An elastodynamic reciprocity theorem-based closed-form solution to second harmonic generation of lamb waves by a fatigue crack: Theory & experimental validation

Lei Xu, Yiyin Su, Kai Wang, Xiongbin Yang, Shenfang Yuan, Zhongqing Su

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)


Characterization of small-scale damage (with dimensions smaller than 1/10 of the probing wavelength) using nonlinear guided ultrasonic waves (GUWs) has been practiced over years, numerically and experimentally. To compensate for the insufficiency of analytical solutions that are able to interpret the underlying physical aspects of nonlinearity in GUWs induced by the small-scale damage and in particular fatigue damage, a new theoretical model based on the elastodynamic reciprocity theorem is developed. The model yields a closed-form solution to the modulation mechanism of a fatigue crack with ‘breathing’ attributes on Lamb wave propagation, gaining insight into the generation of second harmonics in Lamb waves. The model depicts the ‘breathing’ crack as an additional wave source imposing extra forces on crack surfaces that is equivalent to the integral of the stress tensor, and the source interferes with the wavefield of the original probing wave. In a time-frequency domain, this additional wave source is linked to the second harmonic generation in spectra. By virtue of the model, a nonlinear damage indicator, governed by the quantified second harmonic generated by the crack, is defined, to calibrate crack severity quantitatively. Finite element simulation is performed to verify the analytical model and demonstrate its accuracy when used for evaluating damage onset. Proof-of-concept experimental validation is conducted to verify the proportional trend of the damage indicator with respect to damage severity. This elastodynamic reciprocity-driven model and the closed-form solution shed light, from an analytical perspective, on the nonlinear interaction of GUWs with damage of small scale featuring ‘breathing’ attributes.

Original languageEnglish
Article number116226
JournalJournal of Sound and Vibration
Publication statusPublished - 29 Sep 2021


  • Contact acoustic nonlinearity
  • Damage detection
  • Elastodynamic reciprocity
  • Fatigue crack
  • Lamb wave
  • Second harmonic generation

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Acoustics and Ultrasonics
  • Mechanical Engineering

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