# Mathematical Modeling and Computer-Aided Simulation of the Acoustic Response for Cracked Steel Specimens

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. System Mechanism

#### 2.2. Simulation Setup

_{min}is the minimum wavelength of the propagating wave. The simulation time step is defined as:

_{max}is the highest frequency of interest. Table 1 lists the elastic and mechanical properties of the steel for the simulation settings. Abaqus CAE is utilized to develop the 3D model of the setup, and cracks are localized to verify the proposed technique. A mechanical force at the excitation point actuates acoustic waves to imitate the photoacoustic imaging modality and applies a load in the simulation. At the designated sensing points, the generated acoustic waves that travel through the pipe are measured. Data at sensing points are logged and saved for further analyses. The scheme of this study is to develop a numerical model of the data acquired through the PAI setup of steel pipes in pristine conditions with cracks at different locations. The key feature of this idea is to generate ample responses of pipes with known crack locations to approximate the entire pipe by a combination of these responses. Then, when a new response with an unknown damage location is presented to the designed model, it estimates the crack location by interpolating the best match of the data with the existing information.

#### 2.2.1. Simulation Geometry Variations

#### 2.2.2. Different Crack Sizes

#### 2.2.3. Multiple Cracks

#### 2.3. Data Acquisition and Preprocessing

_{1}and a

_{2}are the scaling parameters of sinusoidal functions to model the data, b

_{1}and b

_{2}are the frequencies of these functions, and c

_{1}and c

_{2}are the phases of the observed data. The proposed model of the system incorporates 6 free variables (a

_{1}, b

_{1}, c

_{1}, a

_{2}, b

_{2}, and c

_{2}), whose values describe the response based on the crack locations. To estimate these parameters in each scenario, the proposed model is reformulated in the form of a cost function J, which describes the difference of actual values, a(x), and calculated values by the mathematical model of f(x):

_{j}(j = 1, 2, .., n) are the vertices, x

_{0}is the initial guess, n is the number of free parameters (6 in this study), and e

_{j}is the unit vector in the direction of the jth vertex. The following step is performed to sort the function in ascending order at all vertices:

_{l}is the vertex with the minimum value, x

_{h}is the vertex with the maximum value, and x

_{s}is the vertex with the second highest value of the cost function. Now, the vertex with the highest value is discarded by defining the centroid:

#### Classification

^{®}and simulation responses were provided to the model. The responses were first divided into a training set (70% of the response dataset) and validation set (30% of the response dataset). The training set is used in supervised learning of the SVM model based on the feature set. A response is placed in a row of the training matrix, and a corresponding label entry is made in the output vector representing the target class of that input row. Cross-validation was performed using the validation set.

## 3. Results and Discussions

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Hou, B.; Li, X.; Ma, X.; Du, C.; Zhang, D.; Zheng, M.; Xu, W.; Lu, D.; Ma, F. The cost of corrosion in China. npj Mater. Degrad.
**2017**, 1, 1–10. [Google Scholar] [CrossRef] - Flint, G.; Packirisamy, S.J. Purity of food cooked in stainless steel utensils. Food Addit. Contam.
**1997**, 14, 115–126. [Google Scholar] [CrossRef] - Liu, Q.; Zhu, Y.; Yuan, X.; Zhang, J.; Wu, R.; Dou, Q.; Liu, S. Internet of Things Health Detection System in Steel Structure Construction Management. IEEE Access
**2020**, 8, 147162–147172. [Google Scholar] [CrossRef] - Lo, K.H.; Shek, C.H.; Lai, J.K.L. Recent developments in stainless steels. Mater. Sci. Eng. R Rep.
**2009**, 65, 39–104. [Google Scholar] [CrossRef] - Baddoo, N.R. Stainless steel in construction: A review of research, applications, challenges and opportunities. J. Constr. Steel Res.
**2008**, 64, 1199–1206. [Google Scholar] [CrossRef] - Oluwasola, E.A.; Hainin, M.R.; Aziz, M.M.A. Characteristics and utilization of steel slag in road construction. Jurnal Teknologi
**2014**, 70, 117–123. [Google Scholar] [CrossRef] [Green Version] - Kerouedan, J.; Queffelec, P.; Talbot, P.; Quendo, C.; De Blasi, S.; Le Brun, A. Detection of micro-cracks on metal surfaces using near-field microwave dual-behavior resonator filters. Meas. Sci. Technol.
**2008**, 19, 105701. [Google Scholar] [CrossRef] - Lankford, J.; Kusenberger, F.J. Initiation of fatigue cracks in 4340 steel. Metall. Trans.
**1973**, 4, 553–559. [Google Scholar] [CrossRef] - Liu, S.; Chai, K.; Zhang, C.; Jin, L.; Yang, Q.J. Electromagnetic Acoustic Detection of Steel Plate Defects Based on High-Energy Pulse Excitation. Appl. Sci.
**2020**, 10, 5534. [Google Scholar] [CrossRef] - Otegui, J.; Kerr, H.; Burns, D.; Mohaupt, U.J. Fatigue crack initiation from defects at weld toes in steel. Int. J. Press. Vessel. Pip.
**1989**, 38, 385–417. [Google Scholar] [CrossRef] - Rajan, K.; Narasimhan, K.J. An investigation of the development of defects during flow forming of high strength thin wall steel tubes. Pract. Fail. Anal.
**2001**, 1, 69–76. [Google Scholar] [CrossRef] - Soukup, D.; Huber-Mörk, R. Convolutional neural networks for steel surface defect detection from photometric stereo images. In International Symposium on Visual Computing; Springer: Cham, Switzerland, 2014; pp. 668–677. [Google Scholar]
- ASCE. 2021 Report Card for America’s Infrastructure. American Society of Civil Engineers: Reston, VA, USA, 2021. Available online: https://infrastructurereportcard.org/wp-content/uploads/2020/12/2021-IRC-Executive-Summary-1.pdf (accessed on 25 June 2021).
- Geng, J.; Sun, Q.; Zhang, Y.; Cao, L.; Zhang, W. Studying the dynamic damage failure of concrete based on acoustic emission. Constr. Build. Mater.
**2017**, 149, 9–16. [Google Scholar] [CrossRef] - Gholizadeh, S. A review of non-destructive testing methods of composite materials. Procedia Struct. Integr.
**2016**, 1, 50–57. [Google Scholar] [CrossRef] [Green Version] - Shah, S.G.; Kishen, J.C. Use of acoustic emissions in flexural fatigue crack growth studies on concrete. Eng. Fract. Mech.
**2012**, 87, 36–47. [Google Scholar] [CrossRef] - Ramani, V.; Kuang, K.S.C. Monitoring chloride ingress in concrete using an imaging probe sensor with sacrificial metal foil. Autom. Constr.
**2020**, 117, 103260. [Google Scholar] [CrossRef] - Schlichting, J.; Maierhofer, C.; Kreutzbruck, M.; International, E. Crack sizing by laser excited thermography. NDT E Int.
**2012**, 45, 133–140. [Google Scholar] [CrossRef] - Xu, Y.; Li, S.; Zhang, D.; Jin, Y.; Zhang, F.; Li, N.; Li, H. Identification framework for cracks on a steel structure surface by a restricted Boltzmann machines algorithm based on consumer-grade camera images. Struct. Control. Health Monit.
**2018**, 25, e2075. [Google Scholar] [CrossRef] - Wang, N.; Zhao, Q.; Li, S.; Zhao, X.; Zhao, P. Damage classification for masonry historic structures using convolutional neural networks based on still images. Comput. Aided Civ. Infrastruct. Eng.
**2018**, 33, 1073–1089. [Google Scholar] [CrossRef] - Carden, E.P.; Fanning, P. Vibration based condition monitoring: A review. Struct. Health Monit.
**2004**, 3, 355–377. [Google Scholar] [CrossRef] - Khan, A.; Stanbridge, A.B.; Ewins, D.J. Detecting damage in vibrating structures with a scanning LDV. Opt. Lasers Eng.
**1999**, 32, 583–592. [Google Scholar] [CrossRef] - Walker, S.V.; Kim, J.-Y.; Qu, J.; Jacobs, L.J. Fatigue damage evaluation in A36 steel using nonlinear Rayleigh surface waves. NDT E Int.
**2012**, 48, 10–15. [Google Scholar] [CrossRef] - Sagar, S.P.; Parida, N.; Das, S.; Dobmann, G.; Bhattacharya, D.K. Magnetic Barkhausen emission to evaluate fatigue damage in a low carbon structural steel. Int. J. Fatigue
**2005**, 27, 317–322. [Google Scholar] [CrossRef] - Chang, Y.; Jiao, J.; Liu, X.; Li, G.; He, C.; Wu, B. Nondestructive evaluation of fatigue in ferromagnetic material using magnetic frequency mixing technology. NDT E Int.
**2020**, 111, 102209. [Google Scholar] [CrossRef] - Man, J.; Vystavěl, T.; Weidner, A.; Kuběna, I.; Petrenec, M.; Kruml, T.; Polák, J. Study of cyclic strain localization and fatigue crack initiation using FIB technique. Int. J. Fatigue
**2012**, 39, 44–53. [Google Scholar] [CrossRef] - Ding, X.; Li, W.; Xiong, J.; Shen, Y.; Huang, W. A flexible laser ultrasound transducer for Lamb wave based structural health monitoring. Smart Mater. Struct.
**2020**, 29, 075006. [Google Scholar] [CrossRef] - Kim, Y.-M.; Han, G.; Kim, H.; Oh, T.-M.; Kim, J.-S.; Kwon, T.-H. An Integrated Approach to Real-Time Acoustic Emission Damage Source Localization in Piled Raft Foundations. Appl. Sci.
**2020**, 10, 8727. [Google Scholar] [CrossRef] - Naeimi, M.; Li, Z.; Qian, Z.; Zhou, Y.; Wu, J.; Petrov, R.H.; Sietsma, J.; Dollevoet, R. Reconstruction of the rolling contact fatigue cracks in rails using X-ray computed tomography. NDT E Int.
**2017**, 92, 199–212. [Google Scholar] [CrossRef] - Wang, R.; Liu, F.; Hou, F.; Jiang, W.; Hou, Q.; Yu, L. A non-contact fault diagnosis method for rolling bearings based on acoustic imaging and convolutional neural networks. IEEE Access
**2020**, 8, 132761–132774. [Google Scholar] [CrossRef] - Guldur, B.; Yan, Y.; Hajjar, J.F. Condition assessment of bridges using terrestrial laser scanners. In Structures Congress; American Society of Civil Engineers: Reston, VA, USA, 2015; pp. 355–366. [Google Scholar]
- Rucka, M.; Zima, B.; Kędra, R. Application of guided wave propagation in diagnostics of steel bridge components. Arch. Civ. Eng.
**2014**, 60, 493–516. [Google Scholar] [CrossRef] [Green Version] - Park, D.-G.; Angani, C.S.; Rao, B.; Vértesy, G.; Lee, D.-H.; Kim, K.-H. Detection of the subsurface cracks in a stainless steel plate using pulsed eddy current. J. Nondestruct. Eval.
**2013**, 32, 350–353. [Google Scholar] [CrossRef] - Knitter-Piątkowska, A.; Dobrzycki, A. Application of Wavelet Transform to Damage Identification in the Steel Structure Elements. Appl. Sci.
**2020**, 10, 8198. [Google Scholar] [CrossRef] - Tang, S.; Wang, R.; Han, J. Acoustic Focusing Imaging Characteristics Based on Double Negative Locally Resonant Phononic Crystal. IEEE Access
**2019**, 7, 112598–112604. [Google Scholar] [CrossRef] - Billeh, Y.N.; Liu, M.; Buma, T. Spectroscopic photoacoustic microscopy using a photonic crystal fiber supercontinuum source. Opt. Express
**2010**, 18, 18519–18524. [Google Scholar] [CrossRef] [PubMed] - Granchi, S.; Vannacci, E.; Miris, L.; Onofri, L.; Zingoni, D.; Biagi, E. Spectral Analysis of Ultrasonic and Photo Acoustic Signals Generated by a Prototypal Fiber Microprobe for Media Characterization. Sens. Imaging
**2020**, 21, 1–13. [Google Scholar] [CrossRef] - Minonzio, J.-G.; Cataldo, B.; Olivares, R.; Ramiandrisoa, D.; Soto, R.; Crawford, B.; De Albuquerque, V.H.C.; Munoz, R. Automatic Classifying of Patients with Non-Traumatic Fractures Based on Ultrasonic Guided Wave Spectrum Image Using a Dynamic Support Vector Machine. IEEE Access
**2020**, 8, 194752–194764. [Google Scholar] [CrossRef] - Fitzpatrick, A.; Singhvi, A.; Arbabian, A. An Airborne Sonar System for Underwater Remote Sensing and Imaging. IEEE Access
**2020**, 8, 189945–189959. [Google Scholar] [CrossRef] - Liu, P.; Sohn, H. Numerical simulation of damage detection using laser-generated ultrasound. Ultrasonics
**2016**, 69, 248–258. [Google Scholar] [CrossRef] - Kamran, M.A.; Jeong, M.Y.; Mannan, M. Optimal hemodynamic response model for functional near-infrared spectroscopy. Front. Behav. Neurosci.
**2015**, 9, 151. [Google Scholar] [CrossRef] [Green Version] - Gong, L.; Yu, X.; Wang, J. Curve-Localizability-SVM Active Localization Research for Mobile Robots in Outdoor Environments. Appl. Sci.
**2021**, 11, 4362. [Google Scholar] [CrossRef] - Wang, S.; Echeverry, J.; Trevisi, L.; Prather, K.; Xiang, L.; Liu, Y. Ultrahigh resolution pulsed laser-induced photoacoustic detection of multi-scale damage in CFRP composites. Appl. Sci.
**2020**, 10, 2106. [Google Scholar] [CrossRef] [Green Version] - Li, X.; Shui, G.; Zhao, Y.; Wang, Y.-S. Propagation of Non-Linear Lamb Waves in Adhesive Joint with Micro-Cracks Distributing Randomly. Appl. Sci.
**2020**, 10, 741. [Google Scholar] [CrossRef] [Green Version] - Kazys, R.J.; Mazeika, L.; Sestoke, J. Development of ultrasonic techniques for measurement of spatially non-uniform elastic properties of thin plates by means of a guided sub-sonic A0 mode. Appl. Sci.
**2020**, 10, 3299. [Google Scholar] [CrossRef] - Park, S.-H.; Kim, J.; Song, D.-G.; Choi, S.; Jhang, K.-Y. Measurement of Absolute Acoustic Nonlinearity Parameter Using Laser-Ultrasonic Detection. Appl. Sci.
**2021**, 11, 4175. [Google Scholar] [CrossRef] - Guan, L.; Zou, M.; Wan, X.; Li, Y. Nonlinear Lamb wave micro-crack direction identification in plates with mixed-frequency technique. Appl. Sci.
**2020**, 10, 2135. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Effects of different energies of the excitation laser on the target material causing thermal expansion or ablation/melting.

**Figure 3.**Geometry snapshots of the numerical model designed in Abaqus: (

**a**) cylindrical pipe with partitions visible for excitation and sensing locations, (

**b**) geometry with only edges, (

**c**) sketch of the cylinder radial base, and (

**d**) crack formation and dimensions.

**Figure 7.**Comparison of system response with an original sized crack and a double sized crack at 50% offset (A1O50S).

**Figure 8.**Comparison of system response with a single crack (A1O50S) and a pair of cracks at 50% offset (A1O50M).

**Figure 9.**Comparison of the actual response and optimized mathematical model for the pipe with multiple damage locations indicated via percentage of offset between exciter and sensor locations along axis 1.

**Figure 10.**Comparison of the actual response and optimized mathematical model for the pipe with damage in axis 2.

**Figure 11.**Comparison of the actual response and optimized mathematical model for the pipe with damage in axis 3.

Property | Value |
---|---|

Density | 8207 kg/m^{3} |

Poisson’s Ratio | 0.33 |

Young’s Modulus | 129 GPa |

A | 1 | O | 12 | S |

25 | ||||

37 | ||||

2 | 50 | |||

62 | M | |||

3 | 75 | |||

87 |

Crack (Axis 1) | Amplitude 1 | Frequency 1 KHz | Phase 1 | Amplitude 2 | Frequency 2 KHz | Phase 2 | RMSE |
---|---|---|---|---|---|---|---|

12% | 1.51 | 298.40 | 8.13 | 0.40 | 247.90 | 1.32 | 0.32 |

25% | 1.58 | 313.80 | −1.53 | 0.54 | 220.70 | 12.95 | 0.27 |

37% | 2.31 | 327.70 | −7.57 | 1.30 | 285.60 | −18.40 | 0.40 |

50% | 3.28 | 262.10 | −7.47 | 1.88 | 216.60 | −9.95 | 1.32 |

67% | 3.81 | 254.40 | −4.32 | 2.53 | 221.10 | −11.99 | 1.68 |

75% | 69.22 | 254.50 | −28.17 | 70.27 | 255.50 | −0.34 | 1.24 |

87% | 1.16 | 331.30 | −9.97 | 1.30 | 278.60 | −9.77 | 0.13 |

Crack (Axis 2) | Amplitude 1 | Frequency 1 KHz | Phase 1 | Amplitude 2 | Frequency 2 KHz | Phase 2 | RMSE |
---|---|---|---|---|---|---|---|

12% | 37.39 | 295.8 | −17.3 | 36.42 | 300.8 | 2.785 | 1.47 |

25% | 4.47 | 262.00 | −1.92 | 1.41 | 193.70 | −1.11 | 1.52 |

37% | 5.02 | 273.90 | −7.60 | 3.06 | 321.30 | −5.45 | 1.37 |

50% | 4.29 | 303.40 | 7.49 | 1.54 | 221.50 | 13.11 | 1.17 |

63% | 4.21 | 314.00 | 2.70 | 1.31 | 146.70 | −8.49 | 1.19 |

75% | 2301.00 | 295.60 | −16.93 | 2301.00 | 295.60 | 5.04 | 1.53 |

87% | 5.47 | 293.20 | 7.05 | 2.91 | 357.80 | 7.27 | 1.80 |

Crack (Axis 3) | Amplitude 1 | Frequency 1 KHz | Phase 1 | Amplitude 2 | Frequency 2 KHz | Phase 2 | RMSE |
---|---|---|---|---|---|---|---|

12% | 1.94 | 323.90 | −4.05 | 2.33 | 275.80 | −8.29 | 0.47 |

25% | 1.73 | 329.50 | −7.25 | 3.41 | 284.60 | −11.90 | 0.49 |

37% | 2.02 | 282.90 | 13.71 | 2.08 | 350.10 | 15.17 | 0.49 |

50% | 3.22 | 259.70 | −4.42 | 3.20 | 335.10 | −11.60 | 1.47 |

63% | 5.31 | 262.80 | −5.89 | 1.90 | 207.30 | −10.00 | 2.03 |

75% | 2.49 | 361.60 | −20.10 | 7.42 | 279.50 | −9.80 | 1.41 |

87% | 2.76 | 305.00 | 3.53 | 0.40 | 192.30 | 0.28 | 0.47 |

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**MDPI and ACS Style**

Akbar, A.; Kamran, M.A.; Kim, J.; Jeong, M.Y.
Mathematical Modeling and Computer-Aided Simulation of the Acoustic Response for Cracked Steel Specimens. *Appl. Sci.* **2021**, *11*, 7699.
https://doi.org/10.3390/app11167699

**AMA Style**

Akbar A, Kamran MA, Kim J, Jeong MY.
Mathematical Modeling and Computer-Aided Simulation of the Acoustic Response for Cracked Steel Specimens. *Applied Sciences*. 2021; 11(16):7699.
https://doi.org/10.3390/app11167699

**Chicago/Turabian Style**

Akbar, Arbab, Muhammad Ahmad Kamran, Jeesu Kim, and Myung Yung Jeong.
2021. "Mathematical Modeling and Computer-Aided Simulation of the Acoustic Response for Cracked Steel Specimens" *Applied Sciences* 11, no. 16: 7699.
https://doi.org/10.3390/app11167699