Complete solutions of three-dimensional problems in transversely isotropic media

We present a systematic approach to the derivation of complete solutions for three-dimensional transversely isotropic elastic problems. The class of solutions derived in the paper is obtained by exploiting the entire kernel of the differential operator C expressing the elastic equilibrium problem of transversely isotropic media. In particular, the displacement field corresponding to any specific solution is expressed as linear combination of the columns of the adjoint C of the operator C, weighted by quasi-harmonic potentials or their derivatives. Invoking the generalized Almansi’s theorem, the vector collecting the functions weighting the columns of C can be split into the sum of three vector functions; in turn, their expressions depend upon the complete or partial coincidence of three scalar quantities whose value is function of the constitutive parameters of the medium. In this way, three different sets of solutions in terms of displacements are obtained. The general solution for the three-dimensional medium is then specialized to the significant case of half-spaces, and the solution corresponding to a point load is detailed by proving that two of the general solutions do coalesce. We further derive the displacement, strain and stress fields due to an horizontal point load, and numerical comparisons with existing solutions are illustrated.