Difference solution for a circular tunnel excavated in strain-softening rock mass considering decayed confinement

Recent research regarding the strain-softening behavior of rock masses has demonstrated that the dilatancy angle ψ and the critical softening parameter η∗ are both strongly related to the confining stress. However, in many studies, ψ is assumed to be a constant or to vary linearly, and η∗ is also assumed to be a constant. The assumption of a constant ψ or η∗ is an approximation that does not correctly reflect the variable process in the plastic zone. In this paper, to describe the confinement-dependent characteristics of ψ and η∗ and their effects on the stress and displacement of tunnels in strain-softening rock masses, four dilatancy cases are defined as different combinations of ψ and η∗. First, two dilatancy models, the variable dilatancy model (VDM) and constant dilatancy model (CDM) are introduced. Then, a finite difference method for the strain-softening model is proposed to consider the variation of ψ and η∗ in analyzing the strain-softening behavior of rock masses. The accuracy of this method is verified by comparing the results with those calculated using the methods of Lee and Pietruszczak and Wang et al. Finally, using this proposed method, relevant comparisons are made among the four dilatancy cases for good-quality to poor-quality rock masses to reveal the confinement-dependent effects of ψ and η∗ on the evolution of several parameters, including the softening parameter, the critical softening parameter, the dilatancy coefficient, strength parameters, stress components, ground response curves, and plastic radii. It is concluded that in the plastic zone, ψ and η∗ varies nonlinearly with decreasing confining stress for underground tunnels excavated in different qualities of strain-softening Hoek-Brown rock masses. ψ and η∗ affect each other, and their relationship further influences the transition location from the plastic softening zone to the plastic residual zone. The effects of the confining dependency in ψ and η∗ for good-quality rock masses are less than those for poor quality rock masses. To guarantee simplicity and security in analyzing the stress and displacement of tunnel excavation problems, a constant ψ with a constant η∗ and a variable ψ with a variable η∗ are recommended as analysis models for good-quality and poor-quality rock masses, respectively.