edxrd analysis of stress corrosion cracking in 4140 …
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EDXRD ANALYSIS OF STRESS CORROSION CRACKING IN 4140 STEEL
by
BRANDON SETH BERKE
A Dissertation submitted to the
Graduate School-New Brunswick
Rutgers, The State University of New Jersey
In partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
Graduate Program in Materials Science and Engineering
Written under the direction of
Professor Thomas Tsakalakos
And approved by
________________________
________________________
________________________
________________________
New Brunswick, New Jersey
OCTOBER, 2012
ii
ABSTRACT OF THE DISSERTATION
EDXRD Analysis of Stress Corrosion Cracking In 4140 Steel
by Brandon Seth Berke
Dissertation Director:
Professor Thomas Tsakalakos
The analysis of the strains around the immediate vicinity of a crack tip to
encompass the plastic zone in 4140 steel specimens was undertaken. Each specimen was
subjected to a cyclic loading in air with a KI of 24.2 MPa√m to generate a fatigue crack
around 5.7mm. After a mechanical overload corresponding to a KImax of 38.5 MPa√m in
varying environments (Air, 3.5% NaCl solution simulating sea water and zinc couple to
the 3.5% NaCl solution for generating hydrogen) ranging from inert up to a stress
corrosion cracking (SCC) environment were carried out.
During the overloading process only the specimens in an SCC environment were
subjected to crack activity as expected. For this reason method that hydrogen attacks
steel (hydrogen embrittlement) was a point of curiosity as well. Three SCC specimens
were prepared to stay within different phases: i) no crack growth, ii) onset of crack
growth and iii) crack growth of ~1mm. Two of the specimens (Air and SCC with the
onset of crack growth) were measured by mapping up to all three 3D principle elastic
strains (PES) using a polychromatic high energy synchrotron x-ray probe with photon
energies up to 200 keV in the Laue mode. The PES ε11 and ε22 were directly measured,
iii
while the PES ε33 was obtained from the combined measurement of ε22 and ε23 of the
elastic strain matrix. By using the generalized Hooke’s law and the von Mises yield
criterion for 3D state of stresses in conjunction with the measured PES, maps of the von
Mises equivalent stress (σeq) were obtained. In addition, of the air specimen, the
principle plastic strains and total strains were calculated and used to show the principle
stress fields and equivalent stress field. The results indicated that the PES in the direction
orthogonal to the crack path i.e. the ε22 is sufficient for measuring the plastic zone in
4140 steel.
The use of three models (Linear Elastic Fracture Mechanics (LEFM), Hutchinson-
Rice-Rosengren (HRR) and a blunted crack tip) to define the border of the plastic zone
was employed. These three models were overlaid on the ε22 strain maps. The
juxtaposition of the models with the data showed some varying agreement with the
projected plastic zone size. The models helped estimate that SCC causes the plastic zone
to grow when an SCC crack occurs, due to a lowering of the yield strength of the material
and decreases it ductility.
iv
ACKNOWLEDGMENTS
I would like to thank my family for their continued guidance throughout my life.
At times when I did not think I could push on they gave me the nudge I needed. In my
family my father in particular Dr. Neal S. Berke inspired me on the path to become a
Materials Engineer.
I would like to show my gratitude to my Advisor and co-Advisors Prof. Thomas
Tsakalakos and Dr. E. Koray Akdoğan. Their patience and guidance have been
invaluable. I feel that under their tutelage gave me the experience I will need to tackle
the world. The environment created by Prof. Tsakalakos’ in his research group has been
very pleasant, making my tenure his group at Rutgers very rewarding. Dr. E. Koray
Akdoğan has been like a big brother, giving all sorts of sage advice on everything. The
countless days spent on discussing my project as well as the finer points of life have been
special.
I am extremely lucky to have really awesome research group mates, past and
present: Ms. Kathleen Paige, Mr. İlyas Şavklıyıldız, Mr. Mike McNeley, Ms. Shivani
McGee, Mr. William Paxton, Mr. Scott Silver, Mr. Robert Palmer and Mr. Wu Yi.
Spending long hours of the night at X17-B1 in the National Synchrotron Light Source
(NSLS) and persuading you all to watch Classic 80’s movies as well as our Super Bowl
party (even though my team lost) are memories which I will cherish.
v
I would like to thank my committee members Prof. Armen Khachaturyan and
Prof. Jafar F. Al-Sharab for going the distance to be on my committee. I benefited for
knowing that I had to bring my best to defend before them. I am indebted to the
resources from Prof. Grzegorz Glinka from the University of Waterloo, Canada. Not
only did he come from Canada to attend my thesis conference but also has helped me
understand his pioneering work on the elementary block modeling which is now an
inspiration for my postgraduate work.
I am grateful for having the opportunity to conduct experimental work at the
NSLS and support staff at Brookhaven National Laboratory. Dr. Zhong Zhong, the
spokesperson and senior staff member at NSLS has been instrumental in teaching me all
of the tricks a good graduate student should know at the X17-B1 beamline in the NSLS.
Also, Mr. Richard Greene is herewith gratefully acknowledged for he has been an
invaluable machinist making many of the necessary rigs for our experiments.
I wish to express their gratitude to the financial support provided by the Office of
Naval Research (ONR) under contract N000140710380. I especially thank Dr. A. K.
Vasudevan and Dr. Lawrence Kabacoff of the ONR for his valuable technical feedback
and support to my project. My research was carried out in part at the NSLS of the
Brookhaven National Laboratory, which is supported by the U. S. Department of Energy,
Division of Material Sciences and Division of Chemical Sciences under Contract No.
DE-AC02-76CH00016.
Finally, parts of this dissertation will appear in a publication that is currently
pending: B.Berke et al J. Appl. Phys. (2012).
vi
TABLE OF CONTENTS
ABSTRACT ii
ACKNOWLEDGEMENTS iv
TABLE OF CONTENTS vi
LIST OF TABLES viii
LIST OF FIGURES x
CHAPTER 1: INTRODUCTION 1
1.1 Challenges in Corrosion 1
1.2 4140 Steel 2
1.3 Statement of the Problem 3
CHAPTER 2: STRESS CORROSION CRACKING 5
2.1 Hydrogen Embrittlement 5
2.2 Electrochemistry 6
CHAPTER 3: STRESS/STRAIN ANALYSIS IN A DUCTILE METAL 9
3.1 Converting Measured Strains into Calculated Stresses 9
3.2 Models 12
3.2.1 Linear Elastic Fracture Mechanics 12
3.2.2 Hutchinson-Rice-Rosengren Zones 17
3.2.3 Blunted Crack Tip Model (ρ* Model) 19
vii
3.3 Direct Measurement of Plastic Zones 21
3.3.1 Mapping all 3 elastic strain components 21
3.3.2 Mapping only ε22 23
CHAPTER 4: EXPERIMENTAL PROCEDURE 24
4.1 Compact Tensile Specimen 24
4.2 Synchrotron-Energy Dispersive X-Ray Diffraction 25
4.2.1 All three strain components 31
4.2.2 Just ε22 32
4.2.3 Plane Stress vs. Plane Strain 32
CHAPTER 5: RESULTS 33
5.1 X-Ray Peak Analysis 33
5.1.2 Iron Hydride 34
5.2 Mechanical Measurements 35
5.2.1 Finding the Crack Tip 35
5.2.2 Components of Strain and Stress for Equivalent Stress Plotting 37
5.2.3 All Components of Strain and Stress in Full 2D Space 40
5.2.4 Strain Maps of ε22 44
5.2.5 Model Overlays of the Plastic Zone with ε22 46
5.2.6 Plane Strain and Plane Stress 50
CHAPTER 6: CONCLUDING REMARKS 52
viii
REFERENCES 55
APPENDIX 59
APPENDIX A – Measuring ρ* 60
APPENDIX B – Alternative Contour Plots 62
ix
LIST OF TABLES
Table 1.1 Composition of 4140 Steel 2
Table 2.1 Half-Cell reactions of selected metals. 8
Table 4.1 Specimen designation with their overload environment and crack conditions. 25 Table 5.1 Crack tip location and crack growth in the 4140 CT specimens 37
x
LIST OF FIGURES
Chapter 1
Figure 1.1 Computed stress-strain curve of 4140 steel from the Ramberg-Osgood relationship.
3
Chapter 2
Figure 2.1 Pourbaix diagram of zinc in water, the dashed lines represent the stability range of
water. 7
Chapter 3
Figure 3.1 Schematic representation of the coordinate system describing the stress field at the
crack front. 13
Figure 3.2 Schematic representation showing the Irwin correction of the plastic zone. 16
Figure 3.3 Theoretical plastic zone shapes at the tip of a crack. 17
Figure 3.4 Theoretical plastic zone shapes at the tip of a crack including HRR. 19
Figure 3.5 Theoretical plastic zone shapes at the tip of a crack of LEFM, HRR and ρ*. 21
Chapter 4
Figure 4.1 Graph of the crack open displacement (COD) potentials in Volts over time in a log
scale. 25
Figure 4.2 The schematic of the experimental set-up used in the present study at the X17-B1
superconductor wiggler beamline in the National Synchrotron Light Source, Brookhaven
National Laboratory showing the relative positions of the beam path, gauge volume,
transmitted beam and diffracted beam detectors, and the Bragg angle are depicted. 27
xi
Figure 4.3 The schematic ray diagram of the transmission mode (Laue mode) diffraction
geometry, where the parameters defining the gauge length L such as incident beam slit
size S1 and the Bragg angle 2θ are illustrated. 29
Figure 4.4 Schematic in 3D space of the transition from the plane strain conditions in the center
to the plane stress conditions at the surface. 30
Chapter 5
Figure 5.1 Indexed spectrum of 4140 steel. Note: The first peak (110) is much more intense than
the other two present due to the Bragg angle. 33
Figure 5.2 Indexed spectrum of 4140 steel along with peak profiles of where FeH would be in
respect to the spectrum measured if the composition of FeH was 1%. 35
Figure 5.3 The comparison of the variation of the measured elastic principle ε22 as a function
along the x-axis in all five samples. 36
Figure 5.4 The comparison of the position dependence of the principle strains ε11, ε22 and ε33 in
the immediate vicinity of the crack tip which is located at ~6 mm for 4140 steel loaded in
air (left) and the SCC B specimen subjected to stress corrosion cracking (right). 38
Figure 5.5 The comparison of the position dependence of the principle stresses σ11, σ22, σ33
and the von Mises equivalent stress σeq in the immediate vicinity of the crack tip which is
located at ~6 mm 4140 steel loaded in air (left), and specimen SCC B subjected to stress
corrosion cracking (right).. 39
Figure 5.6 The comparison of the contour plots based on the position dependence of the
calculated von Mises equivalent stress σeq in the immediate vicinity of the crack tip. 40
Figure 5.7 The comparison of the elastic strains in all three principle directions from left to right
ε11, ε22 and ε33. 41
Figure 5.8 The comparison of the plastic strains in all three principle directions from left to right
ε11, ε22 and ε33. 41
xii
Figure 5.9 The comparison of the total strains in all three principle directions from left to right
ε11, ε22 and ε33. 42
Figure 5.10 The comparison of the principle stresses and von Mises equivalent stress starting
from the top left corner and moving clock wise σ11, σ22, σ33 and σeq. 43
Figure 5.11 Strain map of the air sample, which was the control specimen. 44
Figure 5.12 Strain map of the salt water sample, which was the aqueous control sample. 45
Figure 5.13 Strain map of the salt water coupled with zinc sample, which was the SCC sample in
which no cracking occurred. 45
Figure 5.14 Strain map of the salt water coupled with zinc sample, which was the SCC sample in
which the onset of cracking occurred. 46
Figure 5.15 Strain map of the salt water coupled with zinc sample, which was the SCC sample in
which the cracking propagated ~0.9mm. 46
Figure 5.16 Overlay of all three plastic zone models on the air sample 47
Figure 5.17 Overlay of all three plastic zone models on the salt water sample. 48
Figure 5.18 Overlay of all three plastic zone models on the aqueous sample with the zinc couple
to promote SCC. 48
Figure 5.19 Overlay of all three plastic zone models on the aqueous sample with the zinc couple
to promote SCC right at the onset of crack propagation. 49
Figure 5.20 Overlay of the HRR plastic zone model on the aqueous sample with the zinc couple
to promote SCC right after the crack propagation of 0.9 mm. 49
Figure 5.21 The comparison of the plane strain (left) and the plane stress (right) strain map of the
SCC sample that had a 0.9 mm crack propagate. 50
Figure 5.22 Overlay of the three plastic zone models on the aqueous sample with the zinc couple
to promote SCC right after the crack propagation of 0.9mm 51
xiii
APPENDIX A
Figure A.1 1 The comparison near the crack tip of the calculated and measured data of the elastic
ε22. 61
APPENDIX B
Figure B.1 The equivalent stress map presented from Fig. 5.6 with both plots having the same
stress contour scale. 62
1
CHAPTER 1 Introduction
1.1 Challenges in Corrosion
Aggressive environments cause undesirable effects in materials. The definition of
an aggressive environments vary from material to material, yet typically it contains but is
not limited to temperature extremes, strongly basic or acidic, or highly ionic typically
chlorides when metallic engineering materials are considered. Billions of dollars are
spent on repairs caused by corrosion on structural systems such as bridges, coastal
structures, etc.1,2 Therefore prevention is a necessity.
Corrosion damage causes alloys to fail at lower stresses than expected or foreseen
by design. Empirical observations have shown that catastrophic phenomenon called stress
corrosion cracking (SCC) is a formidable phenomenon with dire consequence if not dealt
with properly. The most visible examples of SCC have been structural failures in
airplanes of Aloha Airlines3 in 1988, and Southwest Airlines4 in 2011. Additionally,
there have been numerous suspended ceilings over swimming pools that have had
catastrophic roof collapses which are all attributed to SCC5. The phenomenon is
characterized by the severe decline of the fracture resistance (K) of the material to resist
cracking. In mathematical terms SCC occurs when the KIC of a material is reduced to as
much as 1% of its original value (at this point called the KSCC). This has become an
important phenomenon requiring a thorough understanding since materials that seem to
be performing within boundaries foreseen could catastrophically fail prematurely and
unexpectedly. Hence, a strategy needs to be developed to prevent such catastrophic
failure, which can only be accomplished if and only if one has a quantitative
2
understanding as to how such failures develop from solid mechanics perspective in
relation to microstructural changes.
The microstructural changes accompanying environment changes can be now
measured with the advent of higher powered X-ray sources, enabling one to perform high
X-ray diffraction (XRD) at levels never before possible. The highest energy X-ray
sources used are from wiggler excited synchrotron radiation which is used in the energy
dispersive XRD (known as S-EDXRD or plainly EDXRD) study reported herein. The
primary advantages of EDXRD are that the beam has a much stronger penetration so bulk
samples of appreciable thickness can be investigated and the diffraction mode allows the
collection of an entire spectrum simultaneously by using a fixed diffraction angle and
wide energy spectrum.1,6-22
1.2 4140 Steel
4140 steel is a high strength medium carbon low alloy steel, used in applications
where high strength to weight issues are desired to remove dead weight, especially in
aircrafts. 23 Additionally the alloy is known for very good fatigue properties and
mechanical wearing resistance.24 The alloy composition is presented in Table 1.1.
Table 1.1 Composition of 4140 Steel Element C Cr Mn Mo P Si S
% 0.40 0.8-1.1 0.75-1 0.15-0.25 <0.035 0.15-0.3 <0.040
The Ramberg-Osgood equation (see ahead in section 3.2.2) creates the relevant
cyclic stress-strain curve shown in Fig. 1.1. The data agrees with our value of the yield
stress as measured by a 0.02% offset, which is 646 MPa.23-26
Fre
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4
from the EDXRD measurements and ensuing calculations. The fifth chapter discusses
the results of the investigation of the dissertation. The sixth chapter has concluding
remarks. The appendices show related work on calculating ρ* (APPENDIX A) and
alternative contour plots (APPENDIX B).
5
CHAPTER 2 Stress Corrosion Cracking
Stress corrosion cracking (SCC) is phenomenon where the crack propagation is
induced by environmental factors leading to the corrosion of the material of interest in the
first place.1,2 Specifically, one considers the environmental attack on the material under a
static tensile loading condition in SCC studies. However, one may also consider an
additionally cyclic loading that could lead to a similar problem as exemplified in the case
called corrosion fatigue cracking (CFC). SCC is an extremely formidable phenomenon
since is can cause a material to fail at a much lower stress (as much as 1%) than the
design stress.1,2 Typically, unalloyed metals are more resistant to SCC, however with
CFC all metals are vulnerable.1,2
In marine environments, it is well known that chlorides easily chemically attack
steels while ammonias do the same on brass alloys.1,2 Another common problem in
marine environments for high strength alloys is the hydrogen embrittlement phenomenon
which is exclusively observed in steels with Rockwell C hardness, (HRC) greater than
30.1,2 To combat SCC there are two major options: a) The removal of the load, or b) the
prevention and/or removal of the corrosive environment. Yet, if the environment or load
cannot be changed, heat treatment of the material would be required to create a softening
effect that would remove surface hardness and energy from the material.1,2
2.1 Hydrogen Embrittlement
Hydrogen embrittlement has been a serious concern for corrosion engineers.
Three different theories of hydrogen embrittlement were accepted in the 1970s: “Pressure
Theory”27-29, “Lattice-Interaction Theory”27,30-33, and “Stress-Sorption Theory”.27,34
6
Pressure Theory states that hydrogen gas enters the voids in the metal and builds in
pressure causing fracture. Lattice Interaction Theory states that hydrogen enters the iron
lattice by diffusing into areas of high stress. This interaction weakens lattice bonding.
The least accepted theory, Stress-Sorption Theory states that the surface energy is lower
at the crack surface where hydrogen gets adsorbed.
In the realm of Geology, a new complimentary theory is proposed. Geologists
work with higher pressures, and have observed the interaction of hydrogen and iron at
high pressures (GPa range) leading to an “Iron Hydride Theory”35. The theory states that
high stresses in the material cause the creation of the non-stoichiometric compound iron
hydride, which is a brittle compound the formation of which results in a 17% volume
expansion in the iron lattice (BCC or FCC). It is hypothesized that the said 17%
expansion at the tip of the crack results in needed driving force for crack propagation,
ultimately leading to rupture. The detection of the Fe-H compound is very difficult
because it is expected to be much localized at the tip of the crack and be present in
minute quantities, but with high ultra-high energy x-ray diffraction it may be possible to
detect it –one of the ambitious goals of the current study.
2.2 Electrochemistry
Zinc is typically added to the immersion solution when studying hydrogen
embrittlement in ferrous alloys such as steel. Upon addition into water, zinc can undergo
three reactions (as noted in its respective Pourbaix diagram; See Fig. 2.1) by which it can
dissociate into ions, form an oxide, or form a hydroxide1,2. In the presence of iron it
becomes more prevalent to react, since it is more anodic and thus more reactive than iron.
T
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9
CHAPTER 3 Stress/Strain Analysis in a Ductile Metal
The propagation of an elliptical crack in a ductile metal causes the formation of a
so called plastic zone, which can be induced by static or cyclic loading.37-42 Such plastic
zones form when the local stress at the tip of the crack exceeds the elastic limit of the
metal in question.37-42 In such ductile metals, the elastoplastic behavior at the tip of a
crack is essentially a strain localization phenomenon which constitutes a highly nonlinear
problem in continuum mechanics, posing challenges in the experimental as well as
computational realms.37-42 The aforementioned elastoplastic phenomenon becomes even
more complicated when the said elastic strain localization phenomena takes place in the
presence of a corrosive environment as exemplified by the stress-corrosion-cracking
(SCC).1,2,26-34 From a continuum mechanics perspective, SCC reduces the fracture
toughness of a given material, lowering the barrier for crack propagation thereby.
However, SCC brings an electrochemical dimension to the said elastoplastic
phenomenon, which makes its experimental and theoretical analysis most formidable.
Therefore, as a first line of attack, it is imperative to carry out local profiling of the
underlying elastic strain fields on the relevant length scales in order to assess the 3D state
of stress in the immediate vicinity where the nonlinear phenomena is concentrated, i.e.
plastic zone in the near field of the crack.
3.1 Converting Measured Strains into Calculated Stresses
The state of stress at the tip of an elliptical crack is a 3D state of stress which can
either be plane strain or plain stress depending on where the crack is located with respect
to the surface in the structural member of interest as well as the geometry of the loading.
10
Under such circumstances, the so-called generalized Hooke’s law applies to the measured
elastic strains which are summarized below:
(3.1a)
(3.1b)
, (3.1c)
where E is the Young’s modulus and ν the Poisson’s ratio of the material under
investigation. In Eqs. (1a-c), it was assumed implicitly that the material of interest is a
polycrystal so that one effective Young’s modulus suffices to describe the elastic
response. If all three elastic strain components are measured it becomes possible to solve
for the principle stresses, σ11, σ22 and σ33.
Furthermore, the following relationship can be written if one takes the elastic and
plastic components of strain into account:
(3.2)
In the center of a plate plane strain conditions exists and in the elastic regime where
ε33total=0, the out of plane stress is given by Eq. (3.3), so by assuming plane strain
conditions one may derive the following relationship from Eq. (3.1c):
(3.3)
which is the most commonly encountered state of stress in analyzing crack propagation in
engineering materials.
11
To account for the plastic strains and thus the total strain one needs to use an
extended form of Hooke’s Law called the Hencky25,26,37 equations which are expressed as
follows:
(3.4a)
(3.4b)
(3.4c)
where is a function defined as
(3.5a)
and ′
1′ (3.5b)
The denominator in Eq. (3.5a) is the von Mises yield criterion. Since plastic deformation
in a metal under a tri-axial state of stress requires the use of a more sophisticated yield
criterion such as the von Mises25,26 yield criterion as given by the following expression:
, (3.6)
eq is the so-called equivalent stress which is used in comparing to the yield stress
measured in uniaxial tension (o). The condition for plastic deformation is eq > o. 25,26
The numerator (Eq. 3.5b) has two coefficients where K' is a constant commonly referred
to as the cyclic strength coefficient and n' is the cyclic strain hardening coefficient. For
the 4140 steel considered in this study K’=1640 MPa and n’=0.15.26 This equality (Eq.
12
(3.5b)) stems from the plastic term in the Ramberg-Osgood relationship (models a stress
strain curve by parameters K’ and n’ see ahead in page 16) generally used as a
constitutive law in cyclic yielding of a material.25,26
The plane strain condition in the plastic regime implies
0
. (3.7)
This would seem to enable a simplified measurement scheme by only requiring two
elastic stain components ε11 and ε22. However, solving the said three equations (3.4a-c)
with the assumption of Eq. 3.7 it becomes no longer trivial, but they cannot be done
algebraically. Instead, a numerical method must be used, though there are two solutions
due to the second order polynomial in the solution process, creating complications.
3.2 Models
3.2.1 Linear Elastic Fracture Mechanics
Linear elastic fracture mechanics (LEFM) coupled with the Irwin equations38
describes the stress fields in the vicinity of an elliptical crack. For the case where the
crack caused by applied stresses that is perpendicular to the x-z plane, as shown in Fig.
3.1, one speaks of a Mode I fracture and the equations are written in the form39:
Fcr
w
d
w
F
ob
Figure 3.1 Scrack front.
where r is the
efined as
where σ and
ig 3.1) the
btained from
chematic repr
e distance fro
a the applied
angle θ is z
m Eqs. (3.8a-
resentation o
√
∙
√
∙
om the crack
d stress and
zero, which
-b) as
f the coordin
1
1
k front and K
√
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defines an
nate system d
∙
∙
KI is the stre
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ength, respec
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.
describing the
,
ess intensity
ctively. On t
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(3
(3
factor which
(3
the x-z plane
ress conditio
(3
13
at the
.8a)
.8b)
h is
.9)
e (see
on as
.10)
14
By applying Eq. (3.10) to the von Mises yield criterion (see Eq. (3.6)) one arrives to the
following set of equations:
2 2 (3.11a)
2 (3.11b)
1 2 , (3.11c)
where in the second step we employed our result from Eq. (3.3). If we assume ν=1/3 we
arrive at the following final result for the plane strain condition:
3. (3.12)
This ratio is called the plastic constraint factor and we can derive this same factor for
plane stress condition 0 by utilizing the von Mises yield criterion on the x-z plane
as:
2 2
, (3.13)
which leads to a plastic constraint factor of
1. (3.14)
To calculate the plastic region of the stress field in the vicinity of the crack, Eq.
(3.13) for plane strain is used as shown below
3 ⇒ (3.15)
15
while the same for plane stress is
⇒ (3.16)
In another article37,40 addressing experimental measurements showing values for
the plastic constraint factor well below 3, Irwin recalculated the plastic constrain factor
and arrived at a value of √3. That analysis37,40 resulted in a plastic region radius equal to
. (3.17)
Earlier, there was a lack of accounting of the radius due to the elasticity of the
material in discussing the radius of the plastic zone. Irwin’s correction applied to the
formulas for a plastic zone show a doubled radius resulting in the following expressions:
2 (3.18a)
2 . (3.18b)
The relationships in Eqs. (3.18a-b) are demonstrated in Fig. 3.2 which clarifies the origin
of the factor of 2.
m
st
h
T
v
so
co
Figure 3
The I
mentioned an
tress σy to re
erein. It ha
Therefore, the
Theor
on Mises eq
olve for the
orrections w
3.2 Schematic
Irwin correc
nd graphicall
elax to the yi
as been show
e total plasti
retically, a p
quation (Eq.
e radius, ass
which yields E
c representatio
ction for the
ly shown in
ield stress σy
wn that the
ic zone size i
plot of the pl
(3.6)) with t
suming a Po
Eq. (3.19a) f
on showing th
e plastic zo
Fig. 3.2. T
ys which is v
increase in
is 2rp.
lastic zone c
the elastic so
oisson’s rati
for plane stre
1 c
1
he Irwin corr
one size is
This factor is
very plausibl
the plastic z
can be done
olutions (Eq.
io of 1/3, an
ess and Eq.
cos sin
cos
rection of the
a factor of
s due to the
le for the pro
zone is appr
in two dime
. (3.8)), and
nd accounti
(3.19b) for p
n
sin
plastic zone.
2 as previ
allowance o
oblem consid
roximately r
ensions usin
by rearrangi
ing for the I
plane strain:
(3
(3
16
ously
of the
dered
rp.25,39
ng the
ing to
Irwin
.19a)
.19b)
th
as
b
h
zo
d
Figure
he plane stra
ssumes a yi
ased upon th
3.2.2 H
The L
ardens the m
one size.
islocation di
e 3.3 shows
ain and the p
eld stress of
he experimen
Figure 3.3,
Hutchinson
LEFM model
model canno
Another fea
irection whic
a polar plot
plane stress c
f 646 MPa a
ntal paramet
Theoretical p
n-Rice-Rosen
l from the p
ot account fo
ature lost d
ch is found
t of the comp
conditions, a
and a KI of
ters used in t
plastic zone sh
ngren Zone
revious sect
or it. This c
due to this
at ± 45° from
puted plastic
accounting fo
f 38.5 MPa∙
this study.
hapes at the ti
es
tion has one
causes the u
underestima
m the crack
c zone at a c
for 2 rp. Add
√ for illu
ip of a crack.2
major setba
underestimati
ation is the
k tip plane. H
crack tip for
ditionally Fig
ustrative purp
25,26
ack. If a ma
ion of the p
e account o
Hutchinson4
17
r both
g. 3.3
poses
aterial
plastic
of the
41 and
18
Rice and Rosengren42 came up with refinements of the LEFM to take into account the
Ramberg-Osgood relationship
/
, (3.20)
from Eq. (3.20) the following relationship could be conceived -
/
, (3.21).
which leads to a very complex numerical solution that has been approximated by the
following plastic zone for a value of n’ = 0.15, which is what we see in 4140 steel. The
following figure (Fig. 3.4) shows a polar plot of the plastic zone boundary computed
from the HRR zone of concern in the plane strain condition shown with both conditions
of the LEFM plastic zone size. It should be noted that at θ=0° the difference in rp does
not change much, but the plastic zone does increase in size quite a bit.
cr
b
co
m
to
b
d
Figure 3
3.2.3 B
Furthe
rack tip bein
efore the cra
onsiders its
model accoun
o the radius o
e 2µm, whic
irect experim
.4, Theoretica
Blunted Cr
er advents fr
ng a sharp
ack tip. Ho
elementary
nts for a par
of the blunte
ch has been
mental mean
al plastic zone
ack Tip Mo
from others,
point cause
wever, if a
block then n
rameter ρ*, w
ed crack tip.
corroborate
ns (APPEND
e shapes at th
odel (ρ* Mo
primarily G
s a singular
model accou
new possibi
which is a c
In 4140 ste
ed from expe
DIX A).
he tip of a crac
del)
Glinka and o
rity and all
unts for the
ilities in mod
constitutive m
eel the value
erimental da
ck including
others43-46 ha
previous m
crack tip be
deling exist
material pro
e of ρ* has b
ata from fati
HRR.25,26,41,42
ave noted tha
odels are in
eing rounded
. Ultimately
operty that re
been calculat
igue tests43-4
19
2
at the
nvalid
d and
y this
elates
ted to
46 and
20
A model of a plastic zone based upon ρ* can be devised by using the ρ*
equations44-46 relating to stress and assuming plane strain conditions (Eq. 3.3). The said
equations are:
√
∗cos
√cos 1 sin sin (3.22a)
√
∗cos
√cos 1 sin sin (3.22b)
In these equations the radius scale is shifted on half radius of the ρ* so that the
point corresponding to the crack tip is ρ*/2. The significance of the equations is that at
the crack tip (ρ*/2) the σ11 is 0, and that is what should be expected of a crack.
Conversely in other models the value of σ11 is near infinite at the same point.
The resultant plastic zone size in plane strain is shown in Fig. 3.5 along with the
previous two models (LEFM and HRR) gives a much different plastic zone shape; the
issue arises due to the extra term for the radius in the stress coordinates increasing the
order of the polynomial in the expression to solve for σeq. Further the model is an
improvement that it only models in front of the crack not behind it. Again like the
previous models at θ=0° the size of the plastic zone isn’t much different in all three
models.
Fρ*
3
st
fi
m
w
th
sc
Figure 3.5, Th*.23,24,47,48,50-52
.3 Direct M
3.3.1 M
The m
train localiza
inite elemen
microscopy,53
which are des
he conventio
cattering me
heoretical plan2
Measurement
Mapping al
measurement
ation phenom
nt modeling,
3 hardness
structive me
onal Bragg-
ethods60 are w
ne strain plast
t of Plastic Z
ll 3 elastic st
t of the sai
mena have b
,47-52 and ex
profiling by
ethods. Whi
-Brentano d
well known,
tic zone shape
Zones
train compo
id plastic zo
been attempt
xperimental
y indentatio
ile elastic str
diffraction t
, the penetra
es at the tip o
onents
one associat
ted by comp
methods su
on54,55 and c
rain measure
technique59
ation depth o
of a crack of L
ted with the
putational te
uch as trans
chemical etc
ement in sol
and by sm
of these meth
LEFM, HRR
e aforementi
echniques su
smission ele
ching,55-58 a
lids by the u
mall angle X
hods is mea
21
and
ioned
uch as
ectron
all of
use of
X-ray
sured
22
by the microns which limits their use analysis to the surface and sub-surface.
Furthermore, the measurement of stress fields in the bulk of cracked samples by neutron
diffraction has been possible for some years as well, however, the spatial resolution (~1
mm) is deemed to be too coarse grained for resolving the immediate crack-tip field.61-64
Thanks to the proliferation of high energy polychromatic synchrotron X-ray probes in
recent years, Laue mode (transmission) X-ray diffraction methods are being brought to
bear which enables one to accomplish strain field profiling/mapping of local strain field
gradients on the pertinent short length scales deep in the interior of materials6-10 whose
thickness can be as high as 150 cm in aluminum, 2.54 cm in steel and 0.5 cm in
tungsten.6,7 A seminal elaborations comparing and contrasting polychromatic and
monochromatic elastic strain mapping techniques using X-rays in polycrystalline solids
can be found in Refs. (12-20).12-20
In this dissertation, reported measurements of the 3D state of stress at the tip of a
crack in a ductile metal which has been first subjected to a cyclic load followed by static
overloading under SCC conditions. To accomplish this polychromatic synchrotron X-rays
with photon energies reaching up to 200 keV were used by which 3D elastic strain
mapping in the near field of the crack tip was possible. Specifically, we will show that
the approach pursued in this study enables one to determine all three elastic principle
strains ε11, ε22, and ε33 from which the 3D state of stress (in terms of the principle stresses
σ11, σ22, and σ33) can be computed without any a priori assumptions typically used in
continuum mechanics.25,26 It will also show that by using the von Mises criterion in
conjunction with the said principle stresses, one can analyze plasticity phenomenon and
measure the size of the plastic zone at the tip of the crack.
23
3.3.2 Mapping of ε22 only
In the previous section (3.3.1) the mapping of the near field was done from all
three directions. However due to the limitations of the beamline, as of the time of the
study, only one direction could be measured at a time. This creates a very tedious and
lengthily process of measurements and realignments of the samples three times. From
previous work6-9 and to be proven here one component can be measured instead. The
elastic strain normal to the crack propagation ε22 is sufficient. This can be verified by the
observing the overlay of the models over all of the strain and stress plots, which is
presented in Chapter 5.
24
CHAPTER 4 Experimental Procedure
4.1 Compact Tensile Specimen
Stress corrosion cracking (SCC hereafter) experiments were conducted on
standard compact tensile (CT hereafter) specimens made out of 4140 steel. Firstly, the
specimens were fatigued using a sinusoidal cyclic excitation to a fracture toughness value
(KI) of 24.2 MPa√m which caused the nucleation and subsequent growth of a ~5.7 mm
crack verified by optical means. Then, an in situ static load (~1000 lbs.), corresponding to
KImax of 38.5 MPa√m was applied while the specimens were subjected to three distinct
environments: i) air in which no crack growth was observed after 24 hours, ii) a 3.5%
NaCl aqueous solution in which no crack growth was observed after 20 hours and iii)
3.5% NaCl aqueous solution with Zn couple (hereafter H2O-NaCl-Zn solution) in which
three samples were placed a) no crack growth was observed after 44 minutes, b) the onset
of crack propagation was observed after 147 minutes, and c) after 4.1 hours a crack grew
up to ~1.3 mm according to optical observations. The crack size was concurrently
measured via potential changes from the crack opening displacement (COD) which is
shown in Fig. 4.1. Table 4.1 summarizes the specimen designations of the 4140 steel
investigated in this study and summarizes the data in Fig. 4.1.
Fscin
4
B
T
p
Figure 4.1 Gracale. Note: Tndication of c
Table 4.1 Specim
Air Con
Aqueous C
SCC SCCSCC
.2 Synchrot
The S
Brookhaven N
The said bea
olychromati
aph of the craThe COD is m
rack size.
– Specimen dmen O
ntrol
Control
A
(Se B C
tron-Energy
S-EDXRD m
National La
amline is equ
ic X-ray rad
ack open displmeasured from
designation wOverload En
Air
3.5% NaCl(Sea W
3.5% NaClea Water) with
y Dispersive
measurement
aboratory’s (
uipped with
diation comp
lacement (COm a change in p
with their overnvironment
r
Solution Water)
Solution h a Zn Coupl
e X-Ray Dif
ts were per
(BNL) Natio
h a supercon
prised of ph
OD) potentialpotential and
rload environC
N
N
e
No Onset oCrack
ffraction
rformed at t
onal Synchr
nducting wig
hoton energi
s in Volts oved the potential
nment and craCrack Cond
No growth afte
No growth afte
growth after of crack growgrows 1.3mm
the X17-B1
rotron Light
ggler, enabl
ies as high
er time in a lol itself is not a
ack conditionsdition
er 24h
er 20 h
44 min. wth 147 min.m after 4.1h
beamline o
Source (NS
ling one to u
as 200 keV
25
og an
s.
of the
SLS).
use a
V. As
26
depicted in Fig. 4.2, the SEDXRD technique used in this study is essentially Laue mode
diffraction (also known as transmission mode diffraction), where the so-called gauge
volume (GV hereafter) and/or diffraction volume is fixed in space. In this technique, the
specimen is mounted on an x-y-z stage which is controlled by a micropositioner. Thusly,
the body center of the GV can be placed on any (x,y,z) coordinate inside of the specimen
so as to obtain a diffraction pattern in transmission from a given region, thanks to the
high energy photons available in the incident beam (see Fig. 4.2). The CT specimen
thickness in the transmission direction was less than 10 mm in all of the specimens which
were studied. Therefore, X-ray absorption that would otherwise cause peak shifts in very
thick specimens is of no concern here due to the high photon energies in the incident
beam. As a matter of fact, ferrous alloys as thick as 40 mm can be probed with no
absorption correction using this technique as shown in our previous studies.6-9
FsuNbe
by
w
1
(s
(e
m
la
Figure 4.2 Tuperconducto
National Laboeam and diffr
The g
y combining
where h is th
08m/s), θ is
see Fig. 1b),
eV) is the co
measured qua
attice vector
he schematicor wiggler bratory showin
racted beam d
overning equ
g the Planck-
he Planck’s
the Bragg a
, dhkl is the i
orresponding
antity. Sinc
of the (hkl)
c of the experbeamline in ng the relativ
detectors, and
uation for th
-Einstein rel
constant (4
angle which
nterplanar sp
g scattered e
ce 1hkld hkQ
) reflection,
E
rimental set-uthe National
ve positions od the Bragg an
he S-EDXRD
lation65 with
4.1357×10−15
h is always k
pacing of th
energy from
kl , where Q
the S-EDX
12
hklh
hcE =
d
up used in thl Synchrotro
of the beam pngle is depicte
D technique
the Bragg’s
5 eV), c is t
kept constan
he (hkl) refle
m the (hkl) re
hklQ
is the
XRD method
hkl
csin
he present stuon Light Sopath, gauge ved.
used in this
s law59 as fol
the speed of
nt in an EDX
ection in An
eflection wh
magnitude
d used in this
udy at the X1urce, Brookholume, transm
study is obt
llows:
(4
f light (2.99
XRD experi
ngstroms, an
hich is indee
of the recip
s study mea
27
17-B1 haven mitted
ained
.1)
979 x
iment
d Ehkl
ed the
procal
asures
28
the changes in the d-spacing in the reciprocal space directly, making it extremely accurate
as compared to conventional Bragg-Brentano diffraction methods.6-12 As seen in Fig. 4.3, the gauge volume, which has the shape of a parallelepiped, is
controlled by two parameters: i) the Bragg angle, and ii) the slit 1opening (designated as
S1). In this diffraction method, the higher the Bragg angle the smaller the dimension of
the parallelepiped in the transmission direction which is called the gauge length (L
hereafter) as depicted in Fig. 4.3. The S1 has two openings which are the horizontal and
vertical openings along the x and y axes of the Cartesian coordinate system (see Fig. 4.3
and Fig. 3.1), respectively. Hence, the shape of the footprint of the S1 openings on the
specimen’s face (see Fig. 3.1) is a rectangle whose dimensions can be adjusted as needed.
The Bragg angle used in this study was typically 2θ=4° which resulted in a gauge length
of ~ 1 mm for the typical S1 y-axis opening used, i.e. 100 μm x 100 μm. Therefore, the
strain mapping measurements in the vicinity of the crack tip reported herein appertain to
a plane strain condition.66 That is so because the specimen thickness (t hereafter) is much
smaller than the gauge length, i.e. L << t, as shown in Fig. 3.1.66
Fgesi
sp
co
st
su
p
Figure 4.3 Teometry depiize S1 and the
The f
pecimen’s ce
onstraint tha
train is und
urface meas
lastic zone s
The schematcting where te Bragg angle
following sc
enter is unde
at as long as
der the plan
sures the pla
sizes of plane
ic ray diagrathe parametere 2θ.
chematic in
er the plane s
L is roughly
e strain con
ane stress c
e strain and
am of the trrs defining the
Fig. 4.4 sho
strain condit
y 50% the th
nditions. C
conditions.
stress.
ransmission me gauge lengt
ows how th
tions.66 This
hickness of t
Conversely a
Refer to Fi
mode (Laue th L, such as
he bulk of a
s property all
the plate, the
as well, me
ig. 3.3 to se
mode) diffraincident beam
a compact te
lows a meas
e measureme
easuring nea
ee the respe
29
action m, slit
ensile
suring
ent of
ar the
ective
Fto
ob
w
th
p
ob
ce
g
re
Figure 4.4 Sco the plane str
The e
btained from
where d°hkl is
he interplana
arameters w
btained by p
entered cubi
ermanium d
eference stan
chematic in 3Dress condition
elastic strain
m the measur
the interplan
ar spacing at
which follow
peak fitting a
ic iron(BCC
detector in
ndards includ
hkl
D space of thens at the surfa
n (εhkl) at a
red energy o
nar spacing
t zero stress,
w from Eq. (
a Gaussian p
-Fe hereafte
the 0-200 k
ding Au, Ag
hkld
d
e transition frace.66
given point
of a given (hk
of the (hkl)
, while Ehkl(σ
(1). Here, th
profile shape
er; Im3mspac
keV range
g, Cu, LaB6,
ohkl
ohkl
d
d
rom the plane
t in the spec
kl) reflection
reflection o
σ) and Eohkl a
he exact valu
e function to
ce group). T
has been a
and CeO2.67
ohkl h
ohkl
E E
E
e strain condit
cimen under
n according
,
of interest un
are the corre
ues of Ehkl(σ
o the (110) r
The energy c
ccomplished
7
ohkl
tions in the ce
r interrogati
to
(4
nder stress, d
esponding en
σ) and E°hkl
reflection of
calibration fo
d using a s
30
enter
ion is
.2)
d°hkl is
nergy
were
body
or the
set of
31
4.2.1 Mapping all three strain components
The principle strain components εii (i=1,2,3) of the general 3D strain tensor has
been measured and analyzed using a gauge of 100 x 100 µm. Figure 2 depicts the axes
system used in the measurement of the strain. Specifically, when the specimen (hence
the orientation of the notch) is as shown in Fig. 3.1, the measured principle strain is ε22.
The measurement of the ε11 strain was accomplished by rotating the specimen (and hence
the notch) by π/2 radians counter clockwise. The principle strain ε33 was obtained through
the measurement of ε23 in conjunction with the measured ε22 through the use of the
following relationship,68
(4.3)
where sec(θ)=cos-1(θ) and θ is the inclination of the specimen with respect to the xy-plane
or, equivalently, the counterclockwise rotation around the x-axis (see Fig. 3.1). The angle
of inclination was set to π/3 radians in all ε23 measurements; i.e. θ=60°.
The S-EDXRD endows one with the unique ability to measure the elastic strains
at any given location in the specimen (see Fig. 3.1). In order to ensure that the plain strain
condition is valid, all strain data reported herein was measured in the x-y plane at z=0 as
shown in Fig. 3.1. Once the principle strains εii (i= 1-3) are obtained experimentally, all
of which are elastic strains, the principle stresses σii(i= 1-3) are computed using the
generalized Hooke’s Law therefrom. Then, the von Mises equivalent stress (σeq hereafter)
was computed for each near field coordinate of the crack tip to evaluate the plasticity
phenomenon and the size of the said plastic zone.
233 23 22 22= + + sec ( )
32
4.2.2 Mapping just the elastic ε22 component.
After measuring 2 of the samples (the air and SCC-B samples refer to Table 4.1)
in all three directions in a limited area, as mentioned in section 4.2.1, a more robust
measurement was done in the ε22 strains exclusively for all 5 specimens as mentioned in
section 4.2.2. The exclusive measurement procedure wasn’t interrupted as much with
reorientation of the samples. This allowed for more time to measure the sample much
more finely over a larger area. In the exclusive measurement the gauge was shrunk down
to 40 x 30 µm (w x h) measuring all the way from the notch tip to past the plastic zone
over 5 mm. All samples except for the SCC C sample were measured in a region from
±0.75 mm along the crack from the notch tip to 14 mm past the notch covering the plastic
zone. In the SCC C sample it was known that a larger area would be needed, so the
height over and under the crack was increased to ±1.3 mm.
4.2.3 Plane Stress vs. Plane Strain
Recalling from earlier (Eq. 3.12,14 and 18), it is expected that the plastic zone
near the surface where the plane stress conditions are met should give a plastic zone
roughly 3 times larger. To verify this method two sets of parallel measurements of ε22
were made along the same points along values of z=0 and z=-2.4 mm (See Fig. 3.2). In
those experiments the gauge was set to 20 x 100 µm (w x h) and from the notch tip to
over 5 mm past the crack tip. This was only conducted on the SCC-C sample figuring
that it would have the largest plastic zone in plane strain, thus having an even larger
plastic zone in plane stress.
C
m
F
5
an
b
w
h
n
Fth
CHAPTER 5
This
measurement
e-H) or mec
.1 X-Ray Pe
Figure
ngle used in
e a preferre
what is happ
igher d spac
ear the (200
Figure 5.1 Indhe other two p
5 Results an
chapter org
ts whether u
chanical prop
eak Analysi
e 5.1 shows
n our studies
d textured,
ening is tha
cing, around
) and (211) p
dexed spectrupresent due to
nd Discussio
ganizes the
used to mea
perties (stres
is
a typical n
(2θ ~ 4°), th
which could
at the low B
d 2Å, in the
planes, creat
um of 4140 steo the Bragg an
on
e results of
asure compo
sses and strai
normalized s
he d spacing
d be expecte
ragg angle (
spectrum w
ting a false i
eel. Note: Thngle.
f the meas
ositional “cr
ins).
spectrum of
g values arou
ed in a plat
(2θ<6°) in t
while not am
indication of
he first peak (
surements o
rystallograph
4140 steel.
und the (110
te to some e
the Laue mo
mplifying the
f absorption
110) is much
of the ED
hic” (presen
Due to the
0) plane appe
extent. How
ode amplifie
e higher ene
from texture
h more intense
33
DXRD
ce of
e low
ear to
wever
es the
ergies
e.
e than
34
5.1.2 Iron Hydride
In figure 5.2 a blown up spectrum of 4140 steel is shown with the all of the
respective indexed peaks. In this situation lines corresponding to 1% FeH are drawn in to
show how that SEDXRD is not fully capable of finding FeH in the crack tip that is a
possible contender for a culprit in hydrogen embrittlement. Noting that the most
prominent peak of FeH is very close to the (110) peak of the iron and thus would be
impossible to detect. The secondary peaks appear close to well defined peaks suggesting
that the FeH is not present since all peaks are accounted for. It would seem that FeH
detection which is possible with the stress loads observed35, but in this alloy it is hard if
not impossible to determine. Further all of the peaks are identified in the steel spectrum
as to be expected from its composition. This further strengthens that there are no iron
hydride peaks. The identification of the carbides at the low energies (and thus high d
spacing points), shows again the enhancement of the gain. The first three carbide peaks
agree with the carbon content of the alloy 0.40% and the chromium near 1%, etc.
Fre
5
el
co
sp
p
th
un
Figure 5.2 Indespect to the s
.2 Mechanic
5.2.1 F
In Fig
lastic strain
ompared an
pecimens de
osition in Fi
he ε22 measu
ntil the crac
dexed spectrumspectrum mea
cal Measure
Finding the
g. 5.3, the va
along the x
nd contraste
esignates the
ig. 5.3 revea
urements, the
k expands fr
m of 4140 steasured if the c
ements
e Crack Tip
ariation of th
x-axis (see F
ed in all of
e crack tip (
als that as th
e trough bec
from the ove
eel along withcomposition o
e so-called n
Fig. 3.1) is s
f the specim
see Fig. 5.3
he material c
comes much
erloading (SC
h peak profileof FeH was 1%
near field as
shown, wher
mens. The
). A compa
cracks due t
h sharper as
CC C). The
es of where Fe%.
well as far f
re the said v
e observed
arison of ε22
to SCC. At
the SCC bu
e localized s
eH would be
field ε22 prin
variation in
minimum i
2’s variation
t the minimu
uilds up (SCC
train is incre
35
in
nciple
ε22 is
in all
n with
um in
C B),
eased
du
v
cr
th
5
T
w
an
FalApr~
ue to increa
ariation of ε
rack bluntin
he crack (co
.3 is the wid
The scale of
which hosts t
nd crack gro
Figure 5.3 Thlong the x-ax
A) is shown trogression of6mm.
ased localiz
ε22 in the wak
ng due to SC
mpare Figs
dth of the m
the said wid
the plastic z
owth are sum
he comparisonxis in all five twice for betf SCC. The
zed deforma
ke of the cra
CC is a mani
5.3 for x<6
minima obser
dth, which r
zone at the c
mmarized in
n of the varisamples. The
tter comparistroughs enab
ation in the
ack (x<6 mm
ifestation of
mm). Anot
ved in ε22 in
ranges up to
crack tip. Th
Table 5.1.
ation of the me SCC samplsons of the sble an appro
immediate
m), as illustra
f the increase
ther importa
n the immed
~1 mm, de
he informati
measured elale that did nostrains in nonximate locati
vicinity of
ated in Fig. 5
ed compress
ant feature o
diate vicinity
fines the so
ion for the c
astic principleot have any crn-cracked samion of the cr
the crack.
5.3, indicate
sive strains a
of the data in
y of the crac
-called near
crack tip loc
e ε22 as a funrack activity mples and forack tip starti
36
The
s that
along
n Fig.
ck tip.
field
cation
nction (SCC
or the ing at
37
Table 5.1 Crack tip location and crack growth in the 4140 CT specimens.
Specimen Crack Tip @ yr=0
(mm) Crack Growth
(mm) Air 5.9 0
Salt Water 6.1 0 SCC A 6.0 0 SCC B 5.9 0.3 SCC C 5.9 0.9
5.2.2 Strain Components of Strain and Stress for Equivalent Stress Plotting
For these measurements only two of the specimens were used, the air control and
the specimen at the onset of crack propagation. Figure 5.4 depicts the variation of the
elastic principle strains ε11, ε22, and ε33 as a function of the x-coordinate within the near
field of the crack tip. The ε33 is obtained directly using Eq. 4.3 with measured values of
ε22 and ε23, and with the known value of θ=60° without any a priori assumptions. As
defined in Sect. 4.2, what is dealt with herein is a plain state of strain for which the while
as proven with the data in hand. Furthermore, as can be verified by a comparison of Fig.
5.4a-b, the major increase in the strain due to SCC is for the elastic principle strain ε33
which is why it is of utmost importance to measure with great accuracy and without any a
priori assumptions. Specifically, one observes that the elastic the elastic principle strain
ε33 reaches 0.4% at x~6 mm which is the crack tip. Were it not for the directly obtained
value of ε33, no such assessment could have been made, making the analysis of crack tip
plasticity impossible.
ε3
ai
si
in
th
d
si
eq
M
M
S
st
d
cr
S
H
Figure33 in the immir (left) and t
The c
imultaneous
n Fig. 5.5. N
he immediat
efines the p
ize of the pla
quivalent str
MPa for 4140
MPa for all v
CC specime
tress of 414
eformation i
rack tip (x~6
CC specime
H2O-NaCl-Zn
e 5.4 The commediate vicinit
he SCC B spe
correspondin
solution of
Notably, the
te vicinity o
plastic behav
astic zone. T
ress σeq as d
0 steel.25,26 B
values of x in
en reaches v
40 (646 MP
in the imme
6 mm), one
en B), which
n corrosive e
mparison of thty of the cracecimen subje
ng principle
f the general
σ33 reaches
of the crack
vior due to s
To quantify
defined by th
By taking a g
n the contro
values as hig
Pa) steel is
ediate vicinit
observes th
h is solely att
environment
he position deck tip which icted to stress
stresses (σ1
ized Hooke’
700 MPa w
(see Fig. 5.
stress conce
the plastic b
he von Mise
glance to Fig
l specimen (
gh as 1000
s exceeded
ty of the cra
at the increm
tributed to th
t.
ependence of is located at ~corrosion cra
1, σ22 and σ
’s Law (See
while σ11~100
.5b). It is t
ntration at t
behavior of th
es criterion (
g. 5.5, one cl
(see Fig. 5.5
MPa (see F
in the SCC
ack tip. By
ment in the
he chemical
f the principle~6 mm for 41acking (right)
σ33) were co
e Eq. 3.1) w
0 MPa and σ
this 3D state
the crack tip
he 4140 stee
(See Eq. 3.6
learly observ
5a), whereas
Fig. 5.5b).
C specimen,
comparing
σeq is ~550
attack along
e strains ε11, ε140 steel load).
mputed from
which is pres
σ22= -100 M
e of stress w
p, and hence
el under SCC
6) where σys
ves that σeq <
s the same fo
Hence, the
, causing p
Figs. 5a-b a
MPa (contro
g the crack b
38
22 and ded in
m the
ented
MPa in
which
e, the
C, the
s=646
< 400
or the
yield
plastic
at the
ol vs.
by the
Fth~cr
w
w
el
m
sa
ap
fo
ax
zo
sh
ca
H
in
th
Figure 5.5 Thhe von Mises 6 mm for 41racking (right
Figure
was obtained
with Eqs. (3.1
lastic strain
material (see
ample (subj
ppreciably l
or the SCC s
xis is ~300 µ
one is indee
hould be not
an assume b
However, one
ndeed under
he far field d
he comparisonequivalent st
140 steel loadt).
e 5.6 compa
d from mapp
1, 3.6 and 4.
is not suffi
Fig. 5.6).
jected to ai
ess than the
specimen rev
µm wherein
ed the plast
ted that σeq
both positive
e should not
a 3D compr
develops a c
n of the posittress σeq in thded in air (le
ares the vari
ing of the el
3). It should
cient in ana
As can be o
ir atmosphe
e yield stress
veals a zone
the σeq stres
ic zone bec
>0 for all v
e and negativ
tice that the
ressive state
compressive
tion dependenhe immediate eft), and spec
iation of the
lastic princip
d be noted th
alyzing plast
observed in
ere) does n
s (646 MPa)
e of pseudo-e
ss exceeds th
cause of all
alues of wh
ve values as
plastic zone
of stress.25,2
e elastic stra
nce of the pri vicinity of th
cimen SCC B
e von Mises
ple strains ε
hat the meas
ticity at the
Fig. 5.6a, t
not exceed
). On the ot
elliptical sha
he yield stre
values of (
hile the princ
s per the von
e at the crack
26,39,42 That
ain due to th
inciple stressehe crack tip wB subjected t
s equivalent
ε11, ε22 and ε
surement of
crack tip in
the highest σ
250 MPa,
ther hand, th
ape whose w
ength of 414
(x,y) coordin
ciple stresses
n Mises equa
k tip in the
is so becau
he formation
es σ11, σ22, σ3
which is locato stress corr
stress σeq w
ε23 in conjun
a single prin
n an elastop
σeq in the co
which, in
he σeq stress
width along t
0 steel. The
nates σeq>σy
s σ11, σ22 an
ation (see E
SCC specim
se the mater
n of an expa
39
33 and ated at rosion
which
nction
nciple
plastic
ontrol
turn,
s map
the x-
e said
ys. It
nd σ33
q. 9).
men is
rial in
ansive
p
sa
Fcalodest
by
h
el
w
th
p
lastic region
aid crack op
Figure 5.6 Thealculated vonoaded in air (leformation attress contour
5.2.3 A
In Fig
y side for th
ow the areas
lastic stains
while the ε11
hat the plast
lastic strains
n associated
ening is indu
e comparisonn Mises equivaleft), and SCCt the crack tipscales.
All Compon
gs 5.7-9 the s
he air contro
s of high com
(Fig. 5.7) th
direction is
tic stain val
s are all clos
with crack
uced by stres
n of the contoualent stress σe
C B subjectedp is monitored
nents of Str
strain plots o
ol specimen
mpressive str
he highest c
primarily te
lues are ext
ser to the sam
opening. In
ss corrosion
ur plots basedeq in the imme
d to stress corrd. In APPEND
ain and Str
of all three p
for elastic,
rain are vary
compressive
ensile. In th
tremely sma
me location.
n the particu
cracking.
d on the positediate vicinityrosion crackinDIX B Fig. 5
ress in Full 2
principle stra
plastic and
y in the 2D s
strains are
he plastic str
all, but unlik
. In the plas
ular case con
tion dependeny of the crackng (right) wh
5.6 is shown w
2D Space
ain direction
total strains
space in the
observed in
rains (Fig. 5
ke in the el
stic strains (
nsidered here
nce of the k tip. 4140 sthereby plastic with the same
ns are shown
s. If can be
near field. I
n the ε22 dire
.8) it can be
lastic strains
(Fig 5.9) the
40
e, the
teel
n side
seen
In the
ection
e seen
s, the
e total
st
re
3
co
Fε1
Fε1
5
Dis
tan
cefr
omn
otch
tip
'y'd
irec
tion
(mm
)
trains aren’t
ecalling the
.2 and 3.7) i
omparison to
Figure 5.7 Th11, ε22 and ε33.
Figure 5.8 Th11, ε22 and ε33..7.
5.4 5.6
Dis
tan
ce f
rom
not
ch t
ip y
dir
ecti
on (
mm
)
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6 11plastic
t much diffe
relationship
in conjuncti
o the elastic
e comparison. All three are
e comparison. All three ar
5.8 6.0 6.2
erent from t
of total stra
on with the
strains.
n of the elastie on the same
n of the plastire on the same
6.4
Distance from
5.4 5.6
plastic
the elastic s
ain being the
plastic strai
c strains in ale strain scale.
c strains in ale strain scale
notch tip 'x' directi
5.8 6.0 6.2
strains. Thi
e sum of ela
ins being ord
ll three princi
ll three princiwhich is muc
ion (mm)
6.4 5.4 5.6
plastic
is comes to
astic and pla
ders of mag
iple direction
iple directionch smaller tha
6 5.8 6.0 6.2
o be expecte
astic strains
gnitude smal
ns from left to
ns from left toan the scale in
plastic (%)
2 6.4
-0.000-0.000-0.000-0.000-0.0000.00000.0000.00020.0003
41
ed by
(Eqs.
ler in
o right
o right n Fig.
)
05 04 03 02 01 0 1 2 3
42
Figure 5.9 The comparison of the total strains in all three principle directions from left to right ε11, ε22 and ε33. All three are on the same strain scale as Fig. 5.7. Due to the extremely small values of plastic strain, there is a negligible difference between Figs 5.7 and 5.9.
In Fig 5.10 the stresses plots are posted showing all three principle stresses and
the von Mises equivalent stress. By convention of the von Mises yield criterion the sign
of the stresses is always positive. In this situation the sample never reaches its yield
stress 646 MPa. It can be seen that the stress plots have varying compressive stress
regions, but the σ22 and σeq mark out the largest are that could be considered a “zone”.
total (%)
5.4 5.6 5.8 6.0 6.2 6.4
Dis
tanc
e fr
om n
otch
tip
'y' d
irec
tion
(m
m)
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Distance from notch tip 'x' direction (mm)
5.4 5.6 5.8 6.0 6.2 6.4 5.4 5.6 5.8 6.0 6.2 6.4
-0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04
total
totaltotal
Ffrsa
Figure 5.10 Trom the top lame stress sca
The comparisleft corner anale.
on of the prind moving clo
inciple stressock wise σ11,
es and von M, σ22, σ33 and
Mises equivad σeq. All fou
alent stress stur plots are o
43
tarting on the
ar
m
ε2
ea
In
th
F
w
F
in
cr
at
F
5.2.4
Previo
re the princi
much finer ga
22 were mad
ach sample.
n Fig. 5.12 t
he area arou
igs. 5.13 an
while the cra
urther in the
n Fig. 5.15 th
rack propag
ttack returns
Figure 5.11 St
Strain Map
ous measure
iple strains p
auge (40 x 3
e for all 5 sa
In Fig. 5.11
the effects o
und the plast
d 5.14 when
ck is stable
e SCC sampl
he strains ap
gates an appr
s.
train map of t
ps of ε22
ements (Figs
perpendicula
30 µm) as m
amples. Figu
1 the air con
of salt-water
tic zone is w
n the environ
(Fig. 5.13) a
les microcra
ppear like a l
reciable am
the air sample
s. 5.3-10) in
ar to the cra
mentioned in
ures 5.11-15
trol sample c
can be seen
washed out.
nment has b
and upon th
acking can b
larger field a
ount that th
e which was t
ndicate that t
ack propagat
n section 4.2
5 are the stra
creates a ver
n, primarily
The same
been coupled
e onset of cr
e observed a
as Fig. 5.11
he strain fiel
the control sp
the measure
tion directio
2.2, a higher
ain maps in th
ry good refe
the crack is
phenomeno
d with zinc (
rack propag
at the crack
. This sugge
ld relaxes un
pecimen.
ements of int
on ie ε22. Us
r resolution s
he same sca
erence strain
s washed ou
on is observ
(SCC condit
gation (Fig. 5
tip. Interest
ests that onc
ntil the chem
44
terest
sing a
set of
ale for
map.
ut and
ved in
tions)
5.14).
tingly
ce the
mical
F
Fwm
Figure 5.12 St
Figure 5.13 Stwhich no crackmicrocracks fo
train map of t
train map of tking occurredorming.
the salt water
the salt water d. You can se
sample, whic
coupled withee that the cra
ch was the aq
h zinc sampleack tip is more
queous contro
, which was te jagged (nea
l sample.
the SCC sampar x=6) possib
45
ple in bly
Fwx=
Fwsi
T
S
sa
Figure 5.14 Stwhich the onse
=6) possibly m
Figure 5.15 Stwhich the cracimilar quality
5.2.5 M
From
To prove tha
ection 3.2.
ample. From
train map of tet of crackingmicrocracks f
train map of tcking propagay to the origina
Model Over
section 5.2.
at the measu
In Fig. 5.16
m the overla
the salt water g occurred. Yforming.
the salt waterated ~0.9 mmal control (Fi
rlays of the
.2 the contou
urements we
6 the three m
ays it shows
coupled withYou can see th
r coupled withm. You can se
g. 5.11).
Plastic Zon
ur plots of t
ere justified
models are p
s that the H
h zinc samplehat the crack t
h zinc sampleee that the cra
ne with ε22
the ε22 were
d we overlai
placed on top
HRR model f
, which was ttip is even mo
e, which was ack tip area r
e plotted wit
id the mode
p of the stra
fits the plas
the SCC sampore jagged (ne
the SCC samreturns to app
th high preci
els mention
ain map of th
stic zone the
46
ple in ear
mple in pear in
ision.
ed in
he air
e best
w
al
th
sa
o
sa
pr
K
fr
ex
Fzo
without a nee
long with th
hree models
amples. In F
f the specim
ample that th
reliminary m
Knowing to e
rom SCC to
xponent n’ h
Figure 5.16 Ooomed in high
ed to adjust t
he theme in
of plastic zo
Fig. 5.16 it c
men ie when
he crack pro
measuremen
expect this p
be estimate
has increased
Overlay of allh resolution ε
the neither y
Fig. 5.16, F
one boundari
can be seen
n yr=0 (See F
opagated an
nts suggested
articular ins
ed using the
d in value ap
l three plasticε22 elastic stra
yield stress n
Figs. 5.17, 1
ies over the
that the crac
Fig. 3.1). I
additional 0
d a larger p
tance of a la
models. In
pproaching n
c zone modelain plot from F
nor KI of the
18, 19 and 2
near field co
ck did not fo
In Fig 5.20 t
0.9 mm has
plastic zone,
arger plastic
n addition in
n=1, which is
ls on the air Fig. 5.11.
material at
20 also show
ontour maps
ollow along
the contour
a larger pla
, a larger ar
zone leads t
n Fig. 5.20 it
s the LEFM
sample. The
hand. Follo
w overlays o
s of the rest o
the exact m
map of the
astic zone. S
rea was map
to the yield s
t appears tha
model.
e contour plo
47
owing
of all
of the
middle
SCC
Since
pped.
stress
at the
ot is a
Fa fo
Fto5
Figure 5.17 Ozoomed in h
ollow along y
Figure 5.18 Oo promote SC.13.
verlay of all thigh resolutioyr=0 so closely
Overlay of all CC. The conto
three plastic zn ε22 elastic sy.
three plastic our plot is a z
zone models ostrain plot fro
zone models zoomed in hi
on the salt waom Fig. 5.12.
on the aqueogh resolution
ater sample. T. In this case
ous sample wn ε22 elastic st
The contour pe the crack di
with the zinc ctrain plot from
48
plot is id not
couple m Fig.
Ftore
Ftohian
Figure 5.19 Oo promote SCesolution ε22 e
Figure 5.20 Oo promote SCigh resolutionnd a subseque
Overlay of all CC right at thelastic strain p
Overlay of theCC right aftern ε22 elastic sent measurem
three plastic he onset of crplot from Fig
e HRR plasticr the crack prstrain plot fro
ment of the σys
zone models rack propagat. 5.14.
c zone model ropagation ofom Fig. 5.15.s was estimate
on the aqueotion. The co
on the aqueof 0.9 mm T. In this figued to be depre
ous sample wontour plot is
ous sample wThe contour pure a larger areciated to 400
with the zinc ca zoomed in
with the zinc cplot is a zoomrea was meas0 MPa.
49
couple n high
couple med in sured,
50
5.2.6 Plane Strain and Plane Stress
The differences in plane strain and stress can be observed in figure 5.21 with
distinction in the SCC sample (C) that propagated a 0.9 mm crack. The measurements in
plane strain were made in the middle of the specimen (at z=0) which is where all of the
measurements were made for 22. However in addition as mentioned in the previous
chapter, measurements were made 2.4 mm from the center towards the surface. This
addition enabled to contrast the plastic zone size from the two conditions, where the
plastic constraint factor (Eqs. 3.3,12,14 and 18) suggests a plastic zone 3 times larger
(See Fig. 3.3) in the plane stress conditions.
Figure 5.21 The comparison of the plane strain (left) and the plane stress (right) strain map of the SCC sample that had a 0.9 mm crack propagate. Note that the plane stress plastic zone is much larger. The difference in shape from the Figs. 5.15 and 20 are due to the lower resolution used in this strain map.
The overlays of the models with considering plane stress also have reasonable
agreements. Figure 5.22 shows the zoomed in area (3.5 x 6 mm) in which the three
models of plastic zones are imposed upon. In this case the third model (ρ*) fits closer to
the shape. It is quite possible that if a larger n’ is chosen it would match up with other
data with better results for matching the plastic zone.
Distance from the notch tip "x" direction (mm)
0 1 2 3 4 5 6 7 8
Dis
tan
ce f
rom
th
e n
otch
tip
"y"
dir
ecti
on (
mm
)
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6 7 8
-0.0020 -0.0015 -0.0010 -0.0005 0.0000 0.0005 0.0010
22
Plane Strain Plane Stress
Ftohianup
Figure 5.22 Oo promote SCigh resolutionnd a subsequpon measurem
Overlay of the CC right aftern ε22 elastic suent measuremments from th
three plastic r the crack prstrain plot froment of the σhe earlier plan
zone modelsropagation ofom Fig. 5.21.σys was estimne strain mode
s on the aqueof 0.9mm Th. In this figu
mated to be del overlays (F
ous sample whe contour ploure a larger ardepreciated toFig. 5.20).
with the zinc cots are zoomrea was meas
o ~400 MPa
51
couple med in sured, based
52
CHAPTER 6 Concluding Remarks
(1) A unique high energy synchrotron diffraction method was used by which we
demonstrated the measurement of the elastic principle strains for a 3D state of
strain at the tip of a crack in an elastoplastic material such as 4140 steel. In
addition, it was demonstrated that strain measurements are very effective in
locating the crack tip in a CT specimen.
(2) Although an extremely powerful and versatile technique for measuring
various phenomena at the atomic levels it still has limitations in the context of
SCC. For instance, it was found that it is impossible to determine minute
(trace) phases at the very crack tip despite very high energy x-ray photons.
Hence, we have no conclusive evidence that the FeH phase was present at the
crack tip, causing crack opening due to massive lattice expansion (18%) upon
FeH formation. However, this hypothesis still remains valid and warrants
further investigation.
(3) It was shown that by measuring the three elastic principle strains of a general
3D state of strain, no assumptions were needed in determining the principle
stresses. Typically, the state of strain is assumed to be plain strain in the
continuum mechanics treatments of CT specimens, no such limiting
assumptions were needed which can be otherwise hard to justify.
(4) Also demonstrated was the analysis of the 3D state of near and far field
stress/strain using a computational procedure based on the generalized
Hooke’s law and von Mises equation. Also, it was proven that the
53
measurement of a single principle elastic strain is not sufficient is analyzing
plasticity at the crack tip in an elastoplastic material.
(5) The mapping of the von Mises equivalent stress at the crack tip has shown that
its magnitude can be as high as 1000 MPa which substantially exceeds the
yield stress of 646 MPa. Moreover, it was shown that without consideration
of the 3D state of stress and strain, the computation of the von Mises
equivalent stress is impossible, preventing one the carry out a quantitative
analysis of plasticity phenomena at the crack tip. We have shown that the von
Mises equivalent stress for a 3D state of stress is a viable approach in
establishing the plastic zone size in a given elastoplastic material. Specifically,
this study has shown that size of the plastic zone at the crack tip in 4140 steel,
when subjected to a H2O-NaCl-Zn environment after fatigue with KI 38.5
MPa√m, grows from 100 to 400 µm as the crack grows ~1mm and the σys
lowers from 646 MPa to around 400 MPa.
(6) Despite being able to measure all three strain components, there is good
agreement of the ε22 with the models of plastic zones. However this cannot
give detailed stress information since the primary stresses can only be
calculated in the elastic regime when the first two primary elastic strains (ε11
and ε22) are measured. Otherwise there needs to be an assumption of plane
strain or plane stress conditions to be able to solve for the missing strains and
stresses. Thus the finding a strain map (in this case ε22 only) that has
agreement with the models was a convenience that should be further verified.
54
(7) The models are merely approximations of the underlying physical reality
which can indeed be accessed using EDXRD strain mapping. The models are
convenient in that an approximate yield stress of the material can be made by
overlaying a model with the elastic ε22 strain map. Even with the models
fitting the data really well in many instances, they should not be used as a
crutch when making experimental measurements. Of the three models used
the HRR zone model works the best. Furthermore, it is observed that the
overlay of the HRR zone seems to show how the n’ changes in a material as it
undergoes corrosion, which makes sense because hydrogen embrittlement
would make the material less elastic and by definition have a higher n’ value.
55
REFERENCES
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APPENDIX
60
APPENDIX A -Measuring ρ*
Measuring the parameter ρ* was possible using EDXRD. The same ε22
measurements were made for two samples along the crack path corresponding to θ=0°
using the equations mentioned in Section 3.2.3 (Eq. 3.22). The difference in these
measurements carried out on the Air specimen and the SCC C specimen was the
application of ~1000 lbs load, which would nullify the relaxation on the sample. (Though
a remnant plastic zone is still in effect after the load is removed, thus enabling all of the
aforementioned measurements). Thus the strain measured would satisfy the conditions of
the equations.
The assumption made in Eq. 3.3 is truly valid in the elastic regime of the material,
as in from outside the plastic zone. From inside the plastic zone it becomes somewhat
harder to discern due to the accounting of the Hencky equation. Yet, since we only
measure elastic strains, the error is negligible except near the blunted crack tip. In
summary, the calculation of the elastic strain assumes a negligible plastic component for
the sake of using Hooke’s law so as to use Eq. 3.1b to calculate the expected strain.
Furthermore, for the gauge area one has to realize that the values must be segmented and
the space averaged. In this study the gauge was 60 x 60 µm. Figure A.1 shows a
comparison of the theoretical/calculated and experimental data for both samples.
61
Figure A.1 The comparison near the crack tip of the calculated and measured data of the elastic ε22. The left is for the Air specimen and the right is the SCC C (0.9mm crack growth) specimen.
From here it can be seen that the value of 2µm for ρ* is a good fit. More interestingly, it
can be seen that the parameter ρ* does not change during distressful corrosion conditions.
This was readily seen in the ρ* model overlays as well, especially in 5.22 where the ρ*
fits the best under the plane stress conditions.
Distance from notch tip (mm)
0.0 0.2 0.4 0.6 0.8 1.0
22
(%)
0.0
0.5
1.0
1.5
2.0
2.5
Distance from notch tip (mm)
0.0 0.2 0.4 0.6 0.8 1.0
22
(%)
0.0
0.5
1.0
1.5
2.0
2.522 calculated (%)
22gauge (%)
22 calculated (%)
22 measured (%)
62
APPENDIX B – Alternative Contour Plots
Figure B.1 The equivalent stress map presented from Fig. 5.6 with both plots having the same stress contour scale. The left plot is from the Air specimen and the right is from the SCC B specimen.
5.4 5.6 5.8 6.0 6.2 6.4
Dis
tan
ce f
rom
not
ch t
ip 'y
' dir
ecti
on (
mm
)
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Distance from notch tip 'x' direction (mm)
5.4 5.6 5.8 6.0 6.2 6.4
200 400 600 800 1000
eq (MPa)