Nonlocal Free Vibration Analysis of Perforated Nanobeams Resting on Kerr-Type Elastic Foundation

This study presents an analytical investigation into the free vibration behavior of perforated nanobeams resting on a Kerr-type elastic foundation within the framework of Eringen’s nonlocal elasticity theory. Specifically, Eringen’s nonlocal elasticity theory is employed to inherently capture small-scale effects, while the three-parameter Kerr model is utilized to provide a mathematically consistent representation of shear continuity and realistic surface interactions. In this context, the governing equations of motion for a perforated Euler–Bernoulli nanobeam are derived using Hamilton’s principle, incorporating both the nonlocal parameter and perforation geometric factors, namely, the filling ratio and the number of holes. The resulting equations are solved analytically via the Navier method for simply supported boundary conditions. The results indicate that the Kerr foundation model exhibits an intermediate behavior between the Winkler and Pasternak models, owing to the stiffness-reducing effect of its upper spring layer connected in series. A key finding is the “masking effect,” where high foundation stiffness significantly suppresses the frequency reduction caused by nonlocal small-scale effects. Furthermore, it is observed that in the absence of foundation support, the vibration behavior is governed by the competition between mass reduction and stiffness loss depending on the number of holes; however, foundation dominance stabilizes the system regardless of perforation geometry.