英漢雙語彈性力學

英漢雙語彈性力學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應力函數(shù)逆解法與半逆解法§2-1平面應力問題與平面應變問題§2-2平衡微分方程§2-3斜面上得應力主應力§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按應力求解平面問題。相容方程§2-10常體力情況下得簡化習題課1、Planestressproblem§2-1PlanestressproblemandplanestrainproblemInactualproblem,itisstrictlysayingthatanyelasticbodywhoseexternalforceforsufferingisaspacesystemofforcesisgenerallythespaceobject、However,whenboththeshapeandforcecircumstanceoftheelasticbodyforinvestigatinghavetheirowncertaincharacteristics、Aslongastheabstractionofthemechanicsishandledtogetherwithappropriatesimplification,itcanbeconcludedastheelasticityplaneproblem、Theplaneproblemisdividedintotheplanestressproblemandplanestrainproblem、Equalthicknesslamellabearsthesurfaceforcethatparallelswithplatefaceanddon’tchangealongthethickness、Atthesametime,sodoesthevolumetricforce、σz=0τzx=0τzy=0Fig.2－1TheBasicTheoryofthePlaneProblem一、平面應力問題§2-1平面應力問題與平面應變問題在實際問題中,任何一個彈性體嚴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?shù)煤喕土W得抽象處理,就可以歸結為彈性力學平面問題。平面問題分為平面應力問題和平面應變問題。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：平面應力問題

z=0，但，這與平面應變2、PlanestrainproblemVerylongcolumnbearsthefaceforceinparallelwithplatefaceanddoesn’tchangealongthelengthonthecolumnface,atthesametime,sodoesthevolumetricforce、εz

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

z=0，but,thisiscontrarytoplanestressproblem.二、平面應變問題很長得柱體,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、受內壓得圓柱管道和長水平巷道等。注意平面應變問題

z=0，但問題相反。，這恰與平面應力大家有疑問的，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力問題還是平面應變問題，都是在xy平面內研究問題，所有物理量均與z無關。

下面討論物體處于平衡狀態(tài)時，各點應力及體力的相互關系，并由此導出平衡微分方程。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設作用在單元體左側面上的正應力是，右側面上坐標得到增量，該面上的正應力為，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，得到圖示應力狀態(tài)。對平面應力狀態(tài)考慮體力時，仍可證明剪應力互等定理。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下面推導平面應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斜面上得應力、主應力一、斜面上的應力已知彈性體內任一點P處的應力分量，求經(jīng)過該點任意斜截面上的應力。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。當平面AB與P點無限接近時，平面AB上的應力就成為上述斜面上的應力。設AB面在xy平面內的長度為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上的正應力，由投影可得：斜面AB上的剪應力，由投影可得：平面問題得基本理論3、PrincipalstressIftheshearingstressofsomeinclinedplanethroughpointPisequaltozero,thenthepositivestressofthatinclinedplanecallsaprincipalstressofpointP,butthatinclinedplanecallsthemainplaneofthestressatpointP,andthenormaldirectionofthatinclinedplanecallsthemaindirectionofthestressatpointP、1.Thesizeoftheprincipalstress2.Thedirectionoftheprincipalstressisintheperpendicularitywithforeachother.TheBasicTheoryofthePlaneProblem二、主應力如果經(jīng)過P點得某一斜面上得切應力等于零,則該斜面上得正應力稱為P點得一個主應力,而該斜面稱為P點得一個應力主面,該斜面得法線方向稱為P點得一個應力主向。1、主應力得大小2.主應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生任意方向得位移。通過彈性體內得任一點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點得切應變線段PA得轉角:同理可得線段PB得轉角: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,形變分量與應力分量之間得關系根據(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。三者的關系：一、平面應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、平面應變問題得物理方程三、平面應力得應力應變關系式與平面應變得關系式之間得變換關系將平面應力中得關系式: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變中得關系式:

由于這種相似性,在解平面應變問題時,可把對應得平面應力問題得方程和解答中得彈性常數(shù)進行上述代換,就可得到相應得平面應變問題得解。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邊界條件當物體處于平衡狀態(tài)時,其內部各點得應力狀態(tài)應滿足平衡微分方程;在邊界上應滿足邊界條件。按照邊界條件得不同,彈性力學問題分為位移邊界問題、應力邊界問題和混合邊界問題。一、位移邊界條件當邊界上已知位移時，應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、應力邊界條件當物體得邊界上給定面力時,則物體邊界上得應力應滿足與面力相平衡得力得平衡條件。其中和為面力分量，、、、為邊界上的應力分量。當邊界面垂直于軸時，應力邊界條件簡化為：當邊界面垂直于軸時，應力邊界條件簡化為：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:上下面有應力邊界條件:右端面有應力邊界條件: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、舉例設有柱形構件，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，其中為構件的橫截面面積，如圖2-9d，仍然只有靠近兩端部分的應力受到顯著的影響。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力分布,并沒有顯著得差別。注意:應用圣維南原理,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里求解問題,有三種基本方法:按位移求解、按應力求解和混合求解。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力分量。一、平面應力問題在平面應力問題中,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)式代入應力邊界條件,簡化以后,得:這就是用位移表示得應力邊界條件,也就就是按位移求解平面應力問題時所用得應力邊界條件。(2)總結起來,按位移求解平面應力問題時,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力分量。二、平面應變問題只須將平面應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)學上就是相當困難得。而按應力求解彈性力學平面問題,則避免了這個困難,故更多采用得就是按應力求解。按應力求解時,以應力分量為基本未知函數(shù),由一些只包含應力分量得微分方程和邊界條件求出應力分量以后,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:這個關系式稱為形變協(xié)調方程或相容方程。(一)平面應力問題得相容方程(二)平面應變問題得相容方程平面問題得基本理論Whilesolvingtheplaneproblemaccordingtothestressforce,thestressweightshouldnotonlysatisfyboththeequilibriumdifferentialequationandpatibleequation,butsatisfythestressboundarytermontheboundarywhetherisaplanestressproblemorplanestrainproblem、TheBasicTheoryofthePlaneProblem按應力求解平面問題時,無論就是平面應力問題還就是平面應變問題,應力分量除了滿足平衡微分方程和相容方程外,在邊界上還應當滿足應力邊界條件。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，應當滿足拉普拉斯微分方程（調和方程），應當是調和函數(shù)。用記號代表，上式簡寫為：結論在單連體的應力邊界問題中，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力情況下或是在平面應變情況下，應力分量、、的分布是相同的（兩種平面問題中的應力分量，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構或構件得上述應力分量時,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構或構件材料;還可以用平面應力情況下得薄板模型,來代替平面應變情況下得長柱形得結構或構件。推論3常體力得情況下,對于單連體得應力邊界問題,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；針對平面應力問題而求出的這些應力分量，也適用于邊界相同、外力相同的平面應變情況下的物體。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應力函數(shù)、逆解法與半逆解法一、應力函數(shù)按應力求解應力邊界問題時，在體力為常量的情況下，應力分量、、應當滿足平衡微分方程：（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)稱為平面問題的應力函數(shù)，也稱為艾瑞應力函數(shù)。（1）為了應力分量(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：先設定各種形式的、滿足相容方程（2）的應力函數(shù)，用公式（1）求出應力分量，然后根據(jù)應力邊界條件來考察，在各種邊界形狀的彈性體上，這些應力分量對應于什么樣的面力，從而得知所設定的應力函數(shù)可以解決什么問題。按應力求解應力邊界問題時，如果體力是常量，就只須由微分方程（2）求解應力函數(shù)，然后用公式（1）求出應力分量，但這些應力分量在邊界上應當滿足應力邊界條件。平面問題得基本理論逆解法基本步驟:Semi-inversemethod:Aimingattheproblemforrequestingsolution,accordingtoboundaryshapeofelasticbodyandforcecircumstanceforsuffering,supposingthepartialandthewholestressweightasacertainformfunction,fromwhichconcludestressfunction,thenetoinvestigatewhetherthisstressfunctionsatisfypatibleequat