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  • 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

    c T

    J D V

    ∇ ∇= − ∇ ∇ 

  • 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 Diffusion Direct Exchange Cyclic Exchange

    Interstitial Collinear Non -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 Ideas Supersaturation of vacancies owing to stress Diffusion results Work done on mat’l by tractions Energy dissipated in heat, entropy, and surface area

    Vacancies Supersaturated

    Va c

    a nc

    ie s

    Un d

    e rs

    a tu

    ra te

    d

  • 12.524 Mechanical properties of rocks

    Diffusion Creep

    Nabarro-Herring Creep Lattice

    Coble Creep Grain Boundary

    Monatomic Quasi static Vacancy Increasing length; Poisoining

  • 12.524 Mechanical properties of rocks

    Critical Idea: Tension makes vacancy formation easier.

    d l

    Tension=supersaturation

    ( ) ( ) ( )

    ( ) ( )

    0

    0 exp

    0 exp

    v v

    v

    v

    f f

    f vo

    f v

    G G

    G C C

    kT G

    C C kT

    σ σ

    σ σ

    ∆ = ∆ − Ω

    ∆ = −

     ∆ − Ω = −  

     

    iAtom

  • 12.524 Mechanical properties of rocks

    Gradient in Composition Path Length:

    Boundary: 2xd/4 Lattice: (π/2)x(d/4)

    Concentration Difference

    Quasi-static Approx.

    exp ( )

    exp ( )

    exp ( ) exp( ) exp( )

    2 exp ( )

    fv o

    fv o

    fv o

    fv o

    G C

    kT G

    C kT

    G C

    kT kT kT G

    C C kT 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

    fv o

    fv total L o b

    G CG kT kTJ D C D

    kT kT d d d

    σ σ δ

    π

    ∆ Ω −∆ Ω

    = − − ⋅ + − i

    2vac flux L B dJ l J lδΦ = ⋅ ⋅ + ⋅ ⋅

    2. .) / 2 vac flux

    total L BJ total ave flux J J iid l d δΦ

    = = + ⋅

    .)path path path

    CJ D i L

    ∆ = −

    ∆ 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 1 2

    vf vfB vo Total L vo

    vf B L vo

    L

    G GD CJ D C d kt kT d d kt kT

    G DD C d kt kT d D

    σ δ σ π

    σ πδ π

    ∆ ∆ 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 b

    l t s d d d Ω∆

    = = ⋅ ⋅ ⋅ = ⋅

    ε22 ε11

    ε12

    2

    32 1 2

    B L

    V

    DD d kT d D

    σ πδε π

     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 1 2

    B L

    V

    DD d kT d D

    σ πδε π

     Ω = + 

     

    L VM VD D C=

    2,3 1 dε ∝

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