FIGURE1DaVincishangingbasket2JRLUNDANDJPBYRNEI723T001015723T001015d723 toestablishthetensilestrengthofthewireLeonardoDaVinci1972Parsons1939ParsonspartialtranslationofLeonardoDaVinciste ID: 192146
Download Pdf The PPT/PDF document "ofheterogeneousmaterialpropertiesinrenai..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
ofheterogeneousmaterialpropertiesinrenaissancematerials.IfsuchanexplanationofLeonardo'sapparenterroriscorrect,itraisesthepossibilitythatthemorecomplexnatureofmechanicsformaterialswithheterogeneouspropertiescouldhaveobscuredclassicalmechanicsofmaterialsandthepermutationsofclassicalmechanicsthatwecurrentlyusetoconsidernon-homo-geneousmaterials.ThisshortnoteexaminesLeonardoDaVinci'sstudyofthetensilestrengthofironwire.LEONARDODAVINCI'STENSILETESTSInhisnotebooks(CA,82v-b)LeonardoDaVincidescribesanexperimentforstudyingthetensilestrengthofwire,entitled,``Testingthestrengthofironwiresofvariouslengths''.Intheexperiment,awireofagiventhicknessandlengthwasusedtosuspendabasket.Thebasketwas®lledslowlywithsand,fedfromanadjacentlysuspendedhopper(Fig.1).Whenthewiresuspendingthebasketbreaks,aspringclosesthehopperopening,andthebasketfallsashortdistanceintoahole,soasnottoupsetthebasket.Thesandinthebasketwasthen FIGURE1DaVinci'shangingbasket.2J.R.LUNDANDJ.P.BYRNEI723T001015.723T001015d.723 toestablishthetensilestrengthofthewire(LeonardoDaVinci,1972;Parsons,1939).Parsons'partialtranslationofLeonardoDaVinci'stextfollows(1939):``Theobjectofthistestisto®ndtheloadanironwirecancarry.Attachanironwire2braccialongtosomethingwhichwill®rmlysupportit,thenattachabasketorsimilarcontainertothewireandfeedintothebasketsome®nesandthroughasmallholeplacedattheendofthehopper.Aspringis®xedsothatitwillclosetheholeassoonasthewirebreaks.Thebasketisnotupsetwhilefalling,sinceitfallsthroughaveryshortdistance.Theweightofsandandthelocationofthefractureofthewirearetoberecorded.Thetestisrepeatedseveraltimestochecktheresults.Thenawireof1/2thepreviouslengthistestedandtheadditionalweightitcarriesisrecorded;thenawireof1/4lengthistestedandsoforth,notingtheultimatestrengthandthelocationofthefracture''.CONFLICTWITHCLASSICALMATERIALMECHANICSParsons(1939)notesthattheresultdescribed,whereshorterwiressupportedagreaterweight,con¯ictswiththeclassicaltheoryofmechanicsofmaterials.Classicaltheoryholdsthatthewire'slengthshouldbeirrelevant,sincethestressshouldbethesamealongtheentirelengthofsuchawireandtheweightofthewireshouldbenegligiblecomparedtotheweightinthebasket.Thewire'sdiameter,however,shouldbeimportant,withawireofdoublethediameterhavingatensilestrengthfourtimesgreater.Thisapparentcon¯ictwithclassicalengineeringmechanicsisfundamental.Inthecaseofawireundertension,classicalengineeringmechanicsholdsthatthestressineachunitlengthofthewirewillbeidenticalandthat,therefore,eachunitlengthofwireshouldbehaveidentically.Therefore,theentirelengthofwireshouldhavethesamestrengthasashortlength.However,anincreaseinthediameterofthewireincreasesthecross-sectionalareaofthesupportingwireandthereforereducestheweightthatmustbesupportedbyeachunitareaofthecross-sectionalarea.I.e.,increasingthewire'sdiameterdecreasesthestressinthewire,allowingittosupportagreaterweightbeforesomeultimatefailurestressisreached.Tosolvethisproblemby3LEONARDODAVINCII723T001015.723T001015d.723 classicalengineeringmechanics,oneneedsknowonlythewire'sdiameterandtheultimatestressthewire'smaterialiscapableofsupporting.Thisultimatestressisamaterialpropertythatisassumedtobeconstantthroughoutthematerial.Parsons'explanationoftherecordedresultisthenthatLeonardoDaVinci,inhisnotes,mistakenlyrecordedtheexperimentanditsresults.Parsonssuggeststhatinsteadofmeaning``length'',Leonardomeant``thickness'',andinsteadofmeaning``additional''weight,hemeant``lesser''weightfortheloadcarriedbythesmallerwire.Withthesetwochanges,thetextindicatingtheresultoftheexperimentischangedto:``Thenawireof1/2theprevious[thickness]istestedandthe[lesser]weightitcarriesisrecorded;thenawireof1/4[thickness]istestedandsoforth,notingtheultimatestrengthandthelocationofthefracture''.WhileParsonsnotesthattherearefrequenterrorsinLeonardo'snotebooks,theparticularerrorssuggestedbyParsons,whileplausible,seemunlikely.Twowordsmustbechangedtorelativelyoppositemeanings,andthewordforlengthappearsinseveralplacesinthetext,includingthesectiontitle.ItseemsunlikelythatLeonardoDaVinciactuallystudiedthee€ectsofwirethicknesswhentheentiretestrepeatedlydiscusseswirelength.ANEXPLANATIONAnalternativeexplanationoftheapparenterrorinLeonardo'snotesspringsfromthehomogeneityofthewire'smaterialpropertiesandtheconstancyofthewire'sdiameterassumedbyclassicalengineeringmechanics.TheseassumptionsalsoweremadebyParsons,apparentlyhimselfanengineer.Ifthematerialpropertiesand/orthediameterofthewireareassumedtohavevaryingproperties,thebehaviorofthewirecanbeentirelydi€erent,andinagreementwithLeonardo'snotes.Thisexplanationbasicallyarisesfromtheoldsaying,``Achainisonlyasstrongasitsweakestlink'',or,inthiscase,awireisonlyasstrongasitsweakestcross-section.AssumethattheultimatestrengthofLeonardo'swireisnotconstant.Intermsofclassicalmechanicsofmaterials,thevariationintheultimatestrengthofwireperunitlengthcouldbeduetovariationinactualwirediameterorvariationinthe4J.R.LUNDANDJ.P.BYRNEI723T001015.723T001015d.723 materialproperties.Perhapsbecauseofdefectsorunevennessinthematerialorintheprocessofbeingworkedintowire,someunitlengthsofwirehavedi€erentultimatestrengthsthanotherunitlengths.Thestrengthofalengthofwirewouldthenbethestrengthoftheweakestunitlength.Thelongerthewire,thegreaterthelikelihoodofthewire'slengthcontainingaparticularlyweakunitlength.Underthesecircumstances,longerwirewillbeweaker,onaverage.Thiscanbeshownmorerigorouslywithabitofmathematics.Givenaprobabilitydistributionofthewirematerial'sultimatestress,Pr(su),andtheprobabilitydistributionfortheactualdiameterofthewire,Pr(D),theprobabilityofaunitlengthofwirehavinganultimatestrengthgreaterthanaloadW(unitsofforceorweight)isPr(suD2/4�W),assumingthatthematerial'sultimatestressvariesonlylengthwiseandnotradiallywithinthewire.Ahypotheticaldistribu-tionforPr(W),theprobabilitytheultimatestrengthofaunitlengthofwireisgreaterthansomeloadWisgiveninFigure2.Giventheprobabilitydistributionsforwirediameterandultimatematerialfailurestress,andassumingthatmaterialfailurestressisconstantacrossanygivencross-section,thefollowingequationgivestheprobabilitythataunitlengthofwirecansupportsomeloadW.Pr…W†ˆZsumax0Pr…su†ZDmaxDlPr…D†dDdsu;whereDlˆSQRT(4W/(su))andsumaxistheupperlimitofmaterialfailurestressandDmaxistheupperlimitofwirediameter. FIGURE2Probabilitydistributionforultimatewirestrengthperunitlength.5LEONARDODAVINCII723T001015.723T001015d.723 probabilitythatawireoflengthLwillbeabletosupportmorethansomeloadweighingW,isthesameastheprobabilitythattheallunitlengthsinthewirecansupportthisload.Assumingthattheultimatestrengthofaunitlengthofwireisindependentofneighboringunitlengthsofwire,thentheprobabilityofawireoflengthLcansupportaloadweighingWisgivenby:Pr…WjL†ˆPr…W†L;wherePr(W)istheprobabilitythattheultimatestrengthaunitlengthofwireexceedstheloadWfromFigure2.SincePr(W)islessthanoneforallinterestingloadsonthewire,theprobabilitythatalengthofwireLunitslongcansupporttheloaddecreasessteadilywithincreasingtotalwirelength.Itbecomesincreasinglylikelythatweakcross-sectionsofwirewillexistinwiresoflongerlength.ThisresultwouldagreewithLeonardoDaVinci'sapparent®ndings.Itisanespeciallylikely®ndingsinceLeonardorecommendsrepeatingtheexperimentseveraltimes.Evidently,thestrengthsobtainedfromeachwirelengthwerenotidentical.Thisphenomenonwasformallystudiedinmodernengineeringmechanicsbeginninginthe1920sand1930s.AsreportedbyTimoshenko(1956),theimportanceofmaterialimperfectionsfortheultimatestrengthofmaterialswas®rstpublishedbyGrithintheearly1920s.Thee€ectsofincreasingsize(includingdiameterandlength)onincreasingtheprobabilityofimperfectionsandreductionsinstrengthwasdevelopedstatisticallybyWeibull(1939).ThisbehaviorwasdemonstratedforhighphosphorussteelinexperimentsbyDavidenkovetal.(1947),somewhatsimilartothosebyLeonardoDaVinciandalsoshowingfailurestressdecreasingforlongersamples.Thediscussionshouldnowturntothelikelihoodthatrenaissanceironwirehadnon-constantmaterialpropertiesordiameters.EVIDENCEOFHETEROGENEITYINRENAISSANCEMETALSThematerialpropertiesofwirearesomewhatheterogeneouseventoday.Itiscommon,whenperformingtensilestrengthtestsonironor6J.R.LUNDANDJ.P.BYRNEI723T001015.723T001015d.723 toobtaindi€erentstrengthsforthesamediameterofwire.Consequently,severalreplicatesaretypicallymadeforsuchtesting(ThomasJ.McCann,personalcommunication),somethingLeonardoalsosuggests.Thewire-makingtechniqueusedcouldhaveintroducedhetero-geneitiesinthewire.ThefabricationtechniqueforthewireusedinLeonardo'sexperimentsislikelytohavebeenthedrawingofwire.Here,ironwouldbepulledthroughprogressivelysmallerholesmadeinsteelplatesuntiladesireddiameterwasreached.Thisrequiredgreatforcetobeplacedonthewire.ThistechnologyisdepictedanddiscussedbyBiringuccio(1540).LeonardoDaVinciwasevidentlyfamiliarwiththistechnology.ElsewhereinhisnotebooksLeonardosuggestsmetaldrawingdevicessimilartothosedepictedbyBiringucciofordrawingwire(Heydenreich,1980).Thiswire-makingtechniquecouldhaveintroducedheterogeneousbehaviorinseveralways.Biringuccio'sdiscussionofthemanufactureofironwireindicatestheuseofannealingperiodicallythroughoutthedrawing.Thiswouldhaveinvolvedheatingthewire,followedbysubsequentdrawing.Unevenheatingintheannealingprocessfollowedbyinadequatedrawingcouldintroduceunevenmaterialpropertiesinthewire.Biringuccio'stextalsomentionstheneedtore-shapetheholesinthesteelplatesusedfordrawingwire.Thisindicatesthatthedrawingofwiretendedtodeformthedrawingholesinthesteelplates.Thewearingoftheseholescouldalsointroduceheterogeneitiesinwirecross-sectionasitwasbeingdrawn.Finally,thedrawingofwireasdepictedanddiscussedbyBiringucciooftenrequiredtheuseofaheavyclamptograspthewire®rmlyanddrawitthroughtheholeinthesteelplate.To®rmlygraspthewire,itispossiblethattheclampbitintothewire,deformingthewireandintroducingaweakness.Whenevertheclampwasmoved,tomakethewirelonger,anewweakcross-sectionwouldbeadded.IMPLICATIONSFORTHEHISTORYOFMECHANICSThediscoveryofthemoderntheoryofengineeringmaterialmechanicsrequiredmorethanmeredevelopmentofaconceptualframework7LEONARDODAVINCII723T001015.723T001015d.723 materialgeometry,propertiesofmaterial,andforce.Thedevelopmentofmodernengineeringmechanicsalsorequiredthattheimportantobscuringfactor(ornoise)ofheterogeneousmaterialbehaviorberealized,controlledforexperimentally,andneglectedwithinthebasictheoreticalframework.Thisapproachallowsustotreatheterogeneousmaterialbehaviorasavariationofthesimplerframeworkofhomogeneousmaterialmechanics.LeonardoDaVinci'sexperimentmighto€eraninstancewherethediscoveryofanimportantprincipleofengineeringmechanicswasobscuredbythisfailuretooverlookanelementofrealmaterialbehaviorthatobscuresourunderstandingofthematerial,materialheterogeneity.AcknowledgementsMr.ThomasJ.McCann,aretiredBoeingCo.metallurgist,andProfessorDonBrusharethankedfortheiradviceonthebehaviorofmetals,particularlyonthedrawingofwire.ColinBrownisthankedforhissuggestionsonmodernhistoricaldevelopments.KyranMishisthankedforhisencouragement.ReferencesBiringuccio,Vannoccio(1959)Pirotechnia,translationof1540Italianedition,MITPress,Cambridge,MA,477pp.Duhem,Pierre(1906)TheOriginsofStatics,translatedbyLeneaux,G.F.,Vagliente,V.N.andWagener,G.H.in1991,KluwerAcademicPublishers,Boston,MA,593pp.Heydenreich,Lugwig,H.,BernDibnerandLadislaoReti(1980)LeonardotheInventor,McGraw-HillBookCo.,N.Y.,N.Y.Davidenkov,N.,Shevandin,E.andWittmann,F.(1947)``TheIn¯uenceofSizeontheBrittleStrengthofSteel'',JournalofAppliedMechanics,14,63±67.LeonardoDaVinci(1972)ILibriDiMeccanica,reconstructedfromtheoriginalnotesbyArturoUccelli,KrausReprint,Nendeln,Liechtenstein.Parsons,WilliamB.(1939)EngineersandEngineeringintheRenaissance,MITPress,Cambridge,MA,661pp.Timoshenko,S.(1956)StrengthofMaterialsPartII:AdvancedTheoryandProblems,3rdedition,D.VanNostrandCo.,Inc.,NewYork,NY,pp.394±400.Weibull,W.(1939)``AStatisticalTheoryoftheStrengthofMaterials'',RoyalSwedishInstituteforEngineeringResearch,No.151.8J.R.LUNDANDJ.P.BYRNEI723T001015.723T001015d.723 Civil.Eng.andEnv.Syst.,Vol.00,pp.1±82000OPA(OverseasPublishersAssociation)N.V.ReprintsavailabledirectlyfromthepublisherPublishedbylicenseunderPhotocopyingpermittedbylicenseonlytheGordonandBreachSciencePublishersimprint.PrintedinMalaysia.LEONARDODAVINCI'STENSILESTRENGTHTESTS:IMPLICATIONSFORTHEDISCOVERYOFENGINEERINGMECHANICSJAYR.LUND*andJOSEPHP.BYRNEDepartmentofCivilandEnvironmentalEngineering,UniversityofCalifornia,Davis,CA95616;HonorsProgram,BelmontUniversity,Nashville,TN37212(Received12July2000;In®nalform17July2000)InoneofLeonardoDaVinci'snotebooks,anexperimentisdescribedwherestrengthsintensionaremeasuredforvariouslengthsofwire.Thenotebookindicatesthattheresultsoftheseexperimentswerethatlongerwireswereweakerthanshorterwires.Thisresultde®nesclassicalmechanicsofmaterials.Thiscon¯icthasbeenexplainedasanote-takingfailurebyLeonardo.Thisshortnotedevelopsanalternativeexplanation,basedonthelikelyheterogeneityofthemechanicalpropertiesofthewireandelementaryprobabilitytheory.Thislatterexplanationhasimplicationsforthedicultyanddelaysexperiencedbyearlyinvestigatorsintothemechanicsofmaterials.Keywords:History;Strengthofmaterials;Probability;ImperfectionINTRODUCTIONIttookthousandsofyearsforthelawsofengineeringmechanicstobeuncoveredandunderstoodandLeonardoDaVinci'sworkshowedgreatinsightinthis®eldforitstime(Duhem,1906).However,severalofhisexperiments(c.1500)wouldhavegivengreaterinsightintoengineeringmechanics,buttheirresultswereapparentlyinterpretedincorrectly.Inatleastonecase,theapparentincorrectinterpretationhasanotherexplanation,aphysicalexplanationbasedonthe *Correspondingauthor.e-mail:jrlund@ucdavis.eduI723T001015.723T001015d.723