paper Theoretical and experimental studies of Lamb wave propagation in attenuative composites

1. Theoretical and experimental studies of Lamb wave propagation
in attenuative composites
Mircea Calomfirescu*ab
, Axel S. Herrmanna
a
Faserinstitut Bremen e.V., University of Bremen, Am Biologischen Garten 2, Bremen, Germany
b
University of South Carolina, 300 Main St., Columbia, SC, USA 29208
ABSTRACT
This paper presents a theoretical model for anisotropic wave attenuation in composites. The model has been
implemented in a software called FIBREWAVE in order to predict dispersion and attenuation of S0, A0 and SH0 Lamb
wave modes. The required input data are the complex stiffness matrix coefficients of the unidirectional plies of the
laminate, which have been measured by a laser interferometry method. Complex stiffness data for an unidirectional
CFRP laminates are moreover presented. Satisfactory agreement has been observed between predicted and experimental
group velocities and wave attenuations.
Keywords: Lamb waves, Composites, Viscoelastic Properties, SHM, Wave Attenuation, Damping
.
1. INTRODUCTION
In order to ensure structural integrity and maintain safety aerospace structures have to be inspected. At the present time
there is a variety of non-destructive, traditional inspection techniques (NDT) available such as eddy current, ultrasonics,
thermography and shearography [1]. The major disadvantages of these techniques are related to high (life-cycle) cost,
damage detection sensitivity, time consuming and require the structure to be out of service. Recent years have shown a
range of different NDT for so called “structural health monitoring (SHM)”, which consists of two different approaches:
passive sensing monitoring and active sensing monitoring. The active approach needs both sensors and actuators to
evaluate and monitor the structural health. The passive technique needs only sensors to collect data of structural waves
excited by an external source, such as an impact or a propagating crack. Various SHM systems based on different types
of sensors such as piezoelectric elements, fibre optics and MEMS sensors has already been developed and presented in
laboratory demonstrations at meetings and workshops [2-5]. Most of the presented SHM techniques [6-7], require
experimental training data sets to determine the relation between force histories and the corresponding strain responses
or between failures and sensor signals. Such techniques would benefit of improved theoretical understanding of wave
propagation in anisotropic composites to construct a precise analytical/numerical model without conducting numerous
experimental tests. A crucial element in the Lamb wave propagation in composites is the wave attenuation. The
fundamental understanding and prediction of the anisotropic wave attenuation could be then applied for:
- the design of sensor networks (number and location of sensors), since the material attenuation decides in the
far field how far waves can travel and can be still detected
- the detection of damage, since the attenuation characteristics of a structure increase in general in the presence
of damage [8]
- the prediction of an impact load in impact identification methods, since the amplitude of the measured stress
waves decays depending on the distance between the impact and the location of the sensor.
Therefore the objective of our research was to develop a theoretical model, enabling the prediction of the dispersion and
attenuation behavior in composites with arbitrary lay-ups. The developed theoretical model has been implemented in a
MatLab software and verified by experiments. The material characterization has been conducted at the Laboratoire de
Mecanique Physique (LMP) at the University of Bordeaux [9].
*calomfirescu@faserinstitut.de; phone +49 421 218 9335; fax +49 421 218 3110

2. 2. LAMB WAVES IN COMPOSITES
Lamb waves are a type of ultrasonic waves, which are also known as guided plate waves, due to the fact that they are
guided between two parallel free surfaces, the upper and the lower surface of the plate [13]. For each frequency more
than one wave modes exists. In composites these wave modes are at low frequencies S0, A0 and SH0, the latter one,
denoting the shear horizontal wave. The symmetrical modes are called, S0, S1, S2,…, and the anti-symmetric ones A0,
A1, A2…, starting with the mode that has the lowest frequency for a given wavenumber.
Fig. 1: Low order Lamb wave modes
2.1 Attenuation of Lamb waves
The definition of attenuation in general is the loss of amplitude of an acoustic wave with propagation distance. In the
case of omnidirectional excited waves, there are, as discussed by [11], four main contributing factors: geometric
spreading, material damping, wave dissipation into adjacent media and losses related to velocity dispersion. In the near
field, close to the source, the geometric spreading is the dominant source for wave attenuation, where for plate-like
structures and a point-wise excitation source, the wave amplitude decreases inversely as the square root of the distance
of propagation. The second factor, which is the dominating influence in the far field and the factor to be most discussed
in this paper is the material damping. Here energy is converted into heat and thus extract from the mechanical system.
The last but one attenuation mechanism is related to energy losses into adjacent media, like from a vessel into the
contained fluid are structural attenuation at stiffeners and joints. The final attenuation mechanism, the attenuation due to
dispersion, as well as the last but one mechanisms can be neglected in these studies, since here narrow band, single
frequency Lamb waves for simple structures are considered.
Due to the high material damping, attenuation leads to a decay of the amplitude of Lamb waves in composites much
stronger that in metallic structures. Attenuation is crucial for the wave propagation and detection, since it decides how
far Lamb waves modes can be transmitted with a sufficient signal to noise ratio. Moreover a change in stress wave
attenuation can be used for damage detection, since attenuation increases in the presence of material damages [8].
Besides of the frequency and Lamb wave mode dependency, attenuation in composites is anisotropic, depending on the
direction of wave propagation.
In this research the viscoelastic behavior of the composites is taken into account by considering complex components in
the material’s stiffness matrix of each unidirectional (UD)- ply:
ijijij CiCC ′′⋅+′=*
(1)
where ijC′ contains the storage moduli and
″
ijC the loss moduli. In order to consider arbitrary laminates with UD-plies
layed up in different angles, the stiffness matrix of each ply ( ∗
ijC ) has to be polar transformed in the coordinate system
of the laminate ( *~
ijC ) . Moreover, the complex wave number is introduced and defined by:
kikk ′′⋅+′=*
, (2)
where k ′′ represents the attenuation coefficient with units of dB/unit distance. Thus, a hysteretic model [12] is
considered in this study to represent material damping.

3. The complex stiffness matrix of the laminate according to the applied higher order theory has the form:
(3)
where the coefficients can be obtained by the lamination theory as follows:
/2
* * * * * * 2 3 4
/2
( , , , , ) ( ) (1, , , , )
h
ij ij ij ij ij k
h
A B D F H Q ij z z z z dz
−
= ∫ , (4)
The
*
Q ij terms are the complex reduced stiffness coefficients of each unidirectional ply as defined by:
* *
* *
*
( ,3) (3, )
(3,3)
ij ij
ij ij
ij
C i C j
Q C
C
⎛ ⎞⎛ ⎞⋅
= −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
(5)
The usual coupling between antisymmetric and symmetric modes for a laminate of arbitrary stacking sequence is given
through the stiffness Bij and Fij. In this study these stiffness coefficients are not taken into account. By applying this
simplification only symmetrical lamintes can be calculated by our approach. With the linear strain-displacement
relations, the equations of motion of the higher-order theory can be derived using the principle of virtual displacement
or Hamilton’s principle. After formulation the equations of motions and assuming the solution form as
( )* *
0 0
x y
i k x k y t
u U e
ω⎡ ⎤⋅ + ⋅ −
⎢ ⎥⎣ ⎦
= ,
( )* *
0 0
x y
i k x k y t
v V e
ω⎡ ⎤⋅ + ⋅ −
⎢ ⎥⎣ ⎦
= ,
( )* *
0 0
x y
i k x k y t
w W e
ω⎡ ⎤⋅ + ⋅ −
⎢ ⎥⎣ ⎦
= ,
( )* *
x y
i k x k y t
x xe
ω
ψ
⎡ ⎤⋅ + ⋅ −
⎢ ⎥⎣ ⎦
= Ψ (6)
( )* *
x y
i k x k y t
y ye
ω
ψ
⎡ ⎤⋅ + ⋅ −
⎢ ⎥⎣ ⎦
= Ψ ,
( )* *
x y
i k x k y t
z ze
ω
ψ
⎡ ⎤⋅ + ⋅ −
⎢ ⎥⎣ ⎦
= Ψ ,
( )* *
x y
i k x k y t
x xe
ω
φ
⎡ ⎤⋅ + ⋅ −
⎢ ⎥⎣ ⎦
= Φ ,
( )* *
x y
i k x k y t
y ye
ω
φ
⎡ ⎤⋅ + ⋅ −
⎢ ⎥⎣ ⎦
= Φ
the following relation is obtained:

4. 0
0
0
0
0
0
88
7877
686766
58575655
4847464544
383736353433
281826252422
18171615141211
=
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
⎬
⎫
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎨
⎧
Φ
Φ
Ψ
Ψ
Ψ
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
y
x
z
y
x
W
V
U
L
LL
LLLsym
LLLL
LLLLL
LLLLLL
LLLLLL
LLLLLLL
(7)
The coefficients Lij are a function of the complex stiffness matrix coefficients from equation (3), the wave number and
the frequency. For symmetric laminates equation (7) can be decoupled into symmetric waves:
0
0
0
88
7877
686766
28182622
1817161211
=
⎪
⎪
⎪
⎭
⎪⎪
⎪
⎬
⎫
⎪
⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
Φ
Φ
Ψ
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
y
x
z
V
U
L
LLsym
LLL
LLLL
LLLLL
(8)
and antisymmetric waves:
0
0
55
4544
353433
=
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
Ψ
Ψ
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
y
x
W
Lsym
LL
LLL
(9)
Each root of these relations is related to a certain Lamb-wave mode, for which the dispersian and attenuation
characteristics can be then obtained.
3. SOFTWARE
To gain better understanding of wave propagation and attenuation in composites the theoretic model has been put into
mathematics software. The software was developed in a MatLab environment and predicts the dispersion and
attenuation behavior for composites with arbitrary lay-ups. As input data the complex stiffness matrix of each UD-ply,
the lay-up and the thickness of each ply has to be given. Fig. 2 shows the front panel of the software called
FIBREWAVE, and developed at the Faserinstitut Bremen.

5. Fig. 2: GUI of the software FIBREWAVE
Additional outputs of the software are polar plots for phase and group velocities and attenuation plots at arbitrary
frequencies.
4. MEASUREMENTS
For applying the described theoretical model for the prediction of the dispersion and attenuation behavior the
components of the viscoelastic stiffness matrix of the CFRP laminates has to be investigated. In this study the properties
have been measured at the Laboratoire de Mecanique Physique (LMP) of the University of Bordeaux. The immersion
technique has been developed in the 90’s at by the LMP and described in detail in [9]. The principle of this method
consists in sending and receiving an ultrasonic plane field. The frequency bandwith has to be chosen so that the material
is seen by the waves as a homogeneous medium. The material properties are then obtained numerically inverting the
transmitted ultrasonic fields, obtained for different incident angles.
The viscoelastic stiffness properties (Cij) obtained and used for a 5.1 mm thick unidirectional carbon-epoxy laminate
are shown in table 1.
C11 C22 C33 C12 C13 C23 C44 C55 C66
C’ij 125 ± 8
13.9 ± 0.4
14.5 ± 0.4
6.3 ± 0.6
5.4± 0.4
7.1 ± 0.2
3.7 ± 0.3
5.4 ± 0.4
5.4 ± 0.6
C’’ij 3 ± 1
0.6 ± 0.2
0.6 ± 0.2
0.9 ± 0.6
0.4 ± 0.2
0.23 ± 0.15
0.12 ± 0.06
0.3 ± 0.15
0.5 ± 0.25
Table 1: Results of Cij (in GPa) for the unidirectional CFRP laminate
The intervals of errors obtained with this technique are quite standard for this type of material. Also usual is the fact that
the relative errors are much bigger for the imaginary parts of the stiffness coefficients than for the real parts. The
imaginary part of C11 is particularly hard to measure.
In order to verify the mathematical model and the developed software, experimental pitch-catch measurements have
been performed. The experimental setup consists of a CFRP plate with the overall dimensions of 500 x 500 mm. The
plate was instrumented with piezoelectric waver active sensors (PWAS) with a diameter of 7 mm and a thickness of 200
µm from American Piezo Ceramics APC-850. A 4-channel digital oscilloscope was used to collect the signals from the
PWAS. For collecting dispersion and attenuation data in three propagation angles (0°, 45° and 90°) the experimental

6. setup illustrated in Fig. 3 has been employed. A total number of 10 round PWAS has been applied. In the configuration
illustrated in Fig. 3 PWAS #1 excites waves, which are then collected at the sensors #2, #3 and #4, for 0° propagation.
Fig. 3: Experimental Setup for Lamb wave propagation
The distance between PWAS #1 and #2 was 149 mm, that between #2 and #3, 130 mm and between PWAS #3 and #4,
69 mm. Thus, with this sensor configuration the propagation characteristics in 0°- direction can be analyzed. For the
Lamb wave excitation an HP33120A arbitrary signal generator was used. The excitation applied was a 3-count tone-
burst signal filtered through a Hanning window to excite coherent single-frequency waves [13]. For the measurements
of wave velocities at different frequencies the corresponding times of flight (TOF) has been analyzed. As arrival time
the maximum of the magnitude of the continuous wavelet transform (CWT) coefficients for the corresponding
excitation scale has been used. By this approach it was possible to obtain a clean, smooth envelope, since only the
excitation frequency is considered. The Gabor function was used as mother wavelet. For the experimental evaluation of
material attenuation the phenomenon of geometric spreading was taken into account by correcting the amplitude
decrease in order to obtain the decay due to material damping only. The decrease of wave amplitude due to geometric
spreading has been corrected by the relation 1 d , where d is the distance between the location the amplitudes have
been collected.
Fig. 4 illustrates the wave propagation at 80 kHz center frequency excitation for the 90° propagation direction (matrix
direction). At this frequency the both Lamb waves modes S0 and the A0 can be excited and recognized and distinguished
clearly. The very first, highest peak has to be neglected since it comes from the electro- mechanical coupling and does
not denote any Lamb wave mode. The first Lamb wave signal arriving at the PWAS # 8 at t = 63 µs corresponds to the
S0 symmetrical mode, the latter, arriving at t = 109 µs corresponds to the A0 mode. The attenuation coefficient in
dB/unit distance can be then calculated by the relation of the amplitude ratios and distances according to:
[ ]
2 2
1 1
2 1
ln
20 log
A d
A d
d d
k dB mm e
⎛ ⎞
⋅⎜ ⎟⎜ ⎟
⎝ ⎠−
−
⎛ ⎞
⎜ ⎟
⎜ ⎟′′ = ⋅
⎜ ⎟
⎜ ⎟
⎝ ⎠
(10)
where d1 and d2 are the distances from the excitation source, and A1 and A2 the collected amplitudes. The factor
2 1d d is the correction factor taking into account the geometric spreading due to the point source excitation.

7. Fig. 4: Wave propagation in 90° laminate direction (at 80 kHz)
This single frequency excitation has been performed from 5 kHz to 400 kHz each 10 kHz. Since not for all excitation
frequencies all Lamb wave modes can be clearly and strongly excited and in additionally many cases the measurements
are disturbed by reflections from boundaries not for all frequencies could be performed reliable measurements. Fig. 5
and 6 show the dispersion and attenuation data obtained by the described experimental pitch-catch measurements and
plotted against theoretical, by FIBREWAVE predicted values.
Fig. 5: Predicted and measured dispersion and attenuation for 0° wave propagation in an unidirectional CFRP laminate
The experimental dispersion values for the A0 mode as well as the attenuation of this Lamb wave mode could be
extracted much more reliable than those of the S0 mode for two reasons. Due to the high wave velocity in fibre direction
(~ 9000 m/s for low frequencies) the various echoes between the edges of the 500 mm long sample are too close to each
other in the dime domain, thus preventing to obtain an accurate estimation of its attenuation. Therefore a layer of clay
has been applied on the CFRP plate at the boundaries in order to reduce boundary reflections. Although the application
of clay reduced the reflections significantly, it was still not possible for the relatively small plate to extract reliable
attenuation values for the S0 mode. Despite some uncertainties satisfactory agreement has been observed for the A0
mode for the dispersion as well as for the attenuation behavior. For higher frequencies beyond 180 kHz the A0 mode
could not be excited strongly enough, so that no experimental values are available for this range. Moreover the SH0
mode could not be detected reliably.
Fig. 6 shows the dispersion and attenuation characteristics of the same CFRP unidirectional plate for 90° Lamb wave
propagation. As expected and predicted it shows clearly the lower propagation velocity for A0 and S0 modes in 90°

8. propagation direction in comparison to the 0° direction (fibre direction). In Fig. 6 right the attenuation is plotted against
frequency for the Lamb wave modes A0 and S0. Since in general, as for the most of the previously analyzed CFRP-
laminates observed, the attenuation of S0 is lower than that of A0, for this laminate it is contrary for frequencies beyond
~40 kHz.
Fig. 6: Predicted and measured dispersion and attenuation for 90° wave propagation in an unidirectional CFRP laminate
5. SUMMARY AND CONCLUSIONS
A theoretical model for the propagation of S0, A0 and SH0 modes in viscoelastic composites has been developed and
implemented in a software called FIBREWAVE enabling the prediction of dispersion and attenuation behavior in
composites with arbitrary lay-ups. The viscoelastic (complex) material properties used have been measured by a laser
interferometry method at the University of Bordeaux. Besides some uncertainties good agreement between the
experimental and the predicted values has been observed showing the high wave attenuation in composites in
comparison to metallic structures.
The highly attenuative and anisotropic behavior of composites has to be taken into account for SHM applications in
future studies since it defines how far waves can travel in different directions. In the next steps, after fully experimental
confirmation on much more laminates with different lay-ups, the model will be implemented in the explicit finite-
element code LS-DYNA, which has already been applied and extensively validated at the Faserinstitut Bremen for wave
propagation analysis. The developed attenuation model in combination with a FEA-based optimization method will be
then applied for design optimal sensor network (no. and location of sensors) for more complex structure. Last but not
least viscoelastic material characterization will be conducted with surface bonded PWAS.
ACKNOWLEDGMENTS
The financial support of this research by a grant from the DFG (Deutsche Forschungsgemeinschaft) is gratefully
appreciated. Moreover the author would like to thank Professor Giurgiutiu from the Laboratory for Active Materials and
Smart Structures (LAMSS) at the University of South Carolina for the benefit of his research environment and
interactions and Professor Castaings from Laboratoire de Mecanique Physique (LMP) at the University of Bordeaux for
the material characterization.
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