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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
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............... 29
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
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
Figure 6-6 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. ................................................96
Figure 6-7 Angular profile for the 50 kHz L(0, 2) wave with axisymmetric and phased
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
Figure 6-8 Angular Profile for at the focal point for a bare pipe and a pipe coated with 3
mm bitumen for 50kHz longitudinal waves, showing that coating introduces some
attenuation but that the profile shape stays the same. ................................................98
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
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.............................................................................................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
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). ..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
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. ........................................................................................................................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
Table 6-5 Amplitude gain by focusing with 100 kHz longitudinal waves at a focal point
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)
Reflected L(0,2) L(0,1)
Reflected L(0,2) L(0,1)
Reflected L(0,1) L(0,2)
Reflected L(0,1) L(0,2)
Reflected L(0,1) L(0,2)
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)
Transmitted L(0,2) L(0,1)
Transmitted L(0,2) L(0,1)
Transmitted L(0,2) L(0,1)
Transmitted L(0,2) L(0,1)
Transmitted L(0,1) L(0,2)
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
AxisymmeticPhased Array
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
AxisymmeticPhased Array
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
AxisymmeticPhased Array
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
Propagation Distance (m)
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.
Table 6-4 Amplitude gain by focusing with 100 kHz longitudinal waves at a 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
Propagation Distance (m)
Dis
pla
cem
en
t M
ag
nit
ud
e
No Coating
Epoxy
Bitumen
Focusing results
95
(a) before normalization (b) after normalization
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.
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.
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
Propagation Distance (m)
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
Table 6-5 Amplitude gain by focusing with 100 kHz longitudinal waves at a focal point at
a distance of 3 meters
Amplitude at the focal point Material
Focusing Axisymmetric
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
Figure 6-6 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.
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
Propagation Distance (m)
Dis
pla
cem
en
t M
ag
nitu
de
No Coating
Epoxy
Bitumen
Figure 6-7 Angular profile for the 50 kHz L(0, 2) wave with axisymmetric and phased
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.
Table 6-6 Amplitude gain by focusing with 50 kHz longitudinal waves at a focal point at
a distance of 1.5 meters
Amplitude at the focal point Material
Focusing Axisymmetric
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
Figure 6-8 Angular Profile for at the focal point for a bare pipe and a pipe coated with 3
mm bitumen for 50kHz longitudinal waves, showing that coating introduces some
attenuation but that the profile shape stays the same.
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
αααα1111 /ω/ω/ω/ω
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.
(a) four times 0804.0/2 =ωα
(b) double 0402.0/2 =ωα
(c) original 0201.0/2 =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
αααα2222 /ω/ω/ω/ω
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
αααα2222 /ω/ω/ω/ω
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
αααα2222 /ω/ω/ω/ω
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)
(a) four times 092.0/1 =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)
(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)
(c) original 023.0/1 =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)
(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
αααα1111 /ω/ω/ω/ω
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
αααα1111 /ω/ω/ω/ω
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.
(a) four times 96.0/2 =ωα
(b) two times 48.0/2 =ωα
(c) original 24.0/2 =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
αααα2222 /ω/ω/ω/ω
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
αααα2222 /ω/ω/ω/ω
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
αααα2222 /ω/ω/ω/ω
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
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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