ULTRASONIC GUIDED WAVES FOR DEFECT CHARACTERIZATION …

The Pennsylvania State University

The Graduate School

Department of Engineering Science and Mechanics

ULTRASONIC GUIDED WAVES FOR DEFECT CHARACTERIZATION IN

COMPOSITE STRUCTURES

A Dissertation in

Engineering Science and Mechanics

by

Xue Qi

© 2011 Xue Qi

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

May 2011

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The dissertation of Xue Qi was reviewed and approved* by the following:

Joseph L. Rose Paul Morrow Professor of Engineering Science and Mechanics Dissertation Co-Advisor Chair of Committee

Edward C. Smith Professor of Aerospace Engineering Dissertation Co-Advisor

Bernhard R. Tittmann Schell Professor of Engineering Science and Mechanics

Clifford J. Lissenden Professor of Engineering Science and Mechanics

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

Ultrasonic guided waves have been widely applied to the nondestructive

evaluation (NDE) and structural health monitoring (SHM) of aircraft structures.

Development of the guided wave technique requires an understanding of wave

propagation and scattering principles. The purpose of this research is to advance

transducer design and signal processing by investigating ultrasonic guided wave

interaction with defects. Both isotropic metallic materials and anisotropic fiber-reinforced

composite materials are included.

Two approaches are introduced for damage characterization in composite

laminates. The first technique is to qualitatively predict guided wave scattering at defects

by analyzing wave propagation characteristics. As a sample application, waves in the

trailing edge of a helicopter rotor blade, which is a composite skin/honeycomb half-space

structure, are analyzed. A global matrix method (GMM) is used to determine complex

solutions of both propagating and evanescent waves. The skin/substrate disbond is

measured by leaky guided waves. In another example, ultrasonic guided waves are

applied to detect and characterize delamination defects inside a 23-layer Alcoa Advanced

Hybrid Structural plate. A semi-analytical finite element (SAFE) method is used to

generate dispersion curves and wave structures for the purpose of selecting appropriate

wave modes that are sensitive to the target defect. One guided wave mode and frequency

is chosen as an example to achieve large in-plane particle displacements at regions of

interest. The high sensitivity of the selected guided wave mode and frequency is first

verified in a finite element model. Theoretically driven experiments are then conducted

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and compared with bulk wave measurements. It is shown that guided waves can detect

and characterize deeply embedded damages inside thick multilayer fiber-metal laminates

with suitable mode and frequency selection quite well.

The second technique is based on quantitative calculations of guided wave

scattering in composite plates. A global-local (GL) method is developed to calculate

transmission and reflection coefficients of guided waves scattered by defects. The GL

method is verified on both metallic and composite plates by comparing with FEM

simulation and experimental results. A parametric study is performed on a unidirectional

carbon/epoxy composite plate with a rectangular notch/embedded void. The influence of

defect location, width, height, and composite ply orientation to guided waves is discussed.

To further verify the GL method, an experiment is carried out on a quasi-isotropic

composite plate with delaminations between different plies. The attenuation of through

transmission waves at each defect is calculated in the 500kHz-1MHz range. The

experimental data matches very well with the simulated results.

In summary, this research proves the feasibility of damage characterization in

plate and plate-like composite laminates with guided waves. Mode selection criteria are

presented and applied to different types of defects. An analytical-numerical hybrid

method is developed to simulate guided wave scattering in composites, which is more

efficient than the traditional FE method in terms of calculation and post processing. A

novel simulation based damage characterization algorithm is derived and verified. All of

these techniques provide guidelines for ultrasonic tests on composite structures

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TABLE OF CONTENTS

LIST OF FIGURES ..................................................................................................... viii

LIST OF TABLES ....................................................................................................... xvi

ACKNOWLEDGEMENTS ......................................................................................... xvii

Chapter 1 Introduction ................................................................................................. 1

1.1 Problem statement ......................................................................................... 1 1.2 Background literature .................................................................................... 3

1.2.1 Guided wave propagation theory ......................................................... 3 1.2.2 Guided wave excitation ........................................................................ 8 1.2.3 Guided wave scattering and defect characterization ............................ 9

1.3 Thesis objectives ........................................................................................... 16 1.4 Organization of the thesis .............................................................................. 18

Chapter 2 Guided Wave Propagation Theory in Multilayered Solids ......................... 20

2.1 The global matrix method ............................................................................. 21 2.2 The semi-analytical finite element (SAFE) method ...................................... 24 2.3 Energy velocity and skew angle .................................................................... 27 2.4 Guided wave mode sorting based on orthogonality ...................................... 29 2.5 Guided wave propagation in an anisotropic composite plate ........................ 29 2.6 Summary ........................................................................................................ 37

Chapter 3 Guided Wave Damage Detection of Skin-Substrate Disbond .................... 39

3.1 Guided wave propagation in a composite laminate on a half-space structure ......................................................................................................... 40

3.2 Finite element simulation .............................................................................. 44 3.2.1 SAW generation and propagation ........................................................ 45 3.2.2 Leaky wave generation and propagation ............................................. 47

3.3 Skin and honeycomb disbond measurement ................................................. 51 3.4 Summary ........................................................................................................ 54

Chapter 4 Guided Wave Nondestructive Testing for Delaminations in Hybrid Laminates .............................................................................................................. 56

4.1 Guided wave dispersion curves and wave structures analysis ...................... 56 4.2 Finite element modeling results ..................................................................... 63 4.3 Experimental results ...................................................................................... 68 4.4 Summary ........................................................................................................ 76

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Chapter 5 Theory of Guided Wave Scattering at Defects ............................................ 78

5.1 Finite element theory for guided waves in a transversally uniform structure ......................................................................................................... 79

5.2 The Global-Local (GL) method ..................................................................... 82 5.3 Validation of the GL method with 2D FE simulation ................................... 87 5.4 Validation with experiments and finite element results in literature ............. 92 5.5 Summary ........................................................................................................ 97

Chapter 6 Defect Characterization in Composite Plates .............................................. 98

6.1 Guided wave scattering in a unidirectional composite plate ......................... 99 6.1.1 Verification with ABAQUS ................................................................. 99 6.1.2 Effect of wave propagation direction to wave scattering at a

surface notch .......................................................................................... 105 6.1.3 Effect of void location through the thickness ...................................... 112 6.1.4 Effect of defect size ............................................................................. 115

6.2 Simulation and experiment for guided wave scattering in a quasi-isotropic composite plate ............................................................................... 121 6.2.1 Sample description ............................................................................... 121 6.2.2 Theoretical analysis ............................................................................. 123 6.2.3 Experiments and discussion ................................................................. 131

6.3 Summary ........................................................................................................ 140

Chapter 7 Conclusions and Recommendations ............................................................ 142

7.1 Summary of the research ............................................................................... 142 7.2 Contributions ................................................................................................. 145 7.3 Recommendations for future work ................................................................ 147

7.3.1 Analysis of guided wave scattering in viscoelastic media ................... 147 7.3.2 Numerical study of guided wave scattering with 3-D finite element

method .................................................................................................... 149 7.3.3 Guided wave defect characterization for composite cylinder .............. 151 7.3.4 Testing with phased array transducers ................................................. 153 7.3.5 Numerical and experimental study of the coupling between the

transducer and the host structure ............................................................ 154 7.3.6 Simulation of guided waves in solid structures coupled to infinite

media ...................................................................................................... 155

References .................................................................................................................... 157

Appendix A Ultrasonic Guided Wave Simulation Toolbox Development for Damage Detection in Composite .......................................................................... 169

A.1 The main interface ......................................................................................... 170 A.2 Case study– compare with prior work ........................................................... 173

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A.2.1 Guided wave propagation in IM7/8552 composites ............................ 175 A.2.2 Guided wave propagation in damaged composites .............................. 178

A.3 Summary ........................................................................................................ 182

Appendix B Nontechnical abstract ............................................................................. 184

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LIST OF FIGURES

Figure 1-1. Delamination in a composite laminate (MERL). ...................................... 2

Figure 1-2.Calculated phase velocity, group velocity, energy skew angle and attenuation versus frequency for the 16 layer [(0/45/90/-45)s]2 lamina for ultrasonic guided wave traveling in 0o direction. The solid and dotted lines represent results from the SAFE method and the GMM respectively (Gao H. , 2007). .................................................................................................................... 7

Figure 1-3.FE simulation of longitudinal wave scattering at a crack in a 2D homogeneous isotropic medium. Absorbing region is applied to eliminate boundary reflection (Velichko & Wilcox, 2010). ................................................. 12

Figure 2-1. The coordinate system for guided wave propagation analysis with the GMM. ................................................................................................................... 21

Figure 2-2. In an anisotropic plate, each guided wave mode can be considered as a combination of six partial waves (two longitudinal and four shear waves). ........ 22

Figure 2-3. The coordinate system for wave propagation analysis with the SAFE method. Each element includes three nodes. ........................................................ 25

Figure 2-4. Phase velocity dispersion curves for guided waves propagating along 0° (a); 45° (b); and 90° (c) directions in a unidirectional composite plate. .......... 31

Figure 2-5. Group velocity dispersion curves for guided waves propagating along 0° (a); 45° (b); and 90° (c) directions in a unidirectional composite plate. .......... 32

Figure 2-6. Skew angle dispersion curves for guided waves propagating along 0° (a); 45° (b); and 90° (c) directions in a unidirectional composite plate. .............. 33

Figure 2-7. Displacements of the first three modes of guided waves propagating along 45° direction at 100kHz. (a) Mode 1; (b) Mode 2; (c) Mode 3. ................. 35

Figure 2-8. Stresses of guided waves propagating along 45° at 100kHz. (a) Mode 1; (b) Mode 2; (c) Mode 3. ................................................................................... 36

Figure 2-9. Energy flux of guided wave modes 1-3 propagating along 45° at 100kHz. Note that curves of mode 2 and mode 3 are overlapped. ....................... 37

Figure 3-1. A sandwich structure of a honeycomb core between two skin layers. ...... 39

Figure 3-2. Sketch of a rotor blade trailing edge section. ............................................ 40

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Figure 3-3. Phase velocity dispersion curves for both propagation and non-propagation guided waves in a composite skin/Nomex substrate structure. ........ 41

Figure 3-4. Real parts (a) and imaginary parts (b) of wave numbers for both propagation and non-propagation guided waves in a composite skin/Nomex substrate structure. ................................................................................................ 42

Figure 3-5. Normalized particle displacements of the first (a) and second modes (b) of guided waves in a composite skin/half space structure at 300kHz. ............ 43

Figure 3-6. Sketch of the FE model for guided wave excitation and propagation in a rotor blade trailing edge section (a composite skin/epoxy/Nomex substrate structure). .............................................................................................................. 44

Figure 3-7. Comparison of defected (red) and baseline (blue) signals for through transmission SAWs at 300kHz. ............................................................................ 46

Figure 3-8. Wave structures of mode one at 300kHz calculated from the FEM. ........ 47

Figure 3-9. Comparison of defected (red) and baseline (blue) signals for through transmission leaky waves at 300kHz. ................................................................... 49

Figure 3-10. Time-space spectra of through transmission leaky wave and SAW in (a) an intact structure and (b) a debonding structure.. .......................................... 50

Figure 3-11. A helicopter rotor blade section with surface mounted PZT wafers. ...... 51

Figure 3-12. Sketch of the perfect bonding section (a), the section with a 0.5 inch disbond (b), 1 inch disbond (c), 1.5 inch disbond (d), 2 inch disbond (e), 2.5 inch disbond (f),and the color bar (g). .................................................................. 52

Figure 3-13. (a) Producing a disbond with a knife. (b) Delamination between the composite skin and the Nomex honeycomb. ........................................................ 52

Figure 3-14. Time domain representation of baseline and damaged signal at 300kHz. A 2.5 inches defect is located between the transmitter and the receiver. ................................................................................................................ 53

Figure 3-15. Normalized through transmission energy with transducer No. 1 as the actuator and 2, 5, 10 as receivers. ................................................................... 54

Figure 4-1. Illustration of the materials and layups of a sample Alcoa Hybrid Structural Laminate. ............................................................................................. 58

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Figure 4-2. Defects in the sample hybrid laminate plate. One defect is located between the first and second aluminum layers. The other one is between the second aluminum layer and the FML layer .......................................................... 59

Figure 4-3. Phase velocity (a) and Group velocity (b) dispersion curves for guided waves propagation in 0° direction of the hybrid laminate plates. ...................... 60

Figure 4-4. Wave structures of A2 mode at 450kHz. Note the high in-plane displacements at the first and second bondpreg layers. ........................................ 62

Figure 4-5. Wave structures of SH0 mode at 450kHz. Note the high in-plane displacements at middle bondpreg layers. ............................................................ 62

Figure 4-6. Hybrid analytical FE analysis results on the particle displacement distributions of the A2 mode at 450kHz, (a) in-plane displacement field Ux, (b) out-of-plane displacement field Uz. ................................................................ 64

Figure 4-7. Finite element simulation for guided wave mode selection with an angle beam transducer. ......................................................................................... 66

Figure 4-8. Finite element simulation of. Guided wave interaction with a simulated delamination defect close to the top surface. ....................................... 67

Figure 4-9. Finite element simulation of Guided wave interaction with a simulated delamination defect close to the bottom surface. .................................................. 68

Figure 4-10. Sketch of ultrasonic C-Scan process. ...................................................... 69

Figure 4-11. (a) Sketch of the defect locations in the sample hybrid laminate plate, (b) C-scan image of the laminate plate shows only the defect located within the first bondpreg layer. ........................................................................................ 70

Figure 4-12. Through-transmission guided wave signals recorded using three different transducer configurations. ...................................................................... 71

Figure 4-13. Guided wave scan results for the sample hybrid laminate plate. (a) Guided wave signals and the corresponding transducer positions. (b) Defect locations. ............................................................................................................... 73

Figure 4-14. Short distance guided wave scan setup and the scan route. .................... 73

Figure 4-15. Defect images obtained using the short distance guided wave scans (a) from the top surface, (b) from the bottom surface. ......................................... 75

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Figure 5-1. Guided waves in a transversally uniform structure. x, y and z are along wave propagation direction, shear horizontal direction and plane normal direction respectively. .............................................................................. 78

Figure 5-2. Nine-node plane Lagrange element in Cartesian coordinates. .................. 80

Figure 5-3. Sketch of the GL model in a wave scattering problem. The red lines indicate boundaries of the local and the global regions. ....................................... 83

Figure 5-4. FE model for S0 mode guided wave generation and propagation in a 1.6mm-thick Al plate. ........................................................................................... 87

Figure 5-5. Damping coefficient of the absorbing layers. ........................................... 88

Figure 5-6. Wave number spectrum of the transmitted and reflected guided waves. The boundary reflection is reduced to 20dB. ........................................................ 88

Figure 5-7. Profiles of in-plane (a) and out-of-plane (b) displacements in an intact Al plate calculated with ABAQUS. The incident wave is S0 mode at 500kHz. ... 89

Figure 5-8. Profiles of in-plane (a) and out-of-plane (b) displacements in an intact Al plate calculated with the GL method. The incident wave is S0 mode at 500kHz. ................................................................................................................. 90

Figure 5-9. Profiles of in-plane (a) and out-of-plane (b) displacements in a damaged Al plate calculated with ABAQUS. The incident wave is S0 mode at 500kHz. ................................................................................................................. 91

Figure 5-10. Profiles of in-plane (a) and out-of-plane (b) displacements in a damaged Al plate calculated with the GL method. The incident wave is S0 mode at 500kHz. ................................................................................................... 91

Figure 5-11. Phase velocity (a) and group velocity (b) dispersion curves for guided waves in a steel plate. ............................................................................... 92

Figure 5-12. A0 reflection coefficient from a 0.5mm deep notch in a 3mm thick steel plate. The notch width varies from 0.25 mm to 5.0mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are made within frequency range of 420 to 480kHz. The GL method prediction is at 450kHz. ..................... 94

Figure 5-13. A0 reflection coefficient from a 0.5mm wide notch in a 3mm thick steel plate. The notch depth varies from 0.25 mm to 2.0mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are made within frequency range of 420 to 480kHz. The GL method prediction is at 450kHz. ..................... 94

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Figure 5-14. A0 reflection coefficient from a 0.5mm deep notch in a 3mm thick steel plate. The notch widths are 3.0mm, 4.0mm and 5.0mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are compared with the GL method prediction. ................................................................................................ 95

Figure 5-15. A0 reflection coefficient from a 0.5mm deep notch in a 3mm thick steel plate. The notch widths are 0.5mm, 1.0mm and 2.0mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are compared with the GL method prediction. ................................................................................................ 95

Figure 5-16. A0 reflection coefficient from a 0.5mm wide notch in a 3mm thick steel plate. The notch depths are 0.5mm and 1.5mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are compared with the GL method prediction. ................................................................................................ 96

Figure 5-17. A0 reflection coefficient from a 0.5mm wide notch in a 3mm thick steel plate. The notch depths are 1.0mm and 2.0mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are compared with the GL method prediction. ................................................................................................ 96

Figure 6-1. ABAQUS model for guided wave scattering in a unidirectional composite plate. .................................................................................................... 101

Figure 6-2.FE simulation for guided wave propagation along the 45° direction in (a) intact and (b) damaged unidirectional composite plates. The defect is a notch normal to the wave propagation direction. The incident wave is mode one. ........................................................................................................................ 102

Figure 6-3.Mesh of the GL model of a unidirectional composite plate with a notch at surface. .............................................................................................................. 103

Figure 6-4. Through-transmission energy of guided wave mode one in a unidirectional composite plate with a notch. Solid lines: GL method; Dots: ABAQUS. ............................................................................................................. 104

Figure 6-5. Guided wave transmission and reflection at a surface notch in a unidirectional composite plate. The incident waves are mode one (a); two (b) and three (c) in 45° direction. ............................................................................... 106

Figure 6-6. Guided wave transmission and reflection at a surface notch in a unidirectional composite plate. The incident waves are mode one (a); two (b) and three (c) in 0° direction. ................................................................................. 108

Figure 6-7. Guided wave transmission and reflection at a surface notch in a unidirectional composite plate. The incident waves are mode one (a); two (b) and three (c) in 90° direction. ............................................................................... 109

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Figure 6-8.Transmission (a) and reflection (b) energy profiles for modes 1-3 at 150kHz in a 2.4mm-thick unidirectional composite plate with a 1mm by 0.8mm surface notch. ............................................................................................ 110

Figure 6-9.Transmission (a) and reflection (b) energy profiles for modes 1-3 at 250kHz in a 2.4mm-thick unidirectional composite plate with a 1mm by 0.8mm surface notch. ............................................................................................ 111

Figure 6-10. Mesh of the GL model for a unidirectional composite plate with a void in the subsurface. Here, x is the wave propagation direction. z is through-thickness. y is the transverse direction. ................................................... 112

Figure 6-11. Mesh of the GL model for a unidirectional composite plate with a void in the middle. ................................................................................................ 113

Figure 6-12. Transmission and reflection coefficients for mode 1 with defects at different locations from plate surface to mid-plane, showing periodical modulation with the ratio of crack width to input wavelength, and spikes at some “resonance frequencies”. ............................................................................. 114



Figure 6-15. Sketch of the GL model for a plate with a rectangular void in the mid-plane. ............................................................................................................. 115

Figure 6-16. Transmission (a) and reflection (b) coefficients from a 1mm wide void in a 2.4mm thick composite plate. The horizontal axis is void height in millimeter. This shows that the reflection coefficients monotonously increase with the defect height. ........................................................................................... 117

Figure 6-17. Transmission (a) and reflection (b) coefficients from a 1mm wide void in a 2.4mm thick composite plate, showing that the sensitivity of mode 3 is higher than that of modes 1 and 2, compared at the same wavelength. The horizontal axis is void height in percent of input wavelength. ............................. 118

Figure 6-18. Transmission coefficients of (a) Mode 2 and 3 (b) with a rectangular void in the mid-plane of a 2.4mm thick composite plate. The horizontal axis is void width in percent of input wavelength. The void height is 0.1mm. ........... 119

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Figure 6-21. Illustration of delamination locations. (a): side view; (b): top view. The transducers are placed on the top surface (ply 1). ......................................... 122

Figure 6-22. (a) Phase velocity; (b) group velocity and (c) skew angle dispersion curves for a 16-layer [(0/45/90/-45)S]2 composite plate made of AS4/8552-2 carbon epoxy prepreg. .......................................................................................... 124

Figure 6-23. (a) Mesh of the GL model for delamination between plies 12 and 13. (b) Enlarged picture at the defected area. Blue lines and dots represent element edges and nodes respectively. ................................................................. 126

Figure 6-24. Normalized transmission and reflection energy of guided wave modes 1-4 interacted with defect 1 in the composite specimen. .......................... 127

Figure 6-25. Normalized transmission and reflection energy of guided wave modes 5-8 interacted with defect 1 in the composite specimen. .......................... 128

Figure 6-26. Transmission and reflection coefficients of guided waves to defect 1. .. 129

Figure 6-27. Sensitivity of guided waves to defect 1. .................................................. 131

Figure 6-28. Ultrasonic guided wave test with one pair of angle beam transducers on an quasi-isotropic composite plate composed of 16 plies. .............................. 132

Figure 6-29. Through transmission signals generated and received with a pair of 60º angle beam transducers centered at 820kHz. ................................................. 133

Figure 6-30. Sketch of an angle beam tranducer. ........................................................ 134

Figure 6-31. Source influence spectrum of angle beam transducer with the incident angle at 60 degree. .................................................................................. 135

Figure 6-32. Effective sensitivity to defect 1 for guided waves excited and received with 60 degree angle beam transducers. ................................................ 136

Figure 6-33. Temporal profile and frequency spectrum of incident signals. ............... 137

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Figure 6-34. Theoretical prediction of guided wave attenuation caused by scattering at defect one. The blue and red curves represent attenuation of continuous wave and Hanning-windowed 10-cycle pulse respectively. .............. 138

Figure 6-35. Comparison of experimental and theoretical results of energy attenuation for guided waves at defects 1-3. The incident angle is 60 degree for both the transmitter and the receiver. .............................................................. 139

Figure 7-1. Sketch of the 3-D FE model. ..................................................................... 150

Figure 7-2. The GL model for circumferential waves in a circular tube. .................... 152

Figure A-1. Main interface of the guided wave simulation toolbox. ........................... 170

Figure A-2. Phase velocity comparison for 8 layer CFRP (1.0 mm total thickness) laminate (a) – from (Guo & Cawley, 1993); (b) – using UGWST. ...................... 174

Figure A-3. Comparison of (a) displacements and (b) stresses for an 8 layer (symmetric) CFRP (1.0 mm total thickness) laminate – using UGWST. ............ 175

Figure A-4. Comparison of (a) displacements and (b) stresses for an 8 layer (symmetric) CFRP (1.0 mm total thickness) laminate (Guo & Cawley, 1993). .. 175

Figure A-5. Phase velocity and group velocity dispersion curves of a unidirectional IM7/8552 composite. The angle between wave propagation and fiber direction is (a) 0 degree, (b) 45 degree and (c) 90 degree ..................... 177

Figure A-6. Wave structures of a unidirectional IM7/8552 composite. The angle between wave propagation and fiber direction is 45 degree. ................................ 178

Figure A-7. Phase velocity dispersion curves and wave structures of the selected mode. .................................................................................................................... 178

Figure A-8. Sketch of the FE model. ........................................................................... 179

Figure A-9. Field output of displacement amplitude at frame 39 in the defect free composite (a) and the notched composite (b). ...................................................... 180

Figure A-10. Time history outputs in the defect free model. (a) In-plane displacements at sensors 1-3; (b) Out-of-plane displacements at sensors 1-3; (c) In-plane displacements at sensors 4-6; (d) Out-of-plane displacements at sensors 4-6; ........................................................................................................... 181

Figure A-11. Out-of-plane time history displacements at sensors 4-6 in the defect free composite (a), the composite with a notch (b), and the composite with material degradation (c). ....................................................................................... 182

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LIST OF TABLES

Table 2-1. . Material properties of IM7/977-3 carbon epoxy prepreg ......................... 30

Table 3-1. Material properties of E-Glass/Epoxy unidirectional composite prepreg. .............................................................................................................................. 40

Table 3-2. Material properties of epoxy and Nomex. .................................................. 41

Table 3-3. Parameters of the FE model for surface acoustic wave (SAW) excitation and reception. ....................................................................................... 45

Table 3-4. Parameters of the FE model for the fifth wave mode (Leaky wave) excitation and reception. ....................................................................................... 48

Table 4-1. Materials and layups of a sample Alcoa Hybrid Structural Laminate. ....... 58

Table 6-1. Parameters of the FE models for simulation of guided wave scattering in a unidirectional composite plate ....................................................................... 101

Table 6-2. Material properties of AS4/8552-2 carbon epoxy prepreg ......................... 121

Table A-1. Material properties of transversely isotropic carbon fiber reinforced plastic (CFRP) (Guo & Cawley, 1993). ................................................................ 174

Table A-2. Elastic constants of IM7/8552 unidirectional composite along the fiber direction. ............................................................................................................... 176

Table A-3. FE parameters for simulation of guided wave scattering in a composite plate ....................................................................................................................... 179

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ACKNOWLEDGEMENTS

I would like to express sincere gratitude to my advisor Dr. Joseph L. Rose for the

guidance and support during the Ph.D. period. The knowledge and experience I learned

from him is a great treasure in my life.

I would also like to thank my co-advisor, Dr. Edward Smith. This dissertation

would not have been possible without his scientific and financial support.

In addition, a thank you to Dr. George Zhao at Intelligent Automation, Inc. (IAI),

who supervised my research off campus. From him I learned not only technology, but

also attitude to work.

Thanks are given to my other committee members, Dr. Bernhard R. Tittmann and

Dr. Clifford J. Lissenden for their instructions and helps to my research and the

preparation of this thesis.

Part of this research was funded by the Vertical Lift Consortium, formerly the

Center for Rotorcraft Innovation and the National Rotorcraft Technology Center (NRTC),

U.S. Army Aviation and Missile Research, Development and Engineering Center

(AMRDEC) under Technology Investment Agreement W911W6-06-2-2002, entitled

National Rotorcraft Technology Center Research Program. The author would like to

acknowledge that this research and development was accomplished with the support and

guidance of the NRTC and VLC. The views and conclusions contained in this document

are those of the author and should not be interpreted as representing the official policies,

xviii

either expressed or implied, of the AMRDEC or the U.S. Government. The U.S.

Government is authorized to reproduce and distribute reprints for Government purposes

notwithstanding any copyright notation thereon.

Financial support from the Vertical Lift Consortium (VLC), NASA, Air Force,

IAI, and Alcoa are greatly appreciated. I would also thank FBS Inc. Sikorsky and Pratt &

Whitney for devices and composite specimens. I would like to thank my colleagues in

PSU and IAI. They gave me a lot of assistance in computations and experiments.

Finally, sincere thanks are given to my wife, Xiaofang, and my parents for their

deep love and continuous support over these years.

Chapter 1

Introduction

1.1 Problem statement

Fiber reinforced polymer composites have attracted considerable interest in the

aircraft and aerospace industries due to their attractive mechanical properties and light

weight. Despite their strength and low weight, composite materials are subject to damage

during fatigue, mechanical impact, and aging in a service environment. For example,

delamination is a common damage mode in a composite laminate (shown in Figure 1-1).

Reliable nondestructive inspection and timely maintenance are desired for improving

structure safety and extending structure’s service life. Maintenance of an aircraft structure

can be either time based (Shull, 2002) or condition based (Chang, Prosser, & Schulz,

2002). In time-based maintenance, the state between two scheduled inspections is not

monitored. High safety factors are usually applied in structure remaining life prediction,

which leads to a heavy maintenance schedule and long service time. The Condition Based

Maintenance (CBM) and Structural Health Monitoring (SHM) provide continuous

evaluation of the monitored object with a sensor system (Chang, Prosser, & Schulz,

2002). SHM effectively minimizes the labor and material costs (Raghavan & Cesnik,

2007). Furthermore, it increases the operational availability and mission reliability of

vehicles.

2

Figure 1-1. Delamination in a composite laminate (MERL).

Ultrasonic guided wave techniques have been widely used in inspection of

composite structures (Quaegebeur, Micheau, Masson, & Maslouhi, 2010). Basic guided

wave SHM methodology and application can be found in a recent review of Raghavan

(Raghavan & Cesnik, 2007). Compared with localized methods such as

electromechanical impedance(Kitts & Zaqrai, 2009), ultrasonic bulk waves(Wilcox &

Velichko, 2010), and fiber optical methods (Hiche, Liu, Seaver, & Wei, 2009), etc.,

guided waves is able to inspect/monitor a larger area with only a few sensors; and

compared with global methods such as nonlinear techniques (Shkolnik, Cameron, & Kari,

2008) and mode shape analyses (Stubbs & Kim, 1996), guided waves have a better

damage localization capability (Rose J. L., 1999). Therefore, it has great potential for

applications in composite damage inspection.

3

1.2 Background literature

1.2.1 Guided wave propagation theory

Analytical methods

Ultrasonic guided waves are elastic waves propagating in bounded solid structures.

It is formed by the superposition of bulk wave multi-reflection between waveguide

boundaries. There are infinite combinations of reflected bulk waves at every frequency.

Each combination is a guided wave mode with its unique particle vibration characteristics,

i.e., wave structures. Some of these modes are propagating waves which can theoretically

travel infinite distance along an elastic waveguide. Others are evanescent waves,

attenuating during propagation. Classical guided wave theories and applications are

illustrated in textbooks (Auld, 1990; Rose J. L., 1999).

Ultrasonic guided wave study can be traced back to the 19th century. In 1885,

Lord Rayleigh first used the partial wave technique to solve the surface wave problem

(Rayleigh, 1885) . After that, Horace Lamb studied the wave propagation in an isotropic

solid plate with a free surface (Lamb, 1917). Stoneley, Scholte and Love studied waves at

a solid-solid interface (Stoneley, 1924), solid-liquid interface (Scholte, 1942) and shear

horizontal wave in one layer on a half-space (Love, 1911) respectively. The names of

these waves are the same as their pioneers.

To solve the guided wave problem in a multi-layered anisotropic plate, a transfer

matrix method (TMM) was developed by Thomson (Thomson, 1950) and refined by

Haskell (Haskell, 1953). In this method, a matrix connects wave fields at the top and

4

bottom surfaces of each layer. Multiplying the matrices of all layers generates a transfer

matrix, which hooks up the wave fields at both surfaces of the plate. The displacements

and stresses at any location inside the laminate can be expressed with the transfer matrix

and boundary conditions. The weakness of the transfer matrix method is its instabilities

when the product of frequency and thickness is large (Lowe, Matrix Techniques for

Modeling Ultrasonic Waves in Multilayered Media, 1995). An alternative technique is

the global matrix method (GMM), provided by Knopoff (Knopoff, 1964). In GMM, the

wave fields at all the interfaces and boundaries are assembled together in a single matrix.

This method is robust but relatively computationally slow because of the large matrix.

Some waveguides are composed of a layer on a substrate, e.g. composite or metal

layer on a honeycomb structure. Since the substrate thickness is much larger than the

wavelength, it can be treated as a half infinite space. There are two types of guided waves

in these structures: leaky wave and surface acoustic wave (SAW).

As a non-propagating guided wave, leaky wave loses energy to the substrate

during propagation. Chimenti and Nayfeh studied leaky Lamb wave propagation in a

water covered composite plate with the transfer matrix method (Chimenti & Nayfeh,

1990). Bar-Cohen et al. characterized defects in a composite material with leaky Lamb

waves (Bar-Cohen, Mal, & Chang, 1998). Zhu etc studied leaky Rayleigh and Scholte

waves at the fluid–solid interface subjected to transient point loading (Zhu, Popovics, &

Schubert, 2004). All these works were on water coupled plate or plate-like structures. For

anisotropic thin layer on a half space structure, leaky wave theory (Mourad, Desmet, &

Thoen, 1996) and experiments (Scala & Doyle, 1995) have been reported.

5

Surface acoustic waves (SAW), also named Rayleigh waves, propagate on the

surface of a solid structure. The particle displacements of SAWs attenuate with depth.

Mourad et al. studied Rayleigh waves in thin layers deposited on anisotropic media

(Mourad, Desmet, & Thoen, 1996). Hurley et al. employed the SAW to determine the

anisotropic elastic properties of thin films (Hurley, Tewary, & Richards, 2001). Shuvalov

and Every measured the near-surface elastic properties of solids and thin supported films

with the SAW (Shuvalov & Every, 2002).

Numerical and hybrid methods

Analytical solutions are not available for some guided wave problems, for

instance, inhomogeneous materials, irregular-shaped cross-sections, and coupling of

transducers with substrate structures. Most of these problems were studied by numerical

or analytical-numerical hybrid methods.

Lee and Staszewski’s paper (Lee & Staszewski, 2003) reviewed several numerical

methods such as the finite element method (FEM), the finite difference method (FDM),

the boundary element method (BEM), the finite strip element method (FSM), the spectral

element method (SEM), the mass spring lattice method (MSLM), and the Local

Interaction Simulation Approach (LISA). Besides these methods, an efficient and

scalable parallel finite element code, Internet Parallel Structural Analysis Program

(IPSAP), was developed to solve large scale FE problems with high-performance parallel

software and hardware (Kim, Lee, & Yeo, 2002). Pike and his partners successfully used

IPSAP to perform direct numerical simulation of active fiber composite in a blade

6

structure (Paik, Kim, Shin, & Kim, 2004). Recently, Kim et al. solved PZT-induced

Lamb wave propagation problems with the SEM (Kim, Ha, & Zhang, 2008).

Analytical GMM and TMM solve guided wave propagation problem by root

searching, which could be time consuming for structures with a large number of layers. If

the waveguide includes both ‘hard’ and ‘soft’ materials, such as metal and epoxy, the

matrix becomes ill-conditioned and missing root is likely to happen. Slow 2-D root

searching has to be conducted for non-propagating guided waves, where the wave

number is a complex with both real and imaginary parts.

A semi-analytical finite element (SAFE) method has been developed to address

the above problems. This technique uses an analytical solution in the wave propagation

direction and a numerical solution in the cross section of the waveguide. The dispersion

relationship is obtained by solving an eigenvalue problem. The SAFE method is

computationally stable and efficient, especially for viscoelastic materials.

Early research on the SAFE method was conducted to solve the problems of

guided wave propagation in a laminated orthotropic cylinder (Nelson, Dong, & Kalra,

1971) and a waveguide with an arbitrary but uniform cross-section (Lagasse, 1973). Rose,

Hayashi, and Lee used the SAFE method to study guided wave propagation in rods, rails,

and pipes (Hayashi, Song, & Rose, 2003; Lee C. , 2006). Matt et al. employed the SAFE

method to detect composite wing skin-to-spar bonded joints condition with viscoelastic

damping considered (Matt, Bartoli, & Lanza di Scalea, 2005). Recently, the SAFE

method was used by Mu and Rose to analyze guided wave propagation in hollow

cylinders with viscoelastic coatings (Mu & Rose, 2008).

7

Figure 1-2.Calculated phase velocity, group velocity, energy skew angle and attenuation versus frequency for the 16 layer [(0/45/90/-45)s]2 lamina for ultrasonic guided wave traveling in 0o direction. The solid and dotted lines represent results from the SAFE method and the GMM respectively (Gao H. , 2007).

Figure 1-2 displays dispersion curves of guided waves in viscoelastic composite

plates calculated with both the SAFE method and the traditional GMM (Gao H. , 2007).

It is shown that the SAFE method can accurately simulate guided wave propagation in a

composite waveguide.

8

1.2.2 Guided wave excitation

For modeling guided wave excitation, analytical solutions exist only when the

transducer is compliant enough compared to the substrate, i.e., the transducer can be

simplified as surface tractions. Existing analytical approaches includes the normal mode

expansion method (Santosa & Pao, 1989), integral transform method (Niklasson & Datta,

2002) and Mindlin plate theory (Rose & Wang, 2004). These methods are under the plane

strain assumption for straight-crested plane waves. In reality, most guided wave sources

are circular or rectangular shaped transducers with restricted length and width. A 3-D

formulation is desired to accurately simulate the behavior of transducers. Wilcox and his

colleagues developed a technique to model guided wave excitation in an anisotropic plate

(Velichko & Wilcox, 2007). This method expressed the circular-crested wave field at far-

field with modified 2-D straight-crested solutions, which provided a 3-D solution for the

wave field in a multilayered anisotropic plate due to a harmonic point force. Based on

Wilcox’s researches, the source influence of phased array transducers was studied by Yan

in his Ph.D. thesis (Yan, 2008).

Numerical and hybrid methods have been developed to model the coupling of

transducer and substrate. A hybrid finite element-normal mode expansion technique was

investigated to model Lamb wave emission-reception with surface mounted transducers

(Moulin, Grondel, Assaad, & Duquenne, 2008; Moulin, Assaad, Delebarre, & Grondel,

2000). The region near the excitation source was divided into discrete elements, and the

far field was represented with combinations of continuous wave modes. Electrical

loading and response were obtained from the system. Ha and Chang introduced a

9

numerical model to simulate piezoelectric actuator-induced wave propagation in thin

plates, which integrated spectral elements in the in-plane direction and finite elements in

the thickness direction (Ha & Chang, 2010).

1.2.3 Guided wave scattering and defect characterization

In ultrasonic guided wave inspection, the incident wave is scattered by defects in

its propagation path. The anomaly of transmitted or reflected waves would be measured

and processed to estimate the defect status, such as location, severity and shape.

Numerous researches have been conducted to model guided wave scattering at defects.

Most of them focused on the forward problems, i.e., predicting the scattered wave field

for a particular type and size of defect.

Analytical methods

Analytical solutions only exist for isotropic material and structures with simple

shapes of defects (Pao & Mow, 1973). For example, the Kirchhoff approximation is

available for wave scattering at a defect with slowly varying shape (Schleicher, Tygel,

Ursin, & Bleistein, 2001). If the properties of damaged region are similar to those of the

surrounding medium, the Born approximation can be applied (Gubernatis, Domany,

Krumhansl, & Huberman, 1977). For most realistic applications, numerical techniques

such as finite elements (FE), finite differences (FD) and boundary elements (BE) are

often used.

10

The boundary element method

The boundary element method (BEM) can simulate propagation and scattering of

both bulk waves (Tan, Hirose, Zhang, & Wang, 2005) and guided waves. With the BEM

and the normal mode expansion technique, Rose et al. studied Lamb wave mode

conversions from the edge of a plate (Cho & Rose, 1996) and interaction with surface

breaking defects (Rose, Pelts, & Cho, Modeling of flaw sizing potential with guided

waves, 2000). Zhao et al. developed BEM models for defect characterization with Lamb

waves and SH waves (Zhao & Rose, 2003). A BE-FE hybrid method was used to

simulate guided wave scattering in laminated structures with different types of defects

(Galán & Abascal, 2005). Recently, the BEM was applied to study guided wave

propagation in 2-D bone mimicking plates with microstructural effects (Vavva,

Papacharalampopoulos, V.C.Protopappas, Fotiadis, & Polyzos, 2009). The BEM has

proved to be an efficient and accurate numerical tool. However, it is limited in ability to

model anisotropic materials due to the complexity of the corresponding Green’s functions.

The finite element method

The finite element method (FEM) has been extensively studied within the last few

decades for ultrasonic waves (Thompson, 2006). It is compatible to inhomogeneous

materials and complex structures. Research shows that the FEM is superior to the BEM

in terms of computational efficiency because the former can be finally expressed into a

form of sparse matrices, which significantly reduce the memory requirement (Burnett,

1994; Harari & Hughes, 1992).

11

Eliminating scattering from boundary is one of the major challenges for FEM in

solid acoustics. One solution is to increase the FE model size so that the reflections can

be separated from incident waves. This method greatly increases the computational cost

for time-harmonic analysis and is unavailable in frequency analysis. Another technique is

to remove the reflected acoustic waves at the boundary. Several approaches have been

reported to handle absorbing boundary conditions. First is the viscous damping boundary

method, which eliminates outgoing waves with damping materials in the absorbing

region (Castaings & Lowe, 2008; Velichko & Wilcox, 2010). This technique is suitable

for isotropic medium, as shown in Figure 1-3. In an anisotropic waveguide, such as

composite, different coefficients should be assigned to the damping layer at every

specific direction, which greatly increase the modeling complexity. The second is the

Perfect Matched Layers (PML) method, which forces the wave to decay exponentially in

the absorbing boundary layer (Drozdz, Skelton, Craster, & Lowe, 2007). The third

method is to place infinite elements with a special shape function at the infinite boundary

(Fu & Wu, 2000).

12

Figure 1-3.FE simulation of longitudinal wave scattering at a crack in a 2D homogeneous isotropic medium. Absorbing region is applied to eliminate boundary reflection (Velichko & Wilcox, 2010).

Another challenge faced by the FEM is to deal with the dispersion error induced

by interpolation. The variables, such as displacements and stresses, are accurate on each

finite element node. Values among these nodes are obtained by polynomial interpolations.

It has been verified that at least 8 elements per wavelength are required to guarantee the

accuracy in a linear interpolation (Lowe, Cawley, Kao, & Diligent, 2002). This limitation

dramatically increases the computation cost at high frequencies and small wavelengths

since the global matrix size is 6N × 6N for an N-node system. The dispersion error can be

reduced by using higher-order polynomial approximations. Ha et al. applied high-order

elements in the in-plane direction and linear elements in the thickness direction for

modeling PZT-induced Lamb wave in thin plates (Ha & Chang, 2010). With a 4th order

interpolation, the computation time has been reduced to around 1/26 of that with linear

elements.

13

Some commercial FE software packages, such as ANSYS and ABAQUS

(SIMULIA, 2010), integrate FE codes with a graphic user interface (GUI). Users can

focus on the physics level rather than programming. In Zhang, Luo and Lee’s graduate

thesis, ABAQUS was used to study guided wave propagation and scattering in pipes and

rails (Zhang, 2005; Luo, 2005; Lee C. , 2006). Gao and Yan applied ABAQUS to

simulate guided waves in composite plates (Gao H. , 2007; Yan, 2008).

Algorithms have been developed to process data from FEM. Demma et al. used

the FEM and modal decomposition methods to study the effect of discontinuity to the

fundamental SH wave in a steel plate (Demma, Cawley, & Lowe, 2003). Terrien et al.

investigated corrosion with the FEM and analyzed the scattered Lamb waves with a

normal mode decomposition method (Terrien, Royer, Lepoutre, & Déom, 2007). Wilcox

and his colleague simulated guided wave scattering in a 2-D FE model and described the

far-field scattered amplitude with an S-matrix, which is a function of the incident angle,

scattering angle and frequency (Wilcox & Velichko, 2010). Each S-matrix contained all

the information of an arbitrary-shaped defect. The idea was to build a library of data with

numerous simulations. In real SHM, the S-matrix from field tests can be compared with

the database for defect characterization. This technique has been expended to 3-D,

describing the scattering behavior of bulk and guided waves (Velichko & Wilcox, 2010).

Hybrid methods

Some numerical-analytical hybrid methods have been developed to simulate

guided wave interaction with defects. For example, Goetschel et al. invented a global-

14

local (GL) method, which model the region near defects with FE and the outside region

with the normal mode expansion technique (Goetschel, Dong, & Muki, 1982). Similar

methods have been used to study guided wave scattering in isotropic plates (Al-Nassar,

Datta, & Shah, 1991) and laminated cylinders (Rattanawangcharoen, Zhuang, Shah,

Member, ASCE, & Datta, 1997). Recently, the GL method was integrated with the SAFE

method (Srivastava, Bartoli, Coccia, & Scalea, 2008), employing FE in the local region

and SAFE in the global region.

Conclusion and challenges

Literature review shows that a lot of research has been conducted to improve the

guided wave based damage detection technique. However, there are still many challenges

to be conquered, especially for composite materials. The following lists some of these

issues that will be addressed in this thesis.

1. Existing analytical methods mainly aim at isotropic structures and simple

shapes of defects. Numerical methods, such as FEM and BEM, can handle

anisotropic materials with complex geometries, but lack efficiency in

terms of parameter study and mode selection. A theoretical solution is

desired for accurate and fast simulation of guided wave interaction with

defects in composites.

2. Previous studies of guided wave scattering were mostly conducted on

Rayleigh-Lamb (R-L) type waves and shear horizontal (SH) waves. The

coupling between longitudinal and shear horizontal vibrations was not

15

considered. This simplification may cause error for composites, where

pure Lamb or SH wave only exists at some particular wave propagation

directions.

3. The sensitivity of guided wave modes to defects in composites was mostly

qualitatively analyzed in previous works. Quantitative comparison

between experimentally measured attenuation spectrum and theoretical

prediction was not reported.

4. Previous numerical studies of guided wave scattering were mainly on low

order modes, e.g. A0, S0 and SH0, at low frequencies. It is worth to explore

high frequency region, which could be more sensitive to small defects. For

FE simulation of high frequency modes, the scattered wavelength can be

very small or extremely high (near the cutoff frequency). The former

factor requires small element size. The later one enlarges the model

geometry. Both increase the computational difficulty.

5. It is usually preferable to generate a single mode in guided wave

inspection. However, sometimes two or more modes with similar

excitability are close in dispersion curves and excited at the same time. In

this case, the source influence should be considered and the contribution

of each wave mode should be discussed.

6. For anisotropic thin layer on a half space structure, leaky wave theory and

experiments have been presented separately. However, a theoretically

driven experiment, including both leaky wave propagation characteristics

analysis and experimental verification, is not yet reported.

16

1.3 Thesis objectives

The overall objective of this research is to develop ultrasonic guided wave based

methods and simulation tools for damage detection and characterization in composite

structures. The outcome of this study will be useful to guided wave NDE and SHM. The

detection probability can be improved by choosing suitable wave modes with the

presented qualitative and quantitative approaches. On the other hand, the defect location

and severity can be estimated by comparing the measured transmission/reflection

coefficient with results from the simulation tools. Challenges listed in last section will be

addressed.

Specific objectives of the research are as follows.

1. Obtain guided wave solutions for traction free, defect free plates with the

SAFE method. Perform model sorting with an orthogonality based technique.

Study dispersion relationships and wave structures for both isotropic and

anisotropic structures. Discuss the effect of material orientation on guided

wave propagation characteristics.

2. Obtain guided wave solutions for a composite skin/substrate structure with the

GMM method, considering both propagating and non-propagating modes.

Develop a 2-D root searching algorithm for evanescent waves with complex

wave numbers.

3. Develop a concept driven, feature based technique for guided wave mode

selection. Two types of defects are considered: skin/substrate disbond and

composite ply delamination.

17

4. Develop FE models to simulate the interaction of guided waves with artificial

defects. The half space substrate is to be modeled with infinite elements. This

task also includes design of loading to excite desired wave modes and data

analysis procedures.

5. Design and conduct laboratory experiments on the skin/substrate structure and

hybrid laminates. Develop a signal processing algorithm to generate a damage

distribution image for the hybrid laminate. Compare it with the bulk wave C-

scan result.

6. Develop a numerical-analytical hybrid Global-Local (GL) method to simulate

guided wave transmission and reflection in an isotropic/anisotropic plate. The

coupling between longitudinal and shear horizontal vibrations should be

modeled for composite laminates with arbitrary layups.

7. Verify the GL method on an isotropic plate by comparing with FE simulation.

The stead state dynamic analysis is to be conducted in ABAQUS. The

boundary reflection can be eliminated by defining damping materials in the

absorbing region.

8. Verify the GL method on an isotropic plate by comparing the GL simulation

results with FE and experimental data in literature.

9. Build a 3-D FE model for guided waves propagation and scattering in a

composite plate. Obtain the transmission coefficients of a wave mode to a

surface notch at different frequencies. Compare them with those from the GL

method.

18

10. Parametrically study guided wave scattering in a unidirectional composite

plate with notch/void. The variables include propagation direction, frequency,

wave mode, defect location in plate thickness, defect width and height.

11. Verify the GL method with experiment on a composite plate with artificial

delaminations. Quantitative comparison should be performed between

theoretically predicted attenuation of guided waves and experimental results.

1.4 Organization of the thesis

This thesis is divided into seven chapters. Chapter 1 introduces the objectives of

guided wave NDE/SHM for composites and provides a comprehensive literature review

of the previous guided wave detection strategies. This chapter concludes with the thesis

objectives and organization.

Chapter 2 illustrates the GMM method and the SAFE method for guided wave

propagation in an anisotropic multilayered plate. An orthogonality based mode sorting

technique is introduced. As an example, calculations are performed for guided wave

propagation in a unidirectional composite plate made of IM7/977-3 carbon epoxy

prepregs.

Chapter 3 and 4 introduce a qualitative damage detection approach. In Chapter 3,

the GMM is applied to analyze guided wave propagation in a composite skin/honeycomb

substrate structure. Both propagating and non-propagating solutions are presented. A

non-propagating wave is then selected to measure skin-core disbond in a composite rotor

blade section.

19

In Chapter 4, ultrasonic guided waves are applied to detect delaminations inside a

23-layer Aluminum /composite hybrid plate. The SAFE method generates dispersion

curves and wave structures. A specific guided wave mode is chosen to focus energy at

interested regions. A finite element model simulates the interaction of the selected mode

with defects. Theoretical driven experiments are conducted and compared with the bulk

wave C-scan result.

Chapter 5 and 6 illustrate a novel quantitative approach for damage

characterization in composites with ultrasonic guided waves. Chapter 5 introduces the GL

method for guided wave scattering in isotropic/anisotropic plates. The validity of the GL

method is verified by comparing with 2-D FE simulations and experiments.

In Chapter 6, a 3-D FE model is developed to simulate guided wave scattering at a

surface notch in a composite plate. The transmission coefficients are compared with those

from the GL method. Then, the effects of damage size and location to guided waves are

discussed. A new simulation-based damage detection method is presented and verified

with experiments.

Chapter 7 summarizes the thesis and recommends future research directions.

Two appendices are included in this thesis. Appendix A introduces a guided wave

simulation toolbox developed with LabVIEW. Appendix B is a nontechnical abstract of

this thesis.

Chapter 2

Guided Wave Propagation Theory in Multilayered Solids

Analysis of wave propagation in undamaged traction free structures is the

preliminary requirement for guided wave inspection. It provides basic information, such

as dispersion curves and wave structures. Based on the free wave solution, source

influence and wave scattering can be further investigated.

Many methods have been developed to solve the problem of free wave

propagation in an anisotropic laminated waveguide. Two of them will be illustrated in

this chapter. The first one is the global matrix method (GMM), developed by Knopoff

(Knopoff, 1964). A detailed introduction of the GMM theory and the partial wave

technique can be found in Nayfeh’s textbook (Nayfeh, 1995). Another technique is the

semi-analytical finite element (SAFE) method (Hayashi, Song, & Rose, 2003). The

SAFE method treats the guided wave problem as an eigen value problem. Usually it is

more efficient than the root search in GMM. However, if the thickness of the plate is

much larger than the ultrasonic wavelength, the element number will be too numerous to

guarantee convergence and the GMM is computationally more efficient than the regular

SAFE method.

An orthogonality based method is introduced in this chapter for guided wave

mode differentiation. The traditional continuous based method requires a very small

frequency increment and may cause error when two dispersion curves are close to each

other (Lowe 1995). The orthogonality based mode sorting method, derived from the

21

complex reciprocity relation is robust and suitable for multilayered anisotropic structures

(Mu & Rose, 2008).

2.1 The global matrix method

This section introduces the partial wave technique and the global matrix method

(GMM) (Auld, 1990). Figure 2-1 represents the coordination system for composite

laminates. Guided waves propagate along the x1 direction. x3 is normal to the plate

surface. hn (n=1, 2, 3…N) represents the thickness of each layer. N is the total number of

plies.

Figure 2-1. The coordinate system for guided wave propagation analysis with the GMM.

Eq. 2.1 is the governing equation for wave propagation in a homogeneous elastic

medium. Here Cijkl is the stiffness coefficients of the medium, ρ is the density, and ui is

the particle displacement.

kj

lijkl

i

xxu

Ctu

∂∂∂

=∂

∂ 2

2

2

ρ 2.1

22

The partial wave technique provides a trial solution for Eq. 2.1. Suppose the

guided wave consists of several partial waves, which propagate in the x1-x3 plane, as

shown in Figure 2-2.

Figure 2-2. In an anisotropic plate, each guided wave mode can be considered as a combination of six partial waves (two longitudinal and four shear waves).

The particle displacement of each partial wave is expressed in Eq. 2.2.

( )( )tCxxikUu pll −+= 31exp α 2.2

Where Ul is the coefficient to be determined, k is the wave number of the guided

wave, α is ratio of the partial wave numbers in x3 and x1 directions, Cp is the phase

velocity of the guided wave, t is time. The relationship of k and Cp is:

pCfk /2π= 2.3

Here f is frequency. Substituting Eq. 2.2 into Eq. 2.1 and neglecting the common

term, a Christoffel equation is written as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000

3

2

1

332313

232212

131211

UUU

AAAAAAAAA

2.4

where

23

2255151111 2 pCCCCA ραα −++=

( ) 24556141612 αα CCCCA +++=

( ) 23555131513 αα CCCCA +++=

2244466622 2 pCCCCA ραα −++=

( ) 23445365623 αα CCCCA +++=

2233355533 2 pCCCCA ραα −++=

For a given value of Cp, there are six solutions of α. Each α corresponds to a

nontrivial solution of the vector <U1, U2, U3>. The ratios of U1, U2 and U3 determine the

polarization of the displacement field. The guided wave field can be expressed as a linear

combination of the six partial waves in Eq. 2.5

( )( )∑=

−+=6

131exp

kpklkkl tCxxiUBu αξ ( )3.2.1=l 2.5

The boundary and interface conditions should be satisfied to determine the

weighting coefficients Bk. Eq. 2.6 and Eq. 2.7 are strain-displacement equations and

constitutive equations, respectively. Eq. 2.8 states the boundary and interface conditions

for ultrasonic waves in a traction free plate.

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=l

k

k

lkl x

uxu

S21

2.6

klijklij SC=σ 2.7

σ31, σ32, σ33 = 0 at top and bottom surface 2.8

u1, u2, u3, σ31, σ32, σ33 continuous at layer interfaces

24

Substituting Eq. 2.5 into Eq. 2.8, the boundary and interface conditions can be

expressed as:

0=⋅ BD 2.9

Here D is a 6N by 6N matrix containing ξ and Cp. To obtain non-trivial solutions

of B in Eq. 2.8, the determinant of the matrix D should be zero.

0=D 2.10

Then the relationship between k and Cp can be obtained, which is usually

expressed as a set of dispersion curves. For any pair of k and Cp, there are unique particle

displacements ui, called wave structures. The particle velocity can also be obtained as:

( ) l

ll ui

tu

v ω−=∂∂

= 2.11

2.2 The semi-analytical finite element (SAFE) method

A semi-analytical finite element (SAFE) method is developed to study wave

propagation in anisotropic laminates (Hayashi, Song, & Rose, 2003). In the SAFE

method, the plate is divided into discrete elements in the thickness direction, and waves

in the propagation direction are described with the orthogonal function exp(ikx). Figure

2-3 shows a SAFE model with 3-node elements. The position of each node is expressed

with the coordinate z and mapped into the local coordination system, where the parameter

ξ=-1, 0, 1 corresponds to the three nodes respectively.

25

Figure 2-3. The coordinate system for wave propagation analysis with the SAFE method. Each element includes three nodes.

The displacement, strain, stress and external traction vectors at any point in an

element are shown in Eq. 2.12.

T

T

T

T

2.12

The relationship of these parameters can be expressed with the virtual work

principle in Eq. 2.13, where the superscript T denotes transposed matrices, ρ is density. Γ

and V stand for the outer surface and volume of the element respectively. The three terms,

from left to right, denotes the work done by the external traction, the increment of kinetic

energy and potential energy, respectively.

δ T d δ T ρ d δ T d 2.13

The displacement vector at any point in the element is described using the shape

function N and the nodal displacement vector U

26

2.14

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

2.15

12

112

2.16

2.17

Here Uij is the displacement of node j in the i direction at a certain wave number k

and frequency ω. The strain-displacement relation is written as

exp

,

1 0 00 0 00 0 00 0 00 0 10 1 0

0 0 00 0 00 0 10 1 01 0 00 0 0

2.18

The stress tensor is

2.19

where C is an elastic coefficient matrix. The external traction vector t is

2.20

where T is the nodal external traction vector. Substituting Eq. 2.14, 2.18, 2.19 and

2.20 into Eq. 2.13 gives

27

’’

d

d

d

d

2.21

Where Γ’ stands for the boundary of the 1D element. Considering Eq. 2.21 for all

the elements and overlapping the values of the common nodes, the governing equation

for the total system can be expressed as Eq. 2.22.

ω

ω

T

T

2.22

The wave number k can be solved as eigen value of Eq. 2.22 for a given

frequency.

2.3 Energy velocity and skew angle

The Poynting’s vector is the power flow density at a particular point within the

wave field. The complex form of the Poynting’s vector is defined in (Auld, 1990)

28

2

* σvP •−=

2.23

Here, v* is the conjugation of the particle velocity vector; σ is the stress field

tensor. The total energy density within the wave field is a summation of the kinetic

energy density Ek and the strain energy density Es.

( ) ( )

Trealreals

zyxT

realrealrealk

sk

E

vvvE

EEE

σss:c:s

vvv

•==

++=•==

+=

21

21

2222222 ρρρ

2.24

where s and c are complex tensors of strain and stiffness respectively. The energy

densities include constant terms (Ek0, Es0) and time variation terms with angular

frequency ω.

( )( ) ( )( )( )( ) ( )( )tkxEtkxEEE

tkxEtkxEEE

ssss

kkkk

ωωωω

−+−+=−+−+=

2sin2cos2sin2cos

210

210 2.25

The energy transmission velocity is then expressed as

( )∫∫

+= H

sk

H

x

energydzEE

dzPC

0 00

0

2.26

Here, Px is the component of Poynting’s vector in x direction. For guided waves

propagation in elastic lossless media, energy velocity is the same as group velocity.

In anisotropic media, an ultrasonic wave may not go exactly where it is sent.

Skew angle is the angle between the launch direction and the wave propagation direction.

Based on the energy transmission of a guided wave mode, skew angle is expressed as the

ratio of the energy transmission rate in the y and x directions.

29

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=Φ

∫∫

H

x

H

y

dzP

dzP

0

0atan

2.27

2.4 Guided wave mode sorting based on orthogonality

Mode sorting can be realized either by checking the continuity of the guided wave

modes or can be based on the orthogonality of the wave modes. The former method

requires a very small frequency increment step. To improve computation efficiency and

reduce error in the dispersion curve calculation, we applied the later method for mode

reorganization and mode sorting. The orthogonality of guided wave modes in lossless

waveguides has been demonstrated by Auld (Auld, 1990) through the derivation of the

reciprocity relation in piezoelectric media. The reciprocity relation can be expressed as:

( ) .ˆ41

0

**∫ •−•−=H

nmmnmn dzxP σvσv 2.28

Here H is the total laminate thickness. The orthogonality can be expressed as:

( ) ( )( )

.0

)ˆ21

0

*

⎪⎩

⎪⎨⎧

≠

=•−= ∫nm

nmdzxPH

mnmn

σv 2.29

The orthogonality between wave modes with different frequencies, i.e., exp(-iω1t)

and exp(-iω2t), has also been proven in mathematics (Hayek, 2001).

2.5 Guided wave propagation in an anisotropic composite plate

As an example, guided wave propagation has been studied for a unidirectional

composite plate made from IM7/977-3 carbon epoxy prepregs. To simplify calculation,

30

material damping is not considered in this research. Table 2-1 lists the engineering

properties of the prepreg. The parameters were calculated from the stiffness matrix in

(Neau, Lowe, & Deschamps, 2001). The density is 1.6g/cm3 and the plate thickness is

2.4mm.

Table 2-1. Material properties of IM7/977-3 carbon epoxy prepreg

E1 172 GPa E2=E3 9.80 GPa

G23 3.2 GPa G12=G13 6.1 GPa ν23 0.55

ν12= ν13 0.37

The SAFE method was applied to calculate dispersion curves and wave structures

for all propagating modes below 800kHz. Then the curves were sorted based on the

orthogonality. Suppose 0° is along the fiber direction. Figure 2-4 (a) –(c) display the

phase velocity dispersion curves for guided wave modes 1-7 propagating in 0° 45° and 90°

directions. Figure 2-5 indicates the group velocity dispersion curves. Since the stiffness

constants in fiber direction are much larger than those in the transverse direction, both

phase velocities and group velocities show their maximum values at 0°, and minimum

values at 90°.

Mode 6 shows negative group velocity near the cutoff frequency (see Figure 2-5

(b) and (c)). Special attention should be paid to these regions in mode sorting procedure

because this mode has two phase velocity values at the same frequency (Figure 2-4).

31

(a)

(b)

(c)

Figure 2-4. Phase velocity dispersion curves for guided waves propagating along 0° (a); 45° (b); and 90° (c) directions in a unidirectional composite plate.

32

(a)

(b)

(c)

Figure 2-5. Group velocity dispersion curves for guided waves propagating along 0° (a); 45° (b); and 90° (c) directions in a unidirectional composite plate.

33

(a)

(b)

(c)

Figure 2-6. Skew angle dispersion curves for guided waves propagating along 0° (a); 45° (b); and 90° (c) directions in a unidirectional composite plate.

34

The skew angles of modes 1-7 are calculated and plotted in Figure 2-6 . As we

expected, there is no skew for waves propagating along the symmetric axis (0° and 90°).

In the 45° direction, skew phenomena can be observed for most wave modes, as shown in

Figure 2-6 (b). The skew phenomena should be considered during guided wave tests on

anisotropic materials. Modes with zero or small skew angles are usually selected to

reduce the complexity of the wave propagation and scattering analysis.

Wave structure analysis has been conducted for guided waves propagating in the

45° direction at 100kHz. The displacements versus thickness are drawn in Figure 2-7 (a)-

(c), corresponding to modes 1-3. Here Ux, Uy and Uz represent displacements in the

wave propagation, shear horizontal and the thickness directions respectively. Note that

the shear horizontal displacements are coupled with the longitudinal and shear vertical

components so that each mode includes displacements in all three directions. This is

different from wave propagation in isotropic material or along the symmetric axis, where.

Lamb waves are decoupled with SH waves.

35

(a) (b)

(c)

Figure 2-7. Displacements of the first three modes of guided waves propagating along 45° direction at 100kHz. (a) Mode 1; (b) Mode 2; (c) Mode 3.

Figure 2-8 plots stress distribution through the plate thickness for modes 1-3.

Figure 2-9 shows the x component of the Poynting’s vector for each mode. These curves

include features for evaluation of a particular defect. For instance, a mode with high shear

stress at the interested region may be sensitive to a “kissing bond”, where the top and

bottom surfaces keep contact in the normal direction, and slide freely in the shear

directions.

36

(a) (b)

(c)

Figure 2-8. Stresses of guided waves propagating along 45° at 100kHz. (a) Mode 1; (b) Mode 2; (c) Mode 3.

37

Figure 2-9. Energy flux of guided wave modes 1-3 propagating along 45° at 100kHz. Note that curves of mode 2 and mode 3 are overlapped.

2.6 Summary

This chapter introduces both the GMM and the SAFE methods for analysis of

guided wave propagation in multilayered anisotropic plates. Example study was

conducted on a graphic/epoxy composite plate. Phase velocity and group velocity

dispersion curves were calculated. Wave characteristics along different propagation

directions were analyzed.

Both the SAFE method and the GMM can accurately simulate guided wave

propagation in composite plates. A detailed comparison of dispersion curves calculated

with both methods is illustrated in Gao’s thesis (Gao H. , 2007). In terms of practical

application, the SAFE method treats the guided wave problem as an eigen value problem.

Usually it is more efficient than the root search in GMM. However, if the thickness of the

plate is much larger than the ultrasonic wavelength, numerous elements are required to

guarantee convergence and the calculation becomes very slow. Therefore, both methods

38

are employed in our research. The SAFE method is applied to multilayered plates with

reasonable thicknesses, and the GMM method is used for wave propagation in very thick

plate or half space.

Chapter 3

Guided Wave Damage Detection of Skin-Substrate Disbond

Composite sandwich structures are common in aircraft because of their light

weight and high normal strength. Figure 3-1 illustrates a typical section of a sandwich

structure composed of honeycomb core and skins. The honeycomb is usually treated as a

homogenous material to simplify calculations. There are many numerical (Pahr &

Rammerstorfer, 2004) and experimental (Klos, Robinson, & Buehrle, 2003; Thwaites &

Clark, 1995) studies on honeycomb structures. Schwingshackl et al. reviewed existing

theories and approaches for effective honeycomb material properties (Schwingshackl,

Aglietti, & Cunningham, 2006). In this chapter, the guided wave technique is applied to

detect skin-core disbonds for a composite honeycomb sandwich structure. The influence

of disbond size to through transmission guided wave intensity is discussed.

Figure 3-1. A sandwich structure of a honeycomb core between two skin layers.

40

3.1 Guided wave propagation in a composite laminate on a half-space structure

The trailing edge of a rotor blade is usually a skin/honeycome/skin sandwich

structure. Because its thickness is much larger than the guided wave wavelength, the

honeycomb can be treated as a half-space substrate, as shown in Figure 3-2. Here the skin

is a fiber glass/epoxy composite with the layup sequence of [45/-45/(0)3-45/45]. In our

simplified model, the substrate is composed of homogeneous isotropic Nomex material.

The skins and the honeycomb are glued with epoxy. The average lamina thickness of the

composite skin is 0.127mm. The glue thickness is 0.0635mm. In this research, guided

waves propagate along the 0° fiber direction. Table 3-1 lists the material properties of the

skin (Daniel & Ishai, 2006). Table 3-2 lists the material properties of the glue (Smith,

Wilkinson, & Reynolds, 1974), and the honeycomb (Florens, Balmes, Clero, & Corus).

Figure 3-2. Sketch of a rotor blade trailing edge section.

Table 3-1. Material properties of E-Glass/Epoxy unidirectional composite prepreg.

Density (g/cm3)

E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) υ12 υ23

1.97 41 10.4 4.3 3.5 0.28 0.50

41

Table 3-2. Material properties of epoxy and Nomex.

Density (g/cm3) E (GPa) υ epoxy 1.52 4.46 0.35

Nomex 1.38 9 0.3

Phase velocity dispersion curves for the structure in Figure 3-2 were calculated

with the GMM. The first six wave modes, including both propagating and leaky solutions,

are drawn in Figure 3-3. Because the leaky modes have complex wave numbers, the

mode search in GMM was conducted in both real and imaginary domains.

Figure 3-3. Phase velocity dispersion curves for both propagation and non-propagation guided waves in a composite skin/Nomex substrate structure.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Frequency (MHz)

Phas

e V

eloc

ity (k

m/s

)

Phase Velocity

Mode 1Mode 2Mode 3Mode 4Mode 5Mode 6

42

(a)

(b)

Figure 3-4. Real parts (a) and imaginary parts (b) of wave numbers for both propagation and non-propagation guided waves in a composite skin/Nomex substrate structure.

Figure 3-4 shows the wave numbers of the first six modes. Mode 1 is a non-leaky

surface acoustic wave (SAW). The imaginary part of its wave number is zero. The others

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Frequency (MHz)

Wav

e N

umbe

r (1/

mm

)

Real Part Of Wave Number


0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

Frequency (MHz)

Atte

nuat

ion

(1/m

m)

Imaginary Part Of Wave Number


43

are all leaky modes. When a leaky wave propagates, it attenuates quickly because of

losing energy to the half-space. A disbond or crack can block the energy leaking and

hence reduce the attenuation. The size of the disbond can be evaluated by measuring the

leaky wave amplitude.

The mode selection criterion depends on the SHM requirement. A wave mode

with a small attenuation coefficient can propagate a long distance in the layer/half-space

structure. Another mode with more attenuation does not cover as large a region. However,

it is more sensitive to disbonds than the former mode.

Figure 3-5 shows the wave structures of modes 1 and 2 at 300kHz. The first one is

a SAW with particle displacements decreasing with thickness. The second mode is a non-

propagating leaky wave. The displacement is nontrivial at infinite depth, which means the

wave structure is unstable and leaking energy to the substrate.

(a) (b)

Figure 3-5. Normalized particle displacements of the first (a) and second modes (b) of guided waves in a composite skin/half space structure at 300kHz.

44

3.2 Finite element simulation

The commercial FE software ABAQUS simulates guided wave propagation in a

rotor blade trailing edge section. Figure 3-6 displays a sketch of the system. The length,

width and thickness of the model are 150mm, 15mm and 21.3mm respectively. The

thicknesses of the composite laminas and epoxy are the same as those of the analytical

model in last section. Infinite elements are defined at the bottom boundary to eliminate

reflection.

Figure 3-6. Sketch of the FE model for guided wave excitation and propagation in a rotor blade trailing edge section (a composite skin/epoxy/Nomex substrate structure).

To simulate a comb transducer, normal surface tractions are applied on four

rectangular fingers on the top surface. The distance between two neighbored fingers is

half wavelength of the interested mode. The input signal is a 5 cycle Hanning-windowed

pulse centered at 300kHz. Nodal displacements are recorded at the receiver. The

delamination is simulated by removing elements in the interested region of the epoxy

layer.

45

3.2.1 SAW generation and propagation

In this section, a FE model simulates SAW generation and propagation in both

intact and delaminated structures of a composite laminate on a Nomex substrate. Table

3-3 lists parameters used in the FE model.

Table 3-3. Parameters of the FE model for surface acoustic wave (SAW) excitation and reception.

Structure Lay up sequence of skin [45/-45/(0)3/-45/45] Length (mm) 150 Width (mm) 15

Thickness of skin (mm) 0.9525 Thickness of glue (mm) 0.0635 Thickness of honeycomb

(mm) 9.6745

Thickness of infinite element (mm)

10.6270

Loading pattern Transducer element number

4

Loading element width (mm)

2.0365

Loading element length (mm)

15

Loading direction Surface traction at the edge of the loading elements in ±x3

Excitation signal

Center frequency (MHz) 0.3 Number of cycles 5

Signal window Hanning Finite element

mesh Element size (mm) 0.2037

Element type 8 node brick element

Through transmission waves are depicted in Figure 3-7. The blue and red lines

represent intact and defected structures, respectively. Only in-plane displacement, U1, is

plotted because the piezoelectric disks in our experiment work at radial vibration mode.

For SAW, the amplitude of through transmission waves is attenuated by the defect, as

46

shown in Figure 3-7 at around 55µs. The following wave package at 70µs is caused by

scattering from ends of the delamination.

Figure 3-7. Comparison of defected (red) and baseline (blue) signals for through transmission SAWs at 300kHz.

The FE simulation of particle displacements through thickness is plotted in Figure

3-8. Generally, it shows good agreement with the GMM results in Figure 3-5 (a). The

values of U2 and U3 in the FE modeling are slightly different from those from the GMM,

which is due to reflections from boundaries in the width direction of the 3D FE model.

47

Figure 3-8. Wave structures of mode one at 300kHz calculated from the FEM.

3.2.2 Leaky wave generation and propagation

In this section, a FE model simulates leaky wave excitation and propagation in

both intact and delaminated structures of a composite laminate on a Nomex substrate.

Table 3-3 lists parameters used in the FE model. The distance between two neighbored

fingers is 6.5mm, which is half wavelength of mode 5 at 300kHz.

48

Table 3-4. Parameters of the FE model for the fifth wave mode (Leaky wave) excitation and reception.

Structure Lay up sequence of skin [45/-45/(0)3/-45/45] Length (mm) 300 Width (mm) 30

Thickness of skin (mm) 0.9525 Thickness of glue (mm) 0.0635 Thickness of honeycomb

(mm) 32.3128

Thickness of infinite element (mm)

33.2652

Loading pattern Transducer element number 4 Loading element width (mm) 6.5641 Loading element length (mm) 30

Loading direction Evenly distributed surface traction in x1

Excitation signal Center frequency (MHz) 0.3 Number of cycles 5

Signal window Hanning Finite element

mesh Element size (mm) 0.6564

Element type 8 node brick element

Figure 3-9 depicts through transmission signals for both intact (blue) and

debonding (red) structures. Although the loading is designed for leaky wave (mode 5)

excitation, the SAW is also actuated because of source influence. Comparison shows that

the leaky wave intensity dramatically increases in the defected structure because there is

no energy loss to the substrate in the delaminated region. This feature can be used for

delamination monitoring and characterization.

49

Figure 3-9. Comparison of defected (red) and baseline (blue) signals for through

transmission leaky waves at 300kHz.

Figure 3-10 plots time-space spectra of in-plane displacement u1, for both intact (a)

and delaminated structures (b). The horizontal and vertical axes represent position and

time respectively. Both the leaky wave and the SAW are excited. In the damaged

structure, the disbond prevents energy loss so the leaky wave propagates further than its

counterpart in the intact case. On the other hand, the SAW is not obviously affected by

the defect.

50

(a)

(b)

Figure 3-10. Time-space spectra of through transmission leaky wave and SAW in (a) an intact structure and (b) a debonding structure..

51

3.3 Skin and honeycomb disbond measurement

The test bed is a helicopter rotor blade section with a composite skin/epoxy

glue/Nomex honeycomb structure. PZT disks were bonded on the top surface and

numbered from 1 to 10, as shown in Figure 3-11 and Figure 3-12. The thickness of each

disk is 0.4mm and the diameter is 6.3mm. The distance between two nearby transducers

is 12.7mm. Transducer No 1 was the exciter and all others were receivers.

In the experiment, a Matec Explorer 2 NDT workstation excited and recorded

through transmission signals as baselines. All signals were centered at 300kHz. After that,

disbond was made and the measurement was repeated. Figure 3-13 indicates the

production of the defect. The length of the disbond increased from 0.5 inch to 2.5 inches,

step by step as shown in Figure 3-12.

Figure 3-11. A helicopter rotor blade section with surface mounted PZT wafers.

52

(g)

Figure 3-12. Sketch of the perfect bonding section (a), the section with a 0.5 inch disbond (b), 1 inch disbond (c), 1.5 inch disbond (d), 2 inch disbond (e), 2.5 inch disbond (f),and the color bar (g).

(a) (b)

Figure 3-13. (a) Producing a disbond with a knife. (b) Delamination between the composite skin and the Nomex honeycomb.

4.5 inch12345678910

4.5 inch12345678910

0.5 inch

(a) (b) 4.5 inch

12345678910

1 inch

4.5 inch12345678910

1.5 inch

(c) (d) 4.5 inch

12345678910

2 inch

4.5 inch12345678910

2.5 inch

(e) (f) TransducerSkinHoneycombGlueDebonding

53

Figure 3-14 presents through transmission signals collected with the sensor pair 1-

10. As we expected, the signal amplitude of the damaged structure is greater than that of

the baseline. The through transmission energy was calculated as an integral of the

squared signal amplitudes, and then normalized to the baseline. Figure 3-15 plots the

curves of normalized energy received at transducers No 2, 5 and 10. Generally, when the

disbond is located between the transmitter and the receiver, the through transmission

energy monotonously increases with the defect length; if the disbond is outside the

transmitter – receiver pair, the through transmission energy is not affected by the damage.

Figure 3-14. Time domain representation of baseline and damaged signal at 300kHz. A 2.5 inches defect is located between the transmitter and the receiver.

54

Figure 3-15. Normalized through transmission energy with transducer No. 1 as the actuator and 2, 5, 10 as receivers.

3.4 Summary

The global matrix method was used to study guided wave propagation in a

composite/half-space structure. Solutions were obtained for both leaky waves and SAW.

It is proved that a leaky wave is suitable for detecting subsurface damage, such as a skin-

substrate disbond. The imaginary part of wave number, representing attenuation, is the

key factor for mode selection and transducer design.

A FE model simulates guided waves interaction with disbonds in a laminated

half-space structure. An experiment was conducted on a composite rotor blade section,

which validated the analytical and numerical calculations. Since the through transmission

0 0.5 1 1.5 2 2.5

1

1.5

2

2.5

Debonding length (inch)

Nor

mal

ized

tran

s-th

roug

h en

ergy

Through transmission energy normalized to baseline signal energy

Transducer No. 2Transducer No. 5Transducer No. 10

55

energy monotonously increases with the disbond length, leaky waves can potentially be

applied to characterize damages, and to predict the life of the structure.

Chapter 4

Guided Wave Nondestructive Testing for Delaminations in Hybrid Laminates

It was shown in Chapter 3 that the skin/core disbond can be measured by leaky

guided waves. This technique is limited to the skin/half space structure. A more common

problem is to detect delaminations between plies of composite laminates. In this chapter,

the guided wave technique is employed to test delaminations in a metal/composite hybrid

plate. The wave propagation characterization is analyzed with the semi-analytical finite

element (SAFE) method. Then a guided wave mode-frequency combination is chosen

based on the wave mechanics study. The excitation of the selected guided wave mode

and its interaction with defects in the hybrid plate are simulated with FE modeling.

Numerical simulation verifies the high sensitivity of the selected mode-frequency

combination to the delamination defects. A short range guided wave scan technique is

developed on the basis of wave mechanics studies and finite element analyses. Excellent

delamination detection and imaging characterization results are obtained with the guided

wave scan technique, which demonstrate the feasibility of detecting delamination defects

using guided waves.

4.1 Guided wave dispersion curves and wave structures analysis

The specimen is a hybrid laminate consisting of metal layers and fiber-reinforced

epoxy prepreg developed by Alcoa (Liu, 2006). The materials and layups are listed in

57

Table 4-1 and illustrated in Figure 4-1. As shown, the sample consists of aluminum

sheets, bondpreg layers and a Fiber Metal Laminate (FML) sheet. The bondpreg layer is a

sandwich layer consisting of 2 adhesive layers with 2 S2 glass layers in the middle. The 2

S2 glass layers are unidirectional glass composite layers and are considered as one

transversely isotropic layer in guided wave mechanics studies. The FML layer has both

aluminum and glass layers. The configuration of the FML sheet is shown in Figure 4-1,

too. The total thickness of the laminate plates is 10.8196mm. The elastic constants of the

S-Glass/Epoxy used in the guided wave mechanics studies are E1 = 45GPa, E2 =

E3=11GPa, G12 = G13 = 4.5GPa, G23 = 4.26GPa, υ12 = υ13= υ23 = 0.29. The density ρ = 2.0

g/cm3. The elastic constants and density of epoxy are E = 4.46GPa, υ=0.35, ρ = 1.52

g/cm3. The elastic constants and density of aluminum are E = 71.4GPa, υ=0.34, ρ =

2.7g/cm3.

The sample hybrid laminate plate has two delamination defects that are simulated by

inserting Teflon tapes into the plate during fabrication. One defect is located between the

first and second aluminum layers, i.e., within the first bondpreg. The other one is located

between the second aluminum layer and the FML layer (layers 2 and 3 in Table 4-1), i.e.,

within the second bondpreg. The sketch provided by Alcoa shows defect locations in the

plate (Figure 4-2).

58

Table 4-1. Materials and layups of a sample Alcoa Hybrid Structural Laminate.

Material Thickness (mm) Special Layers Material Thickness (mm)

Layer 1 Al 1.6 Epoxy 0.127 Bondpreg 0.508 Bondpreg S-Glass 0.254 Layer 2 Al 1.6 Epoxy 0.127 Bondpreg 0.508 Al 0.4064 Layer 3 FML 2.3876 S-Glass 0.254 Layer 4 Al 1.6 Al 0.4064 Bondpreg 0.508 FML S-Glass 0.254 Layer 5 Al 1.6 Al 0.4064 S-Glass 0.254

Al 0.4064

Figure 4-1. Illustration of the materials and layups of a sample Alcoa Hybrid Structural Laminate.

59

Figure 4-2. Defects in the sample hybrid laminate plate. One defect is located between the first and second aluminum layers. The other one is between the second aluminum layer and the FML layer

To obtain the guided wave dispersion curves is the prerequisite for mode selection

and transducer design. As introduced in Chapter 2, the semi-analytical finite element

(SAFE) method is more efficient than the global matrix method (GMM) for this

multilayered waveguide, therefore the dispersion relationships and wave structures in this

chapter are obtained with the SAFE method. Figure 4-3 (a) displays the phase velocity

dispersion curves for waves propagating along fiber direction of the S2 glass layers. All

propagating modes below 600kHz are plotted. In this direction, Lamb waves and SH

(Shear Horizontal) waves are decoupled. Figure 4-3 (b) displays the corresponding group

velocity dispersion curves. The first several Lamb waves and SH waves are labeled in

Figure 4-3.

60

(a)

(b)

Figure 4-3. Phase velocity (a) and Group velocity (b) dispersion curves for guided waves propagation in 0° direction of the hybrid laminate plates.

61

It is important to study the wave structure of each mode in detail for the NDT of a

particular defect. Some of the modes focus energy at the top and bottom surfaces, which

are sensitive to surface damages. Some other modes, on the contrary, are suitable for

embedded damage evaluation. The in-plane particle displacements are usually sensitive

to delamination defects since they are generally discontinued at delamination regions.

The out-of-plane displacements, however, could be continued at the crack region,

especially when the top and bottom surfaces of the defect contact with each other. In this

case, the guided wave with out-of-plane vibration in domination will pass through the

defect without much scattering.

Since the target defects are delaminations, we should pay attention to guided wave

modes with high in-plane displacements in the interested region (the first and second

bondpreg layers). The following are two examples illustrating the displacement based

mode selection process. The first one is the A2 mode at 450KHz. Its nodal displacements

are plotted in Figure 4-4. All interfaces are illustrated with vertical black lines. Here, Ux,

Uy, Uz represents displacements in the wave propagation direction, shear horizontal

direction and normal direction respectively. The in-plane displacement, Ux, is maximal at

the first and second bondpreg layers. Therefore it can be employed to measure the defects

in the hybrid plate. The other example is the SH0 mode at 450kHz. There is only in-plane

displacement, Uy, for this wave mode, as shown in Figure 4-5. Since the maximum

values appear in the middle, this mode can be used to detect delaminations in the FML

sheet.

62

Figure 4-4. Wave structures of A2 mode at 450kHz. Note the high in-plane displacements at the first and second bondpreg layers.

Figure 4-5. Wave structures of SH0 mode at 450kHz. Note the high in-plane displacements at middle bondpreg layers.

63

4.2 Finite element modeling results

To verify the guided wave mode selection criteria introduced in the last section,

numerical models were developed to simulate guided wave excitation and scattering with

defects. Only Lamb waves were studied in this research because of their better

excitability (with piezoelectric transducers and couplant) compared with SH waves. 2-D

FE models were built to improve the computation times.

The first step is to validate the wave propagation characteristics from the SAFE

method. The FE model is a 23-layer laminate plate in the x-z plane, as shown in Figure

4-6. The thickness and material properties of each layer are the same as those used in the

SAFE calculation. The length of the model is 200mm which is much larger than the

wavelength. The wave structures of mode A2 at 450kHz were applied at the left side

boundary to generate guided waves. Figure 4-6 (a) and (b) indicate the in-plane

displacement, Ux, and out-of-plane displacement, Uz, respectively. The red and blue

colors represent positive and negative values. Comparing Figure 4-6 with the

displacements from the SAFE method (Figure 4-4) shows very good agreement. This

simulation demonstrates the feasibility to excite a single mode in a FE model, and

verifies the SAFE results.

64

(a)

(b)

Figure 4-6. Hybrid analytical FE analysis results on the particle displacement distributions of the A2 mode at 450kHz, (a) in-plane displacement field Ux, (b) out-of-plane displacement field Uz.

Figure 4-6 displays guided waves excitation by applying suitable wave structures on

the boundary, which is not feasible in practical applications. For realistic measurements,

mode selection is usually carried out by tuning frequency and the wavelength in comb

transducers, or adjusting frequency and the incident angle in angle beam transducers. Our

z

x

z

x

65

experiments were conducted with angle beam transducers because of their convenience

of operation.

In the following FE model, pressure was applied on an angle wedge to simulate

guided wave excitation with an angle beam transducer. The geometry of the angle wedge

and in-plane displacements outputs are illustrated in Figure 4-7. The selected wave mode

is A2 at 450kHz. Compared to the in-plane wave structure calculated with the SAFE

method, shown to the right of the FE model, one can see a good agreement between the

FE simulation and the SAFE method. This modeling verifies the feasibility of generating

the interested wave mode with an angle beam transducer. It also shows that no unwanted

mode is excited with this incident angle/frequency combination. This is because no other

wave mode exists near the selected region in the phase velocity dispersion curves, as

shown in Figure 4-3 (a).

66

Figure 4-7. Finite element simulation for guided wave mode selection with an angle beam transducer.

The interactions of the selected guided wave mode with a delamination defect were

also investigated using FE analysis. The geometry and analysis result of the FE model is

given in Figure 4-8. As shown, a portion (30mm long) of the third epoxy layer from the

top surface of the plate was removed to simulate a delamination defect in the first

bondpreg layer. It is demonstrated that guided wave mode conversion occurs due to the

delamination defect. There are also multiple reflections of guided wave energies in the

delamination region. The guided wave mode conversion and multiple reflections caused

by the delamination defects can be considered as useful features for delamination

detections. Since the receptiblity of the angle beam transducer to each mode is different,

the mode conversion can be detected via through transmission measurements.

67

Figure 4-8. Finite element simulation of. Guided wave interaction with a simulated delamination defect close to the top surface.

In the next FE model, the transducer was mounted on the top but the damage was

near the bottom (Figure 4-9). There was obvious mode conversion just like that in the last

example, which verifies the feasibility of guided waves testing for deeply embedded

damages.

68

Figure 4-9. Finite element simulation of Guided wave interaction with a simulated delamination defect close to the bottom surface.

4.3 Experimental results

Before the guided wave experiment, we conducted C-Scan tests on the plate as a

reference. Figure 4-10 displays the experiment setup of C-Scan. Both the specimen and

the piezoelectric transducer were merged inside water. The ultrasonic transducer was

fixed at one end of a bar. The other end of the bar was attached to a frame. The

movement of the transducer was driven by a motor and controlled by computer. As a

pulse-echo experiment, the transducer played both the transmitter and the receiver. The

focal distance of the transducer was fixed. The focal point could be adjusted with the

height of the transducer.

69

Figure 4-10. Sketch of ultrasonic C-Scan process.

In the C-scan tests, due to the mismatches in acoustical impedances of the

different layers of the specimen, multiple reflections were observed in the ultrasonic

signals. In order to obtain the images of the cross-sections at different depths, the signal

gate was adjusted to different time-of-flight (TOF) ranges. However, because of the

strong reflections at the interfaces between different layers, the amount of ultrasonic

energy that penetrated several layers of the plates was very small compared to the input

energy. As a result, the C-scan tests were not able to exam the laminate layers that were

several layers below the plate surface, even though the signal gate was carefully chosen

in the tests.

A C-scan image of the sample hybrid laminate plate is presented in Figure 4-11.

The sketch of the defect locations provided by Alcoa is also depicted for comparison. As

shown, the defect 1 in the first bondpreg layer is imaged very well. The bulk wave C-scan,

however, misses the defect 2 located within the second bondpreg layer. Apparently, the

70

ultrasonic C-scan technique becomes problematic for defect detections in the hybrid

laminates.

(a) (b)

Figure 4-11. (a) Sketch of the defect locations in the sample hybrid laminate plate, (b) C-scan image of the laminate plate shows only the defect located within the first bondpreg layer.

Initial wave propagation tests were conducted on the hybrid plate with a pair of angle

beam transducers. Through-transmission signals were monitored and recorded with a

toneburst function generator test system. The transducer configurations and the

corresponding recorded guided wave signals are shown in Figure 4-12. In the first test,

both the actuator and the receiver were placed on the same side of the specimen. In the

second measurement, the actuator and the receiver were on different sides. Clear guided

wave packages were observed in the through transmission signals. Placing the receiver

angle beam transducer on the top or bottom surface of the plate did not affect the received

guided wave signal very much. This verified the symmetry in the wave structure as

71

shown in Figure 4-4. The experiment also showed that the guided wave energy was

distributed across the whole thickness of the plate. Unlike that encountered in the bulk

wave C-scan test where deep penetration of the wave energy in the multilayer laminate

was nearly impossible, in a guided wave test, wave energy can cover the whole plate

thickness. It is thus capable to detect the defects located many layers away from the

inspection surface.

Figure 4-12. Through-transmission guided wave signals recorded using three different transducer configurations.

For comparison, normal beam transducers were employed to generate and receive

guided waves. The signal amplitude is much lower than those from angle beam

transducers since the normal beam transducer doesn’t have mode selection ability. This

measurement proved the efficiency on mode selection of the angle beam transducers.

0 100 200-100

-50

0

50

100

Am

plitu

de

Time (us)0 100 200

-100

-50

0

50

100

Time (us)0 100 200

-100

-50

0

50

100

Time (us)

72

Preliminary guided wave measurements were carried out to evaluate the sensitivity

of the selected wave mode to delamination defects. The guided wave propagation

direction was fixed at 0° direction of the S2-glass layer. A pair of angle beam transducers

was used for through transmission measurements. The recorded guided wave signals as

well as the corresponding transducer positions are shown in Figure 4-13 (a). A sketch of

the defect locations is shown in Figure 4-13 (b) for comparison. Amplitude drops and

change of wave package shapes were observed when the transducer pairs were over the

defected region, which was caused by the guided wave mode conversion and the multiple

reflections at the delaminations. This experiment verified that the selected 450kHz A2

mode is sensitive to both top and bottom defects in the sample hybrid laminate plate.

To image the defects in the hybrid plate, we conducted a short distance guided wave

scan with the selected wave mode and angle beam transducers. Figure 4-14 depicts the

experimental setup and the scanning route. During the scan, the distance between the

transmitter and the receiver was fixed at 3cm. The step increment for both directions was

1cm. For the 12”×12” plate, 17 steps were counted along the wave propagation direction

and 27 steps were counted in the normal direction. Therefore, 459 waveforms were

collected for the whole scan.

73

(a) (b)

Figure 4-13. Guided wave scan results for the sample hybrid laminate plate. (a) Guided wave signals and the corresponding transducer positions. (b) Defect locations.

Figure 4-14. Short distance guided wave scan setup and the scan route.

74

The signal difference coefficient (SDC) has been presented by J L Rose’s group as

an ultrasonic signal feature for tomographic image reconstruction (Gao, Yan, Rose, Zhao,

Kwan, & Agarwala, 2005; Royer, Zhao, Owens, & Rose, 2007).

The correlation coefficient between two sets of data, sj and sk, is

( )kj ss

kj ssCovσσ

ρ,

= 4.1

Where the covariance, Cov, is

( ) ( )( ) ( )( )

Nstssts

ssCovN

i kikjijkj

∑ −−=, 4.2

and the standard deviations are

( )( ) ( )( )

22

∑∑ −=−=N

ikiks

N

ijijs stssts

kjσσ 4.3

Then, the signal difference coefficient is expressed as

ρ−=1SDC 4.4

The calculation of SDC has been proven as a good method for capturing the guided

wave mode conversions and the multiple reflections in the defect regions. Selecting SDC

as the tomographic feature assumes that the characteristics of the received waves do not

change if the structure is not modified between the transducers. This approach works very

well with embedded sensors. However, in our measurements the transducers were not

fixed. A small change in the distance between the transducers would cause phase

variation of time domain signals and the SDC increment. Therefore, a Fourier transform

was conducted to the data before the SDC calculation.

75

At first, we applied FFT to all the data from the guided wave scan. In total, 459

frequency domain signals were obtained. The average value of these data was selected as

the reference signal. After obtaining the reference signal, we compared every frequency

domain signal with the reference and got a 27×17 SDC matrix. Since the wedge size is

4cm×6cm each, and the distance between two wedges is 3cm (Figure 4-14), every point

in the SDC matrix corresponds to a 4cm×15cm area on the surface. We projected the

SDC matrix onto the surface and calculated average values in the overlapped region.

Finally, a scanning image was generated for the sample hybrid laminate plate, as shown

in Figure 4-15 (a). With the same procedure, another image was obtained by scanning the

sample hybrid laminate from the other surface. The result is given in Figure 4-15 (b).

Both defects were well detected and located in the scanning images.

(a) (b)

Figure 4-15. Defect images obtained using the short distance guided wave scans (a) from the top surface, (b) from the bottom surface.

76

Compared to the bulk wave C-scan result shown in Figure 4-11 (b), the guided wave

method is able to find both defects, including that located further from the test surface.

The scanning images obtained from the two surfaces of the plate are quite similar. This is

due to the symmetry in wave structures for the guided waves in the hybrid laminates

whose layup sequence is symmetric. The small discrepancy between Figure 4-15 (a) and

(b) is caused by the inconsistency in transducer coupling conditions. This experiment

demonstrates that defects located at any depth in a laminate can be detected with suitable

mode selection.

Figure 4-15 show some noises close to top and bottom edges, which is caused by

boundary reflections. To reduce the effect of boundary, other launch directions, such as

90 degree, should be studied. It is also useful to improve the resolution by refining the

scan step increment.

In through transmission measurement, damage may happen at any point between

the transmitter and the receiver. Therefore, the resolution along wave propagation

direction is much lower than that in the normal direction. To increase the resolution in

wave propagation direction requires less distance between the actuator and the receiver

and more scan steps.

4.4 Summary

In this chapter, a feature based guided wave inspection method is demonstrated on

a metal/composite hybrid plate. The guided wave propagation in the hybrid laminate was

analyzed with the SAFE method. A qualitative mode selecting method was developed

77

based on the wave structure analysis. FE models were built to simulate selected wave

mode excitation and scattering with the delamination defects. A short distance guided

wave scanning method was developed to provide C-scan like images using the guided

wave method. The guided wave method has proven capable of detecting delaminations

located at any depth of the laminate plates which overcomes the limitation of the bulk

wave C-scan method.

Chapter 5

Theory of Guided Wave Scattering at Defects

A global-local (GL) method is introduced in this chapter to quantitatively study

guided wave scattering at defects in isotropic or anisotropic plates. Unlike the traditional

plane strain theory, which doesn’t consider shear horizontal displacement and stress, the

method presented in this chapter includes displacements in all three directions. The only

difference between this method and a total 3-D model is that the waveguide must be

uniform in the transverse direction, as shown in Figure 5-1.

Figure 5-1. Guided waves in a transversally uniform structure. x, y and z are along wave propagation direction, shear horizontal direction and plane normal direction respectively.

The theory of the GL method will be illustrated in the following sections. To

verify the GL theory, guided wave transmission and reflection are analyzed on an

aluminum plate with a surface notch. Both the GL method and the commercial software

ABAQUS are used to calculate the scattering field. The comparison shows very good

79

agreements. The next example is a notched steel plate. Different defect depths and widths

are evaluated with the GL method. The results match very well with simulated and

experimental data.

5.1 Finite element theory for guided waves in a transversally uniform structure

Section 2.2 presents the SAFE method, which employs 1D finite elements

through the plate thickness, with analytical solutions along the wave propagation

direction. The waveguide is assumed to be homogeneous in the plane parallel to the plate

surfaces. This assumption is not satisfied for a structure with a defect, e.g., a plate with

delaminations. Therefore, the 2-D FE method will be applied to address the problem of

wave scattering at defects.

Figure 5-2 illustrates a quadratic 2-D element in Cartesian coordinates. Here x

and y are directions of wave propagation and plate thickness. ξ and η represent local

coordinates. Each element has nine nodes, distributed at points where ξ and η equal to -1,

0 and 1. The variables, such as displacement, stress and strain are accurate on these nodes

and calculated as interpolations of nodal values elsewhere.

80

Figure 5-2. Nine-node plane Lagrange element in Cartesian coordinates.

The virtual work principle for this 2-D model is the same as that of the 1D SAFE

method (Eq. 2.13). But the shape functions and strain-displacement relation are different.

The displacement vector at any position inside an element is expressed in Eq. 5.1. Note

that it is a function of frequency ω, but does not include a certain wave number k like its

counterpart in homogenous media in Eq. 2.14. In fact the values of wave numbers may

change due to abnormity along the wave propagation direction in the waveguide.

5.1

0 00 00 0

0 00 00 0

…0 0

0 00 0

5.2

81

14

1 112 4

14 1 1

12 4

14 1 1

12 4

14 1 1

12 4

12

1 12

12 1 1 212 1 1 212

1 12

1 1

5.3

Here N is the shape function matrix. U is a vector of nodal displacements.

5.4

Uij is the displacement of node j in i direction. The strain-displacement relation is

defined as

exp

, ,

1 0 00 0 00 0 00 0 00 0 10 1 0

0 0 00 0 00 0 10 1 01 0 00 0 0

5.5

The stress vector is

5.6

where C is an elastic coefficient matrix. The external traction vector t is

82

5.7

where T is the nodal external traction vector. Substituting Eq. 5.1, 5.5, 5.6 and 5.7

into Eq. 2.13 gives

’’

M d

d

5.8

Where Γ’ stands for the boundary of the 2-D element. The external force, f, is

expressed as the function of material density, stiffness, frequency and nodal

displacements. The mass matrix, M, and stiffness matrix, K11, will be integrated with the

normal mode expansion technique to study guided wave scattering.

5.2 The Global-Local (GL) method

The GL method was originally developed by Goetschel et al and applied by

Srivastava et al for simulation of Lamb wave scattering at defects (Srivastava, Bartoli,

Coccia, & Scalea, 2008; Goetschel, Dong, & Muki, 1982). We expanded the GL method

to general guided waves in isotropic/anisotropic plates. Figure 5-3 illustrates a wave

scattering problem of arbitrary-shaped defects surrounded by a boundary. Here x is the

wave propagation direction. y and z are transverse and normal directions of the plate.

Inside the boundary is the local region, discretized into finite elements; outside the

boundary is the global region, described by the linear combination of plate waves

83

calculated from the SAFE method. Continuity of displacements and tractions are satisfied

on the boundary. The following gives details of the GL method.

Figure 5-3. Sketch of the GL model in a wave scattering problem. The red lines indicate boundaries of the local and the global regions.

For a straight-crested plane wave propagating along the x direction, the

displacements on the left side boundary can be expressed as the sum of incident and

scattering fields:

5.9

Here uincident and uscattered are vectors of displacements corresponding to the

incident wave and reflected waves on the left boundary. Ain is the incident wave

amplitude. Aj- represents the reflected jth wave mode amplitude. N is the total number of

all possible waves in the structure, including propagating and non-propagating modes.

kin+ and kj

- are wave numbers of the incident and the jth reflected wave mode. x- is the x

coordinate of the left boundary. Фin+ and Фj

+ are mode shapes of incident and positive

travelling through transmission waves, expressed as nodal displacements at all three

84

directions. The mode shape matrix, G-, and mode amplitude coefficient vector, D-, for left

travelling waves are expressed as:

5.10

5.11

Here, the wave number and nodal displacements of each wave mode are

calculated from the SAFE method. Similarly, the displacement vector on the right

boundary is expressed as:

5.12

where Aj+ and kj

+ are amplitude and wave number of the jth transmitted wave

mode. x+ is the x coordinate of the right boundary. The mode shape matrix, G+, and mode

amplitude coefficient vector, D+, for left travelling waves are

5.13

5.14

The nodal forces on left and right boundaries are:

exp 5.15

5.16

Where F+ and F—are nodal force matrices corresponding to D+ and D--.

According to the virtual work principle in Eq. 2.13, the energy increment inside

the boundary is equal to the work of external forces:

85

δ δ 5.17

5.18

5.19

Here, qI and qB are nodal displacement vectors of interior and boundary nodes

respectively. KI and MI are global stiffness and mass matrices illustrated in Eq. 5.8. ω is

the circular frequency. PB is the external force vector applied on the boundary, since there

is no external force inside the local region. Substituting Eq. 5.18 and Eq. 5.19 into Eq.

5.17 yields:

5.20

5.21

Considering Eq. 5.9 to Eq. 5.16, Eq. 5.20 could be written into the following form:

5.22

With

86

0

0

00

00

5.23

If the Degree of Freedom (number of nodes×3) on left and right boundaries is

larger than the total number of scattering wave modes, Eq. 5.22 is an overdetermined

system of linear equations, which could be solved with the least squares method:

T T 5.24

Then the reflection and transmission coefficients of amplitude are:

⁄⁄ 5.25

where i and j correspond to incident and scattering modes respectively. The

normalized reflected and transmitted energy is expressed as:

I ⁄

I ⁄ 5.26

with 12

H· d 5.27

87

5.3 Validation of the GL method with 2D FE simulation

Frequency domain analysis is conducted on guided wave scattering in an Al plate

with both the GL model and the commercial FE software ABAQUS. The objective is to

verify the GL method with ABAQUS. A steady state dynamic analysis was performed,

which provided the steady-state amplitude and phase of the response of a system due to

harmonic excitation at a given frequency. Figure 5-4 illustrates the 2-D plane strain

ABAQUS model of a 1.6mm-thick Al plate. Nodal forces at 500kHz were applied in the

loading region of a straight line through the plate thickness to generate S0 mode. Nodal

displacements were recorded on the plate top surface along the wave propagating path.

To eliminate boundary reflection, two damping regions were put on both sides. The

damping coefficient of each region is a quadratic equation, decreasing with the distance

to the boundary, as plotted in Figure 5-5.

Figure 5-4. FE model for S0 mode guided wave generation and propagation in a 1.6mm-thick Al plate.

88

Figure 5-5. Damping coefficient of the absorbing layers.

The silent boundaries were tested via an intact model. Figure 5-6 displays the

wave number spectrum of transmitted and reflected guided waves, calculated from spatial

domain FFT of the received signals. It is shown that the amplitude of reflection is

reduced to 20dB by employing damping layers.

Figure 5-6. Wave number spectrum of the transmitted and reflected guided waves. The boundary reflection is reduced to 20dB.

89

Figure 5-7 displays the in-plane and out-of-plane displacements profiles for the S0

mode at 500kHz calculated from ABAQUS. Same results were obtained with the GL

model, shown in Figure 5-8.

(a)

(b)

Figure 5-7. Profiles of in-plane (a) and out-of-plane (b) displacements in an intact Al plate calculated with ABAQUS. The incident wave is S0 mode at 500kHz.

90

(a)

(b)

Figure 5-8. Profiles of in-plane (a) and out-of-plane (b) displacements in an intact Al plate calculated with the GL method. The incident wave is S0 mode at 500kHz.

Then both ABAQUS and the GL methods were employed to simulate S0 wave

scattering in a notched plate. The notch size is 2mm by 0.8mm, located on the top surface.

Figure 5-9 displays the steady state dynamic analysis results in ABAQUS. Note that the

S0 mode converts to A0 while passing through the notch. The GL results are displayed in

Figure 5-10. Comparison of Figure 5-9 and Figure 5-10 shows very good agreement.

x (mm)

z (m

m)

Ux

0 2 4 6 8 10 12 14 160

0.51

1.5

-0.200.2

x (mm)

z (m

m)

Uz

0 2 4 6 8 10 12 14 160

0.51

1.5

-0.0500.05

91

(a)

(b)

Figure 5-9. Profiles of in-plane (a) and out-of-plane (b) displacements in a damaged Al plate calculated with ABAQUS. The incident wave is S0 mode at 500kHz.

(a)

(b)

Figure 5-10. Profiles of in-plane (a) and out-of-plane (b) displacements in a damaged Al plate calculated with the GL method. The incident wave is S0 mode at 500kHz.

92

5.4 Validation with experiments and finite element results in literature

In this section, the GL results are compared with the experimental data and the FE

simulation in the literature. Reflection coefficients of A0 wave from a rectangular notch in

a steel plate have been measured by M. J. S. Lowe, etc (Lowe, Cawley, Kao, & Diligent,

2002). The specimen is a 3mm-thick steel plate. Figure 5-11 displays the phase velocity

and group velocity dispersion curves with first three modes noted in the legend. Only the

A0 mode was measured at frequencies from 350 to 550kHz. In this region, there are no

higher order modes, such as A1, S1 and SH1, etc. Existence of high order modes may

increase the experiment difficulty.

(a) (b)

Figure 5-11. Phase velocity (a) and group velocity (b) dispersion curves for guided waves in a steel plate.

In the first measurement, the notch depth was fixed at 0.5mm and the width varied

from 0.25mm to 5.0mm. Reflection coefficients of the A0 mode were measured for each

width. Time domain signals were generated and collected and then transferred to

93

frequency spectra via FFT. The frequency range in the experiments and FE simulation is

from 420 to 450kHz.

For comparison, the GL method was applied on the same structure. In the GL

model, the reflection coefficients were directly calculated in the frequency domain at

450kHz. All the tests and simulation results are drawn in Figure 5-12. The curve from the

GL method matches very well with the FE predictions and the experimental data. It is

also shown that the reflection coefficient periodically changes with the ratio of notch

width to input wavelength, therefore we need to select suitable frequencies besides wave

modes in guided wave testing.

Figure 5-13 displays the experimental and simulation results for a 0.5mm wide

notch with the depth varies from 0.25mm to 2.0mm. Again, results of the GL model

agree with those from the FE simulations and experiments. Figure 5-14 - Figure 5-17

show similar results for different notch sizes in a frequency range from 350 to 550kHz.

The experimental data show multiple values at some frequency points because they were

converted from time-domain signals with overlapped frequency regions. Generally, the

GL results show good agreement with the FE predictions and same trends with

experiments.

94

Figure 5-12. A0 reflection coefficient from a 0.5mm deep notch in a 3mm thick steel plate. The notch width varies from 0.25 mm to 5.0mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are made within frequency range of 420 to 480kHz. The GL method prediction is at 450kHz.

Figure 5-13. A0 reflection coefficient from a 0.5mm wide notch in a 3mm thick steel plate. The notch depth varies from 0.25 mm to 2.0mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are made within frequency range of 420 to 480kHz. The GL method prediction is at 450kHz.

95

Figure 5-14. A0 reflection coefficient from a 0.5mm deep notch in a 3mm thick steel plate. The notch widths are 3.0mm, 4.0mm and 5.0mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are compared with the GL method prediction.

Figure 5-15. A0 reflection coefficient from a 0.5mm deep notch in a 3mm thick steel plate. The notch widths are 0.5mm, 1.0mm and 2.0mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are compared with the GL method prediction.

96

Figure 5-16. A0 reflection coefficient from a 0.5mm wide notch in a 3mm thick steel plate. The notch depths are 0.5mm and 1.5mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are compared with the GL method prediction.

Figure 5-17. A0 reflection coefficient from a 0.5mm wide notch in a 3mm thick steel plate. The notch depths are 1.0mm and 2.0mm. Experiment and FE results (Lowe, Cawley, Kao, & Diligent, 2002) are compared with the GL method prediction.

97

5.5 Summary

A Global-Local (GL) method has been developed to analyze guided wave

reflection/transmission in an anisotropic multilayer plate, based on the work of Srivastava,

A., etc (Srivastava, Bartoli, Coccia, & Scalea, 2008). The GL method was verified by

comparing with the results of finite element (FE) simulations and experiments on an

aluminum plate and a steel plate. For the aluminum case, we simulated the S0 mode Lamb

wave excitation and propagation in both intact and defected structures. The displacements

distributions were calculated with ABAQUS and the GL method. For the steel example,

the reflection coefficient of the A0 mode with different sizes of notches has been

experimentally studied by Lowe, etc and published in JASA (Lowe, Cawley, Kao, &

Diligent, 2002). We quantitatively compared the literature results with those from our GL

model. Very good agreements were obtained for all the cases.

Chapter 6

Defect Characterization in Composite Plates

Defect characterization in composites is complex due to their anisotropic nature.

The 2-D GL theory (Srivastava, Bartoli, Coccia, & Scalea, 2008) has proved to be able to

solve guided wave scattering problem in an isotropic media. However, it is not suitable

for an anisotropic waveguide where Lamb waves are coupled with SH waves. The FE

method can handle wave scattering in anisotropic structures, but it usually requires a

large dimension to eliminate boundary reflection, therefore increasing the computational

cost. Special techniques, such as the viscous damping boundary method (Castaings &

Lowe, 2008), the perfect matched layers method (Drozdz, Skelton, Craster, & Lowe,

2007), and infinite elements (Fu & Wu, 2000), have been presented to simulate “silent

boundaries” in FE models. These methods are incompatible with anisotropic media, or

constrained to low frequency and certain range of wavelength. The GL model introduced

in last chapter provides a novel solution for theoretical analysis of guided wave scattering

in isotropic structures. It is also compatible to anisotropic media. The GL method is more

efficient than the traditional FE method in terms of computation and data processing, and

performs well at high frequency and large wavelength (near the cutoff frequency).

In this chapter, the GL method is applied to a unidirectional composite plate and

compared with a 3-D ABAQUS model. After that, quantitative studies are carried out for

guided wave scattering at defects with different defect sizes and locations. The influence

of wave propagation directions and frequency are also discussed. The transmission and

99

reflection coefficients are plotted and analyzed. Based on the GL method, a new

technique is developed for quantitative damage characterization in composite plates.

Experiments are conducted on a 16-layer quasi-isotropic carbon/epoxy plate with three

artificial defects embedded at different thicknesses. With the proposed technique, all

defects can be detected. Quantitative comparison is made between the attenuation

coefficients measured from experiments and those from GL simulation. These

experiments verify the GL method on an anisotropic multilayered structure and prove the

feasibility of the guided wave damage characterization technique.

6.1 Guided wave scattering in a unidirectional composite plate

6.1.1 Verification with ABAQUS

In this section, the GL method is applied to an anisotropic composite plate. Time

domain FE analysis is conducted with ABAQUS. The results are converted into

frequency spectrum and compared with the GL model.

The waveguide is a unidirectional carbon/epoxy composite plate made of

IM7/977-3 prepreg, as introduced in Section 2.5. To verify the GL model in terms of the

coupling between longitudinal and shear displacements, we choose the wave propagation

direction as 45°. Table 2-1 lists the material properties. The density is 1.6 g/cm3, and the

plate thickness is 2.4mm. Dispersion curves and wave structures have been calculated in

section 2.5 and plotted in Figure 2-4 -Figure 2-9.

100

Figure 6-1 shows the ABAQUS model with a 1mm-wide, 0.8mm-deep surface

notch. The loading region and the receiver point are located at different sides of the

defect. Normal tractions are applied at nodes in the middle thickness of the loading

region. According to reciprocity theory (Auld, 1990), the mode excitability is

proportional to the particle velocity component in the loading direction. In 50-200kHz

range, there are three propagating modes. Only mode one has nontrivial normal

displacement (Uz) and velocity components at middle thickness, as shown in Figure 2-7.

Therefore mode two and three will not be generated in this model.

The input signal is a Hanning windowed 10-cycle pulse. Simulations were

conducted with both intact and defect models, each at seven frequency points. Table 6-1

lists the center frequencies, plate dimensions, mesh sizes and wavelengths of all

simulations. Generally the mesh size is around 1/10 of the wavelength of excited wave

mode., The plate is large enough for guided waves transmitting through the receiver (the

red point in Figure 6-1).

101

Figure 6-1. ABAQUS model for guided wave scattering in a unidirectional composite plate.

Table 6-1. Parameters of the FE models for simulation of guided wave scattering in a unidirectional composite plate

Simulation 1 2 3 4 5 6 7 Frequency (kHz) 50 75 100 125 150 175 200 Plate size (mm) 600 400 300 240 200 171 150 Mesh size (mm) 1.74 1.16 0.87 0.69 0.58 0.49 0.43 Wavelength (mm) 19.17 14.60 11.88 10.04 8.70 7.69 6.89

102

(a)

(b)

Figure 6-2.FE simulation for guided wave propagation along the 45° direction in (a) intact and (b) damaged unidirectional composite plates. The defect is a notch normal to the wave propagation direction. The incident wave is mode one.

Figure 6-2 displays the contour of normal displacements for guided wave mode

one at 45° direction. The skew angle is around 30°, which agrees well with the SAFE

result in Figure 2-6. Normal displacements (Uz) were recorded for both intact and

defected models and converted into frequency domain by FFT. The transmission

103

coefficients were calculated as the ratios of defected and intact signal amplitudes at the

center frequencies.

ABAQUS simulation results are compared with the GL model. Figure 6-3 depicts

the mesh of the GL model for guided waves scattering at a 10mm-wide, 0.8mm-deep

surface notch in a 2.4mm-thick composite plate. The local region is 80mm long, meshed

with 1mm by 0.8mm 9-node quadratic elements. The minimal wavelength is 6.89mm in

the 50-200kHz range. This mesh size guarantees at least 6 quadratic elements in one

wavelength. The distance from the left and right boundaries to the defect is 35mm, which

is larger than the maximal wavelength of mode one (19.17mm), but much less than the

maximal wavelength of mode three (150mm). This may introduce error when mode three

is dominant. To confirm the model is long enough, the total energy was calculated at all

frequencies, which shows good consistency (Figure 6-4). Therefore we keep the current

setting of mesh and model size.

Figure 6-3.Mesh of the GL model of a unidirectional composite plate with a notch at surface.

104

Figure 6-4. Through-transmission energy of guided wave mode one in a unidirectional composite plate with a notch. Solid lines: GL method; Dots: ABAQUS.

The through transmission energy was calculated from Eq. 5.26. Figure 6-4

compares the GL model and ABAQUS results for guided wave mode one through

transmission energy. The ABAQUS data were obtained from time-history analysis, and

then converted into frequency domain. The GL model directly provided the frequency

spectrum of transmission energy. The good agreement proves the ability of the GL

method for accurate simulation of guided wave scattering in an anisotropic media. The

departure of the black dot from the blue line at 125kHz may be due to errors from beam

disperse in the transverse direction in FEM simulation.

The simulation efficiency has been greatly improved by using the GL method.

According to the example in this section, the calculation speed of GL modeling is more

than 10 times faster than the FEA.

105

6.1.2 Effect of wave propagation direction to wave scattering at a surface notch

The guided wave propagation characteristics vary with the propagation direction,

as discussed in section 2.5. This section focuses on the effect of wave propagation

direction to guided wave scattering. Figure 6-5 shows the transmission and reflection

energy for mode 1-3 with a 1mm by 0.8mm surface notch in a unidirectional composite

plate. The propagation direction is 45°. In this direction the longitudinal and shear

displacements are coupled with each other, so mode conversion can be observed between

the three modes. Note that the transmission and reflection amplitudes change periodically

with the frequency, which is the same as the isotropic example in section 5.4. This is

because the incident wave is reflected back and forth between both ends of the defect.

The multi scatterings overlap and construct the final transmission and reflection waves.

For shallow surface notches/thin voids, the maximal transmission and minimal reflection

are observed when the void width is close to integer times of half wavelength, which will

be further illustrated in section 6.1.4. This feature of periodic scattering can be employed

for defect characterization in nondestructive testing. By choosing incident waves with

different wavelengths and analyzing the reflection or transmission coefficients, the defect

size can be estimated.

106

(a)

(b)

(c)

Figure 6-5. Guided wave transmission and reflection at a surface notch in a unidirectional composite plate. The incident waves are mode one (a); two (b) and three (c) in 45° direction.

107

Figure 6-6 and Figure 6-7 display similar results with guided waves propagating

along 0° and 90° directions. In these directions the Lamb waves (mode 1, mode 3) are

decoupled with the SH waves (mode 2). It is shown that the reflection of mode 1 include

mode 3, and vice versa. But mode 1 or 3 cannot be converted into mode 2, because it

doesn’t include the shear horizontal displacement component. It is also shown that the

scattering of mode 3 changes a lot from 0° (Figure 6-6 c) to 90° (Figure 6-7 c) due to its

wavelength variation.

Usually, fatigue cracks grow along the fiber direction in a unidirectional

composite plate. However, some impact induced damages may cut fibers at an angle

between 0° and 90°. Sensor optimization requires analyzing guided wave propagation

and scattering at all directions. As an example, the scattering of guided wave modes 1-3

at a surface notch is studied. Figure 6-8 and Figure 6-9 display the transmission and

reflection energy profiles for all directions at 150kHz and 250kHz respectively. The

effect of propagation direction to wave scattering is more noticeable for mode 1 and 3

than mode 2. Therefore, mode 2 is a suitable choice in guided wave measurements with

multi-propagation directions, e. g. damage inspection with a sparse circular transducer

array.

108

(a)

(b)

(c)


109

(a)

(b)

(c)


110

(a)

(b)

Figure 6-8.Transmission (a) and reflection (b) energy profiles for modes 1-3 at 150kHz in a 2.4mm-thick unidirectional composite plate with a 1mm by 0.8mm surface notch.

111

(a)

(b)

Figure 6-9.Transmission (a) and reflection (b) energy profiles for modes 1-3 at 250kHz in a 2.4mm-thick unidirectional composite plate with a 1mm by 0.8mm surface notch.

112

6.1.3 Effect of void location through the thickness

In previous sections, the defect was a rectangular-shaped material removal located

at the plate surface. This section discusses two different locations. One is embedded close

the surface (Figure 6-10), and the other is at the middle plane (Figure 6-11). The void size

is still 10mm by 0.8mm. Transmission and reflection coefficients were calculated with

the GL method for modes 1-3 from 50 to 200kHz, propagating along 45° direction.

Figure 6-10. Mesh of the GL model for a unidirectional composite plate with a void in the subsurface. Here, x is the wave propagation direction. z is through-thickness. y is the transverse direction.

113

Figure 6-11. Mesh of the GL model for a unidirectional composite plate with a void in the middle.

The results are plotted in Figure 6-12 - Figure 6-14 with the percentage of defect

width to wavelength as the horizontal axis. Different from the scattering of a surface

notch, which displays smooth periodic variation with frequency, the transmissions and

reflections from embedded defects show both periodic modulation and spikes at some

“resonance frequencies”. The positions of these spikes in frequency spectrum are related

to the defect size and the incident wavelength.

The defect location also affects positions of peaks and troughs in scattering curves,

especially for mode one (Figure 6-12). The modulation period is the same as that of a

surface notch. Note that mode 3 is much less sensitive to all three defects than modes 1

and 2, because the wavelength is too large compared with the defect size.

114

Figure 6-12. Transmission and reflection coefficients for mode 1 with defects at different locations from plate surface to mid-plane, showing periodical modulation with the ratio of crack width to input wavelength, and spikes at some “resonance frequencies”.

Figure 6-13. Transmission and reflection coefficients for mode 2 with defects at different locations from plate surface to mid-plane, showing periodical modulation with the ratio of crack width to input wavelength, and spikes at some “resonance frequencies”.

Figure 6-14locations frof crack wi

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115

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116

First, the defect width is fixed at 1mm, and the height changes from 0.05 to

2.2mm. The increment is 0.05mm in 0.05-0.75mm region, and 0.2mm in 0.8-2.2mm

region. The transmission and reflection coefficients vs. defect height are plotted in Figure

6-16 for modes 1-3 at 150kHz and 250kHz. For most modes and frequencies, the

reflection coefficients monotonously increase with the defect height. It is shown that

lower order modes are more sensitive to the defect than higher order modes, and high

frequency waves are more sensitive than their low frequency counterparts. This

phenomenon is easy to understand because the wavelengths of low order modes at high

frequencies are much less than those of high order modes at low frequencies, and the

sensitivity of ultrasonic waves is usually proportional to the ratio of the defect size to

wavelength.

In order to eliminate the influence of wavelength in the comparison, the x axis is

replaced with percentage of defect height to incident wavelength, as depicted in Figure

6-17. It is shown that mode 3 is more sensitive than mode 1 and 2 at the same wavelength

to defect height ratio. This comparison can be applied to inspection with a comb

transducer. Since the wavelength of a comb transducer is fixed, it is important to select a

guided wave mode based on its real sensitivity to the interested defect.

117

(a)

(b)

Figure 6-16. Transmission (a) and reflection (b) coefficients from a 1mm wide void in a 2.4mm thick composite plate. The horizontal axis is void height in millimeter. This shows that the reflection coefficients monotonously increase with the defect height.

118

(a)

(b)

Figure 6-17. Transmission (a) and reflection (b) coefficients from a 1mm wide void in a 2.4mm thick composite plate, showing that the sensitivity of mode 3 is higher than that of modes 1 and 2, compared at the same wavelength. The horizontal axis is void height in percent of input wavelength.

119

Then the void height is fixed, and the width varies from 0.5mm to 10mm with a

0.5mm increment. Figure 6-18 exhibits the transmission coefficients for mode 2 and 3

with a 0.1mm defect at different frequencies from 50kHz to 250kHz. Figure 6-19 and

Figure 6-20 displays transmission coefficients with 0.8mm and 2.2mm defects,

respectively. The horizontal axes indicate percentage of defect width to input wavelength.

Transmitted waves are periodically modulated by the defect and reach their

maximal values when the wavelength is equal to integer times of half void width. At the

same time, they show spikes at some “resonance void widths”. The spike location is

determined by the void width to wavelength ratio, as shown in Figure 6-18 - Figure 6-20.

For thin voids, the spike phenomena overwhelm modulations. When the defect is large in

height, modulations become more obvious than the spikes (Figure 6-20).

(a) (b)

Figure 6-18. Transmission coefficients of (a) Mode 2 and 3 (b) with a rectangular void in the mid-plane of a 2.4mm thick composite plate. The horizontal axis is void width in percent of input wavelength. The void height is 0.1mm.

120

(a) (b)


(a) (b)


The features analyzed in this section can be used in damage characterization. The

transmission coefficient vs. frequency (phase velocity/wavelength) curve can be plotted

by tuning the frequency of the incident wave and measuring the through transmission

waves. The defect length can be estimated from the distance between two neighbor peaks

121

or troughs in the frequency spectrum. The defect height can be estimated by comparing

the amplitudes of the spikes and the modulation.

6.2 Simulation and experiment for guided wave scattering in a quasi-isotropic composite plate

In this section, the GL method is verified with experiments. The test bed is a

quasi-isotropic carbon/epoxy composite plate with three artificial defects at different plies.

Transmission and reflection coefficients are calculated for all propagating modes up to

1MHz. Through transmission tests are performed with angle beam transducers.

Quantitative comparison is carried out between calculated and experimental data of

energy attenuation for all three defects.

6.2.1 Sample description

The specimen is a 12” by 12” quasi-isotropic plate made from AS4/8552-2 carbon

epoxy prepregs. The stack sequence is [(0/45/90/-45)S]2. Table 6-2 lists the engineering

properties of the prepreg used in the calculation. The density is 1.6g/cm3 and the

thickness of each ply is 0.145mm.

Table 6-2. Material properties of AS4/8552-2 carbon epoxy prepreg

E1 141.35 GPa E2=E3 9.58 GPa

G23 3.67 GPa G12=G13 4.89 GPa ν23 0.3

ν12= ν13 0.3

Figure 6-2transducers

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122

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123

defects is about 2”, which is not enough for a transducer. Therefore the actuator and

receiver were placed on left and right sides of the defects. Waves propagate along 0°

direction in the following simulations and experiments.

6.2.2 Theoretical analysis

The SAFE method was employed to calculate dispersion curves. Figure 6-22

illustrates the phase velocity, group velocity and skew angle dispersion curves for all

propagating wave modes below 1MHz. The modes were sorted based on orthogonality.

Note that for this quasi-isotropic plate, the skew angle is not always zero, as shown in

Figure 6-22 (c). The longitudinal and shear displacements of each mode are coupled, so

the waves were labeled as “Mode 1” to “Mode 7” rather than Lamb waves and SH waves

in an isotropic media.

124

(a)

(b)

(c)

Figure 6-22. (a) Phase velocity; (b) group velocity and (c) skew angle dispersion curves for a 16-layer [(0/45/90/-45)S]2 composite plate made of AS4/8552-2 carbon epoxy prepreg.

125

Fundamental waves such as mode 1-3 at low frequencies are usually favorable for

damage detection because of their low-dispersive feature. Besides, there are fewer modes

in the low frequency region. It is easier to excite the selected wave mode without other

unwanted ones. However, low frequency induces longer wavelength, which is incapable

to detect small cracks or delaminations. To improve the sensitivity of guided wave

inspection, high frequency and higher order modes were considered in our measurements.

It is difficult to simulate guided wave scattering at high frequency because the

wavelengths of those low order modes are very small. The mesh size has to be reduced to

guarantee the accuracy. On the other hand, the wavelengths of some high order modes are

very large near their cutoff frequencies. This requires a large dimension in modeling.

Both factors increase the computational burden. The GL method improves the calculating

efficiency by applying quadratic elements and integrating the normal mode expansion

technique at the boundary of the FE model. Therefore, it is capable to handle high

frequency problems.

GL models have been built for guided wave scattering analysis. The simulated

defect locations were defined the same as those of the composite specimen shown in

Figure 6-21(a). Materials were removed at both top and bottom plies of the damaged

interface to make a 5.2um-thick 1”×1” square-shaped void. Figure 6-23 displays an

example mesh of the GL model with a delamination between plies 12 and 13.

126

(a)

(b)

Figure 6-23. (a) Mesh of the GL model for delamination between plies 12 and 13. (b) Enlarged picture at the defected area. Blue lines and dots represent element edges and nodes respectively.

Transmission and reflection energy was calculated based on Eq. 5.26, for all

propagating modes between 0-1 MHz with all three artificial defects. As an example, the

127

results of scattering at defect 1 are shown in Figure 6-24 and Figure 6-25. This example

demonstrates the capability of the GL method of simulating high order guided wave

modes.

Figure 6-24. Normalized transmission and reflection energy of guided wave modes 1-4 interacted with defect 1 in the composite specimen.

128

Figure 6-25. Normalized transmission and reflection energy of guided wave modes 5-8 interacted with defect 1 in the composite specimen.

The transmission and reflection coefficients of all propagating guided waves to

defect 1, calculated from Eq. 5.25, were projected into the phase velocity dispersion

129

curves in Figure 6-26. The transmission coefficients are relatively high in low frequency

region, corresponding to low sensitivity to the defect. This is due to the large wavelength-

to-defect size ratio of low frequency modes. Therefore higher frequencies (>500kHz) are

preferable in guided wave mode selection.

(a)

(b)

Figure 6-26. Transmission and reflection coefficients of guided waves to defect 1.

130

The sensitivity of guided wave mode n to a defect is defined as the square root of

total reflected energy.

, 6.1

Here n and m are incident and reflected wave modes. is the reflected energy

normalized to energy flux in the wave propagation direction, defined in Eq. 5.26.

Figure 6-27 displays the sensitivity of each wave mode to defect 1. It is slightly

different from Figure 6-26 (b) because of mode conversion during reflection. Note that

the modes near cutoff frequencies show high sensitivity. This can be explained as follows:

A small change of frequency-thickness product can cause large variation of phase

velocity for wave modes near the cutoff frequency where the slope of the dispersion

curve is large. So these modes are sensitive to thickness change. The delamination

induces thickness variation, therefore can be measured by these wave modes. However,

the waves at cutoff frequencies are highly dispersive, which causes serious attenuation

during wave propagation. The attenuation is more severe in a viscoelastic composite

laminate. For this reason, these modes were not chosen in our experiments for damage

characterization.

131

Figure 6-27. Sensitivity of guided waves to defect 1.

6.2.3 Experiments and discussion

Experiments were carried out on the composite sample to verify the GL theory.

Figure 6-28 shows the sample plate with ply 1 on the top and ply 16 at the bottom. Three

artificial defects were represented with 1” by 1” Teflon inserts. Defect 1 was put between

plies 2 and 3. Defect 2 and 3 were located between plies 8, 9 and 12, 13 respectively. A

pair of angle beam transducers, composed of two normal beam transducers and two 60

degree angle wedges, was mounted on the plate surface to actuate and receive ultrasonic

guided waves. The input signal was a 10-cycle Hanning-windowed pulse with the center

frequency changed from 500kHz to 1MHz. Through-transmission measurements were

performed at all damaged locations with the actuator and the receiver at each side of the

defect. Signals in normal areas were also collected as baseline data.

132

Figure 6-28. Ultrasonic guided wave test with one pair of angle beam transducers on an quasi-isotropic composite plate composed of 16 plies.

As an instance, Figure 6-29 plots through-transmission waveforms measured at

820 kHz for both intact and damaged regions. There are decreases in signal intensity at

all three defects. The attenuation of through-transmission energy can be used for damage

characterization. However, each waveform in Figure 6-29 is a combination of two or

more modes. The contribution of each mode to attenuation should be determined for

quantitative comparison between theory and experiment.

133

Figure 6-29. Through transmission signals generated and received with a pair of 60º angle beam transducers centered at 820kHz.

The source influence of angle beam transducers was studied to simulate a more

realistic situation. Suppose a normal beam transducer, of width D, is mounted on a wedge

with an incident angle θ, as shown in Figure 6-30.

134

Figure 6-30. Sketch of an angle beam tranducer.

The parabolic distributed loading function can be written as (Rose J. L., 1999)

1 / /2 if | | /2,0 if | | /2, 6.2

where σ0 represents the maximum pressure which appears at the center of the

transducer-wedge interface (α=0). The excitability of the angle beam transducer can be

expressed as

w χ

8Dχ cos

2 sin χD/2χD

cos χD/2 if χ 0

2 D3 cos

if χ 0 6.3

with

sincos . 6.4

Here kν is the incident wave number. kw=ω/cw is wave number in the wedge,

where cw represents the longitudinal wave velocity in the wedge material.

135

Figure 6-31 depicts the excitability of 60° angle beam transducers calcualted from

Eq. 6.3. The transducer width D is 15mm. The longitudinal wave velocity is 2.67 km/s

for the Plexiglas wedge. It is shown that the 60° angle beam transducer tends to excite

modes with phase velocities close to 3.5km/s, especially at high frequencies.

Figure 6-31. Source influence spectrum of angle beam transducer with the incident angle at 60 degree.

The normalized excitability of mode n is expressed as

∑

. 6.5

Here, kνn is the wave number of mode n. Note that w is a function of wave number,

incident angle and frequency. The effective sensitivity, taking into account the source

influence, is then defined as

136

1

∑, 6.6

where n and m represent the incident and transmitted wave modes. is the

normalized through transmission energy defined in Eq. 5.26.

Figure 6-32. Effective sensitivity to defect 1 for guided waves excited and received with 60 degree angle beam transducers.

The normalized energy attenuation coefficient, for through-transmission waves

generated and received with θ degree angle beam transducers, is defined as

10

∑∑ ∑ , 6.7

For an intact structure, the normalized through transmission energy becomes

1 0 , 6.8

137

and becomes zero. It is worth noting that Eq. 6.7 is for single frequency

input, i.e. continuous waves. In real test, the input is a tone burst pulse, as shown in

Figure 6-33. The normalized attenuation coefficient for pulsed input is written as

,,

, 6.9

Where f0 is the center frequency. Amp(f, f0) is the frequency spectrum of the

Hanning-windowed pulse, as depicted inFigure 6-33. Figure 6-34 displays the theoretical

prediction of energy attenuation at defect 1 for guided waves generated and received with

a pair of 60 degree angle beam transducers. The blue and red curves represent attenuation

of continuous wave and Hanning-windowed 10-cycle pulse respectively. The former

attenuation curve was smoothed by applying Eq. 6.9.

Figure 6-33. Temporal profile and frequency spectrum of incident signals.

138

Figure 6-34. Theoretical prediction of guided wave attenuation caused by scattering at defect one. The blue and red curves represent attenuation of continuous wave and Hanning-windowed 10-cycle pulse respectively.

Finally, the theoretical predictions of energy attenuation at all defects are plotted

in Figure 6-35 and compared with experimental data. The later were calculated as the

integration of squared temporal signals of voltage over the damaged regions, normalized

to their counterparts at the intact area. Generally, the quantitative comparison between

theory and experiment shows good agreement. The strong attenuation due to defect 1 at

high frequencies (750-900kHz) is successfully predicted. The relatively low attenuation

to defects 2 and 3, calculated from the GL theory, matches very well with experiments.

For defect 1, there are around ±2dB errors at some frequencies, but the trend is correct.

Possible reason for these errors will be discussed in the following paragraphs.

139

Figure 6-35. Comparison of experimental and theoretical results of energy attenuation for guided waves at defects 1-3. The incident angle is 60 degree for both the transmitter and the receiver.

In this sample application, the numerical-analytical hybrid model successfully

predicts the attenuation of guided waves to delaminations at different plies of a composite

laminate. The experiment verifies the feasibility of this theoretically driven damage

characterization technique. However, the theory is not complete and can be improved in

the following aspects:

1. The attenuation of guided waves in a viscoelastic material is not considered in

our model. According to Gao’s study (Gao H. , 2007), the viscoelasticity

doesn’t change the particle displacements and stresses, so the elastic material

property in this thesis is a reasonable simplification. However, the damping of

each wave mode can be different, especially at high frequency, which may

affect the ratio of scattered waves. To accurately simulate guided wave

140

scattering in composites, the viscoelasticity should be included into the GL

method.

2. The GL model requires the defect to be infinite in transverse direction, which

is different from the realistic 1” by 1” delaminations in our sample. The

scattering at the edges of the defects is not represented by the theoretical

prediction.

3. The skew angle can be up to ±40º in the interested frequency range, as

shown in Figure 6-22 (c). This factor can also affect the accuracy of our

measurement. The effect of skew angle can be reduced by using large

transducers and decreasing the distance between the actuator and the receiver.

6.3 Summary

A GL method was developed to analyze guided wave scattering at defects in

anisotropic composite laminates. Different from isotropic media, the guided wave

velocity and mode shapes change with the propagation direction in anisotropic

composites. In addition, the longitudinal and shear waves are coupled in some directions,

which increase the complexity of wave scattering analysis.

To verify the GL method in anisotropic materials, the transmission and reflection

coefficients were calculated for guided waves scattered by a surface notch in a

unidirectional carbon/epoxy plate. For comparison, time marching FE simulation was

performed in a 3-D ABAQUS model. Transmission coefficients were calculated at a

series of frequencies, which show good agreement with the GL results.

141

The GL method was employed to perform a quantitative study for guided wave

scattering at defects in a unidirectional composite plate. The influence of wave

propagation direction, defect location, width, height and frequency were discussed for

different guided wave modes. It is shown that the transmission and reflection coefficient

periodically changes with the defect width to wavelength ratio, indicating a modulation

of the defect to incident waves. This explains the sensitivity variation with wavelength in

ultrasonic measurements. Besides the modulation, the frequency spectra of transmission

and reflection coefficients show spikes at some resonance frequencies. The spike

amplitude varies with defect location and size. These simulations provide theoretical

guidelines for ultrasonic tests on composite laminates.

Transmission and reflection were studied for a quasi-isotropic composite plate

with three artificial delaminations at different piles. The sensitivity of propagating wave

modes to each defect was calculated in terms of the attenuation of through-transmission

energy. The source influence of angle beam transducers was considered. By choosing

suitable wave modes and frequencies, the defects can be detected and separated from

others. Experiments were conducted to quantitatively verify the GL theory.

Chapter 7 Conclusions and Recommendations

7.1 Summary of the research

This section summaries the research in terms of theoretical and experimental

investigations on damage characterization of composite structures with ultrasonic guided

wave techniques.

Guided wave propagation characteristics were studied with two methods, namely,

the global matrix method (GMM), and the semi-analytical finite element (SAFE) method.

The GMM is suitable for simulation of guided waves in a thick plate or half-infinite

structure with few layers, and the SAFE method is more efficient for multilayered

laminate composed of thin plies.

A feature based guided wave mode selection approach was presented. With an

understanding of guided wave mechanics, the transducer could be designed to excite

wave modes with better sensitivity to the target defect in a composite structure. Specific

features include phase velocity, group velocity, skew angle, attenuation, and mode shape

such as displacement, stress, strain, power flow distribution, etc.

Two types of structural damage were evaluated to demonstrate the qualitative

SHM procedure and to prove the feasibility of this method. The first damage mode is

skin/core disbond in a skin/substrate structure. The leaky waves, which lose energy into

the substrate during propagation, were chosen for disbond detection. Numerical

simulation indicated that disbond blocked wave leaking, and raised the through-

143

transmission energy. Experiments showed the through-transmission wave intensity

monotonically increased with the defect length. The other type of damage is delamination

at layer interfaces of an Aluminum /composite hybrid plate. The in-plane displacement

was chosen as the SHM feature because of its sensitivity to delamination in composite

laminates. Mode conversion was supposed to happen when selected wave modes pass

through the damaged region, which was verified by the FE simulation and the experiment.

The qualitative approach selects wave modes based on wave structures. However,

the scattered waves of two incident modes with similar wave structures can be quite

different (Lowe, Cawley, Kao, & Diligent, 2002). This is because the incident waves are

reflected back and forth between both ends of the defect. The multi scatterings overlap

and construct the final transmission and reflection waves. The intensity of scattered

waves is therefore a function of incident wavelength. A more accurate method is desired

to simulate guided wave scattering.

A global-local (GL) method has been developed to quantitatively study guided

wave interaction with defects. In a GL model, the transmission and reflection coefficients

are calculated in frequency domain. The normal mode expansion method is applied on

the boundaries. This technique is computationally more efficient than time marching and

steady state FE analysis. It has been verified in the following five aspects:

1. The energy consistency of incident and scattered waves were evaluated for all

GL models. The maximum error is about 1% in our simulation.

2. A steady state FE model simulated S0 wave scattering at a notch in an

aluminum plate. Displacements of all nodes around the damaged region were

calculated, which agrees well with the GL results.

144

3. The GL method calculated transmission and reflection coefficients of A0

waves with a surface notch in a steel plate. Different notch width, depth and

frequencies were discussed. The results were quantitatively compared with FE

and experimental data from literature. Very good agreements were obtained

between the GL modeling and the literature results.

4. To verify the GL method for anisotropic material, a 3-D time-marching FE

model simulated guided wave scattering at a surface notch in a unidirectional

composite plate. The transmission coefficients calculated from both the FE

and the GL method matched very well.

5. Finally, the GL theory was verified with experiments on a composite plate

with delaminations at three different locations in the plate thickness. The

attenuation of through transmission guided waves energy was calculated for

each defect. The theoretical and experimental results were quantitatively

compared. Generally, the comparison shows good agreement. There is about

1dB error at some frequencies in the attenuation coefficient spectrum, which

may be caused by simplification of the GL modeling in terms of material

geometry and damping property.

The GL method was employed to analyze guided wave interaction with different

defects. The influence of defect size, location, wave propagation direction, and frequency

to guided waves were studied. According to these simulations, transmitted waves are

periodically modulated by the defect and reach their maximal values when the

wavelength is equal to integer times of half defect width. At the same time, they show

145

spikes at some “resonant defect widths”. The spike location is determined by the defect

width to wavelength ratio.

In conclusion, this thesis presents two guided wave based techniques for damage

detection and characterization in composite structures. The first one is a concept driven,

feature based qualitative method. The validity of this method has been proven with two

examples: inspection of debonding in a composite skin/substrate structure and

delamination in a metal/composite hybrid plate. One limitation of this technique is that

the guided wave mode selection greatly depends on the inspector’s experience. The

second one is a more accurate and reliable technique based on the GL method, which can

quantitatively predict amplitudes of transmitted and reflected guided waves at a defect.

The GL method is about 10 times faster than the traditional FEA, therefore more suitable

for damage characterization and parametric study. In our research, the defects are simple

notches and voids, which can be expanded to more realistic defects, e.g., corrosion and

kissing bond in future.

7.2 Contributions

The major contributions of this thesis are summarized as follows:

1. Derived a numerical-analytical hybrid solution for guided wave scattering at

defects in composites. Considered the coupling between longitudinal, shear

vertical and shear horizontal displacements for the first time. Quantitatively

validated the hybrid method on both isotropic and anisotropic plates.

146

2. Carried out parametric studies on guided wave scattering at defects in

anisotropic laminates. This specific research provides guidelines for guided

wave SHM of composite structures by analyzing the influence of defect size,

location and material orientation to transmission and reflection coefficients.

3. Constructed a novel comprehensive mode selection and data processing

approach based on the quantitative analysis of guided wave propagation and

scattering characteristics. The definition of through-transmission energy

attenuation spectrum and integration of source influence are original

contributions of the thesis.

4. Integrated quadratic elements into the GL model for the first time, which

effectively saves computational time and memory, and improves the accuracy.

This technique can be easily applied to other FE based methods, such as the 2-

D SAFE method for guided wave propagation in an arbitrary cross-section

waveguide.

5. Systematically analyzed transmission and reflection of high order guided

wave modes with defects for the first time. Obtained transmission and

reflection coefficients for all propagating modes below 1MHz, including those

close to the cutoff frequencies. The applying of high order modes can

effectively improve the guided wave inspection sensitivity.

6. Developed a short distance guided wave scanning method for damage

detection and localization in a composite/metal hybrid plate, and

experimentally tested the method. Adopted frequency domain signal

difference coefficient (SDC) as the tomographic feature, which overcame the

147

inaccuracy from guided wave phase shift during transducer movement. No

baseline of previous structural condition is required.

7. Demonstrated the feasibility of composite skin/Nomex substrate disbond

measurement with leaky guided waves. Studied the propagation

characteristics for both propagating and non-propagating waves. Derived a 2-

D root searching technique.

8. Quantitatively compared simulated and measured attenuation spectra for

guided waves scattering at delaminations in a quasi isotropic composite

laminate for the first time.

Besides above contributions, a software toolbox with a graphic-user interface

(GUI) has been developed for simulation of guided wave propagation and scattering in

composites. Detail of this computational tool is illustrated in Appendix A.

7.3 Recommendations for future work

The main recommendations of future works are summarized as follows.

7.3.1 Analysis of guided wave scattering in viscoelastic media

Problem Statement:

The research in this thesis treated composite as elastic material. This assumption

is reasonable at low frequencies. However, in high frequency region the viscoelatic

behavior of composites cannot be neglected. Selecting a wave mode with less attenuation

148

can increase the monitor range. Viscoelasticity also affects wave interaction with defects.

A low damping incident wave may be converted into a high damping one at the defect,

and vice versa.

Objectives:

Study the influence of viscoelasticity to guided wave propagation and scattering

in composite laminates.

Technical Approach:

The SAFE method introduced in this thesis can be applied to viscoelastic media

by replacing the elastic constants in the governing equation (Eq. 2.1) with a complex

stiffness tensor (Eq. 7.1). Both the real part C’, and imaginary part C” are 6 by 6 matrices

(Rose J. L., 1999).

" 7.1

where f0 is the frequency at which elastic properties were measured. The wave

number of viscoelastic waveguide is also a complex shown in Eq. 7.2. Its real part is

related to the phase velocity and the imaginary part to the attenuation.

7.2

Here, Cp is the phase velocity. The displacement vector is expressed as:

, , 7.3

Its amplitude exponentially decreases with wave propagation.

Guided wave scattering in a viscoelastic composite plate can be calculated by

substituting the complex stiffness matrix and complex wave number into the GL program.

149

Note that the incident energy and scattered energy will not be consistent due to

attenuation.

7.3.2 Numerical study of guided wave scattering with 3-D finite element method

Problem Statement:

The GL method presented in the thesis is limited to waveguides with a uniform

structure in the transverse direction. In realistic SHM, the defect can be any shape and

size. Furthermore, the scattered waves can propagate at all directions. A real 3-D model

is desired for accurate description of guided wave scattering in an anisotropic media. The

time-marching FE technique presented in Section 6.1.1 is not appropriate for systematical

study of guided wave interaction with defects due to the following reasons:

1. It is not computational efficient since every moment between guided wave

excitation and scattering has to be simulated.

2. To reduce the effect of boundary reflection, the FE model should be either

enlarged or surrounded by damping materials. Both approaches have great

effect to the speed of explicit dynamic analysis. According to the ABAQUS

user’s manual, introducing 1% damping can reduces the stable time increment

by a factor of 20 (ABAQUS, 2009).

3. 2-D FFT has to been conducted to analyze the component of scattered wave

modes.

It is therefore necessary to conduct FE analysis in frequency domain for defect

characterization with guided waves.

150

Objectives:

Develop 3-D FE models to simulate guided wave scattering in frequency domain.

Verify the simulation results with experiments.

Technical Approach:

This task can be realized by employing the steady state dynamic analysis module

in ABAQUS. Section 5.3 presents a 2-D model for Lamb wave interaction with a surface

notch in an aluminum plate, which can be extended to 3-D and anisotropic waveguide

(Figure 7-1). Damping material should be defined on the boundary to remove reflection.

In steady state analysis, there is no damping penalty in term of stable time increment. So

the calculation speed is not affected. One challenge is to determine the damping

coefficients. In our isotropic simulation, the damping coefficients were manually

assigned to the “silent boundary”, which is impossible for anisotropic materials because

the wave propagation characteristics vary with directions. The optimization toolbox in

MATLAB can be applied to generate damping curves so that boundary reflections are

eliminated for all directions.

Figure 7-1. Sketch of the 3-D FE model.

151

Scattered waves should be received at all directions surrounding the defect, as

shown in Figure 7-1. Both surface and inner displacements are recorded. The waveform

at each direction consists of one or more wave modes, which can be decomposed using

the normal mode expansion technique and the least squares method. Then the scattering

field can be expressed as an S-matrix (Wilcox & Velichko, 2010), with each row related

to an incident angle and column to a scattered angle.

7.3.3 Guided wave defect characterization for composite cylinder

Problem Statement:

Metallic and composite cylinders are extensively used in civil and military

industries. Flaws occur to these structures during manufacture and service. Ultrasonic

guided waves can be used for SHM of circular tubes and pipelines. The propagation of

guided waves in cylinders is similar to those in plates except that many more modes exist

in tubes (Mu & Rose, 2008). Suitable mode selection is therefore critical for guided wave

pipeline inspection.

Objectives:

Develop an analytical-numerical hybrid method to simulate guided wave

scattering in the circumferential direction of a composite cylinder.

Technical Approach:

The SAFE method can be used to study the dispersion behaviors of waves in a

composite cylindrical shell. In this hybrid method, the radial direction of the shell is

152

constructed of finite elements, while the axial and circumferential displacements are

expressed as exponential functions (Xi, Liu, Lam, & Shang, 2000).

The GL method introduced in Section 5.2 can be easily expended to composite

cylinders by substituting the Cartesian coordinates with polar coordinates. The shape

functions should be replaced as well. Figure 7-2 illustrates the GL model for

circumferential wave interaction with a defect in a cylindrical shell. The region close to

the defect is discretized into finite elements. The boundaries are coupled with the SAFE

solutions. Transmission and reflection coefficients of each mode are obtained with the

normal mode expansion technique.

Figure 7-2. The GL model for circumferential waves in a circular tube.

Many composite cylinders consist of woven fibers/epoxy. The effective moduli of

woven layers can be calculated from their counterparts of the unidirectional ply, based on

equations in (Sun & Li, 1988).

Experiments are to be conducted on a composite cylinder with artificial defects.

Frequency and wave mode will be selected based on simulation. Comb or angle beam

153

transducers will be designed and applied to the specimen. The experimental data should

be compared with theoretical results.

7.3.4 Testing with phased array transducers

Problem Statement:

The experiments in Section 4.3 and 6.2.3 were conducted with angle beam

transducers. The mode selections were realized by tuning the center frequency and

adjusting the incident angle. The wedges and normal beam transducers have to be

assembled every time changing the angle, which is inconvenient in the field test.

As an alternative of the angle beam transducer, the phased array technique can be

applied for guided wave actuation and receiving. A phased array transducer consists of

many sensor elements. By applying appropriate phase delays to each element, a guided

wave mode with a particular wavelength is enhanced and others are attenuated. Mode

switching can be performed by tuning input signals from computer, which increases the

system efficiency and liability.

Another benefit of phased arrays is their guided wave beam steering capability. A

constructive interference of waves from phased array elements can be formed in the

desired scanning direction. As a result, the whole structure can be inspected with limited

access positions. An introduction of the phased array transducer can be found in Rose’s

textbook (Rose J. L., 1999). Commercial application and products are illustrated in (GE

Inspection Technologies, 2008)

Objectives:

154

Explore the feasibility of using phased comb arrays for guided wave mode

selection in composite plates. Study guided wave beam steering and beam focusing.

Derive a phased array based technique for damage detection and characterization of

composites.

Technical Approach:

To investigate the source influence of phased array, the excitation of straight-

crested plane waves and circular-crested waves should be simulated with the normal

mode expansion method. Analytical study should be carried out for guided wave

excitations of piston type transducers and leave-in-place piezoelectric disk transducers.

FE models should be built to verify the analytical results.

Based on the guided wave excitation and scattering theory, phased array

transducers can be designed and applied to a composite plate with artificial defects. The

mode selection ability of the phased array system, as well as beam steering and beam

focusing, is to be demonstrated.

7.3.5 Numerical and experimental study of the coupling between the transducer and the host structure

Problem Statement:

In this thesis, guided wave modes were excited by applying tractions on surface or

inner nodes of FE models. The influence of transducer was not considered. However, the

coupling between the transducer and the substrate is essential to wave excitability. For

instance, a PZT disk bonded on the top surface of an aluminum plate can generate both S0

and A0 modes. The intensities of these two waves depend on the amplitudes of normal

155

and radial loads applied to the interface, which is related to the disk size and weight. To

accurately simulate guided wave excitation, transducers should be included into modeling.

Objectives:

Develop numerical models for piezoelectric transducers integrated into the

composite structure. Behaviors of both actuator and receiver should be studied and

verified with experiments.

Technical Approach:

The coupling between the PZT transducer and the composite substrate can be

studied with a hybrid finite element-normal mode expansion technique, which was

originally applied to model Lamb wave emission-reception with surface mounted

transducers (Moulin, Grondel, Assaad, & Duquenne, 2008; Moulin, Assaad, Delebarre, &

Grondel, 2000). The region near the excitation source is divided into discrete elements,

and the far field is represented with combinations of continuous wave modes. Electrical

loading and response can be obtained from the system.

7.3.6 Simulation of guided waves in solid structures coupled to infinite media

Problem Statement:

In many problems of interest the waveguide is surrounded by infinite or semi-

infinite media. Some examples include: steel bars inside concrete, and pipes in soil. This

type of problems cannot be solved with a traditional FE model, which must be terminated

at some finite boundary. A silent boundary is desired to absorb incoming waves.

Objectives:

156

Study guided wave propagation and scattering in structures coupled to infinite

media.

Technical Approach:

Literature survey reveals that there are three major approaches for handling this

type of boundary conditions. First is the viscous damping boundary method, which

eliminates outgoing waves with damping materials in the absorbing region (Castaings &

Lowe, 2008). The second is the Perfect Matched Layers (PML) method, which forces the

wave to decay exponentially in the absorbing boundary layer (Drozdz, Skelton, Craster,

& Lowe, 2007). The third method is to place infinite elements with a special shape

function at the infinite boundary (Fu & Wu, 2000). Methods two and three are

recommended because of their feasibility to model anisotropic structures. The absorbing

layer or infinite elements can be integrated into the 1-D and 2-D SAFE methods for

guided wave propagation analysis, or the GL method for guided wave scattering analysis.

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Appendix A

Ultrasonic Guided Wave Simulation Toolbox Development for Damage Detection in Composite

Ultrasonic guided wave techniques have been proven to be effective in NDT and

SHM of composite structures However, the use of guided waves for composite inspection

is not a trivial task due to the nature of multi-layer, anisotropic, and viscoelastic behavior

of composite materials. Without a clear understanding of the wave mechanics and good

model guidance, the inspection setup is often less than optimal and the results are

questionable.

A software simulation tool, with the objective of rapid assessment of the

feasibility of ultrasonic guided wave NDE method for critical system components

inspections, is highly desired. For this purpose, we have developed a user-friendly

simulation software for ultrasonic guided waves inspection of composite material and

other structures. With a high-fidelity simulation toolbox, users can design and optimize

the inspection procedures for field maintenance, as well as design and incorporate

inspection requirements at the structure/component design stage. The detailed technical

approach and results will be presented in the following sections.

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A.1 The main interface

The SAFE method has been integrated into LabVIEW programs to calculate

guided wave propagation within composites. Figure A-1 displays the main interface of

the toolbox.

Figure A-1. Main interface of the guided wave simulation toolbox.

To calculate guided waves in composite plates, choose “Plate” in the “Select

Geometry” menu. Then click the “Material Input” button in the main interface, a

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“Composite Layup” window will pop up where users can define material properties and

the layup sequence.

The following is the explanation of other input parameters in the main interface.

Start Freq, End Freq, and Freq Increment: The start frequency, end frequency

and frequency increment for dispersion curve calculations;

Start Velo, End Velo: Mininum and maximum values of the phase velocity;

Prop Direction: The angle between the wave propagation direction and the fiber

direction of the top layer.

Display Modes: The maximum number of modes to be plotted on screen;

Elems/Wavelength: The maximum number of discrete elements in a wavelength.

Wavelength is a function of frequency. It is small in the high frequency region, so the

SAFE model needs more elements to convergent. A dynamic element distribution

algorithm is applied in our program, which use fewer elements at low frequencies and

more elements at high frequencies. This method guarantees both accuracy and

computational efficiency.

User can click the “Calculate Phase Velocity” button to start a guided wave

calculation based on input parameters. The “Phase Velocity Dispersion Curves” graph

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presents the progress of dispersion curve calculation and also display the final phase

velocity dispersion curves. During dispersion curve calculation, clicking the “Stop

Calculation” button can stop the calculation at the current frequency point. The program

then sorts the dispersion curves calculated and plots the curves. When the dispersion

curve calculation is finished naturally, the program also automatically sorts the dispersion

curves and displays the results. In both cases, the corresponding group velocity dispersion

curves are plotted in the graph beside the “Phase Velocity Dispersion Curves” graph.

Drag cursors then appear on the dispersion curve graphs and user can drag the one in

‘Phase velocity Dispersion Curve’ graph to select a point of interest on the curves. The

cursor shown in the “Group Velocity Dispersion Curves” graph follows the curser move

in the “Phase Velocity Dispersion Curves” and presents the corresponding group velocity

point. The wave structures and stress distributions of the cursor location are also plotted

in the “Displacement” and “Stress” graphs. The actual values of the frequency point,

mode number, phase velocity, group velocity and skew angle at the cursor position are

displayed at the bottom-left corner. Change the “Frequency” or “Mode” input will also

move cursors in the phase velocity and group velocity graphs. Users can select a specific

point on the dispersion curves for a certain application based on the wave structure and

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stress distribution information. The save buttons at the bottom-left corner can be used to

save phase velocity dispersion curves, group velocity dispersion curves and wave

structures into ASCII files. For dispersion curves, the saved file contains a spreadsheet

with the first column as the frequency and other columns for the values of phase

velocities or group velocities of different wave modes. 'NaN' was used in the saved files

for the velocities of wave modes at frequencies lower than the cutoff frequencies (no

propagating mode exists). For wave structures, the saved file contains a spreadsheet with

the first column as the coordinates in the thickness direction and other columns for

displacement and stress values.

Click the “Finite Element Analysis” button, a FE simulation toolbox will popup,

where users can produce a FE model to simulate guided wave propagation and scattering

in a multilayered plate and view the results.

A.2 Case study– compare with prior work

A comparison study was made with available data for a transversely isotropic

carbon-fiber reinforced plastic (CFRP) laminate (Guo & Cawley, 1993) (Table A-1). The

layup sequence is [(0/90)2s]. The results were found to be in very good agreement with

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prior work (Figure A-2-Figure A-4). Note that the sign difference of Uz in Figure A-3

and Figure A-4 is due to the normalization of Figure A-4.

Table A-1. Material properties of transversely isotropic carbon fiber reinforced plastic (CFRP) (Guo & Cawley, 1993).

ρ 1605 kg/m^3 Ε11 126.6 GPa Ε33 8.7 GPa G13 3.7 GPa ν13 0.306 ν23 0.5

(a) (b)

Figure A-2. Phase velocity comparison for 8 layer CFRP (1.0 mm total thickness) laminate (a) – from (Guo & Cawley, 1993); (b) – using UGWST.

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(a) (b)

Figure A-3. Comparison of (a) displacements and (b) stresses for an 8 layer (symmetric) CFRP (1.0 mm total thickness) laminate – using UGWST.

(a) (b)

Figure A-4. Comparison of (a) displacements and (b) stresses for an 8 layer (symmetric) CFRP (1.0 mm total thickness) laminate (Guo & Cawley, 1993).

A.2.1 Guided wave propagation in IM7/8552 composites

As a case study, we calculated guided wave propagation in a unidirectional

IM7/8552 composite with the toolbox. The thickness of the plate is 4.44mm. The density

is 1.598g/cm3. Table A-2 lists the engineering elastic constants. Phase velocity and group

velocity dispersion curves have been calculated when guided waves propagate along 0

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degree, 45 degree and 90 degree directions (Figure A-5). For 0 degree and 90 degree, SH

waves and Lamb waves are decoupled because the waves propagate along the symmetric

axis. In the case of 45 degree, the wave structure includes displacements in all three

directions (Figure A-6). So there is no independent SH wave or Lamb wave. Figure A-6

also displays the energy velocity and skew angle. The energy velocity, which should be

the same as the group velocity, can be used to check convergence. If the energy velocity

and group velocity are different, user needs to adjust the frequency increment or number

of elements per wavelength. The skew angle of this mode is 43.13 degree, which means

most of energy propagates out of the launch direction.

Table A-2. Elastic constants of IM7/8552 unidirectional composite along the fiber direction.

E1 (GPa)

E2 (GPa)

E3 (GPa)

ν12 ν13 ν23 G12 (GPa)

G13 (GPa)

G23 (GPa)

167.52 10.47 10.47 0.33 0.33 0.47 5.24 5.24 3.56

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(a)

(b)

(c)

Figure A-5. Phase velocity and group velocity dispersion curves of a unidirectional IM7/8552 composite. The angle between wave propagation and fiber direction is (a) 0 degree, (b) 45 degree and (c) 90 degree

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Figure A-6. Wave structures of a unidirectional IM7/8552 composite. The angle between wave propagation and fiber direction is 45 degree.

A.2.2 Guided wave propagation in damaged composites

(a) (b)

Figure A-7. Phase velocity dispersion curves and wave structures of the selected mode.

With the template script and the LabVIEW program developed, it is very

convenient to conduct parametric studies on guided wave propagation and scattering

problems. In this section, we performed a parametric study on a 4-layer composite plate

made of AS4/3501-6 prepreg. The layup sequence is [(0/90)s]. Figure A-7 displays the

dispersion curves and wave structures of the selected mode. A FE model is built with the

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finite element analysis toolbox (Figure A-8). Table A-3 lists parameters in the simulation.

For comparison, composites with no damage, a rectangular void and a rectangular

material degradation have been modeled.

Figure A-8. Sketch of the FE model.

Table A-3. FE parameters for simulation of guided wave scattering in a composite plate

Plate Length

500mm Phase Velocity

6813m/s Wavelength 30mm

Center Frequency

225KHz Sampling Frequency

4.5MHz Number of Cycles

5

Time Period

153μs Transducer Type

Wave Structure

Defect Center

(250mm, 2mm)

Defect Size (17mm, 1mm)

Receiver Location

(100mm, 0mm), (100mm, 1mm), (100mm, 2mm), (350mm, 0mm), (350mm, 1mm), (350mm, 2mm),

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(a)

(b)

Figure A-9. Field output of displacement amplitude at frame 39 in the defect free composite (a) and the notched composite (b).

Figure A-9 shows the field output of displacements in a damage free model and

wave scattering in a notched model. Figure A-10 plots time domain outputs of

displacements in the damage free model. It is important to evaluate the correctness of

these results. First, we checked the group velocity. From the distance between sensors

and the flight time in Figure A-10, the group velocity is calculated as 6.6km/s, which

matches with the result of semi-analytical finite element method (6.5km/s). Second, we

checked the wave structure. According to the wave structure in Figure A-7b, the selected

mode is a symmetric wave with in-plane displacement in dominant. The FEA results in

Figure A-10 agree with these features. Third, the selected mode is in a low-dispersive

region in the dispersion curve (Figure A-7a), so the pulse width doesn’t change much

when the wave propagates (Figure A-10).

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(a)

(b)

(c)

(d)

Figure A-10. Time history outputs in the defect free model. (a) In-plane displacements at sensors 1-3; (b) Out-of-plane displacements at sensors 1-3; (c) In-plane displacements at sensors 4-6; (d) Out-of-plane displacements at sensors 4-6;

Figure A-11 compares out-of-plane time history displacements at sensors 4-6 in

composites with different defects. A wave package at around 80μs can be observed in

Figures b and c, which should be scatted waves from the defect. This feature can be used

for ultrasonic through transmission measurement. It is also shown that the scattering

caused by material degradation is much weaker than that caused by a notch.

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(a)

(b)

(c)

Figure A-11. Out-of-plane time history displacements at sensors 4-6 in the defect free composite (a), the composite with a notch (b), and the composite with material degradation (c).

A.3 Summary

A user-friendly LabVIEW program has been developed for virtual inspection of a

composite plate with ultrasonic guided waves. The system consists of two major

components: guided wave dispersion curves and mode shape calculation; and Finite

Element Simulation. Guided wave propagation characteristics within composite plates are

calculated with a SAFE method, which uses discrete elements in the thickness direction,

and orthogonal function exp(ikx) in wave propagation direction.

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In the FEA module, the user can define parameters related to the specimen,

transducers, receivers, defects and input signals. The program generates a Python script

file, which is then processed by the commercial FEA solver ABAQUS to simulate

ultrasonic wave generation and propagation within composites. The simulation results are

loaded and displayed by our FEA results viewer. The program can also simulate

piezoelectric materials such as PZT and AlN, and study wave generation and propagation

within the transducer coupled specimen.

Appendix B

Nontechnical abstract

Fiber reinforced composite materials are widely used in aerospace structure

because of high strength to weight ratio. Similar to metal, composites suffer damage and

degradation from extreme temperature, humidity, impact and fatigue, etc. Early detection

of these defects is essential to flight safety.

Ultrasonic guided waves are elastic waves propagating in a bounded structure, e.g.

plate and pipe, where the acoustic energy is trapped between top and bottom boundaries.

The energy is enhanced at some particular frequencies and wavelengths, which are so-

called guided wave modes. When guided waves pass through a defect in a composite

waveguide, the energy is scattered. The change of through transmission waves can be

measured for nondestructive damage detection.

In our research, the guided wave propagation characteristics in composites were

analyzed. A qualitative feature based method has been developed to select guided wave

modes with high sensitivity to interested defects. Specifically, this method was applied to

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detect skin-core disbond in a composite helicopter rotor blade and delamination in a thick

composite-metal hybrid plate.

We also derived a numerical tool to quantitatively predict the interaction of

guided waves with defects in composite plates. With accurate simulation of guided wave

scattering, the measurement can be optimized and the defect location and severity can be

estimated. The accuracy of this tool was verified with data from literature, commercial

simulation software and experiments.

VITA Xue Qi

Education

Ph.D., Engineering Science and Mechanics, 2005-2011

Department of Engineering Science and Mechanics, The Penn State University

M.S. Acoustics, 2002-2005

Department of Electronic Science and Engineering, Nanjing University, China

B.S. Physics, 1998-2002

Department for Intensive Instruction (honored), Nanjing University, China

Publications

1. Xue Qi and Xiaoliang Zhao, “Guided wave propagation in solid structures of arbitrary cross-section

coupled to infinite media.” AIP Conf. Proc, 1211, 1681-1688, 2010.

2. Xue Qi, Xiaoliang Zhao and Joseph L. Rose, “Ultrasonic guided wave simulation toolbox development

for damage detection in composite.” AIP Conf. Proc, 1211, 1095-1102, 2010.

3. Xue Qi, Joseph L. Rose and Edward Smith, “Guided wave subsurface damage detection for a composite

on a half-space structure.” AIP Conf. Proc, 1211, 1135-1141, 2010.

4. Fei Yan, Xue Qi, Joseph L. Rose and Hasso Weiland, “ultrasonic guided wave mode and frequency

selection for multilayer hybrid laminates.” Material Evaluation, 68, 169-175, 2010.

5. Fei Yan, Xue Qi, Joseph L. Rose and Hasso Weiland, “Delamination defect detection using ultrasonic

guided waves in advanced hybrid structural elements.” AIP Conf. Proc, 1211, 2044-2051, 2010.

6. Xue Qi, Joseph L. Rose and Chunguang Xu, “Ultrasonic guided wave nondestructive testing for

helicopter rotor blades.” 17th World Conference on Nondestructive Testing, Shanghai, China, 2008.

7. Xue Qi, Shu-yi Zhang, Xiao-bing Mi, Xiuji Shui, and Xiaojun Liu, “Theoretical and experimental study

of photo-modulated reflectivity detections for transparent film/opaque substrate structures.” Applied

Physics A: Materials Science & Processing, 89(2), 537-542, 2007.

8. Xue Qi, Shu-yi Zhang and Xiao-bing Mi, “Three-dimensional analyses of photo-modulated reflectivity

for transparent film/opaque substrate structures” Ultrasonics, 44, e1183-e1185, 2006.

9. Xue Qi, Shu-yi Zhang, Xiao-bing Mi, Xiuji Shui, and Xiaojun Liu, “Thermal conductivity of transparent

thin films/substrates measured by photothermal reflectivity probing method.” J. Phys. IV France, 125, 261-

264, 2005.

Documents

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