On the mechanics of nanobeams on nano-foundations

A plethora of current applicative problems of Nano-Engineering involve nanostructures interacting with elastic media which exhibit significant size effects. Internal elasticity of the nanostructure and external elasticity of the surrounding foundation can be conveniently modelled by two nonlocal continua, provided that the relevant structure-soil problem results to be well-posed. This issue is still an open problem of Engineering Science and is solved in the present study by improving the Eringen–Wieghardt integral elasticity theory for nanobeams on nano-foundations, without having to resort to local/nonlocal mixtures, governed by a higher number of scale parameters, or physically questionable artifices to get mathematical consistency. Nanobeams resting on nano-foundations are modelled by stress-driven and displacement-driven laws providing nonlocal curvatures and reactions by convolution integrals of bending moments and displacements with bi-exponential averaging kernels. Advantageously, the relevant integro-differential problem is reverted to a simpler differential formulation equipped with non-classical boundary conditions. Effectiveness of the proposed nonlocal approach is illustrated by analytically solving beam–soil systems simulating nanocomposites and biological cells