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A robust method for surface wave dispersion in anisotropic semi-infinite periodically layered structures with coating layers

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Abstract

This study presents a robust method towards the computation of dispersion curves of surface waves in an anisotropic semi-infinite periodically layered structure (PLS) coated by a stack of layers. Using the properties of the symplectic transfer matrix to analyze the relationship between the eigenequations of surface waves in a semi-infinite PLS and of its corresponding finite PLS, the eigenfrequencies within the stopbands for surface waves in a semi-infinite PLS are solved by a finite PLS consisting of a sufficient number of unit cells. The eigenvalues of a finite PLS are calculated by combining the precise integration method with the Wittrick–Williams (W-W) algorithm, which makes the present method accurate, efficient and numerically stable. On this basis, using the concept of the eigenvalue count in the W-W algorithm, the eigenfrequencies of surface waves in a semi-infinite PLS coated by a stack of layers are accurately computed for any given wavenumber. Unlike most existing approaches that determine dispersion curves by directly seeking roots from the transcendental eigenequation, the present method takes full advantages of the eigenvalue count to find all eigenfrequencies without the possibility of any being missed. The method can be applicable for surface wave problems in an arbitrarily anisotropic semi-infinite PLS and does not restrict layer number, layer thickness and material properties.

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References

  1. Brûlé S, Javelaud EH, Enoch S, Guenneau S (2014) Experiments on seismic metamaterials: molding surface waves. Phys Rev Lett 112:133901

    Article  Google Scholar 

  2. Wen JH, Wang G, Yu DL, Zhao HG, Liu YZ (2005) Theoretical and experimental investigation of flexural wave propagation in straight beams with periodic structures: application to a vibration isolation structure. J Appl Phys 97:114907

    Article  Google Scholar 

  3. Wang FJ, Lu Y, Lee HP, Ma XJ (2019) Vibration and noise attenuation performance of compounded periodic struts for helicopter gearbox system. J Sound Vib 458:407–425

    Article  Google Scholar 

  4. Hussein MI, Leamy MJ, Ruzzene M (2014) Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Appl Mech Rev 66:040802

    Article  Google Scholar 

  5. Pennec Y, Vasseur JO, Djafari-Rouhani B, Dobrzyński L, Deymier PA (2010) Two-dimensional phononic crystals: examples and applications. Surf Sci Rep 65:229–291

    Article  Google Scholar 

  6. Xie LX, Xia BZ, Liu J, Huang GL, Lei JR (2017) An improved fast plane wave expansion method for topology optimization of phononic crystals. Int J Mech Sci 120:171–181

    Article  Google Scholar 

  7. Trainiti G, Rimoli JJ, Ruzzene M (2015) Wave propagation in periodically undulated beams and plates. Int J Solids Struct 75:260–276

    Article  Google Scholar 

  8. Chen AL, Yan DJ, Wang YS, Zhang CZ (2019) In-plane elastic wave propagation in nanoscale periodic piezoelectric/piezomagnetic laminates. Int J Mech Sci 153:416–429

    Article  Google Scholar 

  9. Fomenko SI, Golub MV, Zhang C, Bui TQ, Wang YS (2014) In-plane elastic wave propagation and band-gaps in layered functionally graded phononic crystals. Int J Solids Struct 51:2491–2503

    Article  Google Scholar 

  10. Langlet P, Hladky-Hennion AC, Decarpigny JN (1995) Analysis of the propagation of plane acoustic waves in passive periodic materials using the finite element method. J Acoust Soc Am 98:2792–2800

    Article  Google Scholar 

  11. Hussein MI, Hamza K, Hulbert GM, Scott RA, Saitou K (2006) Multiobjective evolutionary optimization of periodic layered materials for desired wave dispersion characteristics. Struct Multidiscip Optim 31:60–75

    Article  Google Scholar 

  12. Hussein MI, Hulbert GM, Scott RA (2006) Dispersive elastodynamics of 1D banded materials and structures: analysis. J Sound Vib 289:779–806

    Article  Google Scholar 

  13. Darinskii AN, Shuvalov AL (2018) Surface acoustic waves on one-dimensional phononic crystals of general anisotropy: existence considerations. Phys Rev B 98:024309

    Article  Google Scholar 

  14. Darinskii AN, Shuvalov AL (2019) Existence of surface acoustic waves in one-dimensional piezoelectric phononic crystals of general anisotropy. Phys Rev B 99:174305

    Article  Google Scholar 

  15. Shuvalov AL, Poncelet O, Golkin SV (2009) Existence and spectral properties of shear horizontal surface acoustic waves in vertically periodic half-spaces. Proc R Soc A 465:1489–1511

    Article  MathSciNet  MATH  Google Scholar 

  16. Shuvalov AL, Kutsenko AA, Korotyaeva ME, Poncelet O (2013) Love waves in a coated vertically periodic substrate. Wave Motion 50:809–820

    Article  MATH  Google Scholar 

  17. Kutsenko AA, Shuvalov AL (2013) Shear surface waves in phononic crystals. J Acoust Soc Am 133:653–660

    Article  Google Scholar 

  18. Hu LX, Liu LP, Bhattacharya K (2012) Existence of surface waves and band gaps in periodic heterogeneous half-spaces. J Elast 107:65–79

    Article  MathSciNet  MATH  Google Scholar 

  19. Camley RE, Djafari-Rouhani B, Dobrzynski L, Maradudin AA (1983) Transverse elastic waves in periodically layered infinite and semi-infinite media. Phys Rev B 27:7318

    Article  Google Scholar 

  20. Djafari-Rouhani B, Dobrzynski L, Duparc OH, Camley RE, Maradudin AA (1983) Sagittal elastic waves in infinite and semi-infinite superlattices. Phys Rev B 28:1711

    Article  Google Scholar 

  21. Nougaoui A, Rouhani BD (1987) Elastic waves in periodically layered infinite and semi-infinite anisotropic media. Surf Sci 185:125–153

    Article  Google Scholar 

  22. Pang Y, Liu YS, Liu JX, Feng WJ (2016) Propagation of SH waves in an infinite/semi-infinite piezoelectric/piezomagnetic periodically layered structure. Ultrasonics 67:120–128

    Article  Google Scholar 

  23. Chen S, Lin SY, Wang ZH (2009) Shear horizontal surface acoustic waves in semi-infinite piezoelectrics/metal superlattices. Ultrasonics 49:446–451

    Article  Google Scholar 

  24. Chen S, Zhang YH, Lin SY, Fu ZQ (2014) Study on the electromechanical coupling coefficient of Rayleigh-type surface acoustic waves in semi-infinite piezoelectrics/non-piezoelectrics superlattices. Ultrasonics 54:604–608

    Article  Google Scholar 

  25. Wang L, Rokhlin SI (2001) Stable reformulation of transfer matrix method for wave propagation in layered anisotropic media. Ultrasonics 39:413–424

    Article  Google Scholar 

  26. Huang JK, Liu W, Shi ZF (2017) Surface-wave attenuation zone of layered periodic structures and feasible application in ground vibration reduction. Constr Build Mater 141:1–11

    Article  Google Scholar 

  27. Pu XB, Shi ZF (2017) A novel method for identifying surface waves in periodic structures. Soil Dyn Earthq Eng 98:67–71

    Article  Google Scholar 

  28. Maznev AA, Every AG (2009) Surface acoustic waves in a periodically patterned layered structure. J Appl Phys 106:113531

    Article  Google Scholar 

  29. Wu TT, Huang ZG, Lin S (2004) Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy. Phys Rev B 69:094301

    Article  Google Scholar 

  30. Wu TT, Hsu ZC, Huang ZG (2005) Band gaps and the electromechanical coupling coefficient of a surface acoustic wave in a two-dimensional piezoelectric phononic crystal. Phys Rev B 71:064303

    Article  Google Scholar 

  31. Wu F, Gao Q, Xu XM, Zhong WX (2014) Expectation-based approach for one-dimensional randomly disordered phononic crystals. Phys Lett A 378:1043–1048

    Article  MathSciNet  MATH  Google Scholar 

  32. Khelif A, Adibi A (2016) Phononic crystals fundamentals and applications. Springer, New York

    Book  Google Scholar 

  33. Ren SY (2007) Electronic states in crystals of finite size: quantum confinement of Bloch waves. Springer, New York

    Google Scholar 

  34. Sigalas MM (1997) Elastic wave band gaps and defect states in two-dimensional composites. J Acoust Soc Am 101:1256–1261

    Article  Google Scholar 

  35. Sigalas MM (1998) Defect states of acoustic waves in a two-dimensional lattice of solid cylinders. J Appl Phys 84:3026–3030

    Article  Google Scholar 

  36. Ren SY, Chang YC (2010) Surface states/modes in one-dimensional semi-infinite crystals. Ann Phys 325:937–947

    Article  MathSciNet  MATH  Google Scholar 

  37. Alami M, El Boudouti EH, Djafari-Rouhani B, El Hassouani Y, Talbi A (2018) Surface acoustic waves in one-dimensional piezoelectric-metallic phononic crystal: effect of a cap layer. Ultrasonics 90:80–97

    Article  Google Scholar 

  38. Laude V, Wilm M, Benchabane S, Khelif A (2005) Full band gap for surface acoustic waves in a piezoelectric phononic crystal. Phys Rev E 71:036607

    Article  Google Scholar 

  39. Kutsenko AA (2014) Wave propagation through periodic lattice with defects. Comput Mech 54:1559–1568

    Article  MathSciNet  MATH  Google Scholar 

  40. Wittrick WH, Williams FW (1971) A general algorithm for computing natural frequencies of elastic structures. Q J Mech Appl Math 24:263–284

    Article  MathSciNet  MATH  Google Scholar 

  41. Gao Q, Zhang YH (2019) A novel method for shear horizontal surface waves in periodically layered semi-infinite structures with coating layers. J Sound Vib 450:61–77

    Article  Google Scholar 

  42. Yao WA, Zhong WX, Lim CW (2009) Symplectic elasticity. World Scientific, Singapore

    Book  MATH  Google Scholar 

  43. Zhong WX (2004) On precise integration method. J Comput Appl Math 163:59–78

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhong WX, Williams FW, Bennett PN (1997) Extension of the Wittrick–Williams algorithm to mixed variable systems. ASME J Vib Acoust 119:334–340

    Article  Google Scholar 

  45. Zhong WX, Williams FW (1995) On the direct solution of wave propagation for repetitive structures. J Sound Vib 181:485–501

    Article  Google Scholar 

  46. Singh KV, Ram YM (2002) Transcendental eigenvalue problem and its applications. AIAA J 40:1402–1407

    Article  Google Scholar 

  47. Bateson LE, Kelmanson MA, Knudsen C (1999) Solution of a transcendental eigenvalue problem via interval analysis. Comput Math Appl 38:133–142

    Article  MathSciNet  MATH  Google Scholar 

  48. Wang L, Rokhlin SI (2004) Recursive geometric integrators for wave propagation in a functionally graded multilayered elastic medium. J Mech Phys Solids 52:2473–2506

    Article  MathSciNet  MATH  Google Scholar 

  49. Thomson WT (1950) Transmission of elastic waves through a stratified solid medium. J Appl Phys 21:89–93

    Article  MathSciNet  MATH  Google Scholar 

  50. Sun L, Pan YY, Gu WJ (2013) High-order thin layer method for viscoelastic wave propagation in stratified media. Comput Methods Appl Mech Eng 257:65–76

    Article  MathSciNet  MATH  Google Scholar 

  51. Gao Q, Zhong WX, Howson WP (2004) A precise method for solving wave propagation problems in layered anisotropic media. Wave Motion 40:191–207

    Article  MATH  Google Scholar 

  52. Gao Q, Lin JH, Zhong WX, Howson WP, Williams FW (2006) A precise numerical method for Rayleigh waves in a stratified half space. Int J Numer Methods Eng 67:771–786

    Article  MATH  Google Scholar 

  53. Gao Q, Zhang YH (2019) Stable and accurate computation of dispersion relations for layered waveguides, semi-infinite spaces and infinite spaces. ASME J Vib Acoust 141:031012

    Article  Google Scholar 

  54. Kennedy D, Cheng RKH, Wei S, Arevalo FJA (2016) Equivalent layered models for functionally graded plates. Comput Struct 174:113–121

    Article  Google Scholar 

  55. Banerjee JR, Ananthapuvirajah A (2018) Free vibration of functionally graded beams and frameworks using the dynamic stiffness method. J Sound Vib 422:34–47

    Article  Google Scholar 

  56. Rasolofosaon PNJ, Zinszner BE (2002) Comparison between permeability anisotropy and elasticity anisotropy of reservoir rocks. Geophysics 67:230–240

    Article  Google Scholar 

  57. Mensch T, Rasolofosaon P (1997) Elastic-wave velocities in anisotropic media of arbitrary symmetry-generalization of Thomsen’s parameters ε, δ and γ. Geophys J Int 128:43–64

    Article  Google Scholar 

  58. Nayfeh AH, Chimenti DE (1989) Free wave propagation in plates of general anisotropic media. ASME J Appl Mech 56:881–886

    Article  MATH  Google Scholar 

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Acknowledgements

The authors are grateful for the financial support of the Natural Science Foundation of China (No. 11972107, 91748203) and the Fundamental Research Funds for the Central Universities (DUT2019TD37).

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  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian, 116023, P. R. China

    Yanhui Zhang & Qiang Gao

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  1. Yanhui Zhang
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Correspondence to Qiang Gao.

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Zhang, Y., Gao, Q. A robust method for surface wave dispersion in anisotropic semi-infinite periodically layered structures with coating layers. Comput Mech 67, 1409–1430 (2021). https://doi.org/10.1007/s00466-021-01995-6

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