Solutions For Two Kinds Of The Nonlinear Evolution Equations And Qualitative Analyses Of Solutions

Based on the theory of finite deformation for nonlinear-elastic materials and structures,this work mainly examines the oscillation problems for a vulcanized rubber sealing ring aboutradial direction, and the dynamic stability problems for cylindrical structures (such as a solidcylinder, a cylinder with a micro-void, a cylindrical shell and a cylindrical membrane)composed of incompressible hyper-elastic materials under the assumption of axiallysymmetric deformation. At first, mathematical models of the relative problems have beenformulated. Then, analytic solutions of the problems are obtained by the inverse method andthe incompressibility constraint. Finally, some new theoretical results are proposed by thequalitative analyses of solutions and numerical simulations. The main works are as follows:In Chapter2, the oscillation problems have been examined for a vulcanized rubbersealing ring composed of a class of transversely isotropic incompressible modified Vargamaterials about radial direction, where the sealing ring is subjected to a suddenly appliedradial load at its inner surface. A nonlinear ordinary differential equation that describes theradial motion of the sealing ring has been developed. For the given material and structureparameters, a critical load has been obtained by numerical examples. It is proved that if theapplied load is lower than the critical load, the motion of the rubber ring with time willpresent a nonlinear periodic oscillation, while if it exceeds the critical load, the motion willincrease infinitely with the increasing time and so the rubber ring will be destroyedultimately.In Chapter3, dynamic stability problems are examined for cylindrical structurescomposed of incompressible hyper-elastic materials under the assumption of axiallysymmetric deformation. Firstly the elastic dynamical equation that describes the radialsymmetric motion of the cylindrical structures is reduced to a second order nonlinear ordinarydifferential equation by using the incompressible condition. And then the general solutionsare obtained for cylindrical structures subjected to an applied pressure under the assumptionof axially symmetric deformation. At last, some common conclusions are presented for theradial symmetric motion problems of a cylinder, a cylinder with a micro-void, a cylindricalshell and a cylindrical membrane, such as, conditions of cavity formation at the axial line ofthe cylinder, conditions of presenting nonlinear periodic oscillation and fracture of acylindrical shell or a cylindrical membrane.