## 1 Introduction  
Holes exist widely in engineering structures for either design or manufacture reasons, and the high stress concentration around them has been a serious issue to the engineers and designers for many years. In order to achieve better machine designs, some classical references such as [PERSON] [1], [PERSON] et al. [2; 3], and [PERSON] and [PERSON] [4] summarized and provided various important solutions of stress concentration factor (SCF) in engineering structures. The effects of stress concentration on the fatigue strength and life of structures are of great importance, especially when the structures are designed to carry high loads [5; 6; 7]. Therefore, it has been of great concern to reduce the stress concentration near holes during the structure design and safety maintenance in engineering. Recent research [8; 9; 10; 11] indicates that the stress concentration around the holes can be effectively reduced by introducing a reinforcing ring made by functionally graded material (FGM). FGM is an advanced composite material in which the material elements (composition and structure) change continuously along one or more directions. The continuous elements in FGM can successfully avoid the mismatch of the material properties at the interface [12]. The introduction of the FGM reinforcing ring with radially varying material properties can not only reduce the stress concentration around the hole, but also enhance the connection strength at the interface [13; 14]. Therefore, many studies have been carried out on the analysis of stresses around holes in FGM in the past 20 years. Inspired by the excellent material distributions near the natural holes in a load-bearing bone, [PERSON] et al. [15; 16] first studied and improved the strength and loading capacity of a plate with a circular hole by controlling the radially varying Young's modulus in the plate. The optimization designs were carried out on choosing the proper mathematical functions of the axisymmetric Young's modulus for increased strength. The stress concentrations in an FGM plate with a center circular hole or elliptical hole were numerically analyzed by [PERSON] and [PERSON] [17] and [PERSON] et al. [18], respectively, based on the finite element method. They both considered the variations of Young's modulus along three different directions: radial direction, x direction, and Y direction. By means of the complex variable methods, [PERSON] et al. [19; 20] presented the semi-analytical solutions of the stress fields in the infinite and finite FGM plates containing a circular hole under arbitrary in-plane constant loads. For some specific radial varying functions of [PERSON]'s modulus, [PERSON] et al. [21] obtained the analytical solution of stresses in an FGM plate with a circular hole under uniform tension. [PERSON] et al. [22] analyzed the effects of different radial variations of [PERSON]'s modulus on the stress concentration factor (SCF) around the hole in FGM plate with the method of extended finite element. Considering the cost and difficulty of material preparation, [PERSON] [23] proposed a more feasible method to reduce the stress concentration by inserting a thin FGM ring around the hole. The analytical solution of the stress fields is presented for the plate subjected to far-field uniaxial tension, and the effects of the homogeneous ring and FGM ring on the SCF are compared by numerical examples. Recently, [PERSON] et al. [24] derived the analytical SCF in an FGM plate with a circular hole as Young's modulus and Poisson's ratio change along the radial direction with exponential function or power function, and they laid emphasis on the tailoring problem of the material. Their results show that the desired SCF can be obtained by tailoring the radial variation of Young's modulus, and at the same time, the hoop stress is almost uniformly distributed in the whole plate.