Abstract
A transversely isotropic material in the sense of Green is considered. A complete solution in terms of retarded potential functions for the wave equations in transversely isotropic media is presented. In this paper we reduce the number of potential functions to only one, and we discuss the required conditions. As a special case, the torsionless and rotationally symmetric configuration with respect to the axis of symmetry of the material is discussed. The limiting case of elastostatics is cited, where the solution is reduced to the Lekhnitskii–Hu–Nowacki solution. The solution is simplified for the special case of isotropy. In this way, a new series of potential functions (to the best knowledge of the author) for the elastodynamics problem of isotropic materials is presented This solution is reduced to a special case of the Cauchy–Kovalevski–Somigliana solution, if the displacements satisfy specific conditions. Finally, Boggio's Theorem is generalized for transversely isotropic media which may be of interest to the reader beyond the present application.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
R. Burridge, The singularity on the plane lids of the wave surface of elastic media with cubic symmetry. Q. Appl. Math. 20 (1967) 41–56.
H.A. Elliott, Three dimensional stress distribution in hexagonal aeolotropic crystals. Proc. Camb. Philol. Soc. 44 (1948) 522–533.
M. Eskandari-Ghadi, Impedance function for rigid foundation resting on semi-infinite transversely isotropic media. Ph.D. thesis, University of Tehran (2000).
R.A. Eubanks and E. Sternberg, On the axisymmetric problem of elasticity theory for a medium with transverse isotropy. J. Ration. Mech. Anal. 3 (1954) 89–101.
F.I. Fedorov, Theory of Elastic Waves in Crystals.Plenum, New York 1968.
M.E. Gurtin. On Helmholtz's theorem and the completeness of the Papkovich–Neuber stress functions for infinite domains. Arch. Ration. Mech. Anal. 9 (1962) 225–233.
M.E. Gurtin, The linear theory of elasticity. In: S. Fhügge (ed.), Handbuch der Physik, Vol. Via/2, Mechanics of Solids II, ed C, Truesdell. Springer, Berlin Heidelberg New York (1972) pp. 1–295.
H.C. Hu, On the three dimensional problems of the theory of elasticity of a transversely isotropic body. Sci. Sinica 2 (1953) 145–151.
O.D. Kellogg, Foundation of Potential Theory. Dover Publication Inc. (1953).
S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body. Mir, Moscow (1981).
A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944).
J.H. Michell, The stress in an aeolotropic elastic solid with an infinite plane boundary. Proc. Lond. Math. Soc. 32 (1900) 247–258.
W. Nowacki, The stress function in three dimensional problems concerning an elastic body characterized by transversely isotropy. Bull. Acad. Pol. Sci. 2 (1954) 21–25.
H.B. Phillips, Vector Analysis. Wiley (1933).
R.K.N.D. Rajapakse and Y. Wang, Greens functions for transversely isotropic elastic half-space. J. Eng. Mech. 119(9) (1993) 1724–1746.
E. Sternberg, On the integration of the equation of motion in the classical theory of elasticity. Arch. Ration. Mech. Anal. 6 (1960) 34–50.
E. Sternberg and R.A. Eubanks, On stress functions for eiastokinetics and the integration of the repeated wave equation. Q. Appl. Math. 15 (1957) 149–153.
E. Sternberg and M.E. Gurtin, On the completeness of certain stress fimctions in the linear theory of elasticity. Proc. Forth U.S. Natl. Cong. Appl. Mech. (1962) 793–797.
M. Stippes. Completeness of Papkovich potentials. Q. Appl. Math. 26 (1969) 477–483.
C. Truesdell, Invariant and complete stress functions for general continua. Arch. Ration. Mech. Anal. 4 (1959) 1–29.
M.Z. Wang and W. Wang, Completeness and nonuniqueness of general solutions of transversely isotropic elasticity. Int. J. Solids Struct. 32(374) (1995) 501–513.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Morton E. Gurtin
Rights and permissions
About this article
Cite this article
Eskandari-Ghadi, M. A Complete Solution of the Wave Equations for Transversely Isotropic Media. J Elasticity 81, 1–19 (2005). https://doi.org/10.1007/s10659-005-9000-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-005-9000-x