G. R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate, Journal of Applied Mechanics, vol.24, pp.361-397, 1957.

D. Duprat, Fatigue et mécanique de la rupture des pièces en alliage léger. Technique de l'ingénieur, p.52

H. M. Westergaard, Stresses at a crack, size of the crack and the bending of reinforced concrete, Proc. American Concrete Institute, pp.93-102, 1934.

I. E. Sneddon, The Distribution of Stress in the Neighbourhood of a Crack in an Elastic Solid, Proc. R. Soc. Lond. A, pp.229-260, 1946.
DOI : 10.1098/rspa.1946.0077

M. L. Williams, The Bending Stress Distribution at the Base of a Stationary Crack, Journal of Applied Mechanics, vol.28, issue.1, pp.109-114, 1957.
DOI : 10.1115/1.3640470

H. Tada, P. C. Paris, and G. R. Irwin, The Stress Analysis of Cracks Handbook, 1985.

A. A. Griffith, The Phenomena of Rupture and Flow in Solids, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.221, issue.582-593, pp.163-198, 1921.
DOI : 10.1098/rsta.1921.0006

A. A. Griffith, The theory of rupture, First International Congress of Applied Mechanics, 1924.

E. Orowan, Fracture and strength of solids, Reports on Progress in Physics, vol.12, issue.1, pp.185-232, 1948.
DOI : 10.1088/0034-4885/12/1/309

I. S. Raju, J. C. Newman, and . Jr, Three dimensional finite element analysis of finitethickness fracture specimens. National Aeronautics and Space Administration, 1977.

I. S. Raju, J. C. Newman, and . Jr, Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates, Engineering Fracture Mechanics, vol.11, issue.4, pp.817-829, 1979.
DOI : 10.1016/0013-7944(79)90139-5

S. M. Tavares, P. M. Moreira, S. D. Pastrama, and P. M. De-castro, Stress intensity factors by numerical evaluation in cracked structures, 11 th Portuguese Conference on Fracture, 2008.

N. Couroneau and J. Royer, Simplified model for the fatigue growth analysis of surface cracks in round bars under mode I, International Journal of Fatigue, vol.20, issue.10, pp.711-718, 1998.
DOI : 10.1016/S0142-1123(98)00037-1

N. Couroneau and J. Royer, Simplifying hypotheses for the fatigue growth analysis of surface cracks in round bars, Computers & Structures, vol.77, issue.4, pp.381-389, 2000.
DOI : 10.1016/S0045-7949(00)00030-4

A. B. De-morais, Calculation of stress intensity factors by the force method, Engineering Fracture Mechanics, vol.74, issue.5, pp.739-750, 2007.
DOI : 10.1016/j.engfracmech.2006.06.017

G. V. Guinea, J. Planas, and M. Elices, KI evaluation by the displacement extrapolation technique, Engineering Fracture Mechanics, vol.66, issue.3, pp.243-255, 2000.
DOI : 10.1016/S0013-7944(00)00016-3

X. B. Lin and R. A. Smith, Finite element modelling of fatigue crack growth of surface cracked plates, Engineering Fracture Mechanics, vol.63, issue.5, pp.503-522, 1999.
DOI : 10.1016/S0013-7944(99)00040-5

X. B. Lin and R. A. Smith, Finite element modelling of fatigue crack growth of surface cracked plates, Engineering Fracture Mechanics, vol.63, issue.5, pp.523-540, 1999.
DOI : 10.1016/S0013-7944(99)00041-7

X. B. Lin and R. A. Smith, Finite element modelling of fatigue crack growth of surface cracked plates, Engineering Fracture Mechanics, vol.63, issue.5, pp.541-556, 1999.
DOI : 10.1016/S0013-7944(99)00042-9

R. S. Barsoum, Application of quadratic isoparametric finite elements in linear fracture mechanics, International Journal of Fracture, vol.9, issue.4, p.10, 1974.
DOI : 10.1007/BF00155266

R. D. Henshell and K. G. Shaw, Crack tip finite elements are unnecessary, International Journal for Numerical Methods in Engineering, vol.31, issue.3, pp.495-507, 1975.
DOI : 10.1002/nme.1620090302

G. R. Irwin, Plastic zone near a crack and fracture toughness, Proc. Seventh Sagamore Conf, pp.4-63, 1960.

J. R. Dixon and L. P. Pook, Stress intensity Factors calculated generally by the Finite Element Technique, Nature, vol.3, issue.5215, pp.166-173, 1969.
DOI : 10.1038/224166a0

T. K. Hellen, On the method of virtual crack extensions, International Journal for Numerical Methods in Engineering, vol.85, issue.1, pp.187-207, 1975.
DOI : 10.1002/nme.1620090114

D. M. Parks, A stiffness derivative finite element technique for determination of crack tip stress intensity factors, International Journal of Fracture, vol.187, issue.4, pp.487-502, 1974.
DOI : 10.1007/BF00155252

E. Barati, Y. Alizadeh, and J. A. Mohandesi, J-integral evaluation of austenitic???martensitic functionally graded steel in plates weakened by U-notches, Engineering Fracture Mechanics, vol.77, issue.16, pp.3341-3358, 2010.
DOI : 10.1016/j.engfracmech.2010.08.020

S. Courtin, Propagation de fissures de fatigue dans une géométrie de gorge de vilebrequin en présence de contraintes résiduelles de galetage

S. Courtin, C. Gardin, G. Bézine, and H. Ben-hadj-hamouda, Advantages of the J-integral approach for calculating stress intensity factors when using the commercial finite element software ABAQUS, Engineering Fracture Mechanics, vol.72, issue.14, pp.2174-2185, 2005.
DOI : 10.1016/j.engfracmech.2005.02.003

B. J. Carter, A. Wawrzynek, and A. R. Ingraffea, Universal crack closure integral for SIF estimation, Engineering Fracture Mechanics, vol.60, issue.2, pp.133-146, 1998.

A. C. De-oliveira-miranda, M. A. Meggiolaro, L. F. Martha, T. Pinho-de-castro, and J. , Stress intensity factor predictions: Comparison and round-off error, Computational Materials Science, vol.53, issue.1, pp.354-358, 2012.
DOI : 10.1016/j.commatsci.2011.09.033

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.376-386, 1968.
DOI : 10.1115/1.3601206

B. Moran and C. Shih, A general treatment of crack tip contour integrals, International Journal of Fracture, vol.34, issue.4, pp.295-310, 1987.
DOI : 10.1007/BF00276359

B. Moran and C. F. Shih, Energy release rate along a three dimensional crack front in a thermally stressed body, International Journal of Fracture, vol.30, pp.79-102, 1986.

D. Owen and J. Fa, Engineering fracture of Mechanics: Numerical methods and application, 1983.

J. Lu, Fatigue des alliages ferreux ? Définition et diagrammes. Technique de l'ingénieur

P. Paris, M. Gomez, and W. Anderson, A rational analytic theory of fatigue, Trends Eng, vol.13, pp.9-14, 1961.

P. Paris and F. Erdogan, A Critical Analysis of Crack Propagation Laws, Journal of Basic Engineering, vol.85, issue.4, pp.528-534
DOI : 10.1115/1.3656900

S. Pearson, Initiation of fatigue cracks in commercial aluminium alloys and the subsequent propagation of very short cracks, Engineering Fracture Mechanics, vol.7, issue.2, pp.235-247, 1975.
DOI : 10.1016/0013-7944(75)90004-1

N. E. Frost and J. R. Dixon, A theory of fatigue crack growth, International Journal of Fracture Mechanics, vol.3, issue.4, pp.301-316, 1967.
DOI : 10.1007/BF00182895

S. Suresh and R. O. Ritchie, Propagation of short fatigue cracks, Int. Metal Rev, vol.29, pp.445-476, 1984.

A. J. Mcevily, On the growth of small/short fatigue cracks, JSME Int J, vol.32, pp.181-191, 1989.

K. Minakawa, J. C. Newman, and A. J. Mcevily, A CRITICAL STUDY OF THE CRACK CLOSURE EFFECT ON NEAR-THRESHOLD FATIGUE CRACK GROWTH, Fatigue & Fracture of Engineering Materials and Structures, vol.13, issue.3, pp.359-365, 1983.
DOI : 10.1007/BF00038901

J. L. Breat, F. Mudry, and A. Pineau, SHORT CRACK PROPAGATION AND CLOSURE EFFECTS IN A508 STEEL, Fatigue & Fracture of Engineering Materials and Structures, vol.13, issue.9, pp.349-358, 1983.
DOI : 10.1016/0013-7944(79)90081-X

J. Petit, J. Mendez, W. Berata, L. Legendre, and C. Müller, Influence of the environment on the propagation of short fatigue cracks in a Titanium Alloy, ESIS STP13, Proc. Short Fatigue cracks, pp.235-250, 1992.

J. Petit, Influence of environment on small fatigue crack growth. Small fatigue Cracks: Mechanics, Mechanisms and Applications, pp.167-178, 1999.

M. D. Chapetti, Fatigue propagation threshold of short cracks under constant amplitude loading, International Journal of Fatigue, vol.25, issue.12, pp.1319-1326, 2003.
DOI : 10.1016/S0142-1123(03)00065-3

C. Santus and D. Taylor, Physically short crack propagation in metals during high cycle fatigue, International Journal of Fatigue, vol.31, issue.8-9, pp.1356-1365, 2009.
DOI : 10.1016/j.ijfatigue.2009.03.002

J. Petit and A. Zeghloul, The behaviour of short fatigue crack, pp.163-177, 1986.

U. Krupp, O. Duber, H. Christ, B. Kunkler, P. Koster et al., Propagation mechanisms of microstructurally short cracks???Factors governing the transition from short- to long-crack behavior, Materials Science and Engineering: A, vol.462, issue.1-2, pp.174-177, 2007.
DOI : 10.1016/j.msea.2006.03.159

D. Taylor and J. F. Knott, FATIGUE CRACK PROPAGATION BEHAVIOUR OF SHORT CRACKS; THE EFFECT OF MICROSTRUCTURE, Fatigue & Fracture of Engineering Materials and Structures, vol.2, issue.2, pp.147-155, 1981.
DOI : 10.1016/0013-7944(79)90081-X

W. Elber, Fatigue crack closure under cyclique tension, Engineering Fracture Mechanics, vol.1, pp.163-178, 1970.

S. Suresh, G. F. Zaminski, and R. O. Ritchie, Oxide-Induced Crack Closure: An Explanation for Near-Threshold Corrosion Fatigue Crack Growth Behavior, Metallurgical Transactions A, vol.97, issue.8, pp.1435-1443, 1981.
DOI : 10.1007/BF02643688

G. T. Gray, W. J. Thompson, and A. W. , Roughness-Induced Crack Closure: An Explanation for Microstructurally Sensitive Fatigue Crack Growth, Metallurgical Transactions A, vol.13, issue.2, pp.421-433, 1983.
DOI : 10.1007/BF02644220

J. R. Rice, Mathematical Analysis in the Mechanics of Fracture, chapter 3 of Fracture: an Advanced Treatise, pp.191-311, 1968.

J. R. Rice, The mechanics of crack tip deformation and extension by fatigue. Fatigue crack propagation, ASTM STP, vol.415, pp.247-309, 1967.

R. Mcclung, CRACK CLOSURE AND PLASTIC ZONE SIZES IN FATIGUE, Fatigue & Fracture of Engineering Materials and Structures, vol.982, issue.4, pp.455-468, 1991.
DOI : 10.1016/0022-5096(73)90024-0

W. Elber, The Significance of Fatigue Crack Closure, ASTM STP, vol.486, pp.230-242, 1971.
DOI : 10.1520/STP26680S

N. Walker and C. J. Beevers, A FATIGUE CRACK CLOSURE MECHANISM IN TITANIUM, Fatigue & Fracture of Engineering Materials and Structures, vol.5, issue.1, pp.135-148, 1979.
DOI : 10.1016/0013-7944(72)90074-4

K. Solanki, S. R. Daniewicz, and J. C. Newman-jr, Finite element analysis of plasticity-induced fatigue crack closure: an overview, Engineering Fracture Mechanics, vol.71, issue.2, pp.149-171, 2004.
DOI : 10.1016/S0013-7944(03)00099-7

F. V. Antunes and D. M. Rodrigues, Numerical simulation of plasticity induced crack closure: Identification and discussion of parameters, Engineering Fracture Mechanics, vol.75, issue.10, pp.3101-3120, 2008.
DOI : 10.1016/j.engfracmech.2007.12.009

J. Toribio and V. Kharin, Plasticity-induced crack closure: A contribution to the debate, European Journal of Mechanics - A/Solids, vol.30, issue.2, pp.105-112, 2011.
DOI : 10.1016/j.euromechsol.2010.11.002

N. Jr and J. C. , Finite element analysis of fatigue crack propagation-including the effects of crack closure, 1974.

N. Jr and J. C. , A finite-element analysis of fatigue crack closure, ASTM STP, vol.590, pp.281-301, 1976.

R. G. Chermahini, K. N. Shivakumar, . Newman-jr, and A. F. Blom, Three-Dimensional aspects of plasticity-induced fatigue crack closure, Engineering Fracture Mechanics, vol.34, issue.2, pp.393-401, 1989.
DOI : 10.1016/0013-7944(89)90152-5

R. G. Chermahini, B. Palmberg, and A. F. Blom, Fatigue crack growth and closure behaviour of semicircular and semi-elliptical surface flaws, International Journal of Fatigue, vol.15, issue.4, pp.259-63, 1993.
DOI : 10.1016/0142-1123(93)90374-Y

P. F. De-matos and D. Nowell, Numerical simulation of plasticity-induced fatigue crack closure with emphasis on the crack growth scheme: 2D and 3D analyses, Engineering Fracture Mechanics, vol.75, issue.8, pp.2087-2114, 2008.
DOI : 10.1016/j.engfracmech.2007.10.017

D. F. Sarzosa, B. L. Godefroid, and C. Ruggieri, Fatigue crack growth assessments in welded components including crack closure effects: Experiments and 3-D numerical modeling, International Journal of Fatigue, vol.47, pp.279-291, 2013.
DOI : 10.1016/j.ijfatigue.2012.09.009

F. V. Antunes, D. M. Rodrigues, and R. Branco, An analytical model of plasticity induced crack closure, Procedia Engineering, vol.2, issue.1, pp.1005-1014, 2010.
DOI : 10.1016/j.proeng.2010.03.109

S. Simandjuntak, H. Alizadeh, M. J. Pavier, and D. J. Smith, Fatigue crack closure of a corner crack: A comparison of experimental results with finite element predictions, International Journal of Fatigue, vol.27, issue.8, pp.914-919, 2005.
DOI : 10.1016/j.ijfatigue.2004.11.008

H. Alizadeh, S. Simandjuntak, D. Smith, and M. Pavier, Prediction of fatigue crack growth rates using crack closure finite element analysis, International Journal of Fatigue, vol.29, issue.9-11, pp.1711-1715, 2007.
DOI : 10.1016/j.ijfatigue.2006.12.002

A. Gonzalez-herrera and J. Zapatero, Influence of minimum element size to determine crack closure stress by the finite element method, Engineering Fracture Mechanics, vol.72, issue.3, pp.337-355, 2005.
DOI : 10.1016/j.engfracmech.2004.04.002

A. Gonzalez-herrera and J. Zapatero, Tri-dimensional numerical modelling of plasticity induced fatigue crack closure, Engineering Fracture Mechanics, vol.75, issue.15, pp.4513-4528, 2008.
DOI : 10.1016/j.engfracmech.2008.04.024

H. Alizadeh, D. A. Hills, P. F. De-matos, D. Nowell, M. J. Pavier et al., A comparison of two and three-dimensional analyses of fatigue crack closure, International Journal of Fatigue, vol.29, issue.2, pp.222-231, 2007.
DOI : 10.1016/j.ijfatigue.2006.03.014

R. Branco, F. V. Antunes, and R. F. Martins, Modelling fatigue crack propagation in CT specimens, Fatigue & Fracture of Engineering Materials & Structures, vol.11, issue.6, pp.452-465, 2008.
DOI : 10.1111/j.1460-2695.2008.01241.x

R. Branco, D. M. Rodrigues, and F. V. Antunes, Influence of through-thickness crack shape on plasticity induced crack closure, Fatigue & Fracture of Engineering Materials and Structures, vol.50, issue.2, pp.209-220, 2008.
DOI : 10.1111/j.1460-2695.1992.tb00032.x

R. Branco and F. V. Antunes, Finite element modelling and analysis of crack shape evolution in mode-I fatigue Middle Cracked Tension specimens, Engineering Fracture Mechanics, vol.75, issue.10, pp.3020-3037, 2008.
DOI : 10.1016/j.engfracmech.2007.12.012

S. Simandjuntak, H. Alizadeh, D. J. Smith, and M. J. Pavier, Three dimensional finite element prediction of crack closure and fatigue crack growth rate for a corner crack, International Journal of Fatigue, vol.28, issue.4, pp.335-345, 2006.
DOI : 10.1016/j.ijfatigue.2005.07.034

J. D. Dougherty, J. Padovan, and T. S. Srivatsan, Fatigue crack propagation and closure behavior of modified 1070 steel: Finite element study, Engineering Fracture Mechanics, vol.56, issue.2, pp.189-212, 1997.
DOI : 10.1016/S0013-7944(96)00104-X

S. Park, Y. Earmme, and J. Song, DETERMINATION OF THE MOST APPROPRIATE MESH SIZE FOR A 2-D FINITE ELEMENT ANALYSIS OF FATIGUE CRACK CLOSURE BEHAVIOUR, Fatigue & Fracture of Engineering Materials & Structures, vol.15, issue.4, pp.533-545, 1997.
DOI : 10.1111/j.1460-2695.1997.tb00285.x

S. Roychowdhury and R. H. Dodds, Three-dimensional effects on fatigue crack closure in the small-scale yielding regime - a finite element study, Fatigue <html_ent glyph="@amp;" ascii="&"/> Fracture of Engineering Materials and Structures, vol.15, issue.8, pp.663-673, 2003.
DOI : 10.1016/S0921-5093(00)00971-0

R. C. Mcclung, B. H. Thacker, and S. Roy, Finite element visualization of fatigue crack closure in plane stress and plane strain, International Journal of Fracture, vol.50, issue.1, pp.27-49, 1991.

R. C. Mcclung, On the finite element analysis of fatigue crack closure???1. Basic modeling issues, Engineering Fracture Mechanics, vol.33, issue.2, pp.237-252, 1989.
DOI : 10.1016/0013-7944(89)90027-1

R. C. Mcclung, On the finite element analysis of fatigue crack closure???2. Numerical results, Engineering Fracture Mechanics, vol.33, issue.2, pp.253-272, 1989.
DOI : 10.1016/0013-7944(89)90028-3

K. N. Solanki, Two and three-dimensional finite element analysis of plasticity induced crack closure-a comprehensive parametric study, 2002.

J. D. Skinner and S. R. Danieiwicz, Simulation of plasticity-induced fatigue crack closure in part-through cracked geometries using finite element analysis, Engineering Fracture Mechanics, vol.69, issue.1, pp.1-11
DOI : 10.1016/S0013-7944(01)00115-1

N. Jr and J. C. , A crack opening stress equation for fatigue crack growth, International journal of Fracture, vol.24, pp.131-135, 1984.

N. Jr, J. C. Ruschau, and J. J. , The stress level effect on fatigue-crack growth under constant-amplitude loading, International Journal of Fatigue, vol.29, pp.1608-1615, 2007.

D. S. 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

N. Jr and J. C. , A crack closure model for predicting fatigue crack growth under aircraft spectrum loadings. Methods and Models for predicting fatigue crack growth under random loading, ASTM STP, vol.748, pp.53-84, 1981.

J. Liu, P. Du, X. Liu, and Q. Du, Modelling of Fatigue crack growth Closure considering the Integrative Effect of Cyclic Stress Ratio, Specimen Thickness and Poisson's Ratio, Chinese Journal of Mechanical Engineering, 2012.

X. P. Huang, M. Torgeir, and W. C. Cui, An engineering model of fatigue crack growth under variable amplitude loading, International Journal of Fatigue, vol.30, issue.1, pp.2-10, 2008.
DOI : 10.1016/j.ijfatigue.2007.03.004

J. Codrington and A. Kotousov, A crack closure model of fatigue crack growth in plates of finite thickness under small-scale yielding conditions, Mechanics of Materials, vol.41, issue.2, pp.165-173, 2009.
DOI : 10.1016/j.mechmat.2008.10.002

L. Minh, B. Maitournam, M. H. Doquet, and V. , A cyclic steady-state method for fatigue crack propagation: Evaluation of plasticity-induced crack closure in 3D
URL : https://hal.archives-ouvertes.fr/hal-00725063

J. C. Amazigo and J. W. Hutchinson, Crack-tip fields in steady crack-growth with linear strain-hardening, Journal of the Mechanics and Physics of Solids, vol.25, issue.2, pp.81-97, 1977.
DOI : 10.1016/0022-5096(77)90005-9

Q. S. Nguyen and M. Rahimian, Mouvement permanent d'une fissure en milieu elastoplastique, Journal of Mechanics and applications, vol.5, pp.95-120, 1981.
URL : https://hal.archives-ouvertes.fr/hal-00105515

K. Roe and T. Siegmund, An irreversible cohesive zone model for interface fatigue crack growth simulation, Engineering Fracture Mechanics, vol.70, issue.2, pp.209-232, 2003.
DOI : 10.1016/S0013-7944(02)00034-6

J. Bouvard, J. Chaboche, F. Feyel, and F. Gallerneau, A cohesive zone model for fatigue and creep???fatigue crack growth in single crystal superalloys, International Journal of Fatigue, vol.31, issue.5, pp.868-879, 2009.
DOI : 10.1016/j.ijfatigue.2008.11.002

P. F. De-matos, D. Nowell, and D. , On the accurate assessment of crack opening and closing stresses in plasticity-induced fatigue crack closure problems, Engineering Fracture Mechanics, vol.74, issue.10, pp.1579-1601, 2007.
DOI : 10.1016/j.engfracmech.2006.09.007

L. P. Borrego, F. V. Antunes, J. D. Costa, and J. M. Ferreira, Numerical simulation of plasticity induced crack closure under overloads and high???low blocks, Engineering Fracture Mechanics, vol.95, pp.57-71, 2012.
DOI : 10.1016/j.engfracmech.2012.07.016

J. Z. Zhang and P. Bowen, On the finite element simulation of three-dimensional semi-circular fatigue crack growth and closure, Engineering Fracture Mechanics, vol.60, issue.3, pp.341-360, 1998.
DOI : 10.1016/S0013-7944(98)00016-2

J. D. Skinner, Finite element analysis of plasticity-induced fatigue crack closure in three-dimensional cracked geometries, 2001.

H. Dill and C. R. Saff, Spectrum Crack Growth Prediction Method Based on Crack Surface Displacement and Contact Analyses, ASTM STP 595, pp.306-319, 1976.
DOI : 10.1520/STP33381S

H. Sehitoglu and W. Sun, Modeling of Plane Strain Fatigue Crack Closure, Journal of Engineering Materials and Technology, vol.113, issue.1, pp.31-40, 1991.
DOI : 10.1115/1.2903380

W. Sun and H. Sehitoglu, RESIDUAL STRESS FIELDS DURING FATIGUE CRACK GROWTH, Fatigue & Fracture of Engineering Materials and Structures, vol.45, issue.2, pp.115-128, 1992.
DOI : 10.1007/BF02320416

J. Wu and F. Ellyin, A study of fatigue crack closure by elastic-plastic finite element analysis for constant-amplitude loading, International Journal of Fracture, vol.17, issue.1, pp.43-65, 1996.
DOI : 10.1007/BF00017863

L. W. Wei and M. N. James, A study of fatigue crack closure in polycarbonate CT specimens, Engineering Fracture Mechanics, vol.66, issue.3, pp.223-242, 2000.
DOI : 10.1016/S0013-7944(00)00014-X

M. Kikukawa, M. Jono, and S. Mikami, Fatigue crack propagation and crack closure behaviour under stationary vary loading-test results of Aluminium alloy, Journal of Society on Materials, vol.31, pp.438-487, 1982.

H. F. Bueckner, A novel principle for the computation of stress intensity factors, Journal of Applied Mathematics, vol.50, pp.529-574, 1970.

S. Pommier, A study of the relationship between variable level fatigue crack growth and the cyclic constitutive behaviour of steel, International Journal of Fatigue, vol.23, issue.1, pp.111-118, 2001.
DOI : 10.1016/S0142-1123(01)00165-7

K. Vor, Etude expérimentale et modélisation numérique de la fermeture de fissures longues et courtes dans un acier inoxydable 304L, Thèse ENSMA, 2009.

P. Chea, Etude numérique de la fermeture de fissures dans un acier inoxydable 304L, 2010.

J. Lemaitre and C. J. , Mécanique des matériaux solides, 2001.

K. Vor, C. Gardin, C. Sarrazin-baudoux, and J. Petit, Wake length and loading history effects on crack closure of through-thickness long and short cracks in 304L: Part I ??? Experiments, Engineering Fracture Mechanics, vol.99, pp.266-277, 2013.
DOI : 10.1016/j.engfracmech.2013.01.003

K. Vor, C. Sarrazin-baudoux, C. Gardin, C. Petit, and J. , Wake history effect on closure of short and long fatigue crack in 304L stainless steel, Procedia Engineering, vol.2, issue.1, pp.2327-2336, 2010.
DOI : 10.1016/j.proeng.2010.03.249

K. Vor, C. Gardin, C. Sarrazin-baudoux, and J. Petit, Wake length and loading history effects on crack closure of through-thickness long and short cracks in 304L: Part II ??? 3D numerical simulation, Engineering Fracture Mechanics, vol.99, pp.306-323, 2013.
DOI : 10.1016/j.engfracmech.2013.01.014

M. Arzaghi, C. Gardin, P. Chea, K. Vor, and C. Sarrazin-baudoux, Modélisation sous Abaqus de la fermeture de fissures courtes dans un acier inoxydable 304L. Congrès Français de Mécanique, 2011.

A. Sbitti, Propagation des fissures 2D et 3D planes sous chargement thermomécaniques à amplitude variables, Thèse Université Pierre et Marie Curie, 2009.

D. Camas, J. Garcia-manrique, and A. Gonzalez-herrera, Numerical study of the thickness transition in bi-dimensional specimen cracks, International Journal of Fatigue, vol.33, issue.7, pp.921-928, 2011.
DOI : 10.1016/j.ijfatigue.2011.02.006

J. Garcia-manrique, D. Camas, P. Lopez-crespo, and A. Gonzalez-herrera, Stress intensity factor analysis of through thickness effects, International Journal of Fatigue, vol.46, pp.58-66, 2013.
DOI : 10.1016/j.ijfatigue.2011.12.012

D. Camas, J. Garcia-manrique, and A. Gonzalez-herrera, Crack front curvature: Influence and effects on the crack tip fields in bi-dimensional specimens, International Journal of Fatigue, vol.44, pp.41-50, 2012.
DOI : 10.1016/j.ijfatigue.2012.05.012

N. A. Fleck, Finite element analysis of plasticity-induced crack closure under plane strain conditions, Engineering Fracture Mechanics, vol.25, issue.4, pp.441-449, 1986.
DOI : 10.1016/0013-7944(86)90258-4

Z. Ilievski, Boundary element method (BEM), 2006.

C. Bellis, Méthode des Eléments de Frontière Accélérée en Mécanique de la Rupture 3D, Master 2 Mathématiques et Applications, Laboratoire Jacques-Louis Lions, 2008.

A. I. Wanderlingh, Technical Note: Comparison of boundary element and finite element methods for linear stress analysis???technical program results, Engineering Analysis, vol.3, issue.3, pp.177-180, 1986.
DOI : 10.1016/0264-682X(86)90057-2

D. Danson, C. A. Brebbia, and R. A. Adey, The BEASY system, Advances in Engineering Software (1978), vol.4, issue.2, pp.68-74, 1982.
DOI : 10.1016/S0141-1195(82)80056-9

A. C. Neves, R. A. Adey, W. Baynham, and S. M. Niku, Automatic 3D Crack Growth using BEASY, Computational Mechanics BEASY

N. Moes, J. Dolbow, and T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering, vol.46, issue.1, pp.131-150, 1999.
DOI : 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.3.CO;2-A
URL : https://hal.archives-ouvertes.fr/hal-01004829

H. Bayesteh and S. Mohammadi, XFEM fracture analysis of shells: The effect of crack tip enrichments, Computational Materials Science, vol.50, issue.10, pp.2793-2813, 2011.
DOI : 10.1016/j.commatsci.2011.04.034

P. Laborde, J. Pommier, Y. Renard, and M. Salaün, Méthodes XFEM d'ordre supérieur en mécanique de la rupture, pp.45-57, 2002.

A. Alizada and T. Fries, Simulation of cracks with XFEM and hanging nodes, International Conference on Extended Finite Element Methods ? Recent Developments and Applications XFEM, 2009.

E. Giner, N. Sukumar, J. E. Tarancon, and F. J. Fuenmayor, An Abaqus implementation of the extended finite element method, Engineering Fracture Mechanics, vol.76, issue.3, pp.347-368, 2009.
DOI : 10.1016/j.engfracmech.2008.10.015

J. Shi, D. Chopp, J. Lua, N. Sukumar, and T. Belytschko, Abaqus implementation of extended finite element method using a level set representation for three-dimensional fatigue crack growth and life predictions, Engineering Fracture Mechanics, vol.77, issue.14, pp.2840-2863, 2010.
DOI : 10.1016/j.engfracmech.2010.06.009

A. Philippe, Etude de la propagation de fissures dans un noeud MarkIII

J. C. Morano, Étude sous Code_Aster de la propagation d'une fissure dans une éprouvette CT

J. Hou, M. Goldstraw, S. Maan, and M. Knop, An Evaluation of 3D Crack Growth Using ZENCRACK. Airframes and Engines Division Aeronautical and Maritime Research Laboratory, 1158.

A. R. Maligno, S. Rajaratnam, S. B. Leen, and E. J. Williams, A three-dimensional (3D) numerical study of fatigue crack growth using remeshing techniques, Engineering Fracture Mechanics, vol.77, issue.1, pp.94-111, 2010.
DOI : 10.1016/j.engfracmech.2009.09.017

S. Pommier, Cyclic plasticity and variable amplitude fatigue, International Journal of Fatigue, vol.25, issue.9-11, pp.9-11983, 2003.
DOI : 10.1016/S0142-1123(03)00137-3

L. Taleb and G. Cailletaud, Cyclic accumulation of the inelastic strain in the 304L SS under stress control at room temperature: Ratcheting or creep?, International Journal of Plasticity, vol.27, issue.12, pp.1936-1958, 2011.
DOI : 10.1016/j.ijplas.2011.02.001
URL : https://hal.archives-ouvertes.fr/hal-00652972

A. Standards, Metals test methods and analytical procedures, 1993.

Z. P. Bazant and L. Et-estenssoro, Surface singularity and crack propagation, International Journal of Solids and Structures, vol.15, issue.5, pp.405-426, 1979.
DOI : 10.1016/0020-7683(79)90062-3

M. Sevcik, P. Hutar, M. Zouhar, and L. Nahlik, Numerical estimation of the fatigue crack front shape for a specimen with finite thickness, International Journal of Fatigue, vol.39, pp.75-80, 2012.
DOI : 10.1016/j.ijfatigue.2011.03.010

L. P. Pook, Finite element analysis of corner point displacements and stress intensity factors for narrow notches in square sheets and plates. Fatigue Fract

J. J. Newman and W. Elber, Mechanics of Fatigue Crack Closure, ASTM STP 982, 1988.

J. C. Newman, . Jr, and I. S. Raju, An empirical stress-intensity factor equation for the surface crack, Engineering Fracture Mechanics, vol.15, issue.1-2, pp.185-192, 1981.
DOI : 10.1016/0013-7944(81)90116-8

P. Yu and W. Guo, An equivalent thickness conception for prediction of surface fatigue crack growth life and shape evolution, Engineering Fracture Mechanics, vol.93, pp.65-74, 2012.
DOI : 10.1016/j.engfracmech.2012.06.008

C. Y. Hou, Simultaneous simulation of closure behavior and shape development of fatigue surface cracks, International Journal of Fatigue, vol.30, issue.6, pp.1036-1046, 2008.
DOI : 10.1016/j.ijfatigue.2007.08.020

C. Y. Hou, Simulation of surface crack shape evolution using the finite element technique and considering the crack closure effects, International Journal of Fatigue, vol.33, issue.5, pp.719-726, 2011.
DOI : 10.1016/j.ijfatigue.2010.11.022

Z. Wu, The shape of a surface crack in a plate based on a given stress intensity factor distribution, International Journal of Pressure Vessels and Piping, vol.83, issue.3, pp.168-180, 2006.
DOI : 10.1016/j.ijpvp.2006.01.004

B. R. Davis, P. A. Wawrzynek, and A. Ingraffea, 3-D simulation of arbitrary crack growth using an energy-based formulation ??? Part I: Planar growth, Engineering Fracture Mechanics, vol.115, pp.204-220, 2014.
DOI : 10.1016/j.engfracmech.2013.11.005

C. F. Shih and R. J. Asaro, Elastic-Plastic Analysis of Cracks on Bimaterial Interfaces: Part I???Small Scale Yielding, Journal of Applied Mechanics, vol.55, issue.2, pp.299-316, 1988.
DOI : 10.1115/1.3173676

M. Akamatsu and C. E. , Caractérisation chimique et mécanique de matériaux approvisionnés pour l'étude du comportement en fatigue des aciers inoxydables austénitiques, Note technique interne, 2001.

S. Petitjean, Influence de l'état de surface sur le comportement en fatigue à grande nombre de cycle de l'acier inoxydable 304L, Thèse de doctorat, 2003.

K. Levenberg, A method for the solution of certain non-linear problems in least squares, Quarterly of Applied Mathematics, vol.2, issue.2, pp.164-168, 1944.
DOI : 10.1090/qam/10666

D. W. Marquardt, An Algorithm for Least-Squares Estimation of Nonlinear Parameters, Journal of the Society for Industrial and Applied Mathematics, vol.11, issue.2, pp.431-441, 1963.
DOI : 10.1137/0111030

J. J. Moré, The Levenberg-Marquardt algorithm: Implementation and theory, Lecture Notes in Mathematics, vol.11, pp.105-116, 1978.
DOI : 10.1137/0111030