Inﬂuence of the cover plate thickness on the Lamb wave ... · PDF fileInﬂuence of the...

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


R. Lammering

Institut fur Mechanik, Helmut Schmidt Universitat Hamburg,

39106 Hamburg, Germany


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



























sensor bottom

increasing damping factor

increasing damping factor FEMAPMaterial2:asd

material7















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


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


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


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


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


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


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


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


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


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


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)


123

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