The stress trajectories method for plane plastic problems

This 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.