准晶力学中周期问题的复函数边值问题和积分方程方法

The project focuses on the integral equation (less computational efforts but high precision computation) method to establish some theoretical analysis models for quasicrystal mechanics. The single periodic and doubly periodic defect (including the cracks, rigid lines, holes, inclusions, etc.) mechanics and periodic contact mechanics problems will be considered. Using the related theories and the numerical methods of the (singular) integral equations and developing the numerical method for singular integral equations with the doubly periodic kernel and doubly quasiperiodic kernel, reference data in engineering practice related to safety and reliability requirements of the quasicrystal material will be given, which gives a certain degree of technical support for the design of high-quality quasicrystal materials and further guidance on the design and application of certain quasicrystal materials. In addition, the theoretical description for the deformed state of quasicrystal mechanics requires a combined consideration of interrelated phonon and phason field. The phonon field describes the quasiperiodic arrangement of atoms in the complementary orthogonal space. Owing to the existence of phason field, the elasticity of quasicrystal mechanics is more complex than that of the conventional crystals. Some final governing equations of quasicrystals are multiple harmonic equations. The fundamental solutions of multiple harmonic equations can be expressed in multiple analytic functions of complex variables. And the boundary value problems for multiple analytical functions have absorbed much attention of the researchers in the related fields, and fruitful results have already obtained. So this project also wants to use the latest results in the boundary value problem (compact form and beautiful structure) for multiple analytical functions to the study of the quasicrystal mechanics problems.

本项目应用积分方程方法(计算量小而精度高)建立准晶力学中周期问题的理论分析模型。主要研究准晶(包括一维、二维和三维准晶)的单周期和双周期缺陷(包括裂纹、刚性线、孔洞、夹杂等)力学以及周期接触力学问题，利用已有(奇异)积分方程有关理论和数值方法，并发展双周期核和双准周期核－Weierstrass核奇异积分方程的数值解法，给出工程实际中对准晶材料安全和可靠性要求的参考数据，从而对准晶材料的研究、设计和应用给予一定的理论指导。同时，声子场和相位子场的相互耦合导致准晶弹性理论较经典弹性理论复杂，一些准晶的终态控制方程为多重调和方程(如二维五次十次对称准晶对应四重调和方程，三维二十面体准晶对应六重调和方程)。考虑到多重解析函数的实(虚)部恰是多重调和方程的解，国内外学者在多重解析函数边值问题领域成果丰富，所以本研究尝试将多重解析函数边值问题(形式紧凑而结构优美)领域的最新成果应用于准晶力学的研究。

准晶的发现是80年代凝聚态物理的重大进展之一。独特的原子排列结构， 使得准晶具有硬度高、密度低、耐磨、耐蚀、耐氧化等优良性能，成为一种新型的功能材料和结构材料, 在众多科技领域有广泛的应用前景。由于声子场和相位子场的耦合特性，使得准晶弹性理论比经典弹性理论要复杂，最终控制方程的个数比弹性材料的多，阶数也更高。一些准晶的最终控制方程为多重调和函数(如二维五次、十次对称准晶对应四重调和方程，三维二十面体准晶对应六重调和方程)，由复变函数理论知，多重解析函数的实部或虚部恰好是多重调和函数的解。积分方程方法的优点在于计算量小而且计算精度高，国内外学者在积分方程领域研究成果丰富。本项目利用复变函数和积分方程理论建立准晶力学中非周期和周期问题的理论分析模型。主要研究准晶(包括一维、二维和三维及压电准晶)的单周期和双周期缺陷(包括裂纹、刚性线、孔洞、夹杂等)力学以及非周期和周期接触力学问题。利用已有的复变函数和(奇异)积分方程有关理论及数值方法，同时发展了双周期核和双准周期核－Weierstrass 核奇异积分方程的数值解法，得到准晶若干力学问题的解，给出工程实际中对准晶材料安全和可靠性要求的参考数据，从而对准晶材料的研究、设计和应用给予一定的理论指导。本项目研究的理论意义在于将分析数学领域的复变函数理论和积分方程方法的应用范围推广到更宽更新的领域，是对准晶材料力学理论的有效补充。