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TRANSCRIPT
Finite element modelling and simulation of guided wave
propagation in steel structural members
Tianwei Wang
A thesis submitted in fulfilment of the requirements for the degree of
Master of Engineering (Honours)
School of Computing, Engineering and Mathematics
University of Western Sydney
September, 2014
i
Declaration
Date: September 2014
Author: Tianwei Wang
Title: FINITE ELEMENT MODELLING AND SIMULATION OF GUIDED WAVE
PROPAGATION IN STEEL STRUCTURAL MEMBERS
Degree: Master of Engineering (Honours)
I certify that the work presented in this thesis is, to the best of my knowledge and
belief, original, except as acknowledged in the text, and the material has not been
submitted, either in full or in part, for a degree at this or any other institution.
I certify that I have complied with the rules, requirements, procedures and policy
relating to my higher degree research award of the University of Western Sydney.
------------
Author’s Signature
ii
Abstract
In this thesis, a guided wave-based active structural health monitoring (SHM) system using
PZT actuator/sensor network is developed for non-destructive defect identification for steel
structural members, including steel pipes and plates. The development is conducted
through analytical characterisation of guided wave propagation, wave mode selection,
numerical modelling and simulations of guided wave propagation in plates and pipes
containing defects in the form of notch, hole, and crack. Different finite element models for
the elastic wave propagation in steel structures are developed to determine their
characteristics and used to optimise wave mode and frequency for defect identification. In
order to capture the electromechanical behaviours of the piezoelectric actuators of
interests, exact models and effective models with/without adhesive layer are developed.
Considering the shape and size effects of PZT actuators the finite element models are
devised to determine the discrepancy when different PZT actuators used. Through
employing different dynamic analysis techniques, three finite element modelling methods
are developed and applied into the finite element modelling and simulations and they are
Explicit Dynamic Analysis (EDA), Implicit Dynamic Analysis (IDA) and Combined Implicit-
Explicit Dynamic Analysis (CIEDA). The numerical results show EDA and CIEDA both
perform well in simulating the wave propagation in structures. The developed PZT models,
finite element modelling and dynamic analysis techniques are further employed in two
typical structural members – plates and pipes with/without defects and as a result, the
characteristics of guided wave propagation in the structural members with/without defects
are determined numerically which pave the way to design such a guided wave-based
active structural health monitoring (SHM) system for steel plates and pipes.
iii
Acknowledgements
Throughout my whole journey of master studying, I received plenty of support from
supervisors and family so that I could present this project to the public. First and foremost,
I would like to express my sincere appreciation to my Principal Supervisor, Associate
Professor Richard (Chunhui) Yang, who provided with significant guidance that inspired
me to have these ideas presented in my thesis. His arrangement of weekly meeting
delivered important skills for being a researcher and writing a thesis. Besides, he cared
about my life and made me feel like I have a family in this country.Additionally, I would like
to thank to my co-supervisor, Professor Yang Xiang who supported and encouraged me
during my study.
Finally, I want to thank my family and friends. Their expectations gave me the motivation
and empowerment to proceed.
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Contents
Chapter 1 Introduction ......................................................................................................... 1
1.1 Problem statement .................................................................................................. 1
1.2 Aim and objectives .................................................................................................. 1
1.3 Outline of the thesis ................................................................................................ 2
Chapter 2 Literature Review ................................................................................................ 4
2.1 Introduction ............................................................................................................. 4
2.2 Basics of Structural health monitoring .................................................................... 4
2.2.1 Structural health monitoring (SHM) and non-destructive evaluation (NDE)
method .................................................................................................................. 7
2.2.2 Research on Guided-wave-based SHM ..................................................... 11
2.3 Guided waves ....................................................................................................... 14
2.3.1 Guided waves in plates............................................................................... 15
2.3.2 Guided waves in pipes................................................................................ 19
2.3.3 Velocity ....................................................................................................... 20
2.3.4 Guided wave dispersion in pipes ................................................................ 21
2.4 Methods to simulate the guided wave propagation ............................................... 27
2.5 Piezoelectricity ...................................................................................................... 30
2.6 Signal operation .................................................................................................... 32
2.6.1 Generation of excitation signals .................................................................. 32
2.6.2 Signal processing ....................................................................................... 33
2.7 Summary .............................................................................................................. 33
Chapter 3 Selection and development of effective PZT models ........................................ 35
3.1 Introduction ........................................................................................................... 35
3.2 Piezoelectric effect and constitutive formulas ....................................................... 35
3.3 Effective piezoelectric actuators/ sensors equations ............................................ 39
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3.3.1 The uniform strain model (USM) ................................................................. 39
3.3.2 The pin force model (PFM) ......................................................................... 41
3.3.3 The enhanced pin force model (EPFM) ...................................................... 42
3.3.4 The strain energy model (SEM) .................................................................. 43
3.3.5 The Bernoulli-Euler model (BEM) ............................................................... 44
3.3.6 The Bernoulli-Euler model considering an adhesive layer (BEMA) ............ 46
3.3.7 The effective piezoelectric actuator model (EPM) ...................................... 47
3.4 Development of effective PZT model with an adhesive layer (EPMA) .................. 50
3.5 Case study ............................................................................................................ 53
3.5.1 Mechanistic output comparison of PZT actuator models ............................ 56
3.5.2 Adhesive layer effect on mechanistic output of PZT actuator ..................... 65
3.5.3 Size and shape effect on PZT actuators ..................................................... 71
3.6 Summary .............................................................................................................. 74
Chapter 4 FE-based numerical modelling of elastic wave propagations in plates ............. 75
4.1 Introduction ........................................................................................................... 75
4.2 Elastic waves in plates .......................................................................................... 75
4.2.1 Elastic waves propagation in free plate ...................................................... 77
4.3 FEM of elastic waves in plates.............................................................................. 79
4.3.1 Explicit Dynamic Analysis (EDA) method ................................................... 79
4.3.2 Implicit Dynamic Analysis (IDA) method ..................................................... 81
4.4 Analysis on FEM using different dynamic analysis techniques ............................. 83
4.4.1 FEA model .................................................................................................. 84
4.4.2 Wave mode selection ................................................................................. 85
4.4.3 Five cycle Hanning-window excitation signal .............................................. 87
4.4.4 Some vital parameters used in explicit dynamic procedure ........................ 88
4.4.5 Modelling of effective PZT-Plate model ...................................................... 91
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4.4.6 Modelling of full PZT-Plate model ............................................................... 93
4.4.7 Modelling of co-simulation model ............................................................... 96
4.5 Case study .......................................................................................................... 100
4.6 Summary ............................................................................................................ 107
Chapter 5 FE-based numerical modelling of elastic wave propagations in pipes ............ 108
5.1 Introduction ......................................................................................................... 108
5.2 Elastic waves in pipes ......................................................................................... 108
5.3 Dispersion characteristics of steel pipes ............................................................. 111
5.4 Finite element modelling of guided waves in pipes ............................................. 113
5.4.1. FE modelling using implicit dynamic analysis and co-simulation analysis
.......................................................................................................................... 115
5.4.2 FEM using implicit dynamic analysis ........................................................ 117
5.4.3 FEM using explicit dynamic analysis ........................................................ 120
5.4.4 FEM using co-simulation analysis ............................................................ 123
5.5.5 Results and discussions ........................................................................... 124
5.5 Case study .......................................................................................................... 130
5.5.1 Finite element modelling ........................................................................... 131
5.5.2 Results and discussion ............................................................................. 132
5.6 Summary ............................................................................................................ 137
Chapter 6 Conclusions and recommendations ................................................................ 138
6.1 Conclusions ........................................................................................................ 138
6.2 Recommendations on future work ...................................................................... 139
APPENDIX A: Matlab code for the excitation signal……………………………………...…141
Research outcomes during the study of Master (Honours) degree………………………..142
References……………………………………………………………………………………….143
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Table of Figures
Figure 2-1 Analogy between the nervous system of man and a structure with SHM
(Balageas et al., 2006) .......................................................................................... 5
Figure 2-2 Two possible attitudes of the experimenter defining (a) passive and (b)
active monitoring (Balageas et al., 2006) .............................................................. 7
Figure 2-3 The basic components of SHM (Balageas et al., 2006) .............................. 8
Figure 2-4 Comparison from ultrasonic bulk wave and guided wave, (a) bulk wave, (b)
guided wave (Rose, 2004) ................................................................................... 10
Figure 2-5 Rayleigh wave: (a) Schematic representation (Royer and Dieulesaint,
2000) and (b) Wave on Sagittal plane (Krautkramer and Krautkrâmer, 1990) ..... 16
Figure 2-6 Lamb wave: (a) Anti-symmetric mode and (b) Symmetric mode
(Krautkramer and Krautkrâmer, 1990) ................................................................. 17
Figure 2-7 Displacement distribution of Love waves (Royer and Dieulesaint, 2000) .. 18
Figure 2-8 SH waves: (a) Anti-symmetric mode and (b) Symmetric mode (Giurgiutiu,
2005) ................................................................................................................... 19
Figure 2-9 (a) Group U and phase V velocities and Arrival of a dispersive wave at
different geophones (Sheriff and Geldart, 1995).................................................. 21
Figure 2-10 Phase velocity dispersion curves for 76mm (3 in.) diameter pipe (Lowe et
al., 1998) ............................................................................................................. 23
Figure 2-11 Group velocity dispersion curves for 76 mm (nominal 3 in)-diameter pipe
(Lowe et al., 1998) ............................................................................................... 23
Figure 2-12 Mode shapes for 76mm (3 in.) diameter pipe at 70 kHz: (a) L(0,1); (b):
L(0,2) (Lowe et al., 1998) .................................................................................... 25
Figure 2-13 T(0, 1) mode shape in a 3 inch pipe at 45 kHz Radial and axial
displacements are zero (Demma et al., 2003) ..................................................... 26
Figure 2-14 Peripheral vibration model sketch map (Shen et al.) ............................... 27
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Figure 2-15 (a) Perovskite Structure above the Curie point (before poling) ; (b) below
the Curie point, the crystal was displaying polarization (after poling) (Giurgiutiu,
2007) ................................................................................................................... 31
Figure 3-1 Orthogonal coordinate system and poling direction (Inman and Cudney,
2000) ................................................................................................................... 36
Figure 3-2 Strain distributions through thickness for the uniform strain model ........... 40
Figure 3-3 Strain distributions through thickness for the pin force model ................... 41
Figure 3-4 Strain distributions through thickness for the enhanced pin force model ... 42
Figure 3-5 Strain distributions through thickness for the strain energy model ............ 43
Figure 3-6 Strain distributions through thickness for the Bernoulli-Euler model .......... 44
Figure 3-7 Stress distributions through thickness for Bernoulli-Euler model with an
adhesive layer ..................................................................................................... 46
Figure 3-8 Strain and stress distributions through thickness for the effective model .. 47
Figure 3-9 Strain and stress distributions through thickness for the effective model
with an adhesive layer ......................................................................................... 51
Figure 3-10 Setup of a plate model............................................................................. 53
Figure 3-11 Poling direction for circular PZT disk ....................................................... 55
Figure 3-12 Geometry of the finite element models, (a) Perfect bound PZT-Plate
model, (b) PZT-Plate model with Adhesive layer ................................................ 56
Figure 3-13 Effective displacements for each model .................................................. 61
Figure 3-14 Effective strains for each model .............................................................. 61
Figure 3-15 Effective stresses for each model ............................................................ 62
Figure 3-16 Effective displacement for each model with an adhesive layer ................ 62
Figure 3-17 Effective strain for each model with an adhesive layer ............................ 63
Figure 3-18 Effective stress for each model with an adhesive layer ........................... 63
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Figure 3-19 Effective displacement for different adhesive layer thicknesses, (a)
0.09t mm , (b) 0.1 t mm ,(c) 0.12 t mm . .......................................................... 67
Figure 3-20 Effective strain for different adhesive layer thicknesses, (a) 0.09 t mm , (b)
0.1 t mm , (c) 0.12 t mm . .................................................................................. 69
Figure 3-21 effective stress for different adhesive layer thicknesses, (a) 0.09 t mm , (b)
0.1 t mm ,(c) 0.12 t mm . ................................................................................... 70
Figure 3-22 Geometry of PZT actuators with different sizes ....................................... 72
Figure 3-23 Effective displacements for each model .................................................. 72
Figure 3-24 Effective strains for each model .............................................................. 73
Figure 3-25 Effective stress for each model ............................................................... 73
Figure 4-1 Transverse and longitudinal waves ........................................................... 76
Figure 4-2 Free plate geometry .................................................................................. 78
Figure 4-3 Schematic of a plate model ....................................................................... 84
Figure 4-4 Lamb wave dispersion curve for 1.275 mm thick steel plate, (a) phase
velocity, (b) group velocity. .................................................................................. 86
Figure 4-5 5-count 200 kHz Hanning windows signal ................................................. 87
Figure 4-6 Modulation of a carrier wave by a time window (Giurgiutiu, 2007) ............ 88
Figure 4-7 The flow tree of Abaqus Explicit model...................................................... 91
Figure 4-8 The effective displacements applied in the modes around the disk actuator
............................................................................................................................ 92
Figure 4-9 Lamb wave propagation and scattering at different time instants .............. 93
Figure 4-10 The flow tree of Abaqus Implicit model .................................................... 94
Figure 4-11 Finite element model of the PZT bounded to the plate ............................ 94
Figure 4-12 Lamb wave propagation and scattering at different time instants ............ 95
Figure 4-13 The flow tree of Abaqus co-simulation model .......................................... 96
Figure 4-14 Settings of the co-simulation model ......................................................... 97
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Figure 4-15 Lamb wave propagation and scattering at different time instants ............ 98
Figure 4-16 Comparison of time signals of displacement magnitude at sensor point S1
............................................................................................................................ 99
Figure 4-17 Setup of a plate model with a linear crack ............................................. 101
Figure 4-18 Displacement magnitude of Lamb wave propagation ............................ 102
Figure 4-19 Acquired signals received at S1 ............................................................ 103
Figure 4-20 Displacement magnitude of Lamb wave propagation ............................ 104
Figure 4-21 Acquired signals received at S1 ............................................................ 105
Figure 4-22 Displacement magnitude of Lamb wave propagation ............................ 105
Figure 4-23 Acquired signals received at S1 ............................................................ 106
Figure 5-1 Reference coordinates and characteristic dimensions of the pipe ........... 109
Figure 5-2 Dispersion curves for the steel pipe of a 28 m diameter and a wall
thickness of 1 mm: (a) group velocity and (b) phase velocity. ........................... 113
Figure 5-3 Schematic of a pipeline model ................................................................. 116
Figure 5-4 The flow tree of Abaqus Implicit model .................................................... 118
Figure 5-5 Finite element model of four PZT actuators bounded to the pipe ............ 119
Figure 5-6 Finite element model of six PZT actuators bounded to the pipe .............. 119
Figure 5-7 Finite element model of eight PZT actuators bounded to the pipe .......... 120
Figure 5-8 The flow tree of Abaqus explicit model .................................................... 120
Figure 5-9 Finite element model of four PZT actuators bounded to the pipe ............ 121
Figure 5-10 Finite element model of six PZT actuators bounded to the pipe ............ 122
Figure 5-11 Finite element model of four PZT actuators bounded to the pipe .......... 122
Figure 5-12 The flow tree of Abaqus co-simulation model ........................................ 123
Figure 5-13 Displacement signals form four PZT model ........................................... 125
Figure 5-14 Displacement signals from form four PZT model after Hilbert-Huang
transform ........................................................................................................... 125
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Figure 5-15 Displacement signals from six PZT model in the z-direction ................. 126
Figure 5-16 Displacement signals from six PZT model after Hilbert-Huang transform…
.......................................................................................................................... 126
Figure 5-17 Displacement signals from eight PZT model in the z-direction .............. 127
Figure 5-18 Displacement signals from FE dynamic simulation of eight pipes after
Hilbert-Huang transform .................................................................................... 127
Figure 5-19 Displacement signals from co-simulation method of steel pipes after
Hilbert-Huang Fourier transform with different number of PZT actuators .......... 128
Figure 5-20 Displacement signals from implicit simulation of steel pipes after Hilbert-
Huang transform with different number of PZT actuators .................................. 129
Figure 5-21 Displacement signals from explicit simulation of steel pipes after Hilbert-
Huang transform with different number of PZT actuators ................................. 129
Figure 5-22 FE models of pipe without defect .......................................................... 130
Figure 5-23 FE models of pipes with/without defect: A) no defect; B) hole; C) crack;
and D) notch 3D solid model ............................................................................. 131
Figure 5-24 Disperse curves of a pipe with an outer diameter of 323mm and a wall
thickness of 6.4 mm ........................................................................................... 133
Figure 5-25 Displacement outputs from FE dynamic simulations along the longitudinal
direction of the pipe ........................................................................................... 134
Figure 5-26 Displacement signals from FE dynamic simulations of steel pipes with and
without a defect, ................................................................................................ 134
Figure 5-27 Displacement signals from FE dynamic simulations of steel pipes with and
without a defect after HT-processed .................................................................. 135
Figure 5-28 Displacement signals from FE dynamic simulations of steel pipes with
different shaped PZT actuators ......................................................................... 136
Figure 6-1 The general structure of this thesis ......................................................... 138
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List of Tables
Table 2-1 Estimated time saved on inspection operations by use of SHM (Bartelds,
1997) ..................................................................................................................... 6
Table 2-2 Comparison of NDE and SHM technologies (Adams, 2007) ........................ 8
Table 2-3 Comparison of different NDE methods and their suitability for SHM (Li,
2011, Willcox and Downes, 2000, McCann and Forde, 2001, Adams, 2007) ...... 11
Table 3-1Dimensions and properties of the plate and PZT, respectively.................. 54
Table 3-2 FE models using 3-D solid elements .......................................................... 55
Table 3-3 Material Properties in Abaqus ..................................................................... 55
Table 3-4 The percentage relative errors for each model on effective displacement
aspect .................................................................................................................. 64
Table 3-5 The percentage relative errors for each model on effective strain aspect .. 64
Table 3-6 The percentage relative errors for each model on effective strain aspect .. 65
Table 4-1 Mechanical properties of steel alloy plate ................................................... 84
Table 4-2 The limitation of some vital parameters used in Abaqus/Explicit ................ 91
Table 4-3 Magnitudes of effective displacement at different voltages conditions ....... 92
Table 4-4 Dimensions and properties of the plate and PZT ..................................... 101
Table 5-1 Dimensions and properties of the plate and PZT ..................................... 114
Table 5-2 The method of choosing parameters of FEM model ................................. 116
Table 5-3 Main parameters in dynamic simulations the steel pipe of 3 m length ..... 117
Table 5-4 Magnitudes of effective displacement at different voltages conditions ..... 123
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Nomenclatures
Symbol Term
BEM The Bernoulli-Euler model
BEMA The Bernoulli-Euler model considering an adhesive layer
CIEDA Combined Implicit-Explicit dynamic analysis
EDA Explicit dynamic analysis
EPFM The enhanced pin force model
EPM The effective piezoelectric actuator model
EPMA The effective piezoelectric actuator model with an adhesive layer
FE Finite element
FEA Finite element analysis
FEM Finite element method
FFT Fast Fourier transform
HHT Hilbert-Huang transform
IDA Implicit Dynamic Analysis
NDE Non-destructive evaluation
PFM The pin force model
PZT Lead Zirconate Titanate
SAFEM Semi-analytical finite element method
SEM The strain energy model
SHM Structural health monitoring
USM The uniform strain model
1
Chapter 1 Introduction
1.1 Problem statement
In the modern world, the steel structural members have been widely used and become an
indispensable part in our industry. For example, the steel pipelines are heavily used for
the transportation of oil, natural gas even fresh water. The steel plates are often employed
for structural and construction applications, such as buildings, bridges and vehicles.
However, these structures are easily affected by environmental surroundings, such as
mechanical wear or chemical corrosion for their material properties, thereby weakening
their performance and reducing their service life. It is therefore necessary for people to find
out an accurate system of analysis and diagnosis to regularly inspect these structures for
structural integrity.
In order to improve the steel structural members’ performance and reduce the operational
cost at the same time, some new kinds of structure health monitoring (SHM) systems have
been explored recently by many researchers. One idea of such SHM systems is to employ
the ultrasonic guided waves to monitor their conditions online. This is because the guided
wave testing can offer many advantages that it is low in cost, higher efficiency and can
monitor larger area of the structures. However, there are some limitation on its application,
including the dispersive nature of the waves and the signal processing. Therefore, using
the finite element analysis producers to simulate the guided wave propagation in steel
structural members is significant for providing design guides for such SHM system.
1.2 Aim and objectives
The aim of this thesis is to develop finite element models and dynamic finite element
analysis procedures for guided wave propagations in steel structural members with and
without defects. Applying these proposed FE modelling and FEA procedures, the
2
characterisation of the interaction between the guided waves and crack-like defects in
steel structures is also conducted. In order to achieve this goal, the following objectives
need to be addressed:
1) Understanding the principle of the piezoelectric effects and the relationships between
its input voltage and relative effective displacement / force. As a result, the different
kind of effective PZT models based on different analytical theories are developed.
2) Understanding the basic knowledge about the finite element method and signal
processing. Knowing how to apply a commercial package – Abaqus in the modelling
and simulation of guided waves in plates and pipes is very important.
3) Developing mode dispersion and mode selection by calculating the dispersive
properties of elastic waves propagating in structures.
4) Designing a suitable network of surface-bonded actuators/sensors for signal excitation
and data acquisition.
5) Developing different finite element models and finite element analysis procedures on
elastic wave propagation in structures with or without crack-like defects. In this part, we
also applied the finite element analysis to simulate 3D guided wave propagation in
engineer structures.
Therefore, following with these objectives, this thesis is divided into six main chapters to
address these issues.
1.3 Outline of the thesis
Chapter 2 is the literature review, which gives all the research backgrounds and useful
information about this thesis. In this section, the structural health monitoring is first briefly
introduced. Different non-destructive damage identification techniques, including eddy
current testing, X-ray testing and guided wave testing, are also discussed. After selecting
guided wave testing as a primary technique, this chapter outlines some vital knowledge
3
which will be used in the following chapters. For instance, the general knowledge and
history of development in the piezoelectric sensors/actuators, guided wave types, wave
mode selections and signal processing.
Chapter 3 presents the detailed development for different equivalent effective PZT models,
which can be used to replace the full PZT actuators in a simulation process. As to
compare and validate those models in a case study of a steel plate is conducted with
applying the proposed the effective piezoelectric actuator models (EPM and EPMA). Two
models are identified as the most reliable effective models in the further study.
In Chapter 4, the elastic wave propagation in steel plates is simulated and analysed by
using Abaqus software. Three dynamic analyses techniques, which are Explicit Dynamic
Analysis (EDA), Implicit Dynamic Analysis (IDA) and Combined Implicit-Explicit Dynamic
Analysis (CIEDA), are all applied in the FE model. Furthermore, through a comparison with
using a one simple Plate model which has only one PZT installed, the results form EDA
and CIEDA based models are more reliable and these two dynamic techniques are used in
the further study.
The works presented in Chapter 5 focuses on simulate of the elastic wave propagations in
steel pipes. Similarly with the Chapter 4, three dynamic analyses techniques, which are
EDA, IDA and CIEDA, are still employed. Besides, the relationship of the interaction
between the defects and the elastic wave propagations in pipes are identified based on the
numerical analysis.
In Chapter 6, a review of this thesis is presented, which are summarised and
recommendations are given to suggest potential work.
4
Chapter 2 Literature Review
2.1 Introduction
This chapter presents a brief review of the literatures on SHM techniques, SHM systems
using guided waves, and current research status on guided wave inspection techniques.
In addition, the signal operation and processing for guided wave analysis are also given in
this chapter.
2.2 Basics of Structural health monitoring
Structural health monitoring (SHM) is a process of implementing a damage identification
strategy. Unlike other damage identification methods, a SHM system is associated with
online-global damage identifications, which involve networks of actuators and sensors,
data transmission and computational power. Hence, it can monitor a structurial or
mechanical system and analyse its features over time, thereby determining the in-situ
state of system health (Farrar and Worden, 2007). Such a system could be imaged as a
human nervous system shown in Fig. 2-1. More specifically, the brain can check and
indicate the pain when people get hurt. The sensors (just like nerves in human body) can
indicate the damage, and the central processor (similar to human brain) can analyse the
location and identify the follow-up actions which needed to be taken (Speckmann and
Henrich, 2004).
5
Figure 2-1 Analogy between the nervous system of man and a structure with SHM
(Balageas et al., 2006)
Economic impacts are the primary motivations for SHM. On one hand, the cost of
unnecessary maintenance can be saved greatly by applying the SHM system. This is all
because the structural health monitoring makes the maintenance as as-needed rather than
scheduled (Giurgiutiu, 2007). This change of maintenance would reduce labour costs as
well as improve safety and reliability. For instance, as Honeywell’s Central Maintenance
Computer (CMC) IVHM system integrates a SHM subsystem and it helps air carriers
achieve maintenance cost reduction at 50% to 80% (Chang, 2005). Besides, the SHM
may extend the life-cycle time and reduce its cost for a structure. As Inaudi (2011) pointed
out, because of many bridges are not assessed on their real condition and load-bearing
capacity quantitatively, some of them are labelled as deficient which are routinely repaired
or replaced. According to his estimation, if people use a SHM system on the bridge
inspection, only 60% of the bridges will be deemed to actually require replacement.
The time saved on maintenance is also an important factor that attracted people attention.
Bartelds (1997) provided an example on military aircraft in his report (as shown in Table 2-
1). According to his results, nearly 40% or more can be saved on inspection time by using
a SHM, in which a large portion of time saved on the unscheduled inspection and the flight
6
line inspection. Hence, the SHM should be a usable system and give a mature platform
that can be extend to pipeline inspection.
Table 2-1 Estimated time saved on inspection operations by use of SHM (Bartelds, 1997)
Inspection Type Current Inspection
Time (% of total)
Estimate Potential For smart system
Time saved (% of Total)
Flight Line 16 0.40 6.5
Scheduled 31 0.45 14.0
Unscheduled 16 0.1 1.5
Service instructions 37 0.6 22.0
100 44.0
Generally speaking, the structural health monitoring system can be either active or passive
(Giurgiutiu, 2007). A passive SHM system is primarily measuring various parameters using
passive sensors (loading, stress and performance indicators, etc.) over time and then
determining the state of the structural health from these parameters. Hence, the passive
SHM only acquire information from the structure rather than interact it. The disadvantage
for using a passive SHM is that it does not directly address the crux of the problem. For
instance, it does not directly examine whether the structure has been damaged (Giurgiutiu,
2007). Active SHM is different from passive one, which uses actuators/sensors to interact
with the structure (Giurgiutiu, 2005). Therefore, this method can directly detect the present
state of structural health. Fig. 2-2 presents the difference between these two kinds of SHM.
In summary, the active SHM would be the better choice for structural health detection
either on its economic benefits or on its high performance, reliability and security on
structural health inspection.
7
Figure 2-2 Two possible attitudes of the experimenter defining (a) passive and (b) active
monitoring (Balageas et al., 2006)
2.2.1 Structural health monitoring (SHM) and non-destructive evaluation (NDE)
method
The active SHM approach is similar with the approach taken by non-destructive evaluation
(NDE). In fact, SHM has a common basis with NDE as shown in Fig. 2-3, and several NDE
approaches can be converted or amended into the SHM approach (Balageas et al., 2006).
Both of these methods can be used to identify the damage state for a structure. However,
being different from the traditional NDE structural health monitoring method does online
global damage identification and predict future performance, whereas the NDE is sort of
offline method and can be only used for local inspection (Adams, 2007). The differences
between NDE and SHM can be illustrated in Table 2-2. Considering both of them are used
for damage inspection, the structural health monitoring can be regarded as a
8
complementary method taken by NDE, which can be used to correlate results and perform
more precise inspections of local areas for one structure (Adams, 2007). However,
although there are a large number of NDE techniques available to the damage
identification, only a few of them can be used in the SHM method. The information is listed
as follows.
Figure 2-3 The basic components of SHM (Balageas et al., 2006)
Table 2-2 Comparison of NDE and SHM technologies (Adams, 2007)
Non-destructive Evolution (NDE)
Structural Health Monitoring (SHM)
off-line mostly On-line
Generally implemented locally Generally implemented globally
Time-based maintenance Condition-based maintenance
Baseline generally not available Start with baseline in initial date collection
Equipment is high fidelity and relatively expensive
Equipment has less fidelity but less expensive
Workforce is trained Workforce is not trained
Not integral with host structure Using integral sensing
Visual inspection, as the name implies, is to examine the structure with naked eyes, which
is the most common non-destructive examination (NDE) technique (Allgaier et al., 1993).
Defects can be detected by human sensory systems aided with mechanical enhancements,
9
including fiberscopes, bore scopes, magnifying glasses and mirrors (Spencer, 1996).
Therefore, this technique is simple, easy to apply and usually low in cost. However, it can
be only used on the surface inspection and need operator skills and experiences.
Eddy current testing is using the principal of “electromagnetism” as the basis for
examinations, which means the defects can be detected by the flow of the magnetic field
created (Jones and Pezdirtz, 1972). Then, the results can be provided in simple technical
terms, which are often identified as a go or no go. As the eddy current test is purely
electrical, this system is simple in geometry and portable (Willcox and Downes, 2000).
However, the results produced from this form of testing can be misinterpreted due to the
flux leakage. Furthermore, it is only possible with materials that hold magnetic properties
(Jones and Pezdirtz, 1972).
Radiographic testing, likes x-rays on human skeletal structures, can take x-ray image for
one structure. Hence, the results are presented pictorially and can reveal minor fractures
and blemishes within it (Willcox and Downes, 2000). However, one of the main
disadvantages for this technique is it can only be applied over shot distances, which is
unsuitable for a big structure, i.e., pipeline inspection.
Ultrasonic guided wave testing and ultrasonic bulk wave testing both belong to the
ultrasonic inspection, which employs the high frequency wave that propagates along one
structure and receives reflected back signals to determine the current state of the structure.
The comparison between ultrasonic guided waves and bulk waves is shown in Fig. 2-4.
Rose (2004) gave the briefly fundamental differences between these two methods. The
bulk waves travel within the bulk of material away from boundaries. Whereas, guided
waves travel on the surface of a material or through the thickness of thin materials and
create numbers of wave modes according to their dispersive property (Giurgiutiu, 2007).
Therefore, for the structure with certain boundary conditions, guided waves are more
useful than bulk waves for developing an elastic wave based structural health monitoring.
10
Dye penetrant testing, based on the principal of “capillary action”, is often used for the
detection of surface-breaking defects in non-ferromagnetic materials. It uses series of
processes, including removing all traces of foreign materials from the surface, to
demonstrate the crack as visual form (Willcox and Downes, 2000). Hence, it is a simple
and effective method for surface inspection. However, the cost is much high and
application is narrow and it is in lack of adaptability for steel structures and structural
members.
Acoustic emission is also widely used to detect and locate faults in a structure (Miller and
McIntire, 1987). However, unlike ultrasonic inspection, acoustic emission testing is only
related to captured signals produced by a sudden internal stress in materials and can not
account the source of signals and wave propagation (Huang et al., 1998). Hence, this
passive testing is far away from this project and not considered in this thesis.
Table 2-3 (Li, 2011, Willcox and Downes, 2000, McCann and Forde, 2001, Adams, 2007)
represents and illustrates the differences among NDE techniques as well as their suitability
for SHM. According to these summaries, ultrasonic guided wave testing appears benign
compatibilities with SHM as well as ability to inspect large structures. Therefore, the
following part will illustrate the history and contribution for ultrasonic guided waves
researches.
Figure 2-4 Comparison from ultrasonic bulk wave and guided wave, (a) bulk wave,
(b) guided wave (Rose, 2004)
11
Table 2-3 Comparison of different NDE methods and their suitability for SHM (Li, 2011,
Willcox and Downes, 2000, McCann and Forde, 2001, Adams, 2007)
Techniques Advantages Disadvantages On-line
monitoring Integral with
host structure
Visual inspection
Quick and simple
Surface inspection;
modest skills required
No No
Eddy current testing
Extremely compact and simple results (go/ no-go);
The responses affected easily;
surface inspection
No Yes
Radiographic testing
Pictorial results; suitable for any
material
Not suitable for surface
inspection; short distances
No No
Ultrasonic guided waves
testing
Cross-sectional inspection
Multiple modes Yes Yes
Ultrasonic bulk waves testing
Local and size defects
Very thin sections difficult
to detect No Yes
Dye penetrant testing
Simplicity of operation; best
method for surface
inspection
Only for Surface inspection; large
costs No No
Acoustic emission testing
Volumetric and surface
inspection Passive method Yes Yes
2.2.2 Research on Guided-wave-based SHM
Basically, there are two subsystems for the SHM applying on damage prognosis, which
are an on-board network of actuators and sensors for signal generation and data
acquisition and a central processor to evaluate the structural health, respectively
(Raghavan and Cesnik, 2007). Therefore, the research on this topic was divided into two
parts. The first one is about the piezoelectric sensors and actuators, especially for using
on the interaction between the host structure and the actuators. Apart from experimental
studies, these two parts can be studied using different methods, which are analytical
12
methods and numerical methods, respectively. The more detailed information are listed as
follows:
Piezoelectric sensors/actuators
In order to predict and optimise the behaviour of the piezoelectric actuators/sensors, most
researches have focused on development of analytical models. These models can be
divided into four board categories and discussed as follows.
The first category is based on a uniform strain assumption on both actuators/sensors and
a host plate, likes the uniform strain model. This theory was firstly introduced by Bailey and
Ubbard (1985). Then, Crawley and Anderson (1990) dedicated several times and energies
to the optimization of this theory and built an analytical model. They also summarized and
compared other models with it. However, all of the assumptions were based on the
problem of one-dimensional vibration excitation. Following by this method, Gao et al.
(1998) extended it to three-dimensional problems for composite plate and developed
series of equations for the mid-plane thickness, bending stiffness and Poisson’s ratio.
They also validated the uniform method by comparing the natural frequencies to
experimental natural frequencies and the results were found satisfactory and useful to
future work.
In the second category, strain distribution was assumed as a uniform-linear one, which is
more accuracy on actuators’ performance. According to summarized many researches,
there are two basic models belonging to this category. The first one is the pin force model ,
in which the strain in the actuator was assumed uniform and linear in the host plate
(Alghamdi and Dasgupta, 2001), which was firstly invested by Bailey and Ubbard (1985).
Then, it was further developed by Crawley and De Luis (1987). They used the Rayleigh-
Ritz equation of motion to develop one-dimensional pin force beam model with different
conditions, including surface-bounded and embedded configurations. The ‘strain node’
(“zero crossing point”), which indicated the beam where the strain changes from positive to
13
negative, was defined in their paper. It is the key factor for the follow-up works. The
second classic model is strain energy model. Unlike the pin force model, in which the
‘strain node’ located at the centre of the host plate, the strain energy model maintain that
point at lower surface of the plate (Wang and Rogers, 1991). This model was firstly
invested by Lee (1990) by applying the classical lamination theory on laminated plates.
Wang and Rogers (1991), Dimitriadis et al. (1991) and Chee et al. (1998) implemented this
method in their papers and listed the detailed formulas for different laminated plates.
In the third category, the linear strain distribution assumption provided a good performance
on rectangular shape actuator when the bonding layer is thin (Li and Chen, 2003). many
models belongs to here, which consists of the enhance pin force model, the Bernoulli-
Euler model, etc., More specifically, Chaudhry and Rogers (1994) extended the pin-force
model by assuming an uniform strain distribution in the actuators. This method performs
better than the pin-force model. However, it cannot be applied on thin structures. For the
Bernoulli-Euler model, Crawley and Anderson (1990) developed an analytical model for a
PZT actuator based on this assumption. They also concluded that this method can
accurately predict extensional and bending deformations on thin PZT actuator. However,
one of the weaknesses in this model is builded based on the one-dimensional structure. In
1994, Rogers presented the Bernoulli-Euler rectangular actuator model to determine the
optimal thickness with fixed boundary conditions. For circular PZT actuators, Li and Chen
(2003) applied the linear strain assumption to analyse the passive plate deflection in a
circular PZT with a bounding layer and verified these results with both FEM simulation
results and experiment data. This model can be used to predict the performance of the
PZT actuator.
In the last category, a relatively complex model was developed, which consists of the
equivalent displacement, shear force and moment. Sonti et al. (1995) considered the
equivalent forces and moments in their research and presented the influence of actuator
14
shape on a flat plate. Su and Ye (2005) presented this sort of mathematical model for
describing Lamb waves excited by the piezoelectric actuators. In their work, they used the
equivalent moments and forces methods to calculate the equivalent radial displacement
for one circular PZT. Furthermore, Yang et al. (2006) applied and advanced it to evaluate
Lamb wave propagation in isotropic plates and quasi-isotropic composite laminates,
respectively.
The numerical modelling and simulations to investigate the interaction between
sensors/actuators and host structures can be conducted using different numerical methods.
The most popular and powerful one is finite element (FE) method. Robbins and Reddy
(1991) used four different displacement-based finite element models to invest the
interaction between a bonded PZT actuator and a host beam. They also listed all the
formulas for different models and their results indicated that the finite element method can
be used to evaluate the performance for PZT actuators/sensors. Chaudhry and Rogers
(1994) devised FE models to verify the Bernoulli-Euler model and the pin force model,
using the commercial FEA software- Abaqus. Li and Chen (2003) also applied the FE
method to verify their analytical equation and the results, which is much close to their
experiment data. Yang et al. (2006), Morris and Forster (2000) and Allik and Hughes (1970)
applied FE method in their research. Therefore, the finite element method is employed in
current study and the numerical results are used to verify the analytical results.
2.3 Guided waves
Guided waves method can be regarded as a primary method to identify and determine the
defects existing in the structures or parts in the field of SHM. More specifically, the guided
waves are generated or excited at a certain frequency. Then they travelled, or propagated,
from one region in one space to another. However, in different media at different
frequencies, there are different types of guided waves generated. In fact, when the guided
15
waves propagate in a structure, they transmit changes in the stress and velocity, which
influence the quantitative wave characteristics (Giurgiutiu, 2007). Therefore, the main
characteristics of guided waves can be summarised as frequency, period, phase,
wavelength, wave speed, wave number and amplitude of particle displacement, etc.
In this project, the guided waves propagation in steel plates and pipes will be considered.
Then, this section will be further discussed the guided waves into these two sub-section as
follows.
2.3.1 Guided waves in plates
The theory of elastic wave propagation in plates has been builded up over 100 years.
Lamb and Rayleigh, as pioneers, investigated the wave propagation in isotropic plates with
free boundary conditions (Lamb, 1917, Rayleigh and Lindsay, 1945). In their works, the
Rayleigh-Lamb equations was developed, which identified the relationship between wave
frequency and wave number under certain conditions. Following with their researches,
different types of guided waves were found in plates based on different conditions. Among
them, four basic types of wave motions were found and three of them were named after
the investigators who did great contributions to understand these waves. There are
Rayleigh waves, Love wave, Lamb wave and Shear horizontal (SH) waves. The detailed
information is listed as follows:
Rayleigh waves
Rayleigh waves, as the simplest guided wave, propagate on the reaction-free surface of a
semi-infinite solid (Royer and Dieulesaint, 2000). In these waves, the particle motion is
contained in the vertical surface and their path has the shape of an elliptical rotation (as
shown in Fig. 2-5). There are three typical characterises on these waves propagation: (1)
their propagation occurs close to the body surface; (2) the motion amplitude decreases
rapidly with depth; and (3) the polarization of these waves lies in a plane that is
16
perpendicular to the surface (Giurgiutiu, 2005). Hence, the Rayleigh waves are very
sensitive to one surface features, likes defects, with very litter penetration in the depth of
the solid (Giurgiutiu, 2005). Therefore, it can be used to inspect the surface properties for
a structure.
Figure 2-5 Rayleigh wave: (a) Schematic representation (Royer and Dieulesaint, 2000)
and (b) Wave on Sagittal plane (Krautkramer and Krautkrâmer, 1990)
Lamb waves
Lame waves was first studied in 1917 by Horace Lamb for these wave propagation in
plates having traction free boundary condition (Lamb, 1917). A comprehensive analysis of
Lamb wave was given by Mindlin in 1950, Schoch in 1962, Viktorov in 1967, Graff in 1975,
Rose in 1999 and Royer and Dieulesaint in 2000. Similarly to the Rayleigh waves, Lamb
waves are a combination of both a longitudinal and transverse motion which results in an
elliptical motion (Giurgiutiu, 2005). However, these waves can only be generated in thin
walled structures that means the motion amplitude remain same on both top and bottom
surfaces only. Therefore there are two kinds of Lamb waves may occur in plates: anti-
symmetrical (A0, A1, A2 …) and symmetrical modes (S0, S1, S2 …) as shown in Fig. 2-6.
17
Symmetrical Lamb waves move in a symmetrical fashion about the neutral plane and
stretch/compress the plate in the wave motion direction, which resembled the ‘extensional
modes”. The anti-symmetrical modes are often called the “flexural modes” for their motion
normal direction to the plate. Unlike the Rayleigh waves, the Lamb waves are highly
dispersive and their speed is related to their frequency and plate thickness (Giurgiutiu,
2005). The more detailed information about the dispersion will be illustrated as the
following sections. The Lamb waves can travel a long distances in one structure with only
a little energy loss, hence it is useful to detect the health statements for one structure
rather than only inspecting its surfaces.
Figure 2-6 Lamb wave: (a) Anti-symmetric mode and (b) Symmetric mode (Krautkramer
and Krautkrâmer, 1990)
Love waves
Like the Rayleigh waves, Love waves are another surface waves applied for surface
inspection. This kind of waves was firstly found by Love in 1911 and verified by many
researchers, such as Sezawa and Kanai. Their particle motion is perpendicular to wave
18
propagating direction and parallel to surface of medium. As in the case of Rayleigh waves,
their wave amplitude decrease rapidly with depth (Fig. 2-7).
Figure 2-7 Displacement distribution of Love waves (Royer and Dieulesaint, 2000)
Shear horizontal (SH) waves
Shear horizontal (SH) waves have a shear-type particle motion maintained in the
horizontal plane, as their name explained. These waves can be regarded as more complex
waves based on Love waves, because their particle motions are similar. As their wave
amplitude have no change on both sides for one structure, shear horizontal (SH) waves,
as same as the Lamb waves, have two kinds of modes: anti-symmetrical (A0, A1, A2 …)
and symmetrical modes (S0, S1, S2 …), illustrated in Fig. 2-8 (Giurgiutiu, 2005). These
waves are assumed the particle motion along the z axis and the wave propagation in x
direction. The advantage of applied SH waves on SHM is summarised by Petcher et al.
(2013). For instance, they can be used on curved structures with a small energy loss.
19
Figure 2-8 SH waves: (a) Anti-symmetric mode and (b) Symmetric mode (Giurgiutiu, 2005)
2.3.2 Guided waves in pipes
Many researchers have provided their reviews on guided waves in pipes and extended this
theory further (Silk and Bainton, 1979, Rose, 2004, Gazis, 2005a, Gazis, 2005b,
Giurgiutiu, 2007). More specifically, guided waves in the infinite hollow cylinders were first
studied and found by Chree (1897). Love and Rayleigh also analysed wave propagation in
an isotropic cylinder using shell theories (Love, 1944, Rayleigh and Lindsay, 1945). Gazis
(1958) summarised the comprehensive work on wave propagations in hollow circular
cylinders and symmetrically derived the equations to reinforce the theoretical basis of
elastic waves in pipes. Armenakas, et al. (1969) conducted a detailed discussion on the
wave propagation in cylinder structure and pointed out they can exist multimodally and
disperse at a high frequency range. Recently, lots of attentions have been paid on
theoretical developments for wave propagation in pipes (Lowe et al., 1998b; Galvagni and
Cawley, 2011). There is also having many researchers devoted into experimental study.
To be more exact, Fitch (1963) reported some experimental measurements of both the
axially symmetric and non-axially symmetric longitudinal modes of elastic waves in hollow
and circular cylinders. The final results had much close agreement with the results from
20
Gazis (1969). The propagation of the L (0, 1) mode around tubing was first investigated by
Silk and Bainton (1979) and it proved that the guided waves can be used on the detection
of damage in pipes. Recently, Cawley and Alleyne (1996), Cawley (2002) also conducted
experiments on Lamb wave-based inspection. They also suggested that it is better to
choose the wave modes in a non-dispersive frequency range. Alleyne et al. (1996),
proposed another option, and they used the reflection of L (0, 2) axially symmetric guided
elastic wave on the pipes inspection.
In summary, for the plate’s simulation, Rayleigh waves and Love waves are belonging to
the surface waves, which are widely used on surfaces inspection, whereas, Lamb waves
and SH waves are both more complex waves and common applied on ultrasonic guided
waves testing. The difference between the Lamb waves and SH waves is their
performances. For example, the Lamb waves have a better performance on the plates
inspection, whereas the SH waves are more suitable for curved structure inspection.
Therefore, the Lamb waves are closely related to this project. However, these waves are
more complicated guided plate waves, which have symmetric and anti-symmetric modes.
Both types are quite dispersive. Then, the following section will illustrate the relation
between the velocity and the dispersion for guided waves. For the pipes inspection, the
theory of guided wave propagation in pipes is relatively perfect. However, many
experiments show that the wave mode selection is still a difficult problem. Hence, at
following of section, the mode selection methods will be presented as well as the detailed
information regarding their dispersion.
2.3.3 Velocity
The velocities of waves can be defined in many different ways. Generally speaking, the
wave velocities can be primarily classified into the group velocity and phase velocity. The
group velocity is the velocity with which wave packets travel. In contrast, the phase
21
velocity is the wave speed of the individual waves. Fig. 2-9 shows the detailed information
about the phase velocity and group velocity. More precisely, the velocity of wave train in
contrast with that of the carrier is the phase velocity. The velocity of the envelope is the
group velocity.
Figure 2-9 the relationship between group velocity and phase velocity (Shen and Qin,
2012)
2.3.4 Guided wave dispersion in pipes
The guided waves propagating along the pipes are more complex than those in plates as
well as in shells. Generally speaking, there are three wave modes existing in pipes and
they are
1 Longitudinal axially symmetric modes
L (0, m) (m = 1, 2, 3, 4…)
2 Torsional axially symmetric modes
T (0, m) (m = 1, 2, 3, 4…)
3 Non-axially symmetric modes
F (n ,m) (n = 1, 2, 3, 4…, m = 1, 2, 3, 4…)
The index m represents the number of mode shape across the wall of the pipes. The index
n determines the manner in which the fields generated by the guided wave modes vary
22
with an angular coordinate in the cross-section of pipes. Figs. 2-10 and 2-11 illustrate
both of the phase and group velocity dispersion curves for a pipe of a nominal 3-in
diameter (internal diameter 76 mm and wall thickness 5.5 mm). Hence, the characteristics
of each wave modes will be identified by using this information according to the wave
velocity (speed) and frequency.
Figs. 2-10 and 2-11 illustrate that guided waves at different modes can propagate at a
given frequency simultaneously. Thus, the coherent noise can be generated by the
excitation of unwanted wave modes, considerable efforts have been concentrated on the
generation of a single mode (Demma et al., 2004, Cawley et al., 2003) and for the
sensitivity of a test is defined by function of signal to coherent noise ratio. The velocity of a
particular mode can change with the frequency. It leads to the distortion of wave packet
when traveling along the pipes.
It is ideally convenient to use only one mode in a non-dispersive region. Hence, each wave
modes (L, T and F) should be analysed and then a suitable one can be chosen for in this
research.
23
Figure 2-10 Phase velocity dispersion curves for 76 mm (3 in) diameter pipe (Lowe et al.,
1998)
Figure 2-11 Group velocity dispersion curves for 76 mm (3 in) diameter pipe (Lowe et al.,
1998)
24
L(0, m) modes
In the lower frequency region, there are only two kinds of L(0, m) modes existing. They are
L(0, 1) and L(0, 2) modes (as shown in Fig. 2-11). In order to recognise them better, it is
useful to acknowledge the mode shapes of the displacements in pipes.
Fig. 2-12 (a) shows the L(0, 1) mode which is related to axially symmetric bending in pipe
wall. The radial displacement keeps at same level. However, axial displacement of L(0, 1)
is smaller and has the different directions between inside surface and outside surface.
Then, it is better to appling on locating circumferential cracks. Fig. 2-12 (b) shows L(0, 2)
mode which is predominant by uniform axial motion through wall thickness. The axial
displacement is remaining the same and it has the least energy lost at 70 kHz. Hence, L (0,
2) mode is preferred for defect detection in long pipes. Rose (2004) is also having the
same opinion in his paper. Besides, L(0,1) has a much lower velocity than L(0,2) at the
operating frequency range above 35 kHz, as shown in Fig. 2-10. Therefore, the presence
of reflections of this mode can make interpretation of the results less reliable (Giurgiutiu,
2007).
25
Figure 2-12 Mode shapes for 76mm (3 in.) diameter pipe at 70 kHz: (a) L(0,1); (b): L(0,2)
(Lowe et al., 1998)
T (0, m) modes
T (0, 1) mode is the only one T mode occurred in the frequency range which can be used
for both finite element models and experiments (Fig. 2-11). The mode shape of the T(0, 1)
mode in a 3-inch pipe diameter is shown in Fig. 2-13. This illustrates the profile of the
tangential displacement through the thickness of the pipe wall. No axial or radial
displacement is found in this region in this situation. It can be seen that the tangential
displacements are nearly constant through the wall thickness, which is useful to determine
defects anywhere in the cross section of the pipes and also be potential for defect sizing
(Demma et al., 2003).
26
Figure 2-13 T(0, 1) mode shape in a 3 inch pipe at 45 kHz Radial and axial displacements
are zero (Demma et al., 2003)
To sum up, the torsional mode has the advantage of being non-dispersive across the
whole frequency range and there is no other axially symmetric torsional mode for all tube
thicknesses. The torsional mode is hardly affected by the presence of liquids in the pipe
and preferred detect the longitudinal cracks. However, a disadvantage of this wave mode
to axial features is that the torsional mode reflects relatively strongly from support brackets
that are welded axially along the pipe (Lowe and Cawley, 2006).
F (n, m) modes
Unlike L and T modes, F modes have various vibration directions depending on value of n
(Fig. 2-10). When the value of n = 1, the particle vibration direction is along a single radius
of one pipe. More specifically, F(1, m) mode is a non-axis-symmetric wave mode in which
related to a circumferential angle with three direction displacement components (Shen et
al. 2012). It is clear from Fig. 2-14 that many kinds of F modes appear in 50~100 kHz.
Different wave velocities are accompanying with these wave modes. Hence, it is hard to
recognise and separate the wave modes from a signal aspect. Thereby the F (n, m)
modes are not useful on the pipe inspection (Alleyne et al., 1998).
27
Figure 2-14 Peripheral vibration model sketch map (Shen et al., 2012)
2.4 Methods to simulate the guided wave propagation
There are two general analytical methods to analyse the guided wave propagation in a
structure. There are the transfer matrix method and the global matrix method, respectively.
The transfer matrix method was first developed by Thomson (1950) and advanced by
Haskell (1953), with enhancing this method by a correction. Hence, this Thomoson-Haskell
method is to express the displacements and stresses between top and bottom surfaces by
applying this method to a structure containing only one single layer. Those matrices for
any number of layers could be multiplied and then structural response solutions could be
found by application of the appropriate boundary conditions. Therefore, it is more
effectively to apply this method to evaluate the wave propagation in the multilayer
structures. However, Dunkin (1965) found that this method was affected by a loss in
precision when the layers of large thickness and high frequencies were considered, which
means this process may show numerical instability for large layer thickness and high
28
excitation frequency. He modified these difficulties by using the determinant matrix
decomposition theory and expressing the solution in terms of its Laplace-Fourier
transformation. In 1994, Castaings, et al., improved the transfer matrix by applying the
delta matrix operation in anisotropic materials. Rokhlin and Wang (2002) developed an
efficient and stable recursive compliance/stiffness matrix algorithm based on the transfer
matrix method.
In 1964, Knopoff proposed an alternative method that is the global matrix formulation. In
this formulation, all the layers of the structure were considered and there was no a prior
assumption on the interdependence between sets of equation for each layers, thereby
avoiding instability for results. Lowe (1995) presented a review of the global matrix and the
transfer matrix method and provided detailed information on the both methods. For
commercialised application, Pavlakovic et al. (1997) developed a software packages-
Disperse, which can be used to calculate wave dispersion curves in plates, shells and
pipes. This software was developed based on the global matrix method.
Those two methods are both solving the roots for dispersion equations, thereby occurring
error on the calculating processing, especially consideration of the material viscosity.
Besides, those two analytical methods are only able to address canonical problems,
whereas real-life problem have to be tackled with numerical methods.
Compared to the analytical methods, the numerical ones are more flexible and suitable to
calculate the wave velocity in a structure. The finite element (FE) method is the most
flexible and stable to calculate the wave speed in a structure, thereby many researchers
applied this method to address guided wave problems. For a plate structure, Alleyne and
Cawley (1992) invested a lot of time and effort on this aspect. They characterised the
interaction between individual Lamb waves and a variety of defects in plates through the
FE simulation. The results illustrated that the sensitivity of individual Lamb waves to
particular notches is dependent upon the frequency-thickness product, the mode type, the
29
mode order as well as the geometry of the notch. Therefore, the different Lamb waves
have different sensitive for different defects in the plate. Karunasena et al. (1995)
combined an FE method and wave function to study the Lamb wave reflection in plates
and extracted the results by experimental results. For the pipes inspection, Lin et al. (1998)
applied FE method to analyse the circumferential wave propagation in pipes and evaluated
pipes. They found the relationship between the defect depth and wave transformation
characteristics. However, they only considered the wave propagation along the axial
direction, which means there no torsional waves and flexural waves considered. Alleyne et
al. (1998) also used the FE method to study the reflection of the L (0, 2), axially symmetric
guided wave from notches in pipes. They concluded that the relationship between the
reflection coefficient of this model and the ratio of the circumferential extent of the notch to
the pipe circumference is a linear function. In addition, the results from FE method are
much close to their experimental data.
In summary, one of the advantages of FE method is that it can be used to almost every
materials, including inhomogeneous and anisotropic materials. However, there is a big
demand on the computational resources for FEA. In order to reduce computational costs,
various hybrid approaches that combine analytical methods and finite elements methods
have been developed. The Semi-analytical finite element method (SAFEM) is the most
popular one. Hayashi et al. (2003) calculated the dispersion curves of an arbitrary cross-
section rail by the SAFEM and their experimental data validated the numerically-predicted
dispersion curve. Marzani et al. (2008) summarised the SAFEM formulas for modelling
stress wave propagation in axisymmetric damped waveguide. They concluded that the
benefit for applying SAFEM is no missing root when the eigenvalue problem is addressed.
The results from SAFEM performed stable for the bulk velocities of the materials.
Moreover, compared to FE guided waves simulation, the SAFEM presents enormous
computational, time and memory saving for determining dispersion curves of structures of
30
interests. Hence, in this project, GUIGUW software, based on SAFEM formulations, was
chosen to calculate the wave dispersion curve of steel plates and pipes.
2.5 Piezoelectricity
The word ‘piezoelectricity’ is directly translated from Greek word ‘Piezein’ for pressure
(Jordan and Ounaies, 2001). The piezoelectricity describes the phenomenon of generating
an electric field when the material is subjected to a mechanical stress or conversely,
generating a mechanical strain in response to an applied electric field (Giurgiutiu, 2007).
This phenomenon was first discovered by Jacques Curie and Pierre Curie in 1880. They
found electrification under mechanical pressure of certain crystals, which is including
quartz, cane sugar and Rochelle salt. This linear and reversible phenomenon is called as
the direct piezoelectric effect (Mould, 2007).However, they did not discover the
deformation or stress of those materials under electric field. This important property was
mathematically deduced by Lippmann in 1881 (Jordan and Ounaies, 2001). At the same
year, Curie brothers confirmed this effect, which is referred to the converse piezoelectric
effect (Safari and Akdogan, 2008). A review of the early history of piezoelectricity can be
found in the work of Jordan and Candy (2001). In 1910, the Voigt published a text book
named as ‘Lerbuch der Krisallphic’, which he illustrated the complex electromechanical
relationship in piezoelectric crystals. During the World War I, a piezoelectric ultrasonic
transducer was developed by Langevin et al. in 1950. Whereas, when it was coming to the
Second World War, the piezoelectric effect evolved from just a laboratory curiosity to a
multimillion dollar industry. The barium titanate and lead zirconate titanate (PZT) were first
discovered during this period. The families of those materials exhibits very high dielectric
and piezoelectric properties (Jordan and Ounaies, 2001). Nowadays, the PZT becomes
one of the most widely used piezoelectric materials. Those property can be illustrated
clearly by the well-known Perovskite Structure (Fig. 2-15 a).
31
The Perovskite structure can be simply expressed as 3ABO . It means the corner-sharing
oxygen octahedral are connected together in a regular cubic array with small caption B , in
which can be substituted by Ti, Zr, Sn, Nb etc. The larger cations (Pb, Ba, Sr, Ca, Na etc.)
filling the interstices between octahedrons in the larger A-site (Jordan and Ounaies, 2001).
As the ambient temperature decreases, the lattice shrinks and the symmetric arrangement
is no longer stable. This phenomenon is called as poling and this temperature is named as
Curie temperature. Fig. 2-15 (b) shows the poling within a crystal lattice that becomes
distorted and creates strain or electric dipoles on the both sides (Giurgiutiu, 2007). To be
more exact, if this material is forced into compression or tension, an opposite voltage will
be produced across the electrodes, which is called the direct piezoelectric effect. This
effect is used in the development of piezoelectric sensors. Conversely, when the material
is posited in an electric field, the relative strain will appear on the surfaces, which is
expressed as converse piezoelectric effect. Hence, it is useful in the development of
piezoelectric actuators (Giurgiutiu, 2007).
Figure 2-15 (a) Perovskite Structure above the Curie point (before poling) ; (b) below the
Curie point, the crystal was displaying polarization (after poling) (Giurgiutiu, 2007)
32
2.6 Signal operation
In this section, the signal processing or operation of guided waves is discussed. Hence, in
order to express it clearly, two sub-sections were developed based on the signal transfer
processes.
2.6.1 Generation of excitation signals
As the previous section shows, the guided waves are dispersive. Hence, once traveling at
a long distance, the packet of waves will contain various frequencies, therefore spreading
out and distort. Then, in order to obtain the useful information for the wave propagation
phenomena, the input signal should be limited its bandwidth, which can reduce the
problem of dispersion (Xu and Giurgiutiu, 2007). Generally speaking, there are two kinds
of waveforms used in here: (1) Morlet mother wavelet (Gaussian windowed tone burst)
and (2) Hanning windowed tone burst. These two waveforms both consist of a single-
frequency carrier wave. However, the difference is to use the different window functions to
modulate that amplitude. Many researchers imported these methods in their work. For
instance, on one hand, Lemistre and Balageas (2001) applied Morlet wavelet in composite
structures. Sun et al. (2000) used Morlet transform to analysis the Lamb waves
propagating in an aluminium alloyed pipe. On the other hand, Gao (2007) applied 5 cycles
Hanning windowed tone-burst signal on their simulation and achieved a better result for
guided wave propagation on different composite materials. Giurgiutiu (2007) in his book
advised to employ the Hanning windowed tone-burst signal as an input one and he
continue to utilise this method in his following researches (Giurgiutiu et al., 2004, Giurgiutiu
et al., 2012). Kumar et al. (2010) also suggested that the Hanning windowed signals can
give a narrower bandwidth of frequency band, which can extract the useful information
from this signal more easily.
33
In summary, the Hanning windowed tone burst, as a commonly used excitation signal, was
selected in this project and created by using Matlab software.
2.6.2 Signal processing
Signal processing is also an important in process of extracting useful information from the
data. Currently, 2 Dimensional Fourier transform (2D-FFT) and Hilbert-Huang transform
(HHT) are more primary techniques for signal processing. Alleyne and Cawley (1991)
employed 2-D FFT method to analyse the amplitudes and velocities of propagating Lamb
waves and he point out this method can be used to measure more complex waves, Likes a
mixture of longitudinal and shear waves. Chun-gou et al. (2008) applied the dispersion-
based HHT method to analyse the dispersive waves and demonstrated this method into 龙
bone structure. Hence, the HHT method was selected as the signal processing in this
project. All the results were analysed using a code programme in a Matlab working
environment.
2.7 Summary
This thesis is relative to the structural health monitoring, which is including the guided
wave analysis, the mode selection and the signal operation. In the first place, according to
comparison between the active SHM and passive SHM, it is better to choose the active
one for its economic benefits and high performance on the inspection. Secondly, the
ultrasonic guided wave testing was selected for its ability to inspect a large structure. In
addition, based on the research on SHM, the general knowledge about piezoelectric
sensors /actuators and guided wave propagation is determined as well as their interaction,
including their dispersion, group velocity and phase velocity. Furthermore, the signal
operation is also reviewed to achieve a better result from the simulation. More specifically,
the Hanning windowed tone burst signal is regarded as an input signal for its reliable
34
performance on a narrower bandwidth of frequency band. The Hilbert-Huang transform
(HHT) is selected as a signal processing method to extract the envelope of amplitude
values for better understanding.
35
Chapter 3 Development and selection of effective PZT models
3.1 Introduction
In this chapter, basics of piezoelectric effects and fundamental constitutive formulas are
presented in the first place. Then, the detailed development for different equivalent
effective PZT disk models is presented, including their governing equations to describe
relationships between voltage and displacement. In addition, the enhanced tri-layer
actuator/sensor models based on the force and moment equilibriums are developed.
Lastly, a case study is conducted to show the effective piezoelectric actuator model is in
good agreement with these from a numerical simulation of a full actuator model devised.
3.2 Piezoelectric effect and constitutive formulas
As shown in pervious chapter, the crystal structure of piezoelectrics ceramic can be
expressed as the Perovskite structure, 3ABO (Fig. 2-13). According to Giurgiutiu (2007), a
typical piezoelectric ceramic is 1 3x xPb Zr Ti O , in which A is Pb and B is Ti4+, respectively.
As the piezoelectric material, its piezoelectricity has two effects: a) direct piezoelectric
effect and b) converse piezoelectric effect. Therefore, in order to identify its properties
better, subscripts and superscripts are both used to describe each property coefficient as
other material parameters. Hereby, subscripts are introduced using an orthogonal
coordinate system to illustrate this material and they are integers. 1, 2 and 3, referring to
the three principal axes, whereas 4, 5 and 6 describing three shear effects around 1, 2,
and 3, respectively.
In this project, PZT material was selected as actuators and sensors, which maintains
double sub-scripts. The first one refers to the direction in which an electrical field is
produced on the material, while the second one means the direction of the mechanical
strain that material experiences, as shown in Fig. 3-1.
36
Figure 3-1 Orthogonal coordinate system and poling direction (Inman and Cudney, 2000)
The interaction between the electrical and mechanical variables can by described by a
linear relation in 3D cases (IEEE, 1988); it is a general case that the poling direction is in
the negative ‘3’ direction.
d
i ij j im mD e E d (3.1)
c E
k jk j km md E S (3.2)
where, 3 1D is the electric displacement matrix (Coulomb/m2), 6 1 is the strain vector, 3 1E
is the applied field vector (Volt/m) and 6 1m is the stress vector (N/m2). The
piezoelectric constants are dielectric permittivity, which is 3 3ije (Farad/m). Moreover,
the piezoelectric coefficients 3 6d
imd (Coulomb/N) is electric displacement per unit stress
at a constant electric field and 6 3c
jkd (Coulomb/N or m/Volt) is strain per unit field at a
constant stress as well as the elastic compliance 6 6E
KmS (m2/N) (Zhu, 2010).
37
Eq. (3.2) is standing for the actuator performance, which is normally attached on the
surface of the structures or parts and induces a mechanical strain when applying an
electric field. In contrast, Eq. (3.1) demonstrates a sensor performance which is according
to the direct piezoelectric effect.
Hence, the above-mentioned Eqs. (3.1) and (3.2) can be combined and rewritten as:
d
c E
D Ee d
d S
(3.3)
The matrix of 6 3c
jkd can described as:
31
32
33
24
15
0 0
0 0
0 0
0 0
0 0
0 0 0
d
d
dd
d
d
(3.4)
where the coefficients 31d , 32d and 33d are normal strains in the 1, 2 and 3 directions to an
electric field 3E along the polling direction, respectively. The coefficients 15d and 24d are
shear strains in the 1-3 planes to the electric field 1E and in the 2-3 plane to the electric
field 2E , respectively.
The compliance matrix SE is:
11 12 13
12 22 23
13 23 33
44
55
66
0 0 0
0 0 0
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
E
S S S
S S S
S S SS
S
S
S
(3.5)
38
The permittivity matrix eσ is:
11
22
33
0 0
0 0
0 0
e
e e
e
(3.6)
The stress vector σ is:
1 11
2 22
3 33
4 23
5 31
6 12
(3.7)
where the Eqs. (3.1) and (3.2) can be written in matrix form as:
1 11 12 13 1 31
2 12 22 23 2 32
1
3 13 23 33 3 33
2
4 44 4 24
5 55 5 15
6 66 6
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
S S S S d
S S S S dE
S S S S dE
S S d
S S d
S S
3E
(3.8)
1
2
1 15 11 1
3
2 24 22 2
4
3 31 32 33 33 3
5
6
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0
D d E
D d E
D d d d E
(3.9)
1
1 2 3 2
3
dA
q D D D dA
dA
(3.10)
The electric displacement D is transferred to charge, where the dA1, dA2 and dA3 are the
electrode areas in the 2-3, 1-3 and 1-2 planes, respectively.
/c pV q C
39
The charge q (Coulomb) and the voltage generated across the sensor electrodes Vc are
related by the capacitance Cp of the piezoelectric materials. To apply Eqs. (3.8) and (3.9)
stresses in the material can be calculated.
In summary, the coupled linear electromechanical constitutive relations can be expressed
as:
1 151
2 152
3 31 31 333
31 11 12 1311
31 12 11 1322
33 13 13 3333
15 4423
15 4413
11 1212
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 2
D dq
D dq
D d d dq
d S S S
d S S S
d S S S
d S
d S
S S
1
2
3
11
22
33
23
13
12
E
E
E
(3.11)
3.3 Effective piezoelectric actuators/ sensors equations
In order to develop the robust and cost-effective finite element-based models, researchers
have developed some effective or equivalent models for piezoelectric actuators and/or
sensors based on different theories. In this section, these models are classified into four
categories. They are the uniform strain models, the uniformly-linear strain models, the
linear strain models and more complex models. In order to explain them better, a flat PZT
disk attached in a host structure-a plate with an electric field applied is analysed
analytically and numerically in this section.
3.3.1 The uniform strain model (USM)
The uniform strain model, as its name expressed, has a uniform strain profile along the
poling direction in the piezoelectric actuator and the host structure when the electric filed is
applied. This assumption is acceptable for actuators embedded at the middle of the plate,
but not for surface-bonded ones. However, it was proposed in Crawley and Anderson
40
(1990) and ignored the flexural stiffness of the plate. Hereby, this model can be referred as
the simplest PZT-plate model, as shown in Fig. 3-2.
Figure 3-2 Strain distributions through thickness for the uniform strain model
According to Crawley and de Luis (1987), the strain can be expressed as
2
2
a p
(3.12)
where is the stiffness ratio, given as
p p
a a
E t
E t
(3.13)
and is the free strain caused by the piezoelectricity effect, given by
31
a
Vd
t (3.14)
where pE is the modulus of elasticity of the host plate, aE is the modulus of elastic of the
actuator, pt is the host plate thickness and at represents the actuator thickness, V is the
electrical potential applied across the actuator electrodes and 31d is the piezoelectric
electromechanical coupling.
41
According to Chaudhry and Rogers (1994) , the strain in the actuator is assumed to remain
constant:
aa
aE
(3.15)
Then, apply Eq. (3.12) to (3.15), the stress in the actuator can be derived as
2
a aE
(3.16)
The voltage on the PZT poling surfaces is relative to the stress by
31 a aV g t (3.17)
where 31g is the PZT voltage constant. The voltage is relative to the stress for the uniform
strain models by substituting Eq. (3.16) into (3.17) as
31
2
a aE t gV
(3.18)
3.3.2 The pin force model (PFM)
In the pin force model (Fig. 3-3), the strain is assumed to be uniformly distributed in the
actuator and linearly distributed in the host structure. This model is also termed as the
uniform–linear strain model. The difference between the uniform strain model and the pin
force model is the flexural stiffness term.
Figure 3-3 Strain distributions through thickness for the pin force model
42
The strain in actuator can be assumed as:
3
3
a
(3.19)
Hence, according to Eq. (3.15), the stress in the actuator can be written as
3
a aE
(3.20)
According to Eq. (3.17), the voltage applied to this model is expressed as,
31
3
a aE t gV
(3.21)
3.3.3 The enhanced pin force model (EPFM)
In this model, it is assumed as there is linear distribution of strain in both the actuator and
the host structure, which are illustrated in Fig. 3-4. Thus, the flexural stiffness of the
actuator is taken into account as well.
Figure 3-4 Strain distributions through thickness for the enhanced pin force model
One can write the strain distribution in the actuators as (Chaudhry and Rogers, 1994)
43
2
3
13
a
T
(3.22)
where /p aT t t (3.23)
Thus, based on Eq. (3.15), the stress distribution in the actuator is given by
2
2 2
1
3 1
aaE T
T T
(3.24)
The voltage on its surfaces can be calculated based on Eq. (3.17), as
2
31
2 2
1
3 1
a aE t g TV
T T
(3.25)
3.3.4 The strain energy model (SEM)
The strain energy model was developed by Wang and Rogers (1991). Similarly with the
pin force model, the strain remains constant in the actuator and decreasing linearly
through the host plate. According to Fig. 3-5, there are three assumptions introduced in
this model, which is the linear strain distribution in the host plate, uniform strain in the
actuator and zero strain at the lower surface of the plate.
Figure 3-5 Strain distributions through thickness for the strain energy model
Therefore, according the last assumption, which is the zero-strain case, the uniform strain
in the actuator can be calculated as
44
6
6
a
(3.26)
Hence, the stress for the strain energy model will now be
6
a aE
(3.27)
The voltage between the PZT poling surfaces is
31
6
a aE t gV
(3.28)
3.3.5 The Bernoulli-Euler model (BEM)
In this model, the force and stress analyses of composite materials are used to predict the
stress and strain of the actuator. Perfect bonding between actuator and host structure is
assumed, and thus the PZT and host structure are a compound and can be regarded as a
two-layer composite plate. Hence, the classical laminate plate theory can be applied in
here. The total strain in this model can be regarded as a linear distribution (Crawley and
Anderson, 1990) show in Fig. 3-6, which is written as
0 z (3.29)
where z is the distance measured from the structure neutral axis.
Figure 3-6 Strain distributions through thickness for the Bernoulli-Euler model
The uniform strain 0 can be calculated, shown as
45
01
(3.30)
The curvature is given by
Total
M
EI (3.31)
where
2
aa a p
tM E t t z
(3.32)
2 2
3 31 1
12 2 12 2
p ap p p p a a a a pTotal
t tEI E t E t z E t E t t z
(3.33)
2
2 1
p at tz
(3.34)
46
3.3.6 The Bernoulli-Euler model considering an adhesive layer (BEMA)
In this model, an adhesive layer was considered by using Bernoulli-Euler method.
Generally, the strain distribution will remain the same across the thickness direction (Fig.
3-7), including actuator, adhesive layer and host plate.
Figure 3-7 Stress distributions through thickness for Bernoulli-Euler model with an
adhesive layer
Hence, the strain in any location of this structure is same as previous formula. However,
as the third layer influence, the way to calculate curvature becomes more complex. The
more detailed information is listed as follows,(Yang et al., 2010)
1Total
Mz
EI
(3.35)
2
aa a p b
tM E t t t z
(3.36)
2 2
3 3 3
2
1 1 1
12 2 12 2 12
2
p bp p p p b b b b p a aTotal
aa a p b
t tEI E t E t z E t E t t z E t
tE t t t z
(3.37)
2 2 2
p b ap p b b p a a p b
p p b b a a
t t tE t E t t E t t t
zE t E t E t
(3.38)
47
3.3.7 The effective piezoelectric actuator model (EPM)
This model can be regarded as a perfectly bonded PZT actuator model. Hence, for
convenience of analysis, this structure can be divided into two parts: one is the PZT disk
and another is an isotropic plane. Then, the PZT disk and plate can be assumed as thin
enough. It means that the classical lamination theory is applicable and the stress
distribution in the plate and PZT disk can be assumed linear across the thickness, as
shown in Fig. 3-8.
Figure 3-8 Strain and stress distributions through thickness for the effective model
For the no in-plane external electric field PZT disks 1 2 0E E , the corresponding
constitutive Eq. (3.11) will be changed to
3 3 3 31 rq D E d (3.39)
21
1
ar r a a
a
E
(3.40)
21
1
aa r a
a
E
(3.41)
where E is Young’s module, is Poisson’s ratio and polar coordinate was used in here,
which is ,r .
For free PZT disk actuators without any loading, the induced in-plane strain by using a
voltage across the thickness of this disk can be written as,
48
3131 3
a a
r in
a
dd E V
t (3.42)
For the interface between the host plate and the actuator, the strain must be equal,
I Ia p I
r r r (3.43)
I Ia p I
(3.44)
Then, according to the Laminate plate theory, all the strain components are linear through
the thickness with the same slop.
Ia rr r
n
z zt
(3.45)
Ia
n
z zt
(3.46)
where nt is the distance from the neutral plate to the top surface of the plat, and r and
are the curvatures in polar coordinate.
Hence, substituting the Eqs. (3.45) and (3.46) into (3.40), the stress in the actuators can
be expressed as,
21
1
a I Iar r a a
a n
E z
t
(3.47)
However, for the host plate, the stress and strain relationship can be written as,
1 IpI I I I p
r p r
pE
(3.48)
Combining the Eqs. (3.47) and (3.48), the stress can be expressed as
49
p I
r r
n
z
t (3.49)
a I
r r
n
z
t
(3.50)
1
1
p a
a p
E
E
(3.51)
1
a
a
E
(3.52)
where superscripts p and I refer to the host plate and the interface between the actuator
and host plate, respectively. tn is the distance from the neutral plane to the top surface of
the host plate. By using the shear force and moment equilibrium about the neutral axis of
the combined PZT disk and the host plate. z is measured from the plate neutral surface.
2 2 0n n a
n p n
t t tp a
r rt t t
R dz R dz
(3.53)
2 2 0n n a
n p n
t t tp a
r rt t t
R zdz R zdz
(3.54)
where R is the radius of the PZT disk.
2
2 2 3 2 2 3
3 2
2 3 3 3 3
n n a aI
r
n p n p p n a n a a
t t t t
t t t t t t t t t t
(3.55)
For a small PZT disk, the strain distribution can be considered as I I I
r .The
equivalent radial displacement,rd , along the disk circumference can be obtained by
integration the strain at interface:
0
1R pI I
r r
p
d dr RE
(3.56)
For the effective shear force for the PZT actuator, it can be illustrated as:
22 2n a
n
t ta I
r r r a n a at
n
F R dz R t t t tt
(3.57)
50
For the sensor model, the deformations can only occur at r plane. Hence, considering the
external electric field, the constitutive relation for a PZT sensor is:
3 31 311
ar r
a
Eq d d
(3.58)
If considering the PZT disk is smaller than the size of host plate, the relationship of
2r r rCentre Centre could be accepted.
The electric charges, Q, accumulated on both surface of PZT sensor due to electric displacement
can be regarded as
3
1 1
4 8S
Q qdv q ds
(3.59)
Further the output voltage of the sensor can be obtained
3
out
Q QV
C C
(3.60)
where 3 03
a
a
AC
t
and 2
0 0A r .
Therefore,
31 31 31
3 0 3 0 3 08 1 8 1 4
aa a a a aOutput r rCentre Centre
a a
d t E d t E d tV
(3.61)
3.4 Development of effective PZT model with an adhesive layer (EPMA)
In practice, the adhesive layer is also an important aspect on numerical simulation in PZT
actuators model. Hence, based on the prefect bound PZT actuators model, the full PZT
actuators model can be divided into three sections. There are the PZT disk, the bonding
material, and the host plate, respectively. Similarly, a linear strain distribution across the
thickness direction can also be assumed in this model, shown in Fig. 3-9. Therefore, the
stain at the interface will be continuous and the radius of curvature for each layer shares
the same center.
51
Figure 3-9 Strain and stress distributions through thickness for the effective model with an
adhesive layer
Then, the strain distribution can be assumed as
1Ipp b a I r
r r r r
n n P
zz
t t E
(3.62)
where superscripts p, b and a refer to the host plate, the bonding layer and the PZT
actuator, respectively. tn, is same as the previous model, represent the distance from the
neutral plane to the top surface of the host plate.
For the bounding material, the stress follows:
b I
r r
n
z
t
(3.63)
where
1
1
p b
b p
E
E
(3.64)
For the host plate and actuator disk, the formulas of strain for each component are similar
to Eqs. (3.49) and (3.50), respectively.
Hence, Balancing the shear force and moment in the three-layer structure gives:
52
2 2 2 0n n b n b a
n p n n b
t t t t t tp b a
r r rt t t t t
R dz R dz R dz
(3.65)
2 2 2 0n n b n b a
n p n n b
t t t t t tp b a
r r rt t t t t
R zdz R zdz R zdz
(3.66)
where R is the radius of the PZT disk and tb is the thickness of adhesive layer.
The interface stress can be found from Eqs. (3.56) and (3.57).
2
3 3 3 33 3
3 2 2
2
n a a b a nI
r
n n p n b n n b a n b
t t t t t t
t t t t t t t t t t t
(3.67)
Besides, the equivalent normal displacement, rd , along the edge of disk, can be obtained
as
0
1R pI I
r r
p
d dr RE
(3.68)
For the effective shear force for the PZT actuator, it can be illustrated as,
22 2n
n p
It
p rr r p n p
t tn
F R dz R t t tt
(3.69)
The voltage can also be calculated based on the Eq. (3.61).
53
3.5 Case study
In this section, in order to validate the proposed and existed effective models developed in
pervious sections, their effective displacements, stresses and strains are calculated
analytically based on their equations and compared them with their numerical results. In
addition, the Adhesive layer, the PZT’s shapes and sizes effect are also compared in this
section. Hence, two kinds of models are built and analyzed in here. The first one is a
perfect bound PZT-Plate model and another one is a PZT-Plate model with Adhesive layer.
These models was analysed by using Abaqus/Standard.
Fig. 3-10 shows a setup for the plate model and actuator model. The PZT actuator was
designed to locate at the centre of the plate and added a series of voltage which is range
from 0 V to -20 V. Then, the effective displacement, strain and stress for the actuator can
be calculated from either analytical equations or numerical models. Besides, Table 3-1
gives the plate and PZT’s dimensions and properties.
Figure 3-10 Setup of a plate model
54
Table 3-1 Dimensions and properties of the plate and PZT, respectively
Parameter Value Units
Plate (Steel)
Length 300 mm Width 300 mm
Thickness 1.275 mm Density 7700 2/ mmg
Young’s Modulus 210000 MPa Poisson’s Ratio 0.28
Adhesive Layer (E
silver epoxy)
Thickness 0.1 mm Density 1360 2/ mmg
Young’s Modulus 7629.9 MPa Poisson’s Ratio 0.266
Actuator (PZT-4)
Radius 3.45 mm Thickness 0.5 mm
Density 7500 2/ mmg
Elastic constant 11E
81.3 GPa
Elastic constant 33E
64.5 GPa
Poisson’s ratio 0.33
Elastic constant 11S
1.23E-2 1/ GPa
Elastic constant 33S
1.55E-2 1/ GPa
Charge constant 31d
-1.23E-10 /m V
Charge constant 33d
2.89E-10 /m V
Charge constant 15d
4.96E-10 /m V
Relative dielectric constant 1 1475
Relative dielectric constant 3 1300
Dielectric permittivity 0 8.854E-12 / mF
For the numerical analysis, this model was built and analysed using Abaqus. The plate
and adhesive layer were discretised as 3-D brick element (C3D8R) and the PZT actuator
modelled as piezoelectric brick element (C3D8E). Detailed information about these FE
model is given in Table 3-2. The boundary condition was added via applying a series of
voltage which has a range from 0V to -20V on PZT’s surfaces. The poling direction for the
circular PZT disk is shown in Fig. 3-11. Hence, following with this poling direction, the
55
material properties for PZT should be changed and listed in Table 3-3. Fig.3-12 illustrated
the geometry of the finite element models in Abaqus. One point along the edge of PZT
actuator was regarded as an output point, which the effective displacement, stress and
strain can be obtained in these cases.
Table 3-2 FE models using 3-D solid elements
FE models Number of nodes
Number of elements
Number of D.O.F.
Average size of element (mm)
Plate 482403 320000 1447209 0.75×0.75×0.6375
Adhesive layer 208 357 1071 0.58×0.61×0.05
PZT actuator 336 194 1344 0.65×0.67×0.25
Figure 3-11 Poling direction for circular PZT disk
Table 3-3 Material Properties of PZT defined in Abaqus
56
Figure 3-12 Geometry of the finite element models, (a) Perfect bound PZT-Plate model
(Full PZT model), (b) PZT-Plate model with Adhesive layer
3.5.1 Mechanistic output comparison of PZT actuator models
In this section, all the results are listed and classified into three main aspects: 1) effective
displacement; 2) effective strain; and 3) effective stress. These results are demonstrated
by s series of line graphs and tables about relative errors.
As can be seen from the Fig. 3-13, the values of displacement produced from the edge of
each actuator are declining over these voltage ranges. More specifically, at first at range of
-2.5 V to 0 V those values are nearly the same. The strain energy model owns the largest
displacement, whereas the uniform strain model presents the lowest one. The simulation
result from numerical simulation just occurs in the middle of those values, in which the
difference is around from 0.14e-9 m to 0.2e-9 m. Then, the gaps between those values
become significant at rest of the range. Among them, the numerical result is remaining the
middle value between them. The results from pinforce model and the enhanced pin force
57
model are mainly same from -2.5 V to -10 V. After that, the difference between the pin
force model and the enhanced one keeps the value at 1.2e-9 m. Similarly, the effective
piezoelectric actuator model and the Bernoulli-Euler model are the same at the lower
voltage range, and appear a marked difference at range from -7.5 V to -20 V, which is 1e-9
m. In summary, the strain energy model and the uniform strain model are useless on their
effective displacement aspect. Other models perform well at the lower voltage range,
which should not exceed ±2.5 V.
Figure 3-14 illustrates the effective strains for those models. Basically, the general trend
for these results is close to the effective displacement one, which is linear and increasing
with the growing of voltage. The numerical result of the full model using Abaqus/Standard
presents the largest strain, while the uniform strain model remains the lowest one.
Specifically, at the lower voltage (-2.5 V to 0 V), the FEM results and the strain energy
model are around same and remain the higher value. The Bernoulli-Euler model and the
effective piezoelectric model, as the same to the previous graph, are similar and with the
value of 0.6e-7 from the numerical result. The results from others models, including the pin
force model, the enhanced pin force model and the strain energy model, are the lowest,
which is 2.4e-7. At the higher voltage range, those values are dispersive and their
differences are significant. For instance, at the -20 V, the numerical results show the strain
for one point along the actuator’s edge is 2.55e-6, whereas, the strain energy model
shows this value is 2.37e-6. The effective piezoelectric actuator model and the Benroulli-
Euler model are 1.95e-6 and 1.85e-6, respectively. The pin force model and the enhanced
one are roughly the same, which is 1.54e-6 and 1.51e-6, respectively. The uniform strain
model also remains the lowest value, which is 1.30e-6. In summary, the result from the
strain energy model is much close to the results from its numerical results. The effective
piezoelectric actuator model and the Bernoulli-Euler model can be used on determining
58
the effective strain at the lower voltage range for their minor gap with the numerical result.
Other models are not applicable at the strain determination.
Fig.3-15 shows the effective stresses for each model. Generally, all the effective stress is
increasing with the declining of voltage on different effective models. The numerical results
from the full PZT model (shown in Fig. 3-12) and the effective piezoelectric actuator model
are nearly the same and represent a dramatically rise rate with declining of voltage.
Whereas, the other models behave opposite, which is go up slightly. To be more precise,
at the lower voltage range (from 0 V to ±2.5 V), the average difference between those
values are smaller, which is 8540 Pa. Among them, the effective piezoelectric actuator
model owns the lowest gap from the numerical results, which is 953 Pa. The uniform strain
model has 3617 Pa difference between the numerical results, which is second. Others
present a larger difference, which is about 8500 Pa. For the higher voltage range, the
general trend for the effective piezoelectric model is parallel to the numerical results, which
is about 13347 Pa. Other models extract the results far away from the results of the full
model, as shown in Fig. 3-13. In summary, the effective piezoelectric actuator model is the
best choice to determine its effective stress at the lower voltage range.
Fig.3-16 shows the effective displacements for each model with an adhesive layer. The
values of displacement calculated from each model are descending while increasing the
voltage on different effective models. More specifically, the numerical results of full model
remain the lowest value. In contrast, the Bernoulli-Euler model presents the highest
performance. At the lower voltage range, the difference between the effective piezoelectric
actuator model and the numerical results is about 0.58e-10 m, but this value increases to
1.64e-10 m when compared between the Bernoulli-Euler model and its numerical result.
For the higher voltage range, those gaps both grow dramatically and reached to 1.17e-9 m
and 3.29e-9 m at -20 V, respectively. In summary, for determining the effective
displacement for PZT model with adhesive layer, the effective piezoelectric actuator model
59
reflects well on its performance. Hence, it is better to choose this model at the lower
voltage range to determine the effective displacement.
Fig. 3-17 presents the effective strains for each model with an adhesive layer. Basically,
the general trend of results from the effective piezoelectric actuator model is parallel with
its full model results with only a smaller gap, which is 0.37e-7. By contrast, the trend of
results from Bernoulli-Euler model are far away from the numerical results. To be more
exact, at the lower voltage range, the difference between the numerical results and the
effective piezoelectric ones is 0.34e-8. This value is increasing to 2.73e-8, when compared
with the Bernoulli-Euler one. Therefore, for determine the effective strain, the effective
piezoelectric actuator model is still the best choice.
Fig.3-18 illustrates the effective stresses for each model with an adhesive layer. In this
linear graph, the numerical results located the middle position between the results from the
Bernoulli-Euler model and the effective piezoelectric actuator model. More exactly, at the
lower voltage range, the difference between the full model results and the results from
effective piezoelectric actuator model is about 96 Pa while this value becomes to 131 Pa
when compared to the Bernoulli-Euler model. At the higher voltage range, the gaps
between those values are ascending sharply. For example, the difference between the
numerical results and the effective one is 1939 Pa at -20 V. Whereas, compared to the
results from the Bernoulli-Euler model, the difference reaches to 2602 Pa. Therefore, the
effective piezoelectric actuator model is the best to replace the numerical one on
determining the effective stress.
In conclusion, Table 3-4 shows the percentage relative errors for each model. It is clearly
that effective piezoelectric actuator model performs less percentage relative errors with the
numerical results. More specifically, the effective piezoelectric model has about 10.5%
percent relative error on effective displacement aspect, which is same as the error from
Bernoulli-Euler model. The pin force and the enhanced one reflect well on their
60
performance, which are only about 9% and 10%, respectively. While, the strain energy
model is the worst one, in which the error is about 37.3%. Therefore, the effective
piezoelectric model is at middle level among them. According to Table 3-5, the
performance of the effective piezoelectric model is consistently better than other models
on the effective strain aspect. More precisely, unlike the effective displacement aspect, the
strain energy model turns to be the best one, which has only 8% relative error. In contrast,
the uniform strain model becomes the worst one, which is maintaining 49% relative error.
The effective piezoelectric actuator model still remains a better level, which is about 23%.
Hence, the real strain distribution through the plate thickness would be the same as the
strain energy model demonstrated. The further study is needed in future research. Table
3-6 illustrate the percentage relative errors for each model on effective strain aspect. The
effective piezoelectric actuator model was found the best one, which is having 5% relative
error.
Therefore, it can be seen from the tables that the effective piezoelectric actuator model is
substantially better than other models and in high performance to determine the effective
stresses.
61
-20 -15 -10 -5 00.00E+000
1.00E-009
2.00E-009
3.00E-009
4.00E-009
5.00E-009
6.00E-009
7.00E-009
8.00E-009
9.00E-009
Dis
pla
ce
me
nt (m
)
Voltage(V)
The uniform strain model
The pin force model
The enhanced pin force model
The strain energy model
The Bernoulli-Euler model
The effective piezoelectric actuator model
The numerical results from Abaqus
Figure 3-13 Effective displacements for each model
-20 -15 -10 -5 00.0
2.0x10-7
4.0x10-7
6.0x10-7
8.0x10-7
1.0x10-6
1.2x10-6
1.4x10-6
1.6x10-6
1.8x10-6
2.0x10-6
2.2x10-6
2.4x10-6
2.6x10-6
2.8x10-6
3.0x10-6
Str
ain
E1
1
Voltage (V)
The uniform strain model
The pin force model
The enhanced pin force model
The strain energy model
The Bernoulli-Euler model
The effective piezoelectric actuator model
The numerical results from Abaqus
Figure 3-14 Effective strains for each model
62
-20 -15 -10 -5 0-4.0x10
5
-2.0x105
0.0
Str
ess S
11
(P
a)
Voltage (V)
The uniform strain model
The pin force model
The enhanced pin force model
The strain energy model
The Bernoulli-Euler model
The effective piezoelectric actuator model
The numerical results from Abaqus
Figure 3-15 Effective stresses for each model
-20 -15 -10 -5 00.00E+000
1.00E-009
2.00E-009
3.00E-009
4.00E-009
5.00E-009
6.00E-009
7.00E-009
Dis
pla
ce
me
nt (m
)
Voltage (m)
The Bernoulli-Euler model with an adhesive layer
The effective piezoelectric actuator model an adhesive layer
The numerical results from Abaqus
Figure 3-16 Effective displacement for each model with an adhesive layer
63
-20 -15 -10 -5 00.0
5.0x10-7
1.0x10-6
1.5x10-6
2.0x10-6
Str
ain
E1
1
Voltage (V)
The Bernoulli-Euler model with an adhesive layer
The effective piezoelectric actuator model an adhesive layer
The numerical results from Abaqus
Figure 3-17 Effective strain for each model with an adhesive layer
-20 -15 -10 -5 0
-1.4x104
-1.2x104
-1.0x104
-8.0x103
-6.0x103
-4.0x103
-2.0x103
0.0
2.0x103
4.0x103
Str
ess S
11
(P
a)
Voltage (V)
The Bernoulli-Euler model with an adhesive layer
The effective piezoelectric actuator model an adhesive layer
The numerical results from Abaqus
Figure 3-18 Effective stress for each model with an adhesive layer
64
Table 3-4 The percentage relative errors for each model on effective displacement aspect
Items Voltage -20 -15 -10 -3 -1
Displacement
USM 4.48E-09 (23.94%)
3.36E-09 (23.98%)
2.24E-09 (23.81%)
6.72E-10 (23.9%)
2.24E-10 (23.81%)
PFM 5.32E-09 (9.68%)
3.99E-09 (9.73%)
2.66E-09 (9.52%)
7.98E-10 (9.63%)
2.66E-10 (9.52%)
EPFM 5.23E-09 (11.21%)
3.93E-09 (11.09%)
2.62E-09 (10.88%)
7.85E-10 (11.10%)
2.62E-10 (10.88%)
SEM 8.10E-09 (37.52%)
6.07E-09 (37.33%)
4.05E-09 (37.76%)
1.21E-09 (37.03%)
4.05E-10 (37.76%)
BEM 6.51E-09 (10.53%)
4.88E-09 (10.41%)
3.25E-09 (10.54%)
9.76E-10 (10.53%)
3.25E-10 (10.54%)
EPM 6.73E-09 (10.53%)
5.05E-09 (10.41%)
3.36E-09 (10.54%)
1.01E-09 (10.53%)
3.36E-10 (10.54%)
FEM 5.89E-09 4.42E-09 2.94E-09 8.83E-10 2.94E-10
Table 3-5 The percentage relative errors for each model on effective strain aspect
Items Voltage -20 -15 -10 -3 -1
Strain
USM 1.23E-06 (49.02%)
9.74E-07 (49.01%)
6.49E-07 (49.30%)
1.95E-07 (49.09%)
6.49E-08 (49.30%)
PFM 1.46E-06 (39.61%)
1.16E-06 (39.27%)
7.71E-07 (39.77%)
2.31E-07 (39.69%)
7.71E-08 (39.77%)
EPFM 1.44E-06 (40.39%)
1.14E-06 (40.31%)
7.58E-07 (40.78%)
2.28E-07 (40.47%)
7.58E-08 (40.78%)
SEM 2.23E-06 (7.84%)
1.76E-06 (7.85%)
1.17E-06 (8.59%)
3.52E-07 (8.09%)
1.17E-07 (8.59%)
BEM 1.79E-06 (25.88%)
1.41E-06 (26.18%)
9.43E-07 (26.33%)
2.83E-07 (26.11%)
9.43E-08 (26.33%)
EPM 1.85E-06 (23.53%)
1.46E-06 (23.56%)
9.75E-07 (23.83%)
2.93E-07 (23.50%)
9.75E-08 (23.83%)
FEM 2.42E-06 1.91E-06 1.28E-06 3.83E-07 1.28E-07
65
Table 3-6 the percentage relative errors for each model on effective strain aspect
Items Voltage -20 -15 -10 -3 -1
Stress
USM -307000 (19.21%)
-230000 (19.30%)
-154000 (18.95%)
-46100 (18.98%)
-15400 (18.95%)
PFM -275000 (27.63%)
-206000 (27.72%)
-138000 (27.37%)
-41300 (27.42%)
-13800 (27.37%)
EPFM -277000 (27.11%)
-208000 (27.02%)
-139000 (26.84%)
-41600 (26.89%)
-13900 (26.84%)
SEM -209000 (45.00%)
-157000 (44.91%)
-105000 (44.74%)
-31400 (44.82%)
-10500 (44.74%)
BEM -247000 (35.00%)
-185000 (44.91%)
-124000 (44.74%)
-37100 (44.82%)
-12400 (44.74%)
EPM -360000 (5.26%)
-270000 (5.26%)
-180000 (5.26%)
-54100 (4.92%)
-18000 (5.26%)
FEM -380000 -285000 -190000 -56900 -19000
3.5.2 Adhesive layer effect on mechanistic output of PZT actuator
In order to identify the adhesive layer effects on mechanistic output of PZT actuator, three
validation examples are chosen. In the first validation example, the adhesive layer keeps
the same as the former one, in which the thickness is t 0.1 mm . In the second example,
relatively thin adhesive layer with t 0.09 mm is selected. In the last example, the thickness
is expanded to t 0.12 mm . Those values of thickness are satisfied with the general
thickness for silver epoxy layer (Samir, 2006).
Figs. 3-19, 3-20 and 3-21 demonstrated the effective displacement, stress and strain for
different adhesive layer thicknesses. The black line means these values are calculated by
using Eq. (3.67), while the red line is the results coming from the numerical analysis. The
models built in the Abaqus software are same as the previous ones, including the
boundary conditions.
Fig. 3-19 illustrated the results comparison on effective displacement aspect. It can be
seen from those line graphs that with the increase of the adhesive layer’s thickness, the
difference between the numerical results and analytical results are widening. However,
the effective strain and stress gives opposite phenomenon, which are illustrated in Figs 3-
20 and 3-21. To be more precise, for effective strain, there is a small gap between
66
numerical results and analytical results at a lower thickness condition, which is nearly
about 4.4e-8. While, when it comes to a larger thickness 0.12 t mm , this gap becomes
smaller that is about 3e-8. For the effective stress aspect, as can be seen from Fig.3-21,
the gap between the numerical results and analytical results reduced slightly following with
the increasing of adhesive layer thickness. For instance, the average difference is about
1150 Pa at the thickness 0.09 t mm , this value arrives at 649Pa when the thickness
0.12 t mm .
In summary, the adhesive layer, indeed, exerts significant influence on numerical results
and analytical results. The effective piezoelectric actuator model (EPMA) performs well to
fit with the numerical results, especially on determining the effective strain aspect.
However, this model should also be further improved and developed for the lower
adhesive layer thickness.
-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 00.00E+000
1.00E-009
2.00E-009
3.00E-009
4.00E-009
5.00E-009
Dis
pla
ce
me
nt (m
)
Voltage (V)
EPMA0.9
FEM0.9
(a)
67
-20 -15 -10 -5 00.00E+000
1.00E-009
2.00E-009
3.00E-009
4.00E-009
5.00E-009
Dis
pla
ce
me
nt (m
)
Voltage (V)
EPMA1
FEM1
-20 -15 -10 -5 00.00E+000
1.00E-009
2.00E-009
3.00E-009
4.00E-009
5.00E-009
Dis
pla
ce
me
nt (m
)
Voltage (V)
EPMA1.2
FEM1.2
Figure 3-19 Effective displacement for different adhesive layer thicknesses, (a) 0.09t mm ,
(b) 0.1 t mm ,(c) 0.12 t mm .
(b)
(c)
68
-20 -15 -10 -5 00.0000000
0.0000002
0.0000004
0.0000006
0.0000008
0.0000010
0.0000012
0.0000014S
tra
in E
11
Voltage (V)
EPMA0.9
FEM0.9
-20 -15 -10 -5 00.0000000
0.0000002
0.0000004
0.0000006
0.0000008
0.0000010
0.0000012
0.0000014
Str
ain
E1
1
Voltage (V)
EPMA1
FEM1
(a)
(b)
69
-20 -15 -10 -5 00.0000000
0.0000002
0.0000004
0.0000006
0.0000008
0.0000010
0.0000012
0.0000014
Str
ain
E11
Voltage (V)
EPMA1.2
FEM1.2
Figure 3-20 Effective strain for different adhesive layer thicknesses, (a) 0.09 t mm , (b)
0.1 t mm , (c) 0.12 t mm .
-20 -15 -10 -5 0
-12000
-10000
-8000
-6000
-4000
-2000
0
Str
ess (
Pa)
Voltage (V)
EPMA0.9
FEM0.9
(a)
(c)
70
-20 -15 -10 -5 0
-12000
-10000
-8000
-6000
-4000
-2000
0
Str
ess (
Pa)
Voltage (V)
EPMA1
FEM1
-20 -15 -10 -5 0-12000
-10000
-8000
-6000
-4000
-2000
0
Str
ess (
Pa)
Voltage (V)
EPMA1.2
FEM1.2
Figure 3-21 Effective stress for different adhesive layer thicknesses, (a) 0.09 t mm , (b)
0.1 t mm ,(c) 0.12 t mm .
(b)
(c)
71
3.5.3 Size and shape effect on PZT actuators
In this section, the main purpose is to determine the size and shape effects of PZT
actuators on elastic wave generation. Hence, there were 4 different shapes and sizes
actuators designed. The detailed information is shown in Fig. 3-22. Like previous sections,
the numerical results were obtained and analytical results are used and compared with
each other. For the numerical aspect, the Abaqus was employed to simulate the models.
For the analytical aspect, Eq. (3.55) is applied to calculate the results.
Fig.3-23 shows the effective displacement for different actuators. Basically, the large sizes
of actuator can create more displacement along the edge of actuator. In contrast, the
smaller size of actuators can achieve less displacement. To be more specific, the
analytical results are more closed to the disk actuator with radius of 3.45 mm and square
actuator with length of 9.76 mm. However, this values leave far away from the values that
calculated from smaller square actuator model and larger disk actuator model.
Figs. 3-24 and 3-25 describes the effective strain and stress for different sizes and shapes
of actuators, respectively. As can be seen from those graphs, it is clearly that all results
from analytical analysis are smaller than numerical results. Specifically, the results from
these two kinds of square actuators are nearly same either on effective strain aspect or on
effective stress aspect. However, there is a gap between the results from larger disk
actuator and smaller disk actuator. It shows that the effective strain and stress can be
changed slightly with increasing of dimension of actuators for the square shape actuator.
Besides, these line graphs shows that the results from effective piezoelectric actuator
model (EPM) are more closed to the results from the disk actuator. Then, it is suitable to
use EPM to replace the disk actuator model in this case.
72
Figure 3-22 Geometry of PZT actuators with different sizes
-20 -15 -10 -5 00.00E+000
1.00E-009
2.00E-009
3.00E-009
4.00E-009
5.00E-009
6.00E-009
7.00E-009
8.00E-009
Dis
pla
ce
me
nt (m
)
Voltage (V)
R=3.45mm Disk actuator
R=4.88mm Disk actuator
L=4.88mm Square actuator
L=9.76mm Square actuator
EPM
Figure 3-23 Effective displacements for each model
73
-20 -15 -10 -5 00.0000000
0.0000005
0.0000010
0.0000015
0.0000020
0.0000025
0.0000030
0.0000035
0.0000040
Str
ain
E11
Voltage (V)
R=3.45mm Disk actuator
R=4.88mm Disk actuator
L=4.88mm Square actuator
L=9.76mm Square actuator
EPM
Figure 3-24 Effective strains for each model
-20 -15 -10 -5 0
-400000
-350000
-300000
-250000
-200000
-150000
-100000
-50000
0
Str
ess S
11
(P
a)
Voltage
R=3.45mm Disk actuator
R=4.88mm Disk actuator
L=4.88mm Square actuator
L=9.76mm Square actuator
EPM
Figure 3-25 Effective stress for each model
74
3.6 Summary
In this chapter, six basic piezoelectric effective models were reviewed and developed.
Each model has their disadvantages and advantages. For instance, the uniform strain
models, the pin force model, the enhanced pin force model and the strain energy model
can be only used in a perfect bounded PZT condition. In contrast, the Bernoulli-Euler
model and the effective piezoelectric actuator model can be applied on the model with an
adhesive layer. Hence, these two effective models are much closer to the real situation. In
addition, according to those results listed above, the effective force model performs well
compared with those models. This is because it is average values among those results.
Furthermore, its calculation is convenience for analytical analysis, such as saving
computing time and resources. Hence, it has been selected as another benchmark for the
further simulation.
75
Chapter 4 FE-based numerical modelling of elastic wave propagations in plates
4.1 Introduction
In this chapter, elastic wave propagation in plates is simulated and analysed by using
Abaqus. Two basic solvers, which are Implicit Dynamic Analysis solver (IDA)-
Abaqus/Implicit and Explicit Dynamic Analysis solver (EDA)-Abaqus/Explicit, are
introduced at the beginning of this chapter. However, there is still having several
disadvantages by using them individually. For instance, the implicit dynamic procedure is
regarded not suitable for wave propagation modelling and may lead to inaccurate results.
Then, in order to improve the accuracy of the results, a Combined Implicit-Explicit Dynamic
Analysis (CIEDA), which is using both Abaqus/Implicit and Abaqus/Explicit, is employed.
Those three methods are verified and compared by using a simple PZT-Plate model.
Finally, the detailed development for finite element modeling and the relationship between
the defects and the elastic wave propagation is presented.
4.2 Elastic waves in plates
As mentioned at section 2.3, elastic waves travelling in a free plate can be treated as two
different displacements, which are displacements of the particles though the thickness (y-
direction) and the direction of wave propagation (x-direction). More specifically,
displacements of particles in the x-direction can be regarded as longitudinal waves and
displacement in the y-direction of particles can be considered as the vertical shear waves.
Fig.4-1 illustrates particle motion in transverse and longitudinal waves. The arrows stand
for the direction of wave propagation and the black lines are direction of particle of motion.
76
Figure 4-1 Transverse and longitudinal waves
Wave propagation is relative to the density and elastic properties of a medium. The
longitudinal wave speed is dependent on the Young’s modulus E as follows:
L
EC
(4.1)
Similarly, the transverse (shear) wave speed is characterised by the shear modulus G
as follows:
T
GC
(4.2)
Because of the modulus values for an isotropic material are constant in all directions, the
previous longitudinal and transverse wave speed equations can be replaced with the Lame
constants and . However, when the lateral of a medium, which is parallel to the wave
propagation direction, are much greater than the wavelength, the modulus can be writen
as the plane wave modulus M . The plane wave modulus is equivalent to 2 and
shear modulus is equal to . Then, the new equations for the longitudinal and transvers
wave speeds, along with the plane wave modulus, are listed as follows:
2
=L
MC
(4.3)
2
1-=
1- -2
EM
(4.4)
77
=T
GC
(4.5)
where is poisson’s ratio. The dependence of the Lame constants on E and Poisson’s
ratio, , is expressed as follows:
= =
2 1
EG
(4.6)
=
1-2 1
E
(4.7)
Therefore, the longitudinal and transverse waves are non-dispersive since their wave
speeds are only a function of material properties rather than the frequency.
4.2.1 Elastic waves propagation in free plate
Fig. 4-2 illustrates the geometry of the free plate. For a isotropic plate with traction free
boundary condition, as shown in Fig.4-2, the basic elasticity equations can be expressed
as (Giurgiutiu, 2007) ,
, equations of motion , ,ij j i if u i x y z (4.8)
, ,
1 strain-displacement equations
2ij i j j iu u (4.9)
+2 constitutive equationsij kk ij ij (4.10)
where , , , , , , ij ij i i iju f and denote stresses, strains, displacements, body forces,
the Kronecker date, mass density and Lame’s constants, respectively. Then, after elimiting
the stress and strain in equation of motion by substituting Eq. 4-9, the Navier’s equation of
motion in linear elastodynamics can be obtained as follows:
, ,i jj j ji i iu u f u (4.11)
78
Figure 4-2 Free plate geometry
By using the Helmholtz decomposition, the displacement vector of Eq. 4-11 can be
expressed as a scalar potential and a vector potential that yield two wave equations
for longitudinal and shear waves. For plane strain problems, z is unchanged, the body
force if will be neglected. The results are,
2 2 2
2 2 2 2
1 governing longitudinal waves
Lx z c t
(4.12)
2 2 2
2 2 2 2
1 governing shear waves
Tx z c t
(4.13)
where , L Tc c are longitudinal and shear wave velocities.
Assuming the displacement potential solutions in Eqs. 4-12 and 4-13 in the form
expy i kx t (4.14)
expy i kx t (4.15)
in which , k are wave number and angular frequency, respectively.
According to Eq. 4-12 and 4-13, the equations for unknown functions , can be written
as,
1 2sin cos pyy A py A (4.16)
1 2sin q cos qyy B y B (4.17)
where 2 2 2 2/ Lp c k and 2 2 2 2/ Tq c k .
Then the displacement and stress terms can be rewriten as follows
79
x
du ik
x y dy
(4.18)
y
du ik
y x dy
(4.19)
2y yx
yy L
u uu
x y y
(4.20)
y xyx L
u u
x y
(4.21)
4.3 FEM of elastic waves in plates
The wave propagation problems can be solved by using either implicit or explicit solvers.
These two solvers are parts of the Abaqus package and having their own characteristics
and functions. Generally, Abaqus/Implicit is more efficient for solving nonlinear problems,
whereas Abaqus/Explicit is more suitable to simulate brief transient dynamic problems
(Abaqus, 2011). Although both of them can be used to address the wave propagation
problem, the results from these processes are quite different. Hence, a clear
understanding of their characteristics and functions is helpful to handle a specific problem.
This section presents the numerical implementation in these cases.
4.3.1 Explicit Dynamic Analysis (EDA) method
Explicit dynamic analysis in Abaqus/Explicit is an incremental procedure based on the
central-difference operator and diagonal element mass matrices (Abaqus, 2011). In order
to describe it better, the equation of dynamic equilibrium for finite element model is
presented firstly, which can be expressed as:
M U C U K U F (4.22)
= M KC C M C K (4.23)
where ,U U and U are the acceleration, velocity and displacement respectively. M
means the diagonal lumped mass matrix whose values are determined by the density of
80
the material used. K is the static stiffness matrix whose values are defined by the
Young’s modulus and Poisson’s ratio. C is the viscous damping matrix which is
determined by the Rayleigh damping. MC and KC are the mass and stiffness proportional
damping coefficient, respectively. F is the external force vector.
For the wave propagation, the movement of the wavelet is caused by the application of
displacement or force on each node, which can lead to disruption on the initial equilibrium.
Then, the general wave equation of motion can be expressed by using the explicit central
difference formulation, which is:
N
i
iiN
i
N
i Utt
UU 2
1
2/12/1
(4.24)
N
ii
N
i
N
i UtUU 2/111 (4.25)
where NU is a degree if freedom and subscript ‘i’ refers increment number in the explicit
solution scheme, and t is the time increment (Abaqus, 2011).
From Eqs. (4.24) and (4.25), it is clear that the explicit process is calculated by using the
values from the previous time step, which means the time step t is an important factor to
be selected in here. In addition, in order to make sure one wavelet can be detected within
a time step t , the element size should also be calculated based on different wave speed
when meshing one model. Therefore, three conditions should be satisfied when we use
the explicit dynamic procedure for handling the wave propagation problem.
1. The stable time increment t required for the stability if the central difference operator
is related to the maximum frequency present in system by the inequality.
ft
2 (4.26)
and
81
max
2
max12
f
t (4.27)
where max is the fraction of critical damping in the mode with the highest frequency.
2. The distance travelled by a wave (disturbance) within a time step t should not exceed
the length of the smallest element minL as given by the inequality. If the increment is too big,
the solution could become unstable and diverges rapidly. Then,
c
Lt min (4.28)
where c is the velocity of fastest wave in medium
3. The spatial sampling interval which is equal to the element size should be sufficient to
reconstruct the smallest wavelength of the wave that can exist in the computation domain.
10
minmax
L (4.29)
where is wave length
In conclusion, the efficiency of computation in terms of the geometry and cost of
computation makes the explicit solver more technically appropriate for the wave
propagation problem. However, the explicit dynamic procedure in Abaqus has a big
shortcoming that is lack of piezoelectric elements. Hence, applying displacement or force
on related nodes in contact with piezoelectrics is good choice in Abaqus/Explicit. Therefore,
the effective piezoelectric model, which is introduced in Chapter 3, is helpful to be applied
in here.
4.3.2 Implicit Dynamic Analysis (IDA) method
Unlike the Explicit method, the implicit method can be used to handle either the wave
propagation problems or mechanical-piezoelectric problems. Generally, for wave
propagation method, the Abaqus/Implicit solves this problem in the frequency domain. For
82
the mechanical-piezoelectric problems, the implicit method applies Newton’s method to
solve for static equilibrium Eq. (4.22), which transfers to
uu uM U K U K F (4.30)
uK U K Q (4.31)
where U and U are the acceleration, displacement vector respectively, is the
electric potential vector, K expresses the dielectric stiffness matrix of this piezoelectric
actuator, uK
stands for the piezoelectric coupling matrix, F means the mechanical
force vector, and Q is the electrical charge vector (Abaqus, 2011).
In Abaqus/implicit analysis, an implicit operator, Hilbert-Hughes-Taylor (HHT), is used to
calculate these formulas. Hereby, the operator definition is finished by Newmark formula
for displacement and velocity integration as
2 1
2t t t t t t t
U U tU t U U
(4.32)
1t t t t t tU U t U U
(4.33)
where 21
14
, 1
2 , and
10
3 (4.34)
This operation has controllable numerical damping, which is identified by the parameter .
If 0 , it means there is no damping. If 1
3 , it stand for the damping effect is
significant.
In addition, the time step t should also be limited, which would interrupt the response
prediction for one structure. Hence, there are three factors considered before choosing the
maximum allowable time step. 1) the complexity of the stiffness properties; 2) the type of
structure which will be simulated; and 3) the rate of change of the applied loading.
83
Typically, the maximum time increment versus period ratio should be 1 10t T , which
can achieve reliable results.
In conclusion, the explicit procedure is well-performance on high-speed application,
whereas the implicit procedure is effective when the analysis can be performed in
relatively few time (load) increments. Hence, the explicit procedure is more suitable to
solve time domain wave propagation. However, for the full PZT model, which includes
piezoelectric elements, it cannot be performed using Abaqus/Explicit code. Therefore, the
Abaqus/Stand-Explicit co-simulation can be applied to overcome this deficiency. This
procedure allows implicit solver and explicit solver to obtain the solutions in the full model.
Typically, the output of the piezoelectric actuators will be applied as the input of the
transient dynamic analysis in Abaqus/Explicit. In order to compare these three dynamic
analysis methods better, a simple PZT-Plate model was devised and simulated.
4.4 Analysis on FEM using different dynamic analysis techniques
In this section, the main purpose is to simulate the simple PZT-Plate model using three
different analysis techniques, which are the explicit method, the implicit method and Co-
simulation method, respectively. At the beginning of this section, the detailed information
about this Finite Element (FE) model is illustrated, which is including material properties
and the wave mode selection. In addition, the excitation signal used for simulation is given
as well as the reason for the choice. Based on this input signal, some vital parameters,
such as the element edge length of FE models and the time step, should be calculated.
Then, the detailed development for these calculations is also included here in. Lastly, the
FE modelling processes using different solvers are illustrated.
84
4.4.1 FEA model
In order to verify, compare and demonstrate the difference between the three analysis
techniques, a steel plate was devised to describe the same Lamb wave propagation in
here. A PZT disk actuator (A1) with a diameter of 6.9 mm and a thickness of 0.5 mm
located at the centre on this square flat plate (300 mm×300 mm×1.275 mm), is applied to
excite the lamb wave and (shown in Fig.4-3). In chapter, a steel plate was employed.
Table 4-1 gives the mechanical properties of this steel plate. For the actuator aspect, the
dimension and the material properties are the same as the previous one, which is listed in
Table 3-1. In addition, one sensor point (S1) is located at the right of the actuator. The
distance between the actuator and sensor allows time for wave propagation and changes
in the signal to occur.
Figure 4-3 Schematic of a plate model
Table 4-1 Mechanical properties of steel alloy plate
Thickness d (m)
Young’s modulus E (Gpa)
Poisson’s ratio
Density (kg/m3)
0.001275 210 0.27 7850
85
In Abaqus/CAE, the model was discretised as 3-D brick element (C3D8R) and the PZT
actuator modelled as piezoelectric brick element (C3D8E). This is because the 3D model
can give more information about wave propagation. In addition, in order to save the
computation time and increase the efficiency, the disk sensor is simplified to one point
located on the plate, thus the output is the only the displacement instead of the complex
voltage, which is easy to be analysed after post processing.
4.4.2 Wave mode selection
The next step for modelling is to consider the frequency which will be used. This is
because the guided waves are highly dispersive, containing dependency of phase or
group velocity on wave excitation frequency. Hence, the wave packets of different speed,
even at a unique frequency, can interfere with each other and distort the signal. Therefore,
the better option is to consult the Lamb wave dispersion curve and select a preferable
frequency bandwidth. Fig. 4-4 shows the phase velocity phC and group velocity
gC
dispersion curve for a 1.275 mm thick steel plate, generated by using GUIGUW software
(Marzani et al., 2008).
86
Figure 4-4 Lamb wave dispersion curve for 1.275 mm thick steel plate, (a) phase velocity,
(b) group velocity.
87
According to the Fig. 4-4, a frequency of 200 kHz is selected and used in this simulation.
This is because the guided wave at this frequency locates at a relative non-dispersive
region and has symmetric mode S0, anti-symmetric mode A0. In addition, the group
velocity difference between S0 mode and A0 mode (as shown in Fig.4-4 b) is large enough,
which means those two wave modes may not interfere with each other in this simulation.
4.4.3 Five cycle Hanning-window excitation signal
Once the frequency is determined, it is easier for us to select the excitation signal that
used in this study. A five cycles Hanning window exciting signal was found as the best
option, which has the following advantages: 1) it can limit the frequency bandwidth of the
excitation, thereby reducing the undesired reflections between wave packers; and 2) it can
reduce the energy at a certain frequency other than the excitation or ‘centre’ frequency.
Fig.4-5 provides 5-cycle Hanning window excitation signals at 200 kHz.
0.00000 0.00002 0.00004 0.00006 0.00008 0.00010
-1.0
-0.5
0.0
0.5
1.0
No
rma
lize
d V
Inp
ut
Time (s)
5-cycle 200 kHz
Figure 4-5 5-count 200 kHz Hanning windows signal
88
Generally, the Hanning windows signal comprises of a tone-burst signal and a Hanning
window. To be more exact, the carrier signal is defined as:
c
c
Tt
Ttttg
0
0sin (4.35)
f 2 ,f
T1
, TNT cc (4.36)
where cN is the number of counts of the waves and f is the carrier frequency.
The Hanning window is defined by
c
c
TttN
th
0
1cos15.0 (4.37)
Then, the excitation signal, as shown in Fig. 4-6, is the convolution of carrier signal and
Hanning window. The equation can be expressed as
*F t g t h t (4.38)
Figure 4-6 Modulation of a carrier wave by a time window (Giurgiutiu, 2007)
This excitation signal was created using Matlab, Listed in Appendix A.
4.4.4 Some vital parameters used in explicit dynamic procedure
At the previous sections, the general information about the explicit method has been
provided, especially three conditions should be satisfied before modelling. In this study,
these three conditions are described, analysed and calculated as follows,
Determining maximal element size ( maxL ) of FE models
89
According to spatial sampling criteria, the maximal element size maxL should be small
enough, thereby allowing the smallest wavelength of the wave can exist in the computation
domain. To determine the maximal element size, the following procedure should be
considered.
In the first place, the transverse wave speed TC should be calculated.
= = =
2 1+T
G EC
(4.39)
where G is the shear modulus, is the Lame constants , is the density, E is the Young’s
modulus and is Poisson’s ratio.
According to Table 4-1, the transverse wave speed can be calculated as
9 2
2
210 10 /= = 3245.32 /
2 1+ 2 1 0.27 7850 /T
E N mC m s
kg m
The simulation has been performed at the largest frequency of 2 MHz, which is discussed
in Section 2.3. Then, applying this maximum frequency and the smallest wavelength
min is calculated:
6 6
max 2 2 10 2 10 /f MHz Hz cycles s
min 6
3245.32 /0.0016 1.6
2 10 /
TC m sm mm
f cycles s
(4.40)
From the above results, the maximal element size maxL is calculated using equation
(4.29).
minmax
1.6 mm0.16
10 10L mm
For this simulation, the element size we selected is 0.09 mm, which is smaller than the
criterion of maximal element size maxL .
Hence, this value can be used in the Mesh Module in Abaqus/CAE.
90
Determining time step t
Because of the explicit process is using known values from the previous time step, the
time step t is an important factor for the accuracy of the solution. Hence, it should be
calculated before creating the analysis step in the modelling process. Generally, the
accuracy of the model can be enhanced with increasingly smaller integration time step. It
means if the time step t is too large, the high frequency components could not be
resolved accurately. In contrast, if smaller time step t is used, the more calculation time
could be wasted. Therefore, the suitable time should be found by the following processes
In order to determine the time step t , the Lame’s constant , should be calculated
firstly.
According to Eq. (4.39), the value of can be calculated as
92 9 2210 10
/ 82.68 10 /2 1 2 1 0.27
EN m N m
For another Lame’s constant value , it can be calculated as
92 9 2210 10 0.27
/ 97.06 10 /1 1 2 1 0.27 1 2 0.27
EN m N m
According to results from the Lame constants above, the longitudinal wave speed LC
can be determined as
2LC
(4.41)
Then, the longitudinal wave speed can be calculated based on Table 4-1, which is
9 2 9 2
2
97.06 10 / 2 86.68 10 /25869.28 /
7850 /L
N m N mC m s
kg m
Applying the previous results and Eq (4.28), the time step t can be calculated as
91
min 0.09 15.33
5869.28 /
L mmt ns
c m s
This value can be applied in the Step Module in Abaqus /CAE. The important parameters
used in Abaqus/Explicit are listed in Table 4-2.
Table 4-2 The limitation of some vital parameters used in Abaqus/Explicit
the maximal element size maxL 0.16 mm
the time step t 15.33 ns
4.4.5 Modelling of effective PZT-Plate model
According to section 4.2, the piezoelectric materials cannot be employed using the
Abaqus/Explicit code. Hence, only the related displacements generated by piezoelectrics
can be applied instead of the voltage. According to Chapter 3, one of the most suitable
substitutions for piezoelectric elements is the effective piezoelectric actuator model (EPM).
Hence, this explicit model is the simplest one and the flow tree of this model is illustrated in
Fig. 4-7.
Figure 4-7 The flow tree of Abaqus Explicit model
For the modelling aspect, the dimension of the steel plate is listed in Fig. 4-3 and Table 4-1
shows its material properties. The plate is meshed with 111,719 C3D8R solid elements to
ensure the element size is smaller enough than the maximal element size maxL . After
that, the step is selected as Dynamic, Explicit and the time step is the 10 ns 10 t ns .
According to Table 4-2, this value meets its criterion and this simulation can obtain more
accurate results. For the boundary condition aspect, applying effective displacements on
related nodes in contact with piezoelectric actuator is suitable in Abaqus/Explicit. Then, 16
effective displacements are symmetrically distributed around the actuator as shown in
92
Fig.4-8. The value of effective displacement is determined based on Eq. (3.56). Table 4-3
shows magnitudes of effective displacement based on different voltages. As mentioned at
chapter 3, the effective piezoelectric actuator model has a better performance at a lower
voltage range. Therefore, when voltage is at 2V , the effective displacement
106.73 10d mm is used in here. A series of output of displacement fields for this plate
are shown in Fig. 4-9 for illustrating wave propagation. It is evidence that three kinds of
wave modes are propagating in the plate, there are anti-symmetric mode A0, symmetric
mode S0 and shear horizontal wave mode SH0, respectively.
Figure 4-8 The effective displacements applied in the modes around the disk actuator
Table 4-3 Magnitudes of effective displacement at different voltages conditions
Voltage (V) Effective displacement (m)
93
0 0
1 3.36E-10
2 6.73E-10
3 1.01E-09
4 1.35E-09
5 1.68E-09
Figure 4-9 Lamb wave propagation and scattering at different time instants
4.4.6 Modelling of full PZT-Plate model
In this section, a full PZT-Plate model are used and analysed in Abqus/Standard. Fig.4-10
shows the flow tree of Abaqus Implicit model. In this way, there are two sub-model of
94
piezoelectric actuator and steel plate are defined as parts and assembled together as one
whole model, which is to execute a single job with the Abaqus implicit code. The material
properties for this steel and piezoelectric actuator are given in Tables 4-1 and 3-1,
respectively. All the dimensions are similar as the former one.
Figure 4-10 The flow tree of Abaqus/Implicit model
Fig.4-11 illustrates the finite element model of the PZT that is bounded to the host
structure. Generally, The PZT was meshed with 194 C3D8E solid elements and the Plate
was meshed with 321,194 C3D8 solid ones. These mesh sizes can ensure at least 10
nodes per wavelength. For this boundary condition aspect, an input of 2 V is applied and
this electrical excitation signal given to the PZT is Hanning windows at 200 kHz. The
poling direction for this PZT disk is given in Fig. 3-13.
Figure 4-11 Finite element model of the PZT bounded to the plate
Responses of the plate for the input signal for different time instants are shown in Fig. 4-12.
The responses for the two time instants show that three wave modes (A0, S0 and SH0)
95
are propagating along this surface. Compared with Fig.4-9 extracted from Abaqus/Explicit
model, the Implicit one cannot demonstrate well on the wave modes dispersion, which
contains some noises between the S0 waves and SH0 waves. Hence, the implicit dynamic
produce, just as Giurgiutiu (2005) said, is not suitable for wave propagation modelling and
can cause some inaccurate results.
Figure 4-12 Lamb wave propagation and scattering at different time instants
96
4.4.7 Modelling of co-simulation model
Fig. 4-13 demonstrates the flow tree of Abaqus co-simulation model, which can allow
different solvers to calcualate in the piezoelectric actuator and the host structure. As
shown in Fig. 4-13, the output of piezoelectric analysis will be applied as an input of the
transient dynamic analysis in Abaqus/Explicit. An interaction interface is necessary to
manage the data exchange in every time increment between two solvers, which is passing
the data of force, displacement, strain and stress each other. In addition, these two
analysis jobs are submitted together by creating co-execution. Therefore, this method can
perform well either on wave propagation problems or mechanical-piezoelectric problem.
Figure 4-13 The flow tree of Abaqus co-simulation model
The detailed information about these two kinds of models on this FEM simulation is
illustrated in Fig.4-14. To be more precise, a steel plate is defined as the parent model and
the PZT actuator is defined as a child model. All the material properties and dimensions
are similar as the previous one, which are given in Tables 4-1 and 3-1, respectively. For
the child model, the actuator was meshed with 172 C3D8E solid elements and applied a 2
V 5-cycle Hanning window tone-burst signal with 200 kHz on the both sides of
piezoelectric elements. Then, the induced strain created from the actuator was exchanged
by a co-simulation interface. For the parent model, the steel plate was meshed with
321,194 solid elements and simulated under the Abaqus/Explicit code. This Explicit
procedure was applied to simulate the wave propagation in this plate, when the induced
strain was transferred from the interaction surface.
97
Figure 4-14 Settings of the co-simulation model
Fig.4-15 shows a series of out of displacement fields for this plate. As it can be seen from
these graphs, the shapes of wave models are clearly than the Abaqus Implicit one.
98
Figure 4-15 Lamb wave propagation and scattering at different time instants
4.4.8 Comparison among EDA, IDA and CIEDA
The focus of this section is to check the effectiveness of the three different analysis
methods, which are Explicit Dynamic Analysis method (EDA), Implicit Dynamic Analysis
method (IDA) and Combined Implicit-Explicit Dynamic Analysis method (CIEDA). Fig.4-16
illustrates the comparison of time signals at point S1 for these three models with
displacement loading.
The most significant difference found in Fig.4-16 is that the results from Abaqus/Implicit
code have a short time delay. Two reasonable explanations can be found from literature
and the analysis of the results. In the first place, Giurgiutiu (2007) suggested that the
implicit method is useless to simulate the wave propagation problems. This is because of
the non-stability of wave signal for 3D wave generation by using implicit solver due to
considering the more displacement boundary conditions during the analysis processes.
Secondly, the influence of piezoelectric coefficients on time signals is also significant. It
may spend a short time duration on transferring the voltage to induced strain for the
piezoelectrics.
99
Secondly, the magnitudes of the explicit results were found the smallest ones. This is
because the effective displacement, which we calculated in Chapter 3, is smaller than the
real displacement. This phenomenon has been found in many researches, in which they
replaced them with the normalized displacements. Hence, the effective piezoelectric
actuator model (EPM) should also be enhanced and improved in the further study.
However, the general trend of implicit results is similar with the co-simulation one, which
can be used to illustrate the characteristics of wave propagation in a plate.
0.00000 0.00002 0.00004 0.00006 0.00008 0.00010
-8.00E-011
-6.00E-011
-4.00E-011
-2.00E-011
0.00E+000
2.00E-011
4.00E-011
6.00E-011
8.00E-011
1.00E-010
Dis
pla
ce
me
nt (m
)
Time (s)
Implicit
Explicit
Co-simulation
Figure 4-16 Comparison of time signals of displacement magnitude on z-direction at
sensor point S1
In conclusion, the co-simulation method is more suitable to demonstrate and simulate the
wave propagation in a plate. However, it still has several shortcomings. For instance, it will
expand to fill all the available memory during the analysis processing and is unstable when
calculating some larger models. Therefore, for some simple model, it is better to apply the
co-simulation method to achieve more precise results. However, for the more complex
100
models, the explicit method is more useful and powerful, which is faster and more stable
way to acquire the results.
4.5 Case study
In this section, the main purpose is to use the co-simulate method to determine the wave
propagation in steel plates with different kinds of defects. Generally, there are three kinds
of defects are considered in here: 1) series of a small linear crack, 2) a crack with an angle
of 120o relative to the x-axis; and 3) a circumferential defect.
Fig. 4-17 shows a setup for the plate model and actuator model. A three-dimensional plate
model, which contains eight-node brick solid element, was modelled as 300 mm×300 mm.
The PZT actuator is located at the centre of the plate and charged with a voltage of 2 V.
Besides, there are three other sensor points, as shown in Fig 4-17, was employed to
receive the signal. In order to simply this analysis, the output of these sensors is
displacement rather than relative voltage. Table 4-4 gives the plate and PZT dimensions
and properties. The cracks are introduced with central position of quarter of this plate, in
which the centre point is located at (x = 35.5 mm y = 37.5 mm).
According to Section 4.3, the element size selected in this study is 0.09 mm and the time
step is we applied is 10 ns 10 t ns . The simulated Lamb wave signal is the same as
the previous models, which is 5-cycle Hanning window signal with 200 kHz. Co-simulation
solve is used in here.
101
Figure 4-17 Setup of a plate model with a linear crack
Table 4-4 Dimensions and properties of the plate and PZT
Parameter Value Units
Plate (Steel) Length 300 mm Width 300 mm
Thickness 1.275 mm
Density 7850 3/ mmg
Young’s Modulus 210 GPa
Poisson’s Ratio 0.27
Liner cracks
Length 37.5 mm width 0.7 mm
depth 0.6375 mm 1.275 mm
Angle with x-axis 120 degree
Circumferential defect
Radius 1.5 mm
Actuator (PZT-4)
Radius 3.45 mm Thickness 0.5 mm
Density 7500 2/ mmg
Elastic constant 11E 81.3 GPa
Elastic constant 33E 64.5 GPa
Poisson’s ratio 0.33
Elastic constant 11S 1.23E-2 1/ GPa
Elastic constant 33S 1.55E-2 1/ GPa
Charge constant 31d -1.23E-10 /m V
102
Charge constant 33d 2.89E-10 /m V
Charge constant 15d 4.96E-10 /m V
Relative dielectric constant
1 1475
Relative dielectric constant
3 1300
Dielectric permittivity 0 8.854E-12 / mF
For the crack detection, we used two kinds of linear crack, which are with 0.6375 mm
depth and 1.275 mm depth, respectively. Fig. 4-18 demonstrates the contour of
displacement magnitude of Lamb waves scattered by a crack 1.275 mm in thickness. It is
clearly that the amplitude of refection waves appeared after the guided wave passing
through the crack. However, these reflective waves can be interfered by the S0 wave,
which are created by the PZT actuator.
Figure 4-18 Displacement magnitude of Lamb wave propagation
The acquired displacement responses at S1 without and with cracks 0.6375 mm, 1.275
mm in x-direction listed in Fig.4-19. As can be seen from this graph, the wave modes can
be clearly identified based on their group velocity, as shown in Fig. 4-4. It is clear that the
wave mode at the range of 0.00001 s to 0.000036 s is the S0 wave model, in which the
103
group velocity can be calculated as6
3
5
66.3 10 m5412.24 m/s
1.958 10 7.33 10 sg
LC
t
.
Besides, the group wave velocity at the range of 0.0005 s to 0.0006 s can also be
calculated6
3
5
66.3 10 m1898.63 m/s
5.245 10 1.753 10 sg
LC
t
, which is the A0 mode.
Besides, since the Figs. 4-9, 4-21, 4-23 are poled in the x-direction, it cannot clearly
indicates the SH wave modes.
Fig. 4-19 also shows the differences between these kinds of linear crack models.
Generally, the differences between the results from the cracks models and benchmark
model can be clearly identified in the region of 0.000036 s to 0.000042 s. Hence, this can
be used to determine whether there are linear cracks in a steel plate. Besides, for the
crack of 0.6375 mm, a small time delay appeared in this region, this is because the other
wave models influences, which is reflected from the boundary.
0.00000 0.00002 0.00004 0.00006 0.00008
-4.00E-011
-3.00E-011
-2.00E-011
-1.00E-011
0.00E+000
1.00E-011
2.00E-011
3.00E-011
4.00E-011
Dis
pla
ce
me
nt (m
)
Time (s)
Crack=0.6375 mm
Benckmark
Crack=1.275 mm
Figure 4-19 Acquired signals received at S1 in the x-direction
Fig. 4-20 demonstrates the contour of displacement magnitude of Lamb waves scattered
by a crack of 1.275 mm with an angle of 120o relative to the x-axis. It is found that the
S0 A0
104
amplitude of refection waves appeared after the guided wave passing through the crack.
This magnitude of this reflection is larger than the previous one. Then, this kind of defect
can be determined by sensor S1.
Figure 4-20 Displacement magnitude of Lamb wave propagation
As can be seen from Fig. 4-21, it is clearly found that the magnitudes of displacement
have changed at the range of 0.00003 s to 0.00005 s. This is because that the wave
signals have been reflected back from the boundary of crack. Therefore, the reflection
waves have the same frequency as the input signals. In addition, the maximum magnitude
of this reflection wave is higher than the previous one which could be due to the boundary
effects.
105
0.00000 0.00002 0.00004 0.00006-5.00E-011
-4.00E-011
-3.00E-011
-2.00E-011
-1.00E-011
0.00E+000
1.00E-011
2.00E-011
3.00E-011
4.00E-011
5.00E-011
6.00E-011
Dis
pla
ce
me
nt (m
)
Time (s)
Benchmark
Crack of 1.275mm with 120o
Figure 4-21 Acquired signals received at S1 in the x-direction
Fig. 4-22 demonstrates the contour of displacement magnitude of Lamb waves scattered
by hole defect. It is found that the amplitude of refection waves appeared after the guided
wave passing through the crack. This magnitude of this reflection is the largest one. Then,
this kind of defect can be determined by the magnitude from the sensor point S1.
Figure 4-22 Displacement magnitude of Lamb wave propagation
S0
A0
106
Fig.4-23 demonstrates the acquired signals received at S1. It is clearly can be seen that at
the range of 0.000037 s to 0000048 s, the magnitude of reflection is almost equal to the
input signal. Hence, it can be regarded as identification for the steel structures with a
circumferential defect.
0.00000 0.00002 0.00004 0.00006-5.00E-011
-4.00E-011
-3.00E-011
-2.00E-011
-1.00E-011
0.00E+000
1.00E-011
2.00E-011
3.00E-011
4.00E-011
5.00E-011
Dis
pla
ce
me
nt (m
)
Time (s)
Hole defect with R=3.45mm
Benchmark
Figure 4-23 Acquired signals received at S1 in the x-direction
S0 A0
107
4.6 Summary
In the beginning of this chapter, the basic equations for elastic wave propagation in plates
are described. Using the phase velocity dispersion curves, the wave structure distribution
can be useful for selecting existing excitation signal and its frequency. Besides, three
different dynamic analysis techniques, which are the explicit dynamic analysis (EDA)
method, the implicit dynamic analysis (IDA) method and combined implicit-explicit dynamic
analysis (CIEDA) method, are developed, compared and analysed, respectively. This
section also includes some detailed information about this finite element (FE) and the
excitation signal selection for simulation. Furthermore, in order to compare those three FE
methods, one simple PZT-Plate model has been used. All the results show that the co-
simulation method is more reliable and efficiency. Lastly, the wave propagation in steel
plate with different kinds of defects is determined by using co-simulation method. The
results show the S0 wave model can easy detect the defects on the plate surfaces and the
maximum values demonstrated different types of defects.
108
Chapter 5 FE-based numerical modelling of elastic wave propagations in pipes
5.1 Introduction
This chapter focuses on the finite element modelling and simulation of the elastic wave
propagations in steel pipes. The general formulas for guided wave propagation in pipes
will be developed and given in the first place. Hereby, the dispersion curves for a typical
steel pipe of research interest are achieved, which can demonstrate a reliable frequency
range and wave velocities (speeds). Based on the information, the characteristics of three-
dimensional elastic wave field can be determined and wave mode selection can be
conducted for FE-based numerical simulation. In literature, some researchers have
contributed to the simulations of guided wave propagations in pipes using either
Abaqus/Explicit or Abaqus/Implicit to conduct explicit dynamic analysis and implicit
dynamic analysis, respectively. In this chapter, except for using both explicit and implicit
dynamic analysis techniques, a new dynamic analysis technique – co-simulation method is
developed combining the explicit and implicit dynamic analysis processes. Three dynamic
analysis techniques are compared through several case studies to validate them one
another. Then considering a large steel pipe from reality as a special case, we employ the
explicit method, with applications of equivalent PZT models developed in Chapter 3, to
determine the relationship between the defects and the elastic wave propagation in pipes.
5.2 Elastic waves in pipes
Elastic wave-based structural health monitoring (SHM), which is a process of
implementing an in-situ damage identification and detection, has been developed several
decades and it brings significant influences on non-destructive defect detection. Many
researchers devoted to study it for the guided wave can propagation for a long distance
and with a higher speed in one structure, such as in pipes, thereby detecting the whole
thickness of the wall. However, there are still several technical problems on developing
109
such an SHM system, i.e., wave mode selection and wave propagation in complex
structures.
Gazis (1958) summarised the basic knowledge on the propagation of free harmonic waves
along a hollow circular cylinder and given the equation of motion for an isotropic elastic
wave propagation in solid media, which is:
2 2 2/u u u t (5.1)
where u
is the displacement vector, is the density, and are Lame’s constants,
and is the three-dimensional Laplace operator, respectively. For an anisotropic cylinder
hollow pipe with outer diameter a and inner diameter b, as illustrated in Fig.5-1, the root of
the Eq. (5.1) can be assumed as follows:
Figure 5-1 Reference coordinates and characteristic dimensions of the pipe
cos cosr ru U r n t z (5.2)
sin cosu U r n t z (5.3)
cos sinz zu U r n t z (5.4)
110
where , , r zu u u are radical displacement, circumferential displacement and axial
displacement, respectively. , , r zU r U r U r are amplitudes formed by Bessel
function respectively.
Considering the Helmholtz equation, the vector u
can be divided into dilation scalar
potential and the equivolume vector potential H
, shown as
u H (5.5)
,H F r t (5.6)
In order to satisfy the displacement equations of motion, the potential and the
equivolume vector potential H
must have
2 2 2/pC t (5.7)
2 2 2/sC H H t (5.8)
where /22 pC and /2 sC are the pressure and shear wave speeds,
respectively.
In Eqs. (5.2), (5.3) and (5.4), the wave motion with wave number along z axis was
assumed. According to the stress-strain relation and displacement potentials, each stress
components can be written as
2 2 3
3 12 cos cos
rr
gnf f g g n t z
r r
(5.9)
2
3 3 1 1
2 12 sin cosr
n f nf g g g g n t z
r r r
(5.10)
2 2
1 1 3
1cos sinrz
n n nz f g g g n t z
r r r
(5.11)
where 2222 / pC , 2222 / SC are given by Laplace operator 2 . (5.12)
111
Substitution of Eqs (5.9), (5.10) and (5.11) to (5.7) and (5.8) yields the characteristic
equation, which we can obtain that
11 12 16
21 22 26
61 62 66
0
C C C
C C CD
C C C
(5.13)
Thus, we get a linear system of six homogeneous equations with six unknowns. For
nontrivial solution, the system determinant must vanish, i.e.,
0ijC 6,5,4,3,2,1, ji (5.14)
The coefficients ijC in Eq. (5.13) have complicated algebraic expressions that are not
reproduced here for sake of brevity. Hence, when the wave number 0 , Eq. (5.13)
breaks into product of sub determinants
1 2 0D D (5.15)
where
11 13 14 16
21 23 24 26
1
41 43 44 46
51 53 54 56
C C C C
C C C CD
C C C C
C C C C
and
6562
3532
2CC
CCD (5.16)
Hence, either 1D or 2D is equal to zero. As shown by Gazis (1959), the case of 1 0D
corresponds to plane-strain vibrations L (0, m). Whereas, the case of 2 0D corresponds
to longitudinal shear vibrations T (0, m).
5.3 Dispersion characteristics of steel pipes
The dispersion characteristics of steel pipes can be achieved based on Eqs. (5.13) and
(5.14). In this study we applied a free software GUIGUW (Marzani et al., 2008) to obtain
dispersion curves of interest. Fig. 5-2 plots the (a) group velocity and (b) phase velocity
dispersion curves for a 3-m-long steel pipe with an inner diameter of 28 mm and a
112
thickness of 1 mm. the dispersion curves shows elastic waves propagating in pipes are
highly dispersive, containing different speeds at each excitation frequency. Based on these
curves, we can identify a suitable wave mode and frequency for appropriate guided waves
for the development of the SHM system for steel pipes. As it can be seen from Figs. 5-2 (a)
and (b), there is only one longitudinal wave mode, L(0,1), at the frequencies ranging from
15 to 40 kHz, which means this wave mode at those frequencies locates at a non-
dispersive operation region. Therefore, the longitudinal L(0,1) wave mode at 20 kHz was
chosen in the finite element modelling and simulation of the steel pipe.
113
Figure 5-2 Dispersion curves for the steel pipe of a 28 m diameter and a wall thickness of
1 mm: (a) group velocity and (b) phase velocity.
5.4 Finite element modelling of guided waves in pipes
In this research, three series of finite element models were developed with the
employment of three dynamic analysis techniques: a) Explicit Dynamic Analysis (EDA); b)
Implicit Dynamic Analysis (IDA) and c) Combined Explicit-Implicit Dynamic Analysis
(CEIDA). Considering the PZT actuators two models developed in Chapter 3 were both
used, which are the full PZT model and effective PZT model, respectively. For the full PZT-
based FE model, it is an assembly containing a part of the steel pipe and several parts of
PZT actuators, as shown in Fig. 5-3. For the effective PZT-based FE model, it only has a
part of the steel pipe loaded with effective forces applied in the location of these actuators.
Two steel pipes were selected based on the work done by other researchers (Alleyne and
Cawley,1997). One is 3 m long and with an outer diameter of 30 mm and a wall thickness
of 1 mm and the other is 1.5 m long and has an outer diameter of 646 mm and a wall
thickness of 6.4 mm.
114
Three PZT actuators/sensors were chosen: a) line PZT; b) circular PZT; and c) square
PZT. The main dimensions of the line PZT patch are 10 mm in length, 1 mm in width and 1
mm in thickness as indicated in Table 5.1. The main parameters of the circular and square
ones can be found in Table 3.1 and Fig. 3.22 in Chapter 3. Several PZT actuator/sensor
network were designed to generate and collect signals for damage detection. Basically, a
set of PZT patches is designed to locate one end of the pipe while the other set of PZTs is
located at its other end. For conveniences, these two sets are bonded to the pipe
symmetrically for taking both signal excitation and collection roles as actuators and
sensors if need.
In the developed finite element models, the pipe can be discretised using solid elements or
shell elements. In this study we mainly used solid elements to model the pipe for better
interpreting the wave propagations. Considering the convergence and stabilisation of the
dynamic simulation appropriate mesh sizes have been chosen and employed.
For boundary conditions applied in the models, in the full PZT-based FE models, only a
charge varying with time applied in the PZT patch which acts as the actuator and in the
effective PZT-based FE models, the effective forces or displacements are applied in the
location of the actuators.
Table 5-1 Dimensions and properties of the plate and PZT, respectively
Parameter Value Units
Pipeline (Steel)
Length 300 mm Inner diameter 28 mm
Thickness 1 mm Density 7850 2/ mmg
Young’s Modulus 210000 MPa Poisson’s Ratio 0.27
Adhesive Layer (E
silver epoxy)
Thickness 0.1 mm Density 1360 2/ mmg
Young’s Modulus 7629.9 MPa Poisson’s Ratio 0.266
115
Actuator (PZT-4)
Length 10 mm Width 1 mm
Thickness 1 mm Density 7500 2/ mmg
Elastic constant 11E
81.3 GPa
Elastic constant 33E
64.5 GPa
Poisson’s ratio 0.33
Elastic constant 11S
1.23E-2 1/ GPa
Elastic constant 33S
1.55E-2 1/ GPa
Charge constant 31d
-1.23E-10 /m V
Charge constant 33d
2.89E-10 /m V
Charge constant 15d
4.96E-10 /m V
Relative dielectric constant 1 1475
Relative dielectric constant 3 1300
Dielectric permittivity 0 8.854E-12 / mF
5.4.1. FE modelling using implicit dynamic analysis and co-simulation analysis
As shown in Figure 5.3, the finite element model has two parts: a) Pipe part; and b) PZT
part which can be one or several up to the sensor network used. The sensor is not
necessary to be modelled considering the complexity of the pre-processing and post
processing as well as more CPU time will be spent, not economical. Therefore the sensor
point is use, i.e., for sensor 1, S1 is located on the surface of the pipe with a distance of
500 mm to its left end. While for the line PZT actuators, they are modelled with the exact
size of 10 mm in length and 1 mm in width with the material of PZT-4, which are
symmetrically installed along the circumference with a distance of 500 mm to the right end
of the pipe. In reality, these PZT actuators are bonded on the pipes using a thin silver
epoxy layer (about 0.1 mm thick) and thus this glue layer was also modelled as an
additional part to the PZT strips and patches. Moreover, the direction for polarisation of
PZTs is normal to the pipe surface. Therefore, once an electrical input is applied, the
ceramic patch or trip can expand and cause the actuator to flex on the surface of the pipe.
116
For the line PZT actuators, only the expected longitudinal mode rather than the flexural
modes can be excited when applying such long PZT strips since in their width direction the
generated displacement and force will be very weak and negligible (Alleyne and
Cawley,1997) and thus some unwanted wave modes can be eliminated.
Figure 5-3 Schematic of a pipeline model
For convergence and stabilisation of the dynamic simulation, the main control parameters
for the dynamic simulation should also be chosen carefully and based on the discussion
conducted in Section 4.4.4, the criteria to choose the control parameters of dynamic
simulations are listed in Table 5-2.
Table 5-2 The method of choosing parameters of FEM model
Parameters Formula Description
Total Length of the pipeline (L)
min gL V T
gV is the group velocity, it means the
minimum group velocity can travel along in the pipeline at a certain frequency.
Frequency f - It has been determined at the beginning of this section, which is based on the dispersion curve
117
graphs.
The maximal element size
maxL min
max 10
L
min is the smallest wavelength. maxL
should be small enough, thereby allowing the smallest wavelength of the wave can exist in the computation domain.
The time step t min
min g
Lt
V
It can assure a good precision in calculating and analysis of structures.
The total time T min g
LT
V
This condition is to ensure the sensor can receive signal at least
for one period.
According to Table 5-2, main parameters for the steel pipe of 3 m length were calculated in
Table 5-3.
Table 5-3 Main parameters in dynamic simulations the steel pipe of 3 m length
Parameters Values
maxL 0.005 m
t 8 8 sE
T 0.002 s f 20 kHz
In order to efficiently identify the wave propagations in the pipe and check the
effectiveness of the actuator and sensors, the sensor networks were designed to have the
following three arrangements:
(1) Use the four actuators and four sensors
(2) Use the six actuators and six sensors
(3) Use the eight actuators and eight sensors
5.4.2 FEM using implicit dynamic analysis
In this section, a full PZT-Pipeline model is employed and simulated by using Abaqus/
Implicit method. Fig. 5-4 illustrates the flow tree of Abaqus implicit model, which is divided
118
into three sub-sections. There are piezoelectric actuators, adhesive layer and a steel
pipeline, respectively. All the material property and their dimensions are given in Table 5-1.
Figure 5-4 The flow tree of Abaqus Implicit model
The finite element models of 4 PZT actuators, 6 actuators and 8 PZT actuators are given
in Figs. 5-5, 5-6 and 5-7, respectively. Generally, each PZT actuator is meshed with 8
elements, which is including 4 C3D8E solid elements (PZT) and 4 C3D8 solid elements
(adhesive layer). The pipeline is meshed with 18048 C3D8R solid elements. All those
values are satisfied with the limitation which is listed in Table 5-3. For the boundary
condition aspect, an input of 2 V is applied on the surfaces of PZT actuators and 3-count
20 kHz Hanning windows is used as an electrical excitation signal in this simulation. The
poling direction for these PZT actuators is illustrated in Fig. 5-3. Moreover, Point S1 is a
sensor point, which is located on the pipe’s surface, as shown in Figs. 5-5, 5-6 and 5-7,
with a distance of 495 mm to the left end.
119
Figure 5-5 Finite element model of four PZT actuators bounded to the pipe
Figure 5-6 Finite element model of six PZT actuators bounded to the pipe
120
Figure 5-7 Finite element model of eight PZT actuators bounded to the pipe
5.4.3 FEM using explicit dynamic analysis
Fig. 5-8 demonstrates the flow tree of Abaqus explicit model. As mentioned at section
4.4.5, the mechanical-piezoelectric elements cannot be performed by using the Abaqus
explicit code. Therefore, the related force caused by piezoelectrics can be employed to
instead of the input voltage. According to Chapter 3, it is clearly that the effective PZT
model with an adhesive layer (EPMA) is more suitable for this situation. Then we can use
the Eq (3.69). to calculate the relative force.
Figure 5-8 The flow tree of Abaqus explicit model
On the modelling aspect, the dimension and material properties of the steel pipeline are
the same as the previous one, which is listed in Fig.5-1. The pipeline is meshed with
18048 C3D8R solid elements to make sure the element size is smaller than the maximal
121
element size maxL . Besides, the step in Abaqus simulation is Dynamic, Explicit with 8E-8
s 8 8 t E s time step. For the boundary condition aspect, we apply effective forces on
related nodes, which are symmetrically installed along the circumference and locations
with a distance of 495 mm to the right end of this pipeline. The more detailed information is
listed in Figs. 5-9, 5-10 and 5-11.
Figure 5-9 Finite element model of four PZT actuators bounded to the pipe
122
Figure 5-10 Finite element model of six PZT actuators bounded to the pipe
Figure 5-11 Finite element model of four PZT actuators bounded to the pipe
As mentioned at Chapter 3, the effective piezoelectric actuator model has a better
performance at a lower voltage range. Therefore, the effective force 5.52 4 NeF E ,
which is picked up from Table 5-4, is employed at the voltage of 2 V.
123
Table 5-4 Magnitudes of effective displacement at different voltages conditions
Voltage (V) Effective force (N)
0 0
1 -2.76E-4
2 -5.52E-4
3 -1.10E-3
4 -1.66E-3
5 -2.21E-3
5.4.4 FEM using co-simulation analysis
Fig. 5-12 illustrates the flow tree of Abaqus co-simulation model, which can allow different
solvers to get results in the piezoelectric actuators and the host structure. As shown in
section 4.4.7, the output of the piezoelectric analysis will be used as an input of transient
dynamic analysis in Abaqus/Explicit. Hence, the adhesive layer is regarded as an
interaction interface which manage the date exchange in every time increment between
two solvers, as shown in Fig. 5-12.
Figure 5-12 The flow tree of Abaqus co-simulation model
The detailed information about these two kinds of models on FEM simulation is illustrated
in Fig. 5-3. To be more precise, a steel pipe is defined as the parent model and a series of
PZT actuators are defined as the child model. All the material properties and dimensions
are similarly as the previous one, which are given in Tables 5-1. For the child model, each
actuator is meshed with 4 C3D8E solid elements and applied a 2 V 3-cycle Hanning
window tone-burst signal with 20 kHz on the piezoelectric elements through the electrodes.
Then, the induced strain created from the actuator is exchanged by a co-simulation
interface. For the parent model, the steel pipe is meshed with 18048 C3D8R solid
elements and simulated under the Abaqus Explicit code. This Explicit procedure is applied
124
to simulate the wave propagation in this pipe, when the induced stress is transferred from
the interaction surface.
5.5.5 Results and discussion
In this section, all the results from 4PZT model are given, including those obtained from
the co-simulation model, explicit model and implicit model. Since the frequency is selected
as 20 kHz, only L (0,1) mode exists in the final results. Figs. 5-13, 5-15, 5-17 illustrate the
displacement signals from FE dynamic simulations of steel pipes. As can be seen from
those figures, the general trends for these three results, which are calculated by using co-
simulation method, implicit method and explicit method, are nearly the same. To be more
exact, the maximum values of those displacements are all located at some, which is about
0.00066 s. However, for the implicit simulation, the results signal appears unstable at the
range of 0.0009 s to 0.0014 s. This is because the implicit dynamic produce cannot
demonstrate well on the wave propagation problem.
Figs. 5-14, 5-16 and 5-18 give the displacement signals from FE dynamic simulation of
steel pipes after the Hilbert-Huang transform. From these graphs, it is clearly that the
difference between these three kinds of methods when calculating the wave propagation in
pipes is clear. In the first place, the implicit results have several noises between 0.0009 s
and 0.0014 s. Besides, the results from co-simulation method and explicit method are
nearly the same. It means that these two methods can be substituted for another at some
certain conditions. Lastly, all the group velocities are satisfied with the group velocity L(0,1)
in Fig. 5-2. The computing processes are listed as follows. This means the results from
this simulation is reliable.
2.01 m5082.56 m/s
0.00039547 s
Co simulation Lg t
V
2.01 m5082.17 m/s
0.0003955 s
Explicit Lg t
V
125
Implicit 2.01 m5081.40 m/s
0.00039556 sL
g tV
0.0000 0.0005 0.0010 0.0015 0.0020
-6.00E-012
-4.00E-012
-2.00E-012
0.00E+000
2.00E-012
4.00E-012
6.00E-012
8.00E-012
Dis
pla
ce
me
nt
(m)
Time (s)
Input signal
Co-simulation
Implicit
Explicit
Figure 5-13 Displacement signals from four PZT model in the z-direction
0.0000 0.0005 0.0010 0.0015 0.00200.00E+000
1.00E-012
2.00E-012
3.00E-012
4.00E-012
5.00E-012
6.00E-012
Dis
pla
ce
me
nt (m
)
Time (s)
Input signal
Co-simulation
Explicit
Implicit
Figure 5-14 Displacement signals from four PZT model after Hilbert-Huang transform
126
0.0000 0.0005 0.0010 0.0015 0.0020-1.00E-011
-8.00E-012
-6.00E-012
-4.00E-012
-2.00E-012
0.00E+000
2.00E-012
4.00E-012
6.00E-012
8.00E-012
1.00E-011D
isp
lace
me
nt (m
)
Time (s)
Input signal
Co-simulation
Explicit
Implicit
Figure 5-15 Displacement signals from six PZT model in the z-direction
0.0000 0.0005 0.0010 0.0015 0.00200.00E+000
2.00E-012
4.00E-012
6.00E-012
8.00E-012
1.00E-011
Dis
pla
ce
me
nt (m
)
Time (s)
Input signal
Co-simulation
Explicit
Implicit
Figure 5-16 Displacement signals from six PZT model after HHT Fourier transform
127
0.0000 0.0005 0.0010 0.0015 0.0020-1.50E-011
-1.00E-011
-5.00E-012
0.00E+000
5.00E-012
1.00E-011
1.50E-011
Dis
pla
cem
ent (m
)
Time (s)
Input signal
Co-simulation
Implicit
Explicit
Figure 5-17 Displacement signals from eight PZT model in the z-direction
0.0000 0.0005 0.0010 0.0015 0.00200.00E+000
2.00E-012
4.00E-012
6.00E-012
8.00E-012
1.00E-011
1.20E-011
Dis
pla
cem
ent (m
)
Time (s)
Input signal
Co-simulation
Implicit
Explicit
Figure 5-18 Displacement signals from FE dynamic simulation of eight PZT pipes after
HHT Fourier transform
128
According to compared all the results from pervious section, the influence about the
number of PZT actuators used can be demonstrated. Fig. 5-19, 5-20 and 5-21 illustrate the
displacement signals from different simulation methods compared with different number of
PZT actuators. It is clearly can be found that with the increasing number of PZT actuators,
the maximum value of displacement is also increasing. Besides, according to Fig. 5-20,
the noises which occupied between 0.0009 s to 0.0014 s appear significantly with the
increasing of the number of PZT actuators.
0.0000 0.0005 0.0010 0.0015 0.00200.00E+000
2.00E-012
4.00E-012
6.00E-012
8.00E-012
1.00E-011
Dis
pla
ce
me
nt
(m)
Time (s)
4 PZT
6 PZT
8 PZT
Figure 5-19 Displacement signals from co-simulation method of steel pipes after HHT
Fourier transform with different number of PZT actuators
129
0.0000 0.0005 0.0010 0.0015 0.00200.00E+000
2.00E-012
4.00E-012
6.00E-012
8.00E-012
1.00E-011
1.20E-011D
isp
lace
me
nt
(m)
Time (s)
6PZT
8PZT
4PZT
Figure 5-20 Displacement signals from implicit simulation of steel pipes after HHT Fourier
transform with different number of PZT actuators
0.0000 0.0005 0.0010 0.0015 0.00200.00E+000
2.00E-012
4.00E-012
6.00E-012
8.00E-012
1.00E-011
Dis
pla
ce
me
nt
(m)
Time (s)
4 PZT
6 PZT
8 PZT
Figure 5-21 Displacement signals from explicit simulation of steel pipes after HHT Fourier
transform with different number of PZT actuators
130
5.5 Case study
There are two purposes in this study. One is to obtain the relationship between the wave
propagation and different types of defects in a steel pipe. Another one is to understand the
shape effects of PZT actuators on elastic wave propagation. A steel pipe, which is used in
this simulation, is 1.5 m long and has an outer diameter of 646 mm and a wall thickness of
6.4m. A sensor/ actuator network with 16 PZT actuators/sensors was devised, as shown in
Fig. 5-22. Eight actuators/sensors were installed at each end of pipe and they functioned
as either generating or collecting the signals of elastic waves. For the first group, three
kinds of defects are employed on the pipes, which are: 1) a hole of diameter 12 mm and
depth 5 mm, 2) a crack of length of 120 mm and width 2mm and 3) a notch of length 120
mm, width 12 mm and depth 5 mm as shown in Fig. 5-23. For the second group, two kinds
of actuators, which are square PZT actuators (length=7 mm, thickness=1 mm) and round
actuators (radius=5 mm, thickness= 1mm).
Figure 5-22 FE models of pipe without defect
131
Figure 5-23 FE models of pipes with/without defect: A) no defect; B) hole; C) crack; and D)
notch 3D solid model
5.5.1 Finite element modelling
As for the finite element modelling, the steel pipes were treated as a 3D deformable solid
and meshed with 684,512 C3D8R solid elements, which were chosen properly according
to wave length. Since piezoelectric materials are used to act as actuators and/or sensors
to generate and collect signals of elastic waves, the effective PZT actuator model (EPM),
has been given in Chapter 3, and were further applied in the finite element models of steel
pipes. In the FE models, A1 to A8 are standing for actuators while S1 to S8 are for sensors,
as shown in Fig. 5-22.
The defects added to the pipe model are a hole of diameter 12 mm and depth 5 mm, a
crack of length 120 mm and width 2 mm and a notch of length 120 mm, width 12 mm and
depth 5 mm as shown in Fig. 5-23. Those defects were designed to locate at the middle of
the pipeline and kept aligned with the actuator-sensor pair: A1 and S1.
In addition, to investigate the shape effect of PZT actuators, another two pipe models
without defect were also devised. In these two models, PZT actuator models with round
132
and square shapes were considered respectively, using the effective displacements
applied at relevant nodes around these actuators.
5.5.2 Results and discussion
To investigate the relationship between the guided wave propagation and defects, wave
modes were identified based on their group velocities. According to Fig. 5-24 (b), the
group velocity for L(0, m) mode is the fastest. Then, the wave packet I in Fig. 5-27 should
be longitudinal axial symmetric modes L(0, m). According to simulation results, the group
velocity was calculated as 5278 m/s. According to Fig. 5-24 (b) (Cg = 5400 m/s), it is
evident that wave packet I is L(0, 2). The result is consistent with the experimental result of
5300 m/s. The group velocity which we calculated for wave packet II is 2107.90 m/s. This
value is very closed to the group velocity of L(0,1) at 20 kHz in Fig. 5-24 (a).
Then, the wave packet II is L(0, 1). In addition, the wave packet III in Fig. 5-25 (b) must be
F(n, m) modes. For the wave packet IV, it could be an F(n, m) mode combined with L(0, m)
mode. This is because the amplitude of VI wave packet is the largest. Wave packet II
contains various wave modes and reflection waves. In brief, wave packet I is L(0, 2) mode,
wave packet III is F(n, m) mode and wave packet II is L(0, 1). Therefore the basic
relationship between the guided wave propagation and defects can be determined. First of
all, L(0, 2) mode is hard to detect the longitudinal defects, which means the results from
the crack structure are almost identical to the benchmark ones in wave packet I. Rose
(2004) had the same opinion in his works. Besides, L(0,1) can be used to detect the
circumferential cracks. This is because there are differences between the results from the
model with a notch defect and those from the benchmark model in the region of 0.0006 s
to 0.0008 s. Moreover, the F(n, m) is useful to detect some large defects. According to
wave packet III depicted in Fig. 5-25 (b), it is clearly indicated that the results from the
133
model with a hole defect show there is no F(n, m) mode compared to those from the
benchmark model. Lastly, according to the comparison on the results from the model with
a notch defect and the ones from the benchmark, it can be found that the notch can lead to
waveform transformation, especially for the period between 0.0008 s to 0.001 s.
0 20 40 60 80 1000
2000
4000
6000
8000
Cp (
m/s
)
Frequency (Khz)
L(0,1)
L(0,2)
T(0,1)
0 20 40 60 80 1000
1000
2000
3000
4000
5000
6000
7000
Cg (
m/s
)
Frequency (Khz)
L(0,1)
L(0,2)
T(0,1)
Figure 5-24 Disperse curves of a pipe with an outer diameter of 323mm and a wall
thickness of 6.4 mm
134
Figure 5-25 Displacement outputs from FE dynamic simulations along the longitudinal
direction of the pipe
0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012-3.00E-011
-2.00E-011
-1.00E-011
0.00E+000
1.00E-011
2.00E-011
3.00E-011
Dis
pla
ce
me
nt (m
) U
3
Time (s)
Benchmark
Hole
Crack
Notch
Figure 5-26 Displacement signals from FE dynamic simulations of steel pipes with and
without a defect in the z-direction
135
0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.00120.00E+000
5.00E-012
1.00E-011
1.50E-011
2.00E-011
2.50E-011
3.00E-011
Dis
pla
ce
me
nt
(m)
Time (s)
Benchmark
Hole
Crack
Notch
Figure 5-27 Displacement signals from FE dynamic simulations of steel pipes with and
without a defect after HT-processed
As for the shape effect of PZT actuators on the elastic wave propagation in pipes, Fig. 5-
28 illustrates the displacement outputs in the FE models using different shaped actuators.
It is worth to point out that the square PZT actuator can generate a slightly stronger
excitation in the pipe than the round one with close size, as depicted in Fig.5-28
I
II
III
IV
136
0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012-3.00E-011
-2.00E-011
-1.00E-011
0.00E+000
1.00E-011
2.00E-011
3.00E-011D
ispla
cem
en
t (m
)
Time (s)
Round PZT
Square PZT
0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.00120.00E+000
5.00E-012
1.00E-011
1.50E-011
2.00E-011
2.50E-011
3.00E-011
Dis
pla
ce
me
nt (m
)
Time (s)
Round PZT
Square PZT
Figure 5-28 Displacement signals from FE dynamic simulations of steel pipes with different
shaped PZT actuators
(a)
(b)
137
5.6 Summary
Guide wave propagation in steel pipes and the relationship between the guided wave
propagation and defects have been evaluated and investigated mainly using finite element
modelling and simulations in the present study. The enhanced effective piezoelectric
actuator model has been further applied in the FE models and the results show their
effectiveness for generating elastic waves although a shape effect has been observed.
Experimental results have primarily been used for verification on wave modes only at this
stage and the on-going experimental work will be conducted considering the complexity of
the elastic wave propagation in steel pipes. Based on the numerical results, the
longitudinal axially symmetric modes L(0, 2) cannot be used on the detection of
longitudinal defects while the non-axially symmetric modes F(n, m) are sensitive to
different kinds of defects, which can be focused in future research for more comprehensive
understanding on them.
138
Chapter 6 Conclusions and recommendations 6.1 Conclusions
The purpose of this thesis is to investigate elastic wave propagation in different steel
structures within the subject of non-destructive testing. This involved studying the effective
models of PZT actuators, creating finite element models and signal operating. Fig. 6-1
illustrates the general structure of this thesis. According to this figure, the conclusions for
this thesis can be summarised as follows:
Figure 6-1 The general structure of this thesis
1. The main purpose, which applied Finite element methods to simulate the guided wave
propagation in steel structural members, has been finished.
139
2. In order to figure out what is the most reliable effective PZT model, we use eight
different effective PZT models to compare with each other. For a single PZT actuator,
the results from EPM are the best, which is closed to the real condition of real PZT
actuator. Besides, when we added an adhesive layer under the actuator, the EPMA is
the best choice for its well performance on simulation.
3. In this thesis, we also employ three different FEMs to simulate the elastic wave
propagation in steel structures, which are Explicit Dynamic Analysis (EDA), Implicit
Dynamic Analysis (IDA) and Combined Explicit-Implicit Dynamic Analysis (CEIDA),
respectively. As compared the results from these three simulations, the IDA is the
worst one, which could lead to signal interference during the analysing process.
4. According to simulate the guided wave propagation in steel plates with different types
of defects, we get that the S0 wave model can easy detect the defects on the plate
surfaces and the maximum values demonstrated different types of defects.
5. Based on the numerical results from pipes inspection, we acquire that the longitudinal
axially symmetric modes L(0, m) cannot be used on the detection of longitudinal
defects while the non-axially symmetric modes F(n, m) are sensitive to different kinds
of defects
6.2 Recommendations on future work
This study provides some opportunities for future work. In the first place, the effective
piezoelectric actuator model can be further improved. This is because the real effective
displacements/forces are not parallel to the host structure’s surface. There may exist some
angles or degrees between the surfaces and the effective displacements and forces.
Secondly, the results from the pipes show that there are still having some uncertainties
about the relationship between defects and elastic wave propagation. Hence, it is better to
develop a large model, which needs the supercomputer resources to run the simulations.
140
Besides, additional parametric studies on wave propagation on structures with different
defect locations and severities is highly recommended in the future work, especially those
about damage wave modes. Furthermore, there are still limitations when using the co-
simulation method and the most significant one is about the limitation of physical memory
of a computer used, which may be overcome via using supercomputer resources too.
Nevertheless, many other steel structural members, including beam structures, can be
simulated and studied by using these methods developed in this thesis. Most of all, a
thorough and systematic experimental research should be conducted to further verify the
numerical findings, which unfortunately has not been conducted in this work due to the
time limitation.
141
APPENDIX A: Matlab code for the excitation signal
fs=200000
Nc=5
w=2*pi*fs
T=1/fs
Tc=Nc*T
t=0:Tc/10000:Tc
h=0.5*(1-cos(t.*w/Nc))
g=sin(t.*w)
l=h.*g
k=l'
H=t'
figure,plot(t,l)
save('p','k','-ascii')
save('t','H','-ascii')
142
Research outcomes during the study of Master (Honours) degree
Peer-reviewed Journal Paper:
Wang, T., Yang, C., 2014, Effective models of PZT actuator/sensor for numerical
simulation of elastic wave propagation, Applied Mechanics and Materials (ERA Journal
ID: 124777), 553: 705-710.
Peer-reviewed Conference Paper:
Wang, T., Yang, C., Spray, D., Ye, L., and Xiang, Y., 2014, Evaluation of elastic wave
propagation in steel pipes, Recent Advances in Structural Integrity Analysis -
Proceedings of the International Congress (APCF/SIF-2014) - The International
Congress (APCF/SIF-2014) uniting the Asian-Pacific Conference on Fracture and
Strength 2014 (APCFS-2014) and the International Conference on Structural Integrity
and Failure (SIF-2014), Sydney, December 9-12, 2014, 255-260, Woodhead
Publishing, ISBN: 978-0-08-100203-2.
143
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