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A general dynamic theoretical model of elastic micro-structures with consideration of couple stress effects and its application in mechanical analysis of size-dependent properties

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Abstract

A general and systematic theoretical framework of elastic micro-structures is established with the aid of modified couple stress theory for investigating the size-dependent property in small scale, in which the size-dependence is considered by introducing a material length scale parameter. Mathematically, dynamic governing equations and corresponding boundary conditions are derived and simplified by using single power series expansion for a micro-plate and double power series expansion for a micro-beam. It is demonstrated that this method exhibits extraordinary superiority, i.e., different vibration modes can be extracted easily from artificial truncations. This theoretical model can be reduced to some classical cases, including the Bernoulli–Euler beam, Timoshenko beam, Kirchhoff plate and Mindlin plate, if some specific assumptions are made. After validation, a systematic numerical investigation is carried out, which focuses on the couple stress effect on shear resonance of a cantilever micro-plate. Finally, a methodology for proposing the critical size that distinguishes micro-scale from macro-scale is illustrated in detail.

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Acknowledgements

The authors gratefully acknowledge the supports by Natural Science Foundation of China (11672223 and 11972276), Natural Science Foundation of Shaanxi Province of China (2018JM1039), and Project Funded by China Postdoctoral Science Foundation (2018M640975), Opening Project from the State Key Laboratory for Strength and Vibration of Mechanical Structures (SV2018-KF-36), and the Postgraduate Research Fund of Shaanxi Province of China.

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    Authors and Affiliations

    1. State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, China

      Yilin Qu, Peng Li & Feng Jin

    2. Department of Civil Engineering, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, China

      Peng Li

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    1. Yilin Qu
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    Correspondence to Peng Li or Feng Jin.

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    Peng Li and Feng Jin have contributed equally to this article.

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    Qu, Y., Li, P. & Jin, F. A general dynamic theoretical model of elastic micro-structures with consideration of couple stress effects and its application in mechanical analysis of size-dependent properties. Acta Mech 231, 471–488 (2020). https://doi.org/10.1007/s00707-019-02534-4

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