Acta Mechanica Sinica

, Volume 33, Issue 4, pp 754–762 | Cite as

Interesting effects in harmonic generation by plane elastic waves

Research Paper


The harmonics of plane longitudinal and transverse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a primary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.


Cubic nonlinearity Third harmonic Quadratically cumulative behavior Interface Compensatory wave 



The work was supported by the National Natural Science Foundation of China (Grants 11621062 and 11532001) and the China Scholarship Council (CSC).


  1. 1.
    Hikata, A., Chick, B.B., Elbaum, C.: Dislocation contribution to the second harmonic generation of ultrasonic waves. J. Appl. Phys. 36, 229–236 (1965)CrossRefGoogle Scholar
  2. 2.
    Hikata, A., Sewell Jr, F.A., Elbaum, C.: Generation of ultrasonic second and third harmonics due to dislocations. II. Phys. Rev. 151, 442–449 (1966)CrossRefGoogle Scholar
  3. 3.
    Hikata, A., Elbaum, C.: Generation of ultrasonic second and third harmonics due to dislocations. I. Phys. Rev. 144, 469–477 (1966)CrossRefGoogle Scholar
  4. 4.
    Pruell, C., Kim, J.Y., Qu, J., et al.: Evaluation of plasticity driven material damage using Lamb waves. Appl. Phys. Lett. 91, 231911 (2007)CrossRefGoogle Scholar
  5. 5.
    Cantrell, J.H., Yost, W.T.: Nonlinear ultrasonic characterization of fatigue microstructures. Int. J. Fatigue 23, 487–490 (2001)CrossRefGoogle Scholar
  6. 6.
    Kim, J.Y., Jacobs, L.J., Qu, J., et al.: Experimental characterization of fatigue damage in a nickel-base superalloy using nonlinear ultrasonic waves. J. Acoust. Soc. Am. 120, 1266–1273 (2006)CrossRefGoogle Scholar
  7. 7.
    Frouin, J., Matikas, T.E., Na, J.K., et al.: In-situ monitoring of acoustic linear and nonlinear behavior of titanium alloys during cycling loading. In: Nondestructive Evaluation of Aging Materials and Composites III, Newport Beach, February 8 (1999)Google Scholar
  8. 8.
    Korneev, V.A., Demčenko, A.: Possible second-order nonlinear interactions of plane waves in an elastic solid. J. Acoust. Soc. Am. 135, 591–598 (2014)Google Scholar
  9. 9.
    Chen, Z., Tang, G., Zhao, Y., et al.: Mixing of collinear plane wave pulses in elastic solids with quadratic nonlinearity. J. Acoust. Soc. Am. 136, 2389–2404 (2014)CrossRefGoogle Scholar
  10. 10.
    Tang, G., Jacobs, L.J., Qu, J.: Scattering of time-harmonic elastic waves by an elastic inclusion with quadratic nonlinearity. J. Acoust. Soc. Am. 131, 2570–2578 (2012)CrossRefGoogle Scholar
  11. 11.
    Wang, Y., Achenbach, J.D.: The effect of cubic material nonlinearity on the propagation of torsional wave modes in a pipe. J. Acoust. Soc. Am. 140, 3874–3883 (2016)CrossRefGoogle Scholar
  12. 12.
    Liu, Y., Chillara, V.K., Lissenden, C.J., et al.: Third harmonic shear horizontal and Rayleigh Lamb waves in weakly nonlinear plates. J. Appl. Phys. 114, 114908 (2013)CrossRefGoogle Scholar
  13. 13.
    Lissenden, C.J., Liu, Y., Choi, G.W., et al.: Effect of localized microstructure evolution on higher harmonic generation of guided waves. J. Nondestruct. Eval. 33, 178–186 (2014)CrossRefGoogle Scholar
  14. 14.
    Rénier, M., Gennisson, J.L., Barrière, C., et al.: Fourth-order shear elastic constant assessment in quasi-incompressible soft solids. Appl. Phys. Lett. 93, 101912 (2008)CrossRefGoogle Scholar
  15. 15.
    Chillara, V.K., Lissenden, C.J.: Constitutive model for third harmonic generation in elastic solids. Int. J. Non Linear Mech. 82, 69–74 (2016)CrossRefGoogle Scholar
  16. 16.
    Hamilton, M.F., Ilinskii, Y.A., Zabolotskaya, E.A.: Separation of compressibility and shear deformation in the elastic energy density (L). J. Acoust. Soc. Am. 116, 41–44 (2004)CrossRefGoogle Scholar
  17. 17.
    Zhou, S., Shui, Y.: Nonlinear reflection of bulk acoustic waves at an interface. J. Appl. Phys. 72, 5070–5080 (1992)CrossRefGoogle Scholar
  18. 18.
    Deng, M.: Cumulative second-harmonic generation of Lamb-mode propagation in a solid plate. J. Appl. Phys. 85, 3051–3058 (1999)CrossRefGoogle Scholar
  19. 19.
    Bender, F.A., Kim, J.Y., Jacobs, L.J., et al.: The generation of second harmonic waves in an isotropic solid with quadratic nonlinearity under the presence of a stress-free boundary. Wave Motion 50, 146–161 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nagy, P.B.: Fatigue damage assessment by nonlinear ultrasonic materials characterization. Ultrasonics 36, 375–381 (1998)CrossRefGoogle Scholar
  21. 21.
    Zhang, Z., Nagy, P.B., Hassan, W.: Analytical and numerical modeling of non-collinear shear wave mixing at an imperfect interface. Ultrasonics 65, 165–176 (2016)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Engineering MechanicsZhejiang UniversityHangzhouChina
  2. 2.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA

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