Fractional order theory of thermoelasticity

Sherief, Hany H.; Elsayed, Ahmed; El-Latief, A. M. Abd

doi:10.1016/j.ijsolstr.2009.09.034

Cited by 493 publications

(196 citation statements)

References 24 publications

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“…[5]). In this section the first-order relaxation of thermal energy transfer by means of the Cattaneo model for the local thermal energy exchange is considered as:…”

Section: A Non-equilibrium Model Of Fractional-order Thermal Energy Tmentioning

confidence: 99%

“…Such studies have been further developed toward the use of advanced mathematical tools as the fractional-order calculus [4] to capture memory [5,6] as well as non-local effects [7][8][9]. Indeed fractional (real) order integro-differential operators have been introduced more and more often in several contexts of physics and engineering for their capability to interpolate among the well-known integer-order operators of classical differential calculus [10].…”

Section: Introductionmentioning

confidence: 99%

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Fractional-order theory of heat transport in rigid bodies

Zingales¹

2014

Communications in Nonlinear Science and Numerical Simulation

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a b s t r a c tThe non-local model of heat transfer, used to describe the deviations of the temperature field from the well-known prediction of Fourier/Cattaneo models experienced in complex media, is framed in the context of fractional-order calculus. It has been assumed (Borino et al., 2011 [53], Mongioví and Zingales, 2013 [54]) that thermal energy transport is due to two phenomena: (i) A short-range heat flux ruled by a local transport equation; (ii) A long-range thermal energy transfer proportional to a distance-decaying function, to the relative temperature and to the product of the interacting masses. The distance-decaying function is assumed in the functional class of the power-law decay of the distance yielding a novel temperature equation in terms of a-order Marchaud fractional-order derivative ð0 6 a 6 1Þ. Thermodynamical consistency of the model is provided in the context of Clausius-Plank inequality. The effects induced by the boundary conditions on the temperature field are investigated for diffusive as well as ballistic local heat flux. Deviations of the temperature field from the linear distributions in the neighborhood of the thermostated zones of small-scale conductors are qualitatively predicted by the used fractional-order heat transport model, as shown by means of molecular dynamics simulations.

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“…[5]). In this section the first-order relaxation of thermal energy transfer by means of the Cattaneo model for the local thermal energy exchange is considered as:…”

Section: A Non-equilibrium Model Of Fractional-order Thermal Energy Tmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractional-order theory of heat transport in rigid bodies

Zingales¹

2014

Communications in Nonlinear Science and Numerical Simulation

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“…Non-local models of thermal energy transport have been used in recent physics and engineering applications to describe "small-scale" and/or high-frequency thermodynamic processes [1][2][3][4][5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

“…In more recent years, fractional models of non-local thermal energy transport have been awarded a growing attention [4][5][6][7][8][9][10]. Being capable of interpolating among the integer-order operators of classical differential calculus, fractional operators have been used indeed in several contexts of physics and engineering .…”

Section: Introductionmentioning

confidence: 99%

“…Regarding, in particular, heat transfer problems in rigid as well as elastic bodies, fractional operators have been used in ref. [4][5] to model long-memory effects, and in ref. [6][7][8][9][10] to model long-memory with long-range spatial effects, providing fractional time [4][5][6] and fractional time-space [7][8][9] heat conduction equations.…”

Section: Introductionmentioning

confidence: 99%

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The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors

Zingales

Failla

2015

Communications in Nonlinear Science and Numerical Simulation

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a b s t r a c tIn a non-local fractional-order model of thermal energy transport recently introduced by the authors, it is assumed that local and non-local contributions coexist at a given observation scale: while the first is described by the classical Fourier transport law, the second involves couples of adjacent and non-adjacent elementary volumes, and is taken as proportional to the product of the masses of the interacting volumes and their relative temperature, through a material-dependent, distance-decaying power-law function. As a result, a fractional-order heat conduction equation is derived. This paper presents a pertinent finite element method for the solution of the proposed fractional-order heat conduction equation. Homogenous and non-homogeneous rigid bodies are considered. Numerical applications are carried out on 1D and 2D bodies, including a standard finite difference solution for validation.

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Fractional thermoelectric viscoelastic materials

Ezzat

El-Karamany

2011

J of Applied Polymer Sci

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A new mathematical model of thermoelectro viscoelasticity theory was constructed in the context of a new consideration of heat conduction with fractional order. The state space approach developed earlier by Ezzat was adopted for the solution of a one-dimensional problem in the presence of heat sources. The Laplace-transform technique was used. A numerical method was employed for the inversion of the Laplace transforms. According to the numerical results and their graphs, a conclusion about the new theory was constructed. Some comparisons are shown in figures to estimate the effect of the fractional order parameter on all of the studied fields.

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Fractional order theory of thermoelasticity

Cited by 493 publications

References 24 publications

Fractional-order theory of heat transport in rigid bodies

Fractional-order theory of heat transport in rigid bodies

The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors

Fractional thermoelectric viscoelastic materials

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