Torsional wave propagation in a pre-stressed hyperelastic annular circular cylinder

Shearer, Tom and Abrahams, I. David and Parnell, William J. and Daros, Carlos H. (2013) Torsional wave propagation in a pre-stressed hyperelastic annular circular cylinder. The Quarterly Journal of Mechanics and Applied Mathematics, 66 (4). pp. 465-487.

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Abstract

We consider torsional wave propagation in a pre-stressed annular cylinder. Hydrostatic pressure is applied to the inner and outer surfaces of an incompressible hyperelastic annular cylinder, of circular cross-section, whose constitutive behaviour is governed by a Mooney�Rivlin strain energy function. The pressure difference creates an inhomogeneous deformation field, which modifies the inner and outer radii of the annular cylinder.We wish to determine the effect that this pre-stress, and a given axial stretch, has on the propagation of small-amplitude torsional waves through the medium. We use the theory of small-on-large to deduce the linear wave equation that governs incremental torsional waves and then determine the dispersion relation for the pre-stressed annulus by using an approximation procedure (the Liouville�Green transformation). We show that this scheme compares well to numerical solutions except in regions very close to turning points (of the transformed ordinary differential equation). In particular, we find that the inhomogeneous deformation makes the coefficients of the governing ordinary differential equation spatially dependent and affects the location of the roots of the dispersion relation.We observe that, if the pressure on the outer surface of the annular cylinder is greater (smaller) than that on the inner, then the cut-on frequencies are spaced further apart (closer) than they would be in the stress-free case. This result could potentially be used to tune the propagation characteristics of the cylinder over a range of frequencies.

Item Type: Article
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 74 Mechanics of deformable solids
PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 40 ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS > 46 Continuum mechanics of solids
Depositing User: Dr Tom Shearer
Date Deposited: 08 Dec 2014
Last Modified: 20 Oct 2017 14:13
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2203

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