Rheologies quasi wave number independent in a sphere and splitting the spectral line

Caputo, Michele

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 3, pp. 507-526

Résumé

The solution of the equations which govern the slow motions (for which the inertia forces are negligible) in an elastic sphere is studied for a great variety of rheological models and surface tractions with rotational symmetry (Caputo 1984a). The solution is expressed in terms of spherical harmonics and it is shown that its time dependent component is dependent on the order of the harmonic. The dependence of the time component of the solution on the order of the harmonic number is studied. The problem of causality is then discussed showing that the rheological models defined by strees-strain relations of the generalized Maxwell type (Caputo 1984b), which contain derivatives of real order, are causal. It is also seen that the rheological model based on stress strain relations of the generalized Maxwell type multiplies the number of spectral lines of the free modes of a spherical shell. The same applies also to the rheologies of Voigt, Maxwell and of the standard linear solid.