# Double Crack Damage Identification of Welded Steel Structure Based on LAMB WAVES of S0 Mode

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Theory of Lamb Waves

#### 2.1. Basic Theory of Group Velocity and Phase Velocity

_{1}and ω

_{2}).

_{1}= ω/c

_{1}, k

_{2}= ω/c

_{2}. k

_{1}, k

_{2}are wave numbers; H is amplitude. The formula can be deduced as:

_{p}= ω

_{c}/k

_{c}. Simultaneously, low frequency is defined as group velocity c

_{g}= Δω/Δk, and it is also c

_{g}= dω/dk. By defining group velocity as phase velocity, the formula for calculating group velocity can be obtained as follows:

#### 2.2. Dispersion Curve

_{L}is the velocity of the longitudinal wave, c

_{T}is the velocity of the transverse wave. The output result is shown in Figure 1. In the figure, the frequency is set to the abscissa and the velocity to the ordinate. From the figure, we can see that there was a dispersion effect in the Lamb waves, and there were many modes. Under the excitation of different frequencies, the same plate was subjected to different situations.

## 3. Identification and Imaging of Double Crack Damage Based on the Lamb Waves of the S0 Mode

#### 3.1. Geometric Model for Numerical Simulation

#### 3.2. Excitation Signal

#### 3.3. Mesh Size

_{min}) can be calculated as:

#### 3.4. Elliptical Location Method

_{1}+ r

_{2}; (3) Transmitted from the actuator, the wave reached the receiver after the second damage reflection, and its propagation path was r

_{3}+ r

_{4}; (4) Transmitted by the actuator, the signal first reached damage 1, then reached damage 2, and finally reflected to the receiver. Thus, the path obtained after multiple reflections was r

_{1}+ s + r

_{4}.

#### 3.5. Simulation Results

^{−5}s to 5.0 × 10

^{−5}s, and with a time interval of 5 × 10

^{−6}s.

#### 3.6. Experiment on the Identification of Double Crack Damage

## 4. Damage Identification of Welded Steel Plate Based on S0 Mode

#### 4.1. Numerical Simulation

#### 4.2. Numerical Simulation Results

^{−5}s to 5.0 × 10

^{−5}s with a time interval of 5 × 10

^{−6}s.

#### 4.3. Experiment

#### 4.3.1. Experimental Results

#### 4.3.2. Attenuation Analysis of Signal Energy

## 5. Conclusions

- The damage imaging results of the steel plate with double cracks from numerical simulation and experiments were in good agreement. The damage location shows a high level of accuracy. The mutual verification of finite element method and experiments proved the reliability of the Lamb waves monitoring method.
- When reflected by damages, Lamb waves had amplitude attenuation, which could be reflected in the echo signal. As can be seen from the imaging results of double damages, the differences in the damage length were quite obvious. It was proved that the method is sensitive to the length of damage. This will contribute to research on the length of the structures’ crack in further study.
- When studying damage location in welded steel plates, it is concluded that a part of the Lamb waves would reflect at the weld and would partly pass through the weld. The results of numerical simulation and experiments confirm this inference. The agreement of simulation results and experimental results prove the feasibility of the application of the Lamb wave method in the welded structure.
- By analysing the amplitude and distance of the signal in the welded steel plate, it is concluded that the energy of Lamb wave in the steel plate decreased with the increment of the distance. The welding seam reflected most Lamb waves, and the energy considerably reduced when they passed through the welding seam during the propagation process.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Dispersion curve of plate: (

**a**) phase velocity dispersion curve of 4-mm steel plate, (

**b**) group velocity dispersion curve of 4-mm steel plate.

**Figure 4.**Excitation modes of the Lamb wave: (

**a**) unilateral excitation, (

**b**) symmetric excitation, (

**c**) anti-symmetric excitation.

**Figure 6.**Load simulation in ABAQUS: (

**a**) schematic diagram of excitation point, (

**b**) top view of the excitation load, (

**c**) side view of the excitation load.

**Figure 8.**Schema of Lamb waves propagation in a double damaged steel plate: (

**a**) 2.5 × 10

^{−5}s, (

**b**) 3.0 × 10

^{−5}s, (

**c**) 3.5 × 10

^{−5}s, (

**d**) 4.0 × 10

^{−5}s, (

**e**) 4.5 × 10

^{−5}s, (

**f**) 5.0 × 10

^{−5}s.

**Figure 9.**(

**a**) Received signal at PZT4 of intact steel plate, (

**b**) received signal at PZT4 of damaged steel plate.

**Figure 10.**Damage imaging: (

**a**) damage imaging of the left 80 mm crack, (

**b**) damage imaging of the right 40 mm crack.

**Figure 12.**The damage and PZT layout in experiments: (

**a**) intact steel plate, (

**b**) damaged steel plate.

**Figure 13.**Damage imaging: (

**a**) damage imaging of the left 80 mm crack, (

**b**) damage imaging of the right 40 mm crack.

**Figure 16.**Propagation of Lamb waves in welded steel plates: (

**a**) 2.5 × 10

^{−5}s, (

**b**) 3.0 × 10

^{−5}s, (

**c**) 3.5 × 10

^{−5}s, (

**d**) 4.0 × 10

^{−5}s, (

**e**) 4.5 × 10

^{−5}s, (

**f**) 5.0 × 10

^{−5}s.

E (GPa) | ν | ρ (kg/m^{3}) | d (mm) | c_{L} (m/s) | c_{T} (m/s) |
---|---|---|---|---|---|

206 | 0.25 | 7800 | 4 | 5856 | 3130 |

Transducer | Coordinate (mm) | Transducer | Coordinate (mm) | Transducer | Coordinate (mm) |
---|---|---|---|---|---|

PZT1 | (0,160) | PZT4 | (−160,0) | PZT7 | (−80,−80) |

PZT1’ | (0,160) | PZT4’ | (−160,0) | PZT7’ | (−80,−80) |

PZT2 | (−80,80) | PZT5 | (0,0) | PZT8 | (80,−80) |

PZT2’ | (−80,80) | PZT5’ | (0,0) | PZT8’ | (80,−80) |

PZT3 | (80,80) | PZT6 | (160,0) | PZT9 | (0,−160) |

PZT3’ | (−80,80) | PZT6’ | (160,0) | PZT9’ | (0,−160) |

Transducer (Left Side) | Coordinate (mm) (x,y) | Transducer (Left Side) | Coordinate (mm) (x,y) | Transducer (Right Side) | Coordinate (mm) (x’,y’) |
---|---|---|---|---|---|

PZT1 | (0,50) | PZT3’ | (0,0) | PZT6 | (0,50) |

PZT1’ | (0,50) | PZT4 | (100,0) | PZT7 | (−100,0) |

PZT2 | (−100,0) | PZT4’ | (100,0) | PZT8 | (0,0) |

PZT2’ | (−100,0) | PZT5 | (0,−50) | PZT9 | (100,0) |

PZT3 | (0,0) | PZT5’ | (0,−50) | PZT10 | (0,−50) |

Propagation Path | Distance (mm) | Voltage (V) ×10 ^{−3} | Propagation Path | Distance (mm) | Voltage (V) ×10 ^{−3} |
---|---|---|---|---|---|

PZT1–PZT2 | 112 | 4.014 | PZT1–PZT7 | 380 | 0.210 |

PZT1–PZT3 | 50 | 6.565 | PZT1–PZT8 | 350 | 0.305 |

PZT1–PZT4 | 112 | 3.210 | PZT1–PZT9 | 380 | 0.242 |

PZT1–PZT5 | 100 | 6.112 | PZT1–PZT10 | 300 | 0.672 |

PZT1–PZT6 | 400 | 0.198 |

Propagation Path | Distance (mm) | Displacement (mm) ×10 ^{−9} | Propagation Path | Distance (mm) | Displacement (mm) ×10 ^{−9} |
---|---|---|---|---|---|

PZT1–PZT2 | 112 | 1.076 | PZT1–PZT7 | 380 | 0.2608 |

PZT1–PZT3 | 50 | 4.642 | PZT1–PZT8 | 350 | 0.3105 |

PZT1–PZT4 | 112 | 1.078 | PZT1–PZT9 | 380 | 0.2819 |

PZT1–PZT5 | 100 | 2.679 | PZT1–PZT10 | 300 | 0.4221 |

PZT1–PZT6 | 400 | 0.2283 |

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**MDPI and ACS Style**

Hu, M.; Sun, X.; He, J.; Zhan, Y.
Double Crack Damage Identification of Welded Steel Structure Based on LAMB WAVES of S0 Mode. *Materials* **2019**, *12*, 1800.
https://doi.org/10.3390/ma12111800

**AMA Style**

Hu M, Sun X, He J, Zhan Y.
Double Crack Damage Identification of Welded Steel Structure Based on LAMB WAVES of S0 Mode. *Materials*. 2019; 12(11):1800.
https://doi.org/10.3390/ma12111800

**Chicago/Turabian Style**

Hu, Muping, Xiaodan Sun, Jian He, and Yangyang Zhan.
2019. "Double Crack Damage Identification of Welded Steel Structure Based on LAMB WAVES of S0 Mode" *Materials* 12, no. 11: 1800.
https://doi.org/10.3390/ma12111800