, Volume 53, Issue 13, pp 3415–3435 | Cite as

Nonlinear wave propagation analysis in Timoshenko nano-beams considering nonlocal and strain gradient effects

  • A. Norouzzadeh
  • R. AnsariEmail author
  • H. Rouhi


This article is aimed to investigate the geometrically nonlinear wave propagation of nano-beams on the basis of the most comprehensive size-dependent elasticity theory. To this end, the integral model of nonlocal elasticity theory in the most general form without any simplification in conjunction with the modified strain gradient theory is implemented in the analysis. Also, the Timoshenko beam model is utilized in the presented nonlocal strain gradient elasticity theory. By Hamilton’s principle, the governing integro-partial differential equations of motion are derived. Employing numerical integration and an efficient method called as periodic grid technique, a semi-analytical approach is presented for the solution procedure. To detect the impacts of nonlocality and small scale effects on the nonlinear wave propagation characteristics of beams at nanoscale, adequate numerical examples and comparison studies are presented.


Geometrical nonlinearity Integral model of nonlocal elasticity Modified strain gradient theory Timoshenko nano-beam Nonlinear wave propagation 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of GuilanRashtIran
  2. 2.Department of Engineering Science, Faculty of Technology and Engineering, East of GuilanUniversity of GuilanRudsar, VajargahIran

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