Wave-number-independent theology in a sphere

Caputo, Michele

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 81 (1987) no. 2, pp. 175-207

Résumé

We determine the displacement vector and the relaxation caused by time varying surface tractions on a spherical Earth model whose anelastic properties are described by a class of stress strain relations representing a wide variety of rheologies including the Maxwell model and the type which approximates polycrystalline halite and granite. It is seen that this class of rheologies is almost wave number independent, and that once a solution is found for a boundary condition whose time variation does not depend on the wave number, then an approximate solution is found for any other boundary condition with the same geometry. We then study the particular case of the Burger solid and see that its relaxation is described by three different exponentials, one of which is rigorously wave number independent, whereas the other two are weakly wave number dependent. We also study the general solution of the equations of elasticity for a spherical Earth model with assigned surface tractions when the stress-strain relations are defined using derivatives of fractional order and find a subclass of models having the same relaxation time; we see that the relaxation time is not sufficient to describe the rheology of a medium but that the relaxation of the medium needs a much more detailed description in the time domain. This rheology causes a splitting of the free modes in a set of very close lines in the frequency domain. We discuss the effect of successive glacial loads on the Earth for this subclass of models and find it is possible to still see today the effect of more than one glaciation and that the effect of the last glaciation may be masked by the quasi-fossil effect of the previous ones. A relation between migration of isotherms and rheology is also considered. We show that the reciprocity theorem of Betti is valid also with the stress strain relations of the generalized Maxwell models. We finally find a general solution of the equations which govern the deformation of an elastic sphere with Maxwell rheology.