A review on the nonlocal stress field theory and the physics of nanoscales in nonlocal nanomechanics

This paper presents a review of nonlocal models in nanomechanics particularly in the modeling and analysis of nanostructures based on the nonlocal elastic stress field theory. It begins with a general description of Eringen’s pioneering paper on nonlocal elasticity model and his many subsequent works on nonlocal fluid mechanics, nonlocal electromagnetism, nonlocal thermoelasticity, nonlocal memory-dependent elasticity and nonlocal piezoelectricity, etc. The review goes on with the research works on nonlocal nanostructures from the turn of the century in which many papers were published based on a nonlocal differential constitutive relation instead of the original integral equation in Eringen’s formulation. These works investigated various aspects of nanostructures including bending, vibration, buckling, wave propagation, etc., with specific attention to nanotubes, nanobeams, nanoplates, nanoshells, etc.Subsequently, this paper summarizes the key findings of the research works after 2003. These papers concluded, either directly or indirectly, three critical but overlooked issues in the physics of 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, ie. increasing static deflection, decreasing natural frequency and decreasing buckling load, in virtually all previously published works in this subject although intuition in physics tells otherwise? (ii) the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study for bending of nanobeams, and (iii) the missing of additional boundary conditions which are required in the governing higher-order differential equation. With respect to these queries, this paper offers a new nonlocal elasticity model by applying the original differential constitutive relation and an exact variational principal. It further presents new exact equilibrium conditions, domain governing higher-order differential equation and higher-order boundary conditions. The missing or negligence of higher-order terms in the previous works results in misleading nanoscale effects which predicts completely reverse trends with respect to what the conclusion of this paper tells. The paper 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 substantiated by other approaches in nanostructural models such as molecular dynamic simulations, strain gradient theory, modified couple stress models and experiments.