Creep and Stress Rupture :
Ch. 13 : 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15(optional)
• Definition of Creep and Creep Curve : (13-3)
• def. Creep is the time-dependent plastic strain at constant stress and temperature
Creep curve : Fig. 13-4
• steady-state creep-rate ( εD s or simply εD ) : Temperature and Stress Dependencies
- Fig. 13-6
Fig. 13-8
- total creep curve : ε = εo + εp + εs
εo = instantaneous strain at loading (elastic, anelastic and plastic)
εs = steady-state creep strain (constant-rate viscous creep ) = εD st
εp = primary or transient creep : Andrade-β flow (or 1/3 rd law) : βt1/3
primary or transient creep :
• Andrade-β flow (or 1/3 rd law) : εp = βt1/3 ⇔ problem as t → 0
• Garofalo / Dorn Equation : εp = εt (1 - e-rt ) , r is related to
ε>i
(~1-20)
ε> s
Dorn ⇒ Both primary and steady-state follow similar kinetics
- temperature compensated time (θ = t e- Qc/RT)
- single universal curve with t replaced by θ or εÝst
εD
Or, creep strain ε - εo = εt (1 - e- st ) + εD st ⇔ see Sherby-Dorn (Al), Murty (Zr)
Sherby-Dorn θ-parameter
Creep curves for Al at
Sherby & Dorn (1956)
(3,000 psi) and at three different temperatures
KL Murty
MSE 450
A single curve demonstrating the
validity of θ-parameter
page 1
εÝst
Creep data in Zircaloy at varied temperatures (˚F)
and stresses (ksi) fall into a single curve
demonstrating the validity of Dorn equation
(Murty et al 1976)
(K. L. Murty, M.S. Thesis, 1967)
• Zener-Holloman : Z = εDe Q / RT
• Stress Rupture Test : (13-4) σ vs tr
• Representation of engineering creep / rupture data (13-12, 13-13)
- Figs. 13-17, 13-18
• Sherby-Dorn Parameter :
PS-D = t e-Q/RT
• Larson-Miller Parameter :
PL-M = T (log t + C)
T - Ta
• Manson-Haferd Parameter : PM-H = log t - log t
Fig. 13-19-21
a
--- these parameters are for a given stress and are functions of σ (Fig. 13-20) ---
• Monkman-Grant : εCs t r = Κ
Eq. 13-24
Demonstration of Monkman-Grant
Relationship in Cu (Feltham and Meakin 1959)
KL Murty
MSE 450
page 2
Creep Under Multiaxial Loading
(text 14-14)
Use Levy-Mises Equations in plasticity
1
(σ1-σ2)2 + (σ2-σ3)2 + (σ3-σ1)2
σeff =
2
dεeff
1
[σ1 - 2 (σ2+σ3) ] ,
and dε1 =
σeff
since creep is plastic deformation 1/2 appears as in plasticity.
Similarly, dε2 and dε3.
Dividing by dt, get the corresponding creep-rates,
εÝeff
1
εD 1 =
[σ1 - 2 (σ2+σ3) ], etc.
σeff
One first determines the uniaxial creep-rate equation,
εD s = A σn e-Q/RT
n
and assume the same for effective strain-rate : εD eff = A σeff e-Q/RT
so that
n-1
1
εÝ1 = A σeff e-Q/RT [σ1 - 2 (σ2+σ3)]
etc.
Stress Relaxation
As noted in section 8-11, the stress relaxation occurs when the deformation is held
constant such as in bolt in flange where the constraint is that the total length of the
system is fixed.
σ
εt = εE + εcreep = const. Here, εE = E .
dεt
1 dσ
dσ
Or, dt = - E εD s = - E A σn @ fixed T
Thus dt = 0 = E dt + εD s
Integration from o to t gives,
σf
t
⌠ dσ
 n =-EA⌠
⌡ dt = - E A t
⌡σ
o
σi
100
Data from "HW #8-8"
80
60
σ final or σ (t ) =
σo
[1 + AE (n − 1)σ on −1t ]1 /( n −1)
40
20
0
1000
2000
3000
4000
5000
time, hr
KL Murty
MSE 450
page 3
• Deformation / Creep Mechanisms :
• Introduction - structural changes (13-5)
- Slip (difficult to observe slip lines / folds etc are usually noted)
Subgrains
GBS
- excess (deformation induced) vacancies
• Two important relationships :
Orowan equation : εD =ρbv
and Taylor equation : ρ =
σ2
α2G2b2
• Thermally Activated Dislocation Glide (at low T and/or high strain-rates)
εD = A eBσ e-Qi/RT where Qi is the activation energy for the underlying mechanisms
Peierls mechanism (bcc metals)
Intersection mechanism (fcc and hcp metals)
• Dislocation creep - (lattice) diffusion controlled glide and climb
• Diffusion creep - (viscous creep mechanisms mainly due to point defects) - at low
stresses and high temperatures
• Grain-Boundary Sliding - (GBS) - intermediate stresses in small grained
materials and ceramics (where matrix deformation is difficult)
• Many different mechanisms may contribute and the total strain-rate :
parallel mechanism
series mechanisms
(fastest controls / dominates)
(slower controls / dominates)
εD = ∑ εD
i
εD =
i
1
∑  ε
 i



−1
Slip following creep deformation in α-iron
Uncrept specimen
Crept at 5500 psi to 21.5% strain
(K.L. Murty, MS thesis, Cornell University, 1967)
KL Murty
MSE 450
page 4
• Dislocation Creep :
• Pure Metals / Class-M alloys: Experiments : εD = A σn e-Qc/RT ,n ≈ 5, Qc ≈ QL (QD)
(edge ⊥) glide - climb model Weertman-Climb model (Weertman Pill-Box Model)
• sequential processes
⊥
L = average distance a dislocation glides
h
⊥
tg = time for glide motion
⊥
h = average distance a dislocation climbs
FR
L
Lomer-Cottrell
tc = time for climb
Barrier
∆γ = strain during glide-climb event = ∆γg + ∆γc ≈ ∆γg = ρ b L
h
t = time of glide-climb event = tg + tc ≈ tc = v , vc = climb velocity
c
L
∆γ ρ b L
∴ γD = t = h/v = ρ b ( h ) vc
c
where vc ∝ ∆Cv e-Em/kT , Em = activation energy for vacancy migration
+
-
o
o
o
σV
Here, ∆Cv = Cv - Cv = Cv eσV/kT - Cv e-σV/kT = Cv 2 Sinh( kT )
L
L
σV
∴ εD = α ρ b ( h ) vc = α ρ b ( h ) Cov e-Em/kT 2 Sinh( kT )
At low stresses, Sinh(ξ) ≈ξ so that
Garofalo Eqn.
L
εD = A D (sinhBσ)n
σV
εD = A1 ρ b ( ) Cov e-Em/kT
kT
h
L
L
σV
εD = A1 ρ b ( ) DL
≈
A
ρ
σ
(
2
kT
h
h ) DL
Or εD = Aσ 3 D ⇔ natural creep-law
L
Weertman: h ∝ σ1.5, εD = A σ4.5 D as
experimentally observed in Al
In general εD = A(T) σn Power-law
- n is the stress exponent
{f(xal structure, Γ)}
also known as Norton’s Equation
(n is Norton index)
At high stresses (σ ≥ 10-3 E), Sinh(x) ≈ ex, εD = AH eBσ D (Power-law breakdown)
KL Murty
MSE 450
page 5
Experimental Observations - Dislocation Creep
Fig. 13-13 (Dieter)
(Sherby)
What happens if we keep decreasing the stress, say to a level at and below the τFR?
As σ is decreased ⇒ reach a point when σ ≤ σFR ,
dislocation density would become constant (independent of σ): εD ∝ σ
- viscous creep known as Harper-Dorn creep
Harper-Dorn creep occurs at
σ
ρο ≈ 10-5 , ρo ≈ 106cm-2
E ≤b
2
ln ρ
• H-D creep is observed in large grained
materials (metals, ceramics, etc.)
εD HD = AHD DLσ
1
ln σ
Characteristics of Climb Creep (Class-M) :
• large primary creep regions
1
• subgrain formation (δ ∝ )
σ
2
• dislocation density ∝ σ
• independent of grain size
KL Murty
MSE 450
page 6
• Effects of Alloying : (class-A)
• Solid-solution - decreases rate of glide glide controlled creep although
annihilation due to climb still occurs (micro-creep / viscous glide creep)
viscous glide controlled creep : (decreased creep-rates)
(Al)
εD g = Ag Ds σ3 , Ds is solute diffusion
•
•
•
•
class-M
little or no primary creep
no subgrain formation
ρ ∝ σ2
grain-size independent
(Al-3Mg)
5
1
3
class-A
1
log(stress)
• At low stresses (for large grain sizes), Harper-Dorn creep dominates
⇒ what happens as grain size becomes small ⇐
As grain-size decreases (and at low stresses) diffusion creep due to point defects
becomes important : (due to migration of vacancies from tensile boundaries to
compressive boundaries)
σ
• Nabarro-Herring Creep (diffusion through the lattice) : εD NH = ANH DL 2
d
σ
• Coble Creep (diffusion through grain-boundaries) : εD Co = ACo Db 3
d
Nabarro-Herring Creep vs Coble Creep :
Coble creep for small grain sizes and at
low temperature
NH creep for larger grain sizes and at
high temperatures
Coble
3
1
2
• at very large grain sizes, Harper-Dorn
creep dominates
N-H
1
Harper-Dorn
log (grain-size)
At small grain-sizes, GBS dominates at intermediate stresses and temperatures :
σ2
• εD GBS = AGBS Db 2
d
KL Murty
⇔ superplasticity
MSE 450
page 7
• Effect of dispersoids : Dispersion Strengthening / Precipitate Hardening
- recall Orowan Bowing
• at high temperatures, climb of dislocation loops around the precipitates
controls creep ⇒ εD ppt = Appt D σ8 - 20
Formability Improvement
Rules for Increasing Creep Resistance
• Large Grain Size
(directionally solidified superalloys)
• Small (stable) Equiaxed Grain Size
(superplasticity)
• Low Stacking Fault Energy
(Cu vs Cu-Al alloys)
• Strengthen Matrix
(i.e., increase GBS - ceramics)
• Solid Solution Alloying
(Al vs Al-Mg alloys)
• Stoichiometry
(especially Ceramics)
• Dispersion Strengthening
(Ni vs TD-Ni)
KL Murty
MSE 450
page 8
 1
1 
Summary of Creep Mechanisms: εD t = εD N-H + εD Coble + εD H-D + εD GBS +  + 
 εc ε g 
Dorn Equation :
εkT
σ 
= A 
DEb
E
−1
n
Mechanism
D
n
A
Climb of edge dislocations
DL
5
6x107
(Pure Metals and class-M alloys)
(n function of Xal structure & Γ)*
Low-temperature climb
D⊥
7
2x108
Viscous glide (Class-I alloys - microcreep)
Ds
3
6
Nabarro-Herring
DL
1
b
14 (d )2
Coble
Db
1
b
100 (d )3
Harper-Dorn
DL
1
3x10-10
GBS (superplasticity)
Db
2
b
200 (d )2
DL = lattice diffusivity; Ds = solute diffusivity; D⊥ = core diffusivity;
Db = Grain-Boundary Diffusivity; b = Burgers vector; d = grain size;
Gb
σ2
δ = subgrain size = 10
and
ρ
=
G2b2 where G is the shear modulus
τ
*n increases with
KL Murty
decreasing Γ (stacking-fault energy)
MSE 450
page 9
Deformation Mechanism Maps
• Visual picture of the domains (σ, T) where various mechanisms dominate
Ashby-Map
Lead pipes on a 75-year-old building in southern England
The creep-induced curvature of these pipes is typical
of Victorian lead water piping. (Frost and Ashby)
Other examples :
• W filament (light bulbs)
• turbind blade {Ni-based alloy DS by Ni3(Ti,Al)}
KL Murty
MSE 450
page 10
WEERTMAN PILLBOX MODEL
Pure Metals - Glide faster
Climb-controlled creep (n≈5)
 1
1
εÝt = 
+

εÝg 
 εÝc
−1
Alloys - Glide slower
Glide-controlled creep (n≈3)
Solid Solution Alloys
10
-6
10
-8
Pb 9Sn
d = 0.25 mm
IV
ln (
10
-10
10
-12
γkT
)
Dµ b
III
II
10
-14
10
-16
I
10
Creep Transitions for Alloy Class
KL Murty
-6
10
-5
10
-4
τ
ln ( )
µ
10
-3
10
-2
Murty and Turlik (1992)
MSE 450
page 11