Similarity solution for a spherical or circular opening in elastic-strain softening rock mass

Abstract

This paper deals with the similarity solution for a spherical or circular opening excavated in elastic-strain softening rock mass compatible with a linear Mohr–Coulomb (M–C) or a nonlinear Hoek–Brown (H–B) yield criterion. A similarity solution for stresses and displacement is presented by replacing the partial differential equations from stress equilibrium, constitutive law, and consistency equations with first-order ordinary differential equations. The Runge–Kutta (R–K) method is used to solve those first-order ordinary differential equations. Some measures are discussed to solve numerical instability problems in the use of R–K driver with adaptive steps. For comparison, the simple numerical stepwise procedure for a spherical opening is also presented by modifying the previous procedure for a circular opening. Three data sets are used to show the accuracy and practical application of the proposed methods. The results show the importance in choosing the solver for the system of ordinary differential equations and the initial values in the similarity solution.