Известия РАН. Механика твердого тела, 2022, № 5, стр. 141-149

РАСПРОСТРАНЕНИЕ ПЛОСКОГО УДАРНОГО ФРОНТА В УПРУГОМ СЛОЕ

А. В. Ильяшенко a*

a Московский Государственный строительный университет
Москва, Россия

* E-mail: avi_56@mail.ru

Поступила в редакцию 24.01.2022
После доработки 03.02.2022
Принята к публикации 04.02.2022

Аннотация

Исследуется задача о волновом фронте в анизотропном упругом слое. Показано, что в случае упругой изотропии однородная волна с плоским фронтом в слое возможна лишь в одном частном случае, при нулевом коэффициенте Пуассона. В других случаях для существования волны с плоским фронтом, волна должна быть неоднородной по отношению к трансверсальной координате. Аналитическое решение, обеспечивающее существование плоского ударного волнового фронта, получено впервые.

Ключевые слова: анизотропия, волновой фронт, акустический тензор, упругий слой

Список литературы

  1. Strutt J.W. (Lord Rayleigh) On wave propagating along the plane surface of an elastic solid // Proc. London Math. Soc. 1885. V. 17. P. 4–11.

  2. Farnell G.W. Properties of elastic surface waves // Phys. Acoust. 1970. V. 6. P. 109–166.

  3. Ventura P., Hodre J.M., Desbois J., Solal M. Combined FEM and Green’s function analysis of periodic SAW structure, application to the calculation of reflection and scattering parameters // IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 2001. V. 48. P. 1259–1274.

  4. Synge J.L. Elastic waves in anisotropic media // J. Math. Phys. 1956. V. 35. P. 323–334.

  5. Stoneley R. The propagation of surface elastic waves in a cubic crystal // Proc. Roy. Soc. 1955. A232. P. 447–458.

  6. Stroh A.N. Steady state problems in anisotropic elasticity // J. Math. Phys. 1962. V. 41. P. 77–103.

  7. Lim T.C., Farnell G.W. Search for forbidden directions of elastic surface-wave propagation in anisotropic crystals // J. Appl. Phys. 1968. V. 39. P. 4319–4325.

  8. Lim T.C., Farnell G.W. Character of pseudo surface waves on anisotropic crystals // J. Acoust. Soc. Amer. 1969. V. 45. P. 845–851.

  9. Farnell G.W. Properties of elastic surface waves // Phys. Acoust. 1970. V. 6. P. 109–166.

  10. Bauerschmidt P., Lerch R., Machui J., Ruile W., Visintini G. Reflection and transmission coefficients of SAW in a periodic grating computed by finite element analysis // Proc. IEEE Ultrasonics Symposium. 1990. P. 421–423. https://doi.org/10.1109/ULTSYM.1990.171400

  11. Davies R.M. A critical study of the Hopkinson pressure bar // Phil. Trans. R. Soc. 1948. V. A240. P. 375–457.

  12. Mindlin R.D., Hermann G. A one-dimensional theory of compressive waves in an elastic rod // Proc. First U.S. National Congress Appl. Mech. Chicago, 1955.

  13. Haskell N.A. Dispersion of surface waves on multilayered media // Bull. Seismol. Soc. America. 1953. V. 43. № 1. P. 17–34.

  14. Knopoff L. A matrix method for elastic wave problems // Bull. Seismol. Soc. America. 1964. V. 54. № 1. P. 431–438.

  15. Graff K.F. Wave Motion in Elastic Solids. New York: Dover Publ., 1975. 649 p.

  16. Ting T.C.T. Anisotropic elasticity: theory and applications. New York: Oxford Univ. Press, 1996.

  17. Kravtsov A.V. et al. Finite element models in Lamb’s problem // Mech. Solids. 2011. V. 46. P. 952–959. https://doi.org/10.3103/S002565441106015X

  18. Fortunato D., Hale N., Townsend A. The ultraspherical spectral element method // J. Comput. Phys. 2021. V. 436. P. 110087. https://doi.org/10.1016/j.jcp.2020.110087

  19. Orszag S.A. Spectral methods for problems in complex geometries // J. Comput. Phys. 1980. V. 37. P. 70–92. https://doi.org/10.1016/0021-9991(80)90005-4

  20. Martinsson P. A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method // J. Comput. Phys. 2013. V. 242 P. 460–479. https://doi.org/10.1016/j.jcp.2013.02.019

  21. Babb T., Gillman A., Hao S., Martinsson P.-G. An accelerated Poisson solver based on multidomain spectral discretization // BIT Numer. Math. 2018. V. 58. P. 851–879. https://doi.org/10.1007/s10543-018-0714-0

  22. Chua C., Stoffab P.L. Nonuniform grid implicit spatial finite difference method for acoustic wave modeling in tilted transversely isotropic media // J. Appl. Geophys. 2012. V. 76. P. 44–49.

  23. Falk J., Tessmer E., Gajewski D. Efficient finite-difference modelling of seismic waves using locally adjustable time steps // Geophys. Prospecting. 1998. V. 46. P. 603–616.

  24. Moczo P., Kristek J., Galis M., Pazak P. On accuracy of the finite-difference and finite-element schemes with respect to p-wave to s-wave speed ratio // Geophys. J. Int. 2010. V. 182. P. 493–510.

  25. Antunes A.J.M., Leal-Toledo R.C.P., Filho O.T.S., Toledo, Elson M. Finite difference method for solving acoustic wave equation using locally adjustable time-steps // Procedia Computer Sci. 2014. V. 29. P. 627–636. https://doi.org/10.1016/j.procs.2014.05.056

  26. Oliveira S.A.M. A fourth-order finite-difference method for the acoustic wave equation on irregular grids // Geophys. 2003. V. 68 (2). P. 672–676.

  27. Wu T.W. Boundary Element Acoustics: Fundamentals and Computer Codes. Advances in Boundary Elements. Southampton, Boston: Witpress, 2000.

  28. Silva J.J.R. Acoustic and Elastic Wave Scattering using Boundary Elements. Southampton: Computational Mechanics Publications, 1994.

  29. Wang X., Chen H., Zhang J. An efficient boundary integral equation method for multi-frequency acoustics analysis // Eng. Anal. Boundary Elem. 2015. V. 61. P. 282–286. https://doi.org/10.1016/j.enganabound.2015.08.006

  30. Zhang Y.O., Zhang T., Ouyang H., Li T.Y. SPH simulation of acoustic waves: Effects of frequency, sound pressure, and particle spacing // Math. Problems Eng. 2015. V. 2015. P. 348314. P. 1–7. https://doi.org/10.1155/2015/348314

  31. Wang Sh., Zhang Y.O., Wu J.P. Lagrangian meshfree finite difference particle method with variable smoothing length for solving wave equations // Adv. Mech. Eng. 2018. V. 10 (7). P. 1–12. https://doi.org/10.1177/1687814018789248

  32. Cleckler J., Elghobashi S., Liu F. On the motion of inertial particles by sound waves // Phys. Fluids. 2012. V. 24 (3). P. 033301. https://doi.org/10.1063/1.3696243

  33. Gogoladze V.G. Dispersion of Rayleigh waves in a layer // Publ. Inst. Seism. Acad. Sci. U.R.S.S. 1947. V. 119. P. 27–38.

  34. Thomson W.T. Transmission of elastic waves through a stratified solid medium // J. Appl. Phys. 1950. V. 21 (2). P. 89–93.

  35. Kuznetsov S.V. SH-waves in laminated plates // Quart. Appl. Math. 2006. V. 64 (1). P. 153–165. https://doi.org/10.1090/s0033-569x-06-00992-1

  36. Evans R.B. The decoupling of seismic waves // Wave Motion. 1986. V. 8 (4). P. 321–328.

  37. Kuznetsov S.V. Abnormal dispersion of Lamb waves in stratified media // Z. Angew. Math. Phys. 2019. V. 70. P. 175. https://doi.org/10.1007/s00033-019-1222-z

  38. Ilyashenko A. et al. SH waves in anisotropic (monoclinic) media // Z. Angew. Math. Phys. 2018. V. 69. P. 17. https://doi.org/10.1007/s00033-018-0916-y

  39. Goldstein R.V. Rayleigh waves and resonance phenomena in elastic bodies // J. Appl. Math. Mech. 1965. V. 29 (3). P. 608–619. https://doi.org/10.1016/0021-8928(65)90066-3

  40. Goldstein R.V., Kuznetsov S.V. Long-wave asymptotics of Lamb waves // Mech. Solids. 2017. V. 52. P. 700–707. https://doi.org/10.3103/S0025654417060097

  41. Argatov I., Iantchenko A. Rayleigh surface waves in functionally graded materials – long-wave limit // 2019. Quart. J. Mech. Appl. Math. V. 72 (2) P. 197–211. https://doi.org/10.1093/qjmam/hbz002

  42. Kaplunov J., Prikazchikov D. Asymptotic theory for Rayleigh and Rayleigh-type waves // Adv. Appl. Mech. 2017. V. 50. P. 1–106. https://doi.org/10.1016/bs.aams.2017.01.001

  43. Craster R.V., Joseph L.M., Kaplunov J. Long-wave asymptotic theories: The connection between functionally graded waveguides and periodic media // Wave Motion 2014. V. 51 (4). P. 581–588. https://doi.org/10.1016/j.wavemoti.2013.09.007

  44. Wootton P.T., Kaplunov J., Prikazchikov D. A second order asymptotic model for Rayleigh waves on a linearly elastic half plane // IMA J. Appl. Math. 2020. V. 85 (1). P. 113–131. https://doi.org/10.1093/imamat/hxz037

  45. Djeran-Maigre I. et al. Solitary SH waves in two-layered traction-free plates // Comptes Rendus. Mec. 2008. V. 336 (1–2). P. 102–107. https://doi.org/10.1016/j.crme.2007.11.001

  46. Karman T., Duwez P. The propagation of plastic deformation in solids // J. Appl. Phys. 1950. V. 21. P. 987–994. https://doi.org/10.1063/1.1699544

  47. Knowles J. Impact-induced tensile waves in a rubberlike material // J. Appl. Math. 2002. V. 62. P. 1153–1175. /https://doi.org/10.1137/S0036139901388234

  48. Molinari A., Ravichandran G. Fundamental structure of steady plastic shock waves in metals // J. Appl. Phys. 2004. V. 95. P. 1718–1732. https://doi.org/10.1063/1.1640452

  49. Kuznetsova M., Khudyakov M., Sadovskii V. Wave propagation in continuous bimodular media // Mech. Adv. Mater. Struct. 2021. https://doi.org/10.1080/15376494.2021.1889725

  50. Hafskjold B., Bedeaux D., Kjelstrup S., Wilhelmsen Ø. Theory and simulation of shock waves: Entropy production and energy conversion // Phys. Rev. Ser. E. 2021. V. 104 (1). https://doi.org/10.1103/physreve.104.014131

  51. Ilyashenko A.V. et al. Pochhammer–Chree waves: polarization of the axially symmetric modes // Arch. Appl. Mech. 2018. V. 88. P. 1385–1394. https://doi.org/10.1007/s00419-018-1377-7

  52. Ilyashenko A.V. Pochhammer–Cree Longitudinal Waves: Anomalous Polarization // Mech. Solids. 2019. V. 54. P. 598–606. https://doi.org/10.3103/S0025654419040149

  53. Мокряков В.В. Максимумы напряжений в продольных волнах Похгаммера–Кри // Изв. РАН. МТТ. 2019. № 5. С. 86–103. https://doi.org/10.1134/S057232991905012X

  54. Гаджибеков Т.А., Ильяшенко А.В. Теоретические аспекты применения волн Похгаммера–Кри к задачам определения динамического коэффициента Пуассона // Изв. РАН. МТТ. 2021. № 5. С. 113–126. https://doi.org/10.31857/S0572329921050044

Дополнительные материалы отсутствуют.