ChapterChapter 1313 -- ElasticityElasticity AA PowerPointPowerPoint PresentationPresentation byby PaulPaul E.E. Tippens,Tippens, ProfessorProfessor ofof PhysicsPhysics SouthernSouthern PolytechnicPolytechnic StateState UniversityUniversity
© 2007 ChapterChapter 13.13. ElasticityElasticity BUNGEE jumping utilizes a long elastic strap which stretches until it reaches a maximum length that is proportional to the weight of the jumper. The elasticity of the strap determines the amplitude of the resulting vibrations. If the elastic limit for the strap is
Photo © Vol. 10 exceeded, the rope PhotoDisk/Getty will break. Objectives:Objectives: AfterAfter completioncompletion ofof thisthis module,module, youyou shouldshould bebe ableable to:to:
•• DemonstrateDemonstrate youryour understandingunderstanding ofof elasticityelasticity,, elasticelastic limitlimit,, stressstress,, strainstrain,, andand ultimateultimate strengthstrength.. •• WriteWrite andand applyapply formulasformulas forfor calculatingcalculating YoungYoung’’ss modulusmodulus,, shearshear modulusmodulus,, andand bulkbulk modulusmodulus.. •• SolveSolve problemsproblems involvinginvolving eacheach ofof thethe parametersparameters inin thethe aboveabove objectives.objectives. ElasticElastic PropertiesProperties ofof MatterMatter
AnAn elasticelastic bodybody isis oneone thatthat returnsreturns toto itsits originaloriginal shapeshape afterafter aa deformation.deformation.
Golf Ball Rubber Band Soccer Ball ElasticElastic PropertiesProperties ofof MatterMatter
AnAn inelasticinelastic bodybody isis oneone thatthat doesdoes notnot returnreturn toto itsits originaloriginal shapeshape afterafter aa deformation.deformation.
DoughDough oror BreadBread ClayClay InelasticInelastic BallBall ElasticElastic oror Inelastic?Inelastic?
An elastic collision loses In an inelastic collision, no energy. The deform- energy is lost and the ation on collision is fully deformation may be restored. permanent. (Click it.) AnAn ElasticElastic SpringSpring
AA springspring isis anan exampleexample ofof anan elasticelastic bodybody thatthat cancan bebe deformeddeformed byby stretching.stretching.
AA restoringrestoring forceforce,, FF,, actsacts inin thethe directiondirection oppositeopposite thethe displacementdisplacement ofof thethe FF oscillatingoscillating body.body. x FF == --kxkx HookeHooke’’ss LawLaw WhenWhen aa springspring isis stretched,stretched, therethere isis aa restoringrestoring forceforce thatthat isis proportionalproportional toto thethe displacement.displacement.
FF == -kx-kx x TheThe springspring constantconstant F F kk isis aa propertyproperty ofof thethe k m springspring givengiven by:by: x
TheThe springspring constantconstant kk isis aa measuremeasure ofof thethe elasticityelasticity ofof thethe spring.spring. StressStress andand StrainStrain Stress refers to the cause of a deformation, and strain refers to the effect of the deformation.
The downward force F causes the displacement x.
FF Thus, the stress is the force; x the strain is the elongation. TypesTypes ofof StressStress FF AA tensiletensile stressstress occursoccurs whenwhen equalequal andand oppositeopposite forcesforces areare W directeddirected awayaway fromfrom eacheach other.other. TensionTension
AA compressivecompressive stressstress occursoccurs W whenwhen equalequal andand oppositeopposite F forcesforces areare directeddirected towardtoward F eacheach other.other. CompressionCompression SummarySummary ofof DefinitionsDefinitions
StressStress isis thethe ratioratio ofof anan appliedapplied forceforce FF toto thethe areaarea AA overover whichwhich itit acts:acts: F N lb Stress Units : Pa or A min.22 StrainStrain isis thethe relativerelative changechange inin thethe dimensionsdimensions oror shapeshape ofof aa bodybody asas thethe resultresult ofof anan appliedapplied stress:stress:
Examples:Examples: ChangeChange inin lengthlength perper unitunit length;length; changechange inin volumevolume perper unitunit volume.volume. LongitudinalLongitudinal StressStress andand StrainStrain
ForFor wires,wires, rods,rods, andand bars,bars, therethere isis aa longitudinallongitudinal L A F stressstress F/AF/A thatthat producesproduces aa A changechange inin lengthlength perper unitunit L length.length. InIn suchsuch cases:cases:
F L Stress Strain A L ExampleExample 1.1. AA steelsteel wirewire 1010 mm longlong andand 22 mmmm inin diameterdiameter isis attachedattached toto thethe ceilingceiling andand aa 200200--NN weightweight isis attachedattached toto thethe end.end. WhatWhat isis thethe appliedapplied stress?stress? FirstFirst findfind areaarea ofof wire:wire: D22(0.002 m) L A A F 44 A A = 3.14 x 10-6 m2 L
F 200 N Stress Stress A 3.14 x 10-6 m 2 6.37 x 107 Pa ExampleExample 11 (Cont.)(Cont.) AA 1010 mm steelsteel wirewire stretchesstretches 3.083.08 mmmm duedue toto thethe 200200 NN load.load. WhatWhat isis thethe longitudinallongitudinal strain?strain?
Given:Given: LL == 1010 m;m; LL == 3.083.08 mmmm L 0.00308 m L Srain L 10 m
L Longitudinal Strain 3.08 x 10-4 TheThe ElasticElastic LimitLimit TheThe elasticelastic limitlimit isis thethe maximummaximum stressstress aa bodybody cancan experienceexperience withoutwithout becomingbecoming permanentlypermanently deformed.deformed.
2 m FF 2 m
Okay W F W BeyondBeyond limitlimit Stress A W IfIf thethe stressstress exceedsexceeds thethe elasticelastic limit,limit, thethe finalfinal lengthlength willwill bebe longerlonger thanthan thethe originaloriginal 22 m.m. TheThe UltimateUltimate StrengthStrength TheThe ultimateultimate strengthstrength isis thethe greatestgreatest stressstress aa bodybody cancan experienceexperience withoutwithout breakingbreaking oror rupturing.rupturing.
2 m FF
W F W W Stress A W W IfIf thethe stressstress exceedsexceeds thethe ultimateultimate strengthstrength,, thethe stringstring breaks!breaks! ExampleExample 2.2. TheThe elasticelastic limitlimit forfor steelsteel isis 2.482.48 xx 10108 PaPa.. WhatWhat isis thethe maximummaximum weightweight thatthat cancan bebe supportedsupported withoutwithout exceedingexceeding thethe elasticelastic limit?limit? Recall: A = 3.14 x 10-6 m2 F L A Stress 2.48 x 108 Pa F A A 8 L F = (2.48 x 10 Pa) A
F = (2.48 x 108 Pa)(3.14 x 10-6 m2) FF == 779779 NN ExampleExample 2(Cont.)2(Cont.) TheThe ultimateultimate strengthstrength forfor steelsteel isis 40894089 xx 10108 PaPa.. WhatWhat isis thethe maximaxi-- mummum weightweight thatthat cancan bebe supportedsupported withoutwithout breakingbreaking thethe wire?wire? Recall: A = 3.14 x 10-6 m2 F L A Stress 4.89 x 108 Pa F A A 8 L F = (4.89 x 10 Pa) A
F = (4.89 x 108 Pa)(3.14 x 10-6 m2) FF == 15361536 NN TheThe ModulusModulus ofof ElasticityElasticity
ProvidedProvided thatthat thethe elasticelastic limitlimit isis notnot exceeded,exceeded, anan elasticelastic deformationdeformation (strain)(strain) isis directlydirectly proportionalproportional toto thethe magnitudemagnitude ofof thethe appliedapplied forceforce perper unitunit areaarea (stress)(stress)..
stress Modulus of Elasticity strain ExampleExample 3.3. InIn ourour previousprevious example,example, thethe stressstress appliedapplied toto thethe steelsteel wirewire waswas 6.376.37 xx 10107 PaPa andand thethe strainstrain waswas 3.083.08 xx 1010-4.. FindFind thethe modulusmodulus ofof elasticityelasticity forfor steel.steel.
Stress 6.37 x 107 Pa Modulus L Strain 3.08 x 10-4
ModulusModulus == 207207 xx 101099 PaPa L
ThisThis longitudinallongitudinal modulusmodulus ofof elasticityelasticity isis calledcalled Young’sYoung’s ModulusModulus andand isis denoteddenoted byby thethe symbolsymbol YY.. YoungYoung’’ss ModulusModulus
ForFor materialsmaterials whosewhose lengthlength isis muchmuch greatergreater thanthan thethe widthwidth oror thickness,thickness, wewe areare concernedconcerned withwith thethe longitudinallongitudinal modulusmodulus ofof elasticity,elasticity, oror YoungYoung’’ss ModulusModulus (Y)(Y).. longitudinal stress Young' s modulus longitudinal strain
F / AFL lb Y Units: Pa or L / LAL in.2 ExampleExample 4:4: YoungYoung’’ss modulusmodulus forfor brassbrass isis 8.968.96 xx 101011PaPa.. AA 120120--NN weightweight isis attachedattached toto anan 8 m 88--mm lengthlength ofof brassbrass wire;wire; findfind thethe increaseincrease inin length.length. TheThe diameterdiameter isis 1.51.5 mmmm.. L FirstFirst findfind areaarea ofof wire:wire: 120 N D22(0.0015 m) A A = 1.77 x 10-6 m2 44 FLFL YL or ALAY ExampleExample 4:4: (Continued)(Continued)
YY == 8.968.96 xx 101011 Pa;Pa; FF == 120120 N;N; 8 m LL == 88 m;m; AA == 1.771.77 xx 1010-6 mm2 FF == 120120 N;N; LL == ?? L FLFL YL or 120 N AL AY FL (120 N)(8.00 m) L AY (1.77 x 10-6 m 2 )(8.96 x 1011 Pa)
Increase in length: LL== 0.6050.605 mmmm ShearShear ModulusModulus AA shearingshearing stressstress altersalters onlyonly thethe shapeshape ofof thethe body,body, leavingleaving thethe volumevolume unchanged.unchanged. ForFor example,example, considerconsider equalequal andand oppositeopposite shearingshearing forcesforces FF actingacting onon thethe cubecube below:below: A dd F F ll
TheThe shearingshearing forceforce FF producesproduces aa shearingshearing angleangle TheThe angleangle isis thethe strainstrain andand thethe stressstress isis givengiven byby F/AF/A asas before.before. CalculatingCalculating ShearShear ModulusModulus dd AA StressStress isis F F F ll forceforce perper Stress unitunit area:area: A d TheThe strainstrain isis thethe angleangle Strain expressedexpressed inin radiansradians:: l TheThe shearshear modulusmodulus SS isis defineddefined asas thethe ratioratio ofof thethe shearingshearing stressstress F/AF/A toto thethe shearingshearing strainstrain :: F A TheThe shearshear modulus:modulus: S UnitsUnits areare inin Pascals.Pascals. ExampleExample 5.5. AA steelsteel studstud ((SS == 8.278.27 xx 101010PaPa)) 11 cmcm inin diameterdiameter projectsprojects 44 cmcm fromfrom thethe wall.wall. AA 36,00036,000 NN shearingshearing forceforce isis appliedapplied toto thethe end.end. WhatWhat isis thethe defectiondefection dd ofof thethe stud?stud? D22(0.01 m) ll A 44 dd Area:Area: AA == 7.857.85 xx 1010-5 mm2 FF FA FA Fl Fl Sd; d l Ad AS (36,000 N)(0.04 m) d dd== 0.2220.222 mmmm (7.85 x 10-5 m 2 )(8.27 x 1010 Pa) VolumeVolume ElasticityElasticity NotNot allall deformationsdeformations areare linlinear.ear. SometimesSometimes anan appliedapplied stressstress F/AF/A resultsresults inin aa decreasedecrease ofof volumevolume.. InIn suchsuch cases,cases, therethere isis aa bulkbulk modulusmodulus BB ofof elasticity.elasticity.
Volume stress F A B Volume strain V V
TheThe bulkbulk modulusmodulus isis negativenegative becausebecause ofof decreasedecrease inin V.V. TheThe BulkBulk ModulusModulus
Volume stress F A B Volume strain V V
SinceSince F/AF/A isis generallygenerally pressurepressure PP,, wewe maymay write:write:
PPV B VV/ V
UnitsUnits remainremain inin PascalsPascals (Pa)(Pa) sincesince thethe strainstrain isis unitless.unitless. ExampleExample 7.7. AA hydrostatichydrostatic presspress containscontains 55 litersliters ofof oil.oil. FindFind thethe decreasedecrease inin volumevolume ofof thethe oiloil ifif itit isis subjectedsubjected toto aa pressurepressure ofof 30003000 kPakPa.. (Assume(Assume thatthat BB == 17001700 MPaMPa.).) PPV B VV/ V PV (3 x 106 Pa)(5 L) V B (1.70 x 109 Pa)
Decrease in V; milliliters (mL): VV == -8.82-8.82 mLmL Summary:Summary: ElasticElastic andand InelasticInelastic
AnAn elasticelastic bodybody isis oneone thatthat returnsreturns toto itsits originaloriginal shapeshape afterafter aa deformation.deformation. An elastic collision loses no energy. The deform- ation on collision is fully restored.
AnAn inelasticinelastic bodybody isis oneone thatthat doesdoes notnot returnreturn toto itsits originaloriginal shapeshape afterafter aa deformation.deformation.
In an inelastic collision, energy is lost and the deformation may be permanent. SummarySummary TypesTypes ofof StressStress FF AA tensiletensile stressstress occursoccurs whenwhen equalequal andand oppositeopposite forcesforces areare W directeddirected awayaway fromfrom eacheach other.other. TensionTension
AA compressivecompressive stressstress occursoccurs W whenwhen equalequal andand oppositeopposite F forcesforces areare directeddirected towardtoward F eacheach other.other. CompressionCompression SummarySummary ofof DefinitionsDefinitions
StressStress isis thethe ratioratio ofof anan appliedapplied forceforce FF toto thethe areaarea AA overover whichwhich itit acts:acts: F N lb Stress Units : Pa or A min.22 StrainStrain isis thethe relativerelative changechange inin thethe dimensionsdimensions oror shapeshape ofof aa bodybody asas thethe resultresult ofof anan appliedapplied stress:stress:
Examples:Examples: ChangeChange inin lengthlength perper unitunit length;length; changechange inin volumevolume perper unitunit volume.volume. LongitudinalLongitudinal StressStress andand StrainStrain
ForFor wires,wires, rods,rods, andand bars,bars, therethere isis aa longitudinallongitudinal L A F stressstress F/AF/A thatthat producesproduces aa A changechange inin lengthlength perper unitunit L length.length. InIn suchsuch cases:cases:
F L Stress Strain A L TheThe ElasticElastic LimitLimit
TheThe elasticelastic limitlimit isis thethe maximummaximum stressstress aa bodybody cancan experienceexperience withoutwithout becomingbecoming permanentlypermanently deformed.deformed.
TheThe UltimateUltimate StrengthStrength TheThe ultimateultimate strengthstrength isis thethe greatestgreatest stressstress aa bodybody cancan experienceexperience withoutwithout breakingbreaking oror rupturing.rupturing. YoungYoung’’ss ModulusModulus
ForFor materialsmaterials whosewhose lengthlength isis muchmuch greatergreater thanthan thethe widthwidth oror thickness,thickness, wewe areare concernedconcerned withwith thethe longitudinallongitudinal modulusmodulus ofof elasticity,elasticity, oror YoungYoung’’ss ModulusModulus YY.. longitudinal stress Young' s modulus longitudinal strain
F / AFL lb Y Units: Pa or L / LAL in.2 TheThe ShearShear ModulusModulus dd AA StressStress isis F F F ll forceforce perper Stress unitunit area:area: A d TheThe strainstrain isis thethe angleangle Strain expressedexpressed inin radiansradians:: l TheThe shearshear modulusmodulus SS isis defineddefined asas thethe ratioratio ofof thethe shearingshearing stressstress F/AF/A toto thethe shearingshearing strainstrain :: F A TheThe shearshear modulus:modulus: S UnitsUnits areare inin Pascals.Pascals. TheThe BulkBulk ModulusModulus
Volume stress F A B Volume strain V V
SinceSince F/AF/A isis generallygenerally pressurepressure PP,, wewe maymay write:write:
PPV B VV/ V
UnitsUnits remainremain inin PascalsPascals (Pa)(Pa) sincesince thethe strainstrain isis unitless.unitless. CONCLUSION:CONCLUSION: ChapterChapter 1313 -- ElasticityElasticity