Abstract
Due to the conflict between equilibrium and constitutive requirements, Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest. As an alternative, the stress-driven model has been recently developed. In this paper, for higher-order shear deformation beams, the ill-posed issue (i.e., excessive mandatory boundary conditions (BCs) cannot be met simultaneously) exists not only in strain-driven nonlocal models but also in stress-driven ones. The well-posedness of both the strain- and stress-driven two-phase nonlocal (TPN-StrainD and TPN-StressD) models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded (FG) materials. The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions. By using the generalized differential quadrature method (GDQM), the coupling governing equations are solved numerically. The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.
Highlights
Beam-like micro-/nano-structures are essential parts and are often utilized in the engineering design of micro-/nano-electromechanical systems (MEMSs/NEMSs)
5.1 Validation Since the ill-posedness of the pure nonlocal models exists, and there is no relevant research on higher-order refined beams related to the two-phase nonlocal models, the efficiency and accuracy of the present solution procedure are validated by comparing the results of a degradation case (i.e., ξ = 0.000 1, κ = 0.001) with those of the local elasticity theory
The well-posedness of the two-phase strategy of nonlocal integral models is illustrated by analyzing the scale-effected bending of a curved beam made of functionally graded (FG) materials
Summary
Beam-like micro-/nano-structures are essential parts and are often utilized in the engineering design of micro-/nano-electromechanical systems (MEMSs/NEMSs). In such applications, both experiments and atomistic simulations show that the size-effect must be comprehensively and rigorously considered. Owing to the lack of any inherent length characteristic parameter, the classical elasticity theory fails to address such size-dependent problems. Molecular dynamics (MD) simulations require high computational costs, and experiments are often difficult to implement at a micro-/nano-scale. Several non-classical continuum theories are c The Author(s) 2021. Pei ZHANG and Hai QING suggested to account for the size-effects, thereby overcoming the breakdown of classical elasticity. Among the higher-order continuum theories, the nonlocal elasticity model[1,2,3] is a popular tool
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