## Finite element modeling of acoustic field induced by laser line source near surface defect

Optics Express, Vol. 15, Issue 9, pp. 5512-5520 (2007)

http://dx.doi.org/10.1364/OE.15.005512

Acrobat PDF (665 KB)

### Abstract

A numerical model of acoustic field induced by laser line source near the surface defect is established by finite element method (FEM), where a surface notch of rectangular shape has been introduced to represent the fatigue defect for the convenience of modeling. After calculating numerically the transient displacement distributions, which are generated by the laser irradiation, the ultrasonic wave modes on the surface and in the body of the plate material are presented in details. The longitudinal, transverse and surface acoustic waves (SAWs) excited by laser pulses near surface notch are compared under the situations that the notch depths are different. As the notch depth increases, the directivity of the bulk waves generation changes greatly. The amplitude of the reflected SAW rises observably at the same time, which is observed experimentally when the laser source is shifted near the surface notch in scanning laser line source (SLLS) measurement. Another effect induced by the surface notch is the time lag of the transmitted SAW pulse with respect to the original incident pulse. These phenomena can be explained from the results. The conclusions can be used to surface notch depth evaluation.

© 2007 Optical Society of America

## 1. Introduction

4. J. A. Cooper, R. A. Crosbie, R. J. Dewhurst, A. Mckie, and S. B. Palmer, “Surface acoustic wave interactions with cracks and slots: a noncontacting study using lasers,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control **UFFC 33**, 462–
470 (1986). [CrossRef]

5. Q. Shan and R. J. Dewhurst, “Surface-breaking fatigue crack detection using laser ultrasound,” Appl. Phys. Lett. **62**, 2649–2651 (1993). [CrossRef]

9. Y. Sohn and S. Krishnaswamy, “Mass spring lattice modeling of the scanning laser source technique,” Ultrasonics **39**, 543–551(2002). [CrossRef] [PubMed]

10. I. Arias and J. D. Achenbach, “A model for the ultrasonic detection of surface-breaking cracks by the scanning laser source technique,” Wave Motion **39**, 61–75(2004). [CrossRef]

11. J. F. Guan, Z. H. Shen, J. Lu, X. W. Ni, J. Wang, and B. Xu, “Numerical simulation of the reflected acoustic wave components in the near field of surface defects,” J. Phys. D: Appl. Phys. **39**, 1237–1243(2006). [CrossRef]

12. J. F. Guan, Z. H. Shen, J. Lu, X. W. Ni, J. Wang, and B. Xu, “Finite element analysis of the scanning laser line source technique,” Jpn. J. Appl. Phys. **45**, 5046–5050(2006). [CrossRef]

13. B. Q. Xu, Z. H. Shen, X. W. Ni, J. Lu, and S. Y. Zhang, “Numerical simulation of laser-induced transient temperature field in film-substrate system by finite element method,” International Journal of Heat and Mass Transfer **46**, 4963
–4968(2003). [CrossRef]

14. M. S. Murali and S.H. Yeo, “Process simulation and residual stress estimation of micro-electrodischarge machining using finite element method,” Jpn. J. Appl. Phys. **44**, 5254–5263(2005). [CrossRef]

15. B. Q. Xu, Z. H. Shen, X. W. Ni, J. Lu, and Y. W. Wang, “Finite element modeling of laser-generated ultrasound in coating-substrate system,” J. Appl. Phys. **95**, 2109–2115 (2004). [CrossRef]

16. B. Q. Xu, Z. H. Shen, X. W. Ni, J. J. Wang, J. F. Guan, and J. Lu, “Determination of elastic properties of a film-substrate system by using the neural networks,” Appl. Phys. Lett. **85**, 6161–6163(2004). [CrossRef]

## 2. Theory and numerical method

### 2.1 Thermoelastic theory model

*T*(

*x*,

*y*,

*t*) is the temperature distribution in temporal and spatial domain,

**u**(

*x*,

*y*,

*t*) denotes the displacement vector field and

*k*represents the thermal conductivity.

*β*is the thermoelastic coupling constant and expressed as

*β*= (3

*λ*+ 2

*μ*)

*α*, where

_{T}*α*is the coefficient of linear thermal expansion.

_{T}*λ*and

*μ*are the Lamé constants of the material.

*A*(

*T*) is the optical absorptivity of the specimen surface, and

*I*

_{0}is the incident laser power density.

*f*(

*x*) and

*g*(

*t*) are the spatial and temporal distributions of the laser pulse, respectively. These two functions are

*x*

_{0}is the half irradiation width of the pulsed laser line source,

*t*

_{0}is the rise time of the laser pulse. In this numerical simulation, the laser illuminative region just abuts on the left edge of the notch. The distance between the centre of laser spot and the left edge of the notch is

*x*

_{0}.

**n**is the unit vector normal to the surface,

**I**is the unit tensor, and

**σ**is the stress tensor. In addition to the top surface, the restrictive boundary conditions are adopted on the other three sides of the sample.

### 2.2 Finite element formulation

*C*], the conductivity matrixes [

*K*], the heat source vector {

*q*} can be expressed as

*T*} is the temperature vector, and {

*T*̇} is the temperature rise rate vector.

*M*] and [

*S*] are the mass and stiffness matrixes, and {

*U*} and {

*Ü*} are the displacement and acceleration vectors, respectively; and {

*F*} is the external force vector. For thermoelasticity, the external force vector for each element in the finite element model can be denoted as ∫

_{ext}_{Se}[

*B*]

^{T}[

*D*]{

*ε*}

_{th}*dS*, where [

_{e}*B*]

^{T}is the transpose of the derivative of shape functions, [

*D*] is the material matrix and the thermal strain vector {

*ε*} can be expressed as

_{th}*T*} and {

*T*} are the temperature, reference temperature vectors respectively.

_{ref}12. J. F. Guan, Z. H. Shen, J. Lu, X. W. Ni, J. Wang, and B. Xu, “Finite element analysis of the scanning laser line source technique,” Jpn. J. Appl. Phys. **45**, 5046–5050(2006). [CrossRef]

13. B. Q. Xu, Z. H. Shen, X. W. Ni, J. Lu, and S. Y. Zhang, “Numerical simulation of laser-induced transient temperature field in film-substrate system by finite element method,” International Journal of Heat and Mass Transfer **46**, 4963
–4968(2003). [CrossRef]

## 3. Numerical simulation and results

### 3.1 Numeral simulated acoustic field in defect-free specimen

*t*

_{0}and the half irradiation width of the pulsed laser

*x*

_{0}on the surface are taken to be 10 ns and 50 μm, respectively. The amplitude of the displacement is denoted by the grayscale. The largest displacement appears in the laser irradiation region, which is brought forth by the thermal expansion. In Fig. 3 we can see the wavefronts of the longitudinal waves and shear waves. The Rayleigh waves and the Head waves propagating along the surface of the specimen are also shown in the Fig. 3. The displacement distribution shows symmetrical about the line of the laser incident direction.

### 3.2 Acoustic field induced by a laser near surface notches with various depths

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. Kenderian, B. B. Djordjevie, and R. E. Green, “Point and line source laser generation of ultrasound for inspection of internal and surface flaws in rail and structural materials,”
Res. Nondestr. Eval. |

2. | T. Tanaka and Y. Izawa, “Nondestructive detection of small defect by laser ultrasonics,” SPIE |

3. | T. Tanaka and Y. Izawa, “Nondestructive detection of dmall Internal defects in carbon steel by laser ultrasonics,” Jpn. J. Appl. Phys. |

4. | J. A. Cooper, R. A. Crosbie, R. J. Dewhurst, A. Mckie, and S. B. Palmer, “Surface acoustic wave interactions with cracks and slots: a noncontacting study using lasers,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control |

5. | Q. Shan and R. J. Dewhurst, “Surface-breaking fatigue crack detection using laser ultrasound,” Appl. Phys. Lett. |

6. | A. K. Kromine, P. A. Fomitchov, S. Krishnaswamy, and J. D. Achenbach, “Scanning laser source technique for detection of surface-breaking and sub-surface cracks,” Review of Progress in Quantitative Nondestructive Evaluation |

7. | P. A. Fomitchov, A. K. Kromine, Y. Sohn, S. Krishnaswamy, and J. D. Achenbach, “Ultrasounic imaging of small surface-breaking defects using scanning laser source technique,” Rev. Prog. In Quantitative Nondestructive Eval. |

8. | A. K. Kromine, P. A. Fomitchov, S. Krishnaswamy, and J. D. Achenbach, “Detection of subsurface defects using laser based technique,” Rev. Prog. in Quantitative Nondestructive Evaluation |

9. | Y. Sohn and S. Krishnaswamy, “Mass spring lattice modeling of the scanning laser source technique,” Ultrasonics |

10. | I. Arias and J. D. Achenbach, “A model for the ultrasonic detection of surface-breaking cracks by the scanning laser source technique,” Wave Motion |

11. | J. F. Guan, Z. H. Shen, J. Lu, X. W. Ni, J. Wang, and B. Xu, “Numerical simulation of the reflected acoustic wave components in the near field of surface defects,” J. Phys. D: Appl. Phys. |

12. | J. F. Guan, Z. H. Shen, J. Lu, X. W. Ni, J. Wang, and B. Xu, “Finite element analysis of the scanning laser line source technique,” Jpn. J. Appl. Phys. |

13. | B. Q. Xu, Z. H. Shen, X. W. Ni, J. Lu, and S. Y. Zhang, “Numerical simulation of laser-induced transient temperature field in film-substrate system by finite element method,” International Journal of Heat and Mass Transfer |

14. | M. S. Murali and S.H. Yeo, “Process simulation and residual stress estimation of micro-electrodischarge machining using finite element method,” Jpn. J. Appl. Phys. |

15. | B. Q. Xu, Z. H. Shen, X. W. Ni, J. Lu, and Y. W. Wang, “Finite element modeling of laser-generated ultrasound in coating-substrate system,” J. Appl. Phys. |

16. | B. Q. Xu, Z. H. Shen, X. W. Ni, J. J. Wang, J. F. Guan, and J. Lu, “Determination of elastic properties of a film-substrate system by using the neural networks,” Appl. Phys. Lett. |

**OCIS Codes**

(140.6810) Lasers and laser optics : Thermal effects

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 13, 2006

Revised Manuscript: April 4, 2007

Manuscript Accepted: April 9, 2007

Published: April 20, 2007

**Citation**

Yifei Shi, Zhonghua Shen, Xiaowu Ni, Jian Lu, and Jianfei Guan, "Finite element modeling of acoustic field induced by laser line source near surface defect," Opt. Express **15**, 5512-5520 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5512

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### References

- S. Kenderian, B. B. Djordjevie, and R. E. Green, "Point and line source laser generation of ultrasound for inspection of internal and surface flaws in rail and structural materials," Res. Nondestr. Eval. 13, 189-200 (2001).
- T. Tanaka and Y. Izawa, "Nondestructive detection of small defect by laser ultrasonics," SPIE 3887, 341-348 (2000). [CrossRef]
- T. Tanaka and Y. Izawa, "Nondestructive detection of dmall Internal defects in carbon steel by laser ultrasonics," Jpn. J. Appl. Phys. 40, 1477-1481 (2001). [CrossRef]
- J. A. Cooper, R. A. Crosbie, R. J. Dewhurst, A. Mckie, and S. B. Palmer, "Surface acoustic wave interactions with cracks and slots: a noncontacting study using lasers," IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control UFFC 33, 462-470 (1986). [CrossRef]
- Q. Shan and R. J. Dewhurst, "Surface-breaking fatigue crack detection using laser ultrasound," Appl. Phys. Lett. 62, 2649-2651 (1993). [CrossRef]
- A. K. Kromine, P. A. Fomitchov, S. Krishnaswamy, and J. D. Achenbach, "Scanning laser source technique for detection of surface-breaking and sub-surface cracks," Review of Progress in Quantitative Nondestructive Evaluation 19, 335-342 (2000).
- P. A. Fomitchov, A. K. Kromine, Y. Sohn, S. Krishnaswamy, and J. D. Achenbach, "Ultrasounic imaging of small surface-breaking defects using scanning laser source technique," Rev. Prog. In Quantitative Nondestructive Eval. 21A, 356-362(2002).
- A. K. Kromine, P. A. Fomitchov, S. Krishnaswamy, and J. D. Achenbach, "Detection of subsurface defects using laser based technique," Rev. Prog. in Quantitative Nondestructive Evaluation 20, 1612-1617 (2001).
- Y. Sohn, and S. Krishnaswamy, "Mass spring lattice modeling of the scanning laser source technique," Ultrasonics 39, 543-551 (2002). [CrossRef] [PubMed]
- I. Arias and J. D. Achenbach, "A model for the ultrasonic detection of surface-breaking cracks by the scanning laser source technique," Wave Motion 39, 61-75 (2004). [CrossRef]
- J. F. Guan, Z. H. Shen, J. Lu, X. W. Ni, J. Wang, and B. Xu, "Numerical simulation of the reflected acoustic wave components in the near field of surface defects," J. Phys. D: Appl. Phys. 39, 1237-1243 (2006). [CrossRef]
- J. F. Guan, Z. H. Shen, J. Lu, X. W. Ni, J. Wang, and B. Xu, "Finite element analysis of the scanning laser line source technique," Jpn. J. Appl. Phys. 45, 5046-5050 (2006). [CrossRef]
- B. Q. Xu, Z. H. Shen, X. W. Ni, J. Lu, and S. Y. Zhang, "Numerical simulation of laser-induced transient temperature field in film-substrate system by finite element method," International Journal of Heat and Mass Transfer 46, 4963-4968 (2003). [CrossRef]
- M. S. Murali and S.H. Yeo, "Process simulation and residual stress estimation of micro-electrodischarge machining using finite element method," Jpn. J. Appl. Phys. 44, 5254-5263 (2005). [CrossRef]
- B. Q. Xu, Z. H. Shen, X. W. Ni, J. Lu, and Y. W. Wang, "Finite element modeling of laser-generated ultrasound in coating-substrate system," J. Appl. Phys. 95, 2109-2115 (2004). [CrossRef]
- B. Q. Xu, Z. H. Shen, X. W. Ni, J. J. Wang, J. F. Guan, and J. Lu, "Determination of elastic properties of a film-substrate system by using the neural networks," Appl. Phys. Lett. 85, 6161-6163 (2004). [CrossRef]

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