A. Visa-c and . Sauvey-c, Approximate Analytical Model for Hertzian Elliptical Contact Problems, ASME Journal of Tribology, vol.128, issue.3, pp.660-664, 2006.

A. H. Gartner-f and . Stoltenhoff-t, Bonding Mechanism in Cold Gas Spraying, Acta Materiala, vol.51, issue.15, pp.4379-4394, 2003.

M. Baskaran-b, . S. Srinivasan, and . Maffeo-b, Optimzation of the Fatigue Strength of Materials Due to Shot Peening: A Survey, International Journal of Structural Changes in Solids, vol.2, pp.33-63, 2010.

J. H. Bentall-r, Slip in the rolling contact of two dissimilar elastic rollers, International Journal of Mechanical Sciences, vol.9, issue.6, pp.389-404, 1967.
DOI : 10.1016/0020-7403(67)90043-4

. B. Bhuvaraghan, . M. Srinivasan-s, . Maffeo-b, and . Prakash-o, Analytical Solution for Single and Multiple impacts with Strain-rate Effects for Shot Peening, Computer Modeling in Engineering and Sciences, vol.57, issue.2, pp.2010-137

. M. Bijak-zochowski, Residual stress in some elasto-plastic problems of rolling contact with friction, International Journal of Mechanical Sciences, vol.39, issue.1, pp.15-32, 1997.
DOI : 10.1016/0020-7403(96)00018-5

. R. Bosman and . J. Schipper-d, Running-In of Systems Protected by Additive-Rich Oils, Tribology Letters, vol.26, issue.183, pp.263-282, 2011.
DOI : 10.1007/s11249-010-9713-2

. R. Bosman and . J. Schipper-r, Transition from mild to severe wear including running in effects, Wear, vol.270, issue.7-8, pp.472-478, 2011.
DOI : 10.1016/j.wear.2010.12.009

V. Boucly, N. D. Liu-s, and W. Q. Keer-l, Contact Analyses for Bodies With Frictional Heating and Plastic Behavior, Journal of Tribology, vol.127, issue.2, pp.355-364, 2005.
DOI : 10.1115/1.1843851

URL : https://hal.archives-ouvertes.fr/hal-00368428

B. V. Nélias-v and . Green-i, Modeling of the Rolling and Sliding Contact Between Two Asperities, ASME Journal of Tribology, vol.129, issue.2, pp.235-245, 2007.

L. A. Brandt, Multilevel matrix multiplication and fast solution of integral equations, Journal of Computational Physics, vol.90, issue.2, pp.348-370, 1990.
DOI : 10.1016/0021-9991(90)90171-V

. Bucaille-j.-l, . Gauthier-c, . Felder-e, and . Schirrer-g, The influence of strain hardening of polymers on the piling-up phenomenon in scratch tests: Experiments and numerical modelling, Wear, vol.260, issue.7-8, pp.803-814, 2005.
DOI : 10.1016/j.wear.2005.04.007

C. S. Bhushan-b, A Numerical Three-Dimensional Contact Model for Rrough, Multi-Layered Elastic/Plastic Solid Surfaces, INSA de Lyon, tous droits réservés Bibliography [CAI 05], pp.7-12, 2005.

S. K. Chao-k, Powder???Part Two, Tribology Transactions, vol.20, issue.1, pp.632-642, 2009.
DOI : 10.1080/10402009608983497

C. W. Liu-s and W. Q. , Fast Fourier Transform Based Numerical Methods for Elasto-Plastic Contacts of Nominally Flat Surfaces, ASME Journal of Applied Mechanics, vol.75, issue.1, pp.11022-11023, 2008.

C. W. Wang-q, Thermomechanical Analysis of Elasto-Plastic Bodies in a Sliding Spherical Contact and the Effects of Sliding Speed, Heat Partition, and Thermal Softening, ASME Journal of Tribology, vol.130, issue.4, pp.41402-41403, 2008.

C. W. , W. Q. , and W. F. Keer-l, Three-dimensional Repeated Elasto-Plastic Point Contact, Rolling and Sliding, ASME Journal of Applied Mechanics, vol.75, issue.2, pp.21021-21022, 2008.

C. W. Wang-q, A Numerical Static Friction Model for Spherical Contacts of Rough Surfaces, Influence of Load, Material, and Roughness, ASME Journal of Tribology, vol.131, issue.2, pp.21402-21403, 2009.

C. W. Zhou-k and W. M. Keer-l, Modeling Elasto-Plastic Indentation on Layered Materials Using the Equivalent Inclusion Method, International Journal of Solids and Structures, vol.47, issue.20, pp.2010-2841

. Davies-r, The Determination of Static and Dynamic Yield Stresses Using a Steel Ball, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.197, issue.1050, pp.416-432, 1050.
DOI : 10.1098/rspa.1949.0073

D. D. Nowell-d, Flat and Rounded Fretting Contact Problems Incorporating Elastic Layers, International Journal of Mechanical Sciences, vol.46, issue.11, pp.1653-1657, 2004.

E. I. Kligerman-y and . Kadin-y, Unloading of an Elastic-Plastic Loaded Spherical Contact, ASME/STLE 2004 International Joint Tribology Conference, Parts A and B, pp.3716-3729, 2005.
DOI : 10.1115/TRIB2004-64387

F. P. Nemat-nasser-s, A Universal Integration Algorithm for Rate-Dependant Elastoplasticity, Computers and Structures, vol.59, issue.6, pp.1173-1184, 1996.

F. A. De-campos-v, T. D. De, and . Neto-c, Analytical Modelling for Residual Stresses Produced by Shot Peening, Materials and Design, vol.30, issue.5, pp.1556-1560, 2009.

. A. Galin-l, Contact Problems in the Theory of Elasticity, North Carolina College, Departments of Mathematical and Engineering Research, 1953.

L. Gallego and N. D. , A Comprehensive Method to Predict Wear and to Define the Optimum Geometry of Fretting Surfaces, Journal of Tribology, vol.128, issue.3, pp.476-485, 2006.
DOI : 10.1115/1.2194917

URL : https://hal.archives-ouvertes.fr/hal-00938584

. Gallego-l, Fretting et Usure des Contacts Mécaniques : Modélisation Numérique, 2007.

. L. Gallego, Modeling of Fretting Wear Under Gross Slip and Partial Slip Conditions, Journal of Tribology, vol.129, issue.3, pp.528-535, 2007.
DOI : 10.1115/1.2736436

URL : https://hal.archives-ouvertes.fr/hal-00380751

. L. Gallego and D. S. Fulleringer-b, Multiscale computation of fretting wear at the blade/disk interface, Tribology International, vol.43, issue.4, pp.708-718, 2010.
DOI : 10.1016/j.triboint.2009.10.011

URL : https://hal.archives-ouvertes.fr/hal-00514513

N. L. Gallego, A fast and efficient contact algorithm for fretting problems applied to fretting modes I, II and III, Wear, vol.268, issue.1-2, pp.2010-208
DOI : 10.1016/j.wear.2009.07.019

URL : https://hal.archives-ouvertes.fr/hal-00514514

N. L. Gallego, A fast and efficient contact algorithm for fretting problems applied to fretting modes I, II and III, Wear, vol.268, issue.1-2, pp.2010-202
DOI : 10.1016/j.wear.2009.07.019

URL : https://hal.archives-ouvertes.fr/hal-00514514

. J. Greenwood and . B. Williamson-j, Contact of Nominally Flat Surfaces, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.295, issue.1442, pp.300-319, 1442.
DOI : 10.1098/rspa.1966.0242

Y. Guo and . Barkey-m, FE-Simulation of the Effects of Machining-Induced Residual Stress Profile on Rolling, p.217

C. Jacq, N. D. Lormand-g, and . Girodin-d, Development of a Three-Dimensional Semi-Analytical Elastic-Plastic Contact Code, Journal of Tribology, vol.124, issue.4, pp.653-667, 2002.
DOI : 10.1115/1.1467920

URL : https://hal.archives-ouvertes.fr/hal-00454903

J. R. Green-i and B. M. , Predicting the Coefficient of Restitution of Impacting Elastic-Perfectly Plastic Spheres, Nonlinear Dynamics, vol.60, issue.3, pp.217-229, 2010.

. Jiang-y, . Xu-b, and . Sehitoglu-h, Three-Dimensional Elastic-Plastic Stress Analysis of Rolling Contact, Journal of Tribology, vol.124, issue.4, pp.699-708, 2002.
DOI : 10.1115/1.1491978

J. Y. Farris-t, Spectral Analysis of Two-Dimensional Contact Problems, ASME Journal of Tribology, vol.118, issue.2, pp.320-328, 1996.

K. R. , P. A. , and H. J. Roy-m, Analytical and Experimental Elastic-Plastic Impact Analysis of a Magnetic Storage Head-Disk Interface, ASME Journal of Tribology, vol.131, issue.1, pp.11902-11903, 2009.

G. Kermouche, K. A. , and G. P. , Combined numerical and experimental approach of the impact-sliding wear of a stainless steel in a nuclear reactor, Wear, vol.263, issue.7-12, pp.1551-1555, 2007.
DOI : 10.1016/j.wear.2007.02.015

M. Klemenz, . Schulze-v, and . Rohr-i, Application of the FEM for the prediction of the surface layer characteristics after shot peening, Journal of Materials Processing Technology, vol.209, issue.8, pp.4093-4102, 2009.
DOI : 10.1016/j.jmatprotec.2008.10.001

. I. Kogut and . Etsion-i, Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat, Journal of Applied Mechanics, vol.69, issue.5, pp.657-662, 2002.
DOI : 10.1115/1.1490373

. G. Le, . M. Drissi-habti, I. G. Mauvoisin-g, and L. , Application de l'Indentation Instrumentée Sphérique à l'Estimation des Courbes Locales de Traction d'un Matérieu Métallique, Matériaux et Techniques, 2011.

. J. Leroux and . Fulleringer-b, Contact analysis in presence of spherical inhomogeneities within a half-space, International Journal of Solids and Structures, vol.47, issue.22-23, pp.22-23, 2010.
DOI : 10.1016/j.ijsolstr.2010.07.006

URL : https://hal.archives-ouvertes.fr/hal-00514526

L. Li and C. T. Wu-c.-y, A theoretical model for the contact of elastoplastic bodies, Proceedings of the Institution of Mechanical Engineers, pp.421-431, 2002.
DOI : 10.1243/0954406021525214

L. Y. , F. , and L. J. Xu-k, Finite Element Analysis for Shot Peening Models of Single Shot Impact and Multiple Impingement. Hsi-An Chiao Tung Ta Hsueh, Journal of Xi?an Jiaotong University, vol.41, issue.3, pp.348-352, 2007.

L. F. Li, . J. Pan, and . Sinka-c, Contact laws between solid particles, Journal of the Mechanics and Physics of Solids, vol.57, issue.8, pp.1194-1208, 2009.
DOI : 10.1016/j.jmps.2009.04.012

. Li and . J. Li, Simulation des Réparations par Soudage et Billage d'un Alliage à Base Nickel, 2011.

L. S. and W. Q. Liu-g, A Versatile Method of Discrete Convolution and FFT (DC-FFT) for Contact Analyses, Wear, vol.243, issue.12, pp.101-111, 2000.

. Liu, W. Q. Liu-g, and . Liu-s, A Three-Dimensional Thermal-Mechanical Asperity Contact Model for Two Nominally Flat Surfaces in Contact, Journal of Tribology, vol.123, issue.3, pp.595-602, 2001.
DOI : 10.1115/1.1308044

. Liu, . S. Liu, and . Wang-q, Transient Thermoelastic Stress Fields in a Half-Space, Journal of Tribology, vol.125, issue.1, pp.33-43, 2003.
DOI : 10.1115/1.1501087

. Liu, . S. Liu, and . Wang-q, Elastic Fields due to Eigenstrains in a Half-Space, Journal of Applied Mechanics, vol.72, issue.6, pp.871-878, 2005.
DOI : 10.1115/1.2047598

M. G. , A. R. Alavi, and N. A. , A Three-Dimensional Simulation of Shot Peening Process Using Multiple Shot Impacts, Journal of Materials Processing Technology, pp.164-165, 2005.

M. N. Barber-j and . N. Fawcett-j, The Oblique Impact of Elastic Spheres, INSA de Lyon, tous droits réservés Bibliography [MAW 76], p.101, 1976.

M. N. Barber-j and . N. Fawcett-j, The Role of Elastic Tangential Compliance in Oblique Impact, Journal of Tribology, vol.103, issue.1, p.74, 1981.
DOI : 10.1115/1.3251617

G. S. Meguid and S. J. , Development and Validation of Novel FE Models for 3D Analysis of Peening of Strain-Rate Sensitive Materials, Journal of Engineering Materials and Technology, vol.129, issue.2, pp.271-283, 2007.
DOI : 10.1115/1.2712469

. Mei, The Contact Problem of a Rigid Cylinder on an Elastic Layer, Applied Science Research, vol.18, issue.5, p.353, 1968.

. Mia, . Miao-h, . Larose-s, . Perron-c, and . Levesque-m, On the potential applications of a 3D random finite element model for the simulation of shot peening, Advances in Engineering Software, vol.40, issue.10, pp.1023-1038, 2009.
DOI : 10.1016/j.advengsoft.2009.03.013

M. R. Cheng-d, Thermoelastic Stress in the Semi-Infinite Solid, Journal of Applied Physics, vol.21, issue.9, pp.931-933, 1950.

M. H. Seifried-r and E. P. Kawamura-s, Analysis of Repeated Impacts on a Steel Rod with Visco-Plastic Material Behavior, European Journal of Mechanics / A Solids, vol.30, issue.3, pp.2010-336

M. H. Kawamura-s, Moderately High Speed Impact of Two Identical Spheres, International Journal of Impact Engineering, vol.38, issue.2-3, pp.123-129, 2011.

N. D. , J. C. Lormand-g, . Dudragne-g, and . Vincent-a, A New Methodology to Evaluate the Rolling Contact Fatigue Performance of Bearing Steels with Surface Dents -Application to 32CrMoV13 (Nitrited) and M50 Steels, ASME Journal of Tribology, vol.127, issue.3, pp.611-622, 2005.

N. D. Boucly-v and . Brunet-m, INSA de Lyon, tous droits réservés Bibliography Elastic-Plastic Contact Between Rough Surfaces: Proposal for a Wear or Running-In Model, these.pdf © [T. Chaise], pp.476-485, 2006.

A. D. Nél-07a-]-nélias and . Boucly-v, Rolling of an Elastic Ellipsoid Upon an Elastic-Plastic Flat, Journal of Tribology, vol.129, issue.4, pp.791-800, 2007.
DOI : 10.1115/1.2768078

A. D. Nél-07b-]-nélias and . Boucly-v, A Three-Dimensional Semianalytical Model for Elastic-Plastic Sliding Contacts, Journal of Tribology, vol.129, issue.4, pp.761-771, 2007.
DOI : 10.1115/1.2768076

N. D. Dai-d, Analysis of Surface Tractions in Complex Fretting Fatigue CycLe Using Quadratc Programming, ASME Journal of Tribology, vol.120, issue.4, pp.744-749, 1998.

P. B. Pau and . Hashemi, Contact Pressure on Closely Conforming Elastic Bodies, ASME Journal of Applied Mechanics, vol.48, issue.3, pp.543-548, 1981.

P. W. Bhushan-b, Modelling of Surfaces with a Bimodal Roughness Distribution Proceedings of the institution of mechanical engineers, Part J: Journal of Engineering Tribology, vol.214, issue.5, pp.459-470, 2000.

P. W. Bhushan-b, Three Dimensional Contact Analysis of Layered Elastic/Plastic Solids with Rough Surfaces, Wear, vol.249, issue.9, pp.741-760, 2001.

P. W. Bhushan-b, Sliding Contact Analysis of Layered Elastic: Plastic Solids With Rough Surfaces, ASME Journal of Tribology, vol.124, issue.1, pp.46-61, 2002.

P. I. Keer-l, Fast Methods for Solving Rough Contact Problems: a Comparative Study, ASME Journal of Tribology, vol.122, issue.1, pp.36-40, 2000.

G. A. Popescu-g and . E. Morales-espejel-g, Micro-Plastic Material Model and Residual Fields in Rolling Contact, Journal of ASTM International, vol.3, pp.1-12, 2006.

. Popescu-g, . E. Morales-espejel-g, . Wemekamp-b, and . Gabelli-a, An Engineering Model for Three-Dimensional Elastic-Plastic Rolling Contact Analyses, Tribology Transactions, vol.19, issue.3, pp.387-399, 2006.
DOI : 10.1016/0956-7151(91)90206-G

Z. N. Ren, C. W. , and W. Q. , Plasto-Elastohydrodynamic Lubrication (PEHL) in Point Contacts, Journal of Tribology, vol.132, issue.3, p.31501, 2010.
DOI : 10.1115/1.4001813

J. P. Sainsot, A Numerical Model for Elastoplastic Rough Contact, Cmes-Computer Modeling in Engineering & Sciences, vol.3, issue.4, pp.497-506, 2002.
URL : https://hal.archives-ouvertes.fr/hal-00454931

S. A. Garcia-navas-v and . Gonzalo-o, Surface State of Inconel 718 Ultrasonic Shot Peened: Effect of Processing Time, Material and Quantity of Shot Balls and Distance From Radiating Surface to Sample, Materials and Design, vol.32, issue.4, pp.2010-2213

T. Chaise, Mechanical simulation using a semi analytical method [SCH 02, Characteristics of Surface Layers Produced by Shot Peening. International Conference on Shot Peening, 2002.

. Sei, . Seifried-r, and E. P. Schiehlen-w, Numerical and Experimental Evaluation of the Coefficient of Restitution for Repeated Impacts, International Journal of Impact Engineering, vol.32, pp.1-4, 2005.

. Sei, . Seifried-r, and E. P. Minamoto-h, Viscoplastic Effects Occuring in Impact of Aluminium and Steel Bodies and Their Influence on the Coefficient of Restitution, ASME Journal of Applied Mechanics, vol.77, issue.4, pp.41008-41009, 2010.

. Sim and T. J. Simo, Consistent tangent operators for rate-independent elastoplasticity, Computer Methods in Applied Mechanics and Engineering, vol.48, issue.1, pp.101-118, 1985.
DOI : 10.1016/0045-7825(85)90070-2

S. S. Sherrington-i and L. S. Roberts-e, An Evaluation of the Effect of Simulated Launch Vibrations on the Friction Performance and Lubrication of Ball Bearings for Space Applications, Wear, vol.260, pp.11-12, 2006.

. T. Stoltenhoff, . Borchers-c, and . Gärtner-f, Microstructures and key properties of cold-sprayed and thermally sprayed copper coatings, Surface and Coatings Technology, vol.200, issue.16-17, pp.16-17, 2006.
DOI : 10.1016/j.surfcoat.2005.05.011

S. M. and O. Schreier-h, Image Correlation for Shape, Motion, and Deformation Measurements: Basic Concepts , Theory and Applications, 2009.

R. C. Thompson-j, An Exact Solution for the Superseismic Stage of Dynamic Contact Between a Punch and an Elastic Body, Journal of Applied Mechanics, vol.44, issue.4, p.583, 1977.
DOI : 10.1115/1.3424139

T. C. Ning-z, A Theoretical Model for the Stick/Bounce Behaviour of Adhesive, Elastic-Plastic Spheres, Powder Technology, vol.99, issue.2, pp.154-162, 1998.

. F. Tian-x and . Bhushan-b, A Numerical Three-Dimensional Model for the Contact of Rough Surfaces by Variational Principle, Journal of Tribology, vol.118, issue.1, pp.33-42, 1996.
DOI : 10.1115/1.2837089

N. A. Vincent, J. C. , and R. Y. , Comparison of Fatigue Performances of 32CrMoV13 and M50 Steels in Presence of Surface Dents, Journal of ASTM International, vol.3, issue.2, pp.1-9, 2006.

. L. Vu-quoc and . Zhang-x, An elastoplastic contact force-displacement model in the normal direction: displacement-driven version, Proc. R. Soc. Lond. A, pp.4013-4044, 1999.
DOI : 10.1098/rspa.1999.0488

W. F. Keer-l, Numerical Simulation for three Dimensional Elastic-Plastic Contact with Hardening Behavior, ASME Journal of Tribology, vol.127, issue.3, pp.494-502, 2005.

W. Z. , W. W. Hu-y, and . Wang-h, A Numerical Elastic-Plastic Contact Model for Rough Surfaces, STLE Tribology Transactions, vol.53, issue.2, pp.2010-224

. Wei, . G. Weir, and . Tallon-s, The Coefficient of Restitution for Normal Incident, Low Velocity Particle Impacts, Chemical Engineering Science, vol.60, issue.13, pp.3637-3647, 2005.

]. and L. Thornton-c, Rebound Behaviour of Spheres for Plastic Impacts, International Journal of Impact Engineering, vol.28, issue.9, pp.929-946, 2003.

W. C. and L. L. Thornton-c, Energy Dissipation During Normal Impact of Elastic and Elastic-Plastic Spheres, International Journal of Impact Engineering, vol.32, pp.1-4, 2005.

Y. B. and M. S. Gouldstone-a, Modeling of High Velocity Impact of Spherical Particles, Wear, vol.270, pp.9-10, 2011.

Y. and Y. H. Sanday-s, Elastic Fields in Joined Half-Spaces Due to Nuclei of Strain, Proceedings of the Royal Society of London, Series A, vol.434, pp.503-519, 1991.

Y. and Y. H. Sanday-s, Elastic Fields in Joined Semi-Infinite Solids With an Inclusion, Proceedings of the Royal Society of London, Series A, vol.434, pp.521-530, 1991.

Z. Y. Zolotarevsky-v, . Kligerman-y, and . Etsion-i, Multiple Normal Loading-Unloading Cycles of a Spherical Contact Under Stick Contact Condition, ASME Journal of Tribology, vol.132, issue.4, pp.2010-041401

. Zhang-x and . Vu-quoc-l, Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions, International Journal of Impact Engineering, vol.27, issue.3, pp.317-341, 2002.
DOI : 10.1016/S0734-743X(01)00052-5

Z. K. Wayne, C. W. Keer-l, and W. Q. , A Fast Method for Solving Three-Dimensional Arbitrarily-Shaped Inclusions in a Half Space, Computer Methods in Applied Mechanics and Engineering, vol.198, pp.9-12, 2009.

. Prénoms:-thibaut and T. Pierre, Mechanical simulation using a semi analytical method: from elasto-plastic rolling contact to multiple im- pacts NATURE: Doctorat Numéro d'ordre, pp.2011-0076

. Cette-thèse-présente-le-développement, application de méthodes de calcul semi analytiques (SA) à la modélisation de procédés mécaniques de mise en compression des surfaces. Les méthodes SA, initialement développées pour la simulation des contacts élasto-plastiques, ont l'avantage de temps de calcul très significativement réduits par rapport aux méthodes numériques conventionnelles . Cette méthode est d'abord utilisée pour la simulation d'un procédé connu sous le nom de galetage, avec un chargement de type contact roulant

. Le-procédé-de-billage-ultrasonore-a-Été-tout-particulièrement-Étudié, Dans un premier temps nous nous sommes attachés à la décrire la cinématique des billes dans une chambre fermée, mises et maintenues en mouvement par une sonotrode. L'utilisation de formules analytiques pour l'estimation des coefficients de restitution, lors des nombreux chocs entre les billes ou avec les parois de la chambre, a permis d'affiner le calcul des vitesses moyennes d'impact en fonction des paramètres du procédé, La méthode SA est ensuite utilisée pour déterminer le champ de déformations plastiques induits par les impacts. Enfin une méthode de projection est proposée pour permettre in fine la prédiction des contraintes résiduelles pour des structures fines ou épaisses

. Directeur-de-thèse, Monsieur le Professeur Daniel Nélias Président du jury: Prof. Jean-Louis Chaboche Composition du jury