Dynamics of FG nanobeams on nonlocal medium

Nanotechnology is central in several research fields, from bioengineering to energy harvesting, in which the topic of functionally graded (FG) nanobeams resting on nano-foundations is highly investigated in both static and dynamic contexts. In order to make the structure-foundation problem result to be well-posed, the Eringen–Wieghardt theory has been replaced with a novel nonlocal approach based on stress- and displacement-driven integral constitutive laws. This methodology provides nonlocal elastic curvature and foundation reaction fields as spatial convolutions driven by bending interaction and displacement fields, respectively. In this paper, an analytical approach is described to reverse the relevant integro-differential elastodynamic problem into an equivalent purely differential formulation. The relevant eigenanalysis of nanobeams resting on elastic foundations is carried out. Natural frequencies and mode shapes are evaluated for simple structural schemes of current interest in Nano-Engineering. The outcomes can be valuable for sustainable design and optimization of composite structural nanocomponents of modern small-scale systems.