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Volume 1, Number 1, 2016 , pages 15-33 DOI: https://doi.org/10.12989/mmm.2016.1.1.015 |
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An asymptotic multi-scale approach for beams via strain gradient elasticity: surface effects |
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Jun-Sik Kim
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| Abstract | ||
| In this paper, an asymptotic method is employed to formulate nano- or micro-beams based on strain gradient elasticity. Although a basic theory for the strain gradient elasticity has been well established in literature, a systematic approach is relatively rare because of its complexity and ambiguity of higher-order elasticity coefficients. In order to systematically identify the strain gradient effect, an asymptotic approach is adopted by introducing the small parameter which represents the beam geometric slenderness and/or the internal atomistic characteristic. The approach allows us to systematically split the two-dimensional strain gradient elasticity into the microscopic one-dimensional through-the-thickness analysis and the macroscopic one-dimensional beam analysis. The first-order beam problem turns out to be different from the classical elasticity in terms of the bending stiffness, which comes from the through-the-thickness strain gradient effect. This subsequently affects the second-order transverse shear stress in which the surface shear stress exists. It is demonstrated that a careful derivation of a first strain gradient elasticity embraces \"Gurtin-Murdoch traction\" as the surface effect of a one-dimensional Euler-Bernoulli-like beam model. | ||
| Key Words | ||
| strain gradient elasticity; size effect; surface tension; asymptotic method | ||
| Address | ||
| Jun-Sik Kim: Department of Mechanical System Engineering, Kumoh National Institute of Technology, 61 Daehak-ro, Gumi, Gyeongbuk 730-701, Republic of Korea | ||
