World Journal of Mechanics, 2013, 3, 6-12

http://dx.doi.org/10.4236/wjm.2013.35A002 Published Online August 2013 (http://www.scirp.org/journal/wjm)

Thermoelastic Problem of a Long Annular Multilayered

Cylinder

Yi Hsien Wu1*, Kuo-Chang Jane2

1Department of Information Management, Oriental Institute of Technology, Taipei, Chinese Taipei

2Department of Applied Mathematics, National Chung Hsing University, Taichung, Chinese Taipei

Email: *yhwu@mail.oit.edu.tw

Received May 2, 2013; revised June 2, 2013; accepted June 9, 2013

Copyright © 2013 Yi Hsien Wu, Kuo-Chang Jane. 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

Thermoelastic transient response of multilayered annu lar cylinde rs of infinite leng ths subjected to known inner pressure

and outer surfaces cooling are considered. A method based on the Laplace transformation and finite difference method

has been developed to analyze the thermoelasticity problem. Using the Laplace transform with respect to time, the gen-

eral solutions of the governing equations are obtained in transform domain. The solution is obtained by using the matrix

similarity transformation and inverse Laplace transform. Solutions for the temperature and thermal stress distributions

in a transient state were obtained. It was found that the temperature distribution, the displacement and the thermal

stresses change slightly as time increases.

Keywords: Thermoelastic; Multilayered Annular Cylinders; Laplace Transformation; Finite Difference Method

1. Introduction

A thermal problem arises when the composed materials

are generated by a sudden change in temperature. Shell

structures are widely used in contemporary industries, so

we must take care of the thermal problem. The shell

structures may be affected due to the pressure change or

the various temperature distributions. It is necessary to

solve for temperature or pressure at first.

The dynamic thermoelastic response of circular shell

rapidly change of thermal environments is important for

the design of many engineering structures. Due to the

complexity of the governing equations and the mathe-

matical difficulties associated with the solution, several

simplifications have been used. For example, Sherief and

Anwar [1] discussed the problem of an annular infinitely

long elastic circular. They have neglected both the in ertia

terms and the relaxation effects of the problem. Sherief

and Anwar [2] considered the thermoelasticity problem

of an infinitely long annular cylinder composed of two

different materials with axial symmetry. The solution

was obtained in the Laplace transform domain by using

the potential functio n approach.

The present work deals with the one-dimensional qua-

sistatic coupled thermoelastic problems of an infinitely

long annular multilayered cylinder composed of multi-

layered different materials. The medium has a pressure at

the inner layer, the temperature to be heated at the outer

layer, without body forces and internal heat generation.

Derivatives are approximated by central differences re-

sulting in an algebraic representation of the partial dif-

ferential equation. By taking the Laplace transform with

respect to time, the general solutions in the transform

domain are first obtained. The final solutions in the real

domain can be obtained by inverting the Laplace trans-

form.

2. Formulation

This work deals with the one-dimensional, quasi-static

coupled, thermoelastic problems of an infinitely long

annular cylinder composed of multilayered laminated

materials with axial symmetry under the following as-

sumptions: 1) Materials of each layer are assumed to be

non-homogeneous; 2) Deformation and strain satisfy the

Hooke’s law and small strain theory; 3) The composite

cylinder is constructed of multilayered laminates bonded

together p erfectly; 4) Th e medium is initiall y undisturb ed,

and without body forces and internal heat sources; 5) The

medium is applied by a force, which is the function of

time; 6) The temperature at inner layer and outer layer

are the functions of time.

We now consider an infinitely long annular cylinder

*Corresponding a uthor.

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opyright © 2013 SciRes. WJM