Chinese Journal of Theoretical and Applied Mechani ›› 2016, Vol. 48 ›› Issue (6): 1265-1280.DOI: 10.6052/0459-1879-16-186

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Cai Wei, Chen Wen   

  1. Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 211100, China
  • Received:2016-07-08 Revised:2016-08-24 Online:2016-11-15 Published:2016-11-25
  • Contact: 10.6052/0459-1879-16-186


The existing experimental data indicate that the attenuations of acoustic waves propagating in complex media always exhibit a non-integer power-law dependence on frequency. Such phenomenon is di cult to be characterized by traditional damping wave equation or approximate thermo-viscous wave equation, which can only describe the frequency independent or frequency-squared dependent attenuation, respectively. With the dynamic development and wide applications of fractional calculus, wave equations with fractional derivative terms have been successfully applied to depicting the frequency dependent attenuation. Based on the research achievements of our group, this paper aims at presenting a review of the various fractional derivative wave equations, discussing the corresponding mechanical constitutive relationships and statistical interpretation, and laying the foundation for the in-depth study in the future. The time-and space-fractional derivative wave equations for soft matters are introduced, which can be classified into two groups:the constitutive models and the phenomenological models. The connections and di erences between such models are also discussed. Then, the successful applications of fractional derivative in modeling wave propagation in porous media are also summarized. The statistical interpretation for the power-law dependent exponent covering[0, 2] is presented via linking the space-fractional diffusion equation with Lévy stable distribution. Finally, the key problems in such area for future explorations are highlighted.

Key words:

complex media|fractional derivative|acoustic wave|attenuation|power-law dependent

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