Thermoelastic Analysis in Poro-Elastic Materials Using a TPL Model

Abstract

The main aim of the paper is to study the impact of delay times in a poro-elastic medium using the finite element approach and the three-phase lag thermo-elastic theory. The governing equations were obtained for a three-phase lag model with six delay times. Consideration was given to a one-dimensional application of a poro-elastic half-space. Because of the complex form of the basic equations, the finite element approach was used to solve this problem. Asymmetric and symmetric tensors were used to represent all of the physical quantities. The numerical results were presented in graphical form. The effects of porosity and delays were introduced. Finally, the results were plotted to show the difference between the three-phase delay (TPL) and the Green−Naghdi with and without energy dissipation (GNIII) models.

1. Introduction

Elastic materials consisting of small pores are defined as linear elastic materials with voids. In classical elastic theory, the volumes of these pores are neglected; they play an important role because independent kinematic variables must consider the volume of these pores. Recently, the problems of heat transfer in porous media, essentially with short thermal periods and small-time scales, have been addressed by many researchers. The effective thermal conductivity generated by weakly conducting pores is used to characterize heat conduction for a volume of material with hundreds of holes. Furthermore, the interactions between thermo-elastic liquids and solids, such as water, a domain of poro-thermal elasticity, are receiving more attention because of their multiple applications in related domains, including geophysics. The poro-elastic theory was developed by Biot [1]. Biot [2,3] developed the theoretical foundation for propagating iso-thermal waves in fluid-saturated, poro-elastic mediums for high and low-frequency instances. Many researchers have looked at the challenges of plane and surface wave propagation in liquid-saturated poro-elastic media using Biot’s model. For example, three forms of theories were proposed by Green−Naghdi [4,5,6] (GN-I, GNII, and GN-III).

The constitutive equations of the GN models have been linearized. The classical coupled thermo-elasticity theory is similar to type I, but it has a different name (GN-I). Tzou [7] used the dual-phase-lag models to study the interaction between photons and electrons on a microscopical level. At a microscopical scale, the relaxing source caused various relaxed responses. Tzou [8] confirmed the physical concepts and practicality of the dual-phase-lag model in their experimental results. The elastic theory of a material with a void is one of the most serious generalizations of the classical elastic material model. This model applies to elastic material composed of distributions of tiny holes, with the empty volume included among the kinematic variables. This model has been rationally constructed to generate a perfectly consistent model that can logically accommodate thermal impulse transmissions.

Roychoudhuri [9] presented a three-phase lag thermo-elasticity model (TPL) that encompassed the generalizations. Many researchers, including [10,11,12,13], have contributed to various difficulties in poro-thermo-elastic media after the work of [3]. El-Naggar et al. [14] used a generalized thermo-elasticity model to explore the effects of the initial stress, rotation, magnetic field, and voids on plane waves. The natural frequency of a poro-elastic circular cylinder was investigated by Abbas [15]. Schanz and Cheng [16] investigated the propagations of the transient waves in a one-dimension poro-elastic column. Abbas et al. [17] investigated the impacts of thermal dispersions on free convections in a fluid-saturated porous medium. Saeed et al. [18] used the finite element method to study the thermo-elastic interaction in a poro-elastic medium under the GL model. Sur et al. [19] looked into the influence of memory on the propagation of thermal waves in an elastic material containing voids.

The analytical solutions for time-dependent problems for coupled and nonlinear/linear systems exist only for simple boundary conditions with initial conditions. The finite element scheme is preferable to the time domain problem in order to avert this complication. Procedures for solving deformations concerning this problem using finite element methods have been presented. It is a powerful scheme and has been expanded to solve complex problems in the structural mechanics. Carcione et al. [20] studied the physics and simulations of wave propagation in linear poro-thermo-elastic mediums. Zhou et al. [21] investigated the propagations of thermo-elastic waves in an unsaturated poro-thermo-elastic medium. With fractional thermo-elastic theory, Wen et al. [22] described a circular walled hole’s thermo-mechanical and hydro-dynamic responses in a poro-elastic material. Fractional thermo-elastic applications for a porous asphaltic material were investigated by Ezzat and Ezzat [23]. Singh [24] studied the propagation and attenuation of an elastic wave in a generalized thermo-poro-elastic model. Hussein [25] investigated the impact of porosity on porous media saturated with fluid and exposed to temperature changes. Sur [26] studied how memory responses affected wave propagation in porous asphalts. Alawi [27,28] studied the generalized poro-thermo-elasticity of asphaltic materials. Jangid and Mukhopadhyay [29] established variational principle and continuous dependency findings for the generalized poro-thermo-elasticity theory with a single relaxation parameter. Kumar et al. [30] investigated wave propagation at the welded interface between an elastic solid and an unsaturated poro-thermo-elastic solid. Mirparizi et al. [31] studied one-dimensional electro-magneto-poro-thermo-elastic wave propagation in functionally graded materials with energy dissipation. Abbas and Hobiny [32] studied the thermos-mechanical responses of poro-elastic mediums with two thermal relaxation times. Many researchers [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50] have addressed various problems regarding porous and elastic media under different boundary conditions using generalized thermo-elastics models.

This study uses three-phase-lag theory to investigate the thermo-elastic interaction in poro-elastic media caused by thermal shock. In addition, consideration is given to a one-dimensional application of a poro-elastic half-space. Because of the complicated governing equations, the finite element scheme was used to address these problems. Finally, the outcomes of the field variables were plotted graphically for various estimates of thermal delay times and porosities.

2. Basic Equations

Following the three-phase-lag model [9,13], the basic formulations in the dynamical model of the homogeneous isotropic and thermo-poro-elastic media without body forces and heating sources are written as follows:

$$\mu u_{i,jj}+\left(\lambda +\mu \right)u_{j,ij}-R_{11}{\Theta }_{,i}^s-R_{12}{\Theta }_{,i}^f={\rho }_{11}\frac{{\partial }^2{u}_i}{\partial t^2}+{\rho }_{12}\frac{{\partial }^2{U}_i}{\partial t^2},$$

(1)

$$R{U}_{j,ij}+Q{u}_{j,ij}-R_{21}{\Theta }_{,i}^s-R_{22}{\Theta }_{,i}^f={\rho }_{12}\frac{{\partial }^2{u}_i}{\partial t^2}+{\rho }_{22}\frac{{\partial }^2{U}_i}{\partial t^2},$$

(2)

$$\left[K_{\ast }^s+\left(K^s+K_{\ast }^s{\tau }_{\vartheta }^s\right)\frac{\partial }{\partial t}+K^s{\tau }_{\theta }^s\frac{{\partial }^2}{\partial t^2}\right]{\Theta }_{,ii}^s=\left(1+{\tau }_q^s\frac{\partial }{\partial t}\right)\left(F_{11}\frac{{\partial }^2{\Theta }^s}{\partial t^2}+F_{12}\frac{{\partial }^2{\Theta }^f}{\partial t^2}+T_o{R}_{11}\frac{{\partial }^2{e}_{kk}}{\partial t^2}+T_o{R}_{21}\frac{{\partial }^2ϵ}{\partial t^2}\right),$$

(3)

$$\left[K_{\ast }^f+\left(K^f+K_{\ast }^f{\tau }_{\vartheta }^f\right)\frac{\partial }{\partial t}+K^f{\tau }_{\theta }^f\frac{{\partial }^2}{\partial t^2}\right]{\Theta }_{,ii}^f=\left(1+{\tau }_q^f\frac{\partial }{\partial t}\right)\left(F_{21}\frac{{\partial }^2{\Theta }^s}{\partial t^2}+F_{22}\frac{{\partial }^2{\Theta }^f}{\partial t^2}+T_o{R}_{12}\frac{{\partial }^2{e}_{kk}}{\partial t^2}+T_o{R}_{22}\frac{{\partial }^2ϵ}{\partial t^2}\right).$$

(4)

The constitutive equations

$${\sigma }_{ij}=2\mu e_{ij}+\left(\lambda e_{kk}+Qϵ-R_{11}{\Theta }^s-R_{12}{\Theta }^f\right){\delta }_{ij},$$

(5)

$$\sigma =Q{e}_{kk}+Rϵ-R_{21}{\Theta }^s-R_{22}{\Theta }^f$$

(6)

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

(7)

This model can be summarized as follows:

(i)

The three-phase-lag model is denoted by (TPL)

$$0<{\tau }_{\vartheta }^s<{\tau }_{\theta }^s<{\tau }_q^s \mathrm{and} 0<{\tau }_{\vartheta }^f<{\tau }_{\theta }^f<{\tau }_q^f$$

(ii)

(GNIII) indicate the Green−Naghdi type III model

$${\tau }_{\vartheta }^s={\tau }_{\theta }^s={\tau }_q^s={\tau }_{\vartheta }^f={\tau }_{\theta }^f={\tau }_q^f=0$$

where ${\tau }_{\vartheta }^s,{\tau }_{\theta }^s,{\tau }_q^s,{\tau }_{\vartheta }^f,{\tau }_{\theta }^f,{\tau }_q^f$ are the solid and liquid phases’ thermal delay times, respectively; $K_{\ast }^{\mathrm{s}}$ is the property of the solid material constant in theory; $K_{\ast }^f$ is a property of the fluid material constant in theory; ${\rho }^s$ is the solid-phase density per unit volume of bulk; $K$ is the interphase heating conductions’ interface coefficients; ${\rho }^f$ is the density of liquid-phase per unit volume of bulk;$\sigma$ is the normal stress of the liquid surface; ${\sigma }_{ij}$ is the stress components of the solid surfaces; $\lambda ,\mu , R, Q$ are the poro-elastic materials coefficients; ${\alpha }^s, {\alpha }^f$ are the thermal expansion coefficients; $\beta$ is the material porosity; $T_o$ is the reference temperature; $R_{11}, R_{12}, R_{21}, R_{22}$ are the thermal and mixed coefficients; $C_E^s,C_E^f$ are the specific thermals of the solid phase and liquid phase, respectively;$C_E^f$ is the specific thermal coupling between the liquid–solid phases; $K^s$ is the thermal conductive of the solid; $K^{f\ast }, K^{s\ast }$ are the liquid and the solid thermal conductivities, respectively; $ϵ$ are the strain components of the liquid phase; ${\rho }^{s\ast }, {\rho }^{f\ast }$ are the solid and liquid densities, respectively; ${\alpha }^{sf}, {\alpha }^{fs}$ are the thermo-elastic coupling between the liquid−solid phases; ${\eta }^s$ is the solid thermal viscosity; $\eta$ is the coupling thermal viscosity between the phases; ${\eta }^f$ is the fluid thermal viscosity; ${\rho }_{22}$ is the fluid phase mass coefficient; ${\rho }_{12}$ is the dynamic coupling coefficient; ${\Theta }^f$ is the increment of liquid temperature; ${\Theta }^s$ is the variation of solid temperature; $u_i$ and $U_i$ are the solid and liquid phase displacements, respectively; $e_{ij}$ is the solid phase strain components; $K^f$ is the liquid phase thermal conductivity; and ${\rho }_{11}$ is the solid phase mass coefficient with

• ${\Theta }^f=T^f-T_o$, ${\Theta }^s=T^s-T_o$, $K^s=\left(1-\beta \right)K^{s\ast }$, $K^f=\beta K^{f\ast }$, ${\eta }^s=\frac{{\rho }^s{C}_E^s}{K^s}$, ${\eta }^f=\frac{{\rho }^f{C}_E^f}{K^f}$,
• ${\rho }^s=\left(1-\beta \right){\rho }^{s\ast }$, ${\rho }_{11}={\rho }^s-{\rho }_{12}$, ${\rho }_{22}={\rho }^f-{\rho }_{12}$, ${\rho }^f=\beta {\rho }^{f\ast }$, $R_{21}=3{\alpha }^{s}Q+{\alpha }^{sf}R$,
• $R_{22}={\alpha }^{f}R+3{\alpha }^{sf}Q$, $P=3\lambda +2\mu$, $F_{11}={\rho }^s{C}_E^s$, $F_{22}={\rho }^f{C}_E^f$, $F_{12}=-\left(3{\alpha }^s{R}_{12}+{\alpha }^{fs}{R}_{22}\right)T_o$,
• $F_{21}=-\left(3{\alpha }^{sf}{R}_{11}+{\alpha }^f{R}_{21}\right)T_o$ $\eta =\frac{{\rho }_{12}{C}_E^{sf}}{K}$, $R_{11}=P{\alpha }^s+Q{\alpha }^{fs}$, $R_{12}={\alpha }^{f}Q+P{\alpha }^{sf}$ as in [23].

Now, let us investigate a poro-thermo-elastic material in the $x\ge 0$ regions, as shown in Figure 1 [32]. All of the functions studied in the one-dimensional problem will depend only on the variables of time $t$ and space x. The displacement components are written as follows:

$$u_x=u\left(x,t\right),U_x=U\left(x,t\right), u_y=0,U_y=0, u_z=0, U_z=0$$

(8)

The basic formulations can then be defined as

$$\left(\lambda +2\mu \right)\frac{{\partial }^{2}u}{\partial x^2}+Q\frac{{\partial }^{2}U}{\partial x^2}-R_{11}\frac{\partial {\Theta }^s}{\partial x}-R_{12}\frac{\partial {\Theta }^f}{\partial x}={\rho }_{11}\frac{{\partial }^{2}u}{\partial t^2}+{\rho }_{12}\frac{{\partial }^{2}U}{\partial t^2},$$

(9)

$$Q\frac{{\partial }^{2}u}{\partial x^2}+R\frac{{\partial }^{2}U}{\partial x^2}-R_{21}\frac{\partial {\Theta }^s}{\partial x}-R_{22}\frac{\partial {\Theta }^f}{\partial x}={\rho }_{21}\frac{{\partial }^{2}u}{\partial t^2}+{\rho }_{22}\frac{{\partial }^{2}U}{\partial t^2},$$

(10)

$$\left[K_{\ast }^s+\left(K^s+K_{\ast }^s{\tau }_{\vartheta }^s\right)\frac{\partial }{\partial t}+K^s{\tau }_{\theta }^s\frac{{\partial }^2}{\partial t^2}\right]\frac{{\partial }^2{\Theta }^s}{\partial x^2}=\left(1+{\tau }_q^s\frac{\partial }{\partial t}\right)\left(F_{11}\frac{{\partial }^2{\Theta }^s}{\partial t^2}+F_{12}\frac{{\partial }^2{\Theta }^f}{\partial t^2}+T_o{R}_{11}\frac{{\partial }^{3}u}{\partial t^2\partial x}+T_o{R}_{21}\frac{{\partial }^{3}U}{\partial t^2\partial x}\right),$$

(11)

$$\left[K_{\ast }^f+\left(K^f+K_{\ast }^f{\tau }_{\vartheta }^f\right)\frac{\partial }{\partial t}+K^f{\tau }_{\theta }^f\frac{{\partial }^2}{\partial t^2}\right]\frac{{\partial }^2{\Theta }^f}{\partial x^2}=\left(1+{\tau }_q^f\frac{\partial }{\partial t}\right)\left(F_{21}\frac{{\partial }^2{\Theta }^s}{\partial t^2}+F_{22}\frac{{\partial }^2{\Theta }^f}{\partial t^2}+T_o{R}_{12}\frac{{\partial }^{3}u}{\partial t^2\partial x}+T_o{R}_{22}\frac{{\partial }^{3}U}{\partial t^2\partial x}\right),$$

(12)

$${\sigma }_{xx}=\left(\lambda +2\mu \right)\frac{\partial u}{\partial x}+Q\frac{\partial U}{\partial x}-R_{11}{\Theta }^s-R_{12}{\Theta }^f,$$

(13)

$$\sigma =Q\frac{\partial u}{\partial x}+R\frac{\partial U}{\partial x}-R_{21}{\Theta }^s-R_{22}{\Theta }^f,$$

(14)

3. Application

The initial and boundary conditions are defined by

$$\frac{\partial U\left(x,0\right)}{\partial t}=\frac{\partial u\left(x,0\right)}{\partial t}=0,U\left(x,y,0\right)=u\left(x,0\right)=0,$$

(15)

$${\Theta }^f\left(x,0\right)={\Theta }^s\left(x,0\right)=0, \frac{\partial {\Theta }^f\left(x,0\right)}{\partial t}=\frac{\partial {\Theta }^s\left(x,0\right)}{\partial t}=0,$$

(16)

On the other hand, the boundary conditions may be provided by the following:

(1)

The thermal boundary conditions

$${\Theta }^f\left(0,t\right)=\beta {\Theta }_{o}H\left(t\right), {\Theta }^s\left(0,t\right)={\Theta }_o\left(1-\beta \right)H\left(t\right)$$

(17)

(2)

The mechanical boundary conditions

$$\sigma \left(0,t\right)=0, {\sigma }_{xx}\left(0,t\right)=0$$

(18)

where the Heaviside unity step function is $H\left(t\right)$ and ${\Theta }_o$ is constant. For ease of usage, the non-dimensional variables may be expressed with the following forms:

$$\begin{gathered} \left({\sigma }_{xx}^{\prime },{\sigma }^{\prime }\right)=\frac{\left( {\sigma }_{xx}, \sigma \right)}{\left(\lambda +2\mu \right)},\left(x^{\prime },u^{\prime },U^{\prime }\right)=\eta c\left(x,u,U\right), \left({\Theta }^{s^{\prime }},{\Theta }^{f^{\prime }}\right)=\frac{R_{11}\left({\Theta }^s,{\Theta }^f\right)}{\left(\lambda +2\mu \right)}, \\ \left(t^{\prime },{\tau }_q^{s^{\prime }}, {\tau }_q^{f^{\prime }},{\tau }_{\theta }^{s^{\prime }}, {\tau }_{\theta }^{f^{\prime }},{\tau }_{\vartheta }^{s^{\prime }}, {\tau }_{\vartheta }^{f^{\prime }}\right)=\eta c^2\left(t,{\tau }_q^s, {\tau }_q^f,{\tau }_{\theta }^s, {\tau }_{\theta }^f,{\tau }_{\vartheta }^s, {\tau }_{\vartheta }^f\right), \end{gathered}$$

(19)

where $c=\sqrt{\frac{\lambda +2\mu }{{\rho }_{11}}}$ and $\eta =\frac{F_{11}}{K^s}$.

The above equations may be stated in terms of these dimensionless representations of physical quantities in (19) as follows (the script has been neglected for its appropriateness):

$$\frac{{\partial }^{2}u}{\partial x^2}+f_1\frac{{\partial }^{2}U}{\partial x^2}-\frac{\partial {\Theta }^s}{\partial x}-f_2\frac{\partial {\Theta }^f}{\partial x}=\frac{{\partial }^{2}u}{\partial t^2}+f_3\frac{{\partial }^{2}U}{\partial t^2},$$

(20)

$$f_1\frac{{\partial }^{2}u}{\partial x^2}+f_4\frac{{\partial }^{2}U}{\partial x^2}-f_5\frac{\partial {\Theta }^s}{\partial x}-f_6\frac{\partial {\Theta }^f}{\partial x}=f_7\frac{{\partial }^{2}u}{\partial t^2}+f_8\frac{{\partial }^{2}U}{\partial t^2},$$

(21)

$$\left[{ϵ}_s+\left(1+{ϵ}_s{\tau }_{\vartheta }^s\right)\frac{\partial }{\partial t}+{\tau }_{\theta }^s\frac{{\partial }^2}{\partial {\mathrm{t}}^2}\right]\frac{{\partial }^2{\Theta }^s}{\partial x^2}=\left(1+{\tau }_q^s\frac{\partial }{\partial t}\right)\left(\frac{{\partial }^2{\Theta }^s}{\partial t^2}+f_9\frac{{\partial }^2{\Theta }^f}{\partial t^2}+f_{10}\frac{{\partial }^{3}u}{\partial t^2\partial x}+f_{11}\frac{{\partial }^{3}U}{\partial t^2\partial x}\right),$$

(22)

$$\left[{ϵ}_f+\left(1+{ϵ}_f{\tau }_{\vartheta }^f\right)\frac{\partial }{\partial t}+{\tau }_{\theta }^f\frac{{\partial }^2}{\partial {\mathrm{t}}^2}\right]\frac{{\partial }^2{\Theta }^f}{\partial x^2}=\left(1+{\tau }_q^f\frac{\partial }{\partial t}\right)\left(f_{12}\frac{{\partial }^2{\Theta }^s}{\partial t^2}+f_{13}\frac{{\partial }^2{\Theta }^f}{\partial t^2}+f_{14}\frac{{\partial }^{3}u}{\partial t^2\partial x}+f_{15}\frac{{\partial }^{3}U}{\partial t^2\partial x}\right),$$

(23)

$${\sigma }_{xx}=\frac{\partial u}{\partial x}+f_1\frac{\partial U}{\partial x}-{\Theta }^s-f_2{\Theta }^f,$$

(24)

$$\sigma =f_1\frac{\partial u}{\partial x}+f_4\frac{\partial U}{\partial x}-f_5{\Theta }^s-f_6{\Theta }^f,$$

(25)

$$u\left(x,0\right)=U\left(x,y,0\right)=0,\frac{\partial u\left(x,0\right)}{\partial t}=\frac{\partial U\left(x,0\right)}{\partial t}=0,$$

(26)

$${\Theta }^s\left(x,0\right)={\Theta }^f\left(x,0\right)=0, \frac{\partial {\Theta }^s\left(x,0\right)}{\partial t}=\frac{\partial {\Theta }^f\left(x,0\right)}{\partial t}=0,$$

(27)

$${\Theta }^s\left(0,t\right)=\left(1-\beta \right){\Theta }_{o}H\left(t\right), {\Theta }^f\left(0,t\right)=\beta {\Theta }_{o}H\left(t\right),$$

(28)

$${\sigma }_{xx}\left(0,t\right)=0, \sigma \left(0,t\right)=0,$$

(29)

where

${ϵ}_s=\frac{K_{\ast }^s}{{\eta c}^2{K}^s}$, ${ϵ}_f=\frac{K_{\ast }^f}{{\eta c}^2{K}^f}$, $f_1=\frac{Q}{\lambda +2\mu }$, $f_2=\frac{R_{12}}{R_{11}}$, $f_3=\frac{{\rho }_{12}}{{\rho }_{11}}$, $f_4=\frac{R}{\lambda +2\mu }$, $f_5=\frac{R_{21}}{R_{11}}$, $f_6=\frac{R_{22}}{R_{11}}$,

$f_7=\frac{{\rho }_{21}}{{\rho }_{11}}$, $f_8=\frac{{\rho }_{22}}{{\rho }_{11}}$, $f_9=\frac{F_{12}}{F_{11}}$, $f_{10}=\frac{T_o{R}_{11}^2}{F_{11}\left(\lambda +2\mu \right)}$, $f_{11}=\frac{T_o{R}_{11}{R}_{21}}{F_{11}\left(\lambda +2\mu \right)}$, $f_{12}=\frac{F_{21}}{\eta K^f}$,

$f_{13}=\frac{F_{22}}{\eta K^f}$, $f_{14}=\frac{T_o{R}_{11}{R}_{12}}{\eta K^f\left(\lambda +2\mu \right)}$, $f_{15}=\frac{T_o{R}_{11}{R}_{22}}{\eta K^f\left(\lambda +2\mu \right)}$.

4. Finite Element Method

For the numerical validations, the weak formulations associated with the initial-boundary value problems of the generalized thermo-elastic models were obtained by proceeding over the standard procedure, as in [51,52]. The associated matrix forms were obtained by discretizing the domain occupied by the poro-elastic medium into nonoverlapping finite elements. The sets of independent weight functions consisted of the displacement of solids $\delta u,$ the displacement of liquids $\delta U$, the solid temperature $\delta {\Theta }^s$, and the liquid temperature $\delta {\Theta }^f$, which were fixed. The basic equations were multiplied by independent test functions. After that, the boundary conditions were applied so as to integrate across the spatial domains. Furthermore, the corresponding nodes for the displacement of the liquid, the solid temperature, the solid displacement, and the temperature of the liquid, are expressed as follows

$${\overline{\Theta }}^s={\displaystyle \sum }_{j=1}^n{N}_j{\overline{\Theta }}_j^s\left(t\right), {\overline{\Theta }}^f={\displaystyle \sum }_{j=1}^n{N}_j{\overline{\Theta }}_j^f\left(t\right), \overline{u}={\displaystyle \sum }_{j=1}^n{N}_j{\overline{u}}_j\left(t\right), \overline{U}={\displaystyle \sum }_{j=1}^n{N}_j{\overline{U}}_j\left(t\right),$$

(30)

where $N$ refers to the shape function, while $n$ refers to the number of nodes per element; where the parts of Galerkin’s standard proceed, the test function and the shape function are the same. So,

$$\delta {\overline{\Theta }}^s={\displaystyle \sum }_{j=1}^n{N}_j\delta {\overline{\Theta }}_j^s, \delta {\overline{\Theta }}^f={\displaystyle \sum }_{j=1}^n{N}_j\delta {\overline{\Theta }}_j^f, \delta \overline{u}={\displaystyle \sum }_{j=1}^n{N}_j\delta {\overline{u}}_j, \delta \overline{U}={\displaystyle \sum }_{j=1}^n{N}_j\delta {\overline{U}}_j.$$

(31)

While the other techniques are the derivatives of time that should be determined at the following steps by using the implicit process. Now, the following are the weak formulations for the finite element scheme that are equivalent to the Equations (20)–(23):

$${{\displaystyle \int }}_0^L\frac{\partial \delta u}{\partial x}\left({\sigma }_{xx}\right)dx+{{\displaystyle \int }}_0^L\delta u\left(\frac{{\partial }^{2}u}{\partial t^2}+f_3\frac{{\partial }^{2}U}{\partial t^2}\right)dx=\delta u{\left({\sigma }_{xx}\right)}_0^L,$$

(32)

$${{\displaystyle \int }}_0^L\frac{\partial \delta U}{\partial x}\left(\sigma \right)dx+{{\displaystyle \int }}_0^L\delta U\left(f_7\frac{{\partial }^{2}u}{\partial t^2}+f_8\frac{{\partial }^{2}U}{\partial t^2}\right)dx=\delta U{\left(\sigma \right)}_0^L,$$

(33)

$$\begin{gathered} {{\displaystyle \int }}_0^L\delta {\Theta }^s\left(\left(1+{\tau }_q^s\frac{\partial }{\partial t}\right)\left(\frac{{\partial }^2{\Theta }^s}{\partial t^2}+f_9\frac{{\partial }^2{\Theta }^f}{\partial t^2}+f_{10}\frac{{\partial }^{3}u}{\partial t^2\partial x}+f_{11}\frac{{\partial }^{3}U}{\partial t^2\partial x}\right)\right)dx \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^L\frac{\partial \delta {\Theta }^s}{\partial x}\left[{ϵ}_s+\left(1+{ϵ}_s{\tau }_{\vartheta }^s\right)\frac{\partial }{\partial t}+{\tau }_{\theta }^s\frac{{\partial }^2}{\partial t^2}\right]\frac{\partial {\Theta }^s}{\partial x}dx \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\delta {\Theta }^s{\left(\left[{ϵ}_s+\left(1+{ϵ}_s{\tau }_{\vartheta }^s\right)\frac{\partial }{\partial t}+{\tau }_{\theta }^s\frac{{\partial }^2}{\partial t^2}\right]\frac{\partial {\Theta }^s}{\partial x}\right)}_0^L, \end{gathered}$$

(34)

$$\begin{gathered} {{\displaystyle \int }}_0^L\delta {\Theta }^f\left(\left(\frac{\partial }{\partial t}+{\tau }_o^f\frac{{\partial }^2}{\partial t^2}\right)\left(f_{12}{\Theta }^s+f_{13}{\Theta }^f+n\left(f_{14}\frac{\partial u}{\partial x}+f_{15}\frac{\partial U}{\partial x}\right)\right)\right)dx \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int }}_0^L\frac{\partial \delta {\Theta }^f}{\partial x}\left[{ϵ}_f+\left(1+{ϵ}_f{\tau }_{\vartheta }^f\right)\frac{\partial }{\partial t}+{\tau }_{\theta }^f\frac{{\partial }^2}{\partial t^2}\right]\frac{\partial {\Theta }^f}{\partial x}dx \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\delta {\Theta }^f{\left(\left[{ϵ}_f+\left(1+{ϵ}_f{\tau }_{\vartheta }^f\right)\frac{\partial }{\partial t}+{\tau }_{\theta }^f\frac{{\partial }^2}{\partial t^2}\right]\frac{\partial {\Theta }^f}{\partial x}\right)}_0^L. \end{gathered}$$

(35)

5. Results and Discussions

The physical quantity distribution in a poro-elastic material was investigated. The thermal properties of saturated sandstone at $T_o=27 °\mathrm{C}$ have been written as in Singh [11,53] for the numerical computation:

$$\begin{gathered} \mathrm{R}=0.07435\times {10}^{11}\left(\mathrm{dyne}\right)\left({\mathrm{cm}}^{-2}\right),{\mathsf{\rho }}^{\mathrm{s}\ast }=2.6\left(\mathrm{g}\right)\left({\mathrm{cm}}^{-3}\right),{ \mathrm{K}}^{\mathrm{s}}=0.4\left(\mathrm{cal}\right)\left({\mathrm{cm}}^{-1}\right)\left({\mathrm{s}}^{-1}\right)\left({°\mathrm{C}}^{-1}\right) \\ {\mathrm{K}}^{\mathrm{f}}=0.3\left(\mathrm{cal}\right)\left({\mathrm{cm}}^{-1}\right)\left({\mathrm{s}}^{-1}\right)\left({°\mathrm{C}}^{-1}\right),\mathrm{Q}=0.99663\times {10}^{11}\left(\mathrm{dyne}\right)\left({\mathrm{cm}}^{-2}\right),{ \mathsf{\rho }}^{\mathrm{f}\ast }=0.82\left(\mathrm{g}\right)\left({\mathrm{cm}}^{-3}\right), \\ \mathsf{\lambda }=0.44363\times {10}^{11}\left(\mathrm{dyne}\right)\left({\mathrm{cm}}^{-2}\right),{ \mathsf{\rho }}_{11}=0.002137\left(\mathrm{g}\right)\left({\mathrm{cm}}^{-3}\right) \\ {\mathrm{C}}_{\mathrm{E}}^{\mathrm{f}}=1.9\left(\mathrm{cal}\right)\left({\mathrm{g}}^{-1}\right)\left({°\mathrm{C}}^{-1}\right),{\mathrm{C}}_{\mathrm{E}}^{\mathrm{s}}=2.1\left(\mathrm{cal}\right)\left({\mathrm{g}}^{-1}\right)\left({°\mathrm{C}}^{-1}\right),\mathsf{\mu }=0.2765\times {10}^{11}\left(\mathrm{dyne}\right)\left({\mathrm{cm}}^{-2}\right) \\ {\mathsf{\alpha }}^{\mathrm{f}}={\mathsf{\alpha }}^{\mathrm{s}}={\mathsf{\alpha }}^{\mathrm{fs}}={\mathsf{\alpha }}^{\mathrm{sf}}=0.1\left({\mathrm{cm}}^{-3}\right)\left({\mathrm{g}}^{-1}\right) \end{gathered}$$

The physical quantities calculated numerically for different x values are explained in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7, and are based on the previous dataset. Numerical calculations were carried out for the liquid and solid displacements, the liquid and solid temperatures, and the components of the stress distribution concerning the distances under six thermal delay times in the framework of the poro-thermo-elastic theory. Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 show the four curves predicted by different levels of porosity $\left(\beta =0.15, \beta =0.30\right)$ using the three-phase lag model (TPL) and the Green−Naghdi type III model (GNIII) at time $\left(t=0.2\right)$. Figure 2 displays the variation of solid displacement with respect to distance $x$. It was discovered that the displacement began at zero, satisfying the boundary constraints, and subsequently increased to the maximum values at a specific spot near the surface. Then, based on the values of porosity and the thermal delay times, monotonic declines in displacement to zero at $1.3\le x\le 2$ were observed. Figure 3 depicts the fluid phase displacement changes along distance x. It is obvious that the liquid displacement began at zero, satisfying the boundary constraints, and subsequently increased to maximum values at a specific spot near the surface. According to the values of thermal delay times and porosity, monotonic declines in liquid displacement to zero at $1.3\le x\le 2$ were found. Figure 4 shows the variations of solid temperature via distance x. It was observed that the solid temperature started at its greatest values at $x=0$ and progressively reduced as the distances $x$ grew to zero beyond a wavefront, satisfying the problem’s theoretical boundary requirements. Figure 5 shows how the liquid temperature changes via distance x. It was discovered that the ${\Theta }^f$ value was maximum at $x=0$ and declined with an increasing x distance until they were close to zero at $x=0.3$, depending on the porosity and model type. The effects of thermal delay times and porosity in liquid and solid stresses throughout the distance $x$ are shown in Figure 6 and Figure 7 under the three-phase lag model (TPL) and Green−Naghdi type III model (GNIII). As expected, the porosity and thermal delay times and porosity had the most significant influence on all of the examined fields’ values.

6. Conclusions

The effects of thermal delay times in poro-elastic media have been studied. The variations of temperature, variations of displacement, and distributions of solid and liquid stresses in poro-elastic media have been given using the three-phase-lag poro-thermo-elastic theory. The finite element technique has been used to obtain the numerical solutions for the thermo-elastic interactions in poro-elastic materials subjected to thermal shock loading. The parametric analysis has been finished so that everyone can agree on the best way to choose critical design variables so that new poro-elastic heat conduction models can be understood on a basic level.