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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
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[ ]
[ ]
==
==
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.
AfterMechanicalBehaviorofMaterials W.F.Hosford
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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AfterMechanicalBehaviorofMaterialsbyK.Bowman
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
AfterMechanicalBehaviorofMaterials W.F.Hosford
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
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f
Crystals
InBCCcrystal,slipoccursbypencilglide.
TheTArotatestowardsthe
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LatticeRotationinCompression
Incompression,theslipplanenormal
(SPN)rotatestowardsthecompressionaxis(CA).
AfterMechanicalBehaviorofMaterialsbyK.Bowman
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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
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{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
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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?
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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
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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
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