## 1 Introduction
[21] studied the nonlinear free vibration of a multi-layer polymer composite beam reinforced with non-uniformly distributed GPLs through the thickness by applying [PERSON]'s principle and the von Karman nonlinearity to Timoshenko beam theory. [PERSON] et al. [22] numerically examined the nonlinear vibration of graphene sheets by applying the element-free kp-Ritz method to the nonlocal elasticity theory, which accounts for the size effect. [PERSON] et al. [23] numerically examined the influence of GPL and porosity distributions on the nonlinear natural vibration of FG-GPLRC porous plates supported by the elastic foundations by applying the differential quadrature method (DQM). [PERSON] et al. [24] investigated the temperature-dependent large amplitude vibration of GPLRC-laminated cylindrical panels resting on an elastic foundation using a two-step perturbation technique. [PERSON] [25] proposed a nonlinear formulation for the nonlinear natural vibration of GPLRC-laminated plates using a non-uniform rational B-spline (NURBS)-based isogeometric finite element method. [PERSON] and [PERSON] [26] analytically investigated the influence of porosity coefficient and the foam skeleton distribution on the nonlinear free vibration of graphene-reinforced plates by applying [PERSON]'s principle to the [PERSON] nonlinear plate theory. [PERSON] et al. [27] numerically and parametrically investigated the temperature-dependent nonlinear free vibration characteristics of the cracked FG-GPLRC beams resting on an elastic foundation using the DQM. [PERSON] and [PERSON] [28] analyzed the size-dependent nonlinear free vibration of FG-GPLRC annular sector microplates by applying the isogeometric analysis to a four-variable higher-order SDT. [PERSON] et al. [29] investigated the nonlinear natural vibration of an FG-GPLRC circular plate on the nonlinear elastic foundation using the generalized DQM. [PERSON] and [PERSON] [30] numerically investigated the nonlinear natural vibration of an FG-GPLRC titanium alloy trapezoid plate using the Rayleigh-Ritz method and the direct iterative process. As can be found from the literature survey, previous studies on the nonlinear free vibration of FG-GLRC structures were mostly limited to beams and plates and furthermore assumed that the structures are perfect without any cracks. However, the structural elements used in real applications have various forms such as cylindrical and conical shells, and various abnormal loading conditions such as thermal shock may induce microcracking [31] within GPL-reinforced nanocomposite structures. These undesired cracks damage the surrounding region such that the whole structural strength becomes significantly weakened, implying that the consideration of a crack is essential in structural analyses and designs. Meanwhile, the consideration of a crack in a mesh- or grid-based numerical analysis may frequently suffer from the painstaking crack modeling job [32; 33; 34]. Furthermore, the crack modeling deteriorates the mesh uniformity, which may induce numerical instability and even numerical failure. In this situation, this study aims to determine the large amplitude natural vibration of a porous FG-GPLRC cylindrical panel with a central inclined crack. To avoid the above-mentioned problems in crack meshing, the phase field formulation (PFF) [35; 36] is adopted and a virtual geometry rotation is introduced into the framework of 2-D extended NEM [19; 37]. In other words, the nodes sitting on the crack lines are not separated and the region at the crack-tip in the grid is not centrally refined. Instead, the crack line is represented by the phase field and its virtual rotation to the NEM grid line, and the crack-induced singularity is enhanced by the near-tip singular functions [38]. The large displacement field is expressed by the first-order SDT incorporated with the von Karman geometry nonlinearity. The painstaking definition and manipulation of high-smooth Laplace interpolation (L/I) functions in NEM is relaxed by mapping the cylindrical neutral surface to a rectangular plane.