Acta Mechanica

, Volume 228, Issue 1, pp 333–349

# Exact solutions of three-dimensional equations of static transversely isotropic elastic model

• Yu. A. Chirkunov
• N. F. Belmetsev
Original Paper

## Abstract

We found the conditions under which the system of the equations of the three-dimensional static transversely isotropic elastic model has a gradient of a harmonic function as a partial solution. In this case, parameters of the elasticity modulus tensor satisfy the Gassmann conditions. The Gassmann conditions are widely used in geophysics in the research of transversely isotropic elastic media. We fulfilled a group foliation of the system of the equations of the static transversely isotropic elastic model with the Gassmann conditions with respect to the infinite subgroup generated by the gradient of a harmonic function and contained in a normal subgroup of the main group of this system. We obtained a general solution of the automorphic system. This solution is a three-dimensional analogue of the Kolosov–Muskhelishvili formula. We found the main Lie group of transformations of the resolving system of this group foliation. With the help of this group foliation, we obtained non-degenerate exact solutions of the equations of the static transversely isotropic elastic model with the Gassmann conditions. For the found exact solutions, we have depicted the corresponding deformations arising in an elastic body for particular values of the elastic moduli.

## References

1. 1.
Annin, B.D., Ostrosablin, N.I.: Anisotropy of elastic properties of materials. J. Appl. Mech. Tech. Phys. 49(6), 998–1014 (2008)
2. 2.
Annin, B.D.: A transversely isotropic elastic model of geomaterials. J. Appl. Math. Mech. 4(3), 299–308 (2010)Google Scholar
3. 3.
Annin, B.D., Belmetsev, N.F., Chirkunov, Yu.A.: Group analysis of equations dynamic transversely isotropic elastic model. J. Appl. Math. Mech. 78(5), 529–537 (2014)Google Scholar
4. 4.
Gassmann, F.: Introduction to seismic travel time methods in anisotropic media. Pure Appl. Geophys. II 58, 1–224 (1964)Google Scholar
5. 5.
Goldin, S.V.: Seismic Waves in Anisotropic Media. Publishing House of the Russian Academy of Sciences, Novosibirsk (2008). (in Russian)Google Scholar
6. 6.
Carrier, G.F.: Propagation of waves in orthotropic media. Q. Appl. Math. 2(2), 160–165 (1946)
7. 7.
Chirkunov, Y.A., Khabirov, S.V.: The Elements of Symmetry Analysis of Differential Equations of Continuous Medium Mechanics. NSTU, Novosibirsk (2012). (in Russian)Google Scholar
8. 8.
Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)
9. 9.
Chirkunov, Y.A.: Linear autonomy conditions for the basic lie algebra of a system of linear differential equations. Dokl. Math. 79(3), 415–417 (2009)
10. 10.
Chirkunov, Y.A.: Systems of linear differential equations symmetric with respect to transformations nonlinear in a function. Sib. Math. J. 50(3P), 541–546 (2009)
11. 11.
Chirkunov, Y.A.: Group foliation of the lame equations of the classical dynamical theory of elasticity. Mech. Solids 44(3), 372–379 (2009)
12. 12.
Cartan, H.: Calcul Differentiel. Formes Differentielles. Hermann, Paris (1967)