The Effect of Fractional Time Derivative on Two-Dimension Porous Materials Due to Pulse Heat Flux

Abstract

In the present article, the generalized thermoelastic wave model with and without energy dissipation under fractional time derivative is used to study the physical field in porous two-dimensional media. By applying the Fourier-Laplace transforms and eigenvalues scheme, the physical quantities are presented analytically. The surface is shocked by heating (pulsed heat flow problem) and application of free traction on its outer surface (mechanical conditions) by the process of temperature transport (diffusion) to observe the full analytical solutions of the main physical fields. The magnesium (Mg) material is used to make the simulations and obtain numerical outcomes. The basic physical field quantities are graphed and discussed. Comparisons are made in the results obtained under the strong (SC), the weak (WC) and the normal (NC) conductivities.

1. Introduction

Porous media appear in many forms of environmental, natural, and synthetic implementations and in several technologies. To overcome the first insufficiency in the decoupled thermoelasticity theorem, in 1956, Biot [1] presented the coupled thermoelasticity theorem to control the first insufficiency in the decoupled thermoelastic model, which prognosticates two phenomena not suitable for physical observation. Firstly, the thermal conductivity equation is parabolic, presenting an infinite propagation speed for thermal waves. Secondly, the thermal conductivity equation of this model does not contain an elastic term. Rosencwaig et al. [2] investigated the local thermoelastic deformation of the model caused by excitations.

Biot developed poroelasticity models [1,3,4] for a high–low-frequency range and built upon the coupled thermoelasticity hypothesis to overcome the inconsistency in the uncoupled hypothesis [1]. The heat conduction and elasticity equations in this theory are coupled. However, it includes a drawback of the uncoupled hypothesis in which the heat wave propagates with an infinite velocity that is impractical in nature. Then, to solve the problem of the coupled hypothesis, generalized thermoelasticity models were expanded. It is recognized that there are several generalizations of the thermoelasticity hypothesis, such as that presented by Lord-Shulman [5]. Green-Naghdi [6,7,8] formulated three types of models (GNI, GNII, and GNIII). The constitutive equations of the GN models are linearized, where the first type is similar to the classical coupled thermoelastic model, the second type demonstrates the propagations of thermal signals with finite velocity without energy dissipations, and the third type proposes the finite speed of propagations with energy dissipations. During the second half of the 19th century, it can be said that the complete model of fractional integrals and derivatives was determined. In the context of generalized thermoelastic models, Youssef [9,10] established generalized fractional-order thermoelastic models under strong, normal, and weak conductivities. Sherief et al. [11] presented a new theory by using the thermal conduction law. Ezzat and Elkaramany [12] established another theory for a generalized fractional-order thermoelasticity model based on Taylor’s expansions of time-fractional order. Ezzat et al. [13] studied modeling in a generalized thermoelastic model using a memory-dependent derivative. Marin [14] discussed basic models in elastostatic micropolar media. Saeed et al. [15] used the finite-element scheme to investigate thermoelastic interactions in poroelastic media using the GL model.

Ouyang et al. [16] proposed three-equation models considering local thermal nonequilibrium states. Many authors [17,18,19,20,21,22,23] have discussed the solutions to some problems using various generalized thermoelastic models. Hussein [24] presented a mathematical model for a spherical thermoelastic porous material. Hobiny and Abbas [25] discussed the effect of a dual-phase lag model in two-dimensional porous media. Biswas [26] investigated surface waves in nonlocal, orthotropic thermoelastic porous media. Carini and Zampoli [27] studied porous matrices with three delay times. Abbas and Marin [28] presented the analytical solution of a two-dimensional generalized thermodiffusion problem subjected to a laser pulse. Shekhar [29] studied deformations caused by thermal shock in a porous thermoelastic medium with properties dependent on the reference temperature. Calin et al. [30] investigated improvements in the rigidity of circular-plate composites. Abd-Elaziz et al. [31] discussed the Thomson impacts on thermoelastic porous materials based on Green and Naghdi’s model. Abbas and Kumar [32] investigated deformations induced by a heating source in the micropolar plane under generalized thermoelasticity using the finite-element scheme.

Ellahi et al. [33,34] studied the solutions of different problems under several boundary conditions in porous media. Singh [35] has investigated the wave propagations in a medium with voids under thermoelasticity models. Palani and Abbas [36] discussed the free convections magnetohydrodynamics flow with thermal radiations. Villatoro et al. [37] have applied the perturbation approach upon Laplace transform to get the solution of the heating equation in porous media which consists of gas and solid phases. The discontinuous front was observed in gas temperature because of incompatibility between the initial and boundary conditions, leading to the constant speed for thermal propagation. However, there was a smooth front at the solid temperature using an internal layer of asymptotic approximation. Abbas [38] discussed the natural frequency of a poro-elastic hollow cylinders. Alzahrani et al. [39] used the eigenvalues approaches to investigate the effects of thermal relaxation times in two-dimension porous media under strong, normal and weak thermal conductivities.

The objective of this work is to study the effects of strong, weak, and normal conductivities in a two-dimensional porous medium by using the eigenvalue scheme. By using the eigenvalue technique with Fourier–Laplace transforms on numerical and analytical methods, basic formulations are presented. The nondimensional temperature, displacements, stresses, and volume fraction are obtained and represented graphically. In the calculations, the impacts of strong, normal, and weak thermal conductivities on the considered variables are investigated and compared.

2. Basic Equations

For an isotropic 2D elastic porous material, the basic formulations based on [9,35] models without body force and the heating resources are expressed as:

$$\left(\lambda +\mu \right)u_{j,ij}+\mu u_{i,jj}+b{\phi }_{,i}-{\gamma }_t{\mathsf{\Theta }}_{,i}=\rho \frac{{\partial }^2{u}_i}{\partial t^2},$$

(1)

$$\alpha {\phi }_{,jj}-b{u}_{j,j}-{\zeta }_1\phi -{\omega }_o\frac{\partial \phi }{\partial t}+m\mathsf{\Theta }=\rho \psi \frac{{\partial }^2\phi }{\partial t^2},$$

(2)

$${\left(\left(K^{*}+K\frac{\partial }{\partial t}\right)I^{ϵ-1}{\mathsf{\Theta }}_{,j}\right)}_{,j}=\frac{\partial }{\partial t}\left(\rho c_e\frac{\partial \mathsf{\Theta }}{\partial t}+m{T}_o\phi +{\gamma }_t{T}_o\frac{\partial u_{j,j}}{\partial t}\right),0<ϵ\le 2.$$

(3)

where the integral operator of fractional derivative is expressed as [9]

$$I^{ϵ-1}g\left(\mathit{r},t\right)=\frac{1}{\mathsf{\Gamma }(\in )}{{\displaystyle \int }}_0^t{\left(t-\tau \right)}^{ϵ-1}g\left(\mathit{r},\tau \right)d\tau , \{\begin{array}{cc}0<ϵ<1,& weak conductivity\\ ϵ=1,& normal conductivity\\ 1<ϵ\le 2,& strong conductivity\end{array},$$

(4)

where $\mathsf{\Gamma }(\in )$ is the Gamma function. The stress-displacement equations are defined by [35]

$${\sigma }_{ij}=\mu \left(u_{i,j}+u_{j,i}\right)+\left(\lambda u_{k,k}+b\phi -{\gamma }_t\mathsf{\Theta }\right){\delta }_{ij}.$$

(5)

where $m$ is the coefficient of thermo-void, $\psi$ is the equilibrated inertia, $b$ is the measure of diffusion effects, ${\omega }_o, \alpha ,{\zeta }_1$ are the parameters of voids material, $c_e$ is the specific heat, $K^{*}$ is the model characteristic material constant, $\mathsf{\Theta }=T-T_o$, $T_o$ is the reference temperature, $\rho$ is the material density, ${\sigma }_{ij}$ are the stress components, $K$ is the thermal conductivity coefficient, $\mu ,\lambda$ are the Lame’s parameters, $t$ is the time, $u_i$ are the components of displacement, $i,j,k=1,2,3$, ${\gamma }_t=\left(3\lambda +2\mu \right){\alpha }_t$, ${\alpha }_t$ is the linear coefficient of thermal expansions.

3. Formulations of the Problem

We consider a two-dimension porous material fills the region $0\le x\le \infty , -\infty \le y\le \infty$. By using the Cartesian co-ordinates $\left(x,y,z\right)$, the governing equations with the components of the displacement $\left(u,v,0\right)$ can be given by

$$\left(\lambda +2\mu \right)\frac{{\partial }^{2}u}{\partial x^2}+\left(\lambda +\mu \right)\frac{{\partial }^{2}v}{\partial x\partial y}+\mu \frac{{\partial }^{2}u}{\partial y^2}+b\frac{\partial \phi }{\partial x}-{\gamma }_t\frac{\partial \mathsf{\Theta }}{\partial x}=\rho \frac{{\partial }^{2}u}{\partial t^2}$$

(6)

$$\left(\lambda +2\mu \right)\frac{{\partial }^{2}v}{\partial y^2}+\left(\lambda +\mu \right)\frac{{\partial }^{2}u}{\partial x\partial y}+\mu \frac{{\partial }^{2}v}{\partial x^2}+b\frac{\partial \phi }{\partial y}-{\gamma }_t\frac{\partial \mathsf{\Theta }}{\partial y}=\rho \frac{{\partial }^{2}v}{\partial t^2},$$

(7)

$$\alpha \left(\frac{{\partial }^2\phi }{\partial x^2}+\frac{{\partial }^2\phi }{\partial y^2}\right)-{\omega }_o\frac{\partial \phi }{\partial t}-{\zeta }_1\phi -b\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)+m\mathsf{\Theta }=\rho \psi \frac{{\partial }^2\phi }{\partial t^2},$$

(8)

$$\left(K^{*}+K\frac{\partial }{\partial t}\right)I^{ϵ-1}\left(\frac{{\partial }^2\mathsf{\Theta }}{\partial x^2}+\frac{{\partial }^2\mathsf{\Theta }}{\partial y^2}\right)=\frac{\partial }{\partial t}\left(\rho c_e\frac{\partial \mathsf{\Theta }}{\partial t}+m{T}_o\phi +{\gamma }_t{T}_o\frac{\partial }{\partial t}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)\right),$$

(9)

$${\sigma }_{xx}=\lambda \frac{\partial v}{\partial y}+\left(\lambda +2\mu \right)\frac{\partial u}{\partial x}+b\phi -{\gamma }_t\mathsf{\Theta }, {\sigma }_{xy}=\mu \left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right).$$

(10)

The problem initial conditions are defined by

$$\phi =\frac{\partial \phi }{\partial t}=0,\mathsf{\Theta }=\frac{\partial \mathsf{\Theta }}{\partial t}=0, u=\frac{\partial u}{\partial t}=0, v=\frac{\partial v}{\partial t}=0,t=0.$$

(11)

While, the problem boundary conditions are given by

$${\sigma }_{xx}=0,{\sigma }_{xy}=0,-\left(K^{*}+K\frac{\partial }{\partial t}\right)\frac{\partial \mathsf{\Theta }\left(x,y,t\right)}{\partial x}=q_o\frac{t^2{e}^{-\frac{t}{t_p}}}{16t_p^2}H\left(a-\left|y\right|\right), \frac{\partial \phi }{\partial x}=0$$

(12)

where $q_o$ is a constant, $t_p$ is the time of the flux pulse heating characteristics, and H is the function unit step. For appropriateness, the nondimensional variables can be taken as

$$\begin{gathered} {\mathsf{\Theta }}^{\prime }=\frac{\mathsf{\Theta }}{T_o}, {\phi }^{\prime }=\psi {\eta }^2{c}^2\phi , \left({\sigma }_{xx}^{\prime },{\sigma }_{xy}^{\prime }\right)=\frac{\left({\sigma }_{xx}, {\sigma }_{xy}\right)}{\left(\lambda +2\mu \right)}, \\ \left(t^{\prime },t_p^{\prime }\right)=\eta c^2\left(t,t_p\right), \left(u^{\prime },v^{\prime },x^{\prime },y^{\prime }\right)=\eta c\left(u,v,x,y\right), \end{gathered}$$

(13)

where $\eta =\frac{\rho c_e}{k}$ and $c=\sqrt{\frac{\lambda +2\mu }{\mathsf{\rho }}}$. In these nondimensional terms of the variables in Equation (13), the basic formulations can be written by (the dashes have been neglected for convenience)

$$\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial x^2}+\left(1-r_1\right)\frac{{\partial }^{2}v}{\partial x\partial y}+r_1\frac{{\partial }^{2}u}{\partial y^2}+r_2\frac{\partial \phi }{\partial x}-r_3\frac{\partial \mathsf{\Theta }}{\partial x},$$

(14)

$$\frac{{\partial }^{2}v}{\partial t^2}=\frac{{\partial }^{2}v}{\partial y^2}+\left(1-r_1\right)\frac{{\partial }^{2}u}{\partial x\partial y}+r_1\frac{{\partial }^{2}v}{\partial x^2}+r_2\frac{\partial \phi }{\partial y}-r_3\frac{\partial \mathsf{\Theta }}{\partial y},$$

(15)

$$r_4\frac{{\partial }^2\phi }{\partial t^2}=\frac{{\partial }^2\phi }{\partial x^2}+\frac{{\partial }^2\phi }{\partial y^2}-r_5\frac{\partial \phi }{\partial t}-r_6\phi -r_7\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)+r_8\mathsf{\Theta },$$

(16)

$$\left(r_9+\frac{\partial }{\partial t}\right)\left(\frac{{\partial }^2\mathsf{\Theta }}{\partial x^2}+\frac{{\partial }^2\mathsf{\Theta }}{\partial y^2}\right)=\frac{{\partial }^{ϵ}}{\partial t^{ϵ}}\left(\frac{\partial \mathsf{\Theta }}{\partial t}+r_{10}\phi +r_{11}\frac{\partial }{\partial t}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)\right),$$

(17)

$${\sigma }_{xx}=\left(1-2r_1\right)\frac{\partial v}{\partial y}+\frac{\partial u}{\partial x}+r_2\phi -r_3\mathsf{\Theta }, {\sigma }_{xy}=r_1\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right).$$

(18)

$${\sigma }_{xx}={\sigma }_{xy}=0.0, \left(r_9+\frac{\partial }{\partial t}\right)\frac{\partial \mathsf{\Theta }\left(x,y,t\right)}{\partial x}=-q_o\frac{t^2{e}^{-\frac{t}{t_p}}}{16t_p^2}H\left(a-\left|y\right|\right), \frac{\partial \phi }{\partial x}=0,$$

(19)

where $r_1=\frac{\mu }{\rho c^2}$, $r_2=\frac{b}{\rho \psi {\eta }^2{c}^4}$, $r_3=\frac{{\gamma }_t{T}_o}{\rho c^2}$, $r_4=\frac{\rho c^2\psi }{\alpha }$, $r_5=\frac{{\omega }_o}{\alpha \eta }$, $r_6=\frac{{\zeta }_1}{\alpha {\eta }^2{c}^2}$, $r_7=\frac{b\psi }{\alpha }$, $r_8=\frac{m\psi T_o}{\alpha },$ $r_9=\frac{K^{*}}{\rho c_e{c}^2},$ $r_{10}=\frac{m}{\psi {\eta }^4{c}^{4}k}$, $r_{11}=\frac{{\gamma }_t}{\rho c_e}$.

Now, the Laplace transforms for any function $f\left(x,y,t\right)$, are given by

$$\overline{f}\left(x,y,s\right)={{\displaystyle \int }}_0^{\infty }f\left(x,y,t\right)e^{-st}dt,$$

(20)

however, the Fourier transform for any function $\overline{f}\left(x,y,s\right)$ can be expressed by

$${\overline{f}}^{*}\left(x,q,s\right)={{\displaystyle \int }}_{-\infty }^{\infty }\overline{f}\left(x,y,s\right)e^{-iqy}dy,$$

(21)

Thus, the governing formulations are expressed to obtain the ordinary differential equations with the boundary conditions as follow

$$\frac{d^2{\overline{u}}^{*}}{d{x}^2}=\left(s^2+r_1{q}^2\right){\overline{u}}^{*}-iq\left(1-r_1\right)\frac{d{\overline{v}}^{*}}{dx}-r_2\frac{d{\overline{\phi }}^{*}}{dx}+r_3\frac{d{ \overline{\mathsf{\Theta }}}^{*}}{dx},$$

(22)

$$\frac{d^2{\overline{v}}^{*}}{d{x}^2}=\frac{\left(s^2+q^2\right)}{r_1}{\overline{v}}^{*}-\frac{r_{2}iq}{r_1}{\overline{\phi }}^{*}+\frac{r_{3}iq}{r_1}{ \overline{\mathsf{\Theta }}}^{*}-\frac{iq\left(1-r_1\right)}{r_1}\frac{d{\overline{u}}^{*}}{dx},$$

(23)

$$\frac{d^2{\overline{\phi }}^{*}}{d{x}^2}=r_{7}iq{\overline{v}}^{*}+\left(s^2{r}_4+q^2+r_6+r_{5}s\right){\overline{\phi }}^{*}-r_8{ \overline{\mathsf{\Theta }}}^{*}+r_7\frac{d{\overline{u}}^{*}}{dx},$$

(24)

$$\frac{d^2{ \overline{\mathsf{\Theta }}}^{*}}{d{x}^2}=\frac{r_{11}iq{s}^{\in +1}}{r_9+s}{\overline{v}}^{*}+\frac{r_{10}{s}^{\in }}{r_9+s}{\overline{\phi }}^{*}+\left(q^2+\frac{s^{\in +1}}{r_9+s}\right){ \overline{\mathsf{\Theta }}}^{*}+\frac{s^{\in +1} r_{11}}{r_9+s}\frac{d{\overline{u}}^{*}}{dx},$$

(25)

$${\overline{\sigma }}_{xx}^{*}=iq\left(1-2r_1\right){\overline{v}}^{*}+\frac{d{\overline{u}}^{*}}{dx}+r_2{\overline{\phi }}^{*}-r_3{ \overline{\mathsf{\Theta }}}^{*}.$$

(26)

$${\overline{\sigma }}_{xy}^{*}=r_1\left(\frac{d{\overline{v}}^{*}}{dx}+iq{\overline{u}}^{*}\right),$$

(27)

$${\overline{\sigma }}_{xx}^{*}={\overline{\sigma }}_{xy}^{*}=0,\frac{d{\overline{\phi }}^{*}}{dx}=0,\frac{d {\overline{\mathsf{\Theta }}}^{*}}{dx}=-\frac{q_o{t}_p}{8\left(r_9+s\right){\left(s{t}_p+1\right)}^3}\sqrt{\frac{2}{\pi }}\frac{sin\left(qa\right)}{q},$$

(28)

Now, the vector-matrix differential equations (Equations (22)–(25)) are written as

$$\frac{dM}{dx}=AM,$$

(29)

where $A$ and $M$ are defined as shown in Appendix A. By using the eigenvalues method as in [17,40,41,42,43], the exact solutions for the Equation (29) are obtained. So, the matrix characteristic equation for $A$ are taken the form as follow:

$${\xi }^8-f_1{\xi }^6+f_2{\xi }^4+f_3{\xi }^2+f_4=0,$$

(30)

where $f_1,$ $f_2$, $f_3$ and $f_4$ are determined as in Appendix B. To obtain the solution of Equation (29), the eigenvalue of matrix $A$ and its eigenvectors must be computed, where ${\xi }_1, {\xi }_2, {\xi }_3$, ${\xi }_4$, $-{\xi }_1,- {\xi }_2,- {\xi }_3$ and $- {\xi }_4$ are the eigenvalues which have the corresponding eigenvectors as in Appendix C. Thus, the analytical solutions of Equation (29) are written as:

$$M\left(x,q,s\right)={{\displaystyle \sum }}_{i=1}^4{B}_i{Y}_i{e}^{-{\xi }_{i}x},$$

(31)

The positive exponentials can be discarded, which, due to the conditions of the regulations of the solution at infinity and $B_1, B_2, B_3$, and $B_4$, are constants that are computed using the conditions of the problem boundary. Now, for each function ${\overline{f}}^{*}\left(x, q, s\right)$, the inverse Fourier transform can be expressed by

$$\overline{f}\left(x,y,s\right)=\frac{1}{\sqrt{2\pi }}{{\displaystyle \int }}_{-\infty }^{\infty }{\overline{f}}^{*}\left(x,q,s\right)e^{iqy}dq,$$

(32)

Finally, to get the general solution for the displacement, the stresses components, the changes in volume fraction field of void distributions $\phi$ and the variations of temperature with respect to the distances $x$, $y$ at any time $t$, Stehfest [44] numerical inversion schemes were chosen. In these schemes, the Laplace transform inverse for $\overline{f}\left(x,y,s\right)$ are defined by

$$f\left(x,y,t\right)=\frac{ln\left(2\right)}{t}{{\displaystyle \sum }}_{n=1}^N{V}_n\overline{f}\left(x,y,n\frac{ln\left(2\right)}{t}\right),$$

(33)

where

$$V_n={\left(-1\right)}^{\left(\frac{N}{2}+1\right)}{{\displaystyle \sum }}_{p=\frac{n+1}{2}}^{min\left(n,\frac{N}{2}\right)}\frac{\left(2p\right)!p^{\left(1+\frac{N}{2}\right)}}{p!\left(n-p\right)!\left(\frac{N}{2}-p\right)!\left(2n-1\right)!},$$

(34)

where $N$ is the term number.

4. Numerical Results

For numerical examples, the magnesium mediums can be taken to object of numerical calculations. The parameters values of (Mg) are taken from [45]

$$\begin{gathered} \alpha =3.688\times {10}^{-5}\left(\mathrm{N}\right), {\zeta }_1=1.475\times {10}^{10}\left(\mathrm{N}\right)\left({\mathrm{m}}^{-2}\right),{\mathsf{\omega }}_{\mathrm{o}}=0.0787\times {10}^{-3}\left(\mathrm{N}\right)\left({\mathrm{m}}^{-2}\right)\left({\mathrm{s}}^{-1}\right), \\ \mu =3.278\times {10}^{10}\left(\mathrm{N}\right)\left({\mathrm{m}}^{-2}\right), \lambda =2.17\times {10}^{10}\left(\mathrm{N}\right)\left({\mathrm{m}}^{-2}\right),\rho =1740\left(\mathrm{kg}\right)\left({\mathrm{m}}^{-3}\right), t=0.3, \\ {T}_o=298\left(\mathrm{k}\right), c_e=1040 \left(\mathrm{J}\right)\left({\mathrm{kg}}^{-1}\right)\left({\mathrm{k}}^{-1}\right), a=0.25, \psi =1.753\times {10}^{-15}\left({\mathrm{m}}^2\right), \\ {\alpha }_t=1.98\times {10}^{-6}\left({\mathrm{k}}^{-1}\right),K=1.7 \left(\mathrm{W}\right)\left({\mathrm{k}}^{-1}\right)\left({\mathrm{m}}^{-1}\right), \beta =2.68\times {10}^6\left(\mathrm{N}\right)\left({\mathrm{m}}^{-2}\right)\left({\mathrm{k}}^{-1}\right), \\ b=1.13840\times {10}^{10}\left(\mathrm{N}\right)\left({\mathrm{m}}^{-2}\right),m=2\times {10}^6\left(\mathrm{N}\right)\left({\mathrm{m}}^{-2}\right)\left({\mathrm{k}}^{-1}\right). \end{gathered}$$

The above data have been used to study the strong (SC), the normal (NC) and the weak (WC) conductivities in 2D porous materials by the eigenvalue method. The voids changes in volume fraction field distribution $\phi$, The variations of temperature $\mathsf{\Theta }$, the stresses ${\sigma }_{xx}, {\sigma }_{xy}$ and the of displacement components $u, v$ are studied. The material is considered to be homogeneous two-dimensions media. Figure 1 displays the change in the volume fraction field of void distributions $\phi$ along $x$. It is clear that it reduces with the rising $x$ till attaining zeros. Figure 2 displays the temperature variations via $x$. It is noticed that it starts from heights value according to the application of boundary condition and reduces with the rising $\mathrm{x}$ to come to zeros. Figure 3 depicts the variations of vertical displacement with respect to x which have maximum value on $x=0$ and reduces with the rising $x$. Figure 4 illustrates the horizontal displacement variations $u$ via $x$. It is indicated that it attains maximums negative value and gradually rises till it attains peak value at specific locations in close nearness to $x=0$ and after that reduces to come to zeros. As seen in Figure 4, the displacement changes continuously from negative to positive and after that goes down to zero, which is caused by the combined effect of the traction free bounding surface, thermal expansion and finite heat speed. Figure 5 and Figure 6 show the components of stress variations ${\sigma }_{xy}$ and ${\sigma }_{xx}$ along $x$. It is noticed that the magnitudes of components, permanently begin from zeros which obeyed the boundary condition.

Figure 7 and Figure 8 show the changes in volume fraction field of voids variations $\phi$ and the variations of temperature $\mathsf{\Theta }$ via the distance $y$ when $x=0.4$. It is indicated that the variations of changes in volume fraction field of void and the variations of temperature have maximum values at the length of heating surfaces $\left(\left(\left|y\right|\le 0.4\right)\right)$ after that begin to reduces totally near the edges $\left(\left(\left|y\right|\le 0.4\right)\right)$ where they reduce and reach to zeros values. Figure 9 displays the vertical displacement variation $v$ via $y$. It is noticed that it starts the rising at the start and end of the thermal surfaces $\left(\left|y\right|\le 0.4\right)$, and have small values at the center of the thermal surfaces $\left(\left|y\right|\le 0.4\right)$, after that it begins the rising and reach ultimate value totally near the edges $\left(y=\pm 0.4\right)$, then it reduces to come to zeros. Figure 10 shows the variation of horizontal displacement $u$ along $x$. It is observed that the horizontal displacement has a maximum value at the length of the heating surface $\left(\left|y\right|\le 0.4\right),$ and then it starts to decrease totally near the edges $\left(y=\pm 0.4\right)$, and then reduces to zeros values. The stresses ${\sigma }_{xx}$ and ${\sigma }_{xy}$ with respect to $y$ are presented in Figure 11 and Figure 12.

Finally, Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 explain the variations of all the studied fields along the distance $y$ and the distance $x$ at $t=0.3$. These figures display the predict curves during the strong, the weak and the normal conductivities. Unsurprisingly, you can find that the stages of the strong, normal and weak conductivities have great influences on the values of variables. According to the fractional-order generalized thermoelastic model, we have to construct new classifications for all mediums according to their fractional parameter where these parameters become new indicators of their power to conduct thermal energy.

5. Conclusions

In this work, we studied the impacts of strong, weak, and normal thermal conductivities in porous materials under the generalized fractional-order thermoelastic model. The resulting nondimensional equations were solved by the Fourier–Laplace transformation method and, following that, applying the eigenvalue approach. The significant impacts of the strong, normal, and weak thermal conductivities were discussed for all physical quantities. Accordingly, generalized thermoelastic fractional-order models are considered as an advancement in the study of porous elastic materials.