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02.PhysicalquantitiesrepresentedinCartesiancoordinatesystem
강의명 : 금속유동해석특론 (AMB2039 )
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Objectives§Understandmaterialproperty
§Whendefiningamaterialproperty,onehastolookforapairofstimulus(자극) andresponse(반응).
§Whenyouhaveenoughof‘knowledge’(experimentaldata),youcancorrelatethestimulustotheresponseina‘linear’basis.§ Thatis,Response(asaquantifiablevalue)=PropertytimesStimulus(asaquantifiablevalue)§ Astraightforwardexampleis:
§ Stress=ModulustimesStrain§ Strain=compliancetimesStress
Outline§Mathematicalrepresentationofphysicalquantity§ Scalar§ Vector§ Tensors
§CartesianCoordinatesystem
§Coordinatetransformation§ Rulestotransformscalar§ Vectors§ Tensors
§Materialpropertiesrepresentedastensors
§Physicalrulesrepresentedbytensors
Physicalquantities§Q)Whatisaphysicalquantity?§ *Aphysicalquantityisaphysicalpropertyofaphenomenon,body,orsubstance,thatcanbequantifiedbymeasurement.
§Mechanical“physicalquantities”thatweareinterestedin:§ Displacement,force,andvelocityandsoforth.
§Universelooksthesametoallobservers,regardlessofhowtheymove(relativity)§ Thatmeans,aphysicalquantityshouldnotbeaffectedby‘coordinatesystem’.§ Thatalsomeans,thatthephysicalrules(laws)shouldnotbeaffectedby‘coordinatesystem’.§ Wewilllearnhowthiscanbeachievedbyrepresentingphysicalquantitiesusingtensor.
Directiondependenceinphysicalquantities§Somephysicalquantitiesaredirection-dependent.
§Somematerialpropertiesaredirection-dependent.
§Howtorepresenttheproperties(andotherphysicalquantitiessuchasresponse/stimulus)thataredirection-dependent?§ Scalar§ Vector§ Tensor?
§0th ranktensorisscalar
§1st ranktensorisvector
§Therecanbe2nd ranktensor,3rd andsoforth.
Application:motionsensors§Youcaneasilyfindlightsthatarecontrolledbymotionsensorsatyourhome,restroomsandsoforth.
§Manymotionsensorshavepyroelectric(pyro:temperature)materialssuchasleadtitanate (PbTiO3)andtriglycine sulphate.
§Variable(orstimulus)istheheatinputtothematerial
§Responseofthematerialundersuchastimulusiselectricpolarization
§Theycanbelinearlycorrelatedthroughacertainsetofnumbers:
§𝐩 = #𝐏#%
where𝐩 (lowercase)isthecoefficient(slope)
§and𝐏 (uppercase)isthepolarizationvector.
§Avectorhas‘three’components:
§𝑝' =#()#*
,𝑝+ =#(,#*
,𝑝- =#(.#*
.
§Theaboveisshortenedtoaform:𝑝/ =#(0#*
(wherethesubscripti=1,2,3)
Application:quartzoscillators
Quartzcrystalexhibitsaninterestingmaterialproperty:Whenthequartzcrystalisexposedtoelectricity(electricfield),thecrystaldistorts (strain);Thestimulusandtheresponsecanbelinearlycorrelatedthrough:
𝜀/2 = 𝑑/24𝐸4Thisnotationwithsubscriptwillbediscussedlater(Einsteinsummationconvention).Yes,itisthephysicist.
Whattobediscussed§Response=PropertyxStimulus
§Linearproperties
§Axistransformation(coordinatetransformation;changingthecoordinatesystem;itdoesnotchangethephysicalquantity)
§Scalars,vectors,andtensors
§Tensortransformationrule
§Examples
Mathusedtorelatemicrostructure-property
§Materialpropertyisdefinedonthebasisof‘stimulus’-’response’pairing.
§Astimulusissomethingthatonedoestoamaterial:e.g.,load(force)
§Aresponseissomethingthatistheresultofapplyingastimulus:§ Ifyouapply‘force’thematerialwillchangeitsshape(strain)
§Thematerialpropertyisthe‘connection’betweenthestimulusandtheresponse.
§Thematerialpropertyis‘quantifiable’betweendifferenttypesofphysicalquantities(scalar,vector,tensorsandsoforth).
§Infact,scalarandvectorisaspecifictypeof‘tensors’andtensoristhegeneralizedwaytoexpressthestimulus,responseandeventhematerialproperties.
Stimulus->Property->Response§Mathematicalframeworktodescribetheconnectionamongstimulus,propertyandresponse?
§Thepropertyisequivalenttoafunction(P)andthestimulus(F)andresponse(R)arevariables.Thestimulusisalsocalledafieldbecauseinmanycases,thestimulusisactuallyanappliedelectricalfieldormagneticfield,orpressurefield,orforceofsomekind.
§Theresponse(R)isafunctionofthefieldsoweinsertthesymbolPtodesignatethematerialproperty:
§R=R(F)
§R=P(F)
Scalar,linearproperties
§Inmanyinstances,bothstimulusandresponsearescalar quantities,meaningthatyouonlyneedone
numbertoprescribethem.SpecificHeatisanexampleofascalarproperty.
ExampleaboveshownofYoung’smodulusandShearmodulusversustemperatureforTi-6Al-4V,courtesyofBrianGockel
• Tofurthersimplify,somepropertiesarelinear,whichmeansthattheresponseislinearlyproportionaltothestimulus:R = P ´ F.However,thepropertyisgenerallydependentonothervariables.• Example:elasticstiffnessintension/compression changeswith,orisafunctionoftemperature,whichweindicatebyadding“(T)”afterthesymbolfortheproperty,“P”:
s = E(T) ´ eº R = P(T) ´ F.
Scalars,vectors,tensors§Scalar:=quantitythatrequiresonlyonenumber,e.g.density,mass,specificheat.Equivalenttoazero-ranktensor.
§Vector:=quantitythathasdirectionaswellasmagnitude,e.g.velocity,current,magnetization;requires3numbersorcoefficients(in3D).Equivalenttoafirst-ranktensor.
§Tensor:=quantitythatrequireshigherorderdescriptionsbutisthesamephysicalquantity,nomatterwhatcoordinatesystemisusedtodescribeit,e.g.stress,strain,elasticmodulus;Ageneralconcepttomathematicallyexpressthephysicalquantitiesandtheassociatedmaterialproperties.
Anisotropy§Anisotropy asawordsimplymeansthatsomethingvarieswithdirection.§AnisotropyisfromtheGreek:aniso =different,varying;tropos =direction.§Almostallcrystallinematerialsareanisotropic;manymaterialsareengineeredtotakeadvantageoftheiranisotropy(beercans,turbineblades)§Oldertextsusetrigonometricfunctionstodescribeanisotropybuttensorsofferageneraldescriptionwithwhichitismucheasiertoperformcalculations.§Formaterials,weknowthatsomepropertiesareanisotropic.Thismeansthatseveralnumbers,orcoefficients,areneededtodescribetheproperty- onenumberisnotsufficientfullyquantifytheanisotropicproperty.§Elasticityisanimportantexampleofapropertythat,whenexaminedinsinglecrystals,isoftenhighlyanisotropic.Infact,thelowerthecrystalsymmetry,thegreatertheanisotropyislikelytobe.§Nomenclature: ingeneral,weneedtousetensors todescribefieldsandproperties.Thesimplestcaseofatensorisascalar whichisallweneedforisotropicproperties.Thenext“level”oftensorisavector,e.g.electriccurrent.
§Q)Whichonedoyouthinkismoreanisotropic:cubicorhexagonal-closed-packed.Why?
Coordinatesystemandbasisvectors§앞으로좌표계를설명할때좌표계의근간이되는방향들을 normalvector (즉크기가 1인벡터)로표현.
§Cartesiancoordinatesystem은 orthonormalcoordinatesystem
§서로수직한세 normalvector로표현이가능하다.
𝐞+
𝐞-
𝐞'
그세 normalvector들을 basisvector로칭하겠다.그리고각각 e', e+, e-로나타내겠다.
가령, vectorv는위의 basisvector에평행한성분들로구성할수있다.
𝒗 = 𝑣'𝐞𝟏 + 𝑣+𝐞𝟐 + 𝑣-𝐞𝟑
Lettersinboldfaceisvector;
Physicallawandmaterialproperties§Stressandstrainare‘linearly’correlatedformostmetalswithintheirelasticregime.
§ Q:Howisstressrelatedtostrain?§ Q:CanyouexplainHooke’slaw?
TensorandMatrix(행렬)§ThefirstconfusionIencounteredwhenIlearnedtensorwasmostlyduetotheconfusionofthetensorwithmatrix.§ ItisbecauseIlearned2nd ranktensor,andthe2nd tensorisgenerallywrittenina3x3matrix.
§Remember:matrixisamethodtorepresentcertainvaluesinatableandissovalidonlyfor2nd ranktensors(or2dimensionaltensorsconsistingofsinglerowandsinglecolumn)
§Tensorsarewhattransformliketensors§ Bylearningthewaytensor‘transforms’,you’lllearnthemostimportantaspectoftensor.§ Wewilllearnaboutthistransformationrulesappliedfortensorslater.
§EinsteinusedTensors:§ “Inthe20thcentury,thesubjectcametobeknownas tensoranalysis,andachievedbroaderacceptancewiththeintroductionof Einstein'stheoryof generalrelativityaround1915.Generalrelativityisformulatedcompletelyinthelanguageoftensors.Einsteinhadlearnedaboutthem,withgreatdifficulty,fromthegeometerMarcelGrossmann."
Recap§Materialproperty’connects’thestimulusanditspairedresponse.
§For1Dlinear:§ R=PxS
§Scalarandvectorsarejusttwospecialcasesoftensors– theyare0rankand1st ranktensors,respectively.
§Forthecaseof‘anisotropic’andlinear(thisactuallywillbelearnedlater):§ 𝑅 = 𝑃𝑆§ 𝑅/2 = 𝑃/2𝑆§ 𝑅/2 = 𝑃/24𝑆4§ 𝑅/2 = 𝑃/24C𝑆4C ..Andmore.(We’lllearnaboutthisconventionslater)
Referencesandacknowledgements§References.
§Acknowledgements§ Someoftheslidesarebasedontheslidesofprof.A.D.Rollett @CarnegieMellonUniversity.Hekindlypermittedthereuseofhisslides.
§ SomeimagespresentedinthislecturematerialswerecollectedfromWikipedia.