弹性力学课件二三

9SolutionofSpatialProblems9.1Solutionintermsofdisplacements9.2Infiniteelasticlayerundergravityanduniformpressure9.6Concentratednormallodeonboundaryofasemi-infinitebody9.10Solutionintermsofstresses按应力求解空间问题9.10Solutionintermsofstresses取应力分量为基本未知量，从15个基本方程中消去位移分量和应变分量，得出只包含6个应力分量的方程Takethesixstresscomponentsasthebasicunknownfunction，formulateasystemofdifferentialequationsandboundaryconditionscontainingthestresscomponentsonly，solvefortheseunknownfunctionsandtherebyfindthestraincomponentsbythephysicalequationsandthenthedisplacementcomponentsbythegeometricalequations.平衡方程中只含应力分量，应保留从几何方程和物理方程中导出补充方程Sincethedifferentialequationsofequilibriumdonotinvolvedisplacementsandstrains,itisnecessaryonlytoeliminatethedisplacementandstraincomponentsinthegeometricalandphysicalequationsThegeometricalequations,即同理变形协调方程，也是一组相容方程推导另一组相容方程：因为：所以：同理：这是另一组相容方程通过以上相同的步骤，可以导出无数的相容方程，都是应变分量应该满足的，但如果六个应变分量满足了上述两组相容方程，就可以保证位移分量的存在，即可以由几何方程求位移分量Bysimilarmanipulations,wecanderivemanyothercompatibilityequationswhichthestraincomponentsmustsatisfy.However,itcanbeprovedthatthetwoEqs.aresufficienttoensuretheexistenceofdisplacementcomponentscorrespondingtoagivensetofstraincomponents.将物理方程代入上述相容方程—消除应变分量，得到只含应力分量的相容方程：Substitutingthephysicalequationsintothecompatibilityequationsabove,weobtainthecompatibilityequationsintermsofstresscomponents.利用轿平衡披微分息方程满化简司上述趴六个念相容母方程恩后得结：By渐u脑si确ng血t什he先d吊if肠fe痒re升nt衰ia努l信eq描ua愤ti种on歼sof娱e讯qu灿il拼ib音ri愧um务,悼we没c刷an蛋s盘im浙pl托if释y钳th规eEq榆s.上述六个方程就是密切尔相容方程TheMichell’scompatibilityequations.在体屿力为尤零或股常体烟力情边况下按，相右容方撑程可耍以简怨化为瓜：与（拉挎密方排程）Wh杯en茄b视od惜y穗fo暖rc在es施a坏re宣c挎on链st广an扰t爪or疫e摸qu谢al靠t破oze琴ro盛,T可heMi萌ch坐el昌l’烛sco料mp沃at卖ib戚il遇it码y鹿eq洽ua发ti低on婚s劈燕re肤du示ce撤t煌o减th旱eBe投lt壶ra小mi毙’sco商mp艺at串ib辣il膨it稿y怠eq饭ua叠ti兵on税s世.Th消us归纳奔：按应这力求衡解空贡间问耳题时旗，要雅使6谜个应全力分要量满置足平弃衡微洁分方嫁程和粗上述偶六个害相容赵方程耽，并俊在边担界上弱满足帆应力悉边界慨条件若应骄力分猛量的避表达堡式是x、猎y、晃z的线居性函嫩数，披则相缴容方雷程总恋能满穿足，艰对单终连域暑的应疲力边俊界问历题，卖