NUMERICAL DETERMINATION OF FORCES
ACTING ON MATERIAL INTERFACES:
AN APPLICATION TO RAFTING IN
Ni-SUPERALLOYS
by
Simona Socrate
Laurea in Nuclear Engineering
University of Rome
(Rome, Italy, 1984)
Submitted in partial fulfillment of the
requirements for the degree of Master of Science in
Mechanical Engineering
at the
Massachusetts Institute of Technology
August, 1990
©
Massachusetts Institute of Technology
._
_/. _
Signature of Author_
_
_
_
___
__-_
Department of Mechanical Engineering
August, 1990
Certified by
---
.
-
-
-
David M. Parks
Associate Professor of Mechanical Engineering
Thesis Supervisor
Accepted by
Ain A. Sonin,Chairman
Departmental Committee on Graduate Studies
MASSACHUSETTS INSTIIUTE
OF TECHNOl OGY
APR 2 6 1991
UIWIEAS
ARCHiVES
NUMERICAL DETERMINATION OF FORCES
ACTING ON MATERIAL INTERFACES:
AN APPLICATION TO RAFTING IN
Ni-SUPERALLOYS
by
Simona Socrate
Submitted to the Department of Mechanical Engineering
on August 10, 1990 in partial fulfillment of the requirements
for the Degree of Master of Science in Mechanical Engineering
ABSTRACT
Numerical techniques have been developed to evaluate local driving forces
acting on material interfaces. Since migrations of the interfaces are associated with
modifications of the microstructure, these methods can be applied to predict and
model morphological evolution in multi-phase materials. Here this methodology
has been applied to the study of an evolution of the morphology of 7' precipitates
in Ni-superalloys, which has been termed rafting.
It is shown how predictions of the model agree with available experimental
data, and a comprehensive treatment of the rafting phenomenon is proposed.
Thesis Supervisor: Dr. David M. Parks
Title: Associate Professor of Mechanical Engineering
ACKNOWLEDGEMENTS
I would like to express my sincerest gratitude to Professor David M. Parks for
his guidance and support during this project. Not only his uncommon knowledge
and understanding have always been offered when needed, but, most important,
his generosity and friendship have been an invaluable help through any difficult
situation.
I also extend my thanks to Professors Ali Argon, Rohan Abeyaratne and Mary
Boyce for their help through numerous discussions.
Thanks to my office mates, and to all the Mechanics of Materials research
group, for creating a nice working environment, for being always willing to lend a
hand, for keeping together our computer facility and for watering my plants when
I forget to.
A special thank to Mary Toscano, without whom this manuscript wouldn't
exist. She is nice, patient and amazingly efficient: three qualities that rarely
coexist in one person.
I would also like to thank Leslie Regan and Joan Kravit for having always
been extremely friendly and helpful.
I think I should also thank Dr. R. G. S. P. Stringfellow. Thank you Richy
for having been always at my side, for your paramount help and support and for
having made my life worth living.
Finally, I would like to thank my mother and sister. I will have to do this in
Italian since my mother is still at her third English lesson and my sister has just
got a "C"in her latest English exam.
Grazie mamma e Carlaper avermi sempre aiutato e incoraggiatoanche quando
le mie scelte hanno portatl dolore e sacrifici. Siete sempre nel mio cuore e questo
Master 6 per voi (e per papa se mi pub vedere in qualche modo).
This work was supported by the Ida Green Fellowship and by the M.I.T. Center
for Material Science and Engineering under National Science Foundation Grant
No. DMR-87-19217.
TABLE OF CONTENTS
Page
A BSTR A CT ................................................................. 2
ACKNOW LEDGEMENTS ...................................................
3
TABLE OF CONTENTS .....................................................
4
LIST OF FIGURES .........................................................
6
INTRODUCTION ..................................... ......................
10
1. RAFTING IN ,--y' Ni SUPERALLOYS ...................................
1.1 Introduction ................................
...........................
1.2 Experimental Observations .........................
......................
1.3 Energy Approaches .........................................
.......
1.4 Historical Review of Rafting Models .................................
12
12
17
22
25
2. THE
2.1
2.2
2.3
FORCE ON A MATERIAL INTERFACE ..........................
Introduction ............................. ...........................
The Energy Momentum Tensor in Continuum Mechanics ............
An Expression for the Force on an Interface .........................
30
30
33
36
3. NUMERICAL METHODS ...............................................
3.1 Introduction ....................
................
.......................
3.2 A Direct Approach: The Computer Program POSTABQ .............
3.3 A Domain Integral Approach:
The Computer Program DOMAIN ........................
.........
3.3.1 Historical perspective .....................................
3.3.2 Derivation of a domain integral expression
for the energy perturbation .................................
3.3.3 Finite element implementation ............................
42
42
43
4. ANALYSIS OF RAFTING ......................
.........................
4.1 Introduction ... .....................................................
4.2 Test Case: A Cylindrical Inclusion in an Infinite Matrix ..............
4.3 Problem Description ................................................
4.4 Elastic Analysis ....................................................
4.5 Analysis of the Stress-annealing Transients...........................
4.6 D iscussion ..................................
.......................
46
46
47
50
59
59
60
73
80
93
139
5. CONCLUSIONS ....................................... .................
149
5.1 Summary of Results .......................... .....................
149
5.2 Suggestions for Future Study ........................................ 150
4
TABLE OF CONTENTS
Page
REFEREN CES .............................................................
151
APPENDIX I: The Computer Program POSTABQ .......................... 154
APPENDIX II: The Computer Program DOMAIN ........................... 218
APPENDIX III: The Computer Program GOODIER ......................... 264
APPENDIX IV: ABAQUS Input File for the Test Case: a Cylindrical
Inclusion in an Infinite Matrix ................................... 267
APPENDIX V: Typical ABAQUS Input File for the Elastic Analysis
of a 7-y- 7' Unit Cell ............................................ 275
APPENDIX VI: Typical ABAQUS Input File for the Analysis
of a Stress-Annealing Transient for a 7 - y' Unit Cell ................... 280
LIST OF FIGURES
Page
No.
1.1 Typical microstructure of the fully heat treated single crystal [30] .......... 13
1.2 Lattice misfit and its effect on the internal stress state ....................
14
1.3 Directional coarsening of -y' precipitates [15] ...............................
16
1.4 Effect of the initial microstructure on the coarsening process [24] ........... 19
1.5 Energy approaches ........................................................
24
1.6 A schematic summarizing the effects of stress orientation on the stress
annealed shape of the 7' precipitates [13] .................................
26
1.7 Correspondence between the model problem and the actual
m orphology .............................................................
27
1.8 Map giving the conditions which lead to the lowest total elastic energy for
spheres (S) , plates normal to the stress axis (N) and needles parallel
to the stress axis (P). [11] ...............................................
29
2.1 Interface configuration
32
...........................................................
2.2 Variation in energy associated with a migration of the interface ............ 37
2.3 rn as a driving force for shape evolution
..................................
40
2.4 Schematic plot of the force along the interface for 2-D models .............
41
3.1 Finite element model of a boundary value problem .........................
45
3.2 Definition of the domain of integration ...................................
48
3.3 Finite element discretization of the interface ..............................
51
3.4 Finite element discretization of the domain of integration .................
55
3.5 Perturbation patterns .....................................................
56
3.6 Test for the accuracy of the numerical solution ...........................
58
4.1 Cylindrical inclusion in an infinite matrix .................................
63
4.2 Relation between the profile of rn along the interface and the tendency
toward morphological evolution ...........................................
64
4.3 Parametric study of r. profile for:
6= 0.1%; Em= 100 GPa; 0.4 < Ep/Em
2.0 ...........................
65
4.4 Parametric study of rn profile for:
6 = 0.1%; Em = 100 GPa; 0.1 < Ep/Em < 10.0 .........................
66
4.5 Evolution map for an isotropic cylindric inclusion in an infinite matrix ..... 67
4.6 Parametric study of r. profile for:
S = -0.1%; Em = 100 GPa; 0.4 < Ep/Em
2.0 ...........................
68
4.7 Parametric study of rT profile for:
6 = 0.3%; Em = 100 GPa; 0.4 < Ep/Em < 2.0 ....................
........ 69
4.8 Schematic description of the cylindrical inclusion problem modeled,
with the boundary conditions indicated (ro/L, = 1/20) ..........
........ 70
4.9 Finite element mesh for the cylindrical inclusion problem modeled.
In this figure the bottom mesh ("near-particle" region) fits into
the indicated area of the top mesh ("nominal far-field" region) .... ........ 71
4.10 Comparison of analytical and numerical results
(for Em = 200 GPa; Ep/Em = 0.5; o,
= 1000 MPa; 6 = -0.1%)
... ........ 72
4.11 Finite element discretization of the unit cell ...................... ........ 74
4.12 Comparison of rn profiles obtained with different choices
of mesh and/or numerical method ................................ ........ 75
4.13
profile for the elastic analysis of case 1 (MNMT) ......................
4.14
profile for the elastic analysis of case 2 (MNMC) ....................... 83
4.15
profile for the elastic analysis of case 3 (TCT) ........................
84
4.16
profile for the elastic analysis of case 4 (TCC) ........................
85
4.17
profile for the elastic analysis of case 5 (PLT) .........................
86
4.18
profile for the elastic analysis of case 6 (PHT) ......................... 87
4.19
profile for the elastic analysis of case 7 (MNMTINV) ...................
88
4.20
profile for the elastic analysis of case 8 (MNMTNMs) .................
.89
4.21
profile for the elastic analysis of case 9 (TCTPMs) ..................... 90
4.22
profile for the elastic analysis of case 10 (TCCPMs) .................... 91
4.23
profile for the elastic analysis of case 11 (PLTIso) .....................
82
92
4.24 Contours of all due to misfit only (b = +.56%) for the alloy tested
I
by
lk S.
Miyazaki,
.................................... 9 7
Nakamura
and
Mori
[16]
4.25 Contours of '22 due to misfit only (8 = +.56%) for the alloy tested
by Miyazaki, Nakamura and Mori [16] .................................... 98
4.26 Contours of a21 due to misfit only (b = +.56%) for the alloy tested
by Miyazaki, Nakamura and Mori [16] .................................... 99
4.27 Contours of oir due to misfit only (S = -0.38%) for the alloy tested
by Pollock [25] .................................................. ........
100
4.28 Contours of 022 due to misfit only (b = -0.38%) for the alloy tested
by Pollock [25] ..........................................................
101
4.29 Contours of a1 2 due to misfit only (6 = -0.38%) for the alloy tested
by Pollock [25] .......... ..............................................
102
4.30 Schematic representation of the evolution of the stress and strain
................... 103
fields in the primary stage of the creep transient
4.31 Contours of ecreep
eq for the transient analyzed in case 1 (MNMT)
at creep time t = 0.011 sec ........................................... .. 104
4.32 Contours of
for the transient analyzed in case 1 (MNMT)
at creep time t = 4.46 sec ............................................ ..105
Ecreep
4.33 Contours of Ecreep for the transient analyzed in case 1 (MNMT)
at creep time t = 63.47 sec ........................................... .. 106
4.34 Contours of Ereep
for the transient analyzed in case 1 (MNMT)
-eq
at creep time t =5 50,000 sec ........................................... .. 107
4.35 Contours of -'eepIfor the transient analyzed in case 2 (MNMC)
teq
at creep time t = 0.011 sec ...........................................
..
108
4.36 Contours of -reep for the transient analyzed in case 2 (MNMC)
at creep time t =3 3.86 sec .............................................
..
109
4.37 Contours of Eeep for the transient analyzed in case 2 (MNMC)
at creep time t = 252 sec ............................................. .. 110
4.38 Contours of freep for the transient analyzed in case 2 (MNMC)
at creep time t = 50,000 sec ............
..............................
4.39 Contours of an at the end of the analyzed transient for
case 1 (MNMT) .............
..
..
. ......
... .........
.112
.....
4.40 Contours of a22 at the end of the analyzed transient for
case 1 (MNMT) ............. ............. .. ......
4.41 Contours of all at the end of the analyzed transient for case
4.42 Contours of 0 22 at the end of the analyzed transient for case
4.43 Contours of all at the end of the analyzed transient for case
4.44 Contours of a at the end of the analyzed transient for case
22
4.45 Contours of all at the end of the analyzed transient for case
4.46 Contours of a 22 at the end of the analyzed transient for case
4.47 Contours of all at the end of the analyzedI transient for
case 9 (TCTpMs) ...........
............ 113
(TCT) .... 114
(TCT) .... 115
(PLT) .... 116
(PLT) .... 117
(PHT) .... 118
(PHT) .... 119
e............................ ................
4.48 Contours of a 22 at the end of the analyzed transient for
case 9 (TCTpMs) ...........
.....
.......
.....
....
....
111
.
.
................
120
121
4.49 Contours of a11 at the end of the analyzed transient for
case 2 (MNMC) ............. ...
e...................
................ 122
4.50 Contours of a 22 at the end of the analyzed transient for
case 2 (MNMC) .............
................ 123
......................
•..
....
4.51 Contours of or, at the end of the analyzed transient for
case 4 (TC C ) ..........................................................
124
4.52 Contours of C22 at the end of the analyzed transient for
case 4 (T C C ) ........................................................... 125
4.53 Contours of ao at the end of the analyzed transient for
case 10 (TC CpM s) ......................................................
126
4.54 Contours of a22 at the end of the analyzed transient for
case 10 (TCCpMs) ......................................................
127
4.55 Evolution of the r. profile during the stress-annealing transient for
case 1 (M N M T) .........................................................
128
4.56 Evolution of the r. profile during the stress-annealing transient for
case 2 (M N M C) .....................................
................... 129
4.57 Evolution of the r. profile during the stress-annealing transient for
case 3 (TCT) ....................... ...............................
o ..... 130
4.58 Evolution of the rn profile during the stress-annealing transient for
case 4 (TCC) .......................
..... 131
4.59 Evolution of the rn profile during the stress-annealing transient for
case 5 (PLT) ........................ ............ ....
..... 132
4.60 Evolution of the rn profile during the stress-annealing transient for
case 6 (PHT) ....................... ................ .... t..........
..... 133
4.61 Evolution of the -r profile during the stress-annealing transient for
case 7 (MNMTINv) .................
transient for.. ..... 134
............
4.62 Evolution of the r. profile during the stress-annealing
case 8 (MNMTNMs) ................ ................ transient for ..... 135
n...........
4.63 Evolution of the T. profile during the stress-annealing ...
transient
for
...............................
...............
................
case 9 (TCTpMs) ...................
..............
..
...............
transient for
...............
profile during the stress-annealing transient for
..... 136
4.64 Evolution of the Tr profile during the stress-annealing
case 10 (TCCPMs) ..................
................
..... 137
4.65 Evolution of the Tcase 11 (PLTiso) ......................................................
4.66 A typical creep curve with corresponding micrographs which show
the development of directional coarsening [22] ..........................
4.67 Creep curve and corresponding changes in microstructure [23] ............
138
144
145
4.68 A schematic diagram showing the r. levels at the end of the
primary stage of the creep transient ..................................... 146
4.69 Evolution of Arnl/a for the analyzed creep transients ..................
147
4.70 Comparison of "Type P" and "Type N" evolution
curves for A r /
.a......................................................
.
148
INTRODUCTION
The mechanical and physical properties of any material depend critically on two
parameters. The first is its constitution: the overall composition, the number of phases
and their relative volume fractions and compositions. The second is its microstructure:
the shape, size and distribution of each phase.
The application of external loads to a multi-phase crystalline solid can significantly
alter its microstructure.
Limiting our attention to the most common case of a two-phase (matrix/precipitate) alloy, an external stress can modify the morphology of the precipitates, influence the precipitate coarsening kinetics and alter the relative stability of
the two phases.
The influence of the applied loading conditions on the morphology and stability
of the microstructure depends on the material parameters of the precipitate and of
the matrix, and on the magnitude and nature of the applied loads.
Since many of the properties of the alloy are determined by its microstructure, it
is of technological relevance to be able to predict microstructural development. The
microstructure evolution of the precipitate phase can be understood on the basis of
simple energy arguments. When the morphology of the microstructure evolves, it is
because there is a driving force for the change: the system is trying to lower its total
energy by modifying its configuration.
Obviously, the mere presence of a driving force does not guarantee that a change
will occur: the kinetics of the process will dictate the pace at which the structural
change will eventually take place. Nevertheless, quantifying the driving force for the
morphological evolution is still the essential first step toward a complete modeling of
the phenomenon.
In analyzing the evolution of the crystal microstructure, we can focus our attention on the migration of the interfaces between different phases: an evolution of the
microstructure corresponds to a migration of the interfaces. Thus we can think of the
driving forces for microstructure development as forces acting on the interfaces and
driving their migration.
The objective of this research has been to develop numerical techniques for the determination of the forces acting on material interfaces, and to apply these techniques
to the analysis of morphological evolution in y - -y' Nickel superalloys.
We will first describe, in chapter 1, the phenomenon that we intend to analyze:
an evolution of the morphology of -y' precipitates in Ni-superalloys, which has been
termed rafting. We will present a brief historical review of the experimental observations and discuss the models which have been proposed in the literature.
In chapter 2 we will define, in a more rigorous context, the notion of force on a
material interface, and in chapter 3 we will introduce numerical techniques to evaluate
it.
In chapter 4 we will apply these techniques to the analysis of rafting in 7-- -Y'Nisuperalloys; we will compare the predictions of our model with available experimental
data and present a general discussion of the rafting process.
Finally, in chapter 5, we will draw the conclusions of this research and discuss
some suggestions for future studies.
CHAPTER 1
RAFTING IN 7 - 7y' Ni-SUPERALLOYS
1.1
Introduction
Nickel-base superalloy single crystals are a class of two-phase precipitationstrengthened materials.
The microstructure consists of an fcc nickel solid solution matrix, 7, with Ni3 Al
precipitates, -y'. The precipitate phase, an fcc intermetallic compound, is ordered and
coherent with the matrix and can constitute up to 70 percent of the volume of the
crystal. After complete aging the precipitates are distributed in the 7 matrix as a
stable periodic array of < 100 > aligned cuboids of fairly uniform size, usually in the
range of 0.2 - 0.5 pm in diameter (Figure 1.1)
Typically, a lattice parameter mismatch, 6, exists between the 7 and 7' phases:
6
a., - a,
(1.1)
a,,
where a-, and a, are the lattice parameters of the two phases.
In most commercial alloys the magnitude of the misfit is minimal ([Ib less than
0.5%); nevertheless, since the matrix-precipitate interface is coherent, the lattice misfit introduces a significant internal stress in the crystal (Figure 1.2).
The 7-y- ' misfit, which is temperature-dependent due to the differing coefficients
of thermal expansion in the-two phases, has remarkable effects on the evolution of
the morphology of the y' precipitates.
Nathan [1] observed that, during the aging treatment, the morphology change,
from spherical to cuboid, occurs at smaller sizes of the precipitates for higher levels
of misfit.
But the most dramatic effect of the magnitude and sense of the misfit has been
recognized in a phenomenon, called rafting, which has been observed by various researchers during stress-annealing experiments or during creep tests at elevated temperatures.
At high temperatures - above 9000 C for most commercial alloys - the 7''cuboidal
precipitates become unstable. The cubic 7' particles link together to form rods and/or
plates (rafts).
When the crystal is annealed in the absence of an applied stress, the new lamellar
structure is randomly oriented along the three < 100 > cube directions, without
showing a preferential orientation (Figure 1.3(a)).
Figure 1.1 Typical microstructure of the fully heat treated single crystal [30].
13
S..
it
A
••j
4
ay , > ay ->
.• 0
e oe
&
•
6. >
o
positive misfit alloy
)
O00
ay ,<
O OO
00
0-1
)---(
0
I
*
"
*j
0
*
0
*0
0
0
0
0
*1*
0
0
*0*
0
*
0- 0
*.
*
0
*
I
0 00
---
0
0
0.
@000.
I
000
@000
*. eT.
0
Is0jT10
0 0
0
state of strain and internal
stress due to positive
---.0~~I•_•_oo
misfitting precipitates
00.
"`
0
0
t
0
00
00
6L
F-
0
0
0 0
*
negative misfit alloy
0.00*o
*
, Io .•, 0
-oo
loolo•
0
000]o..
00
*00000..
< 0
0100
0
0 .0 o0*0
)
)--(
ay --
0
0
o 0 0
"l
b·.
Figure 1.2 Lattice misfit and its effect on the internal stress state.
When a tensile or compressive stress is applied along the < 001 > direction, the
morphological evolution of the precipitates exhibits a marked directionality. Two
different types of behavior for the coarsening of the precipitates have been observed:
- Type P (Parallel):
the cuboids coarsen preferentially along the direction of
the applied stress: the precipitates assume the form of
plates which lie parallel to the stress direction. In alloys with small volume fraction of 7y', strings of 7' cubes
can coarsen along the < 001 > direction and form rods
(Figure 1.3(b)).
- Type N (Normal):
the cuboids coarsen preferentially along the directions
normal to the applied stress: the precipitates assume the
form of broad flat plates with their faces normal to the
stressed < 001 > direction (Figure 1.3(c))
Several rafting observations have been reported over the last two decades [29,12,13,15-25]. According to these observations, different materials can coarsen in
opposite directions under the same loading conditions.
The lattice misfit was soon recognized as a key parameter controlling the rafting
behavior of the alloys.
Since the morphology of the precipitates strongly affects the mechanical properties
of the alloy, several studies have been carried out in order to model, predict and control
the 7' morphology evolution.
In the following paragraphs, we give a brief survey of rafting observations, identify
possible approaches to analyze the phenomenon, and discuss the models proposed in
the literature.
(1) Rafted structure after annealing in absence of an applied stress.
(a) Isotropic Eehavior
(2) Rafted structure after stress annealing (the direction of stress is vertical in
the figure)
(b) "Type P" Behavior
(c) "Type N" Behavior
Figure 1.3 Directional coarsening of y' precipitates [15].
1.2
Experimental Observations
Directional coarsening of the y7'precipitates was first observed in commercial Nisuperalloys after prolonged creep exposure [2-9].
These instabilities of the 7' phase were viewed initially with some concern since
it was observed that they were generally associated with a reduction in the creep
resistance of the alloy [5,9,10,11]. This led to a number of theoretical and experimental
studies, toward a better understanding of the rafting phenomenon [12-15]. These
studies, which will be discussed in greater detail in paragraph 1.4, identified somnc of
the major factors related to the stress coarsening behavior, and suggested that rafting
of the 7' precipitates could be essentially eliminated by reducing the 7 - 7' lattice
misfit through careful alloy design.
In the early '80s, Pearson, Kear and Lemkey [18,19] presented an innovative study
where they demonstrated that directional coarsening significantly enhanced the high
temperature creep properties of an experimental alloy with an unusually high negative
value of the misfit (6 = -0.78%).
These results brought about a wave of renewed attention for the rafting phenomenon. A number of investigations were undertaken concerning the actual development, under different testing conditions, of the 7' rafts and their influence on
the creep properties of the crystal [16-25]. The creep loads were generally applied
along the < 001 > crystal direction. While in most studies only tensile loads were
considered, in some experiments the effect of compressive loads was also investigated.
Here we will briefly review some of the results of these observations, together with
the models proposed to explain the observed variation in creep resistance.
1. The rafts begin to form early in primary creep [17-24,27]. The time needed to
attain a fully-developed lamellar structure appears to be influenced by:
(a) the test temperature: the rate of directional coarsening increases when the
test temperature is raised [24];
(b) the applied load : upon increasing the magnitude of the applied load a
hastening of the rafting process is observed [24];
(c) the lattice misfit: under the same conditions of applied load and test temperature, alloys with larger magnitude of misfit exhibit a higher rate of
directional coarsening [27];
(d) the initial microstructure: a fine microstructure with closely-spaced, smallsized 7' precipitates considerably hastens the development of rafts [22].
2. The rafted configuration is very stable [18,20,22,24]. The average thickness of
the rafts is initially very close to the original 7' particle size [17-25]. As the creep
transient progresses, contradictory observations have been reported concerning
the evolution of raft thickness and interlamellar spacing.
Some researchers [22,24,25] report that the thickness of the rafts remains constant up to the onset of tertiary creep and the interlamellar spacing also shows a
similar behavior, while other research groups [18,20,23] have observed a gradual
thickening of the rafted lamellae during steady state creep.
Differences in the alloy composition could be responsible for these discrepancies: alloys with higer levels of refractory elements are characterized by reduced
diffusion rates so that, for these alloys, the thickening of the lamellae might be
hindered [24].
In tertiary creep the rafts become irregularly shaped , lose their perfect alignment and coarsen considerably prior to failure.
3. The initial microstructure prior to testing can drastically affect the resulting
rafted morphology [18, 19, 21, 22]. An ordered, perfectly-aligned structure of
7 ' cuboids promotes a rapid formation of more perfect platelets with a high
aspect ratio. Since the initial thickness of the rafts basically coincides with the
original dimension of the 7' cuboids, a finer initial microstructure will produce
a finer rafted structure. In contrast, if the initial structure is overaged, and is
characterized by irregularly-shaped 7' particles, the raft morphology will appear
very irregular as well, with a very low average aspect ratio (Fig. 1.4).
4. A characteristic feature of the stress-coarsened structure is the presence of networks of dislocations at the -y7- 7' interfaces [17, 18, 19, 21, 23, 25]. These
dislocations are true misfit dislocations since their Burgers' vectors are appropriate for relaxing the internal stress due to the misfit. Merging interfaces of
coarsening cuboids are usually deprived of dislocations [23, 25].
5. Under low stress, at high temperatures, the operative creep mechanism involves
dislocation motion primarily in the 7-y-matrix with the mobile dislocations circumventing the 7-' particles which remain virtually dislocation-free [18, 19, 21,
22, 25]. Thus, -'-rafts with a high aspect ratio provide an ideal structure for
creep resistance because circumvention of the 7' phase is eliminated. Significant creep can occur only by insertion of dislocations through the ordered
intermetallic 7' phase, resulting in dramatically improved creep resistance [18,
19, 22]. Furthermore, the misfit dislocation networks at the 7 - 7' interfaces
act as obstacles to the penetration of the dislocations inside the precipitates.
Under high stresses, at high temperatures, 7' particle shearing tends to become
the prevailing creep mode [18, 19]. For these loading conditions, a directionallycoarsened structure may not be beneficial since one set of 7 - 7' interfaces is
essentially eliminated [24].
(1) Morphology of the precipitates prior to creep test
(a)
(b)
(c)
(2) Rafted structure after 50 hours of creep testing
(a)
(b)
(c)
Figure 1.4 Effect of the initial microstructure on the coarsening process
[241.
We can thus think of two possible explanations for the earlier observations [5, 9, 10,
11] in which rafted crystals did not exhibit an improvement in their creep resistance.
First, in some studies the yield behavior of the alloy was investigated [11] so that the
tests were conducted at stress levels well inside the range in which 7Y'particle shearing
is the predominant mechanism of creep. Second, for the tests conducted in the low
stress regimes, the overaged, irregular and coarser microstructure of the 7' rafts in
these early alloys was easily circumvented by the dislocations in the ' phase [22].
Finally, a number of observations concerning the direction of coarsening of the
7' precipitates under tensile and compressive creep loads applied along the < 001 >
crystal direction are schematically summarized in Table 1.1.
Here, assuming that the direction of the < 001 > applied load is as shown, we
graphically identify a "type N" behavior with a horizontal rectangle and a "type P"
behavior with a vertical rectangle.
The sign of the misfit of the alloy is also indicated. Note that in three cases (a, c, f),
two different signs of the misfit are given. For these cases, the first sign corresponds
to the value given by the authors in the referenced paper, while the second sign
corresponds to the actual sign of the misfit for the alloy at test temperature. In
particular, for the three cases:
(a) Tien and Copley [12, 13] state the value of the misfit of their alloy, Udimet-700,
as +0.02%, as measured by Oblack and Kear [16] at room temperature. This
value should be therefore corrected to obtain the value of the misfit at test
temperature.
In several studies [23, 27, 28], the expansion coefficients of the -' phase have
been found to be lower than those of the -y phase; this is consistent with the
long-range ordered structure of -y' [29]. According to the data of Grose and
Ansell [28], we can infer that the value of the misfit of Udimet-700 at the test
temperature of 954°C is of the order of -0.3%.
(c) The value of the misfit given by Caron and Khan in [21] (+0.14%) has also
been measured at room temperature. Fredholm and Strudel have subsequently
determined that the actual value of the misfit at test temperature is -0.33% [23].
(f) In [17] Carry and Strudel give a negative value for the misfit (- 0.4%). This
value has been corrected in a subsequent study [23] where the actual value of
the misfit has been found to be + 0.38%. The erroneous value reported in [17]
was probably due to an incorrect Burgers' vector sign convention.
Since the sign of the misfit at test temperature plays a fundamental role in the
rafting behavior of the alloys, these misleading indications in the literature have
brought about a substantial confusion both in the interpretation of the experimental
results and in the modeling of the rafting phenomenon.
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1.3
Energy Approaches
The rafting phenomenon is an interesting example of how an externally applied
loading condition can influence the morphology evolution of the microstructure in a
multi-phase material.
The most puzzling point in this phenomenon is the marked directionality of the
coarsening process in presence of an applied stress. We can ask ourselves two basic
questions:
What are the parameters that affect the direction of rafting?
How can we predict, knowing the value of these parameters, the
rafting behavior of the material?
The evolution of precipitate morphology can be qualitatively understood by using
an energy analysis. Rafting is a spontaneous evolution of the microstructure; thus
we can conclude that it must be associated with a decrease in the energy level of the
system.
If we consider the original cuboidal morphology and the final rafted configuration,
the variation in energy, AET, associated with this variation in the microstructure,
can be roughly broken down into three terms:
AE
w
= AEchem +
+ AE,
ETL.
(1.2)
Here, the first two terms, AEch,,em and AEint, account for, respectively, the variation in chemical energy and interfacial energy. They are most certainly important
terms that play a fundamental role in the coarsening process, but they "cannot discern among the three < 100 > crystal directions". In other words, the value of these
two terms will be the same for all the rafted configurations shown in Fig. 1.3: they
are, giving to the word a broader meaning, isotropic terms and cannot account for
the directionality of coarsening under stress annealing. Thus the key to the problem
must lie in the third term, AETL: the variation of the total elastic energy of the
system.
Since this is the only term which is sensitive to the direction of the applied stress,
we can infer that the directionality of rafting is controlled by a tendency to decrease
the total elastic energy EETL which is given by the sum of the elastic energy of the
crystal plus the potential energy of the loading system.
If we express the total energy of the crystal as a function of the precipitate morphology, the misfit, the applied stress, ax, and all the other parameters that characterize the system:
ET =
wT(Q'.morphology , 6, a,,...),
(1.3)
then we can think of two possible approaches to predict the morphology evolution of
the precipitates (Fig. 1.5).
A first approach can be termed an energy minimization approach: we calculate
the finite energy levels for different morphologies of the precipitates relative to some
reference state such as a homogeneous Ni-Al solid solution crystal. We then compare
the energy levels of the different morphologies under a certain loading condition and
infer that the system will evolve toward the morphology that corresponds to the
lowest value of the total energy.
An alternative approach can be termed an energy perturbationapproach: we consider the actual initial configuration with 7'cuboids, and investigate the effect of a
perturbation of the morphology of the precipitates on the total energy. We can thus
determine the driving force:
f
-6ET
-=
T(1.4)
-'..morphology
(1.4)
which is acting on the system, and infer the most likely direction for morphology
evolution.
=
Note that, in both approaches, we implicitly assume that there exists a mechanism,
characterized by suitable kinetics, to accomplish the morphology evolution.
If we compare the two approaches on a schematic "total energy vs.
7'-morphology" graph (Fig. 1.5), the first approach corresponds to the determination of the minimum of the curve, while the second approach corresponds to the
determination of the slope of the curve for the initial morphology.
The first approach seems to be more suitable to study displacive transformations,
where the microstructure instantaneously switches to the lowest energy configuration,
while the second approach appears to be more easily incorporated in a kinetic model
to study diffusive transformations, where the morphology evolution is controlled by
the instantaneous value of the driving force as well as by the kinetics of the diffusion
process.
Historically, in the study of rafting, the energy minimization approach has been
more widely used, as we will briefly discuss in the next paragraph.
In our study we will instead follow'an energy perturbation approach.
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1.4 Historical Review of Rafting Models
The first attempt to explain the rafting behavior through energy considerations
is due to Tien and Copley [13]. They observed the structure of the 7' precipitates in
Udimet-700 and found that the sense and crystallographic orientation of the external
stress influenced the final rafted morphology of the 7' precipitates. Their observations
are graphically summarized in Fig. 1.6. The dotted cubes show the initial orientation
of the -y' cuboids with respect to the direction of stress (vertical in the figure). The
shapes bounded by solid lines represent the aligned plates, parallelepipeds and cuboids
that result from stress annealing.
In the discussion of their observation, Tien and Copley gave only a qualitative theoretical analysis and could not draw quantitative conclusions regarding the influence
of elastic energy on the final particle shape.
A significant step in the understanding of the rafting phenomenon is due to Pineau
[14]. Since the evaluation of the total energy for the actual 7 - Y'microstructure is
extremely arduous, the calculations are drastically simplified by considering a parallel
model problem in which a single, ellipsoidal inclusion, representing the ^' phase, is
embedded in an infinite 7--matrix. The fundamental idea behind this approach is
that there would exist a direct correspondence between the actual problem and the
model problem so that the results of the model problem could be directly extrapolated
to the real microstructure (Fig. 1.7). In other words, if for the model problem an
ellipsoid prolate in the direction of the applied stress minimizes the energy, then a
"type P" behavior is inferred for the actual microstructure, while if an oblate ellipsoid
minimizes the energy, a "Type N" behavior is inferred.
Pineau systematically applies Eshelby's theory and calculates the energy of a solid
containing misfitting coherent precipitates of various shapes subjected to applied
stresses. Three shapes are considered: spheres, plates perpendicular to the stress axis
and needles aligned with the stress axis. Matrix and precipitates are assumed to be
elastically isotropic.
Based on these calculations, the most stable shapes (corresponding to the lowest
elastic energy), have been determined, and the final results are presented in the
graphical form shown in Fig. 1.8.
According to this map, the major factors which affect stress coarsening behavior
are the direction and the value of the applied stress, normalized by the elastic modulus
of the matrix, the ratio between the elastic moduli of the two phases, and the 7 - 7'
misfit. The derivations of Pineau have been generally accepted, for a certain time, as
the most satisfactory treatment of the rafting process.
Unfortunately, as noted by Fredholm and Strudel [23], the model considered by
Pineau leads to predictions that agree with the experimental observations summarized
in Table 1.1, only if it is assumed that the -y' particles are stiffer than the matrix for
all the tested alloys. But for at least two cases [13, 15], we have positive evidence
TIHNSIL
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Figure. 1.6
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.1
A schematic summarizing the effects of stress orientation on the
stress annealed shape of the 7' precipitates [13].
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that the precipitate phase is actually softer than the matrix so that Pineau's map
predictions do not agree with the experimental data.
In order to obtain results more consistent with the experimental observations,
subsequent studies have tried to reduce the number of simplifications in the Pineau
model.
The Eshelby equivalent inclusion approach can be successfully applied to the study
of ellipsoidal inclusions in anisotropic media, and models based on anisotropic elasticity have been derived in a number of studies [15, 31, 35].
The complexity of the problem increases by an order of magnitude when a cuboidal
shape for the precipitate is considered. Faivre [33] has treated the problem of a single
cuboidal inclusion for isotropic media and Chang [35] introduced a method based
upon Green's function techniques to study single cuboidal precipitates in anisotropic
media.
morphology, and the effect
Regarding the actual periodic structure of the y - 7-'
of interaction energy between particles, Johnson [34] has considered the problem of
two spherical inhomogeneities under the influence of an applied stress and studied the
coarsening kinetics of the two particles; Jankowski, Wingo and Tsakalakos [32] have
modeled the periodicity of the microstructure by using a space and time-dependent
Fourier series; finally, Chang [35] has attempted, with partial success, to apply the
Mura trigonometric series method and the finite element method to treat arrays of
inclusions.
Still, no comprehensive model is available that accounts for the actual structure
of the crystal and gives predictions which explain all the experimental results.
The reasons for this failure lie not only in the inherent limitations of the Eshelby
inclusion approach: even a perfect elastic model of the real microstructure would not
yield correct predictions for the rafting behavior.
The essential common flaw for all the model proposed in the literature is that
the inelastic response of the 7-matrix, due to dislocation motion, is always neglected
in the evaluation of the stress and strain fields. As mentioned in paragraph 1.2, a
network of misfit dislocations is always observed at the 7 - 7' interfaces during rafting.
The presence of these dislocations suggest that a localized "creep process" is relieving
the 7 - 7' misfit, altering the state of stress in the crystal. As will be shown in Chapter
4, only a model that takes into account this effect can successfully account for the
rafting behavior of Ni-superalloys.
40
A /mS
Figure. 1.8
Map giving the conditions which lead to the lowest total elastic
energy for spheres (S), plates normal to the stress axis (N) and
needles parallel to the stress axis (P). Ep/Em is the ratio of the
Young's modulus of the precipitate and that of the matrix, UA, is
the applied stress and 6 the misfit between the precipitates and the
matrix. The domains where two or three shapes are indicated for
the particles are those where the difference in the corresponding
elastic energies is less than 0.1 Em b 2 . [11]
CHAPTER 2
THE FORCE ON A MATERIAL INTERFACE
2.1
Introduction
Consider a generic system whose configuration can be identified by a suitable
number of parameters 81,#2....#,. We can express the total energy of the system,
ET, as a function of these parameters:
ET = ET(#,1#2,... #,I... #).
(2.1)
Following the terminology of analytical mechanics and thermodynamics, we can
introduce the notion of generalized force conjugate to the ith parameter,fi, defined as
8E
fA =
T
0T
(2.2)
Thus fi, which we may regard as the force acting on f3i, is the rate of decrease of
the total energy with respect to the parameter /3.
It is important to note that, in this definition, ET is the total energy, i.e., the
energy of the system we are studying plus the energy of the environment with which
it interacts.
We will now restrict our attention to solid bodies with prescribed boundary conditions.
We can expect, in the material of the body, departures from uniformity on various
scales which we may call imperfections. Examples on a microscopic scale are dislocations, foreign atoms, vacant lattice points and grain boundaries. On a macroscopic
scale, there might be inclusions of one phase in another, cavities and cracks.
If the boundary conditions are held constant, the total energy of the system (the
energy of the body plus the potential energy of the loading device) is a function of
the set of parameters necessary to specify the configuration of the imperfections.
Thus, following the general derivation, we can define the force on a gliding dislocation, on an extending crack, or on a growing cavity.
For the particular case of inclusions and precipitates of one phase in another, such
as martensitic plates in ferrite or 7' precipitates in a 7 matrix in Ni-superalloys, if
the two phases are uniform within themselves, we can consider the interface itself to
be the imperfection.
If C* is a reference configuration for the interface, we can characterize any other
configuration, C, in terms of the normal displacement of the interface, &n, with respect
to C* (Fig. 2.1).
Thus the energy level ET is determined by the parameter, ýn, which is a function
of the position X along the interface:
ET= "T(n(())
(2.3)
It is then straightforward to define a normal force acting on the interface,
Tn,
which is work-conjugate to n, as:
rn(V)
-OE
-
T
.=
(2.4)
It is possible to express this generalized force, rn, in an extremely convenient form
using a quantity whose interesting features were first recognized by Eshelby [36], and
which is therefore known as Eshelby's Energy Momentum Tensor (EMT).
In the following paragraphs, which closely follow the derivation presented by Eshelby in [37], we will first describe the mathematical process which generates the
EMT, then we will discuss some physical interpretations and derive an expression for
the force on a material interface.
Reference configuration C*
ET = EoT = •(
= )
.. ::::::
Current configuration C
ET = ET(en(z-))
Figure 2.1 Interface configuration.
2.2 The Energy Momentum Tensor in Continuum
Mechanics
We will derive an expression for the EMT in the framework of finite deformation
theory with a hyperelastic stress-strain relation.
Rectangular Cartesian coordinates X, are us(ed to label the position of material
particles in the initial state.
If ui is the displacement field, the final position of the particle, in the same coordinate system, will be xi so that
uI = x, - Xi.
(2.5)
As a stress measure, we use the nominal Piola Kirchhoff stress, pij, so that, if W
is the Lagrangian strain energy density,
P =(2.6)
01
The equilibrium condition is
+ pbi = Oi,
(2.7)
where bi are the body forces per unit mass (which will include the D'Alembert inertial
force in dynamic problems), and p is the mass density referred to the initial volume.
We will consider the general case in which the strain energy density, W, depends
on the gradient of the displacement field uij, and also explicitly on the Xm:
W = W(uI,,;Xm).
(2.8)
This explicit dependency on Xm allows us to consider material inhomogeneities
(regions with varying elastic constants) as well as states of internal stress characterized
by the presence of eigenstrainsci(XAm).
Here, with the term eigenstrains we indicate all such nonelastic strains as those
associated with thermal expansion, phase transformation, creep, plastic flow, and
lattice misfit [38].
For the elastic calculations of the force on a material interface, we will consider
these strain fields as "frozen" in the material, i.e., the eigenstrain field will depend
only on the location Xm.
Under these conditions it is perfectly equivalent to regard eJ as an extra field
variable or to absorb the eT-dependence of W into its explicit dependence on Xm.
If we substitute Eq. (2.6) in (2.7), we obtain an alternative form of the equilibrium
equations
= -pbi.
OuW
(2,9)
OX, Ou,,,
At this point we need to distinguish the gradient of W, OW/OXi, defined by
-04W
)21
W(X + dk) = W(X) + X--dXj + o(dXi),
(2.10)
from the explicit partial derivative of W with respect to Xj,
OW
X,,) ui~ = const.
(OW)eXp
= OW(u•,j;
XW(uj;X)
j
=Xm
(2.11)
= const. m
.
The gradient of W is thus given by
OW
OW OuC,
Uij
CuXk
Noting that Ui,jk
= Ui,kj
oW
o
Xk
++(
OW
(2.12)
)e
8Xkexp*
and using the rule for differentiating a product, we obtain
oW
0
- (w
(-
-xj
-)
('-)u,k
uikx
OX
OX, OUk,
aXk
Substituting (2.9) and (2.6) in (2.13), we have
0
OW
(Wbjk - PijUi,k) =
-
)exp
oW
+(
p biUi,k.
)xp
(2.13)
(2.14)
We can now introduce the Energy Momentum Tensor, P, whose components are
given by
Pki
W6kj -Pijui,k,
(2.15)
so that eq. (2.14) reads
OPk]
OW
S- (Ok)exp
+ pbiui,k.
(2.16)
Thus the divergence of P vanishes wherever W does not explicitly depend on Xm
and there are no body forces.
Note how, for a more concise notation, here we have incorporated the cT -dependence
of W into its dependence on Xm; if we regard ET as an extra field variable, then
W = WV(ui,j; 7(Xm); Xm).
We can expand (-)xp
(2.17)
into two contributions:
=
OW
Be" ,
XW )inhom + O
OW )expO~~OT,
em O
O~
Xk'
8Wex
(2.18)
where the first term accounts for material inhornogeneities as well as, iii a more general
context, for any effect due to space gradients of elastic constants (e.g., temperature
dependence of elastic constants associated with a temperature gradient).
An interesting interpretation of the physical meaning of P1can be (derived if we
consider an elastic body subject only to surface loading (with no bod(ly forces), containing a certain number of imperfections 11,12,... I,.These imlperfections might be
iniihomogeneities, inclusions with an eigenstrain, point defects, etc. We assuime that,
apart from these imperfections, the remaining material is "good" elastic material
where
8W
(9-k)Cxp = Ok.
(2.19)
If the generic defect I suffers a small translation 6ý, there will be a variation in
the total energy of the system
bET = -bAk,
(2.20)
where fis the generalized force acting on I.
It can be proved [37] that the components of fare given by:
k=
PkjdS,
(2.21)
where S is any surface surrounding I and isolating it from all the other imperfections,
and dS = dSii is the oriented surface element with in being the unit outward normal
to the surface. Note that S can be chosen arbitrarily, without altering f, only as long
as it remains in "good" material where (2.19) holds.
2.3
An Expression for the Force on an Interface
We are now concerned with the special case where the imperfection is represented
by the interface between an inclusion, Q, and an otherwise homogeneous medium
which we will call the matrix, D (Fig. 2.2). We limit our attention to coherent
interfaces, for which the displacement field iUis continuous at the interface. The
inclusion may have elastic constants differing from those of the matrix, and the body
may be subjected to an eigenstrain field, cT, which might be discontinuous across the
interface.
We will derive an expression for the generalized normal force acting on the interface
as a result both of the internal state of stress, due to the J-fieldl, and of the externally
applied loads. This quantity will be a measure of the force driving the migration of
the interface.
Consider a reference configuration for the interface, S*, and an infinitesimal migration to a current configuration S (Fig. 2.2). The migration can be specified by a
small vector 6b at each point of S*.
We want to evaluate the change in the total energy of the system, SET, as a result
of the migration S* -+ S. We can obtain our result with the help of a sequence of
imaginary steps which simulate the migration.
We start with the interface in the S* configuration, a displacement field i* in
the body (i*D in the matrix, iP.n in the inclusion), and we want to end up with the
interface in the S configuration and a displacement field i1 in the body (ilD in the
matrix, u-o in the inclusion).
Step 1. We cut out and remove the "D-material" which lies in the region between
S* and S, and apply suitable surface tractions to the "hole" to prevent relaxation.
The variation in energy for step 1 is
SEbT = - f 6jWDdSj,
(2.22)
where dS = dSin, with in being the unit outward normal to the interface (from Q
to D: see Fig. 2.2), and WD is the elastic energy of the "D-material" that we are
removing.
The displacement on the boundary, S, of D is now u!D + ku* and the traction
is pD(-dSj) + o(d~j).
Step 2. We bring the displacement field at the boundary of D to its final value
9D. The variation in energy is:
° + S~kk )]pidSr.
E2) = - J[uP - (u D
*D
D
.t
(2.23)
II
j
cijTI
•in
migration
of the
E
E
interface
S
Figure 2.2
Variation in energy associated with a migration of the interface.
Step 3. We allow the material that we have removed to transform from "Dmaterial" to "f-rnaterial" and we put it back in place.
The variation in energy is then
63
,oWdS,.
ET) =
0 +
The displacement on the boundary S of [1 is now u11
(2.24)
ku,?'
and the traction is
p}dS1 + o(d4j).
Step 4. We alter the displacement of the boundary of Q to its final value, Uf,
with a variation in energy:
bE = [u
-
(u + 6k)pdS].
(2.25)
Note that if the transformation from "D-material" to "'1-material" brings about
a variation in chemical energy, we should also include a term
6Eem=J6(4(W2 - Wo)dSj,
(2.26)
where (Won - Wo) is the work required to transform a unit volume of unstressed D
into an equal mass of unstressed f.
Furthermore, if a surface energy YTint is associated with the interface, the displacement of the boundary brings about a variation in energy:
bEit = J
iTiitKdS1 ,
(2.27)
with K = 1/R 1 + 1/R 2 , where R1, R2 are the local principal radii of curvature of the
surface.
As previously mentioned (paragraph 1.3), in order to give an explanation for the
directionality of rafting we can actually restrict our attention to the variation of the
total elastic energy 6ETL(= 6E( + 6E) + bE ) + SET) so that we will neglect the
chemical and interface energy terms in the following derivation. Thus, the expression
for the force acting on the interface rn, that we will derive in the following section,
actually gives only the "elastic contribution" to the force on the interface.
Chemical and interface energy terms should, however, be included in more general
applications.
Since U' is continuous across the interface, we have uP = u? and u7D -= iL for
each element dSj. Also, the traction vector pijdSj is continuous at the interface, so
that on adding the four contributions (2.22, 2.23, 2.24, 2.25), we obtain the total
variation:
ERL = -
jS*k{(WD6kj
and, using the definition (2.15),
- PijUik) - (W~6kj - PijUik)}dSj,
(2.28)
ET= -
L-{Pk
- PI}dSj.
(2.29)
It is natural to choose the direction of bCnormal to the interface: 6b = 6~Cni. If we
use the notation [A] to denote the discontinuity of the generic quantity A across the
interface, we can recast Eq. (2.29) in the form
6ETEL= -j 6Sn(nk[Pkjlnj)dS.
(2.30)
If we compare Eq. (2.30) and Eq. (2.4) we can easily see that Eq. (2.30) is
equivalent to the statement that there is an effective normal force per unit area of
the interface of magnitude
rn
1 = nk[Pkjlnj,
(2.31(a))
or, if we substitute expression (2.15) for P,
rn = [W] - til
J,
(2.31(b))
an
where ti = pijnj are the components of the traction vector t at the interface and d/On
denotes spatial differentiation along the normal direction.
Note how, throughout our derivation of rn,, we have never needed to invoke condition (2.19). This means that expressions (2.31) for rn hold also in the presence
of eigenstrains cT and body forces bi. In fact, we will actually use these expressions
to evaluate the elastic driving force for morphology evolution of 7' precipitates in
Ni-superalloys.
We will first perform our calculations for merely elastic fields in which the only
eigenstrain is due to the 7-- 7' lattice misfit, then we will follow the evolution of the
force on the interface during a stress-annealing transient where creep strains set in,
so that the eigenstrain field will be the superposition of the initial misfit field and of
the creep field.
Before concluding this paragraph, it is perhaps appropriate to emphasize that rn
is an extremely convenient measure of the elastic driving force for the evolution of
the inclusion shape.
A positive value of rn (Fig. 2.3) means that the total elastic energy will be reduced
if the interface migrates outward, thus indicating a tendency for the inclusion to grow.
Conversely, a negative value of rn indicates a tendency for the interface to reduce its
volume.
We can construct graphs of the distribution of 7n over the interface; in particular,
if we limit ourselves to 2D-problems, we can plot the value of rn as a function of the
arclength along the interface (Fig. 2.4).
We will introduce numerical methods to calculate rn in Chapter 3, and in Chapter
4 we will apply these methods to the analysis of rafting in 7 - 7' superalloys.
Figure 2.3
Itn as a driving force for shape evolution.
40
Tn
.I
tA
TI
S
Figure 2.4 Schematic plot of the force along the interface for 2-D models .
CHAPTER 3
NUMERICAL METHODS
3.1
Introduction
In Chapter 2 we have defined a quantity, r, which provides direct information on
the driving force for morphology evolution of multi-phase materials. If we examine
expressions (2.31), we can notice how the force on the interface can be obtained
directly in terms of elastic field quantities evaluated at the material interface. This is
indeed an extremely convenient feature of this approach, since we have an instrument,
the Finite Element Method (FEM), that allows us to evaluate the elastic field around
a material interface, for virtually any kind of morphology of the system, material
constants, applied loads and internal stress state (eigenstrain field).
Throughout this thesis, the finite element program ABAQUS [39] has been used,
so that, even if the methods are general, the software has been specifically developed
to postprocess ABAQUS results.
For the present applications, we can limit our attention to small-deformation
analysis, so that we can substitute the nominal Piola Kirchhoff stress, ps,, with the
Cauchy stress aij and the expression for the EMT becomes
Pk, = Wbkj-- aijui,k.
(3.1)
The elastic strain energy density will be given by
W
= V(Cý)
=
]
dj,
(3.2)
where
,-
-
£
(3.3)
is the elastic part of the strain tensor cij:
1
c = 21(uij + uj,i).
(3.4)
In this chapter we will first describe an extremely direct method to evaluate r,
then we will illustrate an alternative method which utilizes a domain integral representation for the energy perturbation 6ETL of exp. (2.30), and discuss its numerical
implementation.
3.2
A Direct Approach: The Computer Program
POSTABQ
Consider a generic boundary value problem (BVP) for which we intend to evaluate
Tr at the interface of an inclusion f0 (Fig. 3.1(a)). We first build a finite element
model for the BVP, discretizing the domain under examination and applying appropriate boundary conditions (Fig. 3.1(b)). This model represents the input for the
structural finite element code, ABAQUS, which will solve the BVP. ABAQUS provides, in output, values at nodes and integration points for the elastic energy density
W, the displacement field Ut and the stress tensor a. Thus, one simple way to calculate
the force on the interface would be to postprocess the ABAQUS output file, evaluating the quantities [W], [ f], and substitute these values in expression (2.31(b)) to
directly obtain rn.
This method has been implemented in the computer program POSTABQ.
The program POSTABQ has been conceived as a multi-purpose postprocessor for
ABAQUS. The current version of the program can handle only 2-D problems with
second-order eight-node isoparametric elements, but has been structured so that its
applicability can be easily extended to cope with 3-D geometries. The user supplies
a set of nodes (by way of their number in the F.E. model) which define a "path"
- in our case the path will be the interface - and the program evaluates the path
geometry: curvilinear coordinate and normal vector to the path, topology and connectivity of the elements around the path, etc. Then the user asks the program to
perform a sequence of operations on a certain number of selected field variables along
the path. These operations can include reading the quantities from the ABAQUS
file and storing them in local arrays, evaluating gradients of scalar and vector fields,
evaluating the components of vectors and second order tensors in a rotated reference
frame, evaluating the dot product of vector and tensor fields with the normal to the
path (thus calculating the normal component of these quantities), and printing the
results of such operations in an ASCII file.
The user can also program a user-subroutine where he can process the quantities
resulting from the previous operations. Thus, in order to calculate rn on the interface,
the user will
(a) provide an ordered list of the nodes along the interface;
(b) require the program to:
* read and store the values of the strain energy density W;
* read the values of the stress field and evaluate the components of the
traction vector
ti = oijnj;
* read the value of the displacement field i, evaluate the gradient of the
displacement field Vu, and evaluate the derivative of iUalong the normal
direction:
9 ui
(c) write a user subroutine where the results of the previous operations are combined
to obtain
S= [W] at all nodes along the interface.
The listing of the program POSTABQ is given in APPENDIX I.
The advantage of this method lies in its simplicity and therefore in the limited
amount of calculations required. Unfortunately, it has one main drawback: since the
evaluation of T directly relies on the determination of elastic field at the interface,
we can expect that our results will be affected by some numerical noise. The area
adjacent to the interface is in fact directly affected by numerical disturbances arising
from the discontinuity of the elastic field. Extremely refined meshes, therefore, are
required to limit this effect. An alternative approach which can overcome this kind
of difficulty will be presented in the next section.
PL
0
3.3 A Domain Integral Approach: The Computer
Program DOMAIN
3.3.1 Historical perspective
The J-integral, introduced by Eshelby [45] together with a number of other pathindependent contour integrals, has quickly become one of the most-used crack parameters in fracture mechanics.
Its role in the context of nonlinear fracture mechanics was introduced by Rice
[40, 41] who has interpreted J as a measure of the intensity of the crack tip field: if
evaluated on a contour placed within the region of HRR dominance [42, 43], the value
of J serves as a unique scaling amplitude for the HRR singular fields. The initiation
of crack growth in plastically deforming bodies can thus be correlated with a critical
value of the J-integral.
The reason for the "success" of the J-integral lies mainly in the mentioned property
of path-independence: the integrand is divergence-free for a material that admits a
strain energy function, so that the J-integral has the same value for all open paths
beginning on one face of the crack and ending on the opposite face. This feature plays
an important role since it allows a direct computation of the strength of crack tip
singularities by evaluating the integral in regions remote from the crack tip, where
the numerical field solution is more reliable.
With the help of weighting functions, the J contour integral can be recast into a
finite domain volume integral. This alternative formulation is naturally compatible
with the finite element method.
In fact, the domain-approach first appeared already cast in its finite element implementation: the Virtual Crack Extension (VCE) technique, introduced by Parks
[47], actually corresponds to a finite element formulation of the domain integral approach. By the VCE technique [48, 49, 50], accurate pointwise values of the energy
release rate can be obtained.
This method has been interpreted in a somewhat more general context by deLorenzi [51, 52], who has obtained a more compact continuum formulation of the
VCE technique.
The use of domain integral techniques in fracture mechanics has been subsequently
addressed in several studies [44, 46, 53, 54], where thermal loads and kinetic energy
have been rigorously incorporated in the formulation.
Here we present a straightforward extension of this methodology, which will allow
us to evaluate the force acting on a material interface.
We are led to this approach by the same reasons which have brought about the
use of contour and domain integrals in fracture mechanics. We must deal with a
structural discontinuity (in our case, the interface; in fracture mechanics, the crack),
which locally affects our numerically-determined field solutions. Thus we need to
develop a formulation in which the desired local quantities are expressed in terms of
the field solution in regions as remote as possible from the discontinuity.
In the next section, 3.3.2, we present a derivation of the domain integral expression
for the energy variation, 6jETL, that corresponds to a migration of the interface. We
then outline in Section 3.3.3 the finite element formulation of the domain integral
method to determine the force on the interface. In Appendix II, we give the listing of
the computer program DOMAIN in which this methodology has been implemented.
3.3.2
Derivation of a domain integral expression for the energy
perturbation
Let S be the interface separating an inclusion 01 and an otherwise homogeneous
medium (Fig. 3.2(a)). If n' is the unit outward normal to the interface, we can define
a perturbation of the interface bý in terms of its normal component 4n as
4V) = 4n(Y) (V),
(3.5)
thus 6~ is a vector quantity defined at all locations X along the interface.
In section 2.3 we have derived an expression for the change in the total energy,
6ETL, associated with the perturbation bý:
6EL -=J6i[Pij]njdS.
(3.6)
Let S+ now be any surface completely surrounding S, and S- any surface completely surrounded by S, as illustrated in Fig. 3.2(b). It is also assumed that the
volume between S+ and S- is simply connected. In order to develop a volume integral
expression for SETL, we first introduce the vector field q'defined over the domain V
enclosed by S- and S+ . The vector field q'is defined so that it satisfies the following
requirements:
(a) q(•')
(b) q-'()
(c) q-~F)
6~('); if XF identifies a point on S;
if X identifies a point on S+or S-;
0;
is a smooth function in V.
(3.7)
Then (3.5) can be written as
6ETL = -
j
qi()[Psj]njdS.
(3.8)
We can recast (3.8) in the form
bEETL = I+ + I-,
(3.9)
where
I+= -
qiPindS
(3.10(a))
I- =
qiPiJnidS,
(3.10(b))
(a)
(b)
Figure 3.2 Definition of the domain of integration.
with Pý+ being the EMT on the "plus side" of the interface (Fig. 3.2(b)) and Pj the
EMT on the "minus side".
Now consider the expression (3.10(b)) for I-; if we indicate with V- the domain
delimited by S- and S, and with OV- the boundary of V-, since ' vanishes on Swe can write:
I-=
L
(3.11)
qiPi- njdS
and, applying the divergence theorem:
I-=
JV,(qiPi'),,dV
(3.12(a))
or
I = v(Pijqi,j + Pijdq1 i)dV.
(3.12(b))
In section 2.2 we have derived expression (2.16) for the divergence of P. If we
substitute (2.18) in (2.16), we obtain
OW 9ET
OW
Pij,j = (Xi)inhom +
+ pbkUk,i
0 CTr-M
(3.13)
Using definition (3.2) for the strain energy density and substituting for the elastic
strain (3.3) we obtain:
OW
OW
-
(3.14)
e_-= - Omn
so that:
Pij,
OW
= ('iX)inhom -'
7
T
mnmn,i
+ pbkuk,i.
If the material within the domain V- is homogeneous ((OW/0X
have:
I-=
j
{Pijqi,j + (pbkuk,i
- amnTmn,i)qij}dV.
(3.15)
j
)inhom = 03), we
(3.16)
We can perform an analogous sequence of operations on I+ :
I+ =
qiPVPmjdS,
(3.17)
where V + is the domain delimited by S and S+,OV + is its boundary, and r' is the
unit outward normal to OV+ (M'= -nf on S).
By applying the divergence theorem and substituting for Pi,j, if the material
within V+ is homogeneous, we obtain:
I+ =
{Pijqi,j + (pbkuk,i
rmnn,)i}dV.
-
(3.18)
Substituting (3.16) and (3.18) in (3.9) and noting that V - V + + V- we obtain
EL = V {Pijqi,j + (pbkUk,i
-
Omnmn,i)qi}dV,
(3.19(a))
or
6EL
=V
T{(Wqi,i=---Okjuk,jqij) + (pbkuk,i - a,mmn,i)qi}dV
(3.19(ng
,
T,,Nteqhow.(3.19(b))
We have thus obtained a volume-integral expression for SEL. Note how, as long
as the q-field and the domain V are chosen so that requirements (3.7) are satisfied,
expressions (3.19) will give us the same value for 6ETL for any choice of the domain
V. This invariance of the domain integral representation of SEEL with respect to
variation in domain size and shape will provide a useful independent check on the
consistency and quality of our numerical calculations. This is indeed another attractive feature of the domain integral approach.
If we substitute expression (2.31(a)) in (2.30) we have
SETL = -
(3.20)
6bn -rndS;
if we equate the right hand side of Eq. (3.20) and (3.19) we have
-
býnrndS =
{(WVqi,i - OkjUk,iqi,j) + (pbkuk,i
-- O'mn
,i)qi}dV.
(3.21)
The desired quantity, rn, appears on the left-hand side, while we can evaluate the
expression on the right-hand side in terms of known field quantities: W, e, u, b, eT.
In the next section, 3.3.3, we will describe how, with an appropriate choice of the
perturbation field (6( and q), we can extract local values of rn from Eq. (3.21).
3.3.3
Finite element implementation
We will discuss a numerical implementation of the domain integral approach, for 2D configurations, within the context of the displacement finite-element method using
biquadratic Lagrangian shape functions.
We define on the interface S, N nodes and M biquadratic (3-node) isoparametric
elements (Fig. 3.3(a)). For each element we can identify an isoparametric coordinate
r -
as shown in Fig. 3.3(b) -
and define the basic shape functions:
s=O
2
3
4
5
nodes (global number)
(a) the interface discretization
N-1
11=-
l11=0
11=1
2
11
(b) 1-D biquadratic element
3
element nodal points
Figure 3.3 Finite element discretization of the interface.
1
(71) = y&(-
1)
N'(q) = (1 - -q)(1 + 1)
1
N3(1) = 2(= + 1).
Throughout our presentation, superscripts refer to the element nodal point number
(as in Fig. 3.3(b)), while subscripts refer to the global node/element number along
S.
We can express the value of any quantity, g, at location q along the element j in
terms of the nodal values of g and of the shape functions:
g(t) = NM()(gi),
(3.22)
where the summation convention is implied also for superscripts and (g')j is the value
of g at node i of element j.
In particular, if a is the curvilinear coordinate along S, we have
s(7) = N'(q)(s')j
(3.23)
and
ds = ds
dy
dN(S')jdl.
(3.24)
dy
We can also write:
rn(q) = N'(i 1 )(rn)j,
b
(3.25)
==n(q)
N'(7)(6ýn)j.
(3.26)
We can then recast Eq. 3.20 in the form
M
6ETL =
k1
{j= 1
Nk(
1
dN m
nk)jN i(7)(n)j- d
m)d},
(3.27)
where we have extended the summation to all the elements of S and we have substituted the area element dS with the line element ds since we are modeling 2-D
problems.
Now consider N different patterns for the perturbation 6n, defined so that, for
the pth perturbationpattern (Fig. 3.5(a)):
(6n) -=1 at the pth node
(8ýn)p = 0 at all other nodes.
If we write Eq. (3.27) for the pth perturbation, only the elements j, to which node
p belongs will give a contribution to the pth perturbation in energy bET , so that we
obtain
{- j_ N (rl)N (q)l('rn)jn--dym
=
6ETLELpI
j(3.28)
m)ipdl'},
(3.28)
(dN
where we are summing only over the elements jp which contain node p and 1 is the
position of node p in element jp,.
We can also write (3.28) as:
EL, =-E•T[in]p[amli
-
ip
1
dNm
f-N()N'()_dr
},
(3.29)
and introduce the notation
.
Bim = -
dNm'
dN(q)N'(r)
dm .N
-1
dq
(3.30)
The quadratic form B" m can be explicitly evaluated for all combinations of indices.
The results are shown in Table 3.1. Eq. (3.29) can then be written in a more compact
form as
ETL = E(ri)jP
(sm)jPBlim.
Eip
(3.31)
If we write the same expression for all the N possible patterns, we obtain a system
of N equations in the N unknown rn nodal values:
MprTn, = SET
,
(3.32(a))
[M]{rI} = {SEL},
(3.32(b))
or, in matrix notation:
where
{bE4 L}
n,
Mr,
is the vector of the energy perturbations, 6EELp,,
is the nodal value of 'nat node r
is the p, r component of the coefficient matrix [M] which is given by:
Mp, = E[Sm.,,.Blim.
(3.33)
jP,r
Here the summation is extended only to the elements jp,r which contain both node r
and node p (if r $ p there will be only one element) and t is the position of node r
in element jp,r.
We can thus easily construct the coefficient matrix [M] from Table 3.1 and the
curvilinear coordinates of the nodes.
Table 3.1
B111
= 1/3
= 1/5
= 8/15
13 1
B
= -1/30
B2 3 1 = -1/15
B 33 1' = -1/15
B1 2 1
B2 2 1
Bijk Values
B 112 = -2/5
B 122 = -4/5
B 222 = 0
B I 13 = 1/15
B 123 = 1/15
B 223 = -8/15
B 132 = 0
B 232 = 4/15
B 2 32 = 2/5
33
B'
= 1/30
B
23
B
3 3=
= -1/5
-1/3
To obtain the values of rn from system (3.32) we need now to evaluate the righthand side vector {bETL}. Consider a finite element discretization of the region surrounding the interface, as shown in Fig. 3.4(a).
We use 2-D isoparametric 8-node elements, for which the nodal point numbers are
shown in the isoparametric (q, ()-space in Fig. 3.4(b). The 2-D domain, V, around the
interface can now be defined by considering a certain number of layers of elements
around the interface as shown in Fig. 3.4(c). We can evaluate numerically each
EETL,,P, integrating expression 3.19(b) over V. We first define the perturbation field
(q )P associated with the pth pattern of interface perturbation (n),p.
We can express
the (fl),p field in terms of nodal values (I k)pand 2-D biquadratic shape functions as
(3.34)
where Afk(', C) are the 2-D shape functions expressed in terms of the isoparametric
coordinates ('7, () (See Table 3.2). The nodal values of the p~th perturbation field,
(q k), can be chosen freely as long as
1. ( k)p
1 . nf, at node p on the interface
)( (
)
-Afk('
)(i"'k))
2. (q k)
0 at all other nodes of the interface
3. (1 k),
0 at all nodes along the boundary of V.
A possible choice for the nodal values (q k)) is shown in Fig. 3.5(b).
Table 3.2
2-D Shape Functions
A(, (;) = (-1/4)(1 - 17)(1 - ()(1 + 77+
AP(, (;)
= (-1/4)(1
()
+ q)(1 - ~)(1 + '1+ C)
A(, () = (-1/4)(1 + 7)(1 + ()(1 + 7 - ()
A (,9()
= (-1/4)(1 - 7)(1 + ()(1 + 1-
A/' 5(, (,) = (1/2)(1 - q)(1 + 77)(1 - ()
A(6(7,C() = (1/2)(1 - ()(1 + 7)(1 + ()
Ar'(n, () = (1/2)(1 - q)(1 + 7i)(1 + ()
AfN(8,() = (1/2)(1 - (C)(1+ (C)(1 - 4)
()
(a)
(b)
bI
.5
A
-1
dl
9L
r
I
NfifWA
-b-
RIOW~x
I
-4
~
i
Ill
~A
h~
~%,
ir~
%,%,%
~
-i
T% - -
N.
A
V~
~
k
%~%~
~
(c)
~f
IV
Figure 3.4 Finite element discretization of the domain of integration.
i'V~
N~
I~I~#~I
ti
in)
(a) Interface perturbati
node P
(b) Perturbation fiel
(only the nodes for
the vector ' k
indicated are pertu
Figure 3.5
Perturbation patterns.
The integral form (3.19a) can now be written as
6EL, =P
J1
V f1
iE[Pij(qj)p
+ {pbkuk,i
f1
-
mn.fmnij}(qji)p]
I J dqdC,
•
(3.35)
where the sum is extended to all elements in V, and I J I is the determinant of the
Jacobian matrix:
axOz y
IJ = I
0
y Ozx(3.36)
(3.36)
The integrand in (3.35) is, for each element, a function of the isoparametric coordinates q, C:
F(j?, C) = Pij(qid)p + {pbkuk,i - -mnmn,i}(qi)p
IJ
1,
(3.37)
so that each of the integrals can be evaluated numerically, e.g., by Gauss quadrature:
f1
L
1
F(, C)dqd( =
lWiWF(7i, C),
(3.38)
where F(qi, Cj) is the value of F at integration point (i,j) and the Wi, Wi are weights
for Gauss quadrature.
Thus, if a finite element analysis has been performed on the body, we can calculate
the values of F at the integration points of the elements inside V, substitute these
values in (3.38) and apply (3.35) to evaluate the right-hand side of system (3.32).
As we have already mentioned in section 3.32, we can use the invariance property
of the domain integral expression for 6ETL as a check for the quality of our numerical
calculations.
We can consider several choices for the domain of integration, V, and/or for the
form of the perturbation field ' (See Fig. 3.6).
For each of these choices we can evaluate the RHS vector {•ETL} and solve the
system (3.32) for the force on the interface {fr}.
Differences in the values of the {I} vectors can be regarded as a measure of the
numerical error in the solution.
This method has been implemented in the computer program DOMAIN for which
a listing of the symbolic code is given in Appendix II.
I
0
0
I-.
DO
0
4.D
0
0
U
Cu
CHAPTER 4
ANALYSIS OF RAFTING
4.1
Introduction
The numerical methods described in Chapter 3 allow us to evaluate the distribution of rn over a material interface. As discussed in Chapter 2, this is a quantity
directly related to the tendency of the interface to migrate and, with the convention
used throughout our derivation, a positive value of rn, at a certain location, indicates
a tendency for the inclusion to grow at that location. If the value of rn along a
precipitate-matrix interface is not uniform, this indicates a tendency for the inclusion
to modify its morphology.
In this chapter we will apply these concepts to the analysis of rafting in y - 7'
Ni-superalloys.
In paragraph 4.2, we will first assess the accuracy of our models by comparing data
obtained by numerical calculations with analytical solutions for the simple case of a
cylindrical inclusion in an infinite matrix. This test case will also provide the opportunity to construct an "evolution map" which interestingly compares with Pineau's
map (see Section 1.4).
In paragraph 4.3 we will then introduce the finite element model, for the - 7'
cuboidal morphology, that we will use to perform the rafting analysis, and we will
identify the set of alloys for which we will carry out our calculations.
In paragraph 4.4 we will present and discuss the results of purely elastic analyses,
where the force on the interface is evaluated for the initial conditions of applied stress
and misfit, and no misfit-relaxation effect due to creep is considered.
In paragraph 4.5 we will follow the evolution of the driving force during the very
fitst stage of primary creep in a simulated stress-annealing experiment.
In paragraph 4.6 we will finally discuss the results, draw our conclusions concerning the rafting process and offer a synoptic interpretation of numerical results and
experimental data.
4.2
Test Case: A Cylindrical Inclusion in an Infinite
Matrix
As a test case, we will consider the very simple problem of a cylindrical isotropic
misfitting inclusion in an infinite isotropic matrix subjected, at infinity, to a state
of uniaxial stress a, normal to the axis of the cylinder (Fig. 4.1(a)). The elastic
solution for this problem has been first determined by Goodier [55].
In order to simplify our calculations, we will assume that the matrix and the
inclusion have the same Poisson ratio v. Then the state of stress and strain at any
location (r/ra; 0), where rO is the radius of the inclusion, can be determined in terms
of:
Ep: the stiffness (Young's modulus) of the precipitate
E,,,: the stiffness (Young's modulus) of the matrix
ao0: the applied stress
6:
the misfit
v:
the Poisson ratio (assumed common to matrix and precipitate)
From the analytical expression for the elastic field, given in [55], the quantity T
can be readily evaluated at all locations along the interface using eq. (2.31(b)).
These calculations have been implemented in the simple program GOODIER,
listed in Appendix III, which evaluates r.at N locations along the interface (N is a
user-defined quantity).
Due to the symmetry of the problem, only one fourth of the geometry needs to
be considered. (We have chosen the quadrant shown in Fig. 4.1(b)).
The values of rn along the interface can be plotted against a normalized arclength
s as shown in Fig. 4.2. The origin for s is at the very top of the particle (see Fig.
4.1(b)) so that, in the graphs in Fig. 4.2, the left part of the graph roughly corresponds
to the "top" of the particle, while the right part corresponds to the "side".
Then, graph 4.2(a), where the average value of % on the "top" is larger than the
average value of r, on the "side", indicates a tendency toward a "type P" rafting
behavior, with rafts aligned in the direction of the applied stress; graph 4.2(b) indicates a tendency toward an isotropic rafting behavior; and graph 4.2(c) indicates
a tendency toward a "type N" rafting behavior with plates forming in the direction
normal to the applied stress.
Since, for this case, the analytical distribution of -acan be readily found, we have
been able to perform a brief parametric study varying the same characteristic quantiLies chosen by Pineau to construct his map (see paragraph 1.4), namely ao/Emb6 and
Ep/Em,,. The results of this study, obtained with a value of the misfit 6 = 0.1%, are
summarized in Fig. 4.3 and, with a broader spectrum of Ep/E,,, values, in Fig. 4.4.
For each small plot two numerical values are given; the first represents the magnitude (in MPa) of the difference between the average values of % on the top and on
the side: this is a measure of the tendency to coarsen with a preferential orientation;
the second value represents the average value of rn (in MPa) along the interface: this
is a measure of the average tendency for the particle to grow in order to decrease the
total elastic energy.
From this set of results, we can sketch the map shown in Fig. 4.5 that can be
compared with Pineau's map of Fig. 1.8. Besides the differences in the geometry of
the problems (spherical inclusion for Pineau; cylindrical inclusion for our test case),
the two maps differ for a fundamental reason: Pineau's map is a stability map: it is
constructed by determining the configuration that minimizes the energy. Our map is
an evolution map: it is constructed by evaluating the tendency toward shape evolution
for the precipitate in the actual initial configuration.
Still, bearing in mind this basic difference, the patterns of the two maps show
striking similarities.
We have also explored the effects of varying the sign and magnitude of the misfit.
A mere change of the sign of the misfit does not affect the results, as can be noted
by comparing Fig. 4.6, obtained with a misfit b = -0.1%, and Fig. 4.3 Conversely, if
we change the magnitude of the misfit, while the pattern of the map is not modified,
the magnitude of the driving force is remarkably altered.
The plots in Fig, 4.7 have been obtained with a misfit 6 = 0.3% : if we compare
the numerical values of rn with those in Fig. 4.3 we can notice that the force on the
interface is increased by almost one order of magnitude. This result should have been
expected since, for a purely elastic problem, rn is a quadratic form in the stress field
so that, if we increase the misfit (and thus ,oo,to keep coo/Em,,b constant) of a factor
3, we can expect that rn will increase of a factor 32 = 9.
After this digression we will now return to our original task of assessing the accuracy of the numerical methods introduced in Chapter 3 to evaluate Tn.
A boundary value problem to be solved using the finite element program ABAQUS
has been defined as shown in Fig. 4.8. Plane-strain, eight-node isoparametric element
have been used. Symmetry conditions are applied at the axes; the nodes along the
top of the domain of analysis are constrained to have equal vertical displacement and
to these nodes the external load ao, is applied; the nodes along the outer edge are
constrained to have equal horizontal displacements.
In this analysis the ratio of the particle radius, ro, to the dimension of the domain
of analysis, Loo, is 1/20. This ratio is low enough to insure "far field" conditions on
the outer boundary of the domain. The finite element mesh used for this problem is
shown in Fig. 4.9. A listing of the ABAQUS input file is given in Appendix IV.
The results of the finite element analysis have been post processed using both
the program POSTABQ and the program DOMAIN to obtain the distribution of Trn
along the interface.
A comparison of analytical and numerical results, for a representative case, is
shown in Fig. 4.10. In general, the domain integral method yields smoother results,
but the agreement with the analytic solution is satisfactory also for the direct method.
No appreciable difference in the results obtained using the domain integral method is
observed when varying the domain of integration, thus confirming the good level of
accuracy of the calculations. However, as we will discuss in the next paragraph, there
exist particular cases in which the direct method can yield more consistent results
than the domain integral method.
a0
(T.
a) Test problem
symmetry
plane
1a(00
t tttttt
b) Representative section
to be modeled
FigureCylindrical
4.1
inclusion in an infinite matrix.
Figure 4.1
Cylindrical inclusion in an infinite matrix.
00
zn profile
i n
IW0
Tendency toward morphological
evolution
(a)
0l
O
0
-
0
Type P
Normalized arclength
onn
2000
'U'
O 1
(bl
100'
0
Isotropic
Normalized arclength
-
4An
O-
(c)
0
0
Type N
Normalized arclength
Figure 4.2 Relation between the profile oftrn along the interface
and the tendency toward morphological evolution.
cz~
W
C4vi
II
C;
6.4
to
.m
0
"
0
LO
No0
0
-
0
6
0
Go
6
0
0
W
0
0
~
0
o.
oW
0
a
0
0
C4
-o0o
;.
0
.
0
0
-o
E -/E
.............
....
.
.....
I .. I II I .. . II I II I
I . II . II . II I
I I I. . .. I I .. II I I II
ii
ii
!
i
!ii!!
!iiii
!i
ii
!i
ii
!iii~iiiii~iiii
ii~~i
ii
ii
1,
12
i
I.I ... I . I . . I. IIII .. I
I.
~i
10
1
0
1
II1III
II
I0
o0o101
No
I oI
I
.
. . .
. . . . . . .
.. . ..
.
A.
-r
.:
: .:. :.:
.:.: :..- :.. : :... . .:P
. ... ........
...... .. . .....
... .. ..
... ....
IzI
.
.
.
.
-25
.
.
-20
.
.
-15
.
.
-10
.
.
.
-5
III
".
I.I.10.1
I1,
I6,
.I
I.I
I.0
IIIIIII.I
01.
1,
10
0.
11I.
1.
I
.0
I
0.
1
0o0 1
1 01Io6
o 110 o0
: -:-'.::
: "' ::.. : :"' -::.-:.
" ::. ::-
.
..........
. . ..... ...
-
.
__
0
5
10
15
20
a=/Er8
GV
m
Figure 4.5
Evolution map for an isotropic cylindric inclusion in an infinite matrix.
67
-ýA
0II
vi
I F
I•o
11
to
VI°
0
I,,l
El
C-4
0
0
-
-
0
0
0
0
o
0
6
0
0
a
c-
'?
V
0
00
-
-
446IC
0oo0
0
0
uy = const
4 t t4t
C
1 190
C
C
ux = const.
C
C
C
C
A
r~r'Zoa g c..- g3
- --
0000000
C
r
Figure 4.8
Schematic description of the cylindrical inclusion problem modeled,
with the boundary conditions indicated ( r" /L,
= 1/20).
L
___r
_ _·_
I
I
9
i
r-
I
-,
I
___
__J
____
i
,1
"o
-04
U-////
1
i
0
-- - A
14
7~WE
id
kr
r-l"""
7
... ...
. -
4
Figure 4.9
Finite element mesh for the cylindrical inclusion problem
modeled. In this Figure the bottom mesh ("near-particle" region)
fits into the indicated area of the top mesh
("nominal far-field" region).
a)
0
'4.4
5.4
V
V
0
V
U
0
0.0
0.0
Normalized arclength
0o
0.0
W.v
V.%
0.0
0.56
Normalized arclength
Figure 4.10
1.0
0.0
0.6
Normal ized arclength
Comparison of analytical and numerical results
(for Em = 200 GPa; Ep/Em = 0.5; o, = 1000 MPa;
b = -0.1%).
4.3
Problem Description
In the previous paragraph we have assessed the reliability of the proposed numerical procedures, by direct confrontation of numerical results and analytical solutions
in a simple test problem. We can now apply these numerical methods to analyze the
morphological evolution of y' precipitates in Ni-superalloys.
The first step will be to construct a finite element model of the array of cuboidal
y' precipitates. We have simplified the problem by considering a 2-D model using
generalized plane strain elements.
These elements permit uniform displacements in the "thickness" direction (normal
to the mesh-plane), thus allowing expansion in the third dimension as well as the
development of normal stresses and strain in this direction.
Due to the symmetry and periodicity of the microstructure, we can confine our
analysis to the "unit cell" depicted in Fig. 4.11(a).
Two different meshes have been generated: a coarse mesh (Fig. 4.11(b)), used
in the "tune-up" stage of the research to obtain first order approximations of the rn
profile with a limited amount of required computational time, and a more refined
mesh (Fig. 4.11(c)) for the final calculations.
The level of numerical noise associated with different choices of mesh and/or
numerical method can be appreciated by comparing the T, profiles in Fig. 4.12.
All of the plots shown in the following paragraphs have been obtained using the
refined mesh. Most of them have been obtained by applying the domain integral
method. However, as previously mentioned, there exist particular cases in which
the level of numerical noise in the domain integral solution is particularly high: if we
consider the integral expression for 6ET in Eq. (3.19 (a)), we can see that bET depends
on the gradient of the transformation strains cT . When we will perform the stressannealing analyses, the transformation strains will be given by the superposition of
misfit strains and creep strains c'eeP. In the very last stage of the analyzed transients,
extremely steep Ic~P-gradients develop around the corners of the -Y'precipitates (see
Figs 4.34 and 4.38). For these particular conditions the finite element mesh is not
refined enough to allow a precise numerical evaluation of the gradient of LcrcP so
that, for these particular cases, the direct method yields more reliable results than
the domain integral approach.
In order to perform a finite element analysis of the y - 7-y'crystal, we need the
following set of data:
- elastic constants of matrix and precipitate;
- misfit at test temperature;
- loading conditions;
(a) the unit cell
b
rv... '~~tj r...
maines
w
poo
Ww
4--
04
i
L
4bo-
symmetry planes
w/L =0.12
p/L =0.18
(b) coarse mesh
(c) refined mesh
Figure 4.11 Finite element discretization of the unit cell.
74
4.4
oS
41
0@
ki
o0
- creep properties;
-
microstructure geometry (dimensions of the 7' cubes and of the 7 channels).
While a simulation of actual 7-7' crystals is of extreme interest because the results
of the analysis can be directly compared with experimental data, we are limited, in
this approach, by the extremely exiguous number of well-characterized alloys available
in the literature. Values for the elastic constants of the 7 and of the 7' phases are
available only for three alloys on which rafting experiments have been carried out.
These are:
a. The negative-misfit alloy studied by Pollock [25]
b. The alloy studied by Tien and Copley [13]
c. The alloys studied by Miyazaki, Nakamura and Mori [15]
For these alloys the loading conditions and the test temperature are available as
well; also the value of the misfit at test temperature is given (alloys (a) and (c)) or
can be evaluated (alloy (b): see the end of paragraph 1.2). In Table 4.1 we have
summarized the characteristics of these alloys, together with the rafting behavior
observed in the experiments. Thus we have only six cases, to which we have assigned
an acronym that we will use to identify each case, that we can use to compare our
results with experimental data.
In order to explore a wider range of possibilities, we have "created" a few hypothetical alloys by altering some of the characteristics of the alloys of Table 4.1.
Altering the characteristics of the alloy studied by Miyazaki Nakamura and Mori,
we have "created" an alloy with positive misfit and soft precipitates by switching
the elastic constants of the two phases, and an alloy with negative misfit and hard
precipitates by changing the sign of the misfit.
For the alloy studied by Tien and Copley, we have explored the effect of assuming
the value of the misfit reported in the paper: in this way we have studied the behavior
of an alloy with soft particle and small positive misfit. Finally, we have explored
the effect of varying the crystal symmetry by considering for both phases isotropic
crystal lattices with elastic constants equivalent to those of the cubic lattices of the
alloy studied by Pollock; we have evaluated an equivalent Young's modulus as
C12 )(3 + (C11 - C12 )/C 1 2 )
2 + (C11 - C12)/12
and an equivalent Poisson's ratio as
Eeq
-
(C1 1
-
1
=(42+ (Ca - C )/C
v' =2
12
12
1ve
where C11 and C12 are the elastic constants of the cubic lattice [58].
(4.1)
(4.2)
0
$4
U,
0
;l
c.e
0f
5>
z
0
d
m0(
00
V-=4
bo
-
V
k
V-4
--
+
"-
+
+
+
-
-I-
00
C.
ýo
+
,I
u
"•T
I
mIII
I
cco
II
II
II
s
I
-444
II
Gc
II
0
00"'
if
j
+
-.4II
°C4
00
Z
II
•.
.
•
C4
II
a
II4
•
o
.
•
-. ,
0
0>1
r•C)
II
• •o
____0
.
11
- 0
,.-,
4.P.
V
0000)0
.-
.5
.
4l)
Wo.. 6
4
M.
I
W
I
The characteristics of these hypothetical alloys and the loading conditions for
which the simulations have been carried out are summarized in Table 4.2
Regarding the creep properties, in the range of loading conditions that will be
explored, the predominant creep mechanism involves dislocation motion primarily
in the 7-matrix. TEM studies [25] show that the 7y'
cuboids are not sheared by
dislocations until the latest stage of steady state creep. Thus in our model of the very
precipitate is allowed to deform only
first stage of the stress-annealing transient, the 7y'
elastically while the matrix material can deform by creep.
Since the < 001 > orientation of the applied load is a multiple slip orientation, a
simple isotropic power-law relation has been used to characterize the creep properties
of the matrix:
c
=Aao".
(4.3)
In all our stress annealing transients the creep properties given by Johnson et al.,
[56] for Ni-6W tested at 854* have been used, with:
n = 4.8
- 4 "8 )
A = 5.4E - 15(s-'MPa
Actually, since we are not particularly interested in "real time" simulation of
the stress-annealing transient, the accuracy of the pre-exponential coefficient A, is
of secondary relevance; the stress exponent, n, is a more important quantity but it
should not exhibit significant modifications when the alloying of the 7 phase is slightly
modified.
Regarding the microstructure geometry, we have relied on the data obtained by
Pollock [25] by TEM study of CMSX3. This choice will not yield an accurate simulation of the Tien and Copley and of the Miyazaki-Nakamura-Mori alloys which are
characterized by a lower volume fraction of the 7' phase. However, we believe that
this inaccuracy will not affect the basic pattern of the results. This point should be,
however, proved by further simulations in which the 7' volume fraction is modified.
According to Pollock's data, the 7' precipitates have an average cube length of
0.45 pm and the matrix passages have an average thickness of 60 nm, giving a 7'
volume fraction of 0.68. The finite element models shown in Fig. 4.11 have been
formulated so as to preserve these dimensions, so that since a generalized plane strain
analysis is done, the volume fraction of the 7' in the model is somewhat higher than
in the real material because the matrix channels parallel to the plane of the mesh
are neglected. While we are aware of the fact that our models still present a certain
degree of inaccuracy, and that the eleven cases that we will analyze do not constitute
a complete parametric study, we will see in the following paragraphs how the results
of our simulations fit in a logical pattern that can yield significant insight to the
rafting phenomenon.
Elastic Analysis
4.4
An elastic analysis for the eleven cases characterized in Tables 4.1 and 4.2, has
been carried out. Both the matrix and the precipitate are only allowed to deform
elastically and coherency at the matrix-precipitate interface is enforced. A typical
ABAQUS input file for this type of analysis is listed in Appendix V. The particular
file listed is relative to case 1 (MNMT), but all the other cases can be easily obtained
by correcting the values of misfit, elastic constants and applied load, according to the
data of Tables 4.1 and 4.2.
The results of the elastic analyses have been postprocessed to obtain the distribution of the force on the interface. Profiles of ra for each of the eleven cases as shown
in Figs. 4.13 through 4.23. A synopsis of the results is given in Table 4.3.
We can notice observing the rTprofiles, that the value of the difference of driving
force on the top and on the side of the particle is always very limited: at most it is
of the order of 0.3 MPa for the "MNM" - cases and it is as low as 0.006 MPa for the
TCTpMs and TCCpMs cases.
If we evaluate the "Pineau's parameters" Ep/Em and a/Emb for all the eleven
cases (for the cubic crystals an equivalent Young's modulus can be calculated using
Eq. (4.1)), we can see how the type of rafting behavior indicated by the rn profiles is
perfectly consistent with the behavior predicted by our simple evolution map of Fig.
4.5.
This suggests that a very similar map could be constructed by performing a parametric study on the actual -y - 7y'
morphology.
Table 4.3 Synopsis of the Elastic Analysis
Case
Case
#
Acronym
Ep/Em
o-/Emb
Rafting
Consistency
Consistency
behavior
with the
with
predicted
evolution map
experimental
by the model
of Fig. 4.5
results
1
MNMT
1.23
0.388
P
YES
YES
2
MNMC
1.23
-0.388
N
YES
YES
3
TCT
0.91
-0.327
P
YES
NO
4
5
TCC
PLT
0.91
0.92
+0.327
-0.148
N
P
YES
YES
NO
NO
6
PHT
0.92
-0.399
P
YES
NO
7
MNMTINV
0.81
+0.314
N
YES
8
MNMTNMs
1.23
-0.388
N
YES
9
TCTPMs
0.91
+4.9
N
YES
10
TCCPMS
0.91
-4.9
P
YES
11
PLTiso
0.92
-0.148
P
YES
Unfortunately, such a map would be totally inadequate to predict the rafting behavior
of the crystals since, as indicated in Table 4.3, our predictions based on elastic calculation of the r, profile are not always in agreement with the available experimental
data.
This leads us toward a more accurate simulation of the actual conditions in which
the rafting behavior is observed, as will be discussed in the next paragraph.
It is interesting to note that if we assume, for the alloy studied by Tien and Copley,
the erroneous value of the misfit given in the paper, the rafting behavior predicted
by our evolution map, and by the T. profiles, is consistent with the experimental
observations.
This unfortunate contingency has represented, in the past, a confusing source of
misplaced confidence in the methods based on purely elastic calculations.
3.0
co
a)
2.5
C4
2.0
O
O
1.5
0.0
0.5
Normalized arclength
Figure 4.13
rn profile for the elastic analysis of case 1 (MNMT).
1.0
1.5
Cd
Q.)
co
1.0
4-4
0.5
on0.0
0.0
0.5
Normalized arclength
Figure 4.14
1rn profile for the elastic analysis of case 2 (MNMC).
1.0
0.70
0.65
04
114
0o
0.60
0.55
0.0
0.5
Normalized arclength
Figure 4.15
rn profile for the elastic analysis of case 3 (TCT).
1.0
1.60
1.55
94)
104
Q)
0
1.50
1 .45
0.0
0.5
Normalized arclength
Figure 4.16
ern profile for the elastic analysis of case 4 (TCC).
1.0
2.0
co
1.5
Q)
A4-
1.0
Q~)
0
0.5
0
0.0
0.0
0.5
Normalized arclength
Figure 4.17
rn profile for the elastic analysis of case 5 (PLT).
1.0
2.0
cd
1.5
Q.)
1.0
C.)
0o
0.5
0.0
0.0
0.5
Normalized arclength
Figure 4.18
rn profile for the elastic analysis of case 6 (PHT).
1.0
Q')
C.)
'4-4
cJ)
4~)
0
C)
0
1I
0.0
0.5
Normalized arclength
Figure 4.19
Fr profile for the elastic analysis of case 7 (MNMINV).
1.0
1.5
Vco
1.0
4-4
6,-p
0(
0.5
0
0
0.0
0.0
0.5
Normalized arclength
Figure 4.20
irn profile for the elastic analysis of case 8 (MNMTNMs)-
1.0
0.048
C)
0.046
Q~)
0.044
0
0.042
0
r=
0.040
0.0
0.5
Normalized arclength
Figure 4.21
rn profile for the elastic analysis of case 9 (TCTPMS).
1.0
f%4 '
-U.U14
-0.016
U
0)
.•.
-0.018
0)
0
-0.020
0
-_n
02p
0.0
0.5
Normalized arclength
Figure 4.22
rn profile for the elastic analysis of case 10 (TCCpMs).
1.0
4
Ain•
1.uo
1.04
C4
1.02
Q)
0
1.00
0
9QR
0.0
0.5
Normalized arclength
Figure 4.23
rn profile for the elastic analysis of case 11 (PLTIso).
1.0
4.5
Analysis of the Stress-Annealing Transients
As previously discussed in paragraph 1.2, TEM observations of - - 7' crystals show
how, in the very first stage of stress-annealing tests, mobile dislocations start to form
and multiply in the 7- matrix. As soon as the crystal is brought to test temperature,
networks of dislocations form at the 7y- -I' interfaces, wrapping the f' cuboids and
relieving most of the shear component of the misfit strain. When an external load is
applied, dislocations glide and climb in the narrow 7 channels so that a macroscopic
creep flow in the 7 matrix is observed.
As a result of this process, the elastic fields in the crystal are dramatica!!y modified.
Therefore a meaningful evaluation of the force on the interface should be ba.ed on
these altered fields.
Even if, considering the scale of the microstructure, an accurate simulation of
the process should be based on a model capable to treat discrete dislocations, some
interesting results can be obtained by considering a simplified macroscopic creep
model for the 7 phase. As we have already mentioned in paragraph 4.3, we have
modeled the creep behavior of the matrix with the simple isotropic power-law relation
(4.3).
We have carried out a simulation of the very first stage of the stress annealing
transients, using the ABAQUS code, and analyzed the corresponding evolution of the
rn profile.
A typical ABAQUS input file for this kind of analysis is listed in Appendix VI.
The particular file listed corresponds to case 1 (MNMT).
As initial conditions for the simulated transients we have considered the elastic
field in the crystal associated with the misfit strain at test temperature, enforcing
perfect coherency at the interfaces. We have then applied the test load, over a short
linear ramp of one second, and let the matrix creep according to equation 4.3.
Typical contour plots of the initialstress state, due to the misfit only, are shown
in Figs. 4.24 through 4.29 for a positive-misfit alloy (Miyazaki-Na'(amura-Mori alloy:
Cases 1,2,7) and for a negative-misfit alloy (Pollock alloy: Case 5).
As schematically indicated in Fig. 4.30(a), the main components of stress in the
matrix, ao in the Figure, are in the planes of the channels (tensile for positive-misfit
alloys, compressive for negative-misfit alloys). The magnitude of the biaxial stress a0o
in the 7ychannels varies with the misfit and the elastic constants of the two phases but
typically is of the order of 400 - 500 MPa in most commercial alloys. The component
of stress normal to the planes of the channels is one order of magnitude smaller and
of opposite sign. The Y' precipitates are almost in a state of hydrostatic stress of the
same magnitude as the normal stress in the 7 channels.
Thus in the very first stage of a stress-annealing transient with an applied stress,
a, in the range 50-200 MPa, the effect of the misfit stresses dominates over the effect
of the applied load, and the 7-ymatrix creeps so as to accommodate the misfit (Fig.
4.30(b)). This stage corresponds to the formation of dislocation networks at the 7 -7'
interfaces.
In Figs. 4.31 through 4.38, a sequence of contour plots of the equivalent creep
strain in the matrix, crP, at subsequent stages of the primary creep transient for
case 1 (MNMT) and case 2 (MNMC) is shown.
The equivalent creep strain in the matrix is defined as:
•creep
eq
where ,
2
i•
re
.2
si
s
P
(4.4)
are the components of the creep strain tensor.
The steep gradient of cc' p in the direction normal to the interface, particularly
noticeable around the corner of the precipitate, is the "continuum" analogy to the
network of dislocations at the interface. The gradient would be even steeper with a
higher creep exponent , n, in expression 4.3.
As the creep transient progresses, and the deviatoric components of the stress
tensor are accommodated by the creep strains, a state of hydrostatic stress builds
up in the matrix channels. Regardless of the sign of the misfit and of the relative
stiffnesses of matrix and precipitates, all the transients evolve toward a stress state,
at the end of the primary creep stage, in which (Fig. 4.30 (c)):
* if a tensile stress is applied, there is a build up of negative pressure in the
horizontal channels ( ',e channels normal to the applied stress) and of positive
pressure in the vertica channels (the channels parallel to the applied stress);
* if a compressive stress is applied, there is a build up of positive pressure in the
horizontalchannels and of negative pressure in the vertical channels.
The magnitude of the stress in the horizontal channels is higher than the applied
stress, while the magnitude of the stress in the vertical channels is much lower than
the applied stress.
In Figs. 4.39 through 4.54, typical contour plots of the stress state at the end
of the primary creep stage are shown for applied tensile loads (Cases 1,3,5,6,9) and
compressive loads (Cases 2,4,6).
As the stress state evolves, the force on the interface is modified as well. Plots
of the evolution of the % profile during the stress annealing transient for the eleven
c, ,es considered are shown in Figs. 4.55 through 4.65. For each case, nine plots are
gi ven of the T, profile at successive times of the transient. As a normalized measure
of the stage of evolution of the creep transient we have chosen the ratio between the
average equivalent creep strain in the matrix and the initial misfit (ZCrMP/6). The
average creep strain in the matrix is evaluated as a volume average, over the 7 phase,
of the equivalent creep strain:
Y=fe dVV,
(4.5)
where V is the volume occupied by the y phase.
We have chosen the parameter 7arP/b rather than a more immediate quantity,
such as the creep time, because it is more directly related to the evolution of the r.
profile, as we will discuss in the next paragraph.
Let's now consider the evolution of the a profiles for the first six cases, relative
to the simulation of stress annealing tests performed on experimental alloys.
We can notice how the profiles quickly evolve toward configurations that show
a marked driving force for directional coarsening. If we compare these graphs with
those obtained in our purely elastic analyses, we can see that the absolute values
of the differences between the force on the top and on the side of the precipitates,
A-, are generally increased by one order of magnitude; however, the most important
result is that the signs of ALT are inverted for several cases so that now for all the
experimental alloys we have perfect agreement between the observed rafting behavior
and the tendency toward directional coarsening that can be inferred from the r.
profiles.
If we now consider the evolution of the r. profiles for the hypothetical alloys of
cases 7 through 11, we can list a number of considerations:
* comparing case 7 (MNMTINV) and case 1 (MNMT) we can see that, if we invert
the relative stiffnesses of matrix and precipitate, the driving force for rafting
remains virtually identical. This result is in antithesis to that of a purely elastic
calculation for which an inversion of the elastic constants brings about a parallel
inversion of the rafting tendency (see map in Fig. 4.5);
and case 1 (MNMT) we see that if
* conversely, comparing case 8 (MNMTNMs)
we change the sign of the misfit the r. profiles result totally reversed;
* we compare now case 9 (TCTpMs) and case 7 (MNMTINV). Both cases are
relative to positive-misfit alloys with soft precipitates but the misfit for case 9
is about 1/20th of the misfit for case 7. We can see that the fourth plot of
Fig. 4.63 (case 9), relative to a stage of the transient for which -rT"P/6= 1.88
shows a profile similar to that of the eighth plot of Fig. 4.61 (case 7), for which
-'"P/b = 1.83. However, the value of ArT for this stage of case 9 is only a few
percent of the correspondent value for case 7;
* comparing case 10 (TCCpMs) and case 9 (TCTpMs) we see that if we invert
the direction of the applied load, the r. profiles are reversed. (This can be also
noted comparing cases 1 and 2 and cases 3 and 4).
* comparing case 11 (PLTIso) and case 5 (PLT) we see that a change in the
symmetry of the crystal brings about only minor changes ip the r. profiles
which are mainly localized around the corners of the precipitate.
In the next paragraph, we will reinterpret these observations, as well as other
features of the r. profiles, within the context of a more general framework and we
will offer a general discussion of the rafting phenomenon.
__
Oii
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-2.00E+02
-1.30E+02
-5.99E+01
+1.00E+01
+8.00E+01
+1.50E+02
+2.20E+02
+2.90E+02
+3.60E+02
+4.30E+02
+5.00E+02
Figure 4.24
Contours of all due to misfit only (6 = +.56%)
for the alloy tested by Miyazaki, Nakamura and Mori [16].
S22
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-2.00E+02
-1.30E+02
-5.99E+01
+1.00E+01
+8.00E+01
+ 1.50E+02
+2.20E+02
+2.90E+02
+3.60E+02
+4.30E+02
+5.00E+02
Figure 4.25 Contours of a22 due to misfit only (6 = +.56%)
for the alloy tested by Miyazaki, Nakamura and Mori [16].
0'12
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-5.00E+02
-4.40E+02
-3.80E+01
-3.20E+02
-2.60E+02
-2.00E+02
-1.40E+02
-8.00E+01
-1.99E+01
+4.00E+01
+1.00E+02
Figure 4.26
Contours of '1 2 due to misfit only (6 = +.56%)
for the alloy tested by Miyazaki, Nakamura and Mori [16].
c' 1
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-6.00E+02
-5.20E+02
-4.40E+02
-3.60E+02
-2.80E+02
-2.00E+02
-1.20E+02
-3.99E+01
-4.00E+01
+1.20E+02
+2.00E+02
Figure 4.27
Contours of ol,1 due to misfit only (6 = -0.38%)
for the alloy tested by Pollock [25].
100
0 22
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-6.00E+02
-5.20E+02
-4.40E+02
-3.60E+02
-2.80E+02
-2.00E+02
-1.20E+02
-3.99E+01
+4.00E+01
+1.20E+02
+2.00E+02
Figure 4.28
Contours of 9 22 due to misfit only (6 = -0.38%)
for the alloy tested by Pollock [25].
101
012
1
2
3
4
Value (MPa)
+7.00E+05
+7.00E+01
+1.40E+02
+2.10E+02
IC)
--
6
7
8
9
10
11
.
Q
-
'
ino
u%iJ-. T
.j,
+3.50E+02
+4.20E+02
+4.90E+02
+5.50E+02
+6.30E+02
+7.00E+02
Figure 4.29
Contours of a12 due to misfit only (6 = -0.38%)
for the alloy tested by Pollock [25].
102
Positive M~isfit Alloys
Negative Misfit Alloys
-q/10
a) initial state of
stress due to
lattice misfit
,c/10O
,/10
b) primary
creep flow
' channels
Y channels
become
become
thicker
thinner
Tensile Load
Compressive load
(T
I II II I I
I II t I I tIII
(c) state of stress
at the end of the
primary stage
of the creep
transient
r,
11111 11111
Figure 4.30
0.15 CF
tI1
II II
Schematic representation of the evolution of the stress and strain fields
in the primary stage of the creep transient.
103
creep
-eq
1
2
3
4
5
6
8
9
10
11
Value
+3.00E-10
+3.00E-04
+6.00E-04
+9.00E-04
+1.20E-03
+1.50E-03
+1.80E-03
+2.10E-03
+2.40E-03
+2.70E-03
+3.00E-03
creep/6
= 0.0853
Figure 4.31
Contours of ecreep for the transient analyzed in case 1 (MNMT)
at creep time t = 0.011 sec.
104
creep
eq
Value
+9.00E-10
2 +9.00E-04
3 +1 .80E-03
4 4+2.70E-03
5 +3.60E-03
6 +4.50E-03
7 +5.40E-03
8 +6.30E-03
9 +7.20E-03
10 +8.10E-03
11 +9.00E-03
-creep/6
=
Figure 4.32
0.98
Contours of cre.ep for the transient analyzed in case 1 (MNMT)
at creep time t = 4.46 sec.
105
c
reep
q
1
2
3
4
5
6
7
8
9
10
11
Value
+3.00E-09
+3.00E-03
+6.00E-03
+9.00E-03
+1.20E-02
+1.50E-02
+ 1.80E-02
+2.10E-02
+2.40E-02
+2.70E-02
+3.00E-02
Tcreep/6
-
Figure 4.33
1.55
Contours of c'reep for the transient analyzed in case 1 (MNMT)
at creep time t = 63.47 sec.
106
eireep
,q
c
1
2
3
4
5
6
7
8
9
10
11
Value
+9.00E-09
+9.00E-03
+1.80E-02
+2.70E-02
+3.60E-02
+4.50E-02
+5.40E-02
+6.30E-02
+7.20E-02
+8.10E-02
+9.00E-02
creep/b = 2.39
Figure 4.34
Contours of creep for the transient analyzed in case 1 (MNMT)
at creep time t = 50,000 sec.
107
clreep
fe
,q
Value
+3.00E-10
2 +3.00E-04
3 +6.00E-04
4 +9.00E-04
5 + 1.20E-03
6 + 1.50E-03
7 +1.80E-03
8 +2.10E-03
9 +2.40E-03
10 +2.70E-03
11 +3.00E-03
1
rreep/6
= 0.085
Figure 4.35
Contours of :Creep for the transient analyzed in case 2 (MNMC)
at creep time t =0.011 sec.
108
creep
Ceq
Value
+1.00E-09
2 +1.00E-03
3 +2.00E-03
4 +3.00E-03
5 +4.00E-03
6 +-5.00E-03
7 +6.00E-03
8 4-7.00E-03
9 +8.00E-03
10 +9.00E-03
11 +1.00E-02
1
reep/S =
Figure 4.36
1.25
Contours oi~caeep for the transient analyzed in case 2 (MNMC)
at creep time t = 3.86 sec.
109
creep
1
2
3
4
5
6
7
8
9
10
11
-creep/
Value
+4.00E-09
+4.00E-03
+8.00E-03
+1.20E-02
+1.60E-02
+2.00E-02
+2.40E-02
+2.80E-02
+3.20E-02
+3.60E-02
+4.00E-02
-=
Figure 4.37
2.23
Contours of c eep for the transient analyzed in case 2 (MNMC)
at creep time t = 252 sec.
110
Value
creep
1 +8.00E-09
2 +8.00E-03
3 +1.60E-02
4 +2.40E-02
5 +3.20E-02
6 +4.00E-02
7 +4.80E-02
8 +5.60E-02
9 +6.40E-02
10 +7.20E-02
11 +8.00E-02
Tureep/&
= 2.71
Figure 4.38
Contours of
reep
for the transient analyzed in case 2 (MNMC)
at creep time t = 50,000 sec.
111
Figure 4.39
Contours of al at the end of the analyzed
transient for case 1 (MNMT).
112
Figure 4.40
Contours of a22 at the end of the analyzed
transient for case 1 (MNMT).
113
c1ni
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-1.00E+02
-7.00E+01
-4.00E+01
-9.99E+00
+2.00E+01
+5.00E+01
+8.00E+01
+1.10E+02
+1.40E+02
+1.70E+02
+2.00E+02
Figure 4.41
crrr~r
rC
irrsrsEs
Contours of all at the end of the analyzed
transient for case 3 (TCT).
114
0
22
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-1.00E+02
-7.00E+01
-4.00E+01
-9.99E+00
+2.00E+01
+5.00E+01
+8.00E+01
+1.iOE+02
+1.40E+02
+1.70E+02
+2.00E+02
Figure 4.42
Contours of U22 at the end of the analyzed
transient for case 3 (TCT).
115
0"11
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-2.00E+01
-1.20E+01
-3.99E+00
-4.00E+00
+1.20E+01
+2.00E+01
+2.80E+01
+3.60E+01I
+4.40E+01
+5.20E+01
+6.00E+01
Figure 4.43
Contours of all at the end of the analyzed
transient for case 5 (PLT).
116
Figure 4.44
Contours of a22 at the end of the analyzed
transient for case 5 (PLT).
117
Figure 4.45
Contours of o11 at the end of the analyzed
transient for case 6 (PHT).
118
0"11
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-1.00E+02
-7.10E+01
-4.00E+01
-9.99E+00
+2.00E+01
+5.00E+01
+8.00E+01
+1.10E+02
+1.40E+02
+1.70E+02
+2.00E+02
Figure 4.46
Contours of 2r2 at ihe end of the analyzed
transient for case 6 (PHT).
119
CT11
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-1.00E+02
-7.10E+01
-4.00E+01
-9.99E+00
+2.00E+01
+5.00E+01
+8.00E+01
+1.10E+02
+1.40E+02
+1.70E+02
+2.00E+02
Figure 4.47
Contours of all at the end of the analyzed
transient for case 9 (TCTPMs).
120
Value (MPa)
1
-1.00E+02
-7.10E+01
2
-4.00E+01
3
-9.99E+00
4
5
+2.00E+01
6
+5.00E+01
71 +8.00E+01
8
+1.10E+02
+1.40E+02
9
+1.70E+02
10
11
+2.00E+02
'22
Figure 4.48
Contours of a22 at the end of the analyzed
transient for case 9 (TCTPAs).
121
o'ii
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-2.00E+02
-1.70E+02
-1.40E+02
-1.10E+02
-8.00E+01
-5.00E+01
-1.99E+01
+1.00E+01
+4.00E+01
+7.00E+01
+1.00E+02
Figure 4.49
Contours of oal at the end of the analyzed
transient for case 2 (MNMC).
122
'22
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-2.00E+02
-1.70E+02
-1.40E+02
-1.10E+02
-8.00E+01
-5.00E+01
-1.99E+01
+1.00E+01
+4.00E+01
+7.00E+01
+1.00E+02
Figure 4.50
Contours of a22 at the end of the analyzed
transient for case 2 (MNMC).
123
011
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-3.00E+02
-2.60E+02
-2.20E+02
-1.80E+02
-1.40E+02
-1.00E+02
-6.00E+01
+1.99E+01
+2.00E+01
+6.00E+01
+1.00E+02
L
i~
\
~8oe~oooed
ý6ý~e
Ii
Figure 4.51
Contours of a11 at the end of the analyzed
transient for case 4 (TCC).
124
Figure 4.52
Contours of 0r22 at the end of the analyzed
transient for case 4 (TCC).
125
"11
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-2.00E+02
-1.70E+02
-1.40E+02
-1.10E+02
-8.00E+01
-5.00E+01
-1.99E+01
+1.00E+01
+4.00E+01
+7.00E+01
+1.00E+02
1 _
_1
llkýa
-
-
__
Q---~L
z,
psý
..
........
.................
L-
Figure 4.53
Contours of all at the end of the analyzed
transient for case 10 (TCCPMS).
126
0'22
1
2
3
4
5
6
7
8
9
10
11
Value (MPa)
-2.00E+02
-1.70E+02
-1.40E+02
-1.10OE+02
-8.00E+01
-5.00E+01
-1.99E+01
+I.00E+01
+4.00E+01
+7.00E+01
+1.00E+02
Figure 4.54
Contours of a22 at the end of the analyzed
transient for case 10 (TCCpMS).
127
2.0
-
U
ES
4,
r
0
0E
o.
he
0.0
0.0
2
1
0
-1
0.0
0.5
1.0
2
2
1
0
0
-2
-1
-4
0.5
0.0
Normalized arclength
Figure 4.55
1.0
0.0
0.5
Normalized arclength
1.0
0.0
0.5
Normalized arclength
Evolution of the rn profile during
the stress-annealing transient for case 1 (MNMT).
128
2.0
v
0
Ei
.'=
¢u
-1
a
-2
@1
n00.
0
0.6
rF-
1
0.0
1.0
I
0
0
0
-I
-1
-2
-2
-3
-3
-1
-2
"/6: 0.125E+01
L-
-q
--
0.0
0.0
1.0
0.0
0.6
1.0
Normalized arclength
0.0
0
-1
-2
-3
-4
-8
0.0
0.5
1.0
Normalized arclength
Figure 4.56
0.0
0.5
Normalized arclength
Evolution of the Tn profile during
the stress-annealing transient for case 2 (MNMC).
129
nU.6%
0.7
0.6
0.0
S0.6
-0.6
0.4
0
-1.0
-0.2
S0.3
-.
0.2
0.5
--I-
4A
1.0
0.0
0.0
0.6
0
-1
w
-2
-2
0.5
0.0
i-
/
-3
1.0
0.0
0.5
1.0
: -0.187E+01
0.0
&
1
0
0
-1
-1
-2
-2
-3
-3
S
0.0
0.5
Normalized arclength
Figure 4.57
1.0
-4
-
0.0
0.5
1.0
0.0
Normalized arclength
Evolution of the rn profile during
the stress-anneaii.ig transient for case 3 (TCT).
130
Normalized arclength
I1.0.U
0.7
..
0
0.5
6)
61
.rl
0.0
0.3
a.
-0.6
0.2
0.5
-
1.0
0.0
0.5
1.0
1.0
0.0
05
1.0
0.0
0.5
1.0
-
0.0
1.6
1.0
0.5
0.0
-1
-0.5
-2
-1
0.0
0.6
Normalized arclength
Figure 4.58
1.0
0.0
0.6
-3
0.0
Normalized arclength
Evolution of the Tn profile during
the stress-annealing transient for case 4 (TCC).
0.5
Normalized arclength
0.5
1.0
0.0
00
05
-0.5
-05
00
-0.5
-- .
I1 .-0
0.6
0.0
1.0
0.0
06
1 .V
0.0
1.0
-
nc
0.5
0.5
0.0
00
0.0
-0.5
-0.5
-0.5
-1 0
1.0
0.5
-1.0
-1.0
0.5
0.0
0.0
0.6
0.0
1.0
0.6
0.5
0.0
0.0
-0.5
-0.5
-0.5
-1.0
0.0
0.5
Normalized arclength
Figure 4.59
1.0
-1.0
0.5
-1.0
0.0
0.5
1.0
0.0
Normalized arclength
Evolution of the rn profile during
the stress-annealing transient for case 5 (PLT).
132
0.5
Normalized arclength
1.0
l
0.6
0
1
V
U
0
0.0
t
V
00
-0.5
-0.6
-1.0
0
0
V
U
I0
-I
-1I
-1
0.0
0.5
1.0
0.5
1.0
0.0
0.5
1.0
00
06
1.0
0.0
1
0
12
U
0
-1
0
12
0
S..
0
-2
-3
-3
0.0
'"1/6
U
1.
0
V
U
0
1 -3
: -0.227E+01
L-
00
1
0
0
V
-1
4.~
-1
-2
0
12
U
I..
0
-2
-3
~z.
-4
0.0
0.5
1.0
Normalized arclength
Figure 4.60
0.0
0.5
Normalized arclength
1.0
0.0
Evolution of the rn profile during
the stress-annealing transient for case 6 (PHT).
133
0.5
Normalized arclength
£i
0.5
0.0
0
u11
0
-0.5
-1.0
0.5
1.0
0.5
1.0
Iu
0
0
- I00
2
i
0.0
Normali zed arclength
Figure 4.61
2
--
0.0
0.5
Normalized arclength
1.0
0.0
0.5
1.0
Normalized arclength
Evolution of the rn profile during
the stress-annealing transient for case 7 (MNMTINV).
134
1.5
0
1.0
0.5
-1
0.0
0.0
0.5
-9
1.0
0.0
0.5
1.0
0.0
0
-1
-1
-2
-2
-3
-3w
-4
--
0.5
1.0
0.0
0.5
1.0
0.0
0
-2
-4
-4
0.0
0.5
1.0
Normalized arclength
Figure 4.62
-4
0.0
-6
0.5
Normalized arclength
1.0
0.5
0.0
Normalized arclength
Evolution of the m, profile during
the stress-annealing transient for case 8 (MNMTNMs).
135
- ý0.00562
-- ý
ý 0.0280
^ ^^n^
0.0080
0.0076
0.0275
00060
0.0270
0.0049
-~--~-
.-- %
0.0
0.5
1.0
0.0070
-`--`-
nn
nr n;;
..0.0285
VM40
0.0
0.5
1.0
1.0
0.0
0.0
.
.J.U
ooe
-0.2
-0.1
0.00
-0.4
-0.056
-0.2
-0.6
oL
ca
0
0.188E+01
-o010
0.0
-06B
-.0
0.5
1.0
0.0
0.6
1.0
0.6
1.0
0.5
1.0
A
I
n00n
rl.0
U
-1
6)
0
0
-1.0
-2
-1.0
-1.5
-1.5
0.0
0.5
Normalized arclength
Figure 4.63
1.0
-3
0.5
0.0
Normalized arclength
1.0
0.0
Normalized arclength
Evolution of the 7n profile during
the stress-annealing transient for case 9 (TCTpAs).
136
A^^Ar
0.004b
^ ^^n
^
4A
-0.014
0,0024
0.0022
-0.015
0.0044
0.0020
0.0043
-0.016
0.0018
0.0042
1.0
-
^A
-0
00016
0.0
-I
0.00UU
0.0
017
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A
^
0.0
A
0.0
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-0.1
-0.2
-0.10
-0.2
-0.4
-0.15
-0.3
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v.r
0.253E+01
L-
-i
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0.0
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0.0
nn
v.0
-0.5
-0.5
-1.0
-1.0
-1.5
-3
-1.5
0.0
0.5
Normalized arclength
Figure 4.64
1.0
0.0
0.5
Normalized arclength
1.0
0.0
0.5
Normalized arclength
Evolution of the rn profile during
the stress-annealing transient for case 10 (TCCPAfS).
137
- In
f%
a
u.e
-~1.0
ns;
0.4
0.9
0.0
.9
0.8
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.
0.0
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.
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-0.5
0
0
ý43 -1.0
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-1.0
0.0
0.5
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0.0
0.5
1.0
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c
^0.6
S0.o0
-0.5
-0.5
-0.5
0
0
-10
-.-.
0.0
0.5
Normalized arclength
Figure 4.65
1.0
10.
--1..,
0.0
0.5
Normalized arclength
1.0
.l~
0.0
0.5
Normalized arclength
Evolution of the 7n profile during
the stress-annealing transient for case 11 (PLTiso).
138
1.0
4.6
Discussion
As we have repeatedly mentioned, experimental observations show that the rafting
process initiates in the primary stage of the creep transient.
Figure 4.66 shows a typical creep curve with corresponding micrographs illustrating the development of rafting as observed by MacKay and Ebert [22]. Figure 4.67
shows a similar curve obtained by Fredholm and Strudel [23] with the corresponding
changes in microstructure.
From these and several other observations [17-24,27], we can infer that the driving
force for directional coarsening sets in early in the transient and that the morphology
evolution ensues within an extent of time which is determined by the kinetics of the
process - in particular by the diffusion properties of the crystal [57]. We can then
expect to be able to explain the tendency toward directional coarsening by analyzing
the state of strain and stress in the crystal at the end of the primary creep.
The primary stage of creep corresponds to the formation of networks of dislocations which relieve the misfit stresses; in our continuum model, we can identify this
stage by comparing the value of the average equivalent creep strain in the matrix
7creep as defined in (4.5), with the initial value of the misfit, 6 : we can assume that
the misfit stresses are relieved when 7rep
and 6 are of the same order of magnitude.
Actually, from the results of our finite element analysis, we can conclude that the
primary stage of creep is completed when ? re"p I/6 2.
Let's now analyze the state of stress and strain at this stage of the transient, and
the corresponding effects on the 7n profile. We recall that the value of - is given by
the expressions (2.31) that we repeat here for convenience:
Tn = [W]-
ti[--].
an
(4.6)
We will first consider the case of an applied tensile load and we will afterward discuss
the case of an applied compressive load.
(a) Applied Tensile Load
It is convenient to independently analyze the conditions on the "top" of the
precipitates (corresponding to the channels normal to the applied load) and on
the "side"" of the precipitates (corresponding to the channels parallel to the
applied load).
* On the top of the precipitates, due to the high level of hydrostatic stress
in the matrix (see Fig. 4.30(c)), the t.[-] term in expression (4.6) largely
dominates. The [W] term ranges between one and five percent of the ti[
[ ]
term for the analyzed alloys.
The value of the component of the traction vector normal to the interface,
tn, is large and positive, while the tangential component is negligible. The
139
magnitude of the normal component of [a], whiclh we will indicate as
[un,n ], is larger than the magnitude of the original misfit since the primary
creep flow acts so as to enhance the jump in 'un,n across the interface,
initially due to the lattice misfit (see Fig. 4.30(b)). Thus for negativemisfit alloys [un,n] is positive so that the force on the interface on the
top of the precipitate results negative. The magnitude of the force scales
with the applied load (which directly affects the magnitude of in) and
with the misfit (which directly affects [un,n ]). As the transient progresses
past the primary stage, and the matrix starts creeping under the effect of
the applied load, the creep flow keeps acting so as to increase [Un,n ] and
therefore rn will assume increasingly larger negative values.
Conversely, for positive-misfit alloys [un,n ] is initially negative so that the
force on the interface on the top of the precipitate results positive. The
magnitude of rn scales again with the applied load and with the misfit. As
the transient progresses past the primary stage, the matrix starts creeping
under the effect of the applied load. This process gives a positive contribution to [un,n] so that the magnitude of rn starts to decrease and, for
T"reP/l -- 4,Tn becomes negative.
* On the side of the precipitates, since the traction vector is approximately
one order of magnitude smaller than that of the top, the two terms in (4.6)
are comparable and the magnitude of the force on the interface is much
lower. The [W] term always gives a negative contribution to rn because
the state of stress in the precipitate gives rise to an elastic energy level
higher than that in the matrix channels, which are essentially in a state of
moderate hydrostatic stress.
The normal component of the traction vector, tn, is negative and the tangential component is negligible.
For negative-misfit alloys, the [Un,n ] term at the end of primary creep is
large and positive (for what concerns this term, the primary creep flow
acts in the same sense of the misfit) so that the (-t [
,])term is positive
and counterbalances the [W] term.
Thus for alloys characterized by a small negative misfit, the [W4] term
initially dominates and -n is negative, while for alloys with large negative
misfit the (-t [0 j]) term dominates and rn is positive (this is normally the
case for most commercial alloys). As the transient progresses, the matrix
material creeps under compressive stress so that negative contributions are
added to [un,n ] and thus to the force 7n.
For positive-misfit alloys, the [Un,n] term at the end of the primary creep
is large and negative so that the (-ti[•]) term is negative as well and
adds up with [W] to give a negative value for rTn. As the transient evolves,
negative contributions are added to [un,n] so that rn assumes increasingly
larger negative values.
140
(b) Applied Compressive Loads
The patterns according to which rn evolves are essentially symmetrical to those
for tensile loads due to the fact that the sign of the traction vector is inverted
* on the top of the precipitates, the ti[ ] term dominates; tn is large and
negative while the tangential component of the traction vector is negligible.
For negative-misfit alloys [Un,n ] at the end of primary creep is positive so
that the force on the interface results positive, and its magnitude scales
with the applied load and the initial value of the misfit.
As the transient evolves, the matrix material creeps under the applied compressive loads and negative contributions are added to [un,n]I so that the
magnitude of Tn starts to decrease and Tn will eventually become negative.
For positive-misfit alloys [un,n ]at the end of the primary creep is negative
so that the force on the interface is negative as well. As the transient progresses past the primary stage of creep, the matrix creeps in compression
under the applied load so that negative contributions are added to [un,n ]
and rn will assume increasingly larger negative values.
* On the side of the precipitates the two terms in (4.6) are comparable and
the magnitude of rn is much lower. The [W] term always gives a negative
contribution to rn. The normal component of the traction vector, tn, is
positive and the tangential component is negligible. For negative-misfit
alloys, the [Un,n ]term at the end of primary creep is positive so that the
(-t[ ]) term is negative and adds up with [W] to give a negative value
for rn. As the transient evolves, rn will assume increasingly larger negative
values due to the positive contributions to [Un,n ]as the matrix creep under
tensile stress.
For positive-misfit alloys, the [un,n] term at the end of primary creep is
negative so that the (-ti ,,]) term counterbalance the [W] term and rn
will be positive or negative depending on the value of the initial misfit. As
the transient evolves negative contributions will be added to rn.
In Fig. 4.68, we give a schematic synoptic diagram showing the levels of Trn
on the top and on the side of the precipitates at the end of the primary stage of
the creep transient for the possible combinations of tensile/compressive load and
positive/negative misit.
From this diagram we can see how under a tensile load a negative-misfit alloy will
tend to exhibit a "type N" rafting behavior, while a positive-misfit alloy will tend to
exhibit a "type P" rafting behavior and vice versa for a compressive load.
These simple patterns are indeed in agreement with all the available experimental
data listed in Table 1.1.
The arrows in Fig. 4.68 indicate how the levels of rn will decrease as the creep
transient evolves toward higher levels of creep strains under the effect of the applied
141
loads. If we consider this further evolution of the 7n profiles we can conclude that we
can always expect to observe a "Type N" rafting behavior for a negative-misfit alloy
under a tensile load and for a positive-misfit alloy under a compressive load. However,
if we consider the case of negative-misfit alloys under compressive loads and positivemisfit alloys under tensile loads, we can expect to observe a marked tendency toward
a "type P" rafting behavior only if we conduct our test so as to maintain the crystal
at a low level of creep for the time needed by the kinetics of the process to accomplish,
at least partially, the morphology evolution.
In fact, if we compare micrographs of "Type P" rafts and "Type N" rafts, the latter
are generally characterized by a higher aspect ratio and a more regular structure.
With regard to the evolution of An, since the magnitude of rn essentially scales
with the misfit, 6, and the applied stress, a, and the sign of Arn
1 changes with the
sign of 6 and o, as shown in Fig. 4.68, we can expect that plots of the normalized
quantity ATn/ab versus the magnitude of T•eep/6 will show similar patterns for all
the different alloys that we have analyzed. These curves are shown in Fig. 4.69, and
it can be noted how all the data correlates within a very narrow band. 1
This result also agrees with the experimental observations that indicate how the
rate of directional coarsening scales with the lattice misfit and the applied stress (see
paragraph 1.2): the process is accelerated when the driving force is increased. The
hastening of the rafting process observed when the test temperature is increased and
when the microstructure is refined, is most probably related to a reduction of the
characteristic diffusion time, namely, to an increase in diffusivity and to a shortening
of the diffusion path.
In Fig. 4.69 we can notice a substantial increase in A7n within the very first stage
of primary creep, when 0.5 < 'reep/6b < 2.0. After this sharp rise, which corresponds
to the period in which the misfit stresses are relieved, the creep flow in the matrix
becomes dominated by the applied stress and the behavior of "Type N" evolutions
and "Type P " evolutions branches: the driving force for "Type N" coarsening keeps
increasing, while the force for "Type P" coarsening will eventually reach a saturation
level.
These trends can be already qualitatively observed in the last portion of the curves
in Fig 4.69. If we compare correspondent pairs of "Type N" and "Type P " plots
(see Fig. 4.70) we can notice that the last section of the "Type P " curves (MNMT,
TCC) dips below the last section of the "Type N" curves (MNMC , TCT) which still
exhibit an upward curvature.
We have chosen not to continue our analysis beyond this stage because experimental observations show that at the end of primary creep, relevant morphological
changes are already occurring: if we extend our analysis, based on the initial cuboidal
'Note that we have not included the data for case 9 (TCTpMs) and case 10 (TCCpMs). This is
because the misfit that we have chosen for this hypothetical alloy is exceedingly small -the misfit
strains and the elastic strains have comparable magnitude. It is obvious how, in the limit of zero
misfit, quantities normalized by the misfit itself lose their significance.
142
shape of the precipitates, to the steady state creep we would not obtain a reliable
simulation of this subsequent stage of the process.
143
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6
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144
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146
'G
b
1
2
Igcreep /
Figure 4.69
3
6
Evolution of Arn/a6 for the analyzed creep transients
147
4
CO
1
2
3
Icreep/16
Figure 4.70
Comparison of "Type P" and "Type N" evolution
curves for Arn/cr•b
148
4
CHAPTER 5
CONCLUSIONS
5.1
Summary of Results
We have (levelope(d numerical techniques, in the framework of the finite elenment
inetholod, which allows us to (evaluate local values of the generalized(l force acting on
a material interface which is work-conjugate withl the normial displaceimentt of the
interface itself.
This quantity is a direct measure of the tendency for the interface to migrate, and
thus of the driving force for mnorp)hological. evolutions of the microstructuire.
We have appliedl these methlod(s to the stIudy of rafting in 1 - 7' Ni-superalloys.
The flexibility of thle proposed methods has readlily allowedl us to closely nmodel
the actual microstructural morl)hology of the alloys and to account for the effects of
applied boundary condlitions, lattice misfit, elastic anisotropy and( inelastic b)ehavior
of the crystals during the stress annealing transients.
We have modeled some experimental alloys, for which we have positively compared
the indications of our model with the experimental (lata, and( we have conducted a
circumscribed parametric study which has allowed us to formulate a more general
interpretation of the rafting phenomenon, which appears to give a satisfactory explanation for all the available experimental observations.
According to the results of our analysis, it is of fundamental importance, in modeling the rafting phenomenon, to consider the effects of the creepl flow in the 7-matrix
and in particular the evolution of the stress and strain fields during the primary creep
stage of the stress-annealing transients.
Earlier attempts to interpret the rafting behavior of these alloys have indeed failed
essentially because these effects were neglected.
In summary, the proposed methodology has proved itself successful for our particular application and appears suitable to be used in the analysis of parallel phenomena
concerning microstructural evolutions in multi-phase materials.
149
5.2 Suggestions for Future Study
We can identify two main topics that we have addressed in our research. The first
is the development of the numerical techniques per se; the second is the analysis of
the rafting phenomenon. Regarding the development of numerical techniques for the
evaluation of local forces acting on maternal interfaces, the first and most immediate
development is the extension of the computer programs to cope with 3-D geometries.
A more ambitious and substantial development, more closely related .o the study of
rafting, would be the implementation of a kinetic model to follow the evolution of
the microstructure morphology. In such a model, the evaluation of the driving force
would represent only one of several steps in the procedure.
With regard to the analysis of the rafting phenomenon, more involved creep models
could be implemented, and the effects of modifications of the volume fraction of the
precipitates could be investigated.
Finally, we should recognize that a continuum model for creep is not entirely
adequate to model the discrete nature of motion and multiplication of dislocations in
the narrow y' channels.
The development of a discrete model for dislocation mechanics conceived so that it
could be interactively superposed to the finite element solutions of continuum elastic
behavior, would represent a substantial contribution to the analysis of this process
as well as of other phenomena characterized by creep processes with a small length
scale.
150
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153
APPENDIX I: THE COMPUTER PROGRAM
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