19th International CODATA Conference
Category: Knowledge Discovery

Mathematical model of continuum with defects in application to geophysical problems

Dr. M.D. Kovalenko (m_kovalenko@trancom.ru), Dr. Yu.L. Rebetsky (y.reb@mail.ru), and Prof. A.D. Gvishiani (gvi@wdcb.ru),
Institute of Physics of the Earth RAS, Russia

Dr. M.Diament (diament@ipgp.jusiieu.fr) and Prof. J.E. Dubois (dubois@ipgp.jusiieu.fr)
Institute de Physique du Globe de Paris, France


Now there is not enough acceptable model of the continuum which takes into account structural defects. Corrspondingly, geophysics should use models and methods which have been developed for studies of metals, that is for homogeneous continuum. However rocks are too different from metals because it has various defects. In this presentation the mathematical model of the continuum which can take into account various defects is offered. At that, it is a question of exact solutions of the elasticity theory. The continuum with defects has interesting properties. In particular, continuum with defects decidedly has self-energy which is not connected to external actions. In such continuum there are a set of conditions with closely related levels of self-energy.  The transition from one condition to another can be connected with very small change of self-energy. It means that such continuum has trigger mechanisms and can collapse spontaneously. The coming into operation of the trigger mechanism can be caused by the various reasons:  external actions, electromagnetic phenomena, phase and chemical transformations and so on. From the mathematical standpoint, condition changes at the point defect appear as a change of solving smoothness at this point. Now we have an opportunity to construct exact solutions for boundary value problems of the dislocation and cracks theory, for boundary value problems of phase transition and some other problems of geophysics.

The special attention is given to point defects, as global changes of the rock energy (for example, earthquakes) actually arise in very small volumes (mathematically - in a point). For example, it is shown that change of the solution smoothness in apex of crack proceeds of its propagation. It results to stress-redistribution, than to change of a boundary conditions on a crack face and at last, to crack propagation. One of the goal tasks of this work is mechanics of a seismic center. Perhaps, it will be possible to connect electromagnetic, chemical and other processes, which precede earthquake, with developed mathematical model. This hope is based on the following reasons. Probably,  each physical process in a point defect is connected with change of the solution smoothness to this defect.