T. Hun, L. , and K. Jhang, Experimental investigation of nonlinear acoustic effect at crack, NDT & E International, vol.42, pp.757-764, 2009.

Y. Shui, I. Yu, and . Solodov, Nonlinear properties of Rayleigh and Stoneley waves in solids, Journal of Applied Physics, vol.64, issue.11, pp.6155-6165, 1988.
DOI : 10.1063/1.342120

T. David and . Blackstock, Propagation of plane sound waves of finite amplitude in nondissipative fluids, The Journal of the Acoustical Society of America, vol.34, issue.1, pp.9-30, 1962.

P. A. Johnson and P. N. Rasolofosaon, Manifestation of nonlinear elasticity in rock: Convincing evidence over large frequency and strain intervals from laboratory studies, The Journal of the Acoustical Society of America, vol.98, issue.5, pp.77-88, 1996.
DOI : 10.1121/1.414229

URL : https://hal.archives-ouvertes.fr/hal-00331039

V. A. Korneev, K. T. Nihei, and L. R. Myer, Nonlinear interaction of plane elastic waves, 1998.
DOI : 10.2172/290877

L. A. Ostrovsky and P. A. Jonhson, Dynamic nonlinear elasticity in geomaterial, Rivista del nuovo cimento, vol.24, issue.7, 2001.

M. Destrade, M. D. Gilchrist, and R. W. Ogden, Third- and fourth-order elasticities of biological soft tissues, The Journal of the Acoustical Society of America, vol.127, issue.4, pp.2103-2106, 2010.
DOI : 10.1121/1.3337232

J. Herrmann, J. Kim, L. J. Jacobs, J. Qu, J. W. Littles et al., Assessment of material damage in a nickel-base superalloy using nonlinear Rayleigh surface waves, Journal of Applied Physics, vol.99, issue.12, pp.99124913-124913, 2006.
DOI : 10.1063/1.2204807

J. Kim, L. J. Jacobs, J. Qu, and J. W. Littles, Experimental characterization of fatigue damage in a nickel-base superalloy using nonlinear ultrasonic waves, The Journal of the Acoustical Society of America, vol.120, issue.3, pp.1266-1273, 2006.
DOI : 10.1121/1.2221557

L. Zarembo and V. Krasil-'nikov, NONLINEAR PHENOMENA IN THE PROPAGATION OF ELASTIC WAVES IN SOLIDS, Soviet Physics Uspekhi, vol.13, issue.6, pp.778-797, 1971.
DOI : 10.1070/PU1971v013n06ABEH004281

F. D. Murnaghan, Finite Deformations of an Elastic Solid, American Journal of Mathematics, vol.59, issue.2, pp.235-260, 1937.
DOI : 10.2307/2371405

Y. Zheng, R. Gr, I. Maev, . Yu, and . Solodov, <i>Review / Syth??se</i> Nonlinear acoustic applications for material characterization: A review, Canadian Journal of Physics, vol.77, issue.12, pp.927-967, 1999.
DOI : 10.1139/cjp-77-12-927

K. Jhang, Erratum to: Nonlinear ultrasonic techniques for nondestructive assessment of micro damage in material: A review, International Journal of Precision Engineering and Manufacturing, vol.18, issue.1, pp.123-135, 2009.
DOI : 10.1007/s12541-017-0018-3

A. A. Shah and Y. Ribakov, Non-linear ultrasonic evaluation of damaged concrete based on higher order harmonic generation, Materials & Design, vol.30, issue.10, pp.4095-4102, 2009.
DOI : 10.1016/j.matdes.2009.05.009

M. A. Breazeale and D. O. Thompson, FINITE???AMPLITUDE ULTRASONIC WAVES IN ALUMINUM, Applied Physics Letters, vol.3, issue.5, pp.77-78, 1963.
DOI : 10.1063/1.1753876

K. Jhang, Applications of nonlinear ultrasonics to the NDE of material degradation . Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on, vol.47, issue.3, pp.540-548, 2000.

K. Jeong, M. A. Na, and . Breazeale, Ultrasonic nonlinear properties of lead zirconate-titanate ceramics, The Journal of the Acoustical Society of America, vol.95, issue.6, pp.3213-3221, 1994.

D. S. Hughes and J. L. Kelly, Second-Order Elastic Deformation of Solids, Physical Review, vol.92, issue.5, pp.1145-1149, 1953.
DOI : 10.1103/PhysRev.92.1145

R. A. Toupin and B. Bernstein, Sound Waves in Deformed Perfectly Elastic Materials. Acoustoelastic Effect, The Journal of the Acoustical Society of America, vol.33, issue.2, pp.216-225, 1961.
DOI : 10.1121/1.1908623

C. George and . Johnson, Acoustoelastic theory for elastic?plastic materials, The Journal of the Acoustical Society of America, vol.70, issue.2, pp.591-595, 1981.

M. Destrade and M. D. Gilchrist, Third- and fourth-order constants of incompressible soft solids and the acousto-elastic effect, The Journal of the Acoustical Society of America, vol.127, issue.5, pp.2759-2763, 2010.
DOI : 10.1121/1.3372624

Y. H. Pao, W. Sachse, and H. Fukuoka, Acoustoelasticity and unltrasonic measurements of residual stresses, Physical Acoutics, vol.17, pp.61-143, 1984.

T. Leon-salamanca and D. F. Bray, Residual stress measurement in steel plates and welds using critically refracted longitudinal (lcr) waves, pp.169-184, 1996.

B. Peter and . Nagy, Fatigue damage assessment by nonlinear ultrasonic materials characterization, Ultrasonics, vol.36, pp.375-381, 1998.

A. Robert, P. A. Guyer, and . Johnson, Nonlinear mesoscopic elasticity : Evidence for a new class of materials, Physics Today, vol.52, issue.4, pp.30-36, 1999.

I. Yu, B. A. Solodov, and . Korshak, Instability, chaos, and "memory" in acoustic-wavecrack interaction. Physical review letters, pp.14303-14306, 2002.

M. Alexei, G. Vitalyi, and C. Bernard, Self-induced hysteresis for nonlinear acoustic waves in cracked material. Physical review letters, pp.124301-124305, 2003.

V. E. Nazarov, A. V. Radostin, L. A. Ostrovsky, and I. A. Soustova, Wave processes in media with hysteretic nonlinearity. Part I, Acoustical Physics, vol.49, issue.3, pp.344-353, 2003.
DOI : 10.1134/1.1574363

E. Veniamin, A. B. Nazarov, and . Kolpakov, Experimental investigations of nonlinear acoustic phenomena in polycrystalline zinc, The Journal of the Acoustical Society of America, vol.107, issue.4, pp.1915-1921, 2000.

V. E. Nazarov, A. V. Radostin, L. A. Ostrovsky, and I. A. Soustova, Wave processes in media with hysteretic nonlinearity: Part 2, Acoustical Physics, vol.49, issue.4, pp.444-448, 2003.
DOI : 10.1134/1.1591300

K. Van-den-abeele and F. Windels, Characterization and Imaging of Microdamage Using Nonlinear Resonance Ultrasound Spectroscopy (NRUS): An Analytical Model, Universality of Nonclassical Nonlinearity, pp.369-388, 2006.
DOI : 10.1007/978-0-387-35851-2_23

J. Ortin, Preisach modeling of hysteresis for a pseudoelastic Cu???Zn???Al single crystal, Journal of Applied Physics, vol.71, issue.3, pp.1454-1461, 1992.
DOI : 10.1063/1.351238

R. A. Guyer, K. R. Mccall, and G. N. Boitnott, Hysteresis, Discrete Memory, and Nonlinear Wave Propagation in Rock: A New Paradigm, Physical Review Letters, vol.74, issue.17, pp.3491-3494, 1995.
DOI : 10.1103/PhysRevLett.74.3491

URL : https://hal.archives-ouvertes.fr/hal-00301808

F. Preisach, Uber die magnetische nachwirkung. Zeitschrift fur Physik, pp.277-302, 1935.

I. Mayergoyz, Mathematical models of hysteresis, IEEE Transactions on Magnetics, vol.22, issue.5, pp.603-608, 1986.
DOI : 10.1109/TMAG.1986.1064347

M. Scalerandi, V. Agostini, P. P. Delsanto, K. E. Van-den-abeele, and P. A. Johnson, Local interaction simulation approach to modeling nonclassical, nonlinear behavoir in solids, 2003.

K. Van-den-abeele, . Schubert, . V-aleshin, J. Windels, and . Carmeliet, Resonant bar simulations in media with localized damage, Ultrasonics, vol.42, issue.1-9, pp.1017-1024, 2004.
DOI : 10.1016/j.ultras.2003.12.021

V. Aleshin and K. Van-den-abeele, Microcontact-based theory for acoustics in microdamaged materials, Journal of the Mechanics and Physics of Solids, vol.55, issue.2, pp.366-390, 2007.
DOI : 10.1016/j.jmps.2006.07.002

O. Buck, W. L. Morris, and J. M. Richardson, Acoustic harmonic generation at unbonded interfaces and fatigue cracks, Applied Physics Letters, vol.33, issue.5, pp.371-373, 1978.
DOI : 10.1063/1.90399

I. Y. Solodov, N. Krohn, and G. Busse, CAN: an example of nonclassical acoustic nonlinearity in solids, Ultrasonics, vol.40, issue.1-8, pp.621-625, 2002.
DOI : 10.1016/S0041-624X(02)00186-5

B. A. Korshak, I. Y. Solodov, and E. M. Ballad, DC effects, sub-harmonics, stochasticity and ???memory??? for contact acoustic non-linearity, Ultrasonics, vol.40, issue.1-8, pp.707-716, 2002.
DOI : 10.1016/S0041-624X(02)00241-X

N. A. Burnham, A. J. Kulik, G. Gremaud, P. J. Gallo, and F. Oulevey, Scanning local-acceleration microscopy, Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures, vol.14, issue.2, pp.794-799, 1996.
DOI : 10.1116/1.588715

URL : https://hal.archives-ouvertes.fr/jpa-00254592

Y. Ohara, T. Mihara, and K. Yamanaka, Effect of adhesion force between crack planes on subharmonic and DC responses in nonlinear ultrasound, Ultrasonics, vol.44, issue.2, pp.194-199, 2005.
DOI : 10.1016/j.ultras.2005.10.006

O. Kolosov and K. Yamanaka, Nonlinear Detection of Ultrasonic Vibrations in an Atomic Force Microscope, Japanese Journal of Applied Physics, vol.32, issue.Part 2, No. 8A, pp.1095-1098, 1993.
DOI : 10.1143/JJAP.32.L1095

K. Yamanaka, T. Mihara, and T. Tsuji, Evaluation of Closed Cracks by Model Analysis of Subharmonic Ultrasound, Japanese Journal of Applied Physics, vol.43, issue.5B, pp.3082-3087, 2004.
DOI : 10.1143/JJAP.43.3082

V. Tournat, V. E. Gusev, and B. Castagnède, Subharmonics and noise excitation in transmission of acoustic wave through unconsolidated granular medium, Physics Letters A, vol.326, issue.5-6, pp.5-6340, 2004.
DOI : 10.1016/j.physleta.2004.04.042

K. Van-den-abeele, P. A. Johnson, and A. Sutin, Nonlinear Elastic Wave Spectroscopy (NEWS) Techniques to Discern Material Damage, Part I: Nonlinear Wave Modulation Spectroscopy (NWMS), Research in Nondestructive Evaluation, vol.43, issue.1, pp.17-30, 2000.
DOI : 10.1029/98GL51231

K. E. Van-den-abeele, J. Carmeliet, J. A. Cate, and P. A. Johnson, Nonlinear Elastic Wave Spectroscopy (NEWS) Techniques to Discern Material Damage, Part II: Single-Mode Nonlinear Resonance Acoustic Spectroscopy, Research in Nondestructive Evaluation, vol.77, issue.1, pp.31-42, 2000.
DOI : 10.1121/1.418198

C. Payan, V. Garnier, J. Moysan, and P. A. Johnson, Applying nonlinear resonant ultrasound spectroscopy to improving thermal damage assessment in concrete, The Journal of the Acoustical Society of America, vol.121, issue.4, pp.125-130, 2007.
DOI : 10.1121/1.2710745

M. Meo, U. Polimeno, and G. Zumpano, Detecting Damage in Composite Material Using Nonlinear Elastic Wave Spectroscopy Methods, Applied Composite Materials, vol.24, issue.7
DOI : 10.1007/s10443-008-9061-7

W. T. Yost, J. H. Cantrell, M. A. Jr, and . Breazeale, Ultrasonic nonlinearity parameters and third???order elastic constants of copper between 300 and 3?????K, Journal of Applied Physics, vol.52, issue.1, pp.126-128, 1981.
DOI : 10.1063/1.328443

M. A. Breazeale and J. Ford, Ultrasonic Studies of the Nonlinear Behavior of Solids, Journal of Applied Physics, vol.36, issue.11, pp.3486-3490, 1965.
DOI : 10.1063/1.1703023

A. James, M. A. Bains, and . Breazeale, Third-order elastic constants of germanium between 300 and 3°k, Phys. Rev. B, vol.13, pp.3623-3630, 1976.

H. John, M. A. Cantrell, and . Breazeale, Ultrasonic investigation of the nonlinearity of fused silica for different hydroxyl-ion contents and homogeneities between 300 and 3°k, Phys. Rev

K. Jeong, . Na, H. John, . Cantrell, T. William et al., Linear and nonlinear ultrasonic properties of fatigued 410 cb stainless steel, Twenty-Second Symposium on Quantitative Nondestructive Evaluation, pp.1347-1360, 1995.

A. A. Xander, L. A. Verbeek, J. M. Ledoux, P. J. Willigers, A. P. Brands et al., Experimental investigation of the pulse inversion technique for imaging ultrasound contrast agents, Journal of the Acoustical Society of America, vol.107, issue.4, pp.2281-2290, 2000.

C. Pruel, J. Kim, J. Qu, and L. J. Jacobs, Evaluation of plasticity driven material damage using Lamb waves, Applied Physics Letters, vol.91, issue.23, p.91, 2007.
DOI : 10.1063/1.2811954

N. De-jong, R. Cornet, and C. T. Lancée, Higher harmonics of vibrating gas-filled microspheres. Part two: measurements, Ultrasonics, vol.32, issue.6, pp.455-459, 1994.
DOI : 10.1016/0041-624X(94)90065-5

K. E. Morgan, M. Averkiou, and K. Ferrara, The effect of the phase of transmission on contrast agent echoes, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, vol.45, issue.4, pp.872-875, 1998.
DOI : 10.1109/58.710539

J. M. Richardson, Harmonic generation at an unbonded interface???I. Planar interface between semi-infinite elastic media, International Journal of Engineering Science, vol.17, issue.1, pp.73-85, 1979.
DOI : 10.1016/0020-7225(79)90008-9

S. Biwa, S. Yamaji, and E. Mastumoto, Quantitative Evaluation of Harmonic Generation at Contacting Interface, AIP Conference Proceedings, p.978, 2008.
DOI : 10.1063/1.2956268

T. Nam, T. Lee, C. Kim, K. Jhang, and N. Kim, Harmonic generation of an obliquely incident ultrasonic wave in solid???solid contact interfaces, Ultrasonics, vol.52, issue.6, pp.778-783, 2012.
DOI : 10.1016/j.ultras.2012.02.008

W. L. Morris, O. Buck, and R. V. Inman, Acoustic harmonic generation due to fatigue damage in high???strength aluminum, Journal of Applied Physics, vol.50, issue.11, pp.506737-6741, 1979.
DOI : 10.1063/1.325917

E. Ballad, S. Y. Vezirov, I. Y. Pfleiderer, G. Solodov, and . Busse, Nonlinear modulation technique for NDE with air-coupled ultrasound, Ultrasonics, vol.42, issue.1-9, pp.1-91031, 2004.
DOI : 10.1016/j.ultras.2003.12.022

D. Dutta, H. Sohn, K. A. Harries, and P. Rizzo, A Nonlinear Acoustic Technique for Crack Detection in Metallic Structures, Structural Health Monitoring, vol.8, issue.3, pp.251-262, 2009.
DOI : 10.1177/1475921709102105

D. Yan, B. W. Drinkwater, and S. A. Neild, Measurement of the ultrasonic nonlinearity of kissing bonds in adhesive joints, NDT & E International, vol.42, issue.5, pp.459-466, 2009.
DOI : 10.1016/j.ndteint.2009.02.002

D. Yan, The detectability of kissing bonds in adhesive joints using non-linear ultrasonic techniques, 2010.

Y. Ohara, K. Kawashima, R. Yamada, and H. Horio, Evaluation of Amorphous Diffusion Bonding by Nonlinear Ultrasonic Method, AIP Conference Proceedings, pp.944-951, 2004.
DOI : 10.1063/1.1711720

Y. W. Mao, Y. Shui, W. Jiang, Z. Lu, and W. Wu, Second???harmonic generation of interface waves, Applied Physics Letters, vol.55, issue.23, pp.552394-2395, 1989.
DOI : 10.1063/1.102026

J. Laura, J. Pyrak-nolte, G. M. Xu, and . Haley, Elastic interface waves propagating in a fracture. Physical review letters, pp.3650-3653, 1992.

J. Laura, N. G. Pyrak-nolte, and . Cook, Elastic interface waves along a fracture, Geophysical Research Letters, vol.14, issue.11, pp.1107-1110, 1987.

S. Biwa, S. Hiraiwa, and E. Mastumoto, Stiffness evaluation of contacting surfaces by bulk and interface waves, Ultrasonics, vol.47, issue.1-4, pp.123-129, 2007.
DOI : 10.1016/j.ultras.2007.08.005

S. Biwa, A. Suzuki, and N. Ohno, Evaluation of interface wave velocity, reflection coefficients and interfacial stiffnesses of contacting surfaces, Ultrasonics, vol.43, issue.6, pp.495-502, 2005.
DOI : 10.1016/j.ultras.2004.09.003

G. Lee, J. , and D. R. Kobett, Interaction of elastic waves in an isotropic solid, The Journal of the Acoustical Society of America, vol.35, issue.1, pp.5-10, 1963.

P. Duffour, M. Morbidini, and P. Cawlay, A study of the vibro-acoustic modulation technique for the detection of cracks in metals, The Journal of the Acoustical Society of America, vol.119, issue.3, pp.1463-1475, 2006.
DOI : 10.1121/1.2161429

D. Donskoy, A. Sutin, and A. Ekimov, Nonlinear acoustic interaction on contact interfaces and its use for nondestructive testing, NDT & E International, vol.34, issue.4, pp.231-238, 2001.
DOI : 10.1016/S0963-8695(00)00063-3

G. L. Gao, D. Y. Li, D. Q. Shi, J. W. Dong, X. D. Shi et al., Nonlinear acoustic characteristics of fatigue microcracks in al alloy plate, Journal of the Minerals, Metals and Materials Society, vol.63, issue.2, pp.77-80, 2011.

J. Kim, V. A. Yakovlev, and S. I. Rokhlin, Surface acoustic wave modulation on a partially closed fatigue crack, The Journal of the Acoustical Society of America, vol.115, issue.5, pp.1961-1972, 2004.
DOI : 10.1121/1.1695012

V. Zaitsev, V. Nazarov, V. Gusev, and B. Castagnede, Novel nonlinear-modulation acoustic technique for crack detection, NDT & E International, vol.39, issue.3, pp.184-194, 2006.
DOI : 10.1016/j.ndteint.2005.07.007

K. Zacharias, E. Balabanidou, I. Hatzokos, I. T. Rekanos, and A. Trochidis, Microdamage evaluation in human trabecular bone based on nonlinear ultrasound vibro-modulation (NUVM), Journal of Biomechanics, vol.42, issue.5
DOI : 10.1016/j.jbiomech.2008.12.018

M. Morbidini, P. Duffour, and P. Cawley, Vibro-Acoustic Modulation NDE Technique. Part 2: Experimental Study, AIP Conference Proceedings, pp.616-623, 2005.
DOI : 10.1063/1.1916732

V. Vyacheslav, A. Kazakov, P. A. Sutin, and . Johnson, Sensitive imaging of an elastic nonlinear wave-scattering source in a solid, Applied Physics Letters, vol.81, issue.4, pp.646-648, 2002.

T. Goursolle, S. Santos, O. Bou-matar, and S. Callé, Non-linear based time reversal acoustic applied to crack detection: Simulations and experiments, International Journal of Non-Linear Mechanics, vol.43, issue.3, pp.170-177, 2008.
DOI : 10.1016/j.ijnonlinmec.2007.12.008

M. Vila, F. Vander-meulen, S. Santos, L. Haumesser, O. Bou et al., Contact phase modulation method for acoustic nonlinear parameter measurement in solid, Ultrasonics, vol.42, issue.1-9, pp.1-91061, 2004.
DOI : 10.1016/j.ultras.2003.12.024

A. J. Croxford, P. D. Wilcox, B. W. Drinkwater, and P. B. Nagy, The use of non-collinear mixing for nonlinear ultrasonic detection of plasticity and fatigue, The Journal of the Acoustical Society of America, vol.126, issue.5, pp.117-122, 2009.
DOI : 10.1121/1.3231451

I. Yu, J. Solodov, K. Wackerl, G. Pfleiderer, and . Busse, Nonlinear self-modulation and subharmonic acoustic spectroscopy for damage detection and location, Applied Physics Letters, vol.84, issue.26, 2004.

W. Lauterborn and E. Cramer, Subharmonic Route to Chaos Observed in Acoustics, Physical Review Letters, vol.47, issue.20, pp.1445-1448, 1981.
DOI : 10.1103/PhysRevLett.47.1445

Y. Ohara, T. Mihara, R. Sasaki, T. Ogata, S. Yamamoto et al., Imaging of closed cracks using nonlinear response of elastic waves at subharmonic frequency, Applied Physics Letters, vol.90, issue.1, pp.11902-011902, 2007.
DOI : 10.1063/1.2426891

V. Schmitz, S. Chakhlov, and W. Müller, Experiences with synthetic aperture focusing technique in the field, Ultrasonics, vol.38, issue.1-8, pp.731-738, 2000.
DOI : 10.1016/S0041-624X(99)00219-X

C. Matsuoka, K. Nakahata, A. Baba, N. Kono, and S. Hirose, COMPARATIVE STUDY ON ULTRASONIC IMAGING METHODS WITH ARRAY TRANSDUCERS, AIP Conference Proceedings, pp.707-714, 2008.
DOI : 10.1063/1.2902731

Y. Ohara, H. Endo, M. Hashimoto, Y. Shintaku, and K. Yamanaka, MONITORING GROWTH OF CLOSED FATIGUE CRACK USING SUBHARMONIC PHASED ARRAY, AIP Conference Proceedings, issue.1, pp.1211-903, 2010.
DOI : 10.1063/1.3362519

Y. Ohara, S. Horinouchi, M. Hashimoto, Y. Shintaku, and K. Yamanaka, Nonlinear ultrasonic imaging method for closed cracks using subtraction of responses at different external loads, Ultrasonics, vol.51, issue.6, pp.51661-666, 2011.
DOI : 10.1016/j.ultras.2010.12.010

P. M. Shankar, P. D. Krishna, and V. L. Newhouse, Subharmonic backscattering from ultrasound contrast agents, The Journal of the Acoustical Society of America, vol.106, issue.4, pp.2104-2110, 1999.
DOI : 10.1121/1.428142

P. Krishna, P. Shankar, and V. Newhouse, Subharmonic generation from ultrasonic contrast agents, Physics in Medicine and Biology, vol.44, issue.3, p.681, 1999.
DOI : 10.1088/0031-9155/44/3/004

D. Euvrard, Résolution numérique des équations aux dérivées partielles, 1994.

G. Dhatt, E. Lefrancois, and G. Touzot, Méthode des éléments finis, 2005.

L. L. Thompson, A review of finite-element methods for time-harmonic acoustics, The Journal of the Acoustical Society of America, vol.119, issue.3, pp.1315-1330, 2006.
DOI : 10.1121/1.2164987

S. Arthur, Méthodes de domaines fictifs d'ordre élevé pour les équations elliptiques et de Navier-Stokes : application au couplage fluide-structure, 2009.

M. Bonnet, Équations intégrales et éléments de frontière : applications en mécanique des solides et des fluides, Eyrolles, 1995.

G. C. Hsiao, Boundary Integral Equations, 2008.

L. A. Ostrovsky, Wave processes in media with strong acoustic nonlinearity, The Journal of the Acoustical Society of America, vol.90, issue.6, pp.3332-3337, 1991.
DOI : 10.1121/1.401444

J. Sinha and M. I. , Simulation of the dynamic response of a cracked beam, Computers & Structures, vol.80, issue.18-19, pp.1473-1476, 2002.
DOI : 10.1016/S0045-7949(02)00098-6

K. Kawashima, R. Omote, T. Ito, H. Fujita, and T. Shima, Nonlinear acoustic response through minute surface cracks: FEM simulation and experimentation, Ultrasonics, vol.40, issue.1-8, pp.611-615, 2002.
DOI : 10.1016/S0041-624X(02)00184-1

M. Krawzuk, Application of spectral beam finite element with a crack and iterative search technique for damage detection. Finite Elements in Analysis and Design, pp.537-548, 2002.

G. G. Adams, Contact modeling ??? forces, Tribology International, vol.33, issue.5-6, pp.431-442, 2000.
DOI : 10.1016/S0301-679X(00)00063-3

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

K. Kendall and D. Tabor, An Ultrasonic Study of the Area of Contact between Stationary and Sliding Surfaces, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.323, issue.1554, pp.321-340, 1554.
DOI : 10.1098/rspa.1971.0108

. Jai-man, R. B. Baik, and . Thompson, Ultrasonic scattering from imperfect interfaces : A quasi-static model, Journal of Nondestructive Evaluation, vol.4, pp.177-196, 1984.

N. Haines, The theory of sound transmission and reflection at contacting surfaces, CEGB Berkeley Nuclear Laboratories, p.4744, 1980.

B. W. Drinkwater, R. S. Dwyer-joyce, and P. Cawley, A Study of the Interaction between Ultrasound and a Partially Contacting Solid--Solid Interface, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.452, issue.1955, pp.4522613-2628, 1955.
DOI : 10.1098/rspa.1996.0139

P. P. Delsanto and M. Scalerandi, A spring model for the simulation of the propagation of ultrasonic pulses through imperfect contact interfaces, The Journal of the Acoustical Society of America, vol.104, issue.5, pp.2584-2591, 1998.
DOI : 10.1121/1.423841

S. Biwa, S. Hiraiwa, and E. Mastumoto, Pressure-Dependent Stiffnesses and Nonlinear Ultrasonic Response of Contacting Surfaces, Journal of Solid Mechanics and Materials Engineering, vol.3, issue.1, pp.10-21, 2009.
DOI : 10.1299/jmmp.3.10

F. J. Margetan, R. B. Thompson, and T. A. Gray, Interfacial spring model for ultrasonic interactions with imperfect interfaces: Theory of oblique incidence and application to diffusion-bonded butt joints, Journal of Nondestructive Evaluation, vol.320, issue.3-4, pp.131-152, 1988.
DOI : 10.1007/BF00565998

F. J. Margetan, R. B. Thompson, J. H. Rose, and T. A. Gray, The interaction of ultrasound with imperfect interfaces: Experimental studies of model structures, Journal of Nondestructive Evaluation, vol.47, issue.3-4, pp.109-126, 1992.
DOI : 10.1007/BF00566403

C. Pecorari, Nonlinear interaction of plane ultrasonic waves with an interface between rough surfaces in contact, The Journal of the Acoustical Society of America, vol.113, issue.6, pp.3065-3072, 2003.
DOI : 10.1121/1.1570437

C. Pecorari and M. Poznic, Nonlinear acoustic scattering by a partially closed surface-breaking crack, The Journal of the Acoustical Society of America, vol.117, issue.2, pp.592-600, 2005.
DOI : 10.1121/1.1850052

J. Kim, A. Baltazar, J. W. Hu, and S. I. Rokhlin, Hysteretic linear and nonlinear acoustic responses from pressed interfaces, International Journal of Solids and Structures, vol.43, issue.21, pp.6436-6452, 2006.
DOI : 10.1016/j.ijsolstr.2005.11.006

D. A. Mendelsohn and J. M. Doong, Transient dynamic elastic frictional contact: A general 2D boundary element formulation with examples of SH motion, Wave Motion, vol.11, issue.1, pp.1-21, 1989.
DOI : 10.1016/0165-2125(89)90009-7

S. Hirose and J. D. Achenbach, Higher harmonics in the far field due to dynamic crack???face contacting, The Journal of the Acoustical Society of America, vol.93, issue.1, pp.142-147, 1993.
DOI : 10.1121/1.405651

S. Hirose, 2-D scattering by a crack with contact-boundary conditions, Wave Motion, vol.19, issue.1, pp.37-49, 1993.
DOI : 10.1016/0165-2125(94)90011-6

O. V. Menshykov, M. V. Menshykova, and I. A. Guz, Effect of friction of the crack faces for a linear crack under an oblique harmonic loading, International Journal of Engineering Science, vol.46, issue.5, pp.438-458, 2008.
DOI : 10.1016/j.ijengsci.2007.12.006

G. E. Stavroulakis, H. Antes, and P. D. Panagiotopoulos, Transient elastodynamics around cracks including contact and friction, Computer Methods in Applied Mechanics and Engineering, vol.177, issue.3-4, pp.427-440, 1999.
DOI : 10.1016/S0045-7825(99)00391-6

A. Tan, S. Hirose, C. Zhang, and C. Wang, A 2D time-domain BEM for transient wave scattering analysis by a crack in anisotropic solids, Engineering Analysis with Boundary Elements, vol.29, issue.6, pp.610-623, 2005.
DOI : 10.1016/j.enganabound.2005.01.012

M. Kögl, S. Hurlebaus, and L. Gaul, Finite element simulation of non-destructive damage detection with higher harmonics, NDT & E International, vol.37, issue.3, pp.195-205, 2004.
DOI : 10.1016/j.ndteint.2003.09.003

J. E. Akin, The generation of elements with singularities, International Journal for Numerical Methods in Engineering, vol.6, issue.6, pp.1249-1259, 1976.
DOI : 10.1002/nme.1620100605

P. R. Heyllger, On conventional and quarter-point mixed elements in linear elastic fracture mechanics, Engineering Fracture Mechanics, vol.31, issue.1, pp.157-171, 1988.
DOI : 10.1016/0013-7944(88)90129-4

Y. Zhang, A fictitious domain method for acoustic wave propagation problems, Mathematical and Computer Modelling, vol.50, issue.3-4, pp.351-359, 2009.
DOI : 10.1016/j.mcm.2009.04.020

E. Bécache, J. Rodriguez, and C. Tsogka, A fictitious domain method with mixed fiinite elements for elastodynamics, Journal of Computational Science, vol.29, issue.3, pp.1244-1267, 2007.

G. Scarella, Etude théorique et numérique de la propagation d'ondes en présence de contact unilatéral dans un milieu fissuré, Mathématiques appliquées, 2004.

J. Haslinger, T. Tozubek, and R. Kucera, Fictitious domain formulations of unilateral problems: analysis and algorithms, Computing, vol.196, issue.1-2, pp.69-96, 2009.
DOI : 10.1007/s00607-009-0026-y

J. Haslinger and Y. Renard, A New Fictitious Domain Approach Inspired by the Extended Finite Element Method, SIAM Journal on Numerical Analysis, vol.47, issue.2, pp.1474-1499, 2009.
DOI : 10.1137/070704435

URL : https://hal.archives-ouvertes.fr/hal-00220482

B. O. Neill, R. G. Maev, and F. Severin, Distortion of shear waves passing through a friction coupled interface, Review of progress in Quantitative Nondestructive Evaluation, pp.1264-1267, 2001.

A. Meziane, A. N. Norris, and A. L. Shuvalov, Nonlinear shear wave interaction at a frictional interface: Energy dissipation and generation of harmonics, The Journal of the Acoustical Society of America, vol.130, issue.4, pp.1820-1828, 2011.
DOI : 10.1121/1.3628663

Y. Wang, G. Yu, and H. Dai, Transmission of elastic waves through a frictional contact interface between two anisotropic dissimilar media, Wave Motion, vol.37, issue.2, pp.137-156, 2003.
DOI : 10.1016/S0165-2125(02)00039-2

L. Baillet and T. Sassi, Simulations num??riques de diff??rentes m??thodes d'??l??ments finis pour les probl??mes de contact avec frottement, Comptes Rendus M??canique, vol.331, issue.11, pp.789-796, 2003.
DOI : 10.1016/j.crme.2003.08.005

M. Jean, The non-smooth contact dynamics method, Computer Methods in Applied Mechanics and Engineering, vol.177, issue.3-4, pp.3-4235, 1999.
DOI : 10.1016/S0045-7825(98)00383-1

URL : https://hal.archives-ouvertes.fr/hal-01390459

M. Cocou, Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity, Zeitschrift fur angewandte Mathematik und Physik ZAMP, pp.1099-1109, 2002.
DOI : 10.1007/PL00012615

M. Cocou and G. Scarella, Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body, Comptes Rendus Mathematique, vol.338, issue.4, pp.341-346, 2004.
DOI : 10.1016/j.crma.2003.12.013

URL : https://hal.archives-ouvertes.fr/hal-00088248

N. Bourago and V. Kukudzhanov, A review of contact algorithms. Mechanics of solids, pp.35-71, 2005.

P. Wriggers, Finite element algorithms for contact problems, Archives of Computational Methods in Engineering, vol.33, issue.4, pp.1-49, 1995.
DOI : 10.1007/BF02736195

J. T. Oden and E. B. Pires, Algorithms and numerical results for finite element approximations of contact problems with non-classical friction laws, Computers & Structures, vol.19, issue.1-2, pp.137-147, 1984.
DOI : 10.1016/0045-7949(84)90212-8

J. O. Hallquist, G. L. Goudreau, and D. J. Benson, Sliding interfaces with contact-impact in large-scale Lagrangian computations, Computer Methods in Applied Mechanics and Engineering, vol.51, issue.1-3, pp.1-3107, 1985.
DOI : 10.1016/0045-7825(85)90030-1

B. Anil, K. Chaudhary, and . Bathe, A solution method for static and dynamic analysis of three-dimensional contact problems with friction, Computers & Structures, vol.24, issue.6, pp.855-873, 1986.

F. J. Gallego and J. J. Anza, A mixed finite element model for the elastic contact problem, International Journal for Numerical Methods in Engineering, vol.18, issue.4, pp.1249-1264, 1989.
DOI : 10.1002/nme.1620280603

N. J. Carpenter, R. L. Taylor, and M. G. Katona, Lagrange constraints for transient finite element surface contact, International Journal for Numerical Methods in Engineering, vol.1, issue.1, pp.103-128, 1991.
DOI : 10.1002/nme.1620320107

URL : https://hal.archives-ouvertes.fr/hal-01389918

R. Glowinski and P. L. Tallec, Numerical Solution of Problems in Incompressible Finite Elasticity by Augmented Lagrangian Methods II. Three-Dimensional Problems, SIAM Journal on Applied Mathematics, vol.44, issue.4, pp.710-733, 1984.
DOI : 10.1137/0144051

C. Juan, P. Simo, R. L. Wriggers, and . Taylor, A perturbed lagrangian formulation for the finite element solution of contact problems, Computer Methods in Applied Mechanics and Engineering, vol.50, issue.2, pp.163-180, 1985.

T. A. Laursen, Formulation and Treatment of Frictional Contact Problems Using Finite Elements, 1992.

D. Vola, . Pratt, M. Jean, and . Raous, Consistent time discretization for dynamical frictional contact problems and complementarity techniques. Revue Européenne des éléments finis, pp.149-162, 1998.

L. Baillet and T. Sassi, Mixed finite element formulation in large deformation frictional contact problem, Revue Europ??enne des ??l??ments Finis, vol.9, issue.2, pp.287-304, 2005.
DOI : 10.1080/03605309208820864

URL : https://hal.archives-ouvertes.fr/hal-00368442

L. Baillet, Du mécanisme au contact ? modélisation par éléments finis, Journées Francophones de Tribologie, 2002.

J. Haslinger and J. C. Nedlec, Approximation of the signorini problem with friction, obeying the coulomb law, Mathematical Methods in the Applied Sciences, vol.17, issue.1, pp.422-437, 1983.
DOI : 10.1002/mma.1670050127

L. Baillet and T. Sassi, Mixed finite element methods for the Signorini problem with friction, Numerical Methods for Partial Differential Equations, vol.331, issue.6, pp.1489-1508, 2006.
DOI : 10.1002/num.20147

URL : https://hal.archives-ouvertes.fr/insu-00355157

P. Coorevits, P. Hild, K. Lhalouani, and T. Sassi, Mixed finite element methods for unilateral problems: convergence analysis and numerical studies, Mathematics of Computation, vol.71, issue.237, pp.711-736, 2001.
DOI : 10.1090/S0025-5718-01-01318-7

G. Peillex, L. Baillet, and Y. Berthier, Strong coupling between finite elements and PML for the simulation of wave propagation in infinite medium, 2008.

D. Broek, Elementary Engineering Fracture Mechanics, 1991.

L. Gray, A. Phan, H. Glaucio, T. Paulino, and . Kaplan, Improved quarter-point crack tip element, Engineering Fracture Mechanics, vol.70, issue.2, pp.269-283, 2003.
DOI : 10.1016/S0013-7944(02)00027-9

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

T. Apel, A. Sändig, R. John, and . Whiteman, Graded Mesh Refinement and Error Estimates for Finite Element Solutions of Elliptic Boundary Value Problems in Non-smooth Domains, Mathematical Methods in the Applied Sciences, vol.19, issue.1, pp.63-85, 1996.
DOI : 10.1002/(SICI)1099-1476(19960110)19:1<63::AID-MMA764>3.0.CO;2-S

S. K. Chan, I. S. Tuba, and W. K. Wilson, On the finite element method in linear fracture mechanics, Engineering Fracture Mechanics, vol.2, issue.1, pp.1-17, 1970.
DOI : 10.1016/0013-7944(70)90026-3

P. Blanloeuil, A. J. Croxford, and A. Meziane, Numerical and experimental study of the nonlinear interaction between a shear wave and a frictional interface, The Journal of the Acoustical Society of America, vol.135, issue.4, 2012.
DOI : 10.1121/1.4868402

S. Liu, A. J. Croxford, S. A. Neild, and Z. Zhou, Effects of experimental variables on the nonlinear harmonic generation technique, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, vol.58, pp.1142-1451, 2011.

J. T. Hunt, M. R. Knittel, and D. Barach, Finite element approach to acoustic radiation from elastic structures, The Journal of the Acoustical Society of America, vol.55, issue.2, pp.269-280, 1974.
DOI : 10.1121/1.1914498

J. T. Hunt, M. R. Knittel, C. S. Nichols, and D. Barach, Finite???element approach to acoustic scattering from elastic structures, The Journal of the Acoustical Society of America, vol.57, issue.2, pp.287-299, 1975.
DOI : 10.1121/1.380459

P. Blanloeuil, A. Meziane, and C. Bacon, Nonlinear interaction of ultrasonic waves with a crack of different orientations, AIP Conference Proceedings, vol.1511, issue.1, pp.99-106, 2013.
DOI : 10.1063/1.4789036

P. Blanloeuil, A. Meziane, and C. Bacon, Numerical study of nonlinear interaction between a crack and elastic waves under an oblique incidence, Wave Motion, vol.51, issue.3, 2012.
DOI : 10.1016/j.wavemoti.2013.10.002

F. R. Rollins, INTERACTION OF ULTRASONIC WAVES IN SOLID MEDIA, Applied Physics Letters, vol.2, issue.8, pp.147-148, 1963.
DOI : 10.1063/1.1753818

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