Monday, Nov. 3, 2003 12.524 Mechanical properties of rocks

Diffusion Creep

Poirier, Chapter 2 and 7, 1985.Gordon, 1985.

12.524 Mechanical properties of rocks

Fick’s First Law: Driving Force

Chemical Diffusion

Thermal diffusion

Electrical conduction

Diffusion creep

µ

σ

i

cT

J DV

∇∇= − ∇ ∇

12.524 Mechanical properties of rocks

Types of Diffusion

Pore fluidRing

SurfaceInterstitial

AmbipolarGrain BoundaryVacancy

CreepPipeSelf-diffusion

InterdiffusionLatticeIsotope

ProcessPathMechanism

12.524 Mechanical properties of rocks

Diffusion mechanisms

BA

Ring DiffusionDirect ExchangeCyclic Exchange

InterstitialCollinearNon -collinear

Vacancy Diffusion

12.524 Mechanical properties of rocks

Vacancy–Assisted Diffusion

W = 3−1( )Da = 0.73Da

The FCC lattice geometry requires

plan view

From Site by Glicksman and Lupulescu, RPI, 2003

For more information, see http://www.rpi.edu/~glickm/diffusion/

Images removed due to copyright considerations.

12.524 Mechanical properties of rocks

Kinetics Equation for Vacancy Diffusion

Coefficient of Diffusivity for Self-diffusion not the same as Coefficient for Vacancy Diffusion

∆ ∆

∆ ∆

i

iexp ( / ) exp ( / )

exp ( ) /

sd v v migration

vo vf v o vm

sd o vf vm

D N D

N G kT D G kT

D G G kT

=

= − −

= − +

12.524 Mechanical properties of rocks

Diffusion Creep in Monatomic Solid

Basic IdeasSupersaturation of vacancies owing to stressDiffusion resultsWork done on mat’lby tractionsEnergy dissipated in heat, entropy, and surface area

Vacancies Supersaturated

Vac

anc

ies

Und

ers

atu

rate

d

12.524 Mechanical properties of rocks

Diffusion Creep

Nabarro-Herring CreepLattice

Coble CreepGrain Boundary

MonatomicQuasi staticVacancyIncreasing length;Poisoining

12.524 Mechanical properties of rocks

Critical Idea: Tension makes vacancy formation easier.

dl

Tension=supersaturation

( ) ( )( )

( ) ( )

0

0exp

0exp

v v

v

v

f f

fvo

fv

G G

GC C

kTG

C CkT

σ σ

σσ

∆ = ∆ − Ω

∆= −

∆ − Ω= −

iAtom

12.524 Mechanical properties of rocks

Gradient in CompositionPath Length:

Boundary: 2xd/4Lattice: (π/2)x(d/4)

Concentration Difference

Quasi-static Approx.

exp ( )

exp ( )

exp ( ) exp( ) exp( )

2 exp ( )

fvo

fvo

fvo

fvo

GC

kTG

CkT

GC

kT kT kTG

C CkT kT

σ

σ

σ σ

σ

∆ − Ω−

∆ + Ω− −

∆ Ω − Ω − − ∆ Ω

∆ = − i

12.524 Mechanical properties of rocks

Fick’s 1st Law

d/2δ

( )2 exp ( )1 22 exp ( )

8 2

fvo

fvtotal L o b

GCG kT kTJ D C D

kT kT d d d

σσ δ

π

∆ Ω−∆ Ω

= − − ⋅ + −i

2vac flux L BdJ l J lδΦ = ⋅ ⋅ + ⋅ ⋅

2. .)/ 2vac flux

total L BJ total ave flux J J iid l d

δΦ= = +

⋅

.)path pathpath

CJ D iL

∆= −

∆Total flux =ΣFlux on each Path:

Plugging ∆C and ∆L into i.) and inserting fluxes in ii.):

12.524 Mechanical properties of rocks

Total Flux on Both Paths

28 2exp exp 2

16 exp 12

vf vfB voTotal L vo

vf BL vo

L

G GD CJ D Cd kt kT d d kt kT

G DD Cd kt kT d D

σ δ σπ

σ πδπ

∆ ∆ Ω Ω = − + − ∆ Ω = − +

12.524 Mechanical properties of rocks

Converting flux to strain

b

b

1

2

Each vacancy that travels through the channel adds layer of depth b

2 2# 2 totaltotal

Jl vacs b bJ bl t s d d d

Ω∆= = ⋅ ⋅ ⋅ =

⋅

ε22 ε11

ε12

2

32 12

BL

V

DDd kT d D

σ πδεπ

Ω= +

12.524 Mechanical properties of rocks

Summary: N-H and Coble Creep

Diffusion Creep Constitutive Law:

Strain rate linear in stress

Other geometries change initial constant

2

32 12

BL

V

DDd kT d D

σ πδεπ

Ω= +

L VM VD D C=

2,31dε ∝