In this paper, the Caputo-Fabrizio fractional derivative is introduced to investigate the one-dimensional consolidation behavior of viscoelastic soils. Using the Caputo-Fabrizio operator, a novel four-element fractional-derivative model is proposed to capture the viscoelastic properties of the soils, and further the one-dimensional consolidation equation is derived to simulate the consolidation behavior of the soils. Using the techniques of eigenfunction expansion and Laplace transform, a series of analytical solutions are derived to calculate the excess pore-water pressure and the average degree of consolidation of the soils. The total vertical stress in the soil is assumed to change linearly with depth, and its distribution patterns are classified to rectangular pattern, trapezoidal pattern and inverse trapezoidal pattern. Four loading types including instantaneous loading, ramp loading, sinusoidal loading and general cyclic loading are considered. Then, a comparison for several special cases is presented to verify the correctness of the proposed solutions through comparing with existing theories. Moreover, two examples considering ramp and sinusoidal loadings are given to study the consolidation behavior of the viscoelastic soils incorporating the Caputo-Fabrizio fractional derivative.

The safety of existing tunnels during the entire under-crossing process of a new shield tunnel is critically important for ensuring the sustainable operation of urban transportation infrastructure. The nonlinear behavior of surrounding soils plays a significant role in the mechanical response of tunnel structures. In order to assess the mechanical response of the existing tunnel more reasonably, this study attempts to propose a novel theoretical solution and calculation method by simultaneously considering the nonlinear characteristics of surrounding soils and the tunneling effects of a new tunnel during its entire under-crossing process. Firstly, the additional stresses acting on the existing tunnel stemming from the tunneling effects of a new shield tunnel during different under-crossing stages are calculated using the typical Mindlin solution, as well as the Loganathan and Poulos solutions. The influences of the additional thrust, friction force, and grouting pressure and the loss of surrounding soils are taken into account. Then, the nonlinear Pasternak foundation model is introduced to characterize the behavior of surrounding soils, and the governing differential equation for the mechanical response of the existing tunnel is derived using the typical Euler–Bernoulli beam model. Subsequently, a novel theoretical solution and calculation approach are established using the finite difference formula and the Newton iteration method for assessing the mechanical response of the existing tunnel. Finally, one case study is performed to illustrate the mechanical behavior of the existing tunnel during the whole under-crossing process of a new shield tunnel, and the validity of the developed solution is verified against both the computed result of finite element simulation and the field measurements. In addition, the influences from the ultimate resistance and reaction coefficient of surrounding soils and those from the vertical distance and intersection angle between existing and newly constructed tunnels are analyzed and discussed in detail.