Theoretical analysis for static bending of circular Euler-Bernoulli beam using local and Eringen's nonlocal integral mixed model

Many studies have shown that Eringen's differential nonlocal model leads to some inconsistencies for the cantilever beams, which indicates the necessity to adopt the original integral nonlocal model. In this paper, Eringen's two-phase local/nonlocal model is applied to capture the size effect in the curved Euler-Bernoulli beam. The governing equations of equilibrium and boundary conditions are derived on the basis of Hamilton's principle. The Fredholm type integral governing equations are transformed to Volterra integral equations of the second kind through simply adjusting the limit of integrals, and solved uniquely through the Laplace transformation with several unknown constants, which are determined through the boundary conditions and extra constrained equations. The Exact solutions for a curved microbeam bending are derived explicitly for different boundary conditions. The present model is validated against the straight Euler beam for large radius circular case. The numerical results from current model show that the consistent softening effect of the two parameters on the bending deflection of the microbeams is observed for all boundary conditions.