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Research Papers

Surface Elasticity Effects Can Apparently Be Explained Via Their Nonconservativeness

[+] Author and Article Information
Noël Challamel1

 INSA de Rennes, Laboratoire de Génie Civil et Génie Mécanique (LGCGM), 20, avenue des Buttes de Coësmes, 35043 Rennes cedex, Francenoel.challamel@univ-ubs.fr

Isaac Elishakoff2

 Florida Atlantic University, Boca Raton, FL 33431-0991elishako@fau.edu

1

Current address: Professor of Civil Engineering, Université Européenne de Bretagne University of South Brittany UBS UBS - LIMATB Centre de Recherche, Rue de Saint Maudé, BP92116 56321 Lorient cedex, France. On sabbatical leave:Mechanics Division, Department of Mathematics University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway.

2

Current address: Florida Atlantic University, Boca Raton, FL 33431-0991.Phone: (407) 367-2729.

J. Nanotechnol. Eng. Med 2(3), 031008 (Jan 11, 2012) (8 pages) doi:10.1115/1.4005486 History: Received June 27, 2011; Revised August 24, 2011; Published January 11, 2012; Online January 11, 2012

Recently, considerable attention has been given to investigating the surface effects on nanoscale materials. These effects can be predominant for small-scale structures, such as nanobeams, nanoplates, and nanoshells. In this paper, surface elasticity effects are considered for small scale beam structures based on the Laplace–Young equation, which results in an equivalent distributed loading term in the beam equation. We show that these effects are explained by their nonconservative nature that can be essentially modeled as a follower tensile loading for inextensible beams. The buckling and vibrations of small scale beams in the presence of surface elasticity effects is studied for various boundary conditions. It is shown that the surface elasticity effects may significantly affect the buckling and vibrations behavior of small scale beams. For clamped-free boundary conditions, we show that the buckling load is reduced compared to the one without this surface effect. This result is consistent with some recent numerical results based on surface Cauchy–Born model and with experimental results available in the literature. It appears that this result cannot be obtained if surface elasticity effects are modeled as a conservative-type loading. For other boundary conditions such as hinge–hinge and clamped–clamped boundary conditions, the results are identical to the ones already published. We explain in this paper the surprising results observed in the literature that surface elasticity effects may soften a nanostructure for some specific boundary conditions (due to the nonconservative nature of its loading application). The same conclusions are obtained for the vibrations of small scale beams with surface elasticity effects, where the natural frequency tends to decrease with surface elasticity effects for clamped-free conditions.

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Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 7

Buckling of hinge–hinge columns with surface elasticity effects; conservative axial load P

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Figure 8

Influence of the surface elasticity effects on the fundamental frequency; clamped-free column

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Figure 9

Fundamental natural frequency versus the (conservative) tension load p0 for the clamped-free column; asymptotic formulae based on (ωω0)2=1+p0

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Figure 2

Buckling of columns with surface elasticity effects; conservative axial load P

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Figure 3

Buckling load p versus the surface elasticity factor p0 for the clamped-free column; asymptotic formulae based on p=1+(1-4π)p0

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Figure 4

Buckling load p versus the (conservative) tension load p0 for the clamped-free column; p=1+p0 (also valid for other boundary conditions)

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Figure 5

Buckling of columns with surface elasticity effects; nonconservative axial load P

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Figure 6

Surface elasticity effects lead to a distributed follower-type loading for nonuniform column; Leipholz’s column with tensile distributed load

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Figure 1

Surface elasticity effects lead to a follower-type loading; Beck’s column with tensile follower load

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