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Thermal Fatigue Damage Assessment in an Isotropic Pipe Using Nonlinear Ultrasonic Guided Waves

Abstract

The use of nonlinear ultrasonic waves has been accepted as a potential technique to characterize the state of material micro-structure in solids. The typical nonlinear phenomenon is generation of second harmonics. Second harmonic generation of ultrasonic waves propagation has been vigorously studied for tracking material micro-damages in unbounded media and plate-like waveguides. However, there are few studies of launching second harmonic guided wave propagation in tube-like structures. Considering that second harmonics could provide useful information sensitive for material degradation condition, this research aims at developing a procedure for detecting second harmonics of ultrasonic guided wave in an isotropic pipe. The second harmonics generation of guided wave propagation in an isotropic and stress-free elastic pipe is investigated. Flexible polyvinylidene fluoride (PVDF) comb transducers are used to measure fundamental wave and second harmonic one. Experimental results show that nonlinear parameters increase monotonically with propagation distance. This work experimentally verifies that the second harmonics of guided waves in pipe have the cumulative effect with propagation distance. The proposed procedure is applied to assessing thermal fatigue damage indicated by nonlinearity in an aluminum pipe. The experimental observation verifies that nonlinear guided waves can be used to assess damage levels in early thermal fatigue state by correlating them with the acoustic nonlinearity.

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References

  1. 1.

    Dace GE, Thompson PB, Brash LJH et al (1991) Nonlinear acoustics, a technique to determine micro-structural changes in material. In: Thompson DO, Chimenti DE (eds) Review of progress in quantitative nondestructive evaluation. Plenum Press, New York, pp 1685–1692

  2. 2.

    Cantrell JH (2003) Fundamentals and application of nonlinear ultrasonic nondestructive evaluation. In: Kundu T (ed) Ultrasonic nondestructive evaluation. CRC press, Florida, pp 363–434

  3. 3.

    Cantrell JH, Yost WT (2001) Nonlinear ultrasonic characterization of fatigue microstructures. Int J Fatigue 23:487–490

  4. 4.

    Li W, Cho Y, Lee J, Achenbach JD (2013) Assessment of heat treated inconel X-750 alloy by nonlinear ultrasonics. Exp Mech 53:775–781

  5. 5.

    Li W, Cho Y, Hyun S (2012) Characteristics of ultrasonic nonlinearity by thermal fatigue. Int J Precis Eng Man 13:935–940

  6. 6.

    Jhang KY (2000) Applications of nonlinear ultrasonics to the NDT of material degradation. IEEE Trans Ultrason Ferroelectr Freq Control 47:540–548

  7. 7.

    Chimenti DE (1997) Guided waves in plates and their use in materials characterization. Appl Mech Rev 50:247–284

  8. 8.

    Bermes C, Kim J-Y, Qu J, Jacobs LJ (2007) Experimental characterization of material nonlinearity using Lamb waves. Appl Phys Lett 90:021901

  9. 9.

    Pruell C, Kim J-Y, Qu J, Jacobs LJ (2007) Evaluation of fatigue damage using nonlinear guided waves. Smart Mater Struct 18:035003

  10. 10.

    Li W, Cho Y, Achenbach JD (2012) Detection of thermal fatigue damage in composites by second harmonic Lamb waves. Smart Mater Struct 21:085019

  11. 11.

    Bermes C, Kim J-Y, Qu J, Jacobs LJ (2008) Nonlinear Lamb waves for the detection of material nonlinearity. Mech Syst Signal Pr 22:638–646

  12. 12.

    Deng M (1999) Cumulative second harmonic generation of Lamb mode propagation in a solid Plate. J Appl Phys 85:3051–3058

  13. 13.

    de Lima WJN, Hamilton MF (2003) Finite-amplitude waves in isotropic elastic Plates. J Sound Vib 265:819–839

  14. 14.

    Deng M (2003) Analysis of second harmonic generation of Lamb modes using modal analysis approach. J Appl Phys 94:4152–4159

  15. 15.

    de Lima WJN, Hamilton MF (2005) Finite amplitude waves in isotropic elastic waveguides with arbitrary constant cross-sectional area. Wave Motion 41:1–11

  16. 16.

    Auld BA (1990) Acoustic fields and waves in solids Vols. I and Vols. II. Wiley, London

  17. 17.

    Graff KF (1991) Wave motion in elastic solids. Dover, New York

  18. 18.

    Hay TR, Rose JL (2002) Flexible PVDF comb transducers for excitation of axi-symmetric guided waves in pipe. Sensor Actuat A: Phys 100:18–23

  19. 19.

    Monkhouse RSC, Wilcox PD, Cawley P (1997) Flexible inter-digital PVDF transducers for the generation of Lamb waves in structures. Ultrasonics 35:489–498

  20. 20.

    Zinck AA, Krishnaswamy S (2010) Ultrasonic nonlinearity measurements on rolled polycrystalline copper. In: Thompson DO, Chimenti DE (eds) Review of progress in quantitative nondestructive evaluation. Melville, New York, pp 1404–1409

  21. 21.

    Landau LD, Lifshitz EM (1986) Theory of elasticity. Pergamon Press, Oxford

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Acknowledgments

We would like to express our gratitude to Professor Jan D. Achenbach from Northwestern University, U S, for his time and advice. This work was supported by the Korea Atomic Energy Research Institute (KAERI) and Radiation Technology R&D program through the National Research Foundation of Korea funded by the Ministry of Science, ICT & Future Planning (2013M2A2A9043241).

Author information

Correspondence to Y. Cho.

Appendix

Appendix

Momentum equation in Lagrangian coordinates can be written as

$$ {\rho}_0\frac{\partial^2\mathrm{u}}{\partial {t}^2}=\nabla \cdot \mathrm{P}, $$
(A1)

where P is a Lagrangian stress tensor, and the expression for the strain energy function for an isotropic solid in terms of displacement derivatives u i ⋅ j , which include the third order elastic constants, was given by L. D. Landau [21] in the following form:

$$ \begin{array}{l}W=\frac{1}{4}\mu {\left({u}_{i\cdot j}+{u}_{j\cdot i}\right)}^2+\frac{1}{2}\lambda {\left({u}_{i\cdot j}\right)}^2+\left(\mu +\frac{1}{4}A\right)\left({u}_{i\cdot j}{u}_{k\cdot i}{u}_{k\cdot j}\right)+\frac{1}{2}\left(\lambda +B\right)\left[{u}_{i\cdot i}{\left({u}_{j\cdot k}\right)}^2\right]\hfill \\ {}+\frac{1}{12}A\left({u}_{i\cdot j}{u}_{j\cdot k}{u}_{k\cdot i}\right)+\frac{1}{2}B\left({u}_{j\cdot k}{u}_{k\cdot j}{u}_{i\cdot i}\right)+\frac{1}{3}C\left({u}_{i\cdot i}^3\right)+\dots \hfill \end{array}, $$
(A2)
$$ {P}_{ij}=\frac{\partial W}{\partial {u}_{i\cdot j}}, $$
(A3)
$$ {P}_{ij}={P}_{ij}^L+{p}_{ij}^{NL}, $$
(A4)
$$ {\varepsilon}_{ij}=\frac{1}{2}\left({\mu}_{i\cdot j}+{\mu}_{j\cdot i}\right), $$
(A5)

where \( \begin{array}{l}{P}_{ij}^L=\lambda {\delta}_{ij}{\varepsilon}_{ii}+2\mu {\varepsilon}_{ij},\hfill \\ {}{P}_{ij}^{NL}=\left(\mu +\frac{A}{4}\right)\left({u}_{l\cdot i}{u}_{l\cdot j}+{u}_{j\cdot l}{u}_{i\cdot l}+{u}_{l\cdot j}{u}_{i\cdot l}\right)\hfill \\ {}+\frac{1}{2}\left(\lambda -\mu +B\right)\left({\left({u}_{l\cdot m}\right)}^2{\delta}_{ij}+2{u}_{i\cdot j}{u}_{l\cdot l}\right)+\frac{A}{4}{u}_{j\cdot I}{u}_{l\cdot i}\hfill \\ {}+\frac{B}{2}\left({u}_{l\cdot m}{u}_{m\cdot l}{\delta}_{ij}+2{u}_{j\cdot i}{u}_{l\cdot l}\right)+C{\left({u}_{l\cdot l}\right)}^2{\delta}_{ij}\hfill \end{array} \)

Substituting equation (A4) into equation (A1), the nonlinear wave equation is obtained as:

$$ \left(\lambda +2\mu \right)\nabla \left(\nabla \cdot \mathrm{u}\right)-\mu \nabla \times \left(\nabla \times \mathrm{u}\right)+\mathrm{f}={\rho}_0\frac{\partial^2\mathrm{u}}{\partial {t}^2}. $$
(A6)

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Li, W., Cho, Y. Thermal Fatigue Damage Assessment in an Isotropic Pipe Using Nonlinear Ultrasonic Guided Waves. Exp Mech 54, 1309–1318 (2014). https://doi.org/10.1007/s11340-014-9882-2

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Keywords

  • Second harmonics
  • Nonlinear guided wave
  • Pipe
  • Thermal fatigue