Higher-harmonic generation analysis in complex waveguides via a nonlinear semianalytical finite element algorithm.

*(English)*Zbl 1264.74133Summary: Propagation of nonlinear guided waves is a very attracting phenomenon for structural health monitoring applications that has received a lot of attention in the last decades. They exhibit very large sensitivity to structural conditions when compared to traditional approaches based on linear wave features. On the other hand, the applicability of this technology is still limited because of the lack of a solid understanding of the complex phenomena involved when dealing with real structures. In fact the mathematical framework governing the nonlinear guided wave propagation becomes extremely challenging in the case of waveguides that are complex in either materials (damping, anisotropy, heterogeneous, etc.) or geometry (multilayers, geometric periodicity, etc.). The present work focuses on the analysis of nonlinear second-harmonic generation in complex waveguides by extending the classical Semianalytical Finite Element formulation to the nonlinear regime, and implementing it into a powerful commercial Finite Element package. Results are presented for the following cases: a railroad track and a viscoelastic plate. For these case-studies optimum combinations of primary wave modes and resonant double-harmonic nonlinear wave modes are identified. Knowledge of such combinations is critical to the implementation of structural monitoring systems for these structures based on higher-harmonic wave generation.

##### MSC:

74J30 | Nonlinear waves in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

PDF
BibTeX
XML
Cite

\textit{C. Nucera} and \textit{F. L. Di Scalea}, Math. Probl. Eng. 2012, Article ID 365630, 16 p. (2012; Zbl 1264.74133)

Full Text:
DOI

##### References:

[1] | Dace, G. E. Thompson, R. B. Brasche, L. J. H. Rehbein, and D. K. Buck, “Nonlinear acoustic, a technique to determine microstructural changes in material,” Review of Progress in Quantitative Nondestructive Evaluation (QNDE), vol. 10, pp. 1685-1692, 1991. |

[2] | C. Bermes, J. Y. Kim, J. Qu, and L. J. Jacobs, “Experimental characterization of material nonlinearity using Lamb waves,” Applied Physics Letters, vol. 90, no. 2, Article ID 021901, 2007. · doi:10.1063/1.2431467 |

[3] | P. Cawley and D. Alleyne, “The use of Lamb waves for the long range inspection of large structures,” Ultrasonics, vol. 34, no. 2-5, pp. 287-290, 1996. |

[4] | J. L. Rose, Ultrasonic Waves in Solid Media, Cambridge University Press, Cambridge, UK, 1999. |

[5] | J. L. Rose, “Standing on the shoulders of giants: an example of guided wave inspection,” Materials Evaluation, vol. 60, no. 1, pp. 53-59, 2002. |

[6] | R. Ahmad, S. Banerjee, and T. Kundu, “Pipe wall damage detection in buried pipes using guided waves,” Journal of Pressure Vessel Technology, Transactions of the ASME, vol. 131, no. 1, pp. 0115011-01150110, 2009. · doi:10.1115/1.3027460 |

[7] | T. Kundu, S. Das, and K. V. Jata, “Health monitoring of a thermal protection system using lamb waves,” Structural Health Monitoring, vol. 8, no. 1, pp. 29-45, 2009. · doi:10.1177/1475921708090558 |

[8] | H. Reis, Nondestructive Testing and Evaluation for Manufacturing and Construction, Hemisphere, New York, NY, USA, 1990. |

[9] | T. Kundu, S. Banerjee, and K. V. Jata, “An experimental investigation of guided wave propagation in corrugated plates showing stop bands and pass bands,” Journal of the Acoustical Society of America, vol. 120, no. 3, pp. 1217-1226, 2006. · doi:10.1121/1.2221534 |

[10] | Z. A. Goldberg, “Interaction of plane longitudinal and transverse elastic waves,” Soviet Physics. Acoustics, pp. 306-310, 1960. |

[11] | L. K. Zarembo and V. A. Krasil’nikov, “Nonlinear phenomena in the propagation of elastic waves in solids,” Soviet Physics USPEKHI, vol. 13, no. 6, pp. 778-797, 1971. |

[12] | J. H. Cantrell, “Quantitative assessment of fatigue damage accumulation in wavy slip metals from acoustic harmonic generation,” Philosophical Magazine, vol. 86, no. 11, pp. 1539-1554, 2006. · doi:10.1080/14786430500365358 |

[13] | J. H. Cantrell and W. T. Yost, “Nonlinear ultrasonic characterization of fatigue microstructures,” International Journal of Fatigue, vol. 23, no. 1, pp. S487-S490, 2001. |

[14] | W. T. Yost and J. H. Cantrell, “The effects of fatigue on acoustic nonlinearity in aluminum alloys,” Proceedings of the IEEE, vol. 2, pp. 947-954, 1992. |

[15] | A. A. Shah and Y. Ribakov, “Non-linear ultrasonic evaluation of damaged concrete based on higher order harmonic generation,” Materials and Design, vol. 30, no. 10, pp. 4095-4102, 2009. · doi:10.1016/j.matdes.2009.05.009 |

[16] | N. Kim, et al., “Nonlinear behaviour of ultrasonic wave at a crack,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 1211 of AIP Conference Proceedings, pp. 313-318, 2010. |

[17] | I. Arias and J. D. Achenbach, “A model for the ultrasonic detection of surface-breaking cracks by the scanning laser source technique,” Wave Motion, vol. 39, no. 1, pp. 61-75, 2004. · Zbl 1163.74309 · doi:10.1016/j.wavemoti.2003.06.001 |

[18] | S. S. Kulkarni and J. D. Achenbach, “Structural health monitoring and damage prognosis in fatigue,” Structural Health Monitoring, vol. 7, no. 1, pp. 37-49, 2008. · doi:10.1177/1475921707081973 |

[19] | C. Bermes, J. Y. Kim, J. Qu, and L. J. Jacobs, “Nonlinear Lamb waves for the detection of material nonlinearity,” Mechanical Systems and Signal Processing, vol. 22, no. 3, pp. 638-646, 2008. · doi:10.1016/j.ymssp.2007.09.006 |

[20] | S. Küchler, T. Meurer, L. J. Jacobs, and J. Qu, “Two-dimensional wave propagation in an elastic half-space with quadratic nonlinearity: a numerical study,” Journal of the Acoustical Society of America, vol. 125, no. 3, pp. 1293-1301, 2009. · doi:10.1121/1.3075597 |

[21] | C. Nucera and F. Lanza Di Scalea, “Nonlinear ultrasonic guided waves for prestress level monitoring in prestressing strands for post-tensioned concrete structures,” Structural Health Monitoring-an International Journal, vol. 10, no. 6, pp. 617-629, 2011. · doi:10.1117/12.880291 |

[22] | W. J. N. De Lima and M. F. Hamilton, “Finite-amplitude waves in isotropic elastic plates,” Journal of Sound and Vibration, vol. 265, no. 4, pp. 819-839, 2003. · doi:10.1016/S0022-460X(02)01260-9 |

[23] | M. Deng, “Analysis of second-harmonic generation of Lamb modes using a modal analysis approach,” Journal of Applied Physics, vol. 94, no. 6, pp. 4152-4159, 2003. · doi:10.1063/1.1601312 |

[24] | A. Srivastava, I. Bartoli, S. Salamone, and F. Lanza Di Scalea, “Higher harmonic generation in nonlinear waveguides of arbitrary cross-section,” Journal of the Acoustical Society of America, vol. 127, no. 5, pp. 2790-2796, 2010. · doi:10.1121/1.3365247 |

[25] | F. D. Murnaghan, Finite Deformation of an Elastic Solid, Dover, New York, NY, USA, 1967. |

[26] | L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Addison-Wesley, London, UK, 1959. · Zbl 0178.28704 |

[27] | I. Bartoli, A. Marzani, F. Lanza di Scalea, and E. Viola, “Modeling wave propagation in damped waveguides of arbitrary cross-section,” Journal of Sound and Vibration, vol. 295, no. 3-5, pp. 685-707, 2006. · doi:10.1016/j.jsv.2006.01.021 |

[28] | B. A. Auld, “Application of microwave concepts to theory of acoustic fields and waves in solids,” IEEE Transactions on Microwave Theory and Techniques, vol. 17, no. 11, pp. 800-811, 1969. |

[29] | M. Deng, Y. Xiang, and L. Liu, “Time-domain analysis and experimental examination of cumulative second-harmonic generation by primary Lamb wave propagation,” Journal of Applied Physics, vol. 109, no. 11, Article ID 113525, 2011. · doi:10.1063/1.3592672 |

[30] | T. Hayashi, W. J. Song, and J. L. Rose, “Guided wave dispersion curves for a bar with an arbitrary cross-section, a rod and rail example,” Ultrasonics, vol. 41, no. 3, pp. 175-183, 2003. · doi:10.1016/S0041-624X(03)00097-0 |

[31] | COMSOL, COMSOL v4.2a Multiphysics User’s Guide2011: COMSOL, Inc. 1166. |

[32] | M. V. Predoi, M. Castaings, B. Hosten, and C. Bacon, “Wave propagation along transversely periodic structures,” Journal of the Acoustical Society of America, vol. 121, no. 4, pp. 1935-1944, 2007. · doi:10.1121/1.2534256 |

[33] | S. S. Sekoyan and A. E. Eremeev, “Measurement of the third-order elasticity constants for steel by the ultrasonic method,” Measurement Techniques, vol. 9, no. 7, pp. 888-893, 1966. · doi:10.1007/BF00998445 |

[34] | E. Onate, Structural Analysis with the Finite Element Method. Linear Statics ?vol. 1, 2009. |

[35] | A. Bernard, M. Deschamps, and M. J. S. Lowe, “Energy velocity and group velocity for guided waves propagating within an absorbing or non-absorbing plate in vacuum,” in Review of Progress in Quantitative NDE, D. O. Thompson and D. E. Chimenti, Eds., vol. 18, pp. 183-190, Plenum Press, New York, NY, USA, 1999. |

[36] | A. Bernard, M. J. S. Lowe, and M. Deschamps, “Guided waves energy velocity in absorbing and non-absorbing plates,” Journal of the Acoustical Society of America, vol. 110, no. 1, pp. 186-196, 2001. · doi:10.1121/1.1375845 |

[37] | C. Cattani and Y. Y. Rushchitskii, Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure, Series on Advances in Mathematics for Applied Sciences, World Scientific, Hackensack, NJ, USA, 2007. |

[38] | S. B. Dong and R. B. Nelson, “High frequency vibrations and waves in laminated orthotropic plates,” Journal of Applied Mechanics, Transactions ASME, vol. 39, no. 3, pp. 739-745, 1972. |

[39] | R. B. Nelson and S. B. Dong, “High frequency vibrations and waves in laminated orthotropic plates,” Journal of Sound and Vibration, vol. 30, no. 1, pp. 33-44, 1973. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.