## 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

The pulsed laser ultrasound technique in solid has attracted increasing attention owing to its wide application potential in nondestructive evaluation and characterization of materials [1–3]. The generation of ultrasound by pulsed laser irradiation provides a number of advantages over the conventional generation by transducers, namely high spatial resolution, non-contact generation and detection of ultrasonic waves.

Generally the reflections (pulse-echo) [4] or the transmissions (pitch-catch) [5] of the ultrasound waves at the edge of the defect are utilized in the surface-breaking defect detection, a main application field of laser ultrasound. Nevertheless, for small defects relative to the wavelength of the laser-generated Rayleigh wave, the reflected amplitude and amplitude changes in the transmission are often too weak to be detected with existing optical detectors. The recently proposed Scanning Laser Source (SLS) technique provides an alternative method which overcomes these size limitations. This technique employs a line-focused high-power laser source which swept across the test specimen and passes over surface-breaking defects [6]. And the changes in the amplitude and frequency content of laser-generated ultrasound when the laser source scans over the defect can be observed in experiments [7, 8].

These changes have been attributed to both near-field scattering and changes in laser generation constraints in defective regions. To ascertain the internal physical mechanism of the SLS technique, a mass spring lattice model (MSLM) [9] has been employed to simulate the SLS signatures, assuming the laser source as shear dipoles on a surface. This assumption cannot predict the body source generated by a pulsed laser in reality. A direct frequency domain boundary element method has been utilized to characterize the presence and size of surface-breaking cracks [10], which is based on the decomposition of the field generated by a laser in a cracked two-dimensional half-space, by virtue of linear superposition, into the incident and scattered fields. This method can reproduce the experimentally obtained results very well in the case of a large notch, and further adjustment of the parameters of the model is required for a better quantitative agreement. A finite element model[11, 12], which can simulate transient temperature field [13, 14] and elastic wave propagation both within elastic media and on a surface [15, 16] , is used to demonstrate the SLS behavior and the results show a good agreement with the experiment [7].

Due to the complexity of the acoustic modes conversion near the defect and the change of the laser irradiation constraints, the finite element method (FEM) is adopted to calculate the whole acoustic field excited by pulsed laser near the surface defect in this paper. By analyzing the acoustic field variation with the varied notch depth, the mechanism of the SLS detection method is discussed in detail. These conclusions are helpful to the development of the surface defect detection technique.

## 2. Theory and numerical method

#### 2.1 Thermoelastic theory model

In our simulation model, we use the notch instead of crack for convenience. The geometry model of laser irradiation near the notch on the top surface of an aluminum plate is schematically shown in Fig. 1. The pulsed laser line source whose axis is parallel to the relatively long direction of the surface notch in the plate is considered and the spatial distribution of the laser beam along X axis is assumed Gaussian and the spatial distribution along Z axis is homogeneous; in addition, the test specimen used is homogeneous, isotropic and linearly elastic. The generation and propagation of elastic waves in the plate can be modeled as a two-dimensional plane strain thermoelastic problem shown in Fig. 2. Ignoring the heat produced by mechanical deformation in our analysis, the so-called thermal stress approximation, the governing equations of the thermal and elastic problems are coupled only one way through the thermal stress term.

The governing equations in the X-Y plane for an isotropic solid are

where*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 _{T}*, where

*α*is the coefficient of linear thermal expansion.

_{T}*λ*and

*μ*are the Lamé constants of the material.

The boundary conditions at the edges of the model are treated as adiabatic, unless there is heat flux on the laser illuminative region in the thermal analysis. Suppose the laser illuminative region is I and the other region is II, the boundary conditions are expressed as

where *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

where *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}.

A traction-free boundary is satisfied on the top surface (y=h) and three internal sides of the surface notch, which can be expressed as

where **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.

Moreover, the initial conditions for the temperature and displacement fields are

#### 2.2 Finite element formulation

The classical thermal conduction equations for finite element with the heat matrix [*C*], the conductivity matrixes [*K*], the heat source vector {*q*} can be expressed as

where {*T*} is the temperature vector, and {*T*̇} is the temperature rise rate vector.

For wave propagation in elastic media, ignoring damping, the governing finite element equation is

where [*M*] and [*S*] are the mass and stiffness matrixes, and {*U*} and {*Ü*} are the displacement and acceleration vectors, respectively; and {*F _{ext}*} is the external force vector. For thermoelasticity, the external force vector for each element in the finite element model can be denoted as ∫

_{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}In above equation {*T*} and {*T _{ref}*} are the temperature, reference temperature vectors respectively.

The procedure employed to determine the solutions of the thermoealstic equations is sequential field-coupling modeling. Namely, the transient temperature gradient field induces the stress and displacement fields, and the effects of the stress and displacement fields on the temperature field are neglectable. Thus, sequential field coupling is considered in this modeling. The details of a complete algorithm have been given in previous papers [12, 13].

## 3. Numerical simulation and results

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

The numeral simulated acoustic field at t=0.486μs is shown in Fig. 3. These results are calculated on the specimen without surface defect and the laser irradiation region is at the centre of the specimen. The laser energy is 0.35 mJ, the pulse rise time *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.

The displacement vectors of these waves are shown in the top right corner in Fig. 3. Each displacement vector is the vector sum of the displacement along X and Y directions in the positions indicated by the characters in the Fig. 3. The direction of the displacement vector of the Rayleigh waves rotates elliptically as a function of the position and the vector direction of the head waves makes an angle with the surface, the propagation direction of the head waves. These are effects that the longitudinal and shear waves are coupled on the boundary of the elastic body. And the vibration directions of the longitudinal and shear waves are also shown in the Fig. 3.

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

When the laser pulse irradiates the edge of the surface notch, the generated acoustic field is much different from the acoustic field in non-defect specimen. Figure 4 shows the acoustic fields induced by laser pulse irradiating the edge of surface notches with various depths at t=0.486μs. In this FEM model the specimen is an aluminum plate with 8 mm length and 6 mm height and on the top surface of the specimen there is a notch with 20 μm width and various depths of d. The positions of the notches are indicated in the Fig. 4. The thermal flux generated by laser pulse irradiates the edge of the surface notch.

The displacement magnitude distributions generated by laser pulses irradiation are denoted by grayscale in Fig. 4. We can observe that the acoustic field varies as the depth of surface notch increasing. Compare to the Fig. 4(a), the case without surface notch, the symmetry of emitting acoustic field is destroyed by the existence of the surface notch. The wave energy of the right side, which is beyond the notch is less than the wave energy of left side, i.e. the amplitudes of the longitudinal wave, shear wave and Rayleigh wave diminish. When notch gets more and more deep, the energy propagating beyond the notch is more and more small and some acoustic modes disappear. At the same time the amplitudes of the waves propagating to the left side are enhanced much more. The directivity of the bulk waves generation changes by the surface notch.

From the Fig. 4 we can also see the phenomenon that two peaks of Rayleigh wave magnitude appear when the notch depth increases up to 80μm. It is shown in Fig. 5 in detail that the waveforms of the acoustic waves received on the top surface of the notches with various depths at the point, which is 1.5 mm away from the laser source. The waves on the surface of the specimen without notch are monopolar. But in the case of the specimen with surface notch, a positive wave peak appears and the waveform becomes bipolar. And the positive peak amplitude increases with the depth of the surface notch. We denote the original negative peak as RW and the positive wave peak as RR. Because the RW wave exists on the specimen without surface notch, so the RW waves are Rayleigh waves propagating directly from the laser illuminative region. Thus the arrival time of the RW waves is constant in the different cases of surface notch with different depths. And the RR waves accompany by the occurrence of the notchs, which indicates that the RR waves are reflected from the surface notches. And the arrival time of the RR waves is not various with the notch depth. This characteristic confirms that the RR waves are just reflected from the left edge of the notch. This phenomenon is also observed in experiments reported in Ref. 8. This change is the foundation of the SLS defect detection technique. But the RR waves cannot be used to measure the depth of the surface notch.

Another effect induced by the surface notch is the time lag of the transmitted SAW pulse with respect to the original incident pulse, as can be clearly seen in Fig. 6. Figure 6 shows the displacement of the surface point at 1.5 mm away from the laser source at the other side of the notch. The first arrived waves, which are denoted with TR, reach the detection point at the same time with the various depths. But the second arrived waves denoted with TP have different delay corresponding to the depths of the notch. The delay varies between 29-143 ns for different notch depth.

The arrival time of the TR waves is constant as the depth of notch increases and the TR waves have the same velocity as the Rayleigh wave in the aluminum sample. Only the amplitude and pulse width change. So this component transmits the surface notch directly.

The arrival times of the TP waves increases when the notch becomes deeper. And when the notch is deeper than 120 μm, we can find that the TR and TP waves separate from each other. We can draw the conclusion that the TP component first propagates though the edge of notch and converts to compressive wave at the bottom of the notch, and then it propagates as the compression wave speed from the bottom of the notch to the surface of sample. The TP waves can be used to measure the depth of the notch.

## 4. Conclusion

The acoustic field generated by pulsed laser near surface defect is investigated by numerical simulation. The finite element method is used to establish the model of transient displacement field. By discussing the displacement distribution in the elastic media, the veracity of this numerical method is proved. Then the acoustic fields induced by laser pulse near surface notch with various depths are calculated. The varieties of the amplitude and directivity of the generated acoustic waves are analyzed though comparison between different situations. The conclusions are helpful to design new measurement methods of the surface defect.

## Acknowledgments

This work is supported by the National Natural Science Foundations of China under Grant No. 60578015 and 60208004, in addition partly supported by the Teaching and Research Award Program for Outstanding Young Professor in Higher Education Institute MOE, P. R. C.

## 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. **13**, 189–200(2001).

**2. **T. Tanaka and Y. Izawa, “Nondestructive detection of small defect by laser ultrasonics,” SPIE **3887**, 341–348 (2000). [CrossRef]

**3. **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]

**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]

**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 **19**, 335–342 (2000).

**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. **21A**, 356–362
(2002).

**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 **20**, 1612–1617(2001).

**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]