Inhomogeneous "longitudinal" plane waves in a deformed elastic material

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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 = n cos k(n.x - ct), 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. In other words, the displacement is of the form u = Re S exp[i (omega S.x - ct)], 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 Bⁿ-ellipsoid, where B is the left Cauchy-Green strain tensor corresponding to the primary pure homogeneous deformation.