## 1 Introduction
[24] investigated the thermal buckling of a size-dependent composite cylindrical nanoshell reinforced with GPLs, which is placed on an elastic foundation using Hamilton's principle with the help of analytical methods. [PERSON] et al. [25] analyzed the thermal buckling of composite laminated annular sector plates, which are reinforced with GPLs by applying the generalized differential quadrature method (DQM) to first-order SDT incorporated with von Karman geometric nonlinearity. [PERSON] and [PERSON] [26] numerically investigated the influences of GPL pattern, plate geometry parameter, and surface temperature profile on the buckling and postbuckling of GPLRC plates under diverse thermal loadings using the isogeometric analysis method. [PERSON] and [PERSON] [27] numerically studied the thermal buckling and postbuckling behaviors of porous GPLRC beams by reflecting the temperature-dependent material properties, for which higher-order SDT and von Karman geometric nonlinearity were combined. [PERSON] [28] numerically investigated the thermal buckling behavior of trapezoidally corrugated FG-GPLRC laminated panels by applying the isogeometric method to higher-order SDT. [PERSON] et al. [29] analytically solved the thermal buckling and postbuckling of FG-RPLRC multilayer microtubes by applying the minimum potential energy principle to a refined higher-order beam theory with von Karman geometric nonlinearity. Most existing studies have focused on the buckling behavior of intact plates, while research on the thermal buckling of cracked plates is relatively limited. Micro-cracking may occur within the nanocomposite-reinforced GPLs when severe thermal loading, such as thermal shock, is applied [30; 31]. The stiffness of the structure is significantly reduced when cracking occurs within the structure because the region in the vicinity of the cracks loses its load-bearing capacity due to crack-induced damage. Therefore, crack-induced damage cannot be ignored in the examination of the mechanical behaviors of GPLRC structures. Meanwhile, in a numerical study using the grid-based numerical approximation method, crack modeling was achieved by dividing the nodes lying on the crack [32]. However, this meshing-based approach to crack modeling is not only laborious but also causes numerical instability [33]. The creation of additional nodes during node separation requires the complex adaptation of node numbering and element connectivity, and furthermore, it de deteriorates the grid uniformity and causes distorted elements. To overcome this numerical problem, a phase-field formulation (PFF) has been introduced and widely adopted in the fracture mechanics community [34; 35; 36; 37]. In this approach, the existence of a crack line is represented by introducing the phase-field variable, which indicates the damage level of the structure, to avoid the numerical instability caused by node separation in the grid-based traditional method. In this context, this study intends to fill the important gap in the study of thermal buckling and postbuckling of FG-GPLRC plate by introducing an effective phase-field formulation to handle an inclined central crack. The numerical method was developed in the framework of the 2D extended natural element method (XNEM), in which the crack-induced singularity is captured by adding the crack-tip singular functions without locally refining the NEM grid [38; 39]. The inclined crack line is basically represented according to the phase field formulation by introducing a virtual rotation of an inclined crack line. The thermally induced large deformation is expressed by first-order SDT incorporated with von Karman nonlinearity, and the troublesome shear locking [40; 41], the deterioration of numerical accuracy for the bending-dominated thin structures, is suppressed by adopting the MITC3+ shell approach and stabilization parameters [42; 43].