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

doi:10.1007/s10483-018-2395-9

Applied Mathematics and Mechanics (English Edition) ›› 2018, Vol. 39 ›› Issue (12): 1805-1824.doi: https://doi.org/10.1007/s10483-018-2395-9

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

Licheng MENG¹, Dajun ZOU¹, Huan LAI¹, Zili GUO¹, Xianzhong HE¹, Zhijun XIE¹, Cunfa GAO²

1. Facility Design and Instrumentation Institute, China Aerodynamics Research and Development Center, Mianyang 621000, Sichuan Province, China;
2. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

收稿日期:2018-02-27 修回日期:2018-07-03 出版日期:2018-12-01 发布日期:2018-12-01
通讯作者: Licheng MENG E-mail:lichengmeng@nuaa.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (No. 11472130)

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

Licheng MENG¹, Dajun ZOU¹, Huan LAI¹, Zili GUO¹, Xianzhong HE¹, Zhijun XIE¹, Cunfa GAO²

1. Facility Design and Instrumentation Institute, China Aerodynamics Research and Development Center, Mianyang 621000, Sichuan Province, China;
2. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received:2018-02-27 Revised:2018-07-03 Online:2018-12-01 Published:2018-12-01
Contact: Licheng MENG E-mail:lichengmeng@nuaa.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (No. 11472130)

摘要/Abstract

摘要： 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.

关键词: direct control, simplified form, absolute stability, necessaryand sufficient condition, nanobeam, internal characteristic length, size effect, axial force, unique solution, nonlocal elasticity

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.

Key words: direct control, simplified form, absolute stability, necessaryand sufficient condition, unique solution, axial force, nonlocal elasticity, internal characteristic length, size effect, nanobeam

中图分类号:

45B05
45K05

Licheng MENG, Dajun ZOU, Huan LAI, Zili GUO, Xianzhong HE, Zhijun XIE, Cunfa GAO. Semi-analytic solution of Eringen's two-phase local/nonlocal model for Euler-Bernoulli beam with axial force[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(12): 1805-1824.

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

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

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