Removal of singularities in Hoek-Brown criterion and its numerical implementation and applications

The representation of the Hoek-Brown criterion is an irregular curved hexagonal pyramid in the principal stress space, which leads to the occurrence of numerical singularities on the edges of the pyramid. With the aim of achieving a physical approximation to the pyramid, this study used both the C1 and C2 smoothing artifices on its sharp edges, and it was found that they were able to successfully eliminate the singularity and ensure the convexity of the yield surface. Meanwhile, to reflect the characteristic of the poor tensile strength of a rock mass, a tension cutoff surface was employed to form the whole modified combined yield surface. To facilitate comprehension and programming, the initial Hoek-Brown criterion and the smoothing and tension cutoff yield functions were all expressed in terms of stress invariants. The fully implicit backward Euler integral regression algorithm was employed to form the consistent stiffness matrix to ensure the high accuracy and fast convergence of numerical computations. In accordance with the failure zone in which a trial stress may fall, it may be pulled back to the initial Hoek-Brown yield surface, the transitional rounding yield surface, the tension cutoff yield surface, or the vertices that are the intersections of the former two with the latter. Furthermore, to facilitate the application of this modified Hoek-Brown criterion, a three-dimensional (3D) user-defined material behavior subroutine was developed in a finite-element program, and its reliability and applicability were verified through the numerical simulations of the triaxial compression and uniaxial tension tests of a rock and the excavation of a rock tunnel.