ALLl12lCHAPTERSlCOVERED
SOLUTIONS MANUAL
l
,TableofContents
l l
Chapterl1.lElasticlResponseloflSolidslChapterl
2.lYieldinglandlPlasticlFlowlChapterl3.lContro
llinglStrength
Chapterl4.lTime-
DependentlDeformationlChapterl5.lFracture
:lAnlOverview
Chapterl6.lElementsloflFracturelMechanicslC
hapterl7.lFracturelToughness
Chapterl8.lEnvironment-
AssistedlCrackinglChapterl9.lCycliclStresslandlSt
rainlFatiguelChapterl10.lFatiguelCracklPropagati
onlChapterl11.lAnalysesloflEngineeringlFailuresl
Chapterl12.lConsequencesloflProductlFailure
,DeformationlandlFracturelMechanicsloflEngineeringlMaterials,l5thled. ProblemlSolutionslp.l1/162
Draftldocument,lCopyrightlR.lHertzberg,lR.lVinci,lJ.lHertzbergl2009
CHAPTER 1 l
Review
1.1 Inl yourl ownl words,l whatl arel twol differencesl betweenl productl testingl andl materiallt
esting?
Possiblelanswerslinclude:l(a)lThelgoallofltheltwolprocedureslisldifferent.lWhereaslproductltesti
nglisldesignltoldeterminelthellifetimeloflalcomponentlunderlconditionslthatlmimiclreal-
lworldluse,lmaterialltestinglislintendedltolextractlfundamentallmateriallpropertieslthatlarelinde
pendentloflthelmaterial’sluse.l(b)lThelspecimenlshapelisldifferent.lProductltestinglmustluselthel
materiallinlthelshapelinlwhichlitlwilllbelusedlinlthelreallproduct.lMaterialltestingluseslidealizedl
specimenlshapesldesignedltolunambiguouslyldeterminelonelorlmorelpropertieslofl thelmateriall
withlthelsimplestlanalysislpossible.
1.2 Whatlareltheldistinguishingldifferenceslbetweenlelasticity,lplasticity,landlfracture?
Elasticitylinvolveslonlyldeformationlthatlislfullylreversiblelwhenlthelappliedlloadlislremovedl(e
venliflitltakesltimeltoloccur).lPlasticitylislpermanentlshapelchangelwithoutlcracking,levenlwhenl
nolloadlexists.lFracturelinherentlylinvolveslbreakingloflbondslandlthelcreationloflnewlsurfaces.
lOftenltwolorlmorelofltheselprocessesltakelplacelsimultaneously,lbutlthelcontributionlofleachlca
nlbelseparatedlfromlthelothers.
1.3 Writel thel definitionsl forl engineeringl stress,l truel stress,l engineeringl strain,l andl truels
trainlforlloadinglalonglalsinglelaxis.
load P
eng =lengineeringlstressl= =l (1-1a)
initiall cross-sectionall area A0
load P
true =ltruelstressl= =l (1-2a)
instantaneousl cross-sectionall area Ai
changelinllengthl llfl −ll0
=lengineeringlstrainl=l =l (1-1b)
eng
initialllength l0
finalllength llf
true =ltruelstrainl=llnl =llnl (1-2b)
initialllength l0
1.4 Underl whatl conditionsl isl Eq.l 1-
4l valid?l Whatl makesl itl nol longerl usefull ifl thoselconditionslarelnotlmet?
Excerptslfromlthislworklmaylbelreproducedlbylinstructorslforldistributionlonlalnot-for-
profitlbasislforltestinglorlinstructionall purposeslonlyltolstudentslenrolledlinlcourseslforlwhichltheltextbooklhaslbeenladopted.l
Anylotherlreproductionlorltranslationlofl thislworklbeyondlthatlpermittedlbylSectionsl107lorl108loflthel1976lUnitedlStateslCo
pyrightlActlwithoutlthelpermissionloflthe
copyrightlownerlislunlawful.
, DeformationlandlFracturelMechanicsloflEngineeringlMaterials,l5thled. ProblemlSolutionslp.l1/162
Draftldocument,lCopyrightlR.lHertzberg,lR.lVinci,lJ.lHertzbergl2009
Pl
=l (ll /lll )l=l (ll /lll )l=l (1+ll l ) (1-4)
true i 0 engl li eng eng
A0 0
Thislexpressionlisltruelwhenlvolumelislconserved.lHowever,litlislonlylusefulliflthelcross-
lsectionallarealislthelsameleveryonelonltheltestlspecimen.lIflthislisn’tlthelcaselthenlthelstresslan
dlstrainlwilllvarylfromlonelpartloflthelspecimenltolanother.
1.5 SketchlFigurel1.3,lcurvel‘b’l(alductilelmetal).lLabellitlwithlthelfollowinglterms,lindicatin
glfromlwhichllocationlonlthelcurveleachlquantitylcanlbelidentifiedlorl extracted:lelasticlre
gion,lelastic-
plasticlregion,lproportionalllimit,ltensilelstrength,lonsetloflnecking,lfracturelstress.
onsetloflnecking
tensilelstrength
fracturelstress
proportionalllimit
elastic-plasticlregion
elasticlregion
stress
strain
1.6 Onlalsinglelsetloflaxes,lsketchlapproximatelatomiclforcelvs.latom-
separationlcurvesllikelthelonelshownlinlFig.l1.4blforltungstenlatltemperatureslofl200,l600
,landl1000lK.lPaylcloselattentionltolthelpointlx0landlthelslopeldF/dxlforleachloflthelcurves
lyouldraw.
Thelkeylfeaturesloflthelplotlarelthelincreasinglx0lspacinglwithlincreasingltemperaturel(i.e.,lwith
lthermallexpansion)landltheldecreasinglslopelassociatedlwithldecreasedlelasticlmodulus.lThelp
lotlislexaggeratedlbutltheltrendslarelreasonable.
F
dF
ld
x 200l K
600l K
1000l K
x
x0l (1000lK)
x0l(600lK)
x0l(200lK)
Excerptslfromlthislworklmaylbelreproducedlbylinstructorslforldistributionlonlalnot-for-
profitlbasislforltestinglorlinstructionall purposeslonlyltolstudentslenrolledlinlcourseslforlwhichltheltextbooklhaslbeenladopted.l
Anylotherlreproductionlorltranslationlofl thislworklbeyondlthatlpermittedlbylSectionsl107lorl108loflthel1976lUnitedlStateslCo
pyrightlActlwithoutlthelpermissionloflthe
copyrightlownerlislunlawful.
SOLUTIONS MANUAL
l
,TableofContents
l l
Chapterl1.lElasticlResponseloflSolidslChapterl
2.lYieldinglandlPlasticlFlowlChapterl3.lContro
llinglStrength
Chapterl4.lTime-
DependentlDeformationlChapterl5.lFracture
:lAnlOverview
Chapterl6.lElementsloflFracturelMechanicslC
hapterl7.lFracturelToughness
Chapterl8.lEnvironment-
AssistedlCrackinglChapterl9.lCycliclStresslandlSt
rainlFatiguelChapterl10.lFatiguelCracklPropagati
onlChapterl11.lAnalysesloflEngineeringlFailuresl
Chapterl12.lConsequencesloflProductlFailure
,DeformationlandlFracturelMechanicsloflEngineeringlMaterials,l5thled. ProblemlSolutionslp.l1/162
Draftldocument,lCopyrightlR.lHertzberg,lR.lVinci,lJ.lHertzbergl2009
CHAPTER 1 l
Review
1.1 Inl yourl ownl words,l whatl arel twol differencesl betweenl productl testingl andl materiallt
esting?
Possiblelanswerslinclude:l(a)lThelgoallofltheltwolprocedureslisldifferent.lWhereaslproductltesti
nglisldesignltoldeterminelthellifetimeloflalcomponentlunderlconditionslthatlmimiclreal-
lworldluse,lmaterialltestinglislintendedltolextractlfundamentallmateriallpropertieslthatlarelinde
pendentloflthelmaterial’sluse.l(b)lThelspecimenlshapelisldifferent.lProductltestinglmustluselthel
materiallinlthelshapelinlwhichlitlwilllbelusedlinlthelreallproduct.lMaterialltestingluseslidealizedl
specimenlshapesldesignedltolunambiguouslyldeterminelonelorlmorelpropertieslofl thelmateriall
withlthelsimplestlanalysislpossible.
1.2 Whatlareltheldistinguishingldifferenceslbetweenlelasticity,lplasticity,landlfracture?
Elasticitylinvolveslonlyldeformationlthatlislfullylreversiblelwhenlthelappliedlloadlislremovedl(e
venliflitltakesltimeltoloccur).lPlasticitylislpermanentlshapelchangelwithoutlcracking,levenlwhenl
nolloadlexists.lFracturelinherentlylinvolveslbreakingloflbondslandlthelcreationloflnewlsurfaces.
lOftenltwolorlmorelofltheselprocessesltakelplacelsimultaneously,lbutlthelcontributionlofleachlca
nlbelseparatedlfromlthelothers.
1.3 Writel thel definitionsl forl engineeringl stress,l truel stress,l engineeringl strain,l andl truels
trainlforlloadinglalonglalsinglelaxis.
load P
eng =lengineeringlstressl= =l (1-1a)
initiall cross-sectionall area A0
load P
true =ltruelstressl= =l (1-2a)
instantaneousl cross-sectionall area Ai
changelinllengthl llfl −ll0
=lengineeringlstrainl=l =l (1-1b)
eng
initialllength l0
finalllength llf
true =ltruelstrainl=llnl =llnl (1-2b)
initialllength l0
1.4 Underl whatl conditionsl isl Eq.l 1-
4l valid?l Whatl makesl itl nol longerl usefull ifl thoselconditionslarelnotlmet?
Excerptslfromlthislworklmaylbelreproducedlbylinstructorslforldistributionlonlalnot-for-
profitlbasislforltestinglorlinstructionall purposeslonlyltolstudentslenrolledlinlcourseslforlwhichltheltextbooklhaslbeenladopted.l
Anylotherlreproductionlorltranslationlofl thislworklbeyondlthatlpermittedlbylSectionsl107lorl108loflthel1976lUnitedlStateslCo
pyrightlActlwithoutlthelpermissionloflthe
copyrightlownerlislunlawful.
, DeformationlandlFracturelMechanicsloflEngineeringlMaterials,l5thled. ProblemlSolutionslp.l1/162
Draftldocument,lCopyrightlR.lHertzberg,lR.lVinci,lJ.lHertzbergl2009
Pl
=l (ll /lll )l=l (ll /lll )l=l (1+ll l ) (1-4)
true i 0 engl li eng eng
A0 0
Thislexpressionlisltruelwhenlvolumelislconserved.lHowever,litlislonlylusefulliflthelcross-
lsectionallarealislthelsameleveryonelonltheltestlspecimen.lIflthislisn’tlthelcaselthenlthelstresslan
dlstrainlwilllvarylfromlonelpartloflthelspecimenltolanother.
1.5 SketchlFigurel1.3,lcurvel‘b’l(alductilelmetal).lLabellitlwithlthelfollowinglterms,lindicatin
glfromlwhichllocationlonlthelcurveleachlquantitylcanlbelidentifiedlorl extracted:lelasticlre
gion,lelastic-
plasticlregion,lproportionalllimit,ltensilelstrength,lonsetloflnecking,lfracturelstress.
onsetloflnecking
tensilelstrength
fracturelstress
proportionalllimit
elastic-plasticlregion
elasticlregion
stress
strain
1.6 Onlalsinglelsetloflaxes,lsketchlapproximatelatomiclforcelvs.latom-
separationlcurvesllikelthelonelshownlinlFig.l1.4blforltungstenlatltemperatureslofl200,l600
,landl1000lK.lPaylcloselattentionltolthelpointlx0landlthelslopeldF/dxlforleachloflthelcurves
lyouldraw.
Thelkeylfeaturesloflthelplotlarelthelincreasinglx0lspacinglwithlincreasingltemperaturel(i.e.,lwith
lthermallexpansion)landltheldecreasinglslopelassociatedlwithldecreasedlelasticlmodulus.lThelp
lotlislexaggeratedlbutltheltrendslarelreasonable.
F
dF
ld
x 200l K
600l K
1000l K
x
x0l (1000lK)
x0l(600lK)
x0l(200lK)
Excerptslfromlthislworklmaylbelreproducedlbylinstructorslforldistributionlonlalnot-for-
profitlbasislforltestinglorlinstructionall purposeslonlyltolstudentslenrolledlinlcourseslforlwhichltheltextbooklhaslbeenladopted.l
Anylotherlreproductionlorltranslationlofl thislworklbeyondlthatlpermittedlbylSectionsl107lorl108loflthel1976lUnitedlStateslCo
pyrightlActlwithoutlthelpermissionloflthe
copyrightlownerlislunlawful.
