02.PhysicalquantitiesrepresentedinCartesiancoordinatesystem

강의명 : 금속유동해석특론 (AMB2039 )

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창원대학교 신소재공학부

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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.