Prof. Tarik Chakkour

What do you mean please by the mathematical elastic model? the Christoffel equation?

In that case, there is an analytical solution up to the hexagonal point group of symmetry, the matrix can be obtained analytically, as for the cubic case.

There is a first solution, the phase velocity, then the group velocity which is the one that most interestest researchers. And also the phase and the group angles.

There are several approximations, and for the P mode, they work pretty well, but, for the 2 transversal modes, the approximate analytical solution works well only along some symmetry axes.

There is a lot of literature on seismic exploration used for oil & gas prospection, books, and papers. Just names are different but for example VTI means hexagonal point group. You can check books on seismic anisotropy to start with and see if they help with your question.

Best Regards.

Thank you, Prof. Tarik Chakkour

Well, I will have to check the monography form where I learned elasticity, but meanwhile, I can mention to you: Landau and Lifshitz, Elasticity Theory Vol VII, a new version with more authors are also available, but the first one is enough for the part you mention which is studied in chapter one parts 1 & 2.

They mention two tensors that are fundamental, the deformation tensor uij and the stress tensor σij, which probably refer to the Cauchy stress problem and the conservation of linear momentum in elasticity for what it should be:

div Tx(c) + f(x) = 0 where [T]Cxyz = σij

Notation varies: see also the Wiki commons articles:

https://en.wikipedia.org/wiki/Cauchy_stress_tensor

Best Regards.