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A size‐dependent Bernoulli‐Euler beam model based on strain gradient elasticity theory incorporating surface effects
Author(s) -
Fu Guangyang,
Zhou Shenjie,
Qi Lu
Publication year - 2019
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201800048
Subject(s) - timoshenko beam theory , elasticity (physics) , boundary value problem , mechanics , surface stress , bernoulli's principle , classical mechanics , mathematics , deflection (physics) , materials science , mathematical analysis , beam (structure) , physics , thermodynamics , optics , composite material
Abstract In this paper, a size‐dependent model for Bernoulli‐Euler beam is developed by using a general strain gradient theory and a surface elasticity theory. The general strain gradient theory is used to capture the strain gradient effects while the surface elasticity theory is applied to explain the surface effects. As comparison, the approximate theories including the couple stress theory and the simplified strain gradient theory are also used to describe the strain gradient effects. The Hamilton's principle is used to derive the governing equations, boundary conditions and initial conditions. The static bending and free vibration behaviour of nanobeams with different boundary conditions are analysed. Compared with the results predicted by the couple stress theory with surface effects and the simplified strain gradient theory with surface effects, because the general strain gradient theory contains the influences of all strain gradients, the deflection predicted by the general strain gradient theory with surface effects is smallest while the natural frequency is highest. The general strain gradient theory reflects size effects more accurately.