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Fractional strain energy and its application to the free vibration analysis of a plate

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Abstract

Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and provides a new approach to non-local mechanics. In this study, a theoretical consideration on a new fractional non-local model is presented based on existence of fractional strain energy. It has two additional free parameters compared to classical local mechanics: (1) a fractional parameter which controls the strain gradient order in the strain energy relation and makes the model more flexible to describe physical phenomena, and (2) a non-local parameter to consider small scale effects in micron and sub-micron scales. The model has been used to obtain a fractional non-local plate theory. Free vibrations of a rectangular simply supported (S–S–S–S) plate has been investigated and the meaning of different parameters, such as fractional and non-local parameters, has been shown. The non-linear governing equation has been solved by the Galerkin method. A simple form of the governing equation and the numerical solution is an advantage of this fractional non-local model. Furthermore, the fractional nonlocal theory is contrasted with the Eringen nonlocal theory to show that fractional one enables to obtain much wider class of solutions.

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Change history

  • 08 September 2018

    In the original article, one of the co-author’s (Ghader Rezazadeh) surname has been published incorrectly. The correct surname should be Rezazadeh.

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Acknowledgements

This work is supported by the National Science Centre, Poland under Grant no. 2017/27/B/ST8/00351.

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Correspondence to Zaher Rahimi.

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The original version of this article was revised: one of the co-author’s (Ghader Rezazadeh) family name has been corrected.

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Rahimi, Z., Shafiei, S., Sumelka, W. et al. Fractional strain energy and its application to the free vibration analysis of a plate. Microsyst Technol 25, 2229–2238 (2019). https://doi.org/10.1007/s00542-018-4087-8

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