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Research Papers

Calculation of the Additional Constants for fcc Materials in Second Strain Gradient Elasticity: Behavior of a Nano-Size Bernoulli-Euler Beam With Surface Effects

[+] Author and Article Information
H. M. Shodja1

Institute for Nanoscience and Nanotechnology, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran; Department of Civil Engineering, Sharif University of Technology, P.O. Box 11155-9313, Tehran, Iranshodja@sharif.edu

F. Ahmadpoor

Department of Civil Engineering, Sharif University of Technology, P.O. Box 11155-9313, Tehran, Iranfatemeh.ahmadpoor@gmail.com

A. Tehranchi

Department of Civil Engineering, Sharif University of Technology, P.O. Box 11155-9313, Tehran, Irantehranchi@alum.sharif.edu

1

Corresponding author.

J. Appl. Mech 79(2), 021008 (Feb 24, 2012) (8 pages) doi:10.1115/1.4005535 History: Received May 05, 2011; Revised June 21, 2011; Posted January 25, 2012; Published February 13, 2012; Online February 24, 2012

In addition to enhancement of the results near the point of application of a concentrated load in the vicinity of nano-size defects, capturing surface effects in small structures, in the framework of second strain gradient elasticity is of particular interest. In this framework, sixteen additional material constants are revealed, incorporating the role of atomic structures of the elastic solid. In this work, the analytical formulations of these constants corresponding to fee metals are given in terms of the parameters of Sutton-Chen interatomic potential function. The constants for ten fcc metals are computed and tabulized. Moreover, the exact closed-form solution of the bending of a nano-size Bernoulli-Euler beam in second strain gradient elasticity is provided; the appearance of the additional constants in the corresponding formulations, through the governing equation and boundary conditions, can serve to delineate the true behavior of the material in ultra small elastic structures, having very large surface-to-volume ratio. Now that the values of the material constants are available, a nanoscopic study of the Kelvin problem in second strain gradient theory is performed, and the result is compared quantitatively with those of the first strain gradient and traditional theories.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

A Bernoulli-Euler beam under lateral loading

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Figure 2

Schematic figure of a cantilever beam on which a concentrated force is acting at its end

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Figure 3

Deflection of a cantilever Al beam

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Figure 4

Bending rigidity of the Al beam

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Figure 5

Normalized displacement field (μa0/P)u2

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