Analysis of an initially stressed functionally graded thermoelastic medium (type III) without energy dissipation

Abstract

The current investigation deals with the deformation of a non-homogeneous thermoelastic half space under hydrostatic initial stress for the Green–Naghdi model III. The medium is supposed to be rotating with a constant angular velocity. The non-homogeneous properties of the material are along the x-direction. At the first instance, the problem has been solved analytically to obtain stress and displacement components. Further, the numerical values of these expressions are evaluated using a computer program for a particular medium. The numerical values obtained are then presented graphically to show the effect of initial stress parameter and non-homogeneity parameter on the quantities.

1 Introduction

The coupled thermoelastic theory was first given by Biot [1] to remove the deficiency of the uncoupled theory which states that the thermal waves propagate with infinite speed. It was observed that the classical theory of thermoelasticity needs modifications. Hence, different theories of thermo-elasticity were established. Lord and Shulman [2] formulated a theory of thermoelasticity by including a flux rate term and another constant called the thermal relaxation time by modifying Fourier’s law of heat conduction. Without affecting Fourier’s law of heat conduction, Green and Lindsay [3] established a theory which depends on the temperature rate by including two constants termed as thermal relaxation times. The two theories mentioned above confirm the finite speed of heat propagation. This wave like thermal disturbance was called as second sound by Chandrasekharaiah [4]. Green and Nagdhi [5,6,7] proposed three new thermoelastic theories known as G-N model of type I, type II, and type III. The type I theory is reduced to the classical theory of thermoelasticity in linearized form. The G-N model of type II predicts the finite velocity for thermal waves without energy dissipation and model of type III allows thermal signals to propagate with finite speed. Barber and Martin-Moran [8] solved the isothermal contact problem of traction free bodies. Marin [9, 10] investigated some important problems in thermoelastic micropolar bodies. Sharma and Marin [11] discussed wave propagation in micropolar thermoelastic solid half space with distinct conductive and thermodynamic temperatures. Marin and Öchsner [12] studied the mixed initial boundary value problem for a dipolar body in the context of the thermoelastic theory proposed by Green and Naghdi. The solution of mixed problems in partial differential equations existing in boundary value problems is given by Marin and Öchsner [13]. Vlase et al. [14] presented a method to simplify the calculus of the eigenmodes of a mechanical system with bars.

Many researchers [15,16,17,18,19] studied the propagation of waves in initially stressed solids for different models. Different problems related to rotating media were investigated by many researchers in the past. Schoenberg and Censor [20], Clarke and Burdness [21], Destrade [22] discussed the effect of rotation on elastic waves. Sharma and Othman [23] discussed the effect of rotation on generalized thermo-viscoelastic Rayleigh waves. Othman and Song [24] studied the effect of rotation in two dimensional problem for a magneto-thermo-elasticity half space under various theories. Ailawalia et al. [25, 26] investigated the effect of rotation on stress and displacement components in thermoelastic half space. Marin et al. [27] constructed numerical estimations of a nonlinear hyperbolic bioheat equation under various boundary conditions for medicinal treatments of tumor cells. Marin et al. [28] studied the mixed initial-boundary value problem in the context of the Moore-Gibson-Thompson theory of thermoelasticity for dipolar bodies. A problem related to Rayleigh waves in an initially stressed two temperature magneto thermoelastic media was studied by Kumari et al. [29]. Hetnarski and Ignaczak [30] reviewed the concept of generalised thermoelasticity. Sharma et al. [31] and Sharma and Chauhan [32] studied the effect of mechanical and thermal sources in an isotropic thermoelastic medium. Sharma et al. [33] discussed the effect of load moving in a homogeneous thermoelastic layer. The problem of moving load response in a thermo-diffusive elastic solid was discussed by Deswal and Choudhary [34].

Over the past few decades, the concept of functionally graded material (FGM) has caught the attention of researchers. FGMs are generally used in high temperature environments. Thermal stresses like sudden heating and cooling can easily being controlled by FGM. Some prominent work [34, 35, 35,36,37,38,39,40,41,42,43] has been done by researchers in the field of functionally graded materials of different types. In the recent couple of years a lot of papers have been published in the field of non-homogeneous thermoelastic media [44,45,46,47,48,49,50].

The present research article deals with the study of deformation of a rotating functionally graded thermoelastic half space under hydrostatic initial stress. The expressions are obtained for displacement, temperature distribution and force stress by using the normal mode technique. The variation of these field variables are calculated numerically. The effect of initial stress and non-homogeneity parameter are shown graphically in the derived expressions using MATLAB.

2 Basic equations

It is considered that the free surface of the initially stressed thermoelastic half space is rotating with a uniform angular velocity of magnitude $\Omega$. For a rotating media, two additional terms defined as centripetal acceleration $\vec {\Omega }\times (\vec {\Omega }\times \vec {u})$ and Coriolis acceleration $2\Omega \times \dot{\vec {u}}$ are included in the displacement equation. All condidered quantities are functions of t, x and z. The thermoelastic medium is under the influence of a mechanical/thermal source of constant magnitude. The model of the problem is such that the free surface lies along x=0 and the x- axis is pointing vertically downwards.

The equation of motion and stress–strain relation for a homogenous, isotropic rotating thermoelastic medium under hydrostatic initial stress and in the absence of body forces and heat sources is given by Montanaro [15], Green and Nagdhi [7] and Schoenberg and Censor [20]:

$$\begin{aligned} t_{ij}= & {} -p(\delta _{ij}+\omega _{ij}) + 2\mu e_{ij}+\lambda e \delta _{ij} -\dfrac{\alpha }{k_{T}}T\delta _{ij}, \end{aligned}$$

(1)

$$\begin{aligned} h_{i}= & {} K \dfrac{\partial T}{\partial x_{i}}, \end{aligned}$$

(2)

$$\begin{aligned} U= & {} \dfrac{\alpha \theta _{0}}{\rho _{0}k_{T}}e_{kk} + C_{e}T, \end{aligned}$$

(3)

$$\begin{aligned} e_{ij}= & {} \frac{1}{2}(u_{i,j}+ u_{j,i}), \end{aligned}$$

(4)

$$\begin{aligned} \omega _{ij}= & {} \frac{1}{2}(u_{j,i}- u_{i,j}) \end{aligned}$$

(5)

$$\begin{aligned} \rho \left[ \ddot{u_{i}}+\vec {\Omega }\times (\vec {\Omega }\times \vec {u})+2(\vec {\Omega }\times \dot{\vec {u}})\right]= & {} \left( \mu - \frac{p}{2}\right) u_{i,kk}+\left( \lambda +\mu +\frac{p}{2}\right) u_{k,ik}-\frac{\alpha }{k_{T}}T_{,i}. \end{aligned}$$

(6)

The heat equation in the context of GN theory of type III is given by:

$$\begin{aligned} K T_{,ii} + k^{*}\dot{T_{,ii}} - \rho C_{e}{\ddot{T}} = \vartheta \theta _{0}\ddot{u_{i,i}}, (i,j=1,2,3) \end{aligned}$$

(7)

where $T=\theta - T_{0}$, $T_{0}$ is the initial temperature, $\lambda$, $\mu$ are the elastic constants termed as Lame’s parameter, U is the specific internal energy measured from the reference state, $\vec {u}$ is the displacement vector with components $u_{i}$, $h_{i}$ are components of heat flux vector, p is the hydrostatic stress parameter, $\alpha$ represents the volume coefficient of thermal expansion, $k_{T}$ is the isothermal compressibility, $\delta _{ij}$ is the Kronecker delta, $\rho _{0}$ is the density, $C_{e}$ is the specific heat, $K(\ge 0)$ is the thermal conductivity and $k^{*}$ is the material constant of the theory. Letting $k^{*}\longrightarrow 0$ in Eq. (7), the heat conduction equation is obtained for type II thermoelastic theory under the GN theory.

In case of a non-homogenous medium, the otherwise assumed constants $\lambda$, $\mu$, K, $k^{*}$, $\rho$, p, $\upsilon$, $\alpha$ depend on space variables. Here $\lambda$, $\mu$, K, $k^{*}$, $\rho$, p, $\upsilon$, $\alpha$ are replaced by $\lambda _{0}\phi (X)$, $\mu _{0}\phi (X)$, $K_{0}\phi (X)$, $k^{*}_{0}\phi (X)$, $\rho _{0}\phi (X)$, $p_{0}\phi (X)$, $\upsilon _{0}\phi (X)$, $\alpha _{0}\phi (X)$ respectively, $\phi (X)$ is taken as a dimensionless function which depends on the space variable $X = (x,y,z)$.

We suppose that the material properties vary along the x-axis only, hence $\phi (X)$ will be further expressed as $\phi (x)$, therefore the field Eqs. (6) and (7) can be written as:

$$\begin{aligned}{} & {} \phi (x)\left[ (\lambda _{0}+2\mu _{0})\dfrac{\partial ^{2}u_{1}}{\partial x^{2}}+\left( \lambda _{0}+\mu _{0}+\frac{p_{0}}{2}\right) \dfrac{\partial ^{2}u_{2}}{\partial x\partial y}+\left( \mu _{0}-\frac{p_{0}}{2}\right) \dfrac{\partial ^{2}u_{1}}{\partial y^{2}}-\frac{\alpha _{0}}{k_{T}}\dfrac{\partial T}{\partial x}\right] \nonumber \\ {}{} & {} \quad + \dfrac{\partial \phi (x)}{\partial x}\left[ -p_{0}+(\lambda _{0}+2\mu _{0})\dfrac{\partial u_{1}}{\partial x}+\lambda _{0}\dfrac{\partial u_{2}}{\partial y}-\frac{\alpha _{0}}{k_{T}}T\right] = \rho _{0}\phi (x)\left[ \dfrac{\partial u_{1}}{\partial t^{2}}-\Omega ^{2}u_{1}-2\Omega \dfrac{\partial u_{2}}{\partial t}\right] , \end{aligned}$$

(8)

$$\begin{aligned}{} & {} \quad \phi (x)\left[ (\lambda _{0}+2\mu _{0})\dfrac{\partial ^{2}u_{2}}{\partial y^{2}}+\left( \lambda _{0}+\mu _{0}+\frac{p_{0}}{2}\right) \dfrac{\partial ^{2}u_{1}}{\partial x \partial y}+\left( \mu _{0}-\frac{p_{0}}{2}\right) \dfrac{\partial ^{2}u_{2}}{\partial x^{2}}-\frac{\alpha _{0}}{k_{T}}\dfrac{\partial T}{\partial y}\right] \nonumber \\ {}{} & {} \quad + \dfrac{\partial \phi (x)}{\partial x}\left[ \left( \mu _{0}- \frac{p_{0}}{2}\right) \dfrac{\partial u_{2}}{\partial x}+\left( \mu _{0}+\frac{p_{0}}{2}\right) \dfrac{\partial u_{1}}{\partial y}\right] = \rho _{0}\phi (x)\left[ \dfrac{\partial u_{2}}{\partial t^{2}}-\Omega ^{2}u_{2}+2 \Omega \dfrac{\partial u_{1}}{\partial t}\right] , \end{aligned}$$

(9)

$$\begin{aligned}{} & {} \quad K_{0}\left[ \phi (x)\triangledown ^{2} T+\dfrac{\partial }{\partial x}\phi (x)\dfrac{\partial T}{\partial x}\right] + k_{0}^{*}\left[ \phi (x)\triangledown ^{2}{\dot{T}}+\dfrac{\partial }{\partial x}\phi (x)\dfrac{\partial {\dot{T}}}{\partial x}\right] \nonumber \\{} & {} \quad - \rho _{0}C_{e}\phi (x)\dfrac{\partial ^{2}T}{\partial t^{2}} = \phi (x)\vartheta _{0}T_{0}\left[ \dfrac{\partial ^{2}}{\partial t^{2}}\left( \dfrac{\partial u_{1}}{\partial x}+\dfrac{\partial u_{2}}{\partial y}\right) \right] . \end{aligned}$$

(10)

The components of the stress tensor are given by:

$$\begin{aligned} t_{xx}= & {} \phi (x)\left[ -p_{0}+(\lambda _{0}+2 \mu _{0})\dfrac{\partial u_{1}}{\partial x}+\lambda _{0}\dfrac{\partial u_{2}}{\partial y}-\dfrac{\alpha _{0}}{k_{T}}T\right] , \end{aligned}$$

(11)

$$\begin{aligned} t_{yy}= & {} \phi (x)\left[ -p_{0}+(\lambda _{0}+2 \mu _{0})\dfrac{\partial u_{2}}{\partial y}+\lambda _{0}\dfrac{\partial u_{1}}{\partial x}-\dfrac{\alpha _{0}}{k_{T}} T\right] , \end{aligned}$$

(12)

$$\begin{aligned} t_{xy}= & {} \phi (x)\left[ \left( \mu _{0}+\dfrac{p_{0}}{2}\right) \dfrac{\partial u_{2}}{\partial x}+\left( \mu _{0}-\dfrac{p_{0}}{2}\right) \dfrac{\partial u_{1}}{\partial y}\right] . \end{aligned}$$

(13)

For the problem under consideration, the initial conditions are:

$$\begin{aligned} u_{i}(x,y,0) = \dot{u_{i}}(x,y,0) = T(x,y,0) = 0. \end{aligned}$$

(14)

3 Non-homogeneity variation

The function $\phi (x)$ is assumed to be of the form $\phi (x)= e^{-nx}$, where n is a dimensionless constant. Hence, the material properties of the medium are exponentially varying along the x direction.

For numerical computations, the dimensionless parameters are introduced in the equations which are defined below:

$$\begin{aligned} x^{'}&=\dfrac{1}{c_{0}\varpi }x,\quad y^{'}=\dfrac{1}{c_{0}\varpi }y,\ u_{i}^{'}=\dfrac{1}{c_{0}\varpi }u_{i},\ t^{'}=\dfrac{t}{\varpi },\\ \Omega ^{'}&=\Omega \varpi ,\ T^{'}=\dfrac{\vartheta _{0}}{\lambda _{0}+2 \mu _{0}}T,\ t_{ij}^{'}=\dfrac{1}{\lambda _{0}+2 \mu _{0}}t_{ij},\ p_{0}^{'}=\dfrac{p_{0}}{\lambda _{0}+2 \mu _{0}},\ \phi (x^{'})=\phi (x), \end{aligned}$$

where $c_{0}^{2}=\frac{\lambda _{0}+2 \mu _{0}}{\rho _{0}}$, $\varpi =\frac{k_0^*}{\rho _0 c_0^2C_e}$.

Substituting the above non-dimensional variables, the field equations and stress components (after suppressing the primes) are obtained in dimensionless form as,

$$\begin{aligned}{} & {} \left( \dfrac{\partial ^{2}u_{1}}{\partial x^{2}}+a_{1}\dfrac{\partial ^{2} u_{2}}{\partial x \partial y}+a_{2}\dfrac{\partial ^{2} u_{1}}{\partial y^{2}}-\beta \dfrac{\partial T}{\partial x}\right) -n\left( -p_{0}+\dfrac{\partial u_{1}}{\partial x}+a_{3}\dfrac{\partial u_{2}}{\partial y}-\beta T\right) \nonumber \\ {}{} & {} \quad = \left[ \dfrac{\partial ^{2}u_{1}}{\partial t^{2}}- 2\Omega \dfrac{\partial u_{2}}{\partial t}- \Omega ^{2}u_{1}\right] , \end{aligned}$$

(15)

$$\begin{aligned}{} & {} \quad \left( \dfrac{\partial ^{2}u_{2}}{\partial y^{2}}+a_{1}\dfrac{\partial ^{2} u_{1}}{\partial x \partial y}+a_{2}\dfrac{\partial ^{2} u_{2}}{\partial x^{2}}-\beta \dfrac{\partial T}{\partial y}\right) -n\left( a_{2}\dfrac{\partial u_{2}}{\partial x}+a_{4}\dfrac{\partial u_{1}}{\partial y}\right) \nonumber \\ {}{} & {} \quad = \left[ \dfrac{\partial ^{2}u_{2}}{\partial t^{2}} + 2\Omega \dfrac{\partial u_{1}}{\partial t}- \Omega ^{2}u_{2}\right] , \end{aligned}$$

(16)

$$\begin{aligned}{} & {} \quad \epsilon _{1}\left( \triangledown ^{2}T-n \dfrac{\partial T}{\partial x}\right) +\epsilon _{2}\frac{\partial }{\partial t}\left( \triangledown ^{2}T-n \dfrac{\partial T}{\partial x}\right) - \dfrac{\partial ^{2}T}{\partial t^{2}} = \epsilon _{3}\dfrac{\partial ^{2}}{\partial t^{2}}\left( \dfrac{\partial u_{1}}{\partial x}+\dfrac{\partial u_{2}}{\partial y}\right) , \end{aligned}$$

(17)

$$\begin{aligned}{} & {} \quad t_{xx}= e^{-nx}\left[ -p_{0}+\dfrac{\partial u_{1}}{\partial x}+a_{3}\dfrac{\partial u_{2}}{\partial y}-\beta T\right] , \end{aligned}$$

(18)

$$\begin{aligned}{} & {} \quad t_{yy}= e^{-nx}\left[ -p_{0}+\dfrac{\partial u_{2}}{\partial y}+a_{3}\dfrac{\partial u_{1}}{\partial x}-\beta T\right] , \end{aligned}$$

(19)

$$\begin{aligned}{} & {} \quad t_{xy}= e^{-nx}\left[ a_{4}\dfrac{\partial u_{2}}{\partial x}+a_{2}\dfrac{\partial u_{1}}{\partial y}\right] . \end{aligned}$$

(20)

4 Problem solution

The solution of the variables may be considered in the form of modes as:

$$\begin{aligned}{}[u_{1},u_{2},T,t_{ij}](x,y,t) = [u_{1}^{*},u_{2}^{*},T^{*},t_{ij}^{*}](x)e^{\omega t+\iota by}. \end{aligned}$$

(21)

Here $\omega$ represents the frequency which is complex, b is the wave number along y direction, and $u_{i}^{*}$, $T^*$, $t_{ij}^{*}$ are the amplitudes of the field quantities.

Using the assumed solutions defined by (21) in (15)–(20), we get

$$\begin{aligned}{} & {} \quad (D^{2}-nD+d_{1})u_{1}^{*}+(d_{2}D+d_{3})u_{2}^{*}-\beta (D-n)T^{*}= -np_{0}, \end{aligned}$$

(22)

$$\begin{aligned}{} & {} \quad (d_{2}D-d_{4})u_{1}^{*}+(a_{2}D^{2}-na_{2}D+d_{5})u_{2}^{*}-d_{6} T^{*}= 0, \end{aligned}$$

(23)

$$\begin{aligned}{} & {} \quad d_{7}Du_{1}^{*}+d_{8}u_{2}^{*}+(d_{9}D^{2}-nd_{9}D-d_{10})T^{*}=0, \end{aligned}$$

(24)

$$\begin{aligned}{} & {} \quad t_{xx}^{*}= e^{-nx}\left[ \dfrac{-p_{0}}{e^{\omega t+\iota by}}+Du_{1}^{*}+a_{3}\iota b u_{2}^{*}-\beta T^{*}\right] , \end{aligned}$$

(25)

$$\begin{aligned}{} & {} \quad t_{yy}^{*}= e^{-nx}\left[ \dfrac{-p_{0}}{e^{\omega t+\iota by}}+\iota bu_{2}^{*}+a_{3}Du_{1}^{*}-\beta T^{*}\right] , \end{aligned}$$

(26)

$$\begin{aligned}{} & {} \quad t_{xy}^{*}= e^{-nx}[a_{4}Du_{2}^{*}+a_{2}\iota bu_{1}^{*]}. \end{aligned}$$

(27)

The expressions $a_{i}(i=1,2,3,4)$, $d_{j}(j=1\ldots 9)$, $\beta$, $\epsilon _{1},\epsilon _{2},\epsilon _{3}$ are given in the appendix.

Eliminating $u_{2}^{*}$, and $T^{*}$ from (22)–(24), a non-homogeneous sixth-order differential equation for $u_{1}^{*}$ is obtained

$$\begin{aligned} (D^{6}+A_{1}D^{5}+A_{2}D^{4}+A_{3}D^{3}+A_{4}D^{2}+A_{5}D+A_{6})u_{1}^{*}= np_{0}d_{6}C_{14}, \end{aligned}$$

(28)

where $A_{i}(i=1..6)$ are listed in the appendix.

The bounded solution of (28), after applying radiation conditions as $x \rightarrow \infty$, is given by

$$\begin{aligned} u_{1}^{*}= & {} \sum _{i=1}^{3}P_{i}(a,\omega )e^{-k_{i}x}+\xi _{1}, \end{aligned}$$

(29)

$$\begin{aligned} u_{2}^{*}= & {} \sum _{i=1}^{3}T_{1i}P_{i}(a,\omega )e^{-k_{i}x}+ J_{1} \xi _{1}, \end{aligned}$$

(30)

$$\begin{aligned} T^{*}= & {} \sum _{i=1}^{3}T_{2i}P_{i}(a,\omega )e^{-k_{i}x}+ J_{2} \xi _{1}. \end{aligned}$$

(31)

The roots $k_{i}$ $(i=1,2,3)$ are obtained by solving Eq. (28) and $P_{i}(a,\omega ) (i=1,2,3)$ are the parameters which depend on a and $\omega$, and the coupling constants $T_{1i}$ and $T_{2i}$ are given by,

$$\begin{aligned} T_{1i}= & {} \dfrac{C_{7}k_{i}^{3}+C_{8}k_{i^{2}}+C_{9}k_{i}-C_{10}}{C_{11}k_{i}^{4}+2nC_{11}k_{i}^{3}+C_{12}k_{i}^{2}-nC_{13}k_{i}+C_{14}}, \\ T_{2i}= & {} \dfrac{d_{7}k_{i}-d_{8}T_{1i}}{k_{i}^{2}d_{9}+nd_{9}k_{i}-d_{10}}. \end{aligned}$$

Under the assumed solution, the stress components are given by:

$$\begin{aligned} t_{xx}^{*}= & {} e^{-nx}[\phi (t,y)- \Sigma _{i=1}^{3}P_{i}e^{-k_{i}x}U_{i}+J_{3}\xi _{1}], \end{aligned}$$

(32)

$$\begin{aligned} t_{yy}^{*}= & {} e^{-nx}[\phi (t,y)+ \Sigma _{i=1}^{3}P_{i}e^{-k_{i}x}V_{i}+J_{4}\xi _{1}], \end{aligned}$$

(33)

$$\begin{aligned} t_{xy}^{*}= & {} e^{-nx}[\Sigma _{i=1}^{3}P_{i}e^{-k_{i}x}W_{i}+a_{2}\iota b \xi _{1}], \end{aligned}$$

(34)

where

$$\begin{aligned} \phi (t,y)= \frac{-p_{0}}{e^{\omega t+\iota by}}. \end{aligned}$$

5 Boundary conditions

In order to evaluate the constants $P_{i}$ (i =1,2,3), the following boundary conditions are applied along the surface $x = 0$,

$$\begin{aligned} a) t_{xx}= & {} -F_{1} e^{\omega t + \iota b y}, \end{aligned}$$

(35)

$$\begin{aligned} b) t_{xy}= & {} 0, \end{aligned}$$

(36)

$$\begin{aligned} c) T= & {} F_{2} e^{\omega t + \iota b y}. \end{aligned}$$

(37)

Here $F_{1}$ and $F_{2}$ are the magnitude of the applied mechanical and thermal source respectively on the free surface.

Using Eqs. (29)–(31) and (32)–(34) in boundary conditions (35)–(37), we get a set of three non-homogeneous equations in three unknowns as:

$$\begin{aligned}{} & {} U_{1}P_{1}+ U_{2}P_{2}+ U_{3}P_{3} = \phi (t,y)+ J_{3} \xi _{1}+P_{1}, \end{aligned}$$

(38)

$$\begin{aligned}{} & {} W_{1}P_{1}+ W_{2}P_{2}+ W_{3}P_{3} = -a_{2}\iota b \xi _{1}, \end{aligned}$$

(39)

$$\begin{aligned}{} & {} T_{21}P_{1}+ T_{22}P_{2}+ T_{23}P_{3} = P_{2}-J_{2} \xi _{1}. \end{aligned}$$

(40)

Solving Eqs. (38)–(40), which can be written in the matrix form as:

$$\begin{aligned} \begin{bmatrix} U_{1}&{}\quad U{2}&{}\quad U_{3}\\ W_{1}&{}\quad W_{2}&{}\quad W_{3}\\ T_{21}&{}\quad T_{22}&{}\quad T_{23}\\ \end{bmatrix} \begin{bmatrix} P_{1}\\ P_{2}\\ P_{3}\\ \end{bmatrix} = \begin{bmatrix} \phi (t,y)+ J_{3} \xi _{1}+P_{1}\\ -a_{2}\iota b \xi _{1}\\ P_{2}-J_{2} \xi _{1}.\\ \end{bmatrix} \end{aligned}$$

(41)

The parameters $P_{i} (i = 1,2,3)$ are defined as follows:

$$\begin{aligned} P_{i} = \dfrac{\vartriangle _{i}}{\vartriangle }, (i=1,2,3), \end{aligned}$$

(42)

where

$$\begin{aligned} \vartriangle= & {} U_{1}[W_{2}T_{23}-W_{3}T_{22}]-U_{2}[W_{1}T_{23}-W_{3}T_{21}]+U_{3}[W_{1}T_{22}-W_{2}T_{21}], \end{aligned}$$

(43)

$$\begin{aligned} \vartriangle _{1}= & {} (\phi (t,y)+ J_{3} \xi _{1}+P_{1})(W_{2}T_{23}-W_{3}T_{22})+a_{2}\iota b \xi _{1}(U_{1}T_{23}-U_{3}T_{22})\nonumber \\ {}{} & {} +(P_{2}-J_{2} \xi _{1})(U_{2}W_{3}-U_{3}W_{2}), \end{aligned}$$

(44)

$$\begin{aligned} \vartriangle _{2}= & {} (\phi (t,y)+ J_{3} \xi _{1}+P_{1})(W_{1}T_{23}-W_{3}T_{21})+a_{2}\iota b \xi _{1}(U_{1}T_{23}-U_{3}T_{21})+\nonumber \\{} & {} (P_{2}-J_{2} \xi _{1})(U_{1}W_{3}-U_{3}W_{1}), \end{aligned}$$

(45)

$$\begin{aligned} \vartriangle _{3}= & {} (\phi (t,y)+ J_{3} \xi _{1}+P_{1})(W_{1}T_{22}-W_{2}T_{21})+a_{2}\iota b \xi _{1}(U_{1}T_{22}-U_{2}T_{21})\nonumber \\{} & {} +(P_{2}-J_{2} \xi _{1})(U_{1}W_{2}-U_{2}W_{1}). \end{aligned}$$

(46)

6 Different cases of non-homogeneous medium

6.1 Rotating thermoelastic solid without hydrostatic initial stress

The expressions (29)–(31) and (32)–(34) in the absence of initial stress are obtained by substituting $p_0=0$.

6.2 Initially stressed thermoelastic medium without rotation

When the medium under consideration is not rotating, the corresponding expressions are reduced by taking $\Omega =0$ in the above mentioned expressions.

7 Homogeneous thermoelastic medium

Taking $n=0$ in particular cases (6.1) and (6.2), the analytical expressions are reduced for:

• (i) Rotating thermoelastic solid without hydrostatic initial stress,

• (ii) Initially stressed thermoelastic medium without rotation.

8 Mechanical and thermal sources

8.1 Mechanical force

For $F_1\ne 0$ and $F_2=0$, the expressions given by (29)–(31), and (32)–(34) are obtained for a mechanical force.

8.2 Thermal source

For $F_1=0$ and $F_2\ne 0$, the expressions given by (29)–(31), and (32)–(34) are obtained for a thermal point source applied along the free surface of the thermoelastic medium.

9 Numerical results

To compliment the theoretical results derived earlier, we now present a numerical example for a particular medium. The numerical results in graphical form represent the variations of physical quantities. The following values of physical constants are considered for numerical computations:

$$\begin{aligned}{} & {} \rho _0=2.7\times 10^3\ \text {kg/m}^3, K_0=2.0595\times 10^3\ \text { J/(m s}^\circ \textrm{C}), E=36.9\times 10^{10}\ \text { N/m}^2, \\{} & {} C_e=0.9878\times 10^3\ \text { J/kg}^\circ \textrm{C}, T_0=20^\circ \textrm{C}, \lambda _0=\frac{E\sigma }{\eta (1+\sigma )(1-2\sigma )},\\{} & {} \mu _0=\frac{E}{2\eta (1+\sigma )}, \eta =1.5, \sigma =0.33, \beta =1.0,\\{} & {} \epsilon _1=0.0198, \epsilon _2=0.05, \epsilon _3=0.057. \end{aligned}$$

The numerical computations are carried out at the instance $t=0.1$. The results for tangential displacement $u_2$, normal displacement $u_1$, tangential force stress $t_{xy}$, normal force stress $t_{xx}$ and temperature distribution T are presented graphically in Figs. 1, 2, 3, 4 and 5 to show the effect of non-homogeneity $(n=0,1,2)$ and hydrostatic initial stress $(p_0=0,1.5)$. Here $\omega =\omega _0+\iota \omega _1$ with $\omega _0=1.5$, $\omega _1=0.2$, $b=1.6$ and $\Omega =1.5$.

Fig. 1

Variation of normal displacement $u_{2}$ with distance x

Fig. 2

Variation of tangential displacement $u_{1}$ with distance x

Fig. 3

Variation of normal stress with distance x

Fig. 4

Variation of tangential stress with distance x

Fig. 5

Variation of temperature distribution with distance x

10 Discussions

The variations of all the quantities are oscillatory in nature and the magnitude of oscillations decrease as we move away from the point of source. In the absence of hydrostatic initial stress, the magnitude of displacement components is very small and hence, the oscillations are less oscillatory. Very close to the point of application of source, the values of displacement components (tangential and normal), for a fixed value of hydrostatic initial stress parameter $p_0$, increase with increase in the non-homogeneity parameter n. These variations of displacement components are shown in Figs. 1 and 2 respectively. The variations of tangential stress and normal stress are significantly oscillatory for a homogeneous thermoelastic medium with hydrostatic initial stress. When the thermoelastic medium is not initially stressed ($p_0=0$), the values of tangential stress increase sharply in the range $0\le x\le 1.0$ and henceforth, these values are close to zero. These variations of stress components are shown in Figs. 3 and 4 respectively. It can be observed from Fig. 5, that the values of the temperature field lie in a large range when the medium is initially stressed. The difference in values of temperature distribution in the thermoelastic without initial stress is observed in the range $0\le x\le 1.5$ but in the remaining range, these values are quite close to each other.

11 Conclusion

The graphical results obtained for the present problem are similar in nature for both type of sources represented by $P_1$ and $P_2$ in magnitude. The authors have presented the results for the mechanical force which conclude that:

• 1. The effect of initial stress and non-homogeneity can be observed on all the quantities.

• 2. Close to the point where the source is applied, the values of displacement components is directly proportional to non-homogeneity parameter n.

• 3. The variations of mechanical stress are more oscillatory for an initially stresses homogeneous thermoelastic medium.

• 4. The values of displacement components and temperature distribution for an initially stressed medium are highly significant.

• 5. As expected from the analytical expressions, the graphical results prove that due to initial stress the body is deformed to a more extent.

Appendix

$$\begin{aligned} D= & {} \dfrac{d}{dx} ,\ a_{1}= \left( \dfrac{\lambda _{0}+\mu _{0}}{\lambda _{0}+2\mu _{0}}+\dfrac{p_{0}}{2}\right) ,\\ a_{2}= & {} \left( \dfrac{\mu _{0}}{\lambda _{0}+2\mu _{0}}-\dfrac{p_{0}}{2}\right) ,\ a_{3}= \left( \dfrac{\lambda _{0}}{\lambda _{0}+2\mu _{0}}\right) ,\\ a_{4}= & {} \left( \dfrac{\mu _{0}}{\lambda _{0}+2\mu _{0}}+\dfrac{p_{0}}{2}\right) ,\ \beta = \dfrac{\alpha _{0}}{k_{T}\vartheta _{0}},\\ d_{1}= & {} \Omega ^{2}-\omega ^{2}-a_{2}b^{2},\ d_{2}= a_{1}\iota b ,\ d_{3}= 2\Omega \omega - na_{3}\iota b ,\ d_{4}= na_{4}\iota b+2\Omega \omega ,\ d_{5}= \Omega ^{2}-\omega ^{2}-b^{2},\\ d_{6}= & {} \beta \iota b ,\ d_{7}= \epsilon _{3}\omega ^{2},\ d_{8}= \epsilon _{3}\omega ^{2}\iota b ,\ d_{9}= \epsilon _{1}+\omega \epsilon _{2} ,\\ d_{10}= & {} b^{2}d_{9}-\omega ^{2} ,\\ C_{1}= & {} d_{6}-\beta d_{2}, C_{2}=-nd_{6}+d_{4}\beta +d_{2}n \beta ,C_{3}=d_{1}d_{6}-d_{4}n \beta , \\ C_{4}= & {} -a_{2}\beta ,C_{5}=d_{2}d_{6}-d_{5}\beta -n^{2}a_{2}\beta ,C_{6}=d_{3}d_{6}+n \beta d_{5}, \\ C_{7}= & {} d_{2}d_{9}, C_{8}=d_{4}d_{9}+nd_{2}d_{9}, C_{9}=nd_{4}d_{9}-d_{2}d_{10}+d_{7}d_{6}, \\ C_{10}= & {} d_{4}d_{10}, C_{11}=a_{2}d_{9}, C_{12}=d_{5}d_{9}+n^{2}a_{2}d_{9}-a_{2}d_{10}, \\ C_{13}= & {} -d_{5}d_{9}+a_{2}d_{10}, C_{14}= d_{5}d_{10}+d_{6}d_{8}. \\ A_{1}= & {} \dfrac{C_{2}C_{11}-2nC_{1}C_{11}+C_{8}C_{4}-2nC_{7}C_{4}}{C_{1}C_{11}-C_{7}C_{4}}, \\ A_{2}= & {} \dfrac{C_{3}C_{11}-2nC_{2}C_{11}+C_{1}C_{12}-C_{9}C_{4}+2nC_{4}C_{8}-C_{5}C_{7}}{C_{1}C_{11}-C_{7}C_{4}}, \\ A_{3}= & {} \dfrac{-2nC_{3}C_{11}+C_{2}C_{11}+nC_{1}C_{13}-C_{4}C_{10}-2nC_{4}C_{9}+C_{5}C_{8}-C_{6}C_{7}}{C_{1}C_{11}-C_{7}C_{4}}, \\ A_{4}= & {} \dfrac{C_{3}C_{12}+nC_{2}C_{13}+C_{1}C_{14}-2nC_{4}C_{10}-C_{9}C_{5}+C_{6}C_{8}}{C_{1}C_{11}-C_{7}C_{4}}, \\ A_{5}= & {} \dfrac{nC_{3}C_{13}+C_{2}C_{14}-C_{5}C_{10}-C_{6}C_{9}}{C_{1}C_{11}-C_{7}C_{4}}, \\ A_{6}= & {} \dfrac{C_{3}C_{14}-C_{6}C_{10}}{C_{1}C_{11}-C_{7}C_{4}}, \\ \xi _{1}= & {} \dfrac{-np_{0}d_{6}C_{14}}{A_{6}}, J_{1}=\dfrac{-C_{10}}{C_{14}}, \\ J_{2}= & {} \dfrac{d_{8}J_{1}}{d_{10}}, U_{i} = k_{i}+a_{3}\iota b T_{1i} + \beta T_{2i}, \\ V_{i}= & {} \iota b T_{1i}-a_{3}k_{i} - \beta T_{2i}, W_{i} = a_{2}\iota b - a_{4}k_{i}T_{1i}. \end{aligned}$$