Creep

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

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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)

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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

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• 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

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• 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)

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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

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• 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

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⇔ superplasticity

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• 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)

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 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)

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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

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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)

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