S. Acharyya, S. Dhar, and J. Chattopadhyay, Evaluation of critical fracture energy parameter Gfr and assessment of its transferrability, Engineering Fracture Mechanics, vol.75, issue.2, pp.253-274, 2008.
DOI : 10.1016/j.engfracmech.2007.03.032

D. A. Akhrass, J. Bruchon, S. Drapier, and S. Fayolle, Integrating a logarithmic-strain based hyperelastic formulation into a three-field mixed finite element formulation to deal with incompressibility in finite-strain elastoplasticity Finite Elements in Analysis and Design, pp.61-70, 2014.

N. Allahverdizadeh, A. Manes, M. Giglio, and A. Gilioli, Geometry Transferability of Lemaitre's Continuum Damage Mechanics Model in the Plane Stress Specimens, Materials Structure & Micromechanics of Fracture Vii, pp.266-270, 2014.
DOI : 10.4028/www.scientific.net/KEM.592-593.266

M. Ambati, T. Gerasimov, and L. De-lorenzis, Phase-field modeling of ductile fracture, Computational Mechanics, vol.92, issue.3???4, pp.1-24, 2015.
DOI : 10.1007/s00466-015-1151-4

F. X. Andrade, J. C. De-sá, and F. A. Pires, A Ductile Damage Nonlocal Model of Integral-type at Finite Strains: Formulation and Numerical Issues, International Journal of Damage Mechanics, vol.8, issue.4, 2011.
DOI : 10.1016/0045-7825(90)90131-5

F. X. Andrade, J. De-sa, and F. M. Pires, Assessment and comparison of non-local integral models for ductile damage, International Journal of Damage Mechanics, vol.47, issue.4, pp.261-296, 2014.
DOI : 10.1016/j.ijplas.2008.09.009

R. J. Asaro, Micromechanics of crystals and polycrystals. Advances in applied mechanics 23, 1983.

H. Askes and L. J. Sluys, Explicit and implicit gradient series in damage mechanics, European Journal of Mechanics - A/Solids, vol.21, issue.3, pp.379-390, 2002.
DOI : 10.1016/S0997-7538(02)01214-7

F. Auricchio, L. C. Beirão-da-veiga, C. Lovadina, A. Reali, R. Taylor et al., Approximation of incompressible large deformation elastic problems: some unresolved issues, Computational Mechanics, vol.102, issue.5, pp.1153-1167, 2013.
DOI : 10.1007/s00466-013-0869-0

Y. L. Bai and T. Wierzbicki, A new model of metal plasticity and fracture with pressure and Lode dependence, International Journal of Plasticity, vol.24, issue.6, pp.1071-1096, 2008.
DOI : 10.1016/j.ijplas.2007.09.004

Y. B. 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.
DOI : 10.1016/j.ijmecsci.2004.02.006

G. I. Barenblatt, The Mathematical Theory of Equilibrium Cracks in Brittle Fracture, Advances in applied mechanics, pp.55-129, 1962.
DOI : 10.1016/S0065-2156(08)70121-2

R. Bargellini, J. Besson, E. Lorentz, and S. Michel-ponnelle, A non-local finite element based on volumetric strain gradient: Application to ductile fracture, Computational Materials Science, vol.45, issue.3, pp.762-767, 2009.
DOI : 10.1016/j.commatsci.2008.09.020

URL : https://hal.archives-ouvertes.fr/hal-00379224

S. Basu and A. A. Benzerga, On the path-dependence of the fracture locus in ductile materials: Experiments, International Journal of Solids and Structures, vol.71, pp.79-90, 2015.
DOI : 10.1016/j.ijsolstr.2015.06.003

K. Bathe, The inf???sup condition and its evaluation for mixed finite element methods, Computers & Structures, vol.79, issue.2, pp.243-252, 2001.
DOI : 10.1016/S0045-7949(00)00123-1

L. Bauvineau, Approche locale de la rupture ductile: application à un acier carbone-manganèse, 1996.

L. Beirão-da-veiga, F. Brezzi, A. Cangiani, G. Manzini, L. Marini et al., BASIC PRINCIPLES OF VIRTUAL ELEMENT METHODS, Mathematical Models and Methods in Applied Sciences, vol.23, issue.01, pp.199-214, 2013.
DOI : 10.1142/S0218202512500492

A. Benallal, R. Billardon, and G. Geymonat, Bifurcation and Localization in Rate-Independent Materials. Some General Considerations, 1993.
DOI : 10.1007/978-3-7091-2712-4_1

A. A. Benzerga, D. Surovik, and S. M. Keralavarma, On the path-dependence of the fracture locus in ductile materials ??? Analysis, International Journal of Plasticity, vol.37, pp.157-170, 2012.
DOI : 10.1016/j.ijplas.2012.05.003

J. Besson, Continuum Models of Ductile Fracture: A Review, International Journal of Damage Mechanics, vol.19, issue.1, pp.3-52, 2010.
DOI : 10.1177/1056789509103482

URL : https://hal.archives-ouvertes.fr/hal-00550957

J. Besson, A Two Length Scale Non-Local Model to Describe Ductile Rupture at Low Stress Triaxiality, The third international conference on comutational modeling of fracture and failure onf materials and structures, 2013.

J. Besson and C. Berdin, Local Approach to Fracture, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00755207

J. Besson and C. Guillemer-neel, An extension of the Green and Gurson models to kinematic hardening, Mechanics of Materials, vol.35, issue.1-2, pp.1-18, 2003.
DOI : 10.1016/S0167-6636(02)00169-2

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.46-47, pp.8259-8284, 2001.
DOI : 10.1016/S0020-7683(01)00167-6

C. Bouchet, B. Tanguy, J. Besson, and S. Bugat, Prediction of the effects of neutron irradiation on the Charpy ductile to brittle transition curve of an A508 pressure vessel steel, Computational Materials Science, vol.32, issue.3-4, pp.294-300, 2005.
DOI : 10.1016/j.commatsci.2004.09.039

URL : https://hal.archives-ouvertes.fr/hal-00154555

F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, 1991.
DOI : 10.1007/978-1-4612-3172-1

W. Brocks and H. Yuan, Numerical investigations on the significance of for large stable crack growth, Engineering Fracture Mechanics, vol.32, issue.3, pp.459-468, 1989.
DOI : 10.1016/0013-7944(89)90317-2

P. Broumand and A. R. Khoei, The extended finite element method for large deformation ductile fracture problems with a non-local damage-plasticity model. Engineering Fracture Mechanics 112-113, pp.97-125, 2013.

M. Brünig, An anisotropic ductile damage model based on irreversible thermodynamics, International Journal of Plasticity, vol.19, issue.10, pp.1679-1713, 2003.
DOI : 10.1016/S0749-6419(02)00114-6

M. Brünig, D. Albrecht, and S. Gerke, Modeling of Ductile Damage and Fracture Behavior Based on Different Micromechanisms, International Journal of Damage Mechanics, vol.17, issue.4, pp.558-577, 2011.
DOI : 10.1016/0001-6160(84)90213-X

M. Brünig, O. Chyra, D. Albrecht, L. Driemeier, and M. Alves, A ductile damage criterion at various stress triaxialities, International Journal of Plasticity, vol.24, issue.10, pp.1731-1755, 2008.
DOI : 10.1016/j.ijplas.2007.12.001

M. Brünig, S. Gerke, and D. Brenner, New 2D-Experiments and Numerical Simulations on Stress-state-dependence of Ductile Damage and Failure, Procedia Materials Science, vol.3, pp.177-182, 2014.
DOI : 10.1016/j.mspro.2014.06.032

T. S. Cao, M. Maziere, K. Danas, and J. Besson, A model for ductile damage prediction at low stress triaxialities incorporating void shape change and void rotation, International Journal of Solids and Structures, vol.63, pp.240-263, 2015.
DOI : 10.1016/j.ijsolstr.2015.03.003

URL : https://hal.archives-ouvertes.fr/hal-01165304

F. Cazes and N. Moës, Comparison of a phase-field model and of a thick level set model for brittle and quasi-brittle fracture, International Journal for Numerical Methods in Engineering, vol.339, issue.1, 2015.
DOI : 10.1002/nme.4886

D. Chapelle and K. Bathe, The inf-sup test, Computers & Structures, vol.47, issue.4-5, pp.537-545, 1993.
DOI : 10.1016/0045-7949(93)90340-J

URL : https://hal.archives-ouvertes.fr/hal-00839728

J. Charmet, Mécanique du solide et des matériaux Elasticité-Plasticité-Rupture. ESPCI- Laboratoire d'Hydrodynamique et Mécanique Physique, pp.113-144, 2005.

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.
DOI : 10.1115/1.3224807

A. Cornec, I. Scheider, and K. Schwalbe, On the practical application of the cohesive model, Engineering Fracture Mechanics, vol.70, issue.14, pp.1963-1987, 2003.
DOI : 10.1016/S0013-7944(03)00134-6

E. Cosserat and F. Cosserat, Théorie des corps déformables, 1909.

K. Danas and P. P. Castañeda, A finite-strain model for anisotropic viscoplastic porous media: I ??? Theory, European Journal of Mechanics - A/Solids, vol.28, issue.3, pp.387-401, 2009.
DOI : 10.1016/j.euromechsol.2008.11.002

URL : https://hal.archives-ouvertes.fr/hal-00491346

J. C. De-sá, P. Areias, and C. Zheng, Damage modelling in metal forming problems using an implicit non-local gradient model, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.48-49, pp.6646-6660, 2006.
DOI : 10.1016/j.cma.2005.02.037

E. A. De-souza-neto, D. Peric, and D. R. Owen, Computational methods for plasticity: theory and applications, 2011.
DOI : 10.1002/9780470694626

E. A. De-souza-neto, F. M. Pires, and D. R. Owen, F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking, International Journal for Numerical Methods in Engineering, vol.4, issue.3, pp.353-383, 2005.
DOI : 10.1002/nme.1187

G. Debruyne, Proposition d'un param??tre ??nerg??tique de rupture pour les mat??riaux dissipatifs, Comptes Rendus de l'Académie des Sciences -Series IIB -Mechanics, pp.785-791, 2000.
DOI : 10.1016/S1620-7742(00)01253-8

S. Dhar, S. Marie, and S. Chapuliot, Determination of critical fracture energy, Gfr, from crack tip stretch, International Journal of Pressure Vessels and Piping, vol.85, issue.5, pp.313-321, 2008.
DOI : 10.1016/j.ijpvp.2007.11.002

D. Luzio, G. Ba?ant, and Z. P. , Spectral analysis of localization in nonlocal and over-nonlocal materials with softening plasticity or damage, International Journal of Solids and Structures, vol.42, issue.23, pp.6071-6100, 2005.
DOI : 10.1016/j.ijsolstr.2005.03.038

D. Pietro, D. A. Ern, and A. , A hybrid high-order locking-free method for linear elasticity on general meshes, Computer Methods in Applied Mechanics and Engineering, vol.283, pp.1-21, 2015.
DOI : 10.1016/j.cma.2014.09.009

URL : https://hal.archives-ouvertes.fr/hal-00979435

D. Dugdale, Yielding of steel sheets containing slits, Journal of the Mechanics and Physics of Solids, vol.8, issue.2, pp.100-104, 1960.
DOI : 10.1016/0022-5096(60)90013-2

U. Eisele and E. Roos, Evaluation of different fracture-mechanical J-integral initiation values with regard to their usability in the safety assessment of components, Nuclear Engineering and Design, vol.130, issue.3, pp.237-247, 1991.
DOI : 10.1016/0029-5493(91)90216-5

K. Enakoutsa and J. Leblond, Numerical implementation and assessment of the GLPD micromorphic model of ductile rupture, European Journal of Mechanics - A/Solids, vol.28, issue.3, pp.445-460, 2009.
DOI : 10.1016/j.euromechsol.2008.11.004

K. Enakoutsa, J. B. Leblond, and G. Perrin, Numerical implementation and assessment of a phenomenological nonlocal model of ductile rupture, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.13-16, pp.1946-1957, 2007.
DOI : 10.1016/j.cma.2006.10.003

R. A. Engelen, M. G. Geers, and F. P. 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.
DOI : 10.1016/S0749-6419(01)00042-0

A. C. Eringen and D. Edelen, On nonlocal elasticity, International Journal of Engineering Science, vol.10, issue.3, pp.233-248, 1972.
DOI : 10.1016/0020-7225(72)90039-0

A. C. Eringen and E. Suhubi, Nonlinear theory of simple micro-elastic solids???I, International Journal of Engineering Science, vol.2, issue.2, pp.189-203, 1964.
DOI : 10.1016/0020-7225(64)90004-7

A. Ern and J. Guermond, Theory and practice of finite elements, 2013.
DOI : 10.1007/978-1-4757-4355-5

H. A. Ernst, Material Resistance and Instability Beyond J-Controlled Crack Growth, ASTM STP, vol.803, pp.191-213, 1983.
DOI : 10.1520/STP37294S

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.3, issue.1, pp.655-680, 2011.
DOI : 10.1016/0045-7949(84)90231-1

URL : https://hal.archives-ouvertes.fr/hal-00619641

S. Forest, Micromorphic Approach for Gradient Elasticity, Viscoplasticity, and Damage, Journal of Engineering Mechanics, vol.135, issue.3, 2009.
DOI : 10.1061/(ASCE)0733-9399(2009)135:3(117)

URL : https://hal.archives-ouvertes.fr/hal-00368014

S. Forest, K. Ammar, and B. Appolaire, Micromorphic vs. phase-field approaches for gradient viscoplasticity and phase transformations, Advances in extended and multifield theories for continua, pp.69-88, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00683083

G. A. Francfort and J. Marigo, Revisiting brittle fracture as an energy minimization problem, Journal of the Mechanics and Physics of Solids, vol.46, issue.8, pp.1319-1342, 1998.
DOI : 10.1016/S0022-5096(98)00034-9

A. Franklin, Comparison between a quantitative microscope and chemical methods for assessment of non-metallic inclusions, J Iron Steel Inst, vol.207, pp.181-186, 1969.

R. Gannon, What really sank the Titanic, Popular Science, vol.246, pp.49-55, 1995.

M. Geers, Finite strain logarithmic hyperelasto-plasticity with softening: a strongly non-local implicit gradient framework, Computer Methods in Applied Mechanics and Engineering, vol.193, issue.30-32, pp.3377-3401, 2004.
DOI : 10.1016/j.cma.2003.07.014

M. G. Geers, R. De-borst, W. A. Brekelmans, and R. H. Peerlings, Strain-based transient-gradient damage model for failure analyses, Computer Methods in Applied Mechanics and Engineering, vol.160, issue.1-2, pp.133-153, 1998.
DOI : 10.1016/S0045-7825(98)80011-X

M. Gologanu, J. Leblond, G. Perrin, and J. Devaux, Recent Extensions of Gurson???s Model for Porous Ductile Metals, 1997.
DOI : 10.1007/978-3-7091-2662-2_2

M. Gologanu, J. B. Leblond, J. Devaux, . Approximate, . Models et al., Approximate models for ductile metals containing non-spherical voids???Case of axisymmetric prolate ellipsoidal cavities, Journal of the Mechanics and Physics of Solids, vol.41, issue.11, pp.1723-1754, 1993.
DOI : 10.1016/0022-5096(93)90029-F

A. P. Gómez, N. Moës, and C. Stolz, Comparison between thick level set (TLS) and cohesive zone models, Advanced Modeling and Simulation in Engineering Sciences, vol.2, pp.1-22, 2015.

M. Grange, J. Besson, and E. Andrieu, An anisotropic Gurson type model to represent the ductile rupture of hydrided Zircaloy-4 sheets, International Journal of Fracture, vol.105, issue.3, pp.273-293, 2000.
DOI : 10.1023/A:1007615513884

P. Grassl, D. Xenos, M. Jirásek, and M. Horák, Evaluation of nonlocal approaches for modelling fracture near nonconvex boundaries, International Journal of Solids and Structures, vol.51, issue.18, pp.3239-3251, 2014.
DOI : 10.1016/j.ijsolstr.2014.05.023

A. A. Griffith, The phenomena of rupture and flow in solids. Philosophical transactions of the royal society of london. Series A, containing papers of a mathematical or physical character, pp.163-198, 1921.

A. L. Gurson, 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.
DOI : 10.1115/1.3443401

B. Halphen and Q. Nguyen, Sur les matériaux standard généralisés, Journal de Mécanique, vol.14, pp.39-63, 1975.

B. S. Henry and A. R. Luxmoore, The stress triaxiality constraint and the Q-value as a ductile fracture parameter, Engineering Fracture Mechanics, vol.57, issue.4, pp.375-390, 1997.
DOI : 10.1016/S0013-7944(97)00031-3

A. E. Huespe, A. Needleman, J. Oliver, and P. J. Sánchez, A finite thickness band method for ductile fracture analysis, International Journal of Plasticity, vol.25, issue.12, pp.2349-2365, 2009.
DOI : 10.1016/j.ijplas.2009.03.005

A. E. Huespe, A. Needleman, J. Oliver, and P. J. Sánchez, A finite strain, finite band method for modeling ductile fracture, International Journal of Plasticity, vol.28, issue.1, pp.53-69, 2012.
DOI : 10.1016/j.ijplas.2011.05.010

G. Huetter, T. Linse, U. Muehlich, and M. Kuna, Simulation of ductile crack initiation and propagation by means of a non-local Gurson-model, International Journal of Solids and Structures, vol.50, issue.5, pp.662-671, 2013.
DOI : 10.1016/j.ijsolstr.2012.10.031

T. J. Hughes, The finite element method: linear static and dynamic finite element analysis, Courier Corporation, 2012.

J. Hutchinson, Plastic stress and strain fields at a crack tip, Journal of the Mechanics and Physics of Solids, vol.16, issue.5, pp.337-342, 1968.
DOI : 10.1016/0022-5096(68)90021-5

J. W. Hutchinson, Fundamentals of the Phenomenological Theory of Nonlinear Fracture Mechanics, Journal of Applied Mechanics, vol.50, issue.4b, pp.1042-1051, 1983.
DOI : 10.1115/1.3167187

G. Hutter, T. Linse, U. Muhlich, and M. Kuna, Simulation of ductile crack initiation and propagation by means of a non-local Gurson-model, International Journal of Solids and Structures, vol.50, issue.5, pp.662-671, 2013.
DOI : 10.1016/j.ijsolstr.2012.10.031

G. Hutter, T. Linse, S. Roth, U. Muhlich, and M. Kuna, A modeling approach for the complete ductile???brittle transition region: cohesive zone in combination with a non-local Gurson-model, International Journal of Fracture, vol.87, issue.4, pp.129-153, 2014.
DOI : 10.1007/s10704-013-9914-4

G. Irwin, Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate, J. Appl. Mech, 1957.

J. Jackiewicz and M. Kuna, Non-local regularization for FE simulation of damage in ductile materials, Computational Materials Science, vol.28, issue.3-4, pp.684-695, 2003.
DOI : 10.1016/j.commatsci.2003.08.024

M. Jirásek, Nonlocal damage mechanics. Revue européenne de génie civil 11, pp.993-1021, 2007.

M. Kailasam and P. P. Castañeda, A general constitutive theory for linear and nonlinear particulate media with microstructure evolution, Journal of the Mechanics and Physics of Solids, vol.46, issue.3, pp.427-465, 1998.
DOI : 10.1016/S0022-5096(97)00095-1

N. Kanetake, M. Nomura, and T. Choh, Continuous observation of microstructural degradation during tensile loading of particle reinforced aluminium matrix composites, Materials Science and Technology, vol.58, issue.395, pp.1246-1252, 1995.
DOI : 10.2320/matertrans1989.32.931

J. Koplik and A. Needleman, Void growth and coalescence in porous plastic solids, International Journal of Solids and Structures, vol.24, issue.8, pp.835-853, 1988.
DOI : 10.1016/0020-7683(88)90051-0

E. Kröner, Elasticity theory of materials with long range cohesive forces, International Journal of Solids and Structures, vol.3, issue.5, pp.731-742, 1967.
DOI : 10.1016/0020-7683(67)90049-2

D. Lassance, D. Fabregue, F. Delannay, and T. Pardoen, Micromechanics of room and high temperature fracture in 6xxx Al alloys, Progress in materials science 52, pp.62-129, 2007.
DOI : 10.1016/j.pmatsci.2006.06.001

L. Delliou and P. , Etat de l'art sur les modèles en déchirure ductile. Note interne EDF R&D, 2012.

J. B. Leblond, Brittle fracture and ductile fracture, Comptes Rendus Acad. Sci. Ser. II-B, vol.326, pp.243-250, 1998.

J. Lemaitre, A Continuous Damage Mechanics Model for Ductile Fracture, Journal of Engineering Materials and Technology, vol.107, issue.1, pp.83-89, 1985.
DOI : 10.1115/1.3225775

J. Lemaitre and R. Desmorat, Engineering damage mechanics: ductile, creep, fatigue and brittle failures, 2005.

J. Lemaitre, R. Desmorat, and M. Sauzay, Anisotropic damage law of evolution, European Journal of Mechanics - A/Solids, vol.19, issue.2, pp.187-208, 2000.
DOI : 10.1016/S0997-7538(00)00161-3

J. Lemaitre and J. Dufailly, Damage measurements, Engineering Fracture Mechanics, vol.28, issue.5-6, pp.643-661, 1987.
DOI : 10.1016/0013-7944(87)90059-2

H. Li, M. W. Fu, J. Lu, and H. Yang, Ductile fracture: Experiments and computations, International Journal of Plasticity, vol.27, issue.2, pp.147-180, 2011.
DOI : 10.1016/j.ijplas.2010.04.001

T. Linse, G. Huetter, and M. Kuna, Simulation of crack propagation using a gradient-enriched ductile damage model based on dilatational strain, Engineering Fracture Mechanics, vol.95, pp.13-28, 2012.
DOI : 10.1016/j.engfracmech.2012.07.004

T. Linse, M. Kuna, and H. W. Viehrig, Quantification of brittle-ductile failure behavior of ferritic reactor pressure vessel steels using the Small-Punch-Test and micromechanical damage models, Materials Science and Engineering: A, vol.614, pp.136-147, 2014.
DOI : 10.1016/j.msea.2014.05.095

P. Longere, A. G. Geffroy, B. Leble, and A. Dragon, Modeling the Transition between Dense Metal and Damaged (Microporous) Metal Viscoplasticity, International Journal of Damage Mechanics, vol.75, issue.7, pp.1020-1063, 2012.
DOI : 10.1016/S0020-7683(01)00087-7

E. Lorentz, Lois de comportement à gradients de variables internes: construction, formulation variationnelle et mise en oeuvre numérique, 1999.

E. Lorentz, A mixed interface finite element for cohesive zone models, Computer Methods in Applied Mechanics and Engineering, vol.198, issue.2, pp.302-317, 2008.
DOI : 10.1016/j.cma.2008.08.006

URL : https://hal.archives-ouvertes.fr/hal-00359363

E. Lorentz and S. Andrieux, Analysis of non-local models through energetic formulations, International Journal of Solids and Structures, vol.40, issue.12, pp.2905-2936, 2003.
DOI : 10.1016/S0020-7683(03)00110-0

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.50-52, pp.5191-5220, 2005.
DOI : 10.1016/j.cma.2004.12.016

URL : https://hal.archives-ouvertes.fr/hal-00089126

E. Lorentz, J. Besson, and V. Cano, Numerical simulation of ductile fracture with the Rousselier constitutive law, Computer Methods in Applied Mechanics and Engineering, vol.197, issue.21-24, pp.1965-1982, 2008.
DOI : 10.1016/j.cma.2007.12.015

URL : https://hal.archives-ouvertes.fr/hal-00268315

E. Lorentz and V. Godard, Gradient damage models: Toward full-scale computations, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.21-22, pp.1927-1944, 2011.
DOI : 10.1016/j.cma.2010.06.025

S. Marie and S. Chapuliot, DUCTILE TEARING SIMULATION BASED ON A LOCAL ENERGETIC CRITERION, Fatigue <html_ent glyph="@amp;" ascii="&"/> Fracture of Engineering Materials and Structures, vol.21, issue.2, pp.215-227, 1998.
DOI : 10.1046/j.1460-2695.1998.00017.x

S. Marie and S. Chapuliot, Ductile crack growth simulation from near crack tip dissipated energy, Nuclear Engineering and Design, vol.196, issue.3, pp.293-305, 2000.
DOI : 10.1016/S0029-5493(99)00306-4

J. Marigo, Plasticité et Rupture, pp.45-48, 2012.

P. Matheron, S. Chapuliot, L. Nicolas, V. Koundy, and C. Caroli, Characterization of PWR vessel steel tearing under severe accident condition temperatures, Nuclear Engineering and Design, vol.242, pp.124-133, 2012.
DOI : 10.1016/j.nucengdes.2011.10.046

J. Mediavilla, R. Peerlings, and M. Geers, A nonlocal triaxiality-dependent ductile damage model for finite strain plasticity, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.33-36, pp.4617-4634, 2006.
DOI : 10.1016/j.cma.2005.10.001

J. Mediavilla, R. H. Peerlings, and M. G. Geers, An integrated continuous???discontinuous approach towards damage engineering in sheet metal forming processes, Engineering Fracture Mechanics, vol.73, issue.7, pp.895-916, 2006.
DOI : 10.1016/j.engfracmech.2005.10.011

J. G. Merkle and H. T. Corten, A J Integral Analysis for the Compact Specimen, Considering Axial Force as Well as Bending Effects, Journal of Pressure Vessel Technology, vol.96, issue.4, pp.286-292, 1974.
DOI : 10.1115/1.3454183

C. Miehe, Variational gradient plasticity at finite strains. Part I: Mixed potentials for the evolution and update problems of gradient-extended dissipative solids, Computer Methods in Applied Mechanics and Engineering, vol.268, pp.677-703, 2014.
DOI : 10.1016/j.cma.2013.03.014

C. Miehe, N. Apel, and M. Lambrecht, Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation based on incremental minimization principles for standard materials, Computer Methods in Applied Mechanics and Engineering, vol.191, issue.47-48, pp.5383-5425, 2002.
DOI : 10.1016/S0045-7825(02)00438-3

N. Moës, C. Stolz, P. E. Bernard, and N. Chevaugeon, A level set based model for damage growth: The thick level set approach, International Journal for Numerical Methods in Engineering, vol.61, issue.1, pp.358-380, 2011.
DOI : 10.1002/nme.3069

N. Moës, C. Stolz, and N. Chevaugeon, Coupling local and non-local damage evolutions with the Thick Level Set model, Advanced Modeling and Simulation in Engineering Sciences, vol.3, issue.1, pp.1-21, 2014.
DOI : 10.1186/s40323-014-0016-2

M. , D. Delliou, P. L. Vincent, W. Sonnefraud, C. Roirand et al., Déchirure ductile et transférabilité des propriétés de ténacité entre éprouvettes et structures, Synthèse des travaux menés dans le cadre des actions STYLE et CoMaDiS, 2014.

T. F. Morgeneyer, T. Taillandier-thomas, L. Helfen, T. Baumbach, I. Sinclair et al., In situ 3-D observation of early strain localization during failure of thin Al alloy (2198) sheet, Acta Materialia, vol.69, pp.78-91, 2014.
DOI : 10.1016/j.actamat.2014.01.033

URL : https://hal.archives-ouvertes.fr/hal-00952176

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.
DOI : 10.1016/j.euromechsol.2007.08.002

A. Needleman, An analysis of tensile decohesion along an interface, Journal of the Mechanics and Physics of Solids, vol.38, issue.3, pp.289-324, 1990.
DOI : 10.1016/0022-5096(90)90001-K

M. Niazi, H. Wisselink, T. Meinders, and C. Horn, Implementation of an anisotropic damage material model using general second order damage tensor, Steel Research International, vol.81, pp.1396-1399, 2010.

K. L. Nielsen and V. Tvergaard, Effect of a shear modified Gurson model on damage development in a FSW tensile specimen, International Journal of Solids and Structures, vol.46, issue.3-4, pp.587-601, 2009.
DOI : 10.1016/j.ijsolstr.2008.09.011

O. , N. Shih, and C. F. , Family of crack-tip fields characterized by a triaxiality parameter?I. Structure of fields, Journal of the Mechanics and Physics of Solids, vol.39, pp.989-1015, 1991.

O. Dowd and N. P. , Applications of two parameter approaches in elastic-plastic fracture mechanics, Engineering Fracture Mechanics, vol.52, issue.3, pp.445-465, 1995.
DOI : 10.1016/0013-7944(95)00033-R

O. Dowd, N. P. Shih, and C. F. , Family of crack-tip fields characterized by a triaxiality parameter???II. Fracture applications, Journal of the Mechanics and Physics of Solids, vol.40, issue.5, pp.939-963, 1992.
DOI : 10.1016/0022-5096(92)90057-9

J. Papasidero, V. Doquet, and D. Mohr, Ductile fracture of aluminum 2024-T351 under proportional and non-proportional multi-axial loading: Bao???Wierzbicki results revisited, International Journal of Solids and Structures, vol.69, issue.70, pp.459-474, 2015.
DOI : 10.1016/j.ijsolstr.2015.05.006

URL : https://hal.archives-ouvertes.fr/hal-01227222

T. Pardoen and J. W. Hutchinson, Micromechanics-based model for trends in toughness of ductile metals, Acta Materialia, vol.51, issue.1, pp.133-148, 2003.
DOI : 10.1016/S1359-6454(02)00386-5

P. C. Paris, M. P. Gomez, and W. E. Anderson, A rational analytic theory of fatigue. The trend in engineering 13, pp.9-14, 1961.

R. Peerlings, R. De-borst, W. Brekelmans, and J. De-vree, GRADIENT ENHANCED DAMAGE FOR QUASI-BRITTLE MATERIALS, International Journal for Numerical Methods in Engineering, vol.23, issue.19, pp.3391-3403, 1996.
DOI : 10.1002/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3.0.CO;2-D

R. Peerlings, M. Geers, R. De-borst, and W. Brekelmans, A critical comparison of nonlocal and gradient-enhanced softening continua, International Journal of Solids and Structures, vol.38, issue.44-45, pp.7723-7746, 2001.
DOI : 10.1016/S0020-7683(01)00087-7

G. Pijaudier-cabot and Z. P. Bazant, Nonlocal Damage Theory, Journal of Engineering Mechanics, vol.113, issue.10, pp.1512-1533, 1987.
DOI : 10.1061/(ASCE)0733-9399(1987)113:10(1512)

L. Poh and S. Swaddiwudhipong, Gradient-enhanced softening material models, International Journal of Plasticity, vol.25, issue.11, pp.2094-2121, 2009.
DOI : 10.1016/j.ijplas.2009.01.003

C. Polizzotto, Nonlocal elasticity and related variational principles, International Journal of Solids and Structures, vol.38, issue.42-43, pp.7359-7380, 2001.
DOI : 10.1016/S0020-7683(01)00039-7

R. Razvan, Some remarks on the history of fracture mechanics, 2009.

J. N. Reddy, C. S. Krishnamoorthy, and K. N. Seetharamu, Finite Element Analysis for Engineering Design, 1988.
DOI : 10.1007/978-3-642-83535-3

F. Reusch, B. Svendsen, and D. Klingbeil, Local and non-local Gurson-based ductile damage and failure modelling at large deformation, European Journal of Mechanics - A/Solids, vol.22, issue.6, pp.779-792, 2003.
DOI : 10.1016/S0997-7538(03)00070-6

F. Reusch, B. Svendsen, and D. Klingbeil, A non-local extension of Gurson-based ductile damage modeling, Computational Materials Science, vol.26, pp.219-229, 2003.
DOI : 10.1016/S0927-0256(02)00402-0

#. Rice, R. , J. Paris, #. , C. et al., Some Further Results of J-Integral Analysis and Estimates, American Society for Testing and Materials, 1973.
DOI : 10.1520/STP49643S

J. Rice and G. Rosengren, Plane strain deformation near a crack tip in a power-law hardening material, Journal of the Mechanics and Physics of Solids, vol.16, issue.1, pp.1-12, 1968.
DOI : 10.1016/0022-5096(68)90013-6

J. R. Rice, A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks, Journal of Applied Mechanics, vol.35, issue.2, pp.379-386, 1968.
DOI : 10.1115/1.3601206

J. R. 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.
DOI : 10.1016/0022-5096(69)90033-7

R. , Q. Moinereau, D. Delliou, P. L. Sonnefraud, C. Vincent et al., Analyse de la déchirure ductile d'une tuyauterie en acier ferritique soumise à un essai de flexion 4 points, 2014.

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.
DOI : 10.1016/0029-5493(87)90234-2

M. K. Samal, M. Seidenfuss, E. Roos, B. K. Dutta, and H. S. Kushwaha, Finite element formulation of a new nonlocal damage model, Finite Elements in Analysis and Design, vol.44, issue.6-7, pp.358-371, 2008.
DOI : 10.1016/j.finel.2007.12.002

M. K. Samal, M. Seidenfuss, E. Roos, B. K. Dutta, and H. S. Kushwaha, A mesh-independent Gurson???Tvergaard???Needleman damage model and its application in simulating ductile fracture behaviour, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol.304, issue.2, pp.283-292, 2009.
DOI : 10.1243/09544062JMES1121

I. Scheider and W. Brocks, Simulation of cup???cone fracture using the cohesive model, Engineering Fracture Mechanics, vol.70, issue.14, pp.1943-1961, 2003.
DOI : 10.1016/S0013-7944(03)00133-4

I. Scheider and W. Brocks, Cohesive elements for thin-walled structures, Computational Materials Science, vol.37, issue.1-2, pp.101-109, 2006.
DOI : 10.1016/j.commatsci.2005.12.042

M. Seidenfuss, M. K. Samal, and E. Roos, On critical assessment of the use of local and nonlocal damage models for prediction of ductile crack growth and crack path in various loading and boundary conditions, International Journal of Solids and Structures, vol.48, issue.24, pp.3365-3381, 2011.
DOI : 10.1016/j.ijsolstr.2011.08.006

P. Sicsic, J. Marigo, and C. Maurini, Initiation of a periodic array of cracks in the thermal shock problem: A gradient damage modeling, Journal of the Mechanics and Physics of Solids, vol.63, pp.256-284, 2014.
DOI : 10.1016/j.jmps.2013.09.003

URL : https://hal.archives-ouvertes.fr/hal-00843625

J. Simo and C. Miehe, Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis and implementation, Computer Methods in Applied Mechanics and Engineering, vol.98, issue.1, pp.41-104, 1992.
DOI : 10.1016/0045-7825(92)90170-O

J. D. Sumpter, An experimental investigation of the T stress approach, ASTM SPECIAL TECHNICAL PUBLICATION 1171, pp.492-492, 1993.

P. F. Thomason, An assessment of the validity of J-controlled crack growth, and the stability of cracks in incremental-plastic/elastic solids, International Journal of Fracture, vol.44, pp.259-281, 1990.

C. Turner, A Re-assessment of Ductile Tearing Resistance. II. Energy Dissipation Rate and Associated R-Curves on Normalised Axes, Retroactive Coverage). ECF 8: Fracture Behaviour and Design of Materials and Structures, pp.951-968, 1990.

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

V. Tvergaard and J. W. Hutchinson, The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, Journal of the Mechanics and Physics of Solids, vol.40, issue.6, pp.1377-1397, 1992.
DOI : 10.1016/0022-5096(92)90020-3

V. Tvergaard and J. W. Hutchinson, Effect of T-Stress on mode I crack growth resistance in a ductile solid, International Journal of Solids and Structures, vol.31, issue.6, pp.823-833, 1994.
DOI : 10.1016/0020-7683(94)90080-9

V. Tvergaard and A. Needleman, Analysis of the cup-cone fracture in a round tensile bar, Acta Metallurgica, vol.32, issue.1, pp.157-169, 1984.
DOI : 10.1016/0001-6160(84)90213-X

V. Tvergaard and A. Needleman, Effects of nonlocal damage in porous plastic solids, International Journal of Solids and Structures, vol.32, issue.8-9, pp.1063-1077, 1995.
DOI : 10.1016/0020-7683(94)00185-Y

K. Washizu, On the variational principles of elasticity and plasticity. M.I.T. Aeroelastic and Structures Research Laboratory, 1955.

A. Weck, D. S. Wilkinson, E. Maire, and H. Toda, Visualization by X-ray tomography of void growth and coalescence leading to fracture in model materials, Acta Materialia, vol.56, issue.12, pp.2919-2928, 2008.
DOI : 10.1016/j.actamat.2008.02.027

URL : https://hal.archives-ouvertes.fr/hal-00433967

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.
DOI : 10.1016/j.ijmecsci.2005.03.003

M. Williams, The Bending Stress Distribution at the Base of a Stationary Crack, Journal of Applied Mechanics, vol.28, issue.1, pp.109-114, 1956.
DOI : 10.1115/1.3640470

L. Xia and C. F. Shih, Ductile crack growth-I. A numerical study using computational cells with microstructurally-based length scales, Journal of the Mechanics and Physics of Solids, vol.43, issue.2, pp.233-259, 1995.
DOI : 10.1016/0022-5096(94)00064-C

L. Xia, C. F. Shih, and J. W. Hutchinson, A computational approach to ductile crack growth under large scale yielding conditions, Journal of the Mechanics and Physics of Solids, vol.43, issue.3, pp.389-413, 1995.
DOI : 10.1016/0022-5096(94)00069-H

L. Xue, Constitutive modeling of void shearing effect in ductile fracture of porous materials, Engineering Fracture Mechanics, vol.75, issue.11, pp.3343-3366, 2008.
DOI : 10.1016/j.engfracmech.2007.07.022

L. Xue and T. Wierzbicki, Numerical simulation of fracture mode transition in ductile plates, International Journal of Solids and Structures, vol.46, issue.6, pp.1423-1435, 2009.
DOI : 10.1016/j.ijsolstr.2008.11.009

Z. Xue, J. Faleskog, and J. W. Hutchinson, Tension???torsion fracture experiments ??? Part II: Simulations with the extended Gurson model and a ductile fracture criterion based on plastic strain, International Journal of Solids and Structures, vol.50, issue.25-26, pp.4258-4269, 2013.
DOI : 10.1016/j.ijsolstr.2013.08.028

Q. D. Yang, M. D. Thouless, and S. M. Ward, Numerical simulations of adhesively-bonded beams failing with extensive plastic deformation, Journal of the Mechanics and Physics of Solids, vol.47, issue.6, pp.1337-1353, 1999.
DOI : 10.1016/S0022-5096(98)00101-X