The initial stress effect on a thermoelastic micro-elongated solid under the dual-phase-lag model

Abstract

This paper's main objective is to study the two-dimensional deformation of the thermoelastic micro-elongated solid with the effect of initial stress on the model of dual-phase-lag and the theory of Lord and Shulman (L-S). The interface of the elastic half-space and the thermoelastic micro-elongated half-space is utilized to apply mechanical force. The solution has been made using the normal mode analysis method. For aluminum epoxy, numerical simulations are done and graphically presented to demonstrate the presence and complete absence of initial stress on the various physical quantities.

1 Introduction

Thermal elasticity is an essential branch of applied mechanics concerned with the thermal effect and its correlation to stresses and strains that occur in elastic bodies [1]. The fundamentals of the thermal elasticity theory were laid down in the first half of the nineteenth century. Duhamel [2] was the first to coin equations for solving thermal elasticity problems in the nineteenth century. In 1885, Neumann [3] reformulated the equations obtained by Duhamel using another method. Their theory was called the theory of uncoupled thermoelasticity. Two equations govern this theory: the equation of heat, which is autonomous of mechanical influences, and the equation of motion, which involves heating as a known function. Two major flaws exist in this theory, the first point to make would be that the mechanical regime of the body does not influence temperature and the second of them: The heat equation that is a partial differential equation which means the speed of heat is infinite and this contradicts the experimental process. In 1956, Biot [4] established the fundamentals of the theory of coupled thermal elasticity. In this theory, the elasticity equations relate to thermal, this is consistent with physical experiments, as any change in temperature leads to the elastic body being excited and vice versa. In many cases, this theory is applicable. In 1967, Lord and Shulman [5] inferred the theory of thermal elasticity generalized in a special case when the medium is isotropic. Othman [6] applied the theory of (L-S) to the two-dimensional problem of general linear thermoelasticity with the elastic modulus dependent on the reference temperature. In this theory, the heat equation, which is in the shape of a hyperbolic partial differential equation, has been developed, and this removes the contradiction of limitless speeds found in the coupled and uncoupled theories. Green and Lindsay [7] suggested the theory of general linear thermoelasticity identified from the temperature, frequency theory of thermoelasticity, which involves the average of temperature in governing equations. Tzou [8, 9] created a Neoteric theory named the DPL model for a heat carrier system wherein the law of Fourier is substituted by an approach to an alteration of the law of Fourier with various phases' lags for the temperature gradient and the heat flux. In this context, valuable information on the development of the elasticity and thermoelasticity theories is presented in [10,11,12,13,14,15,16,17].

Three degrees of freedom are for translation, and one is for micro-elongation in a micro-elongated elastic solid. Material particles, according to the micro-elongation theory, only volumetric micro-elongation can be performed in addition to classical medium deformation. Micro-elongated media also include solid–liquid crystals, structural materials strengthened with shredded elastic fibers, and porous media with pores full to the brim with gas or non-viscous fluid are also examples of the micro-elongated media. In papers [18,19,20,21,22,23], we have observed that there has not been a lot of emphasis on the various effects on micro-elongated thermoelasticity, such as initial stress, as well as comparison relaxation times and their effects on all physical quantities.

Under the model of (DPL), Othman et al. [24] examined the influence of initial stress on semiconductor material with temperature-dependent parameters. After four years, Othman et al. [25] used the previous effects and model to present the previous study on a different material which is a micro-stretch.

This study was created to fill the aforementioned gap. The influence of initial stress on a micro-elongated thermoelastic layer medium utilizing two theories, the DPL model and L-S theory when an elastic layer is above it, is investigated. We began by introducing the governing equations and employing non-dimensions. Second, we used the normal mode to transform the partial differential equations to ordinary differential equations, and then we set the boundary conditions to $z = 0, - h$ to find the constants in the solutions. The plane z = 0 is taken as surface of half-space as shown in Fig. 1. The present analysis is restricted to xz-plane, thus all the field quantities are independent of the space variable, the numerical results are implemented, discussed, and graphed.

Fig. 1

Geometry of the problem

2 Governing equations

The basic, governing equations of a micro-elongated thermoelasticity with initial stress in a (DPL) model can be obtained as [21, 24]

$$\sigma_{ij,j} = \rho u_{i,tt} ,$$

(1)

$$a_{0} \varphi_{,ii} + \beta_{1} T - \lambda_{1} \varphi - \lambda_{0} u_{j,j} = \frac{1}{2}\rho j_{0} \varphi_{,tt} ,$$

(2)

$$k(1 + \tau_{\theta } \frac{\partial }{\partial t})T_{,ii} = (1 + \tau_{q} \frac{\partial }{\partial t})(\rho c_{e} \frac{\partial T}{{\partial t}} + \beta_{0} Tu_{k,kt} ) + \beta_{1} T_{0} \varphi_{,t}$$

(3)

$$\sigma_{ij} = 2\mu \varepsilon_{ij} + (\lambda e - \beta_{0} T + \lambda_{0} \varphi )\delta_{ij} - P(\delta_{ij} - w_{ij} ).$$

(4)

where $w_{ij} = \frac{1}{2}(u_{j,i} - u_{i,j} ).$

From Eqs. (1), (4) for displacement vector ${\varvec{u}}\,(x,z,t) = (u_{1} ,0,u_{3} )$ and $P$ is the initial stress, the equations of motion are given by

$$\mu \nabla^{2} u_{1} + (\lambda + \mu )e_{,x} - \frac{P}{2}(u_{1,zz} - u_{3,xz} ) - \beta_{0} T_{,x} + \lambda_{0} \varphi_{,x} = \rho u_{1,tt}$$

(5)

$$\mu \nabla^{2} u_{3} + (\lambda + \mu )e_{,z} - \frac{P}{2}(u_{3,xx} - u_{1,xz} ) - \beta_{0} T_{,z} + \lambda_{0} \varphi_{,z} = \rho u_{3,tt} .$$

(6)

For simplicity, the following non-dimensional variables are utilized

$$\begin{gathered} x^{\prime}_{i} = \frac{{w^{ * } }}{{c_{1} }}x_{i} ,z^{\prime} = \frac{{w^{ * } }}{{c_{1} }}z,u^{\prime}_{i} = \frac{{w^{ * } \rho c_{1} }}{{\beta_{0} T_{0} }}u_{i} ,u_{i}^{{e^{\prime}}} = \frac{{w^{ * } \rho c_{1} }}{{\beta_{0} T_{0} }}u_{i}^{e} ,t^{\prime} = w^{ * } t,\tau^{\prime}_{\theta } = w^{ * } \tau_{\theta } ,\tau^{\prime}_{q} = w^{ * } \tau_{q} , \hfill \\ \sigma^{\prime}_{ij} = \frac{{\sigma_{ij} }}{{\beta_{0} T_{0} }},\sigma_{ij}^{{e^{\prime}}} = \frac{{\sigma_{ij}^{e} }}{{\beta_{0} T_{0} }},\varphi^{\prime} = \frac{{\lambda_{0} }}{{\beta_{0} T_{0} }}\varphi ,T^{\prime} = \frac{T}{{T_{0} }},P^{\prime} = \frac{P}{\lambda + 2\mu },P^{\prime}_{1} = \frac{{P_{1} }}{{\beta_{0} T_{0} }}, \hfill \\ \end{gathered}$$

(7)

where $w^{ * } = \frac{{\rho c_{1}^{2} c_{e} }}{k},c_{1}^{2} = \frac{\lambda + 2\mu }{\rho }.$

Substituting from Eqs. (7) into Eqs. (2), (3), (5), and (6), we obtain

$$a_{1} \nabla^{2} u_{1} + a_{2} e_{,x} + \frac{P}{2}(u_{1,zz} - u_{3,xz} ) - T_{,x} + \varphi_{,x} = u_{1,tt}$$

(8)

$$a_{1} \nabla^{2} u_{3} + a_{2} e_{,z} + \frac{P}{2}(u_{3,xx} - u_{1,xz} ) - T_{,z} + \varphi_{,z} = u_{3,tt}$$

(9)

$$(\nabla^{2} - a_{4} )\varphi + a_{3} T - a_{5} e = a_{6} \varphi_{,tt}$$

(10)

$$(1 + \tau_{\theta } \frac{\partial }{\partial t})\nabla^{2} T = (1 + \tau_{q} \frac{\partial }{\partial t})[a_{7} T_{,t} + a_{8} e_{,t} ] + a_{9} \varphi_{,t}$$

(11)

3 Normal mode analysis

The solution of the physical variable recognized may be analyzed in normal modes as the following form:

$$[u_{i} ,\varphi ,T,\sigma_{ij} ,u_{i}^{e} ,\sigma_{ij}^{e} ](x,z,t) = [u_{i}^{*} ,\varphi^{*} ,T^{*} ,\sigma_{ij}^{*} ,u_{i}^{e*} ,\sigma_{ij}^{e*} ](z)e^{(\omega t + ibx)} .$$

(12)

where $\omega$ is a complex constant, $i = \sqrt {\, - 1}$, $b$ is wave number in the $x$ direction.

Using Eq. (12) into Eqs. (8)–(11), then we have

$$[\delta_{1} D^{2} - \delta_{2} ]u_{1}^{*} + \delta_{3} Du_{3}^{*} - ibT^{*} + ib\varphi^{*} = 0$$

(13)

$$\delta_{3} Du_{1}^{*} + [\delta_{4} D^{2} - \delta_{5} ]u_{3}^{*} - DT^{*} + D\varphi^{*} = 0$$

(14)

$$- \delta_{6} u_{1}^{*} - a_{5} Du_{3}^{*} + a_{3} T^{*} + [D^{2} - \delta_{7} ]\varphi^{*} = 0$$

(15)

$$\delta_{10} u_{1}^{*} + \delta_{11} Du_{3}^{*} + [ - \delta_{8} D^{2} + \delta_{12} ]T^{*} + \delta_{13} \varphi^{*} = 0$$

(16)

Equations (13)–(16) have a non-trivial solution if the physical quantities' determinant coefficients equal to zero, then we get.

$$(D^{8} - AD^{6} + BD^{4} - CD^{2} + E)\{ u_{1}^{*} (z),u_{3}^{*} (z),T^{*} (z),\varphi ^{*} (z)\} = 0$$

(17)

Equation (18) can be factorized as:

$$(D^{2} - k_{1}^{2} )(D^{2} - k_{2}^{2} )(D^{2} - k_{3}^{2} )(D^{2} - k_{4}^{2} )\{ u_{1}^{*} (z),u_{3}^{*} (z),T^{*} (z),\varphi^{*} (z)\} = 0$$

(18)

where, $k_{n}^{2} ,\,(n = 1,2,3,4)$ are roots of the auxiliary equation of Eq. (18).

The general solutions of Eq. (18) bound as $(\,z\, \to \;\,\infty \,)$ is given by

$$(\,\,u_{1}^{*} \,,\,u_{3}^{*} \,,T^{*} \,,\,\varphi^{*} \,)\,(z) = \sum\limits_{n = 1}^{4} {(\,1\,,\,H_{1n} \,,\,H_{2n} \,,\,H_{3n} \,)\,M_{n} \,e^{{ - k_{n} \,z}} } .$$

(19)

Substituting from Eq. (7) into (4), and with the help of Eq. (19), we obtain the components of stresses.

$$4n$$

(20)

$$\sigma_{xz} (z) = \sum\limits_{n = 1}^{4} {H_{6n} M_{n} e^{{( - k_{n} z{\kern 1pt} + \omega t + ibx)}} } .$$

(21)

where the coefficients $a_{i}$, $\delta_{i}$,$A,B,C,E$, and $H_{in}$ are obtained inAppendix 1 .

The system of governing equations of general elastic medium are given by [26]

$$\sigma_{ij,j}^{e} = \rho^{e} \ddot{u}_{i}^{e} ,$$

(22)

$$\sigma_{ij}^{e} = \lambda^{e} \,u_{k,k}^{e} \,\delta_{ij} + \mu^{e} \,(\,u_{i,j}^{e} + u_{j,i}^{e} \,).$$

(23)

Substituting from Eqs. (7) and (12) into Eq. (22)

$$(l_{3} D^{2} - \ell _{1} )u_{1}^{{e*}} + ibl_{2} Du_{3}^{{e*}} = 0$$

(24)

$$ibl_{2} Du_{1}^{{e*}} + (l_{1} D^{2} - \ell _{2} )u_{3}^{{e*}} = 0.$$

(25)

Eliminating $u_{1}^{e*} ,\,u_{3}^{e*}$ between Eqs. (24) and (25), We obtain

$$(D^{4} - GD^{2} + N)\{ u_{1}^{e*} (z),u_{3}^{e*} (z)\} = 0.$$

(26)

Equation (26) could be factorized as:

$$(D^{2} - r_{1}^{2} )(D^{2} - r_{2}^{2} )\{ u_{1}^{e*} (z),u_{3}^{e*} (z)\} = 0,$$

(27)

where $r_{n}^{2} ,(\,n = 1,2\,)$ are roots of the auxiliary equation of Eq. (27). The solutions of Eq. (27) are of the form:

$$u_{1}^{e*} (z) = \sum\limits_{n = 1}^{2} {R_{n} e^{{ - r_{n} z}} + \sum\limits_{n = 1}^{2} {R_{n + 2} e^{{r_{n} z}} } } ,$$

(28)

$$u_{3}^{e*} \,(z) = \sum\limits_{n = 1}^{2} {\,L_{1n} \,R_{n} \,} e^{{ - r_{n} \,z}} + \sum\limits_{n = 1}^{2} {\,L_{1(n + 2)} \,R_{n + 2} \,e^{{r_{n} \,z}} } .$$

(29)

Substituting from Eqs. (7) and (13) into (23) and with the help of Eqs. (28) and (29), we obtain the components of stresses in an elastic medium

$$\sigma_{xx}^{e*} (z) = \sum\limits_{n = 1}^{2} {L_{2n} R_{n} } e^{{ - r_{n} z}} + \sum\limits_{n = 1}^{2} {L_{2(n + 2)} R_{n + 2} e^{{r_{n} z}} ,}$$

(30)

$$\sigma_{zz}^{e*} \,(z) = \sum\limits_{n = 1}^{2} {\,L_{3n} \,R_{n} } \,e^{{ - r_{n} \,z}} + \sum\limits_{n = 1}^{2} {\,L_{2(n + 2)} \,R_{n + 2} \,e^{{r_{n} \,z}} }$$

(31)

$$\sigma_{xz}^{e*} \,(z) = \sum\limits_{n = 1}^{2} {\,L_{4n} \,R_{n} } \,e^{{ - r_{n} \,z}} + \sum\limits_{n = 1}^{2} {\,L_{4(n + 2)} \,R_{n + 2} \,e^{{r_{n} \,z}} } .$$

(32)

where, the coefficients $l_{i}$,$\ell_{i}$,$G,N$, and $L_{in}$ are given inAppendix 2 .

4 Boundary conditions

The parameters $M_{n\,} ,\,(\,n = 1\,,\,2\,,\,3\,,\,4\,)\,$ and $R_{n\,} ,(\,n = 1\,,\,2\,,\,3\,,\,4\,)\,$ have to be selected such that boundary conditions at the surface are.

$$\begin{gathered} \sigma_{zz} = \sigma_{zz}^{e} - a_{11} P,\sigma_{xz} = \sigma_{xz}^{e} ,u_{1} = u_{1}^{e} ,u_{3} = u_{3}^{e} ,\varphi^{*} = 0,T = fe^{(\omega t + ibx)} ,{\text{at}}\;z = 0, \hfill \\ \sigma_{zz} = \sigma_{zz}^{e} - P_{1} e^{(\omega t + ibx)} - a_{11} P,\sigma_{xz} = 0,{\text{at}}\;z = - h. \hfill \\ \end{gathered}$$

(33)

Where $P_{1}$ is the magnitude of the mechanical force, and $f$ is constant.

The utilization of the expressions of the variables assumed into the boundary mentioned above conditions (33) to obtain the equations that are satisfied with the parameters. And hence, eight equations will be obtained. If the inverse of matrix method is applied to the eight equations, we get then the value constant $M_{n\,} ,\,(\,n = 1\,,\,2\,,\,3\,,\,4\,)\,$ and $R_{n\,} ,(\,n = 1\,,\,2\,,\,3\,,\,4\,).$

$$\left( {\begin{array}{*{20}c} {M_{1} } \\ {M_{2} } \\ {M_{3} } \\ {M_{4} } \\ {R_{1} } \\ {R_{2} } \\ {R_{3} } \\ {R_{4} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {H_{{51}} } & {H_{{52}} } & {H_{{53}} } & {H_{{54}} } & { - L_{{31}} } & { - L_{{32}} } & { - L_{{33}} } & { - L_{{34}} } \\ {H_{{61}} } & {H_{{62}} } & {H_{{63}} } & {H_{{64}} } & { - L_{{41}} } & { - L_{{42}} } & { - L_{{43}} } & { - L_{{44}} } \\ 1 & 1 & 1 & 1 & { - 1} & { - 1} & { - 1} & { - 1} \\ {H_{{11}} } & {H_{{12}} } & {H_{{13}} } & {H_{{14}} } & { - L_{{11}} } & { - L_{{12}} } & { - L_{{13}} } & { - L_{{14}} } \\ {H_{{31}} } & {H_{{32}} } & {H_{{33}} } & {H_{{34}} } & 0 & 0 & 0 & 0 \\ {H_{{21}} } & {H_{{22}} } & {H_{{23}} } & {H_{{24}} } & 0 & 0 & 0 & 0 \\ {H_{{51}} e^{{k_{1} h}} } & {H_{{52}} e^{{k_{2} h}} } & {H_{{53}} e^{{k_{3} h}} } & {H_{{54}} e^{{k_{4} h}} } & { - L_{{31}} e^{{r_{1} h}} } & { - L_{{32}} e^{{r_{2} h}} } & { - L_{{33}} e^{{ - r_{1} h}} } & { - L_{{34}} e^{{r_{2} h}} } \\ {H_{{61}} e^{{k_{1} h}} } & {H_{{62}} e^{{k_{2} h}} } & {H_{{63}} e^{{k_{3} h}} } & {H_{{64}} e^{{k_{4} h}} } & 0 & 0 & 0 & 0 \\ \end{array} } \right)^{{ - 1}} \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ f \\ { - P_{1} } \\ 0 \\ \end{array} } \right).$$

(34)

5 Numerical results and discussion

The analysis is conducted for aluminum epoxy-like material as [19]

$$\begin{array}{*{20}l} {\lambda = 7.59 \times 10^{{10}} N/m^{2} ,\quad \mu = 1.89 \times 10^{{10}} {\mkern 1mu} N/m^{2} ,\quad a_{0} = 0.61 \times 10^{{ - 10}} {\mkern 1mu} N,\quad \rho = 2.19 \times 10^{3} {\mkern 1mu} kg/m^{3} ,} \hfill \\ {\beta _{0} = \beta _{1} = 0,05 \times 10^{5} {\mkern 1mu} N/m^{2} .k,\quad c_{e} = 966{\mkern 1mu} J/kg.k,\quad k = 252\;J/m.s.k,\quad j_{0} = 0.196 \times 10^{{ - 4}} {\mkern 1mu} m^{2} ,} \hfill \\ {\lambda _{0} = \lambda _{1} = 0.37 \times 10^{{10}} {\mkern 1mu} N/m^{2} ,\quad T_{0} = 293{\mkern 1mu} k,P = 0.001,\quad \tau _{\theta } = 1.5 \times 10^{{ - 4}} ,\quad \tau _{q} = 9 \times 10^{{ - 4}} ,\quad \omega = \omega _{0} + i\zeta ,} \hfill \\ {\omega _{0} = - 1.77 \times 10^{{ - 4}} ,\quad \zeta = 3.59 \times 10^{{ - 3}} ,\quad b = 3,h = 0.001,\quad f = 0.025.} \hfill \\ \end{array}$$

The physical constants for elastic medium (granite) as [27]

$$\lambda ^{e} {\mkern 1mu} = 0.884 \times 10^{{10}} {\text{N}}/{\text{m}}^{2} ,\;\mu ^{e} = 1.2667 \times 10^{{10}} {\text{N}}/{\text{m}}^{2} ,\;\rho ^{e} = 2.6 \times 10^{3} {\mkern 1mu} {\text{kg}}/{\text{m}}^{3} ,\;c_{e}^{e} = 720.7{\mkern 1mu} {\text{J}}/{\text{kg.k}},k^{e} = 3.1{\mkern 1mu} {\text{J}}/{\text{m.s.k.}}$$

In this paper, the calculations are carried out to a value dimensionless time $t = 1.02$ in the range $0 \le z \le 2$ on the surface $x = 0.5$. The numerical strategy stated herein is used to distribute horizontal displacement $u_{1}$, micro-elongational scalar $\varphi$ in the range $0 \le z \le 3$, stress components $\sigma_{zz}$, $\sigma_{xz}$ versus $z.$ To examine the influence of the presence and complete absence of initial stress in the model of DPL and the theory of L-S and the effect of the relaxation time over the model of DPL model. This paper introduces the results of the numerical assessment in the shape of a graph. The results are shown in Figs. 2–7 for the mechanical force of magnitude $P_{1} = 15$ for both previous theories.

Fig. 2

Effect of initial stress on the variation of the horizontal displacement $u_{1}$ and the micro-elongated scalar $\varphi .$

5.1 Effect of the initial stress

Figures 2, 3 are graphed to describe the variation of the horizontal displacement $u_{1}$, the micro-elongational scalar $\varphi$, the stress components $\sigma_{zz} ,$$\sigma_{xz}$ versus $z.$ Figure 2 exhibits the effect of initial stress is decreasing in the context of the model of DPL, while the influence of initial stress is increasing in the context of the theory of L-S for two functions the displacement $u_{1}$ and the micro-elongated scalar $\varphi .$ Figure 3 demonstrates that the initial stress has a decreasing effect in the context of both the model of DPL and the theory of L-S on the two components of stress $\sigma_{zz}$ and $\sigma_{xz} .$

Fig. 3

Effect of initial stress on the variation of the stress components $\sigma_{zz}$ and $\sigma_{xz} .$

5.2 Influence of the temperature gradient $\tau_{\theta }$

Figures 4, 5 are represented to exhibit the distribution of the above quantities versus a distance $z$ in the presence of initial stress (i.e., $P\, = \,0.001$) on the DPL model for different phase-lag of the temperature gradient $\tau_{\theta }$ values. These values are as follows at $\tau_{q} \, = \,9 \times 10^{ - 4} ,$ $\tau_{\theta } \, = \,3.5 \times 10^{ - 4} ,\;4.5 \times 10^{ - 4} ,\;7.75 \times 10^{ - 4} .$ Figure 4 clarifies the distribution of the horizontal displacement $u_{1}$ and the micro-elongational scalar $\varphi$ versus $z.$ It is obvious the influence of $\tau_{\theta }$ on displacement $u_{1}$ and the micro-elongational scalar $\varphi$ is increasing and also all curves decrease and converge to zero. Figure 5 represents the distribution of the stress components $\sigma_{zz} ,$ $\sigma_{xz}$ versus $z.$ It is evident that all curves begin from negative values and then decrease over the range $0 \le z \le 1.5,$ then converge to − 78 at $z \ge 1.5$ for the normal stress component $\sigma_{zz} ,$ while curves for tangential stress begin from negative values and then increase over the range $0 \le z \le 1.5,$ then converge to zero at $z \ge 1.5$. It is obvious that $\tau_{\theta }$ has an increasing effect on the normal stress component $\sigma_{zz} ,$ while the influence of $\tau_{\theta }$ on the tangential stress component $\sigma_{xz}$ is decreasing.

Fig. 4

Effect of $\tau_{\theta }$ on the variation of the horizontal displacement $u_{1}$ and the micro-elongated scalar $\varphi .$

Fig. 5

Effect of $\tau_{\theta }$ on the variation of the stress components $\sigma_{zz}$ and $\sigma_{xz} .$

5.3 Influence of the phase-lag of heat flux $\tau_{q}$

Figures 6, 7 are graphed to describe the variation of the previous quantities versus the distance $z$ in the presence of initial stress (i.e., $P\, = \,0.001$) on the (DPL) model for different values of the phase-lag of the heat flux $\tau_{q}$. These values are as follows $\tau_{\theta } \, = \,5 \times 10^{ - 5} ,$$\tau_{q} \, = \,5 \times 10^{ - 4} ,\;7 \times 10^{ - 4} ,\;8 \times 10^{ - 4} .$ Figure 6 describes the variation of the horizontal displacement $u_{1}$ and the micro-elongational scalar $\varphi$ versus $z.$ It is evident that the influence of $\tau_{q}$ on the displacement $u_{1}$ the micro-elongational scalar $\varphi$ is decreasing and also all the curves decrease up to vanish. Figure 7 introduces the variation of the stress components $\sigma_{zz} ,$ $\sigma_{xz}$ versus $z.$ It is evident that all curves begin from negative values and then decrease over the range $0 \le z \le 1.5,$ then converge to − 78 at $z \ge 1.5$ for the normal stress component $\sigma_{zz} ,$ while curves for tangential stress begin from negative values and then increase over the range $0 \le z \le 1.5,$ then converge to zero at $z \ge 1.5$. It is clear that $\tau_{q}$ has a decreasing effect on the normal stress component $\sigma_{zz} ,$ while $\tau_{q}$ has an increasing effect on the tangential stress component $\sigma_{xz} .$

Fig. 6

Effect of $\tau_{q}$ on the variation of the horizontal displacement $u_{1}$ and the micro-elongated scalar $\varphi .$

Fig. 7

Effect of $\tau_{q}$ on the variation of the stress components $\sigma_{zz}$ and $\sigma_{xz} .$

6 Conclusion

The important findings emerged in this study are:

• 1 All the physical quantities approach zero except for the stress components $\sigma_{xx}$ and $\sigma_{zz}$ close to $- 78$ at a significant distance $z.$

• 2 A comparison is made between the model of DPL and the theory of L-S in the presence and absence of initial stress.

• 3 The effect of the initial stress and the micro-elongational scalar play a pivotal role in this study of the thermoelastic medium's deformation.

• 4 The phase-lag of the heat flux and the temperature gradient in the present system have considerable effects on all the physical quantities.

• 5 A decomposition solution to the problem is developed for the thermoelastic micro-elongated layer covered by the finite elastic layer under the initial stress effect.

Appendices

Appendix 1

$$\begin{gathered} a_{1} = \frac{\mu }{{\rho c_{1}^{2} }},\;a_{2} = \frac{\lambda + \mu }{{\rho \,c_{1}^{2} }},\;a_{3} = \frac{{\beta_{1} \lambda_{0} c_{1}^{2} }}{{a_{0} \beta_{0} \,w^{ * 2} }},\;a_{4} = \frac{{\lambda_{1} c_{1}^{2} }}{{a_{0} \,w^{ * 2} }},\;a_{5} = \frac{{\lambda_{0}^{2} }}{{a_{0} \rho \,w^{ * 2} }},\;a_{6} = \frac{{\rho j_{0} c_{1}^{2} }}{{2a_{0} }}, \hfill \\ \,a_{7} = \frac{{\rho c_{e} c_{1}^{2} }}{{kw^{ * } }},\;a_{8} = \frac{{\beta_{0}^{2} T_{0} }}{{k\rho w^{ * } }},\;a_{9} = \frac{{\beta_{0} \beta_{1} T_{0} c_{1}^{2} }}{{k\lambda_{0} w^{ * } }},\;a_{10} = \frac{\lambda }{{\rho c_{1}^{2} }},\;a_{11} = \frac{\lambda \, + \,2\mu }{{\beta_{0} \,T_{0} }},\;a_{12} = \frac{{\mu + \tfrac{1}{2}(\lambda + 2\mu )P}}{{\rho c_{1}^{2} }}, \hfill \\ a_{13} = \frac{{\mu - \tfrac{1}{2}(\lambda + 2\mu )P}}{{\rho \,c_{1}^{2} }},\;\delta_{1} = a_{1} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \frac{P}{2},\;\delta_{2} = a_{1} b^{2} + a_{2} b^{2} + \omega^{2} ,\;\delta_{3} = iba_{2} - \frac{P}{2}ib,\;\delta_{4} = a_{1} {\kern 1pt} + {\kern 1pt} a_{2} , \hfill \\ \delta_{5} = a_{1} b^{2} + \frac{P}{2}b^{2} + \omega^{2} ,\;\delta_{6} = iba_{5} ,\;\delta_{7} = b^{2} \, + \,a_{4} \, + \,a_{6} \omega^{2} ,\;\delta_{8} = 1 + \tau_{\theta } \omega ,\;\delta_{9} = 1 + \tau_{q} \omega , \hfill \\ \delta_{10} = i{\kern 1pt} b{\kern 1pt} a_{8} \omega {\kern 1pt} \delta_{9} ,\;\delta_{11} = a_{8} \omega \,\delta_{9} ,\;\delta_{12} = b^{2} \delta_{8} + a_{7} \omega \,\delta_{9} ,\;\delta_{13} = a_{9} \omega , \hfill \\ \end{gathered}$$

$$A = \frac{ - 1}{{\delta_{1} \delta_{4} \delta_{8} }}\left[ {a_{5} \delta_{1} \delta_{8} - \delta_{1} \delta_{5} \delta_{8} - \delta_{2} \delta_{4} \delta_{8} - \delta_{3}^{2} \delta_{8} - \delta_{1} \delta_{4} \delta_{12} - \delta_{1} \delta_{4} \delta_{7} \delta_{8} } \right]$$

$$B = \frac{1}{{\delta_{1} \delta_{4} \delta_{8} }}\left[ \begin{gathered} \delta_{2} \delta_{11} + \delta_{3} \delta_{10} - a_{3} \delta_{1} \delta_{11} - a_{5} \delta_{2} \delta_{8} - a_{5} \delta_{1} \delta_{12} - a_{5} \delta_{1} \delta_{13} + ib\delta_{3} \delta_{11} - ib\delta_{4} \delta_{10} + \delta_{2} \delta_{5} \delta_{8} \hfill \\ - \delta_{3} \delta_{6} \delta_{8} + \delta_{1} \delta_{5} \delta_{12} + \delta_{2} \delta_{4} \delta_{12} + \delta_{3}^{2} \delta_{12} + \delta_{1} \delta_{7} \delta_{11} + ia_{5} b\delta_{3} \delta_{8} + a_{3} \delta_{1} \delta_{4} \delta_{13} + ib\delta_{4} \delta_{6} \delta_{8} + \delta_{1} \delta_{5} \delta_{7} \delta_{8} \hfill \\ + \delta_{2} \delta_{4} \delta_{7} \delta_{8} + \delta_{3}^{2} \delta_{7} \delta_{8} + \delta_{1} \delta_{4} \delta_{7} \delta_{12} \hfill \\ \end{gathered} \right]$$

$$\begin{gathered} C = \frac{ - 1}{{\delta_{1} \delta_{4} \delta_{8} }}[a_{3} \delta_{2} \delta_{11} + a_{3} \delta_{3} \delta_{10} + a_{5} \delta_{2} \delta_{12} + a_{5} \delta_{2} \delta_{13} + ib\delta_{5} \delta_{10} - \delta_{2} \delta_{5} \delta_{12} - \delta_{2} \delta_{7} \delta_{11} - \delta_{3} \delta_{7} \delta_{10} + \delta_{3} \delta_{6} \delta_{12} \hfill \\ + \delta_{3} \delta_{6} \delta_{13} + iba_{3} \delta_{3} \delta_{11} - iba_{3} \delta_{4} \delta_{10} + iba_{5} \delta_{3} \delta_{12} + iba_{5} \delta_{4} \delta_{13} - a_{3} \delta_{1} \delta_{5} \delta_{13} - a_{3} \delta_{2} \delta_{4} \delta_{13} - a_{3} \delta_{3}^{2} \delta_{13} \hfill \\ - ib\delta_{5} \delta_{6} \delta_{8} - ib\delta_{3} \delta_{7} \delta_{11} + ib\delta_{4} \delta_{7} \delta_{10} - ib\delta_{4} \delta_{6} \delta_{12} - ib\delta_{4} \delta_{6} \delta_{13} - \delta_{2} \delta_{5} \delta_{7} \delta_{8} - \delta_{1} \delta_{5} \delta_{7} \delta_{12} \hfill \\ - \delta_{2} \delta_{4} \delta_{7} \delta_{12} - \delta_{3}^{2} \delta_{7} \delta_{12} ] \hfill \\ E = \frac{1}{{\delta_{1} \delta_{4} \delta_{8} }}[iba_{3} \delta_{5} \delta_{10} + a_{3} \delta_{2} \delta_{5} \delta_{13} - ib\delta_{5} \delta_{7} \delta_{10} + ib\delta_{5} \delta_{6} \delta_{12} + ib\delta_{5} \delta_{6} \delta_{13} + \delta_{2} \delta_{5} \delta_{7} \delta_{12} ] \hfill \\ \end{gathered}$$

$$\begin{gathered} H_{1n} = \frac{{k_{n} \delta_{2} + ibk_{n} \delta_{3} - \delta_{1} k_{n}^{3} }}{{(ib\delta_{4} - \delta_{3} )k_{n}^{2} - ib\delta_{5} }},\;H_{2n} = \frac{{k_{n}^{3} \delta_{11} H_{1n} + a_{5} \delta_{11} k_{n} H_{1n} + \delta_{7} \delta_{11} H_{1n} k_{n} - k_{n}^{2} \delta_{10} - \delta_{6} \delta_{13} + \delta_{7} \delta_{10} }}{{\delta_{7} \delta_{8} k_{n}^{2} + \delta_{12} k_{n}^{2} - \delta_{8} k_{n}^{4} - \delta_{7} \delta_{12} - a_{3} \delta_{13} }}, \hfill \\ H_{3n} = \frac{{a_{5} k_{n} H_{1n} + a_{3} H_{2n} - \delta_{6} }}{{\delta_{7} - k_{n}^{2} }},\;H_{4n} = ib - a_{10} k_{n} H_{1n} - H_{2n} + H_{3n} ,\;H_{5n} = iba_{10} - k_{n} H_{1n} - H_{2n} + H_{3n} , \hfill \\ H_{6n} = iba_{13} H_{1n} - a_{12} k_{n} . \hfill \\ \end{gathered}$$

Appendix 2

$$\begin{gathered} l_{1} = \frac{{\lambda^{e} + 2\mu^{e} }}{{\rho^{e} c_{1}^{e2} }},l_{2} = \frac{{\lambda^{e} + \mu^{e} }}{{\rho^{2} c_{1}^{e2} }},\;l_{3} = \frac{{\mu^{e} }}{{\rho^{e} c_{1}^{e2} }},\;l_{4} = \frac{{\lambda^{e} }}{{\rho^{e} c_{1}^{e2} }},\;\ell_{1} = b^{2} l_{1} + \omega^{2} ,\;\ell_{2} = l_{3} b^{2} + \omega^{2} , \hfill \\ G = \frac{{b^{2} l_{2}^{2} - \delta_{1} l_{1} - \delta_{2} l_{3} }}{{l_{1} l_{3} }},\;N = \frac{{\delta_{1} \delta_{2} }}{{l_{1} l_{3} }},\;L_{1n} = \frac{{l_{3} r_{n}^{2} - \delta_{1} }}{{ibl_{2} r_{n} }},\;L_{1(n + 2)} = \frac{{l_{3} r_{n}^{2} - \delta_{1} }}{{ - \;ibl_{2} r_{n} }},\;L_{2n} = ibl_{1} - r_{n} l_{4} L_{1n} , \hfill \\ L_{2(n + 2)} = ibl_{1} + r_{n} l_{4} L_{1(n + 2)} ,\;L_{3n} = ibl_{4} - l_{1} r_{n} L_{1n} ,\;L_{3(n + 2)} = ibl_{4} - l_{1} r_{n} L_{1(n + 2)} , \hfill \\ L_{4n} = ibl_{3} L_{1n} - l_{3} r_{n} ,\;L_{4(n + 2)} = i{\kern 1pt} b{\kern 1pt} l_{3} L_{1(n + 2)} + l_{3} r_{n} . \hfill \\ \end{gathered}$$