About the influence of pre-stress upon adiabatic perturbations of the Earth

Summary. In this paper we examine the influence of the state of stress in the equilibrium configuration of the Earth (i.e. the pre-stress) upon its adiabatic perturbations. The equations governing these perturbations to the first order (Woodhouse & Dahlen; Dahlen) are re-derived using a Lagrangian approach. Different expressions of the sesquilinear form associated to the elastic-gravitational operator are given. One of these provides a way to extend to hydrostatically pre-stressed solids the criterion of local stability given by Friedman & Schutz for uniformly rotating fluids. Then the propagation in the Earth of seismic wavefronts is considered. It is shown that the nature of these different wavefronts is entirely determined by the quadratic coefficients of the development of the specific internal energy variation, corresponding to isentropic evolution, with respect to the Lagrangian finite deformation tensor. Expressions for the velocities of the various waves are given as functions of incidence angle and pre-stress for orthotropic elastic material. In the particular case where the elastic parameters depend only on one coordinate of a curvilinear system and the axis of orthotropy of the material coincides with the corresponding natural base vector, the elastodynamic equations are reduced to a simple system for a displacement stress vector, using surface operators. In particular for spherical geometry, equations are obtained which generalize to orthotropic pre-stress those given by Alterman et al. and Takeuchi & Saito.