Vol. 3, No. 5, 2008

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The influence of an initial simple shear deformation on long-wave motion in an elastic layer

Svetlana R. Amirova and Graham A. Rogerson

Vol. 3 (2008), No. 5, 831–851

Long wave dispersion in an incompressible elastic layer subject to an initial static simple shear deformation is investigated. Long wave approximations of the dispersion relation associated with zero incremental traction on the faces are derived for both low and high-frequency motion. Comparison of approximate and numerical solutions is shown to provide excellent agreement over a surprisingly large wave number range. Within both the low and high-frequency regimes, the approximations are employed to establish the relative asymptotic orders of the displacement components and hydrostatic pressure. In the high-frequency case, the in-plane component of displacement is shown to be asymptotically larger than the normal component; motion is, therefore, essentially that of thickness shear resonance. The influence of this specific form of initial deformation is, therefore, seemingly minor in respect of long-wave high-frequency motion. However, in the long-wave low-frequency case, considerable differences are noted in comparison with both the classical and previously published prestressed cases. Specifically, both the normal and in-plane displacement components are of the same asymptotic order, indicating the absence of any natural analogue of either classical bending or extension.

elastic waves, dispersion, shear
Received: 31 October 2007
Revised: 16 April 2008
Accepted: 19 April 2008
Published: 1 July 2008
Svetlana R. Amirova
Department of Mathematics
University of Keele
Keele, Staffordshire
United Kingdom
Graham A. Rogerson
Department of Mathematics
University of Keele
Keele, Staffordshire
United Kingdom