Modelling long-term deformation of granular soils incorporating the concept of fractional calculus

Sun, Yifei; Xiao, Yang; Zheng, Changjie; Hanif, Khairul Fikry

doi:10.1007/s10409-015-0490-x

Cited by 14 publications

(7 citation statements)

References 46 publications

(84 reference statements)

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“…However, the physical meanings of these parameters are currently unknown and need further investigation.

\overset{ε}{true}

denotes the strain rate with respect to N , which can be considered as scale invariant and obeys the power law in relation to the number of load cycles . Similar observations can be also found in Figure where the evolutions of the shear and volumetric strain rates of ballast and subballast with the number of load cycles are reported.…”

Section: Fractional Strain Ratesupporting

confidence: 76%

“…In this study, two common definitions, known as the Riemann–Liouville fractional derivative and integral , are used. The Riemann–Liouville fractional derivative of function z ( x ) can be formulated as

{}_{0}{D_{x}^{α}} z ((), x) = \frac{d^{α} z ((), x)}{d x^{α}} = \frac{d}{d x} \frac{1}{Γ ((), 1 - α)} true {true\int}_{0}^{x} \frac{z ((), τ) d τ}{{((), x - τ)}^{α}}, x > 0

where D means derivation; α is the fractional order, ranging from 0 to 1. x denotes the independent variable and can be regarded as the loading time in static test or the loading cycles in cyclic test , for example, the creep test that is usually carried out by controlling the time t and the cyclic triaxial test that is usually carried out by controlling the number of load cycles N . The gamma function Γ(•) is defined as

Γ ((), x) = true {true\int}_{0}^{\infty} e^{- τ} τ^{x - 1} d τ

Γ ((), x + 1) = x Γ ((), x)

…”

Section: Notations and Definitionsmentioning

confidence: 99%

“…It is just treated as an empirical parameter that obtained from curve fitting. However, it was suggested that the fractional order did have some connections with the fractal dimension of the material . It is because of the fractal nature of granular soils that the fractional order strain rate could be better than the traditional integer order strain rate in constitutive modelling.…”

Section: Fractional Strain Ratementioning

confidence: 99%

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Fractional order model for granular soils under drained cyclic loading

Sun¹,

Xiao

2016

Int. J. Numer. Anal. Meth. Geomech.

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The theory of fractional calculus has been successfully applied to model the triaxial behaviour of soils under static loading conditions. However, limited work has been carried out in using the fractional calculus to describe the cyclic behaviour of granular soils. In this paper, a fractional order constitutive model for granular soils under drained cyclic loading is proposed by incorporating the concept of fractional rate for strain accumulation. The fractional rate for strain accumulation is obtained from the analysis of the experimental data by utilizing the fractional calculus. Comparison between the test results and model predictions is presented. The key feature of the proposed model is that it can reasonably characterise the cyclic deformation of granular soils under both low and high loading cycles.

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“…However, the physical meanings of these parameters are currently unknown and need further investigation.

\overset{ε}{true}

Section: Fractional Strain Ratesupporting

confidence: 76%

{}_{0}{D_{x}^{α}} z ((), x) = \frac{d^{α} z ((), x)}{d x^{α}} = \frac{d}{d x} \frac{1}{Γ ((), 1 - α)} true {true\int}_{0}^{x} \frac{z ((), τ) d τ}{{((), x - τ)}^{α}}, x > 0

Γ ((), x) = true {true\int}_{0}^{\infty} e^{- τ} τ^{x - 1} d τ

Γ ((), x + 1) = x Γ ((), x)

…”

Section: Notations and Definitionsmentioning

confidence: 99%

Section: Fractional Strain Ratementioning

confidence: 99%

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Fractional order model for granular soils under drained cyclic loading

Sun¹,

Xiao

2016

Int. J. Numer. Anal. Meth. Geomech.

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“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14] The main idea of using the fractional differentiation lies in the property of the fractional-order derivatives to describe memory effects. In the last 30 years, this subject has been extended in various directions such as fluid dynamics, tribology, electrochemistry, vibrations, finance, the design of optimal control systems, diffusion, geophysics, thermoelectricity, reaction-diffusion equations, signal processing, among others.…”

Section: Introductionmentioning

confidence: 99%

“…In the last 30 years, this subject has been extended in various directions such as fluid dynamics, tribology, electrochemistry, vibrations, finance, the design of optimal control systems, diffusion, geophysics, thermoelectricity, reaction-diffusion equations, signal processing, among others. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] The main idea of using the fractional differentiation lies in the property of the fractional-order derivatives to describe memory effects. Derivatives and integrals of fractional-order consider the system memory, hereditary properties, and nonlocal distributed effects; these effects are essential for describing real-world problems.…”

Section: Introductionmentioning

confidence: 99%

Fractional operator without singular kernel: Applications to linear electrical circuits

Morales‐Delgado

Gómez-Aguilar

Taneco-Hernández

et al. 2018

Circuit Theory & Apps

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This paper presents the analytical solutions of fractional linear electrical systems by using the Caputo-Fabrizio fractional-order operator in Liouville-Caputo sense. This novel operator involves an exponential kernel without singularities. The fractional equations were solved analytically by using the properties of Laplace transform operator, as well as the convolution theorem. To validate the analytical solutions, numerical simulations were carried out. KEYWORDSCaputo-Fabrizio fractional operator, exponential decay law kernel, fractional linear electrical circuits, Laplace transform 2394

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