ULTRASONIC GUIDED WAVES AND WAVE SCATTERING IN

The Pennsylvania State University

The Graduate School

Department of Engineering Science and Mechanics

ULTRASONIC GUIDED WAVES AND WAVE SCATTERING IN

VISCOELASTIC COATED HOLLOW CYLINDERS

A Thesis in

Engineering Science and Mechanics

by

Wei Luo

2005 Wei Luo

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

December 2005

The thesis of Wei Luo was reviewed and approved* by the following:

Joseph L. Rose

Paul Morrow Professor of Engineering Science and Mechanics

Thesis Advisor

Chair of Committee

Bernhard R. Tittmann

Schell Professor of Engineering Science and Mechanics

Clifford J. Lissenden

Associate Professor of Engineering Science and Mechanics

Eduard S. Ventsel

Professor of Engineering Science and Mechanics

Qiming Zhang

Professor of Electrical Engineering

Judith A. Todd

Professor of Engineering Science and Mechanics

P. B. Breneman Department Head Chair

Department of Engineering Science and Mechanics

*Signatures are on file in the Graduate School

iii

ABSTRACT

Over a million miles of piping is used in the USA in almost every industry that

calls for a large scale transportation and distribution of energy or product, like natural gas,

oil, water, etc. Pipeline safety is crucial in that defective pipelines can lead to catastrophic

failure, property damage and high replacement costs. To preserve the integrity of these

pipelines, viscoelastic coatings are widely used on the pipes. However, pipe aging and

exposure to a variety of changing environmental conditions reduces the protection

effectiveness consequently leading to the occurrence of defects. An effective non-

destructive evaluation (NDE) method is needed to provide the current pipeline status to

the pipeline operators for any further decisions on repair or replacement actions.

Ultrasonic guided waves, with a long range propagation capability, are becoming

useful in new solutions for pipeline inspection. It is much more efficient and economical

than other commonly used NDE methods like point-by-point bulk waves and magnetic

flux leakage. Long pipes can be inspected from a simple sensor position. Among the two

methods for the long rang guided wave pipeline inspection, almost decade old

axisymmetric waves and recently developed phased array focusing, the latter presenting

itself with a tremendous improvement in terms of penetration power, detection sensitivity,

and inspection distance. However, guided wave inspection potential in coated pipe has

not yet been studied in detail. Many important questions need answers, like focusing

feasibility in coated pipes, wave scattering possibilities study for effective inspection of

3-D defects, and quantitative evaluation of inspection distances under various coating

conditions. Since a large percentage of the pipelines are covered with viscoelastic

iv

coatings, a thorough study of guided waves in viscoelastically coated pipes is strongly

called for.

In this work, guided wave propagation, scattering and focusing in coated pipes are

studied comprehensively for the first time with numerical, analytical and experimental

methods. A three-dimensional finite element tool utilizing ABAQUS/Explicit was

developed for quantitatively and systematically modeling guided wave behavior in pipes

with different viscoelastic materials. A whole process, from experimental measurement to

theoretical modeling has been established, including in-situ coating property

measurement, transformation of measured properties to model inputs, specific wave

mode generation, and output data processing and analysis. With the help of this new

powerful modeling tool, it is very exciting to find that guided waves can still be focused

very well in a coated pipe for the frequency studied, although there is an amplitude loss

due to the viscoelastic nature of the coating materials. The quantitative evaluation of the

energy increment and the subsequent inspection distance increment from axisymmetric

loading and focusing was studied. Wave scattering from planar and corrosion like defects

were investigated under both axisymmetric and phased array focused loading with both

longitudinal and torsional waves. It was found that axisymmetric waves had a small

possibility of finding small corrosion like defects while focusing had a much higher

chance. Defect sizing potential was also studied based on an observation of the wave

interaction with defects and the mode conversion that occurred thereafter.

v

Moreover, in order to minimize the attenuative effect from the coatings, a

parametric study of coating property effects on wave attenuation was conducted making

use of the attenuation dispersion curves. An improved mathematical root search

algorithm was utilized for highly viscoelastic materials. This lead to a decision on the

best choice of either coating to be used or in the case of existing coatings the best set of

sensor and instrumentation parameters to do the test. Appropriate coating properties,

frequency range, and wave type were recommended for future work in the pipeline

industry. In addition, an experimental method of property measurement for field coating

materials was developed as a means of providing inputs to computer models. It was

found that coating properties had a wide variation suggesting the need of in-situ coating

property measurements for any subsequent modeling work based on field coated pipes.

Finally, a detailed criterion to improve the inspection potential of coated pipes is

recommended.

vi

TABLE OF CONTENT

List of Figures .................................................................................................................... ix

List of Tables .................................................................................................................. xvii

Acknowledgement ......................................................................................................... xviii

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

1.1 Problem Statement .........................................................................................................1

1.2 Literature Review...........................................................................................................5

1.2.1 Guided Waves in Pipes .........................................................................................5

1.2.2 The Viscoelastic Multilayer Pipe..........................................................................8

1.2.3 Numerical Methods...............................................................................................9

1.3 Objectives ....................................................................................................................12

Chapter 2 Guided Waves in Pipes .....................................................................................14

2.1 Introduction..................................................................................................................14

2.2 Guided Wave Theory in Pipes .....................................................................................15

2.2 Dispersion Curves........................................................................................................21

2.3 Wave Structures ...........................................................................................................25

Chapter 3 The Focusing Principle......................................................................................31

3.1 Introduction..................................................................................................................31

3.2 The Normal Mode Expansion Technique ....................................................................32

3.3 Angular Profile for Partial Loading .............................................................................36

3.4 Phased Array Focusing ................................................................................................38

Chapter 4 Finite Element Modeling of Guided Waves......................................................42

4.1 Introduction..................................................................................................................43

4.2 Finite Element Method in Dynamics ...........................................................................43

4.2.1 Basic Theory .......................................................................................................43

4.2.2 ABAQUS Strategy..............................................................................................45

4.3 Guided Wave Propagation Modeling...........................................................................47

vii

4.3.1 Three-dimensional Model Establishment ...........................................................47

4.3.2 NME vs. Boundary Value Problem ....................................................................48

4.3.3 Model Accuracy Consideration ..........................................................................53

4.4 Modeling of Guided Wave Phased Array Focusing ....................................................57

4.5 Summary......................................................................................................................63

Chapter 5 Wave Scattering ................................................................................................65

5.1 Introduction..................................................................................................................65

5.2 Mode Conversion from Two-dimensional Defects......................................................65

5.3 Mode conversion from three-dimensional defects.......................................................73

5.3.1 Planar Defects .....................................................................................................73

5.3.2 Non-planar Defects .............................................................................................79

5.3 Summary......................................................................................................................82

Chapter 6 Finite Element Modeling of Guided Waves in Coated Pipe .............................83

6.1 Introduction..................................................................................................................83

6.2 Damping Induced by Viscoelasticity ...........................................................................83

6.2.1 Rayleigh Damping ..............................................................................................83

6.2.2 Viscoelastic Property Estimation from Acoustic Measurement .........................85

6.3 Wave Propagation in a Coated Pipe.............................................................................89

6.4 Wave Focusing in a Coated Pipe .................................................................................92

6.5 Summary and Rules of Thumb ....................................................................................99

Chapter 7 The Coating Property Effect on Wave Attenuation ........................................100

7.1 Introduction................................................................................................................100

7.2 Guided Waves in Viscoelastic Coated Pipes .............................................................100

7.2.1 Dispersion Curves in Viscoelastic Multilayer ..................................................101

7.2.2 Dispersion Equation Solution ...........................................................................103

7.2.3 Root Searching Algorithm ................................................................................104

7.3 Coating Property Effect Investigation .......................................................................105

7.3.1 Coating Material -- Mereco 303 Epoxy............................................................106

7.3.2 Coating Material -- Bitumastic 50 ....................................................................117

7.4 Summary and Rules of Thumb ..................................................................................124

viii

Chapter 8 Experiments.....................................................................................................127

8.1 Introduction................................................................................................................127

8.2 Attenuation Experiments on Coated Pipelines in Fields............................................127

8.3 Acoustic Property Measurement of Various Sample Coatings................................. 134

8.4 Guided Wave Experiments in Coated Pipe............................................................... 139

8.5 Summary................................................................................................................... 145

Chapter 9 Concluding Remarks ...................................................................................... 146

9.1 Concluding Remarks................................................................................................. 146

9.2 Contributions............................................................................................................. 149

9.3 Future Directions ...................................................................................................... 151

Reference .........................................................................................................................153

Appendix A Dispersion Equation ....................................................................................160

Appendix B Nontechnical Abstract .................................................................................161

ix

LIST OF FIGURES

Figure 1-1 Coating usage history of pipeline industry in North America [ Papavinasam

2005] ........................................................................................................................... 2

Figure 1-2 Field coated pipelines showing differences in field coating conditions ........... 3

Figure 2-1 Cylindrical coordinates of a hollow cylinder ................................................. 14

Figure 2-2 Phase velocity (a) and group velocity (b) dispersion curves of axisymmetric

longitudinal waves ( L(0,m) ) in a 10 inch schedule 40 steel pipe (outer radius =

136.5 mm, thickness = 9.27 mm). Note that the ‘0’ denotes the zero circumferential

order and ‘m’ denotes the mode family order........................................................... 23

Figure 2-3 Phase velocity (a) and group velocity (b) dispersion curve of axisymmetric

torsional waves in a 10 inch schedule 40 steel pipe (outer radius = 136.5 mm,

thickness = 9.27 mm). Note that the ‘0’ denotes the zero circumferential order and

‘m’ denotes the mode family order. .......................................................................... 24

Figure 2-4 Phase velocity dispersion curves of axisymmetric and flexural modes in a 10

inch schedule 40 steel pipe (outer radius = 136.5 mm, thickness = 9.27 mm). ....... 25

Figure 2-5 Wave field distribution in the circumferential direction (from 0 to 360 degree)

for wave modes with zero to the 5th

order, explicitly expressed by the sinusoid

functions in equation (2.11) ...................................................................................... 27

Figure 2-6 Wave structures in the radial direction( )(rU r, )(rUθ , and )(rU z

in equation

(2.11) ) for axisymmetric T(0,1), L(0,1) and L(0,2) as well as the 1st order non-

axisymmetric modes at 40 kHz for a 10 inch schedule 40 steel pipe, showing the

displacement field distribution along the pipe wall. Note T(0,1) has only the )(rUθ

component, L(0,1) and L(0,2) have other two but with )(rU r and )(rU z

dominant,

respectively. For order one, the displacements of the three waves all become 3-

dimensional ............................................................................................................... 28


in equation

(2.11) ) for axisymmetric T(0,1), L(0,1) and L(0,2) as well as the 1st order non-

axisymmetric modes at 100 kHz for a 10 inch schedule 40 steel pipe, showing the

displacement field distribution along the pipe wall. Comparison with Figure 2-6

shows some little variance due to frequency change but are not dramatic............... 29


in equation

(2.11) ) for the L(0,2) family with order from zero to five, at 100 kHz in a 10 inch

x

schedule 40 steel pipe. Note that the percentage of )(rUθ is zero for axisymmetric

L(0,1) and increases as the order becomes higher, while )(rU r and )(rU z

stays

almost the same......................................................................................................... 30

Figure 3-1 Cylindrical coordinates of a hollow cylinder with partial loading of a

transducer with 2L axial length and 02α circumferential coverage angle. .............. 33

Figure 3-2 Amplitude factors for the L(0, 2) and FL(n, 2) modes generated by a 45°

circumferential loading at 100 kHz in a 10 inch schedule 40 steel pipe................... 35

Figure 3-3 Angular profiles of the L(0,2) to FL (9,2) modes generated by a 45°

circumferential loading at 100 kHz in a 10 inch schedule 40 steel pipe, showing the

superposition of wave fields in the circumferential direction for all the generated

modes. Note that the wave field distribution in the circumferential direction changes

with wave propagation distance due to the slightly different phase velocity for each

mode.......................................................................................................................... 37

Figure 3-4 Illustration of a 8-Channel guided wave phased array system........................ 38

Figure 3-5 Angular profiles for a 45° single channel loading and 8-channel focusing with

the 2nd family longitudinal waves, at 50, 100 and 150 kHz, in a 10 inch schedule 40

steel pipe. Note that although angular profiles for a single channel are different at

various frequencies, they can all be focused when using the deconvolution algorithm.

Left column – single channel profile )(θh ; Right column – focused profile using the

calculated amplitude and time delay......................................................................... 41

Figure 4-1 (a) The finite element model for a 10 inch schedule 40 pipe, with mesh size

equal to about half of the wall-thickness; (b) Axial pressure loading at the right end;

the middle plane is used for signal extraction and model analysis ............................48

Figure 4-2 Hanning-windowed tone-burst signals (left column) and their Fourier

Transform (right column) at 50 kHz used as the input loading function for guided

wave model (a) 5 cycles; (b) 10 cycles; (c) 15 cycles ...............................................50

Figure 4-3 Phase and group velocity curves for longitudinal guided waves in a 10 inch

schedule 40 steel pipe. ...............................................................................................51

Figure 4-4 Wave structures of (a) L(0,1) and (b) L(0, 2) modes in a 10” schedule 40 pipe

at 50 kHz, showing the distribution of the axial and radial displacement component

along the pipe wall thickness direction......................................................................52

Figure 4-5 Wave propagation modeling result for the displacement magnitude U, for 50

kHz L(0,2) wave in a 10” schedule 40 steel pipe ......................................................55

xi

Figure 4-6 ABAQUS output of displacement Uz and Ur for nodes 4, 9 and 901 on the

plane with Z = 0.8 m. The measured group velocity was 5.387 mm/ secµ

( theoretical group velocity = 5.345 mm/ secµ )........................................................56

Figure 4-7 Mode decomposition using the normal mode expansion technique, showing

that L [0, 2] was the dominant mode which is consistent with the loading condition

and group velocity measurement. ..............................................................................57

Figure 4-8 Finite element model for a 8-channel phased array focusing in a 10 inch

schedule 40 steel pipe. Loadings are applied with different time delays and

amplitudes to the 8 segments at one end of the pipe..................................................58

Figure 4-9 Eight-channel phased array focusing at zero degrees at a 1.5 meter axial

distance in a 3-meter-long 10 inch schedule 40 pipe at a frequency of 100 kHz,

showing resulting displacement profiles at different focusing steps. Focusing

realizes a significantly higher Energy (5×) compared to the axisymmetric case, and

20× higher compared to partial loading. ....................................................................59

Figure 4-10 Analytical calculation results: angular profile of the displacement field of the

100 kHz L(0,2) mode at a distance of 1.5 m in a 10 inch schedule 40 pipe (a) 45

degree partial loading and (b) 8-segment phase array loading. Note the maximum

displacement amplitude in (b) is about 5.8 times larger than that in (a)....................61

Figure 4-11 FEM numerical calculation results: angular profiles of the displacement field

of the 100 kHz L(0,2) mode at a distance of 1.5 m in a 10 inch schedule 40 pipe (a)

45 degree partial loading and (b) 8-segment phased array loading. Note that the

maximum displacement amplitude in (b) is also about 5.8 times larger than that in

(a). ..............................................................................................................................61

Figure 4-12 Axial profile at zero degrees of the 100 kHz L(0,2) mode in a 10 inch

schedule 40 pipe with 8-segment phased array loading.............................................62

Figure 5-1 Finite element model of a 10 inch schedule 40 pipe with a 5-mm-wide and

30%-through-wall circumferential notch. ..................................................................66

Figure 5-2 Reflected and transmitted wave fields showing the displacement magnitude

compared with the incident wave fields. Note a new mode L(0,1) is converted due to

the existence of the circumferential notch. ................................................................67

Figure 5-3 ABAQUS output of displacement Uz and Ur for nodes 5,8, and 153 on the

reflection analysis plane with Z = 1.1 m....................................................................68

Figure 5-4 ABAQUS output of displacement Uz and Ur for nodes 10,13, and 2136 on the

reflection analysis plane with Z = 1.9 m....................................................................69

xii

Figure 5-5 Mode decomposition of the output result using the normal mode expansion

technique, showing that the L(0, 1) mode was converted in both transmitted and

reflected waves due to the existence of the defect. ....................................................71

Figure 5-6 Mode decomposition of the output result for a series of notches with 10%,

30%, …,90% through wall depths under 50 kHz L(0,2) wave incidence, showing

that the transmitted and reflected L(0, 2) modes have a monotonic relationship with

defect depth, but not true for the converted L(0,1) modes. This suggests the incident

mode L(0,2) is more reliable for defect sizing...........................................................71

Figure 5-7 Transmission and reflection ratio of T(0,1) wave for a series of notches with

10%, 30%, …,90% through wall depths under 50 kHz T(0,1) wave incidence,

showing that the transmitted and reflected T(0, 1) modes have a monotonic

relationship with defect depth. Note there is no mode conversion to other modes,

suggesting the incident mode T(0,1) is simpler for defect sizing. .............................72

Figure 5-8 A finite element model for a saw cut with 50% wall depth, CSA 3.53%, in a

10 inch schedule 40 pipe............................................................................................74

Figure 5-9Wave scattering from a 50% saw cut in a 10 inch schedule 40 pipe (3.53%

CSA) for 100 kHz L(0,2) wave, for axisymmetric loading, (a) in the beginning, (b)

after a while................................................................................................................75

Figure 5-10 signal at the point of zero degrees and 0.6 meters away from the defect on the

reflection path for axisymmetric loading with a 100 kHz L(0,2) wave. The group

velocity comparison shows that the reflected waves are the flexural F(n,2) modes

and the converted flexural F(n,1) mode. ....................................................................76

Figure 5-11Wave scattering from a 50% saw cut in a 10 inch schedule 40 pipe (3.53%

CSA) for the 100 kHz L(0,2) mode, for 8-segment phased array loading, (a) in the

beginning, (b) after a while. .......................................................................................77

Figure 5-12 Signals of (a) zU (b) displacement magnitude at the point 0.6 meter away

from the notch on the reflection path for both axisymmetric and phased array

loading with the 100kHz L(0,2) wave. Note both focusing and axisymmetric waves

can find the defect in this example but focusing increases the reflection amplitude by

about 3 times in this case. ..........................................................................................78

Figure 5-13 A corrosion model with 70% wall depth, width CSA 3.8%, and length CSA

6.4% in a 10 inch schedule 40 pipe............................................................................80

Figure 5-14 Wave scattering signals of (a) zU (b) displacement magnitude at the point 0.6

meters away from the corrosion on the reflection path for axisymmetric and phased

array loading with the 100 kHz L(0,2) input wave, showing that focusing has strong

reflections from corrosion while the defect signals for axisymmetric waves are too

weak to see. ................................................................................................................81

xiii

Figure 6-1 ABAQUS model for guided wave propagation analysis in a 10 inch schedule

40 pipe coated with viscoelastic materials.................................................................90

Figure 6-2 Displacement magnitude vs. propagation distance for the L(0, 2) mode at 30

kHz propagation in a 10 inch schedule 40 pipe. ........................................................91

Figure 6-3 Axial profile for the 100 kHz L(0, 2) wave with axisymmetric and phased

array loading, for no coating, 3 mm epoxy and 3 mm bitumen. Note that focusing

increase the magnitude significantly at the focal point..............................................94

Figure 6-4 Angular Profile for at the focal point for a bare pipe and a pipe coated with 3

mm bitumen for 100 kHz longitudinal waves, showing that coating introduces some

attenuation but that the profile shape stays the same. ................................................95

Figure 6-5 Angular profile for the 100 kHz L(0, 2) wave with axisymmetric and phased

array loading (focal distance equal to 3 m), for no coating, 3 mm epoxy and 3 mm

bitumen. Note that focusing increase the magnitude significantly at the focal point.95





array loading (focal distance equal to 1.5 m), for no coating, 3 mm epoxy and 3 mm

bitumen. Note that focusing increase the magnitude significantly at the focal point.97


mm bitumen for 50kHz longitudinal waves, showing that coating introduces some


Figure 6-9 Focusing experiments on (a) a bare pipe and (b) a tar-coated pipe with a 6mm

deep, 3.26% CSA, 63% through-wall saw cut in both, showing focusing in tar-

coated pipe with torsional waves. ..............................................................................98

Figure 7-1 Cylindrical coordinates of a hollow viscoelastic coated cylinder and

dimensions. ..............................................................................................................101

Figure 7-2 Flow chart for developing dispersion curves of a multilayered structure ......102

Figure 7-3 The curve tracking process to predict the starting point of the searching at the

next frequency by an extrapolation of previous roots..............................................105

Figure 7-4 Attenuation and phase velocity dispersion curves as functions of densities in a

10’’ schedule 40 steel pipe with 1mm Mereco 303 Epoxy coating. Note that the

xiv

original density is 1.08 g/cm3

, other densities values are used for the parametric

study.........................................................................................................................107

Figure 7-5 Attenuation vs. density curves for selected 3 frequencies 0.05MHz, 0,1MHz

and 0.15MHz, showing that attenuation increases almost linearly with an increase of

density. Note that the change is more dramatic for higher frequency. Torsional

mode has a larger attenuation than longitudinal modes...........................................108

Figure 7-7 Attenuation Vs. the longitudinal wave attenuation constant ( ωα /1 ), for 3

frequencies, 0.05MHz, 0.1MHz and 0.15MHz, showing that the attenuation

increases linearly with ωα /1 , and the increase of L(0,2) is less sensitive to the

attenuation constant than L(0,1). The change of ωα /1 has no effect on T(0,1). ....111

Figure 7-8 Attenuation and phase velocity dispersion curves as functions of ωα /2 in a

10’’ schedule 40 steel pipe with 1mm Mereco 303 Epoxy coating. ........................113

Figure 7-9 Attenuation Vs. the shear wave attenuation constant ( ωα /2 ), for 3


increases linearly with ωα /2 , and the higher frequency, the larger the slope, which

means more sensitivity.............................................................................................114

Figure 7-10 Attenuation and phase velocity dispersion curves as functions of lc and tc in

a 10’’ schedule 40 steel pipe with 1mm Mereco 303 Epoxy coating. .....................115

Figure 7-11 Attenuation vs. longitudinal velocity, for 3 frequencies, 0.05MHz, 0.1MHz

and 0.15MHz, showing that the attenuation increases monotonically with velocities,

although not linearly. ...............................................................................................116

Figure 7-12 Attenuation and phase velocity dispersion curves as functions of densities in

a 10’’ schedule 40 steel pipe with 1mm Bitumastic 50 coating. .............................118

Figure 7-13 Attenuation vs. density curves for selected 3 frequencies 0.05MHz, 0,1MHz

and 0.15MHz. Note: similarly with epoxy, the results show that attenuation

increases almost linearly with an increase of the density, and again attenuation is

more sensitive to high frequency. ............................................................................119

Figure 7-14 Attenuation and phase velocity dispersion curves as functions of ωα /1 in a

10’’ schedule 40 steel pipe with 1mm Bitumastic 50 coating. ................................120

Figure 7-15 Attenuation Vs. the longitudenal wave attenuation constant ( ωα /1), for 3


increases nearly linearly with ωα /1, and the attenuation for 0.05MHz is very small

and almost kept the same as ωα /1 changes. L(0,2) is less sensitive than L(0,1). ..121

xv

Figure 7-16 Attenuation and phase velocity dispersion curves as functions of ωα /2in a

10’’ schedule 40 steel pipe with 1mm Bitumastic 50 coating. ................................122

Figure 7-17 Attenuation Vs. longitudinal wave attenuation constant ( ωα /2), for 3


increases monotonically with ωα /2 for 0.05 MHz and 0.1 MHz but not for 0.15

MHz. ........................................................................................................................123

Figure 8-1 Field coated pipelines showing differences in field coating conditions ........128

Figure 8-2 Experimental schematic diagram for coating attenuation measurement with a

normal beam transducer .......................................................................................... 129

Figure 8-3 Results of attenuation experiments using a longitudinal normal beam

transducer on a 30” Schedule 10 pipe coated with 15-year-old bitumen tape: (a)

transducer in the pulse-echo mode; (b) a waveform at 2 MHz with a exponential

curve fit to the echo peaks, where the exponential curve gives the attenuation

constant )(ωα ; (c) attenuation constant vs. frequency, showing that the attenuation

constant is a quasi-linear function of frequency. .................................................... 129

Figure 8-4 Attenuation constant vs. frequency curve of shear waves on the same coated

pipe in Figure 8-3.................................................................................................... 131

Figure 8-5 (a) Normal beam incidence with a shear transducer on a new patched area

with a new bitumen tape; (b) Wave signals showing very low reflected waves due to

the new bonding condition incapable for shear wave transmission........................ 131

Figure 8-6 Experiments using a normal beam longitudinal transducer on a 24’’ schedule

10 pipeline (6.35mm wall thickness) with 3 mm fibered tar coating: (a) coated and

uncoated area; (b) 2 MHz signals with the transducer in pulse-echo mode coupled on

steel surface directly; (c) 2 MHz signals with the transducer in pulse-echo mode on

coating surface directly, showing wave energy can not penetrate the coating due to

the poor bonding. .................................................................................................... 132

Figure 8-7 Guided wave experiments for a 24” schedule 10 pipeline with 3 mm fibrous

tar coating: (a) pitch-catch experiment setup with tar coating in the middle; (b) 540

kHz wave signals for pitch-catch mode with transducers 22 mm apart on steel

surface directly; (c) 540 kHz wave signals for pitch-catch mode with transducers 22

mm apart on steel surface, with tar coating in the middle as shown in (a), showing

some wave energy was attenuated but the major part of the waveform still exists.134

Figure 8-8 Different pipeline coating samples with various thicknesses: (1)-(4) Samples

collected from field studies; (5)-(6) New coated samples. ..................................... 136

Figure 8-9 Through Transmission configuration using a 150 kHz longitudinal transducer

................................................................................................................................. 137

xvi

Figure 8-10 (a) Longitudinal velocity vs. frequency curve for different coating samples;

(b) longitudinal wave attenuation vs. frequency curve showing the attenuation

increases slightly with the frequency. ..................................................................... 137

Figure 8-11 (a) Shear wave velocity vs. frequency curve for different coating samples; (b)

shear wave attenuation vs. frequency curve showing that the attenuation increases

slightly with the frequency...................................................................................... 138

Figure 8-12 Angular profile measurement experiment of a 16” schedule 30 pipe coated

with one ply wax (2 mm in width, 4 feet covered length): (a) Schematic illustration;

(b) transducer array; (c) wax coating. A transducer was used as receiver 10 feet

away from the transmitter. ...................................................................................... 140

Figure 8-13 Comparison of angular profiles for uncoated and wax coated pipe when

using 45 kHz Longitudinal ( L [0, 2] ) focusing at 10 feet at 270 degree, in a 16 inch

Schedule 30 pipe. The coating was one ply, 2 mm wax 4 feet in length. A

piezoelectric angle beam transducer was used as the receiver. .............................. 141

Figure 8-14 Comparison of angular profiles for uncoated and wax coated pipe when

using 45 kHz Torsional ( T[0, 1] ) focusing at 10 feet at 270 degree, in a 16 inch

Schedule 30 pipe. The coating was one ply, 2 mm wax 4 feet in length. A SH EMAT

was used as the receiver. ......................................................................................... 141

Figure 8-15 Attenuation of longitudinal and torsional axisymmetric wave in a 16”

schedule 30 pipe with 28 feet length covered with a ply of wax coating ( 2 feet in

length, 2 mm in thickness). The attenuation values were measured by comparing the

back echo amplitudes for coated pipe and bare pipe. Note torsional waves have

much larger attenuations than longitudinal waves over the frequency range used. 142

Figure 8-16 Non-planar and planar defects for an experimental wave scattering study in a

16’’ schedule 30 pipe: (a) 90% deep corrosion like defect; (b) 63% deep saw cut.143

Figure 8-17 Reflected waveform from a corrosion (90% depth) 11 feet away from the

transducer array for 40 kHz torsional wave in a 16 inch schedule 40 pipe coated with

2 mm wax, using 4-channel phased array focusing. ............................................... 144

Figure 8-18 Reflected waveform from a saw cut (63% depth) 21 feet away from the

transducer array for 40 kHz torsional wave in a 16 inch schedule 40 pipe coated with

2 mm wax, using 4-channel phased array focusing. ............................................... 144

xvii

LIST OF TABLES

Table 2-1 Bessel function types used at different frequency intervals. ............................ 19

Table 4-1 Signal amplitude and time delay for focusing at zero degrees at a 1.5 meter

distance for the 100 kHz L(0,2) mode in a 10 inch schedule 40 pipe........................59

Table 6-1 Elastic and viscoelastic material properties.......................................................88

Table 6-2 Calculated material damping properties............................................................88

Table 6-3 Attenuation introduced by 3 mm epoxy and bitumastic coating.......................91

Table 6-4 Amplitude gain by focusing with 100 kHz longitudinal waves at a focal point

at a distance of 1.5 meters..........................................................................................94


at a distance of 3 meters.............................................................................................96

Table 6-6 Amplitude gain by focusing with 50 kHz longitudinal waves at a focal point at

a distance of 1.5 meters..............................................................................................97

Table 8-1 Densities of coating samples ..........................................................................138

xviii

ACKNOWLEDGEMENTS

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

whom I learned not only the knowledge, the methodology, but also the philosophy to be a

Ph.D. His education will influence and benefit me forever in my future career and life.

Thanks are given to FBS, Inc., PA, Plant Integrity, Inc. UK, and the Department

of Transportation, USA, for their technical and financial supports of the thesis work.

Throughout this endeavor, many colleagues in our lab and friends gave me lots of help

either on experiments, computations or suggestions. Thanks a lot for all of their support.

I also appreciate very much all of the committee members for their kind suggestions and

corrections of my Ph.D. thesis.

Finally, I would like to thank my wife, Yuan, and my parents for their deep love,

care and support over the years.

1

Chapter 1 Introduction

1.1 Problem Statement

Over a million miles of piping is used in the USA for the transportation and

distribution of energy, such as petroleum products, gas, and as conduits for electrical

power. When pipelines are aging, the inspection and monitoring becomes indispensable

because of the high cost of replacement. To preserve the integrity and safety of these

pipelines, a large percentage of them are coated and/or encased and buried underground.

Unfortunately, environmental conditions, aging, and excavation accidents can

compromise the effectiveness of these protective measures. Thus periodic or as necessary

inspections using NDE methods such as ultrasonics are used to evaluate the current in-

situ state of a pipe in terms of its fitness for continued service.

Ultrasonic guided waves, because of their long range inspection ability, are being

used more and more as a very efficient and economical non-destructive evaluation (NDE)

method for pipeline inspection [Rose 2003]. A typical scenario of long range guided

wave inspection is to generate guided waves from one single transducer position, which

will propagate over long distances and then impinge onto any possible defects with the

occurrence of wave scattering. The inspection strategy is to acquire the possible reflected

waves from defects and then to analyze the waves for defect detection, locating, sizing

and characterization. Current long range guided wave techniques for pipeline inspection

include axisymmetric waves, non-axisymmetric waves with partial loading and phased

2

array focusing. Compared with the other two techniques, the focusing technique can

increase energy impingement, locate defects, and greatly enhance the inspection

sensitivity and propagation distance of guided waves. The inspection distance is an

important parameter of great interests to the pipeline industry. The huge mileage of

piping necessarily imposes a high cost factor for inspection, principally arising from

excavation costs and time at the site. The number of excavations is dictated by the range

of the inspection method used. Guided wave phased array focusing, with a tremendous

potential of increasing inspection resolution and distance, reduces the number of

excavations necessary, thus consequently reducing inspection costs.

Viscoelastic coatings, such as bitumen and epoxy, are commonly used for

protection against corrosion in the pipeline industry, thus making the pipeline become a

multilayer structure. Figure 1-1 shows the coating usage history in the pipeline industry

in North America [Papavinasam 2005]. It is seen that coatings have been used in the

0

1

2

3

4

5

6

7

8

1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Year

Coal Tar

Wax

Asphalt

Polyethelene Tape

Extruded Polyolefins

3-Layer

Composite

Fusion Bonded Epoxy

Figure 1-1 Coating usage history of pipeline industry in North America [ Papavinasam 2005]

3

Figure 1-2 Field coated pipelines showing differences in field coating conditions

pipeline industry since the 1930’s and the coating type depends on the date of the coating

application. The presence of coating results in changes of guided wave propagation

characteristics. The viscoelastic nature of coating materials leads to significant

attenuation, consequently reducing the guided wave inspection distance. Figure 1-2

shows pipes coated with different materials, for example, 15-year-old bitumen tape,

epoxy, fibrous coal tar, and bitumen tape. Because of a variation of coating materials and

the complexity of the wave mechanics in a viscoelastic coated multilayered structure,

many aspects and questions on guided wave inspection in coated pipe are still untouched

(a) 15-years-old bitumen tape on a 30-inch pipe (b) Epoxy on a 24-inch pipe

(c) Fibrous coal tar coating on a 24-inch pipe (d) new-bonded bitumen tape on a 10-inch pipe

4

and remain very challenging, including such areas of study as follows:

• Whether axisymmetric waves are adequate for long-rang inspection of defects

without using focusing in a bare pipe and a coated pipe;

• Whether guided wave phased array focusing is still feasible and realizable in a

viscoelastic multilayered pipe;

• Can we provide quantitative estimations of the coating effect on guided wave

inspection ability;

• Can we provide quantitative studies of wave scattering from 3-dimenasional

defects with both axisymmetric waves and focusing for defect sizing and

characterization;

• What optimal coating material should be used in the pipeline industry in order to

have a minimum effect on guided wave inspection ability.

Due to a limitation of experiments because of costs, theoretical and numerical

modeling studies are strongly called for as powerful tools for studying the above aspects

and hence exploring the potential of long range guided wave pipeline inspection.

Modeling is essential considering the many variables that exist. It would be a virtually

impossible task to perform experimental studies to determine the best possible modes,

frequencies, and the focusing parameters of loading length, number of focal spots and so

on. Modeling can bring us closer to the optimal parameter choices for an inspection.

The work presented here is focused on guided wave propagation, scattering and

5

phased array focusing in a viscoelastic coated pipe. A literature survey is given next, and

then the objective of this study is presented.

1.2 Literature Review

1.2.1 Guided Waves in Pipes

Although the first study of guided waves was more than one century ago when

Rayleigh theoretically studied the surface wave, the ultrasonic guided wave technique has

not been applied and developed thoroughly until the middle 1980’s. With the explorations

of many pioneers, the guided wave technique has shown great potential in the field of

NDE because of its long-range testing ability and efficiency compared with the

traditional point-by-point bulk wave methods [Rose 1999]. Guided waves can propagate

in various bounded structures such as plates, cylinders, rods, multi-layers, rails, and

therefore result in different wave behaviors. Some well-known guided waves include

Lamb wave in a plate-like structure, Rayleigh wave in a half space, a Stoneley wave

between the interface of two solid strcutures, etc. The theory of different guided waves

can be found in the work of many pioneers [Viktorov 1967] [Miklowitz 1978]

[Achenbach 1984] [Auld 1990] [Graff 1991] [Nayfeh 1995] [Rose 1999]. Rose gives a

very comprehensive review on the historical development and applications of guided

waves [Rose 2002][Rose 2002].

Guided waves in a hollow cylinder (pipeline) can be divided into two groups. One

is in the circumferential direction, which includes the circumferential Lamb type wave

6

and the circumferential shear horizontal (SH) waves. The first study of circumferential

waves was presented by Cook and Valkenburg [1954] who observed that surface waves

could propagate on an annular surface. Liu [1998] studied the circumferential Lamb-type

wave in a hollow cylinder and derived the wave propagation formulation. Zhao [2004]

and Gridin [2003] studied the dispersion curves of circumferential shear horizontal waves.

Many applications of circumferential guided waves can be found in the reference [Kley

1999][Valle 1999][Qu 2000][Luo 2004].

The other group of guided waves in a hollow cylinder is in the axial direction,

including longitudinal and torsional waves with both axisymmetric and non-

axisymmetric modes which are also called flexural modes historically in the field of

ultrasonics. The main advantage of guided waves in the axial direction is the long range

inspection capability compared with the circumferential guided waves whose propagation

is limited to circumferential geometry. Therefore, guided waves in the axial direction are

the major interests of this study.

Axisymmetric waves were first studied by Ghosh [1923] who presented the

mathematical dispersion equation of longitudinal motion in a hollow cylinder like

structure. It was Gazis who first presented a full solution of all the axisymmetric and

non-axisymmetric wave modes in a hollow cylinder considering a 3-dimensional wave

field in the radial, angular and axial directions [Gazis 1959]. This classic pioneering

study has turned out to be the foundation of the current research in the axial-direction

guided waves in pipelines. Due to their relative simplicity compared to flexural modes,

7

axisymmetric waves were investigated further by many researchers [Zemanek 1972][Silk

1979]. Later on, several experimental methods for axisymmetric wave excitation were

also developed and presented, such as the piezoelectric transducer array [Alleyne and

Cawley 1996], the comb transducer [Quarry and Rose 1999], and the magnetostrictive

sensor [Kwun 1994].

However, quite often, the use of axisymmetric waves is found to be limited. For

example, non-axisymmetric waves will be generated when axisymmetric waves impinge

onto a 3-dimensional defect or a geometry change like an elbow occurs. Moreover, in

many cases, the inspected structure has only limited access for transducer placement

which makes axisymmetric loading impossible, calling for a study of non-axisymmetric

waves generated by partial loading. Ditri and Rose [1992] first reported the calculation

of amplitude factors for all of the excited modes corresponding to a specific partial

loading condition utilizing the normal mode expansion (NME) technique. Li and Rose

[2001a] investigated the field distribution of non-axisymmetric longitudinal waves first

and then came up with an angular profile (overall field distribution for a certain partial

source loading) taking into account the amplitude factors of every excited mode

calculated with Ditri’s method. The angular profile is tunable and therefore can be

focused at an expected distance and circumferential angle by using a phased array

concept and a deconvolution algorithm for calculating the time delays and weights

applied on each array channel [Li and Rose 2001b]. Thereafter, the guided wave phased

array mechanics was expanded further to torsional waves by Sun [2003].

8

1.2.2 The Viscoelastic Multilayer Pipe

All the studies shown above on axial-direction guided waves in pipes are all based

on an elastic single-layer bare pipe. Wave mechanics in a viscoelastic multilayered pipe

is much more complicated. According to the correspondence principle [Christensen

1981], for the time harmonic wave motion, a viscoelastic problem can be solved by

replacing the complex viscoelastic moduli into the solution of the elastic problem.

Therefore, the problem in this work could be simplified to find the dispersion equation of

an elastic multi-layered hollow cylinder followed by the complex moduli replacement.

Wave mechanics in a multilayer wave guide has been studied by many researchers. Lowe

[1995] gave a comprehensive summary and review on ultrasonic waves in multilayered

media. Basically, there are two methods for solving the multi-layer problem: one is the

transfer matrix method which also called Thomson-Haskell method [Thomson

1950][ Haskell 1953]. The other is the global matrix method presented first by Knopoff

[1964]. The basic strategy is to develop the displacement and stress expressions of each

layer first and then to construct and solve a global matrix by applying boundary

conditions and interface continuity conditions. The Global Matrix method is easier to

implement while it is more computation consuming.

Although the multilayer problem has been studied by many investigators, most of

the research is focused on elastic and plate like structures. The difficulty of solving a

viscoelastic multilayer problem is the complex root search in a 3-dimensional space

rather than the 2-dimensional space searching for the elastic case. Xu [2004] presented an

9

algorithm to find propagating modes with the complex wave number for viscoelastic

composite materials. Simonetti [2003] studied the Lamb and SH waves in a plate coated

with a thick and highly attenuative material and also carried out a measurement of the

solid viscoelastic material property.

Axial-direction guided waves in a viscoelastic multilayered pipeline, was first

studied by Barshinger [2004] utilizing the global matrix method. However, due to the use

of a simplified complex root searching algorithm, his study of axisymmetric wave

propagations was limited to very lightly viscoelastic coating materials without the

capabilities of handling highly viscoelastic materials, like bitumen. The algorithm also

had problems for root finding over a low frequency range, say, below 30 kHz. Guided

waves in pipes coated with highly viscoelastic materials have not been studied thoroughly.

1.2.3 Numerical Methods

Wave scattering has been a topic of study for many years in that it is important in

terms of a quantitative evaluation of defects, such as defect sizing, characterization, and

discrimination [Ditri 1994][Shin and Rose 1996]. Due to the complexity of wave

propagation characteristics, numerical methods have been explored to study wave

scattering problems by many investigators. Basically, there are four numerical methods

used in guided wave scattering: the finite element method (FEM), boundary element

method (BEM), the semi-analytical finite element method (SAFEM), and also finite

difference methods.

10

FEM has been applied in examining guided waves by many researchers in the

past two decades. However, FEM requires a large number of elements due to the whole

domain discretization, which is usually limited by the available computational ability.

Therefore, most of the wave scattering studies were focused on either 2-D plate or 2-D

defects in pipes. Some work on Lamb waves and Shear Horizontal (SH) waves in a 2-

dimensional plate can be found from the references [Abduljabbar 1983][Koshiba 1984]

[Alleyne 1992][Chang 1999][Lowe 2002][Diligent 2002]. For the wave scattering

problem in pipes, Cawley’s group has conducted a series of studies on notch- like defects

in pipes. For instance, their studies on the longitudinal wave mode L(0,2) includes the

reflection guided waves from circumferential notches [Alleyne 1998], mode conversion

by a partial circumferential notch [Lowe 1998], variation of reflection factors as a

function of defect geometry and frequency [Cawley 2002]. The interactions of the T(0,1)

torsional wave with cracks and notches were also studied by Demma [2003]. However,

most of the defects they studied are planar 2-dimensional notch defects with sharp edges.

The defects are either with through wall depth and a partial circumference or with full

circumference and partial wall depth. Zhu [2002] reported a FEM model of wave

scattering from corrosion like defects under longitudinal wave impingement with only

one element in the wall thickness direction by manually controlling the element sizes and

shapes in the corrosion area.

Also researchers developed other numerical methods good computational

efficiency, such as BEM and SAFEM. Semi-analytical FEM uses the analytical solution

along the wave propagation direction, thus reducing the problem by one dimension

11

[Cheung 1976]. Some SAFEM wave scattering studies can be found in the literature,

such as, reflection wave analysis from the free edge of a semi-infinite laminated

composite circular cylinder [Rattanawangcharoen 1994], axisymmetric wave reflections

from 2-D cracks and weldments in a steel pipe [Zhuang 1997], Lamb wave scattering

study in a multilayered sandwich-like plate [Galan 2003]. Hayashi and Rose [2003]

simulated phased array focusing in pipes but wave scattering was not involved.

BEM, with the element discretization only on the boundary, thus reducing the

problem by one dimension, has also been studied in guided wave mechanics in plate by

some researchers due to its high computational efficiency. Cho and Rose [1996] first

developed a hybrid BEM to study the Lamb wave mode conversion from the edge of a

plate and the guided wave interaction with surface breaking defects. Zhao [2003] used the

same technique to study crack sizing in a plate utilizing both Lamb and SH waves. Zhao

also studied wave scattering from a through-wall hole in a plate utilizing a 3-D boundary

element normal mode expansion technique. But due to the mathematical complexity of 3-

D BEM, the progress on 3-D wave scattering in plate is very limited. Wave scattering

with BEM for real pipeline structures has not been studied.

Although SAFEM and BEM are very computationally efficient, current studies

are only focused on modeling simple and regular geometries. Modeling 3-dimensional

defects especially with arbitrary shapes is still too complicated due to the utilization of

lots of theoretical and mathematical derivations to reduce the meshing dimensions. With

such an increase of computational power, FEM will be used more and more widely.

12

Although FEM has been studied by many people on pipelines, finite element models of

coated pipes, have not been found.

1.3 Objectives

Based on the challenges presented in the problem statement and the literature

survey, a 3-D finite element method is used in this work to study guided waves in a pipe

coated with viscoelastic materials. ABAQUS/Explicit, a FEM package, is explored for

modeling guided waves three-dimensionally in a bare pipe and also for a coated pipe.

With help from the developed modeling tool, a series of studies are then carried out with

the following objectives:

• To quantitatively study and compare guided wave propagations in pipes with

partial loading, axisymmetric, and phased array focusing;

• To study the feasibility of guided wave phased array focusing in coated pipes as

well as the effects of the viscoelastic coating;

• To study wave scattering from planar and non-planar defects under both

axisymmetric loading and phased array loading, and to find a criteria to increase

the guided wave detection ability of 3-D defects, especially corrosion like defects,

by analyzing the interactions of guided waves with defects;

• To investigate the coating material property effects through the computation of

attenuation dispersion curves for coated pipes, with a goal to recommend a

coating material to the pipeline industry with the least effect on inspection

capability;

13

• To find a criterion to improve the inspection of currently existing coated pipes

based on the knowledge learned from the work with suggestion on sensor,

instrumentation, and software.

14

Chapter 2 Guided Waves in Pipes

2.1 Introduction

Axially-directed guided waves in pipes are of key interest for long range pipeline

inspection due to the unlimited length in the axial direction compared with the

circumferential direction. Gazis [1959] first derived the full 3-dimensional solution of

axial-direction guided waves with displacement fields in the radial, angular and axial

directions. Basically, the solution is acquired by solving the wave propagation equation in

a hollow cylinder with traction free boundary conditions. The problem is equivalent to

solving an eigenvalue and eigenvector problem once a characteristic equation is

established. The eigenvalue solution gives the dispersion curves containing all of the

mode information and the eigenvector gives the wave structure or wave field distribution

in the pipe. Dispersion curves and wave structures are the foundations required for the

studies of guided waves.

Figure 2-1 Cylindrical coordinates of a hollow cylinder.

θe a

b

re

ze

15

A brief derivation of three-dimensional guided wave theory useful in pipes

following Gazis’ paper is given next with some modifications of nomenclature. Some

calculated sample dispersion curves and wave structures are also discussed.

2.2 Guided Wave Theory in Pipes

Figure 2-1 shows the cylindrical coordinates for a hollow cylinder. The Navier’s

equation of motion is expressed in equation (2.1):

( ) 2

22

tu

uu∂∂=⋅∇∇++∇

ρµλµ (2.1)

where λ and µ are Lamé’s constants, ρ is the density, and u

is the displacement field.

Boundary conditions are the 6 traction-free conditions at the inner and outer surfaces as

shown in equation (2.2):

0,

== barrrσ (2.2-1)

0,

== barrθσ (2.2-2)

0,

== barrzσ (2.2-3)

16

Helmholtz decomposition is described in equation (2.3), which separates

longitudinal waves from shear waves in an isotropic structure:

Η×∇+∇= φu (2.3)

where φ is a scalar potential field and Η

is a vector potential field. Then displacement

u

in equation (2.3) can be expanded from equation (2.2) and expressed further as in

equation (2.4):

∂Η∂−

∂Η∂+

∂∂=

zr

rru z

rθ

θφ 1

(2.4-1)

∂Η∂−

∂Η∂−

∂∂=

zrru rz

θφ

θ1 (2.4-2)

∂Η∂−

∂Η∂+Η+

∂∂=

θφ θ

θr

z rr

rzu

1 (2.4-3)

The gauge condition expressed in equation (2.5) is used to separate longitudinal

and shear waves into equations (2.6) and (2.7), respectively [Achenbach 1984].

0=Η⋅∇∇

(2.5)

17

2

222

tcL ∂

∂=∇ φφ (2.6)

2

222

tc

T ∂Η∂=Η∇

(2.7)

where ( ) ρµλ 2+=Lc and ρµ=Tc are the longitudinal and shear wave

velocity, respectively.

The solutions can be assumed as formats in equations (2.8) with separated

expressions in radial, angular and axial directions:

( ) ( )kztnrf += ωθφ coscos (2.8-1)

( ) ( )kztnrg rr +=Η ωθ sinsin (2.8-2)

( ) ( )kztnrg +=Η ωθθθ sincos (2.8-3)

( ) ( )kztnrgz +=Η ωθ cossin3 (2.8-4)

where ( )rf , ( )rgr , ( )rgθ and ( )rg3 are potentials as functions of radius, n is the

circumferential order number representing the static wave pattern in the circumferential

direction, k is the wave number representing the wave propagation in the axial direction.

18

After substituting equation (2.8) into equation (2.6) and (2.7), the following is

obtained:

01 2

2

2

=+−′+′′ ffrn

fr

f α (2.9-1)

01

32

32

2

33 =+−′+″ ggrn

gr

g β (2.9-2)

( )

011

12

12

2

11 =++−′+″ ggr

ng

rg β (2.9-3)

( )

011

22

22

2

22 =+−−′+″ ggr

ng

rg β (2.9-4)

where 2)(1 θggg r −= , 2)(2 θggg r += ; 222 kcL −= ωα , and 222 kcT −= ωβ .

The unknowns ( )rf , ( )rgr , ( )rgθ and ( )rg3 in equation (2.8) can be expressed in

equation (2.10) as the solutions of the Bessel equations in (2.9):

( ) ( )rBWrAZf nn 11 αα += (2.10-1)

( ) ( )rWBrZAg nn 13133 ββ += (2.10-2)

( ) ( ) ( )rWBrZAggg nnr 1111111 222 ββθ ++ +=−= (2.10-3)

( ) ( ) ( )rWBrZAggg nnr 1121122 222 ββθ −− +=+= (2.10-4)

19

Table 2-1 Bessel function types used at different frequency intervals.

Frequency interval α and β Bessel function types

ω<kcL 2α > 0, 2β > 0 ( ) ( ) ( ) ( )rYrJrYrJ ββαα ,,, kckc LT << ω 2α < 0, 2β > 0 ( ) ( ) ( ) ( )rYrJrKrI ββαα ,,, 11

kcT<ω 2α < 0, 2β < 0 ( ) ( ) ( ) ( )rKrIrKrI 1111 ,,, ββαα

where rr αα =1 , rr ββ =1 . Z indicates Bessel functions J or I while W indicates

modified Bessel functions Y or K, which depends on whether α and β are real or

imaginary as shown in Table 2-1.

The next step would be to solve the displacement field in equation (2.4) by

substituting equations (2.10) and (2.8) into equation (2.4). However, there are 8

unknowns in equation (2.10) but only 6 boundary conditions in equation (2.2). The gauge

invariance property is now used to eliminate two unknowns from equation (2.10),

according to the gauge invariance property, any of the terms 1g , 2g or 3g could be set to

zero without affecting the solution generality. Therefore, let 2g = 0, and the displacement

field in equation (2.4) can be expressed as in equation (2.11):

( )[ ] ( ) ( )kztnrUkztnkggrnfu rr +=+++′= ωθωθ coscos)(coscos13 (2.11-1)

( ) ( ) ( )kztnrUkztngkgfrnu +=+

′−+−= ωθωθ θθ cossin)(cossin31 (2.11-2)

( )( ) ( ) ( )kztnrUkztnrgngkfu zz +=+

+−′−−= ωθωθ sincos)(sincos1 11 (2.11-3)

20

where )(rU r , )(rUθ , and )(rU z , as combinations of Bessel functions, are the wave field

distributions (wave structure) in the radial direction [Rose 1999].

From equation (2.11), the stress can be derived and then applied to the six traction

free boundary conditions with a matrix form shown in equation (2.12). See Appendix A

for details on the expression of matrix [C]. The dispersion equation (also called the

characteristic or frequency equation) can therefore be expressed as an solution in

equation (2.13):

0][

163

3

1

1

163

3

1

1

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

66

66

=

=

××

×

×B

AB

A

B

A

C

B

AB

A

B

A

cccccc

cccccccccccc

cccccc

cccccc

cccccc

(2.12)

66×

C = 0 (2.13)

Numerical methods should be used to solve the dispersion equation. A bisection

routine (Rose, 1999) can be used to solve the roots of the wave number k as a function of

frequency ω , which is also a process equivalent to finding the eigenvalue of the

characteristic matrix [C]. More generally, phase velocity k

cpω= is calculated as a

function of frequency ω . After finding an eigenvalue, the corresponding eigenvector

21

composed of A, B, A1, B1, A3 and B3 can be calculated and then substituted into equation

(2.10) and (2.11) for determining the wave structure )(rU r , )(rUθ , and )(rU z , which

gives the wave field distribution along the pipe wall thickness.

2.2 Dispersion Curves

Theoretically, guided waves in pipes include an infinite number of modes

considering the explicit circumferential order n and the implicit family order m.

According to the circumferential order, they can be categorized as axisymmetric waves

for n equal to zero and non-axisymmetric waves for n larger than zero. Taking into

account the displacement component, axisymmetric waves includes longitudinal waves

L(0, m) with two displacement components ru , zu , and the torsional wave T(0, m) with

only the angular displacement θu . When n is larger than zero, according to equation

(2.11), all of the modes have three displacement components ru , θu , and zu . These are

called flexural modes F(n, m). More specifically, if a flexural mode is one with a higher

order with respect to the zero-order torsional mode (from the torsional family), it is called

a torsional flexural mode FT(n, m) [Sun 2003]. Otherwise it is called a longitudinal

flexural mode FL(n, m).

Dispersion curves for a sample 10 inch schedule 40 steel pipe were calculated and

are shown next. Shown in Figure 2-2 are the phase velocity and group velocity

22

dispersion curves for axisymmetric longitudinal waves. In the frequency range from 0 to

0.7 MHz there are six modes L(0,1) to L(0,6). Phase velocity is important in terms of

mode excitation, for example, in using an angle beam transducer or comb transducer

[Rose 2002]. Group velocity is the propagation speed of the energy transport or of the

wave group package. It is one of the primary features used for experimental signal

analysis, for example, in defect location estimation and mode type identification. All of

the curves are dispersive in nature indicating phase velocity as a function of frequency,

although certain sections are flatter and less dispersive, such as in the low frequency

range for the L(0,2) mode which is usually preferred for experiments. Dispersion curves

of axisymmetric torsional modes are shown in Figure 2-3. It can be seen that the

fundamental torsional mode is non-dispersive representing a wave package with

consistent length as it travels along the structure.

Although Axisymmetric waves have the advantages of easy excitation and

relatively simple wave behavior, non-axisymmetric waves are indispensable in cases like

wave scattering from 3-D defects and partial loading due to limited access. More

comprehensive dispersion curves of the same pipe are shown in Figure 2-4, where

axisymmetric longitudinal, torsional, and flexural modes of the first five orders are all

presented. It is seen that in the low frequency range, the difference between the zeroth-

order axisymmetric modes and flexural modes are very apparent. When the frequency

increases, the difference between the flexural modes becomes smaller and smaller. For a

certain family, the slightly different phase velocities of all of the modes in that family are

23

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Frequency (MHz)

Pha

se V

eloc

ity

(mm

/ µµ µµse

c)

L(0,1)

L(0,2)

L(0,3)

L(0,4)

L(0,5) L(0,6)

(a)

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Frequency (MHz)

Gro

up V

eloc

ity

(mm

/ µµ µµse

c)

L(0,1)

L(0,2)

L(0,3)

L(0,4)

L(0,5) L(0,6)

(b)

Figure 2-2 Phase velocity (a) and group velocity (b) dispersion curves of axisymmetric longitudinal waves ( L(0,m) ) in a 10 inch schedule 40 steel pipe (outer radius = 136.5 mm, thickness = 9.27 mm). Note that the ‘0’ denotes the zero circumferential order and ‘m’ denotes the mode family order.

24

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Frequency (MHz)

Pha

se V

eloc

ity

(mm

/ µµ µµse

c)

T(0,1)T(0,2)T(0,3)T(0,4)

(a)

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Frequency (MHz)

Gro

up V

eloc

ity

(mm

/ µµ µµse

c)

T(0,1)T(0,2)T(0,3)T(0,4)

(b)

Figure 2-3 Phase velocity (a) and group velocity (b) dispersion curve of axisymmetric torsional waves in a 10 inch schedule 40 steel pipe (outer radius = 136.5 mm, thickness = 9.27 mm). Note that the ‘0’ denotes the zero circumferential order and ‘m’ denotes the mode family order.

25

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Frequency (MHz)

Pha

se V

eloc

ity

(mm

/ µµ µµse

c)

Flexural (n=1-5)Longitudinal (n=0)Torsional (n=0)

L(0,1)

FL(1-5,1)

FT(1-5,1)

FT(1-5,2)FT(1-5,3)

FT(1-5,4)

L(0,2)

FL(1-5,2)

FL(1-5,3)

L(0,3)

L(0,4)

L(0,5)

FL(1-5,4)

T(0,1)

T(0,4)T(0,3)

T(0,2)

Figure 2-4 Phase velocity dispersion curves of axisymmetric and flexural modes in a 10 inch schedule 40 steel pipe (outer radius = 136.5 mm, thickness = 9.27 mm).

extremely critical in that this represents the physical foundation of the realization of

guided wave phased array focusing. This will be presented later.

2.3 Wave Structures

For long range guided wave inspection, the low frequency range is of primary

interest from a wave attenuation point of view. In this work, frequencies below 100 kHz

are studied in detail. From the dispersion curves in Figure 2.4, only three wave families,

F(n,1), T(n,1) and F(n,2) exist. Wave structure is a representation of the energy

26

distribution in the pipe. Understanding wave structure is prerequisite to studying angular

profiles and focusing.

From equation (2.11), wave structure in the pipe circumference is expressed

explicitly as a sinusoid function. Figure 2.5 shows the wave field distribution for n equal

0 to 5. As discussed before, calculation of wave structure in the radial direction through

an eigenvector of the characteristic matrix is more complicated. Some examples of wave

structure in the radial direction are given in Figure 2.6 to 2.8. Wave structures of three

axisymmetric waves as well as the corresponding first order non-axisymmetric modes at

a frequency of 40 kHz are shown in Figure 2.6 for comparison purposes, in which the

radial displacement is dominant for L(0,1), the axial displacement is dominant for L(0,2)

and the angular displacement is dominant for T(0,1). When the order increases to one, the

displacements of the three waves all become 3-dimensional. The same plots for 100 kHz

are shown in Figure 2.7 in order to find the difference due to the frequency variance. It

can be found that there are little changes due to frequency changes. Figure 2.8 is to

compare the wave structures difference among wave modes with different orders for the

L(0,2) family at 100 kHz. It is seen that the wave structure variation is very little for each

mode among the family. Wave structure analysis will be discussed further in later mode

decomposition for wave scattering.

27

(a) order 0 (b) 1st order

(c) 2nd order (d) 3rd order

(e) 4th order (f) 5th order

Figure 2-5 Wave field distribution in the circumferential direction (from 0 to 360 degree) for wave modes with zero to the 5th order, explicitly expressed by the sinusoid functions in equation (2.11).

28

(a) L(0,2) (b) FL(1,2)

(c) L(0,1) (d) FL(1,1)

(e) T(0,1) (f) FT(1,1)

Figure 2-6 Wave structures in the radial direction( )(rU r , )(rUθ , and )(rU z in equation (2.11) ) for axisymmetric T(0,1), L(0,1) and L(0,2) as well as the 1st order non-axisymmetric modes at 40 kHz for a 10 inch schedule 40 steel pipe, showing the displacement field distribution along the pipe wall. Note T(0,1) has only the )(rUθ component, L(0,1) and

L(0,2) have other two but with )(rU r and )(rU z dominant, respectively. For order one, the displacements of the three waves all become 3-dimensional.

29

(a) L(0,2) (b) FL(1,2)

(c) L(0,1) (d) FL(1,1)

(e) T(0,1) (f) FT(1,1)

Figure 2-7 Wave structures in the radial direction( )(rU r , )(rUθ , and )(rU z in equation (2.11) ) for axisymmetric T(0,1), L(0,1) and L(0,2) as well as the 1st order non-axisymmetric modes at 100 kHz for a 10 inch schedule 40 steel pipe, showing the displacement field distribution along the pipe wall. Comparison with Figure 2-6 shows some little variance due to frequency change but are not dramatic.

30

(a) L(0,2) (b) FL(1,2)

(c) FL(2,2) (d) FL(3,2)

(e) FL(4,2) (f) FL(5,2)

Figure 2-8 Wave structures in the radial direction( )(rU r , )(rUθ , and )(rU z in equation (2.11) ) for the L(0,2) family with order from zero to five, at 100 kHz in a 10 inch schedule 40 steel pipe. Note that the percentage of )(rUθ is zero for axisymmetric L(0,1) and increases as the order becomes higher, while )(rU r and )(rU z stays almost the same.

31

Chapter 3 The Focusing Principle

3.1 Introduction

Ultrasonic bulk wave phased array focusing has been used widely for years in

medical diagnostic ultrasound [Rose 1979][Shung 1992] as well as in later NDE

applications with high-frequency bulk waves [R/D Tech 2004]. Guided wave phased

array focusing is still new due to its complexity caused by its dispersive characteristic and

mode diversity compared with the non-dispersive velocity and much fewer modes for

bulk waves. Different from bulk ultrasonic wave focusing whose time delay is a linear

function of the focal distance, the time delay and amplitude of guided wave focusing are

a non-linear function of the focal distance as well as frequency, pipe geometry, and

excitation condition.

Although it was in 2002 when Li and Rose reported a deconvolution algorithm

realizing guided wave focusing in a hollow cylinder [Li 2002], the initial background

studies were started many years earlier. Ditri first solved the source influence problem by

the normal mode expansion technique showing that amplitude ratios of all of the

generated wave modes under a certain excitation condition can be calculated [Ditri 1992]

[Ditri 1994]. Then Shin obtained the dispersion curves of both axisymmetric and non-

axisymmetric waves as well as wave structures of axisymmetric waves [Shin 1997]. Li

calculated the wave structures of non-axisymmetric waves and subsequently obtained the

32

overall wave field distribution under partial loading conditions by taking into account all

of the excited modes whose amplitude ratios can be calculated based on Ditri’s study [Li

2001]. With the wave field distribution (also called an angular profile associated with the

displacement variation along the circumference of a pipe) calculated under one channel

loading condition, a phased array focusing can be realized thereafter by using a

deconvolution algorithm. All of the studies are based on the three-dimensional guided

wave theory developed by Gazis in 1959. The following sections in this chapter will

focus on this theoretical development process with some technical details.

3.2 The Normal Mode Expansion Technique

There are an infinite number of propagating modes in a hollow cylinder and each

one corresponds to an eigenvalue and eigenvector of the dispersion equation.

Mathematically, these modes are orthogonal to each other in terms of the eigenvector and

therefore referenced as a “normal mode”. The normal mode expansion (NME) technique

is to expand the wave fields generated under certain excitation conditions in terms of an

infinite number of the normal modes in the hollow cylinder [Ditri 1992]. The overall

wave displacement field V can be expressed as the summation of all of the normal

modes:

Veiωt =mn

tinm

nm eVA

,

)(ω (3.1)

33

Figure 3-1 Cylindrical coordinates of a hollow cylinder with partial loading of a transducer with 2L axial length and 02α circumferential coverage angle.

where nmA denotes the amplitude and n

mV denotes the wave structure of the mth family

order and the nth circumferential order.

Particle displacement distribution nmV , dependent on the cylinder radial and angular

axes, can be expressed in equation (3.2) as a composition of three wave field component

in the er , eθ , ez directions as shown Figure 3-1:

)(),( zktinm

nmerV −ωθ = )(rUn

mrnrΘ (nθ)er + )(rUn

mθnθΘ (nθ)eθ + )(rUn

mznzΘ (nθ)ez )( zkti n

me −ω (3.2)

where )(rU represents the radial distribution of the displacement field. The corresponding

expression has been shown in equation (2.11); )( θnΘ denotes the angular distribution of

θe a

b

re

ze

02α

2L

34

the displacement field and corresponds to the sinusoidal functions in equation (2.11).

Generally speaking, equation (3.2) is an overall representation of the three displacement

components in equation (2.11) solved by Gazis’s theory considering both the

circumferential and radial wave structures.

Since the item nmV can be calculated by applying equation (3.2), the next step is to

find out the value of the amplitude factor MnA . The amplitude factor calculation is

dependent on the transducer loading condition. For phased array focusing, mode analysis

of the waves excited by one transducer is important and therefore the partial loading

condition is presented here. Shown in Figure 3-1 is a partial loading by a transducer with

2L axial length and 02α circumferential coverage angle, the traction T applied on the

outer surface of the cylinder is expressed in equation (3.3):

>=>≤≤−≤≤−=−=⋅

0

0021

,,,0,,,)()(

αθαθαθ

brLz

LzLbrzppT rr

ee (3.3)

where )(1 θp and )(2 zp denote the loading amplitude function in the angular and axial

directions, respectively.

35

Figure 3-2 Amplitude factors for the L(0, 2) and FL(n, 2) modes generated by a 45° circumferential loading at 100 kHz in a 10 inch schedule 40 steel pipe.

The amplitude factor can be calculated with the formulation in equation (3.4) [Ditri

1992]:

)(,)(,4

)()( 21

*

zpepP

ebUzA zikn

rnnmm

ziknmrn

m

nm

nm

⋅Θ−=−

+ θ (3.4)

Ω ⋅⋅+⋅−= )(41 ** n

mn

mn

mn

mnn

mm TVTVP ez dσ (3.5)

where “+” means wave propagation in the +z direction, * denote the conjugate operator

and “< >” denotes the inner product; nnmmP is the penetration power [Achenbach 1984] for

36

the mode with the mth family order and nth circumferential order; Ω presents the cross

section. From equation (3.4), it can be seen the amplitude factor of a certain mode can be

increased by matching the loading function p1( )θ with the sinusoidal function nrΘ

consequently increasing their inner product. A sample calculation of amplitude factors

was carried out and shown in Figure 3-2 for L(0, 2) and FL(n, 2) modes generated by a

45° circumferential loading at 100 kHz in a 10 inch schedule 40 steel pipe. It is seen that

the first several modes are dominant. In practice the modes with a circumferential order

up to 10 are usually considered.

3.3 Angular Profile for Partial Loading

With mode amplitude factors calculated by NME with equation (3.4) and the wave

structures calculated using equation (3.2), the overall displacement fields under partial

loading with respect to one transducer of a phased array can be acquired using equation

(3.1). They are usually plotted as angular profiles which represent the wave field

amplitudes as functions of circumferential angle θ at a certain radius (like the outer

surface). Shown in Figure 3-3 are the angular profiles of the L(0,2) to FL (9,2) modes

generated by a 45° circumferential loading at 100 kHz in a 10 inch schedule 40 steel pipe,

representing the superposition of the wave fields from all of the generated modes. The

45° loading angle corresponds to one channel of an 8-segment phased array system. Note

that the angular profiles change with wave propagation distance due to the slight

difference of phase velocity for each mode. They are also functions of pipe size,

37

Figure 3-3 Angular profiles of the L(0,2) to FL (9,2) modes generated by a 45° circumferential loading at 100 kHz in a 10 inch schedule 40 steel pipe, showing the superposition of wave fields in the circumferential direction for all the generated modes. Note that the wave field distribution in the circumferential direction changes with wave propagation distance due to the slightly different phase velocity for each mode.

38

CH1, 1θ =0°

CH8, 8θ =315°

CH7, 7θ =270°

CH6, 6θ =225°

CH5, 5θ =180°

CH4, 4θ =135°

CH3, 3θ =90°

CH2, 2θ =45°

Figure 3-4 Illustration of a 8-Channel guided wave phased array system

frequency and excitation conditions. The angular profile variation along the propagation

distance is very crucial in attaining a focusing possibility, in that the variation provides a

possible energy at any point in the pipe by parameter tuning. Therefore, for a phased

array system, it is possible to adjust the input to each channel in order to realizing a

focused wave field.

3.4 Phased Array Focusing

Figure 3-4 shows an 8-channel guided wave phased array system. For an N channel

phased array system, assume channel one has an angular profile of h(θ) at the focal

distance z with period 2π, and )(θg is the desired focused profile of the phased array

system at the focal distance. Due to the symmetry of the pipeline, each channel has the

same profile but with a rotation due to the circumferential position difference. Therefore,

the angular profile of channel i can be expressed as )( ih θθ − where iθ is the

circumferential position of the channel i and also the angle difference from channel one.

39

As discussed before, the overall angular profile g(θ) at the focal distance z is the

superposition of angular profiles from all of the channels.

−

=

−

=

−==1

0

1

0

)()()()()(N

iii

N

iii hahag θθθθθθ (3.6)

where the )( ia θ is the unknown complex amplitude applied to channel i.

Equation (3.6) happens to be a formulation of convolution and can be expressed

further as in equation (3.7):

)()()()()(1

0

θθθθθθ hahagN

iii ⊗=−=

−

=

(3.7)

The unknown amplitudes )(θa can be solved by deconvolution. After doing a Fourier

Transform in the space domain, the space-domain convolution in equation (3.7) can be

changed to an equivalent product in spatial frequency domain as shown in equation (3.8):

)()()( ωωω HAG = (3.8)

)(/)()( ωωω HGA = (3.9)

40

where )(ωA , )(ωG and )(ωH are the Fourier transforms of )(θa , )(θg , and )(θh ,

respectively. The term )(θa can be finally solved by doing an inverse FFT of equation

(3.9) which is equivalent to the deconvoluation of )(θg and )(θh :

)()())(/)(())(()( 111 θθωωωθ hgGHFFTAFFTa −−− ⊗=== (3.10)

The term )(θa is a complex vector containing the amplitudes for all N channels. The

magnitude and phase iφ of )( ia θ gives the amplitude (voltage level) and time delay

applied on each channel, as shown in equation (3.11):

))(( ii aabsamp θ−= (3.11-1)

ft ii πφ 2/−=∆ (3.11-2)

Sample angular profiles were calculated and shown in Figure 3-5. The 2nd family

longitudinal waves (L(0,2) to FL(9,2)) at 50, 100 and 150 kHz are used in a 10 inch

schedule 40 steel pipe. Angular profiles were plotted for a 45° single channel loading as

well as an 8-channel phased array loading with the calculated amplitude and time delay

as inputs. Note that although angular profiles for a single channel are different at various

frequencies, they can all be focused very well using the deconvolution algorithm. Guided

wave focusing has been realized by using a Teletest system [Sun 2004].

41

(a) Single channel, 50 kHz (b) Focusing, 50 kHz

(c) Single channel, 100 kHz (d) Focusing, 100 kHz

(e) Single channel, 150 kHz (f) Focusing, 150 kHz

Figure 3-5 Angular profiles for a 45° single channel loading and 8-channel focusing with the 2nd family longitudinal waves, at 50, 100 and 150 kHz, in a 10 inch schedule 40 steel pipe. Note that although angular profiles for a single channel are different at various frequencies, they can all be focused when using the deconvolution algorithm. Left column – single channel profile )(θh ; Right column – focused profile using the calculated amplitude and time delay.

42

Chapter 4 Finite Element Modeling of Guided Waves

4.1 Introduction

Analytical studies of guided waves, such as non-axisymmetric wave

propagation and excitation, have brought people much deep understanding of guided

wave characteristics and behavior, and also the consequence of a break-through

accomplishment based on this understanding, such as guided wave phased array

focusing. This development has brought the long range guided wave inspection tool to

a new level. The analytical study is indispensable. However, in lots of cases,

problems are too complex to find analytical solutions, such as wave scattering from 3-

D defects, wave propagation in a wave guide with an irregular cross section, or waves

in viscoelastically coated wave guides. For these cases, numerical modeling is usually

conducted as a means of providing an insight into guided wave problems beyond the

ability of analytical and experimental measures. Further, modeling is essential

considering the many variables that exist. It would be a virtually impossible task to

perform experimental studies to determine the best possible modes, frequencies, and

the focusing parameters of loading length, number of focal spots and so on. Modeling

can bring us closer to the optimal parameter choices for an inspection system design.

Three-dimensional finite element method of structural dynamics is used in this

work to study guided wave propagation, wave scattering from defects, and focusing in

a pipe. Based on the knowledge and understanding of guided wave mechanics,

43

ABAQUS, a finite element software package, is explored for the usage of guided

wave modeling. It turns out that guided wave mechanics is not only helpful but also

necessary for running proper guided wave finite element models. This discussion will

start with an introduction of FEM in Dynamics and ABAQUS/Explicit followed by a

discussion of the modeling strategy, meshing and accuracy, through the establishment

of a sample model. Then the modeling of guided wave phased array focusing is

presented.

4.2 Finite Element Method in Dynamics

4.2.1 Basic Theory For structural problems, if loading is a function of time, structure response is

also dependent on time. If the loading is in a high frequency range or applied

suddenly like a blast loading, the problem requires a dynamics analysis. Starting from

Newton’s second law, the global form of the finite element (FE) governing equation

for dynamics can be acquired in equation (4.1) by using the virtual work principle.

See reference for detailed derivations of equation (4.1) [Cook 2001].

][][][ extRDKDCDM =++ (4.1)

where ][M is the mass matrix, ][C is the damping matrix, ][K is the stiffness matrix,

and extR is the external load, D is the nodal d.o.f. as functions of time, D and

D are the first and second order derivatives of D , respectively. Besides the same

44

stiffness matrix used in statics, there are two additional mass and damping matrices

for dynamics problems.

Dynamics problems can usually be categorized into two types. One is the

wave propagation problem that is related to fast loading and generated modes in a

higher frequency range. The other is the structural dynamics problem which is related

to much slower loading and lower modes, like the vibration or seismic problem. To

obtain the response history of dynamics problems, basically, there are three ways: a

modal method, implicit direct integration, and explicit direct integration [Cook 2001].

The first two methods are suitable for the structural dynamics problem. The explicit

direct integration works best for wave propagation problems due to its lower

computational cost.

Direct integration methods (both explicit and implicit) calculates the response

history through a step-by-step integration in time domain. The finite difference

method is used for the time discretization. Therefore, equation (4.1) can be rewritten

with respect to the nth time step:

next

nnn RDKDCDM ][][][ =++ (4.2)

In the explicit method, the term 1 +nD for the n+1th step can be solved simply and

directly based on the historical results:

,...),,,( 11 −+ = nnnnn DDDDfD (4.3)

45

While for the implicit method, the term 1 +nD is calculated from both historical and

current information:

,...),,,,( 111 nnnnnn DDDDDfD +++ = (4.4)

Therefore, the implicit method requires the solving of a simultaneous system of

equations by iterating, which is computational expensive and requires large storage

space and memory. Methods are called single-step if only one-step historical

information is used in equations (4.3) and (4.4), and two-step if historical information

up to the (n-1)th step is used. A two-step explicit method, a central difference method,

is utilized in ABAQUS/Explicit for the modeling of dynamics.

4.2.2 ABAQUS Strategy ABAQUS has two main packages: ABAQUS/Standard and ABAQUS/Explicit.

The former is based on implicit analysis and while the latter is based on explicit

analysis. Both of them have the capability of solving various problems. An

understanding of their characteristics is helpful in choosing which method to use for a

specific problem. Clearly, ABAQUS/Explicit is the right choice for wave

propagation. For the two-step explicit method used in ABAQUS/Explicit, half-step

central differences, the relationship of displacement, velocity and acceleration can be

obtained from a Taylor series expansion of 1 +nD and nD about time 2t∆

, as

shown in equation (4.5) and (4.6):

46

...6

)2(

2)2(

2

21

3

21

2

21211 +∆

+∆

+∆

+= +++++ nnnnn Dt

Dt

Dt

DD (4.5)

...6

)2(

2)2(

2

21

3

21

2

2121 +∆

−∆

+∆

−= ++++ nnnnn Dt

Dt

Dt

DD (4.6)

Equation (4.7) and (4.8) can be acquired by summing (4.5) and (4.6), and discarding

the items with order higher than 2:

211 ++ ∆=− nnn DtDD (4.7)

211 ++ ∆+= nnn DtDD (4.8)

Similarly, equation (4.9) can be obtained:

nnn DtDD 2121 ∆+= −+ (4.9)

By rewriting the governing equation of motion with velocity lagging by 2t∆

, equation

(4.2) can be expressed as:

next

nnn RDKDCDM ][][][ 21 =++ + (4.10)

Substituting equations (4.8) and (4.9) into (4.10), we obtain the follows:

next

nnn RDKDCDM ][][][ 21 =++ + (4.11)

47

The primary error due to the discarding operation for deriving equation (4.7) is

proportional to 2

2

∆t. This is the reason for using a half step in ABAQUS/Explicit

rather than a full time step in order to reduce the error.

4.3 Guided Wave Propagation Modeling

4.3.1 Three-dimensional Model Establishment

Our work began with a simple 3-D wave propagation model. A sample 10 inch

schedule 40 pipe shown in Figure 4-1(a) was used to test the validity of the model.

The pipe model length is 1.6 m and the pipe wall thickness is 9.27 mm. There are

several types of 3-D element available for use. A Linear 8-node brick element (C3D8)

is used here in order to reduce the total node number as well as the output file size.

The element size is determined by the wavelength which is about 108 mm for the

frequency of 50 kHz used in this model. Usually at least 5-10 elements should be used

in the length of one wavelength in order to guarantee an effective representation of the

wave field change in a wave length. Given a frequency, the phase velocity dispersion

curve could be used here to estimate the wavelength and then to determine the mesh

size range. Another factor for mesh size determination is the model geometry. Like

the pipe model, the wall is usually very thin, consequently requiring a smaller element

size. Considering two elements in the pipe wall thickness, therefore, the element size

is chosen to be about 5 mm for meshing. For the pipe FE models run by other

researchers, the element number from 1 to 3 in the pipe wall thickness were generally

used with very good accuracy [Cawley 2002][Zhu 2002][Demma 2004]. With an

48

Figure 4-1 (a) The finite element model for a 10 inch schedule 40 pipe, with mesh size equal to about half of the wall-thickness; (b) Axial pressure loading at the right end; the middle plane is used for signal extraction and model analysis

element size of 5 mm, the total element number for this model is 99600. The next

important issue is how to apply a loading to generate guided waves.

4.3.2 NME vs. Boundary Value Problem Basically, there are two ways to apply time-dependent loading in order to

generate guided waves. The first one that we can directly think of is to simulate the

transducer loading behavior by defining a proper boundary condition pattern and thus

(a)

Analysis plane for mode analysis

(b)

49

called a boundary value problem. Transducer vibration is different for generating

different wave types. For example, the vibration should be dominant in the

circumferential direction for generating torsional waves. Shown in Figure 4-1(b) is an

example of generating the longitudinal L(0,2) wave. An axial pressure loading is

added to the right end of the pipe model to simulate the generation of the L(0,2) mode

with normal beam transducers. This is based on the fact that the displacement field for

L(0,2) in the low frequency range is dominant by zu which is uniformly distributed in

the wall thickness. Although this method is easy to implement, the drawback is that

some unwanted modes may be generated under some circumstances. This is because

the applied loads cannot always match the wave structure of a certain wave mode.

The second method is to prescribe the displacement of a cross section at the pipe end

with the wave structure of a certain mode. This method can generate a pure and

specific mode by satisfying constraints of the mode. The trade-off is the complicity

which may require displacement definition node by node.

The excitation frequency can be realized by using a windowed sinusoidal

signal as the time-dependent amplitude of the pressure, also called a loading function.

Shown in Figure 4-2 are 50 kHz tone-burst signals and their Fourier Transforms for

different cycle numbers. The tone-burst signals are acquired by filtering continuous

sinusoidal functions with a Hanning window. It can be seen that the frequency band

becomes much narrower as cycle number increases. A narrow frequency band is

helpful for a purer mode excitation. However, too many cycles may result in a long

time span. Usually 5-15 cycles are used. For low frequency and less dispersive waves,

like the L(0,2) mode below 100 kHz shown in Figure 4-3(b), less cycle numbers could

be used for getting a short duration. For more dispersive waves, like in the 250-300

50

kHz range, more cycles and narrow band should be used to reduce the dispersion

effects.

(a)

(b)

(c)

Figure 4-2 Hanning-windowed tone-burst signals (left column) and their Fourier Transform (right column) at 50 kHz used as the input loading function for guided wave model (a) 5 cycles; (b) 10 cycles; (c) 15 cycles.

51

0

1

2

3

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Frequency (MHz)

Pha

se V

eloc

ity

(mm

/mic

rose

c)

L(0,1)

L(0,4)

L(0,3)

L(0,2)

(a) Phase velocity

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Frequency (MHz)

Gro

up V

eloc

ity

(mm

/mic

rose

c)

L(0,1)

L(0,4)

L(0,3)

L(0,2)

x(0.5, 5.345)

(0.5, 2.947)

(b) Group velocity

Figure 4-3 Phase and group velocity curves for longitudinal guided waves in a 10 inch schedule 40 steel pipe.

52

(a) L(0,1)

(b) L(0,2)

Figure 4-4 Wave structures of (a) L(0,1) and (b) L(0, 2) modes in a 10” schedule 40 pipe at 50 kHz, showing the distribution of the axial and radial displacement component along the pipe wall thickness direction.

53

4.3.3 Model Accuracy Consideration In this study, the model accuracy is checked by comparing the modeling result

to the analytical results like group velocity and wave structure. This is also the

method used by many other researchers for FE guided wave modeling [Cawley

2002][Zhu 2002][Lowe 2003]. The phase velocity and group velocity dispersion

curves for a 10 inch schedule 40 pipe are shown in Figure 4-3. The 50 kHz point is

labeled on the L[0, 2] curve with a group velocity of 5.345 mm/ secµ . There are two

axisymmetric modes, L[0,1] and L[0,2], at 50 kHz. Figure 4-4 shows the wave

structure of these two modes at 50 kHz. It can be seen that the Uz component is

dominant for L[0,2] while Ur is dominant for L[0,1]. θU is zero for both

axisymmetric modes. Therefore, due to the axial loading added to the model, it was

expected that L[0,2] would be dominant for the generated wave modes. The modeling

result of the displacement magnitude is shown in Figure 4-5.

In order to analyze the wave fields quantitatively, time-domain signals at

nodes 4, 901, and 9 in the analysis plane of Figure 4-1(b) were extracted and are

shown in Figure 4-6. From these waveforms we can see that the amplitude of Uz

(~10-9) is about ten times larger than that of Ur (~10-10). The amplitude of θU which is

not plotted, is only about 10-13 and therefore is negligible compared with Uz and Ur.

A Fourier transform was performed on the time-domain signals to acquire the

displacement amplitudes in the frequency domain which were then used as inputs for

mode decomposition via the normal mode expansion technique.

54

Formulations of mode decomposition using normal mode expansion technique

are expressed in equations (4.12) to (4.15).

1

2

1

102

01

30201

202

201

102

101

...

×

×

×

=

nn

nL

L

nn

Ln

L

LL

LL

U

U

U

A

A

WW

WW

WW

(4.12)

][]][[ UAW = (4.13)

][][]][[][ UWAWW TT = (4.14)

])[]([])[]([][ 1 UWWWA TT −= (4.15)

Where W is the wave structure; A is the coefficient of each mode; U is the

displacement amplitude in the frequency domain; the subscript denotes the mode and

the superscript n denotes the node number.

For this simple problem, there were only two possible longitudinal modes at

50 kHz. The aim was to calculate the coefficients of these two modes. Basically, only

two conditions can guarantee a solution. For each node, there are two conditions:

calculated Uz and Ur. The information gathered from one node can provide a solution

for this simple axisymmetric wave propagation case. However, in order to acquire a

more general and more accurate solution, more than one node should be considered.

In this case, the system becomes over determined (more equations than unknowns).

The least square method was used to minimize the sum of the squared node errors.

Utilizing the data from all three nodes in Figure 4-6, the mode decomposition was

performed with the results as shown in Figure 4-7. As expected, the result showed that

L[0,2] was dominant having an energy percentage of 99.77%.

55

To confirm the validity of the mode decomposition result, the group velocity

was measured according to the arrival time shown in Figure 4-6. The measured group

velocity was 5.387 mm/ secµ which was very consistent with the theoretical group

velocity of 5.345 mm/ secµ . In addition, when comparing the signals for nodes 4 and

9 which correspond to the outer and inner side of the pipe, respectively, we found that

the signals of Uz at these two nodes have the same phase, while signals of Ur have a

180o phase difference. This phenomena could be explained by the L[0,2] wave

structure shown in Figure 4-4 where the Uz has the same sign on both side of the pipe

but Ur has a negative sign. Moreover, the amplitude distribution of Ur as well as Uz

on the inner and outer side shown in the wave structure can also be seen from the

calculated time-domain signals of node 4 and 9 shown in Figure 4-6.

Figure 4-5 Wave propagation modeling result for the displacement magnitude U, for the 50 kHz L(0,2) wave in a 10” schedule 40 steel pipe.

56

Figure 4-6 ABAQUS output of displacement Uz and Ur for nodes 4, 9 and 901 on the plane with Z = 0.8 m. The measured group velocity was 5.387 mm/ secµ ( theoretical group velocity = 5.345 mm/ secµ )

1

2

Node 4

Node 9

Node 901

Z= 0.8m

57

0.23%

99.77%

0

0.1

0.2

0.30.40.5

0.6

0.7

0.8

0.91

L(0,1) L(0,2)

Ene

rgy

Per

cent

age

Figure 4-7 Mode decomposition using the normal mode expansion technique, showing that L [0, 2] was the dominant mode which is consistent with the loading condition and group velocity measurement.

Although this example model is very simple, the results were very

encouraging in that their validation showed that this modeling method provided an

accurate, effective and powerful tool for further studies on more complex problems

such as mode conversion, wave scattering, and phased array focusing.

4.4 Modeling of Guided Wave Phased Array Focusing

In order to numerically study wave scattering from defects under phased array

loading as well as the coating effect on focusing, a finite element model of an 8-

channel phased array focusing system is first set up in a pipe as shown in Figure 4-8.

Each channel is applied with a loading with different amplitude and time delay.

Figure 4-9 shows the focusing process at zero degrees and at a distance of 1.5 meter

away in a 10 inch schedule 40 pipe with 100 kHz L(0,2) and higher order flexural

waves. The time delay and amplitudes are calculated with the algorithm presented in

Chapter 3 and summarized in Table 4-1. Four steps of the focusing process are plotted

58

in (a) to (d) of Figure 4-9 where quantitative analysis shows that the energy is

increased tremendously (5×) at the focal point compared with axisymmetric loading.

Modeling work also shows that focusing increases the energy by 20 times compared

with partial loading. Therefore, under the loadings with the same strength, the energy

ratio of the three excitation methods is shown as below:

Energy ratio: Epartial : Eaxisymm : Efocusing = 1 : 4 : 20

Figure 4-8 Finite element model for a 8-channel phased array focusing in a 10 inch schedule 40 steel pipe. Loadings are applied with different time delays and amplitudes to the 8 segments at one end of the pipe.

X

Y

Z

Loading area of CH1

Pipe length section

59

(a) Beginning (b) Start focusing

(c) Focusing (d) Beyond focusing

Figure 4-9 Eight-channel phased array focusing at zero degrees at a 1.5 meter axial distance in a 3-meter-long 10 inch schedule 40 pipe at a frequency of 100 kHz, showing resulting displacement profiles at different focusing steps. Focusing realizes a significantly higher Energy (5×) compared to the axisymmetric case, and 20× higher compared to partial loading.

Table 4-1 Signal amplitude and time delay for focusing at zero degrees at a 1.5 meter distance for the 100 kHz L(0,2) mode in a 10 inch schedule 40 pipe

Channel No. 1 2 3 4 5 6 7 8 Amplitude 1.000 0.933 0.769 0.726 0.610 0.726 0.769 0.933 Time Delay ( secµ )

10.364

9.771

7.811

4.558

0.000

4.558

7.811

9.771

60

Note that there are some additional waves with smaller group velocities

behind the expected L(0,2) mode. This occurs because each channel is excited at a

sequence of different times and consequently the actual excitation performs like

partial loading with one channel or a combination of some channels. Therefore some

unwanted modes like the first longitudinal wave family were generated with slower

group velocities, thus forming the “tail” waves. This has been observed by the wave

propagation animation of one channel partial loading. These “tail” waves are

annoying since they could interfere destructively with the reflected wave and

therefore reduce the inspection ability, especially for small or non-planar defects with

weak reflections. Because of linear superposition theory, proper sensor placement can

lead to successful results.

The angular profile at the focusing distance for one channel with 45-degree

partial loading is calculated analytically by our MATLAB program and shown in

Figure 4-10 (a). The angular profile is a superposition of the wave fields from all of

the excited flexural modes under a partial loading condition and each mode has

slightly different phase velocity which results in a wave interference changing along

the distance. Therefore, the angular profile changes with the propagation distance,

providing a physical basis of focusing with phased array loading. The focusing

angular profile is shown in Figure 4-10(b) with an increase of about 5.8 times with

respect to the maximum displacement amplitude.

The angular profiles for the same cases but calculated by the finite element

analysis are shown in Figure 4-11. The result is acquired by extracting the wave

signals from the model in Figure 4-10 and then plotting the signal amplitudes. Great

61

2e-009

4e-009

6e-009

8e-009

300

120

330

150

0

180

30

210

60

240

90 270

1e-008

2e-008

3e-008

4e-008

300

120

330

150

0

180

30

210

60

240

90 270

(a) (b)

Figure 4-10 Analytical calculation results: angular profile of the displacement field of the 100 kHz L(0,2) mode at a distance of 1.5 m in a 10 inch schedule 40 pipe (a) 45 degree partial loading and (b) 8-segment phase array loading. Note the maximum displacement amplitude in (b) is about 5.8 times larger than that in (a).

1e-010

2e-010

300

120

330

150

0

180

30

210

60

240

90 270

5e-010

1e-009

1.5e-009

300

120

330

150

0

180

30

210

60

240

90 270

(a) (b)

Figure 4-11 FEM numerical calculation results: angular profiles of the displacement field of the 100 kHz L(0,2) mode at a distance of 1.5 m in a 10 inch schedule 40 pipe (a) 45 degree partial loading and (b) 8-segment phased array loading. Note the maximum displacement amplitude in (b) is also about 5.8 times larger than that in (a).

62

0.0E+00

2.0E-10

4.0E-10

6.0E-10

8.0E-10

1.0E-09

1.2E-09

0 0.5 1 1.5 2 2.5 3

Propagation Distance (m)

Dis

plac

emen

t Mag

nitu

de

Figure 4-12 Axial profile at zero degrees of the 100 kHz L(0,2) mode in a 10 inch schedule 40 pipe with 8-segment phased array loading.

consistency has been seen from the FEM numerical results and the analytical results.

The reason that the absolute amplitude values in Figure 4-11 differs from those in

Figure 4-10 is that the pressure loading on each segment is different than the loading

condition for the analytical method. However, this is not a serious matter because the

system is linear. Consequently, the same amplitude increase (5.8 times) from partial

loading to phased array loading can be also seen from the numerically calculated

angular profile. The result is extremely encouraging in that, at first, it once again

demonstrates the accuracy of the finite element model, and secondly it indicates the

value of a powerful finite element tool for further research work on wave scattering

and wave attenuation using phased array loading.

The axial profile is also plotted in Figure 4-12 showing the displacement

magnitude along the axial direction at zero degrees. From the plot it can be seen that

the magnitude oscillates a little in the beginning, then goes to stable until a significant

increase occurs at the expected focal distance, and finally drops very fast beyond the

63

focal area due to the energy distribution to other parts in the cross section. Axial

profiles will become a useful representation of the wave attenuation for the

attenuation studies in this report.

4.5 Summary

In this chapter, the finite element method in dynamics was introduced. The

ABAQUS/Explicit package was then explored for modeling guided wave propagation

with two different loading methods. Procedures of data acquisition and analysis using

the output of the FE models have been developed. The model validity and accuracy

were tested by a comparison of theoretical group velocities and wave structures.

Guided wave phased array focusing was also realized using theoretically calculated

time delays and amplitudes. High consistency between modeling results and

theoretical focusing results were observed, which once again indicates the accuracy of

the 3-D modeling. With the help of models, partial loading, axisymmetric loading and

phased array focusing were studied and compared quantitatively with respect to the

wave field distribution and energy.

A powerful 3-D finite element tool for wave propagation, scattering and

focusing study has been developed which will have a high impact on future research

work. This accomplishment provides us with great confidence for further wave

scattering and mode conversion studies with both longitudinal and torsional

axisymmetric and phased array loading. Moreover, it can be seen that finite element

modeling of guided waves is not just only running ABAQUS software, but also

64

requires a deep understanding of wave mechanics as well as the necessary wave

mechanics analytical calculations in providing inputs to the ABAQUS models.

65

Chapter 5 Wave Scattering

5.1 Introduction Three-dimensional finite element modeling of guided wave propagation and

focusing has been developed utilizing ABAQUS/Explicit. The modeling validity and

accuracy was confirmed and tested by wave mechanics studies on group velocity,

wave structure and angular profiles. A wave scattering model for straight pipe is now

available for studying the responses from 2-dimensional and 3-dimensional defects

and any subsequent mode conversions that might occur. In a few words, finite

element modeling provides a powerful tool for studying wave scattering, mode

conversion, and phased array focusing. The model will be used to study cracking,

corrosion, and other defect possibilities. The results will be used as a basis for

designing appropriate data acquisition and signal processing schemes for the best

opportunity for a reliable inspection.

5.2 Mode Conversion from Two-dimensional Defects

In preparation for the discussion on 3-dimensional defects, some 2-

dimensional results are presented. Modeling of a pipe with a two-dimensional defect

is used to study mode conversion phenomenon. As shown in Figure 5-1, a 360o notch

(5 mm wide, 30% through-wall depth) was modeled within a 10 inch schedule 40 pipe.

A pressure loading with a 5-cycle 50 kHz tone burst wave was used. The pipe length

66

Figure 5-1 Finite element model of a 10 inch schedule 40 pipe with a 5-mm-wide and 30%-through-wall circumferential notch.

was 3 meters to provide sufficient wave propagation distance for observation of mode

conversion. Two analysis planes were selected 0.4 meter away from the defect on

both sides to avoid the influence of evanescent modes. Figure 5-2 shows the

displacement magnitude of the propagating waves before and after the wave scattered

from the defect. It is seen that a new mode L[0, 1] arises from the defect. The

converted L[0, 1] mode propagated much slower than L[0, 2], which is consistent

with the group velocity difference between L[0, 1] and L[0, 2] shown in Figure 5-3.

The time-domain signals in the reflection analysis plane are shown Figure 5-3

where the incident L[0, 2] mode and the reflected L[0, 1] and L[0, 2] modes are

labeled above the corresponding signals. A careful comparison will show that the

Transmitted wave analysis plane Z = 1.9 m

Reflected wave analysis plane Z = 1.1 m

A circumferential notch with a 5 mm width and a 30% through-wall depth

67

Figure 5-2 Reflected and transmitted wave fields showing the displacement magnitude compared with the incident wave fields. Note a new mode L(0,1) is converted due to the existence of the circumferential notch.

(a) incident L(0,2) mode

(b) scattered L(0,1) and L(0,2) propagating with different group velocities

Scattered L(0,1) mode

Scattered L(0,2) mode

Wave propagation direction

68

Figure 5-3 ABAQUS output of displacement Uz and Ur for nodes 5,8, and 153 on the reflection analysis plane with Z = 1.1 m.

1

2

Node 8

Node 5

Node 153

Z= 1.1 m

Incident L(0,2)

Incident L(0,2)

Incident L(0,2)

Incident L(0,2)

Incident L(0,2)

Incident L(0,2)

Reflected L(0,2) L(0,1)






69

Figure 5-4 ABAQUS output of displacement Uz and Ur for nodes 10,13, and 2136 on the reflection analysis plane with Z = 1.9 m.

1

2

Node 10

Node 13

Node 2136

Z= 1.9 m

Transmitted L(0,2) L(0,1)






70

variation of signal amplitude and phase of Uz and/or Ur in Figure 5-3 at the outer and

inner sides of the pipe are consistent with the wave structure in Figure 5-4. The

signals in the transmission analysis plane are plotted in Figure 5-4.

Mode decomposition was performed to calculate the transmission and

reflection coefficients. See equations 4.12 to 4.15 for the algorithm of mode

decomposition. Figure 5-5 shows the results. Since the pulse-echo mode is usually

used for long-range pipeline inspection, the reflection coefficients are the more

important ones. From these results, it can be seen that only about 15% of the total

energy is reflected for a 360 o

notch with a 30% through-wall depth. If the defect is

not 360 o

but only 30 o

, there must be very weak energy being reflected. In this case,

the axisymmetric longitudinal waves may not be able to detect the reflected echo.

This shows the necessity of using the guided wave phased-array focusing technique

which focuses energy at any selected inspection point. Similarly, more models were

run for notches with different depths from 10%, 30%, …, 90%. Results are shown in

Figure 5-6 in which the L(0,2) mode has a monotonic relationship with defect depth

while the converted L(0,1) mode does not, suggesting that the original L(0,2) mode is

suitable for defect sizing for this case. Moreover, among the reflected modes, the

converted L[0,1] mode had more energy than that of the L[0, 2] mode for defects with

small depths, indicating that the L[0, 1] mode is more effective than the L[0, 2] mode

for small defect detection in this case. Mode conversion is very valuable in the sense

of providing insight and guidance for pipe experiments and for field inspections.

71

72.2%

12.6%

4.5%10.7%

0%

10%

20%

30%

40%

50%

60%

70%

80%

Transmitted L(0,2) Transmitted L(0,1) Reflected L(0,2) Reflected L(0,1)

Figure 5-5 Mode decomposition of the output result using the normal mode expansion technique, showing that the L(0, 1) mode was converted in both transmitted and reflected waves due to the existence of the defect.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Defect Depth

Tran L02

Tran L01

Refl L02

Refl L01

Figure 5-6 Mode decomposition of the output result for a series of notches with 10%, 30%, …,90% through wall depths under 50 kHz L(0,2) wave incidence, showing that the transmitted and reflected L(0, 2) modes have a monotonic relationship with defect depth, but not true for the converted L(0,1) modes. This suggests the incident mode L(0,2) is more reliable for defect sizing.

72

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Defect Depth

Ene

rgy

Rat

io

Tran T01Refl T01

Figure 5-7 Transmission and reflection ratio of T(0,1) wave for a series of notches with 10%, 30%, …,90% through wall depths under 50 kHz T(0,1) wave incidence, showing that the transmitted and reflected T(0, 1) modes have a monotonic relationship with defect depth. Note there is no mode conversion to other modes, suggesting the incident mode T(0,1) is simpler for defect sizing.

Modeling studies with 50 kHz torsional mode T(0,1) as the incident wave on

the same series of 2-dimensional notches were also carried out. Results in Figure 5-7

show that there are no mode conversions to longitudinal modes. Since there is one

axisymmetric torsional mode at the frequency of 50 kHz, no mode conversion to other

torsional modes occurred. Additionally, it can be seen that the transmission and

reflection ratios have a monotonic relationship with defect depths, indicating a quite

good potential for defect sizing. This phenomenon was also observed by other

researches [Demma 2003]. Further research on 3-D defects with torsional wave

incidence will find out whether mix mode conversion can occur and under what

conditions it occurs.

73

5.3 Mode conversion from three-dimensional defects

5.3.1 Planar Defects In reality, most defects are 3-dimensional, like commonly occurring crack and

corrosion, calling for a 3-dimensional wave scattering model. Shown in Figure 5-8 is

a model for a saw cut with a 50% wall depth and a 3.53% cross sectional area (CSA)

in a 10 inch schedule 40 pipe. The defect model was generated by interacting a plate

(with the same thickness as the saw cut) with the pipe and then cutting the common

part from the pipe. At first, an axisymmetric loading of 100 kHz L(0,2) is considered

as input to the wave scattering model. Figure 5-9 shows the modeling results of wave

scattering upon the saw cut at two different times. Reflected modes are generated as

shown in Figure 5-9(a), and Figure 5-9(b) shows that the waves are separated into two

mode groups after propagating for awhile. Apparently, the reflected waves are not

axisymmetric any more and each mode group should become a superposition of

several flexural modes. By analyzing the group velocity, the two groups are

recognized as the slower flexural waves of the F(n,1) group and the faster flexural

waves of the F(n,2) group. According to the NME study on partial loading which is

similar to mode conversion on partial defects, each group is composed of several

modes with different circumferential orders with slightly different phase and group

velocity. The wave signal at the point at zero degrees and 0.6 meter away from the

defect on the reflection path is shown in Figure 5-10 where the F(n,2) waves

propagate much faster than the F(n,1) waves. Mode decomposition will be carried out

with the NME technique in order to find out the composition of the wave package. It

is also noticeable that the reflected wave groups have a similar circumferential width

74

as that of the defect. The mode decomposition result will vary for different wave

circumferential widths, suggesting the potential of defect circumferential sizing by

NME.

The modeling work is also carried out with an 8-segment phased array loading

and the results are shown in Figure 5-11. The two scattered flexural wave groups are

observed similarly as in the axisymmetric case, but with much larger amplitudes. The

wave signals are plotted in Figure 5-12 with the same results from the axisymmetric

loading case for comparison purposes. It is very encouraging to see that focusing

increases the reflection amplitude dramatically, indicating a potential of higher

sensitivity and stronger inspection ability. Note that the total energy input in the

focused case was the same as in the axisymmetric case.

Figure 5-8 A finite element model for a saw cut with 50% wall depth, CSA 3.53%, in a 10 inch schedule 40 pipe.

75

Figure 5-9 Wave scattering from a 50% saw cut in a 10 inch schedule 40 pipe (3.53% CSA) for 100 kHz L(0,2) wave, for axisymmetric loading, (a) in the beginning, (b) after a while.

Defect

Transmitted modes

Reflected modes

Reflected modes

Defect

Transmitted modes

(a)

(b)

76

-4E-10

-3E-10

-2E-10

-1E-10

0

1E-10

2E-10

3E-10

4E-10

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007

Time (Second)

Am

plit

ude

Incident wave

Reflected F(n,2)

Reflected F(n,1)

Figure 5-10 zU signal at the point of zero degrees and 0.6 meters away from the defect on the reflection path for axisymmetric loading with a 100 kHz L(0,2) wave. The group velocity comparison shows that the reflected waves are the flexural F(n,2) modes and the converted flexural F(n,1) mode.

77

Figure 5-11 Wave scattering from a 50% saw cut in a 10 inch schedule 40 pipe (3.53% CSA) for the 100 kHz L(0,2) mode, for 8-segment phased array loading, (a) in the beginning, (b) after a while.

Defect

Transmitted modes

Reflected modes

(b)

(a)

Reflected modes

Transmitted modes

Defect

78

-8E-10

-6E-10

-4E-10

-2E-10

0

2E-10

4E-10

6E-10

8E-10

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007

Time (Second)

Am

plit

ude

AxisymmeticPhased Array

Incident wave

Reflected F(n,2)

Reflected F(n,1)

(a)

0

1E-10

2E-10

3E-10

4E-10

5E-10

6E-10

7E-10

8E-10

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007

Time (Second)

Am

plit

ude


Incident wave

Reflected F(n,2)

Reflected F(n,1)

(b)

Figure 5-12 Signals of (a) zU (b) displacement magnitude at the point 0.6 meter away from the notch on the reflection path for both axisymmetric and phased array loading with the 100kHz L(0,2) wave. Note both focusing and axisymmetric waves can find the defect in this example but focusing increases the reflection amplitude by about 3 times in this case.

79

5.3.2 Non-planar Defects

To study wave scattering from a non-planar defect, modeling work is

conducted for a corrosion simulation with 70% wall depth, 3.8% width CSA, 6.4%

length CSA in a 10 inch schedule 40 pipe shown in Figure 5-13. Wave signals are

shown in Figure 5-14 where the reflection amplitudes from the axisymmetric loading

are almost zero while those from focusing are still strong enough to see the defect

information. Note there is a wave package called “the 2nd time reflection modes” in

Figure 5-14 (b). According to an observation of the wave propagation animation, this

wave group was generated when the scattered waves propagating in the

circumferential direction in the first-time wave scattering come back and interacts

again with the defect. It is therefore called the second time reflected waves. In other

words, the first wrap-around of the circumferential guided waves scattered during the

first wave scattering process interacts with the defect again resulting in the new

scattered waves in the axial reflection direction. The waves are apparent because the

axial extent of the corrosion is pretty large. However, due to a small axial size of the

notch defect, the second time reflection wave for the previous case is smaller and

therefore could be neglected. Since the amplitude of the 2nd time reflection is even

larger than the direction reflection from the defect in this case, it is very worthwhile

studying the potential of using this phenomenon as a new feature for defect axial

sizing.

80

Figure 5-13 A corrosion model with 70% wall depth, width CSA 3.8%, and length CSA 6.4% in a 10 inch schedule 40 pipe.

81

-8E-10

-6E-10

-4E-10

-2E-10

0

2E-10

4E-10

6E-10

8E-10

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007

Time (Second)

Am

plit

ude


Incident wave

Reflected F(n,2)

Reflected F(n,1)

2nd time Reflected modes

(a)

-1E-10

0

1E-10

2E-10

3E-10

4E-10

5E-10

6E-10

7E-10

8E-10

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007

Time (Second)

Am

plit

ude


Incident wave

Reflected F(n,2)

Reflected F(n,1)

2nd time Reflected modes

(b)

Figure 5-14 Wave scattering signals of (a) zU (b) displacement magnitude at the point 0.6 meters away from the corrosion on the reflection path for axisymmetric and phased array loading with the 100 kHz L(0,2) input wave, showing that focusing has strong reflections from corrosion while the defect signals for axisymmetric waves are too weak to see.

82

5.3 Summary

Wave scattering from defects in a pipe has been studied with a 3-D finite

element model. Studies of 2-D axisymmetric notches show that there is an apparent

mode conversion from the L(0,2) to L(0,1) mode. Mode decomposition using NME

was carried out with a result that the reflected L(0,1) mode is sensitive to small

notches while the reflected L(0,2) mode is suited for defect depth sizing. Wave

scattering study was also conducted on 3-D planar and non-planar defects with both

axisymmetric loading and phased array loading. The similar mode conversion

phenomenon was observed but for non-axisymmetric mode groups whose

circumferential widths were proportional to the defect circumferential width. This

indicates a circumferential sizing potential by looking at the mode decomposition

results which are also dependent on the wave group width according to the NME

technique. Based on the circumferential size, defect depth can be estimated more

accurately by the reflection wave amplitudes. Defect depth estimation is more critical

in that a catastrophic failure might occur when the depth reaches some extent. In a

word, 3-D FE modeling shows the potentials and criteria of defect sizing in axial,

circumferential, and depth directions. Mode conversion from 3-dimensional defects is

quite complex, but a modeling tool is now available to study the wave scattering from

such defects. The results of using the model will impact the selection of inspection

parameters for improved reliability and probability of detection.

Modeling work also shows that focusing increases the defect reflection

amplitude compared with axisymmetric waves, and enhances tremendously the ability

of finding non-planar defects which is difficult to find by axisymmetric waves.

83

Chapter 6 Finite Element Modeling of Guided Waves in Coated Pipe

6.1 Introduction

Guided wave pipeline inspection in the field could encounter many unknown

challenges due to the presence of viscoelastic coatings on the pipe. Most of the

current guided wave research is still based on bare pipes, although viscoelastic coated

pipe study is strongly desired. It has been shown that guided wave focusing increases

the inspection sensitivity tremendously, especially for corrosion like defects.

However, a guided wave focusing possibility and focusing profile changes in a coated

pipe is still unknown, which has intrigued people’s interests recently. For the first

time, finite element modeling of coated pipe is studied in terms of wave propagation

and focusing.

6.2 Damping Induced by Viscoelasticity

6.2.1 Rayleigh Damping

Rayleigh damping is used in ABAQUS to introduce the complex elastic

modulus constants which are caused by viscoelasticity. Rayleigh damping is meant to

reflect physical damping in the actual material. Rayleigh damping is defined by a

damping matrix formed as a linear combination of the mass and the stiffness matrices:

84

][][][ KMC βα += (6.1)

Where [M] is the mass matrix of the model, [K] is the stiffness matrix of the model,

and α and β are damping factors. With Rayleigh damping, the eigenvectors of the

damped system are the same as the eigenvectors of the undamped system. Rayleigh

damping can, therefore, be converted into critical damping fractions for each mode.

For a mode, the fraction of critical damping can be expressed as [Cook 2001]:

22

ωβ

ω

αξ RR += (6.2)

where Rα is the mass proportional Rayleigh damping factor which damps the lower

frequencies and Rβ is the stiffness proportional damping factors which damps the

higher frequencies. The Rα factor simulates the damping caused by the model

movement through a viscous fluid and therefore it is related with the absolute model

velocities. Since the frequency used in this study is in the ultrasonic wave range (>20

kHz), the contribution from the Rα factor is negligible. The Rβ factor defines

damping which is related to the material viscous property and proportional to the

strain rate. The next question is how to connect the Rβ factor with the complex

elastic modulus or the acoustic parameters describing wave propagation and

attenuation. According to a definition in vibration theory, the damping loss factor η

is the ratio between dissipated energy and the input energy. It is expressed as follows

[Sun 1993]:

85

ξη 21

==Q

(6.3)

where Q is the quality factor.

For a time harmonic case, according to the correspondence principle

[Christensen 1981], the stress-strain relationship for a viscoelastic material is changed

by using the complex, viscoelastic modulus. Therefore, the complex elastic modulus

*E can be expressed as:

'''*iEEE += (6.4)

where 'E is the storage modulus which defines the material stiffness, and ''E is the

loss modulus which defines the energy dissipation of the material. Therefore, the

damping loss factor η can be expressed as the ratio of the loss modulus and the

storage modulus [Sun 1993].

ωβωβ

ω

αξη R

RR

E

E≈+=== )

22(22

'

''

(6.5)

'

''2

E

ER

ωω

ξβ =≈ (6.6)

where the approximation sign works for the high frequency range of ultrasonic waves.

6.2.2 Viscoelastic Property Estimation from Acoustic Measurement

For one-dimensional wave propagation in a viscoelastic material, the wave

86

equation is expressed in (6.7), where u is the displacement and c is the complex wave

velocity. See reference for the detailed derivation [Blanc 1993][Barshinger

2001][Christensen 1981].

( ) 2

2

2*2

2 1

dt

ud

icdx

ud

ω= (6.7)

The solution to (6.7) in terms of the attenuation and phase velocity is:

( ) ( ) ( )

−

−−+−− ====tx

ci

xtxkixkxikktixktieAeeAeAeAetxu

ωω

ω

ωαωωω )()())( ''''''*

),(

(6.8)

where k is the wave number, ' and '' indicates the real part and the imaginary part,

respectively. The term xe

)(ωα− introduces the amplitude attenuation and )(ωα is the

attenuation coefficient. Therefore, the imaginary part and the real part of the complex

wave number can be expressed as in equation (6.9) and (6.10), respectively.

( ) ( )

'

*

'

=

=

ω

ω

ω

ω

cck (6.9)

( )

''

*

'' )(

=−=

ω

ωωα

ck (6.10)

Consequently, the complex velocity )(* ωc can be derived from (6.9) and (6.10):

87

( )

( )( )ω

ωα

ω

ωi

c

c

−

=1

1* (6.11)

The velocity is specified to be complex and frequency dependent due to the

viscoelastic material properties. Phase velocity c(ω) and the attenuation constant α(ω)

can be defined from the wave velocity as follows:

( )

1

*1Re)(

−

=

ωω

cc (6.12)

( )

−=

ωωωα

*

1Im)(

c (6.13)

The c(ω) and α(ω) can be measured by experiments and then )(* ωc can be acquired

by equation (6.11). The complex shear modulus *G (also the 2

nd Lamé constant *µ )

is calculated as in equation (6.14):

( )( )

ραω

ωρ

ω

ωα

ωρµ ⋅

−=⋅

−=⋅==

− 2

22

2

2

2

2

2*

2

** 1

ic

ci

ccG (6.14)

where the subscript 2 indicates the variables for shear wave and ρ is the density of

the material.

Young’s modulus is expressed in equation (6.15) :

ραω

ω

αω

αω

αω

αω

ρ ⋅

−⋅

−

−−

−

−−

=⋅⋅

−

−

=

−

−

=

2

22

2

2

2211

1212

2

2211

1212

2*

22

*

1

*

2

2

*

1

*

2

*

2

*

1

*

2

2

*

1

*

2

*

1

43

1

43

1

43

ic

c

cicc

cicc

cicc

cicc

c

c

c

c

c

G

c

c

c

c

E (6.15)

88

Table 6-1 Elastic and viscoelastic material properties

Material 1c

( )sec/km ω

α1 2c

( )sec/km ω

α 2 ρ

( )3/ cmgm

E&C 2057 / Cat9

Epoxy

2.96 0.0047 1.45 0.0069 1.60

Mereco 303

Epoxy

2.39 0.0070 .99 0.0201 1.08

Bitumastic 50

Coating

1.86 0.0230 0.75 0.2400 1.50

Table 6-2 Calculated material damping properties

Damping factor

Rβ Material E* (Pa) G* (Pa) damping

ratio

ωβη R= 30 kHz 50 kHz 100 kHz

E&C 2057 /

Cat9 Epoxy

9.03E9+1.92E8i 3.36E9+6.73E7i 2.12% 1.13E-7 6.76E-8 3.38E-8

Mereco 303

Epoxy

2.95E9+1.92E8i 1.06E9+4.27E7i 3.93% 2.08E-7 1.249E-7 6.25E-8

Bitumastic 50

Coating

2.18E9+7.6E8i 7.66E8+2.85E8i 34.9% 1.85E-6 1.11E-6 5.55E-7

where the subscript 1 indicates the variables for longitudinal wave. Therefore, the

complex modulus can be calculated from the measured velocities and attenuation for

longitudinal and shear waves. See reference [Krautkramer 1990] for the origin of the

derivation of equation (6.14) and (6.15).

Coating materials used for the pipeline industry varies, depending on the

pipeline buried time and the specific condition. Some typical viscoelastic coating

materials are selected for this study and their measured material elastic and

viscoelastic properties are shown in Table 6-1. The damping properties calculated

using equation (6.5)-(6.6) are shown in Table 6-2.

89

6.3 Wave Propagation in a Coated Pipe

With the calculated parameters in Table 6-2 as inputs to the ABAQUS models,

some numerical experiments were carried out for pipes with coatings at several

frequencies 30 kHz, 50 kHz and 100 kHz. The finite element model is shown in

Figure 6-1 with an axial line for signal amplitude analysis with respect to wave

attenuation. Shown in Figure 6-2 are the curves of the displacement magnitude vs.

wave propagation distance for the 30 kHz L(0,2) wave in a 10 inch schedule 40 pipe.

Computations were run on a bare pipe first in order to remove the attenuation from a

bare pipe due to dispersion. The calculated displacement magnitude at each axial

distance was plotted as a point, and then an exponential curve was fit to all of the

points, thus acquiring the term xe

)(ωα− in equation (6.8). The attenuation can be

calculated following equation (6.16) and (6.17):

( ) ( ) ( ) xexex )(69.8log20)(log20dBnAttenuatio 10

)(

10 ωαωαωα −=−== − (6.16)

( ) )(69.8dB/mnAttenuatio ωα−= (6.17)

The numerical experiments were also carried out for other frequencies such as

50 kHz and 100 kHz with all of the results summarized in Table 3.3. The resulting

wave propagation distance under a 50 dB attenuation law, for example, is also

estimated based on the calculated attenuation (dB/m), showing that at a very low

frequency, like 30 kHz, that the wave can propagate much longer than at other

frequencies, regardless of coating type, say epoxy or a bitumastic coating. Results

90

also show that attenuation is a function of material and frequency. For epoxy, which is

less viscoelastic, the

Figure 6-1 ABAQUS model for guided wave propagation analysis in a 10 inch schedule

40 pipe coated with viscoelastic materials.

Axial line for amplitude analysis

Coating

Steel pipe

Coating

Steel pipe

(a) 3D model

(b) 2D projection from one pipe end

91

y = 2E-09e-0.0286x

y = 2E-09e-0.0286x

y = 2E-09e-0.0366x

y = 2E-09e-0.1467x

0.0E+00

5.0E-10

1.0E-09

1.5E-09

2.0E-09

2.5E-09

3.0E-09

0 0.5 1 1.5 2 2.5 3


Dis

pla

cem

en

t M

ag

nitu

de

No Coating

Epoxy

Bitumen

Figure 6-2 Displacement magnitude vs. propagation distance for the L(0, 2) mode at 30

kHz propagation in a 10 inch schedule 40 pipe.

Table 6-3 Attenuation introduced by 3 mm epoxy and bitumastic coating

Material Frequency Attenuation constant

α

Attenuation

(dB/m)

Propagation Distance with

50 dB attenuation (m)

30 kHz 0.008 0.07 714

50 kHz 0.0794 0.69 72.5

Mereco 303

Epoxy

100 kHz 0.144 1.25 40.0

30 kHz 0.118 1.03 48.0

50 kHz 0.739 6.42 7.9

Bitumastic 50

Coating

100 kHz 0.511 4.44 11.3

attenuation is much less than for the bitumastic coating. It is also noticeable that the

attenuation increases monotonically with frequency for epoxy, but not for bitumastic

coating whose attenuation at 50 kHz is higher than that at other frequencies.

Therefore, in order to estimate the guided wave inspection distance of a coated pipe, it

is necessary to analyze the specific situation based on the coating material and various

guided wave parameters.

92

6.4 Wave Focusing in a Coated Pipe

As demonstrated previously, focusing enhances the inspection ability

tremendously for defects in a bare pipe. However, for pipes coated with viscoelastic

materials, it is very crucial to investigate the focusing possibility and the coating

effect which is now studied for the 1st time. Therefore, the FEM modeling study of

phased array focusing in coated pipes is carried out for no coating, a 3 mm epoxy

coating and a 3 mm bitumen coating. The results are shown in Figure 6-3. The axial

profiles for axisymmetric loading are also plotted for comparison purpose. It is very

exciting to see that focusing is realized quite well at the expected distance and that the

signal magnitudes are increased tremendously. As an example, at the focal point, the

focused wave amplitude for the pipe coated with bitumen is even higher than the

wave amplitude in a bare pipe. The quantitative information about the amplitude gain

is tabulated in Table 6-4 in which about 8 dB gain can be acquired from focusing.

Angular profiles at the focal point are also plotted in Figure 6-4 comparing with the

one in a bare pipe. Angular profiles are important because they help decide the

inspection and circumferential sizing resolution potential. It is very nice to see from

the profile after normalization the profile shape stays almost unchanged although

there is some amplitude loss.

The results are shown in Figure 6-4 and Table 6-4, only for the distance of 1.5

meters. Modeling work for a focal distance of 3 meters was also carried out and the

results are shown in Figure 6-5, Figure 6-6 and Table 6-5. Although the gains caused

by focusing are generally smaller than those for the 1.5-meter focal distance, it can be

seen that the gain value is different for the three coating conditions. The gain for the

93

epoxy is much larger than those for the bare pipe and bitumen, indicating focusing

gain is also a function of coating materials besides the distance.

More generally, similar modeling work was also conducted for a different

frequency, 50 kHz, in order to find the generality of the focusing effect in a coated

pipe. Results are shown in Figure 6-7, Figure 6-8 and Table 6-6. Note that 4-channel

focusing was used rather than 8-channel focusing for the 50 kHz frequency. This is

because modes are separated more in a low frequency range as shown in Figure 2-4.

In other words, fewer modes are available with slightly different phase velocities, thus

resulting in fewer channel number. Results show that focusing was realized very well

and gains were from 5 to 6 dB.

A similar phenomenon can be also observed from experiments. Figure 6-9

shows two waveforms obtained using focusing techniques in two 28ft long, 16”

diameter, schedule 30 pipes. In both pipes, guided waves at 30 kHz are focused on a

6mm deep, 3.26% CSA, 63% through-wall saw cut. Results obtained from the

coating-free reference pipe are displayed in (a) and those for the tar coated pipe in (b).

When comparing the two signals it can be seen that there is, approximately, a 14dB

loss due to the tar coating. Although the experiment has a different setting than the

modeling, such as pipe size, frequency, etc, it still provides much confidence on the

modeling results and the corresponding conclusions. Guided wave phased array

focusing is realizable in a coated pipe in spite of the amplitude loss compared to a

bare pipe. Further study will be carried out to investigate any other possible effects

from coating with other frequencies and/or materials and to confirm the conclusion

that has been reached so far.

94

Figure 6-3 Axial profile for the 100 kHz L(0, 2) wave with axisymmetric and phased

array loading, for no coating, 3 mm epoxy and 3 mm bitumen. Note that focusing

increase the magnitude significantly at the focal point.


at a distance of 1.5 meters

Amplitude at the focal point Material

Focusing Axisymmetric

Amplitude

Ratio

Gain

(dB)

No Coating 1.04E-9 3.97E-10 2.62 8.37

Mereco 303

Epoxy

8.18E-10 3.62E-10 2.26 7.08

Bitumastic 50

Coating

4.71E-10 2.01E-10 2.34 7.38

0.E+00

2.E-10

4.E-10

6.E-10

8.E-10

1.E-09

1.E-09

0 0.5 1 1.5 2 2.5 3


Dis

pla

cem

en

t M

ag

nit

ud

e

No Coating

Epoxy

Bitumen

Focusing results

95

(a) before normalization (b) after normalization



attenuation but that the profile shape stays the same.


array loading (focal distance equal to 3 m), for no coating, 3 mm epoxy and 3 mm

bitumen. Note that focusing increase the magnitude significantly at the focal point.

0.E+00

1.E-10

2.E-10

3.E-10

4.E-10

5.E-10

6.E-10

7.E-10

8.E-10

0 0.5 1 1.5 2 2.5 3 3.5 4


Dis

pla

ce

me

nt

Mag

nit

ud

e

No Coating

Epoxy

Bitumen

Focusing results

5e-010

1e-009

1.5e-009

300

120

330

150

0

180

30

210

60

240

90 270

No Coating

Epoxy

Bitumen

0.2

0.4

0.6

0.8

1

300

120

330

150

0

180

30

210

60

240

90 270

No Coating

Epoxy

Bitumen

96


a distance of 3 meters



Amplitude

Ratio

Gain

(dB)

No Coating 4.33-10 2.57E-10 1.68 4.51

Mereco 303

Epoxy

3.53E-10 1.50E-10 2.35 7.43

Bitumastic 50

Coating

8.92 E-10 4.91E-10 1.82 5.20

(a) before normalization (b) after normalization




0.2

0.4

0.6

0.8

1

300

120

330

150

0

180

30

210

60

240

90 270

No Coating

Epoxy

Bitumen

1e-010

2e-010

3e-010

4e-010

5e-010

300

120

330

150

0

180

30

210

60

240

90 270

No Coating

Epoxy

Bitumen

97

0.0E+00

5.0E-10

1.0E-09

1.5E-09

2.0E-09

2.5E-09

3.0E-09

0 0.5 1 1.5 2 2.5 3


Dis

pla

cem

en

t M

ag

nitu

de

No Coating

Epoxy

Bitumen


array loading (focal distance equal to 1.5 m), for no coating, 3 mm epoxy and 3 mm

bitumen. Note that focusing increase the magnitude significantly at the focal point.


a distance of 1.5 meters



Amplitude

Ratio

Gain

(dB)

No Coating 2.45-9 1.32E-9 1.85 5.37

Mereco 303

Epoxy

2.21E-9 1.12E-9 1.97 5.89

Bitumastic 50

Coating

5.69 E-10 2.97E-10 1.92 5.67

98

5e-010

1e-009

1.5e-009

2e-009

2.5e-009

300

120

330

150

0

180

30

210

60

240

90 270

No Coating

Epoxy

Bitumen

0.2

0.4

0.6

0.8

1

300

120

330

150

0

180

30

210

60

240

90 270

No Coating

Epoxy

Bitumen


mm bitumen for 50kHz longitudinal waves, showing that coating introduces some


Figure 6-9 Focusing experiments on (a) a bare pipe and (b) a tar-coated pipe with a

6mm deep, 3.26% CSA, 63% through-wall saw cut in both, showing focusing in tar-

coated pipe with torsional waves.

35kHz 1f1 236in.

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 1000 2000 3000 4000 5000 6000 7000

Backwall Echo

Defect Echo

35kHz 1f1 231in.

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 1000 2000 3000 4000 5000 6000 7000

Backwall Echo

Defect Echo

(a)

(b)

99

6.5 Summary and Rules of Thumb

A two layer 3-D finite element model has been developed to study the coating

effect on guided wave propagation and focusing in a coated pipe. A transformation

algorithm from coating acoustic properties to complex viscoelastic coating properties

has been developed to provide inputs to 3-D FE models. Some rules of thumb are

summarized as follows:

• Under a 50 dB attenuation threshold, as shown in Table 6-3, axisymmetric

longitudinal waves can propagate 714 m, 72.5m and 40 m for 30 kHz, 50 kHz,

and 100 kHz respectively in a pipe coated with epoxy. The propagation

distance reduces to 48 m, 8m and 11 m for a pipe respectively with bitumen

coating. Therefore low frequency is highly recommended.

• It is shown that a viscoelastic coating has no effect on a focusing capability for

the studied frequency, although there is an amplitude loss which is dependent

on the coating viscosity and frequency. This was also confirmed by

experimental studies.

• Phased array focusing with longitudinal waves increases the signal energy by

9 to 16 dB in both bare pipe and coated pipe for the frequencies and distances

studied, indicating a much longer propagation distance and also improved

defect detection sensitivity.

100

Chapter 7 The Coating Property Effect on Wave Attenuation

7.1 Introduction

For guided wave inspection, the valid inspection distance is an important

parameter for uncoated and coated pipe. The distance can effectively be increased

greatly by utilizing the focusing technique. But coating presence can still introduce a

significant reduction in the inspection distance. Since the attenuation is caused by the

viscoelastic coating, it might be possible to find a coating with similar performance

but a smaller attenuation, hence leading to a recommendation it to the pipeline

industry for new work. This could be realized by a parametric study based on a large

number of FEA computations. However, since the FE modeling is a slow and time

consuming process, a faster and easier way for a wave attenuation study is utilized

hence making use of guided wave attenuation dispersion curves.

7.2 Guided Waves in Viscoelastic Coated Pipes

Barshinger [2004] studied axisymmetric guided waves in a viscoelastic

multilayered pipeline utilizing the global matrix method and a simplified complex

root searching algorithm. However, his code was limited to very lightly viscoelastic

coating materials and had problems of root finding in the low frequency range below

30 kHz. In order to overcome the limitations, an improved algorithm is used to find

101

Figure 7-1 Cylindrical coordinates of a hollow viscoelastic coated cylinder and dimensions.

the proper dispersion curves for highly viscoelastic materials.

7.2.1 Dispersion Curves in Viscoelastic Multilayer For a pipe with a viscoelastic coating, the phase velocity dispersion curve,

attenuation dispersion curve and wave structure can be calculated using the global

matrix technique [Lowe 1995]. Figure 7-1 shows a viscoelastic coated pipe.

According to the correspondence principle (Christensen, 1981), under a steady state,

time harmonic assumption, the guided wave propagation solution for a N-layered pipe

with one or more viscoelastic layers can be acquired simply by replacing the elastic

modulus of the relative layers with the complex viscoelastic modulus into the

dispersion equation for a totally elastic N-layered pipe. Therefore, the problem of an

N-layered elastic pipe should be solved first.

θ r

z

r1 r3

r2

Viscoelastic Elastic Layer

102

Figure 7-2 Flow chart for developing dispersion curves of a multilayered structure

The flow chart of the dispersion curve development, for longitudinal waves as

an example, is shown in Figure 7-2. The dispersion equation of an N-layered

viscoelastic pipe can be generated similarly by just using the complex viscoelastic

constants instead of the elastic constants. For longitudinal waves, boundary conditions

are 4 traction-free conditions at the inner and outer surface as in equation (7.1) and

4(N-1) displacement and stress continuity conditions at the (N-1) interfaces as in

equation (7.2):

01,1

=

+= Nrrrrz

rr

σσ

(7.1)

0

11

1

=

−

++ =+=

==

nn rrnLayerrz

rr

r

z

rrnLayerrz

rr

r

z

u

u

u

u

σσ

σσ

, n=1,2,…,N-1 (7.2)

Governing equation + Helmholtz decomposition for the nth layer

Separation of variables

Apply surface traction-free and continuity conditions

Dispersion equation

Eigenvalues: Dispersion curves Eigenvectors: Wave structures

Displacement and stress expression for the nth layer

103

There are a total of 4N boundary conditions for longitudinal waves (2N for torsional

waves). The expression of displacement and stress for a single layer can be obtained

from the guided wave solution by considering circumferential order equal to 0. Each

layer has four unknown coefficients, and thus there are a total of 4N unknowns. The

NN 44 × global matrix can be constructed by applying the 2N boundary conditions

and then the dispersion equation can be acquired as in equation (7.3):

044

=× NN

A (7.3)

7.2.2 Dispersion Equation Solution

A numerical method is needed to solve the dispersion equation and the root

will be the wave number k. For a multi-layer pipe with viscoelastic layers, the wave

number k* become a complex number. The complex root is composed of the real part

Re(k*), which results in the phase velocity dispersion curves, and the imaginary part

Im(k*), leads to the attenuation dispersion curves:

*)/1Re(

1*)Re(

)(ck

cp == ωω (7.4)

*)/1Im(

*)Im()(c

kωωα == (7.5)

Note that both the phase velocity and attenuation constant are functions of the

frequency. Different from the elastic case, the attenuation axis is introduced and

consequently the complex root search has to be carried out in three dimensions:

frequency, velocity and attenuation. It is very difficult to find a root with both the real

104

and imaginary parts equal to zeros, which leads to a four-dimensional search.

Therefore, the minimization of the absolute value of the characteristic matrix

determinant is usually utilized for the complex root searching (Lowe, 1995).

7.2.3 Root Searching Algorithm Shown in Figure 7-3 is a general root searching algorithm used to find the

roots (also the minima of the dispersion equation) in the three-dimensional plane. A

root tracing algorithm was used: First, fix one variable or axis, say frequency, and

then do a fine and full (will be slow though) two-dimensional search in the two-

dimensional plane comprised of the other two axes. Once a root at one frequency 1f is

found, it can be used as an initial guess and a starting point from which a fine 2-D

searching begins for the next step at frequency 2f = 1f + f∆ . The roots solved at these

two frequencies 1f and 2f can then be used to give a linear prediction of the third

root. After finding 3, 4 and more roots, quadratic, cubic and spline extrapolations

could be applied respectively for new root predictions. This method is called root

following or curve tracing, which has been demonstrated to be a general, accurate and

efficient way (Lowe, 1995) and (Xu and Jenot, 2004). A good initial guess could save

searching time and also improve the accuracy tremendously. With the initial guess, a

searching algorithm Muller method could be used to find the minima of the dispersion

matrix around the guess [Mathews 1999]. The Muller method usually takes only

several times to converge to the expected tolerance and is much faster than other

methods such as the Simplex method used in Matlab function “fmin” or “fminbnd”

[refer to Matlab help].

105

Attenuation

Phase velocity 1. Initial point by performing a fine 2-D search at 1f

Linear prediction from roots found at 1f and 2f

cubic or spline extrapolation

2f1f

3f

Quadratic extrapolation

1−nfnf

4f

1+nf Find the exact root of the viscoelastic dispersion equation by Müller method

Attenuation

Phase velocity 1. Initial point by performing a fine 2-D search at 1f

Linear prediction from roots found at 1f and 2f

cubic or spline extrapolation

2f1f

3f

Quadratic extrapolation

1−nfnf

4f

1+nf Find the exact root of the viscoelastic dispersion equation by Müller method

Figure 7-3 The curve tracking process to predict the starting point of the searching at the next frequency by an extrapolation of previous roots.

7.3 Coating Property Effect Investigation With the help of attenuation dispersion curves, the coating material property

was then investigated with respect to the effect on wave attenuation from 20 kHz to

160 kHz for L(0,1), L(0,2) and T(0,1) modes in a 10 inch schedule 40 pipe.

Properties like density, velocity and attenuation constants are considered for the

investigation. Epoxy (light viscoelastic) and bitumastic material (highly viscoelastic)

are used as the basic materials whose properties can be found in Tables 6-1.

Parametric studies were carried out by increasing or decreasing the basic material

property values with a goal to find the property effects on wave attenuation.

106

7.3.1 Coating Material -- Mereco 303 Epoxy

At first, a parametric study is conducted for the coating material of Mereco

303 epoxy. Figure 7-4 shows the results of the density influence on wave attenuation.

The density values used for the study are ½, 1, 2, and 4 times that of the original

epoxy density. Both attenuation and phase velocity dispersion curves are plotted.

The relationships between attenuation vs. density are then summarized in Figure 7-5

for three selected frequencies 50 kHz, 100 kHz and 150 kHz. It is interesting to find

that the attenuation is nearly a linear function of the density for all of the three

axisymmetric modes L(0,1), L(0,2) and T(0,1). Apparently, for higher frequencies,

the line slope is much larger, which indicates a more sensitive relationship with the

density.

It is also worthwhile to see that the torsional wave has a much larger

attenuation value than for longitudinal waves. This can be explained by the

attenuation constant difference in Table 6.1. Those attenuation constants were

measured at high frequencies (MHz) which give directly the viscoelasticity or

complex modulus. The relationship of the wave attenuation constants to the

calculated attenuation dispersion curve is similar to the bulk wave velocity and the

calculated phase velocity dispersion. For a certain viscoelastic material, the shear

wave attenuation is usually higher than the longitudinal attenuation, which was

approved by the measurement of such researchers as Barshinger [2004] and Simonetti

[2003].

107

A. Density influence

-6

-5

-4

-3

-2

-1

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(a) Four times, )/(32.4 3cmg=ρ

-2.5

-2

-1.5

-1

-0.5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(),1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(b) Double, )/(16.2 3cmg=ρ

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(c) Original )/(08.1 3cmg=ρ

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(d) Half, )/(54.0 3cmg=ρ

Figure 7-4 Attenuation and phase velocity dispersion curves as functions of densities in a 10’’ schedule 40 steel pipe with 1mm Mereco 303 Epoxy coating. Note that the original density is 1.08 g/cm3

, other densities values are used for the parametric study.

108

-1.8-1.6-1.4-1.2

-1-0.8-0.6-0.4-0.2

0

0 1 2 3 4 5

ρ (ρ (ρ (ρ (g/cm3))))

Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(a) L(0,1)

-1.6-1.4

-1.2-1

-0.8-0.6

-0.4-0.2

0

0 1 2 3 4 5

ρ (ρ (ρ (ρ (g/cm3))))

Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(b) L(0,2)

-4-3.5

-3-2.5

-2-1.5

-1-0.5

0

0 1 2 3 4 5

ρ (ρ (ρ (ρ (g/cm3))))

Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(c) T(0,1)

Figure 7-5 Attenuation vs. density curves for selected 3 frequencies 0.05MHz, 0,1MHz and 0.15MHz, showing that attenuation increases almost linearly with an increase of density. Note that the change is more dramatic for higher frequency. Torsional mode has a larger attenuation than longitudinal modes.

109

B. Influence of the longitudinal wave attenuation constant ωα /1

Parametric studies on the longitudinal attenuation constant ωα /1 was also

carried out with results summarized in Figure 7-6 and Figure 7-7. Note that the

longitudinal wave attenuation constant only has a contribution to the longitudinal

wave computation but not to the torsional wave. The torsional wave computation only

needs the shear wave attenuation constant ωα /2 . Similarly, only shear modulus or

shear bulk wave velocity is needed for the torsional wave computation. Therefore,

shown in Figure 7-7 are the results only for the longitudinal waves L(0,1) and L(0,2).

Once again, almost linear relationships between attenuation and ωα /1 are observed.

However, from the line slopes, it can be seen that the attenuation of the L(0,2) wave is

less sensitive to ωα /1 than the L(0,1) mode.

110

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

00 0.05 0.1 0.15 0.2

frequency(Mhz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(e) a quarter 00175.0/1 =ωα

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(d) half 0035.0/1 =ωα

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

(c) original 0070.0/1 =ωα

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)C

p(km

/s) L(0,1)

L(0,2)

T(0,1)

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

(b) double 014.0/1 =ωα

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)L(0,1)

L(0,2)

T(0,1)

(a) four times 0070.0/1 =ωα

Figure 7-6 Attenuation and phase velocity dispersion curves as functions of ωα /1 in a 10’’ schedule 40 steel pipe with 1mm Mereco 303 Epoxy coating.

111

-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1

0

0 0.005 0.01 0.015 0.02 0.025 0.03

αααα1111 /ω/ω/ω/ω

Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(a) L(0,1)

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 0.005 0.01 0.015 0.02 0.025 0.03


Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(b) L(0,2)

Figure 7-7 Attenuation Vs. the longitudinal wave attenuation constant ( ωα /1 ), for 3 frequencies, 0.05MHz, 0.1MHz and 0.15MHz, showing that the attenuation increases linearly with ωα /1 , and the increase of L(0,2) is less sensitive to the attenuation constant than L(0,1). The change of ωα /1 has no effect on T(0,1).

112

C. Influence of the shear wave attenuation constant ωα /2

The investigation of the shear wave attenuation constant is shown in Figure

7-8 and Figure 7-9. The attenuation also increases linearly with ωα /2 for the selected

3 frequencies. The influences of ωα /2 on L(0,1) and L(0,2) are very similar. But

torsional waves still have a larger attenuation value than the longitudinal waves.

113

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

-0.25

-0.2

-0.15

-0.1

-0.05

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

Figure 7-8 Attenuation and phase velocity dispersion curves as functions of ωα /2 in a 10’’ schedule 40 steel pipe with 1mm Mereco 303 Epoxy coating.


(b) double 0402.0/2 =ωα


(d) half 01.0/2 =ωα

(e) half 005.0/2 =ωα

114

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.02 0.04 0.06 0.08 0.1


Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(a) L(0,1)

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.02 0.04 0.06 0.08 0.1


Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(b) L(0,2)

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.02 0.04 0.06 0.08 0.1


Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(c) T(0,1)

Figure 7-9 Attenuation Vs. the shear wave attenuation constant ( ωα /2 ), for 3 frequencies, 0.05MHz, 0.1MHz and 0.15MHz, showing that the attenuation increases linearly with ωα /2 , and the higher frequency, the larger the slope, which means more sensitivity.

115

D. Velocity Influence

Due to the relationship of longitudinal wave velocity and shear wave velocity,

the two velocities are changed simultaneously for a parametric study. See the results

in Figure 7-10 and Figure 7-11. A linear relationship between attenuation and velocity

has not been seen but a monotonic relationship still exists. Here the L(0,1) has a

larger attenuation value than the other two waves.

-12

-10

-8

-6

-4

-2

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(a) )/(96.3),/(56.9 skmcskmc tl ==

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(b) )/(98.1),/(78.4 skmcskmc tl ==

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(c) )/(99.0),/(39.2 skmcskmc tl ==

Figure 7-10 Attenuation and phase velocity dispersion curves as functions of lc and

tc in a 10’’ schedule 40 steel pipe with 1mm Mereco 303 Epoxy coating.

116

-10-9-8-7-6-5-4-3-2-10

0 2 4 6 8 10 12

CL(km/s)

Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(a) L(0,1)

-2.5

-2

-1.5

-1

-0.5

0

0 2 4 6 8 10 12

CL(km/s)

Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(b) L(0,2)

-5-4.5

-4-3.5

-3-2.5

-2-1.5

-1-0.5

0

0 2 4 6 8 10 12

CL(km/s)

Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(c) T(0,1)

Figure 7-11 Attenuation vs. longitudinal velocity, for 3 frequencies, 0.05MHz, 0.1MHz and 0.15MHz, showing that the attenuation increases monotonically with velocities, although not linearly.

117

7.3.2 Coating Material -- Bitumastic 50

A parametric study based on the property of a highly viscoelatic bitumastic

material is also conducted as shown in Figure 7-12 through Figure 7-17. The

influences from density and ωα /1 are quite similar with those for an epoxy material,

while the influence from ωα /2 and wave velocity are more complicated. For

example, a monotonic relationship between ωα /2 and attenuation has not been seen

for torsional waves. And for a velocity influence, the relationship is not very clear.

These are probably caused by the complex root searching difficulty when the material

is highly viscoelastic and with larger values of ωα /1 and ωα /2 .

118

A. Density Influence

-300

-250

-200

-150

-100

-50

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(a) four times )/(0.6 3cmg=ρ

-120

-100

-80

-60

-40

-20

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(b) double )/(0.3 3cmg=ρ

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(c) original )/(5.1 3cmg=ρ

-25

-20

-15

-10

-5

00 0.05 0.1 0.15 0.2

frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(d) Half )/(75.0 3cmg=ρ

Figure 7-12 Attenuation and phase velocity dispersion curves as functions of densities in a 10’’ schedule 40 steel pipe with 1mm Bitumastic 50 coating.

119

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8

ρρρρ (g/cm3)

Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(a) L(0,1)

-60

-50

-40

-30

-20

-10

0

0 2 4 6 8

ρρρρ (g/cm3)

Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(b) L(0,2)

-180-160-140-120-100-80-60-40-20

0

0 2 4 6 8

ρρρρ (g/cm3)

Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(c) T(0,1)

Figure 7-13 Attenuation vs. density curves for selected 3 frequencies 0.05MHz, 0,1MHz and 0.15MHz. Note: similarly with epoxy, the results show that attenuation increases almost linearly with an increase of the density, and again attenuation is more sensitive to high frequency.

120

B. Influence of the longitudinal wave attenuation constant ωα /1

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)


-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(b) double 046.0/1 =ωα

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)


-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

(d) half 0115.0/1 =ωα

Figure 7-14 Attenuation and phase velocity dispersion curves as functions of ωα /1 in a 10’’ schedule 40 steel pipe with 1mm Bitumastic 50 coating.

121

-9-8-7-6-5-4-3-2-10

0 0.02 0.04 0.06 0.08 0.1


Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(a) L(0,1)

-12

-10

-8

-6

-4

-2

0

0 0.02 0.04 0.06 0.08 0.1


Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(b) L(0,2)

Figure 7-15 Attenuation Vs. the longitudenal wave attenuation constant ( ωα /1 ), for 3 frequencies, 0.05MHz, 0.1MHz and 0.15MHz, showing that the attenuation increases nearly linearly with ωα /1 , and the attenuation for 0.05MHz is very small and almost kept the same as ωα /1 changes. L(0,2) is less sensitive than L(0,1).

122

C. Influence of shear wave attenuation constant ωα /2

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

-40

-35

-30

-25

-20

-15

-10

-5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)C

p(km

/s) L(0,1)

L(0,2)

T(0,1)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

-40

-35

-30

-25

-20

-15

-10

-5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

-25

-20

-15

-10

-5

00 0.05 0.1 0.15 0.2

Frequency(MHz)

Atte

nuat

ion(

dB/m

)

L(0,1)

L(0,2)

T(0,1)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2

Frequency(MHz)

Cp(

km/s

) L(0,1)

L(0,2)

T(0,1)

Figure 7-16 Attenuation and phase velocity dispersion curves as functions of ωα /2 in a 10’’ schedule 40 steel pipe with 1mm Bitumastic 50 coating.


(b) two times 48.0/2 =ωα


(d) half 12.0/2 =ωα

(e) a quarter 06.0/2 =ωα

123

-9-8-7-6-5-4-3-2-10

0 0.2 0.4 0.6 0.8 1 1.2


Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(a) L(0,1)

-12

-10

-8

-6

-4

-2

0

0 0.2 0.4 0.6 0.8 1 1.2


Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(b) L(0,2)

-30

-25

-20

-15

-10

-5

0

0 0.2 0.4 0.6 0.8 1 1.2


Atte

nuat

ion(

dB/m

)

0.05MHz

0.1MHz

0.15MHz

(c) T(0,1)

Figure 7-17 Attenuation Vs. longitudinal wave attenuation constant ( ωα /2 ), for 3 frequencies, 0.05MHz, 0.1MHz and 0.15MHz, showing that the attenuation increases monotonically with ωα /2 for 0.05 MHz and 0.1 MHz but not for 0.15 MHz.

124

7.4 Summary and Rules of Thumb

Attenuation dispersion curves of guided waves in a viscoelastic coated pipe

were calculated with a goal to investigate the coating property effect on wave

attenuation. A tracking method was used for the root searching of highly viscoelastic

material. The observations from the parametric studies are summarized as follows

with some simple rules of thumb based on the observation:

1) Density effect

Attenuation has a nearly linear increasing relationship with density for both lightly

and highly viscoelastic materials. For example, the attenuation will be doubled if

the density becomes two times larger. The line slope increases with frequency,

indicating that the attenuation is more sensitive to density changes for higher

frequencies.

2) Longitudinal attenuation constant ωα /1 effect

There is also an almost linear increasing relationship between wave attenuation and

longitudinal attenuation constants for both the L(0,1) and L(0,2) waves. For

example, doubling ωα /1 will lead to a doubled attenuation. The longitudinal

attenuation constant has no effect on torsional wave attenuation. For the L(0,1)

mode, attenuation is more sensitive to ωα /1 at higher frequency. For the L(0,2)

mode, the trend of the attenuation change is about the same for all frequencies.

125

3) Longitudinal attenuation constant ωα /2 effect

The effect from ωα /2 depends on material viscoelasticity. For lightly viscoelastic

materials (epoxy), the linear increasing relationship between attenuation and ωα /2

still exists. For highly viscoelastic materials (bitumen), there is a monotonic

relationship between ωα /2 and attenuation only for low frequency (say, less than

100 kHz).

4) Longitudinal and shear wave velocity effect

There is a monotonic (not linear) relationship between velocity and attenuation

only for a lightly viscoelastic material. The relationship for highly viscoelastic

material is unclear.

5) Acoustic Impedance

It has been found that attenuation increases with both density and velocity

whose product is the acoustic impedance. With the concept of impedance

matching, it is easier to explain the relationship between attenuation and density

and velocity. Since generally the acoustic impedance of a coating material is

smaller than that of steel, an increment of density and/or velocity indicates a closer

acoustic impedance matching which leads to larger amount of energy transmitted

into coating material, thus causing higher attenuation.

6) Frequency selection

For all cases, low frequency ( ≤ 50 kHz) exhibits a much lower attenuation than

high frequency. Therefore, low density, low viscosity and low velocity for coating

material selection, and low frequency for inspection, are highly recommended.

126

7) Wave selection

Computation has shown that torsional waves have much larger attenuation than

longitudinal waves in a viscoelastic coated pipe. This is due to the nature that

viscoelastic material is more attenuative for shear waves. Therefore, longitudinal

waves are recommended rather than torsional waves from a wave attenuation

point of view.

127

Chapter 8 Experiments

8.1 Introduction Our previous modeling and experimental studies have shown that attenuation is

dependent upon coating properties. Acoustic coating properties were also used to

estimate the viscoelastic properties integrated into the finite element model for

viscoelastic coated pipes. Therefore, a proper measurement of the coating property is in

great need for future studies of guided waves in coated pipe. Presented here are the

results of field experiments performed on coated pipes.

8.2 Attenuation Experiments on Coated Pipelines in the Field

Some results of lab experiments on coatings performed at FBS, Inc. and

experiments performed at Battelle, Columbus, are reported here. Figure 8-1 (same as

Figure 1-2 ) shows pipes coated with different materials, for example, 15-year-old

bitumen tape, epoxy, fibrous coal tar, and bitumen tape. The bonding conditions are

different for each of these samples. For example, the coal tar coating in (c) and the epoxy

coating in (b) have peeled off, while (a) and (d) have tight bonding conditions. Figure 8-2

shows representation of the configuration used to make attenuation measurements with

longitudinal and shear transducers. The experimental results along with some sample

analysis from the pipe of Figure 8-1(a) are shown in Figure 8-3. Figure 8-3(b) shows the

128

Figure 8-1 Field coated pipelines showing differences in field coating conditions

received longitudinal waveform at frequency 2 MHz along with an exponential curve fit

to the echo peaks. The exponential curve gives the value of attenuation constant. Similar

experiments were conducted for frequencies ranging from 1 MHz to 3 MHZ in 0.25 MHz

steps. The attenuation constants at these frequencies were calculated and plotted in Figure

8-3(c). According to viscoelastic theory, the attenuation constant is a linear function of

frequency. Therefore, a least squares linear fit was used to obtain the trend of the

experimental attenuation constants.

(a) 15-years-old bitumen tape on a 30-inch pipe (b) Epoxy on a 24-inch pipe

(c) Fibrous coal tar coating on a 24-inch pipe (d) new-bonded bitumen tape on a 10-inch pipe

129

Figure 8-2 Experimental schematic diagram for coating attenuation measurement with a normal beam transducer

(a)

(b) (c)

Figure 8-3 Results of attenuation experiments using a longitudinal normal beam transducer on a 30” Schedule 10 pipe coated with 15-year-old bitumen tape: (a) transducer in the pulse-echo mode; (b) a waveform at 2 MHz with a exponential curve fit to the echo peaks, where the exponential curve gives the attenuation constant )(ωα ; (c) attenuation constant vs. frequency, showing that the attenuation constant is a quasi-linear function of frequency.

Transducer working in pulse-echo mode

coating

Steel pipeline

y = 0.0537x + 0.0057

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

1 1.5 2 2.5 3

Frequency (MHz)

Att

enua

tion

con

stan

t (1

/mm

)

130

For wave propagation in a viscoelastic medium, the solution involves not only a

harmonic wave propagation term, but also a decaying exponential term xe )(ωα− causing

wave attenuation as a function of distance. Physically, this expression represents the

exponential attenuation curve in Figure 8-3 (b). The frequency-dependent attenuation

constant )(ωα can be calculated once the exponential fitting curve of a certain frequency

is acquired. The slope of the linear fitting line shown in Figure 8-3(c) presents ωωα /)(

which is a constant for a particular material. This constant is used as an input parameter

for the multi-layer modeling of wave propagation.

Similar attenuation experiments using shear waves were performed with the

results as shown in Figure 8-4. The shear transducer used had a frequency band different

from that of the longitudinal transducer and therefore the frequency tuning range is

different. Figure 8-5(a) shows a newly applied bitumen tape patch on the steel surface of

the pipe. Measurements exactly like those of Figure 8-4 were taken using shear waves,

but with values totally different than those of the original taped inspection. Figure 8-5(b)

shows these latter values. It is interesting when one notes that there are almost no

reflection signals [even with increased gain]. Measurements on the patch coating using

longitudinal waves were also carried out. Values were similar to the values shown in

Figure 8-3(b) except for a 12 dB amplitude loss. The loss is most likely due to the loose

bonding condition of the new tape as compared with that of the 15-year-old tape.

Additionally, the in-plane particle vibration of shear waves coupled with the loose

bonding layer makes wave energy penetration difficult. This variation of amplitude

131

suggests that a three-layer model, pipe, interface, and coating is needed for wave

propagation modeling for coated pipe.

y = 0.0987x + 0.005

0

0.02

0.04

0.06

0.08

0.1

0.12

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Frequency (MHz)

Att

enua

tion

Con

stan

t (1

/mm

)

Figure 8-4 Attenuation constant vs. frequency curve of shear waves on the same coated pipe in Figure 8-3.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 20 40 60 80 100

Time (µsec)

Am

plit

ud

e

(a) (b)

Figure 8-5 (a) Normal beam incidence with a shear transducer on a new patched area with a new bitumen tape; (b) Wave signals showing very low reflected waves due to the new bonding condition incapable for shear wave transmission.

132

(a)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60

Time (µ sec)

Am

plit

ud

e

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 20 40 60 80 100

Time (µ sec)

Am

plit

ude

(b) (c)

Figure 8-6 Experiments using a normal beam longitudinal transducer on a 24’’ schedule 10 pipeline (6.35mm wall thickness) with 3 mm fibered tar coating: (a) coated and uncoated area; (b) 2 MHz signals with the transducer in pulse-echo mode coupled on steel surface directly; (c) 2 MHz signals with the transducer in pulse-echo mode on coating surface directly, showing wave energy can not penetrate the coating due to the poor bonding.

Another example illustrating the importance of considering the bonding layer is

provided by the results of measuring a fibrous coal tar coating. See Figure 8-6. There was

no apparent reflected wave energy. This indicates a very low penetration power which

would somehow disallow guided wave inspection over a few feet. In this case, the

coating should be removed in order to improve the coupling conditions, consequently

increasing the penetration power.

133

A preliminary through transmission guided wave experiment was also conducted

using the configuration shown in Figure 8-7 (a). The transducers were placed 22 mm

apart, in a through transmission mode, and excited at a frequency of 540 kHz. Note that

the transducer was coupled directly onto an exposed steel section. There were no signals

if coupled directly on the fibrous coating. Figure 8-7 (b) shows guided wave propagation

in an exposed portion of steel pipe while (c) shows the results with coating between the

transmitter and receiver. Note some modes with lower energy are attenuated while the

dominant mode propagated with little influence from the tar coating. This is probably due

to the different attenuation responses from the coating for different modes. It is possible

to find certain wave modes less affected by viscoelastic coatings. Coating properties as

well as the bonding conditions of course play important roles.

In summary, as is obvious, not all coatings are the same. Some have very serious

effects on guided wave inspection and some, virtually no effect. There is much more

work that has to be performed to access the impact of coatings and their boundary layers

(coating to pipe interface). Additionally, dispersion curve and wave scattering computer

models require accurate material properties as input to produce realistic (usable) results.

134

(a)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 100 200 300 400 500 600 700 800

Time (µµµµ sec )

Am

plit

ud

e

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 100 200 300 400 500 600 700 800

Time (µsec)

Am

plitu

de

(b) (c)

Figure 8-7 Guided wave experiments for a 24” schedule 10 pipeline with 3 mm fibrous tar coating: (a) pitch-catch experiment setup with tar coating in the middle; (b) 540 kHz wave signals for pitch-catch mode with transducers 22 mm apart on steel surface directly; (c) 540 kHz wave signals for pitch-catch mode with transducers 22 mm apart on steel surface, with tar coating in the middle as shown in (a), showing that some wave energy was attenuated but the major part of the waveform still exists.

8.3 Acoustic Property Measurement of Various Sample Coatings As the foregoing attenuation experiments have shown, bonding conditions have a

significant influence on attenuation. The variation of coating properties is another

important consideration. For example, see Figure 8-8. Consider the 6 coating samples;

(1) through (4) removed from service and (5) and (6), newly applied bitumen tape

Transmitter Receiver

Guided Wave

Bitumin coating

Steel pipeline

135

patches. Measurements were taken on these specimens to investigate the differences

between coating properties. The properties that we evaluated were density, longitudinal

and shear wave velocity, and attenuation. These, in turn, can be used as inputs to the

model of wave propagation. Through-transmission measurements were taken with a pair

of 150 kHz longitudinal transducers as depicted in Figure 8-9. Figure 8-10 shows plots

of longitudinal velocity and attenuation. Shear wave results are shown in Figure 8-11. It

is seen that there are dramatic differences among all the coatings. Even the two new

bitumen tapes having the same thickness and similar appearance, have a significant

variation with respect to longitudinal velocity, shear velocity and shear wave attenuation.

Note the attenuation values are much higher than the attenuation observed in the previous

section. This is most likely due to coupling differences. These values give a relatively

accurate physical comparison of the attenuation variation among the coatings under the

same experimental conditions.

The densities of coating samples were measured with an AccuPyc 1330

Pycnometer, an automatic density analyzer. The results are shown in Table 8-1. The

differences between the measured properties themselves indicate the necessity for in-situ

coating property measurement. These measurements reinforce our assertion that coating

properties show wide variations and that these variations play a role in determining

guided wave penetration power within a pipe.

All of these measured properties including attenuative constant, velocities and

densities will be used as inputs to modeling work, as shown in equations 6.5 to 6.15.

136

(1) Epoxy 1, 1.5 mm (2) Epoxy 2, 0.5 mm

(3) Fibrous Coal Tar, 3 mm (4) 15-year-old Field Bitumen Tape, 1.5 mm

(5) New Bitumen Tape 1, 1 mm (6) New Bitumen Tape 2, 1 mm

Figure 8-8 Different pipeline coating samples with various thicknesses: (1)-(4) Samples collected from field studies; (5)-(6) New coated samples.

137

Figure 8-9 Through Transmission configuration using a 150 kHz longitudinal transducer

0

500

1000

1500

2000

2500

3000

3500

110 130 150 170 190 210 230 250

Frequency (kHz)

Long

itudi

nal V

eloc

ity

(m/s

)

Epoxy 1Epoxy 2coal tar"Field Bitumen Tape"New Bitumen Tape 1New Bitumen Tape 2

(a)

-2

0

2

4

6

8

10

110 130 150 170 190 210 230 250Frequency (kHz)

Att

enua

tion

(dB

/mm

)

Epoxy 1Epoxy 2Coal TarField Bitumen TapeNew Bitumen Tape 1New Bitumen tape 2

Figure 8-10 (a) Longitudinal velocity vs. frequency curve for different coating samples; (b) longitudinal wave attenuation vs. frequency curve showing the attenuation increases slightly with the frequency.

Krautkramer 150 KHz

Transducer

Coating

Couplant

138

0

500

1000

1500

2000

400 500 600 700 800 900 1000

Frequency (kHz)

Shea

r ve

loci

ty (m

/s)

Epoxy 1Epoxy 2Coal TarField Bitumen TapeNew Bitumen Tape 1New Bitumen Tape 2

(a)

0

5

10

15

20

400 500 600 700 800 900 1000

Frequency (kHz)

Att

enua

tion

(dB

/mm

)

Epoxy 1

Epoxy 2

Coal TarField Bitumen Tape

New Bitumen Tape 1

New Bitumen Tape 2

(b)

Figure 8-11 (a) Shear wave velocity vs. frequency curve for different coating samples; (b) shear wave attenuation vs. frequency curve showing that the attenuation increases slightly with the frequency.

Table 8-1 Density measurements of coating samples using AccuPyc 1330

Sample Epoxy 1 Epoxy 2 Coal Tar Field Tape New Tape 1 New Tape 2 ρ ( 3cmg ) 0.9865 1.5490 1.2702 1.1008 1.3087 1.1073

Standard Deviation ( 3cmg )

0.0004 0.0004 0.0004 0.0004 0.0004 0.0001

139

8.4 Guided Wave Experiments in Coated Pipe Guided wave experiments were also carried out on a coated pipe with a goal to

demonstrate some of the observations and conclusions from the theoretical work.

Focusing experiments for coated pipe were first conducted as shown in Figure 8-12. A

phased transducer array was used as a transmitter on a 16” schedule 30 pipe coated with a

viscoelastic wax coating with 2 mm thickness and 4 feet in length. A receiving

transducer was placed 10 feet away from the transmitter to measure the wave field at

various points along the circumference. Both the L(0,2) longitudinal waves and the

T(0,1) torsional waves at 45 kHz were used for the focusing experiment. Shown in

Figure 8-13 are the experimental angular profiles for the 45 kHz longitudinal waves

focused at a 10 feet distance and at 270 degrees in the bare pipe and also in the coated

pipe. An angle beam piezoelectric transducer was used for the longitudinal profile

measurement. It can be seen that wave energy was focused at the expected angle for both

the no coating and wax coating conditions, although there was some amplitude loss due

to the wax coating. Torsional wave angular profiles were measured with a SH EMAT

sensor and the results are shown in Figure 8-14. The wave was once again focused quite

well under the coating condition again with an amplitude loss. These two experiments

agree quite well with the numerical results previously acquired.

140

Figure 8-12 Angular profile measurement experiment of a 16” schedule 30 pipe coated with one ply wax (2 mm in width, 4 feet covered length): (a) Schematic illustration; (b) transducer array; (c) wax coating. A transducer was used as receiver 10 feet away from the transmitter.

Transducer array

Air pump

Wax coating

Steel pipe

(b)

(a)

(c)

Wax coating Transducer array Receiver

141

Figure 8-13 Comparison of angular profiles for uncoated and wax coated pipe when using 45 kHz Longitudinal ( L [0, 2] ) focusing at 10 feet at 270 degree, in a 16 inch Schedule 30 pipe. The coating was one ply, 2 mm wax 4 feet in length. A piezoelectric angle beam transducer was used as the receiver.

Figure 8-14 Comparison of angular profiles for uncoated and wax coated pipe when using 45 kHz Torsional ( T[0, 1] ) focusing at 10 feet at 270 degree, in a 16 inch Schedule 30 pipe. The coating was one ply, 2 mm wax 4 feet in length. A SH EMAT was used as the receiver.

0

0.2

0.4

0.6

0.8

1

No CoatingWax Coating

0o

270o

225o

180o

135o

90o

45o 315

o

0

2

4

6

8

10

12

No CoatingWax Coating

0o

270o

225o

180o

135o

90o

45o 315

o

142

30 40 50 60 70 80 90 100-8

-7

-6

-5

-4

-3

-2

-1

0

Frequency (kHz)

Atte

nuat

ion

(dB

)

longitudinaltorsional

Figure 8-15 Attenuation of longitudinal and torsional axisymmetric wave in a 16” schedule 30 pipe with 28 feet length covered with a ply of wax coating ( 2 feet in length, 2 mm in thickness). The attenuation values were measured by comparing the back echo amplitudes for coated pipe and bare pipe. Note torsional waves have much larger attenuations than longitudinal waves over the frequency range used.

Wave attenuations induced by the wax coating of longitudinal and torsional

waves were also compared. Figure 8-15 shows the attenuation of the longitudinal and

torsional waves in a 30 to 100 kHz frequency range for a 16” schedule 30 pipe (28 feet in

length) coated with the same wax material 2 feet in length. The attenuation was

calculated by comparing the back echo amplitudes for both the coated pipe and bare pipe.

It can be seen that the torsional waves had a much larger attenuation than the longitudinal

waves, which is consistent with the conclusion obtained with the attenuation dispersion

curve computations.

143

(a) (b)

Figure 8-16 Non-planar and planar defects for an experimental wave scattering study in a 16’’ schedule 30 pipe: (a) 90% deep corrosion like defect; (b) 63% deep saw cut.

Wave scattering experiments on a corrosion defect with a 90% depth and a saw

cut with 63% depth in the same pipe covered with the wax coating was carried out.

Waveforms with 40 kHz torsional waves focused at the defect area are shown in Figures

8-17 and 8-18. It is found that the saw cut was much easier to find than the corrosion like

defect, indicating clearly that it was more difficult to be able to find corrosion defects.

Experiments showed that axisymmetric waves cannot find the corrosion even in a bare

pipe while phased array focusing could find the defects in both bare and coated pipes.

This demonstrates the necessity of using phased array focusing for carrying out a valid

defect inspection test.

144

Figure 8-17 Reflected waveform from a corrosion (90% depth) 11 feet away from the transducer array for 40 kHz torsional wave in a 16 inch schedule 40 pipe coated with 2 mm wax, using 4-channel phased array focusing.

Figure 8-18 Reflected waveform from a saw cut (63% depth) 21 feet away from the transducer array for 40 kHz torsional wave in a 16 inch schedule 40 pipe coated with 2 mm wax, using 4-channel phased array focusing.

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 1000 2000 3000 4000 5000 6000 7000

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 1000 2000 3000 4000 5000 6000 7000

Time (µ sec)

Am

plit

ud

e

Reflection from saw cut

Reflection from corrosion

Back wall echo

Back wall echo

145

8.5 Summary Attenuation experiments on in-situ pipe coatings and measurements on various

coating materials have been performed to evaluate the influence of coating properties on

guided wave propagation. It was found that pipe coatings have a large range of property

variation and that bonding condition plays an important role in wave attenuation. In-situ

coating measurements are suggested for attaining coating properties as these can provide

guidelines for inspection design. It is possible to use a 3-layer model to approximate the

influence of bonding conditions on wave propagation. It is also found that coating

removal is needed for some coatings and adhesive characteristics than disallow the

inspection. Further studies will be conducted to improve the accuracy (regarding

representation of in-situ conditions) of the 3-layer model’s “bonding” layer. Guided

wave experiments in coated pipes were also carried out for demonstrating many

conclusions drawn from previous theoretical work. The impact of this work on the

practical side of things is huge as it will provide us with a measure of the capability to

inspect when certain coating conditions are in evidence and tedious trial and error

approaches.

146

Chapter 9 Concluding Remarks

9.1 Concluding Remarks

Long range ultrasonic guided wave inspection of coated pipes has been studied

through the use of numerical, analytical and experimental methods. The approaches

accomplished, the knowledge learned, as well as the remarks acquired are summarized as

follows:

First, the 3-D finite element method of dynamics using the ABAQUS/Explicit

package was successfully studied for modeling guided wave propagation and focusing in

pipes. Guided waves can be modeled by either defining the boundary value

corresponding to the transducer excitation or precisely prescribing the displacement field

or wave structure of a specific wave mode from dispersion curve analysis and elasticity

as input. The latter is highly recommended because of the easy control and generation of

certain desired modes. High consistency between modeling and theoretical results in

terms of group velocities, wave structures and angular profiles proves the validity and

high accuracy of the guided wave FE models. To run proper ABAQUS models of guided

waves, an understanding and necessary calculations of wave mechanics are required for

model inputs as well as result analysis and interpretation including, for example, wave

structure, wave velocities, time delays and amplitudes for focusing.

147

With the established 3-D FE models, wave scattering studies from 2-D defects

and 3-D defects in a pipe were carried out. Mode conversions were observed and defect

sizing potential was studied based on the observations. It was shown that defect

circumferential size could be estimated by mode decomposition using NME. Based on

the circumferential sizing results, the more critical defect depth analysis can be estimated.

Modeling work also shows that focusing increases tremendously the inspection

sensitivity and ability of finding 3-D defects, especially corrosion-like defects which can

be difficult to detect by axisymmetric waves.

In order to study the coating effects on guided wave propagation and focusing in a

coated pipe, a two layer 3-D finite element model was developed. A transformation

algorithm from coating acoustic properties to complex viscoelastic coating properties has

been developed to provide inputs for the 3-D FE models. Wave propagation modeling

for two coating materials shows that wave attenuation increases with frequency and

coating viscoelasticity. It was also interesting to find that a viscoelastic coating has no

effect on the focusing capability for the studied frequency, although there is an amplitude

loss. Phased array focusing with longitudinal waves often increases the signal energy by

9 to 16 dB for the studied frequencies and distances, indicating a much longer

propagation distance and also improved defect detection sensitivity.

Since the wave attenuation is dependent on coating properties, frequency and

wave modes, a parametric study was carried out thoroughly with a goal to understand

current coatings and perhaps to find an optimum coating material with least attenuation.

148

Attenuation dispersion curves of guided waves in a viscoelastic coated pipe were

calculated by using an improved complex root searching algorithm. It was found that

materials with low density, low attenuation constants and low velocities over a low

frequency range were recommended for realizing a low attenuation. Acoustic impedance

matching between coating material and pipe material can be used to explain the

relationship between attenuation, density and velocity. Studies also show that torsional

waves are more attenuative than longitudinal waves.

In addition, coating properties like velocity, density, and attenuation constants of

various field coating materials were measured experimentally and then used as inputs to

finite element wave propagation models. It was found that coating properties show wide

variations and that these variations play a huge role in determining guided wave

penetration power within a pipe. Results demonstrate the needs of in-situ coating

property measurements for any subsequent modeling work based on a field coated pipe.

Based on what we learned so far, a criterion as follows is suggested in order to

improve the inspection potential of coated pipes. When coatings are involved at an

inspection site, some experimentation can be conducted to measure coating properties

which can be used as inputs to computer models to evaluate coating effects on ultrasonic

wave propagation and focusing, and to determine what the effective inspection range is

and how to possibly increase the range by focusing and tuning. The range of the

inspection can dictate a possible smaller number of excavations and consequently reduce

the cost and time incurred by the excavation. A database or library can then be built

gradually for future use when more field inspections are conducted.

149

9.2 Contributions

1. Three-dimensional finite element tool for guided wave modeling

A powerful 3-D finite element tool for modeling guided wave propagation,

scattering and focusing has been developed utilizing ABAQUS/Explicit. Procedures

including mode generation, data acquisition and result analysis have been established

for the modeling and analysis of any wave type in a hollow cylinder. This

accomplishment has a high impact on research work on wave scattering from

arbitrary 3-D defects and mode conversion studies.

2. Three-dimensional defect inspection and sizing potential

Mode conversion from 3-D defects was studied for the first time by 3-D FE

modeling with axisymmetric waves and focusing loading. Circumferential sizing can

be realized by looking at the mode decomposition results and the second time

reflection waves, respectively. The depth can also subsequently be sized with

reflection amplitudes after taking into account the circumferential size estimations.

An accurate sizing of the defect will provide pipeline operators with valuable

information for a judgment of repair to prevent catastrophic accidents. This work was

also the first to conduct quantitative modeling studies in which phased array focusing

was strongly recommended for corrosion like defects which was hard to find by

axisymmetric waves.

150

3. First to use three-dimensional finite element models for Coated Pipe

A two-layer 3-D FE model was developed for the first time for modeling

guided wave propagation and focusing in a coated pipe. Rayleigh damping was

utilized to introduce the damping caused by viscoelastic properties which can be

calculated from the measurable acoustic properties. Wave propagation, scattering and

focusing in pipes coated with any coating materials and any incident wave becomes

possible provided the coating property is measurable.

4. First to Study Focusing in Coated Pipes and the Coating Effects

It was shown by 3-D FE modeling that focusing can be realized quite well in a

pipe with viscoelastic materials for the frequencies studied. Coating introduces

amplitude attenuation but without changing the focused angular profile. Focusing can

possibly realize a 16 dB gain of energy in both coated pipes and bare pipes. This

conclusion is extremely encouraging and useful for further coated pipe inspection

although studies on more waves, frequencies and coating materials are still needed.

5. Criteria on Wave Attenuation Minimization

An improved complex root searching method was used to enable the calculation

of attenuation dispersion curves of guided waves in a highly viscoelatic multilayer

151

pipe. Appropriate coating material, frequency range and wave type were

recommended by a parametric study in order to minimize the wave attenuation.

6. Experimental methods for coating assessment

An experimental method using normal beam transducers was developed to

measure attenuation constants by fitting an exponential curve to the signal echo peaks.

These constants along with other measured properties can be used as inputs of both

ABAQUS modes and analytical wave propagation models in order to produce

realistic modeling results. In a word, a process from experimental measures to

theorectical models has been established as a tool to evaluate the guided wave

inspection potential of in-field coated pipes.

9.3 Future Directions

1. Parametric studies of wave scattering from more 3-D defects with a goal to find a

general rule of defect sizing, characterization and classification is necessary.

Wave scattering from 3-D defects under coating will also be considered in order

to study the coating effect on scattered wave fields.

2. Health monitoring of critical pipeline area with leave-in sensor arrays on a bare

pipe or on the pipe-coating interface of a coated pipe will be done.

3. Wave scattering, focusing and mode conversion studies of a bare elbow or a

coated elbow will be under taken.

152

4. Coating delamination inspection method development with modeling and

experimental work should be studied.

5. Modeling study of the pipe-coating interface in order to more accurately compute

wave attenuation would be useful.

6. Ultrasonic device development for in-situ coating property measurement as inputs

to models would be useful.

153

References

Abduljabbar A., Datta S. K., “Diffraction of horizontally polarized shear waves by

normal edge cracks in a plate”, Journal of Applied Physics, vol. 54, no. 2, pp. 461-472,

1983.

Achenbach J.D., “Wave Propagation in Elastic Solids”, North-Holland Publishing Co.,

New York, NY, 1984.

Alleyne, D.N. and Cawley, P., “The Interaction of Lamb Waves with Defects”, IEEE

Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 39, no. 3, pp.

381-397, 1992.

Alleyne, D.N., Cawley, P., “Long Range Propagation of Lamb Wave in Chemical

Plant Pipework”, Material Evaluation, vol. 45, no. 4, pp. 504-508, 1997.

Alleyne, D. N. and Cawley, P., “The excitation of Lamb Waves in Pipes Using Dry-

coupled Piezoelectric Transducers”, Journal of Nondestructive Evaluation, vol. 15, no.

1, pp. 11-20, 1996.

Alleyne D., Lowe M., Cawley P.,“The reflection of guided waves from

circumferential notches in pipes”. Journal of Applied Mechanics, vol. 65, pp. 635–41,

1998.

Auld, B. A., “Acoustic Fields and Waves in Solids”, Vol. 1 and 2, Second edition;

Kreiger Publishing Co., FL, 1990,

Barshinger, J. N., Rose, J. L., “Guided wave propagation in an elastic hollow cylinder

coated with a viscoelastic material”, IEEE transactions on ultrasonics, ferroelectrics,

and frequency control, vol. 51, no. 11, pp. 1547-1556, 2004.

Barshinger, J.N., Guided wave propagation in pipes with viscoelastic coatings,Ph.D.

Thesis, the Pennsylvania State University, 2001.

154

Blanc R.H., “Transient wave propagation methods for determining the viscoelastic

properties of solids”, Journal of Applied Mechanics, vol. 60, pp. 763-768, 1993.

Cawley P., Lowe M., Simonetti F.,Chevalier C. “The variation of reflection

coefficient of extensional guided waves in pipes from defects as a function of defect

depth, axial extent, circumferential extent and frequency”, Journal of Mechanical

Engineering Science, vol. 216, pp. 1131–41, 2002.

Christensen R.M., “Theory of Viscoelasticity: An Introduction”, Academic Press,

New York, 1981.

Chang, Z., Mal, A., “Scattering of Lamb Waves from a Rivet Hole with Edge Cracks”,

Mechanics of Materials, vol. 31, pp. 197-204, 1999.

Cheung Y. K., “Finite Strip Method in Structural Mechanics”, Pergamon, Oxford,

1976.

Cook, E.G. and Valkenburg H.E., “Surface Waves at Ultrasonic Frequencies”, ASTM

Bulletin, vol. 3, pp. 81-84, 1954.

Cook R.D., Malkus D.S., Plesha M.E., Witt R.J., “Concepts and Applications of

Finite Element Analysis”, J. Wiley & Sons, New York, 2001.

Datta S. K., Al-Nassar Y., and Shah A. H., “Lamb wave scattering by a surface

breaking crack in a plate”, Review of Progress in Quantitative Nondestructive

Evaluation, vol. 10, Plenum Press, New York, pp. 97-104, 1991.

Demma A, Cawley P, Lowe M, Roosenbrand A. “The reflection of the fundamental

torsional mode from cracks and notches in pipes”, Journal of the Acoustical Society

of America, vol. 114, pp. 611–625, 2003.

155

Demma, A., Cawley, P., Lowe, M.J.S., Roosenbrand, A.G. and Pavlakovic, B. “The

reflection of guided waves from notches in pipes,pp a guide for interpreting corrosion

measurements”, NDT&E International, vol. 37, pp 167-180, 2004

Diligent, O., Grahn, T., Bostrom, A., Cawlwy, P., Lowe, M., “The Low-Frequency

Reflection and Scattering of the S0 Lamb Mode from a Circular Through-Thickness

Hole in a Plate: Finite Element, Analytical and Experimental Studies”, Journal of the

Acoustical Society of America, vol. 112, no. 6, pp. 2589-2602, 2002.

Ditri, J. J. and Rose, J. L., “Excitation of Guided Wave Modes in Hollow Cylinders

by Applied Surface Tractions,” Journal of Applied Physics, vol. 72, no. 7, pp. 2589-

2597, 1992.

Ditri, J. J., “Utilization of Guided Waves for the Characterization of Circumferential

Cracks in Hollow Cylinders,” Journal of the Acoustical Society of America, vol. 96,

pp. 3769-3775, 1994.

Galan J. and Abascal R., “Elastodynamic guided wave scattering in infinite plates”,

International journal for numerical methods in engineering, vol. 58, pp. 1091–1118,

2003.

Galan J. and Abascal R., 2004, Remote characterization of defects in plates with

viscoelastic coatings using guided waves, Ultrasonics, vol. 42, pp. 877-882, 2004

Ghosh, J., “Longitudinal Vibtrations of a Hollow Cylinder”, Bulletin of the Calcutta

Mathematical Society, vol. 14, pp. 31-40, 1923-24.

Gridin D., Craster R.V., Fong J., Lowe M.J.S., Beard M., “The high-frequency

asymptotic analysis of guided waves in a circular elastic annulus”, Wave Motion, v 38,

n1, pp. 67-90, 2003.

Graff, K. F., Wave Motion in Elastic Solids, Dover Publications Inc., New York,

1991.

156

Hayashi, T., Sun, Z., Kawashima, K. and Rose, J.L., “Analysis of flexural mode

focusing by a semianalytical finite element method,” Journal of the Acoustical

Society of America, vol. 113, no. 3, pp. 1241-1248, 2003.

Kley, M., C. Valle, L.J. Jacobs, J. Qu and J. Jarzynski, “Development of Dispersion

Curves for Two-layered Cylinders Using Laser Ultrasonics”, Journal of the

Acoustical Society of America, vol. 106, pp. 582-588, 1999.

Koshiba, M., Karakida, S., Suzuki, M., “Finite-Element Analysis of Lamb Wave

Scattering in an Elastic Plate Waveguide”. IEEE Transaction on Sonics and

Ultrasonics, 31, pp.18-25, 1984.

Krautkramer J., Krautkramer H., “Ultrasonic Testing of Materials”, Springer-Verlag,

4th

edition, 1990.

Kwun, H. and Teller, C.M., “Magnetostrictive Generation and Detection of

Longitudinal, Torsional, and Flexural Waves in a Steel Rod,” The Journal of

Acoustical Society of America, vol. 96(2), pp. 1202-1204, 1994.

Li, J. and Rose, J.L., “Excitation and Propagation of Non-axisymmetric Guided

Waves in a Hollow Cylinder”, The Journal of the Acoustical Society of America, Vol.

109(2), 457-464, 2001.

Li, J. and Rose, J.L., “Implementing Guided Wave Mode Control by Use of a Phased

Transducer Array”, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency

Control, vol. 48(3), pp. 761-768, 2002.

Liu, G. and J. Qu, “ Guided Circumferential Waves in a Circular Annulus”, Journal of

Applied Mechanics, vol. 65, pp. 424-430, 1998.

Liu, G.R. and Achenbach, J.D., “A Strip Element Method for Stress Analysis of

Anisotropic Linearly Elastic Solids”, Journal of Applied Mechanics, vol. 61, no. 6, pp.

270-277, 1994.

157

Liu, G.R. and Xi, Z.C., Elastic waves in anisotropic laminates, CRC Press, 2002.

Li, J. and Rose, J.L., “Excitation and Propagation of Non-axisymmetric guided waves

in a Hollow Cylinder”, The Journal of the Acoustical Society of America, vol. 109(2),

457-464, 2001.

Lowe, M.J.S. , “Matrix techniques for modeling ultrasonic waves in multilayered

media”, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol.

42, no. 4, pp. 525-542, 1995.

Lowe M, Diligent O. Low frequency reflection characteristics of the S0 Lamb wave

from a rectangular notch in a plate. Journal of the Acoustical Society of America, vol.

111, pp. 64–74, 2002.

Luo W., Rose J. L. and Kwun H., “Circumferential Shear Horizontal Wave Axial-

Crack Sizing in Pipes”, The Research in Nondestructive Evaluation, vol. 15, no. 4, pp.

1-23, 2004.

Mathews, J.H. and Fink, K.D., Numerical Methods Using MATLAB, Third Edition,

Prentice-Hall, Inc., pp. 92-100, 1999.

Miklowitz, J., “The Theory of Elastic Waves and Waveguides”, North Holland

Publishing Co., New York, 1978.

Nayfeh, A. H., “Wave Propagation in Layered Anisotropic Media With Applications

to Composites”, North-Holland, Elsevier Science B. V., The Netherlands, 1995.

Papavinasam S., Revie R.W., “Standards for Pipeline Coatings”, Workshop on

Advanced Coatings for R&D for Pipelines and Related Facilities, National Institute of

Standards and Technology, Garithersburg, MD, June 9-10, 2005.

Rattanawangcharoen N., Shah A. H., and Datta S. K., “ Reflection of waves at the

free edge of a laminated circular cylinder”, Journal of Applied Mechanics, vol. 61,

323–329, 1994.

158

R/D Tech, “Introduction to Phased Array Ultrasonic Technology Applications: R/D

Tech Guideline”, Quebec City, Canada, R/D Tech, 2004.

Rose, J. L., “Ultrasonic Waves in Solid Media”, Cambridge University Press, 1999.

Rose, J. L., “Basic Physics in Diagnostic Ultrasound”, Jonh Widley & Sons, Inc, 1979.

Rose, J.L., “Standing on the Shoulders of Giants: An Example of Guided Wave

Inspection”, Materials Evaluation, vol. 60, pp. 53-59, 2002.

Rose J.L., “A Baseline and Vision of Ultrasonic Guided Wave Inspection Potential”,

Journal of Pressure Vessel Technology, vol. 124, no. 8, pp. 273-282, 2002.

Shin, H. J., “Non-Axisymmetric Ultrasonic Guided Waves for Tubing Inspection,”

Ph.D. thesis, the Pennsylvania State University, 1997.

Shung K. , Smith M., Tsui B., “Principles of Medical Imaging”, Academic Press,

1992.

Shin, H. J. and Rose, J. L., “Guided Wave Tuning Principles for Defect Detection in

Tubing”, Journal of Nondestructive Evaluation, vol. 17, no. 1, pp. 27-36, 1998.

Silk M.G., and Bainton, K. P., “The propagation in metal tubing of ultrasonic wave

modes equivalent to Lamb waves,” Ultrasonics, vol. 17, pp. 11-19, 1979.

Simonetti F., “Sound propagation in lossless waveguides coated with attenuative

materials”, Ph.D Thesis, Imperial College London, 2003.

Sun, J., Wang C.,”Theroy of Mechanical Noise Control”(in Chinese), Northwestern

Polytechnical University Press, 1993.

159

Sun, Z., Zhang, L., Gavigan, B.J., Hayashi, T., and Rose, J.L., “Ultrasonic Flexural

Torsional Guided Wave Pipe Inspection Potential,” Proceedings, ASME Pressure

Vessel and Piping Division Conference, C. Miyasaka, ed., vol. 456, pp. 29-34, 2003.

Sun, Z., “Phased Array Focusing Wave Mechanics in Tubular Structures”, Ph.D.

thesis, the Pennsylvania State University, 2004.

Valle, C., J. Qu and L.J. Jacobs, “Guided Circumferential Waves in Layered

Cylinders”, International Journal of Engineering Science, vol. 37, pp.1369–1387,

1999.

Viktorov, I. A., “Rayleigh and Lamb Waves—Physical Theory and Applications”,

Plenum Press, New York, NY, 1967.

Xu, W., Jenot F., Ourak M., “Modal Waves Solved in Complex Wave Number,”

Review of Progress in Quantitative Nondestructive Evaluation, American Institute of

Physics, Melville, NY, vol. 24, pp. 156-163, 2004.

Zemanek, J., Jr., “An Experimental and Theoretical Investigation of Elastic Wave

Propagation in a cylinder”, Journal of Acoustical Society of America, vol. 51,no. 1,

pp. 265-283, 1972.

Zhao X., Rose, J.L., “Guided circumferential shear horizontal waves in an isotropic

hollow cylinder”, Journal of the Acoustical Society of America, vol. 115, pp. 1912-

1916, 2004.

Zhu W., “An FEM Simulation for guided elastic wave generation and reflection in

hollow cylinders with corrosion defects”, Journal of Pressure Vessel Technology, vol.

124:108–17, 2002.

Zhuang, W., Shah, A.H. and Datta, S.K., “Axisymmetric guided wave scattering by

cracks in welded steel pipes,” Journal of Pressure Vessel Technology, vol. 119, pp.

401-406, 1997.

160

Appendix A Dispersion Equation

66][

×C = 0

( ) ( )[ ] ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( )[ ] ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( )[ ] ( ) ( )( ) ( ) ( )

( ) ( ) ( )( )[ ] ( ) ( )

( ) ( )( ) ( ) ( )

( )( ) ( )

( ) ( ) ( )( )ankaWc

aWakaaWnc

aWakankaWc

ankaZc

aZakaaZnc

aZakankaZc

aaWaWannc

aWnkaaWakc

aaWnaWnnc

aaZaZannc

aZnkaaZakc

aaZnaZnnc

aaWnaWnnc

akaWnaWakc

aaWaWaknnc

aaZnaZnnc

aZnkaaZakc

aaZaZaknnc

n

nn

nn

n

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

136

11222

11235

112

1134

133

11222

1132

112

11131

111122

26

1112

1225

111124

1112122

23

1112

122

1111121

111116

1112

1215

1111222

14

1112113

1112

112

11111222

11

22

22

212

12

212

212

12

212

212

122

212

212

122

212

βββββλ

αααβ

ββββααλα

βββββββλ

αααββλββ

βββααλα

ββββββλ

αααβββλββββ

ααλαβ

−=−+−=

+−=

−=−+−=

+−=

−−−−=

+−=

+−−=−−−−=

+−=

+−−=−−=

+−=

+−−−=

−−=+−=

+−−−=

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

The fourth to sixth rows can be obtained by replacing a by b in the expressions of the first

three rows.

161

Appendix B Nontechnical Abstract

Pipeline safety plays an important role in the transmission and distribution

of energy, such as in fossil fuel and natural gas pipelines. To preserve the integrity and

safety of these pipelines, a large percentage of them are coated with protective materials.

However, environmental conditions, aging, and excavation accidents can compromise the

effectiveness of these protective measures. Thus the inspection and monitoring become

indispensable due to the high cost of replacement with new pipelines. Periodic or as

necessary non-destructive evaluation (NDE) is required to tell the pipeline operator the

current status of a pipe and whether remedial action is necessary. Methods exist for

detecting cracks in pipelines, but they are either highly costly or with limited inspection

abilities. Ultrasonic guided waves, because of their long range inspection ability, are

being used more and more as a very efficient and economical pipeline inspection method.

An ultrasonic wave is a mechanical wave at frequencies (usually larger

than 20 kHz) higher than the human’s audible frequency range. For civilian applications,

ultrasonic techniques have been used widely in medical diagnostics and NDE of material

and structures for many years. These applications basically utilize ultrasonic waves in

frequency ranges higher than several MHz and so the wave propagation distance is very

limited due to the high attenuation at those frequency ranges. These waves are usually

called bulk waves due to the small wavelength compared to the size of bulk wave

propagation media. When the wavelength is equivalent to media geometry size at low

162

frequency (kHz) range, waves are called guided waves which can propagate along a wave

guide for as long as hundreds of feet. Therefore, guided wave inspection is much more

efficient than the tedious point-by-point bulk wave inspection.

Guided waves in pipes are quite complicated in terms of dispersion and

mode diversity. They can be categorized into axisymmetric including longitudinal and

torsional waves, and non-axisymmetric waves including flexural longitudinal and flexural

torsion waves. Axisymmetric waves and phased array focusing are currently the two

main techniques for long range pipeline inspection. Focusing techniques can increase

energy impingement, locate defects, and enhance greatly inspection sensitivity and

propagation distance of guided waves. A typical scenario of long range guided wave

inspection is to generate guided waves from one single transducer position, which will

propagate with long distance and then impinge onto any possible defects with the

occurrence of wave scattering. The inspection strategy is to acquire the possible reflected

waves from defects and to analyze the waves for defect detection, locating, sizing and

characterization. Therefore, the inspection distance is am important parameter evaluating

the guided wave inspection ability.

However, the viscoelastic nature of coating materials leads to significant

attenuation consequently reducing guided wave inspection distance. Because of the

variation of coating materials and the complexity of the wave mechanics in viscoelastic

multilayered structure, many aspects and questions on guided wave inspection in coated

pipe still remain unknown and very challenging. In this work, guided wave propagation,

163

scattering and phased array focusing in viscoelastic coated pipes were studied for the first

time via numerical method, analytical method as well as some experimental

measurements. A powerful 3-dimensional finite element tool was developed first for the

modeling of any guided wave propagation and focusing in a coated pipe. Wave

scattering studies were then followed on three-dimensional defects with respect to

inspection and sizing potentials. Some exciting results were acquired in which phased

array focusing potentials in coated pipes were demonstrated. Some criteria on wave

attenuation reduction and consequent inspection distance increment were established. A

process from experimental measures to theoretical models has been established as a tool

to evaluate the guided wave inspection potential of in-field coated pipes. Most of the

work has never been studied before and therefore the accomplishments achieved have a

high impact for the future long range guided wave inspection of coated pipe.

VITA

Wei Luo

EDUCATION

Ph.D. Engineering Science and Mechanics, 2001 – 2005 The Pennsylvania State University, University Park, PA M.S. Electrical Engineering, 2002 – 2004 The Pennsylvania State University, University Park, PA M.S. Material Processing Engineering, 1998 – 2001 Tsinghua University, Beijing, China B.S. Mechanical Engineering, 1993 –1998 Tsinghua University, Beijing, China

SELECTED PUBLICATIONS

1. Luo W., Zhao X. and Rose, J. L., “A Guided Wave Plate Experiment for a Pipe,” Journal of Pressure Vessel Technology, vol. 127, no. 8, pp. 345-350, 2005.

2. Luo W. and Rose J.L., Veslor J.V., Mu J., "Phased array focusing with longitudinal waves in a viscoelatic coated hollow cylinder", the 32nd Annual Review of Progress in Quantitative Nondestructive Evaluation, Brunswick, Maine, July 31 - August 5, 2005

3. Luo W. and Rose J.L.,Veslor J.V., Spanner J., "Circumferential guided waves for defect detection in coated pipes", the 32nd Annual Review of Progress in Quantitative Nondestructive Evaluation, Brunswick, Maine, July 31 - August 5, 2005

4. Luo W., Zhao X. and Rose J. L., “Guided wave scattering and mixed mode conversions from 3-dimensional defects”, Review of Progress in Quantitative Nondestructive Evaluation, v24, pp. 105-111, 2005.

5. Luo W., Rose J. L. and Kwun H., “Circumferential SH Wave Axial Crack Sizing in Pipes”, The Research in Nondestructive Evaluation, v15, n4, pp. 1-23, 2004.

6. Luo W. and Rose J.L., “Lamb Wave Thickness Measurement Potential with Angle Beam and Normal Beam Excitation”, Materials Evaluation, vol. 62, no. 8, pp. 860-866, 2004.

7. Luo W., Rose J. L. and Kwun H., “A two dimensional model for crack sizing in pipes”, Review of Progress in Quantitative Nondestructive Evaluation, v23, pp. 187-192, 2004.

8. Luo, W., Rose, J.L., “Guided Wave Thickness Measurement with EMATS,” Insight, v45, n11, pp. 735-739, 2003.

9. Hay, T.R., Luo, W., Rose, J.L., Hayashi, T., 2003, “Rapid Inspection of Composite Skin-Honeycomb Core Structures with Ultrasonic Guided Waves,” Journal of Composite Materials, v37, pp. 929-939, 2003.

10. Chen Y., Luo W., Fu D. and Shi K., “Research on F-Scan Acoustic Imaging of Composite Materials”, Journal of Intelligent Material System and Structures, vol. 12, no. 10, pp. 701-708, 2001.

PROFESSIONAL AFFILIATIONS:

IEEE, student member • Ultrasonic, Ferroelectric and Frequency Control Society, • Signal Processing Society,

ASNT, student member

Documents

ULTRASONIC GUIDED WAVES AND WAVE SCATTERING IN