## 1 Introduction
In recent years, various theoretical and experimental studies have been conducted on the static and dynamic properties of sandwich porous beams and plates to meet various needs [29; 30; 31; 32]. The functionally graded (FG) porous materials are a hot research topic for new advanced composites [33; 34; 35]. The internal pores of FG porous materials show gradient distribution in a certain direction. FG porous materials can be varied by local density to obtain the desired natural frequency range of the structure [25]. Previous researchers have focused on studying the dynamic analysis, static bending, and buckling of FG porous plate and beam structures. For example, [PERSON] et al. [36] investigated the bending response of FG porous plates based on the cubic shear deformation theory. Based on two-variable shear deformation and modified coupled stress theory, [PERSON] et al. [37] investigated the nonlinear bending of FG porous thick circular/annular nano-plates using a generalized differential quadrature method (GDQM). Based on the Timoshenko beam theory, [PERSON] et al. [38] analyzed the static buckling and elastic buckling of shear deformable FG porous beams. In the above study, it can be found that porous beams or porous plates exhibit better mechanical properties than ordinary beams or plates of the same mass. The structure size [39] exhibits non-negligible scale effects when it is at the micro- and nanoscale. Although researchers have performed a great deal of work using classical continuum mechanics, their application on the nanoscale is questionable. Because in classical continuum mechanics, small-scale effects such as surface effects, lattice spacing, van der Waals forces, chemical bonding, and electric forces are neglected. However, both experimental and atomic simulation results show that at the nanoscale, these small-scale effects cannot be ignored [40]. This is because at small sizes, the lattice spacing between atoms becomes important and the interior of the material is no longer homogenized into a continuum [41]. Therefore, for the scale effects of micro- and nanoscale structures, non-classical continuum theories have been proposed by previous authors to evaluate the mechanical properties of micro- and nanostructures, such as surface elasticity theory [42; 43; 44], strain gradient elasticity theory [45; 46; 47; 48], and nonlocal elasticity theory [49; 50]. In the last decade,numerous researchers [51; 52; 53] have studied a large number of size dependencies on micro- and nanostructures based on the nonlocal elasticity theory and strain-gradient elasticity theory. These studies have shown that nonlocal theory can only capture the stiffness-softening effect and the strain gradient theory can only capture the stiffness-hardening effect. Recently, researchers have combined the nonlocal elasticity theory and the strain gradient elasticity theory together to propose a new theory, namely, the nonlocal strain gradient theory (NSGT) [54]. The NSGT combines the advantages of both theories so that not only the strain-gradient stress field but also the non-local stress field can be investigated. In addition, the NSGT is able to describe both the stiffness-hardening and stiffness-softening effects. Based on the NSGT, a large number of researchers have investigated the size-dependent bending, buckling, wave propagation, and vibration of FG micro- and nanoscale structures [18; 55; 56; 57]. For example, [PERSON] et al. [18] investigated the free vibration of piezoelectric sandwich nanobeams with flexoelectricity based on the NSGT. [PERSON] et al. [57] analyzed the size-dependent nonlinear axial postbuckling properties of functionally graded porous micro/nanoplates enhanced by graphene platelets based on the NSGT. [PERSON] et al. [58] analyzed the functional gradient piezoelectric nanoplate vibration using the NSGT. [PERSON] et al. [19] performed a nonlinear electromechanical analysis of micro/nanobeams based on the NSGT taking into account the flexoelectric and piezoelectric effects.