and Creep 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 ( D s or simply D ) : Temperature and Stress Dependencies - Fig. 13-6 Fig. 13-8 - total creep curve : = o + p + s o = instantaneous strain at loading (elastic, anelastic and plastic) s = steady-state creep strain (constant-rate viscous creep ) = D st p = primary or transient creep : Andrade- flow (or 1/3 rd law) : t1/3 primary or transient creep : Andrade- flow (or 1/3 rd law) : p = t1/3 problem as t 0 Garofalo / Dorn Equation : p = t (1 - e-rt ) , r is related to Dorn Both primary and steady-state follow similar kinetics - temperature compensated time ( = t e- Qc/RT) - single universal curve with t replaced by or st
> i (~1-20) > s
D Or, creep strain - o = t (1 - e- st ) + D st see Sherby-Dorn (Al), Murty (Zr)
Sherby-Dorn -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 -parameter
page 1
st 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 = De Q / RT
Stress Rupture Test : (13-4) vs tr Representation of engineering creep / rupture data (13-12, 13-13) - Figs. 13-17, 13-18 Sherby-Dorn Parameter : Larson-Miller Parameter : PS-D = t e-Q/RT PL-M = T (log t + C)
a
Fig. 13-19-21
T - Ta Manson-Haferd Parameter : PM-H = log t - log t
--- these parameters are for a given stress and are functions of (Fig. 13-20) ---
Monkman-Grant : Cs t r =
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 (1-2)2 + (2-3)2 + (3-1)2 eff = 2 deff 1 [1 - 2 (2+3) ] , and d1 = eff since creep is plastic deformation 1/2 appears as in plasticity. Similarly, d2 and d3. Dividing by dt, get the corresponding creep-rates, eff 1 D 1= [1 - 2 (2+3) ], etc. eff One first determines the uniaxial creep-rate equation, D s = A n e-Q/RT
n and assume the same for effective strain-rate : D eff = A eff e-Q/RT n-1 1 1 = A eff e-Q/RT [1 - 2 (2+3)]
so that
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. t = E + creep = const. Here, E = E . dt 1 d d Or, dt = - E D s = - E A n @ fixed T Thus dt = 0 = E dt + D s Integration from o to t gives, f t d n = - E A dt = - E A t o i
100
Data from "HW #8-8"
80
60
final or (t ) =
o n [1 + AE (n 1) o 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 : D =bv and Taylor equation : = 2 2G2b2
Thermally Activated Dislocation Glide (at low T and/or high strain-rates)
D = A eB 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
(fastest controls / dominates)
D = Di
i
series mechanisms
(slower controls / dominates)
D =
1 i
1
Slip following creep deformation in -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 : D = A n 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
= strain during glide-climb event = g + c g = b L h t = time of glide-climb event = tg + tc tc = v , vc = climb velocity
c
L b L D = t = h/v = b ( h ) vc c where Cv vc e-Em/kT , Em = activation energy for vacancy migration Here, Cv = Cv - Cv = Cv eV/kT - Cv e-V/kT = Cv 2 Sinh( kT ) L L V D = b ( h ) vc = b ( h ) Co e-Em/kT 2 Sinh( kT ) v At low stresses, Sinh() so that Garofalo Eqn. L D = A D (sinhB)n V D = A1 b ( ) Co e-Em/kT v kT h L L V D = A1 b ( ) DL A2 ( h ) DL kT h Or D = A 3 D natural creep-law L Weertman: h 1.5, D = A 4.5 D as experimentally observed in Al In general D = A(T) n Power-law - n is the stress exponent {f(xal structure, )}
+ o o o
V
also known as Nortons Equation (n is Norton index)
At high stresses ( 10-3 E), Sinh(x) ex, D = AH eB 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 FR? As is decreased reach a point when FR , dislocation density would become constant (independent of ): D - viscous creep known as Harper-Dorn creep Harper-Dorn creep occurs at 10-5 , o 106cm-2 E b H-D creep is observed in large grained materials (metals, ceramics, etc.)
D HD = AHD DL
ln
1
2
ln
Characteristics of Climb Creep (Class-M) : large primary creep regions 1 subgrain formation ( ) 2 dislocation density independent of grain size
KL Murty
MSE 450
page 6
Effects of Alloying : (class-A) Solid-solution - decreases rate of glide A glide controlled creep although annihilation due to climb still occurs (micro-creep / viscous glide creep) viscous glide controlled creep : (decreased creep-rates)
D g = Ag Ds 3 , Ds is solute diffusion
(Al) class-M 1 1 log(stress) 5 3 (Al-3Mg) class-A
little or no primary creep no subgrain formation 2 grain-size independent
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) Nabarro-Herring Creep (diffusion through the lattice) : D NH = ANH DL 2 d Coble Creep (diffusion through grain-boundaries) : D 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 at very large grain sizes, Harper-Dorn creep dominates
3 1 2 Coble N-H 1 Harper-Dorn
log (grain-size)
At small grain-sizes, GBS dominates at intermediate stresses and temperatures : D GBS = AGBS Db 2 d 2
superplasticity
KL Murty
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 D ppt = Appt D 8 - 20 Rules for Increasing Creep Resistance Large Grain Size
(directionally solidified superalloys)
Formability Improvement 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: D t = D N-H + D Coble + D H-D + D GBS + + c g
1
Dorn Equation :
kT = A DEb E
n
Mechanism D n A Climb of edge dislocations DL 5 6x107 (Pure Metals and class-M alloys) (n function of Xal structure & )* Low-temperature climb D 7 2x108 Viscous glide (Class-I alloys - microcreep) Ds 3 6 Nabarro-Herring Coble Harper-Dorn GBS (superplasticity) DL Db DL Db 1 1 1 2 b 14 (d )2 b 100 (d )3 3x10-10 b 200 (d )2
DL = lattice diffusivity; Ds = solute diffusivity; D = core diffusivity; Db = Grain-Boundary Diffusivity; b = Burgers vector; d = grain size; Gb 2 = subgrain size = 10 and = G2b2 where G is the shear modulus
*n increases with
decreasing (stacking-fault energy)
KL Murty
MSE 450
page 9
Deformation Mechanism Maps
Visual picture of the domains (, 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 (n5)
1 1 t = + g c
1
Alloys - Glide slower Glide-controlled creep (n3)
Solid Solution Alloys
10
-6
10
-8
Pb 9Sn d = 0.25 mm IV
10
-10
III
ln (
kT ) D b
10
-12
II
-14
10
I
10
-16
10
-6
10
-5
ln ( )
10
-4
10
-3
10
-2
Creep Transitions for Alloy Class
Murty and Turlik (1992)
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MSE 450
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