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

AIP Advances - Tập 6 Số 8 - 2016
Y. B. Wang1,2,3, Xi Zhu4,2,5, Hui–Hui Dai4,1,6
12School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
23Department of Mathematics, City University of HongKong, 83 Tat Chee Avenue, Kowloon Tong, HongKong
3ShaoXing University 1 Department of Mathematics, , No.900, ChengNan Avenue 312000, ShaoXing, Zhejiang, China
41Department of Mathematics, ShaoXing University, No.900, ChengNan Avenue 312000, ShaoXing, Zhejiang, China
5Zhongnan University of Economics and Law 2 School of Statistics and Mathematics, , Wuhan 430073, China
6City University of HongKong 3 Department of Mathematics, , 83 Tat Chee Avenue, Kowloon Tong, HongKong

Tóm tắt

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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AIP Advances

Tập 6 Số 8

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