Nonlocal Static and Dynamic Characterization of Nanobeams on Elastic Substrates with Surface Energy Effects

This study proposes a unified beam–substrate model for the static and dynamic characterization of nanobeams resting on elastic substrates, explicitly incorporating size-dependent surface energy effects. The formulation combines the modified strain gradient elasticity theory (MSGET) with the Euler–Bernoulli beam theory to capture nonclassical bulk behavior through three intrinsic length-scale parameters, while the substrate–structure interaction is modeled using a Winkler foundation. Surface energy effects are incorporated via the Gurtin–Murdoch surface elasticity framework. Hamilton’s principle is used to derive the governing equation of motion and the associated boundary conditions, from which closed-form solutions are obtained for bending, buckling, and free vibration. Representative simulations demonstrate pronounced stiffening induced by the combined effects of MSGET nonlocality, surface energy, and foundation stiffness. For the cantilever case, the proposed model predicts a tip-deflection reduction of approximately 64.14% compared with the classical local model. For simply supported systems, the normalized critical buckling load and the normalized first natural frequency can reach up to 27.05 and 5.17, respectively, over the considered parameter ranges. These findings underscore the importance of accounting for small-scale and surface effects in the design and interpretation of the mechanical response of nanobeam–substrate systems.