弹性力学课件三一

2.9SOLUTIONOFPLANEPROBLEMINTERMSOFDISPLACEMENTForthesolutionofanelasticityproblem,wecanproceedinthreedifferentways:1.Takethedisplacementcomponentsasthebasicunknownfunctions,formulateasystemofdifferentialequations

andboundaryconditions

containingthedisplacementcomponentsonly,solvefortheseunknownfunctionsandtherebyfindthestraincomponentsbythegeometricalequationsandthenthestresscomponentsbythephysicalequations.

2.Takethestresscomponentsasthebasicunknownfunctions,formulateasystemofdifferentialequationsandboundaryconditionscontainingthestresscomponentsonly,solvefortheseunknownfunctionsandtherebyfindthestraincomponentsbythephysicalequationsandthenthedisplacementcomponentsbythegeometricequation.3.Takesomeofthedisplacementcomponentsandalsosomeofthestresscomponentsasthebasicunknownfunctions,formulateasystemofdifferentialequationsandboundaryconditionscontainingthestresscomponentsonly,solvefortheseunknownfunctionsandtherebyfindtheotherunknownfunctions.Nowweproceedtoformulatethedifferentialequationsandboundaryconditionsforsolutionofaplaneproblemintermsofdisplacements.Thegeometricequations：

1.Thedifferentialequations

Thephysicalequations(planestressproblem)SubstitutionofgeometricEotheseequationsNow,usingtheserelationsinequilibriumequationsThedifferentialequationsforthesolutionoftheproblemintermsofdisplacements.

2.Boundaryconditions

lx+mxy=Xmy+lxy=YStressboundaryconditionsofaplaneproblemWeobtainthestressboundaryconditionsoftheproblemintermsofdisplacements：Tosumup,Thedisplacecomponentsu(x,y),v(x,y)inaplanestressproblemmustsatisfythroughoutthebodyconsideredandalsosatisfyonthesurfaceofthebody.Foraplanestrainproblem,itisnecessaryinaboveequations.2.10SOLUTIONOFPLANEPROBLEMINTERMSOFSTRESSESThetwodifferentialequationsofequilibriumcontainthestresscomponentsonlyandmaybeusedfortheirsolution.Thethirddifferentialequationcanbeobtainedfromthegeometricalandphysicalequationsbyeliminatingthedisplacementcomponentstherein.ThegeometricalequationsofaplaneproblemareAddingthesecondderivativeofxwithrespecttoyandthesecondderivativeofywithrespecttox,weget

ThecompatibilityequationforstrainThecompatibilityequationforstrainmustbesatisfiedbythestraincomponentsx,yandxytoensuretheexistenceofsingle-valuedcontinuousfunctionsuandvconnectedwiththestraincomponentsbythegeometricalequations.若所选的x、y和xy不满足这个方程，那么，由几何方程中的任意两个所求出的位移分量，将不满足第三个方程例如选x=0，y=0，xy=cxy

不满足相容方程由此应变求位移第三个方程不能满足，所求u，v不存在Byusingphysicalequations,thecompatibilityequationcanbetransformedintoarelationbetweenthestresscomponents.

Substitutionofthephysicalequationsinto

Foraplanestressproblem

Totransformthisequationintoadifferentformmoresuitableforuse,weeliminatetheterminvolvingxy

byusingthedifferentialequationsofequilibrium.

Differentiatingthefirstequationwithrespecttoxandthesecondwithrespecttoy,addingthemupandnotingthatxy=yx,wegetSubstitutingthisintothecompatibilityequationandperformingsomesimplification,weobtainthecompatibilityequationintermsofstresses.Foraplanestrainproblem,anequationsimilartoaboveequationmaybeobtainedsimplybyThe湾re贫sul装ti爆sTh心us狱,1.In荣t必he别s狗ol仪ut默io涉n绳of景a完p叶la条ne污p个ro明bl轨em犹i烤n齿te顽rm血s痕of项s格tr讨es旗se最s,迟t尺he鼠s防tr珍es拒s魔co怖mp华on两en炉ts渗m介us岛t浪sa纤ti助sf传yth澡e赢di斥ff穷er陡en植ti恒al钞e矩qu肆at敏io膀ns钓o未f鄙eq衔ui更li辆br够iu教mandthe拾co占mpa偶tib并ili撇ty李equ旅ati永on.Bes惯ide恼s,欲the瘦ym土ust船sa植tis固fyth斯e润st争re证ss粗b踏ou慨nd纷ar杂y拦co腰nd赶it绸io黎ns.2.Si相nc亭e驼th麦e爪di柜sp猪la默ce曾me辈nt蓝b横ou攻nd壤ar乌y扯co失nd础it终io揪ns瓣c酬an溪b航e劳ex窄pr蚁es钓se驴d哄ne歼it悄he妹r组in讯t灵er讲ms材o筐f近st究re愚ss胜c框om煮po斤ne思nt耐s碍no冈r恩in棋t痒er皆ms绞o苍f塌th瞒ei寒r猛de献ri适va拢ti河ve岸s悼wi消th五r绿es还pe撞ct最t第o蝇th宅e扭co艇or屈di华na罗te爱s,dis倾pla束cem钢ent意bo宫und抽ary卸pr拜obl显emsandmix钓ed蔽bou高nda惹ry搅pro锻ble都msca副nn坟otbe炉s肆ol崭ve副d淘in涉t偿er愉ms炼o皂f棍st红re迈ss肯es王.In雾the绳so磁lut倍ion弯of竭el促ast邻ici翼ty签pro蹄ble匹ms,哗it巷is睛ne品ces洲sar血yt急od飘ist谣ing询uis表hb摄etw伐eensim唯ply逝co蹦nne狐cte洁db梦odi改esandmu码lt恰ip厦ly芦c掉on着ne狠ct代ed芳o暮ne包s.A瞒bo确dy爬i稻s劝sa贫id词t欢o察besi阔mp班ly酬c停on农ne崇ct熄edif麻a扁n密ar踏bi背tr东ar桥y进cl军os短ed乞c抱ur也ve昏l嗽yi烫ng超i非n挖th爪e赖bo机dy虹c列an桂b默e占sh舞ru庄nk纲t去o睁a狼po是in歼t,佳b密y糕co悟nt洁in爸uo匙us秩c丽on趟tr愈ac巧ti纲on瘦,防wi挂th裕ou值t翼pa肺ss刚in捕g绣ou牙ts宋id替e亭it俊s垒bo枕un堤da醋ri伴es泡.So窑li伯d斩bl室oc脏ks弄a界nd侮h例ol惠lo戚w轿sp哀he椒re莲s耻ar寻e晴ex袄am越pl写es涂o虫f径si车mp确ly险c蛇on唱ne慈ct扭ed裂b张od送ie减s.Ot答he冰rw此is业e,旱t蹦he苗b扑od普y猎wi滤ll变b较e屈sa达id含t扫o企bemu巧lt毁ip刃ly推c晒on薯ne掀ct牲ed.Ri应ng俘s科an尤d逐ho汁ll借ow砍c煎yl糠in栽de鸽rs示a虑re费e爬xa育mp移le雕s益of薯m告ul触ti莲pl槐y拥co峡nn堂ec标te酸d名on望es关.In呀the沫ca泳se蛾of朽mul聋tip嘴ly夹con饿nec祥ted仁bo势dy,上th乐ere债mi醋ght约be导so于me却arb概itr胳ary饱fu闻nct搭ion艺sl仅ead乌ing亩to谊mu劝lti拖-va衬lue窝dd利isp沈lac虾eme内nts宇,w很hic弯ha炕re券imp啄oss确ibl勺ei堵na尝co绸nti腊nuo贡us椒bod痛y.绑The稳n,蚁we径hav垂et装oc缎ons友ide截rth述e妻co升nd骂it肠io屯n器of臣s垒in旱gl役e-作va制lu摊ed端d碍is午pl洪ac态em夜en讯tsto骄d退et原er窄mi文ne悦t墙he耐s歼tr娃es饼se泥s.In她p蒸la女ne执p舅ro前bl老em锋s,浮h暴ow台ev找er坊,效we链m们ay寇a想ls控o课br简ie般fl授y谊de款fi跨ne彻asim深ply龟co贯nne柄cte工db统odyas盏o满ne扁w喘it惹h团on菠ly攀o凑ne锁c榨on被ti疲nu邀ou致s污bo扯un膀da怜ry舍a硬nd富amul伶tip参ly济con达nec栽ted穴bo僚dyas闲o遍ne兼w核it饲h旦tw相o子or载m付or对e刚bo编un毁da半ri擦es责.2.1歉1CAS犹EO购FC傍ONS缝TAN掘TB时ODY叙FO抢RCE组SIn伏m雄an织y膛en欢gi吵ne嗽er银in骑g飘pr邀ob河le蹦ms孤,抵th扒e假bo妥dy限f蚁or钉ce奴s握ar酱e侄co晓ns怪ta牙nt究,磁I.遮e.办,疲th唱e还co岭mp疗on孝en拾tsXan萄dYdo饿not薯va剃ry忆wit居hc仁oor图din推ate烧st豪hro仍ugh膜out剃th烂ev滨olu醋me馆of彼the陶wh昼ole型bo唇dy.（the锈gr扁avi兰ty喜for细ces薪,车the腾in做ert盘ia偿for国ces）On因the圆co亏ndi啊tio照no姐fc叶ons间tan慰tb博ody禾fo身rce违s,裳the骗co温mpa些tib坡ili感ty桌equ顶ati拘ons郊wi灰ll严red词uce源to冻th包eh声omo诸gen卖eou餐sd芹iff赌ere龟nti脖al驱equ命ati可onNow挠,the惑di器ffe江ren浅tia汇le锹qua择tio碑ns税of勒equ历ili班bri妹uman疗dthe尿st漏res菜sb享oun务dar戚yc陈ond矮iti平ons,按as速w再el灿l穿asthe称co锦mpa蒸tib让ili撑ty睬equ度ati汉ons,do疮not跌co狡nta青in蜓any纵el默ast取ic涛con枪sta沾ntan梯d扁ar片e坦th厕esa溉me顾f复or躬b畜ot旦hki私nd肚s烫of霜p奔la漠ne粥p到ro冰bl自em府s.He份nc鉴e,缸i衫n苹a芬st某re沿ss短b泥ou斤nd葬ar烘y马pr吨ob滥le临m约fo埋r堂a纤si约mp芳ly泰c蔽on塞ne躲ct涉ed肌b井od冤y勉wi盏th玩a肯c客er台ta食in拒b盟ou烦nd纤ar希y誉an凶d形su搞bj纲ec葱te委d逮to拍c厦er吃ta稼in足e颜xt晒er斑na任l巡寿fo葛rc煌es吴,the款st国res省sc争omp企one唇ntsx,浊y,哪xywi扁ll正b继e剖in脂de射pe蒙nd予en底t麦of暴t简he诞e奇la索st访ic度p比ro挤pe旗rt野ie俱sof剖the迁bo懂dy私and篇ha功ve孟thesa薪me耽d讨is呜tr编ib依ut民io口nin槐bot冤hp津lan阅es奔tre金ss厕con盟dit堡ion薪an脸dp箱lan渐es桨tra均in余con反dit箭ion刘.Thi俘sc蜘onc芦lus苍ion回is监ve亲ryhel脚pfu王li脖nt助he略exp谨eri推men双talana盼lys负is厕of校the把st米res窃ses顶in峡a跪str周uct雅ure厨or唇it株se轿lem旱ent全s.（1）浇可将某续种材料升，某种编状态下堡所求的炭应力分萄量的结导论用于督其他材切料或其央他状态扯（边界根条件，盈外荷载寸相同）（2池）We螺may降us链ea乖mo丑del杜inpl伤an庭e赏st以re孕sscon扔dit支ion趋(a欠th登in始sli嘱ce)ins肠tea嚼do妻fone填inpla余ne而str士ainco误nd贩it旁io葵n叉(a户l白on葡g追cy武li揭nd缴ri叉ca它l枣bo布dy芦).（1）We脏may旅us润ea腹ny您mod很el消mat自eri染al厕co气nve厕nie盾nt卵for成st袜res厕sm庭eas溜ure掏men翅ti请nst明ead傍of去th钳es头tru建ctu懂re贞mat盯eri报al跪on他whi嫂ch持the努me裤asu排rem牌ent诞mi椒ght装be长im溉pos景sib司le.(3)Be梢si辈de槐,步in状t傅he崖c稍as抢e乱of籍a患s据tr逮es划s朱bo帽un奇da音ry鞠p必ro钉bl胳em姓f锤or筑s它im穴pl解y肤co膜nn捷ec新te归d农bo鸦di码es栋s植ub睁je哥ct泳ed皱t损o死co卷ns喊ta唯nt退b息od泻y墨fo饼rc腔es问,税a衡st蛋re夸ss乳a终na朝ly臣si拥s冰fo铺r里th态e尾ac国ti挪on折o遮f鼠th波ebod奔yf弟orc基esma茄ybe迎con鸭ver掩ted叶toth厘e阵an测al经ys折is罩f际or墓t斗he仁a胸ct遇io腥n豆ofsu唱rf坡ac台e炭fo剃rc始es.Th呀e茫st刮re殃ss望c苦om阿po域ne泊nt睁sx,贸y,挨xyaredet熄erm怒ine极db寇yt涨he生dif箩fer蚊ent守ial菠eq辈uat有ion悔s（a）（b）an舒d驶th啦e忧bo计un乡丰da苹ry昌c胆on漂di蜂ti页on之sl(x)s+m(xy)s=Xm(y)s+l(xy)s=YNow牺,w武es浩etx=’x-Xx,微y=’y-Yy,xy=’xyandpro澡cee桐dt素of丝式ind贯th脊ed宏iff角ere衫nti辅al柴equ沿ati凉ons真an或db抵oun堪dar液yc姜ond遗iti饰ons彩wh配ich钱mu砖st滨be侧sat笋isf陷ied西by闻’x,白’y,失’xy.(c)Su残bs财ti弦tu蚕te宜t怠he甲m纵in记to估(陈a)布,犬(b费)词an里d排(c飘)却an勒d隆ob份ta砌inl(’x)s+m(’xy)s=X+lXxm(’y)s+l(’xy)s=Y+mYyWeseethatthedifferentialequationsandboundaryconditionstobesatisfiedby’x,’y,’xymustbethesameasthoseinaproblemwithzerobodyforcesandwithsurfaceforcecomponentsXandYincreasedbylXxandmYy,respectively.Thi洲sc露onc肯lus辅ion晶su峰gge描sts雅a径pro筒ces乏sf朋or黄the残so晓lut沉ion狗ofx,寇y,运xy:Neg医lec升tt殃he引bod辈yf尘orc迅es匆and赚ap惨ply陈fi岁cti具tio吹us旗sur颜fac树ef犁orc技ec沈omp凡one掘ntsX”=lX塌xan雷dY”睁=mYyin味a到dd君it菜io遗n龙to星t冤he师o鞋ri咬gi牛na云l被su泄rf枣ac逆e食fo裳rc票es价;Sol摧ve谱for小th翅es坚tre停ss吹com唐pon大ent雾s,’x,满’y,客’xy,by屑app絮rop胳ria艇te碧met经hod遣s;Fi丝式ndx=悦’x-Xx,疼y=’y-Yy,xy=’xy2.绳12AIR夏Y’S介ST荷RES胜SF棋UNC返TIO宣N.特INV铜ERS袜EM说ETH饮OD著AND假SE悬MI-蒙INV亦ERS绑EM己ETH肢ODisnon予hom菠oge坊neo匀usan佳d,下t秀he解re厌fo抗re片,滩it失s繁ge雨ne苹ra萍l守so坟lu援ti腥on黄m理ay保b辩e芝ex胡pr机es礼se歌d比as栋t柔he未s训um出o舌f顶a侧pa祥rt避ic顾ul泄ar装s躬ol夫ut猪io三n切an普d凶th肥e音ge吉ne磁ra洪l呢so奥lu望ti身on叫o把f挂th捐e传ho敲mo紫ge膏ne袍ou毅s容sy累st顾emWhe犯nb被ody尤fo肿rce云sa侮re注con叙sta筝nt,勿th锄ep来art沈icu折lar布so棉lut它ion质ma凳yb悄et陕ake创na呆sx=-Xx，y=-Yy，xy=0orx=0，辞y=0，苍xy=-Xy-Yxorx=-Xx-Yy，y=-Xx-Yy，xy=0Whi凭ch推sat慨isf拖ye况qua禁tio飞ns(1)(2)Accordingtodifferentialcalculus,for(1),thereexistsacertainfunctionA(x，y)sothat：RewriteSimilarly,(2)ensurestheexistenceofanotherfunctionB(x，y)sothat：Sincexy=yx,wehaveWhichensurestheexistenceofstillanotherfunction(x,y)sothatWe膜obt井ain繁th礼eg识ene关ral傲so由lut纲ion扁of志ho拿mog颂ene墓ous返eq懂uat蝴ion雪s：Now明,t轿he登sup愈erp非osi酸tio画no毒ft矩he朴gen买era忆ls汗olu影tio嫂nw拜ith哭th洽ep等art若icu倡lar掌so控lut睬ion宣yi犹eld叹st狼he团fol为low轧ing导co天mpl押ete们so怖lut邀ion亡:The枣fu径nct戚ion(x葬,y)到is突kn带own盒as亩th福es匙tre怎ss松fun迹cti予on它for显pl射ane败pr逐obl季ems党,o申rthe逮Ai杯ry’箭ss厘tre特ss牙fun脾cti伟on.In忆o穴rd贯er宋f傅or番t扑he委s房诚tr鸦es棚s牧co怠mp破on音en哭ts叮t去o涛sa带ti派sf脆y否th咸e惭co躬mp持at纤ib厚il娇it脏y堤eq丹ua盐ti窃on堵a瓜s彼we辞ll康,述th信e削st要re晚ss坑f堂un邪ct匀io哈n拼mu报st侍s傲at滴is妙fy咱a爆c生er味ta妥in帮e晃qu制at韵io助n.or

2(Xx)=

2(Yy)=0intheconditionofconstantbodyforceorbesimplywrittenas：Wh蒜en芳b埋od轮y示fo社rc防es因a捏re托n是ot承c昌on劫si村de述re用d,名t群he威s打ol轨ut株io菌n兔wi侵ll殃r迷ed而uc平e屑toTh摔us：in裕t竞he笑s垂ol巩ut袄io析n蝇of锁p赌la谢ne拿p飘ro班bl增em榴s眼in籍t告er绑ms挣o烤f姿st寺re乒ss移es程,惑wh习en杨t队he伐b猪od衫y婶fo像rc铅es萝a钢re礼c洒on仗st导an丈t,辨i迎t蜜is竞o捎nl达y茧to梯s姻ol揪ve黄f控or算t幅he莲s汇tr纺es冻s斑fu旧nc朋ti叼on竟f奇ro愚m研th智e多si缝ng孔le姨d残if侦fe挨re界nt桃ia莲l岸eq们ua耐ti研onand甩th序en银fin钥dt乎he载str向ess不co威mpo螺nen宰ts疮byBu伍t百th流es桨e尸st集re度ss竞c菜om未po蛛ne痕nt抚s泡mu回st丸s章at阳is锅fy仗t鼠he境s饱tr妥es叉s彩bo款un确da抚ry瓶c竟on阔di剃ti楚on规.争In暂t娘he询c包as跌e震of超m欲ul岩ti率pl勒y饮co修nn醋ec还te腿d仆bo繁di标es轨,沾th像e谦co黑nd旨it敬io税n仰of妈s贱in绕gl话e-减va猫lu嘉ed熄d助is升pl帜ac类em名en虚ts蹦m施us援t总be意i企ns烫pe闲ct萝ed扫i嘴n削ad季di倾ti新on贼.To移s慎ol闪ve放t蝴he偶p封ar些ti弹al稻d绝if智fe虏re里nt脸ia比l份eq卵ua角ti帜on建s旱of网e统la于st暗ic忠it廊y匠to袜ge拍th曾er累w备it引h旋th悠e欢gi船ve扶n咐bo串un竟da宿ry手c饲on徐di舰ti信on骄s,桶t次he提d墓ir伙ec犁t历me兽th案od禽o为f母so疮lu仓ti栽on很i左s留us仆ua大ll艇y喝im五po染ss缺ib采le疏.逆We熔h俘av表e宵to架u图seth净e虎in腊ve盛rs屋e普me颤th碍odorthe精se赴mi-畜inv处ers妖em底eth疫od.Inthe谨in减ver著se省met候hod,辛so股me谦f叉un国ct体io阁ns竭s油at旋is廉fy姿in扁g芬th弊e既di副ff虹er誉en帽ti乌al晨e呀qu染at裹io坦ns崭a朵re节t蔬ak痛en您a粉nd以e关xa艰mi烈ne提d槽to魂s全ee愤w众ha锹t饲bo榨un士da切ry涨c季on逃di真ti桐on他s危th轮es古e页fu隙nc汽ti劲on械s坝wi颈ll绣s崖at仪is横fy柳a额nd梯t洒he疲re庸by遣t悟o棚kn里ow俊w系ha撞t驰pr配ob骨le鹊ms浸t例he恐y舞ca柜n当so汽lv灭e.In拐the罚ca烧se益of哈sol材uti闸on霜by虏Air产y’s澡st茧res前sf响unc级tio的n,三wese瞧le庭ct拦s纵om魔e皮fu糊nc液ti步ons乱ati启sfy兰ing盟th饿ec竹omp忆ati窗bil昏ity咱eq猾uat绍ion,f养ind贩th让es萄tre定ss巷com难pon裙ent巴s,场and易th弹en盏fin换dt织he甜sur依fac喉ef三orc鸦ec季omp殊one乳nts宵.I忠nt乔his您wa悔y,纤we遭ide爷nti脏fy黄the鹿pr梯obl巧em依whi逮ch铅the瞒st辣res馆sf剧unc仰tio界nc丢an火sol抹ve.Inth种e稿se凑mi疲-i玩nv米er此se梁m毁et气ho耀d,w开ea中ssu头me嫁the薄so疫lut值ion云fo举rt家he记str粉ess聋es胞or东dis此pla徐cem丘ent溜si执na澡gi债ven材pr始obl成em,密ba戒sed赢on毒th良el窜oad倚ing疗co申ndi村tio欺na牧nd肃bou颠nda层ry掏con要dit葵ion哄so肢ft蜡he进pro累ble吊m,进and截th钟en城pro毒cee厚dt矮os这how意th遭at济all花th斯ed泄iff低ere莫nti跳al雾equ直ati生ons半an富db漆oun颂dar扣y