567
Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 567-578
A COUPLED MAGNETO-THERMO-ELASTIC PROBLEM IN A PERFECTLY
CONDUCTING ELASTIC HALF-SPACE WITH THERMAL RELAXATION
S.K. ROY-CHOUDHURI
Department of Mathematics
Burdwan University
West Bengal, India
and
GARGI CHATTERJEE (ROY)
R.B.C. College, Nalhatl
West Bengal, India
(Received January 12, 1988 and in revised form August 14, 1989)
In the present paper we consider the magneto-thermo-elastic wave produced
Here the Lordby a thermal shock in a perfectly conducting elastic half-space.
Shulman theory of thermoelasticity [I] is used to account for the interaction between
ABSTRACT.
the elastic and thermal fields.
The solution obtained in analytical form reduces to
those of Kaliski and Nowacki [2] when the coupling between the temperature and strain
fields and the relaxation time are neglected.
The results also agree with those of
Massalas and DaLamangas [3] in absence of the thermal relaxation time.
KEY WORDS AND PHRASES.
Magneto-thermoelastic wave; Thermal relaxation time.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
magneto-thermo-elastlc
disturbances generated by a thermal shock in a perfectly conducting elastic half-space
It was assumed that both in the medium and in the vacuum
in contact with a vacuum.
there acted an initial magnetic field parallel to the plane boundary of the half-space
Kaliski
and
Nowacki
[2]
investigated
the
problem
of
and there was no influence of coupling between temperature and strain fields.
Later, Massalas and Dalamangas [3] considered the same problem where the coupling
Very recently Chatterjee
between the temperature and strain fields was considered.
and Roy Choudhuri [4] extended the problem [3] in generalized thermo-elasticity of
Green and Lindsay taking into account the two relaxation times.
using
In the present paper we extend the problem [3] in generalized thermoelasticity by
The solutions for
the thermal relaxation time of Lord-Shulman theory [I].
temperature distribution, deformation and perturbed magnetic field in the vacuum are
obtained in analytical form in the first power of the magnetothermo-elastlc coupling
parameter e and relaxation parameter T ’.
In absence of e, T the solutions agree
o
o
with those in [2] and in absence of
’,
the results agree with those in [3].
o
568
S. K. ROY-CHOUDHURI AND G. CHATTERJEE
Surface stress for different times is calculated and graphically presented.
It
believed that this particular problem has not been considered earlier.
2.
PROBLEM FORMULATION.
We assume that a magneto-thermo-elastic wave is produced in an elastic half> 0 due to the thermal shock
where
is a
applied on
space x
0(o,t)--oH(t)
constant and H(t)
there
an
is
the Heavlside function.
is
initial
magnetic
field
Xl--0
0o
We also assume that in both the medla
acting
in
the
of
direction
x3-axls.
The
simplified equations of slowly movlng bodies in electrodynamlcs after llnearlzatlon
are the fol[owlng:
x =--
%
.=o,
where
(2 )
t
c
denotes the electric field,
is the perturbation of the magnetic field,
the initial constant magnetic field,
displacement vector,
and
c
the
is
o
is
velocity of
is
O
is
the current density vector, u denotes the
the magnetic permeability, o is the electric conductlvtty
The displacement equation of motion in thermo-
light.
elasticity including the electromagnetic effect after llnerlzatlon is,
v2 +
IJo
(.) + [(h5
(+.)
x
o
-o
(2.2)
o
Also the modified form of Fourler’s law of heat conduction taking into account the
thermal relaxation time [I] is
pc.( + "o ) + ?ro( +
1/2A)
K
0’it’
(2.3)
(i=1,2,3)
where X, are the Lame’ constants, T is equal to (3k+4) oT, o is the co-efflclent
T
of linear thermal expanslon,
is equal to
are the reference and absolute
T-To; To’T
temperature of the body respectively; K is the co-efflclent of heat conduction; p is
the mass density;
time.
c
is the specific heat at constant volume;
T
The magneto-thermo-elastlc wave propagated in the medium x
depend on x
For
is the relaxation
> 0
is assumed to
and time t.
o
f=
Equations
O
(0,O,H 3) equations (2.1) reduce to
UoH3( o
C
c
o)
tl0
(o,o
E2
(2.2) and (2.3) then lead to
2
X+2p + aop)
2u
_--yx
)0
y--t
,cu
),
.=c
3
x
0).
(2.4)
COUPLED MAGNETO-THERMO-ELASTIC PROBLEM IN A HALF-SPACE
a
cv(- +
where
+
---at2)
z
a2Ul
YTo(-l
+ z
a3u-l-)
x
at
the Alfven wave velocity.
is
u, x
notations u
2
2
2
a
K
x
569
0
2
(2.6)
For convenience, we shall use the
x.
In vacuum the system of equations of electrodynamics are
a2 .)
a2
x
’2
c
2
at
2
o
0
h3
(2.7)
2
a
x ’2
where x’
a
c
2
2
at
o
0
E2
5
x.
The components
Tll
and
Tll of Maxwell’s
stress tensor in elastic medium and in vacuum
are
o
4--’-
Tll
h
and T
3H3
o
o
h3H3"
11
The normal mechanical and thermal stress is
Oll
The boundary conditions to be satisifed are
Oll +
E
2
Z
o
2
(o,t)
3.
Tl
Tll
O, x
x’
0
(2.8)
x
x’
0
(2.9)
0 H(t).
o
(2.10)
SOLUTION OF THE PROBLEM.
To find the solution of the problem we now
introduce the following notations and
non-dlmensional variables
2
2
Cl
o
C2
o
2
ao
2
+ Cl’
CXo
Cto
"--;-’ z---;-,
S. K. ROY-CHOUDHURI AND G. CHATTERJEE
570
K
(2.7) and boundary conditions (2.8)
The equations (2.5)
a2U
)2U
aZ
}2Z
_,
3Z
0,
}2Z
(3.1)
>0
a3u
dU
2
-o,
>o
(3.2)
20
h
o
h3
3
22
2
}- - U
(2.10) become
z
32 U
+
0,
-
131 Oh
3
ab 3
-
>
(3.3)
0
’
0
(3.4)
o,
(3.5)
H(T),
(3.6)
0
Z(o,T)
o
where
Initial conditions in the new variables are
u(:,o)
o, z(:,o)
o, az(;,.o_)
o.
We now introduce a potential function @ defined by
(3.7)
Using (3.7) in (3.1) and then integrating we get
z(,)
32
(--f
)2 )
z2
In
>
O.
(3.8)
COUPLED MAGNETO-THERMO-ELASTIC PROBLEM IN A HALF-SPACE
571
Using (3.7), the equation (3.2) leads to
2z
z
@2
)T
T’o
2z
2
3 i--
T’o
@2
4
O
@2
(3.9)
In the Laplace transform domain the equations (3.8), (3.9) and (3.3) become
a2
(-2-s) ,
2
z(,s)
()C
’o s
-s
z
- -3o
c
3
e Bs
>
’
(3.10)
s(l+’S)o @,2’
>
{
o
>
0
(3.11)
0
(3.12)
In Laplace transform domain, the boundary conditions (3.4) -(3.6) reduce to
+
2
B1f3= o
h3
%_
=o
(3.13)
’
O, {
(3.14)
0
e
Z(o,s)
(3.15)
o
Ellmtnating
from (3.10) and (3.11) we get
@2
(3.16)
s
The equation (3.16) reduces to (31) in [4] on setting
The general solution of (3.16) vanishing at
_X
(,s)
where 1I,
2
+ C2
Cle
e
-X2
is
>
0
(3.17)
are given by the roots of the equation
x4
x2 + 3(+ )
s {+++(+)
o.
(3.18)
Hence
[S{s+l+e+Ts+s) :I:
+ 2(- I+2E:T’ + T’ +
0
0
[(1+2T’2o
2T,)
0
S
+
+
,2o
+
2E’o + 2’o
(1+)2[/2}/2’"
2
2 )s2
(3.19)
S. K. ROY-CHOUDHURI AND G. CHATTERJEE
572
The
(3.19)
equation
For a’
[3].
agrees
with
that
(34)
of
[4]
in
for a’fa*’=
’.
o
0 ’ffi 0, the equations (3.16), (3.19) are in agreement with that of (24) in
Thus the equations (3.1), (3.2), (3.16), (3.19) are more general in the sense
that they incorporate the effect of thermal relaxation time of Lord-Shulman theory.
From (3.10) using (3.17) we have
z(,s)
C
1()‘1
S
C2(
e
s
>
e
O.
(3.20)
From the boundary conditions (3.13) -(3.15) taking into account (3.17) and (3.20)we
obtain a linear algebraic system with respect to C
C
C
1, 2 and 3 as
C1 s2 + C2 s2 +BlC 3
B2S)‘lC + 62s)‘2C 2
2
c1()‘
The
for
constants
,Z, U,
s
2
Ci(iffil,2,3)
h3
+c2(
being
0,
(3.21)
0
(3.22)
O, at {- {’ffi 0
6 C3
s
’
at
0
o
2
(3.23)
"T o s
determined
by
(3.23),
(3.21)
the
solutions
are given by
-h
(s+6162),2) e
-[o
2
s()‘1 1/2 (6182s
O
( {,s, e, )
(s6 +
+
6162),1
6()‘1 +1/2)s
+
e
-1/2
(3.24)
6162)‘1)‘2
(3.25)
({,s
0
e,’)
o
o
h3( ’s, , ’)o ToO
Since E,
into
’o <
e
2
s62 e -I’
2
+
)‘1(s6 + 6162)‘2
B(x + ) + BB2
’>
e
{> 0
O.
for small thermo-elastlc coupllngs, we expand the
functions
Maclaurlan’s
obtain
)‘2(s6 + 6162)‘1
s()‘1-2 (BB2
(3 26)
(3.27)
,,3
series and retain the first two terms in the
series expansion to
COUPLED MAGNETO-THERMO-ELASTIC PROBLEM IN A HALF-SPACE
-s
--S+
e
e
(,s,,r’)--[
o
o
6
2(
+
62
fl e
(6 + 8162) (s-1)2
{
+_e
2 /(sI)
8+61 82)s(s_l
2
573
e
821
(s-l)
2
+
s(s-l)
2
e
2s (/s--- I)
2
2(
s+6! 82)s(s+)
2
I2e- L }]
o
(3.28)
s--
-s
6e
62e
+e{(s-l)
(8+61 62)s
2( +61 )s/’(s- 1)2
2(
+61 ) ,s
2(
(s-l)
2s
s(s-l) (,/-s-1)
(-61 B2)s Vs(s-l)(V-B-I)
2(B+61 82)s(s-l) (s-1)
2(8+61 82)2s(s-1) (s+l)
2
{
2(
8+61 62) 28 -s(s- V--I6
e-
(’6!62
3(’,s,e,
(s-l) 2
+
+61 62),/-s(s_ 1)2
=o
2(’61 2)s,s(s-1) (,’+I)
61 62
2(s-l)
2(s-l)
6 e
e
Zo ’)
2}+’o
2
2
2(
e
B+61 82)/6(s-l)
(3.29)
}]
-m{’
-Ss{’
62 e
(+6182) s(s+l)
82" e-BS
e
662e
2(’6162 )2 s(/’s/1)
s
To’
2(
2]
8+61 B2)(/’s+I)
(3.30)
Taking inverse laplace transform we obtain (Chatterjee (Roy) and Roy Choudhurl [4],
Hetnarskl [5], Oberhettlner and Badll [6]),
S. K. ROY-CHOUDHURI AND G. CHATTERJEE
574
+ (---
)
)erf /-’-]
e
H(r-) +
I
’
)
f2
’
)
fl
’
) + erfc(
8182
2(8+81
2(
5/2 e
II-81B2)
o
U(,I,e,T)
81132
[e (r-) erf
erfc
B
2(
(T--I) e
8+81152
-[ 12) fz(
,
+
+ 4
f4 (, )
)
f,
"
24’--e (--) +
f2(,)-
o
e
r-_
8
)1 + 2(I1"8
T-)H(r-)
I [Tf
2
r-_ )H(
T- )
T- )e
f2 (,T)
(,T)
2(8+8[ 82 If3
(,T)-
+ 2(IBI IB2
8+8182)
-(7 erfc(i_)(r-)
(e
2
+ 2( IB
-I-8182)
, )rc()
(3.31)
8
erfc
H(-) + e{-
2
4.r}]
f
2 (,T)-
r-OH
’+
f
(,T)+ erfc
2
81 82
)+ 2(
I-B 182)
1
’
-) e (1-) erfc --]
(2r--2)e (r-)erfc(i_
2:
/T)
T)
H( r- )
COUPLED
MAGNETO-THERMO-ELASTIC
5
+ -(
-0+(-) 2_2 ](r-) +
82
)
2(
_32
5
(’t- K)erf/-+ gC-K-)e
((r-K) +
2
[-
575
PROBLEM IN A HALF-SPACE
3
(r-K-)
F_
(SrSK-3)e
"
(r-Kl2lerf/- +-3
(r-K) +
(r-K)
(r-K)2-11
2
If (K )
r+K)erfc(_____ +/)
4-: e
2
2(
4-1 82
(r-K-
2
+ ((r-F.) + (r-K) 2)
81 82
2(8+% 82
e
h3(
,,
[.
o
+ [l-2(r-SK’) 2]
2(r-SK’)
82
eCr-sK’)erfc/r-SK
e(r-SK’)erfc/r-SK
"
-
fi(K,T),
[e-erfc(
+
-
2/r-BK’]
+
e
,
{" f2
o
K)H( r )
’
o
(K,’r)
+
%rfc(
---_ + )]
2(r-BK’)
{-
82
{- 2
8+
H(r-SK’)}].
/
(3.32)
H(’r-SK’)
i-1,2,3,4,5 are given by
)
+
H(r-K)}].
e(r-SK’)erfc/r-SK H(r-SK’)} +
where the functions
f i( K, T)
r K)e
’812
K)erf +
0
o
B
/- H(T-K)
)e
"
11
r- K)er
f
(7"r’-7 K- --)e
(-K)-(-K)2+2]H(r-K)} +
erf
r K) erf
[((r-K)
(7-TK-5)e (r-K)
9
-)
81 82
[/7(
’
r- 8K’
(3.33)
S. K. ROY-CHOUDHURI AND G. CHATTERJEE
576
2
3 (’t)
+ 2
eric
+ (2
2
f3
(’T)
eric
w
(2T+-I) e (r+Oerfc(--_ +
e
2
f4 (, )
f em{2
o
f5 (’)
Sem
o
+ [2(z-m)-]
e
Oerf
g-l)e
e
(z--m)erfc(
-) }dm
2
2
where
)
{2
e
4(r-m)
erfx and erfcx denote
[2(T-m)+] e
the error
rb-m)erfc(-
+
/--) }din
function and complementary error
function
respectively.
4.
NUMERICAL RESULT.
The surface stress is given by
Tl
4,z
l-erf 2(I+83
e
%H3
{- 2
s(z+3)
eZ(l-erf/) +
-
+
x
(I-2:2)e ( l-erf)
}
where
If
there
field, H 3
no
is
0,
coupling
0, 83
between
the
electromagnetic
0 and 8 is finite so that
o
TII
0 on
field
and
a0
In presence of the electomagnetlc field and strain field, the surface stress
by
T
o
11
8oH 3
82
4.To
(1+3)
’
X( T, e,
o
strain
is given
COUPLED MAGNETO-THERMO-ELASTIC PROBLEM IN A HALF-SPACE
577
.05.
and
are finite. We take
and
We can assume _B <<
since c >>
3
For numerical calculation we take the material of the half-space to be copper for
If we assume that a representative value of the relaxation
0.0168.
which e
-11
time
is 10
(see [7]), then the non-dimensional thermal wave speed in copper
ao
Co
B3
should be approximately equal to 0.66.
Then
[8]).
’
o
2.3 (For thermal properties and sound wave speed in copper, see ref.
Surface stress X for various values of times
are exhibited In the following
table and also graphically represented.
"r
1.5
2.0
2.5
3.5
4.0
4.5
5.0
5.5
6.0
-IOX
5.4
7.7
11.8
16.5
23.2
25.0
26.8
28.3
30.0
31.5
d/v/s/on
S.K. ROY-CHOUDHUII AND G. CHATTERJEE
578
REFERENCES
1.
LORD, H.W. and SHULMAN, Y., A Generalized Dynamical Theory of Thermoelastlclty,
J. Mech. Phys. Solids 15 (1967).
2.
KALISKI, S. and NOWACKI, W., Excitation of Mechanlcal-electromagnetlc Waves
Induced by a Thermal Shock, Bull. Acad. Polon. Scl., Series Scl. Tech., I,
I0, (1962).
3.
MASSALAS,
4.
CHATTERJEE,
C. and DALAMANGAS, A., Coupled Magneto-thermoelastlc
Elastic Half-space, Lett, Appl. Engg, Scl. 21(2), (1983).
Problem
in
G. (Roy), and ROY CHOUDHURI, S.K., Coupled Magneto-thermoelastlc
Appll. Engg.
Problem In Elastic Half-space with Two Relaxation Times,
Scl. 23(9), (1985).
Le.tt.
5.
HETNARSKI, R.B., Solution of the Coupled Problem of Thermoelasticlty in the Form
of Series of Functions, Arch. Mech. Stos, 4(16), (1964).
6.
BADII, L. and OBERHETTINGER, F., Tables of Laplace Transforms, Sprlnger-Verlag,
1973.
7.
NAYFEH, A.H. and NEMAT-NASSER, S., Electromagento-thermoelastlc Plane Waves in
Solids with Thermal Relaxation, Jr. of Applied Mechanics 39(I) (1972).
8.
AMERICAN INSTITUTE OF PHYSICS Handbook McGraw-Hill, New York, 1957.