University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Dislocations: Stress field and Energy

Stress field of a straight screw dislocationStress field of a straight edge dislocationEnergy of a dislocationEnergy of the core vs. elastic energyEnergy of edge, screw, and mixed dislocationsThermodynamics of dislocations

References:Hull and Bacon, Ch. 4Kelly and Knowles, Ch. 8Friedel, Dislocations, Ch. 2

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Let’s consider a Volterra deformation mode that corresponds to a straight screw dislocation of infinite length

Displacements: 0== yx uu

πθ

=2bu z

)/arctan( xy=θ

- changes from 0 to b as θ goes from 0 to 2π

,xue x

xx ∂∂

=

,y

ue y

yy ∂

∂=

,zue z

zz ∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂

∂==

yu

zu

ee zyzyyz 2

1

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

==zu

xuee xz

xzzx 21

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

==xu

yuee yx

xyxy 21

0===== yxxyzzyyxx eeeee

rb

yxybee zxxz

θπ

−=+π

−==sin

44 22

rb

yxxbee zyyz

θπ

=+π

==cos

44 22

xyxy Ge2=σ yzyz Ge2=σ zxzx Ge2=σ

rGb

yxxGb

zyyzθ

π=

+π=σ=σ

cos22 22

rGb

yxyGb

zxxzθ

π−=

+π−=σ=σ

sin22 22

Stress field of a straight dislocation: Screw dislocations

0 dilatation =++=δ zzyyxx eee

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Stress field of a straight dislocation: Screw dislocationsIn cylindrical coordinates, θσ+θσ=σ sincos yzxzrz

shear strain:

shear stress:

πrb

ddω

z 2θ)θ(==γ θ

ω

rbπθ

=θω2

)(

πrGbG zzz 2

=γ=σ=σ θθθ

Stress field from screw dislocations:pure shear stresses, no tensile or compressive componentscomplete radial symmetry for a dislocation of opposite sign, the signs of all the field components are reversed

θσ+θσ−=σθ cossin yzxzz

θ+θ= sincos yzxzrz eeeθ+θ−=θ cossin yzxzz eee

πrGb

zz 2=σ=σ θθπr

bee zz 4== θθ

where

z

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Stress field of a straight dislocation: Screw dislocations

As r → 0, stresses and strains become infinite ⇒ linear elasticity theory is not applicable in the core regionπr

Gbz 2=σ θπr

bz 2=γ θ

θσ z

rR0r

core region ~2-3 b

screening

30...

2maxGG

π≈τrecall theoretical strength

of a perfect crystal:

3br when max ≤τ>σ θz

10%)( 1.02

>>=γ θ πrb

zalso, at br 5.1≤

R is related to the finite size of the sample and screening by other dislocations

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Stress field of a straight dislocation: Edge dislocationsThe deformations and stresses due to an edge dislocation are more complex but can be analyzed in a manner similar to screw dislocations, by considering the corresponding Volterra deformation mode

The displacement and strains in the z-direction are zero → plane strain deformationSimilarly to screw dislocations,

⎥⎦

⎤⎢⎣

⎡υ−θ

+θπ

=)1(4

2sin2bu x ⎥

⎦

⎤⎢⎣

⎡υ−θ

+υ−υ−

π−=

)1(42cosln

)1(221

2rbu y 0=zu

)1(2 ν−π=

GbD

( ) rD

yxyxDyxx

)2cos2(sin3222

22 θ+θ−=

+

+−=σ

( ) rD

yxyxDyyy

θθ=

+

−=σ

2cossin222

22

)( yyxxzz σ+συ=σ

( ) rD

yxyxDxyxxy

θθ=

+

−=σ=σ

2coscos222

22

0=σ=σ=σ=σ zyyzzxxz

rik1~σ

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Stress field of a straight dislocation: Edge dislocations

- normal stresses that are parallel to b are stronger +

–

compression

tension

xyσ

x

y

|||| yyxx σ>σ

stress field has both dilatational and shear components

0yfor )( 0 ><σ ncompressioxx

0yfor )( 0 <>σ tensionxx

( )222

223yxyxDyxx

+

+−=σ

( )222

22

yxyxDxxy

+

−=σ

xyxy sign with changes and 0for maximum is =σ

xyσ

22121

2 dilatation

yxybeee zzyyxx +υ−

υ−π

−=++=δ

the compression for y > 0 and expansion for y < 0 compensate each other and the average density of the crystal approaches that of the perfect crystal

edge dislocation is characterized by two vectors, b and l, whereas screw dislocation is characterized by one vector (i.e., b || l) → stress field is not symmetric with respect to the axis for edge and symmetric for screw dislocations

Dislocations generate strong stresses (of the order of D at r = b) that are proportional to b and are slowly decreasing with r, σik ~ 1/r

( ) 223)1(2

31 pressure

yxyDp zzyyxx +

υ+=σ+σ+σ−=

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Stress field of a straight dislocation: Edge dislocations

xxσ

peak tension (red) is 5 GPapeak compression (blue) is -5 GPa

Stress fields (σxx) surrounding the core of an edge dislocation in atomistic simulations of Al

-10 nm 0 10 nmWebb, Zimmerman, and Seel, Mathematics

and Mechanics of Solids 13, 221, 2008

x

y

Stress fields (σxx) along y-axis as a function of distance from the core of the edge dislocation

(nm) y

(GPa) xxσ

elasticity theoryatomistic simulations

For mixed dislocations, the elastic field can be obtained by adding the fields of the edge and screw components with corresponding Burgers vectors b1 and b2

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Energy of a dislocation

for screw dislocation:

Energy added to the crystal by a dislocation = elastic strain energy + energy of dislocation core

The elastic strain energy of volume dV is dVedVedWzri zrj

ijijzyxi zyxj

ijijel ∑ ∑∑ ∑θ= θ== =

σ=σ=,, ,,,, ,, 2

121

πrGb

zz 2=σ=σ θθ πr

bee zz 4== θθ

( )0

222

00

ln4

14

00rRLGbdr

rLGbrdreeddzW

R

r

R

rzzzz

L

el π=

π=σ+σθ= ∫∫ ∫∫

π

θθθθ

and

similarly, for edge dislocation:0

2

ln)1(4 r

RLGbWel ν−π=

or, in general: ,ln4 0

2

rR

KLGbWel π

= where K = 1 for screw and K = 1 - ν for edge dislocations

integration is from the surface of the core cylinder with radius r0 to the external surface of the cylindrical sample of radius R

coreeldisl WWW +=

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Energy of a dislocation

,ln4 0

2

rR

KLGbWel π

= where K = 1 for screw and K = 1 - ν for edge dislocations

The total energy Wdisl is proportional to its length L ⇒ energy is minimized when dislocation segments between immobile points (e.g. nodes) are straight

The energy per unit length of an edge dislocation is larger than that of a screw dislocationsince (1 – ν) < 1. For ν ≈ 1/3, Wscrew ≈ 0.66 Wedge ⇒ energy is minimized when dislocation has as large screw component as possible

Relatively weak (logarithmic) dependence of the elastic energy on R- in polycrystalline materials R is limited by the size of the grains- in crystals with many dislocations, the dislocations tend to form configurations in whichsuperimposed elastic fields tend to cancel

example: two parallel screw dislocations of opposite sign separated by distance d from each other

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−≈⎟⎟

⎠

⎞⎜⎜⎝

⎛−=σ θ ||

1||

12||

1||

12 1121 rdrπ

Gbrrπ

Gbz rrrrr

stress generated at r1 from the first dislocation:

1rr

2rr

⎟⎠⎞

⎜⎝⎛≈⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−≈σ θ rd

πrGb

rdπrGb

z 2111

2for large distances, r ≈ |r1| ≈ |r2| >> |d|

dr

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Energy of a dislocation

,ln4 0

2

rR

KLGbWel π

= where K = 1 for screw and K = 1 - ν for edge dislocations

Relatively weak (logarithmic) dependence of the elastic energy on R- in crystals with many dislocations, Rmin is on the order of distance between dislocations

metal) worked-coldheavily a(for m 10~for nm 100m 10~1~ -2147min ρ=ρ −R

taking r0 = 0.5 nm,

sample) a of (size m 10cm 1~ -2max =R

3.5m 105

m 10lnln 10

7

0

min =×

= −

−

rR

8.16m 105

m 10lnln 10

2

0

max =×

= −

−

rR

9ln21

20

maxmin ≈rRRgeometric mean:

only a factor of ~3 variation

LGbLGbWel2

2

21

249

≈π

≈ for screw dislocation

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Energy of a dislocation

Energy of the dislocation core: 31−≈Z

Let’s estimate:

ZLGbWb.r core π≈≈

4 ,51 if

2

0

⎟⎟⎠

⎞⎜⎜⎝

⎛+

π=+= Z

rR

KLGbWWW coreeldisl

0

2

ln4

(Al) GPa 5.25=G

m 103 10−×≈beV 0.6J10

3.6m1027Jm 105.25

2 ,for 19

330-393

≈≈×××

=π

≈= −−GbWbL core

eV 62ln4

3

0

2

−≈α≈⎟⎟⎠

⎞⎜⎜⎝

⎛+

π=+= GbZ

rR

KLGbWWW coreeldisl 5.15.0 −≈α

Two contributions to the total energy: dislocation core (~Z ≈ 1-3) and long-range elastic field (~ln(R/r0) ≈ 5-17). The contribution of elastic energy is larger than the one from the core, i.e., Z < ln(R/r0)

The energy of a dislocation ranges between and⎟⎟⎠

⎞⎜⎜⎝

⎛+

0

minlnr

RZ ⎟⎟⎠

⎞⎜⎜⎝

⎛+

0

maxlnr

RZ

(can change by a factor of ~3 depending on the conditions of screening of the long-range elastic field)

⎥⎦

⎤⎢⎣

⎡πK

LGb4

of unitsin 2

per length of b

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Energy of a dislocation

The total energy Wdisl is proportional to b2 ⇒ dislocations tend to have the smallest possible Burgers vector

Frank’s rule for dislocation reactions:

This rule ignores interaction between the dislocations and assumes that all dislocations are perfect

,~ eWel σ ,~ bσ bGe ~2/σ= 2~ bWel

3br 2b

r

3br

3br 2b

r

1br

1br

321 bbbrrr

=+

if b12 + b2

2 > b32, the reaction is favorable

(π/2 < φ ≤ π)

if b12 + b2

2 < b32, the reaction is unfavorable

(0 ≤ φ < π/2)dissociation of b3 into b1 and b2 is favorable in this case

ϕ

Mixed dislocations: stresses of screw and edge dislocations are “orthogonal” (stress tensors have no non-zero common components) - a mixed dislocation can be decomposed into edge and screw components and the energies of the two components can be added.

⎟⎟⎠

⎞⎜⎜⎝

⎛+

π= Z

rR

KLGbWdisl

0

2

ln4 where

υ−φ

+φ=1sincos1 2

2

K

φ br

edgebr

screwbr

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Thermodynamics of dislocations

The number of dislocations that corresponds to the thermodynamic equilibrium is zero.

The dislocations, however, are almost always present in real materials due to the high kinetic barrier for their removal.

In well-annealed crystals the density of dislocations can be reduced down to ~1010 m-2 (104 mm-2)

for a dislocation loop with radius 3b

What is the equilibrium concentration of dislocations?

Equilibrium concentration of dislocation loops:

K 300at eV 05.0k2S B −=−≈Δ− TT

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−=

TkGNn

Beq exp

STHG Δ−Δ=Δ

eV 623 −≈≈≈Δ GbWH dislper length of b:

eV 4062 33 >π≈π

≈Δ GbbRGbG

N < Na = 6×1023 mol-1number of nucleation sites cannot be larger than the number of atoms:

1-6504-

23 mol 10~300K eV/K 100.86

eV 40exp106exp −⎟⎠⎞

⎜⎝⎛

××−×<⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−=

TkGNn

Beq