Lamb Wave in Composite

Experimental study of Lamb wave propagation in composite laminates

Lei Wang and F. G. Yuan*

Department of Mechanical and Aerospace Engineering, Campus box 7921, North Carolina State University, Raleigh, NC 27695-7921, USA

ABSTRACT

This paper focuses on the existence of higher-order Lamb wave modes that can be observed from piezoelectric sensors by the excitation of ultrasonic frequencies from piezoelectric actuators. Using three-dimensional (3-D) elasticity theory, the exact dispersion relations governed by transcendental equations are numerically solved for an infinite number of possible wave modes. For symmetric laminates, a robust method by imposing boundary conditions on mid-plane and top surface is developed to separate wave modes. Then both phase and group velocity dispersions of Lamb waves in composites are obtained. Meanwhile three characteristic wave curves including velocity, slowness, and wave curves are introduced to analyze the angular dependency of Lamb wave propagation at a given frequency. In the experiments, two surface-mounted piezoelectric actuators are operated corporately to excite either symmetric or anti-symmetric wave modes with narrow banded excitation signals, and a Gabor wavelet transform is used to extract group velocities from arrival times of Lamb wave received by a piezoelectric sensor. In comparison with the results from the theory and experiment, it is confirmed that the higher-order Lamb waves can be excited from piezoelectric actuators and the measured group velocities agree well with those from 3-D elasticity theory.

Keywords: Lamb waves; SH waves; Dispersion relation; Phase velocity; Group velocity; Composite; SHM

1. INTRODUCTION

The integrity and degradation of the composite structures have been traditionally evaluated by Nondestructive Evaluation (NDE)1,2, now being potentially assessed by Structural Health Monitoring (SHM)3, to assure the performance of these structures. Because the guided waves remain confined inside the structure, they can travel over long distances and even in the inaccessible areas, enabling the inspection of a large area with only a limited number of sensors. These properties make them well suited to the ultrasonic inspection of plate-like aircraft components, missile cases, pressure vessel, oil tanks, pipelines, etc. The dispersion relations for an elastic isotropic plate with infinite extent in plane strain state were first derived by Lamb4. With generalization the guided waves propagating along the plane of an elastic plate with traction-free boundaries are usually called Lamb waves. For the long-range damage detection techniques, in theory if the wavelengths of the excitation signals are significantly longer than the sizes of the constituents of composites (fiber diameters and spacing), each lamina can be treated as an equivalent homogeneous orthotropic or transversely isotropic material with the symmetry axis parallel to the fibers. Tauchert and Guzelsu5 measured scattering in boron/epoxy composites at ultrasonic frequencies at which the wavelength is small enough to produce scattering. For extensional waves, scattering began to show when the wavelength is of the same order of magnitude as the diameter of the fiber; however for flexural waves scattering appeared when the wavelength to fiber diameter ratio is about 40. Considering the composite laminates composed of macroscopically homogeneous layers, the wave interactions involve not only the reflection on the surfaces but the reflection and refraction between layers, manifested themselves in the form of resulting waves propagating along the plane of the plate. Besides the velocity being dependent on its direction of propagation, the other significant consequence of elastic anisotropy is the loss of pure wave modes for general propagation directions. The dependence of wave velocity on the direction of propagation implies that the direction of group velocity does not generally coincide with the wave vector (or wave normal). In addition, the distinction between wave mode types in composites is somewhat artificial since three distinct types of wave modes observed in isotropic plates are generally coupled. In practice, symmetric laminates are commonly used in the design of composite structures.

* [email protected]; phone 1 919 515 5947; fax 1 919 515 5934

Smart Structures and Materials 2006: Sensors and Smart Structures Technologies for Civil,Mechanical, and Aerospace Systems, edited by Masayoshi Tomizuka, et al., Proc. of SPIE Vol. 6174,

617442, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.677936

Proc. of SPIE Vol. 6174 617442-1

p\5

Figure 1: Diagram of Lamb waves propagating in a composite laminate

θ

φ

The Lamb waves for symmetric laminates can be separated into symmetric and anti-symmetric modes. For symmetric modes, one type is designated as quasi-extensional (qSn), where the dominant component of the polarization vector is along the propagation direction, and the other type is quasi-horizontal shear (qSH), where the polarization vector is mainly parallel to the plane of the plate. Similarly for the anti-symmetric types of wave modes, the quasi-flexural (qAn) and quasi-horizontal shear (qSH) are generated. For ease of notation, in the paper the prefix quasi is omitted unless stated otherwise. Experimental studies on the determination of dispersive curves have been focused mainly on isotropic plates. The most effective experimental setup to excite and receive multiple modes of Lamb waves is the laser source together with interferometer6,7 because of their superior sensitivity and wide band nature. Moreover the phase velocities of Lamb waves are normally measured by wedge transducer8-10. For SHM using guided waves the frequencies used are typically much lower than in conventional ultrasonic NDE methods, typically above few MHz. If piezoelectric actuators/sensors are used, they are often used in a frequency range below 1 MHz. In addition, the input signals often used are band limited (finite tone bursts or wave packets) to prevent the excitation of many higher frequency modes. Jeong11 investigated the group velocities for the lowest flexural waves in unidirectional and quasi-isotropic composites using Mindlin plate theory and Gabor wavelet transform. For fiber reinforced composites, another complexity of wave attenuation can be caused by the viscoelastic nature of the resin and by scattering from the fibers and other heterogeneities in the material at higher frequencies. In the next Section, the exact dispersion relations of Lamb waves in both composite lamina and laminate will be established from 3-D elasticity theory. Section 3 discusses the physical relations between phase velocity, slowness, and group velocity of Lamb waves; and the characteristic wave curves such as velocity, slowness, and wave curves are introduced to analyze the anisotropy and dispersion of Lamb waves. Then Section 4 investigates the experimental procedure of exciting/receiving Lamb waves in laminates using piezoelectric actuator and sensor, applies a signal processing technique based on Gabor wavelet transform to extract the group velocities arrival times of all wave modes, and compares experimental results with theoretical prediction. Finally, in Section 5 some conclusions and guidelines for the practice of NDE and SHM are drawn.

2. FORMULATION OF LAMB WAVES IN COMPOSITES BY 3-D ELASTICITY THEORY In this Section, a theoretical framework is developed first for displacement expression of plane harmonic waves using 3-D elasticity. In Section 2.1, focus is on the Lamb waves in a single lamina where a compact closed-form dispersion relation can be derived by separating symmetric and anti-symmetrical modes using trigonometric functions through the lamina thickness. Then a modified exponential form in the thickness direction is proposed for deriving the dispersion relation for a composite laminate, with special emphasis on the symmetric laminates. A Cartesian coordinate system is used with z axis normal to the mid-plane of a composite laminate spanned by x and y axes. Two outer surfaces of the laminate are at 2hz ±= . A packet of Lamb waves propagates in an arbitrary directionθ , which is defined counterclockwise relative to the x axis. Each layer of the composite laminate with an arbitrary orientation in the global coordinate system (x, y, z) is considered as a monoclinic material having x-y as a plane of symmetry, the stress-strain relations therefore take the following matrix form:


⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

xy

xz

yz

z

y

x

xy

xz

yz

z

y

x

CCCC

CC

CC

CCCC

CCCC

CCCC

γ

γ

γ

ε

ε

ε

τ

τ

τ

σ

σ

σ

66362616

5545

4544

36332313

26232212

16131211

00

0000

0000

00

00

00

(1)

When the global coordinate system (x, y, z) does not coincide with the principal material coordinate system (x’, y’, z) of each layer but makes an angle φ with the x axis shown in Fig. 1, the stiffness matrix ijC in (x, y, z) system can be

obtained from the lamina stiffness matrix ijC′ in (x’, y’, z) system by using a transformation matrix method12. The lamina is orthotropic or transversely isotropic with respect to the principal material axes in (x’, y’, z) and its lamina stiffness matrix ijC′ can be calculated from the lamina properties iE , ijν , and ijG 13. The linear engineering strain-displacement relations are

xx u,=ε , yy v,=ε , zz w,=ε , yzyz wv ,, +=γ , xzxz wu ,, +=γ , xyxy vu ,, +=γ (2) where subscript comma indicates partial differential; u , v , and w are the displacements in the x , y , and z directions, respectively. The equations of motion in the absence of body forces are governed by:

uzxzyxyxx &&ρττσ =++ ,,, (3a)

vzyzyyxxy &&ρτστ =++ ,,, (3b)

wzzyyzxxz &&ρσττ =++ ,,, (3c) where ρ is the mass density of the lamina, and dot denotes time derivative. The traction-free boundary conditions on the top and bottom surfaces are

0=== yzxzz ττσ , at 2hz ±= (4) Since the Lamb waves travel along the plane of a plate with traction-free boundaries, but are standing waves in the z direction of the plate. The wave motion may be expressed by superposition of plane harmonic waves. Each plane harmonic wave traveling in the direction of wave normal k is represented by

])[()}( ),( ),({} , ,{ tykxki yxezWzVzUwvu ω−+= (5)

where Tyx kk ] ,[=k and its magnitude is pyx ckkk /22 ω=+== k is the wave number.

Substituting Eqs. (5) and (2) in (1) gives the expressions of stresses in each layer:

])[(16131211 )]([ tykxki

xyyxxyxieVkUkCWiCVkCUkC ωσ −+++′−+= (6a)

])[(

26232212 )]([ tykxkixyyxy

yxieVkUkCWiCVkCUkC ωσ −+++′−+= (6b)

])[(

36332313 )]([ tykxkixyyxz

yxieVkUkCWiCVkCUkC ωσ −+++′−+= (6c)

])[(

4544 )]()([ tykxkixyyz

yxeWikUCWikVC ωτ −++′++′= (6d)

])[(

5545 )]()([ tykxkixyxz

yxeWikUCWikVC ωτ −++′++′= (6e)

])[(

66362616 )]([ tykxkixyyxxy

yxieVkUkCWiCVkCUkC ωτ −+++′−+= (6f) where the prime denotes the derivative with respect to z. Substituting Eq. (6) into Eq. (3), the equations of motion of each layer become


0])()[()

()2(

453655132

2666

122

1622

66162

114555

=′+++−++

++−+++′′−′′−

WkCCkCCiVkCkkC

kkCkCUkCkkCkCVCUC

yxyyx

yxxyyxx ρω (7a)

0])()[()2

(])([

4423453622

2226

266

2266612

2164445

=′+++−−++

+++++′′−′′−

WkCCkCCiVkCkkC

kCUkCkkCCkCVCUC

yxyyx

xyyxx

ρω (7b)

0)2(

])()[()])()[(22

44452

5533

4423453645365513

=−+++′′−

′+++−′+++−

WkCkkCkCWC

VkCCkCCiUkCCkCCi

yyxx

yxyx

ρω (7c)

2.1. Lamb waves in a composite lamina In an off-axis lamina, the solutions of Eq. (7) can be simply separated into symmetric and anti-symmetric wave modes, which render the analytical representation particularly simple:

zCWzBVzAUzCWzBVzAU

aaaaaa

ssssss

ξξξξξξ

cos,sin,sinsin,cos,cos

======

(8)

where ξ is a unknown variable to be determined later; moreover the subscripts s and a represent symmetric and anti-symmetric modes, respectively. First substituting the symmetric mode into the equations of motion, Eq. (7) may be expressed in a matrix form

0 2

332313

232

2212

13122

11

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−ΓΓΓ

Γ−ΓΓ

ΓΓ−Γ

s

s

s

C

B

A

ωρ

ωρ

ωρ

(9)

where the bar indicates complex conjugate. The elements in the above matrix defined by I)Γ 2ρω−( are listed as:

255

26616

21111 2 ξCkCkkCkC yyxx +++=Γ (10a)

245

2266612

21612 )( ξCkCkkCCkC yyxx ++++=Γ (10b)

ξ)])()[( 4536551313 yx kCCkCCi +++−=Γ (10c)

244

22226

26622 2 ξCkCkkCkC yyxx +++=Γ (10d)

ξ)])()[( 4423453623 yx kCCkCCi +++−=Γ (10e)

233

24445

25533 2 ξCkCkkCkC yyxx +++=Γ (10f)

where I is a 3×3 identity matrix. Eq. (9) is a standard linear eignevalue problem of a Hermitian matrixΓ . It can be shown if the matrix is positive definite that the eigenvelues 2ρω of Γ are positive and nonzero, furthermore the right eigenvectors follow the orthogonality

relation12. Following the same procedure for the anti-symmetric mode, the resulting matrix becomes I)Γ 2ρω−( . If

Hermitian matrix Γ is positive definite and so isΓ . It can be readily proved that the eigenvalues of the symmetric mode are also the eigenvalues of the anti-symmetric mode. For nontrivial solutions of sA , sB , and sC in Eq. (9), vanishing the determinant of the 33× matrix

I)Γ 2ρω−( yields the following sixth-order polynomial in ξ :

032

24

16 =+++ αξαξαξ (11)


where αi (i = 1, 2, 3) are real-valued coefficients of Cij, k, and 2ωρ . If variable ξ is redefined as ξk , the

coefficients αi are functions of Cij, θ , and 2pcρ . Roots of this equation can be obtained explicitly from known

formulas since the equation can be reduced to a cubic polynomial in terms of 2ξ .

In general there are three positive, nonzero, and discrete jξ (j = 1, 2, 3). For each jξ in symmetric modes, Bs and Cs related to symmetric modes can be expressed in terms of As via Eq. (9) as

sss RAAB ≡ΓΓ−−ΓΓΓΓ−Γ−Γ

=2312

22213

1312232

11

)()(ωρ

ωρ (12a)

sss iSAAC ≡ΓΓ−−ΓΓ

−Γ−Γ−Γ=

23122

2213

222

211

212

)())((

ωρωρωρ

(12b)

and similarly for anti-symmetric modes, Ba = RAa and Ca = -iSAa. With the above equations, the polarization displacement vectors are determined from the three roots. Consequently, the general solution of Eq. (8) is

∑

∑

=

=

−=

=

3

1

3

1

}cos,sin,{sin},,{

}sin,cos,{cos},,{

jjjjjjajaaa

jjjjjjsjsss

ziSzRzAWVU

ziSzRzAWVU

ξξξ

ξξξ (13)

Substituting of Eq. (13) in Eq. (6) and considering Eq. (4), zσ , yzτ , and xzτ are reduced for the symmetric and anti-symmetric modes respectively as

0)cos

sin ,

cos

sin ,

sin

cos() , ,(

3

13212 =

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= ∑=

=j

jj

j

jj

j

jj

j

jhzxzyzz Az

zH

z

zH

z

zH

ξ

ξ

ξ

ξ

ξ

ξττσ (14)

where

)(363323131 jxyjjjyxj RkkCSCRkCkCH ++++= ξ

(15a)

)()( 45442 jxjjyjjj SkCSkRCH +++= ξξ

(15b)

)()( 55453 jxjjyjjj SkCSkRCH +++= ξξ

(15c) The existence of a nontrivial solution of Eq. (14) leads to closed-form dispersion relations as

0)2(tan)()2(tan)()2tan()(

33122322113

2332131231213223332211

=+−++−++−

ϕξϕξϕξ

hHHHHHhHHHHHhHHHHH

(16)

where 0=ϕ and 2π represent anti-symmetric and symmetric Lamb wave modes, respectively. Eq. (16) is a transcendental equation implicitly relating ω to k. For a fixed θ , a numerical iterative root-finding method is employed to compute the admissible ω for a range of k values, leading to dispersion relations of Lamb wave modes in the direction of propagation. Furthermore in general the frequency ω of each mode is single-valued function of k. 2.2. Lamb waves in a composite laminate In formulating Lamb waves in a laminate, the interfaces between layers are assumed to be perfectly bonded. The displacement components of each layer in the z axis Eq. (8) needs to be modified in exponential forms to accommodate the inhomogeneity of the multi-layered laminates.

ziAeU ξ= , ziBeV ξ= , ziiCeW ξ−= (17) Substituting these expressions into the equations of motion, Eq. (7) may be rearranged in a matrix form

0},,{][ 2 =− CBAIΓ ρω (18)


The nontrivial solution for A, B, and C yields the sixth-order polynomial in terms of ξ shown in Eq. (11). The six roots for ξ can be arranged in three pairs as jj ξξ −=+1 , )5,3,1( =j . For each jξ , B and C can be expressed in terms of

A via Eq. (18) as RAB ≡ and SAC −≡ . Consequently, the general solution of Eq. (17) in each lamina is

∑=

−+=6

1

])[( },,1{},,{j

zikjjj

tykxki jyx eSRAeWVU ξω (19)

The interlaminar stress components zσ , yzτ and xzτ in each lamina may be expressed as

∑=

−+=6

1321

])[( } , ,{} , ,{j

zikjjjj

tykxkixzyzz

jyx eHHHAike ξωττσ (20)

Generally, there are two methods, namely transfer matrix method and assemble matrix method, for obtaining the dispersion relations in laminates14,15. Although the procedures of these two methods seem different, they are identical in principle by both satisfying traction-free boundary conditions on the outer surfaces of the laminate and continuity of interface conditions between two adjacent laminas in different manner. Both methods can calculate dispersion curves in a general laminate with an arbitrary stacking sequence. Using Eq. (17), it may be observed that symmetric and anti-symmetric wave modes in general laminates can not be decoupled. However, in designing the composite structures, symmetric laminates are practically used. A robust method is proposed to separate the two types of wave modes by imposing boundary conditions at both top and mid-plane surface. Traction-free boundary conditions on the top surface of the laminate are given by

0} , ,{ 2 ==hzxzyzz ττσ (31)

Because of the symmetric geometry and symmetric material property of the laminate, only half of the laminate needs to be considered and then the following conditions on the stress and displacement components at the mid-plane for symmetric modes are imposed

0} , ,{ 0 ==zxzyzw ττ (32) Likewise, the boundary conditions of anti-symmetric modes at the mid-plane are

0} , ,{ 0 ==zzvu σ (33) By imposing displacement and stress continuity conditions along the interfaces of half layup of an N-layered laminate, total 3N equations are constructed if the assemble matrix method is used. Then set the determinant of the 3N equations to zero, and numerically solve the resulting transcendental equation for the dispersion relations of Lamb waves in symmetric laminates.

3. VELOCITY DISPERSIONS AND CHARACTERISTIC WAVE CURVES

The dispersion relation between ω and k can be symbolically represented by an implicit functional form 0),( =kωG , or 0),( =θω,kG . It is assumed that this relation may be explicitly solved in the form of real roots of )(kW=ω , or

),( θω kW= whose solutions correspond to different wave modes. For plane waves, the phase velocity vector is

defined as k)/( 2kp ω=c and thus its magnitude is kcp ω= . A curve generated by all choices of k from the origin for cp at a given frequency is called velocity curve. The radius vectors of velocity curves in the direction of a given k represent the admissible phase velocities of different wave modes. Similarly by defining a slowness vector s = k/ω, the slowness curve can be simply formed from the velocity curve by geometric inversion, i.e., by the mapping through reciprocal radius. Thus, the slowness vector has the same direction as the phase velocity vector. It is worth of noting that in isotropic materials the phase velocity depends only on the magnitude of the wave vector k; while in anisotropic materials the phase velocity depends on the wave vector k, its magnitude and direction in which the wave propagates. The group velocity is defined by k∂∂== /grad WWkgc . If the closed form of implicit function G can be obtained, the group velocity may be also calculated by


0 0

ω∂∂

∂∂−=GG kc g (34)

In a polar coordinate system, Wkgrad has a radial component k∂∂ /W in the direction of k and an angular

component θ∂∂ k/W perpendicular to k. Using coordinate transformation, the group velocity in a Cartesian coordinate can be attained as

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∂∂∂

∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

θθθ

θθ

k

kc

c

gy

gx

W

W

cossin

sincos (35)

where the subscripts x and y represent the components in x and y axes, respectively. The magnitude of group velocity gc and the angle gθ from the x axis are given by

22gygxg ccc += and

gx

gyg c

c1tan−=θ (36)

The skew angle ϑ (or steering angle16) is defined as θθϑ −= g (37)

The locus of group (ray) velocity vector along all choices of cg from the origin at a given frequency is referred to as wave curve (or wave front curve). It is worth of noting that the radius vector joining the source point to a point on a wave curve represents the distance traveled by the elastic disturbance in unit time. The wave curve therefore gives the locus of wave front, at a unit time, by the disturbance emitted by a point source acting through the origin at time t = 0. Thus wave curves are of great importance in damage detection of SHM or NDE17. An important geometric relation between slowness curve and group velocity direction is introduced for computing wave curves. The dispersion relation of each Lamb wave mode can be expressed as an explicit function of ),( θkW which may be viewed as a conical-like surface in 3-D domain. Furthermore, the slowness curve is geometrically the level surface of ),( θkW at 0),( ωθ =kW . Differentiating both sides of the equation with respect to θ yields

0=∂∂

+∂

∂θθWW

ddk

k (38)

In polar coordinates, the tangent vector of slowness curve 0),( ωθ =kW is proportional

to ),( ⊥ll kddkθ

and group velocity vector in polar

coordinates is ( lk∂

∂W, ⊥∂

∂ lθkW

) where l and

⊥l are the unit vector along and perpendicular to the k direction respectively. It can be concluded from Eq. (38) that the group velocity vector cg is perpendicular to the tangent vector of slowness curve; that is, the group velocity vector cg is parallel to the normal direction of slowness curve as shown in Fig. 2(a). Similarly it can be proved that the wave normal k is parallel to the normal direction of wave curve shown in Fig. 2(b). Based on Eq. (38), Eq. (35) can be rearranged as

gc

θ

ϑ

ωk

gθ

Figure 2: Schematic of geometric relation between wave vector and group velocity vector: (a) slowness curve, (b) wavefront curve

(a) (b)

gc

ϑ

gθ

θ


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∂∂

−

∂∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

θθθ

θθ

ddk

kk

kc

c

gy

gx

W

W

cossin

sincos (39)

Although group velocity expressions in Eqs. (35) and (39) are physically equivalent, they are suitable for different numerical implementations: Eq. (35) should be employed to compute group velocity dispersions along a given wave propagation direction; while Eq. (39) is suitable for calculating wave curves at a given frequency. Note that the geometric relation above is also valid for bulk (non-dispersive) waves. In addition, the polar reciprocal of the slowness curve is the wave curve (i.e., 1=⋅ gcs ) and ϑcosgp cc = 18. However, these relations break down for Lamb waves in lamina and laminates because of the dispersive behavior16.

4. EXPERIMENTAL RESULTS

In this section, the dispersive and anisotropic behavior of Lamb wave propagation in two laminates will be obtained from experiments, and then compared with the theoretical results presented in the previous section. The composite material used in this study is AS4/3502 graphite/epoxy as shown in Table 1. Two laminates [+456/-456]s and [+45/-45/0/90]s are used in the tests and their dimensions are listed in Table 2. Table 1 Material properties of AS4/3502 composite lamina Table 2 Properties of two AS4/3502 laminates

E1 (GPa)

E2 (GPa)

E3 (GPa)

G12 (GPa)

G13 (GPa)

G23 (GPa) ν12 ν13 ν23

ρ (kg/m3)

127.6 11.3 11.3 5.97 5.97 3.75 0.3 0.3 0.34 1578

A pair of lead zirconium titanate (PZT) disks (Navy Type II, PKI-502) serving as actuator is mounted on the opposite side of top and bottom surfaces of the laminate. The diameter and thickness of PZT disks are 6.4 mm and 1.57 mm, respectively. A setup schematic of PZT actuators exciting S and A modes respectively is illustrated in Fig. 3, where PZT actuators are perfectly bonded by Loctite® Extra Time Epoxy and the vertical arrows denote polarity directions and horizontal arrows indicate deformations of PZTs. A piezoelectric sensor (PAC Micro-80) is chosen to receive the response waves due to its wide band and small size (10 mm diameter). Both “induced strain” piezoelectric actuator and sensors utilize the d31 mode to generate and receive stress waves19. Micro-80 is temporally bonded on the laminate surface at 10 cm away from the actuators by 2211 silicone compound vacuum grease coupler. Since the grease coupler is removable, it is convenient to measure the Lamb wave signals at different directions of wave propagation. According to the calibration certificate of Micro-80, the sensor is sensitive in a wider frequency range, typically from 100 kHz up to 1 MHz.

Fig. 4 displays the diagram and actual photo of experimental setup. An Agilent 3220A function generator outputs a five-peaked tone burst signal. Then the signal is sent to TDS420A oscilloscope and K-H7602 amplifier as well, and the peak-to-peak voltage of the excitation signal from the amplifier is kept at 40 V. Then the response signal collected by the sensor is displayed and stored in the digital oscilloscope, whose sampling rate is set to 5 MHz and storage is set to 1000

Stacking sequence Dimension

I [+456/-456]s 46×43×3 mm3

II [+45/-45/0/90]s 41×81×1 mm3

(a)

(b)

Figure 3: Schematic of PZT actuators: (a) symmetric mode excitation, (b) anti-symmetric mode excitation where vertical arrows denote polarity directions and horizontal arrows indicate deformation of PZTs


fl2ObIB

bC

K-H GObOM6L swbiiiei

LD2 t5OV D!02c!IIo2cobe

Hb 325ovEflUCf!OU eueLsfOL

VC{r19O1

26U20L

cowbo2l

.1

2005'1r1

data points for each test. Finally, a computer obtains the collected data via GPIB interface bus and runs Gabor wavelet transform to extract group velocities from arrival times of Lamb modes.

The function generator generates a transient N-peaked tone burst input voltage signal governed by

tfN

tffNtHtHPtV cp

ccpin ππ 2sin)2cos1(])()([)( −−−= (40)

where peak number Np = 5, constant P = 10.25 is signal intensity, fc is the central (dominant) frequency and H(t) is Heaviside step function. It can be found through frequency domain analysis that the frequency components for this excitation signal are mainly concentrated in a small range around the central frequency fc, thus the dispersive effect can be significantly reduced. Furthermore, the frequency band width of the input signal is proportional to fc/Np. Although frequency resolution can be improved by increasing the peak number Np of excitation signal, the time resolution worsens because of Heisenberg’s Uncertainty Principle20. In the tests of group velocity dispersions, the sensor location is fixed (i.e, the direction of wave propagation is given) and fc is varied from 50 kHz to 1 MHz in increments of 50 kHz. This directly corresponds to the time duration from 5 µs to 100 µs. For measuring wave curves for a given fc, the sensor locations are varied so that the actuator-to-sensor orientation changes from 0º (x-direction) to 90º (y-direction) with 10º increment, varying the direction of wave propagation. Note that the distance between sensor and actuator maintains 10 cm for all tests, except at frequency 50 kHz where the distance is 20 cm. In order to enhance the contrast of the arriving wave packets and to increase the precision regarding their arrival times, Gabor wavelet transform is introduced to process the received signals of Lamb waves. The Gabor wavelet transform has been demonstrated as a post signal processing technique for analyzing dispersive waves in SHM11,21-22. The theory and physical interpretation of Gabor wavelet transform are not repeated here. Kishimoto et al.21 have shown that by using the peak of the magnitude of wavelet coefficients the arrival times of the group velocity at each local frequency can be extracted. Accordingly, the procedure of extracting arrival times by Gabor wavelet transform is listed as follows: (1) computing wavelet coefficients of the response signal in time-scale domain, (2) picking up a time slice from the magnitude of wavelet coefficients at the local frequency equal to the central frequency of excitation, (3) storing the time instants of each peak in the time slice, which are associated with arrival times of Lamb modes. The group velocity can be easily attained by dividing the propagation length (actuator-to-sensor distance) by the arrival time.

(a)

(b)

Figure 4: Experimental setup for the measurement of Lamb waves in composites: (a) diagram, (b) actual setup


0 200 400 600 800 10000

1

2

3

4

5

6

7

8

f (kHz)

c g (km

/s)

SH0

S0

S1

SH2

S2

3−D elasticity theoryExperimental results

(a) cg of S modes

0 200 400 600 800 10000

1

2

3

4

5

6

7

8

f (kHz)

c g (km

/s)

A0

A1

SH1

A2 A

3


(b) cg of A modes

Figure 5: Theoretical and experimental results of group velocity dispersion curves

traveling along θ = 30° in the laminate [+456/-456]s

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

cgx

(km/s)

c gy (

km/s

)

SH0

S0


(a) Wave curves of S modes

0 1 2 3 4 50

1

2

3

4

5

cgx

(km/s)

c gy (

km/s

)

A0

SH1

SH1

A1


(b) Wave curves of A modes

Figure 6: Theoretical and experimental results of wave curves in the laminate [+456/-456]s at f = 413 kHz (ωh/cT = 4)

Figs. 5 and 6 show compared results between theoretical prediction and experimental measurement of Lamb waves

in the relatively thick laminate [+456/-456]s. It can be seen that the exact solutions match well with the experimental results for both A and S modes. Even the higher modes such as S1 and A1 can be detected in experiment. While, it is hard to distinct SH0 and S0 modes at very low frequency range (50 kHz ~150 kHz) as shown in Fig. 5(a), because the arrival time difference of the two modes is very small (typically less than 10 µs), and the two modes have already appeared even before the end of the excitation duration. Furthermore, in Fig. 5(a) the exact solution of S2 modes does not agree with the experimental results. The disagreement may result from larger scattering signals from the heterogeneities at high frequency range for the lower wave modes than those from the S2 mode itself. The cusps of the SH0 and SH1 wave curves are observed in these measurements. Moreover, the dispersion of A0 mode in the laminates is weaker beyond 100 kHz as shown in Fig. 5 (b) and this feature is desirable for SHM since the group velocity dispersive curve of A0 is primarily constant especially for the frequency.

Proc. of SPIE Vol. 6174 617442-10

0 200 400 600 800 10000

1

2

3

4

5

6

7

8

f (kHz)

c g (km

/s)

SH0

S0


(a) cg of S modes

0 200 400 600 800 10000

1

2

3

4

5

f (kHz)

c g (km

/s)

A0

SH1

A1


(b) cg of A modes

Figure 7: Theoretical and experimental results of group velocity dispersion curves

traveling along θ = 45° in the laminate [+45/-45/0/90]s

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

cgx

(km/s)

c gy (

km/s

)

SH0

S0


(a) Wave curves of S modes

0 0.5 1 1.5 20

0.5

1

1.5

2

cgx

(km/s)

c gy (

km/s

)

A0


(b) Wave curve of A mode

Figure 8: Theoretical and experimental results of wave curves in the laminate

[+45/-45/0/90]s at f = 500 kHz (ωh/cT = 1.78)

The compared results of Lamb waves in quasi-isotropic laminate [+45/-45/0/90]s are displayed in Figs. 7 and 8. Because the frequency is chosen at 1.78Th cω = which is below the cut-off frequencies of A1 and S1 modes, only the fundamental modes (A0, S0, and SH0) exist. The angular dependence of Lamb waves in the laminate [+45/-45/0/90]s becomes weaker because of quasi-isotropic layup. For the excitation frequencies up to 1 MHz, the eight-layered thinner composite only exhibits fundamental guided waves (SH0, S0, and A0) propagating in the laminate. It can be also found that the exact solutions make good agreements with the experimental measurements for both group velocity dispersions and wave curves. The discrepancy of SH0 and S0 modes in Fig. 7 (a) at very low frequency may result from the same reason as mentioned for laminate [+456/-456]s. The weak dispersion of A0 can also be observed in Fig. 7 (b) when the frequency is higher than 200 kHz.

5. CONCLUSIONS

Exact solutions of Lamb waves in a lamina are established from 3-D elasticity theory, and then they are extended to a general laminate with an arbitrary layup. For symmetric laminates, a robust method by imposing boundary conditions on both mid-plane and top surfaces is developed to decouple the wave modes. Then the physical relations between phase, slowness, and group velocity are discussed, and the characteristic wave curves are introduced to analyze the angular dependence of Lamb wave propagation in composites. In experiments, two piezoelectric actuators operate corporately to excite pure S or A wave modes, and Gabor wavelet transform is adopted to obtain the group velocities of Lamb waves. The induced strain piezoelectric actuator/sensor can capture all the wave modes, not like interferometer only S and A

Proc. of SPIE Vol. 6174 617442-11

modes can be detected16. From the compared results between theoretical predictions and experimental measurements, it can be seen that the proposed methods effectively predict the dispersive and anisotropic behavior of Lamb waves in laminates. It is also suggested that A0 mode Lamb wave is more suitable for SHM, the higher-order Lamb waves can be excited from piezoelectric actuators and the measured group velocities agree well with those from 3-D elasticity theory. Future studies will apply the proposed methods together with piezo-excitation/collection technique to Lamb-wave-based SHM in composites. Due to the deficiency of intensive computation using 3-D elasticity theory, future study may include developing a new higher-order plate theory to significantly reduce computational cost and more accurately approximate the exact solutions. Additionally a wideband laser source and interferometers can be considered to improve experimental accuracy.

ACKNOWLEDGEMENT

This research is supported by the Sensors and Sensor Networks Program from the National Science Foundation (Grant No. CMS-0329878). The authors would like to thank NASA Langley Research Center providing two composite laminates.

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Lamb Wave in Composite