准晶数学弹性理论和某些有关研究的进展(上)

DEVELOPMENT ON MATHEMATICAL THEORY OF ELASTICITY OF QUASICRYSTALS AND SOME RELEVANT TOPICS

本文对固体准晶力学性能和准晶数学弹性,塑性,断裂以及有关研究的进展作了评论,尤其对材料常数和塑性变形行为的测量,一维、二维、三维准晶弹性理论,动力学、非线性、缺陷理论、准晶弹性新型偏微分方程的推导和精确分析解,复分析方法,变分原理和有限元方法,有限差分方法这些宏观问题和它们的数学方法进行了分析,同时对准晶晶格动力学问题的数学理论也作了初步讨论.近来在软物质中发现了12次和18次对称准晶,意义重大,这里也做了初步介绍.文中重点讨论此领域最近这些年来中国科学工作者的工作.

This paper gives an introduction to the development on mathematical theory of elasticity of quasicrystals and some relevant topics, in which the physical framework of elasticity is discussed first, the measuremental data of material constants for quasicrystals in solid phase are one of the most important basis to the theory and applications and are listed in detail. As the fundamental points, the analytic methods and exact solutions for various boundary value or boundary-initial value problems are summarized, the Chinese researchers contributed the effort in this respect. Apart from elasticity, dynamics and plasticity of the novel material are also discussed. At last the paper offers a simple description on a possible theoretical treatment, from the view point of elasticity and hydrodynamics, to quasicrystals in soft matter phase observed very recently.