THE 19TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS

1 Introduction

In recent years advanced composites made out of

polymer matrix combined with glass or carbon fiber

reinforcements have become one of the materials of

choice for many aerospace structural components.

This shift from metallic materials is primarily due to

the tailorability of composite parts for aerospace

applications, which are lighter, stronger, and resist

corrosion and fatigue damage better than traditional

aluminum alloys. Despite enhancements in terms of

specific strength and stiffness, susceptibility to

hidden and Barely Visible Impact Damage (BVID)

in composites is still a major point of concern. These

damages may occur during manufacturing,

maintenance, and in service and generally hide

below the surface, where visual inspections are

limited. The delays encountered during the

development of Boeing’s 787 Dreamliner and

finding of cracks in the wing ribs of the A380 have

highlighted problems with such hidden damages.

If undetected these types of damage may grow

during service and may lead to catastrophic failure

of the structure. To avoid such catastrophic failures,

composite structures have to be inspected at regular

intervals and repaired when damage is detected as a

part of an interval based maintenance schedule [1].

The current maintenance techniques make use of the

Non-Destructive Inspection (NDI) technology for

damage detection and identification on aerospace

structures. Certified technicians are trained on

specific NDI techniques and regularly re-certified to

be qualified to perform inspections of aerospace

structures such as: Ultrasonic Testing (UT), Acoustic

Emission Testing (AE), Electromagnetic Testing

(ET), Leak Testing (LT), Liquid Penetrant Testing

(PT), Magnetic Particle Testing (MT), Radiographic

Testing (RT), Thermal Infrared Testing (TIR) and

Visual Testing (VT). However, NDI requires a high

level of human interaction and it’s intended for

local/focused inspection and in most cases requires

access to the area of interest [2]. In some cases the

word in-situ NDI is being used as a link between

traditional NDI and the new upcoming Structural

Health Monitoring (SHM) techniques. In contrast to

SHM, NDI usually requires direct access to the area

of interest during service of the aircraft, NDI

techniques are used to inspect components and

structures at specific time intervals [3]. These

inspections increase the life cycle cost in two ways,

direct costs associated with the inspections, and

indirect cost induced by having the component

temporarily taken out of service. Therefore, a

reliable and low-cost approach for damage detection

in composites is needed to ensure that the total life

cycle cost does not become a limiting factor for their

use. [4]

The overall lifecycle cost could be reduced with the

use of a Structural Health Monitoring (SHM)

system as a part of a Condition Based Maintenance

(CBM) approach. In a CBM approach, the aircraft is

taken out of service only when potential damage is

detected and verified by the SHM system. In a

typical SHM system, the structure is monitored

using on-board sensors continuously or at discrete

intervals. The data gathered by the sensors can be

processed on-board or sent to the ground control

station for evaluation. The evaluation results are

used to identify damage occurrences and inform the

operator and maintenance personnel about the

location and severity of the damage. This

information in turn can be used to decide the proper

maintenance actions.

There are several methods for detecting and

analyzing damages in composites using SHM

STRUCTURAL HEALTH MONITORING (SHM) OF COMPOSITE

AEROSPACE STRUCTURES USING LAMB WAVES

S. Pant1*, J. Laliberte1, M. Martinez2 1 Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Canada

2 Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands * Corresponding author (shashankpant@cmail.carleton.ca)

Keywords: Structural Health Monitoring, Composite, Acoustic-Ultrasonic, Lamb Waves,

Piezoelectric, Damage Detection, Non-Destructive Evaluation,

systems. One of such techniques is by using

acoustic-ultrasonic Lamb waves, which are

generated and acquired by piezoelectric

actuators/sensors installed on-board the structure.

For the design of a SHM system based on acoustic-

ultrasonic Lamb waves, the understanding of the

underlying physics behind Lamb waves and its

propagation within the host material is essential. The

propagation characteristic of Lamb waves is given in

the form of dispersion curves, illustrating the plate-

mode phase and group velocity as a function of the

frequency-thickness product.

This paper presents the Lamb wave equations for a

monoclinic composite laminate based on 3D linear

elasticity and partial wave techniques for generating

the dispersion curve. The analytical dispersion

curves, which are generated using MatLab-based

software, are compared with the experimental data

for verification.

2 Acoustic-Ultrasonic Lamb Wave

2.1 Lamb Wave Theory

Lamb waves are ultrasonic guided waves that travel

between two parallel free surfaces and are

superposition of longitudinal waves (P-waves) and

shear waves (S-waves). Lamb waves are highly

susceptible to interference on the propagation path

and can travel long distances even in materials with

high attenuation ratio such as composites. Damages

can be detected using the difference between the

phase/group velocity of a Lamb wave on damaged

and un-damaged baseline specimens. The analysis is

performed by measuring the time of flight between

two sensors with a known distance, while observing

the disturbances in the waves between the sensor

and the Lamb wave generator. [5]

Lamb waves exist simultaneously in two modes,

which are symmetric and anti-symmetric and

propagate independently of each other. The motions

of such symmetric and anti-symmetric Lamb waves

are shown in Fig. 1. For a finite plate thickness at

any acoustic frequency, there exist infinite numbers

of such symmetrical and anti-symmetrical Lamb

waves, differing from one another by their phase and

group velocities. Phase velocity is the rate at which

the individual phase of the wave propagates,

whereas group velocity is the rate at which the

overall envelop of the wave propagates. The

propagation characteristic of Lamb waves is given in

the form of dispersion curves, illustrating the plate-

mode phase and group velocity as a function of the

frequency-thickness product (Fig. 2).

Fig. 2: Lamb wave dispersion curve for Al 2024-T6

Propagation of Lamb waves within isotropic

medium is well defined, which is not the case for

composites. The wave propagation in composite is

complex due to anisotropy and their strongly

attenuative and dispersive nature [6]. Parameters of

composite materials such as fiber volume fractions,

layup, type of matrix and reinforcements, strongly

Fig. 1: Propagation of (a) symmetric and (b)

anti-symmetric Lamb wave modes

3

influence the velocity of propagating waves. Waves

in composite plates propagate in each direction with

different velocities; also the shape of the wave front

changes with the frequency [7]. For simplification,

the composite laminates are assumed to have

orthotropic or higher degrees of symmetry in order

to generate dispersion curves. This may not be true,

if the actuators and sensors in an orthotropic or

higher symmetry laminate are installed in a non-

principle direction or the layup is symmetric but not

balanced.

This section provides the derivation of Lamb waves

in a composite laminate for monoclinic symmetry,

which is based on the partial wave techniques. First

the stress-strain relationship for the composite

laminate is provided which is followed by the

derivation of Lamb wave equations.

2.2 Stress-Strain Relationship for Composite

Laminate

Stress strain relationship in the Cartesian co-ordinate

system for an anisotropic solid medium assuming

linear elastic behavior can be written in the tensor

form as:

ij ijkl kl

ij ijkl kl

c

s

(1)

Where, ijklc and ijkls are the stiffness and compliance

tensor respectively; both containing material elastic

constants.

Strain in terms of displacement is given by:

1

2

jiij

j i

uu

x x

(2)

The general equation of motion without considering

body forces can be written as: 2

2

ij i

j

u

x t (3)

Using Eqn. 1 and Eqn. 2; Eqn. 3 can be rewritten in

terms of displacement as: 2 2

2

k iijkl

j l

u uc

x x t (4)

In composite materials, the fibers are oriented at

desired angles for optimal performance. Due to

different fiber orientations, the material can behave

differently in various directions. Depending on how

the fibers are orientated (planes of symmetry);

composite materials can be characterized as triclinic,

monoclinic, orthotropic, transversely isotropic, and

isotropic. The general co-ordinate system used to

describe the planes of symmetry is shown in Fig. 3,

where 3x represents the thickness direction.

Fig. 3: Coordinate system for composite laminate

With respect to Fig. 3, the transformation of the

stiffness tensor ijklc from the local *

ix to global ix

coordinate can be performed by orthogonal

transformation assuming that the rotation takes place

along the thickness axis ( *

3 3;x x ), positive counter-

clockwise. Ref [8] is used for finding the

transformation relationship provided below.

The stiffness matrix can be transformed from local

to global system by:

1*c T c T

(5)

Similarly, the compliance matrix can be transformed

by:

1*s T s T

(6)

Where Ts

and Te

are the stress and strain 6 by 6

transformation matrices as a function of sine and

cosine [8] of the rotation angle shown in Fig. 3.

The stress strain relationship for a material with

monoclinic symmetry can be expanded as:

11 11 12 13 16 11

22 12 22 23 26 22

33 13 23 33 36 33

23 44 45 23

45 5513 13

16 26 36 6612 12

0 0

0 0

0 0

0 0 0 0 2

0 0 0 0 2

0 0 2

c c c c

c c c c

c c c c

c c

c c

c c c c

(7)

2.3 Lamb Wave Equations for Lamina

The derivation provided here is based on the partial

wave techniques. In this technique, the principle of

superposition of three upward and three downward

travelling plane waves is assumed in order to satisfy

the associated boundary conditions. The bounded

upper and lower surface reflects the wave and the

combination of these reflections going towards the

upper or lower interfaces results in the propagating

guided waves (Fig. 4).

With reference to Fig. 4, six waves for each layers

contain two quasi-longitudinal (L+/-), two quasi-

shear vertical (SV+/-) and two quasi-shear

horizontal (SH+/-). Positive and negative sign

denotes that the wave is travelling down or up

respectively.

For the derivation of Lamb wave equations, consider

a plane wave travelling through the plate shown in

Fig. 5, for which the displacement ( iu ) is assumed to

be:

1 1 2 2 3 3( )i i

i k x k x k x tu U e

(8)

Where, iU = displacement amplitude, ik = wave

number, ix = direction, = angular frequency, and

t = time.

The wavevector k , which defines the travelling

direction of the wave and is given by:

1 2 3 12 12 12 12 3[ , , ] [ cos , sin , ]T Tk k k k k k k (9)

Where, 12k = wave vector along 1 2x x plane and

12 = angle with respect to 1x , positive counter

clockwise (Fig. 5).

Magnitude of 12k is given by:

2 2

12 12 1 2

2| |

p

k k k kc

(10)

Where, pc = phase velocity and = wavelength,

Expanding the general partial differential equation

(Eqn. 4), in terms of displacements iu (Eqn. 8), for a

monoclinic material (Eqn. 7) gives:

For 1x direction:

2 2 2 2

1 1 1 1

11 66 55 162 2 2

1 2 3 1 2

2 2 2 2

2 2 2 2

16 26 45 12 662 2 2

1 2 3 1 2

2 2 2

3 3 1

13 55 36 45 2

1 3 2 3

2u u u u

c c c cx x x x x

u u u uc c c c c

x x x x x

u u uc c c c

x x x x t

(11)

For 2x direction:

Fig. 4: Upward and downward travelling partial waves Fig. 5: Thin monoclinic plate with 2h thickness

5

2 2 2 2

1 1 1 1

16 26 45 12 662 2 2

1 2 3 1 2

2 2 2 2

2 2 2 2

66 22 44 262 2 2

1 2 3 1 2

2 2 2

3 3 2

36 45 23 44 2

1 3 2 3

2

u u u uc c c c c

x x x x x

u u u uc c c c

x x x x x

u u uc c c c

x x x x t

(12)

For 3x direction:

2 2

1 1

13 55 36 45

1 3 2 3

22 2

32 2

36 45 23 44 55 2

1 3 2 3 1

2 2 2 2

3 3 3 3

44 33 452 2 2

2 3 1 2

2

u uc c c c

x x x x

uu uc c c c c

x x x x x

u u u uc c c

x x x x t

(13)

Substituting the displacements iu (Eqn. 8) into Eqn.

11 to Eqn. 13; the general equilibrium equation can

be reorganized in the form:

3 0ij jK k U (14)

Where, the elements of ijK is given as:

2 2 2 2

11 11 1 66 2 55 3 16 1 2

2 2 2

12 21 16 1 26 2 45 3 12 66 1 2

13 31 13 55 1 3 36 45 2 3

2 2 2 2

22 66 1 22 2 44 3 26 1 2

23 32 36 45 1 3 23 44 2 3

2 2 2

33 55 1 44 2 33 3

2

2

K c k c k c k c k k

K K c k c k c k c c k k

K K c c k k c c k k

K c k c k c k c k k

K K c c k k c c k k

K c k c k c k

2

45 1 22c k k

(15)

For a non-trivial solution to exist; det( ) 0ijK must

be true. Setting the determinant to be zero gives a

polynomial solution in terms of 3k as:

6 4 2

1 3 2 3 3 3 4 0 D k D k D k D (16)

The coefficients iD are functions of

1 2, , , , ijk k c and are provided in Appendix A.

Reference [9] is used for the subsequent derivation.

Eqn. 16 provides three roots for 2

3k , which

correspond to one quasi-longitudinal and two quasi-

shear modes. Altogether six roots of 3k are present

in two pairs that are negative of each other. Each

pair represents an upward and downward travelling

wave making the same angle with the 1x axis (Fig.

4). Representing the direction vector 3k for each

mode as q, where q=1,2,..6, gives 2 1 ,

4 3 and 6 5 .

Applying traction free boundary conditions, which

require the stresses to be zero at the top and bottom

surfaces of the plate (Fig. 5), gives

13 23 23, , equals zeros at 3x h .

Using the boundary conditions, the stresses can be

simplified in terms of stiffness and displacement as:

31 2 2 1

33 13 23 33 36

1 2 3 1 2

3 32 1

13 45 55

3 2 1 3

3 32 1

23 44 45

3 2 1 3

uu u u uc c c c

x x x x x

u uu uc c

x x x x

u uu uc c

x x x x

(17)

Taking the partial derivative of the displacements

and subbing into Eqn. 17 gives:

13 1 1 23 2 2 33 3 3

36 2 1 36 1 2

1 1 2 2 3 3

1 1 2 2 3 3

1 1 2 2 3 3

33

13

45 2 3 45 3 2 55 3 1 55 1 3

23

44 2 3 44 3 2 45 3 1 45 1

( )

( )

( )

i k x k x k x t

i k x k x k x t

i k x k x k x t

c U k c U k c U k

c U k c U k

ie

ie

c U k c U k c U k c U k

ie

c U k c U k c U k c U k

3

(18)

Defining the displacement component ratios as

2 1q q qV U U and 3 1q q qW U U .

qV and qW in terms of ijK is given as :

11 23 12 13

13 22 12 23

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

q q q q

q

q q q q

K K K KV

K K K K (19)

11 23 12 13

33 12 13 23

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

q q q q

q

q q q q

K K K KW

K K K K (20)

The total displacement can now be written in terms

of qV and qW as:

1 1 2 2 3

1 1 2 2 3

1 1 2 2 3

6( )

1 1

1

6( )

2 1

1

6( )

3 1

1

q

q

q

i k x k x x t

q

q

i k x k x x t

q q

q

i k x k x x t

q q

q

u U e

u V U e

u W U e

(21)

The total stress can also be simplified in terms of

qV , qW , and iq

D as:

1 1 2 2 3

1 1 2 2 3

1 1 2 2 3

6

33 1 1

1

6

13 2 1

1

6

23 3 1

1

( )

( )

( )

q

q

q

q q

q

q q

q

q q

q

i k x k x x t

i k x k x x t

i k x k x x t

i D U e

i D U e

i D U e

(22)

Where, iq

D is given by:

1 1 13 36 2 36 23 33

2 55 1 45 2 55 45

3 45 1 44 2 45 44

q q q q q

q q q q q

q q q q q

D k c c V k c c V c W

D c W k c W k c c V

D c W k c W k c c V

(23)

Using the boundary conditions that stresses must

each go to zero at the top and bottom surfaces of the

plate ( 3x h ); Eqn. 22 can be simplified and

expressed in the matrix form as: _ _ _

11 1 12 1 13 3 14 3 15 5 16 5_ _ _

21 1 22 1 23 3 24 3 25 5 26 5_ _ _

31 1 32 1 33 3 34 3 35 5 36 5_ _ _

11 1 12 1 13 3 14 3 15 5 16 5_ _ _

21 1 22 1 23 3 24 3 25 5 26 5_ _ _

31 1 32 1 33 3 34 3 35 5 36 5

D E D E D E D E D E D E

D E D E D E D E D E D E

D E D E D E D E D E D E

D E D E D E D E D E D E

D E D E D E D E D E D E

D E D E D E D E D E D E

11

12

13

14

15

16

000000

UUUUUU

(24)

Where, _

,

q qik h ik h

q qE e E e

For a symmetric Lamb mode, the displacement

3u (Fig. 1a) is given by:

3 3( ) ( ) u h u h (25)

Subbing, Eqn. 25 as 3u in Eqn. 21 it can be found

that, in order for the relationship to be true, the

conditions for displacement amplitudes should

be, 11 12U U , 13 14U U , and 15 16U U . Using the

displacement amplitude relationship it can be seen

that:

2 1 4 3 6 5

2 1 4 3 6 5

; ;

; ;

V V V V V V

W W W W W W (26)

Using the relationship given in Eqn. 26, the

properties ofijD can be written as:

12 11 14 13 16 15

22 21 24 23 26 25

32 31 34 33 36 35

; ;

; ;

; ;

D D D D D D

D D D D D D

D D D D D D

(27)

Using the aforementioned relationships, Eqn. 24 can

be simplified to:

_ _ _

11 1 11 1 13 3 13 3 15 5 15 5_ _ _ 11

21 1 21 1 23 3 23 3 25 5 25 5 13_ _ _

15

31 1 31 1 33 3 33 3 35 5 35 5

0

D E D E D E D E D E D EU

D E D E D E D E D E D E UU

D E D E D E D E D E D E

(28)

Where the trigonometric identities for qE and _

qE is

given as [10]: _

_

2 2cos( )

2 2 sin( )

qq q q

qq q q

E E C h

E E S i h

(29)

Using identities given in Eqn. 29; Eqn. 28 can be

simplified as:

11 1 13 3 15 5 11

21 1 23 3 25 5 13

31 1 33 3 35 5 15

2 2 2 0

2 2 2 0

2 2 2 0

D C D C D C U

D iS D iS D iS U

D iS D iS D iS U

(30)

In order for a non-trivial solution to exist, the

determinant of Eqn. 30 should be zero, which gives

the symmetric Lamb wave dispersion equation as:

7

11 25 33 11 23 35 1 3 5

35 21 13 13 25 31 3 1 5

31 23 15 15 21 33 5 1 3

cos( )sin( )sin( )

cos( )sin( )sin( )

cos( )sin( )sin( ) 0

D D D D D D h h h

D D D D D D h h h

D D D D D D h h h

(31)

Similar process can be followed for the anti-

symmetric Lamb mode in which, the displacement

3u (Fig. 1b) is given by:

3 3( ) ( ) u h u h (32)

Substituting, Eqn. 32 to 3u in Eqn. 21, it can be

found that, in order for the relationship to be true,

the conditions for displacement amplitudes should

be, 12 11 U U , 14 13 U U , and 16 15 U U .

Following the process as described for symmetric

Lamb modes, the anti-symmetric Lamb wave

dispersion equation can be found as:

11 25 33 11 23 35 1 3 5

35 21 13 13 25 31 3 1 5

31 23 15 15 21 33 5 1 3

sin( )cos( )cos( )

sin( )cos( )cos( )

sin( )cos( )cos( ) 0

D D D D D D h h h

D D D D D D h h h

D D D D D D h h h

(33)

Eqn. 31 and Eqn. 33 are solved numerically to

generate the Lamb wave dispersion curve if the

global material properties of a laminate are

available. If the material properties of the lamina

instead of the laminate are available then Eqn. 31

and Eqn. 33, which describe the wave propagation in

a single layer, need to be expanded to account for

the n-layered lamina at different orientation angle.

The most common methods for solving such a

problem are based on Global Matrix [12], Transfer

Matrix [10], and Laminate Plate Theory [13].

The group velocity gc can then be calculated using

the phase velocity data using: [14] 1

2 ( 2 )( 2 )

p

g p p

dcc c c f h

d f h

(34)

Where, 2

f is the frequency

2.4 MatLab Based Software to Generate

Dispersion Curve

Numerical solution for the propagation of Lamb

wave in anisotropic medium is similar to that of an

isotropic medium as provided in [11]. However, the

longitudinal and transverse velocities are coupled

together in the case of anisotropic medium, which

makes the solution complicated. Also at any given

frequency there exist infinite wavenumbers both real

and imaginary satisfying the Lamb wave equation.

Therefore, Lamb wave equations Eqn. 31 and

Eqn. 33 are solved numerically in a graphical user

interface (GUI) software developed in MatLab to

obtain the dispersion curves. For an n-layered

lamina the software uses the Global Matrix approach

to generate the dispersion curve since it is stable

even at higher frequencies as compared to Transfer

Matrix. The snapshot of the MatLab GUI along with

the description of the major modules is shown in

Fig. 6.

Fig. 6: MatLab-based Lamb wave dispersion software

The software shown in Fig. 6 can generate

dispersion curves for both isotropic and anisotropic

materials. For an n-layered composite, user can enter

the type of material, layup sequence and the

thickness of each layer as shown in Fig. 7

The output of the software is the dispersion curve

plotted in MatLab window similar to the one shown

in Fig. 2 and a text file containing the data regarding

the curve (phase and group velocities vs. frequency-

thickness product) for further analysis.

3 Experimental Verification

3.1 Coupon Preparation and Instrumentation

In order to prove the validity of the presented

method to generate the dispersion curve, one of the

most widely used carbon-fiber prepreg composite in

the aerospace industry was selected. The layup

sequence along with the material type of the

composite specimen is provided in Table 1.

Table 1: Specimen type, layup, and material

Specimen Layup/Thickness Material

Carbon

Epoxy

Prepreg

0,90, 45, 45SYM

Ply Thickness:

0.17 mm

G40-800 /

5276-1

Cytec

Industries

The composite plate was instrumented with nine

lead zirconate titanate (PZT) piezoelectric sensors.

The PZT sensors were acquired from Acellent

System and were permanently bonded onto the

composite plate using M-Bond AE-10 adhesive from

Vishay Micro-Measurements. Fig. 8 shows the

location of the PZT sensors along with the plate

dimensions.

The installed PZTs were excited at different

frequency intervals using five bursts of windowed

sinusoidal waves using ARB-1410 board and

WaveGen1410 software to generate the Lamb wave.

Windowing aids in focusing the excitation energy at

the desired frequency as shown in

Fig. 9

Fig. 8: Specimen dimension and PZT locations

Fig. 7: Snapshot of laminate information window

9

Fig. 9: Actuator signal excited at 200 kHz frequency in

(a) Time domain (b) Frequency Domain The signals were acquired and saved on a TDS 5104

digital oscilloscope. Fig. 10 shows the experimental

setup in which the PZTs installed on the composite

plate are connected to the signal generator and the

oscilloscope. A MatLab-based program was also

developed to process the acquired data and to extract

the relevant phase and group velocities.

As for the numerical solution, the material properties

provided in Table 2 were used to generate the Lamb

wave dispersion curve.

Table 2: G40-800/5276-1 material properties [15]

Elastic Modulus

(GPa) Shear Modulus (GPa)

Poisson’s

Ratio

E11 = 143 G12 = 4.8 µ12 = 0.3

E22 = 9.1 Density = 1650 kg/m3 µ23 = 0.3

The phase and group velocity of the fundamental

symmetric (So) and anti-symmetric (Ao) Lamb waves

were extracted by tracking the peaks of each

individual wave phase and the wave envelope

respectively. The waves were excited and gathered

at three different angles of 0⁰, 45⁰, and 90⁰. The

actuator and sensors used to excite the waves for

different angles are shown in Fig. 8.

3.3 Comparisons of Experimental vs. Theoretical

Dispersion Curves

The experimental and the theoretical results are

plotted in Fig. 11 to Fig. 13.

Fig. 10: Experimental setup to generate and gather

Lamb waves

Fig. 11: Lamb wave dispersion travelling at 0 degree

Fig. 12: Lamb wave dispersion travelling at 45 degrees

Fig. 13: Lamb wave dispersion travelling at 90 degrees

The theoretical results consist of the dispersion

curve generated using the presented 3D linear elastic

approach and the commonly used first-order

classical laminate plate theory (CLPT). For the

experimental data, the group velocity of the slow

moving fundamental anti-symmetric Lamb mode

was extracted. However, for the fast moving

symmetric Lamb mode phase velocity was easier to

extract due to reflection from the boundaries than

the group velocity.

From the plots (Fig. 11 to Fig. 13), it is evident that

the experimental data follows the 3D linear elastic

approach closely as compared to the CLPT. The

CLPT overestimated the fundamental anti-

symmetric mode group velocity by a factor of two at

higher frequency. CLPT also tend to remain constant

for the symmetric wave phase velocity and failed to

predict the drop off of the symmetric wave at higher

frequency especially in 0⁰ (Fig. 11) and 45⁰ (Fig. 12)

propagation angles.

It was also found that the anti-symmetric wave was

dominant at frequencies below 100 kHz. At

frequencies above 200 kHz, the symmetric wave

was the dominant one. Therefore, the plate was

excited between 20 kHz to 100 kHz for Ao mode,

whereas for the So mode it was excited from 200

kHz to 500 kHz. It was difficult to extract any

meaningful data between 100 kHz to 200 kHz due to

the coexistence of both Ao and So modes and the

reflections from the boundaries. Also at frequencies

above 500 kHz the signal was found to be noisy,

hence, it was difficult to extract any relevant

information regarding the Lamb wave dispersion.

3 Conclusions

An exact solution of the Lamb wave based on 3D

linear elasticity for a monoclinic material is

presented. The solution is then numerically solved

by custom developed Matlab-based software to

generate the dispersion curve. The numerical

solution is then compared against the classical

laminate plate theory and the experimental results.

For comparison purposes the fundamental symmetric

and anti-symmetric Lamb wave were excited at

different frequencies using the installed piezoelectric

actuators/sensors. The phase and group velocity data

were extracted for the So and Ao respectively by

tracking the wave peaks/envelope using custom

Matlab-based software.

It was found that the 3D linear elasticity model

followed the experimental curve for both symmetric

and anti-symmetric Lamb modes, whereas the

classical laminate plate theory did not follow the

curve and over predicted both the phase and group

velocities. This proves the validity of the exact

solution based on 3D linear elasticity presented in

this paper.

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Appendix A: Constants for Coefficient (Di)

2

1 55 44 33 45 33D c c c c c

2 2

55 66 33 45 13 55 36 13 44 55 2

12

45 36 13 16 45 33 13 44 11 44 33

2

55 23 55 22 33 26 45 33

2

55 23 44 66 44 33 45 23 36 2

2 2

36 44 45 23

55

2

2

2 2

2 2

2

2 2

2

2

c c c c c c c c c ck

c c c c c c c c c c c

c c c c c c c c

c c c

D

c c c c c c k

c c c c

c c

2 2

6 33 45 36 45 36 16 44 33

45 23 13 55 36 44 13 44 36 2 1

55 36 23 45 66 33 45 12 33

2 2

45 55 44 55 33 44 33

2 4

2

2 4 2

2 2 2

c c c c c c c c

c c c c c c c c c k k

c c c c c c c c c

c c c c c c c

2

11 36 45 11 66 33 13 66 16 36 13

2 2 4

16 36 55 11 36 11 44 55 16 33 1

2

11 45 13 66 55 16 45 13

11 36 44 16 23 55 12 36 55 13 26 55

16 44 13 16 23 1

3

3 1

2 2

2

2 2

2 2 2

4

2 2 2 +

c c c c c c c c c c c

c c c c c c c c c c k

c c c c c c c c

c c c c c c c c c c c c

c c c c c

D

c c

2 36 13 16 12 33 3

2 12

13 26 11 26 33 45 12 13 11 36 23

2

16 45 11 45 23 16 44 55

2

45 13 55 55 66 36 45 11 44 2

12 2

11 33 36 55 44 66 33 16 45 13

2

2 2 2 2

4 2

4

2 2

2

c c c c ck k

c c c c c c c c c c c

c c c c c c c c

c c c c c c c c ck

c c c c c c c c c c

2

16 26 33 12 36 45 16 36 23 12 23 55

66 44 13 66 23 55 12 44 13 26 36 13

2 2

66 23 13 26 45 13 23 11 12 33

2 2 2 2

13 22 45 66 45 12 36 12

12 66 33 16 45 23 16 36 4

2 4 2

2

2 2 2 2

2 2

4 2 2

2 2 2

c c c c c c c c c c c c

c c c c c c c c c c c c

c c c c c c c c c c

c c c c c c c c

c c c c c c c c c

2 2

2 1

4 12 44 55

26 36 55 12 23 13 13 22 55 11 22 33

11 23 44 66 44 55

2

2 2 2

2 4

k k

c c c

c c c c c c c c c c c c

c c c c c c

2 6

11 66 55 16 55 1

2 5

16 12 55 11 66 45 11 26 55 16 45 2 1

12 66 55 11 66 44 16 26 55 11 22 55 2 4

2 12 2

16 12 45 11 26 45 12 55 16 44

2

16

4

55 66 11 55

2 2 2 2

2 2

4

4

c c c c c k

c c c c c c c c c c c k k

c c c c c c c c c c c ck k

c c c c c c c c c c

c c c

D

c c c

4 2

11 66 1

11 45 11 26 16 12 3 2

2 1

55 26 16 55 45 66

2

3 312 45 16 22 55 11 22 45 11 26 44

2 1

16 26 45 12 66 45 16 12 44 26 12 55

11 55 66

2 2 2

2 2 2

2 2 2 2

4

4 2 2

c k

c c c c c ck k

c c c c c c

c c c c c c c c c c ck k

c c c c c c c c c c c c

c c c

2 2 4

1

2

2 2 212 66 44 55 66 55 22 11 22 11 44

2 1

12 66 16 26 16 45 26 45

26 12 45 16 22 45 12 66 44 16 26 44 4 2

2 12 2

11 22 44 66 22 55 26 55 12 44

2 2

4 4

4 4 2 2

k

c c c c c c c c c c ck k

c c c c c c c c

c c c c c c c c c c c ck k

c c c c c c c c c c

2 5

66 22 45 26 12 44 16 22 44 26 45 2 1

16 22 26 12 26 44 3 2

2 1

16 44 45 66 22 45

2 4

16 45 26 2 1

2 2 4

22 44 66 2

2

26 22 44 66 22 66 44

2 2 2 2

2 2 2

2 2 2

2 2 2

c c c c c c c c c c c k k

c c c c c ck k

c c c c c c

c c c k k

c c c k

c c c c c c c

4 2

2

2 6 3 6

66 22 44 26 44 2

k

c c c c c k