英漢雙語彈性力學(xué)演示文稿

英漢雙語彈性力學(xué)演示文稿當(dāng)前第1頁\共有111頁\編于星期四\10點英漢雙語彈性力學(xué)當(dāng)前第2頁\共有111頁\編于星期四\10點第二章平面問題的基本理論當(dāng)前第3頁\共有111頁\編于星期四\10點TheBasicTheoryofthePlaneProblemChapter2TheBasictheoryofthePlaneProblem§2-11Stressfunction.Inversesolutionmethodandsemi-inversemethod§2-1Planestressproblemandplanestrainproblem§2-2Differentialequationofequilibrium§2-3Thestressontheincline.Principalstress§2-4Geometricalequation.Thedisplacementoftherigidbody§2-5Physicalequation§2-6Boundaryconditions§2-7Saint-Venant’sprinciple§2-8Solvingtheplaneproblemaccordingtothedisplacement§2-9Solvingtheplaneproblemaccordingtothestress.Compatibleequation§2-10ThesimplificationunderthecircumstancesofordinaryphysicalforceExerciseLesson當(dāng)前第4頁\共有111頁\編于星期四\10點平面問題的基本理論第二章平面問題的基本理論§2-11應(yīng)力函數(shù)逆解法與半逆解法§2-1平面應(yīng)力問題與平面應(yīng)變問題§2-2平衡微分方程§2-3斜面上的應(yīng)力主應(yīng)力§2-4幾何方程剛體位移§2-5物理方程§2-6邊界條件§2-7圣維南原理§2-8按位移求解平面問題§2-9按應(yīng)力求解平面問題。相容方程§2-10常體力情況下的簡化習(xí)題課當(dāng)前第5頁\共有111頁\編于星期四\10點1.Planestressproblem§2-1PlanestressproblemandplanestrainproblemInactualproblem,itisstrictlysayingthatanyelasticbodywhoseexternalforceforsufferingisaspacesystemofforcesisgenerallythespaceobject.However,whenboththeshapeandforcecircumstanceoftheelasticbodyforinvestigatinghavetheirowncertaincharacteristics.Aslongastheabstractionofthemechanicsishandledtogetherwithappropriatesimplification,itcanbeconcludedastheelasticityplaneproblem.Theplaneproblemisdividedintotheplanestressproblemandplanestrainproblem.

Equalthicknesslamellabearsthesurfaceforcethatparallelswithplatefaceanddon’tchangealongthethickness.Atthesametime,sodoesthevolumetricforce.σz=0τzx=0τzy=0Fig.2－1TheBasicTheoryofthePlaneProblem當(dāng)前第6頁\共有111頁\編于星期四\10點一、平面應(yīng)力問題§2-1平面應(yīng)力問題與平面應(yīng)變問題在實際問題中，任何一個彈性體嚴(yán)格地說都是空間物體，它所受的外力一般都是空間力系。但是，當(dāng)所考察的彈性體的形狀和受力情況具有一定特點時，只要經(jīng)過適當(dāng)?shù)暮喕土W(xué)的抽象處理，就可以歸結(jié)為彈性力學(xué)平面問題。平面問題分為平面應(yīng)力問題和平面應(yīng)變問題。

等厚度薄板，板邊承受平行于板面并且不沿厚度變化的面力，同時體力也平行于板面并且不沿厚度變化。σz=0τzx=0τzy=0圖2－1平面問題的基本理論當(dāng)前第7頁\共有111頁\編于星期四\10點TheBasicTheoryofthePlaneProblemxyCharacteristics：1)Thedimensionoflengthandbreadthisfarlargerthanthatofthickness.2)Theforcealongtheplatefaceforsufferingisthefaceforceinparallelwithplateface,andalongthethicknesseven,thevolumetricforceisinparallelwithplateforceanddoesn’tchangealongthethickness,andhasnoexternalforcefunctiononthesurfacefrontandbackoftheflatpanel.Attention:Planestressproblemz=0,but,thisiscontrarytoplanestrainproblem.當(dāng)前第8頁\共有111頁\編于星期四\10點平面問題的基本理論xy特點：1)長、寬尺寸遠(yuǎn)大于厚度2)沿板邊受有平行板面的面力，且沿厚度均布，體力平行于板面且不沿厚度變化，在平板的前后表面上無外力作用。問題相反。注意：平面應(yīng)力問題z=0，但，這與平面應(yīng)變當(dāng)前第9頁\共有111頁\編于星期四\10點2.Planestrainproblem

Verylongcolumnbearsthefaceforceinparallelwithplatefaceanddoesn’tchangealongthelengthonthecolumnface,atthesametime,sodoesthevolumetricforce.εz

=0τzx=0τzy=0xFig.2－2TheBasicTheoryofthePlaneProblemForexample:dam,circularcylinderpipingbytheinternalairpressureandlonglevellanewayetc.Attention:Planestrainproblemz=0，but,thisiscontrarytoplanestressproblem.當(dāng)前第10頁\共有111頁\編于星期四\10點二、平面應(yīng)變問題

很長的柱體，在柱面上承受平行于橫截面并且不沿長度變化的面力，同時體力也平行于橫截面并且不沿長度變化。εz

=0τzx=0τzy=0x圖2－2平面問題的基本理論如：水壩、受內(nèi)壓的圓柱管道和長水平巷道等。注意平面應(yīng)變問題z=0，但問題相反。，這恰與平面應(yīng)力當(dāng)前第11頁\共有111頁\編于星期四\10點§2-2DifferentialEquationofEquilibriumWhetherplanestressproblemorplanestrainproblem,istheresearchprobleminplanexy,allthephysicsquantityhasnothingtodowithz.Discussbelowthecorrelationbetweenanypointstressandvolumetricforcewhentheobjectisplacedinthestateofequilibrium,andleadanequilibriumdifferentialequationfromhere.FromthelamellashowninFig.2-1,wetakeoutasmallandpositiveparallelepipedPABC,andtakeforanunitlengthinthedirectionaldimensioninz.Fig.2－3Establishingthefunctionofthepositivestressforceinanunitontheleftsideis,thecoordinateontherightsidexgetstheincrement,thepositivestressonthefaceis,spreadingtheformulaabovewillbeTaylor’sseries:TheBasicTheoryofthePlaneProblem當(dāng)前第12頁\共有111頁\編于星期四\10點§2-2平衡微分方程無論平面應(yīng)力問題還是平面應(yīng)變問題，都是在xy平面內(nèi)研究問題，所有物理量均與z無關(guān)。

下面討論物體處于平衡狀態(tài)時，各點應(yīng)力及體力的相互關(guān)系，并由此導(dǎo)出平衡微分方程。從圖2－1所示的薄板取出一個微小的正平行六面體PABC(圖2－3），它在z方向的尺寸取為一個單位長度。圖2－3設(shè)作用在單元體左側(cè)面上的正應(yīng)力是，右側(cè)面上坐標(biāo)得到增量，該面上的正應(yīng)力為，將上式展開為泰勒級數(shù)：平面問題的基本理論當(dāng)前第13頁\共有111頁\編于星期四\10點Afteromittingsmallquantityofthetworankandabovethetworank,canget,atthesametime,,,aregetthestateofstressfromthedrawingshow.Whileconsideringthevolumetricforcetotheplanestressstate,stillprovemutualandequaltheoryofshearingstrength.RegardthecenterDandstraightlineinparallelwiththeshaftofzasthemomentshaft,listtheequilibriumequationofthemomentshaft:Thebothsidesoftheformulaabovedivideget:Cause,Omittingsmallquantityisn’taccounted,canget:TheBasicTheoryofthePlaneProblem當(dāng)前第14頁\共有111頁\編于星期四\10點略去二階及二階以上的微量后便得同樣、、都一樣處理，得到圖示應(yīng)力狀態(tài)。對平面應(yīng)力狀態(tài)考慮體力時，仍可證明剪應(yīng)力互等定理。以通過中心D并平行于z軸的直線為矩軸，列出力矩的平衡方程：將上式的兩邊除以得到：令，即略去微量不計，得：平面問題的基本理論當(dāng)前第15頁\共有111頁\編于星期四\10點Deducetheequilibriumdifferentialequationoftheplanestressproblembelow,listtheequilibriumequationtotheunit:TheBasicTheoryofthePlaneProblem當(dāng)前第16頁\共有111頁\編于星期四\10點下面推導(dǎo)平面應(yīng)力問題的平衡微分方程，對單元體列平衡方程：平面問題的基本理論當(dāng)前第17頁\共有111頁\編于星期四\10點Sortingthemgets:Thesetwodifferentialequationincludethreeunknownfunctions.Therefore,decidingtheproblemofthestressweightisexceedinglyandstaticallydeterminate;Andstillmustconsiderthedeformationanddisplacement,thentheproblemcanbesolved.Fortheplanestrainproblem,thefacesfrontandbackstillhaveButtheydonotaffectcompletelytheestablishesoftheequationabove.Sotheequationaboveappliestwokindsofplaneproblemalike.TheBasicTheoryofthePlaneProblem當(dāng)前第18頁\共有111頁\編于星期四\10點

整理得:

這兩個微分方程中包含著三個未知函數(shù)。因此決定應(yīng)力分量的問題是超靜定的；還必須考慮形變和位移，才能解決問題。對于平面應(yīng)變問題,雖然前后面上還有,但它們完全不影響上述方程的建立。所以上述方程對于兩種平面問題都同樣適用。平面問題的基本理論當(dāng)前第19頁\共有111頁\編于星期四\10點§2-3ThestressontheInclinedPlane.Principalstress1.ThestressontheinclinedplaneHavingknownthestressweightofanypointPinsidetheelasticbody,wetrytogetthestresswhichpassthepointPonthearbitrarilyinclinedcrosssection.FromneighborhoodofpointPtakingaplaneAB,whichisinparallelwiththeinclinedplaneabove,anddrawsasmallsetsquareorthreecolumnPABontwoplaneswhichpasspointPandhaveperpendicularityintheshaftofxandy.WhentheplaneABapproachespointPinfinitely,themeanstressontheplaneABwillbecomethestressontheinclinedplaneabove.

EstablishthelengthofthefaceABintheplanexyisdS,Nistheexteriornormaldirection,anditsdirectioncosineis:TheBasicTheoryofthePlaneProblemFig.2－4當(dāng)前第20頁\共有111頁\編于星期四\10點§2-3斜面上的應(yīng)力、主應(yīng)力一、斜面上的應(yīng)力已知彈性體內(nèi)任一點P處的應(yīng)力分量，求經(jīng)過該點任意斜截面上的應(yīng)力。為此在P點附近取一個平面AB，它平行于上述斜面，并與經(jīng)過P點而垂直于x軸和y軸的兩個平面劃出一個微小的三角板或三棱柱PAB。當(dāng)平面AB與P點無限接近時，平面AB上的應(yīng)力就成為上述斜面上的應(yīng)力。設(shè)AB面在xy平面內(nèi)的長度為dS，厚度為一個單位長度，N為該面的外法線方向，其方向余弦為：平面問題的基本理論圖2－4當(dāng)前第21頁\共有111頁\編于星期四\10點TheprojectionofthewholestressontheinclinedplaneABisXNandYNrespectivelyalongwiththeshaftofxandy.FromthePABequilibriumtermcanget:Divideandget:Samefromandget:

ThepositivestressontheinclinedplaneAB,fromtheprojectioncanget:TheshearingstrengthontheinclinedplaneAB,fromtheprojectioncanget:TheBasicTheoryofthePlaneProblem當(dāng)前第22頁\共有111頁\編于星期四\10點斜面AB上全應(yīng)力沿x軸及y軸的投影分別為XN和YN。由PAB的平衡條件可得：除以即得：同樣由得出：斜面AB上的正應(yīng)力，由投影可得：斜面AB上的剪應(yīng)力，由投影可得：平面問題的基本理論當(dāng)前第23頁\共有111頁\編于星期四\10點3.PrincipalstressIftheshearingstressofsomeinclinedplanethroughpointPisequaltozero,thenthepositivestressofthatinclinedplanecallsaprincipalstressofpointP,butthatinclinedplanecallsthemainplaneofthestressatpointP,andthenormaldirectionofthatinclinedplanecallsthemaindirectionofthestressatpointP.1.Thesizeoftheprincipalstress2.Thedirectionoftheprincipalstressisintheperpendicularitywithforeachother.TheBasicTheoryofthePlaneProblem當(dāng)前第24頁\共有111頁\編于星期四\10點二、主應(yīng)力如果經(jīng)過P點的某一斜面上的切應(yīng)力等于零，則該斜面上的正應(yīng)力稱為P點的一個主應(yīng)力，而該斜面稱為P點的一個應(yīng)力主面，該斜面的法線方向稱為P點的一個應(yīng)力主向。1.主應(yīng)力的大小2.主應(yīng)力的方向與互相垂直。平面問題的基本理論當(dāng)前第25頁\共有111頁\編于星期四\10點§2-4GeometricalEquation.TheDisplacementoftheRigidBodyInplaneproblem,everypointinsidetheelasticbodycanproducethearbitrarilydirectionaldisplacement.TakeanunitPABthroughanypointPinsidetheelasticbody,suchasFig.2-5show.Aftertheelasticbodysuffersforce,thepointP,A,BmovetothepointP′、A′、B′respectively.Fig.2－5一、ThepositivestrainatpointPHerebecauseofsmalldeformation,PAforcausingstretchandshrinkfromtheydirectiondisplacementvisthesmallquantityofahighrankandthissmallquantitymaybeomitted.TheBasicTheoryofthePlaneProblem當(dāng)前第26頁\共有111頁\編于星期四\10點§2-4幾何方程、剛體位移在平面問題中，彈性體中各點都可能產(chǎn)生任意方向的位移。通過彈性體內(nèi)的任一點P，取一單元體PAB，如圖2-5所示。彈性體受力以后P、A、B三點分別移動到P′、A′、B′。圖2－5一、P點的正應(yīng)變在這里由于小變形，由y方向位移v所引起的PA的伸縮是高一階的微量，略去不計。平面問題的基本理論當(dāng)前第27頁\共有111頁\編于星期四\10點Thesamecanget:2.ShearingstrainatpointPThecornerofthelinesegmentPA:ThesamecangetthecornerofthelinesegmentPB:ThusTheBasicTheoryofthePlaneProblem當(dāng)前第28頁\共有111頁\編于星期四\10點同理可求得：二、P點的切應(yīng)變線段PA的轉(zhuǎn)角：同理可得線段PB的轉(zhuǎn)角：所以平面問題的基本理論當(dāng)前第29頁\共有111頁\編于星期四\10點ThereforegetthegeometricalequationoftheplaneproblemFromthegeometricalequationabove,whenthedisplacementweightoftheobjectiscompletelycertain,thedeformationweightiscompletelycertain,uniqueweightcannotbemadesurethoroughly.TheBasicTheoryofthePlaneProblem當(dāng)前第30頁\共有111頁\編于星期四\10點因此得到平面問題的幾何方程：由幾何方程可見，當(dāng)物體的位移分量完全確定時，形變分量即可完全確定。反之，當(dāng)形變分量完全確定時，位移分量卻不能完全確定。平面問題的基本理論當(dāng)前第31頁\共有111頁\編于星期四\10點§2-5ThePhysicalEquationIntheisotropyofthecompleteelasticity,therelationbetweenthedeformationweightandthestressweightisestablishedaccordingtotheHooke’slawasfollows:TheBasicTheoryofthePlaneProblem當(dāng)前第32頁\共有111頁\編于星期四\10點§2-5物理方程在完全彈性的各向同性體內(nèi)，形變分量與應(yīng)力分量之間的關(guān)系根據(jù)虎克定律建立如下：平面問題的基本理論當(dāng)前第33頁\共有111頁\編于星期四\10點Insidetheformula,theEisamodulusofelasticity;theGisastiffnessmodulus;theuisapoissonratio.Therelationofthreeonesabove:1.ThephysicsequationoftheplanestressproblemAndhave:theBasicTheoryofthePlaneProblem當(dāng)前第34頁\共有111頁\編于星期四\10點式中，E為彈性模量；G為剛度模量；為泊松比。三者的關(guān)系：一、平面應(yīng)力問題的物理方程且有：平面問題的基本理論當(dāng)前第35頁\共有111頁\編于星期四\10點2.Thephysicsequationoftheplanestrainproblem3.Thetransformationrelationoftherelationtypebetweenthestressstrainandtheplanestrain.Therelationtypeoftheplanestress:TheBasicTheoryofthePlaneProblem當(dāng)前第36頁\共有111頁\編于星期四\10點二、平面應(yīng)變問題的物理方程三、平面應(yīng)力的應(yīng)力應(yīng)變關(guān)系式與平面應(yīng)變的關(guān)系式之間的變換關(guān)系將平面應(yīng)力中的關(guān)系式：平面問題的基本理論當(dāng)前第37頁\共有111頁\編于星期四\10點ForchangeCangettherelationtypeintheplanestrain:Becauseofthesimilarityofthiskind,whilesolvingplanestrainproblem,thecorrespondingequationoftheplaneproblemandtheelasticconstantintheanswercanbeexchangedasabove,cangetthesolutionofthehomologousplanestrainproblem.TheBasicTheoryofthePlaneProblem當(dāng)前第38頁\共有111頁\編于星期四\10點作代換就可得到平面應(yīng)變中的關(guān)系式：

由于這種相似性，在解平面應(yīng)變問題時，可把對應(yīng)的平面應(yīng)力問題的方程和解答中的彈性常數(shù)進(jìn)行上述代換，就可得到相應(yīng)的平面應(yīng)變問題的解。平面問題的基本理論當(dāng)前第39頁\共有111頁\編于星期四\10點§2-6BoundaryConditionsWhentheobjectisplacedinthestateofequilibrium,itsinternalstateofstressatallpointshouldsatisfytheequilibriumdifferentialequationandalsosatisfytheboundarytermontheboundary.Accordingtothedifferenceoftheboundarycondition,theelasticityproblemisdividedintothedisplacementboundaryproblem,stressboundaryproblemandmixedboundaryproblem.1.DisplacementBoundaryTermWhenthedisplacementhasbeenknownontheboundary,thedisplacementofthepointontheobjectboundaryandtheequaltermofthefixeddisplacementshouldbeestablished.Forexample,ifmakingtheboundaryofthefixeddisplacementis,andhave(onthe):Amongthem,andmeansthedisplacementweightontheboundary,however,andisthecoordinatefunctionwehaveknowtheboundary.TheBasicTheoryofthePlaneProblem當(dāng)前第40頁\共有111頁\編于星期四\10點§2-6邊界條件當(dāng)物體處于平衡狀態(tài)時,其內(nèi)部各點的應(yīng)力狀態(tài)應(yīng)滿足平衡微分方程；在邊界上應(yīng)滿足邊界條件。按照邊界條件的不同，彈性力學(xué)問題分為位移邊界問題、應(yīng)力邊界問題和混合邊界問題。一、位移邊界條件當(dāng)邊界上已知位移時，應(yīng)建立物體邊界上點的位移與給定位移相等的條件。如令給定位移的邊界為，則有（在上）：其中和表示邊界上的位移分量，而和在邊界上是坐標(biāo)的已知函數(shù)。平面問題的基本理論當(dāng)前第41頁\共有111頁\編于星期四\10點2.StressboundarytermWhentheboundaryoftheobjectisgiventosurfaceforce,thenthestressoftheobjectontheboundaryshouldsatisfytheequilibriumtermofforceswiththeequilibriumofthesurfaceforce.Amongthem,andarethesurfaceforceweightsand,,,arethestressweightsontheboundary.Whentheboundaryfaceisinperpendicularityinshaftx,stressboundarytermcanbechangedbrieflyinto:Whentheboundaryfaceisinperpendicularityinshafty,stressboundarytermcanbechangedbrieflyinto:TheBasicTheoryofthePlaneProblem當(dāng)前第42頁\共有111頁\編于星期四\10點二、應(yīng)力邊界條件當(dāng)物體的邊界上給定面力時，則物體邊界上的應(yīng)力應(yīng)滿足與面力相平衡的力的平衡條件。其中和為面力分量，、、、為邊界上的應(yīng)力分量。當(dāng)邊界面垂直于軸時，應(yīng)力邊界條件簡化為：當(dāng)邊界面垂直于軸時，應(yīng)力邊界條件簡化為：平面問題的基本理論當(dāng)前第43頁\共有111頁\編于星期四\10點3.Mixedboundarycondition1.Thedisplacementhasbeenknownonapartofboundariesoftheobject,theresultofwhichhavethedisplacementboundaryterm,theboundariesofotherpartshavethesurfaceforcewehaveknow.Andthenthereshouldbestressboundarytermanddisplacementboundarytermrespectivelyontwopartsoftheboundaries.Theleftsurfaceofthecantilevercontainsdisplacementboundaryterm,suchasshowninFig.2-6.Topandbottomsurfacecontainsstressboundaryterm:Therightsurfacecontainsstressboundaryterm:Fig.2-6TheBasicTheoryofthePlaneProblem當(dāng)前第44頁\共有111頁\編于星期四\10點三、混合邊界條件1.物體的一部分邊界上具有已知位移，因而具有位移邊界條件，另一部分邊界上則具有已知面力。則兩部分邊界上分別有應(yīng)力邊界條件和位移邊界條件。如圖2-6，懸臂梁左端面有位移邊界條件：上下面有應(yīng)力邊界條件：右端面有應(yīng)力邊界條件：圖2-6平面問題的基本理論當(dāng)前第45頁\共有111頁\編于星期四\10點2.Onthesameboundary,therearenotonlystressboundarytermbutdisplacementboundaryterm.Couplersustainstheboundaryterm,suchasshowninFig.2-7.ThealveolusboundarytermshowninFig.2-8.Fig.2-7Fig.2-8TheBasicTheoryofthePlaneProblem當(dāng)前第46頁\共有111頁\編于星期四\10點2.在同一邊界上，既有應(yīng)力邊界條件又有位移邊界條件。如圖2-7連桿支撐邊界條件：如圖2-8齒槽邊界條件：圖2-7圖2-8平面問題的基本理論當(dāng)前第47頁\共有111頁\編于星期四\10點§2-7Saint-VenantPrinciple1.Saint-Venant’sPrincipleIftransformingasmallpartofthesurfaceforceontheboundaryintothesurfaceforcethathasequaleffectbutdifferentdistribution(Themainvectorisequal,soisthemainquadraturetothesamepointaswell),andthenthedistributionofthestressforcenearbywillhaveprominentchanges,buttheinfluencefromthedistantplacecannotbeaccounted.2.GiveExamplesEstablishingthecomponentofthecolumnforms,thecentroidofareaincrosssectionsofbothendssuffersthetensibleforcewhichisequalinsizebutcontraryindirection,suchasshowninFig.2-9a.Iftransforminganorbothendsoftensileforceintotheforceatthesameeffectasthestaticforce,suchasshowninFig.2-9borFig.2-9c,thedistributionofstressforcedrawnonlybybrokenlinehasprominentchanges,whereas,theinfluenceoftherestpartscannotbeaccounted.Ifchangingbothendsoftensileforceintothatofuniformdistributionagain,thegatheringdegreeisequaltoP/AandamongthemAisthecross-sectionareaofthecomponent,suchasshowninFig.2-9d,thereisstillthestressclosetobothendsunderthenoticeableinfluence.TheBasicTheoryofthePlaneProblem當(dāng)前第48頁\共有111頁\編于星期四\10點§2-7圣維南原理一、圣維南原理如果把物體的一小部分邊界上的面力，變換為分布不同但靜力等效的面力（主矢量相同，對于同一點的主矩也相同），那么，近處的應(yīng)力分布將有顯著的改變，但是遠(yuǎn)處所受的影響可以不計。二、舉例設(shè)有柱形構(gòu)件，在兩端截面的形心受到大小相等而方向相反的拉力，如圖2-9a。如果把一端或兩端的拉力變換為靜力等效的力，如圖2-9b或2-9c，只有虛線劃出的部分的應(yīng)力分布有顯著的改變，而其余部分所受的影響是可以不計的。如果再將兩端的拉力變換為均勻分布的拉力，集度等于，其中為構(gòu)件的橫截面面積，如圖2-9d，仍然只有靠近兩端部分的應(yīng)力受到顯著的影響。平面問題的基本理論當(dāng)前第49頁\共有111頁\編于星期四\10點Fig.2-9(a)(b)(c)(d)(e)Underthefourkindsofcircumstancesabove,partsofdistributionofstressforcedistantfrombothendshavenomarkeddifference.Attention:TheapplicationoftheSaint-Venant’sprincipleisbynomeansseparatedfromthetermofEqualEffectofStaticForce.TheBasicTheoryofthePlaneProblem當(dāng)前第50頁\共有111頁\編于星期四\10點圖2-9(a)(b)(c)(d)(e)在上述四種情況下，離開兩端較遠(yuǎn)的部分的應(yīng)力分布，并沒有顯著的差別。注意：應(yīng)用圣維南原理，絕不能離開“靜力等效”的條件。平面問題的基本理論當(dāng)前第51頁\共有111頁\編于星期四\10點§2-8SolvingthePlaneProblemaccordingtothedisplacementTherearethreekindsofbasicmethodstosolvetheprobleminelasticity:thesolutiontotheproblemaccordingtodisplacement,stressforceandadmixture.Whilesolvingproblemsusingdisplacementmethod,weregarddisplacementweightasthebasicfunctionunknown.Aftergettingdisplacementweightfromonlyincludingthedifferentialequationandboundarytermofthedisplacementweight,thengetthedeformationweightusinggeometricalequation,therefore,getthestressweightwiththephysicsequation.1.PlaneStressProblemInplanestressproblem,thephysicsequationis:TheBasicTheoryofthePlaneProblem當(dāng)前第52頁\共有111頁\編于星期四\10點§2-8按位移求解平面問題在彈性力學(xué)里求解問題，有三種基本方法：按位移求解、按應(yīng)力求解和混合求解。按位移求解時，以位移分量為基本未知函數(shù)，由一些只包含位移分量的微分方程和邊界條件求出位移分量以后，再用幾何方程求出形變分量，從而用物理方程求出應(yīng)力分量。一、平面應(yīng)力問題在平面應(yīng)力問題中，物理方程為：平面問題的基本理論當(dāng)前第53頁\共有111頁\編于星期四\10點Fromthreeformulasabovementionedtosolvethestressweight,canget:withthesubstitutionofgeometricalequation,wecangettheelasticityequation:Againequilibriumdifferentialequationwithsubstitutioninformula(a),simplificationhereafter,canget:（a)Thisistheequilibriumdifferentialequationtomeanwiththedisplacement,ie,whensolvingtheplanestressproblemaccordingtodisplacementmethod,weadoptabasicdifferentialequationforneeds.（1）TheBasicTheoryofthePlaneProblem當(dāng)前第54頁\共有111頁\編于星期四\10點由上列三式求解應(yīng)力分量，得：將幾何方程代入，得彈性方程：再將式（a）代入平衡微分方程，簡化以后，即得：（a)這是用位移表示的平衡微分方程，也就是按位移求解平面應(yīng)力問題時所需用的基本微分方程。（1）平面問題的基本理論當(dāng)前第55頁\共有111頁\編于星期四\10點Thestressboundarytermwithsubstitutioninformula(a),simplificationhereafter,canget:Thisisthestressforceboundarytomeanwiththedisplacement,ie,weadopttheboundarytermofthestressforcewhensolvingtheplanestressproblemaccordingtodisplacementmethod.（2）Sumup,whensolvingtheplanestressproblemaccordingtodisplacementmethod,weshouldmakethedisplacementweightsatisfydifferentialequation(1)andcombinetosatisfydisplacementboundarytermorstressboundarytermorstressboundaryterm(2)ontheboundary.Aftergettingdisplacementweight,wecangetthedeformationweightwithgeometricalequationandthengetthestressforceweightwiththephysicsequation.2.Planestrainproblem

Makethesubstitutionbetweenandineachequationoftheplanestrainproblem:TheBasicTheoryofthePlaneProblem當(dāng)前第56頁\共有111頁\編于星期四\10點將（a）式代入應(yīng)力邊界條件，簡化以后，得：這是用位移表示的應(yīng)力邊界條件，也就是按位移求解平面應(yīng)力問題時所用的應(yīng)力邊界條件。（2）總結(jié)起來，按位移求解平面應(yīng)力問題時，要使得位移分量滿足微分方程（1），并在邊界上滿足位移邊界條件或應(yīng)力邊界條件（2）。求出位移分量以后，用幾何方程求出形變分量，再用物理方程求出應(yīng)力分量。二、平面應(yīng)變問題只須將平面應(yīng)力問題的各個方程中和作代換：平面問題的基本理論當(dāng)前第57頁\共有111頁\編于星期四\10點§2-9SolvingthePlaneProblemAccordingtotheStressForce.CompatibleEquantionWhilesolvingtheplaneproblemaccordingtothedisplacement,wemustcombinetwopartialdifferentialequationofthesecondrankstosolvetheproblem,thisisverydifficultonthemathematics.Butwhilesolvingtheplaneproblemaccordingtothestressforce,wecanavoidthisdifficultyandsowhatweadoptmoreistogetthesolutionaccordingtothestressforce.Whilegettingthesolutionaccordingtothestressforce,weregardstressweightasthebasicfunctionunknown.Aftergettingdisplacementweightfromonlyincludingthedifferentialequationandboundarytermofdisplacementweight,thengetthedeformationweightusingphysicsequation,therefore,getthedisplacementweightwithgeometricalequation.CompatibleEquationFromgeometricalequationoftheplaneproblem:TheBasicTheoryofthePlaneProblem當(dāng)前第58頁\共有111頁\編于星期四\10點§2-9按應(yīng)力求解平面問題。相容方程按位移求解平面問題時，必須求解聯(lián)立的兩個二階偏微分方程，這在數(shù)學(xué)上是相當(dāng)困難的。而按應(yīng)力求解彈性力學(xué)平面問題，則避免了這個困難，故更多采用的是按應(yīng)力求解。按應(yīng)力求解時，以應(yīng)力分量為基本未知函數(shù)，由一些只包含應(yīng)力分量的微分方程和邊界條件求出應(yīng)力分量以后，再用物理方程求出形變分量，從而用幾何方程求出位移分量。相容方程由平面問題的幾何方程：平面問題的基本理論當(dāng)前第59頁\共有111頁\編于星期四\10點Canget:ie,Thisrelationtypecallsthedeformationmoderatesequationorcompatibleequation.1.Compatibleequationinplanestressforce2.CompatibleequationinplanestrainforceTheBasicTheoryofthePlaneProblem當(dāng)前第60頁\共有111頁\編于星期四\10點可得：即：這個關(guān)系式稱為形變協(xié)調(diào)方程或相容方程。（一）平面應(yīng)力問題的相容方程（二）平面應(yīng)變問題的相容方程平面問題的基本理論當(dāng)前第61頁\共有111頁\編于星期四\10點Whilesolvingtheplaneproblemaccordingtothestressforce,thestressweightshouldnotonlysatisfyboththeequilibriumdifferentialequationandcompatibleequation,butsatisfythestressboundarytermontheboundarywhetherisaplanestressproblemorplanestrainproblem.TheBasicTheoryofthePlaneProblem當(dāng)前第62頁\共有111頁\編于星期四\10點按應(yīng)力求解平面問題時，無論是平面應(yīng)力問題還是平面應(yīng)變問題，應(yīng)力分量除了滿足平衡微分方程和相容方程外，在邊界上還應(yīng)當(dāng)滿足應(yīng)力邊界條件。平面問題的基本理論當(dāng)前第63頁\共有111頁\編于星期四\10點§2-10TheSimplificationUndertheCircumstancesofOrdinaryPhysicalForceUnderthecircumstancesofordinaryphysicalforce,thecompatibleequationoftwokindsofplaneproblemsissimplifiedas:Therefore,underthecircumstancesofordinaryphysicalforce,shouldsatisfyLaplacedifferentialequation(inharmonywithequation),shouldbeharmonicfunctions.Representwiththemark,theformulaabovecanbesimplifiedas:

ConclusionInthestressboundaryproblemofsingleconnectioniftwoelasticbodieshavethesameboundaryshapeandsuffertheexternalforceofthesamedistribution,andthenstressforcedistribution,,shouldbethesamewhetherthematerialsoftwoelasticbodiesaresameornotandwhethertheyareundertheplanestresscircumstancesorundertheplanestraincircumstances(Twokindsofthestressforceweightintheplaneproblem,thedeformationandthedisplacementareuncertainlythesame).TheBasicTheoryofthePlaneProblem當(dāng)前第64頁\共有111頁\編于星期四\10點§2-10常體力情況下的簡化常體力下，兩種平面問題的相容方程都簡化為：可見，在常體力的情況下，應(yīng)當(dāng)滿足拉普拉斯微分方程（調(diào)和方程），應(yīng)當(dāng)是調(diào)和函數(shù)。用記號代表，上式簡寫為：結(jié)論在單連體的應(yīng)力邊界問題中，如果兩個彈性體具有相同的邊界形狀，并受到同樣分布的外力，那么，不管這兩個彈性體的材料是否相同，也不管它們是在平面應(yīng)力情況下或是在平面應(yīng)變情況下，應(yīng)力分量、、的分布是相同的（兩種平面問題中的應(yīng)力分量，以及形變和位移，卻不一定相同）。平面問題的基本理論當(dāng)前第65頁\共有111頁\編于星期四\10點Inference2Whenmeasuringtheabovestressweightofthestructureorcomponentwiththemethodofexperiment,wecanmakethemodelusingthematerialoftheconvenientmeasurementinordertoreplaceoriginalstructureorcomponentmaterialsoftheinconvenientmeasurement;wealsocanadoptstructureorcomponentoflongcolumnshapeundertheplanestraincircumstances.Inference3Underthecircumstanceofconstantvolumetricforce,forthestressboundaryproblemofsingleconnection,wecanchargethefunctionofthevolumetricforceasthatofthesurfaceforceinordertosolvetheproblemandexperimentmeasurement.Inference1Thestressweight,,thatissolvedaccordingtoanyobjectisalsoapplicabletotheobjectwhichhasthesameboundaryandothermaterialssufferingthesameexternalforce;Thestressweightthatissolvedaccordingtoplanestressproblemisalsoapplicabletotheobjectwhichhasthesameboundaryandthesameexternalforceundertheplanestraincircumstances.TheBasicTheoryofthePlaneProblem當(dāng)前第66頁\共有111頁\編于星期四\10點推論2在用實驗方法測量結(jié)構(gòu)或構(gòu)件的上述應(yīng)力分量時，可以用便于量測的材料來制造模型，以代替原來不便于量測的結(jié)構(gòu)或構(gòu)件材料；還可以用平面應(yīng)力情況下的薄板模型，來代替平面應(yīng)變情況下的長柱形的結(jié)構(gòu)或構(gòu)件。推論3常體力的情況下，對于單連體的應(yīng)力邊界問題，還可以把體力的作用改換為面力的作用，以便于解答問題和實驗量測。推論1針對任一物體而求出的應(yīng)力分量、、，也適用于具有同樣邊界并受有同樣外力的其它材料的物體；針對平面應(yīng)力問題而求出的這些應(yīng)力分量，也適用于邊界相同、外力相同的平面應(yīng)變情況下的物體。平面問題的基本理論當(dāng)前第67頁\共有111頁\編于星期四\10點§2-11StressFunction.InverseSolutionMethodandSemi-InverseMethod1.StressfunctionWhilesolvingthestressboundaryproblemaccordingtothestressforceandwhenthevolumetricforceistheconstantquantity,thestressweight,,shouldsatisfytheequilibriumdifferentialequation:（a）Andcompatibleequation（b)Thesolutiontotheequation(a)includestwoparts:arbitrarilyaparticularsolutionandthegeneralsolutiontothefollowinghomogeneousdifferentialequation.TheBasicTheoryofthePlaneProblem當(dāng)前第68頁\共有111頁\編于星期四\10點§2-11應(yīng)力函數(shù)、逆解法與半逆解法一、應(yīng)力函數(shù)按應(yīng)力求解應(yīng)力邊界問題時，在體力為常量的情況下，應(yīng)力分量、、應(yīng)當(dāng)滿足平衡微分方程：（a）以及相容方程（b)方程（a）的解包含兩部分：任意一個特解和下列齊次微分方程的通解。平面問題的基本理論當(dāng)前第69頁\共有111頁\編于星期四\10點Theparticularsolutionis:Rewritetheformerequationinsidethehomogeneousdifferentialequation(c)as:Accordingtothedifferentialequationtheory,itiscertaintoexistsomefunction,make:（c）（d）（e）（f）TheBasicTheoryofthePlaneProblem當(dāng)前第70頁\共有111頁\編于星期四\10點特解取為：將齊次微分方程（c）中前一個方程改寫為：根據(jù)微分方程理論，一定存在某一個函數(shù)，使得：（c）（d）（e）（f）平面問題的基本理論當(dāng)前第71頁\共有111頁\編于星期四\10點

Similarlyrewritethesecondequationinside(c)as:Itiscertaintoexistsomefunctionaswell,make:（g）（h）Fromtheformula(f)and(h),canget:Thus,itiscertaintoexistsomefunction,make:（i）（j）TheBasicTheoryofthePlaneProblem當(dāng)前第72頁\共有111頁\編于星期四\10點

同樣將（c）中的第二個方程改寫為：也一定存在某一個函數(shù)，使得：（g）（h）由式（f）及（h）得：因而一定存在某一個函數(shù)，使得：（i）（j）平面問題的基本理論當(dāng)前第73頁\共有111頁\編于星期四\10點Maketheformula(i)substituteto(e),(j)to(g),and(i)to(f),thengetthegeneralsolution:（k）Makethegeneralsolution(k)plustheparticularsolution(d),thengetthewholesolutionofthedifferentialequation(a):ThefunctioncallsthestressfunctionoftheplaneproblemandalsocallstheArraystressfunction.Inorderthatthestressweight(1)canalsosatisfythecompatibleequation(b),makeformula(1)substituteformula(b),thenge