Multi‐parametric analysis of strongly inhomogeneous periodic waveguideswith internal cutoff frequencies

Abstract

In this paper, we consider periodic waveguides in the shape of a inhomogeneous string or beam partially supported by an uniform elastic Winkler foundation. A multi-parametric analysis is developed to take into account the presence of internal cutoff frequencies and strong contrast of the problem parameters. This leads to asymptotic conditions supporting non-typical quasi-static uniform or, possibly, linear micro-scale displacement variations over the high-frequency domain. Macro-scale governing equations are derived within the framework of the Floquet-Bloch theory as well as using a high-frequency-type homogenization procedure adjusted to a string with variable parameters. It is found that, for the string problem, the associated macroscale equation is the same as that applying to a string resting on a Winkler foundation. Remarkably, for the beam problem, the macro-scale behavior is governed by the same equation as for a beam supported by a two-parameter Pasternak foundation. Copyright c 0000 John Wiley & Sons, Ltd.Keywords: Periodic waveguide; cutoff frequency; homogenization; contrast; high-frequency IntroductionPeriodic structures with internal cutoff frequencies are of interest for numerous applications: as an example, we mention elastically supported periodic strings and beams [1,2], composite materials [3], phononic crystals [4,5,6] and vibration absorbers in fluid carrying pipes [7]. It is well renowned that a string supported by a Winkler foundation exhibits a cutoffwhere β is Winkler foundation modulus and ρ is the string linear mass density. As it is shown in [9], a two-phase piecewise periodic string supported by a uniform Winkler foundation cannot be treated by the conventional "low-frequency" homogenization method [10,11]. For the latter, the sought for macro-scale homogenized equation is of the same form as the original equation governing the behavior of the periodic system. Besides, a quasi-static uniform variation of the displacement field is retrieved at the micro-scale (see also [12]). On the other hand, the periodically supported string problem can be efficiently treated through a high-frequency asymptotic homogenization procedure, as established in [9,13,14,15]. Within this approach and contrarily to the classical setup, the homogenized macro-scale equation takes, as a rule, a different form than the original equation. Moreover, sinusoidal variations at the micro-scale are found which correspond to the eigenforms of the unit cell. Dynamic homogenization has been the subject of a number of remarkable contributions among which we mention [16,17,18,19,20,21,22] and also [23,24], dealing with the important case of periodic waveguides with contrast properties.In this paper, we show that quasi-static uniform (or, possibly, linear) micro-scale variation over the high-frequency range is still possible for strongly inhomogeneous periodic structures with internal cutoffs. As an example, a two-phase periodic waveguide in the shape of a string or a beam supported by an elastic Winkler foundation is studi...

Asymptotic analysis of an anti-plane dynamic problem for a three-layered strongly inhomogeneous laminate

“…Asymptotic considerations on the subject, not emphasizing characteristic peculiarities of dynamic behaviour, were reported in [12] and [13]. In addition, we cite here recent papers [14,15] devoted to homogenization of high-contrast periodic structures. Similarity of the asymptotic techniques for periodic and thin layered structures was addressed in [16].…”

The lowest vibration spectra of multi-component structures with contrast material properties

“…The first one deals with asymptotic homogenisation of high-contrast periodic media at various level of mathematical rigour, see e.g. [7], [8], [9], and [10]. Another research strand is oriented mainly towards twoor three-component structures.…”

Dispersion of elastic waves in a strongly inhomogeneous three-layered plate

“…In addition, we cite Cherdantsev & Cherednichenko (2012), Smyshlyaev (2009), Figotin & Kuchment (1998, and Kaplunov & Nobili (2016) devoted to homogenization of high-contrast periodic composites. Similarity of the asymptotic procedures underlying multi-layered plate theories and homoge-40 nization for periodic media has been recently reported in Craster et al (2014).…”