The Pennsylvania State University
The Graduate School
Department of Engineering Science and Mechanics
ULTRASONIC GUIDED WAVE MECHANICS
FOR COMPOSITE MATERIAL STRUCTURAL HEALTH MONITORING
A Thesis in
Engineering Science and Mechanics
by
Huidong Gao
2007 Huidong Gao
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
May 2007
The thesis of Huidong Gao was reviewed and approved* by the following:
Joseph L. Rose
Paul Morrow Professor of Engineering Science and Mechanics
Thesis Advisor
Chair of Committee
Bernhard R. Tittmann
Schell Professor of Engineering Science and Mechanics
Clifford J. Lissenden
Associate Professor of Engineering Science and Mechanics
Charles E. Bakis
Professor of Engineering Science and Mechanics
Qiming Zhang
Distinguished Professor of Electrical Engineering
Judith A. Todd
Professor of Engineering Science and Mechanics
P.B.Breneman Department Head Chair
Head of the Department of Engineering Science and Mechanics
*Signatures are on file in the Graduate School
iii
ABSTRACT
The ultrasonic guided wave based method is very promising for structural health
monitoring of aging and modern aircraft. An understanding of wave mechanics becomes
very critical for exploring the potential of this technology. However, the guided wave
mechanics in complex structures, especially composite materials, are very challenging
due to the nature of multi-layer, anisotropic, and viscoelastic behavior.
The purpose of this thesis is to overcome the challenges and potentially take
advantage of the complex wave mechanics for advanced sensor design and signal
analysis. Guided wave mechanics is studied in three aspects, namely wave propagation,
excitation, and damage sensing. A 16 layer quasi-isotropic composite with a [(0/45/90/-
45)s]2 lay up sequence is used in our study.
First, a hybrid semi-analytical finite element (SAFE) and global matrix method
(GMM) is used to simulate guided wave propagation in composites. Fast and accurate
simulation is achieved by using SAFE for dispersion curve generation and GMM for
wave structure calculation. Secondly, the normal mode expansion (NME) technique is
used for the first time to study the wave excitation characteristics in laminated
composites. A clear and simple definition of wave excitability is put forward as a result
of NME analysis. Source influence for guided wave excitation is plotted as amplitude on
a frequency and phase velocity spectrum. This spectrum also provides a guideline for
transducer design in guided wave excitation. The ultrasonic guided wave excitation
characteristics in viscoelastic media are also studied for the first time using a modified
normal mode expansion technique. Thirdly, a simple physically based feature is
developed to estimate the guided wave sensitivity to damage in composites. Finally, a
fuzzy logic decision program is developed to perform mode selection through a
quantitative evaluation of the wave propagation, excitation and sensitivity features.
Numerical simulation algorithms are validated with both finite element analyses
and laboratory experiments. For the quasi-isotropic composite, it is found that the
ultrasonic wave propagation characteristics are not always quasi-isotropic. The
directional dependence is very significant at high frequency and higher order wave modes.
iv
Mode separation between Rayleigh-Lamb type and Shear Horizontal type guided waves
is not possible. In addition, guided wave modes along one dispersion curve line could
have a significant difference in wave structure. Therefore, instead of using traditional
symmetric, antisymmetric, and SH notation, a new notation is used to identify the
dispersion curves in a numerical order. Wave modes with a skew angle larger than 30
degrees can exist in a quasi-isotropic composite plate, which is validated by both FEM
and experiment. At low frequency, the first wave mode has higher sensitivity than that of
the third wave mode. However, the attenuation of the first wave mode is higher than that
of the third wave mode. The mode selection trade-offs are evaluated and
recommendations are provided for guided waves used in long range structural health
monitoring.
v
TABLE OF CONTENTS
NOMENCLATURE ....................................................................................................viii
GLOSSARY AND ABBREVIATIONS......................................................................x
LIST OF FIGURES .....................................................................................................xi
LIST OF TABLES.......................................................................................................xxi
ACKNOWLEDGEMENTS.........................................................................................xxii
Chapter 1 Introduction ................................................................................................1
1.1 Problem statement ..........................................................................................1
1.2 Literature review.............................................................................................6
1.2.1 Ultrasonic guided wave propagation in plates......................................6
1.2.2 Guided wave excitation and scattering.................................................9
1.2.3 Experimental techniques in structural health monitoring of
composites ..............................................................................................11
1.2.4 Summary of the literature review and challenges for further study .....12
1.3 Thesis objectives.............................................................................................13
1.4 A preview of the thesis content ......................................................................14
Chapter 2 Wave propagation theory in composite laminates .....................................17
2.1 Problem formulation.......................................................................................17
2.2 The global matrix method...............................................................................19
2.2.1 The partial wave theory .......................................................................19
2.2.2 Boundary conditions and the global matrix method............................20
2.2.3 Wave field solutions ............................................................................21
2.3 The semi-analytical finite element method....................................................24
2.4 Important derived guided wave properties .....................................................28
2.4.1 Power flow and energy density ...........................................................28
2.4.2 Group velocity and energy velocity ....................................................29
2.4.3 Skew angle...........................................................................................31
2.4.4 Wave field normalization ....................................................................32
Chapter 3 Guided wave propagation in quasi-isotropic composites...........................34
3.1 A numerical model of composite laminates ..................................................34
3.2 Phase velocity dispersion curves ...................................................................36
3.3 Group velocity and energy velocity dispersion curves..................................42
3.4 Skew angle dispersion curves ........................................................................45
vi
3.5 Wave structure analysis .................................................................................47
3.6 Summary.........................................................................................................49
Chapter 4 Guided wave excitation in composites.......................................................51
4.1 Theory............................................................................................................51
4.1.1 The reciprocity relation and mode orthogonality ...............................51
4.1.2 The normal mode expansion technique...............................................54
4.1.3 Source influence on wave excitation...................................................56
4.2 Numerical proof of mode orthogonality in a quasi-isotropic laminate...........58
4.3 Wave mode excitability ..................................................................................60
4.4 Numerical results of source influence ............................................................63
4.4.1 Excitation signal analysis ....................................................................64
4.4.2 Transducer geometry and loading pattern influence ...........................68
4.4.3 Frequency and phase velocity spectrum of a finite source..................71
4.5 Guided wave dispersion.................................................................................74
4.5.1 Dispersion signal reconstruction .........................................................74
4.5.2 Wave mode dispersion coefficient ......................................................78
4.6 Guided wave field simulation........................................................................80
4.6.1 Theory..................................................................................................80
4.6.2 Wave field reconstruction case studies in composite laminates...........81
4.6.2.1 First fundamental wave mode ....................................................81
4.6.2.2 The third fundamental wave mode.............................................88
4.7 Guided wave beam spreading analyses .........................................................92
Chapter 5 Finite element modeling of wave excitation and propagation ...................97
5.1 Theory of a three dimensional FEM...............................................................97
5.2 Wave excitation and propagation case studies in ABAQUS..........................98
5.2.1 Case I: the first wave mode ..................................................................98
5.2.2 Case II: the third wave mode................................................................104
5.2.3 Case III: wave modes with large skew angle .......................................109
5.3 Summary........................................................................................................114
Chapter 6 Guided waves in composites considering viscoelasticity ..........................115
6.1 Dispersion relation derivation.........................................................................115
6.2 Numerical simulation results on wave propagation........................................116
6.3 A new normal mode expansion technique for viscoelastic media..................124
6.4 Numerical simulation results ..........................................................................125
6.5 Summary.........................................................................................................127
Chapter 7 Guided wave sensitivity to damage in composites.....................................129
vii
7.1 Effect of material property degradation on guided wave propagation ..........129
7.1.1 Theoretical study .................................................................................129
7.1.2 Density variation ................................................................................130
7.1.3 Elastic stiffness variation.....................................................................131
7.1.4 Ply thickness variation.........................................................................136
7.2 Guided wave scattering sensitivity ................................................................137
7.3 Summary.........................................................................................................141
Chapter 8 Guided wave mode selection.....................................................................143
8.1 Introduction....................................................................................................143
8.2 Guided wave mode selection rules .................................................................144
8.3 Guided wave long range monitoring potential ...............................................147
Chapter 9 Experimental studies ..................................................................................152
9.1 Wave propagation study with contact transducers .........................................152
9.1.1 Ultrasonic transducers and instruments...............................................152
9.1.2 Experimental phase velocity dispersion curves....................................154
9.1.3 Guided wave group velocity and attenuation studies ..........................160
9.1.4 Guided wave skew angle studies .........................................................163
9.2 Wave excitation with piezoelectric active sensors .........................................165
9.3 Guided wave damage detection with piezoelectric active sensors .................168
9.4 Summary........................................................................................................172
Chapter 10 Conclusions and discussions ....................................................................174
10.1 Summary of the thesis study.........................................................................174
10.2 Specific contributions ...................................................................................177
10.3 Future work...................................................................................................178
References....................................................................................................................179
Appendix A Guided wave imaging techniques in SHM.............................................184
A.1 Signal processing and feature extraction .......................................................184
A.2 Guided wave imaging algorithms ..................................................................185
A.3 Application of imaging techniques in laboratory experiments ......................187
A.3.1 E2 airplane wing crack monitoring .....................................................187
A.3.2 Helicopter component corrosion monitoring.......................................188
A.3.3 Composite delamination monitoring ...................................................189
Appendix B Nontechnical abstract .............................................................................191
viii
NOMENCLATURE
Symbol Meaning
ijklc , c Elastic constant tensor
IJC , C Elastic constant in matrix format
'C Elastic part of stiffness constant
''C Viscous part of stiffness constant
1E , 2E , 12G , 12ν , 23ν Engineering elastic constants of a composite lamina
ρ Density
t Time
ix Position
ih Thickness of each ply
H Total thickness of the laminate
iu , u Displacement
iv , v Particle velocity
ijσ , σ Stress
ijε , ε Strain
ijS Engineering strain
iP , P Poynting’s vector
iT , F Traction
λ Wavelength
f Frequency
ω Angular frequency
pc Phase velocity
gc Group velocity
ix
eV Energy velocity
θ Propagation direction
Φ Skew angle
ξ Wave number
α Attenuation part of wave number
β Complex wave number
mnP Wave mode orthogonality evaluation from complex
reciprocity relation
mnQ Wave mode orthogonality evaluation from real
reciprocity relation
iB , B Coefficient of partial waves in the global matrix
method
D Global matrix in the global matrix method
N Shape functions in FEM
M Mass matrix in FEM
K Stiffness matrix in FEM
x
GLOSSARY AND ABBREVIATIONS
Name Explanation
Dispersion coefficient An attribute of a wave mode describing its extent of dispersion
Dispersion curve A set of curves describing the wave mode dispersion relation of
phase velocity, or some other feature, with respect to frequency
Energy velocity Velocity of wave energy transmission
GMM Global matrix method
Goodness An evaluation of a wave mode describing how much it satisfies
a given set of criteria
Group velocity Velocity of wave package transmission
Mode sensitivity An attribute of a wave mode describing its ability to detect a
particular damage
NDE Nondestructive evaluation
NME Normal mode expansion
Phase velocity Velocity of a wave propagation with constant phase
RAPID Reconstruction algorithm for probabilistic inspection of damage
SAFE Semi-analytical finite element method
SDC Signal difference coefficient
SHM Structural health monitoring
Skew angle Angle between wave energy propagation and wave vector
direction
Source influence The influence of source geometry and signal on guided wave
excitation
Wave excitability An attribute of a wave mode describing its response to wave
excitation
Wave structure Wave field profile along the thickness direction of a particular
displacement, stress, energy, or other features.
xi
LIST OF FIGURES
Figure 1-1: Materials on a Boeing 777 aircraft. (Courtesy NASA Langley
Research Center)...................................................................................................1
Figure 1-2: The trend of using composites in (a) military and (b) civil aircraft ..........2
Figure 1-3: Impact and fatigue damage to composite panels. (a) picture of a
composite panel after low velocity impact (b) ultrasonic C-scan image
showing internal delamination (c) fatigue damage to a composite plate..............2
Figure 1-4: Concept of damage detection using guided waves. (a) pulse echo (b)
through transmission.............................................................................................4
Figure 1-5: A vision of a “Theoretically driven structural health monitoring ”
strategy..................................................................................................................5
Figure 2-1: A coordinate system for wave mechanics study of a multi-layered
structure ................................................................................................................17
Figure 2-2: A sketch of a one dimensional three node isoparametric element...........24
Figure 2-3: A sketch of the power flow in a guided wave mode for the derivation
of energy velocity. ................................................................................................30
Figure 2-4: Sketch of slowness profile and skew angle. (Modified from [Rose
1999])....................................................................................................................32
Figure 3-1: The sketches of the lay-up sequence and the wave propagation in a 16
layer quasi-isotropic composite laminate. Layup sequence is [(0/45/90/-45)s]2 ..35
Figure 3-2: A comparison of phase velocity dispersion curves obtained from two
methods. Continuous lines: SAFE; Blue dots: GMM. Wave propagates in the
0o direction............................................................................................................36
Figure 3-3: Phase velocity dispersion curves for guided wave modes in different
propagation directions. (a) mode 1, (b) mode 2 (c) mode 3. ................................38
Figure 3-4: Angular profiles of the phase velocity dispersion curves at a
frequency of 200kHz, (a) mode 1 (b) mode 2 (c) mode 3. The units in the
radius is km/s. .......................................................................................................39
Figure 3-5: Phase velocity dispersion surfaces of the first three modes, (a) mode 1
(b) mode 2 (c) mode 3. .........................................................................................40
xii
Figure 3-6: Phase velocity dispersion surfaces of mode 4 to mode 6, (a) mode 4
(b) mode 5 (c) mode 6. .........................................................................................41
Figure 3-7: Comparison between the group velocity dispersion curves obtained
from SAFE methods and energy velocity curves from the global matrix
method. Continuous lines: SAFE; Blue dots: Global matrix method. Wave
propagation in 0o direction....................................................................................42
Figure 3-8: Mode 1 group velocity dispersion curves for different propagation
directions...............................................................................................................43
Figure 3-9: Group velocity dispersion curves for different propagation directions
and frequencies. (a) mode 3 (b) mode 4. Note: There are only four lines in (b)
because 0.2 MHz is below the cut-off frequency of mode 4. ...............................44
Figure 3-10: Group velocity dispersion surface. (a) mode 1 (b) mode 2..................44
Figure 3-11: A comparison between skew angle curves obtained from the SAFE
and the GMM. Continuous lines: SAFE; Blue dots: GMM. Wave propagation
in 0 degree direction .............................................................................................45
Figure 3-12: Variation of guided wave skew angle with respect to wave
propagation directions. (a) mode 1-3 at 200kHz (b) mode 1-4 at 1.0 MHz. ......46
Figure 3-13: Skew angle surface of the first wave mode............................................46
Figure 3-14: A comparison of wave structures obtained from the GMM and the
SAFE method. (a) displacement u1, (b) stress σ33. ...............................................48
Figure 3-15: Displacement and stress wave structure for the first wave modes at
0.2 MHz. (a) displacements (b) out of plane stress (c) in plane stress (d)
power flow distribution.........................................................................................49
Figure 4-1: Wave structure components for mode orthogonality validation. (a) v1
(b) v2 (c) v3 (d) σ11 (e) σ12 (f) σ13. ..........................................................................59
Figure 4-2: Particle velocity spectrum in the x1 direction for the wave
propagating in 0o at the surface of the [(0/45/90/-45)s]2 laminate. .......................60
Figure 4-3: Rectified particle velocity spectrum for the wave propagating in 0o at
the surface of the [(0/45/90/-45)s]2 laminate: (a) x1 direction, (b)x2 direction,
(c) x3 direction.......................................................................................................61
xiii
Figure 4-4: Angular profile of wave mode excitability (wave mode particle
velocity at surface) for mode 1 to mode 3 at 200kHz. (a) shear (x1 direction)
loading (b) normal (x3 direction) loading. ............................................................62
Figure 4-5: Phase velocity dispersion curves of guided wave propagation at 0o
with the embedded information of wave mode excitability. Blue sections are
most easily excited with shear loading in the x1 direction; red sections and
black sections correspond to x2 and x3 direction loading respectively. ................63
Figure 4-6: Sample waveforms. (a) 5 cycled tone-burst signal with 1 MHz center
frequency and rectangular window, (b) 5 cycled tone-burst signal with 1
MHz center frequency and Hanning window.......................................................64
Figure 4-7: Amplitude spectra of the 5 cycled tone burst signals with 1MHz
center frequency.(a) Rectangular window, (b) Hanning window.........................65
Figure 4-8: Relation between bandwidth and center frequency under constant
number of cycles (a) Rectangular window (b) Hanning window.........................66
Figure 4-9: Relation between bandwidth and center frequency under constant
pulse width (a) Rectangular window (b) Hanning window..................................67
Figure 4-10: Sketch of transducer loading model. (a) concentrated shear loading
(b) concentrated normal loading (c) evenly distributed normal loading. .............68
Figure 4-11: Spatial domain loading distribution and its corresponding spatial
frequency spectrum of a 3 element linear array with 1mm array element
width. (a) concentrated shear loading (b) concentrated normal loading (c)
evenly distributed normal loading. .......................................................................70
Figure 4-12: Phase velocity spectrum for a 3 element linear array with 1 mm
array element width at 1 MHz. (a) concentrated shear loading (b)
concentrated normal loading (c) evenly distributed normal loading. ...................71
Figure 4-13: Source influence spectrum of a 3 element transducer with 1mm
element width and excited by a 10 cycled tone burst signal with Hanning
window and 1 MHz center frequency. The loading is concentrated shear. ..........72
Figure 4-14: Source influence spectrum of a 3 element transducer with 1mm
element width and excited by a 10 cycled tone burst signal with Hanning
window and 1 MHz center frequency. The loading is concentrated normal. .......73
Figure 4-15: Source influence spectrum of a 3 element transducer with 1mm
element width and excited by a 10 cycled tone burst signal with Hanning
xiv
window and 1 MHz center frequency. The loading is evenly distributed
normal. ..................................................................................................................73
Figure 4-16: Reconstructed signals at 0 mm, 100 mm, 200 mm, 300 mm, and 400
mm away from the excitation source. Source signal is a 5 cycle Hanning
windowed tone burst with 500 kHz center frequency. The wave mode
considered is the first dispersion curve line for guided wave propagating in 0o
of the [(0/45/90-45)s]2 laminate. ...........................................................................75
Figure 4-17: Reconstructed waveforms showing the effect of wave dispersion.
Wave mode considered is the fifth mode line. The excitation signal is a 5-
cycle Hanning windowed tone burst with 600kHz center frequency. ..................76
Figure 4-18: Sections of the phase velocity, group velocity dispersion curves for
the fifth wave mode line along 0 degree propagation direction. The amplitude
spectrum of a 5-cycle Hanning windowed tone burst signal with 600 kHz
center frequency is also plotted. ...........................................................................77
Figure 4-19: First five dispersion coefficient lines for the wave propagation in the
0o direction of the [(0/45/90/-45)s]2 laminate. ......................................................79
Figure 4-20: Mode selection results by the criterion of dispersion coefficient. (a)
less than µs/mm1.0 (b) larger than µs/mm5.0 for the wave propagating in 0o
direction of an [(0/45/90/-45)s]2 composite laminate. Blue dashed lines are
the entire dispersion curve set. Red line sections are the modes that satisfy the
criterion.................................................................................................................80
Figure 4-21: (Frequency)-(Phase velocity) spectrum of a 3 cycle 200 kHz signal
with Hanning window on a 2mm wide element. ..................................................82
Figure 4-22: Wave mode component function of the wave field excited from a
finite source listed in Tab. 4-3 . ...........................................................................82
Figure 4-23: u1 direction wave displacement at four positions....................................83
Figure 4-24: u3 direction displacement at four positions. ............................................84
Figure 4-25: Wave field distribution along the thickness of the [(0/45/90/-45)s]2
structure. (a) u1 , (b) u3...........................................................................................85
Figure 4-26: Comparison between the wave field profile of the excited wave from
a finite source with the wave structure at center frequency. The mode selected
is the first mode at 200 kHz..................................................................................86
Figure 4-27: Wave field snapshots at time equals to 20 µs. (a) u1 , (b) u3...................87
xv
Figure 4-28: (Frequency)-(Phase velocity) spectrum of a 3 cycle 200 kHz signal
with Hanning window on a 16 mm wide element using concentrated shear
loading. Wave propagation direction is 0o............................................................89
Figure 4-29: Wave mode component function of the wave field excited from a
finite source listed in Tab. 4-4. ...........................................................................89
Figure 4-30: Reconstructed wave signal at 0, 20, 40, 60 mm. (a) u1, (b) u3. ...............90
Figure 4-31: Wave field snapshots at a time of 20 µs, (a) u1 , (b) u3..........................91
Figure 4-32: Skew angle and beam spreading curves of the first five wave mode
lines. Structure: [(0/45/90/-45)s]2 laminate with 0.2 mm ply thickness. Wave
vector direction: 0o................................................................................................93
Figure 4-33: Mode selection results by the criterion of a beam spreading angle
less than 5o for the wave propagating in 0
o direction of an [(0/45/90/-45)s]2
composite laminate. Blue dashed lines are the entire dispersion curve set. Red
line sections are the modes that satisfy the criterion. ...........................................94
Figure 4-34: Mode selection results by the criterion of beam spreading angle
larger than 20o for the wave propagating in 0o direction of an [(0/45/90/-
45)s]2 composite laminate. Blue dashed lines are the entire dispersion curve
set. Red line sections are the modes that satisfy the criterion. .............................95
Figure 4-35: Beam spreading dispersion curves of the third mode line for four
excitation wave vector directions. This shows the dependence of beam
spreading on wave launching direction. ...............................................................96
Figure 5-1: A picture of a numerical model in ABAQUS used to efficiently excite
the first guided wave mode at a 200 kHz center frequency..................................98
Figure 5-2: A finite element mesh. (a) the entire model (b) a corner of the model. ...100
Figure 5-3: Top view of the wave field at 20 µs. (a) u1, (b) u3. ...................................101
Figure 5-4: Top view of the wave field at 40 µs. (a) u1 , (b) u3. .................................101
Figure 5-5: Thickness profile of the guided wave at 20 µs excited from a 2 mm
wide transducer element at 200 kHz. (a) u1, (b) u3. ..............................................102
Figure 5-6: Wave signal comparison between the theoretical prediction from
normal mode expansion and finite element modeling. (a) u1, (b) u3. The
black box with dotted line shows the directly excited wave package. .................103
Figure 5-7: Wave field snapshots at 10 µs. (a) u1 field, (b) u3 field. ...........................105
xvi
Figure 5-8: Wave field snapshots at 20 µs: (a) u1 field, (b) u3 field. ...........................106
Figure 5-9: Thickness profile of the guided wave at 20 µs. ........................................107
Figure 5-10: Wave signal comparison between the theoretical prediction from
normal mode expansion and finite element modeling: (a) u1, (b) u3. Black box
with dotted line: excited wave package. ...............................................................108
Figure 5-11: Skew angle dispersion curve of wave propagation in 0o direction of a
quasi-isotropic composite laminate. .....................................................................109
Figure 5-12: Wave excitation (f-cp) spectrum for a 5 element transducer with 3
mm element width and excited with a 10 cycled signal at 720 kHz using a
Hanning window...................................................................................................110
Figure 5-13: Wave mode content curve for the loading described in Tab. 5-3 . .........111
Figure 5-14: Sample wave field snapshots of u3 . (a) 2.5 µs (b) 20 µs ........................112
Figure 5-15: The displacement and power flow wave structures of the guided
wave mode with large skew angle. The mode studied is the fifth wave mode
at a frequency of 0.72 MHz. .................................................................................113
Figure 6-1: (a) Phase velocity dispersion curve and (b) attenuation dispersion
curves obtained from Hysteretic model. ...............................................................118
Figure 6-2: (a) Phase velocity dispersion curve and (b) attenuation dispersion
curves obtained from Kelvin-Voigt model ...........................................................119
Figure 6-3: Wave modes with least attenuation at a given frequency. (a)
Hysteretic model (b) Kelvin-Voigt model. ...........................................................119
Figure 6-4: Comparison of phase velocity dispersion curves between the elastic
model and the Hysteretic viscoelastic model. Dotted line: elastic model, solid
line: viscoelastic model. (a) full set of dispersion curve, (b) magnified curve
shows mode interaction. .......................................................................................120
Figure 6-5: Wave structure comparison between the elastic model and the
viscoelastic model. Wave mode: first mode at 200kHz, u1 displacement. (a)
real part (b) imaginary part. ..................................................................................121
Figure 6-6: A comparison of energy velocity dispersion curve generated from the
elastic and viscoelastic models. (a) elastic model (b) viscoelastic model. ..........122
Figure 6-7: Guided wave feature comparisons from a viscoleastic model. (a)
Wave modes with largest group velocity for a given frequency (b) Wave
modes with smallest attenuation for a given frequency........................................123
xvii
Figure 6-8: Comparison of skew angle dispersion curves obtained from elastic
and viscoelastic models. Dotted line: elastic model. Solid line: Hysteretic
viscoelastic model.................................................................................................123
Figure 6-9: Comparison of wave mode excitability with x1 direction force on the
surface. Dotted line: Elastic model; solid line: Viscoelastic Hysteretic model.
(a) frequency range of 0 to 2 MHz. (b) magnified region of mode interaction. ...126
Figure 6-10: A comparison of wave mode excitability using F1 direction loading
in (a) an elastic and (b) a viscoelastic model. .......................................................127
Figure 7-1: Dispersion curves for guided wave propagation in composite
laminates. Blue dashed line: nominated mass density of IM7/977-3 ρ=1.6
kg/m3 Red line :assumed 10% density reduction ρ=1.44 kg/m
3 ..........................131
Figure 7-2: Figure illustrates the effect of dispersion curve scaling when the
material property degradation introduces 10% stiffness reduction. Blue
dashed line: no stiffness reduction, blue solid line predicted dispersion curve
with stiffness reduction, red dots calculated dispersion curve with stiffness
reduction. ..............................................................................................................132
Figure 7-3: Effect of engineering constant variation on guided wave dispersion
curves. Blue dashed line: nominal material property. Red line: with 10% fiber
direction modulus (E1) reduction of the lamina....................................................133
Figure 7-4: Effect of engineering constant variation on guided wave dispersion
curves. Blue dashed line: nominal material property. Red line: with 10%
transverse modulus (E2) reduction of the lamina..................................................134
Figure 7-5: Effect of engineering constant variation on guided wave dispersion
curves. Blue dashed line: nominal material property. Red line with 10% in
plane shear modulus (G12) reduction of the lamina. .............................................134
Figure 7-6: Effect of engineering constant variation on guided wave dispersion
curves. Blue dashed line: nominal material property. Red line with 10%
Poisson’s ratio(v12) reduction of the lamina. ........................................................135
Figure 7-7: Effect of engineering constant variation on guided wave dispersion
curves. Blue dashed line: nominal material property. Red line with 10%
Poisson’s ratio (v23) reduction of the lamina. .......................................................135
Figure 7-8: Variation of dispersion curves due to ply thickness change of a 16
layer quasi-isotropic composite. Blue lines: 0.2mm Red lines: 0.18mm. ............136
xviii
Figure 7-9: Effect of surface erosion on guided wave phase velocity dispersion of
a 16 layer quasi-isotropic composite laminate. Blue line: all ply thickness 0.2
mm. Red line: first layer thickness reduction of 0.1 mm......................................137
Figure 7-10: Estimated sensitivity spectrum of guided wave modes to
delamination at the first laminate interface of a [(0/45/90/-45)s]2 composite
structure. ...............................................................................................................139
Figure 7-11: Estimated sensitivity spectrum of guided wave modes to
delamination at the 3rd laminate interface of a [(0/45/90/-45)s]2 composite
structure. ...............................................................................................................140
Figure 7-12: Estimated sensitivity spectrum of guided wave modes to
delamination in a [(0/45/90/-45)s]2 composite structure......................................141
Figure 8-1: Goodness function definition for guided wave selection. Evaluates
attenuation characteristic (a) Crisp rule (b) fuzzy rule. ........................................144
Figure 8-2: Mode selection results considering wave mode attenuation. (a) Crisp
mode selection with 0.5dB/mm allowed. (b) Fuzzy selection..............................145
Figure 8-3: The mode selection rule and candidate wave modes for large skew
angle demonstration. (a) High pass filter for the absolute value of skew angle.
(b) Mode selection results.....................................................................................146
Figure 8-4: Guided wave mode selection for the purpose of demonstration large
skew angle. ...........................................................................................................147
Figure 8-5: Guided wave mode selection considering attenuation. (a) selection
rule (b) selection result. ........................................................................................148
Figure 8-6: Guided wave mode selection considering mode dispersion. Less
dispersive modes selected (a) selection rule (b) selection results.........................149
Figure 8-7: Guided wave mode selection considering mode sensitivity (a)
selection rule (b) selection result. .........................................................................149
Figure 8-8: Guided wave mode selection considering wave excitation with
loading in the x1 direction. (a) Selection rule (b) qualified wave modes plotted
in red. ....................................................................................................................150
Figure 8-9: Guided wave mode selection considering wave excitation with
loading in the x3 direction. (a) Selection rule (b) qualified wave modes plotted
in red. ....................................................................................................................150
xix
Figure 8-10: Overall guided wave mode selection considering rules 1 to 4 listed in
Tab. 8-1.................................................................................................................151
Figure 9-1: Test setups for ultrasonic guided wave propagation study. ......................153
Figure 9-2: Integrated ultrasonic testing system. .........................................................153
Figure 9-3: Guided wave signal collected at 200mm position, when the
transmitter is at 100 from the left edge. ................................................................154
Figure 9-4: Ultrasonic guided wave phase velocity dispersion curve for wave
propagating in the 0o of a quasi-isotropic composite laminate. Wave mode
lines are numbered on the dispersion curves. .......................................................155
Figure 9-5: Guided wave signals collected from a linear scan showing edge
multi-mode, edge reflection, and complex interference. (a) Experimental
signals. (b) sketch of the first few wave paths. (1): direct transmission mode 3,
(2) Reflected mode three from left edge, (3) reflected mode 3 from right edge,
(4) direct through transmission of mode 1...........................................................156
Figure 9-6: Frequency and phase velocity spectrum of guided wave signals
shown in Fig. 9-5 . ................................................................................................157
Figure 9-7: A comparison between the experimental dispersion curve and
theoretical dispersion curve for wave propagation along the 0 degree
direction. ...............................................................................................................158
Figure 9-8: Comparison of guided wave modes in the experiment with theoretical
expectation using low attenuation, low skew angle, and excitable rules, and
frequency spectrum of source influence. The result shows that the experiment
meets the expectation............................................................................................160
Figure 9-9: Guided wave signals from a 800kHz transducer. (a) Illustration of
guided wave phase velocity and group velocity in a wave package. (b)
frequency and phase velocity spectrum................................................................161
Figure 9-10: Energy content in the guided wave signal as a function of position
showing wave attenuation.....................................................................................162
Figure 9-11: Experiments to test the effect of energy skew in a quasi-isotropic
composite plate. ....................................................................................................163
Figure 9-12: Guided waves excited from angle wedge to validate the concept the
concept of large skew angle. The black line at the center corresponds to the
xx
wave launching direction. The line in -60 mm position marks the position
where a maximum signal is detected. ...................................................................164
Figure 9-13: Guided wave signals from surface mounted piezoelectric
transducers. Excitation signal 200kHz, pulse width 5 µs. transducer element
widths: (a) 4mm, (b) 6mm, and (c) 8mm..............................................................166
Figure 9-14: Expected guided wave modes from a surface excitation source.
Excitation signal has center frequency 200kHz and 2 cycles...............................167
Figure 9-15: Picture of a plastic put on the top of a composite plate to simulate
damage. Plastic putting dimensions: 10 mm x 10mm x5 mm..............................168
Figure 9-16: Guided wave signals (a) before damage, (b) after damage, and (c) the
difference of the signals in (a) and (b). Transducer: 4mm width. Excitation
signal. 200kHz with 5 µs pulse width...................................................................169
Figure 9-17: Guided wave signals (a) before damage, (b) after damage, and (c)
the difference of the signals in (a) and (b). Transducer: 8mm width.
Excitation signal. 200kHz with 5 µs pulse width. ................................................170
Figure 9-18: Guided wave signals (a) before damage, (b) after damage, and (c)
the difference of the signals in (a) and (b). Transducer: disc. Excitation signal.
350 kHz with 5 µs pulse width. ............................................................................171
Figure A-1: Concept of a ray affect area in RAPID reconstruction ............................186
Figure A-2: Piezoelectric sensors on an aircraft wing panel. ......................................187
Figure A-3: Reconstruction results from 15 micro pulse width data with adaptive
threshold. (a) Reference state , (b) 2mm defect, (c) 3mm defect, (d) 4mm
defect.....................................................................................................................188
Figure A-4: (a) Simulated corrosion damage in an helicopter component,
Corrosion thickness 1/1000 inch, area 1’’ x 1’’. and (b) damage monitoring
results with ultrasonic guided waves and RAPID reconstruction technique........189
Figure A-5: (a) Sensor array on a composite panel for impact damage detection,
(b) sample signals before and after impact showing damage detection. .............190
Figure A-6: Impact damage localization with (a) ultrasonic C-scan (b) guided
wave monitoring with RAPID algorithm. ............................................................190
xxi
LIST OF TABLES
Table 3-1: Material properties of IM7/977-3 unidirectional composite properties .....35
Table 3-2: Phase velocity values at low frequency limit of the dispersion curves
(10 kHz) ................................................................................................................37
Table 4-1: Wave mode orthogonality validation table ...............................................58
Table 4-2: Comparison of theoretical velocities with the velocity values obtained
from the reconstructed signals ..............................................................................77
Table 4-3: A loading design to efficiently excite the first fundamental wave mode...81
Table 4-4: A loading design to efficiently excite the third fundamental wave mode..88
Table 5-1: Model and loading parameters in a finite element simulation ...................99
Table 5-2: A finite element model parameters to excite the 3rd
wave mode ...............104
Table 5-3: Model geometry and loading pattern to demonstrate large skew angle ....110
Table 6-1: Lamina properties of the IM7/977-3 composite used in simulation ..........117
Table 8-1: Proposed mode selection rules for mode selection based on long range
delamination detection in composite laminates. ...................................................148
Table 9-1: Quantitative comparison of wave mode attenuation .................................162
xxii
ACKNOWLEDGEMENTS
I would like to take this opportunity to express my sincere thanks to my advisor,
Dr. Joseph L. Rose, for his guidance and encouragement during the course of my study in
the Pennsylvania State University. His valuable advice and philosophy on technical as
well as personal matters will be a great treasure in my future career and life. I am also
indebted to all other members of my doctoral committee, Dr. Bernhard Tittmann, Dr.
Charles Bakis, Dr. Clifford Lissenden, and Dr. Qiming Zhang for their help in my
research and suggestions on the improvement of this thesis.
Thanks will also be given to FBS. Inc, Intelligent automation Inc., and NAVAIR,
USA, for financial support in the research projects over the years, American society for
nondestructive testing for a fellowship support, and GE Inspection Technologies for
technical support in my study.
I have also benefited from many colleagues in the Ultrasonic NDE lab and friends
in Penn State. Thanks a lot for all of their helps in experiments and computations,
valuable discussions, technical support, and finally valuable comments during this thesis
preparation.
Finally, sincere thanks are given to my parents for their support and
understanding all along the path of my education. I would like to dedicate this thesis to
my lovely wife, Guangfei, thank her for the support and sharing of my happiness and
difficulties.
Chapter 1
Introduction
1.1 Problem statement
Fiber reinforced polymer composites are finding increased applications in the
aircraft and aerospace industries due to their superior mechanical properties and light
weight. Fig. 1-1 shows an example of the application of graphite fiber reinforced
composites in a Boeing 777 aircraft [Chambers 2003]. The application of composites in
military aircraft is even more common than its use in commercial aircraft. The trends of
increased use of composites in military and civil aircraft are illustrated in Fig. 1-2
[Chambers 2003]. It shows that about 40% of the weight in a Lockheed F-22 aircraft is
made of composite materials. The new Boeing 787 aircraft under development will have
50% of composite usage. In addition, a composite fuselage will be used for the first time
in commercial aircraft.
Figure 1-1: Materials on a Boeing 777 aircraft. (Courtesy NASA Langley Research
Center)
2
1965 1970 1975 1980 1985 1990 19950
10
20
30
40
50
60
Year of first flight
We
igh
t p
erc
en
t o
f co
mp
osite
F-14 F-15 F-16
F-18
AV-8B
F-22
1960 1965 1970 1975 1980 1985 1990 1995 2000012345
10
15
20
Year of first flight
We
igt p
erc
en
tag
e o
f co
mp
osite
DC-9 DC-10
747 757 767
737-300 747-400
A300-600
A310 777
A330
MD-90
A320 A321 A322
(a) (b)
Figure 1-2: The trend of using composites in (a) military and (b) civil aircraft
(a) (b)
(c)
Figure 1-3: Impact and fatigue damage to composite panels. (a) Picture of a composite
panel after low velocity impact (b) Ultrasonic C-scan image showing internal
delamination (c) fatigue damage to a composite plate.
3
Despite their strength and low weight, composite materials are subject to damage
during fatigue, mechanical impact, and aging in a service environment. As a simple
example, dropping a ball on a composite plate might cause its fibers to break, the matrix
to crack, and delamination to occur between the layers. Fig. 1-3 (a) shows a photo of a
composite panel after low velocity impact. Although the damage is invisible at the
surface, a big delamination occurs within the panel. Fig. 1-3 (b) shows the image of the
delamination detected using a high frequency ultrasonic c-scan system [Bell 2004].
Fig. 1-3 (c) shows an image of a composite panel after a laboratory fatigue test
[Lissenden et al. 2006]. Fiber breaking and delamination at the top surface can be clearly
observed. These experiments were carried out in the Ultrasonic NDE lab at Penn State
University in collaboration with the Composite Manufacturing Technology Center for
composite panel preparation, the Engineering Nano Characterization Center for C-scan
imaging, and the Axial-Torsion Fatigue Lab for fatigue testing.
Damage in military aircraft and rotorcraft are even more serious due to erosion in
harsh environments, ballistic impact, and fatigue. If the composite material is damaged,
immediate repair would be required. Otherwise, these damages might lead to
malfunctioning or even catastrophic failure of the aircraft.
Traditionally, nondestructive evaluation (NDE) techniques are used to inspect the
structures on a periodic basis. However, significant damages to the structure could occur
during the intervals between inspections. Therefore, to provide early warning and timely
detection of damage, real time structural health monitoring (SHM) techniques are greatly
needed. Several methods currently under intensive research are vibration based methods,
fiber optic based methods and electromechanical impedance based methods, and
ultrasonic guided wave based methods.
Ultrasonic guided waves are the mechanical waves propagating along the
structure under the guidance of its boundaries. In structural health monitoring, waves are
usually excited with surface mounted or embedded piezo-electric transducers. They
propagate along the structure. The existence of damage and material degradation can be
detected by evaluating the guided wave signals. Fig. 1-4 shows the concept of active
damage detection using guided waves. Fig. 1-4 (a) is the pulse echo scenario, where a
4
single sensor is used in both wave generation and receiving. Fig. 1-4 (b) is the through
transmission scenario, where a second transducer is used to receive the signal (from [Gao
et al. 2006] ).
Besides the guided wave based method, some other methods are also under
research and development for structural health monitoring. These include localized
monitoring methods such as electromechanical impedance based methods, fiber optical
methods, and global vibration based methods such as frequency transfer function and
vibration mode shape analyses. Compared with localized monitoring, the ultrasonic
guided wave based method has the capability of monitoring a larger area with a few
sensors. Compared with the global vibration based technologies, ultrasonic guided waves
can provide better sensitivity to localized damage. Therefore, it has great potential for
applications in structure health monitoring.
Sensor
damage
Sensor
damage
Sensor
damage
(a)
SensordamageActuator
SensordamageActuator
(b)
Figure 1-4: Concept of damage detection using guided waves. (a) pulse echo (b) through
transmission.
5
Fig. 1-5 shows our PSU vision of a “Theoretically driven structural health
monitoring” design and process based on ultrasonic guided waves. The entire process is
divided into four levels, namely structural level, system level, sensor level, and physical
level. In the design phase, the structural level health monitoring requirements are input
into system level, sensor level, and physical level. In addition, the structural and material
properties are used in a physical level guided wave mechanics study. The results of the
guided wave mechanics studies are used for sensor, sensor network design, and optimized
testing design. The information from the sensor design will also be used in the system
level design. During the monitoring phase, excitation signals are sent out from
monitoring systems to sensors. The information of structural integrity carried in the
guided wave signal is collected with the sensors. Signal processing and decision making
is then carried out in the system level. Finally, a health assessment of the structure is
reported to meet the monitoring requirements.
Since the use of ultrasonic waves to interrogate damage is the basis of the entire
process, an understanding of ultrasonic guided wave propagation, excitation, and damage
interrogation mechanisms is critical for the advancement of the technology.
Figure 1-5: A vision of a “Theoretically driven structural health monitoring ” strategy
6
1.2 Literature review
A literature survey is presented on the subjects of ultrasonic guided wave
mechanics and the current practice of ultrasonic guided wave based structural health
monitoring. Since the objective of this study is more theoretically oriented, the literature
survey therefore has an emphasis on the theoretical and numerical aspects of wave
mechanics. In Section 1.2.1, early studies of guided waves, ultrasonic waves in multi-
layered media, and waves in composite materials are reviewed. Studies on guided wave
excitation and scattering are reviewed in Section 1.2.2. The state of the art of
experimental studies in structural health monitoring using guided waves is reviewed in
Section 1.2.3. Based on these reviews, challenges to guided wave study in composite
structures are summarized in Section 1.2.4.
1.2.1 Ultrasonic guided wave propagation in plates
Theoretical studies of ultrasonic guided waves can be retrieved back to a century
ago when Lord Rayleigh [Rayleigh 1885] studied surface acoustic waves. After that,
waves in isotropic plates, waves at solid-solid and solid-liquid interface were studied by
Lamb [Lamb 1917], Stoneley [Stoneley 1924], and Scholte [Scholte 1942] respectively.
All these wave types considered in the early studies have their displacement in the
sagittal plane, which is a plane consisting of the wave propagation and thickness
directions. In later research, these types of guided waves are generally referred to as
Rayleigh-Lamb type waves. Another type of wave is called a shear horizontal (SH) wave,
whose displacement is perpendicular to the sagittal plane. Love studied the shear
horizontal waves in a layer on half space, which was later called a Love wave [Love
1911]. Beyond these classic guided wave types, guided waves in other fundamental
geometries, such as rods and hollow cylinders, were also studied. They can be found in
these classical and recent text books [Victorov 1967; Achenbach 1973; Graff 1973; Auld
1990; Rose 1999].
7
Ultrasonic bulk wave propagation in isotropic media can be studied using a
Helmholtz decomposition or a partial wave theory. However, analytical solutions of wave
propagation in an anisotropic media can only be studied using the partial wave technique.
In a multi-layered media, the boundary conditions and interface conditions can be
formulated in two different ways using the partial wave technique. One is a transfer
matrix method, which is first developed by Thomson in 1950 and later refined by Haskell
[Thomson 1950; Haskell 1953]. In this method, a matrix is used to express the wave field
on the top surface of a layer as a function of the field at the bottom surface. A transfer
matrix, obtained by multiplying the matrices in all the layers, is used to describe the
relationship between the wave field at the bottom and the top surfaces of the multi-
layered structure. The final size of the transfer matrix does not increase with the number
of layers in the structure. Therefore, the transfer matrix method is efficient for structures
with many layers. However, it suffers from numerical instabilities when the product of
frequency and thickness is large. Dunkin used a delta operator method to alleviate the
numerical instability of the transfer matrix method [Dunkin 1965]. The other method is
called a global matrix method first used by Knopoff [Knopoff 1964] . A single matrix is
used to assemble all the interface and boundary conditions together. This method does
not have the numerical instability problem as in the transfer matrix method. However, the
global matrix method can be computationally expensive especially when the number of
layers is large.
Around the 1990s, studies on material property characterization for composites
using ultrasonic bulk waves have accelerated the theoretical study of wave propagation in
multilayered media. Nayfeh [Nayfeh 1995] provided a standard solution for transfer
matrix methods in generally anisotropic multilayered media. Experimental works were
carried out on the bulk wave transmission and reflection in unidirectional composites,
and cross ply composites [Chimenti D.E. 1990; Nayfeh et al. 1991]. Hosten, Tittmann,
and Castaings applied the transfer matrix method in viscoelastic material characterization
and bulk wave transmission and reflection studies in unidirectional, cross ply, and quasi-
isotropic composites [Hosten et al. 1987; Hosten 1991; Hosten et al. 1993]. They also
improved the delta operator method in the transfer matrix method [Castaings et al. 1994].
8
Recently, as modifications to the transfer matrix method, compliance matrix and stiffness
matrix methods were developed [Rokhlin et al. 2002].
The development of the transfer matrix method and global matrix method for bulk
wave propagation in composite materials were also adapted for leaky guided wave
analysis in composites. Mal generalized the global matrix method for guided waves in
anisotropic media [Mal 1988]. Chimenti and Nayfeh also studied the leaky Lamb waves
in composite materials including the wave propagation in a [02/902] cross ply composite.
Their experimental results matched well with numerical predictions for the waves
propagating in directions with structural symmetry. However, the reflection spectra in
non-symmetric directions do not match very well with the numerical expectation by
missing predicted reflection minimums. The authors attributed this discrepancy to the
coupling of wave displacements in three directions [Nayfeh 1995]. To be more specific,
we think that this should be related to the energy skew of the guided waves. Leaky
guided waves in composites were also studied by many other researchers; details can be
found in [Dayal et al. 1989; Yoseph Bar-Cohen et al. 1998] and the review paper by
Chimenti [Chimenti 1997]. A comprehensive review and comparison of the transfer
matrix method and the global matrix method for guided waves in multi-layered structures
can be found in Lowe [Lowe 1995]. The global matrix method formulation with an
emphasis for guided wave modal analysis used for long range NDE is addressed in detail
in [Rose 1999].
Besides the analytical models, a semi analytical finite element method (SAFE) is
also used to simulate guided wave propagation. The SAFE method was used for the first
time to study propagating wave modes in an arbitrary but uniform cross section in 1973
[Lagasse 1973]. In the SAFE method, the cross section of the waveguide is discretized
with finite elements and an analytical solution is assumed in the wave propagation
direction. After applying boundary conditions, dispersion curves describing wave
propagation mode possibilities can be obtained. After that, the SAFE method is used by
Dong and his colleagues to calculate both propagation and evanescent guided wave
modes in anisotropic cylinders [Huang et al. 1984]. Datta and his colleagues used the
SAFE method for cross ply composite plates considering wave propagation in the
9
structural symmetry direction [Datta et al. 1990]. Recently, the SAFE method was used
by Garvric, Hayashi, Rose and Lee to study the wave propagation in rods, rails, and pipes
[Gavric 1995; Hayashi et al. 2003; Lee 2006]. Lanza di Scalea and his colleagues
extended the SAFE method for guided waves in composites considering material
damping effect [Matt et al. 2005; Bartoli et al. 2006]. Compared with the transfer matrix
method, the SAFE method does not have the problem of numerical instability. In
addition, the solution of guided wave dispersion curve is obtained by solving an
eigenvalue problem. Therefore, complicated route searching is avoided in SAFE, and
thus eliminated the problem of possible missing roots in matrix based methods. Another
advantage of the SAFE method is its easy extension from an elastic model to a
viscoelastic model as compared with a two dimension root searching method in matrix
based methods.
1.2.2 Guided wave excitation and scattering
Analytical methods for solving ultrasonic guided wave excitation problems
generally falls into two categories, one is an integral transform method and the other is a
normal mode expansion technique. The integral transform method is studied by
transforming the excitation source into frequency and wavenumber domain. After a
harmonic system of equations with source terms is solved, the results are transformed
back into time and spatial domain. The integral transform method is discussed in [Rose
1999] for shear horizontal guided wave excitation in an isotropic plate. Recently,
Giurgiutiu [Giurgiutiu 2005] used the integral transform method to study Lamb wave
excitation in an aluminum plate. The frequency tuning effect of guided wave excitation is
also investigated. [Raghavan et al. 2005] extended the integral transform method into a
three dimensional analysis of guided wave excitation in isotropic plates. Mal and his
students studied the wave excitation phenomenon in unidirectional and cross ply
composites from a localized source especially to simulate the process of impact and
acoustic emission effects using the global matrix method and simplified models using
plate theory [Lih et al. 1995; Mal 2002; Banerjee et al. 2005]. Although the integral
10
transform method can be used to analyze wave excitation from localized sources, the
inverse process of the integral transform is usually very difficult. In addition, the
formulation of the integral transform method is usually very cumbersome.
The normal mode expansion (NME) technique is based on a reciprocity relation in
dynamics. The basic idea of NME is to express the actual wave field as a superposition of
orthogonal guided wave mode solutions. The general theory of NME in an elastic and
piezoelectric plate is described in a classic textbook by Auld [Auld 1990]. Ditri and Rose
used the NME technique for guided wave excitation in isotropic plates and pipes [Ditri et
al. 1992; Ditri et al. 1994]. These analyses are later used as a basis for guided wave
natural focusing and phased array focusing in pipes. The normal mode expansion
technique is closely related to the guided wave propagation mode analysis. Compared
with the integral transform method, a direct physical insight can be obtained from the
process of normal mode expansion.
The guided wave scattering problem is a very important yet difficult problem for
NDE and SHM. Analytical study of guided wave scattering was introduced with an S-
parameter technique in [Auld 1990]. In the past two decades, more and more research on
wave excitation and scattering were carried out using numerical methods such as the
finite difference method (FDM), the boundary element method (BEM), and the finite
element method (FEM). Among a large amount of numerical simulations on wave
scattering, some typical ones related to guided waves in composites are reviewed in the
following. A two-dimensional finite element method for wave propagation and scattering
study in composites was presented by Cawley and Guo [Guo et al. 1993]. A numerical
simulation tool based on the finite different method called LISA was developed and used
to simulate the wave propagation and scattering in a composite plate [Agostini et al.
2003; Lee et al. 2003; Lee et al. 2003]. Recently, commercial finite element software
packages, such as ABAQUS and ANSYS, are also used in the simulation of ultrasonic
guided waves in many structures including composites [Su et al. 2004; Yang et al. 2006].
Although these finite element methods can be used to calculate the guided wave field in
composites, these methods are computationally expensive and usually difficult for
handling large structures. In order to combine the benefit of numerical methods for
11
scattering field calculation and analytical techniques for a physical understanding of
guided waves, a hybrid BEM and NME technique was used by Rose and his colleagues
for isotropic plate structures [Cho et al. 1996; Cho 2000; Zhao et al. 2003]. A similar
method of combining NME with the FEM was recently used to simulate guided wave
excitation in a unidirectional composite plate [Moulin et al. 2000].
1.2.3 Experimental techniques in structural health monitoring of composites
Several decades ago, the need for nondestructive evaluation of composite
materials accelerated the analytical and numerical research of ultrasonic wave
propagation in multi-layered structures. As a result, this research has benefited the NDE
applications with new techniques using bulk waves and leaky lamb waves. In addition, to
water immersion tests, recently air coupled transducers were also used for nondestructive
evaluation of composites.
In recent years, the demand of condition based maintenance has created the need
for real time structural health monitoring and assessment using permanently attached
sensors. The research emphasis is turning toward guided wave monitoring using active
and passive sensing due to its long range monitoring capability. Surface attached and
embedded piezoelectric elements have become a common practice for structural health
monitoring of metallic and composite structures. A detailed review of experimental
techniques can be found in [Sohn et al. 2003; Staszewski et al. 2004]. Although a detailed
review is beyond the scope of this thesis, a non-comprehensive list of research groups
with great contributions in this field are Chang at Stanford University, Giurgiutiu at
University of South Carolina, Inman at Virginia Tech, Boller at Sheffield University, and
Sohn at Carnegie Mellon. Chang and his colleagues developed a SMART layer concept
with piezoelectric disc elements integrated in a polyimide substrate, such that they can be
easily applied to both metallic and composite structures [Lin et al. 2002]. Giurgiutiu and
his students studied the electromechanical impedance method for damage detection in
metallic aircraft panels using piezoelectric wafers [Giurgiutiu et al. 2005] . In addition,
they studied the lamb wave excitation in isotropic plates [Giurgiutiu 2005] and developed
12
piezoelectric sensor phased arrays for guided wave beam steering in an isotropic plate
[Giurgiutiu et al. 2004]. The SMART layer as well as the individual piezoelectric wafer
sensors have been used in many applications including composite plates. Many intelligent
signal processing and statistical analysis techniques have been explored by these
researchers [Staszewski 2002; Staszewski et al. 2004; Yu et al. 2005] . Although there is
a trend of increased consideration of ultrasonic guided wave mechanics in these
experiments, the application of guided wave mechanics in the SHM of composites is still
desperately needed.
1.2.4 Summary of the literature review and challenges for further study
Despite the great application needs, the study of guided wave mechanics in
composite materials still remains very challenging due to the following reasons.
1. Previous studies in composite materials are mostly in bulk wave
propagation and a leaky wave point of view. Mode analyses for long
range guided wave propagation in composites are very few.
2. Previous guided wave studies are mostly on isotropic structures or
anisotropic structures with wave propagation in structural symmetry
directions. Therefore, Rayleigh-Lamb (R-L) type waves are well
separated from the shear horizontal (SH) waves. The guided wave mode
notation, SH, A and S, are used to denote wave mode lines with shear
horizontal, antisymmetric R-L, and symmetric R-L waves. However,
for composite materials with general lay-ups, no distinct separation of
R-L and SH type waves is guaranteed. No structural symmetric
direction for wave propagation is guaranteed. No symmetric plane in
the thickness direction is guaranteed. Therefore, the SH, A, and S
notation system will no longer be universally applicable. In addition,
guided wave modes along a single dispersion curve line could exhibit
significantly different behaviors.
13
3. The complexities of guided wave propagation are significantly related
to the composite lay-up sequence. Unidirectional and cross ply
composites are studied in some previous works. However, only a few
works are reported on guided waves in composite materials with other
stacking sequences.
4. The energy skew effect is very important in guided wave propagation in
anisotropic media, especially composites. However, to the best of our
knowledge, no result is reported on the consideration of guided wave
skew angle except for unidirectional composites.
5. Viscoelasticity of composite materials will significantly affect long
range guided wave propagation potential. In previous literature,
viscoelasticity is considered in guided wave propagation using the
semi-analytical finite element method. However, the study of the
viscoelastic effect on guided wave excitation is not reported. A new
theoretical derivation is required to solve this problem.
6. In order to apply guided wave mechanics as a natural constitution of a
structural health monitoring system, the research of guided wave
mechanics should reach out from complicated mathematics formula to
easily controllable design features, such as the evaluation of the extent
of dispersion, extent of skew, and extent of attenuation of a guided
wave mode.
1.3 Thesis objectives
The objective of this study is to provide a set of simulation methods and tools to
integrate ultrasonic guided wave studies into the theoretically driven structural health
monitoring strategy. For the wave mechanics community, the outcome of this study will
be a deeper understanding of wave propagation and excitation characteristics in
composites. For the structural health monitoring community, the outcome will be some
14
useful tools for sensor design and signal analysis. These tools are also applicable to other
complex structures involving material anisotropy, multiple-layers, and viscoelasticity.
Specific objectives of the research, aimed at conquering the challenges just listed,
are as follows.
1. Develop computer programs to study the guided wave propagation in a
composite structure with an arbitrary lay-up sequence. Use a quasi-
isotropic composite plate as a specific example.
2. Study the skew angle effect of wave propagation in composite plates.
3. Develop computer programs to study the wave excitation principles in
composite laminates. Use the quasi-isotropic laminate as an example.
4. Derive a new theoretical procedure for the wave excitation study in
composites considering material viscoelasticity and implement the new
procedure into computer programs.
5. Develop simple physically based features to evaluate wave dispersion,
excitation, and sensitivity.
6. Comprehensively consider all possible features for guided wave mode
selection and to provide a mode selection framework that is directly
applicable to structural health monitoring design.
7. Develop computer programs for sensor design to achieve appropriate
mode selection and mode control.
8. Use FEM simulation and experiments to validate the performance of
the simulation tools.
1.4 A preview of the thesis content
This thesis contains 10 chapters and two appendices.
Chapter 1 presents the problem statement of guided wave mechanics studies in
composites, a literature review, and a description of the thesis objectives.
Chapter 2 introduces the theories of the global matrix method and the semi-
analytical finite element method in elastic composite laminates. Guided wave mode
15
characteristics including phase velocity, group velocity, energy velocity, skew angle,
displacement and stress distributions, power flow and energy distributions are defined.
Chapter 3 studies the guided wave propagation in a 16 layer quasi-isotropic
composite plate with [(0/45/90/-45)s]2 lay up sequence. Phase velocity dispersion curves,
group velocity dispersion curves, and skew angle dispersion curves for different wave
propagation directions are all obtained. A three dimensional dispersion surface is used to
efficiently express the anisotropy of wave propagation.
Chapter 4 discusses the problem of wave excitation in composites assuming
elastic material properties. Based on the normal mode expansion theory, influence of the
guided wave mode and the excitation source on wave excitation is studied. A new
dispersion coefficient feature is used to study guided wave dispersion phenomenon. A
guided wave beam spreading feature is also defined to evaluate the beam spreading of a
guided wave package due to wave skew. Finally, an algorithm is developed to predict the
transient guided wave field from a finite source.
Chapter 5 presents some numerical validations of the wave propagation and
excitation characteristics in the quasi-isotropic composite plate using finite element
analysis.
Chapter 6 studies the influence of material viscoelasticity on guided wave
propagation and excitation. Attenuation dispersion curves for a composite material are
introduced. A new normal mode expansion technique is derived from a real reciprocity
relation. The wave propagation features are compared in the case of considering
viscoelasticity versus the case of an elastic assumption.
Chapter 7 covers the study of guided wave mode sensitivity to overall material
degradation as well as localized damage. A wave mode sensitivity feature is defined
specifically for delamination detection based on a theory of guided wave scattering.
Chapter 8 introduces a new platform for comprehensive wave mode evaluation
and selection. A goodness value is defined for each guided wave mode under specific
design requirements.
Chapter 9 presents some experimental validations of the theoretical study in terms
of guided wave propagation, excitation, and sensing.
16
Chapter 10 summarizes the thesis and recommends some future research
directions.
Two appendices are included in this thesis. Appendix A is on guided wave
imaging techniques and some experimental studies of SHM. Appendix B is a
nontechnical abstract of this thesis.
Chapter 2
Wave propagation theory in composite laminates
Composite laminates are commonly fabricated by stacking unidirectional
composite prepregs with a desired lay-up sequence. After the composite is properly
cured, a multilayered structure is formed with all the layers bonded together. Therefore, a
composite laminate is commonly modeled as a multilayered medium with elastic and
anisotropic material properties. In this chapter, the problem formulation and solution
techniques using a global matrix method and a semi-analytical finite element method are
presented. Physical understanding of the phase velocity, group velocity, wave structure,
and skew angle of guided waves mode are discussed.
2.1 Problem formulation
Wave propagation in a multi-layered structure are affected by the thickness ( h ),
density ( ρ ), and material elastic properties in each layer. A sketch of a multilayered
structure and the coordinate system for wave propagation are presented in Fig. 2-1.
Figure 2-1: A coordinate system for wave mechanics study of a multi-layered structure
18
The governing equation, constitutive equation, and strain displacement equation
in elasticity are shown in Equation 2.1 , Equation 2.2 , and Equation 2.3, respectively.
Here, ρ is the density of the material, iu is the displacement, ijσ , ijs and ijklc are stress,
strain, and elastic stiffness constants respectively. The indices ( ji, ) refers to the three
coordinate directions in a Cartesian system, .3,2,1, =ji
In the contracted engineering stress and strain format, the constitutive equation is
expressed in Equation 2.4.
The relationship of strains in the engineering format and the tensor format are as follows.
In order to study guided wave propagation, the elastic constants of all the layers
must be expressed in the global coordinate system, (x1, x2, x3). For a composite material,
this can be achieved with a vector and tensor rotation process from lamina properties
[Nayfeh 1995]. Equation 2.6 expresses the rotation procedure for a fist order, a second
order, and a fourth order tensor. The rotation matrix is in Equation 2.7. The tensors
before and after rotation are expressed with prime and without prime, respectively. Here,
θ is the angle of rotation from the original system to the new system. The value of θ is
positive when the rotation is counterclockwise.
kj
lijkl
i
xx
uc
t
u
∂∂
∂=
∂
∂ 2
2
2
ρ (2.1)
klijklij scσ = (2.2)
)(2
1
l
k
k
l
klx
u
x
us
∂
∂+
∂
∂= (2.3)
=
12
13
23
33
22
11
665646362616
565545352515
464544342414
363534332313
262524232212
161514131211
12
13
23
33
22
11
S
S
S
S
S
S
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
σ
σ
σ
σ
σ
σ
(2.4)
≠=
==
)(2
)(
jisS
jisS
ijij
ijij (2.5)
19
2.2 The global matrix method
Based on the theory of elasticity described in section 2.1, guided wave
propagation in multi-layered elastic media is studied in this section using the partial wave
theory and a global matrix method.
2.2.1 The partial wave theory
A partial wave is a harmonic plane wave solution that satisfies Equation 2.1 to
Equation 2.3. Assuming a plane wave propagating in the x1 direction, the wave field is
independent on the x2 coordinate. Equation 2.8 is a trial solution for a partial wave field
[Nayfeh 1995].
Substituting this trial solution into the governing equation and neglecting the common
term ))(exp()( 31
2tcxxii p−+ αξξ , a Christoffel equation is expressed in Equation 2.9 .
'
jiji uu β= ,
'
ijnjmimn σββσ = ,
'
ijklploknjmimnop cc ββββ=
(2.6)
−=
100
0cossin
0sincos
θθ
θθ
βij (2.7)
))(exp( 31 tcxxiUu pll −+= αξ , ( 3,2,1=l ) (2.8)
20
A solution of Equation 2.9 is called a partial wave. For a given value of phase velocity
pc , there are six solutions of α . For each α , there is a nontrivial solution to
),,( 321 UUU .The ratios of 1U , 2U , and 3U define the polarization of the displacement
field. The entire wave field in the layer can be expressed as a linear combination of the
six partial waves in Equation 2.10.
2.2.2 Boundary conditions and the global matrix method
Six partial wave solutions have been obtained for each layer from the analysis of
the Christoffel equation. The next step is to determine the weighting coefficients kB that
satisfy the boundary conditions and interface conditions. For ultrasonic waves in a free
plate, tractions at the top and bottom surfaces are zero. At the layer interfaces, the
displacement and stress components normal to the ),( 21 xx surface must be continuous.
The mathematical formulation of the boundary condition and interface conditions are
expressed in Equation 2.11.
22
33355533
2
3445365623
22
44466622
2
3555131513
2
4556141612
22
55151111
3
2
1
332313
232212
131211
2
)(
2
)(
)(
2
0
0
0
p
p
p
cCCCA
CCCCA
cCCCA
CCCCA
CCCCA
cCCCA
U
U
U
AAA
AAA
AAA
ραα
αα
ραα
αα
αα
ραα
−++=
+++=
−++=
+++=
+++=
−++=
=
(2.9)
))(exp( 31
6
1
tcxxiUBu pklk
k
kl −+=∑=
αξ , ( 3,2,1=l ) (2.10)
21
In the global matrix method, all boundary and interface conditions are assembled into a
single linear system of equations with kB as unknown variables. The dimension of the
global matrix D in Equation 2.12 is N6 by N6 . The 6N unknown variables are
assembled into a vector B .
Wave number ξ and phase velocity pc are two parameters in Equation 2.10. These
parameters are assembled into elements of the D matrix. In order to get non-trivial
solutions of B in Equation 2.12, the determinant of the matrix D must be zero.
Equation 2.13 is a transcendental equation of ξ and pc . The pair of ( pc,ξ ) that satisfies
the equation is called a wave mode. Analytical solutions of ),( pcξ for the transcendental
equation are not easy to obtain. Numerical root searching methods are usually used
instead. Usually, the solutions form a set of curves describing the constrained relation
between ξ and pc . In addition, the change of pc with respect to wave number will lead to
waveform dispersion during guided wave propagation. Therefore, the curves are also
called phase velocity dispersion curves. Using the relation between wave velocity, wave
number, and wave frequency, commonly two dimensional dispersion curves can be
expressed in any two out of the three quantities.
2.2.3 Wave field solutions
For each ( pc,ξ ) pair that satisfies Equation 2.13 , a non-trivial solution of B can
be obtained by solving Equation 2.12. The solutions of B are not unique for the
interfaceslayeratcontinuous,,,,,
surfacebottomandtopat0,,
333231321
333231
σσσ
σσσ
uuu
= (2.11)
0=• BD (2.12)
0=D (2.13)
22
homogeneous linear system of equations. Any scalar multiplication of this B is also a
solution to the equations. The physical meaning of this is that a guided wave mode may
have different amplitude. A detailed description of the wave displacement, particle
velocity and strain in the nth
layer are listed in Equation 2.14 through Equation 2.16.
Constitutive relations Equation 2.2 can be used to obtain the stress field solution.
Equation 2.17 is a general formula for the stress components. Here,
6,...2,1=I are the six stress components in the abbreviated notation. The detailed
expressions of the coefficients are provided in Equation 2.18.
))(exp(])(exp([ 1
6
1)1(6
3 txixiUBun
nk
klkkl ωξαξ −= ∑+−=
( 3,2,1=l ) (2.14)
l
l
l uit
uv )( ω−=
∂
∂= (2.15)
))(exp())(exp()(
))(exp())(exp()()(
))(exp())(exp()(
))(exp())(exp()(
0
)(
1
6
1)1(6
32
1
2
2
112
1
6
1)1(6
331
1
3
3
113
1
6
1)1(6
32
2
3
3
223
1
6
1)1(6
33
3
333
2
222
1
1
111
txixiUBix
u
x
uS
txixiUUBix
u
x
uS
txixiUBix
u
x
uS
txixiUBix
uS
x
uS
uix
uS
n
nk
kkk
n
nk
kkkkk
n
nk
kkkk
n
nk
kkkk
ωξαξξ
ωξαξαξ
ωξαξαξ
ωξαξαξ
ξ
−
=
∂
∂+
∂
∂=
−
+=
∂
∂+
∂
∂=
−
=
∂
∂+
∂
∂=
−
=
∂
∂=
=∂
∂=
=∂
∂=
∑
∑
∑
∑
+−=
+−=
+−=
+−=
(2.16)
))(exp()exp()( 1
6
1)1(6
3 txixiBMin
nk
kkIkI ωξξαξσ −
= ∑
+−=
(2.17)
23
These solutions are harmonic functions of t and 1x . The part within the brackets
are functions of 3x .These depth dependent profiles can be obtained numerically by
evaluating the quantities in the bracket for a given series of 3x . Usually, these terms are
complex values. The amplitude and phase information of the field quantities are within
these complex wave structure values.
kkkkkkkkkk
kkkkkkkkkk
kkkkkkkkkk
kkkkkkkkkk
kkkkkkkkkk
kkkkkkkkkk
UCUUCUCUCUCM
UCUUCUCUCUCM
UCUUCUCUCUCM
UCUUCUCUCUCM
UCUUCUCUCUCM
UCUUCUCUCUCM
26631562463351166
25631552453351155
24631452443341144
23631352343331133
22631252243231122
21631152143131111
)(
)(
)(
)(
)(
)(
+++++=
+++++=
+++++=
+++++=
+++++=
+++++=
ααα
ααα
ααα
ααα
ααα
ααα
(2.18)
2.3 The semi-analytical finite element method
A semi-analytical finite element (SAFE) method is introduced in this section.
Different from the analytical methods, a one dimensional finite element was used to
discretize the composite in the thickness direction. Fig. 2-2 shows a sketch of a 3 node 1
dimensional element. The corresponding isoparametric element is shown on the right side
of Fig. 2-2. All the physical elements are mapped to this element, where the parameters
are -1, 0 and 1 for the three nodes. In this section, the coordinate system is expressed with
(x, y, z).Three degrees of freedom are assigned to each node for the displacements in
three directions. The nodes in each element are designated with number 1, 2, and 3.
Detailed description of one dimentional isoparametric finite element can be found in
[Cook 2001].
Figure 2-2: A sketch of a one dimensional three node isoparametric element
Element 1
Element 2
Node 1, z1
Node 2, z2
Node 3 z3
ξ=-1
ξ=0
ξ=1
25
[ ]
=
3
2
1
321
z
z
z
NNNz ,
Here,
+=
−=
−=
)(2
1
)1(
)(2
1
2
3
2
2
2
1
ξξ
ξ
ξξ
N
N
N
(2.19)
=
z
y
x
z
y
x
z
y
x
z
y
x
u
u
u
u
u
u
u
u
u
NNN
NNN
NNN
u
u
u
3
3
3
2
2
2
1
1
1
321
321
321
000000
000000
000000
(2.20)
The shape functions for position interpolation are listed in Equation 2.19. The
element shape functions are arranged in Equation 2.20. The strain field is calculated
according to Equation 2.3 and Equation 2.5. After the 1x9 vector of nodal solutions are
denoted with ),( txeu , the strain field is expressed in Equation 2.21.
The expressions for the derivative and Jacobian function are in Equation 2.22.
xeexeez
xy
xz
yz
zz
yy
xx
,21,,
010
100
000
000
000
001
000
001
010
100
000
000
ububNuuNε +=
+
=
=
ε
ε
ε
ε
ε
ε
(2.21)
26
Hamilton’s principle is used to derive the finite element formulation of the characteristic
equation.
Here, T, U, and VE are the kinetic energy, strain potential energy, and the energy of the
external source respectively. Equations 2.24 are the equations for these energy
quantities.
Substituting Equation 2.24 into Equation 2.23 , the variation equation can be written as
Equation 2.25 if the external force for an element is only a function of x acting at the
nodes.
Here, the expressions for m , 11k , 12k , 21k ,and 22k are expressed in Equation 2.26
ξξ
ξ,,,
1NNN
Jdz
dz ==
+−−==
3
2
1
2
12
2
1
z
z
z
d
dzJ ξξξ
ξ
(2.22)
0)}({1
0
=+−∫t
tE dtVUTδ (2.23)
∫∫∫= dVTT
uu && ρ2
1
∫∫∫= dVUTCεε
2
1
∫∫ Γ−= dV T
E fu
(2.24)
∫ ∫ =+−−−−1
0
1
0
0)(2
1,22,21,,1211
t
t
x
x
T
exe
T
xee
T
xexe
T
ee
T
ee
T
e dxdtfuukuukuukuukuumu &&δ (2.25)
∫∫= dydzT NNm ρ
∫∫= dydzT
1111 Cbbk
∫∫== dydzTT
212112 Cbbkk
∫∫= dydzT
2222 Cbbk
(2.26)
27
After assembling all the elements, and considering the stress free boundary conditions,
the variational equation can be expressed as Equation 2.27.
Using L to denote the integrand in Equation 2.27, the variational equation can be
evaluated using Equation 2.28 (Euler’s Equation).
The detailed expression for Equation 2.28 is Equation 2.29.
The harmonic wave propagation solution is used in Equation 2.30.
Equation 2.29 becomes Equation 2.31 by omitting the exponential term.
This is equivalent to the linear systems of equations in Equation 2.32.
This is an eigen value problem of ξ when the frequency of the wave mode is given. The
eigen value of the linear system of equations produces dispersion curves for the structure.
∫ ∫ =−−−−1
0
1
0
0)(2
1,22,21,,1211
t
t
x
xxe
T
xee
T
xexe
T
ee
T
ee
T
e dxdtUKUUKUUKUUKUUMU &&δ (2.27)
0)()(,
=∂
∂−
∂
∂−
∂
∂
xnnn u
L
dx
d
u
L
dt
d
u
L
& (2.28)
0)( 11,2112,22 =−−−− UMUKUKKUK &&xxx (2.29)
)(exp(0 txi ωξ −= UU (2.30)
0])([ 0
2
11211222
2 =−+−+ UMKKKK ωξξ i (2.31)
0)( =− QBA ξ
Here,
−−
−=
)(
0
2112
2
11
2
11
KKMK
MKA
iω
ω,
−
−=
22
2
11
0
0
K
MKB
ω
=
0
0
U
UQ
ξ
(2.32)
28
Nodal displacement solutions can be obtained in SAFE from Equation 2.32. The
following procedure can be used to obtain the wave structures of a guided wave mode.
1. Substitute the nodal displacement solution into Equation 2.30 to get
the complete form of the nodal displacement solution as a function of x
and t.
2. Use Equation 2.20 to get the interpolated displacement field.
3. Use Equation 2.21 to get the strain field.
4. The stress field can be obtained from the strain field and the
constitutive equation.
2.4 Important derived guided wave properties
2.4.1 Power flow and energy density
[Auld 1990] studied the power flow distribution within an ultrasonic bulk wave
field and simple guided wave field. In this thesis, the application is extended for multi-
layered composite laminates. Both the real form and complex form Poynting’s vector are
derived based on the wave displacement distribution and stress distribution results.
The real form of Poynting’s vector is defined in Equation 2.33 .
Here, v is the particle velocity vector; σ is the stress field tensor. The real form
Poynting’s vector is a function of ( 31 , xx and t ) since the plane guided wave mode is
assumed to be independent of 2x . The Poynting’s vector is the power flow density at a
particular point within the wave field. This vector usually has three components.
Therefore, although the wave vector is in the 1x̂ direction, the power flow can have 2x̂ and
3x̂ components. This is the reason why energy skew is a very important issue in
anisotropic media.
σvP •−=real (2.33)
29
The complex form of the Poynting’s vector defined in Equation 2.34, is only a function
of 3x .
The energy density within the wave field can be expressed as a combination of the kinetic
energy density and the strain energy density. The expressions are in Equation 2.35.
After substitution of the real solution of particle velocity, stress and strain, it is easy to
separate the energy densities into a constant term and a time variation term with ω2
angular frequency.
2.4.2 Group velocity and energy velocity
Group velocity is the velocity of a wave package. It is defined by Equation 2.37.
For a guided wave mode propagating in the 1x direction, the wave vector 1ˆ)( xk ξ=
r.
Therefore, the group velocity of the wave package in the 1x direction is in Equation 2.38.
2
σvP
•−=
∗
(2.34)
sk EEE +=
)(2
)(22
2
3
2
2
2
1
2vvvE
T
realrealrealk ++=•==ρρρ
vvv
)()(2
1
2
1::
2
1realIrealI
T
realreals SE σ=•== σSscs
(2.35)
))(2sin())(2cos(
))(2sin())(2cos(
12110
12110
txEtxEEE
txEtxEEE
ssss
kkkk
ωξωξ
ωξωξ
−+−+=
−+−+= (2.36)
kd
dcg
rω
= (2.37)
df
dcfc
c
d
dcc
c
d
cd
d
dc
p
p
p
p
p
p
p
g
−
=
−
===22
)(
1
ωω
ω
ωξ
ω
(2.38)
30
Another wave mode quantity is the energy transmission velocity. This can be derived
from the power flow and the energy density.
1xP is the component of the complex Poynting’s vector in the 1x̂ direction. This provides
an average power flow over a time period of T. Within this time period T, the total energy
goes across the plane 01 =x , with thickness H and width W is in Equation 2.39.
The total energy carried in the wave mode within a wave length λ is the integral of
energy density Ek and Es over the volume of the box.
The energy transmitted through the cross-section in time period T can be used to fill the
wave field of length L1, which is expressed in Equation 2.41.
Figure 2-3: A sketch of the power flow in a guided wave mode for the derivation of
energy velocity.
∫=H
xcross dxxPTWE0
33)(1
(2.39)
∫
∫ ∫
∫
+=
−++
−+++=
+=
H
sk
H
sk
sksk
V
sktotal
dxEEW
dxdxtxEE
txEEEEW
dVEEE
0300
03
01122
11100
)(
)))(2sin()(
))(2cos()((
)(
λ
ωξ
ωξλ (2.40)
λ
H
x1 x2x3 W
Px1
31
The energy transmission velocity of the wave mode in the 1x̂ direction is then given in
Equation 2.42.
For the guided wave propagation in elastic media, energy velocity is the same as group
velocity [Auld 1990].
2.4.3 Skew angle
Based on the energy transmission of a guided wave mode, the skew angle is
defined in this thesis as the ratio of energy transmission rate in the 2x and the 1x
directions. The expression is shown in Equation 2.43.
The same as with group velocity, skew angle can also be derived from a spatial
variation of a phase velocity dispersion curve. A slowness value is defined for each wave
mode by taking the reciprocal of the phase velocity value according to Equation 2.44.
For a given frequency and mode index, the value of phase velocity varies with the wave
propagation direction. Therefore, the slowness also varies with direction. Fig. 2-4 shows
the relation between the direction of phase velocity, the direction of power flow, and the
λ
λ∫
∫+
==H
sk
H
x
average
across
dxEEW
dxPTW
E
EL
0300
03
1
)(
1
(2.41)
∫
∫+
==H
sk
H
x
energy
dxEE
dxP
T
LV
0300
03
1
)(
1
(2.42)
∫
∫=Φ
H
x
H
x
dxP
dxPtg
03
03
1
2
)( (2.43)
pcSlowness
1= (2.44)
32
skew angle in a slowness profile. In this figure, the curve is the slowness profile with
respect to wave propagation angle (θ). The dashed line is the tangent of the profile for
direction (θ), The surface normal direction is the actual power flow direction. The angle
between the power flow direction and the wave vector direction is defined as the skew
angle of the guided wave mode.
2.4.4 Wave field normalization
It is indicated in section 2.2.3 that the solution of the weighting coefficients for
the partial waves (B) is not unique. Any constant multiplication of the current solution is
also a solution to the homogeneous linear system of equations. Therefore, in order to
compare the performance between all the guided wave modes, a normalization in the
wave structure solution is necessary. Integrated power flow in the wave vector direction
along a cross section of the entire wave guide is used as a normalization factor.
Equation 2.45 shows the mathematical formulation of the normalization factor.
Equation 2.46 shows the formulation of the normalized wave field quantities.
Figure 2-4: Sketch of slowness profile and skew angle. (Modified from [Rose 1999])
)Re()ˆRe(0
30
31 1∫∫ =•=H
x
H
norm dxPdxxFac P (2.45)
33
After the normalization, the power flow along the 1x direction is 1. The displacement,
velocity, stress and strain are the values corresponding to unit power flow in the
1x direction. Phase velocity, group velocity, energy velocity, and skew angle will not be
affected by the normalization.
normnorm
normnorm
normnorm
normnorm
Fac
Fac
Fac
Fac
σσ
SS
vv
uu
=
=
=
=
normnorm
normnorm
FacEE
Fac
=
= PP (2.46)
Chapter 3
Guided wave propagation in quasi-isotropic composites
The global matrix method (GMM) and the semi-analytical finite element (SAFE)
methods are both implemented in computer programs. The results of these two methods
are presented in this chapter. After comparing their performances, a hybrid technique that
combines the SAFE and GMM techniques is proposed. By using the SAFE for dispersion
curve generation and GMM for wave structure calculation, a fast and accurate simulation
can be performed for any laminated plate structure.
In addition to the phase velocity dispersion curves, the group velocity and skew
angle dispersion curves are also obtained. To account for the anisotropy of composite
laminates, a new dispersion surface concept, as opposed to the commonly used dispersion
curve presentation, is put forward for a better understanding of guided wave propagation
in composite laminates.
3.1 A numerical model of composite laminates
Quasi-isotropic composite laminates are of particular interest as they are
commonly used in aircraft structures. A 16 layer quasi-isotropic composite made of
IM7/977-3 prepreg is studied in this chapter. The average layer thickness of the
composite is 0.2 mm. Therefore, the total thickness of the sample structure is 3.2 mm.
Fig. 3-1 shows a sketch of the lay-up sequence and coordinate systems.
35
The sketch of the lay-up sequence is illustrated on the left side of Fig. 3-1. Two
coordinate systems are defined on the right side of Fig. 3-1. One is the (x, y, z)
coordinate, which is defined according to the fiber directions of the lay-up sequence. The
x direction is in the fiber direction of the first layer. The fiber direction of the second
layer is at 45o. The other coordinate system is the (x1, x2, x3) system, which is associated
with the wave propagation. The x1 direction is the wave vector direction, which is θo in
the (x, y, z) coordinate system. The plane wave propagating along the x1 direction is
independent of x2. After we rotate all the material properties into the (x1, x2, x3)
coordinate system, the theories described in Chapter 2 can be used to generate the
dispersion curves and wave structures for the waves propagating in this direction. By
changing θ from 00 to 180
0, the wave propagation characteristics in all the directions can
be obtained.
Material properties of the IM7/977-3 CFRP lamina are listed in Tab. 3-1. After
[Schoeppner et al. 2001].
Figure 3-1: The sketches of the lay-up sequence and the wave propagation in a 16 layer
quasi-isotropic composite laminate. Layup sequence is [(0/45/90/-45)s]2
Table 3-1: Material properties of IM7/977-3 unidirectional composite properties
Density (g/cm3)
*
E1 (GPa)
E2 (GPa) G12 (GPa) G23 (GPa) ν12 ν23
1.608 172 9.8 6.1 3.2 0.37 0.55 * Measured with mass and volume in our quasi-isotropic specimen
36
3.2 Phase velocity dispersion curves
A comparison of the phase velocity dispersion curves obtained from the GMM
and the SAFE method are shown in Fig. 3-2.The mode points in the global matrix
method are searched with skm /10 8− precision. In the SAFE method, one element is used
for each ply, the error between the points from SAFE method and the points from GMM
are all below skm /10 2− . However, the SAFE technique is more computationally efficient
and robust than the global matrix method. A computer with 3.4 GHz CPU and 3.5 GB
RAM was used in our simulation. For this particular application, the computation time
using SAFE method is 600 s when the step in frequency is 0.002 MHz. However, when
the GMM method is used 6300 s is used when the frequency step is 0.01MHz. Therefore,
the estimated computing speed of SAFE is about 50 times faster than that of the GMM
method in this case.
In a single layer isotropic plate, the wave modes are separated into Lamb waves
and Shear Horizontal (SH) waves. The modes are also numbered as symmetric (S) and
Figure 3-2: A comparison of phase velocity dispersion curves obtained from two
methods. Continuous lines: SAFE; Blue dots: GMM. Wave propagates in the 0o
direction.
37
antisymmetric (A) groups based on the wave structure. However, in Fig. 3-2, there is no
direct separation between Lamb type and SH type waves. The polarization of a wave
field may vary significantly with frequency along each curve. Therefore, the traditional
Lamb wave and SH wave numbering system cannot be applied. In addition, for
unsymmetrical composite laminates, the symmetric (S) and antisymmetric (A) mode
numbering system cannot be applied either. Therefore, a new wave mode system is
created by numbering all the possible mode lines in a numerical order. The first to sixth
mode lines are noted on the dispersion curves in Fig. 3-2. Both mode number and
frequency are needed to specify a wave mode on the dispersion curves.
Ultrasonic wave propagation characteristics are direction dependent in anisotropic
media. The direction dependence is still true for quasi-isotropic composites. Fig. 3-3
shows the dispersion curves of the first three wave modes propagating in the 0o, 45
o, 90
o,
and -45o directions. The extent of direction dependence also varies with frequency. For
example, the low frequency region of the second and third modes does not vary with
direction. However, the low frequency region of the first wave mode varies with direction.
The phase velocity values for a frequency of 10 kHz are listed in Tab. 3-2.
This phenomenon can be explained with a static proximation. The low frequency
limits of the first three modes are equivalent to a bending of the laminate, an in-plane
shear motion, and an in-plane tensile and compression motion. Laminate plate theory
indicates that for quasi-isotropic laminates, the responses are quasi-isotropic under in
plane tension, compression, and shear. However, the response is not quasi-isotropic under
bending.
Table 3-2: Phase velocity values at the low frequency limit of the dispersion curves (10
kHz).
Propagation direction Mode 1 Mode 2 Mode 3
0o
0.618 3.946 6.723
45o 0.603 3.946 6.723
90o 0.577 3.946 6.723
-45o 0.579 3.946 6.723
38
(a)
(b)
(c)
Figure 3-3: Phase velocity dispersion curves for guided wave modes in different
propagation directions. (a) mode 1, (b) mode 2 (c) mode 3.
39
A more efficient way to evaluate the anisotropy of guided wave modes is to plot
the phase velocity with respect to propagation direction. Some sample plots are shown in
Fig. 3-4.
A more general view of the phase velocity dispersion curve is to assemble all the
dispersion curves in different directions into a 3D dispersion surface. The dispersion
surfaces for the first three modes are shown in Fig. 3-5. The dispersion surfaces for mode
4, 5, and 6 are shown in Fig. 3-6. In all these dispersion surfaces, the color corresponds
to the phase velocity value.
(a) (b)
(c)
Figure 3-4: Angular profiles of the phase velocity dispersion curves at a frequency of
200kHz, (a) mode 1 (b) mode 2 (c) mode 3. The units in the radius is km/s.
40
(a)
(b)
(c)
Figure 3-5: Phase velocity dispersion surfaces of the first three modes, (a) mode 1 (b)
mode 2 (c) mode 3.
41
(a)
(b)
(c)
Figure 3-6: Phase velocity dispersion surfaces of mode 4 to mode 6, (a) mode 4 (b) mode
5 (c) mode 6.
42
3.3 Group velocity and energy velocity dispersion curves
The group velocity and the energy velocity are equivalent for an elastic system.
Both of them can be used as a measurement of wave transmission speed along the wave
vector direction. Group velocity is calculated using Equation 2.38, and energy velocity is
calculated using Equation 2.42.
Fig. 3-7 shows a comparison between the group velocity dispersion curve
obtained from the SAFE method and the energy velocity dispersion curve obtained from
the GMM. Again, the correlation of the group velocity and energy velocity curves is very
good. The average error in the data obtained from these two methods is within 0.01 km/s.
In Fig. 3-2, an example of a wave mode interaction region between mode 2 and
mode 3 around 0.4 MHz is shown with a circle. This interaction is further validated in
the group velocity dispersion curve. The results proved that the dispersion curves change
Figure 3-7: Comparison between the group velocity dispersion curves obtained from
SAFE methods and energy velocity curves from the global matrix method. Continuous
lines: SAFE; Blue dots: Global matrix method. Wave propagation in 0o direction.
43
their directions at the mode interaction region. In the group velocity dispersion curves, a
sharp reduction of group velocity is seen for both modes and is a result of mode
interaction.
Similar to the plots shown for the phase velocity dispersion curves, a comparison
of the group velocity dispersion curves of the first wave mode in different propagation
directions are shown in Fig. 3-8.
Shown in Fig. 3-9 are polar plots of the group velocity with respect to the wave
vector direction. The results are given for different modes and frequencies. It is obvious
that for higher order modes the group velocity value of a wave mode is strongly
dependent on propagation direction. In addition, wave propagation behavior is not
symmetric according to the 0o direction. The three dimensional group velocity surfaces
for mode 1 and mode 2 are shown in Fig. 3-10. This describes the wave velocity as a
function of both frequency and propagation angle.
Although only the first several mode surfaces are discussed in this section, other
guided wave modes can be considered in future mode selection processes.
Figure 3-8: Mode 1 group velocity dispersion curves for different propagation directions.
44
(a) (b)
Figure 3-9: Group velocity dispersion curves for different propagation directions and
frequencies. (a) mode 3 (b) mode 4. Note: There are only four lines in (b) because 0.2 MHz is
below the cut-off frequency of mode 4.
(a) (b)
Figure 3-10: Group velocity dispersion surface. (a) mode 1 (b) mode 2
45
3.4 Skew angle dispersion curves
Fig. 3-11 shows skew angle curves for the quasi-isotropic composite laminate.
The continuous lines are obtained from the SAFE method using the derivative of the
slowness curve with respect to the propagation direction. The blue dots are obtained from
the global matrix method using the power flow analysis. (See details in Section 2.4.1 ). It
is very promising that good agreement is achieved between the results from the two
computational methods. This again validated the accuracy of both methods for wave
mechanics studies in composites.
Similar display techniques can be used for skew angle curves as was used for the
phase velocity and group velocity dispersion curves. Fig. 3-12 shows some sample skew
angle curves with respect to propagation direction. It is quite valuable to know that even
for quasi-isotropic laminates, the skew angle of mode 4 at 1MHz ranges from -40o to 40
o.
In addition, at low frequencies, the skew angle is small. When the frequency is increased,
skew angle becomes a very important issue affecting wave propagation. Therefore,
Figure 3-11: A comparison between skew angle curves obtained from the SAFE and the
GMM. Continuous lines: SAFE; Blue dots: GMM. Wave propagation in the 0o direction.
46
understanding the nature of wave skewing is very important to avoid or to employ this
phenomenon.
An example of a three dimensional skew angle surface is shown in Fig. 3-13 for
the first wave mode. The relation between skew angle, frequency and propagation
direction is illustrated.
(a) (b)
Figure 3-12: Variation of guided wave skew angle with respect to wave propagation
directions. (a) Mode 1-3 at 200kHz (b) Mode 1-4 at 1.0 MHz.
Figure 3-13: Skew angle surface of the first wave mode
47
3.5 Wave structure analysis
The wave field profile along the thickness direction of a guided wave mode is
called a wave structure. Wave structures are usually expressed in terms of displacement
or stress. In this thesis, the analysis is extended to particle velocity, strain, energy density,
and power flow density. These are all very important parameters to evaluate the
performance of a wave mode for SHM applications.
Chapter 2 provided two methods for wave structure calculation. In this section,
the performances of these methods are compared. Fig. 3-14 shows the normalized wave
structure for the first wave mode at 0.2 MHz. The blue line is the result from SAFE and
the red line is the result from GMM. For the displacement along the wave propagation
direction (x1), the difference between the red line and the blue line is very small.
However, the stress distribution calculated from SAFE is not as accurate as the
displacement solutions. An interface discontinuity occurs because of the interpolation
process. This is a common result of finite element methods. The GMM produces an
accurate solution for both displacement and stress. The stress continuity condition at the
interface and the stress free boundary conditions are all met.
Fig. 3-15 shows the displacement, stress and power flow wave structures for the
first wave mode at 0.2 MHz. As was expected from the theory, three displacement and
three interface stress terms are continuous at the layer interfaces. Other field quantities
are allowed to have a discontinuity from one layer to the other. Figure 3-15 (a) shows that
the out of plane displacement (u3) is dominant for this mode. Figure 3-15 (b) and Fig. 3-
15 (c) show that the dominant stress components are 13σ and 11σ . The first mode at low
frequency corresponds to the bending motion of the plate. The 0o direction plies at the top
and bottom surface carry most of the tension and compression stress. The average power
flow is shown in Fig. 3-15 (d), the wave energy transmits forward in the plane
dominantly in the x1 direction.
48
-
(a)
(b)
Figure 3-14: A comparison of wave structures obtained from the GMM and the SAFE
method. (a) displacement u1, (b) stress σ33.
49
3.6 Summary
In this chapter, both the global matrix method (GMM) and the semi analytical
finite element (SAFE) method are used to simulate the wave propagation in a quasi-
isotropic composite material. Some observations and conclusions are summarized next.
1. A new hybrid SAFE-GMM dispersion curve generation and wave
structure calculation technique is introduced and is based on a
(a) (b)
(c) (d)
Figure 3-15: Displacement and stress wave structure for the first wave modes at 0.2
MHz. (a) displacements (b) out of plane stress (c) in plane stress (d) Power flow
distribution.
50
performance comparison of the two techniques. Both GMM and SAFE
can be used to generate dispersion curves accurately. For the 16 layer
quasi-isotropic laminate, SAFE with one element per layer provided
comparable results with GMM with less than 0.01 km/s average error in
the dispersion curve generation. However, the computing time of SAFE
is 1/50 that of the GMM using the same computer. In addition,
dispersion curve generation using the SAFE method is based on solving
an eigenvalue problem. This method is computationally stable and does
not have missing roots as the GMM method might have during root
searching. However, a stress discontinuity is observed in wave structure
calculations using the SAFE method. Therefore, the GMM method is
used to calculate wave structures in the hybrid method to obtain
accurate stress distributions.
2. Guided waves propagating in composite structures are much more
complicated than that in isotropic media. A new wave mode numbering
system is introduced in this chapter to clarify the effect of wave mode
interaction and coupling. Even in quasi-isotropic media, the wave
propagation characteristics are highly direction dependent in certain
regions of the dispersion curves. Even in quasi-isotropic material,
guided wave modes with skew angles larger than 30o can occur. These
wave modes will be studied further in the following chapters.
Chapter 4
Guided wave excitation in composites
This chapter discusses wave excitation characteristics in composite laminates.
Section 4.1 introduces the normal mode expansion (NME) theory and guided wave mode
excitability with a finite loading pattern. Section 4.2 is a numerical proof of guided wave
mode orthogonality as a basis for NME. Section 4.3 describes the wave mode excitability
coefficient. Wave mode excitation from a finite source including the frequency spectrum
and phase velocity spectrum are discussed in Section 4.4. In Section 4.5, a wave mode
dispersion coefficient is defined to quantitatively describe the dispersion characteristic of
a guided wave mode. Section 4.6 provides a reconstruction algorithm and results for a
wave field excited from a finite and transient source. A guided wave beam spreading
evaluation is presented in Section 4.7. If not otherwise stated, the examples provided in
this chapter are all for guided wave propagation in the 0o direction of the [(0/45/90/-45)s]2
composite laminate.
4.1 Theory
4.1.1 The The The The reciprocity relation and mode orthogonality
Modal analysis provides information on all guided wave propagation possibilities
within a composite laminate. According to the discussion in Section 2.2.2, any linear
combination of two or more guided wave mode solutions can still satisfy the boundary
conditions and interface continuity conditions in Equation 2.11. Therefore, an actual
wave field within a waveguide can be expressed as a linear combination of guided wave
mode solutions if the wave modes form an orthogonal and complete basis.
52
A complete set of guided wave modal solutions includes all possible solutions of
the characteristic equation (Equation 2.12) in Chapter 2. By solving the guided wave
dispersion curves, we have obtained the solutions ξ that are real and positive for a given
positive value of frequency. These wave modes correspond to the rightward propagating
waves. It is not difficult to prove that for a given wave mode ),( ξf , there is a leftward
propagating mode ),( ξ−f , which also satisfies the condition in Equation 2.13. When the
equation is solved in complex space, there exists other solutions of ξ that are either pure
imaginary or have a complex value. These solutions are called non-propagating modes, or
evanescent wave modes, since their amplitude decays exponentially with respect to x1.
The typical effective distance of evanescent waves is within several millimeters.
Therefore, only the excitation characteristics of propagating modes are studied in this
chapter for purposes of structural health monitoring.
The proof of orthogonality of guided wave modes is important for the application
of the normal mode expansion theory. B. A. Auld [Auld 1990] proved the orthogonality
of guided wave modes in lossless wave guides through the derivation of the reciprocity
relation in piezoelectric media. In composite laminates, the complex reciprocity relation
can be expressed in Equation 4.1 [Auld 1990].
Here, the subscripts I and II denote two wave mode solutions, and s is the compliance
matrix. In addition, the following equation holds when both modes are of the same
frequency.
)()::0
0]([][ *
IIIIII
I
I
IIIIIIIIIIst
FvFvσ
vσvσvσv •+•+
∂
∂−=•−•−•∇ ∗∗∗∗∗
ρ (4.1)
)(exp()( 13 txix mmI ωξ −= vv
)(exp()( 13 txix nnII ωξ −= vv
))(exp()( 13 txix mmI ωξ −= σσ
))(exp()( 13 txix nnII ωξ −= σσ
0=IF
0=IIF
(4.2)
53
Therefore, the reciprocity relation reduces to Equation 4.3.
Integration of Equation 4.3 over a cross section of the waveguide will lead to
Equation 4.4.
where,
For the wave mode solutions obtained earlier, stress free boundary conditions are
satisfied. Therefore, the right hand side of Equation 4.4 is zero. For propagating modes
m and n , nm ξξ ≠ , and, Equation 4.6 holds.
Equation 4.6 is the guided wave mode orthogonality relation for composite laminates.
When the two modes are the same,
The real part of Equation 4.7 is the average power flow across a waveguide section with
unit width in the x2 direction. Equation 4.7 is equivalent to Equation 2.45 derived from
the complex form of Poynting’s vector. The normalization of the wave mode solution by
the average power transmission is discussed in Section 2.4.4 .
For wave modes with different frequency, the orthogonality between )exp( 1tiω−
and )exp( 2tiω− has been proven in mathematics [Hayek 2001] . Therefore, the two mode
solutions are also orthogonal.
0][ ** =•−•−•∇ IIIIII σvσv (4.3)
Hx
xnmmnmnnm xPi==••−•−=− 3
3 03
** ˆ}{4)( σvσvξξ (4.4)
∫ ••−•−=H
dxxP nmmnmn0
31
** ˆ}{4
1σvσv (4.5)
nmforPmn ≠= 0 (4.6)
∫ ••−=H
mmmm dxxP0
31
* ˆ)(2
1σv (4.7)
54
4.1.2 The normal mode expansion technique
Guided wave orthogonality in composite laminates has been validated
theoretically in Section 4.1.1. In addition, the normalization of the guided wave mode
solution has been performed in Section 2.4.4. Therefore, guided wave mode solutions
form an orthonormal basis for an arbitrary wave field in a composite material.
The normal mode expansion (NME) technique is a process of expanding an
arbitrary wave field into combinations of the orthonormal guided wave mode solutions.
When the common time dependent factor )exp( tiω− is suppressed in a harmonic wave
propagation situation, the expansion equation is shown in Equation 4.8.
Substituting Equation 4.8 into the reciprocity relation Equation 4.1, and integrating over
the cross-section of the laminate, Equation 4.9 is obtained.
Here, nnP is the entire power flow in the x1 direction, snf is the surface loading, and vnf is
the body force loading. In general, the loading terms are functions of x1.
The next step is to solve for the mode weighting coefficient function )( 1xan from
Equation 4.9. Given the wave structure solutions obtained for each mode, Equation 4.9
can be treated as an ordinary differential equation. Assuming the loading area is within
[ ]21 LL , the wave propagation in the positive x1 direction must have zero amplitude at
the left side of the source. Equation 4.12 is the solution for rightward propagating wave
modes.
∑=n
nn xxaxx )()(),( 3131 vv
131131ˆ)()(ˆ),( xxxaxxx
n
nn •=• ∑ σσ (4.8)
)()()()(4 111
1
xfxfxaix
P vnsnnnnn +=−∂
∂ξ (4.9)
H
xnnsn xxxxxxxf
03
*
31313
*
13
ˆ}),(),()({)(=
••+•= σvσv (4.10)
∫ •=H
nvn dxxxxxf0
3313
*
1 ),()()( σv (4.11)
55
When the position is outside the source region, )( 1xan is a harmonic wave function of x1
with amplitude as given in Equation 4.13.
For leftward propagating waves, 0<n , the amplitude weighting is given by
Equation 4.14.
When the wave mode solution is normalized according to Section 2.4.4, 1=nnP , and
1=−− nnP . The weighting coefficient is a good measurement of the excitability of a
particular wave mode under the given excitation configuration. The term 2
na is
proportional to the total power flow in the wave propagation direction. The phase angle
of the coefficient corresponds to the phase delay between the wave and the excitation.
When the transducer is mounted on the surfaces of the composite structure, only
surface loading exists. When the wave mode is normalized, the excitability of a guided
wave mode is defined by Equation 4.15.
Take a simple example, when the surface loading is only in the x1 direction on the
top surface, Equation 4.15 simplifies to Equation 4.16.
0)( 1 =xan , 11 Lx ≤ ;
∫ −+
=1
1
)exp(4
)()()exp()( 11
x
Ln
nn
vnsnnn di
P
ffxixa ηηξ
ηηξ , 211 LxL ≤≤ ;
∫ −+
=2
1
)exp(4
)()()exp()( 11
L
Ln
nn
vnsn
nn diP
ffxixa ηηξ
ηηξ , 21 Lx ≥ .
(4.12)
∫ −+
=2
1
)exp(4
)()()( 1
L
Ln
nn
vnsn
n diP
ffxa ηηξ
ηη (4.13)
0)( 1 =− xa n , 21 Lx ≥ ;
∫ −
−−
−−
−− −+
=1
2
)exp(4
)()()exp()(
)()(
11
x
Ln
nn
nvns
nn diP
ffxixa ηηξ
ηηξ , 211 LxL ≤≤ ;
∫ −
−−
−−−− −
+=
1
2
)exp(4
)()()exp()( 11
L
Ln
nn
nvns
nn diP
ffxixa ηηξ
ηηξ , 11 Lx ≤ .
(4.14)
∫ −=2
1
)exp()(4
1 L
Lnsnn difA ηηξη (4.15)
56
Here, the notation has been changed slightly; )(*
1 Hv is the complex conjugate of
the normalized x1 directional velocity at the top surface. It is clear from Equation 4.16
that the excitability of mode ( ξ,f ) is affected by two major factors: the normalized
surface velocity of the wave mode and the Fourier transform of the loading distribution.
When a transient wave is used, the frequency component of the excitation signal would
also be obtained from the Fourier transform of the excitation signal. Combining the time
and spatial domain Fourier transforms lead to a two dimensional Fourier spectrum of the
loading source.
When the loading is in the x2 or x3 direction, the active surface velocity spectrums
for the excitabilities are v2 and v3, respectively. For a linear elastic system, the response of
the structure under multiple loading cases can be considered as linear combinations of all
the loading types. Therefore, the surface velocity spectrum obtained from the mode
analysis is very important for guided wave excitation.
4.1.3 Source influence on wave excitation
Source influence is a very important consideration in wave excitation. Since wave
mode studies are carried out in the ),( ξf and/or ),( pcf domain, it is necessary and
convenient to transform the loading distribution into these domains to illustrate the wave
excitability.
When a rightward propagating wave solution is defined as in Equation 2.14,
where the propagation term is )(exp( 1 txi ωξ − , the inverse 2D Fourier transform can be
written as in Equation 4.17.
The standard 2D FFT expression is given by Equation 4.18.
∫ −=2
1
)exp()(4
)(1
*
1L
LdiT
HvA ηξηη (4.16)
∫ ∫+∞
∞−
+∞
∞−−= ξωωξξω ddtxiFtxF ))(exp(),(),( 11 (4.17)
57
Therefore, a variable transform should be performed before a standard 2D FFT
subroutine is used. The reformatted pairs of Fourier and inverse Fourier transforms are
seen in Equation 4.19.
The transformed 2D FFT would be symmetric about the point (0,0). This means
Equation 4.20.
The first quadrant, where 0,0 >> ξω , corresponds to rightward propagating
waves. The source influence in the ( pcf , ) domain can be easily obtained from the source
influence in the ( ξω, ) domain by using the relations in Equation 4.21.
The general formulation is suitable for any kind of surface loading, including a
single element, linear array, or phased array excitation. The only difference is in the
detailed descriptions of the time and spatial domain input functions. For the case of a
single element or a linear array, the time and frequency domain source term are
separable. The loading function can be expressed as in Equation 4.22.
Here, xF is the spatial domain distribution and tF is the transient excitation signal. In this
case, the 2D FFT can be evaluated by a multiplication of two FFTs.
∫ ∫+∞
∞−
+∞
∞−+= dudvvyuxivuFyxF ))(2exp(),(),( π (4.18)
∫ ∫+∞
∞−
+∞
∞−
−−
−= )
2()
2())(exp()
2,
2(),( 11
π
ξ
π
ωωξ
π
ξ
π
ωddtxiFtxF
∫ ∫∞+
∞−
∞+
∞−−−=
−dtdxtxitxFF 111 ))(exp(),()
2,
2( ωξ
π
ξ
π
ω
(4.19)
),(),( ξωξω FF =−− (4.20)
π
ω
2=f ,
λ
πξ
2= , λ
ξ
ωfc p == (4.21)
)()(),( 11 tFxFtxF tx= (4.22)
58
4.2 Numerical proof of mode orthogonality in a quasi-isotropic laminate
Numerical proof of guided wave mode orthogonality is presented in this section
and is based on the theories discussed in Section 4.1.1. As an example, the velocity and
stress distributions of the three wave modes at a frequency of 200 kHz are shown in
Fig. 4-1.
The values of the real part of mnP for the three wave modes are listed in Tab. 4-1.
The imaginary parts of the mnP are all in the 10-16
scale, which is negligible. The result
proves wave mode orthonormality in the sense of Equation 4.6. Similar proofs can also
be carried out with other guided wave modes.
Table 4-1: Wave mode orthognality validation table
m,n
combination
11 22 33 12 13 14
Pmn 1 1 1 1.53e-006 0.000255 -5.11e-006
59
0 0.5 1 1.5 2 2.5 3 3.5-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Depth (mm)
No
rma
lize
d v
1 (
mm
/s)
Mode 1
Mode 2
Mode 3
0 0.5 1 1.5 2 2.5 3 3.5-10
-8
-6
-4
-2
0
2
4
6
8
10
Depth (mm)
No
rma
lize
d σ
11 (
kP
a)
Mode 1
Mode 2
Mode 3
(a) (d)
0 0.5 1 1.5 2 2.5 3 3.5-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Depth (mm)
No
rma
lize
d v
2 (
mm
/s)
Mode 1
Mode 2
Mode 3
0 0.5 1 1.5 2 2.5 3 3.5-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
Depth (mm)
No
rma
lize
d σ
12 (
kP
a)
Mode 1Mode 2Mode 3
(b) (e)
0 0.5 1 1.5 2 2.5 3 3.5-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Depth (mm)
No
rma
lize
d v
3 (
mm
/s)
Mode 1
Mode 2
Mode 3
0 0.5 1 1.5 2 2.5 3 3.5-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Depth (mm)
No
rma
lize
d σ
13 (
kP
a)
Mode 1Mode 2Mode 3
(c) (f)
Figure 4-1: Wave structure components for mode orthogonality validation. (a) v1 (b) v2
(c) v3 (d) σ11 (e) σ12 (f) σ13.
60
4.3 Wave mode excitability
Normal mode expansion theory indicates that the particle velocity at the surface
describes the mode attributes for wave excitation using surface mounted transducers.
Therefore, wave mode excitability is defined with particle velocity components at the
surfaces. The three components of the surface particle velocity are related to the
excitation forces in the three directions. Surface velocity spectra in the x1 direction of the
first eight wave modes are shown in Fig. 4-2. Phase information of the surface velocity
is indicated in the figure with plus and minus signs. Absolute values of the spectra
indicating the magnitude of the excitability, are shown in Fig. 4-3.
Figure 4-2: Particle velocity spectrum in x1 direction for the wave propagating in 0o at the
surface of the [(0/45/90/-45)s]2 laminate.
61
Some modes are easily excitable with loading in one direction and not so
excitable in other directions. These will help to decide which mode to use when a specific
wave excitation scheme is desired; and also which excitation method to use once a
specific mode is selected. For example, when the frequency is less than 0.4 MHz, the
first, second, and third modes are easily excited with loading in the x3, x2, and x1
directions, respectively. There are also points with zero value in the curves, which means
that the mode is not excitable with the given loading direction.
Surface particle velocity spectra also depends on wave propagation directions.
Fig. 4-4 illustrates several examples of the relationship between surface particle
velocities and their wave propagation direction. Both figures are for the first three modes
(a) (b)
(c)
Figure 4-3: Rectified particle velocity spectrum for the wave propagating in 0o at the
surface of the [(0/45/90/-45)s]2 laminate: (a) x1 direction, (b)x2 direction, (c) x3 direction.
62
at 200 kHz. It is shown that with shear loading, the angular profile of mode 3 is quasi-
circular. Mode 2, which corresponds to a quasi shear-horizontal wave mode, is the least
efficiently excited mode. The excitability profile of mode 1 has a minimum value around
0o and reaches its maximum around 90
o. For normal loading, the first mode is dominant
in all directions.
By comparing the relative wave mode excitability for three directions, we can get
an idea of how to excite a particular wave mode. For example, we can see that the first
mode at 200 kHz is most efficiently excited with a normal loading. If we plot out each
mode according to the direction that it is most efficiently excited, the corresponding
phase velocity dispersion curve is shown in Fig. 4-5.
0.1
0.2
0.3
30
210
60
240
90
270
120
300
150
330
180 0
Mode 1
Mode 2
Mode 3
0.2
0.4
0.6
30
210
60
240
90
270
120
300
150
330
180 0
Mode 1
Mode 2
Mode 3
(a) (b)
Figure 4-4: Angular profile of wave mode excitability (wave mode particle velocity at
surface) for mode 1 to mode 3 at 200kHz. (a) shear (x1 direction) loading (b) normal (x3
direction) loading.
63
4.4 Numerical results of source influence
In this section, the effect of source influence on wave excitation will be discussed
in three steps. First, the relationship between the time domain signal and frequency
spectrum is discussed. Secondly, the relation between spatial domain loading and the
phase velocity spectrum is covered. Finally, the source influence is described with a two
dimensional frequency and phase velocity spectrum.
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (MHz)
Cp
(km
/s)
Blue: x1
Red: x2
Black: x3
Figure 4-5: Phase velocity dispersion curves of guided wave propagation at 0o with the
embedded information of wave mode excitability. Blue sections are most easily excited
with shear loading in the x1 direction; red sections and black sections correspond to x2
and x3 direction loading respectively.
64
4.4.1 Excitation signal analysis
Sinusoidal signals with a rectangular window or Hanning window are usually
used in nondestructive evaluation and structural health monitoring. These signals are
usually called a tone-burst. The relation between center frequency, pulse width (or
number of cycles), and frequency bandwidth are studied. To illustrate the key concepts in
excitation signal design, the rectangular windowed and Hanning windowed tone burst
signals are studied in this work.
Fig. 4-6 shows two examples of excitation signals. The signal in Fig. 4-6 (a) is a
5 cycle tone-burst signal with a 1 MHz center frequency and a rectangular window.
Fig. 4-6 (b) is a 5 cycle Hanning windowed tone-burst signal. The amplitude spectra of
these excitation signals are shown in Fig. 4-7.
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5
Time (µs)
Am
plit
ud
e
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5
Time (µs)
Am
plit
ud
e
(a) (b)
Figure 4-6: Sample waveforms. (a) 5 cycled tone-burst signal with 1 MHz center
frequency and rectangular window, (b) 5 cycled tone-burst signal with 1 MHz center
frequency and Hanning window.
65
Comparing Fig. 4-7 (a) with Fig. 4-7 (b), the effect of using a Hanning window
can be concluded as follows. First, the side lobes in the rectangular windowed signal are
reduced with the Hanning window. Therefore, the mode selection will be more efficient.
Secondly, the center frequency amplitude is reduced. Thirdly, the bandwidth of the main
lobe is increased. The second and the third effects are tradeoffs to the advantage of the
first effect.
In an ultrasonic test, the input signal can be controlled with three parameters,
center frequency, number of cycles, and pulse width. The three parameters are related by
Equation 4.23.
The frequency bandwidth of an excitation signal is closely related to the guided wave
mode selectability and wave package dispersion. Based on the amplitude spectrum of an
excitation signal, the bandwidth can be defined as the frequency range where the
amplitudes decrease by a certain number of decibels. The definition is illustrated in
Equation 4.24. A 6 dB bandwidth is usually used in ultrasonic testing. Fig. 4-8 shows
that the 6 dB bandwidth increases with center frequency when the number of cycles are
0 2 4 6 8 100
0.5
1
1.5
2
2.5
3
Frequency (MHz)
Am
plit
ud
e
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency (MHz)
Am
plit
ud
e
(a) (b)
Figure 4-7: Amplitude spectra of the 5 cycled tone burst signals with 1MHz center
frequency.(a) Rectangular window, (b) Hanning window.
Pulsewidth=Number of Cycles/(Center frequency) (4.23)
66
kept the same. In addition, signal bandwidth decreases with an increase in the number of
cycles.
)(log20 10
peakA
AdB = (4.24)
0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4
Frequency (MHz)
Ba
nd
wid
th (
MH
z)
3 cycle5 cycle7 cycle9 cycle11 cycle
(a)
0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4
Frequency (MHz)
Ba
nd
wid
th (
MH
z)
3 cycle5 cycle7 cycle9 cycle11 cycle
(b)
Figure 4-8: Relation between bandwidth and center frequency under constant number of
cycles (a) Rectangular window (b) Hanning window.
67
Fig. 4-9 shows the relationship between frequency bandwidth and center
frequency when the pulse width is constant. The figure indicates that when the pulse
width is reasonably long (more than 1-2 wavelengths), the bandwidth stays constant with
the increase of excitation center frequency. In the case of a Hanning windowed signal,
this phenomenon is more evident. In the case of the rectangular window, although the
influence of the side lobes introduces oscillations in bandwidth, the overall trend is still a
constant.
0 1 2 3 4 5
0.1
0.15
0.2
0.25
0.3
0.35
Frequency (MHz)
Ba
nd
wid
th (
MH
z)
PW=6µs
PW=8µs
PW=10µs
PW=12µs
PW=14µs
(a)
0 1 2 3 4 50.1
0.15
0.2
0.25
0.3
0.35
Frequency (MHz)
Ba
nd
wid
th (
MH
z)
PW=6µs
PW=8µs
PW=10µs
PW=12µs
PW=14µs
(b)
Figure 4-9: Relation between bandwidth and center frequency under constant pulse
width (a) Rectangular window (b) Hanning window.
68
4.4.2 Transducer geometry and loading pattern influence
According to Equation 4.16 , the Fourier transform of the spatial domain loading
distribution is an important factor affecting wave mode excitability. Discussed in this
section is the wave number spectrum of one dimensional arrays using Fourier transforms.
When a thin piezoelectric wafer transducer is bonded to a host structure, the
interaction between the transducer and the host structure has been mentioned in
[Giurgiutiu 2005]. It shows that the dominant interaction occurs at the edge of the wafer
transducer when the bonding between the transducer and the structure is rigid. The edge
loading is separated into two loading directions, namely the x1 direction and the x3
direction. In addition, distributed loading is also considered in this thesis. Fig. 4-10
shows a sketch of the three loading cases. In order to improve mode control capability,
linear transducer arrays with multiple elements are also considered.
Theoretically, the concentrated loading is expressed as a Dirac delta function. In
the numerical model presented here, loading within a narrow width is considered. For
Figure 4-10: Sketch of transducer loading model. (a) concentrated shear loading (b)
concentrated normal loading (c) evenly distributed normal loading.
69
example, when the element width are 1mm, and 3 elements, the spatial domain
distribution of the loading function F(x), and the corresponding Fourier transform spectra,
are shown in Fig. 4-11 The difference between these three loading models are very
significant in the spatial frequency domain. The concentrated shear loading has maximum
responses at Len /)12( πξ += , where n is any integer. The concentrated normal loading
has maximum responses at Len /)2( πξ = , where n is any integer. For both cases, the
number of side lobes between every two main bands is )1(2 −eN . A zero value exists
between every two lobes. These peaks and zeros can be used to select or avoid a guided
wave mode. For evenly distributed loading, the response is much stronger at low spatial
frequency.
70
0 1 2 3 4 5 6 7 8-1.5
-1
-0.5
0
0.5
1
1.5
x1 (mm)
F (
x 1)
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2
0.25
0.3
0.35
ξ (1/mm)
Am
plit
ud
e s
pe
ctr
um
(a)
0 1 2 3 4 5 6 7 8-1.5
-1
-0.5
0
0.5
1
1.5
x1 (mm)
F(x
1)
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2
0.25
0.3
0.35
ξ (1/mm)
Am
plit
ud
e s
pe
ctr
um
(b)
0 1 2 3 4 5 6 7 8-1.5
-1
-0.5
0
0.5
1
1.5
x1 (mm)
F(x
1)
0 10 20 30 40 50 60 700
0.5
1
1.5
2
2.5
3
3.5
ξ (1/mm)
Am
plit
ud
e s
pe
ctr
um
(c)
Figure 4-11: Spatial domain loading distribution and its corresponding spatial frequency
spectrum of a 3 element linear array with 1mm array element width. (a) concentrated
shear loading (b) concentrated normal loading (c) evenly distributed normal loading.
71
4.4.3 Frequency and phase velocity spectrum of a finite source
In this section, joint time and spatial domain design will be considered. A two
dimensional (f, cp) spectrum will be generated, in which a wave propagation phase
velocity dispersion curve will be mapped to determine the mode excitation capabilities.
The wave number spectra shown in Fig. 4-11 can be transformed into phase
velocity spectra. When the frequency is 1 MHz, the results are shown in Fig. 4-12.
Figure 4-12 shows that only the first few peaks in Fig. 4-11 are mapped into a
phase velocity region of 1 km/s to 20 km/s, which is commonly considered in guided
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
Cp (km/s)
Am
plit
ud
e S
pe
ctr
um
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
Cp (km/s)
Am
plit
ud
e S
pe
ctr
um
(a) (b)
0 5 10 15 200
0.5
1
1.5
2
2.5
3
3.5
Cp (km/s)
Am
plit
ud
e S
pe
ctr
um
(c)
Figure 4-12: Phase velocity spectrum for a 3 element linear array with 1 mm array
element width at 1 MHz. (a) concentrated shear loading (b) concentrated normal loading
(c) evenly distributed normal loading.
72
wave studies. The three loading cases have different characteristics in the phase velocity
spectrum. The normal loading cases are in favor of the high phase velocity region
because they both have peaks at zero wave number. The concentrated shear loading tends
to produce an efficient response at the region where the wavelength is equal to twice the
element width.
The 2-dimensional (f, cp) spectrum of the transducer can be obtained by taking the
frequency spectrum of an excitation signal into consideration. As an example, consider a
transducer with a three-element linear array with 1 mm element widths, where the
excitation signal is a Hanning windowed 10-cycle tone burst with 1 MHz center
frequency. Figure 4-13, Fig. 4-14, and Fig. 4-15 are for the cases of concentrated shear
loading; concentrated normal loading; and evenly distributed normal loading,
respectively. The color in the figure represents the amplitude of the spectrum.
Figure 4-13: Source influence spectrum of a 3 element transducer with 1mm element
width and excited by a 10 cycled tone burst signal with Hanning window and 1 MHz
center frequency. The loading is concentrated shear.
73
Phase velocity dispersion curves are also plotted in the excitation spectra. When
the excitation spectrum hotspot is located on a dispersion curve, the corresponding wave
Figure 4-14: Source influence spectrum of a 3 element transducer with 1mm element
width and excited by a 10 cycled tone burst signal with Hanning window and 1 MHz
center frequency. The loading is concentrated normal.
Figure 4-15: Source influence spectrum of a 3 element transducer with 1mm element
width and excited by a 10 cycled tone burst signal with Hanning window and 1 MHz
center frequency. The loading is evenly distributed normal.
74
mode will be efficiently excited. As an example, Fig. 4-13 indicates an efficient mode
selection at a frequency of 1 MHz and a phase velocity of 2 km/s. However, the two
normal loading cases are more efficient at exciting those wave modes with high phase
velocity. This further indicates that when the number of elements is small, the commonly
used excitation line principle cannot be applied. In a real situation, the interaction
between the surface mounted transducer and the composite structure will be a
combination of the three excitation mechanisms described above. The combined
excitation spectrum will be considered for wave excitation analysis.
4.5 Guided wave dispersion
Although guided wave dispersion is a commonly recognized phenomenon, it is
commonly avoided in guided wave applications to reduce complexity. Some quantitative
evaluation of the wave dispersion characteristics in isotropic media can be found in
[Wilcox et al. 2001; Wilcox 2003] . In this section, the physics of guided wave dispersion
behavior in laminated composites is studied. A new methodology to evaluate guided
wave dispersion is then put forward.
4.5.1 Dispersion signal reconstruction
The major concern in this section is to study the mode dispersion characteristics
through a reconstruction of guided wave signals at different locations. The wave field
will be reconstructed according to Equation 4.25 for the wave excited at x1=0 and
propagating in the positive x1 direction.
Here, W is a general wave field quantity; the relation between ξ and f is given by
the phase velocity dispersion curve obtained in Chapter 3. The Fourier Transform of the
∫−=
2
1
1 )2(
1 )(2),(f
f
ftxidfefAtxW
πξ (4.25)
75
excitation signal is expressed in )( fA . Frequency boundaries used to capture the major
energy component of the excitation signal are 1f and 2f . As an example, when the
excitation signal is a 5-cycle Hanning windowed tone burst with a 500 kHz center
frequency, the reconstructed signal of the first wave mode is calculated. The waveforms
at 0 mm, 100 mm, 200 mm, 300 mm, and 400 mm away from the source are shown in
Fig. 4-16. This is typically a non-dispersive mode. The wave package retains its shape as
it propagates forward. The wave package velocity matches the group velocity at the
center frequency and is found to be 1.86 km/s.
An example of a dispersive wave is shown in Fig. 4-17. The excitation signal is a
5-cycle signal with a 600 kHz center frequency and the wave mode considered is the fifth
mode line for a wave propagating in the 0o direction of the [(0/45/90/-45)s]2 laminate.
0 50 100 150 200 250 300 350 400
-1
0
1
Am
plit
ud
e
0 50 100 150 200 250 300 350 400
-1
0
1
Am
plit
ud
e
0 50 100 150 200 250 300 350 400
-1
0
1
Am
plit
ud
e
0 50 100 150 200 250 300 350 400
-1
0
1
Am
plit
ud
e
0 50 100 150 200 250 300 350 400
-1
0
1
Time (µs)
Am
plit
ud
e 400 mm
300 mm
200 mm
100 mm
0 mm
Figure 4-16: Reconstructed signals at 0 mm, 100 mm, 200 mm, 300 mm, and 400 mm
away from the excitation source. Source signal is a 5 cycle Hanning windowed tone burst
with 500 kHz center frequency. The wave mode considered is the first dispersion curve
line for guided wave propagating in 0o of the [(0/45/90-45)s]2 laminate.
76
Reconstructed waveforms are also computed at positions of 0 mm, 100 mm, 200 mm,
300 mm, and 400 mm. The solid line, dotted line, and dashed line track the time of flight
of the leading edge, trailing edge, and peak amplitude of the wave packages. The velocity
values corresponding to these three lines are listed in Tab. 4-2
Fig. 4-18 is the corresponding phase velocity and group velocity of the wave
modes and the relative amplitude spectrum of the excitation signal. The maximum group
velocity, minimum group velocity, and center frequency group velocity are marked in the
0 50 100 150 200 250 300 350 400
-1
0
1
Am
plit
ud
e 0 mm
0 50 100 150 200 250 300 350 400
-1
0
1
Am
plit
ud
e 100 mm
0 50 100 150 200 250 300 350 400
-1
0
1
Am
plit
ud
e 200 mm
0 50 100 150 200 250 300 350 400
-1
0
1
Am
plit
ud
e 300 mm
0 50 100 150 200 250 300 350 400
-1
0
1
Time (µs)
Am
plit
ud
e 400 mm
Figure 4-17: Reconstructed waveforms showing the effect of wave dispersion. Wave
mode considered is the fifth mode line. The excitation signal is a 5-cycle Hanning
windowed tone burst with 600kHz center frequency.
77
figure with red, blue and black circles. The values of these group velocities are also listed
in Table 4-2 .
Table 4-2 lists the velocity features from the reconstructed signal abstracted from
the leading edge, trailing edge, and the peak amplitude of the waveform.
The results show that the maximum and minimum group velocity obtained from
the gc curve match very well with the leading edge and trailing edge wave velocity. This
proves that the group velocity dispersion curve is directly related to the dispersion
phenomenon of guided wave propagation. Maximum and minimum group velocity values
0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
16
18
20
Frequency (MHz)
Ve
locity
(km
/s)
Phase velocity Group velocity Relative amplitude spectrum
Figure 4-18: Sections of the phase velocity, group velocity dispersion curves for the fifth
wave mode line along 0 degree propagation direction. The amplitude spectrum of a 5-
cycle Hanning windowed tone burst signal with 600 kHz center frequency is also plotted.
Table 4-2: Comparison of theoretical velocities with the velocity values obtained from
the reconstructed signals
Leading edge wave velocity 4.44 km/s
Trailing edge wave velocity 1.16 km/s
Reconstructed
signal
Peak amplitude velocity 4.18 km/s
Maximum group velocity 4.3303 km/s
Minimum group velocity 1.18 km/s
Group velocity
values
Center frequency group velocity 3.70 km/s
78
can be used to predict the duration of a guided wave signal at a given location. However,
there could be a large mismatch between the group velocity at the center frequency and
the predicted wave package velocity from the peak amplitude of the signal for highly
dispersive waves.
4.5.2 Wave mode dispersion coefficient
Traditionally, the dispersion of a guided wave mode is measured by the slope of
the dispersion curve. In order to provide more insight into the guided wave mechanics, a
mode dispersion coefficient is defined in this section by considering the effect of
frequency bandwidth.
In guided wave NDE and SHM applications, the excitation signal is commonly
around 10 microseconds and the corresponding 6dB bandwidth is about 200 kHz. Section
4.5.1 indicates that the guided wave dispersion is affected by the maximum and minimum
group velocity within the frequency range. Therefore, a wave mode dispersion coefficient
is defined in this thesis as seen in Equation 4.26 .
Here, x is the propagation distance from the sensor position to the excitation source. t∆ is
the extra signal spreading due to wave dispersion. MDC is the mode dispersion
coefficient, which is a measurement of pulse spreading in the unit of mm
µs when the
unit of group velocity is µs
mm .
As an example, mode dispersion coefficient curves for the first five mode lines
are shown in Fig. 4-19. The first mode is dispersive at low frequency and becomes non-
dispersive when the frequency is higher. The second and third modes are non-dispersive
[ ]
[ ]
minmax
minmaxmaxmin
min
max
))100100(min(
))100100(max(
gg
gggg
ccgg
ccgg
cc
cc
x
c
x
c
x
x
tMDC
kHzfkHzfcc
kHzfkHzfcc
−=
−
=∆
=
+−=
+−=
(4.26)
79
in the low frequency range, however they both have a highly dispersive region in the
frequency range of 300kHz to 550kHz.
These lines provide a qualitative measurement of the dispersion behavior of a
guided wave mode. One can set a threshold for the acceptable level of dispersion and find
the wave modes satisfying his design criteria. One can also find the wave modes with
minimum or maximum dispersion for a particular testing requirement.
If we set the acceptable level of dispersion to be µs/mm1.0 , which means that the
desired wave package spreading can not exceed sµ20 in a through transmission testing
distance of 200 mm, the acceptable sections of the wave mode in the phase velocity
dispersion curve are shown in Fig. 4-20. On the other hand, if we are interested in the
wave mode regions that are sensitive to thickness changes in the structure, the highly
dispersive region of the dispersion curve are ideal. Wave modes with dispersion
coefficients larger than µs/mm5.0 are shown in Fig. 4-20.
Figure 4-19: First five dispersion coefficient lines for the wave propagation in the 0o
direction of the [(0/45/90/-45)s]2 laminate.
80
4.6 Guided wave field simulation
4.6.1 Theory
Wave excitation from a finite source with time harmonic excitation has been
discussed in section 4.1 using the normal mode expansion technique. In this section, we
will consider the case where the excitation source is an arbitrary time domain signal. In a
most general case, the excitation source can be expressed as a function ),,( 31 txxoσ .
Similar to Equation 4.10 and Equation 4.11 , a general source term of ),( 1 txf n can be
obtained. In order to express the excitation component in the frequency domain, a Fourier
transform of ),( 1 txf n is carried out to get ),( 1 ωxFn . The excited wave components can
be calculated using Equation 4.27.
0 0.5 1 1.5 2
0
2
4
6
8
10
12
14
16
18
20
Frequency (MHz)
Cp
(km
/s)
Figure 4-20: Mode selection results by the criterion of dispersion coefficient. (a) less
than 0.1 mm/µs (b) larger than 0.5 mm/µs for the wave propagating in 0 degree direction
of an [(0/45/90/-45)s]2 composite laminate. Blue dashed lines are the entire dispersion
curve set. Red line sections are the modes that satisfy the criterion.
∫−=
2
1
1
4
),()( 1
L
L
i
nn
nxi
n deP
xFea nn η
ωω ηξξ
(4.27)
81
The total wave field can be reconstructed from these wave mode components using
Equation 4.28.
In Equation 4.28, ),,( 31 txxU is a wave field quantity, )( 3xU n is the wave structure of the
mode, and nξ is the wave number, which is also a function of frequency according to the
dispersion curve.
4.6.2 Wave field reconstruction case studies in composite laminates
4.6.2.1 First fundamental wave mode
A normal loading pattern used to efficiently excite the first fundamental wave
mode at a 200 kHz center frequency is listed in Tab. 4-3 . The f-cp spectrum is plotted in
Fig. 4-21.
∫∑+∞
∞−
−=n
txi
nn dexUatxxU n ωω ωξ )(
3311)()(),,( (4.28)
Table 4-3: A loading design to excite first fundamental wave mode
Structure [(0/45/90/-45)s]2 quasi-isotropic laminate
Wave launching direction 0o
Evenly distributed normal loading on top surface
Loading Element
Number
1
Loading pattern
Element width (mm) 2
Center frequency (MHz) 0.2
Number of cycles 3
Excitation signal
Signal window Hanning
Signal amplitude (kPa) 3
82
The amplitude of the wave mode component function )(ωna is plotted in Fig. 4-
22. The second and the third wave mode are not efficiently excited because they are not
sensitive to normal loading. Therefore, good mode selection is achieved here.
Frequency (MHz)
Ph
ase
ve
locity
(km
/s)
0.5 1 1.5 2
2
4
6
8
10
12
14
16
18
20
1
2
3
4
5
6
7
Figure 4-21: (Frequency)-(Phase velocity) spectrum of a 3 cycle 200 kHz signal with
Hanning window on a 2mm wide element.
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
Frequency (MHz)
To
tal E
xcita
bili
ty
Mode 1
Mode 2
Mode 3
Figure 4-22: Wave mode component function of the wave field excited from a finite
source listed in Tab. 4-3 .
83
The wave field reconstructed at 0 mm, 20 mm, 40 mm, and 80 mm away from the
excitation position is plotted in Fig. 4-23 and Fig. 4-24. Fig. 4-23 shows the x1 direction
component of the displacement. Fig. 4-24 shows the x3 direction component of the
displacement. Both the first and the third wave mode is shown clearly in Fig. 4-23.
However, only the first mode is clearly demonstrated in Fig. 4-23. This matches the wave
structure of the wave mode. The second mode is not clearly identified because its
amplitude is very small.
0 10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
0 10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
0 10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
0 10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
Time (µs)
u1 a
mp
litu
de
(n
m)
60 mm
40 mm
20 mm
0 mm
3rd mode
1st mode
Figure 4-23: u1 direction wave displacement at four positions.
84
0 10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
Time (µs)
u3 a
mp
litu
de
(n
m)
60 mm
40 mm
20 mm
0 mm
1st mode
Figure 4-24: u3 direction displacement at four positions.
85
The wave field distribution across the thickness at the x1= 40mm position is
shown in Fig. 4-25. It is shown that the x1 direction displacement is the largest at the
surfaces and small in the center. The x3 component of the displacement is almost uniform
through the thickness.
To compare the wave field distribution with the wave structure at the center
frequency, the peak-to-peak amplitude of the first wave mode package was extracted and
plotted in Fig. 4-26. In this figure, the wave amplitude values are normalized according to
the maximum value of the x3 amplitude. Wave structures of the first mode at 0.2 MHz are
also plotted in this figure. Fig. 4-26 shows that the wave field distribution generally
matches the wave structure of the mode at the center frequency. However, the details are
not exactly the same because of the contribution from other wave modes.
0 10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
0 10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
0 10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
0 10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
0 10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
Time (µs)
u1 (n
m)
x3=0 mm
x3=0.8 mm
x3=1.6 mm
x3=2.4 mm
x3=3.2 mm
0 10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
0 10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
Time (µs)
u3 (
nm
)
x3=0 mm
x3=0.8 mm
x3=1.6 mm
x3=2.4 mm
x3=3.2 mm
(a) (b)
Figure 4-25: Wave field distribution along the thickness of the [(0/45/90/-45)s]2 structure.
(a) u1 , (b) u3.
86
A wave field at a particular time can also be reconstructed based on the theory
expressed in section 4.6.1. Fig. 4-27 shows two snapshots of the wave field at 20
microseconds, where Fig. 4-27(a) is for u1 and Fig. 4-27(b) is for u3. The wave
distribution is clear in this figure.
0 0.5 1 1.5 2 2.5 3 3.50
0.05
0.1
0.15
0.2
0.25
Depth (mm)
u1 (
nm
)
Field distribution
Wave structure at fc
0 0.5 1 1.5 2 2.5 3 3.50.8
0.85
0.9
0.95
1
1.05
Depth (mm)
u3 (
nm
)
Field distribution
Wave structure at fc
Figure 4-26: Comparison between the wave field profile of the excited wave from a finite
source with the wave structure at center frequency. The mode selected is the first mode at
200 kHz.
87
Propagation direction x1 (mm)
Th
ickn
ess x
3 (
mm
)
0 20 40 60 80 100 120 140 160 180 200
0
0.5
1
1.5
2
2.5
3
(a)
Propagation direction x1 (mm)
Th
ickn
ess x
3 (
mm
)
0 20 40 60 80 100 120 140 160 180 200
0
0.5
1
1.5
2
2.5
3
(b)
Figure 4-27: Wave field snapshots at time equals to 20 µs showing two guided wave
modes. (a) u1 , (b) u3.
88
4.6.2.2 The third fundamental wave mode
An example setup used to excite the third fundamental wave mode is discussed in
this section. The mode excitation setup is listed in Tab. 4-4.
The source spectrum is plotted in Fig. 4-28. The total excitability curves are
plotted in Fig. 4-29. Both the third wave mode and the first wave mode will be excited
from this finite source. However, the energy contained in the third mode will be about 9
times larger than the energy contained in the first mode. Besides this, there will also be a
very small amount of the second wave mode excited.
Table 4-4: A loading design to efficiently excite third fundamental wave mode
Structure [(0/45/90/-45)s]2 quasi-isotropic laminate
Wave launching direction 0o
Concentrated shear force
Loading Element
Number
1
Loading pattern
Element width (mm) 16
Center frequency (MHz) 0.2
Number of cycles 3
Excitation signal
Signal window Hanning
Signal amplitude (kPa) 3
89
Frequency (MHz)
Ph
ase
ve
locity
(km
/s)
0.5 1 1.5 2
2
4
6
8
10
12
14
16
18
20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Figure 4-28: (Frequency)-(Phase velocity) spectrum of a 3 cycle 200 kHz signal with
Hanning window on a 16 mm wide element using concentrated shear loading. Wave
propagation direction is 0 degree.
0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency (MHz)
Excita
bili
ty
Mode 3
Mode 1
Figure 4-29: Wave mode component function of the wave field excited from a finite
source listed in Tab. 4-4.
90
Wave signals reconstructed at 0 mm, 20 mm, 40 mm and 60 mm are shown in Fig. 4-30 .
10 20 30 40 50 60 70 80 90 100-0.05
0
0.05
10 20 30 40 50 60 70 80 90 100-0.05
0
0.05
10 20 30 40 50 60 70 80 90 100-0.05
0
0.05
10 20 30 40 50 60 70 80 90 100-0.05
0
0.05
Time (µs)
u1 a
mp
litu
de
(n
m)
0 mm
20 mm
40 mm
60 mm
(a)
10 20 30 40 50 60 70 80 90 100
-0.02
0
0.02
10 20 30 40 50 60 70 80 90 100
-0.02
0
0.02
10 20 30 40 50 60 70 80 90 100
-0.02
0
0.02
10 20 30 40 50 60 70 80 90 100
-0.02
0
0.02
Time (µs)
u3 a
mp
litu
de
0 mm
20 mm
40 mm
60 mm
(b)
Figure 4-30: Reconstructed wave signal at 0, 20, 40, 60 mm. (a) u1, (b) u3.
91
Wave fields reconstructed at 20 µs is shown in Fig. 4-31
Propagation direction x1 (mm)
Thic
kness x
3 (
mm
)
0 20 40 60 80 100 120 140 160 180 200
0
0.5
1
1.5
2
2.5
3 -0.03
-0.02
-0.01
0
0.01
0.02
0.03
(a)
Propagation direction x1 (mm)
Thic
kness x
3 (
mm
)
0 20 40 60 80 100 120 140 160 180 200
0
0.5
1
1.5
2
2.5
3 -0.03
-0.02
-0.01
0
0.01
0.02
0.03
(b)
Figure 4-31: Wave field snapshots at a time of 20 µs, (a) u1 , (b) u3.
92
4.7 Guided wave beam spreading analyses
Plane wave excitation and propagation was assumed in earlier sections. However,
in real experiments, the transducer is always of a finite length. Therefore, the waves will
be excited and propagating with a finite beam width. The spreading of the beam is related
to a distributed wave front of the excitation and the spreading is due to the skew angle of
the propagating wave. In this section, the effect of beam spreading is studied considering
the variation of skew angle in the wave package.
The skew angle effect of a guided wave mode is discussed in Section 3.4. Section
4.5 indicates that a frequency bandwidth is usually associated with the excitation signals
used in guided wave tests. Therefore, even if all the waves are launched in the same
direction, beam spreading will occur due to the variation of skew angles.
According to the discussion in Section 4.5, the 6dB bandwidth of a µs10 Hanning
windowed pulse is kHz100m . We can also define an attribute of the beam spreading
angle for each guided wave mode using the definition in Equation 4.29.
Here, maxΦ , minΦ , and ∆Φ are the maximum, minimum, and range of skew angle within
the 6dB bandwidth, respectively. Their units are all in degrees.
As an example, the first five mode lines propagating in the 0o direction of the
[(0/45/90/-45)s]2 laminate is shown in Fig. 4-32. Figure (a) is the skew angle dispersion
curve and Figure (b) is the corresponding beam spreading angle curves. This provides a
quantitative measurement of the beam spreading of a given wave mode. Notice that the
skew angle of the mode at the center frequency is not necessarily at the center of the
guided wave beam. Therefore, an examination of the maximum and minimum value of
the skew angle is needed to provide a detailed estimation of the wave beam angle
distribution.
[ ][ ]
minmax
min
max
))100100(min(
))100100(max(
Φ−Φ=∆Φ
+−Φ=Φ
+−Φ=Φ
kHzfkHzf
kHzfkHzf
cc
cc
(4.29)
93
Both skew angle and beam spreading are very important parameters in selecting
an effective wave mode region for a particular guided wave structural health monitoring
0 0.5 1 1.5 2-40
-30
-20
-10
0
10
20
30
Frequency (MHz)
Ske
w a
ng
le (
de
gre
e)
Mode 1Mode 2Mode 3Mode 4Mode 5
(a) Skew angle
0 0.5 1 1.5 20
5
10
15
20
25
30
35
40
45
50
Frequency (MHz)
Be
am
Sp
rea
din
g (
de
gre
e)
Mode 1Mode 2Mode 3Mode 4Mode 5
(b) Beam spreading
Figure 4-32: Skew angle and beam spreading curves of the first five wave mode lines.
Structure: [(0/45/90/-45)s]2 laminate with 0.2 mm ply thickness. Wave vector direction: 0
degree.
94
task. For some cases, one may want to obtain good beam direction control. Reducing the
bandwidth of the frequency spectrum is one solution, however, the usage of a very long
pulse is not possible due to instrument constraints. Therefore, one needs to select those
wave modes with relatively small beam spreading. As an example, the qualified wave
modes with beam spreading less than 5o are shown in Fig. 4-33. The blue dashed lines
are the phase velocity dispersion curves. The qualified wave modes are plotted with red
solid sections in the phase velocity dispersion curves.
There are also cases where a large beam spreading angle maybe desired. One
example is when we want to use a straight linear array to monitoring a comparably large
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (MHz)
Cp
(km
/s)
Figure 4-33: Mode selection results by the criterion of a beam spreading angle less than
5o for the wave propagating in 0 degree direction of an [(0/45/90/-45)s]2 composite
laminate. Blue dashed lines are the entire dispersion curve set. Red line sections are the
modes that satisfy the criterion.
95
range of angles in the structure. For the criterion that the beam spreading be greater than
20o, the qualified wave modes are shown in Fig. 4-34
It must be kept in mind that since the phase velocity dispersion curve varies for
different wave propagation directions, the beam spreading angle is also a function of the
wave launching direction. Fig. 4-35 shows four curves of the third mode line for the
waves launched in 0o, 45
o, 90
o, and -45
o respectively. A large difference is seen for the
highly spread regions. Therefore, if the same mode selection criterion is used different
wave modes will be qualified for wave launching in different directions.
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (MHz)
Cp
(km
/s)
Figure 4-34: Mode selection results by the criterion of beam spreading angle larger than
20o for the wave propagating in 0
o direction of an [(0/45/90/-45)s]2 composite laminate.
Blue dashed lines are the entire dispersion curve set. Red line sections are the modes that
satisfy the criterion.
96
0 0.5 1 1.5 20
5
10
15
20
25
30
35
40
45
50
Frequency (MHz)
Be
am
Sp
rea
din
g (
de
gre
e)
00
450
900
-450
Figure 4-35: Beam spreading dispersion curves of the third mode line for four excitation
wave vector directions. This shows the dependence of beam spreading on wave launching
direction.
97
Chapter 5
Finite element modeling of wave excitation and propagation
5.1 Theory of a three dimensional FEM
Guided wave excitation and propagation can be considered as a high frequency
dynamics problem. Therefore, finite element analysis, which is commonly used in
structural mechanics analysis, can also be applied in wave propagation studies. The
detailed theory of 3D continuum finite element analysis is discussed in textbooks [Cook
2001] and the manuals of finite element analyses software, such as [ANSYS; ABAQUS
2003] .
A finite element method in dynamic analysis is based on the governing equation
in the theory of elasticity (Equation 2.1 ), the constitutive equation (Equation 2.2 ), and
the strain displacement equation (Equation 2.3 ), material mechanical properties,
boundary conditions, and loading conditions. A 3D continuum structure is meshed into
small elements. Interpolation from the nodal solutions is used to approximate the
displacement field of the structure according to shape functions. A finite element
formulation relates the external load to the nodal solution by the expression seen in
Equation 5.1.
Here, D is the nodal degree of freedom, M is the mass matrix, K is the stiffness matrix,
and extR is the external load on the structure, which can be either a volume load, a surface
load, or a point load. Two methods are provided in ABAQUS for transient analyses. One
is an implicit method, in which the nodal solution is calculated from both historical and
current information. The other is an explicit method, in which the nodal solution is
calculated only from historical information. The explicit method is used in this study due
to its computational efficiency [Luo 2005].
extRKDDM =+&& (5.1)
98
5.2 Wave excitation and propagation case studies in ABAQUS
Studies in Chapter 4 indicated that the wave excited from a finite source consists
of a set of wave modes. By adjusting the source parameters, a desired wave mode can be
efficiently excited while the other modes are suppressed. In the following case studies,
3D FEM is used to validate the predictions of wave excitation.
In ABAQUS, the quasi-isotropic composite material is modeled as a multi-
layered structure, with each layer corresponding to a lamina. The material properties used
in this simulation are listed in Tab. 3-1. The materials are oriented such that the fiber
directions are in the 0o, 45
o, 90
o, and -45
o directions. The layers are attached to each other
with rigid bonding.
5.2.1 Case I: the first wave mode
The finite element model in this section is used to study the wave excitation of the
first fundamental wave mode. A numerical model corresponding to the system discussed
in Section 4.6.2.1 is studied. A picture of the numerical model can be seen in Fig. 5-1
and a list of parameters used in the model can be seen in Tab. 5-1. The coordinates shown
in yellow at the center are the local coordinates indicating the fiber orientation of each
lamina.
Figure 5-1: A picture of a numerical model in ABAQUS used to efficiently excite the first
guided wave mode at a 200 kHz center frequency.
X1_c
99
A picture of the finite element mesh is shown in Fig. 5-2. Fig. 5-2 (a) is a mesh of
the entire model. Fig. 5-2 (b) shows the magnified picture of the mesh at one corner,
where the meshing in the thickness and plate surface directions are clearly illustrated.
Table 5-1: Model and loading parameters in a finite element simulation
Lay up sequence [(0/45/90/-45)s]2
L: Length 150 mm
W: Width 70 mm
Structure
d: Thickness 3.2 mm
Evenly distributed normal loading on top surface
Loading Element Number 1
We: width 2 mm
Loading pattern
Le : length 40 mm
X1_c : Right edge of the loading
area to the center of the plate
20 mm
Center frequency 0.2 MHz
Number of cycles 3
Signal window Hanning
Excitation signal
Signal amplitude 3 kPa
Structured mesh with 8 node brick element
mesh element size 0.5mm
Finite element mesh
No. of element per layer 1
100
Fig. 5-3 shows the top view of the wave field at 20 µs. The wave field at 40 µs is
shown in Fig. 5-4. Several observations are listed as follows:
1.Using the rectangular loading, the majority of the wave energy propagates in the
0o and 180
o directions. The wave front is parallel to the length direction of the transducer
element.
2. The skew angle of this wave is small. A large portion of the energy is within
the width of the transducer element during wave propagation. Therefore, the beam
spreading effect is not very significant.
3. The wave package dispersion is not significant.
(a)
(b)
Figure 5-2: A finite element mesh. (a) the entire model (b) a corner of the model.
Thickness
direction
101
(a)
(b)
Figure 5-3: Top view of the wave field at 20 µs. (a) u1, (b) u3.
(a)
(b)
Figure 5-4: Top view of the wave field at 40 µs. (a) u1 , (b) u3.
x1
x2
x1
x2
102
The wave field distribution in the (x1,x3) cross section is shown in Fig. 5-5 when
the time is 20 µs. The x1 direction of the picture starts from the right edge of the loading
element. This profile is compared to Fig. 4-27. The numerical result matches the
expectation based on normal mode expansion.
Besides the field output, the wave signal at a given point can also be obtained
using ABAQUS. To compare the results quantitatively, the results from the theoretical
predictions (Fig. 4-23 and Fig. 4-24 ) are plotted together with the results from finite
element modeling in Fig. 5-6.
(a)
(b)
Figure 5-5: Thickness profile of the guided wave at 20 µs excited from a 2 mm wide
transducer element at 200 kHz. (a) u1, (b) u3.
x1
x3
x1
x3
103
10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
10 20 30 40 50 60 70 80 90 100
-0.05
0
0.05
Time (µs)
u1 a
mp
litu
de
(n
m)
Theoretical prediction
FEm calculation
Theoretical prediction
FEM calculation
x1=20 mm
x1=40 mm
(a)
10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
10 20 30 40 50 60 70 80 90 100
-0.2
0
0.2
Time (µs)
u3 a
mp
litu
de
(n
m)
Theoretical prediction
FEM calculation
Theoretical prediction
FEM calculation
x1=20 mm
x1=40 mm
(b)
Figure 5-6: Wave signal comparison between the theoretical prediction from normal
mode expansion and finite element modeling. (a) u1, (b) u3. The black box with dotted
line shows the directly excited wave package.
104
The black rectangles with dotted lines in Fig. 5-6 indicate the region where the
directly excited wave packages arrive at the sensor point. The waveforms after the
rectangles in the FEM calculation results are reflected waves and scattered waves. The
theoretical predictions match very well with the finite element calculation, quantitatively.
This match validates the results from both methods. As a result, the wave excitability
defined in Chapter 4 is a good measurement of how efficiently a wave mode can be
excited.
5.2.2 Case II: the third wave mode
According to Section 4.6.2.2, a shear loading at the edges of a 16 mm wide
transducer element can be used to efficiently excite the 3rd
wave mode at 200 kHz. The
corresponding finite element model was built in ABAQUS with the parameters listed in
Tab. 5-2.
Table 5-2:A finite element model parameters to excite the 3rd
wave mode
Lay up sequence [(0/45/90/-45)s]2
L: Length (mm) 150
W: Width (mm) 70
Structure
d: Thickness (mm) 3.2
Transducer element number 1
We: Element width (mm) 16
Le : Element length(mm) 40
Loading direction Shear traction in x1
Loading width (mm) 0.5
Loading pattern
Loading amplitude (kPa) 3
Center frequency (MHz) 0.2
Number of cycles 3
Excitation signal
Signal window Hanning
Mapped mesh with 8 node brick element
mesh element size 0.5mm
Finite element mesh
No. of element per layer 1
105
Fig. 5-7 shows snapshots of the wave field from a top view of the structure, i.e.
(x1, x2) plane, at 10 µs. The x1 and x3 direction displacement is plotted in part (a) and part
(b) respectively. The x2 direction displacement is not shown because the amplitude in the
x2 direction is much smaller than the other components. Since the first mode and third
mode are dominated by u3 and u1 respectively, the wave field in two displacement
components will show different distributions. Figure (a) shows a wave with larger
wavelength and group velocity, which is dominated with the third eave mode. Figure (b)
shows a wave mode with smaller wavelength and smaller group velocity, which is
dominated with the first wave mode.
(a)
(b)
Figure 5-7: Wave field snapshots at 10 µs. (a) u1 field, (b) u3 field.
X1
X2
106
The snapshots of the wave field at 20 µs are shown in Fig. 5-8. The third wave
mode has reached the boundaries of the model. Therefore, the reflected waves interfere
with the incident wave. Fig. 5-8 (a) shows that the plane wave characteristics of the third
wave mode are not preserved very well anymore. However, the first wave mode has a
smaller group velocity, it still propagates in its launching direction. The first wave mode
can be seen more evidently from Fig. 5-8 (b) since the first wave field is dominant in the
x3 direction.
(a)
(b)
Figure 5-8: Wave field snapshots at 20 µs: (a) u1 field, (b) u3 field.
X1
X2
107
In order to compare the thickness profile of the finite element analysis with the
theoretical prediction, the wave field snapshots taken at 20 µs are presented in Fig. 5-9.
These results are directly comparable with Fig. 4-31, the only difference is that in Fig. 5-
9, the x1 coordinate is only in the range of [0 95] mm because of the mode size in FEM
calculation.
Comparison of the time domain wave signal is shown for two positions in the
model. One is at 20 mm away from the edge of the loading; the other is 40 mm away
from the loading. The results of the theoretical prediction and numerical simulation are
both plotted in Fig. 5-10 for comparison purpose. Again, a matching of the wave field
validated both methods in the guided wave field analysis.
(a)
(b)
Figure 5-9: Thickness profile of the guided wave at 20 µs.
x1
x3
108
10 20 30 40 50 60 70 80 90 100
-0.04
-0.02
0
0.02
0.04
10 20 30 40 50 60 70 80 90 100
-0.04
-0.02
0
0.02
0.04
Time (µs)
u1 a
mp
litu
de
(n
m)
Theoretical prediction
FEM calculation
Theoretical prediction
FEM calculation
x1=20 mm
x1=40 mm
(a)
10 20 30 40 50 60 70 80 90 100
-0.02
-0.01
0
0.01
0.02
10 20 30 40 50 60 70 80 90 100
-0.02
-0.01
0
0.01
0.02
Time (µs)
u3 a
mp
litu
de
(n
m)
Theoretical prediction
FEM calculation
Theoretical prediction
FEM calculation
x1=20 mm
x1=40 mm
(b)
Figure 5-10: Wave signal comparison between the theoretical prediction from normal
mode expansion and finite element modeling: (a) u1, (b) u3. Black box with dotted line:
excited wave package.
109
5.2.3 Case III: wave modes with large skew angle
It was discussed in Section 3.4 that in quasi-isotropic composite laminates, there
are some wave modes with large skew angles. Fig. 5-11 shows the skew angle dispersion
curves of the quasi-isotropic laminate for wave propagation in the 0o direction. The skew
angle of the fifth wave mode (high lighted purple line) at 0.72 MHz has a skew angle of
-37o. In this section, this phenomenon will be validated with finite element simulation in
ABAQUS.
A wave loading pattern is designed to efficiently excite the guided wave mode on
the fifth mode line at 0.72 MHz. The model geometry and loading parameters are listed
in Tab. 5-3 . The corresponding spectrum of the loading is shown in Fig. 5-12. Predicted
wave mode components are plotted in Fig. 5-13.
0 0.5 1 1.5 2-40
-30
-20
-10
0
10
20
30
40
Frequency (MHz)
Ske
w a
ng
le (
de
gre
e)
1 2
3
4
5
Figure 5-11: Skew angle dispersion curve of wave propagation in the 0o direction of a
quasi-isotropic composite laminate.
110
Table 5-3: Model geometry and loading pattern to demonstrate large skew angle
Lay up sequence (0/45/90/-45)s2
L: Length (mm) 150
W: Width (mm) 70
Structure
d: Thickness (mm) 3.2
Element Number 5
We: Element width (mm) 3
Le : Element length(mm) 40
Loading direction normal traction in x3
Loading pattern
Loading amplitude (kPa) 3
Center frequency (MHz) 0.72
Number of cycles 10
Excitation signal
Signal window Hanning
Mapped mesh with 8 node brick element
mesh element size 0.5mm
Finite element mesh
No. of element per layer 1
Frequency (MHz)
Ph
ase
ve
locity (
km
/s)
0.5 1 1.5 2
2
4
6
8
10
12
14
16
18
20
5
10
15
20
25
30
Figure 5-12: Wave excitation (f-cp) spectrum for a 5 element transducer with 3 mm
element width and excited with a 10 cycled signal at 720 kHz using a Hanning window.
111
The excited wave field at several times are shown in Fig. 5-14. The skew angle of
the wave mode can be estimated from the wave field images. The result turns out to be
-32o. The predicted value of the skew angles is from -30
o to -37
o in the frequency range.
Therefore, the numerical simulation confirms the prediction of the large skew angle
phenomenon in the quasi-isotropic laminate.
Figure 5-13: Wave mode content curve for the loading described in Tab. 5-3 .
112
The reason for the large skew angle can be found by using wave structure
analyses. Fig. 5-15 shows the displacement profile and the power flow profile of the
wave mode (0.72 MHz, mode 5). The skew angle effect cannot be directly observed
from the displacement wave structure. What can be observed is a strong coupling
(a)
(b)
Figure 5-14: Sample wave field snapshots of u3 . (a) 2.5 µs (b) 20 µs
113
between the displacement in three directions. In the power flow wave structure, it is clear
that most of the energy is transmitted in the -45o layers with layer number 4, 5, 12 and 13.
Therefore, this wave mode has a large skew angle with negative value. For the
application in SHM, this wave mode will potentially be sensitive for fiber breakage
detection and matrix cracking in these plies.
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
0 1 2 3
-0.1
-0.05
0
0.05
0.1
Depth (mm)
Dis
pla
ce
me
nt
-1
0
1
-1
0
1
u1
u2
u3
P1
P2
0 1 2 3
-1
0
1
Depth (mm)
Po
we
r flo
w
P3
0o
45o
90o
-45o
Figure 5-15: The displacement and power flow wave structures of the guided wave mode
with large skew angle. The mode studied is the fifth wave mode at a frequency of 0.72
MHz.
114
5.3 Summary
Finite element simulations were used in this chapter to validate the predictions of
the guided wave mechanics theory. Guided wave modes in the quasi-isotropic laminate
with different characteristics are demonstrated. As an example, the first guided wave
mode at a frequency of 200 kHz propagates slower than the third wave mode at the same
frequency. The first wave mode at a frequency of 200kHz are excited efficiently with a
normal loading on the surface of the laminate. The guided wave mode with large skew
angle around -30 degree is also demonstrated.
The theoretical predictions made from the normal mode expansion technique
match very well with the numerical results obtained in the finite element modeling. The
theoretical study using normal mode expansion is computationally more efficient than the
three-dimensional finite element method in ABAQUS. The computing times of the
models presented in this chapter are about 10 hours in average. However, the normal
mode expansion technique only takes several minutes after the dispersion curves are
calculated. After the validation with finite element analysis, the normal mode expansion
technique can be used for guided wave excitation analysis in composite laminates and
elimination the difficulties of trial and error numerical experiments using FEM.
Note that, finite element modeling is still useful for wave excitation and scattering
analyses in irregularly shaped structures and for wave response from unusually shaped
defects.
Chapter 6
Guided waves in composites considering viscoelasticity
In most ultrasonic wave propagation problems, laminated composites are assumed
as elastic. However, at high frequencies, viscoelastic behavior of the material will
introduce significant attenuation to the wave propagation. This effect has been neglected
in the previous chapters. In this chapter, the influence of viscoelasticity will be addressed
for guided wave propagation and excitation in laminated composites. The hybrid SAFE-
GMM method developed in Chapter 2 will be used to generate the dispersion curves and
the wave structures. The normal mode expansion technique will also be modified to study
the wave excitation in lossy composite laminates.
6.1 Dispersion relation derivation
Wave propagation in viscoelastic media can be studied by substituting the elastic
constants in the governing equation (Equation 2.1 ) with a complex stiffness tensor. The
real part corresponds to the energy storage in wave propagation; and the imaginary part
corresponds to the damping introduced by the material viscoelasticity [Bartoli et al.
2006].
In Equation 6.1, both 'C and ''C are 6 by 6 matrices. Two models are used in
modeling the material viscoelasticity. One is the hysteretic model, where ''C is frequency
independent. The other is the Kelvin-Voigt model [Rose 1999], where ''C is a linear
function of frequency. The measurement of ''C at a given frequency 0f is provided as a
6x6 matrix η . Mathematical expressions for these two models are listed in Equation 6.2
and Equation 6.3.
''' CCC i−= (6.1)
116
When the material properties are substituted into the SAFE formulation, the dispersion
curves for the viscoelastic media can be obtained. Different from the wave propagation in
an elastic media, the wave numbers obtained from the solution of eigen function
Equation 2.32 are generally complex. The real part is related to the phase velocity of the
wave mode; and the imaginary part is related to the attenuation.
Therefore, two dispersion curves are needed to describe the guided waves in a
viscoelastic media. One is the phase velocity dispersion curve, and the other is the
attenuation dispersion curve.
The wave structure of a particular guided wave mode can be calculated using the
hybrid SAFE-GMM method by substituting the complex stiffness constant and complex
wave number into the GMM program. The final solution of a guided wave mode can be
expressed as Equation 6.5.
6.2 Numerical simulation results on wave propagation
Numerical simulations for the quasi-isotropic composite are carried out using
material properties listed in Tab. 6-1. The real parts of the elastic constants are the same
as the ones used in Chapter 3. Due to the limitation in obtaining the imaginary part of the
material properties specifically for the IM7/977-3 composite, the properties provided in
[Neau et al. 2001] are used. Although this will not be able to provide exact values of
Hysteretic model: ηCC i−= ' (6.2)
Kelvin-Voigt model: ηCC0
'f
fi−= (6.3)
αω
αξβ ic
ip
+=+= (6.4)
111 )(
3
)(
331 )()(),,(xtxitxi
eexUexUtxxuαωξωβ −−− == (6.5)
117
attenuation for each guided wave mode, a relative comparison between the guided wave
modes can be obtained. The predicted attenuation will also be compared to some
preliminary experiment results in Chapter 9.
Fig. 6-1 shows the phase velocity dispersion curves and attenuation dispersion
curves obtained from the Hysteretic model. In order to show the curves clearly, only
those wave modes with attenuation less than 1 neper/mm and phase velocity less than 20
km/s are plotted.
Table 6-1: Lamina properties of the IM7/977-3 composite used in simulation
Real part (GPa) Imaginary part (GPa)
'
11C 178 ''
11C 8.23
'
12C 8.35 ''
12C 0.65
'
13C 8.35 ''
13C 0.6
'
22C 14.4 ''
22C 0.34
'
23C 8.12 ''
23C 0.25
'
33C 14.4 ''
33C 0.65
'
44C 3.16 ''
44C 0.24
'
55C 6.10 ''
55C 0.28
'
66C 6.10 ''
66C 0.25
Measurement
frequency 0f 2 MHz
Density ρ 1.60 g/cm3
118
The relationship between nepers and decibels is expressed in Equation 6.6.
The typical attenuation value shown in Fig. 6-1 (b) is 0.1 neper/mm, which means the
wave attenuation is 8.69 dB/cm. From this figure, we can also see that the overall trend of
attenuation increases with frequency. However, for a specific mode, the attenuation could
also decrease with the increase of frequency. In addition, at a specific frequency, we can
always find a mode that has the smallest attenuation.
As a comparison, the results obtained from the Kelvin-Voigt model are shown in
Fig. 6-2. Since the reference frequency used in this calculation is 2 MHz, the attenuation
for the wave modes at 2 MHz is the same as the values obtained in the Hysteretic model.
For the frequencies less than 2 MHz, a smaller imaginary part of the stiffness constant is
used according to Equation 6.3 , therefore, these attenuation results are smaller than for
the Hysteretic model.
(a) (b)
Figure 6-1: (a) Phase velocity dispersion curve and (b) attenuation dispersion curves
obtained from Hysteretic model.
dB69.8)(log20neper1 1
10 == dBe (6.6)
119
The wave modes having the least attenuation are highlighted in Fig. 6-3. Similar
modes are identified in both the Hysteretic model and the Kelvin-Voigt model.
Another important issue in the consideration of material viscoelasticity is whether
the introduction of material viscoelasticity will affect the phase velocity dispersion curve.
In order to answer this question, the dispersion curves obtained from the Hysteretic
models and the dispersion curves without considering damping are all plotted in Fig. 6-4.
The results indicate that the phase velocity dispersion curve in the elastic model have a
Figure 6-2: (a) Phase velocity dispersion curve and (b) attenuation dispersion curves
obtained from Kelvin-Voigt model
Figure 6-3: Wave modes with least attenuation at a given frequency. (a) Hysteretic model
(b) Kelvin-Voigt model.
120
good match with the viscoelastic model. This agreement generally validates the
usefulness of phase velocity dispersion curves obtained in the elastic model for guided
wave applications. Differences occur at mode interaction regions. As is shown in Fig. 6-4
(b), the dispersion curves are two separate lines in the elastic model. However, in the
viscoelastic model, the two curves interact with each other.
In the next step, we will compare the wave structures obtained from these two
models. As an example, the first wave mode at 200kHz is considered. The x1 direction
displacement distribution obtained from the elastic model and the Hysteretic viscoelastic
model is plotted in Fig. 6-5. The result indicates that the difference in the real part of u1
solution is very slight for these two models. In the elastic model, the imaginary part of u1
is zero. However, in the viscoelastic model, the u1 wave structure also has an imaginary
part, although its amplitude is only 3% as for that of the real part.
(a) (b)
Figure 6-4: Comparison of phase velocity dispersion curves between the elastic model and
the Hysteretic viscoelastic model. Dotted line: elastic model, solid line: viscoelastic model.
(a) Full set of dispersion curve, (b) Magnified curve shows mode interaction.
121
This tells us that for the wave modes with light attenuation, the wave structure
obtained from the elastic model is still a good approximation. For the wave modes with
large attenuation, larger differences are observed between the wave structures.
With the introduction of attenuation, the definition of guided wave group velocity
will be altered. Equation 6.7 repeats the group velocity definition presented in
Equation 2.38.
When ξ is complex, the derivative of ξ can not be calculated exactly from the
derivative of phase velocity with respect to frequency. However, the energy velocity
defined in Equation 2.42 will still provide a good estimation on the wave energy
transmission in viscoelastic media. The comparison between the energy velocity obtained
from viscoelastic and elastic models are shown in Fig. 6-6. The comparison shows that
the difference only occurs at the mode interaction region. For other modes, the average
(a ) (b)
Figure 6-5: Wave structure comparison between the elastic model and the viscoelastic
model. Wave mode: first mode at 200kHz, u1 displacement. (a) real part (b) imaginary
part.
ξ
ω
d
dcg =
pc
ωξ ≠Q ,
df
dcfc
cc
p
p
p
g
−
≠∴2
(6.7)
122
difference in energy velocity is below 0.01 km/s. Therefore, this information again
validates the general accuracy of using an elastic model for guided wave group velocity
(energy velocity) prediction.
One important feature is observed when we compare the energy velocity
dispersion curve with the minimum attenuation modes shown in Fig. 6-3. We find that
the minimum attenuation modes have strong correlations with the maximum energy
velocity modes. We believe this is a new observation and this can be used in future NDE
and SHM tests. With this rule, the modes with least attenuation can be selected by only
investigating the modes with large group velocity in the elastic model. Fig. 6-7 shows
these two sets of curves side by side for a comparison.
(a ) (b)
Figure 6-6: A comparison of energy velocity dispersion curve generated from the elastic
and viscoelastic model. (a) elastic model (b) viscoelastic model.
123
The comparison between skew angle dispersion curves obtained from the elastic
model and the viscoelastic model are presented in Fig. 6-8. The skew angle prediction
from the elastic model matches the skew angle from the viscoelastic mode for most of the
wave modes. At mode interaction regions, for example the ones indicated with circles, a
comparably large difference is observed from the two models.
(a) (b)
Figure 6-7: Guided wave feature comparisons from a viscoleastic model. (a) Wave modes
with largest group velocity for a given frequency (b) Wave modes with smallest
attenuation for a given frequency.
0 0.5 1 1.5 2-40
-30
-20
-10
0
10
20
30
Frequency (MHz)
Ske
w a
ng
le (
de
gre
e)
Mode 1Mode 2Mode 3Mode 4Mode 5Mode 1Mode 2Mode 3Mode 4Mode 5
Figure 6-8: Comparison of skew angle dispersion curves obtained from elastic and
viscoelastic model. Dotted line: elastic model. Solid line: Hysteretic viscoelastic model.
124
6.3 A new normal mode expansion technique for viscoelastic media
The normal mode expansion technique developed in section 4.1 is based on the
complex reciprocity relation. It is only valid for lossless media. In this chapter, a new
normal mode expansion technique for viscoelastic media will be developed. To the
knowledge of the author, this is the first attempt to solve this problem.
This section will be started with the derivation of wave mode orthogonality in
viscoelastic media from the real reciprocity relation shown in Equation 6.8.
First, we will prove mode orthogonality. Assuming m and n are two wave modes with
one frequency. The wave mode solutions are in a form of Equation 6.5. For stress free
mode solutions, no external force is applied. Therefore both Fm and F
n are zero. The
mode orthogonality equation is in Equation 6.9.
This means, when nm ββ −≠ (i.e. nm −≠ ),
With the wave orthogonality function defined, normal mode expansion can be used to
study the wave excitation characteristics. The equation for mode component is expressed
in Equation 6.11.
The new snf and vnf are defined in Equation 6.12 and Equation 6.13 .
[ ] nmmnmnnm FvFvσvσv −=−•∇ (6.8)
04)( =− mnnm Qi ξξ
∫ ••−•=sectioncross
31ˆ}{
4
1dxxQ nmmnmn σvσv
(6.9)
0=mnQ for nm −≠ (6.10)
)()()()(4 111
1
xfxfxaix
Q vnsnnnnn +=+∂
∂−− β (6.11)
H
ynnsn xxxxxxxxf
033313131
ˆ)}(),(),()({)(=
••−•= σVFv (6.12)
125
Here F is the excitation force and V is the velocity prescription. Solving these equations,
the mode expansion coefficients for wave excitation in viscoelastic media are found as in
Equation 6.14.
Comparing Equation 6.14 with Equation 4.12, the difference is in the wave mode
solutions used to calculate sf and vf . For rightward propagating waves (n>0), the wave
structure information of the leftward propagating wave (n<0) will be used.
Both Equation 6.14 with Equation 4.12 will come to the same solution when an
elastic media is considered. This can be proved with the facts listed in Equation 6.15,
Equation 6.16 , and Equation 6.17.
However, for an attenuative wave mode, Equation 6.15 and Equation 6.16 are not valid.
6.4 Numerical simulation results
Similar to the case of elastic material, the wave mode excitability from a surface
source can be defined as the particle velocity at the surface from a left propagating wave.
Since the wave structure solution is generally complex, the absolute value of the particle
velocity is a good measurement of the excitabilities. Three excitability curves can be
∫ •=H
nvn dxxxxxf0
33131 ),()()( Fv (6.13)
12
)(
)()(
11
211
)(
)()(
11
111
,)exp(4
)()()exp()(
,)exp(4
)()()exp()(
,0)(
2
1
1
xLdiQ
ffxixa
LxLdiQ
ffxixa
Lxxa
L
Ln
nn
nvns
nn
x
Ln
nn
nvns
nn
n
<−+
=
≤≤−+
=
≤=
∫
∫
−
−−
−
−−
ηηβηη
β
ηηβηη
β (6.14)
)()( 33
*xvxv nn −= (6.15)
nn ξβ = (6.16)
nnnn PQ =− )( (6.17)
126
defined for the loading in x1, x2 and x3 directions respectively. As an example, Fig. 6-9
shows the wave excitability of the first three wave modes using loading in the x1
direction. The corresponding excitability in the elastic model is also shown as a
comparison and is represented by the dotted line. Differences are observed in the mode
conversion region and the higher attenuative region.
Mode components from a finite source can be obtained from Equation 6.14.
Fig. 6-10 show the wave mode excitation performance using a 4 mm wide element of x1
direction shear loading. The amplitude of the load is 1000 N. The result shows that at low
frequency, the wave excitabilities are almost not changed. However, for high frequency
and severe damped modes, the wave mode excitability is reduced in the viscoelastic
model.
0 0.5 1 1.5 20
0.05
0.1
0.15
0.2
0.25
Frequency (MHz)
Excita
bili
ty F
1
0.38 0.4 0.42 0.44 0.46 0.480
0.02
0.04
0.06
0.08
0.1
0.12
Frequency (MHz)
Excita
bili
ty F
1
(a ) (b)
Figure 6-9: Comparison of wave mode excitability with x1 direction force on the surface.
Dotted line: Elastic model; solid line: Viscoelastic Hysteretic mdoel. (a) Frequency range
of 0 to 2 MHz. (b) Magnified region of mode interaction.
127
6.5 Summary
Material viscoelasticity of a laminated composite is considered in this chapter.
Several conclusions and observations of wave propagation and excitation are summarized
as follows.
1. With an introduction of material viscoelasticity, an attenuation
dispersion curve can be obtained for guided wave modes. The
attenuation dispersion curve provides a guideline for estimating the
propagation distance of a wave mode.
2. From the attenuation dispersion curve, the wave modes with least
attenuation are studied. The results of the least attenuative modes match
with the wave modes with maximum group velocity.
3. Introducing a weak attenuation does not significantly affect the phase
velocity, group velocity, skew angle and wave structure of a guided
wave mode. Therefore, most of the results obtained from an
approximated elastic model are still valid. However, the introduction of
0 0.5 1 1.5 20
1
2
3
4
5
6
7
8
9
10x 10
4
Frequency (MHz)
Excita
bili
ty
0 0.5 1 1.5 20
1
2
3
4
5
6
7
8
9
10x 10
4
Frequency (MHz)
Excita
bili
ty
(a) (b)
Figure 6-10: A comparison of wave mode excitability using F1 direction loading in (a) an
elastic and (b) a viscoelastic model.
128
attenuation affects the mode interaction and dispersion curve crossing.
At these local regions, the dispersion curves will be modified.
4. The introduction of attenuation into the modeling affects the procedure
of normal mode expansion. A new normal mode expansion formula is
derived in this thesis starting from the real reciprocity relation. With this
formula, the wave excitation characteristics in a viscoelastic composite
can be studied. The result shows that the introduction of attenuation
affects the wave mode coefficients and the excited wave field.
129
Chapter 7
Guided wave sensitivity to damage in composites
Two types of damage could occur in engineering composites from a guided wave
modeling point of view. One is introduced by long term environmental aging and fatigue.
The distributed microscopic fiber breaking and matrix cracking can be modeled as global
material property degradation. Detection of global material property change with guided
waves could provide early warnings to more severe damage types. Another type of
damage is modeled in such a way that a delamination or damage introduced by
mechanical impact appears as a discontinuity in material properties. In this case, wave
scattering phenomenon could be used to detect the damage. Influences of these two types
of modeled damages on guided wave propagation are studied in Section 7.1 and Section
7.2 respectively.
7.1 Effect of material property degradation on guided wave propagation
7.1.1 Theoretical study
Density, stiffness constants, and thickness of the layers are key material properties
affecting guided wave propagation. Equation 7.1 is the governing equation for wave
mode analysis in the semi-analytical finite element formulation.
Details for M , 11K , 12K , 21K , and 22K are listed in Equation 2.26 .
Assuming the density of the material changes by a factor ( a ), the influence on
phase velocity dispersion curve is derived in Equation 7.2 .
0])([ 0
2
11211222
2 =−+−+ UMKKKK ωξξ i (7.1)
130
Therefore, the phase velocity dispersion curve expands and shrinks proportionally with
respect to )0,0(),( =pcf . When the density increases, the dispersion curve shrinks; when
the mass density decreases, the dispersion curve expands.
Similarly, if the elements in the stiffness constant matrix all change together with
a factor (b), the changes in dispersion curves are derived in Equation 7.3
Therefore, the phase velocity dispersion curve also expands with increased stiffness and
shrinks with reduced stiffness.
When the elements of the stiffness constant do not change together, the changes in
guided wave dispersion relation will be more complicated. Numerical simulation is
needed to predict the dispersion curve at degraded states.
7.1.2 Density variation
The relation of the phase velocity dispersion curve to the density is now studied
numerically as a proof to the derivation of Section 7.1.1. The results presented in this
section are for the wave propagation in the 0o direction of a quasi-isotropic laminate. The
nominal density is 1.60 kg/m3. Fig. 7-1 plotted the dispersion curves for the undamaged
material (blue) and the material with 10% density reduction (red). This figure confirms
that with the reduction of density, the dispersion curves expanded in both f and pc axis.
The variation of mass density affects the velocity of each mode as well as the cut-off
frequency of the mode lines.
01 ρρ a= => 01 MM a=
when 01 ξξ = , => 01
1ωω
a= , 01
1f
af = ,
01
1pp c
ac =
(7.2)
01 CC b= => )()( 1 mnmn b KK = , 2,1, =nm
when 01 ξξ = , => 01 ωω b= , 01 fbf = , 01 pp cbc =
(7.3)
131
7.1.3 Elastic stiffness variation
As was predicted in Section 7.1.1 , when all components of the elastic stiffness
vary at the same rate, the dispersion curve corresponding to the degraded structure can be
predicted from the pristine structure using Equation 7.3 . Fig. 7-2 confirms this rule with
a structure whose stiffness matrices are reduced 10% from the nominal elastic stiffness
constants. The blue dashed lines are the dispersion curves for the structure without
stiffness reduction. The blue solid lines are the predicted dispersion curve for the stiffness
reduced structure from proportional scaling. The red dots are the representative points
calculated with a stiffness-reduced model. A good match is observed between the blue
solid lines and the red dots.
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (mm)
Ph
ase
ve
locity
(km
/s)
Blue: Original
Red: 10% reduction
Figure 7-1: Dispersion curves for guided wave propagation in composite laminates.
Blue dashed line: nominated mass density of IM7/977-3 ρ=1.6 kg/m3 Red line :assumed
10% density reduction ρ=1.44 kg/m3 .
132
In real situations, the components of the elastic stiffness matrix do not necessarily
vary together. Therefore, studying the effect of changing a particular engineering
constant is valuable for both damage detection and material property estimation.
Comparison of 10% change of E1, E2, G12, 12ν , and 23ν on the dispersion curves are
shown in Fig. 7-3 through Fig. 7-7 .
These figures indicate that the variation of E1 significantly affects the phase
velocity dispersion curves of u1 dominant wave modes in the cp axis. Therefore, the
second and the third fundamental modes can be used to detect this change efficiently.
Degradation of E2 shifts the u3 dominant wave modes in the frequency axis. The second
and third wave modes around 0.4 MHz could be sensitive modes to detect this change.
Changing G12 affects the cut off frequency of most of the modes especially the u2
dominant wave modes. A broadband excitation and frequency domain analysis would be
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (mm)
Ph
ase
ve
locity
(km
/s)
Figure 7-2: Figure illustrates the effect of dispersion curve scaling when the material
property degradation introduces 10% stiffness reduction. Blue dashed line: no stiffness
reduction, blue solid line predicted dispersion curve with stiffness reduction, red dots
calculated dispersion curve with stiffness reduction.
133
a benefit in detecting this change. Poisson’s ratio 12v only has slight effects on the
dispersion curves. Therefore its values are not very critical in dispersion curve
generation. Poisson’s ratio 23v affects the dispersion curve of some higher order wave
modes at high frequency.
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (mm)
Ph
ase
ve
locity
(km
/s)
Figure 7-3: Effect of engineering constant variation on guided wave dispersion curves.
Blue dashed line: nominated material property. Red line: with 10% fiber direction
modulus (E1) reduction of the lamina.
134
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (mm)
Ph
ase
ve
locity
(km
/s)
Figure 7-4: Effect of engineering constant variation on guided wave dispersion curves.
Blue dashed line: nominated material property. Red line: with 10% transverse modulus
(E2) reduction of the lamina.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
8
10
12
14
16
18
20
Frequency (mm)
Ph
ase
ve
locity
(km
/s)
Figure 7-5: Effect of engineering constant variation on guided wave dispersion curves.
Blue dashed line: nominated material property. Red line with 10% in plane shear
modulus (G12) reduction of the lamina.
135
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
8
10
12
14
16
18
20
Frequency (mm)
Ph
ase
ve
locity
(km
/s)
Figure 7-6: Effect of engineering constant variation on guided wave dispersion curves.
Blue dashed line: nominated material property. Red line with 10% Poisson’s ratio(v12)
reduction of the lamina.
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (mm)
Ph
ase
ve
locity
(km
/s)
Figure 7-7: Effect of engineering constant variation on guided wave dispersion curves.
Blue dashed line: nominated material property. Red line with 10% Poisson’s ratio (v23)
reduction of the lamina.
136
7.1.4 Ply thickness variation
It is commonly recognized in the use of guided waves in single layer isotropic
material that the frequency thickness product is a characteristic quantity in both the phase
velocity and group velocity dispersion curves. However, frequency is used as a variable
throughout this thesis. This is because in a multilayered structure the thickness of each
layer has an effect on the dispersion relation of the structure. The variation of the total
thickness can not uniquely define the behavior of the structure.
In laminated composites, the effect of the variation in the thickness of the prepreg
on the dispersion curves is shown in Fig.7-8. Dispersion curves for two 16 layer
structures with 0.2 mm and 0.18 mm ply thickness are plotted with blue curves and red
curves respectively. The result indicates that the curves are scaled in the frequency axis
only. The most significant effect of the ply thickness change on the dispersion curve is
the shifting of cut off frequencies.
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (mm)
Ph
ase
ve
locity
(km
/s)
Figure 7-8: Variation of dispersion curves due to ply thickness change of a 16 layer
quasi-isotropic composite. Blue lines: 0.2mm Red lines: 0.18mm.
137
A common damage to the composite material is surface erosion. The effect of the
first ply thickness reduction on guided wave dispersion curves are illustrated in Fig. 7-9
In this figure the blue lines are for the structure with ply thickness 0.2 mm. The red lines
are for the structure with the first ply thickness reduced to 0.1 mm. In this case we could
see that the most significant change in the dispersion curve is the shifting of the cut off
frequencies of the higher order modes.
7.2 Guided wave scattering sensitivity
Wave scattering is an important issue in damage detection using ultrasonic guided
waves. The existence of a flaw in a material is usually detected from its echo signal.
However, quantitative analysis of guided wave scattering in composite material is very
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (mm)
Ph
ase
ve
locity
(km
/s)
Figure 7-9: Effect of surface erosion on guided wave phase velocity dispersion of a 16
layer quasi-isotropic composite laminate. Blue line: all ply thickness 0.2 mm. Red line:
first layer thickness reduction of 0.1 mm.
138
difficult. Although finite element analysis can be used to calculate wave scattering for a
given situation, the extrapolating of the results to a different case is difficult. In this
section, we put forward a hypothesis on guided wave scattering sensitivity to damage in
composites based on an analytical wave scattering study.
In [Auld 1990], an S-parameter method is used as an indication of how much
energy of the incident wave is converted into reflected waves and mode-converted
transmission waves. Numerical expression of the S parameter is in Equation 7.4
Here, FS is the surface of the damage, n̂ is the direction normal at the surface of the
damage. The wave field in the undamaged state and the damaged state are denoted with
( v , σ ) and ( 'v and '
σ ) respectively. When we want to study the wave mode scattering
sensitivity, the wave field of the undamaged state is the incident wave mode. In the case
of a delamination, the damaged wave field can be approximated as stress free at the
debonded surfaces ( 0'=σ ). Therefore, Equation 7.5 describes the wave scattering
parameter in the case of delamination.
Equation 7.5 indicates that the wave mode conversion parameter is related to the stress
distribution of the undamaged field, the wave velocity of the damaged field at the
delamination boundary, and the shape of the delamination. The shape of the delamination
is case dependent, and the wave field at the delamination boundary after delamination
occurred is difficult to obtain. However, from the equation, it is quite clear that the
sensitivity of a guided wave mode to detect a delamination is directly related to the stress
distribution (σ ) of the incident wave mode at the position of the delamination.
In the case of a delamination, the surface normal n̂ is in the 3x direction.
Therefore, the following hypothesis is put forward to estimate the sensitivity of a guided
wave mode for delamination without considering the detailed size of the delamination.
dSnS
FS
rNlM ∫ ••−•=∆ ˆ)(4
1 ''
,σvσv (7.4)
dSnS
FS
rNlM ∫ ••−=∆ ˆ)(4
1 '
,σv (7.5)
139
To maximize the sensitivity of a guided wave to a delamination at a certain depth,
the stress components normal to the (x1, x2)plane ( 313233 ,, σσσ ) should be maximized.
Therefore, one sensitivity metric can be formulated as in Equation 7.6.
As an example, when the structure is a [(0/45/90/-45)s]2 laminate, the sensitivity
spectrum of the first six wave mode lines for the wave propagating in the 0o direction are
shown in Fig. 7-10 to Fig. 7-11. In these two cases the delamination is located at the
first interface (between 0o and 45
o) and the third interface (between 90
o and -45
o).
2
31
2
32
2
33 σσσ ++=ySensitivit (7.6)
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
3.5
Frequency (MHz)
Se
nsiti
vity
Mode 1Mode 2Mode 3Mode 4Mode 5
Figure 7-10: Estimated sensitivity spectrum of guided wave modes to delamination at the
first laminate interface of a [(0/45/90/-45)s]2 composite structure.
140
In most cases, the location of the delamination can not be precisely predicted in
structural health monitoring. Therefore, a general evaluation of the sensitivity to
delamination in an arbitrary layer would be helpful. In order to do this, a general
sensitivity definition is presented in Equation 7.7 . Here, N is the total number of layers.
N-1 is the number of interfaces.
Fig. 7-12 is the overall sensitivity estimation of guided waves in the 16 layer
quasi-isotropic laminate.
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
Frequency (MHz)
Se
nsiti
vity
Mode 1Mode 2Mode 3Mode 4Mode 5
Figure 7-11: Estimated sensitivity spectrum of guided wave modes to delamination at the
3rd laminate interface of a [(0/45/90/-45)s]2 composite structure.
1
)(1
2
31
2
32
2
33
−
++
=
∑−
NySensitivit
N
Interfaces
σσσ
(7.7)
141
7.3 Summary
In this chapter, influences of damage on guided wave propagation characteristics
are studied. Long term aging is modeled with a change of composite density, stiffness
constant, and thickness variation. The results on guided wave propagation are
summarized as follows.
1. When the density of a composite decreases by a factor of a , the phase
velocity dispersion curve expands in both frequency and phase velocity
axes by a factor a/1 .
2. When the stiffness of the composite material decreases by a factor of b ,
the phase velocity dispersion curve shrinks in both frequency and phase
velocity axes by a factor of b .
3. Studies of influence of each engineering constant change on guided
wave dispersion curves are also performed. Decreasing E1 shrinks the
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
3.5
Frequency (MHz)
Se
nsiti
vity
Mode 1Mode 2Mode 3Mode 4Mode 5
Figure 7-12: Estimated sensitivity spectrum of guided wave modes to delamination in a
[(0/45/90/-45)s]2 composite structure.
142
dispersion curve on the phase velocity axis. The effect is particularly
significant for u1 dominant wave modes. Decreasing E2 shrinks the
dispersion curve in the frequency axis. The phenomenon is significant
for quasi R-L type wave modes, especially for u3 dominate modes.
Decreasing G12 shrinks the dispersion curve in frequency axes. The
phenomenon is significant for quasi shear horizontal waves. Influence
of Poison’s ratio on dispersion curve is not very significant, especially
at lower frequency.
4. When the thickness of each lamina decreases by a factor of a , the
dispersion curve expands in the frequency axis by a factor of a . The
surface erosion in a composite also affects the dispersion topology by
expanding slightly in the frequency axes and the reduction of phase
velocity at low frequency.
Although the details of the material degradation are not explored in this thesis
study, the rules obtained in this section will be important for selecting a good mode to
detect material degradation. In addition, these rules can also be used in material property
characterization. By detecting the changes in dispersion curves, mistakes in lay-up
sequence could also be detected.
In Section 2, the expected sensitivity of a guided wave mode to localized damages
is studied. The formulation of the sensitivity is based on an analytical study of guided
wave scattering. A new sensitivity definition for guided wave mode selection is proposed
using stresses at the damage surface. Specific results are provided for delamination
detection. They indicate that commonly used fundamental wave modes 3 at low
frequency are not sensitive to delamination.
Chapter 8
Guided wave mode selection
8.1 Introduction
In the previous chapters, many features related to a guided wave mode in
composite materials have been obtained. Chapter 3 studied the phase velocity, group
velocity, and skew angle of a wave mode when the material was assumed elastic. Chapter
4 studied the wave excitation, dispersion, and beam spreading characteristics. Chapter 6
evaluated the viscoelastic effect on the wave propagation and excitation. In addition to all
the features studied in Chapter 3 and Chapter 4, the attenuation characteristic of a guided
wave mode is introduced. Chapter 7 studied the expected wave mode sensitivity to
different kinds of structural damage especially delamination in composites.
How to comprehensively evaluate all these features and select the most suitable
wave mode candidates for an inspection task becomes the next problem to address. A
wave mode might have superior performance in terms of one feature but it is not very
good in terms of another feature. For example, “mode 3” at low frequency has the least
attenuation; therefore it is a candidate for long range inspection. However, its sensitivity
to delamination is not good compared with other modes. There are also two ways of
evaluating a feature. One is a crisp selection method. For example, when the wave mode
skew angle is less than 10 degrees, it is acceptable. When it is larger than 10 degrees, it
will not be selected. Another evaluation method is based on fuzzy analysis. As an
example, when one says a wave propagates “fast”, we are not referring to the group
velocity larger than a specific value to be “fast” and below that to be “slow”. We are
thinking in a way that the larger the value is, the faster it is. In order to have a flexible
and comprehensive evaluation of the guided wave mode features, a set of mode selection
rules will be used.
144
8.2 Guided wave mode selection rules
A mode selection rule is a descriptive criterion that states how a guided wave
feature will satisfy our inspection goal. Two types of rules are considered. One is a crisp
mode selection rule, which means the mode either satisfies the condition or not. The other
is a fuzzy mode selection rule. For both of these cases, a goodness function will be used
to describe how a mode qualifies the requirements.
As an example, we can select those wave modes with an attenuation rate below
0.5 dB/mm to be qualified wave modes for long range monitoring. For this case the
goodness function of attenuation is shown in Fig. 8-1 (a). In a fuzzy case, a wave mode
with an attenuation rate below 0.2 dB/mm will be considered perfect, and the ones above
1 dB/mm will not be acceptable and the goodness ramps in between.
The corresponding mode selection results for the guided wave propagating in the
0o direction of the [(0/45/90/-45)s]2 laminate is shown in Fig. 8-2. The attenuation
dispersion curve from the Hysteretic model described is used. The mode selection result
is superimposed on the phase velocity dispersion curves. In Fig. 8-2 (a), the wave modes
that meet the criterion are plotted in red. In Fig. 8-2 (b), the mode goodness results are
plotted in color with red as one and blue as zero. Basically, the results indicated that
(a) (b)
Figure 8-1: Goodness function definition for guided wave selection. Evaluates
attenuation characteristic (a) Crisp rule (b) fuzzy rule.
145
guided waves at a low frequency are good. If the desired frequency is higher, only some
specific modes can satisfy the requirement.
Enlightened from the filter design concepts in electrical engineering, our
goodness function can also be defined as low path, high path, band path, or band stop
distributions. As an example, if we want to illustrate the possibility of large skew angle
wave modes in a quasi-isotropic composite laminate. We will select those modes with
large skew angles. The rule and the corresponding mode selection results are shown in
Fig. 8-3 .
(a)
(b)
Figure 8-2: Mode selection results considering wave mode attenuation. (a) Crisp mode
selection with 0.5dB/mm allowed. (b) Fuzzy selection.
146
The mode selection with a single criterion has been discussed. However, in a
practical design, many parameters need to be considered to make a decision. The joint
consideration of a group of mode selection criteria can be carried out in the following
steps.
1. Any wave mode that does not satisfy the crisp rules will be excluded
from the pool of candidates.
2. The fuzzy rules can be evaluated in two ways. (a) Average the fuzzy
rules with specified weights for each rule. (b) Choose the minimum
value of goodness from all the rules to be the final goodness value. This
will keep those wave modes with good performance for all the rules.
3. Combine crisp and fuzzy rules.
For example, in order to show the wave modes with large skew angle, the modes
with excessive attenuation will be excluded. If we apply the rules described in Fig. 8-3 (a)
and Fig. 8-1 (a), the results are shown in Fig. 8-4. The fifth wave mode around 0.69
MHz and 5.1 km/s phase velocity will be a good mode to show the concept.
(a) (b)
Figure 8-3: The mode selection rule and candidate wave modes for large skew angle
demonstration. (a) High pass filter for the absolute value of skew angle. (b) Mode
selection results.
147
8.3 Guided wave long range monitoring potential
In structural health monitoring, long range wave propagation potential will be
explored. In order for the waves to propagate long distances, the wave attenuation rate
and wave dispersion coefficient should be small; the wave excitability should be large. In
addition, for appropriate damage detection, the expected sensitivity should be large.
Whether the selected wave mode could be suitable for a practical transducer design is
also an important issue. Here, we will focus on selecting the guided wave modes with
desired performance. First, we will study the mode selection for each criterion. Tab. 8-1
lists some rules to be considered in the design.
Figure 8-4: Guided wave mode selection for the purpose of demonstration large skew
angle.
Selected mode
148
The mode selection performance for each individual rule is studied as follows from
Fig. 8-5 to Fig. 8-9.
Figure 8-5 is a representation of wave mode attenuation characteristics. The result
indicates that wave modes in the low frequency region can propagate a long distance.
Figure 8-6 selected the less dispersive wave modes. Figure 8-7 selected the wave modes
with reasonable sensitivity to delamination. It shows that the low frequency regions of the
second and third wave mode are excluded. Figure 8-8 and Fig. 8-9 shows the wave mode
selection with reasonable excitation performance using x1 and x3 direction loading
respectively. Since in the composite plate the guided wave modes generally have
displacement components in all three directions, traditional Lamb modes and SH modes
Table 8-1: Proposed mode selection rules for mode selection based on long range
delamination detection in composite laminates.
Rule No Feature Rule
1 Attenuation Fuzzy Rule: Low pass [0.01 0.5] dB/mm
2 Dispersion Fuzzy Rule: Low pass [0.1 0.3 ] µs/mm
3 Sensitivity Fuzzy Rule: High pass [0.1 1.5]
4 Excitability F1 Crisp Rule: High pass 0.05
5 Excitability F3 Crisp Rule: High pass 0.1
(a) (b)
Figure 8-5: Guided wave mode selection considering attenuation. (a) selection rule (b)
selection result.
149
cannot be strictly identified. Using x1 and x3 directional loading basically generates those
wave modes with dominant response in x1 and x3 plane. These wave modes can be
recognized as quasi-Lamb mode. We also see that the third mode at low frequency can
be efficiently excited with an x1 directional loading but not with a loading in the x3
direction.
(a) (b)
Figure 8-6: Guided wave mode selection considering mode dispersion. Less dispersive
modes selected (a) selection rule (b) selection results.
Figure 8-7: Guided wave mode selection considering mode sensitivity (a) selection rule
(b) selection result.
150
In order to select a mode that can efficiently detect delamination in a composite
plate with long range coverage, a comprehensive mode selection will be performed to
evaluate the features expressed above. When we set the weight of importance for rule 1, 2,
and 3 to be 0.6, 0.2, and 0.2, and an x1 direction loading (rule 4) is considered, the overall
goodness of the wave modes are plotted in Fig. 8-10. The result indicates that the low
frequency region of mode 1 and 3 are the best choice. However, because of the trade-off
in attenuation and sensitivity, none of the wave modes has absolute preference. In
addition, the mode 5 and 6 around 500 kHz to 900 kHz is also valuable to be explored.
(a) (b)
Figure 8-8: Guided wave mode selection considering wave excitation with loading in the
x1 direction. (a) Selection rule (b) qualified wave modes plotted in red.
(a) (b)
Figure 8-9: Guided wave mode selection considering wave excitation with loading in the
x3 direction. (a) Selection rule (b) qualified wave modes plotted in red.
151
Other regions of the dispersion curve will produce either less propagation distance or
reduced sensitivity.
The results of mode selection provided some general guidelines for guided wave mode
selection. It direct us to the parts of dispersion curves that are most likely going to
produce good testing results. A detailed wave excitation and propagation analysis can be
performed using the simulation tool expressed in Chapter 4. A detailed wave scattering
analysis can also be performed with numerical simulations, such as finite element
analysis.
Figure 8-10: Overall guided wave mode selection considering rules 1 to 4 listed in
Tab. 8-1.
1
2 3
4
Chapter 9
Experimental studies
In this chapter, some experiments on guided wave mechanics in composite
materials are presented to validate the observations in the theoretical study. The
composite material is fabricated from unidirectional composite prepreg purchased from
Cytec Engineered Materials. The laminate was constructed by hand lay up at the
Composite Manufacturing Technology Center (CMTC) and cured in an autoclave in the
Applied Research Lab at Penn State University. Detailed documentation of our composite
preparation process can be seen in [Bell 2004; Noga 2006].
In section 9.1, guided wave phase velocity dispersion curves, wave attenuation
characteristics, and skew angle effects are studied. Section 9.2 studies the wave excitation
in composites using surface mounted piezoelectric transducers. Section 9.3 presented
some preliminary studies on guided wave damage detection. Comparisons are made
between guided wave modes and different transducers.
9.1 Wave propagation study with contact transducers
9.1.1 Ultrasonic transducers and instruments
Fig. 9-1 shows two test setups to excite ultrasonic guided waves in a 16’’ by 16’’
composite plate. The waves are generated using piezoelectric transducers on the left side.
A transducer can be put directly on the plate or through a variable angle wedge. The two
test methods are called normal incidence and oblique incidence, respectively. The
transducer on the right side is used to pick up the ultrasound signal.
153
Fig. 9-2 shows an integrated ultrasonic testing system. Excitation signals are sent
out from the system to the transmitter. The system is also used for data acquisition and
preliminary signal processing.
Figure 9-1: Test setups for ultrasonic guided wave propagation study.
Figure 9-2: Integrated ultrasonic testing system.
Transmitter Receiver
Oblique
incidence
Normal
incidence
154
9.1.2 Experimental phase velocity dispersion curve
Firstly, we will study the wave propagation along the zero degree direction. A
pair of 200 kHz transducers is used in normal incidence. The excitation signal is a 10
micro second pulse with 200 kHz center frequency. When the transmitter is located
100mm from the left edge of the plate, and the receiver is at 200mm, the ultrasonic
guided wave signal is shown Fig. 9-3. The guided wave signal is complicated in the
sense that multi wave packages and multi-modes exist. In addition, multiple reflections
from the edges of the plate are also collected in the signal.
A theoretical phase velocity dispersion curve is shown in Fig. 9-4 with the guided
wave mode lines named in a numerical order. Despite the complexities of the guided
wave signal shown in Fig. 9-3, each wave package within the signal can be designated
with guided wave mode analysis. One method to identify the wave package is to keep
track of the wave propagation by scanning the receiver along the wave path. A series of
guided wave signals are shown in Fig. 9-5 (a). From this figure, we can observe
0 50 100 150 200-15
-10
-5
0
5
10
15
Time (µs)
Am
plit
ud
e
Figure 9-3: Guided wave signal collected at 200mm position, when the transmitter is at
100 from the left edge.
155
multimode propagation, wave reflection, and interference. The first few wave packages
are marked in Fig. 9-5 (a), with the wave paths shown in Fig. 9-5 (b). The wave package
marked with red arrow and number 1 is the direct excited “mode 3”, the wave package
marked with red arrow and number 2 is the “mode 3” sending out to the left and then
reflected from the left edge and propagating to the right. The wave package marked with
red arrow and number 3 is the left propagation wave reflected from the right edge of the
plate. We also notice that, there is a wave package propagating in a much lower velocity.
This is the “mode 1” marked with the blue arrow and number 4. There are also other
reflections occurred in later time.
0 0.5 1 1.5 20
2
4
6
8
10
12
14
16
18
20
Frequency (MHz)
Cp
(km
/s)
1
2
3
4 5
6
7 8 9 10
Figure 9-4: Ultrasonic guided wave phase velocity dispersion curve for wave propagating
in the 0o of a quasi-isotropic composite laminate. Wave mode lines are numbered on the
dispersion curves.
156
Time (µs)
Po
siti
on
(m
m)
0 50 100 150
200
250
300
350
(a) Signals
1
3
2
4
Transmitter Receiver
1
3
2
4
Transmitter Receiver
(b) Sketch
Figure 9-5: Guided wave signals collected from a linear scan showing edge multi-mode,
edge reflection, and complex interference. (a) Experimental signals. (b) sketch of the first
few wave paths. (1): direct transmission mode 3, (2) Reflected mode three from left edge,
(3) reflected mode 3 from right edge, (4) direct through transmission of mode 1.
1 2 3 4
157
The phase velocity of a mode can be obtained by tracking the waves with a
constant phase angle. The estimated value from experiments are 6.45 mm/µs and 1.48
mm/µs for “mode 3” and “mode 1” respectively. The expected values from numerical
simulation are 6.62 mm/µs and 1.5 mm/µs. Therefore, the error between numerical
simulation and experiment is 2.5% and 1.3% respectively.
Besides time domain signal analysis, the ultrasonic signal can also be evaluated in
frequency domain. A two dimensional Fourier transform can be used to convert the time
and spatial domain information into frequency and phase velocity space. A 2D FFT of the
signal is shown in Fig. 9-6. Although the signals are complicated in time domain, they
are easy to analyze in the transformed domain with guided wave theory. Besides the
wave modes consisting most of the energy, some content of higher frequency waves are
also revealed. One drawback of the 2D FFT is the resolution of wavelength is limited by
the step of the linear scan. The step size used in the scan is 5mm. The wave length of the
“mode 1” at 200kHz is about 7.5 mm, which is below the resolvable wavelength.
Frequency (MHz)
Cp
(km
/s)
0 0.5 1 1.50
2
4
6
8
10
12
14
16
18
20
Figure 9-6: Frequency and phase velocity spectrum of guided wave signals shown in
Fig. 9-5 .
158
In order to explore the guided wave propagation possibilities, another four pairs
of transducers with center frequency at 500kHz, 800kHz, 1 MHz, and a broad band
transducer covering 200kHz to 700kHz are used in normal mode incidence test. In
addition, the broad band transducer pair is also used with a variable angle wedge at 0o,
15o, 30
o, 45
o, and 60
o incident angles. A combined (f-cp) spectrum obtained from all the
tests after data fusion is shown in Fig. 9-7. The theoretical dispersion curve of the wave
propagating in the 0o direction is also superimposed in the figure for the purpose of
comparison. A good match between the theory and the experiment is obtained. The match
between the experiment and the theory basically proved two things. One is the credibility
of the theoretical simulation tool. The other is the material property used in the
simulation is close to the actual material property value.
The influence of spatial domain sampling rate on the experimental dispersion
curve generation is also manifested in Fig. 9-7. The wave modes with high frequency
Frequency (MHz)
Cp
(km
/s)
0 0.5 1 1.50
2
4
6
8
10
12
14
16
18
20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 9-7: A comparison between the experimental dispersion curve and theoretical
dispersion curve for wave propagation along the 0 degree direction.
159
and low phase velocity, i.e., small wavelength, are not represented in the experimental
dispersion curve.
The experimental dispersion curve shown in Fig. 9-7 is only a subset of the
theoretical dispersions. Wave dispersion, attenuation, skew, and the source influence can
be used to explain this phenomenon. The strong regions in the experimental dispersion
curve are related to the wave modes with easy excitation with normal loading, small
attenuation, and small skew angle. If we select the guided wave modes with attenuation
less than 0.5dB/mm, skew angle within 010± , and excitability with normal loading larger
than 0.05, using the comprehensive mode selection algorithm, the remaining wave
modes are plotted in Fig. 9-8 together with the experimental dispersion curves. The
wave modes qualified with these rules matched very well with the experiment result.
Experimental curve breaks at two frequencies are also investigated. The reason for this is
the source influence, because our excitation signal is a 6 micro second tone burst with
500kHz center frequency for the broad band transducer. The frequency spectrum have
zeros at 0.33MHz and 0.66 MHz.
160
9.1.3 Guided wave group velocity and attenuation study
Group velocity and attenuation of a guided wave mode can also be extracted from
guided wave linear scan experiments. Fig. 9-9 shows the guided wave signals for the
“mode 8” in the frequency range of 0.7MHz to 0.8 MHz. The transmitter is located at
100mm position, and receivers are at a series of positions from 150mm to 350 mm. A
significant difference between the phase velocity and wave package velocity can be
noticed from the signals. An estimation of these two velocities can be obtained from the
slopes of line 1 and line 2 in the figure. The results are 9.5 km/s and 4.9km/s for phase
velocity and group velocity respectively. The range of theoretical prediction are [9.6
10.6] km/s for phase velocity and [5.1 5.4] km/s for group velocity respectively. The
slight discrepancy between theory and experiment might be attributed to the inexact
Frequency (MHz)
Cp
(km
/s)
0 0.5 1 1.50
2
4
6
8
10
12
14
16
18
20
Figure 9-8: Comparison of guided wave modes in the experiment with theoretical
expectation using low attenuation, low skew angle, and excitable rules, and frequency
spectrum of source influence. The result shows that the experiment total meets the
expectation.
161
material property used in the simulation and also measurement accuracy in the
experiment.
Attenuation information can also be observed from Fig. 9-9. Wave mode
attenuation can be estimated with energy reduction of the wave package. Fig. 9-10 shows
the energy content in the signals as a function of position. The linear fitting of the data in
log scale suggests the attenuation rate of this guided wave mode to be 0.1 dB/mm.
Time (µs)
Po
siti
on
(m
m)
10 20 30 40 50 60 70 80
150
200
250
300
350
Frequency (MHz)C
p (
km
/s)
0 0.5 1 1.50
2
4
6
8
10
12
14
16
18
20
(a) (b)
Figure 9-9: Guided wave signals from a 800kHz transducer. (a) Illustration of guided
wave phase velocity and group velocity in a wave package. (b) frequency and phase
velocity spectrum
1 2
162
Tab. 9-1 listed the results of estimated attenuation of other guided wave modes
from both experimental tests and the predictions from both the Hysteretic model and the
Kelvin-Voigt model.
The result shows that although the imaginary part of the complex stiffness value
used in the simulation is not exact, the attenuation prediction is in an acceptable scale.
The experiment result generally lies in between the attenuation result estimated from the
two models. At low frequency the hysteretic model is more close to experiment; at higher
150 200 250 300 350-25
-20
-15
-10
-5
0
Position (mm)
Re
lativ
e A
mp
litu
de
(d
B)
y = - 0.1*x + 15
Relative Energy linear fitting
Figure 9-10: Energy content in the guided wave signal as a function of position showing
wave attenuation.
Table 9-1: Quntitative comparison of wave mode attenuation
Wave mode Attenuation (dB/mm)
Frequency (kHz) Mode Number Measurement Hysteretic model K-V model
120-200 1 0.11 0.12-0.25 0.01-0.03
120 200 3 0.05 0.03-0.05 0.003-0.004
480-550 5 0.06 0.14-0.19 0.035-0.05
650-750 8 0.099 0.21-0.22 0.07-0.08
720-800 8 0.1 0.21-0.22 0.075-0.085
163
frequency the K-V model are closer. Quantitative analysis of attenuation will be studied
in the future after the viscoelastic material properties for this material are determined.
9.1.4 Guided wave skew angle study
Guided wave modes with large skew angles have been predicted in the theoretical
study and validated in finite element analysis. In this section, an experiment is carried out
to validate the skew angle in composite plates. The experimental setup is shown in
Fig. 9-11. A pair of angle wedge transducer is used to transmit and receive waves. The
wave vector direction is kept in 0 degree, i.e., the direction parallel to fiber direction of
the first layer. The receiving wedge is also kept in the 0 degree direction. The receiver is
then moved along the vertically line 100mm away from the transmitter.
Based on the mode selection results presented in chapter 8.2 , the fifth wave mode
around 690kHz is used. The receiving signals at a series of positions are plotted in
Fig. 9-12. Strong signals are received below the centerline, with a maximum at the -
60mm position. Since the separation of transducers in x direction is 100mm. The skew
Figure 9-11: Experiments to test the effect of energy skew in a quasi-isotropic composite
plate.
100mm Wave vector direction
Transmitter Receiver
x y
164
angle is estimated to be -31 degree. This result validated the prediction value of -33.6
degree.
Time (µs)
Po
sitio
n (
mm
)
0 50 100 150
-150
-100
-50
0
50
100
150
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 9-12: Guided waves excited from angle wedge to validate the concept the concept
of large skew angel. The black line at the center corresponds to the wave launching
direction. The line in -60 mm position marks the position where a maximum signal is
detected.
165
9.2 Wave excitation with piezoelectric active sensors
The theoretical study indicated that source influences are very important in wave
excitation. Influence of piezoelectric element geometry on wave excitation is studied in
this section. The elements used are all 40mm long with widths of 4mm, 6mm, and 8mm.
These elements are attached on the composite in three groups with 200 mm spacing
between the transmitters and the receivers. The piezoelectric material used in the study is
CTS 3203HD.
Fig. 9-13 shows the received signals from these three pairs of transducers. The
excitation signal is a 200kHz tone burst with 5 µs pulse width. The wave package
arriving at around 40 µs is the direct through transmission signal of mode 3. The wave
package arriving about 140 µs is the direct through transmission signal of mode 1. Other
wave packages in between are the reflections from the edge of the plate. Durations of the
wave packages shown in the signals are typically longer than the excitation pulse width.
The ringing of the transducer element after excitation is the major reason for this. Wave
dispersion also contributes to this effect.
Although quantitative analysis of the signal still needs more investigation on the
transmitter and the receiver responses, the effect of the source influence on wave
excitation is already demonstrated in the signals. The relative amplitude between the
direct through transmission mode 3 and mode 1 amplitudes changes with the change of
element width. When the element width is 4mm and 6mm, the amplitudes of mode 1 are
larger than the amplitudes of mode 3. However, when the element is 8mm, the amplitude
of the mode 3 becomes dominant. This phenomenon can be explained with the wave
mode decomposition spectra. In Fig. 9-14, plots (a), (c), and (e) are for the cases of
evenly distributed normal loading; plots (b), (d), and (f) are for the case of concentrated
shear loading at the edge of the transducer element. Plots (a) and (b) are for the 4mm
transducer element; plots (c) and (d) are for the 6mm transducer element; plots (e) and (f)
are for the case of 8 mm transducer element. In both normal loading and concentrated
shear loading cases, the amplitude of mode 1 reduces with the increase of element width;
while the amplitude of mode 3 increases with the increase of element width.
166
0 50 100 150 200
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Time (µs)
Am
plitude
(a)
0 50 100 150 200
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Time (µs)
Am
plit
ude
(b)
0 50 100 150 200
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Time (µs)
Am
plitude
(c)
Figure 9-13: Guided wave signals from surface mounted piezoelectric transducers.
Excitation signal 200kHz, pulse width 5 µs. transducer element width (a) 4mm, (b) 6mm,
and (c) 8mm.
Mode 3 Mode 1
167
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency (MHz)
Excita
bili
ty
Mode 1
Mode 3
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
Frequency (MHz)
Excita
bili
ty
Mode 3
Mode 1
(a ) 4mm transducer normal loading (b) 4mm transducer shear loading
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency (MHz)
Excita
bili
ty
Mode 1
Mode 3
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
Frequency (MHz)
Excita
bili
ty
Mode 3
Mode 1
(c) 6mm transducer normal loading (d) 6mm transducer shearl loading
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency (MHz)
Excita
bili
ty
Mode 1
Mode 3
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
Frequency (MHz)
Excita
bili
ty
Mode 3
Mode 1
(e) 8mm transducer normal loading (f) 8mm transducer shear loading
Figure 9-14: Expected guided wave modes from a surface excitation source. Excitation
signal has center frequency 200kHz and 2 cycles.
168
9.3 Guided wave damage detection with piezoelectric active sensors
A small plastic cube is put on the surface of the plate with double sided tape to
simulate damage. The picture of the plastic cube on the plate is shown in Fig. 9-15. The
dimension of the cube is 10mm by 10mm by 5mm.
The wave mode have significant out of plane field will be sensitive to the attached
damping damage. Therefore, the “mode 1” is expected to have better sensitivity than the
“mode 3”. Shown in Fig. 9-16 are the guided wave signals collected with the 4mm
transducer pair. The excitation is a 200 kHz excitation with 5 micro second pulsewidth.
The result does show that the “mode 1” is much more sensitive to the damping damage.
Figure 9-15: Picture of a plastic putting on the top of a composite plate to simulate
damage. Plastic putting dimensions: 10 mm x 10mm x5 mm.
169
The results from the 8 mm transducer pair is shown in Fig. 9-17. The amplitude
of the “mode 3” is much larger than “mode 1”. However, the difference in the signal is
more significant in the “mode 1” region.
0 50 100 150 200
-0.4
-0.2
0
0.2
0.4
0.6
0 50 100 150 200
-0.4
-0.2
0
0.2
0.4
0.6
0 50 100 150 200-0.1
-0.05
0
0.05
0.1
Time (µs)
Am
plit
ud
e Damaged
Difference signal
No damage
Figure 9-16: Guided wave signals (a) before damage, (b) after damage, and (c) the
difference of the signals in (a) and (b). Transducer: 4mm width. Excitation signal.
200kHz with 5 µs pulse width.
(a)
(b)
(c)
Mode 1
170
Wave mode sensitivity ratio can be estimated from the amplitude value of the
wave modes. In Fig. 9-16, the relative amplitude ratio in the un-damaged case is
(Mode1/Mode3)=0.606/0.314=1.92, and the amplitude ratio of the difference is
Mode1/Mode3=0.163/0.012=13.6. Therefore, the estimated sensitivity difference is
Mode1/Mode3 about 7. The same procedure applied to Fig. 9-17 results in the sensitivity
difference of about 13. Therefore, the experiment indicates that the mode 1 is about ten
times more sensitive to the damage than mode 3. The theoretical estimate of the
sensitivity presented in Chapter 7 indicates that the ratio of mode 1 and mode 3 at 200
kHz is also about 10 times. (See Fig. 7-12 for details). Although this measurement is
only an estimation, it does show a good match between the theory and the experiment.
0 50 100 150 200
-0.5
0
0.5
0 50 100 150 200
-0.5
0
0.5
0 50 100 150 200-0.1
-0.05
0
0.05
0.1
Time (µs)
Am
plit
ud
e
No damage
With damage
Signal difference
Figure 9-17: Guided wave signals (a) before damage, (b) after damage, and (c) the
difference of the signals in (a) and (b). Transducer: 8mm width. Excitation signal.
200kHz with 5 µs pulse width.
(a)
(b)
(c)
Mode 3 Mode 1
171
Small PZT discs have been used in some previous studies to monitor damage in
aging aircraft components and composites [Gao et al. 2004; Lissenden et al. 2006]. The
disc is of 1/4’’ diameter and 10 mils thickness. The radial resonance is 350 kHz. The
results using a pair of disc transducer is shown in the following. In the experiment, a 350
kHz signal with 5 micro second pulsewidth is used. The signals are shown in Fig. 9-18.
In this case, the raw signal is more complicated because more wave modes are involved.
Guided wave “mode 1” still arrives around 140 µs. However, the amplitude is about 20
dB less than the amplitude obtained from the 4mm wide transducer. Therefore, the
amplitude of the signal difference is only 0.014V, which is 21 dB less than the change
obtained with the 4mm transducer.
0 50 100 150 200-0.1
-0.05
0
0.05
0.1
0 50 100 150 200-0.1
-0.05
0
0.05
0.1
0 50 100 150 200-0.02
-0.01
0
0.01
0.02
Time (µs)
Am
plit
ude
un-damaged
damaged
deference
Figure 9-18: Guided wave signals (a) before damage, (b) after damage, and (c) the
difference of the signals in (a) and (b). Transducer: disc. Excitation signal. 350 kHz with
5 µs pulse width.
172
9.4 Summary
The experimental study presented in this chapter covers guided wave propagation,
excitation, and damage detection in composite materials. The results are used to validate
the observations from numerical simulations.
1. Moveable transducers are used in the guided wave propagation study. The
features studied include guided wave phase velocity dispersion curves, guided wave
group velocity, skew angle, and attenuation.
Dispersion curves from theoretical calculation matched with experimental results
for waves propagating in the [(0/45/90/-45)s]2 quasi-isotropic composite laminate. The
strong regions of the reconstructed experimental dispersion curve are observed to be
related to the wave modes with small attenuation, small skew angle, and reasonable
excitability with normal loading from the contact transducer.
An ultrasonic guided wave mode with large skew angle is experimentally
demonstrated and the result matches quantitatively with the theoretical expectation.
Ultrasonic guided wave attenuation is measured for the wave modes with long
range propagation potential. The attenuation is measured to be in the range of 0.05 to 0.1
dB/mm. This indicates that these guided wave modes can generally propagate 0.5 to 1
meter distance if a 50 dB amplitude decrease is allowed. Although the use of a substitute
material damping properties does not provide exact solutions for guided wave attenuation
prediction, the trend and relative relations of wave mode attenuation agrees quite well
with experimental measurement. Therefore, the numerical simulation results can be used
as a guideline for mode selection.
2. Permanently attached transducers are used to study the wave excitation
characteristics of guided waves in a composite. Effects of transducer width on wave
excitation are studied. The experimental results proved the importance of considering
excitation spectrum and wave mode excitability in guided wave excitation.
3. Guided wave damage detection is performed using simulated damping damage
on the surface of the composite. The result indicates that the “mode 1” has about 10 time
173
sensitivity to the damage than the “mode 3”. The result agrees with the numerical
prediction.
4. The performances of rectangular transducers are compared with disc
transducers. At a transmitter receiver distance of 200mm, the 4mm wide rectangular
transducer have a mode selection preference toward “mode 1”, which is sensitive to the
damage. The disc has an excitation preference to other wave modes, that are comparably
insensitive to the damage. The overall signal amplitude is about 10 dB stronger from the
rectangular transducer compared to the disc. Therefore, the difference in the signal
introduced by the damage is 20 dB stronger using the 4mm wide rectangular transducer
than the disc transducer.
Chapter 10
Conclusions and discussions
10.1 Summary of the thesis study
Condition based maintenance is of great interest to the aircraft industry because of
its big payoffs in safety assurance and cost savings. The ultrasonic guided wave based
method emerged as a promising technology due to its long range monitoring potential. In
addition, both surface and interior damage can be detected with ultrasonic guided waves.
Despite the potential benefit of using ultrasonic guided waves for structural health
monitoring, guided wave mechanics in composite structures are very complicated and
hence not been adequately studied prior to this thesis work.
With the completion of this thesis, the state of the art of guided wave based
structural health monitoring will be advanced from a pure experimental process to the
realization of a theoretically driven design and implementation process. With an
understanding of guided wave mechanics, guided wave modes with better sensitivity can
be identified. Rather than merely using the fundamental wave modes, now more guided
wave modes can be selectively used for better sensitivity and/or larger coverage area.
Rather than picking a transducer geometry purely based on experiences, now a transducer
can be design based on the properties of each composite laminate.
The detailed guided wave mechanics are studied in the three important aspects
namely, wave propagation, wave excitation, and wave-damage interaction. Wave
propagation characteristics are studied using two methods. One is a global matrix method
(GMM), and the other is a semi-analytical finite element (SAFE) method. The
comparison of these two methods indicates that the SAFE method is computationally
more efficient than the GMM. It also does not have the problem of missing roots and/or
alias roots. However, the SAFE method is not very accurate for stress field distribution
calculation. Therefore, a hybrid SAFE-GMM method is used to achieve a
175
computationally efficient as well as accurate wave mode analysis. The output of wave
propagation mode analysis is a set of dispersion curves describing the wave propagation
possibilities and their characteristics. They are phase velocity, group velocity, energy
velocity, skew angle, and attenuation. Specific analysis of a guided wave mode can also
be obtained by calculating the displacement, stress, strain, power flow distribution, and
energy distribution in the thickness direction of the laminate.
The normal mode expansion technique is first used in this thesis for a guided
wave excitation study in multi-layered composites. In addition, a new formula of normal
mode expansion is developed to study viscoelastic effects on wave excitation in
composites. Compared with the integral transform method used in some other studies,
this technique provides a clearer and simpler physical insight on guided wave excitation.
In the case of surface excitation, the wave mode excitability is directly related to the
particle velocity at the surface. The influence of the excitation source on wave excitation
is expressed with a frequency and phase velocity spectrum. This spectrum can also be
used in the process of transducer design after a specific wave mode is selected.
Two types of structural damage are evaluated in the thesis. One is general
material property degradation. The influences of surface layer erosion, elastic property
degradation, and layer thickness and density change on guided wave dispersion curves
are studied. The other type of damage is internal flaws such as a delamination at layer
interfaces. A new feature is defined to estimate the sensitivity of a guided wave mode
without considering the detailed shape of a delamination.
Based on the studies of wave propagation, excitation, and sensing, a new guided
wave mode selection platform is developed. Important features of a guided wave mode
include but are not limited to frequency, phase velocity, group velocity, skew angle,
dispersion coefficient, beam spreading, excitability, sensitivity, and attenuation. A set of
crisp and fuzzy reasoning rules are used to evaluate the tradeoffs of each mode. A
goodness value is obtained from the reasoning to represent the qualifications of a wave
mode for the given requirements. As an example, typical wave modes for long range
monitoring are identified from the set of dispersion curves in Chapter 8.
176
Theoretical expectations of wave mechanics studies are validated with numerical
and experimental study. Some observations are listed next.
1. For the quasi-isotropic laminate, there is no wave propagation direction
that Rayleigh-Lamb (R-L) type waves and shear horizontal (SH) waves
can be distinctly separated. Therefore, all the possible guided wave
modes are presented in one set of dispersion curves.
2. In a quasi-isotropic laminate, the “mode 2” and “mode 3” has quasi-
isotropic behavior at low frequency. Other guided wave modes are
direction dependent.
3. Even in quasi-isotropic laminates, the guided wave mode skew effect is
still very significant. A wave mode with skew angle larger than 30o is
predicted in theory and observed in both numerical simulation and
experiment.
4. Guided wave attenuation is a function of wave mode and frequency.
Along a single dispersion curve, attenuation can be both increasing and
decreasing with frequency. Generally, attenuation increases at mode
interaction regions. For a given frequency, a guided wave mode with
the least amount of attenuation can be identified from the attenuation
dispersion curve. Below the cut off frequency of the “mode 4”, “mode
3” has the least attenuation. At higher frequencies, “mode 5” and
“mode 8” has the least attenuation. We observed that the wave modes
with the least attenuation commonly correspond to the wave modes
with the largest group velocity.
5. The wave excitability describes the response of a guided wave mode to
a specific loading direction on the surface. At low frequencies, the
“mode 1” is easily excited with both normal loading and shear loading;
“mode 2” is easily excited with shear horizontal loading; “mode 3” is
easily excited with shear loading in the propagation direction.
6. The first guided wave mode at low frequency is more sensitive to
delamination and surface mass damping than the third mode.
177
10.2 Specific contributions
1. Put forward a hybrid SAFE-GMM method for dispersion curve and wave structure
calculation. The SAFE method is computationally efficient for dispersion curve
calculation and does not have missing roots. The global matrix method can generate
accurate wave structure without stress discontinuities. The hybrid method brings
the advantage of the two methods together.
2. Used normal mode expansion technique for wave excitation analysis in composites
for the first time. Developed a program with the capability of studying the wave
excitation from a finite and transient source.
3. Derived a new normal mode expansion technique based for viscoelastic media.
4. Compared the guided wave propagation and excitation characteristics in the elastic
model and viscoelastic model. The result indicates that the most significant effect of
material viscoelasticity is on the attenuation of the wave mode. Except for the mode
interaction regions, an elastic approximation is accurate for guided wave phase
velocity and energy velocity dispersion curves and wave structure analysis.
5. Put forward a novel and comprehensive mode selection framework based on the
analysis of guided wave characteristics, such as phase velocity, group velocity,
attenuation, skew angle, dispersion, excitability, and sensitivity to specific damage.
6. The guided wave skew effect is systematically studied for the first time.
7. Studied the direction dependency of wave propagation in anisotropic media. A three
dimensional dispersion surface is used to display phase velocity dispersion curves
and group velocity dispersion curves.
8. Studied wave propagation in the [(0/45/90/-45)s]2 composite structure for the first
time. The understanding on the wave mechanics in this specific material is valuable
for SHM sensor design. A summary of the observations is provided in Section 10.1.
The study presented in this thesis has opened a door for many case studies in
structural health monitoring. Besides composites, guided wave mechanics in many other
multi-layered materials can all be studied with the simulation tool.
178
In addition to the contributions in wave mechanics studies, the author has also
contributed to the area of guided wave imaging, and ultrasonic sensor placement
optimization for structural health monitoring. Combining mechanics development with
imaging in the future will be quite useful. New computer algorithms developed for
experimental structural health monitoring damage detection and localization are
presented in Appendix A. These areas of study are at a different level of the “theoretically
driven” SHM strategy, but in the future can be combined with the wave mechanics tool
box presented in this thesis.
10.3 Future work
Towards a broader application of the theoretically driven guided wave structural
health monitoring strategy, some future work recommendations are as follows.
1. Study the behavior of piezoelectric transducers and the interaction between the
transducer and the structure, such that a quantitative input can be used in the wave
excitation modeling.
2. Numerical simulation of guided wave interaction with damage to quantitatively
validate the damage sensitivity of guided wave modes.
3. Quantitative study of composite material viscoelasticity, such that an accurate
material property can be used for each test structure to produce quantitative
information about guided wave attenuation.
4. Design new sensors based on the theoretical guidelines provided in this research for
more SHM applications.
5. Application of the wave mechanics simulation tools on composite materials with
other property and stacking sequences. Application of guided wave mechanics study
will be used in real damage detection such as impact damage and fatigue damage.
6. Improved imaging methodologies combining wave mechanics and physically based
tomography.
References
ABAQUS (2003). ABAQUS User's Manual
Achenbach, J. D. (1973). Wave Propagation in Elastic Solids. Amsterdam, North-
Holland.
Agostini, V., Delsanto, P. P., Genesio, I. and Olivero, D. (2003). "Simulation of Lamb
wave propagation for the characterization of complex structures." IEEE
Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 50(4): 441-
448.
ANSYS ANSYS Software Manual.
Auld, B. A. (1990). Acoustic Fields and Waves in Solids. Malabar, Florida, Krieger
Publishing Company.
Banerjee, S., Prosser, W. and Mal, A. K. (2005). "Calculation of the response of a
composite plate to localized dynamic surface loads using a new wave number
integral method." ASME Journal of Applied Mechanics 72: 18-24.
Bartoli, I., Marzani, A., Lanza di Scalea, F. and Viola, E. (2006). "Modeling wave
propagation in damped waveguides of arbitrary cross-section." Journal of Sound
and Vibration 295: 685-707.
Bell, M. (2004), M.S. Thesis, Ultrasonic guided wave defect detection feasibility in a
carbon/epoxy composite plate, Department of Engineering Science and Mechnics,
Penn State University
Castaings, M. and Hostern, B. (1994). "Delta operator technique to improve the
Thomson-Haskell method stability for propagation in multilayered anisotropic
absorbing plates." Journal of Acoust, Soc. Am 95(4): 1931-1941.
Chambers, J. R. (2003). "Concept to reality: Contributions of the NASA Langley
research center to U.S. Civil aircraft of the 1990s." from
http://oea.larc.nasa.gov/PAIS/Concept2Reality/composites.html.
Chimenti D.E., a. N. A. H. (1990). "Ultrasonic reflection and guided wave propagation in
biaxially laminated composite plates." Journal of Acoust, Soc. Am 87(4): 1409-
1415.
Chimenti, D. E. (1997). "Guided waves in plates and their use in material
characterization." Appl Mech Rev 50(5): 247-284.
Cho, Y. (2000). "Estimation of ultrasonic guided wave mode conversion in a plate with
thickness variation." IEEE Transactions on Ultrasonics, Ferroelectrics, and
Frequency Control 47(3): 591-603.
Cho, Y. and Rose, J. L. (1996). "A boundary element solution for a mode conversion
study on the edge reflection of Lamb waves." Journal of Acoust, Soc. Am 99(4):
2097-2109.
Cook, R. D. (2001). Concepts and Applications of Finite Element Analysis John Wiley &
Sons Inc.
Datta, S. K., Shan, A. H. and Karunasena, W. (1990). "Edge and layering effects in a
multilayered composite plate." Computers and Structures 37(2): 151-162.
180
Dayal, V. and Kinra, V. K. (1989). "Leaky Lamb waves in an anisotropic plate. I: An
exact solution and experiments." Journal of Acoust, Soc. Am 85(6): 2268-2276.
Ditri, J. J. and Rose, J. L. (1992). "Excitation of guided elastic wave modes in hollow
cylinders by applied surface tractions." Journal of Appied Physics 72(7): 2589-
2597.
Ditri, J. J. and Rose, J. L. (1994). "Excitation of guided waves in generally anisotropic
layers using finite source " ASME Journal of Applied Mechanics 61(2): 330-338.
Dunkin, J. W. (1965). "Computation of modal solutions in layered elastic media at high
frequencies." Bull Seismol Soc Am 55: 335-358.
Gao, H. and Rose, J. L. (2006), "Sensor placement optimization in structural health
monitoring using genetic and evolutionary algorithms", Smart Structures and
Materials 2006, San Diego, SPIE.
Gao, H., Shi, Y. and Rose, J. L. (2004), "Guided wave tomography on an aircraft wing
with leave in place sensors", Quantitative Nondestructive Evaluation, Golden, CO,
American Institute of Physics.
Gao, H., Yan, F. and Rose, J. L. (2005), "Ultrasonic guided wave tomography in
structural health monitoring of an aging aircraft wing", ASNT Fall conference,
Columbus, OH.
Gavric, L. (1995). "Computation of propagating waves in free rail using finite element
techniques." Journal of Sound and Vibration 185: 531-543.
Giurgiutiu, V. (2005). "Tuned Lamb wave excitation and detection with piezoelectric
wafer active sensors for structural health monitoring." Journal of Intelligent
Material Systems and Structures 16: 291-305.
Giurgiutiu, V. and Bao, J. (2004). "Embedded-ultrasonics structural radar for in situ
structural health monitroing of thin-wall structures." Structural Health Monitoring
3(2): 121-140.
Giurgiutiu, V. and Zagrai, A. (2005). "Damage detection in thin plates and Electro-
Mechanical impedance method." Structural Health Monitoring 4(2): 99-117.
Graff, K. (1973). Elastic Waves in Solids. New York, Oxford university press.
Guo, N. and Cawley, P. (1993). "The interaction of Lamb waves with delaminations in
composite laminates." Journal of Acoust, Soc. Am 94(4): 2240-2246.
Haskell, N. A. (1953). "Dispersion of surface waves in multilayered media." Bull Seismol
Soc Am 43: 17-34.
Hay, T., Royer, R., Gao, H., Zhao, x. and Rose, J. L. (2006). "A comparison of embedded
sensor lamb wave ultrasonic tomography approaches for material loss detection."
Journal of Smart Structures and Materials 15: 946-951.
Hayashi, T., Song, W. J. and Rose, J. L. (2003). "Guided wave dispersion curves for a bar
with an arbitrary cross-section, a rod and rail example." Ultrasonics 41(2003):
175-183.
Hayek, S. I. (2001). Advanced Mathematical Methods in Science and Engineering. New
York, Basel, Marcel Dekker, Inc.
Hosten, B. (1991). "Reflection and transmission of acoustic plane waves on an immersed
orthotropic and viscoelastic solid layer." Journal of Acoust, Soc. Am 89(6): 2745-
2752.
181
Hosten, B. and Castaings, M. (1993). "Transfer matrix of multilayered absorbing and
anisotropic media. Measurements and simulations of ultrasonic wave propgation
through compisite materials." Journal of Acoust, Soc. Am 94(3): 1488-1495.
Hosten, B., Deschamps, M. and Tittman, B. R. (1987). "Inhomogeneous wave generation
and propagation in lossy anisotropic solids. Application to characterization of
viscoelastic composite materials." Journal of Acoust, Soc. Am 82(5): 1763-1770.
Huang, K. H. and Dong, S. B. (1984). "Propagating waves and edge vibrations in
anisotropic composite cylinders." Journal of Sound and Vibration 96(3): 363-379.
Knopoff, L. (1964). "Matrix method for elastic wave problems." Bull Seismol Soc Am 54:
431-438.
Lagasse, P. E. (1973). "Higher-order finite-element analysis of topographic guides
supporting elastic surface waves." Journal of Acoust, Soc. Am 53(4): 1116-1122.
Lamb, H. (1917). "Waves in elastic plates." Proc. Roy. Soc. A 93: 114-128.
Lee, B. C. and Staszewski, W. J. (2003). "Modeling of Lamb waves for damage detection
in metallic structures: part II: Wave interaction with damage." Smart Materials
and Structures 12: 815-824.
Lee, B. C. and Stszewski, W. J. (2003). "Modeling of Lamb waves for damage detection
in metalic structures: Part I: Wave propagation." Smart Materials and Structures
12: 804-814.
Lee, C. (2006), PhD Thesis, Guided waves in rail for transverse crack detection,
Department of engineering science and mechanics, Penn State Univeristy
Lih, S.-S. and Mal, A. K. (1995). "On the accuracy of approximate plate theories for
wave field calculation in composite laminates." Wave Motion 21: 17-34.
Lin, M. and Chang, F.-k. (2002). "The manufacturing of composite structures with a
built-in network of piezoceramics " Composite Science and Technology 62: 919-
939.
Lissenden, C. J., Yan, F., Hauck, E. T., Noga, D. M. and Ros, J. L. (2006), "Internal
damage detection in a laminated composite plate using ultrasonic guided waves",
Review of Quantitative Nondestructive Evaluation, Portland, Oregon, American
Institute of Physics.
Love, A. E. H. (1911). Some Problems of Geodynamics, Cambridge University Press.
Lowe, M. J. S. (1995). "Matrix techniques for modeling ultrasonic waves in multilayered
media." IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency
Control 42(2): 525-541.
Luo, W. (2005), PhD Thesis, Ultrasonic guided waves and wave scattering in
viscoelastic coated hollow cylinder, Engineering Science and Mechanics Penn
State University
Mal, A. K. (1988). "Guided waves in layered solid with interface zones." Int. J. Eng. Sci
26: 873-881.
Mal, A. K. (2002). "Elastic waves from localized sources in composite lamnates "
International Journal of Solids and Structures 39: 5481-5494.
Matt, H., Bartoli, I. and Lanza di Scalea, F. (2005). "Ultrasonic guided wave monitoring
of composite wing skin-to-spar bonded joints in aerospace structures." Journal of
Acoust, Soc. Am 118(4): 2240-2252.
182
Moulin, E., Assaad, J., Delebarre, C. and Osmont, D. (2000). "Modeling of Lamb waves
generated by integrated transducers in composite plates using a coupled finite
element-normal modes expansion method." Journal of Acoust, Soc. Am 107(1):
87-94.
Nayfeh, A. H. (1995). Wave propagation in layered anisotropic media. Amsterdam
Lausanne New York Oxford Shannon Tokyo, Elsevier.
Nayfeh, A. H. and Chimenti, D. E. (1991). "Elastic wave propagation in fluid-loaded
multiaxial anisotropic media." Journal of Acoust, Soc. Am 89(2): 542549.
Neau, G., Lowe, M. J. S. and Deschamps, M. (2001), "Propagation of Lamb waves in
anisotropic and absorbing plates: Theoretical derivation and experiments",
Review of Progress in Quantitative Nondestructive Evaluation, Brunswick, Maine,
American Institute of Physics.
Noga, D. (2006), Bachelor of Science Thesis, Detection of fatigue damage in fiber
reinforced composite plates using ultrasonic guided waves, Department of
Engineering Science and Mechanics, Penn State University
Raghavan, A. and Cesnik, C. E. S. (2005). "Finite-dimensional pieozoelectric transducer
modeling for guided wave based structural health monitoring." Smart Materials
and Structures 14: 1448-1461.
Rayleigh, L. (1885). "On waves propagating along the plane surface of an elastic solid."
Proc. London Math. Soc. 17(1885): 4-11.
Rokhlin, S. I. and Wang, L. (2002). "Ultrasonic waves in layered anisotropic media:
characterization of multidirectional composites." International Journal of Solids
and Structures 39: 5529-5545.
Rose, J. L. (1999). Ultrasonic Waves in Solid Media, Cambridge University Press.
Rose, J. L., Yan, F. and Hauck, E. (2006). IA composite project final report, Penn state
university.
Schoeppner, G. A., Kim, R. and Donaldson, S. L. (2001), "Steady state cracking of PMCs
at cryogenic temperatures", Proceedings of AIAA/ASME/ASCE/AHS/ASC
Structures, Structural Dynamics, and Materials Conference and Exhibit,, Seattle,
WA, AIAA.
Scholte, J. G. (1942). "On the stonely wave equation." Proc. Kon. Nederl. Akad.
Westensch 45: 20-25.
Sohn, H., Farrar, C. R., Hemez, F. M., Shunk, D. D., Stinemates, D. W. and Nadler, B. R.
(2003). A review of strucutral health monitoring literature: 1996-2001, Los
Alamos National Laboratory
Staszewski, W. J. (2002). "Intelligent signal processing for damage detection in
composite materials." Composite Science and Technology 62: 941-950.
Staszewski, W. J., Boller, C. and Tomlinson, G. R. (2004). Health Monitoring of
Aerospace Structures: Smart Sensor Technologies and Signal Processing, John
Wiley & Sons, Ltd.
Stoneley, R. (1924). "Elastic waves at the surface of sepration of two solids." Proc. Roy.
Soc. 106: 416-428.
Su, Z. and Ye, L. (2004). "Fundamental lamb mode-based delamination detection for
CF/EP composite laminates using distributed piezoelectrics." Structural Health
Monitoring 3: 43-68.
183
Thomson, W. T. (1950). "Transmission of elastic waves through a stratified solid
material." Journal of Appied Physics 21: 89-93.
Victorov, I. A. (1967). Rayleigh and Lamb Waves-Physical Theory and Applications.
New York, Plenum.
Wilcox, P. D. (2003). "A rapid signal processing technique to remove the effect of
dispersion from guided wave signals." IEEE Transactions on Ultrasonics,
Ferroelectrics, and Frequency Control 50(4): 419427.
Wilcox, P. D., Lowe, M. J. S. and Cawley, P. (2001). "Mode and transducer selection for
long range Lamb wave inspection." Journal of Intelligent Material Systems and
Structures 12: 553-565.
Yang, C., Ye, L., Su, Z. and Bannister, M. (2006). "Some aspects of numerical simulation
for Lamb wave propagation in composite laminates." Composite Structures 75:
267-275.
Yoseph Bar-Cohen, Ajit K. Mal and Chang, Z. (1998), "Composite material defects
characterization using leaky Lamb wave dispersion data ", SPIE's NDE of
Materials and Composites II, San Antonio, SPIE.
Yu, L. and Giurgiutiu, V. (2005). "Advanced signal processing for enhanced damage
detection with piezoelectric wafer active sensor." Smart Structures and Systems
1(2): 185-215.
Zhao, X. and Rose, J. L. (2003). "Boundary element modeling for defect characterization
potential in a wave guide." International Journal of Solids and Structures 40:
2645-2658.
.
Appendix A
Guided wave imaging techniques in SHM
A.1 Signal processing and feature extraction
Signal processing and feature extraction are also very critical components to the
success of damage detection in structural health monitoring. In addition to wave mode
analysis features, other statistical features and signal comparison features are also in our
studies.
A brief review of some signal processing techniques used in wave analysis is
described next.
1. Analytical envelope extraction with Hilbert transform. Wave package peaks and
arrival time can be identified from the analytical envelope.
2. Cross correlation. When the prototype of the wave package is available, cross
correlation of the prototype and the detected signal can be used to identify the peak of the
package.
3. Frequency domain analyses. Frequency domain analysis of the signal can be used
to detect a frequency shift during guided wave propagation. The frequency domain
analyses include the generation of Fourier transform and the resulting frequency
spectrum, phase spectrum, power spectrum, and frequency domain filtering.
4. Time frequency analysis. Instead of performing a Fourier transform of the entire
signal, a Short Time Fourier Transform (STFT) and wavelet transform (WT) can be used
to obtain the time frequency spectrum of the signal. STFT and WT can be used in wave
package location and experimental dispersion curve generation. Other applications of
wavelet transforms include wavelet decomposition and wavelet de-noising.
5. Signal comparison features. In structural health monitoring, the essence of damage
detection is to compare the signal of the current state with a predefined reference signal.
The occurrence of damage will introduce changes in the detected signal. When the
185
system being monitored is complex, changes in the signal may be subtle. In this case,
direct extraction of physically based features may be difficult. In this case, features
extracted from signal comparison functions could be helpful for the detect the changes in
the structure. When the sensor is designed according to the theoretical guidelines
determined from wave mechanics studies, the change in the signal will potentially be
more evident.
A signal comparison feature used developed through out the course of this work is
called the signal difference coefficient (SDC) [Gao et al. 2004].
-
Here, f1 is the time domain waveform of the reference signal; f2 is the waveform
of the signal in the subsequent states. The value of the SDC lies within a range of 0 to 1.
An average signal difference over a long time or a specific signal difference related to a
certain wave mode can be extracted. The signal difference coefficient can also be used on
analytical envelopes of guided wave signals.
A.2 Guided wave imaging algorithms
Compared to damage detection, damage localization and assessment are higher level
objectives of structural health monitoring. Based on signal processing, visualization of
the localization of damage can be achieved with computed tomography (CT) algorithms.
Several CT algorithms are investigated in this study, including the back projection (BPJ)
algorithm, a shifting and multi-resolution algebraic reconstruction tomography (SMART)
algorithm, and a reconstruction algorithm for probabilistic inspection of damage
(RAPID).
dtftfdtftf
dtftfftfCS
ff
ff
ff2
22
2
11
2211
21 ))(())((
))(())(()0(21
21
∫∫
∫−−
−−==
σσ
21211 ffff SSDC −=
(1.1)
186
The RAPID technique is widely used in related research projects due to its direct
physical insight and easy implementation. Fig. A-1 shows the concept of the RAPID
algorithm. In the RAPID algorithm, it is assumed that the occurrence of a localized defect
may cause a significant change in the through transmission signal [Gao et al. 2005]. The
probability of defect occurrence at a certain point can be reconstructed from the severity
of the signal change and its relative position to the sensor pair. Using the transmitter and
the receiver as two focal points, a set of ellipses can be drawn within the reconstruction
region. Therefore, all the points that have the same total distance to the two transducers
have the same defect distribution probability. Again, the defect distribution probability on
the direct path is affected by the severity of the change in the signal. By superimposing
the defect probability obtained from all the sensor pairs together, a global defect
distribution signature can be obtained. Basically, if a defect occurs somewhere, a suite of
signals will be affected. As a result, in the final defect probability image, the defect point
will have larger probability than the other points. Therefore, by applying a threshold to
the final defect probability image, the defect location can be depicted from the whole
reconstruction region.
Transmitter Receiver
Direct path
Indirect path
Transmitter Receiver
Direct path
Indirect path
Figure A-1: Concept of a ray affect area in RAPID reconstruction
187
A.3 Application of imaging techniques in laboratory experiments
The RAPID algorithm has been applied in many of our research projects for
damage localization and assessment. Several examples in the aircraft industry are
presented in this section [Gao et al. 2004; Gao et al. 2005; Hay et al. 2006; Rose et al.
2006]. Some other applications include the pipeline critical zone monitoring and elbow
area monitoring.
A.3.1 E2 airplane wing crack monitoring
In this section, a simulated crack around a rivet hole on an E2 aircraft wing skin is
detected and localized with 8 sensors attached on the inner surface of the skin. Fig. A-2
shows the sensors on the wing panel. Fig. A-3 shows the crack localization and
assessment result for a crack length ranging from 2mm to 4mm. The result indicates that
the reconstruction algorithm successfully identified the location of the crack. The
increase in crack length is also indicated in the images as an increase in the change of
color obtained from changes in SDC.
Figure A-2: Piezoelectric sensors on an aircraft wing panel.
3
4
2
1
6
5
7
8
Defect
3
4
2
1
6
5
7
8
3
4
2
1
6
5
7
8
DefectSensor
Copper tap
ground
188
A.3.2 Helicopter component corrosion monitoring
Fig. A-4 shows an application of the RAPID technique for the corrosion
monitoring of a helicopter component. A 1 mil thick metal loss is introduced in the
metallic panel with a 1 inch by 1 inch area. The photo of the simulated corrosion is
Position (mm)
Po
sitio
n (
mm
)
-100 -50 0 50 100
-100
-50
0
50
100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Position (mm)
Po
sitio
n (
mm
)
-100 -50 0 50 100
-100
-50
0
50
100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
(a) (b)
Position (mm)
Po
sitio
n (
mm
)
-100 -50 0 50 100
-100
-50
0
50
100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Position (mm)
Po
sitio
n (
mm
)
-100 -50 0 50 100
-100
-50
0
50
100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
(c) (d)
Figure A-3: Reconstruction results from 15 micro pulse width data with adaptive threshold.
(a) Reference state , (b) 2mm defect, (c) 3mm defect, (d) 4mm defect
Defect
189
shown in Figure (a). Sixteen transducers are used to monitor the corrosion. The
reconstruction result of the corroded area is shown Figure (b). The corroded area shape is
also plotted in the figure to compare with the reconstructed image.
A.3.3 Composite delamination monitoring
Guided wave damage monitoring is also studied with surface mounted transducers.
Sixteen transducers mounted on a 24 ply [0/90]s6 composite made from Hexcel prepreg
are shown in Fig. A-5. Impact damage using 4.23 J of energy was introduced into the
composite panel. Fig. A-5 (b) shows a typical signal across the image path. The wave
package with significant change is circled in the figure. Although the composite material
and layup is different from the quasi-isotropic layup, it was also observed that the change
caused by delamination occured more significantly to the first mode than to the other
guided wave modes.
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
0
0.5
1
1.5
2
Figure A-4: (a) Simulated corrosion damage in an helicopter component, Corrosion
thickness 1/1000 inch, area 1’’ x 1’’. and (b) damage monitoring results with ultrasonic
guided waves and RAPID reconstruction technique
190
(a) (b)
Figure A-5: (a) Sensor array on a composite panel for impact damage detection, (b)
sample signals before and after impact showing damage detection.
Figure A-6: Impact damage localization with (a) Ultrasonic C-scan (b) guided wave
monitoring with RAPID algorithm.
First Mode
Appendix B
Nontechnical abstract
Carbon fiber reinforced polymer composites have found increased application in
the aircraft and aerospace industries due to their superior mechanical properties and light
weight. However, the composite materials are subject to damage from periodic loading,
impact from foreign objects, and aging. This damage might lead to malfunctioning or
even catastrophic failure of the aircraft. Nondestructive evaluation techniques are used to
inspect the structures. However, significant damage to the structure could happen in
between inspection intervals. Therefore, monitoring techniques, capable of providing
timely information on structural health conditions, are needed.
Ultrasonic waves have been successfully used in defect detection by evaluating
wave reflection and transmission. Ultrasonic technologies are also very promising for use
with real time health monitoring. In order to reduce the number of sensors installed on the
structure, ultrasonic guided waves propagating along the structure becomes a natural
choice. Some research work has been carried out using ultrasonic guided waves excited
from piezoelectric transducers to monitor damage in aging aircraft components. However,
the current practices are mostly on an experimental trial and error basis due to the
complexities of guided wave mechanics. An understanding of wave mechanics is needed
to design a structural health monitoring system with better performance.
The purpose of this thesis is to address the complexity of ultrasonic guided wave
mechanics in composite materials, to understand its behavior, and to take advantage of its
complexities for future structural health monitoring. In order to achieve this goal, the
ultrasonic guided wave mechanics study has been separated into three aspects, namely,
wave propagation, excitation, and damage sensing. Commonly used guided wave
features such as phase velocity and group velocity are investigated in this study. Some
features particularly important for composite materials, like guided wave energy skew
and wave propagation direction dependence are systematically studied. In addition, some
192
new features such as wave excitability, wave dispersion coefficient, wave beam
spreading, and damage sensitivity are defined for the purpose of evaluating the
performance of a guided wave mode. All the features defined above are used in a
comprehensive mode selection algorithm to evaluate the performance of a wave mode.
Quasi-isotropic composites are commonly used because of their isotropic
behavior in carrying tensile loads. As an example, a 16 layer quasi-isotropic composite is
studied in this thesis. The application of guided wave NDE to quasi-isotropic composites
is very challenging due to the multilayer, anisotropic, and viscoelastic nature of the plate.
Ultrasonic displacements in three coordinate directions are not separable. Significant
anisotropy and wave skew exists in quasi-isotropic composites. In addition, the
attenuation of ultrasonic waves in composite material is not negligible, it plays a very
important role in the overall mode selection process. A new derivation is required to fully
understand the wave excitation phenomenon in the lossy media.
A numerical simulation tool combines the benefits of many numerical algorithms for
guided wave study. The semi analytical finite element, global matrix method, and normal
mode expansion techniques are used for the numerical prediction of guided wave
propagation and excitation. The performance of the simulation tool is validated with
finite element analysis using ABAQUS. The expectations of guided wave propagation,
excitation and damage detection are also validated with laboratory experiments. Signals
from a prototype transducer element achieved 10 times improvement in sensitivity and 4
times in mode selection capability as compared to a commercial piezo-electric disc used
in previous experiments.
The guided wave simulation tool developed here will have a very broad impact in
structural health monitoring applications. The general model can be applied to
composites with specific lay-up sequences and non-composite structures. In addition to
the application in structural health monitoring and nondestructive evaluation, the
understanding of wave propagation in multi-layered, anisotropic, viscoelastic media will
also be useful in many other areas such as acoustic wave sensors and actuators, and
medical ultrasound, and even seismology.
VITA
HUIDONG GAO
EDUCATION
Ph. D, Engineering Science and Mechanics, 2007
The Pennsylvania State University, University Park, PA, USA
M.S. Acoustics, Department of Electronic Science and Engineering, 2003
Nanjing University, Nanjing, China.
B.S. Physics, Department for Intensive Instruction (honored), 2000
Nanjing University, Nanjing, China.
SELECTED PUBLICATIONS
1. Huidong Gao, Manton J. Guers, and Joseph L. Rose, “Flexible ultrasonic guided wave sensor
development for structural health monitoring”, SPIE proceedings Vol. 6176, 2006.
2. Hudiong Gao, Joseph L. Rose, “Sensor Placement Optimization in Structural Health Monitoring
Using Genetic and Evolutionary Algorithms”, SPIE proceedings, Vol. 6174, 2006.
3. Xiang Zhao, Joseph Rose, and Huidong Gao, "Determination of density distribution in ferrous
powder compacts using ultrasonic tomography," IEEE Trans. Ultrasonics, Ferroelectronics, &
Frequency Control, 53(2), 360-369, 2006.
4. Hay, T.R., Royer, R, Gao, Huidong, ., Zhao, Xiang, Rose, J.L., “A Comparison of Embedded
Sensor Lamb Wave Ultrasonic Tomography Approaches for Material Loss Detection”, Journal of
Smart Structures and Materials, 15, 946-951,2006.
5. H. Gao, and J. L. Rose, "Ultrasonic Sensor Optimization in Structural Health Monitoring Using
Genetic Algorithms", Presented at Review of Progress in Quantitative NDE, Brunswick, Maine,
Aug. 1-5, 2005, Published in Review of Quantitative Nondestructive Evaluation Vol. 25, 2005,
1687-1693.
6. Hui-dong Gao, Shu-yi Zhang, et. al, "Influence of Material parameters on Acoustic Wave
Propagation Modes in ZnO/Si Bi/Layered Structure", IEEE transactions on Ultrasonics,
Ferroelctrics and Frequency Control, 52 (12), 2361-2369, (2005).
7. H. Gao, Y. Shi, J. L. Rose, “Guided Wave Tomography on an Aircraft Wing with Leave in Place
Sensors”, Review of Quantitative Nondestructive Evaluation, 24, 1788-1794, 2004.
8. Huidong Gao and Shuyi Zhang,“Theoretical Analysis of Acoustic Wave Propagation in ZnO /Si
Bi-layered System Using Transfer Matrix Method” Acoustical Science and Technology, 25, pp.
90-94,(2004).
9. X. B. Mi, H.D. Gao, and S.Y. Zhang, "Two dimensional transient thermal analysis of Diamond/Si
strucuture heated by a pulsed circular gausian laser beam", International Journal of Heat and
Mass Transfer, 47, 2481-2485, (2004).
10. Huidong Gao, Liping Cheng, Xiuji Shui and Shuyi Zhang, “Surface Acoustic Wave Actuators
with Modulated Driving Signal”, Technical Acoustics(in Chinese) Vol.21, 38-41(2002).
PROFESSIONAL AFFILIATIONS
American Society for Nondestructive Testing (ASNT) , student member
Institute of Electrical and Electronics Engineers, Inc. (IEEE), student member