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Applied Mathematics and Mechanics

, Volume 40, Issue 4, pp 515–548 | Cite as

Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects

  • Bo ZhangEmail author
  • Huoming Shen
  • Juan Liu
  • Yuxing Wang
  • Yingrong Zhang
Article
  • 75 Downloads

Abstract

In this paper, multi-scale modeling for nanobeams with large deflection is conducted in the framework of the nonlocal strain gradient theory and the Euler-Bernoulli beam theory with exact bending curvature. The proposed size-dependent nonlinear beam model incorporates structure-foundation interaction along with two small scale parameters which describe the stiffness-softening and stiffness-hardening size effects of nanomaterials, respectively. By applying Hamilton’s principle, the motion equation and the associated boundary condition are derived. A two-step perturbation method is introduced to handle the deep postbuckling and nonlinear bending problems of nanobeams analytically. Afterwards, the influence of geometrical, material, and elastic foundation parameters on the nonlinear mechanical behaviors of nanobeams is discussed. Numerical results show that the stability and precision of the perturbation solutions can be guaranteed, and the two types of size effects become increasingly important as the slenderness ratio increases. Moreover, the in-plane conditions and the high-order nonlinear terms appearing in the bending curvature expression play an important role in the nonlinear behaviors of nanobeams as the maximum deflection increases.

Key words

nanobeam nonlocal strain gradient theory two-step perturbation method deep postbuckling 

Chinese Library Classification

O344.1 

2010 Mathematics Subject Classification

74H55 

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Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Bo Zhang
    • 1
    • 2
    Email author
  • Huoming Shen
    • 1
    • 2
  • Juan Liu
    • 1
    • 2
  • Yuxing Wang
    • 1
    • 2
  • Yingrong Zhang
    • 1
    • 2
  1. 1.School of Mechanics and EngineeringSouthwest Jiaotong UniversityChengduChina
  2. 2.Applied Mechanics and Structure Safety Key Laboratory of Sichuan ProvinceChengduChina

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