Abstract: The tunnel excavation and support are continuous processes. A time-varying function of radius is established to simulate the excavation process of a circular tunnel. The general solutions of stresses and displacements of viscoelastic rock mass with elastic support during construction are derived by the Laplace transformation method, which contains the undetermined supporting force. An integral equation for the supporting force is established based on the contact conditions between the rock mass and the support. By means of Boltzmann viscoelastic model, the supporting force can be calculated exactly. The example shows that the displacement is larger when the longitudinal excavation velocity is higher. Besides, the effect of the longitudinal excavation is more clear when the cross-section is excavated faster. If the final tunnel is in the same size and supported immediately at the finishing time but excavated with different velocities, the displacement of cases with high velocity is larger at the beginning and smaller after some time. The final steady displacement is also smaller when excavated faster, but the displacement occurring after supporting is larger. The solutions can be employed to calculate the displacements and stresses of arbitrary time-varying radius. The proposed method is suitable for the analysis of other viscoelastic models.