Second Harmonic Generation of Guided Waves in Circular Cylinders: Analysis of Axisymmetric Torsional and Longitudinal Modes

Generally speaking, linear ultrasonics with bulk waves can detect anomalies of the order of the wavelength [1]. Ultrasonic guided waves can do significantly better by properly selecting a mode with wave structure sensitive to the defect [2]. Nonlinear ultrasonics, where the received signal containing the information of interest is at a different frequency than the emitted signal, can provide sensitivity to micro-structural changes [3]. The generation of measurable higher harmonics is extremely useful because these harmonics are sensitive to the very micro-structural features that cause the nonlinear elasticity, which means that they can be related to not merely macro-scale defects but precursors to them, such as dislocation density, nucleated voids, and precipitate coarsening [4]. Guided waves in cylindrical waveguides have received a lot of research efforts in recent years due to their extensive applications in NDE and SHM [5]. However, much less effort has been applied to the study of higher harmonic generation of guided waves due to the mathematical complexity of nonlinear wave equation and the boundary problem, even fewer study on harmonic generation of guided waves modes in cylindrical waveguides. De Lima and Hamilton [6-7] solved the nonlinear wave equation in cylindrical rods and shells by a normal mode expansion method with the perturbation assumption, then they investigated the cumulative condition for the longitudinal modes. Srivastva and Lanza di Scalea [8-9] studied the higher harmonic generation in rods of longitudinal modes, they claimed that the nature of the primary generating modes restricts the orders that are generated at the higher harmonics. The contribution of this work is that it considers the second harmonic generation in cylindrical waveguides by both torsional and longitudinal modes. It is concluded that only longitudinal mode secondary wave fields can be cumulative along the propagation distance, torsional wave modes cannot be in resonance with any type of fundamental wave fields. Both longitudinal and torsional type fundamental wave fields have the potential to generate cumulative second wave fields. Nonlinear finite element simulation have been made to demonstrate second harmonic generations. Experimental results confirmed the preferred excitation points can generate strongly cumulative second harmonics. Combining a cumulative harmonic with the penetration power of guided waves could be beneficial for nondestructive evaluation and eventually structural health monitoring.

References
1. Dace, G.E., R.B. Thompson, L.J.H. Brasche, D.K. Rehbein and O. Buck, “Nonlinear acoustics, a technique to detection microstructural changes in materials,” Review of progress in quantitative nondestructive evaluation, Vol. 10B 1991. 2. Puthillath, P. and J.L. Rose, “Ultrasonic guided wave inspection of a Titanium repair patch bonded to an Aluminum aircraft skin,” International Journal of Adhesion and Adhesives, 30:566-573, 2010. 3. Cantrell, J.H., “Fundamentals and Applications of Nonlinear Ultrasonic NDE,” Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, (T. Kundu, Ed.), CRC Press, Boca Raton, FL, 2003. 4. Cantrell, J.H., “Substructural organization, dislocation plasticity and harmonic generation in cyclically stressed wavy slip metals,” The Royal Society, A(2004) 460, 757-780. 5. Li, J. and J.L. Rose, “Excitation and Propagation of Non-axisymmetric guided waves in a Hollow Cylinder,” J. Acoust. Soc. Am., 109(2), 457-464, 2001. 6. de Lima, W.J.N. and M.F. Hamilton, “Finite-amplitude waves in isotropic elastic plates,” Journal of sound and vibration, 265(2003) 819-839. 7. de Lima, W.J.N. and M.F. Hamilton, “Finite-amplitude waves in isotropic elastic waveguides with arbitrary constant cross-sectional area,” Wave motion, 41(2005) 1-11. 8. Srivastava, A. and F.L. di Scalea, “On the existence of antisymmetric or symmetric Lamb waves at nonlinear higher harmonics,” Journal of Sound and Vibration, 323(2009) 932-943. 9. Srivastava, A. and F.L. di Scalea, “On the existence of longitudinal or flexural waves in rods at nonlinear higher harmonics,” Journal of Sound and Vibration, 329(2010) 1499-1506. 10. Auld, B.A., Acoustic fields and waves in solids, Robert E. Krieger Publishing Company, Malabar, FL, 1990.
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