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**Unformatted text preview: **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 Dorn ⇒ Both primary and steady-state follow similar kinetics - temperature compensated time (θ = t e- Qc/RT) Ý - single universal curve with t replaced by θ or ε st
> εi (~1-20) > εs
D ε 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
Ý ε 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 = 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 : • 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 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, Ý ε eff 1 D ε1= [σ1 - 2 (σ2+σ3) ], etc. σeff One first determines the uniaxial creep-rate equation, D ε s = A σn e-Q/RT
n and assume the same for effective strain-rate : εD eff = A σeff e-Q/RT n-1 1 Ý ε 1 = A σeff e-Q/RT [σ1 - 2 (σ2+σ3)]
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. 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 = - E A ⌠ dt = - E A t ⌡ ⌡σ o σi
100
Data from "HW #8-8"
80
60
σ final or σ (t ) =
σo n [1 + AE (n − 1)σ 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 : εD =ρbv and Taylor equation : ρ = σ2 α2G2b2
• Thermally Activated Dislocation Glide (at low T and/or high strain-rates)
D ε = 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
(fastest controls / dominates)
D ε =∑ ε Di
i
series mechanisms
(slower controls / dominates)
D ε=
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 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 ) Co e-Em/kT 2 Sinh( kT ) v At low stresses, Sinh(ξ) ≈ξ so that Garofalo Eqn. L D ε = A D (sinhBσ)n σV D ε = A1 ρ b ( ) Co e-Em/kT v kT h L L σV D ε = A1 ρ b ( ) DL ≈ A2 ρ σ ( h ) DL kT h 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)}
+ o o o
σV
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 • H-D creep is observed in large grained materials (metals, ceramics, etc.)
D ε HD = AHD DLσ
ln ρ
1
2
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 A glide controlled creep although annihilation due to climb still occurs (micro-creep / viscous glide creep) viscous glide controlled creep : (decreased creep-rates)
D ε g = Ag Ds σ3 , 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 ρ ∝ σ2 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) 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 • 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 : • εD GBS = AGBS Db 2 d σ2
⇔ 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 ⇒ εD ppt = Appt D σ8 - 20 Rules for Increasing Cr...