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Symplectic analysis for elastic wave propagation in two-dimensional cellular structures

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Abstract

On the basis of the finite element analysis, the elastic wave propagation in cellular structures is investigated using the symplectic algorithm. The variation principle is first applied to obtain the dual variables and the wave propagation problem is then transformed into two-dimensional (2D) symplectic eigenvalue problems, where the extended Wittrick–Williams algorithm is used to ensure that no phase propagation eigenvalues are missed during computation. Three typical cellular structures, square, triangle and hexagon, are introduced to illustrate the unique feature of the symplectic algorithm in higher-frequency calculation, which is due to the conserved properties of the structure-preserving symplectic algorithm. On the basis of the dispersion relations and phase constant surface analysis, the band structure is shown to be insensitive to the material type at lower frequencies, however, much more related at higher frequencies. This paper also demonstrates how the boundary conditions adopted in the finite element modeling process and the structures’ configurations affect the band structures. The hexagonal cells are demonstrated to be more efficient for sound insulation at higher frequencies, while the triangular cells are preferred at lower frequencies. No complete band gaps are observed for the square cells with fixed-end boundary conditions. The analysis of phase constant surfaces guides the design of 2D cellular structures where waves at certain frequencies do not propagate in specified directions. The findings from the present study will provide invaluable guidelines for the future application of cellular structures in sound insulation.

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References

  1. Yan Z.Z., Wang Y.S.: Calculation of band structures for surface waves in two-dimensional phononic crystals with a wavelet-based method. Phys. Rev. B (Condensed Matter and Materials Physics) 78(9), 094306–094311 (2008)

    MathSciNet  Google Scholar 

  2. Zhang X., Cheng G.D.: A comparative study of energy absorption characteristics of foam-filled and multi-cell square columns. Int. J. Impact Eng. 34(11), 1739–1752 (2007)

    Article  Google Scholar 

  3. Liu T., Deng Z.C., Lu T.J.: Bi-functional optimization of actively cooled, pressurized hollow sandwich cylinders with prismatic cores. J. Mech. Phys. Solids 55(12), 2565–2602 (2007)

    Article  MATH  Google Scholar 

  4. Lu T.J.: Heat transfer efficiency of metal honeycombs. Int. J. Heat Mass Transf. 42(11), 2031–2040 (1998)

    Article  Google Scholar 

  5. Gibson L.J., Ashby M.F.: Cellular Solids: Structure and Properties. Tsinghua University Press, Beijing (2003) (in Chinese)

    Google Scholar 

  6. Wang A.J., McDowell D.L.: In-plane stiffness and yield strength of periodic metal honeycombs. J. Eng. Mater. Technol. Trans. ASME 126, 137–156 (2004)

    Article  Google Scholar 

  7. Martinsson P.G., Movchan A.B.: Vibrations of lattice structures and phononic band gaps. Q. J. Mech. Appl. Math. 56, 45–64 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Zhen N., Yan Z.Z., Wang Y.S.: Elastic wave propagation in honeycomb materials. Chin. J. Theor. Appl. Mech. 40(6), 769–775 (2008) (in Chinese)

    Google Scholar 

  9. Ruzzene M., Scarpa F., Soranna F.: Wave beaming effects in two-dimensional cellular structures. Smart Mater. Struct. 12(3), 363–372 (2003)

    Article  Google Scholar 

  10. Jeong S.M., Ruzzene M.: Analysis of vibration and wave propagation in cylindrical grid-like structures. Shock Vib. 11(3-4), 311–331 (2004)

    Google Scholar 

  11. Gonella S., Ruzzene M.: Analysis of in-plane wave propagation in hexagonal and re-entrant lattices. J. Sound Vib. 312(1–2), 125–139 (2008)

    Article  Google Scholar 

  12. Liu S., Xianfan C., Dan H. et al.: Phononic band gaps and vibrations in truss-like materials. Chin. J. Appl. Mech. 24(2), 213–218 (2007)

    Google Scholar 

  13. Jensen J.S.: Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures. J. Sound Vib. 266(5), 1053–1078 (2003)

    Article  Google Scholar 

  14. Gonella S., Ruzzene M.: Homogenization of vibrating periodic lattice structures. Appl. Math. Model. 32(4), 459–482 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhang H.W., Yao Z., Wang J.B. et al.: Phonon dispersion analysis of carbon nanotubes based on inter-belt model and symplectic solution method. Int. J. Solids Struct. 44(20), 6428–6449 (2007)

    Article  MATH  Google Scholar 

  16. Zhong W.X., Williams F.W., Leung A.Y.T.: Symplectic analysis for periodical electro-magnetic waveguides. J. Sound Vib. 267(2), 227–244 (2003)

    Article  Google Scholar 

  17. Zhou M., Zhong W.X., Williams F.W.: Wave propagation in substructural chain-type structures excited by harmonic forces. Int. J. Mech. Sci. 35(11), 953–964 (1993)

    Article  MATH  Google Scholar 

  18. Zhong W.X.: Symplectic Solution Methodology in Applied Mechanics. Higher Education Press, Beijing (2006) (in Chinese)

    Google Scholar 

  19. Zhong W.X., Williams F.W.: The eigensolutions of wave propagation for repetitive structures. Struct. Eng. Mech. 1(1), 47–60 (1993)

    Google Scholar 

  20. Wittrick W.H., Williams F.W.: A general algorithm for computing natural frequencies of elastic structures. Q. J. Mech. Appl. Math. 24(3), 263–284 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  21. Williams F.W., Zhong W.X., Bennett P.N.: Computation of the eigenvalues of wave propagation in periodic substructural systems. J. Vib. Acoust. 115(4), 422–426 (1993)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an, 710072, China

    Xiu-Hui Hou, Zi-Chen Deng, Jia-Xi Zhou & Tie-Quan Liu

  2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Zi-Chen Deng

Authors
  1. Xiu-Hui Hou
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  2. Zi-Chen Deng
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  3. Jia-Xi Zhou
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  4. Tie-Quan Liu
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Correspondence to Zi-Chen Deng.

Additional information

The project was supported by the National Natural Science Foundation of China (10972182, 10772147, 10632030), the National Basic Research Program of China (2006CB 601202), the Doctorate Foundation of Northwestern Polytechnical University (CX200908), the Graduate Starting Seed Fund of Northwestern Polytechnical University (Z200930) and the NPU Foundation for Fundamental Research and the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (GZ0802).

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Hou, XH., Deng, ZC., Zhou, JX. et al. Symplectic analysis for elastic wave propagation in two-dimensional cellular structures. Acta Mech Sin 26, 711–720 (2010). https://doi.org/10.1007/s10409-010-0373-0

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  • DOI: https://doi.org/10.1007/s10409-010-0373-0

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