Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium

•The sinusoidal shear deformation plate theory is used to analyze the bending and vibration of the nanoplates.•The nanoplates are assumed to be embedded in two-parameter elastic foundations and subjected to mechanical and thermal loads.•The governing equations are solved analytically for various boundary conditions.•A detailed parametric study is carried out to highlight the influences of the different parameters on the bending and the frequency of the nanoplates.

In the present paper, the sinusoidal shear deformation plate theory (SDPT) is reformulated using the nonlocal differential constitutive relations of Eringen to analyze the bending and vibration of the nanoplates, such as single-layered graphene sheets, resting on two-parameter elastic foundations. The present SDPT is compared with other plate theories. The nanoplates are assumed to be subjected to mechanical and thermal loads. The equations of motion of the nonlocal model are derived including the plate foundation interaction and thermal effects. The governing equations are solved analytically for various boundary conditions. Nonlocal theory is employed to bring out the effect of the nonlocal parameter on the bending and natural frequencies of the nanoplates. The influences of nonlocal parameter, side-to-thickness ratio and elastic foundation moduli on the displacements and vibration frequencies are investigated.