Buckling and free vibration analyses of functionally graded timoshenko nanobeams resting on elastic foundation

In the present work, a novel nonlocal finite element model is presented for functionally graded (FG) Timoshenko nanobeams resting on a size-dependent elastic foundation. In contrast to the previous major studies, the size-dependent effects of both nanobeam and elastic foundation are taken into account simultaneously and modeled with the equivalent stress-driven two-phase local/nonlocal differential model equipped with two constitutive boundary conditions. The weak form of governing equations is derived and the higher-order variables in the additional external forces are eliminated with the aid of the constitutive boundary conditions. A finite element formulation based on the differential nonlocal constitutive relations is developed for buckling and free vibration analysis of FG nanobeams. Several comparative studies are conducted to verify the efficiency and accuracy of the proposed nonlocal finite element method (FEM). Considering the nonlocality of the elastic foundation, the effects of two-phase local/nonlocal elasticity on critical buckling load and vibration frequency of FG Timoshenko nanobeam are investigated in detail with different gradient index, nonlocal parameter, local volume fraction and buckling as well as vibration orders under different boundary conditions.