All-Fiber Optic Ultrasonic Structural Health Monitoring System

All-Fiber Optic Ultrasonic Structural Health Monitoring System Vladimir Kochergin,1* Kevin Flanagan,1 Zhong Shi,1 Michael Pedrick,1 Blake Baldwin,1 Thomas Plaisted,1 Balakishore Yellampalle,1 Eugene Kochergin,2 Larry Vicari1 1 Luna Innovations, Inc., 3157 State Street, Blacksburg, VA, USA 24060 2 Donetsk National Technical University, Donetsk, Ukraine 83001 ABSTRACT Structural Health Monitoring (SHM) is required for early detection of damage in structural components to improve the safety, reduce the cost, and increase the performance and efficiency of aircrafts. Currently available techniques have a number of deficiencies prohibiting wide spread of SHM in aerospace applications. In this contribution we will present the initial results of development at Luna Innovations of an all-fiber optic ultrasonic airframe SHM system that will be able to address the deficiencies of solutions suggested/developed to date. In this contribution we will present the details on design, development and testing of the prototype fiber optic SHM system. Keywords: structural health monitoring, nondestructive evaluation, fiber optic, ultrasonic.

1. INTRODUCTION Nondestructive evaluation (NDE) and structural health monitoring (SHM) are increasingly used1 in evaluation of material state of various components, for example, aircraft components, bridges, highways, etc. due to promises of NDE and SHM technologies to detect and identify the invisible damage in the components before complete failure.2 A number of NDE and SHM techniques have been developed. The popular techniques for real-time SHM are distributed ultrasonic sensing (electronic) and fiber optic sensing. Ultrasonic sensing offers a wealth of information and straightforward damage detection. It employs a number of piezoelectric actuators and piezoelectric sensors in mechanical contact with the DUT (Device Under Test).3 Piezoelectric actuators excite acoustic vibrations in the DUT, while piezoelectric sensors pick up the transmitted, reflected, and scattered acoustic vibrations from the DUT, indicating the presence and structure of the defects.4 However, ultrasonic sensors and actuators require separate electrical wiring (resulting in expensive, heavy and complex, EMI-subjective solutions), and performe poorly in harsh environments.5 Fiber optic sensors, on the other hand, offer stability in harsh environments, high multiplexibility and EMI immunity.6,7 Although fiber optic sensors for detecting the acoustic waves have already been developed,8 no practical fiber optic actuators have been developed or even proposed to date. In this paper, we present the initial results of design, fabrication and testing of fiber-optic acoustic actuator. The described fiber optic actuator is employing the phenomenon of laser ultrasound generation9 to convert the laser pulses in the fiber into acoustic waves in the DUT’s structure. A portion of the fiber cladding is replaced by graphite-epoxy composite to provide localized absorption of the pulsed laser light propagating in the fiber. The absorbed radiation is causing local temperature increase in the graphite-epoxy composite, resulting in subsequent thermal expansion, which in turn, becomes an efficient, multiplexible source of acoustic waves. It will be further shown that by combining the developed actuator and fiber optic ultrasonic sensors an all-fiber optic NDE system can be realized.

2. MODELING The drawing of the fiber optic acoustic actuator is provided in Fig. 1(left). We assumed that DUT is made of Al, and absorption material to be graphite-filled epoxy. Material parameters for standard multimode fiber were used. To define the governing equations for modeling the thermo-elastic generation of ultrasound,10 two coupled physical processes should be accounted: 1) thermal expansion and heat conduction due to optical absorption of laser pulse; 2) stress in the material resulting in acoustic wave propagation due to thermal expansion in which the source term is temperature dependant. The governing equations are shown below used in the presented model:

Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2009 edited by Masayoshi Tomizuka, Chung-Bang Yun, Victor Giurgiutiu, Proc. of SPIE Vol. 7292 72923D · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.815315 Proc. of SPIE Vol. 7292 72923D-1

k∇ 2T − Cρ

∂T = q(r , z, t ) ∂t

ρ

(1)

1 q ( r , z , t ) = σEo2 e − 2 k − z f (r ) g (t ) 2

r r r ∂ 2u = μ∇ 2u + (λ + μ )∇(∇ ⋅ u ) + β∇T 2 ∂t

f (r ) = e

(3)

⎛ 2r 2 −⎜⎜ 2 ⎝ ωo

⎞ ⎟ ⎟ ⎠

(2)

(4)

⎛ t ⎞

⎛ t 2 ⎞ −⎜⎜ t ⎟⎟ g (t ) = ⎜⎜ 2 ⎟⎟e ⎝ o ⎠ ⎝ to ⎠

(5)

where u and T are the material displacement and material temperature respectively, κ is the thermal conductivity, σ is the electrical conductivity, C is the specific heat coefficient, ρ is the mass density, α is the linear expansion coefficient, µ and λ are the first and second lame’ parameters respectively, β is defined as α(3λ+2µ), E0 is the laser wave amplitude, kis 1/skin depth (parameter for optical absorption), ω0 is the optical beam waist and t0 is the time constant that characterizes the optical pulse. In our model we have neglected the heat produced from mechanical deformation in Eq. (1).11 In addition, we have approximated the graphite/epoxy and aluminum to be perfectly elastic and isotropic and have assumed Eq. (2) to be parabolic. Such approximations were previously shown to provide good agreement with experiments for thermo-elastic generation of ultrasound.12 Graphite parameters used in the model were obtained from the literature.13,14 The value of the electrical conductivity for used graphite-epoxy used was (1S/m).15 The governing equations were implemented into COMSOL finite element analysis (FEA) software. 2-D axial symmetry model of the actuator was developed. The boundary and initial conditions were applied to the system (actuator and plate) in a straightforward manner. The initial acoustic displacement throughout the plate was set to zero. Since the chosen geometry exhibits axial symmetry, the axial center was held fixed to prevent free body motion, but was otherwise represented with traction free boundary conditions. The initial temperature for the system was set to a constant 300K in addition to a zero heat flux condition which is appropriate considering that the heat generated by thermo-elastic absorption is generated from within the system and not from a heat flux through the surface of the system.11

Core Nanocomposite coating

3

xl 0_I 0

Static Pulse Peak (after pulse onset)

E 2.5 0 2

E 1.5 (0

0

I

E

II)

o 0.5 (0

DUT material

.

Cladding

0

-0i5

0.55

0.6

0.65

0.7 0.75

0.8

0.85

0.9

0.95

1

r-displacement distance from actuator (m)

xI03

Fig. 1. Left: Drawing of the fiber optic acoustic actuator. Right: Plots of the modeled acoustic displacements at 10ns time steps from 20ns to 120ns after the absorption of laser pulse.

The illustration of the modeled geometry is provided in Fig. 2 (right): The pulse is observed as it propagates through the aluminum plate. The modeling results are presented in Fig. 1 (right) and Fig. 2(left). There are two relevant time regimes: the first time regime (see Fig. 1 right) the acoustic pulse is a result of the initial thermal expansion resulting from the incident laser pulse. The pulse peak can be seen to rise within 20ns and appears static beyond 60ns. In the second time regime (see Fig. 2 left), the front edge of the pulse can be seen at a distance L from the acoustic source at a propagation time of 500ns. This corresponds to an acoustic wave velocity of ~ 7000 m/s. A second component of the pulse can also be seen which lags behind the front edge pulse and corresponds to a wave speed of ~ 4800 m/s. These two wave velocities agree reasonably well with the longitudinal and shear wave velocities for aluminum respectively.16

Proc. of SPIE Vol. 7292 72923D-2

The acoustic wave speeds were approximated through the following relationships: L = 3.5mm, Δt= 500ns, v=L/Δt, where L corresponds to the distance traversed by the front edge of the pulse (see Fig. 2 right). C(10_10

Frcnt Edge of PuIs.e

Graphite Act iator

I

I

point of observation

Aluminum Plate

" radial slice L ___________ 5

1

1.5

2

2.5

3.5

3

4

4.5

r-displacement distance from actuator (m)

Fig. 2. Left: Plots of the modeled acoustic displacements at equal time steps of 100ns from 100ns to 500ns. In this time regime, the front edge of the pulse can be seen at a distance L from the acoustic source at a propagation time of 500ns. This corresponds to an acoustic wave velocity of ~ 7000 m/s (close to the longitudinal wave velocity for aluminum). Right: Illustration of modeled geometry. The radial displacements was evaluated on the surface of an aluminum plate, perpendicular to the plate, at a distance of 1” from the source, similar to the measurement setup with PZT transducer.

The decay of acoustic pulse amplitude as it propagates through the Aluminum is visible. This decay is mainly due to reshaping of the pulse as it propagates. The simulation of an NDE measurement (i.e. a measurement from a PZT transducer yielding displacement information) was made by evaluating the radial displacement on the surface of an aluminum plate, perpendicular to the plate, at a distance of 1” from the source (see Fig. 2 right). The displacement was evaluated by COMSOL in equal time steps of 100ns from 9μs to 10μs as shown in Fig. 3. The peak-to-peak acoustic displacement with the r-coordinate at 0.5 x 10-3m. was predicted to be 12 x 10-12m. As will be shown in later section this number agrees well with experimental findings, validating the developed model.

E i.s .-.

1

E 0.5

E

0

0

5

-1

05

1

1.5

2

2.5

radial distance from acoustic source (m)

3

xlO -3

Fig. 3. Simulated plot of acoustic displacements along a radial slice of aluminum at 1” from in equal time steps of 100ns from 9μs to 10μs. The displacements correspond to left edge of the plot with the r-coordinate evaluated at 0.5 x 10mm.

3. EXPERIMENT We selected C60 fullerene-filled epoxy as an absorbing (transducing) material in fiber optic acoustic atuator. We used Luna nanoWorks synthesized C60 fullerenes with purity greater than 99% for this purpose. The epoxy resin Duralco 4460 was selected on the basis of its viscosity and thermal resistance (600 cps and maximum service temperature of 315°C). The fullerenes were combined with the epoxy resin to achieve a mixture that would possess the desired optical properties while maintaining its processability. The fullerenes were loaded into the epoxy, along with a 2wt% loading of


dispersant (E-sperse 130) to aid in breaking up the agglomeration of fullerenes within the resin. The sample components were mixed in a Flaktec Speedmixer. Both 1mm diameter and 200um diameter multimode optical fibers were used for actuator fabrication for different experiments. The portion of the fiber cladding (and, in some experiments, the portion of the core as well) was removed by polishing in specifically fabricated jig that allowed the cladding to be removed in a uniform and controlled manner. A graphite-epoxy application jig was constructed in such a way as to allow a uniform distribution of graphite-epoxy that matches the cladding diameter of the fiber (see Fig. 4 left) thereby improving the mechanical integrity of the fiber. The photo of fabricated acoustic actuator is provided in Fig. 4 right, where the portion of the fiber with absorptive coating is clearly visible.

Fig. 4. Left: Illustration of an actuator cross-section. Right: Photo of fabricated fiber optic acoustic actuator.

During the actual testing, experiments were conducted with 1064nm Nd:YAG Pulsed laser with a pulse rate of 20Hz with 50mJ per pulse (7ns long) resulting in average output power of 1W coupled in the 1mm diameter fiber (~200mW power was detected in 200um diameter fiber). Up to 5 actuators were distributed along a single fiber at a separation of 5cm. An aluminum plate with dimensions of 30 x 33 cm by approximately 2.4 mm thick was used for proof-of-concept testing. In the plate, several step defects were machined, ranging in depth from 10% to 80% of full plate thickness where each defect was 30 mm long by 6.3 mm wide. The actuator portion of the fiber optic source was laid down on the aluminum plate and coupled via grade 40 Sonotrace Ultrasonic couplant. The acoustic signal was received by a Panametrics 1MHz / .5 X 1 PZT transducer with the polarization set parallel to the normal of the plate surface. The received signal was amplified approximately 20 dB with a custom preamp. A photo of the experimental setup is shown in Fig. 5 (left). Source Fiber

1615141312111098765432

xx------ xxxxxxxxx.,

-

____________ Detects

Piezo

FiD.

-

:

.-

____

-

+ 11

-

II

r

-

ctuator

1j1

Msurm,ent Locabons

t ,-'

0

Actuators

x xxxxxxxxxxxxxx Aluminum Plate

Fig. 5. Left: Photos of the experimental setup used to validate fiber-optic generation of ultrasound in an aluminum plate. The actuator portion of the fiber optic source was laid down on the aluminum plate where the resulting acoustic signal was received by a PZT transducer and sent to an oscilloscope so that the time dependent waveform could be captured. Right: Illustration of measurement layout for distributed measurements. The acoustic signals were measured every 2 cm (16 per side) at a distance of 3cm away from the line of actuators at the defect and non-defect sides of the plate.

Acoustic signals were measured every 2 cm (16 per side) at a distance of 3cm away from the line of actuators for both the defect and non-defect sides of the plate as illustrated in Fig. 5 (not to scale). For the defect side, the measurements were made such that the defects were between the actuators and the measurement location. Such a testing provided a reliable qualitative assessment of the defect so detected. In order to demonstrate all-fiber optic NDE, an all fiber-optic acoustic sensor was developed. This was accomplished by using the Extrinsic Fabry Perot Interferometer (EFPI) type fiber optic sensor. EFPI sensors are based on a distance measurement using a low-finesse Fabry-Perot cavity formed between the polished end face of an optical fiber (R1) and a reflective transducer surface (R2). The distance measurement (deflection of diaphragm) is obtained by analyzing the


resulting interference pattern. This deflection information together with known elastic properties of the diaphragm yielded the acoustic detection. The mass of the sensor is on the order of 1.0μg thus enabling a fast response. For interrogation of the EFPI sensors, the Hyperscan system from Luna Innovations was used. The Hyperscan system uses two different wavelength lasers to generate two optical signals that are π/2 phase shifted resulting in quadrature. In this way, one signal always has a linear response with the change in gap. Perturbations in pressure (caused by acoustic waves in our case) will drive the sensor through the interference fringes as measured by a photo-detector, which are followed up by data processing to determine the gap length at high sampling rates (up to 2MHz).

Acoustic Amplitude (V)

For all-fiber optic acoustic NDE experiments, the EFPI sensor was mounted on the surface of the aluminum plate such that normal displacements to the plate surface would be measured (analogous to the PZT transducer). Ultrasonic gel was applied between the plate and EFPI sensor to ensure coupling of the acoustic signal. The fiber optic acoustic actuator was used as a source. For comparison, the measurements were performed with both a PZT sensor (as a reference) and an EFPI sensor. The results are presented in Fig. 6. Fig. 6 left shows the two signals overlaid on the same time scale. The EFPI sensor yielded an acoustic signal that is very similar to the reference PZT transducer. The signal-to-noise levels of the two sensors are comparable. The small difference in structure between the two acoustic signals reveals a different responsivity between the two types of sensors. Unlike the EFPI, the PZT sensor is known to be bandwidth limited.

II, -0.

0.5

1

U)

CD

-0.5

3

-I

-0.81.5

Time (sec)

2

2.5

3

3.5

4

x

Fig. 6. Left: Comparison of measured acoustic displacement amplitudes for a PZT sensor (dark gray line), and EFPI sensor (gray line), reveals a similar signal to noise ratio and signal structure in both cases. Right: Measured temporal acoustic displacement amplitude from a Panametrics 1MHz / .5 X 1 transducer. Measurements were taken at position #1 (see figure 5), which is only 1” away from the first actuator.

4. RESULTS AND DISCUSSION In order to compare the predictions of developed model with experimental results, we needed to transform the measured data (voltage from PZT transducer) into the displacement information by using the piezo-coupling coefficient that corresponds to one of the terms in the piezo strain matrix. Depending on the type of piezo crystal and its orientation will determine which term is applicable to measurement. In our case we have a Panametrics 1MHz / .5 X with piezo coupling coefficient for PZT-5A of D33 = 3.6 x 10-10m/V.17 As mentioned before, the COMSOL model was setup to evaluate the radial displacement on the surface an aluminum plate, perpendicular to the plate, at a distance of 1” from the source (see Fig. 2 right). The measurement that most closely corresponds to the model is measurement #1 on the non-defect side of the plate (see Fig. 5 right). The peak displacement corresponds to the peak voltage which was measured roughly to be 1.6 V peak-to-peak ( as shown in Fig. 6 right). Using the piezo coupling coefficient and taking amplification (factor of 130) into account, the measured displacement is found to be ~ 4.4 x 10-12 m whereas the model predicted a value of ~ 12 x 10-12 m. These values are within the order of magnitude agreement, the accuracy that can be realistically expected for such a model. Perhaps the most compelling evidence to validate the COMSOL model comes from analyzing the time and frequency domains of the measured and simulated waveforms respectively. A side-by-side comparison of the measured and simulated waveforms for the time and frequency domains is shown in Fig. 7. Close match between the predicted and observed results is clear. The deviation between the modeling and experiment is probably due to the bandwidth-limited detection in experiment.


Measured Acoustic Pulse (time domain)

10-fl

EXPERIMENT

COMSOL Acoustic Pulse (time domain)

MODELING -

0.80.60.4-

00

0

fr

-0.4-

-0.6-0.

-0.8-0.8-0.5

0.5

1

1.5

2

2.5

4

3.5

Time (sec)

0.5

35

2.5

2

1.5

Time (sec)

10

10_j

COMSOL Acoustic Pulse (frequency domain)

Measured Acoustic Pulse (frequency domain)

MODELING 5 6-

a. 2-

1.5

2.5

2

3

Frequency (Hz)

3.5

4

4.5

5

io

Fig. 7: Comparison of the measured and simulated acoustic data for the time and frequency domains. Left: Time and frequency domains of the measured acoustic data. Right: Time and frequency domains of the simulated acoustic data.

For demonstration of defect detection the following procedure was utilized: acoustic signals were measured symmetrically about the line of actuators for both the defect and non-defect sides of the plate in 2 cm increments at a distance of 3cm away from the line of actuators (see Fig. 5). Then, the time domain data was transformed into the timefrequency domain via Fourier transformation, so the spectrogram was obtained. The spectrogram was then re-plotted in a form of Group velocity as a function of a frequency thickness product (thickness here refers to the aluminum plate thickness). Presented in this way, the data shows the mode dispersion. In order to show frequency contrast (i.e. defect detection), the difference in the dispersion data spectral density from between adjacent sets of points was calculated (i.e. one point located on a non-defect side at 3cm away from the line of actuators and the other point located on the direct opposite side set behind the defects) as: DP = Pd − Pnd Pd Pnd

, where Pd and Pnd is the spectral power density on defect

and non-defect sites, and DP is the difference in spectral power density between defect and non-defect sites. The DP calculated for measurement position 15 and 1 is shown in Fig. 8. The correlation between the detected defects with the actual defects is clearly shown, validating the NDE capabilities of the suggested fiber optic technique. Tin order to retrieve the specific information about the defects from measured data a modal analysis should be employed. By removing the source term in Eq. (2), one can obtain: r r r ∂ 2u ρ 2 = μ∇ 2 u + (λ + μ )∇(∇ ⋅ u ) (6) ∂t This is the homogeneous acoustic wave equation that describes the acoustic wave propagation. Eq. (6) can be decoupled into two wave equations for a scalar and vector potential respectively. This is done by first allowing the function u to be composed of the sum of the gradient of a scalar potential and the rotor of a vector potential. Then, by setting the divergence of the vector potential to zero) and re-arranging terms, then from Eq. (10) it follows:

ρ

∂ 2ϕ − (λ + 2μ ) ∇ 2ϕ = 0 ∂t 2

(

)

(7)

ρ

∂ 2ψ − μ ∇ 2ψ = 0 ∂t 2

(


)

(8)

r r r u = ∇ϕ + ∇ ×ψ

(9)

ρ

CL =

ρ μ

CT =

(10)

(λ + 2μ ) (11)

By applying traction free (normal and shear stresses are zero at the surfaces) boundary conditions to the top and bottom surface of the plate leads to two characteristic equations as shown below: tan(qh) 4k 2 qp =− tan( ph) q2 − k 2

(

p2 =

ω2 C L2

)

2

(symmetric)

(

−k2

)

q2 − k 2 tan(qh) =− tan( ph) 4k 2 pq

(12)

(14)

q2 =

ω2 CT2

2

(antisymmetric)

−k2

(13)

(15)

Eqs. (14) and (15), represent the dispersion relations for acoustic plate waves. Eqs. (12) and (13)correspond to a discrete set of allowed acoustic modes which satisfy the plate boundary conditions.

Frequency Thickness Product (MHZ *mm)

Frequency Thickness Product (MHz*mm)

Fig. 8: The difference in spectral power density (DP) between defect and non-defect sites is compared for two different regions of the DUT. Top: DP is calculated for measurement position 15 (a non-defect region) (see Fig. 5). Bottom: DP is calculated for measurement position 1 (a defect region).

The dispersion curves show the allowed acoustic modes. Fig. 9 left is a plot of theoretical group velocity vs. FTP for aluminum where the solid lines are to indicate symmetric modes whereas the dotted lines are to indicate anti-symmetric modes. These dispersion curves can be used for NDE analysis to retrieve the information on defects in the DUT. As shown in Fig. 9 center and left. In Fig. 9 center the dispersion data is shown for acoustic waves generated in aluminum from a fiber optic actuator without the presence of a defect. In Fig. 9 right the dispersion data is shown with a defect extending 80% deep into aluminum plate between the actuator and receiver.

0

0 Frequency Thickness Product (MHz*mm)

0

2

4

6

8

10


0

2

4

6

8

10


Fig. 9: Right: Theoretical group velocity dispersion curves for an aluminum plate. The solid lines correspond to symmetric modes while the dotted lines correspond to anti-symmetric modes. Center and Left: Experimental data overlaid on theoretical dispersion curves for aluminum (lines). Center: Data was collected with no defect between actuator and receiver. Right: Data was collected with 80% defect between actuator and receiver.


The location of defect is shown as the mode redistribution (area of the plot highlighted by a box in the right plot). It follows that the introduction of the defect resulted in coupling of initially excited symmetric mode into anti-symmetric mode. Providing us with the some information on defect depth/size.

5. CONCLUSIONS This paper presented a design and realization all-fiber optic acoustic NDE system on the basis of fiber optic acoustic actuator. The model of the physical processes involved in acoustic generation by fiber optic actuator were developed and validated by experiment. Fabrication of fiber-optic actuator was established and optimize to provide sufficient intensity of acoustic generation for NDE experiments. An all-fiber optic generation and detection of acoustic waves in device under test was demonstrated. Finally, defect detection was demonstrated in both a qualitatively and quantitatively. The proposed methodology is expected to find a wide range of applications spaning from NDE and SHM of structural components in aircrafts, ships and civil structures to monitoring of plasma and nuclear equipment due to inherent EMI immunity and harsh environment compatibility of the fiber optic actuators and sensors. Demonstrate multiplexing of a number of actuators on the same fiber promises the low cost per actuation point of the system.

ACKNOWLEDGEMENTS The authors would like to acknowledge the support of NASA, through NRA Contract # NNL08AA26C.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Komatsuzaki S., et al., “Small-diameter optical fiber and high-speed wavelength interrogator for FBG/PZT hybrid sensing system”, Proc. of SPIE Vol. 6530, 65300O, (2007). Petculescu G., and Achenbach J.D., “Schedule Based NDT and Structural Health Monitoring of Safety Critical Composite Structures”, Materials Evaluation, 65 (7), pp. 731-739 (2007). Sundaresan M., et al., “Evaluation Of A Scalable Structural Health Monitoring System Based on Acoustic Emission Sensing,” European Workshop on SHM, Munich Germany, July (2004). Devillers D. et al, “Interaction of Lamb waves with defects in composite sandwich structures”, Proc. of ESTI Aeronauticos, pp. 629-638, (2000). Zhang S., “High temperature piezoelectric materials for actuators and sensors,” SPIE Proc. vol. 5761, pp. 279286 (2006). Fielder R.S., and Stinson-Bagby K.L., “High-Temperature Fiber Optic Sensors for Harsh Environment Applications,” SPIE Photonics East, (2003). Fielder R.S., et al., “Harsh-Environment Fiber Optic Sensors for Structural Monitoring Applications,” SPIE International Symposium on Smart Structures and Materials, (2004). Sun,C. S., "Multiplexing of fiber-optic acoustic sensors in Michelson interferometer configuration." Optics Letters 28(12), pp. 1001-1003 (2003). Aussel J.D., LeBrun A., and Baboux, J.C., “Generating Acoustic Waves by Laser: Theoretical and Experimental Study of the Emission Source”, Ultrasonics, 26(5), pp. 245-255 (1988). Arias I., and Achenbach J.D., “Thermoelastic generation of ultrasound by line-focused laser irradiation,” Int. J. of Solids and Structures 40, pp. 6917-6935 (2003). Shi Y., Shen Z., Ni X., Lu J., and Guan J., “Finite element modeling of acoustic field induced by laser line source near surface defect”, Opt. Express, 15(9) , pp. 5512-5520 (2007). Sanderson T., Ume C., and Jarzynski J., "Hyperbolic heat equations in laser generated ultrasound models”, Ultrasonics, 33(6), pp. 415-421 (1995). Nayar H A., The Metal Databook. McGraw-Hill, New York, (1997). Metals Handbook, Vol.2, Properties and Selection: Nonferrous Alloys and Special-Purpose Materials, ASM International 10th Ed. (1990). Foley M., Ton-That C., and Kirkup L., “Electrical Properties of Pure and Oxygen-Intercalated Fullerene Films”, Proc. 31st Annual Condensed Matter and Materials Meeting, (2007). Shull P.J., Nondestructive Evaluation, Theory, Techniques and Applications, Marcel Dekker, Inc., (2002). Hooker M.W., “Properties of PZT-Based Piezoelectric Ceramics Between 150 and 250 Degrees Celsius”, NASA/CR-1998-208708, pp. 30 (1998).
