有限大八次准晶薄板刚性夹杂问题的应力分析

Stress Analysis of Plane Problems in Two-Dimensional Octagonal Quasicrystals Containing an Elliptic Rigid Inclusion

摘要: 准晶因其性能良好而广泛应用于发动机等设备的表面涂层中。由于准晶材料非常脆，因而对准晶部件应力集中处的应力场分析已引起广泛关注。本文利用复边界元法研究八次对称二维准晶中的椭圆形刚性夹杂问题。首先，通过推广的Stroh公式推导出在集中点力作用下，含椭圆刚性夹杂八次对称二维准晶平面弹性问题的Green函数。其次，基于不计体力的平衡方程和椭圆形刚性夹杂问题所对应的边界条件，构建边界积分方程，最后，通过Guass数值积分公式离散该边界积分方程并求解，分别获得了声子场和相位子场的孔边应力值。数值实例讨论了椭圆形刚性夹杂所引起的孔边应力变化，与含孔洞问题相比，内部夹杂的存在使基体孔边应力集中处的应力值减弱。

Abstract: Quasicrystals are widely used in surface coatings for devices such as engines due to their excellent properties. Because quasicrystal materials are very brittle, the analysis of the stress field at stress concentration areas of quasicrystal components has attracted widespread attention. The plane elastic problem of octagonal symmetric two-dimensional quasicrystals containing an elliptical rigid inclusion is considered by using the complex boundary element method. First, Green’s functions are obtained utilizing the extended Stroh formalism under concentrated force. Second, based on the equilibrium equation satisfying body force free and the boundary conditions corresponding to the elliptical rigid inclusion problem, a boundary integral equation is constructed, which is discretized and solved using the Guass’s formula of numerical integration. The stresses of the phonon field and the phason field on the boundary of an elliptic hole are obtained. The effect of the elliptical rigid inclusion on the stress is also discussed by comparing with the problem of containing an elliptic hole, and the presence of the internal inclusion weakens the stress values at the stress concentration points.