By
Chaiyod Soontrapa
B.S., Economics, Mathematics, and Physics
Carnegie Mellon University, 2001
Submitted to the Department of Materials Science and Engineering
In Partial Fulfillment of the Requirements for the Degree of
Master of Science in Materials Science and Engineering at the
Massachusetts Institute of Technology
September 2005
( 2005 Massachusetts Institute of Technology. All Rights Reserved.
Signature of Author I
Deparment of Materials Science and Engineering
September 2005
Certified by
Certified by
Ronald G. Ballinger
Professor of Nuclear Engineering and Materials Science and Engineering
Thesis Supervisor
.7
Kenneth C. Russell
Professor of Materials Science and Engineering and Nuclear Engineering
Thesis Reader
Accepted by
7 ,_,,,,_ I
MASSACHUSETTS INSTi?Miil
OF TECHNOLOGY
SEP 2 9 2005 l I I IADIc
&.. , I e4L,--c7--~~"~~- H"nd
Ceder
Chair, Departmental Committee on Graduate Students
4
'qCbIS

By
Chaiyod Soontrapa
Submitted to the Department of Materials Science and Engineering
In Partial Fulfillment of the Requirements for the Degree of
Master of Science in Materials Science and Engineering
This study explores the possibility of extending the Ph.D. work of Yan Xu on copper-tin alloys (University of Pennsylvania, 1999) to model stress accelerated grain boundary oxidation (SAGBO) in INCOLOY alloy 908. The steady state model involves the embrittlement along the grain boundary due to oxygen diffusion with the concentration gradient and the stress field ahead of the crack tip as the driving forces. As oxygen forms brittle phases with the segregates in the grain boundary, it reduces the cohesive strength of the grain boundary and causes intergranular cracking in the material.
The extensions to the original model include (1) dependence of oxygen concentration at crack tips on oxygen partial pressure and (2) a new creep law specific to nickel-based superalloys. While the steady state model correctly indicates temperature as one of three leading factors in SAGBO, it fails to capture the effects of the two remaining factors: applied loading and oxygen partial pressure.
Thesis Supervisor: Ronald G. Ballinger
Title: Professor of Nuclear Engineering and Materials Science and Engineering
3

4

1. Introduction ....................................................................................
1.1 Brief review of stress corrosion cracking ...................................................
1.2 Stress accelerated grain boundary oxidation (SAGBO) in INCOLOY alloy 908
2. Cohesive Zone Model ................................................................
2.1 General theory ..............................................................................
2.2 Extension to the original model ...................................... ..........................
3. Numerical Analysis ..................................................................
3.1 Normalization of the governing equations ..............................................
3.2 Numerical evaluation of the integral equations .......................................
3.3 Numerical solution procedure ...................................... ......................
4. Results .......................................................................................
5. Conclusions and Future Research .........................................................
Appendix ...........................................................................................
Reference ...........................................................................................
7
15
15
22
34
37
50
27
27
29
6
6
51
57
5

Nickel-iron alloys are susceptible to embrittlement when exposed to oxygen at immediate temperatures relative to their melting temperatures. The immediate temperatures usually range around one-third or one-half of their melting temperatures.
When put in service these nickel-based alloys also frequently encounter high stresses and fatigue loadings. Synergic effects of stresses (loading or residual), high temperature, and oxygen can induce intergranular cracking in these alloys. This type of cracking belongs to a class generally known as stress corrosion cracking.
1.1 Brief review of stress corrosion cracking
Stress Corrosion Cracking (SCC) is caused by the combined effects of tensile stress and a corrosive environment. In most cases, the stress corrosion cracking is pathspecific such as a path along the grain boundary. SCC failures have been observed in pressure vessels, turbines, and other highly stressed components. The stresses that cause
SCC need not come from the service alone. Highly localized residual stresses resulting from part production including welding can also be a factor in SCC.
SCC only occurs when all of its favorable conditions are met. This requires a matching of a material with a specific environment at above a particular tensile stress level. Examples of pairing between materials and environments are shown in Table 1-1
[10]. Nuclear industry concerns about SCC focus on structural components made of stainless steels and superalloys that are in service for a long time. In high temperature
6

environment and presence of oxygen, chloride aggravates SSC in stainless steel components and hydrogen embrittlement jeopardizes the parts made of superalloys like
X-750 and Inconel 718 (a cathodically driven form of SCC).
Table 1-1 Matched pairs in Stress Corrosion Cracking
Brass
Stainless steel
High strength steel
Ammonia
Chlorides (aqueous)
Hydrogen
1.2 SAGBO in INCOLOY alloy 908
INCOLOY alloy 908 has been selected as a candidate for the structural material for International Thermonuclear Experimental Reactor (ITER). ITER will be the first fusion reactor that can produce thermal energy at the same scale of commercial electricity-generation power stations. Based on the Tokamak design, the major confinement components of ITER are superconducting magnetic coils (18 Toroidal Field
(TF) coils, 6 Poloidal Field (PF) coils, a Central Solenoid (CS) coil, and Correction
Coils). The magnet coil consists of Nb
3
Sn superconducting cables installed inside a jacket. One choice for the jacket material for the CS coil is INCOLOY alloy 908. This type of superconducting magnet coil is called "Cable-In-Conduit-Conductor (CICC)."
The fabrication process for CICC is described as "wind then react." Figure 1-1 shows the fabrication path of a conduit in various stages. After many sections of conduits are welded together, the filaments containing separated Nb and Sn (in a bronze matrix)
7

are pulled in. Then the jacket is compacted along with the enclosed ductile Nb and Sn cables into a smaller square cross-sectional geometry. At this stage the conduit experiences residual tensile stresses. Following the cable bending and winding stage, a heat treatment (usually650 °C for 200 hours) is performed so that Nb and Sn react together to form the brittle Nb
3
Sn.
PD-Mcutak t M(@(-4qn- 4(?imUR
WELDING
CABLE PULLING and
TUBE REDUCTION
CABLE BENDING and
WINDING
AGING
Figure 1-1 Fabrication steps of CICC
8

Stress Accelerated Grain Boundary Oxidation (SAGBO)
Stress Accelerated Grain Boundary Oxidation (SAGBO) in low thermal expansion superalloys such as those listed in Table 1-1 usually occurs in the temperature range 550 °C to 880 °C in environment containing oxygen. Oxygen penetrates along the grain boundaries and cracking occurs by cracking of oxides formed along the grain boundaries or by direct oxygen embrittlement-the two dominant mechanisms that have been proposed. In engineering applications SAGBO is classified as the intergranular mode because intergranular effects are very dominant. The term "SAGBO" first appeared in the work of McMahon and Coffin [3]. They proposed a SAGBO mechanism that oxidation of grain boundaries precedes crack growth. Bricknell and Woodford however suggested another theory that oxygen embrittlement of grain boundaries precedes crack growth [4]. Figure 1-2 compares the two SAGBO theories, which are summarized below.
Table 1-1 Alloy compositions in weight percent
Mo:
903 Bal. 38.0 15.0 1.4 0.9 0.5 0.2 3.0 --- 0.03
903A Bal. 37.5 14.0 1.6 0.02 --- --- Nb+Ta=5.0 0.1 ---
905 Bal. -- 48.9 0.33 1.59 0.04 --- 0.05 4.7 --- 0.01
908 Bal. 4.12 48.7 --- 1.54 1.1 --- 0.09 3.04 --- 0.01 0.17
9

0 a.
Grain Boundary
O2 a.
Grain Boundary b. b.
..... ~
C. C.
A. McMahon and Coffin B. Bricknell and Woodford
Figure 1-2 Schemetic comparison between the proposed SAGBO models of (A)
McMahon and Coffin and (B) Bricknell and Woodford
SAGBO Mechanisms
1. Oxidation Precedes Crack Growth (Figure 1-2A)
An oxide starts to form at the crack tip and preferentially continue down along the grain boundary. The wedge-shaped oxide slowly cracks the grain boundary apart. Oxygen then diffuses to the newly exposed crack tip and the process is repeated. In fact the term
"SAGBO" originated from this mechanism.
10

2. Embrittlement Precedes Crack Growth (Figure 1-2B)
Oxygen diffuses down the grain boundary and embrittles the grain boundary, possibly by brittle phase formation, solute segregation, and removal of strengthening precipitates.
The susceptibility to tension of the grain boundary then increases dramatically. This model assumes residual tensile stresses exist. As the grain boundary weakens, the local tensile stresses then crack the boundary apart. Another succession of oxygen diffusion takes place at the head of the crack tip and embrittles the next section of the grain boundary. The cycle is then repeated.
SAGBO Conditions in INCOLOY Alloy 908
SAGBO occurs in INCOLOY alloy 908 if three conditions are met simultaneously:
1. tensile stress > 200 MPa
2. quantity of oxygen present > 0.1 ppm
3. temperature range 450 °C - 850 °C
The heat treatment process of the superconducting coils (Nb
3
Sn) perfectly fulfills the SAGBO conditions in terms of temperature and stress. If oxygen is present, failure will occur. However, if any one of the conditions is avoided, there will be no SAGBO.
The temperature during the heat treatment is fixed at 650 °C and it cannot be changed.
Otherwise Nb and Sn will not react to form the superconducting wire. So only two parameters, tensile stress and amount of oxygen present, remain as the variables.
11

SAGBO Remedies
The solutions to SAGBO fall into two groups, depending on the SAGBO condition to be eliminated. The first group deals with the residual tensile stress during the production of the conduits. The second group specifically targets the oxygen issue.
1. Solution to residual tensile stress
If the tensile stress is reduced below 200 MPa then there will be no surface cracking. Surface finish modifications such as shot peening can effectively remove any residual tensile stress on the surface. However, the work pieces can only be shot-peened on accessible surfaces. Any irregular or hollow pieces like conduits make the inner surface inaccessible for shot peening.
2. Reduce oxygen content
The very direct way to deal with this is to do the heat treatment in vacuum.
However, a careful monitoring the environment is required because the conduit heat treatment must be done continuously for 200 hours. Alternatively, several feasible remedies exist for SAGBO that affect the oxygen reactions with the alloy internally.
These solutions fall into three categories:
2.1. Chemistry Modification of Alloy
Modifying alloy chemistry helps improve SAGBO resistance to some degree. For example, oxygen getters like silicon retard the kinetics of embrittlement. A weight
12

percent increase of silicon in the alloy would improve SAGBO resistance [19]. However adverse effects on needed properties of superalloys such as thermal expansion coefficient and yield strength may result from composition variations of superalloys. In Nb
3
Sn magnet applications where operating temperature range is narrow, any compromise in thermal expansion coefficients of the alloys may be found unacceptable. Mismatch in thermal expansion between the superconductor and INCOLOY alloy 908 sheath will degrade the superconductor performance.
2.2. Microstructure Modification by Heat Treatment Process
A few ways to improve SAGBO resistance include overaging, increasing grain size, and changing grain orientation. Floreen and Kane reported that overaging Inconel
718 helped improve its SAGBO resistance [5]. The larger grain size and grain orientation transpose result in longer fatigue life. As the grains grow larger, crack growth changes from intergranular to transgranular.
2.3. Thermo-mechanical Process
Repeated cold rolling and annealing on alloys promote coincidence site lattice
(CSL) boundaries [6]. These CSL boundaries possess a special ability to resist SAGBO.
More than 50% CSL of the total grain boundaries can be obtained by doing a series of cold deformation and recrystallization-annealing steps performed within specific limits of deformation, temperature, and annealing time. Materials produced by this process exhibit significantly improved resistance to high temperature degradation (e.g. creep, hot corrosion, etc.), enhanced weldability, and high cycle fatigue resistance. However,
13

processing to achieve high concentration of CSL boundaries is not possible for the CICC application.
1.3 Thesis Objective
This thesis extends the Ph.D. work of Yan Xu on copper-tin alloys (University of
Pennsylvania, 1999) [2] to model stress accelerated grain boundary oxidation in
INCOLOY alloy 908. The steady state model involves embrittlement along the grain boundary due to oxygen diffusion with the concentration gradient and the stress field ahead of the crack tip as the driving forces. Oxygen impurities along the grain boundary weaken the cohesive strength of the grain boundary and result in intergranular decohesion. This thesis extends the original model to include (1) dependence of oxygen concentration at crack tips on oxygen partial pressure and (2) a new creep law specific to nickel-based superalloys. The reasons for these modifications are that the original model assumes (1) abundance of oxygen at crack tips and (2) a creep law that requires an empirical constant from experiments, which is not available for INCOLOY alloy 908.
14

The term "cohesive zone" means the region ahead of the crack tip where the material suffers from creep and diffusion of impurities such as oxygen ingress. The region is sometimes called the "damaged zone" to reflect the material damage involving nucleation, growth, and coalescence of voids in the damage zone especially in ductile material. The idea of the cohesive zone model originates from the works of Dugdale [27] and Barenblatt [28]. Section 2.1 summarizes the formulation of the model for oxygen embrittlement in the cohesive zone described by Xu [2]. Then Section 2.2 presents the extensions to the original model.
2.1 Formulation of the Cohesive Model
Consider a two-dimensional semi-infinite crack as shown in Figure 2-1. In a steady state, the crack propagates along the plane y = 0 in the positive x direction with a constant crack growth speed v and the crack tip always lies at x = 0 (assume that the frame of reference x-y also moving together with the crack tip at the same speed v).
Further assume that the material behaves elastically everywhere except inside the cohesive zone. In addition, the cohesive zone is confined in a finite region of length L and thickness h where the conditions for small scale yielding apply.
15

Figure 2-1 The cohesive zone along the grain boundary
Under small scale yielding conditions, the applied stress d
4 inside the cohesive zone can be written as a function of the applied stress intensity factor KA and the distance x in front of the crack tip r A
A K
K
A
(2-1)
The well-known equation of the normal stress c'(x,t) resulting from the dislocation density p(x,t) in a cracked body has been derived in previous works [11], [12]. Given the small scale yielding, the equation for oD(x,t) is
Cr (x,t) = E
[
(4 ,t) i-f r d: u( (2-2)
For plane strain: E* = E/(1-v
2
), E = Young's modulus, and for plane stress E* = E.
Equation 2-2 can be simplified further to
16

4(X,
o FX x- r d;jx (2-3)
Note that the integral signs in Equations 2-2 and 2-3 are denoted by I. This type of integrals is called a Cauchy principal value integral and is defined as follows. Suppose
a < c < b and f(x, is unbound in the vicinity of = c that any integral in the form b g(x) = j f (x, )d: is an improper integral.
a
Therefore the total normal stress in the cohesive zone resulting from the applied stress and the dislocation stress oP(x,t) is a(x,t) = KA E*
72=,r 4 o
(,t)d
Jx x-
(2-4)
Notice that inside the cohesive zone the values of o(x,t) are unbounded when x = 0 or x =
L when L represents the length of the cohesive zone. Thus o(O,t) and u(L,t) must vanish so that the stress is finite inside the cohesive zone. This leads to the Dugdale condition
[13]
KA E* ( t)d
- 2
L U( t)
7r f
' d-~d (2-5)
Substituting KA = E * P j 't d in Equation 2-4 results in 22;047
5(x,t) =- i(
4r'o o(x,t) = E*L J
4;oJ x -
+-f J-
4;ro X- -
(2-6) d; cr(x,t)
"d5 (2-7)
17

#(x, t) =
4 iL-X x(,t)d
E*7izo -x
Equations 2-7 and 2-8 represent the stress and dislocation density in the cohesive zone respectively.
So far the equations describing the stress and dislocation density in the cohesive zone are obtained. However, each equation contains two unknown variables and cannot be solved. Another relation is needed in order to transform either Equations 2-7 or 2-8 into one equation with one unknown. This requires a constitutive law for either stress
o(x, t) or dislocation density u(x, t). Norton's creep law can describe the relation of the strain rate as a function of stress. Then the only remaining tasks are to find a relation to link the strain rate to the dislocation density and transform either Equations 2-7 or 2-8 into one equation with one unknown.
One definition of the dislocation density p(x,t) can be written in the form [2]:
,u(xt t)=_ a5, (xt = -2 au (x, t) ax ax
(2-9) where S(x,t) = 2uy(x,t) is the cohesive zone opening as shown in Figure 2-2. In addition, the cohesive zone opening rate a is related to the strain rate
£ by:
= h
(2-10) where h is the thickness of the cohesive zone. To proceed further, the relation between the strain rate
£ and the stress o(x,t) in the cohesive zone is assumed to obey a Norton's creep law: t 1]
17
(2-11)
18

where B is a constant and r* is a reference stress, say the tensile stress of the material.
.
8(x, t)
1 x
Figure 2-2 Opening of the cohesive zone
In the case of a crack opening with speed v in the x direction where the crack tip always stay at x = 0, the cohesive zone opening is described by [13]
(, t)= -v-3(x, t) + ax a
S(x,t)
,at
When a steady state is assumed, the above equation reduces to
(2-12)
(2-13) ax
By applying Equation 2.2.9, the above expression becomes
6(x) = vt(x)
Combining Equations 2-10, 2-11, and 2-14 yields
(2-14)
19

I(X)
6(x) hh VU(X) = 1] Lo*i
Therefore the dislocation density as a function of stress in a form of the power law is
(2-15)
Va(X) = B ( ) - 1 (2-16) where B = Bh.
To reflect the effect of oxygen embrittlement, the damage function is applied to
Equation 2-16. The damage function can be defined as a simple Kachanov-type damage function [2] f(X)
= I
C(X) (2-17) where C(x) is the oxygen concentration ahead of the crack tip at a location x and Co is the oxygen concentration at the crack tip. Replacing r* with fo*, the modified power law in Equation 2-16 is then
(2-18)
Equation 2-18 postulates that the oxygen concentration causes decohesion of the grain boundary. When the concentration is zero, the grain boundary maintains its normal strength r*. However, when impurities present, the strength of the grain boundary reduces tofDo*. The degree of damage depends on the exponent k in the damage function Equation 2-18 [15]. The value of k ranges from zero to infinity. When k equals to zero, the strength of the grain boundary remains unchanged. When k equals to infinity, the grain boundary fails wherever there exists the oxygen ingress.
Finally, combining Equations 2-8 and 2-18 yields
20

4v 'L x
E ro L- -x
(2-19) BI (x) 1I
J or equivalently
E* $ ( t)
4zo0 x-: d = (
B
+1 (2-20)
Both of Equations 2-19 and 2-20 are in a form of one equation with one unknown.
Specifically, the unknowns in Equations 2-19 and 2-20 are and respectively. Solving both Equations 2-19 and 2-20 requires a numerical method because their closed form solutions do not exist.
Based on Figure 2-1, the oxygen diffusion equation along the grain boundary with the presence of the stress is [17] ac a at x
(
D ac DbQo aox
,-- T au
(2-21) where
Db = oxygen diffusion coefficient along the grain boundary
LA = atomic volume of the alloy kB = Boltzman constant
, = stress along the grain boundary
Under a steady state with a cracking speed v, the time-dependent in Equation 2-21 is given by [16] dC at
Substituting Equation 2-22 in 2-21 gives dC ax
(2-22)
21

d D0C f!DfI± DbQaCSl dx b ax
Db, f kBT o
Ox
If
Db and ado not depend on C, the closed form solution for Equation 2-23 is [17]
(2-23)
C(x) = exp- x + r(x)
Db kB
IA
A
2 fexpL
-
0
Db kBr
The constants Al and A
2
C(oo) = O0. The results are Al = Co exp(-oo) and A
2
= O0. Accordingly the closed form solution of Equation 2-24 becomes [2]
(2-24)
C(x) = C o exp -- xx (a(x)- cO) (2-25)
2.2 Extension to the original model
Section 2.1 completes the formulation of the original model with the abundant source of oxygen. Thus oxygen concentrations at crack tips remain constant at all times.
However it is generally known that the oxygen concentration at the crack tip depends on the partial pressure of oxygen. Sieverts' law describes the concentration of dissolved oxygen in equilibrium with gaseous oxygen is related to the partial pressure P of oxygen
[18]
C = sP
2
To derive Equation 2-26, first consider the equilibrium between gaseous oxygen and
(2-26) dissolved oxygen
1
2
2
(gas)
*->
O(solution) (2-27) with the equilibrium constant
22

K = aC p2
(2-28) where a is the thermodynamic activity constant of the dissolved oxygen.
By rearranging the terms, Equation 2-28 appears in the same form as Equation 2-26 with
K
S = -.
a
Let C and P be the concentration and partial pressure at the crack tip at a reference state, say 0.21 atm at 650 °C. If the parameter s does not vary with concentration then it follows from Equation 2-26 that:
C r
Co L
PoE
POJ
Therefore Equation 2-25 can be rewritten as
C(x)
C(x) = C Po exp- + ((X)-
Db kBT
(2-29)
(2-30)
In addition to the modified equation for oxygen concentration, this study introduces one particular form of the power contains parameters that have physical basis
[14]
= AD[ YSFE U -C1 effGb E
The above equation can be rewritten as:
£7 = A YSFE
.Z ADeff a7
E Gb
L
07 r
Equivalently,
(2-31)
(2-32)
23

BL -1] (2-33) and BAef l [bJ where
A = Structure-dependent parameter
Deff = Bulk diffusion coefficient
= Reference stress
E = Young's modulus
YSFE = Stacking fault energy
G = Shear modulus b = Burger's vector
A is a function of the volume fraction (Vf) of y' and y":
A =
34(2-34)
Vf+ 0.005
Note that the exponent has been changed to 34.1 from the original value of 28.7. The exponent value of 28.1 does not yield any solution to Equations 2-19 or 2-20. Therefore an adjustment is made by varying the exponent until the first solution is found and the new exponent turns out to be 34.1. Table 2-3 shows the various parameter values of the standard condition used in the numerical calculations. Many of the parameter values appear in the Incoloy alloy 908 data handbook [20]. The burger's vector is given by
Lattice Parameter
2
24

Table 2-1 Parameter values for the standard condition
Mesh points
Creep exponent
Creep special exponent
Damage function exponent
Gas constant (J/mol-K)
Boltzmann constant (J/K)
Temperature (K)
Melting temperature (K)
Volume fraction of y'
Cracking speed (m/s)
Cohesive zone thickness (m)
Applied stress (Pa)
Tensile stress (Pa)
Young's modulus (Pa)
Stacking fault energy (J/m 2
)
Shear modulus (Pa)
Burger's vector (m)
Specimen pre-cracked length (m)
Specimen width (m) kB
T h
Tr
Yf
V b a
W m kl
R
N n
E
E
SFE
G
4
100
3
2
8.3145
1.38 x 10
-
23
650 + 273.15
1650
0.2
1 x 10 ' 6
1 x 10
- 3
650 x 106
1000 x 106
185 x 109
1 X 10
-3
70 x 10 9
2.5 x 10
-
1
°
1.905x 10
-3
6.35x 10
-3
25

This study assumes the test specimen is a rectangular plate of width W with a through-thickness, single edge crack of length a subjected to a tensile stress o. The height H of the specimen is greater than or equal to 2W. The stress intensity factor at the crack tip is given by [21]
(2-35)
W
= 1.99
W W
- 38.48() + 53.85-) t tJ t t
Figure 2-3 Single-edge-cracked tension specimen
26

Since closed form solutions do not exist for Equations 2-19 and 2-20, they must be solved numerically. The Gaussian Quadrature formula becomes useful in the numerical calculation. Equations 2-19 and 2-20 fit into one of the special cases in the
Gaussian Quadrature formula, which requires normalization of the integration interval to
[-1, 1]. To reduce complexity in the problem, other important parameters in equations are normalized as well. The final form of Equations 2-19 and 2-20 become the summations of various terms and must be solved iteratively for a solution at each mesh point. Since
Equations 2-19 and 2-20 are equivalent, Equation 2-20 was chosen to be evaluated for its solution in this thesis due to its ease of calculation (the exponent 1/n term in Equation
2.20 vs. the exponent n term in Equation 2-19). The sections below follow closely to the numerical treatment by Xu [2] except where noted.
3.1 Normalization of the governing equations
A special case in the Gaussian Quadrature formula requires the integration interval to be [-1, 1]. Therefore the variables and end points in Equation 2-20 must be changed accordingly. Define the new variables .x and r as x = --
L
1 and ; =
L
1.
The new variables for steady-state stress and dislocation density are u(x)=
0'* and
27

4z *
O' * with =-. Also define the dimensionless cracking speed as
4z * v
V =
B
Therefore Equations 2-19 and 2-20 become:
7r i1 lX
1I ci+~3~d f= fD
1In d;
= fD (al (X
(3-1)
(3-2)
Define the dimensionless stress intensity factor as =
KA
The Dugdale condition
O'*'~ 'L in Equation 2-5 becomes
- =ix d
I d;-
=- d;
(3-3)
Thus the length of the cohesive zone can be calculated once fi is known by using
L= K
(3-4)
The dimensionless cohesive zone opening is defined as a (x)= =2J(k
(3-5)
Consider Equation 2-30 for the oxygen concentration in the cohesive zone and define the vL following new variables: = -- ,
2D b
= kBT
, and o = . By normalizing the
* concentration at any point in the cohesive zone, Equation 2-30 becomes
C
C( = *P°,/
(PO
(3-6)
28

And Equation 2-17 for the damage function is now: fD (1-x (37)
3.2 Numerical evaluation of the integral equations
Take Equation 3-2 as the equation of interest:
___d
= fD (+/(ix))n
It is impossible to obtain a closed form solution for Equation 3-2. The Gaussian
Quadrature formula provides the best numerical estimate of an integral, which the numerical analysis for Equation 3-2 will be based on. The formula has a general form of: b a
W (x)f(x)dx=
N i= i f (xi) (3-8) where the function W(x) represents the weight function and the function fx) is estimated using the Lagrange Interpolating Polynomial.
The Lagrange Interpolating Polynomial is described as follows. Suppose there are N+I points given by yo=f(xo), yl=flxl), y2=f(x
2
),..., YN =flxN). A polynomial of degree N passing through the N points is called a Lagrange interpolating polynomial.
The polynomial PNI(x) has the form:
N f (x) (x)= li (x)f(x i=l i where
(3-9)
( (TN (X)
4
(xx X( X
29

;TN (Xi) = (i - Xl Xxi -X2)...(Xi - Xi-l XXi- Xi+ -XN ) or explicitly
(X-X, X-
X2)..(X-xi
Xx-xi+,) ...
(X XN )
ii W = (
(Xi XI
XX -XI) * (X -I XX
XXi - X)" (Xi - Xi-I
)(
XXi - XiJ .. (Xi - XN)
Note that li(xj)=
{ i=j
For a special case, if the weight W(x)= -- l-X
2 over [-1, 1] then Equation 3-8 is written explicitly as [1] f f(x)dx= f
(Xi)+
( 1X(
U f
(2n)
(3-10) where x i
=cos (2i - 1,i=,2,...,N and the last term represents the error on the integral approximated by the summation. Note that in this case wi=
N
The special case for the Gaussian Quadrature formula has the integrand in the b
Cauchy principal value of the form W(x) g(x)dx, a < X < b. Remember that the g equation of interest is in the form:
)(
-1i+ d; or specifically the term f I
I
-+<x3-
So if W()= 1¾2 then
W -"
30

Assume X is distinct from xj, x2, ... ,XN. The Lagrange polynomial of g(x) can be written as g(x)= -li(X
) x- g(x )+ i=l i
-A
;
N x) ffN(2) g(A)
P fW(x)
1 ii(x)x g(xi)+ g()] dx g(xi) W(x i=1 Xi -- a
.TN a
(dx
X-i b
Applying the Gaussian Quadrature formula to the term W(x)i (x)dx, the above a expression becomes
Recall that li (xj)
={ i=I Xi I =1
Wji (Xi +
7rN (A) a
,i=j
,ij=i
X-A(
N
The term ' w i
(x ) reduces to w for i = j. This dictates by the main summation with j=l the index i. And the final result is
N i=,
Wg(') xi (),
JW(x )dx x-.A
Therefore, b
P W(x) g(x) dx
=
N z
Wi i=1 a g(x..) + g() jW(x)Xdx
N ( ) X-
31

Consider the last term on the right hand side of the above expression
Given the weight function W(x)as W(x)=
41-X 2 and [a, b] = [-1, 1]. The function zT
N
(x) can be written as
N (x)- 2= lTN
(x)
2N1 where TN(x)= cos(Ncos-' x) with x i
= os((2i - ) i=1, 2,
i
N
The function TN (x) is called the Chevbyshev function of the first kind. As a result, g_ g
I
_ g({)
2NI TN(A)1
I
I
-
N-
TN )
2____ x-A
Note that the following relation exists between the Chevbyshev functions of the first kind
TN(x) and the second kind UN_ (x) for -1 < X < 1 [22].
-
TN
() dx
1_x 2 X-2 rU
N-1 (A)
0
,N>2
,N=O
Therefore, g(A)
JwNx)dx
;N (A,)_ x - I
=
=
TUN () g(Z)
TN ()
This leads to:
I
1
N
1 g(x)dx = wi i=l g(x_ xi,-,
Recall that the original integral equation of interest is
32

d-- =f (u())n
+1] w( ) I
then g(0)= i7i().
l
So,
= -1
,1 ,\ +- X d
N j=1 N _i +
TN ()
X--;
Finally,
=1N r
-
-T i TN( )/
1
)+fD ( ()
+1] =0
Xu numerically solved the above equation in her thesis. However, the expression can be further simplified so that the Chevbyshev functions disappear. The Chevbyshev function of the second kind UN_ (X) can also be expressed as [23] sin(Ncos -' x) sincos-' x)
_ sin(Ncos-' x)
_x2
Therefore, sin(N cos-' x)
1
UNI_ (X)
Ultimately,
TN (X)
_1-x cos cos-' x)
)),( )+ fD[
_-x 2 %(
))In + = 0
(3-11)
Ni r j=
1
1 l()+ztan(Ncosl(x
N ~i-x3
Equation 3-11 was numerically solved for in this thesis.
]i
33

3.3 Numerical solution procedure
Equation 3-11 approximates the original integral equation. To proceed further in numerical evaluation, Iet (2i -1)7 The value of each , say xk, can also be selected as = cos(2k - 1) with M
X
N so that each xk is distinct from each Xi per the assumption made earlier. This leads to TN
(k ) = cos(N cos' -k )= 0 for all xk.
Equation 3-11 then reduces to
N j= N
Hi i k
( ) + f, [('FU(3k ))n + 1= 0
Although the calculation is less complex, another problem arises. There are M equations from xk to be solved for N unknowns from hi but M # N. Therefore k cannot be selected as k = cos ( with M N. In order to make k distinct from :i, x~k can be simply x5k
= rk
2
_k1.
Since :i
E
[-1, 1], define r0 = 1 at the cohesive zone tip.
As mentioned earlier the dislocation density vanishes at the cohesive zone tip. Hence,
Wo = 0. With k 's and Z 's as defined, the N equations are readily available to be solved for N unknowns for (Xk ).
Let q/(k ) - [v/u(xk )]n be a stress-like variable [15]. Therefore Equation 3-11 becomes
+ BkV
(3-12)
34

where = )
C k + Ik-l
2
Ai -
Bk = tan(N cos (k))
When finding a set of the solutions for the k 's, some of the /k 's may be negative.
Since V/(Xk
=
)] fD fD
1 > 0 at all time in the cohesive zone, another condition must be imposed in order to ensure that the stresses in the cohesive zone are always greater than or equal to zero. Therefore Equation 3-12 with the new restriction is
N
+1XEk AkiAY , Bk +v Yk +f)=0 for k
20
/k
= 0 for k
<
0
To find the solutions, first letfD = 1 at all xk 's and simultaneously solve the N equations. Once the solutions are found, a new set offD's will be determined becausefD depends on C which in turns depends on
Wk.
By iteratively solving the set of N equations with the updatedfD's, the numerical solutions to Equation 3.1.10 can be found at the desired convergent point. In this thesis, the mathematic program "Maple" version 9 is used in finding the solution to Equation 3-12. The actual Maple code written for this thesis is shown in Appendix.
35

After obtaining the stresses at all k
's, the length of the cohesive zone and the opening of the cohesive zone at the crack tip can be calculated by transforming Equations
3-4 and 3-5 into Riemann sums. The cohesive zone length is given by z
(3-13) where Aj = 5 - Ti-
And the cohesive zone opening at the crack tip is given by
N
() = 2ir *L,(iE ) i=l
The results from the numerical calculations are shown in the next chapter.
(3-14)
36

This chapter presents the results from the extended model on INCOLOY alloy
908 (the original model by Xu was applied to copper-tin alloys). Figures 4-2 to 4-4 show the results from varying different parameters and the actual numerical values are given in
Tables 4-1 to 4-6. The parameters of interest are temperature, stacking fault energy, cracking speed, thickness of cohesive zone, partial pressure of oxygen and applied load.
The normal stress is scaled relative to the tensile stress of the material. The partial pressure of oxygen is also scaled relative to a reference partial pressure, say partial pressure of oxygen in the air at 1 atmosphere. Figures 4-5 to 4-10 present the normal stresses resulting from the parameter variations.
The results for various temperatures show that higher temperature leads to
· a weaker cohesive zone
· a longer length of the damage zone
* a wider cohesive zone opening at the crack tip
The above results are expected because higher temperature eases the oxygen ingress into the grain boundary due to the increased mobility of oxygen atoms. Therefore damage due to oxygen embrittlement becomes much more severe as the temperature rises.
For the various values of stacking fault energy, the grain boundary becomes strongest at the lowest stacking fault energy. The cohesive zone length is also smallest at the lowest value of stacking fault energy. The grain boundary with high stacking fault energy provides an easy path for oxygen ingress. Therefore the degree of oxygen
37

embrittlement very much depends on the level of stacking fault energy. In alloys with high stacking fault energy, dislocations are permitted to cross-slip easily when they encounter obstacles such as other dislocations or second-phase particles [26]. This promotes slip-band deformation and large plastic zone at crack tips, resulting in crack initiations and propagations. Alloys with low stacking fault energy constrain dislocations to move in a more planar fashion. The results also show that crack tips do not become wider as stacking fault energy increases. Therefore damage should occur in a form of very fine cracking at high stacking fault energy. This type of damage is hard to detect and very catastrophic.
For the various cracking speeds, the results indicate that as the cracking speed increases the degree of damage decreases. There is less oxygen ingress along the grain boundary as suggested by the shorter length of the damage zone, and the grain boundary is able to retain its strength. The crack tip opening is also smaller at a higher cracking speed. These results agree with what Huges found that the degree of the damage in the alloy depends on the loading frequency [24].
For the various cohesive zone thicknesses, the calculations confirm that the degree of the oxygen embrittlement damage intensifies as the oxygen ingress extends across the bulk material. The results from the thicker cohesive zone are identical to the results from the higher temperature in terms of damaging trend. That is, either high temperature or thick cohesive zone leads to deleterious impacts on the alloy (weaker grain boundary, larger damage zone, wider crack tip opening). It is also important to note that perhaps, in this model, there exists a lower limit on how thin the cohesive zone can
38

be. The model did not give any solution for the cohesive zone thickness roughly below one micron.
The results for partial pressure and applied load imply that these two parameters do not play very important roles in this oxygen embrittlement model. The higher applied load leads to a longer cohesive zone and a larger crack tip opening. However, the alloy is able retain its strength somewhat. On the other hand, partial pressure of oxygen does not seem to affect the alloy at all. The values for normal stress, cohesive zone length, and cohesive zone opening at the crack tip remain virtually unchanged as the oxygen partial pressure varies.
The results also show that in all cases the cohesive zone lengths are in the order of
1 millimeter. This agrees very well with the published data on various superalloys.
Figure 4-1 shows the damage zone sizes in some superalloys from the paper by Liu et al.
[25]. They also claimed that the formation of the damage zone is a thermally-activated process.
2
E
1.5
.d
U) c 1
E 0.5
Q
0
500 600 700
Temperature (C)
800
Figure 4-1 Temperature effect on the damage zone sizes of the alloys
39

o
0
12a
0 z z
E
M
0
500 600 700
Temperature (C)
800 90 0
3.0
2.5 .............
2.0 ............
1.5 ............
0.5
-............
M0.0
0.0 0.2 0.4 0.6 0.8
Stacking Fault Energy (J/m
2
)
1.0 1.2
5.0
M 4.5- ------------------------
4.0 ..........................
03.5
E 2.5
2.0
Z 1.5 .............
M 1.0
E .5 - - - - - - - - - - - - - - - - - - - - - - - - -
M 0.0
O,.OE+00 1.OE-05 1.2E-05
Cracking Speed (m/s)
12
0
*
0
10 _____________
E z
2
E
0X
0.000 0.002 0.004 0.006 0.008 0.010 0.012
Cohesive Zone Thickness (m)
0
, 2.895
i 2.890
2
2.885
2.880
' 2.870
E 2.865
2.860
M 2.855
0.0 0.2 0.4 0.6 0.8 1.0
Oxygen Partial Pressure (Po/Po*)
1.2
-
2.72
2.71
2.70--
e 2.69
z 2.68
E
E 2.67
: 2.66
400 450 500 550 600
Applied Load (MPa)
650 700
Figure 4-2 Effects of different parameters on maximum normal stress
40

t:.OC>UJ c _ _rn _
2.4E-03 -- -
" 2.2E-03-- -
0 2.OE-03 -- -
C 1.8E-03
0
N 1.6E-03
- - -
·*i, 1.4E-03-
0 o 1.2E-03-
1 -
"-- L-K --
500
- - - - - - - - -
600 700
Temperature (C)
800 900
0.0035
E 0.0030
-
C
, 0.0025
-j 0.0020
N 0.0015
> 0.0010
V- - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
0 o 0.0005
0.0000
0 0.2 0.4 0.6 0.8
Stacking Fault Energy (J/m
2
)
1 1.2
1.7E-03 a)
--
0 a)
E r1.5E-03 -- -
1.5E-03 .
.…..........
_. _ _ _ _
…_
_ .
__
1.3E-03 - - - - - - - - - - - -- - a) a 9
0
0
1.1E-03-
1.OE-03
0.OE+00 2.0E-06 4.0E-06 6.OE-06 8.0E-06 1.0E-05 1.2E-05
Cracking Speed (m/s)
1.7E-03
E 1.6E-03- _
, 1.5E-03 -- -
- - - - - - - - - o 1.3E-03 - --------------__-.
> 1.2E-03
1.1E-03 _
1.OE-03
0.000 0.002 0.004 0.006 0.008 0.010 0.012
Cohesive Zone Thickness (m)
1.402E-03
1.401E-03 I-
c
N
> a o
' 1.400E-03
c 1.399E-03-
1.398E-03
1.397E-03-
1.396E-03-
1.395E-03
1.394E-03
------------ ----------
1.393E-03
1.392E-03
--------------------- ----
1.391E-03
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Oxygen partial Pressue (Po/Po,)
1 f n-
E 1.4E-03-
- - - - - - - - - - - - - - - - - - - c 1.2E-03-
- - - - - - - - - - - - - - - -
- - - -
0 1.0E-03-
-J
- - - - - - - - - - - - - - - - - c 8.OE-04
0
N 6.0E04
- - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - o 4.OE-04 -
- - - - - - - - - - - - - - - - - - - - - - - - - - o 2.0E-04
- - - - - - - - - - - - - - - - - - - - - - - - - - -
O.OE+00
400 450 500 550 600 650 700
Applied Load (MPa)
Figure 4-3 Effects of different parameters on cohesive zone length
41

3.00E-05
2.0E-05 -- -
' 1.50E05-
0.
o 5.00E-06
0.OOE+00
500 600 700
Temperature (C)
800
_
900
1.00E-01l -... ___ __ _-----....
0
I
1.005-07
- -
--.........-
, 1.00E-09
------ --
- -..--.....----------
1.00E11 --
.
1.OOE-15
0 0.2 0.4 0.6 0.8 1 1.2
Stacking Fault Energy
E
0) r ct a
-
0
1.80E-05
1.60E-05- _ _ _ _ _ _ _ _
1.40E05- __
1.20E-05-
1.00E05
- - - - - - - -
- - - - - - - - -
8.00E-06 - - - - - - - - -
6.00E06 f___________
0 is
4.00E06-
2.00E06-
0.00E+00
Cracking Speed (m/s)
1.60E-05
- ------
E 1.00E05 -
O 8.0006 ..........-----------.
__ -------_
..
2 4.00E06 - . .
.
2.00E-06- ........................
0.00E+00
0
- - - - - - - -
0.002 0.004 0.006 0.008 0.01 0.012
Cohesive Zone Thickness (m)
1.08E-05
-
1.07E-05 c 1.06E-05 a o 1.03E-05
1.02E-05
0 0.2 0.4 0.6 0.8 1
Oxygen Partial Pressure (Po/Po*)
1.2
1.20E-05
^ 1.00 05 -- - - - - - - - - - - - - - - - -
- --------
O 6.0006 fr- 4.00E-06 ........................
____________
2.00E-06 - - - - - - - - - -
0.00E+00
400 450 500 550 600 650 700
Applied Load (MPa)
Figure 4-4 Effects of different parameters on crack tip opening
42

;
E
0 z
0 6
I6.
Cn
X 5
4
3
2
8
7
10
9
I
0
0 0.1 0.2 0.3 0.4 0.5 0.6
Distance from crack tip
0.7 0.8 0.9 1 r-
Figure 4-5 Normal stress resulting from variation in operating temperature
>-
-: It
2.5
2
0
1.5
1
0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6
Distance from crack tip
0.7 0.8 0.9 1
Figure 4-6 Normal stress resulting from variation in stacking fault energy
43

C,
Cn a)
3.5
3
Cl)
2.5
E
5
4.5
4
1.5
1
0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6
Distance from crack tip
0.7 0.8 0.9 1
Figure 4-7 Normal stress resulting from variation in cracking speed
8
0
0)
(0
-a
E
0
6
4
121
10
--- h=1*10A(-5) m
h=1*10 A(-4) m
h=1*10^(-3) m
- - h=1*10^(-2)m
2
.
~ ~
.%1"L--
0 0.1 0.2 0.3 0.4 0.5 0.6
Distance from crack tip
0.7 0.8 0.9 1
Figure 4-8 Normal stress resulting from variation in thickness of cohesive zone
44

u,
0.
3
2.5
3-
2.5 -
(0
(0
2
-
E
1.5
1-
P02 = 0.0001
P02 = 0.001
-
-- P02 = 0.01
--- P02 = 0.1
P02 = 1
0.5
0
0
__ T
0.1
I
0.2
I
0.3
I
0.4
1
0.5
I
0.6
Distance from crack tip
I
0.7
I
0.8
I
0.9
Figure 4-9 Normal stress resulting from variation in partial pressure of oxygen
2..
9 ..........
-..
--
Load = 550 MPa
Load = 650 MPa
--- Load = 750 MPa
0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6
Distance from crack tip
0.7 0.8 0.9
Figure 4-10 Normal stress resulting from variation in applied load
45

Table 4-1 Variations in temperature
Exposed Maximum Normal Cohesive Zone
Temperature (C)
550
Stress
9.608873
Length (m)
0.00118
650 2.715062 0.001390
750
850
1.499979
1.187387
0.00189
0.00235
Crack Tip Opening (m)
2.81 x 10-6
1.08 x 10 5
-
1.87 x 10-5
2.40 x 10 -5
Table 4-2 Variations in stacking fault energy
Stacking Fault
Energy (J/m 2 )
0.001
Maximum Normal
Stress
2.715062
Cohesive Zone
Length (m)
0.001392
Crack Tip Opening (m)
1.08 x 10
-5
0.01
0. 1
1
1.268728
1.046251
1.008175
0.002249
0.002786
0.002913
2.10 x 10
-8
3.90 x 10
'
12
4.02 x 10 '
Table 4-3 Variations in cracking speed
Cracking Speed Maximum Normal Cohesive Zone Crack Tip Opening (m)
(m/s)
10
°
Stress
4.547205
Length (m)
0.001260 8.24 x 10-6
To w
0.001392
0.001602
1.08 x 10-5
1.29 x 10-5
46

Table 4-4 Variations in thickness of cohesive zone
Cohesive Zone
Thickness (m)
2x 10
- 5
10
-5
10
-4
Maximum Normal Cohesive Zone Crack Tip Opening (m)
Stress Length (m)
11.135956
7.312862
4.284449
0.001095
0.001157
0.001244
3.47 x 10-6
4.40 x 10 -6
10-3
10-2
2.715062
1.910933
0.001392
0.001632
7.05 x 10-6
1.08 x 10
-5
1.48 x 10
-5
Table 4-5 Variations in partial pressure of oxygen
Oxygen Partial
Pressure Ratio
0.001
Maximum Normal
Stress
2.888739
Cohesive Zone
Length (m)
0.001401
0.01
0.1
2.885988
2.879393
0.001400
0.001398
1
.
.
2.858693 0.001390
....
.
Crack Tip Opening (m)
1.03 x 10-
'
1.04 x 10
-5
1.05 x 10
-5
1.08 x 10
- 5
47

Table 4-6 Variations in applied load
Applied Load
(MPa)
450
Maximum Normal Cohesive Zone
Stress Length (m)
2.664526 0.000663
550
650
2.693720 0.000995
2.715062 0.001390
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~....
Crack Tip Opening (m)
5.34 x 10
-6
7.82 x 10
-
6
1.08 x 10 -5
The results show that four parameters, namely temperature, stacking fault energy, cracking speed, and thickness of cohesive zone, strongly affect the cohesive zone strength. The other two parameters, partial pressure of oxygen and applied load, do not exhibit significant impacts on the cohesive zone strength. Recall the three conditions for
SAGBO in INCOLOY alloy 908 that (1) tensile stress > 200 MPa, (2) quantity of oxygen presented > 0.1 ppm, and (3) temperature range 450 °C - 850 °C. According to the steady state model, temperature stands out as the most influential factor. Applied loading does not have a very significant effect, and oxygen partial pressure does not seem to take any effect at all. This hints that the model cannot capture the effects of oxygen partial pressure and applied loading.
One possible reason for the model's deficiency arises from the model's steady state assumption. Given the time to reach equilibrium, oxygen concentration profiles inside the cohesive zone will approach a constant value, the solubility limit of the alloy, regardless of the initial oxygen concentration conditions. As a result, the damage function will yield the same damage profile for all different oxygen partial pressures.
48

Consequently, all cases with different oxygen partial pressures will have the same results on cohesive zone strength, damage zone length, and crack tip opening.
In addition to the effects of oxygen partial pressure, the steady state model also fails to demonstrate the impacts of applied loading. The applied loading is directly related to the applied stress intensity (Equation 2-1), which in turn influences the cohesive zone length as shown in Equation 3-4. Therefore, in the steady state model, applied loading only has a direct impact on the cohesive zone length, as confirmed by the calculation results. Although the applied stress intensity factor has an explicit relation with the stress inside the cohesive zone as shown in Equation 2-5, the resulting cohesive zone strengths from the model do not show noticeable differences for various applied loadings. This finding can be attributed to the fact that, instead of Equation 2-5, the calculations only employ Equations 2-19 or 2-20 in which the dependence on the applied stress intensity disappears.
49

The results from the extended model show that the strength of INCOLOY alloy
908 strongly correlates with temperature, stacking fault energy, cracking speed, and thickness of cohesive zone and weakly correlates with applied load. The partial pressure of oxygen does not seem to have any correlation at all. Yet, it is known that oxygen plays a dominant role in SAGBO, not significantly less than what the temperature does as suggested by the model. This discrepancy may arise from the model itself. Recall that the creep is assumed to be in steady state. In fact it may always be transient when the oxygen resource is limited. Therefore the extended model inadequately explains the
SAGBO mechanism. This leads to a need to develop a fully time dependent cohesive zone model that is expected to give a better correlation on the oxygen and applied loading effects.
50

51

restart: with(CurveFitting):
N:=100: #Number of mesh pts n:=4: #Creep exponent m:=3: #Second exponent in creep eqn kl :=0. 1: #Exponent in damage fcn
R:=8.3145: #Gas constant
T:=650+273.15: #Operating temp
Tm:=1650: #Melting temperature
Vf:=0.2: #fraction of alpha prime phase
A_:=10.0A34.1/(Vf+0.005): #Constant in creep eqn
Dmt:=19.06*exp(-338826/(R*T)): #Diff coeff in matrix
Dgb:=10A(-4)*exp(-9.35*(Tm/T)): #Diff coeff in grain boundary v_:=1*10(-6): #Cracking speed h:=10A(-3): #Cohesive zone thickness s :=650*10A6: #Load sO_:=1000*10A6: #Yield strength
E:=185*10^9: #Young's modulus
SFE:= 10^(-3): #Stacking fault energy
G:=70* 10^9: #Shear modulus b:=2.5* 10(- 10): #Berger vector
B_:=A_*Dmt*h*(sO_/E)An*(SFE/(G*b))Am; #B variable in creep eqn v:=4*evalf(Pi)*(sO_/E)*vJB_; #Scaled cracking speed
Q:=1.5* 10^(-29): #Atomic volume aa:=0.001905: #Pre-cracked length
WW:=0.25*0.0254: #Specimen thickness f_KA:= 1.99-0.41 *(aa/WW)+18.7*(aa/WW)A2-38.48*(aa/WW)A3+53.85*(aa/WW)A4:
KA:=s_*sqrt(aa)*fKA; #Stress intensity factor
P02:=1.0: #Relative partial pressure of oxygen
#The following code serves as the phaser.
#When a system of equations is solved, Maple gives the solution as
{x[1]=a[1],x[2]=a[2],... }
#This code helps extracting the values of a[1], a[2], etc and assign them to x[1], x[2], etc.
sol:=[]:
WWXans:=Table(): assnWWXans:= proc(sol,WWXans) local i, j, stringtemp, target,temp, templ, temp2, temp3, temp4, num, numl, num2, match 1, match2, tl, t2:
52

unassign('i'): unassign('j'): for i from I t:o N do stringtemp:=WWX[i]; target:=convert(stringtemp,string); forj from I by 1 to N do temp:=convert(sollj] ,string); num:=length(temp); temp l :=substring(temp, 1..6); temp2:=substring(temp, 1. .7); temp3:=substring(temp, 1. .8); if substring(templ,6) = "]" then match 1 :=templ; match2:=substring(temp, 1O..num); elif substring(temp2,7) = "]" then match 1 :=temp2; match2:=substring(temp,l 1..num); elif substring(temp3,8) = "]" then match 1 :=temp3; match2:=substring(temp, 12..num); end if; num2:=length(match2); if match 1 = target then if (match2[num2-2] = "e") then tl :=substring(match2, 1..num2-3); t2:=substring(match2,num2- 1..num2); break; else
WWXans [i] :=convert(match2,decimal, 10); break; end if; end if; end do: end do: end proc: unassign('i'):
WWX[O]:=O: x:=table([seq(evalf(cos((2*i- )*Pi/(2*N))),i=1 ..N)]):
53

x[0]:=l: x[N+1]:=-l: unassign('i'): r:=table([seq((x[i]+x[i- 1])/2,i= 1 ..N)]): unassign('i'):
B:=table([seq(evalf(Pi)*tan(N*arccos(r[i])),i= 1. .N)]): unassign('i'): unassign('k'): for k from 1 by 1 to N do for i from 1 by 1 to N do
A[k,i] :=evalf(sqrt( 1-x[i])/(x[i]-r[k])): end do: end do: unassign('k'): for k from 1 by 1 to N do f[k]:=1.00:
WWXguess[k]:=1: end do: unassign('k'): for k from 1 by 1 to N do unassign('i'):
A 1 [k] :=evalf(sqrt( 1 +r[k])*evalf(Pi)/N*sum(A[k,i] *(WWX[i])n,i= 1. .N)): eqn[k]:=A1 [k]+B[k]*(0.5*(WWX[k]+WWX[k-
1 ]))^n+v*f[k] end do:
*(0.5*(WWX[k]+WWX[k-1 ])+1 )=0: unassign('i'): li:={ seq(eqn[i],i=1 ..N) }: unassign('i'): li2:={ seq(WWX[i]=WWXguess[i],i=1 ..N) }: sol:=fsolve(li,li2): assnWWXans(sol,WWXans):
WWXans[0]:=0: unassign('k'): for k from I by 1 to N do
WWXoldans[k] :=WWXans[k]:
54

end do:
WW[k]:=0.5*(WWXans[k]+WWXans[k-1]); unassign('i'):
B 1 :=1/evalf(Pi)*sum((WW[i]+ 1 )/sqrt( 1-r[i])*(r[i- ]-r[i]),i= 1..N):
L:= 1/evalf(Pi)* (KA/(sO_*B 1))A2; a:=v_*L/(2*Dgb): b:=Q*sO_/(kB*T): unassign('k'): for k from 1 by 1 to N do
CC[k]:=PO2^0.5*evalf(exp(-a*(r[k]+l)+b*(WW[k]-WW[N]))): f[k]:=( 1-CC[k])Akl; end do: count:=0: unassign('k'): for k from I by 1 to N do
WWXold[k] :=0: end do: while sqrt(surn((WWXold[i]-WWXans[i])A2,i=1..N)) > 10^(-6) and count < 100 do unassign('k'): for k from 1 by 1 to N do
WWXold[k] :=WWXans [k]: end do: unassign('i'): subs 1: =seq(WWX[i]=WWXans[i],i=1..N): unassign('k'): for k from 1 by 1 to N do unassign('i'): end do:
Al [k] :=evalf(sqrt(1 +r[k])*evalf(Pi)/N*sum(A[k,i] *(WWX[i])An,i= 1. .N)); eqn[k]:=A 1 [k]+B[k]*(0.5*(WWX[k]+WWX[k-
(0.5 *(WWX[k]+WWX [k- 1 ])+1)=0: unassign('i'): li:= { seq(eqn[i],i=l 1..N) }: sol :=fsolve(li,liguess):
55

assnWWXans(sol,WWXans):
WWXans[O] :=O: unassign('k'): for k from 1 by 1 to N do end do: if WWXans[k] < 0 then WWXans[k]:=0 end if:
WW[k] :=0.5 * (WWXans[k]+WWXans[k- 1]); unassign('i'):
B 1:=1/evalf(Pi)*sum((WW[i]+ 1 )/sqrt( 1-r[i])*(r[i-1 ]-r[i]),i= 1..N):
L:= 1/evalf(Pi)*(KA/(sO_*B 1))A2: a:=v_*L/(2*Dgb): b:=Q*s0_/(kB*T): unassign('k'): for k from 1 by 1 to N do
CC[k] :=PO2A0.5*evalf(exp(-a*(r[k]+1)+b*(WW[k]-WW[N- 1]))): f[k]:=(1-CC[k])Akl; end do: count:=count+1: print("count=" ,count); end do: unassign('i'):
11 :=[seq([(x[N-i+ 1]+ 1)/2,WW[N-i+1 ]+ 1 ],i=0..N)];
12:=[seq([(x[N-i+1]+1)/2,CC[N-i+1]],i=0..N)];
13:=[seq([(x[N-i+ 1 ]+1)/2,f[N-i+1]],i=0..N)]; plot(ll,labels=["Distance from the crack tip","Normal stress"],labeldirections=[HORIZONTAL, VERTICAL]); plot(12,labels=["Distance from the crack tip","Oxygen concentration"],labeldirections=[HORIZONTAL, VERTICAL]); plot(13);
56

[1] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, p. 98, Orlando:
Academic Press (1984).
[2] Y. Xu, Crack Propagation Associated with Stress-Assisted Diffusion of Impurities under Creep Conditions. Ph.D. Thesis, University of Pennsylvania (1999).
[3] C.J. McMahon and L.F. Coffin, Jr., "Mechanisms of Damage and Fracture in
High Temperature, Low-Cycle Fatigue of a Cast Nickel-Based Superalloys",
Metallurgical Transactions, 1, 3443 (1970).
[4] R.H. Bricknell and D.A. Woodford, "Grain Boundary Embrittlement of the Iron-
Base Superalloy IN903A", Metallurgical Transactions A, 12A, 1673 (1981).
[5] S. Floreen and R.H. Kane, "Effects of Environment on High Temperature Fatigue
Crack Growth in a Superalloy", Metallurgical Transactions A, 10A, 1745 (1979).
[6] Lehockey, et al., "Metallurgical method for processing nickel- and iron-based superalloys", US Patent Number 6,129, 795, 10 October 2000
[7] R.W. Staehle, "Lifetime Prediction of Materials in Environments", in Uhlig's
Corrosion Handbook 2nd Edition (R.W. Revie Ed.), Wiley, New York (2000).
[8] R.N. Parkins, "Stress Corrosion Cracking", in Uhlig's Corrosion Handbook 2nd
Edition (R.W. Revie Ed.), Wiley, New York (2000).
[9] H.J. DeBruyn, K. Lawson and E.E. Heaver, "On-Line Monitoring Using
Electrochemical Noise measurement in CO-CO2-H20 Systems", in ASTM STP 1277
"Electrochemical Noise Measurement for Corrosion Applications", American Society for
Testing and Materials, West Conshohocken, PA (1996).
57

[10] National Physical Laboratory, "Stress Corrosion Cracking" in Guides to Good
Practice in Corrosion Control, Middlesex, UK (2000).
[11] J. V. Vitek, Journal of the Mechanics and Physics of Solids, 24, 263 (1976).
[12] J. H. Riedel, Journal of the Mechanics and Physics of Solids, 24, 277 (1976).
[13] Fager, et al., "Aspects of Cohesive Zone Models and Crack Growth in Rate-
Dependent Materials," International Journal of Fracture, 52, 119.
[14] A. P. Miodownik et al., "Modelling of Creep in Nickel Based Superalloys"
Parsons 2003: Engineering in Turbine Machinery, Power Plant and Renewables, p. 779,
Maney Publishing, London (2003).
[15] L.-O Fager and J. L Bassani, "Stable crack growth in rate-dependent materials with damage." J. Engng Mater. Tech., Trans. ASME, 115, 252 (1993).
[16] F. A. McClintock and J. L. Bassani, "Problems in environmentally-affected creep crack growth" in Three-dimensional Constitutive Relations and Ductile Fracture (J.
Zarka and S. Nemat-Nasser Eds), pp. 123-145, North Holland (1981).
[17] J. L. Bassani, unpublished work (1981).
[18] R. A. Oriani, J. P. Hirth, and M. Smialowski, Hydrogen Degradation of Ferrous
Alloys, William Andrew Publishing/Noyes (1985).
[19] S. K. Min, Ph.D. Thesis, MIT (2001).
[20] L. S. Toma, M. M. Steeves, and R. P. Reed, Incoloy Alloy 908 Data Handbook,
Plasma Fusion Center, MIT (1994).
[21] S. Suresh, Fatigue of Materials, Cambridge University Press (2001).
[22] M. Abramowitz and I. Stegun (eds), Handbook of Mathematical Functions, New
York: Dover (1972).
58

[23] Chevbyshev functions on Mathworld.com.
[24] R. Hughes, S.M. Thesis, MIT (1999).
[25] X. Liu, L. Ma, and K.-M. Chang, "Time-dependent crack growth behaviors of five superalloys" in Superalloys 718, 625, 706 and various derivatives: proceedings of the International Symposium on Superalloys 718, 625, 706 and Various Derivatives (E.A.
Loria Ed.), (2001).
[26] "Effect of Metallurgical Variables on Fatigue" on www.Kev-to-Steel.com.
[27] D. S. Dugdale, "Yielding of steel sheets containing slits," Journal of the
Mechanics and Physics of Solids, 8, 100.
[28] G. I. Barenblatt, "The mathematical theory of equilibrium cracks in brittle fracture," Advances in Applied Mechanics, 7, 55.
59