Finite element formulation for free vibration of the functionally graded curved nonlocal nanobeam resting on nonlocal elastic foundation

In this work, the influence of elastic foundation on a size-dependent free vibration of functionally graded (FG) curved Euler-Bernoulli nanobeam is investigated on the basis of two-phase local/nonlocal models. The governing equation and standard boundary conditions are derived through Hamilton’s principle. The integral constitutive equation is equivalently transformed into differential forms with the corresponding constitutive boundary conditions. The axial force, bending moment, and react force due to foundation are explicitly expressed with respect to displacement variables. With the aid of the constitutive boundary conditions, the possibility of flexibly meeting higher-order variables is achieved. A finite element formulation based on the differential form of the two-phase nonlocal elasticity is utilized to discretize the nanobeam, and a general eigenvalue equation is obtained about the vibration frequency. The efficiency and accuracy of the proposed finite element model are validated by comparison with the results in the literature. The influences of nonlocal parameters, Winkler elastic parameter, central angle of the curved nanobeam, and length–height ratio on the vibration frequencies are studied for different boundary conditions.