Homogenization of subwavelength free stratified edge of viscoelastic media including finite size effect

2023;
: pp. 10–29
https://doi.org/10.23939/mmc2023.01.010
Received: August 04, 2022
Accepted: November 09, 2022
1
University Hassan II, Ens
2
University Mohammed V, Ensam
3
Solid Mechanics Laboratory, Ecole Polytechnique
4
University Hassan II, Ensam

This paper proposes the homogenization for a stratified viscoelastic media with free edge.  We consider the effect of two-dimensional periodically stratified slab over a semi-infinite viscoelastic ground on the propagation of shear waves hitting the interface.  Within the harmonic regime, the second order homogenization and matched-asymptotic expansions method is employed to derive an equivalent anisotropic slab associated with effective boundary and jump conditions for the displacement and the normal stress across an interface.  The reflection coefficients and the displacement fields are obtained in closed forms and their validity is inspected by comparison with direct numerics in the case of layers associated with Neumann boundary conditions.

  1. Cioranescu D., Donato P.  An Introduction to Homogenization.  No. 17 in Oxford Lecture Series in Mathematics and Its Applications.  Oxford, New York, Oxford University Press (1999).
  2. Li Q., Chen W., Liu S., Wang J.  A novel implementation of asymptotic homogenization for viscoelastic composites with periodic microstructures.  Composite Structures.  208, 276–286 (2019).
  3. Marigo J.-J., Maurel A.  Homogenization models for thin rigid structured surfaces and films.  The Journal of the Acoustical Society of America.  140 (1), 260–273 (2016).
  4. Marigo J.-J., Pideri C.  The Effective Behavior of Elastic Bodies Containing Microcracks or Microholes Localized on a Surface.  International Journal of Damage Mechanics.  20 (8), 1151–1177 (2011).
  5. Marigo J.-J., Maurel A, Pham K., Sbitti A.  Effective Dynamic Properties of a Row of Elastic Inclusions: The Case of Scalar Shear Waves.  Journal of Elasticity.  128 (2), 265–289 (2017).
  6. Delourme B.  High-order asymptotics for the electromagnetic scattering by thin periodic layers.  Mathematical Methods in the Applied Sciences.  38 (5), 811–833 (2015).
  7. Delourme B., Haddar H., Joly P.  Approximate Models for Wave Propagation across Thin Periodic Interfaces.  Journal de Mathématiques Pures et Appliquées.  98 (1), 28–71 (2012).
  8. Bonnet–Bendhia A. S., Drissi D., Gmati N.  Simulation of Muffler's Transmission Losses by a Homogenized Finite Element Method.  Journal of Computational Acoustics.  12 (03), 447–474 (2004).
  9. Marigo J.-J., Maurel A.  Second Order Homogenization of Subwavelength Stratified Media Including Finite Size Effect.  SIAM Journal on Applied Mathematics.  77 (2), 721–743 (2017).
  10. Marigo J.-J., Maurel A.  Supplementary Materials: Second Order Homogenization of Subwavelength Stratified Media Including Finite Size Effect.  13 (2017).
  11. Borcherdt R. D.  Viscoelastic Waves in Layered Media.  Cambridge, Cambridge University Press (2009).
  12. Maurel A., Pham K.  Multimodal method for the scattering by an array of plates connected to an elastic half-space.  The Journal of the Acoustical Society of America.  146 (6), 4402–4412 (2019).
  13. Gumerov N. A., Duraiswami R.  Fast Multipole Methods for the Helmholtz Equation in Three Dimensions.  Elsevier Series in Electromagnetism. 171–223 (2004).
  14. Marigo J.-J., Maurel A.  An Interface Model for Homogenization of Acoustic Metafilms.  World Scientific Handbook of Metamaterials and Plasmonics. 599–645 (2017).
  15. Petit R.  A Tutorial Introduction.  In: Petit R. (eds) Electromagnetic Theory of Gratings.  Topics in Current Physics, vol. 22. Springer, Berlin, Heidelberg (1980).  Petit R, (auth.), Petit P. R. (eds.).
  16. Maurel A., Félix S., Mercier J.-F., Ourir A.  Effective birefringence to analyze sound transmission through a layer with subwavelength slits.  Comptes Rendus Mécanique.  343 (12), 612–621 (2015).
  17. Lalanne P., Lemercier-Lalanne D.  Depth dependence of the effective properties of subwavelength gratings.  Journal of the Optical Society of America A.  14 (2), 450–459 (1997).
  18. Abdelmoula R., Marigo J.-J.  The effective behavior of a fiber bridged crack.  Journal of the Mechanics and Physics of Solids.  48 (11), 2419–2444 (2000).
  19. David M., Marigo J.-J., Pideri C.  Homogenized Interface Model Describing Inhomogeneities Located on a Surface.  Journal of Elasticity.  109 (2), 153–187 (2012).
Mathematical Modeling and Computing, Vol. 10, No. 1, pp. 10–29 (2023)