Abstract
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.
Highlights
IntroductionAs a branch of mathematics, the fractional calculus deals with the generalization of integrals and derivatives to all real (and even complex) orders [1,2]
As a branch of mathematics, the fractional calculus deals with the generalization of integrals and derivatives to all real orders [1,2]
In order to capture the rheological properties of the soils, the Caputo-Fabrizio operator is adopted to develop a novel four-element fractional-derivative model, and one-dimensional consolidation model is establish to simulate the complex consolidation process of the viscoelastic soils
Summary
As a branch of mathematics, the fractional calculus deals with the generalization of integrals and derivatives to all real (and even complex) orders [1,2]. Afterwards, in order to capture the rheological properties of the soil, some viscoelastic models including Maxwell model, Kelvin-Voigt model, Merchant model, four-element model and generalized Kelvin-Voigt model have been introduced to extend the one-dimensional consolidation model of the soil [13,18,19,20,21]. The concept of fractional calculus has been introduced to develop the viscoelastic model for describing the rheological phenomena of materials. To solve the singularity of the kernel of power function, Caputo and Fabrizio recently put forward a new fractional derivative using the kernel of exponential function [27,28]. In order to capture the rheological properties of the soils, the Caputo-Fabrizio operator is adopted to develop a novel four-element fractional-derivative model, and one-dimensional consolidation model is establish to simulate the complex consolidation process of the viscoelastic soils. A comparison for several special cases is presented to verify the correctness of the proposed solutions, and two examples are given to investigate the consolidation behavior of the viscoelastic soils
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