Internat. J. Math. & Math. Sci.
VOL. 14 NO. 3 (1991) 587-594
587
A PROBLEM OF THERMAL SHOCK WITH THERMAL RELAXATION
D. RAMAMURTHY and A.V. MANOHARA SHARMA
Department of Mathematics
University College of Science
Osmanla University
Hyderabad 500 007
A.P., India
(Received May 12, 1988)
ABSTRACT.
plane
The problem of a semi-infinite medium subjected to thermal shock on its
boundary
expressions
is
for
solved
using
temperature,
the
strain
generalized
theory
and
are
stress
of
thermoelastlclty.
The
The
are
presented.
results
exhibited graphically and compared with previous results.
KEY WORDS AND PHRASES.
Generalized theory, relaxation time,
Lame’s elastic constants,
diffusion equation, relaxation constant.
1980 AMS SUBJECT CLASSIFICATION CODE.
73U.
INTRODUCT ION.
Thermoelastic problems are solved using the governing dynamical equations for the
displacement and temperature.
These equations are two partial differential equations.
Equation of motion:
Pui--
( + )
uj,ij
+
(3k +
Bui,jj
(I.I)
2)aT,i.
Equation of energy:
k
T,ll
p
CeT +
(3 + 2) a
To k"
(1.2)
The first equation is of wave typ and the other is of diffusion type.
For an
isotropic, homogeneous elastic body subjected to a shock, the latter equation shows
that the disturbance will be felt instantaneously at distances far from its source.
As
the
equations
displacement.
are
coupled
this
effect
will
be
felt
in
both
temperature
and
Such a behavior is physically inadmissible and contradicts the existing
theory of heat conduction.
Baumlster and
Many researchers, for example Morse and Feshbach [I], Boley [2],
in the
Hamill [3] have discussed this paradox and suggested some modifications
of
Lord and Shulman [4] proposed the Generalized theory
governing equations
wave-relaxation
thermal
of
thermoelasticity, where the time lag needed for the onset
588
D. RAMAMURTHY AND A,V.M. SHARMA
time
is
considered
equation.
Based on
is
this
modified
authors (Fox [5], Ignaczak
Here
the
welt-known
and
as
theory,
[6], Sherief
the
modified
coupled
considerable work is
[7],
and Anwar
heat
conduction
being done by many
and Choudhurl and Sain [8]).
problem of an isotropic,
homogeneous half-space subjected Lo thermal
shock on its plane boundary is solved using the Laplace transform technique.
The
equations concerning the generalized theory of thermoelasticlty are used to solve the
said problem.
a
heating,
The boundary condition for temperature
more
realistic
situation.
s
in the form of exponential
After
effect[ng the Laplace inversion, the
expressions for temperature, strain and stress are obtained.
As a special case, the
results due to Danilovskaya [9] for step-type boundary condftlons and that of
Sternburg et al [I0] for ramp type boundary condition can be obta|ned.
Further by
setting
relaxation constant
zero,
to
the
results due
to
Dalmaruya et
al
[I l] are
obtalned.
2.
FORMULATION OF THE PROBLEM.
Consider an Isotropic homogeneous half space, subjected to a thermal disturbance
-
on its boundary. The governing equatlons of the generalized theory of thermoelastlclty
for the one dimensional case, are
k
-Sx-
axx
T
where k, O, C
T
(+
PC e
(I +
u
2U)----
_2T2)
T
+ (31 + 2)
8t
32u
3u
To ( + o 8xSt -)
(31 + 2) o (T-T)
o
(2.2)
32 u
T
x2
(2.1)
(2.3)
t2
,
T are thermal conductivity, density, specific heat, coefficient of
e
o
linear thermal expansion and the relaxation time, respectively.
I and
are the well
known
Lame’s elastic constants.
Here
T,
Oxx
and
u
are
temperature,
stress
and
displacement, respectively.
The inltal and boundary conditions are
T(x,O)
T
u(x,0)
0
T(0 t)
T
u’(0,t)
x
o
>
0
T
(u__)
3t t--0’ (-) t=0
o
0
(2.4)
[l-exp(-t/t o )]
(31 + 2)aT
o
(I + 2) [I -exp
(-t/to)]"
(2.5)
In the above,
and t o are constants.
The step type boundary condition is
obtained when t
i.e.
0,
T(0,t)
H(t), and the ramp type boundary condition is
o
obtained by expanding the exponential type and neglecting the higher order terms.
Here H(t) is Heavlside unit step function.
To,
The regularity boundary conditions are
T(x,t), u(x,t),
xx
(x,t)
0 as x
.
(2.6)
THERMAL SHOCK WITH THERMAL RELAXATION
589
Introducing the following non-dimensional var[able.q
0 Ce
2)I/2
+
z
(
e
------,o
p
k
S
o
O
(2.1)
e
-k--
0
(3 + ) a
xx
t,
ro
(-----)
(3, + 2V)a T
O
k
e_]
u
stress, and displacement respectlvely in equations
temperature,
time,
(2.7)
p C
+ 21a 3/2
[p
for distance,
p C
T
T
U
2
%+
x, Y
(2.3), we get
e"
e’
U"
@’
(u’ + u’),
B
,
(2.8abc)
E=U’
where ’dot’ and ’dash’ denote differentiation with respect to y and z respectively and
(3 +P 21J_) PCe TO’
-"’--
I
Here B and e are
respectively.
a2
2
T
(3 + 2U)
O
(X + 21) p C e
"
constant
and
In (2.8a), the strain acceleration term
(")
the
thermoelastlc
u
(-)y=O
U(z,O)
.e) y=O’
--y
0
(2.9)
--
exp(-y/’ O
9(z,y), U(z,y), r.(z,y) 0
where
3.
is
O
U’(0,y)
r’ O
.B
(2.6) reduce to
exp (-y/T’)
EKO,y)
constant,
can be ignored as the product
Now our initial and boundary conditions (2.4)
e(z,O)
coupling
in the intermediate and room temperature.
or
much less than
relaxation
X._+
2p
P
Ce
t
as z-
O
SOLUTION OF THE PROBLEM.
In order to obtain the solutions for the temperature (e) and strains (U’), we
U, from the first two equations of (2.8), namely
first eliminate
.....
.....
"-
e
(1 +)
U
(1 + ) U"
(1 +g) e"+e+
(1
/
) U"
/
U
/
e =o
(3.1)
U
(3.2)
0
Applying the Laplace transform to (3.1), (3.2) and (2.9), one obtafns
590
.....
.....
Here O,
D. RAMAMURTHY AND A.V.M. SHARMA
p[p(l+B) +
(I+)]
"
+
p2(p+Bp2)
0
p[p(I+B) +
(1+)]
"
+
p2(p+13p2)
O.
U are Laplace
Lransforms
(3.3)
U
of 0 and
respectively
and
p
Is
the
transform
parameter, and
e (0,p)
’/p
O
O
’(0,p)
I/pp
(z,0)
(z,0)
(3.4)
0.
In the above,
po
’
=p
+ I.
O
From (3.3) and (3.4), we get
0
A
exp
U
BI
exp (-a
(-ct2z)
lz)
+ A exp
2
(-a2z)
+ B
(-a2z)
exp
2
where
al, 2
BI, 2
D[e
--Po
(a21_ a22)
p--
+ Bp
2
2 [(l+13)p + (1+)
al,2
2
Z’o PP + p
:1:
a2,! /p]
[((1+8)p +
(1+-)) 2
4pp]
1/2
i-
We consider a special case in which the relaxation constant (B) is expressed in
For this value of B, we get
terms of the coupling constant e, i.e.
l+e
2
a
p
2
+ p/I3, a2
2
IBp
2
(3.6)
and then the transformed temperature and strain become
z,p)
U’(z,p)
N(p)
[(-z
N(p)[[e +
+ (I +
where
p)exp(-al z)-(p2"r’(o l-IB)+’Po)eXp(-a2z)
p(l-B-z)
p’op)
exp
BoZ’p
(-a2z)]
2
exp
(3.7)
(-al z)
(3.8)
THERMAL SHOCK WITH THERMAL RELAXATION
N(p)
b2)
+
PPo (p
591
(l-B)
and
b
2
S)
S(I
Inverting (3.7) and (3.8), we get
CII(O)
(R)
+ C
+
2
+ C3
exp(-y/) l(I/z)
C4 1 (l/z)
+ C
exp(-b2y)l(b 2)
l (b2) + C62(1/’) + C72(b 2)
5
and
U’
ZlI() + K2I(I/z )
K3I(b 2)
+
+
I (b2) + K6(I/z)
+ K5
(3.10)
K4I(I/z)
K72(b 2)
+
where
Y
z exp[(w-
l(w)
o
S) S
(I
C
C3 =-S(b2 Z’o
C
5
-C3/DI,
K
D
C
+
C
K1 Cl’ K2
K
DID2D3 b2
3
K
b2 DID
2
DID2’
6
D 2,
b2(Z’o
exp
b2S D2D 3 exp
7
b2S D2D3/
2
1/(1-b2"),
[I
F(z,) H(-z)d
)
5
l(W)
)]
C4 "-C2/DI
C
S)
(-b2y)/
B)2,
D
3
=-D3.D2/(I-B)
7
DID2
exp(-y/’)
K
D
6
B)
--D2/(b2" (1
S)
I/(144 S
Ix(Z’/2S)/DIZ’
H(y-z)
O,
0
1,
y>z
<
y
<
D
o
Here I
exp(-z/2S)
2)
2)
2(w)
[!
expI(/z
z
and
Z’
2
(l-B)2 + "o
exp[(z-y)w]] H(y-z)lw,
F(z,y)
-b2(z’o
,
K4
(l-B),
(-z/2B),
(1
2
2 I/2
(y2-z)
is the modified Bessel function of fi[t kiod.
y)wll H(y-/z)/w
R. RAMAMURTHY AND A.V.M. SHARMA
592
The
Eor
expression
be
can
stress
(2.8c),
f-ore
obtained
(3.9)
0,
0, recovers the results due to Dan[tovskay.i [9] and seLLing
recovers the results of Datmaruya Ill].
Taking
4.
(3.10).
and
’o
RESULTS ND DISCUSSION.
results
The
-
temperature,
for
and
strain
numerically and exhibited graphically in figures
relaxation constant (8) given by
I/(l+).
stress
distributions
ate
evaluated
to 3, for a particular value of the
The transport of thermal energy In the
process or a wave Like process depends on the
It was observed that at low temperatures the
magnitude of the relaxation constant.
magnitude of the relaxation constant becomes signif[cant and the energy equal ion
medium,
predicts
i.e.
a
either
wave-type
a
diffusion
The
phenomenon.
magnitudes
of
the
coupling and
relaxatlon
constants were calculated over a range of Intermediale and high temperatures by Lord
are smaller than unity for moat of the
[|2].
The values of the coupling paramter
materials.
ller for the computation, the relaxation constant (8) was taken as 0.98 and 0.76
(the corresponding values of e are 0.I and 0.31 respectively) and the values 0.25,
and 2 for the ’. The tlme-dependence of the non-dimensional temperature (0),
0.5
o
strain (U’), and stress (Y.)are depleted as a function of non-dlmenslonal time y at
2 for
the non-dlmenslonal distance z
’
o
0.5.
As the relaxation constant increases the correspoading components of temperature,
strain and stress decrease. The gap between these corresponding parameters increases,
It may be
due to the effect of relaxation time unlike that of coupled theory.
coupled
of
case
In
entloned that a similar phenomena was observed by Dalmaruya Ill]
theory.
Moreover
due
to
presence
of
Heavfslde
unit
step
function,
discontinuities can occur in temperature and stress at the wavefronts y
the
two
z and
y-
z and the corresponding accouatlc velocity (v I) and thermal velocity (v 2) at
and I/ respectively. Last but not least, the results obtained
the wavefronts are
here which including the effect of the relaxation time are more general.
ACKNOWLEDGEMENT.
The
second
author wishes
to
thank the
Councll
of
Scientific and
Research, New Delhl, for the flnanclal assistance given for his work.
Industrlal
593
THERMAL SHOCK WITH THERMAL RELAXATION
54.3
0
2
3
4
5
6
7
S
9
TIME
Fig.
Fig.
I.
Time dependence of
temperature at z
2. Time dependence of strain at z
2.
2.
I0
11
12
594
R. RAHAHURTHY AND A.V.M. SHARHA
3
4
5
6
7
8
9
10
11
-0.37
-2.77
Fig.
3.
Time dependence of
stress at z
2.
REFERENCES
I.
MORSE, P.M. and FESHBACH, H., Methods of Theoretical Physics, McGraw-Hill, New
York, 1953.
2.
BOLEY, B.A., The Analysis of Problems of Heat Conduction, Pergamon Press, New
York, 1964.
BAUMISTER, K.J. and HAMILL, T.D., Hyperbolic Heat Conduction Equation, Jnl. of
Heat Transfer, ASME Trans. 91 (1969), 543-548.
4. LORD, H.W. and SHULMAN, Y., A Generallzed Dynamical Theory of Thermoelastlcity,
J. Mech. Phys. Solids 15 (1967), 299-309.
5. FOX, N., Generallzed Thermoelastlcity, Int. J. Engg. Scl. 7 (1969), 437-445.
6. IGNACZAK, J., A strong discontinuity wave in Thermoelastlcity with Relaxation
Times, Jnl. of Thermal stresses 8 (1985), 25-40.
7. SHERIEF, H.H. and ANWAR, N., Problem in Generalized Thermoelastlclty., Jnl. of
Thermal Stresses 9 (1986), 165-181.
8. ROY CHOUDHURI, S.K. and SAIN, G., Instantaneous Heat Sources...With Thermal
relaxation., Ind. J. Pure. Appl. Math. 13 (1982), 1340-1353.
9. DANILOVSKAYA, V.I., Thermal Stresses in an Elastic Half-space
Prikl. Mat.
Mech. 14 (1950), 816-820.
I0. STERNBERG, E. and CHAKRAVORTHY, J.G., On Inertia Effects in a Transient
Thermoelastic Problem, Trans. ASME. Set. E. 26 (1959), 503.
II. DAIMARUYA, M. and NATTO, N., A Transient Coupled Thermoelastic Problem in the
Semi-lnflnlte Medium. Bull. JSME. 16 (1973), 1494-1505.
12. LORD, H.W., Ph.D. Dissertation, North Western University, Evanstan, 1966.
3.