Simulationsofplasticprocessing:

elasto-plasticdeformationPart1:tensorsandcontinummechanicsTeacher:法法

QQ號：2306847727，office423-2A區SimulationsofplasticprocessIntroductionInthisclass,westudymaterialsatamacroscopicscale(宏觀尺度):betweendislocation(位錯)andsample(樣品)Weusemathematics,mechanics,materialspropertiesHongguanchiduIntroductionInthisclass,wePlan(2weeks,6classes)Part1:tensorsandcontinuummechanics 張量和連續介質力學Part2:elasticdeformation

彈性變形Part3:plasticdeformation 塑性變形ZhanglianghelianxvjiezhilixueSuxingbianxingtanxingbianxingPlan(2weeks,6classes)PartContentMaterialelement

材料單元Stresstensor

應力張量Straintensor

應變張量Yinglizhangliang

ContentMaterialelement材料單元YSeveralyearsago,youhavelearnNewtonianmechanic:thesolidswereundeformable,andsometimemodeledasmaterialpoint.Inmaterialscience,westudythemechanicalbehavior(力學行為)insideadeformablebodyofmaterialbodyofmaterial=sample,beam,mechanicpiece,bridge…MaterialelementLixuexingweiSeveralyearsago,youhaveleMaterialelementThematerialisacontinuum(連續體)i.e.thematteriscontinuouslydistributedinthebody(novoid,nocracks(無空位，無裂紋)Sothebodycanbesub-dividedintoinfinitesimalelements(無窮小的單位元)WukongweiWuliewenWuqiongxiaodedanweiyuanlianxvtiMaterialelementThematerialiMaterialelementThebodyofmaterialisdividedintomaterialelementsThesematerialelementsarealsocalledinfinitesimalelements(forderivation?/?x)orrepresentativevolumeelementsEachmaterialelementcanhavedifferentproperties（性能）andmechanicalresponse(stress,strain…)MaterialelementThebodyofmaMaterialelementMaterial=body=solidcutthematerialwithavirtualplane(虛擬的平面)Materialelement(1faceisintheplane)X0YZBasisxvniHowtodefine1element:MaterialelementMaterial=body=MaterialelementWewanttostudyin3dimensions,souse3planesdefinedwiththebasis:X0YZBasisPlanenormaltoYPlanenormaltoZPlanenormaltoXMaterialelementWewanttostuMaterialelementdxX0YZdydzWestudythebehaviorof1elementofthematerial,ofdimension1×1×1.elementsizehasnounit,butthesizeoftheelementmuchbiggerthantheatoms(原子),tobeacontinuum.Thesizedependonwhatwewanttostudy,ofwhataccuracyweneed.YuanziMaterialelementdxX0YZdydzWesMaterialelementInmechanic,thematerialelementcontainmanygrainsIncrystalplasticity(晶體的塑性),thematerialelementissmallerthanagrainIncontinuousmaterials,theforcesarerelatedtothebody’sdeformationthroughconstitutiveequations(本構方程)Theinternalforcesarecontinuousandthematerialdisplacementisaderivablefunction(可導的函數)JingtidesuxingBengoufangchengKedaodehanshuMaterialelementInmechanic,tStresstensorWesawthatthebodyofmaterialcanbesubdividedintomaterialelementsWewanttostudywhathappentothebodyofmaterialwhensomeforcesareappliedtothebodySowemuststudywhathappentoeachelementofthematerialWestudyhowtheforcesaretransmittedtotheelementsStresstensorWesawthatthebStresstensorHypothesisonappliedforces:Thesolidisinequilibrium(donotmove)SomeforcesPareappliedonthesolidEquilibriumofforcesandmomentofforces/Torques(應力與動量平衡)P1P2P4P3P1+P2+P3+P4=0Equilibriuminrotation:M1+M2+M3+M4=0(Mi=Pi^OA,PiappliedonA)X0YZYingliyudongliangpinghengStresstensorHypothesisonappStresstensorOriginofthestresstensor:solidandinternalstressWecutthesolid(bodyofmaterial)alongaplaneP1P2P4P3X0YZVirtualplanetocutthematerialVirtualsurfaceinsidethematerialStresstensorOriginofthestrStresstensorFristheforceappliedontheleftpartbytherightpartExternalforcescanbeappliedonpoints,surfaces,orvolume;butinternalforcesaresurfaceforces.TheforcesFrandFlareequivalentonthe2sidesofthesolidP1P2P4P3FlrlFrStresstensorFristheforceaStresstensorTheforceFr=-(P1+P2)isdiscomposedintoanormalcomponentFn(垂直分量)andashearcomponentFt(剪切分量)intheplane(tangentialforce)Incompression,thenormalforceFnisnegativeFrl-FrlP1P2FnFtChuizhifenliangJianqiefenliangStresstensorTheforceFr=-(P1StresstensorOnthesurfacedSoftheelement,wehavetheforceperunitsurfaceσr=stress(應力)TheforceontherightfaceoftheelementdS=dz×dyis:∫S

σr×dS=∫y∫zσr×dy×dz=-Fr-FrlFnFtσrσnσtσlX0YZdSStresstensorOnthesurfacedSStresstensorIftheinternalforceishomogeneous(均勻的)onthesurfaceS,F=S×σUsuallythestressdistributioninthematerialisnothomogeneousForcomplexcases,wemayuseFiniteElementsprograms(有限元軟件)tosolvetheconstitutiveequations(forexampletheHookelaw)andcomputethestress(andstrain)inalltheelements.JunyunYouxianyuanruanjianStresstensorIftheinternalfStresstensorWeapplythevirtualcutanalysisonthe3facesofthematerialelement:weget3vectorof stress=force/surfaceσrσtσfStresstensorWeapplythevirtStresstensorWediscomposeσt,σr,andσf,into3componentinthebasis:weget9componentsforthestresses.σyzσzzσzyσxzσxyσxxσzxσyyσyx0XZYσrσtσfStresstensorWediscomposeσt,StresstensorNormalstresses(正應力)σii,i=x,y,zShearstresses

(剪切應力)σij,i,j=x,y,z,i≠jσxyisthestresscomponentinthedirectionxontheplaneofnormaly.σyzσzzσzyσxzσxyσxxσzxσyyσyx0XZYStresstensorNormalstresses(StresstensorThestresstensorisusedtocomputetheforceFonafaceofnormaln(andsurfaceS=1):F(n)=σ×n=×SoistheforceonthefacenormaltoX:σr=-Frσxx

σxy

σxz

σyx

σyy

σyz

σzx

σzy

σzznxnynzσxx

σyx

σzxStresstensorThestresstensorσzyσxyσyyStresstensorTheStresstensoronamaterialelementdependonthepositionσ=σ(x,y,z)isafunctionofthepositionx,y,zoftheelementStressesonfacesY：0XZYσzy+?σzy/?y*dyσxy+?σxy/?y*dyσyy+?σyy/?y*dyσzyσxyσyyStresstensorTheStreStresstensorStressesonfacesXandZIncontinuummechanics,thestressesarecontinuous.FiniteElementmethodsuseequilibriumprinciplestocomputeheterogeneous(不均勻的)stressesanddeformationsσzx+?σzx/?x*dxσxx+?σxx/?x*dxσyx+?σyx/?x*dxσzz+?σzz/?z*dzσxz+?σxz/?z*dzσyz+?σyz/?z*dz0XZYStresstensorStressesonfacesStresstensorThestressesthroughthematerialelementmustbeinequilibrium:Equilibriuminforces: div(σ)=0Equilibriuminmomentofforces(rotation):

σ=σT

thetensorissymmetric(σij=σji)?σxx/?x+?σxy/?y+?σxz/?z=0?σyx/?x+?σyy/?y+?σyz/?z=0?σzx/?x+?σzy/?y+?σzz/?z=0StresstensorThestressesthroStresstensorWhateveristhestresstensor,thereisabasisinwhichthereisnoshear:

=>Inthisbasis,thestressarecalled‘Principalstresses’0XZY0e1e3e2σI000σII000σIIIσxx

σxy

σxz

σyx

σyy

σyz

σzx

σzy

σzzStresstensorWhateveristhesStresstensorChangeofbasis(=referential,coordinatesystem):simpleshear=planestress0XZY0e1e3Y=e2σzx=10MPaσxz=10MPaσ11=10MPaσ33=-10MPaσXYZ=00100001000σ123=100000000-10StresstensorChangeofbasis(Demonstration:ExpressamatrixintoanotherbasisTransformationMatrixfrom(XYZ)to(e1e2e3):T= (expresseiinsideXYZ)σ123=T-1σxyzT1/√2 0 -1/√20 1 01/√2 0 1/√2Demonstration:ExpressamatriStresstensorPressurep(球應力): p=Stressdeviatortensor(應力偏張量)s:s=σ–pId=Deviatoricofσ=>s11+s22+s33=0Thestressdeviatortensorisusefulbecausepressuredonotproduceplasticdeformationσxx+σyy+σzz3σxx-pσxy

σxz

σyx

σyy-pσyz

σzx

σzy

σzz-pyinglipianzhangliangQiuyingliStresstensorPressurep(球應力)StresstensorEquivalentvonMisesstress(等效應力)σ=σVM=√3/2(sxx2+syy2+szz2+2×sxy2+2×sxz2+2×syz2)AlsocalledEquivalenttensilestress

becauseinuniaxialtensionp=σxx/3:

σVM=√3/2(σxx-p)2+p2+p2)=√3/2(4/9+1/9+1/9)σxx =σxxUsetocomparedifferentstressstates(應力狀態)UsewiththevonMisesequivalentstrain:εVM=√2/3(εxx2+εyy2+εzz2+2×εxy2+2×εxz2+2×εyz2)DengxiaoyingliYinglizhuangtaiStresstensorEquivalentvonMiStresstensorVonMisesstressinplanestresses(平面應力)p=(σx+σy)/3σ=σVM=√3/2((σx-p)2+(σy-p)2-p2)=√σx2+σy2-σxσyDrawσVM=constantσxσxσyσyσxσyσVM=σx=σyσx=-σy=σVM/√3StresstensorVonMisesstressStresstensorVonMisesstressinprincipalstressbasis(i.e.ifi≠j,σij=0)TheconstantvonMisesstressisacylinder(圓柱體)tiltedat45oofftheaxisσyσxσz(111)YuanzhutiStresstensorVonMisesstressStresstensorComputeσαandταasafunctionofα

Mohrcircleinplanestressταismaximumforα=45oσxxσyyσαταταα2ασxxσyyσασα=σxsin2α+σycos2ατα=σysinαcosα-σxcosαsinαStresstensorComputeσαandταchangeofbasisforMohrcircleT= σxyz=σ123=T-1σxyzT=sinα -cosα 0cosα sinα 00 0 1σx 0 00 σy 00 0 0

σxsin2α+σycos2α σysinαcosα-σxcosαsinα 0 σysinαcosα-σxcosαsinα σxcos2α+σysin2α 00 0 0changeofbasisforMohrcirclStraintensorWesawthebodyofmaterialsandthematerialelementsaresubmittedtoforces.Inmaterialscience,asinthereallife,thematerialscandeformWeneedtomeasureandquantifythesedeformationsUsuallythesedeformationsareverysmall…StraintensorWesawthebodyoStraintensorOriginofthestraintensor:thedisplacementofmaterialDisplacementfield(位移場)u(x,y,z)foreachpointX(x,y,z)ofthematerialu=eyezbeforeafteru(x,y,z)0X(x,y,z)exuxuyuzWeiyichangStraintensorOriginofthestrStraintensorDisplacementgradienttensor(位移梯度張量)

G=grad(u)=Thesymmetric(對稱的)partofGis

ε=1/2(G+GT)εisthestraintensor(ordeformationtensor)Theanti-symmetric(非對稱的)partofGisω=1/2(G-GT)ωistherotationtensor(orspintensor)?ux/?x?ux/?y?ux/?z?uy/?x?uy/?y?uy/?z?uz/?x?uz/?y?uz/?zWeiyitiduzhangliangFeiduichengdeStraintensorDisplacementgradStraintensorε=?ux1?ux?uy1?ux?uz

?x2?y?x2?z?x1?uy?ux?uy1?uy?uz

2?x?y?y2?z?y1?uz?ux1?uz?uy?uz2?x?z2?y?z?z

(+)

(+)(+)

(+)(+)

(+)Straintensor?uxStraintensorPrincipalstrain:asthestraintensorissymmetric,thereisacoordinatesystem(basis，坐標系)wherei≠j=>εij=0Inthisbasisεxx,εyyandεzzarecalltheprincipalstrain(eigenvaluesofthematrix，特征值)

=YXe1e2ZuobiaoxiTezhengzhiStraintensorPrincipalstrain:StraintensorExamplesofstraintensor:Pureshearu=ux=α×y G=ε=Planestraintensionu=

ε=0α/20α/200000-ε000ε0000YXε×(x-a)ε×(y-b)0e2e1(a,b,0)sin(α)≈tan(α)≈α(inradian)0α0000000StraintensorExamplesofstraiStrainalongtheelongationdirection‘y’forhomogeneousdeformation:uy=(lf-l0)/l0×yGyy==(lf-l0)/l0=εyyStraintensoryxl0lf?uy?yStrainalongtheelongationdiTotalstrainalongauniformelongationpath:Currentlengthlatt,l+dlatt+dtε=dl/listheelongationduringdtεtotal=∫lilfdl/l=[ln(l)]lilf=ln(lf)-ln(li)=ln()StraintensorlflillilfTotalstrainalongauniformeStraintensorChangeofvolume(2D):V=dx×dy×dzdV=[(dx+εxxdx)(dy+εyydy)dz-dxdydz]/dxdydz ≈εxx+εyy(weneglectεxx×εyy)dxdyεxxdxεyydyεxx×dy×dzεxx×dy×dzStraintensorChangeofvolumeStraintensorUniformdeformation(均勻變形):thestraintensorisconstantthroughthematerial(donotdependon(x,y,z))εisthesumoftheelasticandplasticstrainInplasticdeformation,thevolume

isconstant(體積不變),soεxx+εyy+εzz=0Sheardoesnotproduceanychangeofvolume.StraintensorUniformdeformatiSimulationsofplasticprocessing:

elasto-plasticdeformationPart1:tensorsandcontinummechanicsTeacher:法法

QQ號：2306847727，office423-2A區SimulationsofplasticprocessIntroductionInthisclass,westudymaterialsatamacroscopicscale(宏觀尺度):betweendislocation(位錯)andsample(樣品)Weusemathematics,mechanics,materialspropertiesHongguanchiduIntroductionInthisclass,wePlan(2weeks,6classes)Part1:tensorsandcontinuummechanics 張量和連續介質力學Part2:elasticdeformation

彈性變形Part3:plasticdeformation 塑性變形ZhanglianghelianxvjiezhilixueSuxingbianxingtanxingbianxingPlan(2weeks,6classes)PartContentMaterialelement

材料單元Stresstensor

應力張量Straintensor

應變張量Yinglizhangliang

ContentMaterialelement材料單元YSeveralyearsago,youhavelearnNewtonianmechanic:thesolidswereundeformable,andsometimemodeledasmaterialpoint.Inmaterialscience,westudythemechanicalbehavior(力學行為)insideadeformablebodyofmaterialbodyofmaterial=sample,beam,mechanicpiece,bridge…MaterialelementLixuexingweiSeveralyearsago,youhaveleMaterialelementThematerialisacontinuum(連續體)i.e.thematteriscontinuouslydistributedinthebody(novoid,nocracks(無空位，無裂紋)Sothebodycanbesub-dividedintoinfinitesimalelements(無窮小的單位元)WukongweiWuliewenWuqiongxiaodedanweiyuanlianxvtiMaterialelementThematerialiMaterialelementThebodyofmaterialisdividedintomaterialelementsThesematerialelementsarealsocalledinfinitesimalelements(forderivation?/?x)orrepresentativevolumeelementsEachmaterialelementcanhavedifferentproperties（性能）andmechanicalresponse(stress,strain…)MaterialelementThebodyofmaMaterialelementMaterial=body=solidcutthematerialwithavirtualplane(虛擬的平面)Materialelement(1faceisintheplane)X0YZBasisxvniHowtodefine1element:MaterialelementMaterial=body=MaterialelementWewanttostudyin3dimensions,souse3planesdefinedwiththebasis:X0YZBasisPlanenormaltoYPlanenormaltoZPlanenormaltoXMaterialelementWewanttostuMaterialelementdxX0YZdydzWestudythebehaviorof1elementofthematerial,ofdimension1×1×1.elementsizehasnounit,butthesizeoftheelementmuchbiggerthantheatoms(原子),tobeacontinuum.Thesizedependonwhatwewanttostudy,ofwhataccuracyweneed.YuanziMaterialelementdxX0YZdydzWesMaterialelementInmechanic,thematerialelementcontainmanygrainsIncrystalplasticity(晶體的塑性),thematerialelementissmallerthanagrainIncontinuousmaterials,theforcesarerelatedtothebody’sdeformationthroughconstitutiveequations(本構方程)Theinternalforcesarecontinuousandthematerialdisplacementisaderivablefunction(可導的函數)JingtidesuxingBengoufangchengKedaodehanshuMaterialelementInmechanic,tStresstensorWesawthatthebodyofmaterialcanbesubdividedintomaterialelementsWewanttostudywhathappentothebodyofmaterialwhensomeforcesareappliedtothebodySowemuststudywhathappentoeachelementofthematerialWestudyhowtheforcesaretransmittedtotheelementsStresstensorWesawthatthebStresstensorHypothesisonappliedforces:Thesolidisinequilibrium(donotmove)SomeforcesPareappliedonthesolidEquilibriumofforcesandmomentofforces/Torques(應力與動量平衡)P1P2P4P3P1+P2+P3+P4=0Equilibriuminrotation:M1+M2+M3+M4=0(Mi=Pi^OA,PiappliedonA)X0YZYingliyudongliangpinghengStresstensorHypothesisonappStresstensorOriginofthestresstensor:solidandinternalstressWecutthesolid(bodyofmaterial)alongaplaneP1P2P4P3X0YZVirtualplanetocutthematerialVirtualsurfaceinsidethematerialStresstensorOriginofthestrStresstensorFristheforceappliedontheleftpartbytherightpartExternalforcescanbeappliedonpoints,surfaces,orvolume;butinternalforcesaresurfaceforces.TheforcesFrandFlareequivalentonthe2sidesofthesolidP1P2P4P3FlrlFrStresstensorFristheforceaStresstensorTheforceFr=-(P1+P2)isdiscomposedintoanormalcomponentFn(垂直分量)andashearcomponentFt(剪切分量)intheplane(tangentialforce)Incompression,thenormalforceFnisnegativeFrl-FrlP1P2FnFtChuizhifenliangJianqiefenliangStresstensorTheforceFr=-(P1StresstensorOnthesurfacedSoftheelement,wehavetheforceperunitsurfaceσr=stress(應力)TheforceontherightfaceoftheelementdS=dz×dyis:∫S

σr×dS=∫y∫zσr×dy×dz=-Fr-FrlFnFtσrσnσtσlX0YZdSStresstensorOnthesurfacedSStresstensorIftheinternalforceishomogeneous(均勻的)onthesurfaceS,F=S×σUsuallythestressdistributioninthematerialisnothomogeneousForcomplexcases,wemayuseFiniteElementsprograms(有限元軟件)tosolvetheconstitutiveequations(forexampletheHookelaw)andcomputethestress(andstrain)inalltheelements.JunyunYouxianyuanruanjianStresstensorIftheinternalfStresstensorWeapplythevirtualcutanalysisonthe3facesofthematerialelement:weget3vectorof stress=force/surfaceσrσtσfStresstensorWeapplythevirtStresstensorWediscomposeσt,σr,andσf,into3componentinthebasis:weget9componentsforthestresses.σyzσzzσzyσxzσxyσxxσzxσyyσyx0XZYσrσtσfStresstensorWediscomposeσt,StresstensorNormalstresses(正應力)σii,i=x,y,zShearstresses

(剪切應力)σij,i,j=x,y,z,i≠jσxyisthestresscomponentinthedirectionxontheplaneofnormaly.σyzσzzσzyσxzσxyσxxσzxσyyσyx0XZYStresstensorNormalstresses(StresstensorThestresstensorisusedtocomputetheforceFonafaceofnormaln(andsurfaceS=1):F(n)=σ×n=×SoistheforceonthefacenormaltoX:σr=-Frσxx

σxy

σxz

σyx

σyy

σyz

σzx

σzy

σzznxnynzσxx

σyx

σzxStresstensorThestresstensorσzyσxyσyyStresstensorTheStresstensoronamaterialelementdependonthepositionσ=σ(x,y,z)isafunctionofthepositionx,y,zoftheelementStressesonfacesY：0XZYσzy+?σzy/?y*dyσxy+?σxy/?y*dyσyy+?σyy/?y*dyσzyσxyσyyStresstensorTheStreStresstensorStressesonfacesXandZIncontinuummechanics,thestressesarecontinuous.FiniteElementmethodsuseequilibriumprinciplestocomputeheterogeneous(不均勻的)stressesanddeformationsσzx+?σzx/?x*dxσxx+?σxx/?x*dxσyx+?σyx/?x*dxσzz+?σzz/?z*dzσxz+?σxz/?z*dzσyz+?σyz/?z*dz0XZYStresstensorStressesonfacesStresstensorThestressesthroughthematerialelementmustbeinequilibrium:Equilibriuminforces: div(σ)=0Equilibriuminmomentofforces(rotation):

σ=σT

thetensorissymmetric(σij=σji)?σxx/?x+?σxy/?y+?σxz/?z=0?σyx/?x+?σyy/?y+?σyz/?z=0?σzx/?x+?σzy/?y+?σzz/?z=0StresstensorThestressesthroStresstensorWhateveristhestresstensor,thereisabasisinwhichthereisnoshear:

=>Inthisbasis,thestressarecalled‘Principalstresses’0XZY0e1e3e2σI000σII000σIIIσxx

σxy

σxz

σyx

σyy

σyz

σzx

σzy

σzzStresstensorWhateveristhesStresstensorChangeofbasis(=referential,coordinatesystem):simpleshear=planestress0XZY0e1e3Y=e2σzx=10MPaσxz=10MPaσ11=10MPaσ33=-10MPaσXYZ=00100001000σ123=100000000-10StresstensorChangeofbasis(Demonstration:ExpressamatrixintoanotherbasisTransformationMatrixfrom(XYZ)to(e1e2e3):T= (expresseiinsideXYZ)σ123=T-1σxyzT1/√2 0 -1/√20 1 01/√2 0 1/√2Demonstration:ExpressamatriStresstensorPressurep(球應力): p=Stressdeviatortensor(應力偏張量)s:s=σ–pId=Deviatoricofσ=>s11+s22+s33=0Thestressdeviatortensorisusefulbecausepressuredonotproduceplasticdeformationσxx+σyy+σzz3σxx-pσxy

σxz

σyx

σyy-pσyz

σzx

σzy

σzz-pyinglipianzhangliangQiuyingliStresstensorPressurep(球應力)StresstensorEquivalentvonMisesstress(等效應力)σ=σVM=√3/2(sxx2+syy2+szz2+2×sxy2+2×sxz2+2×syz2)AlsocalledEquivalenttensilestress

becauseinuniaxialtensionp=σxx/3:

σVM=√3/2(σxx-p)2+p2+p2)=√3/2(4/9+1/9+1/9)σxx =σxxUsetocomparedifferentstressstates(應力狀態)UsewiththevonMisesequivalentstrain:εVM=√2/3(εxx2+εyy2+εzz2+2×εxy2+2×εxz2+2×εyz2)DengxiaoyingliYinglizhuangtaiStresstensorEquivalentvonMiStresstensorVonMisesstressinplanestresses(平面應力)p=(σx+σy)/3σ=σVM=√3/2((σx-p)2+(σy-p)2-p2)=√σx2+σy2-σxσyDrawσVM=constantσxσxσyσyσxσyσVM=σx=σyσx=-σy=σVM/√3StresstensorVonMisesstressStresstensorVonMisesstressinprincipalstressbasis(i.e.ifi≠j,σij=0)TheconstantvonMisesstressisacylinder(圓柱體)tiltedat45oofftheaxisσyσxσz(111)YuanzhutiStresstensorVonMisesstressStresstensorComputeσαandταasafunctionofα

Mohrcircleinplanestressταismaximumforα=45oσxxσyyσαταταα2ασxxσyyσασα=σxsin2α+σycos2ατα=σysinαcosα-σxcosαsinαStresstensorComputeσαandταchangeofbasisforMohrcircleT= σxyz=σ123=T-1σxyzT=sinα -cosα 0cosα sinα 00 0 1σx 0 00 σy 00 0 0

σxsin2α+σycos2α σysinαcosα-σxcosαsinα 0 σysinαcosα-σxcosαsinα σxcos2α+σysin2α 00 0 0changeofbasisforMohrcirclStraintensorWesawthebodyofmaterialsandthematerialelementsaresubmittedtoforces.Inmaterialscience,asinthereallife,thematerialscandeformWeneedtomeasureandquantifythesedeformationsUsuallythesedeformationsareverysmall…StraintensorWesawthebodyoStraintensorOriginofthestraintensor:thedisplacementofmaterialDisplacementfield(位移場)u(x,y,z)foreachpointX(x,y,z)ofthematerialu=eyezbeforeafteru(x,y,z)0X(x,y,z)exuxuyuzWeiyichangStraintensorOriginofthestrStraintensorDisplacementgradienttensor(位移梯度張量