## 1 Introduction  
Functionally graded materials (FGMs) are increasingly used in many fields due to their continuous change of material properties and functions in spatial locations. They are able to survive in a harsh working environment, without losing their properties and without failing during service [1; 2; 3; 4]. The continuous variation in the microstructure of FGMs can well mitigate the mismatch of material properties at the interface, which can effectively decrease the stress concentration as compared to the existing material interfaces [5]. The fast development of FGMs has been able to reduce the stress concentration near various shaped holes [6; 7; 8]. In the past decade, many works have analyzed the problems of stress concentration around a hole in FGMs [9; 10; 11; 12]. The axisymmetric problem in planes was first studied in [13; 14; 15], where the stress concentration near a circular hole in a FGMs plate with radial arbitrary material constants was considered. Shortly thereafter, [PERSON] and [PERSON] [16] numerically investigated the non-axisymmetric problem of stress concentration in FGMs panels with a circular hole under uniaxial tension by using the multiple isoparametric finite element formulation. [PERSON] et al. [17; 18] solved the stress field of a FGMs plate with a circular hole under two different loading conditions, arbitrary constant loads and uniform heat flow by using complex variable theory. [PERSON] et al. [19] derived the analytical solution for the stress concentration factor (SCF) around a circular hole in an infinite inhomogeneous plate. Considering the manufacturing feasibility, [PERSON] [20] proposed a method to reduce the stress concentration by inserting an FG ring around the hole, and presented an analytical calculation of stress field in an isotropic plate with a circular hole coated by an FG ring. [PERSON] [21; 22] analyzed the SCF and stress-gradients due to a circular hole in a radial FGMs panel subjected to anti-plane shear loading and obtained closed expressions for the stresses and displacements. Moreover, they introduced a novel definition for the SCF in FGMs panels with geometrical discontinuities in general, and analyzed the circular hole in particular. Recently, [PERSON] and [PERSON] [23] derived the solution for stress and moments around a circular or elliptical hole in an infinite FGMs plate, where the material properties change with the plate thickness. [PERSON] et al. [24] use different radial FGMs to reduce the SCF around a central circular hole in an infinite plate by using the extended finite element method. All of the abovementioned work was conducted for FGMs plates with a circular or elliptical hole. In fact, in addition to circles and ellipses, holes with complex geometrical shapes, including triangles, squares, rectangles, etc., are also very common in engineering structures. The stress concentrations around these complex shaped holes are more obvious than that of circular or elliptical holes [25; 26; 27; 28; 29; 30]. In the case of homogeneous plates, many works have been reported on the analyses of stress concentration around irregular holes. For example, [PERSON] [31] systematically studied the problems of stress concentration near various shaped holes in a homogeneous plate based on complex variable theory. Later, related analytical and numerical methods were further developed to solve the stress field of more complex shaped holes [32; 33; 34; 35]. More recently, [PERSON] and [PERSON] [36; 37] presented a general solution to calculate stresses around hypocycloidal and hypotrochoidal holes in an infinite plate using [PERSON]'s complex variable method. [PERSON] and [PERSON] [38] investigated the stress distributions in a finite metallic plate having different shaped holes, and discussed the effect of bluntness, the size of the hole and the shape of the hole on the stress concentration. To the best of the authors' knowledge, little work can be found for the studies of complex shaped holes in FGMs.