Creep is the progressive time dependent deformation of a material

Typical creep curve showing the three steps of creep. Solid line Constant load test. Dashed line constant stress test. The equation is Andrade’s empirical representation of the creep curve, !_", $, %&' ( are constants.

I II III

Primary creep Secondary creep Tertiary creep

Fracture

Constant stress

!_"

')

'* = ,-&.*%&*

Time t

Strain, /

) = !_" 1 + $*^2/4 5⁶⁷

Creep is the progressive time dependent deformation

of a material

Creep

There are a number of differing creep mechanisms; the operative or dominant one,

usually determined by the level of applied stress and

temperature which creep operates

There have been many engineering empirical approaches and equations developed to “predict” creep behavior as a function of stress level and time, but we will not discuss these.

In the 1970’s Mike Ashby developed so-called deformation mechanism maps which is a convenient way to categorize creep and begin our discussion.

M.F. Ashby, Acta Metall. Vol. 20, pp. 887-897 (1972).

Creep

The way in which σ_s (shear stress),T (temperature) and ̇"

(strain rate) are related for

materials (a) when σ_s and T are prescribed and (b) when strain rate and T are prescribed, for low temperatures (top), high

temperatures (middle) and very high temperatures (bottom).

σ_s& T prescribed ̇# & % prescribed

Deformation mechanisms, divided into five groups.

Collapse at the ideal strength - (flow when the ideal shear strength is exceeded).

Low-temperature plasticity by dislocation glide - (a) limited by a lattice resistance (or Peierls' stress); (b) limited by discrete obstacles; (c) limited by phonon or other drags; and (d) influenced by adiabatic heating.

Low-temperature plasticity by twinning.

Power-law creep by dislocation glide, or glide-plus-climb —(a) limited by glide processes; (b) limited by lattice-diffusion controlled climb (“high-

temperature creep”); (c) limited by corediffusion controlled climb (“low-

temperature creep”); (d) power-law breakdown, (the transition from climb-plus- glide to glide alone); (e) Harper-Dorn creep; (f) creep accompanied by dynamic recrystallization.

Diffusional Flow -(a) limited by lattice diffusion (“Nabarro-Herring creep”); (b) limited by grain boundary diffusion (“Coble creep”); and (c) interface-reaction controlled diffusional flow.

Coble Creep

Nabarro –

Herring Creep Dislocation Creep

Dislocation Glide Theoretical Strength

Homologous Temperature, T/T_m

0 0.2 0.4 0.6 0.8 1.0

Applied Stress,

s

/G

10^-8 10^-6 10^-4

10^-2 10^-0

Generic Deformation Mechanism Map

Mechanisms of Creep Deformation

Dislocation Glide: This involves dislocation motion on slip planes over barriers by stress-assisted thermal activation. This mechanism dominates at high

stress in the range,

s µ / ³ 10

^-2

Low-temperature plasticity limited by discrete obstacles. The strain-rate is determined by the kinetics of obstacle cutting.

2

6

1 ˆ

10 /s

s o

s o

F kT b bv

g g s

t a s

g b

µ

é D æ ö ù

= ê ë - ç è - ÷ ø ú û æ ö

= ç ÷ »

è ø

is the "athermal flow strength"— the shear strength in the absence of thermal energy.

ˆ b / t µ = r

∆F is the total free energy (the activation energy) required to overcome

the obstacle without aid from external stress.

a ^and b are constants,

n

is an attempt frequency.

Low-temperature plasticity limited by a lattice resistance. The strain-rate is determined by the kinetics of kink nucleation and propagation.

Dislocation Glide

4 / 3

2 3 / 4

exp 1

ˆ

s p s

p

F kT

s s

g g µ t

ì D é ù ü

æ ö ï æ ö ï

= ç è ÷ ø - í ï î ê ê ë - ç è ÷ ø ú ú û ý ï þ

Dislocation Creep – power- law creep

Mechanisms of Creep Deformation

n

eff s

D b A kT g µ s

µ æ ö

= ç ÷

è ø

n

can vary anywhere between about 3 – 8.

Mechanisms of Creep Deformation

Diffusion Creep: Nabarro – Herring and Coble Creep

Nabarro – Herring Creep is controlled by lattice diffusion and Coble creep is controlled by grain boundary diffusion.

Nabarro-Herring creep and its rate scales as D/d². At lower temperatures, grain-boundary diffusion takes over;

the flow is then called Coble creep, and scales as Db/d³.

2

42

1

s

eff

b eff

kTd D D D D

d D g s

pd

= W

é ù

= ê ë + ú û

D

- the lattice diffusivity

D

_b - the GB diffusivity

d

- the width of the GB

d

- the grain size

W

- the atomic volume

The construction of a deformation-mechanism map. The field boundaries are the loci of points at which two mechanisms have equal rates.

Deformation Mechanism Map

Pure silver in grain size 1 mm.

Deformation Mechanism Map

Chromium with a grain size of 100 µm.

Mechanisms of Creep Deformation

Dislocation Glide

stress s =

stress internal

I

=

s

(due to work hardening, precipitates)

stress friction

stress;

effective

* = s - s

_I

=

s

stress flow

* + =

= s s

_I

s

γ ! = strain rate

γ ! = γ

_o

^exp − Δ ^G ( σ − σ

_I

)

k

_B

T

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ = γ

_o

^exp − Δ ^G ( ) σ ^* k

_B

T

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ = γ

_o

^exp − Δ ^G ( ) τ ^* k

_B

T

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥

τ ^*

Thermally activated glide

Reaction coordinate, A=x•l*

glide resistance, t b (force/length)

x

₀

x* x

₀

’

ú û ê ù

ë

é D -

= k T

G

B

) exp (

*

* 0

g t g ! !

*) ( t D G

A*

b t *

x

l*

x

₀

x

₀

’ x*

A*

x t

h

l

Orowan Equation g r l b

r l

=

dislocation density

average distance moved by the dislocations

Taking the time derivative, the strain rate is given by,

γ ! = ρ ^{b d} λ

dt = ρ ^bv

Dislocation motion is thermally activated:

*) ) exp( (

v

* ) exp(

v

_{0 0}

T k G T

k l G

B B

n D = - ^D t

=

*) ) exp( (

v

₀

T k

b G

B

r t

g ! = - ^D

l

_o

= distance moved by dislocation following activation

! = vibration frequency Velocity in γ ! = ρ ^{b d} λ

dt = ρ ^bv

Activation energies for obstacles of strength, f(x) =constant

b t *

A

b

0

* t

eff

b t

0K) (T

activation ermal

th * : stress in presence of t >

0K) (T

activation hermal

t obstacle in absence of

₀

* : stress required to overcome

= t

*) ( t D G

* A

0 0

0

0

0

( *) * ( * *) * * (1 * )

*

* (1 * )

*

G A b A b

G

t t t t t

t t

t

D = - = -

= D -

Activation energies for obstacles of strength, f(x) = parabola-shaped

b t *

A

b

0

* t

eff

b t

*) ( t D G

* A

2 / 3 0

0

)

* 1 *

(

*

*)

( t

t = D - t

D G G

Temperature-dependent flow

stress ^{= ê} _ë ^{é - D G} _{k T} ^{- ú} _û ^ù

B

I

) exp (

0

s e s

e ! !

] )) /

* ln(

( 1 [

*

₀ ^2/3

0

0

e e

s s

s ! !

G T k

_B

I

+ - D

=

2 / 3 0

0

)

* 1 *

(

*

*)

( t

t ^{= D -} t

D G G

) / ln(

*)

( t k T e ! e !

₀

G =

_B

D

athermal plateau: determined by the internal stress;

obstacles that can’t be overcome by thermal activation

µ

s / µ = shear modulus

µ s_I /

µ

s */ athermal

plateau

0 T

)]

/

* ln(

1 [

*

₀

0

0

e e

s s

s ! !

G T k

_B

I

+ - D

=

* ) 1 *

(

*

*) (

0

0

t

t = D - t

D G G

) / ln(

*)

( t k T e ! e !

₀

G =

_B

D

µ

s / µ = shear modulus

µ s_I /

µ

s */ athermal plateau

0 T

Effect of temperature on flow stress

Effect of Al concentration solid solution on flow stress in Ag-Al solid solution alloy.

Solid solution gives rise to both a friction (temperature- dependent) and internal (tem- perature independent) stress.

Note that flow stress of pure Ag depends weakly on T

Activation areas

*

*

* l x A »

x

l*

x

₀

x

₀

’ x*

A*

l* : activation length

x* : activation distance (depends on the type of obstacle).

Measurement of A*

ú û ê ù

ë

é D -

= k T

G

B

) exp (

*

* 0

g t g ! !

* *

* 1

*

) /

ln(

, ,

0

G A

b b

T k

T T

B

º

¶ D

= ¶

¶

¶

g

g

t

t g g ! !

A* º activation area

e

g

t ln

,

/

*

T B

b T A k

¶ !

= ¶

t g ! t

Experimental measurement of A*:

Derivative of the flow stress with

respect to strain rate

Measured activation areas

Haasen plot from nanoindentation on molybdenum thin films.

In bcc metals the activation area varies little with flow stress.

Diffusion Creep: Nabarro – Herring and Coble Creep

The general equation describing the equilibrium vacancy concentration on planar surfaces of a stressed solid is,

( ) ^{1 2 1 ( )}

²

¹

ln / 1

2 2

o

v v kk kk ij ij v kk

kT c c P

E E E

n n n

µ ^{æ -} s ^{ö é} s ⁺ s s h s ^ù

D = = - W + ç è ÷ ø - W - ê ë + - ú û

/ 3

o v

v

c P h

-

= DW W

equilibrium vacancy concentration on the surface of an unstressed solid normal stress on the surface of interest

vacancy relaxation

Diffusion Creep: Nabarro – Herring and Coble Creep

Consider a cylindrical bar loaded as shown below in uniaxial tension:

s

zz

s

zz

The surface under stress will have the vacancy formation energy reduced by an amount given by

s

_zz

^W .

Consequently the equilibrium vacancy concentration on this surface will be given by

( ) ^{( )}

²

ln / 1

2

o zz

v v zz v

kT c c

E s h s

= W + + W

zz

P

s = -

Diffusion Creep: Nabarro – Herring and Coble Creep

The vacancy concentration on the traction free (P = 0) cylindrical surface,

rr qq

0

s = s =

( )

²

ln /

2

o zz

v v v zz

kT c c

E

s h s

= W + W

When vacancy relaxation is neglected

( )

²

ln /

2

o zz

v v zz

kT c c

E s s

= W + W

( )

²

ln /

2

o zz

v v

kT c c

E

= s W

Loaded surface

Traction – free surface

Diffusion Creep: Nabarro – Herring and Coble Creep

s

L L

s -

s s

-

µ s D = W µ s

D = - W

D h

Einstein mobility relation,

v MF = D ;

M F

kT x

µ

= = - D D

The vacancy velocity

2

D D

v kT x kT L

µ s

D W

= - =

D

Diffusion Creep: Nabarro – Herring and Coble Creep

This is related to the deformation of the bar.

2

h L D

v t t kT L s

D D W

= = =

D D

This can be converted to a strain rate by dividing by L,

2

/ 2

L L D

t t kT L

g s

D = D = W

D D

Nabarro Herring Creep

γ ! = D kT

2 σΩ

L

²

Grain Boundary Diffusion

Diffusion along grain boundaries is more rapid than normal lattice diffusion.

Packing density in a grain boundary is less than the perfect lattice so atoms can change places more easily.

d

Diffusion Creep: Nabarro – Herring and Coble Creep

Isotope diffusion experiments indicate that

10

6

~ /

_L

GB

D

D ^where Q

_GB

» 60 KJ / mole

and Q

_L

» 100 KJ / mole

When is GB diffusion important?

Consider the following model of steady state diffusion:

d

J

_GB

J

_L

d

d >> : is the GB width ~ 0.5nm d d

Depends upon GB width δ, grain size d and also D

_GB

/ D

_L

.

Diffusion Creep: Nabarro – Herring and Coble Creep

Suppose there are N

_T

atoms per unit time diffusing through a grain of width d with boundary width δ.

2 2

; ; ;

( )

GB

T L

T L GB T L GB

N

N N

N N N J J J

d d d d d

= + = = =

+

( ^{d + d} )

²

^J

_T

^{= d}

²

^J

_L

^{+ d d J}

_GB

d J d

J

_T

@ J

^L

+

^GB

d

from Fick’s 1

^st

law – assuming steady state

dx D dC

dx dC d

D d

J

_T

D

^{L GB}

÷ = -

_app

ø

ç ö è

æ +

-

= d

Diffusion Creep: Nabarro – Herring and Coble Creep

÷ ø ç ö

è + æ

= D D d

D

_{app L GB}

d

or D d

D D

D

L GB L

app

_{=1 +} d

So obviously Grain boundary diffusion is important for

d D D

_GB

d >

_L

Now recall

m d

nm µ d

1000 1

~

5 . 0

~ -

So that

3

6

10

10

~

^-

-

^-

÷ ø ç ö

è æ

d

d

Defining an apparent diffusion coefficient

Diffusion Creep: Nabarro – Herring and Coble Creep

Typically

D

_L

D

_GB

(δ/d) D

_app

At low temperature there is an important contribution of D

_GB

to the flux.

The cross-over occurs at ~ 0.75 T

_mp

.

and ÷÷ ~ 0 . 5

ø çç ö

è æ

L GB

Q Q

D

_GB

1 / T log D

Diffusion Creep: Nabarro – Herring and Coble Creep

÷ ø ç ö

è + æ

= D D d

D

_{app L GB}

d

Diffusion Creep: Nabarro – Herring and Coble Creep

If we insert the equation for the apparent diffusion coefficient (" ≡ $) into the Nabarro-Herring creep equation for D, we obtain,

Note that for Coble Creep (Grain Boundary Diffusion) the strain rate scales with 1/L³ whereas for lattice diffusion it scales as 1/L².

γ ! = D

_L

kT

2 σΩ

L

²

1 + D

_GB

δ D

_L

L

⎛

⎝⎜

⎞

⎠⎟

Creep Crack growth

x y

Consider a “blunted “ Mode I crack in plane stress undergoing a diffusively

driven growth process. We are interested in calculating the steady state growth velocity of this crack.

Diffusional creep occurs owing to the difference in chemical potential between the crack tip surface and crack flank. This difference in chemical potential exists over the radius

r

of the crack tip.

r

For a finite opening of crack, the crack surface is free of tractions. Recall that for a stressed surface, the chemical potential is

2 ( )

0 0

=

=

=

yy yy

xx xy

K f

s r q

p s

s

free surface

r q

vacancy flow atom flow

Creep Crack growth

( ) ^{1 2 1 ( )}

²

¹

ln / 1

2 2

o

v v kk kk ij ij v kk

kT c c P

E E E

n n n

µ ^{æ -} s ^{ö é} s ⁺ s s h s ^ù

D = = - W + ç è ÷ ø - W - ê ë + - ú û

Neglecting lattice relaxation the equilibrium vacancy concentration on the crack surface at the crack tip,

r = 0 , q = 0

^is,

( )

²

ln /

2

o yy

v v

Curvature Term Stress Effects

kT c c

E

µ s g k

D = = - W + W

The second term on the rhs of this equation is a Gibbs-Thomson term. Crack growth occurs owing to the chemical potential difference between a vacancy at the crack tip and the unstressed crack flank. The equation describing the surface flux can be constructed either by consideration of the atom flux or vacancy flux. The

difference in chemical potential for vacancies on the surface of the stressed crack gives rise to a flow of surface atoms with an average velocity described by the

Einstein mobility relation,

Creep Crack growth

= -

^s

( )

s

D d

V kT ds

µ D

Note that when

^D µ = , 0

2

2 0,

yy

E

µ s gk

D = - W + W =

one obtains a Griffith-like equation for diffusive crack growth,

s

yy

⁼ ( ² E g r ^/ )

^{1 2/}

^.

For

s

yy

^< ( ² E g r ^/ )

^{1 2/} the crack will fill in with atoms owing to the capillary term.

Substituting for

D µ

in the mobility relation and multiplying the resultant equation by

N

_s, the number of lattice sites per unit area, yield and equation for the surface flux, (# of vacancies per unit length per unit time)

2

2

æ ö

= = - ç ç è - + ÷ ÷ ø

s s yy

s s s

D N d J V N

kT ds E

s gk

W

Creep Crack growth

2

2

æ ö

- +

ç ÷

ç ÷

è

^yy

ø , d

ds E

s gk

The part of this equation involving the differential

can be approximated by separately considering the difference in stress level

between the crack tip and the crack flank

s

_yy

= 0 ; k = 0

over a distance

( ^{p / 2} ) ^{r ,}

which results in,

Conservation of matter requires that

r V

_n

= J

_s

W

where

V

_n is the velocity of the crack. Making this substitution we obtain,

J

_s

≅ − D

_s

N

_s

Ω kT

2 π

1

ρ ⁻

σ

²_yy

2E + γκ

⎛

⎝ ⎜ ⎞

⎠ ⎟

V

_n

≅ D

_s

N

_s

Ω

²

kT

2 π

1 ρ

²

σ

_yy²

2E − γκ

⎛

⎝ ⎜ ⎞

⎠ ⎟