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Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions
Int. J. Engng Sci. Vol. 33, No. 14, pp. 2011-2020, 1995
Pergamon 0020-7225(95)00050-X
Copyright ~ 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/95 $9.50+ 0.00
F U N D A M E N T A L S O L U T I O N IN T H E R M O E L A S T I C I T Y W I T H TWO R E L A X A T I O N TIMES F O R C Y L I N D R I C A L REGIONS MAGDY A. E Z Z A T Department of Mathematics, Faculty of Science, University of Alexandria, Alexandria, Egypt
(Communicated by E. S. SUHUBI) Abstract--The solution of the problem of determining stress and temperature distributions with a continuous line source of heat in an infinite elastic body governed by the equations of generalized thermoelasticity with two relaxation times are obtained by using the Hankel and Laplace transform techniques. Inverse transforms are obtained in an approximate manner for small values of time. Numerical results for a particular case are given.
INTRODUCTION
The theory of coupled thermoelasticity was formulated by Biot [1] to eliminate the paradox inherent in the classical uncoupled theory that elastic changes have no effect on the temperature. Unfortunately, the heat equations for any of the two theories, though different, are of the diffusion type predicting infinite speeds of propagation for heat waves contrary to physical observations. The equations of generalized thermoelasticity with one relaxation time for a homogeneous isotropic medium were derived by Lord and Shulman [2]. Dhaliwal and Sherief [3] obtained the corresponding equations for a general anisotropic medium. These equations admit the so-called second sound effect in solids; i.e. they predict finite speeds of propagation for heat and displacement disturbances. Because of the complicated nature of these equations, few attempts have been made to solve them. Sherief and Dhaliwal [4] solved a thermal shock problem, Sherief [5] solved a spherically symmetric problem with a point source. Both of these problems are valid for short times. Recently, Sherief and Ezzat has obtained the fundamental solution for this theory valid for all times [6]. Another generalization to the coupled theory of thermoelasticity is what is known as the theory of thermoelasticity with two relaxation times. MUller [7] in a review of thermodynamics of thermoelastic solids has proposed an entropy production inequality, with the help of which, he considered restrictions on a class of constitutive equation. A generalization of this inequality was proposed by Green and Laws [8]. Green and Lindsay obtained an explicit version of the constitutive equations in [9]. These equations were also obtained independently by Suhubi [10]. This theory also predicts finite speeds of propagation as in Lord-Shulman's theory. It differs from the latter in that Fourier's law of heat conduction is not violated if the body under consideration has a center of symmetry. Erbay and ~uhubi [11] have studied wave propagation in infinite cylinders. Ignaczak studied a strong discontinuity wave [12] and obtained a decomposition theory for this theory [13]. Dhaliwal and Rokne have solved a thermal shock problem in [14]. Sherief has obtained the fundamental solutions for generalized thermoelasticity with two relaxation times for point source of heat [15]. 2011
2012
M.A.
EZZAT
FORMULATION
OF T H E P R O B L E M
We shall consider an infinite isotropic homogeneous elastic space. A cylindrically symmetric line source of heat of intensity Q(r, t) is located along the z-axis. A similar problem was considered by Sherief and Anwar [16] in the context of generalized thermoelasticity with one relaxation time. The governing equations for generalized thermoelasticity theory with two relaxation times, in the absence of body forces, can be written as [9]. __-32u~
{
(,~ + ~)uj,ij + ~.,.. - y~T, +
vOT~I
-~-//
(1)
OT oCT Oe kT. = pCE ~t + pCnz at--y+ yTo-~ - pQ.
(2)
P 0t 2 --
The energy equation has the form
The constitutive equations are given by
(
(3)
o~i~ = Ae6ij + 21xe~j - y T - To + v Ot / 'j
where 6~j is the Kroneker delta function and e = u u is the dilatation. In the above equations a comma denotes material derivatives and the summation convention is used. We note that due to cylindrical symmetry the only non-vanishing displacement component is the radial one ur = u(r, t). Using the non-dimensional variables: r* = cl rlr,
u* = cl flu,
o'* = a J l z ,
Z = c27qr,
t* = c~rlt,
and
0 = ( T - To)/To
v* = c~rlv ,
Q* = p Q / k T o ~ % ] ,
equations (1), (2) and (3) take the following form (dropping the asterisks for convenience) 2 0 2 u __ 2 0 0ll 0 (0 + V30~, u __ /3 - ~ - /3 ~r (--~r + r ) - b or _ -~-t/
V 2 0 = -00 -+ at O'rr=[3
10(02)
-g--- r r at e + r Or
-Q,
(4) (5)
20. ~r+(/32-2)U--b ( 0 + vOO r
~r,v, = (/3 z - 2 ) o u + / 3 2 u Or r
Ot]'
b ( 0+
at~
(6)
O'zz : Orz,P : O'zr : Orr~ = 0
where ~7 2 -
02
--
-- Or 2 +
1 O r
Or"
The initial conditions are taken as
u(r, t)l,=o = ~ (r, t)l,=o = 0,
Ou
(7a)
O'rrl,:0 = aO'~rot ,=o = o.e4,1,=° = 3cr~q,at ,=o = 0,
(7b)
O(r, t)l,= o = aO(r,ot t) ,=o = 0.
(7c)
Fundamental solution in thermoelasticity
2013
The boundary conditions are assumed to be
O,
O'rr, O'tbtO, 0"">
as
r--~ ~.
(8)
Introducing the thermoelastic potential function 6 defined by the relation 06 u = --. Ox
(9)
Equations (4), (5) and (6), give
{
O26 V26 - a~O + vO0], Ot2 ~ / O0
020
V20=--+ Ot
--
z Ot 2
+
(10)
atb gV2-_ -z-~- Q, #t
(11)
o',, =/32 026 Ot2
2 06 r Or '
(12a)
= /32 026
026
(12b)
o',q,
~ t z - 2 or 2 .
where a = b//3 2. The initial conditions (7) expressed in terms of ~, become at t = 0
6
06 Ot
026=0,
(13)
Ot2
while the boundary conditions (8) can be expressed as 6,
06 Or
and
--026 "
ar2---~u,
as
r---~oo.
(14)
S O L U T I O N IN T H E L A P L A C E T R A N S F O R M D O M A I N From now we shall consider a continuous linear heat source given by
O= Qo xr 6(r)H(t),
(15)
where Qo is a constant and 6@) and H(t) are the Dirac delta function and Heaviside unit step function, respectively. We now introduce the Laplace transform defined by the formula f(p) =
fo
e -p' dt.
Applying this transform to equations (10)-(12), and using the initial conditions (13), we obtain ( v 2 _ p2),
= a(1 + w ) O ,
V2/9 = p ( 1 + rp)O + gpV2~
O'rr =/32p2(~
Qo6(r) 2xrp
2 Oq~ r Or'
Or2 •
(16) (17) (18a) (18b)
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M.A. EZZAT
Eliminating 0 between equations (16) and (17), we obtain the following fourth-order partial differential equation satisfied by {~74
-
-
V2[p(1 + ~) +p2(1 + z"+ Ev)]
+pZ(p +p2.t.)}t~ =
qo6(r)(1 + vp)
(19)
rp where ¢ = ga and qo = aQo/2~r. This equation can be written in the form (V: - k2)(V 2 where
k2)qb(r, p) =
qo6(r)(1 + rp
vp)
(20)
k,, k2 are the positive roots of the characteristic equation k 4 - p { ( 1 + E) + p ( 1 + r +
ve)}k 2 +p3(1 + ~:p) = 0.
(21)
Hence, k, and k2 are given by
k,.2 = ~ [(1 + e) + p ( 1 + z + Ev) + ((1 + e)2p 2 + 2{(1 + e ) ( r + ev) + (e - 1)2}p3 + {1 + ( r + ev) 2 - 2(z
-
ev)}p4)l/2]) 1/2.
(22)
Applying the Hankel transform; defined by }(~) = Ho[f(r)l =
yorJo(ar)f(r) dr,
where Jo is the Bessel function of the first kind of order zero, to both sides of equation (20), we get after some manipulations (a 2 +
k~)(a z + k~)(b(a, p ) = - qo(1 + vp)
(23)
P Using the formula for the inverse Hankel transform, namely
f(r) = H o ' [ f ( c t ) ] =
OtJo(ar)f(a) dt~,
we obtain from equation (23)
dp(r,p)
vp) p(k2_ k2 ) [Ko(k~r) - Ko(kzr)],
= qo(1 +
(24)
where K0 is the modified Bessel function of the second kind of order zero.
I N V E R S I O N OF T H E L A P L A C E T R A N S F O R M S Let us now determine inverse transforms for the case of small values of time (large values of p). This method was used by Hetnarski [17] to obtain the fundamental solution for the coupled thermoelasticity problem and by Sherief [15] to obtain the fundamental solutions for generalized thermoelasticity with two relaxation times for point source of heat. Let
f(p) = 1 { [ 1 + ~ + p ( 1 + z + ev)] 2 - 4 p ( 1 + rp)} '`e, P
Fundamental
s o l u t i o n in t h e r m o e l a s t i c i t y
2015
It then follows f r o m e q u a t i o n (22) that kf, 2 = P [1 q" ff + p ( 1 + "r "[- EV) q ' - p f ( p ) ] .
(25)
E x p a n d i n g f ( p ) in a M a c l a u r i a n series of p o w e r s of 1/p and neglecting t e r m s of fifth and higher order, we obtain
f(p) = A
B 2eC + -Ap - - ~ A3p2
2eBC ASp3
eCD 2A7p 4 ,
(26)
where A = [(1 + r + ev) 2 - 4r] 'a, B = (1 + e)(1 + r + ev), C = 1 + (1 + e ) ( v -
r),
D = (1 + e)ZA 2 - 5B 2. Substituting f r o m (26) into e q u a t i o n (25), we can write k~Z2 in the f o r m 4 k 2 _ 2 ~ a0 i - P jZ,op-7 ,
(27)
i = 1, 2,
where (1 + r + ev + A ) al0 -
(1 + r + E v - A ) ,
2
a2o =
[(1 + e)A - B]
[(1 + e)A + B] all
--
a12
= - -e C A3,
a21 =
,
2A
2
2A
a22 =
--a12 ,
,
a23 =
--a13 ,
eCD 4A 7 ,
a24 =
-al4.
eBC a l 3 --
a14-
A5
E x p a n d i n g 1 / f ( p ) in a Maclaurin's series and retaining t e r m s of O(1/p 4) only, we get
f(P)
(28)
j=oPj
where
1 bo = ~ ,
B bl
--
A3,
b2 =
B e - 2eC A5
B(2eC - B 2) ,
b3 -
A7
3D 2 + ,
b4
--
10B 4
8A 9
F r o m equations (25) and (28), it follows that 1
k2
_
1
k 2 p2f(p)
_~ bj = j op2+~ /.j
(29)
Using similar e x p a n s i o n techniques, we obtain bi2
bi3
ki = bioP + bil + --p -~ p2, E$ 33-14-B
i=1,2,
(30)
M. A. E Z Z A T
2016
where bit) = aio1/2, ai)
bil -
2bio '
4aioai2 - a2l hi2 -
8aiobio 8a2ioai3 -- 4aioail ai2 + a31
16a2iobi °
bi3 =
TEMPERATURE
DISTRIBUTION
Substituting from equation (24) into equation (16), we get 0 =
ap(k~qo- k~) [Ko(k)r)(K~ -p2) _ Ko(k2r)(k~ -p2)].
(31)
We substitute from equations (27) and (29) in the above equation and use equation (30) in the form ki =
bioP + b i l ,
i = 1, 2.
We arrive at 2
3
Ci j
0 = qo ~ ~ (_l)i) a i=l j=O
Ko(biopr + bilr),
F
(32)
where j-I
qo = bo(a..- 1), and
cq = bj(a~o - 1) + ~ bka~(j-k),
i = 1, 2,
j = 1, 2, 3.
(33)
k-O
Taking the inverse Laplace transform of both sides of equation (32) and making use of the shift property, it can be shown that
_ ~ ~ .... o=q° (-1)'-'C°expt-ait)L
,[ Ko(b~opr)] [~=a-~'J'
a i=~ j=o
where ai = b~j/b~o, i = 1, 2. Using expansion techniques and keeping terms up to order p-4 only, we arrive at
o=q---~°~ ~ ( - 1 ) ~ - ) C q e x p ( - a i t ) L - ' [ K ° ~ p r ) ~ a i=Jj=o
L
ff
j (-Jk2)(
k=o
cq/k ] -p/ j.
(34)
From [18] we obtain
L-'[Ko(b,orp)] = H(t - bmr)(t 2 - ~2 O iOr 2,-,n ) , L L
p2
j
p :
J
b,or>Cosh1(2 ) io g
1
t
1'
t L-'[Ko(~rP)]=H(t-bi°r)[(~
+t(bi°r)21c°sh-'( t - f - l' - { 1 7 2 4\bior/
3
b 2 l~-~t + ( i ° ) /]X/ t 2 -
], (35)
Fundamentalsolution in thermoelasticity
2017
The inverse Laplace transform in equation (34) has been evaluated by using the convolution theorem and equation (35). Upon substitution of the resulting expressions and equation (35) in equation (34), we finally arrive at
[
0 = q_..OOai=l ~ ( - 1 ) i - l n ( t -- b i ° r ) e x p ( - ° t i t )
(')
Oil cosh-' ~
}
- Oi2(t 2 - b2or2) 1/2 ,
(36a)
where
bZior2] 1 1 2 23 0,, = [Cio(1 + - ~ - - / + ~ c i i b i i b , o rz + ~ c,obior J
~4g-+b31r2~+cil(l+ 3bzi' +[t cio(,,;o 4,,io, TrO 2[ciob~l
ciibii
+ t t 2-~/2o + bio +
2]
+ ~ ci2biobil r +
ci3bi20r2]
"3[ci°b311-ciib2il +cizbiz +ci3] 2bio 6 J
(36b)
+ t IL~ 6blo L 3 -1- 2b/2o
and
)(
r {bil+ b3il
0i2 = [ci°t~io
b2r2\]
.Uil,
~io r z + c i ' l
21
b~,, 2cilbil ] - tlcm ~ + + Ciz bio ~
3 r 4L
1
+ -~ ci2biobil
2 1 2 2] r + ~ ciabii~r
17 ~rc,oO,~, + 3¢,-,b,~, + 3ci2bi, ] "~ Ci4 +3g/L-g~,~o biZo bio "
(36c)
STRESS D I S T R I B U T I O N Substituting from equation (24) into equations (18) we get upon using the recurrence relations of the modified Bessel functions of the second kind, the equations o% _ - _-7,_,_2-qo(1 + v p ) ~ ( - 1 ) i-I [~kiKl(kir)+[3Zp2Ko(kir)],
(37)
p(k, - k~),=,
_
~'~
qo(l+
vp)~
p(k2--k~) i=l
(_l)i_l[_2kiKi(kir)+([32,Z_2k2i)Ko(kir)] r
'
(38)
where K~ is the modified Bessel function of the second kind of order one. Following the same procedure as used for the temperature distribution, the inverse Laplace transform of equation (37) can be written in the form
qo i~= 1 (_ Orrr ~ --a
l )i_l H(t _ bior)exp(_otit)[ [3Zco(tz - b~or2)_l/2
+ .rCOS '(,--' / \ oior]
2\1/2"] J,
Oior )
+
(39a)
where 1
2
~rilr = ~{2fl [c,(4 + b~lr 2) + 2czbmb;,r 2 + c3b2or2] - a,o(8b;1 + b31r2) - ~,i b,-o(8 + 3b~,r 2) - 2ai2bi. b~or2 - a;3b3or2}
+41 [32 c~
'+ b]~r~]b,o/ + c2(4 + 3bZ~r2) + 3c3b;~b;or 2 + c4b20r2 43/ob/2,
- 4(2ailbil + alzbio) + 2+r .zfLP ~z{clbZil 1 [,-~/2o
lt3
+6
bio J
czbilbio+c3)
2 cib3l' c2b21 -
3ailb~lbio 3aizbii - a i 3 b i o - - c4)],
c3bil
+
(39b)
2018
M.A. EZZAT
If
[ 13
. 6b21~
tri2. = ~ ~OtiOkb~or2+ ~io } + 6(2otilbil + oli2bio) - /32[c,(~+
~)
+ c2(9 + 3b21r2) + 3c3biobilr2 + c4b2ior2]}
t f [24b, 13b31] 3a;1 +~ (8 + 13b~lr 2) + 39ai2bil + 13ai3bio + -24 - ~ Ot/o~b/---~2 + --ETbio } bior __ 18 2/3 [ c,b21bE+2czbn+bio c3)}
t 2 12~iob~t + 2 4 a n b . d-~-~ 2 2 bior b 3ior2
3 3c2b~1 . 3c3bil - T ~ Jr- - - - - ~ - I " "1- C4 bio blo Dio
12a/2 11/3 F -bio r2 +
+ 0ti__._23~ t 3 (aiob;13 3ot/2b/21 F 3oti2bil ~ + ~ r 2 \ ~io bi3o b;o b,o]
(39c)
where
Co = vbo,
cj = vb/+ bj_ 1
otij = :~ ckbi(i_,) , k=O
j = 1, 2, 3, 4,
i = 1, 2,
(39d)
j = O, 1, 2, 3.
Similarly, the inverse transform of equation (38) yields
try,e, = qo ~ ( - I )i-I H( t - bior)exp(-otit) (/32Co - 2/3/o)(/2 - o;or" 2 2x--I/2) /=1 1
l
--~+ trill' c o s h - (\bior/
~2
2~1/2]
O'/2q,(/2- o,or ,
(40a)
J,
where
1
tr,,. = ~ [(/32c, - 2/3.)(8 + 2b21r 2) + 4(/32c2 - 2/3;2)b. b;or 2 + 2(/32c3 - 2/3;3)b;2r2 3 2) + ailbio(8 + 3b/21r2) + 30ti2bnb20 r2 + oti3b30r2] + aio(8bil + bilr
t[
+ ~ (/32c, - 2/3.) + (/32C 4 --
t2 [
(4bi~l
+ b~tr2~ bio / + (/32c2 - 2/3i2)(4 + 3bElr2) + 3(/32c3_ 2/3i3)biobnr2
2/3i4)bEor2 + 4a;°b21 + 8t~il b;1 + 4a;2bio] bio b2 bii
bil
aiob31
+ -2 [ (/32¢1 -- 2fl11) ~ . + 2(fl2C2 -- 2/3;2) ~/o + (fl2¢3 - 2~i3) b/o + --b,2o
+ 3ailb21 + 3ai2b. + aiabio] d bio [ 3 b~, bil ] + ~t3 (/32c, - 2/3il) b~13+ 3(/32c2 - 2/3;2) ~ + 3(/32c3 - 2/3,-3) b/o + (fl2c4 - 2/3,4) , (40b) b;o 0"i2¢, =
1 [(/32CI -- 2/3il )( ~ + b b i o 31r2~+/3(/32C2 -- 2/3/2)(3 + b 21r2) + 3(/32C3--" 2/3i3)bi°bil r2 18 + -~-;0 6b21~ + (/32c4 - 2/3i4)b2i0rE + aio(~-T~_2 ! + 12t~/i bil + 60ti2bio ] \ O;or t [ bEI b;i + - - 118(/32cl - 2/3;i) ~i- + 36(/32c2 - 2/3,2) + 18(/32c3 - 2/3;3) 24 bio ~/o
2019
Fundamental solution in thermoelasticity
5.0
0
4.5 4.0 3.5 3.0 2.5
25 20 15
1.5 1.0
IO ~rr
5
0.5 0
I
I
0.2
0.4
\
I 0.6
I
I
0.8
I
I
0.6
0.8
r -10 Fig. 2. Radial stress distribution.
Fig. 1. Temperature distribution
{24bil 13b31~ 3otil ] + aiok b2.~r2 + ---ZT-bio] + ~ (8 + 13b2~r 2) + 39a~2b~, + 1 3 a , 3 b ~ t2 [
,~ ~ ~ b 3il +
+ 36-- 11(/32c' - "~")~2o +
33(/32c3 -
hi._22._
33(/32c2 - 2/3'2)b2o
2fli3) ~io + 11(fl2c4 - 2fli4) -t 12ai°b 3 2 i2
24atilbil
12ai2]
÷
b ior
t 3 (a,ob 3, 3c~i,b,2 + ~ + + ~ r 2 k b~4or2 b~o
J
3ai2bi,+a,3] b20 bio]'
(4Oc)
and J
fllj = ~, Ckai(y-,), k=O
i = 1, 2,
NUMERICAL
] = O, 1, 2, 3, 4.
(40d)
RESULTS
The problem was solved for the special case of the copper material (~ =0.0168 and f12 = 3.94). The results for the temperature distribution, the radial stress component and for the tangential stress component are shown in Figs 1-3. The value of time t = 0.1 was taken throughout the computations. It was noted that when Zo, v and ~ are all equal the stress components for the two generalized thermoelasticity theories (generalized thermoelasticity
2 0 -2 0¥¥
I
I
I
I
I
0.2
0.4
0.6
0.8
1.0
r
-4
-6 -8
-10 -12 -14 --
Fig. 3. Transverse stress distribution.
2020
M.A. EZZAT
with one relaxation time theory and generalized thermoelasticity with two relaxation times theory) are identical though the temperature for the uncoupled
and coupled
d i s t r i b u t i o n is t o t a l l y d i f f e r e n t . W e n o t e also t h a t
theories the temperature
a n d t h e stress c o m p o n e n t s
have
n o n - v a n i s h i n g ( t h o u g h s m a l l ) v a l u e s f o r a n y v a l u e o f r > 0 w h e n t > 0. T h i s d e m o n s t r a t e s fact o f i n f i n i t e s p e e d s o f p r o p a g a t i o n .
the
T h i s is n o t t h e c a s e f o r b o t h o f g e n e r a l i z e d t h e o r i e s
w h e r e t h e f u n c t i o n s a r e i d e n t i c a l l y z e r o w h e n r > t/b2o. T h e e f f e c t o f t h e h e a t s o u r c e is c o n f i n e d to a b o u n d e d
but time-dependent
region of space surrounding the source. This means that we
h a v e a s i g n a l p r o p a g a t i n g w i t h a finite s p e e d . The temperature
d i s t r i b u t i o n a n d t h e s t r e s s e s crrr a n d ~ry,,~ h a v e i n f i n i t e d i s c o n t i n u i t y at t h e
o r i g i n d u e t o t h e p r e s e n c e o f t h e h e a t s o u r c e a n d t w o finite j u m p s at r = t / b l o a n d r -- t/b2o.
REFERENCES
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
M. BLOT, J. Appl. Phys. 27, 240 (1956). H. LORD and Y. SHULMAN, J. Mech. Phys. Solid 15, 299 (1967). R. DHALIWAL and H. SHERIEF, Q. Appl. Math. 33, 1 (1980). H. SHERIEF and R. DHALIWAL, J. Therm. Stress 4, 407 (1981). H. SHERIEF, J. Therm. Stress 9, 151 (1984). H. SHERIEF and M. EZZAT, J. Therm. Stress 17, 75 (1984). I. MULLER, Archs Rat. Mech. Anal. 41, 319 (1971). A. E. GREEN and N. LAWS, Archs Rat. Mech. Anal. 45, 47 (1972). A. E. GREEN and K. A. LINDSAY, J. Elast. 2, 1 (1972). E. S. ~UHUBI, Thermoelastic solids. In Continum Physics (Edited by A. C. ERINGEN), Vol. II, Chap. 2. Academic Press, New York (1975). S. ERBAY and E. S. SUHUBI, J. Therm. Stress 9, 279 (1986). J. IGNACZAK, J. Therm. Stress 8, 25 (1985). J. IGNACZAK, J. Therm. Stress 1, 41 (1978). R. DHALIWAL and J. ROKNE, J. Therm. Stress 11, 257 (1989). H. SHERIEF, Int. J. Engng Sci. 30, 861 (1992). H. SHERIEF and M. ANWAR, J. Therm. Stress 9, 165 (1986). R. HETNARSKI, Archs Mech. Stosow. 16, 23 (1964). F. OBERHETTINGER and L. BADI1, Tables o f Laplace Transforms. Springer, New York (1973).
(Received 11 April 1995; accepted 7 June 1995)
NOMENCLATURE
A, # = at= 132 = "y= p= r, ~/,, z = c2 = ai = 0% = eij= cE =
Lam6's constants coefficient of linear thermal expansion (X + 2~)/~ (3A + 2/.z)ot t density cylindrical polar coordinates (A + 2tx)/p square of velocity of longitudinal wave components of displacement vector components of the stress tensor components of strain tensor specific heat at constant strain
k = thermal conductivity = pCE/k t = time T = absolute temperature T, = reference temperature chosen so that I(T - To)/Tol<< 1 b = "yTo/~ g = y/k~ 7 Q = intensity of heat source per unit mass r, v = relaxation times ro = relaxation time for Lord-Shulman theory
Pergamon 0020-7225(95)00050-X
Copyright ~ 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/95 $9.50+ 0.00
F U N D A M E N T A L S O L U T I O N IN T H E R M O E L A S T I C I T Y W I T H TWO R E L A X A T I O N TIMES F O R C Y L I N D R I C A L REGIONS MAGDY A. E Z Z A T Department of Mathematics, Faculty of Science, University of Alexandria, Alexandria, Egypt
(Communicated by E. S. SUHUBI) Abstract--The solution of the problem of determining stress and temperature distributions with a continuous line source of heat in an infinite elastic body governed by the equations of generalized thermoelasticity with two relaxation times are obtained by using the Hankel and Laplace transform techniques. Inverse transforms are obtained in an approximate manner for small values of time. Numerical results for a particular case are given.
INTRODUCTION
The theory of coupled thermoelasticity was formulated by Biot [1] to eliminate the paradox inherent in the classical uncoupled theory that elastic changes have no effect on the temperature. Unfortunately, the heat equations for any of the two theories, though different, are of the diffusion type predicting infinite speeds of propagation for heat waves contrary to physical observations. The equations of generalized thermoelasticity with one relaxation time for a homogeneous isotropic medium were derived by Lord and Shulman [2]. Dhaliwal and Sherief [3] obtained the corresponding equations for a general anisotropic medium. These equations admit the so-called second sound effect in solids; i.e. they predict finite speeds of propagation for heat and displacement disturbances. Because of the complicated nature of these equations, few attempts have been made to solve them. Sherief and Dhaliwal [4] solved a thermal shock problem, Sherief [5] solved a spherically symmetric problem with a point source. Both of these problems are valid for short times. Recently, Sherief and Ezzat has obtained the fundamental solution for this theory valid for all times [6]. Another generalization to the coupled theory of thermoelasticity is what is known as the theory of thermoelasticity with two relaxation times. MUller [7] in a review of thermodynamics of thermoelastic solids has proposed an entropy production inequality, with the help of which, he considered restrictions on a class of constitutive equation. A generalization of this inequality was proposed by Green and Laws [8]. Green and Lindsay obtained an explicit version of the constitutive equations in [9]. These equations were also obtained independently by Suhubi [10]. This theory also predicts finite speeds of propagation as in Lord-Shulman's theory. It differs from the latter in that Fourier's law of heat conduction is not violated if the body under consideration has a center of symmetry. Erbay and ~uhubi [11] have studied wave propagation in infinite cylinders. Ignaczak studied a strong discontinuity wave [12] and obtained a decomposition theory for this theory [13]. Dhaliwal and Rokne have solved a thermal shock problem in [14]. Sherief has obtained the fundamental solutions for generalized thermoelasticity with two relaxation times for point source of heat [15]. 2011
2012
M.A.
EZZAT
FORMULATION
OF T H E P R O B L E M
We shall consider an infinite isotropic homogeneous elastic space. A cylindrically symmetric line source of heat of intensity Q(r, t) is located along the z-axis. A similar problem was considered by Sherief and Anwar [16] in the context of generalized thermoelasticity with one relaxation time. The governing equations for generalized thermoelasticity theory with two relaxation times, in the absence of body forces, can be written as [9]. __-32u~
{
(,~ + ~)uj,ij + ~.,.. - y~T, +
vOT~I
-~-//
(1)
OT oCT Oe kT. = pCE ~t + pCnz at--y+ yTo-~ - pQ.
(2)
P 0t 2 --
The energy equation has the form
The constitutive equations are given by
(
(3)
o~i~ = Ae6ij + 21xe~j - y T - To + v Ot / 'j
where 6~j is the Kroneker delta function and e = u u is the dilatation. In the above equations a comma denotes material derivatives and the summation convention is used. We note that due to cylindrical symmetry the only non-vanishing displacement component is the radial one ur = u(r, t). Using the non-dimensional variables: r* = cl rlr,
u* = cl flu,
o'* = a J l z ,
Z = c27qr,
t* = c~rlt,
and
0 = ( T - To)/To
v* = c~rlv ,
Q* = p Q / k T o ~ % ] ,
equations (1), (2) and (3) take the following form (dropping the asterisks for convenience) 2 0 2 u __ 2 0 0ll 0 (0 + V30~, u __ /3 - ~ - /3 ~r (--~r + r ) - b or _ -~-t/
V 2 0 = -00 -+ at O'rr=[3
10(02)
-g--- r r at e + r Or
-Q,
(4) (5)
20. ~r+(/32-2)U--b ( 0 + vOO r
~r,v, = (/3 z - 2 ) o u + / 3 2 u Or r
Ot]'
b ( 0+
at~
(6)
O'zz : Orz,P : O'zr : Orr~ = 0
where ~7 2 -
02
--
-- Or 2 +
1 O r
Or"
The initial conditions are taken as
u(r, t)l,=o = ~ (r, t)l,=o = 0,
Ou
(7a)
O'rrl,:0 = aO'~rot ,=o = o.e4,1,=° = 3cr~q,at ,=o = 0,
(7b)
O(r, t)l,= o = aO(r,ot t) ,=o = 0.
(7c)
Fundamental solution in thermoelasticity
2013
The boundary conditions are assumed to be
O,
O'rr, O'tbtO, 0"">
as
r--~ ~.
(8)
Introducing the thermoelastic potential function 6 defined by the relation 06 u = --. Ox
(9)
Equations (4), (5) and (6), give
{
O26 V26 - a~O + vO0], Ot2 ~ / O0
020
V20=--+ Ot
--
z Ot 2
+
(10)
atb gV2-_ -z-~- Q, #t
(11)
o',, =/32 026 Ot2
2 06 r Or '
(12a)
= /32 026
026
(12b)
o',q,
~ t z - 2 or 2 .
where a = b//3 2. The initial conditions (7) expressed in terms of ~, become at t = 0
6
06 Ot
026=0,
(13)
Ot2
while the boundary conditions (8) can be expressed as 6,
06 Or
and
--026 "
ar2---~u,
as
r---~oo.
(14)
S O L U T I O N IN T H E L A P L A C E T R A N S F O R M D O M A I N From now we shall consider a continuous linear heat source given by
O= Qo xr 6(r)H(t),
(15)
where Qo is a constant and 6@) and H(t) are the Dirac delta function and Heaviside unit step function, respectively. We now introduce the Laplace transform defined by the formula f(p) =
fo
e -p' dt.
Applying this transform to equations (10)-(12), and using the initial conditions (13), we obtain ( v 2 _ p2),
= a(1 + w ) O ,
V2/9 = p ( 1 + rp)O + gpV2~
O'rr =/32p2(~
Qo6(r) 2xrp
2 Oq~ r Or'
Or2 •
(16) (17) (18a) (18b)
2014
M.A. EZZAT
Eliminating 0 between equations (16) and (17), we obtain the following fourth-order partial differential equation satisfied by {~74
-
-
V2[p(1 + ~) +p2(1 + z"+ Ev)]
+pZ(p +p2.t.)}t~ =
qo6(r)(1 + vp)
(19)
rp where ¢ = ga and qo = aQo/2~r. This equation can be written in the form (V: - k2)(V 2 where
k2)qb(r, p) =
qo6(r)(1 + rp
vp)
(20)
k,, k2 are the positive roots of the characteristic equation k 4 - p { ( 1 + E) + p ( 1 + r +
ve)}k 2 +p3(1 + ~:p) = 0.
(21)
Hence, k, and k2 are given by
k,.2 = ~ [(1 + e) + p ( 1 + z + Ev) + ((1 + e)2p 2 + 2{(1 + e ) ( r + ev) + (e - 1)2}p3 + {1 + ( r + ev) 2 - 2(z
-
ev)}p4)l/2]) 1/2.
(22)
Applying the Hankel transform; defined by }(~) = Ho[f(r)l =
yorJo(ar)f(r) dr,
where Jo is the Bessel function of the first kind of order zero, to both sides of equation (20), we get after some manipulations (a 2 +
k~)(a z + k~)(b(a, p ) = - qo(1 + vp)
(23)
P Using the formula for the inverse Hankel transform, namely
f(r) = H o ' [ f ( c t ) ] =
OtJo(ar)f(a) dt~,
we obtain from equation (23)
dp(r,p)
vp) p(k2_ k2 ) [Ko(k~r) - Ko(kzr)],
= qo(1 +
(24)
where K0 is the modified Bessel function of the second kind of order zero.
I N V E R S I O N OF T H E L A P L A C E T R A N S F O R M S Let us now determine inverse transforms for the case of small values of time (large values of p). This method was used by Hetnarski [17] to obtain the fundamental solution for the coupled thermoelasticity problem and by Sherief [15] to obtain the fundamental solutions for generalized thermoelasticity with two relaxation times for point source of heat. Let
f(p) = 1 { [ 1 + ~ + p ( 1 + z + ev)] 2 - 4 p ( 1 + rp)} '`e, P
Fundamental
s o l u t i o n in t h e r m o e l a s t i c i t y
2015
It then follows f r o m e q u a t i o n (22) that kf, 2 = P [1 q" ff + p ( 1 + "r "[- EV) q ' - p f ( p ) ] .
(25)
E x p a n d i n g f ( p ) in a M a c l a u r i a n series of p o w e r s of 1/p and neglecting t e r m s of fifth and higher order, we obtain
f(p) = A
B 2eC + -Ap - - ~ A3p2
2eBC ASp3
eCD 2A7p 4 ,
(26)
where A = [(1 + r + ev) 2 - 4r] 'a, B = (1 + e)(1 + r + ev), C = 1 + (1 + e ) ( v -
r),
D = (1 + e)ZA 2 - 5B 2. Substituting f r o m (26) into e q u a t i o n (25), we can write k~Z2 in the f o r m 4 k 2 _ 2 ~ a0 i - P jZ,op-7 ,
(27)
i = 1, 2,
where (1 + r + ev + A ) al0 -
(1 + r + E v - A ) ,
2
a2o =
[(1 + e)A - B]
[(1 + e)A + B] all
--
a12
= - -e C A3,
a21 =
,
2A
2
2A
a22 =
--a12 ,
,
a23 =
--a13 ,
eCD 4A 7 ,
a24 =
-al4.
eBC a l 3 --
a14-
A5
E x p a n d i n g 1 / f ( p ) in a Maclaurin's series and retaining t e r m s of O(1/p 4) only, we get
f(P)
(28)
j=oPj
where
1 bo = ~ ,
B bl
--
A3,
b2 =
B e - 2eC A5
B(2eC - B 2) ,
b3 -
A7
3D 2 + ,
b4
--
10B 4
8A 9
F r o m equations (25) and (28), it follows that 1
k2
_
1
k 2 p2f(p)
_~ bj = j op2+~ /.j
(29)
Using similar e x p a n s i o n techniques, we obtain bi2
bi3
ki = bioP + bil + --p -~ p2, E$ 33-14-B
i=1,2,
(30)
M. A. E Z Z A T
2016
where bit) = aio1/2, ai)
bil -
2bio '
4aioai2 - a2l hi2 -
8aiobio 8a2ioai3 -- 4aioail ai2 + a31
16a2iobi °
bi3 =
TEMPERATURE
DISTRIBUTION
Substituting from equation (24) into equation (16), we get 0 =
ap(k~qo- k~) [Ko(k)r)(K~ -p2) _ Ko(k2r)(k~ -p2)].
(31)
We substitute from equations (27) and (29) in the above equation and use equation (30) in the form ki =
bioP + b i l ,
i = 1, 2.
We arrive at 2
3
Ci j
0 = qo ~ ~ (_l)i) a i=l j=O
Ko(biopr + bilr),
F
(32)
where j-I
qo = bo(a..- 1), and
cq = bj(a~o - 1) + ~ bka~(j-k),
i = 1, 2,
j = 1, 2, 3.
(33)
k-O
Taking the inverse Laplace transform of both sides of equation (32) and making use of the shift property, it can be shown that
_ ~ ~ .... o=q° (-1)'-'C°expt-ait)L
,[ Ko(b~opr)] [~=a-~'J'
a i=~ j=o
where ai = b~j/b~o, i = 1, 2. Using expansion techniques and keeping terms up to order p-4 only, we arrive at
o=q---~°~ ~ ( - 1 ) ~ - ) C q e x p ( - a i t ) L - ' [ K ° ~ p r ) ~ a i=Jj=o
L
ff
j (-Jk2)(
k=o
cq/k ] -p/ j.
(34)
From [18] we obtain
L-'[Ko(b,orp)] = H(t - bmr)(t 2 - ~2 O iOr 2,-,n ) , L L
p2
j
p :
J
b,or>Cosh1(2 ) io g
1
t
1'
t L-'[Ko(~rP)]=H(t-bi°r)[(~
+t(bi°r)21c°sh-'( t - f - l' - { 1 7 2 4\bior/
3
b 2 l~-~t + ( i ° ) /]X/ t 2 -
], (35)
Fundamentalsolution in thermoelasticity
2017
The inverse Laplace transform in equation (34) has been evaluated by using the convolution theorem and equation (35). Upon substitution of the resulting expressions and equation (35) in equation (34), we finally arrive at
[
0 = q_..OOai=l ~ ( - 1 ) i - l n ( t -- b i ° r ) e x p ( - ° t i t )
(')
Oil cosh-' ~
}
- Oi2(t 2 - b2or2) 1/2 ,
(36a)
where
bZior2] 1 1 2 23 0,, = [Cio(1 + - ~ - - / + ~ c i i b i i b , o rz + ~ c,obior J
~4g-+b31r2~+cil(l+ 3bzi' +[t cio(,,;o 4,,io, TrO 2[ciob~l
ciibii
+ t t 2-~/2o + bio +
2]
+ ~ ci2biobil r +
ci3bi20r2]
"3[ci°b311-ciib2il +cizbiz +ci3] 2bio 6 J
(36b)
+ t IL~ 6blo L 3 -1- 2b/2o
and
)(
r {bil+ b3il
0i2 = [ci°t~io
b2r2\]
.Uil,
~io r z + c i ' l
21
b~,, 2cilbil ] - tlcm ~ + + Ciz bio ~
3 r 4L
1
+ -~ ci2biobil
2 1 2 2] r + ~ ciabii~r
17 ~rc,oO,~, + 3¢,-,b,~, + 3ci2bi, ] "~ Ci4 +3g/L-g~,~o biZo bio "
(36c)
STRESS D I S T R I B U T I O N Substituting from equation (24) into equations (18) we get upon using the recurrence relations of the modified Bessel functions of the second kind, the equations o% _ - _-7,_,_2-qo(1 + v p ) ~ ( - 1 ) i-I [~kiKl(kir)+[3Zp2Ko(kir)],
(37)
p(k, - k~),=,
_
~'~
qo(l+
vp)~
p(k2--k~) i=l
(_l)i_l[_2kiKi(kir)+([32,Z_2k2i)Ko(kir)] r
'
(38)
where K~ is the modified Bessel function of the second kind of order one. Following the same procedure as used for the temperature distribution, the inverse Laplace transform of equation (37) can be written in the form
qo i~= 1 (_ Orrr ~ --a
l )i_l H(t _ bior)exp(_otit)[ [3Zco(tz - b~or2)_l/2
+ .rCOS '(,--' / \ oior]
2\1/2"] J,
Oior )
+
(39a)
where 1
2
~rilr = ~{2fl [c,(4 + b~lr 2) + 2czbmb;,r 2 + c3b2or2] - a,o(8b;1 + b31r2) - ~,i b,-o(8 + 3b~,r 2) - 2ai2bi. b~or2 - a;3b3or2}
+41 [32 c~
'+ b]~r~]b,o/ + c2(4 + 3bZ~r2) + 3c3b;~b;or 2 + c4b20r2 43/ob/2,
- 4(2ailbil + alzbio) + 2+r .zfLP ~z{clbZil 1 [,-~/2o
lt3
+6
bio J
czbilbio+c3)
2 cib3l' c2b21 -
3ailb~lbio 3aizbii - a i 3 b i o - - c4)],
c3bil
+
(39b)
2018
M.A. EZZAT
If
[ 13
. 6b21~
tri2. = ~ ~OtiOkb~or2+ ~io } + 6(2otilbil + oli2bio) - /32[c,(~+
~)
+ c2(9 + 3b21r2) + 3c3biobilr2 + c4b2ior2]}
t f [24b, 13b31] 3a;1 +~ (8 + 13b~lr 2) + 39ai2bil + 13ai3bio + -24 - ~ Ot/o~b/---~2 + --ETbio } bior __ 18 2/3 [ c,b21bE+2czbn+bio c3)}
t 2 12~iob~t + 2 4 a n b . d-~-~ 2 2 bior b 3ior2
3 3c2b~1 . 3c3bil - T ~ Jr- - - - - ~ - I " "1- C4 bio blo Dio
12a/2 11/3 F -bio r2 +
+ 0ti__._23~ t 3 (aiob;13 3ot/2b/21 F 3oti2bil ~ + ~ r 2 \ ~io bi3o b;o b,o]
(39c)
where
Co = vbo,
cj = vb/+ bj_ 1
otij = :~ ckbi(i_,) , k=O
j = 1, 2, 3, 4,
i = 1, 2,
(39d)
j = O, 1, 2, 3.
Similarly, the inverse transform of equation (38) yields
try,e, = qo ~ ( - I )i-I H( t - bior)exp(-otit) (/32Co - 2/3/o)(/2 - o;or" 2 2x--I/2) /=1 1
l
--~+ trill' c o s h - (\bior/
~2
2~1/2]
O'/2q,(/2- o,or ,
(40a)
J,
where
1
tr,,. = ~ [(/32c, - 2/3.)(8 + 2b21r 2) + 4(/32c2 - 2/3;2)b. b;or 2 + 2(/32c3 - 2/3;3)b;2r2 3 2) + ailbio(8 + 3b/21r2) + 30ti2bnb20 r2 + oti3b30r2] + aio(8bil + bilr
t[
+ ~ (/32c, - 2/3.) + (/32C 4 --
t2 [
(4bi~l
+ b~tr2~ bio / + (/32c2 - 2/3i2)(4 + 3bElr2) + 3(/32c3_ 2/3i3)biobnr2
2/3i4)bEor2 + 4a;°b21 + 8t~il b;1 + 4a;2bio] bio b2 bii
bil
aiob31
+ -2 [ (/32¢1 -- 2fl11) ~ . + 2(fl2C2 -- 2/3;2) ~/o + (fl2¢3 - 2~i3) b/o + --b,2o
+ 3ailb21 + 3ai2b. + aiabio] d bio [ 3 b~, bil ] + ~t3 (/32c, - 2/3il) b~13+ 3(/32c2 - 2/3;2) ~ + 3(/32c3 - 2/3,-3) b/o + (fl2c4 - 2/3,4) , (40b) b;o 0"i2¢, =
1 [(/32CI -- 2/3il )( ~ + b b i o 31r2~+/3(/32C2 -- 2/3/2)(3 + b 21r2) + 3(/32C3--" 2/3i3)bi°bil r2 18 + -~-;0 6b21~ + (/32c4 - 2/3i4)b2i0rE + aio(~-T~_2 ! + 12t~/i bil + 60ti2bio ] \ O;or t [ bEI b;i + - - 118(/32cl - 2/3;i) ~i- + 36(/32c2 - 2/3,2) + 18(/32c3 - 2/3;3) 24 bio ~/o
2019
Fundamental solution in thermoelasticity
5.0
0
4.5 4.0 3.5 3.0 2.5
25 20 15
1.5 1.0
IO ~rr
5
0.5 0
I
I
0.2
0.4
\
I 0.6
I
I
0.8
I
I
0.6
0.8
r -10 Fig. 2. Radial stress distribution.
Fig. 1. Temperature distribution
{24bil 13b31~ 3otil ] + aiok b2.~r2 + ---ZT-bio] + ~ (8 + 13b2~r 2) + 39a~2b~, + 1 3 a , 3 b ~ t2 [
,~ ~ ~ b 3il +
+ 36-- 11(/32c' - "~")~2o +
33(/32c3 -
hi._22._
33(/32c2 - 2/3'2)b2o
2fli3) ~io + 11(fl2c4 - 2fli4) -t 12ai°b 3 2 i2
24atilbil
12ai2]
÷
b ior
t 3 (a,ob 3, 3c~i,b,2 + ~ + + ~ r 2 k b~4or2 b~o
J
3ai2bi,+a,3] b20 bio]'
(4Oc)
and J
fllj = ~, Ckai(y-,), k=O
i = 1, 2,
NUMERICAL
] = O, 1, 2, 3, 4.
(40d)
RESULTS
The problem was solved for the special case of the copper material (~ =0.0168 and f12 = 3.94). The results for the temperature distribution, the radial stress component and for the tangential stress component are shown in Figs 1-3. The value of time t = 0.1 was taken throughout the computations. It was noted that when Zo, v and ~ are all equal the stress components for the two generalized thermoelasticity theories (generalized thermoelasticity
2 0 -2 0¥¥
I
I
I
I
I
0.2
0.4
0.6
0.8
1.0
r
-4
-6 -8
-10 -12 -14 --
Fig. 3. Transverse stress distribution.
2020
M.A. EZZAT
with one relaxation time theory and generalized thermoelasticity with two relaxation times theory) are identical though the temperature for the uncoupled
and coupled
d i s t r i b u t i o n is t o t a l l y d i f f e r e n t . W e n o t e also t h a t
theories the temperature
a n d t h e stress c o m p o n e n t s
have
n o n - v a n i s h i n g ( t h o u g h s m a l l ) v a l u e s f o r a n y v a l u e o f r > 0 w h e n t > 0. T h i s d e m o n s t r a t e s fact o f i n f i n i t e s p e e d s o f p r o p a g a t i o n .
the
T h i s is n o t t h e c a s e f o r b o t h o f g e n e r a l i z e d t h e o r i e s
w h e r e t h e f u n c t i o n s a r e i d e n t i c a l l y z e r o w h e n r > t/b2o. T h e e f f e c t o f t h e h e a t s o u r c e is c o n f i n e d to a b o u n d e d
but time-dependent
region of space surrounding the source. This means that we
h a v e a s i g n a l p r o p a g a t i n g w i t h a finite s p e e d . The temperature
d i s t r i b u t i o n a n d t h e s t r e s s e s crrr a n d ~ry,,~ h a v e i n f i n i t e d i s c o n t i n u i t y at t h e
o r i g i n d u e t o t h e p r e s e n c e o f t h e h e a t s o u r c e a n d t w o finite j u m p s at r = t / b l o a n d r -- t/b2o.
REFERENCES
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
M. BLOT, J. Appl. Phys. 27, 240 (1956). H. LORD and Y. SHULMAN, J. Mech. Phys. Solid 15, 299 (1967). R. DHALIWAL and H. SHERIEF, Q. Appl. Math. 33, 1 (1980). H. SHERIEF and R. DHALIWAL, J. Therm. Stress 4, 407 (1981). H. SHERIEF, J. Therm. Stress 9, 151 (1984). H. SHERIEF and M. EZZAT, J. Therm. Stress 17, 75 (1984). I. MULLER, Archs Rat. Mech. Anal. 41, 319 (1971). A. E. GREEN and N. LAWS, Archs Rat. Mech. Anal. 45, 47 (1972). A. E. GREEN and K. A. LINDSAY, J. Elast. 2, 1 (1972). E. S. ~UHUBI, Thermoelastic solids. In Continum Physics (Edited by A. C. ERINGEN), Vol. II, Chap. 2. Academic Press, New York (1975). S. ERBAY and E. S. SUHUBI, J. Therm. Stress 9, 279 (1986). J. IGNACZAK, J. Therm. Stress 8, 25 (1985). J. IGNACZAK, J. Therm. Stress 1, 41 (1978). R. DHALIWAL and J. ROKNE, J. Therm. Stress 11, 257 (1989). H. SHERIEF, Int. J. Engng Sci. 30, 861 (1992). H. SHERIEF and M. ANWAR, J. Therm. Stress 9, 165 (1986). R. HETNARSKI, Archs Mech. Stosow. 16, 23 (1964). F. OBERHETTINGER and L. BADI1, Tables o f Laplace Transforms. Springer, New York (1973).
(Received 11 April 1995; accepted 7 June 1995)
NOMENCLATURE
A, # = at= 132 = "y= p= r, ~/,, z = c2 = ai = 0% = eij= cE =
Lam6's constants coefficient of linear thermal expansion (X + 2~)/~ (3A + 2/.z)ot t density cylindrical polar coordinates (A + 2tx)/p square of velocity of longitudinal wave components of displacement vector components of the stress tensor components of strain tensor specific heat at constant strain
k = thermal conductivity = pCE/k t = time T = absolute temperature T, = reference temperature chosen so that I(T - To)/Tol<< 1 b = "yTo/~ g = y/k~ 7 Q = intensity of heat source per unit mass r, v = relaxation times ro = relaxation time for Lord-Shulman theory