On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection

Abstract

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams: (i) why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise? (ii) the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and (iii) the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, domain governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary conditions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.

Wave propagation in carbon nanotubes: nonlocal elasticity-induced stiffness and velocity enhancement effects

“…Without rigorous validation, the classical equilibrium equations or equations of motion for beam and shell models were adopted completely for all nonlocal static and dynamic problems. Such directly extended nonlocal models, termed the partial nonlocal stress models, results in two fundamental suspicions that: (a) in many cases of study the nanoscale effect is surprisingly missing in the ultimate analytical solution, for instance the bending of a cantilever nanotube with point force at its end; and (b) the no-existence of any higher-order boundary conditions associated with the higher-order differential equation of motion [Lim 2008;2009;2010]. The second statement above simply implies that the partial nonlocal stress models derives a higher-order equation of motion but, unfortunately, without the corresponding higherorder boundary conditions which is obviously inconsistent.…”

Vibration analysis of three-layered nanobeams based on nonlocal elasticity theory

“…Eringen's nonlocal elasticity can be classified into a differential nonlocal form or an integral nonlocal form. Detailed review of both forms is discussed by Lim (2010).…”

Effect of structural parameter on stress concentration in a linearly elastic body

“…Many publications describe investigations of structured media [4][5][6][7][8][9]. There are many ways to account for the internal structure of a medium, for example, inclusion of additional internal variables in models, or application of the non-Archimedean analysis and description of multi-scale hierarchical space, etc.…”