# Functionally Graded Plate Fracture Analysis Using the Field Boundary Element Method
###### Abstract
Moreover, it is possible to evaluate other aspects of fractures in a microscale approach through anisotropic 2D models. For cracked FGMs with general geometry and loading conditions, advanced numerical methods must be applied because of the mathematically complex nature of the governing partial differential equations with variable coefficients, and because the most available analytical methods can only be successfully applied to cracked FGMs with very simple geometry and loading conditions. A study that considered the plane elasticity problem for an inhomogeneous medium containing a crack and with the derivative of the displacement of the crack surface as a density function with a simple Cauchy-type kernel was developed in [23]. The singular nature of the crack tip stress field in a nonhomogeneous medium having a shear modulus was not affected with a discontinuous derivative as shown in [24]. Periodic surface cracking and the associated problem of stress and energy release and the problem of stress concentration and delamination crack initiation and growth under residual or thermal stress were investigated in [25]. The elastic stress and displacement fields near a crack tip in a two-dimensional inhomogeneous cracked body were derived using an extension of the [PERSON] eigenfunction expansion technique in [26]. An inhomogeneous elastic medium containing a crack arbitrarily oriented to the direction of the property gradient was considered in [27]. Local homogeneity and the small-scale inelasticity of brittle materials, where the toughness was taken to be independent of direction, was investigated in [28; 29]. Moreover, the discretized material property variation assigning different homogeneous elastic properties to each element was used in [30]. Several numerical simulations by varying the location of the crack in the graded region for different material gradients were employed in [31]. The mixed-mode stress intensity factors obtained employing the domain integrals as a postprocessing step in the extended finite element method (XFEM) have been provided in [32; 33; 34; 35]. A multiple iso-parametric finite element method to compute stress intensity factor was employed in [36; 37]. A numerical crack analysis of functionally graded two-dimensional materials using an integral boundary equation method linked to the Galerkin method to overcome hypersingular integral boundary equations was performed by [38; 39]. Extensive transient dynamic crack analyses for a functionally graded material (FGM) by using a hypersingular time-domain boundary integral equation method were presented in [40; 41; 42; 43]. A layer discretization technique to approximate material inhomogeneity and a so-called generalized boundary element method based on the Kelvin solution was used in a numerical examination in [44]. The symmetric Galerkin boundary element method (SGBEM) using nonuniform rational B-splines (NURBS) for both geometry and field variables approximation was presented by [45; 46; 47]. Other numerical solutions have been investigated recently concerning the vibration analysis of nanocomposite conical shells [48]. In the present paper, the plate problem under in-plane loading is analyzed in small-displacement theory. An analysis of large displacements in the bending of tapered plate and shells and FGM cylindrical shells where the variability of the material develops toward the thickness as well as in the middle plane of the plate can be found in [49; 50; 51; 52; 53; 54]. This paper is organized as follows: In Section 2, the mathematical framework of the applied model, geometry, variation of mechanical properties, applied loading, and the fundamentals of integral equations for FGMs are presented. Section 3 details the result of the present FBEM in terms of displacements along x and y directions highlighting the differences between the FGM cases and the homogeneous one. In addition, an intercase comparison is shown regarding the stress intensity factors (SIFs) of mode I.