Study On The Existence And Uniqueness Of Solutions Of The Dynamic Equations Of Nonlinear Elastic Bars

The research on the existence and uniqueness of solutions of nonlinear partial differential equations is one of main branches of nonlinear dynamics. At present, since the variety of non-linearity leads to the complexity of forms of non-linear equations, no general method is available for solving non-linear equations like in the linear case although some approaches may be applicable in very broad context. As a result, it is difficult or even impossible to acquire analytic solutions for most non-linear equations. Thus, without concrete analytic solutions to an equation, numerical methods have to be employed or properties of the equation are figured out based on the examination of characteristics of the equation. However, one frequently ignores the existence and uniqueness of solutions in the process of seeking numerical solutions. Instead, one or more modes are chosen to investigate the property of solutions of the equation. In doing so, rationality for simplifying an infinite-dimensional system into a finite -dimensional system cannot be ensured; or even worse, incorrect conclusions may result in. Consequently, the research on the existence and uniqueness of solutionsof nonlinear equations is a prerequisite and theoretic foundation for justifying numerical solutions. In view of this, we carry out some studies on the existence and uniqueness of solutions of the dynamics equations of a nonlinear elastic bar by means of the Sobolev space. Our main work includes the following.1. The current research methods concerning the existence and uniqueness of solutions of nonlinear partial differential equations are summarized and commented.2. Under some certain initial and boundary conditions of the equations for nonlinear viscid beam, by applying Galerkin method, the existence and uniqueness of global weak solution and its continuous dependence on the initial conditions are shown. How to select types of basis functions in the Sobolev space is illustrated. Moreover, the existence and uniqueness of the strong solution and the existence of classical solution of the nonlinear viscid beam equation are proved. Necessary conditions for the existence of classical solution are presented.Since the equations investigated in this thesis are very general, the obtained results contain many others in the literature as special cases.3. By applying Galerkin method, the existence of weak solution of dynamics equations with constant coefficients for physical nonlinear bar of three order is firstly verified.4. Under some certain initial and boundary conditions of the equations of A physical nonlinear bar with one end fixed and another end subjected to an axial exponential velocity tension, An exploration on the existence of weak solution of dynamics equations with variable coefficients for physical nonlinear bar of third order is made.5. A physical nonlinear bar with one end fixed and another end subjected to an axial exponential velocity tension was studied. Galerkin method was applied to analyze the response and the effect of truncation orders on the computation results.