Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer

doi:10.1080/17455030.2018.1457230

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Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer

Journal Article•DOI•

Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer

Mitali Bachher, Nantu Sarkar¹•Institutions (1)

University of Calcutta¹

02 Oct 2019-Waves in Random and Complex Media (Taylor & Francis)-Vol. 29, Iss: 4, pp 595-613

TL;DR: In this paper, a nonlocal theory of generalized thermoelastic materials with voids based on Eringen's nonlocal elasticity and Caputo fractional derivative is established.

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Abstract: A new nonlocal theory of generalized thermoelastic materials with voids based on Eringen’s nonlocal elasticity and Caputo fractional derivative is established. The one-dimensional form of the above...

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Journal Article•DOI•

Plane waves in nonlocal thermoelastic solid with voids

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Nantu Sarkar¹, Sushil K. Tomar²•Institutions (2)

University of Calcutta¹, Panjab University, Chandigarh²

22 Jan 2019-Journal of Thermal Stresses

TL;DR: In this paper, the propagation of time harmonic plane waves in an infinite nonlocal thermoelastic solid having void pores was studied, and the effects of frequency, void parameters, thermal parameter and nonlocality have been studied numerically on the phase speeds, attenuation coefficients and specific losses of all the propagating waves.

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Abstract: This work is concerned with the propagation of time harmonic plane waves in an infinite nonlocal thermoelastic solid having void pores. Three sets of coupled dilatational waves and an independent transverse wave may travel with distinct speeds in the medium. All these waves are found to be dispersive in nature, but the coupled dilatational waves are attenuating, while transverse wave is nonattenuating. Coupled dilatational waves are found to be influenced by the presence of voids, thermal field and elastic nonlocal parameter. While the transverse wave is found to be influenced by the nonlocal parameter, but independent of void and thermal parameters. For a particular model, the effects of frequency, void parameters, thermal parameter and nonlocality have been studied numerically on the phase speeds, attenuation coefficients and specific losses of all the propagating waves. All the computed results obtained have been depicted graphically and explained.

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68 citations

Cites background from "Nonlocal theory of thermoelastic ma..."

...Other symbols have their usual meanings and borrowed from [50]....
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...Field equations and constitutive relations Within the framework of Eringen’s theory of nonlocal elasticity [2], the constitutive relations for thermoelastic solid with voids are given by [50] 1 e2r2 ð Þtij 1⁄4 t ij 1⁄4 2leij x ð Þ þ kekk x ð Þ þ b/ x ð Þ ch x ð Þ 1⁄2 dij; (1) 1 e2r2 ð Þhi 1⁄4 hij 1⁄4 a/;i x ð Þ; (2) 1 e2r2 ð Þg 1⁄4 g 1⁄4 s _ / x ð Þ n/ x ð Þ bekk x ð Þ þmh x ð Þ; (3) 1 e2r2 ð Þqg 1⁄4 qg ð Þ 1⁄4 cekk x ð Þ þ ah x ð Þ þm/ x ð Þ; (4) where the quantities tL ij; h L i ; g L and ðqgÞ correspond the local thermoelastic solid with voids....
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Journal Article•DOI•

Waves in dual-phase-lag thermoelastic materials with voids based on Eringen’s nonlocal elasticity

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Sudip Mondal, Nihar Sarkar, Nantu Sarkar¹•Institutions (1)

University of Calcutta¹

09 Apr 2019-Journal of Thermal Stresses

TL;DR: The main idea of as mentioned in this paper is to extend Eringen's theory of nonlocal elasticity to generalized thermoelasticity with dual-phase-lag and voids.

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Abstract: The main idea of the present work is to extend Eringen’s theory of nonlocal elasticity to generalized thermoelasticity with dual-phase-lag and voids. Then we study the propagation of time harmonic ...

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65 citations

Cites background from "Nonlocal theory of thermoelastic ma..."

...This shows that x ¼ xc acts as a cutoff frequency for the existing transverse wave, a conclusion in accordance with that earlier mentioned by Sarkar and Tomar [14]....
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...(24), we see that the speed of transverse wave in thermoelastic medium with voids reduces to the classical transverse wave speed, a result recently obtained by Sarkar and Tomar [14] in the relevant medium when r 6¼ 1....
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...Within the framework of Eringen’s theory of nonlocal elasticity [1], the constitutive relations for thermoelastic solid with voids are given by [12,14] 1 e2r2 ð Þsij 1⁄4 sij 1⁄4 2leij þ kekk þ b/ ch ð Þdij; (1) 1 e2r2 ð Þhi 1⁄4 hij 1⁄4 a/;i; (2) 1 e2r2 ð Þg 1⁄4 g 1⁄4 s _ / n/ bekk þmh; (3) 1 e2r2 ð Þqg 1⁄4 qg ð Þ 1⁄4 cekk þ ahþm/; (4)...
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...Bachher and Sarkar [12] established a nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer....
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...Biswas and Sarkar [13] reported fundamental solution of the steady oscillations equations in porous thermoelastic medium with dual-phase-lag model....
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Journal Article•DOI•

A novel model of nonlocal thermoelasticity with time derivatives of higher order

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Ahmed E. Abouelregal¹•Institutions (1)

Mansoura University¹

30 Jul 2020-Mathematical Methods in The Applied Sciences

62 citations

Journal Article•DOI•

Transient response in a thermoelastic half-space solid due to a laser pulse under three theories with memory-dependent derivative

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Sudip Mondal, P. K. Pal¹, Mridula Kanoria¹•Institutions (1)

University of Calcutta¹

01 Jan 2019-Acta Mechanica

TL;DR: In this paper, a novel mathematical model of generalized thermoelasticity was proposed to investigate the transient phenomena due to the influence of a non-Gaussian pulsed laser type heat source in a stress free isothermal half-space in the context of Lord-Shulman (LS), dual-phase lag (DPL), and three phase lag (TPL) theories simultaneously.

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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 of a non-Gaussian pulsed laser type heat source in a stress free isothermal half-space in the context of Lord–Shulman (LS), dual-phase lag (DPL), and three-phase lag (TPL) theories of thermoelasticity simultaneously. The memory-dependent derivative is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. Employing Laplace transform as a tool, the problem has been transformed to the space-domain, and it is then solved analytically. To get back all the thermophysical quantities as a function of real time, we use two Laplace inversion formulas, viz. Fourier series expansion technique (Honig in J Comput Appl Math10(1):113–132, 1984) and Zakian method (Electron Lett 6(21):677–679, 1970). According to the graphical representations corresponding to the numerical results, a comparison among LS, DPL, and TPL model has been studied in the presence and absence of a memory effect simultaneously. Moreover, the effects of a laser pulse have been studied in all the thermophysical quantities for different kernels (randomly chosen) and different delay times. Then, the results are depicted graphically. Finally, a comparison of results, deriving from the two numerical inversion formulas, has been made.

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49 citations

Journal Article•DOI•

Computational analysis of an infinite magneto-thermoelastic solid periodically dispersed with varying heat flow based on non-local Moore–Gibson–Thompson approach

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Ahmed E. Abouelregal¹, Hamid Mohammad-Sedighi², Ali Heidari Shirazi², Mohammad Malikan³, Victor A. Eremeyev³, Victor A. Eremeyev⁴ - Show less +2 more•Institutions (4)

Mansoura University¹, Shahid Chamran University of Ahvaz², Gdańsk University of Technology³, University of Cagliari⁴

29 Mar 2021-Continuum Mechanics and Thermodynamics

TL;DR: In this article, a computational analysis is conducted to study a magneto-thermoelastic problem for an isotropic perfectly conducting half-space medium, where the medium is subjected to a periodic heat flow in the presence of a continuous longitude magnetic field.

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Abstract: In this investigation, a computational analysis is conducted to study a magneto-thermoelastic problem for an isotropic perfectly conducting half-space medium. The medium is subjected to a periodic heat flow in the presence of a continuous longitude magnetic field. Based on Moore–Gibson–Thompson equation, a new generalized model has been investigated to address the considered problem. The introduced model can be formulated by combining the Green–Naghdi Type III and Lord–Shulman models. Eringen’s non-local theory has also been applied to demonstrate the effect of thermoelastic materials which depends on small scale. Some special cases as well as previous thermoelasticity models are deduced from the presented approach. In the domain of the Laplace transform, the system of equations is expressed and the problem is solved using state space method. The converted physical expressions are numerically reversed by Zakian’s computational algorithm. The analysis indicates the significant influence on field variables of non-local modulus and magnetic field with larger values. Moreover, with the established literature, the numerical results are satisfactorily examined.

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45 citations

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References

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Book•

Wave propagation in elastic solids

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Jan Drewes Achenbach

01 Jan 1962

TL;DR: In this article, the linearized theory of elasticity was introduced and the elasticity of a one-dimensional motion of an elastic continuum was modeled as an unbound elastic continuum.

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Abstract: Preface Introduction 1 One-dimensional motion of an elastic continuum 2 The linearized theory of elasticity 3 Elastodynamic theory 4 Elastic waves in an unbound medium 5 Plane harmonic waves in elastic half-spaces 6 Harmonic waves in waveguides 7 Forced motions of a half-space 8 Transient waves in layers and rods 9 Diffraction of waves by a slit 10 Thermal and viscoelastic effects, and effects of anisotrophy and non-linearity Author Index Subject Index

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4,133 citations

"Nonlocal theory of thermoelastic ma..." refers methods in this paper

...Laplace transform [45] together with an eigenvalue approach [33,46] technique is employed to obtain the closed form solutions in the transform domain....
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Journal Article•DOI•

On nonlocal elasticity

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A. Cemal Eringen¹, A. Cemal Eringen², D.G.B. Edelen², D.G.B. Edelen¹•Institutions (2)

Princeton University¹, Lehigh University²

01 Mar 1972-International Journal of Engineering Science

TL;DR: In this article, a theory of non-local elasticity is presented via the vehicles of global balance laws and the second law of thermodynamics via the use of a localized Clausius-Duhem inequality and a variational statement of Gibbsian global thermodynamics.

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2,201 citations

Journal Article•DOI•

Nonlocal polar elastic continua

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A. Cemal Eringen¹•Institutions (1)

Princeton University¹

01 Jan 1972-International Journal of Engineering Science

TL;DR: In this article, a continuum theory of non-local polar bodies is developed for nonlinear micromorphic elastic solids, and the balance laws and jump conditions are given.

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1,788 citations

Journal Article•DOI•

Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids

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Yuriy A. Rossikhin, Marina V. Shitikova

01 Jan 1997-Applied Mechanics Reviews

681 citations

"Nonlocal theory of thermoelastic ma..." refers background in this paper

...In the last few years, fractional calculus has been successfully applied to extend generalized thermoelasticity theories to fractional order generalized thermoelasticity by Bachher et al. [33,34], Povstenko [35,36], Sherief [37], Youssef [38], Ezzat and his co-workers [39–43] etc. Rossikhin and Shitikova [44] applied fractional calculus to various problems of mechanics of solids....
[...]
...Rossikhin and Shitikova [44] applied fractional calculus to various problems of mechanics of solids....
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Journal Article•DOI•

Fractional heat conduction equation and associated thermal stress

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Yuriy Povstenko¹•Institutions (1)

Pedagogical University¹

15 Dec 2004-Journal of Thermal Stresses

TL;DR: A quasi-static uncoupled theory of thermoelasticity based on the heat conduction equation with a time-fractional derivative of order α is proposed in this article.

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Abstract: A quasi-static uncoupled theory of thermoelasticity based on the heat conduction equation with a time-fractional derivative of order α is proposed. Because the heat conduction equation in the case 1≤α≤2 interpolates the parabolic equation (α = 1) and the wave equation (α = 2), the proposed theory interpolates a classical thermoelasticity and a thermoelasticity without energy dissipation introduced by Green and Naghdi. The Caputo fractional derivative is used. The stresses corresponding to the fundamental solutions of a Cauchy problem for the fractional heat conduction equation are found in one-dimensional and two-dimensional cases.

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482 citations

"Nonlocal theory of thermoelastic ma..." refers background in this paper

...In the last few years, fractional calculus has been successfully applied to extend generalized thermoelasticity theories to fractional order generalized thermoelasticity by Bachher et al. [33,34], Povstenko [35,36], Sherief [37], Youssef [38], Ezzat and his co-workers [39–43] etc. Rossikhin and Shitikova [44] applied fractional calculus to various problems of mechanics of solids....
[...]
...[33,34], Povstenko [35,36], Sherief [37], Youssef [38], Ezzat and his co-workers [39–43] etc....
[...]