scispace - formally typeset

Journal ArticleDOI

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

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

AbstractA 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...

...read more


Citations
More filters
Journal ArticleDOI
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 ...

50 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]....

    [...]

  • ...(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....

    [...]

  • ...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)...

    [...]

  • ...Bachher and Sarkar [12] established a nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer....

    [...]

  • ...Biswas and Sarkar [13] reported fundamental solution of the steady oscillations equations in porous thermoelastic medium with dual-phase-lag model....

    [...]

Journal ArticleDOI
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.

43 citations

Journal ArticleDOI
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.

40 citations


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

  • ...Other symbols have their usual meanings and borrowed from [50]....

    [...]

  • ...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....

    [...]

Journal ArticleDOI
TL;DR: This review includes the last researches on bending, buckling, and vibration of nano-plates, nano-beams, nanorods, and nanotubes which were investigated by non-local elasticity theory and nonlocal strain gradient theory.
Abstract: Nanotechnology is one of the pillars of human life in the future. This technology is growing fast and many scientists work in this field. The behavior of materials in nano size varies with that in macro dimension. Therefore scientists have presented various theories for examining the behavior of materials in nano-scale. Accordingly, mechanical behavior of nano-plates, nanotubes nano-beams and nano-rodes are being investigated by Non-classical elasticity theories. This review includes the last researches on bending, buckling, and vibration of nano-plates, nano-beams, nanorods, and nanotubes which were investigated by non-local elasticity theory and nonlocal strain gradient theory. Great scholars have written valuable reviews in the field of nanomechanics. Therefore, given a large number of researches and the prevention of repetition, the articles in the past year are reviewed.

37 citations


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

  • ...[70] M. Bachher, N. Sarkar, Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer, Waves in Random and Complex Media, pp. 1-19, 2018....

    [...]

  • ...Bachher and Sarkar [70] established a new nonlocal theory of generalized thermoelastic materials with voids based on Eringen’s nonlocal elasticity and Caputo fractional derivative to study transient wave propagation in an infinite thermoelastic material....

    [...]

Journal ArticleDOI

36 citations


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

  • ...where n = 5 and αj and Kj are complex numbers and given as [45,46]....

    [...]


References
More filters
Book
01 Jan 1962
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

4,124 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....

    [...]

Journal ArticleDOI
Abstract: Via the vehicles of global balance laws and the second law of thermodynamics, a theory of nonlocal elasticity is presented. Constitutive equations are obtained for the nonlinear theory, first through the use of a localized Clausius-Duhem inequality and second through a variational statement of Gibbsian global thermodynamics.

1,870 citations

Journal ArticleDOI
Abstract: A continuum theory of nonlocal polar bodies is developed. Both the micromorphic and the non-polar continuum theories are incorporated. The balance laws and jump conditions are given. By use of nonlocal thermodynamics and invariance under rigid motions, constitutive equations are obtained for the nonlinear micromorphic elastic solids. The special case, nonpolar, nonlocal elastic solids, is presented.

1,525 citations

Journal ArticleDOI

592 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....

    [...]

Journal ArticleDOI
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.

416 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....

    [...]