Abstract
Methods of deriving equations describing effective models of layered periodic media are presented. Elastic and fluid media, as well as porous Biot media, may be among these media. First, effective models are derived by a rigorous method, and then some operations in the derivation are replaced by simpler ones providing correct results. As a consequence, a comparatively simple and justified method of deriving equations of an effective model is established. In particular, this method allows us to simplify to a degree and justify the derivation of an effective model for media containing Biot layers; this method also produces equations of an effective model of a porous layered medium intersected by fractures with slipping contacts. Bibliography: 15 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. E. Backus, “Longwave elastic anisotropy produced by horizontal layering,”J. Geophys. Res.,67, 4427–4400 (1962).
E. Sanchez-Palencia, “Nonhomogeneous media and vibration theory,”Lect. Notes in Phys.,127 (1980).
N. S. Bakhvalov, and G. P. Panasenko,Averaging of Processes in Inhomogeneous Media. Mathematical Problems of Mechanics of Composite Materials [in Russian], Nauka, Moscow (1984).
L. A. Molotkov, “Equivalence of periodic layered and transversely isotopic media,”Zap. Nauchn. Semin. LOMI,89, 213–233 (1979).
L. A. Molotkov, “New method for deriving the equations of an effective averaged model of periodic media,”Zap. Nauchn. Semin. LOMI,195, 82–102 (1991).
M. Schoenberg, and F. Muir, “A calculus for finely layered anisotropic media,”Geophysics,54, 581–589 (1989).
T. J. Plona, K. W. Winkler, and M. Schoenberg, “Acoustic waves through alternating fluid/solid layers,”J. Acoust. Soc. Am.,81, 1227–1234 (1987).
R. H. Tatham, M. D. Matthews, K. K. Sekharan, C. J. Wade, and L. M. Liro, “A physical model study of shear-wave splitting and fracture intensity,”Geophysics,57, 647–652 (1992).
L. A. Molotkov, and A. V. Bakulin, “The effective model of a fractured medium with fractures described by surfaces of displacements of discontinuities,”Zap. Nauchn. Semin. POMI,218, 118–137 (1994).
C-J. Hsu and M. Schoenberg, “Elastic waves through a simulated fractured medium,”Geophysics,58, 964–977 (1993).
S. Gelinsky and S. Shapiro, “Poroelastic effective media model for fractured and layered reservoir rock,” in:The 65th Annual International Meeting, Society of Exploration Geophysicists (1995), pp. 922–925.
M. A. Biot, “Theory of propagation of elastic waves in fluid-saturated porous solid. I. Low-frequency range,”J. Acoust. Soc. Am.,28, 168–178 (1956).
L. A. Molotkov and A. V. Bakulin, “Effective models of stratified media containing porous Biot layers,”Zap. Nauchn. Semin. POMI,239, 140–163 (1997).
L. A. Molotkov and A. E. Khilo, “Investigation of one- and multi-phase effective models describing periodic media,”Zap. Nauchn. Semin. LOMI,140, 105–122 (1984).
L. A. Molotkov and A. E. Khilo, “On an effective model describing a layered periodic medium with slide contacts on the interfaces,”Zap. Nauchn. Semin. LOMI,210, 192–212 (1994).
Additional information
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998 pp. 219–243.
Translated by L. A. Molotkov.
Rights and permissions
About this article
Cite this article
Molotkov, L.A. On methods of deriving equations describing effective models of layered media. J Math Sci 102, 4275–4290 (2000). https://doi.org/10.1007/BF02673858
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02673858