英漢雙語彈性力學(xué)

英漢雙語彈性力學(xué)第二章平面問題得基本理論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§2-9Solvingtheplaneproblemaccordingtothpatibleequation§2-10ThesimplificationunderthecircumstancesofordinaryphysicalforceExerciseLesson平面問題得基本理論第二章平面問題得基本理論§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斜面上得應(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常體力情況下得簡化習(xí)題課1、Planestressproblem§2-1PlanestressproblemandplanestrainproblemInactualproblem,itisstrictlysayingthatanyelasticbodywhoseexternalforceforsufferingisaspacesystemofforcesisgenerallythespaceobject、However,whenboththeshapeandforcecircumstanceoftheelasticbodyforinvestigatinghavetheirowncertaincharacteristics、Aslongastheabstractionofthemechanicsishandledtogetherwithappropriatesimplification,itcanbeconcludedastheelasticityplaneproblem、Theplaneproblemisdividedintotheplanestressproblemandplanestrainproblem、Equalthicknesslamellabearsthesurfaceforcethatparallelswithplatefaceanddon’tchangealongthethickness、Atthesametime,sodoesthevolumetricforce、σz=0τzx=0τzy=0Fig.2－1TheBasicTheoryofthePlaneProblem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)格地說都就是空間物體,她所受得外力一般都就是空間力系。但就是,當(dāng)所考察得彈性體得形狀和受力情況具有一定特點時,只要經(jīng)過適當(dāng)?shù)煤喕土W(xué)得抽象處理,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得面力,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平面問題得基本理論TheBasicTheoryofthePlaneProblemxyCharacteristics:1)Thedimensionoflengthandbreadthisfarlargerthanthatofthickness、2)Theforcealongtheplatefaceforsufferingisthefaceforceinparallelwithplateface,andalongthethicknesseven,thevolumetricforceisinparallelwithplateforceanddoesn’tchangealongthethickness,andhasnoexternalforcefunctiononthesurfacefrontandbackoftheflatpanel、Attention:Planestressproblemz=0,but,thisiscontrarytoplanestrainproblem.平面問題得基本理論xy特點:1)長、寬尺寸遠大于厚度2)沿板邊受有平行板面得面力,且沿厚度均布,體力平行于板面且不沿厚度變化,在平板得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2、PlanestrainproblemVerylongcolumnbearsthefaceforceinparallelwithplatefaceanddoesn’tchangealongthelengthonthecolumnface,atthesametime,sodoesthevolumetricforce、εz

=0τzx=0τzy=0xFig、2－2TheBasicTheoryofthePlaneProblemForexample:dam,circularcylinderpipingbytheinternalairpressureandlonglevellanewayetc、Attention:Planestrainproblem

z=0，but,thisiscontrarytoplanestressproblem.二、平面應(yīng)變問題很長得柱體,在柱面上承受平行于橫截面并且不沿長度變化得面力,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平面問題得基本理論如:水壩、受內(nèi)壓得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)力大家有疑問的，可以詢問和交流可以互相討論下，但要小聲點§2-2DifferentialEquationofEquilibriumWhetherplanestressproblemorplanestrainproblem,istheresearchprobleminplanexy,allthephysicsquantityhasnothingtodowithz、Discussbelowthecorrelationbetweenanypointstressandvolumetricforcewhentheobjectisplacedinthestateofequilibrium,andleadanequilibriumdifferentialequationfromhere、FromthelamellashowninFig、2-1,wetakeoutasmallandpositiveparallelepipedPABC,andtakeforanunitlengthinthedirectionaldimensioninz、Fig.2－3Establishingthefunctionofthepositivestressforceinanunitontheleftsideis,thecoordinateontherightsidexgetstheincrement,thepositivestressonthefaceis,spreadingtheformulaabovewillbeTaylor’sseries:TheBasicTheoryofthePlaneProblem§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)：平面問題得基本理論Afteromittingsmallquantityofthetworankandabovethetworank,canget,atthesametime,,,aregetthestateofstressfromthedrawingshow.Whileconsideringthevolumetricforcetotheplanestressstate,stillprovemutualandequaltheoryofshearingstrength.RegardthecenterDandstraightlineinparallelwiththeshaftofzasthemomentshaft,listtheequilibriumequationofthemomentshaft:Thebothsidesoftheformulaabovedivideget:Cause,Omittingsmallquantityisn’taccounted,canget:TheBasicTheoryofthePlaneProblem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：平面問題得基本理論Deducetheequilibriumdifferentialequationoftheplanestressproblembelow,listtheequilibriumequationtotheunit:TheBasicTheoryofthePlaneProblem下面推導(dǎo)平面應(yīng)力問題得平衡微分方程,對單元體列平衡方程:平面問題得基本理論Sortingthemgets:Thesetwodifferentialequationincludethreeunknownfunctions.Therefore,decidingtheproblemofthestressweightisexceedinglyandstaticallydeterminate;Andstillmustconsiderthedeformationanddisplacement,thentheproblemcanbesolved.Fortheplanestrainproblem,thefacesfrontandbackstillhaveButtheydonotaffectcompletelytheestablishesoftheequationabove.Sotheequationaboveappliestwokindsofplaneproblemalike.TheBasicTheoryofthePlaneProblem

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。平面問題得基本理論§2-3ThestressontheInclinedPlane、Principalstress1.ThestressontheinclinedplaneHavingknownthestressweightofanypointPinsidetheelasticbody,wetrytogetthestresswhichpassthepointPonthearbitrarilyinclinedcrosssection.FromneighborhoodofpointPtakingaplaneAB,whichisinparallelwiththeinclinedplaneabove,anddrawsasmallsetsquareorthreecolumnPABontwoplaneswhichpasspointPandhaveperpendicularityintheshaftofxandy.WhentheplaneABapproachespointPinfinitely,themeanstressontheplaneABwillbecomethestressontheinclinedplaneabove.

EstablishthelengthofthefaceABintheplanexyisdS,Nistheexteriornormaldirection,anditsdirectioncosineis:TheBasicTheoryofthePlaneProblemFig.2－4§2-3斜面上得應(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：平面問題得基本理論圖2－4TheprojectionofthewholestressontheinclinedplaneABisXNandYNrespectivelyalongwiththeshaftofxandy.FromthePABequilibriumtermcanget:Divideandget:Samefromandget:

ThepositivestressontheinclinedplaneAB,fromtheprojectioncanget:TheshearingstrengthontheinclinedplaneAB,fromtheprojectioncanget:TheBasicTheoryofthePlaneProblem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：平面問題得基本理論3、PrincipalstressIftheshearingstressofsomeinclinedplanethroughpointPisequaltozero,thenthepositivestressofthatinclinedplanecallsaprincipalstressofpointP,butthatinclinedplanecallsthemainplaneofthestressatpointP,andthenormaldirectionofthatinclinedplanecallsthemaindirectionofthestressatpointP、1.Thesizeoftheprincipalstress2.Thedirectionoftheprincipalstressisintheperpendicularitywithforeachother.TheBasicTheoryofthePlaneProblem二、主應(yīng)力如果經(jīng)過P點得某一斜面上得切應(yīng)力等于零,則該斜面上得正應(yīng)力稱為P點得一個主應(yīng)力,而該斜面稱為P點得一個應(yīng)力主面,該斜面得法線方向稱為P點得一個應(yīng)力主向。1、主應(yīng)力得大小2.主應(yīng)力的方向與互相垂直。平面問題得基本理論§2-4GeometricalEquation、TheDisplacementoftheRigidBodyInplaneproblem,everypointinsidetheelasticbodycanproducethearbitrarilydirectionaldisplacement、TakeanunitPABthroughanypointPinsidetheelasticbody,suchasFig、2-5show、Aftertheelasticbodysuffersforce,thepointP,A,BmovetothepointP′、A′、B′respectively、Fig、2－5一、ThepositivestrainatpointPHerebecauseofsmalldeformation,PAforcausingstretchandshrinkfromtheydirectiondisplacementvisthesmallquantityofahighrankandthissmallquantitymaybeomitted、TheBasicTheoryofthePlaneProblem§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)生任意方向得位移。通過彈性體內(nèi)得任一點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點得正應(yīng)變在這里由于小變形,由y方向位移v所引起得PA得伸縮就是高一階得微量,略去不計。平面問題得基本理論Thesamecanget:2、ShearingstrainatpointPThecornerofthelinesegmentPA:ThesamecangetthecornerofthelinesegmentPB:ThusTheBasicTheoryofthePlaneProblem同理可求得:二、P點得切應(yīng)變線段PA得轉(zhuǎn)角:同理可得線段PB得轉(zhuǎn)角:所以平面問題得基本理論ThereforegetthegeometricalequationoftheplaneproblemFromthegeometricalequationabove,whenthedisplacementweightoftheobjectispletelycertain,thedeformationweightispletelycertain,uniqueweightcannotbemadesurethoroughly、TheBasicTheoryofthePlaneProblem因此得到平面問題得幾何方程:由幾何方程可見,當(dāng)物體得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。平面問題得基本理論§2-5ThePhysicalEquationIntheisotropyofthepleteelasticity,therelationbetweenthedeformationweightandthestressweightisestablishedaccordingtotheHooke’slawasfollows:TheBasicTheoryofthePlaneProblem§2-5物理方程在完全彈性得各向同性體內(nèi),形變分量與應(yīng)力分量之間得關(guān)系根據(jù)虎克定律建立如下:平面問題得基本理論Insidetheformula,theEisamodulusofelasticity;theGisastiffnessmodulus;theuisapoissonratio、Therelationofthreeonesabove:1、ThephysicsequationoftheplanestressproblemAndhave:theBasicTheoryofthePlaneProblem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)力問題得物理方程且有:平面問題得基本理論2、Thephysicsequationoftheplanestrainproblem3、Thetransformationrelationoftherelationtypebetweenthestressstrainandtheplanestrain、Therelationtypeoftheplanestress:TheBasicTheoryofthePlaneProblem二、平面應(yīng)變問題得物理方程三、平面應(yīng)力得應(yīng)力應(yīng)變關(guān)系式與平面應(yīng)變得關(guān)系式之間得變換關(guān)系將平面應(yīng)力中得關(guān)系式:平面問題得基本理論ForchangeCangettherelationtypeintheplanestrain:Becauseofthesimilarityofthiskind,whilesolvingplanestrainproblem,thecorrespondingequationoftheplaneproblemandtheelasticconstantintheanswercanbeexchangedasabove,cangetthesolutionofthehomologousplanestrainproblem、TheBasicTheoryofthePlaneProblem作代換就可得到平面應(yīng)變中得關(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)得平面應(yīng)力問題得方程和解答中得彈性常數(shù)進行上述代換,就可得到相應(yīng)得平面應(yīng)變問題得解。平面問題得基本理論§2-6BoundaryConditionsWhentheobjectisplacedinthestateofequilibrium,itsinternalstateofstressatallpointshouldsatisfytheequilibriumdifferentialequationandalsosatisfytheboundarytermontheboundary、Accordingtothedifferenceoftheboundarycondition,theelasticityproblemisdividedintothedisplacementboundaryproblem,stressboundaryproblemandmixedboundaryproblem、1、DisplacementBoundaryTermWhenthedisplacementhasbeenknownontheboundary,thedisplacementofthepointontheobjectboundaryandtheequaltermofthefixeddisplacementshouldbeestablished.Forexample,ifmakingtheboundaryofthefixeddisplacementis,andhave(onthe):Amongthem,andmeansthedisplacementweightontheboundary,however,andisthecoordinatefunctionwehaveknowtheboundary.TheBasicTheoryofthePlaneProblem§2-6邊界條件當(dāng)物體處于平衡狀態(tài)時,其內(nèi)部各點得應(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。按照邊界條件得不同,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)。平面問題得基本理論2、StressboundarytermWhentheboundaryoftheobjectisgiventosurfaceforce,thenthestressoftheobjectontheboundaryshouldsatisfytheequilibriumtermofforceswiththeequilibriumofthesurfaceforce、Amongthem,andarethesurfaceforceweightsand,,,arethestressweightsontheboundary.Whentheboundaryfaceisinperpendicularityinshaftx,stressboundarytermcanbechangedbrieflyinto:Whentheboundaryfaceisinperpendicularityinshafty,stressboundarytermcanbechangedbrieflyinto:TheBasicTheoryofthePlaneProblem二、應(yīng)力邊界條件當(dāng)物體得邊界上給定面力時,則物體邊界上得應(yīng)力應(yīng)滿足與面力相平衡得力得平衡條件。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：平面問題得基本理論3、Mixedboundarycondition1、Thedisplacementhasbeenknownonapartofboundariesoftheobject,theresultofwhichhavethedisplacementboundaryterm,theboundariesofotherpartshavethesurfaceforcewehaveknow、Andthenthereshouldbestressboundarytermanddisplacementboundarytermrespectivelyontwopartsoftheboundaries、Theleftsurfaceofthecantilevercontainsdisplacementboundaryterm,suchasshowninFig、2-6、Topandbottomsurfacecontainsstressboundaryterm:Therightsurfacecontainsstressboundaryterm:Fig、2-6TheBasicTheoryofthePlaneProblem三、混合邊界條件1、物體得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平面問題得基本理論2、Onthesameboundary,therearenotonlystressboundarytermbutdisplacementboundaryterm、Couplersustainstheboundaryterm,suchasshowninFig、2-7、ThealveolusboundarytermshowninFig、2-8、Fig、2-7Fig.2-8TheBasicTheoryofthePlaneProblem2、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平面問題得基本理論§2-7Saint-VenantPrinciple1、Saint-Venant’sPrincipleIftransformingasmallpartofthesurfaceforceontheboundaryintothesurfaceforcethathasequaleffectbutdifferentdistribution(Themainvectorisequal,soisthemainquadraturetothesamepointaswell),andthenthedistributionofthestressforcenearbywillhaveprominentchanges,buttheinfluencefromthedistantplacecannotbeaccounted、2、GiveExamplesEstablishingtheponentofthecolumnforms,thecentroidofareaincrosssectionsofbothendssuffersthetensibleforcewhichisequalinsizebutcontraryindirection,suchasshowninFig、2-9a、Iftransforminganorbothendsoftensileforceintotheforceatthesameeffectasthestaticforce,suchasshowninFig、2-9borFig、2-9c,thedistributionofstressforcedrawnonlybybrokenlinehasprominentchanges,whereas,theinfluenceoftherestpartscannotbeaccounted、Ifchangingbothendsoftensileforceintothatofuniformdistributionagain,thegatheringdegreeisequaltoP/AandamongthemAisthecross-sectionareaoftheponent,suchasshowninFig、2-9d,thereisstillthestressclosetobothendsunderthenoticeableinfluence、TheBasicTheoryofthePlaneProblem§2-7圣維南原理一、圣維南原理如果把物體得一小部分邊界上得面力,變換為分布不同但靜力等效得面力(主矢量相同,對于同一點得主矩也相同),那么,近處得應(yīng)力分布將有顯著得改變,但就是遠處所受得影響可以不計。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。平面問題得基本理論Fig.2-9(a)(b)(c)(d)(e)Underthefourkindsofcircumstancesabove,partsofdistributionofstressforcedistantfrombothendshavenomarkeddifference、Attention:TheapplicationoftheSaint-Venant’sprincipleisbynomeansseparatedfromthetermofEqualEffectofStaticForce、TheBasicTheoryofthePlaneProblem圖2-9(a)(b)(c)(d)(e)在上述四種情況下,離開兩端較遠得部分得應(yīng)力分布,并沒有顯著得差別。注意:應(yīng)用圣維南原理,絕不能離開“靜力等效”得條件。平面問題得基本理論§2-8SolvingthePlaneProblemaccordingtothedisplacementTherearethreekindsofbasicmethodstosolvetheprobleminelasticity:thesolutiontotheproblemaccordingtodisplacement,stressforceandadmixture、Whilesolvingproblemsusingdisplacementmethod,weregarddisplacementweightasthebasicfunctionunknown、Aftergettingdisplacementweightfromonlyincludingthedifferentialequationandboundarytermofthedisplacementweight,thengetthedeformationweightusinggeometricalequation,therefore,getthestressweightwiththephysicsequation、1、PlaneStressProblemInplanestressproblem,thephysicsequationis:TheBasicTheoryofthePlaneProblem§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得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:平面問題得基本理論Fromthreeformulasabovementionedtosolvethestressweight,canget:withthesubstitutionofgeometricalequation,wecangettheelasticityequation:Againequilibriumdifferentialequationwithsubstitutioninformula(a),simplificationhereafter,canget:（a)Thisistheequilibriumdifferentialequationtomeanwiththedisplacement,ie,whensolvingtheplanestressproblemaccordingtodisplacementmethod,weadoptabasicdifferentialequationforneeds、（1）TheBasicTheoryofthePlaneProblem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)這就是用位移表示得平衡微分方程,也就就是按位移求解平面應(yīng)力問題時所需用得基本微分方程。（1）平面問題得基本理論Thestressboundarytermwithsubstitutioninformula(a),simplificationhereafter,canget:Thisisthestressforceboundarytomeanwiththedisplacement,ie,weadopttheboundarytermofthestressforcewhensolvingtheplanestressproblemaccordingtodisplacementmethod、(2)Sumup,whensolvingtheplanestressproblemaccordingtodisplacementmethod,weshouldmakethedisplacementweightsatisfydifferentialequation(1)andbinetosatisfydisplacementboundarytermorstressboundarytermorstressboundaryterm(2)ontheboundary、Aftergettingdisplacementweight,wecangetthedeformationweightwithgeometricalequationandthengetthestressforceweightwiththephysicsequation、2、Planestrainproblem

Makethesubstitutionbetweenandineachequationoftheplanestrainproblem:TheBasicTheoryofthePlaneProblem將(a)式代入應(yīng)力邊界條件,簡化以后,得:這就是用位移表示得應(yīng)力邊界條件,也就就是按位移求解平面應(yīng)力問題時所用得應(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：平面問題得基本理論§2-9SolvingthePlaneProblemAccordingtotheStrespatibleEquantionWhilesolvingtheplaneproblemaccordingtothedisplacement,wemustbinetwopartialdifferentialequationofthesecondrankstosolvetheproblem,thisisverydifficultonthemathematics、Butwhilesolvingtheplaneproblemaccordingtothestressforce,wecanavoidthisdifficultyandsowhatweadoptmoreistogetthesolutionaccordingtothestressforce、Whilegettingthesolutionaccordingtothestressforce,weregardstressweightasthebasicfunctionunknown、Aftergettingdisplacementweightfromonlyincludingthedifferentialequationandboundarytermofdisplacementweight,thengetthedeformationweightusingphysicsequation,therefore,getthedisplacementweightwithgeometricalequation、patibleEquationFromgeometricalequationoftheplaneproblem:TheBasicTheoryofthePlaneProblem§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)立得兩個二階偏微分方程,這在數(shù)學(xué)上就是相當(dāng)困難得。而按應(yīng)力求解彈性力學(xué)平面問題,則避免了這個困難,故更多采用得就是按應(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)力分量得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得幾何方程:平面問題得基本理論Canget:ie,Thisrelationtypecallsthedeformationmoderatesequationorpatibleequation、1、patibleequationinplanestressforce2、patibleequationinplanestrainforceTheBasicTheoryofthePlaneProblem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)力問題得相容方程(二)平面應(yīng)變問題得相容方程平面問題得基本理論Whilesolvingtheplaneproblemaccordingtothestressforce,thestressweightshouldnotonlysatisfyboththeequilibriumdifferentialequationandpatibleequation,butsatisfythestressboundarytermontheboundarywhetherisaplanestressproblemorplanestrainproblem、TheBasicTheoryofthePlaneProblem按應(yīng)力求解平面問題時,無論就是平面應(yīng)力問題還就是平面應(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。平面問題得基本理論§2-10TheSimplificationUndertheCircumstancesofOrdinaryPhysicalForceUnderthecircumstancesofordinaryphysicalforce,thepatibleequationoftwokindsofplaneproblemsissimplifiedas:Therefore,underthecircumstancesofordinaryphysicalforce,shouldsatisfyLaplacedifferentialequation(inharmonywithequation),shouldbeharmonicfunctions.Representwiththemark,theformulaabovecanbesimplifiedas:

ConclusionInthestressboundaryproblemofsingleconnectioniftwoelasticbodieshavethesameboundaryshapeandsuffertheexternalforceofthesamedistribution,andthenstressforcedistribution,,shouldbethesamewhetherthematerialsoftwoelasticbodiesaresameornotandwhethertheyareundertheplanestresscircumstancesorundertheplanestraincircumstances(Twokindsofthestressforceweightintheplaneproblem,thedeformationandthedisplacementareuncertainlythesame).TheBasicTheoryofthePlaneProblem§2-10常體力情況下得簡化常體力下,兩種平面問題得相容方程都簡化為: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）。平面問題得基本理論Inference2Whenmeasuringtheabovestressweightofthestructureorponentwiththemethodofexperiment,wecanmakethemodelusingthematerialoftheconvenientmeasurementinordertoreplaceoriginalstructureorponentmaterialsoftheinconvenientmeasurement;wealsocanadoptstructureorponentoflongcolumnshapeundertheplanestraincircumstances、Inference3Underthecircumstanceofconstantvolumetricforce,forthestressboundaryproblemofsingleconnection,wecanchargethefunctionofthevolumetricforceasthatofthesurfaceforceinordertosolvetheproblemandexperimentmeasurement、Inference1Thestressweight,,thatissolvedaccordingtoanyobjectisalsoapplicabletotheobjectwhichhasthesameboundaryandothermaterialssufferingthesameexternalforce;Thestressweightthatissolvedaccordingtoplanestressproblemisalsoapplicabletotheobjectwhichhasthesameboundaryandthesameexternalforceundertheplanestraincircumstances.TheBasicTheoryofthePlaneProblem推論2在用實驗方法測量結(jié)構(gòu)或構(gòu)件得上述應(yīng)力分量時,可以用便于量測得材料來制造模型,以代替原來不便于量測得結(jié)構(gòu)或構(gòu)件材料;還可以用平面應(yīng)力情況下得薄板模型,來代替平面應(yīng)變情況下得長柱形得結(jié)構(gòu)或構(gòu)件。推論3常體力得情況下,對于單連體得應(yīng)力邊界問題,還可以把體力得作用改換為面力得作用,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。平面問題得基本理論§2-11StressFunction、InverseSolutionMethodandSemi-InverseMethod1、StressfunctionWhilesolvingthestressboundaryproblemaccordingtothestressforceandwhenthevolumetricforceistheconstantquantity,thestressweight,,shouldsatisfytheequilibriumdifferentialequation:（a）Andpatibleequation（b)Thesolutiontotheequation(a)includestwoparts:arbitrarilyaparticularsolutionandthegeneralsolutiontothefollowinghomogeneousdifferentialequation、TheBasicTheoryofthePlaneProblem§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)得解包含兩部分:任意一個特解和下列齊次微分方程得通解。平面問題得基本理論Theparticularsolutionis:Rewritetheformerequationinsidethehomogeneousdifferentialequation(c)as:Accordingtothedifferentialequationtheory,itiscertaintoexistsomefunction,make:（c）（d）（e）（f）TheBasicTheoryofthePlaneProblem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）平面問題得基本理論

Similarlyrewritethesecondequationinside(c)as:Itiscertaintoexistsomefunctionaswell,make:(g)(h)Fromtheformula(f)and(h),canget:Thus,itiscertaintoexistsomefunction,make:（i）（j）TheBasicTheoryofthePlaneProblem

同樣將(c)中得第二個方程改寫為: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）平面問題得基本理論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),thenget:（1）Theformulaabovecanbesimplified:TheBasicTheoryofthePlaneProblem將式(i)代入(e),式(j)代入(g),并將式(i)代入(f),即得通解:（k）將通解(k)與特解(d)疊加,即得微分方程(a)得全解:函數(shù)稱為平面問題的應(yīng)力函數(shù)，也稱為艾瑞應(yīng)力函數(shù)。（1）為了應(yīng)力分量(1)同時也能滿足相容方程(b),將(1)代入式(b),即得:上式可簡化為:平面問題得基本理論Orspreadingtheformulais:Furthersimplificationis:(2)2、Inversesolutionmethodandsemi-inversemethodInversesolution:thefirststepistosetupmultiformstressfunctionwhichsatisfythecompatibleequation(2),andgetthestressweightwiththeformula(1),theninvestigateaccordingtothestressboundaryterm.Ontheelasticbodyineverykindofshape,thesestressweightscorrespondenceinwhatkindsofsurfaceforce,fromwhichweknowthatthestressfunctionforsettingupcansolvewhatkindsofproblem.Whilesolvingthestressboundaryproblemaccordingtostressforce,ifthevolumetricforceisconstantquantity,wemayonlyconsultthedifferentialequation(2)tosolvethestressfunction,andthengetthestressweightwiththeformula(1),butthesestressweightsshouldsatisfythestressboundarytermontheboundary.TheBasicTheoryofthePlaneProblemThebasicstepofinversesolutionmethod:或者展開為:進一步簡寫為:(2)二、逆解法與半逆解法逆解法：先設(shè)定各種形式的、滿足相容方程（2）的應(yīng)力函數(shù)，用公式（1）求出應(yīng)力分量，然后根據(jù)應(yīng)力邊界條件來考察，在各種邊界形狀的彈性體上，這些應(yīng)力分量對應(yīng)于什么樣的面力，從而得知所設(shè)定的應(yīng)力函數(shù)可以解決什么問題。按應(yīng)力求解應(yīng)力邊界問題時，如果體力是常量，就只須由微分方程（2）求解應(yīng)力函數(shù)，然后用公式（1）求出應(yīng)力分量，但這些應(yīng)力分量在邊界上應(yīng)當(dāng)滿足應(yīng)力邊界條件。平面問題得基本理論逆解法基本步驟:Semi-inversemethod:Aimingattheproblemforrequestingsolution,accordingtoboundaryshapeofelasticbodyandforcecircumstanceforsuffering,supposingthepartialandthewholestressweightasacertainformfunction,fromwhichconcludestressfunction,thenetoinvestigatewhetherthisstressfunctionsatisfypatibleequat