Generalized Thermo-Diffusion Interaction in an Elastic Medium under Temperature Dependent Diffusivity and Thermal Conductivity

Abstract

The purpose of this work is to investigate, within the context of extended thermo-diffusion theory, the transient thermo-diffusion responses for a half-space with variable thermal conductivity and diffusivity. The half-bounding space’s surface is traction-free and exposed to a time-dependent thermal shock, but the chemical potential is believed to be a known function of time. Because the nonlinear equations are complicated, the finite element technique is applied to solve these equations. Numerical outcomes are produced and graphically illustrated. The effects of varying thermal conductivity and diffusivity on the response are studied using parameter studies. Using the results of this study, researchers hope to understand better how thermo-mechanical fields interact in real materials. By ignoring the new parameter, a comparison of numerical results and analytical cases is produced, and the behavior of physical quantities for numerical solutions is studied to ensure that the proposed technique is accurate.

1. Introduction

Most technical materials used in rockets, nuclear reactors, and other current technological concerns are subjected to high temperatures or temperature gradients. The material properties do not stay constant in such circumstances. Engineering applications such as drying porous materials, heating components and nuclear reactors now more increasingly rely on understanding a material’s changing thermal conductivity and diffusivity. Changing material characteristics due to temperature is well-documented [1,2,3,4]. The mechanical behaviors of material in a high-temperature environment is influenced by the temperature dependency of material characteristics (e.g., the thermal conductivity). According to Godfrey [5], the thermal conductivity of ceramic decreases by 45 percent as the temperature rises from 1 to 400 degrees Celsius. The diffusivity, on the other hand, is generally proportional to the concentration of the diluting component [6]. Most systems have this kind of concentration dependency. A linear or exponential equation may typically be used to estimate thermal conductivity and diffusivity. Mechanical behaviour with varied material qualities must be investigated. Biot [7] developed the coupled thermoelastic theory to address the contradiction inherent in the old uncoupled model that elastic changes have no influence on temperature. In contrast to scientific findings, both theories’ heat equations are of the diffusion type, suggesting unlimited propagation velocities for heat waves. Lord and Shulman [8] integrated the idea of thermal relaxation time into the traditional Fourier law of heat conduction with their extended thermoelasticity theory (LS). Temperature is important in the diffusion process in the preceding cases, and it is critical to study the relationships between the temperature, strain, concentration, and so on. Nowacki [9,10,11] was the first to develop the thermoelastic diffusion theory for elastically deformed solid structures, and he obtained several significant corollaries, findings, and field-governing equations. The linked thermoelastic diffusion models unrealistically predict an unlimited speed of thermal wave propagation. The generalized thermoelastic diffusions model under the Lord–Shulman theory was initially developed by Sherief et al. [12], by including diffusions relaxation parameters into the well-known Fick’s law of mass diffusions in order to overcome this unrealistic finding. Burchuladze [13] studied the non-stationary problem of generalized elasto-thermo-diffusions for inhomogeneous materials. This universal model is said to be capable of bridging the gap between microscopic and macroscopic techniques while also addressing a broad variety of heat transfer models. In an axisymmetric heat supply plate, Kumar et al. [14] investigated the impact of thermal and diffusions phase-lags. Sharma et al. [15] investigated thermodiffusion in an infinite material with a cylindrical hole. The analytical solutions of a two-dimensional generalized thermoelastic diffusions problem produced by a laser pulse was investigated by Abbas and Marin [16]. Othman and Eraki [17] explored the generalized magneto-thermoelastic with diffusion in a half-space subject to initial stress. The fractional thermo-diffusions models under four thermal delay times were examined by Abouelregal et al. [18]. Abbas et al. [19] applied the finite element method to investigate the response of thermal sources in a transversely isotropic thermoelastic diffusion plane. Lotfy [20] studied the effects of varying thermal conductivity under the photo-thermal diffusions process of semiconducting mediums. For higher-order time-fractional four-phase-lag generalized thermoelastic diffusions theories, Molla and Mallik [21] introduced the variational principle, uniqueness, and reciprocity models. Abbas and Seingh [22] investigated the impacts of the rotating thermo-elastic plane with diffusions using the finite element technique. Using Kirchhoff’s transform, Bajpai et al. [23] have discussed the transient responses of a thermo-diffusion elastic thick circular plate with varying diffusivity and thermal conductivity. Deswal et al. [24] studied the reflections of plane waves at the free surface of a magneto-thermo-elastic material under variable diffusivity and thermal conductivity using Kirchhoff’s transform. Thermal diffusivity and thermal conductivity of two-temperatures thermoelastic diffusions plates with variable diffusion have been studied in detail by Sharma et al. [25] using Kirchhoff’s transform. Xue et al. [26] applied the Kirchhoff’s transform to investigate the generalized thermo-diffusions bi-layered structure under varying thermal conductivity and diffusivity. In recent years, generalized thermoelastic models have posed a number of problems, as seen in [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42].

This study investigates the effects of varying thermal conductivity and diffusivity without Kirchhoff’s transform in an elastic medium. The numerical solutions for the explored fields have been created by using the finite element technique. The outcomes are then shown graphically. According to the numerical study, the variable thermal conductivity and diffusivity without Kirchhoff’s transform offers a restricted velocity for the propagation of both mechanical and thermal waves. By ignoring the new parameter, a comparison between analytical cases and numerical results is produced, and the behavior of physical quantities for numerical solutions is studied to ensure the correctness of the proposed technique.

2. Materials and Methods

The basic formulations for a homogeneous isotropic elastic material without a heat source and body force are investigated using the theory of generalized thermoelastic diffusion based on Sherief et al. [12].

The motion equations:

$$\mu u_{i,jj}+\left(\lambda +\mu \right)u_{j,ij}-{\gamma }_1{T}_{,i}-{\gamma }_2{C}_{,i}=\rho \frac{{\partial }^2{u}_i}{\partial t^2}.$$

(1)

The mass diffusion equation:

$${\left(D\left(T\right)P_{,i}\right)}_{,i}=\left(\frac{\partial }{\partial t}+{\tau }_1\frac{{\partial }^2}{\partial t^2}\right)C.$$

(2)

The equation of heat conduction:

$${\left(K\left(T\right)T_{,i}\right)}_{,i}=\left(\frac{\partial }{\partial t}+{\tau }_o\frac{{\partial }^2}{\partial t^2}\right)\left(\rho c_{e}T+{\beta }_i{T}_o{u}_{j,j}+a{T}_{o}C\right).$$

(3)

The constitutive equations are given by

$${\sigma }_{ij}=2\mu e_{ij}+\left[\lambda e-{\beta }_1\left(T-T_0\right)-{\beta }_{2}C\right]{\delta }_{ij},$$

(4)

$$P=-{\beta }_2{e}_{kk}+bC-a\left(T-T_0\right),$$

(5)

$$e_{ij}=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right),$$

(6)

where ${\beta }_1=\left(3\lambda +2\mu \right){\alpha }_t$ and ${\beta }_2=\left(3\lambda +2\mu \right){\alpha }_c$, ${\alpha }_c$ and ${\alpha }_t$ are the coefficients of linear diffusion and thermal expansions, respectively, $T$ is the temperature, $c_e$ is the specific heat at constant strain, $\rho$ is the density of the material, $T_0$ is the reference temperature, $\lambda ,\mu$ are the Lame’s constants, the variable $t$ is the time, $K$ is the thermal conductivity in particular, is impacted by temperature and may be varied, ${\tau }_1$ is the diffusion relaxation time, ${\delta }_{ij}$ is the Kronecker symbol, ${\tau }_o$ is the thermal relaxation time, The tensor ${\sigma }_{ij}$ are the stress components tensor, $e_{ij}$ are the components of strain tensor, $u_i$ are the components of displacement vector and P is the chemical potential. the constant $D$ is the diffusion coefficient in particular, is impacted by temperature and may be varied, the thermo-diffusion and diffusive effects are measured by $a$ and $b$, respectively, the variable $C$ represents the diffusive material concentration in the elastic material. Let us consider an unbounded isotropic elastic medium and let us consider the state of the medium depends only on $x$ and the time variable $t$. We thus obtain the displacement components that can be given by [12]:

$$u_x=u\left(x,t\right),u_y=0,u_z=0.$$

(7)

From Equation (7), Equations (1)–(5) can be written as:

$$\left(\lambda +2\mu \right)\frac{{\partial }^{2}u}{\partial x^2}-{\beta }_1\frac{\partial T}{\partial x}-{\beta }_2\frac{\partial C}{\partial x}=\rho \frac{{\partial }^{2}u}{\partial t^2},$$

(8)

$$\frac{\partial }{\partial x}\left(D\left(T\right)\frac{\partial p}{\partial x}\right)=\left(\frac{\partial }{\partial t}+{\tau }_1\frac{{\partial }^2}{\partial t^2}\right)C.$$

(9)

$$\frac{\partial }{\partial x}\left(K\left(T\right)\frac{\partial T}{\partial x}\right)=\left(\frac{\partial }{\partial t}+{\tau }_o\frac{{\partial }^2}{\partial t^2}\right)\left(\rho c_{e}T+{\beta }_i{T}_o\frac{\partial u}{\partial x}+a{T}_{o}C\right)$$

(10)

$${\sigma }_{xx}=\left(\lambda +2\mu \right)\frac{\partial u}{\partial x}-{\beta }_1\left(T-T_0\right)-{\beta }_{2}C,$$

(11)

$$P=-{\beta }_2\frac{\partial u}{\partial x}+bC-a\left(T-T_0\right),$$

(12)

As a linear function of temperature, determine the variable thermal conductivity and diffusivity [23] such that

$$K\left(T\right)=K_o\left(1+K_{n}T\right),$$

(13)

$$D\left(T\right)=D_o\left(1+k_{n}T\right),$$

(14)

where $K_o$ are the thermal conductivity when $T=T_o$, $D_o$ is the diffusion coefficient when $T=T_o$ and $K_n\le 0$ is the non-positive parameter.

3. Initial and Boundary Conditions

The initial conditions can be defined as:

$$T\left(x,0\right)=0,\frac{\partial T\left(x,0\right)}{\partial t}=0, C\left(x,0\right)=0,\frac{\partial C\left(x,0\right)}{\partial t}=0, u\left(x,0\right)=0,\frac{\partial u\left(x,0\right)}{\partial t}=0.$$

(15)

While the boundary conditions are stated by:

$${\sigma }_{xx}\left(0,t\right)=0, P\left(0,t\right)=P_{s}H\left(t\right), T\left(0,t\right)=T_{s}H\left(t\right),$$

(16)

where $T_s,P_s$ are constant and $H\left(t\right)$ denotes the Heaviside step unit functions. Accordingly, the non-dimensionality of variables may be described in this manner:

$$\left(x^{\prime },u^{\prime }\right)=c\eta \left(x,u\right),\left(t^{\prime },{\tau }_o^{\prime },{\tau }_1^{\prime }\right)=\eta c^2\left(t,{\tau }_o,{\tau }_1\right),T^{\prime }=\frac{{\beta }_1\left(T-T_0\right)}{\lambda +2\mu },{\sigma }_{xx}^{\prime }=\frac{{\sigma }_{xx}}{\lambda +2\mu },C^{\prime }=\frac{{\beta }_{2}C}{\lambda +2\mu },P^{\prime }=\frac{P}{{\beta }_2},$$

(17)

with $c^2=\frac{\lambda +2\mu }{\rho } \mathrm{and} \eta =\frac{\rho c_e}{K_o}.$

The governing equations in the non-dimensional of physical quantities in Formulation (13) are stated as (after deleting the superscript $\prime$ for appropriateness)

$$\frac{{\partial }^{2}u}{\partial x^2}-\frac{\partial T}{\partial x}-\frac{\partial C}{\partial x}=\frac{{\partial }^{2}u}{\partial t^2},$$

(18)

$$\frac{\partial }{\partial x}\left(\left(1+K_{n}T\right)\frac{\partial p}{\partial x}\right)=s_1\left(\frac{\partial }{\partial t}+{\tau }_1\frac{{\partial }^2}{\partial t^2}\right)C,$$

(19)

$$\frac{\partial }{\partial x}\left(\left(1+K_{n}T\right)\frac{\partial T}{\partial x}\right)=\left(\frac{\partial }{\partial t}+{\tau }_o\frac{{\partial }^2}{\partial t^2}\right)\left(T+s_2\frac{\partial u}{\partial x}+s_{3}C\right),$$

(20)

$${\sigma }_{xx}=\frac{\partial u}{\partial x}-T-C,$$

(21)

$$P=-\frac{\partial u}{\partial x}+s_{4}C-s_{5}T,$$

(22)

where $s_1=\frac{\left(\lambda +2\mu \right)}{D_o\eta {\beta }_2{\beta }_2}$, $s_2=\frac{{\beta }_1{\beta }_1{T}_o}{\left(\lambda +2\mu \right)\eta K_o}$, $s_3=\frac{a{T}_o{\beta }_1}{{\beta }_2\eta K_o}$, $s_4=\frac{b}{{\beta }_2{\beta }_2}\left(\lambda +2\mu \right)$, $s_5=\frac{a}{{\beta }_2{\beta }_1}\left(\lambda +2\mu \right)$.

4. Numerical Method

In this part, the finite element method (FEM) is presented to condense the formulations of thermoelastic diffusions based on thermal relaxation times ${\tau }_o$ and ${\tau }_1$ in an unbounded medium. The finite element approach is the approach of choice for linear/nonlinear systems in a variety of domains. This approach is a potent technique generally used to achieve the numerical solution of complex problems. Here, the finite element approach is used to solve the non-linear Equations (18)–(20) given the starting condition (15) and boundary conditions (16). The solutions for different issues under deference generalized thermoelasticity models were published by Abbas et al. [43,44]. The conventional technique may be used to derive the finite element formulations of thermoelastic diffusion. The generalized thermoelastic model’s non-dimension weak formulations are developed. The temperature $\delta T$, the displacement $\delta u$, and the concentration $\delta C$ are defined as a series of independent test functions. The boundary conditions are utilized to integrate across the spatial domain after the controlling formulations are multiplied by separate weighting functions. The use of the boundary condition is made possible by integrating parts and using the divergence theorem to lower the spatial derivative’s order. The unknown variables and accompanying test functions are approximated using the same shape function using the Galerkin approach.

$$u={{\displaystyle \sum }}_{j=1}^n{N}_j{u}_j\left(t\right),C={{\displaystyle \sum }}_{j=1}^n{N}_j{C}_j\left(t\right),T={{\displaystyle \sum }}_{j=1}^n{N}_j{T}_j\left(t\right),$$

(23)

$$\delta u={{\displaystyle \sum }}_{j=1}^n{N}_j\delta u_j,\delta C={{\displaystyle \sum }}_{j=1}^n{N}_j\delta C_j, \delta T={{\displaystyle \sum }}_{j=1}^n{N}_j\delta T_j,$$

(24)

where $N$ denotes the shape functions and $n$ denotes the number of nodes per element. The master elements’ local coordinates are assumed to be in the range $\left[1, -1\right]$. One-dimensional quadratic components are employed in this case, and they are stated as follows:

$$\begin{array}{ccc}{N}_1=\frac{1}{2}\left({\chi }^2+\chi \right),& N_1=1-{\chi }^2,& N_3=\frac{1}{2}\left({\chi }^2-\chi \right)\end{array},$$

(25)

The weak formulations for the finite element method that corresponds to (14) and (15) are now as follows:

$${{\displaystyle \int }}_0^L\frac{\partial \delta u}{\partial x}\left(\frac{\partial u}{\partial x}-T-C\right)dx+{{\displaystyle \int }}_0^L\delta u\left(\frac{{\partial }^{2}u}{\partial t^2}\right)dx=\delta u{\left(\frac{\partial u}{\partial x}-T-C\right)}_0^L,$$

(26)

$$\begin{gathered} {{\displaystyle \int }}_0^L\frac{\partial \delta C}{\partial x}\left(\left(1+K_{n}T\right)\left(-\frac{{\partial }^{2}u}{\partial x^2}+s_4\frac{\partial C}{\partial x}-s_5\frac{\partial T}{\partial x}\right)\right)dx+{{\displaystyle \int }}_0^L\delta C{s}_1\left(\frac{\partial C}{\partial t}+{\tau }_1\frac{{\partial }^{2}C}{\partial t^2}\right)dx \\ =\delta C{\left(\left(1+K_{n}T\right)\left(-\frac{{\partial }^{2}u}{\partial x^2}+s_4\frac{\partial C}{\partial x}-s_5\frac{\partial T}{\partial x}\right)\right)}_0^L, \end{gathered}$$

(27)

$${{\displaystyle \int }}_0^L\frac{\partial \delta T}{\partial x}\left(\left(1+K_{n}T\right)\frac{\partial T}{\partial x}\right)dx+{{\displaystyle \int }}_0^L\delta T\left(\frac{\partial }{\partial t}+{\tau }_o\frac{{\partial }^2}{\partial t^2}\right)\left(T+s_2\frac{\partial u}{\partial x}+s_{3}C\right)dx=\delta T{\left(\left(1+K_{n}T\right)\frac{\partial T}{\partial x}\right)}_0^L$$

(28)

Finally, implicit techniques should be used to calculate temporal derivatives of unknown variables.

5. The Validation of the Numerical Method

Now, analytical solutions for particular instances are offered to verify the finite element approach. When $K_n$ is equal to zero, the numerical solutions are confirmed by comparing them to the analytical solutions. Consequently, Equations (18)–(22) with the initial and boundary conditions may be written as

$$\frac{{\partial }^{2}u}{\partial x^2}-\frac{\partial T}{\partial x}-\frac{\partial C}{\partial x}=\frac{{\partial }^{2}u}{\partial t^2},$$

(29)

$$\frac{{\partial }^{2}P}{\partial x^2}=s_1\left(\frac{\partial }{\partial t}+{\tau }_1\frac{{\partial }^2}{\partial t^2}\right)C,$$

(30)

$$\frac{{\partial }^{2}T}{\partial x^2}=\left(\frac{\partial }{\partial t}+{\tau }_o\frac{{\partial }^2}{\partial t^2}\right)\left(T+s_2\frac{\partial u}{\partial x}+s_{3}C\right),$$

(31)

$${\sigma }_{xx}=\frac{\partial u}{\partial x}-T-C,$$

(32)

$$P=-\frac{\partial u}{\partial x}+s_{4}C-s_{5}T,$$

(33)

$$T\left(x,0\right)=0,\frac{\partial T\left(x,0\right)}{\partial t}=0, C\left(x,0\right)=0,\frac{\partial C\left(x,0\right)}{\partial t}=0, u\left(x,0\right)=0,\frac{\partial u\left(x,0\right)}{\partial t}=0.$$

(34)

$${\sigma }_{xx}\left(0,t\right)=0, P\left(0,t\right)=P_{s}H\left(t\right), T\left(0,t\right)=T_{s}H\left(t\right),$$

(35)

Using the Laplace transforms to solve problems (29)–(35)

$$\overline{f}\left(x,s\right)=L\left[f\left(x,t\right)\right]=\underset{0}{\overset{\infty }{{\displaystyle \int }}}f\left(x,t\right)e^{-st}dt.$$

(36)

As a result, we may derive the following equations:

$$\frac{d^2\overline{u}}{d{x}^2}=s^2\overline{u}+\frac{d\overline{C}}{dx}+\frac{d\overline{T}}{dx},$$

(37)

$$\frac{d^2\overline{C}}{d{x}^2}=s_6\overline{C}+s_7\overline{T}+s_8\frac{d\overline{u}}{dx},$$

(38)

$$\frac{d^2\overline{T}}{d{x}^2}=s_9\overline{C}+s_{10}\overline{T}+s_{11}\frac{d\overline{u}}{dx} ,$$

(39)

$${\overline{\sigma }}_{xx}=\frac{d\overline{u}}{dx}-\overline{T}-\overline{C},$$

(40)

$$\overline{P}=-\frac{d\overline{u}}{dx}+s_4\overline{C}-s_5\overline{T},$$

(41)

$${\overline{\sigma }}_{xx}\left(0,t\right)=0,\overline{T}\left(0,t\right)=\frac{T_1}{s}, \overline{P}\left(0,t\right)=\frac{P_1}{s},$$

(42)

where $s_6=\frac{\left(s_1\left(s+{\tau }_1{s}^2\right)+\left(1+s_5\right)\left(s+{\tau }_o{s}^2\right)s_3\right)}{\left(s_4-1\right)}$, $s_7=\frac{\left(1+s_5\right)\left(s+{\tau }_o{s}^2\right)}{\left(s_4-1\right)}$, $s_8=\frac{\left(s^2+\left(1+s_5\right)\left(s+{\tau }_o{s}^2\right)s_2\right)}{\left(s_4-1\right)}$, $s_9=\left(s+{\tau }_o{s}^2\right)s_3$, $s_{10}=\left(s+{\tau }_o{s}^2\right)$, $s_{11}=\left(s+{\tau }_o{s}^2\right)s_2$.

The solutions of Equations (37)–(39) are obtained by using the eigenvalues technique as described in [41,45,46,47,48]. As a result, the matrices-vectors may be written as

$$\frac{dF}{dx}=AF,$$

(43)

where $F={\left[\begin{array}{cccccc}\overline{u}& \overline{C}& \overline{T}& \frac{d\overline{u}}{dx}& \frac{d\overline{C}}{dx}& \frac{d\overline{T}}{dx}\end{array}\right]}^T$ and $A=\left[a_{ij}\right]=0, i,j=1\dots 6$, excepting $a_{14}=a_{25}=a_{36}=a_{45}=a_{46}=1$, $a_{41}=s^2$, $a_{52}=s_6$,$a_{53}=s_7$, $a_{54}=s_8$, $a_{62}=s_9$, $a_{63}=s_{10}$, $a_{64}=s_{11}$.

Consequently, the matrix $A$’s characteristic equation may be written as follows:

$${ϵ}^6-f_3{ϵ}^4+f_2{ϵ}^2+f_1=0,$$

(44)

where $f_1=a_{62}{a}_{41}{a}_{53}-a_{63}{a}_{41}{a}_{52}$, $f_2=a_{41}{a}_{63}+a_{52}{a}_{63}+a_{63}{a}_{45}{a}_{54}+a_{41}{a}_{52}-a_{53}{a}_{62}-a_{62}{a}_{46}{a}_{54}+a_{46}{a}_{52}{a}_{64}-a_{64}{a}_{45}{a}_{53}$, $f_3=a_{52}+a_{41}+a_{54}{a}_{45}+a_{63}+a_{64}{a}_{46}$.

The solutions of Equation (43) can be given as

$$F\left(x,s\right)={{\displaystyle \sum }}_{i=1}^3\left(B_i{X}_i{e}^{-{ϵ}_{i}x}+B_{i+1}{X}_{i+1}{e}^{{ϵ}_{i}x}\right),$$

(45)

where $B_1, B_2,B_3, B_4$,$B_5$ and $B_6$ are constants that may be calculated using the boundary conditions of problem. To obtain the final solutions of concentration, displacement, temperature, the chemical potential and the stress distribution, the Stehfest [49] may be employed as a numerical inversions approach.

6. Results and Discussion

The medium was collected for numerical computations, and the problem parameters were selected as described in [50]

$${\alpha }_c=1.98\times {10}^{-4}\left({\mathrm{kg}}^{-1}\right)\left({\mathrm{m}}^3\right),D=0.85\times {10}^{-8}\left(\mathrm{kg}\right)\left(\mathrm{s}\right)\left({\mathrm{m}}^{-3}\right),\rho =8954\left(\mathrm{kg}\right)\left({\mathrm{m}}^{-3}\right), R=1,$$

$${\alpha }_t=17.8\times {10}^{-6}\left({\mathrm{k}}^{-1}\right), b=0.9\times {10}^6\left({\mathrm{m}}^5\right)\left({\mathrm{kg}}^{-1}\right)\left({\mathrm{s}}^{-2}\right),a=1.2\times {10}^4\left({\mathrm{m}}^2\right)\left({\mathrm{k}}^{-1}\right)\left({\mathrm{s}}^{-2}\right),$$

$$T_o=293\left(\mathrm{k}\right),K=3.68\times {10}^2\left(\mathrm{kg}\right)\left(\mathrm{m}\right)\left({\mathrm{s}}^{-3}\right)\left({\mathrm{k}}^{-1}\right),c_e=383.1\left({\mathrm{m}}^2\right)\left({\mathrm{s}}^{-2}\right)\left({\mathrm{k}}^{-1}\right), T_1=1,$$

$$\lambda =7.76\times {10}^{10}\left(\mathrm{kg}\right)\left({\mathrm{m}}^{-1}\right)\left({\mathrm{s}}^{-2}\right),\mu =3.86\times {10}^{10}\left(\mathrm{kg}\right)\left({\mathrm{m}}^{-1}\right)\left({\mathrm{s}}^{-2}\right).$$

The computations are performed at time $t=0.3$. Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 show the computed physical quantities (numeral) along the distance $x$ using the extended thermoelastic theory with two thermal and diffusion relaxation times based on the aforementioned constants list. Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5 show the impacts of varying thermal conductivity and diffusivity on the variations in temperature, the displacement, the concentration, the chemical potential, and the distributions along the distances $x$ when the relaxation times remind be constants ${\tau }_o={\tau }_1=0.05$. Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 display the impacts of thermal and diffusion relaxation times on the variations of temperature, the displacement, the concentration, the chemical potential and the distributions with respect to the distance $x$ when the varying thermal conductivity and diffusivity depend on temperature when $K_n=-0.5$. Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 show the comparison between the numerical solutions (finite element method) and the analytical solutions (Laplace transform and eigenvalue approach) when ${\tau }_o={\tau }_1=0.05$ and $K_n=0$. The numerical results of temperature variations, the displacement variations, the variations in concentration, the chemical potential variation and the stress variations with the findings of those analytical results along the distance $x$ demonstrate a high level of agreement. Figure 1, Figure 6 and Figure 11 show the variations in temperature via the distance $x$. It is noticed that the temperature has maximum values $T=1$ at the surface $x=0$ to obey the boundary conditions of the problem, after that it gradually decreases with the increase in the distances $x$ to close to zeros. Figure 2, Figure 7 and Figure 12 display the displacement variations versus the distance $x$. It was found that the displacement reaches a maximum negative value, grows gradually until it reaches peak values at a specific area near to the surface, and then falls continuously to zero. Figure 3, Figure 8 and Figure 13 depict the variations in concentration as a function of the distance $x$. It is observed that the concentration has maximum values at the surface $x=0$ then it gradually decreases with the increase in the distances $x$ to close to zeros. Figure 4, Figure 9 and Figure 14 display the variations in stress versus the distance $x$. It is seen that the stress starts at zero, which meets the problem boundary condition, and increases steadily up to positive peak values and after that lowers steadily up to negative peak values then increases again to near zero. Figure 5, Figure 10 and Figure 15 show the variations in chemical potential along the distance $x$. It is seen that the chemical potential has maximum values of $P=1$ at the surface $x=0$, in accordance with the problem’s boundary conditions, and then progressively approaches zero as $x$ increases. As expected, it can be found that the variable thermal conductivity and diffusivity have important effects on the values of all the physical quantities. As predicted, the thermal and diffusion relaxation times significantly influence the importance of all physical quantities. Finally, it can be concluded from the numerical results that using the generalized thermoelastic diffusion model with varying thermal conductivity is an important phenomenon that significantly impacts the distribution of physical quantities.