**1. Introduction**

Functionally graded materials (FGMs) arose from the realization that it was necessary to meet ultra-high temperature and cryogenic requirements. The goals of strength, flexibility, and fatigue resistance inspired the early research. This aim in achieving a smooth and perfect spatial variation is most effectively met by gathering different materials with those favorable characteristics, thereby avoiding the detriment effect such as stress concentration and residual stress found in discrete interface. The gradual change in material properties from the original composite has been shown to increase efficiency by mitigating failure and maintaining the intended benefits of merging two or more different materials. Functionally graded materials are found in nature, for example in bones, teeth, wood, and bamboo [1]. As a result of the microstructure and mechanical properties of functionally graded materials (FGMs) that different from their corresponding from the conventional composites, they have to be used in many engineering, military, medical and space sciences applications. Numerical methods have been used and applied in the

analysis of fracture problems in these materials because they have the ability to give realistic results [2–11]. Popovich et al. [12], Zhao et al. [13] studied fatigue load that exerted on certain types of functionally graded materials. Most recently, digital image correlation technique was used with numerical verification on a stepwise functionally graded material made of glass and epoxy to find the path growth of the crack and the stress concentration values [14, 15]. On the other hand, the optical method was used to analyze the crack path in a material made continuously from the graded materials and the stress concentration factor and T-stress were also calculated [16]. One of these methods that have been adopted in this research is the extended weak Galerkin formulation-element free method (XEFGM). The elementfree Galerkin method with the sub-triangle technique to enhance the accuracy of the Gauss squared near the crack to determine stress intensity factors in isotropic and anisotropic materials can give accurate results in analyzing fracture structures. The incompatible interaction integral technique is applied to extract stress intensity factors (SIFs). The method of sub-triangulation is used to hence the discontinuity terms. With low degrees of freedom, results can be obtained that are highly consistent with those of the relevant reference literature.
