Fractional calculus derivation of a rate-dependent PPR-based cohesive fracture model: theory, implementation, and numerical results

Giraldo-Londoño, Oliver; Paulino, Gláucio H.; Buttlar, William G.

doi:10.1007/s10704-018-00334-w

Cited by 24 publications

(2 citation statements)

References 69 publications

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“…Fractional derivative is suitable for describing some substantial problems than the regular derivative, and it has been applied widely to establish mathematical models of realistic problems. Fractional calculus has been used to establish the constitutive relations of viscoelastic materials and anomalous diffusions . Sin et al introduced the fractional K‐BKZ model to viscoelastic fluid for the study of unsteady flow between parallel plates, and the fractional Cattaneo–Christov flux model was used by Liu et al to study the diffusion in comb frame.…”

Section: Introductionmentioning

confidence: 99%

Unsteady free convection of second‐grade nanofluid with a new time‐space fractional heat conduction

Shen

Chen

2019

Heat Trans

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The Caputo and Caputo-Fabrizio derivative are applied to study a second-grade nanofluid over a vertical plate. A comparative analysis is presented to study the unsteady free convection of a second-grade nanofluid with a new time-space fractional heat conduction. The governing equations with mixed time-space fractional derivatives are non-dimensionalized and solved numerically, and a comparison between the Caputo and the Caputo-Fabrizio models is made. It is found that the temperature is higher for the Caputo-Fabrizio fractional model than the Caputo model, but the higher velocity only exists near the vertical plate for the Caputo-Fabrizio model than the Caputo model. Moreover, the velocity for the Caputo model will exceed the Caputo-Fabrizio model as y evolves. K E Y W O R D SCaputo fractional derivative, Caputo-Fabrizio fractional derivative, heat transfer, second-grade nanofluid | INTRODUCTIONThe second-grade fluid is a popular non-Newtonian fluid, and has been used in petroleum industry, chemical industry, light industry, and many other sectors. The second-grade fluid model was first configured by Coleman and Noll, 1 and the heat transfer of the second-grade fluid under various physical conditions has been investigated widely. [2][3][4][5][6] Fractional derivative is suitable for describing some substantial problems than the regular derivative, and it has been applied widely to establish mathematical models of realistic problems. Fractional calculus has been used to establish the constitutive relations of viscoelastic materials and anomalous diffusions. 7-11 Sin et al 12 introduced the fractional K-BKZ model to viscoelastic fluid for the study of unsteady flow between parallel plates, and the fractional Cattaneo-Christov flux model was used by Liu et al 13 to study the diffusion in comb frame. As the nanofluid has better heat transfer efficiency than conventional fluids, 14 the heat transfer of nanofluid flow under various physical conditions has been investigated. [15][16][17] Shen et al 18 proposed a renovated Buongiorno's model with fractional differential equation to study the heat transfer characteristics of Sisko nanofluid. It is clear to note that the researchers mostly utilized Caputo and Riemann-Liouville derivatives with singular kernels.The Caputo-Fabrizio derivative with an unsingular kernel was given by Caputo and Fabrizio, 19 which was based on the exponential function to eliminate the singular kernel. Ali et al 20 used this derivative to investigate magnetohydrodynamics free convection flow on a static vertical plate for generalized Walters'-B fluid, and Asjad et al 6 studied the feature of Caputo and Caputo-Fabrizio fractional derivatives for an unsteady second-grade fluid with Newtonian heating.In this paper, a new time-space fractional heat conduction is introduced into the energy equation to investigate the heat transfer of second-grade nanofluids on a vertical plate by using Tiwari and Das's model. 21 The Caputo and Caputo-Fabrizio derivatives are applied to get numerical solutions of the d...

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Section: Introductionmentioning

confidence: 99%

Unsteady free convection of second‐grade nanofluid with a new time‐space fractional heat conduction

Shen

Chen

2019

Heat Trans

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“…Tang et al [55] developed a variable order rock creep model, with damage evolution as an exponential function of time. Recently, Giraldo-Londoño et al [22] developed a two-parameter, two-dimensional (2-D) rate-dependent cohesive fracture model.…”

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A Thermodynamically Consistent Fractional Visco-Elasto-Plastic Model with Memory-Dependent Damage for Anomalous Materials

Suzuki¹,

Zhou²,

D’Elia³

et al. 2019

Preprint

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We develop a thermodynamically consistent, fractional visco-elasto-plastic model coupled with damage for anomalous materials. The model utilizes Scott-Blair rheological elements for both visco-elastic/plastic parts. The constitutive equations are obtained through Helmholtz free-energy potentials for Scott-Blair elements, together with a memory-dependent fractional yield function and dissipation inequalities. A memory-dependent Lemaitre-type damage is introduced through fractional damage energy release rates. For time-fractional integration of the resulting nonlinear system of equations, we develop a first-order semi-implicit fractional return-mapping algorithm. We also develop a finite-difference discretization for the fractional damage energy release rate, which results into Hankel-type matrix-vector operations for each time-step, allowing us to reduce the computational complexity from O(N 3 ) to O(N 2 ) through the use of Fast Fourier Transforms. Our numerical results demonstrate that the fractional orders for visco-elasto-plasticity play a crucial role in damage evolution, due to the competition between the anomalous plastic slip and bulk damage energy release rates.

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Role of bi-order Atangana–Aguilar fractional differentiation on Drude model: an analytic study for distinct sources

2021

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Fractional calculus derivation of a rate-dependent PPR-based cohesive fracture model: theory, implementation, and numerical results

Cited by 24 publications

References 69 publications

Unsteady free convection of second‐grade nanofluid with a new time‐space fractional heat conduction

Unsteady free convection of second‐grade nanofluid with a new time‐space fractional heat conduction

A Thermodynamically Consistent Fractional Visco-Elasto-Plastic Model with Memory-Dependent Damage for Anomalous Materials

Role of bi-order Atangana–Aguilar fractional differentiation on Drude model: an analytic study for distinct sources

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