Fractional order models for the homogenization and wave propagation analysis in periodic elastic beams

Element‐free Galerkin method for a fractional‐order boundary value problem

“…Note that the computational cost involved following f‐EFG is lower than that of the FE solvers for fractional‐order governing differential equations 28 . This is true even for a given number of nodes $$$ n $$$, but as observed in Section 4.1 the convergence via f‐EFG is achieved for a lower number of nodes.…”

New prospects in non-conventional modelling of solids and structures

“…The fractional-order finite element method developed by the authors in a previous study (see references in ref. [1]) is used to simulate the response of the periodic beam. The fractional-order framework is validated, at both band-pass and band-gap frequencies, by direct comparison against the response of a periodic beam solved via the traditional finite element method based on integer-order equations.…”

“…Remarkably, the fractional-order homogenization approach proves capable of representing the dynamic behaviour of a bi-material periodic beam within the first few frequency band gaps, which is a strength of the proposed method over classical homogenization techniques whose range of validity is limited to the lowfrequency regime. A discussion on potential improvements is also given [1]. Lazopoulos and Lazopoulos [2] propose a novel fractional approach to continuum mechanics, based on the definition of a fractional Λ -derivative and an associated Λ-space to introduce consistent Λ-fractional strain measures, as well as corresponding stress and displacement measures.…”

“…The first group of articles concerns fractional models [1][2][3][4]. Patnaik et al [1] propose a fractional-order homogenization approach to model the elastic wave propagation in a bi-material periodic beam, taken as a representative of a one-dimensional elastic metamaterial.…”

“…The first group of articles concerns fractional models [1][2][3][4]. Patnaik et al [1] propose a fractional-order homogenization approach to model the elastic wave propagation in a bi-material periodic beam, taken as a representative of a one-dimensional elastic metamaterial. The authors assume the homogenized beam is governed by a fractional-order equation analogue of the Euler-Bernoulli beam equation, where fractionalorder kinematic relations involving a space-fractional Riesz-Caputo derivative are defined over a nonlocal horizon.…”