R. Foerch, J. Besson, G. Cailletaud, and P. Pilvin, Polymorphic constitutive equations in finite element codes, Computer Methods in Applied Mechanics and Engineering, vol.141, issue.3, pp.355-372, 1997.

J. Besson and R. Foerch, Large scale object-oriented finite element code design, Computer Methods in Applied Mechanics and Engineering, vol.142, issue.1, pp.165-187, 1997.

J. R-le-riche, J. Gaudin, and . Besson, An object-oriented simulation-optimization interface, Computers & Structures, vol.81, issue.17, pp.1689-1701, 2003.

E. Team and P. Galon, Europlexus: a computer program for the finite element simulation of fluid-structure systems under transient dynamic loading. User's manual, vol.02, 2016.

M. Halilovi?, M. Vrh, and B. ?tok, NICE h : a higher-order explicit numerical scheme for integration of constitutive models in plasticity, Engineering with Computers, vol.29, issue.1, pp.55-70, 2013.

A. Hillerborg, M. Modéer, and P. Petersson, Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement and concrete research, vol.6, issue.6, pp.773-781, 1976.

P. Zden?k, B. H. Ba?ant, and . Oh, Crack band theory for fracture of concrete, Matériaux et Construction, vol.16, issue.3, pp.155-177, 1983.

P. Ladeveze, On a damage mechanics approach. Mechanics and mechanisms of damage in composites and multi-materials, pp.119-141, 1989.

O. Allix and J. Deü, Delayed-Damage Modelling for Fracture Prediction of Laminated Composites under Dynamic Loading, Engineering Transactions, vol.45, issue.1, 1997.

G. Pijaudier, -. Cabot, and Z. P. Ba?ant, Nonlocal damage theory, Journal of Engineering Mechanicsasce -J ENG MECH-ASCE, vol.113, 1987.

G. Pijaudier-cabot, Z. P. Ba?ant, and M. Tabbara, Comparison of various models for strain-softening, vol.5, pp.141-150, 1988.

R. H. Peerlings, R. Borst, W. A. De, J. H. Brekelmans, . Vree et al., Gradient enhanced damage for quasi-brittle materials, International Journal for Numerical Methods in Engineering, vol.39, issue.19, pp.3391-3403, 1996.

S. Forest, Micromorphic approach for gradient elasticity, viscoplasticity, and damage, Journal of Engineering Mechanics, vol.135, issue.3, pp.117-131, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00368014

D. Banabic, Plastic Behaviour of Sheet Metal, Sheet Metal Forming Processes, pp.27-140, 2010.

H. É. Tresca, Mémoire sur l'écoulement des corps solides soumis à de fortes pressions, 1864.

R. V. Mises, Mechanik der festen Körper im plastisch-deformablen Zustand, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp.582-592, 1913.

. Av-hershey, The plasticity of an isotropic aggregate of anisotropic face centered cubic crystals, J. Appl. Mech, vol.21, issue.3, pp.241-289, 1954.

W. F. Hosford, A Generalized Isotropic Yield Criterion, Journal of Applied Mechanics, vol.39, issue.2, pp.607-609, 1972.

D. C. Drucker, United States Office of Naval Research, and United States Navy Department Bureau of Ships. The Relation of Experiments to Mathematical Theories of Plasticity, Division of Applied Mathematics). Division of Applied Mathematics, 1948.

A. Mendelson, Plasticity; theory and application. Macmillan series in applied mechanics, 1968.

P. Ludwik, Elemente der technologischen Mechanik, 1909.

W. Ramberg and . William-r-osgood, Description of stress-strain curves by three parameters, 1943.

E. Voce, The Relationship between Stress and Strain for Homogeneous Deformation, Journal of the Institute of Metals, vol.74, pp.537-562, 1948.

N. L. Maoût, S. Thuillier, and P. Y. Manach, Aluminum alloy damage evolution for different strain paths -application to hemming process, Engineering Fracture Mechanics, vol.76, issue.9, pp.1202-1214, 2009.

C. Christian, D. Roth, and . Mohr, Effect of strain rate on ductile fracture initiation in advanced high strength steel sheets: Experiments and modeling, International Journal of Plasticity, vol.56, pp.19-44, 2014.

R. Gordon, . Johnson, H. William, and . Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, Proceedings of the 7th International Symposium on Ballistics, vol.21, pp.541-547, 1983.

H. Zhao, A constitutive model for metals over a large range of strain rates identification for mild-steel and aluminium sheets, Materials Science and Engineering: A, vol.230, issue.1-2, pp.95-99, 1997.

J. Frank, R. W. Zerilli, and . Armstrong, Dislocation mechanics based constitutive relations for material dynamics calculations, Journal of Applied Physics, vol.61, issue.5, pp.1816-1825, 1987.

G. I. Taylor and H. Quinney, The latent energy remaining in a metal after cold working, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, vol.143, issue.849, pp.307-326, 1934.

C. Trong-son, Models for ductile damage and fracture prediction in cold bulk metal forming processes: a review, International Journal of Material Forming, vol.10, issue.2, pp.139-171, 2017.

T. Wierzbicki, Y. Bao, Y. Lee, and Y. Bai, Calibration and evaluation of seven fracture models, International Journal of Mechanical Sciences, vol.47, issue.4-5, pp.719-743, 2005.

Y. Bai and T. Wierzbicki, A comparative study of three groups of ductile fracture loci in the 3d space, Engineering Fracture Mechanics, vol.135, pp.147-167, 2015.

A. Frank and . Mcclintock, A criterion for ductile fracture by the growth of holes, Journal of Applied Mechanics, vol.35, issue.2, pp.363-371, 1968.

J. Rice and D. M. Tracey, On the ductile enlargement of voids in triaxial stress fields, Journal of the Mechanics and Physics of Solids, vol.17, issue.3, pp.201-217, 1969.

G. R. Johnson and W. H. Cook, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Engineering Fracture Mechanics, vol.21, issue.1, pp.31-48, 1985.

Y. Bai and T. Wierzbicki, Application of extended Mohr-Coulomb criterion to ductile fracture, International Journal of Fracture, vol.161, issue.1, pp.1-20, 2010.

C. A. Coulomb, Essai sur une application des règles de maximis & minimis à quelques problèmes de statique, relatifs à l'architecture. De l'Imprimerie Royale, p.1776

O. Mohr, Abhandlungen aus dem Gebiete der technischen Mechanik, 1914.

D. Mohr and S. J. Marcadet, Micromechanically-motivated phenomenological Hosford-Coulomb model for predicting ductile fracture initiation at low stress triaxialities, International Journal of Solids and Structures, pp.40-55, 2015.

K. Pack and D. Mohr, Combined necking & fracture model to predict ductile failure with shell finite elements, Engineering Fracture Mechanics, vol.182, pp.32-51, 2017.

K. Pack, T. Tancogne-dejean, M. Gorji, and D. Mohr, Hosford-coulomb ductile failure model for shell elements: Experimental identification and validation for dp980 steel and aluminum 6016-t4, International Journal of Solids and Structures, p.151, 2018.

Y. Bao and T. Wierzbicki, On fracture locus in the equivalent strain and stress triaxiality space, International Journal of Mechanical Sciences, vol.46, issue.1, pp.81-98, 2004.

L. M. Kachanov, Time of the rupture process under creep conditions, Izv. Akad. Nauk. S.S.R. Otd. Tech. Nauk, vol.8, pp.26-31, 1958.

J. Lemaitre and J. L. Chaboche, Mechanics of Solid Materials, 1985.

J. Lemaitre, A Continuous Damage Mechanics Model for Ductile Fracture, Journal of Engineering Materials and Technology, vol.107, issue.1, pp.83-89, 1985.

N. Bonora, On the effect of triaxial state of stress on ductility using nonlinear CDM model, International Journal of Fracture, vol.88, issue.4, pp.359-371, 1997.

L. Arthur, Gurson and others. Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile media, Journal of engineering materials and technology, vol.99, issue.1, pp.2-15, 1977.

V. Tvergaard, Influence of voids on shear band instabilities under plane strain conditions, International Journal of Fracture, vol.17, issue.4, pp.389-407, 1981.

V. Tvergaard, A. Needleman, D. , and D. , Analysis of the Cup-cone Fracture in a Round Tensile Bar. Lyngby. Technical university of Denmark, Danish center for applied mathematics and mechanics, 1983.

C. C. Chu and A. Needleman, Void Nucleation Effects in Biaxially Stretched Sheets, Journal of Engineering Materials and Technology, vol.102, issue.3, pp.249-256, 1980.

A. Dalloz, J. Besson, A. Gourgues-lorenzon, T. Sturel, and A. Pineau, Effect of shear cutting on ductility of a dual phase steel, Engineering Fracture Mechanics, vol.76, issue.10, pp.1411-1424, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00399528

V. Tvergaard, On localization in ductile materials containing spherical voids, International Journal of Fracture, vol.18, issue.4, pp.237-252, 1982.

J. Faleskog, X. Gao, C. Fong, and . Shih, Cell model for nonlinear fracture analysis -i. micromechanics calibration, International Journal of Fracture, vol.89, issue.4, pp.355-373, 1998.

J. Besson, D. Steglich, and W. Brocks, Modeling of crack growth in round bars and plane strain specimens. International journal of solids and structures, vol.38, pp.8259-8284, 2001.

K. Nahshon and J. W. Hutchinson, Modification of the Gurson Model for shear failure, European Journal of Mechanics -A/Solids, vol.27, issue.1, pp.1-17, 2008.

K. Lau-nielsen and V. Tvergaard, Ductile shear failure or plug failure of spot welds modelled by modified Gurson model, Engineering Fracture Mechanics, vol.77, issue.7, pp.1031-1047, 2010.

G. Rousselier, Ductile fracture models and their potential in local approach of fracture, Nuclear engineering and design, vol.105, issue.1, pp.97-111, 1987.

A. K. Pickett, T. Pyttel, F. Payen, F. Lauro, N. Petrinic et al., Failure prediction for advanced crashworthiness of transportation vehicles, International Journal of Impact Engineering, vol.30, issue.7, pp.853-872, 2004.

A. Ockewitz, D. Sun, H. Klamser, and D. Malcher, Damage modelling of automobile components of aluminium materials under crash loading, 2006.

J. Marzbanrad, S. Norozi, and A. Ahmadi, Damage modeling in crashworthiness of dual-phase steel, 2013.

J. Besson, Continuum models of ductile fracture : A review, International Journal of Damage Mechanics, vol.19, pp.3-52, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00550957

C. Defaisse, M. Mazière, L. Marcin, and J. Besson, Ductile fracture of an ultra-high strength steel under low to moderate stress triaxiality, Engineering Fracture Mechanics, vol.194, pp.301-318, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01780585

I. S. Sarraf, A. Jenab, K. P. Boyle, and D. E. Green, Effect of rate-dependent constitutive equations on the tensile flow behaviour of DP600 using Rousselier damage model, Materials & Design, vol.117, pp.267-279, 2017.

L. Malcher, F. M. Andrade-pires, and J. M. César-de-sá, An extended GTN model for ductile fracture under high and low stress triaxiality, International Journal of Plasticity, vol.54, pp.193-228, 2014.

W. Jiang, Y. Li, and J. Su, Modified GTN model for a broad range of stress states and application to ductile fracture, European Journal of Mechanics -A/Solids, vol.57, pp.132-148, 2016.

A. Ahmed, J. Benzerga, and . Besson, Plastic potentials for anisotropic porous solids, European Journal of Mechanics-A/Solids, vol.20, issue.3, pp.397-434, 2001.

Y. Zhang, E. Lorentz, and J. Besson, Ductile damage modelling with locking-free regularised gtn model, International Journal for Numerical Methods in Engineering, vol.113, issue.13, pp.1871-1903, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01758768

B. Tanguy and J. Besson, An extension of the Rousselier model to viscoplastic temperature dependent materials, Int. J. Fracture, vol.116, issue.1, pp.81-101, 2002.

A. Needleman and J. R. Rice, Limits to ductility set by plastic flow localization, Mechanics of Sheet Metal Forming, pp.237-267, 1978.

B. Tanguy, J. Besson, R. Piques, and A. Pineau, Ductile-brittle transition of a A508 steel characterized by the Charpy impact test. Part-II: modelling of the Charpy transition curve, Eng. Fract. Mech, vol.72, pp.413-434, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00142085

B. Tanguy, T. T. Luu, G. Perrin, A. Pineau, and J. Besson, Plastic and damage behavior of a high strength X100 pipeline steel: experiments and modelling, Int. J. of Pressure Vessels and Piping, vol.85, issue.5, pp.322-335, 2008.

R. R. Queiroz, F. G. Cunha, and B. M. Gonzalez, Study of dynamic strain aging in dual phase steel, Mater. Sci. Engng A, vol.543, pp.84-87, 2012.

J. R. Thomas and . Hughes, Generalization of selective integration procedures to anisotropic and nonlinear media, International Journal for Numerical Methods in Engineering, vol.15, issue.9, pp.1413-1418

J. Garaud, S. Feld-payet, F. Bettonte, A. Tireira, Y. L. Sant et al., escale : plateforme logicielle pour le dialogue essai -calcul, 13e colloque national en calcul des structures, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01908509

E. Diamantopoulou, W. Liu, C. Labergere, H. Badreddine, K. Saanouni et al., Micromorphic constitutive equations with damage applied to metal forming, International Journal of Damage Mechanics, vol.26, issue.2, pp.314-339, 2017.
URL : https://hal.archives-ouvertes.fr/hal-02279328

J. C. Simo and M. Ortiz, A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations, Computer Methods in Applied Mechanics and Engineering, vol.49, issue.2, pp.221-245, 1985.

M. Ortiz and J. C. Simo, An analysis of a new class of integration algorithms for elastoplastic constitutive relations, International Journal for Numerical Methods in Engineering, vol.23, issue.3, pp.353-366, 1986.

J. L. Lagrange, Traité de la résolution des équations numériques de tous les degrés, p.1826

V. Alireza, S. Amirkhizi, and . Nemat-nasser, A framework for numerical integration of crystal elastoplastic constitutive equations compatible with explicit finite element codes, International Journal of Plasticity, vol.23, issue.10, pp.1918-1937, 2007.

S. Sloan, A. Abbo, and D. Sheng, Refined explicit integration of elastoplastic models with automatic error control, Erratum. Engineering Computations, vol.19, pp.594-594, 2002.

H. Miroslav, A robust explicit integration of elasto-plastic constitutive models, based on simple subincrement size estimation. Engineering Computations, vol.34, pp.1774-1806, 2017.

J. Besson, D. Steglich, and W. Brocks, Modeling of crack growth in round bars and plane strain specimens, International Journal of Solids and Structures, vol.38, issue.46, pp.8259-8284, 2001.

O. Allix, The bounded rate concept: A framework to deal with objective failure predictions in dynamic within a local constitutive model, International Journal of Damage Mechanics, vol.22, issue.6, pp.808-828, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01745205

M. Radioss-theory, Large Displacement Finite Element Analysis, 2011.

F. Casadei, V. Aune, G. Valsamos, and M. Larcher, Accounting for large membrane strains in q4gs and t3gs elements in europlexus, vol.01, p.2016

S. W. Sloan, Substepping schemes for the numerical integration of elastoplastic stress-strain relations, International journal for numerical methods in engineering, vol.24, issue.5, pp.893-911, 1987.

P. Zden?k, T. B. Ba?ant, and . Belytschko, Wave propagation in a strain& softening bar:exact solution, Journal of Engineering Mechanics, vol.111, issue.3, pp.381-389, 1985.

A. Needleman, Material rate dependence and mesh sensitivity in localization problems, Computer Methods in Applied Mechanics and Engineering, vol.67, issue.1, pp.69-85, 1988.

R. Borst, L. J. Sluys, H. Muhlhaus, and J. Pamin, Fundamental issues in finite element analyses of localization of deformation, Engineering Computations, vol.10, issue.2, pp.99-121, 1993.

A. Suffis, Développement d'un modèle d'endommagement à taux de croissance contrôlé pour la simulation robuste de ruptures sous impacts, 2004.

G. Court, Consistent prediction of ductile failure under dynamic loads : bounded rates plastic exhaustion model. Theses, École normale supérieure de Cachan -ENS Cachan, 2006.
URL : https://hal.archives-ouvertes.fr/tel-00341991

M. Chambart, Anisotropic damage and dynamic behavior of reinforced concrete structures until failure. Theses, École normale supérieure de Cachan -ENS Cachan, 2009.
URL : https://hal.archives-ouvertes.fr/tel-00436617

F. H. Wu and L. B. Freund, Deformation trapping due to thermoplastic instability in one-dimensional wave propagation, Journal of the Mechanics and Physics of Solids, vol.32, issue.2, pp.119-132, 1984.

C. Giry, Modélisation objective de la localisation des déformations et de la fissuration des structures en béton soumises à des chargements statiques ou dynamiques, Jacky et Dufour, Frédéric Matériaux, 2011.

R. Faria, J. Oliver, and M. Cervera, A strain-based plastic viscous-damage model for massive concrete structures, International Journal of Solids and Structures, vol.35, issue.14, pp.1533-1558, 1998.

V. Dias-da and S. , A simple model for viscous regularization of elasto-plastic constitutive laws with softening, International Journal for Numerical Methods in Biomedical Engineering, vol.20, issue.7, pp.547-568, 2004.

M. S. Niazi, H. H. Wisselink, and T. Meinders, Viscoplastic regularization of local damage models: revisited, Computational Mechanics, vol.51, issue.2, pp.203-216, 2013.

R. Kluge, Duvaut, g./lions, j. l., inequalities in mechanics and physics. (aus dem französischen), grundlagen aus den mathematischen wissenschaften 219. berlin-heidelberg-new york, springer-verlag 1976. xvi, 397 s., 28 abb., dm, vol.98, pp.583-583, 1979.

G. Pijaudier-cabot, K. Haidar, and J. Dubé, Nonlocal damage model with evolving internal length, International Journal for Numerical and Analytical Methods in Geomechanics, vol.28, pp.633-652, 2004.
URL : https://hal.archives-ouvertes.fr/hal-01007119

G. Pijaudier, -. Cabot, and A. Benallal, Strain localization and bifurcation in a nonlocal continuum, International Journal of Solids and Structures, vol.30, issue.13, pp.1761-1775, 1993.
URL : https://hal.archives-ouvertes.fr/hal-01718249

A. B. Roy, . Engelen, G. D. Marc, F. P. Geers, and . Baaijens, Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour, International Journal of Plasticity, vol.19, issue.4, pp.403-433, 2003.

R. H. Peerlings, M. G. Geers, R. Borst, and W. A. Brekelmans, A critical comparison of nonlocal and gradient-enhanced softening continua, International Journal of Solids and Structures, vol.38, issue.44, pp.7723-7746, 2001.

H. Mühlhaus and E. C. Alfantis, A variational principle for gradient plasticity, International Journal of Solids and Structures, vol.28, issue.7, pp.845-857, 1991.

D. René, H. Borst, and . Mühlhaus, Gradient-dependent plasticity: Formulation and algorithmic aspects, International Journal for Numerical Methods in Engineering, vol.35, issue.3, pp.521-539, 1992.

M. Mazière and S. Forest, Strain gradient plasticity modeling and finite element simulation of lüders band formation and propagation, Continuum Mechanics and Thermodynamics, vol.27, issue.1, pp.83-104, 2015.

E. Lorentz and A. Benallal, Gradient constitutive relations: numerical aspects and application to gradient damage, Computer Methods in Applied Mechanics and Engineering, vol.194, issue.50, pp.5191-5220, 2005.
URL : https://hal.archives-ouvertes.fr/hal-01718363

E. Lorentz and V. Godard, Gradient damage models: Toward full-scale computations, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.21, pp.1927-1944, 2011.

V. Davaze, N. Vallino, S. Feld-payet, B. Langrand, and J. Besson, Plastic and fracture behavior of a dual phase steel sheet under quasi-static and dynamic loadings. (submitted), 2019.

J. Besson, C. N. Mccowan, and E. S. Drexler, Modeling flat to slant fracture transition using the computational cell methodology, Engineering Fracture Mechanics, vol.104, pp.80-95, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00830626

A. Suffis, A. A. Ton, A. Lubrecht, and . Combescure, Damage model with delay effect: Analytical and numerical studies of the evolution of the characteristic damage length, International Journal of Solids and Structures, vol.40, issue.13, pp.3463-3476, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00454712

K. Saanouni and M. Hamed, Micromorphic approach for finite gradient-elastoplasticity fully coupled with ductile damage: Formulation and computational aspects, International Journal of Solids and Structures, vol.50, issue.14, pp.2289-2309, 2013.
URL : https://hal.archives-ouvertes.fr/hal-02279323

S. Feld-payet, J. Besson, and F. Feyel, Finite element analysis of damage in ductile structures using a nonlocal model combined with a three-field formulation, International Journal of Damage Mechanics, vol.20, issue.5, pp.655-680, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00619641

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

C. Christian, D. Roth, and . Mohr, Ductile fracture experiments with locally proportional loading histories, International Journal of Plasticity, vol.79, pp.328-354, 2016.

A. E. , H. Love, and G. Howard-darwin, Xvi. the small free vibrations and deformation of a thin elastic shell, Philosophical Transactions of the Royal Society of London. (A.), vol.179, pp.491-546, 1888.

E. Reissner, The effect of transverse shear deformation on the bending of elastic plate models, J. Appl. Mech, vol.12, pp.69-76, 1945.

R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech, vol.18, pp.31-38, 1951.

S. P. Venkateshan and P. Swaminathan, Chapter 9 -numerical integration, Computational Methods in Engineering, pp.317-373, 2014.

S. Vlachoutsis, Shear correction factors for plates and shells, International Journal for Numerical Methods in Engineering, vol.33, issue.7, pp.1537-1552, 1992.

M. Brunet, F. Morestin, and H. Walter, Damage identification for anisotropic sheet-metals using a non-local damage model, International Journal of Damage Mechanics, vol.13, issue.1, pp.35-57, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00450600

M. Brunet, F. Morestin, and H. Walter-leberre, Failure analysis of anisotropic sheet-metals using a non-local plastic damage model, Journal of Materials Processing Technology, vol.170, issue.1, pp.457-470, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00368447