Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer
Citations
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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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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References
445 citations
"Nonlocal theory of thermoelastic ma..." refers background or methods in this paper
...The fractional order Cattaneo type heat conduction law [33,37] for local thermoelastic solids can be obtained by substituting ε = 0 in the above generalization and the Cattaneo type heat conduction law can be recovered by puttingm = 1 and ε = 0 in (18)....
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...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....
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...[33,34], Povstenko [35,36], Sherief [37], Youssef [38], Ezzat and his co-workers [39–43] etc....
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361 citations
"Nonlocal theory of thermoelastic ma..." refers background in this paper
...Equations of motion for an isotropic nonlocal thermoelastic solid with voids are given by [27]...
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...Following Iesan [27], Bachher et al. [33] and Challamel et al. [15], here, we proposed an Eringen-type differential model for the nonlocal generalization of fractional order Cattaneo type heat conduction law for thermoelastic material with voids as ( 1 − ε2∇2 )( q + τ0 ∂ mq ∂tm ) = k∇θ , (18) whereq is the heat flux vector, τ0 is the relaxation time parameter, k is thermal conductivity, m (0 < m ≤ 1) is the fractional order parameter and ∂m ∂tm f (x , t) = ⎧⎪⎨ ⎪⎩ f (x , t) − f (x , 0), m → 0, I(1−m) ∂f (x ,t) ∂t , 0 < m < 1, ∂f (x ,t) ∂t , m = 1, with Imf (x , t) = 1 (m) ∫ t 0 (t − s)m−1f (x , s)ds, where ( · ) is the well-known Gamma function....
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...Note that, in the absence of nonlocality (ε = 0) andm = 1, these equations reduce to those of homogeneous isotropic local thermoelastic solid with voids earlier derived by Iesan [27]....
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...For linear thermoelastic solid with voids, the equations and relations are already developed by Iesan [27]....
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...In the linear thermoelasticity theory with voids [27], the energy equation has the form...
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"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....
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...[33,34], Povstenko [35,36], Sherief [37], Youssef [38], Ezzat and his co-workers [39–43] etc....
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