材料科学基础课后习题答案5Word下载.docx
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材料科学基础课后习题答案5Word下载.docx
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FCCstructurewith(111)shown
Wenotethatatomslyingalongfacediagonalstouchandarecrystallographicallyequivalent.Therefore,theshortestvectorconnectingequivalentpositionsis½
facediagonal.Forexample,onesuchvectoris
asshownin(111).
A.Thelengthofthisvectoris
B.Byinspection,thesizeofthevectoris2Cuatomradii.
C.Slipoccursinthemostdenselypackedplanewhichisofthetype{111}.ThesearethesmoothestplanesandcontainthesmallestBurgersvector.Thismeansthatthedislocationsmoveeasilyandtheenergyislow.
3.GIVEN:
∣b∣=0.288nminAg
Findlatticeparameter
RecalltheAgisFCC.ForFCCstructurestheBurgersvectoris½
afacediagonalasshown.Weseethat
4.A.FCCstructure
The(111)planeisshowninaunitcellwithallatomsshown.Atomstouchalongfacediagonals.The(111)planeisthemostcloselypacked,andthevectorsshownconnectequivalentatomicposition.Thus
etc.Theningeneral
B.ForNaC1
Weseethattheshortestvectorconnectingequivalentpositionsis
asshown.Thisdirectionliesinboththe{100}and{110}planesandbotharepossibleslipplanes.However{110}aretheplanesmostfrequentlyobservedastheslipplanes.Thisisbecauserepulsiveinterionicforcesareminimizedontheseplanesduringdislocationmotion.Thusweexpect1/2<
110>
Burgersvectorsand{110}slipplanes.
5.GIVEN:
Mocrystal
ao=0.314nm
Determinethecrystalstructure.
IfMowereFCC,then
but|b|=0.272⇒MoisnotFCC.
AssumingMoisBCC,then
ThustheBurgersvectorisconsistentwithMobeingBCC.
6.FIND:
Isthefracturesurfaceinionicsolidsroughorsmooth?
Cleavagessurfacesofionicmaterialsaregenerallysmooth.Onceacrackisstarted,iteasilypropagatesinastraightlineinaspecificcrystallographicdirectiononaspecificcrystallographicplane.Ceramicfracturesurfacesareroughwhenfailureproceedsthroughthenoncrystallineboundariesbetweensmallcrystals.
7.GIVEN:
BCCCrwith|b|=0.25nm
Findlatticeparametera
ASSUME:
forBCCstructure
fromtheformulaforthemagnitudeofavector:
8.GIVEN:
Normalstressof123MPaappliedtoBCCFein[110]direction
Resolvedshearin[101]on(010)
Recallthattheresolvedshearstressisgivenby:
τ=σcosθcosφ
(1)
whereθ=anglebetweenslipdirectionandtensileaxis;
φ=anglebetweennormaltoslipplaneandtensileaxis
Thus
9.GIVEN:
Stressin[123]directionofBCCcrystal
FindthestressneededtopromoteslipifτcR=800psi.Theslipplaneis(1
0)andslipdirectionis[111].
Recallτ=σcosθcosφ
(1)
θ=[123][111]
[123]⋅[111]=∣[123]∣∣[111]∣cosθ
φ=[123][1
0]
[123]⋅[1
0]=∣[123]∣∣[1
0]∣cosφ
10.Burgersvectorslieintheclosestpackeddirectionssincethedistancebetweenequivalentcrystallographicpositionsisshortestintheclose-packeddirections.ThismeansthattheenergyassociatedwiththedislocationwillbeminimumforsuchdislocationssincetheenergyisproportionaltothesquareoftheBurgersvector.
11.Closepackedplanesareslipplanessincethesearethesmoothestplanes(onanatomiclevel)andwouldthenbeexpectedtohavethelowestcriticalresolvedshearstress.
12.GIVEN:
Dislocationlieson(1
1)paralleltointersectionof(1
1)and(111)withBurgersvectorparallelto[
0].StructureisFCC.
A)Burgersvectorofdislocationand,B)Characterofdislocation.
A)SincethestructureisFCC,theBurgersvectorisparallelto<
andhasmagnitude
ForaBurgersvectorparallelto[
0]thescalarmultipliermustbea/2.Thus
=a/2[
0].B)Wemustdeterminethelinedirectionofthedislocation.FromthediagramweseethattheBVandlinedirectionareat60owhichmeansthedislocationismixed.
13.GIVEN:
Dislocationreactionbelow:
Showitisvectoriallycorrectandenergeticallyproper.
Thesumofthex,y&
zcomponentsontheLHSmustbeequaltothecorrespondingcomponentontherighthandside.
xcomponent(LHS)=xcomponent(RHS)
ycomponent(LHS)=ycomponent(RHS)
zcomponent(LHS)=zcomponent(RHS)
Energy:
Thereactionisenergeticallyfavorableif|b1|2+|b2|2>
|b3|3
Thusthereactionisfavorablesince
14.GIVEN:
DislocationinFCC
Parallelto[
01]i.e.
=[
01]
Characterandslipplane
Characterisfoundbyanglebetween
.Note
=-1+0+1=0.Thus
.Since
thedislocationispureedge.
Tofindtheslipplanewenotethatthecrossproduceof
givesavectorthatisnormaltotheplaneinwhich
lie.Thisvectorsoformedhasthesameindicesastheplanesincewehaveafundamentallycubicstructure.
Weseefromthediagramthatthesevectorslieon(010).
Thus,wehavetheplane(0
0)whichisthesameasthe(010)plane.Thisdoesnotmovebyglidesinceplanesofthekind{100}arenotslipplanesfortheFCCstructure.
15.FCCmetalsaremoreductilethanBCCorHCPbecause:
1)thereisnoeasymechanismfornucleationofmicrocracksinFCCasthereisforBCCandHCP;
2)thestressesforplasticdeformationarelowerinFCCduetothe(generally)smootherplanes.ThismeansthatthemicrocracksthatforminBCC&
HCPwillhavehighstressestendingtomakethempropagate.
16.Forasimplecubicsystem,thelowestenergyBurgersvectorsareofthetype<
001>
sincethisistheshortestdistanceconnectingequivalentatomicpositions.ThismeansthattheenergyislowestsincethestrainenergyisproportionaltothesquareoftheBurgersvector.
17.GIVEN:
At.wt.0=16
At.wt.Mg=24.32SamestructureasNaCl
ρ=3.65g/cm3
FindlengthofBurgersVectorinMgO
ThestructureofMgOisshownschematicallybelowalongwiththeshortestBurgersvector.Tosolvetheproblemwefirstnotethatwerequirethelatticeparameterao.Wecantakeasub-sectionoftheunitcell(cross-hatchedcube)whoseedgeis
unitslong.
Wecancalculatethetotalmassofthiscubeandthevolumeandcalculatethedensity.Sincethemassisknownandthedensityisknown,thevolumemaybecalculatedfromwhichaomaybeextracted.
Wenotethatthereare40=ionsand4Mg++ionslocatedatthecorners.However,anionatacornerissharedby8suchcubes.Thus,wehave½
O=and½
Mg++ionsinourcube.
18.GIVEN:
Criticalresolvedshearstress(0.34MPa),slipsystem(111)[
10],andtensileaxis[101]
Appliedstressatwhichcrystalbeginstodeformandcrystalstructure.
(A)Thesituationisshownbelow
τcrss=σcosθ⋅cosφ
θ=anglebetweentensileaxisandslipdirection
θ=anglebetweentensileaxisandnormaltoslipplane
φ=[111][101]θ=[101][
[111]⋅[101]=∣[111]∣∣[101]∣cosφ[101]⋅[110]=∣[101]∣∣[110]∣cosθ
(B):
Tohavea{111}<
slipsystem,thematerialmusthaveanFCCstructure.
19.GIVEN:
τcrss=55.2MPa,(111)[
01]slipsystem,[112]tensileaxis
Findthehighestnormalstressthatcanbeappliedbeforedislocationmotioninthe[10
]direction.
Thesituationisshownbelow.Essentiallytheproblemreducestofindingthevalueofthetensilestresswhenthecriticalresolvedshearisreached.
τcrss=σcosθ⋅cosφ
θ=[112][
01]φ=[112]{111}
B.WouldhaveexactlythesamestressforaBCCmetal(φ&
θwouldbeinterchanged).
20.GIVEN:
σatyield=3.5MPa;
(111)[1
0]slipsystem[1
1]tensileaxis
Computeτcrss
τcrss=σcosθcosφ
θ=[1
1][1
0]φ=[1
1][111]
21.ItemEdgeScrew
Lineardefect?
YesYes
ElasticDistortion?
Glide?
Climb?
YesNo
Cross-slip?
NoYes
BurgersVector(BV)⊥toline//toline
Uniqueslipplane?
Offset//toBV//toBV
Motion//toBV⊥toBV
22.GIVEN:
BCCmetalwithτcrss=7MPa[001]tensileaxis.
(a)Slipsystemthatwillbeactivatedand(b)normalstressforplasticdeformation.
RecallthatforBCCmetalstheusualslipsystemis<
111>
{110}.Deformationoccursontheplaneanddirectionforwhichcosθ⋅cosφisamaximumsincethiswillhavethemaximumresolvedshearstress.Thesituationisshownbelow.
(Notethattheslipdirectionsareshownshortenedinthisview)
Possibleslipsystemsarelistedbelow:
sketch(also[1
1]on(011))(also[
11]on(0
1))(also[
1]on(101))
similartoplanesshowninsketch.Also[111]on(
01)
Weseebyinspectionthattheresolvedshearduetoatensileforcein[001]willallbethesame.Theresolvedshearonallother{110}<
systemsiszero.
B.Tocomputethenormalstressattheonsetofplasticdeformationwewillconsider(011)[
1]
τcrss=σcosθcosφ=7
θ=[001][
1];
cosφ=[001][011]
Noteifweconsidered(101)[
1]wewouldhave
andwewouldo
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