A semi-continuum-based bending analysis for extreme-thin microano-beams and new proposal for nonlocal differential constitution

摘要

A modified semi-continuum Euler beam model with relaxation phenomenon is developed and the bending deformation of extreme-thin beam with microano-scale thickness is presented. The external loads including concentrated force and uniformly distributed loads are considered and different boundary constraints are analyzed. The explicit solutions of bending deflection are determined using the minimum potential energy principle. The appropriate normalizations of semi-continuum deflections are obtained by using the corresponding classical bending deflections. It is more reasonable than previous studies using the semi-continuum deflection with 1.0 relaxation coefficient as the normalized denominator. Different trends of bending deflections emerge and they are caused by different surface relaxation properties. The comparisons of semi-continuum, classical continuum and nonlocal continuum models show good agreement unless the thickness is too small. Further the trial function, the position of concentrated force and the equivalent Young's modulus are discussed in detail. Relation between the equivalent Young's modulus and thickness is developed via a relative simple approach. It is seen that the equivalent Young's modulus may increase or decrease with increasing the thickness which is associated with some previous controversial ideas. Such surface physical phenomena correspond to two opposite nonlocal elasticity models and subsequently a new proposal for differential constitutive equation of nonlocal continuum theory is put forward. Additionally, a nonlinear semi-continuum model is proposed and the elastic carrying capacity is predicted. The nonlinear bending deflection of extreme-thin beam is analyzed accordingly. (C) 2017 Elsevier Ltd. All rights reserved.