A geometric generalization of Hooke's law

Abstract

A geometrical version of Hooke's law for prestressed materials is established using techniques suggested by general relativistic elasticity theory. The concept of dragged (D) tensor fields is used to present a simple coordinate independent form of Hook's law for prestressed materials: P̄ij = D Pij + Cijkl ηkl, where Pij is the initial stress, Cijkl are the second order elastic coefficients, and ηkl is the strain. The effects of initial stresses on Hooke's law have recently been given empirical significance by Wallace and the work here agrees with and simplifies his formalism. The elastic behavior of neutron stars, which requires a general relativistic treatment, is a case in which large prestresses of the type considered here are as important as the elastic modulii themselves. The version of Hooke's law developed here is the classical limit of the general relativistic theory of stellar elasticity. A simple derivation of the seismic response of a Newtonian self-gravitating body to elastic perturbations is presented.