含功能梯度材料加强环的任意几何形状孔附近应力集中分析

基于复变函数理论，结合保角变换技术研究含功能梯度材料（FGM）加强环的任意几何形状孔附近应力集中。采用分层均匀化方法，给出了远场均布载荷作用下材料参数沿孔周法线方向任意变化的FGM加强环内的复势函数解。通过数值算例，详细讨论了加强环内杨氏模量不同变化规律对三角形、正方形、矩形等各种几何形状孔附近应力分布的影响。结果表明：通过在孔周衬入FGM加强环并合理选择加强环内材料参数的递变规律，可以有效缓解各种几何形状孔附近的应力集中。同时通过一些特例与已有文献比对验证了本文结果的正确性。

Stress Concentration Analysis of an Arbitrarily Shaped Hole Coated with a Functionally Graded Layer

A general solution of the stress concentration in a homogeneous plate with an arbitrary shape hole coated by a functionally graded layer is presented under the remote uniform loads. With using the method of piece-wise homogeneous layers, the functionally graded layer in which the material properties change continuously along the normal direction of the hole is approximately decomposed to N homogeneous layers. When N is chosen to be large enough, the material constants in each layer can be regarded as unchanged. By means of the technique of conformal mapping, N homogeneous layers in -plane are transformed into N concentric circular rings in -plane, and then the complex potentials in each circular ring and the plate in -plane can be given in the form of series with unknown coefficients based on the theory of the complex variable functions. The stress and displacement continuous conditions on the interfaces of each homogeneous layer are used to produce a set of linear equations containing all the unknown coefficients. Through solving these linear equations, the complex potentials can be finally obtained in each layer and the plate. Numerical results of stress distribution around the holes with various shapes including circle, ellipse, triangle, square, rectangle etc. are presented for different varying Young’s modulus. It is shown that the stress concentrations around the elliptical and rectangle holes are more obvious than those of circular and square holes, respectively, and the most obvious one is triangle hole. Moreover, it is most important that the influence of the gradient exponent of Young’s modulus on the stress distributions is noticeable for all shape holes, and the stress concentrations decrease remarkably as the exponent value increases. Therefore, it can be concluded that the existence of the functionally graded layer influence obviously the stress distribution around the holes with various shapes, and the stress concentration can be effectively reduced by choosing proper change ways of the normal elastic properties in the functionally graded layer.