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Bending of circular nanoplates with consideration of surface effects

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Abstract

Surface effects play a significant role in affecting mechanical properties of micro- and nanosized materials and structures. This paper studies the bending of nanoplates with consideration of surface effects. Surface effects including surface elasticity and surface residual stress are incorporated into the conventional Kirchhoff theory of thin plates. Two typical cases including a concentrated force at the plate center and uniformly distributed loading over a plate surface are analyzed. Explicit expressions for the deflection of simply supported and clamped circular nanoplates are obtained. Bending moments at the plate center exhibit logarithmic singularity for a concentrated force at the center. When ignoring the surface effects, the classical deflections of circular thin plates are recovered. A comparison of the deflections with and without the surface effects clarifies a significant influence of surface effects on the bending behaviors of nanoplates. Surface effects diminish the bending moments and enhance the load-carrying capacity of a nanoplate. Singularity of elastic fields at the plate center is discussed. The obtained results provide helpful guidelines for design and application of graphene and other microscopic structures.

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Acknowledgements

This study was funded by the National Natural Science Foundation of China (No. 11672336) and the Open Foundation of State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, PRC (No. GZ15204).

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Authors and Affiliations

  1. School of Civil Engineering, Central South University, Changsha, 410075, People’s Republic of China

    Ying Yang, Jiaqi Zou & Xian-Fang Li

  2. State Key Laboratory of Structural Analysis for Industrial Equipment and Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Kang Yong Lee & Xian-Fang Li

Authors
  1. Ying Yang
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  2. Jiaqi Zou
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  3. Kang Yong Lee
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  4. Xian-Fang Li
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Correspondence to Xian-Fang Li.

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Appendix 1

Appendix 1

The following asymptotic expressions for the zeroth- and first-order Bessel functions of imaginary argument are used [27]

$$\begin{aligned} I_{0}\left( \beta \right)&=1+\frac{\beta ^{2}}{4}+\frac{\beta ^{4}}{64}+O(\beta ^{6}), \end{aligned}$$
(A1)
$$\begin{aligned} I_{1}\left( \beta \right)&=\frac{\beta }{2}+\frac{\beta ^{3}}{16} +\frac{\beta ^{5}}{384}+O(\beta ^{6}), \end{aligned}$$
(A2)
$$\begin{aligned} K_{0}\left( \beta \right)&=-\left( \gamma +\ln \frac{\beta }{2}\right) +\frac{\beta ^{2}}{4}\left( 1-\gamma -\ln \frac{\beta }{2}\right) +\frac{\beta ^{4}}{128}\left( 3-2\gamma -2\ln \frac{\beta }{2}\right) +O(\beta ^{6}), \end{aligned}$$
(A3)
$$\begin{aligned} K_{1}\left( \beta \right)&=\frac{1}{\beta }+\frac{\beta }{4}\left( -1+2\gamma +2\ln \frac{\beta }{2}\right) +\frac{\beta ^{3}}{64}\left( -5+4\gamma +4\ln \frac{\beta }{2}\right) +O(\beta ^{4}). \end{aligned}$$
(A4)

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Yang, Y., Zou, J., Lee, K.Y. et al. Bending of circular nanoplates with consideration of surface effects. Meccanica 53, 985–999 (2018). https://doi.org/10.1007/s11012-017-0760-8

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  • DOI: https://doi.org/10.1007/s11012-017-0760-8

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