# Fatigue Damage Evaluation Using Nonlinear Lamb Waves with Quasi Phase-Velocity Matching at Low Frequency

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Considerations

## 3. Experimental Details

#### 3.1. Low Cycle Fatigue Test

_{max}, was set to be 349.8 MPa, corresponding to 1.1 yield stress of the pristine specimen, which was experimentally measured prior to testing. One specimen was left undamaged and served as a reference, while the other one was cyclically loaded to fracture and the fatigue life was determined to be approximately 31,098 cycles. To produce specimens with various fatigue damage, the loading of other 11 specimens was stopped at certain cycles, which are 3000, 6000, 9000, 11,000, 13,000, 15,000, 17,000, 19,000, 21,000, 23,000 and 25,000 cycles, respectively. The test condition and fatigue life fraction of specimens are listed in Table 2.

#### 3.2. Nonlinear Lamb Wave Measurements

_{1}and A

_{2}, as illustrated in Figure 4. Several attempts were performed to confirm the STFT parameters for guaranteeing both the frequency and time resolutions. Note that the time delay in the wedges was considered for the dispersion curves of group-velocity. The relative acoustic nonlinearity parameter was then calculated as A

_{2}/A

_{1}

^{2}, which is proportional to the absolute acoustic nonlinearity parameter and can effectively characterize the evolution of material degradation.

_{2}with the square of the fundamental amplitude A

_{1}

^{2}. The accumulation of ultrasonic nonlinearity was then investigated for the Lamb mode pair S0-s0. Nonlinear ultrasonic measurements were conducted on the intact specimen at various propagation distances between the wedges from 50 mm to 170 mm with an interval of 10 mm by moving the receiving wedge transducer away from the transmitting wedge transducer. Nonlinear ultrasonic measurements were repeated five times at each propagation distance by completely removing and then reattaching the wedge transducer assembly to the plate.

## 4. Simulation Deployments

^{−9}s, much smaller than the time resolution $\Delta d/c$, where c is the group-velocity of the target Lamb mode. As shown in Figure 5, the primary Lamb mode S0 was excited uniformly from the left surface of the aluminum alloy plate at 300 kHz, as the in-plane displacement of the low frequency S0 mode distributes almost linearly through the thickness. The excitation signal is formulated as U = U

_{0}A(t) B(Y), where U

_{0}is the excitation displacement amplitude of 3.5 × 10

^{−4}mm, corresponding to a stress amplitude of a few MPa, is typical for Lamb wave propagation in solids; A(t) is the temporal waveform of a 20-cycles Hanning-windowed sinusoidal tone burst; and B(Y) is the thickness profile of the excitation displacement. Receivers were placed at nodes located 20–600 mm away from the excitation source with an interval of 20 mm to pick up the in-plate displacements at upper surface of the plate. STFT was performed on the received signals to extract the amplitudes of the primary and second harmonic Lamb waves and calculate the relative acoustic nonlinearity parameter with the propagation distance. Consequently, the cumulative generation of second harmonics was expected to be validated with the propagation distance for the low frequency Lamb pair S0-s0.

## 5. Results and Discussions

#### 5.1. Cumulative Generation of Second Harmonics

_{2}/A

_{1}

^{2}, with the propagation distance for the Lamb mode pair S0-s0 with the excitation frequency of 300 kHz. The A

_{2}/A

_{1}

^{2}oscillates along the propagation distance in a sinusoidal behavior with an oscillation spatial period of 439.65 mm owing to the quasi phase-velocity matching condition. The oscillation amplitude and spatial period are consistent between the simulation results and the theoretical analysis, as the simulation data points locate exactly on the curve of perturbation analysis, which was theoretically acquired by Equations (1) and (2). The A

_{2}/A

_{1}

^{2}grows cumulatively with the propagation distance within half the oscillation spatial period, ${z}_{n}$. Note that ${z}_{n}$ could be calculated using Equation (3), which is inversely proportional to the deviation of the phase velocities between the primary and second harmonic Lamb waves.

#### 5.2. Fatigue Damage Evaluation

_{2}/A

_{1}

^{2}, as a function of the fatigue cycles and the fatigue life fraction. For an easy comparison, the A

_{2}/A

_{1}

^{2}were normalized with respect to their minimum value on the specimen with 3000 fatigue cycles. A mountain shape curve between the normalized A

_{2}/A

_{1}

^{2}and the fatigue life is observed. The normalized A

_{2}/A

_{1}

^{2}increases significantly with cyclic loading in the early stage of fatigue life. After about 0.65 fatigue life consumed, a peak value of normalized A

_{2}/A

_{1}

^{2}is reached, which grows to nearly 112%. After this peak point, a slight decrease of normalized A

_{2}/A

_{1}

^{2}is observed with the fatigue life. The error bars of the standard deviation are determined by repeating the measurements five times, which may be ascribed to the coupling conditions. The slight deviation of individual measured data from the fitting curve may be attributed to the stress control in the servohydraulic testing system, as its load capacity is relatively large for the load level in this work. Similar observation was reported for the Lamb mode pairs with the exact phase-velocity matching [32]. Consequently, the low frequency mode pair S0-s0 is found to be effective to quantitatively evaluate the evolution of the fatigue damages.

_{2}/A

_{1}

^{2}obtained in simulations with the increasing third elastic constants, αA, αB and αC. The growth of the A

_{2}/A

_{1}

^{2}with the material degradation was numerically illustrated in the early stage of fatigue life. Although the absolute increase of the A

_{2}/A

_{1}

^{2}varies with the propagation distance, the relative increase of the A

_{2}/A

_{1}

^{2}with respect to the initial value (α = 1) keeps consistent. As the relative increase of the A

_{2}/A

_{1}

^{2}is independent on the propagation distance, it may be an effective parameter to characterize the material degradation, namely, the normalized A

_{2}/A

_{1}

^{2}. Also the consistent oscillation spatial period was observed owing to the constant parameters in Equation (3), as the second-order elastic constants are assumed to remain unchanged in the simulations.

_{2}/A

_{1}

^{2}with respect to the fatigue damage is essentially ascribed to the microstructural evolution, such as dislocation, vacancy, dislocation cell and wall. According to previous studies [26,36], in the initial stage of fatigue life, the dislocation density increases significantly and vacancy clusters are generated due to the dislocation glide in the persistent slip bands. The dislocation-induced nonlinearity is proportional to the dislocation density based on the dislocation models [36,37], while the acoustic nonlinearity caused by the vacancies is considerably smaller than that by the dislocations [36]. Therefore, the significant increase of normalized A

_{2}/A

_{1}

^{2}dominantly attributed to the dislocation density. In the last stage of fatigue life, the dislocation density remains a relatively stable level, and the dislocation cells and walls are formed, while the vacancy clusters coalesce as a precursor to the nucleation and growth of microcracks [36]. The nonlinearity induced by the microcracks is found to decrease with the width of the microcracks, which dominantly contributes to the slight decrease of normalized A

_{2}/A

_{1}

^{2}[38,39].

#### 5.3. Analyses of Sensitivity

_{2}/A

_{1}

^{2}with the fatigue cycles and the fatigue life fraction for the mode pairs S0-s0 and S1-s2, respectively. While the mountain shape curves with peak values of normalized A

_{2}/A

_{1}

^{2}at about 0.65 fatigue life are simultaneously observed for both mode pairs, the normalized A

_{2}/A

_{1}

^{2}of mode pair S1-s2 is relatively larger than that of mode pair S0-s0 at each fatigue cycle. At the peak points, the normalized A

_{2}/A

_{1}

^{2}reaches to nearly 155% for mode pair S1-s2 and nearly 112% for mode pair S0-s0. The mode pair S0-s0 is relatively less sensitive to the fatigue damage than the mode pair S1-s2 with exact phase-velocity matching.

#### 5.4. Effect of Frequency

_{2}/A

_{1}

^{2}with the fatigue cycles for the mode pair S0-s0 at the excited frequency of 300 kHz, 350 kHz and 400 kHz, respectively. At a lower frequency, the normalized A

_{2}/A

_{1}

^{2}grows rapidly with the fatigue cycles, while the mountain shape curves with peak values of normalized A

_{2}/A

_{1}

^{2}at about 0.65 fatigue life are simultaneously observed at all three excited frequencies. The mode pair S0-s0 at a relatively lower frequency are found to be more sensitive to the fatigue damage.

## 6. Conclusions

_{2}/A

_{1}

^{2}and the fatigue life was observed with the peak point at about 0.65 fatigue life. Compared with the mode pair S1-s2 satisfying exact phase-velocity matching condition, which was generally used to characterize the material degradation in pervious researches, the mode pair S0-s0 is found to be relatively less sensitive to the fatigue damage. Since the quasi phase-velocity matching condition was satisfied in a certain frequency range for the mode pair S0-s0 owing to the less dispersive property, the effect of excited frequency on the fatigue damage evaluation was further explored in experimental measurements. The mode pair S0-s0 at a lower excited frequency was found to be more sensitive to the fatigue damage. Consequently, the results show that the low frequency mode pair S0-s0 can be used to effectively detect the fatigue damage, and a relatively lower excited frequency is preferred in a certain frequency range.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations/Nomenclature

Nomenclature | |||

A/B/C | third order elastic constants | k | wave number |

A_{1} | amplitude of primary wave | N | fatigue cycles |

A_{2} | amplitude of second harmonic wave | R | stress ratio |

${a}_{n}(z)$ | magnitude of nth double frequency Lamb wave | U_{0} | excitation displacement amplitude |

A(t) | temporal waveform | w | angular velocity |

B(Y) | thickness profile | v | Poisson’s ratio |

c | group velocity | z | propagation distance |

c_{l} | longitudinal wave velocity | z_{n} | half the oscillation spatial period |

${c}_{p}^{\omega}$ | phase velocity of primary wave | α | scale factor |

${c}_{p}^{2\omega}$ | phase velocity of second harmonic wave | β | absolute nonlinearity parameter |

c_{t} | transverse wave velocity | ρ | density |

Δd | element size | σ_{max} | maximum stress |

E | elasticity modulus | σ_{0.2} | yield stress |

Abbreviations | |||

SHG | second harmonic generation | FE | finite element |

DFLW | double frequency Lamb wave | STFT | short time Fourier transform |

## References

- Chillara, V.K.; Lissenden, C.J. Review of nonlinear ultrasonic guided wave nondestructive evaluation: Theory, numerics, and experiments. Opt. Eng.
**2016**, 55, 011002. [Google Scholar] [CrossRef] - Jhang, K.-Y. Nonlinear ultrasonic techniques for nondestructive assessment of micro damage in material: A review. Int. J. Precis. Eng. Manuf.
**2009**, 10, 123–135. [Google Scholar] [CrossRef] - Deng, M. Analysis of second-harmonic generation of Lamb modes using a modal analysis approach. J. Appl. Phys.
**2003**, 94, 4152. [Google Scholar] [CrossRef] - De Lima, W.J.N.; Hamilton, M.F. Finite-amplitude waves in isotropic elastic plates. J. Sound Vib.
**2003**, 265, 819–839. [Google Scholar] [CrossRef] - De Lima, W.J. Harmonic Generation in Isotropic Elastic Waveguides. Ph.D. Thesis, The University of Texas at Austin, Austin, TX, USA, January 2000. [Google Scholar]
- Zhao, J.; Chillara, V.K.; Ren, B.; Cho, H.; Qiu, J.; Lissenden, C.J. Second harmonic generation in composites: Theoretical and numerical analyses. J. Appl. Phys.
**2016**, 119, 064902. [Google Scholar] [CrossRef] - Matsuda, N.; Biwa, S. Phase and group velocity matching for cumulative harmonic generation in Lamb waves. J. Appl. Phys.
**2011**, 109, 094903. [Google Scholar] [CrossRef] [Green Version] - Chillara, V.K.; Lissenden, C.J. Interaction of guided wave modes in isotropic weakly nonlinear elastic plates: Higher harmonic generation. J. Appl. Phys.
**2012**, 111, 124909. [Google Scholar] [CrossRef] - Bender, F.A.; Kim, J.-Y.; Jacobs, L.J.; Qu, J. The generation of second harmonic waves in an isotropic solid with quadratic nonlinearity under the presence of a stress-free boundary. Wave Motion
**2013**, 50, 146–161. [Google Scholar] [CrossRef] - Liu, M.; Wang, K.; Lissenden, C.; Wang, Q.; Zhang, Q.; Long, R.; Su, Z.; Cui, F. Characterizing hypervelocity impact (HVI)-induced pitting damage using active guided ultrasonic waves: From linear to nonlinear. Materials
**2017**, 10, 547. [Google Scholar] [CrossRef] [PubMed] - Rauter, N.; Lammering, R.; Kühnrich, T. On the detection of fatigue damage in composites by use of second harmonic guided waves. Compos. Struct.
**2016**, 152, 247–258. [Google Scholar] [CrossRef] - Metya, A.K.; Ghosh, M.; Parida, N.; Balasubramaniam, K. Effect of tempering temperatures on nonlinear Lamb wave signal of modified 9Cr-1Mo steel. Mater. Charact.
**2015**, 107, 14–22. [Google Scholar] [CrossRef] - Lim, H.; Sohn, H. Necessary conditions for nonlinear ultrasonic modulation generation given a localized fatigue crack in a plate-like structure. Materials
**2017**, 10, 248. [Google Scholar] [CrossRef] [PubMed] - Srivastava, A.; Lanza di Scalea, F. On the existence of antisymmetric or symmetric Lamb waves at nonlinear higher harmonics. J. Sound Vib.
**2009**, 323, 932–943. [Google Scholar] [CrossRef] - Sun, X.; Liu, X.; Liu, Y.; Hu, N.; Zhao, Y.; Ding, X.; Qin, S.; Zhang, J.; Zhang, J.; Liu, F.; et al. Simulations on Monitoring and Evaluation of Plasticity-Driven Material Damage Based on Second Harmonic of S0 Mode Lamb Waves in Metallic Plates. Materials
**2017**, 10, 827. [Google Scholar] [CrossRef] [PubMed] - Wan, X.; Tse, P.W.; Xu, G.H.; Tao, T.F.; Zhang, Q. Analytical and numerical studies of approximate phase velocity matching based nonlinear S0 mode Lamb waves for the detection of evenly distributed microstructural changes. Smart Mater. Struct.
**2016**, 25, 045023. [Google Scholar] [CrossRef] - Zuo, P.; Zhou, Y.; Fan, Z. Numerical and experimental investigation of nonlinear ultrasonic Lamb waves at low frequency. Appl. Phys. Lett.
**2016**, 109, 021902. [Google Scholar] [CrossRef] - Matsuda, N.; Biwa, S. Frequency dependence of second-harmonic generation in Lamb waves. J. Nondestruct. Eval.
**2014**, 33, 169–177. [Google Scholar] [CrossRef] [Green Version] - Chillara, V.K.; Lissenden, C.J. Nonlinear guided waves in plates: A numerical perspective. Ultrasonics
**2014**, 54, 1553–1558. [Google Scholar] [CrossRef] [PubMed] - Zhu, W.; Xiang, Y.; Liu, C.J.; Deng, M.; Xuan, F.Z. Symmetry properties of second harmonics generated by antisymmetric Lamb waves. J. Appl. Phys.
**2018**, 123, 104902. [Google Scholar] [CrossRef] [Green Version] - Xiao, H.; Shen, Y.; Xiao, L.; Qu, W.; Lu, Y. Damage detection in composite structures with high-damping materials using time reversal method. Nondestruct. Test. Eval.
**2018**, 33, 329–345. [Google Scholar] [CrossRef] - Chronopoulos, D. Calculation of guided wave interaction with nonlinearities and generation of harmonics in composite structures through a wave finite element method. Compos. Struct.
**2018**, 186, 375–384. [Google Scholar] [CrossRef] - Gomes, G.F.; Mendéz, Y.A.D.; da Silva Lopes Alexandrino, P.; da Cunha, S.S.; Ancelotti, A.C. The use of intelligent computational tools for damage detection and identification with an emphasis on composites—A review. Compos. Struct.
**2018**, 196, 44–54. [Google Scholar] [CrossRef] - Nagy, P.B. Fatigue damage assessment by nonlinear ultrasonic materials characterization. Ultrasonics
**1998**, 36, 375–381. [Google Scholar] [CrossRef] - Cantrell, J.H.; Yost, W.T. Nonlinear ultrasonic characterization of fatigue microstructures. Int. J. Fatigue
**2001**, 23, 487–490. [Google Scholar] [CrossRef] - Zhang, J.; Xuan, F.-Z. Fatigue damage evaluation of austenitic stainless steel using nonlinear ultrasonic waves in low cycle regime. J. Appl. Phys.
**2014**, 115, 204906. [Google Scholar] [CrossRef] - Kim, J.-Y.; Jacobs, L.J.; Qu, J.; Littles, J.W. Experimental characterization of fatigue damage in a nickel-base superalloy using nonlinear ultrasonic waves. J. Acoust. Soc. Am.
**2006**, 120, 1266. [Google Scholar] [CrossRef] - Jhang, K.-Y.; Kim, K.-C. Evaluation of material degradation using nonlinear acoustic effect. Ultrasonics
**1999**, 37, 39–44. [Google Scholar] [CrossRef] - Cantrell, J.H. Dependence of microelastic-plastic nonlinearity of martensitic stainless steel on fatigue damage accumulation. J. Appl. Phys.
**2006**, 100, 063508. [Google Scholar] [CrossRef] [Green Version] - Deng, M.; Pei, J. Assessment of accumulated fatigue damage in solid plates using nonlinear Lamb wave approach. Appl. Phys. Lett.
**2007**, 90, 121902. [Google Scholar] [CrossRef] - Pruell, C.; Kim, J.-Y.; Qu, J.; Jacobs, L.J. Evaluation of fatigue damage using nonlinear guided waves. Smart Mater. Struct.
**2009**, 18, 035003. [Google Scholar] [CrossRef] - Zhu, W.; Xiang, Y.; Liu, C.J.; Deng, M.; Xuan, F.Z. A feasibility study on fatigue damage evaluation using nonlinear Lamb waves with group-velocity mismatching. Ultrasonics
**2018**, 90, 18–22. [Google Scholar] [CrossRef] [PubMed] - Auld, B.A. Acoustic Fields and Waves in Solids; Wiley: New York, NY, USA, 1973. [Google Scholar]
- Sewell, G. The Numerical Solution of Ordinary and Partial Differential Equations; John Wiley & Sons: Hoboken, NJ, USA, 2005. [Google Scholar]
- Matlack, K.H.; Kim, J.Y.; Jacobs, L.J.; Qu, J. Review of second harmonic generation measurement techniques for material state determination in metals. J. Nondestruct. Eval.
**2014**, 34, 273. [Google Scholar] [CrossRef] - Cantrell, J.H. Substructural organization, dislocation plasticity and harmonic generation in cyclically stressed wavy slip metals. Proc. R. Soc. A
**2004**, 460, 757–780. [Google Scholar] [CrossRef] - Hikata, A.; Chick, B.B.; Elbaum, C. Dislocation contribution to the second harmonic generation of ultrasonic waves. J. Appl. Phys.
**1965**, 36, 229. [Google Scholar] [CrossRef] - Wan, X.; Zhang, Q.; Xu, G.; Tse, P.W. Numerical simulation of nonlinear Lamb waves used in a thin plate for detecting buried micro-cracks. Sensors
**2014**, 14, 8528–8546. [Google Scholar] [CrossRef] [PubMed] - Jiao, J.; Meng, X.; He, C.; Wu, B. Nonlinear Lamb wave-mixing technique for micro-crack detection in plates. NDT E Int.
**2017**, 85, 63–71. [Google Scholar] [CrossRef] - Bermes, C.; Kim, J.-Y.; Qu, J.; Jacobs, L.J. Nonlinear Lamb waves for the detection of material nonlinearity. Mech. Syst. Signal Process.
**2008**, 22, 638–646. [Google Scholar] [CrossRef] - Xiang, Y.; Deng, M.; Xuan, F.Z.; Liu, C.J. Effect of precipitate-dislocation interactions on generation of nonlinear Lamb waves in creep-damaged metallic alloys. J. Appl. Phys.
**2012**, 111, 104905. [Google Scholar] [CrossRef] - Xiang, Y.; Zhu, W.; Deng, M.; Xuan, F.Z.; Liu, C.J. Generation of cumulative second-harmonic ultrasonic guided waves with group velocity mismatching Numerical analysis and experimental validation. Europhys. Lett.
**2016**, 116, 34001. [Google Scholar] [CrossRef]

**Figure 3.**(

**a**) Schematic of the experimental setup for the nonlinear ultrasonic measurements; and (

**b**) photographic illustration of the experiment.

**Figure 4.**Typical frequency-time spectrogram of received signal with dispersion curves of group-velocity.

**Figure 6.**Relative acoustic nonlinearity parameter A

_{2}/A

_{1}

^{2}with respect to the propagation distance. The A

_{2}/A

_{1}

^{2}was acquired in perturbation analysis and simulations for mode pair S0-s0 with excitation frequency of 300 kHz.

**Figure 7.**Relative acoustic nonlinearity parameter A

_{2}/A

_{1}

^{2}with respect to the propagation distance. The A

_{2}/A

_{1}

^{2}was experimentally measured for mode pair S0-s0 with excitation frequency of 300 kHz.

**Figure 8.**Normalized relative acoustic nonlinearity parameter A

_{2}/A

_{1}

^{2}with respect to the fatigue cycles for mode pair S0-s0 with excitation frequency of 300 kHz.

**Figure 9.**Relative acoustic nonlinearity parameter A

_{2}/A

_{1}

^{2}versus propagation distance with increasing third-order elastic constants up to αA, αB and αC for mode pair S0-s0 with excitation frequency of 300 kHz. α is the scale factor.

**Figure 10.**Comparison of normalized A

_{2}/A

_{1}

^{2}versus fatigue cycles between mode pair S0-s0 with excitation frequency of 300 kHz and mode pair S1-s2 with excitation frequency of 1.81 MHz.

**Figure 11.**Comparison of normalized A

_{2}/A

_{1}

^{2}versus fatigue cycles for S0-s0 mode pairs with excitation frequency of 300 kHz, 350 kHz and 400 kHz.

Si | Fe | Cu | Mn | Mg | Cr | Zn | Ti | Al |
---|---|---|---|---|---|---|---|---|

0.085 | 0.176 | 1.654 | 0.069 | 2.611 | 0.200 | 5.768 | 0.0438 | Bal. |

Specimen | σ_{max} (MPa) | R | N (cycles) | Life Fraction (%) |
---|---|---|---|---|

1 | 349.8 | 0.1 | 3000 | 9.65 |

2 | 349.8 | 0.1 | 6000 | 19.29 |

3 | 349.8 | 0.1 | 9000 | 28.94 |

4 | 349.8 | 0.1 | 11,000 | 35.37 |

5 | 349.8 | 0.1 | 13,000 | 41.80 |

6 | 349.8 | 0.1 | 15,000 | 48.23 |

7 | 349.8 | 0.1 | 17,000 | 54.67 |

8 | 349.8 | 0.1 | 19,000 | 61.10 |

9 | 349.8 | 0.1 | 21,000 | 67.53 |

10 | 349.8 | 0.1 | 23,000 | 73.96 |

11 | 349.8 | 0.1 | 25,000 | 80.39 |

ρ (kg/m^{3}) | E (GPa) | v | σ_{0.2} (GPa) | c_{l} (m/s) | c_{t} (m/s) | A (GPa) | B (GPa) | C (GPa) |
---|---|---|---|---|---|---|---|---|

2757.82 | 73.1 | 0.34 | 318 | 6372.70 | 3146.18 | −351.2 | −149.4 | −102.8 |

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## Share and Cite

**MDPI and ACS Style**

Zhu, W.; Xiang, Y.; Liu, C.-j.; Deng, M.; Ma, C.; Xuan, F.-z.
Fatigue Damage Evaluation Using Nonlinear Lamb Waves with Quasi Phase-Velocity Matching at Low Frequency. *Materials* **2018**, *11*, 1920.
https://doi.org/10.3390/ma11101920

**AMA Style**

Zhu W, Xiang Y, Liu C-j, Deng M, Ma C, Xuan F-z.
Fatigue Damage Evaluation Using Nonlinear Lamb Waves with Quasi Phase-Velocity Matching at Low Frequency. *Materials*. 2018; 11(10):1920.
https://doi.org/10.3390/ma11101920

**Chicago/Turabian Style**

Zhu, Wujun, Yanxun Xiang, Chang-jun Liu, Mingxi Deng, Congyun Ma, and Fu-zhen Xuan.
2018. "Fatigue Damage Evaluation Using Nonlinear Lamb Waves with Quasi Phase-Velocity Matching at Low Frequency" *Materials* 11, no. 10: 1920.
https://doi.org/10.3390/ma11101920