Deformation of Single Crystals

8/13/2019 Deformation of Single Crystals

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DeformationofSingleCrystals

Whenasinglecrystalisdeformedunder

atensilestress,itisobservedthat

plasticdeformationoccursbyslipon

welldefinedparallelcrystalplanes.

Sectionsofthecrystalsliderelativetooneanother,changingthegeometryof

thesampleasshowninthediagram.

Slipalwaysoccursonaparticularsetof

crystallographicplanes,knownasslip

planes.Slipalwaystakesplacealongaconsistentsetofdirectionswithin

theseplanes thesearecalledslip

directions.Thecombinationofslip

planeandslipdirectiontogethermakes

upaslipsystemslipsystem.


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Slipsystemsareusually

specifiedusingtheMillerindex

notation.Forexample,cubic

closepackedmetalsslipon

Theslipdirectionmustlieintheslipplane.

{ }111011

Slipoccursbydislocationmotion.Tomovedislocations,acertain

stressmustbeappliedtoovercome

theresistancetodislocation

motion.

Slipoccurswhentheshearstressactingintheslipdirectionontheslip

planereaches

some

critical

value.

This

critical

shear

stress

is related

to

thestressrequiredtomovedislocationsacrosstheslipplane.


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Thetensileyieldstressofamaterialistheappliedstressrequiredto

startplasticdeformation ofthematerialunderatensileload.We

wanttorelatethetensilestressappliedtoasampletotheshearstressthatactsalongtheslipdirection.

coscoscoscos

cos

cos

___

_____====

A

F

A

F

planeslipofarea

planesliponactingforceresolvedR

ItisfoundthatthevalueofRatwhichslipoccursinagivenmaterialwithspecifieddislocationdensity andpurityisaconstant,knownas

thecriticalresolved

shear

stress C.ThisisSchmid's Law.

ThequantityCosCos isknownastheSchmid Factor(M)Thetensilestressatwhichthematerialstarttoslipistheyieldstrength.

coscosYC

=


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Inagivencrystal,theremaybemanyavailableslipsystems.As the

tensileloadisincreased,theresolvedshearstressoneachsystem

increasesuntileventuallyCisreachedononesystem.Thecrystalbeginstoplasticallydeformbysliponthissystem, known

astheprimaryslipsystem.Thestressrequiredtocausesliponthe

primaryslipsystemistheyieldstressofthesinglecrystal.Astheloadisincreasedfurther,Cmaybereachedonotherslipsystems;thesethenbegintooperate.

FromSchmid's Law,itisapparentthattheprimaryslipsystemwill

bethesystemwiththegreatestSchmid factor(M).

MYC =


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Experimental measurements showedthat

AtRTthemajorsourceforplasticdeformationisthedislocationmotionthroughthecrystallattice.

Dislocationmotionsoccursonfixedcrystalplanes(slipplanes)

infixedcrystallographicdirections(correspondingtotheBurgers

vectorofthedislocationthatcarriestheslip)

ThecrystalstructureofmetalsisnotalteredbytheplasticflowVolumechangesduringplasticflowarenegligible

BasicConsiderationsBasicConsiderations

Experimentaltechnique

Uniaxial TensionorCompression


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SchmidsLaw

Initialyieldstressvariesfromsampletosampledependingon,amongseveralfactors,thepositionofthecrystallatticerelativeto

theloadingaxis.

Itistheshearstressresolvedalongtheslipdirectiononthe slip

planethatinitiatesplasticdeformation.

Yieldwillbeginonaslipsystemwhentheshearstressonthis

systemfirstreachesacriticalvalue(criticalresolvedshearstress,crss),independentofthetensilestressoranyothernormalstressonthelatticeplane.

E.Schmid &W.Boas(1950),PlasticityofCrystals,Hughes&Co.,London.


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ResolvedShearStress

=s n=coscos

c= c

coscos

Soft orientation,

with slip plane at

45to tensile axis

Hard orientation,

with slip plane at

~90to tensile axis


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Slipinsinglecrystals

Criticalresolvedshearstress(crss):minimumshearstressrequiredtoinitiateslip.

Thisiswhenyieldingbegins(i.e.yieldstrength)

Conditionfordislocationmotion: R>

crss

max)cos(cos

crssy=

Crystalorientationcanmakeiteasyordifficulttomovedislocations.

R= 0=90

R= /2=45=45

R= 0=90

a)

Whathappensincasesaandc(w.r.t.plasticdeformation)?

b) c)


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Example

Atensilestressof5kPaisappliedparalleltothe[432]directionina

cubiccrystal.Findtheshearstresses,,onthe(111)planeinthe[011]direction.

Solution

FindtheSchmidtsfactorfortheslipsystem

kPakPa

CosCosM

Cos

Cos

76.15352.0

352.0

6565.0229

5

011234

]011[]432[

536.0

329

5

111234

]111[]432[

222222

222222

=====

=

=++++

=

=

=

++++

=

000


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Therelationshipbetweentheappliedstress

[applied]andtheshearstress issimplyanexampleofatensorrotation.Foratensilestress:

=

=

1

1

1

000

0

'

00

000

000

z

y

x

tensile

EmployingtheEuler

rotationtransformation

][][ 21 =a

tensile

tensileaa

=

=

cossinsin 2

3313

Becausea13isthecosineoftheanglebetweenx

andz,wecannowset

AfterMechanicalBehaviorofMaterialsbyK.Bowman

2sinsincos ==

coscos=


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TherelationshipoftheSchmid factortotheaxialstrainandthe

shearstrainisgivenby: coscos=

Where isthestrainalongthedirectionoftheappliedstressandistheglidestrain.

ThestrainsforanspecificslipsystemcanbeexpressedbySD=[uvw]

andSPN=(hkl)

++

+

+

++

=

lwlvkwhwlu

lvkwkv

kuhv

hwlukuhv

hu

ij

22

22

22

Where isthemagnitudeofthesimpleshearintheslipsystem.


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Example

Acubiccrystalissubjectedtoastressstate

x

=15kPa,y

=0z

=7.5kPa,yz

=zx

=xy

=0,where

x=[100],y=[010]andz=[001].Whatistheshear

stressonthe(111)[101]slipsystem?

7.54

45

90

2

1

=

=

=

kPa

5.700

000

0015

+ sinsinsincoscossincossinsincoscoscos


13/52

[ ]

[ ]

==

==

25.1116.206.3

16.275.33.5

06.33.550.7

]][][[

707.0408.0577.0

0816.0577.0

707.0408.0577.0

5.700

000

0015

707.00707.0

408.0816.0408.0

577.0577.0577.0

]][][[

T

T

aa

aa

[ ]

=

=

+

+

=

+

+

=

707.00707.0

408.0816.04086.0

577.0577.0577.0

45cos045sin

7.54cos45sin7.54sin7.54cos45cos

7.54sin45sin7.54cos7.54sin45cos

][

45cos90cos45sin90sin45sin

7.54cos45sin7.54cos90cos45cos90sin7.54sin7.54cos90sin45cos90cos7.54sin

7.54sin45sin7.54sin90cos45cos90sin7.54cos7.54sin90sin45cos90cos7.54cos

][

coscossinsinsin

cossincoscoscossinsincossincoscossin

sinsinsincoscossincossinsincoscoscos

11

221122112

221122112

a

a

a

Theshearstressintheslipdirectionisinthex1planeandz1 direction=3.06kPa

[ ] [ ]TaTaT ='

E l A FCC C i bj d i i l l d l h [112]


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Example:

IfCRSSis50MPa, whatisthetensilestressatwhichCuwillstarttodeformplastically?

smalleststressto

causeslip

(yielding)

TheinitialSlipSystems(plane,direction)arethen (1 11)[101], (11 1)[011]

Slip

Plane, n

Slip Dir.,

scos =

n

| || n| cos=

s

|

||s

|

M=cos cos )(MPa

(111) [0 1 1][ 1 01]

[ 1 10]

+2 2 / 3 3 / 6

3 / 6

0

6 / 9

6 / 9

0

184

184

undef

( 1 11) [0 1 1][101]

[110 ]

2 /3

3 / 6 3 / 2

3 / 3

6 /18 6 / 66 / 9

367

122

184

(1 1 1) [011][ 1 01][110 ]

2 /3 3 / 2 3 / 6

3 / 3

6 / 6

6 /186 / 9

122

367

184

(11 1 ) [011][101]

[ 1 10]

0 3 / 2

3 / 2

0

0

0

0

undefundefundef

AnFCCCuissubjectedtoauniaxial loadalongthe[112]

direction.Whatismostlikelyinitialslipsystem?


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StrainHardeningofFCCCrystals.StrainHardeningofFCCCrystals.ShearStressShearStressShearStrainCurvesShearStrainCurves

Atypicalshearstressshearstrain

curveforasinglecrystalshows

threestages ofworkhardening:

StageI=easy

glidewithlow

hardeningrates;

StageIIwithhigh,constant

hardeningrate,nearlyindependent

oftemperatureorstrainrate;

StageIIIwithdecreasinghardening

rateandverysensitivetotemperatureandstrainrate.

Theextendofeasyglideinacrystaldependsonitsorientation,

thepresenceofdislocations(defects)andonthetemperature.


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StageII:

Theshearstressneededtocontinueplasticdeformationbeginsto

increaseinanalmostlinearfashion.Thereisextensivework

hardening( G/300).

Thisstagebeginswhenslipisinitiatedonmultipleslipsystems.

Workhardeningisduetointeractionsbetweendislocationsmovingonintersectingslipplanes.

StageI:

Afteryielding,theshearstressforplastic

deformationisessentiallyconstant.Thereislittleornoworkhardening.

Thisistypicalwhenthereisasingleslip

systemoperative.Dislocationsdonot

interactmuchwitheachother.Easy

glide

Activeslipsystemisonewithmaximum

Schmid factor(i.e.,M=coscos)


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StageIII:

Thereisadecreasingrateofwork

hardening. Thisdecreaseisduetoanincreasein

thedegreeofcrossslipresultingina

parabolicshapetothecurve.

EffectofTemperature:

IncreasingTresultsinadecreaseintheextentofStageIandStageII.

WHY?

StageI: Initiationofsecondaryslipsystemsiseasier

StageII: Crossslipiseasier

StackingFault

Energy

(SFE):

FCCmetal:AdecreaseintheSFEcausesadecreaseincrossslip which

increasesthestressneededfortheStageIItoStageIIItransition.

Example:CuZn:Cu30at.%ZnhaslowSFE,extendsStageIItohighstresslevels.

I fl f t i


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Influenceofstressaxis

orientationinthestress

straincurve

Thestressaxisorientation

controlsthenumberofactive

slipsystems.

Recall:SlipoccurswhentheSchmid factorismaximum.

Moreslipsystemsmeansa

harder material.

Slipsteps Luder bands

Slipsteps(fromexitofdislocationsfromthecrystal)onthesurfaceof

compressedsinglecrystalofNb.


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Easyglideisgreaterinorientationsforwhichtheresolvedshear

stressonotherpotentialsystemsislow.Easyglidedoesnotoccurs

inFCCcrystalsorientedinsuchamannerthatslipoccurssimultaneouslyonmanyslipsystems.

NoeasyglideisobservedinBCCsinglecrystals.HCPsinglecrystals

exhibitextensiveeasyglideintensiontests(e.g.Zn,Cd,Mg).

AfterMechanicalBehaviorofMaterials W.F.Hosford

Slip Systems


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SlipSystems

infccmaterials

ForFCCmaterialsthereare12slip

systems(with+andshear

directions:

Four{111}planes,eachwith

threedirections

[Khan]

Thecombinationofslipplane{a,b,c,d}andslipdirection{1,2,3}thatoperateswithineachunit

triangleisshowninthefigure


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TensileDeformationofFCCCrystals

ForallorientationsofFCCcrystalswithinthisstereographictriangle,

theSchmid Factor(M)forslipinthe[101]directiononthe(111)

planeishigherthanthatforanyotherslipsystem.

Theslipsystem[101](111)isknownastheprimaryslipsystem.

If th t il i (TA) i t d l i i th t hi t i l


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Ifthetensileaxis(TA)isrepresentedaslyinginanyotherstereographictriangle,

theslipelementsmaybefoundbyexaminingtheremotecornersofthethree

adjacenttriangles.Thedirectioninoneoftheadjacenttrianglesisthe

normalto

the

slip

plane

and

the

direction

in

another

adjacent

triangle

is

the

slipdirection.

ThereisanorientationdependenceoftheSchmid Factorinthe

stereographictriangle.


The highest value of M=0 5 is obtained when the tensile axis lies on


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Ifthetensileaxis(TA)liesonboundaryofthestereographictriangle,

twoslipsystemsareequallyfavored.

Forexample,ifTAisonthe[100]

[111]

boundary,thesecondsystemisthe[110](111),whichiscalledtheconjugatesystem.

ThehighestvalueofM=0.5isobtainedwhenthetensileaxislieson

thegreatcirclebetweentheslipdirectionandtheslipplanenormal

withanglesof45degrees.

For example if TA is on the [100][111] boundary the second system is


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Forexample,ifTAisonthe[100][111]boundary,thesecondsystemis

the[110](111),whichiscalledtheconjugatesystem.

Twoothersystemshavenames[101](111)isthe

criticalslipsystemand

thesystemthatsharestheslipdirectionwiththe

primarysystem[101](11

1)iscalledthecrossslip

system.

Atthecorners,thereare

four,sixoreightequally

favoredslipsystems.


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Tensile Deformation of BCC Crystals


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TensileDeformationofBCCCrystalsTheslipdirectioninBCCmetalsisalwaysinthedirectionofclose

packing,,invariousslipplanes{110},{123}and{112}.G.ITaylordescribedtheslipinBCCcrystalsaspencilglideslip.Notethatthebasicorientationtriangleisdividedintotworegionswithdifferentslip

direction.


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LatticeRotationinTension

Duringtensiletesting,theendsofthetensilebarare

constrained.Thusthecrystalplanescannotglidefreely.Theyareforcedtorotatetowards

thetensileaxis(i


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p p g p g

Theglideshearstressandshearstraincanbedeterminedfromthe

initialorientationoftheslipplane(o)andslipdirection(o)andthe

extensionofthespecimen(Li/Lo).

Crystalrotationcanbetracedwiththeaidofastereographic

projection.

Lattice Rotation in Tension for FCC Crystals


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LatticeRotationinTensionforFCCCrystals

Slipnormallycausesagraduallatticerotationororientationchange.

Thegradualrotationororientationchangeoftheslipsystemwith

respecttotheTA,canberepresentedbykeepingtheslipsystem

fixedinspaceandrotatingtheTA.

Slip causes translation of point P


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SlipcausestranslationofpointP

paralleltotheslipdirectiontoanew

positionP.PointsCandCareconstructedby

extendingtheSDthroughOand

constructingperpendicularsfromPand

Ptotheextensionoftheslipdirection.

ThedistancePC=PC.Substituting:

+=

+==

===

1

sinsin

1sin

sin

sin''sin

o

o

o

oo

l

l

lCPlPC


Where istheengineeringstrain.Therefore theanglebetweentheTAandtheSDdecreasesduringtension,i.e.theSDrotatestowards

theTA.

FCC Geometry of Slip Systems


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FCCGeometryofSlipSystems

Infcccrystals,theslipsystemsare

combinationsofslipdirections

(theBurgersvectors)and{111}slip

planes.

Similarly,slipsdoesnotchangethe


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distancebetweentwoparallelslip

planes.

+=

===

1

coscos

cos''

cos

0

lBP

lPB

o

o

Theshearstrainassociatedwiththeslipsystemis:

Therefore,theanglebetweenthe

SPNandtheTAincreasesduringtension.

o

o

PB

OC

PB

OC

PB

OCOC

PB

PP

coscos

coscos

'''

=

=

==

ForanFCCcrystalorientedinthebasicstereographictriangle, theTA


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y g p g ,

willrotatetowardsthe[101]directionuntilitreachesthe[100][111]

boundary.Atthispoint,slipstartsonthe[110](111)systemwhichismost

favoredintheconjugatetriangle.Withequalslipofthetwosystems,

thenetrotationoftheTAistowardsthe[211]orientation.The

rotationalongthe[100][111]boundarybecomesslowastheTA

approachesthe[211]direction,whichisthestableendorientation.

LatentHardeningandOvershooting

T k i l l d i d i i h h T il A i


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TakeasinglecrystalandorienteditwithrespecttotheTensileAxis

(TA)insuchawaythatonlyasingleslipsystem( )isactive.Then,strainthecrystaluntila,Pstrainisreached.Now,changetheorientationoftheTAsuchthatasuccessive

secondarytesttakespartonthesamespecimen.Theneworientation

oftheTAwillactivateapreviouslylatentslipsystem().


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LatticeRotationinTensionforBCC


38/52

f

Crystals

InBCCcrystal,slipoccursbypencilglide.

TheTArotatestowardsthe


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LatticeRotationinCompression

Incompression,theslipplanenormal

(SPN)rotatestowardsthecompressionaxis(CA).



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ForaFCCcrystal,theCArotatestowardsthe[111]untilitreaches


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the[100][110]boundary.Thenslipoccurssimultaneouslyonthe

(111)and(111)planes,whichwillcauseanetrotationtowardsthe[110]stableendorientation.

ForaBCCcrystalwithpencilglideslipsystems,theCArotates away


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fromtheactiveslipdirection.IftheCAliesintheregionA,itwillend

uprotatingtowardsthe[111]endstableorientation,whereasif theCAliesinitiallyinregionB,itwillrotatetowardsthe[100]end

orientation.

HCPSlipPlanesandDirectionsPrincipal slip system can depend on c/a and relative orientation of


43/52

{0001}planesinthedirectionof

Slipsystems: 1x3=3

planesinthedirectionof

Slipsystems:

3x

1=1

Principalslipsystemcandependonc/aandrelativeorientationof

loadtoslipplanes

{101 0}

planesin

the

direction

of

Slipsystems: 6x1=6

{10 1 1}

c/a1.6333(ideal)

c/a1.6333(ideal)

hcpZincsinglecrystal

AdaptedfromFig.

7.9,Callister6e.

AdaptedfromFig.

7.8,Callister6e.

Cd,Zn,Mg,Ti,Be

Ti

Mg,Ti

DeformationofDeformationofPolycrystalsPolycrystals


44/52

Monocrystals Polycrystals

Theyareelasticallyandplastically

anisotropic.

Theycanundergodeformationin

onesingleslipsystem.

Intheabsenceoftexturecanbe

treatedasisotropicmaterial.

Deformationononlyoneslipsystem

isnotpossiblebecausevariousgrainshavetobecompatible.

Itisinherentlyinhomogeneous(it

variesfromgraintograin.Dislocationmovementishindered

becauseitisrestrictedtoonegrain.

Whenacrystalissurroundedbyothercrystalsofdifferent

crystallographicorientation,deformationofthecrystalcannot

startattheprimarysystemasthestraintakingplaceneedtobe

compatibleattheboundarywiththestrainintheothercrystals(nodiscontinuitiesalongthegrainboundary).

Deformation of polycrystals


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Microscopephotographofactualshearoffsetsin

differentgrains,onsurfaceofacopperbar.

Deformationofpolycrystals

Slipoccursinwelldefinedcrystallographic

planeswithineachgrain,butmorethan

oneslipplaneispossibleandlikely.

Indifferentgrains,theslipplaneswillhave

differentorientationsbecauseoftherandom

natureofthecrystalorientations.

Singlevs.polycrystal


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R= /2=45=45

Singlecrystal polycrystalline

e.g.

y=2crss

R = /2

=45 =45

Centergrain

Foreachgrain,

max)cos(cos

crssy=

Butandaredifferentforeachgrain.

Whichwillrequiremorestresstoslipsinglecrystalorthe

centergraininpolycrystal?


47/52

Sliplinesafter10%deformation

AfterGrainSizeandSolidSolutionStrengtheningin

MetalsATheoreticalandExperimentalStudyby

Dilip Chandrasekaran (DoctoralThesis)

Plastic Deformation in Polycrystals


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PlasticDeformationinPolycrystals

Before,

undeformedequiaxial grains The

plastic

deformation

has

producedelongatedgrains

Fiveindependentslipsystemsarerequiredtoproduceageneral

h t i i t l b li


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homogeneousstraininacrystalbyslip.

Forapolycrystallinematerialtohaveappreciableductility,eachofitsgrainsmustbeabletoundergothesameshapechangeasthe

entirebody.Thatis,eachgraininapolycrystal mustdeformwith

thesameexternalstrainsasthewholepolycrystal.

,, and , , , 12312321

Thenumberofindependentslipsystemsisequaltothenumberofstraincomponentsthatcanbeaccommodatedbyslip

Thethirdnormalstrainisnotindependentbecause: 213 =Ifamaterialhaslessthanfiveindependentslipsystems,apolycrystal

ofthatmaterialwillhaveverylimitedductilityunlessother

deformationmechanism(e.g.twinning)suppliestheaddedfreedomnecessary.

Foranindividualgrain,thisamountstoanimposedsetofstrains

alongthecrystalaxes.

Plastic deformation within an individual grain

PlasticDeformationinBiCrystals


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Plasticdeformationwithinanindividualgrain

isconstrainedbytheneighboringgrains. Becausestrainsalonggrainboundaries

mustbethesameforeachgrain,thegrains

willdeforminacooperativemanner.Ifthisis

notthecase,catastrophicfailureoccurs.Requiredtomaintaincontinuityof

thegrainboundary

Sinceingeneralonegrain(eitherAorB)willhaveahigherresolved

shearstress(RSS),theplasticdeformationofthatgrainwillberestricted.THUS:

Higheryieldstressforpolycrystal versussinglecrystal.

Greaterworkhardeningforpolycrystal versussinglecrystal.

Thisisthebasisfortexture hardening.

Sinceplasticdeformationofasinglegrainis

restrainedbyitsneighboringgrain,apolycrystalline

materialwillhaveanintrinsicallygreaterresistancetoplasticflowthanwouldasinglecrystal.

RvaluesinPolycrystallineMaterial

R l d fi d th ti f idth t thi kwidthR

=


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Rvaluesaredefinedastheratioofwidthtothickness

strainsintensiontestsoftexturedmaterials(e.g.sheets).thickness

R

=

TocalculateRvaluesinatexturedmaterialitisnecessaryto:

(a)identifythemoststressedslipsystem;

(b)calculatethestrainsinthex,yandzcoordinatesystemintermsof

theshearstrain ontheslipsystem.

Example:Consideracoppersheetwitha(011)[211]texture.PredicttheR

valuesinatensiontestparalleltothe[211]priorrollingdirection.

Solution

Placethecoordinatesystem[ ]

[ ]

[ ] tionness Direc.....Thickz_axisnh Directio......Widty_axis

onng Directi.....Rollix_axis

110

111

112

=

=

=

CopperisaFCCmaterial.Themost

favoredslipsystemwouldbe[ ] [ ]011111110111 and


52/52

Considerthefirstslipsystemandatensionappliedalongthe[211]

direction

[ ] [ ]

[ ][ ]

6

1

18

2

111112

111112cos

12

3

101112

101112cos

coscos

222222

222222

=

=++++

=

=++++

=

=

x

x

x

xxx

[ ][ ]

[ ] [ ]

6

1

6

2

111110

111110cos

4

1

101110

101110cos

coscos

222222

222222

=

=++++

=

=++++

=

=

z

z

z

zzz

[ ][ ]

[ ] [ ]

0

9

3

111111

111111

cos

06

0

101111

101111cos

coscos

222222

222222

==++++

=

==++++

=

=

y

y

y

yyy

0===z

y

thickness

widthR

Deformation of Single Crystals

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