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Elastic wave propagation and vibration characteristics of diamond-shaped metastructures

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Abstract

In this work, the elastic wave propagation behavior of the proposed diamond-shaped metastructure is investigated analytically based on the 8 degrees of freedom mass-spring discretized structure and numerically by finite element method (FEM), and an experiment was carried out with the 3D printed specimen. Analytically, the dispersion relation of the metastructure is derived based on Bloch’s theorem, and the transmittance of the metastructure with finite periods is investigated. Also, the extreme cases of the lattice structure parameters, such as the stiffness and mass, are investigated to examine different configuration effects. When the inner spring’s stiffness tends to infinity or infinitesimal, which is the rigid connection or string connection, it opens a lower boundary of 6.26 Hz with infinitesimal stiffness. Then cases with some specific masses tending to infinity or infinitesimal, representing fixed or missing mass, are discussed, which opens a lower boundary of 19.9 Hz with the largest mass. Also, tuning the FEM model parameters opens lower bandgaps, where a similar trend can be observed from the theoretical and simulation studies. Finally, the frequency response of the finite element solution is validated by conducting a vibration experiment on the 3D printed specimen, and a good agreement can be observed. The proposed metastructure can be utilized in the design of vibration isolators.

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References

  1. Schürch, P., Philippe, L.: Composite metamaterials: types and synthesis. Ref. Mod. Mater. Sci. Mater. Eng. 30, 1–12 (2021). https://doi.org/10.1016/b978-0-12-803581-8.11750-3

    Article  Google Scholar 

  2. Spadoni, A., Ruzzene, M., Gonella, S., Scarpa, F.: Phononic properties of hexagonal chiral lattices. Wave Motion 46(7), 435–450 (2009). https://doi.org/10.1016/j.wavemoti.2009.04.002

    Article  MathSciNet  MATH  Google Scholar 

  3. Fang, N., et al.: Ultrasonic metamaterials with negative modulus. Nat. Mater. 5(6), 452–456 (2006). https://doi.org/10.1038/nmat1644

    Article  Google Scholar 

  4. Hu, X., Ho, K.M., Chan, C.T., Zi, J.: Homogenization of acoustic metamaterials of Helmholtz resonators in fluid. Phys. Rev. B Condens. Matter. Mater. Phys. 77(17), 2–5 (2008). https://doi.org/10.1103/PhysRevB.77.172301

    Article  Google Scholar 

  5. Lee, S.H., Park, C.M., Seo, Y.M., Wang, Z.G., Kim, C.K.: Acoustic metamaterial with negative density. Phys. Lett. Sect. A General Atomic Solid State Phys. 373(48), 4464–4469 (2009). https://doi.org/10.1016/j.physleta.2009.10.013

    Article  Google Scholar 

  6. Argatov, I.I., Guinovart-Díaz, R., Sabina, F.J.: On local indentation and impact compliance of isotropic auxetic materials from the continuum mechanics viewpoint. Int J Eng Sci 54, 42–57 (2012). https://doi.org/10.1016/j.ijengsci.2012.01.010

    Article  Google Scholar 

  7. Chen, C.P., Lakes, R.S.: Micromechanical analysis of dynamic behavior of conventional and negative poisson’s ratio foams. J. Eng. Mater. Technol. 118(3), 285–288 (1996). https://doi.org/10.1115/1.2806807

    Article  Google Scholar 

  8. Liu, W., Wang, N., Luo, T., Lin, Z.: In-plane dynamic crushing of re-entrant auxetic cellular structure. Mater Des 100, 84–91 (2016). https://doi.org/10.1016/j.matdes.2016.03.086

    Article  Google Scholar 

  9. Roohani-Esfahani, S.I., Newman, P., Zreiqat, H.: Design and fabrication of 3d printed scaffolds with a mechanical strength comparable to cortical bone to repair large bone defects. Sci. Rep. 6(January), 1–8 (2016). https://doi.org/10.1038/srep19468

    Article  Google Scholar 

  10. Wang, G., Yu, D., Wen, J., Liu, Y., Wen, X.: One-dimensional phononic crystals with locally resonant structures. Phys. Lett. Sect. A General Atom Solid State Phys. 327(5–6), 512–521 (2004). https://doi.org/10.1016/j.physleta.2004.05.047

    Article  MATH  Google Scholar 

  11. Grima, J.N., Gatt, R., Alderson, A., Evans, K.E.: On the potential of connected stars as auxetic systems. Mol. Simul. 31(13), 925–935 (2005). https://doi.org/10.1080/08927020500401139

    Article  Google Scholar 

  12. Chang, I.L., Liang, Z.X., Kao, H.W., Chang, S.H., Yang, C.Y.: The wave attenuation mechanism of the periodic local resonant metamaterial. J. Sound Vib. 412, 349–359 (2018). https://doi.org/10.1016/j.jsv.2017.10.008

    Article  Google Scholar 

  13. Boucherif, A.: One dimensional lattice dynamics. Acta Mater. 1(1), 1–11 (2014)

    MathSciNet  Google Scholar 

  14. Dove, M.T.: Introduction to the theory of lattice dynamics. École thématique de la Société Française de la Neutronique 12, 123–159 (2011). https://doi.org/10.1051/sfn/201112007

    Article  Google Scholar 

  15. Jensen, J.S.: Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures. J. Sound Vib. 266(5), 1053–1078 (2003). https://doi.org/10.1016/S0022-460X(02)01629-2

    Article  Google Scholar 

  16. Hussein, M.I., Leamy, M.J., Ruzzene, M.: Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook. Appl. Mech. Rev. 66(4), 201 (2014). https://doi.org/10.1115/1.4026911

    Article  Google Scholar 

  17. Muhammad, C., Lim, W.: Ultrawide bandgap by 3D monolithic mechanical metastructure for vibration and noise control. Arch. Civ. Mechan. Eng. 21(2), 1–11 (2021). https://doi.org/10.1007/s43452-021-00201-x

    Article  Google Scholar 

  18. Tian, X., Chen, W., Gao, R., Liu, S.: Merging bragg and local resonance bandgaps in perforated elastic metamaterials with embedded spiral holes. J. Sound Vib. 500, 116036 (2021). https://doi.org/10.1016/j.jsv.2021.116036

    Article  Google Scholar 

  19. Li, H., Li, Y., Li, J.: Negative stiffness devices for vibration isolation applications: a review. Adv. Struct. Eng. 23(8), 1739–1755 (2020). https://doi.org/10.1177/1369433219900311

    Article  Google Scholar 

  20. Miranda, E.J.P., Nobrega, E.D., Rodrigues, S.F., Aranas, C., Dos Santos, J.M.C.: Wave attenuation in elastic metamaterial thick plates: analytical, numerical and experimental investigations. Int. J. Solids Struct. 204–205, 138–152 (2020). https://doi.org/10.1016/j.ijsolstr.2020.08.002

    Article  Google Scholar 

  21. Mizukami, K., Funaba, K., Ogi, K.: Design and three-dimensional printing of carbon-fiber-composite elastic metamaterials with inertial amplification mechanisms. J. Sound Vib. 513, 116412 (2021). https://doi.org/10.1016/j.jsv.2021.116412

    Article  Google Scholar 

  22. Li, Yingli, Yan, Gengwang: Vibration characteristics of innovative reentrant-chiral elastic metamaterials. Europ. J. Mech. A/Solids 90, 104350 (2021). https://doi.org/10.1016/j.euromechsol.2021.104350

    Article  MathSciNet  MATH  Google Scholar 

  23. Wen, S., Xiong, Y., Hao, S., Li, F., Zhang, C.: Enhanced bandgap properties of an acoustic metamaterial beam with periodically variable cross-sections. Int. J. Mech. Sci. 166, 2020 (2019). https://doi.org/10.1016/j.ijmecsci.2019.105229

    Article  Google Scholar 

  24. Goldsberry, B.M., Haberman, M.R.: Negative stiffness honeycombs as tunable elastic metamaterials. J. Appl. Phys. 123(9), 91711 (2018). https://doi.org/10.1063/1.5011400

    Article  Google Scholar 

  25. Zhou, J., Pan, H., Cai, C., Xu, D.: Tunable ultralow frequency wave attenuations in one-dimensional quasi-zero-stiffness metamaterial. Int. J. Mech. Mater. Des. 7, 285 (2020). https://doi.org/10.1007/s10999-020-09525-7

    Article  Google Scholar 

  26. Khajehtourian, R., Hussein, M.I.: Dispersion characteristics of a nonlinear elastic metamaterial. AIP Adv. 4(12), 124308 (2014). https://doi.org/10.1063/1.4905051

    Article  Google Scholar 

  27. W. P. Murphy, “Structural stability ZdeneÏ k P. BazÏ ant*.” [Online]. Available: www.elsevier.com/locate/ijsolstr

  28. Qi, D., Ren, Z., Qu, Z.: Valley-protected topological interface state of the elastic wave: From discrete model to multistable mechanical metamaterials. J. Sound Vib. 529, 116908 (2022). https://doi.org/10.1016/j.jsv.2022.116908

    Article  Google Scholar 

  29. Xu, R., He, Y., Li, X., Lu, M., Chen, Y.: Snap-fit mechanical metamaterials. Appl. Mater. Today. 30, 101714 (2023). https://doi.org/10.1016/j.apmt.2022.101714

    Article  Google Scholar 

  30. Ha, C.S., Lakes, R.S., Plesha, M.E.: Design, fabrication, and analysis of lattice exhibiting energy absorption via snap-through behavior. Mater Des 141, 426–437 (2018). https://doi.org/10.1016/j.matdes.2017.12.050

    Article  Google Scholar 

  31. Lin, Q., Zhou, J., Wang, K., Xu, D., Wen, G., Wang, Q.: Three-dimensional quasi-zero-stiffness metamaterial for low-frequency and wide complete band gap. Compos Struct. 307, 116656 (2023). https://doi.org/10.1016/j.compstruct.2022.116656

    Article  Google Scholar 

  32. H. Sonnerlind, What is the difference between plane stress and plane strain. In: What is the difference between plane stress and plane strain. 2021.

  33. Jin, W., Guo, H., Sun, P., Wang, Y.S.: Numerical investigation of discrepancies between two-dimensional and three-dimensional acoustic metamaterials. Frontiers in Mater. 8, 51675324 (2021). https://doi.org/10.3389/fmats.2021.759740

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (12172383). Thanks are due to Gengwang Yan for his assistance with experiments, numerical analysis and valuable discussion.

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Authors and Affiliations

  1. Key Laboratory of Traffic Safety on Track (Central South University), Ministry of Education, School of Traffic and Transportation Engineering, Central South University, Changsha, 410075, China

    Yingli Li, Ahmed Opeyemi Jamiu & Muhammad Zahradeen Tijjani

  2. Joint International Research Laboratory of Key Technology for Rail Traffic Safety, Central South University, Changsha, 410075, China

    Yingli Li

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  1. Yingli Li
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Contributions

YL: Supervision, Methodology, Funding acquisition, Concepetualization, Writing. JAO: Writing, Validation, Software, Investigation, Formal analysis, Data curation, Visualization. TMZ: Writing, Validation, Data curation.

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Correspondence to Yingli Li.

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Appendix

Appendix

There are some zero-frequency branches observed in Fig. 3, which attributes to the symmetrical arrangement of the unit cell. Obviously, the first two orders of zero-frequency branches found in the band structure Fig. 3b of the unit cell with the configuration angle of 45° are ascended to higher frequencies due to the reduced symmetries with the configuration angles of 60°, 36°, or 30°, as compared in Fig. 

Fig. 17
figure 17

Dispersion relation of the mass-spring model with configuration angle, (a) 60°, (b) 45°, (c) 36°, and (d) 30°

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17. In addition, the band structure of the unit cell with high symmetry (the configuration angle is 45°) shows near-zero group velocity for the 4,5-order branches, while this phenomenon is not found in other unit cells Fig. 17.

The eigenmodes corresponding to the band structures for the unit cells with various angles and symmetries are presented in Figs. 3c and

Fig. 18
figure 18figure 18

The eigenmodes at points Γ and M for the dispersion curve for the unit cells with different configuration angles, (a) 60°, (b) 36°, and (c) 30°

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18 to investigate the reason for the zero-frequency curves. The rigid body motion can be seen in eigenmodes for the first two-order curves of Fig. 3c. For the unit cell with the configuration angle of 45°, the restoring force between masses m1 and m2 is zero through Eqs. (1) and (2) (i.e., u1 = u2 = 0 and v1 = v2 = 0) for branch one, which results in the relative motion between masses m1 and m2 not found. While, for branch two, the restoring force between masses m3 and m4 are zero from Eqs. (3) and (4) (i.e., u3 = u4 = 0 and v3 = v4 = 0) which results in the relative motion between masses m3 and m4 is not found. Therefore, the eigenmodes with the relative motion of masses m1 and m2 for curve one and the relative motion of masses m3 and m4 for curve two cannot occur, leading to zero-frequency branches. As a result of altering the configuration angle and symmetries of the unit cells, there will be no zero-frequency branch.

At points Γ1 and M1, it was observed that masses m1 and m2 had zero deformation while masses m3 and m4 had deformation for the 45° unit cell. For the 60° unit cell, the deformation of all the masses are observed to be in the same direction for both Γ1 and M1, indicating a rigid body deformation for both points of symmetry. At point Γ1, it can be noticed that all of the masses for the 36° unit cell are deforming in the same direction. While at point M1, the structure exhibits shearing deformation, masses m1 and m2 deform in the same direction, and masses m3 and m4 deform in the same direction, showing that equal deformation of the masses acts parallel and opposed to one another. The deformation of the masses for 30° structures was observed to be in phase with one another, which illustrates a rigid body deformation.

At points Γ2 and M2, masses m1 and m2 exhibit deformation for the 45° unit cell in the same direction, whereas masses m3 and m4 exhibit no deformation for the 45° structure. There is a rigid body deformation at both points of symmetry for the 60° unit cell, as seen by noticing that all of the masses deform in the same direction at both Γ2 and M2. For 36° unit cell, it can be seen that all the masses are pointing in the same direction at point Γ2, but shearing deformation is visible at point M2, where masses m1 and m2 are pointing in the same direction and masses m3 and m4 are pointing in the same direction. There is a rigid body deformation at both points of symmetry for the 30° unit cell, as seen by the fact that all of the masses deform in the same direction for both Γ2 and M2.

The 45° unit cell exhibits a shearing deformation at points Γ3 and M3, where masses m1 and m2 deform in the same direction and masses m3 and m4 deform in the same direction. The deformation shows how the masses are moving in opposite but parallel directions. The 60° structure exhibits compression deformation, as a result of two masses acting towards one another and being directed along the same line, with masses m1 and m2 deforming in the same direction and masses m3 and m4 deforming in the same direction. The 36° unit cell illustrates rigid body motion by the deformation of all the masses simultaneously in the same direction. At points Γ3 and M3, the 30° unit cell exhibits stretching and compression, respectively. For the structure that undergoes stretching, masses m1 and m2 deform in the same direction and masses m3 and m4 deform in the same direction, as a result of two masses acting away from each other and directed along the same line, making the structure to be stretching.

The 45° unit cell, as well as all other configuration angles exhibit stretching deformation at points Γ4 and M4, with the exception of the 60° unit cell, which exhibits compression of masses m1 and m3. At points Γ5 and M5, stretching deformation can be noticed for all the structures including the 45° unit cell, with the exception of the 36° and 30° structures, where compression of masses m1 and m2 was observed. All configuration angles indicate that adjacent masses at points Γ6 and M6 are deforming in opposition to one another to create a standing wave. At points Γ7 and M7, all configuration angles exhibit a similar deformation, with m1 and m3 deforming in the same direction, while masses m2 and m4 deform in the same direction. The deformation behavior at points Γ8 and M8 is identical for the structures of various configuration angles, where all the masses deform in the same direction, with the exception of the Γ8 point at a 30° structure, where m2 and m4 are 90° out of phase with the adjacent masses (Fig. 19 ).

Fig. 19
figure 19

Experimental setup

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There are three distinct symmetry points in the dispersion curves of the symmetric structure with a configuration angle of 45°, and these are the Γ, X, and M points. When the configuration angle is changed to 60°, 36°, and 30°, the new points of symmetry are Γ, X, M, and Y. The dispersion curves typically reach maximum or minimum at these locations.

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Li, Y., Jamiu, A.O. & Tijjani, M.Z. Elastic wave propagation and vibration characteristics of diamond-shaped metastructures. Arch Appl Mech 93, 3921–3946 (2023). https://doi.org/10.1007/s00419-023-02468-3

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