Ideas on Interpreting the Phenomenology of Superalloy Creep
A key problem in creep (in general) is explaining the stress and
temperature dependencies of the creep and linking it to the
underlying physical micro-mechanisms.
Weertman has argued the natural form of creep has the form:
In superalloys,
n>3
Qc>QD
How do we rationalize and link to underlying physics??
!
˙ " = k Deff
#
E
$
% &
'
( )
3
Plan
1. Generalize the world of creep
2. Identify and define ‘obstacle controlled creep’
3. Consider phenomenology of 3 somewhat understood cases:
1. Oxide Dispersion Strengthened Metals
2. Pure Metals
3. Metal Matrix Composites
4. Relate this to single xtal superalloy creep
1) Viscous Flow -- Thermal and stress activation of volume defects
gives flow with Newtonian viscosity. (e.g., glass above Tg)
2) Diffusional Creep -- Diffusion flux between surface defects (grain
boundaries) gives flow with Newtonian character and activation energy
of diffusion.
3) Defect Drag Deformation -- Viscous drag of dislocations, or defects
captured by dislocations controls strain rate. Can be Non-Newtonian.
(e.g., Class I creep, Nabarro climb creep, Mills screw dislocations dragging jogs)
4) Obstacle Controlled Deformation -- Dislocation mobility limited
by immobile obstacles. (e.g., Common plasticity, Artz-Rösler DS model, Class II creep, etc.)
Note: Some mechanisms may be mixtures (e.g., grain boundary sliding)!
Limiting Classes of Plastic Flow Mechanisms
Climb Controlled Creep -- A Myth
It is often not the thermally activated climb of dislocations over
obstacles that controls plasticity (and gives its rate dependence), as
commonly assumed.
Dislocations tend to stick to obstacles. Other dislocations are
common obstacles.
The faster of two processes will take place:
1) Dislocations will de-pin from obstacles to accommodate flow.
- or -
2) The obstacle field will coarsen, so the pin spacing increases,
letting dislocations past.
Consider Obstacle Controlled Deformation (from 0 < T < Tm).
Creep is governed by 4 fairly independent processes:
1) Dislocation bypass or release from obstacles,
2) Increase in dislocation density with flow,
3) Recovery -- coarsening the obstacle field (some obstacles disappear!)
4) Load-shedding from soft to hard regions.
Using only these elements the major phenomenology of 1-D plastic
deformation can be recovered, even over a wide range of homologous
temperature, and this can be applied to many materials.
More Generalizations
Case I -- Oxide Dispersion Strengthened Metals
-- Very stable, very fine oxides in metal matrix.
-- Shows threshold stress and activation energy for creep far greater than
that for self-diffusion.
-- Strength is strongly related to Orowan Strength.
-- For a long, long time a key question was why don’t the dislocations
simply climb over the dispersoids??
-- 1988 Rösler and Arzt proposed a model of flow that is based on attractive
interaction between a particle and dislocation.
Other mechanisms are all better understood than obstacle controlled
deformation.
Obstacle Controlled Deformation
Rösler and Arzt have a theoretically
appealing and predictive model for
dispersion hardened materials.
In DS materials obstacles can be large (stick
dislocations to incoherent dispersoids) and
they are stable.
In simple metals, obstacles may recover
away...
Increasing
Obstacle Size
From Rösler and Arzt (1990)
Thermal activation
over obstacles
Stress activation
over obstacles
k= fraction of energy/length
compared to as in matrix
Discrete Obstacles
Thermodynamics and kinetics of slip...
Good way to link micro and constitutive, but under-utilized,
Gives limits of what can be thermally activated.
Distance
Forc
e
!bl"G*
athermal breaking force
"F*
.1 µb3 Solute interaction
.5-2 µb3 Work hardening
10 µb3 Precipitate interaction
Rules of thumb for !F*-->(see Frost & Ashby)
Diffusion Activ. Energy
~ 0.25µb3
Often diffusion is much more rapid than dislocation release!!
Dislocations may bow around strong obstacles rather than being released.
!G* = !F * 1"fobs
ˆ k
# $ %
& ' (
p#
$ %
&
' (
q
Pslip = exp!"G*
kT
# $ %
& ' (
Case II -- Pure Metals
-- Activation energies for creep and self diffusion similar
-- n ~5
-- can we rationalize as obstacle controlled??
-- In ODS alloys, obstacle field is fixed. What fixes obstacle density in pure
metals??
Creep of Simple Pure Metals
Overall Trend for Pure Metals (Sherby, 1962)No accepted model of pure metal like deformation.
The Dorn equation is commonly used:
n=5
˙ ! ss = A Exp"Qc
kT
#
$ % &
' ( )
µ
*
+ , ,
-
. / /
n
• We note similar patterns in many metals:
! BCC
! FCC
! HCP
! Class II alloys, etc.
Q=QD, n= 4-6, similar strength scaling
• Most models are mechanistic. This is likely
inappropriate since these materials are all
quite different in many details.
• We need a universal explanation
(which does not depend on details of
cross-slip or climb, etc.)
˙ !
D
"/E
5
1
Self Selection of Steady-State Length Scale
Takeuchi & Argon 1976
Transient behavior shows effects of
structural refinement and coarsening
Ste
ady
sta
te s
ub
gra
in s
ize
/ b
Sherby and Burke (1968)
Applied stress / shear modulus
Steady state subgrain size roughly
inversely proportional to creep stress
A Postulate
It is often not climb bypass that controls plasticity (and gives
its rate dependence), as commonly assumed.
Many obstacles are too large to be activated past.
Instead, often thermally-activated coarsening that eliminates
obstacles and releases dislocations to glide. Coarsening is
where the main time and temperature dependence are!
What Provides High Temperature Strength? Junctions!
Example from McLean
(1960) of network formation
in iron; <111> type
react to <100>type.
From Honeycombe.
Networks by intersection are ubiquitous.
These attractive junctions are very hard to break by stress and thermal activation!
!F* >>1µb3
They cannot move by glide, but can be eliminated by recovery-like diffusion-assisted mechanisms.
Bulatov, et. al., Nature 2006,
Dislocation Multi-Junctions (demonstrated in BCC Nb)
!
"#
"$= M #c
Good compilation in:
J. J. Gilman Micromechanics
of Flow in Solids, 1969.
Dislocation Density Accumulation in Simple Annealed Metals
1E+10
1E+11
1E+12
1E+13
1E+14
1E+15
1E+16D
islo
ca
tio
n D
en
sity
(m
^-2
)
0.001 0.01 0.1 1
Plastic Strain
assumed density
Orlova; Fe polyCreep
Ag Polyxtal
Lawley Mo Xtal 300K
Hordon Cu xtal polySlip
Hordon Al Xtal Polyslip
Hordon - Cu Xtal Polyslip
Bailey polyxtal Cu
Livingston Cu Xtal, multislip
Edington Poly V 300 & 400 C
Edington Poly V 20 & 230 C
c=0
c=0.5
!
" = M# 1$ c( ) + "o
1$c( )[ ]1
1$c
Dis
loca
tio
n D
ensi
ty (
m-2
)
Assumed Recovery Form
!
d "mc( ) = K M T( ) dt
This form is widely found and studied in modern coarsening literature.
mc is coarsening exponent. It is determined by mechanistic path
M(T) is the mobility for the rate controlling process.
For simple grain growth (Burke and Turnbull, 1950):
! mc=2
! M(T)=Grain boundary mobility
However, Humphreys* notes: “There is little evidence for m=2, even in very
pure materials”. Typically the coarsening exponent is between 3 and 5.
We assume there is some pattern that is approximately common to many metals…
*in Recrystallization and Related Annealing Phenomena (1995).
Kinetic Mechanisms & Scaling of Velocity
Viscous Boundary Motion Viscous Line Motion LR Diff. / Boundary Motion
LR Diff. / Line Motion Boundary Diff. / Bdy. Motion Pipe Diff. / Line Motion
!
V "d#
dt"M$P
P1P2
!
V "d#
dt"M
dF
dL
!
V "d#
dt"D$P%
#
#
!
V "d#
dt"D
dF
dL
$
#
!
V "d#
dt"Db$P%
#2
!
V "d#
dt"D
dF
dL
$
#3
Jvac
Rate Step \ Geom.
Viscous Motion mc = 2
M(T)
mc = 3
Md(T)
Long-Range
Lattice Diffusionmc = 3
Dl(T)
mc = 4
Dl(T)
Diffusion Along
Feature Defectmc = 4*
Dpath(T)
mc = 6
Dpath(T)
Scaling Analysis of Possible Recovery Paths
System morphology
Ra
te C
on
tro
llin
g S
tep
!
d "mc( ) = K R T( ) dt
!
E
V"#$1
!
E
V"#$2
Grains or subgrains 3-D Dislocation Network
* Pipe diffusion increases mc by one vs. bulk diffusion because area fraction of dislocation scales with 1/#.
If lattice diffusion controls coarsening we expect mc=3 or 4 with a diffusional activation energy.
A Proposal. . .
!
d "mc( ) = K D T( ) dt
mc=3 - 4 and diffusion is the rate limiting step (the activation energy for self-
diffusion is expected).
K is similar for many structural metals and is an important element in the near-
universal scaling of metal creep.
1E-06
1E+00 1E+01 1E+02 1E+03 1E+04 1E+05
Annealing time (s)
400 C
350 C
325 C
300 C
275 C
250 C
m=3
Sm
all
Su
bgra
in S
ize
(µm
)
Replotting of Data
Y. Huang and F. J. Humphreys, “Subgrain Growth and Low Angle Boundary Mobility in
Aluminum Crystals of Orientation {110}<001>”, Acta Mater., 48, 2017-2030 (1999).
1E-06
1E
-1
7
1E
-1
6
1E
-1
5
1E
-1
4
1E
-1
3
1E
-1
2
1E
-1
1
Normalized time (time•D(T))
400 C
325 C
300 C
275 C
250 C
Sm
all
Su
bgra
in S
ize
(µm
)
m=3
K= 4•105
Integration of the Processes
for Power-Law Creep
!
"ss =2 K D(T ) g
2#2c
M ˙ $
%
& '
(
) *
1
2+mc #2c
!
˙ " = B D T( )#
µ
$
% &
'
( )
n
=2 K g
2*2c( )b s( )
* 2+mc *2c( )
M
$
%
& &
'
(
) ) D T( )
#
µ
$
% &
'
( )
2+mc *2c( )
!
d"
d#= M" c
!
" =g
#
!
d "mc( ) = K D T( ) dt
Simple Balance of Refinement and Coarsening
RefinementCoarsening
Steady-state length scale
Stress exponent, n=2+mc-2c ~ 5, and strain rate scales with self diffusivity.
De-pinning (Taylor Equation)
!
" =sµb
#
Phenomenon Equation {Range} &/or
Assumed
value
Structure
parameter
(! or ")
#
! =g
"
g={1-20}
g=4
Athermal
Yield
#
$ f =sµb
!
s={0.1 – 1.0}
s=0.5
Dislocation
Accumulation
#
d"
d%= M" c M~2 x1015 m-2
c=0
Recovery
(structural
coarsening)
#
d !mc( ) = K M T( ) dt K= 10-6 m
m=3
M(T)=Dl(T)
Other
constants
b=0.3nm
1E+10
1E+11
1E+12
1E+13
1E+14
1E+15
1E+16
Dis
loca
tio
n D
en
sity
(m
^-2
)
0.001 0.01 0.1 1
Plastic Strain
assumed density
Orlova; Fe polyCreep
Ag Polyxtal
Lawley Mo Xtal 300K
Hordon Cu xtal polySlip
Hordon Al Xtal Polyslip
Hordon - Cu Xtal Polyslip
Bailey polyxtal Cu
Livingston Cu Xtal, multislip
Edington Poly V 300 & 400 C
Edington Poly V 20 & 230 C
Dis
loca
tion
Den
sity
(m
-2)
1E-06
1E
-1
7
1E
-1
6
1E
-1
5
1E
-1
4
1E
-1
3
1E
-1
2
1E
-1
1
Normalized time (time•D(T))
400 C
325 C
300 C
275 C
250 C
!
˙ "
D(T)=
2 K g2#2c( ) b s( )
# 2+mc #2c( )
M
$
% & &
'
( ) ) *
µ
$
% & '
( )
2+mc #2c( )
Full Model Summary
Subgra
in s
ize,
# (
m)
“A model for creep based on microstructural length scale
evolution” G.S. Daehn, H. Brehm, H. Lee, B.S. Lim, Mat
Sci & Engr A, 387-89, pp 576-584, (2004).
The Model is Predictive WITHOUT Input from Creep
All inputs are taken from non-creep experiments,
and show correspondence to known creep trends
!
˙ "
D(T), cm
#2
!
"
E
!
˙ "
D(T)=
2 K g2#2c( ) b s( )
# 2+mc #2c( )
M
$
% & &
'
( ) ) *
µ
$
% & '
( )
2+mc #2c( )
!
˙ "
D(T)= 210
9 #
µ
$
% & '
( )
5
mc=3, K=10-6 m, g=4, b=0.3nm,
s=0.5, c=0, M=2 1015
Prediction
Case III - Metal Matrix Composites
-- SiC particles in Al
-- inter-particle spacing too open to provide significant Orowan Strengthening
-- These are much stronger than expected....
-- Why? What controls flow??
For details see:
See H. Brehm and G. S. Daehn, “A Framework for Modeling
Creep in Pure Metals”, Met. Mat. Trans., 33A, 363-371 (2002).
Slip versus Coarsening Activation Energies
A slightly more comprehensive model of the type analyzed here using:
shows if !F* for flow is significantly greater than QD then the activation energy
for coarsening is rate controlling; and the activation energy for creep is that for
diffusion.
Experimental Studies on Al-SiC composites support this view…
Imagine coarsening is inhibited by the particles...
!G* = !F * 1"fobs
ˆ k
# $ %
& ' (
p#
$ %
&
' (
q
Pslip = exp!"G*
kT
# $ %
& ' (
10-9
10-8
10-7
10-6
10-5
10-4
10-3
Applied Stress (MPa)
2124Al-20%SiCw.400°C
2124Al-20%SiCw.375°C
2124Al-15%SiCw.375°C
2124Al-15%SiCw.425°C
2124Al.400°C (Nieh et al.)
2124Al-F.400°C
Al-4%Al2O3.400°C (Milicka et al.)
1
3.3
1
3.9
1
10.5
1
21
10-2
10 100
Str
ain
Rate
(s
-1)
300002000010000000.00
0.01
0.02
0.03
Time (sec)
Str
ain
36.67 MPa
35.10 MPa
33.77 MPa
26.97 MPa
49.0 MPa
2124Al-15%SiCw Composite375°C Isothermal Creep
‘Creep Resistant’ Material (Al-SiC, Chen&Daehn 1993)
Little transient on loading
Composite stress exponent ismuch greater than five!!
Compositebehavior
Matrix Behavior
MechanicsSolution
300002500020000150001000050000
10-7
10-6
10-5
2124Al-15%SiCw
Time (sec)
Str
ain
Ra
te (
s
)
-1
n=
12.7
13.9
12.5
12.9
13.7
12.8
13.2
12.3
14.4
12.7
S1
S2
S1
S2
S1
S2
S1
S2 S2
S1 S1
S1=26.3 MPa
S2=29.2 MPa
0
0.004
0.008
0.012
0.016
0.02
0 50000 100000 150000 200000 250000 300000
Time (sec)
!1
!1
!2
Tru
e S
train
2124Al-15%SiCw CompositeMultiple Creep Tests at 375°C
!1=25.3 MPa
!2=28.6 MPa
Al-SiC Transients Show No Stress-Based Structure Change
350300250200150100500100
200
300
400
500
600
700
800
6061Al-20%SiCp: Nieh et al (57)
2124Al-20%SiCw: N&S (59)
2124Al-20%SiCw: this study
2124Al-15%SiCw: this study
6061Al-30%SiCp: Park et al (62)
Applied Effective Stress (MPa)
Act
ivati
on
En
ergy (
KJ/m
ole
)
Al self diffusion, Qc=142 KJ/mole
(400-430°C)
(400-425°C)
(350-400°C)
(375-390°C)
(345-405°C) (149-204°C)
(232-363°C)
(274-302°C)
(345-405°C)
Composite Activation Energy > Self Diffusion (Chen)
Y-C Chen, PhD Thesis, Ohio State University, 1993.
Here, obstacle depinning rather than coarsening controls creep.
(PM)
Observations: QC>QSD, n>5, strong interactions between
dislocations and SiC particles inhibit coarsening....
350300250200150100500100
200
300
400
500
600
700
800
6061Al-20%SiCp: Nieh et al (57)
2124Al-20%SiCw: N&S (59)
2124Al-20%SiCw: this study
2124Al-15%SiCw: this study
6061Al-30%SiCp: Park et al (62)
Applied Effective Stress (MPa)
Act
ivat
ion
En
ergy
(K
J/m
ole)
Al self diffusion, Qc=142 KJ/mole
(400-430°C)
(400-425°C)
(350-400°C)
(375-390°C)
(345-405°C) (149-204°C)
(232-363°C)
(274-302°C)
(345-405°C)
Y-C Chen, PhD Thesis, Ohio State University, 1993.
Here, obstacle depinning rather than coarsening controls creep.
Case IV - single xtal superalloys
-- Many models would lead us to expect:
Activation energies for creep like those for self diffusion
Low stress exponents.
-- Not clear how we model strength
-- How do we rationalize big picture?
Qc >> QD
n>5
Transient behavior not clear
Stress dependencies of Q and n not clear.
Strong suggestion that dislocation link-
length may scale with strength...
Superalloy creep is complex!
dislocation and phase
morphology change in manner
that is not self similar.
Both dislocation nets and
particles are subject to
coarsening.
Both phases subject to
deformation
Concluding Remark
We can identify a class of behavior of obstacle controlled creep and explain the
behavior of several different ‘creep mechanisms’.
The high stress exponent and activation energies of superalloy creep strongly
suggest that dislocations are breaking free from strong pins and this controls
creep rate. Strong pin means DG* > 0.25 µb3.
Coarsening also takes place, but at a rate too slow to accommodate creep.
If this is true, we expect:
constant-structure like transients,
Qc will decrease with increasing stress,
strength should largely scale with (largest link length)-1.
We are a long way from a real quantitative creep model.
Can Framework Be Extended??
Phenomenon How to Treat?
Harper-Dorn Creep Recognition that coarsening leads to limiting
maximum network dimensions.
Power-Law Breakdown Enhanced recovery at very small length scales due to
dislocation annihilation.
Varied stress exponent Naturally varies somewhat due to changes in
hardening and coarsening details.
Low temp. plasticity Naturally arises from this framework.
Qc not equal QsdQc > Qd if obstacles not eliminated by coarsening,
Qc < Qd if obstacles are relatively weak.
Anelasticity Load shedding
Bauschinger Effect Load shedding
Primary Creep Load shedding
in f
ram
ewo
rk
