LECTURE STRESS 9 - STRAIN RELATIONS Stress Strain Relations:The Hook's law,states that within the elastic limits the stress is proportionalto the strain since for most materialsit is impossible to describethe entirestress~straincurvewith simple mathematicalexpression,in anygivenproblemthe behaviorof the materialsis representedbyan idealizedstress ~ straincurve,whichemphasizesthose aspectsof the behaviorswhichare most importantis that particularproblem. (i) Linearelastic material: Alinear elasticmaterialis one in whichthe strainis proportionalto stressas shownbelow: L: Line-any Eidzlc material Thereare also othertypesof idealizedmodelsof materialbehavior. (ii) RigidMaterials: it is the one whichdonotexperienceanystrainregardlessofthe appliedstress. (iii) Perfectlyp|astic(non-strainhardening): L Aperfectlyplastici.e non-strainhardeningmaterialis shownbelow: (iv) RigidPlastic materia|(strainhardening): Rib: Arigid plasticmateriali.e strain hardeningis depictedin thegure below: (v) ElasticPerfectlyPlastic material: L Theelasticperfectlyplasticmaterialis havingthe characteristicsas shownbelow: (vi)Elastic Plasticmaterial: Theelasticplasticmaterialexhibits a stressVs straindiagramas depictedinthegure below: 0' ElasticStress strainRelations: Previously stress- strainrelationswere considered forthe specialcaseof a uniaxialloadingi.e. onlyone component of stressi.e.the axialor normal component of stresswas comingintopicture.in this sectionwe shall generalizethe elasticbehavior, so as to arriveat the relationswhichconnectall the six components ofstresswiththesixcomponents ofelasticstress.Futher, we wouldrestrict overselves tolinearlyelasticmaterial. Beforewritingdowntherelations letus introduce a termISOTROPY ISOTROPIC: iftheresponseofthematerialis independent oftheorientation oftheloadaxisofthesample,thenwe saythatthe materialis isotropic or in otherwords we cansaythatisotropy ofa materialina characteristics, whichgivesus theinformation thatthe properties arethesameinthethreeorthogonal directions xy; onthe otherhandiftheresponseis dependenton orientation it is knownas anisotropic. Examplesofanisotropic materials, whoseproperties aredifferent indifferentdirections are (i) Wood (ii) Fibrereinforced plastic (iii)Reinforced concrete HOMOGENlUS:Amaterial is homogenous if it has the same composition throughour body.Hencethe elasticproperties are the same at everypointin the body. However, theproperties neednotto be thesame in all thedirection forthematerialto be homogenous. isotropic materialshavethe sameelasticproperties in allthe directions. Therefore, thematerialmustbe bothhomogenous andisotropic in ordertohavethelateralstrainsto be sameat everypointina particular component. Generalized Hook's Law:Weknowthatforstressesnotgreaterthan theproportional limit. : '3 : _lelaterall E '53:-iia|| Theseequationexpresses therelationship betweenstressandstrain(Hook's law)foruniaxialstateofstressonlywhenthestressis notgreaterthantheproportional limit.In orderto analyzethe deformational effectsproduced byall thestresses,we shallconsiderthe effectsof oneaxialstressat a time.Sincewe presumably are dealingwithstrainsoftheorderofonepercentorless.Theseeffectscanbe superimposed arbitrarily. Thefigurebelowshowsthegeneraltriaxialstateofstress. Letusconsidera casewhenoxaloneis acting.itwillcausean increaseindimension inX-direction whereasthedimensions inyandzdirection willbe decreased. Thereforethe resultingstrainsin threedirectionsare at Ur EyE,EI-|.LEY;Ez-|_L.Ey U _ I. _ U3. _ UI Ey'¬uE:('_i-'-EiE2'_i-LE Nowletus considerthe stress 62actingalone,thus the strains producedare U2 ez¬,eyo _ . _ u.ez.eI- |.LEz o o E2-¬2.53: l-LEz.E,.c" l-I-E2 Thusthe total strain in any one direction is -5 |-'- Ex'EI_E[:U3r+5z:l In a similar manner, the total strain in the y and 2 directions become 5,}:EE(a_E +52 E2 =UE2_E(UI we (2) (3) in the followinganalysisshear stresseswere not considered.it can be shownthat for an isotropicmaterial's a shear stress will produceonlyits correspondingshear strain and will not influence the axial strain. Thus, we can write Hook's law for the individual shear strains and shear stresses in the following manner. r,c.,= E I3 (4) Ty:=3* (5) T =T3 (6) isotropicmaterials.Whenthese equationsare used as written,the strains can be completelydeterminedfrom knownvaluesof the stresses.To engineersthe plane stress situationis ofmuch relevance( i.e.dz = rxz= ryz= 0 ), Thusthenthe aboveset ofequationsreducesto E,=_~-_v E E 5 Ey_:ir|-[BI Ul-"3 'l E2:HiJand$W: E E G Their inverse relations can he also determined and are given as E :3I = (_1_|-L2) [EI +|.Le] \,r E E +|.LE V ilwtzliy 3) U = TXr=G'7'rr Hook's law is probablythe most well knownand widelyused constitutiveequationsfor an engineeringmaterials.However,we can not say that all the engineering materialsare linearelasticisotropicones. Becausenowin the presenttimes,the newmaterialsare beingdevelopedevewday.Manyuseful materialsexhibitnonlinear responseand are notelastictoo. PlaneStress: in manyinstancesthe stress situationis less complicatedfor exampleifwe pull one long thin wire ofuniform sectionand examine~ small parallepiped wherex axiscoincideswiththe axisofthe wire 3/ Z Eng 5iyar W X So if we take the xy planethenox ,oy ,rXywill be the onlystress componentsactingon the parrallepiped.This combinationof stress componentsis calledthe plane stress situation Aplane stress maybe dened as a stress conditionin whichall componentsassociatedwith a givendirection( i.e the zdirectionin this example)are zero 52:12: 'rzY=U Planestrain: if we focus our attentionon a bodywhose particlesall lie in the same planeand whichdeformsonly in this plane.This deformsonly in this plane.This typeof deformationis calledas the planestrain,so for such a situation. e Z: yzx= yzy= 0 and the non- zerotermswouldbe e X,e y & yxy i.e. if straincomponentsex, ey andyxyand angle6 are specied,the straincomponentsex,e yandyxy'with respectto some otheraxescan be determined. ELASTIC CONSTANT S in consideringthe elasticbehaviorof an isotropicmaterialsunder,normal,shear and hydrostaticloading,we introducea total offour elasticconstantsnamelyE, G, K, and y. ltturns outthat notall ofthese are independentto the others.in fact,givenanytwoofthem,the othertwo can be foundout.Letus dene these elasticconstants (i) E = Young's Modulus of Rigidity = Stress/strain (ii) G = ShearModulusor Modulusof rigidity = Shearstress /Shear strain (iii) y= Possionsratio = lateralstrain/longitudinalstrain (iv)K = BulkModulusof elasticity = Volumetricstress /Volumetricstrain