A finite strain solution for strain-softening rock mass around circular roadways

Abstract This paper presents a semi-analytical finite strain solution for circular roadways excavated in a strain-softening rock mass with a non-associated flow rule. The evolutions of both strength parameters and Young’s modulus are considered, and the finite strains are applied to describe both the elastic and plastic behaviors with the logarithmic strain in Lagrangian coordinates. By transforming the derivative of displacement with a new variable that is correlated with displacement, the stress and displacement in the elastic region are obtained in the form of integral equations. Considering the assumption of constant material parameters within a very small region, the plastic region is divided into a number of concentric annuli, whose closed-form solutions of displacement and stress can be recursively obtained. The solution is validated by similarity solutions and numerical simulation. Both the roadway wall displacement and thickness of the plastic (residual) region appear to be smaller than those of the small strain solution. For the elasto-perfectly-plastic rock mass, the finite strain solution is approximately the same as that of small strain solution. However, for the strain-softening rock mass, with the decrease of Young’s modulus, the thickness of plastic and residual regions shows the first increase and then decrease trend, finally approaching nil; the difference between the thickness of softening and residual region gradually decreases. The decreasing Young’s modulus deterioration coefficient and increasing dilation angle significantly produce the displacement. Both the displacement and the thickness of plastic and residual regions decrease with the increase of Poisson’s ratio. When Young’s modulus is large, the tangential stress gradually decreases with the increasing radius. When Young’s modulus is small, the tangential stress firstly increases to a maximum value, then decreases gradually. Different from the small strain solutions, the roadway wall displacement of finite strain solutions cannot be larger than the excavation radius, which is in compliance with the practical underground engineering condition. The predicted displacement and thickness in both plastic and residual regions are in accordance with the field test results.