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Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

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Abstract

Eringen’s two-phase local/nonlocal model is applied to an Euler-Bernoulli nanobeam considering the bending-induced axial force, where the contribution of the axial force to bending moment is calculated on the deformed state. Basic equations for the corresponding one-dimensional beam problem are obtained by degenerating from the three-dimensional nonlocal elastic equations. Semi-analytic solutions are then presented for a clamped-clamped beam subject to a concentrated force and a uniformly distributed load, respectively. Except for the traditional essential boundary conditions and those required to be satisfied by transferring an integral equation to its equivalent differential form, additional boundary conditions are needed and should be chosen with great caution, since numerical results reveal that non-unique solutions might exist for a nonlinear problem if inappropriate boundary conditions are used. The validity of the solutions is examined by plotting both sides of the original integro-differential governing equation of deflection and studying the error between both sides. Besides, an increase in the internal characteristic length would cause an increase in the deflection and axial force of the beam.

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Authors and Affiliations

  1. Facility Design and Instrumentation Institute, China Aerodynamics Research and Development Center, Mianyang, 621000, Sichuan Province, China

    Licheng Meng, Dajun Zou, Huan Lai, Zili Guo, Xianzhong He & Zhijun Xie

  2. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Cunfa Gao

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  1. Licheng Meng
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  2. Dajun Zou
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  3. Huan Lai
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Project supported by the National Natural Science Foundation of China (No. 11472130)

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Meng, L., Zou, D., Lai, H. et al. Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force. Appl. Math. Mech.-Engl. Ed. 39, 1805–1824 (2018). https://doi.org/10.1007/s10483-018-2395-9

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  • DOI: https://doi.org/10.1007/s10483-018-2395-9

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