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Transient generalized thermoelastic waves in transversely isotropic medium with a cylindrical hole
0020-7225/87 $3.OO+O.lXl Pergamon Journals Ltd
Inc. J. Engng Sci. Vol. 25, No. 4, pp. 463-471, 1987 Printed in Great Britain
TRANSIENT GENERALIZED THERMOELASTIC WAVES IN TRANSVERSELY ISOTROPIC MEDIUM WITH A CYLINDRICAL HOLE JAGAN NATH SHARMA Department of Mathematics, Regional Engineering College, Hamirpur 177001, India Abstract-The distribution of temperature, displacement, and stress in an infinite homogeneous transversely isotropic elastic solid having a cylindrical hole has been investigated by taking (i) unit step in stress and zero temperature change, and (ii) unit step in temperature and zero stress, at the boundary of the cylindrical hole. The Laplace transform on time has been used to obtain the solutions. Because of the short duration of the second sound effects, the small time approximations have been considered. The temperature and stress are found to be discontinuous at the wave fronts in case(i) whereas these quantities are continuous in case (ii). However, as expected, the displacement is found to be continuous in both the cases.
1. INTRODUCTION Recently, the generalized theory of thermoelasticity [l] has been extended to anisotropic solids by Dhaliwal and Sherief [2]. The propagation of plane harmonic waves in a generalized thermoelastic homogeneous transversely isotropic medium has been discussed by Singh and Sharma [3]. The propagation of generalized thermoelastic plane harmonic waves in homogeneous anisotropic media was studied by Sharma and Sidhu [4], Chattopadhyay et al. [S] studied the transient solutions due to (i) a step input of stress and zero temperature, and (ii) a step input of temperature and zero stress, on the boundary of the surface of the cylindrical hole in a transversely isotropic thermoelastic medium in the context of coupled thermoelastic theory. In the present article the transient thermoelastic waves have been studied in a homogeneous transversely isotropic infinite medium having a cylindrical hole in the context of generalized theory of thermoelasticity [2].
2. FORMULATION
OF
THE
PROBLEM
We consider an infinitely extended homogeneous transversely isotropic thermoelastic medium having an infinite cylindrical hole of radius a. Let the origin of the cylindrical coordinate system (I, 8, z) be at the axis of the cylindrical hole. We consider the case of radial symmetry so that the non-zero displacement component u = u(r, t). Then the governing field equations of motion and heat conduction in the absence of body forces and heat sources are: Cl 1u.m
K(T,,, + r-‘T,,) -
+
tr-w,,- 81T, = Pi
pC,(F+ T,T) =
TOj?l[ti,, + r-Q
(1-l) + Z&i,, + r-‘ii)]
(1.2)
where B1 = (c 11 + cl&xl + c13a3, cij are isothermal elastic parameters, K is the thermal conductivity, p, c, and to are respectively the density, specific heat at constant strain and thermal relaxation time. The comma notation is used for spatial derivatives and superposed dot represents time derivatives. We define the quantities If = w*&,, z;, = w*to,
t’ = w*t,
u’ =
pw*v,uf~, To, w* = cllCJK,
E = 8:Tol~Cec,,, 463
T’ = T/T,, v; = ClllP
(2)
464
J. N. SHARMA
Here E is the thermoelastic coupling constant, w* is the characteristic medium and up is the velocity of the QL-wave. Introducing the quantities (2) in eqns (l), we obtain u,,, +
T,,, + r-IT,,
- I.-~u - ii = T9,
r-h,,
- (?+
z,,?+) = ~{ti,, + r-‘f
+ Z&
frequency of the
(3.1) + r-Iii)}
(3.2)
where we have suppressed the dashes. The boundary of the cylindrical hole i.e. r = a is given by I
=
w*a/v,
= tj
(say)
(4)
We assume that the medium is at rest and undisturbed initially. The initial and regularity conditions can be written as u=O=T,att=O,r>q,$=O,atr=O and u=O=T,fort=O,whenr+cc
(5)
We take two types of boundary conditions as S,,(& t) = H(r),
(6)
T(rl, r) = 0
and
S,,(%t) = 0,
Th t) = W
(7)
where S,, = u,, + clzu/cllr
- T
(8)
is the dimensionless form of the stress in the radial direction. 3. SOLUTION OF THE PROBLEM Applying the Laplace transform defined by $(r,p) =
Oc+(r,t)CP’dt
(9)
s0 with respect to time, to eqns (3) and using (5), we obtain {D(D + r - ‘) - p2}u = DT
(10.1)
{(D + r- ‘)D - rp2}T = erp’(D + r-‘)ii
(10.2)
where D = d/dr and r = r. + p-l. Simplifying eqns (lo), we get C{D(D + r -I)}’ - (m: + m$)D(D + r-l) + mf&Jii [{(D + r-1)D}2
- (mf + m$D(D + r-‘)mfm:]T=
= 0
(11)
0
(12)
465
Transient generalized thermoelastic waves
where m$ and rn: are the roots of the quadratic equation. m4 - p(A., + l&m2 a1 = (1 + E),
+ rp4 = 0
(13)
a2 = 1 + 70 + &?g
(14)
Solving eqns (11) and (12) and using (5), we obtain
where K,(mir) and K,-,(mir)are modified Bessel’s functions of order one and zero respectively. Using eqns (15) and (M), in eqns (lo), we get Bi =
i=
(p2- mF)Ag'mi,
1,2
(17)
Case I: Step input of stress and constant temperature
We consider a constant stress of magnitude unity suddenly applied on the boundary of the cylind~c~ hole and the temperature at the boundary is kept constant. The boundary conditions in this case are given by eqns (6), which with the help of (9) becomes
grr= Dti
+ c12ii,Jc11r-
T=
-$
T(r,p)=O,
atr=rj
(18)
Equations (15), (16), (17) and (18) lead to
where
A = m2tp2 - m%Amltf)Cm&(m24+ bKlr;,(m2tlYvl - ml(p2 - m%tw)CmlKdml~) + ~&h~Ytll~ Case 11: Step input of temperature
b = (~11- CI~YC~ 1
(20)
and zero stress
We consider a constant temperature of magnitude unity suddenly applied on the boundary of the cylindrical hole and the boundary is kept stress free. The boundary conditions in this case are given by eqns (7), which with the help of (9) transformed to
srr= DU + c12ii,k,,
-
T= 0,
T(r,p) = l/p,
at r = q
(21)
Equations (15), (16), (17) and (21), provide us
AZ= -CmlKotml?J+ ~Kltml~Y~l/pA where A is again given by eqn (20). es x:&P
(22.2)
466
J. N. SHARMA 4. SMALL
TIME-APPROXIMATIONS
Because
the second sound effects are short lived, Green and Lindsay [6] and Green f7], therefore the thermal relaxation time z. is small. So we concentrate our attention on small time-approximations, that is we take p large. The roots m,, i = 1,2 of eqn (13) are
Expanding binomially and retaining positive sign only, we get % =
Pfvi
4i
+
i= I,2
(24)
* (2; - 42,)“2] - r’z
(25.1)
+
o(l/Pl~
where vi.2 = $[;z,
d 1.2 = [a, + (il,,& - 2),&J; - 4ro)‘/2]/2@[&
t (A$ - 4r0)1’2]‘f2
(25.2)
Also
where R = 1; - 4r, = (1 + &To- To)2 “t-4EZ$> 0 Again (1 + erg + ro)’ > R so that ut < u2 Thus v1 corresponds to the speed of the slowest wave and u2 to that of the fastest wave. As a consequence of this, the points of the solid for which T > vzt do not experience any disturbance. From eqns (25) and (26) we see that as r. -+ O,o, -+ 1 and v2 + co. But r. = 0 corresponds to the case of the coupled theory of thermoelasticity, which predicts an infinite speed of heat propagation. We conclude that the wave propagating with speed ol must be the elastic wave influenced by the thermal field. Since u1 < v2, the elastic wave follows the thermal wave. The modified Bessel’s function K”(z) has the asymptotic expansion [83:
&td
”
0 1 E
e-” l +
(4n2 - 12) + (482 - 12)(4n2 - 32) + ... 2!(8zQ2 lf(8z)
1
(27)
For Case I, using expansions (24) and (27) in (19) and then in (15) to (17), we obtain
e-mi(r-q)[(v: -
(vz
-
vf)/v,(v~ - l)p2 + (A, + 3(v: - v:)/S~(v: - 1)) -$
v3u2(v~
-
1)~’ + {A, + 3(vj - ~~)/8~(v~-
1 1
(28)
Transient generalized themmelastic waves
T=
461
01’2e-mlc-v) 5
_
[(of - 1)(v: - V:)/u:(u: - 1)p
- {(v: - l)C& + (0: - @8tl(u: -
1)lh
+ 41u + a@: - 4>/w: - N/P21 +;
112 ,-m2(r-s)[(u; _ I)(0: - u:)/u;(u: - 1)p
0
- ((0: +
42u
l)C~,
+
+
4H4
(u: -
-
4YW4
a44
-
l))/u2
WP21
(29)
where 1, = (u2 - u,){(8b - l)( ulu2 + 1) + u: - 1}/8r/u,(u: - 1) + 2&l$(l 12
=
(~1
- u:,/u,(u; - 1)3 + &(u: + u’,,
(30.1)
~2){(8b - 1)(~1~2 + 1) + U: - l)/8~~2(0~ - 1)
-
(30.2)
+ 2&0:(1 - u:)/u,(u: - 1)3 + &(u: + u:, Now the stress in the transformed form is given by S,kYP) =
EwKAml~) +
+
~~,(~,~)/~I~,
bK,(m,r)/r]A*
-
-
Cm,kdm*~)
(31)
T
Using eqns (24), (27) and (29), we get S,,(r,p) = - (~~2e-m1(rPq)[(2
+ (u; - u;)/8+;
+ (:r*
- 1)p
1
0: )
- l)}/ul - “l(lu+(;;““‘,
e~m2~r~v’[(2 - u$o(: - u$$(uf
- 1)p
r#~*+ (1 - 8b)/8r-i12/u, - $41 + u:)(u: - u:)/u,(u; - 1)
+
-
- uf)(uf - u$/uf(u:
(4
-
NJ.2
+
(4
-
WM~:
-
Wul
II
(32)
-j
Inverting the Laplace transform, we obtain u(r, t) = (&r2emd1R[(u:
- f&t - R/u,)H(t - R/u,)/u,(uf
- 1)
1
- R/u,)/~~(u; - 1)
e-42R[(u: + 3($
-
- uf)(t - R/u,)H(t
- R/u2)/u1(u$ - 1)
~:)(t - R/~2)*fZ(t - R/u~)/~v(u~ - l)]
(33)
468
J. N. SHARMA
-($Jze-+u;- 1xv: -
T@,t) =
- {(u: - l){A + (0: - 6)/8&
v:)H(t - R,v,),v:(o; - 1) - l)}lu,
1 +(&~2e-@'2R[fu; + +I(1 + u:)(u: - u;)/u,(u: - l)}(t - R/u,)H(t - R/u,
l)(u; - uf)H(t - R,u2),u: - 1)
- {(u: - l)C~, + (VI - o:YWd
- 1)1/~2
+ 42(1 + u:)(u: - u:)/u,(u: - l)}(t - R/u,)H(t - R/u,)
e-41R[(uf
1
(34)
- u$o( - uf)H(t - R/u,)/uf(u~ - 1)
+ ($1 + (1 - 8b)/8(R + q) - A,/u, - (u: - I)[& + (u: - u;)/8q(u: - ~)I/vI - AU + o:Xu: - u%,Cd - l))@ - WWW
- R/M
e-b2R[(u$ - uf)(2 - u:)H(t - R/u,)u:(uf - 1) + (42 + (1 - 8b)/8(R + q) - A,/02 - &(l + u$o(: - u:,/u,(u: - 1)
x
Cl2 + (U: -
U:YfW:
-
W~II(~ - W2VW
- R/u,)1
(35)
where R = I - q. For Case II, using expansions (24) and (27) in (22) and then in (15) to (17), we get
‘Z-m1”-q’[U2(Uf
_
-
U;)/p4 + {3U,U2(Uf
-
u;)/8r - ny}/p5]
y 1’2 e-m2”-q’[u,(u~ - uf)p4 + {3u,u,(u~ - uf)/8r - At}/p5],
0r
T(r,p) =
l/2
0 3
(36)
e-m1(r-q)[u2(u:
- u$(uf - l)/u,p3
- {~XU?- 1)/u, + C~,(U: + 1) + (0: - 1)/8rlu2(u: - dI}/~~l _
g 1’2
e-m2”-q)[ul(u~
0I C42(U:
+
+ 1) + (U: -
- uf)(u: - l)/u2p3 - {Ay(u: - 1)/u, W3rlU,(U;
-
u:))/P~I,
(37)
112 S,,@,P)
=
0
;
e-m1(r-q)[u2(uf - u:)(2 - u~)/ulp3{A~(u~ - 2)/u,
+ v,(vf - v:)[~,(v: _
+ 2) + (8b - 2 + vf)/8rl)/p41
y 1’2e-m2(‘-q)[ul(u$ 0r
- uf)(2 - u9/u2p3 + {Az(u: - 2)/u2
+ u,(u$ - u:)[~~(u: + 2) + (8b - 2 + u:)/8rl)/p41
(38)
Transient generalized themoelastic
469
waves
where A: = u,(u: - u;)(u~ + u: + u,u,)/~~(u, + u2) - 2~,u,u: + q&u:(u: + u:,
(39.1)
A.: = u,(u; - u;)(u: + u: + u,uJ/8~(0, + u2) - 2&u2u: + &u:(u: + u:,
(39.2)
Inverting the Laplace transform, we obtain 112 ? ~ n(r, t) = R + ‘t e-01R[u2(uf - u$(c - R/u~)~J!Z(~- R/u,)/6 ( > + {3U,U,(U: - u:)/8(R + q) - A:}@- R/u~)~H(~ - R/u,)/241 - ? ( R+rl )
-
l/2 e-42R[u1(u$ - t$(t - R/u~)~H(~ - R/u,)/6
+ {3U,U,(U;- u:)/8(R + q) - $}(t - R/u~)~H(~ - R/u,)/241 r/2 ? T(r, t) = R + ‘1) embR[u2(uf - I$)($ - l)(t - R/tQ’H(t ( > - {WU: - 1)/U, + +
-
@J:
-
1)/W
+
u2c4
-
Amy:
+
(40)
- R/u,)/2u,
1)
tt)l# - WJ3Wt - WMI
112
- rl ( R+tt >
-
Rlu2)/2~2
+
(6
-
embzR[ul(u: - u:)(u: - l)(t - R/u~)~ZYI(~
-
1)/W
{Mu:
+
-
lVu2
+
0~:
~)l)(t- WA3Wt -
e-41R[u2(u:
-
u:,C&(u:
+
1)
Rlu2)Pl
(41)
- u:)(2 - of)@ - R/u~)~H(~ - R/u,)/2u,
+ ; {n:(u: - 2)/u, + U,(U: - U:)[&(U: + 2) + (8b - 2 + u:)/8(R + q)l}(t - R/uA3H(t- R/u,)1 e-@‘2R[uI(us - ~$2 - u$o( - R/u~)~H(~
- R/u,)/2u, + ;{A:@; - 2)/u, + u,(u; - u:>[#,(u: -t + (8b - 2 + u:)/8(R + q)]}(t - R/r~~)~If(t - R/u,)]
2) (42)
5. DISCUSSION OF THE RESULTS The short time solutions obtained above show that they consist of two waves that is the dilatational wave and the thermal wave travelling with velocities u1 and u2 respectively. The terms containing H(t - R/u,) represent the contribution of the elastic wave in the vicinity of its wave front R = u,t, and terms with H(t - R/u,) represent contribution of the thermal wave in the vicinity of its wave front R = u2t. It is observed that the deformation is continuous on both the wave fronts in Case I as well as Case II. The temperature and
470
J. N. SHARMA
stress are also found to be continuous at both the wave fronts in Case II. These quantities are found to be discontinuous at both the wave fronts in Case I. The discontinuities are:
These jumps in temperature and stress decay exponentially. 6. PARTICULAR CASES (i) When the thermal relaxation time T,, = 0. This case corresponds to the Conventional Coupled theory of thermoelasticity and we have: 11= 1 +&,
A2 = 1,
VI = 1,
02 = co,
$1 = 612,
42
=
00
It is observed that temperature k continuous at both the wave fronts and the stress is continuous at the thermal wave front, but discontinuous at the elastic wave front. This discontinuity is given by
which agrees with the corresponding result obtained by Chattopadhyay et al. [S]. (ii) When the strain and thermal fields are uncoupled to each other. In this case the coupling constant E is identically zero and we have ‘A1= 1,
12 = 1 + zo,
VI = 1,
02 = l/Jr;,
$1 =O,
42
=
v2/2
It is observed that the temperature and stress have infinite jump at the thermal wave front. The temperature is continuous at the elastic wave front but the stress is discontinuous through the quantity. (‘% -
%)R=qt
=
Cd@
+
dl"'
In this case the jump in the stress at the elastic wave front vanishes as the radial distance increases. Acknowkxfgement-The author is thankful to Dr R. S. Sidhu, Department of Mathematics, Guru Nanak Dev University, Amritsar for discussions.
REFERENCES VI H. W. LORD and Y. SHULMAN, A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299-309 (1967). PI R. S. DHALIWAL and H. H. SHERIEF, Generalised thermoelasticity for anisotropic media. Q. Appl. Math. 38, l-8 (1980). E31H. SINGH and J. N. SHARMA, Generalized thermoelastic waves in transversely isotropic media. J. Acoust. Sot. Am. 77, 1046-1053 (1985).
Transient generalized thermoelastic waves
471
[4] J. N. SHARMA and R. S. SIDHU, On the propagation of plane harmonic waves in anisotropic generalized thermoelasticity. ht. J. EngngSci. 24, 1511-1516 (1986). [S] A. CHATTOPADHYAY, A. KESHRI and S. BASE, A coupled thermoelastic problem for an infinite aelotropic medium having a cylindrical hole. Indian J. Pure Appl. Math. 16(7), 807-823 (1985). [6] A. E. GREEN and K. A. LINDSAY, Thermoelasticity. J. Elasticity 2, l-7 (1972). [7] A. E. GREEN, A note on linear thermoeksticity. Mathematih 19, 69-75 (1972). [S] G. N. WATSON, Theory of Bessel Functions, 2nd edn, p. 202. Cambridge University Press (1980). (Receioed 1 July 1986)
Inc. J. Engng Sci. Vol. 25, No. 4, pp. 463-471, 1987 Printed in Great Britain
TRANSIENT GENERALIZED THERMOELASTIC WAVES IN TRANSVERSELY ISOTROPIC MEDIUM WITH A CYLINDRICAL HOLE JAGAN NATH SHARMA Department of Mathematics, Regional Engineering College, Hamirpur 177001, India Abstract-The distribution of temperature, displacement, and stress in an infinite homogeneous transversely isotropic elastic solid having a cylindrical hole has been investigated by taking (i) unit step in stress and zero temperature change, and (ii) unit step in temperature and zero stress, at the boundary of the cylindrical hole. The Laplace transform on time has been used to obtain the solutions. Because of the short duration of the second sound effects, the small time approximations have been considered. The temperature and stress are found to be discontinuous at the wave fronts in case(i) whereas these quantities are continuous in case (ii). However, as expected, the displacement is found to be continuous in both the cases.
1. INTRODUCTION Recently, the generalized theory of thermoelasticity [l] has been extended to anisotropic solids by Dhaliwal and Sherief [2]. The propagation of plane harmonic waves in a generalized thermoelastic homogeneous transversely isotropic medium has been discussed by Singh and Sharma [3]. The propagation of generalized thermoelastic plane harmonic waves in homogeneous anisotropic media was studied by Sharma and Sidhu [4], Chattopadhyay et al. [S] studied the transient solutions due to (i) a step input of stress and zero temperature, and (ii) a step input of temperature and zero stress, on the boundary of the surface of the cylindrical hole in a transversely isotropic thermoelastic medium in the context of coupled thermoelastic theory. In the present article the transient thermoelastic waves have been studied in a homogeneous transversely isotropic infinite medium having a cylindrical hole in the context of generalized theory of thermoelasticity [2].
2. FORMULATION
OF
THE
PROBLEM
We consider an infinitely extended homogeneous transversely isotropic thermoelastic medium having an infinite cylindrical hole of radius a. Let the origin of the cylindrical coordinate system (I, 8, z) be at the axis of the cylindrical hole. We consider the case of radial symmetry so that the non-zero displacement component u = u(r, t). Then the governing field equations of motion and heat conduction in the absence of body forces and heat sources are: Cl 1u.m
K(T,,, + r-‘T,,) -
+
tr-w,,- 81T, = Pi
pC,(F+ T,T) =
TOj?l[ti,, + r-Q
(1-l) + Z&i,, + r-‘ii)]
(1.2)
where B1 = (c 11 + cl&xl + c13a3, cij are isothermal elastic parameters, K is the thermal conductivity, p, c, and to are respectively the density, specific heat at constant strain and thermal relaxation time. The comma notation is used for spatial derivatives and superposed dot represents time derivatives. We define the quantities If = w*&,, z;, = w*to,
t’ = w*t,
u’ =
pw*v,uf~, To, w* = cllCJK,
E = 8:Tol~Cec,,, 463
T’ = T/T,, v; = ClllP
(2)
464
J. N. SHARMA
Here E is the thermoelastic coupling constant, w* is the characteristic medium and up is the velocity of the QL-wave. Introducing the quantities (2) in eqns (l), we obtain u,,, +
T,,, + r-IT,,
- I.-~u - ii = T9,
r-h,,
- (?+
z,,?+) = ~{ti,, + r-‘f
+ Z&
frequency of the
(3.1) + r-Iii)}
(3.2)
where we have suppressed the dashes. The boundary of the cylindrical hole i.e. r = a is given by I
=
w*a/v,
= tj
(say)
(4)
We assume that the medium is at rest and undisturbed initially. The initial and regularity conditions can be written as u=O=T,att=O,r>q,$=O,atr=O and u=O=T,fort=O,whenr+cc
(5)
We take two types of boundary conditions as S,,(& t) = H(r),
(6)
T(rl, r) = 0
and
S,,(%t) = 0,
Th t) = W
(7)
where S,, = u,, + clzu/cllr
- T
(8)
is the dimensionless form of the stress in the radial direction. 3. SOLUTION OF THE PROBLEM Applying the Laplace transform defined by $(r,p) =
Oc+(r,t)CP’dt
(9)
s0 with respect to time, to eqns (3) and using (5), we obtain {D(D + r - ‘) - p2}u = DT
(10.1)
{(D + r- ‘)D - rp2}T = erp’(D + r-‘)ii
(10.2)
where D = d/dr and r = r. + p-l. Simplifying eqns (lo), we get C{D(D + r -I)}’ - (m: + m$)D(D + r-l) + mf&Jii [{(D + r-1)D}2
- (mf + m$D(D + r-‘)mfm:]T=
= 0
(11)
0
(12)
465
Transient generalized thermoelastic waves
where m$ and rn: are the roots of the quadratic equation. m4 - p(A., + l&m2 a1 = (1 + E),
+ rp4 = 0
(13)
a2 = 1 + 70 + &?g
(14)
Solving eqns (11) and (12) and using (5), we obtain
where K,(mir) and K,-,(mir)are modified Bessel’s functions of order one and zero respectively. Using eqns (15) and (M), in eqns (lo), we get Bi =
i=
(p2- mF)Ag'mi,
1,2
(17)
Case I: Step input of stress and constant temperature
We consider a constant stress of magnitude unity suddenly applied on the boundary of the cylind~c~ hole and the temperature at the boundary is kept constant. The boundary conditions in this case are given by eqns (6), which with the help of (9) becomes
grr= Dti
+ c12ii,Jc11r-
T=
-$
T(r,p)=O,
atr=rj
(18)
Equations (15), (16), (17) and (18) lead to
where
A = m2tp2 - m%Amltf)Cm&(m24+ bKlr;,(m2tlYvl - ml(p2 - m%tw)CmlKdml~) + ~&h~Ytll~ Case 11: Step input of temperature
b = (~11- CI~YC~ 1
(20)
and zero stress
We consider a constant temperature of magnitude unity suddenly applied on the boundary of the cylindrical hole and the boundary is kept stress free. The boundary conditions in this case are given by eqns (7), which with the help of (9) transformed to
srr= DU + c12ii,k,,
-
T= 0,
T(r,p) = l/p,
at r = q
(21)
Equations (15), (16), (17) and (21), provide us
AZ= -CmlKotml?J+ ~Kltml~Y~l/pA where A is again given by eqn (20). es x:&P
(22.2)
466
J. N. SHARMA 4. SMALL
TIME-APPROXIMATIONS
Because
the second sound effects are short lived, Green and Lindsay [6] and Green f7], therefore the thermal relaxation time z. is small. So we concentrate our attention on small time-approximations, that is we take p large. The roots m,, i = 1,2 of eqn (13) are
Expanding binomially and retaining positive sign only, we get % =
Pfvi
4i
+
i= I,2
(24)
* (2; - 42,)“2] - r’z
(25.1)
+
o(l/Pl~
where vi.2 = $[;z,
d 1.2 = [a, + (il,,& - 2),&J; - 4ro)‘/2]/2@[&
t (A$ - 4r0)1’2]‘f2
(25.2)
Also
where R = 1; - 4r, = (1 + &To- To)2 “t-4EZ$> 0 Again (1 + erg + ro)’ > R so that ut < u2 Thus v1 corresponds to the speed of the slowest wave and u2 to that of the fastest wave. As a consequence of this, the points of the solid for which T > vzt do not experience any disturbance. From eqns (25) and (26) we see that as r. -+ O,o, -+ 1 and v2 + co. But r. = 0 corresponds to the case of the coupled theory of thermoelasticity, which predicts an infinite speed of heat propagation. We conclude that the wave propagating with speed ol must be the elastic wave influenced by the thermal field. Since u1 < v2, the elastic wave follows the thermal wave. The modified Bessel’s function K”(z) has the asymptotic expansion [83:
&td
”
0 1 E
e-” l +
(4n2 - 12) + (482 - 12)(4n2 - 32) + ... 2!(8zQ2 lf(8z)
1
(27)
For Case I, using expansions (24) and (27) in (19) and then in (15) to (17), we obtain
e-mi(r-q)[(v: -
(vz
-
vf)/v,(v~ - l)p2 + (A, + 3(v: - v:)/S~(v: - 1)) -$
v3u2(v~
-
1)~’ + {A, + 3(vj - ~~)/8~(v~-
1 1
(28)
Transient generalized themmelastic waves
T=
461
01’2e-mlc-v) 5
_
[(of - 1)(v: - V:)/u:(u: - 1)p
- {(v: - l)C& + (0: - @8tl(u: -
1)lh
+ 41u + a@: - 4>/w: - N/P21 +;
112 ,-m2(r-s)[(u; _ I)(0: - u:)/u;(u: - 1)p
0
- ((0: +
42u
l)C~,
+
+
4H4
(u: -
-
4YW4
a44
-
l))/u2
WP21
(29)
where 1, = (u2 - u,){(8b - l)( ulu2 + 1) + u: - 1}/8r/u,(u: - 1) + 2&l$(l 12
=
(~1
- u:,/u,(u; - 1)3 + &(u: + u’,,
(30.1)
~2){(8b - 1)(~1~2 + 1) + U: - l)/8~~2(0~ - 1)
-
(30.2)
+ 2&0:(1 - u:)/u,(u: - 1)3 + &(u: + u:, Now the stress in the transformed form is given by S,kYP) =
EwKAml~) +
+
~~,(~,~)/~I~,
bK,(m,r)/r]A*
-
-
Cm,kdm*~)
(31)
T
Using eqns (24), (27) and (29), we get S,,(r,p) = - (~~2e-m1(rPq)[(2
+ (u; - u;)/8+;
+ (:r*
- 1)p
1
0: )
- l)}/ul - “l(lu+(;;““‘,
e~m2~r~v’[(2 - u$o(: - u$$(uf
- 1)p
r#~*+ (1 - 8b)/8r-i12/u, - $41 + u:)(u: - u:)/u,(u; - 1)
+
-
- uf)(uf - u$/uf(u:
(4
-
NJ.2
+
(4
-
WM~:
-
Wul
II
(32)
-j
Inverting the Laplace transform, we obtain u(r, t) = (&r2emd1R[(u:
- f&t - R/u,)H(t - R/u,)/u,(uf
- 1)
1
- R/u,)/~~(u; - 1)
e-42R[(u: + 3($
-
- uf)(t - R/u,)H(t
- R/u2)/u1(u$ - 1)
~:)(t - R/~2)*fZ(t - R/u~)/~v(u~ - l)]
(33)
468
J. N. SHARMA
-($Jze-+u;- 1xv: -
T@,t) =
- {(u: - l){A + (0: - 6)/8&
v:)H(t - R,v,),v:(o; - 1) - l)}lu,
1 +(&~2e-@'2R[fu; + +I(1 + u:)(u: - u;)/u,(u: - l)}(t - R/u,)H(t - R/u,
l)(u; - uf)H(t - R,u2),u: - 1)
- {(u: - l)C~, + (VI - o:YWd
- 1)1/~2
+ 42(1 + u:)(u: - u:)/u,(u: - l)}(t - R/u,)H(t - R/u,)
e-41R[(uf
1
(34)
- u$o( - uf)H(t - R/u,)/uf(u~ - 1)
+ ($1 + (1 - 8b)/8(R + q) - A,/u, - (u: - I)[& + (u: - u;)/8q(u: - ~)I/vI - AU + o:Xu: - u%,Cd - l))@ - WWW
- R/M
e-b2R[(u$ - uf)(2 - u:)H(t - R/u,)u:(uf - 1) + (42 + (1 - 8b)/8(R + q) - A,/02 - &(l + u$o(: - u:,/u,(u: - 1)
x
Cl2 + (U: -
U:YfW:
-
W~II(~ - W2VW
- R/u,)1
(35)
where R = I - q. For Case II, using expansions (24) and (27) in (22) and then in (15) to (17), we get
‘Z-m1”-q’[U2(Uf
_
-
U;)/p4 + {3U,U2(Uf
-
u;)/8r - ny}/p5]
y 1’2 e-m2”-q’[u,(u~ - uf)p4 + {3u,u,(u~ - uf)/8r - At}/p5],
0r
T(r,p) =
l/2
0 3
(36)
e-m1(r-q)[u2(u:
- u$(uf - l)/u,p3
- {~XU?- 1)/u, + C~,(U: + 1) + (0: - 1)/8rlu2(u: - dI}/~~l _
g 1’2
e-m2”-q)[ul(u~
0I C42(U:
+
+ 1) + (U: -
- uf)(u: - l)/u2p3 - {Ay(u: - 1)/u, W3rlU,(U;
-
u:))/P~I,
(37)
112 S,,@,P)
=
0
;
e-m1(r-q)[u2(uf - u:)(2 - u~)/ulp3{A~(u~ - 2)/u,
+ v,(vf - v:)[~,(v: _
+ 2) + (8b - 2 + vf)/8rl)/p41
y 1’2e-m2(‘-q)[ul(u$ 0r
- uf)(2 - u9/u2p3 + {Az(u: - 2)/u2
+ u,(u$ - u:)[~~(u: + 2) + (8b - 2 + u:)/8rl)/p41
(38)
Transient generalized themoelastic
469
waves
where A: = u,(u: - u;)(u~ + u: + u,u,)/~~(u, + u2) - 2~,u,u: + q&u:(u: + u:,
(39.1)
A.: = u,(u; - u;)(u: + u: + u,uJ/8~(0, + u2) - 2&u2u: + &u:(u: + u:,
(39.2)
Inverting the Laplace transform, we obtain 112 ? ~ n(r, t) = R + ‘t e-01R[u2(uf - u$(c - R/u~)~J!Z(~- R/u,)/6 ( > + {3U,U,(U: - u:)/8(R + q) - A:}@- R/u~)~H(~ - R/u,)/241 - ? ( R+rl )
-
l/2 e-42R[u1(u$ - t$(t - R/u~)~H(~ - R/u,)/6
+ {3U,U,(U;- u:)/8(R + q) - $}(t - R/u~)~H(~ - R/u,)/241 r/2 ? T(r, t) = R + ‘1) embR[u2(uf - I$)($ - l)(t - R/tQ’H(t ( > - {WU: - 1)/U, + +
-
@J:
-
1)/W
+
u2c4
-
Amy:
+
(40)
- R/u,)/2u,
1)
tt)l# - WJ3Wt - WMI
112
- rl ( R+tt >
-
Rlu2)/2~2
+
(6
-
embzR[ul(u: - u:)(u: - l)(t - R/u~)~ZYI(~
-
1)/W
{Mu:
+
-
lVu2
+
0~:
~)l)(t- WA3Wt -
e-41R[u2(u:
-
u:,C&(u:
+
1)
Rlu2)Pl
(41)
- u:)(2 - of)@ - R/u~)~H(~ - R/u,)/2u,
+ ; {n:(u: - 2)/u, + U,(U: - U:)[&(U: + 2) + (8b - 2 + u:)/8(R + q)l}(t - R/uA3H(t- R/u,)1 e-@‘2R[uI(us - ~$2 - u$o( - R/u~)~H(~
- R/u,)/2u, + ;{A:@; - 2)/u, + u,(u; - u:>[#,(u: -t + (8b - 2 + u:)/8(R + q)]}(t - R/r~~)~If(t - R/u,)]
2) (42)
5. DISCUSSION OF THE RESULTS The short time solutions obtained above show that they consist of two waves that is the dilatational wave and the thermal wave travelling with velocities u1 and u2 respectively. The terms containing H(t - R/u,) represent the contribution of the elastic wave in the vicinity of its wave front R = u,t, and terms with H(t - R/u,) represent contribution of the thermal wave in the vicinity of its wave front R = u2t. It is observed that the deformation is continuous on both the wave fronts in Case I as well as Case II. The temperature and
470
J. N. SHARMA
stress are also found to be continuous at both the wave fronts in Case II. These quantities are found to be discontinuous at both the wave fronts in Case I. The discontinuities are:
These jumps in temperature and stress decay exponentially. 6. PARTICULAR CASES (i) When the thermal relaxation time T,, = 0. This case corresponds to the Conventional Coupled theory of thermoelasticity and we have: 11= 1 +&,
A2 = 1,
VI = 1,
02 = co,
$1 = 612,
42
=
00
It is observed that temperature k continuous at both the wave fronts and the stress is continuous at the thermal wave front, but discontinuous at the elastic wave front. This discontinuity is given by
which agrees with the corresponding result obtained by Chattopadhyay et al. [S]. (ii) When the strain and thermal fields are uncoupled to each other. In this case the coupling constant E is identically zero and we have ‘A1= 1,
12 = 1 + zo,
VI = 1,
02 = l/Jr;,
$1 =O,
42
=
v2/2
It is observed that the temperature and stress have infinite jump at the thermal wave front. The temperature is continuous at the elastic wave front but the stress is discontinuous through the quantity. (‘% -
%)R=qt
=
Cd@
+
dl"'
In this case the jump in the stress at the elastic wave front vanishes as the radial distance increases. Acknowkxfgement-The author is thankful to Dr R. S. Sidhu, Department of Mathematics, Guru Nanak Dev University, Amritsar for discussions.
REFERENCES VI H. W. LORD and Y. SHULMAN, A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299-309 (1967). PI R. S. DHALIWAL and H. H. SHERIEF, Generalised thermoelasticity for anisotropic media. Q. Appl. Math. 38, l-8 (1980). E31H. SINGH and J. N. SHARMA, Generalized thermoelastic waves in transversely isotropic media. J. Acoust. Sot. Am. 77, 1046-1053 (1985).
Transient generalized thermoelastic waves
471
[4] J. N. SHARMA and R. S. SIDHU, On the propagation of plane harmonic waves in anisotropic generalized thermoelasticity. ht. J. EngngSci. 24, 1511-1516 (1986). [S] A. CHATTOPADHYAY, A. KESHRI and S. BASE, A coupled thermoelastic problem for an infinite aelotropic medium having a cylindrical hole. Indian J. Pure Appl. Math. 16(7), 807-823 (1985). [6] A. E. GREEN and K. A. LINDSAY, Thermoelasticity. J. Elasticity 2, l-7 (1972). [7] A. E. GREEN, A note on linear thermoeksticity. Mathematih 19, 69-75 (1972). [S] G. N. WATSON, Theory of Bessel Functions, 2nd edn, p. 202. Cambridge University Press (1980). (Receioed 1 July 1986)