## 1 Introduction  
In recent decades, sandwich structures increasingly attracted the interest of most researchers and designers, because of their outstanding engineering performances, i.e., high flexural stiffness-to-weight ratio, corrosion resistance, thermal and acoustic insulation [1]. Sandwich structures are typically an assemblage of high strength layers, and low-density cores, that can be made of metals, composites, or fiber-metal-laminate, foam, honeycomb or functionally graded materials (FCMs) [2; 3; 4; 5]. More specifically, FGM-based structures serve as bi-phase beams, plates and shells, whose properties vary continuously throughout their thickness or length, as observed in some natural FGMs, such as bamboo trees, teeth, bones and human skin. Nowadays, it is common to find many structural examples of straight and/or curved members with an optimized design and sustainable properties [6; 7]. In such a context, the vibration and buckling behavior represents a double aspect to account for a proper selection of the reinforcement phase during a design process. From this perspective, a rapid development of advanced theoretical strategies and computational methods was observed in the literature to simulate the behavior of FGMs and sandwich structures, even considering the effect of different environmental conditions. Among some relevant works on the topic, [PERSON] and [PERSON] [8] studied the vibrational response of FGM plates using the Runge-Kutta method, whose model considered the possible effect of a thermomechanical coupled loading condition. A thermomechanical post-buckling study of simply supported FGM plates was performed by [PERSON] [9], with the addition of piezoelectric fiber-reinforced composites for actuators, sensors and active damping devices. In line with the previous work, [PERSON] and [PERSON] [10] numerically studied the post-buckling behavior of sandwich plates with FGM skins, while determining the sensitivity of the global response to the power-law index, foundation parameters and imperfections. A homogenization technique based on a Mori-Tanaka scheme was recently combined to a first-order shear deformation theory by [PERSON] and [PERSON] [11] for an accurate evaluation of the thermal-dependent frequency and buckling response of FGM beams, accounting for the micro-mechanical interaction among particles within mixtures. An analytical formulation based on the first order beam approach was also proposed in [12] to investigate the thermal buckling resistance of simply supported FGM beams with a parabolic-concave thickness variation and temperature-dependent material properties, whose solution was determined numerically based on a finite difference method. A refined zigzag theory was proposed instead by [PERSON] et al. [13] to formulate reliable and computationally efficient finite elements suited for large-scale analyses of FG structures. In a further work, [PERSON] et al. [14] applied a three-dimensional linear theory of elasticity along with a series of Chebyshev polynomials multiplied by appropriate functions to study the free vibration of FGM sandwich rectangular plates with simple and clamped supports, while accurately solving the problem according to the Ritz method with a rapid rate of convergence. Further efforts in the same direction can be found in Refs. [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26], where different higher-order formulations and computational approaches were recently implemented to solve different vibration [15; 16; 17; 18; 19; 20] and buckling [21; 22; 23; 24; 25; 26] problems. More specifically, in Ref. [15], the authors focused on the size-dependent computation of the natural frequency in Euler-Bernoulli microbeams, as provided by a modified couple stress theory and Hamilton principle, accounting for simply supported and cantilever beams.