STRUCTURAL HEALTH MONITORING USING LAMB WAVES ACTUATED BY
SURFACE-MOUNTED PIEZOELECTRIC MATERIALS
By
Lassaad Mhamdi

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree
MASTER OF SCIENCE
Civil Engineering
2011

ABSTRACT
STRUCTURAL HEALTH MONITORING USING LAMB WAVES ACTUATED BY
SURFACE-MOUNTED PIEZOELECTRIC MATERIALS
By
Lassaad Mhamdi
Piezoelectric materials have gained wide acceptance in the world and have been
successfully used in a myriad of applications (industrial, engineering, etc.) thanks to their
potential and ability of electromechanical conversion. The research in this thesis focuses on
exploring the capabilities of surface-mounted piezoelectric (PZT) patches in exciting and
detecting lamb waves in order to assess the integrity of a sample of aluminum plates. Therefore,
numerical and analytical formulations of the impedance of an unbonded PZT patch were
conducted and two different S.H.M techniques: the ultrasonic nondestructive technique of pitchcatch and the traditional technique of phased arrays were studied. First, a free PZT patch was
modeled both analytically and numerically in order to see how its impedance varies depending
on its mechanical parameters. The modeling can help understanding the electromechanical
behavior of the patch and control its quality before using it in SHM tasks. Second, the technique
of pitch catch was used in order to study the effect of crack depth on the propagation of lamb
waves. For this reason, experiments on aluminum plates with defects of variable depths were
conducted and results show significant variations of the energy of the wave as function of the
crack depth. Finally, the technique of phased arrays, using a linear array of piezoelectric patches,
was investigated in order to focus the lamb waves in specific directions and enhance the
detection of small notches manufactured in the aluminum plates. Results showed that the simple
phased array made of six simple piezoelectric transducers was able to detect the notches
predefined on the aluminum plates.

AKNOWLEDGEMENTS

In the name of Allah, the Most Beneficent, the Most Merciful

First and foremost, may all praise and glory be to Almighty “Allah” whom without his blessings
and guidance, I would not be able to finish this thesis. Alhamdulillah, Thank you Allah for
giving me the patience and the strength.
I would like to express my deepest gratitude and my most considerate acknowledgements to both
my advisor, Dr. Nizar Lajnef and my co-advisor Dr. Jung Wuk Hong for their patient guidance,
and their priceless, insightful and knowledgeable advices. Their incessant support and help have
been of a great importance to me through this work. Most considerate acknowledgements and
deepest gratitude are also due to Dr. Lalita Udpa for her support and help and for accepting to
serve as a member of my committee.
Thanks are due to all the wonderful people and colleagues I had the great opportunity to know
during the years I spent at Michigan State University. Without their generous support and help,
this work would not be possible.
I am very grateful for all the members of my loving family and all my dearest friends for their
encouragements. Words cannot express my gratitude for their continuous emotional support.

iii

TABLE OF CONTENTS
LIST OF TABLES ....................................................................................................................... vi
LIST OF FIGURES .................................................................................................................... vii
Chapter 1
Introduction ....................................................................................................................................1
1. Literature Review: .....................................................................................................................5
1.2. Magnetostrictive Transducers: ............................................................................................7
1.3. Fiber Optic Sensors: ............................................................................................................9
1.4. Piezoelectric Sensors: .......................................................................................................11
1.5. Issues in Structural Health Monitoring: ............................................................................14
1.6. Phased Array Structural Health Monitoring: ....................................................................15
2. Thesis Overview: .....................................................................................................................15
2.1. Research Objectives: .........................................................................................................15
Chapter 2
Piezoelectric Materials:Properties and Characteristics ...........................................................17
1. Introduction to Piezoelectricity: ...............................................................................................17
2. Understanding of the Piezoelectric Effect: ..............................................................................18
3. Piezoelectric Materials: ............................................................................................................20
3.1. Polarization of Piezoelectric Materials: ............................................................................21
4. Piezoelectric Constitutive Equations: ......................................................................................22
5. Electrical Impedance of a Piezoelectric Patch: ........................................................................24
5.1. Introduction: ......................................................................................................................24
5.2. Analytical Formulation of the Electrical Impedance of a PZT: ........................................24
5.3. Numerical Modeling of an Unbonded PZT Patch: ...........................................................26
5.3.1. Properties of the Material: ......................................................................................27
5.3.2. Geometry and Meshing of the Numerical Model: ..................................................29
5.4. Impedance Plots: ...............................................................................................................29
5.5. Parametric Study: ..............................................................................................................30
Conclusion: ....................................................................................................................................51
Chapter 3
Lamb Waves: General Overview ...............................................................................................53
1. Introduction to Lamb Waves: ..................................................................................................53
1.1. Elastic Waves in Solids and Structures:............................................................................53
2. Lamb Waves: ...........................................................................................................................56
2.1. Historical Overview: .........................................................................................................56
2.2. Theoretical Backgrounds: .................................................................................................57
2.2.1. Governing Equation: ...............................................................................................57
2.2.2. Symmetric Modes: ..................................................................................................59
2.2.3. Antisymmetric Modes:............................................................................................59
2.2.4. Dispersion of Lamb waves: ....................................................................................60
2.3. Lamb waves in Structural Health Monitoring: .................................................................62
iv

3. Conclusion: ..............................................................................................................................63
Chapter 4
Damage Detection in Aluminum plates......................................................................................64
1. Introduction: .............................................................................................................................64
2. Ultrasonic Health Monitoring Techniques: .............................................................................65
2.1. Pulse-Echo Technique: .....................................................................................................65
2.2. Pitch-Catch Technique: .....................................................................................................65
3. Ultrasonic Health Monitoring of Finite Aluminum Plates: .....................................................67
3.1. Geometrical and Electro-mechanical Properties:..............................................................68
3.2. Aluminum Plate of Thickness 2mm: ................................................................................69
3.2.1. Experimental Set-up: ..............................................................................................69
3.2.2. Results and Discussions: .........................................................................................71
3.3. Aluminum Plate of Thickness 4mm: ................................................................................72
3.3.1. Experimental Set-up: ..............................................................................................73
3.3.2. Actuator-Sensor-Notch Configuration: ...................................................................74
3.3.3. Actuator-Notch-Sensor Configuration: ...................................................................78
4.Conclusion: .................................................................................................................................80
Chapter 5
Phased Array Health Monitoring ...............................................................................................81
1. Introduction: .............................................................................................................................81
2. Ultrasonic Method of Phased Array: .......................................................................................83
2.1. Concept: ............................................................................................................................83
2.2. Principles: .........................................................................................................................83
3. Piezoelectric Array for Inspection of Aluminum Plates: .........................................................84
3.1. Description: .......................................................................................................................84
3.2. Experimental Setup: ..........................................................................................................85
3.3. Time Delays: .....................................................................................................................87
3.4. Healthy Plate: ....................................................................................................................88
3.5. Defective Plates: ...............................................................................................................89
3.6. Results and Discussions: ...................................................................................................89
4. Conclusion: ..............................................................................................................................96
Chapter 6
Conclusion ....................................................................................................................................97
APPENDIX A ...............................................................................................................................99
APPENDIX B .............................................................................................................................104

v

LIST OF TABLES

Table 2.01: Forms of piezoelectric constitutive relations ............................................................. 24
Table 2.02: Properties and expressions of the parameters used in the analytical model .............. 26
Table 2.03: Geometrical and Mechanical properties of the analytical and numerical models ..... 28
Table 2.04: Variation of the Coefficient d51 ................................................................................ 31
Table 2.05: Variation of the Coefficient d32 ................................................................................ 34
Table 2.06.a: Variation of the Coefficient d33 (d33 increases) ..................................................... 36
Table 2.06.b: Variation of the Coefficient d33 (d33 decreases) .................................................... 36
Table 2.07: Variation of the Coefficient S12 ................................................................................ 39
Table 2.08: Variation of the Coefficient S13 ................................................................................ 40
Table 2.09: Variation of the Coefficient S23 ................................................................................ 42
Table 2.10: Variation of the Coefficient S22 ................................................................................ 44
Table 2.11: Variation of the Coefficient S33 ................................................................................ 45
Table 2.12: Parametric Study and effects on the numerical impedance of the PZT ..................... 50
Table 4.01: Electromechanical properties of the SM410 material ............................................... 68
Table 4.02: Geometrical properties of the aluminum plates ......................................................... 72
Table 4.03: Velocities of the fundamental modes S0 and A0 in the aluminum plates. ................ 75
Table 5.01: Matrix of signals collected from each plate............................................................... 86

vi

LIST OF FIGURES

Figure 1.01: Crash of Shuttle Columbia on March 14, 2003 (www.life.com) ............................... 4
Figure 1.02: Inspection of aircrafts (www.life.com) ...................................................................... 4
Figure 1.03: Smart Magnetoresistive Sensor .................................................................................. 8
Figure 1.04: Fiber Bragg Grating method..................................................................................... 10
Figure 2.01: Coupling effects between Elasticity and Electricity................................................. 18
Figure 2.02: Molecular Modeling of Piezoelectricity Effect: (a) Unagitated Molecule, (b)
Molecule under external stress (c) Polarization Effect ............................................ 19
Figure 2.03: Polarization of the piezoelectric material and generation of electric field ............... 19
Figure 2.04: Different natural and made-up piezoelectric materials ............................................ 21
Figure 2.05: Polarization of a Ceramic Piezoelectric Material ..................................................... 25
Figure 2.06: Other forms of the piezoelectric constitutive equations ........................................... 25
Figure 2.07: Z-polarized Piezoelectric Patch under Electric Voltage........................................... 25
Figure 2.08: ANSYS FEM Model of the Piezoelectric Patch ...................................................... 29
Figure 2.09: Analytical and Numerical Impedance of the Unbounded PZT Patch ...................... 30
Figure 2.10: Effects of the coefficients d51, d42, S44 and S55 on the impedance of the PZT ....... 33
Figure 2.11: Effect of the strain coefficient d32 on the impedance of the PZT ............................ 35
Figure 2.12.a: Effect of the coefficient d33 on the impedance of the PZT (d33 increases) .......... 36
Figure 2.12.b: Effect of the coefficient d33 on the impedance of the PZT (d33 decreases).......... 36
Figure 2.13: Effect of the Variation of the compliance coefficient of S12 ................................... 39
Figure 2.14: Effect of the Variation of the compliance coefficient of S13 ................................... 41
Figure 2.15: Effect of the Variation of the compliance coefficient of S23 ................................... 43
Figure 2.16: Effect of the Variation of the compliance coefficient of S22 ................................... 44

vii

Figure 2.17: Effect of the Variation of the compliance coefficient of S33 ................................... 46
Figure 2.18: Effect of the variation of the damping coefficient ζ on the impedance of the PZT . 47
Figure 2.19: Effect of the variation of the meshing module ......................................................... 49
Figure 2.20: Reduction of the error between the analytical and numerical results....................... 51
Figure 3.01: Propagation of (a) Longitudinal waves, (b) Shear waves ........................................ 55
Figure 3.02: Propagation of Rayleigh waves. ............................................................................... 56
Figure 3.03: Schematic of Lamb wave propagation in symmetric mode ..................................... 59
Figure 3.04: Schematic of Lamb wave propagation in antisymmetric mode ............................... 59
Figure 3.05: Presentation of a Morlet wave in the time and frequency domains ......................... 60
Figure 3.06: Phase velocity in aluminum plates for symmetric lamb wave modes ...................... 61
Figure 3.07: Phase velocity in aluminum plates for antisymmetric lamb wave modes ................ 62
Figure 4.01: Schematics of the pulse-echo technique ................................................................... 66
Figure 4.02: Schematics of the pitch-catch technique .................................................................. 66
Figure 4.03: Experimental set-up of a standard pitch-catch configuration ................................... 67
Figure 4.04: Piezoelectric wafer used in the experiments ............................................................ 68
Figure 4.05: Experimental setup ................................................................................................... 70
Figure 4.06: Input signal for lamb wave actuation: Morlet wavelet ............................................. 70
Figure 4.07: Output signal sensed by the receiver ........................................................................ 71
Figure 4.08: Geometry of an aluminum plate containing a predefined notch .............................. 73
Figure 4.09: (a) Defective plates and (b) Healthy plate in an Actuator-Sensor-Notch
configuration ................................................................................................................................. 74
Figure 4.10: Input signal and Sensed signals at 250 KHz. ........................................................... 77
Figure 4.11: Time – Frequency plot at 250 KHz for different crack depths (a) 40%T, (b)
60%T,(c)80%T.............................................................................................................................. 78
Figure 4.12: (a) Defective plates and (b) Healthy plate in an Actuator -Notch-Sensor
configuration. ................................................................................................................................ 79

viii

Figure 4.13: S0 mode at 250 KHz for deferent crack depths........................................................ 80
Figure 5.01: Beam focusing on misoriented cracks ...................................................................... 83
Figure 5.02: Beam forming technique .......................................................................................... 84
Figure 5.03: Sketch of the phase array on the healthy plate ......................................................... 85
Figure 5.04: Sketch of the phase array on a defective plate ......................................................... 86
Figure 5.05: Sketch of the phased array for a random location (D, θs) ........................................ 88
Figure 5.06: Sketch of the phased array and corresponding time delays for the healthy plate .... 88
Figure 5.07: Sketch of the phased array and corresponding time delays for a defective plate ..... 89
Figure 5.08: Signals Received by the Sensors on the healthy plate.............................................. 90
Figure 5.09: Signals Received by the Sensors for the plate with 40% defect depth..................... 91
Figure 5.10: Signals Received by the Sensors for the plate with 60% defect depth..................... 92
Figure 5.11: Signals Received by the Sensors for the plate with 80% defect depth..................... 93
Figure 5.12: Total Signal Received by the array on the healthy plate .......................................... 94
Figure 5.13: Total Signal Received by the array (40% defect depth) ........................................... 95
Figure 5.14: Total Signal Received by the array (60% defect depth) ........................................... 95
Figure 5.15: Total Signal Received by the array (80% defect depth). .......................................... 96
Figure B.01: Experimental signals at 100 KHz .......................................................................... 105
Figure B.02: Experimental signals at 125 KHz. ......................................................................... 106
Figure B.03: Experimental signals at 150 KHz. ......................................................................... 107
Figure B.04: Experimental signals at 175 KHz. ......................................................................... 108
Figure B.05: Experimental signals at 200 KHz. ......................................................................... 109
Figure B.06: Experimental signals at 225 KHz. ......................................................................... 110
Figure B.07: Experimental signals at 250 KHz. ......................................................................... 111
Figure B.08: Experimental signals at 275 KHz. ......................................................................... 112
Figure B.09: Experimental signals at 300 KHz. ......................................................................... 113
ix

Figure B.10: Time– frequency representation at 100 KHz for deferent crack depths ................ 114
Figure B.11: Time – frequency representation at 125 KHz for deferent crack depths ............... 114
Figure B.12: Time – frequency representation at 150 KHz for deferent crack depths ............... 115
Figure B.13: Time – frequency representation at 175 KHz for deferent crack depths ............... 115
Figure B.14: Time – frequency representation at 200 KHz for deferent crack depths ............... 116
Figure B.15: Time – frequency representation at 225 KHz for deferent crack depths ............... 116
Figure B.16: Time – frequency representation at 250 KHz for deferent crack depths ............... 117
Figure B.17: Time – frequency representation at 275 KHz for deferent crack depths ............... 117
Figure B.18: Time – frequency representation at 300 KHz for deferent crack depths ............... 118
Figure B.19: S0 mode at 100 KHz for deferent crack depths ..................................................... 119
Figure B.20: S0 mode at 125 KHz for deferent crack depths ..................................................... 119
Figure B.21: S0 mode at 150 KHz for deferent crack depths ..................................................... 120
Figure B.22: S0 mode at 175 KHz for deferent crack depths ..................................................... 120
Figure B.23: S0 mode at 200 KHz for deferent crack depths ..................................................... 121
Figure B.24: S0 mode at 225 KHz for deferent crack depths ..................................................... 121
Figure B.25: S0 mode at 275 KHz for deferent crack depths ..................................................... 122
Figure B.26: S0 mode at 300 KHz for deferent crack depths ..................................................... 122

x

Chapter 1
Introduction
Repair, maintenance and rehabilitation of structures in different fields (aerospace engineering,
maritime engineering, civil engineering, mechanical engineering, etc.) and sectors (government,
private and commercial) are usually tasks that necessitate both considerable amounts of time and
highly varying costs depending on the age of structures and on their importance. The United
States spends yearly a huge amount of money, more than $200 billion, for only maintenance
tasks (maintain buildings, bridges, aircraft, railroads, and other infrastructure). Maintenance of
commercial US aircrafts, for example, is estimated to a quarter of operating costs.
Approximately 200000 bridges in the US national inventory need either repair (because of
structural deficiencies, deteriorations, reduced bearing capacities, etc.) or replacement (because
of geometrical non conformity to new standards). Costs of reparation and maintenance in 2004,
of some deficient and obsolete bridges were estimated to $12 billion as reported by the Federal
Highway Administration in 2006 (FHWA). All those high costs of reparation and maintenance
aim at reducing as maximum as possible the risk for the general public since the high number of
damaged structures put the human safety in concern and any possible failure of a frequently used
structure (aircraft, bridge, building, etc.) may lead to catastrophic results (loss of human lives,
loss of the structure, etc.). Many examples of catastrophic failures of daily used structures show

1

the big need for more advanced and accurate inspection methods that effectively reveal on time
all the appropriate safety concerns. The most horrific bridge failure in the United States was the
collapse in December 5, 1967 of the silver bridge which caused death of 46 people, serious
injuries for 9 persons and struck fear across the US. Investigation of the wreckage of the bridge
showed that the cause of the collapse was the failure of a single eyebar in a suspension chain,
due to a small defect of 0.1 inch depth. A less horrific bridge failure, as measured in terms of loss
of life was the collapse, on August 1, 2007, of the Mississippi River Bridge in Minneapolis
which resulted in the loss of 13 lives and the injury of 145 persons. The National Transportation
Safety Board (NTSB) cited in the investigation that a design flaw was likely the cause of the
collapse. Both collapses were caused by small structural faults. In Train tracks, faults caused also
approximately 1080 train accidents in 2005 which resulted in approximately $220 Million of
equipment damages and $114 Million of track damages as reported by the Federal Railroad
Administration in its annual report of 2005 (FHWA). In aviation industry, huge amount of
money is spent yearly on maintenance of aircrafts which make the probability of failure less
compared to the probability of failure in other industries. Despite all the effort made in
inspection (high cost, long time of inspection, etc.), the frequency at which aircrafts are inspected
is limited which may increase the chance of not detecting faults that may, depending on their
severity, result in dramatic accidents that cause more and more costs and send more lives to
death. The National Transportation Safety Board recorded two fatal accidents in 2006 due to
structural faults, which resulted in the loss of 50 lives (NTSB).
All those dramatic accidents due to tiny structural blemishes put in evidence the great need for
developing new accurate and reliable inspection techniques that can be implemented on different

2

structures (bridges, aircrafts, railroads, buildings, etc.) in order to considerably enhance structural
and human safety and reduce maintenance costs.
A massive scientific effort has been devoted during the two last decades to build up novel
reliable techniques for the diagnosis and assessment of structures. Researchers have been trying
to come up with new techniques that will grant efficient inspections, reduce considerably the cost
of maintenance, eliminate unscheduled maintenance, address problems related to aging structures
and extend the service life of expensive structures. The scientific endeavor has led to the
development of many methods of inspection, the most important technique that has gained a
momentum in use in industrial communities and much interest in scientific community is the
Structural Health Monitoring (SHM). This technique has provided world-wide confidence and
satisfaction leading to its acceptance as prominent and efficient approach of structural
evaluation. Its concept is straightforward: every structure has sensitive features that are subjected
to changes as the structure degrades and gets old with time, the idea behind the Structural Health
Monitoring is to measure the changes of those sensitive structural features, collect and save
periodical measurements and extract important information (dynamic signature of the structure,
properties, etc.) in order to give a diagnosis and a prognosis (residual life, evolution of cracks
and damage, etc.) of the structure at every moment during its service life.
SHM offered a significant boost to the area of health monitoring introducing powerful
techniques of monitoring that improve the detection of damages, help reducing the maintenance
costs, help reducing common design and technical errors and help enhancing the overall safety of
structures. The most important potential of the technique, however, is that it enables continuous
monitoring of structures. A SHM system mounted on a bridge, for example, could be used for
incessant control of the structure, without interrupting the traffic and could help detecting small

3

flaws in their beginning stages of growth. On aging aircrafts, SHM packages could be used to
detect possible problems, help limiting unscheduled maintenance and help repairing the
structure, when necessary, based on its condition. Many other SHM schemes have been
successfully used for the inspection of other infrastructures

(buildings, railroads, etc.). In all

cases, safety of the structures could be enhanced and costs of maintenance could be reduced.
SHM is an area of growing interest due to its advantageous capabilities which put it in the
forefront of the enabling technologies. Implementation of some SHM systems on existing
structures, however, still faces many challenges such as the development of autonomous
vibration transducers, self-powered inspecting systems, low power wireless systems.

(a) Shuttle Columbia During launching

(b) Wreckage of the crashed shuttle

Figure 1.01: Crash of Shuttle Columbia on March 14, 2003 (www.life.com)
“For interpretation of the references to color in this and all other figures, the reader is referred to
the electronic version of this thesis.”

(a) Inspection of an aircraft

(b)

Robot inspection of a fuselage

Figure 1.02: Inspection of aircrafts (www.life.com)
4

1. Literature Review:
Tremendous amount of research has been conducted in the area of structural health monitoring.
Many scientists excelled in this field and published valuable works in books and papers. The
following is a literature review on the most important contributions that have been achieved on
the area of structural health monitoring over the past few decades with an emphasis on works
that involved the development of novel effective methods and schemes, especially methods
which involve the use of Lamb waves.

1.1.

SHM Overview:

During the last few decades, there has been a lot of interest, among researchers, in the field of
Structural Health Monitoring; this growing interest has led to valuable advancement in
inspection techniques. In fact, many inspection tasks that have been hard to carry out two
decades ago due to different difficulties (complex structures, very small faults, etc.) are now
feasible to accomplish due to the innovative brilliant and enabling techniques the SHM has
brought. Many online monitoring operations that have been difficult to conduct, two decades
ago, using the ordinary nondestructive evaluation (NDE) methods are now easy to perform using
smart monitoring devices permanently mounted on structures.

Several others methods of

inspection have been put in practice and have showed a great success for the industrial
community. The following is a chronological overview of the most important historical events
and technological breakthroughs that led to the acceptance, among the world, of structural health
monitoring as the preeminent most proficient inspection scheme.
The real beginning of the research on the structural health monitoring, started with the first
IWSHM (International Workshop on Structural Health Monitoring) launched in 1997 by

5

Professor Fu-Kuo Chang and held at Stanford University. This workshop received an
overwhelming success and was followed by a second workshop in 1999 where researchers
emphasized on the identification of promising key technologies for industrial monitoring
applications. From that time, researchers have devoted huge amounts of their times for the
understanding of the concepts related to SHM and the development of new ideas, formulations
and experiments in order to give answer to aroused questions related to inspection limitations.
We count today more several journals and conferences throughout the world where big parts are
devoted to the field of structural health monitoring. Many ideas and techniques that have been
first proposed during conferences or first cited in journals can be now found fully documented on
books, articles and review articles. The published documentation show that the first attempts on
the development of new SHM techniques were first performed in order to find ways for the
damage identification of structures under ambient and forced vibration. Doebling et al [10]
presented the first review article on this issue where damage identification of structural and
mechanical systems can be performed through the measurement of changes of vibration
characteristics. The same authors published in the Shock and Vibration Digest in 1998 a revised
and more condensed version of their 1996 review article. In 2001, Sohn et al [47] published an
article in which they presented a general review of structural health monitoring from 1996 to
2001. Many others researchers excelled in the field and published many articles and books
related to the field of structural health monitoring such as the book by Inman et al [26], the book
by Balageas et al [06] the book by Giurgiutiu [21] in which he fully documented his work on the
health monitoring of structures using Piezoelectric Wafer Active Sensors (PWAS), etc.
Ample literature on diversified methods and techniques related to the field of structural health
monitoring is available on different manuscripts (books, articles, reports, reviews, web, etc). In

6

this thesis, we emphasize on methods that involve the use of smart actuators and sensors for
assessment and damage identification of structures, especially techniques that involve use of
piezoelectric materials. Those techniques have gained wide acceptance due to the simplicity in
implementation, efficiency and due to the numerous enabling features they present. In the
following, a succinct description is given to highlight the importance of two main types of smart
materials (magnetostrictive materials and fiber optic materials) and a more elaborated description
is given to illustrate the efficiency of using piezoelectric materials for monitoring applications.
The three smart materials greatly contribute to the evolution of the structural health monitoring
area.

1.2. Magnetostrictive Transducers:
Ferroelectric and ferromagnetic materials (cobalt, nickel, iron, etc.) have the ability to undergo
changes in size and/or shape in the presence of a magnetic field, this physical property is known
as Magnetostriction. A magnetostrictive material undergoes changes in dimension when placed
under a time varying magnetic field; it enlarges with positive magnetostriction and shrinks with
negative magnetostriction, this effect is called the magnetostrictive effect. Inversely, a
magnetostrictive material undergoes changes in its magnetic properties when subjected to a time
varying stress field; this effect is referred to as the Villari effect. Both effects have been of great
utility in structural health monitoring related tasks. The magnetostrictive effect has been used for
actuating purposes; changes in shape of a ferroelectric-made transducer under a magnetic field
induce vibrations in the structure on which the actuator is mounted, this will generate a wave that
will propagates along the structure. The inverse magnetostrictive effect (Villari effect) has been
used for sensing purposes; the same transducer is subjected to changes on its magnetic effect due

7

to the presence of a propagating wave in the test structure. The changes in the magnetic
properties of the transducer are recorded and saved in order to be analyzed. For example, the
evaluation, using magnetostrictive materials, of the current state of a structure, is based on the
comparison between baseline measurements from the structure in its healthy state and
measurement from the structure in its actual state. Magnetostrictive actuators are first used to
induce vibrations in the structure and magnetostrictive sensors are used to collect the dynamic
response of the structure. The comparison between the two recorded signals is carried out using
the mathematical processing technique of damage index in order to quantify the damage in the
structure.
Permanents magnets

Preload springs

Output
rod
Coil

Terfenol-D drive rod

Figure 1.03: Smart Magnetostrictive Sensor

Many structural health monitoring tasks have been achieved using magnetostrictive materials.
Weber et al [53] developed and characterized a method for the health monitoring of adhesive
joints by incorporating magnetostrictive filler into the polymeric matrix. Light et al [32]
developed a thin-nickel-strip magnetostrictive guided-wave sensor in order to monitor the
growth of cracks and corrosion wall losses in piping and plates. Ghosh et al [14] proposed a new
numerical method for the structural assessment of laminated composite beams with embedded
magnetostrictive patches. Kwun et al [29] studied the feasibility of magnetostrictive sensors for
8

the detection of corrosion near fasteners in aircraft wings and similar structures. Giurgiutiu et al
[15] conducted an experimental study of magnetostrictive tagged composite strain response for
health monitoring purposes.

1.3.

Fiber Optic Sensors:

Optical fibers are thin, flexible, transparent cables that act as waveguides to transmit the light.
They are made of plastic or glass cores and outer claddings which allow the transmission of the
light to be carried only through the length and not through the lateral surfaces of the cables.
Optical fibers have gained wide acceptance in the research and industrial communities due to
their enabling capabilities and have been widely approved for several engineering applications.
They have been successively applied in structural health monitoring applications due to their
great ability of sensing and consequently detecting changes and damages in structures. The
concept behind the technique of optical fiber is straightforward; an optical fiber embedded into a
structure to be evaluated, represent a thin light pipe; changes and damages in the structure to be
evaluated result directly on changes on the properties of the light (intensity, wavelength, phase,
etc.. By sensing the variation of those properties, one can collect data that contain the
information about the changes and the possible damages that are happening in the structure.
A typical fiber optic-based experiment involves embedding one optical fiber into a structure to
be evaluated, use a source of light to generate a wave in one end of the optical fiber and use a
photo-detector to sense the light in the other end. Three main techniques are used to sense the
light, each rely on measuring the variation of one different property. The first method is based on
sensing the intensity of the light and comparing the amount of light generated by the source to
the amount captured by the photo-detector. Any damage in the structure will result on the

9

variation of the amount of light transmitted through the optical fiber. The second method is based
on the variation of the phase of the light. It resembles to the first method except that, in this
method measurement are effectuated on the phase and not on the intensity of the light. The third
conventional method relies on the measurement of the variation of the wavelength; this method
is commonly known as Fiber Bragg Grating (FBG) (Figure 1.04). In this method, the form of the
cable is different from the standard geometry; the plastic or glass core of the optical fiber
contains equally spaced carves. When the wave encounter a carve, a small band of light will
reflect back to the source. The presence of any damage in the structure will manifest in a shift in
the reflected wavelength due to the presence of induced strain in carves. Measurements of the
wavelength variations, helps quantifying the damages in the structure.

Figure 1.04: Fiber Bragg Grating method
Many researchers conducted studies on the practicability of using fiber optic based sensors for
the health monitoring of structures. Rippet et al [45] showed the ability of an integrated simple
microbend optical sensor to sense information on the elastic energy released by a host structure
whenever damage is being introduced to it. Zhou et al [57] proposed a Fiber Bragg Grating
sensor for the health monitoring of civil infrastructures. Ecke et al [11] used a network of

10

multiplexed system of FGP strain and temperature sensors for the health monitoring of
spacecrafts. Several other publications have contributed to the development of the fiber optic
sensing technique. Measures [39] documents good contributions in his book and give an
excellent review of optical sensors for health monitoring applications.

1.4.

Piezoelectric Sensors:

Piezoelectric materials have been markedly used in a myriad of engineering applications due to
the remarkable characteristics they present. They are known for their great inherited ability of
electromechanical coupling, a property that allow the conversion of mechanical stimuli to
electrical energy and inversely the conversion of electrical stimuli to a mechanical output.
Piezoelectric materials are among the most frequently used materials for structural health
monitoring. In most SHM related applications, transducers made of piezoelectric materials are
surface mounted on host structures and used as sensors and/ or actuators depending on the SHM
task. A piezoelectric material can perform as an actuator to induce vibrations in the host
structures, as a sensor to sense vibrations and stress waves or as both an actuator and a sensor at
the same time. In piezoelectric vibration-related SHM schemes, we distinguish two major
techniques; the first technique is based on the measurement of the electrical impedance of the
piezoelectric and the second technique is based on the propagation of waves in the host structure
to which the piezoelectric is surface-mounted. The former is commonly known as impedance
technique, the latter is termed wave propagation-based technique. For the impedance technique,
the piezoelectric transducer is used to excite the host structure with a high frequency and sense
its structural response. Any variation of the mechanical impedance of the host structure, due to
the presence of damage, will generate a change on the impedance of the piezoelectric transducer

11

(Liang et al [31] ). Damages in the structure are quantified based on the comparison of a baseline
measurement of the structure with current measurements. Both measurements are carried out on
the mechanical impedance of the host structure. Many authors contributed to the development
and the improvement of the piezoelectric impedance-based technique. Park et al [41] presented a
general review of the impedance technique, the theory behind it and stated results of some past
experiments. Giurgiutiu et al [18] used the impedance technique for the detection of damages in
aircraft structures. Grisso et al [22] applied the impedance technique with the aim to monitor the
thermal protection systems on space shuttles. Park et al [42] also discussed the possibility of
using the impedance technique for the detection of cracks in concrete structures. The second
vibration-based technique that involves the use of piezoelectric transducers is the technique of
wave propagation. This method of SHM differs from the impedance based technique in the
concept, in the theory and in the applications. In a wave based technique, a piezoelectric device
attached to the surface of a structure generates a stress wave when fed with an electrical input.
The induced stress wave will propagate through the structure and interact with possible present
defects and flaws. Comparison between baseline measurements and current measurements helps
quantifying the damage and conducting a successful interrogation of the structure. The method of
wave propagation has been in the forefront of the methods used for the nondestructive evaluation
of structures. Perhaps lamb waves stand as the best proof for the success of wave based structural
health monitoring techniques. Those waves have gained a momentum in use in the research and
industrial communities due to their enabling capabilities. They are elastic waves capable of
propagating for long distances in thin solid and shell structures. They remain constrained
between the upper and the bottom surfaces of the structure, hence termed guided waves. A
detailed description of lamb waves, including historical backgrounds, theoretical backgrounds,

12

will be furnished in the third chapter of this thesis. Lamb waves have been extensively studied by
many researchers and have been used in myriad of structural health monitoring applications. One
among the first uses of lamb waves was to study the influence of the homogeneity of materials
on the propagation behavior of the wave. Researchers then realized the sensitivity of lamb waves
to the presence of damages in structures and components and started to use them for the
detecting of possible flaws and cracks. Worlton et al [55] conducted the first ultrasonic testing
using lamb waves in order to locate possible cracks in isotropic structures. They used two
transducers, one as actuator and the other as receiver. Worlton and his group used the hypothesis
of infinite surface because the formulation of lamb waves for semi-infinite and finite plates were
not developed yet at that time. It was until a decade later when such an assumption was proposed
by Torvik et al [49] who paved the way for other researchers interested in lamb waves to study
the reflection from boundaries and the conversion of modes. Raghavan et al [44] presented a
good review in which they gave a grasp of the lamb wave theory, discussed the use of several
smart materials (magnetostrictive transducers, fiber optic transducers, piezoelectric transducers)
to generate stress Lamb waves in isotropic plates, and included some examples of lamb wave
based inspection in several areas (civil infrastructures, pipelines, aerospace, turbines, etc.).
Another good review on nondestructive health monitoring based on wave propagation was
presented by Giurgiutiu et al [20] in 2005. The authors enclosed a brief historical overview of
NDE, a theoretical overview of lamb waves and discussed their use for health mentoring
applications along with piezoelectric transducers. Some lamb waves techniques (pulse-echo and
pitch-catch) used to identify and detect flaws and delaminations in composite structures were
also discussed by the authors. Piezoelectric based Lamb waves were exploited by Giurgiutiu et al
[18] in order to detect cracks near rivets on the panel of an aging aircraft in a pulse-echo

13

configuration. Wang et al [51] introduced an active damage detection technique based on lamb
wave propagation actuated by piezoelectric transducers in order to locate cracks and damages in
an isotropic plate. Kim et al [28] proposed a new methodology based on lamb waves in order to
detect damages in civil infrastructures (bridges, buildings, etc.). The technique does not rely on
the use of baseline measurements to interrogate the health of the structure, rather on the
polarization properties of the piezoelectric devices. Many other researchers combined
experimental work with theoretical and/or numerical analysis in order to prove the efficiency of
piezo-actuated waves damage detection techniques. Liu et al [34] investigated on the use of the
finite element method to simulate the propagation of lamb waves actuated by Piezoelectric
Wafer Actuated Sensors (PWAS). Lu et al [38] proposed also both an experimental study and a
numerical modeling using the finite element method for the quantitative assessment of throughthickness crack sizes in a finite aluminum plate. Han et al [25] compared computational,
experimental and theoretical results of the response of piezo-actuated lamb wave models in a thin
aluminum plate in order to study the effect of variable isothermal and thermal gradient
conditions on the propagation of the waves and hence on the detection of damages and cracks.

1.5.

Issues in Structural Health Monitoring:

Many issues present barriers for judicious practical health monitoring. In many cases, several
conventional techniques of structural health monitoring have been employed successfully. In
some other cases however, many conventional techniques could not be adapted effectively. Such
circumstances put in evidence the need for more advanced techniques in order to resolve the
challenges. In lamb wave based health monitoring, two major techniques (Pitch-catch, Pulseecho) have been used to interrogate, qualify and quantify damages in simple isotropic and

14

composite structures. They have been satisfactory enough for certain structures. For many other
structures with complex geometries, pitch-catch and pulse-echo are not suitable for efficient
detection of damages and cracks in structural components. Complex structures usually present
variable sections, geometries and shapes. It is impossible to mount the piezoelectric devices on
some narrow, curved and angled parts and locations (welded joints, hard to reach locations, etc.).
This makes the detection of cracks within those regions very cumbersome. Therefore, there has
been an urgent need for the improvement of the conventional techniques based on guided wave
inspection.

1.6. Phased Array Structural Health Monitoring:
In the quest for more reliable guided-wave propagation methods, the inspection using phased
arrays is being actively pursued by many researchers. Phased array is an ultrasonic wave
propagation-based technique, suitable for almost all tests where conventional methods have been
traditionally used. It has been used in wide range of industrial applications (aerospace, pipelines,
tubular manufacturing and maintenance and in other general manufacturing purposes). It uses an
assembly of transducers arranged in a predefined form (one dimensional array, two dimensional
arrays) in order to steer and focus all the waves induced by the elements of the array in one
direction.

2. Thesis Overview:
2.1. Research Objectives:
Based on the literature review given above, one can underline the importance of the field of
structural health monitoring for the assessment of structures. This field encompasses a number of
15

interactive technologies and advanced procedures combined together in order to offer systems
that can potentially interrogate structures by identifying possible damages and characterizing
structural performances. The implementation of those systems on different structures can reduce
the cost associated with inspection and maintenance as well as increase safety and save lives.
The research in this thesis is conducted in order to prove the efficiency of using simple
piezoelectric transducers for the detection of predefined notches manufactured on thin aluminum
plates. The first main part of the thesis focuses on studying the impedance of the piezoelectric
materials, that will be used in the experiments, by plotting analytical and numerical results and
compare them. This study is of a great importance for checking the healthy state of the
piezoelectric transducers and judge about their efficiency before using them in real experiments.
The second part is devoted to the study of two basic techniques of ultrasonic structural health
monitoring mainly the pitch catch technique and the phased array technique in order to show
their efficiency in detecting the defects predefined on the aluminum plates. Both techniques will
be presented in details in the following chapters and results of experimental testing will be
analyzed and discussed.

16

Chapter 2
Piezoelectric Materials:
Properties and Characteristics
1. Introduction to Piezoelectricity:
Etymologically, electricity by pressure or Piezoelectricity, as widely known, takes its root from
the Greek word “piezein” which means “to squeeze” or “to press”. It characterizes the ability of
certain materials (notably crystals and certain ceramics) to produce an electric potential in
response to an applied mechanical stress and, conversely, generate a stress in response to an
electric stimulus. The electric to mechanical conversion is called “direct piezoelectric effect” and
the reversible effect is termed “converse piezoelectric effect”. All piezoelectric materials
available in nature exhibit both the direct and the converse piezoelectric effects. The direct
piezoelectric effect was discovered and initiated by the brothers Pierre and Jacques Curie. The
Curies, who first observed this unusual property, combined their understanding of crystals with
their knowledge in pyroelectricity and demonstrated in 1880 the direct piezoelectric effect using
crystals of quartz and Rochelle salt. Subsequently, the converse process was predicted by the
mathematician Gabriel Lippmann in 1881 and then immediately confirmed by the Curies.
Although discovered in in 1880, the first industrial applications of this fundamental breakthrough

17

started in 1940-1950. Since then, piezoelectricity has been used in many fields mainly in
electromechanical engineering applications: piezoelectric materials were notably used as
electromechanical transducers such as echographic probes, sensors and actuators.

Direct
Piezoelectric
effect

Elasticity

Electricity
Converse
Piezoelectric
effect

Figure 2.01: Coupling effects between Elasticity and Electricity
Recently, piezoelectric materials are used in myriads of engineering and industrial applications
(power harvesting devices, ultrasonic probes, smart actuators and sensors for structural health
monitoring, etc.).

2. Understanding of the Piezoelectric Effect:
When a piezoelectric material is subjected to a mechanical stress, intermolecular distances and
forces vary. This variation manifests at the surface of the element by generation of small
electrical dipoles. This process is called the piezoelectric effect and can be better understood by
the explanation of its microscopic origins. A molecular model illustrating the piezoelectric effect
is, illustrated in Figure 2.02. When the molecule is unperturbed, the centers of its positive and
negative charges coincide and the external effects of the charges cancel out in a reciprocal way.
As a result, the molecule remains neutral (Figure 2.02 (a)). However when the molecule is
disturbed due to an external force or pressure applied on the material, its internal structure
18

deforms which causes the separation of the positive and negative centers (Figure 2.02 (b) ), thus
small dipoles appears between two neighbors molecules (Figure 2.02 (c) ). Inside the material the
dipoles cancel mutually and charges appears only on the surface thus the material is polarized
and an electric field is generated.

Figure 2.02: Molecular Modeling of Piezoelectricity Effect: (a) Unagitated Molecule, (b)
Molecule under external stress (c) Polarization Effect

Figure 2.03: Polarization of the piezoelectric material and generation of electric field

19

3. Piezoelectric Materials:
We distinguish many materials that exhibit piezoelectric effects. Natural and artificial
piezoelectric materials include quartz, Rochelle salt, Topaz, Tourmaline, Barium Titanate and
Lead Zirconate Titanate. Each material responds differently to external stimuli (stress, pressure,
electric field, etc.). Some of the materials, especially man-made materials containing Lead,
raised big environmental concerns regarding toxicity. As a result, necessary iniatives have been
taken in order to manufacture innovative lead-free materials, safe and friendly for the
environment.
Many piezoelectric materials have gained wide applications in industry. The Lead-Zirconate
Titanate Pb(Zr,Ti)O3 and the Polyvinilidene Fluoride (PVDF) are among the most preferred
candidates due to their actuating and sensing capabilities. The Pb(Zr,Ti)O3 (PZT) is a synthetic
compound formed by small ceramic particles obtained by calcination and grinding of different
oxides. During the polarization step, the material is placed in a silicon oil bath at a temperature
150 oC and subjected to an electrical field of 3KV/mm for one minute. The Pb(Zr,Ti)O3 (PZT) is
known for its higher piezoelectric constant and can be used to manufacture many electronic
components (ultrasonic transducers, ceramic resonators, ceramic capacitors, etc.) however it is
very brittle. Unlike, Pb(Zr,Ti)O3 (PZT), PVDF is a plastic material, It can be synthesized from
the gaseous VDF monomer via a radical or controlled polymerization process followed by a melt
casting process. When poled, PVDF exhibits efficient piezoelectric properties which make it
very useful for many industrial applications. PVDF is a very flexible material, it allows the
health monitoring of curved structures.

20

Single crystals piezoelectrics
(Quartz, Lead Magnesium Niobate)

Ceramic piezoelectrics
(Lead Zirconate Titanate, Lead Titanate)

Polymers piezoelectrics
(Polyvinilidene Fluoride (PVDF)

Composite piezoelectrics
(Ceramic polymer, ceramic glass)

Figure 2.04: Different natural and made-up piezoelectric materials

3.1.

Polarization of Piezoelectric Materials:

Polarization in electricity is commonly known as the rearrangement of the intermolecular
distances and the shifting of positive and negative electric charges in opposite directions within a
dielectric material under an external electric field. This physical property is applied for almost all
piezoelectric materials which are not naturally polarized. For ceramic piezoelectrics, for
example, the polarization is achieved under an appropriate electric voltage in order to guarantee
that the material remains poled and electric dipoles are arranged. The polarization of a

21

piezoelectric material will be altered when the temperature of the material reaches the Curie
temperature.

Figure 2.05: Polarization of a Ceramic Piezoelectric Material

4. Piezoelectric Constitutive Equations:
The constitutive equations of piezoelectricity represent the fundamental relations that describe
the electromechanical coupling between elasticity and electricity. Mathematical relations are
derived from linear field equations (Equations of motion, Gauss’s, kinematic relations,
Maxwell’s Law) in order to model the direct and the converse piezoelectric effects. Field linear
equations are as follows:
Equations of Motion in Elasticity:

Di,i – q = 0

Gauss’s Law in Electricity:
Kinematic relations in Elasticity:
Maxwell’s Law:

Tij , j + fi = ρ üi

2Sij = (ui, j + u j , i )
Ei = −φ, i

The constitutive equations in the stress-charge form are given as follows:

Tij = cijkl Skl − ekij Ek
Dk = ekij Sij + ε ki Ei
where: the Scalars, vectors and tensor quantities are as follows:
22

fi : Mechanical Body forces ( N / m3 )

ui: Mechanical Displacement (m)
φ : Electric Field Potential (V=J/C)

q: Electric Body Charge ( C / m3 )

Sij : Electric Field Potential (m/m)

ρ: Mass Density (Kg/m )

Ei : Electric Field Potential (V/m)

cijkl : Elastic Material Constants ( N / m 2 )

3

Tij : Stress tensor ( N / m 2 )

eij : Piezoelectric stress constants ( C / m2 )

Di : Electric Displacement ( C / m2 )

ε ij : Dielectric Constant (F/m)

Those constitutive equations can also be written the forms below:

Strain-Charge Form

Strain-Voltage Form

S = sE = 0 : T + d T ⋅ E

S = sD = 0 : T + g T ⋅ E

T = cD = 0 : T + g T ⋅ D

D = d : T + εT =0 ⋅ E

E = − g : T + ε −1 ⋅ E

E = − g : T + ε −1 ⋅ D

Stress-Voltage form

Figure 2.06: Other forms of the piezoelectric constitutive equations
s: Compliance Matrix
q: Piezoelectric Stress matrix (Stress-Voltage Form)
d: Piezoelectric Strain matrix (Strain-Voltage Form)

g: Piezoelectric Strain matrix (Strain-Voltage Form)
Table.2.01 gives a summary of the four different forms of the piezoelectric constitutive
equations:

23

Table 2.01: Forms of piezoelectric constitutive relations

5. Electrical Impedance of a Piezoelectric Patch:
5.1. Introduction:
When an unbonded PZT patch is driven by an alternating electric voltage, a mechanical strain is
produced and electrical charges are generated in the surface electrodes of the PZT. The ratio of
the applied voltage to the induced current (resulting from the accumulations of electrical charges
on the top and bottom electrodes) is defined as the electrical impedance of the PZT and can be
measured by commercially available impedance analyzers. In this part of the thesis, a finite
element model of an unbonded PZT was built using the Finite Element Multipurpose Software
ANSYS and impedance data was extracted and compared to results obtained from available
analytical formulation. The modeling of the unbonded piezoelectric patch can be of good utility
for the understanding of the electromechanical behavior of the PZT patch and can also be useful
for controlling the qualities of the patch before using it in structural health monitoring tasks.

5.2. Analytical Formulation of the Electrical Impedance of a PZT:
Giurgiutiu et al [16] developed the expression of the electrical impedance of a free PWAS
(Piezoelectric Wafer Actuated Sensor) driven by an electrical voltage V ( t ) = Veiwt between the
24

top and bottom surface electrodes. The PZT is polarized in the thickness direction and its length
l, its width w and its thickness t satisfy the condition l >> b >> t

by assuming uncoupled

motions and hence unidirectional motion in the longitudinal direction.

Figure 2.07: Z-polarized Piezoelectric Patch under Electric Voltage
The developed expression of the electrical impedance is given by:



 1  

 tan  2 λl   
1
2

 
1 − k31 1 −
Z=
1
jwc 

λl 



2





−1

(Eq1)

The above equation is obtained after developing the expression of the electrical displacement D3 ,
the electric charge Q and the electric current I as follows:

d
u ' 
2 
D3 = d31T1 + ε 33 E3 = 31 ( s11 − d31E3 ) + ε 33 E3 = ε 33 E3 1 − k31 1 −

s11

 d31E3  


l /2 b

Q=

∫ ∫ D3dx1dx2

and I =

−l / 2 0

where:

dQ
= iwQ
dt

u ' ( l / 2 ) − u ' ( −l / 2 ) tan ( λ l / 2 )
=
λl / 2
d31El

All the parameters and their expressions are summarized in Table 2.02:

25

Property

Symbol Expression

ε 33

l×w
t

Capacitance

C

Electromechanical Coupling Factor

k31

s11ε 33

Wave Number

λ

w
c

Wave Speed

c

1
ρ s11

Dielectric constant

ε 33

*******

Compliance, in plane

s11

*******

In plane strain coefficient

d31
w

*******

Angular Frequency

ρ

Density

d31

*******
*******

Table 2.02: Properties and expressions of the parameters used in the analytical model

5.3. Numerical Modeling of an Unbonded PZT Patch:
A coupled-field finite element analysis is conducted using the finite element code ANSYS in
order to simulate the behavior of a free PZT-5A piezoelectric patch under an electrical voltage
and extract appropriate data in order to plot the graph of the electrical impedance and compare
the results to the analytical results obtained from the expression (Eq1). 3-D brick coupled-field
finite elements are used in the analysis. These elements are eight-noded elements with six
degrees of freedom in each node and an extra degree of freedom which represent the electrical
DOF. In this study, we account only for the three displacements degrees of freedom X, Y and Z
and the electrical degree of freedom V so we used 3-D coupled-field SOLID5 elements available
26

in ANSYS. For each displacement degree of freedom X, Y and Z, in a SOLID5 element,
corresponds a reaction force RX, RY and RZ respectively and for the electrical DOF V
corresponds an electrical charge which represent the reaction force that we need to extract from
ANSYS in order to plot the impedance graph. For this purpose, a harmonic coupled field
analysis is performed on a PZT-5A by sweeping the frequency from 0 to 1MHz and data are
collected and postprocessed. The electrical charge Q extracted from ANSYS is related to the
electric impedance Z by the following expression:

Z=

V
V
V
=
=
I iwQ i 2π f ( Re ( Q ) + i Im ( Q ) )

(Eq2)

Where: f, Re (Q) and Im (Q) represent, respectively, the frequency, the real part of the electrical
charge and the imaginary part of the electrical charge
The harmonic analysis was performed in a frequency range of [0, 1MHz]. Applying a voltage of
magnitude 1V on the top electrode, the expression of the Impedance reduces to the following:

Z=

1
i 2π f ( Re ( Q ) + i Im ( Q ) )

(Eq3)

The frequency, the real and the imaginary parts of the electrical charge are extracted in the form
of a table from the time history output of the harmonic analysis.

5.3.1. Properties of the Material:
The material used in this study is the Zirconate Titanate (PZT-5A). It has the following
mechanical properties:

27

0
0
0 
0 −171
 16.4 −5.74 −7.22
 0




0
0
0 
0 −171
 −5.74 16.4 −7.22
 0
2
 −7.22 −7.22 18.8
 0
0
0
0 
0
374 
−12 m
−12 C
; d =
S =
 ×10
 ×10
0
0
47.5
0
0 
0 
N
N
 0
 0 584
 0
 584 0
0
0
0
47.5
0 
0 




 0



0
0
0
0
44.3 
0
0 

 0
where: S and d represent, respectively, the compliance matrix and the piezoelectric strain matrix.
3

The density of the material is ρ=7500Kg/m . The properties of the Z-polarized PZT patch used
for the analytical and numerical models are summarized in Table 2.03
Variable
Length (mm)
Width (mm)
Thickness (mm)

Stiffness
Coefficients

Coupling
Coefficients

Permittivity
Coefficients

Analytical model Numerical Model
20
20
1
1
0.5
0.5
S11
S22
S33
S12
S13
S44
S55
S66
d13
d23
d33
d42
d51

ε11
ε 22
ε 33

Damping Coefficient

-12

16.4x10
********
********
********
********
********
********
********

-12

-171x10
********
********
********
********
********
********
1700 ε0
0

-12

16.4x10
-12
16.4x10
-12
18.8x10
-12
-5.74x10
-12
-7.22x10
-12
44.3x10
-12
47.5x10
-12
47.5x10
-12
-171x10
-12
-171x10
-12
374x10
-12
584x10
-12
584x10
1730 ε0
1730 ε0
1700 ε0
0.0001

Table 2.03: Geometrical and Mechanical properties of the analytical and numerical models

28

5.3.2. Geometry and Meshing of the Numerical Model:
The numerical model is built, meshed and processed using the software ANSYS (see ANSYS
code in Appendix A). The body of the model was meshed into 10000 hexahedral mapped
elements. Each element is a coupled-field cube of edge length 0.1mm. No boundary conditions
are assigned to the model and the PZT is subjected to only a potential difference V in the
thickness direction as indicated in Figure 2.07. The bottom electrode was grounded while the top
electrode was subjected to a voltage which has maximum amplitude of 1V.

Figure 2.08: ANSYS FEM Model of the Piezoelectric Patch

5.4.

Impedance Plots:

The numerical data (frequency, real and imaginary parts of the electrical charge) was extracted
from ANSYS and processed using MATLAB in order to plot the graph of the impedance. The
analytical formulation was also coded on MATLAB (see Matlab code in Appendix A) and
appropriate mechanical properties were provided in the code in order to plot the graph of the
analytical impedance. For all the geometrical and mechanical parameters summarized in Table
2.03, the plot in Figure 2.09 was obtained for both the numerical impedance and the analytical
impedance.

29

Interpretation:

The two curves have the same trend and same number of resonance and anti-resonance peaks but
they present some clear differences. As depicted from the plot, the variation between the
amplitude of the numerical impedance and the amplitude of the analytical impedance increases
with the frequency so does the error between the resonance and anti-resonance peaks of the two
curves. A first apparent reason of this variation is that the assumption of uncoupled motions of
the PZT patch in the analytical formulation cannot be perfectly assumed in the numerical
simulation thereby coupling cannot be totally ignored.

Figure 2.09: Analytical and Numerical Impedance of the Unbounded PZT Patch

5.5.

Parametric Study:

In order to find other possible reasons behind the analytical and numerical variations, we run
several finite element simulations using the built model in Figure 2.08 where in each simulation

30

we vary only one parameter (except S11 and d31 which remain constant in all simulations) and
keep all the other parameters unchangeable. No variations at all were made on the analytical
model. The idea behind this parametric study is to study the possible variations of the numerical
impedance and specify the parameters most affecting those variations. For all the simulations
performed in this parametric study, the curve of the analytical result remains the same as no
variation at all was applied to the analytical model.

Variation of the piezoelectric strain coefficient d51:
This section aims at studying the effect of variation of the coefficient d51 on the numerical
impedance of the piezoelectric patch. For this purpose we run four different numerical
simulations in which the coefficient d51 was increased by, respectively, 0%, 25%, 50% and 75%
from its initial value while all the other parameters were kept unchanged. The variations of the
coefficient d51 are presented in Table 2.04 as follow:

Initial value (d51) I New value (d51) N α =
Run 1
Run 2
Run 3
Run 4

-12

584x10
-12
584x10
-12
584x10
-12
584x10

( d51 ) N − ( d51 )I
( d51 )I

-12

584x10
-12
730x10
-12
876x10
-12
1022x10

Table 2.04: Variation of the Coefficient d51

31

0%
25%
50%
75%

(%)

Interpretation:
Once the simulations carried out, the numerical data was extracted from ANSYS, separately, for
each simulation and then processed on Matlab. The plots of the numerical results corresponding
to the four different simulations are presented in Figure 2.08 along with the plot of the analytical
model. It appears from the plot that the curves corresponding to the four different simulations are
exactly the same with no clear variations neither on the amplitude of the impedance nor on the
number and/or the frequencies of the peaks. As a result, we can conclude that the coefficient d51
does not have any effect on the impedance of the PZT hence it is not responsible for the errors
between the numerical model and the analytical model.

Variation of the piezoelectric strain coefficient d42:
Similar to d51, d42 was varied four times from its initial value with 0%, 25%, 50% and 75%
respectively. Four simulations were run on ANSYS for the model in Figure 2.06 and the results
showed no variations at all on the numerical impedance; neither the amplitude nor the peaks
(number and frequencies) were affected. We got the exact plots of Figure 2.10. As a result, the
variation of the coefficient d42 does not have any effect on the impedance of the PZT. Hence, the
coefficient d42 is, like d51, not responsible for the errors between the numerical and the
analytical models.

Variation of the compliance coefficient S44:
Similar to d51 and d42, S44 was varied four times from its initial value with 0%, 25%, 50% and
75% respectively. Four simulations were run on ANSYS for the model in Figure 2.08 and the
32

results showed no variations at all on the numerical impedance, neither the amplitude nor the
peaks (number and frequencies) were affected. We got the exact plots of Figure 2.10. As a result,
the variation of the coefficient S44 does not have any effect on the impedance of the PZT.

Variation of the compliance coefficient S55:
Similar to d51, d42 and S44, S55 was varied four times from its initial value with 0%, 25%, 50%
and 75% respectively. Four simulations were run on ANSYS for the model in Figure 2.08 and
the results showed no variations at all on the numerical impedance, neither the amplitude nor the
peaks (number and frequencies) were affected. We got the exact plots of Figure 2.10. As a result,
the variation of the coefficient S44 does not have any effect on the impedance. Hence, the
coefficient S55 is not responsible for the errors between the numerical and the analytical models.

Figure 2.10: Effects of the coefficients d51, d42, S44 and S55 on the impedance of the PZT

33

Variation of the piezoelectric strain coefficient d32:
This section aims at studying the effect of variation of the coefficient d32 on the numerical
impedance of the piezoelectric patch. For this purpose we run four different simulations using
the model in Figure 2.08 where the initial negative value of the coefficient d32 was decreased by
0%, 25%, 50% and 75%. The variations are listed in Table 2.05.

Initial value (d32) I New value (d32) N
Run 1
Run 2
Run 3
Run 4

-12

-171x10
-12
-171x10
-12
-171x10
-12
-171x10

β=

( d32 )N − ( d32 ) I
( d32 ) I

-12

-171x10
-12
-213.75x10
-12
-256.5x10
-12
-299.25x10

(%)

0%
25%
50%
75%

Table 2.05: Variation of the Coefficient d32
The electrical charge was extracted from ANSYS for each simulation and processed on Matlab.
The plots of the numerical impedance for the four different simulations and the plot of the
analytical impedance are presented in Figure 2.11.

Interpretation:
Figure 2.11 shows that each of the four curves corresponding to the four numerical simulations is
different from the curve of the analytical impedance both in amplitude and in frequencies of
peaks of resonance and anti-resonance. However, the four curves of the numerical simulations
have the same shape, same number of peaks but they present a slight difference in the amplitude
of the impedance. The zoomed area shows that the amplitude of the impedance for the case of
75% is less than the impedance in the case of variation by 50% and both are less than the case of
34

variation by 25%. So an increase in the percent variation of the coefficient d32 which, in fact,
corresponds to a decrease in its negative initial value engenders a decrease in the amplitude of
the impedance. Hence, the coefficient d32 has an effect on the numerical impedance of the PZT
and the effect is manifested by only a decrease of the amplitude when the value of d32 decreases.

Figure 2.11: Effect of the strain coefficient d32 on the impedance of the PZT

Variation of the piezoelectric strain coefficient d33:
This section aims at studying the effect of the variation of the coefficient d33 on the numerical
impedance of the piezoelectric patch. For this purpose we run four different numerical
simulations using the FEM model in Figure 2.08 where the initial positive value of the
coefficient d33 increased by 0%, 25%, 50% and 75%. The variations are listed in Table 2.06

35

Initial value (d33) I New value (d33) N
-12

Run 1
Run 2
Run 3

374x10
-12
374x10
-12
374x10
-12
374x10

Run 4

-12

374x10
-12
467.5x10
-12
561x10
-12
654.5x10

ν=

( d33 ) N − ( d33 ) I
( d33 ) I

(%)

0%
25%
50%
75%

Table 2.06.a: Variation of the Coefficient d33 (d33 increases)
The electrical charge was extracted from ANSYS for each simulation and processed on Matlab.
The plots of the numerical impedance for the four different simulations and the plot of the
analytical impedance are presented in Figure 2.12.

Figure 2.12.a: Effect of the coefficient d33 on the impedance of the PZT (d33 increases)

Interpretation:
Figure 2.12.a shows a clear variation on the amplitude of the impedance between the four
numerical curves. As shown in Table 2.06, when the percent variation on the initial value of the

36

coefficient d33 increases the value of d33 increases and this engenders a decrease in the amplitude
of the impedance. In order to see if the error between the amplitudes of the numerical impedance
and the analytical impedance reduces when we decrease the initial value of the coefficient d33,
we run four more runs where the initial positive value of d33 decreases by 25%, 50%, 75% and
100%. The variations are summarized in table
Initial value (d33) I New value (d33) N
Run 1
Run 2
Run 3
Run 4
Run 5

-12

374x10
-12
374x10
-12
374x10
-12
374x10
-12
374x10

ν=

-12

374x10
-12
280.5x10
-12
187x10
-12
93.5x10
0

( d33 ) N − ( d33 ) I
( d33 ) I

(%)

0%
-25%
-50%
-75%
-100%

Table 2.06.b: Variation of the Coefficient d33
The plots of the different cases are presented in Figure 2.12.b.

Interpretation:

From the plots of the numerical simulations for the five different runs were d33 was decreased by
-0%, -25%, -50%, -75% and -100%, we see that the impedance first increased when a variation
by 25% was applied to d33, then slightly increased from the case of -25% to the case of 50% but
decreased from the case of -50% to the case of -75% although remained slightly larger than the
case -0%. When d33 was decrease by -100%, we see that the impedance further decreases and
become less than the impedance in the case of no variation (0%). Runs for some other percentage
variations (-60%, -55 %...) between -50% and -75% showed that the impedance does not
37

decrease more than the case of 50% variation. No variation was recorded on the number and
frequencies of the peaks.

Figure 2.12.b: Effect of the coefficient d33 on the impedance of the PZT (d33 decreases)

To conclude, we can say that the coefficient d33 has a major effect on the numerical impedance
of the PZT and the effect is manifested by only a clear variation of the amplitude.

Variation of the compliance coefficient S12:
This section aims at studying the effect of the variation of the coefficient S12 on the numerical
impedance of the piezoelectric patch. For this purpose we run four different numerical
simulations on the FEM model in Figure 2.08 where we decreased the initial negative value of
S12 by 0%, 25%, 50% and 75% respectively. The variations are listed in Table 2.07.
38

Initial value (S12) I New value (S12) N
Run 1
Run 2
Run 3
Run 4

-12

-5.74x10
-12
-5.74x10
-12
-5.74x10
-12
-5.74x10

γ=

( S12 ) N − ( S12 ) I
( S12 ) I

-12

-5.74x10
-12
-7.175x10
-12
-8.61x10
-12
-10.045x10

(%)

0%
25%
50 %
75%

Table 2.07: Variation of the Coefficient S12
The plots of the numerical impedance for the four different simulations and the plot of the
analytical impedance are presented in Figure 2.13

Figure 2.13: Effect of the Variation of the compliance coefficient S12

Interpretation:
Figure 2.13 shows that the curves of the numerical impedance, for the four cases, do not present
any clear variation on the amplitude however it shows a variation on the frequencies of the
39

peaks. In fact, the zoomed area in the plot shows that the peaks of resonance and anti-resonance
are shifted more and more to the left when the value of the compliance coefficient S12 decreases
by 25%, 50% and 75%. Hence, one way of reducing the errors between the peaks in the
numerical model and the analytical model while keeping the amplitude unchangeable is by
increasing the initial value of the compliance coefficient S12.

Variation of the compliance coefficient S13:
This section aims at studying the effect of the variation of the coefficient S13 on the numerical
impedance of the piezoelectric patch. For this purpose we chose to decrease the initial negative
value of S13 by 0%, 25%, 50% and 75% as shown in Table 2.08. We could only run the cases of
variations by 0%, 25% and 50% and collect the required data from ANSYS for each case. The
case of variation by 75% could not be performed because the new value of S13 gave a nonpositive definite compliance matrix and the finite element job aborted with error. The variations
made on the coefficient S13 are presented in Table 2.08.

Initial value (S13) I New value (S13) N
Run 1
Run 2
Run 3
Run 4

-12

-7.22x10
-12
-7.22x10
-12
-7.22x10
-12
-7.22x10

ξ=

( S13 ) N − ( S13 )I
( S13 )I

-12

-7.22x10
-12
-9.025x10
-12
-10.83x10
-12
-12.635x10

Table 2.08: Variation of the Coefficient S13

40

0%
25%
50 %
75%

(%)

The plots of the numerical impedance for the four different simulations and the plot of the
analytical impedance are presented in Figure 2.14

Figure 2.14: Effect of the Variation of the compliance coefficient S13

Interpretation:
Figure 2.14 shows that the curves of the numerical impedance for the four cases do not present
any clear variation on the amplitude however it shows a variation on the frequencies of the
peaks. In fact, the zoomed area in the plot shows that the peaks of resonance and anti-resonance
are slightly shifted to the left when the value of the compliance coefficient S12 decreases by 25%
and 50%. We remark that the effect of the compliance coefficient S13 on the numerical
impedance is the same as the effect of the coefficient S12. In fact, decreasing one of the two
coefficients engenders a shift to the left of the peaks of resonance and anti-resonance but the
41

effect of S12 is clearer than the effect of S13. For both coefficients, we do not see any effect on
the amplitude of the impedance.

Variation of the compliance coefficient S23:
This section aims at studying the effect of the variation of the coefficient S23 on the numerical
impedance of the piezoelectric patch. For this purpose we chose to decrease the initial negative
value of S23 by 0%, 25%, 50% and 75% as shown in Table 2.09. We could only run the cases of
variations by 0%, 25% and 50% and collect the required data from ANSYS for each case. The
case of variation by 75% could not be performed because the new value of S13 gave a nonpositive definite compliance matrix and the finite element job aborted with error. The variations
made on the coefficient S23 are presented in Table 2.09.
Initial value (S23) I New value (S23) N
Run 1
Run 2
Run 3
Run 4

-12

-7.22x10
-12
-7.22x10
-12
-7.22x10
-12
-7.22x10

η=

( S23 ) N − ( S23 )I
( S23 ) I

-12

-7.22x10
-12
-9.025x10
-12
-10.83x10
-12
-12.635x10

(%)

0%
25%
50 %
75%

Table 2.09: Variation of the Coefficient S23
The plots of the numerical impedance for the four different simulations and the plot of the
analytical impedance are presented in Figure 2.15.

Interpretation:
Figure 2.15 shows that the curves of the numerical impedance for the four cases do not present
any clear variation on the amplitude however it shows a variation on the frequencies of the
peaks. In fact, the zoomed area in the plot shows that the peaks of resonance and anti-resonance
42

are slightly shifted to the right when the value of the compliance coefficient S23 decreases by
25% and 50%. We remark that the effect of the compliance coefficient S23 on the numerical
impedance is opposite to both the effects of the coefficients S12 and S13. In fact, decreasing the
coefficient S23 engenders a shift to the right of the peaks of resonance and anti-resonance
whereas decreasing either S12 or S13 causes the peaks to shift to the left.

For the three

coefficients, we do not see any effect on the amplitude of the impedance.

Figure 2.15: Effect of the Variation of the compliance coefficient S23

Variation of the compliance coefficient S22:
This section aims at studying the effect of the variation of the coefficient S22 on the numerical
impedance of the piezoelectric patch. For this purpose we run four different numerical

43

simulations on the FEM model in Figure 2.08 where the initial positive value of the compliance
coefficient S22 is increased by 0%, 25%, 50% and 75%. The variations on S22 are presented in
Table 2.10.
Initial value (S22) I New value (S22) N
Run 1
Run 2
Run 3
Run 4

-12

16.4 10
-12
16.4 10
-12
16.4 10
-12
16.4 10

θ=

( S22 )N − ( S22 ) I
( S22 ) I

-12

16.4 10
-12
20.5 10
-12
24.6 10
-12
28.7 10

(%)

0%
25%
50 %
75%

Table 2.10: Variation of the Coefficient S22
The plots of the numerical impedance for the four different simulations and the plot of the
analytical impedance are presented in Figure 2.16.

Figure 2.16: Effect of the Variation of the compliance coefficient S22
44

Interpretation:
Figure 2.16 shows that the compliance coefficient S22 has a very slight effect on the impedance
of the PZT. We can barely see in the zoomed area a very slight shift of the peaks where the
frequencies slightly decrease when the coefficient S22 is increased by 25%, 50% and 75%. The
plot also does not show any variation on the amplitude of the impedance. To conclude, we can
say that the coefficient S22 has a very small effect on the impedance of the PZT compared to
S13, S23 and especially to S12.

Variation of the compliance coefficient S33:
This section aims at studying the effect of the variation of the coefficient S33 on the numerical
impedance of the piezoelectric patch. For this purpose we run four different numerical
simulations on the FEM model in Figure 2.06 where the initial positive value of the compliance
coefficient is increased by 0%, 25%, 50% and 75%. The variations are summarized in Table
2.11.

Initial value (S33) I New value (S33) N
Run 1
Run 2
Run 3
Run 4

-12

18.8x10
-12
18.8x10
-12
18.8x10
-12
18.8x10

σ=

( S33 ) N − ( S33 ) I
( S22 ) I

12

18.8x10-12
23.5x10

-12

28.2x10
-12
32.9x10

Table 2.11: Variation of the Coefficient S33

45

0%
25%
50 %
75%

(%)

The plots of the numerical impedance for the four different simulations and the plot of the
analytical impedance are presented in Figure 2.17.

Figure 2.17: Effect of the Variation of the compliance coefficient S33

Interpretation:
Figure 2.17 shows that the compliance coefficient S33 has a very slight effect on the impedance
of the PZT. Similarly as in Figure 2.16, we can barely see in the zoomed area a very slight shift
of the peaks where the frequencies slightly decrease when the coefficient S33 is increased by
25%, 50% and 75%. The plot does not show any variation on the amplitude of the impedance. To
conclude, we can say that the coefficient S33 has an effect on the impedance of the PZT similar
to the effect of S22.
46

Variation of the damping coefficient:
This section aims at studying the effect of the damping coefficient on the numerical impedance
of the piezoelectric patch. For this reason, we run three different simulations on the FEM model
in Figure 2.08 where the damping coefficient ζ was given the values 0.01% in the first
simulation, 1% in the second simulation and 10% in the third simulation. In the three
simulations, no compliance coefficient or piezoelectric strain coefficient was varied. The data in
the three simulations was also saved from the output of ANSYS in order to plot the curves of the
numerical impedance for the three cases. The corresponding plots of the three simulations and
the plot of the analytical impedance are presented in Figure 2.18.

Figure 2.18: Effect of the variation of the damping coefficient ζ on the impedance of the PZT

47

Interpretation:
Figure 2.18 shows that the number of peaks of resonance and anti-resonance decreases from the
curve of the numerical impedance when the damping coefficient ζ increases. For a very light
damping ζ=0.01%, we count 7 different peaks of resonance and anti-resonance. This number
decreases to only two peaks when the damping coefficient ζ was increased to 1% and to only one
peak when ζ was increased to 10%. So the variation of the damping coefficient has a direct effect
on the number of peaks of resonance and anti-resonance but not on the amplitude of the
impedance. In fact, as we can see in the fourth subplot (three cases together), the impedance
curves perfectly match and no variation is recorded on the frequencies of occurrence of the peaks
(this can be clearly seen for the first and second peaks). As a result, we can conclude that the
variation of the damping coefficient ζ has no effect on the frequency shift between the analytical
and numerical curves.

Variation of the meshing module:
In this part of the work, the numerical model was tested for two different cases; case of a coarse
meshing and case of a refined meshing. In both cases, the geometrical and mechanical properties
of the piezoelectric patch were kept constant. A light damping ζ=0.01% was applied to the
model. The two conducted simulations aim at studying the effect of the meshing module on the
numerical impedance of the PZT patch. For the coarse mesh, the model was meshed into 640
cubic elements each of edge length equal to 0.25 mm however for the refined mesh, the model
was meshed into 10000 cubic elements each of edge length 0.1 mm. The plots showing the
impedance curves for the two cases are presented in the Figure 2.19.

48

Interpretation:
The plots in Figure 2.19 show that both the coarse mesh and the refined gave very similar results.
No major effect can be seen on the numerical impedance. We only see very small shifts on the
frequencies of the peaks especially at higher frequencies. Concerning the number of peaks and
the amplitude of the impedance, they were not affected by the variation of the number of
elements in the model. So the refinement of the model did not have a big effect on the numerical
impedance.

Figure 2.19: Effect of the variation of the meshing (number of elements)

49

Summary of the parametric study.
The parametric study conducted in this work showed that many coefficients (compliance
coefficients, strain coefficients, damping coefficient) have variable effects on the numerical
impedance of the PZT. Some coefficients have a direct effect on the amplitude, some others
affect the number of peaks and some other coefficients affect the frequencies of occurrence of
the peaks. Table 2.12 gives a detailed summary of this parametric study.
Type of the effect on the impedance

Parameters

Nature

Effect only on the amplitude

d32, d33
ζ

Strain Coefficients

Effect only on the number of peaks

Damping Coefficient

Effect only on the frequencies of the peaks S , S , S , S ,
12 13 23 22
No effects
d ,d ,S ,S
42

51

44

55

Compliance Coefficients
Mixed

Table 2.12: Parametric Study and effects on the numerical impedance of the PZT

In order to find a best configuration that minimizes the error between the numerical impedance
and the analytical impedance, we run several finite element simulations on the FEM model in
Figure 2.08 where in each simulation we vary at least two different coefficients and keep the rest
unchanged. The best case was obtained when we increase the strain coefficient S12 by 40% of
the initial value and set all the strain coefficients equal to zero except d13 which remains
constant. A light damping of 0.01% was applied to the model. The curve of the impedance for
this case (Best case) is plotted with both the curve of the analytical impedance and the curve of
the numerical impedance where no variations are made on all the coefficients (Initial case). The
curves are presented in Figure 2.20.

50

We can see that the selected numerical configuration gives results very comparable to the
analytical ones. In fact, the error on the amplitude between the analytical curve and the new
numerical curve (green) reduced considerably and the peaks of both curves have now almost the
same frequencies and no shifts can be seen even at higher frequencies.

Figure 2.20: Reduction of the error between the analytical and numerical results

Conclusion:
This work was conducted in order to study the electrical impedance of an unbounded
piezoelectric patch driven by a constant electric voltage applied on its surface electrodes. Two
models are proposed and compared in this study; an analytical model found in the literature for
which the displacement motions are assumed uncoupled and the piezoelectric patch was
supposed to experience a one dimensional motion along its length and a numerical model built
using the finite element software ANSYS. The two models have the same mechanical and
51

geometrical characteristics. The comparison between the curves of impedance of the two models
shows clear errors. A parametric study was conducted in order to see the causes of those errors.
It is found that the compliance coefficients affect the impedance by shifting the peaks of
resonance and anti-resonance and that the piezoelectric strain coefficients have a direct effect on
the amplitude of the impedance. Those interpretations were successfully used to find a better
configuration where the error between the analytical model and the numerical model reduced
considerably.

52

Chapter 3
Lamb Waves: General Overview
1. Introduction to Lamb Waves:
Waves are disturbances (perturbations) that propagate through or on the surfaces of material
media (solid, liquid, and gas) at variable speeds functions of the elastic and internal properties of
the media. They represent ideal means of transportation of the information which made them at
the center of preoccupation of many researchers.
Waves are classified into different categories based on their natures, their origins and their
modes of propagation in elastic media. We distinguish:

1.1. Elastic Waves in Solids and Structures:
Longitudinal Waves:

Represent a common type of elastic waves that can propagate through solids, liquids and gases.
They are called longitudinal waves because they induce vibration of particles in the direction of
propagation. Those waves are also known as P-waves which stand for primary waves (fastest to
be recorded) or for pressure waves (result from an alternation of compressions and rarefactions).
The velocity of P-waves in elastic homogeneous solid media is given by:

53

cp =

cp =

cp =

λ + 2µ
. For a 3-dimentionnal free solid media
ρ
1

E

1− v ρ
2

E

ρ

. For a 2-dimentionnal plate with free top and bottom surfaces

. For a 1-dimentionnal slender bar

vE


(1 − 2v )(1 + v ) 
 : Lamé constants
E

µ=

2 (1 + v )


λ=

 E :Young`s Modulus

v: Poisson`s ratio
 ρ : Density


Shear Waves:

Represent another common type of elastic waves. A shear wave, or also S-wave, occurs in an
elastic medium when it is subjected to a periodic shear and results in alternating transverse
motion of particles (displacement of particles is perpendicular to the direction of propagation of
the wave). The velocity of s-waves in elastic homogeneous solid media is given by:
cs =

µ
ρ

Flexural Waves:

Flexural waves are deformations that propagate in bars and in flat or curved plates due to the
presence of bending actions. On their propagations, flexural waves subject the bar or plate to
vibrational movements that induce a motion perpendicular to the axis of the bar or to the plane of
the plate. The speed of flexural waves in a simple beam is given by:
h E 
cF = w 
 2 3ρ 




54

where:
w=2πf: angular frequency and h: height of the beam
The expression above shows a dependence of the velocity of the flexural wave on the frequency
of vibration of the beam. This property is inherent for flexural waves and is known as
Dispersion.

Figure 3.01: Propagation of (a) Longitudinal waves, (b) Shear waves

Rayleigh Waves:

Rayleigh waves are a common type of surface waves. They propagate near the surface of a solid
media and result in elliptical motions of the particles in a vertical plane parallel to the direction
of propagation. The existence of these waves was predicted by John William Strutt (Lord
Rayleigh) in 1885. The amplitude of vibration decreases with an increase in the thickness of the
media.
55

Figure 3.02: Propagation of Rayleigh waves.

2. Lamb Waves:
2.1. Historical Overview:
Lamb waves are elastic perturbations capable of propagating relatively long distances in solid
plates and laminated structures. They are also known as guided plate waves because they remain
guided between two free parallel surfaces. Horace Lamb [30] presented, in an outstanding paper,
his work on elastic waves. He was the first one to formulate the dispersion equations of waves
propagating in elastic infinite plates. In homage to his person, those waves are now called “Lamb
waves”. Since then, many researchers have showed a big endeavor for studying these particularly
important elastic waves and great contributions have been achieved leading to very promising
technologies. Lamb waves have several enabling capabilities which made them, in an early stage,
a hub in both research and applications. One among the earliest uses of lamb waves was to verify
the homogeneity of materials. Researchers realized later the sensitivity of lamb waves to the
presence of damages in structures and components and started to use them for the detection of
possible flaws and cracks. Worlton [55] conducted the first ultrasonic testing using lamb waves
in order to investigate on the presence of possible damages in a surface.

He used two

transducers, one as actuator and the other as receiver. Worlton used the hypothesis of infinite
surface because the formulation of lamb waves for semi-infinite and finite plates were not
56

developed yet at that time. It was until a decade later when such an assumption was proposed by
Torvik [49] who paved the way for other researchers interested in lamb waves to study the
reflection from boundaries and the conversion of modes.
The massive amount of research devoted to the understanding of the behavior of lamb waves
when propagating in elastic media has led to prominent and reliable inspection techniques. Their
use in engineering applications has been growing considerably, especially in the area of
structural health monitoring. A Main reason is that their propagation in damaged structures is
altered by the presence of defects (attenuation of the amplitude, reflection from the defect…). By
analyzing the output signals and comparing them to baseline data, one can locate, qualify and
quantify all the flaws and imperfections (disbands, corrosion, fatigue cracks, misalignment,
delaminations, etc.) in any structure to evaluate re. Another important reason is that the use of
lamb waves will not subject the structure to any harm and the evaluation is totally
nondestructive. Many other reasons also made from lamb waves an ideal tool in structural health
monitoring.

2.2.

Theoretical Backgrounds:

2.2.1. Governing Equation:
A fundamental grasp of the theory of lamb waves and their basics is given here in order to
achieve better understanding of their concepts and their fundamental characteristics (frequency,
wave speed, wavelength, etc.) and to, later, ease the interpretation of experimental and numerical
results. Historical drafts show that, the English mathematician Horace Lamb was the first
researcher who developed the equations for the dispersion of elastic waves in elastic media.

57

After Sir Lamb`s outstanding breakthrough, the theory of lamb waves has been extensively
studied and completely documented in several scientific references (journals, books, etc.).
Lamb wave can be described in terms of two scalar potentials φ and ψ. The potentials satisfy the
wave equation as follows:
 ∂ 2φ ∂ 2φ
1 ∂ 2φ
 2 λ + 2µ
 2+ 2 = 2 2
c p = ρ
∂y
c p ∂t
 ∂x


where: 
 2
2
2
1 ∂ψ
∂ ψ ∂ ψ
c 2 = µ
+
=
 2
 s ρ

∂x
∂y 2 c 2 ∂t 2
p



The propagation of lamb waves in an isotropic and homogeneous plate assumes the general
governing equation (Eq. 3.1) ([Su and Ye] [48]).
tan ( qd )
4k 2 pq µ
=
=0
tan ( pd )
λ k 2 + λ p 2 + 2µ p 2 k 2 − q 2

(

Where:

p2 =

w2
c2
p

)(

− k 2 ; q2 =

w2
2
cs

)

Eq 3.1

− k2

λ , k , w, cL and cT represent respectively the wavelength, the wavenumber, the angular
frequency, the longitudinal and the shears wave velocities.
The equation above equation describes the motion of the particles in the elastic media due to
perturbations caused by the input waveform (elastic displacement). The restoring forces between
particles (Han et al) [25] will cause the wave to propagate through the whole media. Lamb
waves propagate in different modes which are classified based on the arrangement of the general
governing equation. This equation is split into two sub-equations: one equation describes the
motion of particles in symmetric modes and the other describes their motion in antisymmetric
modes.
58

2.2.2. Symmetric Modes:
Symmetric modes are designated S0, S1, S2, etc., they propagate through the thickness in a
symmetrical way about the median plane of the structure as illustrated in Figure 3.03.

Figure 3.03: Schematic of Lamb wave propagation in symmetric mode
The equation that governs the motion of particles in symmetric modes is given by:

(

k 2 − q2
tan ( qh )
=−
tan ( ph )
4k 2 pq

)

2.2.3. Antisymmetric Modes:
Antisymmetric modes are designated A0, A1, A2, etc., they propagate through the thickness but
in an asymmetrical way about the median plane of the structure as shown in Figure 3.04

Figure 3.04: Schematic of Lamb wave propagation in antisymmetric mode
The equation that governs the motion of particles in antisymmetric modes is given by:
tan ( qh )
4k 2 pq
=−
tan ( ph )
k 2 − q2

(

)

During propagation, lamb waves are subjected to various parametrical modifications (changes in
direction due to reflection from boundaries and cracks, attenuation in amplitudes, etc.), such
59

modifications are not very important in the wave analysis and are easy to interpret. Lamb waves
are also known to be dispersive waves: the speed of the wave depends on the input frequency as
well as the on the thickness of the elastic medium the wave propagates through.

2.2.4. Dispersion of Lamb waves:
In digital signal processing, any signal that can be represented as time-varying amplitude has an
equivalent frequency spectrum in the frequency domain. The signal in Figure 3.05 (a) represents
the shape of a Morlet wave in the time domain, its corresponding frequency spectrum shown in
Figure 3.05 (b) was obtained using the technique of Fast Fourier Transform.

Figure 3.05: Presentation of a Morlet wave in the time and frequency domains

It shows that the wave is made of a central frequency with many side frequencies. If each
frequency propagates at the same constant velocity when the wave propagates through an elastic
medium, the medium is non-dispersive and the propagation of the wave is non-dispersive which
means that the wave is not subjected to any deformation. However, when each frequency
propagates at a different velocity when the wave propagates in an elastic medium, the elastic

60

medium is dispersive and the propagation of the wave is dispersive which means that the wave is
subjected to deformation while propagating. This phenomenon is called dispersion.
Almost all waves in nature are dispersive (light, sound, sea waves, etc.). Lamb waves are also
very dispersive; they propagate in elastic media with different velocities. The wave speed
depends on the frequency as well as on the thickness of the medium the wave propagates
through. Such dependence can be seen in plots called dispersion curves which represent, for both
symmetric and antisymmetric modes, the variation of the phase and group velocities as function
of the product of the frequency of vibration times the thickness of the structure on which lamb
waves are to be actuated. As we can see from the dispersion curves for both symmetric modes
(Figure 3.06) and antisymmetric modes (Figure 3.07), an increase either on the frequency of
vibration or on the thickness of the elastic medium will result in an increase on the number of
excited modes. Usually, it is advisable to pick a range of frequencies and a range of thicknesses
for which only the two fundamental modes S0 and A0 are excited, this helps considerably the
interpretation of output signals. If more modes are considered, the analysis of output signals will
be cumbersome and more complicated as modes can overlap easily, can be converted and can
generate new different modes when interacting with defects, etc.).

Figure 3.06: Phase velocity in aluminum plates for symmetric lamb wave modes
61

Figure 3.07: Phase velocity in aluminum plates for antisymmetric lamb wave modes

2.3.

Lamb waves in Structural Health Monitoring:

Lamb waves have received great attention since they were discovered and have been extensively
used in several engineering and industrial applications. In structural health monitoring, lamb
waves have gained a special interest as good alternative for techniques that rely on conventional
ultrasonic transducers. In fact, compared to conventional techniques which are tedious and timeconsuming, use of lamb waves has reduced considerably the time of inspection of structures by
offering the advantage of scanning very large surfaces in very short times and the ability to scan
curved, hidden and hard to reach areas. Due to these enabling capabilities, lamb waves have been
used for detection of all kinds of damages (cracks, delaminations, corrosion, etc.). Alleyne and
Cawley (1990 [01], 1991 [02], and 1992 [03]) used conjointly the 2-dimensional Fourier
transform and the finite element method in order to study the interaction of lamb waves with
simulated damages. They showed that the sensitivity of lamb waves to the presence of notches
depends on many factors such as the thickness frequency product and the geometry of the notch.
Guo et al. [24] studied the interaction of the fundamental symmetric mode S0 with a horizontal
delamination in a composite material. Lowe et al [35], [36] studied the behavior of the lamb
62

wave mode A0 in the presence of notches of different lengths and depths and in the presence of
vertical cracks. Later Lowe and Diligent [37] studied the case of S0. These studies paved the way
for research on the two fundamental modes. Many researchers used different other numerical
techniques (Finite difference, Boundary Element method, etc.) in order to characterize the
behavior of lamb waves when interacting with different types of damages in different structures.
Grondel [23] contributed to the optimization of inspection techniques based on lamb wave base
for structural health monitoring of aircrafts. Bourasseau [08] studied the propagation of lamb
waves in sandwich structures. Tua et al [50] proposed a comprehensive method based on the
time of flight in order to locate and determine linear cracks on homogeneous aluminum plates.
They showed that the method which consists on sweeping systematically a square grid of PZT
transducers and analyzing recorded signals based on times of flight is good in detecting linear
cracks and ascertain their locations.

3. Conclusion:
In this chapter, focus was placed on the presentation, description and understanding of the theory
of lamb waves. An in-depths review about the history, the formulations and the use of lamb
waves as an efficient tool for the assessment of several structures in, both, engineering and
industrial fields was also given. It appears through the literature review presented that the lamb
waves have been gaining much interest due to the capabilities they present in the technological
developments of new reliable techniques for nondestructive inspection.

63

Chapter 4
Damage Detection in Aluminum plates
1. Introduction:
Plates and plate-like structures are extensively used in all fields of engineering because of the
distinct advantages they present (high load-carrying capacities, light weights, effectiveness in
structural engineering, reduced prices, etc.). They are used in aircrafts, missiles, machines, civil
infrastructures (bridges, pavements, hydraulic structures, buildings, etc.). Due to the importance
of such structures, full inspection of plates and plate like structures before use (following
manufacturing process) and during use (under normal working conditions) is a necessity in order
to avoid any detrimental damages due to failure of one or more parts. For this reason, many
inspection techniques have been used in order to detect all possible forms of damages in plates
(cracks, delaminations, disband, etc.). Shang et al [46] used the technique of double-exposure
holography for the inspection and assessment of plates containing locally thinned regions as
artificial defects. Kannappan et al [27] developed an analytical solution based on the variation of
natural frequencies in order to detect and characterize damage in simply supported plates. They
used values of stored strain energies computed from the deformed mode shapes in order to locate
and characterize the damage. Many other schemes have been used for the characterization of
damages in plates and plate-like structures.

64

During the last two decades, inspection schemes based on structural health monitoring have been
extensively adopted for the detection of damages in plates and plate like structures. Ultrasonic
techniques (pitch-catch, pulse-echo, time reversal, etc.) based on the use of piezoelectric actuated
lamb waves have been among the leading ones. In this work, a brief description of the concept of
pulse-echo and the concept of pitch catch will be presented and attention will be specifically
given to technique of pitch-catch. Analysis of experiments conducted on aluminum plates using
piezoelectric mounted transducers will be discussed in depth.

2. Ultrasonic Health Monitoring Techniques:
2.1. Pulse-Echo Technique:
The pulse-echo is an ultrasonic nondestructive technique that involves the use of one transducer
as pulser/receiver to actuate signals through a medium and sense reflected signals in order to
detect flaws and cracks in simple and composite structures. The concept is straightforward: a
high frequency sound is actuated into a medium to evaluate, which generates a wave that
propagates through the material. When the wave interacts with a discontinuity, a part of the
energy will reflect back from the surface of the discontinuity. The receiver detects the reflected
energy and transforms it to an electric signal that can be analyzed in order to detect damages.

2.2.

Pitch-Catch Technique:

The pitch catch technique is an ultrasonic method that involves a set-up of a transmitter and
receiver located in the same side of the structure within a distance from each others. This
technique has been widely used to detect many types of defects (delaminations, inclusions, inhomogeneities, flaws, etc.). It is based on a simple concept; waves actuated by the transmitter
65

propagate through the material at a velocity c and reach the receiver after traveling for a time
t=d/c, known as the time of flight. Reflections from the boundaries and from defects will also
reflect back and reach the receiver at different times. Many researchers used the pitch-catch
configuration for several purposes. Bar-Cohen et al [07] used leaky Lamb waves in a pitch-catch
arrangement in order to detect and characterize flaws and defects in a composite material. Park et
al [43] introduced a sensor network system using four pairs of sensors in a pitch catch
configuration in order to identify possible cracks in a welded zone of a steel truss member. They
focused on the generation of the fundamental mode A0.

Figure 4.01: Schematics of the pulse-echo technique

Figure 4.02: Schematics of the pitch-catch technique
66

3. Ultrasonic Health Monitoring of Finite Aluminum Plates:
This part of the chapter details experiments carried in the nondestructive testing lab at Michigan
State University for the health monitoring of finite aluminum plates. Assessment was based on
the technique of pitch catch with surface-mounted lead zirconate titanate (PZT) piezoelectric
wafers. All the experiments conducted involved the use of the following equipments:
1. Arbitrary Function Generator

4. Piezoelectric patches

2. Digital Oscilloscope

5. Aluminum plate

3. Computer

6. Permabond 910

In each experiment, the aluminum plate was cleaned and then mounted on a table. Second,
piezoelectric patches were cut to custom dimensions and then bonded to the surface of the plate
in predefined configurations. Third, an input signal, with appropriate shape, frequency and
amplitude, was generated using the function generator and sent to the transmitter. Finally, the
digital oscilloscope attached to the sensor was used to record the output signal and store it in the
computer. The figure below summarizes the experimental set-up followed in all the experiments.

Figure 4.03: Experimental set-up of a standard pitch-catch configuration

67

3.1. Geometrical and Electro-mechanical Properties:
The Lead Zirconate Titanate (PZT) piezoelectric patches used in the experiments were purchased
from STEMINC (STEINER & MARTINS, INC), they have the following properties:
3

Geometry: square PZT patches of dimensions: 10x10x0.55 mm

Figure 4.04: Piezoelectric wafer used in the experiments

Electro-mechanical properties:
The piezoelectric patches used in all the experiments are made of the ceramic material SM410
with the Electro-mechanical properties summarized in Table 4.01.
Property

Unit

Symbol

Value

Kp

0.64

Kt

0.45

K31

0.36

10
m/v

d33

500

d31

-210

-3

g33

23.3

g31

-10.3

Y33

5.6

Y11

6.5

Qm

60

ε33/ ε0

2000

Tc

320

R

7.7

Electromechanical
coupling
coefficient
-12

Piezoelectric
constant

10
Vm/N
10

Elastic Constant
Mechanical
Quality Factor
Dielectric
constant
Curie temperature
Density

10

2

N/m

@1Khz
o

C
3

g/cm

Table 4.01: Electromechanical properties of the SM410 material
68

3.2. Aluminum Plate of Thickness 2mm:
Aluminum is a commonly used material in engineering and industrial applications (aerospace,
civil, mechanical, etc.). It has specific chemical, mechanical and physical properties much like
3

any other metal (steel, zinc, copper, etc). It is very light, with a weight of 2.7 g/cm which
represent approximately the third of the weight of steel. The use of aluminum in manufacturing
applications reduces the weight of structures and components hence the energy consumption
while maintaining high load capacities (for example in aircrafts design, less weight results in less
fuel consumption). Aluminum presents also many others properties; it is highly resistive to
corrosion, electrically and thermally conductive, adaptable to different applications (easy
modification of the compositions of its alloys) and environment friendly (easily recycled).
Moreover, wave propagation in aluminum is easily understood.
For all above reasons, we selected aluminum testing sample plates for our experiments. In this
experiment, we used a 2mm thick aluminum plate with length and width both equal to 600mm.
The plate is free of any visible crack.

3.2.1. Experimental Set-up:
Two piezoelectric wafers were bonded to the surface of the plate distant of 20 cm center-tocenter using the Cyanoacrylate glue Permabond 10. The arbitrary function generator was used to
generate and send an input signal to one of the wafers (acting as transmitter) in order to generate
a mechanical lamb wave in the plate. The digital oscilloscope was directly connected to the
second piezoelectric wafer (acting as a receiver) in order to visualize and record the output
signal. Once the wave is sensed and captured by the oscilloscope, data was transferred to the
computer via GPIB and gathered for further signal processing.
69

Figure 4.05: Experimental setup
A Morlet wavelet is used in this experiment as an input signal (Figure 4.06). The wavelet was
built up in the computer, fed to the function generator and saved in the memory for further uses.
The wavelet is centered at a frequency of 500 KHz. The frequency was chosen such that
minimum number of modes is activated with sufficient amplitudes to propagate for the required
distance. Due to the limitation of the arbitrary function generator, a peak to peak voltage of 10V
was applied to the transmitter.

Figure 4.06: Input signal for lamb wave actuation: Morlet wavelet

70

3.2.2. Results and Discussions:
Lamb waves were generated in the 2 mm plate in a standard pitch-catch technique. After the
wave propagates for a 20 cm, it reaches the receiver where it is sensed saved and analyzed. The
Figure 4.07 shows the output signal sensed by the receiver.
The analysis of the output signal shows the presence of the two first fundamental modes S0 and
A0 with extra packets due to the reflection of the wave from the edges of the plate. At the
frequency of 500 KHz, the symmetric mode S0 appears to be more dominant than the
antisymmetric mode A0.

Figure 4.07: Output signal sensed by the receiver

The result confirms a fact related to lamb waves in thin structures: at higher ranges of frequency,
S0 is dominant however at lower ranges of frequency A0 is dominant. Such property of lamb
waves is important as it helps deciding which mode is to be used for the detection of cracks in a
structure. Figure 4.07 shows also that the symmetric mode reaches the sensor earlier than the
antisymmetric mode A0 which means that S0 is faster than A0. This actually can be seen on the
71

dispersion curves for any thickness-frequency product (f.d). In fact, for a given thicknessA0
S0
frequency product, correspond two values of the velocities V ph and V ph for respectively S0

mode and A0 mode. Since we know the distance between the two piezoelectric patches, we can
easily compute the time each packet of the wave takes to travel the required distance between the
PZT actuator and the PZT sensor. This time is known as the time of flight.

3.3.

Aluminum Plate of Thickness 4mm:

This part of the chapter is devoted to study the effect of the presence of cracks in aluminum
plates on the propagation of lamb waves. For this reason, four aluminum plates of grade 3003-0 SH were used in the experiments. One plate is healthy (without any defect) and the three other
plates with predefined notches. The notches are of rectangular shape with similar lengths, similar
widths but with different depths. The table below summarizes the dimension of the plates and the
notches.
Plate Dimensions
Length

width

thickness

Notch Dimensions
length

width depth (d)

Plate 1
Plate 2

0
600 mm 600 mm

4 mm

20 mm 2 mm

1.6 mm

Plate 3

2.4 mm

Plate 4

3.2 mm

Table 4.02: Geometrical properties of the aluminum plates
The notches were introduced with depths equal to 40%, 60% and 80% of the plate thickness
respectively for plates 2, 3 and 4

72

Figure 4.08: Geometry of an aluminum plate containing a predefined notch

3.3.1. Experimental Set-up:
Similar to the case of the 2mm plate described above, piezoelectric wafers were bonded to the
surface of each plate using the Cyanoacrylate glue Permabond 10. The arbitrary function
generator was used to generate and send an input signal to the piezoelectric transmitter in order
to generate elastic lamb waves. The digital oscilloscope was directly connected to the
piezoelectric receiver in order to visualize and record the output signal. Once the wave is sensed
and captured by the oscilloscope, data is transferred to the computer via GPIB and gathered for
further signal processing.
All the experiments in this section are based on the ultrasonic technique of pitch-catch. Two
piezoelectric patches were attached on each plate, one acting as a receiver and the other as a
sensor. The piezoelectric transducers were placed in the same positions for the four plates and
two configurations (based on the location of the piezoelectric patches with respect to the location
of the notch) were tested.

73

3.3.2. Actuator-Sensor-Notch Configuration:
In this configuration, the sensor was placed between the notch and the actuator at a distance of
15 cm. This configuration aims at studying the reflection of the incident lamb waves from the
notch. A sketch of the experiment is shown in Figure 4.09.
Measurements were performed on the four aluminum plates at variable frequencies (frequency of
the input signal). A range of frequencies from 100 KHz to 300 KHz with an increment of 25
KHz was selected in order to see, the behavior of lamb waves when interacting with the
introduced notches.

Figure 4.09: (a) Defective plates and (b) Healthy plate in an
Actuator-Sensor-Notch configuration

Results and discussions:
The data gathered from all the plates at different frequencies, are saved in the computer, and are
presented in form of plots showing the output signals collected from the piezoelectric sensors.
Using group velocity dispersion curves for aluminum; we extract the velocities of S0 and A0
modes at each frequency. The table below summarizes the computed velocities.

74

Frequency (KHz)

100 125 150 175 200 225 250 275 300
Velocity of S0 (ms ) 5320 5300 5270 5200 5170 5100 5050 4950 4850
-2

-2

Velocity of A0 (ms ) 2850 2950 3000 3050 3080 3100 3105 3110 3112
Table 4.03: Velocities of the fundamental modes S0 and A0 in the aluminum plates.
The velocities were used to determine times of flights of the different packets of the signal
•

Incident symmetric wave: S0

•

Incident asymmetric wave: A0

•

Reflection of S0 from the crack

•

Reflection of A0 from the crack

•

Reflection of S0 from the close edge

•

Reflection of A0 from the close edge

In the next plots, the following notations are used:
•

TS0: Time of flight of the incident S0

•

TA0: Time of flight of the incident A0

•

TS0 (Defect): Time of flight of the scattered S0

•

TA0 (Defect): Time of flight of the scattered A0

•

TS0 (Edge): Time of flight of S0 reflected from the edge of the plate

•

TA0 (Edge): Time of flight of A0 reflected from the edge of the plate.

The time of flight of the lamb wave is equal to the ratio of the distance travelled by the wave to
reach the sensor to the velocity of the wave: t f =

d
V

The experimental configuration described above was tested at nine different frequencies (100 to
300 KHz with an increment of 25 KHz. The Output Data for each excitation frequency was
gathered from the oscilloscope, processed in Matlab and presented in form of signals showing

75

the response of the sensors on the four sample plates at that excitation frequency. Figure 4.10
shows the plot of the signals sensed in the four plates when the actuators were excited by a 250
KHz Morlet wave. Plots of all the others frequency cases can be seen in Appendix B. In all the
plots, the region that correspond to the defect is limited by TS0(Defect) which marks the time
when the incident signal reflects back from the defect and TS0(Edge) which corresponds to the
time when the signal reflects from the closest edge of the plate.
Figure 4.10 shows that the signal in the region of the defect becomes more intense and its
amplitude increases when the percent defect increases. This is explained by the fact that the
defect reflects more energy when it is deeper and only a smaller part of the wave can propagate.

Plots in the Time-Frequency Domain.
Analysis of the signals at different frequencies was also carried out in the time-frequency
domain. Plots showing the intensity of the lamb wave were presented in order to interpret the
influence of the crack on the propagation of the wave. Figure 4.11 shows the time-frequency
plot of the signals sensed in the defected plates at 250 KHz excitation frequency. The plots of the
others frequency cases can be seen in Appendix B.
The box in Figure 4.11 delimits the region of the signal where the input signal interacts with the
defect. It can be seen in this region that the colors become more intense and more spread in the
plot when the percent depth of the defect increases. This can be explained by the fact that the
defect reflects more energy of the incident signal when its depth increases.

76

Figure 4.10: Input signal and Sensed signals at 250 KHz.

77

Figure 4.11: Time – Frequency plot at 250 KHz for different crack depths
(a) 40% T (b) 60% T (c) 80% T

3.3.3. Actuator-Notch-Sensor Configuration:
In this configuration, the notch is located between the actuator and the sensor. This aims at
studying the effect of the notch depth on the amplitude of the incident wave. A sketch of the
experiment is represented in Figure 4.12 where the actuator is placed at the center of the
aluminum plate in order to avoid undesirable reflections from close edges and get clear incident
signals without interference with backscattered waves.
Measurements on this experiment were also performed on the four aluminum plates in a
frequency range ranging from 100 KHz to 300 KHz with an increment of 25 KHz. For all the
frequency cases, focus was only placed on the symmetric mode S0 of incident wave. The idea is
to study the effect of the defect depth on the amplitude of the signal.

78

Figure 4.12: (a) Defective plates and (b) Healthy plate in an Actuator -Notch-Sensor
configuration.

Results and Discussions:
The experiment sketched in Figure 4.12 was carried out at different frequencies. The output data
was gathered from the digital oscilloscope, saved in the computer then processed for use
purposes. Only the region of the signal resultant from the propagation of the incident wave in
the symmetric mode S0 is of concern. For this reason, the output signals sensed on the plates
were chopped, in each frequency case, and regions of interest saved for further use.
Figure 4.13 illustrates the result of this experiment for the case of an exciting frequency equal to
250 KHz. It shows the variation of the amplitude of the signal as function of the defect depth.
The coefficient R represents the depth of the notch measured in percent of the plate thickness. R
takes the values 0, 40%, 60% and 80% which corresponds to the depths 0 mm (pristine plate),
1.6 mm, 2.4 mm and 3.2mm respectively. We clearly see that the amplitude decreases when the
depth increases. The interpretation of this finding is based on the fact that the amplitude of the
signal is proportional to the amount of energy transmitted through the defect. In fact, the defect

79

transmits less energy of the incident wave when its depth increases and a weaker signal with
smaller amplitude is obtained. The figures corresponding to the other frequency cases are
appended in Appendix B
600
R=0%
R=40%
R=60%
R=80%

Amplitude (mV)

400
200
0
-200
-400
-600

0

50

100
150
Time (micro-sec)

200

250

Figure 4.13: S0 mode at 250 KHz for deferent crack depths

4. Conclusion:
In this chapter, experiments based on the ultrasonic technique of pulse-echo were conducted on
four aluminum plates with manufactured notches of variable depths in order to characterize the
effect of defects on the behavior of piezoelectric actuated lamb. First, a simple experiment used
as proof of concept was conducted on a 2 mm thick aluminum plate in order to check the quality
of the piezoelectric patches and to ease the understanding of lamb waves before conducting the
major experiments. Second, experiments were carried out on 4 mm thick aluminum plates in
order to study the effect of the notch depth on the behavior of lamb waves. Results show that for
notches with same lateral dimensions and different depths, the energy of the transmitted signal
decreases when the depth of the notch increases.

80

Chapter 5
Phased Array Health Monitoring
1. Introduction:
Piezoelectric materials have been of great advantages in Structural health monitoring
applications. They enable quick, efficient and less costly inspection operations. During the last
few years, piezoelectric materials have been combined with more advanced nondestructive
techniques in order to enhance the assessment of structures and solve inspection problems that
traditional techniques could not solve. As a result, many smart techniques have been developed
and successfully used in diverse applications.
In ultrasonics, the technique of phased arrays stands among the best advanced techniques that
rely on the use of smart piezoelectric elements in emission and sensing applications for
inspection of structures.

It is based on a concept and principles similar to the radar, sonar,

seismology, oceanography and medical imaging (Lines et al [33]): a set of probes made of the
same material and arranged in a desired form (linear, circular, matrix, etc.) are driven either
independently or simultaneously in order to emit and receive focused waves that can be used to
inspect a structure. The probes used in the phased array are actuated in different manners
(different modifications of the directivity pattern of the array) in order to form moving beams,
81

focused beams and or deflected beams that propagate in desired directions. This procedure is
known as beam forming and it aims at steering the maximum sensitivity of the array in a desired
direction of particular interest in order to obtain the maximum amplitude of the received signal.
Phased arrays technique has gained wide interests in both the industrial and the research
communities and has been successfully applied as an alternative for ultrasonic conventional
techniques. In fact, the phased array ultrasonic technique presents major advantages over
ordinary and conventional ultrasonic techniques. For example, it helps reducing the inspection
times by performing fast electronic scanning and eliminating the need for mechanical scanning.
As a result the reliability of measurements is considerably improved and variations and changes
due to losses of the contact between the probe and the structure are decreased. Phased arrays
technique has also the ability to control the ultrasonic beam and steer it in a desirable direction
which helps optimizing inspection operations whereas in conventional technique, the wave
cannot be steered and it has only one orientation.
Due to its enabling capabilities, the phased array ultrasonic technique has been the subject of
research for many scientists. Deutsch et al [09] presented one of the earliest works which
investigates on the use of the phased array technique for lamb wave. They developed a time
reversal based method in order to focus a uniform linear array on a defect. Wilcox et al [54]
proposed an array approach using ceramic disc of circular shape in order to inspect large areas of
thick plates. Giurgiutiu et al [17] developed a new approach of health monitoring using phased
array technique. They proposed an experimental set up which they called “embedded-ultrasonic
structural radar” (EUSR). This setup involves the use of a linear array of piezoelectric wafer
actuated sensor (PWAS) fired in round-bond manner so that to produce a virtual beamforming in
order to detect two manufactured defects in a thin aluminum plate. Hong et al [52] conducted a

82

numerical study on which they successfully implement a virtual lamb wave phased array using
16 piezoelectric patches for damage detection purposes.

Many others works have been

performed in order to study the applicability of the ultrasonic method phased array in health
monitoring tasks.

2. Ultrasonic Method of Phased Array:
2.1. Concept:
In ultrasonics, a phased array is a set of probes driven independently or simultaneously with
variable time delays in such a way that the directivity or the propagation pattern of the array is
altered. This aims at intensifying the propagation in a desired direction (Figure 5.01) and
suppressing it in unwanted directions.

Figure 5.01: Beam focusing on misoriented cracks

2.2. Principles:
In a phased array experiment, the probes are fired independently or simultaneously with variable
time delays so that to steer the beam in a desired direction Figure 5.02. When the incident wave
front resulting from the constructive beam forming reaches a defect, a part of the energy will
83

reflect back to the array and reach the elements at different times. In order to get the total
scattered signal due to the actuating array, appropriate time delays should be applied to the
signals sensed at each element.

Figure 5.02: Beam forming technique

3. Piezoelectric Array for Inspection of Aluminum Plates:
3.1. Description:
Giurgiutiu et al [17] developed the technique of Embedded-ultrasonics structural radar (EUSR)
by which they proved that a linear array of piezoelectric wafer active sensors permanently
attached to an aluminum plate can perform as good as any conventional ultrasonic phased array
apparatus in the detection of cracks and flaws in an aluminum plate. This technique is easy to
implement, cheap (available and cheap piezoelectric materials), efficient and rapid.
The technique is based on a virtual scan of the structure on which the array of piezoelectric wafer
active sensors is mounted or embedded. The virtual scan is achieved by activating the elements
of the array independently in a round-robin fashion and collecting the data from all the elements.
An assembly of the gathered data is then generated and analyzed using signal processing

84

methods in order to synthesize the virtual transmitted and received waves and apply
beamforming theory for the detection of cracks and damages.
In this thesis, a method based on the same concept of the EUSR technique was used for the
inspection of aluminum plates with variable crack depths. The method uses also a linear array of
piezoelectric wafers active sensor in order to actuate and sense lamb waves in the four aluminum
plates. The difference, in this thesis, is that the assumption of far field and Omni-directional
wave in Girgiutiu’s work was not followed. Rather, a different formulation for time delays was
adopted.

3.2. Experimental Setup:
Ultrasonic Phased arrays experiments were conducted on the four aluminum plates used in the
previous experiments. Each plate was implemented with an array of 6 piezoelectric elements of
length, width and thickness equal to 10 mm, 10 mm and 0.5 mm thick respectively. For the
Healthy plate, the objective was to focus the beam on a point located at one of the edges (Figure
4.04). The beam was steered at an angle of 90 degrees from the axis of the array.

Figure 5.03: Sketch of the phase array on the healthy plate
85

For the defective plates, the objective was to focus the beam on the manufactured defect located
at a focal distance d=20 cm from the axis of the array (Figure 5.04)

Figure 5.04: Sketch of the phase array on a defective plate

In order to generate a virtual phased array signal, all the elements of the array were actuated
separately in a round-band fashion (Giurgiutiu and Bao 2004) which means we actuate only one
element each round and receive signals from all the others element. At the end we collect all the
signals and arrange them in a matrix form in order to apply the necessary delays.

T0

T2

T3

T4

T5

T1R0

R0

T1

T2R0

T3R0

T4R0

T5R0

T2R1

T3R1

T4R1

T5R1

T3R2

T4R2

T5R2

T4R3

T5R3

R1

T0R1

R2

T0R2

T1R2

R3

T0R3

T1R3

T2R3

R4

T0R4

T1R4

T2R4

T3R4

R5

T0R5

T1R5

T2R5

T3R5

T5R4
T4R5

Table 5.01: Matrix of signals collected from each plate
86

In the Table 5.01, the cell in row i and column j contains the signal Ti R j recorded by the element
receiver ‘Rj’ when the element transducer ‘i’ is fired.

3.3. Time Delays:
In a traditional phased array experiment, all the probes in the array are fired simultaneously with
different time delays in order to steer the beam at a specific direction. In this experiment, we
actuated the elements of the array separately and without delays. So, we applied appropriate time
delays in order to focus the beam on the notch.
The required time delays were computed using a general focusing technique proposed by Azar et
al [05]. The expression was successfully used by Hong et al [52] to numerically implement a
virtual phased piezoelectric array
1
1

 ( i − N ) d  2
2 
  Nd 2
2
( i − N ) d sin θ  
D 
Nd

+2
sin θ s  − 
ti =  

 −2

s
c   D 
D
D
D

 











ti: required time delay for the i

th

element of the array.

D: distance from the center of the array to the point where the beam is to be focused at.
c: is the wave speed, θs: steering angle
N: number of actuating elements in the array.

µ = N ሺܰ 2
n=0, …, N-1: array index, N ܰ (ൌ -1) /െ 1ሻ/2
Using the proposed equation we compute the required time delays for the elements of the array
on the healthy plate and the time delays for the elements of the arrays on the defective plates.

87

Figure 5.05: Sketch of the phased array for a random location (D, θs)

3.4. Healthy Plate:
The beam was focused on one edge of the plate at the distance D=25 cm and a steering angle
θs=0. The experiment and the corresponding timing delays are shown on Figure 5.06.

Figure 5.06: Sketch of the phased array and corresponding time delays for the healthy plate

88

3.5. Defective Plates:
The beam on the defective plates were focused on the defect located on a focal distance D=20
cm and a steering angle θS = 0 degrees. The sketch of the experiment and the corresponding
timing delays are shown on Figure 5.07.

Figure 5.07: Sketch of the phased array and corresponding time delays for a defective plate

3.6. Results and Discussions:
Once all the elements of the array fired and the signals sensed, we proceed to the construction of
the virtual beamforming by applying the appropriate time delays to the signals received by each
piezoelectric element.
The total signal Rj received by the element j; j ∈{0,1, 2,3, 4,5} is given by the expression:

Rj =

5

∑ Ti R j

i =0
i≠ j

89

The summation is computed for each element of the array after all the necessary time delays are
applied to the corresponding signals TiRj. The plots in Figures 5.08, 5.09, 5.10 and 5.11 illustrate
the total signals received by the elements of the array for the healthy plate, the plate with 40%
defect, the plate with 60%defect and the plate with 80% defect respectively.

Figure 5.08: Signals Received by the Sensors on the healthy plate

90

Figure 5.09: Signals Received by the Sensors for the plate with 40% defect depth

91

Figure 5.10: Signals Received by the Sensors for the plate with 60% defect depth

92

Figure 5.11: Signals Received by the Sensors for the plate with 80% defect depth

After applying the time delays and obtaining the total signal in each element of the array, it is
now easy to find the final signal sensed by the array. This can be easily done by simply
adding all the tolal signals obtained on the elements of the array. For example, for the case of
the healthy plate, all the signals in Figure 5.08 summed up together form the final signal

93

sensed by the array on the plate (Figure 5.12). Similar procedure was applied for three the
defective plate.

Figure 5.12: Total Signal Received by the array on the healthy plate

The plot in figure 5.12 shows that the total signal on the healthy plate is formed of two main
wave packets: a first packet represented by an initial bang in which are included both the
symmetric mode S0 and the anti-symmetric mode A0 of the incident wave and a second region
which contains the reflection of the incident wave from the edges of the plate. Those packets can
also be seen on the signals of the defective plates but the main difference is that each signal
sensed on a defective plate contains one more packet which represents the reflection of the
incident wave from the defect. Those packets are depicted by the plots in Figures 5.13, 5.14 and
5.15.

94

Figure 5.13: Total Signal Received by the array (40% defect depth)

Figure 5.14: Total Signal Received by the array (60% defect depth)

95

Figure 5.15: Total Signal Received by the array (80% defect depth).

4. Conclusion:
In this chapter, experiments based on the technique of ultrasonic phased arrays were conducted
on four aluminum plates, one healthy and three defective plates with manufactured notches of
variable depths. The idea was to illustrate the principle of phased arrays and see its efficiency in
detecting the manufactured notches present on the plates. For this reason, a virtual beamforming
method, for which an array of six piezoelectric transducers actuated in a round-bond manner, is
applied in order to characterize the behavior of lamb waves in presence of the manufactured
defects. Signals are collected separately and appropriate delays are applied in order to form the
total signal sensed by the phased array. Results show that the phased array was successfully used
to detect the manufactured damages on all the defective plates. This makes from the technique of
virtual beamforming used with an array of simple piezoelectric transducers a good experimental
alternative for the detection, the qualification and the quantification of more complex and
misoriented damages present in different daily used structures.
96

Chapter 6
Conclusion
This thesis discusses some main properties and applications of piezoelectricity and lamb waves
in structural health monitoring. In a first part, numerical and analytical simulations of an
unbonded piezoelectric patch were conducted in order to plot the impedance and study its
variation with the variation of the compliance and strain parameters of the patch. The results of
the experiments showed that the compliance coefficients have an effect on the frequencies of
peaks of resonance and antiresonance while the strain coefficients have an effect on the
amplitude of the impedance. The comparison between the numerical results and the analytical
results showed some differences between the curves of impedance. Those differences were
considerably, but not fully, reduced when some variations were applied to some of the
parameters. Such a study can be of good utility for the understanding of the electromechanical
behavior of the patch and can also be useful for controlling its quality before using it in
experiments.
In the second part of the thesis, experiments on the wave propagation in finite aluminum plates
were conducted in order to show the efficiency of lamb waves in the detection of defects present
in finite aluminum plates with predefined notches of different depths. For this reason small
piezoelectric transducers have been used as both actuators and sensors in order to generate lamb

97

waves in the aluminum plates. The idea behind this first experiment was to show the effect of the
notch depth on the propagation of energy transported by the wave. It has been shown that the
amount of energy transmitted through the notch decreases with the increase in the depth.
The propagation of lamb waves actuated by a phased array of piezoelectric transducers was also
investigated in this thesis. The main idea behind the experiment was to focus the wave in a
specific direction in order to detect defects manufactured on three aluminum plates. The results
show that the technique of virtual beamforming used on an array of simple piezoelectric was
successful in detecting the defects pre-installed on the aluminum plates.

98

APPENDIX A

\

99

ANSYS CODE USED FOR THE NUMERICAL SIMULATIONS:

fini
/CLEAR,START
! START NEW MODEL.
/FILNAME,Impedance
/TITLE,Impedance Analysis PZT-5A
/CONFIG,NRES,1e+6
/PREP7
csys,0
/UNITS,SI
! ****************************************************************************
/com USER DIMENTIONS OF PZT
L=20e-3
W=1e-3
thick=0.5e-3
mesh_size_l=200
mesh_size_w=10
mesh_size_thick=5
! a mesh of 0.1mmX0.1mmx0.1mm

! length of the PZT
! width of the PZT
! thickness of the PZT
! No of elements in length
! No of elements in width
! No of elements in thickness

/com Compliance matrix
! ****************************************************************************
/com PZT-5A PIEZO ELEMENT Z-POLARIZED STIFFNESS MAT PROP
/COM, MATERIAL MATRICES (POLAR AXIS ALONG Z-AXIS)
/COM,
/COM, [S11 S12 S13 0 0 0 ]
[ 0 0 e31]
[eps11 0
0 ]
/COM, [S12 S22 S23 0 0 0 ]
[ 0 0 e32]
[ 0 eps22 0 ]
/COM, [S13 S12 S33 0 0 0 ]
[ 0 0 e33]
[ 0
0 eps33]
/COM, [ 0 0 0 S66 0 0 ]
[ 0 0 0]
/COM, [ 0 0 0 0 S44 0 ]
[ 0 e42 0 ]
/COM, [ 0 0 0 0 0 S55]
[e51 0 0 ]
TB,ANEL,1,1,21,1
TBDATA,1,16.4E-12,- 5.74E-12,-7.22E-12
TBDATA,7, 16.4E-12, -7.22E-12
TBDATA,12, 18.8E-12
TBDATA,16, 44.3E-12
TBDATA,19,47.5E-12
TBDATA,21, 47.5E-12

! S11,S12,C13
! S22,S23
! S33
! S66
! S44
! S55

/com PIEZO MATRIX PIEZOELECTRIC STRESS MATRIX [d], [C/N]
![d] matrix is associated with the compliance matrix [S]
TB,PIEZ,1,,,1
! DEFINE PIEZ. TABLE
TBDATA,3, -171E-12
! d31 PIEZOELECTRIC CONSTANT
TBDATA,6, -171E-12
! d32 PIEZOELECTRIC CONSTANT
100

TBDATA,9, 374E-12
TBDATA,14, 584E-12
TBDATA,16, 584E-12
!TBLIST

! d33 PIEZOELECTRIC CONSTANT
! d42 PIEZOELECTRIC CONSTANT
! d51 PIEZOELECTRIC CONSTANT

/com PERMITTIVITY AT CONSTANT STRAIN
EMUNIT,EPZRO,8.85E-12
! FREE-SPACE PERMITTIVITY 8.85E-12 F/M.
MP,PERX,1,173
! PERMITTIVITY (X DIRECTION) relative permittivity
MP,PERY,1,1730
! PERMITTIVITY (Y DIRECTION) relative permittivity
MP,PERZ,1,1700
! PERMITTIVITY (Z DIRECTION) relative permittivity
/com Density
MP,DENS,1,7750
! Density for material 1
SAVE
!
******************************************************************************
*****
! 3-D COUPLED-FIELD SOLID5,
ET,1,SOLID5,3
MAT,1
TYPE,1
/com START BUILDING THE MODEL
BLOCK,0,L,0,W,0,thick,
! Define volume block,x1,x2,y1,y2,z1,z2
lsel,s,line,,1
LESIZE,1, , ,mesh_size_w, , , , ,1
lsel,s,line,,2
LESIZE,2, , ,mesh_size_l, , , , ,1
lsel,s,line,,9
LESIZE,9, , ,mesh_size_thick, , , , ,1
VMESH,1 ! mesh volume 1
lsel,all
NSEL,S,LOC,Z,0
! SELECT NODES AT TOP ELECTRODE nsel- select a subset of nodes
CM,bottom_elec,NODE ! CREATE A COMPONENT FOR THE BOTTOM SURFACE
CP,1,VOLT,ALL
! COUPLE VOLT DOF ON NODES OFF BOTTOM SURFACE
*GET,N1,NODE,,NUM,MIN ! GET NODE ON ELECTRODE i,e Pick a node in this group
ALLSEL
NSEL,S,LOC,Z,thick ! SELECT NODES AT TOP ELECTRODE nsel-select a subset of nodes
CM,top_elec,NODE
! CREATE A COMPONENT FOR THE TOP SURFACE
CP,2,VOLT,ALL
! COUPLE VOLT DOF ON NODES OFF TOP SURFACE
*GET,N2,NODE,,NUM,MIN ! GET NODE ON ELECTRODE i,e Pick a node in this group
ALLSEL
101

FINISH
! Finish pre-processing
! ****************************************************************************
!/STATUS,SOLU
/SOLU
! ENTER THE SOLUTION PROCESSOR
/COM, BOUNDARY CONDITIONS
D,bottom_elec,VOLT,0
! GROUND BOTTOM ELECTRODE
D,top_elec,VOLT,1.0
! APPLY 1 VOLT TO TOP ELECTRODE
ALLSEL
! ****************************************************************************
/SOLU
EQSLV,Sparse
ANTYP,HARMIC,NEW
HROPT,FULL
HROUT,ON
OUTRES,ALL,NONE
OUTRES,RSOL,ALL
HARFRQ,0,1e+6
! Frequency range
dmprat,0.0001
! damping coefficient
NSUBST,1000
! number of subsets for each set
KBC,1
ALLSEL
/STATUS,SOLU
! Provide a solution status summary
SOLVE
SAVE
FINISH
! Exit active processor
!/COM EXIT,NOSAVE
!
******************************************************************************
*****
/POST26
! Enter post processor
/UI,COLL,1
! Activates specified GUI dialog boxes
RFORCE,2,N2,AMPS
! STORE CHARGES Q ON ELECTRODE,! reaction data

Matlab Code used to compare the numerical to the analytical results
102

%% analytical and numerical impedance of a PZT (l>>b>>h): 1-D assumption
function [ ]=impedance
%% analytical impedance
clear all
clc
l=20e-3;
%length of PZT
b=1e-3;
%width of PZT
h=0.5e-3;
%thickness of PZT
s11=16.4e-12;
%compliance coefficient s11
d31=-171e-12;
% piezoelectric coupling d31
ep0=8.854e-12;
ro=7750;
%density
eps33=1700*ep0;
% relative permittivity
Cap=(eps33*b*l)/h;
% Capacitance of the PZT
k31=sqrt((d31^2)/(s11*eps33)); % coupling factor
f=1000:1000:1e+6;
%frequency range
w=2*pi*f;
c=sqrt(1/(ro*s11));
%wave speed
gamma=w/c;
%wave number
phi=0.5*gamma*l;
Za=(1./(i*w*Cap)).*((1-(k31)^2*(1-(tan(phi)./phi)))).^(-1); % analytical impedance
figure (1)
plot(f,log10(abs(Za)),'r')
title('IMPEDANCE')
xlabel('Frequency(Hz)')
ylabel('Magnitude');
legend('---')
hold on
%% numerical impedance
load hong.txt
% save the output of ansys under the name imped.txt, edit it (delte the text
part) and load it in matlab
f=hong(:,1);
% frequency
rl=hong(:,2);
% real part
im=hong(:,3);
% imaginary part
Y=(2*pi*i).*f.*(rl+1i*im); % admittance
Zn=1./Y;
% impedance
plot(f,log10(abs(Zn)),'b')
h = legend('Analytical Impedance','Numerical Impedance',2);
end
%%

103

APPENDIX B

104

Figure B.01: Experimental signals at 100 KHz

105

Figure B.02: Experimental signals at 125 KHz.

106

Figure B.03: Experimental signals at 150 KHz.

107

Figure B.04: Experimental signals at 175 KHz.

108

Figure B.05: Experimental signals at 200 KHz.

109

Figure B.06: Experimental signals at 225 KHz.

110

Figure B.07: Experimental signals at 250 KHz.

111

Figure B.08: Experimental signals at 275 KHz.

112

Figure B.09: Experimental signals at 300 KHz.

113

Figure B.10: Time– frequency representation at 100 KHz for deferent crack depths
(a) 40% TW (b) 60% TW (c) 80% TW

Figure B.11: Time – frequency representation at 125 KHz for deferent crack depths
(a) 40% TW (b) 60% TW (c) 80% TW
114

Figure B.12: Time – frequency representation at 150 KHz for deferent crack depths
(a) 40% TW (b) 60% TW (c) 80% TW

Figure B.13: Time – frequency representation at 175 KHz for deferent crack depths
(a) 40% TW (b) 60% TW (c) 80% TW

115

Figure B.14: Time – frequency representation at 200 KHz for deferent crack depths
(a) 40% TW (b) 60% TW (c) 80% TW

Figure B.15: Time – frequency representation at 225 KHz for deferent crack depths
(a) 40% TW (b) 60% TW (c) 80% TW

116

Figure B.16: Time – frequency representation at 250 KHz for deferent crack depths
(b) 40% TW (b) 60% TW (c) 80% TW

Figure B.17: Time – frequency representation at 275 KHz for deferent crack depths
(a) 40% TW (b) 60% TW (c) 80% TW

117

Figure B.18: Time – frequency representation at 300 KHz for deferent crack depths
(a) 40% TW (b) 60% TW (c) 80% TW

118

100
R=0%
R=40%
R=60%
R=80%

Amplitude (mV)

50

0

-50

-100

0

50

100

150

200
250
Time (micro-sec)

300

350

400

Figure B.19: S0 mode at 100 KHz for deferent crack depths

200
R=0%
R=40%
R=60%
R=80%

150

Amplitude (mV)

100
50
0
-50
-100
-150
-200

0

100

200

300
Time (micro-sec)

400

500

Figure B.20: S0 mode at 125 KHz for deferent crack depths

119

600

250
R=0%
R=40%
R=60%
R=80%

200
150

Amplitude (mV)

100
50
0
-50
-100
-150
-200
-250

0

50

100

150

200
Time (micro-sec)

250

300

350

400

Figure B.21: S0 mode at 150 KHz for deferent crack depths

400
R=0%
R=40%
R=60%
R=80%

300

Amplitude (mV)

200
100
0
-100
-200
-300
-400

0

50

100

150

200
250
Time (micro-sec)

300

350

400

Figure B.22: S0 mode at 175 KHz for deferent crack depths

120

450

500
R=0%
R=40%
R=60%
R=80%

400

Amplitude (mV)

300
200
100
0
-100
-200
-300
-400

0

50

100

150

200
Time (micro-sec)

250

300

350

400

Figure B.23: S0 mode at 200 KHz for deferent crack depths

500

Amplitude (mV)

R=0%
R=40%
R=60%
R=80%

0

-500

0

50

100

150
Time (micro-sec)

200

250

Figure B.24: S0 mode at 225 KHz for deferent crack depths

121

300

800
R=0%
R=40%
R=60%
R=80%

600

Amplitude (mV)

400
200
0
-200
-400
-600
-800

0

100

200

300
Time (micro-sec)

400

500

600

Figure B.25: S0 mode at 275 KHz for deferent crack depths

800
R=0%
R=40%
R=60%
R=80%

600

Amplitude (mV)

400
200
0
-200
-400
-600
-800

0

50

100

150

200
250
Time (micro-sec)

300

350

Figure B.26: S0 mode at 300 KHz for deferent crack depths

122

400

BIBLIOGRAPHY

123

BIBLIOGRAPHY

[01] Alleyne, D.N., Cawley, P. (Documentation) (1990), “A 2-dimensional Fourier transform
method for the quantitative measurement of Lamb modes”, IEEE Ultrasonics Symposium,
1990, p. 1143-1146.
[02] Alleyne, D.N., Cawley, P. (1991), “The measurement and prediction of Lamb wave
interaction with defects”, IEEE Ultrasonics Symposium, 1991, p. 855-857.
[03] Alleyne, D.N., Cawley, P. (1992), “The interaction of Lamb waves with defects. IEEE
Transactions on Ultrasonics”, Ferroelectrics (Azar, 2000) (Balageas) (Bar-Cohen, 1998)
and frequency control. 39 N° 3, 1992, p. 381-397.
[04] ANSYS Documentation: http://www.kxcad.net/ansys/ANSYS/ansyshelp/ansys.set.html.
[05] Azar et al (2000) Azar, L., Shi, Y. and Wooh, S. C. (2000) “Beam focusing behavior of
linear phased arrays” NDT&E Int. 33 189–98.
[06] Balageas, D., Fritzen, C.-P., and Gemes, A. (2006), “Structural Health Monitoring”, ISTE,
London, England.
[07] Bar-Cohen, Y., Mal, A., Chang, Z. (1998) “Composite material defects characterization
using leaky lamb wave dispersion data”. SPIE`s NDE Techniques for Aging Infrastructure
& Manufacturing. N: 3396-25.
[08] Bouraszeau, N. (2001), “(Bourasseau, 2001)”. Thèse de doctorat, 2001, Université de
Valenciennes et du Hainaut Cambrèsis. N° d’ordre : 01-13.
[09] Deutsch, W. A. K., Cheng, A. and Achenbach, J. D. (1997) “Self-focusing of Rayleigh
waves and Lamb waves with a linear phased array”. Research in Nondestructive
Evaluation, 9(2):81–95, 1997.
[10] Doebling, S. W., Farrar, C. R., Prime, M. B., and Shevitz, D. W. (1996), “Damage
identification and health monitoring of structural and mechanical systems from changes in
their vibration characteristics: A literature review”, Tech. Rep. Report Number LA-13070MS, Los Alamos National Laboratory, Los Alamos, NM.
[11] Ecke, W., Latka, I., Willsch, R., Reutlinger, A., Graue, R. (2001), “Fibre optic sensor
network for spacecraft health monitoring”, Journal of Measurement Science and
Technology, Volume 12 Number 7, (2001 (Ecke, 2001)), pp 974–980
[12] FHWA (2005), “Railroad Safety Statistics (FHWA, 2005) 2005 Annual Report“.
[13] FHWA (2006), “2006 Status of the Nation's Highways, Bridges, and Transit: Conditionsand
Performance”. http://www.fhwa.dot.gov/policy/2006cpr/pdfs.htm
124

[14] Ghosh, D. P. and Gopalakrishnan, S., (2003) “Structural health monitoring in a composite
beam using magnetostrictive material through a new FE formulation”, Proc. SPIE, Vol.
5062, 704 (2003); doi:10.1117/12.514687
[15] Giurgiutiu, V., Jichi, F., Rogers, C. A., Quattrone, R., Berman, J. B., Kamphaus, J. (1999),
“Experimental Study of Magnetostrictive Tagged Composite Strain Sensing Response for
Structure Health Monitoring” 2nd International Workshop of Structural Health
Monitoring, Sept. 8-10, 1999, Stanford University.
[16] Giurgiutiu, V. and Zagrai, A.N. (2000) “Characterization of Piezoelectric Wafer Active
sensors”. Journal of Intelligent Material Systems and Structures, Vol. 11 – December
2000.
[17] Giurgiutiu, V. and Bao, J. (2002) “Embedded Ultrasonic Structural Radar for the
Nondestructive Evaluation of Thin-Wall Structures”. In Proceedings of the 2002 ASME
International Mechanical Engineering Congress, pages 17–22, 2002.
[18] Giurgiutiu, V., Zagrai, A., and Bao, J. (2004), “Damage identification in aging aircraft
structures with piezoelectric wafer active sensors,” Journal of Intelligent Material Systems
and Structures, Vol. 15, No. 9-10, pp. 673–687.
[19] Giurgiutiu (2004) “Health Monitoring With Piezoelectric Wafer Active Sensors”. p-247
[20] Giurgiutiu, V. and Cuc, A. (2005), “Embedded non-destructive evaluation for structural
health monitoring, damage detection, and failure prevention”, Shock and Vibration Digest,
Vol. 37, No. 2, pp. 83 – 105.
[21] Giurgiutiu, V. (2008), “Structural Health Monitoring With Piezoelectric Wafer Active
Sensors”, Academic Press, London, England.
[22] Grisso, B. L. and Inman, D. J. (2006), “Impedance-based structural health monitoring of
thermal protection systems,” Proceedings of SPIE - The International Society for Optical
Engineering, Nondestructive Evaluation and Health Monitoring of Aerospace Materials,
Composites, and Civil Infrastructure V, February 26 - March 2, San Diego, CA, Vol. 6176,
p. 61760M.
[23] Grondel, S. (2000), “Contribution à l’optimisation du contrôle santé intégré par ondes de
Lamb : Application à la surveillance de structures aéronautiques”. Thèse de doctorat, 2000,
Université de Valenciennes et du Hainaut Cambrèsis. N° d’ordre : 00-34.
[24] Guo, N. Cawley, P. (1993), “The interaction of Lamb waves with delaminations in
composite laminates” Journal of Acoustical Society of America. 94 (4), 1993, p. 22402246.
[25] Han, S., Palazotto, A.N., Asce, F., and Leakeas, l., (2009) “Finite-element analysis of lamb
wave propagation in a thin aluminum plate” Journal of Aerospace Engineering
10.1061/ASCE 0893-1321 (2009) 22:2(185).
125

[26] Inman, D. J., Farrar, C. R., Lopes Jr., V., and Steffen Jr., V. (2005), “Damage Prognosis:
For Aerospace, Civil and Mechanical Systems”, John Wliey & Sons, Chichester, West
Sussex, England.
[27] Kannappan, L., Shankar, K. (2007), “Non destructive inspection of plates using frequency
measurements”, 5th Australasian Congress on Applied Mechanics, ACAM 2007 10-12
December 2007, Brisbane, Australia.
[28] Kim, S.B., Sohn, H. (2007), “Instantaneous reference-free crack detection based
polarization characteristics of piezoelectric materials”, Journal of Smart Materials and
Structures. Vol. 16 (2007) 2375–2387
[29] Kwun, H., Light, G. M., Kim, S., and Spinks, R. L. (2002), “Magnetostrictive sensor for
active health monitoring in structures,” Proceedings of SPIE - The International Society for
Optical Engineering, Smart Nondestructive Evaluation for Health Monitoring of Structural
and Biological Systems, March 17-21, San Diego, CA, Vol. 4702, pp. 282–288.
[30] Lamb, H. (1917) “On waves in an elastic plate” Proc. R. Soc. London, Ser. A, 93(648),
114-128.
[31] Liang, C., Sun, F., and Rogers, C. (1994), “Coupled electro-mechanical analysis of adaptive
material systems - determination of the actuator power consumption and system energy
transfer”, Journal of Intelligent Material Systems and Structures, Vol. 5, No. 1, pp. 12 – 20.
[32] Light, G.M., Kwun, H., Kim, S. Y., Spinks, R. L. (2003) ,“Health Monitoring of Piping and
Plate Using the Magnetostrictive Sensor (MsS) Guided-Wave”, 2nd Middle East
Nondestructive Testing Conference and Exhibition Proceedings, December 8-10, 2003.
[33] Lines, D. and Dickson, K. (1999) “Optimization of high-frequency array technology for
lap-joint inspection”. Proceedings of the 3rd Joint Conference on Aging Aircraft.
[34] Liu, W. and Giurgiutiu, V., (2007), “Finite Element Simulation of Piezoelectric Wafer
Active Sensors for Structural Health Monitoring with Coupled-Filed Elements”, Sensors
and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2007.
[35] Lowe, M.J.S., Cawley, P., Kao, J-Y. Diligent, O. Prediction and measurement of the
reflection of the fundamental anti-symmetric Lamb wave from cracks and notches. Review
of Progress in Quantitative Nondestructive Evaluation, 509 (1), 25-30 July 1999, p. 193200.
[36] Lowe, M.J.S., Cawley, P., Kao, J.Y., Diligent, O. (2002), “The low frequency reflection
characteristics of the fundamental antisymmetric Lamb wave A0 from a rectangular notch
in a plate”, J. Acoust. Soc. Am. 112 (6), 2002, p 2612-2622.
[37] Lowe, M.J.S., Diligent, O., “Low-frequency reflection characteristics of the S0 Lamb wave
from a rectangular notch in a plate”. J. Acoust. Soc. Am. 111 (1), 2002, p 64-74.

126

[38] Lu, L., Ye, L., Su, Z., Yang, C. (2008), “Quantitative assessment of through-thickness
crack size based on Lamb wave scattering in aluminum plates”, NDT&E International 41
(2008) 59–68.
[39] Measures, R. M. (2001), “Structural Monitoring with Fiber Optic Technology”, Academic
Press, San Diego, CA.
[40] NTSB. “Aviation Accidents” http://www.ntsb.gov/
[41] Park, G., Sohn, H., Farrar, C. R., and Inman, D. J. (2003), “Overview of piezoelectric
impedance-based health monitoring and path forward,” The Shock and Vibration Digest,
Vol. 35, No. 6, pp. 451–463.
[42] Park, S., Ahmad, S., Yun, C. B., and Roh, Y. (2006), “Multiple crack detection of concrete
structures using impedance-based structural health monitoring techniques”, Experimental
Mechanics, Vol. 46, No. 5, pp. 609 – 618.
[43] Park, S., Chung-Bang, Y., Yongrae, R. (2007) “Damage diagonistics on a welded zone of a
steel truss member using an active sensing network system”. NDT& E international ISSN
0963-8695, vol. 40, pp. 71-76.
[44] Raghavan, A. and Cesnik, C. E. (2007), “Review of guided-wave structural health
monitoring”, Shock and Vibration Digest, Vol. 39, No. 2, pp. 91 – 114.
[45] Rippert, L., Papy, J-M., Wevers, M., Van Huffel, S. (2002), “, Fibre optic sensor for
continuous health monitoring in CFRP composite materials”, Smart Structures and
Materials, Proceedings of SPIE Vol. 4693 (2002), pp. 312 – 323.
[46] Shang, H. M., Tay, T. E., Lwin, M. (1992), “Holographic Inspection of Plates with Regions
of Local Thinning as Defects”, Journal of Nondestructive Evaluation, Voume 11, No. 2,
1992.
[47] Sohn, H., Farrar, C., Hemez, F., Czarnecki, J., Shunk, D., Stinemates, D., and Nadler, B.
(2001), “A review of structural health monitoring literature 1996-2001,” Tech. Rep. Report
Number LA-13976-MS, Los Alamos National Laboratory, Los Alamos, NM.
[48] Su, Z., Ye, L., Lu, Y. (2006), “Guided Lamb waves for identification of damage in
composite structures: a review”. Journal of Sound and Vibration. No 295 (2006), 753-780.
[49] Torvik, P.J. (1967), “Reflection of wave trains in semi infinite plates”, J. Acoust. Soc. Am.
(1967), 99(4), pp. 346-353.
[50] Tua, P.S., Quek, T., Wang, Q. (2004), “Detection of cracks in plates using piezo-actuated
Lamb waves”. Journal of Smart Materials and Structures. 13 (2004) 643–660.
[51] Wang, L., Yuan, F. G. (2007), “Active damage localization technique based on energy
propagation of Lamb waves”, Smart Structures and Systems, Vol. 3, No. 2 (2007) 201-217.

127

[52] W. Liu, and Hong, J.W., (2010) “Three dimensional Lamb Wave propagation excited by a
phased piezoelectric array” Journal of Smart Materials and Structures, Vol:19 N0 8.
[53] Weber, Y. B., Schweitzer, D., Bar, H., Shilo, D. (2008), “Health Monitoring of Adhesive
Joints using Magnetostrictive Fillers”, Proceedings of the 9th Biennial ASME Conference
on Engineering Systems Design and Analysis ESDA 2008.
[54] Wilcox, P. D., Lowe, M. J. S. and Cawley, P. (2000) “Lamb and SH wave transducer arrays
for the inspection of large areas of thick plates”. In Annual Review of Progress in QNDE
volume 19A, pages 1049 – 1056, 2000.
[55] Worlton, D.C. (1957), “Ultrasonic Testing with Lamb Waves”, Nondestructive Testing,
15:218 222.
[56] Zhongqing S., Ye, L., “Identification of Damage Using Lamb Waves: From Fundamentals
to Applications”, Lecture Notes in Applied and Computational Mechanics, Vol. 48.
[57] Zhou, Z. and Ou, J. (2004) “Development of FBG sensors for structural health monitoring
in civil infrastructures”. Proceeding of North American Euro-Pacific Workshop “Sensing
Issues in Civil Structural Health Monitoring”, 2004, Waikiki Beach, Oahu, Hawaii, USA.

128