A well-posed Euler-Bernoulli beam model incorporating nonlocality and surface energy effect  

作　　者：Xiaowu ZHU Li LI 

机构地区：[1]School of Statistics and Mathematics,Zhongnan University of Economics and Law,Wuhan 430073,China [2]State Key Laboratory of Digital Manufacturing Equipment and Technology,School of Mechanical Science and Engineering,Huazhong University of Science and Technology,Wuhan 430074,China

出　　处：《Applied Mathematics and Mechanics(English Edition)》2019年第11期1561-1588,共28页应用数学和力学（英文版）

基　　金：Project supported by the National Natural Science Foundation of China(No.51605172);the Natural Science Foundation of Hubei Province of China(No.2016CFB191);the Fundamental Research Funds for the Central Universities(Nos.2722019JCG06 and 2015MS014)

摘　　要：This study shows that it is possible to develop a well-posed size-dependent model by considering the effect of both nonlocality and surface energy,and the model can provide another effective way of nanomechanics for nanostructures.For a practical but simple problem (an Euler-Bernoulli beam model under bending),the ill-posed issue of the pure nonlocal integral elasticity can be overcome.Therefore,a well-posed governing equation can be developed for the Euler-Bernoulli beams when considering both the pure nonlocal integral elasticity and surface elasticity.Moreover,closed-form solutions are found for the deflections of clamped-clamped (C-C),simply-supported (S-S) and cantilever (C-F) nano-/micro-beams.The effective elastic moduli are obtained in terms of the closed-form solutions since the transfer of physical quantities in the transition region is an important problem for span-scale modeling methods.The nonlocal integral and surface elasticities are adopted to examine the size-dependence of the effective moduli and deflection of Ag beams.This study shows that it is possible to develop a well-posed size-dependent model by considering the effect of both nonlocality and surface energy, and the model can provide another effective way of nanomechanics for nanostructures. For a practical but simple problem(an Euler-Bernoulli beam model under bending), the ill-posed issue of the pure nonlocal integral elasticity can be overcome. Therefore, a well-posed governing equation can be developed for the Euler-Bernoulli beams when considering both the pure nonlocal integral elasticity and surface elasticity. Moreover, closed-form solutions are found for the deflections of clamped-clamped(C-C), simply-supported(S-S) and cantilever(C-F) nano-/micro-beams. The effective elastic moduli are obtained in terms of the closed-form solutions since the transfer of physical quantities in the transition region is an important problem for span-scale modeling methods. The nonlocal integral and surface elasticities are adopted to examine the size-dependence of the effective moduli and deflection of Ag beams.

关 键 词：NONLOCAL integral ELASTICITY BENDING SIZE-DEPENDENCE EFFECT surface ELASTICITY 

分 类 号：O343.7[理学—固体力学] O343.9[理学—力学]