The stress trajectories method for plane plastic problems

Abstract

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.

Plastic stress field reconstruction based on stress orientations data

“…First order stress indicators of the World Stress Map (WSM) project database [Heidbach et al, 2008] are utilized to evaluate the magnitudes of intraplate stresses. The numerical method is based on the stress trajectory concept and presents further development ES4001 haderka and galybin: plastic stress field reconstruction ES4001 of the approaches reported by Galybin and Mukhamediev [2004] for elastic solids and by Haderka and Galybin [2011] for plastic media. Methods widely accepted for stress modeling are based on finite element analyses in view of gravitational potential energy (GPE) predictions [Bird, 1989[Bird, , 1999England and Molnar, 1997;Humphreys and Coblentz, 2007].…”

“…It is explained by the fact that the classical formulations assume specification of boundary stresses while the majority of data are given on the stress orientations, therefore restricting the straightforward application of the conventional techniques. In the approach proposed by Haderka and Galybin [2011] the orientations of principal stresses are considered as one of the possible boundary conditions for solving the BVP of plane plasticity. This opened a possibility to model non-elastic stress fields without knowing reliable information on the boundary stress magnitudes.…”

“…In other words, there exist regions of the reconstructed stress trajectories grids located outside of the domain of dependence. The implications of this fact are discussed in great detail in Haderka and Galybin [2011]. This approach is abbreviated as ST (stress trajectories) further on.…”

“…We further use both approaches in turn and introduce the ST-SL alternations method (for detail see [Haderka and Galybin, 2011]). It is shown how slip lines and stress trajectories built by the respective approaches are utilized for further extension of the stress field.…”

“…A method for solution of (9) along the stress trajectories for a general, explicit yield criterion has been developed by Haderka and Galybin [2011]. Here we focus on the MohrCoulomb criterion (3) and the boundary conditions specified in terms of orientations of the principal stress and their normal derivatives ( , ).…”