Nonlocal Beam-Foundation Problems with Discontinuities

A broad spectrum of nanomechanical applications involves small-scale structures interacting with nanofoundations. The elastostatics of nanobeams on nonlocal media is here investigated in the presence of discontinuities, representing the most interesting condition in solid and structural mechanics. Indeed, beam-foundation problems usually involve concentrated loading systems, internal kinematic constraints, or nonsmooth geometric and elastic properties. To model the size-dependent behavior of nanobeams, the Eringen strain-driven theory is replaced with a stress-driven approach. The Wieghardt reaction-driven integral law of an elastic medium is replaced with an alternative two-phase nonlocal foundation theory. Such a model requires that transverse displacements are convex combination of local and nonlocal phases governed by Winkler and Wieghardt responses, respectively. The proposed methodology overcomes conflict issues with both kinematic compatibility and equilibrium conditions, ensuring that the governing structural problem is well posed. A computationally effective differential formulation is provided by extending previous contributions in nonlocal continuum mechanics to the case in which nonsmooth fields are involved. Structural problems of nanobeams on nonlocal foundations involving nonsmooth fields are finally analyzed and solved, providing benchmark results in nanomechanics.