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Research Papers

Finite Element Modeling of Beams With Surface Energy Effects

[+] Author and Article Information
C. Liu

Department of Mechanical Engineering, University of British Columbia, Vancouver, V6T 1Z4, Canada

R. K. N. D. Rajapakse

Faculty of Applied Science, Simon Fraser University, Burnaby, V5A 1S6, Canadarajapaks@sfu.ca

A. S. Phani

Department of Mechanical Engineering, University of British Columbia, Vancouver, V6T 1Z4, Canadasrikanth@mech.ubc.ca

J. Appl. Mech 78(3), 031014 (Feb 16, 2011) (10 pages) doi:10.1115/1.4003363 History: Received February 27, 2010; Revised December 28, 2010; Posted January 05, 2011; Published February 16, 2011; Online February 16, 2011

A finite element formulation of a nonclassical beam theory based on the Gurtin–Murdoch model for continua with deformable elastic surfaces is presented. The governing equations for thin and thick beams are used together with a weighted residual formulation to explicitly obtain the beam stiffness and mass matrices. Numerical solutions for selected test cases are compared with the analytical results available in literature for beam static deflections, natural frequencies, and buckling loads. The modified bending stiffness corresponding to the present model agrees closely with a recently reported rigorous solution. The maximum influence of surface energy effects is observed for cantilever beams. The finite element scheme provides an efficient tool to analyze, design, and predict the mechanical response of beam elements encountered in nanoelectromechanical systems and other nanoscale devices.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 8

(a) Free-body diagram of a beam element with surface residual stress τ0 on both top and bottom surfaces. (b) State of stress on the interfaces and in the bulk. The surfaces are exaggerated for clarity.

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Figure 1

A prismatic beam with length L and height H set in Cartesian coordinates

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Figure 2

Free-body diagram of an incremental beam (bulk) element subjected to a compressive force F and a distributed load q(x, t)

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Figure 3

Two-node beam element

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Figure 4

Three-node beam element

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Figure 5

Normalized deflections of thin simply supported beams. The results are from FEM, surface elastic model, and classical theory. (a) Al beam under distributed load, (b) Al beam under point load, (c) Si beam under distributed load, and (d) Si beam under point load.

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Figure 6

Normalized deflections of thick simply supported beams. The results are from FEM, surface elastic model, and classical theory. (a) Al beam under distributed load, (b) Al beam under point load, (c) Si beam under distributed load, and (d) Si beam under point load.

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Figure 7

Mode shape of a Si simply supported beam

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