P. Ballard, D. Van, K. Deperrois, A. Papadopoulos, and Y. V. , HIGH CYCLE FATIGUE AND A FINITE ELEMENT ANALYSIS, Fatigue & Fracture of Engineering Materials and Structures, vol.2, issue.4, pp.397-411, 1995.
DOI : 10.1016/0093-6413(77)90007-6

A. Banvillet, T. Palin-luc, and S. Lasserre, A volumetric energy based high cycle multiaxial fatigue citerion, International Journal of Fatigue, vol.25, issue.8, pp.755-769, 2003.
DOI : 10.1016/S0142-1123(03)00048-3

URL : https://hal.archives-ouvertes.fr/hal-01372060

G. Baudry, Tenue en service des composants mécaniques ? quelle endurance pour quelle pièce et quelle procédé, compte rendu des 22 e Journées de Printemps, Fatigue et procédés de fabrication, 2003.

S. Beretta, A. Blarasin, M. Endo, T. Giunti, and Y. Murakami, Defect tolerant design of automotive components, International Journal of Fatigue, vol.19, issue.4, pp.319-333, 1997.
DOI : 10.1016/S0142-1123(96)00079-5

S. Beretta and Y. Murakami, SIF and threshold for small cracks at small notches under torsion, Fatigue <html_ent glyph="@amp;" ascii="&"/> Fracture of Engineering Materials and Structures, vol.23, issue.2, pp.97-104, 2000.
DOI : 10.1046/j.1460-2695.2000.00260.x

A. Bernasconi and I. V. Papadopoulos, Efficiency of algorithms for shear stress amplitude calculation in critical plane class fatigue criteria, Computational Materials Science, vol.34, issue.4, 2005.
DOI : 10.1016/j.commatsci.2005.01.005

T. Billaudeau, Fatigue multiaxiale des matériaux à défauts: mécanismes et critère d'endurance, 2002.

H. D. Bui, Etude de l'évolution de la frontière du domaine élastique avec l'écrouissage et relations de comportement élasto-plastique des métaux cubiques, Thèse de doctorat ès-sciences physiques, 1969.

F. Cano, A. Constantinescu, and H. Maitournam, Crit??re de fatigue polycyclique pour des mat??riaux anisotropes??: application aux monocristaux, Comptes Rendus M??canique, vol.332, issue.2, pp.115-124, 2004.
DOI : 10.1016/j.crme.2003.11.005

J. L. Chaboche and O. Jung, Application of a kinematic hardening viscoplasticity model with thresholds to the residual stress relaxation, International Journal of Plasticity, vol.13, issue.10, pp.785-807, 1998.
DOI : 10.1016/S0749-6419(97)00066-1

M. D. Chapetti, T. Kitano, T. Tagawa, and T. Miyata, Two small-crack extension force concept applied to fatigue limit of blunt notched components., International Journal of Fatigue, vol.21, issue.1, pp.77-82, 1991.
DOI : 10.1016/S0142-1123(98)00057-7

T. Coupez, A mesh improvement method for 3D automatic remeshing

. Weatherill, editor, Numerical Grid Generation in Computational Fluid Dynamics and Related Fields, pp.615-626, 1994.

T. Coupez, Stable stabilized finite element for 3D forming calculation, 1995.

T. Coupez, Parallel adaptative remeshing in 3D moving mesh finite element, Numerical Grid Generation in Computational Field Simulations, pp.783-792, 1996.

J. Courbon, A. Vincent, G. Lormand, P. Daguier, and D. Girodin, Influence du fibrage sur les propriétés de fatigue des aciers à roulement, 2003.

B. Crossland, Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel, Proceedings of the International Conference on Fatigue of Metals, 1956.

D. Van, K. Griveau, B. Message, and O. , On a new multiaxial fatigue limit criterion : Theorie and application, Biaxial and multiaxial fatigue, pp.479-496, 1989.

F. De-k70-]-de-kazinczy, Effect of small defects on the fatigue properties of mediumstrength cast steel, Journal of the Iron and Steel Institute, pp.851-855, 1970.

J. Demurger, R. Forestier, B. Kieber, and P. Lasne, Deep rolling process simulation: impact of kinematic hardening on residual stresses, 2006.

A. Deperrois, Sur le calcul de limites d'endurance des aciers, Thèse de Doctorat, 1991.

D. Mcdiarmid, A GENERAL CRITERION FOR HIGH CYCLE MULTIAXIAL FATIGUE FAILURE, Fatigue & Fracture of Engineering Materials and Structures, vol.853, issue.65, pp.429-53, 1991.
DOI : 10.1016/0020-7403(70)90091-3

E. Haddad, M. H. Topper, T. H. Smith, and K. N. , Prediction of non propagating cracks, Engineering Fracture Mechanics, vol.11, issue.3, pp.573-584, 1979.
DOI : 10.1016/0013-7944(79)90081-X

M. Emmerich, A. Giotis, M. Özdemir, T. Bäck, and K. Giannakoglou, Metamodelassisted evolution strategies, Parallel Problem Solving from Nature VII, Proc. Inte'I Conf, 2002.

M. Endo and Y. Murakami, Effects of an Artificial Small Defect on Torsional Fatigue Strength of Steels, Journal of Engineering Materials and Technology, vol.109, issue.2, pp.123-129, 1987.
DOI : 10.1115/1.3225951

M. Endo, Effects of Artificial Small Defect and Graphite on Torsional Fatigue Strength of Ductile Cast Iron., Journal of the Society of Materials Science, Japan, vol.45, issue.1, pp.16-20, 1996.
DOI : 10.2472/jsms.45.16

M. Endo, Effect of a small defect on the fatigue strength of annealed JIS-S35C steel under combined axial-torsional stress, Proceedings JSME Conference, pp.357-358, 1998.

A. Ekberg and P. Sotkovszki, Anisotropy and rolling contact fatigue of railway wheels, International Journal of Fatigue, vol.23, issue.1, pp.29-43, 2001.
DOI : 10.1016/S0142-1123(00)00070-0

W. N. Findley, A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending, Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry, Series B, vol.81, 1959.

J. F. Flavenot and N. Skalli, L'épaisseur de la couche critique ou une nouvelle approche du calcul en fatigue des structures soumises à des sollicitations multiaxiales, pp.15-25, 1983.

J. F. Flavenot and J. Lu, Prediction of the residual stress relaxation during fatigue loading and taking the residual-stress in multiaxial fatigue criteria into account, Proceedings of 2 nd international conferenceon residual stress, pp.784-790, 1988.

R. Forestier, Développement d'une méthode d'identification de paramètres par analyse inverse couplée avec un modèle éléments finis 3D, 2004.

B. Gärtener, Smallest Enclosing Balls a Fast and Robust C++ Implementation, 1999.

M. Gaucheron, Mise en forme de l'acier par estampage, Techniques de l'Ingénieur, traité M3200, 1998.

Y. Germain, Modélisation par éléments finis d'écoulement viscoplastique avec frottement, Thèse de Doctorat, 1985.

H. J. Gough and H. V. Pollard, The strength of metals under combined alternating stresses, Proceedings of the Institution of Mechanical Engineers, 1935.
DOI : 10.1243/PIME_PROC_1935_131_008_02

F. Guiu and R. N. Stevens, THE NUCLEATION AND GROWTH OF SHORT FATIGUE CRACKS BOTH AT PLAIN SURFACES AND NOTCHES, Fatigue & Fracture of Engineering Materials and Structures, vol.5, issue.6, pp.625-635, 1990.
DOI : 10.1016/0001-6160(85)90241-X

. Hankins-becker, Further experiments on the effect of surface condition on the fatigue resistance of steels, J. Iron Steel Institute, vol.133, pp.399-425, 1936.

J. E. Hoffmann, D. Löhe, and E. Macherauch, Influence of machining residual stresses on the bending fatigue behaviour of notched specimens of Ck 45 in different heat treating states, 1986.

T. H. Hopp and C. P. Reeve, An algorithm for computing the minimum covering sphere in any dimension, 1996.

J. Husson, Pratique du forgeage, Pyc Edition, 1975.

G. Inglebert, T. Hassine, E. Legrand, and G. Louradour, Adaptation et fatigue de composants m??caniques sous chargement complexe, M??canique & Industries, vol.1, issue.6, pp.603-608, 2000.
DOI : 10.1016/S1296-2139(00)01069-1

J. M. Jachmich, Tenue en service des bielles: influence de la peau de forge, 2004.

M. Kage, K. J. Miller, and R. A. Smith, FATIGUE CRACK INITIATION AND PROPAGATION IN A LOW-CARBON STEEL OF TWO DIFFERENT GRAIN SIZES, Fatigue & Fracture of Engineering Materials and Structures, vol.10, issue.8, pp.763-774, 1992.
DOI : 10.1016/0020-7403(77)90011-X

A. Kuntsevich and F. Kappel, SolvOpt the Solver for Local Nonlinar Optimization Problems, 1997.

C. D. Liu, M. N. Bassim, S. Lawrence, and . St, Dependence of the fatigue limit of rail steels on stress intensity factor near inclusions, Engineering Fracture Mechanics, vol.50, issue.2, pp.301-307, 1995.
DOI : 10.1016/0013-7944(94)00197-P

J. Lu, Fatigue des alliages ferreux ? Facteurs d'influence, Techniques de l'Ingénieur, traité BM 5043, 2002.

E. Mamiya and J. Araujo, Fatigue limit under multiaxial loadings: on the definition of the equivalent shear stress, Mechanics Research Communications, n° 29, pp.141-151, 2001.

G. Marquis and D. Socie, Long-life torsion fatigue with normal mean stresses, Fatigue <html_ent glyph="@amp;" ascii="&"/> Fracture of Engineering Materials and Structures, vol.52, issue.4, pp.293-300, 2000.
DOI : 10.1046/j.1460-2695.1998.00459.x

T. Matake, Fatigue criterion for notched and unnotched specimens under combined stress state, pp.993-997, 1980.

M. R. Mitchell, Review of the Mechanical Properties of Cast Steels With Emphasis on Fatigue Behavior and the Influence of Microdiscontinuities, Journal of Engineering Materials and Technology, vol.99, issue.4, pp.99-329, 1977.
DOI : 10.1115/1.3443549

K. Mocellin, Contribution à la simulation numérique tridimensionnelle du forgeage à chaud : étude du contact et calcul multigrille, 1999.

F. Morel, A critical plane approach for life prediction oh high cycle fatigue under multiaxial variable amplitude loading, Int. J. Fatigue, vol.22, pp.2-101, 2000.

Y. Murakami and M. Endo, Quantitative evaluation of fatigue strength of metals containing various small defects or cracks, Engineering Fracture Mechanics, vol.17, issue.1, pp.1-15, 1983.

Y. Murakami, M. Endo, . K. Ed, E. R. Miller, and . De-los-rios, The behaviour of short fatigue cracks, pp.275-293, 1986.

Y. Murakami, S. Kodomas, and S. Konuma, Quantitative evaluation of effects of non-metallic inclusions on fatigue strength of high strength steels. I: Basic fatigue mechanism and evaluation of correlation between the fatigue fracture stress and the size and location of non-metallic inclusions, International Journal of Fatigue, vol.11, issue.5, pp.291-298, 1989.
DOI : 10.1016/0142-1123(89)90054-6

Y. Murakami, Y. Uemura, Y. Natsume, and S. Miyakawa, Effect of mean stress on the fatigue strength of high-strength steels containing small defects or nonmetallic inclusions., Transactions of the Japan Society of Mechanical Engineers Series A, vol.56, issue.525, p.1074, 1990.
DOI : 10.1299/kikaia.56.1074

Y. Murakami, K. Kawakami, and W. E. Duckworth, Quantitative evaluation of effects of shape and size of artificially introduced alumina particles on the fatigue strength of 1.5Ni???Cr???Mo (En24) steel, International Journal of Fatigue, vol.13, issue.6, pp.489-499, 1991.
DOI : 10.1016/0142-1123(91)90485-H

Y. Murakami and M. Endo, Effects of defects, inclusions and inhomogeneities on fatigue strength, International Journal of Fatigue, vol.16, issue.3, 1994.
DOI : 10.1016/0142-1123(94)90001-9

Y. Murakami, K. Tahahashi, M. Takada, and T. Toryiama, Quantitative Evaluation of Effect of Artificial Small Defects on Torsional Fatigue Strength., Transactions of the Japan Society of Mechanical Engineers Series A, vol.64, issue.618, pp.271-277, 1998.
DOI : 10.1299/kikaia.64.271

O. Orowan, Theory of the Fatigue of Metals, Proceedings of the Royal Academy, 1939.
DOI : 10.1098/rspa.1939.0055

L. Oulladji, A. Janka, J. A. Désidéri, and A. Dervieux, Optimisation aérodynamique par algorithmes génétiques hybrides: application à la réduction d'un critère de bang sonique, 2003.

T. Palin-luc and S. Lasserre, An energy based criterion for high cycle multiaxial fatigue, European Journal of Mechanics - A/Solids, vol.17, issue.2, pp.237-251, 1998.
DOI : 10.1016/S0997-7538(98)80084-3

URL : https://hal.archives-ouvertes.fr/hal-01376427

I. V. Papadopoulos, Fatigue polycyclique des métaux: une nouvelle approche, Thèse de Doctorat, Ecole Nationale des Ponts et Chaussées, 1987.

I. V. Papadopoulos, A new criterion of fatigue strength for out-of-phase bending and torsion of hard metals, International Journal of Fatigue, vol.16, issue.6, pp.337-384, 1994.
DOI : 10.1016/0142-1123(94)90449-9

I. V. Papadopoulos and V. P. Panoskaltsis, Invariant formulation of a gradient dependent multiaxial high-cycle fatigue criterion, Engineering Fracture Mechanics, vol.55, issue.4, pp.513-528, 1996.
DOI : 10.1016/S0013-7944(96)00047-1

I. V. Papadopoulos, Long life fatigue under multiaxial loading, International Journal of Fatigue, vol.23, issue.10, pp.839-849, 2001.
DOI : 10.1016/S0142-1123(01)00059-7

I. V. Papadopoulos, P. Davoli, A. Bernasconi, M. Filippini, and S. Foletti, Independence of the torsional fatigue limit upon a mean shear stress, International Journal of Fatigue, vol.25, issue.6, pp.471-480, 2003.

J. S. Park, S. J. Kim, K. H. Kim, S. H. Park, and C. S. Lee, A microstructural model for predicting high cycle fatigue life of steels, International Journal of Fatigue, vol.27, issue.9, 2005.
DOI : 10.1016/j.ijfatigue.2005.01.013

R. E. Peterson, Metal Fatigue, pp.193-306, 1959.

A. Pokorny and J. Pokorny, Inclusions non métalliques dans l'acier, Techniques de l'Ingénieur, traité M220, 1985.

A. Pokorny and J. Pokorny, Action du corroyage sur la structure de coulée des aciers, 1997.

P. Rabbe, H. P. Lieurade, and A. Galtier, Essais da fatigue. Partie I, techniques de l'Ingénieur, traité M4170, 2000.

J. Robert, Contribution à l'étude de la fatigue multiaxiale sous sollicitation périodiques ou aléatoires, Thèse de Doctorat, 1992.

G. Savaidis, A. Savaidis, G. Tsamasphyros, and C. Zhang, On size and technological effects in fatigue analysis and prediction of engineering materials and components, International Journal of Mechanical Sciences, vol.44, issue.3, pp.521-543, 2002.
DOI : 10.1016/S0020-7403(01)00105-9

G. G. Sines, J. L. Sines, and . Waisman, Metal Fatigue, Mc-Graw-Hill, pp.145-149, 1959.

G. Sines, Fatigue Criteria Under Combined Stresses or Strains, Journal of Engineering Materials and Technology, vol.103, issue.2, pp.82-90, 1981.
DOI : 10.1115/1.3224995

R. K. Singh, A. K. Singh, E. Prassad, and N. , Texture and mechanical properties anisotropy in Al-Mg-Si-Cu alloy, Material Science and Engineering A227 Experimental measurement and finite element simulation of the interaction between residual stresses and mechanical loading, Internationl Journal of Fatigue, vol.23, pp.114-122, 2000.

R. A. Smith and K. J. Miller, Prediction of fatigue regimes in notched components, International Journal of Mechanical Sciences, vol.20, issue.4, pp.201-206, 1978.
DOI : 10.1016/0020-7403(78)90082-6

M. A. Torres and H. J. Voorwald, An evaluation of shot peening, residual stress and stress relaxation on the fatigue life of AISI 4340 steel, International Journal of Fatigue, vol.24, issue.8, pp.877-886, 2002.
DOI : 10.1016/S0142-1123(01)00205-5

K. Tanaka, Y. Nakai, and A. Yamashit, Fatigue growth threshold of small cracks

K. Traoré, Simulation thermomécanique du laminage circulaire. Développement d'une formulation quasi-Eulérienne tridimensionnelle sur une architecture parallèle, 2001.

S. Usami, S. Shida, and F. Eng, ELASTIC?PLASTIC ANALYSIS OF THE FATIGUE LIMIT FOR A MATERIAL WITH SMALL FLAWS, Fatigue & Fracture of Engineering Materials and Structures, vol.760, issue.3, p.471, 1979.
DOI : 10.1016/0022-5096(60)90013-2

B. Weber, B. Keunmeugne, J. C. Clement, and J. L. Robert, Improvements of multiaxial fatigue criteria computation for a strong reduction of calculation duration, Computational Materials Science, vol.15, issue.4, pp.381-399, 1999.
DOI : 10.1016/S0927-0256(98)00129-3

B. Weber, Fatigue miltiaxiale des structures industrielles sous chargement quelconque, Thèse de Doctorat, Institut National des Sciences Appliquées de Lyon, 1999.

D. Shang, D. Wang, M. Li, and W. Yao, Local stress???strain field intensity approach to fatigue life prediction under random cyclic loading, International Journal of Fatigue, vol.23, issue.10, pp.903-910, 2001.
DOI : 10.1016/S0142-1123(01)00051-2

C. Yeung and D. W. Hoeppner, The effect of grain flow orientation on the fretting fatigue characteristics of forged AMS6415 steel, Wear, vol.104, issue.3, pp.189-201, 1985.
DOI : 10.1016/0043-1648(85)90047-X

G. R. Zadnoazizi, Cintrage des tôles épaisses : analyse mécanique et influence sur la transition fragile-ductile d'un acier faiblement allié, 1983.

J. L. Zhou and A. L. Tits, Nonmonotone line search for minimax problems, Journal of Optimization Theory and Applications, vol.14, issue.3, pp.455-475
DOI : 10.1007/BF00939377