彈性力學(xué)課件二三

9SolutionofSpatialProblems9.1Solutionintermsofdisplacements9.2Infiniteelasticlayerundergravityanduniformpressure9.6Concentratednormallodeonboundaryofasemi-infinitebody9.10Solutionintermsofstresses按應(yīng)力求解空間問題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以上相同的步驟，可以導(dǎo)出無數(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.利用平衡微分分方程化簡(jiǎn)上上述六個(gè)相容容方程后得：：Byusingthedifferentialequationsofequilibrium,wecansimplifytheEqs.上述六個(gè)方程就是密切爾相容方程TheMichell’scompatibilityequations.在體力為零或或常體力情況況下，相容方方程可以簡(jiǎn)化化為：（拉拉密方程）Whenbodyforcesareconstantorequaltozero,TheMichell’scompatibilityequationsreducetotheBeltrami’scompatibilityequations.Thus歸納：按應(yīng)力求解空空間問題時(shí)，，要使6個(gè)應(yīng)應(yīng)力分量滿足足平衡微分方方程和上述六六個(gè)相容方程程，并在邊界界上滿足應(yīng)力力邊界條件若應(yīng)力分量的的表達(dá)式是x、y、z的線性函數(shù)，，則相容方程程總能滿足，