Development of a Numerical Procedure for Mixed Mode K-Solutions and Fatigue Crack Growth in FCC Single Crystal Superalloys

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DEVELOPMENT OF A NUMERICAL PROCEDURE FOR MIXED MODE
K-SOLUTIONS 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 high-quality 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 Tae-Joong 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 iv-I .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

1-1 Direction cosines of material coordinate axes (xyz) with universal co-
ordinate axes (x'y'z') . . . . . . 8

1-2 Direction cosine of (x', y', z') with (x, y, z) coordinate axes, when x'
axis is aligned along [213] orientation. ....... . . 16

3-1 Geometrical and material properties of the two specimens analyzed 28

6-1 The geometry and loading condition of the three Brazilian Disk speci-
m en tested . . . . . . . . 68

6-2 Krs8 for 12 primary slip systems with increasing crack length for spec-
im en 95830 . . . . . . . . 75

6-3 K.rs for 12 primary slip systems with increasing crack length for spec-
im en 96842 . . . . . . . . 76

6-4 Krss for 12 primary slip systems with increasing crack length for spec-
im en 98C 21 . . . . . . . 77














LIST OF FIGURES
Figure page

1-1 Schematic of the 7' precipitate in a 7 matrix .. . .... 2

1-2 Cuboidal 7' precipitates (0.35-0.6 pm) in PWA1480. . .... 3

1-3 Temperature capability of superalloys with approximate year of in-
troduction . . . . . . . .. 4

1-4 Aeroengine blades are nominally oriented in the (001) orientation. 4

1-5 Material coordinate system (xyz) relative to universal coordinate sys-
tem (x'y'z') . . . . . . . 11

1-6 Two separated portions of a single crystal showing a model for calcu-
lating the resolved shear stress in a single-crystal specimen. . 11

1-7 Slip lines and fracture plane in experimental tensile test specimen
tested by Materials Science and Engineering Department, UF. 12

1-8 Primary (Close-pack) and Secondary (Nonclose-pack) slip directions
on the octahedral planes for an FCC i-I 1 . ...... 14

1-9 Cube slip planes and slip directions for an FCC crystal. . ... 15

3-1 Crack tip nodal displacement of isotropic elastic material . ... 24

3-2 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

3-3 Brazilian disk having center crack lying in {111} slip plane and aligned
along [T21] direction . . . . . . 28

3-4 Rectangular specimen 'A' meshed with solid95 element (left) and the
geometrical details (right) . . . . . 29

3-5 Brazilian disk (Specimen 'B') with center crack lying in (111) plane
and oriented along [10l] direction . . . . 29

4-1 Brazilian disk meshed with triangular element having isotropic prop-
erties (left) and zoomed view of crack tip (right). . .... 31









4-2 (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

4-3 Arrangement of quarter-point wedge elements along segment of crack
front with nodal lettering convention. ....... . . 34

4-4 The symmetric meshing of BD specimen with solid95 element and
enlarged picture of crack tip elements ..... . . 34

5-1 K, versus Crack Length/Width ratio for [10t] and [T21] orientation
of Specimen 'A' at 0 00. . . . . . 37

5-2 KI1 versus Crack Length/Width ratio for [101] and [T21] orientation
of Specimen 'A' at 0 00 . . . . . ... 38

5-3 KII, versus Crack Length/Width ratio for [101] and [121] orienta-
tion of Specimen 'A' at 0 00 . . . . .... 38

5-4 K, versus crack angle with force for [10t] and [12t] orientation of
Specimen 'A' at 2a/W 0.4. . . . . . 39

5-5 KH versus crack angle with force for [10t] and [T21] orientation of
Specimen 'A' at 2a/W 0.4. . . . . . 39

5-6 KmI, versus crack angle with force for [101] and [T21] orientation of
Specimen 'A' at 2a/W 0.4. . . . . . 40

5-7 K, versus Crack Length/Diameter ratio for [101] and [121] orienta-
tion of Specimen 'B' at 00 . . . . . 41

5-8 KI1 versus Crack Length/Diameter ratio for [10t] and [12t] orienta-
tion of Specimen 'B' at 00 . . . . . 42

5-9 KmI versus Crack Length/Diameter ratio for [10t] and [121] orien-
tation of Specimen 'B' at 00. . . . . 42

5-10 K, versus crack angle with force for [101] and [T21] orientation of
Specimen 'B' at 2a/W 0.55. . . . . ... 43

5-11 K11 versus crack angle with force for [101] and [121] orientation of
Specimen 'B' at 2a/W 0.55. . . . . ... 43

5-12 Km, versus crack angle with force for [101] and [T21] orientation of
Specimen 'B' at 2a/W 0.55 . . . . . 44

5-13 Half meshed model of Brazilian disk specimen and the crack coordi-
nate system . . . . . . . 45









5-14 Variation of SIF K, along isotropic BD specimen thickness at differ-
ent Crack angle . . . . . . ..

5-15 Variation of SIF K, along orthotropic BD specimen thickness at dif-
ferent Crack angle . . . . . . ..

5-16 Variation of SIF KI1 along isotropic BD specimen thickness at differ-
ent Crack angle . . . . . . ..

5-17 Variation of SIF KI1 along orthotropic BD specimen thickness at dif-
ferent Crack angle . . . . . . ..

5-18 Variation of SIF K1I, along isotropic BD specimen thickness at dif-
ferent Crack angle . . . . . . ..

5-19 Variation of SIF KII, along orthotropic BD specimen thickness at
different Crack angle. . . . . . . .

5-20 a)Unsymmetry about mid plane for crack oriented along {111}(101);
b) Symmetry for crack lying along {111}(121) . . .

5-21 Variation of SIF K, along BD specimen thickness at different Crack
angle for [10T] orientation . . . . . ..

5-22 Variation of SIF K, along BD specimen thickness at different Crack
angle for [T21] orientation . . . . . ..

5-23 Variation of SIF K11 along BD specimen thickness at different Crack
angle for [101] orientation . . . . . ..

5-24 Variation of SIF Kn along BD specimen thickness at different Crack
angle for [T21] orientation . . . . . ..

5-25 Variation of SIF KmI, along BD specimen thickness at different Crack
angle for [101] orientation . . . . . ..

5-26 Variation of SIF KII, along BD specimen thickness at different Crack
angle for [T21] orientation . . . . . ..

5-27 Variation of SIF K, along BD specimen thickness at different Crack
Length/Diameter ratio for [lOT] orientation. . . . .

5-28 Variation of SIF K, along BD specimen thickness at different Crack
Length/Diameter ratio for [121] orientation. . . . .

5-29 Variation of SIF K11 along BD specimen thickness at different Crack
Length/Diameter ratio for [101] orientation. . . . .


5-30 Variation of SIF K1 along BD specimen thickness
Length/Diameter ratio for [T21] orientation. ..


at different Crack









5-31 Variation of SIF K11, along BD specimen thickness at different Crack
Length/Diameter ratio for [10l] orientation.. . . 57

5-32 Variation of SIF K11, along BD specimen thickness at different Crack
Length/Diameter ratio for [121] orientation.. . . 57

6-1 Schematic Fatigue Crack Growth Curve ..... . . 59

6-2 Fatigue Crack Length versus Applied Cycles. Fracture is Indicated
by the x . . . . . . . . 61

6-3 Microscopic slip observed on two {111} slip planes inclined 520 and
-380 to the starter notch . . . . . 62

6-4 Details of crack tip displacements and stresses at a distance r and 0
from the crack tip in the crack coordinate system . . 64

6-5 Burgers vector b is along slip direction (011) and slip plane direc-
tion is normal vector n along (111) . . . . 65

6-6 Fatigue crack growth rate as a function of AKrs8 and AK for 2-D
rectangular specimen . . . . . . 65

6-7 Crack growth of BD specimen 95830 with no. of cycles . ... 69

6-8 Crack growth of BD specimen 96842 with no. of cycles . ... 69

6-9 Crack growth of BD specimen 98C21 with no. of cycles . ... 69

6-10 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

6-12 Crack growth rate of BD specimen 98C21 with increasing crack length 70

6-13 Trace of primary slip planes on the plane normal to the crack plane 71

6-14 Crack growth on {111} slip plane can be observed for 96842 BD spec-
im en . . . . . . . . 73

6-15 Fatigue crack growth rate of 3 specimens 95830, 96842, 98C21, as a
function of AK ms . . . . . . 74

6-16 Fatigue crack growth rate of 3 specimens 95830, 96842, 98C21, as a
function of AKr . . . . . . . 74

A-1 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
K-SOLUTIONS AND FATIGUE CRACK GROWTH IN FCC SINGLE CRYSTAL
SUPERALLOYS

By

Srikant R lii i-n-

August 2005

C!i ,i: N. K. Arakere
M i"r Department: Mechanical and Aerospace Engineering

Fatigue-induced 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 nickel-base 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 mixed-mode

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 mixed-mode loading

conditions, using three-dimensional 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 mode-mixity 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-, iron-nickel-, and cobalt-base 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 (FCC-austenitic) 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 high-temperature 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 1-1: 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. 1-1 shows a schematic diagram of the 7' phase within the

7 matrix [1] and Fig. 1-2 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 Ni-Cr-Al system. These alloys exhibit better

high temperature properties than p, i, i il -1Iiiw. wrought or cast alloys (Fig. 1-3

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 1-2: Cuboidal 7' precipitates (0.35-0.6 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' i-I 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 nickel-base

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 / MAR-M*22 MAR-M l2S CMSX-2
MAR1-200 M21 M246 M-2 +HlDS 0
SVuM21 M248 MA|-M-200 )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 Introduced-M-509 Ime72
auction. [3]
X750 N80AWrought
.A[-- Co'bas.

OD at rest* DO S aund SC



1940 195 19110 1070 1 98D09

Year Introduced


Figure 1-3: Temperature capability of superalloys with approximate year of intro-
duction. [3]


Figure 1-4: 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. 1-5 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

orthogonally-anisotropic 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. 1-5 on

page 11, Table 1-1 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 1-2 on page 16, Fig. 1-8 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 1-5: Material coordinate system (xyz) relative to universal coordinate sys-
tem (x'y'z')


Figure 1-6: Two separated portions of a single crystal showing a model for calcu-
lating the resolved shear stress in a single-crystal specimen. [7]



















Figure 1-7: 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. 1-8 on page 14). In addition, there are six possible slip

directions in the three cube planes, as shown in Fig. 1-9 on page 15. There are 30

possible slip systems in an FCC crystal (Table 1-2 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, t-4


Figure 1-8: Primary (Close-pack) and Secondary (Nonclose-pack) slip directions on
the octahedral planes for an FCC ( i -I 1 [-)]


Secondary:~ 0 7T6


T4
r17




















Plaie2


Figure 1-9: Cube slip planes and slip directions for an FCC -i iv-l [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, i--I 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 iv-I ,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

mixed-mode 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 2-D anisotropic cracks to evaluate SIF

under mixed-mode loading condition. Fractal finite element method (FFEM) [19]

was used to calculate Mode I and II SIFs for 2-D 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 2-D 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 2-D problems which included

generalized plane stress, generalized plane strain and anti-plane problems.

Denda and Marante [21] developed a crack tip singular element (CTSE)

for the general anisotropic solids in 2-D 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 2-D multiple curvilinear cracks in general anisotropic solids, but as it was

based on plane strain assumption, it could not be incorporated into a generalized

3-D 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 2-D

model. Likewise, Sosa and Eischen [12] calculated SIF for a plate containing a

through crack subjected to bending loads using the J integral. Two-D 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 mixed-mode 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 anti-plane 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 anti-plane

problems with cracks of any geometric shapes. It did not give a complete solution

under mixed-mode loading conditions, but it did give an idea about the behavior of

K111 under anti-plane shear loading.

Shih et al. [24] have calculated SIFs for 2-D 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 3-D 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 3-D anisotropic elastic material based on the model

of Shih et al. The computed SIFs, when compared with 2-D anisotropic bodies

with known exact solutions yielded an error of 6-1'-. 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 3-D

anisotropic cracked solids. Denda [36] used BEM to determine mixed mode SIFs

(KI, K11 and K111) of 3-D 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 3-D anisotropic material is

unavailable. The objective of the present work is to model a 3-D 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 center-cracked 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. 3-1 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 3-D model to calculate SIF for mixed-

mode loading. It can be shown that mixed mode stress intensity factors can be
























z


Figure 3-1: 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. 4-3 on page 34.

TI is the natural co-ordinate system value defined as


2z
S- -( + 1)
L2


L2 is the length of the element along the crack front (Fig. 4-2 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[ /1-2 (/1P2
Re[ 12 (pijq2
1^1-92 ^'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 Mixed-Mode Loading. Two

specimen geometries were used to investigate the effects of mode mixity at the

crack tip. One was a rectangular tension specimen (Fig. 3-4 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. 3-5 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 3-2: 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. 3-2 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 3-1 on page 28.

Where,




















Crack Cordinate System


Material Coordibue System aligned


Material Coordinate System to Crack Cooninate System

Figure 3-3: Brazilian disk having center crack lying in {111} slip plane and aligned
along [12t] direction

Table 3-1: Geometrical and material properties of the two specimens analyzed

Rectangular Specimen 'A' (Fig. 3-4) Brazilian Disk B (Fig. 3-5)
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 0.2-0.8
F 4.448KN 2a/W 0.1-0.9 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. 3-3 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 3-4: Rectangular specimen 'A' meshed with solid95 element (left) and the
geometrical details (right)


Figure 3-5: 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 3-1 on page 28 were modeled, meshed and analyzed for

different crack lengths, crack angles, material properties and i i v-I 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 4-1: 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. 3-5 on page 29, 3-3 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. 3-4 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. 4-1), which is a quadratic isoparametric

brick element with 8 corner nodes and 12 midside nodes (Fig. 4-2 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 two-dimensional geometry the 8-node quadratic isoparametric

element can be made to simulate the 1//r singularity in stress. That is done

by placing the mid-side 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 mid-side nodes between quarter and mid-point 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.

Two-dimensional triangular elements [30], formed by collapsing one side of

rectangular elements, give better results for stress intensity factor. Also, it has



















N
-JR


Figure 4-2: (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 three-dimensional case, 20-node isoparametric elements were used. 20-node

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. 4-2. Fig. 4-2(a) shows a 20 node isoparametric element in universal

coordinate system, and Fig. 4-2(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. 4-4 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. 3-1 on page 24),

and the crack direction was aligned along family of either (101) or (121) directions.


































Figure 4-3: Arrangement of quarter-point wedge elements along segment of crack
front with nodal lettering convention. [31]













--I-.LL".LI-I" '











Figure 4-4: 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 <-12-1>
-*-6Anisotropic Solution for <-12-1>
"- Isotropic Solution for <10-1>
5000 Anisotropic Solution for <10-1>

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 5-1: 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. 5-1 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. 5-2 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.

5-3 on page 38) because the relative displacement of the two crack faces along the

z-direction (Fig. 4-3 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 <-12-1-

120_ Isotropic Solution for <-12-1>
Anisotropic Solution for <10-1>
100 6. Isotropic Solution for <10-1>


~0


0.1 0.2 0.3 0.4 0.5 0.6 .7 0.8 0


Crack Length / Width ratio


Figure 5-2: 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 <-12-1"
- 4- Isotropic Solution for <-12-1>

- Anisotropic Solution for <10-1>
- *- Isotropic Solution for <10-1>


Crack Length / Width Ratio


Figure 5-3: K11, versus Crack Length/Width ratio for [101] and [121] orientation

of Specimen 'A' at 00.
















--Anisotropic Solution for <-12-1:
- ** Isotropic Solution for <-12-1>
Anisotropic Solution for <10-1>
- *- Isotropic Solution for <-12-1>


0 10 20 30 40 50 60 70 80
Crack angle with Force (degree)


Figure 5-4: 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 5-5: 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 <-12-1>
S-*- Anisotropic Solution for <-12-1>
-1000- Isotropic Solution for <10-1>
"-10 J -A-Anisotropic Solution for <10-1>

S-1500


-2000


-2500
Crack angle with Force (degree)

Figure 5-6: K111 versus crack angle with force for [101] and [121] orientation of
Specimen 'A' at 2a/W 0.4.


from the plots (Fig. 5-4 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. 5-5 on page 39).

The value of K111 for the [10t] orientation is negative (Fig. 5-6 on page 40) as the

relative nodal displacements of the crack faces along the z-direction 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. 5-7, 5-8 on page 42, 5-9, 5-10 on page 43,

5-11, 5-12 on page 44), under fatigue loading for same AK values. Even though










25000

-*-Anisotropic Solutions for <10-1>
20000 .- Isotropic Solution for <10-1>
20000 --
SAnisotropic Solutions for <-12-1>
*- Isotropic Solutions for <-12-1>

'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 5-7: 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 <10-1>
Isotropic Solution for <10-1>

-4000 Anisotropic Solutions for <-12-1
*- Isotropic Solutions for <-12-1>


Crack Length / Diameter ratio


Figure 5-8: 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 <10-1>

- Isotropic Solution for <10-1>

- Anisotropic Solutions for <-12-1

- *- Isotropic Solutions for <-12-1>


-14000


Crack Length / Diameter ratio


Figure 5-9: 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 <10-1>

-25000 Anisotropic Solutions for <-12-1>
-25000 -
*- Isotropic Solutions for <-12-1>

-30000
Crack angle with Force (deg)


Figure 5-10: 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 -12-1
10 20 30 40 50 60 70 80 90








-10000 Anisotropic Solutions for <10-1>

Anisotropic Solutions for <-12-1>
@- Isotropic Solutions for <-12-1>
-15000
Crack angle with Force (deg)


Figure 5-11: KI1 versus crack angle with force for [101] and [T2T] orientation of
Specimen 'B' at 2a/W 0.55.










20000 Anisotropic Solutions for <10-1>
.B- Isotropic Solution for <10-1>
15000Anisotropic Solutions for <-12-1>
15000 .- -Isotropic Solutions for <-12-1>


10000


5000



0 10 20 30 40 50 60 70 80 90

-5000


-10000
Crack angle with Force (deg)

Figure 5-12: 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. 5-15 on page 47, 5-17 on page 48, 5-19 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 5-13: 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. 5-14, 5-15 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 mid-plane (Figs. 5-14, 5-15 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. 5-13 on

page 45, 6-4 on page 64). The absolute value of Kt reached maximum at ~ 300

for both the specimens(Figs. 5-16, 5-17 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. 5-16, 5-17 on page 48). Higher K7). values were observed for the










10
-*- Angle 00
S-U- 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 5-14: Variation of SIF K, along isotropic BD specimen thickness at differ-
ent Crack angle.


orthotropic specimen than for isotropic specimen (Figs. 5-18, 5-19 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. 5-13 on page 45), which was

equal and opposite to each other. The absolute value of K1I reaches maximum

between 30-360 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.

5-18, 5-19 on page 49). The average value of K1II is zero across the thickness for

both specimen.

From Figs. 5-21, 5-22 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
-e-Angle -12
--- Angle -18
-+- Angle -24


I N


0.2 0.4 0.6 0.8 1


Normalized Thickness


Figure 5-15:
ferent Crack






2

0

-2

-4

o -61


S-8

I -10


Variation of SIF K, along orthotropic BD specimen thickness at dif-
angle.


Normalized Thickness


Figure 5-16: 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


------


S-Angle uu
-U-Angle -06
-s-Angle 12
-x-Angle 18
-Angle 24
- -Angle -30
-Angle -36
-Angle 42
--Angle -48
-Angle -54
-*-Angle -60
--Angle -66
-4-Angle -72
-Angle -78
--Angle -84


Normalized Thickness


Figure 5-17: Variation of SIF KI1 along orthotropic BD specimen thickness at

different Crack angle.


-*- Angle -00
--- Angle 06
-e-Angle 12
-Angle 18
-+- Angle 24


0.2 0.4


Normalized Thickness


Figure 5-18: Variation of SIF KII, along isotropic BD specimen thickness at differ-

ent Crack angle.










25 -*- Angle 00
-2- Angle 06
20 -s-Angle 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 5-19: 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. 5-10 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. 5-13 on page

45) is alvb--, greater than those at the crack edges (plane 1, 5, Fig. 5-13). 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. 5-23, 5-24

on page 53), as also illustrated in Fig. 5-11 on page 43. K11 then starts slowly

decreasing with further increases in crack angle, as was observed in Fig. 5-11. It

can be seen that at one of the faces (Plane 1, thickness = 0; Fig. 5-21 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. 5-23 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. 5-22 on

page 52, 5-24 on page 53, 5-26 on page 54)(KI, Kl and K1il) are symmetric














14

12



10 -*- Angle 00
8 --Angle 06
-e-Angle- 12
6 -Angle- 18
S- -- Angle 24
4




2
II 0.2 .OA S 0.8 1 1.2



-4-

-6
Normalized Thickness


Figure 5-21: 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 5-22: Variation of SIF K, along BD specimen thickness at
angle for [121] orientation.


different Crack






















Angle 30
--- Angle 36
-a- Angle 42
.94Angle -48
S--Angle 60
--Angle 60
-6 -- -Angle -66
Angle 72
M- 8 --Angle -78
| -u-Angle 84

-10


-12


-14
Normalized Thickness


Figure 5-23: 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
-4-Angle -24
5 -^ -Angle -30
*a --Angle -36
-Angle 42
-Angle- 48
---6 Angle 54
-4-Angle -60
^ -h-Angle -66
SAngle -72
M .- -8-Angle-78
|--o-Angle -84


-10



-12
Normalized Thickness


Figure 5-24: 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 5-25: Variation of SIF KII, along BD specimen thickness at different Crack
angle for [o10] orientation.


-*- Angle 00
-- Angle 06
-e-Angle 12
--Angle -18
-- Angle 24


0.2 0.4


Normalized Thickness


Figure 5-26: 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 5-27: 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 5-28: 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 5-29: 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
-a-Ratio 0.5
-- Ratio 0.6
-*- Ratio 0.7
-*-Ratio 0.8
-0- Ratio 0.8


Normalized Thickness


Figure 5-30: 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






S-30


-40


-50
Normalized Thickness


Figure 5-31: 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 5-32: Variation of SIF Ksss along BD specimen thickness at different Crack
Length/Diameter ratio for [12T] orientation.















Sl > S2 > S3


Applied cycles, N


Figure 6-2: 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 [44-48]. 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 i1-I 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 Ni-base single crystals [49-53] 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. 6-3 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 6-3: 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. (6-4 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 i-I 1 orientation relative to the crack surface.

For a given crystal orientation and crack geometry, the angle 0 is equal to the angle










t-x


Crack Surface

Figure 6-4: 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)
r--O

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. 6-5 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 2-D

model and checked the validity of this parameter by plotting it against da/dN data

(Fig. 6-6 on page 65). Also AK for mode I has been plotted against da/dN to




















n
4 {111}


Figure 6-5: 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.. MPoV-rm


Figure 6-6: Fatigue crack growth rate as a function of AKrs8 and AK for 2-D
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. 6-4 on

page 64 and Fig. 6-13 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 r-12 -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 iv-I 1 superalloy.


/' [nj]


K,

KIl

Kil


(6.14)


K/rss
































0 10 20 30 40 50 60
No. of Cycles, N in Millions


Figure 6-7: 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 6-8: 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 6-9: 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 6-10: 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 6-11: Crack growth rate of BD specimen 96842 with increasing crack length




4.E-04 I


3.E-04

S3.E-04

2.E-04

2.E-04

1.E-04


2 4 6 8 10
Crack length, a in mm


12 14


Figure 6-12: Crack growth rate of BD specimen 98C21 with increasing crack length


9.E-05

8.E-05

7.E-05

6.E-05

5.E-05

*9 4.E-05

3.E-05









The mixed mode SIFs were calculated at those points shown in the Figs.
(6-10, 6-11, 6-12 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. 6-13.


(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 6-13: 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 6-1 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 6-2, 6-3 on page 76 and Table 6-4 on page 77),

as observed in the experiment test results (Fig. 6-14 on page 73). The calculated

AKr.s was plotted against da/dN on a log-log scale to check the validity of the

model (Fig. 6-16 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 log-log 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 6-15: Fatigue crack growth
function of AKrms


rate of 3 specimens 95830, 96842, 98C21, as a


1.E-03


A A


1.E-04




1.E-05




1.E-06


1.E-07 I
1.OE+00


o 95830
n 96842
A 98C21


1.OE+01 1.OE+02
AK.,. MPa m-1/2


1.OE+03


Figure 6-16: Fatigue crack growth rate of 3 specimens 95830, 96842, 98C21, as a
function of AKrss


1.E-03


1.E-04


A 95830
* 96842

o 98C21


1.E-05


1.E-06



1.E-07 --
1.0OE+00


1.0OE+01

AKms, MPa m-1/2


1.0OE+02









75


.g J, L-- b c- 1 CM c. o, ,, o I- -






~C/2




-' I













a o
C- I




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- v ri L'- icx -2mc




















cl i,.... i.... i.... ,I' c-: CV -0 M
cmom -,"




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I^
(LI.






cm





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CHAPTER 7
CONCLUSIONS

The goal was to estimate the fatigue life of single crystal nickel-base 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 non-zero 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 x-axis.


Ix + aXdy dx axdy + TsY + Txydy\ dx Txydx + Xdxdy = 0 (A.1)

PAGE 3

Thisdissertationistheresultof5yearsofworkwherebyIhavebeenaccom-paniedandsupportedbymanypeople.ItisapleasantaspectthatIhavenowtheopportunitytoexpressmygratitudetoallofthem.TherstpersonIwouldliketothankismysupervisor,Dr.NagarajArakere.Ihavebeenunderhissupervisionsince2000whenImovedtotheUniversityofFlorida.Hisenthusiasmandintegralviewonresearchandhismissionforproviding"onlyhigh-qualityworkandnotless,"hasmadeadeepimpressiononme.Iowehimlotsofgratitudeforhavingmeshownthiswayofresearch.IwouldalsoliketothanktheothermembersofmyPhDcommitteewhotooktheeortinreadingandprovidingmewithvaluablecommentsonearlierversionsofthisthesis:Dr.AshokKumar,Dr.BhavaniSankar,Dr.FereshtehEbrahimiandDr.JohnZiegert.Ithankthemall.MycolleaguesintheFatigueandTribology(FAT)LabErikKnudsen,JeLeismer,ShadabSiddiquiandTae-JoongYUallgavemethefeelingofbeingathomeatwork.ThefriendlyambienceofFATlabalwayskeptmeinacheerfulmood.ThediscussionsandtheinteractionswithShadabandJehadadirectimpactonthenalformandqualityofthisthesis.IwouldliketothankShadabforprovidingmetipsthathelpedmealotwheneverIwentotrack.Itwasnotpossibletocontinuemyworkwithouthiscompanionshipandcheerfulsupport.Erik,JeandAmitojLikharihavereadpartsofmythesisandprovidedmevaluablecomments.AlexanderPachecohashelpedmetremendouslyingettingthethesisincorrectformat.Myfriendandcolleaguesincemyundergraddays, iii

PAGE 4

iv

PAGE 5

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

PAGE 6

Table page 1{1Directioncosinesofmaterialcoordinateaxes(xyz)withuniversalco-ordinateaxes(x0y0z0) .......................... 8 1{2Directioncosineof(x0;y0;z0)with(x;y;z)coordinateaxes,whenx0axisisalignedalong[213]orientation. ................. 16 3{1Geometricalandmaterialpropertiesofthetwospecimensanalyzed .. 28 6{1ThegeometryandloadingconditionofthethreeBrazilianDiskspeci-mentested ................................ 68 6{2Krssfor12primaryslipsystemswithincreasingcracklengthforspec-imen95830 ................................ 75 6{3Krssfor12primaryslipsystemswithincreasingcracklengthforspec-imen96842 ................................ 76 6{4Krssfor12primaryslipsystemswithincreasingcracklengthforspec-imen98C21 ............................... 77 vi

PAGE 7

Figure page 1{1Schematicofthe0precipitateinamatrix .............. 2 1{2Cuboidal0precipitates(0.35-0.6m)inPWA1480. .......... 3 1{3Temperaturecapabilityofsuperalloyswithapproximateyearofin-troduction. ............................... 4 1{4Aeroenginebladesarenominallyorientedintheh001iorientation. .. 4 1{5Materialcoordinatesystem(xyz)relativetouniversalcoordinatesys-tem(x0y0z0) ............................... 11 1{6Twoseparatedportionsofasinglecrystalshowingamodelforcalcu-latingtheresolvedshearstressinasingle-crystalspecimen. .... 11 1{7SliplinesandfractureplaneinexperimentaltensiletestspecimentestedbyMaterialsScienceandEngineeringDepartment,UF. ... 12 1{8Primary(Close-pack)andSecondary(Nonclose-pack)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{1Braziliandiskmeshedwithtriangularelementhavingisotropicprop-erties(left)andzoomedviewofcracktip(right). .......... 31 vii

PAGE 8

................. 33 4{3Arrangementofquarter-pointwedgeelementsalongsegmentofcrackfrontwithnodalletteringconvention. ................. 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 1orienta-tionofSpecimen'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 1orienta-tionofSpecimen'B'at=00. .................... 41 5{8KIIversusCrackLength/Diameterratiofor10 1and 1orienta-tionofSpecimen'B'at=00. .................... 42 5{9KIIIversusCrackLength/Diameterratiofor10 1and 1orien-tationofSpecimen'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{13HalfmeshedmodelofBraziliandiskspecimenandthecrackcoordi-natesystem ............................... 45 viii

PAGE 9

............................ 46 5{15VariationofSIFKIalongorthotropicBDspecimenthicknessatdif-ferentCrackangle. ........................... 47 5{16VariationofSIFKIIalongisotropicBDspecimenthicknessatdier-entCrackangle. ............................ 47 5{17VariationofSIFKIIalongorthotropicBDspecimenthicknessatdif-ferentCrackangle. ........................... 48 5{18VariationofSIFKIIIalongisotropicBDspecimenthicknessatdif-ferentCrackangle. ........................... 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!bisalongslipdirectionh011iandslipplanedirec-tionisnormalvector!nalongh111i 65 6{6FatiguecrackgrowthrateasafunctionofKrssandKfor2-Drectangularspecimen. ......................... 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{14Crackgrowthonf111gslipplanecanbeobservedfor96842BDspec-imen ................................... 73 6{15Fatiguecrackgrowthrateof3specimens95830,96842,98C21,asafunctionofKrms 74 6{16Fatiguecrackgrowthrateof3specimens95830,96842,98C21,asafunctionofKrss 74 A{1Forcesactingonarectangularelementwithdimensiondxdy 81 x

PAGE 11

Fatigue-inducedfailuresinaircraftgasturbineandrocketengineturbopumpbladesandvanesareapervasiveproblem.Turbinebladesandvanesrepresentperhapsthemostdemandingstructuralapplicationsduetothecombinationofhighoperatingtemperature,corrosiveenvironment,highmonotonicandcyclicstresses,longexpectedcomponentlifetimesandtheenormousconsequenceofstructuralfailure.Singlecrystalnickel-basesuperalloyturbinebladesarebeingutilizedinrocketengineturbopumpsandjetenginesbecauseoftheirsuperiorcreep,stressrupture,meltresistance,andthermomechanicalfatiguecapabilitiesoverpolycrystallinealloys.Thesematerialshaveorthotropicpropertiesmakingthepositionofthecrystallatticerelativetothepartgeometryasignicantfactorintheoverallanalysis.Computationofstressintensityfactors(SIFs)andtheabilitytomodelfatiguecrackgrowthrateatsinglecrystalcrackssubjecttomixed-modeloadingconditionsareimportantpartsofdevelopingamechanisticallybasedlifepredictionforthesecomplexalloys. xi

PAGE 12

TheresolvedshearstressintensitycoecientoreectiveSIF,Krss,canbecomputedasafunctionofcracktipSIFsandtheresolvedshearstressonprimaryslipplanes.ThemaximumvalueofKrssandKrsswasfoundtodeterminethecrackgrowthdirectionandthefatiguecrackgrowthraterespectively.Thefatiguecrackdrivingforceparameter,Krss,formsanimportantmultiaxialfatiguedamageparameterthatcanbeusedtopredictlifeinsuperalloycomponents. xii

PAGE 13

Asuperalloyisagroupofnickel-,iron-nickel-,andcobalt-basemetallicalloyswhichcanbeusedathightemperatures,ofteninexcessof0.7oftheabsolutemeltingtemperature.Exceptionalcreepandhightemperatureoxidationresistancearetwooftheprimedesigncriteriaofsuperalloys,butapartfromthesepropertiestheyexhibitacombinationofhighstrengthathightemperature,stressruptureresistance,toughnessandmetallurgicalstability,usefulthermalexpansioncharacteristicsandstrongresistancetothermalfatigue.Theyexhibitexceptionalcorrosionresistanceatroomtemperature.ThehightemperaturestrengthofNi-basedsuperalloysdependsonastable,facecenteredcubic(FCC)matrix,combinedwitheitherprecipitationstrengtheningand/orsolidsolutionhardening.Iron,nickel,andcobaltaregenerallyfacecenteredcubic(FCC-austenitic)incrystalstructurewhentheyarethebasisforsuperalloys.However,thenormalroom-temperaturestructuresofironandcobaltelementalmetalsarenotFCC.BothironandcobaltundergotransformationsandbecomeFCCathightemperaturesorinthepresenceofotherelementsalloyedwithironandcobalt.Nickel,ontheotherhand,isFCCatalltemperatures.ThemicrostructureofanFCCsuperalloyconsistsofanaustenitic(phase)matrixandawidevarietyofsecondaryphase.Themostcommonsecondaryphasesare0andmetalcarbides.Themicrostructureconsistsof~60percentbyvolumeof0precipitatesinamatrix.Thesealloysobtaintheirhigh-temperaturestrengthfromthepresenceofthe0precipitatephasewithintheprimarymatrix.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. NickelbasesinglecrystalsuperalloysareprecipitationstrengthenedcastmonograinsuperalloysbasedontheNi-Cr-Alsystem.Thesealloysexhibitbetterhightemperaturepropertiesthanpolycrystallinewroughtorcastalloys(Fig. 1{3 onpage 4 ).Inhightemperatureapplicationgrainboundariesaretypicallytheweaklink,whichprovidepassagesfordiusionandoxidation,whichresultsin

PAGE 15

Figure1{2: Cuboidal0precipitates(0.35-0.6m)inPWA1480. failuresatthislocation.Grainboundarystrengthenersareaddedtothealloychemistrytoincreasecapability,whichresultsinloweringthemeltingpointofthealloy. Removalofgrainboundariesandgrainboundarystrengtheningelementsraisetheincipientmeltingtemperatureofthealloyby150oFandresultinimprovedhightemperaturefatigueandcreepcapabilities[ 4 ].Thisincreaseinmelttemperaturepermitshigherheattreatmenttemperaturethatinturnyieldsimprovedcreepcapability.Unlikethemorecommonlyusedpolycrystallinealloys,thesesinglecrystalsuperalloysareorthotropicandhavehighlydirectionalmaterialpropertieswithexceptionalthermomechanicalfatiguepropertiesathightemperature.Themostcommonprimarygrowthdirectionforthenickel-basesuperalloysistheh001idirection.Thisisnotonlythemosteasilygrownbutalsoisthedirectionwiththemostdesirablecombinedstrengthproperties.Thisisadvantageousbecausemanypartsarecast,orgrown,ratherthanmachinedfromalargersinglecrystalsamplewithaspecicorientation.

PAGE 16

Figure1{3: Temperaturecapabilityofsuperalloyswithapproximateyearofintro-duction.[ 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,whichiscalledorthogonally-anisotropicor,forbrevity,orthotropic.Thereforecubicsym-metrycanbedescribedwiththreeindependentconstantsdesignatedastheelasticmodulus,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).

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[Q]=266666666666666421222322323121221222322323121221222322323121211223323+3213+3112+2111223323+3213+3112+2111223323+3213+3112+213777777777777775(1.8) Table1{1: Directioncosinesofmaterialcoordinateaxes(xyz)withuniversalcoor-dinateaxes(x0y0z0) y z x0

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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

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dislocations.Plasticdeformationcorrespondstothemotionoflargenumbersofdislocations.Theprocessbywhichplasticdeformationisproducedbydislocationmotioniscalledslip.Slipinmetalcrystalsoftenoccursonplanesofhighatomicdensityincloselypackeddirectionsbecausetheseplanesgenerallycorrespondtothelowestpossibleenergyforslip.Theplaneonwhichthemotionoccursiscalledslipplane.Thereisapreferredplane(slipplane)andapreferreddirectionwithinthatplane(slipdirection).Thecombinationofslipplaneandslipdirectioniscalledaslipsystem.Thepreferredslipplaneisthatwiththegreatestplanardensity.Thepreferredslipdirectionisthatwiththegreatestlineardensity. Slipmechanism .Inasinglecrystalspecimen,theextantofslipdependsonthemagnitudeoftheshearingstress.Magnitudeoftheresolvedstressdependsonthemagnitudeoftheappliedforce,itsorientationrelativetotheslipplaneandslipdirection.Thereforethestress-strainbehaviorofamaterial,whichisafunctionofthenumberofactivatedslipsystems,varieswithorientation.AsliplineorstepisobservedinexperimentsinvolvingpolishedsinglecrystalsspecimensasshowninFig. 1{7 onpage 12 .TheslipplanesofhighestatomicdensitiesarethersttogetactivatedonanFCCsinglecrystalspecimenandarecalledtheprimaryoctahedralslipsystems. InFig. 1{6 onpage 11 ,Fistheappliedforce,Aisthecross-sectionalareaofthespecimen,istheanglebetweenthetensileaxisandthenormaltotheslipplaneandistheanglebetweentheslipdirectionandthetensileaxis.Theresolvedshearstressalongtheslipdirectioncanbecalculatedas Thefavoredslipsystemhasthelargestresolvedshearstresscalledcriticalresolvedshearstress.Ithasbeenobservedexperimentallythatasinglecrystalwill

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Figure1{5: Materialcoordinatesystem(xyz)relativetouniversalcoordinatesys-tem(x0y0z0) Figure1{6: Twoseparatedportionsofasinglecrystalshowingamodelforcalcu-latingtheresolvedshearstressinasingle-crystalspecimen.[ 7 ]

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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(Close-pack)andSecondary(Nonclose-pack)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.TheknowledgeofSIFisnecessarytopredictthegrowthofafatiguecrackortodeterminetheresidualstrengthofacrackedstruc-ture.Thisfactorcharacterizestheintensityofthestresseldintheneighborhoodofthecracktipanddependssubstantiallyonspecimengeometry,materialproper-ties,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,tocalculateSIFsforcrackssubjectedtomixed-modeloadingconditionsinisotropicelasticsolids.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 ]studiedvariouskindsof2-DanisotropiccrackstoevaluateSIFundermixed-modeloadingcondition.Fractalniteelementmethod(FFEM)[ 19 ]wasusedtocalculateModeIandIISIFsfor2-Danisotropicplate.ThevariationoftheSIFswithmaterialpropertiesandorientationsofacrackwaspresented.ItwasshownthatSIFswerenotsensitivetothevariationofshearmodulus.HwuandLiang[ 20 ]usedremoteboundarydatatocalculateSIFsfor2-Danisotropicmaterial.IteliminatedtheerrorinSIFcalculation,causedbyabruptchangeinthestressesnearcracktips,byndingequivalentformulationforSIFbyusingonlyremoteboundaryresponses(displacements,stressesandstrains),cooperatingwiththenecessarygeometricdata.Aspecialboundaryelementwasdevelopedwhichremovedtherequirementofmeshingaroundthecrackboundary.Through

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thisboundaryelement,alltheinternalstressesandstrainscouldbeexpressedintermsofdisplacementsandtractionsontheboundariesexcludingthehole,crackandinclusionboundaries.Theseresultscouldbeappliedtoanykindoflinearanisotropicmaterialsbutwasrestrictedtothe2-Dproblemswhichincludedgeneralizedplanestress,generalizedplanestrainandanti-planeproblems. DendaandMarante[ 21 ]developedacracktipsingularelement(CTSE)forthegeneralanisotropicsolidsin2-Dwiththebuiltinp 22 ]forcombinedmode(IandII)SIFsincaseofplanar,rectilinear,anisotropicstructuresusinga12nodesingularniteelementwasaccurate,butwasrestrictedtoa2-Dmodel.Likewise,SosaandEischen[ 12 ]calculatedSIFforaplatecontainingathroughcracksubjectedtobendingloadsusingtheJintegral.Two-DeightnodeelementwasselectedforthispurposewhereKII=KIII=0.MewsandKuhn[ 23 ]usedtheGreen'sfunctionapproachtocalculatemixedmodeSIFwithoutanycrackdiscretizationinanisotropicplate,whichusedanasymptoticdisplacementeldatthecracktip.TheycalculatedSIFsforplatehavingmultiplecracks,forvariousinclinations,andfoundSIFsveryclosetothatofShihetal.[ 24 ].Ishikava[ 14 ]presentedtheideaofthestrainenergyreleaserate(virtualcrackextensionmethod)tocalculateSIFformodeIandII,anddescribedSIFcalculationbasedononlyoneuseofvirtualcrackextension. HuangandKardomateas[ 25 ]describedthecontinuousdislocationtechniquetocalculatemixed-modestressintensityfactorsinananisotropicinnitestrip.ThismethodwaslimitedinitsusetoModeIandIISIFs,anditwasfoundtobe

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mostsuitableforcracksofrelativelysmalldimensions.ThemethodwasveriedbycalculatingSIFsforisotropicmaterialandcomparingtheresultswithreadilyavailableformulasgiveninTadaetal.[ 26 ],whichresultedinexcellentagreement.Sunetal.[ 27 ]usedtheboundaryelementmethod(BEM)toanalyzecrackedanisotropicbodiesunderanti-planeshear.Thenewboundaryformulationuseddislocationdensityasanunknownonthecracksurface,andKIIIwasdeterminednearthecracktip.Theequationandmethodcouldbedirectlyusedforanti-planeproblemswithcracksofanygeometricshapes.Itdidnotgiveacompletesolutionundermixed-modeloadingconditions,butitdidgiveanideaaboutthebehaviorofKIIIunderanti-planeshearloading. Shihetal.[ 24 ]havecalculatedSIFsfor2-DisotropicmaterialsusingquarterpointelementnodaldisplacementsatthecracktipbasedonFiniteElementMethod(FEM).TheModesI,IIandIIIhavebeendecoupledbecauseoftheisotropicnatureofthematerial.Sihetal.[ 28 ]denedSIFsasafunctionofstressatthecracktip.FollowingtheworkofShihetal.,IngraeaandManu[ 29 ]showedhowtocomputeSIFfrom3-Dquarterpointnodaldisplacement,forcrackedisotropicelasticbodiesforallthreemodes.Theyusedaquarterpointisoparametricelement,whichhasbeenaccurateincomputingSIF[ 30 ].SaoumaandSikiotis[ 31 ]introducedanisotropyintheabovemodel[ 29 ]andproposedamethodtocalculateSIFfor3-DanisotropicelasticmaterialbasedonthemodelofShihetal.ThecomputedSIFs,whencomparedwith2-Danisotropicbodieswithknownexactsolutionsyieldedanerrorof6-12%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 ]usedsingledomainBEMtocalculatedmixedmodeSIFsforbothboundedandunbounded3-Danisotropiccrackedsolids.Denda[ 36 ]usedBEMtodeterminemixedmodeSIFs(KI,KIIandKIII)of3-Danisotropicmaterialwithmultiplecracks.ItaddressedtheissueofcouplingeectofthethreemodesoffracturecontrolledbyModeI,IIandIIISIFs.Thoughveryaccurate,theseformulationswerebasedontheplanestrainassumption. AlthoughasubstantialbodyofliteraturedescribescomputationofSIF,ageneralizednumericalsolutiontocalculateSIFfor3-Danisotropicmaterialisunavailable.Theobjectiveofthepresentworkistomodela3-DorthotropicspecimenhavingathroughcrackandtocalculatetheSIFsforallthreemodesassuminglinearelasticpropertiesatthecracktip.AmathematicalmodelhasbeendevelopedtocalculatemixedmodeSIFforFCCsinglecrystalorthotropicmaterialfordierentorientations,whichisalsoapplicabletogeneralizedanisotropicmaterial.Lookingatthecracktipnodaldisplacementsitispossibletocalculatestressintensityfactorforanycrystallographicorientationofthematerial.Itwasobservedthatthematerialorthotropyresultsincoupledcracktip(x,y,

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z)displacements,leadingtotheinterdependenceofSIFsforModesI,II,andIII.Resultsarepresentedforacenter-crackedBDspecimen,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 ).Theinterde-pendenceofdisplacementsinanisotropicmaterialsduetoshearcouplingresultsincoupledstressintensityfactors. Anisotropywasincorporatedinthe3-DmodeltocalculateSIFformixed-modeloading.Itcanbeshownthatmixedmodestressintensityfactorscanbe 23

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Figure3{1: Cracktipnodaldisplacementofisotropicelasticmaterialforcalcula-tionofStressIntensityFactorsofModeI;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. SingleCrystalSpecimenGeometriesUsedforMixed-ModeLoading .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 ) BrazilianDisk-B(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: Braziliandiskmeshedwithtriangularelementhavingisotropicproper-ties(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.Fortwo-dimensionalgeometrythe8-nodequadraticisoparametricelementcanbemadetosimulatethe1=p 40 ].LynnandIngraea[ 41 ]haveshowncomputationofSIFforisotropicmaterialwithhighaccuracycanbeachievedbyplacingmid-sidenodesbetweenquarterandmid-pointoftheelementsatthecracktip.Thisinformationfurtherledtothedevelopmentofthetransitionelement,whichresultedinimprovedSIFcalculationswhentheratioofthequarterpointelementandcracklengthisdecreased. Quadraticisoparametricelementswith20nodeshaveproventobeexcellentelementsforthecalculationofSIFforelasticcrackproblems[ 31 ],whenthemid-sidenodesareputnearthecracktipatthequarterpoint.Alsothiselementhasinversesquareroot(1 42 ]hasshowntheeectofquarterpointsingularandtransitionelement'ssizeonseveralSIFcomputationtechniques.ItwasdeterminedthatcomputedSIFwasnotsensitivetothesingularelementinternalangleandsizeexceptforthequarterpointdisplacementtechniqueandthedisplacementcorrelationtechnique.Thereforenermesharoundthecracktipgivesbetterresultsforthequarterpointdisplacementmethod. Two-dimensionaltriangularelements[ 30 ],formedbycollapsingonesideofrectangularelements,givebetterresultsforstressintensityfactor.Also,ithas

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Figure4{2: (a)20nodeisoparametricelementinnaturalcoordinatesystemand(b)Quarterpointsingularelementwiththe=1facecollapsedinlocalCartesiancoordinatesystem.[ 31 ] beenfoundthatrectangularquadraticisoparametricelements(andbrickelements)havesingularstiness(singulartotalstrainenergy)atthecracktip[ 30 ].Forthethree-dimensionalcase,20-nodeisoparametricelementswereused.20-nodebrickelementscollapseandformwedgeshapedelementsatthecracktip.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: Arrangementofquarter-pointwedgeelementsalongsegmentofcrackfrontwithnodalletteringconvention.[ 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 )becausetherelativedisplacementofthetwocrackfacesalongthez-direction(Fig. 4{3 onpage 34 )wasnegative.Thisissimplyaconsequenceofsignconventiondenition.However,themagnitudeoftheSIFfor10 1isfoundtobegreaterthanfor 1.Thisvalidatesthattheh101ifamilyofslipdirectionsareprimaryoreasyslipdirectionsascomparedtothoseofh121i. ThechangeinSIFswiththecrackangle''wasanalyzedforbothorienta-tions.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 )astherelativenodaldisplacementsofthecrackfacesalongthez-directionwerenegative.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,non-negligiblevaluesofKIIIwerefound,andtheyvariedwithrespecttotheappliedforce,cracklengthandcrackangle.

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Figure5{8: 1and 1orienta-tionofSpecimen'B'at=00. Figure5{9: 1and 1orienta-tionofSpecimen'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). InordertocheckthedierenceinSIFofisotropicBDspecimenandor-thotropicBDspecimenwithchangeinloadanglewithcrack,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: HalfmeshedmodelofBraziliandiskspecimenandthecrackcoordi-natesystem forboththespecimenweresignicantlydierent,becauseofthematerialprop-erties.Thecrackclosureanglewasthesame(200,Figs. 5{14 5{15 onpage 47 ).NegativeKIwasfoundbecausenocontactelementswereusedatthecrackinterface.Afterreachingthecrackclosureangle,furtherincreaseinforceangleleadtopenetrationofcrackfaces,resultinginnegativeKI.Attheedges,KIwaslessthanthattowardsmid-plane(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: VariationofSIFKIalongisotropicBDspecimenthicknessatdier-entCrackangle. 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.TheabsolutevalueofKIIIreachesmaximumbetween300-360anditis15Kpsip 5{18 5{19 onpage 49 ).TheaveragevalueofKIIIiszeroacrossthethicknessforbothspecimen. FromFigs. 5{21 5{22 onpage 52 ,itcanbeseenthatKIdecreaseswithanincreaseincrackangleandcrackclosureangle(whereKIisalmostzero)was

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Figure5{15: VariationofSIFKIalongorthotropicBDspecimenthicknessatdif-ferentCrackangle. Figure5{16: VariationofSIFKIIalongisotropicBDspecimenthicknessatdier-entCrackangle.

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Figure5{17: VariationofSIFKIIalongorthotropicBDspecimenthicknessatdierentCrackangle. Figure5{18: VariationofSIFKIIIalongisotropicBDspecimenthicknessatdier-entCrackangle.

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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)andreachesmaximumatanangle300-360,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,acrackcanchangeitsdirectionofgrowthasdierentcombinationsofmixed-modeloadingcanexistfrommultiaxialloads.Inordertoobtainthecrackgrowthrate,da=dN;orcyclestofailureusingParisequation( 6.1 ),undermixed-modeloading,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.SeveralStudieshavebeenconductedonFCGofNi-basesinglecrystals[ 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 ]usedtheabovemodeltocalculateKrssfor2-Dmodelandcheckedthevalidityofthisparameterbyplottingitagainstda=dNdata(Fig. 6{6 onpage 65 ).AlsoKformodeIhasbeenplottedagainstda=dNto

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Figure6{5: Burgersvector!bisalongslipdirectionh011iandslipplanedirectionisnormalvector!nalongh111i FatiguecrackgrowthrateasafunctionofKrssandKfor2-Drectangularspecimen.[ 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: ThegeometryandloadingconditionofthethreeBrazilianDiskspeci-mentested 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=dNonalog-logscaletocheckthevalidityofthemodel(Fig. 6{16 onpage 74 ).AfterKrssreaches10MPap 59 ]

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Figure6{14: Crackgrowthonf111gslipplanecanbeobservedfor96842BDspeci-men AsimilarnatureofKrmscanbeseeninFig. 6{15 onpage 74 .Theadvan-tageofKrssliesintheabilitytopredicttheactualmicroscopicfatiguefracturemechanismsandalsoittakesintoconsiderationtheorientationofthegrain.TheKrssalsoisamultiaxialfatiguestressparameter,incorporatingresolvedshearstressonprimaryslipsystems.

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Figure6{15: Fatiguecrackgrowthrateof3specimens95830,96842,98C21,asafunctionofKrms Fatiguecrackgrowthrateof3specimens95830,96842,98C21,asafunctionofKrss

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Table6{2: 1]-6.45-6.59-6.95-7.18-7.58-7.61-7.64(111)[0 11]1.911.972.132.222.402.422.43(111)[1 10]-4.54-4.63-4.83-4.96-5.18-5.19-5.21 ( 11 1)[10 1]-1.17-1.18-1.21-1.23-1.27-1.27-1.27( 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]-2.99-3.02-3.08-3.11-3.15-3.15-3.16( 1 11)[1 10]-4.91-4.99-5.20-5.33-5.55-5.57-5.59

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Table6{3: 1]-6.11-5.95-6.20-6.39-6.56-6.75-6.91(111)[0 11]1.031.121.291.511.711.932.10(111)[1 10]-5.08-4.83-4.91-4.88-4.85-4.81-4.81 ( 11 1)[10 1]-1.61-1.48-1.46-1.39-1.33-1.25-1.21( 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]-0.17-0.020.130.360.560.790.96(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]-4.66-4.25-4.15-3.86-3.60-3.29-3.10( 1 11)[1 10]-5.68-5.37-5.44-5.37-5.30-5.22-5.19

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Table6{4: 1]-12.42-14.40-16.41-17.54-21.82-30.65-37.37(111)[0 11]1.822.623.504.026.1010.7514.46(111)[1 10]-10.60-11.78-12.91-13.52-15.72-19.91-22.91 ( 11 1)[10 1]-3.45-3.65-3.82-3.90-4.14-4.41-4.48( 11 1)[110]5.805.745.505.324.341.53-0.86( 11 1)[011]9.249.399.329.228.485.943.62 (1 1 1)[110]-0.68-0.160.460.832.386.029.01(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]-10.09-10.52-10.78-10.88-10.99-10.37-9.50( 1 11)[1 10]-11.91-13.14-14.29-14.90-17.09-21.12-23.96

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Thegoalwastoestimatethefatiguelifeofsinglecrystalnickel-basesuperal-loys,basedonfatiguecrackdrivingforceparameter,forwhichanumericalmodelwasdevelopedtocomputemixedmodeSIFsatthecracktip.Theresultsobtainedcanbesummarizedas 1. Ananalyticalmethodhasbeendevelopedforthecalculationofallthethreemodesofstressintensityfactorsasafunctionofcrystallographicorientationforanorthotropicmaterial,whichcanbeappliedtoanyanisotropicmaterial,ifallthematerialconstantsareknown.Nopriorassumptionofplanestressorplanestrainwasmadeindevelopingthistheory. 2. ModeI,(KI),wasalwaysfoundtobegreaterfor[10 1]thanforthe[ 12 1]orientation.Foranorthotropicmaterial,KIwasfoundtobenon-zeroatcrackclosureduetothecouplingofthenodaldisplacements,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,wascalculatedforthethreeBraziliandiskspecimenandwasplottedonlog-logscalewithda=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: Forcesactingonarectangularelementwithdimensiondxdyinequi-libriumcondition. 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

PAGE 102

where, Assumingthatalltheroots(k)aredistinct,then Thegeneralintegralisequaltoanarbitraryfunctionoftheargumentx+6yandisdenotedbyfv6(x+6y). Therefore, Integratingtheaboveequation,weget

PAGE 103

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

PAGE 105

Basedontheaboveformulas,thegeneralexpressionsforthecomponentsofstressescanbedenedas, Bysubstitutingaboveequationsandintegratingtheresultingequation,wendthefunctionsU,V,Was (A.49) .........fork=1,2

PAGE 107

Thestateofstressandthedisplacementeldnearacracktipisgivenby[ 28 ], 1. Inplane-symmetricloading Re1 Re1 (B.2) 2. Inplane-skewsymmetricloading 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)

PAGE 109

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

PAGE 110

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 ),themixed-modestressintensityfactorscanbefoundas 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)

PAGE 111

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 ),duetomixed-modeloading,thestateofstressatanypointaroundthecracktipcanbegivenas where,

PAGE 112

and3istherootofthecharacteristicequation[ 26 ] Alsozcanbeexpressedfrom A.6 basedonplanestrainassumption(z=0)as Where, Fromequations( B.20 )and( B.21 )itcanbewritten, (B.22)

PAGE 113

EquatingthecoecientsofKI;KIIandKIIIinequation( B.16 )and( B.22 ),itcanbefound Forplanestresscondition(z=0).

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[1] J.Moroso.Eectofsecondarycrystalorientationonfatiguecrackgrowthinsinglecrystalnickelturbinebladesuperalloys.Master'sthesis,UniversityofFlorida,May1999. [2] DanielP.DelucaandC.Annis.Fatigueinsinglecrystalnickelsuperalloys.TechnicalReportFR23800,OceofNavalResearch,DepartmentoftheNavy,August1995. [3] J.R.Davis.ASMSpecialtyHandbook:Heat-ResistantMaterials.ASMInternational,MaterialsPark,Ohio,1997. [4] D.C.StouerandL.T.Dame.InelasticDeformationofMetals:Models,Mechanicalproperties,andMetallurgy.JohnWiley&Sons,NewYork,1996. [5] S.Gunturi.Hightemperaturealloys.http://www.msm.cam.ac.uk/mech-prop/sats.html,UniversityofCambridge,May2002. [6] S.G.Lekhnitskii.TheoryofElasticityofanAnisotropicElasticBody.Holden-DayInc.,1963. [7] G.Dieter.MechanicalMetallurgy.McGraw-HillScience/Engineering,3rdedition,1986. [8] F.EbrahimiandL.Forero.Theeectofmicrostructureonstrainratesensitivityofnickel.InSymposiumonFatigueofHighTemperatureMaterials,pages17{22,Seattle,February2002.TMSAnnualMeeting. [9] G.R.SwansonandN.K.Arakere.Eectofcrystalorientationonanalysisofsingle-crystal,Nickel-basedturbinebladesuperalloys.TechnicalPublicationNASA/TP{2000-210074,NASA,MarshallSpaceFlightCenter,February2000. [10] MasayukiKamayaandTakayukiKitamura.Stressintensityfactorsofmicrostructurallysmallcrack.InternationalJournalofFracture,0:113,2003. [11] J.R.Rice.Apathindependentintegralandtheapproximateanalysisofstrainconcentrationbynotchesandcracks.JournalofAppliedMechanics,35:379{386,1968. [12] H.A.SosaandJ.W.Eischen.Computationofstressintensityfactorsforplatebendingviaapath-independentintegral.EngineeringFractureMechanics,25(4):451{462,1986. 121

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SrikantRanjanwasborninDaltongunjtowninBihar,India,in1977.Alwaysatopperduringhisschooling,heactivelyparticipatedandbaggedseveralaccoladesatvariouslevelsofextracurricularactivities(elocution,essaywriting,scienceexhibition,quizcompetition,dramacompetitionetc.).AfterhisschoolinghejoinedthepremierengineeringinstituteofIndia,IndianInstituteofTechnology,Kharagpur(IIT-KGP)in1995.Thereheactivelyparticipatedin,andorganizedvarioussportsandculturalevents,andwasInter-IITathleticscaptaininhisnalyear.HewasawardedIIT-Blue,ahighesthonor,givenforoutstandingperformanceinsportsactivities.HegothisB.Tech(Honors)degreein1999inManufacturingScienceandEngineeringandjoinedMechanicalEngineeringDepartmentatUniversityofFloridainMay,2000,onaproject,thenfundedbyNASA,topursueMasterofScience.UnderthetutelageofDr.NagarajK.Arakere,hecontinuedhismaster'sprogramtowardsPhDandwishestocontinuehisresearchworkforindustrialapplications. 126