A. Abdulle and P. Henning, Chapter 20 -Multiscale Methods for Wave Problems in Heterogeneous Media, Handbook of Numerical Methods for Hyperbolic Problems, vol.18, pp.545-576, 2017.

J. D. Achenbach and A. N. Norris, Loss of specular reflection due to nonlinear crack-face interaction, Journal of Nondestructive Evaluation, vol.3, issue.4, pp.229-239, 1982.

I. V. Andrianov, V. V. Danishevs'kyy, H. Topol, and D. Weichert, Homogenization of a 1D nonlinear dynamical problem for periodic composites, ZAMM -Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, vol.91, issue.6, pp.523-534, 2011.

Y. C. Angel and J. D. Achenbach, Reflection of ultrasonic waves by an array of microcracks, Review of Progress in Quantitative Nondestructive Evaluation, pp.83-89, 1985.

R. Assier, M. Touboul, B. Lombard, and C. Bellis, High-frequency homogenisation in periodic media with imperfect interfaces
URL : https://hal.archives-ouvertes.fr/hal-02615237

J. Baik and R. B. Thompson, Ultrasonic scattering from imperfect interfaces: A quasi-static model, J. Nondestruct. Eval, vol.4, issue.3-4, pp.177-196, 1984.

S. C. Bandis, A. C. Lumsden, and N. R. Barton, Fundamentals of rock joint deformation, International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, vol.20, issue.6, pp.249-268, 1983.

A. Bensoussan, J. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, 2011.

D. Broda, W. J. Staszewski, A. Martowicz, T. Uhl, and V. V. Silberschmidt, Modelling of nonlinear crack-wave interactions for damage detection based on ultrasound-a review, Journal of Sound and Vibration, vol.333, issue.4, pp.1097-1118, 2014.

R. Cornaggia and B. B. Guzina, Second-order homogenization of boundary and transmission conditions for one-dimensional waves in periodic media, International Journal of Solids and Structures, pp.88-102, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02297483

R. Cornaggia and B. Lombard, Transient waves in 1D periodic media : second-order homogenization of transmission conditions and source points

C. D. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2005.

E. Godlewski and P. A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, 1996.

V. Gusev, B. Castagnède, and A. Moussatov, Hysteresis in response of nonlinear bistable interface to continuously varying acoustic loading, Ultrasonics, vol.41, issue.8, pp.643-654, 2003.

Z. Hashin, Thermoelastic properties of fiber composites with imperfect interface, Mechanics of Materials, vol.8, pp.333-348, 1990.

Z. Hashin, Thin interphase/imperfect interface in elasticity with application to coated fiber composites, J. Mech. Phys. Solids, vol.50, issue.12, pp.2509-2537, 2002.

K. L. Johnson, Contact Mechanics, 1985.

S. Junca and B. Lombard, Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave, SIAM Journal on Applied Mathematics, vol.70, issue.3, pp.735-761, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00339279

S. Junca and B. Lombard, Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave, SIAM Journal on Applied Mathematics, vol.70, issue.3, pp.735-761, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00339279

S. Junca and B. Lombard, Interaction between periodic elastic waves and two contact nonlinearities, Mathematical Models and Methods in Applied Sciences, vol.22, issue.04, p.1150022, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00549051

J. Kevorkian and D. L. Bosley, Multiple-scale homogenization for weakly nonlinear conservation laws with rapid spatial fluctuations, Studies in Applied Mathematics, vol.101, issue.2, pp.127-183, 1998.

A. Lamacz, Dispersive effective models for waves in heterogeneous media, Mathematical Models and Methods in Applied Sciences, vol.21, issue.09, pp.1871-1899, 2011.

P. D. Lax, Hyperbolic systems of conservation laws ii, Comm. Pure Appl. Math, vol.10, issue.4, pp.537-566, 1957.

P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys, vol.5, issue.5, pp.611-613, 1964.

F. Lebon and R. Rizzoni, Asymptotic behavior of a hard thin linear elastic interphase: An energy approach, International Journal of Solids and Structures, vol.48, issue.3, pp.441-449, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00589975

F. Lebon and R. Rizzoni, Higher order interfacial effects for elastic waves in one dimensional phononic crystals via the Lagrange-Hamilton's principle, European Journal of Mechanics -A/Solids, vol.67, pp.58-70, 2018.

R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, 2002.

C. Licht, F. Lebon, and A. Léger, Dynamics of elastic bodies connected by a thin adhesive layer, Ultrasonic Wave Propagation in Non Homogeneous Media, vol.128, pp.99-110, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00574684

B. Lombard, Modélisation numérique de la propagation et de la diffraction d'ondes mécaniques, 2010.

B. Lombard and J. Piraux, How to incorporate the spring-mass conditions in finite-difference schemes, SIAM J. Sci. Comput, vol.24, issue.4, pp.1379-1407, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00004812

B. Lombard and J. Piraux, Modeling 1-D elastic P-waves in a fractured rock with hyperbolic jump conditions, J. Comput. App. Math, vol.204, pp.292-305, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00095896

J. C. López-realpozo, R. Rodríguez-ramos, R. Guinovart-díaz, J. Bravo-castillero, L. Fernández et al., Effective properties of non-linear elastic laminated composites with perfect and imperfect contact conditions, Adv. Mater. Struc, vol.15, issue.5, pp.375-385, 2008.

S. Nakagawa, K. T. Nihei, and L. R. Myer, Stop-pass behavior of acoustic waves in a 1D fractured system, J. Acoust. Soc. Am, pp.40-50, 2000.

H. Nassar, Q. He, and N. Auffray, Willis elastodynamic homogenization theory revisited for periodic media, Journal of the Mechanics and Physics of Solids, vol.77, pp.158-178, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01112307

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.

L. J. Pyrak-nolte, L. R. Myer, and N. G. Cook, Anisotropy in seismic velocities and amplitudes from multiple parallel fractures, J. Geophys. Res, pp.345-358, 1990.

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.

S. I. Rokhlin and Y. J. Wang, Analysis of boundary conditions for elastic wave interaction with an interface between two solids, J. Acoust. Soc. Am, vol.89, issue.2, pp.503-515, 1991.

E. Sanchez-palencia, Non-Homogeneous Media and Vibration Theory, 1978.

F. Santosa and W. W. Symes, A dispersive effective medium for wave propagation in periodic composites, SIAM Journal on Applied Mathematics, vol.51, issue.4, pp.984-1005, 1991.

M. Schoenberg, Elastic wave behavior across linear slip interfaces, J. Acoust. Soc. Am, vol.68, issue.5, pp.1516-1521, 1980.

I. Sevostianov, R. Rodriguez-ramos, R. Guinovart-diaz, J. Bravo-castillero, and F. J. Sabina, Connections between different models describing imperfect interfaces in periodic fiber-reinforced composites, International Journal of Solids and Structures, vol.49, issue.13, pp.1518-1525, 2012.

I. Y. Solodov, Ultrasonics of non-linear contacts: propagation, reflection and NDE-applications, Ultrasonics, vol.36, issue.1, pp.383-390, 1998.

J. R. Willis, Effective constitutive relations for waves in composites and metamaterials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.467, pp.1865-1879, 2011.

J. R. Willis, Variational principles for dynamic problems for inhomogeneous elastic media, Wave Motion, vol.3, issue.1, pp.1-11, 1981.