准晶材料接触问题的复变函数方法

The discovery of quasicrystals is one of the major advances for condensed matter state physics in 1980s. Owing to its unique arrangement of atoms, quasicrystals have many desirable properties, such as high hardness, low density, corrosion resistance, wear resistance and oxidation resistance, which enable quasicrystals to become a new type of functional materials and structural materials and to be widely used in engineering and other fields. The theoretical and experimental analysis of contact problems can not only offer suggestions to security and reliability design but also provide theoretical basis and technical support for research, design, preparation and experiment of high quality quasicrystalline materials, then guide their development. Due to the coupling characters of phone field and phason field, the elasticity of quasicrystals is more complex than that of the conventional elastic materials, the number of final governing equations are more than that of the conventional elastic materials, and the order of final governing equations are higher than that of the conventional elastic materials. The final governing equations of some quasicrystals are multiple harmonic equations, knowing by complex variable theory, the fundamental solutions of multiple harmonic equations can be expressed in multiple analytic functions of complex variables and the complex variable method is one of the most effective methods to solve the contact problems. Thus, this project tries to discuss contact problems in quasicrystals with the aid of the complex variable method. The solutions of the stated problems will be given. Numerical examples will be used to illustrate interesting results with some practical valuable conclusions obtained.

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, 在众多科技领域有着广泛的应用前景. 准晶接触问题的理论和实验分析可为工程实际中准晶材料安全性和可靠性需求提供参考依据, 对高品质准晶材料的研究、设计、制备提供理论和技术支持, 进而指导准晶材料的研发. 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，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，且复变函数方法是解决接触问题的有效方法之一. 因此，本项目尝试用复变函数方法研究准晶材料的几类接触问题，得到接触问题物理场场变量的解, 并对所得结果进行数值分析研究, 进而获得有实际应用价值的结论.

本项目主要研究了(压电)准晶及(压电)准晶涂层材料的接触问题。对于准晶半平面的接触问题，探讨了一维六方、二维六方准晶和二维八次的有限摩擦和半平面粘结接触问题，得到了平底刚性压头作用下接触应力和接触位移的显式表达式。数值算例用于分析摩擦系数对接触应力，声子场、相位子场弹性常数、耦合系数对接触应力和接触位移的影响，讨论了接触应力与接触位移之间的关系。对于带裂纹准晶半平面的接触问题，研究了带任意形状裂纹的二维十次和三维二十面体准晶的无摩擦接触，得到了应力函数的封闭解, 并给出了裂纹端点处应力强度因子和压头下方准晶体表面任意点处接触应力的显式表达式。从压头下方接触应力的表达式可以看出, 接触应力在压头边缘和裂纹端点处具有奇异性。数值算例给出了单个平底刚性压头无摩擦压入带单个垂直裂纹和水平裂纹的准晶下半平面的结果。对于准晶涂层的接触问题，研究了带电压头作用在一维和二维六方压电准晶涂层上的有限摩擦接触问题，得到了应力的一般解。数值结果揭示材料参数和载荷对接触行为的影响。同时，本项目还拓展研究了(压电)准晶及热电材料的缺陷问题和热应力问题。本项目的研究成果扩展了经典弹性理论中接触问题的结果，是对(压电)准晶材料及其他多场耦合材料接触力学理论的有效补充。数值算例分析结果可以给出工程实际中对(压电)准晶材料安全性和可靠性要求的一些参考数据，对高品质的准晶材料的研究、设计、制备及实验提供理论依据和技术支持，进而指导某些准晶材料的设计。