CHARACTERIZATION OF PLASTICALLY DEFORMED STEEL
UTILIZING EMAT ULTRASONIC VELOCITY MEASUREMENTS
P. D. Panetta1, B. Francini1, M. Morra1, G. A. Alers2, and K. Johnson1
Pacific Northwest National Laboratory, Richland, WA 99352
EMAT Consulting, San Luis Obispo, CA 93401
2
ABSTRACT. There is a desire to characterize plastically deformed regions in structures to monitor
their integrity. Of particular importance is the accurate prediction of the lifetime of damaged
pipelines due to outside force. In order to accurately predict the remaining life it is essential to
accurately determine the degree stress and strain in the damaged region for input into fracture
mechanics models. Currently, determination of the degree of stress and strain in damaged regions
utilizing ultrasonic velocity measurements is complicated by the inherent texture in the materials and
the difficulty in separating these effects from the stress and strain contributions. We will report
ultrasonic velocity measurements on plastically deformed steel specimens to elucidate the state of
damage. Specifically, we have found the shear wave birefringence and SH wave velocity are
sensitive to the degree of plastic deformation. Ultrasonic results will be compared with finite
element modeling calculation of the stress and strain distributions.
INTRODUCTION
Within the gas pipeline community, the characterization and detection of third party
damage is of paramount importance. Several events occur when a pipe is mechanically
damaged. Impacts from construction equipment cause dents and gouges to the pipe and the
associated large plastic deformations and cold-working of the pipe surface. When the
indenting force is removed, the dent can be partially rererounded by the internal pressure in
the pipe. This partial removal of the dent is significant because it can place the coldworked surface in a high tensile stress state, which frequently cracks the surface. The
presence of the gouge-in-dent dramatically increases the potential for crack initiation by (1)
providing a geometric irregularity that concentrates the stresses, and (2) the cold-worked
steel in the gouge now has a reduced ductility which can easily initiate cracks.
Commonly used techniques for nondestructively determining the stress and strain in
damaged materials typically rely on ultrasonic velocity measurements. Ultrasonic velocity
is very sensitive to damage induced stress and strain and offers the potential for accurate
quantitative determination of stress and strain as inputs into fracture mechanics models.
This powerful combination offers a prognostic capability that is not currently available to
pipeline inspection companies and utilities because of the lack of ability to accurately
measure stress and strain. The main drawbacks include the fact that velocity is affected by
competing sources of velocity shifts due to microstructural effects such as texture,
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
1516
temperature variations, the necessity for very precise time measurements, and low spatial
resolution. Despite these drawbacks, ultrasonic measurements are valuable because they
allow one to obtain information about the stress and strain in the interior of the materials as
a function of depth. Typical ultrasonic measurement equipment is relatively inexpensive,
portable, quick to set up, and the ultrasonic velocity data is rich with information. For
these reasons the use of ultrasonic measurements to determine stress and strain has been an
active area of research for many years.
The ultrasonic velocities are related to the stress and texture by well-known
equations. The relationship is governed by the acoustoelastic stress constants, three
orientation distribution coefficients that describe the texture and the bulk and shear moduli.
Algebraic equations with these seven unknowns can be compared with velocity
measurements if enough measurements can be taken.
For a longitudinal wave propagating in the thickness or the z-direction:
pVp2z = B + ^ - oca +12.761W40o
3
(1)
For a shear wave propagating in the z-direction and polarized in the axial direction:
pVs2a = G - PC - [6.381W400 +10.089W420 ]
(2)
For a shear wave propagating along the z-direction and polarized in the circumferential or c
direction:
2
(3)
PVS C = G-jt7-[6.3810^ -10.089^420]
In these equations, p is the density of the steel alloy, B and G are the bulk modulus
and the shear modulus, respectively, of a hypothetical pipeline steel sample in which the
grains are randomly oriented (the texture is zero), a, p, and y, are the acoustoelastic
constants that are related to the Muraghan third-order elastic constants (TOEC) of the steel,
and W4oo and W42o are the orientation distribution coefficients (ODCs) that describe the
preferred orientation of the grains (the texture) in the particular section for the pipeline
under study. Equations 1,2, and 3 represent three equations and five unknowns- B, G, o,
W40o, and W42o- The three ultrasonic wave velocities are considered as known because
they can be directly measured. The three acoustoelastic constants (a,|3, and y) could be
measured in laboratory calibration experiments and the density p , can be taken from
handbook values because it is a microstructure insensitive property of the steel involved.
Clearly, more equations are needed to deduce the stress and texture contributions
separately.
When considering applications to characterizing dents and gouges in pipes it is
important to realize that the stress and strain exhibit spatial variations and depth gradients.
There are several different ultrasonic methods that are appropriate for this application,
including through thickness measurements to characterize spatial gradients and waves with
penetration depths that can be varied to characterize depth gradients. Thompson et al. [1]
thoroughly reviewed the past several decades of research that utilized ultrasonic velocity
measurements to characterize the stress and strain of materials. They reviewed the
considerable progress in developing the scientific foundation underlying the techniques and
highlighted specific results that have promise for various applications.
One such result that is promising for characterizing inhomogeneous through-plate
stress distributions is the work by King and Fortunko [2] based on the velocity
measurements of horizontally polarized shear waves that were incident to the surface at
shallow, grazing angles. The theoretical development exploits the relative insensitivity of
the grazing shear wave propagation mode to texture and other microstructural anisotropies,
compared to other wave propagation modes including bulk and Rayleigh modes. These
1517
results help to establish the fundamental scientific foundation and point to the need to
improve spatial resolution for practical applications.
Results by Thompson et al [3] applied many different types of waves generated by
EMATs to accurately characterize the texture in aluminum and copper plates. Their
experimental results agreed well with theoretical predictions for these plates under stress
free conditions. In addition, the texture parameters obtained ultrasonically agreed well
with x-ray determinations. Thompson et al. presented the theory and applications of the
measurements of stress and texture in biaxially stressed specimens. [4]. Their
measurements agreed well with theoretical predictions and form the scientific basis for
characterizing biaxial stress states.
Initial measurements performed by the authors have shown that through thickness
measurements were strongly correlated with the strain in pipeline steels and experiments
were performed which indicate that the stress and texture may have separable contributions
to the velocity measurements. [5,6] Results were obtained while uniaxially loading a
pipeline steel sample, which showed systematic changes in the shear wave birefringence
during and after loading, which increased with increasing strain. Compression tests were
also performed on the pipe and showed systematic differences in the birefringence between
compression and tension that could easily be distinguished [6].
EXPERIMENTAL METHODS
Velocity (Longitudinal, Ravleigh, Shear Horizontal)
The longitudinal wave velocity was measured at the same locations as the shear
wave birefringence. The velocity was determined from the time difference between
successive echoes using the pulse echo overlap technique. For the Rayleigh wave velocity
measurements, three electromagnetic acoustic transducers (EMATs) were used, one as a
transmitter and two EMATs as receivers, see Figure 1. The SH-wave measurements were
also performed in a pitch catch arrangement with one EMAT as the transmitter and one
EMAT as the receiver, see Figure 1. The time of flight for the pitch catch measurements
was also determined utilizing the pulse echo overlap method.
Shear Wave Birefringence
For practical detection and characterization of the plastically deformed regions in a
structure such as a pipeline, the thickness cannot be assumed constant. Thus, it is desirable
to use a measurement that is independent of the thickness. Therefore, the shear wave
birefringence was measured, defined as the percent difference in time between the fast and
slow mode for shear waves polarized along the principal axis of the material. The
measurement was performed utilizing a rotating EMAT from Sonic Sensors, with the result
being the time of flight between two echoes as a function of angle. Results are shown in
Figure 2 for several levels of stress as will be described in the next section.
R
P — R
SH Wave EMATs
~~~
R
Rayleigh Wave EMATS
FIGURE 1. A schematic of the SH and Rayleigh wave EMAT configurations.
1518
25.05
25.05
25.05
Max − Min
Birefringe
nce = - Max — Min
Birefringence
Average
Max
− Min
Average
Birefringence =
Average
67 ksi
25.00
Arrival time (us)
Arrival time (us)
25.00
25.00 4
24.95
24.95
67 ksi
24.95
24.90
24.90
0 ksi
24.90
47 ksi
33 ksi
47 ksi
27 ksi
33 ksi
13 ksi
27 ksi
13 ksi
0 ksi
24.85
24.85
0
24.85
90 90
180180
270270
Angle
from
axial
direction
Angle
axial
direction
(degrees)
90 from
180 (degrees)
270
0
360360
360
Angle from axial direction (degrees)
FIGURE
2. The
timetime
of flight
as aasfunction
of angle
for for
thethe
pressurized
pipe
specimen.
FIGURE
2. The
of flight
a function
of angle
pressurized
pipe
specimen.
FIGURE 2. The time of flight as a function of angle for the pressurized pipe specimen.
1
1
0.8
0.6
0.4
0.2
0.8 0.8
15%
0.8
0.6
10%
5%
Unloading
15% Unloading
Plastic
Plastic
Strain
Strain
Effect
Effect
0.6 0.6
10%
5%
0.4
0.4 0.4
Load
Load
Elastic
Elastic
Stress
Stress
Effect
Effect
0.2 0.2
0.2
Loading
Initial Initial
Loading
0
0
0
0
20 2020
40
40 40
60
60 60
Tensile
TensileStress
Stress(ksi)
(ksi)
Tensile Stress (ksi)
80 80
0.0
Echo #4
Echo
#4
0.0
0
5%
Plastic
Deformation
5%5%
Plastic
Deformation
Plastic
Deformation
10%
Tensile
Stress
10%
Tensile
Stress
+Cσ
Ca
+DεDe
B0B=B0 0=A=A+A+Cσ
+ +Dε
Birefringence
1.2
Birefringence
1.2
Relative Birefringence
Relative Birefringence
RESULTS
RESULTS
RESULTS
Measurements
Measurementswere
wereperformed
performedononseveral
severalspecimens,
specimens, including
including dogbones
dogbones
Measurements
were
performed
on
several
specimens,
including
dogbones
measuring
36
inches
x
6
inches
with
a
gage
width
of
4
inches,
cut
from
steel
measuring 36 inches x 6 inches with a gage width of 4 inches, cut froma a½”Vi"thick
thick
steel
measuring
36
inches
x
6
inches
with
a
gage
width
of
4
inches,
cut
from
a
½”
thick
steel
plate.
These
dogbones
were
deformed
in
a
tensile
machine
to
15%
plastic
deformation.
plate. These dogbones were deformed in a tensile machine to 15% plastic deformation.
plate.
These
dogbones
were
deformed
in a deformation
tensile
machine
toshown
15%
deformation.
Results
of of
the
shear
wave
birefringence
during
areare
ininFigure
3,3,where
Results
the
shear
wave
birefringence
during
deformation
shownplastic
Figure
where
Results
of
the
shear
wave
birefringence
during
deformation
are
shown
in
Figure
3, while
where
thethe
shear
wave
birefringence
as as
a function
stress
shear
wave
birefringence
a functionof of
stressis isplotted
plottedduring
duringloading
loadingand
and
while
the shearAlso
wave
astheoretical
atheoretical
function predictions
ofpredictions
stress is for
plotted
during
loading
and while
unloading.
inbirefringence
Figure
3 are
plastic
deformation
the
unloading.
Also
in
Figure
3 are
for5%
5%
plastic
deformation
the
unloading.
Also in Figure
3asareantheoretical
fordetermined
5% plastic from
the
constants
in the
equation,
shown
inset
of of
thepredictions
graph
constants
in the
equation,
shown
as an
inset
the
graphwere
were
determineddeformation
froma afitfittotothe
the
constants in the equation, shown as an inset of the graph were determined from a fit to the
experimental
data.
While
thethe
fit fit
is quite
good,
thethe
terms
A,A,
CC
and
areare
complex
experimental
data.
While
is quite
good,
terms
andD D
complexfunctions
functions
experimental data. While the fit is quite good, the terms A, C and D are complex functions
of of
materials
parameters
such
as as
elastic
materials
parameters
such
elasticmoduli,
moduli,texture
textureparameters,
parameters,and
andacoustoelastic
acoustoelastic
of materials parameters such as elastic moduli, texture parameters, and acoustoelastic
constants.
details
calculation,
publicationbybyThompson
Thompsonetetal.al.[1].
[1].The
The
constants.
ForFor
details
of of
thethe
calculation,
seesee
thethe
publication
constants. For details of the calculation, see the publication by Thompson et al. [1]. The
birefringence
first
decreased
as
the
load
was
applied,
followed
by
and
increase
birefringence
first
decreased
as
the
load
was
applied,
followed
by
and
increase
birefringence first decreased as the load was applied, followed by and increase in
inin
birefringence
at
about
60
ksi
after
the
materials
began
to
plastically
deform.
The
birefringence
at
about
60
ksi
after
the
materials
began
to
plastically
deform.
The
birefringence at about 60 ksi after the materials began to plastically deform. The
birefringence
continued
during
unloading
untilthethe
theload
loadwas
wasremoved.
removed. The
The
birefringence
continued
to to
increase
during
unloading
birefringence
continued
to increase
increase
during
unloadinguntil
until
load
was
removed.
The
change
ininthe
as
initial
load
was
increased
was
due
the
acoustoelastic
change
in the
birefringence
as the
initial
load
was
increased
was
due
tototo
the
acoustoelastic
change
thebirefringence
birefringence
as the
the
initial
load
was
increased
was
due
the
acoustoelastic
0
10
10
20
20
20
30
40
30
40
5050
30
40
50
TensileStress
Stress(ksi)
(ksi)
Tensile
Tensile Stress (ksi)
Echo #3
Echo
#3
60
60
70
70
FIGURE3.3.The
Theshear
shearwave
wavebirefringence
birefringence as
as aa function
function of
for the
dogbones
for various
levels
of
FIGURE
of stress
stress
dogbones
various
levels
FIGURE
3. The shear
wave birefringence
as a function
of stress
for for
thethe
dogbones
forfor
various
levels
of of
plasticstrain..
strain..
plastic
plastic strain..
1519
Change in Birefringence
Change in Birefringence
0.7
0.7
0.6
0.6
Core 10 Core 10
PNNL Data
PNNL Data
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
A36 Steel
A36 Steel
NIST Data
NIST Data
0
0
00
0.05
0.1
0.15
0.2
0.05
0.1
0.15
0.2 0.2
0.05
0.1
0.15
Plastic strain (mm/mm)
Plastic
strain
(mm/mm)
Plastic
strain
(mm/mm)
FIGURE 4. The change in the birefringence as a function of plastic strain for two different steel alloys.
FIGURE 4.
birefringence
as aas
function
of plastic
strain strain
for twofor
different
steel alloys.
FIGURE
4. The
Thechange
changeininthethe
birefringence
a function
of plastic
two different
steel alloys.
effect. Assuming that the specimen was stress free after unloading, the change in the
birefringence
before and the
after unloading
was
due free
to a after
change
in the texture
accompanied
effect. Assuming
was
stress
unloading,
the change
in the in the
effect.
Assumingthat
that thespecimen
specimen
was
stress free
after
unloading,
the change
bybirefringence
a residual strain.
The
change
in the birefringence
as
a function
of texture
plastic accompanied
strain for two
before
and
after
unloading
was
due
to
a
change
in
the
birefringence
before shown
and after
unloading
was due to a change
in the texture accompanied
different
steel alloys
in Figure
The changeasina the
birefringence
the
by a residual
strain. isThe
change
in the 4.
birefringence
function
of plasticincreased
strain forastwo
by
a
residual
strain.
The
change
in
the
birefringence
as
a
function
ofincreased
plastic
degree
of plastic
strainis increased
indicating
thatchange
the birefringence
is directly
relatedstrain
thefor two
different
steel alloys
shown in Figure
4. The
in the birefringence
astothe
different
steel
alloys
is
shown
in
Figure
4.
The
change
in
the
birefringence
increased
degree
in the specimens.
the physics
and related
mathematical
degreeofof plastic
plastic strain
strain increased
indicating Understanding
that the birefringence
is directly
to the as the
degree
of
plastic
strain
increased
indicating
that
the
birefringence
is
directly
related
relationships
of thisstrain
correlation
an importantUnderstanding
goal of this work.
degree of plastic
in theisspecimens.
the physics and mathematical to the
degree
of
plastic
strain
in
the
specimens.
Understanding
themeasured
physics as
and
mathematical
relationships
of this
correlationwere
is ancomplete,
importantthe
goalbirefringence
of this work.was
After the
deformations
at several
relationships
of this
isthe
anaxis
important
goal of this
work.
After
the
deformations
were
complete,
was measured
several5,
locations
separated
by correlation
1 inch along
of the birefringence
dogbone.
Results
are shownasinatFigure
locations
separated
by 1 inchtoalong
the
axis
of the
dogbone.
Resultsshow
are
shown
in Figure
5, several
After
thedeformed
deformations
were
complete,
theThe
birefringence
was
as the
at
for
the dogbones
5%,
10%
and
15%.
results
anmeasured
increase
in
for
the
dogbones
deformed
to
5%,
10%
and
15%.
The
results
show
an
increase
in
the
birefringence
in the gage
wherethe
theaxis
mostofdeformation
hasResults
occurred,
locations separated
by 1region
inch along
the dogbone.
arerelative
showntointhe
Figure 5,
birefringence
in the
gage region
the most
deformation
occurred,
to thein in the
grip
regions.
Also
evident
are small
regions
ofand
residual
asresults
indicated
byrelative
aan
decrease
for
the dogbones
deformed
to where
5%,
10%
15%.stress
Thehas
show
increase
gripbirefringence
regions. Also
evident
regions
of residual
stress
as indicated
by a decrease
in to the
the
Figure
5. small
These
results
are in
qualitative
agreement
with
FEM
birefringence
in in
the
gage are
region
where
the
most
deformation
has
occurred,
relative
the
birefringence
in
Figure
5.
These
results
are
in
qualitative
agreement
with
FEM
predictions,
which
also
show
small
regions
of
residual
stress
at
the
locations
where
the
grip regions. Also evident are small regions of residual stress as indicated by a decrease in
predictions, which also show small regions of residual stress at the locations where the
the birefringence in Figure 5. These results are in qualitative agreement with FEM
predictions, which also show small regions of residual stress at the locations where the
Before Unloading
1.0
Dogbone
Relative Birefringence
0.8
1.0
After Unloading
After Unloading
After Unloading
Before Unloading
15%
0.6
10%
0.4
0.8
Relative Birefringence
Before Unloading
Dogbone
5%
0.2
15%
0.6
0.0
0.40
5
15
10%
10
20
25
30
35
-0.2
Axial Stress
Axial Stress
Axial Stress
Axial Stress
Region with/ residual
Regionstress
with residual
stress
5%
Position
0.2
0.0
0
5
Regions with
residual stress
10
15
20
25
30
Axial Strain
Axial Strain
35
-0.2
Axial Stress
Axial Stress
Axial Strain
FIGURE 5. The shear wavePosition
birefringence (left) as a function of position
and FEM
Region(1”
withseparation)
residual
FIGURE 5. The shear
wavewith
birefringence (left) as a function of position
(1"
separation)
and FEM
Regions
stress
predictions
of
the
axial
stress
and
strain.
predictions of the axial
stress
and strain.
residual
stress
FIGURE 5. The shear wave birefringence (left) as a function of position (1” separation) and FEM
predictions of the axial stress and strain.
1520
Birefringence
Birefringence
0.5
0.4
0.3
Variability with
location (unloaded)
0.2
1.6 MHz
0.1
0.0
20
40
60
20
40Stress (ksi)
60
Circunferential
Circunferential Stress (ksi)
0
Axial propagation
!0.2995
0.2996
I0.2994
0.2995 -
1.88 MHz
1.88MHz
I0.2993
0.2994 j,0.2992
0.2993 -
Circumferential
L-Wave Velocity (cm/us)
Load
0.289
0.288
0.287
Unload
0.286
0.285
Cicumferential
0.284
0.283
10
20
30
40
50
60
0.29900-
10
10
SH-Wave
Axial
0.290
"0.2990
0.2991 -
80
SH-Wave
0.291
SH-Wave Velocity (cm/us)
0.2997
0.2998
,0.2996
0.2997 i
j>0.2991
0.2992 -
0
0
Rayleigh
Rayleigh
0.2998
Rayleigh Velocity (cm/us
Birefringence
0.6
70
30
40
Stre ss (ksi)
30
40
20
50
60
70
50
L-Wave
0.5836
L-Wave
0.5832
5 MHz
0.5828
0.5824
0.5820
0.12% change
0.5816
0.5812
0
0
Stress (ksi)
20
20
20
40
40
Stress (ksi)
60
60
80
80
Stress (ksi)
FIGURE 6. The velocity results while pressurizing the pipe with water in the elastic region to produce
FIGURE 6. The velocity results while pressurizing the pipe with water in the elastic region to produce
a biaxial stress state.
a biaxial stress state.
birefringence
on
birefringence decreased.
decreased. These
These results
results are
are also
also inin agreement
agreement with
with measurement
measurement on
specimens
performed
by
Panetta
and
Alers
[6]
specimens performed by Panetta and Alers [6]
Measurements
Measurementswere
werealso
alsoperformed
performedon
onportions
portions of
of natural
natural gas
gas pipelines,
pipelines, provided
provided
bybyPacific
Gas
and
Electric.
These
pipe
specimens
were
approximately
6
Pacific Gas and Electric. These pipe specimens were approximately 6 feet
feet in
in length,
length, 22
22
inches
end and
and
inchesinindiameter,
diameter,with
witha awall
wallthickness
thicknessofof3/8
3/8inch.
inch. The
The pipe
pipe was
was capped
capped on
on each
each end
pressurized
and
pressurizedtotoininthe
theelastic
elasticregion
regiontotoproduce
produce aa biaxial
biaxial stress
stress state.
state. Longitudinal
Longitudinal and
birefringence
through
birefringencemeasurements
measurementswere
wereperformed
performed by
bypropagating
propagating the
the ultrasonic
ultrasonic waves
waves through
ehehthickness
thicknessofofthe
thepipe
pipewall
wallthickness,
thickness,while
while Rayleigh
Rayleigh and
and SH-waves
SH-waves were
were propagated
propagated
along
alongthethepipe
pipeaxis
axisand
andininthe
thecircumferential
circumferentialdirections.
directions. Results
Results as
as aa function
function of
of stress
stress are
are
shown
in
Figure
6
for
the
birefringence,
Rayleigh
wave,
SH-wave,
and
longitudinal
shown in Figure 6 for the birefringence, Rayleigh wave, SH-wave, and longitudinal waves.
waves.
Qualitatively,
Qualitatively,the
thechanges
changesasasa afunction
functionofofstress
stressagree
agreewith
with published
published results
results [1]
[1] with
with the
the
SH-wave
SH-wavepropagating
propagatingininthe
thecircumferential
circumferentialdirection
direction showing
showing aa change
change that
that is
is larger
larger than
than
expected
expecteddue
duetotothe
thelack
lackofofadequate
adequatemagnetization
magnetization of
of the
the pipe
pipe wall.
wall Work
Work is
is currently
currently
underway
underwaytotoovercome
overcomethese
thesemagnetization
magnetization effects
effects and
and will
will be
be the
the topic
topic of
of aa future
future
publication.
publication.
CONCLUSIONS
CONCLUSIONS
Theresults
resultsshow
showthat
thatthe
theultrasonic
ultrasonicvelocity
velocity isis sensitive
sensitive to
to small
small changes in the
The
stressand
andstrain
strainofofalloys.
alloys.The
Thechange
changeininthe
thebirefringence
birefringence before
before and
and after loading shows
stress
directcorrelation
correlationwith
withthe
thedegree
degreeofofplastic
plastic strain
strain inin the
the materials.
materials. The
The ultrasonic
ultrasonic
a adirect
velocitiesininpipe
pipespecimens
specimenswere
weremeasured
measuredwhile
whilepressured
pressuredtotoobtain
obtain aa biaxial
biaxial stress
stress state.
velocities
The
experimentalresults
resultsare
areininqualitative
qualitativeagreement
agreementwith
withtheoretical
theoreticalpredictions.
predictions.
The
experimental
1521
ACKNOWLEDGMENTS
The authors would like to thank Kayte Judd for performing key experimental
measurements. The authors would also like to thank Dan Kerr from Pacific Gas and
Electric for pipe specimens and access to their facilites. This work was supported by the
Department of Energy's Natural Gas Infrastructure Reliability Program, operated by the
National Energy Technology Laboratory. Pacific Northwest National Laboratory is
operated for the U.S. Department of Energy by Battelle under Contract DE-AC0676RLO18310.
REFERENCES
1. R.B. Thompson, W.-Y. Lu, and A.V. Clark, Jr. 1996. "Ultrasonic Methods." Chapter
7 in Handbook of Measurement of Residual Stresses. Eds. Dr. Jian Lu, pp. 149-178,
The Fairmont Press, Inc., Lilburn, Georgia.
2. R.B. King, and C.M. Fortunko. 1984. Surface-residual-stress evaluation using
horizontally polarized shear waves. J. Appl Phys. 55(11), p. 3978.
3. R.B. Thompson, J.F. Smith, S.S. Lee, and G.C. Johnson. 1989. "A comparison of
ultrasonic and x-ray determinations of texture in thin Cu andAl plates" Metallurgical
Transactions A, Volume 20A, pp. 2431 -2447.
4. R.B. Thompson, S.S. Lee, and J.F. Smith. 1986. "Angular dependence of ultrasonic
wave propagation in a stressed, orthorhombic continuum: theory and application to the
measurement of stress and texture." J. Acoust. Soc. Am. 80(3):921-931.
5. G.A. Alers and J.D. McColskey, "Measurement of residual stress in bent pipelines." in
Review of Progress in Quantitative Nondestructive Evaluation, Vol. 21, eds .D.O.
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6. P.D. Panetta and G.A. Alers, "Characterization of plastically deformed steel utilizing
ultrasonic velocity and attenuation measurements", in Review of Progress in
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Chimenti, American Institute of Physics, 2000, pp. 1494-1500. PNNL-SA-33946,
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