Upload
vonga
View
238
Download
3
Embed Size (px)
Citation preview
ORIGINAL PAPER
Influence of the cover plate thickness on the Lamb wavepropagation in honeycomb sandwich panels
S. M. H. Hosseini • U. Gabbert • R. Lammering
Received: 11 April 2012 / Revised: 19 November 2012 / Accepted: 13 December 2012 / Published online: 28 December 2012
� Deutsches Zentrum fur Luft- und Raumfahrt e.V. 2012
Abstract Within this paper, the guided Lamb wave
propagation in thin honeycomb sandwich panels is studied.
The Lamb waves are excited by thin piezoelectric (PZT)
patch actuators glued to the surface of the plate, and the
signals are received by similar PZT sensor patches. Such
actuator and sensor systems can be used for a cost-effective
online health monitoring of structures. In homogeneous
plates, Lamb waves propagate with symmetrical and anti-
symmetrical modes. However, the propagation in hetero-
geneous media is not as clear and depends on the exciting
frequency, the material properties, and the geometry of the
structure. In this paper, the influence of the geometrical
properties of honeycomb plates on the Lamb wave propa-
gation is studied. For this purpose detailed 3-D finite ele-
ment calculations are performed, which result in very time
consuming computations. Consequently, also simplified
models are developed to reduce the computing time with-
out losing the quality of the results. To this end, the hon-
eycomb core material is replaced by a homogeneous layer
with orthotropic mechanical properties. The homogenized
properties are calculated numerically using a homogeni-
zation technique based on the representative volume ele-
ment method. The comparison of the results received with
the two different approaches has shown that the simplified
models are in a good agreement with the extended models
for a certain range of exciting frequencies and geometrical
properties only. The wave propagation on the top and
bottom surfaces is also compared in order to show how
deep the waves can travel inside the structure.
Keywords Guided waves � Honeycomb sandwich plate �Finite element method
1 Introduction
The application of high frequency guided Lamb waves in
thin-walled structures is a challenging technique in indus-
try to receive information about the health state of a
structure (structural health monitoring—SHM). Such Lamb
waves can be simply excited and received by a network of
thin piezoelectric (PZT) patches glued to the surface of the
structure. In recent years a lot of papers have been pub-
lished dealing with the interaction of Lamb waves with
different types of damages in metallic structures as well as
in composite materials [2]. The high sensitivity of ultra-
sonic waves with respect to small structural changes and
the low costs of health monitoring systems built from
piezoelectric patches, make such systems very attractive
for industrial applications. The wave fields are definitely
changed by structural damages. But, unfortunately, there
are a lot of open questions, especially regarding the
application of ultrasonic waves in layered composite
structures. In such structures creeping mode conversions,
small reflections at inner boundaries and at small thickness
changes, amplitude reductions due to material damping,
etc., have been observed also in undamaged structures.
Consequently, in such cases it is complicated to estimate
structural changes (e.g. the type, the size, and the position
S. M. H. Hosseini (&) � U. Gabbert
Institut fur Mechanik, Otto-von-Guericke Universitat
Magdeburg, 39106 Magdeburg, Germany
e-mail: [email protected]
U. Gabbert
e-mail: [email protected]
R. Lammering
Institut fur Mechanik, Helmut Schmidt Universitat Hamburg,
39106 Hamburg, Germany
e-mail: [email protected]
123
CEAS Aeronaut J (2013) 4:69–76
DOI 10.1007/s13272-012-0054-8
of damage) reliably and with a sufficient accuracy. To
overcome such problems a large amount of papers are
dealing with different aspects of the computer-assisted
design of reliable SHM systems [1, 5]. The situation
becomes even more complex and complicated if the wave
propagation is studied in honeycomb sandwich panels.
Only a few papers have been found dealing with this
special case of a heterogeneous material system (see for
instance [9, 11, 12]).
In the paper by Song et al. [9] a 3-D finite element
model is used to investigate the propagation of guided
waves excited by a PZT actuator/sensor system in a hon-
eycomb panel. In order to reduce the computational effort
the authors have also used a simplified model with a
homogenized core layer. A good agreement of the 3-D
model and simplified model is obtained, if the central
frequency of the exciting signals is relatively low (5 kHz).
In higher frequency ranges (from 40 until 90 kHz) there is
no agreement between both models. Interesting is that the
authors found again a good agreement between both
models if the exciting frequency is about 100 kHz; only the
amplitudes are different. The experimental and the simu-
lation results are always in a good agreement. In another
study by Hosseini and Gabbert [3] also a 3-D finite element
method is used to study the wave propagation in honey-
comb sandwich panels. In this study two different honey-
comb sandwich panels with different geometrical
properties are considered. It is shown that the ultrasonic
waves propagate mainly either in the top layer or more in
the core layer depending on the thickness relations and the
properties of the respective materials.
In the present paper, the main aim is to investigate the
influence of changes of the cover plate thickness (tp) (cf.
Fig. 1) of the honeycomb sandwich structure on the wave
propagation. In addition to a 3-D finite element model of the
honeycomb structure, two other models are also considered.
First, a simplified model with homogenized material prop-
erties in the core layer is evaluated, and the aim is to study
the influence of geometry and material properties individ-
ually on the wave propagation as well as reducing the cal-
culation time. And secondly, a single plate model is also
considered; in this model the cover plate of the honeycomb
structure is taken as the single plate and the rest of com-
ponents of the sandwich structure are omitted. In this case,
the influences of the core and bottom face are neglected.
Comparing the results from the extended honeycomb model
and single plate model one can see clearly the influence of
the core layer and cover plate on the wave propagation. The
paper is organized as follows: At first the finite element
modeling of the sandwich panel is presented, and the data of
the test cases are given; the test cases are aimed to study the
influence of the changes in thickness of the cover plate and
also changes in the central frequency of the exciton signal
on the wave propagation in the honeycomb sandwich panel.
Then the methodology to receive the phase and group
velocity from the finite element results is discussed. In the
next section the calculated results are given and general
conclusions are drawn. The paper concludes with a sum-
mary and outlook to further investigations.
2 Lamb wave propagation in honeycomb sandwich
plates
2.1 FEM modeling
Honeycomb sandwich panels consist of two plates on top
and on bottom with a core material in the middle. The
thickness of the plates is normally much higher than the
thickness of the honeycomb material. Thus, to be able to
include the symmetrical as well as the asymmetrical wave
modes it is obvious to model the cover plates with 3-D
finite elements. 2-D finite shell elements are sufficiently
accurate to model the honeycomb cell structures. For
modeling the piezoelectric actuators and sensors, 3-D
electromechanical coupled solid elements are applied. In
addition, in case of the simplified model, the honeycomb
core layer is also modeled with 3-D solid finite elements
with orthotropic material properties. For a proper estima-
tion of the orthotropic material parameters also a homog-
enization method based on representative volume element
(RVE) is applied [8]. The RVE is a sample volume of a
heterogenous material which is large enough to represent
effectively all microstructural heterogeneities of the
structure. After applying the periodic boundary conditions
to the RVE model, several tensile tests are implemented to
evaluate the mechanical properties of the RVE in different
directions. In these cases the average stress is calculated,
dividing the resulting tractions on the borders of RVE by
the surface area [4].
Z
X Y
t p
Hei
ght
Cell sizet h
Honeycomb core layer
Fig. 1 The sandwich panel design and geometrical properties
(cf. Table 1)
70 S. M. H. Hosseini et al.
123
In the test example, the piezoelectric sensor is located in
parallel to the actuator on both top and bottom layers. The
sensors are glued to the structure in a distance of 180 mm
from the actuator in the x direction (cf. Fig. 2). The bottom
nodes of the PZT elements are considered as grounded. The
exciting signal is an electric voltage in form of a half-cycle
narrow band tone burst [9], which is applied to the top
nodes of the actuator (t is time, fc is central frequency and
H(t) is the Heaviside step function) as:
Vin ¼ V½HðtÞ � Hðt � 3:5=fcÞ� 1� cos2pfct
3:5
� �sin 2pfct:
ð1ÞTo represent the conductivity of the copper layer on the
top layer of the sensor, all nodes on the top layer of the
sensor are tied together. Symmetric boundary conditions
are applied to reduce the model size. Also non-reflecting
boundary conditions [5], are applied to reduce the wave
reflections from free borders of the plate. The damping
factors of the elements are additionally increased gradually
from non-damped elements to elements with high damping
factors at the free borders, by applying an exponential
function. These procedures make sure that the results are not
influenced by the boundaries of the test specimen (cf. Fig. 2).
The element size of the applied finite element mesh has
been evaluated to guarantee numerical results with high
accuracy. It has been proved that a mesh size smaller than
one tenth of the wavelength results in solutions with suf-
ficient accuracy. Song et al. [9] have shown that results
calculated with this mesh size also match well with
experimental results. In a case study several honeycomb
sandwich panels with different thickness of the cover plate
(tp) from 0.5, 0.75, 1, 1.25, 1.5, 1.75 and 2 mm have been
analyzed to investigate the influence of the cover plate
thicknesses on the wave propagation. By increasing the
thickness of the honeycomb core unit cells (th) more energy
will transmit to the sensors, therefore, rather large value of
1.48 mm has been considered for th in order to receive a
clear response from the sensors. The rest of the geometrical
properties are shown in Table 1 (see also Fig. 1). Table 2
shows the material properties of the skin plates and the
honeycomb core materials.
The material properties of the PZT actuators and sensors
are presented in [9]. The dielectric matrix ½e� and the pie-
zoelectric matrix [e], are, respectively,
½e� ¼6:450 0
6:45 0
Symmetry 5:62
24
3510�9 ðC V�1 m�1Þ;
½e� ¼
0 0 �5:20 0 �5:20 0 15:10 0 0
0 12:7 0
0 12:7 0
26666664
37777775ðC m�2Þ;
and the stiffness matrix is
The calculations are performed using the commercial
finite element package ANSYS 11.0.
Z
YXactuator
symmetric boundary con-dition
sensor top
FEMAPMaterial2:asd
material7
FEMAPMaterial2:asd_1
FEMAPMaterial2:asd_2
FEMAPMaterial2:asd_3
FEMAPMaterial2:asd_4
FEMAPMaterial2:asd_5
FEMAPMaterial2:asd_6
FEMAPMaterial2:asd_7
FEMAPMaterial2:asd_8
FEMAPMaterial2:asd_9
FEMAPMaterial2:asd_10
FEMAPMaterial2:asd_11
FEMAPMaterial2:asd_12
FEMAPMaterial2:asd_13
FEMAPMaterial2:asd_14
FEMAPMaterial2:asd_15
FEMAPMaterial2:asd_16
FEMAPMaterial2:asd_17
FEMAPMaterial2:asd_18
FEMAPMaterial2:asd_19
FEMAPMaterial2:asd_20
FEMAPMaterial2:asd_21
FEMAPMaterial2:asd_22
FEMAPMaterial2:asd_23
FEMAPMaterial2:asd_24
FEMAPMaterial2:asd_25
FEMAPMaterial2:asd_26
FEMAPMaterial2:asd_27
sensor bottom
increasing damping factor
increasing damping factor FEMAPMaterial2:asd
material7
FEMAPMaterial2:asd_1
FEMAPMaterial2:asd_2
FEMAPMaterial2:asd_3
FEMAPMaterial2:asd_4
FEMAPMaterial2:asd_5
FEMAPMaterial2:asd_6
FEMAPMaterial2:asd_7
FEMAPMaterial2:asd_8
FEMAPMaterial2:asd_9
FEMAPMaterial2:asd_10
FEMAPMaterial2:asd_11
FEMAPMaterial2:asd_12
FEMAPMaterial2:asd_13
FEMAPMaterial2:asd_14
ties
Fig. 2 The orientation of the PZT elements in a single plate model.
In addition, symmetric boundary condition and non-reflecting
boundary condition are shown
½c� ¼
13:9 6:78 7:43 0 0 0
13:9 7:43 0 0 0
11:5 0 0 0
3:56 0 0
2:56 0
Symmetry 2:56
26666664
37777775
1010 ðPaÞ:
Lamb wave propagation 71
123
2.2 Methodology
The Lamb waves propagate along the media with different
modes with different group velocities and wave-lengths. A
continuous wavelet transform (CWT) based on the
Daubechies wavelet D10 is used to calculate the time of
flight for each mode [10, 12]. Using the time of the flight
and knowing the distance between sensor and actuator (in
our test cases 180 mm) one can calculate the group
velocity for each mode [10]. The phase velocity and the
wave length of each mode can be determined using a fast
Fourier transform algorithm. The phase velocity can be
expressed in terms of the frequency using the following
equation:
tðf Þ ¼ 2pfL
½/ðf Þ � /0�ð2Þ
where f is the frequency, /0 is the exciting phase function,
/(f) is the received phase function, and L stands for the
axial distance between the actuator and the sensor, [7].
Dividing the phase velocity by the frequency will give the
wavelength [6]. It must be mentioned that a mode with a
wavelength ‘‘a’’ is only able to determine damages bigger
than ‘‘a’’ [6]. The evaluation of the finite element results is
performed with help of the software package MATLAB�.
3 Results
To show the influence of changes of the cover plate
thickness (tp) (cf. Fig. 1) of the honeycomb sandwich
structure on the wave propagation, the results will be
presented in the following order:
1. Voltage and displacement based time signal: the
electrically excited ultrasonic wave is received at the
piezoelectric sensor again in form of an electric
voltage signal measured at the top nodes of the
piezoelectric finite elements, called voltage signal. In
addition, the out of plane displacement (in z direction)
of a single node located on the sensor is also evaluated,
called nodal displacement signal (cf. Fig. 3). The
voltage (or out of plane displacement) of a specific
node for different time increments called time based
signal. In this part different time signals based on
nodal voltage and out of plate displacement are
described and compared.
2. Wave field: to visualize the wave transformation in the
structure, a 2-D and 3-D wave field snapshot is
presented in this part.
3. Group velocity: as a first property of wave propagation
in a structure the group velocity of different modes in
different structures is presented.
4. Wave length: using wave lengths one can predict
which kind of damage can be detected by each mode in
a specific structure.
In all mentioned parts, results from three different
models are compared, a 3-D model of the honeycomb
structure, a simplified model with a homogenized core
layer and a single cover plate. In different models changes
of the cover plate thickness (tp) are applied. In addition, the
influence of changes in the central exciting frequency is
also evaluated.
3.1 Voltage and displacement based time signal
In this part the signals based on voltage and out of plane
displacement are compared. Figure 3 compares the sensor
signals in the time domain calculated with different exci-
tation frequencies from 5 to 150 kHz. In the first row the
responded voltage signals at the sensor patch, and in the
second row the z-displacement at the reference node on top
of the sensor patch are shown. It can be seen in the low
frequency range that both results are not matching.
Table 1 Geometrical properties of the sandwich panel (units are in
mm)
Skin plate Honeycomb cell [9] PZT actuator/sensor [9]
Length Width Cell size Height Radius Thickness
290 124 4.8 15 3.17 0.7
Table 2 Material properties of the plate and honeycomb cells
Young’s modulus (GPa) Poisson’s ratio Density (kg m-3)
Skin plate (aluminum alloy T6061 [9])
70 0.33 2,700
Ex = Ey (GPa) Ez (GPa) txy and tyz = txz Gxy (GPa) Gyz = Gxz (GPa) Density (kg m-3)
Honeycomb cell (HRH-36-1/8-3.0) [8]
2.46 3.4 0.3 0.94615 1.154 50
72 S. M. H. Hosseini et al.
123
But with increasing central frequency of the exciting signal
a better agreement can be observed for all models.
3.2 Wave field
A 3-D snapshot of the wave field uz in the sandwich panel
with the plate layer of 1.75 and 0.75 mm (tp) is shown in
Fig. 4. It can be seen that the group velocity and the wave
length is nearly the same for the A0 mode. On the other
hand, Fig. 5 shows the 2-D wave forms in a quarter of the
plate calculated with the three different models; the central
frequency of 150 kHz and a cover plate thickness 0.5 mm
(tp) is used.
As the conclusion in this part it must be mentioned that
in Fig. 4 one can see that for a different thickness of the
skin plate (tp) the wave still can travel inside the structure
-1.0
-0.5
0.0
0.5exciting
S0 S0
A0S0
A0
-1.0
-0.5
0.0
0.5
Nor
mal
ized
Vol
tage
(-)
Nor
mal
ized
Dis
plac
emen
t(-)
0.00025 0.00050 0.00075
S0
A0
A0
S0
A0 S0
A0
0.00005 0.00010 0.00015 0.00003 0.00005 0.00008 0.00003 0.00005 0.00008
Time (s) Time (s) Time (s) Time (s)
HoneycombSingleplateSimplifiedmodel
exciting
exciting exciting
exciting
exciting
excitingF = 40 kHzF = 5kHz
F = 5kHz F = 40 kHz
F = 100 kHz
F = 100 kHz
F = 150 kHz
F = 150 kHzexciting
(a) Time signal obtained based on nodal voltage of nodes on free surface of PZT sensor
(b) Time signal obtained based on out of plane nodal displaciment of nodes on free surface of PZT sensor
Fig. 3 Responded time domain voltage signals based on voltage and
out plane displacement. The results are plotted for honeycomb (solidline), simplified model (big dashed) and single plate (small dashed).
The thickness of the cover plate is (tp) 0.5 mm and the honeycomb
thickness (th) is 1.48 mm. The sensor is 180 mm far from the actuator
Z
(a)
(b)-9.615e-010-7.693e-010-5.772e-010-3.850e-010-1.929e-010-7.260e-013 1.914e-010 3.836e-010 5.757e-010 7.679e-010 9.600e-010
Inc: 200Time: 4.4e-005
A0
X Y
S
A0
S
Fig. 4 The wave field of uz in sandwich panel structure, the exciting
signal with central frequency of 100 kHz has been generated. In the
model a tp is 1.75 mm and in part b tp is 0.75 mm. In both cases th is
1.48 mm
-9.600e-010-7.680e-010-5.760e-010-3.840e-010-1.920e-010 0.000e+000 1.920e-010 3.840e-010 5.760e-010 7.680e-010 9.600e-010
Inc: 250Time: 5.500e-005
X
Y
Displacement Z (m)
Honeycomb Simplified Single plate
A0
A0
A0
Sensor
Actuator
SSS
Fig. 5 The uz wave field on top surface of honeycomb, simplified and single plate structures. Central frequency of the exciting signal is 150 kHz,
tp is 0.5 mm and th is 1.48 mm
Lamb wave propagation 73
123
distinguishably. However, a pervious study by Hosseini
and Gabbert [3] has shown that for a weaker honeycomb
core and thicker cover plate of the sandwich panel, the
wave mostly propagates on the top plate. In addition, it has
been shown in Fig. 5 that the wave form is similar in all
three models. However, the group velocity of the traveling
wave is higher in the single plate compare with the two
other models.
3.3 Group velocity
Figure 6 shows the influence of changes in the exciting
frequency on the group velocity in models with specific
geometrical properties. The results are calculated on the
top and on the bottom plates for different frequency ranges.
In Fig. 7 the influence of the cover plate thickness on the
group velocity of the A0 mode on the top and on the bottom
layer is presented for all three different models. Only the
mode A0 is shown in the figure, because the S0 mode is
hardly seen in models with higher plate thicknesses. It must
be mentioned that the presented results are fitted linearly.
To summarize the results in this part, one can see in
Fig. 6 that both S0 and A0 propagate slower on the bottom
plate in all cases. As expected, the S0 mode propagates
faster than A0. In Fig. 7 it is obvious that the group velocity
of the single plate model is not influenced by the plate
thickness; the solutions calculated with the two other
models show more significant influence of the thickness.
All three models indicate that the wave propagates faster
on the top layer than on the bottom layer.
3.4 Wave length
Figure 8 shows the wave length of different modes calcu-
lated for the top and the bottom layer in dependence of the
frequency. In Fig. 9 the wave length of the anti-symmetric
modes calculated from the nodal and voltage signals of the
top surface sensors are compared. The cover plate thick-
ness of the honeycomb structure (tp) is changing from 0.5
till 2 mm and the central frequency of the exciting signal is
100 kHz.
As the result Fig. 8 shows that the wavelength of the A0
mode in all there models are in a good agreement. In Fig. 9
it can also be seen that the results from the voltage and
nodal signals are in a same range. Also, it is clear that by
increasing of tp, the wavelength is decreasing in both top
and bottom surfaces in all models.
0
1000
2000
3000
4000
Gro
upve
loci
ty(m
/s)
0 100 200 300 400
Centeral exciting frequency (kHz)
HoneycombSingle plateSimplified modelTop surface Bottom surface
A0
S0
S0
A0
0
1000
2000
3000
4000
Gro
upve
loci
ty(m
/s)
0 100 200 300 400
Centeral exciting frequency (kHz)
th p= 1.48, t = 0.5 mm th p= 1.48, t = 0.5 mm
Fig. 6 The influence of changes in the central exciting frequency on the group velocity
0
1000
2000
3000
4000
Gro
upve
loci
ty(m
/s)
0.50 0.75 1.00 1.25 1.50 1.75 2.00
Plate’s thickness, t (mm)p
Top surface
A0 Honeycomb
Single plate
Simplified model Bottom surface
A0
0
1000
2000
3000
4000
Gro
upve
loci
ty(m
/s)
0.50 0.75 1.00 1.25 1.50 1.75 2.00
Plate’s thickness, t (mm)p
S0
S0
Fexciting = 100 kHz Fexciting = 100 kHz
th = 1.48 mm th = 1.48 mm
Fig. 7 Dependency of the group velocity on the cover plate thickness, which is changing from tp equal 0.5–2 mm
74 S. M. H. Hosseini et al.
123
4 Summary and outlook
The Lamb wave propagation in honeycomb sandwich
panels excited and received with piezoelectric actuators
and sensors has been analyzed with three different 3-D
finite element models. In the detailed 3-D model the hon-
eycomb core layer is modeled with shell type elements. In
the simplified model, the core layer has been homogenized
to an orthotropic layer which is also modeled with 3-D
finite elements. In the third model, the honeycomb plate is
modeled as a single layer where the influences of the core
and bottom face are refused. It has been shown that for the
investigated parameter variations the thickness of the skin
plates (tp) does not play a significant role to prevent waves
travel inside the structure. Also, it has been shown that the
wave form in all three models is similar. However, by
comparing the group velocity values in the honeycomb
sandwich panel and single plate; it has been observed that
the honeycomb core layer in the extended honeycomb
model causes lower group velocity values on the bottom
surface compared to the top surface, while in the single
plate model group velocity on top and bottom layers are
nearly the same. Furthermore, it has been figured out that
the responded voltage signals and nodal displacements at
the sensor are nearly the same. If the central frequency of
the exciting signal is increasing, also a better agreement
between the three different models could be observed. The
comparison of the group velocities calculated with
the three different models has shown that both the S0 and
the A0 mode propagate slower on the bottom plate. For
different thicknesses of the skin plate (tp), the group
velocities calculated with the different models are only
slightly changing. The wave length of the different modes
has also been calculated. It has been shown that when tp is
increasing the wavelength is decreasing in both top and
bottom surfaces in all models. The wavelengths are eval-
uated with the voltage signals of the sensor as well as with
the normal displacements of a nodal point on top of the
sensor. Both results are in a good agreement in all models.
To understand better the energy transmission phenom-
ena through the honeycomb structure further investigations
are required to receive a global map for the energy trans-
mission and the wave length changes inside the structure in
dependence of the thicknesses tp/th and the applied fre-
quency range. In addition, further studies are also needed to
evaluate the influence of the other geometrical properties
on the wave propagation, such as cell size and cell height,
as well as the sandwich panel material properties on the
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Wav
ele
ngth
(m)
0 100 200 300 400
Centeral exciting frequency (kHz)
A0
S0
Top surface
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Wav
ele
ngth
(m)
0 100 200 300 400
Centeral exciting frequency (kHz)
A0
S0
Bottom surface
Honeycomb
Single plate
Simplified model
th p= 1.48, t = 0.5 mm th p= 1.48, t = 0.5 mm
Fig. 8 Wave length of different modes on the top layer (left) and on the bottom layer (right) based on changes of central exciting frequency from
5 to 400 kHz
Voltage signal
A0 mode
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
Wav
ele
ngth
(m)
0.50 0.75 1.00 1.25 1.50 1.75 2.00
Plate’s thickness, t (mm)p
HoneycombSingle plateSimplified modelNodal signal
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
Wav
ele
ngth
(m)
0.50 0.75 1.00 1.25 1.50 1.75 2.00
Plate’s thickness, t (mm)p
A0 modeTop surfaceTop surface
Fexciting = 100 kHz Fexciting = 100 kHz th = 1.48 mmth = 1.48 mm
Fig. 9 Wave length as function of the plate thickness
Lamb wave propagation 75
123
wave propagation in such a structure. Besides numerical
studies also experiments are under progress to improve the
development of health monitoring systems.
Acknowledgments Hereby the authors appreciate the financial
support of the German Research Foundation (GA 480/13).
References
1. Ahmad, Z., Gabbert, U.: Influence of material variations in
composite plates on lamb wave propagation and edge reflection.
In: Allix, O., Wriggers, P. (eds.) Proceedings of the European
Conference on Computational Mechanics—ECCM 2010 (2010)
2. Boller, C., Chang, F.K., Fujino, Y.E.: Encyclopaedia of Structural
Health Monitoring, vol. 1–5. Wiley, New York (2008)
3. Hosseini, S.M.H., Gabbert, U.: Analysis of guided lamb wave
propagation (GW) in honeycomb sandwich panels. Proc. Appl.
Math. Mech. 10, 11–14 (2010)
4. Kari, S.: Micromechanical Modelling and Numerical Homoge-
nization of Fibre and Particle Reinforced Composites. VDI
Verlag, Germany (2007)
5. Liu, G.R., Quek Jerry, S.S.: A non-reflecting boundary for ana-
lyzing wave propagation using the finite element method. Finite
Elem. Anal. Des. 39, 403–417 (2003)
6. Paget, A.: Active health monitoring of aerospace composite
structures by embedded piezoceramic transducers. Department of
Aeronautics Royal Institute of Technology, Sweden (2001)
7. Sachse, W., Pao, Y.: On the Determination of Phase and Group
Velocities of Dispersive Waves in Solids. Physics and Astronomy
Classification Scheme, PACS, USA (1977)
8. Semkat, M.: Diploma thesis. Institut fur Mechanik, Fakultat fur
Maschinenbau, Otto-von-Guericke-Universitat Magdeburg, Ger-
many (2009)
9. Song, F., Huang, G.L., Hudson, K.: Guided wave propagation in
honeycomb sandwich structures using a piezoelectric actuator/
sensor system. Smart Mater. Struct. 18, 125,007–125,015 (2009)
10. Song, F., Huang, G.L., Kim, J.H., Haran, S.: On the study of
surface wave propagation in concrete structures using a piezo-
electric actuator/sensor system. Smart Mater. Struct. 17,
055,024–055,032 (2008)
11. Swartz, A., Backman, D., Flynn, E.: Guided wave propagation in
honeycomb sandwich structures using a piezoelectric actuator/
sensor system. Los Alamos Dynamics Summer School, Los
Alamos National Laboratory, Los Alamos, USA (2006)
12. Ungethuem, A., Lammering, R.: Impact and damage localization
on carbon-fibre-reinforced plastic plates. In: Casciati, F., Giord-
ano, M. (eds.) Proceedings 5th European Workshop on Structural
Health Monitoring, Sorrento, Italy (2010)
76 S. M. H. Hosseini et al.
123