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DISLOCATIONSDISLOCATIONS
Edge dislocation Screw dislocation
Dislocations in crystals
Introduction to DislocationsD. Hull and D.J. Bacon
Pergaon Press! O"#ord $%&'()
Recommended website
*tt+,--www.t#.uni/iel.de-atwis-aat-de#0en-
Further reading
Theory of Dislocations J. P. Hirt* and J. Lot*e
1c2rawHill! New 3or/ $%&4')
Advanced reading (comprehensive)
1ATE5IALS SCIENCE1ATE5IALS SCIENCE
66EN2INEE5IN2EN2INEE5IN2
Anandh Subramaniam & Kantesh Balani
1aterials Science and Engineering $1SE)
Indian Institute o# Tec*nology! 7an+ur 89'9%4
Email anandh!iit"#ac#in$ %R& home#iit"#ac#in'anandh
AN INT5OD:CTO53 EBOO7 AN INT5OD:CTO53 EBOO7
art of
http''home#iit"#ac#in'anandh'E*boo"#htm
A Learner’s GuideA Learner’s Guide
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As seen ;e#ore dislocations are %D $line) de#ects
T*e role o# dislocations goes far beyond
T*ey +lay an i+ortant role in a ?ariety o# de#oration +rocesses li/e cree+! #atigue and #racture
T*ey can +lay a =constructi?e role> in crystal growt*
T*ey can +ro?ide s*ort circuit +at*s #or di##usion $+i+e di##usion)
:nderstanding t*e i+ortance o# dislocations in aterial ;e*a?iour cannot ;e
o?erstated@ *ence it is ?ery i+ortant to t*oroug*ly understand t*e structure and ;e*a?iour o# dislocations
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T*e i+ortance o# understanding dislocations and t*eir e##ect on aterial ;e*a?iour cannot ;e o?erstated.
T*oug* o#ten t*e i+ortance o# dislocations in t*e conte"t o# +lasticde#oration ;y sli+ is *ig*lig*ted its role in aterials science is #ar greater.$T*e ne"t slide s*ows soe o# t*ese roles).
In t*is conte"t it is i+ortant to note t*at e?en in crystalline aterials t*ere arealternate ec*aniss o# +lastic de#oration $as s*own in an u+coing slide)
@ Twinning also ;eing an i+ortant one. T*e i+ortant t*ing to ;e /e+t in ind is t*e role o# dislocations in wea"ening
crystals $ta/en u+ a#ter t*e a;o?e entioned slides).
Dislocations
Plastic de#oration
@ +eranent de#oration t*at reains w*en all e"ternal loading-constraints are reo?ed Sli+ @ is a tec*nical ter re#erring to +lastic de#oration caused ;y dislocations t*e =#irst ste+> o# t*e +rocess is t*e sall sur#ace ste+ w*ic* is created w*en a dislocation lea?es a crystal Twinning
@ +rocess ;y w*ic* one +art o# t*e crystal gets related to anot*er +art! ;y a syetry o+erator $usually a irror) w*ic* is not a syetry o+erator o# t*e crystal.
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Consider a dislocation in in an in#inite crystal
Ta/e into account #inite crystal e##ects
Consider interaction o# dislocations wit* ot*er de#ects
Pat* to understanding t*e role o# Dislocations in aterial ;e*a?iour
+tress fields$ strain fields$ energy etc#
Free surfaces$ grain boundaries etc#
! ! ! ! ! ...d ,, yy ,y ,, ,y E σ σ τ ε ε
Interactions with other dislocations$ interstitials$ precipitates etc#
Static and dynaic e##ects and interactions s*ould ;e included
- Dynaic e##ects include, (Altered) +tress field of a moving dislocation
Interactions evolving in time
Collecti?e ;e*a?iour and e##ects o# e"ternal constrains
&ong range interactions . collective behaviour . e,ternal constraints--
Though these points are
written as a se/uence
many of these have to be
considered in parallel
0ote the above step by step method may often not be the most practical one and there are techni/ues which ta"e up collective behaviour directly
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Slip
5ole o# Dislocations
FractureFatigue Creep
Diffusion
(Pipe)
Structural
2rain ;oundary(low angle)
Inco*erent Twin
Seico*erent Inter#aces
Disc o# ?acancies edge dislocation
Crystal Growth(Screw dislocation)
0ote +tructural dislocations can also play a role in deformation and "inetic processes
and more122
Cree+ec*aniss in
crystallineaterials
Dislocation cli;
acancy di##usion
Crosssli+
2rain ;oundary sliding
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Sli+
$Dislocationotion)
Plastic Deformation in Crystalline aterials
Twinning P*ase Trans#oration Cree+ 1ec*aniss
2rain ;oundary sliding
acancy di##usion
Dislocation climb
Ot*er 1ec*aniss
0ote lastic deformation in amorphous materials occur by other mechanisms including flow (viscous fluid) and
shear banding
T*oug* +lasticity ;y sli+ is t*e ost i+ortant ec*anis o# +lastic de#oration! t*ere areot*er ec*aniss as well (plastic deformation here means permanent deformation in theabsence of e,ternal constraints),
2rain rotation
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Fea/ening o# a crystal ;y t*e +resence o# dislocations
To cause +lastic de#oration ;y s*ear ( all of plastic deformation by slip re/uire shear stresses at the microscopic scale- ) one can ?isualiGe a +lane o# atos
sliding +ast anot*er $#ig ;elow) T*is reuires stresses o# t*e order o# 2Pa $calculation in t*e ne"t slide)
But ty+ically crystals yield at stresses 1Pa
⇒ T*is i+lies t*at =soet*ing> ust ;e wea/ening t*e drastically It was +ostulated in %&9sK and con#ired ;y TE1 o;ser?ations in %&9s! t*at
t*e agent res+onsi;le #or t*is wea/ening are dislocations
E?en i# one does a +ure unia"ial tension test wit* t*e tension a"is along t*e G a"is! e"ce+t#or t*e *oriGontal and t*e ?ertical +lanes all ot*er +lanes =#eel> s*ear stresses on t*e
As to *ow t*is atoic sli+ is connected to acrosco+ic +eranent s*a+e c*anges will ;e
considered later K By Taylor! Orowan and Polyani
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Plastic de#oration o# a crystal ;y s*ear
S
*
e
a
r
i
n
g
s
t
r
e
s
s
$
)
τ
D i s + l a c e e n t
Sinusoidal relations*i+
5ealistic cur?e
τm
Let us consider t*e s*earing o# an entire +lane o# atos o?er one anot*er@ causing +lastic de#oration ;y s*ear
+tarting configuration Final configuration
Entire row of atoms sliding past another
The shear stressdisplacement curve
loo"s as shown in the
diagram on the right
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=
b
,+inm
π τ τ
8As a #irst a++ro"iation t*e stressdis+laceent cur?e can ;e written as
a ,33 == γ τ At sall ?alues o# dis+laceent
Hoo/e>s law s*ould a++ly
=
b
,m
π τ τ
8⇒ Mor sall ?alues o# "-;
a
b3m
π τ
8=Hence t*e a"iu s*ear
stress at w*ic* sli+ s*ould occur
8m
3τ
π :I# ; a
8m
, ,3
a b
π τ
= ÷
8m
3τ
π :
T*e t*eoretical s*ear stress will ;e int*e range 9 GPa~
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8m3
τ π :
T*e s*ear odulus o# etals is in t*e range 89 %9 2Pa
D!S"#CA$!#%S
Actual s*ear stress is 9. %9 Pa (e,perimentally determined)
I.e. $S*ear stress)t*eoretical %99 × $S*ear stress)e"+eriental
Dislocations severely weaken the crystal
F*is/ers o# etals (single crystal free of dislocations$ Radius 45− 6
m) can approach theoretical shear strengthsF*is/ers o# Sn can *a?e a yield strengt* in s*ear %9 −8 2 (457 times bul" +n)
T*e t*eoretical s*ear stress will ;e in
t*e range 9 GPa~
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As we *a?e seen ;e#ore dislocations can +lay di?erse /inds o# role in aterialsstructure and its ;e*a?iour
Per*a+s t*e ost i+ortant o# t*ese is t*e wea/ening o# t*e crystal in t*e +resence o#dislocations
Mro a slide ;e#ore we /now t*e +at* to understanding t*e role o# dislocations inaterials in?ol?es t*eir interactions wit* ot*er dislocations and de#ects +resent in t*eaterial $and t*e e?olution o# t*e syste wit* tie-de#oration)8 This path will include the 9hardening: of the crystal$ i#e strengthening of the
wea"ened crystal
8 In this conte,t it will be noted that many dislocations will interact with each other
and there will be a strengthening effect
As late as %&9 t*e reason ;e*ind t*is wea/ening o# t*e crystal was not clear (to imagine that this was the post Relativity$ post ;uantum
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T*e analogy usually gi?en to understand t*e role o# dislocations in wea/ening a crystal is t*eone o# =+ulling a car+et>.
I# one tries to +ull an entire car+et $a long and wide one)! ;y sliding it against t*e #loor! t*ee##ort reuired is large.
Howe?er! i# a =;u+> is ade in t*e car+et $as in t*e #igure in t*e #ollowing slide) and t*is ;u+ is o?ed
across t*e lengt* o# t*e car+et! t*en t*e car+et o?es #orward ;y a sall distance $as +ro?ided ;y t*e ;u+).
T*e #orce reuired to o?e t*e ;u+ will ;e considera;ly sall as co+ared to t*e #orcereuired to +ull t*e entire car+et.
By creating and o?ing a series o# ;u+s successi?ely t*e car+et can ;e o?ed #orward =;it ;y ;it>. (3raphic on ne,t slide)#
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T * e = c a
r + e t . + u
l l i n g > a
n a l o g y
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T*e olterra Dislocation
T*e continuu conce+t o# a dislocation $and ot*er de#ects) was +ro+osed ;y ito olterra in %&9.
His ideas and calculations ;ased on *is ideas +redate t*eir a++lication to crystals. Howe?er!continuu calculations ;ased on olterra>s idea are used e?en today to understand t*e ;e*a?iour o#
dislocations in crystals. Continuu calculations o# stress #ields! dis+laceent #ields etc. related to dislocations are #ound to ;e
?alid to wit*in a #ew atoic s+acing $i.e. t*e continuu descri+tion #ails only wit*in a;out atoicdiaeters-Burgers ?ector).
In t*is c*a+ter t*e stress #ields o# dislocations s*own are ;ased on elastic continuu t*eories.
Continuu i+lies t*at we are not =worried> a;out atos (Antonym → Discretum)# T*is is a =strange> as+ect, we *a?e used elasticity t*eory to calculate t*e stress #ields o# t*e ?ery $ost i+ortant) agent res+onsi;le #or
+lastic de#oration
=ontinuum description of a dislocation
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Deformations of a hollow cylinder showing the formation of various defects (>olterra constructions)
erfect cylinder
+crew dislocation
Edge dislocations
D i s l o c a t i o n s D
i s c l i n a t i o
n s
Disclinations
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I# one loo/s at a sa+le o# Aluinu under a TE1! one usually #inds cur?ed dislocationlines ⇒ :sually dislocations *a?e a i"ed c*aracter and Edge and +crew dislocations aret*e ideal e,tremes.
Not only t*is! t*e c*aracter o# t*e dislocation $i.e t*e +ercentage o# screw and +ercentageo# edge c*aracter) will c*ange #ro +osition to +osition along t*e dislocation line.
Howe?er! under s+ecial circustances Pure Edge! Pure Screw or a 1i"ed Dislocationwit* a #i"ed +ercentage o# edge c*aracter can #or.(e#g# in 3e+i epita,ial films on +i substrate 65° misfit dislocations form* i#e# the dislocation lines are straight with
the angle between b and t being 65° )
8 more about these aspects will be considered later
T*e edge dislocation is easier to ?isualiGe and *ence any o# t*e conce+ts regardingdislocations will ;e illustrated using t*e e"a+le o# t*e +ure edge dislocation.
DG
D!S"#CA$!#%S
!'D SC()
Dislocation linesTE< micrograph
showing dislocation lines
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Slipped
part
of the
crystal
Unslipped
part
of the
crystal
Dislocation can ;e consideredas a ;oundary ;etween t*esli++ed and t*e unsli++ed
+arts o# t*e crystal lying o?era sli+ +lane
- this is ?ust a way of visuali@ation and often the slipped and unslipped regions may not be distinguished
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A dislocation *as associated wit* it two ?ectors,
linendislocatiot*ealongnt ?ectorunit tangeAt →
?ector BurgersT*e ; →
T*e Burgers ?ector is li/e t*e =SO:L o# a dislocation>. It =can ;e de#ined> e?en i#
t*ere is no dislocation in t*e crystal $it is t*e s*ortest lattice translation ?ector #or a
#ull-+er#ect dislocation)! it de#ines e?ery as+ect o# t*e dislocation $its stress #ields! energy!
etc.) and e"+resses itsel# e?en in t*e =deat*> o# t*e dislocation $i.e. w*en t*e dislocation
lea?es t*e crystal and creates a ste+ o# *eig*t =;>).
Burgers ?ector is t*e s*ortest translation ?ector $#or a #ull-+er#ect dislocation) and can ;e
deterined ;y t*e Burgers circuit $coing u+).
Hence! Burgers ?ector is an in?ariant #or a crystal! w*ile t*e line ?ector is not. I#
one loo/s at a transission electron icrogra+* s*owing a dislocation line inAluiniu! it will not ;e straig*t− i.e. t*e line ?ector is not #i"ed $usually).
Dislocation linesTE< micrograph
showing dislocation lines
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Burgers *ector dge dislocation
Deterination o# Burgers ?ector in a dislocated crystal using 5ig*t Hand Minis* to Start 5ule $5HMS)
In a +er#ect crystal a/e a circuit $e.g. as in t*e #igure s*own, ' atoic ste+s to rig*t! R
down! ' le#t 6 R u+). T*e circuit is Right anded . Do t*e sae in t*e sae in t*e dislocated crystal. T*e =issing lin/> $using soe con?entionli/e 5HMS) is t*e Burgers ?ector.
5HMS,
5ig*t Hand Minis* to Start con?ention Note: the circuit is drawn away from the dislocation line
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Soe odels o# Edge Dislocation
ideo, Edge Dislocation
1odel using agnetic ;alls
(not that accurate2)
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T*e edge dislocation is NOT t*e =e"tra *al#+lane>! it is neither t*e =issing *al#+lane>@ it is t*e line ;etween t*e =e"tra> and t*e =issing> *al#+lanes.
T*e regions #ar away #ro t*e dislocation line are +er#ect @ all t*e =de#oration> isconcentrated around t*e dislocation line.
Howe?er! t*e stress #ield o# t*e dislocation is a =long range> #ield.
:nderstanding t*e Edge dislocation
Note, T*e Burgers ?ector *as ;een drawn away #or t*e
dislocation line $soeties it ay ;e drawn closeto dislocation line #or con?enience).
T*e dislocation line is ;etween t*e =issing> and=e"tra> *al#+lane.
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dge dislocation
t
;
Dislocation line
O#ten to ?isualiGe t*e edge dislocation! only t*e e"tra =*al#>+lane and sli+ +lane ares*own. T*e reaining crystal is *idden away.
T*e intersection o# t*e e"tra *al#+lane and sli+ +lane can ;e ?isualiGed as t*e dislocation
line $one o# t*e two +ossi;le directions is re+resents t*e line ?ector s*own in ;luecolour).
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Dislocation can ;e considered as t*e ;oundary ;etween t*e sli++ed and t*eunsli++ed +arts o# t*e crystal lying o?er a sli+ +lane.
T*e intersection o# t*e e"tra *al#+lane o# atos wit* t*e sli+ +lanede#ines t*e dislocation line (for an edge dislocation)#
Direction and agnitude o# sli+ is c*aracteriGed ;y t*e Burgers ?ector o# t*e dislocation(A dislocation is born with a Burgers vector and e,presses it even in its death2)#
T*e Burgers ?ector can ;e deterined ;y t*e Burgers Circuit. 5ig*t *and screw $#inis* to start) con?ention is usually used #or deterining t*e
direction o# t*e Burgers ?ector. As t*e +eriodic #orce #ield o# a crystal reuires t*at atos ust o?e
#ro one euili;riu +osition to anot*er ⇒ b ust connect onelattice +osition to anot*er (for a full dislocation)#
Dislocations tend to *a?e as sall a Burgers ?ector as +ossi;le.
Dislocations are noneuili;riu de#ects and would lea?e t*e crystal i# gi?en ano++ortunity.
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Screw dislocation
0otes
The figure shows a Right anded +crew (R+) dislocation (R+ is structurally distinct from &+)#
As for the edge dislocation the Burgers circuit has to be drawn far away from the dislocation line#
Sli+ Plane
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1odel o# Screw Dislocation
T*oug* it is di##icult to understand anyt*ing #ro t*e +*oto o# t*e odel
1i d di l ti Di l i i h i d d d h
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As we *ad noted! e"ce+t in s+ecial circustances! dislocations *a?e i"ed edge andscrew c*aracter.
In a cur?ed dislocation t*e edge and screw c*aracter c*ange #ro +oint to +oint.
Ty+ically in a dislocation loo+ only =+oints> *a?e +ure edge or +ure screw c*aracter Edge, b ⊥ tScrew, b t.
1i"ed dislocations
b
Dislocations with mi,ed edge and screw character
b
t ectors de#ining a dislocation
?eEdge
−?eEdge
5HS
LHSli+ Plane
Red line is the loop
L t id = t > # l
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Pure EdgePure screw
S
E
E,cept for points + and E the remaining portion of the dislocation line has a mi,ed character
Let us consider a =uarter> o# a loo+
Ed d S t t* = l> t t t* ## ti B t
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Edge and Screw co+onents, t*e =usual> way to get t*e e##ecti?e Burgers ?ector
T*e b ?ector is resol?ed into co+onents, = +arallel to t> @ screw co+onent and= +er+endicular to t> @ edge co+onent
Co+onents o# t*ei"ed dislocation at P
Screw Co+onent
Edge co+onent $ )b +in θ
$ )b =os θ
Edge component
+crew component
Ed d S t di## t t i li t* i t ti # t* ## ti * l# l
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Edge and Screw co+onents, di##erent way to ?isualiGe t*e orientation o# t*e e##ecti?e *al#+lane
Instead o# resol?ing t*e b ?ector i# t*e t ?ector is resol?ed to #ind t*e edge and screw co+onents
Mor an edge dislocation t*e e"tra *al#+lane contains t*e t ?ector @ ;y resol?ing t*e t ?ector t*e edge co+onent o# t*e t ?ector t+Sinθ lies in t*e e##ecti?eU *al#+lane $Migure ;elow)
-0ote For a mi,ed dislocation there is no distinct 9half*plane:
Edge and Screw co+onents t*e e##ecti?e *al#+lane, a =crude> anology to understand t*e orientation o# t*e e"tra *al# +lane
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Edge and Screw co+onents t*e e##ecti?e *al# +lane, a crude anology to understand t*e orientation o# t*e e"tra *al# +lane
Assue water is #lowing #ro le#t to rig*t onto a rigid cur?ed wall $in red colour ;elow). T*e green +ortiono# t*e wall =#eels> only s*ear stresses! w*ile t*e aroon +ortion #eels only noral stresses $o# agnitude ;).
A +oint 1 $in t*e cur?ed segent) #eels ;ot* noral and s*ear stresses. T*e e##ecti?e +art w*ic* #eelsnoral stresses is oriented ?ertically wit* agnitude ;Sinθ.
M
1otion o# Dislocations
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Two /inds o# otion o# a dislocation are +ossi;le, 2lide and Cli;. Mirst we consider glide otion. Dislocations may o?e under an e"ternally a++lied #orce $resulting in stress
inside t*e aterial o#ten casually re#erred to =a++lied stress>). At t*e local le?el s*ear stresses on t*e sli+ +lane can only dri?e dislocations. T*e iniu stress reuired to o?e a dislocation is called t*e PeierlsNa;arro
$PN) stress or t*e Peierls stress or t*e Lattice Mriction stress (i#e the e,ternally 9applied stress: may even be purely tensile but on the slip plane shear stresses must act in order to move the dislocation)#
Dislocations ay also o?e under t*e in#luence o# ot*er internal stress #ields (e#g#those from other dislocations$ precipitates$ those generated by phase transformations etc#)# Dislocations are attracted to #reesur#aces $and inter#aces wit* so#ter aterials)
and ay o?e ;ecause o# t*is attraction @ t*is #orce is called t*e Iage Morce. In any case the eierls stress must be e,ceeded for the dislocation to move#
T*e ?alue o# t*e Peierls stress is di##erent #or t*e edge and t*e screw dislocations. T*e #irst ste+ o# +lastic de#oration can ;e considered as t*e ste+ created w*ent*e dislocation o?es and lea?es t*e crystal.@ One sall ste+ #or t*e dislocation! ;ut a giant lea+ #or +lasticityU.
F*en t*e dislocation lea?es t*e crystal a ste+ o# *eig*t =;> is created @ wit* it
all t*e stress and energy stored in t*e crystal due to t*e dislocation is relie?ed.
1otion o# Dislocations
Clic/ *ere to /now ore a;out Peierls StressClic/ *ere to /now ore a;out Peierls Stress
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otion of dge
dislocation
Conser?ati?e(3lide)
Nonconser?ati?e(=limb-)
Mor edge dislocation, as b ⊥ t @ t*ey de#ine a +lane @ the slip plane# Cli; in?ol?es addition or su;traction o# a row o# atos ;elow t*e *al# +lane
V ?e cli; W cli; u+ @ reo?al o# a +lane o# atos
V −?e cli; W cli; down @ addition o# a +lane o# atos. I+ortance o# cli;, cli; +lays an i+ortant role in any ways.
I# an edge dislocation is =stuc/> at soe o;stacle on a particular slip plane! t*en it cancontinue to glide i# cli; can cli; =a;o?e> t*at sli+ +lane. T*is way cli; +lays ani+ortant role in #acilitating continued sli+. acancy concentration in t*e crystal can decrease due to ?e cli;.
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Edge Dislocation 2lide
Sur#ace ste+(atomic dimensions)
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1otion o# ascrew
dislocationleading to aste+ o# b
2ra+*ics ideo, 1otion o# Screw Dislocation2ra+*ics ideo, 1otion o# Screw Dislocation
0ote +chematic diagrams
F* t* di l ti l t* t l t*
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Sur#ace ste+
X F*en t*e dislocation lea?es t*e crystal! t*estress #ield associated wit* it is relie?ed.
X Howe?er! it costs soe energy to create t*ee"tra sur#ace corres+onding to t*e ste+.
Are t*ese ste+s ?isi;leY
T*ese ste+s ;eing o# atoic diensions are not ?isi;le in o+tical icrosco+es. Howe?er! i# anydislocations o+erate on t*e sae sli+ +lane t*en a ste+ o# n; $n %99s%999s) is created w*ic* can e?en ;eseen in an o+tical icrosco+e (called the slip lines).
Sur#ace ste+s $sli+ lines)?isi;le in a Scanning
Electron 1icrogra+* Sli+ lines $w*ic* are crystallogra+*ic ar/ers)=re#lecting across> a twin ;oundary in Cu
Dislocations lea?ing t*e sli+ +lane
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As it was o;ser?ed t*e =#irst ste+> o# +lastic de#oration is t*e otion o# a dislocationlea?ing t*e crystal $or to soe ot*er inter#ace ;ounding t*e crystal) @ leading to t*e#oration o# a ste+.
Mor continued +lastic de#oration it is necessary t*at dislocations continue to o?e andlea?e t*e crystal. Hence! any i+edients to t*e otion o# a dislocation will lead to=*ardening> o# t*e crystal and would =stall> +lastic de#oration (the pinning of adislocation).
Once a dislocation *as ;een +inned it can eit*er =;rea/ down t*e ;arrier> or =;y+ass> t*e ;arrier.
By+assing t*e ;arrier can ta/e +lace ;y ec*aniss li/e, Cli; Cross Sli+ Mran/5ead ec*anis Z.
In cli; and cross sli+ t*e dislocation lea?es-c*anges its =current> sli+ +lane and o?es toanot*er sli+ +lane t*us a?oiding t*e ;arrier
Howe?er! t*ese +rocesses $cli; and cross sli+) can occur inde+endent o# t*e +inning o#
t*e dislocation
Dislocations lea?ing t*e sli+ +lane
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Dislocation lea?ing-c*angingt*e sli+ +lane
Screw dislocation
Edge dislocation Cli;
Cross Sli+
0on*conservative-
involves mass transport
=onservative
-=onservative climb is also possible22 8 by motion of prismatic edge loop on the slip plane
In climb an edge dislocation moves to an ad?acent parallel plane$ but in cross slip a screw dislocation
moves to a plane inclined to the original plane#
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Climb of dge Dislocation
Positi?e cli; Removal of a row of atoms
Negati?e cli; Addition of a row of atoms
5eo?al o# a row o# atos leads to a decrease in ?acancy concentration in t*e crystal and negati?e cli;leads to an increase in ?acancy concentration in t*e crystal.
Screw dislocation, Cross Sli+
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Screw dislocation, Cross Sli+
T*e dislocation is s*own crosssli++ing #ro t*e ;lue +lane to t*e green +lane
Let t*e dislocation ;e o?ing on SP% $as t*e resol?ed s*ear stress is a"iu on Sli+Plane% $SP%)).
T*e #igures ;elow s*ow t*e cross sli+ o# a screw dislocation line #ro SP% to Sli+ +lane8
$SP8). T*is ay occur i# t*e dislocation is =+inned> in sli+ +lane%. Mor suc* a +rocess to occur t*e 5esol?ed S*ear Stress on SP8 s*ould ;e at least greatert*an t*e Peierls stress(often stresses higher than the eierls stress has to be overcome due to the presence of other stress fields)#
It is to ;e noted t*at SP% 6 SP8 are $usually) crystallogra+*ically eui?alent! i.e. i# SP% is$%%%)CCP Crystal t*en SP8 can ;e $%%%)CCP Crystal.
How does +lastic de#oration ;y sli+
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How does +lastic de#oration ;y sli+occurY
Munda C*ec/
T*e #irst ste+ o# +lastic de#oration ;y sli+ $at t*e #undaental le?el) is t*e otion o# a
dislocation lea?ing t*e crystal.
By e"ternally a++lied #orce $or soe ot*er eans) stress *as to ;e =generated> wit*in t*ecrystal.
T*e sli+ +lane s*ould #eel s*ear stresses.
T*e s*ear stress s*ould e"ceed t*e =Critical 5esol?ed S*ear Stress $C5SS)> or Peierls
stress. T*e dislocation s*ould lea?e t*e crystal creating a sur#ace ste+ o# *eig*t =;>.
T*e +rocess a*ead o# t*is w*ic* leads to an ar;itrary s*a+e c*ange is co+licated and we will
deal wit* a +art o# it later.
F*ere can a dislocation line endY
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Dislocation line cannot end inside t*e crystal (abruptly) T*e dislocation line,
• Ends on a #ree sur#ace o# t*e crystal
• Ends on an internal sur#ace or inter#ace• Closes on itsel# to #or a loo+• Ends in a node
A node is t*e intersection +oint o# ore t*an two dislocations T*e ?ectoral su o# t*e Burgers ?ectors o# dislocations eeting at a
node W 9
F*ere can a dislocation line endY
F*at a;out t*e introduction o# a uarter +lane o# atos doesn>t t*e dislocation line end inside t*e crystalY
As seen in t*e #igure ;elow t*ere are two sections to t*e dislocation line ending on #ree sur#ace o# t*e crystal and *encenot inside t*e crystal.
Munda C*ec/
Positi?e and Negati?e dislocations
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As we *a?e seen w*en t*ere are two are ore ED2E dislocations in a sli+ +laneone o# t*e is assigned a ?e sign and t*e ot*er one a −?e sign (done arbitrarily)
In t*e case o# screw dislocations t*e 5ig*t Handed Screw $5HS) Dislocation is+tructurally Distinct #ro t*e Le#t Handed Screw $LHS) Dislocations In t*e case o# 5HS dislocation as a cloc/wise circuit $Burgers) is drawn t*ena *elical +at* leads into t*e +lane o# t*e
Positi?e and Negati?e dislocations
Energy o# dislocations
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Energy o# dislocations
T*e +resence o# a dislocation distorts t*e ;onds and costs energy to t*e crystal. Hence!dislocations *a?e distortion energy associated wit* t*e
T*e energy is e"+ressed as Energy +er unit lengt* o# dislocation line
@ :nits, [J-\ Edge @ Co+ressi?e and tensile stress #ields
Screw @ S*ear stress #ields T*e energy o# a dislocation can a++ro"iately ;e calculated #ro linear elastic t*eory.
T*e distortions are ?ery large near t*e dislocation line and t*e linear elastic descri+tion#ails in t*is region @ called t*e Core o# t*e dislocation (estimates of this region range from b to Cb depending on the crystal in /uestion). T*e structure and energy o# t*e core*as to ;e co+uted t*roug* ot*er et*ods and t*e energy o# t*e core is a;out %-%9 t*etotal energy o# t*e dislocation.
T*e #orula gi?en ;elow gi?es reasona;le a++ro"iation o# t*e dislocation energy.
Energy o# dislocation Elastic Nonelastic $Core)E ~E/10
8%I
8
d E 3bElastic Energy o# a dislocation - unit lengt* 2 @ $µ) s*ear odulus
; @ ,b,
As it costs energy to +ut a dislocation in a crystal,
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⇒ Dislocations will *a?e as sall a b as +ossi;le
Dislocations(in terms of lattice translation)
Mull
Partial
b @ Mull lattice translation
b @ Mraction o# lattice translation
gy + y Dislocations tend to *a?e as sall a b as +ossi;le T*ere is a line tension associated wit* t*e dislocation line Dislocations ay dissociate into Partial Dislocations to reduce t*eir energy
8%I8
d E 3b
8
9 8 ln( $% )
edge
d 3b E
bγ
π ν + ÷ −
Anot*er #orula #or t*e energy $Edge dislocation)
γ 9 siGe o# t*e control ?olue R9;
8
9 8 ln(
screw
d
3b E
b
γ
π
+ ÷
Core contri;ution
Dissociation o# dislocations
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Dissociation o# dislocations
Consider t*e reaction, 8b @ b b
C*ange in energy,
Initial energy ;e#ore s+litting into +artials, 2$8;)8-8 W 82;8
Energy a#ter s+litting into +artials, 8[2$;)8-8\ W 2$;)8
5eduction in energy W 82;8 2;8 W 2;8.
⇒ T*e reaction would ;e #a?ora;le.
Dislocations ay dissociate to reduce t*eir energy
Note t*at t*is e"a+le is considered #or illustration +ur+ose only (here a full dislocation is not splitting into partials)
Stress Mields o# Dislocations
dge dislocationThe self stresses
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An edge dislocation in an in#inite ;ody *as co+ressi?e stress #ield a;o?e (the region ofthe e,tra half*plane) and tensile stress #ield ;elow (the region of the missing half*plane)t*e sli+ +lane
T*ese stress #ields will ;e altered in a #inite ;ody Asyetric +osition o# t*e dislocation in t*e crystal will also alter t*e stress #ield
descri;ed ;y t*e standard euations (as listed below) T*e core region is ignored in t*ese euations (which hence have a singularity at , 5$ y 5)
$Core ;eing t*e region w*ere t*e linear t*eory o# elasticity #ails)
O;?iously a real aterial cannot ;ear suc* =singular> stresses
T*e interaction o# t*e stress #ields o# t*e dislocations wit*, $i) t*ose originating #roe"ternally a++lied #orces and $ii) ot*er internal stress #ields @ deterines t*e otion o#dislocation @ leading to any as+ects o# ec*anical ;e*a?iour o# aterials
Stress Mields o# Dislocations
1ore a;out stress #ields o# dislocations1ore a;out stress #ields o# dislocations
8 8
8 8 8
$ )
8 $% ) $ )
+=
− + ,,
3b y , y
, y
σ
π ν 8 8
yy 8 8 8
$ ) 8 $% ) $ )
− −=
− +3b y , y
, yσ
π ν
8 8
"y 8 8 8
$ ) W
8 $% ) $ ) ,y
3b , , y
, yσ τ
π ν
− −=
− +
stress #ields
The material is considered isotropic (two elasticconstants only* E . ν or 3 . ν )
8 in reality crystals are anisotropic w#r#t to the
elastic properties
gThe self stresses
dge dislocation
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Note t*at t*e region near t*e dislocation *as stresses o# t*e order o# 2Pa
T*ese stresses are t*e sel# stresses o# t*e dislocation and a straig*t dislocation line cannoto?e under t*e action o# sel# stresses alone (in an infinite body)
A dislocation interacts with other defects in the material via these 9long range: stress fields
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Note t*at t*e stresses near t*e dislocation line reac*es ?alues in 2Pa. 3ield stressis usually in %99s o# 1Pa. How does a aterial not yield under t*e e##ect o# suc**ig* stressesY
Munda C*ec/
T*e +redoinant ec*anis o# +lastic de#oration is sli+! w*ic* in?ol?es otion o# a
large nu;er o# dislocations $ultiately lea?ing inter#aces). T*e stresses descri;ed *ere are due to t*e =dislocation itsel#> $t*e causati?e agent #or +lastic
de#oration ;y sli+) and t*ey are t*e elastic stress fields associated wit* t*e dislocation.
Interaction ;etween dislocations dge dislocation
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Here we only consider elastic interactions ;etween edge dislocations on t*e sae sli+ +lane.
T*is can lead to Attracti?e and 5e+ulsi?e interactions.
To understand t*ese interaction we need to consider Positi?e and Negati?e edgedislocations. I# a single edge dislocation is +resent in a aterial it can ;e called eit*er +ositi?e or negati?e. I# two $or ore) dislocations are +resent on t*e sae sli+ +lane! wit*t*e e"tra *al#+lane on two di##erent sides o# t*e sli+ +lane! t*en one o# t*e is +ositi?eand t*e ot*er negati?e.
T*e +icture $region o# attraction and re+ulsion) gets a little =detailed> i# t*e twodislocations are ar;itrarily oriented. [See Stress Mields o# Dislocations\
Interaction ;etween dislocations
Positi?e edge dislocation Negati?e edge dislocation
ATT5ACTION Can coe toget*er and cancel one anot*er
5EP:LSION
Dislocations in CCP Crystals
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Sli+ syste @ ^%%9! _%%%` Per#ect dislocations can s+lit into +artials (+hoc"ley partials considered first) to reduce
t*eir energy
T*e dissociation into +artials lea?es a Stac/ing Mault ;etween t*e two +artials on t*e sli+ +lane
T*e two +artials re+el eac* ot*er and want to ;e as #ar as +ossi;le @ ;ut t*is leads to alarger #aulted area $leading to an increase in energy) @ de+ending on t*e stac/ing #aultenergy t*ere will ;e an euili;riu se+aration ;etween t*e +artials
T*e S*oc/ley +artial *as Burgers ?ector o# t*e ty+e, $%-4) [8%%\ ty+e.T*is is an i+ortant ?ector in t*e CCP crystals! as ?ectors o# t*is #aily connect B site to C site and ?ice?ersa.
Mor a +ure edge dislocation in a CCP crystal t*e =e"tra *al#+lane> consists o# two atoic +lanes. T*e +artial dislocations consist o# one =e"tra> atoic +lane eac* $;ut t*e Burgers?ector o# t*e +artial is not +er+endicular to t*e dislocation line as in t*e case o# t*e +er#ect edge dislocation).
Dislocations in CCP Crystals
- ill be considered in the topic on GD defects
Figures in coming slides
CCP
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8
%8%%
4b =
%%8%
4b =
%
%%%9
8b =
$%%%)
$%%%)
+lip plane
$%%%)
%[%%9\
8
÷ $%%%)
%[%8%\
4
÷
$%%%)
%[8%%\
4
÷ → +
;%8 $;8
8 ;8)
b
CCP
S*oc/ley Partials
$%%%)
+ome of the atoms are omitted for clarity$ Full vectors (blue vector) . (green vector) shown in the left figure
$%%%)
%[%8%\
4 ÷
$%%%)
%[8%%\
4
÷
8 88 8 8
8
% 8 % 4 %S S
4 4 4b
+ += = = ÷ ÷ ÷ ÷
r
$;88 ;8) W %-4 %-4 W %-
Energy of the dislocation is proportional to bG#
As the energy of the system is reduced on dissociation into partials the perfect dislocation will split into two partials#
Per#ect EdgeDislocation andS*oc/ley Partials
8%% %8%
Pure edge dislocation CCP
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The e,tra* Hhalf plane consists of two 9planes: of atoms2
S* /l P ti l
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S*oc/ley Partials
erfect edge dislocation (9full: Burgers
vector) with two atomic 9e,tra*half: planes
artial dislocations each with one atomic
9e,tra*half: plane
Clic/ *ere to see *ow two dislocations gi?e rise to one dislocationClic/ *ere to see *ow two dislocations gi?e rise to one dislocation
Here one dislocation s+lits into two +artials
Dislocation loo+s and Mran/ Partial dislocations
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Dislocation loo+s and Mran/ Partial dislocations
Mored ;y insertion or reo?al o# a disc o# atos #ro t*e $%%%) +lane. T*e $%%%) crystal +lane consists o# t*ree atoic +lanes and t*e lattice translation ?ector
along [%%%\ *as a agnitude o# √ $t*e distance ;etween t*e atoic +lanes along [%%%\ is
√-). T*e +ac/ing along t*is direction is, ABCABCABCZ 5eo?al o# a disc @ Intrinsic #ault
Insertion o# a disc @ E"trinsic #ault bMran/ Partial W [%%%\-
As b is not on a sli+ +lane $a e;er o# t*e _%%%` #aily) t*e dislocation cannot o?e
conser?ati?ely $i.e. wit*out ass trans+ort) @ is a Sessile Dislocation $as o++osed to a2lissile dislocation $w*ic* can o?e! e.g. t*e S*oc/ley +artials)). E"cess ?acancies $uenc*edin or #ored ;y irradiation) can #or an intrinsic #ault $t*ese
ay *a?e *e"agonal s*a+e in soe cases). T*is s*ows t*at a dislocation loo+ can *a?e a co+letely edge c*aracter ;ut ne?er a
co+letely screw c*aracter.
Fran" partial loop bounding a stac"ing fault in == crystal
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Intrinsic
E"trinsic
Stac/ing #aults
Two ;rea/s introduced intot*e stac/ing seuence
Per#ect region
Maulted region
Mran/ +artial loo+s wit* bW%- [%%%\
J444K
These lines are pro?ection of (444) planes
BCC P d di l ti%
[%%8\ ÷ Di l ti li t
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BCC Pure edge dislocation
$%%9)
% [% % %\8
÷
$%%%)$%%9)!
[%%8\8
÷
$%%%)
$%%9)
Sli+ +lane
E"tra *al# +lane
Burger>s ?ector
Dislocation line ?ector
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Moration o# dislocations $in t*e ;ul/ o# t*e crystal)
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Due to accidents in crystal growt* #ro t*e elt. 1ec*anical de#oration o# t*e crystal. Nucleation o# dislocation.
Hoogenous nucleation o# a dislocation reuired *ig* stresses $2-%9). Stress concentrators in t*e crystal can aid t*e +rocess.
Dislocation density increases due to +lastic de#oration ainly ;y ulti+lication o# +ree"isting dislocations.
Dislocation density re#ers to t*e lengt* o# dislocation lines in a ?olue o#aterial @ *ence t*e units are [-\ (it is better not to cancel the 9m: in the numerator and the denominator and write as 'mG as the
units m'm7
is more physical2)# Annealed crystal, dislocation density $ρ) %94 %9%9 -. Cold wor/ed crystal, ρ %9%8 %9%( -. As t*e dislocation density increases t*e crystal ;ecoes stronger $ore a;out t*is later).
Ty+ical ?alues o# Dislocation Density
0ote in this conte,t it is noteworthy that screw dislocations can actually play a role in crystal
growth 8 =onstructive role of dislocations
Burgers ?ectors o# dislocations in cu;ic crystals
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1onoatoic MCC ^%%91onoatoic BCC ^%%%
1onoatoic SC ^%99
NaCl ty+e structure ^%%9
CsCl ty+e structure ^%99
DC ty+e structure ^%%9
Crystallography determines the Burgers 3ector
fundamental lattice translational vector lying on the slip plane
“Close packed volumes tend to remain close packed,close packed areas tend to remain close packed &close packed lines tend to remain close packed”
A rule of thumb can be evolved as follows
=lose pac"ed in this conte,t implies 9better bonded:
Sli+ systes
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Crystal Slip plane4s5 Slip direction
MCC _%%%` ^%%9
HCP $999%) ^%% 89
BCC 0ot close pac"ed
_%%9`! _%%8`! _%8` ^%%%
0o clear choice of slip plane
⇒ Fa?y sli+ linesAnisotro+ic sli+ ;e*a?iour
A co;ination o# a sli+ direction $b) lying on a sli+ +lane is called a sli+ syste. T*is isdescri;ed in ters o# a #aily o# directions and a #aily o# +lanes.
In close +ac/ed crystals it is a close +ac/ed direction lying on a close +ac/ed +lane.
In BCC crystals t*ere are any +lanes wit* siilar +lanar atoic density @ t*ere is noclear c*oice o# sli+ +lane.
T*ere ig*t ;e ore t*an one active sli+ syste in soe crystals (e#g# B== crystals below)#8 the active slip system gives rise to plastic deformation by slip#
E?en i# t*ere is only one sli+ syste is acti?e at low te+erature! ore sli+ systes ay ;ecoe acti?e at *ig* te+eratures @ +olycrystalline aterials w*ic* are ;rittle at roo
te+erature ay ;ecoe ductile at *ig* te+eratures.
Jogs and 7in/s Defect in a defect2
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A straig*t dislocation line can *a?e a ;rea/ in it o# two ty+es, A
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Edge dislocation Screw Dislocation
Jog Edge c*aracter Edge c*aracter 7in/ Screw C*aracter Edge c*aracter
Jogs and 7in/s, C*aracter Ta;le
Jogs and 7in/s in a screw dislocation will *a?e edge c*aracter.
Jog in a Edge dislocation *as Edge c*aracter and 7in/ in a edge dislocation *as screwc*aracter.
Jogs
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T*e +resence o# a
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Two straig*t dislocation can intersect to lea?e Jogs and 7in/s in t*e dislocation line T*ese e"tra segents in a dislocation line cost energy and *ence reuire wor/ done ;y t*e
e"ternal #orce ⇒ lead to *ardening o# t*e aterial
(Additional stress as compared to the stress re/uired to glide the dislocation line isre/uired to form the Oog'Pin")
Mour ty+es o# interactions are considered ne"t.
EdgeEdge Intersection% erpendicular Burgers vector
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T*e
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Bot* dislocations are /in/ed Edge Dislocation% (Burgers vector b% ) → 7in/ $Screw c*aracter) → Lengt* b8 Edge Dislocation8 (Burgers vector b8 ) → 7in/ $Screw c*aracter) → Lengt* b% T*e /in/s can glide
EdgeScrew Intersection erpendicular Burgers vector
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Edge Dislocation (Burgers vector b% ) → Jog $Edge C*aracter) → Lengt* b8 Screw Dislocation (Burgers vector b8 ) → 7in/ $Edge C*aracter) → Lengt* b%
Screw Screw Intersection( erpendicular Burgers vector
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I+ortant #ro +lastic de#oration +oint o# ?iew Screw Dislocation (Burgers vector b% ) → Jog $Edge C*aracter) → Lengt* b8 Screw Dislocation (Burgers vector b8 ) → Jog $Edge C*aracter) → Lengt* b/ Bot* t*e
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T*e stress #ield o# a dislocation can interact wit* t*e stress #ield o# +oint de#ects. De#ects associated wit* tensile stress #ields are attracted towards t*e co+ressi?e region
o# t*e stress #ield o# an edge dislocation (and vice versa)#
Solute atos can segregate in t*e core region o# t*e edge dislocation (formation of the=ottrell atmosphere) 8 *ig*er stress is now reuired to o?e t*e dislocation (the system is in alow energy state after the segregation and higher stress is re/uired to 9pull: the dislocation out of the energy well)#
Hig*er #ree?olue at t*e core o# t*e edge dislocation aids t*is segregation +rocess. De#ects associated wit* s*ear stress #ields $*a?ing a nons+*erical distortion #ield e.g.
interstitial car;on atos in BCC Me) can interact wit* t*e stress #ield o# a screw
dislocation.
acancies are attracted to t*e co+ressi?e regions o# an edge dislocation and are re+elled#ro tensile regions
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+tress values in 3a
σ""
Position o# t*e Dislocation line → into t*e +lane
Tensile Stresses
Co+ressi?e Stresses
5 stress lineacancies $
)
A t t r a c t e d
R e p e l l e
d
0o interaction
#ro tensile regions T*e ;e*a?iour o# su;stitutional atos saller t*an t*e +arent atos is siilar to t*at o#
t*e ?acancies Larger su;stitutional atos are attracted to t*e tensile region o# t*e edge dislocation and
are re+elled #ro t*e co+ressi?e regions Interstitial atos $associated wit* co+ressi?e stress #ields) are attracted towards t*e
tensile region o# t*e edge dislocation and are re+elled #ro t*e co+ressi?e region o# t*estress #ield
Suary o# edge dislocation +oint de#ect interactions
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Point De#ect Tensile 5egion Co+ressi?e 5egion
acancy 5e+elled Attracted
Interstitial Attracted 5e+elled
Saller su;stitutional ato 5e+elled Attracted
Larger Su;stitutional atos Attracted 5e+elled
3ield Point P*enoenon
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3ield Point P*enoenon
+chematic
T*e interaction o# interstitial car;on atos wit* edge dislocations (interaction of stress fields of dislocations with solute atoms) @ leading to t*eir segregation to t*e core o# t*eedge dislocations is res+onsi;le #or t*e 3ield Point P*enoenon seen in t*e tensile test o#ild steel s+eciens
Interstitial Atom at the core
1odel ade o# agnetic ;alls(core segregation)
1ore a;out t*is in t*e c*a+ter on +lasticity
DislocationPreci+itate Interactions
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Dislocations can interact wit* t*e stress #ields o# +reci+itates. 1o?ing dislocations can,
A glide t*roug* co*erent +reci+itates @ s*earing t*e +reci+itate.B ;ow around inco*erent +reci+itates! lea?ing loo+s as t*ey ;y+ass two +reci+itatesw*ic* act li/e +inning centres #or t*e dislocations (8 leading to an increase in thedislocation density)#
Bot* t*ese +rocesses need an a++lication o# *ig*er stresses $assuing an *arder +reci+itate) @ lead to t*e strengt*ening o# t*e aterial.
Seico*erent +reci+itates *a?e inter#acial is#it dislocations w*ic* +artially relie?e t*e
co*erency strains. (These dislocations are structural dislocations)#Co*erent +reci+itate note t*at t*e lattice
+lanes are continuous across t*e +reci+itate
- Though the word coherent is used as an ad?ective for the precipitate* actually what is meant is that the interface iscoherent (or semi*coherent if we tal" about a semi*coherent precipitate)
SeiCo*erent inter#ace
Clic/ *ere to /now ore a;out inter#acesClic/ *ere to /now ore a;out inter#aces
A 2lide t*roug* co*erent +reci+itates
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Preci+itate +articleb
;
Double Ended Mran/5ead SourceB Bow around inco*erent +reci+itates
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A dislocation can ;e +inned ;etween two inco*erent +reci+itate +articles $or in ot*erways)! t*us *indering t*e otion o# t*e dislocation.
Mor otion o# t*e dislocation! leading to +lasticity t*e dislocation *as to ;y+ass t*e +inned
segent! under t*e action o# t*e a++lied stress $s*ear stress on t*e sli+ +lane dri?es t*eotion). T*e dislocation ta/es a series o# con#igurations $as s*own t*e in t*e #igures) under t*e
action o# t*e a++lied stress @ leading to t*e #oration o# a dislocation loo+ (leaving theoriginal pinned segment)#
T*is leads to an increase in dislocation density (one of the mechanisms by which
dislocation density increases with plastic deformation)# As t*e original segent is retained t*e =source> $Mran/5ead source) can o+erate
re+eatedly #oring a loo+ eac* tie. As t*e +ree"isting loo+s would o++ose t*e #oration o# t*e ne"t loo+ $re+ulsi?e stresses
dislocations o# t*e sae sign)! *ig*er stresses are reuired to o+erate t*e source eac* tie. Till t*e #oration o# t*e *al#loo+ $seicircle)! increasingly *ig*er stresses are reuired.
A#ter t*is t*e +rocess occurs down*ill in stresses. T*e a"iu stress $τa") reuired to o+erate t*e source t*us corres+onds to t*e
#oration o# t*e *al#loo+ $wit* radius r in).
a"
in
I
8
3b
r
τ
Initial con#iguration A B
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Dislocation line segent +innedat A and B ;y +reci+itates
6 Pinning could 4also5 be caused by7
Dislocation in t*e +lane o# t*e +a+er intersects dislocations in ot*er +lanes
Anc*ored ;y i+urity atos or +reci+itate +articles
Dislocation lea?es t*e sli+ +lane at A and B
A++lication o# stress on dislocation segent τ
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A B
Force - τ b
τ
Line tensionBowing
As the dislocation line gets curved the energy of the system increases⇒ wor" has to be done by
e,ternal stresses to cause this e,tension# &ine tension (opposes the shear stress on the slip plane ( τ )# At a given stress there might be a
balances of forces leading to a curved geometry of the dislocation line# Further e,tension of the dislocation line occurs by increasing the stress#
8
8
%I 3bengthenergy ' l n Dislocatio γ =
dsblinendislocatioon Force τ =
θ γ θ
γ d8
dsin8
= forcetension &ine
dsbd τ θ γ =Mor euili;riu in cur?ed con#iguration
r
3b
8τ
τa"
⇒ r in
L%
and − segents coe toget*er and annul eac* ot*er
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τ b
τ b
τ b
I n c r e a s i n g s t r e s s
semicircle8
corresponds to ma,imum stress re/uired
to e,pand the loop
After this decreasing stress is re/uired to e,pand the loop
Direction of dislocation motion
is ⊥ to the dislocation line
(e,cept at A and B)
8
%
(
Original segent
New loo+ created
8
Fran9 (ead dislocation source ;
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Fran9:(ead dislocation source ;
Can o+erate #ro a single source +roducing a loo+ eac* tie
T*is loo+ +roduces a sli+ o# %b eac* tie on t*e sli+ +lane
T*e a"iu ?alue o# s*ear stress reuired is w*en t*e ;ulge ;ecoes a seicircle$r in W L-8) @ τa" 2;-L⇒ τ as L i.e. T*e longest segents o+erate #irstV F*en t*e long segents get io;iliGed s*orter segents o+erate wit* increasing stress
⇒ wor" hardening
I# t*e dislocation loo+s /ee+ +iling u+ on t*e sli+ +lane t*e ;ac/ stress will o++ose t*ea++lied stress
F*en t*e bac"*stress τa" t*e source will cease to o+erate
Dou;le ended M5 sources *a?e ;een o;ser?ed e"+erientally t*ey are not #reuent ⇒ ot*erec*aniss ust e"ist
a" -3b &τ α = α W 9. #or edge dislocations and α W %. #or screw dislocations.
Dislocation Mree sur#ace Interaction @ Conce+t o# Iage Morces
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A dislocation near a #ree sur#ace $in a seiin#inite ;ody) e"+eriences a #orce towards t*e#ree sur#ace! w*ic* is called t*e iage #orce. T*is is a ty+e o# Con#igurational Morce (i#e# force e,perienced when the energy is lowered by a change in configuration of a system)
T*e #orce is called an =iage #orce> as t*e #orce can ;e calculated assuing an negati?e*y+ot*etical dislocation on t*e ot*er side o# t*e sur#ace (figure below)# T*e #orce o#attraction ;etween t*e dislocations $ 6 −) is gi?es t*e iage #orce. T*e aterial +ro+erties are identical t*roug*out.
I# t*e iage #orce e"ceeds t*e Peierls stress t*en t*e dislocation can s+ontaneously lea?et*e crystal! wit*out a++lication o# e"ternal stresses
Hence! regions near a #ree sur#ace and nanocrystals can ;ecoe s+ontaneouslydislocation #ree. In nanocrystals due to t*e +ro"iity o# ore t*an one sur#ace! anyiages *a?e to ;e constructed and t*e net #orce is t*e su+er+osition o# t*ese iage #orces.
A hypothetical negative dislocation is assumed to e,ist across the free*surface for the calculation of the force (attractive)
e,perienced by the dislocation in the pro,imal presence of a free*surface
d 3b F image
)%$(
8
ν π −−=
Doain de#orations in Nanocrystals in t*e +resence o# dislocations
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Dislocation near a #ree sur#ace in a seiin#inite ;ody can de#or t*e sur#ace. T*is is asall de#oration as s*own in t*e #igure ;elow #or t*e case o# an edge dislocation.
In nanocrystals (e#g# the "ind shown in the figure below) t*e doain can ;end-;uc/le in
t*e +resence o# dislocations (the figure shows the effect of an edge dislocation in a plate8 a screw dislocation leads to the twisting of * for e,ample * a cylindrical domain) T*is is elastic de#oration in t*e +resence o# dislocations in a nanoaterial
Hence! we can *a?e reversible plastic deformation due to elasticity222
%9;
Dislocation in a t*in =+late> o#Al leading to its ;ending
5ole o# dislocations in crystal growt* =onstructive role of dislocations2
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Crystals grown under low su+ersaturation $%) t*e growt* rate is considera;ly#aster t*an t*at calculated #or an ideal crystal
In an ideal crystal sur#ace t*e di##iculty in growt* arises due to di##iculty in t*enucleation o# a new onolayer
I# a screw dislocation terinates on t*e sur#ace o# a crystal t*en addition o#atos can ta/e +lace around t*e +oint w*ere t*e screw dislocation intersects t*esur#ace (the step) @ leading to a s+iral $actually s+rial *eli") growt* staircase +attern
2rowt* s+iral on t*e sur#ace o# crystalliGed +ara##in wa"
5ole o# dislocations in +*ase trans#oration
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Dislocations can act as *eterogeneous nucleation sites during +*ase trans#oration. Dislocations ay dictate t*e orientation and or+*ology o# t*e second +*ase.
T*e stain associated wit* t*e dislocation ay ;e +artly relie?ed ;y t*e #oration o# a
second +*ase. T*e strain associated wit* t*e trans#oration $t*e Es*el;y strain) ay ;e accoodated
;y +lastic #low $ediated ;y dislocations).
1ental Picture o# a dislocation
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Let us consider t*e ?arious ways o# understanding t*e dislocation @ T*e di##erent +ers+ecti?es
Association wit* translational syetry
As a line de#ect
Distortion o# ;onds @ region o# *ig* energy
Increase in entro+y o# t*e syste
Mree ?olue at t*e core @ +i+e di##usion
Core o# dislocation 6 its geoetry @ Peierls Stress Stress 6 Strain #ields
Interaction wit* ot*er de#ects
5ole in sli+
5ole as a structural de#ect
Munda C*ec/ F*y are dislocations noneuili;riu de#ectsY
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It is clear #ro t*e a;o?e euation t*at i# a con#iguration gi?es an entro+y ;ene#it $i.e. ∆S is +ositi?e) t*en t*at state will ;e sta;iliGed at soe te+erature$e?en i# t*e ent*al+y cost is ?ery *ig* #or t*at con#iguration)
In t*e +resent case, it costs an energy o# 2;8-8 +er unit lengt* o# dislocation line
introduced into t*e crystal ;ut! t*is gi?es us a con#igurational entro+y ;ene#it $ast*is dislocation can e"ist in any e/uivalent +ositions in t*e crystal)
T*is i+lies t*at t*ere ust ;e te+erature w*ere dislocations can ;ecoe sta;lein t*e crystal $ignoring t*e c*ange in t*e energy cost wit* te+erature #or now)
%nfortunately this temperature is above the melting point of all "nown materials
Hence! dislocations are not sta;le t*erodynaic de#ects in aterials
- Including positional$ electronic$ rotational . vibrational multiplicity of states
V T*e energy reuired to create 7in/s and Jogs o# lengt* =;> is /0@ t*ese can ;e created ;y t*eral #luctuations
∆2 W ∆H − T ∆S ?e #or dislocations
Munda C*ec/ How are crystals wea/enedY T*e two ste+ +rocess
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As we *a?e seen t*e +rocess o# sliding $;y s*ear) o# an =entire +lane> o# atos can ;e reduced toa 9line*wise: +rocess ;y dislocationsV t*is leads to a s*ear stress reduction o# a #ew orders o# agnitude
T*is +ro;le can ;e #urt*er ;ro/en down $in diension and energy) to t*e #oration andigration o# /in/ +airs along t*e dislocation line (usually occurs to screw components of dislocations in B==metals)
V t*is can #urt*er lead to t*e reduction in stress reuired #or dislocation otion
=ontinued1
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V In BCC etals and 2e t*erally assisted #oration o# /in/ +airs can cause sli+ at stresses
τ ^ τPN =ontinued1
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D!S"#CA$!#%S
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5ando Structural
Distinct #ro =2eoetrically Necessary Dislocations> $2ND)
As entioned ;e#ore, Structural de#ects +lay a ?ery di##erent role in aterial ;e*a?iour as co+ared to 5ando Statistical De#ectsU (non*structural)
Structural dislocations ay ;e associated wit* a ;oundary and *ence t*eir role in +lasticity ay ;e ?ery di##erent #ro rando $statistically storedU) dislocations
O#ten a related ter 2eoetrically Necessary Dislocations is used in literature(we have intentionally avoided the use of the term here)
Structural dislocations include, Dislocations at low angle grain ;oundaries $will ;e discussed along wit* ot*er 8D de#ects) @ res+onsi;le #or a tilt or twist
Dislocation at seico*erent inter#aces@ res+onsi;le #or atc*ing is#it $;etween ad
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Tilt
1is#it
Structural dislocations can accoodate Twistbetween two crystals
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T*ere are two distinct uestions we can as/,
f%
I# you already *a?e a dislocation *ow to deterine t*e Burgers ?ectorYf8 F*at deterines t*e Burgers ?ectorY
T*e answer to f% is ;y constructing a Burgers circuit
T*e answer to f8 is, Crystallogra+*y @ t*e Burgers ?ector is t*e s*ortest latticetranslation ?ector $#or a +er#ect-#ull dislocation)
Sol3ed
?ample
In a cu;ic crystal a dislocation line o# i"ed c*aracter lies along t*e [%%8\direction. Burgers ?ector W [%%9\. F*at are t*e edge and screw co+onents o#
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pt*e Burgers ?ectorY F*ic* is t*e sli+ +laneY
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$on [%%9\)
$on [%%8\)
Sol3ed
?ample
In a CCP crystal is t*e dislocation reaction s*own ;elow #easi;le energeticallyYF*at is t*e signi#icance o# t*e ?ectors on t*e 5HS o# t*e reactionY
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p
[ ] [ ]% % %
%%9 8% % %8%8 4 4
→ +
88 8
8
%
% % %S S
8 8b
+= = ÷ ÷
T*is is o# t*e #or ;%
@ ;8
;
T*e dislocation reaction is #easi;le i#,8 8 8
% 8 Jb b b
> + As the energy of a dislocation (per unit length of the dislocation line is proportional to bG
88 8 8
8
8
8 % % %S S
4 4b
+ += = ÷ ÷
% % % %
8 4 4 J
> + = ÷
⇒ t*e dislocation reaction is #easi;le (i#e# the full dislocation can lower its energy by splitting into partials)
88 8 8
8
J
% 8 % %S S
4 4b
+ += = ÷
÷
T*e ?ectors on t*e 5HS lie on t*e $%%%) close +ac/ed +lane in a CCPcrystal and t*ey connect B to C sites and C to B sites res+ecti?ely.Eui?alent ?ectors (belonging to the same family) are s*own in t*e#igure on t*e rig*t.
Sol3ed
?ample
F*at is t*e iage #orce e"+erienced ;y an edge dislocation at a distance o# %99;#ro t*e #ree sur#ace o# an sei in#inite Al crystalY Is t*is #orce su##icient to o?et* di l ti i t* t t* P i l M $ P i l St × ;) 8 × %9 ( N-
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t*e dislocation gi?en t*at t*e Peierls Morce $W Peierls Stress × ;) W 8. × %9−( N-
8
Iage( $% )
3b F
d π ν
−=
−
Data #or Al, a9 W (.9( ]! Sli+ syste, ^%%9_%%%`! ; W √8a9-8 W 8.'4 ]! 2 W 84.%' 2Pa! ν W 9.('
& %9J
Iage
$84.%' %9 )$8.'4 %9 )&.% %9 -
( $% 9.J(')$%99)
F 0 m
π
−−− × ×= = ×
−
8
Iage( $% )%99
3b F
bπ ν
−=
− −?e sign i+lies an attraction towards t*e #ree sur#ace
As Miage MPeierls ⇒ t*at t*e dislocation will s+ontaneously o?e to t*e sur#ace (creating a step) under t*e
action o# t*e iage #orce! wit*out t*e a++lication o# an e"ternally a++lied stress.
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