# Surface Rust Detection Using Ultrasonic Waves in a Cylindrical Geometry by Finite Element Simulation

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## Abstract

**:**

## 1. Introduction

## 2. Finite Element Simulation

#### 2.1. Intact Steel Rod Model

#### 2.2. Corroded Steel Rod Models

## 3. Research Hypotheses and Approach

#### 3.1. Hypotheses of Ultrasonic Wave Propagation in Intact and Corroded Rod Models

- In the intact steel rod model, the time domain radial displacement $u\left(t\right)$ is collected at R. The first ultrasonic wave packet is the one propagating along the path ${\overrightarrow{s}}_{1}$ at a velocity of ${c}_{1}$ and arriving at time ${t}_{1}$. The second ultrasonic wave packet propagates along the path ${\overrightarrow{s}}_{2}$ and arrives at time ${t}_{2}$ with a velocity of ${c}_{2}$. Both the first and the second wave packets are surface waves (fundamental mode of Rayleigh waves). These surface waves were chosen for surface rust detection, rather than bulk waves [28] and guided waves [29], because bulk waves attenuate much faster than surface waves, and guided waves require multiple sensors to be in place.
- In the corroded steel rod models (CM1~CM5), the ultrasonic waves propagating along the path ${\overrightarrow{s}}_{1}$ are affected by the presence of surface rust. As shown in Figure 3c, some of the ultrasonic waves propagate through the surface rust and arrive at time ${t}_{1}^{\prime}$ (i.e., ${t}_{1}^{\prime}$ > ${t}_{1}$, since the ultrasonic wave velocity is slower in rust than it is in steel).
- Some of the ultrasonic waves are scattered from the surface rust and propagate along the path ${\overrightarrow{s}}_{4}$. Time ${t}_{2}^{\prime}$ is the total time of flight (TOF) of the scattered ultrasonic wave propagating along path $({\overrightarrow{s}}_{3},{\overrightarrow{s}}_{4})$ (${t}_{2}^{\prime}={t}_{3}+{t}_{4}$). The propagation velocities of ultrasonic waves on path ${\overrightarrow{s}}_{1}$ and path $({\overrightarrow{s}}_{3},{\overrightarrow{s}}_{4})$ are respectively ${c}_{1}^{\prime}$ and ${c}_{2}^{\prime}$.
- In Figure 3d, path ${\overrightarrow{s}}_{8}$ is the path of ultrasonic waves diffracted by the surface rust (${\overrightarrow{s}}_{8}={\overrightarrow{s}}_{6}+d+{\overrightarrow{s}}_{7}$). TOF of these ultrasonic waves is ${t}_{8}$ (i.e., ${t}_{8}={t}_{6}+{t}_{d}+{t}_{7}$).
- Higher frequencies are affected more than lower frequencies by the presence of surface rust. This is because the effective depth of each frequency is approximately its wavelength [30]. With a ‘shallow’ effective depth, higher frequencies interact with the surface rust more than lower frequencies.

#### 3.2. Damage Detection Algorithm

#### 3.2.1. Damage Detection

- Generate/introduce ultrasonic waves at transmitter T of model IM.
- Collect the time domain radial displacement $u\left(t\right)$ at receiver R.
- Apply STFT to $u\left(t\right)$ in order to convert it to its spectrogram $U(t,f)$.
- In the spectrogram $U(t,f)$, show the half-power contour at −3 dB from the maximum amplitude of the first wave packet.
- Determine the centroid of the half-power contour for the first wave packet by finding its coordinates $({t}_{c},{f}_{c})$ in the spectrogram $U(t,f)$.
- The centroid frequency ${f}_{c}$ of this FE simulation is thus found. For the intact model (IM), ${f}_{c}={f}_{c,i}$.
- Repeat the steps for an artificially corroded model. For corroded models, ${f}_{c}={f}_{c,c}$.

#### 3.2.2. Damage Localization

#### 3.2.3. Damage Quantification

**length estimation**criterion in our algorithm.

**width estimation**criterion in our algorithm.

**thickness estimation**criterion in our algorithm.

## 4. Simulation Results

#### 4.1. Time Domain Response

#### 4.2. Time-Frequency Domain Response

#### 4.3. Surface Rust Detection

#### 4.4. Surface Rust Localization

#### 4.5. Surface Rust Quantification

## 5. Conclusions

- The presence of surface rust can be detected by the reduction of centroid frequency of the first wave packet in the STFT spectrogram of corroded steel rod models.
- The location of surface rust is estimated by finding the difference in arrival time (TOF) between helically propagating ultrasonic waves and scattered ultrasonic waves (due to surface rust).
- The length of surface rust can be predicted by calculating the difference in TOF between longitudinally propagating ultrasonic waves of intact and corroded steel rod models. This difference in TOF is related to the longitudinal dimension (length) of surface rust.
- The width of surface rust can be determined by calculating the difference in TOF of the first wave packet between intact and corroded steel rods in the STFT spectrogram at a fixed frequency (e.g., 1 MHz in this paper).
- The thickness of surface rust can be estimated by utilizing the second-order derivative of the first wave packet of corroded steel rod models.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Designed loading function in the time domain and; (

**b**) designed loading function in the frequency domain.

**Figure 3.**(

**a**) Mercator projection of intact steel rod model surface; (

**b**) Mercator projection of corroded steel rod model surface; (

**c**) side view of ultrasonic wave propagation paths around the surface rust; and (

**d**) top view of ultrasonic wave propagation paths around the surface rust.

**Figure 4.**(

**a**) Time domain radial displacement $u\left(t\right)$ collected at receiver R; (

**b**) spectrogram $U(t,f)$ obtained by applying short-time Fourier transform (STFT) to $u\left(t\right)$; (

**c**) contours at half-power level of the first wave packet (${f}_{c}$ is the centroid of the contour for the first wave packet).

**Figure 5.**Time domain radial displacement $u\left(t\right)$ in (

**a**) intact model (IM); (

**b**) corroded model 1 (CM1); (

**c**) model CM2; (

**d**) model CM3; (

**e**) model CM4; and (

**f**) model CM5.

**Figure 6.**(

**a**) Time domain radial displacement $u\left(t\right)$ at transmitter T of model IM and (

**b**) spectrogram obtained by applying STFT to $u\left(t\right)$.

**Figure 7.**Spectrogram of (

**a**) model IM; (

**b**) model CM1; (

**c**) model CM2; (

**d**) model CM3; (

**e**) model CM4; and corroded steel rod models at receiver R(f) model CM5.

**Figure 12.**(

**a**) Spectrogram of model IM; (

**b**) spectrogram of model CM1; (

**c**) spectrogram of model CM4; (

**d**) spectrogram of model IM at 1 MHz; (

**e**) spectrogram of model CM1 at 1 MHz; (

**f**) spectrogram of model CM4 at 1 MHz.

**Figure 13.**(

**a**) Ridge of the first wave packets and; (

**b**) second-order derivatives of the first wave packets.

Material | Density (kg/m${}^{3}$) | Young’s Modulus (MPa) | Poisson’s Ratio |
---|---|---|---|

Steel | 7850 | 210,000 | 0.3 |

Rust | 2610 | 500 | 0.3 |

Model | Surface Rust Location ${\mathit{s}}_{3}$ (mm) | Surface Rust Length d (mm) | Surface Rust Width w (mm) | Surface Rust Thickness h (mm) |
---|---|---|---|---|

CM1 | 4 | 2 | 2.2 | 1 |

CM2 | 6 | 2 | 2.2 | 1 |

CM3 | 4 | 4 | 2.2 | 1 |

CM4 | 4 | 2 | 4.4 | 1 |

CM5 | 4 | 2 | 2.2 | 0.5 |

Model | ${\mathit{f}}_{\mathit{c}}$ (MHz) | $\mathbf{\Delta}{\mathit{f}}_{\mathit{c}}$ (MHz) |
---|---|---|

IM | 1.00 | 0 |

CM1 | 0.83 | 0.17 |

CM2 | 0.83 | 0.17 |

CM3 | 0.82 | 0.18 |

CM4 | 0.82 | 0.18 |

CM5 | 0.80 | 0.20 |

Model | Predicted (mm) | Actual (mm) | Error (%) | |
---|---|---|---|---|

Location, ${s}_{3}$ | CM1 | 3.86 | 4 | 3.5 |

CM2 | 5.91 | 6 | 1.5 | |

CM3 | 2.92 | 3 | 2.6 | |

Length, d | CM1 | 1.97 | 2 | 1.5 |

CM2 | 3.69 | 4 | 7.75 | |

Width, w | CM1 | 2.36 | 2.2 | 7.27 |

CM4 | 4.2 | 4.4 | 4.54 | |

Thickness, h | CM1 | 0.98 | 1 | 2 |

CM5 | 0.53 | 0.5 | 6 |

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## Share and Cite

**MDPI and ACS Style**

Tang, Q.; Du, C.; Hu, J.; Wang, X.; Yu, T.
Surface Rust Detection Using Ultrasonic Waves in a Cylindrical Geometry by Finite Element Simulation. *Infrastructures* **2018**, *3*, 29.
https://doi.org/10.3390/infrastructures3030029

**AMA Style**

Tang Q, Du C, Hu J, Wang X, Yu T.
Surface Rust Detection Using Ultrasonic Waves in a Cylindrical Geometry by Finite Element Simulation. *Infrastructures*. 2018; 3(3):29.
https://doi.org/10.3390/infrastructures3030029

**Chicago/Turabian Style**

Tang, Qixiang, Cong Du, Jie Hu, Xingwei Wang, and Tzuyang Yu.
2018. "Surface Rust Detection Using Ultrasonic Waves in a Cylindrical Geometry by Finite Element Simulation" *Infrastructures* 3, no. 3: 29.
https://doi.org/10.3390/infrastructures3030029