Statics and dynamics of nanorods embedded in an elastic medium: Nonlocal elasticity and lattice formulations

This paper is focused on the static and the dynamic behaviour of an axial lattice (with direct neighbouring interaction) loaded by some distributed forces and in interaction with an elastic medium. Some exact analytical solutions are provided both in static and in dynamic settings, for the finite lattice system under general boundary conditions including fixed- and free-end boundary conditions. A nonlocal rod model based on the introduction of one additional length scale, is then constructed by continualization scheme of the lattice difference equations, to capture the scale effects associated with the lattice spacing. The continualized nonlocal model coincides with a phenomenological Eringen's nonlocal model, except eventually for the boundary conditions. These new continualized nonlocal boundary conditions are derived from the end lattice boundary conditions. The enriched nonlocal wave equation augmented by the elastic medium interaction has a spatial derivative which coincides with the local wave equation, thus avoiding the need of higher-order boundary conditions. The static and the dynamic responses of the equivalent nonlocal bar are also analytically studied and compared to the lattice problem. It is shown that the nonlocal solution efficiently fits the lattice one, both in static and in dynamic settings. The nonlocal model can be also introduced from variational arguments, thus leading to a nonlocal optimal Rayleigh quotient. For very high frequencies, the nonlocal model is corrected by a two-length scale model, which is shown to capture efficiently the frequency spectra of the lattice model for all frequency range.