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

Xiaowu ZHU, Li LI

doi:10.1007/s10483-019-2541-5

Applied Mathematics and Mechanics >

2019 , Vol. 40 >Issue 11: 1561 - 1588

DOI: https://doi.org/10.1007/s10483-019-2541-5

Articles

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

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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

Received date: 2019-03-25

Revised date: 2019-05-29

Online published: 2019-10-28

Supported by

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

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Abstract

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.

Key words： nonlocal integral elasticity; bending; size-dependence effect; surface elasticity

Cite this article

Xiaowu ZHU, Li LI . A well-posed Euler-Bernoulli beam model incorporating nonlocality and surface energy effect[J]. Applied Mathematics and Mechanics, 2019 , 40(11) : 1561 -1588 . DOI: 10.1007/s10483-019-2541-5

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