## 1 Introduction  
Sandwich structures, generally made of a soft core and two hard face sheets, are largely used in the aerospace, oil, gas, and petrochemical industries, due to their enhanced mechanical properties, namely, a high strength-to-weight ratio and a high resistance to heat, humidity, and noise [1; 2; 3; 4; 5]. Hence, in recent decades, much attention has been paid to the mechanical behavior of these structures [6; 7; 8; 9; 10; 11; 12]. Based on the available literature, it seems that the geometry of the layers, the mechanical properties of the constituents, and the geometrical properties of the whole structure can have a meaningful effect on the static and dynamic behavior of sandwich structures [13; 14; 15; 16; 17; 18; 19; 20]. The presence of some reinforcing layers in sandwich structures represents one important issue to consider for a general improvement of their mechanical properties [21; 22; 23; 24]. Nowadays, with the advancement of nanotechnology, carbon nanotubes (CNTs) and graphene sheets (GSs) are two alternative options for the reinforcement of structures, due to their extraordinary properties. This has led to an extensive research on the behavior of sandwich structures reinforced with nanocomposites [25; 26; 27; 28]. Among different reinforcement possibilities, graphene nanoplatelets (GPLs) provide a uniform reinforcedassembly, as well as the easiest manufacturing process, as discussed in [29; 30; 31; 32]. Graphene is a monolayer structure of carbon atoms with extraordinary electrical, mechanical, thermal, and optical properties [33; 34; 35], which make it very attractive for high-tech device applications, such as micro/nano-electromechanical systems [36]. Graphene and its derivatives--namely, GPLs--are increasingly applied as a reinforcement material in many nanocomposite structures [37; 38; 39; 40; 41; 42; 43]. This justifies the large attention paid in the literature to the mechanical behavior of structures reinforced with graphenic materials [44; 45; 46; 47; 48; 49; 50; 51; 52; 53]. Despite the extensive literature available on the behavior of composite structures reinforced with nanostructures, there is a general lack of works focusing on the nonlinear dynamic and vibration behavior of sandwich beams reinforced by GPLs. This is here investigated for thick polymer sandwich beams with face sheets reinforced by GPLs, in a context where the reduced weight of polymers and the high strength of GPLs can provide remarkable properties in the equivalent composite structure. A novel reinforcement model is proposed herein, which considers the functionality of the GPLs distribution throughout the thickness of the face sheets, and a constant total amount of the reinforced material. A higher-order laminated beam theory is applied to include the shear and rotation effects on the thick GPL-reinforced sandwich beam, where the nonlinear governing equations of the problem are solved in a straightforward manner by means of the multiple timescales method. The main advantage of the present method is that it can cover weak or strong nonlinearities with possible damping effects. The method is demonstrated to be very simple and accurate with respect to other existing predictions and theories from the literature. The reinforcement phase varies along the thickness according to a power-law distribution, whereby the effective material properties of the nanocomposite beam are determined by means of the Halpin-Tsai micromechanics model and the rule of mixtures. The nonlinear partial equations of motion are derived by the Hamilton's principle, in accordance with the third-order shear deformation theory and the Von Karman strain-displacement relationships. We then apply [PERSON]'s approach to discretize the nonlinear differential equations of motion, while determining the frequency equations by means of the multiple timescales method.