Condensed Matter Physics - Dislocationsmetal.elte.hu/~groma/mater_phys/materphys7.pdf ·...

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Condensed Matter Physics Dislocations Istv · an Groma ELTE April 10, 2019 Istv · an Groma, ELTE Condensed Matter Physics, Dislocations 1/32

Transcript of Condensed Matter Physics - Dislocationsmetal.elte.hu/~groma/mater_phys/materphys7.pdf ·...

  • Condensed Matter PhysicsDislocations

    István Groma

    ELTE

    April 10, 2019

    István Groma, ELTE Condensed Matter Physics, Dislocations 1/32

  • Plastic deformation

    Ideal shear

    István Groma, ELTE Condensed Matter Physics, Dislocations 2/32

  • Plastic deformation

    Ideal shear

    F (x) = F0

    ∣∣∣∣sin(πa x)∣∣∣∣ = σa2

    Small deformationγ =

    xa, σ = µ

    xa

    So

    F0 = µa2

    π

    From thisσf ∼=

    µ

    π∼= 30GPa

    Experimentσf ∼= 100MPa

    István Groma, ELTE Condensed Matter Physics, Dislocations 3/32

  • MDD

    Polányi, Orován, Taylor (1934)

    Vito Volterra (1905)

    István Groma, ELTE Condensed Matter Physics, Dislocations 4/32

  • MDD

    Polányi, Orován, Taylor (1934)

    Vito Volterra (1905)

    István Groma, ELTE Condensed Matter Physics, Dislocations 4/32

  • MDD

    Polányi, Orován, Taylor (1934)

    Vito Volterra (1905)

    István Groma, ELTE Condensed Matter Physics, Dislocations 4/32

  • Dislocation

    Glide: the number of atom is conservedClimb: the number of atom is not conserved, requires high temperature.

    István Groma, ELTE Condensed Matter Physics, Dislocations 5/32

  • Dislocation

    Field theory of dislocations

    ∂i uj := βij = βpij + β

    eij

    dudr

    := β̂ = β̂e + β̂p

    Dislocation density tensor

    αij = eikl∂kβplj = −eikl∂kβ

    elj α̂ = ∇× β̂

    p = −∇× β̂e

    bj =∫

    Aαij dAi

    bj = −∫

    Aeikl

    ∂rkβelj dAi = −

    ∮βeij dsi = −

    ∮duj .

    István Groma, ELTE Condensed Matter Physics, Dislocations 6/32

  • Dislocation

    Individual dislocation

    αij = li bjδ(ξ) (α̂ = l ◦ bδ(ξ))Plastic distortion

    βpij (~r) = ni bkδ(ζ)

    Multiple with the elastic constants and take its div:

    ∂riCijkl

    ∂uk∂rl

    =∂

    ∂riCijklβ

    pkl = fj

    (div Ĉ

    dudr

    = div Ĉβ̂p = f)

    Displacement field of a screw dislocation

    u3 =b

    2πϕ

    Stress field of a straight dislocation σ ∝ b/r

    σ11 =µb

    2π(1− ν)y(3x2 + y2)(x2 + y2)2

    σ22 =µb

    2π(1− ν)y(x2 − y2)(x2 + y2)2

    σ12 =µb

    2π(1− ν)x(x2 − y2)(x2 + y2)2

    István Groma, ELTE Condensed Matter Physics, Dislocations 7/32

  • Energy of a dislocation

    σ ∝ b/r ı́gy E ∝ b2

    2(

    b2

    )2< b2

    b = aiFcc b = 12 (1, 1, 0)Bcc b = 12 (1, 1, 1)

    István Groma, ELTE Condensed Matter Physics, Dislocations 8/32

  • Dislocation-dislocation interaction

    Elastic energy

    E =12

    ∫σij∂i uj dV

    Let us

    σij = σextij + σ

    dislij

    ui = uexti + udisli

    leads to

    E =12

    ∫ [σextij ∂i u

    extj + σ

    extij ∂i u

    dislj + σ

    dislij ∂i u

    extj + σ

    dislij ∂i u

    dislj

    ]dV

    Since

    σij = Cijkl∂i uj

    E = Eext + Edisl +∫σextij ∂i u

    dislj dV

    István Groma, ELTE Condensed Matter Physics, Dislocations 9/32

  • Dislocation dislocation interaction

    Eint =∫σextij ∂i u

    dislj dV

    Partial integration

    Eint =∫∂i (σ

    extij u

    dislj )dV −

    ∫(∂iσ

    extij )u

    dislj dV

    Since

    ∂iσextij = 0

    Eint =∫∂i (σ

    extij u

    dislj )dV =

    ∫(σextij u

    dislj )dni

    István Groma, ELTE Condensed Matter Physics, Dislocations 10/32

  • Dislocation-dislocation interaction

    u-nak b has a jump on the cut surface!

    Eint = bj

    ∫fσextij dni

    For the moving line

    4Eint = bj∮

    vσextij (4r × dl)i

    4Eint =∮

    v(dl × (σ̂ext b))i 4 ri

    Peack Koehler force

    f = (σ̂ext b)× l

    István Groma, ELTE Condensed Matter Physics, Dislocations 11/32

  • Dislocation-dislocation interaction

    For glide only, we need

    fglide = [(σ̂ext b)× l]bb

    = nσ̂ext b = τb ahol n =bb× l

    Edge dislocation

    f = b1b2µ

    2π(1− ν)x(x2 − d2)(x2 + d2)2

    István Groma, ELTE Condensed Matter Physics, Dislocations 12/32

  • Dislocation multiplication

    Frank-Read source

    Flow stress (Taylor relation)

    τf = αµb√ρ

    István Groma, ELTE Condensed Matter Physics, Dislocations 13/32

  • Work hardening by precipitate

    Line tension τl = El/R

    Critical stress

    τc =∝1l

    István Groma, ELTE Condensed Matter Physics, Dislocations 14/32

  • Dislocation patterning

    Cell structure PSB structure

    István Groma, ELTE Condensed Matter Physics, Dislocations 15/32

  • Size effect

    Torsion test Microhardness

    István Groma, ELTE Condensed Matter Physics, Dislocations 16/32

  • Size effect

    Torsion test Microhardness

    Local plasticity

    τclass(γ, γ̇, ...)

    István Groma, ELTE Condensed Matter Physics, Dislocations 16/32

  • Size effect

    Torsion test Microhardness

    Nonlocal plasticity

    τ(γ, γ̇, ...) = τclass(γ, γ̇, ...) + l2µd2

    d~r2γ

    István Groma, ELTE Condensed Matter Physics, Dislocations 16/32

  • Pillar compression

    10-2

    10-1

    100

    S=s L (Γ+Θ)/(Eb)

    10-5

    10-4

    10-3

    10-2

    10-1

    100

    101

    102

    P(S

    )

    Data from Dimiduk et al.L=1.5µm, bending

    L=0.5µm, load control

    L=0.5µm, strain control

    L=0.8µm, strain control

    exp(-(S/0.6)2)/S

    3/2

    István Groma, ELTE Condensed Matter Physics, Dislocations 17/32

  • Pillar compression

    10-2

    10-1

    100

    S=s L (Γ+Θ)/(Eb)

    10-5

    10-4

    10-3

    10-2

    10-1

    100

    101

    102

    P(S

    )

    Data from Dimiduk et al.L=1.5µm, bending

    L=0.5µm, load control

    L=0.5µm, strain control

    L=0.8µm, strain control

    exp(-(S/0.6)2)/S

    3/2

    István Groma, ELTE Condensed Matter Physics, Dislocations 17/32

  • Pillar compression

    10-2

    10-1

    100

    S=s L (Γ+Θ)/(Eb)

    10-5

    10-4

    10-3

    10-2

    10-1

    100

    101

    102

    P(S

    )Data from Dimiduk et al.L=1.5µm, bending

    L=0.5µm, load control

    L=0.5µm, strain control

    L=0.8µm, strain control

    exp(-(S/0.6)2)/S

    3/2

    István Groma, ELTE Condensed Matter Physics, Dislocations 17/32

  • Relevant quantities

    Plastic deformation

    d~Rd~r

    = 1̂ +d~ud~r

    = F̂ e · F̂ p

    Definitions

    F̂ e,p = 1̂ + β̂e,p

    F̂ e · F̂ e∗ = 1̂ + 2�̂e

    CurlF̂ p = α̂

    István Groma, ELTE Condensed Matter Physics, Dislocations 18/32

  • Relevant quantities

    Plastic deformation

    d~Rd~r

    = 1̂ +d~ud~r

    = F̂ e · F̂ p

    Definitions

    F̂ e,p = 1̂ + β̂e,p

    F̂ e · F̂ e∗ = 1̂ + 2�̂e

    CurlF̂ p = α̂

    István Groma, ELTE Condensed Matter Physics, Dislocations 18/32

  • Small deformations

    Deformations

    ∂i uj := βij = βpij + β

    eij

    d~ud~r

    := β̂ = β̂e + β̂p

    Nye’s dislocation density tensor

    αij = eikl∂kβplj = −eikl∂kβ

    elj α̂ = ∇× β̂

    p = −∇× β̂e

    Is it that we learned in school?

    bj =∫

    Aαij dAi

    bj = −∫

    Aeikl

    ∂rkβlj dAi = −

    ∮βij dsi = −

    ∮dui .

    For a single dislocation

    αij = li bjδ(ξ)(α̂ =~l ◦ ~bδ(ξ)

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 19/32

  • Small deformations

    Deformations

    ∂i uj := βij = βpij + β

    eij

    d~ud~r

    := β̂ = β̂e + β̂p

    Nye’s dislocation density tensor

    αij = eikl∂kβplj = −eikl∂kβ

    elj α̂ = ∇× β̂

    p = −∇× β̂e

    Is it that we learned in school?

    bj =∫

    Aαij dAi

    bj = −∫

    Aeikl

    ∂rkβlj dAi = −

    ∮βij dsi = −

    ∮dui .

    For a single dislocation

    αij = li bjδ(ξ)(α̂ =~l ◦ ~bδ(ξ)

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 19/32

  • Small deformations

    Deformations

    ∂i uj := βij = βpij + β

    eij

    d~ud~r

    := β̂ = β̂e + β̂p

    Nye’s dislocation density tensor

    αij = eikl∂kβplj = −eikl∂kβ

    elj α̂ = ∇× β̂

    p = −∇× β̂e

    Is it that we learned in school?

    bj =∫

    Aαij dAi

    bj = −∫

    Aeikl

    ∂rkβlj dAi = −

    ∮βij dsi = −

    ∮dui .

    For a single dislocation

    αij = li bjδ(ξ)(α̂ =~l ◦ ~bδ(ξ)

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 19/32

  • Small deformations

    Deformations

    ∂i uj := βij = βpij + β

    eij

    d~ud~r

    := β̂ = β̂e + β̂p

    Nye’s dislocation density tensor

    αij = eikl∂kβplj = −eikl∂kβ

    elj α̂ = ∇× β̂

    p = −∇× β̂e

    Is it that we learned in school?

    bj =∫

    Aαij dAi

    bj = −∫

    Aeikl

    ∂rkβlj dAi = −

    ∮βij dsi = −

    ∮dui .

    For a single dislocation

    αij = li bjδ(ξ)(α̂ =~l ◦ ~bδ(ξ)

    )István Groma, ELTE Condensed Matter Physics, Dislocations 19/32

  • Incompatibility

    Incompatibility operator

    (IncÂ)ij = −eikmejln∂k∂l Amn Inc = −∇× Â×∇

    Inc(∂i fj + ∂j fi ) ≡ 0 Inc[

    d~fd~r

    ]sim

    ≡ 0

    Incompatibility field

    ηij = −eikmejln∂

    ∂rk

    ∂rl�emn

    (inc �̂e = η̂

    )

    ηij =12

    (ejln

    ∂rlαim + eiln

    ∂rlαjm

    ) (η̂ = [α̂×∇]sym

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 20/32

  • Incompatibility

    Incompatibility operator

    (IncÂ)ij = −eikmejln∂k∂l Amn Inc = −∇× Â×∇

    Inc(∂i fj + ∂j fi ) ≡ 0 Inc[

    d~fd~r

    ]sim

    ≡ 0

    Incompatibility field

    ηij = −eikmejln∂

    ∂rk

    ∂rl�emn

    (inc �̂e = η̂

    )

    ηij =12

    (ejln

    ∂rlαim + eiln

    ∂rlαjm

    ) (η̂ = [α̂×∇]sym

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 20/32

  • Incompatibility

    Incompatibility operator

    (IncÂ)ij = −eikmejln∂k∂l Amn Inc = −∇× Â×∇

    Inc(∂i fj + ∂j fi ) ≡ 0 Inc[

    d~fd~r

    ]sim

    ≡ 0

    Incompatibility field

    ηij = −eikmejln∂

    ∂rk

    ∂rl�emn

    (inc �̂e = η̂

    )

    ηij =12

    (ejln

    ∂rlαim + eiln

    ∂rlαjm

    ) (η̂ = [α̂×∇]sym

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 20/32

  • Stress

    Linear elasticity

    σij = Lijkl�ekl(σ̂ = L̂ : �̂e

    )

    Elastic deformation↔ Stress

    ηij = −eikmejln∂

    ∂rk

    ∂rlL−1mnopσop

    (inc(

    L̂−1σ̂)= η̂

    )

    Is it enough?

    inc

    [d~fd~r

    ]sym

    ≡ 0

    Equilibrium condition

    ∂riσij = 0 (div σ̂ = 0)

    István Groma, ELTE Condensed Matter Physics, Dislocations 21/32

  • Stress

    Linear elasticity

    σij = Lijkl�ekl(σ̂ = L̂ : �̂e

    )Elastic deformation↔ Stress

    ηij = −eikmejln∂

    ∂rk

    ∂rlL−1mnopσop

    (inc(

    L̂−1σ̂)= η̂

    )

    Is it enough?

    inc

    [d~fd~r

    ]sym

    ≡ 0

    Equilibrium condition

    ∂riσij = 0 (div σ̂ = 0)

    István Groma, ELTE Condensed Matter Physics, Dislocations 21/32

  • Stress

    Linear elasticity

    σij = Lijkl�ekl(σ̂ = L̂ : �̂e

    )Elastic deformation↔ Stress

    ηij = −eikmejln∂

    ∂rk

    ∂rlL−1mnopσop

    (inc(

    L̂−1σ̂)= η̂

    )Is it enough?

    inc

    [d~fd~r

    ]sym

    ≡ 0

    Equilibrium condition

    ∂riσij = 0 (div σ̂ = 0)

    István Groma, ELTE Condensed Matter Physics, Dislocations 21/32

  • Stress

    Linear elasticity

    σij = Lijkl�ekl(σ̂ = L̂ : �̂e

    )Elastic deformation↔ Stress

    ηij = −eikmejln∂

    ∂rk

    ∂rlL−1mnopσop

    (inc(

    L̂−1σ̂)= η̂

    )Is it enough?

    inc

    [d~fd~r

    ]sym

    ≡ 0

    Equilibrium condition

    ∂riσij = 0 (div σ̂ = 0)

    István Groma, ELTE Condensed Matter Physics, Dislocations 21/32

  • Stress

    Linear elasticity

    σij = Lijkl�ekl(σ̂ = L̂ : �̂e

    )Elastic deformation↔ Stress

    ηij = −eikmejln∂

    ∂rk

    ∂rlL−1mnopσop

    (inc(

    L̂−1σ̂)= η̂

    )Is it enough?

    inc

    [d~fd~r

    ]sym

    ≡ 0

    Equilibrium condition

    ∂riσij = 0 (div σ̂ = 0)

    István Groma, ELTE Condensed Matter Physics, Dislocations 21/32

  • Second order stress function

    σij = −eikmejln∂

    ∂rk

    ∂rlχmn (σ̂ = inc χ̂)

    for isotropic materials

    χ′ij =1

    (χij −

    ν

    1 + 2νχkkδij

    )χij = 2µ

    (χ′ij +

    ν

    1− νχ′kkδij

    )

    ∇4χ′ij = ηij(∇4χ̂′ = η̂

    ).

    gauge condition

    ∂riχ′ij = 0

    (div χ̂′ = 0

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 22/32

  • Second order stress function

    σij = −eikmejln∂

    ∂rk

    ∂rlχmn (σ̂ = inc χ̂)

    Due to the identity

    div inc ≡ 0 div σ̂ = 0

    for isotropic materials

    χ′ij =1

    (χij −

    ν

    1 + 2νχkkδij

    )χij = 2µ

    (χ′ij +

    ν

    1− νχ′kkδij

    )

    ∇4χ′ij = ηij(∇4χ̂′ = η̂

    ).

    gauge condition

    ∂riχ′ij = 0

    (div χ̂′ = 0

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 22/32

  • Second order stress function

    σij = −eikmejln∂

    ∂rk

    ∂rlχmn (σ̂ = inc χ̂)

    ηij = eikmejlneoqv epuw L−1mnop

    ∂rk

    ∂rl

    ∂rq

    ∂ruχvw

    (η̂ = inc

    (L̂−1inc χ̂

    ))

    for isotropic materials

    χ′ij =1

    (χij −

    ν

    1 + 2νχkkδij

    )χij = 2µ

    (χ′ij +

    ν

    1− νχ′kkδij

    )

    ∇4χ′ij = ηij(∇4χ̂′ = η̂

    ).

    gauge condition

    ∂riχ′ij = 0

    (div χ̂′ = 0

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 22/32

  • Second order stress function

    σij = −eikmejln∂

    ∂rk

    ∂rlχmn (σ̂ = inc χ̂)

    for isotropic materials

    χ′ij =1

    (χij −

    ν

    1 + 2νχkkδij

    )χij = 2µ

    (χ′ij +

    ν

    1− νχ′kkδij

    )

    ∇4χ′ij = ηij(∇4χ̂′ = η̂

    ).

    gauge condition

    ∂riχ′ij = 0

    (div χ̂′ = 0

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 22/32

  • Second order stress function

    σij = −eikmejln∂

    ∂rk

    ∂rlχmn (σ̂ = inc χ̂)

    for isotropic materials

    χ′ij =1

    (χij −

    ν

    1 + 2νχkkδij

    )χij = 2µ

    (χ′ij +

    ν

    1− νχ′kkδij

    )

    ∇4χ′ij = ηij(∇4χ̂′ = η̂

    ).

    gauge condition

    ∂riχ′ij = 0

    (div χ̂′ = 0

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 22/32

  • 2D case

    Stress function→ stress (∂/∂z = 0)

    σ11 = −∂2χ

    ∂y2, σ22 = −

    ∂2χ

    ∂x2, σ12 =

    ∂2χ

    ∂x∂y, χ ≡ χ33

    σ23 = −∂φ

    ∂x, σ13 =

    ∂φ

    ∂y, φ = −

    ∂χ23

    ∂x+∂χ31

    ∂y

    Field equationsEdge dislocation

    ∇4χ =2µ

    1− ν

    (b1

    ∂y− b2

    ∂x

    )(ρd+ − ρd−)

    Shrew dislocation

    ∇2φ = µb3(ρd+ − ρd−)

    István Groma, ELTE Condensed Matter Physics, Dislocations 23/32

  • 2D case

    Stress function→ stress (∂/∂z = 0)

    σ11 = −∂2χ

    ∂y2, σ22 = −

    ∂2χ

    ∂x2, σ12 =

    ∂2χ

    ∂x∂y, χ ≡ χ33

    σ23 = −∂φ

    ∂x, σ13 =

    ∂φ

    ∂y, φ = −

    ∂χ23

    ∂x+∂χ31

    ∂y

    Field equationsEdge dislocation

    ∇4χ =2µ

    1− ν

    (b1

    ∂y− b2

    ∂x

    )(ρd+ − ρd−)

    Shrew dislocation

    ∇2φ = µb3(ρd+ − ρd−)

    István Groma, ELTE Condensed Matter Physics, Dislocations 23/32

  • 2D case

    Stress function→ stress (∂/∂z = 0)

    σ11 = −∂2χ

    ∂y2, σ22 = −

    ∂2χ

    ∂x2, σ12 =

    ∂2χ

    ∂x∂y, χ ≡ χ33

    σ23 = −∂φ

    ∂x, σ13 =

    ∂φ

    ∂y, φ = −

    ∂χ23

    ∂x+∂χ31

    ∂y

    Field equationsEdge dislocation

    ∇4χ =2µ

    1− ν

    (b1

    ∂y− b2

    ∂x

    )(ρd+ − ρd−)

    Shrew dislocation

    ∇2φ = µb3(ρd+ − ρd−)

    István Groma, ELTE Condensed Matter Physics, Dislocations 23/32

  • 2D stress

    ∇4G = δ(r) G(~r) =1

    8πr2 ln(r)

    σ11 =µb

    2π(1− ν)y(3x2 + y2)(x2 + y2)2

    σ22 =µb

    2π(1− ν)y(x2 − y2)(x2 + y2)2

    σ12 =µb

    2π(1− ν)x(x2 − y2)(x2 + y2)2

    István Groma, ELTE Condensed Matter Physics, Dislocations 24/32

  • Variational approaches I.

    Classical elasticity F (�̂) = F ([d~u/d~r

    ]sim

    σ̂ =δFδ�

    δFδui

    = ∂jδFδ�ij

    = ∂jσij = 0

    Plasticity

    Q[�̂e, χ̂] = F (�̂)−∫

    [χ̂ : (Inc�̂e − η̂)]dV

    δQδ�̂e

    = σ̂ − Incχ̂ = 0,δQδχ̂

    = Inc�̂e − η̂ = 0

    István Groma, ELTE Condensed Matter Physics, Dislocations 25/32

  • Variational approaches I.

    Classical elasticity F (�̂) = F ([d~u/d~r

    ]sim

    σ̂ =δFδ�

    δFδui

    = ∂jδFδ�ij

    = ∂jσij = 0

    Plasticity

    Q[�̂e, χ̂] = F (�̂)−∫

    [χ̂ : (Inc�̂e − η̂)]dV

    δQδ�̂e

    = σ̂ − Incχ̂ = 0,δQδχ̂

    = Inc�̂e − η̂ = 0

    István Groma, ELTE Condensed Matter Physics, Dislocations 25/32

  • Variational approaches II.

    Gibbs free energy G(σ̂)

    −δGδσ̂

    = �̂e

    Consider

    P[χ̂] = G(Incχ̂ = σ̂) +∫

    [χ̂ : η̂)]dV

    One obtains

    δPδχ̂

    = IncδGδσ̂

    + η̂ = −Inc�̂e + η̂ = 0,

    Dislocation dislocation interaction (Peach-Koehler force)

    Fi = eikl tkσlmbm = −dP

    dr ji

    István Groma, ELTE Condensed Matter Physics, Dislocations 26/32

  • Variational approaches II.

    Gibbs free energy G(σ̂)

    −δGδσ̂

    = �̂e

    Consider

    P[χ̂] = G(Incχ̂ = σ̂) +∫

    [χ̂ : η̂)]dV

    One obtains

    δPδχ̂

    = IncδGδσ̂

    + η̂ = −Inc�̂e + η̂ = 0,

    Dislocation dislocation interaction (Peach-Koehler force)

    Fi = eikl tkσlmbm = −dP

    dr ji

    István Groma, ELTE Condensed Matter Physics, Dislocations 26/32

  • Variational approaches II.

    Gibbs free energy G(σ̂)

    −δGδσ̂

    = �̂e

    Consider

    P[χ̂] = G(Incχ̂ = σ̂) +∫

    [χ̂ : η̂)]dV

    One obtains

    δPδχ̂

    = IncδGδσ̂

    + η̂ = −Inc�̂e + η̂ = 0,

    Dislocation dislocation interaction (Peach-Koehler force)

    Fi = eikl tkσlmbm = −dP

    dr ji

    István Groma, ELTE Condensed Matter Physics, Dislocations 26/32

  • Plain strain (2D) case

    P[χ] =∫ [−

    D2(4χ)2 + bχ∂yκ)

    ]dV

    with D = (1− ν)/2µ, and

    κ =∑

    j

    ±δ(~r −~rj )

    Leading to

    D42 χ = b∂y

    ∑j

    ±δ(~r −~rj )

    István Groma, ELTE Condensed Matter Physics, Dislocations 27/32

  • Core regularization

    Nonlocal term

    Gnonlocal[σ] = G0 − b2∫

    Nijklmn(∂iσjk )∂lσmndV ,

    2D isotopic material

    P[χ] =∫ {−

    1− ν4µ

    [|4χ|2 + a2|∇4χ|2

    ]+ bchi∂2δ(r)

    }d2r

    Leading To

    42χ− a243χ =2bµ

    1− ν∂2δ(r)

    χ =ba

    4πDyr

    [−2K1

    (ra

    )+ 2

    ar+

    raln

    (ra

    )]

    István Groma, ELTE Condensed Matter Physics, Dislocations 28/32

  • Core regularization

    Nonlocal term

    Gnonlocal[σ] = G0 − b2∫

    Nijklmn(∂iσjk )∂lσmndV ,

    2D isotopic material

    P[χ] =∫ {−

    1− ν4µ

    [|4χ|2 + a2|∇4χ|2

    ]+ bchi∂2δ(r)

    }d2r

    Leading To

    42χ− a243χ =2bµ

    1− ν∂2δ(r)

    χ =ba

    4πDyr

    [−2K1

    (ra

    )+ 2

    ar+

    raln

    (ra

    )]

    István Groma, ELTE Condensed Matter Physics, Dislocations 28/32

  • Core regularization

    Nonlocal term

    Gnonlocal[σ] = G0 − b2∫

    Nijklmn(∂iσjk )∂lσmndV ,

    2D isotopic material

    P[χ] =∫ {−

    1− ν4µ

    [|4χ|2 + a2|∇4χ|2

    ]+ bchi∂2δ(r)

    }d2r

    Leading To

    42χ− a243χ =2bµ

    1− ν∂2δ(r)

    χ =ba

    4πDyr

    [−2K1

    (ra

    )+ 2

    ar+

    raln

    (ra

    )]

    István Groma, ELTE Condensed Matter Physics, Dislocations 28/32

  • Core regularization

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    -20 -15 -10 -5 0 5 10 15 20

    τxy

    (x,0

    ) [

    Gb

    /a]

    x [a]

    Classical stress fieldStress field with core regularisation

    0.02

    0.1

    1

    1 10 50

    István Groma, ELTE Condensed Matter Physics, Dislocations 29/32

  • Role of elastic anharmonicity

    σij = Lijkl�kl + Kijklmn�kl�mn

    Ganharm[σ] := G0 +∫ [

    Cijklmnσijσklσmn +O(σ4ij )]

    dV

    2D edge dislocation

    G[χ] :=∫ [−

    D2(4χ)2 + �a

    D2

    3(4χ)3

    ]d2

    Extra energy

    V1(r) =(b1 − b2)b2

    16π2D

    [y(3x2 + y2)

    r4+ 4

    yr2

    ln

    (rr0

    )].

    István Groma, ELTE Condensed Matter Physics, Dislocations 30/32

  • Role of elastic anharmonicity

    Ganharm[σ] := G0 +∫ [

    Cijklmnσijσklσmn +O(σ4ij )]

    dV

    2D edge dislocation

    G[χ] :=∫ [−

    D2(4χ)2 + �a

    D2

    3(4χ)3

    ]d2

    Extra energy

    V1(r) =(b1 − b2)b2

    16π2D

    [y(3x2 + y2)

    r4+ 4

    yr2

    ln

    (rr0

    )].

    István Groma, ELTE Condensed Matter Physics, Dislocations 30/32

  • Role of elastic anharmonicity

    Ganharm[σ] := G0 +∫ [

    Cijklmnσijσklσmn +O(σ4ij )]

    dV

    2D edge dislocation

    G[χ] :=∫ [−

    D2(4χ)2 + �a

    D2

    3(4χ)3

    ]d2

    Extra energy

    V1(r) =(b1 − b2)b2

    16π2D

    [y(3x2 + y2)

    r4+ 4

    yr2

    ln

    (rr0

    )].

    István Groma, ELTE Condensed Matter Physics, Dislocations 30/32

  • Role of elastic anharmonicity

    Ganharm[σ] := G0 +∫ [

    Cijklmnσijσklσmn +O(σ4ij )]

    dV

    2D edge dislocation

    G[χ] :=∫ [−

    D2(4χ)2 + �a

    D2

    3(4χ)3

    ]d2

    Extra energy

    V1(r) =(b1 − b2)b2

    16π2D

    [y(3x2 + y2)

    r4+ 4

    yr2

    ln

    (rr0

    )].

    István Groma, ELTE Condensed Matter Physics, Dislocations 30/32

  • Time evolution

    Dislocation density tensor

    αij = eikl∂

    ∂rkβ

    plj

    (α̂ = ∇× β̂p

    )

    Evolution equation

    α̇ij + eikl∂

    ∂rkjlj = 0

    (˙̂α+∇× ĵ = 0

    )where

    ĵ = − ˙̂βp

    Physical meaning

    dbjdt

    = −∮

    Ljij dsi

    For individual dislocation (Orowan’s equation)

    jik = eilm ll vmbkδ(ξ)(̂

    j = (~l ◦ ~bδ(ξ))× ~v)

    István Groma, ELTE Condensed Matter Physics, Dislocations 31/32

  • Time evolution

    Evolution equation

    α̇ij + eikl∂

    ∂rkjlj = 0

    (˙̂α+∇× ĵ = 0

    )where

    ĵ = − ˙̂βp

    Physical meaning

    dbjdt

    = −∮

    Ljij dsi

    For individual dislocation (Orowan’s equation)

    jik = eilm ll vmbkδ(ξ)(̂

    j = (~l ◦ ~bδ(ξ))× ~v)

    István Groma, ELTE Condensed Matter Physics, Dislocations 31/32

  • Time evolution

    Evolution equation

    α̇ij + eikl∂

    ∂rkjlj = 0

    (˙̂α+∇× ĵ = 0

    )where

    ĵ = − ˙̂βp

    Physical meaning

    dbjdt

    = −∮

    Ljij dsi

    For individual dislocation (Orowan’s equation)

    jik = eilm ll vmbkδ(ξ)(̂

    j = (~l ◦ ~bδ(ξ))× ~v)

    István Groma, ELTE Condensed Matter Physics, Dislocations 31/32

  • Time evolution

    Evolution equation

    α̇ij + eikl∂

    ∂rkjlj = 0

    (˙̂α+∇× ĵ = 0

    )where

    ĵ = − ˙̂βp

    Physical meaning

    dbjdt

    = −∮

    Ljij dsi

    For individual dislocation (Orowan’s equation)

    jik = eilm ll vmbkδ(ξ)(̂

    j = (~l ◦ ~bδ(ξ))× ~v)

    István Groma, ELTE Condensed Matter Physics, Dislocations 31/32

  • Displacement field

    Plastic deformation

    ∂uj∂ri

    = βeij + βpij

    (d~ud~r

    = β̂e + β̂p)

    Multiply with L̂ and take its div

    ∂riLijkl

    ∂uk∂rl

    =∂

    ∂riLijklβ

    pkl

    (div L̂

    d~ud~r

    = div L̂β̂p)

    Taking the time derivative

    ∂riLijkl

    ∂u̇k∂rl

    =∂

    ∂riLijkl jkl

    (div L̂

    d~̇ud~r

    = div L̂̂j

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 32/32

  • Displacement field

    Plastic deformation

    ∂uj∂ri

    = βeij + βpij

    (d~ud~r

    = β̂e + β̂p)

    Multiply with L̂ and take its div

    ∂riLijkl

    ∂uk∂rl

    =∂

    ∂riLijklβ

    pkl

    (div L̂

    d~ud~r

    = div L̂β̂p)

    Taking the time derivative

    ∂riLijkl

    ∂u̇k∂rl

    =∂

    ∂riLijkl jkl

    (div L̂

    d~̇ud~r

    = div L̂̂j

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 32/32

  • Displacement field

    Plastic deformation

    ∂uj∂ri

    = βeij + βpij

    (d~ud~r

    = β̂e + β̂p)

    Multiply with L̂ and take its div

    ∂riLijkl

    ∂uk∂rl

    =∂

    ∂riLijklβ

    pkl

    (div L̂

    d~ud~r

    = div L̂β̂p)

    Taking the time derivative

    ∂riLijkl

    ∂u̇k∂rl

    =∂

    ∂riLijkl jkl

    (div L̂

    d~̇ud~r

    = div L̂̂j

    )

    István Groma, ELTE Condensed Matter Physics, Dislocations 32/32