JOURNAL OF APPLIED PHYSICS 108, 123101 共2010兲
Thermoviscoelastic finite element modeling of laser-generated ultrasound in viscoelastic plates Hong-xiang Sun1,2 and Shu-yi Zhang1,a兲 1
Lab of Modern Acoustics, Institute of Acoustics, Nanjing University, Nanjing 210093, China Faculty of Science, Jiangsu University, Zhenjiang 212013, China
2
共Received 27 August 2010; accepted 27 October 2010; published online 16 December 2010兲 Laser-generated ultrasound in a thin composite plate with thermoviscoelastic property has been studied quantitatively. According to thermoviscoelastic theory, considering the viscoelastic and thermophysical properties of materials, a numerical model for the laser-generated Lamb waves is established in the frequency domain by using a finite element method. It is confirmed that the temperature and displacement fields calculated in the frequency domain coincide well with those obtained in the time domain. In the numerical simulations of thermoviscoelastically generated Lamb waves, the effects of viscoelastic and elastic stiffness moduli, and the thickness of the materials have been taken into account in details. The characteristics of the Lamb waves in the numerical results agree well with the features of the disperse curves. The results show that the finite element method in this paper provides a useful technique to characterize mechanical properties of composite materials. © 2010 American Institute of Physics. 关doi:10.1063/1.3520675兴 I. INTRODUCTION
Due to the noncontact feature and ability of broadband signal generation, laser ultrasound technique 共LUT兲 has demonstrated its great potential for nondestructive evaluation and material characterization. When a solid is illuminated by a laser pulse, absorption of the laser pulse results in a localized increased temperature, which in turn causes thermal expansion and generates ultrasonic waves in the solid. Various ultrasonic waves can be excited, for example, longitudinal and transverse waves,1,2 surface acoustic waves,3–7 and even Lamb waves in thin plates8–12 in the regime of thermoelastic excitation. Most researches are reported for the wave propagation in purely elastic materials, ignoring viscoelasticity. Nevertheless, the materials with polymers or polymer-based matrix composites, which have been widely used in aircrafts, spacecraft, and other industries, possess isotropic or anisotropic viscoelastic properties that can strongly affect the propagation of ultrasonic waves. Therefore, the LUT in the application of nondestructive evaluation of composite materials has attracted more and more attention. In order to determine the properties of a composite material by LUT, the ultrasonic signals generated by the laser source in such a material must be well understood. However, the characteristics of the laser-generated ultrasound depend strongly not only on the optical penetration, thermal diffusion, and elastic and geometrical features of the material, but also on the property of the material viscoelasticity, so that the ultrasonic signals in the composite material are very difficult to be interpreted. On the theoretical study, several methods have been proposed for investigating thermoelastically generated ultrasound, essentially using Laplace–Hankel transformations, Green functions, and eigenfunction expansions. Based on temporal Laplace and two-dimensional spatial Fourier transformations, Spicer et al.8 presented a theoretical fora兲
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mulation for laser generation of ultrasonic waves in an isotropic thin plate, and the effectiveness of this formulation was shown with the evaluation of the plate thickness and modulus by direct comparison of the theoretical results with the experimental waveforms. Dubois et al.13 established a numerical model to describe the thermoelastic generation of ultrasonic waves in an orthotropic plate. However, all these theoretical works involved an inverse Laplace integral, which is usually evaluated by the calculus of residues. The evaluation of the inverse Laplace integral in this approach involves the understanding of the intricate behavior of dispersion relations with real as well as complex wave numbers and its physical sense is hard to be understood. Rose1 treated the point source as a surface center of expansion and obtained a formal solution using Green function formalism, without taking the thermal diffusion of the material into account. Due to the difficulty in finding analytical solution of Green’s functions, it is even more difficult to find the solution at the locations far from the surface source. Using the eigenfunction expansion method, Cheng et al.10,14 investigated not only the thermoelastic generation of longitudinal, transverse, and surface acoustic waves in thick samples but also the excitation of the Lamb wave modes along arbitrary directions in thin plates. This approach is suitable for waveform analyses of the excitation of transient Lamb waves in thin plates because only the contributions of several lower modes need to be calculated. However, as the sample thickness increases, not only the higher harmonic components of the lowest modes, but also the higher order modes, have to be taken into consideration. Thus, the computations will be quite complicated. Generally, all of the studies introduced above, the material viscoelasticity was neglected, and no work has been published so far on the investigation for modeling thermoviscoelasticlly laser-generated ultrasound in viscoelastic materials. Constitutive relations developed in the frequency
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domain and complex moduli that represent the viscoelastic material properties can properly describe the dynamic behavior of these media15–17 and several cases of viscoelastic wave propagation in plates made of dissipative materials have been modeled successfully.18–21 Nevertheless, these models are restricted to investigate the viscoelastic waves generated by transducers which cannot generate broadband ultrasonic signals with high spatial resolution. Due to the complexity of the generation of thermoviscoelastic waves, numerical methods will be much more suitable in dealing with the complicated processes, especially in processes involving various thermophysical parameters. The finite element 共FE兲 method, solving the problems in the frequency domain, adopted in this paper, has many advantages. First of all, it is versatile due to the flexibility in modeling complicated geometries and the feasibility of obtaining full field numerical solutions. In addition, the FE-code in this method supplies stationary solutions for a small number of frequency components representing a given temporal excitation. This approach considerably speeds up the computation by avoiding the numerical temporal differentiation and decreasing the variable number used for each calculation. In comparison to the classical FE method22–24 that uses numerical routines to produce time marching solutions and requires hundreds or thousands of iterative calculations, the FE method proposed here is much less time and memory consuming since it consists in solving the thermoviscoelastic coupled equations for a limited number of frequencies that constitute the frequency spectrum of a temporal laser source. Moreover, the complex moduli representing the viscoelastic property introduced in this method can effectively model the thermoviscoelastic wave attenuation in composite materials. The work25 has successfully brought in complex moduli as material viscoelasticity for modeling laser-generated Lamb waves in thin composite plates, ignoring thermophysical properties of the material. However, since it replaced the laser source with an equivalent stress set of stress boundary conditions, temperature evolutions and higher frequencies cannot be obtained. In this paper, based on thermoviscoelastic theory, considering the viscoelastic and thermophysical properties of materials, laser-generated thermoviscoelastic Lamb waves in thin composite plates have been studied quantitatively. II. BASIC THEORY AND NUMERICAL METHOD A. Thermoviscoelastic theory model
We consider an infinite homogeneous isotropic thermally conducting viscoelastic plate in the undisturbed state having a thickness h, see Fig. 1. The coordinates x, y, and z of the model are chosen to be parallel to the principal axes of the sample and y-axis is the optical axis of the incident laser radiation. The pulsed laser line source is assumed to be focused along the z-axis, and this situation can be modeled as a two-dimensional thermoelastic plane strain problem as shown in Fig. 2, where the symmetrical left half is omitted. In our model, we neglect the mechanical heat sources in the heat equation. The classical thermal conduction equation can be expressed as
FIG. 1. Geometry of theoretical model.
k
冉
冊
2T 2T T = Q, 2 + 2 − Cp x y t
共1兲
where k is the thermal conductivity, is the density, Cp is the specific heat, T 共x , y , t兲 represents the temperature distribution at time t, and Q is the heat input. The heat input can be described as Q = Q0 f共t兲g共x兲h共y兲,
共2兲
where Q0 is the incident laser energy and f共t兲 and g共x兲 are the temporal and spatial profiles of the incident laser, respectively, which can be represented as f共t兲 =
g共x兲 =
冉 冊 冉 冊
t t , 2 exp − t0 t0 1
2 exp −
2a0
x2 , a20
共3兲
共4兲
where t0 is the rise time and a0 is the beam radius. As a function of the depth y, the heat deposition due to the laser pulse is assumed to decay exponentially within the plate h共y兲 = ␥ exp共− ␥y兲,
共5兲
where ␥ is the optical absorption coefficient 共1 / ␥ is the optical penetration depth兲. When the plate surface is irradiated by the pulsed laser line source in the thermoelastic regime, a displacement field is generated due to the thermal expansion caused by the deposited heat. The motion is assumed to take place in three dimensions x , y , z and the displacements in the x , y , z directions are u , v , w, respectively. We consider the problem in the xoy plane and the equations of motion can then be expressed in terms of the displacements u and v as C11
2u T 2u 2u 2v −  − 2 = 0, 2 + C66 2 + 共C12 + C66兲 x t x y x y
FIG. 2. Cross section of sample.
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2v T 2v 2v 2u − − + C + 共C + C 兲  = 0, 66 12 66 y t2 y2 x2 x y 共6兲
where Cij, the components of the real stiffness matrix, represent the elastic stiffness moduli,  is the thermoelastic coupling constant and is expressed as  = 共C11 + 2C12兲␣T, where ␣T is the coefficient of linear thermal expansion. However, these formalisms do not take the material viscoelasticity into account. An efficient way to model the material viscoelasticity is to solve thermoelastic coupled Eqs. 共1兲–共6兲 in the frequency domain, by using Fourier transforms of the variables, instead of solving in the time domain. The thermoviscoelastic coupled equations are written as follows:
冉
冊
2Tˆ 2Tˆ ˆ, + − iCpTˆ = Q k x2 y 2
共7兲
ˆ = Q Fˆ共兲g共x兲h共y兲, Q 0
共8兲
Fˆ共兲 =
ⴱ C11
1 , 共1 + it0兲2
sample, Lab is the length of the AR, xab is the starting position of the AR, and A is a coefficient that may be adjusted to minimize the acoustic impedance mismatch between the RP and the AR.
B. FE formulation
For thermoviscoelastic coupled problem solved in the frequency domain, the above partial differential equations, Eqs. 共7兲–共10兲, must be coupled in the following FE-code form:27 ˆ + ␣U ˆ 兲 − aU ˆ = f, ⵜ · 共c ⵜ U
ˆ to be where the symbol ⵜ = 共 / x , / y兲 and the variable U ˆ determined corresponds to the vector 共T , uˆ , vˆ 兲. The coefficients c and ␣ are 3 ⫻ 3 matrixes composed of nine submatrices, that are,
共9兲 c=
2ˆ ˆ 2uˆ 2vˆ ⴱ u ⴱ ⴱ ⴱ T − + 2uˆ + C + 共C + C 兲  66 12 66 x2 y2 x y x
= 0, ˆ u ⴱ v ⴱ v ⴱ ⴱ ⴱ T − + 2vˆ + C + 共C + C 兲  C11 66 12 66 y2 x2 x y y 2ˆ
2ˆ
2ˆ
共10兲
= 0,
where Tˆ共x , y , 兲, Fˆ共兲, uˆ共x , 兲, and vˆ 共y , 兲 are the Fourier transforms of T共x , y , t兲, f共t兲, u共x , t兲, and v共y , t兲, respectively, is the angular frequency, and ⴱ is the thermoviscoelastic ⴱ ⴱ coupling constant and defined as ⴱ = 共C11 + 2C12 兲␣T. The ⴱ moduli Cij in Eq. 共10兲 can be defined as complex quantities if the material is viscoelastic:Cⴱij = C⬘ij + iC⬙ij, in which the real and imaginary parts represent the elastic and viscoelastic stiffness moduli, respectively. This allows viscoelastic an absorbing region 共AR兲 to be added at the edge of the sample, so that the model can represent the behavior of a structure that is infinitely long, while still using a small number of mesh elements, see Fig. 2. As described in previous works with the AR,20,25,26 the AR is rendered more and more viscoelastic as the distance away from the region of propagation 共RP兲 increases, i.e., the region of the generation, propagation, and detection of ultrasonic waves. Specifically, the AR has the same density and elastic properties as that of the RP, but its viscoelasticity, i.e., the imaginary parts of the complex moduli, gradually increases according to the following law:
冋冉
= C⬘ij + i A CⴱAR ij
兩xab − x兩 Lab
冊
3
C⬘ij + C⬙ij
册
x 苸 共L − Lab,L兴, 共11兲
where C⬘ij and C⬙ij represent the material properties in the RP, is the complex moduli in the AR, L is the length of the CⴱAR ij
共12兲
␣=
冋 册 冋 册 冋 册 冋 册冋 册冋 册 冋 册冋 册冋 册 冋册 冋册冋册 冋 册冋册冋册 冋 册冋册冋册
冤 冤
k 0
0 0
0 0
0 k
0 0
0 0
0 0
ⴱ C11
0
0
ⴱ C12
0 0
0
ⴱ C66
ⴱ C66
0
0 0
0
ⴱ C66
ⴱ C66
0
0 0
ⴱ C12
0
0
ⴱ C22
0
0
0
0
0
0
0
0
0
0
0
0
0 0
−
ⴱ
0
0
−
ⴱ
冥
冥
,
,
共13兲
共14兲
the coefficient a is a 3 ⫻ 3 matrix described as a=
冤
i C p
0
0
0
− 2
0
0
−
0
2
冥
,
共15兲
and the coefficient f is a 3 ⫻ 1 matrix given by ˆ 0 0兴T . f = 关Q
共16兲
The boundary conditions are assumed to be adiabatic in the thermal analysis. In addition, stress-free boundary conditions are satisfied at the two parallel surfaces y = 0 and y = h in the structural analysis. Therefore, the boundary conditions are expressed as the Neumann type, which can be described as ˆ 兲 = g, ˆ + ␣U n · 共c ⵜ U
共17兲
where n is the unit vector normal to the surface and the coefficient g is a 3 ⫻ 1 matrix given as g = 关0 0 0兴T .
共18兲
Moreover, the initial condition for the temperature is set as 300 K and the initial displacement and velocity are null.
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H.-x. Sun and S.-y. Zhang TABLE I. Properties of materials used in models. Physical properties Thermal conductive coefficient 共W m Density 共kg m−3兲 Thermal capacity 共J kg−1 K−1兲 Thermal expansion coefficient 共K−1兲
−1
K 兲 −1
k Cp ␣T Cⴱ11 C*66 Cⴱ22 Cⴱ12
Complex stiffness modulus 共GPa兲
III. NUMERICAL RESULTS AND DISCUSSION A. Material and laser parameters
Based on the above-described theories, the thermoviscoelastically generated Lamb waves are calculated in a thin plate with 15 mm length and 0.1 mm thickness. Besides, the length of the AR is 5 mm, the starting position of the AR is xab = 10 mm, and the coefficient A is 50. The properties of the material used in the calculation are listed in Table I. As shown in Fig. 2, the pulsed laser source is located at the position x = 0 on the top surface of the sample, the incident laser energy is 1.8 mJ. Two radii of the laser spot, 100 and 300 m, are adopted on the sample surface. The pulse rise time and the optical penetration depth into the sample are taken to be 10 ns and 40 m, respectively. B. Model of simulation
The spatial resolution of the FE model is critical for the convergence of these numerical results. So, choosing an adequate element size is very important for the stability and accuracy of the solution. Triangular element with quadratic behavior is used to mesh the model and the element size Le is determined by the criterion28 Le ⱕ
1 C , 10 f max
共19兲
where C represents the surface acoustic wave speed of the medium and f max is the highest frequency in the ultrasonic wave, which can be evaluated by29 f max =
冑2C a0
,
共20兲
where a0 is the radius of the laser pulse spot. In the model, the maximum element sizes are generally arranged to be 20 m and 60 m for a0 = 100 m and 300 m, respectively, whereas the element size is 5 m near the heataffected zone, which is much finer for obtaining the transient temperature field accurately. The frequency step ⌬f is also very important to make the FE calculation efficient and accurate. The frequency step ⌬f is determined by the criterion given as30 ⌬f ⱕ
1 , 2td
共21兲
where td is the upper bound of time interval of the displacement fields.
Elastic material
Composite material
160 2700 900 23.0⫻ 10−6 112 27
16.2 1580 1500 27.0⫻ 10−6 14.9共1 + i0.03兲 3.8共1 + i0.03兲 Cⴱ11 Cⴱ11 − 2Cⴱ66
ˆ 共x , y , 兲 in Eq. 共8兲 is used as the exThe heat source Q citation source of the FE model in the frequency domain and the frequency content of the pulse Fˆ共兲 is confined to 0 ⱕ f ⱕ 10 MHz in this study. So, mainly the lower-frequency components of the symmetric mode 共S0兲 and asymmetric mode 共A0兲 are excited in such a thin plate. The upper bound of time interval is chosen as td = 10 s, therefore, the size of the frequency step is arranged to be ⌬f = 0.02 MHz from Eq. 共21兲, and the frequency range is divided into 500 seriate frequency components. With these schemes, the generation and propagation of the guided modes are covered. In the calculation, a number of 500 stationary analysis runs for the corresponding laser source amplitudes are performed, with the method of using a single parametric solution in the frequency domain. The temperatures and complex displacements are predicted in both directions x and y for the whole set of nodes. The temporal responses are then reconstructed for several node positions, by applying an inverse fast Fourier transform 共FFT兲 for the set of temperatures and complex displacements predicted for the 500 frequency components. C. Numerical results and discussions
As the first example, to prove the validity of the thermoviscoelastic FE model in the frequency domain, a classical FE model in the time domain22–24 is established to compare the temporal solution with the frequency solution by using inverse FFT. In this calculation, the properties of the elastic material in Table I are used as the input data. The governing equations and the temporal heat source Q共x , y , t兲, Eqs. 共1兲–共6兲, are adopted in the classical FE model, and the other numerical conditions are the same as those for the frequency solution. As the driving force of the ultrasonic waves, an accurate determination of the temperature field is vital to accurate prediction of laser-generated ultrasonic waves. Taking account of the effects of thermal conduction and optical penetration, as well as the radius and duration of the laser source, the temperature and displacement fields in the elastic plate are calculated from the temporal solution and the frequency solution by using inverse FFT, respectively. The temperature evolution at the irradiation center of the sample with different laser radii is plotted in Fig. 3. As shown in Fig. 3, when the laser radius is 100 m 关Fig. 3共b兲兴, the temperature increases more rapidly and reaches a higher maximum than that when the laser radius is 300 m 关Fig. 3共a兲兴.
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FIG. 3. Surface temperature evolution at laser irradiation center with laser radii of 共a兲 300 m and 共b兲 100 m in elastic plate, using temporal solution 共—兲 and inverse FFT of frequency solution 共- -兲.
FIG. 5. Normal surface displacement at source-receiver distance of 6 mm with laser radii of 共a兲 300 m and 共b兲 100 m in nondamping 共- -兲 and damping composite plates 共—兲.
The normal surface displacements at a source-receiver distance of 6 mm are shown in Fig. 4. The initial arrival is the S0 mode and followed by the A0 mode with larger displacement amplitude, which means the energy of Lamb waves, is mainly in the A0 mode. It can be seen from Fig. 4共a兲 that the S0 mode shows a nondispersive spike and the generated signal frequency is relatively low. However, as shown in Fig. 4共b兲, the higher-frequency components can be excited, and the S0 and A0 modes show more clear dispersive characteristics and the frequencies are higher than those in Fig. 4共a兲. This is because the frequency of Lamb waves is related to the size of the laser radius. The smaller the laser radius is, the higher the frequency is. In addition, it shows from Figs. 3 and 4 that both solutions are almost in superposition. This indicates that excellent agreement is obtained between the two time traces calculated by either the time 共solid lines兲 or frequency solution 共dashed lines兲. Moreover, the frequency solution is obtained through the method of a single parametric solution, so as to cause the huge reduction in the time and memory of the computation due to the diminution in number of iterations. In order to investigate the effects of the viscoelasticity of the material on wave propagation with different laser radii, the properties of the composite material in Table I are used as the input data. In the calculation, two different cases are modeled in the frequency domain. First, the composite material 共except the AR兲 is considered to be a purely elastic material, using its real elastic moduli as the input data. Second, the properties of viscoelasticity are adopted 共complex moduli are used as the input data兲. Figure 5 presents the normal surface displacements with different laser radii at a
source-receiver distance of 6 mm. By comparing the transient responses of the viscoelastic plate with those of the purely elastic plate, we can clearly see that the amplitudes of the elastic waves are much larger than those of the viscoelastic waves, and the amplitude ratio of the viscoelastic wave to the elastic wave in the higher frequencies is smaller than that in the lower frequencies. This can be explained that the amplitudes of Lamb waves are attenuated largely by the material viscoelasticity and the attenuation of the higher frequencies is stronger than that of the lower frequencies. So, neglecting material viscoelasticity can lead to greatly erroneous results since the viscoelastic time trace is significantly different than the elastic one. Next, the normalized displacements vˆ with different laser radii are monitored along the x direction on the top surface of the plate. Figure 6 shows the huge effect of viscoelasticity on the amplitude of the S0 and A0 modes coexisting at the frequency of 6 MHz, which is a higher frequency in Lamb waves. It shows that the signal amplitude decreases gradually in the RP, i.e., from x = 0 to 10 mm, and is minute when entering the AR if the viscoelasticity is taken into account. As would be expected, in the AR, i.e., from x = 10 to 15 mm, the amplitude drastically diminishes and tends toward zero, indicating that no wave propagates backward along the RP to influence the incident wave. It can be seen from Fig. 6 that the amplitude is almost zero at the frequency of 6 MHz in the RP when the laser radius is 300 m, while the amplitude is much larger when the laser radius is
FIG. 4. Normal surface displacement at source-receiver distance of 6 mm with laser radii of 共a兲 300 m and 共b兲 100 m in elastic plate, using temporal solution 共—兲 and inverse FFT of frequency solution 共- -兲.
FIG. 6. Normalized normal surface displacement vˆ at 6 MHz with different laser radii along x direction in composite plate.
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FIG. 9. Normalized surface normal displacement vˆ along x direction at 6 MHz in nondamping 共- -兲 and damping composite plates 共—兲, with viscoelastic rates of 共a兲 0.5, 共b兲 1.0, and 共c兲 1.5. FIG. 7. Normal surface displacement with different viscoelastic rates at source-receiver distance of 6 mm in composite plate.
100 m. This suggests that the frequency of the signal for a0 = 100 m are higher than that for a0 = 300 m. Furthermore, it also clearly shows from Figs. 5共a兲 and 5共b兲, when the laser radius is 100 m, more frequency components can be excited, and the frequencies of the signal are higher, which shows much stronger attenuation than that when the laser radius is 300 m. Therefore, the generation of the thermoviscoelastic waves with high frequencies is great important in studying mechanical characteristics of the composite material and smaller laser radius should be taken into account when simulating those ultrasound waves with high frequencies. In our following calculations, the laser radius is arranged to be 100 m. We also calculated the normal surface displacements with different viscoelastic moduli at a source-receiver distance of 6 mm. In this case, the viscoelastic rate m is gradually increased by varying the imaginary parts of the complex moduli:Cⴱij = C⬘ij + i共mC⬙ij兲, from 0.5 to 1.5 of the nominal values given in Table I. The transient responses are shown in Fig. 7. It clearly shows that the velocities and the dispersive characteristics of the S0 and A0 modes are almost the same in the different viscoelastic plates, which indicates that the viscoelasticity has no influence on the velocities of the Lamb waves. This phenomenon is also verified by the dispersion curve in Fig. 8共a兲. In addition, the amplitudes of the S0 and A0 modes are decreased gradually with the increase in the rate m, which suggests that the attenuation of Lamb waves is related to the extent of the viscoelasticity. The higher the viscoelasticity is, the stronger the wave attenuation is. The effects of different viscoelastic moduli are also shown in the attenuation curve in Fig. 8共b兲, in which the increase in the material viscoelas-
FIG. 8. Lamb waves dispersion curve for composite plate with different viscoelastic rates, 共a兲 phase velocity and 共b兲 attenuation.
ticity causes a marked increase in the attenuation of the S0 and A0 modes. It also can be seen from Fig. 8共b兲 that the attenuation of the higher-frequency signal is stronger than the lower-frequency one in the same viscoelastic material, which coincides well with the conclusion obtained in Fig. 5. Figure 9 shows the normalized displacements vˆ with different viscoelastic moduli along the x direction on the top surface of the plates. From the comparison between the normalized displacements in the viscoelastic plates 共solid lines兲 and those in the purely elastic plate 共dashed lines兲, we can find that the attenuation of the displacement in the higher viscoelastic plate is much stronger than that in the lower viscoelastic plate, which is in excellent agreement with the results attained in Figs. 7 and 8共b兲. The normal surface displacements with different elastic stiffness moduli at a source-receiver distance of 6 mm are plotted in Fig. 10. In this example, the complex moduli are set to be Cⴱij = nC⬘ij + iC⬙ij and the elastic rate n is varied from 0.75 to 1.25. As shown in Fig. 10, the velocities of the S0 and A0 modes increase gradually with the increase in the elastic rate n. This is because the velocity of Lamb waves is relevant to the elastic stiffness modulus. The greater the elastic stiffness modulus is, the faster the wave velocity is. Such a phenomenon can also be observed from the dispersion curve in Fig. 11共a兲 that the velocities of both modes increase with the increase in the rate n. Moreover, it can be seen from Fig. 10 that the amplitudes of the S0 and A0 modes increase gradually with the increasing of n, which indicates that the attenuation of Lamb waves is also related to the elastic stiffness modulus. The greater the elastic stiffness modulus is, the larger the wave amplitude is, and the weaker the wave attenuation is. To verify that, the attenuation curves for the three cases have been calculated and the S0 and A0 modes
FIG. 10. Normal surface displacement with different elastic rates at sourcereceiver distance of 6 mm in composite plate.
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FIG. 11. Lamb waves dispersion curve for composite plate with different elastic rates, 共a兲 phase velocity and 共b兲 attenuation.
are plotted in Fig. 11共b兲. The attenuation of both modes decreases gradually with the increasing n, which coincides well with the conclusion obtained in Fig. 10. To investigate the influences of the plate thickness on the laser-generated thermoviscoelastic Lamb waves, the normal surface displacements are calculated in three plates with the thicknesses h of 0.05 mm, 0.10 mm, and 0.15 mm, respectively. The transient responses at a source-receiver distance of 6 mm are plotted in Fig. 12. It is shown from Fig. 12 that the thickness of the plate has remarkable influence on Lamb waves. The S0 and A0 modes show less dispersive characteristics and the frequencies of them decrease gradually with the increase in the thickness h. Such a phenomenon results from the fact that the dispersive nature of the S0 and A0 modes is related to the frequency and thickness. The thinner the thickness is, the higher the wave frequency is, and the clearer the dispersive characteristic is. The effect of the thickness is also shown in the wave number dispersion curve as shown in Fig. 13共a兲. It can be found that the S0 and A0 modes show more clear dispersive characteristics with the decrease in the thickness h. In addition, through the comparison of the displacements in the viscoelastic plates 共solid lines兲 with those in the elastic plates 共dashed lines兲, we can see that the attenuation of the viscoelastic wave in the thinner plate is much stronger than that in the thicker plate. Because the higher frequencies can be excited in the thinner plate and the attenuation of the higher frequencies is much stronger than that of the lower frequencies. The attenuation curves for the three cases are given in Fig. 13共b兲. The attenuation of the S0 and A0 modes also decreases with the increase in the thickness h, which is in very good agreement with the conclusion obtained in Fig. 12.
FIG. 13. Lamb waves dispersion curve for composite plate with different plate thicknesses, 共a兲 real wave number and 共b兲 attenuation.
studied quantitatively in the frequency domain using the FE method, taking account of the viscoelastic and thermophysical properties. It is proved that the temperature and displacement fields calculated in the frequency domain coincide well with those obtained in the time domain. The numerical results indicate that the increase in the viscoelastic rate causes a marked increase in the attenuation of the S0 and A0 modes and the attenuation of the higherfrequency signal is stronger than the lower-frequency one. In addition, the velocities of the S0 and A0 modes increase with the increase in the elastic rate, while the attenuation of both modes decreases. Moreover, when the plate thickness increases, the S0 and A0 modes show less dispersive characteristics and the frequencies of them decrease and so does the attenuation of the S0 and A0 modes. The characteristics of the Lamb waves in the numerical results agree well with the features of the disperse curves. It indicates that the FE method in this paper will provide a useful technique to characterize mechanical properties of composite materials. The further investigation to determine the properties of composite materials by using an inverse method is in progress. ACKNOWLEDGMENTS
This work is supported by National Natural Science Foundation of China under Grant No. 11074125, the Major Project of the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant No. 10KJA140006, and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant No. 08KJB140003. L. R. F. Rose, J. Acoust. Soc. Am. 75, 723 共1984兲. F. A. McDonald, Appl. Phys. Lett. 56, 230 共1990兲. 3 T. W. Murray, S. Krishnaswamy, and J. D. Achenbach, Appl. Phys. Lett. 74, 3561 共1999兲. 4 A. Cheng, T. W. Murry, and J. D. Achenbach, J. Acoust. Soc. Am. 110, 848 共2001兲. 5 Y. Matsuda, H. Nakano, and S. Nagai, Appl. Phys. Lett. 89, 171902 共2006兲. 6 X. Jian, Y. Fan, R. S. Edwards, and S. Dixon, J. Appl. Phys. 100, 064907 共2006兲. 7 A. Moura, A. M. Lomonosov, and P. Hess, J. Appl. Phys. 103, 084911 共2008兲. 8 J. B. Spicer, A. D. W. McKie, and J. W. Wagner, Appl. Phys. Lett. 57, 1882 共1990兲. 9 T.-T. Wu and Y.-H. Liu, Ultrasonics 37, 405 共1999兲. 10 J. C. Cheng and S. Y. Zhang, Appl. Phys. Lett. 74, 2087 共1999兲. 11 H. Al-Qahtani and S. K. Datta, J. Appl. Phys. 96, 3645 共2004兲. 12 C. Prada, O. Balogun, and T. W. Murray, Appl. Phys. Lett. 87, 194109 1
IV. CONCLUSIONS
Thermoviscoelastic generation of ultrasound in thin composite plates subjected to laser illumination has been
FIG. 12. Normal surface displacement at source-receiver distance of 6 mm with plate thicknesses of 共a兲 0.05, 共b兲 0.10, and 共c兲 0.15 mm in nondamping 共- -兲 and damping composite plates 共—兲.
2
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