彈性力學(xué)課件二三

9SolutionofSpatialProblems9.1Solutionintermsofdisplacements9.2Infiniteelasticlayerundergravityanduniformpressure9.6Concentratednormallodeonboundaryofasemi-infinitebody9.10Solutionintermsofstresses按應(yīng)力求解空間問(wèn)題9.10Solutionintermsofstresses取應(yīng)力分量為基本未知量，從15個(gè)基本方程中消去位移分量和應(yīng)變分量，得出只包含6個(gè)應(yīng)力分量的方程Takethesixstresscomponentsasthebasicunknownfunction，formulateasystemofdifferentialequationsandboundaryconditionscontainingthestresscomponentsonly，solvefortheseunknownfunctionsandtherebyfindthestraincomponentsbythephysicalequationsandthenthedisplacementcomponentsbythegeometricalequations.平衡方程中只含應(yīng)力分量，應(yīng)保留從幾何方程和物理方程中導(dǎo)出補(bǔ)充方程Sincethedifferentialequationsofequilibriumdonotinvolvedisplacementsandstrains,itisnecessaryonlytoeliminatethedisplacementandstraincomponentsinthegeometricalandphysicalequationsThegeometricalequations,即同理變形協(xié)調(diào)方程，也是一組相容方程推導(dǎo)另一組相容方程：因?yàn)椋核裕和恚哼@是另一組相容方程通過(guò)以上相同的步驟，可以導(dǎo)出無(wú)數(shù)的相容方程，都是應(yīng)變分量應(yīng)該滿足的，但如果六個(gè)應(yīng)變分量滿足了上述兩組相容方程，就可以保證位移分量的存在，即可以由幾何方程求位移分量Bysimilarmanipulations,wecanderivemanyothercompatibilityequationswhichthestraincomponentsmustsatisfy.However,itcanbeprovedthatthetwoEqs.aresufficienttoensuretheexistenceofdisplacementcomponentscorrespondingtoagivensetofstraincomponents.將物理方程代入上述相容方程—消除應(yīng)變分量，得到只含應(yīng)力分量的相容方程：Substitutingthephysicalequationsintothecompatibilityequationsabove,weobtainthecompatibilityequationsintermsofstresscomponents.利用轎平衡披微分息方程滿化簡(jiǎn)司上述趴六個(gè)念相容母方程恩后得結(jié)：By漸u腦si確ng血t什he先d吊if腸fe癢re升nt衰ia努l信eq描ua憤ti種on殲sof娛e訊qu燦il拼ib音ri愧um務(wù),悼we沒(méi)c刷an蛋s盤im浙pl托if釋y鉗th規(guī)eEq榆s.上述六個(gè)方程就是密切爾相容方程TheMichell’scompatibilityequations.在體嶼力為尤零或股常體煙力情邊況下按，相右容方撐程可耍以簡(jiǎn)怨化為瓜：與（拉挎密方排程）Wh杯en茄b視od惜y穗fo暖rc在es施a壞re宣c挎on鏈st廣an擾t爪or疫e摸qu謝al靠t破oze琴ro盛,T可heMi萌ch坐el昌l’燭sco料mp沃at賣ib戚il遇it碼y鹿eq洽ua發(fā)ti低on婚s劈燕re膚du示ce撤t煌o減th旱eBe投lt壺ra小mi斃’sco商mp藝at串ib辣il膨it稿y怠eq飯ua疊ti兵on稅s世.Th消us歸納奔：按應(yīng)這力求衡解空貢間問(wèn)耳題時(shí)旗，要雅使6謎個(gè)應(yīng)全力分要量滿置足平棄衡微潔分方嫁程和粗上述偶六個(gè)害相容趙方程耽，并俊在邊擔(dān)界上弱滿足帆應(yīng)力悉邊界慨條件若應(yīng)驕力分猛量的避表達(dá)堡式是x、獵y、晃z的線居性函嫩數(shù)，披則相繳容方雷程總戀能滿穿足，艱對(duì)單終連域暑的應(yīng)疲力邊俊界問(wèn)歷題，賣