The Pennsylvania State University

The Graduate School

Department of Engineering Science and Mechanics

ULTRASONIC GUIDED WAVE PROPAGATION ACROSS WAVEGUIDE

TRANSITIONS APPLIED TO BONDED JOINT INSPECTION

A Dissertation in

Engineering Science and Mechanics

by

Padmakumar Puthillath

2010 Padmakumar Puthillath

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

December 2010

The dissertation of Padmakumar Puthillath was reviewed and approved* by the following:

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

Cliff. J. Lissenden Professor of Engineering Science and Mechanics Dissertation Co-Advisor

Bernhard R. Tittmann Schell Professor of Engineering Science and Mechanics

Edward C. Smith Professor of Aerospace Engineering

Judith A. Todd Professor of Engineering Science and Mechanics P. B. Breneman Department Head Head of the Department of Engineering Science and Mechanics

*Signatures are on file in the Graduate School

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ABSTRACT

Adhesively bonded joints are increasingly used in safety critical applications such

as load bearing elements in aerospace structures. The quality of surface preparation is

crucial to the strength of an adhesive joint and also the adhesives are susceptible to

environmental conditions and undergo degradation. The possibility of the presence of

defects and interfacial weakness make nondestructive inspection techniques a valuable

tool to evaluate the structural reliability of the adhesive joints.

Two problems related to the inspection of aircraft adhesive joints – the adhesive

repair patches and adhesive skin-stringer joints are nondestructively evaluated using

ultrasonic guided waves in this thesis. The former is a life extending patch bonded at

defect locations on aircraft while the latter is a structural stiffener found in skin and

wings in aircraft. Ultrasonic guided waves are thickness resonances that propagate under

stress-free boundary conditions in plate-like structures called as waveguides. The guided

waves display frequency dispersion or velocity variation with frequencies that are

depicted using dispersion curves. Based on the nature of the cross-sectional displacement

distribution at each point on the dispersion curves, the guided waves are categorized into

modes. Selection of modes for sensitivity to interfacial weakness in a waveguide, and

understanding the mode conversion at geometric discontinuities or transitions in

waveguides is a major challenge to the inspection using guided waves.

The objective of this thesis is to enhance the understanding of guided wave

interaction with waveguide transitions and interfacial defects in bonded assemblies in

order to develop a field implementable solution for nondestructive inspection.

At the beginning, the problem of inspecting aircraft adhesive repair patches

applied to epoxy bonded aluminum – titanium joint is presented. Weakness at the

aluminum-epoxy interface and bulk defects in epoxy, simulated on repair patch samples

prepared in the lab, were successfully detected and sized by selecting modes with large

in-plane displacement at the interface of interest.

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The problem of inspecting adhesive skin-stringer joints requires an understanding

of the guided wave mode conversion and scattering at a waveguide transition in addition

to the need to detect interfacial weakness in the stringer joint region. The existence and

unique displacement characteristics for modes in a bonded joint that envelope or appear

near to modes in one of the adherends, observed in the study of skin-stringer joints, is

reported in the thesis. These modes are coined here as ‘mode pairs’ and show a matching

trend in the phase velocity vs. frequency curve with the adherend. A quantitative model

combining the Semi-Analytical Finite Element (SAFE) and Normal Mode Expansion

(NME) was developed to handle guided wave scattering at a waveguide transition.

Further, a qualitative model using wavestructure matching coefficient was also developed

to determine the mode conversion for a single mode incidence at the waveguide

transition.

The models showed equal excitation of the mode pairs and up to 100% energy

transmission for matching group velocity vs. frequency curve. Using a commercial Finite

Element package numerical experiments were conducted that agreed with the hybrid

model and the wavestructure matching model. Fourier transform based signal processing

algorithms for orthogonal decomposition of guided wave data into constituent normal

modes, directional and mode matching filters, computationally efficient element-less

receiver and wavestructure data processing were developed. The processing of numerical

data revealed the instantaneous mode formation at a source and also at a transition thus

enabling guided wave inspection of the edge of a bonded joint.

The mode conversion and scattering study were coupled with the interfacial in-

plane displacement parameter to form a combination parameter called Effectiveness

index. Higher effectiveness index modes (>0.3) successfully detected simulated

interfacial weakness in the skin-stringer joints prepared in the lab.

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TABLE OF CONTENTS

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

LIST OF TABLES.......................................................................................................xix

ACKNOWLEDGEMENTS.........................................................................................xx

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

1.1 Problem Statement..........................................................................................1 1.2 Structural Adhesive joints and Mechanical testing ........................................3 1.3 Literature Review ...........................................................................................6

1.3.1 Wave propagation modeling in structures............................................6 1.3.2 Simulation of wave propagation in waveguides...................................9 1.3.3 Nondestructive Inspection of Adhesive Joints .....................................10 1.3.4 Ultrasonic guided wave inspection of adhesive step-lap and stringer

joints .......................................................................................................18 1.3.5 Nonlinear ultrasonic techniques for adhesive bond inspection ............19

1.4 Challenges for further study ...........................................................................22 1.5 Thesis Objectives............................................................................................23 1.6 Contents of this thesis.....................................................................................24

Chapter 2 Analysis of guided wave propagation in plate-like structures and their transmission ..........................................................................................................26

2.1 Introduction.....................................................................................................26 2.2 Wave propagation modeling in plate-like structures ......................................27 2.3 Guided wave propagation in plate-like structures ..........................................28 2.4 Guided wave dispersion in a waveguide – an example ..................................33

2.4.1 Wavestructure.......................................................................................37 2.4.2 Power flow concepts.............................................................................41

2.5 Guided wave propagation in a bonded plate...................................................43 2.5.1 Material based phase velocity zones ....................................................45 2.5.2 Mode Pairs............................................................................................45

2.6 Summary.........................................................................................................50

Chapter 3 Finite Element Modeling and Analysis of wave propagation through waveguides ...........................................................................................................52

3.1 Introduction to numerical computation techniques in wave propagation.......52 3.2 FE Theory and implementation to simulate wave propagation using

ABAQUS.......................................................................................................54 3.3 FE for modeling wave propagation in infinite domains .................................57

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3.4 A brief review of the guided wave mode identification techniques ...............59 3.5 FE modeling of waveguide transitions in adhesive joints using ABAQUS ...61

3.5.1 Model of the adhesive bond .................................................................62 3.5.2 FE model of the adhesive joint transition.............................................64 3.5.3 Guided wave excitation using boundary conditions.............................64 3.5.4 Oblique incidence guided wave generation and reception ...................66

3.6 Some numerical experiments and data processing .........................................68 3.6.1 Processing single point data – Short time Fourier transform ...............69 3.6.2 Processing small line data on the surface of waveguide – Phased

addition...................................................................................................72 3.6.3 Processing large line data on the surface of waveguide –

Wavenumber filtering ............................................................................75 3.6.4 Processing line data across the waveguide thickness –

Wavestructure data .................................................................................87 3.7 Summary.........................................................................................................94

Chapter 4 Ultrasonic Guided Wave Inspection of Adhesive Repair Patches .............96

4.1 Introduction.....................................................................................................96 4.2 Literature on adhesive repair patch inspection ...............................................97 4.3 Ultrasonic guided wave propagation through a repair patch ..........................102 4.4 Lamb wave mode selection ............................................................................106

4.4.1 Displacement wavestructure.................................................................107 4.4.2 Interfacial in-plane displacement for defect sensitivity........................109 4.4.3 Influence of adhesive thickness............................................................110 4.4.4 Interface selectivity of a defect sensitive mode....................................114

4.5 Experimental Work.........................................................................................116 4.5.1 Fabrication of repair patch samples with controlled interface

conditions ...............................................................................................116 4.5.2 Mechanical Testing ..............................................................................121 4.5.3 Ultrasonic water immersion C-Scan.....................................................123

4.6 Design of sensor configuration for selective excitation of modes..................126 4.6.1 Wedge loading and source influence study ..........................................126 4.6.2 Pitch-catch data using wedge transducer..............................................129

4.7 Summary.........................................................................................................137

Chapter 5 Guided Wave Inspection of Adhesive Skin-Stringer Joints.......................139

5.1 Introduction.....................................................................................................139 5.2 Literature on guided wave propagation across stringer joints and other

waveguide transitions ....................................................................................143 5.3 Guided wave propagation and dispersion in bonded joints ............................149 5.4 Guided wave mode conversion at a transition................................................152 5.5 Hybrid model for guided wave scattering at a transition................................153

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5.5.1 SAFE ....................................................................................................153 5.5.2 Scattering at a waveguide transition using Normal Mode Expansion..158 5.5.3 Case study: Analysis of waveguide transitions using the hybrid

SAFE-NME............................................................................................161 5.5.4 Reciprocity check .................................................................................165

5.6 Dispersion curve matching and wavestructure based analysis .......................168 5.6.1 Inverse wavestructure matching coefficient .........................................178 5.6.2 Mode Transfer Function for guided wave propagation across

waveguide transitions.............................................................................180 5.7 Finite Element evaluation of transmission across a waveguide transition .....183 5.8 Guided wave mode sensitivity to interfacial defects in bonding....................185

5.8.1 Inspection chart for a Stringer Joint .....................................................186 5.8.2 Observations from Effectiveness charts ...............................................191 5.8.3 Conclusions from mode effectiveness study ........................................193

5.9 Experiments on skin-stringer joints ................................................................194 5.9.1 Fabrication of skin-stringer adhesive joint samples .............................194 5.9.2 Ultrasonic oblique incidence guided wave inspection test bed ............196 5.9.3 Ultrasonic guided wave inspection of stringer joints ...........................198

5.9.3.1 Inspection at mode-frequency locations with low effectiveness index (EAB) ................................................................199

5.9.3.2 Inspection at mode-frequency locations with high effectiveness index (EAB) ................................................................201

5.10 Summary.......................................................................................................206

Chapter 6 Summary, Contributions and Future Directions ........................................209

6.1 Summary of this thesis work ..........................................................................209 6.2 Specific contributions and their impact ..........................................................213 6.3 Future research................................................................................................215

6.3.1 Inspection of composite skin-stringer joints.........................................215 6.3.2 Modeling waveguides with a continuous transition .............................216 6.3.3 Inspection of skin-stringer joints using sensors mounted on stringer

surface ....................................................................................................217 6.3.4 Sensor design for optimal mode generation .........................................219 6.3.5 Nonlinear ultrasonic waves for damage detection in bonded joints.....219 6.3.6 Wave propagation across waveguides with multiple branches ............220

References....................................................................................................................222

Appendix A Two-Dimensional Fast Fourier Transform (2DFFT) .............................231

Appendix B Non-technical abstract ............................................................................233

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LIST OF FIGURES

Figure 1-1: Normal incidence reflection (blue) and transmission (black) factors - Good interface (solid), adhesive weakness (+) and cohesive weakness (>).........15

Figure 1-2: Schematic of interfacial spring model used to model an adhesive interface ................................................................................................................16

Figure 1-3: An adhesive step-lap joint and a simplified skin-stringer joint ................18

Figure 2-1: A general multi-layered structure with the coordinate system. ................28

Figure 2-2: Phase and group velocity dispersion curves for Lamb and SH type waves in a 2 mm thick aluminum plate. ...............................................................35

Figure 2-3: Phase and group velocity dispersion curves for Lamb wave modes in a 2 mm aluminum plate. The symmetric and antisymmetric modes have been labeled on the charts. ............................................................................................37

Figure 2-4: Displacement, stress and Poynting’s vector for wave propagation along an aluminum (2 mm thick) waveguide. The first column is for a0 mode propagation and the second column for s0 mode propagation at 0.3 MHz...........40

Figure 2-5: Superposition of phase and group velocity dispersion curves for aluminum plate (2 mm) and bonded aluminum joint. The modes in aluminum are labeled using an alphabet and a subscript whereas the modes in the bond are numbered incrementally. ................................................................................44

Figure 2-6: Wavestructure of the mode 2 at 3 MHz. The peak displacements are within the epoxy layer. At this frequency, the waveguide is similar to an embedded epoxy layer within aluminum half-spaces...........................................46

Figure 2-7: The normalized displacement wavestructure for s0 mode in aluminum (left) and mode 2 in bonded aluminum (right) at 200 kHz are shown. The match between the displacement components in both waveguides can be clearly seen. ..........................................................................................................47

Figure 2-8: The normalized displacement wavestructure for s0 mode in aluminum (left) and for mode 3 (right top) and mode 4 (right bottom) in bonded aluminum at 800 kHz are shown. The wavestructures match very well. .............48

Figure 2-9: The normalized displacement wavestructure for s2 mode in aluminum (top centre) and for mode 10 (middle row) and mode 11 (bottom row) in bonded aluminum. The match between the displacement components in both waveguides can be clearly seen. ...........................................................................49

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Figure 3-1: Model of solid-solid interface using (a) normal and tangential stiffnesses (spring model) with springs controlling interface strength, and (b) three layered model of adhesive with the layers 1 and 3 being used to model interfacial weakness..............................................................................................63

Figure 3-2: 2-D finite element model of a simplified skin-stringer adhesive joint. The rectangular elements are plane strain elements having 4 nodes. The thickness of the adhesive layer has been exaggerated for visual clarity of the bonded layup.........................................................................................................64

Figure 3-3: Methods for excitation of guided wave modes within a FE model. (a) Wedge loading, (b) comb loading with simultaneous or time-delayed inputs to the transducers and (c) Loading at the edge with wave-structure of the desired guided wave mode....................................................................................66

Figure 3-4: Schematic of the time delay based loading/receiving to simulate an oblique incidence loading/reception by a transmitter-receiver (T/R). The triangle showing the horizontal spacing between measurement location (dx), the oblique loading/reception angle (θi) and the delay length (dl) is shown on the right. A Hanning weight is also included to make simulation close to the practical case.........................................................................................................67

Figure 3-5: Schematic of the geometry for numerical experiment #1. Time delays were used to simulate 40°incidence wave impingement from water. The measurement node on the surface of the 2 mm thick aluminum plate. ................70

Figure 3-6: Amplitude vs. time plot of a guided wave mode (a0) propagating in 2 mm thick aluminum is shown on the top (black line) along with its Hilbert transform based envelope (blue line). On the bottom plot the short time Fourier transform of the RF waveform is shown with superimposed white lines that correspond to the appropriately scaled group velocity dispersion curves for a 2 mm aluminum plate. The data is from numerical simulation using ABAQUS. ...................................................................................................71

Figure 3-7: Schematic of the geometry, loading and measurement set used in numerical experiment #2. The wavestructure of s0 mode at 300 kHz is used for guided wave excitation. The measurement nodes at the surface of aluminum are also shown. ....................................................................................73

Figure 3-8: Phased addition based simulated received waveforms at different angles from –π/2 to π/2 radians (-90° to 90°) in an acrylic wedge. The positive angles are measuring the incident waves and the negative angles the waves reflected from the waveguide transition. The radial lines are the time axis. .......................................................................................................................74

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Figure 3-9: FE model of an adhesive step-lap joint along with the data collection and processing scheme. A wave-structure based displacement loading is provided on the left end of the geometry to create single mode incidence at the joint. ................................................................................................................77

Figure 3-10: Displacements measured at the incident side of the FE geometry. The wave incident at the bonded joint gets reflected and the same wave undergoes further reflection. The incident, reflected and the re-reflected waves overlap at several locations on the geometry.............................................78

Figure 3-11: Displacements recorded from the FE solution to wave propagation across a single step-lap joint. ................................................................................79

Figure 3-12: Frequency (ω) -wave number (k) plots corresponding to the incident, reflected and the transmitted wave fields obtained by transformation of displacements using the 2DFFT. The reflected field is the result of directional filtering obtained after removing re-reflected waves............................................81

Figure 3-13: An example of the windowing used for defining guided wave mode matching number filter is shown for the incident mode. The white lines mark the lower and higher cut-off values of wave number at every frequency value...82

Figure 3-14: Guided wave mode matching filters with rectangular weighting (left) and Gaussian weighting (right).............................................................................83

Figure 3-15: (a) Reflected and (b) transmitted waveforms separated into constituent modes using the guided wave mode matching filter. .........................84

Figure 3-16: Guided wave modes transmitted across the overlap region in a step-lap joint for an s0 mode incidence.........................................................................86

Figure 3-17: Ultrasonic loading function for exciting guided wave mode(s) and the data collection scheme used in the FE model. ................................................88

Figure 3-18: Schematic of the geometry, loading and measurement set used in numerical experiment #4. A 5 element comb loading on the surface at 0.5 MHz for 3 cycles generates a0 mode. The measurement nodes across the cross-section of aluminum are also shown. ..........................................................89

Figure 3-19: Fourier transform based scheme for extracting wavestructure data at a cross-section (located at x1) using the displacement from FE at the nodal points u(x2,t) at that cross-section. ‘u’ includes both in-plane (u1) and out-of-plane (u2) components. .........................................................................................90

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Figure 3-20: The magnitude of displacement (U) and phase (φ) of components (both in-plane and out-of-plane) at a frequency of 0.5 MHz across the thickness of the 2 mm thick aluminum sample when excited by a 5 element comb load with λ = 4.7 mm and a 3 cycle Hanning windowed pulse at 0. 5 MHz. The values were calculated from cross-sectional nodes located at the end of the comb source. ........................................................................................91

Figure 3-21: FE wavestructure values after incorporating phase correction. The extracted wavestructure matches the a0 mode wavestructure at 0.5 MHz. ..........92

Figure 3-22: Wavestructure data extracted from the FE model of the aluminum to epoxy bonded aluminum transition. The wavestructure is very close to that of mode 3 in bonded aluminum joint. .......................................................................93

Figure 4-1: Top: Cracks in the upper attachment flange in the longeron of an F-16. Bottom: Titanium (0.5 mm) repair patch bonded at the crack location on the longeron. [Modified from source: t’Hart and Boogers, 2002]........................97

Figure 4-2: Material layers in a typical aircraft adhesive repair patch. The aluminum layer represents the aircraft skin on which the titanium repair patch has been bonded using epoxy adhesive. The coordinate system with two representative conventions is also shown. ............................................................103

Figure 4-3: Lamb wave phase velocity dispersion curves for the aircraft adhesive repair patch: Aluminum (3.175 mm)-Epoxy (0.66 mm)-Titanium (1.6 mm). The Lamb wave modes are identified with numbers............................................104

Figure 4-4: Lamb wave group velocity dispersion curves for the aircraft adhesive repair patch: Aluminum (3.175 mm)-Epoxy (0.66 mm)-Titanium (1.6 mm). The Lamb wave modes are identified with numbers............................................106

Figure 4-5: Lamb wave dispersion curves for aluminum-epoxy-titanium adhesive repair patch and two wave structures or cross-sectional displacement profiles (at locations 1 and 2 on the dispersion curves). The dotted lines demarcate the aluminum, epoxy and the titanium regions, with aluminum being at the bottom. A larger in-plane displacement (ux) at the aluminum-epoxy interface can be noticed at location 2. .................................................................................108

Figure 4-6: Amplitude map of the in-plane displacement at the aluminum-epoxy interface superimposed over the Lamb wave dispersion curves for the adhesive aircraft repair patch comprised of a titanium patch (1.6 mm) bonded using epoxy (0.66 mm) onto an aluminum skin (3.175 mm). ..............................109

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Figure 4-7: Lamb wave phase velocity dispersion curves for the aircraft adhesive repair patch: Aluminum (3.175 mm)-Epoxy (t mm)-Titanium (1.6 mm). The value of t varies from 0.4318 mm to 0.8636 mm centered at 0.6604 mm............112

Figure 4-8: Amplitude map of the in-plane displacement at the aluminum-epoxy interface superimposed over the Lamb wave dispersion curves for the adhesive aircraft repair patch comprised of a titanium patch (1.6 mm) bonded using epoxy (t mm) onto an aluminum skin (3.175 mm). The values of t are (a) 0.4318 mm and (b) 0.8636 mm. ......................................................................113

Figure 4-9: Amplitude map of the in-plane displacement at the titanium-epoxy interface superimposed over the Lamb wave dispersion curves for the adhesive aircraft repair patch comprised of a titanium patch (1.6 mm) bonded using epoxy (t mm) onto an aluminum skin (3.175 mm). t varies clockwise as 0.4318 mm, 0.6604 mm and 0.8636 mm..............................................................115

Figure 4-10: Key steps employed in fabricating epoxy bonded titanium-aluminum adhesive repair patch. The arrows guide the process from the beginning to end.........................................................................................................................119

Figure 4-11: Temperature and pressure conditions used in the autoclave cure of the adhesively bonded repair patch. The adhesive – EA9696 dictates the cure profile....................................................................................................................120

Figure 4-12: Side view of the ASTM 3165 tensile test specimen cut from the bonded repair patch sample. Notches were machined through either side of the test specimen to create a 0.5” overlap.............................................................121

Figure 4-13: Static test results on representative ASTM 3165 test specimens. The overlap length was 0.5” for all specimens. The width of all specimens was 1” except the weak repair sample where the width was 0.5”. ...................................122

Figure 4-14: A typical RF waveform collected from ultrasonic immersion C-scan of the repair patch sample. The time windows are marked using numeric labels. ....................................................................................................................124

Figure 4-15: Ultrasonic water immersion C-Scan amplitude image for adhesive repair patch sample at time gates 4, 5 and 6 corresponding to the interface signal. Gates 3 and 5 show some contour lines which corresponds to the thickness variation in the adhesive. Gates 4 and 6 show the defects more clearly. In gate 6 image, the circular regions seen are blend outs or machined cavities. They have not been included in this thesis work....................................125

Figure 4-16: Geometric influence of a 6 mm diameter transducer mounted on a 10° acrylic angle beam wedge and supplied with a 2.5 MHz tone burst input

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voltage for 5 cycles on the range of phase velocities and frequencies excited. The white lines are the Lamb wave phase velocity dispersion curves for the repair patch. ..........................................................................................................129

Figure 4-17: Hilbert transformed ultrasonic guided wave RF waveform from frequency sweep experiment in pitch-catch mode using piezoelectric transducers mounted on a variable angle beam wedge set to an angle of 10°. Maximum energy transfer occurs in the frequency range of 2-3 MHz corresponding to mode 18 in the repair patch. .....................................................131

Figure 4-18: Variation in the energy transmission across regions with simulated interfacial weakness in the aircraft adhesive repair patch specimen. A pair of variable angle beam wedges set to 10° of incidence and reception in pitch-catch configuration was used to collect the frequency swept tone burst signals transmitted along a short distance of the repair patch sample. The collected signals were squared and summed to obtain the energy quantity. The results are presented in a normalized energy scale. .........................................................132

Figure 4-19: Schematic of the ultrasonic Lamb wave pitch-catch measurement configuration on the adhesively bonded repair patch. The wedge angles are 10° from the vertical and the transmitter (T) and the receiver (R) are both commercial piezoelectric transducers rated at 2.25 MHz. The wedges were separated by a distance of around 38 mm.............................................................133

Figure 4-20: Amplitude vs. time chart or the RF waveform measured by placing 10° wedge mounted 2.25 MHz transducers across the region to be inspected is shown for the different simulated bond interface conditions in the repair patch samples is shown on the left. The corresponding frequency content, obtained using Fast Fourier Transform (FFT) is shown on the right column. It can be noticed that there is a significant amplitude based difference between the repair patch samples with simulated interfacial conditions. ...........................134

Figure 4-21: Guided wave scan (G* scan) images (using mode 18) obtained from a linear scan using 10° wedge with 2.25 MHz ultrasonic transducers mounted on top and the whole assembly oriented in pitch-catch mode across the region with simulated interface conditions. The discontinuity in the first arriving wave seen from the above G* scan image approximately spans the length of the defect at the aluminum-epoxy interface..........................................................136

Figure 5-1: Internal view of the fuselage of a Boeing Dreamliner. The longitudinal stringers can be either adhesively bonded to the skin or co-cured with the skin. [Source: Boeing webpage - http://www.boeing.com/]...................139

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Figure 5-2: Schematic of a simplified skin-stringer adhesive joint used in this study. The thickness of the adhesive layer has been exaggerated for visual clarity of the bonded layup. ..................................................................................141

Figure 5-3: A typical epoxy bonded aluminum step-lap joint. The thickness of the epoxy layer is shown exaggerated for visual clarity.............................................142

Figure 5-4: Discrete waveguide transition found in adhesive skin-stringer joints and adhesive step-lap joint....................................................................................142

Figure 5-5: Superposition of phase and group velocity dispersion curves for aluminum plate (2 mm) and bonded stringer joint (aluminum 2 mm-epoxy 0.3 mm – aluminum 2 mm). The modes in aluminum are labeled using an alphabet and a subscript whereas the modes in the bond are numbered incrementally. .......................................................................................................150

Figure 5-6: One dimensional discretization of the thickness of a general layered waveguide. The 3 node isoparametric element used in discretization is shown in the inset.............................................................................................................154

Figure 5-7: A zoomed view of the discretized transition region between waveguides 1 and 2 for analysis using the hybrid-SAFE-NME method is shown. The nodes are marked with black circles. The portion of the interface satisfying the continuity conditions and the free boundary are shown.................159

Figure 5-8: Amplitude reflection and transmission factors for in-plane displacement computed using the hybrid-SAFE-NME method for an abrupt step change and a bonded lap joint. ......................................................................162

Figure 5-9: A typical skin-stringer joint with the discretized cross-section at the transition. The aluminum layers are 2 mm thick and the epoxy bond layer is 0.3 mm thick. The region on the left and right sides of the transition are labeled as A and B respectively............................................................................163

Figure 5-10: Energy partitioning among modes in the bonded stringer for transmission past the transition from aluminum (A) to the bonded stringer (B) are shown as intensity maps. The case of incidence of modes 1 (a0), 2 (s0), 3 (a1) 4 (s1), 5 (s2) and 6 (a2) in waveguide A are shown........................................164

Figure 5-11: Energy partitioning among modes in the aluminum for transmission past the transition from bonded stringer (B) to aluminum (A) are shown as intensity maps. The case of incidence of modes 1 to 6 in waveguide B is shown....................................................................................................................166

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Figure 5-12: Reciprocity checks for the hybrid analytical SAFE calculations for mode scattering at a transition. The dotted lines correspond to

jBAT1

and the

colored lines correspond to1AB j

T ...........................................................................167

Figure 5-13: A discrete waveguide transition found in aircraft skin-stringer joints is shown. For convenience, the waveguide where the wave is incident is defined here as the primary waveguide. The geometry after transition is referred to here as the secondary waveguide. The solid vertical line is used to establish the demarcation between the primary and secondary waveguides artificially and also to denote the interface Г common to waveguides A and B ..168

Figure 5-14: Wavestructure matching coefficient ρAB(1,mB) for mode 1 (a0) incidence from the aluminum aircraft skin (A) onto the bonded stringer joint region (B) is shown superimposed on the phase velocity dispersion curves of the bonded stringer joint (B). The color scale value varies from 0 to 2 and represents the value of ρAB(1,mB). .........................................................................172

Figure 5-15: Energy partitioned wavestructure matching coefficient ( )1PABρ i.e. for

mode 1 (a0) incidence from the aluminum aircraft skin (A) onto the bonded stringer joint region (B) is shown superimposed on the phase velocity dispersion curves of the bonded stringer joint. The color scale value is set to an auto scale so that the individual contributions are clear. .................................174

Figure 5-16: (a) Wavestructure matching coefficient ρAB(2,mB) and (b) its energy partitioned form ( )2P

ABρ (bottom plot) for mode 2 (s0) incidence from the aluminum aircraft skin (A) onto the bonded stringer joint region (B) are shown superimposed on the phase velocity dispersion curves of the bonded stringer joint. Note the difference in color scales due to the different maximum values...................................................................................................176

Figure 5-17: The energy partitioned wavestructure matching coefficients corresponding to the propagation of aluminum modes 3-8 (A) into the bonded stringer (B) are shown superimposed on the phase velocity dispersion curves of the bonded stringer joint. Note the variation in the intensity scales due to the redistribution of energy into the normal modes at each frequency.................177

Figure 5-18: The discrete waveguide transition found in aircraft skin-stringer joints is shown. The primary and secondary waveguides are labeled. The solid vertical line is used to establish the demarcation between the primary and secondary waveguides artificially and also to denote the interface Г common to waveguides B and A. .........................................................................178

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Figure 5-19: The energy partitioned wavestructure matching coefficients ( )BPBA mρ

corresponding to the propagation of modes 1-6 from the bonded stringer (B) into the aluminum skin (A) are shown superimposed on the phase velocity dispersion curves for an aluminum plate (2 mm). Note the variation in the intensity scales due to the redistribution of energy into the normal modes at each frequency. .....................................................................................................179

Figure 5-20: Waveguide transitions (A-B and B-A) in a bonded stringer joint along with the proper labels to denote the regions. The solid vertical lines are used to establish the demarcation between the different waveguide regions in the stringer joint. ГL and ГR denote the left and right interfaces common to waveguides A and B. ............................................................................................181

Figure 5-21: Guided wave mode transfer function for a stringer joint for different modes propagating in the primary waveguide (A) through the secondary waveguide (B) to the waveguide A, are shown superimposed on the phase velocity dispersion curves for an aluminum plate (2 mm). ..................................182

Figure 5-22: Snapshots from the FE model showing the interaction of s0 mode at 300 kHz with the transition from waveguide A to waveguide B. ........................184

Figure 5-23: Snapshots from the FE model showing the interaction of s1 mode at 2.36 MHz with the transition from waveguide A to waveguide B. ......................185

Figure 5-24: Intensity map of the in-plane displacement at the aluminum-epoxy interface superimposed over the Lamb wave dispersion curves for the adhesive stringer joint comprised of two epoxy bonded aluminum plates (2 mm).......................................................................................................................186

Figure 5-25: Effectiveness index EAB of the first two modes in aluminum (a0 and s0 respectively) for inspection of the interfacial defects in the bonding across the waveguide transition in a stringer joint is shown mapped on the dispersion curves of the bonded stringer region. The dotted red line marks the mode in aluminum shown in the parenthesis on the plot title. .............................188

Figure 5-26: Effectiveness index EAB of the modes 3 and 4 in aluminum (a1 and s1 respectively) for inspection of the interfacial defects in the bonding across the waveguide transition in a stringer joint is shown mapped on the dispersion curves of the bonded stringer region. The dotted red line marks the mode in aluminum shown in the parenthesis on the plot title. ...........................................189

Figure 5-27: Effectiveness index EAB of the modes 5 and 6 in aluminum (s2 and a2 respectively) for inspection of the interfacial defects in the bonding across the waveguide transition in a stringer joint is shown mapped on the dispersion

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curves of the bonded stringer region. The dotted red line marks the mode in aluminum shown in the parenthesis on the plot title. ...........................................190

Figure 5-28: Dimensioned sketch of the aluminum skin-stringer adhesive joint sample fabricated at Penn State University. .........................................................195

Figure 5-29: Ultrasonic guided wave excitation methods – comb loading (wavelength spaced piezoelectric loading) and variable angle beam acrylic wedge with mounted piezoelectric transducer......................................................196

Figure 5-30: A Goniometer with an ultrasonic immersion transducer attached to it. The Goniometer permits orienting the transducer at any angle of incidence (< 50°) desired. The incidence angles are measured from the vertical for experiments...........................................................................................................197

Figure 5-31: Ultrasonic oblique incidence pitch-catch inspection in a water immersion mode. Each of the goniometers holding the transducers can be moved independently of the other along the line joining the two transducers. ....198

Figure 5-32: RF signals and their fast Fourier transforms obtained from transmission measurements for an s0 mode generated using a tone burst input of 0.5 MHz for 5 cycles and oblique incidence at 16° in water. (a) Aluminum (2mm) plate, (b) good stringer joint, (c) stringer joint with adhesive (Teflon) weakness and (d) stringer joint with cohesive (Teflon) weakness. ......................200

Figure 5-33: RF signals and their fast Fourier transforms obtained from transmission measurements for a Rayleigh wave generated using a tone burst input of 5 MHz for 5 cycles and oblique incidence at 32° in water. (a) Aluminum (2mm) plate, (b) good stringer joint, (c) stringer joint with adhesive (Teflon) weakness and (d) stringer joint with cohesive (Teflon) weakness. ..............................................................................................................201

Figure 5-34: RF signals and their fast Fourier transforms obtained from transmission measurements for an a1 mode generated using a tone burst input of 2.3 MHz for 5 cycles and oblique incidence at 36° in Plexiglas wedge. (a) Aluminum (2mm) plate, (b) good stringer joint, (c) stringer joint with adhesive (Teflon) weakness and (d) stringer joint with cohesive (Teflon) weakness. ..............................................................................................................202

Figure 5-35: Geometric influence of loading on the range of phase velocities and frequencies excited. The color intensity shows the strength of the ultrasonic excitation of a 12.5 mm diameter transducer oriented at an angle of 14° and supplied with a 1.5 MHz tone burst input voltage for 5 cycles. The white lines are the Lamb wave phase velocity dispersion curves for aluminum (2 mm).......................................................................................................................204

xviii

Figure 5-36: RF signals and their fast Fourier transforms obtained from transmission measurements for s1 and a1 mode generated using a tone burst input of 1.5 MHz for 5 cycles and oblique incidence at 14° in water. (a) Aluminum (2mm) plate, (b) good stringer joint, (c) stringer joint with adhesive (Teflon) weakness and (d) stringer joint with cohesive (Teflon) weakness. ..............................................................................................................205

Figure 6-1: A constant stiffness metal-composite joint. ..............................................217

Figure 6-2: Schematic of the inspection of a stringer joint using sensors mounted on top of the joint..................................................................................................218

Figure 6-3: A portion of the bridge truss from the failed Minnesota bridge. The gusset plate encircled is buckled under load. The joint formed by the elements of the truss can be considered as a complex waveguide with multiple connections for a guided wave analysis. [www.minnesota.publicradio.org/display/web/2008/11/12/ntsb_bridge] ............220

Figure A-1: The ω−k space with the four quadrants. Quadrants I and III and, quadrants II and IV are related. ............................................................................232

xix

LIST OF TABLES

Table 1-1: Advantages and limitations of adhesive bonding.......................................4

Table 1-2: Limiting values of interfacial spring stiffness............................................16

Table 2-1: Guided wave phase and group velocities for modes s2, 10 and 11 at different frequencies used in this study ................................................................48

Table 3-1: Definition of the directional filters.............................................................80

Table 4-1: Wave propagation velocities and the computed elastic modulus values for materials used in this study .............................................................................105

Table 4-2: Summary of the mode and frequency (~2.5 MHz) combination with larger in-plane displacement at the aluminum-epoxy interface for different epoxy thicknesses in the titanium (1.6 mm) – epoxy (t mm) – aluminum (3.175 mm) bonded media. Locations with phase velocity close to 15 km/s has been tabulated.................................................................................................114

Table 4-3: Average value of the shear strength of adhesive (MPa) obtained from ASTM 3165 tests on specimens fabricated with different simulated interface conditions..............................................................................................................122

xx

ACKNOWLEDGEMENTS

I would like to express my gratitude to my advisor Dr. Joseph Rose, whose

guidance on both technical and professional fronts was invaluable. His teachings on life

will provide me guidance for the rest of my career. Thanks also to my co-advisor Dr.

Cliff Lissenden. His thought provoking questions has always made me more thorough in

my research. The encouragement and support for my work from both my advisors

through the length of the degree program was a key to the successful completion of this

thesis. Without their continuous financial support, realization of this thesis would have

been impossible. I would like to express my thanks to Dr. Bernhard Tittmann for

supporting my research and also extending his lab facilities to me. Thanks also to Dr. Ed

Smith for the support and encouragement that he extended to me and for the good

suggestions.

I thank NASA Aircraft Aging and Durability Project under agreement number

NNX07AB41A for financial support to my research on stringer joints over a major part

of my PhD term. Thanks also to Air Force Robins for supplying materials and sharing

their knowledge for preparing repair patch samples. Thanks to FBS, Inc (State College

PA) and Intelligent Automation Inc (Rockville, MD) for directly supporting the repair

patch work. The support from the research group FBS, Inc., especially Dr. Mike Avioli,

is also acknowledged. The timely help from Dr. Jose Galan, University of Seville, with

the work on hybrid model is also greatly appreciated.

I am grateful to several groups/labs at Penn State for supporting different parts of

the experimental portion of my work. These include Dr. Charles Bakis (Composites lab),

Dr. Tom Juska and Mr. Chris Rachau (Applied Research Lab), Dr. Barbara Shaw

(Corrosion lab), Mr. C. Baird (Learning factory), Dr. Cliff Lissenden (Material testing

lab), Dr. Bernhard Tittmann (Nanoimaging lab), Dr. Joseph Rose (Ultrasonics lab) and

Dr. Ed Smith (Rotorcraft Centre of Excellence). I would like to thank all the students at

the Ultrasonics lab for all their help through out my stay at Penn State.

Above all I would like to thank my parents and sisters for their support and

encouragement throughout my education.

Chapter 1

Introduction

1.1 Problem Statement

Engineering structures are generally multi-component assemblies which need to

be joined together to form a single load bearing arrangement. The art of joining materials

for structural applications started as an extension of the wood based structural design.

From the use of mortise and tenon joints with tapered pegs as connectors, the design of

structural joints developed in sync with the development in the metallurgical and

manufacturing technologies to accommodate mild steel sections joined using bolted

connections and later using rivets. With the growth of the aerospace industry, the

demands on joining technologies also increased due to the need to efficiently join

structural members in an aircraft to form a single load bearing flight-capable structure

and also the use of polymer matrix composites. There are many joining techniques like

threaded fasteners, riveting, welding, and bonding that can be used in structural joints, of

which the adhesive bonding technology is getting greater acceptance and implementation

in the recent decades due to some specific advantages that it provides.

In aircraft, adhesives are employed in many structural joints, most important

being the attachment of stringers to fuselage and wing skins. The adhesive joints are also

used for bonding life extending patches at defect locations in aircrafts. This provides a

cheaper alternate to replacement of parts and at a lower downtime. Structural adhesives

2

withstand load in the structure and contribute to its strength and stiffness. Their structural

integrity is crucial for the operational safety of the aircraft. The material used in the

adhesive undergoes degradation under the environmental conditions resulting in

weakness of the joint. Also, there are defects that occur at the adhesive-adherend

interface that result in interfacial weakness of the joint. For these reasons, the inspection

of adhesive joints is critical. As composite materials are used more and more as primary

structural components, the need to inspect or monitor adhesively bonded joints between

composite laminates will increase further. For in-situ monitoring, a non-destructive

inspection technique becomes important.

Ultrasonic wave propagation through a structure depends on the elastic properties

of the medium, and hence it can be used to characterize the mechanical properties.

Ultrasonic wave propagation based methods are nondestructive techniques and have been

studied for the last few decades [Rose 2002] as a possible tool for inspection of

adhesives. The techniques adopted in ultrasonic non-destructive evaluation of adhesive

bonds are detailed in the review article by Rose [2002]. Broadly, the methods can be

classified into the ones that use bulk waves for inspection and the ones that use guided

Lamb-type waves. While the bulk wave based methods inspect only a local region and

are not feasible for timely inspection of large aerospace structures, the use of guided

waves enables a global inspection in addition to providing multiple modes for inspection.

The guided wave approach becomes an attractive choice.

Since guided wave propagation is dependent on the waveguide geometry, its use

for inspection requires an understanding of the frequency dependent characteristics of the

waveguide and the displacement distributions corresponding to the multiple modes

3

possible. Also, since the guided waves undergo mode conversion at geometric

discontinuities or transitions like those that exist in a structural adhesive joint, it is

important to quantify this effect using wave mechanics studies before developing

strategies for inspection.

1.2 Structural Adhesive joints and Mechanical testing

Adhesives are mostly chemical polymers which can be applied to the surfaces of

two materials to join them in order to withstand separation after consolidation. The

materials being joined are called the adherends or substrates.

The ASTM D 907 provides definition of the terms related to adhesive bonding as

listed below:

1. Adhesive – It is a substance capable of holding materials together by surface

attachment.

2. Structural adhesive – a bonding agent used for transferring required loads

between adherends exposed to service environments typical for the structure

involved.

3. Adherends – a body held to another body by an adhesive

4. Adhesion – the state in which two surfaces are held together by interphase forces.

In mechanical adhesion, the adhesive provides interlocking action which holds

together adherends.

5. Adhesive joint – the location at which two adherends are held together with

adhesive

Adhesive bonding has several advantages that make it an attractive alternate to

other mechanical joining techniques and also some limitations that make inspection

4

techniques important to ascertain its reliability after manufacture and while in service.

These are shown in Table 1-1.

Adhesives are strong in shear and weak in peel stress (i.e. normal separation

stresses in the through thickness direction). For the design of an adhesive joint, to

theoretically compute the stress distribution within the joint, to compare the performance

of different adhesives and the joint fabrication techniques, mechanical test results are

used as a benchmark. There are several mechanical testing methods to characterize

adhesives. They can be grouped into four categories – shear, tension, peel and fracture

toughness.

1. Shear test – The lap shear tests result in non-uniform stress distribution in the

adhesive layer. Generally, the measured experimental value is the averaged value of

the load to failure divided by the bonded area. The fact that the stresses at the end of

Table 1-1: Advantages and limitations of adhesive bonding

ADVANTAGES LIMITATIONS

Results in light weight, strong and stiff structures

Strength is sensitive to the extent of surface preparation of the adherends, and may need heat and temperature to cure

Uniformity of load distribution, hence low stress concentration and excellent fatigue strength

Environmental degradation because of moisture, presence of chemicals and other severe service conditions

Can join dissimilar materials like metal to composite and can be used to efficiently join thin metal/composite sheets

Temperature range in service is limited by the glass-transition temperature of the polymeric adhesive.

Being a non-conducting layer, it prevents galvanic corrosion when used to join reactive metals

Residual stresses can result when joining two metals with different coefficients of thermal expansion

Adhesives are resistant to corrosion Weak under peel stresses Can be used to create complex joints Disassembly is not possible without damage

to the joint Damps structural vibration and absorbs shock Reliable inspection techniques do not exist Good finish of joints

5

the bonded area are higher and that there is peel stress also present in the adhesive

is known, but not taken into account during calculation of the failure stress. Some

of the shear testing standards are - ASTM D 1002 Single lap joint in metals, ASTM

D 3528 Double lap joint in metals, ASTM D 3165 Laminated assemblies in metals,

ASTM D 5573-94 Composite joint testing, ASTM D 5656 Thick adherend metal

lap-shear test etc.

2. Tensile test – Tensile modulus and tensile strength of the adhesives can be

determined using some ASTM standards like - ASTM D 897-78 Butt joint, ASTM

D 2095-72 Sandwich butt joint, ASTM D 1344-78 Cross-lap joint.

3. Peel test – It can be used to determine the peel resistance of adhesive bonds between

relatively flexible adherends and between a relatively flexible adherend and a rigid

adherend e.g. ASTM D 903-49 180 degree peel test, ASTM D 1781-76 Climbing

drum peel test, ASTM D 1876-72 T Peel test.

4. Fracture toughness test – The adhesive joints can fail due to crack propagation in the

bondline or along the adhesive-adherend interface or in a combination of both these

methods. Fracture toughness tests provide the energy release rates of adhesives

sandwiched between substrates.

ASTM D 1062-78 provides comparative cleavage strengths of adhesive bonds

sandwiched between metal adherends.

ASTM D 3433-75 tapered double cantilever beam (DCB) test helps in determining

the fracture strength in cleavage of adhesive bonds which can help in design

improvements.

Based on the mechanical tests (ASTM D 907-08), the failure of the adhesive bond system

could be of three types

1. Adhesion failure – separation of adhesive-adherend interface leading to rupture

of the adhesively bonded assembly.

2. Cohesion failure or cohesive failure – separation of the adhesive in an adhesively

bonded joint leading to rupture of the adhesive and also the bonded assembly.

6

Substrate failure – The case of failure of the adherend in the adhesive joint. This

is a possibility in the case of composite co-cured joints where the material holding the

different plies and the material connecting different sections are the same give rise to the

possibility of the failure of the composite laminates because of delamination or matrix

cracking.

1.3 Literature Review

A brief review of the literature on modeling guided wave propagation in

structures, modeling and analysis of mode conversion in waveguide transitions, adhesive

joint inspection techniques is presented.

1.3.1 Wave propagation modeling in structures

The foundational work by Lord Rayleigh in 1885 on wave propagation in semi-

infinite elastic half-space paved the way for the analysis of wave propagation in

multilayered media. In 1917, Lamb studied the wave propagation in a free isotropic

elastic plate. Later, Stoneley generalized the single interface problem to describe the

waves traveling in the interface between two elastic solids, followed by Scholte who

studied a solid-liquid interface [Nayfeh 1995].

The first major work on the interaction of wave with a multi-layered media is due

to Thomson’s Transfer Matrix method [Thomson 1950] which involved the formulation

of a matrix to transfer the displacements and stresses from one interface to the other

7

resulting in a condensed matrix relating the first and the last interface. Haskell’s

correction to the formulation by Thomson resulted in the name Thomson-Haskell method

[Haskell 1953]. The numerical instability at large frequency-thickness values due to the

presence of decaying and growing components in the same matrix, was a major drawback

of this method. This problem was avoided by the use of a global matrix containing the

expressions for continuity at every interface [Knopoff 1964] resulting in a stable but

computationally slow formulation. Dunkin [Dunkin 1965] used a rearrangement of the

material properties to ensure stability at the cost of simplicity in formulation. Further

modifications to this were done by Levesque and Piche [Levesque and Piche 1992,

Castaings and Hosten 1994]. In order to analyze the wave propagation in anisotropic

periodic multilayered structures, Potel and Belleval [1993] proposed the Floquet wave

formulation. Balasubramaniam [2000] proposed a numerical truncation algorithm, which

limits the exponential terms and improves the stability of the transfer matrix based

formulation without any need for reformulation or any additional increase in

computational time. A recursive matrix approach relating the stresses at the top and

bottom of a layer to the displacements at the top and bottom of the layer called the

Stiffness-Matrix method was proposed by Wang and Rokhlin [2001] for unconditional

stability of the matrix formulation. The global matrix method (GMM) being simple for

implementation, yet intuitive is implemented in this study.

Waveguides and transitions

8

In this study, any solid medium with a smaller overall thickness compared to

length is considered as a waveguide for an analysis in the ultrasonic frequency range (i.e.

> 20 kHz). Combination of waveguides mechanically coupled to each other by means of

welding, adhesive bond, riveting etc is also referred to as a waveguide. A location along a

waveguide where there is a cross-section change is referred to as a transition.

Guided wave dispersion curves and the wave structure based analysis shows the

wave propagation characteristics in a waveguide. Important implicit assumptions for the

dispersion based analysis are that the cross-section of the waveguide is constant, the

material properties of the waveguide do not vary along the length of the waveguide and

boundary conditions are uniform throughout the length of the waveguide. Hence the most

common waveguide - an infinite plate can be handled as a one-dimensional problem in

wave mechanics. Bounded structures like rod, pipe, bar etc can be considered as two-

dimensional waveguides and the analysis of wave propagation through them in the form

of dispersion curves also implicitly considers a prismatic structure without any variation

in the material or boundary condition along the wave propagation direction.

Restricting this discussion to the case of a one-dimensional waveguide, there are

still a lot of engineering structures like the fuselage of an aircraft which come under this

sub-domain of waveguides.

Some other assumptions normally made in the waveguide dispersion analysis

such as stress-free boundaries and continuity of traction at the interfaces in the waveguide

material can be handled by applying elasticity principles.

9

Also, a dispersion based analysis does not answer the question of the length of

waveguide needed for the formation of a stable wave mode, whose possibility of

existence is predicted by the dispersion curve computation. This question becomes

relevant in the case of wave propagation analysis through structures having abrupt

transitions like stringers attached to an aircraft fuselage, ribs and spars attached to wing

skin etc involve joints where finite length waveguides are mechanically coupled by

means of joints such as adhesive joints. For the analysis of the wave propagation through

such structures, the simple analysis based on dispersion curves will not be sufficient

because of the cross-sectional variation in the waveguide geometry. Current literature

fails to address this issue.

1.3.2 Simulation of wave propagation in waveguides

Simulation is an excellent tool to visualize physical phenomena and to get an

insight into the problem without actually preparing samples or conducting experiments.

In the field of ultrasonic nondestructive evaluation, the use of simulation tools helps us to

understand the wave interaction with geometry, defects to aid inspection.

There are several numerical computation techniques that can be used to solve

structural mechanics problems like stress wave propagation hence forming the basis for

guided wave propagation based studies.

The boundary element method (BEM) can be applied to problems whose

governing equation has a Green’s function solution readily available. Cho and Rose have

10

applied BEM in the study of guided wave propagation and interaction with defects in

isotropic plates and also quantified the mode conversion effect [Cho and Rose 1996, Cho

and Rose 2000]. Material in-homogeneity and complications in geometry render this

method more cumbersome in application, which also explains why this technique is still

not widely employed [Brebbia and Dominguez 1989]. Techniques employing special

elements like the spectral element method require extra effort in formulation of the

element stiffness matrix to handle any non-standard geometry [Gopalakrishnan and

Doyle 1995]. The finite difference method is probably the closest to finite elements (FE)

in terms of implementation. The difficulty in handing complex geometry and the

difficulty associated with raising the order of the difference expression restricts the

application of this technique in wave propagation problems. The FE method is one of the

most used methods for simulation of guided wave propagation in structures. The ease of

modeling and formulation using the FE, availability of large number of commercial

software packages capable of handling material models and elements to model the

physics of the problem makes the FE method more preferable over the other numerical

techniques.

1.3.3 Nondestructive Inspection of Adhesive Joints

Defects in adhesive joints are classified broadly into cohesive and adhesive.

Micro level defects like voids, porosity, cracks and density gradients in an adhesive layer

causes weakness in the whole layer of the adhesive and are known as cohesive weakness.

Over-curing or under-curing, environmental degradation due to the presence of

11

temperature or humidity cause the cohesive weakness. Interfacial weakness due to the

presence of a disbond or complete breakage of bond between adhesive and adherend,

kissing bond are classified as adhesive weakness. The surface preparation plays an

important role in the bonding quality. Disbonds are caused due to poor surface

preparation or the incomplete removal of adhesive backing film (in case of film

adhesives).

There are several non-destructive techniques that can be applied towards

inspection of adhesive joints. They are presented in the next few sections.

Visual Inspection

Visual inspection of the adhesive fillet is considered a good indicator of the

adhesive quality. The presence of moisture in the fillet hints towards the presence of

porosity in the joint. Similarly the absence of fillet may indicate the presence of internal

void spaces. [Adams et al. 1997]

Optical holography and shearography

In the optical holography technique, the image from an object before and after

stressing is compared. The presence of defects will produce changes to the interference

pattern from the reflected laser light. The experiment should be conducted in a vibration

free environment. Optical shearography is very similar to the holography technique, but

is not sensitive to vibration unlike holography. The commonly used loads are in the form

of a pressure load (vacuum or positive) or heat. These techniques are successful in

detecting defects in honeycomb structures. [Niu 1993, Adams et al. 1997]

12

Radiography

This technique uses radioactive X-Rays or neutron sources and is ideal for the

inspection of composite materials. Metal bonds cannot be inspected using this method.

The propagation of X-rays through a material, like a composite, causes a part of the

energy to be absorbed and a part transmitted. The transmitted rays are collected on a

sensitive film. With the advent of advanced sensing technologies, digital recording of the

measurement can be made. Some applications of radiography include inspection of

honeycomb structures for core damage, misalignment, presence of foreign objects and

porosity. By using multiple receivers, a tomographic arrangement can also be realized for

three-dimensional (3-D) reconstruction of the interior of the structure. A serious safety

related disadvantage of this technique is the use of radioactive rays.

Thermal Imaging

Supplying heat to an area to be inspected and observing the temperature profile

using a thermal imaging camera is useful in applications like finding delamination in

composite laminates and leak detection in thermal ducts. This happens primarily due to a

difference in the thermal conductivity created due to the presence of a defect [Niu 1993].

Inspection of bonded metallic structures using thermal imaging is a challenging problem

because the high thermal conductivity of the metal requires the use of very short bursts of

energy.

Acoustic emission

This is a passive technique dependent on the measurement of the energy released

13

during the creation of free surfaces like formation of cracks. In order to make this an

active technique, load must be applied to an actual structure and the measurements should

be recorded using microphone or piezoelectric transducer.

Tap testing

Localized excitation of a structure using light impacts with a spherical nosed

impactor followed by measurement of the impact duration using accelerometer is called

mechanical tap testing. If the accelerometer is replaced by human ear, and the audible

resonant sound is used, the technique is called acoustic tap testing. Damage in sandwich

structures can be detected using these techniques.

Mechanical Impedance analysis

In this method, the stiffness of a structure is measured using contact probes. The

stiffness of a structure (mechanical impedance) changes with the presence of defects.

Piezoelectric crystals spaced within a single holder and making measurements in pitch-

catch mode is used in this inspection technique.

Ultrasonic techniques

Ultrasonic inspection techniques offer a wide range of choices for inspection

strategies.

1. Bulk waves based approach

14

The reflection or transmission of ultrasound at normal incidence from an

adhesively bonded sample was found to produce spectral shifts when the adhesive is

degraded (Figure 1-1 ). This was found to occur at very high frequencies of incidence and

the sensitivity of this method was found to increase with frequency [Pilarski and Rose

1988 a, b]. This method is clearly not feasible for measurement in attenuative media.

Ultrasonic A-, B- and C-scan based techniques using parameters like the amplitude,

phase or attenuation of the signal and time of flight can also be found in the literature.

These techniques are cumbersome and hence impractical for large area inspection. They

are successful in volumetric defect detection, but fail to detect the kissing bond kind of

defect.

The advantage of introducing shear energy at the interface [Pilarski and Rose

1988 a] encouraged researchers to attempt oblique incidence of ultrasonic wave at angles

larger than the first critical angle of the adherend media.

The ultrasonic energy reflection and transmission as a function of the angle of

incidence and frequency of inspection can be theoretically obtained by forming a matrix

of equations satisfying the interfacial continuity conditions for stresses. This technique of

using the spectral response of signals collected at oblique incidence is also known as

angle beam ultrasonic spectroscopy (ABUS) [Lavrentyev and Rokhlin 1994].

Mechanical models for interfaces often employ springs to model the interfacial

connection between adhesive and adherend [Baik and Thompson 1985, Pilarski and Rose

1988]. See Figure 1-2 .

15

Most of these models employ a normal and a tangential spring to model the

discontinuity of the displacements and continuity of the stresses. The expression for the

stress at the interface can expressed in terms of the displacement discontinuity as shown

by the Equation 1.1

where σ is the stress and K the interfacial stiffness which relate to the displacement

discontinuity u∆ .

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Ref

lect

ion/

Tran

smis

sion

Coe

ffic

ient

Water-Al-Epoxy-Al-Water

Figure 1-1: Normal incidence reflection (blue) and transmission (black) factors - Good interface (solid), adhesive weakness (+) and cohesive weakness (>).

uK ∆=σ (1.1)

itydiscontinuntDisplacemestiffnessTangential

stiffnessNormal113333

=∆==

∆=∆=

uKK

uKuK

T

N

TN σσ

(1.2)

16

Equation 1.2 relates the stresses to the displacement jumps.

In the literature, we can find the values of the spring stiffness KN and KT

employed to simulate different interface conditions for studying ultrasonic reflection and

transmission. The extreme cases of spring stiffness are listed in Table 1-2 .

An actual adhesive joint is supposed to possess interfacial spring stiffness in

between those of the welded and debonded case. Ultrasonic inspection strategy was thus

framed based on the sensitive range of angle and frequency found from theoretical study.

2. Guided wave based approach

For practical purposes, a low frequency of ultrasonic inspection is made possible

with the use of plate waves – Lamb type waves and SH waves, or in general, guided

Figure 1-2: Schematic of interfacial spring model used to model an adhesive interface

Table 1-2: Limiting values of interfacial spring stiffness

Sl. No. Interface condition KN KT

1 Perfect bonding (Welded) ∞ ∞

2 Perfect slip interface ∞ 0

3 Perfect debonding 0 0

17

waves. Guided waves are comprised of both longitudinal and shear motion and possess

multi-modal nature at every value of frequency, thus providing multiple points on the

phase velocity dispersion curves holding a potential for inspection.

Using a global matrix approach for forming the characteristic matrix for a

waveguide, Pilarski and Rose [1992] incorporated the interfacial spring stiffness

conditions listed in Table 1-2, to obtain the dispersion curves for a bonded joint. The

sensitivity of the dispersion curves to the variation in the bulk adhesive properties as well

as the properties at the interface was clearly demonstrated. Each point on the phase

velocity dispersion curve has a unique cross-sectional displacement variation and energy

distribution. From the infinite possibilities provided by the Lamb type waves, it was

shown by Pilarski and Rose [1992] that by a suitable definition of mode selection criteria;

say by power flow criteria at the interfacial region, it is possible to narrow down to a few

inspection points.

It was shown by Mal et al. [1989] that oblique incidence of ultrasonic waves on a

specimen in a water immersion mode results in the generation of Lamb-type waves

within the specimen, which leak into the surrounding fluid, interfere with the specularly

reflected waves to produce a leaky Lamb wave (LLW) spectra, which has characteristics

of Lamb wave dispersion curves. The LLW spectra were found to be sensitive to the bulk

and interfacial properties of an adhesive joint and hence useful for inspection of adhesive

joints.

18

A review article by Rose [2002] gives the details of the different schemes for

measurement using ultrasonic bulk-waves, Lamb waves, leaky waves and also very high

frequency techniques like scanning acoustic microscopy.

1.3.4 Ultrasonic guided wave inspection of adhesive step-lap and stringer joints

The most commonly studied adhesive joint configurations are the step-lap joint and the

stringer joint (Figure 1-3) because of their simple geometry and/or widespread use in

aerospace structures. A Lamb wave based approach is preferred for inspection of such

joints because of the large area inspection possibility and the availability of multiple

mode combinations. In the literature we can find many articles related to wave

propagation in these bonded joints. Rokhlin [1991] studied the wave propagation in the

bonded geometries and suggested guidelines for the inspection. These guidelines are the

same as the ones suggested by Rose [1999]. They are stated below for completeness.

(i) Select a wave mode and frequency such that the wave excited in a single layer is

still sensitive to the interface conditions when it enters the bonded region.

(ii) Find Lamb wave modes which have to be excited in the single layer such that it

gets transformed to the required mode within the interface region

Figure 1-3: An adhesive step-lap joint and a simplified skin-stringer joint

19

Lowe and co-workers used finite elements and experiments to study the

propagation of the first three Lamb wave modes across a step-lap joint [Lowe et al.

2000]. They computed the transmission coefficient of different Lamb wave modes for

variation in the geometry of the joint. Later, di Scalea et al. [2004] studied the

propagation of the fundamental antisymmetric mode (a0) across a step-lap joint for

different states of adhesive – namely the fully cured, poorly cured and slip interface.

They measured the energy transmission for these different bonding conditions. Matt et al.

[2005] extended this work for the case of composite joints and found that there is an

increase in transmission in the case of a bond with defects.

None of these works provide an approach to determine modes that when

introduced in the skin of an aircraft can convert to modes sensitive to the interface within

the stringer region. So an understanding of mode conversion at a waveguide transition is

clearly lacking. The literature also lacks clarity with regards to selection of modes that

are sensitive to the interface of a layered media like the adhesive-adherend interface in a

bonded stringer or a bonded repair patch.

1.3.5 Nonlinear ultrasonic techniques for adhesive bond inspection

The presence of defects in materials can produce frequency components higher

than the excitation source which can be measured under controlled conditions in

experiments.

A very common non-linear effect found is the acousto-elastic effect where the

20

stress influences the wave propagation speed in a material [Rose 1999].

Hikata et al. [1963] have reported that a higher amplitude single frequency

longitudinal wave propagating through a material can have different velocities

corresponding to compression and rarefaction phases resulting in pulse distortion. This

distortion is related to the changes in the third-order elastic constants due to the presence

of defects. One of the reasons for the nonlinear elasticity of a solid can thus related to the

interatomic separations and the interatomic forces acting between the atoms in a crystal

lattice. [Kundu 2004, Kim et al 2008a]

The generation of higher harmonics due to the single frequency high amplitude

pulse propagation through a material with defects has thus been observed by a lot of

researchers and is a classic case of nonlinear behavior. This behavior can be attributed to

two different categories of defects –

(1) Defects at the microstructure level

(2) Macroscopic defects

The microscopic defects or anharmonicity in crystals like dislocations, secondary

phases, grain boundaries etc. result in the generation of harmonics and sidebands in the

frequency spectrum of a propagated signal and is known is nonlinear elastic wave

spectroscopy [Cantrell and Yost 2000, Kim et al 2005, Kim et al 2008b].

The presence of macroscopic defects like cracks, disbonds and delamination

result in a non-linear stress-strain relationship at the local regions. A high amplitude

ultrasonic wave propagating through such defects will cause opening and closing of the

21

defects and the contact of the free surfaces thus resulting in the harmonic generation. This

is also known as contact acoustic nonlinearity. [Solodov et al. 2002, Biwa et al. 2004,

Kawashima et al. 2006]. This technique is more relevant to the detection of defects in

adhesively bonded regions.

Recently there have been attempts by researchers to extend the nonlinear concepts

to the guided wave propagation [Bermes et. al. 2007]. They found that symmetric Lamb

wave modes which have same phase velocity at frequencies of ω and 2ω, termed as

synchronous modes, can be used for nonlinear measurements. The generation of the

symmetric wave mode at 2ω in a plastically deformed sample when a symmetric wave

mode at a frequency of ω propagates through it, demonstrated the nonlinear guided wave

measurement idea.

The acoustic nonlinearity parameter β is experimentally determined by means of

an amplitude ratio of harmonics, given in Equation 1.3

where k is the longitudinal wave number, X the propagation distance, f(ω) is a frequency

dependent function, and A1 and A2 are the amplitudes of the first and second harmonic

displacement measurements. [Bermes et. al. 2007].

)(82

1

22 ωβ f

AA

Xk= (1.3)

22

1.4 Challenges for further study

From the literature on wave propagation in structural waveguides, it is clear that

an understanding of the wave mode formation at a finite length waveguide transition is

lacking. The following issues need to be addressed:

1. the length (or characteristic length) required for the formation of a stable wave

mode i.e. a stable wave structure with time, as a wave mode enters from a part

of waveguide into another at a transition such as step change.

2. Mode selection for sensitivity to an interface of choice.

3. Understanding the guided wave mode behavior at a waveguide transition

4. the mode transmission/conversion efficiency of a waveguide

Addressing some of the questions can help us in better handling of real wave

guide transitions – such as the discrete step transition in an adhesive step-lap and skin-

stringer joint.

An answer to the issue 1 in the list will be of interest in addressing the

inspectability of the most critical region in an adhesive bonded joint – the start/end of the

adhesive or the transition region. The literature on adhesive joints [Adams et al. 1997]

shows that the end-effects in an adhesive joint result in larger shear stresses in that region

making defects in this region critical to the reliability of the joint. If a stable wave mode

formation requires some length of travel, then it will not be able to inspect the transition

region.

Since a continuous variation in cross-section cannot be addressed clearly in terms

of guided wave propagation, it is to be determined from research if the terms ‘modes’ in

23

the conventional guided wave sense is still relevant in such geometries as put forth in

issue 2.

Addressing the guided wave mode transmission efficiency will help in

formulation of guided wave inspection approaches for adhesive joints, keeping the defect

sensitivity also in mind.

Also, there are some specific challenges in the adhesive bonded systems that need

further research. For example, the problem of detection of kissing bonds in adhesive

joints has not been addressed in the current literature.

Most of the works in the literature handle the adhesive joint inspection based on a

specific geometry like the step-lap joint and the skin-stringer joint. In real structures there

are more complicated joints like the scarf joint. So it is required to establish an

understanding of the guided wave behavior at waveguide transitions for ready extension

into complicated transitions such as changes in the geometry of the adherend like the ply

drop, and the gradual transition in the waveguide geometry like in the scarf joint,

handling of the spew in a joint.

1.5 Thesis Objectives

The goal of this thesis work is to develop a general framework to handle

ultrasonic guided wave propagation across waveguide transitions and apply it to the

practical problem of adhesive joint inspection using ultrasonic guided waves.

The specific objectives of the thesis will be to:

24

1. Understand guided wave mode formation at transitions and determine the effect

of near-field or stable mode formation length

2. Develop an understanding of guided wave mode conversion and transmission at a

waveguide transition and formulate guided wave mode ‘transfer functions’ to

define the transmission efficiency of a waveguide for different guided wave

modes. This will help in theoretically defining conditions for larger energy

transfer across a joint and hence in forming inspection strategies

3. Efficient numerical modeling of waveguide transitions and algorithms for

processing the data

4. Determine mode selection for interfacial defect or weakness sensitivity in an

adhesive joint

5. Apply the knowledge gained from objectives 1 to 4 in different representative

adhesive joint geometries – like in an adhesive repair patch and an adhesive skin-

stringer joint

1.6 Contents of this thesis

This thesis has 6 chapters.

Chapter 1 presents the problem statement, a brief review of the literature and the

thesis objectives. Chapter 2 covers the theoretical foundation for guided wave

propagation study in a linearly elastic anisotropic layered media using partial wave

method and global matrix method for calculating characteristic curves – dispersion

curves - for a waveguide. Chapter 3 describes numerical modeling of waveguides and

waveguide transitions, efficient methods for exciting guided wave mode within them.

Signal processing algorithms that handle the output from numerical models is presented.

The problem of near field is also attempted by developing a new processing technique.

25

Chapter 4 handles the guided wave inspection problem in a continuous bonded joint- an

adhesive repair patch. The selection of modes for sensitivity to the interface in a general

layered media is developed. In Chapter 5, a framework for understanding the guided

wave interaction with a waveguide transition is analyzed. Mode conversion and mode

sensitivity for efficient inspection of the bonded stringer region is presented. Chapter 6

provides the summary of the entire thesis and also suggests future research directions that

can be handled based on this research or that remains to be attempted in the area of

adhesive bond testing.

Chapter 2

Analysis of guided wave propagation in plate-like structures and their transmission

2.1 Introduction

Engineering structures use laminated construction e.g. bonded components,

laminated composites which possess different levels of symmetry. Ultrasonic

nondestructive inspection of such structures demands an understanding of the ultrasonic

wave propagation through them. The ultrasonic bulk waves and the guided waves are just

two regimes of propagation of the stress wave in a media. The bulk waves correspond to

the case where the wavelength is small compared to waveguide dimensions. In the case

of guided waves, the wavelength is comparable or higher than the waveguide thickness.

Ultrasonic guided waves are special in this regard because of their thickness

coverage in a waveguide and interaction with the boundaries also, thus permitting

interrogation of the whole thickness and measurement using sensors deployed at the

surface of the waveguide.

There is much work reported in the literature regarding wave propagation in a

layered media. In this chapter the wave propagation characteristics in the form of

dispersion curves will be analytically evaluated by implementing the developments from

the literature. Some physical insights into the solutions, not found elsewhere in the

27

literature are provided. A hybrid analytical technique is proposed for handling the

geometry changes.

2.2 Wave propagation modeling in plate-like structures

A historical perspective of the progress in the field of wave propagation and

dispersion was covered in section 1.3.1. The theory of guided waves in solid media is

also covered by several classical books. Popular approaches to solve waveguide

dispersion problem include potential theory based approach, partial wave approach and

Semi-Analytical Finite Element (SAFE) approach.

The potential based approach can be found in Achenbach [1968], Miklowitz

[1978], Auld [1990], Graff [1991] and Rose [1999]. The partial wave approach to solve

for dispersion in waveguides can be found in Auld [1990], Nayfeh [1995] and Rose

[1999]. Nayfeh [1995] formulates a general approach for obtaining dispersion curves in a

general layered anisotropic media and further extends it to handle piezoelectric material

also. SAFE approach is not yet available in any standard textbooks on wave propagation.

Research by Galan and Abascal [2002], Hayashi [2003], Matt et al. [2005] show the

application of SAFE.

In a recent book, Datta and Shah [2009] applied a stiffness based approach,

similar to that applied in laminate analysis to solve dispersion in waveguides.

28

2.3 Guided wave propagation in plate-like structures

Guided waves in a solid media are elastic waves guided by the boundaries of the

medium through which they are propagating. The classical problem as studied by Lamb

considers the propagation of wave within a finite thickness isotropic layer having stress-

free boundary conditions. The medium through which the guided wave propagates is

called a waveguide. [Rose 1999].

A general layered anisotropic media is presented in Figure 2-1 along with a

coordinate system. For wave propagation in layered media, the use of term Lamb waves

is not a suitable one. It is better to employ the term Lamb-type wave to still refer to the

plane-strain solution to the problem. In a general sense the use of ‘guided wave’ to

represent wave propagation in a general media is more suitable and would account for the

shear horizontal (SH) waves also.

Figure 2-1: A general multi-layered structure with the coordinate system.

t1

t2

t3

tN

x1

x3 x2

29

The equilibrium equations are given by

where jiσ is the stress tensor with indices j and i represent the plane and the direction of

its action, ρ is the density of the material, bi is the body force per unit volume and ui is

the particle displacement.

The stress-strain relations (constitutive equations) and the strain-displacement

relations can be written

where ijklC is the elastic stiffness tensor and ijε is the tensorial strain.

Substituting the stress-strain relations and the strain-displacement relations for a

homogenous elastic medium in the equilibrium equations, and employing the symmetry

of the stiffness matrix, we obtain the governing equation for the wave propagation

problem. ( Equation 2.3)

The stress-strain equations are rewritten using contracted engineering stress and

strains

iijji ub &&ρρσ =+, (2.1)

( )ijjiij

klijklij

uu

C

,,21 +=

=

ε

εσ (2.2)

ijklijkl uuC &&ρ=, (2.3)

30

where γ is the engineering shear strain. The stiffness tensor can be transformed using

rotation matrix [Nayfeh 1995] to handle wave propagation in an anisotropic material.

For the case of plane wave propagation along the x1 direction, a trial solution for

particle displacement ui is assumed by following the partial wave method (Equation 2.5)

where ξ is the wave number of the propagating guided wave, α is the ratio of

wave number components along the x3 and x1 directions, cp is the phase velocity of the

guided wave, t is the time and Ul is the coefficient to be determined. Theξ , cp and the

frequency ( f ) are related by

( )jiCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

ijij ≠=

=

εγγγγεεε

σσσσσσ

212

31

23

33

22

11

665646362616

565545352515

464544342414

363534332313

262524232212

161514131211

12

31

23

33

22

11

(2.4)

( ){ }tcxxiUu pll −+= 31exp αξ (2.5)

pp ccf ωπξ == 2 (2.6)

31

Another explanation of the displacement assumption is to consider the solution to

be a combination of a thickness resonance term and a propagating term. Hence a guided

wave can also be defined as a propagating thickness resonance.

Substituting the partial wave solution into the equilibrium equations, we obtain a

system of three coupled equations given in Equation 2.7

Nontrivial solutions for {U} require that the determinant of [K] vanish. This

results in a 6th degree polynomial equation inα , each of whose roots represent either an

upward propagating or a downward propagating partial wave. The partial waves in

themselves are bulk waves. The wave field in each material layer is thus obtained by

summation of the partial waves components of L (longitudinal), SV (shear vertical) and

SH (shear horizontal)

The values of α can be used to calculate the displacement component ratios or the

polarization vector for the x2 and x3 directions. The strain-displacement relations and the

[ ]{ } { }

23445365623

23555131513

24556141612

2233355533

2244466622

2255151111

)(

)(

)(

2

2

20

αααααα

ρααραα

ραα

CCCCK

CCCCK

CCCCK

cCCCK

cCCCK

cCCCKUK

P

P

P

+++=

+++=

+++=

−++=

−++=

−++==

(2.7)

( ){ }∑=

−+=6

131exp

kpklkkl tcxxiUBu αξ (2.8)

32

stress-strain relations can be used to obtain the stress values in terms of the displacement

solution

Combining the boundary conditions and the continuity conditions together, we

obtain a system of linear equations in terms of the unknown coefficient Bk. For an N

layered system, the system of equations can be represented by a 6N x 6N matrix – the D

matrix.

This method of matrix assembly is called the global matrix method. By

employing a root searching algorithm for different values of pc , we can obtain the value

of ξ to form the pair ( )pc,ξ . These ( )pc,ξ pairs result in characteristic curves for a

waveguide called the phase velocity dispersion curves. The solution {B} for every

( )pc,ξ value provides the wave structure i.e. the cross-sectional displacement/stress

solution. The ( )pc,ξ values with similar wave structure are classified as a guided wave

mode.

( ) ( ){ } ( ){ }

( ) ( ){ } ( ){ }

( ) ( ){ } ( ){ }txiUCUUCUCUCUCBi

txiUCUUCUCUCUCBi

txiUCUUCUCUCUCBi

kkkkkkkkkkk

kkkkkkkkkkk

kkkkkkkkkkk

ωξαααξσ

ωξαααξσ

ωξαααξσ

−+++++=

−+++++=

−+++++=

∑

∑

∑

=

=

=

1

6

1236313523433311333

1

6

1246314524433411423

1

6

1256315524533511513

exp

exp

exp

(2.9)

[ ]{ } 0=BD (2.10)

33

From the phase velocity dispersion curves, we can compute the group velocity

dispersion curves

2.4 Guided wave dispersion in a waveguide – an example

An isotropic aluminum plate, 2 mm thick, is considered in this section as an

example of a simple waveguide. Detailed study of some guided wave propagation aspects

are also covered in this section.

Determination of the material properties of aluminum for calculation of dispersion

curves was carried out using ultrasonic bulk waves [Rose 1999]. For isotropic materials,

the constitutive stiffness matrix reduces to

The entries within the constitutive matrix can be related to the Lame’s constants

and ultrasonic wave velocity

ξω

ddcg = (2.11)

( )jiCCC

CC

CCCCCCCCCC

ijij ≠=−=

=

εγ

γγγεεε

σσσσσσ

22

000000000000000000000000

441112

12

31

23

33

22

11

44

44

44

111212

121112

121211

12

31

23

33

22

11

(2.12)

34

Engineering moduli can also be expressed in terms of the Lame’s constants

The Lamb wave motion and the SH wave motion are decoupled from each other

in the case of an isotropic plate. The phase and group velocity dispersion curves for Lamb

and SH waves are shown in Figure 2-2.

In the case of aluminum plate, being a mid-plane symmetric geometry, the modes

can be categorized into and identified as symmetric and antisymmetric. The symmetric or

antisymmetric nature of the in-plane displacement wavestructure or the cross-sectional

displacement distribution is the basis for the classification into symmetric and

antisymmetric modes.

ρµ

ρµλ

µµλ

=+=

=+=

SL cc

CC

;2

;2 4411

(2.13)

( )( )

ratio sPoisson'modulusShear

elasticity of ModulusG

211

====

−+=

υ

µυυυλ

EG

E

(2.14)

35

Figure 2-3 shows the phase and group velocity dispersion curves for Lamb wave

modes in a 2 mm thick aluminum plate. The modes are categorized as symmetric or anti

symmetric depending on the symmetry or asymmetry respectively in the variation of the

in-plane displacement across the waveguide thickness. The modes are named with

alphabet ‘a’ for anti symmetric and ‘s’ for symmetric along with an integer subscript that

0 1 2 3 40

5

10

15

20

Frequency (MHz)

Phas

e V

eloc

ity (k

m/s

)

Lamb wavesSH Waves

0 1 2 3 40

1

2

3

4

5

6

Frequency (MHz)

Gro

up V

eloc

ity (k

m/s

)

Lamb wavesSH Waves

Figure 2-2: Phase and group velocity dispersion curves for Lamb and SH type waves in a2 mm thick aluminum plate.

36

increases from 0. The modes with a subscript zero are called the fundamental modes. The

phase velocity (cp) varies in the range of [ ]∞Sc for most modes, except for modes a0

and s0. For a0, the range of variation of cp is [ ]Rc0 , where cR is the Rayleigh velocity in

aluminum. The cp varies in the range [ ]Rplate cc for s0 mode, cplate being the plate wave

velocity.

The value of cg is bounded on the higher side by the longitudinal bulk wave

velocity. On the lower side, the value can be negative corresponding to the generation of

a backward propagating mode. For example, the s1 mode terminates at a phase velocity

around 10 km/s and at a frequency of ~1.42 MHz. The back-propagating part of the s2

mode starts at the same frequency and becomes forward propagating. In the complex

wavenumber frequency domain this becomes clearer. Several authors in the past have

also shown these phenomena, e.g. Graff [1991], Galan and Abascal [2005a].

37

2.4.1 Wavestructure

Wavestructure refers to the variation in the displacement, particle velocity, strain,

stress or the power flow within the waveguide cross-section.

0 1 2 3 40

5

10

15

20

Frequency (MHz)

Phas

e V

eloc

ity (k

m/s

)

a0

s0

a1

s1

s2a2

s3

a3

s modesa modes

0 1 2 3 40

1

2

3

4

5

6

Frequency (MHz)

Gro

up V

eloc

ity (k

m/s

)

a0

s0

a1 s1

s2

a2

s3

a3a4

s modesa modes

Figure 2-3: Phase and group velocity dispersion curves for Lamb wave modes in a 2 mmaluminum plate. The symmetric and antisymmetric modes have been labeled on thecharts.

38

The solution in the form of ( )pc,ξ is used to compute the displacement and other

variables

From this the velocity can be computed using

The expression for the strains is then

Using the constitutive equations, the expression for stress can be written.

( ){ }∑=

−+=6

131exp

kpklkkl tcxxiUBu αξ (2.15)

ll

l uituv ω−=∂∂= (2.16)

{ } ( ){ }

{ } ( ){ }

( ) { } ( ){ }

{ } ( ){ }

−

−+

−

−=

∑

∑

∑

∑

=

=

=

=

6

1132

6

11331

6

1132

6

1133

1

12

31

23

33

22

11

expexp

expexp

expexp

expexp

0

kkkk

kkkkkk

kkkkk

kkkkk

txixiUBi

txixiUUBi

txixiUBi

txixiUBi

ui

ωξξαξ

ωξξααξ

ωξξααξ

ωξξααξ

ξ

γ

γ

γ

ε

ε

ε

(2.17)

39

Where the value of Mlk is

The displacement and stress wavestructures for the fundamental symmetric and

antisymmetric modes at 0.3 MHz are given in Figure 2-4. The mid-plane symmetry of the

in-plane displacement component (ux or u1) in the case of the s0 mode and the anti-

symmetric nature of ux in the case of the a0 mode can be clearly seen. The component of

displacement along uy is zero. It can also be noted from Figure 2-4 that the stress free

boundary conditions are satisfied by the traction in the ‘z’ (or ‘3’) direction.

{ } ( ){ }∑=

−=6

113 expexp

kkklkl txixiBMi ωξξαξσ (2.18)

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) kkkkkkkkk

kkkkkkkkk

kkkkkkkkk

kkkkkkkkk

kkkkkkkkk

kkkkkkkkk

UCUUCUCUCUCMUCUUCUCUCUCMUCUUCUCUCUCMUCUUCUCUCUCMUCUUCUCUCUCMUCUUCUCUCUCM

26631662463361166

25631552453351155

24631452443341144

23631352343331133

22631252243231122

21631152143131111

+++++=+++++=+++++=+++++=+++++=+++++=

αααααααααααα

(2.19)

40

-1 -0.5 0 0.5 10

0.5

1

1.5

2

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 0.30 MHz, 2.01 km/s

uxuyuz

-1 -0.5 0 0.5 10

0.5

1

1.5

2

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 0.30 MHz, 5.40 km/s

uxuyuz

-1 -0.5 0 0.5 10

0.5

1

1.5

2

σ

Posi

tion/

Thic

knes

s (m

m)

Stress at 0.30 MHz, 2.01 km/s

σ11σ22σ33σ23σ13σ12

-1 -0.5 0 0.5 10

0.5

1

1.5

2

σ

Posi

tion/

Thic

knes

s (m

m)

Stress at 0.30 MHz, 5.40 km/s

σ11σ22σ33σ23σ13σ12

-1 -0.5 0 0.5 10

0.5

1

1.5

2

Poynting Vector

Posi

tion/

Thic

knes

s (m

m)

Poynting Vector at 0.30 MHz, 2.01 km/s

PxPyPz

-1 -0.5 0 0.5 10

0.5

1

1.5

2

Poynting Vector

Posi

tion/

Thic

knes

s (m

m)

Poynting Vector at 0.30 MHz, 5.40 km/s

PxPyPz

Figure 2-4: Displacement, stress and Poynting’s vector for wave propagation along analuminum (2 mm thick) waveguide. The first column is for a0 mode propagation and the second column for s0 mode propagation at 0.3 MHz.

41

2.4.2 Power flow concepts

The complex form of Poynting’s vector is defined as the dot product of velocity

vector and the stress tensor

In general the Poynting’s vector is a complex quantity. The physical meaning of

the real part of the Poynting’s vector is related to the instantaneous energy flow in a

waveguide. The Poynting’s vector has three components – the in-plane and the two out-

of-plane components, given by

The value of P2 is zero for a plane-strain analysis in the case of isotropic media

because the displacement component is zero in that direction. In the case of anisotropic

layered media, the value of P2 is non-zero. The physical meaning of the components of

Poynting’s vector is described below.

2σvP

* •−= (2.20)

( )

∫∫

∫

∫

==

=

−=

tx

tx

tx

tmm

dxPPdxPP

dxPP

dxxvP

3332

31

31*

1

32

1

;

;

.Re21

Similarly

shortinor

σ

(2.21)

42

(a) The component of Poynting’s vector along the direction of wave propagation

(Px1) represents the energy transfer along the waveguide. The value of P1 will be

non-zero and positive for the propagating waves. In the case of backward

propagating modes, the value of P1 is negative indicating that the energy flows

opposite to the direction of wave propagation.

(b) In the case of anisotropic layered media, P2 is non-zero and its value represents

the extent of energy flow in the direction normal to the wave propagation

direction but in-plane with respect to the sample boundaries. The inverse tangent

of the ratio of the P2 to P1 represents the skew angle in the composite for that

particular propagation direction

(c) The component of Poynting’s vector along the cross-section (Px3) represents the

energy movement through the thickness. Physically it represents the thickness

resonance component in the guided wave propagation. The value of P3 equals

zero for the case of isotropic and anisotropic media without attenuation and with

stress free boundaries (Figure 2-4). The value of Px3 across the thickness becomes

dominant over the Px1 at higher phase velocity values indicating a larger thickness

resonance and a lower group velocity. This explains the existence of physically

unrealizable velocities in the dispersion curves and also the phenomena of cut-off

frequency. Hence, in the case of a waveguide with non-attenuating material layup

and also for a waveguide with non-leaky boundaries, the guided wave energy is

exchanged among the layers in the waveguide. The energy distribution varies over

the time period represented by the distance to move over a wavelength space

within the sample.

= −

1

21tanPPϕ (2.22)

43

The transfer of energy within a waveguide surrounded by leaky media is

accurately represented by the energy velocity and not the group velocity. Bernard et al.

[2001] show that the energy velocity in the case of leaky boundaries will have

components both along of the waveguide and also across the thickness. The group

velocity and energy velocity are distinct for the case of waveguide with attenuating

layers.

2.5 Guided wave propagation in a bonded plate

Guided wave dispersion in an epoxy (0.3 mm) bonded aluminum (2 mm) to

aluminum (2 mm) adhesively bonded joint is studied in this section.

The phase and group velocity dispersion curves in a bonded aluminum joint and

an aluminum layer are compared in Figure 2-5. The modes in aluminum are labeled as

symmetric or antisymmetric with a numeric subscript. The bonded joint is mid plane

symmetric but still the modes are numbered instead of being labeled alphabetically for

convenience. It is clearly seen from the dispersion curves in Figure 2-5 that the number of

modes in the bonded joint at frequencies above 150 kHz is always greater than that in

aluminum.

Some observations that can be made from the study of dispersion curves in

Figure 2-5 and also wavestructures (not shown in detail here) are provided in the sections

that follow.

44

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

Frequency (MHz)

Phas

e V

eloc

ity (k

m/s

)

a0

s0

a1

s1

s2

a2

s3

a3

1

2

3

4

5

6

7 8

9

10

11

12

13

14

15

16

17

18

19

al modesbond modes

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

Frequency (MHz)

Gro

up V

eloc

ity (k

m/s

)

a0

s0

a1

s1

s2

a2

s3

a3 a4

1

2

3

4

5 6

78 9

1011

12 1314

15

16

1718

1920

21

al modesbond modes

Figure 2-5: Superposition of phase and group velocity dispersion curves for aluminumplate (2 mm) and bonded aluminum joint. The modes in aluminum are labeled using an alphabet and a subscript whereas the modes in the bond are numbered incrementally.

45

2.5.1 Material based phase velocity zones

The dispersion curve can be divided into two major zones of phase velocity

values based on the material lay-up – epoxy dominated zone and aluminum dominated

zone. The epoxy layer embedded between the aluminum layers becomes closer to a finite

layer embedded between half-spaces at larger frequencies. The range of phase velocity

values below the Rayleigh velocity (s0 and a0 at higher frequencies) in aluminum

represents the epoxy dominated zone in the aluminum bonded joint. The wavestructure of

mode 2 at 3 MHz in Figure 2-6 shows clearly energy confined to the epoxy layer. The

phase velocity range from [ ]∞Rc is the aluminum dominated zone. Similarly, group

velocity zones can be demarcated. But the existence of zero or negative group velocities

corresponding to higher phase velocity points and backward propagating modes implies

that this division is not very stringent on a group velocity representation. The energy gets

trapped within the epoxy layer in the epoxy dominated zone.

2.5.2 Mode Pairs

A closer look at the phase velocity dispersion curves in Figure 2-5 shows the

existence of modes within the bonded joint that superimpose with those of aluminum. In

most cases it can be observed that there are at least two modes in the bond that overlap or

bound or lie closer (i.e. with a very small difference in phase velocity value) to that in

aluminum. The modes within the bonded joint which are similar to a mode in aluminum

are termed as “mode pairs” in this thesis.

46

Observations from a detailed analysis of the wavestructures of such mode pairs in

waveguide set of aluminum and the bonded aluminum are shown with few representative

cases of modes.

a) Modes 2 to 4 and s0: The mode 2 and s0 overlap till ~225 kHz. The phase and

group velocity of mode 2 reduces beyond 225 kHz with the least cg at 330 kHz.

The wavestructures of the modes 2 and s0 at 200 kHz are compared in Figure 2-7.

At 200 kHz, the values of cg are 5.21 km/s for mode 2 and 5.42 km/s for the s0

mode. Beyond 225 kHz, modes 3 and 4 overlap with the s0 mode. A comparison

of the wavestructure of modes 3 and 4 with the wavestructure of the s0 mode is

presented for a frequency of 800 kHz in Figure 2-8. The cg of s0 mode (4.43 km/s)

lies between the cg of modes 3 (4.04 km/s) and 4 (4.74 km/s). Modes 3 and 4

hence for a mode pair with respect to the s0 mode.

b) Modes 10 and 11 and mode s2: In the frequency range from around 1.7 MHz to

2.5 MHz, the s2 mode is enveloped by the modes 10 and 11 in the bonded joint in

the phase velocity dispersion curves. The modes have nearly equal group velocity

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 3.00 MHz, 2.17 km/s

uxuyuz

Figure 2-6: Wavestructure of the mode 2 at 3 MHz. The peak displacements are withinthe epoxy layer. At this frequency, the waveguide is similar to an embedded epoxy layerwithin aluminum half-spaces.

47

also. The wavestructure of the s2 mode and modes 10 and 11 are provided in

Figure 2-9. The wavestructures at the same phase velocity (~12.57 km/s) for all

the modes is presented on the centre and left column of Figure 10. The

wavestructures for the case of the same frequency are compared in the centre and

right column of the Figure 2-9. The frequency, phase velocity, and the group

velocity values for the s2 mode and modes 10 and 11 are listed in Table 2-1 for

both the cases studied. From Figure 2-9, it can be seen that the wavestructures of

modes 10 and 11 are identical for a frequency value at the bottom layer. The

nature of the displacement within the epoxy layer is distinct for modes 10 and 11.

While it is symmetric in the case of mode 11, the displacement is antisymmetric

for the mode 10. Comparing the displacement in the aluminum layers, it can be

observed that the displacement profile in these layers match exactly for mode 11,

but has a phase shift in the case of mode 10. Modes 10 and 11 hence form a mode

pair with respect to the s2 mode.

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 0.20 MHz, 5.42 km/s

uxuyuz

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 0.20 MHz, 5.21 km/s

uxuyuz

Figure 2-7: The normalized displacement wavestructure for s0 mode in aluminum (left) and mode 2 in bonded aluminum (right) at 200 kHz are shown. The match between thedisplacement components in both waveguides can be clearly seen.

48

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 0.80 MHz, 5.07 km/s

uxuyuz

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 0.80 MHz, 5.15 km/s

uxuyuz

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 0.80 MHz, 5.35 km/s

uxuyuz

Figure 2-8: The normalized displacement wavestructure for s0 mode in aluminum (left) and for mode 3 (right top) and mode 4 (right bottom) in bonded aluminum at 800 kHz areshown. The wavestructures match very well.

Table 2-1: Guided wave phase and group velocities for modes s2, 10 and 11 at different frequencies used in this study

Mode Frequency Phase Velocity

Group Velocity

[MHz] [km/s] [km/s] s2 1.98 12.57 3.14

1.9 12.57 2.95 10 1.98 11.14 3.08 2.02 12.56 2.97 11 1.98 13.46 2.88

49

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 1.98 MHz, 12.57 km/s

uxuyuz

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 1.90 MHz, 12.57 km/s

uxuyuz

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 1.98 MHz, 11.14 km/s

uxuyuz

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 2.02 MHz, 12.56 km/s

uxuyuz

-1 -0.5 0 0.5 10

1

2

3

4

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Displacements at 1.98 MHz, 13.46 km/s

uxuyuz

Figure 2-9: The normalized displacement wavestructure for s2 mode in aluminum (top centre) and for mode 10 (middle row) and mode 11 (bottom row) in bonded aluminum.The match between the displacement components in both waveguides can be clearly seen.

50

Similar analysis was also performed for other mode pairs like modes 5 and 6 with

respect to a1 mode, modes 7 and 8 with respect to mode s1, modes 12 and 13 with respect

to mode a2, and modes 16 and 18 with respect to mode a3. The phase shift in the

displacement profile in the top and bottom layers for the mode pairs was observed in all

these cases.

It is noted that the mode pairs in general can change with respect to frequency

value. So mode pairs exist within a frequency range. The mode pairs mentioned above

were not identified precisely with respect to the frequency as it was only meant to point

to the existence of the mode pairs over the range of frequency and for the symmetric and

antisymmetric modes in the aluminum waveguide.

It is expected that the mode pairs will have almost equal chance of generation by

mode conversion at the transition from the aluminum to the bonded waveguide for the

incidence of a matching wave mode from the aluminum waveguide. A linear combination

of mode pairs with equal weighting function will hence cause perfect cancellation of

displacements within the top layer of the bonded waveguide.

2.6 Summary

In this chapter the fundamental approach to solving the dispersion relations for

wave propagation in a linearly elastic anisotropic waveguide with stress-free boundaries

using the partial wave approach was presented. Wave propagation in an aluminum plate

waveguide was solved also using the partial wave technique. The concepts related to the

51

power flow in a waveguide were presented along with the examples of wavestructures for

aluminum plate waveguide. The guided wave dispersion curves for an aluminum-epoxy-

aluminum bonded waveguide were presented along with a comparison with the

dispersion curves for aluminum.

The following observations were made:

1. The power flow in a waveguide has three components – two in-plane and one out-

of-plane. The out-of-plane power flow corresponds to the thickness resonance

component of guided waves and sums to zero for a non-attenuative lay up of

materials having stress-free boundaries. The two in-plane components provide

energy skew in anisotropic media.

2. The dispersion in bonded aluminum joints was studied. The phase velocity

dispersion space was found to have two distinct zones based on the material

layers. The low phase velocity region i.e. with phase velocity below the Rayleigh

velocity in aluminum was found to be dependent on the epoxy layer alone with

the energy trapped in that layer. Above the Rayleigh velocity in aluminum, the

dispersion curves in the bonded joint are dominated by displacement components

in the aluminum layers.

3. Comparing the dispersion curves for aluminum and bonded aluminum, it was

found that for every mode in aluminum, there are modes in bonded aluminum

with matching phase velocity and matching group velocity in one of the

aluminum layers. A new term called – Mode pair – was coined to describe the

behavior of the modes. The modes have identical displacement wavestructure in

one of the aluminum layers in the bonded aluminum joint. The phase of the

displacement was reversed in the other aluminum layer in the case of one of the

modes in the mode pair. It is expected that the modes constituting the mode pair

have equal chances of generation under mode conversion.

Chapter 3

Finite Element Modeling and Analysis of wave propagation through waveguides

3.1 Introduction to numerical computation techniques in wave propagation

There are several numerical computation techniques that can be used to solve

structural mechanics problems like stress wave propagation that forms the basis for

guided wave propagation studies. Numerical techniques enable modeling of guided wave

propagation and mode conversion across discontinuities in a waveguide e.g. defects or

transitions where analytical approaches are either applicable only to simplified

geometries or not even possible. There are several numerical tools that can be applied to a

general class of problems that include wave propagation across waveguide transitions.

These include finite element method (FEM), boundary element method (BEM), finite

difference time domain method (FDTD), semi-analytical finite element (SAFE) etc.

The boundary element method (BEM) can be applied to problems whose

governing equation has a Green’s function solution readily available. Cho and Rose have

applied BEM in the study of guided wave propagation and interaction with defects in

isotropic plates and also quantified the mode conversion effect [Cho and Rose 1996, Cho

and Rose 2000]. They used the normal mode expansion theory for this. Hayashi and

Endoh [2000] later developed a visualization method for Lamb wave motion from the

combined solution obtained from BEM and normal mode expansion. Material in-

homogeneity and complications in geometry render this method more cumbersome in

53

application, which also explains why this technique is still not widely employed [Brebbia

and Dominguez 1989]. The finite difference method is probably the closest to finite

elements (FE) in terms of implementation. The difficulty in handing complex geometry

and the difficulty associated with raising the order of the difference expression restricts

the application of this technique in wave propagation problems. The FE method is the

most widely used method for simulation of guided wave propagation in structures. The

ease of modeling and formulation using the FE, availability of a large number of

commercial software capable of handling material models and elements to model the

physics of the problem makes the FE method more preferable over the other numerical

techniques. Finite-element based solution to Lamb wave scattering can be found in Auld

[1990], Ditri [1996], Hasegawa [1986] and many other publications. The use of FE along

with modal expansion can be seen in Ditri [1996] and Al-Nasser [1991]. Wave

propagation modeling using FE software packages like ABAQUS, COMSOL and

FINEL, and processing the output data using a Fourier transform based technique – two

dimensional fast Fourier transform (2DFFT) can be found in numerous works in the

literature [Puthillath et al. 2008, Hosten and Castaings 2003, Alleyne and Cawley 1991].

Processing the FE output using orthogonal decomposition is also employed in some

works [Moreau et al. 2006, Castaings et al. 2002].

The semi-analytical finite element (SAFE) framework employs an analytical

function along the wave propagation direction [Hayashi and Rose 2003, Hayashi et al.

2003, Mu and Rose 2008]. Combination of SAFE with BEM can be found in Galan and

54

Abascal [2002, 2005] and Song et al [2005]. Combination of FE and SAFE has also been

reported in literature [Hayashi et al. 2009].

Techniques employing special functions like the spectral element method also

exist. The spectral element method, as explained in Doyle [1997], uses Fourier

transforms to convert the partial differential wave equation into linear equations.

Techniques employing special elements like the spectral element method require extra

effort in formulation of the element stiffness matrix to handle any non-standard geometry

[Gopalakrishnan and Doyle 1995].

In this chapter, a hybrid analytical finite element (FE) method is employed to

handle wave propagation problems. The hybrid analytical FE approach uses the

theoretical inputs from wave mechanics calculations like dispersion curves and mode

excitation as input to FE model and the FE result analyzed based on physical parameters

using signal processing approach.

3.2 FE Theory and implementation to simulate wave propagation using ABAQUS

Ultrasonic wave propagation is a mechanics problem involving high frequency

loading that requires the solution of the hyperbolic wave propagation equation

(Equation 2.3). Simulation of ultrasonic wave propagation through a structure using finite

elements was implemented by researchers in the late 1980’s [Ludwig and Lord 1988].

The FE simulation of wave propagation is performed here using the dynamic explicit

solver in ABAQUS, a commercial software package [ABAQUS 2006].

55

Finite elements involve discretization of model geometry into smaller

computational units called elements that satisfy certain assumed displacement variations.

The nodal displacements are related to the element displacement using shape functions.

Applying the concept of minimization of the total energy, the wave equation reduces to a

matrix form:

where [M] is the mass matrix, [K] is the stiffness matrix of the structure. {R} is the

external load, { }U&& is the nodal acceleration vector and { }U is the nodal displacement

vector.

ABAQUS offers both implicit and explicit solvers for handling the dynamic wave

propagation problem. The flexibility in computation in terms of the geometry, cost of

computation in terms of the time and efficiency of computation makes an explicit solver

more suited for wave propagation problem. In the explicit dynamic analysis procedure,

the equations of motion are integrated using the explicit central difference formulation

{ } { } { }RUKUM =+ ][][ && (3.1)

( ) ( )

( ) ( ) ( )N

ii

Ni

Ni

Ni

iiN

i

N

i

UtUU

Utt

UU

+

++

−

−

+

∆+=

∆+∆+=

2111

)(1

21

21 2

&

&&&&

(3.2)

56

where NU is a degree of freedom and subscript ‘i’ refers to the increment number in the

explicit solution scheme. [ABAQUS 2006].

For computational stability of the explicit dynamic procedure, the element size

and the computational time step used in explicit time marching should satisfy certain

conditions

(1) Courant criteria [Taflove et al. 2000] i.e. the distance traveled by a wave

(disturbance) within a time step ∆t should not exceed the length of the smallest

element Lmin

(2) Spatial sampling criteria – the spatial sampling interval, which is equal to the

element size should be sufficient to reconstruct the smallest wavelength of the

wave that can exist in the computational domain

From the literature on wave propagation in solid media, we find that the value of

n = 10 is normally used in FE simulations.

(3) Nyquist criteria – the stable time increment ∆t required for the stability of the

central difference operator is related to the maximum frequency present in the

system ( maxω )

mediumtheinwavefastesttheofVelocity

minmin

=

≤∆⇒≤∆

cc

LtLtc (3.3)

2integeranis

minmax

>

≤

nn

L λ (3.4)

57

A larger model size significantly increases the computation time and memory

requirements and hence practical simulations use only a finite computational domain with

boundaries that absorb the outgoing waves thus simulating the infinite physical domain.

3.3 FE for modeling wave propagation in infinite domains

Perfectly Matched Layers (PML) are non-physical computational domains with

special governing equations obtained on a modification of equations for an elastic

medium using special functions called stretching functions. With a proper choice of the

stretching function, the interface reflection between the model domain and the PML can

be avoided. [Basu and Chopra 2003]. PML’s are stated to be effective for all wave

incidence angles irrespective of their frequency content. Becache et al. have proved

theoretically that the stability of PML can be expressed in terms of the elastic stiffness

values of the material that it bounds; especially for anisotropic media [Becache et al.

2003]. The complex wave structure in guided waves renders the PML ineffective.

The review article by Givoli [2004] gives details about the work on non-reflecting

boundary conditions (NRBC) where special boundary conditions are provided to avoid or

reduce the boundary reflections. Givoli also mentions that there is no general NRBC that

can ensure a stable solution which is also efficient and easy to implement.

max

2ω

≤∆t (3.5)

58

Recently researchers have suggested the use of absorbing regions defined using

physical material parameters attached to the model geometry such that the wave

attenuates within the absorbing region [Castaings et al. 2004; Drozdz et al. 2006]. Drozdz

et al. stated that the addition of the absorbing region reduces the stable time increment in

a time marching solution thus increasing the solution time without promising a total

elimination of the boundary reflections. The use of frequency domain based solution was

proposed as an alternate to the time based solution. The frequency domain solution

simulates the harmonic response of an infinite physical domain and is also very natural

for the implementation of absorbing conditions through the frequency dependent

imaginary part of the modulus. Since there is no optimal method for determining the

damping parameter, Drozdz et al. proposed the following guidelines:

(i) The length of the absorbing region should be three times the largest wavelength in

the model

(ii) The value of the damping should be gradually increased along the length of the

absorbing region, and away from the model. This will ensure minimal reflection

at the boundary between the model and the absorbing region.

This method also requires careful evaluation at frequencies near to the mode cut-

off value because the length of the absorbing region is determined based on the largest

wavelength present in the model.

The FE method also provides special elements to simulate the infinite domain.

ABAQUS provides “infinite elements” with built-in damping for longitudinal and shear

waves [ABAQUS 2006] to affect damping along a single direction. These elements,

defined as special regions at the ends of the model, help to provide quiet boundaries for

59

waves impinging orthogonally on this boundary. Since the guided wave propagation

involves displacement components in all three coordinate directions, the infinite elements

do not completely avoid the boundary reflections. Also, in the case of anisotropic layered

media, the infinite elements do not perform efficiently. This aspect of modeling in FE has

been studied in this work and a signal processing algorithm has been developed

(presented in a later section) to negate the boundary effects.

The FE solution to the wave propagation problem is in the form of the stress and

displacement fields over the geometry of the structure. In order to quantify the FE

solution in terms of the guided wave modes that are excited and the modes that result

from the interaction of the excited modes with the structure, we need techniques to

process the results from the FE solution.

3.4 A brief review of the guided wave mode identification techniques

In guided wave numerical simulations/experiments (i.e. numerical/actual

experiments), identification of the guided wave modes that exist in the model/sample

becomes very important in order to determine the contribution of each mode. This will

help us in understanding the influence of scatterers on mode conversion.

Depending on the type or methodology of data collected several processing

techniques can be employed. For example, in experiments where only a single point data

is recorded, a short time Fourier transforms (STFT) can be useful. An STFT employs a

moving window and transforms signal segments to give a time-frequency representation

60

that can help detect arrival of different modes in an ultrasonic guided wave signal, which

can be correlated to the group velocity dispersion curves for the modes existing in the

waveguide. Another prominent approach is to collect time data at periodic spatial

locations and applying two successive Fourier transforms (2D FT) – one in time

(temporal, t) and other in space (spatial, x) to obtain the wave number (k)-frequency (ω)

dispersion plot. Since every waveguide has a characteristic k-ω plot, the plots resulting

from 2D FT can be used to characterize a waveguide.

Since the measurements from the FE model and experiments are discrete in

nature, the Fast Fourier Transform (FFT) algorithm is used for computation of the Fourier

domain information [Alleyne and Cawley 1991]. Following the lines of Alleyne and

Cawley, several researchers have adopted the two-dimensional Fast Fourier Transform

(2D FFT) technique as a first step to analyze the data from FE and experiments [Hayashi

and Kawashima 2002; Castaings et al. 2004; Hosten and Castaings 2006; Drozdz et. al.

2006; Hosten et al. 2007]. The 2D FFT decomposes the waveform data into orthogonal

modes in k-ω domain. Hayashi and Kawashima [2002] used the information in k-ω to

define an analytical dispersion curve based wave number filter to obtain single mode

information and used an inverse 2D FFT to obtain the individual mode signals. Castaings

and co-workers [Castaings et al. 2004; Hosten and Castaings 2006; Drozdz et. al. 2006;

Hosten et al. 2007] used either time domain or frequency domain solution for the wave

propagation problem. They employed COMSOL, an equation based multi-physics solver

to perform the frequency domain solution. The frequency domain solution requires only a

single transform to obtain the k-ω information.

61

The use of the orthogonality property of guided wave modes to determine the

modal amplitudes as a function of frequency is also being employed by some researchers

[Moreau et al. 2006, Drozdz et al. 2006]. This method uses the cross-sectional

displacement and stress values from a FE model and analytical method to compute the

amplitude reflection and transmission factors for the modes and is hence not practical in

experimental measurements.

Using angle beam wedge based measurements, it is possible to experimentally

determine the contributions from the guided wave modes existing in a waveguide [Rose

1999]. This must be done with an understanding of the source influence study presented

in Rose [1999].

3.5 FE modeling of waveguide transitions in adhesive joints using ABAQUS

Structural adhesive joints like lap joints and stringer joints are typical waveguides

involving transition from a single waveguide to a bonded joint and back to a similar or

dissimilar single waveguide. The single waveguide can be an isotropic layer or a

laminated composite.

Understanding the guided wave mode conversion at the transition between

waveguides becomes important for formulating approaches for nondestructive inspection.

The region of interest in the adhesively bonded joints is the adhesive and its bond with

the adherends. Hence the FE model of adhesive should be capable of handling the

adhesive completely. The model of the adhesive is thus very important.

62

ABAQUS is a general purpose FE solver. The explicit time marching solver is

employed in all the FE simulations. The details of this were already mentioned in section

3.2 of this chapter.

3.5.1 Model of the adhesive bond

There are two possible approaches for modeling the adhesive-adherend bonding

which can be implemented in the numerical studies.

1. Combination of longitudinal and transverse springs at the interface between the solids

bonded together [Rose 1999, Castaings 2005]: In the Figure 3-1a, KN and KT denotes

the normal and tangential stiffness of the interface between the two solids – epoxy

and aluminum in this study. These stiffness values can be related to the jump in the

displacement as given by

The interfacial spring stiffness values can be varied suitably to simulate the interfacial

weakness in the bonding. The uses of interfacial springs reduce the computational

load while performing FE calculations, since the elements are replaced by kinematic

relations. ABAQUS provides cohesive elements for use in cases where the adhesive

thickness is very small compared to the adherend thickness [ABAQUS 2008].

2. Use of thin layers to create the interface and model the weakness of bond with the

properties of the interface layer [Castaings 2006]: The adhesive layer (epoxy) is

divided into three parts along its thickness direction (direction 3), as shown in

Figure 3-1b. The layers 1 and 3 in the divided adhesive are given properties such that

.directions coordinate therefer to 3''and1''SubscriptitydiscontinuntDisplaceme

StiffnessStress

and 333131

=∆==

∆=∆=

uK

uKuK NT

σσσ

(3.6)

63

the normal and tangential strengths can be designed for the interface. The interface

material can be declared as orthotropic. A reduction in the stiffness values of the

interface layers simulates the interfacial weakness in the adhesive bond.

Figure 3-1b shows the three layered adhesive model, which was implemented in

this work, to provide a means to account for interfacial weakness between adhesive

(epoxy) and adherend (aluminum).

Figure 3-1: Model of solid-solid interface using (a) normal and tangential stiffnesses(spring model) with springs controlling interface strength, and (b) three layered model ofadhesive with the layers 1 and 3 being used to model interfacial weakness.

(b) Three layered adhesive

Layer 2 (0.14 mm thick)

Layer 1 (0.03 mm thick)

Layer 3 (0.03 mm thick)

(a)

3

1

Solid 1 (Epoxy)

Solid 2 (Aluminum)

KN KT

64

3.5.2 FE model of the adhesive joint transition

A 2-D plane strain model of the adhesive joint was constructed using ABAQUS.

In order to discretize the geometry of the model for the finite element solution,

continuum plane strain elements were used with the element size approximately a tenth

of the wavelength computed based on the smallest wave velocity in the medium at the

loading frequency considered. The explicit solver available in ABAQUS was employed

to obtain the wave propagation through the 2-D geometry. The discretized model is

shown in Figure 3-2. The adhesive region was modeled using the three layer adhesive

model as explained previously.

3.5.3 Guided wave excitation using boundary conditions

In the FE evaluation, we have to use certain boundary conditions in the form of

displacement or pressure for exciting guided wave modes. In order to keep the analysis

Figure 3-2: 2-D finite element model of a simplified skin-stringer adhesive joint. The rectangular elements are plane strain elements having 4 nodes. The thickness of theadhesive layer has been exaggerated for visual clarity of the bonded layup.

Aluminum

Epoxy

Zoomed version of a typical FE mesh at the joint

65

simple, a single mode excitation is preferred. There are different approaches for exciting

a single guided wave mode within a structure for an FE analysis, as shown in Figure 3-3.

1. Wedge loading – A load applied on an angle wedge enables excitation of modes

with a particular phase velocity depending on the wedge angle, at a value of

frequency. (Figure 3-3a).

2. Comb loading – Figure 3-3b shows the comb type loading, which consists of equi-

spaced transducers with spacing equal to the wavelength of the wave mode to be

excited. The transducer inputs can be provided with some time-delays to achieve

beam steering.

3. Wave structure loading – Providing displacement inputs at the cross-section can be

used to generate a single wave mode within the FE geometry (Figure 3-3c). The

geometric length required for this kind of loading is the least when compared to the

wedge and comb loading and is hence preferred for FE simulations. In this work, a

wave structure based loading has been used.

Piezoelectric elements have not been simulated in the numerical study of guided

wave excitation using wedge and comb loading in this thesis. The generation of guided

wave modes is achieved by using either wavestructure loading (Figure 3-3c) or using a

pressure/displacement loading on the region over the wedge or the comb loading region.

66

3.5.4 Oblique incidence guided wave generation and reception

In this thesis, an oblique incidence transducer loading or reception is numerically

modeled by synthetically providing time shifted loading over the spatial span of the

transducer.

The time delay to simulate element less wedge loading/reception is given by,

The time delay for the nth location on the grid from the start (assuming that first

element has zero delay) then becomes

Figure 3-3: Methods for excitation of guided wave modes within a FE model. (a) Wedgeloading, (b) comb loading with simultaneous or time-delayed inputs to the transducers and (c) Loading at the edge with wave-structure of the desired guided wave mode.

( )iLL c

dxcdldt θsin== (3.7)

67

A Hanning weighting function over a length N can be written as

Incorporating the Hanning weighted spatial variation to the time delayed loading

the expression for the loading amplitude at the nth grid point from the start becomes

( )iLL

n cdxn

cdlndt θsin== (3.8)

( ) NnN

nnw ≤≤

−= 0;2cos1

21 π

(3.9)

Figure 3-4: Schematic of the time delay based loading/receiving to simulate an obliqueincidence loading/reception by a transmitter-receiver (T/R). The triangle showing the horizontal spacing between measurement location (dx), the oblique loading/reception angle (θi) and the delay length (dl) is shown on the right. A Hanning weight is alsoincluded to make simulation close to the practical case.

( )nn dttN

naA +

−= ωπ sin2cos1

2 (3.10)

68

The advantages of such an approach are:

1. Reduction in the computation: By avoiding the medium of the wedge or any

surrounding media, there is reduction in the computational effort.

2. Avoiding the multiple reflections in the wedge: A wedge based loading can result

in internal reflections from the wedge entering the waveguide. This can in some

result in signals overlapping with the defect echoes and making an interpretation

of the simulation difficult.

3. A change in media can be accommodated by mere change in the time delays. So a

change from an acrylic wedge to an oblique loading in water can be performed

very easily.

4. Phasing of the delays properly enables measurement of the waves traveling in

both directions separately. This is a very big advantage because in simulation or

in experiments using finite size samples, the edge echoes can be filtered out

without the use of any special absorbing or silent boundary conditions.

This technique is adaptable to different loadings like a wedge, comb and phased array

transducer. The measurement approach using equi-spaced points is applicable to laser

interferometer based measurements and also any point transducer (e.g. Pinducer by

Valpey-Fisher) based measurements.

3.6 Some numerical experiments and data processing

In this work, several different data processing algorithms were applied for

processing the data from numerical experiments. For performing numerical experiments

and data extraction from ABAQUS, ‘Python’ based scripting was used. Signal processing

work was carried out by writing functions using MATLAB, a proprietary software

69

development package. Many of these are applicable to experimental measurements done

using smaller aperture receivers like lasers or point transducers. The techniques have

been presented mostly by addressing the task of processing data from numerical

experiments for quantification of the modal content or other aspects related to the guided

wave propagation within a waveguide. The techniques are classified, based on the extent

of data required for applying the processing method into

a) point data based processing – using short time Fourier transform

b) line data based processing.

(i) Small line data on the surface of waveguide – using phased addition technique

developed in this work

(ii) Larger line data set on the surface of waveguide or at a depth parallel to the

surface of the waveguide – using wave number filtering technique

(iii) Cross-sectional data sets from a waveguide – using Fourier decomposition

with phase correction

Each of the approaches is explained in detail in the next few sections.

3.6.1 Processing single point data – Short time Fourier transform

A short time Fourier transform (STFT) is obtained by computing the Fourier

transform of a windowed region h(t) in a single RF signal x(t)

STFT is essentially a moving window Fourier transform.

( ) ( ) ( ) τττω ωτ dethxtX j−−= ∫, (3.11)

70

Numerical Experiment #1:

A time delayed ultrasonic pressure loading is applied to the top surface of a 2 mm

thick aluminum plate to simulate an excitation of the a0 mode. The excitation is provided

at an angle of 40° with delays corresponding to water based loading (Figure 3-5).

An RF signal collected on top of a 2 mm thick aluminum plate from a FE

simulation is shown superimposed along with its Hilbert transform based envelope (blue

line) in Figure 3-6. The bottom plot in Figure 3-6 shows the result from the STFT in the

form of an intensity plot along with the group velocity dispersion curves for aluminum

rescaled and represented in the form of a velocity vs. time plot in white lines. The

rescaling has been done with the knowledge of the distance of propagation of the mode.

From the STFT representation it can be concluded that the first antisymmetric mode (a0)

is propagating through the waveguide.

Figure 3-5: Schematic of the geometry for numerical experiment #1. Time delays wereused to simulate 40°incidence wave impingement from water. The measurement node onthe surface of the 2 mm thick aluminum plate.

71

The data from the STFT can be further localized in time and frequency using

reassignment techniques that are available in the literature. Some observations on the use

of STFT in guided wave simulation are listed here.

(a) In the case of guided wave propagation at some frequencies, the STFT cannot

resolve the exact mode. For example if the intensity representation in Figure 3-5

was located at a frequency value of ~ 1 MHz, the first three modes in aluminum

coincide perfectly there or in other words have the same group velocity even though

phase velocity value corresponding to this are different. By a mere change in the

angle of reception using a wedge, it will be possible to identify the mode perfectly.

0 10 20 30 40-1

0

1

Time (µs)

Nor

mal

ised

Am

plitu

de RF Waveform

0 10 20 30 40

0

1

2Freq

uenc

y (M

Hz)

Time (µs)

Short time Fourier transform

Figure 3-6: Amplitude vs. time plot of a guided wave mode (a0) propagating in 2 mm thick aluminum is shown on the top (black line) along with its Hilbert transform basedenvelope (blue line). On the bottom plot the short time Fourier transform of the RF waveform is shown with superimposed white lines that correspond to the appropriatelyscaled group velocity dispersion curves for a 2 mm aluminum plate. The data is fromnumerical simulation using ABAQUS.

72

(b) Resolving a higher order guided wave mode also becomes difficult due to the time-

frequency spread of the STFT intensity. A reassignment of weights can solve this

problem.

(c) The use of STFT for mode identification beyond waveguide transitions becomes

more difficult because a transition results in mode conversion that in most cases

also results in a change in the group velocity. Using multiple single point

measurements, i.e. before and after the transition and adjusting for the time of

propagation before the transition, and with the knowledge of the distance of

propagation in the second waveguide, it will be possible to properly identify a mode

in the waveguide after the transition.

(d) The role of STFT is limited to signals that are band limited i.e. the frequency

bandwidth is not too wide. This is because of the use of a fixed width moving

window for computing STFT. For example a laser based guided wave generation

can result in a very large bandwidth like few kHz to roughly 1 MHz and hence the

time-frequency localization will be poor using an STFT. Wavelet based techniques

with variable scaling for a selected wavelet is more effective in such cases.

3.6.2 Processing small line data on the surface of waveguide – Phased addition

A new processing technique developed in this thesis is the use of phased addition

to simulate an element-less transmitter and receiver (described in section 3.5.4 in this

chapter). It was shown earlier that a time delayed input can be used to excite guided

waves within a waveguide. In general by employing time delays given by

( )

[ ]22

sin

angleReceiver ππθ

θ

−∈

==

r

rLL

n cdxn

cdlndt

(3.12)

73

We can add in-phase signals over a range of angles from –π/2 to π/2 to simulate a

receiver wedge (made from any material).

Numerical Experiment #2:

The FE model of a stepped transition in a waveguide was created using aluminum

(2 mm) and bonded aluminum (aluminum 2 mm- epoxy 0.3 mm – aluminum 2 mm). s0

mode in aluminum at 300 kHz was excited within the aluminum plate using

wavestructure loading (Figure 3-7). A set of nodes on the surface of aluminum were

selected for recording the out-of-plane displacement value. The measurement distance

was chosen to be 25.4 mm which is a typical dimension of an ultrasonic transducer

operating at this frequency.

The output data was then subject to phased addition over angles from –π/2 to π/2

(-90° to 90°) with a time delay corresponding to that of an acrylic angle beam wedge. The

result is presented in Figure 3-8. The radial lines in the semi-polar plot represent time

Figure 3-7: Schematic of the geometry, loading and measurement set used in numericalexperiment #2. The wavestructure of s0 mode at 300 kHz is used for guided wave excitation. The measurement nodes at the surface of aluminum are also shown.

74

axis. The positive angles correspond to the wedge orientation for measuring the incident

waves and the negative angles for measuring the reflected waves. The dominant wave

mode in the wavefront incident at the transition is the s0 mode at ~30°. This ensures that

there was indeed a single mode incident at the transition as desired. Among the reflected

waves, the dominant one is at ~ -52° which corresponds to the a0 mode at that frequency.

The very weak signal at 30° at the end of the time scale corresponds to the re-reflected

wavefront that travels back towards the transition.

Figure 3-8: Phased addition based simulated received waveforms at different angles from –π/2 to π/2 radians (-90° to 90°) in an acrylic wedge. The positive angles are measuringthe incident waves and the negative angles the waves reflected from the waveguidetransition. The radial lines are the time axis.

75

The advantages of a phased addition based reception were provided in detail in an

earlier section. For completeness it is summarized here:

(a) Reduction in the computational effort by avoiding modeling wedges

(b) Avoiding multiple reflections from wedge

(c) Flexibility to change medium of the wedge and also the dimension of the

transducer. The influence of the ultrasonic receiver is built into the calculations. It

is thus helpful in the design of experimental configuration.

(d) Ability to measure waves traveling in each direction. This permits finite sized

models for simulation without the need to introduce silent boundaries.

There are some physical constraints to the range of possible angles while constructing

angle beam wedges for actual measurements. The phased addition approach in simulation

does not have any such limitation. The biggest advantage is in actual measurements using

laser vibrometer where data are collected at a few equispaced points normal to the surface

can be used to simulate the response for an angular reception.

3.6.3 Processing large line data on the surface of waveguide – Wavenumber filtering

Knowledge of the guided wave propagation features as summarized in a phase

velocity dispersion plot and signal processing techniques are combined to obtain a novel

tool for filtering guided wave data i.e. separating the contributions of each guided wave

mode.

A guided wave filtering algorithm based on wave number information obtained

from two-dimensional Fourier transform (2DFFT) based filtering is proposed here. The

76

guided wave phase velocity dispersion curve and the wave number –frequency plot are

inter-convertible and hence equivalent. The wave number dispersion plot is employed in

the design of the mode filter.

The displacement and/or stress values from the FE solution are collected at

equally spaced points on the geometry at equal intervals of time over the range of time

interval in the solution and a spatial distance of 6-10 λ. The data set corresponds to a

periodic and a discrete one. Applying two successive Fourier transforms (2DFFT) – one

in time (temporal, t) and other in space (spatial, x) to obtain the wave number (k)-

frequency (ω) dispersion plot [Appendix A].

Numerical Experiment #3:

Figure 3-9 shows the schematic of the data collection from the FE model. The

data is collected from the top of the plate. This is similar to an experimental measurement

on an accessible portion on the surface. The time based data from the left side of the joint

has the incident and the reflected displacement profile as shown in Figure 3-10. The

complete picture of displacements recorded from the FE model is presented along with

the geometry of the adhesive joint in Figure 3-10 and Figure 3-11 .

The wave number filter developed here has two parts – a Directional Filter and a

Guided Wave Mode Matching Filter (GWMMF) – implemented in the same order. The

directional filter helps in separation of the displacement distribution into wave packets

traveling in different directions –say left traveling and right traveling. Once the wave

77

packets have been separated based on their direction of propagation, the mode filter is

applied to obtain the modal contribution along the propagation directions.

Directional Filter

In the brief review of the literature presented, several techniques to handle finite

computational geometry for simulating an infinite domain effect have been listed. These

techniques focus on numerical modeling aspects alone. In real structures, also, we can

have regions where the boundary effect will hinder an inspection, such as a stiffened

plate with multiple stiffeners periodically spaced. Also in real structures with relatively

Figure 3-9: FE model of an adhesive step-lap joint along with the data collection and processing scheme. A wave-structure based displacement loading is provided on the left end of the geometry to create single mode incidence at the joint.

Equi-spaced points for measuring reflected and transmitted displacements

Incident side Transmitted side

Measure displacements u[t, x]

2-D Fourier Transform

[ ]

+−−

=

−

=∑ ∑= M

mkN

njN

n

M

m

extuMN

kUωπ

ω21

0

1

0

,1],[

Guided wave mode filtering and energy partitioning

(Reflection and transmission coefficient for each mode)

78

smaller geometry, it is not always possible to create any improvement to the data analysis

without causing changes to the structure like absorbing tapes, clay etc. The directional

filter developed here will help avoid problems like edge reflections associated with finite

geometry in the FE model.

It can be noted from Figure 3-10 that the incident, reflected and the re-reflected

portions of the waves overlap at several locations on the geometry. This hinders the data

processing. In order to clearly distinguish between the waves based on their direction of

propagation i.e. from left to right and from right to left, the property of 2DFFT is used to

separate signals in the k-ω space [Appendix A]. The transform results in both positive

Figure 3-10: Displacements measured at the incident side of the FE geometry. The wave incident at the bonded joint gets reflected and the same wave undergoes furtherreflection. The incident, reflected and the re-reflected waves overlap at several locations on the geometry.

x axis distance (mm)

Tim

e ( µ

s)

Typical signals recorded on incident side

20 40 60 80 100 120

0

50

100

150

200 -4

-3

-2

-1

0

1

2

3

4

5x 10-10

Incident wave front

Reflected wave front

Re-reflected wave front

79

and negative values of k, which corresponds to opposite directions of wave propagation.

This was verified using simulated waveforms.

Employing concepts similar to masking in digital image processing, directional

filters are defined here by selectively masking diagonally placed quadrant pairs (i.e. I and

III, II and IV) separately (Table 3-1). Employing the directional filters MF and MB,

provided in Table 3-1, we get two sets of data – one corresponding to the forward

propagating wave (left to right in the geometry in Figure 3-9) and the other corresponding

to the backward propagating wave (right to left in the geometry in Figure 3-9).

Figure 3-11: Displacements recorded from the FE solution to wave propagation across asingle step-lap joint.

80

The reflected waves are separated from the re-reflected ones by using the directional

filter in Equation 3.13.

The result is shown in Figure 3-12. The re-reflected field does not have much

significance to the analysis and hence is not shown.

Guided wave mode matching filter (GWMMF)

The separated incident, reflected and transmitted fields in k-ω space, obtained

after applying the directional filter needs to be further processed to identify the different

modes and also to quantify their contribution. The technique developed here is along the

lines of Hayashi and Kawashima [2002] but focuses on the quantification of modal

contributions in reflected and transmitted signals as a function of frequency. Hayashi and

Kawashima [2002] were interested in obtaining the individual mode signals from

transmission measurements using inverse Fourier transform in two-dimensions (inverse

Table 3-1: Definition of the directional filters

Directional filters

Forward filter Backward filter

[ ]

<>><

=elsewhere,1

0,00,0

,0, ω

ωω k

kkM F [ ]

<<>>

=elsewhere,1

0,00,0

,0, ω

ωω k

kkM B

[ ] [ ][ ] [ ]ωω

ωω,x,:waveBackward,x,:waveForward

kMkUWkMkUW

BB

FF

==

(3.13)

81

2D FFT). Since the implementation of the inverse 2D FFT is a single step processing

from the mode filtering, the time domain waveforms are also presented here.

Based on the theoretical wave number dispersion curves, a wave number window

function [ ]im kH ω, is defined for each mode ‘m’ and at each value of frequency ‘ωi’ in

the frequency range ‘ω ’ where the dispersion is considered.

Figure 3-12: Frequency (ω) -wave number (k) plots corresponding to the incident, reflected and the transmitted wave fields obtained by transformation of displacementsusing the 2DFFT. The reflected field is the result of directional filtering obtained afterremoving re-reflected waves.

82

where lok and hik represent the lower and higher cut-off values for the band-pass wave

number filter and m represents any of the possible guided wave modes at a frequency of

ωi. Using the windowing function [ ]im kH ω, , we can isolate each guided wave mode as

shown in Figure 3-13 for the incident mode.

Wave number window width is adjusted to accommodate the wave number spread

obtained in numerical and experimental work. In the case of numerical simulation and

also in experiments, the finite size of the loading, spacing between measurement points

and the bandwidth of the loading determine the wave number-frequency spread. The

wave number windowed signal can be weighted with either a rectangular function i.e.

unit weights within the window or with Gaussian weights (Figure 3-14).

[ ] [ ]

≡∈∀≤≤

=otherwise

kkkkkkkH

iihilo

im

,0,;,1

,m,ωωωω (3.14)

Windowed Incident wave mode

Wavenumber (1/mm)

Freq

uenc

y (M

Hz)

0 1 2 3 4 50

0.5

1

1.5

2

0.5

1

1.5

2

2.5

x 10-9

Figure 3-13: An example of the windowing used for defining guided wave mode matching number filter is shown for the incident mode. The white lines mark the lowerand higher cut-off values of wave number at every frequency value.

83

The wave number filtered data has the modal contributions of each mode.

Additionally we can apply the inverse 2DFFT technique to obtain the waveform

corresponding to each mode in the incident, reflected and transmitted fields as shown in

Figure 3-15. The incident wave being single mode in nature, does not present any special

challenge towards data processing.

Figure 3-14: Guided wave mode matching filters with rectangular weighting (left) andGaussian weighting (right).

84

Figure 3-15: (a) Reflected and (b) transmitted waveforms separated into constituent modes using the guided wave mode matching filter.

85

Computation of the reflection and transmission coefficients

From the data obtained from the GWMMF, we can compute the reflection and

transmission coefficient for each mode (Equation 3.15). The reflection and transmission

coefficients are obtained by normalizing the spectra of the reflected or transmitted wave

mode with respect to the spectra of the incident wave. The 2D plane strain model

implemented in FE gives two displacement components – in directions normal and

tangential to the boundary of the plate. Since we use both components of displacement

while specifying a guided wave mode in terms of its wave structure, the values

corresponding to each component of displacement were summed up in the k-ω space for

the respective filtered modes.

The results have been presented in Puthillath et al. [2008].

Figure 3-16 shows the result from partition of energy between the guided wave

modes for the case of an s0 mode incident at the joint. The a0, s0, a1 and s1 modes that

result from this incidence are also shown in the figure. The a0 mode intensity

compliments the s0 intensity. In the ranges where the s0 mode intensity is higher, like the

frequency range between 1.2 MHz and 1.6 MHz, the a0 mode intensity is lower.

( ) ( )( )

( )( )

mode(s)waveguidedIncidentmode waveGuided

signalofpartincidenttheofspectraLinearsignalofpartdtransmittetheofspectraLinear

Frequency,,FactoronTransmissi

*

*

==

==

=

=

mmUU

kUkUTF

I

T

Im

Tm

m

ωω

ωωωω

(3.15)

86

A brief discussion on processing of line data from the surface of a waveguide is

provided below. The following points are worth mentioning:

(a) 2DFFT based processing requires equi-spaced data over a few wavelengths (6-10 λ)

from the surface of the waveguide. This can be used in actual experiments also. The

processed result provides the phase velocity dispersion curves of the waveguide

where the measurements are being taken. This approach is useful in obtaining the

guided wave dispersion curves in actual waveguides where the complete geometry

and material information is not available for analytical computation or wherever the

analytical computation is not possible.

(b) The in-plane and out-of-plane displacement can be selectively used for numerical

simulation data to highlight the contribution of different modes.

(c) The data used as input to the 2DFFT is larger than that for a phased addition

approach. It is possible to filter out the back reflection data using this technique,

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

Frequency (MHz)

Tran

smis

sion

Coe

ffic

ient

A0S0A1S1

Figure 3-16: Guided wave modes transmitted across the overlap region in a step-lap joint for an s0 mode incidence.

87

which makes it similar to the phased addition approach. The need for much data can

be a limitation in some cases.

(d) Analysis of scattering and mode conversion can be easily performed using the 2DFFT

and the mode matching filtering scheme developed in this thesis.

(e) In numerical experiments, the 2DFFT based data collection can be performed on a

linear equi-spaced data obtained from any cross-sectional position within a

waveguide. Using this approach it is possible to monitor the interaction of guided

wave mode with embedded flaws like delamination in a composite or debonding in a

bonded joint.

3.6.4 Processing line data across the waveguide thickness – Wavestructure data

Guided wave propagation is characterized by cross-sectional variation of

displacements and stresses. It is possible to obtain the cross-sectional distribution of

displacement or stress from an FE model to improve the understanding of guided wave

propagation within a waveguide. Unlike the signal processing approaches presented so

far, the method presented in this section is not a practical one.

Fourier based decomposition of cross-sectional data obtained from a waveguide

along with phase correction is carried out in this section. The hybrid FE approach

attempted here is expected to help in understanding the aspects of wave propagation

across transitions in any waveguide. The method is specifically applied here to

understand wave propagation across a transition from adherend to bonded region in a

stringer joint viz. the formation of modes within the joint region and the length of travel

within the bonded region for the formation of a stable mode supported within that region.

88

There are no published literature, known to the author at this point, on the mode

formation length for a mode entering the transition. A Semi-Analytical Finite Element

(SAFE) approach to understanding guided wave mode conversion at a transition assumes

the presence of the normal mode. This does not account for any near field distance.

The schematic of the data collection scheme designed for understanding the mode

formation immediately after a transducer, is presented in Figure 3-17. The cross-sectional

lines (vertical lines) in Figure 3-17 are used for extracting the wavestructure data from

the FE model. The data are collected over a length of 3-5 λ. The linear data set for

extracting the 2DFFT data is also shown to highlight the difference in the amount of data

collection required.

The new data collection scheme requires a processing method for decomposing

the displacement profile into the constituent modes. The resulting data are expected to

correspond to the wavestructure at that mode and frequency combination. Displacement

waveform at every node point at a cross-sectional line segment in the FE model is

decomposed into its frequency content using numerical implementation of Fourier

transform for discrete time signals – Fast Fourier Transform (FFT).

x3 x1

x2

Wavestructure measurement (3 to 5 λ)

Equi-spaced measurement nodes for computing 2DFFT (6 to 10 λ)Loading

function

x3 x1

x2

x3 x1

x2

x3 x1

x2

Wavestructure measurement (3 to 5 λ)

Equi-spaced measurement nodes for computing 2DFFT (6 to 10 λ)Loading

function

Figure 3-17: Ultrasonic loading function for exciting guided wave mode(s) and the datacollection scheme used in the FE model.

89

Numerical Experiment #4:

As an example, the case of comb loading with λ = 4.7 mm, 5 elements, excited

with a 3 cycle Hanning windowed pulse centered at a frequency of 0.5 MHz on a 2 mm

thick aluminum plate was solved using Abaqus Explicit (Figure 3-18). This loading

configuration is expected to generate a0 mode in aluminum.

The data processing scheme is presented in Figure 3-19. The absolute value of the

magnitude of the in-plane (U1) and out-of-plane (U2) displacements for f0 = 0.5 MHz (or

ω0 = 2πf0 ) measured at x1 = 0 from the excitation source, i.e. the edge of the comb

loading, are shown with their respective phase values (φU1 and φU2) in Figure 3-20. It can

be observed from Figure 3-20 that φU1 undergoes a sign change across the thickness of

the sample whereas φU2 does not. The theoretical wavestructure has one component of the

displacement as symmetric and another as anti-symmetric. Hence an artificial sign

change is introduced into the in-plane displacement wavestructure from FE (seen in the

top-left plot on the Figure 3-20) and presented in Figure 3-21.

Figure 3-18: Schematic of the geometry, loading and measurement set used in numerical experiment #4. A 5 element comb loading on the surface at 0.5 MHz for 3 cyclesgenerates a0 mode. The measurement nodes across the cross-section of aluminum are also shown.

90

It was observed that the wavestructure from the comb loading measured at the

edge of the comb, corresponds to the a0 mode. We can conclude that a comb source,

designed for a mode and frequency combination possible in a specific waveguide,

instantaneously forms the guided wave mode. Though not reported here, the case of an

acrylic angle beam wedge loading (29°, 3 cycles at 0.5 MHz – for s0 mode) and oblique

incidence loading in water using an immersion transducer (40°, 3 cycles at 0.5 MHz – for

a0 mode) were also studied for mode generation in a 2 mm thick aluminum plate. The

loading using acrylic angle beam wedge did not result in a single mode generation. Using

acrylic wedge, a0 generation is not possible because the phase velocity of a0 mode at 0.5

MHz is smaller than the longitudinal velocity in acrylic. The mode generation using

oblique incidence in water was pure a0 and similar to that of comb load.

Fast Fourier Transform

u(x2,t) |x1

Linear Spectrum

|U(x2,ω)||x1

ω0

Wavestructure U(x2,ω0,φ) |x1

Phase φ(x2,ω) |x1

Phase correction

Fast Fourier Transform

u(x2,t) |x1

Linear Spectrum

|U(x2,ω)||x1

ω0

Wavestructure U(x2,ω0,φ) |x1

Phase φ(x2,ω) |x1

Phase correction

Phase correction

Figure 3-19: Fourier transform based scheme for extracting wavestructure data at a cross-section (located at x1) using the displacement from FE at the nodal points u(x2,t) at that cross-section. ‘u’ includes both in-plane (u1) and out-of-plane (u2) components.

91

The numerical experiments so far have shown that the wavestructure information

extracted can provide the details of the mode generated with a waveguide. In all the cases

it was observed that the mode is generated within the first 1 mm of the excitation point.

From all this it can be concluded that the near field distance for formation of a stable

guided wave packet does not exist and that a stable mode is formed instantaneously

within a waveguide. A study was also conducted using a numerical model of aluminum

to bonded aluminum transition. The observation lines were marked across the thickness

of the three layered bonded joint. Employing the phase corrected wavestructure data

processing scheme developed here, the wavestructure data was extracted (Figure 3-22).

-1 -0.5 0 0.5 10

0.5

1

1.5

2

Normalised displacement

Thic

knes

s (m

m)

U1

-2 -1 0 1 20

0.5

1

1.5

2

Phase of displacement

Thic

knes

s (m

m)

φU1

-1 -0.5 0 0.5 10

0.5

1

1.5

2

Normalised displacement

Thic

knes

s (m

m)

U2

-0.04 -0.03 -0.02 -0.010

0.5

1

1.5

2

Phase of displacement

Thic

knes

s (m

m)

φU2

Displacement and Phase at 0 mm from source; Frequency = 0.5 MHz

Figure 3-20: The magnitude of displacement (U) and phase (φ) of components (both in-plane and out-of-plane) at a frequency of 0.5 MHz across the thickness of the 2 mm thick aluminum sample when excited by a 5 element comb load with λ = 4.7 mm and a 3 cycleHanning windowed pulse at 0. 5 MHz. The values were calculated from cross-sectional nodes located at the end of the comb source.

92

The extracted wavestructure in Figure 3-22 is close to that of mode 3 in the

bonded aluminum joint at 0.5 MHz (wavestructure of mode 3 at 0.8 MHz is shown in

Figure 2-8). An orthogonal decomposition was not required in these cases and hence not

implemented.

The results shown in Figure 3-22 also point to the immediate formation of the

wavestructure within the bonded joint when the guided wave mode is incident from the

aluminum. This resolves the concern regarding the sensitivity of a generated guided wave

mode to defects located at the edge of a bonded joint – i.e. the location of maximum

shear and peel stresses under an applied mechanical loading. Hence it can be concluded

that mode(s) incident from the aluminum side, that converts to the desired interface

sensitive modes will be sensitive to the entire bondline.

-1 -0.5 0 0.5 10

0.5

1

1.5

2

Normalised amplitude

Thic

knes

s (m

m)

Displacement at 0 mm from source; Frequency = 0.5 MHz

U1U2

Figure 3-21: FE wavestructure values after incorporating phase correction. The extractedwavestructure matches the a0 mode wavestructure at 0.5 MHz.

93

The wavestructure extraction work developed and some major findings are

summarized below.

(a) The technique even though not useful in practical measurements, can be used to

extract wavestructure information. By employing orthogonal decomposition similar

to that in the literature, it is possible to find the contribution of each mode in the

transmitted wave.

(b) The influence of a comb, wedge (water and acrylic based) and wavestructure

loading in the generation of guided wave within a waveguide was studied. It was

found that the formation of modes within a waveguide is almost instantaneous and

hence there is no near-field distance involved.

(c) Generation of guided wave modes across a waveguide transition was also studied. It

was found that the mode formation is almost instantaneous. This has a big practical

implication as it can be stated that the interface sensitive modes generated within

the bonded waveguide by means of mode incidence from the aluminum skin will be

Figure 3-22: Wavestructure data extracted from the FE model of the aluminum to epoxybonded aluminum transition. The wavestructure is very close to that of mode 3 in bondedaluminum joint.

94

sensitive to the entire length of the bonded interface. So it is in principle possible to

detect defects located at the edge of a bonded joint where the peel and shear stresses

are a maximum.

3.7 Summary

In this chapter a hybrid analytical finite element framework and the necessary

signal processing techniques for meaningful interpretation of the FE data was either

newly developed or improvised beyond the techniques existing in the literature or

implemented from the literature.

The main work in this chapter is summarized below:

a) A short time Fourier transform (STFT) based signal processing for mode

identification in a waveguide from a single point data was implemented. This

technique can not be easily extended to the case of waveguide transition because of

the change in the velocity with a change in the waveguide. So it has limited

applicability.

b) A new signal processing technique – phased addition approach - applicable to linear

data set collected from the surface of the numerical model or the actual sample was

developed. Using this technique, element-less simulation of a wedge, comb or

phased array loading is possible. By suitably tailoring the time delays central to this

approach, it is possible to separate the incident and reflected waves. This

contributes to receiver size based data collection and ability to reduce

computational effort by avoiding the wedge material and the need for silent

boundary conditions.

c) A modification to the existing two dimensional Fourier transform (2DFFT) based

analysis of equi-spaced data was performed by developing directional and

95

wavenumber filters in order to separate out the waves propagation in a waveguide

both in terms of the direction of their propagation and also into the constituent

modes.

d) A new processing technique to extract wavestructure data from a time marching FE

simulation was performed. This work pointed to the instantaneous mode formation

within a waveguide both near the source and also at a waveguide transition. The

most important implication is the sensitivity of any interface sensitive modes within

the bonded waveguide to the defects at the edges of the bondline – a critical and

high stress region in a bonded joint.

Chapter 4

Ultrasonic Guided Wave Inspection of Adhesive Repair Patches

4.1 Introduction

Aircraft and other load bearing and safety critical structures are subject to in-

service loading like fatigue, thermal and chemical environments that can initiate points of

weakness within the structure, such as fatigue cracks, corrosion, and delamination, thus

leading to a reduction in their service life. In the aircraft industry, especially in military,

aging induced structural weaknesses are mitigated using appropriate repairs because

replacement is prohibitive in terms of time and cost [Pyles 2003]. Repairs can be

performed using mechanically fastened or adhesively bonded patches. In comparison to

mechanical fastened repairs, a bonded repair produces minimal alteration to the

aerodynamic contours, results in weight savings in addition to avoiding the stress raisers

associated with bolt/rivet holes.

Adhesively bonded metal or composite sheets are employed as repair patches in

several aerospace applications [Baker et al. 2009, t’Hart and Boogers, 2002]. An example

of crack in the upper attachment flange in the longeron of an F-16 and the view after

repair using a titanium patch is shown in Figure 4-1. The structural integrity of a titanium

repair patch bonded to the aluminum skin of an aircraft is studied in this chapter.

Titanium has a higher strength-to-weight ratio and good temperature and corrosion

resistance. Typical of adhesive bonding, the repair patches also have a possibility of an

97

occurrence of cohesive (bulk) and adhesive (or interfacial) weaknesses. This will result in

less than expected life extension to the aircraft. Once repairs are complete, it is

imperative that some nondestructive inspection is performed to ascertain their quality.

The use of ultrasonic guided waves for nondestructive inspection of adhesive repair

patches is the subject of study in this chapter.

4.2 Literature on adhesive repair patch inspection

Baker et al. [2009] have recently demonstrated a method for in-situ local

monitoring of an adhesive repair patch using piezoelectric film strain sensors bonded to

the patch and the skin. They have also considered the use of an optical measurement

approach using optical fibers, which requires embedment of the Bragg grating fibers

Figure 4-1: Top: Cracks in the upper attachment flange in the longeron of an F-16. Bottom: Titanium (0.5 mm) repair patch bonded at the crack location on the longeron.[Modified from source: t’Hart and Boogers, 2002]

98

within the repair patch structure. The primary defect types considered are disbonds.

These approaches provide very good local inspection, but require an extensive coverage

of the repair patch using either strain gages or optical fibers to obtain a global inspection.

Thermal wave imaging has been used by Aglan et al. [1999] in order to detect and

size disbonds in boron-epoxy repairs on aluminum skin of an aircraft. This is a non-

contact inspection approach, which requires very high speed digitization and

sophisticated image processing capabilities. For inspection of metallic adhesive repair

patches such as a titanium based patch, the higher thermal conductivity in the metal

necessitates the use of milliseconds or smaller duration flash heating sources along with

an expensive infrared camera, thus pointing to other inspection techniques like ultrasonic

guided waves as a more viable approach.

Researchers have also studied piezoelectric based inspection approaches. Chiu et

al. [2000] have studied the changes to the impedance of a bonded piezoceramic element

in the presence of a disbond. They have also studied a transfer-function based approach

by studying transmission from a single actuator to multiple other actuators bonded to the

repair area. This method is more of a Structural Health Monitoring (SHM) approach and

requires the use of multiple sensors bonded to the surface of the structure that may not be

feasible in the field. Lopes et al. [2000] have formulated impedance-based approaches to

be used in combination with Artificial Neural Networks for detection, localization and

also characterization of damage to structures.

99

Ihn and Chang [2004] and Kumar et al. [2006] have employed a smart

piezoelectric transducer layer (SMART layer from Accelent Technologies) to surround

the repair patch region in order to image weakness in bonding on the lines of structural

health monitoring. The choice of the piezoelectric transducers governs the wave mode

generation aspects in the waveguide. The focus in these works is towards the

development of a SHM system by permanent bonding of sensors to the structure.

Researchers have also reported the use of commercially available bond testing

equipment like the 2100 Bondascope [Baker et al. 2009], and Fokker Bond tester [t’Hart

and Boogers 2002] for bond quality inspection. These are also point by point inspection

tools similar to the conventional ultrasonic C-scan type approach [Rose, 1999] and are

successful primarily in detecting cohesive weakness in the bonding.

Ultrasonic inspection employs elastic wave propagation in structures for non-

destructive assessment of the quality of an adhesive joint. Ultrasonic bulk waves

generating shear at the interface of adhesive joints under oblique incidence have been

shown by Pilarski and Rose [1988] to represent a reasonable approach for inspection. The

bulk wave approach is local and is cumbersome for large area inspection. Additionally, it

requires a very large frequency (>10 MHz) for inspection. These techniques are still

point-by-point inspection approaches, requiring longer inspection times.

Ultrasonic inspection of a particular type of interfacial weakness in mechanically

contacting surfaces called a kissing bond has been studied by several researchers. Jiao

and Rose [1991] considered a kissing bond to be an interface that does not transfer shear

100

and successfully employed an oblique incidence shear wave technique to detect oil

contaminated interfaces in bonded structure. Nagy [1991] used ultrasonic C-scan using

high frequency (50 MHz) probe to detect release agent contaminated areas that were

considered to be kissing bonds. Brotherhood et al. [2002] employed an ultrasonic C-scan

using a 10 MHz probe to successfully detect the presence of a contaminating layer at

aluminum-adhesive interface.

Ultrasonic guided waves are structural resonances that propagate at specific

frequency and phase velocity combinations primarily influenced by the geometry and

material properties of the medium where the wave is propagating with stress free

boundary conditions [Rose, 1991]. Both longitudinal and shear motion can be generated

along the bonded region, thus meeting the conditions of the shear incidence method

developed by Pilarski and Rose [1988], yet leading to a simpler ultrasonic test method

compared to oblique incidence. For a choice of waveguide material, guided waves

demonstrate geometric dispersion phenomena where the phase velocity varies as a

function of the frequency. Guided waves can propagate longer distances thus enabling

inspection of the entire length from the place of its actuation to the place of sensing. With

an appropriate choice of wave modes for inspection, guided waves has been successful in

different aircraft bond inspection scenarios [Pilarski and Rose 1988, Rose et al. 1996].

The requirement for modeling capability has limited the wider applicability of guided

waves in the past, but computational efficiency has advanced to the point that faster

modeling is now possible.

101

Pilarski and Rose [1992] employed dispersion curve shifts and a criterion

combining the cross-sectional displacements and power flow to determine the suitability

of a mode for adhesive bond inspection. It was observed by Pilarski and Rose [1992] that

the modes with larger in-plane displacement did not necessarily have a larger frequency

shift while considering the dispersion curve shifts. Rose and co-workers [1996] have

successfully demonstrated that by selecting the guided wave modes corresponding to a

good overlap between the individual plate dispersion curves - aluminum and boron epoxy

forming a step joint at the repair area, a good defect sensitivity can be obtained. Many

works, both in the past and present, have concentrated on the inspection of adhesive skin-

stringer joints and step-lap joints [Rose and Ditri 1992, Rose et al. 1995, Rokhlin 1991].

The mode selection criterion in these works is governed by a change in the adhesive bond

geometry. This will be presented in the next chapter.

A bonded adhesive joint such as a repair patch can be considered according to the

terminology used in this thesis as a continuous waveguide. In this chapter, a theoretical

study is carried out where the guided wave phase velocity dispersion curves are used in

conjunction with wave structures to determine optimal conditions for inspection of

adhesive and cohesive weakness in continuous waveguides. Epoxy bonded aluminum -

titanium repair patches were prepared with interfacial weakness conditions simulated by

using teflon inserts and other surface variation techniques. The inspection technique

presented here is applicable to the inspection of bonded repair patches under the

condition that both the transmitter and the receiving transducers rest on the bonded joint.

The optimal guided wave mode was generated in the bonded sample using an ultrasonic

102

transducer mounted on an acrylic angle beam wedge. The difference in transmission in

terms of the signal content was successfully analyzed and used to discriminate between

the defective and non-defective regions in the structure. This work has been recently

reported by Puthillath and Rose [2010a, 2010b, 2009].

Section 4.3 covers the guided wave theory, Section 4.4 covers mode selection and

mode generation aspects, Section 4.5 deals with the sample preparation, Section 4.6

provides details of the experiments and a discussion on the experimental observations.

Conclusions are provided in Section 4.7.

4.3 Ultrasonic guided wave propagation through a repair patch

The equation governing the wave propagation in a structure (Navier’s equation),

is obtained by combining the stress-strain relations and the strain-displacement relations

and is written in a index notation in Equation 4.1 as given in section 2.3 .

where ui is the wave displacement, Cijkl is the elastic stiffness tensor , and ρ the density of

the material. The dot in the superscript refers to the time derivative and the comma in the

subscript corresponds to a spatial derivative. In the case of a multilayered media, like the

adhesive repair patch shown in Figure 4-2, Equation 4.1 is valid in each layer. By

representing the displacement in terms of a harmonic trial solution with a multiplier of

ijklijkl uuC &&ρ=, (4.1)

103

the form ( ){ }tcxxik p−+ 31exp α , the dispersion relation can be solved [Rose, 1999] for a

wave propagating along x1 and having a wave number k and phase velocity cp. The wave

number in the direction perpendicular to the wave propagation direction or x3 is

represented with the inclusion of the ratio α [Nayfeh 1995].

Guided wave dispersion curves are typically generated by extracting the eigen

values of the matrix formulated using the global matrix method [Rose, 1999] after

incorporating the free boundary conditions and the interfacial continuity conditions

(Equation 4.2) for the waveguide structure. The general outline for the solution of the

guided wave propagation in a free waveguide is provided in Chapter 2.

Figure 4-2: Material layers in a typical aircraft adhesive repair patch. The aluminum layerrepresents the aircraft skin on which the titanium repair patch has been bonded usingepoxy adhesive. The coordinate system with two representative conventions is alsoshown.

( )3,2,1interfacestheatcontinuousare,

surfacesbottomandtoptheat0

3

3

=

=

iu ii

i

σσ

(4.2)

x3 (3)

x1 (1)

x2 (2)

t1

t2

t3

Aluminum

Epoxy Titanium

104

Ultrasonic guided wave dispersion or variation in wave propagation velocity with

frequency of the wave for the aircraft adhesive repair patch – i.e. aluminum skin (3.175

mm thick) with an epoxy (0.66 mm thick) bonded titanium (1.6 mm thick) repair patch is

shown in Figure 4-3.

The material layers are assumed to be isotropic. The material properties for all

material layers were determined using ultrasonic bulk wave velocity measurements

[Rose, 1999] and are listed in Table 4-1. These material properties and thicknesses were

used in the solution process.

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

Frequency (MHz)

Phas

e V

eloc

ity (k

m/s

)

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

17

18

19

20

21

22

23

24

25 27

Figure 4-3: Lamb wave phase velocity dispersion curves for the aircraft adhesive repairpatch: Aluminum (3.175 mm)-Epoxy (0.66 mm)-Titanium (1.6 mm). The Lamb wave modes are identified with numbers.

105

The guided wave dispersion curves in Figure 4-3 correspond to the plane strain

solution to the elasticity problem and are hence referred to as the Lamb-type wave

dispersion curves. The term Lamb-type is used so as to distinguish from the conventional

Lamb’s problem of wave propagation in a stress-free isotropic layer. Each point on the

Lamb-type wave dispersion curve has a cross-sectional vibration pattern (the eigen vector

corresponding to the eigen value at that point) called the wavestructure. All the numbered

lines in Figure 4-3 – called the Lamb-type wave modes- are obtained by joining points

with similar wavestructure. Conventionally the Lamb wave modes are given alphabetical

prefixes - ‘a’ for antisymmetric and ‘s’ for symmetric based on their symmetry with

respect to the mid-plane. Since the repair patch does not possess mid-plane symmetry, the

prefixes have been dropped and an incremental numbering is used to identify the modes.

The wavestructures result in the sensitivity of different points on the dispersion curves to

defects at different depths in the waveguide and hence the ability to solve infinite defect

detection problems using ultrasonic Lamb-type waves.

For the sake of completeness the group velocity dispersion curves for the repair

patch - Aluminum (3.175 mm)-Epoxy (0.66 mm)-Titanium (1.6 mm) are shown in

Table 4-1: Wave propagation velocities and the computed elastic modulus values formaterials used in this study

Material Density Longitudinal Velocity

Shear Velocity

Young’s Modulus

Poisson’s ratio

[kg/m3] [m/s] [m/s] [GPa]

Aluminum 2700 6219.61 3104.36 69.43 0.334

Epoxy 1104 2249.04 908.65 2.56 0.402

Titanium 4430 5890.31 3087.74 110.70 0.310

106

Figure 4-4. The locations with lower value of group velocity in Figure 4-4 correspond to

a large value of phase velocity.

4.4 Lamb wave mode selection

The primary challenge while using the Lamb wave solutions in a practical

inspection problem is to determine the mode and frequency combination which can

provide optimal defect sensitivity. Additionally it is preferred to have minimum number

of modes excited within the structure so as to keep the experimental interpretation simple.

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

Frequency (MHz)

Gro

up V

eloc

ity (k

m/s

)

1

2

3

4

5

6 7

8

9

1011

12

1314

15

161718

19 2021

22

2324 25 2627 28

29

Figure 4-4: Lamb wave group velocity dispersion curves for the aircraft adhesive repairpatch: Aluminum (3.175 mm)-Epoxy (0.66 mm)-Titanium (1.6 mm). The Lamb wave modes are identified with numbers.

107

The challenge in this problem of adhesive repair patch inspection is to detect

defects at the metal-epoxy bond interface. It is assumed in this study that defects are

mostly located at the aluminum-epoxy interface as a consequence of the repair process.

The structure also imposes some constraints to the inspection – only the titanium side is

accessible for placing the probes. In addition to this, experience and the literature also

point to the possibility of thickness variation in the bond line. A frequency-thickness

product based scaling cannot be extended to handle this situation because variation in the

thickness of a single layer will imply a new waveguide.

4.4.1 Displacement wavestructure

Figure 4-5 shows the Lamb wave dispersion curves for the adhesive repair patch

along with the displacement wavestructures or normalized cross-sectional displacement

distributions for two neighboring modes (modes 17 and 18) on the dispersion curves at

the same phase velocity value (14.37 km/s). The modes are picked for demonstration

purpose only. The different material layers viz. aluminum, epoxy and titanium are shown

separated by dotted black lines on Figure 4-5. The solid lines – black and blue show

respectively the out-of-plane (uz) and the in-plane (ux) displacements. It can be seen from

Figure 4-5 that the in-plane displacement at the aluminum-epoxy interface is much larger

at location 2 (2.54 MHz, 14.37 km/s) compared to location 1 (2.31 MHz, 14.37 km/s). A

larger in-plane displacement at the aluminum-epoxy interface is expected to make the

mode 18 more sensitive to defects located at that interface than mode 17.

108

Figure 4-5: Lamb wave dispersion curves for aluminum-epoxy-titanium adhesive repair patch and two wave structures or cross-sectional displacement profiles (at locations 1 and 2on the dispersion curves). The dotted lines demarcate the aluminum, epoxy and the titanium regions, with aluminum being at the bottom. A larger in-plane displacement (ux) at the aluminum-epoxy interface can be noticed at location 2.

-1 -0.5 0 0.5 10

1

2

3

4

5

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Wave Structure at 2.54 MHz, 14.37 km/s

uxuyuz

2

-1 -0.5 0 0.5 10

1

2

3

4

5

Displacement Amplitude

Posi

tion/

Thic

knes

s (m

m)

Wave Structure at 2.31 MHz, 14.37 km/s

uxuyuz

1

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

14

16

18

20

Frequency (MHz)

Phas

e V

eloc

ity (k

m/s

)

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

17

18

19

20

21

22

23

24

25 27

1 2

109

4.4.2 Interfacial in-plane displacement for defect sensitivity

For sensitivity to aluminum-epoxy interface defects, especially kissing bond type

defects, Lamb wave modes with large in-plane displacements at the aluminum-epoxy

interface were selected. The intensity plot in Figure 4-6 shows the variation of the

normalized in-plane displacement amplitude (ux) at the aluminum-epoxy interface along

with the Lamb wave dispersion curves. The normalization was done for per mode. The

mode numbers are marked in white letters. The red region in the intensity plot shows the

locations on the dispersion curves having large in-plane displacement at the aluminum-

epoxy interface for that mode.

Frequency (MHz)

Phas

e ve

loci

ty (k

m/s

)

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

17

18

19

20

21

22

23

24

25 27

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 4-6: Amplitude map of the in-plane displacement at the aluminum-epoxy interface superimposed over the Lamb wave dispersion curves for the adhesive aircraft repair patch comprised of a titanium patch (1.6 mm) bonded using epoxy (0.66 mm) onto an aluminum skin(3.175 mm).

110

From the amplitude plot in Figure 4-6, we can identify several regions where the

in-plane displacement at the aluminum-epoxy interface is high (red in color). The region

around mode 2 with large in-plane displacement is too low in frequency and hence it is

not considered for reasons of resolution. Also the gradient of the displacement across the

thickness of the repair patch is not large. Regions on modes 11, 14 are surrounded by a

large number of modes. Guided wave excitation studies using finite dimensional sources

supplied with a finite duration electrical pulse, performed by Rose and Ditri [Rose, 1999],

points to the possibility of generating guided waves over a range of phase velocity and

frequency and not a single phase velocity and frequency point. This makes the interface

sensitive regions on modes 11 and 14 not as attractive as a solution.

Mode 18 has a high interfacial in-plane displacement around 2.5 MHz, which is

expected to provide good sensitivity to adhesive weakness. It looks attractive because the

spacing between mode 18 and its neighboring modes is higher along the frequency axis

which can provide simplicity in interpretation of the experimental signals. Also, the

wavelength associated with this mode at around 2.5 MHz range is lower, which implies

better resolution. The phase velocity for this mode at ~2.5 MHz is around 15 km/s.

4.4.3 Influence of adhesive thickness

A big challenge in practical adhesive bond inspection problems is the change in

the bond-line thickness. Use of cured epoxy strips, glass beads, shim-stocks etc. are

generally made to control the thickness of the bond line. Despite this, a local variation in

adhesive thickness is possible. In the guided wave based inspection scenario, a change in

111

bond line thickness implies a change in the thickness of just one layer of the layered

waveguide – implying a new problem to be solved. This is a limitation of the guided

wave technique. Additionally the adhesive repair patch is not mid-plane symmetric

structure. These reasons rule out the possibility of a frequency-thickness scaling to handle

a change in thickness of the adhesive layer in the repair patch.

The effect of variation in the thickness of the bond-line on the Lamb wave phase

velocity dispersion curves is shown in Figure 4-7. The thicknesses of the adhesive layer

are selected such that the extreme limits correspond to a 100 % variation in the thickness

(0.4318 mm to 0.8636 mm). The median thickness (0.6604 mm) value is the average of

the extreme thicknesses.

It can be noticed that with an increase in the thickness of the adhesive layer, the

dispersion curves shift to the lower frequency region with the addition of new modes

also. By careful study it was found that the number of modes that can exist in the

waveguide changes from 27 to 30 to 32 with the increase in epoxy layer thickness. This

was expected because change in the dimension of a single layer in a multi-layered plate

implies a new waveguide problem.

Since each thickness of the adhesive layer implies a new waveguide dispersion

problem, the mode and frequency combination chosen for a guided wave based

inspection should be applicable for this range to be a simple and practically

implementable solution. In order to verify if the choice of ~2.5 MHz and 15 km/s is

112

applicable for the lower and upper bounds of adhesive thickness, the interfacial in-plane

displacement profile at the aluminum-epoxy interface were also plotted (Figure 4-8).

Mode 16 in the case of repair patches with 0.4318 mm thick epoxy (Figure 4-8a)

and mode 19 in the case of repair patch with 0.8636 mm thick epoxy (Figure 4-8b) have a

large in-plane displacement at the aluminum-epoxy interface for a frequency of around

2.5 MHz. The phase velocity in both the cases is also ~ 15 km/s. Hence it is confirmed

that the frequency-phase velocity combination of 2.5 MHz and 15 km/s is sensitive across

the assumed range of variation in epoxy thickness.

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

14

16

18

20

Frequency (MHz)

Phas

e V

eloc

ity (k

m/s

)

Epoxy thickness = 0.4318 mmEpoxy thickness = 0.6604 mmEpoxy thickness = 0.8636 mm

Figure 4-7: Lamb wave phase velocity dispersion curves for the aircraft adhesive repairpatch: Aluminum (3.175 mm)-Epoxy (t mm)-Titanium (1.6 mm). The value of t varies from 0.4318 mm to 0.8636 mm centered at 0.6604 mm.

113

Frequency (MHz)

Phas

e ve

loci

ty (k

m/s

)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23 25

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequency (MHz)

Phas

e ve

loci

ty (k

m/s

)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 4-8: Amplitude map of the in-plane displacement at the aluminum-epoxy interface superimposed over the Lamb wave dispersion curves for the adhesive aircraft repair patchcomprised of a titanium patch (1.6 mm) bonded using epoxy (t mm) onto an aluminum skin (3.175 mm). The values of t are (a) 0.4318 mm and (b) 0.8636 mm.

(a)

(b)

114

Table 4-2 summarizes the interface sensitive mode, frequency and phase velocity

for different thicknesses of epoxy shown in Figure 4-6 and Figure 4-8. It is also noted

that even though the selected frequency-phase velocity combination of 2.5 MHz-15 km/s

has large in-plane displacement at the aluminum-epoxy interface, its value is not the same

in all these cases.

4.4.4 Interface selectivity of a defect sensitive mode

While selecting guided wave modes for sensitivity to a defect at a specific cross-

sectional location in a multi-layered waveguide, it also needs to be ensured that the

chosen phase velocity-frequency combination is sensitive only to defects at the desired

location in the cross-section. In the repair patch inspection case, for example, since the

primary focus of the study is to detect defects at the aluminum skin – epoxy interface and

the inspection is performed from the surface of the titanium layer, it should be ensured

that a defect at the titanium-epoxy interface does not prevent detection of a defect located

Table 4-2: Summary of the mode and frequency (~2.5 MHz) combination with larger in-plane displacement at the aluminum-epoxy interface for different epoxy thicknesses inthe titanium (1.6 mm) – epoxy (t mm) – aluminum (3.175 mm) bonded media. Locations with phase velocity close to 15 km/s has been tabulated.

Epoxy Thickness

t

Mode Number Frequency Phase

Velocity

Normalized displacement at aluminum-epoxy

bondline [mm] [MHz] [km/s] |ux| |uz|

0.4318 16 2.535 15.032 0.753 0.215

0.6604 18 2.532 15.052 0.855 0.270

0.8636 19 2.507 15.066 0.570 0.152

115

at the aluminum-epoxy interface. The intensity chart in Figure 4-9 shows the interfacial

in-plane displacement at the titanium-epoxy interface in the repair patch for the median

and extreme thickness values of the epoxy layer. It is clear from the Figure 4-9 and

Table 4-2 that the mode at 2.5 MHz and 15 km/s (marked on the plot) has a very low

value of in-plane displacement (ux) at the titanium-epoxy interface. So this phase velocity

– frequency combination is selectively sensitive to only the interface of interest. This

verification again confirms the choice of mode at 2.5 MHz and ~15 km/s is optimal for

inspecting the aluminum-epoxy interface.

Frequency (MHz)

Phas

e ve

loci

ty (k

m/s

)

16

0 1 2 3 40

5

10

15

20

Frequency (MHz)

Phas

e ve

loci

ty (k

m/s

)

18

0 1 2 3 40

5

10

15

20

0

0.5

1

Frequency (MHz)

Phas

e ve

loci

ty (k

m/s

)

19

0 1 2 3 40

5

10

15

20

Figure 4-9: Amplitude map of the in-plane displacement at the titanium-epoxy interface superimposed over the Lamb wave dispersion curves for the adhesive aircraft repair patchcomprised of a titanium patch (1.6 mm) bonded using epoxy (t mm) onto an aluminum skin (3.175 mm). t varies clockwise as 0.4318 mm, 0.6604 mm and 0.8636 mm.

116

It should be noted that the choice of modes with selective interface sensitivity is

possible in this case because the waveguide is not mid-plane symmetric. In the case of a

mid-plane symmetric waveguide, the same mode will have sensitivity to two cross-

sectional locations located equidistant from the mid-plane.

4.5 Experimental Work

4.5.1 Fabrication of repair patch samples with controlled interface conditions

Surface preparation forms an important part in any adhesive bonding procedure

and determines the quality of the adhesive joint. The surface preparation procedure

adopted for the preparation of the repair patch was based on the standard directions

provided by many technical experts. According to Davis and Bond [1999], in addition to

the requirement of clean bonding surfaces in the adherends, a higher level of chemical

activity favoring chemical bond formation between the adhesive and the adherend

ensures integrity of the bond over the service life of the joint.

All the steps, except surface polishing, were carried out in a dust free room used

for laying up composites. Polishing the surfaces to be bonded was carried out in a

machine shop. The steps followed in adhesive joint fabrication with their relevance are

listed below:

1. Solvent degreasing: This is the first step and important step in surface preparation

that ensures the removal of the contaminants. The bonding surfaces of aluminum

(3.175 mm thick) and titanium (1.6 mm thick) were degreased using acetone.

117

2. Surface polishing: This step results in a chemically active surface for bonding. It

can be carried out by chemical etching or mechanical abrasion. In the case of

aluminum, the oxide layer is removed by this step, thus exposing fresh and

chemically active aluminum surface for bonding. In this work, mechanical abrasion

was employed. For polishing the bonding surfaces of the aluminum and titanium

plates, a 2-inch Scotch-Brite Roloc ultrafine surface conditioning disc pads (3M

Inc. UF4000) was fitted to the end of a vertical milling machine and used as the

abrasive tool. The plate samples to be polished were fixed on to the horizontal slide

of the milling machine. With an appropriate contact between the Roloc pad and the

plate, determined from practice, polishing was carried out by providing a relative

motion between the rotating tool and the plate by means of a cross-feed.

3. Polished surfaces were cleaned using a dry wipe followed by a wipe containing

acetone.

4. Sol-gel coat: This is another prebond surface treatment that improves adhesion for

bonding metal. Sol-gel promotes enhanced adhesion between the metal and the

primer by means of chemical interaction. This is a safer alternate to the use of

hazardous chemicals like phosphoric acid (anodizing) and sulphuric acid-sodium

dichromate etching. AC-130 sol-gel (AC Tech Inc.) was applied using a bristle

brush in a radially-outward direction on the polished surfaces of the metallic plates.

The sol-gel was mixed and left for 30 minutes before use. The sol-gel employed

here can be used to treat any metallic surface to be adhesively bonded. After

allowing 30 minutes time for fusing, a heat gun was used to fuse the sol-gel by

moving the heat source from once corner to another in a slow fashion. The surface

becomes dull when the sol-gel is fused.

5. Application of primer - A thin layer of water based corrosion inhibiting adhesive

primer BR 6747-1 (Cytec Engineered Material, Inc.) was applied using a foam

brush on the fused sol-gel. After around 30 minutes of recommended waiting time

the heat gun was used for fusing the primer. BR 6747 is environmentally friendly

and provides better mechanical properties and corrosion resistance.

118

6. Application of adhesive - Hysol EA9696 is a modified epoxy film adhesive that

according to its manufacturer (Henkel Corporation) has the following features - low

temperature cure, uniform flow under pressure and temperature, excellent

environmental resistance and longer shop out-time. EA9696 can be used for all

metal to metal bonding applications and provides a bond with high toughness and is

employed in the fabrication of repair patches for this study. EA 9696 was taken out

from the cold storage (maintained at around 0°F) and allowed enough time to attain

room temperature before using it in bonding. The big roll of sheet film adhesive

was then cut into 12” x 12” size sheets. The carrier film and protective covering on

both sides of the film epoxy were removed before placing the sheet over the region

to be bonded.

7. Simulating defects: Both adhesive and cohesive defects were simulated in the repair

patch samples.

a) Adhesive defects: In samples where adhesive weakness was to be simulated; a

teflon film (0.5” x 0.5”) was placed at the surface of aluminum to create a

region of disbond. Another method for creating an adhesive defect, developed

for this work, is by placing a small piece of bubble wrap (unbroken) at the

aluminum-epoxy interface and placing the film adhesive layers. By sealing the

film epoxy layer over and around the bubble wrap, a sealed air pocket was

created. Mold release wax (PA0801 from PTM&W Inc.) was used to artificially

replicate a kissing bond condition. The wax defect is created by applying molten

wax (kept molten by hot air from heat gun) using a clean flat metal strip.

b) Cohesive defect: Cohesive weakness case was simulated by placing teflon

between the plies of the adhesive.

8. Vacuum bagging and cure: After the lay-up is done, the plates are transferred to a

vacuum bag and the whole assembly was cured in an auto-clave. The curing cycle

used is

a) Apply full vacuum – around 29 inches of mercury (gage)

b) Increase temperature at 3 °F/min to 130 °F

119

c) Reduce vacuum to 12 inches of mercury (gage)

d) Increase temperature at 3 °F/min to 250 °F

e) Hold at 250 °F for 2 hours

f) Cool at 5 °F/min and 12 inches of mercury (gage) to room temperature.

9. After curing, the sample was taken out of the autoclave.

The steps from 1 to 7 took around 4 hours to complete, mainly due to the manual

work in step 2. The key steps involved in fabrication of the adhesive repair patch samples

are shown in. Figure 4-10.

The autoclave curing was performed with control over only the temperature and

pressure. The temperature and pressure profiles maintained within the autoclave are

shown in Figure 4-11. Except for the absence of an inert gas environment within the

Figure 4-10: Key steps employed in fabricating epoxy bonded titanium-aluminum adhesive repair patch. The arrows guide the process from the beginning to end.

120

autoclave, the rest of the steps are similar to that employed in curing prepreg composite

layups. The curing step took around 5 hours to complete.

The bonded plates have a curvature due to the differential thermal expansions of

the bi-metallic joint at the adhesive curing temperatures. The actual aircraft repairs are

performed on field by enclosing the repair region in a heat blanket and applying

mechanical loads in addition to the vacuum pressure.

Figure 4-11: Temperature and pressure conditions used in the autoclave cure of theadhesively bonded repair patch. The adhesive – EA9696 dictates the cure profile.

121

4.5.2 Mechanical Testing

From the 12” x 12” bonded sample, 1” x 5” or 0.5” x 5” coupons were cut using

the water-jet cutting method. Notches were created on the bonded sample with the help of

a band saw, one on each side and spaced 0.5” apart from the sample centre line and in

opposite directions according to ASTM guidelines. The resulting sample resembles a

single lap-joint with a 0.5” overlap length (Figure 4-12).

Mechanical testing was performed according to the guidelines in ASTM 3165

using a mechanical testing machine (MTS 810) on samples with different simulated

interface conditions. The MTS 810 was used to load the samples in tension at a cross-

head speed of 0.25” per minute. Representative load-displacement results, for a good

bond, weak repair and adhesively weak bond, obtained from mechanical testing are

presented in Figure 4-13. The width of the weak repair sample in Figure 4-13 was 0.5”

whereas the other samples were 1” wide.

Figure 4-12: Side view of the ASTM 3165 tensile test specimen cut from the bondedrepair patch sample. Notches were machined through either side of the test specimen tocreate a 0.5” overlap.

Titanium

Aluminum Notches

Adhesive

122

The average values of the shear strength of the adhesive, obtained from ASTM

3165 tests on repair patch specimens with simulated interfacial weakness is presented in

Table 4-3.

0 1 2 3 4 5 6 70

2

4

6

8

10

12

Displacement (mm)

Load

(kN

)

Load-displacement Curve: Repair Patch Samples

Good bondWeak repairAdhesively weak bond

Figure 4-13: Static test results on representative ASTM 3165 test specimens. The overlaplength was 0.5” for all specimens. The width of all specimens was 1” except the weakrepair sample where the width was 0.5”.

Table 4-3: Average value of the shear strength of adhesive (MPa) obtained from ASTM3165 tests on specimens fabricated with different simulated interface conditions

Type of Sample Shear Strength of Adhesive

[MPa]

Good bond 34.39

Weak repair 24.55

Adhesively weak bond 4.42

123

From Table 4-3, it can be observed that the good bond region in the repair patch

sample is indeed stronger than any simulated weaknesses. The results from the ASTM

3165 tests point to the reliability of the fabrication procedure implemented and also

provide support to the methods of simulating weaknesses in the bonding.

4.5.3 Ultrasonic water immersion C-Scan

Ultrasonic water immersion C-Scan tests were performed by using a 25 MHz

transducer with 1.5” focal length held at normal incidence to the repair patch sample. The

curvature of the bonded repair patch sample prevents maintaining the constant vertical

separation between the ultrasonic immersion transducer and the plate resulting in the

movement of the front wall echo in the C-scan. This problem becomes more severe when

the whole area is covered in a single scan. A front follower time gate was used to ensure

that all the other time gates corresponding to the interface or defect reflections retain their

position relative to the front wall reflected signal. Several time gates were used because

the position of the defects with respect to the top surface varies from defect to defect.

A typical A-scan signal or RF waveform is presented in Figure 4-14. The

horizontal numbered lines are the time gates and also amplitude thresholds that are used

along with the C-Scan to obtain amplitude images as shown in Figure 4-14.

124

From the C-Scan image in Figure 4-15 the location and size of the simulated

defects in repair patch sample can be seen. The contour lines in Figure 4-15a and c

correspond to thickness variation in the repair patch samples due to its curvature.

C-Scan is not a practical option due to the large investment in time owing to the

point-by-point nature of measurement and also the need for a large frequency transducer

for measurement. Additionally, not all interface conditions are detected using this

technique.

Figure 4-14: A typical RF waveform collected from ultrasonic immersion C-scan of the repair patch sample. The time windows are marked using numeric labels.

125

(a) Gate 3 (b) Gate 4

(c) Gate 5 (d) Gate 6

Figure 4-15: Ultrasonic water immersion C-Scan amplitude image for adhesive repair patch sample at time gates 4, 5 and 6 corresponding to the interface signal. Gates 3 and 5show some contour lines which corresponds to the thickness variation in the adhesive.Gates 4 and 6 show the defects more clearly. In gate 6 image, the circular regions seen are blend outs or machined cavities. They have not been included in this thesis work.

126

4.6 Design of sensor configuration for selective excitation of modes

There are different sensor configurations that can be applied for exciting guided

waves within a waveguide. These include normal incidence, oblique incidence using

acrylic wedge or water delay, comb transducer with or without time delay. The activation

can also be using piezoelectric transducers or Electro Magnetic Acoustic Transducers

(EMAT’s), or magnetostrictive materials or laser pulse etc.

4.6.1 Wedge loading and source influence study

Among the different techniques for the generation of Lamb waves in plate-like

structures [Rose, 1999], an acrylic angle beam wedge with a piezoelectric transducer

mounted on its top was selected due to simplicity in its construction and simplicity in

performing experiments.

In the case of an angle beam wedge excitation, the theoretical excitation line runs

parallel to the frequency axis on the phase velocity dispersion curves. The coincidence

angle, i.e. the angle corresponding to the phase velocity in the dispersion curves for

acrylic wedge based wave incidence (θinc) is computed using the expression for Snell’s

law provided in Equation 4.3. This equation assumes a critical incidence of wave within

the plate where the guided wave mode is to be generated.

= −

p

winc c

c1sinθ (4.3)

127

where cp is the phase velocity of the mode at the selected frequency and cw is the

ultrasonic bulk longitudinal wave velocity in the wedge material.

Using Equation 4.3, the angle of incidence (θinc), in an acrylic wedge, for

generating the Lamb wave mode with maximum in-plane displacement at a frequency of

2.53 MHz and with phase-velocity ~15 km/s was calculated to be 10°. The ultrasonic

loading for the generation of all the aluminum-epoxy interface sensitive modes listed in

Table 4-2 is 10°. It is worth mentioning that the wave velocity in the wedge imposes a

lower limit to the phase velocity of the guided wave mode that can be generated in the

experiment.

The wedge angle computed will successfully generate the selected guided wave

mode if the correct frequency of excitation source is used. The mode generation angle

calculations have plane-wave or infinite source assumptions built in. Since all practical

wave generating sources are finite in dimension, the geometry of the excitation source

influences the range of phase velocity that gets excited when used with an angle beam

wedge. The transducer loading function determines the frequency spread of excitation.

This geometric influence of angle beam wedge with mounted piston type transducer is

captured by the Equation 4.4 from Rose [1999].

( )

( )( )

( )( )incwn

inc

incwn

piston kk

Dkk

Fθ

θθ

sincos2

sinsin2

m

m

=±

(4.4)

128

where k represents the wavenumber, D the diameter of the transducer and θinc the

incidence angle using the angle beam wedge. The subscripts n and w correspond to nth

Lamb wave mode and wedge respectively.

The influence of the diameter of an ultrasonic transducer (6.35 mm) and the

loading conditions viz. the excitation frequency (2.5 MHz), the number of cycles of tone-

burst (5 cycles) and the wedge angle (10°), on the range of phase velocity values excited

was determined using Equation 4.4 and represented as an intensity distribution in

Figure 4-16. The white lines are the Lamb wave dispersion curves for the repair patch

(same as in Figures 2, 3 and 4). It can be observed that mode 18 at cp ≈ 15 km/s and f ≈

2.53 MHz, can be generated with maximum intensity with the selected transducer loading

arrangement. Due to the frequency bandwidth of the excitation, neighboring modes will

also be generated, but with a weaker intensity.

129

4.6.2 Pitch-catch data using wedge transducer

The initial experiment involved the use of a pair of variable angle beam acrylic

wedges adjusted to an incidence angle of 10° and arranged in a pitch-catch configuration.

The broadband piezoelectric transducers mounted on variable angle beam acrylic wedges,

set for an incidence and reception angle of 10° was arranged to correspond to a pitch-

catch configuration, and supplied with a tone burst excitation pulse at frequencies ranging

from 1 MHz to around 3 MHz in steps of 50 kHz.

Figure 4-16: Geometric influence of a 6 mm diameter transducer mounted on a 10° acrylic angle beam wedge and supplied with a 2.5 MHz tone burst input voltage for 5 cycles on therange of phase velocities and frequencies excited. The white lines are the Lamb wave phasevelocity dispersion curves for the repair patch.

Frequency (MHz)

Phas

e ve

loci

ty (k

m/s

)

0.5 1 1.5 2 2.5 3 3.5 4

5

10

15

20

25

0

0.5

1

1.5

2

2.5

3

130

The Hilbert transformed ultrasonic guided wave RF waveforms collected from

this frequency sweep experiment by placing the transmit-receive pair across regions with

simulated interfacial conditions is shown in Figure 4-17. It can be seen that the maximum

energy transfer occurs in the frequency range of 2-3 MHz, which corresponds to the

generation of mode 18 in the repair patch sample. A reduction in the amplitude of the

transmitted RF waveform between a good bond and simulated weakness in bond can be

observed.

The squared summation of the pitch-catch transmitted RF waveform amplitude at

an excitation frequency has the dimensions of energy. The variation of the transmitted

energy across regions in the repair patch sample with simulated interfacial conditions is

presented as a function of the excitation frequency in Figure 4-18.

This step was performed to verify the ability to generate the desired Lamb wave

mode (mode 18) at the selected angle and frequency combination and also to check for

sensitivity of the mode 18 to the defects introduced in bonding. From Figure 4-18 it can

be clearly seen that in the range of frequencies from 2.2-2.8 MHz, it is possible to

distinguish between the adhesive defects and the cohesive defects and a good bond

region. The mode generated is sensitive to the simulated defects in a wider range of

frequencies because the input was in the form of a tone burst having a finite bandwidth.

This provides an additional capability in handling the repair patch samples having a small

variation in the thickness of the epoxy layer alone without changing the ultrasonic Lamb

type wave inspection arrangement.

131

Frequency (MHz)

Tim

e (µ

s)Good bond

1 1.5 2 2.5 3

10

20

30

40

50

Frequency (MHz)

Tim

e (µ

s)

Adhesive (Teflon) defect

1 1.5 2 2.5 3

10

20

30

40

50

Frequency (MHz)

Tim

e (µ

s)

Adhesive (Air gap) defect

1 1.5 2 2.5 3

10

20

30

40

50

Frequency (MHz)

Tim

e ( µ

s)

Adhesive (Wax) defect

1 1.5 2 2.5 3

10

20

30

40

50

Frequency (MHz)

Tim

e ( µ

s)

Cohesive (Teflon) defect

1 1.5 2 2.5 3

10

20

30

40

50

0

0.5

1

1.5

2

Figure 4-17: Hilbert transformed ultrasonic guided wave RF waveform from frequencysweep experiment in pitch-catch mode using piezoelectric transducers mounted on avariable angle beam wedge set to an angle of 10°. Maximum energy transfer occurs in the frequency range of 2-3 MHz corresponding to mode 18 in the repair patch.

132

Two fixed angle acrylic wedges were then fabricated with an incidence angle of

10° from the vertical axis, such that a commercially available ultrasonic transducer – 2.25

MHz, 6.35 mm in diameter – from Krautkramer, could be mounted on them. The fixed

angle wedges were arranged in pitch-catch configuration similar to the variable angle

beam wedges (Figure 4-19) and were held together such that the separation between them

is maintained constant and with a small but equal lift-off from the repair patch surface. A

layer of water between the wedges and the plate couples the ultrasonic energy uniformly

into the sample and hence eliminates the influence of coupling from the measurements.

1 1.5 2 2.5 30

100

200

300

400

500

600

700

800

900

Frequency (MHz)

Ener

gy (V

2 )

Energy Transmission in Repair Patch Sample, Wedge angle: 10°

Good bond Adhesive (Teflon) defect Adhesive (Air gap) defect Adhesive (Wax) defect Cohesive (Teflon) defect

Figure 4-18: Variation in the energy transmission across regions with simulated interfacial weakness in the aircraft adhesive repair patch specimen. A pair of variableangle beam wedges set to 10° of incidence and reception in pitch-catch configuration was used to collect the frequency swept tone burst signals transmitted along a short distance of the repair patch sample. The collected signals were squared and summed to obtain theenergy quantity. The results are presented in a normalized energy scale.

133

The wedges were made smaller in size in order to avoid the effects of plate curvature on

the measurements.

The RF signals collected by this wedge arrangement across the simulated

adhesive bond defects and their frequency content (obtained from fast Fourier transform)

are shown in Figure 4-20. The excitation of the desired mode, besides using appropriate

input angle and frequency, was also verified by means of a group velocity measurement.

It can be seen from Figure 4-20 that using the amplitudes of the RF waveforms, we can

distinguish between the different interface quality conditions simulated in the repair

patches. This is in line with our observations using variable angle wedges. The fixed

angle wedges have one less interface compared to the variable angle acrylic wedges,

which implies a more efficient transfer of energy from the transmitting transducer into

Figure 4-19: Schematic of the ultrasonic Lamb wave pitch-catch measurement configuration on the adhesively bonded repair patch. The wedge angles are 10° from the vertical and the transmitter (T) and the receiver (R) are both commercial piezoelectrictransducers rated at 2.25 MHz. The wedges were separated by a distance of around 38mm

t2

t1

t3

Aluminum

Epoxy Titanium

T R

x3

x1

x2

134

the repair patch sample. This stage of implementation is much closer to a field deployable

configuration for nondestructive inspection. A commercially usable EMAT or roller

probe could be developed based on this work, but it is not discussed here to retain focus

on the guided wave based theoretical development and its experimental validation.

In the absence of coupling based issues, by using a constant thickness layer of

water as couplant, variations in the signal amplitude due to experimental measurement is

50 60 70 80 90 100-0.2

0

0.2

Good bond

50 60 70 80 90 100-0.2

0

0.2

Adhesive (Teflon) defect

50 60 70 80 90 100-0.2

0

0.2

Adhesive (Air gap) defect

50 60 70 80 90 100-0.2

0

0.2

Adhesive (Wax) defect

50 60 70 80 90 100-0.2

0

0.2

Time (µs)

Cohesive (Teflon) defect

1.5 2 2.5 3 3.50

20

40

Good bond

1.5 2 2.5 3 3.50

20

40

Adhesive (Teflon) defect

1.5 2 2.5 3 3.50

20

40

Adhesive (Air gap) defect

1.5 2 2.5 3 3.50

20

40

Adhesive (Wax) defect

1.5 2 2.5 3 3.50

20

40

Frequency (MHz)

Cohesive (Teflon) defect

Figure 4-20: Amplitude vs. time chart or the RF waveform measured by placing 10°wedge mounted 2.25 MHz transducers across the region to be inspected is shown for thedifferent simulated bond interface conditions in the repair patch samples is shown on theleft. The corresponding frequency content, obtained using Fast Fourier Transform (FFT)is shown on the right column. It can be noticed that there is a significant amplitude baseddifference between the repair patch samples with simulated interfacial conditions.

135

avoided. An amplitude ratio based parameter is being used here for classification. It has

been found from experiments, across multiple repair patch samples with similar or

different simulated interface conditions, that within a sample the transmission will be

highest in case of a non defective or good bonded region and the least in case of a region

with cohesive weakness. Hence a ratio of amplitudes with respect to the highest measured

on a sample can be used to classify regions into good region, adhesively weak region or

cohesively weak region. With this idea we proceed from a single measurement to a linear

scan type approach.

Using the same angle beam wedge configuration, keeping the separation between

the transmitter and receiver wedges fixed at 38 mm, a linear scan termed as G* scan, for

guided wave scan using the defect sensitive mode (mode 18), was performed manually by

acquiring RF waveforms at every 2 mm spaced points along the scan direction. Small

variations in the amplitude (1-2 dB) and shape of the transmitted signal could be

observed at locations within a good region, probably due to thickness changes in the

adhesive. The results from the G* scan are presented in Figure 4-21 in the form of

intensity plots that closely resemble a conventional B-Scan representation.

The discontinuity in the first arriving wave packet spans approximately the length

of the defect introduced at the aluminum-epoxy interface. The G* scan images in

Figure 4-21 also support the results in Figure 4-20. The G* scan image is useful for

obtaining an estimate of the size of the defect. Combining the information contained in

the RF waveform and in the G* scan, it is possible to identify and size regions of

weakness in the bonding of a repair patch.

136

0 10 20 30

0

10

20

30

40

Tim

e ( µ

s)

Position (mm)

Good Bond

0

0.5

1

1.5

2

2.5

3

0 10 20 30

0

10

20

30

40

Tim

e ( µ

s)

Position (mm)

Adhesively weak bond (Teflon)

0

0.5

1

1.5

2

2.5

3

0 10 20 30

0

10

20

30

40

Tim

e ( µ

s)

Position (mm)

Adhesively Weak bond (Air gap)

0

0.5

1

1.5

2

2.5

3

0 10 20 30

0

10

20

30

40

Tim

e ( µ

s)

Position (mm)

Adhesively Weak bond (Wax)

0

0.5

1

1.5

2

2.5

3

0 10 20 30

0

10

20

30

40

Tim

e ( µ

s)

Position (mm)

Cohesively Weak bond (Wax)

0

0.5

1

1.5

2

2.5

3

Figure 4-21: Guided wave scan (G* scan) images (using mode 18) obtained from a linearscan using 10° wedge with 2.25 MHz ultrasonic transducers mounted on top and the whole assembly oriented in pitch-catch mode across the region with simulated interfaceconditions. The discontinuity in the first arriving wave seen from the above G* scan imageapproximately spans the length of the defect at the aluminum-epoxy interface.

137

4.7 Summary

In this chapter, a theoretically driven guided wave inspection procedure has been

systematically developed for improved detection of adhesive and cohesive defects in the

case of bonded adhesive repair patches used in aircraft. A titanium (1.6 mm) repair patch

bonded to an aluminum (3.175 mm) aircraft skin using epoxy was studied. The defects of

interest were assumed to be primarily at the aluminum-epoxy interface. It was also

assumed that the inspection technique should work with only access to the titanium

surface, a practical constraint in the case of aircraft repair patches.

The theoretical study consisted of ultrasonic guided wave mode selection. Guided

wave modes with large in-plane displacement at the aluminum-epoxy interface were

selected from a study of the wave structures for sensitivity to the adhesive defects at that

interface. Among the several possible frequency – phase velocity (f-cp) combinations

with large in-plane displacement at the aluminum-epoxy interface, 2.5 MHz -15 km/s was

found to be one of the optimal ones due to the following reasons:

1. This f-cp choice ensured a sufficiently large in-plane displacement value for a large

range of thickness of the adhesive layer – from 0.4318 mm to 0.8636 mm, which

more than covers the thickness variation range found in practice in the case of the

repair patch. Hence a single experimental tool could be used in all cases.

2. The f-cp combination allowed selective sensitivity to only the desired interface i.e.

aluminum-epoxy interface. This implies that the presence of a defect at the

titanium-epoxy interface will not prevent the detection of a defect at the aluminum-

epoxy interface. This is a big advantage of the guided wave based inspection over

the conventional bulk wave inspection (e.g. the C-Scan) where the presence of a

138

defect shields or shadows the subsequent defects located at the same spatial location

but at different depths.

3. This f-cp choice ensured that the mode generated within the repair patch is separated

from its neighboring modes, hence ensuring a simple signal for analysis.

The influence of a practical guided wave excitation source –like an angle beam

wedge in generating the desired mode was studied along with the additional requirement

that the least amount of neighboring modes are excited. In the case of aluminum-epoxy-

titanium bonding and for the selected layer thicknesses, excitation of mode 18 at ~2.53

MHz, using a 10° acrylic angle beam wedge was found to be optimal.

Samples prepared with simulated defects – both adhesive and cohesive defects,

primarily at the aluminum-epoxy interface, were tested using the mode 18 selected from

the theoretical study. A fixed angle beam wedge pair with 10° incidence angle and 2.5

MHz excitation frequency, kept in a pitch-catch mode, was shown to be successful in

discriminating between the adhesive and cohesive weaknesses and a good joint.

A very important direction from the work presented in this chapter is the

establishment of a systematic process of identifying and representing frequency – phase

velocity combinations having large in-plane displacement at an interface of interest. This

forms the first step for many inspection scenarios in multi-layered materials. e.g.

laminated composites where defects such as delamination are to be detected.

Chapter 5

Guided Wave Inspection of Adhesive Skin-Stringer Joints

5.1 Introduction

In aircraft structures, stiffeners are attached to the skin to enable the skin to

withstand compressive and tensile loads that occur during flight. Riveted or adhesively

joined skin-stringer joints are found in the fuselage and the wing structures in an aircraft.

For example, Boeing 787 uses composite stringers cured together with the wing skin to

provide longitudinal stiffness to the wing. There are several designs for aircraft stringers.

An aircraft fuselage with adhesively bonded/co-cured stringers can be seen in Figure 5-1.

Figure 5-1: Internal view of the fuselage of a Boeing Dreamliner. The longitudinal stringers can be either adhesively bonded to the skin or co-cured with the skin. [Source: Boeing webpage - http://www.boeing.com/]

140

Typical issues related to the adhesive bonding like the adhesive (or interfacial)

and cohesive (or bulk) weaknesses exist in the case of the adhesive skin-stringer joints.

The aircraft structures are also subject to aerodynamic loads in flight and chemical or

environmental conditions like moisture, de-icing liquids etc. The adhesive bonds

deteriorate under these situations, thus raising the need for a nondestructive inspection

technique like ultrasonic inspection.

Most inspections using ultrasonic guided waves rely on a wave being propagated

from the skin region, on one side of the stringer joint, across the bonded region to the

skin on the other side of the stringer joint. For the sake of completeness it is also stated

here that only the outer surface of aircraft skin is accessible for inspection without timely

and costly tear down. As long as the skin is a mid-plane symmetric structure, which is

always true for an aluminum skin and usually true for laminated composite skin, it does

not matter on which side of the skin the wave is introduced. This reasoning eliminates the

configurational issues in experimental demonstrations.

The simplified skin-stringer adhesive joint considered in this work has an

aluminum stringer adhesively bonded to an aluminum skin as shown in Figure 5-2. The

wide base layer represents the skin and the narrow layer (top layer) is the stringer. The

thickness of the adhesive layer is typically around 5% of the joint thickness [Nagy and

Adler 1989]. It is shown exaggerated in Figure 5-2 for visual clarity of the bonded

structure. Typically, an epoxy based adhesive is found in most aerospace adhesive joints

and the same has been employed in this work also. The terms epoxy and adhesive are

interchangeably used in this work.

141

The medium supporting the propagation of a guided wave is called a waveguide.

Wave propagation through a waveguide is characterized by the geometric dispersion

phenomenon which is expressed in the form of phase velocity variation as a function of

frequency. A typical dispersion calculation assumes that the finite thickness prismatic

waveguide extends infinitely in two directions, which is true in the case of plate-like

structures such as an aircraft skin. The distinction is made here between the plate and

plate-like structure to highlight the fact that the aircraft skin is not flat but curved.

However, the plate dispersion calculations are still valid in the ultrasonic frequency

regimes because of the large radius of curvature of the fuselage.

A noticeable feature of the stringer joint is the discrete step that exists in the

waveguide, which is referred to in this thesis as a discrete waveguide transition. A major

challenge in the stringer joint inspection problem thus arises from the fact that an abrupt

thickness change along with a variation in the material elastic modulus is involved for

wave propagation across the bonded region. Stringer joints used in aircraft are assumed to

be longer than at least one wavelength corresponding to the ultrasonic frequencies used in

testing. With such an assumption, the problem of wave propagation across stringer joints

can be approached by studying two separate sets of dispersion curves – one for the skin

Figure 5-2: Schematic of a simplified skin-stringer adhesive joint used in this study. The thickness of the adhesive layer has been exaggerated for visual clarity of the bonded layup.

Aluminum

Epoxy

142

and the other for the bonded stringer region [Rokhlin 1991]. Such a transition can also be

found in adhesive step-lap joints (Figure 5-3).

The fundamental difference between the stringer joint and the step-lap joints is the

member that receives the energy. In the case of the stringer joint, the ultrasonic energy is

introduced and received through the same member. In step-lap joints, the energy sent

from one member has to travel through the bonding to the other member to be received.

Figure 5-4 shows the discrete transition which is common in both the skin-stringer

and step-lap adhesive joints. Analytical calculations break down at the discontinuity in

the waveguide geometry, thus highlighting the need for a hybrid-analytical approach or a

numerical approach to study wave propagation across the stringer joint.

Figure 5-3: A typical epoxy bonded aluminum step-lap joint. The thickness of the epoxy layer is shown exaggerated for visual clarity.

Figure 5-4: Discrete waveguide transition found in adhesive skin-stringer joints and adhesive step-lap joint.

x2 x1

x3

Aluminum

Epoxy

Aluminum

Epoxy

143

The inspection of the stringer joint requires guided wave modes from those

theoretically permissible in the bonded assembly that are sensitive to the adhesive and

cohesive defects that may be present. An additional challenge involves finding the modes

that should be generated in the skin so as to form the defect sensitive mode(s) in the

bonded region. The mode conversion from the stringer to the skin is also important.

This chapter is divided into 10 sections. Beyond section 5.1, the introduction, the

section 5.2 reviews the literature on the guided wave propagation across discrete

waveguide transitions. Section 5.3 covers the theoretical solutions to guided wave

propagation in the skin and the bonded regions. In section 5.4, the methods for handling

mode conversion at the discrete transition are listed. Section 5.5 covers the hybrid

method combining Semi Analytical Finite Element (SAFE) formulation and Normal

Mode Expansion (NME) for guided wave mode scattering analysis at a transition. In

section 5.6, a wavestructure matching based analysis is developed. In section 5.7, some

results from Finite Element simulations are presented. Section 5.8 covers guided wave

mode selection aspects for defect sensitivity. Section 5.9 covers the fabrication of stringer

joint samples, experimental design and results. The summary of this chapter is provided

in section 5.10.

5.2 Literature on guided wave propagation across stringer joints and other waveguide transitions

Guided wave propagation across stringer joints becomes significant because of

the discrete transition in the geometry at the bond location. There have been studies in the

144

literature where the influence of the geometry transition on the aspects of guided wave

mode propagation has been studied either numerically or by experiments.

One of the first and only instance of an analytical approach to understand the

guided wave mode behavior at the discrete transition found in both the adhesive step-lap

joint and the adhesive skin-stringer joint was performed by Rokhlin [1991]. He employed

the Wiener-Hopf technique [Noble 1988] to theoretically study the mode conversion at

the step discontinuity in adhesive step-lap joint and skin-stringer joint with two different

interfacial conditions – slip interface and rigid interface. Rokhlin studied the mode

generated in the bonded region in the two types of discrete transitions under s0, a0 and a1

mode incidence from the primary waveguide. Using Plexiglas angle beam wedge

mounted piezoelectric transducers; he studied the amplitude and time shift during the

adhesive curing process and corroborated his observations with the theoretical results.

Rokhlin suggested that the edge effects i.e. the difference in geometry between the step-

lap and skin-stringer configurations did play a role under certain conditions of Lamb-type

wave propagation.

While the previous work relied on the mathematical framework to study and infer

the guided wave behavior at a discrete transition, the first major work highlighting the

importance of the study of the wavestructure was proposed by Rose and co-workers

[1992]. Pilarski and Rose [1992] proposed the use of wavestructure based criteria for

guided wave mode selection for different inspection situations against the use of just the

dispersion curve shift-based mode selection concepts prevalent in the literature then.

145

They employed combination criteria where the interfacial displacement and power flow

were used together to predict the sensitivity of a mode to interfacial weakness condition.

Further works by Rose and co-workers involved experimental components where

the mode selection was put to test. Rose and Ditri [1992] employed the s2 mode at 4.72

MHz mm generated using 17 degree Plexiglas wedges, to test an adhesive aluminum

step-lap joint successfully. A larger out-of-plane displacement component for the s2

mode, at 4.72 MHz mm, at the upper and lower surfaces of the aluminum observed from

the study of wavestructure meant better excitability and additionally the mode also

showed sensitivity to the bond conditions. Larger reflected signal amplitudes in pulse-

echo testing and a smaller amplitude transmission in through-transmission testing were

found to occur at disbonds simulated by lack of adhesive and Teflon inserts.

Rose et al. [1995a] developed a double spring hopping probe (DSHP) for manual

measurement of guided wave transmission across adhesive step-lap and skin-stringer

joints made from 1 mm aluminum plates. They employed a variable angle beam probe

having frequencies 2 MHz and 4 MHz to generate s0 and a1 modes respectively in the

skin and to successfully detect debonding in the joints.

Rose and co-workers [1995b] employed the DSHP to study the transmission of a1

and s1 modes with time as the adhesive cures in an adhesive step-lap joint formed using 1

mm thick aluminum plates. They report a constant transmission of the a1 mode during

and after cure, and a reduction in the s1 transmission with cure time. Using features from

the wavestructure of the a1 and the s1 modes, Rose et al. conclude that since a1 has a

146

larger out-of-plane displacement, its leakage does not change with cure whereas the s1

mode with a larger in-plane displacement is sensitive to interface condition in the step-lap

joint and hence has a higher leakage as the adhesive cures.

Ditri [1996] analyzed the transmission and reflection of energy when an SH wave

interacts with a step change in modulus or geometry such as the one across a joint

between two finite thickness isotropic waveguides. He employed collocation of the

traction free boundary conditions and interface continuity conditions at different points

across the cross-section of the joint between the waveguides. A very important

contribution by Ditri [1996] is laying the mathematical foundation of the

coupling/noncoupling of wave modes at waveguide transitions based on the orthogonality

of the displacement fields in the waveguides forming the transition region. According to

Ditri, any mode in the secondary waveguide whose displacement field distribution is

orthogonal to the displacement field distribution of the incident mode will not be

generated by the incident mode. Mathematically, displacement orthogonality is

represented by the summation of the product of the respective displacement components

across the connecting region between the primary and the secondary waveguide. Zero or

low value of the orthogonality expression implies linear independence of the

displacement fields and noncoupling of wave modes.

Chang and Mal [1995] employed a global local Finite Element method that

combines Finite Elements in the step-lap region and global functions representing the

time-harmonic wave in the frequency domain at locations away from the joint region to

study the guided wave mode conversion, reflection and transmission aspects. Mal et al.

147

[1996] carried out experiments and found a less than ideal match between the

experiments and the predictions from the global local FE results.

Lowe and coworkers [2000] performed an elaborate numerical study using finite

element analysis to understand the mode conversion and transmission of Lamb modes

across an adhesive step-lap joint along with the influence of the bond geometry when an

s0 mode is incident from one side of the joint. They used 2DFFT in order to analyze the

mode conversion aspects associated with the step transition in the lap joint. They observe

that a higher energy is carried by the first order modes in the bond region and point

towards the role of wavestructure matching as the reason for that. This again can be

reasoned by the mode coupling theorem due to Ditri [1996]. Additionally they

corroborate with the observation of Rokhlin [1991] that the energy transmission becomes

fairly constant for joints more than around 15 mm in length.

Demma et al. [2003] also studied the scattering of SH wave modes from step

discontinuity and cracks using FE and modal decomposition approaches.

di Scalea et al. [2004] compliment the work done by Lowe et al. [2000] by

experimentally studying the generation and transmission of a0 mode across an aluminum

step-lap joint. The wavestructure based explanation to the mode conversion is also in the

line of that by Lowe et al [2000]. They observed that the fundamental modes in the bond

region play a major role in energy transfer from an a0 mode incidence at the joint. They

used air-coupled transducers to measure transmission of laser generated guided waves

across a step-lap joint. The role of the guided wave modes in the single and bonded plate

148

in detecting weakness simulated in the samples, using improper adhesive-hardener mix

ratio, and using encapsulated water, was performed.

Matt et al. [2005] also studied the wave propagation along and across a cross-ply

composite spar attached to a quasi-isotropic skin using 1D SAFE based calculation. They

found the most sensitive point for inspection to be the point where mode coupling

occurred between the fundamental symmetric and the first order asymmetric mode in the

bonded region of the joint resulting in a larger interlayer energy transfer. They employed

the ratio of the power flow in a subset of the total number of layers to the total cross-

sectional power flow in order to determine the strength of energy transmission for

different bonding conditions. They also defined an excitation factor based on the cross-

correlation between the incident mode and the mode within the bonded region in order to

compute sensitivity of the modes to the bonding condition.

Song et al. [2005] developed a hybrid approach combining Finite Element (FE)

and Boundary Element (BE) to study the guided wave mode conversion at a step-lap

welded steel connection. From their parametric study varying the overlap length, they

observed that the incident mode, frequency range and overlap length all influence the

transmission and reflection of guided waves. Since the mode conversions in a shear

horizontal wave interaction are less complicated, Song et al. developed a multi-reflection

approach to study shear horizontal wave interaction with step-lap transition.

Puthillath, Yan and Rose [2007] employed a finite element based approach, which

is actually another method of invoking continuity in this case, to compute the amplitude

149

and energy reflection and transmission across step change in a material and also the case

of step change in geometry and modulus found in both adhesive lap-joint and adhesive

skin-stringer joint.

Puthillath et al. [2008] performed a parametric study, similar to the one by Song

et al. [2005], using commercial FE software (ABAQUS) to study wave propagation

across an aluminum step-lap adhesive joint. They studied the variation in the geometry,

material properties and defect location on the guided wave mode conversion. They

developed and employed signal processing techniques based on 2DFFT, tailored to the

guided wave data analysis to perform to perform energy partitioning among the various

modes within the waveguide.

5.3 Guided wave propagation and dispersion in bonded joints

The phase and group velocity dispersion behavior in the aluminum skin and the

aluminum-epoxy-aluminum bonded waveguide were studied in section 2.5. The

dispersion curves for aluminum skin (2 mm thick) and the bonded stringer region

(aluminum (2 mm) – epoxy (0.3 mm) – aluminum (2 mm)) are recalled and presented for

completeness. The Figure 5-5 shows the superimposed phase and group velocity

dispersion curves for the two waveguides – aluminum (or waveguide A) and bonded

aluminum joint (or waveguide B).

150

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

Frequency (MHz)

Phas

e V

eloc

ity (k

m/s

)

a0

s0

a1

s1

s2

a2

s3

a3

1

2

3

4

5

6

7 8

9

10

11

12

13

14

15

16

17

18

19

al modesbond modes

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

Frequency (MHz)

Gro

up V

eloc

ity (k

m/s

)

a0

s0

a1

s1

s2

a2

s3

a3 a4

1

2

3

4

5 6

78 9

1011

12 1314

15

16

1718

1920

21

al modesbond modes

Figure 5-5: Superposition of phase and group velocity dispersion curves for aluminum plate (2 mm) and bonded stringer joint (aluminum 2 mm-epoxy 0.3 mm – aluminum 2 mm). The modes in aluminum are labeled using an alphabet and a subscript whereas the modes in the bond are numbered incrementally.

151

The existence of modes in aluminum that overlaps those in the bonded stringer

can be noticed from Figure 5-5. There are several mode pairs in the bonded stringer that

overlap or bound or lie closer to that in aluminum. Some mode pairs are modes 2 to 4 and

s0; modes 5-7 and a1; modes 7-9 and s1; and, modes 10, 11, 14 and s2. The sets of modes

forming the mode pair varies with frequency. But it can be stated that at any given

frequency there are at least two modes in the bonded stringer that follow a single mode in

aluminum. The mode pairs have similar wavestructure in the one of the aluminum layers

forming the bonded stringer. There was a phase shift to the wavestructure in the other

aluminum layer in the case of one of the modes in the mode pair. The mode pairs also

travel with similar group velocity with their matching mode in the aluminum.

The existence of mode pairs and their special features with respect to the phase

and group velocity matching and similarity in wavestructure with the corresponding

mode in aluminum is expected to be very useful. A more fundamental approach can be

made to the interaction of a guided wave mode at a transition. The following hypothesis

can be drawn from fundamental principles:

For every mode incident from the waveguide on one side of a transition, modes

with matching phase velocity vs. frequency (implies an almost similar

wavestructure) in the waveguide beyond the transition has a higher possibility of

getting excited on transmission across the transition.

As a consequence, the following can also be stated:

(a) The mode pairs are expected to have almost the same tendency to get excited.

152

(b) From the energy balance laws, it is expected that the mode incident at the

transition will transmit higher amount of energy to modes that have similar group

velocity (energy velocity in an attenuative waveguide). So a mismatch between

the group velocities will imply a larger reflected energy or a small transmitted

energy.

5.4 Guided wave mode conversion at a transition

Stringer joint inspection using ultrasonic guided waves involves excitation of a

guided wave mode in the primary waveguide or the skin region which then propagates

into the secondary waveguide or the bonded stringer region. A guided wave mode

propagating from the skin to the stringer undergoes mode conversion at the waveguide

transition to form new modes that can exist within the secondary waveguide at that

frequency.

Hence understanding the mode conversion at a transition is undertaken in the next

few sections, not only because it is an interesting problem in itself, but also due to the

fact that this will help in formulation of optimal input mode(s) that will result in the

generation of a sensitive guided wave mode within the stringer region.

A study of mode conversion that occurs at the discrete waveguide transitions is

attempted in this thesis using three different approaches

(a) a hybrid analytical approach

(b) dispersion curve and wavestructure based analysis, and

(c) a numerical study using Finite Elements

153

Inherent in each of these approaches is the orthogonality of the guided wave

modes. While it is explicitly applied in (a), orthogonality is borne out by the other

assumptions in the other approaches (b and c). The approaches (a) and (b) are explained

in detail in the succeeding sections. The FE approach explained in detail in Chapter 3 is

also used to provide some verification of the validity of the hypothesis.

5.5 Hybrid model for guided wave scattering at a transition

A hybrid model combining the Semi-Analytical Finite Element (SAFE) and

Normal Mode Expansion (NME) for analysis of guided wave scattering at a transition is

presented in this section.

5.5.1 SAFE

SAFE framework as developed in Galan and Abascal [2002], Hayashi [2003] and

several other publications has tremendous advantage in handling guided wave

propagation across simple or complex cross-sections having a prismatic geometry.

Essentially, SAFE applies an exponential propagating term to handle the wave

propagation along the waveguide and a FE discretization across the cross section. The

fundamental equations for SAFE computation are based on the same principles as FE.

Here SAFE is employed to understand guided wave scattering at a waveguide transition.

The governing equation for the wave propagation problem from virtual work

principle [Hayashi et al. 2003] is:

154

Wave propagation through plate geometries, assuming a plane strain condition,

requires only a one dimensional model (Figure 5-6). The cross-section of the plate is

divided into three node elements as shown below. The rest of this section follows closely

Hayashi et al [2003], Gao [2007] and Yan [2008].

The shape functions for such a discretization are given in terms of the local

coordinates expressed in terms ofξ :

( ) ∫∫∫ +=ΓΓ V

T

V

TT dVdVd σεuutu δρδδ && (5.1)

Figure 5-6: One dimensional discretization of the thickness of a general layered waveguide. The 3 node isoparametric element used in discretization is shown in the inset.

[ ]

( )( )( )ξξ

ξ

ξξ

+

−

−

=

=

=

=

2

2

2

3

2

1

3

2

1

321

21

1

21

where;

N

N

N

z

z

z

NNNz

(5.2)

155

The displacement approximation corresponding to plane harmonic wave

propagation along the x direction for the jth element is:

where

The subscript for the displacement u corresponds to the direction and node

number in that order. The expression for strains can be written in terms of the

displacement gradients

where

( ) ( )tij

ωξ −= expUNu (5.3)

[ ]

( ) ( )

[ ] ( )ikxuuuuuuuuu

tiNNN

NNNNNN

uuu

Tzyxzyxzyx

Tzyx

exp

exp000000

000000000000

333222111

321

321

321

=

−

=

=

U

N

u

ωξ

(5.4)

[ ]uLLL

ε

∂∂+

∂∂+

∂∂=

=

zyx zyx

Txyzxyzzyyxx γγγεεε

(5.5)

=

=

=

000001010100000000

;

001000100000010000

;

010100000000000001

zyx LLL

(5.6)

156

By substituting the displacement approximation (Equation 5.4) into the expression

for strain (Equation 5.5) we obtain

The comma subscript refers to differentiation of the function with respect to the

succeeding variable – x, y or z.

Employing the generalized Hooke’s law, the stress-strain relations can be written

as

The traction is expressed in terms of the shape functions in the Equation 5.9

Substituting the equations for stresses, strains, displacements and traction into the

virtual work equation, the final expression will be

( ) ( )

xxyyzz

jtiikxik

,2,,1

21

;

exp

NLBNLNLB

UBBε

=+=

−+= ω

(5.7)

[ ]matrix stiffnessMaterial=

=

=

Cσ

CεT

xyzxyzzzyyxx σσσσσσσ

(5.8)

[ ]( ) )exp( tiikxTN

tttj

Tzyx

ωξ −=

=t (5.9)

157

On element assembly, the global equation becomes

The geometry dependent matrices K1, K2, K3 and M are each 3N x 3N for N nodes

across the waveguide thickness. The unknown nodal displacement vector U is 3N x 1.

The Equation 5.11 is rewritten as

( )

( )

ξρ

ξ

ξ

ξ

ξ

ω

d

d

dC

d

d

wherekik

jTj

jTj

jTjTj

jTj

jT

jjjjjjj

∫∫∫∫∫

−

−

−

−

−

=

=

−=

=

=

−++=

1

1

1

1 223

1

1 12212

1

1 111

1

1

23

221

;

;

;

;

NNM

BCBK

BBBCBK

BCBK

tNNf

UMUKKKf

(5.10)

( ) fUMUKKK =−++ 23

221 ωkik (5.11)

( )

=

=

−−

=

−−

=

=−

fp

k

Bi

pk

0;

;0

0;

0

3

21

22

1

21

UU

Q

KMK

KMKMK

A

UBA

ωω

ω

(5.12)

158

The size of the matrices A and B are 6N x 6N which results in 6N eigen values for

each wave number k at each frequency ω. While the eigen values provide the dispersion

solution, the eigen vectors provide the wavestructures.

5.5.2 Scattering at a waveguide transition using Normal Mode Expansion

Normal Mode Expansion (NME) assumes the completeness of guided wave

modes i.e. any displacement can be expressed in terms of the displacement

wavestructures of the modes present in the waveguide at that frequency [Kirrmann 1995].

A hybrid method combining SAFE and NME is developed here in order to

understand the guided wave mode conversion effect at waveguide transitions. While

considering the wave propagation as a 1-D plane strain problem, evaluation of mode

conversion at a waveguide transition is reduced to the analysis of a single line at the

interface of the waveguides at the transition (Figure 5-7). The common cross-sectional

line at the transition is divided into linear elements. For convergence of the solution, the

dimension of the linear elements (Le) in every material layer is calculated using the

Equation 5.13 [Galan and Abascal 2002]. In simple words, the Equation 5.13 means that

there should be at least 10 elements per shear wavelength in the waveguide material.

<

max

2101

ωπ T

ecL (5.13)

159

The approach to calculate the mode conversions in reflection and transmission for

every mode incidence from one of the waveguides is approached following the

framework laid by Cho and Rose [1996], Rose [1999], Galan and Abascal [2002] and

Song et al. [2005].

A single mode incidence is simulated by providing the correct displacement and

stress profile to the nodes corresponding to the incident mode at the common junction Г1

in the transition. At the transition in geometry, the stress and displacement continuity

conditions at the common boundary (Г1) are imposed, and also the stress-free conditions

at the free surface (Г2) (Figure 5-7). The resultant scattered displacement and stress

profiles satisfy:

Figure 5-7: A zoomed view of the discretized transition region between waveguides 1 and 2 for analysis using the hybrid-SAFE-NME method is shown. The nodes are marked with black circles. The portion of the interface satisfying the continuity conditions and the free boundary are shown.

Discretized interface between the waveguides

Continuity of stresses and displacements

Free boundary

Waveguide A

Waveguide B

Г1

Г2

x2 x1

x3

x

160

The superscripts ‘I’, ‘R’ and ‘T’ refer to the incident, reflected and the transmitted

fields respectively.

At each frequency, NME [Auld 1990, Rose 1999 and Puthillath et al. 2007] is

carried out to express the displacements in terms of all the possible modes existing in

both the waveguides. The coefficients for the wave modes in the NME are the unknowns

to be determined.

The reflected and transmitted power can be determined from the stress and

displacement fields using Poynting’s vector calculations, enabling energy balance

calculations to be performed. Any deviation from the balance in energy is a measure of

the error in the calculation.

} 231

13

0 Γ∈=

Γ∈

=+=+

x

xuuu

Ti

Ti

Ri

Ii

Tij

Rij

Iij

σ

σσσ

(5.14)

( ) ( ) ( )

( )( )( ) iU

imu

A

iNm

UuA

i

mi

m

imi

mN

m

directionalongnt/stressdisplacemeResultantdirectionalongmodeofnt/stressDisplaceme

tcoefficienUnknown

s)(Direction3,2,1modes...,2,1

1

==

=

==

=∑=

ωωω

ωωω

(5.15)

161

5.5.3 Case study: Analysis of waveguide transitions using the hybrid SAFE-NME

Case #1: Comparison of a step change in thickness with step change in thickness and

material

As a first case in the study of waveguide transitions, two similar geometries are

analyzed using the hybrid SAFE-NME method. A step change in aluminum from 1 mm

to 2.2 mm and an adhesive lap which involves a change in geometry and a change in the

waveguide material for the same thickness change were chosen for the study. The lap-

joint considered here has a geometry transition from a 1 mm thick aluminum plate to a

bonded structure comprised of two 1 mm thick aluminum plates bonded using an epoxy

(0.2 mm thick), as shown in Figure 5-8.

Using the hybrid method, the in-plane displacement amplitude reflection and

transmission factors were computed as a function of frequency, at both the geometry

transitions for the case of a single mode (s0 mode) incidence from the aluminum side.

From the results presented in Figure 5-8, it can be observed that the bonded step-lap joint

shows the formation of reflection factor peaks, whereas the abrupt step change shows a

flat distribution of the reflection and transmission factors. The features in the case of the

bonded joint are attributed to the formation of new modes on the transmitted side. Two

waveguides with the same thickness but a different material arrangement can thus be

easily distinguished using guided wave mode based measurement.

162

Figure 5-8: Amplitude reflection and transmission factors for in-plane displacement computed using the hybrid-SAFE-NME method for an abrupt step change and a bonded lap joint.

163

Case Study #2: Skin-stringer transition

A schematic of the skin-stringer configuration used in this case study, with

assumed realistic dimensions of the adhesive and adherend, is shown in the Figure 5-9.

The case of incidence of each mode in aluminum (A) into the bonded stringer (B)

is simulated using analytical inputs- displacement and stress wavestructures and

following the procedure explained in the previous section. The transmission from A to B

is given primary importance.

The energy partitioned transmitted energy (TAB) for the case of incidence of

modes 1-6 in aluminum i.e. a0, s0, a1, s1, s2 and a2 is shown in Figure 5-10. From the TAB

for a0 mode, it can be seen that at low frequencies (f < 400 kHz), the first mode in B

carries the major share of energy. Beyond 400 kHz, the modes 1 and 2 share almost the

same amount of energy. It can also be observed that the energy gets divided again among

all the modes at higher frequencies (> 2 MHz). This occurs around the limiting phase

velocity region of cT and cR. In all these cases the energy balance check was performed.

The maximum error value was observed to be 0.54% for s1 mode incidence at 2.709

Figure 5-9: A typical skin-stringer joint with the discretized cross-section at the transition. The aluminum layers are 2 mm thick and the epoxy bond layer is 0.3 mm thick. The region on the left and right sides of the transition are labeled as A and B respectively.

Aluminum

Epoxy

Discretized waveguide transition

AB

y x

z

164

MHz.

Figure 5-10: Energy partitioning among modes in the bonded stringer for transmission past the transition from aluminum (A) to the bonded stringer (B) are shown as intensity maps. The case of incidence of modes 1 (a0), 2 (s0), 3 (a1) 4 (s1), 5 (s2) and 6 (a2) in waveguide A are shown.

165

From the TAB charts in Figure 5-10, it can be seen that the mode pairs share almost

equal amount of energy among all the transmitted modes. This validates the hypothesis 1

i.e. phase matching modes have a higher chance of getting excited at a waveguide

transition.

Calculations for TBA i.e. transmission of modes into A for a mode incidence from

B were also performed. Some results are shown in Figure 5-11.

Interesting observations can be made from Figure 5-11. For example, the

incidence of modes 1 and 2 from waveguide B result in the generation of a0 mode in

waveguide A. Modes 3 and 4 in waveguide B generate strong s0 mode in waveguide A.

This and many more observations can be explained in terms of the phase velocity

matching.

5.5.4 Reciprocity check

A reciprocity check was also performed in this hybrid analytical SAFE study of

guided wave mode scattering at a waveguide transition. The reciprocal relations from

elasticity [Achenbach 2003] are expressed in Equation 5.16 for the case of wave

incidence at the waveguide transition

In words, it means that the transmission factor of mode j in waveguide B when

ijji ABBA TT = (5.16)

166

mode i is incident from waveguide A must equal the transmission factor of mode i in

waveguide A when mode j is incident from waveguide B.

Figure 5-11: Energy partitioning among modes in the aluminum for transmission past the transition from bonded stringer (B) to aluminum (A) are shown as intensity maps. The case of incidence of modes 1 to 6 in waveguide B is shown.

167

A sample result from reciprocity check is shown in Figure 5-12. The transmission

coefficients for the first 6 modes in waveguide B for the case of the incidence of mode 1

i.e. a0 mode in waveguide A is plotted with dots. The colored lines show the formation of

the a0 mode in waveguide A for the incidence of the first 6 modes from waveguide B. It

can be seen that the two sets of plots are coincident. This was an expected result because

the energy based check revealed energy balance in the calculations.

Figure 5-12: Reciprocity checks for the hybrid analytical SAFE calculations for mode scattering at a transition. The dotted lines correspond to

jBAT1

and the colored lines

correspond to1AB j

T .

168

5.6 Dispersion curve matching and wavestructure based analysis

In the case of discrete waveguide transitions such as an aircraft skin-stringer joint,

it is important for an ultrasonic guided wave inspection to be able to transfer the energy

from one cross-section to another (transmission) efficiently. For convenience, the

waveguide where the wave is incident is defined here as the primary waveguide (A). The

waveguide geometry after transition is referred to as the secondary waveguide (B). In the

case of the simplified aircraft skin-stringer joint, shown in Figure 5-13, the skin region is

the primary waveguide. The bonded stringer region then becomes the secondary

waveguide. It is also assumed here that both waveguides A and B have finite thickness

and are infinitely long in the x1 and x2 directions. The interface common to the

waveguides A and B is denoted by Г.

The problem of finding guided wave modes that efficiently transfer energy from

the primary waveguide to the secondary waveguide is being addressed in this thesis.

Figure 5-13: A discrete waveguide transition found in aircraft skin-stringer joints is shown. For convenience, the waveguide where the wave is incident is defined here as the primary waveguide. The geometry after transition is referred to here as the secondary waveguide. The solid vertical line is used to establish the demarcation between the primary and secondary waveguides artificially and also to denote the interface Г common to waveguides A and B

Aluminum

Epoxy

x2 x1

x3 Primary waveguide (Aircraft skin)

Secondary waveguide (Bonded stringer)

A B

Г

169

A match between the displacement wavestructures of the primary waveguide and

the region of the secondary waveguide to which it is connected is expected to ensure

efficient transfer of energy across the transition. The subset of the secondary waveguide

connected to the primary waveguide is actually an extension of the aircraft skin itself.

Since that is a part of the secondary waveguide, and has its characteristic guided wave

dispersion curves, a distinction is being made here. In this regard, a correlation

coefficient is used to establish the wavestructure matching. Correlation is a mathematical

quantity used to establish statistical relationship between two variables.

A wavestructure matching coefficient ( ( )Γ3,, xmm

ji BAρ or Γji BAρ is defined here

as the summation of the absolute values of the correlation between the components of

wavestructure of the ith mode in the primary waveguide (aircraft skin or A in Figure 5-13)

to the wavestructure of the jth mode in the secondary waveguide over the length of the

connecting region Г (see Figure 5-13). It is mathematically expressed using:

( ) [ ][ ] [ ]

[ ][ ] [ ]

dein waveguinumber Mode3,2,1

3

122

3

1223

)()(

)()())((,,,

=

=

∑

∑

=

=ΓΓ

−−−=

−−−−==

ml

uEuEuuE

uEuEuuExmm

lj

lj

lil

il

jl

il

jl

il

lj

lj

lAl

Al

jl

jl

il

il

BABA jiji

µµµµ

µµµµρωρ

(5.17)

170

The wavestructure matching coefficient is concisely represented as ABρ for

convenience. To specify mode numbers in each waveguide ( )BAAB mm ,ρ will also be

used. The condensed terms in the above equation are re-written in expanded form and

defined as

The value of the correlation coefficient can vary from -1 to 1, with the extremes

corresponding to a perfect match between the trends in the displacement across the cross-

section. The negative value, in this case, specifies a phase mismatch in the displacement

distribution. The net wavestructure matching coefficient (ρAB) is obtained on summation

of the absolute values of the wavestructure correlation components corresponding to the

displacement components – the in-plane and the out-of-plane in the case of the Lamb-

type wave motion. The maximum value that the wavestructure matching coefficient can

take, at any point on the dispersion curve over the prescribed interface Г is 2 since in the

case of Lamb-type waves in a layered isotropic media u2 = 0.

Even though not direct, the numerator of the simplified expression for ρAB i.e.

[ ]( )jl

il

jl

iluuE µµ− can be shown to be an alternate means of expressing the orthogonality of

the wavestructures – a check for guided wave mode coupling according to Ditri [1996].

( )

mean arithmetic theiswhere,][][ :nExpectatio

along position and

offrequency circular aatdirection thin nt Displaceme

3,2,1

,

3

3

µµ

ω

ω

ll

l

ll

uE

x

lu

lxuu

=

Γ

=

=

≡

(5.18)

171

The noncoupling condition for modes, according to Ditri [1996], is expressed in terms of

the wavestructure orthogonality condition, consistent with the current notation for

variables

If modes are orthogonal; it mathematically means that the modes are linearly

independent and hence

The Equation 5.19 in [Ditri 1996] thus implies that ρAB = 0 for noncoupling of

guided wave modes. This completes the proof that the wavestructure matching coefficient

developed here is equivalent to the mode noncoupling condition given by Ditri [1996].

Additionally, ρAB satisfies the reciprocity, a consequence of the commutative property of

each term in the wavestructure matching coefficient.

The wavestructure matching coefficient is computed for all the modes in the

secondary waveguide for a single wave mode incident from the primary waveguide. The

wavestructure correlation coefficient for mode 1 (a0) incident from the waveguide A onto

direction thicknessWaveguides waveguideconnected ebetween thInterface

3

3 0

=

=Γ

=∫Γ

x

dxuu jl

il

(5.19)

[ ] [ ] [ ]

[ ] ][][since

0

ll

AB

jl

il

jl

il

uE

uEuEuuE

µρ

=

=⇒

=

(5.20)

172

the waveguide B (i.e. ρAB(1,mB)) is represented uniquely in Figure 5-14. The intensity

scale represents the numerical value of the wavestructure correlation coefficient.

From the plot in Figure 5-14, it can be seen that the value of ρAB(1,mB) is

maximum for mode 1 in waveguide B (i.e. mB = 1) below the cut-off frequency of mode 3

in the secondary waveguide. Beyond the cut-off frequency of mode 4, the higher value of

the ρAB(1,2) in addition to that of ρAB(1,1) can be observed.

Employing the normality of the guided wave modes, a plot of the partial

contribution of each mode at a frequency when compared with all the possible modes at

that frequency becomes more meaningful in understanding the energy partitioning among

modes. The energy partitioned expression of the wavestructure matching coefficient for

Figure 5-14: Wavestructure matching coefficient ρAB(1,mB) for mode 1 (a0) incidence from the aluminum aircraft skin (A) onto the bonded stringer joint region (B) is shown superimposed on the phase velocity dispersion curves of the bonded stringer joint (B). The color scale value varies from 0 to 2 and represents the value of ρAB(1,mB).

173

every incident mode mA forming a mode mo in the secondary waveguide ( PBA ji

ρ ) incident

from the primary waveguide at every frequency ω and along Г is calculated by the

Equation 5.21. The definition of PBA ji

ρ is similar to the excitation factor defined by Matt

et al. [2005].

A plot of energy partitioned wavestructure matching coefficient for a particular

mode mA propagating into waveguide B is denoted concisely as ( )APAB mρ . The plot

of ( )1PABρ , i.e. for mode 1 incidence, is shown in Figure 5-15. This agrees with the energy

partitioning based result from hybrid SAFE-NME technique shown in Figure 5-10.

A reduction in the value of ( )1PABρ at higher frequencies is due to the

redistribution of energy into the larger number of modes possible at those frequencies.

The plots in Figure 5-14 and Figure 5-15 provide proof for the observations made by di

Scalea et al. [2004] that for a0 incidence, the primary energy carrying mode is the first

fundamental mode in the secondary waveguide at the lower range of frequencies (<400

kHz). The contribution of the second fundamental mode in the secondary waveguide

increases at frequencies above 400 kHz.

ωω

ρ

ρρ

ω

at B dein wavegui modes ofnumber Total

frequencyCircular 21

fπNB

N

jBA

BAPBA B

ji

ji

ji

===

=

∑=

(5.21)

174

Another interesting observation that can be made from Figure 5-14 is the higher

magnitude of ρAB(1,mB) for mB values of 5, 6; and 12, 13 in addition to mB = 1. The

frequency range where higher magnitude of ρAB(1,mB) is observed also correspond to the

points in both the phase and the group velocity dispersion curves where the modes in

waveguide A (modes 1 (a0), 3 (a1) and 6 (a2)) overlap with the modes in waveguide B

(modes 1; 5, 6; and 12, 13) when the dispersion curves are superimposed. This

observation from the wavestructure matching coefficient gives credibility to the phase

velocity overlap based concept that is considered to be useful for optimal energy transfer

across a waveguide transition.

Figure 5-15: Energy partitioned wavestructure matching coefficient ( )1PABρ i.e. for mode

1 (a0) incidence from the aluminum aircraft skin (A) onto the bonded stringer joint region (B) is shown superimposed on the phase velocity dispersion curves of the bonded stringer joint. The color scale value is set to an auto scale so that the individual contributions are clear.

175

A plot of ρAB(2,mB) for mode 2 (s0) incidence from the primary waveguide to the

secondary waveguide is shown in Figure 5-16a. The energy partitioned wavestructure

matching coefficient for s0 mode incidence ( ( )2PABρ ) is shown in Figure 5-16b. From the

plot in Figure 5-16a and b, it can be observed that mode 2 in the secondary waveguide

carries the maximum energy till the cut-off frequency of mode 3. Modes 3 and 4 become

the dominant energy carrying modes from 400 kHz onwards. This agrees with the

observation made by Lowe et al. [2000]. From the plot in Figure 5-16a, it can be

additionally observed that the magnitude of ρAB(2,mB) is higher for mB values of 3, 4; 7-

11 that correspond to the overlap of the mode 1 (s0); and modes 4 (s1) and 5 (s2) in

waveguide A with the modes 3, 4; and 7-11 in waveguide B. Again this shows that the

symmetric mode incidence has a higher tendency to excite modes in the waveguide B

matching with the symmetric modes in waveguide A.

The wavestructure matching coefficient is more helpful in understanding the

mode creation in the waveguide B. Since there has to be energy conservation to handle

the re-distribution among the normal modes at every frequency, the energy partitioned

values are more useful. The energy partitioned wavestructure matching coefficient values

for the aluminum modes 3-8 (a1, s1, s2, a2, s3, a3 in that order) are shown in Figure 5-17.

The wavestructure matching coefficient and the energy partitioned expression

together provide a quick picture of the energy distribution among the various modes

generated in the secondary waveguide for each mode incident from the primary

waveguide. This combined with the intensity line display provides an easy guideline for

the mode selection aspects in the stringer joint inspection problem.

176

Figure 5-16: (a) Wavestructure matching coefficient ρAB(2,mB) and (b) its energy partitioned form ( )2P

ABρ (bottom plot) for mode 2 (s0) incidence from the aluminum aircraft skin (A) onto the bonded stringer joint region (B) are shown superimposed on the phase velocity dispersion curves of the bonded stringer joint. Note the difference in color scales due to the different maximum values.

(a)

(b)

177

Figure 5-17: The energy partitioned wavestructure matching coefficients corresponding to the propagation of aluminum modes 3-8 (A) into the bonded stringer (B) are shown superimposed on the phase velocity dispersion curves of the bonded stringer joint. Note the variation in the intensity scales due to the redistribution of energy into the normal modes at each frequency.

178

The wavestructure matching coefficients shown in Figure 5-14 to Figure 5-17

agree well with the energy based SAFE-NME results in Figure 5-10. This validates the

wavestructure matching coefficient based approach.

5.6.1 Inverse wavestructure matching coefficient

In the stringer joint problem, there is another waveguide transition involved –

from the bonded stringer (waveguide B) to the aircraft skin (waveguide A) as seen in

Figure 5-18. For analyzing this transition, the bonded stringer is the primary waveguide

and the aircraft skin the secondary waveguide. Employing the framework for computing

the wavestructure matching coefficient developed in the section above, it is possible to

compute ρBA(mB,mA) called the inverse wavestructure matching coefficient here that

corresponds to wave mode propagation from the waveguide B to waveguide A.

Figure 5-18: The discrete waveguide transition found in aircraft skin-stringer joints is shown. The primary and secondary waveguides are labeled. The solid vertical line is usedto establish the demarcation between the primary and secondary waveguides artificiallyand also to denote the interface Г common to waveguides B and A.

Aluminum

Epoxy

Primary waveguide (Bonded stringer)

Secondary waveguide (Aircraft skin)

x2 x1

x3 B

AГ

179

Figure 5-19: The energy partitioned wavestructure matching coefficients ( )BPBA mρ

corresponding to the propagation of modes 1-6 from the bonded stringer (B) into the aluminum skin (A) are shown superimposed on the phase velocity dispersion curves for an aluminum plate (2 mm). Note the variation in the intensity scales due to the redistribution of energy into the normal modes at each frequency.

180

ρBA(mB,mA) thus handles the mode conversion from the bonded stringer to that of

the aircraft skin. The energy partitioned wavestructure matching coefficient for every

incident mode mB forming modes in the secondary waveguide ( ( )BPBA mρ ) is presented in

Figure 5-19. From the plots in Figure 5-19, the larger share of the energy carried by the

fundamental modes (a0 and s0) in aluminum for the propagation of the first four modes

from the bonded stringer is evident.

Again it can be noted that the trends in Figure 5-19 follow closely the trends

obtained by SAFE-NME based energy partitioning in Figure 5-11.

5.6.2 Mode Transfer Function for guided wave propagation across waveguide transitions

The wavestructure matching coefficient based analysis for the two step transitions

in the stringer joint problem can be combined to obtain a guided wave mode conversion

model or a mode transfer function for the stringer joint (Figure 5-20).The length of the

waveguide B here is limited, which is not accounted for in the model. It is conveniently

assumed here that the length of the bonded region is at least a few times longer than the

wavelength of the mode propagating through that region.

The mode transfer function for any mode mA propagating from the primary

waveguide (A) to the bonded stringer (B) across interface ГL and back into waveguide A

after traveling along the bonded joint and across the interface ГR at a frequency ω

181

( ( )ω

ρ kAABA mm , ) is expressed in terms of the two wavestructure matching coefficients in

the Equation 5.22.

The expression for ( )ω

ρ kAABA mm , takes into account the propagation of a mode

from waveguide A to waveguide B, its mode conversion, and also the propagation of

modes thus generated back to waveguide A. The multiple reflections within the stringer

are ignored in this calculation. The guided wave mode transfer functions for the first 6

modes in aluminum are provided in Figure 5-21.

Figure 5-20: Waveguide transitions (A-B and B-A) in a bonded stringer joint along with the proper labels to denote the regions. The solid vertical lines are used to establish thedemarcation between the different waveguide regions in the stringer joint. ГL and ГRdenote the left and right interfaces common to waveguides A and B.

( )

frequencyCircular

B dein wavegui modes ofnumber Total

A dein wavegui modethk

1,

=

=

=

=

Γ= Γ∑

ω

ωω

ρρρ

B

k

N

m

Rkj

B

Li

PAB

N

jj

PBAkAABA mm

(5.22)

x2

Aluminum

Epoxy

A B

A ГL ГR x1

x3

182

Figure 5-21: Guided wave mode transfer function for a stringer joint for different modespropagating in the primary waveguide (A) through the secondary waveguide (B) to thewaveguide A, are shown superimposed on the phase velocity dispersion curves for analuminum plate (2 mm).

183

From the plots in Figure 5-21, it can be seen that for each case of mode

propagating in the primary waveguide A, the contribution of the same mode in the

secondary waveguide A is high for most cases. The wavestructure matching coefficient

and the mode transfer function together provide an understanding of the mode

conversions and the energy transfer across a stringer joint.

5.7 Finite Element evaluation of transmission across a waveguide transition

The time domain FE approach presented in Chapter 3 using ABAQUS is used to

perform two case studies. The incidence of a non phase matching mode and a phase

matching mode from waveguide A at the transition from A to B were simulated.

Wavestructure loading based mode excitation was used to generate a single mode in

waveguide A. It is also noted that since the wavestructure of any Lamb-type mode varies

with frequency, its variation over the frequency bandwidth of loading is not captured by

the FE input. Hence this is an approximation. On the same note it is also stated that the

bandwidth spread of loading is naturally present in all real experiments where finite

duration loading is the only feasible option.

The results from the following cases are presented:

1. Non-phase matching mode incidence - s0 mode at 300 kHz – In this case the s0

mode in waveguide A does not perfectly superimpose with the modes in waveguide

B (Figure 5-5).

2. Phase matching mode incidence – s1 mode at 2.36 MHz – In this case the s1 mode

in waveguide A matches with the modes 8 and 9 in waveguide B (Figure 5-5).

184

The snapshots of the wave propagation and interaction with the transition for the

two cases mentioned above are presented in Figure 5-22 and Figure 5-23. It can be seen

from the non-phase matching mode incidence case that there is energy reflected from the

transition between waveguides A and B. In the perfect phase matching case it can be

observed that there is nearly 100 % transmission from waveguide A to B.

The FE result for non-phase matching mode incidence shown in Figure 5-22

agrees with the SAFE-NME result for TAB(2) (Figure 5-10) and wavestructure matching

result ( )2PABρ (Figure 5-16). Similarly the FE result for phase matching mode incidence

in Figure 5-23 agrees with the SAFE-NME result for TAB(4) (Figure 5-10) and

wavestructure matching result (Figure 5-17).

Figure 5-22: Snapshots from the FE model showing the interaction of s0 mode at 300 kHz with the transition from waveguide A to waveguide B.

185

5.8 Guided wave mode sensitivity to interfacial defects in bonding

The adhesive bonding in the stringer joint is prone to defects similar to that found

in the repair patches (Chapter 4) – the adhesive and cohesive types. The adhesive or the

interfacial defects are the critical ones because they are difficult to detect from

conventional ultrasonic measurements. The adhesively bonded stringer region considered

in this study is mid-plane symmetric which implies that studying one of the two

adherend-adhesive interfaces is sufficient and it is still applicable to the inspection of the

complete stringer for most defect cases.

Proceeding on the lines of the approach adopted for the repair patch inspection

problem in Chapter 4, the interfacial in-plane displacement profile is shown

superimposed with the dispersion curves for the bonded stringer in Figure 5-24.

Place Figure Here

Figure 5-23: Snapshots from the FE model showing the interaction of s1 mode at 2.36 MHz with the transition from waveguide A to waveguide B.

186

The red regions in the plot in Figure 5-24 show the mode-frequency locations

with large in-plane displacement at the aluminum-epoxy interface.

5.8.1 Inspection chart for a Stringer Joint

A theoretically driven framework for stringer joint inspection is laid here by

combining the wavestructure matching coefficient that characterizes mode conversion at

waveguide transitions, and the interface sensitive modes in the bonded region of the

stringer.

Frequency (MHz)

Phas

e ve

loci

ty (k

m/s

)

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 5-24: Intensity map of the in-plane displacement at the aluminum-epoxy interface superimposed over the Lamb wave dispersion curves for the adhesive stringer jointcomprised of two epoxy bonded aluminum plates (2 mm).

187

The effectiveness index of a mode in the aircraft skin (mode mA in waveguide A)

for use in detection of interfacial defects in the bonded region of the stringer (waveguide

B), at a frequency ( fπ2=ω ) is defined by ( )ωAAB mE (Equation 5.23). The effectiveness

index of the guided wave modes is thus a map of the interface sensitive mode and

frequency combinations in the stringer joint region (waveguide B) for a single mode

excitation in the aircraft skin (waveguide A).

A map of the effectiveness index of each mode that can propagate in the aircraft

skin for use in the inspection of the stringer joint can be created based on the above

equation. In the intensity line plots on Figure 5-25 - Figure 5-27, the effectiveness of the

first six modes in the aircraft skin is mapped onto the dispersion curves of the bonded

stringer joint. Also shown superimposed using dotted lines are the modes in aluminum

that result in this distribution in the bonded stringer.

( ) ( ) ( )

interface theofLocation

interface adherend-adhesive at thent displaceme plane-In

at B dein wavegui modes ofnumber Total

2frequencyCircular

B waveguidein the mode thj

A waveguidein the Mode

1

11,

ω

ω

ωωω ρ

fπ

d

N

m

m

d

j

N

jj

djA

PABAAB

u

mummmE

B

A

B

=

=

=

=

=

=

=

∑=

×=

(5.23)

188

Figure 5-25: Effectiveness index EAB of the first two modes in aluminum (a0 and s0respectively) for inspection of the interfacial defects in the bonding across the waveguidetransition in a stringer joint is shown mapped on the dispersion curves of the bondedstringer region. The dotted red line marks the mode in aluminum shown in the parenthesis on the plot title.

189

Figure 5-26: Effectiveness index EAB of the modes 3 and 4 in aluminum (a1 and s1respectively) for inspection of the interfacial defects in the bonding across the waveguidetransition in a stringer joint is shown mapped on the dispersion curves of the bondedstringer region. The dotted red line marks the mode in aluminum shown in theparenthesis on the plot title.

190

Figure 5-27: Effectiveness index EAB of the modes 5 and 6 in aluminum (s2 and a2respectively) for inspection of the interfacial defects in the bonding across the waveguide transition in a stringer joint is shown mapped on the dispersion curves of the bondedstringer region. The dotted red line marks the mode in aluminum shown in theparenthesis on the plot title.

191

The superimposed mode in the waveguide A shown in Figure 5-25 - Figure 5-27

the helps in identifying the frequency range where the mode sensitive to the interface

condition is excited for the case of a single mode propagating from the aircraft skin.

5.8.2 Observations from Effectiveness charts

Some observations drawn by comparing the effectiveness index values of the

modes generated in the waveguide B, for a single mode incidence from waveguide A

(Figure 5-25 - Figure 5-27), are listed here:

(a) Propagation of mode 1 (a0) into the bonded stringer, the first mode in the

waveguide B has a low effectiveness – i.e. EAB(1,1) is low (<0.3) over the range

of frequencies. At extremely low frequencies (< 100 kHz) EAB(1,2) has a high

value (~0.7). It can be seen that the 5th mode in the waveguide B, that actually

overlaps with the a1 mode in waveguide A also gets excited due to the mode

conversion from the incident a0 in the range of frequencies around 1.5 MHz.

Similarly a high value of effectiveness can be noticed around 2.5 MHz that

corresponds to modes 12 and 13 in the waveguide B and mode 6 (a2) in

waveguide A. This can also be explained based on the dispersion curve matching

or the wavestructure matching coefficient values.

(b) For mode 2 (s0) propagating into the bonded stringer, the EAB(2,2) in the

waveguide B is high (~0.8) at frequencies below 200 kHz. The value of EAB(2,3)

is high (~0.5-0.6) in the frequency range of roughly 350-800 kHz, which implies a

sensitivity of the s0 mode to the interfacial defects in the stringer. This point is

again at the phase velocity overlap region. This agrees with the results presented

in Rose et al. [1995a] using the s0 mode at 1.455 MHz mm. This frequency

thickness product corresponds to approximately 727 kHz for a 2 mm aluminum

plate used in the current work.

192

(c) For the case of mode 3 (a1) propagating into the bonded stringer, the value of

EAB(3,6) is high in the range of frequencies from 800 kHz to 2.2 MHz, averaging

around, with the highest value of 0.59 at around 1.45 MHz. Again the difference

between the phase velocities of the mode 3 in waveguide A and modes 6 and 7 in

waveguide B is small. A successful parameter choice for stringer joint inspection

in Rose et al. [1995b] using the a1 mode at 3.525 MHz mm implies a frequency

value of 1.725 MHz, which is within the sensitive range found from the

effectiveness index value.

(d) Mode 4 (s1) propagating into the bonded stringer results in a large value of

EAB(4,8) and EAB(4,9), approximately around 0.4, in the frequency range of 1.5 – 3

MHz compared to all the other modes generated in waveguide B at that

frequency. Hence mode 4 in waveguide A at frequencies from 1.5 MHz – 3 MHz

is suitable for the inspection of the stringer joint. Here it can also be seen that the

frequency values with high effectiveness index is spread over a portion where

phase and group velocity matching exists between the dispersion curves of the

waveguide A and B. Based on the phase and group velocity matching the larger

effectiveness index switches from mode 8 to mode 9 in waveguide B. (Modes 8

and 9 form a mode pair).

(e) Mode 5 (s2) propagating into the bonded stringer has a high value of EAB(5,11)

around 3 MHz. The sensitivity of this mode is not as high as some of the previous

modes. Ditri and Rose (1992) have made use of this mode to inspect step-lap

joints successfully. There are some other modes with mode index < 11 that show

a large value of EAB(5). These modes have both a higher wavestructure matching

coefficient and a larger sensitivity to the interface and are theoretically possible in

the waveguide because they form a complete set of solution at that frequency

owing to the normality and completeness of the guided wave modes.

(f) Mode 6 (a2) propagating into the bonded stringer has a high value of EAB(6,13),

around 0.6, in the range of frequencies from 2.6-3.4 MHz. This is again a case

having good phase velocity match and a reasonable group velocity match. This

193

region in the dispersion curves has not been reported for its sensitivity in any

published literature known to the author at this time.

5.8.3 Conclusions from mode effectiveness study

The procedure established here, using the effectiveness index based combination

parameter, for locating modes propagating in the primary waveguide that easily couple to

the secondary waveguide and are also sensitive to the specific regions in the cross-section

of the secondary waveguide is thus not only able to explain the parameter selection in

most of the published literature in the area of guided wave inspection of skin-stringer

joints, but also go beyond and provide many such inspection possibilities. This

generalized physical insight driven approach can be easily extended to further higher

frequencies.

Though qualitatively the modes and frequency choices reported in the literature

match with this model, experiments will bring a closure in the form of a practical proof to

this hypothesis.

The procedure established here for systematically analyzing the coupling modes

at a waveguide transition, followed by an analysis to understand the sensitivity of the

coupled modes to the required defect is expected to be adaptable to different inspection

scenarios like inspection of composite skin-stringer joints, corrosion at a coupler in a

pipe, etc. The composite joints for example would also need an understanding of wave

skew effects. Practical issues in implementation of a guided wave based testing process

194

like excitation or reception over a defect location can also be analyzed and explained

using the approach established here.

Even though most of the results can also be explained in terms of the phase and

group velocity matching, both the interfacial in-plane displacement profile in the bonded

joint and the wavestructure matching together influence the localization of the

effectiveness index over the dispersion space. The role of group velocity matching is also

important as it points to another concept – non stagnation of energy at a transition.

5.9 Experiments on skin-stringer joints

The details of the skin stringer joint samples fabricated and, their inspection using

ultrasonic guided waves is provided in this section.

5.9.1 Fabrication of skin-stringer adhesive joint samples

Aluminum stringer joints were prepared by bonding 12” x 2” aluminum strip (2

mm thick) on the surface of a 12” x 12” aluminum plate (2 mm thick). Sheet adhesive

(EA9696) was used as the adhesive. A dimensioned sketch of the aluminum stringer joint

is shown in Figure 5-28.

Prior to bonding, the surfaces to be bonded were polished using fine grit abrasive

disc pads (Roloc disc pads from 3M Inc.), cleaned with acetone, coated with sol-gel,

followed by primer. Aerospace grade epoxy adhesive - EA9696 available in the form of

195

sheets was selected as the bonding agent. Two layers of this sheet adhesive were cut to

the dimensions of the bonding area and stacked upon the region to be bonded before

placing the aluminum stringer on top of the adhesive layers. The whole assembly was

vacuum bagged and cured in an autoclave at a temperature of 250 °F for around 3 hours.

Different interface conditions were simulated by introducing different materials,

having a principal dimension of ~ 0.5 in., at either the aluminum-epoxy interface or

within the plies of epoxy. An adhesive weakness condition was simulated by the

introduction of teflon at the aluminum-epoxy interface. In order to simulate a cohesive

weakness condition, a piece of teflon was placed between plies of the epoxy.

Figure 5-28: Dimensioned sketch of the aluminum skin-stringer adhesive joint sample fabricated at Penn State University.

196

5.9.2 Ultrasonic oblique incidence guided wave inspection test bed

There are different methods for excitation of guided waves in plate-like structures

like angle wedge based loading, comb loading (Figure 5-29).

Oblique incidence of ultrasonic bulk waves, a theoretically constant phase

velocity excitation, was implemented using Plexiglas variable angle-beam wedges and

oblique incidence loading in water (comparable to a wedge loading). Despite not being a

practical field implementable solution, the water based loading was implemented

(Figure 5-30) due to the uniformity of ultrasonic coupling and manipulation simplicity of

the angle of incidence and the reception of guided waves. The goniometer permits

orienting the transducer at any angle of incidence (< 50°) desired. The Plexiglas based

variable angle beam wedges also permitted incidence angles up to around 50°.

Figure 5-29: Ultrasonic guided wave excitation methods – comb loading (wavelength spaced piezoelectric loading) and variable angle beam acrylic wedge with mountedpiezoelectric transducer.

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The fabricated ultrasonic oblique incidence pitch-catch inspection test bed for

testing adhesive skin-stringer joints is shown in Figure 5-31. The experimental

arrangement comprises of two goniometers holding ultrasonic immersion transducers

positioned in a pitch-catch arrangement. The goniometers were themselves mounted on

separate screw rods with position control knobs, thus enabling positioning independent of

the other.

The stringer joint to be inspected was placed such that the transmission across the

stringer joint can be measured (Figure 5-31)A pair of broad band immersion transducers

was used. The transmitter was excited using a tone-burst source and the receiving signal

was collected using the receiver facing the transmitter and positioned at the same angle as

the transmitter.

Figure 5-30: A Goniometer with an ultrasonic immersion transducer attached to it. TheGoniometer permits orienting the transducer at any angle of incidence (< 50°) desired.The incidence angles are measured from the vertical for experiments.

Goniometer

Ultrasonic immersion transducer

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5.9.3 Ultrasonic guided wave inspection of stringer joints

Based on the wavestructure matching model results and effectiveness index

calculations, mode and frequency locations were identified on the aluminum skin

dispersion curves that propagate and mode convert to interface sensitive modes within the

bond region. Modes found not suitable were also selected for confirming validity of the

model and also for demonstration purposes. Experiments were conducted both in a water

immersion mode and also using Plexiglas variable angle beam wedges. The signals from

the aluminum plate, the good stringer joint and the stringer joint with adhesive and

cohesive weakness, were compared for each incident mode.

The variable wedges were used for modes with higher out-of-plane displacement

in order to avoid the effects of leakage on the signal. Since the measurement approach

Figure 5-31: Ultrasonic oblique incidence pitch-catch inspection in a water immersion mode. Each of the goniometers holding the transducers can be moved independently ofthe other along the line joining the two transducers.

Stringer joint

Goniometer - 1 Goniometer - 2 Ultrasonic immersion

transducers

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adopted here is comparative in nature, even when some leakage is present, its effect is

ignored while comparing transmitted signals across different bonding cases.

5.9.3.1 Inspection at mode-frequency locations with low effectiveness index (EAB)

Set 1: s0 mode incidence at 500 kHz

The s0 mode at 500 kHz has a low effectiveness index (EAB~0.2) as seen from

Figure 5-25. The s0 mode was excited the using water immersion mode at a frequency of

500 kHz and an angle of incidence of 16° computed using Snell’s law based coincidence

angle calculations.

The RF signals and their frequency domain information computed using fast

Fourier transforms (FFT) is presented in Figure 5-32. From Figure 5-32 it can be seen

that there is very little difference between the measured transmission across a good

stringer joint and that corresponding to the propagation along the aluminum plate

obtained with the same spacing between the transmitter and the receiver. The energy is

concentrated more in the skin and hence this parameter set is insensitive to the adhesive

bond.

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Set 2: Rayleigh wave incidence at 5 MHz

The range of frequencies beyond 3 MHz corresponding to a phase velocity equal

to cR has a low effectiveness index, around 0.1, (Figure 5-25). Based on the coincidence

angle calculation, for Rayleigh wave generation at 5 MHz, an incidence angle of 32° in

water is required. The transmission data collected at this input condition are presented in

Figure 5-33.

The results in Figure 5-33 also validate this result from the model. Again the

mode is not able to even confirm the presence or absence of a stringer bond. Hence, a

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Figure 5-32: RF signals and their fast Fourier transforms obtained from transmission measurements for an s0 mode generated using a tone burst input of 0.5 MHz for 5 cyclesand oblique incidence at 16° in water. (a) Aluminum (2mm) plate, (b) good stringer joint,(c) stringer joint with adhesive (Teflon) weakness and (d) stringer joint with cohesive (Teflon) weakness.

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mode-frequency combination having very low effectiveness index results in a

transmission data insensitive to the bonding.

5.9.3.2 Inspection at mode-frequency locations with high effectiveness index (EAB)

Set 1: a1 mode at 2.3 MHz

The higher effectiveness index for a1 mode (EAB~0.3) around 2 MHz can be seen

from Figure 5-26. A tone burst input of 2.3 MHz was supplied for 5 cycles to a

transducer mounted on 36° Plexiglas wedge to generate the a1 mode in the 2 mm thick

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Figure 5-33: RF signals and their fast Fourier transforms obtained from transmission measurements for a Rayleigh wave generated using a tone burst input of 5 MHz for 5cycles and oblique incidence at 32° in water. (a) Aluminum (2mm) plate, (b) goodstringer joint, (c) stringer joint with adhesive (Teflon) weakness and (d) stringer joint with cohesive (Teflon) weakness.

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aluminum plate. Single mode generation in the plate was verified by observing the

measured waveform and also from the measurement of group velocity.

From the RF signals shown in Figure 5-34, it can be seen that the frequency

content in the signal is between 1.91 MHz and 2.3 MHz. From the wavestructure

matching coefficient, it is known that for this input, modes 6 and 7 get excited within the

bonded joint. By comparing the phase and group velocity dispersion curves for the

aluminum skin and the bonded stringer, it was also observed that the modes 6 and 7 in the

bonded stringer have a very good match with the a1 mode (mode 3) in the aluminum skin.

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Figure 5-34: RF signals and their fast Fourier transforms obtained from transmissionmeasurements for an a1 mode generated using a tone burst input of 2.3 MHz for 5 cycles and oblique incidence at 36° in Plexiglas wedge. (a) Aluminum (2mm) plate, (b) goodstringer joint, (c) stringer joint with adhesive (Teflon) weakness and (d) stringer joint withcohesive (Teflon) weakness.

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From Figure 5-34, it can be seen that the amplitude of the wave propagation in the

aluminum skin is higher than that across the stringers. This is an instantaneous

verification for the presence/absence of the stringer in the wave travel path. A higher

transmission can also imply a totally debonded stringer. The amplitude transmitted across

the stringer is higher in case of a good bond and also a cohesively weak bond than an

adhesively weak bond. Since the group velocities of the modes formed within the bonded

joint being almost equal to that in the aluminum skin, there is no velocity shift observed

between the different waveforms shown in Figure 5-34.

Set 2: a1 and s1 mode incidence at 1.5 MHz

Again, from Figure 5-26, the high effectiveness index (~0.5) for incident a1 and s1

modes can be seen. By suitably tuning the transducer loading for guided wave generation,

it is possible to generate multiple modes in a structure with a single source. Supplying a

1.5 MHz tone-burst pulse for 5 cycles to a broad band ultrasonic immersion transducer

2.25 MHz transducer (Ø0.5” or 12.7 mm), held at 14° from the vertical the theoretical

excitation zone due to the influence of the source geometry is provided in Figure 5-35.

The source influence charts point to the generation of a1 and s1 modes in aluminum plate

(2 mm).

The RF signals collected in a pitch-catch mode are presented in Figure 5-36 for

the different cases – namely the aluminum plate, good stringer joint, bad stringer joint

with adhesive and cohesive weaknesses respectively.

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Analyzing the RF signals collected at spatial separation in an aluminum plate (2

mm), along with those shown in Figure 5-34, it is found that the two mode packets

correspond to – s1 and a1 mode propagation respectively. This agrees with the source

influence results in Figure 5-35.

It is very clear from Figure 5-36 that the selected frequency-angle combination is

able to discriminate between an aluminum plate and bonded stringer in terms of both the

amplitude of transmission and velocity of the mode.

Figure 5-35: Geometric influence of loading on the range of phase velocities andfrequencies excited. The color intensity shows the strength of the ultrasonic excitation ofa 12.5 mm diameter transducer oriented at an angle of 14° and supplied with a 1.5 MHztone burst input voltage for 5 cycles. The white lines are the Lamb wave phase velocitydispersion curves for aluminum (2 mm).

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The s1 mode centered at 1.97 MHz (1.8-2.1 MHz), forms the first mode packet in

the signal and propagates with cg ≈ 4.47 km/s. The wavestructure matching coefficient

points to the conversion of the s1 mode into mode 8 and 9 in the bonded regions that have

cg ≈ cgAl. That explains the reduction in the speed of travel through the bond region.

The a1 mode in the RF signal exists in a range of frequencies centered at 1.6 MHz

(1.5-1.76 MHz) and propagates with cg ≈ 3.6 km/s. While propagating through the

bonded region, the a1 mode converts to modes 6 and 7 that have cg ≈ cgAl. That explains

the reduction in the speed of travel through the bond region.

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Figure 5-36: RF signals and their fast Fourier transforms obtained from transmissionmeasurements for s1 and a1 mode generated using a tone burst input of 1.5 MHz for 5cycles and oblique incidence at 14° in water. (a) Aluminum (2mm) plate, (b) goodstringer joint, (c) stringer joint with adhesive (Teflon) weakness and (d) stringer jointwith cohesive (Teflon) weakness.

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From the RF signals and their fast Fourier transforms shown in Figure 5-36, we

can conclude that the modes selected – a1/s1 at 1.5 MHz and 14° are sensitive to the

interface condition. The amplitude becomes a clear distinguishing criterion here. All

these observations agree with the effectiveness index charts EAB(3) and EAB(4) in

Figure 5-26. An indirect observation is that the angle of incidence of 14° corresponds to

the first critical angle in bulk wave sense.

Experimentally it has been verified that using the high effectiveness index points

from the dispersion curves, we can generate interface sensitive modes.

5.10 Summary

In this chapter, the complete analysis of ultrasonic guided wave propagation

across an aircraft skin-stringer joint made of aluminum was presented. A hybrid model

using Semi-Analytical Finite Element method and scattering analysis using Normal Mode

Expansion was presented. Energy balance and reciprocity based verification were

performed to confirm the reliability of this method. A new model for understanding the

guided wave behavior across a transition based on the wavestructure matching coefficient

was proposed. The wavestructure matching coefficient was shown to be equivalent to

mode coupling theory in waveguides. Good agreement was found between the hybrid

SAFE-NME model and the wavestructure matching coefficient model. Combining the

wavestructure matching coefficients at two transitions in the case of a skin-stringer joint

and ignoring the multiple reflections within a stringer region, a waveguide transfer

function was formed. A parameter combining the wavestructure matching coefficient and

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the bond defect sensitive interfacial in-plane displacement parameter – called the

effectiveness index was proposed as a tool for quick selection of the mode-frequency

choices for inspection of the interfacial bonding in a skin-stringer joint. Experiments

were successfully conducted on the samples prepared with simulated interfacial defects in

bonding using appropriate mode-frequency choice from the effectiveness charts. Modes

with effectiveness higher than 0.3 were found to be sensitive to the interfacial condition

in the skin-stringer joint samples.

A hypothesis based on fundamental principles was validated using the modeling

techniques developed in this chapter. The validated hypothesis is

For every mode incident from the waveguide on one side of a transition, modes

with matching phase velocity vs. frequency (implies an almost similar

wavestructure) in the waveguide beyond the transition has a higher possibility of

getting excited on transmission across the transition.

As a consequence, the following statements derived from the hypothesis were also

validated:

(a) The mode pairs have almost the same tendency to get excited.

(b) A mode incident at the transition will transmit higher amount of energy to

modes that have similar group velocity (energy velocity in an attenuative

waveguide). A mismatch between the group velocities results in a larger

reflected energy or a small transmitted energy.

Some observations and conclusions are listed below:

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1. From the hybrid SAFE-NME model, it was verified that the mode pairs share

almost equal energy when generated in the secondary waveguide for a matching

wave mode input from the primary waveguide.

2. The phase and group velocity points that overlap in two connected waveguides

have a higher wavestructure correlation and hence are more likely to be generated

by mode conversion with an appropriately matching input mode-frequency

combination.

3. Group velocity matching between incident mode and the modes generated due to

mode conversion at the transition enables efficient transfer of guided wave

energy. This can be reasoned by energy flux conservation or non-stagnation of

energy.

4. Using the effectiveness index, it possible to relate the successful use of different

mode-frequency points reported in the literature and also generalize it to find

more sensitive points.

5. The procedure established here for systematically analyzing the coupling modes

at a waveguide transition, followed by an analysis to understand the sensitivity of

the coupled modes to the required defect is expected to be adaptable to different

inspection scenarios like inspection of composite skin-stringer joints, corrosion at

a coupler in pipe.

Chapter 6

Summary, Contributions and Future Directions

6.1 Summary of this thesis work

This thesis aims to establish the theoretical foundations for ultrasonic guided

wave based inspection of adhesive joints having two types of bonded joints – continuous

and discrete.

Adhesively bonded joints are increasingly employed as load bearing members in

engineering structures. They offer a stronger, stiffer and lighter joint with relatively lower

stress concentration when compared with the conventional jointing techniques like

riveting and welding. The adhesive joints have a higher fatigue life, more flexibility in

handling complicated geometry and different material combinations. Major limitations in

using the adhesive joints are the need for high quality surface preparation and

susceptibility to environmental conditions during operations. The adhesive or interfacial

weakness and cohesive or bulk weakness in the adhesive joints necessitate the use of

reliable inspection. The ultrasonic nondestructive guided wave inspection approach has

several advantages like potentially infinite inspection points (frequency-phase velocity

pairs), long range and hidden structure inspection capability that make this technique

quite versatile.

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In the first chapter, the background on the need for inspection of adhesive joints

and the non-destructive characterization approaches was listed and the goals for the thesis

work were laid out after a brief review of related literature. Two specific problems related

to aircraft adhesive joints – adhesive repair patches and adhesively bonded skin-stringer

joints were identified where the new theoretical understanding is expected to aid

inspection.

In the second chapter, the foundational theory for understanding and representing

guided wave behavior in waveguides – dispersion was laid out following a partial wave

theory based global matrix approach that is well documented in the literature. A

comparison was made between the dispersion curves of aluminum plate and aluminum-

epoxy-aluminum bonded configuration considered as an example of a stringer joint found

in aircraft adhesive joints. The comparison revealed the existence of epoxy dominated

modes that confined the energy to within the epoxy layer. A new term – Mode Pair – was

coined to describe the existence of modes in the bonded joint that envelope or appear

close to the aluminum modes and have nearly the same phase and group velocity. The

mode pairs were found to have identical cross-sectional displacement distribution or

wavestructure in the base aluminum and a phase reversed displacement in the top

aluminum. It was also noted that the generation of each of the modes in the mode pair

was expected to be similar for the case of generation after mode conversion on incidence

from the aluminum waveguide – which is the typical discrete transition in a waveguide.

In Chapter 3, the numerical experimentation techniques using explicit time

marching algorithm in ABAQUS, a commercial Finite Element (FE) package, along with

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analytically determined inputs were introduced. The analytical inputs helped in the

generation of specific guided wave modes in the structure simulated. The signal

processing techniques for processing the FE data that are equally applicable to

experimental measurements using point measurement devices were introduced. The point

data and line data – both parallel to wave propagation and across the cross-section

collected from FE were processed using techniques like short time Fourier transform

(STFT), two dimensional Fourier transform (2DFFT) coupled with a guided wave mode

matching filter, phased addition, and wavestructure decomposition techniques. The mode

filtering algorithm is unique in terms of its constituent directional and mode matching

filtering schemes that enable finite domain simulation without the need for silent

boundaries in FE. The phased addition based processing developed in this thesis enables

processing smaller data with the ability to separate directional data. The wavestructure

processing scheme enabled numerically determining the absence of near field distance for

mode formation. This has big implications on the ability to detect defects located near the

edge of a bonded joint – a higher peel stress location.

In Chapter 4, a theoretically driven guided wave inspection procedure for

systematically detecting defects in bonded joints is established. It was proposed and

proved experimentally, using epoxy bonded aluminum-titanium samples with simulated

defects, that the choice of modes with larger in-plane displacement at the interface of

interest will enable detection of weakness in bonded repair patches.

In Chapter 5, the foundational work for handling guided wave mode conversion

and scattering at a waveguide transition was developed. A hybrid method combining the

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Semi-Analytical Finite Element (SAFE) and Normal Mode Expansion (NME) was

developed to handle guided wave scattering at a waveguide transition. This method

showed quantitatively the energy partitioning among modes traveling across the

transitions found in a skin-stringer joint. Energy balance and reciprocity based checks

confirmed the validity of the SAFE-NME procedure. A wavestructure matching

coefficient was developed to qualitatively determine the mode generation for incidence of

modes at the junction of two waveguides. Using a single mode incidence, the wave

propagation and mode conversion in a skin-stringer joint was determined. The qualitative

approach using wavestructure matching was found to agree well with the SAFE-NME

based results. The mode conversion and scattering study was extended to determine the

net transmission across the stringer joint. This was coupled with the interfacial in-plane

displacement parameter to form a combination parameter called Effectiveness index.

Higher effectiveness index modes (>0.3) were found to be successful in experimental

detection of interfacial weakness in the skin-stringer joints. The validity of the hypothesis

that “for every mode incident from the waveguide on one side of a transition, modes with

matching phase velocity vs. frequency (implies an almost similar wavestructure) in the

waveguide beyond the transition has a higher possibility of getting excited on

transmission across the transition” was verified using the SAFE-NME approach,

wavestructure matching coefficient approach and numerical studies using FE. Two

important statements derived from this hypothesis, namely the higher tendency of

generation of modes pairs and larger energy transfer with the generation of group

velocity matching modes was successfully tested using the models developed to handle

waveguide transition.

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6.2 Specific contributions and their impact

1. Near field effect at waveguide transition and near guided wave source: The

numerical modeling and signal processing revealed the instantaneous formation of

guided wave mode near an ultrasonic excitation point and also at a structural

transition. The numerical technique employed is unique and the conclusion drawn is

significant as it implies the ability of guided waves to interrogate defects located at

the edge of an adhesive joint which is a critical region due to higher shear and peel

stresses.

2. Improvement of 2DFFT and development of guided wave mode filtering: The

improvement to the two-dimensional fast Fourier transform based processing

technique by the addition of a directional filter and a mode matching filter is

theoretically driven and tailored for the guided wave application. The key

contribution in terms of directional filtering and inverse transforming to determine

the guided wave mode signals and their transmission coefficients is helpful in

understanding the guided wave transmission and also helpful in modeling finite

domains numerically without the need for silent boundaries.

3. Phase addition approach to process small length line date from numerical

experiments and real experiments: The time delay based modeling of transducer

both in the transmission and reception is computationally favorable because it not

only avoids the need for creation and meshing of the wedge, but also enables post-

processing finite data, much smaller than that in 2DFFT to obtain the guided wave

data with the influence of the receiver dimension built-in. This is unique and

powerful and applicable to point measurement using lasers or pinducers in real

experiments

4. Procedure for determining selective interface sensitive modes: The development of

systematic approach in determining guided wave modes with sensitivity to an

interface in a layered media by employing modes with high interfacial in-plane

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displacement is expected to provide a good direction for mode selection for

anisotropic layered media also.

5. Understanding and coining the term ‘Mode Pairs’: The observation of the existence

and unique displacement characteristics for modes in a bonded joint that envelope

or appear near to modes in one of the adherends is not found elsewhere in the

literature. The coining of the term ‘Mode Pairs’ was done in this thesis to describe

such modes in the bonded joint. The near equal energy partitioning among the mode

pairs for a phase matching input was checked using a SAFE-NME hybrid model.

6. Guided wave mode behavior at a transition: The development of the phase and

group velocity matching approach with wavestructure matching coefficient is a

unique and a big step in understanding and analyzing guided wave mode conversion

at discrete transitions in waveguides. The method has reciprocity built-in and also

agrees well with the energy based hybrid SAFE-NME formulation for guided wave

scattering at a waveguide transition. The factors like efficient energy transfer using

group and phase velocity matching modes is very intuitive and expressible in terms

of fundamental energy conservation laws. The wavestructure matching based study

is concept driven and computationally in-expensive and yet very powerful. This can

be extended to study different transitions such as pipe joints or couplers.

7. Combination criteria for finding inputs forming sensitive modes at a transition:

Combining the wavestructure matching coefficient with the interface sensitive in-

plane displacement map provided a map that captures the correct mode input that

convert to interface sensitive modes within a transition.

The contributions in this thesis can be grouped into the following categories:

1. Numerical experimentation and signal processing: Phased addition, mode

matching filter and wavestructure decomposition

2. Modeling: Guided wave behavior at a joint transition using phase, group and

wavestructure matching modes, energy transmitting modes, interface sensitive

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modes, semi-analytical method based proof for the guided wave behavior at

waveguide transitions

3. Conceptual: The immediate mode formation, existence of mode pairs, higher

possibility of generation of phase velocity vs. frequency matching modes on mode

conversion at a transition and the tendency of larger energy transfer by group

velocity matching modes.

4. Experimental parameter determination: The use of interface sensitive modes for

repair patch and combination to determine high quality test parameters.

6.3 Future research

Future researches that can employ the contributions from this thesis and also

extend this work are briefly explained in the following sections.

6.3.1 Inspection of composite skin-stringer joints

The systematic method for handling waveguide transitions and determining

interface sensitive modes developed in Chapter 5 can be directly applied to composite

joints.

The wavestructure matching coefficient based model or SAFE-NME approach for

handling mode conversion and scattering at waveguide transitions established in Chapter

5 with examples of isotropic joints can be applied to the problem of composite joint

inspection. A sample problem to determine mode reflection from a composite joint

transition is presented in Puthillath et al. [2007].

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Determination of the material properties of the composite material is a necessary

initial step. Anisotropic nature of composite material layup results in more than two

independent elastic stiffness values. There are several methods for characterizing

anisotropic laminates using ultrasonic measurement approaches. Measurement of

ultrasonic bulk wave velocity using cube cutting method [Rose 1999] is a direct but

destructive approach that requires material cubes cut along different directions. Leaky

Lamb waves can also be used in characterizing the elastic stiffness values of a composite

laminate [Puthillath et al. 2010 c].

Once the material elastic stiffness values for the anisotropic laminate are known,

the dispersion curves and wavestructures for different directions of wave propagation can

be calculated using the standard approach explained in Chapter 2. Partial wave approach

and SAFE are two standard approaches that can be applied for calculation of the

dispersion curves. Further analysis to determine mode conversion, scattering and the

determination of the interface sensitive modes can be performed by following the

procedure developed in Chapter 5.

6.3.2 Modeling waveguides with a continuous transition

In very general terms it is expected that the approach developed here and the

conceptual understanding gained in guided wave energy coupling can be employed in

other potential problems like bends in pipe, metal composite constant stiffness joints etc.

The schematic of a constant stiffness metal-composite joint is shown in Figure 6-1. The

complex joint in Figure 6-1 is the case of a combination of several discrete transitions.

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In the case of continuous transition waveguide, e.g. a joint between a tapered

adherend and a constant thickness adherend, the continuous slope involved can

approximated by several small length constant thickness adherends making it similar to

Figure. Further analysis will then follow from the work in Chapter 5.

6.3.3 Inspection of skin-stringer joints using sensors mounted on stringer surface

The approach to stringer joint inspection reported in Chapter 5 of this thesis is

based on the concept of mounting sensors on the surface of the skin, on either side of the

stringer. A plane strain analysis that accounts for the transition, developed in this thesis,

can handle this situation theoretically. In a practical implementation, line of sight sensors

(for isotropic adherends) transmits and receives guided wave modes across the stringer.

This approach still requires either a scan or a beam steering approach to cover the

entire joint. A scan requires a physical movement of sensors or switching from many

bonded sensors. The beam steering approach will need an expensive hardware with time

delay controls.

Figure 6-1: A constant stiffness metal-composite joint.

Constant stiffness joint

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In this context, a new inspection approach using sensors mounted directly on top

of the stringer joint is introduced with the schematic in Figure 6-2.

With the help of 3D modeling approach, the propagation of waves along the

length of the joint (x2 direction in waveguide B) and its attenuation because of leakage

into the skin region (waveguides AL and AR) can be studied. Specifically, the waveguides

AL and AR can be modeled with absorbing boundary conditions by introducing material

damping that increases along the ± x1 direction on either side of the waveguide B away

from the joint. Castaings and Lowe [2008] show the implementation of SAFE approach

with semi-infinite conditions for a similar geometry.

It is expected that the modes will a higher energy transfer when a defect is present

in the bondline.

Figure 6-2: Schematic of the inspection of a stringer joint using sensors mounted on top of the joint.

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6.3.4 Sensor design for optimal mode generation

This thesis lays the foundation for understanding the interaction of guided wave

modes with a waveguide transition and the selection of modes with sensitivity to

interface condition in an adhesive joint. The design of sensor mounting for the generation

of the desired interface sensitive mode in the experiments reported in this thesis is only

done for concept demonstration purposes. A practical field implementation will require

the use of bonded sensors which provide mode control. The work on PWAS by

Giurgiutiu [2007] and group, the work on PVDF based transducers by Monkhouse et al.

[1998], and the work on piezocomposite sensors by Gachagan et al. [2005] and group

provide the necessary background on sensor design and implementation.

6.3.5 Nonlinear ultrasonic waves for damage detection in bonded joints

The use of bulk and guided waves for inspection of damage in adhesively bonded

joints will provide an opportunity for improved sensitivity to defects and possibly an

early detection tool to detect progressive degradation in the joint due to aging related

issues.

The present work on mode transfer across a transition and mode selection for

defect sensitivity and can be combined with the nonlinear Lamb wave inspection

approach by Bermes et. al. [2007] to form a nonlinear inspection method for inspection of

bonded waveguides with transition.

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6.3.6 Wave propagation across waveguides with multiple branches

Trusses used in bridges and roof supports are mechanical load bearing structures.

A photograph of the gusset plate region from the failed Minnesota bridge is shown in

Figure 6-3. The gusset plate buckled under load and resulted in the collapse of the bridge.

Bridges are an ideal candidate for the implementation of ultrasonic guided wave

based inspection. For the analytical study, a bridge joint can be considered as a

waveguide with multiple branches. Using the SAFE-NME model or the wavestructure

matching coefficient based model developed in Chapter 5, it is possible to understand the

behavior of guided wave modes in waveguides with multiple bolted or welded

connections that bear load. Understanding the scattering from the rivet holes will be

challenge in this problem.

Figure 6-3: A portion of the bridge truss from the failed Minnesota bridge. The gussetplate encircled is buckled under load. The joint formed by the elements of the truss canbe considered as a complex waveguide with multiple connections for a guided waveanalysis. [www.minnesota.publicradio.org/display/web/2008/11/12/ntsb_bridge]

221

In order to handle the guided wave interaction with a complex joint, the hybrid

BEM-NME approach reported in Galan and Abascal [2003, 2005 a] can also be applied.

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Appendix A

Two-Dimensional Fast Fourier Transform (2DFFT)

Two-dimensional Fast Fourier Transform is the computationally efficient version

of the Discrete Fourier transform in two dimensions and can be written as given in

Equation A.1

where u is a digitized distribution of a physical quantity (like displacement, stress etc.) as

a function of time (t) and space (x). The term MN1 is the normalization constant.

],[ ωkU is obtained on transforming the data from the space-time coordinate to the wave

number (k)-frequency (ω) coordinate.

The inverse transform is obtained by interchanging the u and U terms, changing

the signs on the exponential function and appropriate changes to the indices and constants

in the above expression.

The property of the exponential function makes ],[ ωkU have complex conjugate

values in the diagonal quadrants i.e. quadrants I and III and quadrants II and IV

(Figure A-1). Hence the value of U at ),( ωk is a complex conjugate of the value

at ),( ω−− k .

[ ]

+−−

=

−

=∑ ∑= N

nMmkjM

m

N

nenmu

MNkU

ωπω

21

0

1

0,1],[ (A.1)

232

Figure A-1: The ω−k space with the four quadrants. Quadrants I and III and, quadrantsII and IV are related.

ω

k

Appendix B

Non-technical abstract

In aircraft and other mechanical load bearing structures, the use of glued joints is

becoming a popular alternate to riveted and welded joints. Glued or adhesively bonded

joint provides stronger connections with higher degrees of uniformity in the load

distribution. These joints are susceptible to weakness especially at the interface between

the adhesive and the material being joined because of manufacturing issues or operational

issues. Detecting the weakness in adhesive joints becomes a very critical for ensuring

safety of the structure.

Using ultrasonic waves - i.e. mechanical disturbances propagating at frequencies

above 20,000 cycles/s, it is possible to interrogate mechanical structures non-

destructively. Ultrasonic waves propagating through thin plate-like structures under

certain conditions show behavior similar to light rays in an optical fiber cable. The plate

structure guides the wave energy. These waves are called ultrasonic guided waves.

Ultrasonic guided wave propagation studies through two adhesive joints found in aircraft

- adhesive repair patches and adhesive skin stringer joints with the aim of determining the

interfacial weakness in the bonding are reported in this thesis.

Ultrasonic guided waves propagating along a structure have a displacement

variation across the thickness of the structure - termed as wavestructure. Depending on

the shape of the wavestructure, the guided waves are grouped into modes.

234

The repair patches are glued life extending patches attached at damage locations

in aircraft like C130, F16 etc. to strengthen the weak zones. An approach to select guided

wave inspection parameters for successfully detecting interfacial weakness in adhesive

joints is provided in this thesis.

The skin-stringer joints provide mechanical stiffening to the fuselage skin, wings

on aircraft. The challenge faced in this case is to determine the conversion of ultrasonic

guided wave modes because of the geometry change in a joint such that it is sensitive to

the interfacial conditions in bonding. A concept driven qualitative model for determining

conversion of a guided wave mode into other possible modes is developed and

successfully verified with an energy based numerical model in this thesis. Combining the

conversion of guided wave modes and their sensitivity to interface, a set of experimental

configurations effective in inspection is determined and successfully verified using

experiments.

VITA

Padmakumar Puthillath

EDUCATION PhD, Engineering Science Mechanics, 2010, The Pennsylvania State University, University Park, PA, USA M.S., Mechanical Engineering, 2005, Indian Institute of Technology Madras, Chennai, India B.Tech., Mechanical Engineering, 2002, College of Engineering, University of Calicut, Palakkad, India

AWARDS Thomas and June Beaver Award, Penn State, Spring 2010 Sabih and Güler Hayek Graduate Scholarship in ESM, Penn State, Fall 2009 Student paper competition in ESM Today, Third prize, Spring 2008

PUBLICATIONS 1. P. Puthillath and J. L. Rose. (2010). Ultrasonic guided wave inspection of a Titanium repair patch

bonded to an Aluminum aircraft skin. International Journal of Adhesion and Adhesives 30:566-573. 2. P. Puthillath, C. J. Lissenden, J. L. Rose (2010). Theoretically driven parameter selection for

ultrasonic guided wave inspection of adhesive bonding. 16th US National Congress of Theoretical and Applied Mechanics, June 27-July 2, 2010 at Pennsylvania State University, University Park, PA, USA.

3. P. Puthillath, C. V. Krishnamurthy and K. Balasubramaniam. (2010). Hybrid inversion of elastic moduli of composite plates from ultrasonic transmission spectra using PVDF Plane Wave Sensor. Composites B: Engineering 41:8-16.

4. P. Puthillath, and J. L. Rose. (2010). Aircraft Bond Repair Patch Inspection using Ultrasonic Guided Waves. Review of Progress in Quantitative Nondestructive Evaluation. American Institute of Physics Conference Proceedings 1211: 247-252.

5. P. Puthillath, H. Kannajosyula, C. J. Lissenden and J. L. Rose. (2009). Ultrasonic guided wave inspection of adhesive joints: a parametric study for a step-lap joint. Review of Progress in Quantitative Nondestructive Evaluation American Institute of Physics Conference Proceedings 1096: 1127-1133.

6. H. Kannajosyula, P. Puthillath, C. J. Lissenden and J. L. Rose. (2009). Interface Waves for SHM of Adhesively Bonded Joints. 7th International Workshop on Structural Health Monitoring 2009. Sept. 9-11, 2009 at Stanford University.

7. P. Puthillath, F. Yan, H. Kannajosyula, C. J. Lissenden, J. L. Rose and C. Xu. (2008). Inspection of adhesively bonded joints using ultrasonic guided waves. World Conference of Nondestructive Evaluation Oct. 25-28, 2008, Beijing, China.

8. P. Puthillath, F. Yan, C. J. Lissenden and J. L. Rose. (2008). Ultrasonic guided waves for the inspection of adhesively bonded joints. Review of Progress in Quantitative Non-destructive Evaluation, American Institute of Physics Conference Proceedings 975: 200-206.

9. P. Puthillath, K. Balasubramaniam and C. V. Krishnamurthy. (2006). Determination of transmission spectra using ultrasonic NDE. Transactions of the Indian Institute of Metals, 59: 181-184.

10. G. Swamy, P. Puthillath and K. Balasubramaniam. (2004). An LMS Classifier For Detecting Disbonds Between Metal Composite Interface Using Ultrasonic A-Scan Data. Journal of Non-Destructive Testing and Evaluation 3:35-40.

PROFESSIONAL AFFILIATIONS Student membership in American Society of Mechanical Engineers (ASME), American Society for Nondestructive Testing (ASNT), American Institute of Aeronautics and Astronautics (AIAA) and Institute of Electrical and Electronics Engineers (IEEE).