## 1 Introduction
[25] investigated the transverse vibrations and steady-state responses of axially moving viscoelastic piezoelectric nanostructures containing the thermal effect and the instable behaviors of axially non-uniformly moving viscoelastic piezoelectric nanoplates. On the other hand, the nanosystems are often embedded in a medium or resting on a foundation in many nanotechnology applications. In order to characterize the surrounding effects, [PERSON] and [PERSON] [26] studied the vibration of orthotropic single layered graphene sheets embedded in Winkler-type and Pasternak-type foundations with the differential quadrature method employed to solve the governing differential equations for various boundary conditions. [PERSON] et al. [27] developed a model for sigmoid functionally graded material (S-FGM) nanoplates embedded in a Pasternak elastic medium based on a modified couple stress theory and presented an analytical solution for buckling analysis of S-FGM nanoplates. [PERSON] et al. [28] presented a buckling analysis of double-orthotropic nanoplates based on nonlocal elasticity theory with an assumption that two nanoplates are bonded by an internal elastic medium and surrounded by external elastic foundation. [PERSON] and [PERSON] [29] proposed the thermo-electro-mechanical vibration characteristics of a piezoelectric-nanoplate system (PNPS) subjected to a non-uniform voltage distribution and embedded in a Pasternak elastic medium. The problem was solved by using the differential quadrature method (DQM) and results showed that the natural frequencies are quite sensitive to the non-uniform and nonlocal parameters. [PERSON] et al. [30] also investigated the effect of initial stress on the vibration responses for a double-piezoelectric-nanoplate system based on the nonlocal Kirchhoff plate theory, where the Pasternak-type elastic foundation model was used to account for the effect of shearing between the two nanoplates. [PERSON] et al. [31] studied the vibration of a nonlocal Euler-Bernoulli beam embedded in a Pasternak elastic foundation. An efficient sixth-order finite difference discretization was adopted for various boundary conditions and the effects of nonlocal parameter, slenderness ratiosand boundary conditions on the dynamic characteristics of the beam were discussed. [PERSON] and [PERSON] [32] presented transient behavior of simply-supported orthotropic single-layered graphene sheet (SLGS) resting on orthotropic visco-Pasternak foundation subjected to dynamic loads. Researches show that the structural and foundation damping coefficients are effective parameters on the transient response, particularly for large damping coefficients. [PERSON] et al. [33] used the micromechanics method to consider the small-scale effect in the structural equations of a piezoelectric polymeric nanocomposite panel embedded in the Pasternak-type elastic substrate. [PERSON] et al. [34] studied the nonlinear buckling of straight concrete columns armed with single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation. The column was modeled with Euler-Bernoulli beam theory. The characteristics of the equivalent composite were determined using the Mori-Tanaka model. The foundation around the column was simulated with spring and shear layer. [PERSON] et al. [35] dealt with the free vibration and biaxial buckling of double-magneto-electro-elastic nanoplate systems (DMEENPS) subjected to initial external electric and magnetic potentials with the assumption that the two nanoplates are bonded with each other using a visco-Pasternak medium and are also limited to the external elastic substrate. [PERSON] and [PERSON] [36] investigated the transient thermos-electro-mechanical vibration and bending analysis of a functionally graded piezoelectric nanosheet resting on visco-Pasternak foundation. The effects of some important parameters were studied on the fundamental frequencies of the system and the maximum deflection of the sheet under various thermal and electrical loadings.