Guided wave propagation in carbon composite laminate using

Guided wave propagation in carbon composite laminate using piezoelectric wafer active sensors

M. Gresil*a, V. Giurgiutiu a

aLAMSS, Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA

ABSTRACT

Attenuation of Lamb waves, both fundamental symmetric and anti-symmetric modes, propagating through carbon fiber reinforced polymer (CFRP) was modeled using the multi-physics finite element methods (MP-FEM) and compared with experimental results. Composite plates typical of aerospace applications were used and provide actuation using integrated piezoelectric wafer active sensors (PWAS) transducer. The MP-FEM implementation was used to combine electro active sensing materials and structural composite materials. Simulation results obtained with appropriate level of Rayleigh damping are correlated with experimental measurements. Relation between viscous damping and Rayleigh damping were presented and a discussion about wave attenuation due to material damping and geometry spreading have been led. The Rayleigh damping model was used to compute the wave damping coefficient for several frequency and for S0 and A0 mode. The challenge has been examined and discussed when the guided Lamb wave propagation is multi-modal.

Keywords: Rayleigh damping, Carbon fiber, Piezoelectric wafer active sensors, Finite element method, Guided Lamb waves

1. INTRODUCTION Lamb waves are ultrasonic waves which propagate through thin plate-like structures and are employed for structural health monitoring (SHM) applications. Features of Lamb waves used for SHM are Time-of-Flight, mode conversion/generation, change in amplitude/attenuation, velocity, etc. [1]. Attenuation is often neglected in wave propagation analysis because of complexities involved in modeling. In the case of fiber reinforced polymer (FRP), one of the main complexities involved for the propagation of guided waves is the attenuation effect.

Damping or attenuation mechanisms in FRP materials are different from those in conventional metals and alloys. The different sources of energy dissipation in FRP are:

(a) Viscoelastic nature of matrix and/or fiber materials: the major contribution to composite damping is due to matrix [2]. However, the fiber damping must be also included in the analysis for carbon and Kevlar fibers having high damping as compared to other types of fiber [2].

(b) Damping due to interphase: interphase [3] is the region adjacent to fiber surface all along the fiber length, interphase possesses a considerable thickness and its properties are different from those of embedded fibers and bulk matrix.

(c) Damping due to damage: this is mainly of two types: (i) Frictional damping due to slip in unbound regions between fiber and matrix interface or delaminations (ii) Damping due to energy dissipation in the area of matrix cracks, broken fibers etc.

Increase in damping due to matrix-fiber interface slip is reported to be many fold [4] and is more sensitive to damage than stiffness [5].

(d) Visco-plastic damping: at large amplitudes of vibration/high stress levels, especially thermoplastic composite materials exhibit non-linear damping due to the presence of high stress and strain concentration that exists in local regions between fibers [6].

(e) Thermo-elastic damping: it is due to cyclic heat flow from the region of compressive stress to the region of tensile stress in the composite [6-8].

*[email protected]; phone 1 803-777-0619; fax 1 803-777-0106; http://www.me.sc.edu/Research/lamss/

Health Monitoring of Structural and Biological Systems 2013, edited by Tribikram Kundu, Proc. of SPIEVol. 8695, 869525 · © 2013 SPIE · CCC code: 0277-786X/13/$18 · doi: 10.1117/12.2009254

Proc. of SPIE Vol. 8695 869525-1

The types of matrix and fiber materials, fiber length, curing temperature, laminate configuration, etc. are all factors that can greatly affect the energy dissipation properties of the material. Values for damping are often found in literature with poor reference to the method used for measurement, the environmental conditions, and the characteristics of the material selected. This complicates the task of comparing and validating experimental results.

Castaings and Hosten [9] measured the attenuation of Lamb waves, employing air-coupled ultrasonic transducers, in isotropic and viscoelastic materials. Berthelot and Sefrani [10] showed that vibration damping is a function of frequency and fiber orientation. Hadi and Ashton [11] investigated different fiber volume fractions, and observed that the damping behavior is improved with increased fiber volume fraction. Berthelot and Sefrani [10] studied the effect of the fiber orientation on the damping coefficient. A maximum value for the loss factor was observed at around 60º in glass fiber reinforced polymer (GFRP) composites, and around 30º in Kevlar fiber composites. Hadi and Ashton [11] observed that maximum damping was at fiber orientations of around 30º for the unidirectional E-GFRP. They relate the reason of attaining maximum damping at 30º to the fact that total strain energy is dominated by in plane shear strain energy, which is maximum at this fiber orientation angle. A further effect is the stacking sequence studied by Maher [12], who showed that changing orientation of outer laminates has a significant effect on damping, different than that of inner laminates. Mahi [13] studied various cases of fiber orientations and stacking sequences and showed that for unidirectional GFRP, maximum loss factors occurred in the range of 30°-90° fiber orientation, while the loss factor for taffeta or serge composites is distributed symmetrically versus fiber orientation. Damping in taffeta and serge composites is accordingly significantly higher than cross ply types, which can be attributed to friction between warp and weft fibers [13]. He and Liu [14] studied the effect of fiber-matrix interface on damping, and showed that as we improve adhesion between them damping capabilities are reduced.

Hu et al. [15] while proposing a technique for identifying the location of a delamination in a cross-ply laminated composite using the S0 mode, determined the attenuation coefficients of a Carbon Fiber Reinforced Plastic (CFRP) material by matching the amplitudes to the experimental data in a numerical model. Numerical modeling of guided wave propagation helps on understanding the interaction phenomenon between the material damage and the guided waves. The finite element method (FEM) approach is traditionally used for modeling elastic wave propagation [16-23]. Many authors modeled Lamb wave propagation through FRP media without considering the damping effect [15, 23-25]. However, if material attenuation is also incorporated in the FEM model, then SHM techniques based on the change in amplitude can be understood better and implemented effectively. For this reason, the prediction of the attenuation of Lamb wave in FRP is a critical issue and needs to be developed and known in coordination with the experimental data. The damping model is implemented in many commercial software (eq. ABAQUS), in order to obtain results for numerically sensitive structural systems. Viscous damping is the only dissipation mechanism that is linear and thus low cost, so that the motivation to use it in models is very strong. To simulate viscous damping in FEM modeling, stiffness-proportional or mass-proportional damping can be introduced in the time domain models of most FEM packages. The most common viscous damping description is the Rayleigh damping model, where a linear combination of mass and stiffness matrices is used. This Rayleigh damping is defined as [26]

[ ] [ ] [ ]C M Kα β= + (1)

where α and β are the mass and stiffness proportionality coefficients. The mass proportional damping coefficient α introduces damping forces caused by the absolute velocities of the model and so simulates the idea of the model moving through a viscous “medium” [27]. The stiffness proportional damping coefficient β introduces damping proportional to the strain rate, which can be thought of as damping associated with the material itself [27].

Bert [28] and Nashif et al. [29] did a survey on the damping capacity of fiber reinforced composites and found that composite materials generally exhibit higher damping than metallic materials. Composite damping mechanisms and methodology applicable to damping analysis are described in references [27-28]. Gibson et al. [30] and Sun et al. [31, 32] assumed viscoelasticity to describe the behavior of material damping of composites. Gibson et al. [33] used the modal vibration response measurements to characterize, quickly and accurately the mechanical properties of fiber-reinforced composite materials and structures. Koo et al. [34] studied the effects of transverse shear deformation on the modal loss factors as well as the natural frequencies of composite laminated plates by using the finite element method based on the shear deformable plate theory.

In the present work, a Rayleigh damping model is used to study the damping of the guided Lamb wave in CFRP composite laminates. We use composite plates typical of aerospace applications and provide actuation using integrated


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The piezoelectric material properties were assigned to the PWAS as described in [42]. The CFRP mechanical properties in the MP-FEM model are presented in Table 2.

3. GUIDED WAVE ATTENUATION DUE TO MATERIAL DAMPING FOR 2-D PROPAGATION This section will present the relation between viscous damping and Rayleigh damping in order to develop the wave attenuation due to material damping for 2D wave propagation. In preliminary work [43], we defined the damping ratio as

2ckm

ζ = (3)

Consider the 1-dof damped system consisting of a spring, k , mass, m , and dashpot damper, c .

In FEM modeling, energy dissipation is introduced through the Rayleigh damping concept, i.e. Eq. (1). To illustrate the connection between Rayleigh damping, Eq.(1), and the viscous damping, Eq.(3), we have

12 n

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Lamb waves can have 1-D propagation (straight-crested Lamb waves) or 2-D propagation (circular-crested Lamb waves). Conventional ultrasonic transducers generate straight-crested Lamb waves. This is done through either an oblique impingement of the plate with a tone-burst through a coupling wedge. Circular-crested Lamb waves are not easily excited with conventional transducers. However, circular-crested Lamb waves can be easily excited with PWAS transducers, which have omnidirectional effects and generate circular-crested Lamb waves propagating in a circular pattern. Circular-crested Lamb waves have the same characteristic equation and the same across the thickness Lamb modes as the straight-crested Lamb waves. The main difference between straight-crested and circular-crested Lamb waves lies in their space dependence, which is a trough harmonic function in the first case and through Bessel functions in the second case. In spite of this difference, the Rayleigh-Lamb frequency equation, which was developed for straight-crested Lamb waves, also applies to the circular-crested Lamb waves. Hence, the wave speed and group velocity of circular-crested are the same of straight-crested Lamb waves.

Figure 6a shows the circular-crested Lamb waves in a plate generated by a PWAS bonded on a composite plate. This snapshot is obtained using the MP-FEM simulation described in section 2.6. The attenuation of Lamb wave may be due to many different factors, the four most important ones are [46]: (a) Geometric spreading of the circular wave front; (b) Material damping due to internal energy dissipation; (c) Wave dispersion; (d) Dissipation into adjacent media for plates submerged in liquid. The geometric spreading describes the loss of amplitude due to the growing length of a wave front departing into all directions from a source as illustrated in the MP-FEM simulation of Figure 6a. Figure 6b shows how the signal received at different distance from the source decreases in amplitude due to geometric spreading. The effect of the geometric spreading slight magnitude attenuation with distance which may be confused with “damping”. The attenuation due to wave dispersion is a result of a frequency dependence of wave velocities. As a particular wave is composed of different frequencies, these frequency components travel differently in time leading to a dispersion of the


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f the geometrand elapsed tim

γ ω is the wa

amping coeffic

AS

g the apparenrough a plate w

tween the T-P

(a)

n of circularl received at

wave amplitur , and the o

rsus the distanignal received

ry spreading me, t , is give

ave speed. Fin

cient η can b

t amplitude ofwith material

( , )r tφ

PWAS and the

crested plat different dis

spreadi

ude with propother due to stnce, we propod versus the di

i.e. 1 r fiten by

ally we obtain

(12

be determined

R-PWAS

f the wave pacdamping, we

1 rA e er

η−=

e R-PWAS, A

e wave propastance from ting attenuatio

agation (i.e. wtructural damposed to curve istance.

t very well th

0r c t=

n

( )2α βω η+ =

d by curve fit

cket. In the cacan write

( )i t re ω γ−

is the magnitu

agation in a pthe source dion only;

wave attenuatiping, re η− . Ufitting these v

he simulated

0cη

tting the law

ase of single-fr

ude of the sign

(b)

plate; (b) 2Dsplays the ef

ion) consist osing the magnvalues. Figure

data. The rel

in Eq. (6) wit

frequency wav

gnal.

D MP-FEM daffect of the g

of two componnitude of the se 7 shows the

lation betwee

ith experimen

ve propagation

(6)

amping-less eometric

nents, one duesignal receivede magnitude o

n propagating

(7)

(8)

tally observed

n

)

e d f

g

)

)

d


5

4.5

4

3.5

3a.)vÇ 2 2.5

a 22

1

0 -

00 al

r (m)0.15

Figure 7: C

The essentialcoefficient, ηcan be easilyexperiments imuch as a smdiscuss, a cirspreading; andone alone andifficulty assomost of the taddress this, wguided wave

In our previodescribed in experimental MP-FEM resnot follow thamplitude. Idtone burst exc

To compare modes, the stFigure 9a shoand the strucmode which should apply wave packageperform the scan be expres

urve fitt ing oge

l parameter reη . The determy traced to anis very difficumall PWAS trcular wave

nd (b) materialnd the concurociated with ttime. This latwe developedsignals using

ous work [43Eq. (6) whichdata was obtults. Howeve

he law describdeally, Eq. (6)citation and w

and understantudy was alsoows the attenutural dampingfollows the cto a single f

es. Dispersionstudy in termsssed as

of the MP-FEeometric spre

equired to estimination of η n exponentialult and quite imtransducer. Bufront attenuatl damping. Alrrent effect ofthe multi-modtter aspect mad a methodolog

a curve-fitting

], we showedh combines gtained for η =r, we were subed by Eq. (6) should apply

wave packages

nd the effect o performed atuation curve fg with a wavecurve of only frequency excn induced attens of wave pac

EM results weading and w

4. RESULimate the Lamis straight for decay re η− . Hmpractical. Mut, in this casted due to a lthough our fof geometric spdal character oay be mitigategy for extractig approach.

d that the mageometry spre

15= . Moreovurprised by th6) [43]. This y to a single fs.

of geometric t the excitatiofitting for S0 e damping costructural dam

citation, but thnuation is harcket total ener

without dampiwith curve fit

LTS AND DImb wave attenrward in straigHowever, the

Much easier is se, the guided

combinationocus is interespreading mustof the Lamb wed by exploiting a value fo

agnitude of thading, 1 r ,er, this wave

he anomalous was maybe

frequency exc

spreading anon frequency omode which

oefficient 6η =mping, with ahis impracticard to observe argy instead of

ing. In this cting with (A

ISCUSSIONnuation due toght-crested gue generation othe generationd waves front of at least tst is to determt be also cons

waves where Sthe PWAS m

r the damping

he S0 mode t, with the strudamping coebehavior of thdue to the dicitation, but t

nd material daof 150 kHz wfollows the c6 . Figure 9b a wave dampial in experimand has been f maximum w

ase the atten

0.5259; 0η= =

N o material damided waves beof 1-D guidedn of guided wt is not straigtwo confound

mine the matersidered as illuS0 and A0 momode tuning pg coefficient η

tuned at 300 uctural dampifficient was inhe A0 mode tispersion effechis impractica

amping on cowhen both S0 urve with comshows the daing coefficienents using thelittle studied;

wave amplitud

nuation is onl0) .

mping is the wecause the wad Lamb wave

waves from a “ght but circulading effects: rial damping, ustrated in Eqodes coexist sproperties [36η from the 2-D

kHz decreasing, re η− ; a gin good agreemtuned at 90 kHcts that modial in experim

ombined the Sand A0 mode

mbined geomamping curve nt 14.5η = . Idhe tone burst

one alternativde. The wave

ly due to the

wave dampingave attenuationes in physica

“point” sourcear. As already(a) geometricthis cannot be

q. (6). Anothesimultaneously6]. In order toD propagation

ses as the lawgood fit to thement with ouHz, which didifies the wave

ments using the

S0 and the A0es are presentetry spreadingfitting for A0

deally, Eq. (6excitation andve would be topacket energy

g n al e, y c e r y o n

w e

ur d e e

0 t. g 0 ) d o y


1

0

2

( )t

tE x t dt∝ ∫ (9)

Where, ( )x t is the time signal received by the T-PWAS, 0 1andt t are the time window where the wave packet of interest is.

(a) (b)

Figure 8: Experimental curve fit ting at different distance at 150 kHz for: (a) the S0 mode (Geometry spreading + structural damping) with ( 0.02; 6)A η= = ; (b) the A0 mode (structural damping) with ( 0.06; 14.5)A η= = .

Figure 9a shows the attenuation curve fitting using the energy of the S0 mode wave packet which follows the curve with combined geometry spreading and the structural damping with a wave damping coefficient 18.138η = . Figure 9b shows the damping curve fitting for A0 mode which follows the curve with combined geometry spreading and the structural damping only structural damping, with a wave damping coefficient 16.008η = .

(a)

(b)

Figure 9: Experimental curve fit ting at different distance at 150 kHz for: (a) the S0 mode with( 0.5638; 18.138)A η= = ; (b) the A0 mode with ( 0.3645; 16.008)A η= = .

Knowing the wave damping coefficient η (using the magnitude and not the energy curve fitting), and the group velocity for a given frequency, we can plot Eq. (8) which gives the mass proportional damping α versus the stiffness proportional damping β . The difficulty appears when the guided wave propagation is multi-modal, i.e. at 150 kHz.

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

r (m)

Mag

nitu

de (V

)

Geometric spreadingStructural dampingGeometric spreading+Structural dampingExp. S0 at 150 kHz

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

r (m)M

agni

tude

(V)

Geometric spreadingStructural dampingGeometric spreading+Structural dampingExp. A0 at 150 kHz

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.5

1

1.5

2

2.5

3

3.5

r (m)

Ene

rgy

Geometric spreadingStructural dampingGeometric spreading+Structural dampingExp. S0 at 150 kHz

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.5

1

1.5

2

2.5

r (m)

Ene

rgy

Geometric spreadingStructural dampingGeometric spreading+Structural dampingExp. A0 at 150 kHz

Proc. of SPIE Vol. 8695 869525-10

6.E+04

5.E+04

4.E+04

1.0 *U4

0. E+00

1.E-07

R

Figure 10 shonot intersect,

Figure 10:

The MP-FEMcoefficient. Apeak as show2.5, both S0 simulation anproportional excitation frethan the S0 mHowever, forthe experimen

Figure 11b shPWAS placecorresponding, the MP-FEMFEM signal rto the scatteriR-PWAS as d

ows the α β−i.e. we canno

Model of α

M simulation wA 150kHz threwn on Figure 2

and A0 modnd experimentcoefficient β

equency of 15mode and its r this stiffnessnt value, i.e. th

hows the comed at 100 mg for the A0 mM signal for threceived betweing effect by described on t

β curves at 15t find a comm

versus β (R

was carried ouee-tone burst

2b was applieddes are presental electric sign

86.10β −= whi0 kHz, two gumagnitude is proportional he damping co

mparison betwmm from themode attenuatihe S0 and the een the S0 andthe other bon

the MP-FEM

50 kHz for S0 mon combinati

Rayleigh dam

ut on a CFRPmodulated by

d to the top sunt at this freqnal measured ich is correspouided wave msmaller than coefficient βoefficient is to

ween the MP-e T-PWAS wion as shown A0 modes ar

d the A0 modnded PWAS osnapshot on F

and A0 modeion of α and

ping) for the

P laminate desy a Hanning wurface of the Tquency. Figurat R-PWAS p

onding for thmodes are pres

the S0 magn86.10β −= , the

oo big to simu

-FEM simulatwith the stiffin Figure 10.

re in good agrde is different on the guided Figure 12.

e. It is apparenβ that simult

e CFRP lamin

scribed in thewindow with T-PWAS. Duere 11a showsplaced at 100 e S0 mode atsent, S0 and Aitude as descr

e MP-FEM maulate the exper

tion and expefness proportWith this stifeement with tthan the experwave propaga

nt in Figure 1taneously satis

nate at 150 kH

section 2.6 toa 20 Volts m

e to the tuning the comparimm from the

ttenuation as sA0. The A0 mribed on the tagnitude for thrimental value

erimental electional coefficffness proportithe experimenrimental signaation path bet

0, that the αsfies the A0 a

Hz for S0 an

o determine tmaximum ampg effect describison between T-PWAS witshown in Figu

mode is considtuning curve ithe A0 mode ie.

ctric signal mcient 2.1β =

ional coefficient signal. Howal. This signaltween the T-P

β− curves doand S0 modes.

nd A0 mode.

the attenuationplitude peak tobed on sectionthe MP-FEM

th the stiffnesure 10. At thederably slowein section 2.5is smaller than

measured at R80− which i

ent 82.10β −=wever, the MPl is maybe duePWAS and the

o .

n o n

M s e r

5. n

R-s 8

-e e

Proc. of SPIE Vol. 8695 869525-11

)3

)2

)1

.04

.03

.02

.01

o

.01

.02

_; a

SO

A A

AO

Time

AO

Time

-MP_FEM

_ E-24

s)

-MP-FEM

V'';,'0004-4

S)

_.p

Po.e.00P6.<

2.E-04

Figure 1

Using the enversus the recthe distance. well the simumodes respec

Geometric sptransducer. Inabsorption orthe S0 mode, the attenuatiodamping begi

11: Comparistransmitter

ergy of the Mceived distancThe curve of

ulated data whctively. These

preading and n the far fieldr conversion o at 40 mm the

on is around 4ins to dominat

son between r PWAS at 15

MP-FEM signce. Figure 13athe geometry

hen the wave dvalues are in

structural dad (beyond 50

of sound energe attenuation i.24 dB, i.e. strte the geometr

the experime50 kHz; (a) w

nal received asa and Figure 1

spreading, i.edamping coefrelatively goo

amping are thmm ) the at

gy to heat. Theis around 8.29ructural dampric spreading

(a)

(b)

ental and the with 0andα =

s shown on F3b show the ee. 1 r , and fficient is equaod agreement

he dominant ttenuation is de attenuation c9 dB, i.e. geomping. For the Ais around 4.48

MP-FEM rec86.10β −= ; (b

Figure 11b, wenergy of the the other due

al to 21.7η =with the expe

source of attdominated bycoefficient wimetry spreadinA0 mode, the t8 dB.

ceived signal) with 0aα =

we proposed todamped MP-F

e to structural 7674 and to ηrimental value

tenuation in ty the structurath units of dBng and structutransition dist

l at 100 mm 8and 2.10β −=

o curve fittingFEM signal redamping, i.e.

15.4752η = , es.

the near fieldal damping, mB/unit can be cural damping, tance at which

from the

g these valueeceived versu re η− , fit veryfor S0 and A0

d close to themay be due tocalculated. Foand at 60 mm

h the structura

s s y 0

e o r

m al

Proc. of SPIE Vol. 8695 869525-12

Figure 12: Snapshot of the guided Lamb wave propagation showing the scattering due to the bonded PWAS on

the guided wave propagation path between the T-PWAS and the R-PWAS.

(a)

(b)

Figure 13: MP-FEM curve fitting at different distance at 150 kHz for: (a) the S0 mode with( 0.7415; 21.7674)A η= = ; (b) the A0 mode with ( 0.3248; 15.4752)A η= = .

5. CONCLUSION This paper has presented numerical and experimental results on the use of guided wave for SHM in viscoelastic composite material. The challenge has been discussed to simulate with high accuracy guided wave propagation in composite materials. Guided wave simulation was conducted with the multi-physics finite element method approach in a multi-layer composite material. The simulated electrical signal was excited at a T-PWAS and measured at a R-PWAS using the MP-FEM capability. Relation between viscous damping and Rayleigh damping were presented and a discussion about wave attenuation due to material damping and geometry spreading have been led. The Rayleigh damping model was used to compute the wave damping coefficient for S0 and A0 mode. The multi-modal guided wave propagation was also examined by excitation at 150 kHz where both the A0 and S0 modes are present. We found that the reduction in energy for S0 and A0 modes packet obey a law that incorporates both geometry spreading and structural damping. It was found that the mass and/or stiffness proportional damping coefficient are different for each mode.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.5

1

1.5

2

2.5

3

3.5

4

r (m)

Ener

gy

Geometric spreadingStructural dampingGeometric spreading+Structural dampingMP-FEM S0 at 150 kHz

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.5

1

1.5

2

2.5

r (m)

Ene

rgy

Geometric spreadingStructural dampingGeometric spreading+Structural dampingMP-FEM A0 at 150 kHz

T-PWAS R-PWAS

No scattering (no PWAS on the propagation path)

Scattering due to the PWAS on the propagation path

Proc. of SPIE Vol. 8695 869525-13

Nonetheless, using these coefficients, we were able to simulate these two modes using our MP-FEM approach with high accuracy. Further research needs to be done to better understand the simulation of material damping in guided wave propagation in anisotropic composite materials and the determination of attenuation of dispersive waves for structural health monitoring application.

6. ACKNOLEDGMENTS This work was supported by Office of Naval Research through the Naval International Cooperative Opportunities in Science and Technology Program (Grant no. N00014-09-1-0364; Program sponsor Dr. Ignacio Perez). Dr. Nik Rajic from the Defence Science and Technology Organization is thankfully acknowledged for having provided the composite material.

REFERENCES

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Proc. of SPIE Vol. 8695 869525-14

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Proc. of SPIE Vol. 8695 869525-15

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Guided wave propagation in carbon composite laminate using