Izvestiya, Earth Physics Vol. 27, No. 5, 1991 UDC 551.240 Retrieving Field of Stress Tensor in Crystal Blocks SH. A. MUKHAMEDIYEV Shmidt Institute of Earth Physics, USSR Academy of Sciences The new geomechanical problem of retrieval of the stress field tensor on the basis of data on the trajectories of the principal stress axes is defined and in- vestigated. The properties of the medium need not be reduced to specific form for solution of the problem. It was clarified that the closed system of equations for the problem belongs to systems of the hyperbolic type. The characteristic surfaces coincide either with the surfaces of operation of the principal stresses or with the surfaces formed by the trajectories of one of the principal axes. The types of developing boundary value problems and problems in the practical use of the developed method are examined. The possibilities of formulating and optimiz- ing work on carrying out measurements in shafts and boreholes for the purpose of retrieving and monitoring the stress field tensor in both near-surface and in deep layers of the crust are discussed. The expediency of using additional tectono- physic information is analyzed. 1. Introduction A knowledge of the spatial distribution of the stress tensor in the crust and the evolution of this distribution with time is of enormous im- portance for both theoretical and practical re- search in the earth sciences. Depths of only a few kilometers are accessible for in situ stress measurements [1, 2]. The direct extrapolation of these measurements to deeper horizons is unrelia- ble. Moreover, theoretical estimates of crustal stresses are frequently highly dependent on the used mechanical-mathematical models. The new geomechanical problem of retrieval of the full stress tensor in some volume x °f the crust is formulated and investigated in this arti- cle. The solution is based on information on the orientation of the principal stresses in x axes. This information is drawn, by means of known meth- ods, from data on the focal mechanisms of earth- quakes, from data on special characteristics of striae and friction planes [3-8, and others]. In order to retrieve the modern distribution of the stress tensor it is possible to use data on orien- tation of the axes of paleostresses in the most recent deposits [9]. As in [10], in formulating the problem we in- voke equilibrium conditions. However, this ends the similarity to [ 10], in which, unfortunately, the problem is not fully formulated. The approach set forth below does not require additional infor- mation on the numbering of the principal axes with respect to the magnitude of the stresses and on the distribution of the Lode-Nadai coefficient in x» that is, all that information which expli- citly or implicitly is based on assumptions con- cerning the properties and special features of de- formation of the geophysical medium. (The matter of the expediency of obtaining and using such in- formation is examined below in Section 7.) The article is devoted to a discussion of fundamental approaches to solution of the problem and there- fore satisfaction of the following conditions is assumed: the orientation of the principal stress axes is determined at each point in the x region; the trajectories of the axes and the other func- tions entering into the formulation are quite smooth. In addition, a smallness of deformations and rotations occurring in x from the moment of formation of fault dislocations, from which the principal axes are retrieved, to the moment of ob- servation, is assumed. No assumptions are made concerning the rheological properties of the medi- um, on the nature of the inhomogeneities continu- ously distributed in it and on anisotropy. It has been demonstrated that the used system of equations belongs to systems of the hyperbolic type well studied in the theory of the equations of mathematical physics [11,-12]. The character- istics of the system are the trajectories of the principal stress axes; in a three-dimensional case the characteristic surfaces coincide either with the surfaces of operation of the principal stress- es or with the surfaces formed by the trajectories of one of the principal axes. The system of equa- tions makes it possible to formulate a number of boundary value problems: problem with initial conditions, characteristic problem, mixed problem. The diversity of the types of problems and a know- ledge of the characteristics will make possible 0001-4354/90/2705-0004$18.00/1 370