Numerical analysis of localization phenomena with application in deep boreholes

As boreholes are drilled deeper into the Earth's crust, increasing problems are encountered with failure of the borehole wall (breakout) due to the existing high stresses at great depth. In this thesis we have investigated this failure process numerically in a large scale finite element computation, carried out on the supercomputer Cray-2.;In the first part of this thesis rock is described by the equations of deformation theory of plasticity for a rigid-plastic, cohesive-frictional, incompressible material. The dependency of the borehole stability on the radius of the borehole (scale effect) was examined by extending the constitutive equations to continuum model with microstructure. These constitutive equations have been successfully applied for the case of hydrostatic far-field stress where the derived eigenvalue problem was solved semi-analytically, based on conventional solution methods and traditional perturbation methods. The detected possible bifurcation of the solution and the existence of the scale effect pursued the research further to post bifurcation analysis for more realistic constitutive modeling and loading conditions.;In the second part, rock is modelled by the elastoplastic constitutive equations of a flow theory of plasticity for cohesive-frictional, hardening-softening, dilatant material. These constitutive equations were fitted on true stress-strain data from triaxial compression tests on rock specimens. For the numerical solution of equations, we have developed a non-linear finite element code for a polar (Cosserat) continuum. The ill-posed boundary value problem of borehole stability in strain softening rock, was regularized since the Cosserat-continuum introduces a length scale into the problem and it numerically assures convergency of the elastoplastic code in the softening regime. Continuation methods with a systematic computer-aided analysis are used to investigate how solutions, their existence and their uniqueness change as geostatic stresses change. Results are presented for the case of hydrostatic as well non-hydrostatic far-stress fields. The present analysis enables us to model what is usually called a 'progressive' failure mechanism. Also due to the existence of an internal length in the constitutive model, 'small' holes fail at higher external stresses than 'large' holes. This 'scale' effect and the computed failure modes are in a very good agreement with the experiment.