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

• Published in: AIP advances Vol. 6; no. 8; pp. 085114 - 085114-22
• Main Authors: Wang, Y. B., Zhu, X. W., Dai, H. H.
• Format: Journal Article
• Language: English
• Published: Melville American Institute of Physics 01.08.2016; AIP Publishing LLC
• Subjects: ASYMPTOTIC SOLUTIONS; BEAMS; BOUNDARY CONDITIONS; Cantilever beams; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPARATIVE EVALUATIONS; DIFFERENTIAL EQUATIONS; Euler-Bernoulli beams; EXACT SOLUTIONS; Integral equations; Mathematical models; MICROSTRUCTURE; Reduction; SIMULATION
• DOI: 10.1063/1.4961695
• ISSN: 2158-3226

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.