Abstract
In an elastic medium, it was proved that the stiffness tensor is symmetric with respect to the exchange of the first pair of indices and the second pair of indices, but the proof does not apply to a viscoelastic medium. In this paper, we thus derive the representation theorem for viscoelastic waves in a medium with a non-symmetric stiffness matrix. The representation theorem expresses the wave field at a receiver, situated inside a subset of the definition volume of the viscoelastodynamic equation, in terms of the volume integral over the subset and the surface integral over the boundary of the subset. For the given medium, we define the complementary medium corresponding to the transposed stiffness matrix. We define the frequency-domain complementary Green function as the frequency-domain Green function in the complementary medium. We then derive the provisional representation theorem as the relation between the frequency-domain wave field in the given medium and the frequency-domain complementary Green function. This provisional representation theorem yields the reciprocity relation between the frequency-domain Green function and the frequency-domain complementary Green function. The final version of the representation theorem is then obtained by inserting the reciprocity relation into the provisional representation theorem.
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References
Aki K. and Richards P., 1980. Quantitative Seismology. W.H. Freeman and Co, San Francisco, CA.
Carcione J.M., 2015. Wave Fields in Real Media. Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media. Elsevier, Amsterdam
Červený V., 2001. Seismic Ray Theory. Cambridge Univ. Press, Cambridge, U.K.
Christensen R.M., 1971. Theory of viscoelasticity. An Introduction. Academic Press, New York
de Hoop A.T., 1995. Handbook of Radiation and Scattering of Waves. Academic Press, London, U.K.
Fabrizio M. and Morro A., 1988. Viscoelastic relaxation functions compatible with thermodynamics. J. Elasticity, 19, 63–75
Fabrizio M. and Morro A., 1992. Mathematical Problems in Linear Viscoelasticity. SIAM, Philadelphia, PA.
Gurtin M.E. and Herrera I., 1965. On dissipation inequalities and linear viscoelasticity. Quart. Appl. Math., 23, 235–245
Kamenetskii E.O., 2001. Nonreciprocal microwave bianisotropic materials: Reciprocity theorem and network reciprocity. IEEE Trans. Antennas Prop., 49, 361–366
Klimeš L., 2018. Frequency-domain ray series for viscoelastic waves with a non-symmetric stiffness matrix. Stud. Geophys. Geod., 62, 421–431
Rogers T.G. and Pipkin A.C., 1963. Asymmetric relaxation and compliance matrices in linear viscoelasticity. Z. Angew. Math. Phys., 14, 334–343
Thomson C.J., 1997. Complex rays and wave packets for decaying signals in inhomogeneous, anisotropic and anelastic media. Stud. Geophys. Geod., 41, 345–381
Acknowledgements
The suggestions by Aleksey Stovas and an anonymous reviewer made it possible for me to improve the paper. The research has been supported by the Czech Science Foundation under contract 20–06887S, and by the members of the consortium “Seismic Waves in Complex 3-D Structures” (see “http://sw3d.cz”).
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Klimeš, L. Representation theorem for viscoelastic waves with a non-symmetric stiffness matrix. Stud Geophys Geod 65, 53–58 (2021). https://doi.org/10.1007/s11200-020-0158-2
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DOI: https://doi.org/10.1007/s11200-020-0158-2