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Journal ArticleDOI

Thermoelastic solutions for thermal distributions moving over thin slim rod under memory-dependent three-phase lag magneto-thermoelasticity

03 May 2020-Mechanics Based Design of Structures and Machines (Taylor & Francis)-Vol. 48, Iss: 3, pp 277-298
TL;DR: In this paper, a mathematical model of generalized thermoelasticity was proposed to investigate the transient phenomena due to the influence of the Caputo fractional derivative on generalized thermasticity.
Abstract: Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence ...
Citations
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Dissertation
01 Mar 2009
TL;DR: In this paper, the relationship between these transforms and their properties was discussed and some important applications in physics and engineering were given, as well as their properties and applications in various domains.
Abstract: Integral transforms (Laplace, Fourier and Mellin) are introduced with their properties, the relationship between these transforms was discussed and some important applications in physics and engineering were given. ااااااا دقل مت ضارعتسإ ةساردو ل ةيلماكتلا تليوحتلا لك ، سلبل تلوحت نم روف ي ر نيليمو عم ةشقانم كلذكو ،اهنم لك صاوخ و صئاصخ ةقلعلا ةشقانم مت هذه نيب طبرلاو و ،تليوحتلا مت ميدقت تاقيبطتلا ضعب تليوحتلا هذهل ةمهملا يف تلاجم ءايزيفلا ةسدنهلاو.

383 citations

Journal ArticleDOI
TL;DR: In this paper, coupled plasma, thermal and elastic waves within an orthotropic infinite semiconducting medium in context of photothermal transport process having a spherica spore was studied.
Abstract: This article highlights on the study of coupled plasma, thermal and elastic waves within an orthotropic infinite semiconducting medium in context of photothermal transport process having a spherica...

92 citations

Journal ArticleDOI
TL;DR: The current work deals with the study of a thermo-piezoelectric modified model in the context of generalized heat conduction with a memory-dependent derivative with the aim of verifying the effect of memory presence, graded material properties, time-delay, Kernel function, and the thermosexual response on the physical fields.
Abstract: The current work deals with the study of a thermo-piezoelectric modified model in the context of generalized heat conduction with a memory-dependent derivative. The investigations of the limited-length piezoelectric functionally graded (FGPM) rod have been considered based on the presented model. It is assumed that the specific heat and density are constant for simplicity while the other physical properties of the FGPM rod are assumed to vary exponentially through the length. The FGPM rod is subject to a moving heat source along the axial direction and is fixed to zero voltage at both ends. Using the Laplace transform, the governing partial differential equations have been converted to the space-domain, and then solved analytically to obtain the distributions of the field quantities. Numerical computations are shown graphically to verify the effect of memory presence, graded material properties, time-delay, Kernel function, and the thermo-piezoelectric response on the physical fields.

38 citations


Cites result from "Thermoelastic solutions for thermal..."

  • ...It has been found that the nature of distributions in all physical field in the current study and the corresponding outcomes of generalized thermoelasticity with a single memory-dependent derivative relaxation parameter is consistent with the existence of the physical field variable distribution for both Lord and Shulman as derived from Ezzat and El-Bary [41] and Mondal and Kanoria [52]....

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  • ...Mallik and Kanoria [21] considered the thermoelastic interaction in functionally graded an unbounded solid because of varying heat source....

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Journal ArticleDOI
TL;DR: In this article, heat transport and temperature variations within biological tissues and body organs are modelled and analyzed for medical thermal therapeutic applications, such as hyperthermography and hyper-thermodynamics.
Abstract: Modeling and understanding heat transport and temperature variations within biological tissues and body organs are key issues in medical thermal therapeutic applications, such as hypertherm...

31 citations


Cites background from "Thermoelastic solutions for thermal..."

  • ...So, the common derivative d dt can be seen as the limit of Dx as x ! 0 [53, 54]....

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Journal ArticleDOI
TL;DR: In this paper, the derivation of fundamental equations in generalized thermoelastic diffusion with four lags and higher-order time-fractional derivatives is studied, and the equations of the heat equation are derived.
Abstract: The present work is devoted to the derivation of fundamental equations in generalized thermoelastic diffusion with four lags and higher-order time-fractional derivatives. The equations of the heat ...

30 citations


Cites background from "Thermoelastic solutions for thermal..."

  • ...…have been considered by many investigations (Abbas 2015; Deswal, Kalkal, and Sheoran 2016; Mishra, Sharma, and Sharma 2017; Othman and Eraki 2017; Xiong and Niu 2017; Biswas, Mukhopadhyay, and Shaw 2019; Davydov, Zemskov, and Akhmetova 2019; Mondal, Sur, and Kanoria 2019; Mondal and Kanoria 2019)....

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References
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Book
02 Mar 2000
TL;DR: An introduction to fractional calculus can be found in this paper, where Butzer et al. present a discussion of fractional fractional derivatives, derivatives and fractal time series.
Abstract: An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.

5,201 citations


"Thermoelastic solutions for thermal..." refers methods in this paper

  • ...…Fourier law of heat conduction by using the fractional calculus by Caputo (1967), Caputo and Fabrizio (2015), Carpinteri and Mainardi (1997), Carpinteri and Mainardi (2014), Podlubny (1998), Hilfer (2000), Ignaczak and Ostoja-Starzewski (2009), Debnath and Bhatta (2007), and Abbas (2014a)....

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Book
01 Jan 1999
TL;DR: In this article, the authors present a method for computing fractional derivatives of the Fractional Calculus using the Laplace Transform Method and the Fourier Transformer Transform of fractional Derivatives.
Abstract: Preface. Acknowledgments. Special Functions Of Preface. Acknowledgements. Special Functions of the Fractional Calculus. Gamma Function. Mittag-Leffler Function. Wright Function. Fractional Derivatives and Integrals. The Name of the Game. Grunwald-Letnikov Fractional Derivatives. Riemann-Liouville Fractional Derivatives. Some Other Approaches. Sequential Fractional Derivatives. Left and Right Fractional Derivatives. Properties of Fractional Derivatives. Laplace Transforms of Fractional Derivatives. Fourier Transforms of Fractional Derivatives. Mellin Transforms of Fractional Derivatives. Existence and Uniqueness Theorems. Linear Fractional Differential Equations. Fractional Differential Equation of a General Form. Existence and Uniqueness Theorem as a Method of Solution. Dependence of a Solution on Initial Conditions. The Laplace Transform Method. Standard Fractional Differential Equations. Sequential Fractional Differential Equations. Fractional Green's Function. Definition and Some Properties. One-Term Equation. Two-Term Equation. Three-Term Equation. Four-Term Equation. Calculation of Heat Load Intensity Change in Blast Furnace Walls. Finite-Part Integrals and Fractional Derivatives. General Case: n-term Equation. Other Methods for the Solution of Fractional-order Equations. The Mellin Transform Method. Power Series Method. Babenko's Symbolic Calculus Method. Method of Orthogonal Polynomials. Numerical Evaluation of Fractional Derivatives. Approximation of Fractional Derivatives. The "Short-Memory" Principle. Order of Approximation. Computation of Coefficients. Higher-order Approximations. Numerical Solution of Fractional Differential Equations. Initial Conditions: Which Problem to Solve? Numerical Solution. Examples of Numerical Solutions. The "Short-Memory" Principle in Initial Value Problems for Fractional Differential Equations. Fractional-Order Systems and Controllers. Fractional-Order Systems and Fractional-Order Controllers. Example. On Viscoelasticity. Bode's Analysis of Feedback Amplifiers. Fractional Capacitor Theory. Electrical Circuits. Electroanalytical Chemistry. Electrode-Electrolyte Interface. Fractional Multipoles. Biology. Fractional Diffusion Equations. Control Theory. Fitting of Experimental Data. The "Fractional-Order" Physics? Bibliography. Tables of Fractional Derivatives. Index.

3,962 citations


"Thermoelastic solutions for thermal..." refers background or methods in this paper

  • ...One can refer to Podlubny (1998) for a survey of applications of fractional calculus....

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  • ...…Fourier law of heat conduction by using the fractional calculus by Caputo (1967), Caputo and Fabrizio (2015), Carpinteri and Mainardi (1997), Carpinteri and Mainardi (2014), Podlubny (1998), Hilfer (2000), Ignaczak and Ostoja-Starzewski (2009), Debnath and Bhatta (2007), and Abbas (2014a)....

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Journal ArticleDOI
TL;DR: In this paper, a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges has been investigated by introducing fractional derivatives in the stressstrain relation, and a rigorous proof of the formulae to be used in obtaining the analytic expression of Q is given.
Abstract: Summary Laboratory experiments and field observations indicate that the Q of many non-ferromagnetic inorganic solids is almost frequency independent in the range 10-2-107 cis, although no single substance has been investigated over the entire frequency spectrum. One of the purposes of this investigation is to find the analytic expression for a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges. This will be obtained by introducing fractional derivatives in the stressstrain relation. Since the aim of this research is also to contribute to elucidating the dissipating mechanism in the Earth free modes, we shall treat the dissipation in the free, purely torsional, modes of a shell. The dissipation in a plane wave will also be treated. The theory is checked with the new values determined for the Q of spheroidal free modes of the Earth in the range between 10 and 5 min integrated with the Q of Rayleigh waves in the range between 5 and 0.6 min. Another check of the theory is made with the experimental values of the Q of the longitudinal waves in an aluminium rod in the range between lo-’ and 10-3s. In both checks the theory represents the observed phenomena very satisfactorily. The time derivative which enters the stress-strain relation in both cases is of order 0.15. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. In a homogeneous isotropic elastic field the elastic properties of the substance are specified by a description of the strains and stresses in a limited portion of the field since the strains and stresses are linearly related by two parameters which describe the elastic properties of the field. If the elastic field is not homogeneous nor isotropic the properties of the field are specified in a similar manner by a larger number of parameters which also depend on the position.

3,372 citations


"Thermoelastic solutions for thermal..." refers background in this paper

  • ...This definition of fractional derivative appears to be appropriate for mechanical phenomena related to plasticity, fatigue and with electromagnetic hysteresis by Caputo and Fabrizio (2015). A memory-dependent derivative (MDD) has been introduced by Wang and Li (2011) and Yu, Hu, and Tian (2014). The first order (f1⁄4 1) of function f which is simply defined in an integral form of a common derivative with a kernel function Kðt nÞ (chosen arbitrarily) on a slipping interval [t – s, t] in the form...

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  • ...This definition of fractional derivative appears to be appropriate for mechanical phenomena related to plasticity, fatigue and with electromagnetic hysteresis by Caputo and Fabrizio (2015). A memory-dependent derivative (MDD) has been introduced by Wang and Li (2011) and Yu, Hu, and Tian (2014)....

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Journal ArticleDOI
TL;DR: In this article, a generalized dynamical theory of thermoelasticity is formulated using a form of the heat transport equation which includes the time needed for acceleration of heat flow.
Abstract: In this work a generalized dynamical theory of thermoelasticity is formulated using a form of the heat transport equation which includes the time needed for acceleration of the heat flow. The theory takes into account the coupling effect between temperature and strain rate, but the resulting coupled equations are both hyperbolic. Thus, the paradox of an infinite velocity of propagation, inherent in the existing coupled theory of thermoelasticity, is eliminated. A solution is obtained using the generalized theory which compares favourably with a known solution obtained using the conventional coupled theory.

3,266 citations


"Thermoelastic solutions for thermal..." refers background in this paper

  • ...Among these theories, the first is held by Lord and Shulman (1967) who acquired a wave-type heat equation by hypothesizing another law of heat equation to take the frame q x; t þ sqð Þ ¼ Krh x; tð Þ: (1) Another modified heat conduction law, namely dual-phase lag (DPL) model, given by Tzou (1995)…...

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  • ...At the same time, it contains another constant sq that goes about as a relaxation time in generalized thermoelasticity by Lord and Shulman (1967)....

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  • ...Very recently, several problems in generalized thermoelasticity in the context of memorydependent derivative have been reported in the following literatures by Ezzat, El-Karamany, and El-Bary (2017b), Shaw and Mukhopadhyay (2017), Lotfy and Sarkar (2017), Kant and Mukhopadhyay (2018), Sur, Pal, and Kanoria (2018), Sur and Kanoria (2017), Purokait et al....

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Journal ArticleDOI
TL;DR: In this article, a unified treatment of thermoelasticity by application and further developments of the methods of irreversible thermodynamics is presented, along with a new definition of the dissipation function in terms of the time derivative of an entropy displacement.
Abstract: A unified treatment is presented of thermoelasticity by application and further developments of the methods of irreversible thermodynamics. The concept of generalized free energy introduced in a previous publication plays the role of a ``thermoelastic potential'' and is used along with a new definition of the dissipation function in terms of the time derivative of an entropy displacement. The general laws of thermoelasticity are formulated in a variational form along with a minimum entropy production principle. This leads to equations of the Lagrangian type, and the concept of thermal force is introduced by means of a virtual work definition. Heat conduction problems can then be formulated by the methods of matrix algebra and mechanics. This also leads to the very general property that the entropy density obeys a diffusion‐type law. General solutions of the equations of thermoelasticity are also given using the Papkovitch‐Boussinesq potentials. Examples are presented and it is shown how the generalized coordinate method may be used to calculate the thermoelastic internal damping of elastic bodies.

2,287 citations


"Thermoelastic solutions for thermal..." refers background in this paper

  • ...The traditional coupled dynamic thermoelasticity theories predict the infinite speed of thermal waves, which depend on the mixed parabolic-hyperbolic governing equations of Biot (1956) but the predicted infinite speed of the thermal wave is not an acceptable phenomenon....

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  • ...The traditional coupled dynamic thermoelasticity theories predict the infinite speed of thermal waves, which depend on the mixed parabolic-hyperbolic governing equations of Biot (1956) but the predicted infinite speed of the thermal wave is not an acceptable phenomenon. The infinite speed of thermal waves results in an instantaneous effect on the body, no matter how far it is from the point of heat source applied. The classical theory of thermoelasticity has been generalized and changed into different thermoelastic models that keep running under the mark of “hyperbolic thermoelasticity.” At present, there are a few theories of the hyperbolic thermoelasticity. It ensures the finite speed of wave propagation. Among these theories, the first is held by Lord and Shulman (1967) who acquired a wave-type heat equation by hypothesizing another law of heat equation to take the frame...

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