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Transient thermal stresses on a hollow circular disk under an instantaneous point heat source
NUCLEAR ENGINEERING AND DESIGN 24 (1973) 440-450. NORTH-HOLLAND PUBLISHING COMPANY
TRANSIENT
THERMAL
STRESSES ON A HOLLOW CIRCULAR
UNDER AN INSTANTANEOUS
DISK
POINT HEAT SOURCE
Yoitiro TAKEUTI and Naotake NODA
Mechanical Engineering, Shizuoka University, Johoku, Hamamatsu, Japan Received 2 January 1973
This paper is concerned with a treatment of the asymmetrical transient thermoelastic problem for a hollow circular disk under an instantaneous point heat source at an arbitrary point on the disk. The derivation of temperature and thermal stress components is outlined. The analysis is developed by the technique of Laplace transform.
1. Introduction The thermal stress problems of point heat source in both semi-infinite space and finite bodies have long been of interest to many authors. Recently the transient thermoelastic problem of a solid circular disk due to an instantaneous point heat source has been solved by one of present authors [1]. More recently the same problem has been solved by Hsu [2]. The case of a hollow circular disk, however, introduces more difficulty into the mathematical analysis owing to the dislocation problem in the multiply-connected body. Therefore no solution for the same kind of thermoelastic problem of the hollow circular disk has been obtained so far. As a general analytical treatise for a multiply-connected region in plane thermoelasticity, the problem dealt with in this paper is to find the distribution of temperature and thermas stresses of a hollow circular disk, insulated on both flat faces, subjected to an instantaneous point heat sorce. The solutions obtained from this analysis may also be applied to the plane strain case of an infinite cylinder subjected to a longitudinal line heat source along its axial direction.
2. Analysis For the hollow circular disk having radii a and b in the presence o f an instantaneous point heat source of strength Q on the arbitrary point (rl, 13) of the disk, as shown in fig. 1, the solution of the heat conduction equation by using the Green's function can be expressed b y [3]:
T(r, O, t) = G(r, 0, tlrl,/3, 0) + T 1 (r, 0, t) .
(1)
where G(r, 0, tlrl,/3, O) is the Green's function for an infinite medium at time t and position (r, 0) when an instantaneous point heat source is applied at (rl,~) and t = O, and the function T 1 (r, 0, t) has to satisfy the unsteady heat conduction equation and must vanish for t = O. The equation for T 1 is
1 ~T1
AT 1
K 3t
where
~
1 ~2
A = ~2 + 1 3 ; + - - - Or2 r r 2 30 2
in which the following initial and boundary conditions with thermal insulation on both flat faces hold;
(2)
Y. Takeuti. N. Noda, Thermal stresses on a hollow circular disk
441
×
Fig. l. Instantaneous point heat source on the arbitrary point (rl, ~) on the hollow circular disk. T 1 (r, 0,0) = 0 , ~T ar
haT=O
(3) on
r=a,
~T --+hbT=O ar
on
r=b,
(4)
and K is the thermal diffusivity, and ha, h b are inner and outer surface heat transfer coefficients, respectively. The derivation of the Green's function of this case can be obtained [4] in the form G(r, O, tlrl,~, O) = (Q/47rKt) e x p ( - R2/4Kt)
(5)
in which R2
r 2 + r~ - 2rr 1 cos(0 - / ~ ) .
Let the Laplace transform of the function G and T 1 be: (G*, T o = f 0
(G, TI) e -pt dt .
(6)
Then from eq. (6) G * = (Q/2zr~) K 0 (qR) ,
(7)
where q = px/-P-~-and K 0 is the modified Bessel function of order zero. Eq. (7) can be expressed
= (Q/2rr~) {I0 (qr)Ko (qrl) + 2 ~ /n (qr)Kn(qrl)c°s n(O -/3)} n=l
as
(for r < r l )
G*
(8) = (Q/27rK) (K 0 (qr) I 0 (qrl) + 2 ~ K n (qr) 1n (qrl) cos n (0 -/3)} (for r > rl) , n=l
where In, K n are the modified Bessel functions of order n. The Laplace transform of eq. (2) can be shown as
442
Y. Takeuti, N. Noda, Thermal stresses on a hollow circular disk
82T~
1 aT~
1 ~)2T~
-++ - q2T~ Or2 r ~-r r 2 302
(9)
Considering the symmetry for 0 =/3, the solution of this equation is T~= A ; lo ( qr)+ B ; Ko ( qr) +
tI
n=l
t
{An n (qr) +B n K,, (qr)} cos n(0 -/3)
(10)
Then we have T*=T~+G*
(11)
and Laplace transforms of eqs. (4) are 0T*
---Dr h a T*= 0,
3T*
--+hbT*ar = 0 .
(12)
From eqs. (8), (10), (11) and (12), we obtain
r <5o>, <
T*=
(qr 1)K 0 (qb)-I 0 ( q b ) K 0 (qrl)} ]
2rK L [hahb {I0 (qb)K 0 ( q a ) - I 0 (qa)K 0 (qb)} + haq (I 0 (qa)K l (qb)+ I 1 (qb)K 0 (qa)) × [ha (I0 (qr)K 0 ( q a ) - I 0 (qa)K 0 (qr)} + q (I l (qa)K 0 (qr)+ K 1 (qa)I 0 (qr)}] + hbq(I 1 (qa)K 0 ( q b ) + l 0 (qb)K 1 (qa)}
q2{I 1 (qa)K 1 (qb)
I 1 (qb)K 1 (qa)}]
[q{/(qrl)Kn_ l ( qb) + K n (qrl)in_l ( q b ) } - (h b - n / b ) ( I (qrl) K n (qb) - Kn(qrl)ln (qb)}] [(ha + n/a) (h b - n / b ) { I (qa)K n (qb) I n (qb)K n (qa)} + (h a + n/a)q (I n (qa)Kn_ t (qb)+ In_ 1 ( q b ) K n (qa)}
+2 E
X [q{I n (qr) Kn_ 1 (qa) + In_ 1 (qa) K n ( q r ) } - (h a + n/a) {In (qa) K n (qr) - K n (qa)l n (qr)}] + (h b - n/b) q {in_ 1 (qa) K n (qb) + I n (qb) K n_ 1 (qa)} - q2 (I n- 1 (qa) K n - 1 (qb) - I n 1 (qb) Kn_ 1 (qa)}]
× cos n(0 -/3)1 (for r < r l ) ,
(13)
T* = Q r [q{K0 (qrl)ll (qa)+I° (qrl)K1 (qa)}+ha{lo (qrl)Ko (qa) - K 0 ( q r l ) l 0 (qa)}
× [hb {I0 (qb)K 0 ( q r ) - I 0 (qr)K 0 (qb))+ q {I0 (qr)K 1 (qb)I 1 (qb)K 0 (qr)}] + hbq {I1 (qa) K 0 (qb) + I 0 (qb) K 1 ( q a ) ) - q2 {I1 (qa) K 1 (qb) - I 1 (qb) K 1 (qa)}] oo
+2 ~
n= 1
[q{Kn (qrl)in-1 ( q a ) + I (qrl)Kn_ 1 (qa)}+(h a +n/a){I n (qrl)K n (qa) - K n (qrl)/(qa))] [(ha +n/a)(h b - n / b ) { l n (qa)K n (qb)
I n (qb)K n (qa)}+(h a +n/a) q{In (qa)Kn_ 1 (qb)
443
Y. Takeuti, N. Noda, Thermal stresseson a hollow circulardisk
X [q(l n (qr) K n_ 1 (qb) +I n_ 1 (qb) K n (qr) } + (h b - n/b) (In (qb) K n (qr) - In (qr) K n (qb)]'] + In~ 1 (qb) K. (qa)) + (h b - n/b) q(I n_ 1 (qa) K. (qb) +I n (qb) K n_ 1 (qa)) - q 2
cos n(0 - ~)
J
(14)
× (In_ 1 (qa)Kn_ 1 (qb)-In_ 1 (qb)Kn_ 1 (qa))] (for r>rl) .
From the inversion of the Lapalace transform of eqs. (13)and (14), we obtain T=½Q ~ [EOk(r)/Dok] exp(-g.2kt)+Q ~ ~ [Enk(r)/Dnk ] exp(--ra2kt) c°sn(O-~) k=l n=l k=l
(for r
oc
00
T = ½ Q ~ [ffOk(r)/Dok] e x p ( - K a 2 k t ) + Q ~ ~ [~k(r)/Dnk ] exp(--K"2kt) cosn(O--~) k=l n=l k=l
(for r>rl). (16)
where Eok (r) = aOk [aOk {JO (aOk rl) Y1 (aOk b) - J1 (aOk b) YO (aOk rl)) + hb (JO (°~Okb) YO (aOk rl)-- Jo (aOk rl) YO (aOk b)) X [ha(J 0 (aOkr) YO (aOka) -Jo (aOka) YO (a0kr))+ aOk(Jo (aOkr) Y1 (aOka)-J1 (aOka) YO (%kr))] ' E nk (r) : ¢Xnk[Olnk{Jn (°ink rl) Yn- 1 (°ink b) - Jn- 1 (°~nkb) Yn (°ink rl)) + (h b - n/b) {]. (%k rl) Y, (%k b) - ~ (%k b) Y, (%k rl))] × [%k 7 . (%kO ~ - ~(%k a) --~ - 1 (%k a) Y, (%k r)) -(ha + nla)(Jn (¢Xnkr) Yn (ank a) : In (ank a) Yn (°~nkr))] '
E0k (r) = Ot0k [Ot0k(J0 (°~0krl) Y1 (°~0ka) -- J1 (°t0k a) Y0 (°t0k rl)) + ha (Jo (°lOkrl) Y0 (~X0ka)-- J0 (°t0k a) Y0 (°t0k rl )) X [Ot0k(do (~Ok r) Y1 (aOk b) - J1 (OlOkb) YO (aOk r)) - h h (Jo (°tOkr) YO (aOk b)- Jo (aOk b) YO (c~Okr)] Enk (r) = ~nk [°lnk(dn (°ink rl) Y n-1 (°~nka) --Jn-1 (°Lnka) Yn (°~nkr l))+(ha + n/a) ('In (°Lnka) Yn (O~nkr1) - Jn (ankrl) Yn (°lnk a))] [¢Xnk(dn (°~nkr) Yn-1 (°~nkb) - Jn-1 (°~nkb) Yn (O~nkr)) + (h b - n/b)(,l n (Unkr) Yn (C~nkb) -,I n (C~nkb) Yn (ank r))] , Dok = ~Ok (bhb - aha) ~J1 (aOk a) Yl (uOk b) - J1 (aOk b) Yl (~Oka)) + aOk (bh a -ah b) {Jo (uOk a) YO (~Ok b) -
Jo (aOk b) Yo (c~Oka)) + (ahah b + c~2k b) (J1 (uOka) YO (~Okb) - Jo (~Ok b) Yl (aOk a))
+ (bhah b + ¢x2k a) (Jo (°~Oka) Y1 (aOk b) - J1 (aOk b) Yo (°lOka)} ,
Y. Takeuti, N. Noda, Thermal stresses on a hollow circular disk
444
Dnk = (( 2n/ank) (ha + n/a) (h b - n/ b ) + (h a + n/a) bank - (h b - n/b )aank } (Jn (~nk a) Y~(ank b) -J
(°tnkb) Yn (anka)}- (b(ha +n/a) (h b - n/b) +aoflk} ('In (anka) Yn 1 ( a n k b ) - J n
- (a(h a + ,/a) (h b - ,/b) +
(L,
(ank a) Y (%k b ) - J (ank b) Y
l (ankb) Y (anka)}
(anka)
+ ank (bhb -aha) (Jn-I (ank a) Y - 1 (ank b) - J n - 1 (ank b) Y - 1 (°~nka)) ' and aOk for each value of k is the root of the equation
hahb (Jo (ota) YO (orb) - Jo ( ° & ) ) - h a a ( J o (ota) Yl (o&) - Jl (ab) YO (aa)) + hb Ot(J1 (ozt) YO (orb)-Jo(o
(17)
and Otnk for each value of k is the root of the equation;
(h a + n/a) ( h b - n / b ) (am ( (:~a) Yn ( ozb) - ,In ( ab ) Yn ( aa ) ) + (h a + n/a) o~(J n ( aa ) Yn -1 ( o~b) - ,In- 1 ( o~b) Yn ( oal) } ( h b -- n / b ) o<{,In- 1 ( ota) Yn ( ozb) - Jn ( ab ) Y _ 1 ( o
(18) in which,In and Yn are the Bessel function of order n. From eqs. (17) and (18) Eok (r) = ffOk (r), Enk (r) = ffnk (r), thus putting
Cok (r) = Eok (r) = ffOk (r),
Cnk (r) = Enk (r) = ffnk (r) ,
(19)
we obtain the temperature distribution in the whole region:
T=½Q ~
[Cok(r)/Dok ] exp(--Ka2kt)+ Q ~
k=l
~
[
(20)
n=l k=l
The governing equation in quasi-static two-dimensional thermoelasticity is
AAx=-KAT
,
(21)
where X = thermal stress function, K = material constant, i.e., K = aE for plane stress problem of point heat source, K = a E / ( 1 - u ) for plane strain problem o f line heat source. E = Young's modulus, u = Poisson's ratio. Stress components can be expressed as
i,j= 1,2
o0.= (86 A -- 0i~/) X ,
(when i = / n o t to be summed),
(22)
where o 6 are stress components and 86 is Kronecker's delta. Boundary conditions are
X=ClX+C2Y+C 3 ,
3x/Or=c 1 cos(r,x)+c 2 cos (r,y)
X = Ox/Or = 0 Michell's conditions are
forr=a ,
(23)
for r = b ,
(24)
1I. Takeuti, N. Noda, Thermalstresses on a hollow circulardisk 2~r f o
445
27r al~r(A×+KT)rdO=O,
f o
(y313r-xalraO) (A X +KT) r dO = 0 ,
21T
fo
(xO/ar +ya/rO0) (AX + KT) r dO = 0 .
(25)
In the present problem, it is convenient to split the solution of eq. (21) into the bi-harmonic function and the the particular solution; these will be denoted by Xc and Xp, respectively. Considering the form of eq. (21), Xp is applicable to this problem:
oo
~
[Cnk (r) I ~2k Dnk] exp ( - K c~2k ,) cos n(0 -/7) . k=l
n=l k=l (26)
Because of the symmetry, Xc may be represented in the form Xc =A 0 + B0 In r + for2 +Dor2 In r + (A 1r + B 1r - 1 + C1 r 3 + D 1r In r) cos (0 -/7)
+ ~ (Anrn +Bn r-n + Cnrn+2 +Dn r-n+2) cos n(0 - / 3 ) . n=l
(27)
The thermal stress function X may therefore be obtained as the sum of above two functions. Substituting eqs. (26), (27) and (20) into eqs. (25), we obtain D O = D1 = 0 .
(28)
Considering the symmetry, the rearranged boundary conditions of eqs. (23) and (24) may be written as
X=c'lacos(O-/7)+c 3 ,
3xlar=c' 1 cos(0-17)
forr=a,
X=aX/ar=0
forr=b .
(29)
Substituting eqs. (26) and (27) into eq. (29) and comparing the coefficients of equivalent terms, we can determine the unknown constants. Therefore the final form of the stress expression may be determined after substituting these values of the constants. The stress components for this problem are
° rr = r -2 X'oo + r - l x,r '
°oo = X,rr ,
%0 = - ( r - I ×'0)'r "
(30)
Thus stress components may be expressed by %o = 2C0 - B o r - 2 + 2(Blr-3 + 3Clr) cos (0 -/7)
+ ~ (n(n - 1)Anrn-2 + n(n + 1)gn r - h i 2 + (n + 1) (/'/+ 2) Cnrn + (n - 1) ( r / - 2 ) O n r - n ) c o s n (0 -/7) n=2
+ IKQ ]0 leo7(r)/<~o2~okl exp(-,<-2~
n=l k=l
(31)
Y. Takeuti,N. Noda, Thermalstresseson a hollow circulardisk
446
Orr = 2C 0 + B 0 r - 2 - 2 ( B l r - 3 - C l r ) cos (0 -[3) - ~ n=2 + (n - 2) (n + 1) Cnrn + (n -- 1)
1KQ ~
[Cok(r) /ra2kDok]
{n(n- 1)An rn-2 +n(n + 1)Bnr-n-2
(n + 2)Dnr-n ) cos n(O /3)
exp(-
Ka2k ,)
k=l
+KQ ~
~
[{rCnk(r)-n2Cnk (r)}/r2a2kDnk]exp(-K~2 k t)cos n ( 0 - / 3 ) .
(32)
n=l k=l
Oro=- 2(Blr-3-Clr)sin(O-/3)+ ~ n{(n-1)An rn-2-(n+ l)Bnr n-2 +(n+ l)Cnrn n=2 - (n - 1) Dnr
n) sin (0 -/3) + KQ ~
[n{rCnk(r)-Cnk (r))/r2a2kDnl~] exp(--Ka2kt)
sin
n(O --/3),
(33)
n=l k=l in which C0k (r) -- dCok (r) / d r ,
C0k (r) = d2Cok (r) / d r 2
C'nk (r) = dCnk (r) / dr ,
C'nk (r) =d2Cnk (r) / dr 2
3. Numerical results Numerical examples are carried out for the following two cases, and throughout the c o m p u t a t i o n s the ratio of outer and inner radius has been taken as b/a = 3. Dimensionless quantities are depend as follows:
r =a2 T/Q, PO=b/a,
O0 =a2%o/K Q , Pl =rl/a'
pp =a2orr/KQ,
td =Kt/a2 '
pO =a2ero/KQ, p =r/a,
Ha =aha' Hb = ahb •
ll°°, 1.0-
07
0.5-
a e=rt
p
p
;~
3
O=O
Fig. 2. Variation of temperature on diameter for 0 = 0, n, when an point heat source is applied at the inner region (Pl = 2,/3 = 0).
447
Y. Takeuti, N. Noda, Thermal stresses on a hollow circular disk
\ 0.5-
te'°'°~=
;-'--., ~ ,
td°O'3
/ td-o.o5
l\~\ p,,o ~I~/~ ,
,
I , td-O.3
td'°'5
,
I -0.5-
Fig. 3. H o o p stress d i s t r i b u t i o n o n 0 = 0 , rr w h e n 01 = 2 , / 3 = 0 .
tu=0.05 -0.6-0.4-0.2" 0 e=O
2 P
5
Fig. 4. R a d i a l stress d i s t r i b u t i o n o n 0 = 0 w h e n o 1 = 2,/3 = 0.
21 ~'~'
O.
I["
J
•
0
-o.4
Fig. 5. V a r i a t i o n o f 0 0 a n d PP a l o n g t h e circle w i t h r a d i u s P = 2 w h e n P l
=
2, t3 = 0.
448
Y. Takeuti, N. Noda, Thermal stresses are hollow circular disk
0.6- , ~ td=O-05 --p-I 0.4
td'O.I
---p= 3
,&
0.2- .
-0.3
-0.1 Fig. 6. D i s t r i b u t i o n o f O0 on the b o t h b o u n d a r i e s w h e n o ] = 2, ~ = O, (p = 1 is i n n e r b o u n d a r y a n d p = 3 is o u t e r b o u n d a r y ) .
T
td-o,05
1.5[~td- O. 0 7'
1.0-
0.5-0.3
td=0.5 ,td=l.O
I
P e= rt
P
3
O=O
Fig. 7. V a r i a t i o n o f t e m p e r a t u r e on d i a m e t e r for 0 = 0, ~r. w h e n an p o i n t heat s o u r c e is a p p l i e d at the end o f d i a m e t e r (p I = 3, p = 0).
te=O.07 t d .0. I
0"3 0.2t
t¢.0.3 td-0.5
0,1 p~ ' I td-O.I j g= rt -0.1 -0.2
e=
-0.3 Fig. 8. H o o p stress d i s t r i b u t i o n on 0 = 0, n, w h e n ,o I = 3, t3 = 0.
Y. Takeuti, N. Noda, Thermal stresses on a h o l l o w circular disk
0"05t/ /
/"--~t,-O.O
7
/1._...~ ~ tct°O'l // .~r
X ~ \/t,,-o.~ \ /~/td.l. 0
°~/-'~
~/vtd" °3
/
449
3
Fig. 9. Radial stress distribution on o = 0 when o 1 = 3, ¢~= 0.
0"51
-
I
td-0.5
-02- ~ Fig. 10. Distribution of 00 on the both boundaries when ol = 3, 0 =0.
0.05-~,/td =007
0 I "\td,O,5~td.O, p ~ t d = 0 . 3 e= =/a ~'~
-0.0,5
5
td=Ol
td-0.07
Fig. l l. Hoop stress distribution on 0 = 7r/2 when p 1 = 3,/3 = 0. Case 1. An instantaneous p o i n t heat source is applied at the inner region o f the disk at time t = 0.
The foregoing solution will be illustrated by the following values: p 1 = 2, P0 = 3, H a = H b = oo The numerical results o f the dimensionless temperature and stresses are illustrated in fig. 2 through fig. 6. Fig. 5 shows the variation o f the stresses along the circle with radius p = 2. Fig. 6 illustrates the distribution of 00 on the b o t h boundaries. C a s e 2. An instantaneous point heat source is applied at the outer b o u n d a r y o f the disk at time t = 0. In this case we take
450
Y. Takeuti, N. Noda, T h e r m a l stresses o n a h o l l o w circular disk
Pl = 3 " P 0 = 3 ' H a = H b = 1 . Similarly as b e f o r e the d i s t r i b u t i o n s o f t h e d i m e n s i o n l e s s t e m p e r a t u r e and stresses are illustrated in fig. 7 t h r o u g h fig. 11.
References [i ] 12] [3] [4 ]
Y. Takeuti, Thermal stresses in a circular disk due to an instantaneous point heat source, Bull. JSME, I 0 - 3 9 (1967), 423. T.R. Hsu, Thermal shock on a finite disk due to an instantaneous point heat source, Trans. ASME,E, 36--1 (1969), 113. H.S. Carslaw, and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed. Oxford, London, (1959), Chap. 14. W. Nowacki, Thermoelasticity, Pergamon (1962).
TRANSIENT
THERMAL
STRESSES ON A HOLLOW CIRCULAR
UNDER AN INSTANTANEOUS
DISK
POINT HEAT SOURCE
Yoitiro TAKEUTI and Naotake NODA
Mechanical Engineering, Shizuoka University, Johoku, Hamamatsu, Japan Received 2 January 1973
This paper is concerned with a treatment of the asymmetrical transient thermoelastic problem for a hollow circular disk under an instantaneous point heat source at an arbitrary point on the disk. The derivation of temperature and thermal stress components is outlined. The analysis is developed by the technique of Laplace transform.
1. Introduction The thermal stress problems of point heat source in both semi-infinite space and finite bodies have long been of interest to many authors. Recently the transient thermoelastic problem of a solid circular disk due to an instantaneous point heat source has been solved by one of present authors [1]. More recently the same problem has been solved by Hsu [2]. The case of a hollow circular disk, however, introduces more difficulty into the mathematical analysis owing to the dislocation problem in the multiply-connected body. Therefore no solution for the same kind of thermoelastic problem of the hollow circular disk has been obtained so far. As a general analytical treatise for a multiply-connected region in plane thermoelasticity, the problem dealt with in this paper is to find the distribution of temperature and thermas stresses of a hollow circular disk, insulated on both flat faces, subjected to an instantaneous point heat sorce. The solutions obtained from this analysis may also be applied to the plane strain case of an infinite cylinder subjected to a longitudinal line heat source along its axial direction.
2. Analysis For the hollow circular disk having radii a and b in the presence o f an instantaneous point heat source of strength Q on the arbitrary point (rl, 13) of the disk, as shown in fig. 1, the solution of the heat conduction equation by using the Green's function can be expressed b y [3]:
T(r, O, t) = G(r, 0, tlrl,/3, 0) + T 1 (r, 0, t) .
(1)
where G(r, 0, tlrl,/3, O) is the Green's function for an infinite medium at time t and position (r, 0) when an instantaneous point heat source is applied at (rl,~) and t = O, and the function T 1 (r, 0, t) has to satisfy the unsteady heat conduction equation and must vanish for t = O. The equation for T 1 is
1 ~T1
AT 1
K 3t
where
~
1 ~2
A = ~2 + 1 3 ; + - - - Or2 r r 2 30 2
in which the following initial and boundary conditions with thermal insulation on both flat faces hold;
(2)
Y. Takeuti. N. Noda, Thermal stresses on a hollow circular disk
441
×
Fig. l. Instantaneous point heat source on the arbitrary point (rl, ~) on the hollow circular disk. T 1 (r, 0,0) = 0 , ~T ar
haT=O
(3) on
r=a,
~T --+hbT=O ar
on
r=b,
(4)
and K is the thermal diffusivity, and ha, h b are inner and outer surface heat transfer coefficients, respectively. The derivation of the Green's function of this case can be obtained [4] in the form G(r, O, tlrl,~, O) = (Q/47rKt) e x p ( - R2/4Kt)
(5)
in which R2
r 2 + r~ - 2rr 1 cos(0 - / ~ ) .
Let the Laplace transform of the function G and T 1 be: (G*, T o = f 0
(G, TI) e -pt dt .
(6)
Then from eq. (6) G * = (Q/2zr~) K 0 (qR) ,
(7)
where q = px/-P-~-and K 0 is the modified Bessel function of order zero. Eq. (7) can be expressed
= (Q/2rr~) {I0 (qr)Ko (qrl) + 2 ~ /n (qr)Kn(qrl)c°s n(O -/3)} n=l
as
(for r < r l )
G*
(8) = (Q/27rK) (K 0 (qr) I 0 (qrl) + 2 ~ K n (qr) 1n (qrl) cos n (0 -/3)} (for r > rl) , n=l
where In, K n are the modified Bessel functions of order n. The Laplace transform of eq. (2) can be shown as
442
Y. Takeuti, N. Noda, Thermal stresses on a hollow circular disk
82T~
1 aT~
1 ~)2T~
-++ - q2T~ Or2 r ~-r r 2 302
(9)
Considering the symmetry for 0 =/3, the solution of this equation is T~= A ; lo ( qr)+ B ; Ko ( qr) +
tI
n=l
t
{An n (qr) +B n K,, (qr)} cos n(0 -/3)
(10)
Then we have T*=T~+G*
(11)
and Laplace transforms of eqs. (4) are 0T*
---Dr h a T*= 0,
3T*
--+hbT*ar = 0 .
(12)
From eqs. (8), (10), (11) and (12), we obtain
r <5o>, <
T*=
(qr 1)K 0 (qb)-I 0 ( q b ) K 0 (qrl)} ]
2rK L [hahb {I0 (qb)K 0 ( q a ) - I 0 (qa)K 0 (qb)} + haq (I 0 (qa)K l (qb)+ I 1 (qb)K 0 (qa)) × [ha (I0 (qr)K 0 ( q a ) - I 0 (qa)K 0 (qr)} + q (I l (qa)K 0 (qr)+ K 1 (qa)I 0 (qr)}] + hbq(I 1 (qa)K 0 ( q b ) + l 0 (qb)K 1 (qa)}
q2{I 1 (qa)K 1 (qb)
I 1 (qb)K 1 (qa)}]
[q{/(qrl)Kn_ l ( qb) + K n (qrl)in_l ( q b ) } - (h b - n / b ) ( I (qrl) K n (qb) - Kn(qrl)ln (qb)}] [(ha + n/a) (h b - n / b ) { I (qa)K n (qb) I n (qb)K n (qa)} + (h a + n/a)q (I n (qa)Kn_ t (qb)+ In_ 1 ( q b ) K n (qa)}
+2 E
X [q{I n (qr) Kn_ 1 (qa) + In_ 1 (qa) K n ( q r ) } - (h a + n/a) {In (qa) K n (qr) - K n (qa)l n (qr)}] + (h b - n/b) q {in_ 1 (qa) K n (qb) + I n (qb) K n_ 1 (qa)} - q2 (I n- 1 (qa) K n - 1 (qb) - I n 1 (qb) Kn_ 1 (qa)}]
× cos n(0 -/3)1 (for r < r l ) ,
(13)
T* = Q r [q{K0 (qrl)ll (qa)+I° (qrl)K1 (qa)}+ha{lo (qrl)Ko (qa) - K 0 ( q r l ) l 0 (qa)}
× [hb {I0 (qb)K 0 ( q r ) - I 0 (qr)K 0 (qb))+ q {I0 (qr)K 1 (qb)I 1 (qb)K 0 (qr)}] + hbq {I1 (qa) K 0 (qb) + I 0 (qb) K 1 ( q a ) ) - q2 {I1 (qa) K 1 (qb) - I 1 (qb) K 1 (qa)}] oo
+2 ~
n= 1
[q{Kn (qrl)in-1 ( q a ) + I (qrl)Kn_ 1 (qa)}+(h a +n/a){I n (qrl)K n (qa) - K n (qrl)/(qa))] [(ha +n/a)(h b - n / b ) { l n (qa)K n (qb)
I n (qb)K n (qa)}+(h a +n/a) q{In (qa)Kn_ 1 (qb)
443
Y. Takeuti, N. Noda, Thermal stresseson a hollow circulardisk
X [q(l n (qr) K n_ 1 (qb) +I n_ 1 (qb) K n (qr) } + (h b - n/b) (In (qb) K n (qr) - In (qr) K n (qb)]'] + In~ 1 (qb) K. (qa)) + (h b - n/b) q(I n_ 1 (qa) K. (qb) +I n (qb) K n_ 1 (qa)) - q 2
cos n(0 - ~)
J
(14)
× (In_ 1 (qa)Kn_ 1 (qb)-In_ 1 (qb)Kn_ 1 (qa))] (for r>rl) .
From the inversion of the Lapalace transform of eqs. (13)and (14), we obtain T=½Q ~ [EOk(r)/Dok] exp(-g.2kt)+Q ~ ~ [Enk(r)/Dnk ] exp(--ra2kt) c°sn(O-~) k=l n=l k=l
(for r
oc
00
T = ½ Q ~ [ffOk(r)/Dok] e x p ( - K a 2 k t ) + Q ~ ~ [~k(r)/Dnk ] exp(--K"2kt) cosn(O--~) k=l n=l k=l
(for r>rl). (16)
where Eok (r) = aOk [aOk {JO (aOk rl) Y1 (aOk b) - J1 (aOk b) YO (aOk rl)) + hb (JO (°~Okb) YO (aOk rl)-- Jo (aOk rl) YO (aOk b)) X [ha(J 0 (aOkr) YO (aOka) -Jo (aOka) YO (a0kr))+ aOk(Jo (aOkr) Y1 (aOka)-J1 (aOka) YO (%kr))] ' E nk (r) : ¢Xnk[Olnk{Jn (°ink rl) Yn- 1 (°ink b) - Jn- 1 (°~nkb) Yn (°ink rl)) + (h b - n/b) {]. (%k rl) Y, (%k b) - ~ (%k b) Y, (%k rl))] × [%k 7 . (%kO ~ - ~(%k a) --~ - 1 (%k a) Y, (%k r)) -(ha + nla)(Jn (¢Xnkr) Yn (ank a) : In (ank a) Yn (°~nkr))] '
E0k (r) = Ot0k [Ot0k(J0 (°~0krl) Y1 (°~0ka) -- J1 (°t0k a) Y0 (°t0k rl)) + ha (Jo (°lOkrl) Y0 (~X0ka)-- J0 (°t0k a) Y0 (°t0k rl )) X [Ot0k(do (~Ok r) Y1 (aOk b) - J1 (OlOkb) YO (aOk r)) - h h (Jo (°tOkr) YO (aOk b)- Jo (aOk b) YO (c~Okr)] Enk (r) = ~nk [°lnk(dn (°ink rl) Y n-1 (°~nka) --Jn-1 (°Lnka) Yn (°~nkr l))+(ha + n/a) ('In (°Lnka) Yn (O~nkr1) - Jn (ankrl) Yn (°lnk a))] [¢Xnk(dn (°~nkr) Yn-1 (°~nkb) - Jn-1 (°~nkb) Yn (O~nkr)) + (h b - n/b)(,l n (Unkr) Yn (C~nkb) -,I n (C~nkb) Yn (ank r))] , Dok = ~Ok (bhb - aha) ~J1 (aOk a) Yl (uOk b) - J1 (aOk b) Yl (~Oka)) + aOk (bh a -ah b) {Jo (uOk a) YO (~Ok b) -
Jo (aOk b) Yo (c~Oka)) + (ahah b + c~2k b) (J1 (uOka) YO (~Okb) - Jo (~Ok b) Yl (aOk a))
+ (bhah b + ¢x2k a) (Jo (°~Oka) Y1 (aOk b) - J1 (aOk b) Yo (°lOka)} ,
Y. Takeuti, N. Noda, Thermal stresses on a hollow circular disk
444
Dnk = (( 2n/ank) (ha + n/a) (h b - n/ b ) + (h a + n/a) bank - (h b - n/b )aank } (Jn (~nk a) Y~(ank b) -J
(°tnkb) Yn (anka)}- (b(ha +n/a) (h b - n/b) +aoflk} ('In (anka) Yn 1 ( a n k b ) - J n
- (a(h a + ,/a) (h b - ,/b) +
(L,
(ank a) Y (%k b ) - J (ank b) Y
l (ankb) Y (anka)}
(anka)
+ ank (bhb -aha) (Jn-I (ank a) Y - 1 (ank b) - J n - 1 (ank b) Y - 1 (°~nka)) ' and aOk for each value of k is the root of the equation
hahb (Jo (ota) YO (orb) - Jo ( ° & ) ) - h a a ( J o (ota) Yl (o&) - Jl (ab) YO (aa)) + hb Ot(J1 (ozt) YO (orb)-Jo(o
(17)
and Otnk for each value of k is the root of the equation;
(h a + n/a) ( h b - n / b ) (am ( (:~a) Yn ( ozb) - ,In ( ab ) Yn ( aa ) ) + (h a + n/a) o~(J n ( aa ) Yn -1 ( o~b) - ,In- 1 ( o~b) Yn ( oal) } ( h b -- n / b ) o<{,In- 1 ( ota) Yn ( ozb) - Jn ( ab ) Y _ 1 ( o
(18) in which,In and Yn are the Bessel function of order n. From eqs. (17) and (18) Eok (r) = ffOk (r), Enk (r) = ffnk (r), thus putting
Cok (r) = Eok (r) = ffOk (r),
Cnk (r) = Enk (r) = ffnk (r) ,
(19)
we obtain the temperature distribution in the whole region:
T=½Q ~
[Cok(r)/Dok ] exp(--Ka2kt)+ Q ~
k=l
~
[
(20)
n=l k=l
The governing equation in quasi-static two-dimensional thermoelasticity is
AAx=-KAT
,
(21)
where X = thermal stress function, K = material constant, i.e., K = aE for plane stress problem of point heat source, K = a E / ( 1 - u ) for plane strain problem o f line heat source. E = Young's modulus, u = Poisson's ratio. Stress components can be expressed as
i,j= 1,2
o0.= (86 A -- 0i~/) X ,
(when i = / n o t to be summed),
(22)
where o 6 are stress components and 86 is Kronecker's delta. Boundary conditions are
X=ClX+C2Y+C 3 ,
3x/Or=c 1 cos(r,x)+c 2 cos (r,y)
X = Ox/Or = 0 Michell's conditions are
forr=a ,
(23)
for r = b ,
(24)
1I. Takeuti, N. Noda, Thermalstresses on a hollow circulardisk 2~r f o
445
27r al~r(A×+KT)rdO=O,
f o
(y313r-xalraO) (A X +KT) r dO = 0 ,
21T
fo
(xO/ar +ya/rO0) (AX + KT) r dO = 0 .
(25)
In the present problem, it is convenient to split the solution of eq. (21) into the bi-harmonic function and the the particular solution; these will be denoted by Xc and Xp, respectively. Considering the form of eq. (21), Xp is applicable to this problem:
oo
~
[Cnk (r) I ~2k Dnk] exp ( - K c~2k ,) cos n(0 -/7) . k=l
n=l k=l (26)
Because of the symmetry, Xc may be represented in the form Xc =A 0 + B0 In r + for2 +Dor2 In r + (A 1r + B 1r - 1 + C1 r 3 + D 1r In r) cos (0 -/7)
+ ~ (Anrn +Bn r-n + Cnrn+2 +Dn r-n+2) cos n(0 - / 3 ) . n=l
(27)
The thermal stress function X may therefore be obtained as the sum of above two functions. Substituting eqs. (26), (27) and (20) into eqs. (25), we obtain D O = D1 = 0 .
(28)
Considering the symmetry, the rearranged boundary conditions of eqs. (23) and (24) may be written as
X=c'lacos(O-/7)+c 3 ,
3xlar=c' 1 cos(0-17)
forr=a,
X=aX/ar=0
forr=b .
(29)
Substituting eqs. (26) and (27) into eq. (29) and comparing the coefficients of equivalent terms, we can determine the unknown constants. Therefore the final form of the stress expression may be determined after substituting these values of the constants. The stress components for this problem are
° rr = r -2 X'oo + r - l x,r '
°oo = X,rr ,
%0 = - ( r - I ×'0)'r "
(30)
Thus stress components may be expressed by %o = 2C0 - B o r - 2 + 2(Blr-3 + 3Clr) cos (0 -/7)
+ ~ (n(n - 1)Anrn-2 + n(n + 1)gn r - h i 2 + (n + 1) (/'/+ 2) Cnrn + (n - 1) ( r / - 2 ) O n r - n ) c o s n (0 -/7) n=2
+ IKQ ]0 leo7(r)/<~o2~okl exp(-,<-2~
n=l k=l
(31)
Y. Takeuti,N. Noda, Thermalstresseson a hollow circulardisk
446
Orr = 2C 0 + B 0 r - 2 - 2 ( B l r - 3 - C l r ) cos (0 -[3) - ~ n=2 + (n - 2) (n + 1) Cnrn + (n -- 1)
1KQ ~
[Cok(r) /ra2kDok]
{n(n- 1)An rn-2 +n(n + 1)Bnr-n-2
(n + 2)Dnr-n ) cos n(O /3)
exp(-
Ka2k ,)
k=l
+KQ ~
~
[{rCnk(r)-n2Cnk (r)}/r2a2kDnk]exp(-K~2 k t)cos n ( 0 - / 3 ) .
(32)
n=l k=l
Oro=- 2(Blr-3-Clr)sin(O-/3)+ ~ n{(n-1)An rn-2-(n+ l)Bnr n-2 +(n+ l)Cnrn n=2 - (n - 1) Dnr
n) sin (0 -/3) + KQ ~
[n{rCnk(r)-Cnk (r))/r2a2kDnl~] exp(--Ka2kt)
sin
n(O --/3),
(33)
n=l k=l in which C0k (r) -- dCok (r) / d r ,
C0k (r) = d2Cok (r) / d r 2
C'nk (r) = dCnk (r) / dr ,
C'nk (r) =d2Cnk (r) / dr 2
3. Numerical results Numerical examples are carried out for the following two cases, and throughout the c o m p u t a t i o n s the ratio of outer and inner radius has been taken as b/a = 3. Dimensionless quantities are depend as follows:
r =a2 T/Q, PO=b/a,
O0 =a2%o/K Q , Pl =rl/a'
pp =a2orr/KQ,
td =Kt/a2 '
pO =a2ero/KQ, p =r/a,
Ha =aha' Hb = ahb •
ll°°, 1.0-
07
0.5-
a e=rt
p
p
;~
3
O=O
Fig. 2. Variation of temperature on diameter for 0 = 0, n, when an point heat source is applied at the inner region (Pl = 2,/3 = 0).
447
Y. Takeuti, N. Noda, Thermal stresses on a hollow circular disk
\ 0.5-
te'°'°~=
;-'--., ~ ,
td°O'3
/ td-o.o5
l\~\ p,,o ~I~/~ ,
,
I , td-O.3
td'°'5
,
I -0.5-
Fig. 3. H o o p stress d i s t r i b u t i o n o n 0 = 0 , rr w h e n 01 = 2 , / 3 = 0 .
tu=0.05 -0.6-0.4-0.2" 0 e=O
2 P
5
Fig. 4. R a d i a l stress d i s t r i b u t i o n o n 0 = 0 w h e n o 1 = 2,/3 = 0.
21 ~'~'
O.
I["
J
•
0
-o.4
Fig. 5. V a r i a t i o n o f 0 0 a n d PP a l o n g t h e circle w i t h r a d i u s P = 2 w h e n P l
=
2, t3 = 0.
448
Y. Takeuti, N. Noda, Thermal stresses are hollow circular disk
0.6- , ~ td=O-05 --p-I 0.4
td'O.I
---p= 3
,&
0.2- .
-0.3
-0.1 Fig. 6. D i s t r i b u t i o n o f O0 on the b o t h b o u n d a r i e s w h e n o ] = 2, ~ = O, (p = 1 is i n n e r b o u n d a r y a n d p = 3 is o u t e r b o u n d a r y ) .
T
td-o,05
1.5[~td- O. 0 7'
1.0-
0.5-0.3
td=0.5 ,td=l.O
I
P e= rt
P
3
O=O
Fig. 7. V a r i a t i o n o f t e m p e r a t u r e on d i a m e t e r for 0 = 0, ~r. w h e n an p o i n t heat s o u r c e is a p p l i e d at the end o f d i a m e t e r (p I = 3, p = 0).
te=O.07 t d .0. I
0"3 0.2t
t¢.0.3 td-0.5
0,1 p~ ' I td-O.I j g= rt -0.1 -0.2
e=
-0.3 Fig. 8. H o o p stress d i s t r i b u t i o n on 0 = 0, n, w h e n ,o I = 3, t3 = 0.
Y. Takeuti, N. Noda, Thermal stresses on a h o l l o w circular disk
0"05t/ /
/"--~t,-O.O
7
/1._...~ ~ tct°O'l // .~r
X ~ \/t,,-o.~ \ /~/td.l. 0
°~/-'~
~/vtd" °3
/
449
3
Fig. 9. Radial stress distribution on o = 0 when o 1 = 3, ¢~= 0.
0"51
-
I
td-0.5
-02- ~ Fig. 10. Distribution of 00 on the both boundaries when ol = 3, 0 =0.
0.05-~,/td =007
0 I "\td,O,5~td.O, p ~ t d = 0 . 3 e= =/a ~'~
-0.0,5
5
td=Ol
td-0.07
Fig. l l. Hoop stress distribution on 0 = 7r/2 when p 1 = 3,/3 = 0. Case 1. An instantaneous p o i n t heat source is applied at the inner region o f the disk at time t = 0.
The foregoing solution will be illustrated by the following values: p 1 = 2, P0 = 3, H a = H b = oo The numerical results o f the dimensionless temperature and stresses are illustrated in fig. 2 through fig. 6. Fig. 5 shows the variation o f the stresses along the circle with radius p = 2. Fig. 6 illustrates the distribution of 00 on the b o t h boundaries. C a s e 2. An instantaneous point heat source is applied at the outer b o u n d a r y o f the disk at time t = 0. In this case we take
450
Y. Takeuti, N. Noda, T h e r m a l stresses o n a h o l l o w circular disk
Pl = 3 " P 0 = 3 ' H a = H b = 1 . Similarly as b e f o r e the d i s t r i b u t i o n s o f t h e d i m e n s i o n l e s s t e m p e r a t u r e and stresses are illustrated in fig. 7 t h r o u g h fig. 11.
References [i ] 12] [3] [4 ]
Y. Takeuti, Thermal stresses in a circular disk due to an instantaneous point heat source, Bull. JSME, I 0 - 3 9 (1967), 423. T.R. Hsu, Thermal shock on a finite disk due to an instantaneous point heat source, Trans. ASME,E, 36--1 (1969), 113. H.S. Carslaw, and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed. Oxford, London, (1959), Chap. 14. W. Nowacki, Thermoelasticity, Pergamon (1962).