Response of a tunnel deeply embedded in a viscoelastic medium

Abstract

This paper presents a three-dimensional energy-based solution for the time-dependent response of a deeply embedded and unsupported semi-infinite tunnel of circular cross-section. The tunnel is taken to be excavated quasi-instantaneously from an infinite rock body that initially exhibits an isotropic stress state and that is made up of a homogeneous, isotropic and viscoelastic material. The viscoelastic behaviour is modelled by means of Burger's model, and the rock is taken to behave volumetrically linear elastic and to exhibit exclusively deviatoric creep. This viscoelastic problem is transformed into the Laplace domain, where it represents a quasi-elastic problem. The displacement fields in the new solution are taken to be the products of independent functions that vary in the radial and longitudinal directions. The differential equations governing the displacements of the system and appropriate boundary conditions are obtained using the principle of minimum potential energy. The solutions for these governing equations in the Laplace domain are then obtained analytically and numerically using a one-dimensional finite difference technique. The results are then transformed back into the time domain using an efficient numerical scheme. The accuracy of the new solution is comparable with that of a finite element analysis but requires much less computation effort.