One-dimensional dispersion phenomena in terms of fractional media
Institute of Structural Engineering, Poznań University of Technology, Piotrowo 5 street, 60-969, Poznań, Poland
2 Department of Continuum Mechanics and Structural Analysis, University Carlos III of Madrid, 28911, Madrid, Spain
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Accepted: 24 August 2016
Published online: 19 September 2016
It is well know that structured solids present dispersive behaviour which cannot be captured by the classical continuum mechanics theories. A canonical problem in which this can be seen is the wave propagation in the Born-Von Karman lattice. In this paper the dispersive effects in a 1D structured solid is analysed using the Fractional Continuum Mechanics (FCM) approach previously proposed by Sumelka (2013). The formulation uses the Riesz-Caputo (RC) fractional derivative and introduces two phenomenological/material parameters: 1) the size of non-local surrounding lf, which plays the role of the lattice spacing; and 2) the order of fractional continua , which can be devised as a fitting parameter. The results obtained with this approach have been compared with the reference dispersion curve of Born-Von Karman lattice, and the capability of the fractional model to capture the size effects present in the dynamic behaviour of discrete systems has been proved.
© The Author(s), 2016