The balance of energy in earthquake faulting

Summary An earthquake is modelled kinematically by specifying the tangential slip history on a fault surface which expands within a uniformly rotating, self-gravitating, slightly anelastic earth model. The total amount of energy released by such an idealized earthquake is the sum of three distinct quantities: kinetic energy of rotation, gravitational potential energy and thermodynamic elastic internal energy. The first two of these quantities may also be interpreted as the work done throughout the earth model against the action of the apparent centrifugal and real gravitational body forces respectively. The total energy released by an earthquake fault is in general considerably smaller than any of its three individual constituents, since the work performed against body forces is very nearly balanced by the work performed against the initial hydrostatic pressure in the earth model. The smallest individual constituent is the change in the kinetic energy of rotation of the earth model, which may be as much as two orders of magnitude larger than the total energy released, even though the corresponding change in the angular velocity of rotation due to the redistribution of mass is extremely small. The total energy released by an earthquake fault may also be expressed in terms only of the final static displacement and the initial and final static traction on the fault surface itself. This alternative representation of the energy change is explicitly independent of both the rotation and the self-gravitation of the earth model. All of the energy released by an earthquake fault must be dissipated somewhere within the earth model. Energy may be dissipated during faulting either in heating on the walls of the fault surface, where work must generally be done against the action of the frictional traction acting to resist slip, or at the instantaneously expanding boundary of the fault surface, where some energy may be required to overcome cohesion and where there may be additional heating. The remainder of the energy released, which is generally referred to as the seismic energy, is dissipated both during and subsequent to faulting by the slight bodily friction which must be assumed to exist throughout the entire volume of any physically realizable earth model. The seismic energy may also be expressed in terms only of the displacement and incremental traction histories on the instantaneous fault surface during the course of faulting. This alternative representation of the seismic energy is explicitly independent of both the rotation and the self-gravitation of the earth model, and so therefore is the seismic efficiency, which is defined to be the ratio of the seismic energy to the total energy released. Classical formulae for the total energy released by an earthquake fault, the seismic energy and the seismic efficiency are based not only upon the neglect of rotation and self-gravitation, but also upon the assumption that the initial hydrostatic pressure and deviatoric stress are infinitesimal quantities; those classical formulae, upon which many seismological applications depend, are justified if the initial deviatoric stress at the hypocentre is small compared to the hypocentral rigidity.