The Propagation Of Nonlinear Waves In A Kind Of Phase-transforming Materials

The elastic wave is a kind of nonlinear waves and can be viewed as a transmission form of stress or strain in the elastic medium under an external force or a disturbance. Wave motion is ubiquitous in nature and is a kind of important form of movement of matter.During this process.the wave propagation is accompanied by the delivery of vibration energy. The elastic waves are essentially described by hyperbolic equations. In the higher dimensions case, the corresponding problems are quite difficult to be solved.The system of equations governing the dynamic of one-dimension elastic rods is hyperbolic, while for the elastic bar that composed of phase-transforming materials. the corresponding system of equations is hyperbolic-elliptic coupled. In this thesis, we study the propagation of nonlinear waves in a kind of phase-transforming materi-als. The phase-transforming materials are simulated by a fully nonlinear stress-strain function, which changes from concave to convex as the strain becomes larger and larger. It is well known that the initial boundary value problems corresponding to the impact problem of phase-transforming materials are not well posed for all levels of loading. In this thesis, we concern the propagation of nonlinear waves in detail and study the global structure stability of nonlinear waves. There are three Chapters in this thesis. Chapter 1 introduces the brief histories and present situations of the topic on the propagation of nonlinear waves, and then presents the main results and the organization of the thesis. In Chapter 2, we concern the propagation of reflected and transmitted waves in a composite elastic bar. By utilizing a fully nonlinear stress-strain function to model the phase-transforming materials, we succeed in constructing a physically admissible solution of the wave propagation problem. In Chapter 3,we study the global structure stability of nonlinear waves in a semi-infinite one-dimensional elastic bar composed of phase-transforming materials. For the physically admissible solutions to the initial boundary value problems of partial differential equations, we prove the global structure stability of nonlinear waves by using the theory on the clas-sical free boundary problem and the maximally dissipative kinetics. In other words, when the initial data is disturbed suitably, the original initial boundary value problems still have the corresponding physically admissible solutions with the same structure.