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

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.