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 Title:
 Development of a Numerical Procedure for Mixed Mode KSolutions and Fatigue Crack Growth in FCC Single Crystal Superalloys
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 2008
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 Subjects / Keywords:
 Coordinate systems ( jstor )
Fatigue ( jstor ) Fracture mechanics ( jstor ) Geometric angles ( jstor ) Geometric planes ( jstor ) Heat resistant alloys ( jstor ) Shear stress ( jstor ) Sine function ( jstor ) Single crystals ( jstor ) Stress intensity factors ( jstor )
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 University of Florida
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 7/30/2007

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DEVELOPMENT OF A NUMERICAL PROCEDURE FOR MIXED MODE
KSOLUTIONS AND FATIGUE CRACK GROWTH IN FCC SINGLE CRYSTAL
SUPERALLOYS
By
SRIKANT RANJAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2005
To
My parents Prof. Yamuna Prasad and Mrs. AM ,ti, Devi for aiv, loving me.
ACKNOWLEDGMENTS
This dissertation is the result of 5 years of work whereby I have been accom
panied and supported by many people. It is a pleasant aspect that I have now the
opportunity to express my gratitude to all of them.
The first person I would like to thank is my supervisor, Dr. N1 ,1i ij Arakere.
I have been under his supervision since 2000 when I moved to the University of
Florida. His enthusiasm and integral view on research and his mission for providing
"only highquality work and not less," has made a deep impression on me. I owe
him lots of gratitude for having me shown this way of research.
I would also like to thank the other members of my PhD committee who took
the effort in reading and providing me with valuable comments on earlier versions
of this thesis: Dr. Ashok Kumar, Dr. Bhavani Sankar, Dr. Fereshteh Ebrahimi and
Dr. John Ziegert. I thank them all.
My colleagues in the Fatigue and Tribology (FAT) LabErik Knudsen, Jeff
Leismer, Shadab Siddiqui and TaeJoong YUall gave me the feeling of being at
home at work. The friendly ambience of FAT lab alvb, kept me in a cheerful
mood. The discussions and the interactions with Shadab and Jeff had a direct
impact on the final form and quality of this thesis. I would like to thank Shadab
for providing me tips that helped me a lot whenever I went off track. It was not
possible to continue my work without his companionship and cheerful support.
Erik, Jeff and Amitoj Likhari have read parts of my thesis and provided me
valuable comments. Alexander Pacheco has helped me tremendously in getting
the thesis in correct format. My friend and colleague since my undergrad d ,v
Guruditta Golani, have been of great help in the last d4,v, of thesis writing to meet
all the crucial deadlines.
I feel a deep sense of gratitude for my loving father and mother who formed
part of my vision and taught me to be a good human being first and then anything
else. I thank my three great loving brothers, Nishant Kumar R li, i i Prashant
Agrawal and Ravikant R ,lii in for ahv,x bestowing their love on me and alvi
believing in me. I thank my brother Prashant Agrawal for constantly encouraging
me to pursue excellence and providing me emotional and financial support in my
hardest time. He has been a constant source of motivation and inspiration since my
childhood d,4 I am fortunate to have a big brother like him.
Last but not least, I would like to express my gratitude to my friend Prashant
Kumar Singh from my undergrad d i, who took time out of his busy schedule to
provide unwavering support and help when I needed most.
TABLE OF CONTENTS
page
ACKNOW LEDGMENTS ............................. iii
LIST OF TABLES ................................. vi
LIST OF FIGURES ................................ vii
ABSTRACT .... ............................ xi
CHAPTER
1 INTRODUCTION .............................. 1
2 COMPUTATION OF STRESS INTENSITY FACTORS FOR SINGLE
CRYSTAL: LITERATURE REVIEW .................. 17
3 SIF EQUATION FORMULATION FOR MIXED MODE LOADING 23
4 MODELING AND MESHING ........................ 30
5 MIXED MODE SIF: RESULTS AND DISCUSSION .......... 36
6 FCG GROWTH IN FCC SINGLE CRYSTAL MATERIALS ....... 58
6.1 Isotropic m materials .. .. .. .. .. ... .. .. .. .. .. .. 58
6.2 FCC single ivI .1 m materials ..................... 61
7 CONCLUSIONS ............................... 78
APPENDIX
A DISPLACEMENT FIELD EVALUATION FOR AN ANISOTROPIC
ELASTIC SOLID ............................. 81
B DETAILS OF SIF EQUATION FORMULATION ............ 95
C ANSYS PROGRAM ............................. 102
REFERENCES .. .. ... .. .. .. .. .. .. ... .. .. .. .. ... 121
BIOGRAPHICAL SKETCH ............................ 126
LIST OF TABLES
Table page
11 Direction cosines of material coordinate axes (xyz) with universal co
ordinate axes (x'y'z') . . . . . . 8
12 Direction cosine of (x', y', z') with (x, y, z) coordinate axes, when x'
axis is aligned along [213] orientation. ....... . . 16
31 Geometrical and material properties of the two specimens analyzed 28
61 The geometry and loading condition of the three Brazilian Disk speci
m en tested . . . . . . . . 68
62 Krs8 for 12 primary slip systems with increasing crack length for spec
im en 95830 . . . . . . . . 75
63 K.rs for 12 primary slip systems with increasing crack length for spec
im en 96842 . . . . . . . . 76
64 Krss for 12 primary slip systems with increasing crack length for spec
im en 98C 21 . . . . . . . 77
LIST OF FIGURES
Figure page
11 Schematic of the 7' precipitate in a 7 matrix .. . .... 2
12 Cuboidal 7' precipitates (0.350.6 pm) in PWA1480. . .... 3
13 Temperature capability of superalloys with approximate year of in
troduction . . . . . . . .. 4
14 Aeroengine blades are nominally oriented in the (001) orientation. 4
15 Material coordinate system (xyz) relative to universal coordinate sys
tem (x'y'z') . . . . . . . 11
16 Two separated portions of a single crystal showing a model for calcu
lating the resolved shear stress in a singlecrystal specimen. . 11
17 Slip lines and fracture plane in experimental tensile test specimen
tested by Materials Science and Engineering Department, UF. 12
18 Primary (Closepack) and Secondary (Nonclosepack) slip directions
on the octahedral planes for an FCC iI 1 . ...... 14
19 Cube slip planes and slip directions for an FCC crystal. . ... 15
31 Crack tip nodal displacement of isotropic elastic material . ... 24
32 The (101) and (121) family of slip directions are superimposed on
{ 111} plane showing how the hexagonal crack front is delineated
by the slip directions . . . . . . 27
33 Brazilian disk having center crack lying in {111} slip plane and aligned
along [T21] direction . . . . . . 28
34 Rectangular specimen 'A' meshed with solid95 element (left) and the
geometrical details (right) . . . . . 29
35 Brazilian disk (Specimen 'B') with center crack lying in (111) plane
and oriented along [10l] direction . . . . 29
41 Brazilian disk meshed with triangular element having isotropic prop
erties (left) and zoomed view of crack tip (right). . .... 31
42 (a) 20 node isoparametric element in natural coordinate system and
(b) Quarter point singular element with the ( = 1 face collapsed
in local Cartesian coordinate system. ....... . . 33
43 Arrangement of quarterpoint wedge elements along segment of crack
front with nodal lettering convention. ....... . . 34
44 The symmetric meshing of BD specimen with solid95 element and
enlarged picture of crack tip elements ..... . . 34
51 K, versus Crack Length/Width ratio for [10t] and [T21] orientation
of Specimen 'A' at 0 00. . . . . . 37
52 KI1 versus Crack Length/Width ratio for [101] and [T21] orientation
of Specimen 'A' at 0 00 . . . . . ... 38
53 KII, versus Crack Length/Width ratio for [101] and [121] orienta
tion of Specimen 'A' at 0 00 . . . . .... 38
54 K, versus crack angle with force for [10t] and [12t] orientation of
Specimen 'A' at 2a/W 0.4. . . . . . 39
55 KH versus crack angle with force for [10t] and [T21] orientation of
Specimen 'A' at 2a/W 0.4. . . . . . 39
56 KmI, versus crack angle with force for [101] and [T21] orientation of
Specimen 'A' at 2a/W 0.4. . . . . . 40
57 K, versus Crack Length/Diameter ratio for [101] and [121] orienta
tion of Specimen 'B' at 00 . . . . . 41
58 KI1 versus Crack Length/Diameter ratio for [10t] and [12t] orienta
tion of Specimen 'B' at 00 . . . . . 42
59 KmI versus Crack Length/Diameter ratio for [10t] and [121] orien
tation of Specimen 'B' at 00. . . . . 42
510 K, versus crack angle with force for [101] and [T21] orientation of
Specimen 'B' at 2a/W 0.55. . . . . ... 43
511 K11 versus crack angle with force for [101] and [121] orientation of
Specimen 'B' at 2a/W 0.55. . . . . ... 43
512 Km, versus crack angle with force for [101] and [T21] orientation of
Specimen 'B' at 2a/W 0.55 . . . . . 44
513 Half meshed model of Brazilian disk specimen and the crack coordi
nate system . . . . . . . 45
514 Variation of SIF K, along isotropic BD specimen thickness at differ
ent Crack angle . . . . . . ..
515 Variation of SIF K, along orthotropic BD specimen thickness at dif
ferent Crack angle . . . . . . ..
516 Variation of SIF KI1 along isotropic BD specimen thickness at differ
ent Crack angle . . . . . . ..
517 Variation of SIF KI1 along orthotropic BD specimen thickness at dif
ferent Crack angle . . . . . . ..
518 Variation of SIF K1I, along isotropic BD specimen thickness at dif
ferent Crack angle . . . . . . ..
519 Variation of SIF KII, along orthotropic BD specimen thickness at
different Crack angle. . . . . . . .
520 a)Unsymmetry about mid plane for crack oriented along {111}(101);
b) Symmetry for crack lying along {111}(121) . . .
521 Variation of SIF K, along BD specimen thickness at different Crack
angle for [10T] orientation . . . . . ..
522 Variation of SIF K, along BD specimen thickness at different Crack
angle for [T21] orientation . . . . . ..
523 Variation of SIF K11 along BD specimen thickness at different Crack
angle for [101] orientation . . . . . ..
524 Variation of SIF Kn along BD specimen thickness at different Crack
angle for [T21] orientation . . . . . ..
525 Variation of SIF KmI, along BD specimen thickness at different Crack
angle for [101] orientation . . . . . ..
526 Variation of SIF KII, along BD specimen thickness at different Crack
angle for [T21] orientation . . . . . ..
527 Variation of SIF K, along BD specimen thickness at different Crack
Length/Diameter ratio for [lOT] orientation. . . . .
528 Variation of SIF K, along BD specimen thickness at different Crack
Length/Diameter ratio for [121] orientation. . . . .
529 Variation of SIF K11 along BD specimen thickness at different Crack
Length/Diameter ratio for [101] orientation. . . . .
530 Variation of SIF K1 along BD specimen thickness
Length/Diameter ratio for [T21] orientation. ..
at different Crack
531 Variation of SIF K11, along BD specimen thickness at different Crack
Length/Diameter ratio for [10l] orientation.. . . 57
532 Variation of SIF K11, along BD specimen thickness at different Crack
Length/Diameter ratio for [121] orientation.. . . 57
61 Schematic Fatigue Crack Growth Curve ..... . . 59
62 Fatigue Crack Length versus Applied Cycles. Fracture is Indicated
by the x . . . . . . . . 61
63 Microscopic slip observed on two {111} slip planes inclined 520 and
380 to the starter notch . . . . . 62
64 Details of crack tip displacements and stresses at a distance r and 0
from the crack tip in the crack coordinate system . . 64
65 Burgers vector b is along slip direction (011) and slip plane direc
tion is normal vector n along (111) . . . . 65
66 Fatigue crack growth rate as a function of AKrs8 and AK for 2D
rectangular specimen . . . . . . 65
67 Crack growth of BD specimen 95830 with no. of cycles . ... 69
68 Crack growth of BD specimen 96842 with no. of cycles . ... 69
69 Crack growth of BD specimen 98C21 with no. of cycles . ... 69
610 Crack growth rate of BD specimen 95830 with increasing crack length 70
6 11 Crack growth rate of BD specimen 96842 with increasing crack length 70
612 Crack growth rate of BD specimen 98C21 with increasing crack length 70
613 Trace of primary slip planes on the plane normal to the crack plane 71
614 Crack growth on {111} slip plane can be observed for 96842 BD spec
im en . . . . . . . . 73
615 Fatigue crack growth rate of 3 specimens 95830, 96842, 98C21, as a
function of AK ms . . . . . . 74
616 Fatigue crack growth rate of 3 specimens 95830, 96842, 98C21, as a
function of AKr . . . . . . . 74
A1 Forces acting on a rectangular element with dimension dx x dy . 81
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DEVELOPMENT OF A NUMERICAL PROCEDURE FOR MIXED MODE
KSOLUTIONS AND FATIGUE CRACK GROWTH IN FCC SINGLE CRYSTAL
SUPERALLOYS
By
Srikant R lii in
August 2005
C!i ,i: N. K. Arakere
M i"r Department: Mechanical and Aerospace Engineering
Fatigueinduced failures in aircraft gas turbine and rocket engine turbopump
blades and vanes are a pervasive problem. Turbine blades and vanes represent
perhaps the most demanding structural applications due to the combination of
high operating temperature, corrosive environment, high monotonic and cyclic
stresses, long expected component lifetimes and the enormous consequence of
structural failure. Single crystal nickelbase superalloy turbine blades are being
utilized in rocket engine turbopumps and jet engines because of their superior
creep, stress rupture, melt resistance, and thermomechanical fatigue capabilities
over p., ,i ii~l 111iiw. alloys. These materials have orthotropic properties making the
position of the crystal lattice relative to the part geometry a significant factor in
the overall analysis. Computation of stress intensity factors (SIFs) and the ability
to model fatigue crack growth rate at single crystal cracks subject to mixedmode
loading conditions are important parts of developing a mechanistically based life
prediction for these complex alloys.
A general numerical procedure has been developed to calculate SIFs for a
crack in a general anisotropic linear elastic material subject to mixedmode loading
conditions, using threedimensional finite element analysis (FEA). The procedure
does not require an apriori assumption of plane stress or plane strain conditions.
The SIFs KI, KII, and KIII are shown to be a complex function of the coupled
3D crack tip displacement field. A comprehensive study of variation of SIFs as a
function of i l11. graphic orientation, crack length, and modemixity ratios is
presented, based on the 3D elastic orthotropic finite element modeling of tensile
and Brazilian Disc (BD) specimens in specific crystal orientations. Variation of SIF
through the thickness of the specimens is also analyzed.
The resolved shear stress intensity coefficient or effective SIF, Krs, can be
computed as a function of crack tip SIFs and the resolved shear stress on primary
slip planes. The maximum value of Krs and AK,.s was found to determine the
crack growth direction and the fatigue crack growth rate respectively. The fatigue
crack driving force parameter, AKs8, forms an important multiaxial fatigue
damage parameter that can be used to predict life in superalloy components.
CHAPTER 1
INTRODUCTION
A superalloy is a group of nickel, ironnickel, and cobaltbase metallic
alloys which can be used at high temperatures, often in excess of 0.7 of the
absolute melting temperature. Exceptional creep and high temperature oxidation
resistance are two of the prime design criteria of superalloys, but apart from these
properties they exhibit a combination of high strength at high temperature, stress
rupture resistance, toughness and metallurgical stability, useful thermal expansion
characteristics and strong resistance to thermal fatigue. They exhibit exceptional
corrosion resistance at room temperature. The high temperature strength of Ni
based superalloys depends on a stable, face centered cubic (FCC) matrix, combined
with either precipitation strengthening and/or solid solution hardening. Iron,
nickel, and cobalt are generally face centered cubic (FCCaustenitic) in crystal
structure when they are the basis for superalloys. However, the normal room
temperature structures of iron and cobalt elemental metals are not FCC. Both
iron and cobalt undergo transformations and become FCC at high temperatures
or in the presence of other elements allo, d with iron and cobalt. Nickel, on the
other hand, is FCC at all temperatures. The microstructure of an FCC superalloy
consists of an austenitic (7 phase) matrix and a wide variety of secondary phase.
The most common secondary phases are 7' and metal carbides. The microstructure
consists of ~60 percent by volume of 7' precipitates in a 7 matrix. These alloys
obtain their hightemperature strength from the presence of the 7' precipitate
phase within the primary 7 matrix. The 7' precipitate is a face centered cubic
(FCC) structure and composed of the intermetallic compound Ni3Al, with Al
atoms occupying the corners and Ni the faces of the unit cell. The 7' precipitate is
Figure 11: Schematic of the 7' precipitate in a 7 matrix
suspended within the 7 matrix, which is also of FCC structure and comprised of Ni
with cobalt, chromium, tungsten, and tantalum in solution. Typically the 7 phase
consists of the elements from groups V, VI and VII of the periodic table. The role
of 7' phase within the solid solution strengthened alloy is to increase the resistance
to dislocation motion. The magnitude of the strengthening effect of 7' is governed
by the degree of 7/ 7' mismatch. The greater the mismatch, the higher is the
strength increment (Fig. 11 shows a schematic diagram of the 7' phase within the
7 matrix [1] and Fig. 12 on page 3 shows a picture of cuboidal 7' precipitate in 7
matrix [2]). Hence shearing of the precipitate is required in order for a dislocation
to move through the 7 matrix. Shearing of the 7' precipitate relative to the 7
matrix requires significantly higher energy, resulting in a stronger material.
Nickel base single crystal superalloys are precipitation strengthened cast
monograin superalloys based on the NiCrAl system. These alloys exhibit better
high temperature properties than p, i, i il 1Iiiw. wrought or cast alloys (Fig. 13
on page 4). In high temperature application grain boundaries are typically the
weak link, which provide passages for diffusion and oxidation, which results in
Figure 12: Cuboidal 7' precipitates (0.350.6 pm) in PWA1480.
failures at this location. Grain boundary strengtheners are added to the alloy
chemistry to increase capability, which results in lowering the melting point of the
alloy.
Removal of grain boundaries and grain boundary strengthening elements
raise the incipient melting temperature of the alloy by 150 F and result in
improved high temperature fatigue and creep capabilities [4]. This increase
in melt temperature permits higher heat treatment temperature that in turn
yields improved creep capability. Unlike the more commonly used pf1l. i' iI 11 11,
alloys, these single crystal superalloys are orthotropic and have highly directional
material properties with exceptional thermo mechanical fatigue properties at high
temperature. The most common primary growth direction for the nickelbase
superalloys is the (001) direction. This is not only the most easily grown but also
is the direction with the most desirable combined strength properties. This is
advantageous because many parts are cast, or grown, rather than machined from a
larger single crystal sample with a specific orientation.
4
Mechanical ng
DIOollonal ioying crys
strctus \ MAIMBO CMSX 10
Cast lloy 4,W and Nb / IN59 S
0 / MARM*22 MARM l2S CMSX2
MAR1200 M21 M246 M2 +HlDS 0
SVuM21 M248 MAM200 )HIM 97
1N100 T A
C .0 I w N I1 ,at
a i ng1 71, 0 1 N77 ,1 I NIP
B  I W rIWrought
X5stUNy 0i P. A Nit NI bNe
001940 19 i0 m 1960 19170 130 19
Year IntroducedM509 Ime72
auction. [3]
X750 N80AWrought
.A[ Co'bas.
OD at rest* DO S aund SC
1940 195 19110 1070 1 98D09
Year Introduced
Figure 13: Temperature capability of superalloys with approximate year of intro
duction. [3]
Figure 14: Aeroengine blades are nominally oriented in the (001) orientation.[5]
[aij] is the matrix of 36 elastic coefficients, of which only 21 are independent,
since [aij] = [agj]. Depending on the the material structure, the crystalline
material di pl'i different forms of geometric symmetry. There are 32 forms
of geometric symmetry of crystals, which can be further divided into seven
crystal systems called syngony: 1) triclinic, 2) monoclinic, 3) rhombic, 4)
tetragonal, 5) trigonal, 6) hexagonal, and 7) cubic [6]. For an isotropic
material the mechanical properties (E, v, G) are the same at each point of the
material because it can have only two independent elastic constants as
all = a22 a= 33 a12 a= 13 a= 23 a44 a= 55 a= 66 = 2(all a12) (1.2)
and the rest of the coefficients of deformations are zero.
all a12 a12 0 0 0
a12 all a12 0 0 0
] a12 a12 all 0 0 0
[a,1 (1.3)
0 0 0 a44 0 0
0 0 0 0 a44 0
0 0 0 0 0 a44
An isotropic material subjected to multiaxial lti:. under mechanical
equilibrium, has three principal stresses. These principal stresses act on
orthogonal planes, which are free of shear stresses. The Von Mises and Tresca
criterions are two of the most widely used yield criteria for ductile isotropic
materials.
Anisotropic Elastic Deformation of FCC Single Crystal: The generalized
Hooke's law for a homogeneous anisotropic elastic body in a cartesian
coordinate system (x, y, z with origin at point 0, Fig. 15 on page 11) is
given by equation (1.4) [6]
{} = [ai]{} (.
(1.4)
The elastic properties of FCC i ,I J1 exhibit cubic syngony; i.e., it has
three orthogonal planes of elastic symmetry at every point, which is called
orthogonallyanisotropic or, for brevity, orthotropic. Therefore cubic sym
metry can be described with three independent constants designated as the
elastic modulus, shear modulus, and Poisson's ratio [4] and hence [a1j] can be
expressed as shown in equation (1.3), in the material coordinate system (FCC
crystal axes are parallel to x, y and z coordinate axes) and the coefficient of
deformation is given as
1 1 Vyx 1 xy (
a11 = t, a44 C a12 F (1.5)
Exx Gyx Exx Eyy
The elastic constants in the generalized Hooke's law of an anisotropic body,
[aij], vary with the direction of the coordinate axes. For orientations other
than the (x, y, z) axes, the [a1j] matrix varies with the crystal orientation. In
the case of an isotropic body the constants are invariant in any orthogonal
coordinate system. Consider a Cartesian coordinate system (x', y', z) that has
rotated about the origin 0 of (x, y, z). The elastic constant matrix [at] in
the (x', y', z') coordinate system that relates {e'} and {oa'} is
{c'} = '.]{'} (1.6)
where [at] is given by the following transformation
6 6
] [Q][a] [Q] = a mQm (ij = 1,2, ......,6) (1.7)
rrn in 1
where [Q] is the transformation matrix, which is a function of the direction
cosines between the(x, y, z) and (x', y', z') coordinate axes (Fig. 15 on
page 11, Table 11 on page 8). Here a, /3 and 7 are the direction cosines of
the material coordinate system relative to the universal coordinate system
(specimen coordinate system).
2a2 a3
2020 3
2/32/33
272773
3273 + /3372
72a3 + 73a2
a2/3 + a332
2 ,1
2/33/31
27371
/3173 + /3371
7103 + 730a1
a1/33 + a3/31
2a1a2
2/31/32
27172
/3172 + /3271
71a2 + 72al
a10/32 + a2/31
Table 1 1: Direction
dinate axes (x'y'z')
cosines of material (< : :: : axes (. ) with universal coor
x y 
X / 'i (a2 ( 2
S71 72 .3
Also the transformation of the stress and strain tensors between the material
and specimen coordinate systems is necessary for implementing the failure
theories. The components of stresses and strains in the (x', y', z') system in
terms of the (x, y, z) system are given by
{}
fa /
[Q'] {}7 ; {'} [ 1{
[Q]I 1 {1} [Q] {1'}
IQ(] W1 { ' [Od fc'
(1.9)
2
1
71
/171
710a1
al/3i
2
a2
f3J
7
2
2
7202
/3272
72 a2
a 2/32
2
a3
33
73
73a3
(1.8)
where
01 a2 a3 0203 (' "'1 0102
/3 3 13 /32/33 /33/31 /31/32
712 71 2 2 73 7273 7371 7172
2/3171 2/3272 2/33<73 +0372 /Y173 + 371 /3172 +0271
271ai 272a2 273a3 72a3 + 73a2 7la3 + 73al 71a2 + 72al
2a1/31 2a2/32 2a3/33 a233 + a332 a13 + a331 a1/32 + a231
(1.10)
The transformation matrix [Q] is orthogonal and hence [Q] [Q] [Q,].
Now the resolved shear stresses on the 12 primary octahedral slip systems
(Table 12 on page 16, Fig. 18 on page 14), denoted by ri, T2,..., 712 can
be readily obtained using the transformation given by equation (1.13) [4].
Only the primary octahedral slip systems get activated at room temperature
whereas high temperature is required for the activation of the other slip
systems.
Inelastic Deformation of FCC Single Crystals: Slip and twinning are two
main factors responsible for inelastic deformation in metals. The main
reason for deformation of crystalline metals is the propagation of dislocations
through the metal's lattice when temperatures are less than 0.5 of the
absolute melting temperature.
At higher temperatures, deformation occurs by dislocation climb (which is
a diffusion controlled process). Twinning, a rotation of atoms in the lattice
structure, is not as important as strains produced by this mechanism are very
small as compared to slip and climb. On an atomic scale, plastic deformation
involves the net movement of large numbers of atoms in response to an
applied stress. Actual strength to deform a metal is much lower than that
predicted for theoretically perfect ( i I J1 This difference is explained by
y <010>
yO
z <001>
z'
x <100>
Figure 15: Material coordinate system (xyz) relative to universal coordinate sys
tem (x'y'z')
Figure 16: Two separated portions of a single crystal showing a model for calcu
lating the resolved shear stress in a singlecrystal specimen. [7]
Figure 17: Slip Lines and Fracture Plane in experimental tensile test specimen
tested by Materials Science and Engineering Department, UF. [8]
slip when the resolved shear stress on the slip plane reaches the critical resolved
shear stress for that material. To calculate the stress needed to exceed this level
(TCRSS) is calculated by
TCRSS
cos Q. cos A
a will be a minimum when = 45
Therefore when a load is applied slip systems get activated based on Schmid
factor, m. where
m = cos 4. cos A (1.12)
This behavior is known as Schmid's law.
The value of critical resolved shear stress depends chiefly on the material
composition and temperature. It is also a function of applied load and direction,
crystal structure and specimen geometry. During the application of a load to an
FCC single crystal specimen, the first planes to get activated are the planes of high
atomic densities and called the primary octahedral slip systems. Each octahedral
plane has six slip directions associated with it. Three of these are termed easy
slip or primary slip directions and the other three are secondary slip directions.
Thus, there are 12 primary and 12 secondary slip directions associated with the
four octahedral planes (Fig. 18 on page 14). In addition, there are six possible slip
directions in the three cube planes, as shown in Fig. 19 on page 15. There are 30
possible slip systems in an FCC crystal (Table 12 on page 16).
The resolved shear stress on the 12 primary slip systems, based on kinematic
relations, can be shown to be
51 1 0 1 1 0 1
72 0 1 1 1 1 0
73 1 1 0 0 1 1
4 1 0 1 1 0 1 aJ
5 1 1 0 0 1 1 o7yy
76 1 0 1 1 1 1 0 azz
< > => (1.13)
76 1 1 0 0 1 1 (7Y
S0 1 1 1 1 0 az1
79 1 0 1 1 0 1 o7z
710 0 1 1 1 1 0
711 1 0 1 1 0 1
T12 1 1 0 0 1 1
Plane 1
F rim lry: "l, T2, 0
Secondary: 013, 014, 15
Ulu C16
T6 r
Plane 2
Primary: :4, 5, 6
S radnrarv ,l16 ,17. rl8
 100
Plane 3
Frirm i r. ,7., T ,
Secondary: t 20, 2 ,21
100
Plane 4
Primary: 0 1, r 21 :2
Secondary: C22, r'z, t4
Figure 18: Primary (Closepack) and Secondary (Nonclosepack) slip directions on
the octahedral planes for an FCC ( i I 1 [)]
Secondary:~ 0 7T6
T4
r17
Plaie2
Figure 19: Cube slip planes and slip directions for an FCC i ivl [9]
Plane I
Table 1 2: Direction cosine of (x, ',
is aligned along [213 orientation.
z') with (x, y, z) coordinate axes, when x' axis
" Number ... FI ...., "1.. Direction
Octahedral :: a/2 {111} (110) Primary :, Directions
I (Iln [IOT
2(111) 101]
2 (111) (011
3 (111) [110]
4 (111) 101]
5 (111) [110]
6 (111) [OI11]
7 (111) [110o]
8 (111) [011]
9 (111) [101]
10 (111) [011]
Octahedral :. a/2{ 111 }(112) Secondary .': Directions
13 (111) 121]
14 (111) 211]
15 (111) 1112
16 (111) [1211
17 (111) [1121
18 (111) [211]
19 (111T) [t112]
(100) [211]
21 (111) [121]
22 (111) [211]
(I ) 121
24 (111) 1121
Cubic i. a/2{100}(110) Cube 1 *.Directions
25 (100) 011
(100) 01T1
27 (010) 101
(010) [101]OT
(001) 1101
____________(001) [110]
CHAPTER 2
COMPUTATION OF STRESS INTENSITY FACTORS
FOR
SINGLE CRYSTAL : LITERATURE REVIEW
Stress intensity factor (SIF) about a crack tip p.1i' a significant role in the
propagation of the crack. The SIF is a measure of intensity near the crack tip
under linear elastic conditions. The knowledge of SIF is necessary to predict the
growth of a fatigue crack or to determine the residual strength of a cracked struc
ture. This factor characterizes the intensity of the stress field in the neighborhood
of the crack tip and depends substantially on specimen geometry, material proper
ties, external loads and crack size. In cases with idealized geometry and loads the
SIF can be found in handbooks for isotropic elastic materials. In the case of more
complex structures the SIF has to be computed by numerical methods.
Isotropic materials have uniform material properties in all directions, whereas
anisotropic materials are directional in nature (i.e., they have different material
properties in different directions). Because of this property SIF is direction
invariant for isotropic materials while SIF varies in anisotropic materials due to
change in material coordinate system, while keeping all the other parameters same.
A study of the effect of anisotropy and inhomogeneity on the SIF using the
FEA was done by K rn,v and Kitamura [10]. It was found that in a pcli, iI 1
specimen, the SIF is influenced by the
deformation constraint due to .,I.i Ient < I ,I 1 only for very small cracks; whereas
for anisotropic single
and the magnitude deviates from that in the isotropic body for any size crack. But
as crack size increases in the pf,1 ivI ,1 materials, SIF tends to converge to values
seen in homogeneous/isotropic bodies.
Many methods have been proposed, to calculate SIFs for cracks subjected to
mixedmode loading conditions in isotropic elastic solids. Some commonly used
methods are J integral [11, 12], virtual crack extension [13, 14], modified crack
closure integral and displacement extrapolation methods [15] etc. None of these
proposed methods are able to provide the complete solution for all the three modes
(Mode I, II and III) of SIF for anisotropic material.
Atkinson et al. [16] presented the idea of calculating mixed mode SIF using
Fredholm equation transformation. They used a center cracked Brazilian disc
(BD) test specimen made of isotropic material. The mode mixity ratio for the BD
specimen is a function of the crack angle with respect to the load vector. Results
were presented for varied crack angles and hence mode mixity ratios. Small crack
approximation was also taken and the results were found to be in accordance with
Awaji and Sato [17], but it did not incorporate anisotropy in the model and was
limited to Mode I and II SIFs.
Su and Sun [18] studied various kinds of 2D anisotropic cracks to evaluate SIF
under mixedmode loading condition. Fractal finite element method (FFEM) [19]
was used to calculate Mode I and II SIFs for 2D anisotropic plate. The variation
of the SIFs with material properties and orientations of a crack was presented. It
was shown that SIFs were not sensitive to the variation of shear modulus. Hwu
and Liang [20] used remote boundary data to calculate SIFs for 2D anisotropic
material. It eliminated the error in SIF calculation, caused by abrupt change in
the stresses near crack tips, by finding equivalent formulation for SIF by using
only remote boundary responses (displacements, stresses and strains), cooperating
with the necessary geometric data. A special boundary element was developed
which removed the requirement of meshing around the crack boundary. Through
this boundary element, all the internal stresses and strains could be expressed
in terms of displacements and tractions on the boundaries excluding the hole,
crack and inclusion boundaries. These results could be applied to any kind of
linear anisotropic materials but was restricted to the 2D problems which included
generalized plane stress, generalized plane strain and antiplane problems.
Denda and Marante [21] developed a crack tip singular element (CTSE)
for the general anisotropic solids in 2D with the built in V/ and 1/,r singular
stress behavior at its tip, which provided the SIF as an integral part of the main
solution, no post processing was required. It was an ideal fracture analysis tool
for 2D multiple curvilinear cracks in general anisotropic solids, but as it was
based on plane strain assumption, it could not be incorporated into a generalized
3D anisotropic model. The solution provided by Heppler and Hansen [22] for
combined mode (I and II) SIFs in case of planar, rectilinear, anisotropic structures
using a 12 node singular finite element was accurate, but was restricted to a 2D
model. Likewise, Sosa and Eischen [12] calculated SIF for a plate containing a
through crack subjected to bending loads using the J integral. TwoD eight node
element was selected for this purpose where K = K111 = 0. Mews and Kuhn
[23] used the Green's function approach to calculate mixed mode SIF without any
crack discretization in an isotropic plate, which used an ., iiii 1l tic displacement
field at the crack tip. They calculated SIFs for plate having multiple cracks, for
various inclinations, and found SIFs very close to that of Shih et al. [24]. Ishikava
[14] presented the idea of the strain energy release rate (virtual crack extension
method) to calculate SIF for mode I and II, and described SIF calculation based
on only one use of virtual crack extension.
Huang and Kardomateas [25] described the continuous dislocation technique
to calculate mixedmode stress intensity factors in an anisotropic infinite strip.
This method was limited in its use to Mode I and II SIFs, and it was found to be
most suitable for cracks of relatively small dimensions. The method was verified
by calculating SIFs for isotropic material and comparing the results with readily
available formulas given in Tada et al. [26], which resulted in excellent agreement.
Sun et al. [27] used the boundary element method (BEM) to analyze cracked
anisotropic bodies under antiplane shear. The new boundary formulation used
dislocation density as an unknown on the crack surface, and K111 was determined
near the crack tip. The equation and method could be directly used for antiplane
problems with cracks of any geometric shapes. It did not give a complete solution
under mixedmode loading conditions, but it did give an idea about the behavior of
K111 under antiplane shear loading.
Shih et al. [24] have calculated SIFs for 2D isotropic materials using quarter
point element nodal displacements at the crack tip based on Finite Element
Method (FEM). The Modes I, II and III have been decoupled because of the
isotropic nature of the material. Sih et al. [28] defined SIFs as a function of
stress at the crack tip. Following the work of Shih et al., Ingraffea and Manu
[29] showed how to compute SIF from 3D quarter point nodal displacement, for
cracked isotropic elastic bodies for all three modes. They used a quarter point
isoparametric element, which has been accurate in computing SIF [30]. Saouma
and Sikiotis [31] introduced anisotropy in the above model [29] and proposed a
method to calculate SIF for 3D anisotropic elastic material based on the model
of Shih et al. The computed SIFs, when compared with 2D anisotropic bodies
with known exact solutions yielded an error of 61'. for K, and K11. However the
orientation of the elastic constants was not incorporated. The expression of K111
was not correct as it resulted in almost zero value, irrespective of the geometry,
orientation and the load applied to an anisotropic component.
Dhondt [32] analyzed two methods, interaction integral method (IINT) and
the quarter point element stress method (QPES), to calculate SIF for single edge
notch specimen (SEN) and a slant crack in a 3 point bending (3PB) specimen
for isotropic material. For the regular mesh in the SEN specimen both methods
yielded similar results. For the irregular mesh in the 3PB specimen the QPES
methods seemed to be more robust and accurate. Later Guido [33] presented a
method to calculate SIF for anisotropic material using the finite difference method
along an arbitrary crack. Although the model was more robust to irregular meshes
in comparison to interaction integral method, this method was limited to single
edge specimen and corner crack specimen.
Tweed et al. [34] used the specific case of an edge crack of an anisotropic
elastic half space under generalized plane strain conditions to determine the stress
intensity factors using integral transform techniques. Pan and Yuan [35] used single
domain BEM to calculated mixed mode SIFs for both bounded and unbounded 3D
anisotropic cracked solids. Denda [36] used BEM to determine mixed mode SIFs
(KI, K11 and K111) of 3D anisotropic material with multiple cracks. It addressed
the issue of coupling effect of the three modes of fracture controlled by Mode I, II
and III SIFs. Though very accurate, these formulations were based on the plane
strain assumption.
Although a substantial body of literature describes computation of SIF, a
generalized numerical solution to calculate SIF for 3D anisotropic material is
unavailable. The objective of the present work is to model a 3D orthotropic
specimen having a through crack and to calculate the SIFs for all three modes
assuming linear elastic properties at the crack tip. A mathematical model has been
developed to calculate mixed mode SIF for FCC single crystal orthotropic material
for different orientations, which is also applicable to generalized anisotropic
material. Looking at the crack tip nodal displacements it is possible to calculate
stress intensity factor for any crystallographic orientation of the material. It
was observed that the material orthotropy results in coupled crack tip (x, y,
z) displacements, leading to the interdependence of SIFs for Modes I, II, and
III. Results are presented for a centercracked BD specimen, with two specific
crystallographic orientations. The crack plane for the first specimen is (111)
and the crack direction is [o101. For the second specimen the crack plane is
again (111), while the crack direction is [121]. There are important reasons for
choosing these two specific BD specimen configurations. These two crack directions
typically represent the fastest and the slowest crystallographic crack growth rates,
respectively, on the { 111} family of octahedral planes, and hence have important
implications on estimating fatigue crack growth life for single crystal components
[2, 37]. Even if cracks nucleate on other planes, because of local influence from
intrinsic defects such as micropores, carbide particles and undissolved eutectics,
they tend to migrate to the octahedral planes, in the primary slip directions ((101)
family of directions), since they represent the paths of least resistance for crack
propagation [2]. Results presented show that SIF values are consistently higher for
the (101) crack direction, compared to the (121) direction, for same crack angles
and loads.
CHAPTER 3
SIF EQUATION FORMULATION FOR MIXED MODE LOADING
The stress intensity solution, for all the three modes for isotropic materials,
using crack tip nodal displacements method is given by [38]:
K, = P 2 [4(vB _D) +vE V]
Kil P /2 [4(UB_ UD) + E UC] (3.1)
K +1 L
Here u, v and w are the displacements of the nodes B, C, D and E at the crack tip
along x, y and z directions respectively. L is the length of the element at the crack
tip normal to the crack front (Fig. 31 on page 24).
These equations show that all three stress intensity factors KI, K1 and K111
are decoupled. For anisotropic materials it is not possible to use the same equation,
as their material properties are direction dependent. Because of this directional
dependence the stress intensity factor for the same material changes according to
the orientation of the crack plane with material orientation.
As has been discussed earlier, an orthotropic material has three independent
elastic constants, E, v and G. The elastic constants in material coordinate system
get transformed to specimen coordinate system by equation (1.7). The interde
pendence of displacements in anisotropic materials due to shear coupling results in
coupled stress intensity factors.
Anisotropy was incorporated in the 3D model to calculate SIF for mixed
mode loading. It can be shown that mixed mode stress intensity factors can be
z
Figure 31: Crack tip nodal displacement of isotropic elastic material for calcula
tion of Stress Intensity Factors of Mode I, II and III
computed by equation (3.2) [28] [29]1.
K,
K i [B]1 [A] 2l
Kill
Where L1 is the element length along the crack face and {A} is given as
2uB uc + 2UE uF +UD + T(4UB + UC + 4UE UF)+
2(UF + UC 2D)
2VB vc + 2VE F+ VD + Tl(VB + VC + 4VE VF)+
2WB Wc + 2WE WpF + WD + T(4WB + WCE + 4WE WF)+
lr>2(wF + wc 2wD)
(3.2)
(3.3)
1 The details of the equation are given in Appendix B
Here u, v and w are the nodal displacements of nodes B, C, D, E and F at the
crack tip relative to B', C', D', E', and F' as shown in Fig. 43 on page 34.
TI is the natural coordinate system value defined as
2z
S ( + 1)
L2
L2 is the length of the element along the crack front (Fig. 42 on page 33) and
[B]1 is defined as
Re [' (q2
Re[ '(piq2
qi)] Re [ P (p2
291)2]' Re[ / (1P2
12P)] I
0
0
/C44C55 C(5
(3.5)
As it is apparent from the [B]1 matrix that K, and KI1 are only
(function of u and v) and KII, is function of w only.
D is the determinant of the equation
coupled SIFs
Re\[ (pip2
D R[ /12 (/1P2
Re[ 12 (pijq2
1^192 ^'li
/2Pl)] Re[ P2 (P2
2qi)] Re[ 1 2 (q2
pi)]
qil)]
/i1 and P2 are the roots of the fourth order characteristic equation
all/4 2aI6/p3 + (2a12 + a66)2 2a26p + a22 = 0
and given by
pij aj + i30,and fj > 0
(as pij can be only a complex no.[6]2 )
2 The fourth order equation formulation was followed from the work of Lekhnit
skii [6], refer to Appendix A.
(3.4)
[B]1
(3.6)
(3.7)
pj = aipj + a12 a16/j
qj a= 12 +a22 a26 (3.8)
For Plane stress,
1 1 V12 1
all E ; a22 22 a21 a12 ; a66 1 (3.9)
Ell E22 El1 G12
For Plane strain,
7 ,"I.,
aj = aj (3.10)
a33
Equation (3.2) is used to calculate the SIF at the crack tip by displacement
method; one of the most used methods to get the value accurately. Finite Element
Method was used to calculate the displacements at the crack tip. The commercial
software, ANSYS, was used for FEA modeling. The crack tip nodal displacements
were then extracted from FEA model and fed to the analytical equations explained
above to calculate all the three modes of SIFs.
Single Crystal Specimen Geometries Used for MixedMode Loading. Two
specimen geometries were used to investigate the effects of mode mixity at the
crack tip. One was a rectangular tension specimen (Fig. 34 on page 29) with
a center crack loaded such that the crack lied in the {111} plane. The crack
directions used were (101) and (121) family of directions.
The second specimen modeled is a round Brazilian disk (BD) specimen, loaded
in compression. This specimen with center crack has a mode mixity at the crack
tip, which varies as function of the crack angle '0', shown in Fig. 35 on page 29.
The crack lies on the {111} plane and crack directions used are (101) and (121)
family of directions. These specimen and crack orientations have been checked very
carefully, based on experimentally observed fatigue crack growth rates (FCGR)
[39]. At low temperature (< 4270C(), the dominant mode of FCG in FCC single
crystal superalloys is crystallographic crack propagation on octahedral planes
Figure 32: The (101) and (121) family of slip directions are superimposed on
{111} plane showing how the hexagonal crack front is delineated by the slip
directions. [37]
[37]. Extensive FCGR studies have shown that the fastest crack growth rates
are observed for cracks propagating on the {111} plane, in the (101) family of
directions. A slower FCGR has been observed with the crack on the {111} plane,
in the (121) family of directions. Hence the FCGR on the {111} plane, in these
two directions typically provide the fastest and the slowest ( il I 11 graphic crack
propagation rates. This variation in FCGR in the (101) and (121) directions,
on the {111} is illustrated in dramatic fashion in Fig. 32 on page 27. A penny
shaped crack on the {111} plane evolved into a hexagonal crack front, because
of different crack growth in the (101) and (121) directions. These six fastest
crack growth directions (hexagonal corners) were found to be (101) family of slip
directions on {111} plane.
Details of the two specimens analyzed are presented in Table 31 on page 28.
Where,
Crack Cordinate System
Material Coordibue System aligned
Material Coordinate System to Crack Cooninate System
Figure 33: Brazilian disk having center crack lying in {111} slip plane and aligned
along [12t] direction
Table 31: Geometrical and material properties of the two specimens analyzed
Rectangular Specimen 'A' (Fig. 34) Brazilian Disk B (Fig. 35)
E 106.2GPa H 20.32cm E 106.2GPa W 2.794cm
G 108.2GPa W 5.08cm G 108.2GPa t 0.254cm
v 0.4009 t 1.016cm v 0.4009 2a/W 0.20.8
F 4.448KN 2a/W 0.10.9 F 4.448KN
W  Width or Diameter of the specimen,
H  Height of the Specimen,
t  Thickness of the specimen,
F  Uniaxial load applied,
 Angle of inclination between crack and load applied.
Fig. 33 on page 28 shows that the crack plane lies on (111) plane and is directed
along (121) direction. This slip system has been explained with the help of
octahedral slip planes.
w
[10T]
\ 2a
121]
A
.  (9 e)
H
F
Figure 34: Rectangular specimen 'A' meshed with solid95 element (left) and the
geometrical details (right)
Figure 35: Brazilian disk (Specimen 'B') with center crack lying in (111) plane
and oriented along [101] direction
F
t/
t 
Crack Plane
CHAPTER 4
MODELING AND MESHING
In order to study the effect of orthotropy on SIFs, specimens with geometries
mentioned in Table 31 on page 28 were modeled, meshed and analyzed for
different crack lengths, crack angles, material properties and i i vI orientations. A
rectangular specimen with isotropic properties and without any crack was modeled,
and its accuracy was ascertained. A far field stress was calculated analytically
and was found to be matched with meshed FEA result. The same model was used
with a through thickness center crack and SIF at the crack tip was calculated and
verified with standard table.
A Brazilian disk with center crack having isotropic properties was modeled
and compared with the SIFs extracted from FEA model. The values were verified
with the available tabulated values for the isotropic Brazilian disk. Equation (3.1)
was used to calculate the SIFs to verify the equation used by FEM software. As
expected, the FEA results were found to be in excellent agreement with published
results [26]. The Brazilian disk was selected because of its ability to vary mode
mixity at the crack tip by varying load angle '0'.
The next modeling was done using the orthotropic rectangular specimen
with single edge crack and through thickness center crack. The orthotropic stress
intensity factors calculated through analytical method were also compared with
isotropic results obtained by FEA software. The difference between these results
are discussed. Likewise the orthotropic Brazilian disk, with center crack, was
modeled and a comprehensive parametric ,a i1 ,~i i was performed.
The analysis was done for the following cases:
Figure 41: Brazilian disk meshed with triangular element having isotropic proper
ties (left) and zoomed view of crack tip (right).
The model was checked for errors using isotropic (G ) and or
thotropic (G E m) material properties while keeping all other parameters
same;
The crack length was changed from minimum to maximum in 10 steps;
The crack angle with the applied force direction was changed from 00 to 900;
The material coordinate systems used were {111} (101) and {111} (121).
{111} being the crack plane and (101), (121) being the crack growth direc
tions on {111} plane (Figs. 35 on page 29, 33 on page 28).
After defining the geometry of the material in FEA software, free meshing was
done on one of the surfaces (front or back), and was swept through the thickness.
The mesh density was optimized to save time and effort (Fig. 34 on page 29). The
crack tip element size was also varied to check the accuracy of the result. Once
standardized, the same model was used for various other analyses. The FEA model
was meshed using solid95 element (Fig. 41), which is a quadratic isoparametric
brick element with 8 corner nodes and 12 midside nodes (Fig. 42 on page 33).
The details of this element have been discussed in the following section.
Crack Tip Element. A number of special crack tip finite elements have been
developed applicable to the displacement method, also the hybrid method has been
used in developing special crack tip elements. These special crack tip elements
contain a singularity off the strain field at the crack tip equal to the theoretical
singularity. For twodimensional geometry the 8node quadratic isoparametric
element can be made to simulate the 1//r singularity in stress. That is done
by placing the midside nodal coordinates on any side (connected to the crack
tip) at the quarter point .,.1i 'ent to the crack tip [40]. Lynn and Ingraffea [41]
have shown computation of SIF for isotropic material with high accuracy can be
achieved by placing midside nodes between quarter and midpoint of the elements
at the crack tip. This information further led to the development of the transition
element, which resulted in improved SIF calculations when the ratio of the quarter
point element and crack length is decreased.
Quadratic isoparametric elements with 20 nodes have proven to be excellent
elements for the calculation of SIF for elastic crack problems [31], when the mid
side nodes are put near the crack tip at the quarter point. Also this element has
inverse square root ( ) singularity, which provides a stress field in agreement
with the theoretical stress singularity of linear fracture mechanics. While meshing
the model, the size of the element was of significant importance. Tarasovs [42]
has shown the effect of quarter point singular and transition element's size on
several SIF computation techniques. It was determined that computed SIF was not
sensitive to the singular element internal angle and size except for the quarter point
displacement technique and the displacement correlation technique. Therefore finer
mesh around the crack tip gives better results for the quarter point displacement
method.
Twodimensional triangular elements [30], formed by collapsing one side of
rectangular elements, give better results for stress intensity factor. Also, it has
N
JR
Figure 42: (a) 20 node isoparametric element in natural coordinate system and (b)
Quarter point singular element with the ( = 1 face collapsed in local Cartesian
coordinate system. [31]
been found that rectangular quadratic isoparametric elements (and brick elements)
have singular stiffness (singular total strain energy) at the crack tip [30]. For
the threedimensional case, 20node isoparametric elements were used. 20node
brick elements collapse and form wedge shaped elements at the crack tip. One
of the element faces collapses at the crack tip to form the wedge shape as shown
in the Fig. 42. Fig. 42(a) shows a 20 node isoparametric element in universal
coordinate system, and Fig. 42(b) depicts its quarter point singular version with
the = 1 face collapsed in local Cartesian coordinate system.
This element is the best crack tip element for FEA solution of mixed mode
SIF analysis of orthotropic materials [24] [42]. Around the crack tip the size of the
elements should be very small to reduce the error in calculating SIFs. A typical
example of crack tip element length (Li) is 0.002in. The enlarged figure of the
meshing around the crack tip can be seen in Fig. 44 on page 34. All the nodes
had to be aligned along the material coordinate system to get the desired result.
Two different orientations (101) and (121) (on {111} plane) were used to
analyze the effect on SIF. The crack lies on the {111} plane (Fig. 31 on page 24),
and the crack direction was aligned along family of either (101) or (121) directions.
Figure 43: Arrangement of quarterpoint wedge elements along segment of crack
front with nodal lettering convention. [31]
I.LL".LII" '
Figure 44: The symmetric meshing of BD specimen with solid95 element and
enlarged picture of crack tip elements
The resolved shear stress in the direction of (101) is maximum and in the direction
of (121) is minimum on the {111} plane as the Schmid's factor is maximum along
(101) and minimum along (121) direction. Since the maximum and minimum
resolved shear stress occur along the (101) and (121) directions respectively, these
directions are a family of primary or 'easy slip' directions and secondary or 'hard
slip' directions, on the {111} plane.
7000
6 *[ Isotropic Solution for <121>
*6Anisotropic Solution for <121>
" Isotropic Solution for <101>
5000 Anisotropic Solution for <101>
4000
3000
2000 *
1000
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Crack Length Width ratio
Figure 51: K, versus Crack Length/Width ratio for [101] and [T21] orientation of
Specimen 'A' at 00.
gradual increase in K, was observed with an increase in the 2a/W ratio. Also
the values of K, for the [101] orientation were ahbv~ found to be greater than
the corresponding values for the [121] orientation (Fig. 51 on page 37), at the
same far field stress. The same trend was also observed for K11, though the values
were very small for both the orientations (Fig. 52 on page 38). This is because
the loading is normal to the (111) plane, thus inducing zero shear in the plane of
the crack. Therefore K11 is expected to be very small and within the numerical
calculation error. The values of K111 for the [101] orientation are negative (Fig.
53 on page 38) because the relative displacement of the two crack faces along the
zdirection (Fig. 43 on page 34) was negative. This is simply a consequence of
sign convention definition. However, the magnitude of the SIF for [101] is found to
be greater than for [121]. This validates that the (101) family of slip directions are
primary or easy slip directions as compared to those of (121).
The change in SIFs with the crack angle '0' was analyzed for both orienta
tions. K, decreases for both orientations with increasing crack angle as is evident
140 *Anisotropic Solution for <121
120_ Isotropic Solution for <121>
Anisotropic Solution for <101>
100 6. Isotropic Solution for <101>
~0
0.1 0.2 0.3 0.4 0.5 0.6 .7 0.8 0
Crack Length / Width ratio
Figure 52: KI1 versus Crack Length/Width ratio for [10t] and [121] orientation
of Specimen 'A' at 00.
200
0
T 300
r
400
500
600
700
800
.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
 Anisotropic Solution for <121"
 4 Isotropic Solution for <121>
 Anisotropic Solution for <101>
 * Isotropic Solution for <101>
Crack Length / Width Ratio
Figure 53: K11, versus Crack Length/Width ratio for [101] and [121] orientation
of Specimen 'A' at 00.
Anisotropic Solution for <121:
 ** Isotropic Solution for <121>
Anisotropic Solution for <101>
 * Isotropic Solution for <121>
0 10 20 30 40 50 60 70 80
Crack angle with Force (degree)
Figure 54: K, versus crack angle
Specimen 'A' at 2a/W 0.4.
700
600
500
400 
300
200 
100 
with force for [10t] and [121] orientation of
0 10 20 30 40 50
Crack angle with Force (degree)
60 70 80
Figure 55: KI1
Specimen 'A' at
versus crack angle with force for [iot] and [121]
2a/W 0.4.
orientation of
3000
2500
S2000
5 1500
1000
500
0
500
50 
10 20 30 40 50 60 70 80
500 4* Isotropic Solution for <121>
S* Anisotropic Solution for <121>
1000 Isotropic Solution for <101>
"10 J AAnisotropic Solution for <101>
S1500
2000
2500
Crack angle with Force (degree)
Figure 56: K111 versus crack angle with force for [101] and [121] orientation of
Specimen 'A' at 2a/W 0.4.
from the plots (Fig. 54 on page 39), but the value of K, is alvb, greater for
[10t] than that for the [12t] orientation. For isotropic material, K, becomes zero
upon crack closure at a certain angle. For an orthotropic material, it can be seen
that K, is still nonzero due to the coupling of the nodal displacements. The value
of K1 initially increases with an increase in Q due to an increase in shear stress,
which after reaching a maximum value starts decreasing (Fig. 55 on page 39).
The value of K111 for the [10t] orientation is negative (Fig. 56 on page 40) as the
relative nodal displacements of the crack faces along the zdirection were negative.
However, it was observed that the magnitude of K111 for [10t] was ah ,v greater
than for the [121] orientation.
From the Brazilian Disk specimen (Specimen 'B'), it can be seen that the
magnitude of SIF for the [10t] orientation is ahv greater than the corresponding
values of the [121] orientation, thus enabling a crack to move faster on the [101]
plane than on the [T21] plane (Figs. 57, 58 on page 42, 59, 510 on page 43,
511, 512 on page 44), under fatigue loading for same AK values. Even though
25000
*Anisotropic Solutions for <101>
20000 . Isotropic Solution for <101>
20000 
SAnisotropic Solutions for <121>
* Isotropic Solutions for <121>
'5 15000
10000
................
0
0 
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Crack Length / Diameter ratio
Figure 57: K, versus Crack Length/Diameter ratio for [10] and [121] orientation
of Specimen 'B' at Q 00.
the magnitude of Kn1 for the [121] orientation is greater than that for the [101]
orientation, it may not affect the effective SIF very much, as the magnitude of K111
for the [101] orientation is far greater than that for the [121] orientation.
The values of K, for the [101] orientation rapidly decrease as compared to
the [121] orientation and reach crack closure angle at around 180, whereas the
crack closure angle is 300 for the [172] orientation. The magnitude of K/i for
[121] is alv, greater than that for the [101] orientation, but the difference is far
less in comparison to the magnitude of K111 for the [121] and [t10] orientations.
Following the work of Sauma and Sikiotis [31], the calculated values for Knli were
found to be negligible and after much analysis and rederiving of relevant equations,
the equation given by Sauma and Sikiotis was found to be wrongly formulated. On
the other hand, equation (3.5) correctly gave rise to all the three modes of SIFs. As
a result of the coupling of displacements at the crack tip due to anisotropy, non
negligible values of K111 were found, and they varied with respect to the applied
force, crack length and crack angle.
5
1000
5 2000
*~0
0~~ S p
U
U
2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
3000 *Anisotropic Solutions for <101>
Isotropic Solution for <101>
4000 Anisotropic Solutions for <121
* Isotropic Solutions for <121>
Crack Length / Diameter ratio
Figure 58: KI1 versus Crack Length/Diameter ratio for [10t1 and [121] orienta
tion of Specimen 'B' at 0.
2000
0
0
2000

4000
6000
r
8000
10000
12000
2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
$Anisotropic Solutions for <101>
 Isotropic Solution for <101>
 Anisotropic Solutions for <121
 * Isotropic Solutions for <121>
14000
Crack Length / Diameter ratio
Figure 59: K11 versus Crack Length/Diameter ratio for [101] and [121] orienta
tion of Specimen 'B' at Q 0.
A,_
43
15000
.. *. *
10000 *" '' 
5000 .. 
S 0
0 10 30 40 50 60 70 80 90
5000
10000
15000
20000 _ Isotropic Solution for <101>
25000 Anisotropic Solutions for <121>
25000 
* Isotropic Solutions for <121>
30000
Crack angle with Force (deg)
Figure 510: K, versus crack angle with force for [10T] and [T12] orientation of
Specimen 'B' at 2a/W 0.55.
10000
5000 * q
0
5000 
*Isotropic Solutions for 121
10 20 30 40 50 60 70 80 90
10000 Anisotropic Solutions for <101>
Anisotropic Solutions for <121>
@ Isotropic Solutions for <121>
15000
Crack angle with Force (deg)
Figure 511: KI1 versus crack angle with force for [101] and [T2T] orientation of
Specimen 'B' at 2a/W 0.55.
20000 Anisotropic Solutions for <101>
.B Isotropic Solution for <101>
15000Anisotropic Solutions for <121>
15000 . Isotropic Solutions for <121>
10000
5000
0 10 20 30 40 50 60 70 80 90
5000
10000
Crack angle with Force (deg)
Figure 512: K111 versus crack angle with force for [10lt and [F21] orientation of
Specimen 'B' at 2a/W 0.55.
The BD specimen SIFs (KI, K1i and K1il) were further analyzed along the
thickness for the [10Lt and [T21] crack orientations with change in crack length
and crack angle. They were calculated at 5 different points along the thickness at
the crack front as shown in the Fig. 5 13 on page 45. A plane stress assumption
was made at the surface of the BD specimen (Plane 1 & 5) and a plane strain
assumption was made at all the interior points (Plane 2, 3 & 4).
In order to check the difference in SIF of isotropic BD specimen and or
thotropic BD specimen with change in load angle with crack, under similar loading
condition and specimen geometry, two models were analyzed. The orthotropic
BD specimen analyzed had the material coordinate system aligned with specimen
coordinate system. Therefore a symmetry was expected in the SIF values across
the thickness (Figs. 515 on page 47, 517 on page 48, 519 on page 49). For
isotropic material, properties of pure nickel was used. The trends of KI, K11 and
K111, because of change in force angle with crack, for orthotropic BD specimen
and isotropic BD specimen were found to be similar, but the magnitude of SIFs
calculated x
Crack tip front
Figure 513: Half meshed model of Brazilian disk specimen and the crack coordi
nate system
for both the specimen were significantly different, because of the material prop
erties. The crack closure angle was the same (~ 200, Figs. 514, 515 on page
47). Negative K, was found because no contact elements were used at the crack
interface. After reaching the crack closure angle, further increase in force angle lead
to penetration of crack faces, resulting in negative K1. At the edges, K, was less
than that towards midplane (Figs. 514, 515 on page 47). K1,max was found to
be 12 Kpsivin for orthotropic BD specimen, whereas it was less than 10 Kpsivin
for isotropic BD specimen. The values of K11 can be seen to be negative because
of the sign convention followed at the crack tip coordinate system (Fig. 513 on
page 45, 64 on page 64). The absolute value of Kt reached maximum at ~ 300
for both the specimens(Figs. 516, 517 on page 48), then starts slowly decreasing
towards zero. The absolute Knlmax was 18 Kpsi for orthotropic BD specimen,
whereas it was 14 Kpsi in for isotropic BD specimen. The variation of K1 across
the thickness was more distinguished for the orthotropic case than for the isotropic
specimen (Fig. 516, 517 on page 48). Higher K7). values were observed for the
10
* Angle 00
SU Angle 06
 Angle 12
 Angle 18
S 6  Angle 24
4 4
0
( 0.2 0.4 0.6 0.8 1 1.2
2
4 Normalized Thickness
Figure 514: Variation of SIF K, along isotropic BD specimen thickness at differ
ent Crack angle.
orthotropic specimen than for isotropic specimen (Figs. 518, 519 on page 49).
For load aligned with crack plane, KIII was found to be zero across the thickness.
Since K1II is a function of out of plane relative displacements (normal to the disk
plane) of nodes at the crack tip and because of perfect symmetry in loading and
geometry of the material, it ought to zero. But as the force angle was introduced,
because of poisson's effect, there were small out of plane relative displacements on
both sides of the mid plane of the disk (plane 3, Fig. 513 on page 45), which was
equal and opposite to each other. The absolute value of K1I reaches maximum
between 30360 and it is ~ 15Kpsiin for isotropic and ~ 22Kpsivn for or
thotropic specimen. Among all the SIFs K1I is more dominant at the crack edges
at higher force angles, whereas it is zero at the midplane for all the angles (Figs.
518, 519 on page 49). The average value of K1II is zero across the thickness for
both specimen.
From Figs. 521, 522 on page 52, it can be seen that K, decreases with
an increase in crack angle and crack closure angle (where K, is almost zero) was
*Angle 00
 Angle 06
eAngle 12
 Angle 18
+ Angle 24
I N
0.2 0.4 0.6 0.8 1
Normalized Thickness
Figure 515:
ferent Crack
2
0
2
4
o 61
S8
I 10
Variation of SIF K, along orthotropic BD specimen thickness at dif
angle.
Normalized Thickness
Figure 516: Variation of SIF KI1 along isotropic BD specimen thickness at differ
ent Crack angle.
1.2
2
0
2 i
*a 
12
4'
16
18
0.2 0.4 0.6 0.8 1

SAngle uu
UAngle 06
sAngle 12
xAngle 18
Angle 24
 Angle 30
Angle 36
Angle 42
Angle 48
Angle 54
*Angle 60
Angle 66
4Angle 72
Angle 78
Angle 84
Normalized Thickness
Figure 517: Variation of SIF KI1 along orthotropic BD specimen thickness at
different Crack angle.
* Angle 00
 Angle 06
eAngle 12
Angle 18
+ Angle 24
0.2 0.4
Normalized Thickness
Figure 518: Variation of SIF KII, along isotropic BD specimen thickness at differ
ent Crack angle.
25 * Angle 00
2 Angle 06
20 sAngle 12
Angle 18
+ Angle 24
15 Angle 30
 Angle 36
10 ~ Angle 42
x Angle 48
5 x Angle 54
0* Angle 60
 Angle 66
0 Angle 72
S2 04 Angle 78
5s  Angle 84
10 
15
20
25
Normalized Thickness
Figure 519: Variation of SIF K111 along orthotropic BD specimen thickness at
different Crack angle.
reached at ~ 180 for the [10t] orientation, whereas it was ~ 300 for the [121]
orientation, as was observed by Fig. 510 on page 43. K, can be seen symmetric
for [t21 orientation across the thickness, whereas it is not for the case of [10T]
orientation. In general K, inside the surface, (plane 2, 3, 4, Fig. 513 on page
45) is alvb, greater than those at the crack edges (plane 1, 5, Fig. 513). The
absolute value of K11 increases and becomes maximum at an angle ~ 240 for the
[10t] orientation, whereas for the [T21] orientation it is ~ 360 (Figs. 523, 524
on page 53), as also illustrated in Fig. 511 on page 43. K11 then starts slowly
decreasing with further increases in crack angle, as was observed in Fig. 511. It
can be seen that at one of the faces (Plane 1, thickness = 0; Fig. 521 on page
52), absolute values of K, are alvb greater than those at the other face (Plane 5,
thickness = 1) of the BD specimen for the [10l] orientation. But absolute values
of Kl (Fig. 523 on page 53) are alvb, greater at Plane 5, than those at Plane
1 for the [101] orientation. It is interesting to observe that SIFs (Figs. 522 on
page 52, 524 on page 53, 526 on page 54)(KI, Kl and K1il) are symmetric
14
12
10 * Angle 00
8 Angle 06
eAngle 12
6 Angle 18
S  Angle 24
4
2
II 0.2 .OA S 0.8 1 1.2
4
6
Normalized Thickness
Figure 521: Variation of SIF K, along BD specimen thickness at different Crack
angle for [101] orientation.
12
0.8 1
F 0.2
* Angle 00
 Angle 06
Angle 12
Angle 18
 Angle 24
 Angle 30
Normalized Thickness
Figure 522: Variation of SIF K, along BD specimen thickness at
angle for [121] orientation.
different Crack
Angle 30
 Angle 36
a Angle 42
.94Angle 48
SAngle 60
Angle 60
6  Angle 66
Angle 72
M 8 Angle 78
 uAngle 84
10
12
14
Normalized Thickness
Figure 523: Variation of SIF KI along BD specimen thickness at different Crack
angle for [101 orientation.
0.2 0.4 0.6 0.8 1 1.2
_ "^a ~^ *Angle 00
2 Angle 06
2 Angle 12
Angle 18
4Angle 24
5 ^ Angle 30
*a Angle 36
Angle 42
Angle 48
6 Angle 54
4Angle 60
^ hAngle 66
SAngle 72
M . 8Angle78
oAngle 84
10
12
Normalized Thickness
Figure 524: Variation of SIF KI along BD specimen thickness at different Crack
angle for [12t] orientation.
* Angle 00
Angle 06
 Angle 12
60
Normalized Thickness
Figure 525: Variation of SIF KII, along BD specimen thickness at different Crack
angle for [o10] orientation.
* Angle 00
 Angle 06
eAngle 12
Angle 18
 Angle 24
0.2 0.4
Normalized Thickness
Figure 526: Variation of SIF KII, along BD specimen thickness at different Crack
angle for [121] orientation.
40
35
.5 30
j 25
15
10
5
0
0.2 0.4 0.6 0.8
Normalized Thickness
Figure 527: Variation of SIF K, along BD specimen thickness
Length/Diameter ratio for [101] orientation.
30
25
.a
S20
S15
10
5
A
CU
I
1 1.2
at different Crack
*Ratio 0.3
 Ratio 0.4
Ratio 0.5
e Ratio 0.6
 Ratio 0.7
* Ratio 0.8
 
0.6
Normalized Thickness
Figure 528: Variation of SIF K, along BD specimen thickness at
Length/Diameter ratio for [121 orientation.
different Crack
5
i 4
S3
S2
0
1
0.6 0.8
Normalized Thickness
Figure 529: Variation of SIF KI1 along BD specimen thickness at different Crack
Length/Diameter ratio for [101] orientation.
2
3
S4 
6
7
*Ratio 0.3
1  Ratio 0.4
aRatio 0.5
 Ratio 0.6
* Ratio 0.7
*Ratio 0.8
0 Ratio 0.8
Normalized Thickness
Figure 530: Variation of SIF KI1 along BD specimen thickness at different Crack
Length/Diameter ratio for [T21] orientation.
 Ratio 0.3
 Ratio 0.4
 Ratio 0.5
e Ratio 0.6
 Ratio 0.7
* Ratio 0.8
0.2 0.4
* Ratio 0.3
 Ratio 0.4
10
S30
40
50
Normalized Thickness
Figure 531: Variation of SIF Ksss along BD specimen thickness at different Crack
Length/Diameter ratio for [101] orientation.
2 * Ratio 0.3
 Ratio 0.4
1.5  Ratio 0.5
e Ratio 0.6
Ratio 0.7
1  Ratio 0.8
0
0' 1 0.2 0.4 1 1.2
0.5
1
1.5
Normalized Thickness
Figure 532: Variation of SIF Ksss along BD specimen thickness at different Crack
Length/Diameter ratio for [12T] orientation.
Sl > S2 > S3
Applied cycles, N
Figure 62: Fatigue Crack Length versus Applied Cycles. Fracture is Indicated by
the x. [43]
given initial crack size, the life to fracture depends on the magnitude of the applied
stress and the fracture resistance of the material.
6.2 FCC single < I 1 materials
Fatigue crack driving force parameter AK, as described in previous section is
well suited for isotropic materials, but it may not be the best parameter to use for
very large grain or single crystal alloys. Earlier studies have shown that for crack
size smaller than the grain size or of the same order of magnitude, threshold stress
intensities are lower and FCG rate is accelerated in comparison to long cracks when
compared at similar values of AK [4448]. This shows that the parameter AK
does not incorporate microstructure and the grain orientation and the associated
deformation mechanisms in controlling the FCG behavior.
An FCC single < i i1I J1 alloy offers the best opportunity to study in detail the
effect of microstructure and the deformation mechanisms on the FCG behavior.
Several Studies have been conducted on FCG of Nibase single crystals [4953] and
all of these studies have shown that FCG is highly sensitive to the orientation of
the crystal and that the crack plane is crystallographic and follows a single slip
plane or a combination of slip planes. Since shear decohesion on a slip plane is
caused by dislocation motion, many researchers have r . 1. 1 that the resolved
shear stress acting on the active slip plane ahead of a crack tip must be responsible
for the propagation of the fatigue crack [51, 54, 55]. The dislocation motion is
controlled by the forces at the crack tip, which is directly related to the resolved
shear stress on the slip plane. Therefore the rate of shear decohesion must be
related to the resolved shear stress 'intensity' (RSSI) at the crack tip. The active
shear decohesion plane or planes must be the slip planes) with a high RSSI. If
the RSSI on a plane is much higher than all the other slip planes, then the plane
must be the primary plane for shear decohesion and the slip plane becomes the
crack plane. But if the resolved shear stresses on two or more of the slip systems
are comparable, then the shear decohesion will take place on all of those slip planes
and the macro crack plane will not follow a single slip plane. This was shown
by Telesman and Ghosn [56] as shown in the Fig. 63 on page 62. Even though
macroscopic crack was observed along (001) plane inclined 70 to the starter notch,
the microscopic slip was observed on 111 slip planes, inclined 520 and 380 to the
starter notch.
,' 50 pm
Figure 63: Microscopic slip observed on two {111} slip planes inclined 520 and
380 to the starter notch. [56]
If a crack surface is a slip plane, it is logical that the crack growth rate on that
slip plane will correlate with its RSSI.
Chen and Liu [57] proposed a crack driving force parameter for correlating
FCG data, which is based on the resolved shear stresses on the active slip planess.
This parameter may be better than AK for the correlation of FCG data since it
takes into consideration of the deformation mechanisms, grain orientation and the
actual crack path.
The resolved shear stress field of a slip system is defined by its intensity
coefficient, which can be calculated once the Mode I, II and III crack tip fields are
obtained.
The resolved shear stress is given by [58]
Tss = bibjjj (6.7)
where bi and b are the Burgers vector and its magnitude; nj is the unit normal
vector of the slip plane; and ryj is the crack tip stress tensor field given by [57]
[1yi] [Kf1 (0) + K1fi' (0) + Ki f i' (0)] (6.8)
where r and 0 are the local polar coordinates at the crack tip as shown in
the Fig. (64 on page 64); fij(0) are the angular component of the stress field.
Substituting equation (6.8) into equation (6.7), the resolved shear stress is
1
s ] [KIf' (0) + Ki f i (0) + Ki if"(0)][n ] (6.9)
where b' and nj are the unit Burgers vectors and unit normal vectors of the
slip planes respectively.
The above equation indicates that Trss preserves the 1//r singularity, and the
intensity of T,ss, is dependent on the (i iI 1 orientation relative to the crack surface.
For a given crystal orientation and crack geometry, the angle 0 is equal to the angle
tx
Crack Surface
Figure 64: Details of crack tip displacements and stresses at a distance r and 0
from the crack tip in the crack coordinate system
between the trace of a particular slip plane on the plane normal to the slip plane
and the horizontal axis. The intensity of Trs, is linearly proportional to the quantity
resolved shear stress intensity coefficient, K.s which for a given slip system can be
defined as the limiting value of the resolved shear stress, T.rs, multiplied by 27r,
as r approaches zero [56, 57]
Krs, = limia 8.v2r (6.10)
rO
where r is the distance of the crack tip and Tr~, is defined as the projection
of the stress tensor [oa] on a plane whose outward normal is S in the direction of
slip b (Fig. 65 on page 65). The two distinct advantages in using K.rs are: (1)
the dependency of Tr, on r is eliminated; (2) the angle 0 has a definite physical
ri ii. which is directly related to the orientation of the slip system.
Telesman and Ghosn [56] used the above model to calculate AKrs for 2D
model and checked the validity of this parameter by plotting it against da/dN data
(Fig. 66 on page 65). Also AK for mode I has been plotted against da/dN to
n
4 {111}
Figure 65: Burgers vector b is along slip direction (011) and slip plane direction
is normal vector S along (111)
/ExtrapoLation of
/ Paris region
,o 3 4 6 810 20 0
AK, AKr.. MPoVrm
Figure 66: Fatigue crack growth rate as a function of AKrs8 and AK for 2D
rectangular specimen. [56]
show similarity between both the curves. They look virtually identical and it can
be seen that a linear relationship exists between AKs88 and da/dN, which is in
accordance with the Paris law in region II. But in this method mode III SIF was
not considered as the specimen was not tested under complete multiaxial fatigue
loading.
State of stress on a slip plane, under mixed mode loading, whose trace on a
plane normal to crack plane makes an angle 0 with the horizontal axis (Fig. 64 on
page 64 and Fig. 613 on page 71), can be defined as1
cr dil d12 0
y d21 d22 0
K,
tz 1 d31 d32 d33 (6.
Ty 0 0 d43
Ki
TZ 0 0 d53
Ty d61 d62 0
where,
b b d1F2 22 1 ( p
d11 = Re r12 t P1 d12 = e Pt (P2 j
LI [2 b2 bl [1l p2 b2 bl
1 Re t 1 1 : 2] 1 (I 1)
d21 Re ld22 Re 
11 2 b2 b 1l [1 2 b2 bl
0 Plane Stress
d3j dj1a13 + d2ja23 + d6j36 Plane Strain1, 2
a33 Plane Strain
833
1 The details of the equation derivation can be seen in Appendix B.
d33 {
0
d43a34 + d53a35
a33
d43= Re b]
d61 Re !12 / /1
cL(1 [12 bl b2 n
6i V/cos(0) + sin(0)
Plane Stress
Plane Strain
d53 = Re [b]
d62 Re 11 (I /12)1
62 = 3
P1 [2 bl b2
i 1,2,3
/il and /2 are the two roots with positive imaginary parts as defined by the
equation
all/4 2a16 /3 + (2a12 + a66)_2 2a26p + a22 = 0
(6.12)
and /P3 is the root of the characteristic equation [26]
a55/2 2a45/ + a44 0
(6.13)
It should be noted that the compliance constants (ai,3(i 1, 2,... 6)) used for
the expression (d3,j(j 1,2, 3)) are the ones as given by equation (3.9).
The above equations, when used in conjunction with equations (6.7) & (6.10)
gives Krs. on all the 12 slip systems for FCC single i ivI 1 superalloy.
/' [nj]
K,
KIl
Kil
(6.14)
K/rss
0 10 20 30 40 50 60
No. of Cycles, N in Millions
Figure 67: Crack growth of BD specimen 95830 with no. of cycles
12 n
10
6
4
0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
No. of Cycles, N in Millions
Figure 68: Crack growth of BD specimen 96842 with no. of cycles
16 n
14
12
10
6
4
0 0.1 0.2 0.3 0.4 0.5 0.6
No. of Cycles, N in Millions
Figure 69: Crack growth of BD specimen 98C21 with no. of cycles
4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
Crack Length, a in mm
Figure 610: Crack growth rate of BD specimen 95830 with increasing crack length
4 5 6 7 8 9
Crack length, a in mm
10 11 12
Figure 611: Crack growth rate of BD specimen 96842 with increasing crack length
4.E04 I
3.E04
S3.E04
2.E04
2.E04
1.E04
2 4 6 8 10
Crack length, a in mm
12 14
Figure 612: Crack growth rate of BD specimen 98C21 with increasing crack length
9.E05
8.E05
7.E05
6.E05
5.E05
*9 4.E05
3.E05
The mixed mode SIFs were calculated at those points shown in the Figs.
(610, 611, 612 on page 69) for all the three specimens on the mid plane of the
Brazilian disk specimens, assuming plane strain assumption. AKrms was calculated
using the following equation in order to compare with AKss.
AKrms = AKj + AK}, + AK}JJ
(6.15)
K..s was calculated using the equation (6.10), where state of stress obtained
from equation (6.11) was multiplied by Schmid's factor for FCC single i i,l I as
defined in equation (1.13) to get the resolved shear stress intensity on all the 12
primary slip systems. The trace of the two slip planes (111) and (111) were found
to be making equal angles with the (111) plane (67.780), while the trace of the
plane (111) was making 00 with the crack plane (111) as shown in the Fig. 613.
(111) {1To0}
Plane along depth {1 1 2}
Trace of Slip Planes 11) Trace
SC Trace Plane
y 0 (Plane normal to
0= 670 Crack Plane)
0 = 670
z Trace of Slip Planes
Crack Surface ( 1 1)
Figure 613: Trace of primary slip planes on the plane normal to the crack plane
The load ratio R (min load/max load) was 0.1 (Table 61 on page 68), as the
load on the BD specimen alv,x compressive. As SIFs are linearly proportional to
the load applied, therefore Krs. was multiplied with R to get the AKrss.
The max K,.s was found to be on the (111) slip plane for all the three
specimens (First row of Tables 62, 63 on page 76 and Table 64 on page 77),
as observed in the experiment test results (Fig. 614 on page 73). The calculated
AKr.s was plotted against da/dN on a loglog scale to check the validity of the
model (Fig. 616 on page 74). After AK,.r reaches 10 MPa /m a linear plot
can be seen, which might correspond to region II (Paris region) where the crack
growth rate is directly proportional to the the applied AK, on a loglog basis.
But not enough data is available to support the theory.2 However below a AK,.rs
of 8 MPa\/m an accelerated crack growth can be seen between 5 MPa\/m and
7 MPa /m, in the threshold region; region or stage I. The lines drawn through
those points yield AKth on the AK,.rs axis, which is called fatigue threshold
stress intensity factor. The average AKth of the three specimens has been taken
( 4.3MPa V/), at the given load ratio R(0.1). Below this value (AKr.s < AKt,
crack growth either does not occur by cyclic loading i. e. da/dN = 0 and specimen
can have infinite life or it grows at undetectable rates. The value of AKth can be
affected by alterations in the material microstructure, load ratio, environment and
crack size 3 However the cyclic stress intensity factor (AK, r.) is not the only load
parameter to control the FCG rate, because da/dN, at a given AKrs, can increase
with increase in load ratio 'R' as
max (1 R)
2 Because of the proprietary nature of specimens, only three specimens' data
were made available. But we are expecting some more data in June'05 to verify our
work.
3 The effect of these parameters on cyclic stress intensity factor is beyond the
scope of my research. However the detail can be found in 'Fatigue of Materials' by
S. Suresh [59]
00
* o
0
*0
Figure 615: Fatigue crack growth
function of AKrms
rate of 3 specimens 95830, 96842, 98C21, as a
1.E03
A A
1.E04
1.E05
1.E06
1.E07 I
1.OE+00
o 95830
n 96842
A 98C21
1.OE+01 1.OE+02
AK.,. MPa m1/2
1.OE+03
Figure 616: Fatigue crack growth rate of 3 specimens 95830, 96842, 98C21, as a
function of AKrss
1.E03
1.E04
A 95830
* 96842
o 98C21
1.E05
1.E06
1.E07 
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CHAPTER 7
CONCLUSIONS
The goal was to estimate the fatigue life of single crystal nickelbase superal
loys, based on fatigue crack driving force parameter, for which a numerical model
was developed to compute mixed mode SIFs at the crack tip. The results obtained
can be summarized as
1. An analytical method has been developed for the calculation of all the three
modes of stress intensity factors as a function of crystallographic orientation
for an orthotropic material, which can be applied to any anisotropic material,
if all the material constants are known. No prior assumption of plane stress
or plane strain was made in developing this theory.
2. Mode I, (KI), was ah. found to be greater for [10T] than for the [121]
orientation. For an orthotropic material, K, was found to be nonzero at
crack closure due to the coupling of the nodal displacements, whereas for an
isotropic material, it was zero.
3. The magnitude of K1i for [121] was found to be alvb greater than that for
[101] orientation, but the difference was not much.
4. Mode III SIF (Kill) existed because of the coupling of displacements at
the crack tip due to anisotropy. K111 for [101] was found to be much bigger
than that for [121] orientation for the Brazilian disk specimen. This p .1i 
an important role in calculation of effective K to predict the living of an
anisotropic material.
5. The K, and K111 values calculated were much higher for [101] orientation,
than those for [121] orientation. Therefore a high value of K.rs can be
expected for the [101] direction than that for the [121] direction, if we ignore
required to get the accurate value of parameter C and n of the Paris law. The
equation obtained can be used to calculate the life of the material.
11. The method developed is not related to any specific geometry. Therefore
it can be used for any kind of geometry and for any general anisotropic
material, to calculate mixed mode SIFs at the crack tip and therefore crack
growth rate and hence life of any material can be predicted.
12. Additional experimental data are required to study FCG on {111} plane,
(101) orientation. More data are needed in region II to get the accurate
result to predict lifing. Also experimental data for crack plane orientations
other than slip planes, e.g., (100), can be used to check the validity of the
model.
Possible Directions for Future Work are given below:
Incorporation of crack tip plasticity.
FCG testing under mixed mode loading for a wider range of specimen
geometries and orientations.
FCG testing at high temperatures.
APPENDIX A
DISPLACEMENT FIELD EVALUATION FOR AN ANISOTROPIC ELASTIC
SOLID UNDER GENERALIZED PLANE STRAIN CONDITION
  dx
f)Cx
STxy+ dx
x +y dy
aTy+ aoydy
Figure A 1: Forces acting on a rectangular element with dimension dx x dy in equi
librium condition.
A generalized plane strain condition has been adopted for an anisotropic
solid applicable to that of an FCC single crystal. [6] Let us consider a rectangular
pipe like element with dimension dx, dy, 1. We assume that this body is under
equilibrium under various forces. If body forces are X, Y per unit volume, then the
equation of equilibrium of force along x and y direction can be written as followed,
Along xaxis.
Ix + aXdy dx axdy + TsY + Txydy\ dx Txydx + Xdxdy = 0 (A.1)

Full Text 
PAGE 3
Thisdissertationistheresultof5yearsofworkwherebyIhavebeenaccompaniedandsupportedbymanypeople.ItisapleasantaspectthatIhavenowtheopportunitytoexpressmygratitudetoallofthem.TherstpersonIwouldliketothankismysupervisor,Dr.NagarajArakere.Ihavebeenunderhissupervisionsince2000whenImovedtotheUniversityofFlorida.Hisenthusiasmandintegralviewonresearchandhismissionforproviding"onlyhighqualityworkandnotless,"hasmadeadeepimpressiononme.Iowehimlotsofgratitudeforhavingmeshownthiswayofresearch.IwouldalsoliketothanktheothermembersofmyPhDcommitteewhotooktheeortinreadingandprovidingmewithvaluablecommentsonearlierversionsofthisthesis:Dr.AshokKumar,Dr.BhavaniSankar,Dr.FereshtehEbrahimiandDr.JohnZiegert.Ithankthemall.MycolleaguesintheFatigueandTribology(FAT)LabErikKnudsen,JeLeismer,ShadabSiddiquiandTaeJoongYUallgavemethefeelingofbeingathomeatwork.ThefriendlyambienceofFATlabalwayskeptmeinacheerfulmood.ThediscussionsandtheinteractionswithShadabandJehadadirectimpactonthenalformandqualityofthisthesis.IwouldliketothankShadabforprovidingmetipsthathelpedmealotwheneverIwentotrack.Itwasnotpossibletocontinuemyworkwithouthiscompanionshipandcheerfulsupport.Erik,JeandAmitojLikharihavereadpartsofmythesisandprovidedmevaluablecomments.AlexanderPachecohashelpedmetremendouslyingettingthethesisincorrectformat.Myfriendandcolleaguesincemyundergraddays, iii
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iv
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page ACKNOWLEDGMENTS ............................. iii LISTOFTABLES ................................. vi LISTOFFIGURES ................................ vii ABSTRACT .................................... xi CHAPTER 1INTRODUCTION .............................. 1 2COMPUTATIONOFSTRESSINTENSITYFACTORSFORSINGLECRYSTAL:LITERATUREREVIEW .................. 17 3SIFEQUATIONFORMULATIONFORMIXEDMODELOADING .. 23 4MODELINGANDMESHING ........................ 30 5MIXEDMODESIF:RESULTSANDDISCUSSION ........... 36 6FCGGROWTHINFCCSINGLECRYSTALMATERIALS ....... 58 6.1Isotropicmaterials .......................... 58 6.2FCCsinglecrystalmaterials ..................... 61 7CONCLUSIONS ............................... 78 APPENDIX ADISPLACEMENTFIELDEVALUATIONFORANANISOTROPICELASTICSOLID ............................. 81 BDETAILSOFSIFEQUATIONFORMULATION ............. 95 CANSYSPROGRAM ............................. 102 REFERENCES ................................... 121 BIOGRAPHICALSKETCH ............................ 126 v
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Table page 1{1Directioncosinesofmaterialcoordinateaxes(xyz)withuniversalcoordinateaxes(x0y0z0) .......................... 8 1{2Directioncosineof(x0;y0;z0)with(x;y;z)coordinateaxes,whenx0axisisalignedalong[213]orientation. ................. 16 3{1Geometricalandmaterialpropertiesofthetwospecimensanalyzed .. 28 6{1ThegeometryandloadingconditionofthethreeBrazilianDiskspecimentested ................................ 68 6{2Krssfor12primaryslipsystemswithincreasingcracklengthforspecimen95830 ................................ 75 6{3Krssfor12primaryslipsystemswithincreasingcracklengthforspecimen96842 ................................ 76 6{4Krssfor12primaryslipsystemswithincreasingcracklengthforspecimen98C21 ............................... 77 vi
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Figure page 1{1Schematicofthe0precipitateinamatrix .............. 2 1{2Cuboidal0precipitates(0.350.6m)inPWA1480. .......... 3 1{3Temperaturecapabilityofsuperalloyswithapproximateyearofintroduction. ............................... 4 1{4Aeroenginebladesarenominallyorientedintheh001iorientation. .. 4 1{5Materialcoordinatesystem(xyz)relativetouniversalcoordinatesystem(x0y0z0) ............................... 11 1{6Twoseparatedportionsofasinglecrystalshowingamodelforcalculatingtheresolvedshearstressinasinglecrystalspecimen. .... 11 1{7SliplinesandfractureplaneinexperimentaltensiletestspecimentestedbyMaterialsScienceandEngineeringDepartment,UF. ... 12 1{8Primary(Closepack)andSecondary(Nonclosepack)slipdirectionsontheoctahedralplanesforanFCCcrystal. ............. 14 1{9CubeslipplanesandslipdirectionsforanFCCcrystal. ........ 15 3{1Cracktipnodaldisplacementofisotropicelasticmaterial ....... 24 3{2Theh101iandh121ifamilyofslipdirectionsaresuperimposedonf111gplaneshowinghowthehexagonalcrackfrontisdelineatedbytheslipdirections. .......................... 27 3{3Braziliandiskhavingcentercracklyinginf111gslipplaneandalignedalong 1direction .......................... 28 3{4Rectangularspecimen`A'meshedwithsolid95element(left)andthegeometricaldetails(right) ....................... 29 3{5Braziliandisk(Specimen`B')withcentercracklyingin(111)planeandorientedalong10 1direction .................. 29 4{1Braziliandiskmeshedwithtriangularelementhavingisotropicproperties(left)andzoomedviewofcracktip(right). .......... 31 vii
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................. 33 4{3Arrangementofquarterpointwedgeelementsalongsegmentofcrackfrontwithnodalletteringconvention. ................. 34 4{4ThesymmetricmeshingofBDspecimenwithsolid95elementandenlargedpictureofcracktipelements ................. 34 5{1KIversusCrackLength/Widthratiofor10 1and 1orientationofSpecimen'A'at=00. ....................... 37 5{2KIIversusCrackLength/Widthratiofor10 1and 1orientationofSpecimen'A'at=00. ....................... 38 5{3KIIIversusCrackLength/Widthratiofor10 1and 1orientationofSpecimen'A'at=00. .................... 38 5{4KIversuscrackanglewithforcefor10 1and 1orientationofSpecimen'A'at2a=W=0:4. ..................... 39 5{5KIIversuscrackanglewithforcefor10 1and 1orientationofSpecimen'A'at2a=W=0:4. ..................... 39 5{6KIIIversuscrackanglewithforcefor10 1and 1orientationofSpecimen'A'at2a=W=0:4. ..................... 40 5{7KIversusCrackLength/Diameterratiofor10 1and 1orientationofSpecimen'B'at=00. .................... 41 5{8KIIversusCrackLength/Diameterratiofor10 1and 1orientationofSpecimen'B'at=00. .................... 42 5{9KIIIversusCrackLength/Diameterratiofor10 1and 1orientationofSpecimen'B'at=00. ................... 42 5{10KIversuscrackanglewithforcefor10 1and 1orientationofSpecimen'B'at2a=W=0:55. ..................... 43 5{11KIIversuscrackanglewithforcefor10 1and 1orientationofSpecimen'B'at2a=W=0:55. ..................... 43 5{12KIIIversuscrackanglewithforcefor10 1and 1orientationofSpecimen'B'at2a=W=0:55. ..................... 44 5{13HalfmeshedmodelofBraziliandiskspecimenandthecrackcoordinatesystem ............................... 45 viii
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............................ 46 5{15VariationofSIFKIalongorthotropicBDspecimenthicknessatdifferentCrackangle. ........................... 47 5{16VariationofSIFKIIalongisotropicBDspecimenthicknessatdierentCrackangle. ............................ 47 5{17VariationofSIFKIIalongorthotropicBDspecimenthicknessatdifferentCrackangle. ........................... 48 5{18VariationofSIFKIIIalongisotropicBDspecimenthicknessatdifferentCrackangle. ........................... 48 5{19VariationofSIFKIIIalongorthotropicBDspecimenthicknessatdierentCrackangle. .......................... 49 5{20a)Unsymmetryaboutmidplaneforcrackorientedalongf111gh101i;b)Symmetryforcracklyingalongf111gh121i 50 5{21VariationofSIFKIalongBDspecimenthicknessatdierentCrackanglefor10 1orientation. ...................... 52 5{22VariationofSIFKIalongBDspecimenthicknessatdierentCrackanglefor 1orientation. ...................... 52 5{23VariationofSIFKIIalongBDspecimenthicknessatdierentCrackanglefor10 1orientation. ...................... 53 5{24VariationofSIFKIIalongBDspecimenthicknessatdierentCrackanglefor 1orientation. ...................... 53 5{25VariationofSIFKIIIalongBDspecimenthicknessatdierentCrackanglefor10 1orientation. ...................... 54 5{26VariationofSIFKIIIalongBDspecimenthicknessatdierentCrackanglefor 1orientation. ...................... 54 5{27VariationofSIFKIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor10 1orientation. ............. 55 5{28VariationofSIFKIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor 1orientation. ............. 55 5{29VariationofSIFKIIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor10 1orientation. ............. 56 5{30VariationofSIFKIIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor 1orientation. ............. 56 ix
PAGE 10
1orientation. ............. 57 5{32VariationofSIFKIIIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor 1orientation. ............. 57 6{1SchematicFatigueCrackGrowthCurve ................. 59 6{2FatigueCrackLengthversusAppliedCycles.FractureisIndicatedbythex. ................................. 61 6{3Microscopicslipobservedontwof111gslipplanesinclined520and380tothestarternotch ....................... 62 6{4Detailsofcracktipdisplacementsandstressesatadistancerandfromthecracktipinthecrackcoordinatesystem .......... 64 6{5Burgersvector!bisalongslipdirectionh011iandslipplanedirectionisnormalvector!nalongh111i 65 6{6FatiguecrackgrowthrateasafunctionofKrssandKfor2Drectangularspecimen. ......................... 65 6{7CrackgrowthofBDspecimen95830withno.ofcycles ........ 69 6{8CrackgrowthofBDspecimen96842withno.ofcycles ........ 69 6{9CrackgrowthofBDspecimen98C21withno.ofcycles ........ 69 6{10CrackgrowthrateofBDspecimen95830withincreasingcracklength 70 6{11CrackgrowthrateofBDspecimen96842withincreasingcracklength 70 6{12CrackgrowthrateofBDspecimen98C21withincreasingcracklength 70 6{13Traceofprimaryslipplanesontheplanenormaltothecrackplane 71 6{14Crackgrowthonf111gslipplanecanbeobservedfor96842BDspecimen ................................... 73 6{15Fatiguecrackgrowthrateof3specimens95830,96842,98C21,asafunctionofKrms 74 6{16Fatiguecrackgrowthrateof3specimens95830,96842,98C21,asafunctionofKrss 74 A{1Forcesactingonarectangularelementwithdimensiondxdy 81 x
PAGE 11
Fatigueinducedfailuresinaircraftgasturbineandrocketengineturbopumpbladesandvanesareapervasiveproblem.Turbinebladesandvanesrepresentperhapsthemostdemandingstructuralapplicationsduetothecombinationofhighoperatingtemperature,corrosiveenvironment,highmonotonicandcyclicstresses,longexpectedcomponentlifetimesandtheenormousconsequenceofstructuralfailure.Singlecrystalnickelbasesuperalloyturbinebladesarebeingutilizedinrocketengineturbopumpsandjetenginesbecauseoftheirsuperiorcreep,stressrupture,meltresistance,andthermomechanicalfatiguecapabilitiesoverpolycrystallinealloys.Thesematerialshaveorthotropicpropertiesmakingthepositionofthecrystallatticerelativetothepartgeometryasignicantfactorintheoverallanalysis.Computationofstressintensityfactors(SIFs)andtheabilitytomodelfatiguecrackgrowthrateatsinglecrystalcrackssubjecttomixedmodeloadingconditionsareimportantpartsofdevelopingamechanisticallybasedlifepredictionforthesecomplexalloys. xi
PAGE 12
TheresolvedshearstressintensitycoecientoreectiveSIF,Krss,canbecomputedasafunctionofcracktipSIFsandtheresolvedshearstressonprimaryslipplanes.ThemaximumvalueofKrssandKrsswasfoundtodeterminethecrackgrowthdirectionandthefatiguecrackgrowthraterespectively.Thefatiguecrackdrivingforceparameter,Krss,formsanimportantmultiaxialfatiguedamageparameterthatcanbeusedtopredictlifeinsuperalloycomponents. xii
PAGE 13
Asuperalloyisagroupofnickel,ironnickel,andcobaltbasemetallicalloyswhichcanbeusedathightemperatures,ofteninexcessof0.7oftheabsolutemeltingtemperature.Exceptionalcreepandhightemperatureoxidationresistancearetwooftheprimedesigncriteriaofsuperalloys,butapartfromthesepropertiestheyexhibitacombinationofhighstrengthathightemperature,stressruptureresistance,toughnessandmetallurgicalstability,usefulthermalexpansioncharacteristicsandstrongresistancetothermalfatigue.Theyexhibitexceptionalcorrosionresistanceatroomtemperature.ThehightemperaturestrengthofNibasedsuperalloysdependsonastable,facecenteredcubic(FCC)matrix,combinedwitheitherprecipitationstrengtheningand/orsolidsolutionhardening.Iron,nickel,andcobaltaregenerallyfacecenteredcubic(FCCaustenitic)incrystalstructurewhentheyarethebasisforsuperalloys.However,thenormalroomtemperaturestructuresofironandcobaltelementalmetalsarenotFCC.BothironandcobaltundergotransformationsandbecomeFCCathightemperaturesorinthepresenceofotherelementsalloyedwithironandcobalt.Nickel,ontheotherhand,isFCCatalltemperatures.ThemicrostructureofanFCCsuperalloyconsistsofanaustenitic(phase)matrixandawidevarietyofsecondaryphase.Themostcommonsecondaryphasesare0andmetalcarbides.Themicrostructureconsistsof~60percentbyvolumeof0precipitatesinamatrix.Thesealloysobtaintheirhightemperaturestrengthfromthepresenceofthe0precipitatephasewithintheprimarymatrix.The0precipitateisafacecenteredcubic(FCC)structureandcomposedoftheintermetalliccompoundNi3Al,withAlatomsoccupyingthecornersandNithefacesoftheunitcell.The0precipitateis 1
PAGE 14
Figure1{1: Schematicofthe0precipitateinamatrix suspendedwithinthematrix,whichisalsoofFCCstructureandcomprisedofNiwithcobalt,chromium,tungsten,andtantaluminsolution.TypicallythephaseconsistsoftheelementsfromgroupsV,VIandVIIoftheperiodictable.Theroleof0phasewithinthesolidsolutionstrengthenedalloyistoincreasetheresistancetodislocationmotion.Themagnitudeofthestrengtheningeectof0isgovernedbythedegreeof/0mismatch.Thegreaterthemismatch,thehigheristhestrengthincrement(Fig. 1{1 showsaschematicdiagramofthe0phasewithinthematrix[ 1 ]andFig. 1{2 onpage 3 showsapictureofcuboidal0precipitateinmatrix[ 2 ]).Henceshearingoftheprecipitateisrequiredinorderforadislocationtomovethroughthematrix.Shearingofthe0precipitaterelativetothematrixrequiressignicantlyhigherenergy,resultinginastrongermaterial. NickelbasesinglecrystalsuperalloysareprecipitationstrengthenedcastmonograinsuperalloysbasedontheNiCrAlsystem.Thesealloysexhibitbetterhightemperaturepropertiesthanpolycrystallinewroughtorcastalloys(Fig. 1{3 onpage 4 ).Inhightemperatureapplicationgrainboundariesaretypicallytheweaklink,whichprovidepassagesfordiusionandoxidation,whichresultsin
PAGE 15
Figure1{2: Cuboidal0precipitates(0.350.6m)inPWA1480. failuresatthislocation.Grainboundarystrengthenersareaddedtothealloychemistrytoincreasecapability,whichresultsinloweringthemeltingpointofthealloy. Removalofgrainboundariesandgrainboundarystrengtheningelementsraisetheincipientmeltingtemperatureofthealloyby150oFandresultinimprovedhightemperaturefatigueandcreepcapabilities[ 4 ].Thisincreaseinmelttemperaturepermitshigherheattreatmenttemperaturethatinturnyieldsimprovedcreepcapability.Unlikethemorecommonlyusedpolycrystallinealloys,thesesinglecrystalsuperalloysareorthotropicandhavehighlydirectionalmaterialpropertieswithexceptionalthermomechanicalfatiguepropertiesathightemperature.Themostcommonprimarygrowthdirectionforthenickelbasesuperalloysistheh001idirection.Thisisnotonlythemosteasilygrownbutalsoisthedirectionwiththemostdesirablecombinedstrengthproperties.Thisisadvantageousbecausemanypartsarecast,orgrown,ratherthanmachinedfromalargersinglecrystalsamplewithaspecicorientation.
PAGE 16
Figure1{3: Temperaturecapabilityofsuperalloyswithapproximateyearofintroduction.[ 3 ] Figure1{4: Aeroenginebladesarenominallyorientedintheh001iorientation.[ 5 ]
PAGE 17
Currentlytheprimarydirection(growthdirection)canbecontrolledrelativelywell,butstillhasanallowabledeviationof15oinanydirection.Althoughitispossibletoreducethisdeviationbysimplydiscardingthosepartswithlargervariations,thecostinwastedmaterialswouldbetoogreatandwouldresultinasignicantpercentageofscrapedparts.Thematerialpropertiesvarygreatlyastheprimarydirectiondeviatesfromthe<001>orientationanditisimportanttoquantifytheeectofotherorientationsonmaterialproperties.Thebladesusedinaeroenginesarenominallyorientedinthe<001>orientation(Fig. 1{4 onpage 4 )asthisorientationhasthebestcombinationofmechanicalproperties.Howeverinpractice,duetothedicultiesencounteredincastingandduetocostconsiderations,bladesthatareorientedawayfromthe<001>orientationbyupto15oareused.Alsointheoperatingenvironmentthebladesaresubjectedtoloadinginavarietyoforientationsduetothehotimpinginggasescausingbendingandtorsion.Theformationofhotspotsleadstothermalgradients,whichalsocontributetomultiaxialloading.Thesecondaryorientationcanalsobecontrolledduringcasting;howevermostmanufacturerschoosetoignorethiscontroltoachievegreaterproductivity.Thematerialcanbeexaminedaftercastingtodeterminesecondaryorientation,andifaparticularsecondaryorientationhasbenecialmaterialpropertiesforthespecicapplication,thepartcanbefurthermanufacturedbycuttingfromthecastpiece. TheDeformationMechanismsofcrystallinematerialscanbedescribedasbelow:
PAGE 18
[aij]isthematrixof36elasticcoecients,ofwhichonly21areindependent,since[aij]=[aji].Dependingonthethematerialstructure,thecrystallinematerialdisplaysdierentformsofgeometricsymmetry.Thereare32formsofgeometricsymmetryofcrystals,whichcanbefurtherdividedintosevencrystalsystemscalledsyngony:1)triclinic,2)monoclinic,3)rhombic,4)tetragonal,5)trigonal,6)hexagonal,and7)cubic[ 6 ].Foranisotropicmaterialthemechanicalproperties(E;;G)arethesameateachpointofthematerialbecauseitcanhaveonlytwoindependentelasticconstantsas andtherestofthecoecientsofdeformationsarezero. [aij]=2666666666666664a11a12a12000a12a11a12000a12a12a11000000a44000000a44000000a443777777777777775(1.3) Anisotropicmaterialsubjectedtomultiaxialloading,undermechanicalequilibrium,hasthreeprincipalstresses.Theseprincipalstressesactonorthogonalplanes,whicharefreeofshearstresses.TheVonMisesandTrescacriterionsaretwoofthemostwidelyusedyieldcriteriaforductileisotropicmaterials. 1{5 onpage 11 )isgivenbyequation( 1.4 )[ 6 ]
PAGE 19
TheelasticpropertiesofFCCcrystalsexhibitcubicsyngony;i.e.,ithasthreeorthogonalplanesofelasticsymmetryateverypoint,whichiscalledorthogonallyanisotropicor,forbrevity,orthotropic.Thereforecubicsymmetrycanbedescribedwiththreeindependentconstantsdesignatedastheelasticmodulus,shearmodulus,andPoisson'sratio[ 4 ]andhence[aij]canbeexpressedasshowninequation( 1.3 ),inthematerialcoordinatesystem(FCCcrystalaxesareparalleltox;yandzcoordinateaxes)andthecoecientofdeformationisgivenas TheelasticconstantsinthegeneralizedHooke'slawofananisotropicbody,[aij],varywiththedirectionofthecoordinateaxes.Fororientationsotherthanthe(x;y;z)axes,the[aij]matrixvarieswiththecrystalorientation.Inthecaseofanisotropicbodytheconstantsareinvariantinanyorthogonalcoordinatesystem.ConsideraCartesiancoordinatesystem(x0;y0;z0)thathasrotatedabouttheoriginOof(x;y;z).Theelasticconstantmatrixa0ijinthe(x0;y0;z0)coordinatesystemthatrelatesf0gandf0gis wherea0ijisgivenbythefollowingtransformation [a0ij]=[Q]T[aij][Q]=6Xm=16Xn=1amnQmiQmj(i;j=1;2;::::::;6)(1.7) where[Q]isthetransformationmatrix,whichisafunctionofthedirectioncosinesbetweenthe(x;y;z)and(x0;y0;z0)coordinateaxes(Fig. 1{5 onpage 11 ,Table 1{1 onpage 8 ).Here,andarethedirectioncosinesofthematerialcoordinatesystemrelativetotheuniversalcoordinatesystem(specimencoordinatesystem).
PAGE 20
[Q]=266666666666666421222322323121221222322323121221222322323121211223323+3213+3112+2111223323+3213+3112+2111223323+3213+3112+213777777777777775(1.8) Table1{1: Directioncosinesofmaterialcoordinateaxes(xyz)withuniversalcoordinateaxes(x0y0z0) y z x0
PAGE 21
where [Q]=266666666666666421222323311221222323311221222323311221122223323+3213+3112+2121122223323+3213+3112+2121122223323+3213+3112+213777777777777775(1.10) Thetransformationmatrix[Q]isorthogonalandhence[Q]1=[Q]T=[Q0]. Nowtheresolvedshearstressesonthe12primaryoctahedralslipsystems(Table 1{2 onpage 16 ,Fig. 1{8 onpage 14 ),denotedby1;2;:::;12canbereadilyobtainedusingthetransformationgivenbyequation( 1.13 )[ 4 ].Onlytheprimaryoctahedralslipsystemsgetactivatedatroomtemperaturewhereashightemperatureisrequiredfortheactivationoftheotherslipsystems. Athighertemperatures,deformationoccursbydislocationclimb(whichisadiusioncontrolledprocess).Twinning,arotationofatomsinthelatticestructure,isnotasimportantasstrainsproducedbythismechanismareverysmallascomparedtoslipandclimb.Onanatomicscale,plasticdeformationinvolvesthenetmovementoflargenumbersofatomsinresponsetoanappliedstress.Actualstrengthtodeformametalismuchlowerthanthatpredictedfortheoreticallyperfectcrystals.Thisdierenceisexplainedby
PAGE 22
dislocations.Plasticdeformationcorrespondstothemotionoflargenumbersofdislocations.Theprocessbywhichplasticdeformationisproducedbydislocationmotioniscalledslip.Slipinmetalcrystalsoftenoccursonplanesofhighatomicdensityincloselypackeddirectionsbecausetheseplanesgenerallycorrespondtothelowestpossibleenergyforslip.Theplaneonwhichthemotionoccursiscalledslipplane.Thereisapreferredplane(slipplane)andapreferreddirectionwithinthatplane(slipdirection).Thecombinationofslipplaneandslipdirectioniscalledaslipsystem.Thepreferredslipplaneisthatwiththegreatestplanardensity.Thepreferredslipdirectionisthatwiththegreatestlineardensity. Slipmechanism .Inasinglecrystalspecimen,theextantofslipdependsonthemagnitudeoftheshearingstress.Magnitudeoftheresolvedstressdependsonthemagnitudeoftheappliedforce,itsorientationrelativetotheslipplaneandslipdirection.Thereforethestressstrainbehaviorofamaterial,whichisafunctionofthenumberofactivatedslipsystems,varieswithorientation.AsliplineorstepisobservedinexperimentsinvolvingpolishedsinglecrystalsspecimensasshowninFig. 1{7 onpage 12 .TheslipplanesofhighestatomicdensitiesarethersttogetactivatedonanFCCsinglecrystalspecimenandarecalledtheprimaryoctahedralslipsystems. InFig. 1{6 onpage 11 ,Fistheappliedforce,Aisthecrosssectionalareaofthespecimen,istheanglebetweenthetensileaxisandthenormaltotheslipplaneandistheanglebetweentheslipdirectionandthetensileaxis.Theresolvedshearstressalongtheslipdirectioncanbecalculatedas Thefavoredslipsystemhasthelargestresolvedshearstresscalledcriticalresolvedshearstress.Ithasbeenobservedexperimentallythatasinglecrystalwill
PAGE 23
Figure1{5: Materialcoordinatesystem(xyz)relativetouniversalcoordinatesystem(x0y0z0) Figure1{6: Twoseparatedportionsofasinglecrystalshowingamodelforcalculatingtheresolvedshearstressinasinglecrystalspecimen.[ 7 ]
PAGE 24
Figure1{7: SlipLinesandFracturePlaneinexperimentaltensiletestspecimentestedbyMaterialsScienceandEngineeringDepartment,UF.[ 8 ] slipwhentheresolvedshearstressontheslipplanereachesthecriticalresolvedshearstressforthatmaterial.Tocalculatethestressneededtoexceedthislevel(CRSS)iscalculatedby=CRSS willbeaminimumwhen==45o ThisbehaviorisknownasSchmid'slaw. Thevalueofcriticalresolvedshearstressdependschieyonthematerialcompositionandtemperature.Itisalsoafunctionofappliedloadanddirection,crystalstructureandspecimengeometry.DuringtheapplicationofaloadtoanFCCsinglecrystalspecimen,therstplanestogetactivatedaretheplanesofhighatomicdensitiesandcalledtheprimaryoctahedralslipsystems.Eachoctahedralplanehassixslipdirectionsassociatedwithit.Threeofthesearetermedeasysliporprimaryslipdirectionsandtheotherthreearesecondaryslipdirections.Thus,thereare12primaryand12secondaryslipdirectionsassociatedwiththefouroctahedralplanes(Fig. 1{8 onpage 14 ).Inaddition,therearesixpossibleslipdirectionsinthethreecubeplanes,asshowninFig. 1{9 onpage 15 .Thereare30possibleslipsystemsinanFCCcrystal(Table 1{2 onpage 16 ).
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Theresolvedshearstressonthe12primaryslipsystems,basedonkinematicrelations,canbeshowntobe
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Figure1{8: Primary(Closepack)andSecondary(Nonclosepack)slipdirectionsontheoctahedralplanesforanFCCcrystal.[ 9 ]
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Figure1{9: CubeslipplanesandslipdirectionsforanFCCcrystal.[ 9 ]
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Table1{2: Directioncosineof(x0;y0;z0)with(x;y;z)coordinateaxes,whenx0axisisalignedalong[213]orientation. SlipNumber SlipPlane SlipDirection OctahedralSlipa/2f111gh110i 123456789101112 (111)(111)(111)( 11 1)( 11 1)( 11 1)(1 1 1)(1 1 1)(1 1 1)( 1 11)( 1 11)( 1 11) [10 1][0 11][1 10][10 1][110][011][110][0 11][101][011][101][1 10] OctahedralSlipa/2f111gh112i 131415161718192021222324 (111)(111)(111)( 11 1)( 11 1)( 11 1)(1 1 1)(1 1 1)(1 1 1)( 1 11)( 1 11)( 1 11) [ 12 1][2 1 1][ 1 12][121][1 1 2][ 2 11][ 11 2][211][ 1 21][ 21 1][1 2 1][112] CubicSlipa/2f100gh110i 252627282930 (100)(100)(010)(010)(001)(001) [011][01 1][101][10 1][110][ 110]
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Stressintensityfactor(SIF)aboutacracktipplaysasignicantroleinthepropagationofthecrack.TheSIFisameasureofintensitynearthecracktipunderlinearelasticconditions.TheknowledgeofSIFisnecessarytopredictthegrowthofafatiguecrackortodeterminetheresidualstrengthofacrackedstructure.Thisfactorcharacterizestheintensityofthestresseldintheneighborhoodofthecracktipanddependssubstantiallyonspecimengeometry,materialproperties,externalloadsandcracksize.IncaseswithidealizedgeometryandloadstheSIFcanbefoundinhandbooksforisotropicelasticmaterials.InthecaseofmorecomplexstructurestheSIFhastobecomputedbynumericalmethods. Isotropicmaterialshaveuniformmaterialpropertiesinalldirections,whereasanisotropicmaterialsaredirectionalinnature(i.e.,theyhavedierentmaterialpropertiesindierentdirections).BecauseofthispropertySIFisdirectioninvariantforisotropicmaterialswhileSIFvariesinanisotropicmaterialsduetochangeinmaterialcoordinatesystem,whilekeepingalltheotherparameterssame. AstudyoftheeectofanisotropyandinhomogeneityontheSIFusingtheFEAwasdonebyKamayaandKitamura[ 10 ].Itwasfoundthatinapolycrystalspecimen,theSIFisinuencedbythecrystalorientationatthecracktipandthedeformationconstraintduetoadjacentcrystalsonlyforverysmallcracks;whereasforanisotropicsinglecrystaltheSIFstronglydependsonthecrystalorientation,andthemagnitudedeviatesfromthatintheisotropicbodyforanysizecrack.But 17
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ascracksizeincreasesinthepolycrystalmaterials,SIFtendstoconvergetovaluesseeninhomogeneous/isotropicbodies. Manymethodshavebeenproposed,tocalculateSIFsforcrackssubjectedtomixedmodeloadingconditionsinisotropicelasticsolids.SomecommonlyusedmethodsareJintegral[ 11 12 ],virtualcrackextension[ 13 14 ],modiedcrackclosureintegralanddisplacementextrapolationmethods[ 15 ]etc.Noneoftheseproposedmethodsareabletoprovidethecompletesolutionforallthethreemodes(ModeI,IIandIII)ofSIFforanisotropicmaterial. Atkinsonetal.[ 16 ]presentedtheideaofcalculatingmixedmodeSIFusingFredholmequationtransformation.TheyusedacentercrackedBraziliandisc(BD)testspecimenmadeofisotropicmaterial.ThemodemixityratiofortheBDspecimenisafunctionofthecrackanglewithrespecttotheloadvector.Resultswerepresentedforvariedcrackanglesandhencemodemixityratios.SmallcrackapproximationwasalsotakenandtheresultswerefoundtobeinaccordancewithAwajiandSato[ 17 ],butitdidnotincorporateanisotropyinthemodelandwaslimitedtoModeIandIISIFs. SuandSun[ 18 ]studiedvariouskindsof2DanisotropiccrackstoevaluateSIFundermixedmodeloadingcondition.Fractalniteelementmethod(FFEM)[ 19 ]wasusedtocalculateModeIandIISIFsfor2Danisotropicplate.ThevariationoftheSIFswithmaterialpropertiesandorientationsofacrackwaspresented.ItwasshownthatSIFswerenotsensitivetothevariationofshearmodulus.HwuandLiang[ 20 ]usedremoteboundarydatatocalculateSIFsfor2Danisotropicmaterial.IteliminatedtheerrorinSIFcalculation,causedbyabruptchangeinthestressesnearcracktips,byndingequivalentformulationforSIFbyusingonlyremoteboundaryresponses(displacements,stressesandstrains),cooperatingwiththenecessarygeometricdata.Aspecialboundaryelementwasdevelopedwhichremovedtherequirementofmeshingaroundthecrackboundary.Through
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thisboundaryelement,alltheinternalstressesandstrainscouldbeexpressedintermsofdisplacementsandtractionsontheboundariesexcludingthehole,crackandinclusionboundaries.Theseresultscouldbeappliedtoanykindoflinearanisotropicmaterialsbutwasrestrictedtothe2Dproblemswhichincludedgeneralizedplanestress,generalizedplanestrainandantiplaneproblems. DendaandMarante[ 21 ]developedacracktipsingularelement(CTSE)forthegeneralanisotropicsolidsin2Dwiththebuiltinp 22 ]forcombinedmode(IandII)SIFsincaseofplanar,rectilinear,anisotropicstructuresusinga12nodesingularniteelementwasaccurate,butwasrestrictedtoa2Dmodel.Likewise,SosaandEischen[ 12 ]calculatedSIFforaplatecontainingathroughcracksubjectedtobendingloadsusingtheJintegral.TwoDeightnodeelementwasselectedforthispurposewhereKII=KIII=0.MewsandKuhn[ 23 ]usedtheGreen'sfunctionapproachtocalculatemixedmodeSIFwithoutanycrackdiscretizationinanisotropicplate,whichusedanasymptoticdisplacementeldatthecracktip.TheycalculatedSIFsforplatehavingmultiplecracks,forvariousinclinations,andfoundSIFsveryclosetothatofShihetal.[ 24 ].Ishikava[ 14 ]presentedtheideaofthestrainenergyreleaserate(virtualcrackextensionmethod)tocalculateSIFformodeIandII,anddescribedSIFcalculationbasedononlyoneuseofvirtualcrackextension. HuangandKardomateas[ 25 ]describedthecontinuousdislocationtechniquetocalculatemixedmodestressintensityfactorsinananisotropicinnitestrip.ThismethodwaslimitedinitsusetoModeIandIISIFs,anditwasfoundtobe
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mostsuitableforcracksofrelativelysmalldimensions.ThemethodwasveriedbycalculatingSIFsforisotropicmaterialandcomparingtheresultswithreadilyavailableformulasgiveninTadaetal.[ 26 ],whichresultedinexcellentagreement.Sunetal.[ 27 ]usedtheboundaryelementmethod(BEM)toanalyzecrackedanisotropicbodiesunderantiplaneshear.Thenewboundaryformulationuseddislocationdensityasanunknownonthecracksurface,andKIIIwasdeterminednearthecracktip.Theequationandmethodcouldbedirectlyusedforantiplaneproblemswithcracksofanygeometricshapes.Itdidnotgiveacompletesolutionundermixedmodeloadingconditions,butitdidgiveanideaaboutthebehaviorofKIIIunderantiplaneshearloading. Shihetal.[ 24 ]havecalculatedSIFsfor2DisotropicmaterialsusingquarterpointelementnodaldisplacementsatthecracktipbasedonFiniteElementMethod(FEM).TheModesI,IIandIIIhavebeendecoupledbecauseoftheisotropicnatureofthematerial.Sihetal.[ 28 ]denedSIFsasafunctionofstressatthecracktip.FollowingtheworkofShihetal.,IngraeaandManu[ 29 ]showedhowtocomputeSIFfrom3Dquarterpointnodaldisplacement,forcrackedisotropicelasticbodiesforallthreemodes.Theyusedaquarterpointisoparametricelement,whichhasbeenaccurateincomputingSIF[ 30 ].SaoumaandSikiotis[ 31 ]introducedanisotropyintheabovemodel[ 29 ]andproposedamethodtocalculateSIFfor3DanisotropicelasticmaterialbasedonthemodelofShihetal.ThecomputedSIFs,whencomparedwith2Danisotropicbodieswithknownexactsolutionsyieldedanerrorof612%forKIandKII.Howevertheorientationoftheelasticconstantswasnotincorporated.TheexpressionofKIIIwasnotcorrectasitresultedinalmostzerovalue,irrespectiveofthegeometry,orientationandtheloadappliedtoananisotropiccomponent. Dhondt[ 32 ]analyzedtwomethods,interactionintegralmethod(IINT)andthequarterpointelementstressmethod(QPES),tocalculateSIFforsingleedge
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notchspecimen(SEN)andaslantcrackina3pointbending(3PB)specimenforisotropicmaterial.FortheregularmeshintheSENspecimenbothmethodsyieldedsimilarresults.Fortheirregularmeshinthe3PBspecimentheQPESmethodsseemedtobemorerobustandaccurate.LaterGuido[ 33 ]presentedamethodtocalculateSIFforanisotropicmaterialusingthenitedierencemethodalonganarbitrarycrack.Althoughthemodelwasmorerobusttoirregularmeshesincomparisontointeractionintegralmethod,thismethodwaslimitedtosingleedgespecimenandcornercrackspecimen. Tweedetal.[ 34 ]usedthespeciccaseofanedgecrackofananisotropicelastichalfspaceundergeneralizedplanestrainconditionstodeterminethestressintensityfactorsusingintegraltransformtechniques.PanandYuan[ 35 ]usedsingledomainBEMtocalculatedmixedmodeSIFsforbothboundedandunbounded3Danisotropiccrackedsolids.Denda[ 36 ]usedBEMtodeterminemixedmodeSIFs(KI,KIIandKIII)of3Danisotropicmaterialwithmultiplecracks.ItaddressedtheissueofcouplingeectofthethreemodesoffracturecontrolledbyModeI,IIandIIISIFs.Thoughveryaccurate,theseformulationswerebasedontheplanestrainassumption. AlthoughasubstantialbodyofliteraturedescribescomputationofSIF,ageneralizednumericalsolutiontocalculateSIFfor3Danisotropicmaterialisunavailable.Theobjectiveofthepresentworkistomodela3DorthotropicspecimenhavingathroughcrackandtocalculatetheSIFsforallthreemodesassuminglinearelasticpropertiesatthecracktip.AmathematicalmodelhasbeendevelopedtocalculatemixedmodeSIFforFCCsinglecrystalorthotropicmaterialfordierentorientations,whichisalsoapplicabletogeneralizedanisotropicmaterial.Lookingatthecracktipnodaldisplacementsitispossibletocalculatestressintensityfactorforanycrystallographicorientationofthematerial.Itwasobservedthatthematerialorthotropyresultsincoupledcracktip(x,y,
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z)displacements,leadingtotheinterdependenceofSIFsforModesI,II,andIII.ResultsarepresentedforacentercrackedBDspecimen,withtwospeciccrystallographicorientations.Thecrackplanefortherstspecimenis(111)andthecrackdirectionis10 1.Forthesecondspecimenthecrackplaneisagain(111),whilethecrackdirectionis 1.ThereareimportantreasonsforchoosingthesetwospecicBDspecimencongurations.Thesetwocrackdirectionstypicallyrepresentthefastestandtheslowestcrystallographiccrackgrowthrates,respectively,onthef111gfamilyofoctahedralplanes,andhencehaveimportantimplicationsonestimatingfatiguecrackgrowthlifeforsinglecrystalcomponents[ 2 37 ].Evenifcracksnucleateonotherplanes,becauseoflocalinuencefromintrinsicdefectssuchasmicropores,carbideparticlesandundissolvedeutectics,theytendtomigratetotheoctahedralplanes,intheprimaryslipdirections(h101ifamilyofdirections),sincetheyrepresentthepathsofleastresistanceforcrackpropagation[ 2 ].ResultspresentedshowthatSIFvaluesareconsistentlyhigherfortheh101icrackdirection,comparedtotheh121idirection,forsamecrackanglesandloads.
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Thestressintensitysolution,forallthethreemodesforisotropicmaterials,usingcracktipnodaldisplacementsmethodisgivenby[ 38 ]: +1r L[4(vBvD)+vEvC]KII= +1r L[4(uBuD)+uEuC] (3.1) +1r L[4(wBwD)+wEwC] Hereu,vandwarethedisplacementsofthenodesB,C,DandEatthecracktipalongx;yandzdirectionsrespectively.Listhelengthoftheelementatthecracktipnormaltothecrackfront(Fig. 3{1 onpage 24 ). TheseequationsshowthatallthreestressintensityfactorsKI,KIIandKIIIaredecoupled.Foranisotropicmaterialsitisnotpossibletousethesameequation,astheirmaterialpropertiesaredirectiondependent.Becauseofthisdirectionaldependencethestressintensityfactorforthesamematerialchangesaccordingtotheorientationofthecrackplanewithmaterialorientation. Ashasbeendiscussedearlier,anorthotropicmaterialhasthreeindependentelasticconstants,E;andG.Theelasticconstantsinmaterialcoordinatesystemgettransformedtospecimencoordinatesystembyequation( 1.7 ).Theinterdependenceofdisplacementsinanisotropicmaterialsduetoshearcouplingresultsincoupledstressintensityfactors. Anisotropywasincorporatedinthe3DmodeltocalculateSIFformixedmodeloading.Itcanbeshownthatmixedmodestressintensityfactorscanbe 23
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Figure3{1: CracktipnodaldisplacementofisotropicelasticmaterialforcalculationofStressIntensityFactorsofModeI;IIandIII 3.2 )[ 28 ][ 29 ] WhereL1istheelementlengthalongthecrackfaceandfAgisgivenas 2(4uB+uC+4uEuF)+1 22(uF+uC2uD)2vBvC+2vEvF+vD+1 2(4vB+vC+4vEvF)+1 22(vF+vC2vD)2wBwC+2wEwF+wD+1 2(4wB+wC+4wEwF)+1 22(wF+wC2wD)9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;(3.3)
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Hereu;vandwarethenodaldisplacementsofnodesB,C,D,EandFatthecracktiprelativetoB',C',D',E',andF'asshowninFig. 4{3 onpage 34 L2+1)(3.4)L2isthelengthoftheelementalongthecrackfront(Fig. 4{2 onpage 33 )and[B]1isdenedas[B]1=266664Re[i 12(q2q1)]1 12(p2p1)]1 12(1q22q1)]1 12(1p22p1)]1 Asitisapparentfromthe[B]1matrixthatKIandKIIareonlycoupledSIFs(functionofuandv)andKIIIisfunctionofwonly. 12(1p22p1)]Re[i 12(p2p1)]Re[i 12(1q22q1)]Re[i 12(q2q1)](3.6)1and2aretherootsofthefourthordercharacteristicequation andgivenby 6 ] 6 ],refertoAppendixA.
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ForPlanestress, ForPlanestrain, Equation( 3.2 )isusedtocalculatetheSIFatthecracktipbydisplacementmethod;oneofthemostusedmethodstogetthevalueaccurately.FiniteElementMethodwasusedtocalculatethedisplacementsatthecracktip.Thecommercialsoftware,ANSYS,wasusedforFEAmodeling.ThecracktipnodaldisplacementswerethenextractedfromFEAmodelandfedtotheanalyticalequationsexplainedabovetocalculateallthethreemodesofSIFs. SingleCrystalSpecimenGeometriesUsedforMixedModeLoading .Twospecimengeometrieswereusedtoinvestigatetheeectsofmodemixityatthecracktip.Onewasarectangulartensionspecimen(Fig. 3{4 onpage 29 )withacentercrackloadedsuchthatthecrackliedinthef111gplane.Thecrackdirectionsusedwereh101iandh121ifamilyofdirections. ThesecondspecimenmodeledisaroundBraziliandisk(BD)specimen,loadedincompression.Thisspecimenwithcentercrackhasamodemixityatthecracktip,whichvariesasfunctionofthecrackangle'',showninFig. 3{5 onpage 29 .Thecrackliesonthef111gplaneandcrackdirectionsusedareh101iandh121ifamilyofdirections.Thesespecimenandcrackorientationshavebeencheckedverycarefully,basedonexperimentallyobservedfatiguecrackgrowthrates(FCGR)[ 39 ].Atlowtemperature(<4270C),thedominantmodeofFCGinFCCsinglecrystalsuperalloysiscrystallographiccrackpropagationonoctahedralplanes
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Figure3{2: Theh101iandh121ifamilyofslipdirectionsaresuperimposedonf111gplaneshowinghowthehexagonalcrackfrontisdelineatedbytheslipdirections.[ 37 ] [ 37 ].ExtensiveFCGRstudieshaveshownthatthefastestcrackgrowthratesareobservedforcrackspropagatingonthef111gplane,intheh101ifamilyofdirections.AslowerFCGRhasbeenobservedwiththecrackonthef111gplane,intheh121ifamilyofdirections.HencetheFCGRonthef111gplane,inthesetwodirectionstypicallyprovidethefastestandtheslowestcrystallographiccrackpropagationrates.ThisvariationinFCGRintheh101iandh121idirections,onthef111gisillustratedindramaticfashioninFig. 3{2 onpage 27 .Apennyshapedcrackonthef111gplaneevolvedintoahexagonalcrackfront,becauseofdierentcrackgrowthintheh101iandh121idirections.Thesesixfastestcrackgrowthdirections(hexagonalcorners)werefoundtobeh101ifamilyofslipdirectionsonf111gplane. DetailsofthetwospecimensanalyzedarepresentedinTable 3{1 onpage 28 Where,
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Figure3{3: Braziliandiskhavingcentercracklyinginf111gslipplaneandalignedalong 1direction Table3{1: Geometricalandmaterialpropertiesofthetwospecimensanalyzed RectangularSpecimen'A'(Fig. 3{4 ) BrazilianDiskB(Fig. 3{5 ) 2a=W 2a=W Fig. 3{3 onpage 28 showsthatthecrackplanelieson(111)planeandisdirectedalong 1direction.Thisslipsystemhasbeenexplainedwiththehelpofoctahedralslipplanes.
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Figure3{4: Rectangularspecimen`A'meshedwithsolid95element(left)andthegeometricaldetails(right) Figure3{5: Braziliandisk(Specimen`B')withcentercracklyingin(111)planeandorientedalong10 1direction
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InordertostudytheeectoforthotropyonSIFs,specimenswithgeometriesmentionedinTable 3{1 onpage 28 weremodeled,meshedandanalyzedfordierentcracklengths,crackangles,materialpropertiesandcrystalorientations.Arectangularspecimenwithisotropicpropertiesandwithoutanycrackwasmodeled,anditsaccuracywasascertained.AfareldstresswascalculatedanalyticallyandwasfoundtobematchedwithmeshedFEAresult.ThesamemodelwasusedwithathroughthicknesscentercrackandSIFatthecracktipwascalculatedandveriedwithstandardtable. ABraziliandiskwithcentercrackhavingisotropicpropertieswasmodeledandcomparedwiththeSIFsextractedfromFEAmodel.ThevalueswereveriedwiththeavailabletabulatedvaluesfortheisotropicBraziliandisk.Equation( 3.1 )wasusedtocalculatetheSIFstoverifytheequationusedbyFEMsoftware.Asexpected,theFEAresultswerefoundtobeinexcellentagreementwithpublishedresults[ 26 ].TheBraziliandiskwasselectedbecauseofitsabilitytovarymodemixityatthecracktipbyvaryingloadangle''. Thenextmodelingwasdoneusingtheorthotropicrectangularspecimenwithsingleedgecrackandthroughthicknesscentercrack.TheorthotropicstressintensityfactorscalculatedthroughanalyticalmethodwerealsocomparedwithisotropicresultsobtainedbyFEAsoftware.Thedierencebetweentheseresultsarediscussed.LikewisetheorthotropicBraziliandisk,withcentercrack,wasmodeledandacomprehensiveparametricanalysiswasperformed. Theanalysiswasdoneforthefollowingcases: 30
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Figure4{1: Braziliandiskmeshedwithtriangularelementhavingisotropicproperties(left)andzoomedviewofcracktip(right). 3{5 onpage 29 3{3 onpage 28 ). AfterdeningthegeometryofthematerialinFEAsoftware,freemeshingwasdoneononeofthesurfaces(frontorback),andwassweptthroughthethickness.Themeshdensitywasoptimizedtosavetimeandeort(Fig. 3{4 onpage 29 ).Thecracktipelementsizewasalsovariedtochecktheaccuracyoftheresult.Oncestandardized,thesamemodelwasusedforvariousotheranalyses.TheFEAmodelwasmeshedusingsolid95element(Fig. 4{1 ),whichisaquadraticisoparametricbrickelementwith8cornernodesand12midsidenodes(Fig. 4{2 onpage 33 ).Thedetailsofthiselementhavebeendiscussedinthefollowingsection.
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CrackTipElement .Anumberofspecialcracktipniteelementshavebeendevelopedapplicabletothedisplacementmethod,alsothehybridmethodhasbeenusedindevelopingspecialcracktipelements.Thesespecialcracktipelementscontainasingularityothestraineldatthecracktipequaltothetheoreticalsingularity.Fortwodimensionalgeometrythe8nodequadraticisoparametricelementcanbemadetosimulatethe1=p 40 ].LynnandIngraea[ 41 ]haveshowncomputationofSIFforisotropicmaterialwithhighaccuracycanbeachievedbyplacingmidsidenodesbetweenquarterandmidpointoftheelementsatthecracktip.Thisinformationfurtherledtothedevelopmentofthetransitionelement,whichresultedinimprovedSIFcalculationswhentheratioofthequarterpointelementandcracklengthisdecreased. Quadraticisoparametricelementswith20nodeshaveproventobeexcellentelementsforthecalculationofSIFforelasticcrackproblems[ 31 ],whenthemidsidenodesareputnearthecracktipatthequarterpoint.Alsothiselementhasinversesquareroot(1 42 ]hasshowntheeectofquarterpointsingularandtransitionelement'ssizeonseveralSIFcomputationtechniques.ItwasdeterminedthatcomputedSIFwasnotsensitivetothesingularelementinternalangleandsizeexceptforthequarterpointdisplacementtechniqueandthedisplacementcorrelationtechnique.Thereforenermesharoundthecracktipgivesbetterresultsforthequarterpointdisplacementmethod. Twodimensionaltriangularelements[ 30 ],formedbycollapsingonesideofrectangularelements,givebetterresultsforstressintensityfactor.Also,ithas
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Figure4{2: (a)20nodeisoparametricelementinnaturalcoordinatesystemand(b)Quarterpointsingularelementwiththe=1facecollapsedinlocalCartesiancoordinatesystem.[ 31 ] beenfoundthatrectangularquadraticisoparametricelements(andbrickelements)havesingularstiness(singulartotalstrainenergy)atthecracktip[ 30 ].Forthethreedimensionalcase,20nodeisoparametricelementswereused.20nodebrickelementscollapseandformwedgeshapedelementsatthecracktip.OneoftheelementfacescollapsesatthecracktiptoformthewedgeshapeasshownintheFig. 4{2 .Fig. 4{2 (a)showsa20nodeisoparametricelementinuniversalcoordinatesystem,andFig. 4{2 (b)depictsitsquarterpointsingularversionwiththe=1facecollapsedinlocalCartesiancoordinatesystem. ThiselementisthebestcracktipelementforFEAsolutionofmixedmodeSIFanalysisoforthotropicmaterials[ 24 ][ 42 ].AroundthecracktipthesizeoftheelementsshouldbeverysmalltoreducetheerrorincalculatingSIFs.Atypicalexampleofcracktipelementlength(L1)is0:002in.TheenlargedgureofthemeshingaroundthecracktipcanbeseeninFig. 4{4 onpage 34 .Allthenodeshadtobealignedalongthematerialcoordinatesystemtogetthedesiredresult. Twodierentorientationsh101iandh121i(onf111gplane)wereusedtoanalyzetheeectonSIF.Thecrackliesonthef111gplane(Fig. 3{1 onpage 24 ),andthecrackdirectionwasalignedalongfamilyofeitherh101iorh121idirections.
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Figure4{3: Arrangementofquarterpointwedgeelementsalongsegmentofcrackfrontwithnodalletteringconvention.[ 31 ] Figure4{4: ThesymmetricmeshingofBDspecimenwithsolid95elementandenlargedpictureofcracktipelements
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Theresolvedshearstressinthedirectionofh101iismaximumandinthedirectionofh121iisminimumonthef111gplaneastheSchmid'sfactorismaximumalongh101iandminimumalongh121idirection.Sincethemaximumandminimumresolvedshearstressoccuralongtheh101iandh121idirectionsrespectively,thesedirectionsareafamilyofprimaryor`easyslip'directionsandsecondaryor`hardslip'directions,onthef111gplane.
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CalculationofSIFforanisotropicmaterialsisacrucialstepinordertopredicttheFCGbehaviorofsinglecrystalsuperalloys.ThenumericalproceduredevelopedhereisnotmerelyspecictotheFCCsinglecrystal.ItcanalsobeusedforcomputingtheSIFforanygeneralizedanisotropicmaterialsubjecttomixedmodeloading.Thedisplacements,calculatedusingFEA,wereappliedtotheSIFmodeldevelopedfortheanisotropicmaterial(inourcase,orthotropicmaterial).Itisevidentfromequations( 3.2 )and( 3.3 ),thatthecalculatedSIFsareafunctionofthecracktipnodaldisplacements,whichareafunctionofthestressesdevelopedaroundthecracktip,whichinturnareafunctionoftheappliedforce,geometryofthespecimenandtheorientationofthematerialcoordinatesystemwithrespecttothecrackcoordinatesystem. Thelengthoftherectangularspecimen(Specimen`A')wassucientlylongincomparisontothethicknessandwidthofthespecimen,thusyieldingauniformfareldstress.Therefore,aplanestrainassumptionwasusedinequation( 3.10 )inordertocalculatetheSIFatthemidplaneoftherectangularspecimen.Thevalidityofthemodelwasrstcheckedwithisotropicmaterialproperties.TheanalyticalmodelresultedinKI,KIIandKIIIvalues,whichwereidenticaltotheFEAresultsbasedonisotropicassumptions. Afterverifyingthemodel,theeectofchangeinratio(from10%to80%)ofcracklengthandwidth(2a=W)wasexaminedonSIFsforspecimen`A'(Fig. 3{4 onpage 29 ).ThenormalizedvaluesofSIFshavenotbeenusedbecausetheFracturetoughness,Kc,forthegivenmaterialisnotcharacterizedunambiguously,asafunctionoforientation.Forbothoftheabovementionedorientations,a 36
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Figure5{1: 1and 1orientationofSpecimen'A'at=00. gradualincreaseinKIwasobservedwithanincreaseinthe2a=Wratio.AlsothevaluesofKIforthe10 1orientationwerealwaysfoundtobegreaterthanthecorrespondingvaluesforthe 1orientation(Fig. 5{1 onpage 37 ),atthesamefareldstress.ThesametrendwasalsoobservedforKII,thoughthevalueswereverysmallforboththeorientations(Fig. 5{2 onpage 38 ).Thisisbecausetheloadingisnormaltothe(111)plane,thusinducingzeroshearintheplaneofthecrack.ThereforeKIIisexpectedtobeverysmallandwithinthenumericalcalculationerror.ThevaluesofKIIIforthe10 1orientationarenegative(Fig. 5{3 onpage 38 )becausetherelativedisplacementofthetwocrackfacesalongthezdirection(Fig. 4{3 onpage 34 )wasnegative.Thisissimplyaconsequenceofsignconventiondenition.However,themagnitudeoftheSIFfor10 1isfoundtobegreaterthanfor 1.Thisvalidatesthattheh101ifamilyofslipdirectionsareprimaryoreasyslipdirectionsascomparedtothoseofh121i. ThechangeinSIFswiththecrackangle''wasanalyzedforbothorientations.KIdecreasesforbothorientationswithincreasingcrackangleasisevident
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Figure5{2: 1and 1orientationofSpecimen'A'at=00. Figure5{3: 1and 1orientationofSpecimen'A'at=00.
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Figure5{4: 1and 1orientationofSpecimen'A'at2a=W=0:4. Figure5{5: 1and 1orientationofSpecimen'A'at2a=W=0:4.
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Figure5{6: 1and 1orientationofSpecimen'A'at2a=W=0:4. fromtheplots(Fig. 5{4 onpage 39 ),butthevalueofKIisalwaysgreaterfor10 1thanthatforthe 1orientation.Forisotropicmaterial,KIbecomeszerouponcrackclosureatacertainangle.Foranorthotropicmaterial,itcanbeseenthatKIisstillnonzeroduetothecouplingofthenodaldisplacements.ThevalueofKIIinitiallyincreaseswithanincreaseinduetoanincreaseinshearstress,whichafterreachingamaximumvaluestartsdecreasing(Fig. 5{5 onpage 39 ).ThevalueofKIIIforthe10 1orientationisnegative(Fig. 5{6 onpage 40 )astherelativenodaldisplacementsofthecrackfacesalongthezdirectionwerenegative.However,itwasobservedthatthemagnitudeofKIIIfor10 1wasalwaysgreaterthanforthe 1orientation. FromtheBrazilianDiskspecimen(Specimen'B'),itcanbeseenthatthemagnitudeofSIFforthe10 1orientationisalwaysgreaterthanthecorrespondingvaluesofthe 1orientation,thusenablingacracktomovefasteronthe10 1planethanonthe 1plane(Figs. 5{7 5{8 onpage 42 5{9 5{10 onpage 43 5{11 5{12 onpage 44 ),underfatigueloadingforsame4Kvalues.Eventhough
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Figure5{7: 1and 1orientationofSpecimen'B'at=00. themagnitudeofKIIforthe 1orientationisgreaterthanthatforthe10 1orientation,itmaynotaecttheeectiveSIFverymuch,asthemagnitudeofKIIIforthe10 1orientationisfargreaterthanthatforthe 1orientation. ThevaluesofKIforthe10 1orientationrapidlydecreaseascomparedtothe 1orientationandreachcrackclosureangleataround180,whereasthecrackclosureangleis300forthe 1orientation.ThemagnitudeofKIIfor 1isalwaysgreaterthanthatforthe10 1orientation,butthedierenceisfarlessincomparisontothemagnitudeofKIIIforthe 1and10 1orientations.FollowingtheworkofSaumaandSikiotis[ 31 ],thecalculatedvaluesforKIIIwerefoundtobenegligibleandaftermuchanalysisandrederivingofrelevantequations,theequationgivenbySaumaandSikiotiswasfoundtobewronglyformulated.Ontheotherhand,equation( 3.5 )correctlygaverisetoallthethreemodesofSIFs.Asaresultofthecouplingofdisplacementsatthecracktipduetoanisotropy,nonnegligiblevaluesofKIIIwerefound,andtheyvariedwithrespecttotheappliedforce,cracklengthandcrackangle.
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Figure5{8: 1and 1orientationofSpecimen'B'at=00. Figure5{9: 1and 1orientationofSpecimen'B'at=00.
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Figure5{10: 1and 1orientationofSpecimen'B'at2a=W=0:55. Figure5{11: 1and 1orientationofSpecimen'B'at2a=W=0:55.
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Figure5{12: 1and 1orientationofSpecimen'B'at2a=W=0:55. TheBDspecimenSIFs(KI,KIIandKIII)werefurtheranalyzedalongthethicknessforthe10 1and 1crackorientationswithchangeincracklengthandcrackangle.Theywerecalculatedat5dierentpointsalongthethicknessatthecrackfrontasshownintheFig. 5{13 onpage 45 .AplanestressassumptionwasmadeatthesurfaceoftheBDspecimen(Plane1&5)andaplanestrainassumptionwasmadeatalltheinteriorpoints(Plane2,3&4). InordertocheckthedierenceinSIFofisotropicBDspecimenandorthotropicBDspecimenwithchangeinloadanglewithcrack,undersimilarloadingconditionandspecimengeometry,twomodelswereanalyzed.TheorthotropicBDspecimenanalyzedhadthematerialcoordinatesystemalignedwithspecimencoordinatesystem.ThereforeasymmetrywasexpectedintheSIFvaluesacrossthethickness(Figs. 5{15 onpage 47 5{17 onpage 48 5{19 onpage 49 ).Forisotropicmaterial,propertiesofpurenickelwasused.ThetrendsofKI,KIIandKIII,becauseofchangeinforceanglewithcrack,fororthotropicBDspecimenandisotropicBDspecimenwerefoundtobesimilar,butthemagnitudeofSIFs
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Figure5{13: HalfmeshedmodelofBraziliandiskspecimenandthecrackcoordinatesystem forboththespecimenweresignicantlydierent,becauseofthematerialproperties.Thecrackclosureanglewasthesame(200,Figs. 5{14 5{15 onpage 47 ).NegativeKIwasfoundbecausenocontactelementswereusedatthecrackinterface.Afterreachingthecrackclosureangle,furtherincreaseinforceangleleadtopenetrationofcrackfaces,resultinginnegativeKI.Attheedges,KIwaslessthanthattowardsmidplane(Figs. 5{14 5{15 onpage 47 ).KI;maxwasfoundtobe12Kpsip 5{13 onpage 45 6{4 onpage 64 ).TheabsolutevalueofKIIreachedmaximumat300forboththespecimens(Figs. 5{16 5{17 onpage 48 ),thenstartsslowlydecreasingtowardszero.TheabsoluteKII;maxwas18Kpsip 5{16 5{17 onpage 48 ).HigherKIIIvalueswereobservedforthe
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Figure5{14: VariationofSIFKIalongisotropicBDspecimenthicknessatdierentCrackangle. orthotropicspecimenthanforisotropicspecimen(Figs. 5{18 5{19 onpage 49 ).Forloadalignedwithcrackplane,KIIIwasfoundtobezeroacrossthethickness.SinceKIIIisafunctionofoutofplanerelativedisplacements(normaltothediskplane)ofnodesatthecracktipandbecauseofperfectsymmetryinloadingandgeometryofthematerial,itoughttozero.Butastheforceanglewasintroduced,becauseofpoisson'seect,thereweresmalloutofplanerelativedisplacementsonbothsidesofthemidplaneofthedisk(plane3,Fig. 5{13 onpage 45 ),whichwasequalandoppositetoeachother.TheabsolutevalueofKIIIreachesmaximumbetween300360anditis15Kpsip 5{18 5{19 onpage 49 ).TheaveragevalueofKIIIiszeroacrossthethicknessforbothspecimen. FromFigs. 5{21 5{22 onpage 52 ,itcanbeseenthatKIdecreaseswithanincreaseincrackangleandcrackclosureangle(whereKIisalmostzero)was
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Figure5{15: VariationofSIFKIalongorthotropicBDspecimenthicknessatdifferentCrackangle. Figure5{16: VariationofSIFKIIalongisotropicBDspecimenthicknessatdierentCrackangle.
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Figure5{17: VariationofSIFKIIalongorthotropicBDspecimenthicknessatdierentCrackangle. Figure5{18: VariationofSIFKIIIalongisotropicBDspecimenthicknessatdierentCrackangle.
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Figure5{19: VariationofSIFKIIIalongorthotropicBDspecimenthicknessatdierentCrackangle. reachedat180forthe10 1orientation,whereasitwas300forthe 1orientation,aswasobservedbyFig. 5{10 onpage 43 .KIcanbeseensymmetricfor 1orientationacrossthethickness,whereasitisnotforthecaseof10 1orientation.IngeneralKIinsidethesurface,(plane2,3,4,Fig. 5{13 onpage 45 )isalwaysgreaterthanthoseatthecrackedges(plane1,5,Fig. 5{13 ).TheabsolutevalueofKIIincreasesandbecomesmaximumatanangle240forthe10 1orientation,whereasforthe 1orientationitis360(Figs. 5{23 5{24 onpage 53 ),asalsoillustratedinFig. 5{11 onpage 43 .KIIthenstartsslowlydecreasingwithfurtherincreasesincrackangle,aswasobservedinFig. 5{11 .Itcanbeseenthatatoneofthefaces(Plane1,thickness=0;Fig. 5{21 onpage 52 ),absolutevaluesofKIarealwaysgreaterthanthoseattheotherface(Plane5,thickness=1)oftheBDspecimenforthe10 1orientation.ButabsolutevaluesofKII(Fig. 5{23 onpage 53 )arealwaysgreateratPlane5,thanthoseatPlane1forthe10 1orientation.ItisinterestingtoobservethatSIFs(Figs. 5{22 onpage 52 5{24 onpage 53 5{26 onpage 54 )(KI,KIIandKIII)aresymmetric
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aboutthemidplaneforthe 1orientation.Thisisduetosymmetryobservedforthecracklyingalongh121idirectionthanforthatlyingalongh101idirectiononf111gplaneasshowninFig. 5{20 .Butunliketheorthotropicspecimenaligned Figure5{20: a)Unsymmetryaboutmidplaneforcrackorientedalongf111gh101i;b)Symmetryforcracklyingalongf111gh121i 5{26 onpage 54 5{18 onpage 48 )asthevalueofmaximumKIIIisalmostthesame(14:5Kpsip 1orientatedBDspecimen,ascanbeclearlyseeninFig. 5{12 onpage 44 .ThevaluesofKIII(Fig. 5{25 onpage 54 )forthe10 1orientationatoneofthefaces(Plane1,thickness=0)isalwaysgreaterthanthoseattheotherface(Plane5,thickness=1)andreachesmaximumatanangle300360,sameasisotropicandorthotropicspecimen,butthevalueisfargreater(65Kpsip
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showninFig. 5{12 onpage 44 isat250,becausethevalueswerecalculatedatanosettothemidplane,whichcanbeveriedfromtheosetvaluetakenfromFig. 5{25 onpage 54 Withincreaseincracklength/diameterratio,at00crackangle,thevalueofKI,for10 1and 1orientations,increasesthroughoutthethickness(Figs. 5{27 5{28 onpage 55 and 5{7 onpage 41 ).TheproleofKIismoredistinguishedwithincreaseincracklengthinFig. 5{27 onpage 55 ,whereasinFig. 5{28 onpage 55 ,KIisalmostuniformacrossthethicknessofthespecimen.Butwithincreaseincracklength,KIatthecrackedgesbecomesmorethanKIvaluesinsidethespecimen,uniqueto 1orientation.ThevalueofKIIontheotherhandincreasesforthe10 1orientation(Fig. 5{29 onpage 56 )anddecreasesfor 1orientation(Fig. 5{30 onpage 56 )withincreaseincracklength/diameterratioasillustratedinFig. 5{8 onpage 42 .WhereKIIissymmetricforthe10 1orientation,itdecreasesacrossthicknessforthe10 1orientationandbecomesnegative.Also,itisinterestingtoobservethatthevalueofKIIdoesnotchangeatthickness0:9withchangeincracklength.KIIIforthe10 1orientation(Fig. 5{31 onpage 57 )wasfoundtobecontinuouslydecreasing,exceptatthickness=1,withincreaseincracklength/diameterratio(canbeclearlyseeninFig. 5{9 onpage 42 ).Forthe 1orientation(Fig. 5{32 onpage 57 ),KIIIissymmetricacrossthethicknessallthetime(becauseofthesymmetryaboutthemidplane,asshowninFig. 5{20 onpage 50 )anditincreasesinvaluewithincreaseincracklength/widthratio.Atthemidplaneitisalwayszero,whichmeansthatthereisnooutofplanedisplacementatthemidplane. Usingtheaboveanalysiswecanndtheproleofthecrackgrowthinsideaspecimen,whichcanbecrucialforthelifeassessmentofananisotropicmaterial.
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Figure5{21: VariationofSIFKIalongBDspecimenthicknessatdierentCrackanglefor10 1orientation. Figure5{22: VariationofSIFKIalongBDspecimenthicknessatdierentCrackanglefor 1orientation.
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Figure5{23: VariationofSIFKIIalongBDspecimenthicknessatdierentCrackanglefor10 1orientation. Figure5{24: VariationofSIFKIIalongBDspecimenthicknessatdierentCrackanglefor 1orientation.
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Figure5{25: VariationofSIFKIIIalongBDspecimenthicknessatdierentCrackanglefor10 1orientation. Figure5{26: VariationofSIFKIIIalongBDspecimenthicknessatdierentCrackanglefor 1orientation.
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Figure5{27: VariationofSIFKIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor10 1orientation. Figure5{28: VariationofSIFKIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor 1orientation.
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Figure5{29: VariationofSIFKIIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor10 1orientation. Figure5{30: VariationofSIFKIIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor 1orientation.
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Figure5{31: VariationofSIFKIIIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor10 1orientation. Figure5{32: VariationofSIFKIIIalongBDspecimenthicknessatdierentCrackLength/Diameterratiofor 1orientation.
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6.1 Isotropicmaterials Thefracturemechanicsapproachconsidersonlythepartofthecomponentlifeafterthenucleationofadefectoraw.Inthe1960's,Parisdemonstratedtheusefulnessoflinearelasticfracturemechanics(LEFM)incharacterizingfatiguecrackgrowth.Parisdeterminedthatarelationshipexistedbetweenthecrackextensionperloadcycleandtherangeoftheappliedstressintensityfactor,K.TheformofthisrelationshipisapowerlawformandistodaywidelyknownastheParisLaw. dN=C:Km(6.1) whereCandmareconstantsdeterminedfromttingexperimentaldata.HeremistheslopeoftheregionIIfatiguecrackgrowthrate(FCGR)curve(Fig. 6{1 onpage 59 ,regionII)andCisthecoecientfoundbyextendingthestraightlinetoK=1MPap Intheory,applicationofthislawtoafatiguecrackgrowthproblemcanbeperformed,givenenoughinformation,toapproximatelypredictthenumberofcyclesuntilacriticalcracksizeisreached.TheParisLawservedasabasisfortheconstructionofmorecomplicatedlawswhichhaveattemptedtoaccountforthreshold,crackclosure,stressratioandothereects.Atypical(schematic)fatiguecrackgrowthcurveisshowninFig. 6{1 onpage 59 .Thiscurveindicatesthethreeregionsoffatiguecrackgrowth.InregionIlittleornocrackgrowthoccursduetothelowvalueofstressintensityfactorrange.RegionIIindicatesa 58
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powerlawrelationshipoftheParisLawtype.Inthisregionfatiguecrackgrowthcorrespondstothestablemacroscopiccrackgrowththatistypicallycontrolledbytheenvironment(load,geometry,temp.etc).RegionIIIisaregionofrapidcrackpropagationandimminentfracture.ThisregioniscontrolledprimarilybyfracturetoughnessKcorKIc,whichinturndependsonthemicrostructure,meanstress,andenvironment. Figure6{1: SchematicFatigueCrackGrowthCurve TypicallyModeI(crackopeningmode)FCGisthepredominantmodeofmacroscopicfatiguecrackgrowthwhileModeII(shearingmode)andModeIII(tearingmode)usuallyhaveonlysecondordereectsonbothcrackdirectionandcrackgrowthrates.Undermixedmodeloadingcondition,acrackcanchangeitsdirectionofgrowthasdierentcombinationsofmixedmodeloadingcanexistfrommultiaxialloads.Inordertoobtainthecrackgrowthrate,da=dN;orcyclestofailureusingParisequation( 6.1 ),undermixedmodeloading,severalformsofequivalentstressintensityfactorshavebeenproposed.Anequivalentstressintensityfactorrange,Kq;basedoncracktipdisplacementswasproposedby
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Tanakaas: Kq=K4I+8K4II+8K4III Anotherformofequivalentstressintensityfactorbasedontheequivalenceofenergyreleaserate,G,andthestressintensityfactor,K,fornominalelasticloadingisgivenas: Kq=K2I+K2II+(1+)K2III0:5(6.3) FatiguelifeforthroughthethicknesscracksisfoundbyintegratingParisinregionIIas: WhereKforanisotropicmaterial,undertheapplicationofuniformtensilestressrange,;canbecomputedusingtherelation, K=Cp WhereCisgeometricalconstant. Theequations( 6.4 )and( 6.5 )canbeusedtocalculatethefatiguelifefromanassumedinitialcracksize0toacriticalcracksizeaf: (m2)CmCm()mm=2(1 (a0)(m2)=21 (af)(m2)=2);form6=2(6.6) Fig. 6{2 onpage 61 showsschematicallythreecracklengthversusappliedcyclecurvesforthreeidenticaltestspecimenssubjectedtodierentrepeatedstresslevelswithS1>S2>S3.Alltestspecimenscontainedthesameinitialcracklength,a0,andineachtesttheminimumstresswaszero(R=0).Itcanbeobservedthatwithhigherstressesthecrackgrowthrates,da=dN,arehigheratagivencracklength,andthefatiguecrackgrowthlife(Nf)isshorter.Thecracklengthsatfractureareshorterathigherstresslevels.Thisconcludesthatfora
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Figure6{2: FatigueCrackLengthversusAppliedCycles.FractureisIndicatedbythex.[ 43 ] giveninitialcracksize,thelifetofracturedependsonthemagnitudeoftheappliedstressandthefractureresistanceofthematerial. 6.2 FCCsinglecrystalmaterials FatiguecrackdrivingforceparameterK,asdescribedinprevioussectioniswellsuitedforisotropicmaterials,butitmaynotbethebestparametertouseforverylargegrainorsinglecrystalalloys.Earlierstudieshaveshownthatforcracksizesmallerthanthegrainsizeorofthesameorderofmagnitude,thresholdstressintensitiesarelowerandFCGrateisacceleratedincomparisontolongcrackswhencomparedatsimilarvaluesofK[ 44 { 48 ].ThisshowsthattheparameterKdoesnotincorporatemicrostructureandthegrainorientationandtheassociateddeformationmechanismsincontrollingtheFCGbehavior. AnFCCsinglecrystalalloyoersthebestopportunitytostudyindetailtheeectofmicrostructureandthedeformationmechanismsontheFCGbehavior.SeveralStudieshavebeenconductedonFCGofNibasesinglecrystals[ 49 { 53 ]andallofthesestudieshaveshownthatFCGishighlysensitivetotheorientationofthecrystalandthatthecrackplaneiscrystallographicandfollowsasingleslipplaneoracombinationofslipplanes.Sincesheardecohesiononaslipplaneis
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causedbydislocationmotion,manyresearchershavesuggestedthattheresolvedshearstressactingontheactiveslipplaneaheadofacracktipmustberesponsibleforthepropagationofthefatiguecrack[ 51 54 55 ].Thedislocationmotioniscontrolledbytheforcesatthecracktip,whichisdirectlyrelatedtotheresolvedshearstressontheslipplane.Thereforetherateofsheardecohesionmustberelatedtotheresolvedshearstress'intensity'(RSSI)atthecracktip.Theactivesheardecohesionplaneorplanesmustbetheslipplane(s)withahighRSSI.IftheRSSIonaplaneismuchhigherthanalltheotherslipplanes,thentheplanemustbetheprimaryplaneforsheardecohesionandtheslipplanebecomesthecrackplane.Butiftheresolvedshearstressesontwoormoreoftheslipsystemsarecomparable,thenthesheardecohesionwilltakeplaceonallofthoseslipplanesandthemacrocrackplanewillnotfollowasingleslipplane.ThiswasshownbyTelesmanandGhosn[ 56 ]asshownintheFig. 6{3 onpage 62 .Eventhoughmacroscopiccrackwasobservedalong(001)planeinclined70tothestarternotch,themicroscopicslipwasobservedon111slipplanes,inclined520and380tothestarternotch. Figure6{3: Microscopicslipobservedontwof111gslipplanesinclined520and380tothestarternotch.[ 56 ]
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Ifacracksurfaceisaslipplane,itislogicalthatthecrackgrowthrateonthatslipplanewillcorrelatewithitsRSSI. ChenandLiu[ 57 ]proposedacrackdrivingforceparameterforcorrelatingFCGdata,whichisbasedontheresolvedshearstressesontheactiveslipplane(s).ThisparametermaybebetterthanKforthecorrelationofFCGdatasinceittakesintoconsiderationofthedeformationmechanisms,grainorientationandtheactualcrackpath. Theresolvedshearstresseldofaslipsystemisdenedbyitsintensitycoecient,whichcanbecalculatedoncetheModeI,IIandIIIcracktipeldsareobtained. Theresolvedshearstressisgivenby[ 58 ] wherebiandbaretheBurgersvectoranditsmagnitude;njistheunitnormalvectoroftheslipplane;andijisthecracktipstresstensoreldgivenby[ 57 ] [ij]=1 whererandarethelocalpolarcoordinatesatthecracktipasshownintheFig.( 6{4 onpage 64 );fij()aretheangularcomponentofthestresseld.Substitutingequation( 6.8 )intoequation( 6.7 ),theresolvedshearstressis wherebniandnjaretheunitBurgersvectorsandunitnormalvectorsoftheslipplanesrespectively. Theaboveequationindicatesthatrsspreservesthe1=p
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Figure6{4: Detailsofcracktipdisplacementsandstressesatadistancerandfromthecracktipinthecrackcoordinatesystem betweenthetraceofaparticularslipplaneontheplanenormaltotheslipplaneandthehorizontalaxis.Theintensityofrssislinearlyproportionaltothequantityresolvedshearstressintensitycoecient,Krss,whichforagivenslipsystemcanbedenedasthelimitingvalueoftheresolvedshearstress,rss,multipliedbyp 56 57 ] whereristhedistanceofthecracktipandrssisdenedastheprojectionofthestresstensor[]onaplanewhoseoutwardnormalis!ninthedirectionofslip!b(Fig. 6{5 onpage 65 ).ThetwodistinctadvantagesinusingKrssare:(1)thedependencyofrssonriseliminated;(2)theanglehasadenitephysicalmeaning,whichisdirectlyrelatedtotheorientationoftheslipsystem. TelesmanandGhosn[ 56 ]usedtheabovemodeltocalculateKrssfor2Dmodelandcheckedthevalidityofthisparameterbyplottingitagainstda=dNdata(Fig. 6{6 onpage 65 ).AlsoKformodeIhasbeenplottedagainstda=dNto
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Figure6{5: Burgersvector!bisalongslipdirectionh011iandslipplanedirectionisnormalvector!nalongh111i FatiguecrackgrowthrateasafunctionofKrssandKfor2Drectangularspecimen.[ 56 ]
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showsimilaritybetweenboththecurves.TheylookvirtuallyidenticalanditcanbeseenthatalinearrelationshipexistsbetweenKrssandda=dN,whichisinaccordancewiththeParislawinregionII.ButinthismethodmodeIIISIFwasnotconsideredasthespecimenwasnottestedundercompletemultiaxialfatigueloading. Stateofstressonaslipplane,undermixedmodeloading,whosetraceonaplanenormaltocrackplanemakesananglewiththehorizontalaxis(Fig. 6{4 onpage 64 andFig. 6{13 onpage 71 ),canbedenedas where,
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and3istherootofthecharacteristicequation[ 26 ] Itshouldbenotedthatthecomplianceconstants(ai;3(i=1;2;:::6))usedfortheexpression(d3;j(j=1;2;3))aretheonesasgivenbyequation( 3.9 ). Theaboveequations,whenusedinconjunctionwithequations( 6.7 )&( 6.10 )givesKrssonallthe12slipsystemsforFCCsinglecrystalsuperalloy.
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ThreeBDspecimenofPWA1422weretested(Table 6{1 onpage 68 )tocorrelatethefatiguecrackgrowthdatawiththecalculatedKrss.ThecrackgrowthofthesethreespecimenswiththeapplicationofcyclicloadhasbeenshownintheFigs.( 6{7 6{8 6{9 onpage 69 ).Asitcanbeseenthattherearenumerouspointsalongthepath.Forthespecimen95830(Fig. 6{7 onpage 69 ),itcanbeseenthatthecrackgrowthstoppedafterreaching8:8mm,whichwasduetocrackclosure.Whereasadeclineincrackgrowthcanbeseenforspecimen96842(Fig. 6{11 onpage 69 )aftercracklengthreaches6:5mm.Specimen98C21hasaslowcrackgrowthinthebeginning,butlateranacceleratedgrowthisobservedtillcracklengthreaches8:4mm.Thecurvettingwasnecessaryinordertocalculateda=dNaccuratelyatanygivenpoint.Nosinglecurvetcouldproperlytforentirepath.Thereforeacombinationofseveralequationsforcurvetwasrequiredtogetthedesiredresult.Thecorrespondingda=dNobtainedfromthoseequationsforthethreespecimenshavebeenshowninFigs.( 6{10 6{11 6{12 onpage 69 ).Thecrackgrowthratedeclinesaftercrackgrowsto6:5mmforspecimens95830and96842,whichisunusual. Table6{1: ThegeometryandloadingconditionofthethreeBrazilianDiskspecimentested W(mm)t(mm)a(mm)
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Figure6{7: CrackgrowthofBDspecimen95830withno.ofcycles Figure6{8: CrackgrowthofBDspecimen96842withno.ofcycles Figure6{9: CrackgrowthofBDspecimen98C21withno.ofcycles
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Figure6{10: CrackgrowthrateofBDspecimen95830withincreasingcracklength Figure6{11: CrackgrowthrateofBDspecimen96842withincreasingcracklength Figure6{12: CrackgrowthrateofBDspecimen98C21withincreasingcracklength
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ThemixedmodeSIFswerecalculatedatthosepointsshownintheFigs.( 6{10 6{11 6{12 onpage 69 )forallthethreespecimensonthemidplaneoftheBraziliandiskspecimens,assumingplanestrainassumption.KrmswascalculatedusingthefollowingequationinordertocomparewithKrss. Krms=q 6.10 ),wherestateofstressobtainedfromequation( 6.11 )wasmultipliedbySchmid'sfactorforFCCsinglecrystalasdenedinequation( 1.13 )togettheresolvedshearstressintensityonallthe12primaryslipsystems.Thetraceofthetwoslipplanes( 11 1)and(1 1 1)werefoundtobemakingequalangleswiththe(111)plane(67:780),whilethetraceoftheplane( 1 11)wasmaking00withthecrackplane(111)asshownintheFig. 6{13 Figure6{13: Traceofprimaryslipplanesontheplanenormaltothecrackplane TheloadratioR(minload/maxload)was0.1(Table 6{1 onpage 68 ),astheloadontheBDspecimenalwayscompressive.AsSIFsarelinearlyproportionaltotheloadapplied,thereforeKrsswasmultipliedwithRtogettheKrss.
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ThemaxKrsswasfoundtobeonthe(111)slipplaneforallthethreespecimens(FirstrowofTables 6{2 6{3 onpage 76 andTable 6{4 onpage 77 ),asobservedintheexperimenttestresults(Fig. 6{14 onpage 73 ).ThecalculatedKrsswasplottedagainstda=dNonaloglogscaletocheckthevalidityofthemodel(Fig. 6{16 onpage 74 ).AfterKrssreaches10MPap 59 ]
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Figure6{14: Crackgrowthonf111gslipplanecanbeobservedfor96842BDspecimen AsimilarnatureofKrmscanbeseeninFig. 6{15 onpage 74 .TheadvantageofKrssliesintheabilitytopredicttheactualmicroscopicfatiguefracturemechanismsandalsoittakesintoconsiderationtheorientationofthegrain.TheKrssalsoisamultiaxialfatiguestressparameter,incorporatingresolvedshearstressonprimaryslipsystems.
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Figure6{15: Fatiguecrackgrowthrateof3specimens95830,96842,98C21,asafunctionofKrms Fatiguecrackgrowthrateof3specimens95830,96842,98C21,asafunctionofKrss
PAGE 87
Table6{2: 1]6.456.596.957.187.587.617.64(111)[0 11]1.911.972.132.222.402.422.43(111)[1 10]4.544.634.834.965.185.195.21 ( 11 1)[10 1]1.171.181.211.231.271.271.27( 11 1)[110]1.000.990.950.920.850.850.85( 11 1)[011]2.172.172.162.162.122.122.12 (1 1 1)[110]0.830.870.981.051.181.181.19(1 1 1)[0 11]3.073.093.103.123.113.113.11(1 1 1)[101]3.903.964.084.164.284.294.30 ( 1 11)[011]1.911.972.132.222.402.422.43( 1 11)[101]2.993.023.083.113.153.153.16( 1 11)[1 10]4.914.995.205.335.555.575.59
PAGE 88
Table6{3: 1]6.115.956.206.396.566.756.91(111)[0 11]1.031.121.291.511.711.932.10(111)[1 10]5.084.834.914.884.854.814.81 ( 11 1)[10 1]1.611.481.461.391.331.251.21( 11 1)[110]2.592.292.141.841.561.231.02( 11 1)[011]4.203.773.603.232.892.492.22 (1 1 1)[110]0.170.020.130.360.560.790.96(1 1 1)[0 11]5.424.904.734.313.923.473.17(1 1 1)[101]5.254.894.864.674.484.264.13 ( 1 11)[011]1.031.121.291.511.711.932.10( 1 11)[101]4.664.254.153.863.603.293.10( 1 11)[1 10]5.685.375.445.375.305.225.19
PAGE 89
Table6{4: 1]12.4214.4016.4117.5421.8230.6537.37(111)[0 11]1.822.623.504.026.1010.7514.46(111)[1 10]10.6011.7812.9113.5215.7219.9122.91 ( 11 1)[10 1]3.453.653.823.904.144.414.48( 11 1)[110]5.805.745.505.324.341.530.86( 11 1)[011]9.249.399.329.228.485.943.62 (1 1 1)[110]0.680.160.460.832.386.029.01(1 1 1)[0 11]11.8512.1812.2712.2411.739.487.33(1 1 1)[101]11.1712.0212.7213.0714.1115.5016.33 ( 1 11)[011]1.822.623.504.026.1010.7514.46( 1 11)[101]10.0910.5210.7810.8810.9910.379.50( 1 11)[1 10]11.9113.1414.2914.9017.0921.1223.96
PAGE 90
Thegoalwastoestimatethefatiguelifeofsinglecrystalnickelbasesuperalloys,basedonfatiguecrackdrivingforceparameter,forwhichanumericalmodelwasdevelopedtocomputemixedmodeSIFsatthecracktip.Theresultsobtainedcanbesummarizedas 1. Ananalyticalmethodhasbeendevelopedforthecalculationofallthethreemodesofstressintensityfactorsasafunctionofcrystallographicorientationforanorthotropicmaterial,whichcanbeappliedtoanyanisotropicmaterial,ifallthematerialconstantsareknown.Nopriorassumptionofplanestressorplanestrainwasmadeindevelopingthistheory. 2. ModeI,(KI),wasalwaysfoundtobegreaterfor[10 1]thanforthe[ 12 1]orientation.Foranorthotropicmaterial,KIwasfoundtobenonzeroatcrackclosureduetothecouplingofthenodaldisplacements,whereasforanisotropicmaterial,itwaszero. 3. ThemagnitudeofKIIfor[ 12 1]wasfoundtobealwaysgreaterthanthatfor[10 1]orientation,butthedierencewasnotmuch. 4. ModeIIISIF(KIII)existedbecauseofthecouplingofdisplacementsatthecracktipduetoanisotropy.KIIIfor[10 1]wasfoundtobemuchbiggerthanthatfor[ 12 1]orientationfortheBraziliandiskspecimen.ThisplaysanimportantroleincalculationofeectiveKtopredictthelingofananisotropicmaterial. 5. TheKIandKIIIvaluescalculatedweremuchhigherfor[10 1]orientation,thanthosefor[ 12 1]orientation.ThereforeahighvalueofKrsscanbeexpectedforthe[10 1]directionthanthatforthe[ 12 1]direction,ifweignore 78
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thedierenceofKIIvalues.ThiswillresultinhighKrss,leadingtofastcrackgrowthratefor[10 1]orientationthanthatfor[ 12 1]orientation. 6. SIFscalculatedfor[ 12 1]crackorientationwasfoundtobesymmetric,whichwasduetosymmetryofh121iaboutthemidplaneasshowninFig. 5{20 onpage 50 .Whereasduetounsymmetricalnatureofh101iorientation,thecalculatedSIFsvariedalongthethicknessfor[10 1]crackorientation.Thisvariationcanverywellbeusedtopredictthecrackgrowthproleacrossthethickness. 7. Thecrackclosurefor[10 1]orientationwasfoundtobe180whereasitwas300for[ 12 1]orientationfortheBraziliandiskspecimen,whichshowstheimportanceofknowledgeofsecondaryorientation. 8. Assheardecohesioniscausedbydislocationmotion,itwasexpectedthattheorientationofthecrackplanemustberelatedtotheactiveslipplane(s).Thefatiguedamageparameter,Krsswascalculatedforallthe12primaryslipsystemsforallthe3Braziliandiskspecimens.TheKrss;maxwasfoundonthef111gplane,whichalsohappenedtobethecrackgrowthplaneforallthe3BDspecimens.Thereforeforasingleactiveslipplane,themaximumresolvedshearstressprovidestheprimarydrivingforcefordislocationmotion,thesheardecohesionprocessleadingtocrackgrowthprocess. 9. TheparameterKrssiswellsuitedinidentifyingtheactivecrackplaneaswellaspredictingthemicroscopiccrackgrowthdirection.Itisalsoaneectivemultiaxialcrackdrivingforceparameter. 10. Thefatiguedamageparameter,Krss,wascalculatedforthethreeBraziliandiskspecimenandwasplottedonloglogscalewithda=dNandalinearregion,regionII(Parisregion)wasobtained.ButtheexperimentaldatausedtocalculateKrsswerenotsucientinnumberinregionII.Moredatais
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requiredtogettheaccuratevalueofparameterCandnoftheParislaw.Theequationobtainedcanbeusedtocalculatethelifeofthematerial. 11. Themethoddevelopedisnotrelatedtoanyspecicgeometry.Thereforeitcanbeusedforanykindofgeometryandforanygeneralanisotropicmaterial,tocalculatemixedmodeSIFsatthecracktipandthereforecrackgrowthrateandhencelifeofanymaterialcanbepredicted. 12. AdditionalexperimentaldataarerequiredtostudyFCGonf111gplane,h101iorientation.MoredataareneededinregionIItogettheaccurateresulttopredictling.Alsoexperimentaldataforcrackplaneorientationsotherthanslipplanes,e.g.,h100i,canbeusedtocheckthevalidityofthemodel. PossibleDirectionsforFutureWorkaregivenbelow:
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FigureA{1: Forcesactingonarectangularelementwithdimensiondxdyinequilibriumcondition. AgeneralizedplanestrainconditionhasbeenadoptedforananisotropicsolidapplicabletothatofanFCCsinglecrystal.[ 6 ]Letusconsiderarectangularpipelikeelementwithdimensiondx;dy;1.Weassumethatthisbodyisunderequilibriumundervariousforces.IfbodyforcesareX;Yperunitvolume,thentheequationofequilibriumofforcealongxandydirectioncanbewrittenasfollowed, Alongxaxis. 81
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Alongyaxis Theabovetwoequationsresultas, (A.3) WhereXandYarebodyforces(thisforceactsonentirebodyandconstantthroughoutthebody.e.g.gravitationalforce,magneticforceetc.),whichcanbedenedasaderivativeofpotentialfunction U @xY=@ U @y ThecomponentsofstressesaredenotedasXnandYn(perunitarea)andonlydependonthetwocoordinatesxandy.Nowthefundamentalsystemofequationsofequilibriumtodeterminethestateofstressanddeformationofanelasticbody(boundedbycylindricalsurface)undertheinuenceofbodyforcesandstresseswhicharedistributedalongthesurfacecanbegivenas: (A.5) Theequation( A.4 )isaresultofsummationofforcesalongX,YandZaxesrespectively.
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ThegeneralizedHooke'slawsare A.6 )isafunctionofxandyonlyforthegivencondition. As @zyz=@v @z+@w @y @x+@u @z A.6 ),keepinginmindthatthesearefunctionofxandyonly. Usingthe2ndequationofequation( A.7 )inintegrationofthe4thequationofequation( A.6 )weget @y+za41x+a42y+:::+a46xy@W0 Similarlywiththehelpofthe3rdequationofequation( A.7 )andthe5thequationofequation( A.6 )wecannd @x+z(a51x+a52y+:::+a56xy@W0
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Where andU0;V0;W0arearbitraryfunctionofxandyonly. Asequations( A.8a A.8b A.8c )shouldsatisfytherestofthethreeequationsofequation( A.6 ),letussubstitutetheminequation( A.6 ). @x=z2 @x2+z@ @xa51x+a52y+:::+a56xy@W0 @y=z2 @y2+z@ @ya41x+a42y+:::+a46xy@W0 @y+@v @x=z2@2D @y@x+z[@ @ya51x+a52y+:::+a56xy@W0 @xa41x+a42y+:::+a46xy@W0 Usingequation( A.6 )andequation( A.10 )tocomparethecoecientsofz2;zandz0;asthesearefunctionsofxandyonly,thereforeateverypointofzitmustbesameforsamexandy,henceallthecoecientsofz2;zshouldbezero;weget @x2=0;@2D @y2=0;@2D @y@x=0(A.11) @xa51x+a52y+:::+a56xy@W0 @ya41x+a42y+:::+a46xy@W0 @ya51x+a52y+:::+a56xy@W0 @xa41x+a42y+:::+a46xy@W0 (A.12)
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A.11 )wecansaythatitisalinearfunctionofxandyonly.D=A1x+B1y+C1 Furtherwithoutanylossofgeneralitywecanwrite WhereA;B;Carenewarbitraryconstants. Fromequation( A.9 )andequation( A.14 )wecanwrite Wecanwriteequation( A.12 )intheform @x=0;@N @y=0;@M @y+@N @x=0(A.16) Where
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Integrating1stand2ndequationsofequation( A.16 )knowingthattheyarefunctionofxandyonly Where; ;!1;!2arethenewarbitraryconstants. Using3rdequationofequation(13)andequation(15)weget= A.18 )andusingequation( A.17 )wecanwrite WecanwriteU0;V0;W0asafunctionofxandywithoutanylossofgeneralityas Usingequations( A.8a A.19 A.20 )wecanwrite
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Heretheconstants!1;!2;!3;u0;v0;w0characterizeonlyrigiddisplacementsofthebodyand Substitutingexpressionforequation( A.15 )intoequation( A.13 )wecanwrite@U @x=a11x+a12y+a13[Ax+By+C1 @x=a11a13 @x=11x+12y+14yz+15xz+16xy+a13(Ax+By+C)(A.22a) Similarly @x=21x+22y+24yz+25xz+26xy:::+a23(Ax+By+C)@U @y+@V @x=61x+62y+64yz+65xz+66xy+:::a63(Ax+By+C) (A.22b) Where Similarlysubstitutingequation( A.15 )intoequation( A.19 )weget @x=51x+52y+54yz+55xz+56xy+a53(Ax+By+C)+ @y=41x+42y+44yz+45xz+46xy+a43(Ax+By+C)
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Thestraincompatibilityrelationforplanestrain,whichisvalidforsmallstrainsisgivenas, LetusintroducetwostressfunctionsF(x;y)and(x;y),suchthat @y2+ @x2+ @x@y;xz=@ Usingequation( A.25 )andequation( A.6 ),wecanwrite @y4+@2U @y2+12@4F @y2@x2+@2U @y214@3 @x@y3@2y @x2@y2+@2U @x2+22@4F @x4+@2U @x224@3 @x3@y@2xy @x@y3+@2 @y@x+26@4F @y@x3+@2 @y@x26@3 @x2@y2 UsingEquation( A.24 )andequation( A.26 ),wecanfurtherwriteL4F+L3=(12+22)@2U @x2+(16+26)@2U @x@y(11+12)@2U @y2L3F+L2=2 U @x(15+25)@ U @y (A.27)
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Where Nowthegeneralsolutionofthesystemofequationscanbewrittenas (A.29) Inthepresenceofbodyforces, Fromtheabovetwoequations, (L4L2L23)F0=0(A.31) Theabovesixthorderoperatorcanbebrokenintolinearoperatorsas, Where xk x(A.33) andkaretherootsoftheequationwhichcorrespondstothedierentialequation
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where, Assumingthatalltheroots(k)aredistinct,then Thegeneralintegralisequaltoanarbitraryfunctionoftheargumentx+6yandisdenotedbyfv6(x+6y). Therefore, Integratingtheaboveequation,weget
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ThearbitraryfunctioncanbedenedintermsofF0and0inthefollowingmanner, (A.41) Nowfromequationaandb whereakandbkarearbitraryconstants. Nowtherearefourtheorems,whichareimportantforthetheoryoftheequilibriumofabodyboundedbyacylindricalsurface. Theorem1:Theequationl4()=0cannothaverealroots. Theorem2:Theequationl2()=0cannothaverealroots. Theorem3:Theequationl4()l2()l23=0cannothaverealroots. Theorem4:Theequationa1142a163+(2a12+a66)22a26+a22=0cannothaverealroots. Basedonthethirdtheorem,wecanshowthatthenumbersarealwayscomplexorpurelyimaginary. Thegeneralexpressionforthestressfunctioncannowbeshownas,
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HereFk(zk)areanalyticfunctionsofthecomplexvariableszk=x+ky(k=1,2,3)and1;2;3arethecomplexnumbersequalto (A.44) Nowweintroducenewfunctionsofthecomplexvariablezk,k(zk)=F0k(zk)k=1;2 TakingthederivativeofFandwithrespecttoxandyrespectivelyresultsas, x=2Re[1(z1)+2(z2)+33(z3)]+F0 y=2Re[11(z1)+22(z2)+333(z3)]+F0 x=2Re[11(z1)+22(z2)+3(z3)]+0
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Basedontheaboveformulas,thegeneralexpressionsforthecomponentsofstressescanbedenedas, Bysubstitutingaboveequationsandintegratingtheresultingequation,wendthefunctionsU,V,Was (A.49) .........fork=1,2
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Thestateofstressandthedisplacementeldnearacracktipisgivenby[ 28 ], 1. Inplanesymmetricloading Re1 Re1 (B.2) 2. Inplaneskewsymmetricloading Re1 Re1 (B.4) 95
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3. AntiplaneShearloading Rep c45+3c44 where, and3istherootofthecharacteristicequation[ 26 ] 28 ]c442+2c45+c55=0 Therelativecrackfacedisplacementeldequation(topfacerelativetothebottomface,i.e.primewithrespecttounprimeasshowninFig. 4{3 onpage 34 )forthecracktipelementisgivenbyManuandIngraeaas[ 29 ], 2(4vB+vC+4vEvF)+1 22(vF+vC2vD)]q L1)+[(1)(2vBvC)(1+)(2vEvF)]r L1(B.7)
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2(4uB+uC+4uEuF)+1 22:::(uF+uC2uD)]q L1)+[(1)(2uBuC)(1+)(2uEuF)]r L1(B.8) 2(4wB+wC+4wEwF)+1 22:::(wF+wC2wD)]q L1)+[(1)(2wBwC)(1+)(2wEwF)]r L1(B.9) Thedisplacementeldofthecrackfaces(=1800)nearthecracktip,duetomixedmodeloading,canbegivenbysuperposingthedisplacementequations( B.2 B.4 B.6 )as 12(1p22p1)]Re[i 12(p2p1)]0Re[i 12(1q22q1)]Re[i 12(q2q1)]0001 Equationbasedonthelikepowersofrasabove(i.e.p B.7 B.8 B.9 )as
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2(4uB+uC+4uE:::uF)+1 22(uF+uC2uD)2vBvC+2vEvF+vD+1 2(4vB+vC+4vE:::vF)+1 22(vF+vC2vD)2wBwC+2wEwF+wD+1 2(4wB+wC+4wE:::wF)+1 22(wF+wC2wD)9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;r L1(B.11) Equatingequations( B.10 )and( B.11 ),themixedmodestressintensityfactorscanbefoundas WherefAgisgivenas 2(4uB+uC+4uEuF)+:::1 22(uF+uC2uD)2vBvC+2vEvF+vD+1 2(4vB+vC+4vEvF)+:::1 22(vF+vC2vD)2wBwC+2wEwF+wD+1 2(4wB+wC+4wEwF)+:::1 22(wF+wC2wD)9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;(B.13)
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and[B]1isdenedas[B]1=2666664Re[i 12(q2q1)]1 12(p2p1)]1 12(1q22q1)]1 12(1p22p1)]1 andDisthedeterminantoftheequation 12(1p22p1)]Re[i 12(p2p1)]Re[i 12(1q22q1)]Re[i 12(q2q1)](B.15) Let, thensuperposingthestresses,giveninequations( B.1 B.3 B.5 B.16 ),duetomixedmodeloading,thestateofstressatanypointaroundthecracktipcanbegivenas where,
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and3istherootofthecharacteristicequation[ 26 ] Alsozcanbeexpressedfrom A.6 basedonplanestrainassumption(z=0)as Where, Fromequations( B.20 )and( B.21 )itcanbewritten, (B.22)
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EquatingthecoecientsofKI;KIIandKIIIinequation( B.16 )and( B.22 ),itcanbefound Forplanestresscondition(z=0).
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SrikantRanjanwasborninDaltongunjtowninBihar,India,in1977.Alwaysatopperduringhisschooling,heactivelyparticipatedandbaggedseveralaccoladesatvariouslevelsofextracurricularactivities(elocution,essaywriting,scienceexhibition,quizcompetition,dramacompetitionetc.).AfterhisschoolinghejoinedthepremierengineeringinstituteofIndia,IndianInstituteofTechnology,Kharagpur(IITKGP)in1995.Thereheactivelyparticipatedin,andorganizedvarioussportsandculturalevents,andwasInterIITathleticscaptaininhisnalyear.HewasawardedIITBlue,ahighesthonor,givenforoutstandingperformanceinsportsactivities.HegothisB.Tech(Honors)degreein1999inManufacturingScienceandEngineeringandjoinedMechanicalEngineeringDepartmentatUniversityofFloridainMay,2000,onaproject,thenfundedbyNASA,topursueMasterofScience.UnderthetutelageofDr.NagarajK.Arakere,hecontinuedhismaster'sprogramtowardsPhDandwishestocontinuehisresearchworkforindustrialapplications. 126

