The Dynamics Research On The Problem Of Some Defects About One-dimensional Hexagonal Quasicrystals

Quasicrystals (QCs) are both a novel structure of solids and a kind of new materials discovered in recent more than twenty years. It`s discovery breaks with people's traditional concept that the solids are classified into crystals and uncrystals. A theoretical description of the deformed state of QCs requires a combined consideration of interrelated phonon and phason field. The phonon field describes the motion of lattices in physical space, while the phason field describes the quasiperiodic arrangement of atoms in the complementary orthogonal space, which interact with one another. Owing to the existence of phason field, the elasticity of QCs is more complex than of the conventional crystals. As far as the elasticity and defects problem of QCs are concerned, there are some mathematical methods having been proposed such as Green function method,Fourier transform method,perturbation method and the complex variable function method and so on. This paper researched some kinds of elasticity and defect problems by using the complex variable function method in one dimensional QCs. The structure of this paper is :The frist chapter is an introduction. In this chapter, the find of the QCs, the classification and the function of the QCs, and the general elasticity and defect problem of QCs are introduced. Some concrete defect problems of one dimensional QCs are researched in the second chapter. For the first section, the basic elasticity theory of one dimensional hexagonal QCs is introduced. Its elasticity problem is divided into two independent problems in which the first one is similar to the classical elasticity problem and the second one's control equations are two harmonic equations. For the second section, the basic dynamics theory of one dimensional hexagonal QCs is introduced. Its elasticity problem is divided into two independent problems in which the first one is plane elasticity problem of common hexagonal QCs and the second one is anti-plane problem of quasiperiodic elasticity space. The former is researched in classical elasticiy, the later's dynamics problem is controled by two wave equations. For the third sections, the problem of the dynamics semi-infinite crack in one dimensional hexagonal QCs. Utilizing the tools of conformal mapping and cauchy integral formula and so on, the dynamics SIFs at the crack tips are obtained, which can be reduced to the known results under the condition of limitation ; For the fourth and the fifth section, the dynamics problem of semi-infinite crack in a strip and two semi-infinite collinear cracks in a strip are researched in one– dimensional hexagonal quasicrystals. For the sixth and the seventh section, the problem of the static and dynamics of a finite width strip with a single edge crack are researched in one– dimensional hexagonal quasicrystals. The exact solutions of SIFs at the crack tip are obtained by using the method which is same as the third section, when the speed v→0, both the dynamics crack can be turned into a static crack. Under the condition of limition, some other results can be obtained.The third chapter in which the general work is summarized in this paper is a summarization.