Exact and asymptotic bending analysis of microbeams under different boundary conditions using stress‐derived nonlocal integral model

Abstract

AbstractEringen's nonlocal differential elasticity is widely applied to model nano‐ and micro‐structures, although previous studies have shown some inconsistences, e.g., the nonlocal parameter maybe has no effect on the deflection of Euler‐Bernoulli and Timoshenko beams subjected to some kinds of boundary and loading conditions. In this paper, the static bending analysis of Euler‐Bernoulli and Timoshenko beams is performed with the application of stress‐driven nonlocal integral model. The Fredholm type integral constitutive equations of the first kind are transformed to Volterra integral equations of the first kind through simply adjusting the limit of integrals, and the general solutions to the deflection as well as bending moment and so on are derived and obtained explicitly through solving the integro‐differential governing equations with the Laplace transformation. Exact solutions are derived explicitly for different loading and boundary conditions, especially for the paradoxical beam problem while using nonlocal differential model. The analytical and asymptotic expressions of the beam deflections obtained in this paper are validated against to those existing analytical and numerical results in literature. It is clearly established that, with the application of the stress‐driven nonlocal integral model, a consistent toughening effect can be obtained for both Euler‐Bernoulli and Timoshenko beams.