A Research On The Problem Of Some Defects About One Dimensional Quasicrystals Plane Elasticity

Quasicrystals (QCs) are both a novel structure of solids and a kind of new materials discovered in recent two decades. Its discovery break 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 that 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 first chapter is an introduction. In this chapter, the found 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 problem of two semi-infinite collinear cracks in a strip is investigated in one dimensional hexagonal QCs. Utilizing the tools of conformal mapping and Cauchy integral formula and so on, the SIFs at the crack tips are obtained, which can be reduced to the known results under the condition of limitation; For the third and the fourth section, the problems of arc crack and parabolic crack are researched in one dimensional hexagonal QCs. The exact solutions of SIFs at the crack tip are obtained by using the method which is same as the last section, under the condition of limitation, some other results can be obtained.The third chapter investigated the elasticity problem of one dimensional trigonal QCs. Because of the speciality of the elasticity constants, the elasticity problem of one dimensional trigonal QCs is more complicated. The governing equation of point group 3m for one dimensional trigonal QCs is deduced by introducing potential functions in this chapter. Its governing equation is a 8-ordre partial differential equation with constant coefficients. At the same time, the stress components and the displacement components are expressed by the potential functions. At last, the complex variable function expressions of the governing equation and each displacement component and stress component are obtained.The fourth chapter in which the general work is summarized in this paper is a summarization.