Journal of Elasticity

, Volume 75, Issue 2, pp 147–165 | Cite as

Inhomogeneous ‘longitudinal’ plane waves in a deformed elastic material

  • Michel Destrade
  • Michael Hayes
Article
  • 67 Downloads

Abstract

By definition, a homogeneous isotropic compressible Hadamard material has the property that an infinitesimal longitudinal homogeneous plane wave may propagate in every direction when the material is maintained in a state of arbitrary finite static homogeneous deformation. Here, as regards the wave, ‘homogeneous’ means that the direction of propagation of the wave is parallel to the direction of eventual attenuation; and ‘longitudinal’ means that the wave is linearly polarized in a direction parallel to the direction of propagation. In other words, the displacement is of the form u = ncos k(n · xct), where n is a real vector. It is seen that the Hadamard material is the most general one for which a ‘longitudinal’ inhomogeneous plane wave may also propagate in any direction of a predeformed body. Here, ‘inhomogeneous’ means that the wave is attenuated, in a direction distinct from the direction of propagation; and ‘longitudinal’ means that the wave is elliptically polarized in the plane containing these two directions, and that the ellipse of polarization is similar and similarly situated to the ellipse for which the real and imaginary parts of the complex wave vector are conjugate semi-diameters. In other words, the displacement is of the form u = ℜ{S exp iω(S · xct)}, where S is a complex vector (or bivector). Then a Generalized Hadamard material is introduced. It is the most general homogeneous isotropic compressible material which allows the propagation of infinitesimal ‘longitudinal’ inhomogeneous plane circularly polarized waves for all choices of the isotropic directional bivector. Finally, the most general forms of response functions are found for homogeneously deformed isotropic elastic materials in which ‘longitudinal’ inhomogeneous plane waves may propagate with a circular polarization in each of the two planes of central circular section of the \(\mathbb{B}\) n -ellipsoid, where \(\mathbb{B}\) is the left Cauchy-Green strain tensor corresponding to the primary pure homogeneous deformation.

Mathematics Subject Classifications (2000)

74B20 74J05 74J10 

Key words

nonlinear elasticity small on large inhomogeneous waves polarisation constitutive equations 

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Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Michel Destrade
    • 1
  • Michael Hayes
    • 2
  1. 1.Laboratoire de Modélisation en MécaniqueCNRS, Université Pierre et Marie Curie (UMR 7607)Paris Cedex 05France
  2. 2.Mechanical EngineeringUniversity College DublinBelfield, Dublin 4Ireland

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