Fractional viscoelastic analytical solution for the ground displacement of a shallow tunnel based on a time-dependent unified displacement function

Abstract

A time-dependent unified displacement function expressed by a Fourier series is used as the displacement boundary condition of the tunnel cross-section (BCTC). Abel element expressed by the fractional differential is adopted to reflect the rheological behaviours of the surrounding rock. Using Laplace transform and the inverse transformation, the relaxation modulus of the Fractional Generalized Kelvin constitutive model for one-dimensional (1D) is obtained. The effects of the fractional order and viscous material parameters on the relaxation modulus are analysed. The correspondence principle and complex variable method are adopted to develop the elastic analytical solution into the fractional viscoelastic analytical solution of the ground displacement. Two factors are considered to capture the time effect of ground deformation: a time-dependent displacement BCTC and time-dependent mechanical parameters. A parameter study is conducted to investigate the time effect of the ground displacement. The deformation parameters of the tunnel cross-section are first calibrated by a part of the field data for the Longdong tunnel and Gaoqiao tunnel. Based on the calibrated model, the proposed method can well predict ground displacement due to the time effect.