and L. P. Kubin, Local strain hardening and nonuniformity of plastic deformation, Acta Metallurgica, vol.34, pp.2455-2466, 1986.

J. D. Eshelby, W. T. Read, and W. Shockley, Anisotropic elasticity with applications to dislocation theory, Acta Metallurgica, vol.1, pp.251-259, 1953.

J. D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion and related problems, Proceedings of the Royal Society London A, vol.241, pp.376-396, 1957.

P. Franciosi and S. Berbenni, Heterogeneous crystal and poly-crystal plasticity modeling from a transformation field analysis within a regularized Schmid law, Journal of the Mechanics and Physics of Solids, vol.55, p.2265, 2007.

S. Forest, F. Barbe, and G. Cailletaud, Cosserat modelling of size effects in the mechanical behaviour of polycrystals and multiphase materials, International Journal of Solids and Structures, vol.37, pp.7105-7126, 2000.

P. Franciosi, M. Berveiller, and A. Zaoui, Latent hardening in copper and aluminium single crystals, Acta Metallurgica, vol.28, pp.273-283, 1980.

J. M. Finney, , 1974.

S. Forest, Micromorphic approach for gradient elasticity, viscoplasticity, and damage, Journal of Engineering Mechanics, vol.135, pp.117-131, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00368014

C. Fressengeas, Mécanique des Champs de Dislocations, Hermes Science, 2010.

F. C. Frank and A. N. Stroh, On the theory of kinking, Proceedings of the Physical Society B, vol.65, pp.811-821, 1952.
DOI : 10.1088/0370-1301/65/10/311

A. Ghaderi and M. R. Barnett, Sensitivity of deformation twinning to grain size in titanium and magnesium, Acta Materialia, vol.59, pp.7824-7839, 2011.
DOI : 10.1016/j.actamat.2011.09.018

P. H. Gammon, H. Kiefte, M. J. Clouter, and W. W. Denner, Elastic constants of artificial and natural ice samples by brillouin spectroscopy, Journal of Glaciology, vol.29, pp.433-460, 1983.
DOI : 10.3189/s0022143000030355

URL : https://www.cambridge.org/core/services/aop-cambridge-core/content/view/0C027E622189AE838C8FC631371BD94C/S0022143000030355a.pdf/div-class-title-elastic-constants-of-artificial-and-natural-ice-samples-by-brillouin-spectroscopy-div.pdf

N. P. Gurao, R. Kapoor, S. Suwas, ;. Grennerat, M. Montagnat et al., Experimental characterization of the intragranular strain field in columnar ice during transient creep, Elastic properties of fcc Fe, vol.59, pp.3145-3155, 2011.

D. Gloagen, G. Oum, V. Legrand, J. Fajoui, and S. Branchu, Micromechanical modelling of monotonic loading of cp ?-ti : Correlation between macroscopic and microscopic behaviour, Acta Materialia, vol.61, pp.5779-5790, 2013.

J. Gemperlova and V. Paidar, Stresses in deformed cubic bicrystals with a tilt < 001 > grain boundary, Journal of Physics B, vol.35, pp.351-354, 1985.

J. Gemperlova, V. Paidar, and F. Kroupa, Compatibility stresses in deformed bicrystals, Journal of Physics B, vol.39, pp.427-446, 1989.

C. F. Gu, L. S. Toth, and M. Hoffman, Twinning effects in a polycrystalline magnesium alloy under cyclic deformation, Acta Materialia, vol.62, pp.212-224, 2014.
DOI : 10.1016/j.actamat.2013.09.048

URL : https://hal.archives-ouvertes.fr/hal-01513736

P. Gudmundson, A unified treatment of strain gradient plasticity, Journal of the Mechanics and Physics of Solids, vol.52, pp.1379-1406, 2004.
DOI : 10.1016/j.jmps.2003.11.002

I. Gutierrez-urrutia and D. Raabe, Dislocation and twin substructure evolution during strain hardening of an Fe-22 wt.% Mn-0.6 wt.% C TWIP steel observed by electron channeling contrast imaging, Acta Materialia, vol.59, pp.6449-6462, 2011.
DOI : 10.1016/j.actamat.2011.07.009

J. Gong and A. J. Wilkinson, Anisotropy in the plastic flow properties of singlecrystal ?-titanium determined from micro-cantilever beams, Acta Materialia, vol.57, pp.5693-5705, 2009.

E. O. Hall, The deformation and aging of mild steel. Part III : discussion and results, Proceedings of the Physical Society of London, vol.64, pp.747-753, 1951.

S. V. Harren, The finite deformation of rate-dependent polycrystals. II. A comparison of the self-consistent and taylor methods, Journal of the Mechanics and Physics of Solids, vol.39, pp.361-383, 1991.
DOI : 10.1016/0022-5096(91)90017-i

T. Hoc, B. Devincre, and L. Kubin, Deformation stage i of fcc crystals : Constitutive modelling, Evolution of Deformation Microstructures in 3D, pp.43-59, 2004.

F. Hecht, New development in freefem++, Journal of Numerical Mathematics, vol.20, pp.251-266, 2012.
DOI : 10.1515/jnum-2012-0013

URL : https://hal.archives-ouvertes.fr/hal-01476313

W. F. Hosford, R. L. Fleischer, and W. A. Backofen, Tensile deformation of aluminium single crystals at low temperatures, Acta Metallurgica, vol.8, pp.187-199, 1960.
DOI : 10.1016/0001-6160(60)90127-9

R. E. Hook and J. P. Hirth, The deformation behavior of isoaxial bicrystals of Fe3%Si, Acta Metallurgica, vol.15, pp.535-551, 1967.

J. P. Hirth, The influence of grain boundaries on mechanical properties, Metallurgical Transactionss, vol.3, pp.3047-3063, 1972.

J. P. Hirth and J. Lothe, Theory of dislocations, 1982.

H. Hernández, T. J. Massart, R. H. Peerlings, and M. G. Geers, Towards an unconditionally stable numerical scheme for continuum dislocation transport. Modelling and Simulation in, Materials Science and Engineering, vol.23, pp.85013-85014, 2015.

A. Heinz and P. Neumann, Crack initiation during high cycle fatigue of an austenitic steel, Acta Metallurgica et Materialia, vol.38, pp.1933-1940, 1990.

P. Van-houtte, Simulation of the rolling and shear texture of brass by the taylor theory adapted for mechanical twinning, Acta Metallurgica, vol.26, pp.591-604, 1978.

S. G. Hong, S. H. Park, and C. S. Lee, Role of 10-12 twinning characteristics in the deformation behavior of a polycrystalline magnesium alloy, Acta Materialia, vol.58, pp.5873-5885, 2010.

R. Hill and J. R. Rice, Constitutive analysis of elastic-plastic crystals at arbitrary strain, Journal of the Mechanics and Physics of Solids, vol.20, pp.401-413, 1972.

T. Hirouchi and Y. Shibutani, Mechanical responses of copper bicrystalline micro pillars with ?3 coherent twin boundaries by uniaxial compression tests, Materials Transactions, vol.55, pp.52-57, 2014.

T. Hochrainer, S. Stanfeld, M. Zaiser, and P. Gumbsch, Continuum dislocation dynamics : Towards a physical theory of crystal plasticity, Journal of the Mechanics and Physics of Solids, vol.63, pp.167-178, 2014.

Y. Huang, A user material subroutine incorporating single crystal plasticity in the abaqus finite element program, pp.1-21, 1991.

P. J. Imrich, C. Kirchlechner, and G. Dehm, Influence of inclined twin boundaries on the deformation behavior of Cu micropillars, Materials Science and Engineering A, vol.642, pp.65-70, 2015.

P. J. Imrich, C. Kirchlechner, C. Motz, and G. Dehm, Differences in deformation behavior of bicrystalline Cu micropillars containing a twin boundary or a large-angle grain boundary, Acta Materialia, vol.73, pp.240-250, 2014.

V. L. Indenbom, Internal stresses in crystals, Theory of Crystal Defects, pp.257-274, 1966.

P. A. Juan, S. Berbenni, M. R. Barnett, C. N. Barnett, and L. Capolungo, A double inclusion homogenization scheme for polycrystals with hierarchal topologies : application to twinning in Mg alloys, International Journal of Plasticity, vol.60, pp.182-196, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01503457

S. R. Kalidindi, Incorporation of deformation twinning in crystal plasticity models, Journal of the Mechanics and Physics of Solids, vol.46, pp.267-290, 1998.

L. P. Kubin and B. Devincre, Mesoscopic simulations of dislocations and plasticity, Materials Science and Engineering A, pp.234-236, 1997.

L. Kubin, B. Devincre, and T. Hoc, Modeling dislocation storage rates and mean free paths in face-centered cubic crystals, Acta Materialia, vol.56, pp.6040-6049, 2008.

L. Kubin and Y. Estrin, Evolution of dislocation densities and the critical conditions for the Portevin-Le Châtelier effect, Acta Materialia, vol.38, pp.697-708, 1990.

J. Koike, N. Fujiyama, D. Ando, and Y. Sutou, Roles of deformation twinning and dislocation slip in the fatigue failure mechanism of AZ31 Mg alloys, Scripta Materialia, vol.63, pp.747-750, 2010.

N. Kheradmand, Grain boundary-dislocation interaction : A local investigation via micron-sized bicrystals, 2012.

R. Kral and P. Lukac, Modelling of strain hardening and its relation to the onset of Portevin-Le Châtelier effect in Al-Mg alloys, Materials Science Engineering A, pp.786-789, 1997.

M. Knezevic, R. A. Lebensohn, O. Cazacu, B. Revil-baudard, G. Proust et al.,

M. E. Vogel and . Nixon, Modeling bending of ?-titanium with embedded polycrystal plasticity in implicit finite elements, Materials Science and Engineering A, vol.564, pp.116-126, 2013.

U. F. Kocks and H. Mecking, Physics and phenomenology of strain hardening : the FCC case, Progress in Materials Science, vol.48, pp.171-273, 2003.

A. M. Kosevich, Crystal dislocations and the theory of elasticity (Chap. 1), In Dislocations in Solids, vol.1, pp.33-141, 1979.

E. Kröner, Kontinuumstheorie der Versetzungen und Eigenspannungen, 1958.

E. Kröner, Zur plastischen verformung des vielkristalls, Acta Metallurgica, vol.9, pp.155-161, 1961.

E. Kröner, Continuum theory of defects, Physics of defects, vol.35, pp.215-315, 1981.

U. F. Kocks, C. N. Tomé, and H. R. Wenk, Texture and Anisotropy, 1998.

L. P. Kubin-;-l, X. H. Li, P. J. An, P. Imrich, Z. J. Zhang et al., Microcompression and cyclic deformation behaviors of coaxial copper bicrystals with a single twin boundary, Reviews on the Deformation Behavior of Materials, vol.4, pp.199-202, 1982.

H. Li, C. J. Boehler, T. R. Bieler, and M. A. Crimp, Analysis of the deformation behavior in tension and tension-creep of Ti-3Al-2.5v (wt pct) at 296 K and 728 K (23?c23?c and 455?C455?C) using in situ SEM experiments, Metallurgical and Materials Transactions A, vol.45, pp.6053-6066, 2014.

H. Liang and F. P. Dunne, Gnd accumulation in bi-crystal deformation : Crystal plasticity analysis and comparison with experimentss, International Journal of Mechanical SciencesE, vol.51, pp.326-333, 1966.

E. D. Levine, Deformation mechanisms in titanium at low temperatures. Transactions of the Metallurgical Society of AIME, vol.236, pp.1558-1565, 1966.

T. Leffers and R. K. Kay, The brass-type texture and its deviation from the coppertype texture, Progress in Materials Science, vol.54, pp.351-396, 2009.

R. A. Lebensohn, A. K. Kanjarla, and P. Eisenlohr, An elasto-viscoplastic formulation based on fast Fourier transforms for the predictions of micromechanical fields in polycrystalline materials, International Journal of Plasticity, pp.59-69, 2012.

T. Lebedkina and M. Lebyodkin, Effect of deformation geometry on the intermittent plastic flow associated with the portevin-le chatelier effect, Acta Materialia, vol.56, pp.5567-5574, 2008.

Z. X. Long, T. M. Liu, Y. G. Wu, and Y. Zhang, Improving the anisotropy of rolled Mg-3Al-1Zn alloy by pre-strain and annealing, Materials Science and Engineering A, vol.616, pp.240-245, 2014.

J. Luster and J. M. Morris, Compatibility of deformation in two-phase ti-al alloys : dependence on microstructure and orientation relationships, Metallurgical Materials Transactions A, vol.26, pp.1745-1756, 1995.

H. Li, D. E. Mason, T. R. Bieler, C. J. Boehler, and M. A. Crimp, Methodology for estimating the critical resolved shear stress ratios of ?-phase Ti using EBSD-based trace analysis, Acta Materialia, vol.61, pp.7555-7567, 2013.

L. C. Lim and R. Raj, Interaction between lattice and grain boundary dislocations and their role in the mechanical properties of interfaces, Journal de Physique, vol.46, pp.581-595, 1985.

T. C. Lee, I. M. Robertson, and H. K. Birnbaum, Anomalous slip in an FCC system, Journal of Ultramicroscopy, vol.29, pp.212-216, 1989.

T. C. Lee, I. M. Robertson, and H. K. Birnbaum, An in situ transmission electron microscope deformation study of the slip transfer mechanisms in metals, Metallurgical Transactions A, vol.21, pp.2437-2447, 1990.

L. Lu, Y. F. Shen, X. H. Chen, L. H. Qian, and K. Lu, Ultrahigh strength and high electrical conductivity in copper, Science, vol.304, p.422, 2004.

R. A. Lebensohn and C. N. Tomé, A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals : Application to zirconium alloys, Acta Metallurgica et Materialia, vol.41, pp.2611-2624, 1993.

R. A. Lebensohn, C. N. Tomé, and P. Pontecastaneda, Self-consistent modelling of the mechanical behaviour of viscoplastic polycrystals incorporating intragranular field fluctuations, Philosophical Magazine, vol.87, pp.4287-4322, 2007.

D. Mordehai, E. Clouet-ans, M. Fivel, and M. Verdier, Introducing dislocation climb by bulk diffusion in dislocation dynamics, Philosophical Magazine, vol.88, pp.899-925, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00354629

C. Mareau and S. Berbenni, An affine formulation for the self-consistent modeling of elasto-viscoplastic heterogeneous materials based on the translated fields method, International Journal of Plasticity, vol.64, pp.134-150, 2015.

J. Mach, A. J. Beaudoin, and A. Acharya, Continuity in the plastic strain rate and its influence on texture evolution, Journal of the Mechanics and Physics Solids, vol.58, pp.105-128, 2010.

M. Montagnat, J. R. Blackford, S. Piazolo, L. Arnaud, and R. A. Lebensohn, Measurements and full-field predictions of deformation heterogeneities in ice, Earth and Planetary Science Letters, vol.305, pp.153-160, 2011.
URL : https://hal.archives-ouvertes.fr/insu-00605029

A. Molinari, G. R. Canova, and S. Ahzi, A self consistent approach of the large deformation polycrystal viscoplasticity, Acta Metallurgica, vol.35, pp.2983-2994, 1987.

M. Montagnat, T. Chauve, F. Barou, A. Tommasi, B. Beausir et al., Analysis of dynamic recrystallization of ice from ebsd orientation mapping, Frontiers in Earth Science, vol.3, pp.1-11, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01515284

E. Martin, L. Capolungo, L. Jiang, and J. J. Jonas, Variant selection during secondary twinning in Mg-3, Acta Materialia, vol.58, pp.3970-3983, 2010.

C. Maurice, J. H. Driver, R. Fortunier, ;. Madec, B. Devincre et al., On solving the orientation gradient dependency of high angular resolution ebsd, Ultramicroscopy, vol.113, pp.1879-1882, 2003.
URL : https://hal.archives-ouvertes.fr/emse-00830371

A. Marchenko, M. Mazì-ere, S. Forest, and J. L. Strudel, Crystal plasticity simulation of strain aging phenomena in ?-titanium at room temperature, International Journal of Plasticity, vol.85, pp.1-33, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01358092

S. N. Monteiro and R. E. Reed-hill, An empirical analysis stress-strain curves of titanium, Metallurgical Transactions, vol.4, pp.1011-1015, 1973.

A. Ma, F. Roters, and D. Raabe, On the consideration of interactions between dislocations and grain boundaries in crystal plasticity finite element modelingTheory, experiments, and simulations, Acta Materialia, vol.54, pp.2181-2194, 2006.

P. Mussot, C. Rey, and A. Zaoui, Grain boundary sliding and strain compatibility, Res Mechanica, vol.14, pp.69-79, 1985.

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, pp.69-94, 1998.
URL : https://hal.archives-ouvertes.fr/hal-01282728

T. Mura, Continuous distribution of moving dislocations, Philosophical Magazine, vol.89, pp.843-857, 1963.

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

M. C. Miguel, A. Vespignani, S. Zapperi, J. Weiss, and J. R. Grasso, Intermittent dislocation flow in viscoplastic deformation, Nature, vol.410, pp.667-671, 2001.

M. Montagnat, J. Weiss, P. Duval, H. Brunjail, and J. G. Sevillano, The heterogeneous nature of slip in ice single crystals deformed under torsion, Philosophical Magazine, vol.86, pp.4259-4270, 2011.
URL : https://hal.archives-ouvertes.fr/insu-00375732

R. Masson and A. Zaoui, Self-consistent estimates for the ratedependentelastoplastic behaviour of polycrystalline materials, Journal of the Mechanics and Physics of Solids, vol.47, pp.1543-1568, 1999.

S. Naka, Etude des mécanismes de déformation plastiquè a basse température de monocristaux de titane ?, 1983.

E. V. Nesterova, B. Bacroix, and C. Teodosiu, Microstructure and texture evolution under strain-path changes in low-carbon interstitial-free steel, Metallurgical and Materials Transactions A, vol.32, pp.2527-2538, 2001.
DOI : 10.1007/s11661-001-0042-9

M. E. Nixon, O. Cazacu, and R. A. Lebensohn, Anisotropic response of high-purity ?-titanium : Experimental characterization and constitutive modeling, International Journal of Plasticity, vol.26, pp.516-532, 2010.
DOI : 10.1016/j.ijplas.2009.08.007

H. Neuhäuser, Slip-line formation and collective dislocation motion, In Dislocations in Solids, issue.6, 1983.

P. R. Neumann-;-p, Proceedings of the 7th International Fatigue Congress, p.107, 1999.

M. Niewczas, Lattice correspondence during twinning in hexagonal close-packed crystals, Acta Materialia, vol.58, pp.5848-5857, 2010.
DOI : 10.1016/j.actamat.2010.06.059

S. Nemat-nasser, W. G. Guo, and J. Y. Cheng, Mechanical properties and deformation mechanisms of a commercially pure titanium, Acta Materialia, vol.47, pp.3705-3720, 1999.
DOI : 10.1016/s1359-6454(99)00203-7

URL : http://www.ceam.ucsd.edu/documents/papers/mechanicalproperties.pdf

S. Nemat-nasser and M. Obata, Rate-dependent finite elasto-plastic deformation of polycrystals, Proceedings of the Royal Society of London A, vol.407, pp.343-375, 1986.
DOI : 10.1098/rspa.1986.0101

P. Neumann and A. Tönnessen, Proceedings of the 3rd International Conference on Fatigue and Fatigue Thresholds, vol.3, 1987.

J. F. Nye, ;. Ohashi, R. I. Barabash, J. W. Pang, G. E. Ice et al., X-ray microdiffraction and strain gradient crystal plasticity studies of geometrically necessary dislocations near a ni bicrystal grain boundary, International Journal of Plasticity, vol.1, pp.920-941, 1953.

A. E. Omri, A. Fennan, F. Sidoroff, and A. Hihi, Elastic-plastic homogenization for layered composites, European Journal of Mechanics A/Solids, vol.19, p.585, 2000.

Y. Okada, T. Hondoh, and S. Mae, Basal glide of dislocations in ice observed by synchrotron radiation topography, Philosophical Magazine, vol.79, pp.2853-2868, 1999.

D. Peirce, R. J. Asaro, A. Needleman, ;. Paquin, S. Berbenni et al., Micromechanical modeling of the elastic-viscoplastic behaviour of polycrystalline steels, International Journal of Plasticity, vol.31, pp.1267-1302, 1983.

S. Puri, A. Das, and A. Acharya, Mechanical response of multicrystalline thin films in mesoscale field dislocation mechanics, Journal of the Mechanics and Physics Solids, vol.59, pp.2400-2417, 2011.

N. J. Petch, The cleavage strength of polycrystals, Journal of Iron Steel Institute, vol.174, pp.25-28, 1953.

S. H. Park, S. G. Hong, J. H. Lee, and C. S. Lee, Multiple twinning modes in rolled Mg-3Al-1Zn alloy and their selection mechanism, Materials Science and Engineering A, vol.532, pp.401-406, 2012.
DOI : 10.1016/j.msea.2011.11.003

P. Peralta and C. Laird, Compatibility stresses in fatigued bicrystals : dependence on misorientation and small plastic deformations, Acta Metallurgica, vol.45, pp.5129-5143, 1997.
DOI : 10.1016/s1359-6454(97)00184-5

P. Peralta, L. Llanes, J. Bassani, and C. Laird, Deformation from twin-boundary stresses and the role of texture : application to fatigue, Philosophical Magazine A, vol.70, pp.219-232, 1994.

Q. S. Pan, Q. H. Lu, L. Lu, ;. S. Piazolo, M. Montagnat et al., Single crystal elastic constants of high-manganese transformation-and twinninginduced plasticity steels determined by a new method utilizing nanoindentation, Materials Science and Engineering A, vol.61, pp.134-139, 2013.

L. Priester, ;. Poty, J. Raulot, H. Xu, C. Schuman et al., Classification of the critical resolved shear stress in the hexagonal-closepacked materials by atomic simulation : Application to ?-zirconium and ?-titanium, Les joints de grains de la théoriè a l'ingénierie, vol.110, pp.14905-14906, 2006.

A. Paquin, H. Sabar, and M. Berveiller, Integral formulation and selfconsistent modelling of elastoviscoplastic behavior of heterogeneous materials, Archive of Applied Mechanics, vol.69, pp.14-35, 1999.

M. Philippe, M. Serghat, P. Van-houtte, and C. Esling, Modelling of texture evolution for materials of hexagonal symmetry-II : Application to zirconium and titanium ? or near ? alloys, Acta Metallurgica et Materialia, vol.43, pp.1619-1630, 1995.

P. Peralta, A. Schober, and C. Laird, Elastic stresses in anisotropic bicrystals, Materials Science and Engineering A, vol.169, pp.43-51, 1993.

G. Proust, C. N. Tomé, and G. C. Kaschner, Modeling texture, twinning and hardening evolution during deformation of hexagonal materials, L. Rémy. Twin-slip interaction in F.C.C. crystals. Acta Metallurgica, vol.55, pp.711-714, 1976.

R. Raj and M. F. Ashby, On grain boundary sliding and diffusional creep, Metallurgical Transactions, vol.2, pp.1113-1127, 1971.

A. Roy and A. Acharya, Size effects and idealized dislocation microstructure at small scales : Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics : Part II, Journal of the Mechanics and Physics of Solids, vol.54, pp.1711-1743, 2006.

T. Richeton and S. Berbenni, Relations between twin and slip in parent lattice due to kinematic compatibility at interfaces, International Journal of Solids and Structures, vol.49, pp.1355-1364, 2012.

T. Richeton and S. Berbenni, Effects of heterogeneous elasticity coupled to plasticity on stresses and lattice rotations in bicrystals : a Field Dislocation Mechanics viewpoint, European Journal of Mechanics A/Solids, vol.37, pp.231-247, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01501437

T. Richeton and S. Berbenni, From bicrystals to spherical inclusions : A superposition method to derive analytical expressions of stress fields in presence of plastic strain gradients, International Journal of Solids and Structures, vol.51, pp.794-807, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01503881

T. Richeton, S. Berbenni, M. Berveiller-;-f.-roters, P. Eisenlohr, L. Hantcherli et al., Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling : Theory, experiments, applications, Acta Materialia, vol.57, pp.1152-1211, 2009.

T. Richeton, L. T. Le, T. Chauve, M. Bernacki, S. Berbenni et al., Modelling the transport of geometrically necessary dislocations on slip systems : application to single-and multicrystals of ice, Modelling and Simulation in Materials Science and Engineering, vol.25, pp.236-243, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01479768

A. Roth, Etude multi-´ echelles de l'hétérogénéité et de l'anisotropie de la déformation plastique : cas des aciers TWIP et du Ti-?, 2014.

A. Roy, S. Puri, and A. Acharya, Phenomenological mesoscopic field dislocation mechanics, lower-order gradient plasticity, and transport of mean excess dislocation density. Modelling and Simulation in, Materials Science and Engineering, vol.15, pp.167-180, 2007.

C. Rey and G. Saada, The elastic field of periodic dislocation networks. Philosophical Magazine, vol.33, pp.825-841, 1976.

T. Richeton, I. Tiba, S. Berbenni, and O. Bouaziz, Analytical expressions of incompatibility stresses at ?3(111) twin boundaries and consequences on single-slip promotion parallel to twin plane, Philosophical Magazine, vol.95, pp.12-31, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01515288

T. Richeton, G. F. Wang, and C. Fressengeas, Continuity constraints at interfaces and their consequences on the work hardening of metal-matrix composites, Journal of the Mechanics and Physics of Solids, vol.59, pp.2023-2043, 2011.

T. Richeton, J. Weiss, and F. Louchet, Breakdown of avalanche critical behaviour in polycrystalline plasticity, Nature Materials, vol.4, pp.465-469, 2005.
URL : https://hal.archives-ouvertes.fr/insu-00374916

C. Rey and A. Zaoui, Slip heterogeneities in deformed aluminum bicrystals, Acta Metallurgica, vol.28, pp.687-697, 1980.

C. Rey, A. Zaoui, ;. S. Sun, B. L. Adams, C. Shet et al., Mesoscale investigation of the deformation field of an aluminium bicrystal, Scripta Materialia, vol.30, pp.501-508, 1982.

A. A. Salem, S. R. Kalidindi, and R. G. Doherty, Strain hardening regimes and microstructure evolution during large strain compression of high purity titanium, Scripta Materialia, vol.46, pp.419-423, 2002.

A. A. Salem, S. R. Kalidindi, and R. G. Doherty, Strain hardening of titanium : role of deformation twinning, Acta Materialia, vol.51, pp.4225-4237, 2003.
DOI : 10.1016/s1359-6454(03)00239-8

A. A. Salem, S. R. Kalidindi, R. G. Doherty, and S. L. Semiatin, Strain hardening due to deformation twinning in ?-titanium, Mechanisms. Metallurgical and Materials Transactions A, vol.37, pp.259-268, 2006.
DOI : 10.1007/s11661-006-0171-2

A. A. Salem, S. R. Kalidindi, and S. L. Semiatin, Strain hardening due to deformation twinning in ?-titanium : Constitutive relations and crystal-plasticity modeling, Acta Materialia, vol.53, pp.3495-3502, 2005.
DOI : 10.1016/j.actamat.2005.04.014

S. Stupkiewicz and H. Petryk, Modelling of laminated microstructures in stressinduced martensitic transformations, Journal of the Mechanics and Physics of Solids, vol.50, pp.2303-2331, 2002.
DOI : 10.1016/s0022-5096(02)00029-7

A. N. Stroh, Dislocations and cracks in anisotropic elasticity, Philosophical Magazine, vol.3, pp.625-646, 1958.
DOI : 10.1080/14786435808565804

C. Shearwood and R. Withworth, The velocity of dislocations in ice, Philosophical Magazine, vol.64, pp.289-302, 1991.

Z. Shen, R. H. Wagoner, W. A. Clark, ;. Shi, Y. D. Zhang et al., Dislocation and grain boundary interactions in metals, Acta Metallurgica, vol.36, pp.3231-3242, 1988.
DOI : 10.1016/0001-6160(88)90058-2

T. Lecomte and . Richeton, On the selection of extension twins with low schmid factors in a deformed Mg alloy, Acta Materialia, vol.83, pp.17-28, 2015.

T. Takeuchi, Work hardening of copper single crystals with multiple glide orientations, vol.16, pp.629-640, 19754.
DOI : 10.2320/matertrans1960.16.629

URL : https://www.jstage.jst.go.jp/article/matertrans1960/16/10/16_10_629/_pdf

G. I. Taylor, Plastic strain in metals, Journal of the Institute of Metals, vol.62, pp.307-324, 1938.

I. Tiba, Effets des interfaces cristallines sur les champs mécaniques en plasticité cristalline et conséquences sur le glissement dans des micro-piliers bi-cristallins, 2015.

C. N. Tomé, R. A. Lebensohn, and U. F. Kocks, A model for texture development dominated by deformation twinning : application to Zirconium alloys, Acta Metallurgica et Materialia, vol.39, pp.2667-2680, 1991.

K. Tanaka and T. Mori, Note on volume integrals of the elastic field around an ellipsoidal inclusion, Journal of elasticity, vol.2, pp.199-200, 1972.
DOI : 10.1007/bf00125528

V. Taupin, T. Richeton, J. Chevy, C. Fressengeas, J. Weiss et al.,

;. I. Miguel, T. Tiba, C. Richeton, H. Motz, S. Vehoff et al., Incompatibility stresses at grain boundaries in ni bicrystalline micropillars analyzed by an anisotropic model and slip activity, Acta Materialia, vol.56, pp.227-238, 2008.

V. Taupin, S. Varadhan, J. Chevy, C. Fressengeas, A. J. Beaudoin et al., Effects of size on the dynamics of dislocations in ice single crystals, Physical Review Letters, vol.99, pp.155507-155508, 2007.
URL : https://hal.archives-ouvertes.fr/insu-00377504

N. Thompson and N. Wadsworth, Philosophical Magazine A, vol.1, p.113, 1956.

S. N. Varadhan, A. J. Beaudoin, and C. Fressengeas, Dislocation transport using an explicit Galerkin/least-squares formulation. Modelling and Simulation in, Materials Science and Engineering, vol.14, pp.1245-1270, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00119274

P. R. Van-beers, G. J. Mcshane, V. G. Kouznetsova, and M. G. Geers, Grain boundary interface in strain gradient crystal plasticity, Journal of the Mechanics and Physics of Solids, vol.61, pp.2659-2679, 2013.

M. Verdier, M. Fivel, and I. Groma, Mesoscopic scale simulation of dislocation dynamic in fcc metals : Principle and applications. Modelling and Simulation in, Materials Science and Engineering, vol.6, pp.755-770, 1998.

H. Vehoff, C. Laird, and D. J. Duquette, The effect of hydrogen and segregation on fatigue crack nucleation at defined grain boundaries in nickel bicrystals, Acta Metallurgica, vol.35, pp.2877-28860, 1987.

H. Vehoff, A. Nykyforchyn, and R. Metz, Fatigue crack nucleation at interfaces, Materials Science and Engineering A, pp.387-389, 2004.
DOI : 10.1016/j.msea.2003.12.094

W. Voigt, Lehrbuch Der Kristallphysik. B.G. Teubner, 1928.

Y. N. Wang and J. C. Huang, Texture analysis in hexagonal materials, Materials Chemistry and Physics, vol.81, pp.11-26, 2003.
DOI : 10.1016/s0254-0584(03)00168-8

J. R. Willis, Second order effects of dislocations in anisotropic crystals, International Journal of Engineering Science, vol.5, pp.171-190, 1968.

A. J. Wilkinson, Measurement of elastic strains and small lattice rotations using electron backscatter diffraction, Ultramicroscopy, vol.62, pp.237-247, 1996.

J. L. Warwick, N. G. Jones, K. M. Rahman, and D. Dye, Lattice strain evolution during tensile and compressive loading of cp ti, Acta Materialia, vol.60, pp.6720-6731, 2012.
DOI : 10.1016/j.actamat.2012.08.042

X. Wu, S. R. Kalidindi, C. Necker, and A. A. Salem, Prediction of crystallographic texture evolution and anisotropic stress-strain curves during large plastic strains in high purity ?-titanium using a Taylor-type crystal plasticity model, Acta Materialia, vol.55, pp.423-432, 2007.

C. J. Wilson, D. S. Russell-head, H. M. Sim, ;. S. Wang, C. Schuman et al., The application of an automated fabric analyzer system to the textural evolution of folded ice layers in shear zones, Metallurgical and Materials Transactions A, vol.37, pp.421-430, 2003.

H. Wang, P. D. Wu, C. N. Tomé, and Y. Huang, Dislocation mediated variant selection for secondary twinning in compression of pure titanium, Journal of the Mechanics and Physics of Solids, vol.58, pp.59-70, 2010.

H. K. Yang, V. Doquet, and Z. F. Zhang, Micro-scale measurements of plastic strain field, and local contributions of slip and twinning in TWIP steels during in situ tensile tests, Materials Science and Engineering A, vol.672, pp.7-14, 2016.

M. Yamasaki, K. Hagihara, S. I. Inoue, J. P. Hadorn, and Y. Kawamura, Crystallographic classification of kink bands in an extruded Mg-Zn-Y alloy using intragranular misorientation axis analysis, Acta Materialia, vol.61, pp.2065-2076, 2013.

S. Zaefferer, A study of active deformation systems in titanium alloys : dependence on alloy composition and correlation with deformation texture, Materials Science and Engineering A, vol.344, pp.20-30, 2003.

Z. Zhang, T. S. Jun, T. B. Britton, and F. P. Dunne, Determination of Ti-6242 ? and ? slip properties using micro-pillar test and computational crystal plasticity, Journal of the Mechanics and Physics of Solids, vol.95, pp.393-410, 2016.
DOI : 10.1016/j.jmps.2016.06.007

URL : https://doi.org/10.1016/j.jmps.2016.06.007

Y. T. Zhu, X. Z. Liao, X. L. Wu, Y. T. Zhu, X. L. Wu et al., Dislocation-twin interactions in nanocrystalline fcc metals, Progress in Materials Science, vol.57, pp.812-821, 2011.
DOI : 10.1016/j.actamat.2010.10.028

URL : http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1159&context=usarmyresearch

C. Zambaldi, Y. Yang, T. R. Bieler, and D. Raabe, Orientation informed nanoindentation of a-titanium : Indentation pileup in hexagonal metals deforming by prismatic slip, Journal of Materials Research, vol.27, pp.356-367, 2012.

Y. Shi, F. Zhang, T. Wagner, P. Richeton, J. Juan et al., Sequential double extension twinning in a magnesium alloy : Combined statistical and micromechanical analyses, Acta Materialia, vol.96, pp.333-343, 2015.
DOI : 10.1016/j.actamat.2015.06.029

URL : https://hal.archives-ouvertes.fr/hal-01514963