a b s t r a c tThis paper presents a numerical method for the determination of the full stress tensor in two-dimensional plastic bodies. The method is developed for the Cauchy boundary value problem and uses the principal directions as one of the boundary conditions. The second condition is formulated in terms of the mean or Tresca stress or via the normal derivative of the principal directions. The latter is important for geophysical applications. The method employs the finite-difference scheme, however, in contrast to the conventional approaches (that build a network of slip lines), it builds a pattern of two orthogonal families of the stress trajectories. As a result, the solution can be found in some areas lying outside the characteristic triangle for the hyperbolic problems. Whereas this solution lies outside the domain of dependence, established by the slip lines, numerical experiments are conducted to establish whether the trajectories field accurately approximates the real stress field. This analysis is further used to introduce the concept of alternations of the solutions based on the slip lines and the stress trajectories, allowing significant extension of the domain where the plastic stress state can be identified.The method is not limited to any specific yield criterion; however it has been verified for the Tresca and Mohr-Coulomb criteria for which solutions obtained by conventional approaches are available. Possible applications for geomechanics problems are reported, in particular, for modelling of regional stresses in the Earth's crust.

This study presents an alternative approach for identification of damaged zones near excavations. The approach is based on ideal plastic solutions but in contrast to the classical case it deals with the Cauchy's problem only by alternating classical solutions for slip zones with solutions for stress trajectories followed by conversion of the latter into slip grids. Comparisons with classical solutions are discussed.

The purpose of this study is to compare different rheological models of tectonic plates for the same set of input data. First, the 2D elastic tectonic stress field in Antarctic plate is modelled by employing experimental data on principal stress orientations. Then the determination of the 2D stress field by assuming the Antarctic lithospheric plate to be in fully plastic state follows. Boundary conditions for the plastic model are specified from the elastic solution. A finitedifference type approach based of alternation of the Cauchy boundary value problems for the construction of the slip lines and the principal stress trajectories is applied to obtain distribution of the mean stress within Antarctic plate. The significance of rheology is underpinned by comparing the elastic and plastic fields of the mean stress.

This paper presents an investigation of the applicability of the stress trajectories concept and the stress trajectories -slip lines alternations method to geomechanical problems. We extend our approach introduced for the stress analysis of two-dimensional plastic bodies to the problem of the stress reconstruction in plastic regions of the lithosphere. The method is developed for the Cauchy boundary value problem and utilizes the data on principal directions as one of the boundary conditions. For this purpose the first order stress indicators of the World stress map (WSM) project database (release 2008) are utilized in computations. The set of considered boundary conditions is supplemented by the normal derivatives of the stress orientations. Complete formulation of the problem involves a yield condition. Although the general approach is not limited to a specific yield criterion, present calculations are performed for the Mohr-Coulomb criterion. Applications of the method include the stress reconstructions in three regions of the Earth's crust (Swiss Alps, Tibetan plateau and Eastern Anatolia). The continuous boundary conditions are derived by an averaging method applied to the discrete data in immediate vicinity of the starting boundary. Thereafter, for the chosen strength parameters of the Mohr-Coulomb theory (friction angle and cohesion), the unique grids of stress trajectories and slip lines are determined. These fields are further compared against the WSM data available inside the regions. The computations are made for different strength parameters in order to provide the best fit to the data. The results of the analysis are presented as two plane fields: the map of normalized mean stresses and the grid of corresponding trajectories of principal directions. The normalization parameter is unknown (it represents an initial value of the mean stress in a single node of the boundary), which is a consequence of non-uniqueness of the stress reconstruction problem based on the data on stress orientations alone. The reconstructed stress orientations are compared with the observations from the WSM database.