The gravito-elastodynamics of a pre-stressed elastic earth

Abstract

SUMMARY The notion that the self-gravitation problem of a perfectly elastic solid body involves small strains from a reference state with arbitrary large initial stresses is reconfirmed and extended. It is shown that the Lagrangian, from which the equations of motion, the boundary conditions and Poisson's equation can be derived in Eulerian or Lagrangian coordinates by the Variational Principle, can be written in a general form which incorporates the various stress measures (Piola-Kirchhoff, Cauchy). Especially for an Eulerian description it is shown that the derived equations of motion and boundary conditions lead to a complete set of mutually orthogonal seismic normal modes. The Lagrangian which Geller (1988) gives is re-evaluated. It appears that in his considerations Geller neglected finite pre-stresses and only accounted for gravitational contributions. This, in general, is not correct. Though the static equilibrium equation has no unique solution for the initial stress components, this does not imply that in physical situations there would not be a specified initial pre-stress. Geller's statement that the asymmetric stress tensor Woodhouse & Dahlen (1978) employ in their Lagrangian necessarily leads to non-conservation of angular momentum is not valid. This stress tensor is a first Piola-Kirchhoff stress tensor, which is a so-called two-point tensor, associating two vector fields defined in different coordinate systems.