In geotechnical engineering, the consolidation of unsaturated soil is a common issue of great interest. Considering the multi-layered property and impeded drainage boundary condition of the soil stratum in real engineering, this study aimed to develop a general semi-analytical solution for assessing the one-dimensional (1D) consolidation behavior of multi-layered unsaturated soil that is subjected to a general impeded drainage boundary condition and a time-dependent loading. To achieve the final solution, the proposed consolidation system is firstly decoupled and solved in the Laplace domain. Then, the semi-analytical solutions for the excess pore-air pressure and excess pore-water pressures as well as the soil settlement are formulated. The Crump method is employed to provide their final results in the time domain. The correctness of the derived solutions was verified against the available analytical and numerical solutions, and excellent agreements were found for the two comparisons. Moreover, two studied examples are presented to illustrate the 1D consolidation behavior of multi-layered unsaturated soil and the influences stemming from the impeded drainage parameters are discussed.
The consolidation of soil is one of the most common phenomena in geotechnical engineering. Previous studies for the axisymmetric consolidation of unsaturated soil have usually idealized the boundary conditions as fully drained and absolutely undrained, but the boundaries of unsaturated soil are actually impeded drainage in most practical situations. In this study, we present a general analytical solution for predicting the axisymmetric consolidation behavior of unsaturated soil that incorporates impeded drainage boundary conditions in both the radial and vertical directions simultaneously. The impeded drainage boundary is modeled using the third kind boundary, and it can also realize fully drained and absolutely undrained ones by changing the drainage parameter. A general analytical solution is developed to predict the excess pore-air and pore-water pressures as well as the average degree of consolidation in an unsaturated soil stratum using the common methods of eigenfunction expansion and Laplace transform. The newly developed solution is expressed in the product of the terms of time, depth, and radius, which are derived using Laplace transform, usual Fourier, and Fourier-Bessel series, respectively. The eigenfunctions and eigenvalues are evaluated from the impeded drainage boundaries in both radial and depth dimensions. Then, the correctness of the proposed analytical solution is verified against the existing analytical solution for the case of traditional boundaries and against the finite difference solution for the case of general impeded drainage boundaries, and excellent agreements are obtained. Finally, the axisymmetric consolidation behavior of unsaturated soil involving impeded drainage boundaries is demonstrated and analyzed, and the effects of the drainage parameters are discussed. The results indicate that the larger drainage parameter generally expedites the dissipations of the excess pore pressures and further promotes the soil settling process. As the drainage parameter increases, its influence gradually diminishes and even can be neglected when it is larger than 100. The general analytical solution and findings of this study can help for better understanding the axisymmetric consolidation behavior of the unsaturated soil stratum in the ground improvement project with vertical drains as well as the gas-oil gravity drainage mechanism in the naturally fractured reservoirs.
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.
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