Vol. 6, No. 1-4, 2011

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ISSN: 1559-3959
Propagation of waves in an incompressible transversely isotropic elastic solid with initial stress: Biot revisited

Ray W. Ogden and Baljeet Singh

Vol. 6 (2011), No. 1-4, 453–477
Abstract

In this paper, the general constitutive equation for a transversely isotropic hyperelastic solid in the presence of initial stress is derived, based on the theory of invariants. In the general finite deformation case for a compressible material this requires 18 invariants (17 for an incompressible material). The equations governing infinitesimal motions superimposed on a finite deformation are then used in conjunction with the constitutive law to examine the propagation of both homogeneous plane waves and, with the restriction to two dimensions, Rayleigh surface waves. For this purpose we consider incompressible materials and a restricted set of invariants that is sufficient to capture both the effects of initial stress and transverse isotropy. Moreover, the equations are specialized to the undeformed configuration in order to compare with the classical formulation of Biot. One feature of the general theory is that the speeds of homogeneous plane waves and surface waves depend nonlinearly on the initial stress, in contrast to the situation of the more specialized isotropic and orthotropic theories of Biot. The speeds of (homogeneous plane) shear waves and Rayleigh waves in an incompressible material are obtained and the significant differences from Biot’s results for both isotropic and transversely isotropic materials are highlighted with calculations based on a specific form of strain-energy function.

Keywords
hyperelasticity, initial stress, residual stress, transverse isotropy, invariants, plane waves, surface waves, Biot's theory
Milestones
Received: 4 June 2010
Revised: 25 July 2010
Accepted: 8 August 2010
Published: 28 June 2011
Authors
Ray W. Ogden
Department of Mathematics
University of Glasgow
University Gardens
Glasgow
G12 8QW
United Kingdom
School of Engineering
University of Aberdeen
King’s College
Aberdeen
AB24 3UE
United Kingdom
Baljeet Singh
Department of Mathematics
Postgraduate Government College
Sector 11
Chandigarh 160011
India