Edge Simulation Procedure_13.9

7/21/2019 Edge Simulation Procedure_13.9

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Pile up fow chart:

System PropertiesDuring deormation o a metal, many dierent mechanisms would operate.In the simulation, the ollowing mechanisms would be dealt with as ollows

1. nnihilation: whene!er two oppositely signed dislocations are spaced by 1 burgers !ector,

annihilation would occur.". #ross$slip: when a certain ratio o cross$slip orce to glide orce is e%ceeded, cross$slip

would occur at a certain probability&. Dislocation generation '(ran) *ead source+: sources are acti!ated with a certain

probability when a certain stress is reached.

he dipole consists o " oppositely signed dislocations with burgers !ector -1 and $1. heabo!e distance between the dislocations is adopted such that ater the generation, the

total resol!ed shear stress between the two dislocations would balance the attracti!e orce

between the " dislocationsri) !an der /iessen and lan 0eedleman, Discrete dislocation plasticity: a simple planar

model2 modeling simul. 3ater. Sci. eng. &'1445+ 674$8&5Some authors adopt a /aussian distribution o acti!ation stress.#urrently I use a random between

Should I use /aussian distribution99!erage nucleation stress is ta)en as ;.;1'normaliuires thermal acti!ation, assuming *P, climb is not !ery prominent

?oundary conditions:o simulate tensile loading o a specimen with a small t@d ratio, "D dislocation dynamics is

employed.In the loading direct


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1. In the loading direction, periodicity boundary condition is employed.Since the uppermost and bottom side are also grain boundary, periodicity boundary

condition is adopted so that pile$up orce at the uppermost grain is transerred to the

bottom grain, and the pile$up orce at the bottom grain is transerred to the top grain.". he ABS and *BS are ree suraces, dislocations can escape at the ree surace

he e%istence o a ree surace would introduce an image stress on the dislocations. In the

short range, such an image stress would be attraction in nature. So dislocations are

accelerated towards the ree surace.&. Chen coating or other substance is present, there would also be image stress. Depending

on the elastic modulus o the oreign coating, the image stress may either be attracti!e or

repulsi!e in the short range.

nnihilation:

dge dislocations cannot cross slip, so the chance o meeting oppositely signed dislocations

within too short a distance is !ery small, so the annihilation distance is set to be 5b. nnihilation

re>uires " oppositely signed dislocation on parallel slip plane

Dislocation /eneration

peration o (ran) *ead Sources:

he (ran) *ead sources would be acti!ated when the resultant stress on them e%ceed a certain!alue at a probability. he operating stress o the (ran) *ead sources ollows a /aussian

distribution, e.g.


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he stresses that would be elt by a (ran) *ead source include the

1+ e%ternal applied stress

"+ dislocation stress Eelds&+ pile$ups=+ image stress

nd when the stress is high enough, there is a certain probability, say 5F, o generation o a

dislocation dipole.

he dipole would glide on e%actly the same slip plane and in opposite directions. In order that the

" closely spaced dislocations o the dipole would not immediately annihilate, they would be

spaced at Anucapart, where . his ensures the nucleation stress would Gust

balance the attraction stress between them.

Dislocation Dynamics:

he ollowing mechanisms would operate:

Pinning:

I there are no obstacles, dislocations would not stop midway in the grain, but a cloud o dierent

types o dislocations would actually cluster in the middle o the grain when low stress is applied

Pile$up at grain boundaries:

cross the grain boundary, the slip plane orientation changes abruptly, so dislocations would not

pass through the grain boundary

/lide:

*esults:1. Helocity o dislocations

*emar)s:


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nits:1. (orce". ime: is normali


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*emar)s:he stress Eeld or an edge dislocation would be dierent rom that o screw dislocations.o change rom screw to edge simulation, se!eral changes ha!e to be made:

1. stress Eeld o an edge dislocation". no cross$slip&. image stress Eeld=. annihilation distance 'lower probability o annihilation, so perhaps setting a larger

annihilation distance would better account or the disappearance o dislocationsIn reality, there would be Enite climb acti!ities, but climb are associated with diusion hence

re>uire ele!ated temperature. s the simulation is assuming room temperature, >uasi$statis

loading, it is assumed there are no climb acti!ities. Bowe!er, the assumption o a 6b

annihilation distance may already account or short distance climb$annihilation.

(ran) *ead source:

he current problem with the (ran) read source is that it e%erts no stress to the neighbouring

dislocations. In reality, howe!er, a dislocation source must be associated with some stress

Eled, it is either a orest dislocation intersecting the "$D plane perpendicularly, or a

dislocation line pinned at " points, either o them would ha!e stress Eeld around them, sowhile they get acti!ated by dislocations gliding close to them , they e%ert some orce on the

dislocation as well, hindering dislocation mo!ement.

(or edge dislocation, dislocation$dsilocation interaction stress Eeld is gi!en by:

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(or a dislocation gliding along the %$a%is, the glide orce by this shear stress would be:

Chen the slip plane does not align with the %$a%is, the % and y 'or in !ector orm + in the

ormula ha!e to be corrected by a rotation matri%

Chere N is the angle that the slip plane ma)es with the positi!e %$a%is. hen, the corrected a and

y would be gi!en by

he two slip planes in each grain would ma)e an angle o with the positi!e %$a%is:

(or , NJslipplane angle

(or , Jacos'$

(or positi!e N, rotation is in anticloc)wise sense. (or negati!e theta, rotation is in cloc)wise

sense.

Slip plane 1

Slip plane "

%

y


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Strain:

Image stress:

(or a crystal containing dislocations, the near dislocation stress and the strain energy at arbitrary

point ta)e up Enite !alue. Bowe!er, when approaching the ree surace, it is intuiti!e to imagine

that the strain energy and stress would decrease. his would tend to pull out the near$surace

dislocations.

he image orce is computed by adding in an image dislocation o opposite sign on the other side

o the ree surace.

#onsidering the image orce by image dislocation on the original dislocation, simply substitute

%J"d, yJ; into the abo!e ormula:

Stress Eelds o the original and the image dislocation


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he abo!e is wrong because tractions do not !anish, should be sol!ed by an array o image

dislocations

d is the horiual


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method re>uires the deEnition o the boundary condition in the problem Erst. (inite element

method is then applied. See section " o ri) !an der /iessens wor) 'p.641$+.

pplied strss:

In the system, the applied stress is a normal stress applied along the !ertical direction. he shear

stress on the dislocation has to be resol!ed according to the orientation o the slip plane.

(inal stage in the simulation o edge dislocation dynamics o 1;;;; steps

Cith all types o orces summed up, a dislocation would mo!e with some Enite !elocity. he

!elocity depends on the orce or stress elt:

where is the resultant stress on each indi!idual dislocations, is a reerence stress,

is an e%ponent which depend on the structure o the metal, usually large or cc and smaller or

bcc and hcp.

Depending on the sign o the orce, the dislocation would glide on the original slip plane in the

positi!e or negati!e direction. he displacement in each time step is hence gi!en by , where

time is normali


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lso, the magnitude o each indi!idual step should be limited. his is because too large a glide

step would ma)e a dislocation s)ip too many dislocations, (ran) *ead sources and so on. he

dynamics o such dislocations would hence not be realistic.

Initialisation:

Parameters include:

*egion:

he " dimensions o the region are set in terms o b.

/rains:

he number o rows and columns o grains

he Erst slip plane orientation in each grain is randomi


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Surace Aayer ect:

#onsider a grain with the bottom edge being on the ree surace:

#ombined ect

s loading proceed, dislocations glide and mo!e to the respecti!e e>uilibirum positions. he

dislocation density o a grain in a simple "D edge dislocation simulation would be as ollows:

where a dar)er color represents a higher dislocation density, and a lighter color represents a

lower dislocation density. 0ote that the o!erlapping region where dislocations o dierent slip

planes

and e!en

0egati!e edge dislocations on slip plane

1 glide towards surace

0egati!e edge dislocations on slip plane

" glide towards surace

?oth positi!e and negati!e edge

dislocations on slip plane 1 glide towards

surace due to image stress ?oth positi!e and negati!e edge

dislocations on slip plane " glide towardssurace due to image stress


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dierent signs would escape would beo thelightest color, representing the lowest dislocation

density.

I (ran) reade sources are o Enite number, as loading proceeds and stress increases, the

dislocation density would be be as uniorm as the abo!e color scheme would depict. *ather,

there would be a shiting o the dislocation density towards the grain boundaries, resulting in a

cluster o dislocations at the grain boundary.

#olor Scheme in the simulation to represent dierent stress state:

sing continuous or discrete color scheme99

Discrete is easier to use, plus, the stress map is intended to gi!e a general picture o the stress

state rather than the precise distribution o stress, so discrete color scheme is good enough

#an also use grascale, because easy to recognise the brightness o the gray color


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Shear stress Eeld o a single dislocation, with hori


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he stress ditribution map has a constraint: in order not to calculate interaction orce between

dislocations too ar apart, there is a cut$o distance beyond which " dislocations would not

interact. S, in the ollowing Egure, the stress distribution is incomplete.

#ompared to the ollowing '-burgers !ector+

Perhaps, plot the stress map or the normal stress. nd because the stress applied is normal, we

are also interested in seeing the distribution o normal stress inside the material throughout the

simulation.


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he ormula or the normal stress:

In %,y$coordinates:

In polar coordinates:

#an usethe polar coordinate Erst, then use the angle to correct the stress 'sine theta to gi!e

stress in the !ertical direction, cosine theta to gi!e stress in the hori


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point orce calculated this way can be thought o as actinbg at the center o the mesh. Since the

?oussines> solution sol!es the stress Eeld around a point orce, the stress on the dislocations can

be ound.

3.#. (i!el and /.*. #ano!a, De!eloping rigorous boundary conditions to simulations o discrete

dislocation dynamics, 3odelling Simul. 3ater. Sci. ng. '1444+ 85&$867

(i!el and #ano!a '1444+ how to use the ?oussines> solution to sol!e or the image stress in the

case o &D dislocation dynamics. he ree surace is Erst discreti


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his deals with the case where the point orce applied is !ertical to the surace

ter )nowing the principal stresses, the shear stress that could be resol!ed at the dislocation

glide planes can be ound by using the 3ohrs circle.

y

x

R

y

x

x

y

yx

yx


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In the glide plane o the ithdislocation, the resol!ed shear stress would then depend on the

normal stress:

(or (or

(or

Chere , and are the principal stresses. Since it is the shear stress in the glide plane that

mo!es the edge dislocations, the corresponding shear stress ha!e to be resol!e by plane stress

transormation.

?y assuming the problem as a plane stress problem, the ollowing conditions are assumed:

, ,

(or "D case:

*ecognising dislocations are line deects but not point deects, the dislocation orce should be

thought o as acting uniormly on the surace along the dislocationline, hence the image stress insuch )ind o "D dislocation dynamics simulation should also be line loading in nature. 'plane

strain99+

y

x

y

yx

yx

x

x

y

yx

yx

x


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is the shear component which can mo!e dislocations along

*eer to Contact Mechanicsby R.A. Tohnson, section "." and ".&, which describe concentrated

line loading, both normal and tangential.

(or normal orce, the stresses are the same as that listed abo!e. (or tangential line orce, the

stress Eeld also radial, is similar to that o normal orce, but rotated through 4;U.


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he negati!e sign ensures the radial normal stress is compressi!e or NVJW, and tensile or W@"

VNVJW.

o con!ert the stresses into %


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he stress Eeld in this coordinate systemn would then be transormed by ha!ing < replaced by %and % replaced by y, so the contributions rom normal line orce an dtangential line orce would

now become:

0ormal Aine (orce angential Aine (orce

ncorrected shear strs 'in the coordinates o the dislocation which is tilted with respect to the

system coordinates:

Stress (ield with *espect to the Dislocation #oordinates:

y

x

R

y

x

x

y

yx

yx


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ngle

anticloc)wise rom

positi!e %$a%is+

;

O%

Oy

Y%y

Stress (ield with *espect to the System #oordinates 'i.e. the %, y are aligned with the Systems

%y$ a%es+:

O%or ; and pi@=


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O%or pi@" and &pi@=

Y%yor ; and pi@=

Y%yor pi@" and &pi@=

Oyor ; and pi@=


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Oyor pi@" and &pi@=


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lso, prepare a ew !ideos o preset dislocations to see how they interact with each other

Dealing with Image Stress: 15@6$17@6@";1"

Image Stress is the imaginary stress Eeld applied e%ternally to the surace to satisy certain

boundary conditions.

In the case o a ree surace, the boundary condition would be traction ree: i.e. solution. In the case o the line loading employed in "D simulation, the stress

distribution is sol!ed by (lamant '174"+. Strip loading may also reach the answer, but that would

be a little bit more complicated.

o better mimic the real case, the grids ha!e to be !ery Ene along the ree surace, otherwise the

surace would b erepresented by !eryew points, and not all local regions would eel a realistsic

stress Eeld.

he boo) 3echanical Properties o engineered materials2 mentioned, e!en or hori


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Insight:

he siue to the glide plane under consideration, that

would multiply the eect o dislocations by generating e!en more dislocations.

+ I escape is aster than multiplication:

Could result in a net decrease in dislocation density o!er time. (urthermore, the dislocation

sources in the orm o pinned dislocations would decrease 'simply consider obli>ue planes arealso e%hausted o dislocations+. So the surace grain become rather depleted o dislocations, and

are >uite resistent to deormation. hus, to catch up with the strain rate, the increase in stress

re>uired is >uite large. he process ma)e be " step 'decrease in stress because o less pinning

and then increase+ or simply single step 'increase in stress re>uired+

?+ I escape is slower than multiplication

n increase in the number o dislocations might gi!e rise to " opposite results: 1+ i the

dislocation density is not too high, an incresase in dislocation density would decrease the stress

re>uired to meet the strain "+ i the dislocation density is too high, dislocations would get pinned

!ery easily, ma)ing them immobile.

#onsider a "D case

Dislocation escape inter!al:

ssuming there are n dislocations gliding towards the surace, the inter!al between escape

would be d@'"n!+

3ultiplication inter!al:

#onsider only (ran) *ead source and orest dislocation sources:

Chere (ran) *ead source operation speed may be in terms o areal sweep rate.

(ran) *ead sources: reerences:

1. .R. (aradGian, A.B. (riedman and D.#. #hr


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*e. 1 de!eloped a continuum le!el simulation o dislocation dynamics. It also e%plores the

dependence o (ran) *ead source operation stress on !arious parameters: source length,

orientation and the chosen stress dependence o the !elocity 'which may in turn depend on the

identity o the metal+.

he operation time would depend on the stress abo!e the critical stress.

Si


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Dislocation loops would orm around the sessile orest dislocations.

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Edge Simulation Procedure_13.9