M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, American Journal of Physics, vol.34, issue.2, 1972.
DOI : 10.1119/1.1972842

J. D. Achenbach, Wave Propagation in Elastic Solids, Journal of Applied Mechanics, vol.41, issue.2, 1973.
DOI : 10.1115/1.3423344

L. Achimastos, Appréciation de l'´ etat structurel d'une chaussée souplè a partir des observations de dégradations de surface, 1998.

G. Alfano, L. Rosati, and N. Valoroso, A numerical strategy for finite element analysis of no-tension materials, International Journal for Numerical Methods in Engineering, vol.48, issue.2, pp.317-350, 2000.
DOI : 10.1002/(SICI)1097-0207(20000530)48:3<317::AID-NME868>3.0.CO;2-C

J. Alias, La voie ferrée -Techniques de construction et d'entretien. Eyrolles,deuxì eme edition, 1984.

L. Andersen, S. R. Nielsen, and P. H. Kirkegaard, FINITE ELEMENT MODELLING OF INFINITE EULER BEAMS ON KELVIN FOUNDATIONS EXPOSED TO MOVING LOADS IN CONVECTED CO-ORDINATES, Journal of Sound and Vibration, vol.241, issue.4, pp.587-604, 2001.
DOI : 10.1006/jsvi.2000.3314

D. Aubry, D. Clouteau, and A. Modaressi, Interaction dynamique sol-structure, Calcul tridimensionnel en géotechnique Presse des Ponts et Chaussées, 1998.

E. Azanka, Ecoulements granulaires bidimensionnels sur un plan incliné, Ecole Nationale des Ponts et Chaussées, 1998.

K. Bagi, Stress and strain in granular assemblies, Mechanics of Materials, vol.22, issue.3, pp.165-177, 1996.
DOI : 10.1016/0167-6636(95)00044-5

M. C. Bakker, M. D. Verweij, B. J. Kooij, and H. A. Dieterman, The traveling point load revisited, Wave Motion, vol.29, issue.2, pp.119-135, 1999.
DOI : 10.1016/S0165-2125(98)00029-8

M. Balsan, Pour un modèle mathématique de la voie ferrée moderne, Ecole Nationale des Ponts et Chaussées, 1980.

M. Bats-villard, Influence des défaults de liaison sur le dimensionnement et le comportement de chaussées, 1991.

P. M. Belotserkovskiy, ON THE OSCILLATIONS OF INFINITE PERIODIC BEAMS SUBJECTED TO A MOVING CONCENTRATED FORCE, Journal of Sound and Vibration, vol.193, issue.3, pp.705-712, 1996.
DOI : 10.1006/jsvi.1996.0309

P. M. Belotserkovskiy, Forced oscillations of infinite periodic structures. Applications to railway track dynamics. Vehicle System Dynamics Supplement, pp.85-103, 1998.

V. Bodin, Comportement du ballast des voies ferrées soumisesàsoumises`soumisesà un chargement vertical et latéral, 2001.

V. Bodin, P. Tamagny, K. Sab, and P. Gautier, Fréquence critique de sollicitation lors du tassementàtassement`tassementà grand nombre de cycles d'un milieu granulaire. Colloque Physique et Mécanique des Matériaux Granulaires. Champs-sur-Marne, pp.291-296, 2000.

M. Bonnet, Equations intégrales etélémentsetéléments defrontì ere : applications en mécanique des solides et des fluides, 1995.

C. S. Chang and J. Gao, Non-linear dispersion of plane wave in granular media, International Journal of Non-Linear Mechanics, vol.30, issue.2, pp.111-128, 1995.
DOI : 10.1016/0020-7462(94)00039-D

C. S. Chang and J. Gao, Wave Propagation in Granular Rod Using High-Gradient Theory, Journal of Engineering Mechanics, vol.123, issue.1, pp.52-59, 1997.
DOI : 10.1061/(ASCE)0733-9399(1997)123:1(52)

Y. Chen and Y. Huang, Dynamic stiffness of infinite Timoshenko beam on viscoelastic foundation in moving co-ordinate, International Journal for Numerical Methods in Engineering, vol.2, issue.1, pp.1-18, 2000.
DOI : 10.1002/(SICI)1097-0207(20000510)48:1<1::AID-NME858>3.0.CO;2-G

J. Chung and G. M. Hulbert, A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-?? Method, Journal of Applied Mechanics, vol.60, issue.2, pp.371-375, 1993.
DOI : 10.1115/1.2900803

D. Clouteau, G. Degrande, and G. Lombaert, Modélisation des vibrations induites par le trafic routier er ferroviaire. Actes duCinquì eme Colloque National en Calcul des Structures, GIENS, 2001.

F. Collino and P. B. Monk, Optimizing the perfectly matched layer, Computer Methods in Applied Mechanics and Engineering, vol.164, issue.1-2, pp.157-171, 1998.
DOI : 10.1016/S0045-7825(98)00052-8

M. Cuomo and G. Ventura, A complementary energy formulation of no tension masonry-like solids, Computer Methods in Applied Mechanics and Engineering, vol.189, issue.1, pp.313-339, 2000.
DOI : 10.1016/S0045-7825(99)00298-4

URL : https://hal.archives-ouvertes.fr/hal-00875989

F. C. De-barros and J. E. Luco, Response of a layered viscoelastic half-space to a moving point load, Wave Motion, vol.19, issue.2, pp.189-210, 1994.
DOI : 10.1016/0165-2125(94)90066-3

F. C. De-barros and J. E. Luco, Stresses and displacements in a layered half-space for a moving line load, Applied Mathematics and Computation, vol.67, issue.1-3, pp.103-134, 1995.
DOI : 10.1016/0096-3003(94)00056-A

G. and D. Piero, Constitutive equation and compatibility of the external loads for linear elastic masonry-like materials, Meccanica, vol.90, issue.3, pp.150-162, 1989.
DOI : 10.1007/BF01559418

R. M. Delgado and R. C. Santos, Modelling of railway bridge-vehicle interaction on high speed tracks, Computers & Structures, vol.63, issue.3, pp.511-523, 1997.
DOI : 10.1016/S0045-7949(96)00360-4

H. Dieterman and A. Metrikine, Critical Velocities of a Harmonic Load Moving Uniformly Along an Elastic Layer, Journal of Applied Mechanics, vol.64, issue.3, pp.596-600, 1997.
DOI : 10.1115/1.2788934

H. Dieterman and A. Metrikine, Steady-state displacements of a beam on an elastic halfspace due to an uniformly moving constant load, Eur. J. Mech., A/Solids, vol.16, issue.2, pp.295-306, 1997.

H. A. Dieterman and A. Metrikine, The equivalent stiffness of a half-space interacting with a beam. Critical velocities of a moving load along the beam, Eur. J. Mech. A/Solids, vol.15, issue.1, pp.67-90, 1996.

J. Dinkel and H. Grundmann, Winkler parameters for railway dynamics derived from 3D halfspace analysis, Structural Dynamics -EURODYN. Balkema, 1999.

J. Duran, Sables, poudres et grains, Eyrolles, 1997.

T. Ekevid, M. X. Li, and N. Wiberg, Adaptive finite element analysis of wave propagation under moving loads induced by high-speech trains, European Congres on Computational Methods in Applied Sciences and Engineering -ECCOMAS2000, 2000.

T. Ekevid and N. Wiberg, On wave propagation problems associated to high-speed trains -Linear theory, European Conference on Computational Mechanics, 1975.

L. Fr´ybafr´yba, Vibration of Solids and Structures under Moving Loads, 1999.

S. Garvilov, Non-stationary problems in dynamics of a string on an elastic foundation subjected to a moving load, Journal of Sound and Vibration, vol.222, issue.3, pp.345-361, 1999.

M. Géradin and D. Rixen, Théorie des vibrations. ApplicationàApplication`Applicationà la dynamique des structures, 1996.

J. Ghabussi and R. Barbosa, Three-dimensional discrete element method for granular materials, International Journal for Numerical and Analytical Methods in Geomechanics, vol.5, issue.7, pp.451-472, 1990.
DOI : 10.1002/nag.1610140702

V. T. Granik and M. Ferrari, Microstructural mechanics of granular media, Mechanics of Materials, vol.15, issue.4, pp.301-322, 1993.
DOI : 10.1016/0167-6636(93)90005-C

H. Grundmann, M. Lieb, and E. Trommer, The response of a layered half-space to traffic loads moving along its surface, Archive of Applied Mechanics (Ingenieur Archiv), vol.69, issue.1, pp.55-67, 1999.
DOI : 10.1007/s004190050204

L. Gry, Modélisation du comportement dynamique d'une voie TGV pour la réduction du bruit de roulement, 1995.

N. Guérin, Approche expérimentale et numérique du comportement du ballast des voies férrées, Ecole Nationale des Ponts et Chaussées, 1996.

J. Hamet, Railway noise : Use of Timoshenko model in rail vibration studies, Acta Acustica, vol.85, pp.54-62, 1999.

M. S. Hardy, The generation of waves in infinite structures by moving harmonic loads, Journal of Sound and Vibration, vol.180, issue.4, pp.637-644, 1995.
DOI : 10.1006/jsvi.1995.0104

K. Henchi, M. Fafard, G. Dhatt, and M. Talbot, DYNAMIC BEHAVIOUR OF MULTI-SPAN BEAMS UNDER MOVING LOADS, Journal of Sound and Vibration, vol.199, issue.1, pp.33-50, 1997.
DOI : 10.1006/jsvi.1996.0628

H. M. Hilber, T. J. Hughes, and R. L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Engineering & Structural Dynamics, vol.7, issue.3, pp.283-292, 1977.
DOI : 10.1002/eqe.4290050306

T. J. Hughes, The Finite Element Method -Linear Static and Dynamic Finite Element Analysis, 2000.

T. J. Hughes and G. M. Hulbert, Space-time finite element methods for elastodynamics: Formulations and error estimates, Computer Methods in Applied Mechanics and Engineering, vol.66, issue.3, pp.339-363, 1987.
DOI : 10.1016/0045-7825(88)90006-0

G. M. Hulbert, Time finite element methods for structural dynamics, International Journal for Numerical Methods in Engineering, vol.3, issue.2, pp.307-331, 1992.
DOI : 10.1002/nme.1620330206

H. H. Hung and Y. B. Yang, Elastic waves in visco-elastic half-space generated by various vehicle loads. Soil Dynamics and Earthquake Engineering, pp.1-17, 2001.

K. L. Johnson, Contact mechanics, 1992.

C. J. Jones, X. Sheng, and M. Petyt, SIMULATIONS OF GROUND VIBRATION FROM A MOVING HARMONIC LOAD ON A RAILWAY TRACK, Journal of Sound and Vibration, vol.231, issue.3, pp.739-751, 2000.
DOI : 10.1006/jsvi.1999.2559

D. V. Jones, D. L. Houedec, A. T. Peplow, and M. Petyt, Ground vibration in the vicinity of a moving harmonic rectangular load on a half-space, European Journal of Mechanics - A/Solids, vol.17, issue.1, pp.153-166, 1998.
DOI : 10.1016/S0997-7538(98)80069-7

D. V. Jones and M. Petyt, Ground vibration in the vicinity of a strip load: An elastic layer on an elastic half-space, Journal of Sound and Vibration, vol.161, issue.1, pp.1-18, 1993.
DOI : 10.1016/0022-460X(93)90257-C

D. V. Jones and M. Petyt, Ground Vibration in the Vicinity of a Rectangular Load On a Half-Space, Journal of Sound and Vibration, vol.166, issue.1, pp.141-159, 1993.
DOI : 10.1006/jsvi.1993.1288

P. H. Kirkegaard, S. R. Nielsen, S. Krenk, and L. Kellezi, Radiation of Air-Borne Noise in Non-Homogeneous Wind and Temperature Fields using FEM Analysis, Structural Dynamics -EURODYN '99, 1999.

K. Knothe and Y. Wu, Receptance behaviour of railway track and subgrade, Archive of Applied Mechanics (Ingenieur Archiv), vol.68, issue.7-8, pp.457-470, 1998.
DOI : 10.1007/s004190050179

A. W. Kok, Lumped impulses, discrete displacements and moving load analysis, HERON, vol.42, issue.1, 1997.

A. W. Kok, Finite element models for the steady state analysis of moving loads, HERON, vol.45, issue.1, pp.53-61, 2000.

S. Krenk, L. Kellezi, S. R. Nielsen, and P. H. Kirkegaard, Finite elements and transmitting boundary conditions for moving loads, Structural Dynamics - EURODYN '99, 1999.

V. Krylov, Generation of ground vibrations by superfast trains, Applied Acoustics, vol.44, issue.2, pp.149-164, 1995.
DOI : 10.1016/0003-682X(95)91370-I

V. Krylov, Vibrational impact of high???speed trains. I. Effect of track dynamics, The Journal of the Acoustical Society of America, vol.100, issue.5, pp.3121-3134, 1996.
DOI : 10.1121/1.417123

V. Krylov, , 3121???3134 (1996)], The Journal of the Acoustical Society of America, vol.101, issue.6, p.3810, 1997.
DOI : 10.1121/1.419488

V. Krylov, A. R. Dawson, M. E. Heelis, and A. C. Collop, Rail movement and ground waves caused by high-speed trains approaching track-soil critical velocities, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, vol.49, issue.1, pp.107-116, 2000.
DOI : 10.1243/0954409001531379

T. Krzyzynski, T. Popp, and H. Kruse, The dynamics of raiway tracks based on the theory of periodic structures, ZAMM -Z. Angew. Math. Mech, vol.78, p.1, 1998.

P. and L. Tallec, IntroductionàIntroduction`Introductionà la dynamique des structures. Edition Ellipses, 2000.

G. Lefeuvre-mesgouez and D. L. Houedec, Propagation d'ondes dans un massif soumisàsoumis`soumisà des charges roulantes se déplaçantdéplaçant`déplaçantà vitessé elevée, 14e Congrès Français de Mécanique, 1999.

G. Leufeuve-mesgouez, Propagation d' ondes dans un massif soumisàsoumis`soumisà des charges se déplaçantdéplaçant`déplaçantà vitesse constante, 1999.

X. D. Li and N. Wiberg, Implementation and adaptivity of a space-time finite element method for structural dynamics, Computer Methods in Applied Mechanics and Engineering, vol.156, issue.1-4, pp.211-229, 1998.
DOI : 10.1016/S0045-7825(97)00207-7

M. Lieb and B. Sudret, A fast algorithm for soil dynamics calculations by wavelet decomposition, Archive of Applied Mechanics (Ingenieur Archiv), vol.68, issue.3-4, pp.147-157, 1998.
DOI : 10.1007/s004190050152

G. Lombaert, G. Degrande, and D. Clouteau, Numerical modelling of free field traffic-induced vibrations, Soil Dynamics and Earthquake Engineering, vol.19, issue.7, pp.473-488, 2000.
DOI : 10.1016/S0267-7261(00)00024-5

M. Lucchesi, C. Padovani, and G. Pasquinelli, On the numerical solution of equilibrium problems for elastic solids with bounded tensile strength, Computer Methods in Applied Mechanics and Engineering, vol.127, issue.1-4, pp.37-56, 1995.
DOI : 10.1016/0045-7825(95)00816-4

F. Maddalena and M. Ferrari, Viscoelasticity of granular materials, Mechanics of Materials, vol.20, issue.3, pp.241-250, 1995.
DOI : 10.1016/0167-6636(94)00064-6

S. Melin, Wave propagation in granular assemblie [80] A. Metrikine and H. Dieterman. Instability of vibration of a mass moving uniformly along an axial compressed beam on a viscoelastic foundation, Physical Review Journal of Sound and Vibration, vol.49, issue.2015, pp.567-576, 1994.

A. Metrikine and K. Popp, Vibration of a periodically supported beam on an elastic half-space, European Journal of Mechanics - A/Solids, vol.18, issue.4, pp.679-701, 1999.
DOI : 10.1016/S0997-7538(99)00141-2

A. Metrikine and A. C. Vrouwenvelder, SURFACE GROUND VIBRATION DUE TO A MOVING TRAIN IN A TUNNEL: TWO-DIMENSIONAL MODEL, Journal of Sound and Vibration, vol.234, issue.1, pp.43-66, 2000.
DOI : 10.1006/jsvi.1999.2853

A. V. Metrikine and H. A. Dieterman, Lateral vibrations of an axially compressed beam on an elastic half-space due to a moving lateral load, European Journal of Mechanics - A/Solids, vol.18, issue.1, pp.147-158, 1999.
DOI : 10.1016/S0997-7538(99)80008-4

H. Mühlhaus and F. Oka, Dispersion and wave propagation in discrete and continuous models for granular materials, International Journal of Solids and Structures, vol.33, issue.19, pp.2841-2858, 1996.
DOI : 10.1016/0020-7683(95)00178-6

V. H. Nguyen, D. Duhamel, and B. Nedjar, Calcul de structures soumisesàsoumises`soumisesà une charge mobile par la méthode desélémentsdeséléments finis. XVème Congrès Français de Mécanique, pp.3-7, 2001.

V. H. Nguyen, D. Duhamel, and B. Nedjar, A no-tension model for granular material, 2nd European Conference on Computational Mechanics -ECCM2001, 2001.

M. Olsson, On the fundamental moving load problem, Journal of Sound and Vibration, vol.145, issue.2, pp.299-307, 1991.
DOI : 10.1016/0022-460X(91)90593-9

X. Oviedo, Etude du comportement du ballast par un modèle micromécanique, Ecole Nationale des Ponts et Chaussées, 2001.

X. Oviedo, K. Sab, and P. Gautier, Etude du comportement du ballast ferroviaire par un modèle micromécanique. Colloque Physique et Mécanique des Matériaux Granulaires. Champs-sur-Marne, pp.339-344, 2000.

G. Pan, H. Okada, and S. N. Atluri, Nonlinear transient dynamic analysis of sol-pavement interaction under moving load : a couple BEM-FEM approach. Engineering Analysis with Boundary Element, pp.99-112, 1994.

T. Panzeca and G. D. Piero, Constitutive equations for no-tension materials, Meccanica, vol.9, issue.4, pp.88-93, 1988.
DOI : 10.1007/BF01556706

B. Picoux and D. L. Houedec, Propagation d'ondesàondes`ondesà la surface d'un massif sousmisàsousmis`sousmisà un effort latéral et vertical dûdû`dûà un traffic ferroviaire, XVème Congrès Français de Mécanique, 2001.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C - The Art of Scientific Computing, 1992.

V. Profillidis, La voie ferrée et sa fondation -modélisation mathématique, Ecole Nationale des Ponts et Chaussées, 1983.

K. M. Rassmusen, S. R. Nielsen, and P. H. Kirkegaard, Boundary element method solution in the time domain for a moving time-dependent force, Computers & Structures, vol.79, issue.7, pp.691-701, 2001.
DOI : 10.1016/S0045-7949(00)00175-9

C. Renard and P. Orsero, Influence de l'imperfection de la voie sur le comportement dynamique du pont au passage d'un train, 1999.

J. R. Rieker, Y. Lin, and M. W. Trethewey, Discretisation consideration in moving load finite element beam models. Finite Element in Analysis and Design, pp.129-144, 1996.

M. H. Sadd, J. Gao, and A. Shukla, Numerical analysis of wave propagation through assemblies of elliptical particles, Computers and Geotechnics, vol.20, issue.3-4, pp.323-343, 1997.
DOI : 10.1016/S0266-352X(97)00009-8

R. Sauvage and G. Larible, La modélisation par " ´ eléments finis " des couches d'assise de la voie ferrée, 1982.

X. Sheng, C. J. Jones, and M. Petyt, GROUND VIBRATION GENERATED BY A HARMONIC LOAD ACTING ON A RAILWAY TRACK, Journal of Sound and Vibration, vol.225, issue.1, pp.3-28, 1999.
DOI : 10.1006/jsvi.1999.2232

X. Sheng, C. J. Jones, and M. Petyt, GROUND VIBRATION GENERATED BY A LOAD MOVING ALONG A RAILWAY TRACK, Journal of Sound and Vibration, vol.228, issue.1, pp.129-156, 1999.
DOI : 10.1006/jsvi.1999.2406

A. Shulka, M. H. Sadd, and H. Mei, Experimental and computational modelling of wave propagation in granular materials, Experimental Mechanics, vol.30, issue.3, 1991.

A. S. Suiker, C. S. Chang, R. De-borst, and C. Esveld, Surface waves in a stratified half space with enhanced continuum properties. Part 1: Formulation of the boundary value problem, European Journal of Mechanics - A/Solids, vol.18, issue.5, pp.749-768, 1999.
DOI : 10.1016/S0997-7538(99)00108-4

A. S. Suiker, C. S. Chang, R. De-borst, and C. Esveld, Surface waves in a stratified half space with enhanced continuum properties. Part 2: Analysis of the wave characteristics in regard to high-speed railway tracks, European Journal of Mechanics - A/Solids, vol.18, issue.5, pp.769-784, 1999.
DOI : 10.1016/S0997-7538(99)00107-2

A. S. Suiker, A. V. Metrikine, and R. Borst, DYNAMIC BEHAVIOUR OF A LAYER OF DISCRETE PARTICLES, PART 2: RESPONSE TO A UNIFORMLY MOVING, HARMONICALLY VIBRATING LOAD, Journal of Sound and Vibration, vol.240, issue.1, pp.19-39, 2001.
DOI : 10.1006/jsvi.2000.3203

C. Wassgren, Vibration of Granular Materials, 1997.

J. P. Wolf and C. Song, Finite-Element Modelling of Unbounded Media, 1996.

A. R. Wolfert, H. A. Dietermann, and A. V. Metrikine, PASSING THROUGH THE ???ELASTIC WAVE BARRIER??? BY A LOAD MOVING ALONG A WAVEGUIDE, Journal of Sound and Vibration, vol.203, issue.4, pp.597-606, 1997.
DOI : 10.1006/jsvi.1996.0866

J. Wu, A. R. Whittaker, and M. P. , The use of finite element techniques for calculating the dynamic response of structures to moving loads, Computers & Structures, vol.78, issue.6, pp.789-799, 2000.
DOI : 10.1016/S0045-7949(00)00055-9

Y. Yang and H. Hung, A 2.5D finite/infinite element approach for modelling visco???elastic bodies subjected to moving loads, International Journal for Numerical Methods in Engineering, vol.15, issue.11, pp.1317-1336, 2001.
DOI : 10.1002/nme.208

URL : https://hal.archives-ouvertes.fr/hal-00608655

H. R. Yerli, B. Temel, and E. Kiral, Multi-wave transient and harmonic infinite elements for two-dimensional unbounded domain problems, Computers and Geotechnics, vol.24, issue.3, pp.185-206, 1999.
DOI : 10.1016/S0266-352X(99)00003-8

Y. Zhang and J. Ballman, Two techniques for the absorption of elastic waves using an artificial transition layer, Wave Motion, vol.25, issue.1, pp.15-33, 1997.
DOI : 10.1016/S0165-2125(96)00030-3

Y. Zhu, A. Shukla, and M. H. Sadd, The effect of microstructural fabric on dynamic load transfer in two dimensional assemblies of elliptical particles, Journal of the Mechanics and Physics of Solids, vol.44, issue.8, pp.441283-1303, 1996.
DOI : 10.1016/0022-5096(96)00036-1

H. S. Zibdeh and R. Rackwitz, MOVING LOADS ON BEAMS WITH GENERAL BOUNDARY CONDITIONS, Journal of Sound and Vibration, vol.195, issue.1, pp.85-102, 1996.
DOI : 10.1006/jsvi.1996.0405

O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method Solid and Fluid Mechanics Dynamics and Non-linearity, 1991.