International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 2 Issue: 4 | July-2015 www.irjet.net p-ISSN: 2395-0072
© 2015, IRJET.NET- All Rights Reserved Page 230
RESPONSE OF GENERALIZED THERMOELASTIC INFINITE MEDIUM DUE
TO PERIODICALLY VARYING HEAT SOURCES
Nihar Sarkar
Purba Banbania Bhagabati Vidhyamandir,
Habra, 24-PGS (N), West Bengal, India
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Abstract - A one-dimensional problem for a
homogeneous isotropic thermoelastic infinite medium
subjected to a periodically varying heat sources on
the boundary of the space is considered in the context of
Lord & Shulaman model of linear theory of generalized
thermoelasticity. The Laplace transform together with
an eigenvalue approach technique is used to find the
solutions for the field variables in transformed domain.
The transformed solutions are inverted using the
Zakian algorithm. Numerical results for the
temperature, displacement and stress distributions are
presented graphically and analyzed.
Key Words: Lord & Shulman model, Periodically
varying heat sources, Laplace transform, Eigenvalue
approach.
1. INTRODUCTION
Biot [1] introduced the theory of coupled thermoelasticity
(CTE) to overcome the first shortcoming in the classical
uncoupled theory of thermoelasticity where it predicts
two phenomena not compatible with physical
observations. First, the equation of heat conduction of this
theory does not contain any elastic terms. Second, the heat
equation is of a parabolic type, predicting infinite speeds
of propagation for heat waves. The governing equations
for Biot theory are coupled, eliminating the first paradox
of the classical theory. However, both theories share the
second shortcoming since the heat equation for the
coupled theory is also parabolic.
Lord & Shulman [2] (LS model) attempt to eliminate the
paradox of infinite velocity of thermal disturbances
inherent in CTE theory [1]. This model is based on a
modified Fourier’s law of heat conduction but in addition a
single relaxation time was considered. This theory was
extended by Dhaliwal & Sherief [3] to include the
anisotropic case. In LS model, the thermal signal
propagates with finite speed. The heat conduction
equation in this model is of hyperbolic type and is closely
connected with the theories of second sound.
Saleh [4] have studied a one-dimensional problem in
generalized thermoelasticity subjected to a heat sources.
Youssef [5] studied a two-temperature generalized
thermoelastic medium subjected to a moving heat source
and ramp-type heating by state-space approach. Youssef
[6] also studied generalized thermoelastic infinite medium
with cylindrical cavity subjected to moving heat source.
Othman et al. [7] studied transient disturbance in a half-
space under generalized magneto-thermoelasticity with
internal heat source.
In the present research, we consider a one-dimensional
problem for a thermoelastic infinite medium in the context
of LS model of generalized thermoelasticity subjected to a
periodically varying heat sources. The Laplace transform
together with an eigenvalue approach [8, 9, 10] technique
is used to find the solutions for the field variables in
transformed domain. The transformed solutions are
inverted using the Zakian algorithm [11]. Numerical
results for the temperature, displacement and stress
distributions are presented graphically and analyzed.
2. BASIC EQUATIONS AND FORMULATION OF THE
PROBLEM
Following Lord & Shulman [2], the governing equations
for a homogeneous isotropic thermoelastic material can be
written in the following form:
The constitutive equations are:
0 0 ,
2 ( - ) , (1)
. (2)
ij ij kk ij
E kk
e e
T C Tu
Fourier’s law in the theory of generalized fractional heat
conduction is taken from [25] as:
0 ,
. (3)
i
i i
q
q k
t