Higher order interface conditions for piezoelectric spherical hollow composites: Asymptotic approach and transfer matrix homogenization method

This work is devoted to a comparison of different methods determining stop-bands in 1D and 2D periodic heterogeneous media. For a 1D case, the well-known dispersion equation is studied via asymptotic approach. In particular, we show how homogenized solutions can be obtained by elementary series used up to any higher-order. We illustrate and discuss a possible application of asymptotic series regarding parameters other than wavelength and frequency. In addition, we study antiplane elastic shear waves propagating in the plane through a spatially infinite periodic composite material consisting of an infinite matrix and a square lattice of circular inclusions. In order to solve the problem, a homogenization method matched with asymptotic solution on the cell with inclusion of the large volume fracture is proposed and successfully applied. First and second approximation terms of the averaging method provide the estimation of the first stop-band. For validity and comparison with other approaches, we have also applied the Fourier method. The Fourier method is shown to work well for relatively small inclusions, i.e., when the inclusion-associated parameters and matrices slightly differ from each other. However, for evidently contrasting structures and for large inclusions, a higher-order homogenization method is advantageous. Therefore, a higher-order homogenization method and the Fourier analysis can be treated as mutually complementary.

Download Full-text

Study of wave dispersion in periodic composites by higher-order asymptotic homogenization method

PAMM ◽

10.1002/pamm.200700882 ◽

2007 ◽

Vol 7 (1) ◽

pp. 1090403-1090404

Author(s):

Igor V. Andrianov ◽

Vladimir I. Bolshakov ◽

Vladyslav V. Danishevs'kyy ◽

Dieter Weichert

Keyword(s):

Wave Dispersion ◽

Homogenization Method ◽

Higher Order ◽

Asymptotic Homogenization ◽

Periodic Composites ◽

Asymptotic Homogenization Method

Download Full-text

Acoustic theory of the many-bladed contra-rotating propeller: analysis of the effects of blade sweep on wake interaction noise

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.172 ◽

2019 ◽

Vol 868 ◽

pp. 385-427 ◽

Cited By ~ 4

Author(s):

M. J. Kingan ◽

A. B. Parry

Keyword(s):

Special Functions ◽

Higher Order ◽

Stationary Points ◽

Asymptotic Approach ◽

Azimuthal Mode ◽

Special Cases ◽

Wake Interaction ◽

Radiated Sound ◽

Interior Points ◽

Mode Order

An analytical model is presented for the wake interaction tones produced by a contra-rotating propeller. We re-cast the usual far-field radiation formulae as a double integral over a nominal propeller source annulus. Assuming that the number of blades on both propellers is large, we evaluate the integral asymptotically in terms of its leading-order contributions from interior stationary or boundary critical points which represent the specific locations on the propeller annulus that dominate the sound radiation. The asymptotic approach is powerful producing results in the form of one-line algebraic formulae that contain no integrals or special functions yet remain accurate. The asymptotics show that sweep is not necessarily beneficial and can cause the blade design to become critical for particular tones and directions in terms of a continuum of interior points distributed along a line on the propeller source annulus producing a higher-order result and thus an enhanced radiated sound field. The paper also shows how the interior points are completely consistent with the sub- or super-critical gust response of a swept blade. Tones with low and zero azimuthal mode order are treated as special cases and the asymptotics show that, as the mode order reduces, the radiated sound becomes concentrated around the flight axis where even higher-order solutions are possible, including rings and annuli of stationary points around the propeller annulus. Full numerical calculations confirm the accuracy of the asymptotic approach.

Download Full-text

First-principle description of acoustic radiation of shear flows

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2019.0077 ◽

2019 ◽

Vol 377 (2159) ◽

pp. 20190077 ◽

Cited By ~ 2

Author(s):

Xuesong Wu ◽

Zhongyu Zhang

Keyword(s):

Acoustic Waves ◽

Shear Flows ◽

Acoustic Radiation ◽

Higher Order ◽

Far Field ◽

Sound Waves ◽

Leading Order ◽

Mean Flow ◽

Asymptotic Approach ◽

Flow Distortion

As a methodology complementary to acoustic analogy, the asymptotic approach to aeroacoustics seeks to predict aerodynamical noise on the basis of first principles by probing into the physical processes of acoustic radiation. The present paper highlights the principal ideas and recent developments of this approach, which have shed light on some of the fundamental issues in sound generation in shear flows. The theoretical work on sound wave emission by nonlinearly modulated wavepackets of supersonic and subsonic instability modes in free shear flows identifies the respective physical sources or emitters. A wavepacket of supersonic modes is itself an efficient emitter, radiating directly intensive sound in the form of a Mach wave beam, the frequencies of which are in the same band as those of the modes in the packet. By contrast, a wavepacket of subsonic modes radiates very weak sound directly. However, the nonlinear self-interaction of such a wavepacket generates a slowly modulated mean-flow distortion, which then emits sound waves with low frequencies and long wavelengths on the scale of the wavepacket envelope. In both cases, the acoustic waves emitted to the far field are explicitly expressed in terms of the amplitude function of the wavepacket. The asymptotic approach has also been applied to analyse generation of sound waves in wall-bounded shear flows on the triple-deck scale. Several subtleties have been found. The near-field approximation has to be worked out to a sufficiently higher order in order just to calculate the far-field sound at leading order. The back action of the radiated sound on the flow in the viscous sublayer and the main shear layer is accounted for by an impedance coefficient. This effect is of higher order in the subsonic regime, but becomes a leading order in the transonic and supersonic regimes. This article is part of the theme issue ‘Frontiers of aeroacoustics research: theory, computation and experiment’.

Download Full-text

Novel Formulations for Higher-Order Bounds on Effective Transverse Elastic Moduli of Three-Phase Cylindrical Fiber Reinforced Composites

Volume 8: Mechanics of Solids, Structures and Fluids ◽

10.1115/imece2012-85357 ◽

2012 ◽

Author(s):

Yu-Fu Ko ◽

J. W. Ju

Keyword(s):

Elastic Properties ◽

Elastic Moduli ◽

Homogenization Method ◽

Higher Order ◽

Two Phase ◽

Special Event ◽

Three Phase ◽

Cylindrical Fiber ◽

Theoretical Predictions ◽

Order Bounds

A higher-order multiscale structure for three-phase composites containing randomly located yet unidirectionally aligned circular fibers is proposed to predict effective transverse elastic moduli based on the probabilistic spatial distribution of circular fibers, the pairwise fiber interactions, and the ensemble-area multi-level homogenization method. Specifically, the two inhomogeneity phases feature distinct elastic properties and sizes. In the special event, two-phase composites with same elastic properties and sizes of fibers are studied. Two non-equivalent micromechanical formulations are considered to derive effective transverse elastic moduli of two-phase composites leading to new higher-order bounds. Furthermore, the effective transverse elastic moduli for an incompressible matrix containing randomly located and identical circular rigid fibers and voids are derived. It is demonstrated that significant improvements in the singular problems and accuracy are achieved by the proposed methodology. Numerical examples and comparisons among our theoretical predictions, available experimental data, and other analytical predictions are rendered to illustrate the potential of the present method.

Download Full-text