Distribution of Temperature and Thermal Stresses in Unidirectional Rod with Moving Point Heat Source

Abstract

The present paper comprises the quasi-stationary, non-homogeneous thermoelastic problem with a second kind boundary condition in two-dimensional rod of isotropic material. The unidirectional rod is examined with the condition that ambient and initial temperature is zero. The rod has been observed under the activity of moving heat source located at $x^{{\prime }}$ moving with constant velocity along x-axis. Heat conduction equation is evaluated by using integral transform technique. The three materials, viz. aluminum, copper, and brass, have been studied, and the same are analyzed numerically and graphically for their respective thermal stresses. For aluminum, the change is observed from maximum to minimum stress. The response of copper to change in temperature is linear. The curve of copper shows constant value of stress because of its larger coefficient of thermal expansion. Tensile stress is reduced in copper. Minimum stress is observed in brass that indicates the hardness and tensile strength of brass.

1 Introduction

Various changes in shape and sizes of solids have been found because of the variation in temperature. The elastic behavior of material in terms of stresses which are sensitive to temperatures is responsible for the same. Different materials show changes in heat when flows through it. Here is an attempt to study the thermal stresses in unidirectional rod. The temperature-dependent properties have been studied in the field of aerodynamics heating. The generation of intense thermal stresses decreases the strength of the structure of high-velocity aircraft [1]. The circular plate with upper surface at zero temperature w. r. t. lower surface has been studied by keeping circular edge thermally insulated by considering steady-state thermal stresses along with axisymmetric temperature distribution [2]. With reference to [3], it discussed the problem of thermal deflection of associated axisymmetry with fixed and simply supported edges, when heated circular plate. [4] has studied the circular plate with lower surface at zero temperature w. r. t. upper surface when momentary temperature provided through periphery of circle for quasi-static thermal stresses. Khobragade and Wankhede [5] analyzed thermoelastic problem of a thin rectangular plate by assuming an inverse unsteady state. Quasi-static thermoelastic problem of an infinitely long circular cylinder has been solved by Gaikwad and Ghadle [6]. Gaikwad and Ghadle [7] discussed non-homogeneous heat conduction problem and its thermal deflection which causes the generation of internal heat in a thin hollow circular disk. Patil et al. [8] solved semi-infinite rectangular slab with internal heat source using integral transform technique with the determination of the distribution of temperature, thermal parameters, and displacement at any point. Solanke and Durge [9] have determined distribution of temperature and thermal stresses with second kind boundary conditions in thin rectangular plate with moving line heat source by using integral transform technique and Green’s theorem. Thakare et al. [10] have evaluated thermal stresses due to internal moving point heat source in thin rectangular plate by integral transform technique. Solanke and Durge [11] have solved problem of quasi-stationary thermo-elastic characteristic with moving heat source in unidirectional Robin’s rod. Gaikwad [12] used Hankel transform technique to evaluate two-dimensional steady-state temperature distribution in thin circular plate due to the generation of uniform internal energy, and it has been presented graphically. Ahire and Ghadle [13] elaborate on three-dimensional unsteady-state temperature distribution in thin rectangular plate with moving point heat source.

A tool of integral transform technique has found the most suitable way to obtain the solution of various new general-purpose numerical methods. Due to its simple form, one can implement it to reveal parameters like variation of temperature. This technique has a lot of advantages as compared to other methods.

The analysis in the present paper constructs an effective solution and helps to study the thermal stresses in a unidirectional rod with internal moving line heat source.

The remainder of this paper elaborates the details of the determination of temperature and thermal stresses in unidirectional rod defined as $0 \le x \le a, 0 \le y \le b$. Heat conduction equation has been solved using integral transform technique. Result is composed of the terms in the form of infinite series. It has been determined by numerically and graphically.

2 Formulation of the Problem

Consider a rod of length confined in the region $R{:}\ 0 \le x \le a,0 \le y \le b$. The unidirectional rod is studied with an ambient temperature zero when initial temperature is also zero. The rod is kept under the instantaneous moving heat source located at the point $x^{{\prime }}$, which is moving with constant velocity u along x-axis. Atomic interaction due to moving heat source causes the generation of heat. The differential equation of heat conduction which contains heat generation term results in the distribution of temperature in two-dimensional rod.

The temperature distribution of the rod defined in [14] is given by

$$ \frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{g}{k} = \frac{1}{\alpha }\frac{\partial T}{\partial t} $$

(1)

where k is thermal conductivity and $\alpha$ is thermal diffusivity of the material of the plate.

The volumetric moving heat source in rectangular coordinates $\left( {x^{{\prime }} , y} \right)$ and produces the spontaneous heat at time $t^{{\prime }}$ which is given by

$$ g\left( {x,y,t} \right) = g_{P}^{i} \delta \left( {x - x^{{\prime }} } \right)\delta \left( y \right)\delta \left( {t - t^{{\prime }} } \right) $$

(2)

where $g_{P}^{i}$ is instantaneous heat source.

Hence, Eq. (1) becomes

$$ \frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{1}{k}g_{P}^{i} \delta \left( {x - x^{'} } \right)\delta \left( y \right) = \frac{1}{\alpha }\frac{\partial T}{\partial t} $$

(3)

The following are the initial and boundary conditions:

$$ \left[ T \right]_{t = 0} = 0 $$

(4)

$$ \left[ {\frac{\partial T}{\partial x}} \right]_{x = 0} = 0 $$

(5)

$$ \left[ {\frac{\partial T}{\partial x}} \right]_{{x = {\text{a}}}} = 0 $$

(6)

$$ \left[ {\frac{\partial T}{\partial y}} \right]_{y = 0} = 0 $$

(7)

$$ \left[ {\frac{\partial T}{\partial y}} \right]_{y = b} = 0 $$

(8)

$$ \chi = \chi_{c} + \chi_{p} $$

(9)

where $\chi$ in Eq. (9) is thermal stress function and $\chi_{c}$ is the C.F. and $\chi_{p}$ is P.I.

$\chi_{c }$ and $\chi_{p}$ are governed by equations [15],

$$ \left( {\frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }}} \right)^{2} \chi_{c} = 0 $$

(10)

and

$$ \left( {\frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }}} \right)^{2} \chi_{p} = - \alpha E\varGamma $$

(11)

At $\varGamma = T - T_{0}$, where $T_{0}$ is initial temperature. Component of stress functions [15] for thin plate is given by

$$ \sigma_{xx} = \frac{{\partial^{2} \chi }}{{\partial y^{2} }} $$

(12)

$$ \sigma_{yy} = \frac{{\partial^{2} \chi }}{{\partial x^{2} }} $$

(13)

$$ \sigma_{xy} = - \frac{{\partial^{2} \chi }}{\partial x\partial y} $$

(14)

with boundary conditions $\sigma_{yy} = 0 , \sigma_{xy} = 0 \;{\text{at}}\;y = b$ Eqs. (1)–(14) represent the statement of the problem.

3 Solution of the Problem

The solution of the moving heat source problem is obtained by conveniently assuming that the coordinate system move with the source. A new coordinate $x^{{\prime }}$ defined by Ozisik [14] is used to gain the same.

$$ x^{{\prime }} = x - ut $$

$$ \frac{{\partial^{2} T}}{{\partial \xi^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{g_{P}^{i} \delta \left( {x - x^{'} } \right)\delta \left( y \right)}}{k} = \frac{1}{\alpha }\left[ {\frac{\partial T}{\partial t} - u\frac{\partial T}{\partial \xi }} \right] $$

(15)

$$ \frac{{\partial^{2} T}}{{\partial \xi^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{g_{P}^{i} \delta \left( {x - x^{'} } \right)\delta \left( y \right)}}{k} = \frac{ - u}{\alpha }\left[ {\frac{\partial T}{\partial \xi }} \right] $$

(16)

$$ T\left( {\xi ,y} \right) = \theta \left( {\xi ,y} \right)e^{{ - \left( {\frac{u}{2\alpha }} \right)\xi }} $$

$$ \frac{{\partial^{2} \theta }}{{\partial \xi^{2} }} + \frac{{\partial^{2} \theta }}{{\partial y^{2} }} - \left( {\frac{u}{2\alpha }} \right)^{2} \theta + \frac{{g_{P}^{i} \delta \left( {x - x^{'} } \right)\delta \left( y \right)}}{k} = 0 $$

(17)

Applying finite Fourier cosine transform twice defined in [16] Eq. (16) becomes,

$$ \overline{\overline{T}} = \frac{{g_{P}^{i} }}{kQ}\left( {1 - e^{{\frac{\alpha }{u}Q\xi }} } \right) $$

(18)

where $Q = \left( {\frac{{m^{2} \pi^{2} }}{{a^{2} }} + \frac{{n^{2} \pi^{2} }}{{b^{2} }}} \right).$

Taking inverse finite Fourier cosine transform twice,

$$ T = \frac{4}{ab}\mathop \sum \limits_{m,n = 1}^{\infty } \frac{{g_{P}^{i} }}{kQ}\left( {1 - e^{{\frac{\alpha }{u}Q\xi }} } \right){ \cos }\frac{m\pi \xi }{a}{ \cos }\frac{n\pi y}{b} $$

(19)

$$ T = \frac{4}{ab}\mathop \sum \limits_{m,n = 1}^{\infty } \frac{{g_{P}^{i} }}{{k\left( {\frac{{m^{2} \pi^{2} }}{{a^{2} }} + \frac{{n^{2} \pi^{2} }}{{b^{2} }}} \right)}}\left( {1 - e^{{\frac{\alpha }{u}Q\xi }} } \right){ \cos }\frac{m\pi \xi }{a}{ \cos }\frac{n\pi y}{b} $$

(20)

$$ \varGamma = T - T_{0} $$

$$ \varGamma = \frac{4}{ab}\mathop \sum \limits_{m,n = 1}^{\infty } \frac{{g_{P}^{i} }}{{k\left( {\frac{{m^{2} b^{2} + n^{2} a^{2} }}{{a^{2} b^{2} }}} \right)\pi^{2} }}\left( {1 - e^{{\frac{\alpha }{u}Q\xi }} } \right){ \cos }\frac{m\pi \xi }{a}{ \cos }\frac{n\pi y}{b} $$

$$ \varGamma = \mathop \sum \limits_{m,n = 1}^{\infty } \frac{{4abg_{P}^{i} }}{{k\left( {m^{2} b^{2} + n^{2} a^{2} } \right)\pi^{2} }}\left( {1 - e^{{\frac{\alpha }{u}Q\xi }} } \right){ \cos }\frac{m\pi \xi }{a}{ \cos }\frac{n\pi y}{b} $$

(21)

$$ \upchi_{c} = \mathop \sum \limits_{m = 1}^{\infty } y\left[ {c_{1} e^{{\frac{m\pi y}{a}}} + c_{2} e^{{ - \frac{m\pi y}{a}}} } \right]{ \cos }\left( {\frac{m\pi x}{a}} \right) + y\left[ {c_{3} e^{{\frac{m\pi y}{a}}} + c_{4} e^{{ - \frac{m\pi y}{a}}} } \right]{ \sin }\left( {\frac{m\pi x}{a}} \right) $$

(22)

$$ \chi_{p} = - \alpha E\varGamma $$

$$ \chi_{p} = \mathop \sum \limits_{m,n = 1}^{\infty } \frac{{ - 4a^{3} b^{3} \alpha Eg_{P}^{i} }}{{\left( {m^{2} b^{2} + n^{2} a^{2} } \right)^{2} \pi^{4} }}\left( {1 - e^{{\frac{\alpha }{u}Q\xi }} } \right)\cos \frac{m\pi \xi }{a}\cos \frac{n\pi y}{b} $$

(23)

$$ \chi = \chi_{c} + \chi_{p} $$

$$ \begin{aligned}\upchi & = \mathop \sum \limits_{m = 1}^{\infty } y\left[ {c_{1} e^{{\frac{m\pi y}{a}}} + c_{2} e^{{ - \frac{m\pi y}{a}}} } \right]{ \cos }\left( {\frac{m\pi x}{a}} \right) + y\left[ {c_{3} e^{{\frac{m\pi y}{a}}} + c_{4} e^{{ - \frac{m\pi y}{a}}} } \right]{ \sin }\left( {\frac{m\pi x}{a}} \right) \\ & \quad + \mathop \sum \limits_{m,n = 1}^{\infty } \frac{{ - 4a^{3} b^{3} \alpha Eg_{P}^{i} }}{{\left( {m^{2} b^{2} + n^{2} a^{2} } \right)^{2} \pi^{4} }}\left( {1 - e^{{\frac{\alpha }{u}Q\xi }} } \right)\cos \frac{m\pi \xi }{a}\cos \frac{n\pi y}{b} \\ \end{aligned} $$

(24)

4 Determination of Stress Function

Using Eq. (24) in Eqs. (12)–(14), we get,

$$ \begin{aligned} \sigma_{xx} & = \mathop \sum \limits_{m,n = 1}^{\infty } \left\{ {\left[ {2\frac{m\pi }{a}\left( {c_{1} e^{{\frac{m\pi y}{a}}} - c_{2} e^{{ - \frac{m\pi y}{a}}} } \right) + y\frac{{m^{2} \pi^{2} }}{{a^{2} }}\left( {c_{1} e^{{\frac{m\pi y}{a}}} + c_{2} e^{{ - \frac{m\pi y}{a}}} } \right)} \right]{ \cos }\left( {\frac{m\pi x}{a}} \right)} \right. \\ & \quad \left. { + \left[ {2\frac{m\pi }{a}\left( {c_{3} e^{{\frac{m\pi y}{a}}} - c_{4} e^{{ - \frac{m\pi y}{a}}} } \right) + y\frac{{m^{2} \pi^{2} }}{{a^{2} }}\left( {c_{3} e^{{\frac{m\pi y}{a}}} + c_{4} e^{{ - \frac{m\pi y}{a}}} } \right)} \right]{ \sin }\left( {\frac{m\pi x}{a}} \right)} \right\} \\ & \quad - \frac{{4a^{3} b\alpha Eg_{P}^{i} }}{{\left( {m^{2} b^{2} + n^{2} a^{2} } \right)^{2} \pi^{2} }}n^{2} \left( {1 - e^{{\frac{\alpha }{u}Qx}} } \right)\cos \left( {\frac{m\pi x}{a}} \right)\cos \left( {\frac{n\pi y}{b}} \right) \\ \end{aligned} $$

(25)

$$ \begin{aligned} \sigma_{yy} & = \frac{{ - m^{2} \pi^{2} y}}{{a^{2} }}\left\{ {\left[ {c_{1} e^{{\frac{m\pi y}{a}}} + c_{2} e^{{ - \frac{m\pi y}{a}}} } \right]{ \cos }\left( {\frac{m\pi x}{a}} \right)} \right. \\ & \quad - \left. {\frac{{ym^{2} \pi^{2} }}{{a^{2} }}\left[ {c_{3} e^{{\frac{m\pi y}{a}}} + c_{4} e^{{ - \frac{m\pi y}{a}}} } \right]{ \sin }\left( {\frac{m\pi x}{a}} \right)} \right\} \\ & \quad + \frac{{4\alpha Ea^{3} b^{3} g_{P}^{i} \left( {1 - e^{{\frac{\alpha }{u}Qx}} } \right)}}{{\pi^{{42{\text{os}}}} \left( {a^{2} n^{2} + b^{2} m^{2} } \right)^{2} }}{ \cos }\left( {\frac{m\pi x}{a}} \right){ \cos }\left( {\frac{n\pi y}{b}} \right)\left[ {\frac{{\alpha^{2} Q^{2} }}{{u^{2} }} - \frac{{m^{2} \pi^{2} }}{{a^{2} }}} \right] \\ \end{aligned} $$

(26)

$$ \begin{aligned} \sigma_{xy} & = \mathop \sum \limits_{m,n = 1}^{\infty } \left\{ {\left[ {\left( {\frac{m\pi }{a} + y\frac{{m^{2} \pi^{2} }}{{a^{2} }}} \right)c_{1} e^{{\frac{m\pi y}{a}}} + \left( {\frac{m\pi }{a} - y\frac{{m^{2} \pi^{2} }}{{a^{2} }}} \right)c_{2} e^{{ - \frac{m\pi y}{a}}} } \right]{ \sin }\left( {\frac{m\pi x}{a}} \right)} \right. \\ & \quad + \left. {\left[ {\left( { - \frac{m\pi }{a} - y\frac{{m^{2} \pi^{2} }}{{a^{2} }}} \right)c_{3} e^{{\frac{m\pi y}{a}}} + \left( {\frac{m\pi }{a} + y\frac{{m^{2} \pi^{2} }}{{a^{2} }}} \right)c_{4} e^{{ - \frac{m\pi y}{a}}} } \right]{ \cos }\left( {\frac{m\pi x}{a}} \right)} \right\} \\ & \quad - \frac{{4a^{2} b^{2} \alpha Eg_{P}^{i} mn}}{{\left( {m^{2} b^{2} + n^{2} a^{2} } \right)^{2} \pi^{2} }}\left( {1 - e^{{\frac{\alpha }{u}Qx}} } \right)\sin \left( {\frac{m\pi x}{a}} \right)\sin \left( {\frac{n\pi y}{b}} \right) \\ \end{aligned} $$

(27)

Using the boundary condition $\sigma_{yy} = 0$ and $\sigma_{xy} = 0 \;{\text{at}}\;y = b,$ we get

$c_{3} = 0,\, c_{4} = 0$, $c_{1} = \frac{{2\alpha E\varphi a^{2} \left( {u^{2} + 2m^{2} } \right)}}{{bm^{2} u^{2} }}e^{{\frac{ - m\pi b}{a}}}$, $c_{2} = \frac{{2\alpha E\varphi a^{4} }}{{bm^{2} }}e^{{\frac{m\pi b}{a}}}$

where $\varphi = \frac{{g_{P}^{i} (a^{4} \alpha^{2} n^{2} + a^{2} b^{2} \alpha^{2} m^{2} - u^{2} a^{2} b^{2} m^{2} }}{{\pi^{4} m^{2} \left( {a^{2} n^{2} + b^{2} m^{2} } \right)^{2} }}$

$$ \begin{aligned} \varphi & = \frac{2.3562}{15.5485}{\text{e}}^{{ - 23.8 \times 10^{ - 6} \times 12.4912t}} \left\{ {\int {\left( {{\text{e}}^{{23.8 \times 10^{ - 6} \times 12.4912t}} + 23.8 \times 10^{ - 6} \left( { - 0.9721} \right.} \right.} } \right. \\ & \quad \left. {\left. { - 1.2639t} \right)} \right) - \left. {\int {23.8 \times 10^{ - 6} \left( {\left( { - 0.9721 - 1.2639{\text{t}}} \right)} \right){\text{d}}t} } \right\} \\ \end{aligned} $$

$$ \begin{aligned} \sigma_{xx} & = \mathop \sum \limits_{m,n = 1}^{\infty } \left\{ {\left[ {\left( {\frac{2m\pi }{a} + y\frac{{m^{2} \pi^{2} }}{{a^{2} }}} \right)\frac{{2\alpha E\varphi a^{2} \left( {u^{2} + 2m^{2} } \right)}}{{bm^{2} u^{2} }}e^{{\frac{ - m\pi b}{a}}} e^{{\frac{m\pi y}{a}}} } \right.} \right. \\ & \quad \left. {\left. { + \left( { - \frac{2m\pi }{a} + y\frac{{m^{2} \pi^{2} }}{{a^{2} }}} \right)\frac{{2\alpha E\varphi a^{4} }}{{bm^{2} }}e^{{\frac{m\pi b}{a}}} e^{{ - \frac{{{\text{m}}\uppi{\text{y}}}}{a}}} } \right]{ \cos }\left( {\frac{m\pi x}{a}} \right)} \right\} \\ & \quad - \frac{{4a^{3} b\alpha Eg_{P}^{i} }}{{\left( {m^{2} b^{2} + n^{2} a^{2} } \right)^{2} \pi^{2} }}n^{2} \left( {1 - e^{{\frac{\alpha }{u}Qx}} } \right)\cos \left( {\frac{m\pi x}{a}} \right)\cos \left( {\frac{n\pi y}{b}} \right) \\ \end{aligned} $$

$$ \begin{aligned} \sigma_{yy} & = \frac{{ - m^{2} \pi^{2} y}}{{a^{2} }}\left\{ {\left[ {\frac{{2\alpha E\varphi a^{2} \left( {u^{2} + 2m^{2} } \right)}}{{bm^{2} u^{2} }}e^{{\frac{ - m\pi b}{a}}} e^{{\frac{{{\text{m}}\uppi{\text{y}}}}{a}}} } \right.} \right. \\ & \quad \left. {\left. { + \frac{{2\alpha E\varphi a^{4} }}{{bm^{2} }}e^{{\frac{m\pi b}{a}}} e^{{ - \frac{m\pi y}{a}}} } \right]{ \cos }\left( {\frac{m\pi x}{a}} \right)} \right\} \\ & \quad + \frac{{4\alpha Ea^{3} b^{3} g_{P}^{i} \left( {1 - e^{{\frac{\alpha }{u}Qx}} } \right)}}{{\pi^{{42{\text{os}}}} \left( {a^{2} n^{2} + b^{2} m^{2} } \right)^{2} }}\cos \left( {\frac{m\pi x}{a}} \right)\cos \left( {\frac{n\pi y}{b}} \right)\left[ {\frac{{\alpha^{2} Q^{2} }}{{u^{2} }} - \frac{{m^{2} \pi^{2} }}{{a^{2} }}} \right] \\ \end{aligned} $$

$$ \begin{aligned} \sigma_{xy} & = \mathop \sum \limits_{m,n = 1}^{\infty } \left\{ {\left[ {\left( {\frac{m\pi }{a} + y\frac{{m^{2} \pi^{2} }}{{a^{2} }}} \right)\frac{{2\alpha E\varphi a^{2} \left( {u^{2} + 2m^{2} } \right)}}{{bm^{2} u^{2} }}e^{{\frac{ - m\pi b}{a}}} e^{{\frac{m\pi y}{a}}} } \right.} \right. \\ & \quad \left. {\left. { + \left( {\frac{m\pi }{a} - y\frac{{m^{2} \pi^{2} }}{{a^{2} }}} \right)\frac{{2\alpha E\varphi a^{4} }}{{bm^{2} }}e^{{\frac{m\pi b}{a}}} e^{{ - \frac{m\pi y}{a}}} } \right]{ \sin }\left( {\frac{m\pi x}{a}} \right)} \right\} \\ & \quad - \frac{{4a^{2} b^{2} \alpha Eg_{P}^{i} mn}}{{\left( {m^{2} b^{2} + n^{2} a^{2} } \right)^{2} \pi^{2} }}\left( {1 - e^{{\frac{\alpha }{u}Qx}} } \right)\sin \left( {\frac{m\pi x}{a}} \right)\sin \left( {\frac{n\pi y}{b}} \right) \\ \end{aligned} $$

5 Numerical Results

Physical Properties of Metals

Properties	Aluminum	Copper	Brass
Modulus of elasticity (E)	$0.675 \times 10^{11}$	$1.23 \times 10^{11}$	$0.970 \times 10^{11}$
Coefficient of thermal expansion (lb/lb/℃) ($\alpha$)	$23.8 \times 10^{ - 6}$	$16.2 \times 10^{ - 6}$	$16.7 \times 10^{ - 6}$
Thermal diffusivity (cm2/s) (k)	0.530	0.940	0.310

where a = 3, b = 1, h = 0.5, m = 1, n = 1,$g_{P}^{i} = 1 ,x^{{\prime }} = 1.5$, t = 1, $t^{{\prime }} = 1.5$,

$$ \lambda_{l} = 15.5485 ,\;p_{l} \left( {0.2} \right) = 2.3562 ,\;\varphi = 0.2056, \;Q = 12.4912, \;\emptyset = - 4.5920 $$

5.1 Aluminum Rod

$$ T = 0.5358\left\{ {\left[ {e^{{ - 2.9729 \times 10^{ - 4} t}} } \right]\cos \left( {\pi y} \right)} \right\}\sin \left( {\frac{\pi x}{5}} \right) $$

$$ \begin{aligned} \sigma_{xx} & = \mathop \sum \limits_{m = 1}^{\infty } \left\{ {\left[ {\left( {\pi + \frac{{\pi^{2} }}{4}y} \right)3.7057 \times 10^{12} e^{{\frac{\pi y}{2}}} } \right.} \right. \\ & \left. {\left. {\quad + \left( { - \pi + \frac{{\pi^{2} }}{4}y} \right)3.8623 \times 10^{14} e^{{\frac{ - \pi y}{2}}} } \right]{ \cos }\left( {\frac{\pi x}{2}} \right)} \right\} \\ & \quad - 1.0205 \times 10^{11} { \cos }\left( {\frac{\pi x}{2}} \right)\cos \left( {\pi y} \right) \times \left( {1 - {\text{e}}^{2.9362x} } \right) \\ \end{aligned} $$

$$ \begin{aligned} \sigma_{xy} & = \mathop \sum \limits_{m = 1}^{\infty } \left\{ {\left[ {\left( {\frac{\pi }{2} + \frac{{\pi^{2} }}{4}y} \right){ \sin }\left( {\frac{\pi x}{2}} \right)2.3591 \times 10^{12} e^{{\frac{\pi y}{2}}} } \right.} \right. \\ & \left. {\left. {\quad - \left( {\frac{\pi }{2} - \frac{{\pi^{2} }}{4}y} \right){ \sin }\left( {\frac{\pi x}{2}} \right)3.8623 \times 10^{14} e^{{\frac{ - \pi y}{2}}} } \right]} \right\} \\ & \quad + 5.0362 \times 10^{11} { \sin }\left( {\frac{\pi x}{2}} \right)\sin \left( {\pi y} \right) \times \left( {1 - {\text{e}}^{2.9362x} } \right) \\ \end{aligned} $$

$$ \begin{aligned} \sigma_{yy} & = \left( {1 - {\text{e}}^{2.9362x} } \right)\left\{ {\left[ {\left( {1 - 2.9362 \times 10^{11} } \right)ye^{{\frac{\pi }{2}}} + 9.5298 \times 10^{14} ye^{{\frac{ - \pi }{2}}} } \right]\cos \left( {\frac{\pi x}{2}} \right)} \right\} \\ & \quad + 2.5181 \times 10^{11} \left\{ {\cos \left( {\pi y} \right)} \right\}\cos \left( {\frac{\pi x}{2}} \right) \\ \end{aligned} $$

5.2 Copper Rod

$$ T = 0.5356\left\{ {\left[ {{\text{e}}^{{ - 2.0860 \times 10^{ - 4} t}} } \right]{ \cos }\left( {\pi y} \right)} \right\}\sin \left( {\frac{\pi x}{5}} \right) $$

$$ \begin{aligned} \sigma_{xx} & = \mathop \sum \limits_{m = 1}^{\infty } \left\{ {{ \cos }\left( {\frac{\pi x}{2}} \right)} \right\}\left( {1 - {\text{e}}^{4.71x} } \right)\left( { - 1.4968 \times 10^{8} - 1.1756 \times 10^{8} y} \right)e^{\pi y/ 2} \\ & \quad + \left( {1.2253 \times 10^{9} - 1.56 \times 10^{10} } \right)\cos \left( {\pi y} \right) \times \left( { - 1.245 \times 10^{10} } \right) \\ \end{aligned} $$

$$ \begin{aligned} \sigma_{xy} & = \left\{ {\left[ {\left( {\left( {1 - {\text{e}}^{4.71x} } \right){ \sin }\left( {\frac{\pi x}{2}} \right)\left( {2.4674y + 1.5707} \right) \times 0.0298 \times 10^{11} e^{{\frac{\pi }{2}}} } \right.} \right.} \right. \\ & \quad \left. {\left. {\left. { + \left( {\frac{\pi }{2} - \pi^{2} y} \right)e^{{ - \frac{\pi }{2}}} } \right)0.04966 \times 10^{11} } \right] + 0.06225 \times 10^{11} \times { \sin }\left( {\pi y} \right)} \right\} \\ \end{aligned} $$

$$ \begin{aligned} \sigma_{yy} & = \left\{ {\left[ {\left( {\left( {1 - {\text{e}}^{4.71x} } \right){ \cos }\left( {\frac{\pi x}{2}} \right)\left( {1.1756 \times 10^{8} \times y \times e^{{\frac{\pi y}{2}}} } \right) + 1.2253 \times 10^{10} e^{{\frac{ - \pi y}{2}}} } \right)} \right]} \right. \\ & \quad \left. { - 1.5563 \times 10^{9} { \cos }\left( {\pi y} \right)} \right\} \\ \end{aligned} $$

5.3 Brass Rod

$$ T = 0.5358\left\{ {\left[ {{\text{e}}^{{ - 4.1603 \times 10^{ - 5} t}} } \right]{ \cos }\left( {\pi y} \right)} \right\}\sin \left( {\frac{\pi x}{5}} \right) $$

$$ \begin{aligned} \sigma_{xx} & = \sum\limits_{m = 1}^{\infty } {\left\{ {\left[ {\left( {\pi + \frac{{\pi^{2} }}{4}y} \right)3.7645 \times 10^{12} e^{{\frac{\pi y}{2}}} } \right.} \right.} \\ & \quad \left. {\left. { + \left( { - \pi + \frac{{\pi^{2} }}{4}y} \right)3.9235 \times 10^{14} e^{{\frac{ - \pi y}{2}}} } \right]{ \cos }\left( {\frac{\pi x}{2}} \right)} \right\} \\ & \quad - 1.0367 \times 10^{11} { \cos }\left( {\frac{\pi x}{2}} \right)\cos \left( {\pi y} \right) \times \left( {1 - {\text{e}}^{2.9362x} } \right) \\ \end{aligned} $$

$$ \begin{aligned} \sigma_{xy} & = \left\{ {\left[ {\left( {\left( {1 - {\text{e}}^{4.71x} } \right)\left( { - { \sin }\left( {\frac{\pi x}{2}} \right)} \right)\left( {2.4674y + 1.5707} \right) \times 2.3966 \times 10^{12} \times e^{{\frac{\pi y}{2}}} } \right.} \right.} \right. \\ & \quad \left. {\left. {\left. { + \left( {\frac{\pi }{2} - \frac{{\pi^{2} y}}{4}} \right)e^{{ - \frac{\pi y}{2}}} { \sin }\left( {\frac{\pi x}{2}} \right)} \right) \times 3.9235 \times 10^{14} } \right]{ \sin }\left( {\pi y} \right){ \sin }\left( {\frac{\pi x}{2}} \right)} \right\} \\ \end{aligned} $$

$$ \begin{aligned} \sigma_{yy} & = \left\{ {\left( {1 - {\text{e}}^{2.0603x} } \right)\left( { - \frac{{\pi^{2} }}{4}y} \right)\left( {3.7645 \times 10^{12} \times e^{{\frac{\pi y}{2}}} } \right)} \right. \\ & \quad \left. { + (3.9235 \times 10^{14} e^{{\frac{ - \pi y}{2}}} - 2.5580 \times 10^{11} { \cos }\left( {\pi y} \right) \times { \cos }\left( {\frac{\pi x}{2}} \right)} \right\} \\ \end{aligned} $$

6 Graphical Interpretation

Figure 1: Temperature gradually increases and becomes stable for small duration. Temperature decreases slightly after attaining the stable value. As length increases, temperature also increases with internal moving heat source.

Fig. 1

Length versus temperature

Figure 2: Stress for aluminum is maximum, and for brass, it is minimum. But for copper, it remains constant. Aluminum is very sensitive to change in temperature as compared to copper and brass. Hence, stress shown along x-axis for aluminum is more than copper and brass.

Fig. 2

x versus thermal stress (Sigma xx)

Figure 3: Copper has a very small variation in stress while aluminum and brass have more moderate change in their stresses. Aluminum is the most affected material as compared to copper and brass. Maximum stress is produced in aluminum. As the stress is compared in brass and copper, it is observed that stress is minimum in copper.

Fig. 3

x versus thermal stress (Sigma yy)

Figure 4: As depicted in graph, thermal stress in aluminum varies from its maximum to minimum. For copper, it remains constant along the length while brass has its minimum to its maximum.

Fig. 4

x versus thermal stress (Sigma xy)

7 Discussion

The two-dimensional non-homogeneous heat conduction problems in a unidirectional rod have been discussed here. Numerical computations are performed for a unidirectional rod composed of brass, aluminum, copper. The source of heat $g\left( {x, y, z, t} \right)$ is an instantaneous moving heat source and has strength $g_{P}^{i}$. The temperature-dependent elastic behavior is noted for various temperature and thermal stresses.

8 Conclusion

In this paper, we achieved the solution of non-homogeneous thermoelastic problem by using integral transform techniques numerically. The outcome of this analysis contains the terms in infinite series. Three materials brass, aluminum, copper were examined under unsteady-state temperature with initial value zero. In these three materials, thermal stresses observed for aluminum are maximum while brass and copper lie under the aluminum. These were carried out with initial zero temperature. When heat is provided to the rod, thermal stresses change along with x-axis as well as y-axis.

These results can be applied in engineering problems, such as industrial machines which come in the vicinity of the heating such as the central shaft of a machine, big turbines, the roll of rolling mill, and practical applications in aircraft structures.

Terminology

Notation	Meaning
$T \left( {x,y,t} \right)$	Temperature of the rod (℃)
$g \left( {x,y,t} \right)$	Volumetric energy of heat source (W cm−3)
k	Thermal diffusivity (cm2 s−1)
E	Modulus of elasticity
λ	Thermal conductivity
α	Coefficient of thermal expansion
$g_{p}^{i}$	Point heat source