ultrasonic guided wave mechanics for composite …

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

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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.

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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.

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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

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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

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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

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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.

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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

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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

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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



window and 1 MHz center frequency. The loading is concentrated normal. .......73



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


curves. Blue dashed line: nominal material property. Red line: with 10%

transverse modulus (E2) reduction of the lamina..................................................134


curves. Blue dashed line: nominal material property. Red line with 10% in

plane shear modulus (G12) reduction of the lamina. .............................................134


curves. Blue dashed line: nominal material property. Red line with 10%

Poisson’s ratio(v12) reduction of the lamina. ........................................................135


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


delamination at the 3rd laminate interface of a [(0/45/90/-45)s]2 composite

structure. ...............................................................................................................140


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

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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

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10

20

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40

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60

Year of first flight

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erc

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osite

F-14 F-15 F-16

F-18

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10

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Year of first flight

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osite

DC-9 DC-10

747 757 767

737-300 747-400

A300-600

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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



center frequency. The loading is concentrated normal.



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)


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


Number of cycles 3

Excitation signal


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


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)


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)


(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


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


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)


FEM calculation


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


Number of cycles 3

Excitation signal


Mapped mesh with 8 node brick element


Finite element mesh


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)


FEM calculation


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)


FEM calculation


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


Number of cycles 10

Excitation signal


Mapped mesh with 8 node brick element


Finite element mesh


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)


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)


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)


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)


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


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


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


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


1

3

2

4


(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


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


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


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


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.

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.

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

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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)

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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.


Direct path

Indirect path


Direct path

Indirect path

Figure A-1: Concept of a ray affect area in RAPID reconstruction

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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

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Defect

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DefectSensor

Copper tap

ground

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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)

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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

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0

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0.5

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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

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(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

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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

ultrasonic guided wave mechanics for composite …

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