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Archive of Applied Mechanics
Published in: Archive of Applied Mechanics 4/2021

04-01-2021 | Original

An uncoupled theory of FG nanobeams with the small size effects and its exact solutions

Authors: Y. L. Pei, L. X. Li

Published in: Archive of Applied Mechanics | Issue 4/2021

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Abstract

Due to unphysical coupling induced by the material inhomogeneity, FG (functionally graded) nanobeam problems were formulated in a very complex way so that they cannot be analytically solved. In this paper, an uncoupled theory is proposed for FG nanobeams considering their small size effects. First, with the aid of the neutral axis, the axial displacement is expressed in terms of generalized displacements for FG nanobeams. Based on the nonlocal strain gradient theory, the generalized stresses and strains are accordingly defined and uncoupled constitutive relations are derived. Based on the principle of virtual work, an uncoupled theory is eventually established, including governing equations and boundary conditions. Within the present framework, analytical solutions to FG nanobeams are obtained for the first time for general boundary conditions. These solutions not only re-evaluate the previous results but shed light on the small size effects of FG nanobeams.
previous article Study of frequency shift and thermoelastic damping in transversely isotropic nano-beam with GN III theory and two temperature
next article Free vibration analysis of an aluminum beam coated with imperfect and damaged functionally graded material

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Appendix
Available only for authorised users
Footnotes
1
Only defined in reference to a given axis, rotation (hence moment) is frame indifferent, and therefore physically objective.
 
2
As regards the case of \(l_{c} = l\), see the detailed discussion in “Appendix B”.
 
3
The present theory is applicable to the five different FG nanobeams and the four common displacement modes in “Appendix A”.
 
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Metadata
Title
An uncoupled theory of FG nanobeams with the small size effects and its exact solutions
Authors
Y. L. Pei
L. X. Li
Publication date
04-01-2021
Publisher
Springer Berlin Heidelberg
Published in
Archive of Applied Mechanics / Issue 4/2021
Print ISSN: 0939-1533
Electronic ISSN: 1432-0681
DOI
https://doi.org/10.1007/s00419-020-01849-2

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