弹性力学课件四

2TheoryofPlaneProblems2—1PlaneStressandPlaneStrain2—2DifferentialEquationsofEquilibrium2—3

GeometricalEquations.Rigid-bodyDisplacement2—4PhysicalEquations2—5StressataPoint2—6BoundaryConditions.

Saint-Venant’sPrinciple2—7SolutionofPlaneProbleminTermsofDisplacements2—8SolutionofPlaneProbleminTermsofStresses2—9CaseofConstantBodyForces.StressFunction*SOLUTIONOFPLANEPROBLEM*Forthesolutionofanelasticityproblem,wecanproceedinthreedifferentways:1.Takethedisplacementcomponentsasthebasicunknownfunctions,formulateasystemofdifferentialequationsandboundaryconditionscontainingthedisplacementcomponentsonly,solvefortheseunknownfunctionsandtherebyfindthestraincomponentsbythegeometricalequationsandthenthestresscomponentsbythephysicalequations.

2.Takethestresscomponentsasthebasicunknownfunctions,formulateasystemofdifferentialequationsandboundaryconditionscontainingthestresscomponentsonly,solvefortheseunknownfunctionsandtherebyfindthestraincomponentsbythephysicalequationsandthenthedisplacementcomponentsbythegeometricequation.3.Takesomeofthedisplacementcomponentsandalsosomeofthestresscomponentsasthebasicunknownfunctions,formulateasystemofdifferentialequationsandboundaryconditionscontainingthestresscomponentsonly,solvefortheseunknownfunctionsandtherebyfindtheotherunknownfunctions.2—7按位移求解平面问题平面问题的基本未知量有

x，y，xy，x，y，xy，u，v，根据基本方程即可求解。SolutionofPlaneProbleminTermsofDisplacements求解方法有：按位移求解；按应力求解；混合求解P26按位移求解：以位移分量为基本函数，由只含位移分量的微分方程和边界条件求出位移分量后，再求其他的未知量。Takethedisplacementcomponentsasthebasicunknownfunction，formulateasystemofdifferentialequationsandboundaryconditionscontainingthedisplacementcomponentsonly，solvefortheseunknownfunctionsandtherebyfindthestraincomponentsbythegeometricalequationsandthenthestresscomponentsbythephysicalequations.

导出按位移求解的微分方程和边界条件

1、微分方程

(differentialequations)

Formulatethedifferentialequationsandboundaryconditionsforthesolutionofaplaneproblemintermsofdisplacements

几何方程：geometricalequations物理方程（平面应力问题）physicalequations（planestressproblem）将几何方程代入物理方程SubstitutionofgeometricEqs.intotheseequations将上述方程代入平衡微分方程Now,usingtheserelationsinequilibriumequations按位移求解的平衡微分方程拉密方程在平面问题中的应用Thedifferentialequationsforthesolutionoftheproblemintermsofdisplacements.2、飞边界根条件(Bo建un个da虚ry甘c霉on拘di扣ti旅on固s)l

x+mxy=Xm

y+l

xy=Y平面问题的应力边界条件（Stressboundaryconditionsofaplaneproblem）按位挽移求订解时拣的应固力边秋界条碎件为抹：用位移表示的应力边界条件。Weobtainthestressboundaryconditionsoftheproblemintermsofdisplacements.归纳(To绸s障um嘉u区p)笋,按位千移求赠解平顾面问跳题，肝要使烈位移起分量骆满足控拉密少方程第和边设界条卖件，厕求出略位移净后，垒可用底物理扇方程裙求应蛾力，匠用几积何方构程求产变形淋。Th塞e僵di陕sp撕la夜ce风c珠om项po浸ne的nt婶s丸u(谷x,兵y)鄙,障v(蔑x,拿y)瘦i杜n稠a虑pl唐an胆e苏st跌re制ss叛p互ro著bl琴em市m墓us封t捷sa币ti蜓sf悼yth匠ro枕ug珠ho罗ut侨t悠he键b中od偶y怨co叠ns溜id垫er慰edan盘d虾al抵so根s背at妈is边fyon现t娃he蒸s帐ur枯fa蚁ce盛o袍f娱th子e骄bo停dy种.Fo中r贼a撤pl悄an渠e驻st赚ra除in固p发ro乡丰bl毙em后,晋it乓i伟s众ne捞ce傅ss丘ar迹yin振a赛bo捡ve矩e寺qu盐at目io彼ns段.对平面应变问题，只许将上述方程中的；将即可。2—胖8袭按应江力求翼解平撤面问震题鸭相容塌方程P2妖8So罚lu吩ti盐on继o纵f肿Pl困an蓄e贝Pr滑ob牛le独m阅in孩T乡丰er唐ms籍o况f琴St理re裕ss繁es按应力求解：以应力分量为基本函数，由只含应力分量的平衡微分方程和相容方程及边界条件求出应力分量后，再求其他的未知量。Takethestresscomponentsasthebasicunknownfunction，formulateasystemofdifferentialequationsandboundaryconditionscontainingthestresscomponentsonly，solvefortheseunknownfunctionsandtherebyfindthestraincomponentsbythephysicalequationsandthenthedisplacementcomponentsbythegeometricalequations.含应力分量，需保留Thetwodifferentialequationsofequilibriumcontainthestresscomponentsonlyandmaybeusedfortheirsolution.需建立补充方程相容方程Thethirddifferentialequationcanbeobtainedfromthegeometricalandphysicalequations.相容方程（变形协调方程）Compatibilityequation1、平面问题的几何方程Thegeometricalequationsofaplaneproblemare2、将

x对y的二阶导数和y对x的二阶导数相加Addingthesecondderivativeof

xwithrespecttoyandthesecondderivativeof

ywithrespecttox,weget

相容方程Compatibilityequation应变氧分量

x、

y和xy必须由满足还这个挥方程驴，才性能保隐证位境移分锻量u，谷v的存礼在。若所舒选的

x、

y和xy不满吉足这毙个方吼程，竹那么刮，由旁几何占方程枕中的承任意肿两个勉所求开出的蓝位移吼分量笋，将域不满稼足第蛾三个烈方程值。例如选

x=0，

y=0，

xy=cxy

不满足相容方程由此指应变敲求位押移Th光e夜co辛mp狗at仗ib瞧il菠it贫y兴eq龄ua必ti责on岔f枪or舟s县tr些ai巧n擦mu赞st言b存e忌sa劫ti沟sf毯ie钉d伤by撞t卧he耕s杜tr芦ai恶n啊co啊mp求on唯en清ts

x,

yan画d

xyto普e日ns搞ur差e久th秧e敏ex狮is宣te抵nc阻e黑of事s辰in叼gl挽e-帜va垃lu亭ed格c冤on盏ti维nu郑ou证s年fu款nc抬ti厕on西suan主dvco柄nn疤ec悠te定d承wi诸th梦t屠he须s乡丰tr蹈ai眨n扶co滤mp行on宽en移ts感b决y厘th粮e统ge盘om希et礼ri姜ca伶l凶eq越ua友ti傅on堂s.第三但个方酱程不项能满蛋足，部所求u，春v不存遭在用应爬力表代示的丙相容拖方程Co蝇mp袍at猜ib助il艇it粒y赠eq慰ua组ti琴on更i宁n升te挎rm坑s示of尾s袜tr妥ai府nBy壤u娱si香ng宵p产hy轻si也ca潜l渴eq矮ua歇ti绳on尊s,络t占he今c没om奸pa贫ti端bi极li蝇ty或e民qu酷at局io款n约ca纲n棋be竟t咏ra蛾ns逗fo怕rm理ed芳i沙nt彩o守a钟re马la阻ti旧on描b校et近we厦en翁t奇he禽s刷tr纤es血s抱co石mp枪on受en断ts蜂.将物理方程代入

平面应力问题

foraplanestressproblemSubstitutionofthephysicalequationsinto简化皂上述留相容子方程杰（利奇用平挎衡方成程）To斤t湖ra产ns娃fo凯rm招t灵hi漆s私eq号ua剪ti黄on握i馋nt胀o腿a私di踏ff皆er运en杀t番fo模rm森m由or势e候su熊it枕ab许le抄f求or栋u包se融,待we似e签li搁mi仁na魂te示t满he锐t首er戏m岩in刘vo彼lv扁in泡g

xyby欣u嫩si闪ng诉t靠he霉d茧if甩fe勤re站nt仪ia全l连eq聚ua状ti扣on术s衡of淹e干qu净il笔ib省ri柳um惹.用应力表示的相容方程thecompatibilityequationintermsofstresses.Di屿ff蹈er恼en业ti权at招in染g慈th零e圾fi宽rs毒t沈eq杏ua絮ti帐on灰w脱it网h缘瑞re鼻sp覆ec猾t亡toxan愤d首th促e谷se拖co宇nd紧w睁it调h妈re爆sp贡ec格t捕toy,赚ad宴di司ng争t液he元m洽up波a销nd策n烤ot冲in岩g泥th门at

xy=yx,紫we援g驼et(将却两式董分别浙对x及y求导怨，并壮相加宫得)(将其代入相容方程，并简化后，得)Substitutingthisintothecompatibilityequationandperformingsomesimplification,weobtain平面应变问题的相容方程CompatibilityequationintermsofstrainCo鸡mp抄at霜ib化il和it牙y美eq奖ua讽ti趋on易f扇or屡a苦p蜡la损ne授s艇tr相ai酸n斑pr吧ob穷le装mForaplanestrainproblem,anequationsimilartoaboveequationmaybeobtainedsimplybyTh延e驱re畏su黎lt衣i筐s归纳黄：1、按应力求解平面问题，要求应力分量必须满足平衡微分方程和相应的相容方程，在边界上还要满足应力边界条件（P29）。Inthesolutionofaplaneproblemintermsofstresses，thestresscomponentsmustsatisfythedifferentialequationsofequilibriumandcompatibilityequationinthecaseofplacestrain.Besides,theymustsatisfythestressboundaryconditions.2、由于位移边界条件无法用应力分量或其导数来表示，所以对位移边界条件或混合边界条件，不可能按应力求解得出精确解。Sincethedisplacementboundaryconditionscanbeexpressedneitherintermsofstresscomponentsnorintermsoftheirderivativeswithrespecttothecoordinates,displacementboundaryproblemsandmixedboundaryproblemscannotbesolvedintermsofstresses.对应概力边核界问锐题，强应力絮分量嗽满足替了平本衡微各分方辨程、傅相应窑的相拜容方村程和切应力砍边界溜条件乓，其千应力恰分量算就能足完全塑确定拜？在多者连体爱中，暂要完可全确君定位喜移分牙量，羽还必曲须利咸用“到位移称须为距单值刺”这旁个条穴件No，还必须考虑弹性体是否单连体Inthesolutionofelasticityproblems，itisnecessarytodistinguishbetweensimplyconnectedbodiesandmultiplyconnectedones.多连体：有两个或两个以上连续边界的物体，如：有孔口的物体单连体：只有一个连续边界的物体simplyconnectedbodies:anarbitraryclosedcurvelyinginthebodycanbeshrunktoapoint,bycontinuouscontraction,withoutpassingoutsideitsboundaries.Otherwise,thebodywillbesaidtobemultiplyconnected.In锣t脸he舍c贼as谜e豆of闯m喂ul霜ti当pl慌y北co细nn鄙ec晨te串d掘bo根dy煮,蹲th茎er急e胳mi据gh漠t笋be石s恢om广e裤ar丛bi芝tr占ar仿y馒fu鼠nc露ti修on脖s宏le轿ad新in雀g版to膏m凭ul匪ti骑-v鸟al念ue捉d板di棵sp祝la营ce季me鸟nt菜s,悬w此hi伪ch畜a殖re劣i丹mp圈os霸si岔bl寨e筋in员a农c蓝on响ti郑nu挂ou智s何bo密dy迷.欠Th纷en城,脆we角h弹av卖e剃to点c绞on殖si纯de暂rth月e滔co灵nd碧it归io仙n职of泛s萍in父gl膝e-株va乓lu嘉ed救d言is圈pl开ac庭em奥en叠tsto会d乞et召er属mi键ne似t姑he奇s燃tr偏es陵se敏s.In智p睁la壁ne孟p福ro刑bl柔em国s,渠h柜ow孤ev桶er虎,灿we木m副ay娃a断ls糟o帖br炊ie孩fl赶y相de远fi买ne暑asi常mp拘ly谁c尚on山ne章ct登ed介b倍od挤yas掏o瞎ne话w瓣it曲h获on所ly宅o永ne范c哭on刃ti乐nu抹ou叛s床bo向un剧da咐ry粗a添nd腔amu六lt滑ip来ly逢c煎on洗ne蓬ct开ed拌b亡od弯yas野o绕ne胶w拦it博h幼tw反o同or斧m架or戚e存bo穿un歪da婶ri训es午.2—激9捉常摆体力贵情况派下的缴简化体力劲不随尿坐标性而变软化（抓重力罩、惯世性力踪蝶）应力筋分量束应满逃足：（a）（b）Ca御se愉o艳f筒Co增ns哄ta叉nt眼B晌od夏y姿Fo狸rc绘esIn尽m财an该y勇en经gi粪ne绵er泼in该g籍pr致ob场le众ms宜,t倘he涝b炊od达y塑fo骂rc贡es贝a敏re悄c恭on直st议an蔑t.On暗t玻he拴c社on是di孟ti郊on亭o禁f搞co马ns烘ta雷nt鸟b茅od呼y秤fo淘rc膜es性,迁th晚e词co胆mp黎at秩ib鸦il尤it度y纹eq欺ua斜ti妥on扣s教wi短ll烂r妈ed聋uc校e适to晒t肉he染h好om狱og扇en容eo羞us兽d奴if立fe宪re悬nt哲ia爷l毕eq端ua枝ti塘on上述方程中不含材料常数，所以对两类平面问题都适用.Nowthedifferentialequationsofequilibriumandthestressboundaryconditions,aswellasthecompatibilityequation,donotcontainanyelasticconstantandarethesameforbothkindsofplaneproblems.只要弹性体（单连体）边界相同，外载相同，不管是何种材料，也不管是平面应力状态或平面应变状态，应力分布是相同的（位移及变形是否相同？）Inastressboundaryproblemforasimplyconnectedbodywithacertainboundaryandsubjectedtocertainexternalforces,thestresscomponentswillhavethesamedistributioninbothplanecondition.Th炕is拍c切on怒cl你us蒸io俭n倦is霜v散er迷y滨he启lp德fu懒l敲in星t成he依e困xp丹er紫im纹en铜ta肾l肝an预al传ys缓is咽.（1）可将某种材料，某种状态下所求的应力分量的结论用于其他材料或其他状态（边界条件，外荷载相同）Wemayuseanymodelmaterialconvenientforstressmeasurementinsteadofthestructurematerialonwhichthemeasurementmightbeimpossible.（a）（2）在实验中，可以用便于测量的材料来制造模型；或用平面应力情况下的薄板来代替平面应变情况下的长柱体.Wemayuseamodelinplanestresscondition(athinslice)insteadofoneinplanestraincondition(alongcylindricalbody).（b）Th询e认st照re怖ss门c变om希po被ne岔nt内s典ar累e法d烈et莲er歼mi抱ne贱d撕by毯t债he怒d绑if庙fe凑re齐nt独ia亡l候eq吼ua折ti深on析s:（a）isno做nh愁om起og馅en奏eo旋usan屑d,毙t革he风re把fo裹re杠,封it扫s经ge械ne励ra获l血so胡lu咸ti爷on拳m霜ay俊b汤e导ex甲pr联es瓜se锐d玩as宪t拜he柿s覆um题o胀f充a搂pa幸rt箭ic材ul竿ar飘s刺ol补ut纺io妨n欢an骨d辞th布e标ge革ne存ra并l波so触lu睛ti虾on亭o剧f踢th学e倡ho职mo撤ge诸ne赵ou望s编sy闹st兔em考察嗽（a）肝，其解黎由非绢齐次抗方程驼的特典解+齐次粗方程咽的通淘解特解设为：

x=-Xx，y=-Yy，xy=0提（c爆）或况

x=0宪，y=0掌，xy=-Xy-Yx或船

x=-Xx-Yy，y=-Xx-Yy，xy=0只要垃能满每足方学程即届可取（c）式求齐冷次方怖程的苦通解将方程变为(1)(2)满足（1），必存在一个函数A（x，y）使得：Accordingtodifferentialcalculus,for(1),thereexistsacertainfunctionA(x，y)sothat：Re询wr经it或e同理熔满足（2怕），必偿存在坚一个博函数B（朵x，愧y）使得陡：SoSi唤mi吨la批rl肃y,纽奉(芬2)蜘e极ns虽ur省es童t饲he册e遗xi狱st色en迈ce差o米f酸an神ot款he由r智fu坊nc截ti凶on都B怜(x惠，y创)北so肺t蹈ha值t：必存在一个函数（x,y），且Whichensurestheexistenceofstillanotherfunction(x,y)sothat所以温，齐屑次方芽程的蚀通解堪为：We弯o拣bt框ai孩n骆th煎e见ge粥ne京ra青l四so敏lu以ti跪on肠o摊f颂ho炸mo竿ge厕ne炉ou腊s势eq蜓ua趋ti逝on径s：平衡歌微分气方程峰的解胆为：（c）No描w,泽t落he仅s岔up歼er眨po杰si难ti鸟on杰o蹦f毫th第e怖ge际ne器ra跌l豆so候lu跪ti逝on欲w合it碍h豆th桂e厚pa畏rt扔ic摇ul云ar体s你ol急ut悦io征n症yi递el遗ds而t创he宅f葱ol系lo蚂wi得ng械c群om巾pl缩慧et献e列so刊lu修ti让on掠:Th喊e敬fu份nc歇ti测on（x,蹄y）is谈k境no白wn牢a染s驰th穿e菌st凭re需ss毫f充un泽ct拒io兔n摧fo茅r暂pl吓an伶e季pr勾ob千le赠ms膛,盯orth台e拌Ai初ry沃’s短s获tr装es喘s测fu清nc侮ti度on.（x,y）称为平面问题的应力函数—艾瑞应力函数Withanyfunction（x,y）,thestresscomponentssodefinedalwayssatisfythedifferentialequations.Thisfunction（x,y）isknownasthestressfunctionortheAiry’sstressfunction.

应力分量除满足平衡微分方程外，还必须同时满足相容方程，所以将（c）代入相容方程Inorderforthestresscomponentstosatisfythecompatibilityequationaswell,thestressfunctionmustsatisfyacertainequation.OrTh缠is芝i羽s宰th串e固co状mp腹at域ib莲il推it摆y鲁eq初ua垄ti顿on图i酸n咸te童rm期s宴of呆t闭he腐s献tr尘es殊s挡fu拉nc殊ti陕on.

2(Xx)=

2(Yy)=0intheconditionofconstantbodyforce此方程为双调和方程，写为：or投b弊e刃si湖mp冈ly黎w尼ri撇tt令en无a冲s：若不计体积力，即X=0；Y=0Whenbodyforcesarenotconsidered,thesolutionwillreducetoTh物us：in翼t另he甜s奖ol尝ut挖io军n判of我p命la买ne梨p狐ro踢bl其em私s士in奔t异er障ms摊o绢f蒸st救re恒ss撇es冒,岭wh顿en鹿t变he尿b符od促y苍fo懂rc堆es汽a枪re需c寻on钓st医an碍t,鉴i茫t回is师o飘nl漠y额to蹦s攻ol谋ve状f青or块t卫he膨s绒tr颠es馋s掘fu迹nc穗ti滨on嚼f腾ro猎m咐th捉e移si千ng迎le梨d香if挡fe颤re猾nt四ia绳l茅eq沸ua梢ti堪onan子d育th在en祸f腔in晚d扯th遭e取st湖re骡ss朗c悦om首po杜ne点nt腾s泡byBu换t饶th帆es锦e替st锤re掀ss灿c游om绒po孟ne凤nt平s友mu啦st杨s殿at顽is狮fy谱t蕉he锐s先tr稳es熔s胶bo俘un巩da庭ry必c担on红di恶ti征on朱.圆In坡t屈he索c竭as形e崖of习m播ul庄ti吴pl绿y图co莫nn变ec尖te铺d蜂bo禽di猎es搭,岸th志e骡co灰nd贪it绝io款n猪of吴s壳in竖gl检e-照va闪lu辟ed峡d凝is都pl章ac坚em雄en纪ts惧m梳us感t夺be穿i共ns膝pe贸ct垃ed屠i益n肿ad呀di宁ti蚁on荡.归纳难：按应力求解平面问题时，如果体力是常量，则由求解应力函数，然后按求应力分量，这些应力分量在边界上满足应力边界条件，在多连体中，还须考虑位移单值条件.To馒s庄ol刃ve芒t榴he摸p蚀ar盟ti影al匪d送if远fe段re划nt辣ia树l萝eq霉ua蚀ti分on践s悦of吊e制la瘦st吓ic卧it批y督to脏ge己th遗er培w静it栗h耕th强e海gi猎ve配n括bo棒un艰da杀ry肉c惕on赵di牢ti专on挖s,无t深he伟d估ir肚ec卵t屯me涉th论od搁o饺f张so大lu坡ti险on鸽i施s校us昼ua例ll租y气im报po蝇ss啊ib女le杜.凝We贡h灯av堂e欠to安u陕seth印e觉in读ve拣rs盈e稿me易th宵odorth固e最se议mi炕-i宰nv故er柄se扯m宋et裂ho域d.Inth猴e荣in旷ve绩rs嚷e获me秀th龟od,郑so沸me族f风un失ct多io察ns廉s欢at损is延fy昂in敬g帆th偶e泡di拼ff还er啄en立ti壶al饰e暴qu政at断io群ns朝a视re视t涉ak既en厕a基nd佳e妹xa仅mi逃ne崇d墙to油s武ee依w抛ha些t胡bo洗un仓da财ry轻c康on柄di膜ti祥on壤s屯th亿es标e君fu驱nc充ti变on姜s虾wi昼ll五s答at谁is泳fy假a桑nd滚t痒he披re改by惑t欢o多kn峰ow蹈w扶ha蓝t炮pr横ob排le阶ms倘t娘he菊y卧ca话n速so违lv浅e.In巾t劳he岛c蹈as我e众of精s陡ol纹ut汪io赌n伍by垮A炸ir读y’放s久st叙re叔ss卖f秆un释ct谱io始n,拘w师ese属le诸ct皮s踩om痛e呜fu概nc那ti摆on

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