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Numerical simulation of laser-generated ultrasound in non-metallic material by the finite element method
ARTICLE IN PRESS
Optics & Laser Technology 39 (2007) 806–813 www.elsevier.com/locate/optlastec
Numerical simulation of laser-generated ultrasound in non-metallic material by the finite element method Jijun Wanga,b, Zhonghua Shena,, Baiqiang Xub, Xiaowu Nia, Jianfei Guana, Jian Lua a
Deptartment of Appl. Physics, Nanjing University of Science and Technology Nanjing, 210094, People’s Republic of China b Faculty of Science, Jiangsu University, Zhenjiang, 212013, People’s Republic of China Received 19 August 2005; received in revised form 27 January 2006; accepted 27 January 2006 Available online 29 March 2006
Abstract The use of a pulsed laser for the generation of the elastic waves in non-metallic materials in the thermoelastic regime is investigated by using finite element method (FEM), taking into account not only thermal diffusion and the finite spatial and temporal shape of the laser pulse, but also optical penetration and the temperature dependence of material properties. The optimum finite element model is established based on analysis of two important parameters, meshing size and time step, and the stability of solution. Temperature distributions and temperature gradient fields in non-metallic material for different time steps are obtained, this temperature field is equivalent to a bulk force source to generate ultrasonic wave. The laser-generated ultrasound waveforms at the epicenter and surface acoustic waveforms (SAWs) are obtained and the influence of optical penetration into the material on the temperature field and the ultrasound waveforms are analyzed. The numerical results indicate that the heat penetration into non-metallic material is caused mainly by the optical penetration, and the ultrasound waveforms, especially the shape of the precursor, are strongly dependent on the optical penetration depth into non-metallic material. r 2006 Elsevier Ltd. All rights reserved. Keywords: Laser-generated ultrasound; Finite element method; Non-metallic material
1. Introduction Due to its non-contact feature and the ability of broadband signal generation, laser ultrasound has demonstrated its great potential for non-destructive evaluation. When a solid is illuminated with a laser pulse, absorption of the laser energy results in an increased localized temperature, which in turn causes thermal expansion and generates ultrasonic waves in the solid. Much work has been reported for metallic specimen [1–6]. Scrubby et al. [1] studied early the problem of laser-generated ultrasound using a point source approximation. Rose [2] treated the point source as a surface center of expansion (SCOE) and obtained a formal solution, neglecting the thermal diffusion. McDonald [5] and Sanderson [6] improved this model, taking into account both thermal diffusion and the finite spatial and temporal shape of the laser pulse, and Corresponding author.
E-mail address: [email protected] (Z. Shen). 0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.01.009
obtained good results agreement with experiment for metals. Recently, Laser-generated ultrasound in non-metal material, such as ceramics, neutral glass, polymer and composite materials, arise much interest [7–12]. In the case of a metallic specimen, the optical absorption occurs at the surface layer, resulting in a surface-exciting source, the heat penetration is caused mainly by heat diffusion. For the non-metallic materials, the laser beam can penetrate the specimen to some finite depth and induces a bulk-thermal source. Due to the optical penetration, the characteristics of the laser-generated ultrasound will be significantly different from that in metals. Among the theoretical studies, most search for analytical solutions. Based on Laplace–Hankel transform, neglecting the thermal diffusion and taking into account optical penetration into the non-metallic material, Telschow et al. [7] presented a twodimensional calculation to analysis the generation of ultrasonic waves in non-metallic material, and the results illustrated that the shape of the precursor is strongly
ARTICLE IN PRESS J. Wang et al. / Optics & Laser Technology 39 (2007) 806–813
depend on the optical penetration and the thermal properties of the material; Bai et al. [9] solved the thermoelastic wave equation, obtained analytic expressions of the wave displacement components in far-field, and presented the directivity patterns of both longitudinal and shear waves. Dubois et al. [11] calculated the surface displacements by the absorption of a laser pulse by an orthotropic medium to optimize the laser generation of ultrasound in various materials. The inverse transformation is usually obtained by a numerical inverse method, so it is quite difficult to get the analytical inverse transformation solutions. To get analytical solution, simplifying of condition was taken into account. It is also difficult to deal with the complicated geometry specimen, especially the cracks in specimen. Moreover, during the laser generation of the ultrasonic waves, the thermophysical properties of the irradiated materials vary with the rising temperature, the analytical methods neglected the temperature dependence of the parameters of materials. Due to the complexity of the generation of the thermoelastic waves excited by bulk-thermal source in nonmetallic materials, numerical methods will be much more suitable in dealing with such complicated processes, especially the process where the material parameters are temperature dependent. The finite element method (FEM) is versatile due to its flexibility in modeling complicated geometry and its capability in obtaining full field numerical solutions and the evolution of temperature, thermal stress and displacement fields in non-metallic material in detail. In our previous paper, the transient temperature field and laser-generated ultrasound induced by the pulse laser in single metal plate have been investigated by using finite element method [13], demonstrating the feasibility of modeling laser ultrasound using the thermoelastic equations. In this article, the models of the generation of thermoelastic ultrasonic wave in non-metallic material are established by using FEM, taking into account not only thermal diffusion and the finite spatial and temporal shape of the laser pulse, but also optical penetration and the temperature dependence of material properties. The temperature distributions and temperature gradient fields in non-metallic material for different time steps are calculated accurately by using FEM, then, the ultrasonic waveforms at the epicenter and surface acoustic waves (SAWs) are obtained by coupling the temperature field and the stress field. The influence of the optical penetration depth on the temperature field and the ultrasonic waveforms is analyzed.
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by the target material, and the other is reflected, then a non-uniform temperature field will be generated near the surface, followed by the local thermal expansion, this thermal expansion induces ultrasonic waves. The geometry of laser irradiation on a non-metal specimen is schematically shown in Fig. 1. Due to the thermoelastic mechanism, the thermal conduction and thermal-elastic equation can be descried as [14] Kr2 T rC
@T ¼ q, @t
(1)
ðl þ 2mÞrðr UÞ mr r U r
@2 U ¼ að3l þ 2mÞrT, @t2 (2)
where T is the temperature distribution, q is heat source, K, r and C are the thermal conductive coefficient, density and thermal capacity, respectively; U is the time-dependent displacement, a is the thermoelastic expansion coefficient, l and m are the Lame´ constants, respectively. The spatial mode of the laser beam is assumed to be a Gauss distribution, so a cylindrical coordinate system is adopted. The heat source q can be expressed as q ¼ bð1 RÞI 0 ebz f ðrÞgðtÞ,
(3)
where b is the optical absorption coefficient (1/b is the optical penetration depth), I0 is the incident laser light intensity, R is the optical reflectivity of the specimen, f(r) and g(t) are the spatial and temporal distribution of the laser pulse, respectively. These two functions can be written as 2 r f ðrÞ ¼ exp 2 , (4) a0 gðtÞ ¼
t t exp , t0 t0
(5)
where a0 is the radius of the laser spot, t0 is the rise time of the laser pulse. It is assumed that the specimen surfaces are thermally insulated and mechanically unconstrained, and the temperature elevation and the mechanical displacement as well as their derivatives with respect to time are zero at t ¼ 0. Z
Laser beam
R
2. Theoretical model and numerical method 2.1. Theory of transient temperature field and laser ultrasound by the thermoelastic mechanism When the non-metal specimen is illuminated by a laser pulse with the energy less than the melting threshold of the specimen, a certain amount of the light energy is absorbed
Non-metal material
Fig. 1. Schematic diagram for laser irradiating specimen plate.
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2.2. Numerical method
expressed as [16]
The classical thermal conduction equation for finite elements with the heat capacity matrix [C], conductivity matrix [K], heat flux vector {p1}, and heat source vector {p2} can be expressed as
Le ¼
_ ¼ fp1 g þ fp2 g, ½KfTg þ ½CfTg
lmin , (9) 20 where Le is the element length and lmin is the shortest wavelength of interest.
(6)
3. Numerical results and discussions
_ where {T} is the temperature vector, and fTgis the temperature rise rate vector. For wave propagation, ignoring damping, the governing equation is
3.1. Laser and materials parameters
€ þ ½KfUg ¼ fFext g, ½MfUg
(7)
where [M] is mass matrix, [K] is the stiffness matrix, {U} is € is the acceleration vector and the displacement vector, fUg {Fext} is the action of the laser pulse. For thermoelasticity, R the external force vector for an element is e ½BT ½Dfe0 g dV , where {e0}is the thermal strain vector, [B]T is the transpose of the derivative of the shape functions, and [D] is the material matrix. Eq. (7) is solved using the Newmark time integration method [15]. The process to solve the finite element equation is discretizing the plane with axis symmetry into non-verlapping finite elements, expressing the nodal displacement in terms of the shape functions. Because the mass matrix and the stiffness matrix are sparse, symmetric and positive, it will take much time and large computing capability to solve Eq. (7). Temporal and spatial resolution of the finite element model is critical for the convergence of these numerical results. Choosing an adequate integration time step is very important for the accuracy of the solution. In general, the accuracy of the model can be increased with increasingly smaller integration time steps. With too long time steps, the high frequency components are not resolved accurately enough. On the other hand, too small time steps are a waste of computing time, therefore, a compromise must be found. Dt ¼
1 , 20f max
(8)
where fmax is the highest frequency of interest. But when the rise time of the laser pulse is in the order of ns, a time step in Eq. (8) might not provide sufficient temporal resolution. The time step has to decrease to Dt ¼ 1=ð180f max Þ [16]. Thus, The time step can be determined by estimating the highest frequency of the laser-generated ultrasound waves. Also, the needed time step can alternatively be related to the time the fastest possible wave needs to propagate between successive nodes in the mesh. The element is defined by four nodes with two degrees of freedom at each node in cylindrical coordinate system. In general, the rule of element size is that there are more than 10 nodes in a wavelength. To ensure the propagation of energy between two successive nodes in the mesh, the element length should be fine enough, so the propagating waves are spatially resolved. The recommendation can be
On the basis of the theories described above, the thermoelastically generated waves in non-metal material are calculated. The laser energy is 0.35 mJ. The pulse rise time t0 and the radius of the laser pulse spot on the system are taken to be 10 ns and 50 mm, respectively. The radius and the thickness of the specimen are 2 cm and 5 mm, respectively. The finite element model is constructed, and the element length is 20 mm. The property of semitransparent neutral glass used in the calculation is listed in Table 1. 3.2. Transient temperature and temperature gradient fields A thermoelastic volume source inducing by a laser pulse is originated from a non-uniform temperature field, so an accurate determination of the temperature field is vital to accurate prediction of laser-generated ultrasonic waves. In this research the temperature distributions in nonmetal material were computed for each time step. Fig. 2 shows the temperature evolution at the center of laser irradiation, in which the optical penetration depth is 20 mm. The results show that the temperature rises rapidly during heating by laser, while the cooling process is very slowly, compared with the process of an alumina plate in the similar specimen geometry size and laser parameters [13]. It is because the thermal conductive coefficient of nonmetallic material is smaller than that of metal in one or two order. The temperature distributions with various optical penetration depths at 2 ms are shown in Fig. 3(a)–(c). These figures show the influence of the optical penetration
Table 1 Properties of neutral glass used in the calculation Physical properties
Neutral glass
The optical reflectivity R Thermal conductive coefficient K(W m1 K1)
0.79 1.83 105T+0.012, 300oTo800
Density r(kg m3) Thermal capacity C(J kg1K1) Poission’s ratio Young’s modulus (Pa) Thermal expansion coefficient
1.2 105T+0.021, 800oTo Tm 2400 840 0.23 6.0457 1010 0.63 106
Tm is the melting point of the material.
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600
809
R (µm)
Neutral Glass Al
0
20
40
60
80
500
Z (µm)
Temperature (K)
550 -20
450
(a)
-40 R (µm)
400 0
20
0
40
40
60
80
100
350 300 0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
-20
2.0
Time (µs) Fig. 2. Temperature vs time at the center of laser irradiation in the neutral glass plate and the aluminum plate, respectively.
depth on the temperature field. It shows clearly that there is more volume to be heated by the absorbed laser energy in the non-metallic material than that in the metallic material. Non-uniform temperature distribution is observed as a result of the Gaussian laser power distribution and optical penetration. Fig. 3(a) is the transient temperature distribution, neglecting the optical penetration and taking only the thermal diffusion effects into consideration. It can be seen that the thermal diffusion is only about 15 mm in the axial direction and less than 100 mm in the radial direction. The optical penetration depths in Fig. 3(b)–(c) are 20 and 80 mm, respectively, and the depths of the heated zone in the axial direction are about 60 and 240 mm, respectively, whereas the dimensions of the heated zone in the radial direction are almost same in these three cases. One can see that the heat penetration induced by a pulse laser into the non-metal material is much larger than that neglecting the optical penetration and the volume of the heated zone is much larger too, so the volume of ultrasonic source increases with the increase of optical penetration depth. Fig. 3 indicates that the heat penetration into the specimen is caused by both optical absorption and thermal diffusion, and the subsurface thermal source arises mainly from thermal diffusion in metals, but the bulk-thermal source results mainly from the optical penetration in non-metal material. Non-uniform temperature field causes the temperature gradient field. The temperature gradient fields have also been calculated. Fig. 4(a) and (b) show the horizontal temperature gradient (TGX) and the vertical temperature gradient (TGY), respectively, in which the optical penetration depth is 20 mm at 2 ms. It can be seen that both the horizontal temperature gradient and vertical temperature gradient achieves the maximum value at the specimen surface. The maximum horizontal temperature gradient is at about 35 mm from the center of laser irradiation, where the spatial distribution of laser intensity varies sharpest,
Z (µm)
0.2
-40
-60 (b) R (µm) 80
-20 -40 -60 -80 -100 Z (µm)
0.0
-120 -140 -160 -180 -200 -220 -240
(c) Fig. 3. Contour plots of temperature at 2 ms near the heated affected zone in the neutral glass plate: (a) The penetration depth is 0 mm, Tð0; 0Þ ¼ 611 K, Tð20; 0Þ ¼ 559 K, Tð40; 0Þ ¼ 455 K, Tð60; 0Þ ¼ 369 K, Tð80; 0Þ ¼ 323 K, Tð100; 0Þ ¼ 300 K; (b) The penetration depth is 20 mm, Tð0; 0Þ ¼ 387 K, Tð20; 0Þ ¼ 372 K, Tð40; 0Þ ¼ 351 K, Tð60; 0Þ ¼ 317 K, Tð80; 0Þ ¼ 311 K, Tð100; 0Þ ¼ 300 K; (c) The penetration depth is 80 mm, Tð0; 0Þ ¼ 322 K, Tð20; 0Þ ¼ 318 K, Tð40; 0Þ ¼ 311 K, Tð60; 0Þ ¼ 307 K, Tð80; 0Þ ¼ 302 K, Tð100; 0Þ ¼ 300 K.
and the maximum vertical temperature gradient is at the center of the laser irradiation. The temperature gradient field results in thermal expansion in non-metallic material. The horizontal expansion
ARTICLE IN PRESS J. Wang et al. / Optics & Laser Technology 39 (2007) 806–813
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R (µm) 20
40
60
0.20 80
100
-20 Z (µm)
S
120 Displacement (nm)
0
-40
0.15 L
0.10 0.05 0.00
(a)
-60 -0.05 R (µm) 0
20
40
60
80
100
0.0
0.5
1.0
1.5 Time (µs)
2.0
2.5
Fig. 5. Normal waveform at the epicenter in the neutral glass plate, the penetration depth is 20 mm.
Z (µm)
-20
-40
-60 (b) Fig. 4. Contour plots of temperature gradient at time ¼ 2ms near the heated affected zone in the neutral glass plate, the penetration depth is 20 mm: (a) horizontal temperature gradient (TGX). TGXð0; 0Þ ¼ 1:9 105 , TGXð20; 0Þ ¼ 1:1 106 , TGXð40; 0Þ ¼ 1:7 106 , TGXð60; 0Þ ¼ 7:4 105 , TGXð0; 0Þ ¼ 4:4 105 , TGXð100; 0Þ ¼ 1:7 105 ; (b) vertical temperature gradient (TGY), TGYð0:0Þ ¼ 3:4 106 , TGYð20:0Þ ¼ 2:9 106 , TGYð40:0Þ ¼ 1:9 106 , TGYð60:0Þ ¼ 8:9 105 , TGYð80:0Þ ¼ 3:5 105 , TGYð100:0Þ ¼ 1:2 105 .
induced by the horizontal temperature gradient is constrained by the surrounding unheated material, exciting the horizontal thermal stress. The vertical thermal stress results from the vertical temperature gradient. As would be expected, the longitudinal waves are caused by the vertical thermal stress; shear and surface acoustic waves are generated by the horizontal stress [13]. 3.3. Laser-generated ultrasound 3.3.1. Normal waveform at the epicenter and near-epicenter normal waveform The ultrasonic waveform predicted by the FEM at the epicenter of a 5 mm thickness neutral glass plate are shown in Fig. 5, and the features of the ultrasonic waveform are good agreement with the results reported by Edwards [10]. Well-defined features that occur at the longitudinal and shear arrival times are exhibited, and a significant feature of the laser-generated elastic waveform is a precursor (sharp spike), signaling the arrival of the longitudinal wave. The first arrival time at 0.89 ms corresponds to the direct
longitudinal wave (L). The second arrival time at 1.575 ms corresponds to the direct shear wave (S). The amplitude of the shear wave is considerably larger, compared to the amplitude of the longitudinal wave. This is due to the bulk force source created by the thermo-elastic generation mechanism contains a large shear component. There is a distinguishing feature in the longitudinal component of the ultrasonic waveform at the epicenter. The precursor (sharp spike) is a dipolar pulse. This is caused by the vertical thermal stress during the laser heating and subsequent surface expansion. As the laser energy is deposited, the temperature of the heated zone rises, and the surface irradiated by laser starts expand. The associated thermal stress induced by the vertical temperature gradient causes a negative-going spike (in the coordinate system shown in Fig. 1), which is relieved by the surface expansion. The magnitude of this spike is strongly affected by the thickness (hence mass) of the expanding surface layer [5]. The vertical stress caused by the vertical temperature gradient acts on bottom of the heated zone, generating the primary longitudinal pulse. This pulse radiates down and up simultaneously in the specimen. Reflection from the surface leads to the positivegoing spike, shown in Fig. 6. The time delayed of the positive-going spike is related to the optical penetration depth. The near-epicenter displacements are shown in Fig. 7. It can be seen that the arrival time of shear wave maximum increases with the increase of the distance from the epicenter. The early arrival time of the shear displacement, corresponding to the fringe of the bulk source, does not reflect the finite extent of the source. It takes more time for the signal generated by the center of heated region to transmit to the receiver location with the increase of the distance from the epicenter, so the arrival time of shear wave maximum is certainly delayed in time. It can also be seen that the longitudinal amplitude decreases with
ARTICLE IN PRESS J. Wang et al. / Optics & Laser Technology 39 (2007) 806–813
Fig. 6. Schematic diagram of reflection from the surface of the longitudinal wave.
0.14
0.5mm 1.0mm 1.5mm
0.12
Displacement (nm)
0.10 0.08
811
bulk source. The three kinds of waves overlap in the near field [Fig. 8(a)]. The sP wave is separated from the Rayleigh wave gradually with the increased source-receiver distances while sS wave still couple with the Rayleigh wave in the source–receiver distances of interest [Fig. 8(b)]. The three kinds of waves are separated in the far field completely [Fig. 8 (c) and (d)]. Such a phenomenon results from the fact that the propagation velocity of the Rayleigh wave is much smaller than that of the sP and near equals that of the sS. Due to the small propagation distance and finite pulse width of the three waveforms, such a velocity difference cannot split the three waveforms in the near field. The velocity of the sP wave, sS wave and the Rayleigh wave are 5390m/s, 3190m/s, and 2940m/s, respectively, measured from Fig. 8. It is in good agreement with the theoretical and experiment results reported [9,17].
0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 0.6
0.8
1.2 1.0 Time (µs)
1.4
1.6
1.8
Fig. 7. Near-epicenter normal waveform, the epicenter-receiver distance is 0.5, 1.0 and 1.5 mm, respectively.
increasing distances from the epicenter and becomes wider. The amplitude of the thermoelastic wave depends on r1 in the case of spherical wave propagation, where r is the distance from the source to the receiver location. A bulk source can be regarded as a collection of point sources. As a result of many point sources working concurrently, the longitudinal amplitude in the near field is greater than that in the far field. As the receiving distance increases, the influence of some bulk sources of certain sizes on the longitudinal amplitude will become smaller and smaller. Therefore, in the far field, the source can be approximately viewed as a point source. The acoustic pulse’s widening is due to high frequency diffraction.
3.3.3. The influence of the optical penetration depth on the ultrasonic waveform Under the same irradiating laser energy, the influence of the optical penetration depth into material on the ultrasonic waveform at the epicenter has been calculated, shown in Fig. 9, and the optical penetration depth is 20, 40 and 80 mm, respectively. The precursor amplitude increases obviously with the increase of optical penetration depth. As previously described, the precursor is excited by the vertical thermal stress during the laser heating and subsequent surface expansion, and the magnitude is strongly affected by the thickness (hence mass) of the expanding surface layer. When the optical penetration depth increases, the thickness and mass of the heated zone will enlarge, making the precursor amplitude greater. It can also be seen from Fig. 9 that the spike width increases with the increase of optical penetration depth. Under the same irradiating laser energy, the influence of the optical penetration depth into material on the SAWs has been calculated, shown in Fig. 10. There is no difference obviously in the features of waveform, except the Rayleigh wave amplitude. It is because the SAWs can only propagate at surface with very small depth, so the axial depth of bulk source affects the waveforms very little directly. 4. Conclusion
3.3.2. Laser-generated surface acoustic wave The waveforms of the surface normal displacements at different source-receiver distances are shown in Fig. 8, in which the optical penetration depth is 20 mm. There are three main features in the surface acoustic waves (SAWs). One is the surface skimming longitudinal wave denoted by sP, which is an outward displacing unipolar wave. The other two are the surface shear wave front denoted by sS and the main initially negative-going dipolar Rayleigh wave denoted by R. The sP and sS waves mark the intersection with the surface of that longitudinal wave and shear wave fronts, respectively, which originate from the
A numerical modeling for thermoelastic generation of ultrasound excited by a pulsed laser in non-metallic material is presented by employing the finite element method, taking into account not only thermal diffusion and the finite spatial and temporal shape of the laser pulse, but also optical penetration and the temperature dependence of material properties. The temperature distributions and temperature gradient fields in non-metallic material for different time steps are calculated. This temperature gradient distribution is equivalent to a bulk force source. The numerical results
ARTICLE IN PRESS J. Wang et al. / Optics & Laser Technology 39 (2007) 806–813
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sS
0.10
0.06
sP R Displacement (nm)
Displacement (nm)
0.5mm
0.00 -0.05 -0.10 -0.15
sP
R 2.0mm
0.02 0.00 -0.02 -0.04 -0.06 -0.08
-0.20
-0.10 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (µs) (
(a)
0.0
0.5
1.0
1.5
0.02 sP
0.02
sS
2.5
R
R
sP
0.01
4.0mm
2.0
Time (µs)
(b)
0.04
sS 8.0mm
0.00
0.00
Displacement (nm)
Displacement (nm)
sS
0.04
0.05
-0.02 -0.04 -0.06
-0.01 -0.02 -0.03 -0.04 -0.05
-0.08 0.0
0.5
1.0
(c)
1.5 Time (µs)
2.0
2.5
3.0
1.0
1.5
2.0
(d)
2.5 Time (µs)
3.0
3.5
4.0
Fig. 8. Surface vertical displacements at different source–receiver distance: (a) 0.5; (b) 2; (c) 4; (d) 8 mm.
0.06
0.16 20µm 40µm 80µm
Displacement (nm)
0.12 0.10
20µm 40µm
0.04 Displacement (nm)
0.14
L
0.08 0.06
S
0.04 0.02 0.00 -0.02
80µm
0.02 0.00 -0.02 -0.04 -0.06
-0.04 -0.08
-0.06 -0.08
-0.10 0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (µs) Fig. 9. Comparison of normal waveform at the epicenter in the neutral glass plate, the penetration depths are 20, 40 and 80 mm, respectively.
0.0
0.2
0.4
0.6 Time (µs)
0.8
1.0
Fig. 10. Comparison of surface vertical displacements, the optical penetration depth is 20 and 80 mm, respectively.
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indicate that the heat penetration into the non-metallic material is caused by both optical absorption and thermal diffusion, and the optical penetration is responsible mainly for the heat penetration into non-metal material. The laser-generated ultrasound waveforms, ultrasonic waveform at the epicenter and surface acoustic waveform, are obtained. The numerical results indicate that ultrasonic waveforms are related to the optical and thermal properties of the non-metallic material. A significant feature of the longitudinal wave is that the precursor is a dipolar pulse, and the precursor amplitude is strongly dependent on the optical penetration depth into non-metallic material. There is much information of the non-metallic material in ultrasonic waveforms, so the feature of the ultrasonic waveform indicates the potential interest of a contactless laser-based technique for the non-destructive determination and the nondestructive evaluation of non-metallic materials. Acknowledgements This work is supported by National Natural Science Foundation of China under Grant No. 60208004 and Natural Science Foundation of Jiangsu University of No. 05JDG007, and partly supported by the Teaching and Research Award Program for Outstanding Young Professor in Higher Education Institute. MOE, P. R. C. References [1] Scruby CB, Dewhurst RJ, Hutchins DA, Palmer SB. Quantitative studies of thermally generated elastic waves in laser-irradiated metals. J Appl Phys 1980;51:6210–6. [2] Rose LRF. Point-source representation for laser-generated ultrasound. J Acoust Soc Am 1984;75(3):723–32.
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[3] Dewhurt RJ, Edwards C, Mckie ADW, Palmer SB. Estimation of the thickness of thin metal sheet using laser generated ultrasound. Appl Phys Lett 1987;51(4):1066–8. [4] Bresse LF, Hutchins DA. Transient generation by a wide thermoelastic source at a solid surface. J Phys D 1993;26:329–48. [5] McDonald FA. On the precusor in laser-generated ultrasound waveforms in metals. Appl Phys Lett 1990;56(3):230–2. [6] Sanderson T, Ume C, Jarzynski J. Longitudinal wave generation in laser ultrasonics. Ultrasonic 1998;35:553–61. [7] Telschow KL, Conant RJ. Optical and thermal parameter effects on laser-generated ultrasound. J Acoust Soc Am 1990;88(3). [8] Tayer GS, Hutchins DA, Edwards C, Palmer SB. TEA-CO2 laser generation of ultrasound in non-metals. Ultrsonics 1990;28:343. [9] Bai YH, Pei LW, Fang QP, et al. Experimental studies of directivity patterns of laser generated ultrasound in neutral glasses. Ultrsonics 1995;33(6):429–36. [10] Edwards C, Stratoudaki T, Palmer SB. A new thermoelastic source model for non-metals. In: Thompson DO, Chimenti DE, editors. Review of progress quantitative nondestructive evaluation, Vol. 22. New York: Plenum Press; 2003. p. 326–32. [11] Dubois M, Enguehard F, Bertrand L. Modeling of laser thermoelastic generation of ultrasound in an orthotropic medium. Appl Phys Lett 1994;64(5):554–6. [12] Dubois M, Lorraine PW, Venchiarutti B, et al. Optimization of thermal profile and optical penetration depth for laser-generation of ultrasound in polymer-matrix. In: Thompson DO, Chimenti DE, editors. Review of progress quantitative nondestructive Evaluation. New York: Plenum Press; 2000. p. 287–94. [13] Xu BQ, Shen ZH, Ni XW, Lu J. Numerical simulation of lasergenerated ultrasound by the finite element method. J Appl Phys 2004;95(4):2116–22. [14] Shen ZH, Zhang SY, Cheng JC. Theoretical study on SAW generated by laser pulse in solids. Anal Sci 2001;17:204–7. [15] Reddy JN. An introduction to the finite element method. 2nd ed. New York: McGraw-Hill; 1993. [16] Xu BQ, Shen ZH, Ni XW, Lu J. Thermal and mechanical finite element modeling of laser-generated ultrasound in coating-substrate system. Optics Laser Technol 2006;38(3):138–45. [17] Rose JL. Ultrasonic waves in solid media. Cambridge, UK: Cambridge University Press; 1999.
Optics & Laser Technology 39 (2007) 806–813 www.elsevier.com/locate/optlastec
Numerical simulation of laser-generated ultrasound in non-metallic material by the finite element method Jijun Wanga,b, Zhonghua Shena,, Baiqiang Xub, Xiaowu Nia, Jianfei Guana, Jian Lua a
Deptartment of Appl. Physics, Nanjing University of Science and Technology Nanjing, 210094, People’s Republic of China b Faculty of Science, Jiangsu University, Zhenjiang, 212013, People’s Republic of China Received 19 August 2005; received in revised form 27 January 2006; accepted 27 January 2006 Available online 29 March 2006
Abstract The use of a pulsed laser for the generation of the elastic waves in non-metallic materials in the thermoelastic regime is investigated by using finite element method (FEM), taking into account not only thermal diffusion and the finite spatial and temporal shape of the laser pulse, but also optical penetration and the temperature dependence of material properties. The optimum finite element model is established based on analysis of two important parameters, meshing size and time step, and the stability of solution. Temperature distributions and temperature gradient fields in non-metallic material for different time steps are obtained, this temperature field is equivalent to a bulk force source to generate ultrasonic wave. The laser-generated ultrasound waveforms at the epicenter and surface acoustic waveforms (SAWs) are obtained and the influence of optical penetration into the material on the temperature field and the ultrasound waveforms are analyzed. The numerical results indicate that the heat penetration into non-metallic material is caused mainly by the optical penetration, and the ultrasound waveforms, especially the shape of the precursor, are strongly dependent on the optical penetration depth into non-metallic material. r 2006 Elsevier Ltd. All rights reserved. Keywords: Laser-generated ultrasound; Finite element method; Non-metallic material
1. Introduction Due to its non-contact feature and the ability of broadband signal generation, laser ultrasound has demonstrated its great potential for non-destructive evaluation. When a solid is illuminated with a laser pulse, absorption of the laser energy results in an increased localized temperature, which in turn causes thermal expansion and generates ultrasonic waves in the solid. Much work has been reported for metallic specimen [1–6]. Scrubby et al. [1] studied early the problem of laser-generated ultrasound using a point source approximation. Rose [2] treated the point source as a surface center of expansion (SCOE) and obtained a formal solution, neglecting the thermal diffusion. McDonald [5] and Sanderson [6] improved this model, taking into account both thermal diffusion and the finite spatial and temporal shape of the laser pulse, and Corresponding author.
E-mail address: [email protected] (Z. Shen). 0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.01.009
obtained good results agreement with experiment for metals. Recently, Laser-generated ultrasound in non-metal material, such as ceramics, neutral glass, polymer and composite materials, arise much interest [7–12]. In the case of a metallic specimen, the optical absorption occurs at the surface layer, resulting in a surface-exciting source, the heat penetration is caused mainly by heat diffusion. For the non-metallic materials, the laser beam can penetrate the specimen to some finite depth and induces a bulk-thermal source. Due to the optical penetration, the characteristics of the laser-generated ultrasound will be significantly different from that in metals. Among the theoretical studies, most search for analytical solutions. Based on Laplace–Hankel transform, neglecting the thermal diffusion and taking into account optical penetration into the non-metallic material, Telschow et al. [7] presented a twodimensional calculation to analysis the generation of ultrasonic waves in non-metallic material, and the results illustrated that the shape of the precursor is strongly
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depend on the optical penetration and the thermal properties of the material; Bai et al. [9] solved the thermoelastic wave equation, obtained analytic expressions of the wave displacement components in far-field, and presented the directivity patterns of both longitudinal and shear waves. Dubois et al. [11] calculated the surface displacements by the absorption of a laser pulse by an orthotropic medium to optimize the laser generation of ultrasound in various materials. The inverse transformation is usually obtained by a numerical inverse method, so it is quite difficult to get the analytical inverse transformation solutions. To get analytical solution, simplifying of condition was taken into account. It is also difficult to deal with the complicated geometry specimen, especially the cracks in specimen. Moreover, during the laser generation of the ultrasonic waves, the thermophysical properties of the irradiated materials vary with the rising temperature, the analytical methods neglected the temperature dependence of the parameters of materials. Due to the complexity of the generation of the thermoelastic waves excited by bulk-thermal source in nonmetallic materials, numerical methods will be much more suitable in dealing with such complicated processes, especially the process where the material parameters are temperature dependent. The finite element method (FEM) is versatile due to its flexibility in modeling complicated geometry and its capability in obtaining full field numerical solutions and the evolution of temperature, thermal stress and displacement fields in non-metallic material in detail. In our previous paper, the transient temperature field and laser-generated ultrasound induced by the pulse laser in single metal plate have been investigated by using finite element method [13], demonstrating the feasibility of modeling laser ultrasound using the thermoelastic equations. In this article, the models of the generation of thermoelastic ultrasonic wave in non-metallic material are established by using FEM, taking into account not only thermal diffusion and the finite spatial and temporal shape of the laser pulse, but also optical penetration and the temperature dependence of material properties. The temperature distributions and temperature gradient fields in non-metallic material for different time steps are calculated accurately by using FEM, then, the ultrasonic waveforms at the epicenter and surface acoustic waves (SAWs) are obtained by coupling the temperature field and the stress field. The influence of the optical penetration depth on the temperature field and the ultrasonic waveforms is analyzed.
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by the target material, and the other is reflected, then a non-uniform temperature field will be generated near the surface, followed by the local thermal expansion, this thermal expansion induces ultrasonic waves. The geometry of laser irradiation on a non-metal specimen is schematically shown in Fig. 1. Due to the thermoelastic mechanism, the thermal conduction and thermal-elastic equation can be descried as [14] Kr2 T rC
@T ¼ q, @t
(1)
ðl þ 2mÞrðr UÞ mr r U r
@2 U ¼ að3l þ 2mÞrT, @t2 (2)
where T is the temperature distribution, q is heat source, K, r and C are the thermal conductive coefficient, density and thermal capacity, respectively; U is the time-dependent displacement, a is the thermoelastic expansion coefficient, l and m are the Lame´ constants, respectively. The spatial mode of the laser beam is assumed to be a Gauss distribution, so a cylindrical coordinate system is adopted. The heat source q can be expressed as q ¼ bð1 RÞI 0 ebz f ðrÞgðtÞ,
(3)
where b is the optical absorption coefficient (1/b is the optical penetration depth), I0 is the incident laser light intensity, R is the optical reflectivity of the specimen, f(r) and g(t) are the spatial and temporal distribution of the laser pulse, respectively. These two functions can be written as 2 r f ðrÞ ¼ exp 2 , (4) a0 gðtÞ ¼
t t exp , t0 t0
(5)
where a0 is the radius of the laser spot, t0 is the rise time of the laser pulse. It is assumed that the specimen surfaces are thermally insulated and mechanically unconstrained, and the temperature elevation and the mechanical displacement as well as their derivatives with respect to time are zero at t ¼ 0. Z
Laser beam
R
2. Theoretical model and numerical method 2.1. Theory of transient temperature field and laser ultrasound by the thermoelastic mechanism When the non-metal specimen is illuminated by a laser pulse with the energy less than the melting threshold of the specimen, a certain amount of the light energy is absorbed
Non-metal material
Fig. 1. Schematic diagram for laser irradiating specimen plate.
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2.2. Numerical method
expressed as [16]
The classical thermal conduction equation for finite elements with the heat capacity matrix [C], conductivity matrix [K], heat flux vector {p1}, and heat source vector {p2} can be expressed as
Le ¼
_ ¼ fp1 g þ fp2 g, ½KfTg þ ½CfTg
lmin , (9) 20 where Le is the element length and lmin is the shortest wavelength of interest.
(6)
3. Numerical results and discussions
_ where {T} is the temperature vector, and fTgis the temperature rise rate vector. For wave propagation, ignoring damping, the governing equation is
3.1. Laser and materials parameters
€ þ ½KfUg ¼ fFext g, ½MfUg
(7)
where [M] is mass matrix, [K] is the stiffness matrix, {U} is € is the acceleration vector and the displacement vector, fUg {Fext} is the action of the laser pulse. For thermoelasticity, R the external force vector for an element is e ½BT ½Dfe0 g dV , where {e0}is the thermal strain vector, [B]T is the transpose of the derivative of the shape functions, and [D] is the material matrix. Eq. (7) is solved using the Newmark time integration method [15]. The process to solve the finite element equation is discretizing the plane with axis symmetry into non-verlapping finite elements, expressing the nodal displacement in terms of the shape functions. Because the mass matrix and the stiffness matrix are sparse, symmetric and positive, it will take much time and large computing capability to solve Eq. (7). Temporal and spatial resolution of the finite element model is critical for the convergence of these numerical results. Choosing an adequate integration time step is very important for the accuracy of the solution. In general, the accuracy of the model can be increased with increasingly smaller integration time steps. With too long time steps, the high frequency components are not resolved accurately enough. On the other hand, too small time steps are a waste of computing time, therefore, a compromise must be found. Dt ¼
1 , 20f max
(8)
where fmax is the highest frequency of interest. But when the rise time of the laser pulse is in the order of ns, a time step in Eq. (8) might not provide sufficient temporal resolution. The time step has to decrease to Dt ¼ 1=ð180f max Þ [16]. Thus, The time step can be determined by estimating the highest frequency of the laser-generated ultrasound waves. Also, the needed time step can alternatively be related to the time the fastest possible wave needs to propagate between successive nodes in the mesh. The element is defined by four nodes with two degrees of freedom at each node in cylindrical coordinate system. In general, the rule of element size is that there are more than 10 nodes in a wavelength. To ensure the propagation of energy between two successive nodes in the mesh, the element length should be fine enough, so the propagating waves are spatially resolved. The recommendation can be
On the basis of the theories described above, the thermoelastically generated waves in non-metal material are calculated. The laser energy is 0.35 mJ. The pulse rise time t0 and the radius of the laser pulse spot on the system are taken to be 10 ns and 50 mm, respectively. The radius and the thickness of the specimen are 2 cm and 5 mm, respectively. The finite element model is constructed, and the element length is 20 mm. The property of semitransparent neutral glass used in the calculation is listed in Table 1. 3.2. Transient temperature and temperature gradient fields A thermoelastic volume source inducing by a laser pulse is originated from a non-uniform temperature field, so an accurate determination of the temperature field is vital to accurate prediction of laser-generated ultrasonic waves. In this research the temperature distributions in nonmetal material were computed for each time step. Fig. 2 shows the temperature evolution at the center of laser irradiation, in which the optical penetration depth is 20 mm. The results show that the temperature rises rapidly during heating by laser, while the cooling process is very slowly, compared with the process of an alumina plate in the similar specimen geometry size and laser parameters [13]. It is because the thermal conductive coefficient of nonmetallic material is smaller than that of metal in one or two order. The temperature distributions with various optical penetration depths at 2 ms are shown in Fig. 3(a)–(c). These figures show the influence of the optical penetration
Table 1 Properties of neutral glass used in the calculation Physical properties
Neutral glass
The optical reflectivity R Thermal conductive coefficient K(W m1 K1)
0.79 1.83 105T+0.012, 300oTo800
Density r(kg m3) Thermal capacity C(J kg1K1) Poission’s ratio Young’s modulus (Pa) Thermal expansion coefficient
1.2 105T+0.021, 800oTo Tm 2400 840 0.23 6.0457 1010 0.63 106
Tm is the melting point of the material.
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600
809
R (µm)
Neutral Glass Al
0
20
40
60
80
500
Z (µm)
Temperature (K)
550 -20
450
(a)
-40 R (µm)
400 0
20
0
40
40
60
80
100
350 300 0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
-20
2.0
Time (µs) Fig. 2. Temperature vs time at the center of laser irradiation in the neutral glass plate and the aluminum plate, respectively.
depth on the temperature field. It shows clearly that there is more volume to be heated by the absorbed laser energy in the non-metallic material than that in the metallic material. Non-uniform temperature distribution is observed as a result of the Gaussian laser power distribution and optical penetration. Fig. 3(a) is the transient temperature distribution, neglecting the optical penetration and taking only the thermal diffusion effects into consideration. It can be seen that the thermal diffusion is only about 15 mm in the axial direction and less than 100 mm in the radial direction. The optical penetration depths in Fig. 3(b)–(c) are 20 and 80 mm, respectively, and the depths of the heated zone in the axial direction are about 60 and 240 mm, respectively, whereas the dimensions of the heated zone in the radial direction are almost same in these three cases. One can see that the heat penetration induced by a pulse laser into the non-metal material is much larger than that neglecting the optical penetration and the volume of the heated zone is much larger too, so the volume of ultrasonic source increases with the increase of optical penetration depth. Fig. 3 indicates that the heat penetration into the specimen is caused by both optical absorption and thermal diffusion, and the subsurface thermal source arises mainly from thermal diffusion in metals, but the bulk-thermal source results mainly from the optical penetration in non-metal material. Non-uniform temperature field causes the temperature gradient field. The temperature gradient fields have also been calculated. Fig. 4(a) and (b) show the horizontal temperature gradient (TGX) and the vertical temperature gradient (TGY), respectively, in which the optical penetration depth is 20 mm at 2 ms. It can be seen that both the horizontal temperature gradient and vertical temperature gradient achieves the maximum value at the specimen surface. The maximum horizontal temperature gradient is at about 35 mm from the center of laser irradiation, where the spatial distribution of laser intensity varies sharpest,
Z (µm)
0.2
-40
-60 (b) R (µm) 80
-20 -40 -60 -80 -100 Z (µm)
0.0
-120 -140 -160 -180 -200 -220 -240
(c) Fig. 3. Contour plots of temperature at 2 ms near the heated affected zone in the neutral glass plate: (a) The penetration depth is 0 mm, Tð0; 0Þ ¼ 611 K, Tð20; 0Þ ¼ 559 K, Tð40; 0Þ ¼ 455 K, Tð60; 0Þ ¼ 369 K, Tð80; 0Þ ¼ 323 K, Tð100; 0Þ ¼ 300 K; (b) The penetration depth is 20 mm, Tð0; 0Þ ¼ 387 K, Tð20; 0Þ ¼ 372 K, Tð40; 0Þ ¼ 351 K, Tð60; 0Þ ¼ 317 K, Tð80; 0Þ ¼ 311 K, Tð100; 0Þ ¼ 300 K; (c) The penetration depth is 80 mm, Tð0; 0Þ ¼ 322 K, Tð20; 0Þ ¼ 318 K, Tð40; 0Þ ¼ 311 K, Tð60; 0Þ ¼ 307 K, Tð80; 0Þ ¼ 302 K, Tð100; 0Þ ¼ 300 K.
and the maximum vertical temperature gradient is at the center of the laser irradiation. The temperature gradient field results in thermal expansion in non-metallic material. The horizontal expansion
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R (µm) 20
40
60
0.20 80
100
-20 Z (µm)
S
120 Displacement (nm)
0
-40
0.15 L
0.10 0.05 0.00
(a)
-60 -0.05 R (µm) 0
20
40
60
80
100
0.0
0.5
1.0
1.5 Time (µs)
2.0
2.5
Fig. 5. Normal waveform at the epicenter in the neutral glass plate, the penetration depth is 20 mm.
Z (µm)
-20
-40
-60 (b) Fig. 4. Contour plots of temperature gradient at time ¼ 2ms near the heated affected zone in the neutral glass plate, the penetration depth is 20 mm: (a) horizontal temperature gradient (TGX). TGXð0; 0Þ ¼ 1:9 105 , TGXð20; 0Þ ¼ 1:1 106 , TGXð40; 0Þ ¼ 1:7 106 , TGXð60; 0Þ ¼ 7:4 105 , TGXð0; 0Þ ¼ 4:4 105 , TGXð100; 0Þ ¼ 1:7 105 ; (b) vertical temperature gradient (TGY), TGYð0:0Þ ¼ 3:4 106 , TGYð20:0Þ ¼ 2:9 106 , TGYð40:0Þ ¼ 1:9 106 , TGYð60:0Þ ¼ 8:9 105 , TGYð80:0Þ ¼ 3:5 105 , TGYð100:0Þ ¼ 1:2 105 .
induced by the horizontal temperature gradient is constrained by the surrounding unheated material, exciting the horizontal thermal stress. The vertical thermal stress results from the vertical temperature gradient. As would be expected, the longitudinal waves are caused by the vertical thermal stress; shear and surface acoustic waves are generated by the horizontal stress [13]. 3.3. Laser-generated ultrasound 3.3.1. Normal waveform at the epicenter and near-epicenter normal waveform The ultrasonic waveform predicted by the FEM at the epicenter of a 5 mm thickness neutral glass plate are shown in Fig. 5, and the features of the ultrasonic waveform are good agreement with the results reported by Edwards [10]. Well-defined features that occur at the longitudinal and shear arrival times are exhibited, and a significant feature of the laser-generated elastic waveform is a precursor (sharp spike), signaling the arrival of the longitudinal wave. The first arrival time at 0.89 ms corresponds to the direct
longitudinal wave (L). The second arrival time at 1.575 ms corresponds to the direct shear wave (S). The amplitude of the shear wave is considerably larger, compared to the amplitude of the longitudinal wave. This is due to the bulk force source created by the thermo-elastic generation mechanism contains a large shear component. There is a distinguishing feature in the longitudinal component of the ultrasonic waveform at the epicenter. The precursor (sharp spike) is a dipolar pulse. This is caused by the vertical thermal stress during the laser heating and subsequent surface expansion. As the laser energy is deposited, the temperature of the heated zone rises, and the surface irradiated by laser starts expand. The associated thermal stress induced by the vertical temperature gradient causes a negative-going spike (in the coordinate system shown in Fig. 1), which is relieved by the surface expansion. The magnitude of this spike is strongly affected by the thickness (hence mass) of the expanding surface layer [5]. The vertical stress caused by the vertical temperature gradient acts on bottom of the heated zone, generating the primary longitudinal pulse. This pulse radiates down and up simultaneously in the specimen. Reflection from the surface leads to the positivegoing spike, shown in Fig. 6. The time delayed of the positive-going spike is related to the optical penetration depth. The near-epicenter displacements are shown in Fig. 7. It can be seen that the arrival time of shear wave maximum increases with the increase of the distance from the epicenter. The early arrival time of the shear displacement, corresponding to the fringe of the bulk source, does not reflect the finite extent of the source. It takes more time for the signal generated by the center of heated region to transmit to the receiver location with the increase of the distance from the epicenter, so the arrival time of shear wave maximum is certainly delayed in time. It can also be seen that the longitudinal amplitude decreases with
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Fig. 6. Schematic diagram of reflection from the surface of the longitudinal wave.
0.14
0.5mm 1.0mm 1.5mm
0.12
Displacement (nm)
0.10 0.08
811
bulk source. The three kinds of waves overlap in the near field [Fig. 8(a)]. The sP wave is separated from the Rayleigh wave gradually with the increased source-receiver distances while sS wave still couple with the Rayleigh wave in the source–receiver distances of interest [Fig. 8(b)]. The three kinds of waves are separated in the far field completely [Fig. 8 (c) and (d)]. Such a phenomenon results from the fact that the propagation velocity of the Rayleigh wave is much smaller than that of the sP and near equals that of the sS. Due to the small propagation distance and finite pulse width of the three waveforms, such a velocity difference cannot split the three waveforms in the near field. The velocity of the sP wave, sS wave and the Rayleigh wave are 5390m/s, 3190m/s, and 2940m/s, respectively, measured from Fig. 8. It is in good agreement with the theoretical and experiment results reported [9,17].
0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 0.6
0.8
1.2 1.0 Time (µs)
1.4
1.6
1.8
Fig. 7. Near-epicenter normal waveform, the epicenter-receiver distance is 0.5, 1.0 and 1.5 mm, respectively.
increasing distances from the epicenter and becomes wider. The amplitude of the thermoelastic wave depends on r1 in the case of spherical wave propagation, where r is the distance from the source to the receiver location. A bulk source can be regarded as a collection of point sources. As a result of many point sources working concurrently, the longitudinal amplitude in the near field is greater than that in the far field. As the receiving distance increases, the influence of some bulk sources of certain sizes on the longitudinal amplitude will become smaller and smaller. Therefore, in the far field, the source can be approximately viewed as a point source. The acoustic pulse’s widening is due to high frequency diffraction.
3.3.3. The influence of the optical penetration depth on the ultrasonic waveform Under the same irradiating laser energy, the influence of the optical penetration depth into material on the ultrasonic waveform at the epicenter has been calculated, shown in Fig. 9, and the optical penetration depth is 20, 40 and 80 mm, respectively. The precursor amplitude increases obviously with the increase of optical penetration depth. As previously described, the precursor is excited by the vertical thermal stress during the laser heating and subsequent surface expansion, and the magnitude is strongly affected by the thickness (hence mass) of the expanding surface layer. When the optical penetration depth increases, the thickness and mass of the heated zone will enlarge, making the precursor amplitude greater. It can also be seen from Fig. 9 that the spike width increases with the increase of optical penetration depth. Under the same irradiating laser energy, the influence of the optical penetration depth into material on the SAWs has been calculated, shown in Fig. 10. There is no difference obviously in the features of waveform, except the Rayleigh wave amplitude. It is because the SAWs can only propagate at surface with very small depth, so the axial depth of bulk source affects the waveforms very little directly. 4. Conclusion
3.3.2. Laser-generated surface acoustic wave The waveforms of the surface normal displacements at different source-receiver distances are shown in Fig. 8, in which the optical penetration depth is 20 mm. There are three main features in the surface acoustic waves (SAWs). One is the surface skimming longitudinal wave denoted by sP, which is an outward displacing unipolar wave. The other two are the surface shear wave front denoted by sS and the main initially negative-going dipolar Rayleigh wave denoted by R. The sP and sS waves mark the intersection with the surface of that longitudinal wave and shear wave fronts, respectively, which originate from the
A numerical modeling for thermoelastic generation of ultrasound excited by a pulsed laser in non-metallic material is presented by employing the finite element method, taking into account not only thermal diffusion and the finite spatial and temporal shape of the laser pulse, but also optical penetration and the temperature dependence of material properties. The temperature distributions and temperature gradient fields in non-metallic material for different time steps are calculated. This temperature gradient distribution is equivalent to a bulk force source. The numerical results
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sS
0.10
0.06
sP R Displacement (nm)
Displacement (nm)
0.5mm
0.00 -0.05 -0.10 -0.15
sP
R 2.0mm
0.02 0.00 -0.02 -0.04 -0.06 -0.08
-0.20
-0.10 0.0
0.2
0.4
0.6
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1.4
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Time (µs) (
(a)
0.0
0.5
1.0
1.5
0.02 sP
0.02
sS
2.5
R
R
sP
0.01
4.0mm
2.0
Time (µs)
(b)
0.04
sS 8.0mm
0.00
0.00
Displacement (nm)
Displacement (nm)
sS
0.04
0.05
-0.02 -0.04 -0.06
-0.01 -0.02 -0.03 -0.04 -0.05
-0.08 0.0
0.5
1.0
(c)
1.5 Time (µs)
2.0
2.5
3.0
1.0
1.5
2.0
(d)
2.5 Time (µs)
3.0
3.5
4.0
Fig. 8. Surface vertical displacements at different source–receiver distance: (a) 0.5; (b) 2; (c) 4; (d) 8 mm.
0.06
0.16 20µm 40µm 80µm
Displacement (nm)
0.12 0.10
20µm 40µm
0.04 Displacement (nm)
0.14
L
0.08 0.06
S
0.04 0.02 0.00 -0.02
80µm
0.02 0.00 -0.02 -0.04 -0.06
-0.04 -0.08
-0.06 -0.08
-0.10 0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (µs) Fig. 9. Comparison of normal waveform at the epicenter in the neutral glass plate, the penetration depths are 20, 40 and 80 mm, respectively.
0.0
0.2
0.4
0.6 Time (µs)
0.8
1.0
Fig. 10. Comparison of surface vertical displacements, the optical penetration depth is 20 and 80 mm, respectively.
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indicate that the heat penetration into the non-metallic material is caused by both optical absorption and thermal diffusion, and the optical penetration is responsible mainly for the heat penetration into non-metal material. The laser-generated ultrasound waveforms, ultrasonic waveform at the epicenter and surface acoustic waveform, are obtained. The numerical results indicate that ultrasonic waveforms are related to the optical and thermal properties of the non-metallic material. A significant feature of the longitudinal wave is that the precursor is a dipolar pulse, and the precursor amplitude is strongly dependent on the optical penetration depth into non-metallic material. There is much information of the non-metallic material in ultrasonic waveforms, so the feature of the ultrasonic waveform indicates the potential interest of a contactless laser-based technique for the non-destructive determination and the nondestructive evaluation of non-metallic materials. Acknowledgements This work is supported by National Natural Science Foundation of China under Grant No. 60208004 and Natural Science Foundation of Jiangsu University of No. 05JDG007, and partly supported by the Teaching and Research Award Program for Outstanding Young Professor in Higher Education Institute. MOE, P. R. C. References [1] Scruby CB, Dewhurst RJ, Hutchins DA, Palmer SB. Quantitative studies of thermally generated elastic waves in laser-irradiated metals. J Appl Phys 1980;51:6210–6. [2] Rose LRF. Point-source representation for laser-generated ultrasound. J Acoust Soc Am 1984;75(3):723–32.
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[3] Dewhurt RJ, Edwards C, Mckie ADW, Palmer SB. Estimation of the thickness of thin metal sheet using laser generated ultrasound. Appl Phys Lett 1987;51(4):1066–8. [4] Bresse LF, Hutchins DA. Transient generation by a wide thermoelastic source at a solid surface. J Phys D 1993;26:329–48. [5] McDonald FA. On the precusor in laser-generated ultrasound waveforms in metals. Appl Phys Lett 1990;56(3):230–2. [6] Sanderson T, Ume C, Jarzynski J. Longitudinal wave generation in laser ultrasonics. Ultrasonic 1998;35:553–61. [7] Telschow KL, Conant RJ. Optical and thermal parameter effects on laser-generated ultrasound. J Acoust Soc Am 1990;88(3). [8] Tayer GS, Hutchins DA, Edwards C, Palmer SB. TEA-CO2 laser generation of ultrasound in non-metals. Ultrsonics 1990;28:343. [9] Bai YH, Pei LW, Fang QP, et al. Experimental studies of directivity patterns of laser generated ultrasound in neutral glasses. Ultrsonics 1995;33(6):429–36. [10] Edwards C, Stratoudaki T, Palmer SB. A new thermoelastic source model for non-metals. In: Thompson DO, Chimenti DE, editors. Review of progress quantitative nondestructive evaluation, Vol. 22. New York: Plenum Press; 2003. p. 326–32. [11] Dubois M, Enguehard F, Bertrand L. Modeling of laser thermoelastic generation of ultrasound in an orthotropic medium. Appl Phys Lett 1994;64(5):554–6. [12] Dubois M, Lorraine PW, Venchiarutti B, et al. Optimization of thermal profile and optical penetration depth for laser-generation of ultrasound in polymer-matrix. In: Thompson DO, Chimenti DE, editors. Review of progress quantitative nondestructive Evaluation. New York: Plenum Press; 2000. p. 287–94. [13] Xu BQ, Shen ZH, Ni XW, Lu J. Numerical simulation of lasergenerated ultrasound by the finite element method. J Appl Phys 2004;95(4):2116–22. [14] Shen ZH, Zhang SY, Cheng JC. Theoretical study on SAW generated by laser pulse in solids. Anal Sci 2001;17:204–7. [15] Reddy JN. An introduction to the finite element method. 2nd ed. New York: McGraw-Hill; 1993. [16] Xu BQ, Shen ZH, Ni XW, Lu J. Thermal and mechanical finite element modeling of laser-generated ultrasound in coating-substrate system. Optics Laser Technol 2006;38(3):138–45. [17] Rose JL. Ultrasonic waves in solid media. Cambridge, UK: Cambridge University Press; 1999.