NPTEL __ Mechanical Engineering - Strength of Materials
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4/28/2015 Lecture 35 http://nptel.ac.in/courses/112107146/lects%20&%20picts/image/lect35/lecture35.htm 1/5 LECTURE 35 THEORIES OF ELASTIC FAILURE While dealing with the design of structures or machine elements or any component of a particular machine the physical properties or chief characteristics of the constituent materials are usually found from the results of laboratory experiments in which the components are subject to the simple stress conditions. The most usual test is a simple tensile test in which the value of stress at yield or fracture is easily determined. However, a machine part is generally subjected simultaneously to several different types of stresses whose actions are combined therefore, it is necessary to have some basis for determining the allowable working stresses so that failure may not occur. Thus, the function of the theories of elastic failure is to predict from the behavior of materials in a simple tensile test when elastic failure will occur under any conditions of applied stress. A number of theories have been proposed for the brittle and ductile materials. Strain Energy: The concept of strain energy is of fundamental importance in applied mechanics. The application of the load produces strain in the bar. The effect of these strains is to increase the energy level of the bar itself. Hence a new quantity called strain energy is defined as the energy absorbed by the bar during the loading process. This strain energy is defined as the work done by load provided no energy is added or subtracted in the form of heat. Some times strain energy is referred to as internal work to distinguish it from external work ‘W'. Consider a simple bar which is subjected to tensile force F, having a small element of dimensions dx, dy and dz. The strain energy U is the area covered under the triangle A three dimension state of stress respresented by 1 , 2 and 3 may be throught of consisting of two distinct state of stresses i.e Distortional state of stress Deviatoric state of stress and dilational state of stress Hydrostatic state of stresses.
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Transcript of NPTEL __ Mechanical Engineering - Strength of Materials
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4/28/2015 Lecture35
http://nptel.ac.in/courses/112107146/lects%20&%20picts/image/lect35/lecture35.htm 1/5
LECTURE35
THEORIESOFELASTICFAILURE
Whiledealingwiththedesignofstructuresormachineelementsoranycomponentofaparticularmachinethephysicalpropertiesorchiefcharacteristicsoftheconstituentmaterialsareusuallyfoundfromtheresultsof laboratoryexperimentsinwhichthecomponentsaresubjecttothesimplestressconditions.Themostusualtestisasimpletensiletestinwhichthevalueofstressatyieldorfractureiseasilydetermined.
However,amachinepartisgenerallysubjectedsimultaneouslytoseveraldifferenttypesofstresseswhoseactionsarecombinedtherefore,itisnecessarytohavesomebasisfordeterminingtheallowableworkingstressessothatfailuremaynotoccur.Thus,thefunctionofthetheoriesofelasticfailureistopredictfromthebehaviorofmaterialsinasimpletensiletestwhenelasticfailurewilloccurunderanyconditionsofappliedstress.
Anumberoftheorieshavebeenproposedforthebrittleandductilematerials.
StrainEnergy:Theconceptofstrainenergyisoffundamentalimportanceinappliedmechanics.Theapplicationoftheloadproducesstrain in thebar.Theeffectof thesestrains is to increasetheenergy levelof thebar itself.Henceanewquantitycalledstrainenergyisdefinedastheenergyabsorbedbythebarduringtheloadingprocess.Thisstrainenergyisdefinedastheworkdonebyloadprovidednoenergyisaddedorsubtractedintheformofheat.SometimesstrainenergyisreferredtoasinternalworktodistinguishitfromexternalworkW'.ConsiderasimplebarwhichissubjectedtotensileforceF,havingasmallelementofdimensionsdx,dyanddz.
ThestrainenergyUistheareacoveredunderthetriangle
Athreedimensionstateofstressrespresentedby1,2and3maybethroughtofconsistingoftwodistinctstateofstressesi.eDistortionalstateofstress
Deviatoricstateofstressanddilationalstateofstress
Hydrostaticstateofstresses.
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Thus,Theenergywhichisstoredwithinamaterialwhenthematerialisdeformedistermedasastrainenergy.ThetotalstrainenergyUr
UT=Ud+UH
UdisthestrainenergyduetotheDeviatoricstateofstressandUHisthestrainenergyduetotheHydrostaticstateofstress.Futher,itmaybenotedthatthehydrostaticstateofstressresultsinchangeofvolumewhereasthedeviatoricstateofstressresultsinchangeofshape.
DifferentTheoriesofFailure:Thesearefivedifferenttheoriesoffailureswhicharegenerallyused
(a)MaximumPrincipalstresstheory(duetoRankine)
(b)Maximumshearstresstheory(GuestTresca)
(c)MaximumPrincipalstrain(Saintvenant)Theory
(d)Totalstrainenergyperunitvolume(Haigh)Theory
(e)ShearstrainenergyperunitvolumeTheory(VonMises&Hencky)
Inallthesetheoriesweshallassume.
Yp=stressattheyieldpointinthesimpletensiletest.
1,2,3thethreeprincipalstressesinthethreedimensionalcomplexstateofstresssystemsinorderofmagnitude.
(a)MaximumPrincipalstresstheory:
Thistheoryassumethatwhenthemaximumprincipalstressinacomplexstresssystemreachestheelastic limitstressinasimpletension,failurewilloccur.
Thereforethecriterionforfailurewouldbe
1=yp
Foratwodimensionalcomplexstresssystem1isexpressedas
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Wherex,yandxyarethestressesintheanygivencomplexstresssystem.
(b)Maximumshearstresstheory:
Thistheorystatesthattehfailurecanbeassumedtooccurwhenthemaximumshearstressinthecomplexstresssystemisequaltothevalueofmaximumshearstressinsimpletension.
Thecriterionforthefailuremaybeestablishedasgivenbelow:
Forasimpletensioncase
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(c)MaximumPrincipalstraintheory:
ThisTheoryassumesthatfailureoccurswhenthemaximumstrainforacomplexstateofstresssystembecomesequalstothestrainatyieldpointinthetensiletestforthethreedimensionalcomplexstateofstresssystem.
Fora3dimensionalstateofstresssystemthetotalstrainenergyUtperunitvolumeinequaltothetotalworkdonebythesystemandgivenbytheequation
(d)Totalstrainenergyperunitvolumetheory:
Thetheoryassumesthatthefailureoccurswhenthetotalstrainenergyforacomplexstateofstresssystemisequaltothatattheyieldpointatensiletest.
Therefore,thefailurecriterionbecomes
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Itmaybenotedthatthistheorygivesfairbygoodresultsforductilematerials.
(e)Maximumshearstrainenergyperunitvolumetheory:
This theorystates that thefailureoccurswhenthemaximumshearstrainenergycomponent for thecomplexstateofstresssystemisequaltothatattheyieldpointinthetensiletest.
Hencethecriterionforthefailurebecomes
Asweknowthatageneralstateofstresscanbebrokenintotwocomponentsi.e,
(i)Hydrostaticstateofstress(thestrainenergyassociatedwiththehydrostaticstateofstressisknownasthevolumetricstrainenergy)
(ii)DistortionalorDeviatoricstateofstress(Thestrainenergyduetothisisknownastheshearstrainenergy)
Asweknowthatthestrainenergyduetodistortionisgivenas
Thisisthedistortionstrainenergyforacomplexstateofstress,thisistobeequaledtothemaximumdistortionenergyinthesimpletensiontest.Inordertogetwemayassumethatoneoftheprincipalstresssay(1)reachestheyieldpoint(yp)ofthematerial.Thus,puttinginaboveequation2=3=0wegetdistortionenergyforthesimpletesti.e
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