A discussion on the physics and truth of nanoscales for vibration of nanobeams based nonlocal elastic stress field theory

Two critical but overlooked issues in the physics of nonlocal elastic stress field theory for nanobeams are discussed: (i) why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, ie. increasing static deflection, decreasing natural frequency and decreasing buckling load, in virtually all previously published works in this subject (a total of 51 papers known since 2003) although intuition in physics tells otherwise? and (ii) the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study for bending of nanobeams. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, the exact equilibrium conditions, domain governing differential equation and boundary conditions for vibration of nanobeams are derived for the first time. These new equations and conditions involve essential higher-order terms which are missing in virtually all nonlocal models and analyses in previously published works in statics and dynamics of nonlocal nano-structures. Such negligence of higher-order terms in these works results in misleading nanoscale effects which predicts completely incorrect, reverse trends with respect to what the conclusion of this paper tells. Effectively, for the first time this paper not only discovers the truth of nanoscale, as far as nonlocal elastic stress modelling for nanostructures is concerned, on equilibrium conditions, governing differential equation and boundary conditions but also reveals further the true basic vibration responses for nanobeams with various boundary conditions. It also concludes that the widely accepted equilibrium conditions of nonlocal nanostructures currently are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an equivalent nonlocal bending moment. The conclusions above are illustrated by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.