Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams

Strain-driven and stress-driven integral elasticity models are formulated for the analysis of the structural behaviour of fuctionally graded nano-beams. An innovative stress-driven two-phases constitutive mixture defined by a convex combination of local and nonlocal phases is presented. The analysis reveals that the Eringen strain-driven fully nonlocal model cannot be used in Structural Mechanics since it is ill-posed and the local-nonlocal mixtures based on the Eringen integral model partially resolve the ill-posedeness of the model. In fact, a singular behaviour of continuous nano-structures appears if the local fraction tends to vanish so that the ill-posedness of the Eringen integral model is not eliminated. On the contrary, local-nonlocal mixtures based on the stress-driven theory are mathematically and mechanically appropriate for nanosystems. Exact solutions of inflected functionally graded nanobeams of technical interest are established by adopting the new local-nonlocal mixture stress-driven integral relation. Effectiveness of the new nonlocal approach is tested by comparing the contributed results with the ones corresponding to the mixture Eringen theory.