A Caputo variable-order fractional damage creep model for sandstone considering effect of relaxation time

Liu, Xiaolin; Li, Dejian; Han, Chao; Shao, Yiming

doi:10.1007/s11440-021-01230-9

Cited by 32 publications

(16 citation statements)

References 31 publications

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“…In mathematical analysis, a fractional derivative is a derivative of any arbitrary order, real or complex, which first appeared in the 1690s 49 . For the reason of simple mathematical form, 50,51 the definition of fractional derivative from Caputo 52 and Caputo and Mainardi 53 was presented in this study. At the interval

t\epsilon [ {a,b} ]

, the left and right Caputo fractional derivatives of a function of order

\alpha (\alpha &gt; 0)

, denoted by

{}_a^CD_t^\alpha f( t )

and

{}_t^CD_b^\alpha f( t )

, respectively, are

\begin{equation} a^CD_t^\alpha f\left( t \right) = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} \displaystyle{\frac{1}{{{{\Gamma}}\left( {n - \alpha } \right)}}\mathop \int_a^t \displaystyle\frac{{{f^{\left( n \right)}}\left( \tau \right)}}{{{{\left( {t - \tau } \right)}^{\alpha + 1 - n}}}}d\tau , n - 1 &lt; \alpha &lt; n}\\[18pt] \displaystyle{\frac{{{d^n}}}{{d{t^n}}}f\left( t \right), \alpha = n} \end{array} \right.

…”

Section: Review Of Pd Theory and Fractional Derivativementioning

confidence: 99%

“…In mathematical analysis, a fractional derivative is a derivative of any arbitrary order, real or complex, which first appeared in the 1690s. 49 For the reason of simple mathematical form, 50,51 the definition of fractional derivative from Caputo 52 and Caputo and Mainardi 53 was presented in this study. At the interval 𝑡𝜖[𝑎, 𝑏], the left and right Caputo fractional derivatives of a function of order 𝛼(𝛼 > 0), denoted by 𝐶 𝑎 𝐷 𝛼 𝑡 𝑓(𝑡) and 𝐶 𝑡 𝐷 𝛼 𝑏 𝑓(𝑡), respectively, are…”

Section: Definition Of Fractional Derivativementioning

confidence: 99%

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The peridynamic Drucker‐Prager plastic model with fractional order derivative for the numerical simulation of tunnel excavation

Zhang

Zhou

2022

Num Anal Meth Geomechanics

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In this study, the peridynamic Drucker-Prager plastic model with fractional order derivative is proposed to investigate the plastic behavior of surrounding rocks around tunnels, in which the Caputo fractional derivative is employed due to its mathematical simplicity. Instead of utilizing just one parameter for the typical nonassociated flow rule, such as dilation angle, multiple parameters, such as fractional order and interval of fractional derivative, are used to specify the direction of plastic deformation, and the fractional derivative based-typical flow rule is proposed. As a result, compared with the traditional peridynamic Drucker-Prager plastic model, the proposed method is a more generalized model including the typical nonassociated flow rule. Besides, based on the PD force density, the strategy of exertion of initial stress is proposed to simulate in-situ stress in rocks. Taking a block subjected to compression as an example, the impacts of various factors, such as fractional order and interval of fractional derivative, are investigated. A numerical simulation of tunnel excavation in rocks is carried out and the numerical results obtained by the proposed method are verified by comparing with FEM results.

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t\epsilon [ {a,b} ]

, the left and right Caputo fractional derivatives of a function of order

\alpha (\alpha &gt; 0)

, denoted by

{}_a^CD_t^\alpha f( t )

and

{}_t^CD_b^\alpha f( t )

, respectively, are

\begin{equation} a^CD_t^\alpha f\left( t \right) = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} \displaystyle{\frac{1}{{{{\Gamma}}\left( {n - \alpha } \right)}}\mathop \int_a^t \displaystyle\frac{{{f^{\left( n \right)}}\left( \tau \right)}}{{{{\left( {t - \tau } \right)}^{\alpha + 1 - n}}}}d\tau , n - 1 &lt; \alpha &lt; n}\\[18pt] \displaystyle{\frac{{{d^n}}}{{d{t^n}}}f\left( t \right), \alpha = n} \end{array} \right.

…”

Section: Review Of Pd Theory and Fractional Derivativementioning

confidence: 99%

Section: Definition Of Fractional Derivativementioning

confidence: 99%

The peridynamic Drucker‐Prager plastic model with fractional order derivative for the numerical simulation of tunnel excavation

Zhang

Zhou

2022

Num Anal Meth Geomechanics

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show abstract

“…Considering that the material will be damaged in the process of stress relaxation, and the material parameters will change with time, it is advisable to define the deterioration law 16,17 as…”

Section: The Fractional Order Zener Model For the Frrsmentioning

confidence: 99%

Constitutive modeling of stress relaxation for fabric reinforced rubber composites

Wang

Mao

2023

Journal of Composite Materials

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Establishing an unsteady fractional order Zener model with degradation parameters has significant meaning for predicting stress relaxation of fabric reinforced rubber composites. Based on the experimental results, an integer order Zener model was deduced to demonstrate the stress relaxation behaviors of fabric reinforced rubber composites and pure rubber. Due to the limitations, however, the integer order Zener model could not explain the stress relaxation behavior of the fabric reinforced rubber composites (the FRR) very well. Therefore, a Zener model was developed based on the viscoelasticity theory and the fractional calculus theory to characterize the stress relaxation behavior, but errors still existed in the initial and final stages. With more reasonable physical interpretation, an unsteady fractional order Zener model degraded by parametric aging with 7 parameters is proposed in this paper to accurately capture the stress relaxation behavior of the FRR, whose correlation coefficient is over 99.99%. The parameter sensitivities of the seven-parameter model were analyzed. Present study may provide a critical step for designing and evaluating the fabric reinforced rubber composites.

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“…As a direct expansion of the fractional derivative, the variable-order fractional derivative is recognized as a powerful tool for modeling complex physical phenomena . Significant advances have been made by using variable-order fractional derivatives in developing the constitutive models of materials , and creep models of rocks. ,,− …”

Section: Introductionmentioning

confidence: 99%

Modeling Approaches to Permeability of Coal Based on a Variable-Order Fractional Derivative

et al. 2023

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In the past decades, many permeability models have been proposed to characterize the coal permeability evolution in the elastic state. Considering that the coal near the mining working face is often in the plastic or post-peak failure state, it is significant to characterize the coal permeability evolution in the plastic and post-peak failure stages for safe mining. In this study, two permeability models are developed using a variable-order fractional derivative to characterize the coal permeability evolution during the whole process of elastic, plastic, and post-peak failure stages. The results indicate that both models have the ability to better describe the coal permeability evolution during the whole process in areas near the mining working face. In addition, the physical and mechanical interpretation of the variable-order function is given as an indicator of fracture development, indicating the sensitivity of stress variation to coal fracture development. Moreover, the relationships among the effective stress, damage variable, and post-peak permeability of coal are discussed.

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A Caputo variable-order fractional damage creep model for sandstone considering effect of relaxation time

Cited by 32 publications

References 31 publications

The peridynamic Drucker‐Prager plastic model with fractional order derivative for the numerical simulation of tunnel excavation

The peridynamic Drucker‐Prager plastic model with fractional order derivative for the numerical simulation of tunnel excavation

Constitutive modeling of stress relaxation for fabric reinforced rubber composites

Modeling Approaches to Permeability of Coal Based on a Variable-Order Fractional Derivative

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