(1.3.4)-ch0-4 Deformation and strain材料力學(xué)材料力學(xué)

Chapter0Introduction

MechanicsofMaterialsDeformationandStrainWelcometomechanicsofmaterials.Inthelastvideo,forcesandstresseswerediscussed,thisvedioisgoingtodiscussdeformationsandstrains.DeformationTransformationofabodyfromitsoriginalconfigurationtoanewconfiguration.ElasticdeformationPlasticdeformationDeformationisthetransformationofabodyfromitsoriginalconfigurationtoanew

configuration.Itincludeschangesofsizeandshape.Deformationcouldbeclassifiedintoelasticdeformationandplasticdeformation.Elasticdeformationisatransitorydimensionalchangethatexistsonlywhiletheinitiatingstressisappliedanddisappearsimmediatelyuponremovalofthestress.Plasticdeformationisadimensionalchangethatdoesnotdisappearwhentheinitiatingstressisremoved.②Shear①AxialtensionＦzy⑤Combindeddeformation③Tortion④BendingBasicdeformationtypesInthiscourse,deformationcanbeclassifiedintofourcategories,*axialtension,*axialcompression,*shear,*torsion*and*bending.PPNormalstrainShearstrainSizechangeετShapechangeStrainAsaforementioned,deformationincludessizechangeandshapechange.Sizechangeischaracterizedbynormalstrain,denotedbyσ.Shapechangeischaracterizedbyshearstrain,denotedbyε.Nowlet’slookatthedefinitionsofthetwostrains.NormalstrainUndeformedDeformedΔsΔs’。

。。。Dimensionless＋:elongation－:contractionNormalstrainnn’Averagenormalstrain*Formembersbeforedeformation,*let’stakealinesegmentonthemember,andtheoriginallengthofthesegmentisΔs,*afterdeformation,*thelengthofthesegmentbecomesΔs’.WhentheΔsapproacheszero,thetwopointsapproachthesamepoint,*andthenormalstrainatthispointisdefinedspecificallyalongtheorientationoftheoriginallinen,thatisthelimitofΔs’-ΔsdividedbyΔs.Fromthisexpression,itcanbeseenthatthedenominatorandnumeratorbothhaveunitsoflength,therefore,*normalstrainisdimensionless.Sinceεdescribesthesizechange,*apositiveεindicateselongation,*whileanegativeεindicatescontraction.*Bytheway,Δs’-ΔsdividedbyΔsisdefinedasaveragenormalstrain.Shearstrainntn’t’UndeformedDeformedShearstrainInradianChangeintherightangleShearstrain,again,*formembersbeforedeformation,*let’sdrawapairofaxisnandt,nrepresentsnormal,trepresentstangent,thesetwoaxisarealwaysperpendiculartoeachother.*Alongthesetwoaxis,therearetwoshortlengthonthemember.*Afterdeformation,thetwoaxisdeformed,andthetwoshortlengthsdeformedwiththem.*Thisanglenowbecomesθ’.Whenthetwoshortlengthsapproachzero,*shearstrainisdefinedfortheirintersectionpoint,γnt,equalstohalfPI-θ’.*Thesubsrciptionntforγindicatesthisshearstrainisspecificallydefinedfortheorientationofntaxis.Inotherwords,atthesamepoint,shearstrainisdifferentfordifferentorientation.*Theangleherearebothradian,sotheunitsofγisalsoradian.*γsimplydescribesthechangeintherightangle.Whenγispositive,indicatestheanglebecomessmaller,whilenegtiveγindicatingtheanglebecomeslarger.StresselementzxyNormalstress:σx

,

σy

and

σzTriaxialShearstress:τxy

,

τyz

and

τzxNormalstress:σxand

σyShearstress:τxy

BiaxialσyStrainUndeformedzxyΔzΔxΔyDeformedzxyπ/2Normalstrain:εx

,

εy

and

εzShearstrain:γxy

,

γyz

and

γzxγxyεx(1+εx)ΔxSimilarly,*foranundeformedparticlerepresentedbythiscubeelement,ithasthreelength,*ΔxΔxalongxdirection,*Δyalongydirectionand*Δzalongzdirection.*Andallanglesarerightangle.*Thedeformationofthisparticlecanalsobefullycharacterizedbysixstraincomponents,*threenormalstrain,εx,εyandεz,*thatdescibethesizechangealongx,yandzdirection,respectively,*andthreeshearstrain,γxy,γyzandγzx,*thatdescribetheanglechangewithinxyplane,yzplaneandzxplane,respectively.Notethat,strainsarerelatedtotheforcesactingonthecube,whichareknownasstress,normalstressorshearstress.Therefore,afterdeformation,thiselementhasanewsize,*forthelengthalongxdirection,theoriginallength

Δxchangedinto

(1+εx)multipliedbyΔx.Andnewlengthalongyandzdirectionscanbeobtainedsimilarly.Fortheanglechangesofthiselement,*withinthexyplane,theanglehalfπnowchangedintohalfπ-γxy.and

halfπ+γxy.γxyistheshear

strainassociatedwithxyplane.Andangleswithintheyzplaneandzxplanehavesimilarchanges.Example1:Determine①theaveragestrainalongab;②anglechangebetweeenlineabandad.250200adcba’0.025γLet’slookatexample1.itasksyoutodeterminetheaveragestrainalongabandanglechangebetweeenlineabandad.Forthefirstquestion,*fromthedefinitionofaveragestrain,itisequaltothefinallengthΔs‘-originallengthΔs,dividedbytheoriginallengthΔs.Therefore,*itequalstoa’b-ab,dividedbyab,*equalsto0.025dividedby200,*resultingin125times10tominus6.Forthesecondquestion,theanglechangebetweenadandab,*tha