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Analysis of the vibrational mode spectrum of a linear chain with spatially exponential properties

Michelitsch, T.M., Maugin, G.A., Nowakowski, A.F. and Nicolleau, F.C.G.A. (2009) Analysis of the vibrational mode spectrum of a linear chain with spatially exponential properties. International Journal of Engineering Science, 47 (2). pp. 209-220. ISSN 0020-7225


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We deduce the dynamic frequency-domain-lattice Green's function of a linear chain with properties (masses and next-neighbor spring constants) of exponential spatial dependence. We analyze the system as discrete chain as well as the continuous limiting case which represents an elastic I D exponentially graded material. The discrete model yields closed form expressions for the N x N Green's function for an arbitrary number N = 2,...,infinity of particles of the chain. Utilizing this Green's function yields an explicit expression for the vibrational mode density. Despite of its simplicity the model reflects some characteristics of the dynamics of a I D exponentially graded elastic material. As a special case the well-known expressions for the Green's function and oscillator density of the homogeneous linear chain are contained in the model. The width of the frequency band is determined by the grading parameter which characterizes the exponential spatial dependence of the properties. In the limiting case of large grading parameter, the frequency band is localized around a single finite frequency where the band width tends to zero inversely with the grading parameter. In the continuum limit the discrete Green's function recovers the Green's function of the continuous equation of motion which takes in the time domain the form of a Klein-Gordon equation. (C) 2008 Elsevier Ltd. All rights reserved.

Item Type: Article
Copyright, Publisher and Additional Information: © 2009 Elsevier. This is an author produced version of a paper subsequently published in the International Journal of Engineering Science. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: Linear chain; Dynamic lattice Green's function; Oscillator density; Lattice dynamics; Graded materials; Continuum limit; Klein-Gordon equation
Institution: The University of Sheffield
Academic Units: The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Mechanical Engineering (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 06 Apr 2009 13:51
Last Modified: 08 Feb 2013 16:58
Published Version: http://dx.doi.org/10.1016/j.ijengsci.2008.08.011
Status: Published
Publisher: Elsevier
Refereed: Yes
Identification Number: 10.1016/j.ijengsci.2008.08.011
URI: http://eprints.whiterose.ac.uk/id/eprint/8489

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