Dual-phase-lag model on microelongated thermoelastic rotating medium
Mohamed I. A. Othman^{1} , Sarhan Y. Atwa^{2} , E. E. M. Eraki^{3} , Mohamed F. Ismail^{4}
^{1, 3}Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt
^{2, 4}Department of Engineering Mathematics and Physics, Higher Institute of Engineering, El Shorouk Academy, P.O. Box 11837, Cairo, Egypt
^{1}Corresponding author
Journal of Engineering and Thermal Sciences, Vol. 2, Issue 1, 2022, p. 13-26.
https://doi.org/10.21595/jets.2022.22597
Received 17 April 2022; accepted 12 May 2022; published 30 June 2022
The dual-phase-lag (DPL) model is applied to study the influence of rotation on a two-dimensional micro-elongated thermoelastic medium problem. Mechanical force along with the layer of the elastic half-space interface and micro-elongated thermoelastic half-space is applied. The analytic expressions for displacement component, temperature distribution, micro-elongational scalar and stress components have been derived and represented graphically. The rotation has been studied in the presence DPL model and Lord-Shulman theory.
Keywords: thermoelasticity, micro-elongation, rotation, normal mode, elastic solid.
1. Introduction
A microelongated elastic solid possesses four degrees of freedom: three for translation and micro-elongation. In micro-elongation theory, the material particles can perform only volumetric micro-elongation in addition to classical deformation of the medium. The material points of such a medium can stretch and contract independently of their translations. Solid-liquid crystals, composite materials reinforced with chopped elastic fibres porous media with pores filled with non-viscous fluid or gas can be categorized as micro-elongated medium. The variation of periodical heat source response in a functionally graded micro-elongated medium was discussed by [1, 2]. The plane strain problem in a thermoelastic micro-elongated solid with an overlying infinite non-viscous fluid was discussed by [3]. More interesting problems have been studied about the thermoelastic micro-elongated solid in different cases [4-9].
In the generalized theories, the governing equations involve thermal relaxation times and they are of a hyperbolic type. The extended thermoelasticity theory by [10-14] which introduces one relaxation time in the thermoelastic process and the temperature-rate dependent theory of thermoelasticity by [15], which takes into account two relaxation times are two well-established generalized theories of thermoelasticity.
A new model called the dual-phase-lag model for the heat transport mechanism in which Fourier’s law is replaced by an approximation to the modification of Fourier’s law with two different time translations for the heat flux and the temperature gradient was investigated by [16-20]. The effect of thermal loading due to laser pulse in generalized thermoelastic medium with voids in the dual-phase-lag model was studied by [21]. A dynamic problem in thermoelastic solid using a dual-phase-lag model with an internal heat source was explained by [22]. The effect of rotation on micropolar generalized thermoelasticity with two-temperatures and the thermal laser pulse using a DPL model has been discussed by [23, 24].
In this paper, the effect of rotation on a two-dimensional micro-elongated thermoelastic medium problem has been discussed. The normal mode analysis is used to derive the expressions for the considered variables for a DPL model of thermoelasticity and the variances of the considered variables are represented graphically.
2. Formulation of the problem
The system of prevailing equations of a microelongated thermoelastic solid with rotation, in the DPL model, as [4, 7] (Fig. 1):
where the displacement vector $u(x,z,t)=u({u}_{1},0,{u}_{3})$, and the rotation $\mathrm{\Omega}=(0,\mathrm{\Omega},0)$ then the equations of motion are given by:
For simplification, we shall use the following non-dimensional variables:
${t}^{\text{'}}={w}^{*}t,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{{\tau}^{\text{'}}}_{\theta}={w}^{*}{\tau}_{\theta},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{{\tau}^{\text{'}}}_{q}={w}^{*}{\tau}_{q},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\sigma \text{'}}_{ij}=\frac{{\sigma}_{ij}}{{\beta}_{0}{T}_{0}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\sigma}_{ij}^{e\text{'}}=\frac{{\sigma}_{ij}^{e}}{{\beta}_{0}{T}_{0}},$
${\phi}^{\text{'}}=\frac{{\lambda}_{0}}{{\beta}_{0}{T}_{0}}\phi ,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{T}^{\text{'}}=\frac{T}{{T}_{0}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\mathrm{\Omega}}^{\text{'}}=\frac{\mathrm{\Omega}}{{w}^{*}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{{P}^{\text{'}}}_{1}=\frac{{P}_{1}}{{\beta}_{0}{T}_{0}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{w}^{*}=\frac{\rho {c}_{1}^{2}{c}_{e}}{k},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{c}_{1}^{2}=\frac{\lambda +2\mu}{\rho}.$
Fig. 1. Geometry of the problem
The displacement potentials $\mathrm{\Phi}(x,z,t)$ and $\mathrm{\Psi}(x,z,t)$ which relate to displacement components have been introduced, we obtain:
From Eqs. (7) and (8) into Eqs. (2), (3), (5) and (6), we obtain:
3. Normal mode analysis
The solution of the considered physical variable can be decomposed in terms of normal modes as the following form:
where, $\omega $ is a complex constant, $i=\sqrt{-1}$, $b$ is the wave number in the $x$ direction.
Using Eq. (13) into Eqs. (9)-(12), then we have:
Eqs. (14-17) have a non-trivial solution if the determinant coefficients of the physical quantities equal to zero, then we get:
The coefficients ${a}_{i}$, $A$, $B$, $C$, $E$ and ${H}_{in}$ are given in Appendix A1.
Eq. (18) can be factorized as:
where, ${k}_{n}^{2}$, ($n=$1, 2, 3, 4) are roots of the characteristic equation of Eq. (19).
The general solutions of Eq. (19) bound as $(z\to \infty )$ is given by:
Substituting from Eq. (20) into Eq. (8) we obtain the components of displacements:
Substituting Eqs. (7) and (13) into (4) and by using of Eqs. (20)-(22) we get:
The equations of motion and stress components in an elastic medium are given by [7]:
Substituting from Eqs. (7) and (13) into Eq. (24):
Eliminating ${u}_{1}^{e\mathrm{*}}$, ${u}_{3}^{e\mathrm{*}}$ between Eqs. (26) and (27), we obtain:
Eq. (28) can be factorized as:
where, ${r}_{n}^{2}$, ($n=$1, 2) are the roots of the characteristic equation of Eq. (29), the solutions of Eq. (29) are:
From Eqs. (7) and (13) into (25) and by using the Eqs. (30) and (31), we obtain the components of stresses in an elastic medium:
where, the coefficients ${l}_{i}$, ${\delta}_{i}$, $G$, $N$ and ${L}_{in}$ are given in Appendix A2
4. The boundary conditions
The parameters ${M}_{n}$, ($n=$ 1, 2, 3, 4) and ${R}_{n}$, ($n=$1, 2, 3, 4) can be determined by using the boundary conditions at the surface are [3-7]:
${\sigma}_{zz}={\sigma}_{zz}^{e}-{P}_{1}{e}^{\left(\omega t+ibx\right)},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\sigma}_{xz}=0,\mathrm{}\mathrm{}\mathrm{}\mathrm{a}\mathrm{t}z=-h,$
where, ${P}_{1}$ is the magnitude of the mechanical force.
The using of the expressions of the variables considered in the past boundary conditions Eq. (35), to get the equations that are satisfied with the parameters. And hence, eight equations will be gained. The inverse matrix method is applied to the eighth equation, to obtain the value of ${M}_{n}$ ($n=$1, 2, 3, 4) and ${R}_{n}$ ($n=$1, 2, 3, 4):
5. Numerical results and discussion
The analysis is conducted for aluminum epoxy-like material as [24]: $\lambda =$ 7.59×10^{10} N/m^{2}, $\mu =$1.89××10^{10} N/m^{2}, ${a}_{0}=$ 0.61×10^{-10} N, $\rho =$ 2.19×10^{3} kg/m^{3}, ${\beta}_{0}={\beta}_{1}=$ 0,05×10^{5} N/(m^{2}.k), ${c}_{e}=$ 966 J/(kg.k), $k=$ 252 J/(m.s.k), ${j}_{0}=$ 0.196×10^{-4} m^{2}, ${\lambda}_{0}={\lambda}_{1}=$ 0.37×10^{10} N/m^{2}, ${T}_{0}=$ 293 k, ${\tau}_{\theta}=$ 0.02 s, ${\tau}_{q}=$ 0.5 s, $\omega ={\omega}_{0}+i\zeta $, ${\omega}_{0}=$ 3.56, $\zeta =$ – 4.81, $b=$ 8, $h=$ 1×10^{-6}.
The physical constants for elastic medium (granite) as [5]: ${\lambda}^{e}=$ 0.884×10^{10} N/m^{2}, ${\mu}^{e}=$1.2667×10^{10} N/m^{2}, ${\rho}^{e}=$ 2.6×10^{3} kg/m^{3}, ${c}_{e}^{e}=$720.7 J/(kg.k), ${k}^{e}=$ 3.1 J/(m.s.k.)
In light of the results of this paper, the computations are conducted for the value of non-dimensional time $t=$ 0.01, in the range of $0\le z\le 0.4$ on the surface $x=$ 3.01. The numerical strategy stated herein is utilized for the distribution of horizontal displacement ${u}_{1}$, the vertical displacement ${u}_{3}$, the temperature $T$, the micro-elongational scalar $\phi $, the stress components ${\sigma}_{xx}$, ${\sigma}_{zz}$ and ${\sigma}_{xz}$ with distance $z.$ To study the influence of rotation on the solution in the DPL model and the L-S theory and the effect of phase-lag of heat flux and phase-lag of temperature gradient on the solution in the DPL model, this paper introduces the results of the numerical assessment in the form of graphs. The results are shown in Figs. 2-15 for the mechanical force with magnitude ${P}_{1}=1$ for the DPL model and the L-S theory.
Fig. 2. Distribution of the horizontal displacement ${u}_{1}$ with the distance $z$
Fig. 3. Distribution of the vertical displacement ${u}_{3}$ with the distance $z$
Fig. 4. Distribution of the temperature $T$ with distance $z$
Fig. 5. Distribution of the micro-elongational scalar $\phi $ with the distance $z$
Fig. 6. Distribution of the force stress component ${\sigma}_{xx}$ with the distance $\mathit{z}$
Fig. 7. Distribution of the force stress component ${\sigma}_{zz}$ with the distance $\mathit{z}$
Fig. 8. Distribution of the force stress component ${\sigma}_{xz}$ with horizontal distance $\mathit{z}$
5.1. Influence of rotation
Figs. 2-8 show comparison between the displacement components ${u}_{1}$, ${u}_{3}$, the temperature $T$, the micro-elongational scalar $\phi $ and the force stresses components ${\sigma}_{xx}$, ${\sigma}_{zz}$, ${\sigma}_{xz}$ for various values of $\mathrm{\Omega}$$(\mathrm{\Omega}=$ 0.2, 0.5) for the DPL model and the L-S theory. Fig. 2 represents the distribution of the horizontal displacement ${u}_{1}$with the distance $z$. It is observed that ${u}_{1}$ increases with the increase of rotation for the two theories. In the (DPL) model and the L-S theory, the values of the horizontal displacement ${u}_{1}$ for $\mathrm{\Omega}=$ 0.2 are small as opposed to those for $\mathrm{\Omega}=$ 0.5. Fig. 3 illustrates the variation of the vertical displacement ${u}_{3}$ against the distance $z$, it is observed that the effect of rotation $\mathrm{\Omega}$ is inversely proportional to the value of the vertical displacement ${u}_{3}$ in the DPL model and the L-S theory i.e the rotation $\mathrm{\Omega}$ has a decreasing effect. Fig. 4 describes the distribution of the temperature $\mathrm{T}$ with the distance $z$. In this figure, all curves start from a positive value and then converge to zero with large values of the distance $z$ and fulfill the boundary condition. It is obvious that the values of the temperature $T$ increases with increase of rotation for the two theories. Fig. 5 shows the variation of the micro-elongational scalar $\phi $ against the distance $z$, it is clear that the values of $\phi $ start from zero and decrease to a minimum then increase up to vanish. It is obvious that the values of $\phi $ increase with the decrease of rotation for two theories, and satisfies the boundary condition. Fig. 6 is plotted to describe the distribution of the stress components ${\sigma}_{xx}$ with the distance $z$. In this figure, all curves begin from a positive value, then decrease to a minimum and increase up to vanishes at large values of $z$. The effect of rotation $\mathrm{\Omega}$ is directly proportional to the value of the stress component ${\sigma}_{xx}$ in both the DPL model and the L-S theory. Fig. 7 exhibits the variation of the stress component ${\sigma}_{zz}$ against the distance $z$. In the DPL model, the effect of different values of rotation is hardly visible. It is shown that the influence of rotation is inversely proportional to the value of the stress component ${\sigma}_{zz}$ in the L-S theory. Fig. 8 compares among the two different values of rotation for the DPL model and the L-S theory. It is obvious that the value of the stress component ${\sigma}_{xz}$ increases as the rotation decreasing.
Fig. 9. Distribution of the horizontal displacement ${u}_{1}$ with the distance $\mathit{z}$
Fig. 10. Distribution of the vertical displacement ${u}_{3}$ with the distance $\mathit{z}$
Fig. 11. Distribution of the temperature $T$ with distance $\mathit{z}$
Fig. 12. Distribution of the micro-elongational scalar $\phi $ with the distance $z$
Fig. 13. Distribution of the force stress component ${\sigma}_{xx}$ with the distance $z$
Fig. 14. Distribution of the force stress component ${\sigma}_{zz}$ with the distance $z$
Fig. 15. Distribution of the force stress component ${\sigma}_{xz}$ with the distance $z$
5.2. Influence of the phase-lag of the heat flux and the phase-lag of temperature gradient
Figs. 9-15 show comparison between the displacement components ${u}_{1}$, ${u}_{3}$, the temperature $T$, the micro-elongational scalar $\phi $ and the force stress components ${\sigma}_{xx}$, ${\sigma}_{zz}$, ${\sigma}_{xz}$, when it comes to thermoelasticity the DPL model for different values of the temperature gradient phase-lag ${\tau}_{\theta}$, such as ${\tau}_{\theta}=$ 0.002, 0.47 at ${\tau}_{q}=$1, $\mathrm{\Omega}=$0.5 and for different values of the heat flux phase-lag ${\tau}_{q}=$ 0.0333, 0.99 at ${\tau}_{\theta}=$ 0.02, $\mathrm{\Omega}=$ 0.5. Fig. 9 depicts that the phase-lag of temperature gradient and the heat flux have an increasing effect on the magnitude of the horizontal displacement ${u}_{1}$, whereas in fig. 15 they have a decreasing effect on the magnitude of the stress component ${\sigma}_{xz}$ in the range 0.$05\le z\le 0.4.$ Figs. 10 and 13 show that the phase-lag of the heat flux has an increasing effect on the vertical displacement ${u}_{3}$ over the range $0\le z\le 0.11$ and on the stress component ${\sigma}_{xx}$ over the range $0\le z\le 0.4,$ whereas, the phase-lag of the temperature gradient has a decreasing influence on both. Figs. 11 and 14 in that order, exhibit that the temperature $T$, and the stress component ${\sigma}_{zz}$ are inversely proportional to the value of ${\tau}_{q}$ for $z>0.$ Fig. 12, explains that the value of micro-elongational scalar $\phi $ increases with the increase of ${\tau}_{q}$ and ${\tau}_{\theta}.$
5.3. The 3D surface curves
Figs. 16-21 are representing the 3D surface curves for the physical quantities, i.e., the horizontal displacement component ${u}_{3}\text{,}$ the micro-elongational scalar $\phi $ and the stress components ${\sigma}_{xx}$, ${\sigma}_{xz}$, for the DPL model by keeping in mind the effect of rotation $\mathrm{\Omega}=$ 0.5. The importance of these figures is that they have been utilized to study the dependence of previous physical quantities on both components of distance.
Fig. 16. 3D curve distribution of the horizontal displacement ${u}_{1}$versus distances at $\mathrm{\Omega}=0.5{\tau}_{\theta}=$ 0.02, ${\tau}_{q}=$ 0.5
Fig. 17. 3D curve distribution of the vertical displacement ${u}_{3}$ versus distances at $\mathrm{\Omega}=$ 0.5, ${\tau}_{\theta}=$ 0.02, ${\tau}_{q}=$ 0.5
Fig. 18. 3D curve distribution of the micro-elongational scalar $\phi $ versus distances at $\mathrm{\Omega}=$ 0.5, ${\tau}_{\theta}=$ 0.02, ${\tau}_{q}$ = 0.5$$
Fig. 19. 3D curve distribution of the force stress component ${\sigma}_{xx}$ versus distances at $\mathrm{\Omega}=$ 0.5, ${\tau}_{\theta}=$ 0.02, ${\tau}_{q}=$ 0.5
Fig. 20. 3D curve distribution of the force stress component ${\sigma}_{zz}$ versus distances at $\mathrm{\Omega}=$ 0.5, ${\tau}_{\theta}=$ 0.02, ${\tau}_{q}=$ 0.5
Fig. 21. 3D curve distribution of the force stress component ${\sigma}_{xz}$ versus distances at $\mathrm{\Omega}=$ 0.5, ${\tau}_{\theta}=$ 0.02, ${\tau}_{q}=$ 0.5
6. Conclusions
According to the results of this work, one can see that the effect of rotation, the micro-elongational scalar and the applied boundary conditions play a major role in the study of thermoelastic medium deformation. The effect of rotation and the micro-elongational scalar has an obvious influence on all physical quantities. All the physical quantities, converge to zero very steeply with the distance $z$ increases. A comparison between the DPL model and the L-S theory is conducted. An analytical solution depended on normal mode analysis of the problem on thermoelastic micro-elongated layer by encircling finite elastic under influence of the rotation has been developed and used it.
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