Analytical solutions of static bending of curved Timoshenko microbeams using Eringen's two-phase local/nonlocal integral model

To eliminate the paradox that Eringen's differential nonlocal model leads to some inconsistencies for the cantilever beams, the original integral nonlocal model restarts to attract a lot of attention. In this paper, Eringen's two-phase local/nonlocal integral model is utilized to predict the size-effect on curved Timoshenko microbeams. The governing equations and corresponding boundary conditions are derived via Hamilton's principle. By using the Laplace transform technique and merely adjusting the limit of integrals, the integral constitutive equations are transformed from Fredholm type into Volterra integral equations of the second kind and solved uniquely containing several unknown constants, which are determined through the boundary conditions and extra constrained equations from integral constitutive relationships. The analytical solutions are derived explicitly and are validated against the straight Timoshenko beam for the large-radius curved case. The results show a consistent softening effect of two nonlocal parameters on the bending behavior of the curved Timoshenko microbeams under different boundary conditions.