Effect of Hydrostatic Initial Stress on a Rotating Half-Space in the Context of a Two-Relaxation Power-Law Model

Abstract

The simple and refined Lord–Shulman theories, the simple and refined Green–Lindsay theories as well as the coupled thermoelasticity theory were all employed to investigate the deformation of a rotating thermoelastic half-space. The present medium is subjected to initial pressure, bounded by hydrostatic initial stress and rotation. A unified heat conduction equation is presented. The normal mode strategy is applied to get all analytical expressions of temperature, stresses, and displacements. Some outcomes are tabulated to validate the present refined theories with the simple and classical thermoelasticity theories. The effect of hydrostatic initial stress was investigated on all field quantities of the rotating thermoelastic half-space with and without initial pressure. Two- and three-dimensional plots are illustrated in the context of refined theories to discuss the behaviors of all variables through the coordinate axes. Some particular cases of special interest have been deduced from the present investigation.

1. Introduction

In most of the discussed problems, the structure does not subject to the action of external forces. It is interesting for such structures to deal with what is called the hydrostatic initial stresses. The initial stresses affect a lot of static and dynamic phenomena of elastic structures. The general treatment of the hydrostatic initial stresses in rotating structures has been studied by many investigations such as Ailawalia and Narah [1], Ailawalia et al. [2], Abd-Alla and Abo-Dahab [3], and Saeedm et al. [4]. Fahmy and El-Shahat [5] presented a general treatment of the transient thermoelastic stresses in a rotating nonhomogeneous anisotropic solid under compressive initial stress. Ailawalia et al. [6] studied the deformation of a rotating generalized thermoelastic medium with two temperatures under hydrostatic initial stress subjected to different types of sources. Kumar and Kumar [7] studied wave propagation in transversely isotropic generalized thermoelastic half-space with voids under initial stress. Li et al. [8] studied the effects of rotation on the model of generalized thermoelastic diffusion for a thermally, isotropic and electrically conducting half-space with rotation at the uniform angular velocity. Sheoran et al. [9] established a 2D model of rotating half-space initially stressed with micro temperature. In fact, some investigators have restricted their attention to the rotating problems of half-space with diffusions like Li and Zhang [10], Deswal and Hooda [11], or the propagation of plane waves with two temperatures like Singh and Bijarnia [12] or the magnetothermoelastic interaction for a rotating half-space like Das et al. [13].

One of the most important generalized thermoelasticity theories is that of Lord and Shulman (L–S) [14]. Othman and Abbas [15,16] discussed the propagation of plane waves in fiber-reinforced, rotating thermoelastic half-space proposed by L–S under hydrostatic initial stress. Abo-Dahab [17] discussed the magneto-thermoelastic response of half-space subjected to initial stress in the context of the L–S theory. Recently, Kaur et al. [18] presented a mathematical treatment to study the disturbance due to mechanical and thermal sources in 2D transversely isotropic magneto thermoelastic solids under initial stress via the L–S theory. Kutbi and Zenkour [19] discussed the thermomechanical waves in an axisymmetric rotating disk in the context of different thermoelasticity models. In addition to the classical thermoelasticity (CTE) theory and the L–S theory, Lotfy and El-Bary [20] used the Green–Lindsay (G–L) theory to discuss the magneto-thermoelastic interactions in half-space with two temperatures under the influence of initial stress.

Some modifications are made depending on the fractional order theory of generalized thermoelasticity. Deswal et al. [21] studied magneto-thermoelastic interactions in an initially stressed isotropic homogeneous half-space in the context of the fractional order theory of generalized thermoelasticity. Santra et al. [22] presented a 3D problem for a rotating half-space under a time-dependent and traction-free heat source. Sarkar and Lotfy [23] solved the problem of thermoelastic interactions in a half-space medium under hydrostatic initial stress in the context of a fractional order heat conduction model with the two-temperature theory. Abouelregal and Zenkour [24,25,26] discussed the generalized thermoelastic problems for infinite fiber-reinforced thick plate and viscoelastic microbeams under initial stress.

The second most important generalized thermoelasticity theory is that of Green and Naghdy (G–N) [27,28,29]. Chattopadhyay and Biswas [30] used the G–N theory of type II (without energy dissipation) to deal with a boundary value problem of an isotropic elastic half-space. Othman and Song [31] and Atwa and Othman [32] constructed the equations of generalized thermoelastic isotropic and homogeneous half-space under hydrostatic initial stress in the context of the G–N theory of types II and III. Abbas and Zenkour [33] presented the effect of rotation and initial stress on thermal shock problems for a fiber-reinforced anisotropic half-space using the G–N thermoelastic theory. Said [34] used the three-phase-lag model as well as the G–N theory without energy dissipation to discuss the deformation of a two-temperature generalized-magneto thermoelastic medium under hydrostatic initial stress and rotation. Othman et al. [35] studied the rotation of initially stressed thermoelastic medium with voids subjected to thermal loading due to laser pulse. Xiong and Tian [36] discussed the transient thermoelastic responses of a semi-infinite fiber-reinforced porous body with the initial stress based upon the G–N thermoelastic theory of type II. Abo-Dahab et al. [37] used the G–N theory to investigate a 2D problem for a half-space under a magnetic field and a time-dependent heat source.

The influence of hydrostatic initial stress on the reflected wave for the incidence of P and SV waves from a generalized thermoelastic half-space has been studied by Singh [38], Abo-Dahab and Mohamed [39], and Sarkar et al. [40]. Singh and Chakraborty [41] studied the impact of the magnetic field and initial stress on the reflection of a plane magneto-thermoelastic wave from the boundary of a solid half-space. The basic dynamical equations of thermoelasticity in the presence of the Lorentz force and the heat conduction equation of the Green–Lindsay theory have been solved to derive the wave equations for the reflected P-, SV-, and thermal waves. Othman et al. [42] presented a model of the equations of generalized magneto-thermoelasticity in an isotropic elastic medium with two temperatures under the effect of initial stress. Othman and Eraki [43] studied the effect of initial stress and magnetic field on thermoelastic interactions in half-space under mechanical and thermal loads. Yuan and Jiang [44] presented the initial stresses and rotation impacts on the wave reflection in the pyroelectric half-plane using the wave approach. Abo-Dahab et al. [45] discussed the influence of the rotation, thermal field, initial stress, gravity field, electromagnetic, and voids on the reflection of the P wave under three models of generalized thermoelasticity.

In the current article, an investigation of the influence of the rotation, initial pressure, and hydrostatic initial stress, on the field quantities of half-space is discussed. Here, we deduce the classical, simple, and refined Lord–Shulman, as well as the simple and refined Green–Lindsay theories of generalized thermoelasticity. The problem has been solved and some outcomes for the temperature, displacements, and stresses are tabulated. For the sake of the obvious, some plots for all field quantities are presented.

2. Basic Governing Equations

In the orthogonal coordinate system $XOZ$, we consider a two-dimensional isotropic half-space defined in the region $\Omega =\{0\le x<\infty ,-\infty <z<\infty \}$ subject to traction-free boundary $x=0$, $z$-axis is vertically downwards, and $xz$ along the free surface of the half-space.

2.1. Heat Conduction Equation

The heat conduction equation of the medium is represented by

$$k{\nabla }^2\theta =\rho c_E{𝒟}_1\frac{\partial \theta }{\partial t}+\gamma T_0{𝒟}_1^m\frac{\partial e}{\partial t},$$

(1)

where ${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial z^2}$; $\theta =T-T_0$, in which $T$ is the temperature above the reference temperature $T_0$; $t$ is the time; $\rho$ is the density of the medium; $c_E$ is the specific heat at constant strain, $k$ is the coefficient of thermal conductivity; $e=e_{kk}=u_{k,k}$ is the volumetric strain in which $u_i$ is the displacement components; $\gamma ={\alpha }_t\left(3\lambda +2\mu \right)$ is the thermoelastic coupling parameter in which $\lambda$, $\mu$ are Lame’s constants and ${\alpha }_t$ is the thermal expansion coefficient; and ${𝒟}_1$, ${𝒟}_1^m$ are time-derivative operators that are expressed by [46,47,48,49,50]

$${𝒟}_1=1+{{\displaystyle \sum }}_{n=1}^N{\tau }_1^n\frac{{\partial }^n}{\partial t^n},{𝒟}_1^m=1+{\delta }_{1m}{{\displaystyle \sum }}_{n=1}^N{\tau }_1^n\frac{{\partial }^n}{\partial t^n},N\ge 1,$$

(2)

in which ${\tau }_1$ is the first relaxation time and ${\delta }_{1m}$ is a Kronecker’s delta.

2.2. The Stress-Displacement-Temperature Relations

The constitutive equations of the theory of generalized thermoelasticity are given as

$${\sigma }_{ij}=\left(\lambda e-\gamma {𝒟}_2^m\theta -𝓅\right){\delta }_{ij}+2\mu e_{ij}-𝓅{\omega }_{ij},$$

(3)

where ${\sigma }_{ij}$ is the stress tensor; $e_{ij}$ is the strain tensor; ${\omega }_{ij}$ is referred to as the rotation tensor; ${\delta }_{ij}$ is the Kronecker’s delta; $𝓅$ is the initial pressure, and ${𝒟}_2^m$ is a time-derivative operator that is expressed by

$${𝒟}_2^m=1+{\delta }_{2m}{{\displaystyle \sum }}_{n=1}^N{\tau }_2^n\frac{{\partial }^n}{\partial t^n},N\ge 1,$$

(4)

in which ${\tau }_2$ is the second relaxation time and ${\delta }_{2m}$ is a Kronecker’s delta.

The strain and rotation tensors can be expressed in terms of the displacement $u_i$ as

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

(5)

Then, Equation (3) yields

$${\sigma }_{xx}=\left(\lambda +2\mu \right)\frac{\partial u}{\partial x}+\lambda \frac{\partial w}{\partial z}-\gamma {𝒟}_2^m\theta -𝓅,$$

(6)

$${\sigma }_{zz}=\left(\lambda +2\mu \right)\frac{\partial w}{\partial z}+\lambda \frac{\partial u}{\partial x}-\gamma {𝒟}_2^m\theta -𝓅,$$

(7)

$${\sigma }_{xz}=\left(\mu -\frac{𝓅}{2}\right)\frac{\partial u}{\partial z}+\left(\mu +\frac{𝓅}{2}\right)\frac{\partial w}{\partial x},$$

(8)

$${\sigma }_{zx}=\left(\mu +\frac{𝓅}{2}\right)\frac{\partial u}{\partial z}+\left(\mu -\frac{𝓅}{2}\right)\frac{\partial w}{\partial x},$$

(9)

where the displacements $u_i\equiv \left(u,0,w\right)$.

2.3. Equations of Motion

The equations of motion of the medium are represented by

$$\left(\lambda +2\mu \right)\frac{{\partial }^{2}u}{\partial x^2}+\left(\mu +\frac{𝓅}{2}\right)\frac{{\partial }^{2}u}{\partial z^2}+\left(\lambda +\mu -\frac{𝓅}{2}\right)\frac{{\partial }^{2}w}{\partial x\partial z}-\gamma {𝒟}_2^m\frac{\partial \theta }{\partial x}=\rho \left(\frac{{\partial }^{2}u}{\partial t^2}-{\Omega }^{2}u+2\Omega \frac{\partial w}{\partial t}\right),$$

(10)

$$\left(\lambda +2\mu \right)\frac{{\partial }^{2}w}{\partial z^2}+\left(\mu +\frac{𝓅}{2}\right)\frac{{\partial }^{2}w}{\partial x^2}+\left(\lambda +\mu -\frac{𝓅}{2}\right)\frac{{\partial }^{2}u}{\partial x\partial z}-\gamma {𝒟}_2^m\frac{\partial \theta }{\partial z}=\rho \left(\frac{{\partial }^{2}w}{\partial t^2}-{\Omega }^{2}w-2\Omega \frac{\partial u}{\partial t}\right).$$

(11)

3. Formulation of the Problem

To transform the above equations into non-dimensional forms, we shall use the following non-dimensional variables defined as

$$\begin{gathered} \left\{x^{\prime },z^{\prime },u^{\prime },w^{\prime }\right\}=\frac{c_0}{\eta }\left\{x,z,u,w\right\},\left\{t^{\prime },{\tau }_1^{\prime },{\tau }_2^{\prime }\right\}=\frac{c_0^2}{\eta }\left\{t,{\tau }_1,{\tau }_2\right\},{\theta }^{\prime }=\frac{\gamma \theta }{\rho c_0^2}, \\ \left\{{\sigma }_{ij}^{\prime },{𝓅}^{\prime }\right\}=\frac{1}{\rho c_0^2}\left\{{\sigma }_{ij},𝓅\right\},{\Omega }^{\prime }=\frac{\eta }{c_0^2}\Omega , \end{gathered}$$

(12)

in which

$$\eta =\frac{k}{\rho c_E},c_0=\sqrt{\frac{\lambda +2\mu }{\rho }}.$$

(13)

The non-dimensional forms of stresses can then be written as (omitting the primes for convenience)

$${\sigma }_{xx}=\frac{\partial u}{\partial x}+c_1\frac{\partial w}{\partial z}-{𝒟}_2^m\theta -𝓅,$$

(14)

$${\sigma }_{zz}=\frac{\partial w}{\partial z}+c_1\frac{\partial u}{\partial x}-{𝒟}_2^m\theta -𝓅,$$

(15)

$${\sigma }_{xz}=c_2\frac{\partial u}{\partial z}+c_3\frac{\partial w}{\partial x},$$

(16)

$${\sigma }_{zx}=c_3\frac{\partial u}{\partial z}+c_2\frac{\partial w}{\partial x}.$$

(17)

Also, the governing equations of motion and heat conducting equation are given by

$$\frac{{\partial }^{2}u}{\partial x^2}+c_3\frac{{\partial }^{2}u}{\partial z^2}+c_4\frac{{\partial }^{2}w}{\partial x\partial z}-{𝒟}_2^m\frac{\partial \theta }{\partial x}=\frac{{\partial }^{2}u}{\partial t^2}-{\Omega }^{2}u+2\Omega \frac{\partial w}{\partial t},$$

(18)

$$\frac{{\partial }^{2}w}{\partial z^2}+c_3\frac{{\partial }^{2}w}{\partial x^2}+c_4\frac{{\partial }^{2}u}{\partial x\partial z}-{𝒟}_2^m\frac{\partial \theta }{\partial z}=\frac{{\partial }^{2}w}{\partial t^2}-{\Omega }^{2}w-2\Omega \frac{\partial u}{\partial t},$$

(19)

$${\nabla }^2\theta ={𝒟}_1\frac{\partial \theta }{\partial t}+ϵ{𝒟}_1^m\frac{\partial e}{\partial t},$$

(20)

where

$$c_1=\frac{\lambda }{\lambda +2\mu },c_2=\frac{\mu }{\lambda +2\mu }-\frac{𝓅}{2},c_3=\frac{\mu }{\lambda +2\mu }+\frac{𝓅}{2},c_4=c_1+c_2,ϵ=\frac{{\gamma }^2{T}_0}{\rho c_E\left(\lambda +2\mu \right)}.$$

(21)

4. Solution of the Problem

The solution of the considered physical variable can be decomposed in terms of normal modes in the following form:

$$\left\{u,w,\theta ,{\sigma }_{ij}\right\}\left(x,z,t\right)=\left\{u^{*},w^{*},{\theta }^{*},{\sigma }_{ij}^{*}\right\}\left(x\right){\mathrm{e}}^{\omega t+𝒾bz},$$

(22)

where $\omega$ is a complex frequency constant, $𝒾=\sqrt{-1}$, $b$ is the wave number in the $z$-direction, $u^{*}$, $w^{*}$, ${\theta }^{*}$ and ${\sigma }_{ij}^{*}$ are the amplitudes of the field quantities.

Substituting from Equation (22) in Equations (18)–(20), we get

$$\frac{{\mathrm{d}}^2{u}^{*}}{\mathrm{d}{x}^2}-c_5{u}^{*}+𝒾b{c}_4\frac{\mathrm{d}{w}^{*}}{\mathrm{d}x}-c_6{w}^{*}-{\overline{𝒟}}_2^m\frac{\mathrm{d}{\theta }^{*}}{\mathrm{d}x}=0,$$

(23)

$$c_3\frac{{\mathrm{d}}^2{w}^{*}}{\mathrm{d}{x}^2}-c_7{w}^{*}+𝒾b{c}_4\frac{\mathrm{d}{u}^{*}}{\mathrm{d}x}+c_6{u}^{*}-𝒾b{\overline{𝒟}}_2^m{\theta }^{*}=0,$$

(24)

$$\frac{{\mathrm{d}}^2{\theta }^{*}}{\mathrm{d}{x}^2}-c_8{\theta }^{*}-c_9\frac{\mathrm{d}{u}^{*}}{\mathrm{d}x}-𝒾b{c}_9{w}^{*}=0,$$

(25)

where

$$c_5=b^2{c}_3+{\omega }^2-{\Omega }^2,c_6=2\Omega \omega ,c_7=b^2+{\omega }^2-{\Omega }^2,c_8=b^2+\omega {\overline{𝒟}}_1,c_9=ϵ\omega {\overline{𝒟}}_1^m,$$

(26)

in which

$$\begin{array}{c}{\overline{𝒟}}_1=1+{\displaystyle \sum }_{n=1}^N{\tau }_1^n{\omega }^n\\ {\overline{𝒟}}_1^m=1+{\delta }_{1m}{\displaystyle \sum }_{n=1}^N{\tau }_1^n{\omega }^n\\ {\overline{𝒟}}_2^m=1+{\delta }_{2m}{\displaystyle \sum }_{n=1}^N{\tau }_2^n{\omega }^n\end{array}\},\hspace{1em}N\ge 1.$$

(27)

Simplifying the system of differential Equations (23)–(25), one can get the unified form

$$\left(\frac{{\mathrm{d}}^6}{\mathrm{d}{x}^6}-A_2\frac{{\mathrm{d}}^4}{\mathrm{d}{x}^4}+A_1\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}-A_0\right)\left\{u^{*},w^{*},{\theta }^{*}\right\}=0,$$

(28)

where the coefficients $A_i$, $i=0,1,2$ are given by

$$\begin{array}{c}{A}_0=\frac{1}{c_3}\left[\left(c_5{c}_7+c_6^2\right)c_8+b^2{c}_5{c}_9{\overline{𝒟}}_2^m\right],\\ A_1=\frac{1}{c_3}\left[\left(c_5+c_8+c_9{\overline{𝒟}}_2^m\right)c_7+\left(c_3{c}_5-b^2{c}_4^2\right)c_8+b^2{c}_9{\overline{𝒟}}_2^m\left(1-2c_4\right)+c_6^2\right],\\ A_2=c_5+c_8+c_9{\overline{𝒟}}_2^m+\frac{c_7-b^2{c}_4^2}{c_3}.\end{array}$$

(29)

Presenting ${\xi }_i,$ $\left(i=1,2,3\right)$ into Equation (25), one obtains

$$\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}-{\xi }_1^2\right)\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}-{\xi }_2^2\right)\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}-{\xi }_3^2\right)\left\{u^{*},w^{*},{\theta }^{*}\right\}=0,$$

(30)

where ${\xi }_i^2$ are the roots of

$${\xi }^6-A_2{\xi }^4+A_1{\xi }^2-A_0=0,$$

(31)

with

$$\begin{array}{c}{\xi }_1^2=\frac{4A_2{A}_3+4\left(1-𝒾\sqrt{3}\right)\left(3A_1-A_2^2\right)-\left(1+𝒾\sqrt{3}\right)A_3^2}{12A_3},\\ {\xi }_2^2=\frac{4A_2{A}_3+4\left(1+𝒾\sqrt{3}\right)\left(3A_1-A_2^2\right)-\left(1-𝒾\sqrt{3}\right)A_3^2}{12A_3},\\ {\xi }_3^2=\frac{{\left(A_2+A_3\right)}^2+3\left(A_2^2-4A_1\right)}{6A_3},\end{array}$$

(32)

in which

$$\begin{array}{c}{A}_3^3=108A_0-36A_1{A}_2+8A_2^3+12A_4,\\ A_4^2=3A_0\left(27A_0+4A_2^3-18A_1{A}_2\right)+3A_1^2\left(4A_1-A_2^2\right).\end{array}$$

(33)

Equation (26) has a solution under the regularity conditions: $u^{*},w^{*},{\theta }^{*}\to 0$ as $x\to \infty$. Therefore, its general solution, which is bounded at infinity, is provided by

$$\left\{u^{*},w^{*},{\theta }^{*}\right\}={{\displaystyle \sum }}_{j=1}^3\left\{1,{\overline{\xi }}_j,{\stackrel{=}{\xi }}_j\right\}{B}_j{\mathrm{e}}^{-{\xi }_{j}x},$$

(34)

where $B_j$ $\left(j=1,2,3\right)$ are some parameters in terms of $b$, $\omega$; and ${\overline{\xi }}_j$, ${\stackrel{=}{\xi }}_j$ are some different parameters related to $B_j$. It is to be noted that, in the above solution, there are three terms containing exponentials of growing in nature in the space variables $x$ that have been discarded. Substituting Equation (33) into Equations (23) and (24), we obtain the relations:

$${\overline{\xi }}_j=\frac{𝒾b\left(c_4-1\right){\xi }_j^2-c_6{\xi }_j+𝒾b{c}_5}{c_3{\xi }_j^3-\left(c_7-b^2{c}_4\right){\xi }_j-𝒾b{c}_6},\hspace{1em}{\stackrel{=}{\xi }}_j=-\frac{c_3{\xi }_j^4-\left(c_7+c_3{c}_5-b^2{c}_4^2\right){\xi }_j^2+c_5{c}_7+c_6^2}{\left[c_3{\xi }_j^3-\left(c_7-b^2{c}_4\right){\xi }_j-𝒾b{c}_6\right]{\overline{𝒟}}_2^m}.$$

(35)

So, the final expressions for displacements and temperature can be expressed as

$$\left\{u,w,\theta \right\}={{\displaystyle \sum }}_{j=1}^3\left\{1,{\overline{\xi }}_j,{\stackrel{=}{\xi }}_j\right\}{B}_j{\mathrm{e}}^{\omega t-{\xi }_{j}x+𝒾bz},$$

(36)

Also, from the above equations and Equations (14)–(17), one can obtain the final expressions for stresses as

$$\left\{{\sigma }_{xx},{\sigma }_{zz}\right\}={{\displaystyle \sum }}_{j=1}^3\left\{{\zeta }_j,{\overline{\zeta }}_j\right\}{B}_j{\mathrm{e}}^{\omega t-{\xi }_{j}x+𝒾bz}-𝓅,$$

(37)

$$\left\{{\sigma }_{xz},{\sigma }_{zx}\right\}={{\displaystyle \sum }}_{j=1}^3\left\{{\widehat{\zeta }}_j,{\check{\zeta }}_j\right\}{B}_j{\mathrm{e}}^{\omega t-{\xi }_{j}x+𝒾bz,},$$

(38)

where

$$\begin{array}{c}{\zeta }_j=𝒾b{c}_1{\overline{\xi }}_j-{\overline{𝒟}}_2^m{\stackrel{=}{\xi }}_j-{\xi }_j,{\overline{\zeta }}_j=𝒾b{\overline{\xi }}_j-{\overline{𝒟}}_2^m{\stackrel{=}{\xi }}_j-c_1{\xi }_j,\\ {\widehat{\zeta }}_j=𝒾b{c}_2-c_3{\xi }_j{\overline{\xi }}_j,{\check{\zeta }}_j=𝒾b{c}_3-c_2{\xi }_j{\overline{\xi }}_j.\end{array}$$

(39)

5. Boundary Conditions

In addition to the regularity conditions of the solution at infinity which are used in Equation (33), we further get the displacements, stresses, and temperatures in terms of arbitrary parameters $B_j$, ($j=1,2,3$). Such parameters are to be determined from the following boundary conditions.

5.1. Mechanical Conditions

The boundary of the half-space $x=0$ has no traction elsewhere, i.e.,

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

(40)

where $f\left(z,t\right)={\sigma }_0{\mathrm{e}}^{\omega t+𝒾bz}$ in which ${\sigma }_0$ represents the magnitude of the hydrostatic initial stress.

5.2. Thermal Condition

The half-space is isolated from the thermal field at the boundary $x=0$

$${\frac{\partial \theta }{\partial x}|}_{x=0}=0.$$

(41)

Using the expressions of the stresses ${\sigma }_{xx}$, ${\sigma }_{xz}$ and the temperature $\theta$ in the above boundary conditions, we can obtain the following equations satisfied with the parameters:

$$\left\{\begin{array}{c}{B}_1\\ B_2\\ B_3\end{array}\right\}={\left[\begin{array}{ccc}{\zeta }_1& {\zeta }_2& {\zeta }_3\\ {\widehat{\zeta }}_1& {\widehat{\zeta }}_2& {\widehat{\zeta }}_3\\ {\xi }_1{\stackrel{=}{\xi }}_1& {\xi }_2{\stackrel{=}{\xi }}_2& {\xi }_3{\stackrel{=}{\xi }}_3\end{array}\right]}^{-1}\left\{\begin{array}{c}{𝓅}^{*}-{\sigma }_0\\ 0\\ 0\end{array}\right\}.$$

(42)

So, all variable quantities like temperature, displacement, and stresses will be easily given.

6. Different Thermoelasticity Theories

The closed-form solution is already given for the refined generalized thermoelasticity theory in the context of a one/two-relaxation power-law model. The heat conduction equation that appeared in Equation (1) contains at least three generalized thermoelasticity theories. These theories contain the coupled thermoelasticity (CTE) theory [51], the simple Lord and Shulman (L–S) thermoelasticity theory [14] as well as the simple Green and Lindsay (G–L) thermoelasticity theory [52]. So, one can summarize the above special cases from Equations (1) and (3) as documented here:

• The coupled thermoelasticity (CTE) theory is given when ${𝒟}_1={𝒟}_1^m={𝒟}_2^m=1$ [51];
• The simple L–S theory is given by setting $m=1$, $N=1$ and ${\tau }_1\ge 0$ [15];
• The simple G–L theory is given by setting $m=2$, $N=1$ and ${\tau }_2\ge {\tau }_1\ge 0$ [52];
• The refined L–S theory is given by setting $m=1$, $N>1$ and ${\tau }_1\ge 0$;
• The refined G–L theory is given by setting $m=2$, $N>1$ and ${\tau }_2\ge {\tau }_1\ge 0$.

7. Numerical Results and Discussions

In this section, some illustrated and tabulated results have been computed numerically. The physical parameters used in the calculations are given at $T_0=273\mathrm{K}$ as follows [11]:

$$\begin{array}{c}\lambda =7.76\times {10}^9{\mathrm{N}\mathrm{m}}^{-2},\mu =3.86\times {10}^{10}{\mathrm{N}\mathrm{m}}^{-2},\rho =8954{\mathrm{kg}\mathrm{m}}^{-3},\\ k=150{\mathrm{W}\mathrm{m}}^{-1}{\mathrm{K}}^{-1},c_E=383.1{\mathrm{J}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1},{\alpha }_t=3.78\times {10}^{-4} {\mathrm{K}}^{-1}.\end{array}$$

(43)

Numerical results are obtained (except where otherwise indicated) for $p=10$, 0; $t=0.5$; ${\sigma }_0=9$, 5; $\omega =0.95+0.05𝒾$; $b=\pi /3$ and $\Omega =2.5$.

7.1. Validation

Table 1. Effect of initial stress ${\sigma }_0$ on the temperature $\theta$ and displacements $u$ and $w$ of the rotating half-space with initial pressure ($𝓅=10$).

Theory		${\mathit{\sigma }}_0 = 9$			${\mathit{\sigma }}_0 = 5$
		$\mathit{\theta }\left(3,1\right)$	$\mathit{u}\left(1,0.5\right)$	$\mathit{w}\left(3,0.5\right)$	$\mathit{\theta }\left(3,1\right)$	$\mathit{u}\left(1,0.5\right)$	$\mathit{w}\left(3,0.5\right)$
CTE		1.890364	2.996973	1.847246	1.365965	2.409226	1.434999
L–S	simple	1.832256	2.604654	1.462894	1.346071	2.090542	1.138687
	$N=2$	1.822702	2.563813	1.423560	1.341577	2.056822	1.107933
	$N=3$	1.821655	2.560139	1.419911	1.341048	2.053746	1.105058
	$N=4$	1.821568	2.559885	1.419648	1.341003	2.053531	1.104849
	$N=5$	1.821562	2.559872	1.419633	1.340999	2.053519	1.104837
G–L	simple	1.361227	2.563666	1.418260	0.999593	2.056354	1.104261
	$N=2$	1.296335	2.514292	1.369209	0.953542	2.015296	1.065907
	$N=3$	1.289281	2.509351	1.364012	0.948486	2.011112	1.061815
	$N=4$	1.288682	2.508974	1.363583	0.948053	2.010785	1.061475
	$N=5$	1.288642	2.508951	1.363554	0.948024	2.010765	1.061453

,

Table 2. Effect of initial stress ${\sigma }_0$ on the stresses of the rotating half-space with initial pressure ($𝓅=10$).

Theory		${\mathit{\sigma }}_0 = 9$			${\mathit{\sigma }}_0 = 5$
		${\mathit{\sigma }}_{\mathit{x}}\left(0.5,1.5\right)$	${\mathit{\sigma }}_{\mathit{z}}\left(0.5,1\right)$	${\mathit{\sigma }}_{\mathit{x}\mathit{z}}\left(2,1.5\right)$	${\mathit{\sigma }}_{\mathit{x}}\left(0.5,1.5\right)$	${\mathit{\sigma }}_{\mathit{z}}\left(0.5,1\right)$	${\mathit{\sigma }}_{\mathit{x}\mathit{z}}\left(2,1.5\right)$
CTE		4.223049	5.080442	9.613937	2.280856	0.804530	7.509281
L–S	simple	3.158542	5.097106	8.721411	1.553304	0.934364	7.011773
	$N=2$	3.032111	5.089880	8.615998	1.463906	0.942645	6.950218
	$N=3$	3.019749	5.088609	8.605439	1.455013	0.943041	6.943900
	$N=4$	3.018817	5.088472	8.604620	1.454332	0.943040	6.943399
	$N=5$	3.018761	5.088461	8.604569	1.454291	0.943038	6.943367
G–L	simple	3.027316	5.096085	8.568865	1.462244	0.948727	6.913201
	$N=2$	2.867656	5.085937	8.424122	1.348946	0.958633	6.824960
	$N=3$	2.849907	5.083973	8.407134	1.336130	0.959136	6.814227
	$N=4$	2.848387	5.083733	8.405606	1.335016	0.959128	6.813234
	$N=5$	2.848284	5.083712	8.405498	1.334939	0.959124	6.813161

,

Table 3. Effect of initial stress ${\sigma }_0$ on the temperature $\theta$ and displacements $u$ and $w$ of the rotating half-space without initial pressure ($𝓅=0$).

Theory		${\mathit{\sigma }}_0 = 9$			${\mathit{\sigma }}_0 = 5$
		$\mathit{\theta }\left(3,1\right)$	$\mathit{u}\left(1,0.5\right)$	$\mathit{w}\left(3,0.5\right)$	$\mathit{\theta }\left(3,1\right)$	$\mathit{u}\left(1,0.5\right)$	$\mathit{w}\left(3,0.5\right)$
CTE		0.630417	0.279556	1.603015	0.350231	0.155309	0.890564
L–S	simple	0.819987	0.282788	1.574989	0.455548	0.157104	0.874994
	$N=2$	0.848295	0.282858	1.571541	0.471275	0.157143	0.873078
	$N=3$	0.851151	0.282830	1.571224	0.472862	0.157128	0.872902
	$N=4$	0.851368	0.282825	1.571202	0.472982	0.157125	0.872890
	$N=5$	0.851381	0.282824	1.571200	0.472989	0.157125	0.872889
G–L	simple	0.629805	0.283580	1.571955	0.349892	0.157544	0.873308
	$N=2$	0.629652	0.283818	1.567726	0.349807	0.157677	0.870959
	$N=3$	0.629666	0.283804	1.567293	0.349814	0.157669	0.870718
	$N=4$	0.629670	0.283800	1.567260	0.349817	0.157666	0.870699
	$N=5$	0.629671	0.283799	1.567258	0.349817	0.157666	0.870699

and

Table 4. Effect of initial stress ${\sigma }_0$ on the stresses of the rotating half-space without initial pressure ($𝓅=0$).

Theory		${\mathit{\sigma }}_0 = 9$			${\mathit{\sigma }}_0 = 5$
		${\mathit{\sigma }}_{\mathit{x}}\left(0.5,1.5\right)$	${\mathit{\sigma }}_{\mathit{z}}\left(0.5,1\right)$	${\mathit{\sigma }}_{\mathit{x}\mathit{z}}\left(2,1.5\right)$	${\mathit{\sigma }}_{\mathit{x}}\left(0.5,1.5\right)$	${\mathit{\sigma }}_{\mathit{z}}\left(0.5,1\right)$	${\mathit{\sigma }}_{\mathit{x}\mathit{z}}\left(2,1.5\right)$
CTE		0.677300	0.445521	2.406474	0.376278	0.247511	1.336930
L–S	simple	0.761076	0.449179	2.380849	0.422820	0.249544	1.322694
	$N=2$	0.774165	0.450059	2.378418	0.430092	0.250033	1.321343
	$N=3$	0.775488	0.450134	2.378250	0.430827	0.250075	1.321250
	$N=4$	0.775587	0.450138	2.378243	0.430882	0.250077	1.321246
	$N=5$	0.775593	0.450138	2.378243	0.430885	0.250077	1.321246
G–L	simple	0.769812	0.448273	2.378538	0.427673	0.249040	1.321410
	$N=2$	0.785880	0.448914	2.375733	0.436600	0.249397	1.319852
	$N=3$	0.787678	0.448944	2.375531	0.437599	0.249413	1.319739
	$N=4$	0.787827	0.448942	2.375522	0.437682	0.249412	1.319735
	$N=5$	0.787837	0.448941	2.375522	0.437687	0.249412	1.319735

provide the temperature $\theta$, the displacements $u$ and $w,$ and the stresses using the CTE, simple and refined L–S, and G–L theories. Table 1 and Table 2 show the effect of initial stress ${\sigma }_0$ on all variables of the rotating half-space with initial pressure ($𝓅=10$). Table 3 and Table 4 show the same results of the rotating half-space without initial pressure ($𝓅=0$). 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, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26 and Figure 27 show further results of all variables and two- and three-dimensional plots. The present tables may be used to help other investigators in their investigations. From such tables, we concluded that:

• The refined L–S and G–L theories are developed with $N$ equals from 2 to 5. Nevertheless, the simple L–S and G–L theories are essentially provided when $N=1$.
• The incredibly accurate results are generated when $N=5$.
• The outcomes of the refined L–S and G–L theories are decreased as $N$ increases and may be unchanged when $N\ge 5$.
• For both values of the initial stress ${\sigma }_0$, the temperature and displacements of the simple and refined L–S theories are greater than the corresponding ones of the simple and refined G–L theories and less than the results of the CTE theory. This is already shown in Table 1 for the half-plane with initial pressure ($𝓅=10$).
• Table 2 shows that the results of ${\sigma }_z$ during all theories are very close to each other, especially when ${\sigma }_0=9$. The results of the CTE theory are the largest ones for other stresses ${\sigma }_x$ and ${\sigma }_{xz}$.
• As shown in Table 3 for the rotating half-space without initial pressure ($𝓅=0$), the temperatures due to the simple and refined L–S theories are the greatest ones. However, the temperatures due to the simple and refined G–L theories are very close to those due to the CTE theory.
• The results of horizontal displacement $u$ due to the simple and refined L–S and G–L theories are very close and both of them are greater than those of the CTE theory.
• The results of vertical displacement $w$ due to the simple and refined L–S and G–L theories are very close and both of them are smaller than those of the CTE theory.
• Table 4 shows that the results of the normal ${\sigma }_z$ and shear ${\sigma }_{xz}$ stresses during all theories are very close to each other. The results of the CTE theory are the smallest ones for the transverse normal stress ${\sigma }_z$ and the largest ones for the transverse shear stress ${\sigma }_{xz}$. However, the results of the longitudinal stress ${\sigma }_x$ are due to the simple and refined G–L theories being the largest ones.

7.2. The 2D Applications

A time $t=0.5$ is used in all figures to demonstrate the effect of the initial stress ${\sigma }_0$ on all variables of the rotating half-space with and without initial pressures. The outcomes of the CTE theory are presented in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. Accordingly, all of these plots are shown about the refined L–S and G–L theories with $N=5$. All results are shown across the $x$-axis with $z=1.5$ for temperature and displacements while $z=1$ for stresses.

Figure 1 depicts the temperature of the rotating half-space with initial stresses ($𝓅=10$) matching all theories. Similar plots for other variables of the rotating half-space with initial stresses ($𝓅=10$) are illustrated in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7. However, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show comparable plots of all variables of the rotating half-space without initial stresses ($𝓅=0$).

As shown in Figure 1, the temperature $\theta$ of CTE and refined L–S theories are close for both values of initial stress ${\sigma }_0$ along with the $x$-axis. The amplitude of the temperature wave due to the refined G–L theory is the smallest one.

Figure 2 shows that the horizontal displacement $u$ of the refined L–S and G–L theories are close for both values of initial stress ${\sigma }_0$ and along with the $x$-axis. The amplitude of the horizontal displacement wave due to the CTE theory is the largest one.

Figure 3 shows that the vertical displacement $w$ of the refined L–S and G–L theories are very close for both values of initial stress ${\sigma }_0$ and along with the $x$-axis. The two refined theories may yield the same vertical displacement $w$. The amplitude of the vertical displacement wave due to the CTE theory is the largest one.

Once again, the longitudinal normal stress ${\sigma }_x$ of the refined L–S and G–L theories in Figure 4 are very close for both values of initial stress ${\sigma }_0$ and along with the $x$-axis. The amplitude of the ${\sigma }_x$ wave due to the CTE theory is the largest one. Also, it is clear that when ${\sigma }_0=9$ the longitudinal normal stress ${\sigma }_x$ has the greatest wave amplitude compared with when ${\sigma }_0=5$.

Figure 5 shows that the transverse normal stress ${\sigma }_z$ of all theories are very close for both values of initial stress ${\sigma }_0$ and along with the $x$-axis. In some places, the amplitude of the ${\sigma }_z$ wave due to the CTE theory is the largest one. Also, it is clear that when ${\sigma }_0=9$ the transverse normal stress ${\sigma }_z$ has the greatest wave amplitude compared with when ${\sigma }_0=5$.

In Figure 6, the results of transverse shear stress ${\sigma }_{xz}$ are due to the refined L–S and G–L theories being very close for both values of initial stress ${\sigma }_0$ and along with the $x$-axis. The amplitude of the transverse shear stress ${\sigma }_{xz}$ wave due to the CTE theory being the largest one.

It is interesting here to present the plot of the transverse shear stress ${\sigma }_{zx}$ of the rotating half-space with initial pressure ($𝓅=10$) in Figure 7. It is clear that there is no symmetry between the graphs of both ${\sigma }_{zx}$ and ${\sigma }_{xz}$ when the initial pressure is included. The results of transverse shear stress ${\sigma }_{zx}$ are due to the refined L–S and G–L theories being close for both values of initial stress ${\sigma }_0$. The amplitude of the transverse shear stress ${\sigma }_{zx}$ wave due to the CTE theory being the largest one.

The temperature $\theta$ of the rotating half-space without initial pressure ($𝓅=0$) due to the CTE and refined G–L theories are close for both values of initial stress ${\sigma }_0$, as shown in Figure 8. The amplitude of the temperature wave due to the refined L–S theory is the largest one. This is not the same as the case of the rotating half-space with initial pressure ($𝓅=10$).

Figure 9 shows that the horizontal displacement $u$ of the rotating half-space without initial pressure ($𝓅=0$) is due to all theories being close for both values of initial stress ${\sigma }_0$ along with the $x$-axis. The amplitude of the horizontal displacement wave due to the CTE theory is the smallest one.

Figure 10 shows that the vertical displacement $w$ of the rotating half-space without initial pressure ($𝓅=0$) is due to all theories being close for both values of initial stress ${\sigma }_0$ along with the $x$-axis. When enlarging some portions, we found that the amplitude of the horizontal displacement wave due to the CTE theory is the smallest one.

In Figure 11, Figure 12 and Figure 13, the longitudinal normal stress ${\sigma }_x$, the transverse normal stress ${\sigma }_z$, and on the transverse shear stress ${\sigma }_{xz}$ of the rotating half-space without initial pressure ($𝓅=0$) are illustrated. The stresses due to all theories are close for both values of initial stress ${\sigma }_0$ along with the $x$-axis. When enlarging some portions, we found that the amplitudes of the ${\sigma }_x$ and ${\sigma }_z$ waves due to the CTE theory is the smallest one, while the amplitude of the ${\sigma }_{xz}$ wave due to the CTE theory being the largest one. Once again, there is no need for plotting the transverse shear stress ${\sigma }_{zx}$ of the rotating half-space without initial pressure ($𝓅=0$) since it is the same as ${\sigma }_{xz}$ due to symmetry.

7.3. The 3D Applications

Additional 3D figures are given to demonstrate the effect of the initial stress ${\sigma }_0$ on all variables of the rotating half-space with and without initial pressures. The outcomes are presented due to the refined L–S and G–L theories with $N=5$. All results are shown across the $x$- and $z$-axis when the initial stress ${\sigma }_0=9$, $\Omega =2.5$, and $t=0.5$.

Figure 14 and Figure 15 depict the temperature of the rotating half-space with and without the initial stresses matching both refined thermoelasticity theories. When $𝓅=10$, the temperature has maximum values along the $z$-axis at a fixed value of $x$. However, when $𝓅=0$, the temperature has a maximum value at $z=0$. Also, the temperature vanishes rapidly when $𝓅=10$ compared to when $𝓅=0$.

Figure 16 and Figure 17 illustrate the horizontal displacement $u$ of the rotating half-space with and without the initial stress matching both refined thermoelasticity theories. The maximum values of $u$ are occurring when $z=0$. The horizontal displacement $u$may vanish when $x\to 6$ for $𝓅=10$. However, it may rapidly vanish when $x\to 4$ for $𝓅=0$. The absolute values of $u$ of the rotating half-space with the initial stress ($𝓅=10$) are greater than the corresponding ones of the rotating half-space without the initial stress ($𝓅=0$).

Figure 18 and Figure 19 illustrate the transverse displacement $w$ of the rotating half-space with and without the initial stress matching both the L–S and G–L refined thermoelasticity theories. The maximum values of $w$ are occurring when $z=0$. The transverse displacement $w$ may vanish when $x\to 6$ for $𝓅=10$. However, it may rapidly vanish when $x\to 4$ for $𝓅=0$. The wavelength of $w$ of the rotating half-space with the initial stress ($𝓅=10$) is greater than the corresponding one of the rotating half-space without the initial stress ($𝓅=0$).

Figure 20 and Figure 21 illustrate the longitudinal normal stress ${\sigma }_x$ of the rotating half-space with and without the initial stress matching both the L–S and G–L refined thermoelasticity theories. The maximum values of ${\sigma }_x$ are occurring at the origin. The longitudinal normal stress ${\sigma }_x$ may rapidly vanish when $x\to 4$ for the rotating half-space without the initial stress ($𝓅=0$). The wavelength of ${\sigma }_x$ of the rotating half-space when $𝓅=10$ is greater than the corresponding one of the rotating half-space when $𝓅=0$.

Figure 22 and Figure 23 illustrate the transverse normal stress ${\sigma }_z$ of the rotating half-space with and without the initial stress matching both the L–S and G–L refined thermoelasticity theories. The maximum values of ${\sigma }_z$ are occurring at the origin of the rotating half-space without the initial stress. The transverse normal stress ${\sigma }_z$ may rapidly vanish when $x\to 4$ for the rotating half-space without the initial stress ($𝓅=0$). The wavelength of ${\sigma }_z$ of the rotating half-space when $𝓅=10$ is greater than the corresponding one of the rotating half-space when $𝓅=0$.

Figure 24 and Figure 25 illustrate the transverse shear stress ${\sigma }_{xz}$ of the rotating half-space with and without the initial stress matching both the L–S and G–L refined thermoelasticity theories. The transverse shear stress ${\sigma }_{xz}$ is already vanished at $z=0$ according to the boundary condition. The maximum values of ${\sigma }_{xz}$ are occurring when $z=0$. The transverse shear stress ${\sigma }_{xz}$ may vanish when $x\to 6$ for $𝓅=10$. However, it may rapidly vanish when $x\to 4$ for $𝓅=0$. The wavelength of ${\sigma }_{xz}$ of the rotating half-space when $𝓅=10$ is greater than the corresponding one of the rotating half-space when $𝓅=0$.

Finally, Figure 26 and Figure 27 illustrate the transverse shear stress ${\sigma }_{zx}$ of the rotating half-space with and without the initial stress matching both the L–S and G–L refined thermoelasticity theories. It is interesting that the plots of transverse shear stress ${\sigma }_{zx}$ are the same as for ${\sigma }_{xz}$ due to the symmetry when $𝓅=0$ only. The transverse shear stress ${\sigma }_{zx}$ may vanish when $x\to 6$ for $𝓅=10$. However, it may rapidly vanish when $x\to 4$ for $𝓅=0$. The wavelength of ${\sigma }_{zx}$ of the rotating half-space with initial pressure when $𝓅=10$ is greater than the corresponding one of the rotating half-space without initial pressure.

8. Conclusions

Two refined thermoelasticity theories are presented in the context of the L–S and G–L theories. In addition, the simple L–S and G–L theories as well as the CTE theory are applied to get the field quantities of a rotating half-space. Such medium is subjected to initial pressure as well as is bounded by initial stress. A unified form of the heat conduction equation is presented to be applied to treat all models considered here. The effect of the initial stress and initial pressure were both investigated. Several examples and applications were provided to compare the results of all theories, regardless of whether or not the rotating half-space is subject to initial pressure.

As long as the refined L–S and G–L theories are used, the outcomes are more accurate. All field variables have wave amplitudes for the high value of the initial stress twice those for its small value. So, the inclusion of the initial stress is very sensitive to the variation of all variables. Also, the results of all field quantities vanish with the inclusion of initial pressure rapidly without the inclusion of it. In addition, the wavelengths of all outcomes of the rotating half-space with the inclusion of initial pressure are greater than the corresponding ones of the rotating half-space without the inclusion of initial pressure.