Analysis of dispersion and propagation properties in a periodic rod using a space-fractional wave equation

Abstract

This study explores the use of fractional calculus as a possible tool to model wave propagation in complex, heterogeneous media. While the approach presented could be applied to predict field transport in a variety of inhomogeneous systems, we illustrate the methodology by focusing on elastic wave propagation in a one-dimensional periodic rod. The governing equations describing the wave propagation problem in inhomogeneous systems typically consist of partial differential equations with spatially varying coefficients. Even for very simple systems, these models require numerical solutions which are computationally expensive and do not provide the valuable insights associated with closed-form solutions. We will show that fractional calculus can provide a powerful approach to develop comprehensive mathematical models of inhomogeneous systems that can effectively be regarded as homogenized models. Although at first glance the mathematics might appear more complex, these fractional order models can allow the derivation of closed-form analytical solutions that provide excellent estimations of the systems' dynamic responses. Equally important, these solutions are valid in a frequency range that goes largely beyond the well-known homogenization limit of traditional integer order approaches, therefore providing a possible route to high-frequency homogenization. More specifically, this study focuses on the analyses of the dispersion and propagation properties of a periodic medium under single-tone harmonic excitation and illustrates the methodology to obtain a space-fractional wave equation capable of capturing the behavior of the physical system. The fractional wave equation and its analytical solution are compared with numerical results obtained via a traditional finite element method in order to assess their validity and evaluate their performance. It is found that the resulting fractional differential models are, in their most general form, of complex and frequency-dependent order. arXiv:1808.07422v1 [physics.class-ph]

A generalized fractional-order elastodynamic theory for non-local attenuating media

“…The RC derivatives in equation (3.1) have lower and upper bounds at the respective terminals of the horizon of non-locality. When the fractional derivative has a lower bound of −∞, −normal∞ CDxαfalse[ebxfalse]=bαebx, the solution kernel of the fractional wave equation can be chosen in the form of exponential functions [39] corresponding to propagating plane waves. When lower bounds other than −∞ are chosen, then solution kernels based on Mittag–Leffler functions are appropriate.…”

“…When lower bounds other than −∞ are chosen, then solution kernels based on Mittag–Leffler functions are appropriate. However, under proper assumptions for the interval of the fractional derivative [39] both kernels satisfy the same dispersion relations. Similar comment holds for the upper bound.…”

“…In the context of wave propagation, fractional models have helped modelling dissipation in lossy media using time-fractional derivatives [16,35–37]. More recently, space-fractional derivatives have been employed to capture attenuation in media exhibiting long-range cohesive forces [29,34,38] and even bandgaps in periodic media [39].…”

Applications of variable-order fractional operators: a review

“…The many characteristics of fractional operators have sparked, in recent years, much interest in fractional calculus and produced a plethora of applications with particular attention to the simulation of physical problems. Areas that have seen the largest number of applications include the formulation of constitutive equations for viscoelastic materials [1][2][3][4], transport processes in complex media [4][5][6][7][8][9][10][11], mechanics [12][13][14][15], non-local elasticity [16][17][18][19], plasticity [20][21][22], modelorder reduction of lumped parameter systems [23] and biomedical engineering [24][25][26]. These studies have typically used constant-order (CO) fractional operators.…”

Variable-order particle dynamics: formulation and application to the simulation of edge dislocations

“…The unique set of properties of fractional operators have determined a surge of interest in exploring their possible applications. Among the areas that have rapidly developed, we encounter the modelling of viscoelastic materials [1][2][3], transport processes in complex media [4][5][6][7][8][9][10][11][12], non-local elasticity [13][14][15][16][17][18][19] and model-order reduction of lumped parameter systems [20]. These applications mostly focused on constant-order (CO) fractional operators.…”