Fractional viscoelastic solution of stratum displacement of a shallow tunnel under the surface slope condition

Abstract

The unified displacement function (UDF) is presented to describe the deformation behaviours of the tunnel profile along with time under the surface slope condition. Based on the discrete Fourier method, the third-order UDF in the physical plane is expanded to the Laurent series in the complex variable plane. The complex variable method is employed to derive the elastic analytical solution of stratum displacement, when the third-order UDF is taken as the displacement boundary condition of tunnel cross-section (DBCTC). The proposed elastic solution agrees well with the results of the finite element method for the consistent model, which verifies the correctness of the proposed analytical solution. Combining the corresponding principle and fractional Generalized Kelvin viscoelastic constitutive model, the fractional viscoelastic solution under the surface slope condition is determined. The time effect of stratum displacement is presented in two aspects: time-dependent DBCTC and time-dependent material parameters. The parameter analysis is performed to investigate influences of deformation modes of the third-order UDF, slope angle, tunnel radius and fractional order on the time effect of stratum vertical and horizontal displacement