热场与压电效应下准晶板弯曲和振动精确解的辛方法

With low porosity, low adhesion, good thermal conductivity properties compared to traditional materials, the quasicrystal materials are widely used as surface materials and the solar industry materials. Based on the generalized Hooke theorem of phenomenological constitutive, the scholars studied the bending and vibration of quasicrystal plates structures through the existing research. However, the current research based on the Lagrange systems with one class of variable could not obtain the exact solutions and the associated solution procedures are raising the order of the equations while reducing the number of unknows. Moreover, the consideration in the industrial design of quasicrystal plates with cracks, mixed boundary conditions and thermal and piezoelectric effects have not been found in the current literature. The present project set state vectors as basic variables and establish the Hamiltonian system composed of original and dual variables thought introducing the energy function with thermal field and piezoelectric effect, then the corresponding boundary conditions are obtained. The symplectic method is a kind of efficient and accurate method with rational analysis attributed the bending and free vibration of quasicrystal plates to the first order differential equations, and obtained the completed solution space without introducing any trial functions or guess functions. The solution of mechanical behavior of quasicrystal plates in Hamiltonian systems has important theoretical significance for supply furnishing the effective method for analysis and calculation and providing the theoretical foundation to optimization design and reliability analysis of piezoelectric quasicrystal components.

与传统材料相比，准晶材料以其孔隙率低、粘附性低、导热性好的特性被广泛应用于表面材料、太阳能工业材料。现有的研究中，学者们采用基于唯象本构的广义Hooke定理讨论一维和二维准晶板的弯曲和振动。然而这些研究都是基于一类变量的Lagrange体系，求解思想均为减少未知量个数提高方程阶数，无法得到问题的精确解。且现有文献尚未考虑工业设计中准晶板含裂纹、混合边界条件、热场和压电效应等复杂情形。本项目考虑含热场与压电场作用下准晶板的能量泛函，以状态向量为基本变量，建立了原变量和对偶变量的二类变量的Hamilton体系，并得到相应的边界条件。作为一种高效精确的理性分析方法，辛方法将准晶板弯曲和振动问题归结为一阶微分方程组，无需引入试函数即可得到问题完备的解空间。在Hamilton体系中研究准晶板的弯曲与屈曲，可提供准晶材料研究的有效分析与计算手段，并为准晶压电元件的优化设计和可靠性分析给予的理论依据。

准晶体材料是一种新型的工业材料，具有高强度、高硬度、摩擦系数低、导热性好等特性，一直广泛运用于表面薄膜、复合材料增强相、太阳能工业材料。在目前的研究中，基于Landau的唯象本构理论将其划分为两个相互耦合的场：声子场和相位子场。该准晶弹性模型解决了准晶体长程有序却缺少平移对称问题，却使得本构关系中方程组数量增加难以求得其精确解。本项目基于原变量（声子场和相位子场位移）和对偶变量（应力）为变量矩阵，力求突破以往一类变量的、以复变函数理论和双三角函数为主要的试函数方法，将辛体系引入准晶结构的弹性力学问题研究中，获得相关问题精确的解空间。在本项目执行期间，项目组成员求解了如下准晶弹性理论问题：首先获得了一维六面体准晶材料反平面断裂问题的辛弹性体系，在该体系中得到了应力强度因子和裂纹尖端应力场分布。其次我们得到了一维六面体功能梯度压电准晶体条反平面断裂问题的精确解空间。在对偶变量形成的移位哈密顿体系中首次得到的功能梯度材料断裂问题的解，并考虑了裂纹长度、指数参数对应力强度因子的影响。本项目的研究进一步扩展辛体系在弹性力学中的应用范围，具有重要的方法论意义。