英汉双语弹性力学演示文稿

英汉双语弹性力学演示文稿当前1页，总共111页。英汉双语弹性力学当前2页，总共111页。第二章平面问题的基本理论当前3页，总共111页。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当前4页，总共111页。平面问题的基本理论第二章平面问题的基本理论§2-11应力函数逆解法与半逆解法§2-1平面应力问题与平面应变问题§2-2平衡微分方程§2-3斜面上的应力主应力§2-4几何方程刚体位移§2-5物理方程§2-6边界条件§2-7圣维南原理§2-8按位移求解平面问题§2-9按应力求解平面问题。相容方程§2-10常体力情况下的简化习题课当前5页，总共111页。1.Planestressproblem§2-1PlanestressproblemandplanestrainproblemInactualproblem,itisstrictlysayingthatanyelasticbodywhoseexternalforceforsufferingisaspacesystemofforcesisgenerallythespaceobject.However,whenboththeshapeandforcecircumstanceoftheelasticbodyforinvestigatinghavetheirowncertaincharacteristics.Aslongastheabstractionofthemechanicsishandledtogetherwithappropriatesimplification,itcanbeconcludedastheelasticityplaneproblem.Theplaneproblemisdividedintotheplanestressproblemandplanestrainproblem.

Equalthicknesslamellabearsthesurfaceforcethatparallelswithplatefaceanddon’tchangealongthethickness.Atthesametime,sodoesthevolumetricforce.σz=0τzx=0τzy=0Fig.2－1TheBasicTheoryofthePlaneProblem当前6页，总共111页。一、平面应力问题§2-1平面应力问题与平面应变问题在实际问题中，任何一个弹性体严格地说都是空间物体，它所受的外力一般都是空间力系。但是，当所考察的弹性体的形状和受力情况具有一定特点时，只要经过适当的简化和力学的抽象处理，就可以归结为弹性力学平面问题。平面问题分为平面应力问题和平面应变问题。

等厚度薄板，板边承受平行于板面并且不沿厚度变化的面力，同时体力也平行于板面并且不沿厚度变化。σz=0τzx=0τzy=0图2－1平面问题的基本理论当前7页，总共111页。TheBasicTheoryofthePlaneProblemxyCharacteristics：1)Thedimensionoflengthandbreadthisfarlargerthanthatofthickness.2)Theforcealongtheplatefaceforsufferingisthefaceforceinparallelwithplateface,andalongthethicknesseven,thevolumetricforceisinparallelwithplateforceanddoesn’tchangealongthethickness,andhasnoexternalforcefunctiononthesurfacefrontandbackoftheflatpanel.Attention:Planestressproblemz=0,but,thisiscontrarytoplanestrainproblem.当前8页，总共111页。平面问题的基本理论xy特点：1)长、宽尺寸远大于厚度2)沿板边受有平行板面的面力，且沿厚度均布，体力平行于板面且不沿厚度变化，在平板的前后表面上无外力作用。问题相反。注意：平面应力问题z=0，但，这与平面应变当前9页，总共111页。2.Planestrainproblem

Verylongcolumnbearsthefaceforceinparallelwithplatefaceanddoesn’tchangealongthelengthonthecolumnface,atthesametime,sodoesthevolumetricforce.εz

=0τzx=0τzy=0xFig.2－2TheBasicTheoryofthePlaneProblemForexample:dam,circularcylinderpipingbytheinternalairpressureandlonglevellanewayetc.Attention:Planestrainproblemz=0，but,thisiscontrarytoplanestressproblem.当前10页，总共111页。二、平面应变问题

很长的柱体，在柱面上承受平行于横截面并且不沿长度变化的面力，同时体力也平行于横截面并且不沿长度变化。εz

=0τzx=0τzy=0x图2－2平面问题的基本理论如：水坝、受内压的圆柱管道和长水平巷道等。注意平面应变问题z=0，但问题相反。，这恰与平面应力当前11页，总共111页。§2-2DifferentialEquationofEquilibriumWhetherplanestressproblemorplanestrainproblem,istheresearchprobleminplanexy,allthephysicsquantityhasnothingtodowithz.Discussbelowthecorrelationbetweenanypointstressandvolumetricforcewhentheobjectisplacedinthestateofequilibrium,andleadanequilibriumdifferentialequationfromhere.FromthelamellashowninFig.2-1,wetakeoutasmallandpositiveparallelepipedPABC,andtakeforanunitlengthinthedirectionaldimensioninz.Fig.2－3Establishingthefunctionofthepositivestressforceinanunitontheleftsideis,thecoordinateontherightsidexgetstheincrement,thepositivestressonthefaceis,spreadingtheformulaabovewillbeTaylor’sseries:TheBasicTheoryofthePlaneProblem当前12页，总共111页。§2-2平衡微分方程无论平面应力问题还是平面应变问题，都是在xy平面内研究问题，所有物理量均与z无关。

下面讨论物体处于平衡状态时，各点应力及体力的相互关系，并由此导出平衡微分方程。从图2－1所示的薄板取出一个微小的正平行六面体PABC(图2－3），它在z方向的尺寸取为一个单位长度。图2－3设作用在单元体左侧面上的正应力是，右侧面上坐标得到增量，该面上的正应力为，将上式展开为泰勒级数：平面问题的基本理论当前13页，总共111页。Afteromittingsmallquantityofthetworankandabovethetworank,canget,atthesametime,,,aregetthestateofstressfromthedrawingshow.Whileconsideringthevolumetricforcetotheplanestressstate,stillprovemutualandequaltheoryofshearingstrength.RegardthecenterDandstraightlineinparallelwiththeshaftofzasthemomentshaft,listtheequilibriumequationofthemomentshaft:Thebothsidesoftheformulaabovedivideget:Cause,Omittingsmallquantityisn’taccounted,canget:TheBasicTheoryofthePlaneProblem当前14页，总共111页。略去二阶及二阶以上的微量后便得同样、、都一样处理，得到图示应力状态。对平面应力状态考虑体力时，仍可证明剪应力互等定理。以通过中心D并平行于z轴的直线为矩轴，列出力矩的平衡方程：将上式的两边除以得到：令，即略去微量不计，得：平面问题的基本理论当前15页，总共111页。Deducetheequilibriumdifferentialequationoftheplanestressproblembelow,listtheequilibriumequationtotheunit:TheBasicTheoryofthePlaneProblem当前16页，总共111页。下面推导平面应力问题的平衡微分方程，对单元体列平衡方程：平面问题的基本理论当前17页，总共111页。Sortingthemgets:Thesetwodifferentialequationincludethreeunknownfunctions.Therefore,decidingtheproblemofthestressweightisexceedinglyandstaticallydeterminate;Andstillmustconsiderthedeformationanddisplacement,thentheproblemcanbesolved.Fortheplanestrainproblem,thefacesfrontandbackstillhaveButtheydonotaffectcompletelytheestablishesoftheequationabove.Sotheequationaboveappliestwokindsofplaneproblemalike.TheBasicTheoryofthePlaneProblem当前18页，总共111页。

整理得:

这两个微分方程中包含着三个未知函数。因此决定应力分量的问题是超静定的；还必须考虑形变和位移，才能解决问题。对于平面应变问题,虽然前后面上还有,但它们完全不影响上述方程的建立。所以上述方程对于两种平面问题都同样适用。平面问题的基本理论当前19页，总共111页。§2-3ThestressontheInclinedPlane.Principalstress1.ThestressontheinclinedplaneHavingknownthestressweightofanypointPinsidetheelasticbody,wetrytogetthestresswhichpassthepointPonthearbitrarilyinclinedcrosssection.FromneighborhoodofpointPtakingaplaneAB,whichisinparallelwiththeinclinedplaneabove,anddrawsasmallsetsquareorthreecolumnPABontwoplaneswhichpasspointPandhaveperpendicularityintheshaftofxandy.WhentheplaneABapproachespointPinfinitely,themeanstressontheplaneABwillbecomethestressontheinclinedplaneabove.

EstablishthelengthofthefaceABintheplanexyisdS,Nistheexteriornormaldirection,anditsdirectioncosineis:TheBasicTheoryofthePlaneProblemFig.2－4当前20页，总共111页。§2-3斜面上的应力、主应力一、斜面上的应力已知弹性体内任一点P处的应力分量，求经过该点任意斜截面上的应力。为此在P点附近取一个平面AB，它平行于上述斜面，并与经过P点而垂直于x轴和y轴的两个平面划出一个微小的三角板或三棱柱PAB。当平面AB与P点无限接近时，平面AB上的应力就成为上述斜面上的应力。设AB面在xy平面内的长度为dS，厚度为一个单位长度，N为该面的外法线方向，其方向余弦为：平面问题的基本理论图2－4当前21页，总共111页。TheprojectionofthewholestressontheinclinedplaneABisXNandYNrespectivelyalongwiththeshaftofxandy.FromthePABequilibriumtermcanget:Divideandget:Samefromandget:

ThepositivestressontheinclinedplaneAB,fromtheprojectioncanget:TheshearingstrengthontheinclinedplaneAB,fromtheprojectioncanget:TheBasicTheoryofthePlaneProblem当前22页，总共111页。斜面AB上全应力沿x轴及y轴的投影分别为XN和YN。由PAB的平衡条件可得：除以即得：同样由得出：斜面AB上的正应力，由投影可得：斜面AB上的剪应力，由投影可得：平面问题的基本理论当前23页，总共111页。3.PrincipalstressIftheshearingstressofsomeinclinedplanethroughpointPisequaltozero,thenthepositivestressofthatinclinedplanecallsaprincipalstressofpointP,butthatinclinedplanecallsthemainplaneofthestressatpointP,andthenormaldirectionofthatinclinedplanecallsthemaindirectionofthestressatpointP.1.Thesizeoftheprincipalstress2.Thedirectionoftheprincipalstressisintheperpendicularitywithforeachother.TheBasicTheoryofthePlaneProblem当前24页，总共111页。二、主应力如果经过P点的某一斜面上的切应力等于零，则该斜面上的正应力称为P点的一个主应力，而该斜面称为P点的一个应力主面，该斜面的法线方向称为P点的一个应力主向。1.主应力的大小2.主应力的方向与互相垂直。平面问题的基本理论当前25页，总共111页。§2-4GeometricalEquation.TheDisplacementoftheRigidBodyInplaneproblem,everypointinsidetheelasticbodycanproducethearbitrarilydirectionaldisplacement.TakeanunitPABthroughanypointPinsidetheelasticbody,suchasFig.2-5show.Aftertheelasticbodysuffersforce,thepointP,A,BmovetothepointP′、A′、B′respectively.Fig.2－5一、ThepositivestrainatpointPHerebecauseofsmalldeformation,PAforcausingstretchandshrinkfromtheydirectiondisplacementvisthesmallquantityofahighrankandthissmallquantitymaybeomitted.TheBasicTheoryofthePlaneProblem当前26页，总共111页。§2-4几何方程、刚体位移在平面问题中，弹性体中各点都可能产生任意方向的位移。通过弹性体内的任一点P，取一单元体PAB，如图2-5所示。弹性体受力以后P、A、B三点分别移动到P′、A′、B′。图2－5一、P点的正应变在这里由于小变形，由y方向位移v所引起的PA的伸缩是高一阶的微量，略去不计。平面问题的基本理论当前27页，总共111页。Thesamecanget:2.ShearingstrainatpointPThecornerofthelinesegmentPA:ThesamecangetthecornerofthelinesegmentPB:ThusTheBasicTheoryofthePlaneProblem当前28页，总共111页。同理可求得：二、P点的切应变线段PA的转角：同理可得线段PB的转角：所以平面问题的基本理论当前29页，总共111页。ThereforegetthegeometricalequationoftheplaneproblemFromthegeometricalequationabove,whenthedisplacementweightoftheobjectiscompletelycertain,thedeformationweightiscompletelycertain,uniqueweightcannotbemadesurethoroughly.TheBasicTheoryofthePlaneProblem当前30页，总共111页。因此得到平面问题的几何方程：由几何方程可见，当物体的位移分量完全确定时，形变分量即可完全确定。反之，当形变分量完全确定时，位移分量却不能完全确定。平面问题的基本理论当前31页，总共111页。§2-5ThePhysicalEquationIntheisotropyofthecompleteelasticity,therelationbetweenthedeformationweightandthestressweightisestablishedaccordingtotheHooke’slawasfollows:TheBasicTheoryofthePlaneProblem当前32页，总共111页。§2-5物理方程在完全弹性的各向同性体内，形变分量与应力分量之间的关系根据虎克定律建立如下：平面问题的基本理论当前33页，总共111页。Insidetheformula,theEisamodulusofelasticity;theGisastiffnessmodulus;theuisapoissonratio.Therelationofthreeonesabove:1.ThephysicsequationoftheplanestressproblemAndhave:theBasicTheoryofthePlaneProblem当前34页，总共111页。式中，E为弹性模量；G为刚度模量；为泊松比。三者的关系：一、平面应力问题的物理方程且有：平面问题的基本理论当前35页，总共111页。2.Thephysicsequationoftheplanestrainproblem3.Thetransformationrelationoftherelationtypebetweenthestressstrainandtheplanestrain.Therelationtypeoftheplanestress:TheBasicTheoryofthePlaneProblem当前36页，总共111页。二、平面应变问题的物理方程三、平面应力的应力应变关系式与平面应变的关系式之间的变换关系将平面应力中的关系式：平面问题的基本理论当前37页，总共111页。ForchangeCangettherelationtypeintheplanestrain:Becauseofthesimilarityofthiskind,whilesolvingplanestrainproblem,thecorrespondingequationoftheplaneproblemandtheelasticconstantintheanswercanbeexchangedasabove,cangetthesolutionofthehomologousplanestrainproblem.TheBasicTheoryofthePlaneProblem当前38页，总共111页。作代换就可得到平面应变中的关系式：

由于这种相似性，在解平面应变问题时，可把对应的平面应力问题的方程和解答中的弹性常数进行上述代换，就可得到相应的平面应变问题的解。平面问题的基本理论当前39页，总共111页。§2-6BoundaryConditionsWhentheobjectisplacedinthestateofequilibrium,itsinternalstateofstressatallpointshouldsatisfytheequilibriumdifferentialequationandalsosatisfytheboundarytermontheboundary.Accordingtothedifferenceoftheboundarycondition,theelasticityproblemisdividedintothedisplacementboundaryproblem,stressboundaryproblemandmixedboundaryproblem.1.DisplacementBoundaryTermWhenthedisplacementhasbeenknownontheboundary,thedisplacementofthepointontheobjectboundaryandtheequaltermofthefixeddisplacementshouldbeestablished.Forexample,ifmakingtheboundaryofthefixeddisplacementis,andhave(onthe):Amongthem,andmeansthedisplacementweightontheboundary,however,andisthecoordinatefunctionwehaveknowtheboundary.TheBasicTheoryofthePlaneProblem当前40页，总共111页。§2-6边界条件当物体处于平衡状态时,其内部各点的应力状态应满足平衡微分方程；在边界上应满足边界条件。按照边界条件的不同，弹性力学问题分为位移边界问题、应力边界问题和混合边界问题。一、位移边界条件当边界上已知位移时，应建立物体边界上点的位移与给定位移相等的条件。如令给定位移的边界为，则有（在上）：其中和表示边界上的位移分量，而和在边界上是坐标的已知函数。平面问题的基本理论当前41页，总共111页。2.StressboundarytermWhentheboundaryoftheobjectisgiventosurfaceforce,thenthestressoftheobjectontheboundaryshouldsatisfytheequilibriumtermofforceswiththeequilibriumofthesurfaceforce.Amongthem,andarethesurfaceforceweightsand,,,arethestressweightsontheboundary.Whentheboundaryfaceisinperpendicularityinshaftx,stressboundarytermcanbechangedbrieflyinto:Whentheboundaryfaceisinperpendicularityinshafty,stressboundarytermcanbechangedbrieflyinto:TheBasicTheoryofthePlaneProblem当前42页，总共111页。二、应力边界条件当物体的边界上给定面力时，则物体边界上的应力应满足与面力相平衡的力的平衡条件。其中和为面力分量，、、、为边界上的应力分量。当边界面垂直于轴时，应力边界条件简化为：当边界面垂直于轴时，应力边界条件简化为：平面问题的基本理论当前43页，总共111页。3.Mixedboundarycondition1.Thedisplacementhasbeenknownonapartofboundariesoftheobject,theresultofwhichhavethedisplacementboundaryterm,theboundariesofotherpartshavethesurfaceforcewehaveknow.Andthenthereshouldbestressboundarytermanddisplacementboundarytermrespectivelyontwopartsoftheboundaries.Theleftsurfaceofthecantilevercontainsdisplacementboundaryterm,suchasshowninFig.2-6.Topandbottomsurfacecontainsstressboundaryterm:Therightsurfacecontainsstressboundaryterm:Fig.2-6TheBasicTheoryofthePlaneProblem当前44页，总共111页。三、混合边界条件1.物体的一部分边界上具有已知位移，因而具有位移边界条件，另一部分边界上则具有已知面力。则两部分边界上分别有应力边界条件和位移边界条件。如图2-6，悬臂梁左端面有位移边界条件：上下面有应力边界条件：右端面有应力边界条件：图2-6平面问题的基本理论当前45页，总共111页。2.Onthesameboundary,therearenotonlystressboundarytermbutdisplacementboundaryterm.Couplersustainstheboundaryterm,suchasshowninFig.2-7.ThealveolusboundarytermshowninFig.2-8.Fig.2-7Fig.2-8TheBasicTheoryofthePlaneProblem当前46页，总共111页。2.在同一边界上，既有应力边界条件又有位移边界条件。如图2-7连杆支撑边界条件：如图2-8齿槽边界条件：图2-7图2-8平面问题的基本理论当前47页，总共111页。§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当前48页，总共111页。§2-7圣维南原理一、圣维南原理如果把物体的一小部分边界上的面力，变换为分布不同但静力等效的面力（主矢量相同，对于同一点的主矩也相同），那么，近处的应力分布将有显著的改变，但是远处所受的影响可以不计。二、举例设有柱形构件，在两端截面的形心受到大小相等而方向相反的拉力，如图2-9a。如果把一端或两端的拉力变换为静力等效的力，如图2-9b或2-9c，只有虚线划出的部分的应力分布有显著的改变，而其余部分所受的影响是可以不计的。如果再将两端的拉力变换为均匀分布的拉力，集度等于，其中为构件的横截面面积，如图2-9d，仍然只有靠近两端部分的应力受到显著的影响。平面问题的基本理论当前49页，总共111页。Fig.2-9(a)(b)(c)(d)(e)Underthefourkindsofcircumstancesabove,partsofdistributionofstressforcedistantfrombothendshavenomarkeddifference.Attention:TheapplicationoftheSaint-Venant’sprincipleisbynomeansseparatedfromthetermofEqualEffectofStaticForce.TheBasicTheoryofthePlaneProblem当前50页，总共111页。图2-9(a)(b)(c)(d)(e)在上述四种情况下，离开两端较远的部分的应力分布，并没有显著的差别。注意：应用圣维南原理，绝不能离开“静力等效”的条件。平面问题的基本理论当前51页，总共111页。§2-8SolvingthePlaneProblemaccordingtothedisplacementTherearethreekindsofbasicmethodstosolvetheprobleminelasticity:thesolutiontotheproblemaccordingtodisplacement,stressforceandadmixture.Whilesolvingproblemsusingdisplacementmethod,weregarddisplacementweightasthebasicfunctionunknown.Aftergettingdisplacementweightfromonlyincludingthedifferentialequationandboundarytermofthedisplacementweight,thengetthedeformationweightusinggeometricalequation,therefore,getthestressweightwiththephysicsequation.1.PlaneStressProblemInplanestressproblem,thephysicsequationis:TheBasicTheoryofthePlaneProblem当前52页，总共111页。§2-8按位移求解平面问题在弹性力学里求解问题，有三种基本方法：按位移求解、按应力求解和混合求解。按位移求解时，以位移分量为基本未知函数，由一些只包含位移分量的微分方程和边界条件求出位移分量以后，再用几何方程求出形变分量，从而用物理方程求出应力分量。一、平面应力问题在平面应力问题中，物理方程为：平面问题的基本理论当前53页，总共111页。Fromthreeformulasabovementionedtosolvethestressweight,canget:withthesubstitutionofgeometricalequation,wecangettheelasticityequation:Againequilibriumdifferentialequationwithsubstitutioninformula(a),simplificationhereafter,canget:（a)Thisistheequilibriumdifferentialequationtomeanwiththedisplacement,ie,whensolvingtheplanestressproblemaccordingtodisplacementmethod,weadoptabasicdifferentialequationforneeds.（1）TheBasicTheoryofthePlaneProblem当前54页，总共111页。由上列三式求解应力分量，得：将几何方程代入，得弹性方程：再将式（a）代入平衡微分方程，简化以后，即得：（a)这是用位移表示的平衡微分方程，也就是按位移求解平面应力问题时所需用的基本微分方程。（1）平面问题的基本理论当前55页，总共111页。Thestressboundarytermwithsubstitutioninformula(a),simplificationhereafter,canget:Thisisthestressforceboundarytomeanwiththedisplacement,ie,weadopttheboundarytermofthestressforcewhensolvingtheplanestressproblemaccordingtodisplacementmethod.（2）Sumup,whensolvingtheplanestressproblemaccordingtodisplacementmethod,weshouldmakethedisplacementweightsatisfydifferentialequation(1)andcombinetosatisfydisplacementboundarytermorstressboundarytermorstressboundaryterm(2)ontheboundary.Aftergettingdisplacementweight,wecangetthedeformationweightwithgeometricalequationandthengetthestressforceweightwiththephysicsequation.2.Planestrainproblem

Makethesubstitutionbetweenandineachequationoftheplanestrainproblem:TheBasicTheoryofthePlaneProblem当前56页，总共111页。将（a）式代入应力边界条件，简化以后，得：这是用位移表示的应力边界条件，也就是按位移求解平面应力问题时所用的应力边界条件。（2）总结起来，按位移求解平面应力问题时，要使得位移分量满足微分方程（1），并在边界上满足位移边界条件或应力边界条件（2）。求出位移分量以后，用几何方程求出形变分量，再用物理方程求出应力分量。二、平面应变问题只须将平面应力问题的各个方程中和作代换：平面问题的基本理论当前57页，总共111页。§2-9SolvingthePlaneProblemAccordingtotheStressForce.CompatibleEquantionWhilesolvingtheplaneproblemaccordingtothedisplacement,wemustcombinetwopartialdifferentialequationofthesecondrankstosolvetheproblem,thisisverydifficultonthemathematics.Butwhilesolvingtheplaneproblemaccordingtothestressforce,wecanavoidthisdifficultyandsowhatweadoptmoreistogetthesolutionaccordingtothestressforce.Whilegettingthesolutionaccordingtothestressforce,weregardstressweightasthebasicfunctionunknown.Aftergettingdisplacementweightfromonlyincludingthedifferentialequationandboundarytermofdisplacementweight,thengetthedeformationweightusingphysicsequation,therefore,getthedisplacementweightwithgeometricalequation.CompatibleEquationFromgeometricalequationoftheplaneproblem:TheBasicTheoryofthePlaneProblem当前58页，总共111页。§2-9按应力求解平面问题。相容方程按位移求解平面问题时，必须求解联立的两个二阶偏微分方程，这在数学上是相当困难的。而按应力求解弹性力学平面问题，则避免了这个困难，故更多采用的是按应力求解。按应力求解时，以应力分量为基本未知函数，由一些只包含应力分量的微分方程和边界条件求出应力分量以后，再用物理方程求出形变分量，从而用几何方程求出位移分量。相容方程由平面问题的几何方程：平面问题的基本理论当前59页，总共111页。Canget:ie,Thisrelationtypecallsthedeformationmoderatesequationorcompatibleequation.1.Compatibleequationinplanestressforce2.CompatibleequationinplanestrainforceTheBasicTheoryofthePlaneProblem当前60页，总共111页。可得：即：这个关系式称为形变协调方程或相容方程。（一）平面应力问题的相容方程（二）平面应变问题的相容方程平面问题的基本理论当前61页，总共111页。Whilesolvingtheplaneproblemaccordingtothestressforce,thestressweightshouldnotonlysatisfyboththeequilibriumdifferentialequationandcompatibleequation,butsatisfythestressboundarytermontheboundarywhetherisaplanestressproblemorplanestrainproblem.TheBasicTheoryofthePlaneProblem当前62页，总共111页。按应力求解平面问题时，无论是平面应力问题还是平面应变问题，应力分量除了满足平衡微分方程和相容方程外，在边界上还应当满足应力边界条件。平面问题的基本理论当前63页，总共111页。§2-10TheSimplificationUndertheCircumstancesofOrdinaryPhysicalForceUnderthecircumstancesofordinaryphysicalforce,thecompatibleequationoftwokindsofplaneproblemsissimplifiedas:Therefore,underthecircumstancesofordinaryphysicalforce,shouldsatisfyLaplacedifferentialequation(inharmonywithequation),shouldbeharmonicfunctions.Representwiththemark,theformulaabovecanbesimplifiedas:

ConclusionInthestressboundaryproblemofsingleconnectioniftwoelasticbodieshavethesameboundaryshapeandsuffertheexternalforceofthesamedistribution,andthenstressforcedistribution,,shouldbethesamewhetherthematerialsoftwoelasticbodiesaresameornotandwhethertheyareundertheplanestresscircumstancesorundertheplanestraincircumstances(Twokindsofthestressforceweightintheplaneproblem,thedeformationandthedisplacementareuncertainlythesame).TheBasicTheoryofthePlaneProblem当前64页，总共111页。§2-10常体力情况下的简化常体力下，两种平面问题的相容方程都简化为：可见，在常体力的情况下，应当满足拉普拉斯微分方程（调和方程），应当是调和函数。用记号代表，上式简写为：结论在单连体的应力边界问题中，如果两个弹性体具有相同的边界形状，并受到同样分布的外力，那么，不管这两个弹性体的材料是否相同，也不管它们是在平面应力情况下或是在平面应变情况下，应力分量、、的分布是相同的（两种平面问题中的应力分量，以及形变和位移，却不一定相同）。平面问题的基本理论当前65页，总共111页。Inference2Whenmeasuringtheabovestressweightofthestructureorcomponentwiththemethodofexperiment,wecanmakethemodelusingthematerialoftheconvenientmeasurementinordertoreplaceoriginalstructureorcomponentmaterialsoftheinconvenientmeasurement;wealsocanadoptstructureorcomponentoflongcolumnshapeundertheplanestraincircumstances.Inference3Underthecircumstanceofconstantvolumetricforce,forthestressboundaryproblemofsingleconnection,wecanchargethefunctionofthevolumetricforceasthatofthesurfaceforceinordertosolvetheproblemandexperimentmeasurement.Inference1Thestressweight,,thatissolvedaccordingtoanyobjectisalsoapplicabletotheobjectwhichhasthesameboundaryandothermaterialssufferingthesameexternalforce;Thestressweightthatissolvedaccordingtoplanestressproblemisalsoapplicabletotheobjectwhichhasthesameboundaryandthesameexternalforceundertheplanestraincircumstances.TheBasicTheoryofthePlaneProblem当前66页，总共111页。推论2在用实验方法测量结构或构件的上述应力分量时，可以用便于量测的材料来制造模型，以代替原来不便于量测的结构或构件材料；还可以用平面应力情况下的薄板模型，来代替平面应变情况下的长柱形的结构或构件。推论3常体力的情况下，对于单连体的应力边界问题，还可以把体力的作用改换为面力的作用，以便于解答问题和实验量测。推论1针对任一物体而求出的应力分量、、，也适用于具有同样边界并受有同样外力的其它材料的物体；针对平面应力问题而求出的这些应力分量，也适用于边界相同、外力相同的平面应变情况下的物体。平面问题的基本理论当前67页，总共111页。§2-11StressFunction.InverseSolutionMethodandSemi-InverseMethod1.StressfunctionWhilesolvingthestressboundaryproblemaccordingtothestressforceandwhenthevolumetricforceistheconstantquantity,thestressweight,,shouldsatisfytheequilibriumdifferentialequation:（a）Andcompatibleequation（b)Thesolutiontotheequation(a)includestwoparts:arbitrarilyaparticularsolutionandthegeneralsolutiontothefollowinghomogeneousdifferentialequation.TheBasicTheoryofthePlaneProblem当前68页，总共111页。§2-11应力函数、逆解法与半逆解法一、应力函数按应力求解应力边界问题时，在体力为常量的情况下，应力分量、、应当满足平衡微分方程：（a）以及相容方程（b)方程（a）的解包含两部分：任意一个特解和下列齐次微分方程的通解。平面问题的基本理论当前69页，总共111页。Theparticularsolutionis:Rewritetheformerequationinsidethehomogeneousdifferentialequation(c)as:Accordingtothedifferentialequationtheory,itiscertaintoexistsomefunction,make:（c）（d）（e）（f）TheBasicTheoryofthePlaneProblem当前70页，总共111页。特解取为：将齐次微分方程（c）中前一个方程改写为：根据微分方程理论，一定存在某一个函数，使得：（c）（d）（e）（f）平面问题的基本理论当前71页，总共111页。

Similarlyrewritethesecondequationinside(c)as:Itiscertaintoexistsomefunctionaswell,make:（g）（h）Fromtheformula(f)and(h),canget:Thus,itiscertaintoexistsomefunction,make:（i）（j）TheBasicTheoryofthePlaneProblem当前72页，总共111页。

同样将（c）中的第二个方程改写为：也一定存在某一个函数，使得：（g）（h）由式（f）及（h）得：因而一定存在某一个函数，使得：（i）（j）平面问题的基本理论当前73页，总共111页。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当前74页，总共111页。将式（i）代入（e），式（j）代入（g），并将式（i）代入（f），即得通解：（k）将通解（k）与特解（d）叠加，即得微分方程（a）的全解：函数称为平面问题的应力函数，也称为艾瑞应力函数。（1）为了应力分量（1）同时也能满足相容方程（b），将（1）代入式（b），即得：上式可简化为：平面问题的基本理论当前75页，总共111页。Orspreadingtheformulais:Furthersimplificationis:（2）2.Inversesolutionmethodandsemi-inversemethodInversesolution:thefirststepistosetupmultiformstressfunctionwhichsatisfythecompatibleequation(2),andgetthestressweightwiththeformula(1),theninvestigateaccordingtothestressboundaryterm.Ontheelasticbodyineverykindofshape,thesestressweightscorrespondenceinwhatkindsofsurfa