## 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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## 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).

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## Appendix

### Appendix

Momentum equation in Lagrangian coordinates can be written as

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:

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:

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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