Post on 16-Mar-2020
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