Stress–strain curve - Wikipedia

In engineering and materials science, a stress-strain curve for a material gives the relationship between stress and strain. It is obtained by gradually ... Stress–straincurve FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Conceptinengineering and materialsscience Stress-straincurvetypicalofalowcarbonsteel. Stress-straincurveforatensiletest Forbroadercoverageofthistopic,seeStress-strainanalysis. Inengineeringandmaterialsscience,astress-straincurveforamaterialgivestherelationshipbetweenstressandstrain.Itisobtainedbygraduallyapplyingloadtoatestcouponandmeasuringthedeformation,fromwhichthestressandstraincanbedetermined(seetensiletesting).Thesecurvesrevealmanyofthepropertiesofamaterial,suchastheYoung'smodulus,theyieldstrengthandtheultimatetensilestrength. Contents 1Definition 1.1Engineeringstressandstrain 1.2Truestressandstrain 2Stages 3Classification 3.1Ductilematerials 3.2Brittlematerials 4Seealso 5References Definition[edit] Generallyspeaking,curvesrepresentingtherelationshipbetweenstressandstraininanyformofdeformationcanberegardedasstress-straincurves.Thestressandstraincanbenormal,shear,ormixture,alsocanbeuniaxial,biaxial,ormultiaxial,evenchangewithtime.Theformofdeformationcanbecompression,stretching,torsion,rotation,andsoon.Ifnotmentionedotherwise,stress-straincurvereferstotherelationshipbetweenaxialnormalstressandaxialnormalstrainofmaterialsmeasuredinatensiontest. Engineeringstressandstrain[edit] Considerabaroforiginalcrosssectionalarea A 0 {\displaystyleA_{0}} beingsubjectedtoequalandoppositeforces F {\displaystyleF} pullingattheendssothebarisundertension.Thematerialisexperiencingastressdefinedtobetheratiooftheforcetothecrosssectionalareaofthebar,aswellasanaxialelongation: σ = F A 0 {\displaystyle\sigma={\tfrac{F}{A_{0}}}} ε = L − L 0 L 0 = Δ L L 0 {\displaystyle\varepsilon={\tfrac{L-L_{0}}{L_{0}}}={\tfrac{\DeltaL}{L_{0}}}} Subscript0denotestheoriginaldimensionsofthesample.TheSIunitforstressisnewtonpersquaremetre,orpascal(1pascal=1Pa=1N/m2),andstrainisunitless.Stress-straincurveforthismaterialisplottedbyelongatingthesampleandrecordingthestressvariationwithstrainuntilthesamplefractures.Byconvention,thestrainissettothehorizontalaxisandstressissettoverticalaxis.Notethatforengineeringpurposesweoftenassumethecross-sectionareaofthematerialdoesnotchangeduringthewholedeformationprocess.Thisisnottruesincetheactualareawilldecreasewhiledeformingduetoelasticandplasticdeformation.Thecurvebasedontheoriginalcross-sectionandgaugelengthiscalledtheengineeringstress-straincurve,whilethecurvebasedontheinstantaneouscross-sectionareaandlengthiscalledthetruestress-straincurve.Unlessstatedotherwise,engineeringstress-strainisgenerallyused. Truestressandstrain[edit] Thedifferencebetweentruestress-straincurveandengineeringstress-straincurve Duetotheshrinkingofsectionareaandtheignoredeffectofdevelopedelongationtofurtherelongation,truestressandstrainaredifferentfromengineeringstressandstrain. σ t = F A {\displaystyle\sigma_{\text{t}}={\tfrac{F}{A}}} ε t = ∫ δ L L {\displaystyle\varepsilon_{\text{t}}=\int{\tfrac{\deltaL}{L}}} Herethedimensionsareinstantaneousvalues.Assumingvolumeofthesampleconservesanddeformationhappensuniformly, A 0 L 0 = A L {\displaystyleA_{0}L_{0}=AL} Thetruestressandstraincanbeexpressedbyengineeringstressandstrain.Fortruestress, σ t = F A = F A 0 A 0 A = F A 0 L L 0 = σ ( 1 + ε ) {\displaystyle\sigma_{\text{t}}={\tfrac{F}{A}}={\tfrac{F}{A_{0}}}{\tfrac{A_{0}}{A}}={\tfrac{F}{A_{0}}}{\tfrac{L}{L_{0}}}=\sigma(1+\varepsilon)} Forthestrain, δ ε t = δ L L {\displaystyle\delta\varepsilon_{\text{t}}={\tfrac{\deltaL}{L}}} Integratebothsidesandapplytheboundarycondition, ε t = ln ⁡ ( L L 0 ) = ln ⁡ ( 1 + ε ) {\displaystyle\varepsilon_{\text{t}}=\ln\left({\tfrac{L}{L_{0}}}\right)=\ln(1+\varepsilon)} Soinatensiontest,truestressislargerthanengineeringstressandtruestrainislessthanengineeringstrain.Thus,apointdefiningtruestress-straincurveisdisplacedupwardsandtothelefttodefinetheequivalentengineeringstress-straincurve.Thedifferencebetweenthetrueandengineeringstressesandstrainswillincreasewithplasticdeformation.Atlowstrains(suchaselasticdeformation),thedifferencesbetweenthetwoisnegligible.Asforthetensilestrengthpoint,itisthemaximalpointinengineeringstress-straincurvebutisnotaspecialpointintruestress-straincurve.Becauseengineeringstressisproportionaltotheforceappliedalongthesample,thecriterionforneckingformationcanbesetas δ F = 0 {\displaystyle\deltaF=0} . δ F = σ t δ A + A δ σ t = 0 {\displaystyle\deltaF=\sigma_{\text{t}}\,\deltaA+A\,\delta\sigma_{\text{t}}=0} − δ A A = δ σ t σ t {\displaystyle-{\tfrac{\deltaA}{A}}={\tfrac{\delta\sigma_{\text{t}}}{\sigma_{\text{t}}}}} ThisanalysissuggestsnatureoftheUTSpoint.TheworkstrengtheningeffectisexactlybalancedbytheshrinkingofsectionareaatUTSpoint. Aftertheformationofnecking,thesampleundergoesheterogeneousdeformation,soequationsabovearenotvalid.Thestressandstrainattheneckingcanbeexpressedas: σ t = F A neck {\displaystyle\sigma_{\text{t}}={\tfrac{F}{A_{\text{neck}}}}} ε t = ln ⁡ ( A 0 A neck ) {\displaystyle\varepsilon_{\text{t}}=\ln\left({\tfrac{A_{0}}{A_{\text{neck}}}}\right)} Anempiricalequationiscommonlyusedtodescribetherelationshipbetweentruestressandtruestrain. σ t = K ( ε t ) n {\displaystyle\sigma_{\text{t}}=K(\varepsilon_{\text{t}})^{n}} Here, n {\displaystylen} isthestrain-hardeningcoefficientand K {\displaystyleK} isthestrengthcoefficient. n {\displaystylen} isameasureofamaterial'sworkhardeningbehavior.Materialswithahigher n {\displaystylen} haveagreaterresistancetonecking.Typically,metalsatroomtemperaturehave n {\displaystylen} rangingfrom0.02to0.5.[1] Stages[edit] Aschematicdiagramforthestress-straincurveoflowcarbonsteelatroomtemperatureisshowninfigure1.Thereareseveralstagesshowingdifferentbehaviors,whichsuggestsdifferentmechanicalproperties.Toclarify,materialscanmissoneormorestagesshowninfigure1,orhavetotallydifferentstages. Thefirststageisthelinearelasticregion.Thestressisproportionaltothestrain,thatis,obeysthegeneralHooke'slaw,andtheslopeisYoung'smodulus.Inthisregion,thematerialundergoesonlyelasticdeformation.Theendofthestageistheinitiationpointofplasticdeformation.Thestresscomponentofthispointisdefinedasyieldstrength(orupperyieldpoint,UYPforshort). Thesecondstageisthestrainhardeningregion.Thisregionstartsasthestressgoesbeyondtheyieldingpoint,reachingamaximumattheultimatestrengthpoint,whichisthemaximalstressthatcanbesustainedandiscalledtheultimatetensilestrength(UTS).Inthisregion,thestressmainlyincreasesasthematerialelongates,exceptthatforsomematerialssuchassteel,thereisanearlyflatregionatthebeginning.Thestressoftheflatregionisdefinedastheloweryieldpoint(LYP)andresultsfromtheformationandpropagationofLüdersbands.Explicitly,heterogeneousplasticdeformationformsbandsattheupperyieldstrengthandthesebandscarryingwithdeformationspreadalongthesampleattheloweryieldstrength.Afterthesampleisagainuniformlydeformed,theincreaseofstresswiththeprogressofextensionresultsfromworkstrengthening,thatis,densedislocationsinducedbyplasticdeformationhampersthefurthermotionofdislocations.Toovercometheseobstacles,ahigherresolvedshearstressshouldbeapplied.Asthestrainaccumulates,workstrengtheninggetsreinforced,untilthestressreachestheultimatetensilestrength. Thethirdstageistheneckingregion.Beyondtensilestrength,aneckformswherethelocalcross-sectionalareabecomessignificantlysmallerthantheaverage.Theneckingdeformationisheterogeneousandwillreinforceitselfasthestressconcentratesmoreatsmallsection.Suchpositivefeedbackleadstoquickdevelopmentofneckingandleadstofracture.Notethatthoughthepullingforceisdecreasing,theworkstrengtheningisstillprogressing,thatis,thetruestresskeepsgrowingbuttheengineeringstressdecreasesbecausetheshrinkingsectionareaisnotconsidered.Thisregionendsupwiththefracture.Afterfracture,percentelongationandreductioninsectionareacanbecalculated. Classification[edit] Stress-straincurveforbrittlematerialscomparedtoductilematerials. Itispossibletodistinguishsomecommoncharacteristicsamongthestress-straincurvesofvariousgroupsofmaterialsand,onthisbasis,todividematerialsintotwobroadcategories;namely,theductilematerialsandthebrittlematerials.[2]: 51  Ductilematerials[edit] Ductilematerials,whichincludestructuralsteelandmanyalloysofothermetals,arecharacterizedbytheirabilitytoyieldatnormaltemperatures.[2]: 58  Lowcarbonsteelgenerallyexhibitsaverylinearstress-strainrelationshipuptoawelldefinedyieldpoint(Fig.1).ThelinearportionofthecurveistheelasticregionandtheslopeisthemodulusofelasticityorYoung'smodulus. Manyductilematerialsincludingsomemetals,polymersandceramicsexhibitayieldpoint.Plasticflowinitiatesattheupperyieldpointandcontinuesatthelowerone.Atloweryieldpoint,permanentdeformationisheterogeneouslydistributedalongthesample.Thedeformationbandwhichformedattheupperyieldpointwillpropagatealongthegaugelengthattheloweryieldpoint.Thebandoccupiesthewholeofthegaugeattheludersstrain.Beyondthispoint,workhardeningcommences.Theappearanceoftheyieldpointisassociatedwithpinningofdislocationsinthesystem.Forexample,solidsolutioninteractswithdislocationsandactsaspinandpreventdislocationfrommoving.Therefore,thestressneededtoinitiatethemovementwillbelarge.Aslongasthedislocationescapefromthepinning,stressneededtocontinueitisless. Aftertheyieldpoint,thecurvetypicallydecreasesslightlybecauseofdislocationsescapingfromCottrellatmospheres.Asdeformationcontinues,thestressincreasesonaccountofstrainhardeninguntilitreachestheultimatetensilestress.Untilthispoint,thecross-sectionalareadecreasesuniformlybecauseofPoissoncontractions.Thenitstartsneckingandfinallyfractures. Theappearanceofneckinginductilematerialsisassociatedwithgeometricalinstabilityinthesystem.Duetothenaturalinhomogeneityofthematerial,itiscommontofindsomeregionswithsmallinclusionsorporositywithinitorsurface,wherestrainwillconcentrate,leadingtoalocallysmallerareathanotherregions.Forstrainlessthantheultimatetensilestrain,theincreaseofwork-hardeningrateinthisregionwillbegreaterthantheareareductionrate,therebymakethisregionhardertobefurtherdeformthanothers,sothattheinstabilitywillberemoved,i.e.thematerialshaveabilitiestoweakentheinhomogeneitybeforereachingultimatestrain.However,asthestrainbecomelarger,theworkhardeningratewilldecreases,sothatfornowtheregionwithsmallerareaisweakerthanotherregion,thereforereductioninareawillconcentrateinthisregionandtheneckbecomesmoreandmorepronounceduntilfracture.Aftertheneckhasformedinthematerials,furtherplasticdeformationisconcentratedintheneckwhiletheremainderofthematerialundergoeselasticcontractionowingtothedecreaseintensileforce. Thestress-straincurveforaductilematerialcanbeapproximatedusingtheRamberg-Osgoodequation.[3]Thisequationisstraightforwardtoimplement,andonlyrequiresthematerial'syieldstrength,ultimatestrength,elasticmodulus,andpercentelongation. Brittlematerials[edit] Brittlematerials,whichincludecastiron,glass,andstone,arecharacterizedbythefactthatruptureoccurswithoutanynoticeablepriorchangeintherateofelongation,[2]: 59 sometimestheyfracturebeforeyielding. Brittlematerialssuchasconcreteorcarbonfiberdonothaveawell-definedyieldpoint,anddonotstrain-harden.Therefore,theultimatestrengthandbreakingstrengtharethesame.Typicalbrittlematerialslikeglassdonotshowanyplasticdeformationbutfailwhilethedeformationiselastic.Oneofthecharacteristicsofabrittlefailureisthatthetwobrokenpartscanbereassembledtoproducethesameshapeastheoriginalcomponentastherewillnotbeaneckformationlikeinthecaseofductilematerials.Atypicalstress-straincurveforabrittlematerialwillbelinear.Forsomematerials,suchasconcrete,tensilestrengthisnegligiblecomparedtothecompressivestrengthanditisassumedzeroformanyengineeringapplications.Glassfibershaveatensilestrengthstrongerthansteel,butbulkglassusuallydoesnot.Thisisbecauseofthestressintensityfactorassociatedwithdefectsinthematerial.Asthesizeofthesamplegetslarger,theexpectedsizeofthelargestdefectalsogrows. Seealso[edit] Elastomers Planestraincompressiontest Strengthofmaterials Stress-strainindex Tensometer Universaltestingmachine References[edit] ^Courtney,Thomas(2005).Mechanicalbehaviorofmaterials.WavelandPress,Inc.pp. 6–13. ^abcBeer,F.;Johnston,R.;Dewolf,J.;Mazurek,D.(2009).Mechanicsofmaterials.NewYork:McGraw-Hillcompanies. ^"MechanicalPropertiesofMaterials". 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