Elastic waves blocking in a metamaterial with disordered cascade of doubly-periodic compliant disc-shaped inclusions

Keywords

elastic metamaterial, doubly-periodic compliant inclusions, multiple lattices, wave reflection and transmission coefficients, boundary integral equations method, wide-spacing model.

Cite as

Mykhas’kiv V. V. and Zhbadynskyi I. Ya. Elastic waves blocking in a metamaterial with disordered cascade of doubly-periodic compliant disc-shaped inclusions. Physicochemical Mechanics of Materials. 2025. 61(3), 091-098.

https://doi.org/10.15407/pcmm2025.03.091

Abstract

A three-dimensional problem of time-harmonic longitudinal wave propagation in an elastic matrix containing doubly-periodic arrays of disc-shaped compliant inclusions, arranged in a finite set of non-equidistant parallel square lattices, is studied. The calculation of wave reflection and transmission coefficients in such a metamaterial is based on a generalization of the wide-spacing model to account for varying distances between inclusion lattices in a formed cascade. The relevant quantities are obtained via the numerical solution of a boundary integral equation defined in the lattice unit cell. The results refer to wave incidence on a three-plane cascade of doubly-periodic inclusions (with cracks as a particular case) and the identification of suppression zones in the frequency spectrum of wave penetration, including the effect of complete locking, depending on the distances in the cascade, lattice compactness and stiffness of the scatterers.

References

1. Q. Lu, X. Li, X. Zhang, M. Lu, and Y. Chen, “Perspective: acoustic metamaterials in future engineering,” Engineering, 17, 22-30 (2022). https://doi.org/10.1016/j.eng.2022.04.020
2. K. Qiu, C. Fei, and W. Zhang, “Band-gap design of reconfigurable phononic crystals with joint optimization,” Mech. Adv. Mater. Struct., 31, Is. 3, 1-17 (2024). https://doi.org/10.1080/15376494.2022.2116662
3. Y. Li, C. Wu, Y. Peng, and X. Jiang, “Bandgap mechanisms and wave characteristics analysis of a three-dimensional elastic metastructure,” Int. J. Struct. Integr., 14, Is. 4, 564-582 (2023). https://doi.org/10.1108/IJSI-09-2022-0118
4. H. Isakari, K. Niino, H. Yoshikawa, and N. Nishimura, “Calderon’s preconditioning for periodic fast multipole method for elastodynamics in 3D,” Int. J. Numer. Method Eng., 90, Is. 4, 484-505 (2012). https://doi.org/10.1002/nme.3332
5. J. Zhao, Y. Li, and W. K. Liu, “Predicting band structure of 3D mechanical metamaterials with complex geometry via XFEM,” Comput. Mech., 55, Is. 4, 659-672 (2015). https://doi.org/10.1007/s00466-015-1129-2
6. Z. T. Nazarchuk, D. B. Kuryliak, V. V. Mykhaskiv, and V. F. Chekurin, Mathematical Modeling of Physical Fields Interaction with Material Defects, Prostir-M, Lviv (2018).
7. D. B. Kuryliak, Z. T. Nazarchuk, M. V. Voitko, and Y. P. Kulynych, “Diffraction of SH-waves on the interface defect in the joint of an elastic layer and a half space,” Mater. Sci., 57, No. 5, 612-625 (2022). https://doi.org/10.1007/s11003-022-00587-w
8. V. V. Mykhas’kiv, I. Ya. Zhbadynskyi, and Ch. Zhang, “On propagation of time-harmonic elastic waves through a double-periodic array of penny-shaped cracks,” Eur. J. Mech. A/Solids, 73, 306-317 (2019). https://doi.org/10.1016/j.euromechsol.2018.09.009
9. M. V. Golub, V. V. Kozhevnikov, S. I. Fomenko, E. A. Okoneshnikova, Y. Gu, Z. Y. Li, and D. J. Yan, “Advanced spectral boundary integral equation method for modeling wave propagation in elastic metamaterials with doubly periodic arrays of rectangular crack-like voids,” Eng. Anal. Bound. Elem., 161, 126-138 (2024). https://doi.org/10.1016/j.enganabound.2024.01.023
10. I. Ya. Zhbadynskyi, and V. V. Mykhas’kiv, “Acoustic filtering properties of 3D elastic metamaterials structured by crack-like inclusions,” in: XXIIIrd Int. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED) (September 24-28, Tbilisi, Georgia) (2018), pp. 145-148. https://doi.org/10.1109/DIPED.2018.8543137
11. A. Bostrom, P. Bovik, and P. A. Olsson, “Сomparison of exact first order and spring boundary conditions for scattering by thin layers,” J. Nondestr. Eval., 11, 175-184 (1992). https://doi.org/10.1007/BF00566408
12. P. A. Martin, Multiple Scattering: Interaction of Time-harmonic Waves with <I>N</I> Obstacles, Cambridge University Press., Cambridge (2006). https://doi.org/10.1017/CBO9780511735110
13. C. M. Linton and P. McIver, Handbook of Mathematical Techniques for Wave/Structure Interactions, Chapman & Hall/CRC, London (2001). https://doi.org/10.1201/9781420036060
14. D. A. Sotiropoulos, and J. D. Achenbach, “Ultrasonic reflection by a planar distribution of cracks,” J. Nondestr. Eval., 7, 123-129 (1988). https://doi.org/10.1007/BF00565997