Analytical Solutions for the Construction of Deeply Buried Circular Tunnels with Two Liners in Rheological Rock

The construction of underground tunnels is a time-dependent process. The states of stress and strain in the ground vary with time due to the construction process. Stress and strain variations are heavily dependent on the rheological behavior of the hosting rock mass. In this paper, analytical closed-form solutions are developed for the excavation of a circular tunnel supported by the construction of two elastic liners in a viscoelastic surrounding rock under a hydrostatic stress field. In the solutions, the stiffness and installation times of the liners are accounted for. To simulate realistically the process of tunnel excavation, a time-dependent excavation process is considered in the development of the solutions, assuming that the radius of the tunnel grows from zero until its final value according to a time-dependent function to be specified by the designers. The integral equations for the supporting pressures between rock and first liner are derived according to the boundary conditions for linear viscoelastic rocks (unified model). Then, explicit analytical expressions are obtained by considering either the Maxwell or the Boltzmann viscoelastic model for the rheology of the rock mass. Applications of the obtained solutions are illustrated using two examples, where the response in terms of displacements and stresses caused by various combinations of excavation rate, first and second liner installation times, and the rheological properties of the rock is illustrated.