Indian Journal of Materials Science

Volume 2016 (2016), Article ID 7563215, 7 pages

http://dx.doi.org/10.1155/2016/7563215

## Three-Dimensional Unsteady State Temperature Distribution of Thin Rectangular Plate with Moving Point Heat Source

^{1}Department of Applied Science, PVG’S College of Engineering, Nashik, Maharashtra, India^{2}Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India

Received 30 March 2016; Accepted 20 July 2016

Academic Editor: Dnyaneshwar S. Patil

Copyright © 2016 Yogita M. Ahire and Kirtiwant P. Ghadle. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with the study of thermal stresses in thin rectangular plate subjected to point heat source which changes its place along -axis. Governing heat conduction equation has been solved by using integral transform technique. Results are obtained in the form of infinite series. As a special case, aluminum plate has been considered and results for thermal stresses have been computed numerically and graphically.

#### 1. Introduction

Material properties are dependent on change in temperature. The properties like elasticity and stresses at various temperatures have been studied. These nonisothermal problems of theory of elasticity have attracted the attention of many. The temperature dependent properties are focused on various fields like aerodynamics heating which produces intense thermal stresses reducing the strength of structure of high velocity aircraft [1]. Steady state thermal stresses with axis symmetric temperature distribution in a circular plate subjected to the upper surface with respect to zero temperature on the lower surface and thermally insulated circular edge have been determined by [2]. On fixed and simply supported edges [3] has calculated thermal deflection of associated axis symmetrically heated circular plate. Reference [4] has considered quasi-static thermal stresses in a thin circular plate due to transient temperature applied along the edge of a circle on the upper face with respect to lower face at zero temperature and a thermally insulated fixed circular edge. Reference [5] studied an inverse unsteady state thermoelastic problem of a thin rectangular plate. Quasi-static thermoelastic problem of an infinitely long circular cylinder has been calculated by [6]. Temperature distribution, thermal functions, and displacement at any point of semi-infinite rectangular slab with internal heat source using integral transform technique are solved by [7]. Using integral transform technique and Green’s theorem [8] has determined temperature distribution and thermal stresses by taking second kind boundary condition in thin rectangular plate with moving line heat source. Reference [9] has determined thermal stresses on thin rectangular plate by integral transform with internal moving point heat source. Reference [10] determines temperature distribution, displacement, and thermal stresses of a thin circular plate due to uniform internal energy generation using Hankel transform technique graphically.

Integral transform technique is a powerful tool to solve various new general purpose numerical methods and can be applied to any multidimensional problem to get an approximate solution. This is the easiest way to find parameters like variation of temperature, and so forth. This method is better than other methods.

Attempt is made to determine effective solution and study of thermal stresses in a thin rectangular plate with internally moving heat point source.

Present paper elaborates on determination of temperature and thermal stresses in a thin rectangular plate defined as , , and where and is thickness which is very small. Using integral transform technique the governing heat conduction equation is solved. Results are obtained in the form of infinite series. It has been computed numerically and graphically.

#### 2. Formulation of the Problem

We consider three-dimensional thin rectangular plate under steady state temperature defined in region : , , and , where and is thickness which is very small. The plate is subjected to the motion of moving point heat source at the point . Under these realistic prescribed conditions, temperature and thermal stresses in a thin rectangular plate are required to be determined.

The temperature distribution of the rectangular plate defined in [11] is given bywhere is thermal conductivity and is thermal diffusivity of the material of the plate.

Consider an instantaneous moving heat source at point and release its heat spontaneously at time . Such volumetric moving heat source in rectangular coordinates is given bywhere is instantaneous point heat source.

Hence (1) becomesInitial and boundary conditions are given by

Thermal stress function is , where complementary function is and is particular integral. and are governed by equations,Since plate is thin, is negligible and , where is initial temperature. Components of stress functions [12] are given by with boundary conditions and at .

Equations (1) to (8) represent the statement of the problem.

#### 3. Solution of the Problem

Applying finite Fourier cosine transform, finite Fourier sine transform [13], and Marchi-Fasulo transform [14], using boundary conditions (4), we get where .

Taking inverse Marchi-Fasulo transform [14], finite Fourier sine transform, and finite Fourier cosine transform [13],And ,

As we change the values of , and from to we get infinite terms of this solution which is nothing but infinite series.

#### 4. Determination of Stress Function

Using (15) in (6)–(8) we getUsing the boundary conditions and at , we get

#### 5. Numerical Results

Let , , , and , cm, cm, andwhere , , , ,