A Generalized Heat Conduction Model of Higher-Order Time Derivatives And

www.nature.com/scientificreports OPEN A generalized heat conduction model of higher‑order time derivatives and three‑phase‑lags for non‑simple thermoelastic materials Ahmed E. Abouelregal1,2*, K. M. Khalil1,3, F. A. Mohammed1,4, M. E. Nasr1,3, Adam Zakaria1,5 & Ibrahim‑Elkhalil Ahmed1,6 In the current work, a new generalized model of heat conduction has been constructed taking into account the infuence of the microscopic structure into the on non-simple thermoelastic materials. The new model was established on the basis of the system of equations that includes three‑phase lags of higher-order and two diferent temperatures, namely thermodynamic and conductive temperature. The two-temperature thermoelastic model presented by Chen and Gurtin (Z Angew Math Phys 19(4):614–627, 1968) and some other previous models have been introduced as special cases from the proposed model. As an application of the new model, we studied the thermoelastic interactions resulting from sudden heating in an isotropic solid subjected to external body force. The infuence of the discrepancy parameter and higher‑order of the time‑derivative has been discussed. This work will enable future investigators to gain insight into non-simple thermoelasticity with diferent phase delays of higher‑order in detail. List of symbols , µ Lamé’s constants αt Termal expansion coefcient Ce Specifc heat γ = (3 + 2µ)αt Termal coupling parameter T0 Environmental temperature θ = T − T0 Temperature increment T Absolute temperature u Displacement vector e = divu Cubical dilatation σij Stress tensor eij Strain tensor q Heat fux vector K Termal conductivity ρ Material density Q Heat source t Te time δij Kronecker’s delta function 1Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat, Saudi Arabia. 2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 3Mathematics Department, Faculty of Science, Benha University, P.O. 13518, Benha, Egypt. 4Department of Mathematics, South Valley University, Qena, Egypt. 5Department of Mathematics, Faculty of Science, University of Kordofan, El-Obeid, Sudan. 6Departement of Mathematics, Faculty of Science and Technologe, Shendi University, Shendi, Sudan. *email: [email protected] SCIENTIFIC REPORTS | (2020) 10:13625 | https://doi.org/10.1038/s41598-020-70388-1 1 Vol.:(0123456789) www.nature.com/scientificreports/ ϕ Conductive temperature τq Phase lag of heat fux τθ Phase lag of temperature τϕ Phase lag of ∇ϕ τ0 Termal relaxation time m, n, p Higher orders S Te entropy Duhamel (1837) was the frst to suggest entering the coupling term in the heat equation and proposing the coupling between the temperature and the deformation. However, the heat equation was not well established by the thermodynamic process. Biot 1 attempted the thermodynamic justifcation for this equation. Te heat equation includes, in this case, the dilatation term depend on the thermodynamics of the non-inverse process. Te classical heat conduction theory in thermoelastic solids is based on the assumption that the heat fow is proportional to the temperature gradient. Based on this assumption, the heat equation is governed by a parabolic system of partial diferential equations, which predicts that the thermal disturbance in the material will immediately afect all points of the body. Te phenomenon of infnite velocity of heat waves conficts with physical phenomena. To overcome this defect, generalized thermodynamic theories were introduced. Lord and Shulman2 introduced a generalized theory of thermoelasticity that provides one relaxation time and thus the system that governs the heat equation has turned into a hyperbolic type. Green and Lindsay 3 proposed another model involved two relaxation times. Te following generalization of the theory of thermoelasticity is known as the dual-phase-delay model which was improved by Tzou3 and Chandrasekharaiah4. Tzou3 refected a constitutive equation to explain the lagging behavior in the heat conduction in solid materials. Tzou5,6 presented a thermoelastic model including dual- phase-delay to the heat fux vector and the temperature gradient. Te two-temperature theory (2TTE), proposed by Chen and Gurtin7, Gurtin and Williams8,9 and Chen et al.10,11, based on two distinct temperatures; conductive ϕ and the thermodynamic θ temperatures. According to this theory, the diference between these two temperatures is proportional to the applied heat source. Also, in the absence of the heat source, the two temperatures are equal 12. Based on this theory of thermoelasticity with two-temperature, several authors have studied several problems of thermoelasticity13–18. Quintanilla19 discussed the structural stability, existence, and spatial behavior of the solution of some problems in 2TT. For isotropic and homogeneous bodies, Youssef20 introduced the theory of thermoelasticity with relaxation time and two- temperature. Mukhopadhyay et al.21 also, extended generalized thermoelasticity with two temperatures and dual-phase-lag. Te two temperatures theory has attracted a lot of attention in the recent years 22–26. Recently, many eforts have been introduced to modify the classical heat conduction law. In one of these eforts, Abouelregal27–29 introduced some generalized models of heat conduction including higher-order time- derivative. Earlier, Chiriţă30 investigated the high-order Fourier law to illustrate the lagging performance of heat transfer. Te current paper is concerned with a generalized model that combines the two-temperature theory and the heat conduction of higher-order time-derivatives with two-phase-lags. To further study the accuracy of the current model, the model was applied to study the interaction in an isotropic solid exposed to external body force and due to sudden heating. Some special cases of concern were also deduced from the proposed model. For further clarifcation and comparison, the numerical results were tabulated and illustrated graphically. Te efects of the parameters of temperature distinction, phase delay, and high-orders on all feld variables inside the material have been studied. Te numerical results obtained in this paper were found to be as good as with the results in the current literature. Also, the results and observations also showed that the analytical solutions correspond well with numerical solutions. Derivation of the modifed model Te Fourier’s law 31, is the closest model representing heat conduction, which assumes that there is a linear rela- tionship between the heat fow q and the temperature gradient ∇θ on the following form: q(x, t) =−K∇θ(x, t) (1) Te following relations are achieved by the increment in entropy S divq + Q =−ρT0S˙ (2) ρT0S = ρCeθ + γ T0divu (3) From Eqs. (2) and (3), we obtain the energy equation as ∂θ ∂ ρCe + γ T0 (divu) =−divq + Q (4) ∂t ∂t Among the governing equations in the two-temperature model (2TT), in addition to the energy Eq. (4), the equation that connecting the two temperatures θ and ϕ is given by7–12: 2 θ = 1 − a∇ ϕ (5) which, a > 0 is the temperature distinction parameter (two-temperature). SCIENTIFIC REPORTS | (2020) 10:13625 | https://doi.org/10.1038/s41598-020-70388-1 2 Vol:.(1234567890) www.nature.com/scientificreports/ Te classical Fourier’s law (1) has been replaced, according to Quintanilla29 with q(x, t) =−K∇ϕ(x, t) (6) In the DPL model, the heat Eq. (6) is modifed by introducing two phase-lags as11: q x, t + τq =−K∇ϕ x, t + τϕ (7) In the above equation, τq and τϕ are respectively the phase lags of the heat fux and conductive temperature gradient. Introducing the phase lag of the temperature τθ , in addition to the phase lag τq of conductive temperature, we get the following relation 2 θ(x, t + τθ ) = 1 − a∇ ϕ x, t + τϕ (8) Taylor-series-expansion is applied to both sides of Eqs. (7) and (8) and maintain the terms up to a suitable higher-order of time-deferential (m , n, p ) in τq , τϕ , and τθ respectively, to acquire m r r n r r τq ∂ τϕ ∂ 1 + q =−K 1 + ∇ϕ (9) r! ∂tr r! ∂tr = = r 1 r 1 p r r n r r τ ∂ τϕ ∂ 1 + θ θ = 1 − a∇2 1 + ϕ r! ∂tr r! ∂tr (10) r=1 r=1 It was highlighted that the proposed models associated with the higher-order time-diferential have been extensively considered in many papers regarding their thermodynamic consistency and also with regard to well- presented issues and stimulating stability 32–37. If we adjoin Eq. (9) with the energy Eq. (4), we get n τ r ∂r m τ r ∂r ϕ 2 q ˙ K 1 + ∇ ϕ = 1 + ρCeθ + γ T0e˙ − ρQ (11) r! ∂tr r! ∂tr r=1 r=1 By studying a system of equations similar to the foundational equations of type (9) or (10), Chiriţă et al.35 explain that there are some restrictions to choosing the higher orders m, n, and p , for example when m ≥ 5 leads to an unstable system, and therefore cannot describe a real physical state. In addition, the feld equations, the constitutive relations and the train–displacement relation to thermoelastic isotropic materials at uniform environmental temperature T0 are: σij = 2µeij + δij eij − γθ (12) 2eij = uj,i + ui,j (13) µui,jj + ( + µ)uj,ij − γθ,i + Fi = ρu¨i (14) Application to the model As to achieve the accuracy of the presented model, we are now studying a thermoelastic body that is exposed to thermal shock and is afected by an external force. It is assumed that all feld variables depend only on the distance x and instant time t . Ten displacement components have the form ux = u(x, t), uy = 0, uz = 0. (15) Te non-zero strain is given by ∂u(x, t) e = (16) ∂x Te components of the external body strength can be chosen

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