Homogenization of the Helmholtz problem in the presence of a row of viscoelastic inclusions

: pp. 899–917
Received: April 15, 2023
Revised: June 19, 2023
Accepted: June 22, 2023
University Hassan II, Ens
University Mohammed V, Ensam
University Hassan II, Ens
Solid Mechanics Laboratory, Ecole Polytechnique

We propose a homogenization method based on a matched asymptotic expansion technique to obtain the effective behavior of a periodic array of linear viscoelastic inclusions embedded in a linear viscoelastic matrix.  The problem is considered for shear waves and the wave equation in the harmonic regime is considered.  The obtained effective behavior is that of an equivalent interface associated to jump conditions, for the displacement and the normal stress at the interface.  The transmission coefficients and the displacement fields are obtained in closed forms and their validity is inspected by comparison with direct numerics in the case of a rectangular inclusions.

  1. Hubert J. S., Sanchez-Hubert J.  Introduction aux Méthodes Asymptotiques et à l'homogénéisation: Application à La Mécanique Des Milieux Continus.  Masson (1992).
  2. Bensoussan A., Lions J.-L., Papanicolaou G.  Asymptotic Analysis for Periodic Structures.  American Mathematical Society (2011).
  3. Lapine M., McPhedran R. C., Poulton C. G.  Slow Convergence to Effective Medium in Finite Discrete Metamaterials.  Physical Review B.  93 (23), 235156 (2016).
  4. 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).
  5. Marigo J.-J., Maurel A.  An Interface Model for Homogenization of Acoustic Metafilms.  World Scientific Series in Nanoscience and Nanotechnology, World Scientific Handbook of Metamaterials and Plasmonics. 599–645 (2017).
  6. 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 (2011).
  7. David M., Marigo J.-J., Pideri C.  Homogenized Interface Model Describing Inhomogeneities Located on a Surface.  Journal of Elasticity.  109 (2), 153–187 (2012).
  8. 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).
  9. Pham K., Maurel A., Marigo J.-J.  Revisiting Imperfect Interface Laws for Two-Dimensional Elastodynamics.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.  477 (2245), 20200519 (2021).
  10. Pham K., Maurel A., Marigo J.-J.  Two scale homogenization of a row of locally resonant inclusions – the case of anti-plane shear waves.  Journal of the Mechanics and Physics of Solids.  106, 80–94 (2017).
  11. 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).
  12. Delourme B., Haddar H., Joly P.  On the Well-Posedness, Stability and Accuracy of an Asymptotic Model for Thin Periodic Interfaces in Electromagnetic Scattering Problems.  Mathematical Models and Methods in Applied Sciences.  23 (13), 2433–2464 (2013).
  13. Ourir A., Gao Y., Maurel A., Marigo J.-J.  Homogenization of Thin and Thick Metamaterials and Applications.  Borja, Alejandro Lucas, InTech (2017).
  14. 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).
  15. Belemou R., Sbitti A., Marigo J.-J., Tsouli A.  Homogenization of subwavelength free stratified edge of viscoelastic media including finite size effect.  Mathematical Modeling and Computing.  10 (1), 10–29 (2023).
  16. Belemou R., Sbitti A., Marigo J.-J., El Amri H.  Homogenization of the Helmholtz Problem with Layered Viscoelastic Media Including Finite Size Effect.  IAENG International Journal of Applied Mathematics. 53 (1), 282–293 (2023).
  17. 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).
  18. Marigo J.-J., Maurel A.  Two-Scale Homogenization to Determine Effective Parameters of Thin Metallic–Structured Films.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.  472 (2192), 20160068 (2016).
  19. Maurel A., Marigo J.-J., Ourir A.  Homogenization of ultrathin metallo-dielectric structures leading to transmission conditions at an equivalent interface.  Journal of the Optical Society of America B.  33 (5), 947–956 (2016).
  20. Abdelmoula R., Coutris M., Marigo J.-J.  Comportement asymptotique d'une interphase élastique mince.  Comptes Rendus de l'Académie des Sciences – Series IIB – Mechanics-Physics-Chemistry-Astronomy.  326 (4), 237–242 (1998).
  21. Rizzoni R., Dumont S., Lebon F., Sacco E.  Higher order model for soft and hard elastic interfaces.  International Journal of Solids and Structures.  51 (23–24), 4137–4148 (2014).
  22. Rizzoni R., Dumont S., Lebon F.  On Saint Venant–Kirchhoff Imperfect Interfaces.  International Journal of Non-Linear Mechanics.  89, 101–115 (2017).
  23. Mercier J.-F., Marigo J.-J., Maurel A.  Influence of the neck shape for Helmholtz resonators.  The Journal of the Acoustical Society of America.  142 (6), 3703–3714 (2017).
  24. Maurel A., Marigo J.-J., Mercier J.-F., Pham K.  Modelling resonant arrays of the Helmholtz type in the time domain.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.  474 (2210), 20170894 (2018).
  25. Lebon F., Rizzoni R.  Asymptotic Behavior of a Hard Thin Linear Elastic Interphase: An Energy Approach.  International Journal of Solids and Structures.  48 (3), 441–449 (2011).
  26. Dumont S., Rizzoni R., Lebon F., Sacco E.  Soft and hard interface models for bonded elements.  Composites Part B: Engineering.  153, 480–490 (2018).
  27. Lebon F., Rizzoni R.  Higher Order Interfacial Effects for Elastic Waves in One Dimensional Phononic Crystals via the Lagrange–Hamilton's Principle.  European Journal of Mechanics – A/Solids.  67, 58–70 (2018).
  28. Capdeville Y., Marigo J.-J.  A Non-periodic two scale asymptotic method to take account of rough topographies for 2-D elastic wave propagation.  Geophysical Journal International.  192 (1), 163–189 (2013).
  29. Cioranescu D., Donato P.  An Introduction to Homogenization. Oxford Lecture Series in Mathematics and Its Applications. Vol. 17, Oxford University Press, Oxford, New York (1999).
  30. Borcherdt R. D.  Viscoelastic Waves in Layered Media.  Cambridge University Press, Cambridge (2009).
  31. Pham K., Maurel A., Mercier J.-F., Félix S., Cordero M. L., Horvath C.  Perfect Brewster transmission through ultrathin perforated films.  Wave Motion.  93, 102485 (2020).
  32. Gumerov N., Duraiswami R.  Fast Multipole Methods for the Helmholtz Equation in Three Dimensions.  Elsevier (2004).
  33. Petit R.  Electromagnetic Theory of Gratings.  Topics in Current Physics. Springer-Verlag, Berlin, Heidelberg (1980).
  34. 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).
  35. Abdelmoula R.  The effective behavior of a fiber bridged crack.  Journal of the Mechanics and Physics of Solids.  48 (11), 2419–2444 (2000).
  36. Delourme B.  High-order asymptotics for the electromagnetic scattering by thin periodic layers.  Mathematical Methods in the Applied Sciences.  38 (5), 811–833 (2015).
Mathematical Modeling and Computing, Vol. 10, No. 3, pp. 899–917 (2023)