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Journal ArticleDOI

One-dimensional problem of a fractional order two-temperature generalized thermo-piezoelasticity

01 Aug 2014-Mathematics and Mechanics of Solids (SAGE Publications)-Vol. 19, Iss: 6, pp 672-693
TL;DR: In this article, the authors derived the thermoelastic stress, strain and conductive temperature in a piezoelastic half-space body in which the boundary is stress free and subjected to thermal loading in the context of the fractional order generalized thermelasticity theory (2TT).
Abstract: This paper is concerned with the determination of the thermoelastic stress, strain and conductive temperature in a piezoelastic half-space body in which the boundary is stress free and subjected to thermal loading in the context of the fractional order two-temperature generalized thermoelasticity theory (2TT). The two-temperature three-phase-lag (2T3P) model, two-temperature Green–Naghdi model III (2TGNIII) and two-temperature Lord–Shulman (2TLS) model of thermoelasticity are combined into a unified formulation introducing unified parameters. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain that is then solved by the state-space approach. The numerical inversion of the transform is carried out by a method based on Fourier series expansion techniques. The numerical estimates of the quantities of physical interest are obtained and depicted graphically. The effect of the fractional order parameter, two-temperature and electric field on...
Citations
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Journal ArticleDOI
TL;DR: In this article, a new model of magneto-thermoelasticity theory has been constructed in the context of a new consideration of heat conduction with memory-dependent derivative.
Abstract: In this work, a new model of magneto-thermoelasticity theory has been constructed in the context of a new consideration of heat conduction with memory-dependent derivative. One-dimensional application for a perfect electrically conducting half-space of elastic material, which is thermally shocked, in the presence of a constant magnetic field has been solved by using Laplace transform technique. According to the numerical results and its graphs, conclusion about the new theory of magneto-thermoelasticity has been constructed and compared with dynamic classical coupled theory.

68 citations

Journal ArticleDOI
TL;DR: In this paper, an ultrafast fractional magneto-thermoelasticity model utilizing the modified hyperbolic heat conduction model with fractional order is formulated to describe the thermo-elastic behavior of a thin perfect conducting metal film irradiated by a femtosecond laser pulse.
Abstract: An ultrafast fractional magneto-thermoelasticity model utilizing the modified hyperbolic heat conduction model with fractional order is formulated to describe the thermoelastic behavior of a thin perfect conducting metal film irradiated by a femtosecond laser pulse. Some theorems of generalized thermoelasticity follow as limit cases. The temporal profile of the ultrafast laser was regarded as being non-Gaussian. An analytical–numerical technique based on the Laplace transform was used to solve the governing equations and the time histories of the temperature, displacement, stress, strain, and induced electric/magnetic fields in a gold film were analyzed. Some comparisons have been shown in figures to estimate the effects of some parameters on all the studied fields.

43 citations


Cites methods from "One-dimensional problem of a fracti..."

  • ...Recently, the determination of the thermoelastic stress, strain and conductive temperature in a piezoelastic half-space body in the context of the fractional order two-emperature generalized thermoelasticity theory are obtained by Kanoria and Islam [16]....

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Journal ArticleDOI
TL;DR: In this paper, the authors presented a refined multi-phase-lags theory for a half-space medium with the inclusion of gravity, where the wave propagation of a gravitated piezo-thermoelastic half-spa...
Abstract: This work presents a refined multi-phase-lags theory for thermoelastic response of half-space medium with the inclusion of gravity. The wave propagation of a gravitated piezo-thermoelastic half-spa...

25 citations

Journal ArticleDOI
TL;DR: In this article, the authors investigated the transient phenomena in a homogeneous isotropic three-dimensional medium whose surface is subjected to a time-dependent thermal loading and is free of traction, in the context of two-temperature three-phase-lag thermoelastic model with non-local fractional operators.

22 citations

References
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Journal ArticleDOI
TL;DR: A numerical inversion method for Laplace transforms, based on a Fourier series expansion developed by Durbin [5], is presented in this article, where the disadvantage of the inversion methods of that type, the encountered dependence of discretization and truncation error on the free parameters, is removed by the simultaneous application of a procedure for the reduction of the Discretization error, a method for accelerating the convergence of the Fourier Series and a procedure that computes approximately the "best" choice of the free parameter.

1,044 citations


"One-dimensional problem of a fracti..." refers background or methods in this paper

  • ...The inversion of the transformed solutions is carried out numerically applying a method based on Fourier series expansion technique [78]....

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  • ...It can be shown that [78] the sequence ε1,1, ε3,1, ε5,1, ....

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  • ...Expanding the function h(x, t) = e−dtf (x, t) in a Fourier series in the interval [0, 2T], we obtain the approximate formula [78] f (x, t) = f∞(x, t) + ED (74)...

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  • ...The values of d and T are chosen according to the criterion outlined in Honig and Hirdes [78]....

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  • ...Next, the ε-algorithm is used to accelerate convergence [78]....

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Journal ArticleDOI
TL;DR: In this paper, a review of contributions to the theory of thermoelasticity with thermal relaxation and the temperature-rate dependent thermodynamic properties is presented, and a glance is made at the new theory which includes the so-called dual-phase-lag effects.
Abstract: This review article is a continuation of a previous article by the author, Thermoelasticity with second sound: A review, which appeared in this journal in March, 1986 (Appl Mech Rev 39(3) 355-376). Here, attention is focused on papers published during the past 10-12 years. Contributions to the theory of thermoelasticity with thermal relaxation and the temperature-rate dependent thermoelasticity theory are reviewed. The recently developed theory of thermoelasticity without energy dissipation is described, and its characteristic features highlighted. A glance is made at the new thermoelasticity theory which includes the so-called dual-phase-lag effects. There are 338 references.

1,019 citations


"One-dimensional problem of a fracti..." refers background in this paper

  • ...In the context of the linearized version of this theory [18], a theorem on the uniqueness of solutions has been established by Chandrasekharaiah [19, 20]....

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  • ...Problems concerning generalized theories such as ETE and TRDTE have been studied by Chandrasekharaiah [14] and Ignaczak [15]....

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  • ...The fifth generalization of the thermoelasticity theory is known as the dual-phase-lag thermoelasticity developed by Tzou [28] and Chandrasekharaiah [29]....

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  • ...Chandrasekharaiah [42] has generalized Mindlin’s theory of thermopiezoelectricity to account for the finite speed of propagation of thermal disturbances....

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Journal ArticleDOI

833 citations


"One-dimensional problem of a fracti..." refers background in this paper

  • ...In the context of the linearized version of this theory [18], a theorem on the uniqueness of solutions has been established by Chandrasekharaiah [19, 20]....

    [...]

  • ...Problems concerning generalized theories such as ETE and TRDTE have been studied by Chandrasekharaiah [14] and Ignaczak [15]....

    [...]

  • ...The fifth generalization of the thermoelasticity theory is known as the dual-phase-lag thermoelasticity developed by Tzou [28] and Chandrasekharaiah [29]....

    [...]

  • ...Chandrasekharaiah [42] has generalized Mindlin’s theory of thermopiezoelectricity to account for the finite speed of propagation of thermal disturbances....

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BookDOI
07 Nov 2014
TL;DR: In this article, the authors describe the application of the Fourier Transform in the context of fractional calculus and apply it to the problem of finite differential equations in the complex plane.
Abstract: INTEGRAL TRANSFORMS Brief Historical Introduction Basic Concepts and Definitions FOURIER TRANSFORMS AND THEIR APPLICATIONS Introduction The Fourier Integral Formulas Definition of the Fourier Transform and Examples Fourier Transforms of Generalized Functions Basic Properties of Fourier Transforms Poisson's Summation Formula The Shannon Sampling Theorem Gibbs' Phenomenon Heisenberg's Uncertainty Principle Applications of Fourier Transforms to Ordinary Differential Eqn Solutions of Integral Equations Solutions of Partial Differential Equations Fourier Cosine and Sine Transforms with Examples Properties of Fourier Cosine and Sine Transforms Applications of Fourier Cosine and Sine Transforms to Partial DE Evaluation of Definite Integrals Applications of Fourier Transforms in Mathematical Statistics Multiple Fourier Transforms and Their Applications Exercises LAPLACE TRANSFORMS AND THEIR BASIC PROPERTIES Introduction Definition of the Laplace Transform and Examples Existence Conditions for the Laplace Transform Basic Properties of Laplace Transforms The Convolution Theorem and Properties of Convolution Differentiation and Integration of Laplace Transforms The Inverse Laplace Transform and Examples Tauberian Theorems and Watson's Lemma Exercises APPLICATIONS OF LAPLACE TRANSFORMS Introduction Solutions of Ordinary Differential Equations Partial Differential Equations, Initial and Boundary Value Problems Solutions of Integral Equations Solutions of Boundary Value Problems Evaluation of Definite Integrals Solutions of Difference and Differential-Difference Equations Applications of the Joint Laplace and Fourier Transform Summation of Infinite Series Transfer Function and Impulse Response Function Exercises FRACTIONAL CALCULUS AND ITS APPLICATIONS Introduction Historical Comments Fractional Derivatives and Integrals Applications of Fractional Calculus Exercises APPLICATIONS OF INTEGRAL TRANSFORMS TO FRACTIONAL DIFFERENTIAL EQUATIONS Introduction Laplace Transforms of Fractional Integrals Fractional Ordinary Differential Equations Fractional Integral Equations Initial Value Problems for Fractional Differential Equations Green's Functions of Fractional Differential Equations Fractional Partial Differential Equations Exercises HANKEL TRANSFORMS AND THEIR APPLICATIONS Introduction The Hankel Transform and Examples Operational Properties of the Hankel Transform Applications of Hankel Transforms to Partial Differential Equations Exercises MELLIN TRANSFORMS AND THEIR APPLICATIONS Introduction Definition of the Mellin Transform and Examples Basic Operational Properties Applications of Mellin Transforms Mellin Transforms of the Weyl Fractional Integral and Derivative Application of Mellin Transforms to Summation of Series Generalized Mellin Transforms Exercises HILBERT AND STIELTJES TRANSFORMS Introduction Definition of the Hilbert Transform and Examples Basic Properties of Hilbert Transforms Hilbert Transforms in the Complex Plane Applications of Hilbert Transforms Asymptotic Expansions of One-Sided Hilbert Transforms Definition of the Stieltjes Transform and Examples Basic Operational Properties of Stieltjes Transforms Inversion Theorems for Stieltjes Transforms Applications of Stieltjes Transforms The Generalized Stieltjes Transform Basic Properties of the Generalized Stieltjes Transform Exercises FINITE FOURIER SINE AND COSINE TRANSFORMS Introduction Definitions of the Finite Fourier Sine and Cosine Transforms and Examples Basic Properties of Finite Fourier Sine and Cosine Transforms Applications of Finite Fourier Sine and Cosine Transforms Multiple Finite Fourier Transforms and Their Applications Exercises FINITE LAPLACE TRANSFORMS Introduction Definition of the Finite Laplace Transform and Examples Basic Operational Properties of the Finite Laplace Transform Applications of Finite Laplace Transforms Tauberian Theorems Exercises Z TRANSFORMS Introduction Dynamic Linear Systems and Impulse Response Definition of the Z Transform and Examples Basic Operational Properties The Inverse Z Transform and Examples Applications of Z Transforms to Finite Difference Equations Summation of Infinite Series Exercises FINITE HANKEL TRANSFORMS Introduction Definition of the Finite Hankel Transform and Examples Basic Operational Properties Applications of Finite Hankel Transforms Exercises LEGENDRE TRANSFORMS Introduction Definition of the Legendre Transform and examples Basic Operational Properties of Legendre Transforms Applications of Legendre Transforms to Boundary Value Problems Exercises JACOBI AND GEGENBAUER TRANSFORMS Introduction Definition of the Jacobi Transform and Examples Basic Operational Properties Applications of Jacobi Transforms to the Generalized Heat Conduction Problem The Gegenbauer Transform and its Basic Operational Properties Application of the Gegenbauer Transform LAGUERRE TRANSFORMS Introduction Definition of the Laguerre Transform and Examples Basic Operational Properties Applications of Laguerre Transforms Exercises HERMITE TRANSFORMS Introduction Definition of the Hermite Transform and Examples Basic Operational Properties Exercises THE RADON TRANSFORM AND ITS APPLICATION Introduction Radon Transform Properties of Radon Transform Radon Transform of Derivatives Derivatives of Radon Transform Convolution Theorem for Radon Transform Inverse of Radon Transform Exercises WAVELETS AND WAVELET TRANSFORMS Brief Historical Remarks Continuous Wavelet Transforms The Discrete Wavelet Transform Examples of Orthonormal Wavelets Exercises Appendix A Some Special Functions and Their Properties A-1 Gamma, Beta, and Error Functions A-2 Bessel and Airy Functions A-3 Legendre and Associated Legendre Functions A-4 Jacobi and Gegenbauer Polynomials A-5 Laguerre and Associated Laguerre Functions A-6 Hermite and Weber-Hermite Functions A-7 Hurwitz and Riemann zeta Functions Appendix B Tables of Integral Transforms B-1 Fourier Transforms B-2 Fourier Cosine Transforms B-3 Fourier Sine Transforms B-4 Laplace Transforms B-5 Hankel Transforms B-6 Mellin Transforms B-7 Hilbert Transforms B-8 Stieltjes Transforms B-9 Finite Fourier Cosine Transforms B-10 Finite Fourier Sine Transforms B-11 Finite Laplace Transforms B-12 Z Transforms B-13 Finite Hankel Transforms Answers and Hints to Selected Exercises Bibliography Index

805 citations

Journal ArticleDOI
TL;DR: In this paper, a new Theorie der warmeleitung is presented, in which zwei Temperaturen auftreten, beseitigt einige Pathologien der klassischen Theorie.
Abstract: In dieser Arbeit entwickeln wir eine neue Theorie der Warmeleitung. Diese Theorie, in der zwei Temperaturen auftreten, beseitigt einige Pathologien der klassischen Theorie.

617 citations