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

doi:10.1007/s10483-019-2541-5

Applied Mathematics and Mechanics (English Edition) ›› 2019, Vol. 40 ›› Issue (11): 1561-1588.doi: https://doi.org/10.1007/s10483-019-2541-5

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

Xiaowu ZHU¹, Li LI²

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

收稿日期:2019-03-25 修回日期:2019-05-29 发布日期:2019-10-28
通讯作者: Li LI E-mail:lili_em@hust.edu.cn
基金资助:
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)

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

Xiaowu ZHU¹, Li LI²

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:2019-03-25 Revised:2019-05-29 Published:2019-10-28
Contact: Li LI E-mail:lili_em@hust.edu.cn
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)

摘要/Abstract

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.

关键词: nonlocal integral elasticity, bending, size-dependence effect, surface elasticity

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

中图分类号:

74K10
74S30

Xiaowu ZHU, Li LI. A well-posed Euler-Bernoulli beam model incorporating nonlocality and surface energy effect[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(11): 1561-1588.

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

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

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