## 1 Introduction
The numerical results were evaluated comparatively with respect to the straight configuration, assumed as the reference geometry (i.e., for an infinite radius of curvature), and the main effect of the nonlocal parameter, length scale, power-law exponent, boundary condition, and slenderness ratio were evaluated on the buckling load of curved nanobeams. A free vibration study of curved sandwich nanobeams was also performed by [PERSON] et al. [19], according to the [PERSON]'s nonlocal theory, whereby a meaningful decrease of the natural frequency was observed for increasing opening angles of the nanostructures. A further study of the statics and dynamics of curved single-walled carbon nanotubes was approached by [PERSON] et al. [20], based on the nonlocal theory, whose problem was solved using [PERSON]'s method. A different approach based on the sinusoidal shear deformation theory was applied by [PERSON] and [PERSON] [21; 22] for the analysis of a sandwich microbeam and nanoplate, respectively. Some additional static and dynamic analyses of curved beams at the nano- and macro-scales were presented by [PERSON] and [PERSON] [23], as well as by [PERSON] and [PERSON] [24], respectively. A nonlocal elasticity solution and wave propagation analysis of nanoplates and nanorods were proposed by [PERSON] and [PERSON] [25; 26], whereas the sinusoidal shear deformation theory was applied for the study of the transient response of curved beams [27]. A large variety of size-dependent theories of elasticity have been applied recently in literature to study the mechanics of nanostructures, including [PERSON]'s nonlocal models [28; 29; 30; 31], modified couple stress theories [32; 33; 34; 35; 36], and nonlocal strain gradient laws [37; 38]. Many further coupled nonlinear problems involving composite nanostructures can be found in previous studies [39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49]. Based on the available literature on the vibration and bending response of beams and plates reinforced with GPLs at the macroscale [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17], as well as on the curved beams at the macro- and nano-scales [18; 19; 20; 21; 22; 23; 24; 25; 26; 27], here we propose a combined study of curved nanobeams reinforced with nanoplatelets. In mechanical problems of technical interest, the elastic equilibrium is generally defined in bounded structural domains, so that suitable constitutive boundary conditions have to be prescribed to ensure the equivalence between integral and differential equations [33]. It has been recently proven that constitutive boundary conditions are in contrast with equilibrium requirements, so that [PERSON]'s strain-driven integral model leads to ill-posed structural problems, and therefore it is inapplicable to nanostructures. These difficulties have been overcome by adopting the innovative stress-driven nonlocal integral formulation of elasticity recently proposed in a previous study [33] and applied to static, thermal, vibration, and buckling problems in several papers. Herein, we tackle the problem according to [PERSON]'s nonlocal differential model and the first order shear deformation theory, where the vibration response of FG polymer composite curved nanobeams reinforced with GPLs is analyzed in detail, as useful for practical engineering applications. The Halpin-Tsaimodel and the rule of mixtures are here applied to compute the effective Young's modulus, density, and Poisson's ratio of the FG curved nanobeam, respectively. The analytical solution of the problem is here analyzed parametrically for varying boundary conditions, reinforcement distributions, weight fractions, number of layers, and Pasternak foundation parameters, while evaluating the sensitivity of the vibration global response.