A lumped model for a seismic source

Abstract

A lumped mechanical model is proposed and solved. This model is dynamically equivalent, in the mean field approximation, to a faulted lithosphere. The transition from the continuous system (modelled according to W. H. Prescott and A. Nur) to the lumped one, has been made by preserving all the relevant features exhibited by the continuous system. In particular the coupling between different components of the stress and strain tensors is suitably taken into account. The dynamics of the lumped system depends on six control parameters fixed by the physical properties of the continuous system. Three of the control parameters are dimensionless and describe: the seismic wave quality factor of the medium ( Q ), a parameter related to the vertical change of the medium rigidity ( K ) and one related to the kinematics at the bottom boundary of the seismogenic layer ( β ). The others are forces (in the lumped model) and correspond to the static ( Ф S ) and dynamic ( Ф D ) friction stresses at the fault plane, and to the plastic yield stress of the medium ( Ф P ). The dynamics of the lumped model has an analytical solution, so it is possible to investigate the physical properties exhibited by the model as it evolves. The main results are: (i) the system exhibits a seismic or an aseismic steady state solution, depending on the control parameter values; (ii) the type of the steady state solution which occurs depends on bulk properties of the medium as well as on local (fault) properties; (iii) in the case where the steady state solution is seismic, the dependence of the characteristic quantities of the seismic cycle on the control parameters is obtained.