准晶数学弹性力学和缺陷力学

对准晶数学弹性理论的基本概念和基本框架作了介绍，在此基础上分别针对目前已经发现的几类一维准晶、二维准晶和三维准晶讨论了其数学弹性的理论体系．为了求解准晶弹性的边值问题或初值一边值问题，还必须发展相应的方法论．物理工作者在研究准晶位错弹性问题中发展了Ｇｒｅｅｎ函数方法．针对一维与二维准晶弹性中几类问题提出了分解与叠加程序，这一程序的使用，使极其复杂的准晶弹性问题得到简化，进而引进位移函数或应力函数，把数目。庞大的准晶弹性基本方程化成一个或少数几个高阶偏微分方程，进一步使求解步骤大为简化．对三维立方准晶弹性也采用了类似步骤使求解过程大为简化．在以上化简的基础上，发展了准晶弹性的边值问题或初值一边值问题的复交函数方法和 Ｆｏｕｒｉｅｒ分析方法，求得了一系列准晶位错问题和裂纹问题的分析解（古典解）．在研究准晶弹性的边值问题古典解的同时，也讨论了同这些边值问题相对应的变分问题和广义解（弱解）以及这种弱解的数值方法──有限元法．在物理学家工作基础上开展的这些工作可以看作对经典数学弹性理论和方法、经典Ｖｏｌｔｅｒｒａ位错理论、普通结构材料断裂力学和经典有限元的某些发展．此外，还把一维六方准晶弹性动力学的结果与统计物理的某些

MATHEMATICAL THEORY OF ELASTICITY AND DEFECTS OF QUASICRYSTALS

This paper gives a review on the basic concepts and the basic frame of the theory of elasticity for quasicrystals, then, the theoretical system of the elasticity of some kinds of one-, two- and three-dimensional quasicrystals are discussed. To solve boundary value problems or initial-boundary value problems, the methodology of quasicrystal elasticity should be developed. Physicists used the Green function method in treating dislocation problems of quasicrystals. A decomposition and superposition procedure was suggested in this paper for solving some elasticity problems of one-and two-dimensional quasicrystals. Huge numbers of the basic equations of the elasticity are reduced to a single or a few partial differential equations of higher order by this procedure. A similar work for cubic quasicrystals has also been done. The function theory and the Fourier analysis approaches are developed for dislocation and crack problems of the quasicrystalline materials. In the mean time, the variational problem and the weak solution for quasicrystal elasticity are studied, the numerical implementation of the weak solution-finite element analysis is offered. These are considered as an extension of the classical theory of elasticity, classical Volterra dislocation theory, conventional fracture mechanics and classical finite element. In addition, we developed elastodynamics of one-dimensional hexagonal quasicrystal, and the results are used to discuss the specific heat and thermodynamics functions of the quasicrystals.