# Combination of Phase Matching and Phase-Reversal Approaches for Thermal Damage Assessment by Second Harmonic Lamb Waves

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## Abstract

**:**

## 1. Introduction

## 2. General Considerations

#### 2.1. Principle of Phase-Reversal Approach

**u**of ultrasonic wave propagation can be taken as the sum of the fundamental wave ${\mathbf{u}}_{(1)}$ and the second harmonic ${\mathbf{u}}_{(2)}$:

_{1}(i.e., formally ${A}_{1}=\left|[{{\mathbf{u}}_{(1)}(y)|}_{y=h}]\right|$). If the Lamb wave mode pair selected (primary and double frequency Lamb waves) satisfies the conditions of phase velocity matching and nonzero power flux [11,12], the amplitude of second harmonic generated will grow with propagation distance. Accordingly, the solution to the second harmonic accompany propagation of ${\mathbf{u}}_{(1)}$ can formally be given by ${\mathbf{u}}_{(2)}={\mathbf{u}}_{(2)}(y)z\mathrm{exp}[\mathrm{j}(2k\begin{array}{c}z\end{array}-2\omega \begin{array}{c}t\end{array})]$, where the corresponding field function ${\mathbf{u}}_{(2)}(y)$ is proportional to the square of ${\mathbf{u}}_{(1)}(y)$ [11,12,33,34,35]. Similarly, at the surface of the plate, the amplitude of second harmonic generated can formally be given by ${A}_{2}=\left|{[{\mathbf{u}}_{(2)}(y)|}_{y=h}]\right|z$. The total field of Lamb wave propagation at the surface of the plate, including the fundamental wave and the second harmonic generated, is formally given by,

_{2}is proportional to the square of A

_{1}, and generally the magnitude of the former is much smaller than that of the latter.

#### 2.2. Acoustic Nonlinear Parameter

#### 2.3. Selection of Lamb Wave Mode Pair

## 3. Specimens and Experimental Setup

#### 3.1. Specimens

#### 3.2. Experimental Setup

## 4. Measurement, Results and Discussion

_{1}) and the second harmonic generated (A

_{2}), as well as ${\beta}_{R}$ = ${A}_{2}/{A}_{1}^{2}$, can be completely determined. Similarly, for different spatial separation z in Figure 4, the corresponding ${\beta}_{R}$ can also be determined. Figure 7b shows that ${\beta}_{R}$ grows with propagation distance. Accordingly, the acoustic nonlinear response measured should be attributed to the material nonlinearity. As shown in our earlier work [34], the slope ratio of ${\beta}_{R}$ with wave propagation distance is used to represent the relative acoustic nonlinearity parameter to minimize nonlinearity from couplant or instrument.

_{1a}) can be obtained from its amplitude-frequency curve (similar to that shown in Figure 9a. So does the primary-wave amplitude (denoted by A

_{1b}) of the other signal. For determination of ${\beta}_{R}={A}_{2}/{A}_{1}^{2}$, A

_{1}is calculated through $({A}_{1a}+{A}_{1b})/2$, which may inhibit the influences of the instrumental transient response or coupling conditions in repeated measurements (note the nonzero amplitude for the primary wave in Figure 9b). Meanwhile, A

_{2}is acquired through the amplitude- frequency curve of the superposition signal similar to that shown in Figure 8c (see the second-harmonic amplitude in Figure 9b). Referring to the measurement process described here, the corresponding ${\beta}_{R}$ can be determined for the specimens with different damage levels. Specifically, the phase-reversal nonlinear Lamb wave approach proposed is applied for tracking the thermal induced degradation in the stainless steel plates.

**/**${S4|}_{2f=7.70\mathrm{MHz}}$ (see Figure 3) and following the phase-reversal approach described above, the value of ${\beta}_{R}$ can readily be determined for the specimens subjected to different thermal loadings. Figure 10a,b shows the variation of normalized ${\beta}_{R}$ for the specimens, respectively, subjected to different thermal loading temperatures with two hours’ holding time and different thermal loading time at the temperature of 750 °C.

_{g}) of Lamb wave in the specimens with and without thermal loadings are also provided to compare the sensitivity of linear and nonlinear parameter. It is clearly shown that the differences of wave velocity can be neglected, which indicates that the linear Lamb wave based evaluation technique is insufficiently to evaluate the thermal induced degradation in early stage. In this study, all measured linear and nonlinear parameters are normalized by the values of the raw material to display only the relative change. Consequently, these results indicate that nonlinear ultrasonic parameters are much more promising with significantly improved sensitivity over linear parameters for early stage detection of thermal damages in this specimen.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Superposition of two Lamb wave signals in reverse phase; (

**a**) without and (

**b**) with material nonlinearity induced waveform distortion.

**Figure 3.**(

**a**) Phase velocity and (

**b**) group velocity dispersion curves for Lamb waves in the 201 stainless steel with a thickness of 1.35 mm.

**Figure 5.**Typical excited ultrasonic signals in (

**a**) time domain and (

**b**) frequency spectra of the original signals.

**Figure 6.**Two time-domain signals detected under different propagation distance z

_{1}= 30 mm and z

_{2}= 40 mm.

**Figure 7.**(

**a**) Amplitude-frequency curve of the time-domain signal detected at z = 50 mm, (

**b**) variation of normalized ${\beta}_{R}$ versus propagation distance.

**Figure 8.**(

**a**,

**b**) the time-domain signals detected by Rx at z = 30 mm; (

**c**) the superposition signal of the signals (

**a**,

**b**).

**Figure 9.**The amplitude-frequency curves measured by the conventional approach (

**a**) and the phase-reversal approach (

**b**).

**Figure 10.**Variation of normalized for the specimens subjected to (

**a**) thermal loading temperature with two hours’ holding time (by the reference of ${\beta}_{R}$ = 0.197, that of the specimen subjected 650 °C thermal loading), and (

**b**) different thermal loading time at the temperature of 750 °C (by the reference of ${\beta}_{R}$ = 0.363, that of the specimen subjected 6 hours’ thermal loading).

**Figure 11.**Comparison of the sensitivity of acoustic nonlinearity and linear parameters for the specimens subjected to (

**a**) thermal loading temperature with two hours’ holding time (by the reference of ${\beta}_{R}$ = 0.197, C

_{g}= 3.346 mm/μs that of the specimen subjected 650 °C thermal loading), and (

**b**) different thermal loading time at the temperature of 750 °C (by the reference of ${\beta}_{R}$ = 0.363, C

_{g}= 3.332 mm/μs that of the specimen subjected 6 hours’ thermal loading).

C | Si | Mn | Cr | N | P | Ni | Cu | Fe |
---|---|---|---|---|---|---|---|---|

≤0.15 | ≤0.75 | 5.5–7.5 | 16–18 | ≤0.25 | ≤0.06 | 3.5–5.5 | 2.3 | Remain |

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**MDPI and ACS Style**

Li, W.; Hu, S.; Deng, M.
Combination of Phase Matching and Phase-Reversal Approaches for Thermal Damage Assessment by Second Harmonic Lamb Waves. *Materials* **2018**, *11*, 1961.
https://doi.org/10.3390/ma11101961

**AMA Style**

Li W, Hu S, Deng M.
Combination of Phase Matching and Phase-Reversal Approaches for Thermal Damage Assessment by Second Harmonic Lamb Waves. *Materials*. 2018; 11(10):1961.
https://doi.org/10.3390/ma11101961

**Chicago/Turabian Style**

Li, Weibin, Shicheng Hu, and Mingxi Deng.
2018. "Combination of Phase Matching and Phase-Reversal Approaches for Thermal Damage Assessment by Second Harmonic Lamb Waves" *Materials* 11, no. 10: 1961.
https://doi.org/10.3390/ma11101961