12.3 Stress, Strain, and Elastic Modulus - Lumen Learning

Bulk strain is the response of an object or medium to bulk stress. Here, the elastic modulus is called the bulk modulus. Bulk stress causes a change in the ... Skiptomaincontent UniversityPhysicsVolume1 12StaticEquilibriumandElasticity Searchfor: 12.3Stress,Strain,andElasticModulus LearningObjectives Bytheendofthissection,youwillbeableto: Explaintheconceptsofstressandstrainindescribingelasticdeformationsofmaterials Describethetypesofelasticdeformationofobjectsandmaterials Amodelofarigidbodyisanidealizedexampleofanobjectthatdoesnotdeformundertheactionsofexternalforces.Itisveryusefulwhenanalyzingmechanicalsystems—andmanyphysicalobjectsareindeedrigidtoagreatextent.Theextenttowhichanobjectcanbeperceivedasrigiddependsonthephysicalpropertiesofthematerialfromwhichitismade.Forexample,aping-pongballmadeofplasticisbrittle,andatennisballmadeofrubberiselasticwhenacteduponbysquashingforces.However,underothercircumstances,bothaping-pongballandatennisballmaybouncewellasrigidbodies.Similarly,someonewhodesignsprostheticlimbsmaybeabletoapproximatethemechanicsofhumanlimbsbymodelingthemasrigidbodies;however,theactualcombinationofbonesandtissuesisanelasticmedium. Fortheremainderofthischapter,wemovefromconsiderationofforcesthataffectthemotionofanobjecttothosethataffectanobject’sshape.Achangeinshapeduetotheapplicationofaforceisknownasadeformation.Evenverysmallforcesareknowntocausesomedeformation.Deformationisexperiencedbyobjectsorphysicalmediaundertheactionofexternalforces—forexample,thismaybesquashing,squeezing,ripping,twisting,shearing,orpullingtheobjectsapart.Inthelanguageofphysics,twotermsdescribetheforcesonobjectsundergoingdeformation:stressandstrain. Stressisaquantitythatdescribesthemagnitudeofforcesthatcausedeformation.Stressisgenerallydefinedasforceperunitarea.Whenforcespullonanobjectandcauseitselongation,likethestretchingofanelasticband,wecallsuchstressatensilestress.Whenforcescauseacompressionofanobject,wecallitacompressivestress.Whenanobjectisbeingsqueezedfromallsides,likeasubmarineinthedepthsofanocean,wecallthiskindofstressabulkstress(orvolumestress).Inothersituations,theactingforcesmaybeneithertensilenorcompressive,andstillproduceanoticeabledeformation.Forexample,supposeyouholdabooktightlybetweenthepalmsofyourhands,thenwithonehandyoupress-and-pullonthefrontcoverawayfromyou,whilewiththeotherhandyoupress-and-pullonthebackcovertowardyou.Insuchacase,whendeformingforcesacttangentiallytotheobject’ssurface,wecallthem‘shear’forcesandthestresstheycauseiscalledshearstress. TheSIunitofstressisthepascal(Pa).Whenonenewtonofforcepressesonaunitsurfaceareaofonemetersquared,theresultingstressisonepascal: [latex]\text{onepascal}=1.0\,\text{Pa}=\frac{1.0\,\text{N}}{1.0\,{\text{m}}^{2}}.[/latex] IntheBritishsystemofunits,theunitofstressis‘psi,’whichstandsfor‘poundpersquareinch’[latex]({\text{lb/in}}^{2}).[/latex]Anotherunitthatisoftenusedforbulkstressistheatm(atmosphere).Conversionfactorsare [latex]\begin{array}{c}1\,\text{psi}=6895\,\text{Pa}\enspace\text{and}\enspace1\,\text{Pa}=1.450\,×\,{10}^{-4}\text{psi}\\1\,\text{atm}=1.013\,×\,{10}^{5}\text{Pa}=14.7\,\text{psi.}\end{array}[/latex] Anobjectormediumunderstressbecomesdeformed.Thequantitythatdescribesthisdeformationiscalledstrain.Strainisgivenasafractionalchangeineitherlength(undertensilestress)orvolume(underbulkstress)orgeometry(undershearstress).Therefore,strainisadimensionlessnumber.Strainunderatensilestressiscalledtensilestrain,strainunderbulkstressiscalledbulkstrain(orvolumestrain),andthatcausedbyshearstressiscalledshearstrain. Thegreaterthestress,thegreaterthestrain;however,therelationbetweenstrainandstressdoesnotneedtobelinear.Onlywhenstressissufficientlylowisthedeformationitcausesindirectproportiontothestressvalue.Theproportionalityconstantinthisrelationiscalledtheelasticmodulus.Inthelinearlimitoflowstressvalues,thegeneralrelationbetweenstressandstrainis [latex]\text{stress}=\text{(elasticmodulus)}\,×\,\text{strain.}[/latex] Aswecanseefromdimensionalanalysisofthisrelation,theelasticmodulushasthesamephysicalunitasstressbecausestrainisdimensionless. Wecanalsoseefrom(Figure)thatwhenanobjectischaracterizedbyalargevalueofelasticmodulus,theeffectofstressissmall.Ontheotherhand,asmallelasticmodulusmeansthatstressproduceslargestrainandnoticeabledeformation.Forexample,astressonarubberbandproduceslargerstrain(deformation)thanthesamestressonasteelbandofthesamedimensionsbecausetheelasticmodulusforrubberistwoordersofmagnitudesmallerthantheelasticmodulusforsteel. TheelasticmodulusfortensilestressiscalledYoung’smodulus;thatforthebulkstressiscalledthebulkmodulus;andthatforshearstressiscalledtheshearmodulus.Notethattherelationbetweenstressandstrainisanobservedrelation,measuredinthelaboratory.Elasticmoduliforvariousmaterialsaremeasuredundervariousphysicalconditions,suchasvaryingtemperature,andcollectedinengineeringdatatablesforreference((Figure)).Thesetablesarevaluablereferencesforindustryandforanyoneinvolvedinengineeringorconstruction.Inthenextsection,wediscussstrain-stressrelationsbeyondthelinearlimitrepresentedby(Figure),inthefullrangeofstressvaluesuptoafracturepoint.Intheremainderofthissection,westudythelinearlimitexpressedby(Figure). ApproximateElasticModuliforSelectedMaterials Material Young’smodulus [latex]×\,{10}^{10}\text{Pa}[/latex] Bulkmodulus [latex]×\,{10}^{10}\text{Pa}[/latex] Shearmodulus [latex]×\,{10}^{10}\text{Pa}[/latex] Aluminum 7.0 7.5 2.5 Bone(tension) 1.6 0.8 8.0 Bone(compression) 0.9 Brass 9.0 6.0 3.5 Brick 1.5 Concrete 2.0 Copper 11.0 14.0 4.4 Crownglass 6.0 5.0 2.5 Granite 4.5 4.5 2.0 Hair(human) 1.0 Hardwood 1.5 1.0 Iron 21.0 16.0 7.7 Lead 1.6 4.1 0.6 Marble 6.0 7.0 2.0 Nickel 21.0 17.0 7.8 Polystyrene 3.0 Silk 6.0 Spiderthread 3.0 Steel 20.0 16.0 7.5 Acetone 0.07 Ethanol 0.09 Glycerin 0.45 Mercury 2.5 Water 0.22 TensileorCompressiveStress,Strain,andYoung’sModulus Tensionorcompressionoccurswhentwoantiparallelforcesofequalmagnitudeactonanobjectalongonlyoneofitsdimensions,insuchawaythattheobjectdoesnotmove.Onewaytoenvisionsuchasituationisillustratedin(Figure).Arodsegmentiseitherstretchedorsqueezedbyapairofforcesactingalongitslengthandperpendiculartoitscross-section.Theneteffectofsuchforcesisthattherodchangesitslengthfromtheoriginallength[latex]{L}_{0}[/latex]thatithadbeforetheforcesappeared,toanewlengthLthatithasundertheactionoftheforces.Thischangeinlength[latex]\text{Δ}L=L-{L}_{0}[/latex]maybeeitherelongation(whenLislargerthantheoriginallength[latex]{L}_{0})[/latex]orcontraction(whenLissmallerthantheoriginallength[latex]{L}_{0}).[/latex]Tensilestressandstrainoccurwhentheforcesarestretchinganobject,causingitselongation,andthelengthchange[latex]\text{Δ}L[/latex]ispositive.Compressivestressandstrainoccurwhentheforcesarecontractinganobject,causingitsshortening,andthelengthchange[latex]\text{Δ}L[/latex]isnegative. Ineitherofthesesituations,wedefinestressastheratioofthedeformingforce[latex]{F}_{\perp}[/latex]tothecross-sectionalareaAoftheobjectbeingdeformed.Thesymbol[latex]{F}_{\perp}[/latex]thatwereserveforthedeformingforcemeansthatthisforceactsperpendicularlytothecross-sectionoftheobject.Forcesthatactparalleltothecross-sectiondonotchangethelengthofanobject.Thedefinitionofthetensilestressis [latex]\text{tensilestress}=\frac{{F}_{\perp}}{A}.[/latex] Tensilestrainisthemeasureofthedeformationofanobjectundertensilestressandisdefinedasthefractionalchangeoftheobject’slengthwhentheobjectexperiencestensilestress [latex]\text{tensilestrain}=\frac{\text{Δ}L}{{L}_{0}}.[/latex] Compressivestressandstrainaredefinedbythesameformulas,(Figure)and(Figure),respectively.Theonlydifferencefromthetensilesituationisthatforcompressivestressandstrain,wetakeabsolutevaluesoftheright-handsidesin(Figure)and(Figure). Figure12.18Whenanobjectisineithertensionorcompression,thenetforceonitiszero,buttheobjectdeformsbychangingitsoriginallength[latex]{L}_{0}.[/latex](a)Tension:Therodiselongatedby[latex]\text{Δ}L.[/latex](b)Compression:Therodiscontractedby[latex]\text{Δ}L.[/latex]Inbothcases,thedeformingforceactsalongthelengthoftherodandperpendiculartoitscross-section.Inthelinearrangeoflowstress,thecross-sectionalareaoftheroddoesnotchange. Young’smodulusYistheelasticmoduluswhendeformationiscausedbyeithertensileorcompressivestress,andisdefinedby(Figure).Dividingthisequationbytensilestrain,weobtaintheexpressionforYoung’smodulus: [latex]Y=\frac{\text{tensilestress}}{\text{tensilestrain}}=\frac{{F}_{\perp}\text{/}\,A}{\text{Δ}L\,\text{/}\,{L}_{0}}=\frac{{F}_{\perp}}{A}\,\frac{{L}_{0}}{\text{Δ}L}.[/latex] Example CompressiveStressinaPillar Asculptureweighing10,000Nrestsonahorizontalsurfaceatthetopofa6.0-m-tallverticalpillar(Figure).Thepillar’scross-sectionalareais[latex]0{\text{.20m}}^{2}[/latex]anditismadeofgranitewithamassdensityof[latex]{2700\,\text{kg/m}}^{3}.[/latex]Findthecompressivestressatthecross-sectionlocated3.0mbelowthetopofthepillarandthevalueofthecompressivestrainofthetop3.0-msegmentofthepillar. Figure12.19Nelson’sColumninTrafalgarSquare,London,England.(credit:modificationofworkbyCristianBortes) Strategy Firstwefindtheweightofthe3.0-m-longtopsectionofthepillar.Thenormalforcethatactsonthecross-sectionlocated3.0mdownfromthetopisthesumofthepillar’sweightandthesculpture’sweight.Oncewehavethenormalforce,weuse(Figure)tofindthestress.Tofindthecompressivestrain,wefindthevalueofYoung’smodulusforgranitein(Figure)andinvert(Figure). Solution Thevolumeofthepillarsegmentwithheight[latex]h=3.0\,\text{m}[/latex]andcross-sectionalarea[latex]A=0.20\,{\text{m}}^{2}[/latex]is [latex]V=Ah=(0.20\,{\text{m}}^{2})(3.0\,\text{m})=0.60\,{\text{m}}^{3}.[/latex] Withthedensityofgranite[latex]\rho=2.7\,×\,{10}^{3}\,{\text{kg/m}}^{3},[/latex]themassofthepillarsegmentis [latex]m=\rhoV=(2.7\,×\,{10}^{3}\,{\text{kg/m}}^{3})(0.60\,{\text{m}}^{3})=1.60\,×\,{10}^{3}\text{kg}.[/latex] Theweightofthepillarsegmentis [latex]{w}_{p}=mg=(1.60\,×\,{10}^{3}\text{kg})(9.80\,{\text{m/s}}^{2})=1.568\,×\,{10}^{4}\text{N.}[/latex] Theweightofthesculptureis[latex]{w}_{s}=1.0\,×\,{10}^{4}\text{N},[/latex]sothenormalforceonthecross-sectionalsurfacelocated3.0mbelowthesculptureis [latex]{F}_{\perp}={w}_{p}+{w}_{s}=(1.568+1.0)\,×\,{10}^{4}\text{N}=2.568\,×\,{10}^{4}\text{N.}[/latex] Therefore,thestressis [latex]\text{stress}=\frac{{F}_{\perp}}{A}=\frac{2.568\,×\,{10}^{4}\text{N}}{0.20\,{\text{m}}^{2}}=1.284\,×\,{10}^{5}\text{Pa}=\text{128.4kPa.}[/latex] Young’smodulusforgraniteis[latex]Y=4.5\,×\,{10}^{10}\text{Pa}=4.5\,×\,{10}^{7}\text{kPa}.[/latex]Therefore,thecompressivestrainatthispositionis [latex]\text{strain}=\frac{\text{stress}}{Y}=\frac{128.4\,\text{kPa}}{4.5\,×\,{10}^{7}\text{kPa}}=2.85\,×\,{10}^{-6}.[/latex]   Significance Noticethatthenormalforceactingonthecross-sectionalareaofthepillarisnotconstantalongitslength,butvariesfromitssmallestvalueatthetoptoitslargestvalueatthebottomofthepillar.Thus,ifthepillarhasauniformcross-sectionalareaalongitslength,thestressislargestatitsbase. CheckYourUnderstanding FindthecompressivestressandstrainatthebaseofNelson’scolumn. ShowSolution [latex]206.8\,\text{kPa};\,4.6\,×\,{10}^{-5}[/latex] Example StretchingaRod A2.0-m-longsteelrodhasacross-sectionalareaof[latex]0.30\,{\text{cm}}^{2}.[/latex]Therodisapartofaverticalsupportthatholdsaheavy550-kgplatformthathangsattachedtotherod’slowerend.Ignoringtheweightoftherod,whatisthetensilestressintherodandtheelongationoftherodunderthestress? Strategy Firstwecomputethetensilestressintherodundertheweightoftheplatforminaccordancewith(Figure).Thenweinvert(Figure)tofindtherod’selongation,using[latex]{L}_{0}=2.0\,\text{m}.[/latex]From(Figure),Young’smodulusforsteelis[latex]Y=\,2.0\,×\,{10}^{11}\text{Pa}.[/latex] Solution Substitutingnumericalvaluesintotheequationsgivesus [latex]\begin{array}{ccc}\hfill\frac{{F}_{\perp}}{A}&=\hfill&\frac{(550\,\text{kg})(9.8\,{\text{m/s}}^{2})}{3.0\,×\,{10}^{-5}\,{\text{m}}^{2}}=1.8\,×\,{10}^{8}\text{Pa}\hfill\\\hfill\text{Δ}L&=\hfill&\frac{{F}_{\perp}}{A}\,\frac{{L}_{0}}{Y}=(1.8\,×\,{10}^{8}\text{Pa})\,\frac{2.0\,\text{m}}{2.0\,×\,{10}^{11}\text{Pa}}=1.8\,×\,{10}^{-3}\text{m}=\,1.8\,\text{mm.}\hfill\end{array}[/latex] Significance Similarlyasintheexamplewiththecolumn,thetensilestressinthisexampleisnotuniformalongthelengthoftherod.Unlikeinthepreviousexample,however,iftheweightoftherodistakenintoconsideration,thestressintherodislargestatthetopandsmallestatthebottomoftherodwheretheequipmentisattached. CheckYourUnderstanding A2.0-m-longwirestretches1.0mmwhensubjectedtoaload.Whatisthetensilestraininthewire? ShowSolution [latex]5.0\,×\,{10}^{-4}[/latex] Objectscanoftenexperiencebothcompressivestressandtensilestresssimultaneously(Figure).Oneexampleisalongshelfloadedwithheavybooksthatsagsbetweentheendsupportsundertheweightofthebooks.Thetopsurfaceoftheshelfisincompressivestressandthebottomsurfaceoftheshelfisintensilestress.Similarly,longandheavybeamssagundertheirownweight.Inmodernbuildingconstruction,suchbendingstrainscanbealmosteliminatedwiththeuseofI-beams(Figure). Figure12.20(a)Anobjectbendingdownwardexperiencestensilestress(stretching)intheuppersectionandcompressivestress(compressing)inthelowersection.(b)Eliteweightliftersoftenbendironbarstemporarilyduringlifting,asinthe2012Olympicscompetition.(creditb:modificationofworkbyOleksandrKocherzhenko) Figure12.21SteelI-beamsareusedinconstructiontoreducebendingstrains.(credit:modificationofworkby“USArmyCorpsofEngineersEuropeDistrict”/Flickr) Aheavyboxrestsonatablesupportedbythreecolumns.Viewthisdemonstrationtomovetheboxtoseehowthecompression(ortension)inthecolumnsisaffectedwhentheboxchangesitsposition. BulkStress,Strain,andModulus Whenyoudiveintowater,youfeelaforcepressingoneverypartofyourbodyfromalldirections.Whatyouareexperiencingthenisbulkstress,orinotherwords,pressure.Bulkstressalwaystendstodecreasethevolumeenclosedbythesurfaceofasubmergedobject.Theforcesofthis“squeezing”arealwaysperpendiculartothesubmergedsurface(Figure).Theeffectoftheseforcesistodecreasethevolumeofthesubmergedobjectbyanamount[latex]\text{Δ}V[/latex]comparedwiththevolume[latex]{V}_{0}[/latex]oftheobjectintheabsenceofbulkstress.Thiskindofdeformationiscalledbulkstrainandisdescribedbyachangeinvolumerelativetotheoriginalvolume: [latex]\text{bulkstrain}=\frac{\text{Δ}V}{{V}_{0}}.[/latex] Figure12.22Anobjectunderincreasingbulkstressalwaysundergoesadecreaseinitsvolume.Equalforcesperpendiculartothesurfaceactfromalldirections.Theeffectoftheseforcesistodecreasethevolumebytheamount[latex]\text{Δ}V[/latex]comparedtotheoriginalvolume,[latex]{V}_{0}.[/latex] Thebulkstrainresultsfromthebulkstress,whichisaforce[latex]{F}_{\perp}[/latex]normaltoasurfacethatpressesontheunitsurfaceareaAofasubmergedobject.Thiskindofphysicalquantity,orpressurep,isdefinedas [latex]\text{pressure}=p\equiv\frac{{F}_{\perp}}{A}.[/latex] WewillstudypressureinfluidsingreaterdetailinFluidMechanics.Animportantcharacteristicofpressureisthatitisascalarquantityanddoesnothaveanyparticulardirection;thatis,pressureactsequallyinallpossibledirections.Whenyousubmergeyourhandinwater,yousensethesameamountofpressureactingonthetopsurfaceofyourhandasonthebottomsurface,oronthesidesurface,oronthesurfaceoftheskinbetweenyourfingers.Whatyouareperceivinginthiscaseisanincreaseinpressure[latex]\text{Δ}p[/latex]overwhatyouareusedtofeelingwhenyourhandisnotsubmergedinwater.Whatyoufeelwhenyourhandisnotsubmergedinthewateristhenormalpressure[latex]{p}_{0}[/latex]ofoneatmosphere,whichservesasareferencepoint.Thebulkstressisthisincreaseinpressure,or[latex]\text{Δ}p,[/latex]overthenormallevel,[latex]{p}_{0}.[/latex] Whenthebulkstressincreases,thebulkstrainincreasesinresponse,inaccordancewith(Figure).Theproportionalityconstantinthisrelationiscalledthebulkmodulus,B,or [latex]B=\frac{\text{bulkstress}}{\text{bulkstrain}}=-\frac{\text{Δ}p}{\text{Δ}V\,\text{/}\,{V}_{0}}=\text{−}\text{Δ}p\,\frac{{V}_{0}}{\text{Δ}V}.[/latex] Theminussignthatappearsin(Figure)isforconsistency,toensurethatBisapositivequantity.Notethattheminussign[latex](–)[/latex]isnecessarybecauseanincrease[latex]\text{Δ}p[/latex]inpressure(apositivequantity)alwayscausesadecrease[latex]\text{Δ}V[/latex]involume,anddecreaseinvolumeisanegativequantity.Thereciprocalofthebulkmodulusiscalledcompressibility[latex]k,[/latex]or [latex]k=\frac{1}{B}=-\frac{\text{Δ}V\,\text{/}\,{V}_{0}}{\text{Δ}p}.[/latex] Theterm‘compressibility’isusedinrelationtofluids(gasesandliquids).Compressibilitydescribesthechangeinthevolumeofafluidperunitincreaseinpressure.Fluidscharacterizedbyalargecompressibilityarerelativelyeasytocompress.Forexample,thecompressibilityofwateris[latex]4.64\,×\,{10}^{-5}\text{/atm}[/latex]andthecompressibilityofacetoneis[latex]1.45\,×\,{10}^{-4}\text{/atm}.[/latex]Thismeansthatundera1.0-atmincreaseinpressure,therelativedecreaseinvolumeisapproximatelythreetimesaslargeforacetoneasitisforwater. Example HydraulicPress Inahydraulicpress(Figure),a250-litervolumeofoilissubjectedtoa2300-psipressureincrease.Ifthecompressibilityofoilis[latex]2.0\,×\,{10}^{-5}\,\text{/}\,\,\text{atm},[/latex]findthebulkstrainandtheabsolutedecreaseinthevolumeofoilwhenthepressisoperating. Figure12.23Inahydraulicpress,whenasmallpistonisdisplaceddownward,thepressureintheoilistransmittedthroughouttheoiltothelargepiston,causingthelargepistontomoveupward.Asmallforceappliedtoasmallpistoncausesalargepressingforce,whichthelargepistonexertsonanobjectthatiseitherliftedorsqueezed.Thedeviceactsasamechanicallever. Strategy Wemustinvert(Figure)tofindthebulkstrain.First,weconvertthepressureincreasefrompsitoatm,[latex]\text{Δ}p=2300\,\text{psi}=2300\,\text{/}\,14.7\,\text{atm}\approx\,160\,\text{atm},[/latex]andidentify[latex]{V}_{0}=\,250\,\text{L}.[/latex] Solution Substitutingvaluesintotheequation,wehave [latex]\begin{array}{cc}\text{bulkstrain}=\frac{\text{Δ}V}{{V}_{0}}=\frac{\text{Δ}p}{B}=k\text{Δ}p=(2.0\,×\,{10}^{-5}\text{/atm})(160\,\text{atm})=0.0032\hfill\\\text{answer:}\enspace\text{Δ}V=0.0032\,{V}_{0}=0.0032(250\,\text{L})=0.78\,\text{L.}\hfill\end{array}[/latex] Significance Noticethatsincethecompressibilityofwateris2.32timeslargerthanthatofoil,iftheworkingsubstanceinthehydraulicpressofthisproblemwerechangedtowater,thebulkstrainaswellasthevolumechangewouldbe2.32timeslarger. CheckYourUnderstanding Ifthenormalforceactingoneachfaceofacubical[latex]1{\text{.0-m}}^{3}[/latex]pieceofsteelischangedby[latex]1.0\,×\,{10}^{7}\text{N},[/latex]findtheresultingchangeinthevolumeofthepieceofsteel. ShowSolution 63mL ShearStress,Strain,andModulus Theconceptsofshearstressandstrainconcernonlysolidobjectsormaterials.Buildingsandtectonicplatesareexamplesofobjectsthatmaybesubjectedtoshearstresses.Ingeneral,theseconceptsdonotapplytofluids. Sheardeformationoccurswhentwoantiparallelforcesofequalmagnitudeareappliedtangentiallytooppositesurfacesofasolidobject,causingnodeformationinthetransversedirectiontothelineofforce,asinthetypicalexampleofshearstressillustratedin(Figure).Sheardeformationischaracterizedbyagradualshift[latex]\text{Δ}x[/latex]oflayersinthedirectiontangenttotheactingforces.Thisgradationin[latex]\text{Δ}x[/latex]occursinthetransversedirectionalongsomedistance[latex]{L}_{0}.[/latex]Shearstrainisdefinedbytheratioofthelargestdisplacement[latex]\text{Δ}x[/latex]tothetransversedistance[latex]{L}_{0}[/latex] [latex]\text{shearstrain}=\frac{\text{Δ}x}{{L}_{0}}.[/latex] Shearstrainiscausedbyshearstress.Shearstressisduetoforcesthatactparalleltothesurface.Weusethesymbol[latex]{F}_{\parallel}[/latex]forsuchforces.Themagnitude[latex]{F}_{\parallel}[/latex]persurfaceareaAwhereshearingforceisappliedisthemeasureofshearstress [latex]\text{shearstress}=\frac{{F}_{\parallel}}{A}.[/latex] Theshearmodulusistheproportionalityconstantin(Figure)andisdefinedbytheratioofstresstostrain.ShearmodulusiscommonlydenotedbyS: [latex]S=\frac{\text{shearstress}}{\text{shearstrain}}=\frac{{F}_{\parallel}\text{/}\,A}{\text{Δ}x\,\text{/}\,{L}_{0}}=\frac{{F}_{\parallel}}{A}\,\frac{{L}_{0}}{\text{Δ}x}.[/latex] Figure12.24Anobjectundershearstress:Twoantiparallelforcesofequalmagnitudeareappliedtangentiallytooppositeparallelsurfacesoftheobject.Thedashed-linecontourdepictstheresultingdeformation.Thereisnochangeinthedirectiontransversetotheactingforcesandthetransverselength[latex]{L}_{0}[/latex]isunaffected.Sheardeformationischaracterizedbyagradualshift[latex]\text{Δ}x[/latex]oflayersinthedirectiontangenttotheforces. Example AnOldBookshelf Acleaningpersontriestomoveaheavy,oldbookcaseonacarpetedfloorbypushingtangentiallyonthesurfaceoftheverytopshelf.However,theonlynoticeableeffectofthiseffortissimilartothatseenin(Figure),anditdisappearswhenthepersonstopspushing.Thebookcaseis180.0cmtalland90.0cmwidewithfour30.0-cm-deepshelves,allpartiallyloadedwithbooks.Thetotalweightofthebookcaseandbooksis600.0N.Ifthepersongivesthetopshelfa50.0-Npushthatdisplacesthetopshelfhorizontallyby15.0cmrelativetothemotionlessbottomshelf,findtheshearmodulusofthebookcase. Strategy Theonlypiecesofrelevantinformationarethephysicaldimensionsofthebookcase,thevalueofthetangentialforce,andthedisplacementthisforcecauses.Weidentify[latex]{F}_{\parallel}=50.0\,\text{N},\,\text{Δ}x=15.0\,\text{cm},[/latex][latex]{L}_{0}=180.0\,\text{cm},[/latex]and[latex]A=\text{(30.0cm)}\,\text{(90.0cm)}=2700.0\,{\text{cm}}^{2},[/latex]andweuse(Figure)tocomputetheshearmodulus. Solution Substitutingnumbersintotheequations,weobtainfortheshearmodulus [latex]S=\frac{{F}_{\parallel}}{A}\,\frac{{L}_{0}}{\text{Δ}x}=\frac{50.0\,\text{N}}{2700.0\,{\text{cm}}^{2}}\,\frac{180.0\,\text{cm}\text{.}}{15.0\,\text{cm}\text{.}}=\frac{2}{9}\,\frac{\text{N}}{{\text{cm}}^{2}}=\frac{2}{9}\,×\,{10}^{4}\frac{\text{N}}{{\text{m}}^{2}}=\frac{20}{9}\,×\,{10}^{3}\text{Pa}=\text{2.222kPa.}[/latex] Wecanalsofindshearstressandstrain,respectively: [latex]\begin{array}{c}\frac{{F}_{\parallel}}{A}=\frac{50.0\,\text{N}}{2700.0\,{\text{cm}}^{2}}=\frac{5}{27}\,\text{kPa}=\text{185.2Pa}\hfill\\\frac{\text{Δ}x}{{L}_{0}}=\frac{15.0\,\text{cm}}{180.0\,\text{cm}}=\frac{1}{12}=0.083.\hfill\end{array}[/latex] Significance Ifthepersoninthisexamplegavetheshelfahealthypush,itmighthappenthattheinducedshearwouldcollapseittoapileofrubbish.Muchthesameshearmechanismisresponsibleforfailuresofearth-filleddamsandlevees;and,ingeneral,forlandslides. CheckYourUnderstanding ExplainwhytheconceptsofYoung’smodulusandshearmodulusdonotapplytofluids. ShowSolution Fluidshavedifferentmechanicalpropertiesthanthoseofsolids;fluidsflow. Summary Externalforcesonanobject(ormedium)causeitsdeformation,whichisachangeinitssizeandshape.Thestrengthoftheforcesthatcausedeformationisexpressedbystress,whichinSIunitsismeasuredintheunitofpressure(pascal).Theextentofdeformationunderstressisexpressedbystrain,whichisdimensionless. Forasmallstress,therelationbetweenstressandstrainislinear.Theelasticmodulusistheproportionalityconstantinthislinearrelation. Tensile(orcompressive)strainistheresponseofanobjectormediumtotensile(orcompressive)stress.Here,theelasticmodulusiscalledYoung’smodulus.Tensile(orcompressive)stresscauseselongation(orshortening)oftheobjectormediumandisduetoanexternalforcesactingalongonlyonedirectionperpendiculartothecross-section. Bulkstrainistheresponseofanobjectormediumtobulkstress.Here,theelasticmodulusiscalledthebulkmodulus.Bulkstresscausesachangeinthevolumeoftheobjectormediumandiscausedbyforcesactingonthebodyfromalldirections,perpendiculartoitssurface.Compressibilityofanobjectormediumisthereciprocalofitsbulkmodulus. Shearstrainisthedeformationofanobjectormediumundershearstress.Theshearmodulusistheelasticmodulusinthiscase.Shearstressiscausedbyforcesactingalongtheobject’stwoparallelsurfaces. ConceptualQuestions Note:Unlessstatedotherwise,theweightsofthewires,rods,andotherelementsareassumedtobenegligible.Elasticmoduliofselectedmaterialsaregivenin(Figure). Whycanasquirreljumpfromatreebranchtothegroundandrunawayundamaged,whileahumancouldbreakaboneinsuchafall? ShowSolution Incontactwiththeground,stressinsquirrel’slimbsissmallerthanstressinhuman’slimbs. Whenaglassbottlefullofvinegarwarmsup,boththevinegarandtheglassexpand,butthevinegarexpandssignificantlymorewithtemperaturethandoestheglass.Thebottlewillbreakifitisfilleduptoitsverytightcap.Explainwhyandhowapocketofairabovethevinegarpreventsthebottlefrombreaking. Athinwirestrungbetweentwonailsinthewallisusedtosupportalargepicture.Isthewirelikelytosnapifitisstrungtightlyorifitisstrungsothatitsagsconsiderably? ShowSolution tightly Reviewtherelationshipbetweenstressandstrain.Canyoufindanysimilaritiesbetweenthetwoquantities? Whattypeofstressareyouapplyingwhenyoupressontheendsofawoodenrod?Whenyoupullonitsends? ShowSolution compressive;tensile Cancompressivestressbeappliedtoarubberband? CanYoung’smodulushaveanegativevalue?Whataboutthebulkmodulus? ShowSolution no Ifahypotheticalmaterialhasanegativebulkmodulus,whathappenswhenyousqueezeapieceofit? Discusshowyoumightmeasurethebulkmodulusofaliquid. Problems The“lead”inpencilsisagraphitecompositionwithaYoung’smodulusofapproximately[latex]1.0\,×\,{10}^{9}\text{N}\,\text{/}\,{\text{m}}^{2}.[/latex]Calculatethechangeinlengthoftheleadinanautomaticpencilifyoutapitstraightintothepencilwithaforceof4.0N.Theleadis0.50mmindiameterand60mmlong. ShowSolution 0.3mm TVbroadcastantennasarethetallestartificialstructuresonEarth.In1987,a72.0-kgphysicistplacedhimselfand400kgofequipmentatthetopofa610-m-highantennatoperformgravityexperiments.Byhowmuchwastheantennacompressed,ifweconsiderittobeequivalenttoasteelcylinder0.150minradius? Byhowmuchdoesa65.0-kgmountainclimberstretchher0.800-cmdiameternylonropewhenshehangs35.0mbelowarockoutcropping?(Fornylon,[latex]Y=1.35\,×\,{10}^{9}\text{Pa}\text{.)}[/latex] ShowSolution 9.0cm Whenwaterfreezes,itsvolumeincreasesby9.05%.Whatforceperunitareaiswatercapableofexertingonacontainerwhenitfreezes? Afarmermakinggrapejuicefillsaglassbottletothebrimandcapsittightly.Thejuiceexpandsmorethantheglasswhenitwarmsup,insuchawaythatthevolumeincreasesby0.2%.Calculatetheforceexertedbythejuicepersquarecentimeterifitsbulkmodulusis[latex]1.8\,×\,1{0}^{9}N\,\text{/}\,{m}^{2},[/latex]assumingthebottledoesnotbreak. ShowSolution [latex]4.0\,×\,{10}^{2}\,{\text{N/cm}}^{2}[/latex] Adiskbetweenvertebraeinthespineissubjectedtoashearingforceof600.0N.Finditssheardeformation,usingtheshearmodulusof[latex]1.0\,×\,1{0}^{9}\,{\text{N/m}}^{2}.[/latex]Thediskisequivalenttoasolidcylinder0.700cmhighand4.00cmindiameter. Avertebraissubjectedtoashearingforceof500.0N.Findthesheardeformation,takingthevertebratobeacylinder3.00cmhighand4.00cmindiameter.Howdoesyourresultcomparewiththeresultobtainedintheprecedingproblem?Arespinalproblemsmorecommonindisksthaninvertebrae? ShowSolution [latex]0.149\,\text{μm}[/latex] Calculatetheforceapianotunerappliestostretchasteelpianowireby8.00mm,ifthewireisoriginally1.35mlonganditsdiameteris0.850mm. A20.0-m-tallhollowaluminumflagpoleisequivalentinstrengthtoasolidcylinder4.00cmindiameter.Astrongwindbendsthepoleasmuchasahorizontal900.0-Nforceonthetopwoulddo.Howfartothesidedoesthetopofthepoleflex? ShowSolution 0.57mm Acopperwireofdiameter1.0cmstretches1.0%whenitisusedtoliftaloadupwardwithanaccelerationof[latex]2.0\,{\text{m/s}}^{2}.[/latex]Whatistheweightoftheload? Asanoilwellisdrilled,eachnewsectionofdrillpipesupportsitsownweightandtheweightofthepipeandthedrillbitbeneathit.Calculatethestretchinanew6.00-m-longsteelpipethatsupportsa100-kgdrillbitanda3.00-kmlengthofpipewithalinearmassdensityof20.0kg/m.Treatthepipeasasolidcylinderwitha5.00-cmdiameter. ShowSolution 8.59mm Alargeuniformcylindricalsteelrodofdensity[latex]\rho=7.8\,{\text{g/cm}}^{3}[/latex]is2.0mlongandhasadiameterof5.0cm.Therodisfastenedtoaconcretefloorwithitslongaxisvertical.Whatisthenormalstressintherodatthecross-sectionlocatedat(a)1.0mfromitslowerend?(b)1.5mfromthelowerend? A90-kgmountainclimberhangsfromanylonropeandstretchesitby25.0cm.Iftheropewasoriginally30.0mlonganditsdiameteris1.0cm,whatisYoung’smodulusforthenylon? ShowSolution [latex]1.35\,×\,{10}^{9}\text{Pa}[/latex] Asuspenderrodofasuspensionbridgeis25.0mlong.Iftherodismadeofsteel,whatmustitsdiameterbesothatitdoesnotstretchmorethan1.0cmwhena[latex]2.5\,×\,{10}^{4}\text{-kg}[/latex]truckpassesbyit?Assumethattherodsupportsalloftheweightofthetruck. Acopperwireis1.0mlonganditsdiameteris1.0mm.Ifthewirehangsvertically,howmuchweightmustbeaddedtoitsfreeendinordertostretchit3.0mm? ShowSolution 259.0N A100-Nweightisattachedtoafreeendofametallicwirethathangsfromtheceiling.Whenasecond100-Nweightisaddedtothewire,itstretches3.0mm.Thediameterandthelengthofthewireare1.0mmand2.0m,respectively.WhatisYoung’smodulusofthemetalusedtomanufacturethewire? Thebulkmodulusofamaterialis[latex]1.0\,×\,{10}^{11}\,{\text{N/m}}^{2}.[/latex]Whatfractionalchangeinvolumedoesapieceofthismaterialundergowhenitissubjectedtoabulkstressincreaseof[latex]{10}^{7}\,{\text{N/m}}^{2}\text{?}[/latex]Assumethattheforceisapplieduniformlyoverthesurface. ShowSolution 0.01% Normalforcesofmagnitude[latex]1.0\,×\,{10}^{6}\text{N}[/latex]areapplieduniformlytoasphericalsurfaceenclosingavolumeofaliquid.Thiscausestheradiusofthesurfacetodecreasefrom50.000cmto49.995cm.Whatisthebulkmodulusoftheliquid? Duringawalkonarope,atightropewalkercreatesatensionof[latex]3.94\,×\,1{0}^{3}N[/latex]inawirethatisstretchedbetweentwosupportingpolesthatare15.0mapart.Thewirehasadiameterof0.50cmwhenitisnotstretched.Whenthewalkerisonthewireinthemiddlebetweenthepolesthewiremakesanangleof[latex]5.0\text{°}[/latex]belowthehorizontal.Howmuchdoesthistensionstretchthesteelwirewhenthewalkeristhisposition? ShowSolution 1.44cm Whenusingapencileraser,youexertaverticalforceof6.00Natadistanceof2.00cmfromthehardwood-eraserjoint.Thepencilis6.00mmindiameterandisheldatanangleof[latex]20.0\text{°}[/latex]tothehorizontal.(a)Byhowmuchdoesthewoodflexperpendiculartoitslength?(b)Howmuchisitcompressedlengthwise? Normalforcesareapplieduniformlyoverthesurfaceofasphericalvolumeofwaterwhoseradiusis20.0cm.Ifthepressureonthesurfaceisincreasedby200MPa,byhowmuchdoestheradiusofthespheredecrease? ShowSolution 0.63cm Glossary bulkmodulus elasticmodulusforthebulkstress bulkstrain (orvolumestrain)strainunderthebulkstress,givenasfractionalchangeinvolume bulkstress (orvolumestress)stresscausedbycompressiveforces,inalldirections compressibility reciprocalofthebulkmodulus compressivestrain strainthatoccurswhenforcesarecontractinganobject,causingitsshortening compressivestress stresscausedbycompressiveforces,onlyinonedirection elasticmodulus proportionalityconstantinlinearrelationbetweenstressandstrain,inSIpascals normalpressure pressureofoneatmosphere,servesasareferencelevelforpressure pascal(Pa) SIunitofstress,SIunitofpressure pressure forcepressinginnormaldirectiononasurfaceperthesurfacearea,thebulkstressinfluids shearmodulus elasticmodulusforshearstress shearstrain straincausedbyshearstress shearstress stresscausedbyshearingforces strain dimensionlessquantitythatgivestheamountofdeformationofanobjectormediumunderstress stress quantitythatcontainsinformationaboutthemagnitudeofforcecausingdeformation,definedasforceperunitarea tensilestrain strainundertensilestress,givenasfractionalchangeinlength,whichoccurswhenforcesarestretchinganobject,causingitselongation tensilestress stresscausedbytensileforces,onlyinonedirection,whichoccurswhenforcesarestretchinganobject,causingitselongation Young’smodulus elasticmodulusfortensileorcompressivestress LicensesandAttributions CClicensedcontent,SharedpreviouslyOpenStaxUniversityPhysics.Authoredby:OpenStaxCNX.Locatedat:https://cnx.org/contents/[email protected]:[email protected]:CCBY:Attribution.LicenseTerms:Downloadforfreeathttp://cnx.org/contents/[email protected] Previous Next