Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model

Y. B. Wang; X. W. Zhu; H. H. Dai

doi:10.1063/1.4961695

journal article Open Access Aug 01, 2016

Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model

Y. B. Wang X. W. Zhu H. H. Dai

AIP Advances Vol. 6 No. 8 · AIP Publishing

View at Publisher Save 10.1063/1.4961695

Abstract

Though widely used in modelling nano- and micro- structures, Eringen’s differential model shows some inconsistencies and recent study has demonstrated its differences between the integral model, which then implies the necessity of using the latter model. In this paper, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. Firstly, a reduction method is proved rigorously, with which the integral equation in consideration can be reduced to a differential equation with mixed boundary value conditions. Then, the static bending problem is formulated and four types of boundary conditions with various loadings are considered. By solving the corresponding differential equations, exact solutions are obtained explicitly in all of the cases, especially for the paradoxical cantilever beam problem. Finally, asymptotic analysis of the exact solutions reveals clearly that, unlike the differential model, the integral model adopted herein has a consistent softening effect. Comparisons are also made with existing analytical and numerical results, which further shows the advantages of the analytical results obtained. Additionally, it seems that the once controversial nonlocal bar problem in the literature is well resolved by the reduction method.

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Metrics

171

Citations

31

References

Details

Published: Aug 01, 2016
Vol/Issue: 6(8)
License: View

Authors

Y

Y. B. Wang

ShaoXing University 1 Department of Mathematics, , No.900, ChengNan Avenue 312000, ShaoXing, Zhejiang, China

X

X. W. Zhu

Zhongnan University of Economics and Law 2 School of Statistics and Mathematics, , Wuhan 430073, China

H

H. H. Dai

City University of HongKong 3 Department of Mathematics, , 83 Tat Chee Avenue, Kowloon Tong, HongKong

Funding

National Natural Science Foundation of China Award: 11472147

City University of Hong Kong Award: CityU11303015

Zhongnan University of Economics and Law Award: 31541411205

Shaoxing University Award: 20145002

Cite This Article

Y. B. Wang, X. W. Zhu, H. H. Dai (2016). Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. AIP Advances, 6(8). https://doi.org/10.1063/1.4961695

Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model

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