一维六方准晶中椭圆孔边裂纹的静态与动态分析

STATIC AND DYNAMIC ANALYSIS FOR AN ELLIPTIC HOLE WITH AN EDGE-CRACK IN ONE-DIMENSIONAL HEXAGONAL QUASI-CRYSTALS

通过构造保角映射函数,借助复变函数方法,研究了一维六方准晶中椭圆孔边裂纹的反平面剪切问题,给出了Ⅲ型裂纹问题的应力强度因子的解析解.当椭圆的长、短半轴以及裂纹长度变化时,所得结果不仅可以还原为Griffith裂纹的情形,而且得到孔边裂纹问题、T型裂纹问题和半无限平面边界裂纹问题的应力强度因子的解析解.就声子场而言,这些解与经典弹性的结果完全一致.接着对椭圆孔边裂纹的动力学问题进行了研究,并得到了Ⅲ型动态应力强度因子的解析解.当裂纹速度V→0时,动力学解还原为静力学解.这些解在科学与工程断裂中有着潜在的应用价值.

Using the complex variable function method and the technique of conformal mapping,anti- plane shear problems of an elliptic hole with an edge-crack in one-dimensional hexagonal quasi-crystals are studied. The analytic solutions of the stress intensity factor （SIFs） for mode Ⅲ are obtained. With changes in the ratio of major/minor axes of elliptic holes and crack lengths, the present results can be reduced to the Griffith crack solution, and furthermore,many new results can be obtained as well,including a circular hole with an edge-crack,＂T-shaped＂ crack and half-plane with an edge-crack, etc. As far as the phonon field is concerned, these results are shown to be in good agreement with the classical results. Moreover, dynamic problems about an elliptic hole with an edge-crack are also considered and the analytic solutions of the dy- namic SIFs are found out. When the crack extension velocity V→0, dynamic solutions can reduce to the corresponding static ones. These solutions are with great potential in science and engineering applications.