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Asymmetrical transient thermoelastic problems in a composite hollow circular cylinder
Nuclear Engineering and Design 45 (1978) 159-172 © North-Holland Publishing Company
159
ASYMMETRICAL TRANSIENT THERMOELASTIC PROBLEMS IN A COMPOSITE HOLLOW CIRCULAR CYLINDER Y. TAKEUTI and Y. TANIGAWA Department o f Mechanical Engineering, University o f Osaka Prefecture, Mozu, SakaL Osaka 591, Japan. Received 19 July 1977
The present paper is concerned with the transient thermal stresses in a bonded composite hollow circular cylinder under an arbitrary asymmetrical heat supply. Numerical work is carried out using Laplace transforms and is given for the composite cylinder made of the different materials under three, band heat sources on its outer surface.
1. Introduction In a previous paper [ 1], transient thermal stresses in a bonded, composite, hollow, circular cylinder made of different materials were obtained for the problem under an axisymmetrical temperature distribution. In cases of actual engineering problems, however, we often encounter cases of unsymmetrical heating, So it is necessary to consider the problems due to arbitrary heating in a circumferential direction. The present paper is concerned with the corresponding results for the composite, hollow, circular cylinder under an arbitrary temperature distribution due to a heat supply on the outer surface expressed by the form of arbitrary function of angle. Numerical work was carried out for the transient thermal stress distribution under three band heat sources on the outer surface.
2. Analysis 2.1. H e a t c o n d u c t i o n p r o b l e m
Consider a homogeneous, isotropic, composite, hollow, circular cylinder, as shown in fig. 1, having radii a and b. The temperature distribution on the outer surface of the composite cylinder can be expressed as an arbitrary function f ( O ) with respect to the coordinate 0, which is periodically distributed with an interval of 2 n / k . The inner boundary remains zero. The expression for the temperature condition on the outer boundary is given by f(0) =
(1)
~ a m cos m k O , m=O
in which ao =
(0) dO ,
0
am = -~- o o
f ( O ) cos m k O d O .
Throughout the paper, lower case indices i(1,2) are associated with inner and outer cylinder, respectively. The heat conduction equation of the transient state of the composite cylinder takes the form Ti, t = g i A T i ,
(i = 1,2),
(2)
160
Y. Takeuti and Y. Tanigawa i Thermoelastic problems in hollow cylinder /
~T=fl(e)
~2
~0
/ I Fig. 1. Composite hollow c~cular cylinder with different materials.
where, AT= T,r r +r-lT, r +r-2T, oo . Throughout the paper, a comma denotes a partial differentiation with respect to a variable. The initial and the boundary conditions are Ti = 0 ,
t=O,
(3)
TI = 0 ,
r=a,
(4)
T 1 = Tl , 3`1 Tl,r = 3`2T2,r
r =b ,
(5)
T 2 =f(O)
r = c.
(6)
In order to obtain the solution of eq. (2), we may introduce the Laplace transformation with respect to time. Performing the Laplace transforms on eq. (2) and eqs. (4)-(6) under the consideration of the initial condition (3), we obtain
T;,r+r-IT;r+r-2Ti~oo--q~r;=o,
(7)
in which the Laplace transform of a function is denoted by an asterisk (*) and qi =
•
(8)
Considering the symmetry for 0 = 0, the solution of the fundamental equation (7), with the aid of the separation of variables, is given by t~
T7 = ~
(-4niln(qir) + Bn,Kn(q:) ) cos nO.
(9)
n=O
After substitution of the boundary conditions, we can determine the above integral constants A'ni and Bni, and substituting these integral constants into eq. (9) again, we obtain the required temperature solution in the subsidiary domain as follows ao
T; = ~ am~k2 m=O P ~ ( l m k ( q l a ) K m k ( q l r ) -- I m k ( q l r ) K m k ( q l a ) ) COS mkO , oa
T~ = m~=0 plAI am cosmk0 {ink ~ (3`2 - 3"1)[Imk(q~b) Kink(q2 r) -- Imk(q2r) Kmk(q2b)] X [Imk(qla ) Kmk(ql b) - l m k ( q l b) Kmk(qla)]
(10)
Y. Takeutiand Y. Tanigawa/ Thermoelasticproblems in hollow cylinder
161
+ Xlq I [Imk(q2b ) Kmk(q2r) - Im~(q2r ) Kmk(q ~b)] [Imk(qla ) Kink+ 1(q 1b) + Imk+ 1 ( q l b ) Kmk(q la)]
+ k~qz [lm~+z (q~b) Km~(q~r) + Im~(q~r) Km~+ l(q~b)] [Im~(q~a) Km~(qlb) - Imk(q~b) Km~(q~a)l I , 1 ,(11)
in which IAI
mk =--b-(k2 -
kl )[Im~(qla ) Kmk(qlb ) - Imk(q 1b) Km~(qla)] [Imk(q~b ) Km~(q~c) - Imk(q~c ) Kmk(q2b)]
+ Xlq I [Imk(q2b) Kmk(q2c) -Imk(q2c) Kmk(q2b)] [[mk(qla) Kmk+l(ql b) +Im~+l(qlb) Kmk(qla)] + X2q2 [lmk(ql a) Kmk(ql b) - Imk(qlb) Kmk(qla)] [Imk+ l(q2 b) Kink(q2 c) +Imk(q~c) Kink+ 1(qzb)] " (12) Using the residue theorem, we may perform the inverse Laplace transformation on eqs. (10) and (11), and find the temperature solution in the dimensionless form ~o k l n p + 2X ~ a-m (p-ink -- pmk) cos mkO -- 4 X ~ ~ e-Wins 2 r ~nAms -- COS mkO T ~ ° ' 0' r) = (h - 1) In b + In cm=~Amk ~r m=o s=~ wmsCms (13)
~,2(,o,O,r)=a-o{(h-1)lnb+ln,p} "(X - 1) In b- + In c
~
a-m
+ m=l Amk--[i~mk(P/b)-mk + -~mk(p/b)mk] X cos mkO
e- t o 2ms~ gmBms cos mkO , m=O s= 1 b.)msCms
+2 55
(14)
in which the following dimensionless variables are introduced:
Ti = Tt/T' , -b = b/a ,-5 = c/a, t9 = r/a,
K=~,
~=~2[XI,Z=KIt/a2 ,
am =am/T"
(15)
and the eigenvalue cornsfor each value m are determined as the positive root of the following characteristicequation.
mk(x- 1)[:mk(~o)Ym~(~o)- :mk(~o) Ymk(~o)l[:,.k(K~CO)Ym~(~o)-:mk(K~) Ym~(~O)] + -b(~o[Jmk(~) Ymk+ 1(-b¢~) - Jmk+ 1(b-w) Ymk(¢O)] [Jmk(K-b~o) Ymk(KC-60)-- Jmk(K-C60) Ymk(K-b~)l - x~
[:mk(~) Ymk(b~) -- Jmk(b~) Ymk(O~)] [Jm~+ 1( K ~ ) Ymk(~o~) - J m k ( K ~ ) Ym~+ 1( K ~ ) ] = 0 , (16)
and ~mk, ~mk, 7trek and Ams, Bins, Cms are given by the following expressions;
~mk = (1 + X)-b -ink ÷ (1 - X)-b mk ,
(17)
-~mk = - ( 1 - ~) ~ - m k _ (1 -I-~,)-~mk,
(18)
Amk = ~mk(C /ff )-ink + -~mk(Cf6) mk,
(19)
Ams = Jmk(C.Oms) gmk(P6Oms) -- Jmk(PbJms) Ymk(~Oms),
(20)
Bm~ = m k ( X - 1)[Jmk(~ms) YmkQ~¢Oms)- Jm~(-t;~ms) Ym~(~Oms)] [Jm~(~-t~¢Oms) Ym~(~P6Oms) -
Jmk(KP6Oms) Ymk(Kbc~ms)]
Y. Takeuti and Y. Tanigawa/ Thermoelastic problems in hollow cylinder
162
+ btoms [Jmk(tom,) rink+ 1(btoms) - Jmk÷ l(btoms)Ymk (toms)] X [Jmk(Kbtoms) Ymk (KPtoms) --Jmk(KOtoms) Ymk(KF)torns)] - Xr-btoms [Jmk(toms)rmk(-btoms) -Jmk(btoms)Ymk(toms)]
X [Jmk+ 1(Kbtoms)Ymk(KPtoms)--Jmk(KPtoms)Ymk+ 1 (Kbtom•)] ,
(21)
Crns = -b=( 1 - )~g=)toms [Jm k ( toms)Ym k (IT"toms) -- Jm k (t)toms) Ym k(toms) ] X
[Jmk(Kbtoms)Ymk(r-(toms) --Jmk(Ke~,~rns) Ymk(Kbtorns)]
+ Kb(1 + X) mk [Jmk(toms) Ymk(btoms) - JmkQJtoms) Ymk(toms)]
X [Jmk+ I (gbtoms) Ymk(t~Cb.)ms) -- Jmk(KC~ms) rmk+ 1 (gbtoms)] + KF(1 - k) mk[Jmk(toms ) Ymk(-btoms) - Jmk(btorns) Ymk(toms)]
X [Jmk(K-btoms) Ymk+ l(liCtorns) -- Jmk+ 1(t~Ctoms) rmk(K-btoms)]
+ )~K2-b'(torns[Jmk(toms) Ymk(-btoms) -- Jmk~toms) Ymk(toms)] X [Jmk+ 1(K-btoms) Ymk+ 1(KCtorns) -- Jmk+ 1(KCtoms) Ymk+ I (t~b-toms)] - b(1 + X) mkiJmk(toms ) Ymk+ 1(btoms) - Jmk+ 1(btoms) Ymk(toms)]
x [Jmk(~btoms) Ymk(K~toms)- Jmk(K~ms) Ymk(~bto,.s)] + (1 - ~.) mk [Jmk+ 1(toms) Yrnk(-btoms) - Jmk(-btoms) rink+ 1(toms)]
x [Jmk(Kbtoms)rm~(K~toms)-Jm~(K~toms) rmk(Kbtoms)] -- -btoms [Jmk+ 1(toms) rmk+ 1(btoms) - Jmk+ 1(btoms) Ymk+ 1(tOms)]
x [Jmk(Kbtoms) Ymk(K~toms)- Jmk(~toms) Ymk(K-btom~)] -
-
Kb-2 (1
-- ~)
toms [Jmk(toms) Ymk+ 1(-btoms) - Jmk+ 1(-btoms) Ymk(toms)]
x [Jmk+, (Kt"toms) Ym,~(K~Oms) -- Jm,~(K~toms) Ymk+ 1(~btoms)] -
-
Kbc-toms [Jmk(toms) Ymk+ I (btoms) - Jmk+ 1(btoms) Ymk(torns) ]
X [Jmk (Kbtoms) rink+ 1(KCtoms) -- Jmk+ 1(K-(toms) Ymk(K-btoms)]
+ X~-btoms [Jmk+l (toms) Ymk(btoms) -- Jmk(bto.,s) Ym~+ 1(toms) ] X [Jmk+l (Kbtoms) Ymk(KC-toms) -- Jmk(KCtoms) Ymk+ l(Kbtoms) ] •
(22)
Thus, the solution of the temperature distribution is obtained entirely.
2.2. Transient thermoelastic problem The fundamental equation governing the two-dimensional asymmetrical thermal stress problems may be expressed in the known form,
AAXi = - K i A r i ,
(i = 1,2).
(23)
Now, the fundamental eq. (23) can be resolved into two differential equation systems,
AXi + KiTi = Pi ,
(24)
AP i = 0 .
(25)
Y. Takeuti and Y. Tanigawa/ Thermoelastic problems in hollow cylinder
163
On the other hand, Michell's conditions expressed in the following three integral relations are introduced so as to satisfy the single-valuedness of displacements and rotation in doubly-connected region, 2~r
2r¢
f rei, rdO = 0 , o
21r
f [(sin O)re~r-(cosO)?i,o] d0 = 0 , o
f [(cosO)rei, r+(sinO)?t,o] d 0 = 0 .
(26)
Using the method of separation of variable, the solution of Laplace's eq. (25) can easily be obtained, and the solution satisfying the conditions (26) can be expressed as follows,
Pi = Bot + Co~(O - ~r) In r + DoiO + A i f cos 0 + (Clir - Cot) sin 0 +~
[(Anirn + Bni r-n) cos nO + (Chit n + Ontr -n) sin nO] .
(27)
ri=2
On the other hand, Poisson's eq. (24) can be treated in the same manner as eq. (25), then the complementary solution Xci of eq. (24) can be expressed as Xa = aoi In r + boi + Coco In r + doiO + ~
[(anirn + bntr -n) cos nO + (Chitn + dni r-n) sin nO] ,
(28)
n=l
and taking into account of the solution (13), (14) and (27), the particular solution Xpt of eq. (24) is expressed as
×pt = ~Boi r2 + ~Col(O - n) r2(ln r - 1) + ¼Doir20 + ~A xir3 cos 0 + (~Clir _ 15Cot) r 2 sin 0
r2~--~[(AnirnBni n - 1 r-n ) cos n0 + (~I r n
+-~-n__/=2L\ n + 1
nDni Cir-
n)sinn Ol +~0 i,
(29)
in which
~l=-Kl'a
+
..~ 2 [- ~-oXP2(ln p _ 1) ~m~=l~raP2 ( p-ink pink) L4((--ff~-~ln~--+lnc) ~ n k \ink---l* + m k + l cosrnkO
= = e-t°ms r (omsCms COSink07_J'
~2 = - K I T ' a 2 [ ?tOp-z(Inp-- 1 " ( X - - 1 ) l n b ) 4 (-(~- -- ~ + In ~
-
(30) 1 ~mp21U~mk~/-b)-7 k " 4 m = l Amk [ mk-l*
~mk(P/b)m~ m--k-+l -jcosmkO
2 amBms e -t°msz ~ cos mk . ¢.OmsCms
(31)
Thus, the stress function Xl are determined entirely. The stress components for this problem are
°rri = r-2Xi, oo + r-lXi, r ,
Oooi =
Xt, rr,
OrOi= --(r--l~,O ),r •
(32)
Substituting eqs. (28) and (29) into eq. (32), and taking into account of the symmetrical property for 0 = O, stress components may be expressed by the following relations,
orrl = r-2aol - r -2 ~ n[(n - 1)an1 rn + (n + 1) bnl r-n] cos nO n=l
Y. Takeuti and Y. Tanigawa/ Thermoelasticproblems in hollow cylinder
164
oo
+ { B o l + ¼A11r cos 0 - ,{ ~
[(n - 2 ) A . i r n - (n + 2 ) B n l r - h I cos nO
n.=2
+ K _,1- ~-oX(1 - 2 In p) k x"x dm , 1 ) l n b + l n - c } - - 2 m ~ = l A m k [(mk + 2 ) p - m ~ + ( m k - 2 )
pmk]
llL4{(x-
+ -4Xm~Os~l -~r = =
e-tOmsr 2 ~~mp [ m k-2( m k ms
Oool = - r - 2 a o l + r -~ ~
¢osmkO
--, - 1)Ares +p~msAms] cosmk O],
(33)
r~s
n[(n - 1)an1 rn +(n + 1)bn~r -~] cosnO
~=1
+~B01 +lAllrCOSO +~ C [ ( # +2)An1 ~n - ( n - 2 ) B n l r - n l cosn0 n=2 ~m
~,[+ K11 I-
KoX(1 + 2 In P) L 4{(X 1) In + In
4X ff
~-~°2-'r ~mP-a =
=
Orol =r -2 ~
}
~ Km +-- ~ ------ [(ink* - 2) p -ink + (ink + 2) pink] cos mkO 2 m=l Amk ([ink(ink
1)-p
*)ms] Ams + PC°m~tms } c°s m k
(34)
(.Oms~ms
nI(n-
1)anlr n - (n + 1 ) b n , r -n] sinnO + ~ A l x r s i n O + ~ ~ n ( A n , r n + B n l r - n ) s i n n O
n=l
n=2
_
7r m=
= e-Wins r W3s~ms [ ( m k - 1)Ams - PWm~4ms] sinmk
,
(35)
1
n[(n - 1)an2r n + (n + 1) bn2r -n] cos nO
orr2 = r-2a02 - r - 2 ~ n=l
+ ~Bo2 + ~A 12r cos 0 - ~ ~
[(n - 2) An2r n - (n + 2) Bn2r - n ] cos nO
n=2
+K2T,t -
~o[2(X-1)lnK-l+Zlnpl
I ~
~ m k ( m k - 2)(o/b) mk ] cos mkO -
e-°ams r ~
m= 0002
=
- r - 2 ao2 + r -2 ~
~m
=
[ink(ink - 1) Bins + ~PwmsBms ] cos m k
Wms~ mS
n [(n - 1)an2r n + (n + 1)bn2r -n] cos nO
n=l
+ ~Bo= + ~]a,~r cos o + ~ ~C [(,, + 2) a . 2 e ' - (. - 2) Bn=r-"] cos,,0 n=2
(36) J
,
Y. Takeuti and Y. Tanigawa/ Thermoelasticproblems in hollow cylinder +
1o.1
_
oo
- ( m k + 2) ~mk(p~) rak] cosmkO + 2
165
--
oo
--
,O--2.
~_~ ~.~e_~Om: ~
~ 2 m =o s= I
C°TnsCrns
{[ink(ink- 1)
r2-2co 2 1~ P
msJ
ms
(37)
+ KpCOms~ms} cos mkO], ara2 = r -2 ~ n[(n - 1) an2rn - (n + 1) bn2r -n] sin nO + ~A 12r sin 0 + ~ ~ nOn2r n + Bn2r -n) sin nO n=l. n=2 + K2 T'
I-
~
= ~
l~mk(P/-b ) -ink• + ~mk@l-b ) mk] sin mkO
= ~o2 r -amp-2mk ~-] __2 ~ ~ e - ms ~ [ ( m k - 1)Bms--Kp~msBms ] sinmkvj + g 2 m=O s=1 ¢.omsCms
(38)
where, 74m~ = 1mk(~OmD Ymk +~(P~OmD -- Jmk* , (P~OmD Ymk( ~OmD ,
(39)
Bms = m k ( ~ - 1)[Jmk(COms) Ymk(bwms)-Jmk6~Oms) Ymk(COms)]
x [Jm~(KF~OmDYm~+ ~(K/X°m,). Jm~* 1(~:~Om.) Ym~(~F~OmD] + -b~°ms [Jmk(~°ma) Ymk+ 16Corns) -- Jm~+ ~(-booms) Ym~(COms)] X [:mk(K-b~Oms) Ymk+ 1(gPC.Oms) Jmk+ 1(KP('Oms) Ymk(K-bC'°ms)] -
-
-
~r-b~o,.s[Jm~(~Oms) Ym~6Wm~) - Jm~6~Oms) Ym~(COms)]
X [:mk* 1(gbc-°ms) Ymk+ 1(KPC.Oms)-- ]mk+ 1(KPC.Oma)Ymk* 1(Kb~ms)] •
(40)
In the above expressions (30), (31) and (33)-(38), it is necessary to be arranged in the case of mk equal unity. Then, we may rewrite these equations in such case as follows; 1) For eqs. (30) and (31),
1/(mk-1)*
-+1 - 2 1 n r ,
2) For eqs. (33), (34), (36) and (37),
mk* ~ O,
(41)
3) For eq. (35),
p-ink* ._>2/p , 4) For eq. (38),
~o/~)-m~* -, 2~o/~)-~. Thus, the expressions for stress components are obtained entirely, and the integral constants are determined so as to satisfy the boundary conditions. For the composite cylinder, the boundary conditions at the interface are given
by the stress components and the displacements. Then, we have to derive the expressions for displacements. The
166
Y. Takeuti and Y. Tanigawa / Thermoelastic problems in hollow cylinder
stress-strain relation sv for the plane strain problems are expressed in the known forms; 1 erri = ~ei (Orri - VeiO00i) + (XeiTi '
1
e°°i = ~
( ° ° ° i - Vei°rri) + °leiTi '
2 eroi = Eei (1 + Vei) oro i ,
(42)
in which
1/Eei = (1 - v~)/Ei , Vei = vi/(1 - vi) , Otei= (1 + vi) ~i "
(43)
On the other hand, strain components in terms of the displacements can be expressed as follows;
eooi = r - l ( u r i
erri = Uri, r ,
+ Uoi, o ) ,
eroi = UOi,r - r - l u o i + r - l u r i , O •
(44)
Substituting eq. (44) into eq. (42), and using the expressions for the stress components (33)-(38), displacements
Uri and Uoi can be given by the following relations, EelUrl = -(1 + Vel) r-laol + (1 + vel ) r-2bll cos 0 - (1 + Vel ) r -1 ~ n(anlr n - bnl r-n) cos nO n=2
+ (1 -- v e l ) B o t r + (1 --83re1 ) A l l r2 cos 0
r 5{ 1 1 } 4n=2 -fiT-1 [n - 2 + vel(n + 2)] Anxrn +n - l [n + 2 + vel(n - 2)] Bnl r-n cosn0 oo
+(I+Vel)KIT,[_
~'orX(l-21np) = 4(g7~
+
4La~ .
.
EelttOl
2 amp-' e -wrest . .
.
m=O
~n b + l n c " }
(mkAras
X ~ gmr ( m k - 2 * p-ink + _m_k + 2 pmk) cos mkO 2 m = 1 m----~--k m k - 1 mk+l - ' ) cos mk p(,Om#lms
(45)
+ C1 cos 0 ,
=
= (1 +/"el)
r-2bll sin 0 + (1
+ Pel) r - I
~
l'l(anl rn + bnl r-n) sin nO + ~ A l l
r2 sin 0
n=2
r
S,
1
. n=2
+ (1 +
1
(n +4 +nUel)Anlr n
-
Uel)KiT'[~m~=l -amrmk( ~
4La 5
5
/1' m=0 s=l
e
-¢°2 rYZ'nP-1 ms
3 -- t.OmsCms
n-1
1"
(n-
4 +nVel)Bnlr -n sinnO
#-rak +m~+lpmk
) sin mkO
mkAms sinmkO] - C l s i n 0
J
Ee2ur2 = - (1 + re2) r-Xa02 + (1 + re2) r-2b12 cos 0 - (1 + re2) r -l e~
X ~
n=2
n(an2 rn - bn2 r - n ) cos nO +
- ' 4r n =~ 2 [ ~ 1
12
re2 B02 r + -1 - - 3re2 A 12 r2 cos 0
8
{n-2+Ve2(n+2)}An2r n+ n -l 1 {n+2+Ve2(n - 2)}Bn2r- nl XcosnO
K T' V~-°r{2(X - 1) In b - 1 + 2 In p }
(46)
Y. Takeuti and Y. Tanigawa / Thermoelastic problems in hollow cylinder
167
oo
T' 1 ~ ~m r ( mk - 2" ~mkCO/~)._mk + mk + 2 -~mk(19/-b)mk} cos mkO -4 m~=a'--~--mk[ mk------~--I mk +-----~ 1
-- ~.--~- ~ 5 e -- C°ms'r 3"''''=~ (mk~ms_t~pCOms~ms) CosmkO] + ~ 2 c o s O 2 ffmp-l /¢ m=Os=l ~omsC ras
(47)
oo
Ee2uo2 =.(1 + re2 ) r-2 bl2 sin 0 + (1 + Ve2)r -1 ~ n(an2r n + bn2r -n) sin nO + 5 + re2 A12r 2 sin 0 n=2 8
+4n@2{n~(n+4+nVe2)An2rn-
1s - - -i~ ( nn- 4 +n n VOe 2 ) B n n 2-r - n1}
+ (1 + re2 ) K2T'
"amr mk Amk
,(p/-b)-mk
55
'
ol
+
K m =0 $= 1
2 amO e-Wmsr 3 -- - m k B m s s i n m k (.OrasCm$
~mk (p/-~)mk sin mkO ink+ 1
-C2sinO.
(48)
In the above expressions for displacements, we have to transform these expressions for mk equal unity. Then, we may rewrite in such case as follows; 1) For eq. (45),
{(ink - 2)~(ink - 1)}* -~ 2 In p , 2) For eq. (46),
1~(ink - 1)* -+ 2(1 - In a ) ,
(49)
3) For eq. (47),
{(mk - 2)/(mk - l) }* ~ 2 ln(p/b), 4) For eq. (48),
l/(mk-
1)* -~2{1 - In(p/b)}.
Thus, the expressions for displacements are obtained entirely. Next, we may consider the boundary conditions in order to determine the integral constants. For the composite cylinder, we can assume that there in no free axial displacement of the outer cylinder relative to the inner one. It follows from this assumption that at the interface, the stress normal to the interface and the shear stress along the one, and the displacements are continuous. Then, the boundary conditions available for the asymmetrical thermoelastic problems are given as follows,
Orr1 = 0 ,
OrO1 = 0 ,
r=a,
Orrl =Orr2,OrOl =Ore2, Url =Ur2, UOl =u02 ,
r=b ,
tIrr2 = 0 ,
r =c .
°rO 2 = 0 ,
(50)
The procedure to determine the unknown integral constants can be classified into three cases as shown in the following manner. 1)n=0 In the case of n = 0, four integral constants aoi and Boi can be determined from the following relations,
Y. Takeuti and Y. Tanigawa / Thermoelastic problems in hollow cylinder
168 Orr 1 = 0 ,
r=a,
Orr 1 = Orr 2 , Url = Ur2 ,
r=b,
Orr 2 = 0 ,
r=c,
(51)
and the remaining boundary conditions are satisfied identically. 2)n=mk
= 1,i.e., m = 1 a n d k = 1 .
In the case o f n = m k = 1, integral constants to be determined are b l i , A li and C'i. However, among those constants, Ci are included in the expressions for the displacements. Then, only the constants b 1,' and A li have to be determined in calculating the thermal stress components. First of all, we may find that in this case the boundary conditions for normal stress Orr is identically equivalent to the statement for the shear stress or0. On the other hand, integral constants Ci can be eliminated from the boundary conditions with respect to the radial and circumferential displacements. Thus, the integral constants b l i and A 1i can be determined from the boundary conditions shown as follows, Orr 1 = 0 ,
r=a,
Orr 1 = Orr 2 , (Url + UOl ) = (Ur2 + U 0 2 ) ,
r =b ,
Orr2 = 0 ,
r=c.
(52)
3)n = mk > 1 In the case of n = m k > 1, integral constants ani, bni, A n i and Bni can be determined from the boundary conditions (50). And in the other case n =~ m k , solutions become trivial. Thus, all the integral constants become determined values if the temperature solution is given. For the present problem, we consider only plane strain, then the corresponding axial thermal stress component becomes (53)
Ozzi ='l)i(Orri + O00i ) -- c~iEiTi .
3. Numerical results and conclusion
The foregoing analytical solution will now be applied to a composite cylinder made from alluminum alloy and steel in which the inner material is steel. For illustrative example, it will be assumed that the temperature distribution on the outer surface is given by the band heat source T' as shown in fig. 2. Then, the coefficients am denoted
T'
Fig. 2. Temperature distribution on the outer surface.
Y. Takeutiand Y. Tanigawa/ Thermoelasticproblems in hollow cylinder
169
in eq. ( I ) b e c o m e
k-ST' a°-
27r '
2 T' mk-~ am = - - s i n - m~r
(54)
2
In the above expression, k denotes the number of the periodically distributed band heat source on the outer surface, and ~ is the angle. The computation is carried out for the following data;
-b=b/a=2.0,
g=c/a=3.0,
~k=)k2/~k1 =0.54/0.11 = 4 . 9 0 9 ,
K=qKl/K2 =(0.116/0.94) U= = 0 . 3 5 1 ,
(5s) e = ct=/ei = 2 3 . 1 / 1 2 . 0 = 1.925,
v=v=/vl =0.34/0.3 = 1.133,
E=E=/EI = 0 . 7 2 / 2 . 2 = 0 . 3 2 7 3 .
The temperature and thermal stress distributions are shown in figures for several dimensionless times with the parameter k = 3 and several values of ~. In figs. 3 - 6 , the temperature and thermal stress distributions are shown against the ratio p for several dimensionless time r with respect to ff = 20 °. As an another numerical example, these results for ~ = 120 ° are shown in fig. 7. In the case of ~- = 120 °, the band heat source distributes uniformly on the outer surface, then the problem considered can be treated as an axi-symmetrical one. From the results, it is clearly seen that the discontinuity o f the hoop stress distribution arises at the interface of the composite cylinder, and also, the discontinuity of the gradient of the temperature and radial stress distributions appear at the interface. The hoop and axial stress distributions on the specific points of the composite cylinder are shown against the di-
1-01-
.
i# -
1.0L iV-
%p.,\
T,T"
UOO,F
0.5
°,,\~
~.o4 ~.~.o -,.o
~ -
a.o
I ..¢/
01.0 ~_
iliT, a%.ooo
%!o I
ÜIA
_
pc=
I k~ /~Y'/-Wo °
t?-o.~I-
~
i
~'.o' ~
II~;,~E \ \
~ L / / - ~u
oo
o LJ~ ) ' ' ~ - 2 ; °
128:
~o ~
..... 2 . ~
Le=40~60 °
I\\
J
--
1.o
~:~s~T°~ >
\\~
7
1!0 -
~'~\~
-<:o.o~,~:~o:
Fig. 3. Temperature and thermal stress distributions for ~ = 20° at r = 0 . 0 1 .
"
|
0"=,/k,T'
\
\
"t:O.04, ~=20 °
Fig. 4. Temperature and thermal stress distributions for ~ = 20 ° at r = 0.04.
Y. Takeuti and Y. Tanigawa I Thermoelastic problems in hollow cylinder
170
'° I
I~
""
/ o.~f-
Z.U / 0 6 0 @u
O~!o~ o
60'
~/
I
,,
~!o •
~"
.
°
t
o ~ ~ .
070
0.1-
01.0
°
%0°
,!o -"~.o~
O-oo/k,T"
\\ 2 0
0=-
o 3_0£
-o_-ooJ'~, e :20° i k
~.
~o_ o.,-o..;~:; ~ , ~
. ~,.
f :~t?i:o -o.~I
-o.,~;~~
O,.o_~_~.°
.o---e:,-"
~
2o~
,
\
~. <~
'x '-c= 0"16,, ~-= 20°
-0.II-
Fig. 5. Temperature and thermal stress distributions for ~ = 20 ° at r = 0.16.
o.2~-
/
:
,._ _:~o
-c:oo, ~. =20 o
Fig. 6. Temperature and thermal stress distributions for ~ = 20 ° atr=~.
~.o
o.~
//..-'~.~o.
o.~
y
"
i
-0.5~-
__40
,:>u~',
/
,.o~-
~0=20
"~ ,
~,L ~ ~ o o o.o2~-i ~ .-o:oo
j..~
o.-,^ i.u
O../k,T'
.-~f
,;I
• 03o
_o.,"
l--
--. i
f':
6
\\\Z ;;!
oo
/
T/T"
l,--
0.02
:%
04
~,,'~'" Fo:
,;'7~ ~ /..,#
o . ~ J ~ ~ I.U
'°!
/
~,.= 12 0 °
Fig. 7. Temperature and thermal stress distributions for ~ = 120 °.
1.0
O-ee2/kJ"
-
178 : ~ " Fig. 8. Hoop stress distribution for several values of ~.
Y. Takeuti and Y. Tanigawa / Thermoelastic problems in hollow cylinder
171
O'zz/k~T'
..~ 0.2
~
0 -0.2 -0.4
-0.6
0J -0-2
~ - ~ _ _
O'zz,l klT
"~.,.~
,2 = 20°
o~= 8 0 ° ~
= ov
t
o I
t -0.2 -0.4
~
z
O.C)1 ~ . . . ~ _~_ ~zz,/k,T
~ 0!1
~ = 20I!0 °
" ~ . ~ . _ ~
(0~=~ ~'=2) 120 ° J
Fig. 9. Axial stress distribution for several values of ~.
mensionless time r for several values o f ~ in figs. 8 and 9, respectively. From the results shown in these figures, it is concluded that the temperature and stress distributions of the composite cylinder become too complicated in accordance with the material property and the ratio of the radius. However, we may reduce the maximum stresses by taking into account the quality of materials and the ratio of the radius of the composite cylinder.
Nomenclature a, b, c -b,~
= radius of inner, interface and outer surface of cylinder. = nondimensional radius (b = b/a, ~ = c / a ) ,
am
= Fourier coefficients off(0) (am = a m / T ' ) ,
constants which appear in temperature distributions, ani, bni, Ani, Bni = constants which appear in stress and displacement components, el~ f(o)
Young's modulus (E = E 2 / E x ) , = strain components, = Fourier expansion of temperature on the outer surface,
In, Kn j,,,y,,
= modefied Bessel functions of first and second kind, = Bessel functions of first and second kind,
Ki
= material constant i.e~, K = aE for plane stress problem, K = aE/(1 - u) for plane strain problem,
k p
= number of the band heat source, = parameter of Laplace transforms,
qi
= x/p/Ki,
El
=
172 r, O, z T' Ti t ur, Uo
t~i ri hi ~i p ai/ r X/ Wms
Y. Takeuti and Y. Tanigawa / Thermoelastic problems in hollow cylinder -
= cylindrical coordinates, = temperature o f b a n d heat source o n the outer surface, = temperature change (Ti = T i l T ' ) , = time, = radial and circumferential displacement, = angle o f the b a n d heat source on the outer surface, = linear expansion coefficient (c~ = t~2/~ 1), = t h e r m a l diffusivity (K = Vrrl/K2), = thermal conductivity (~ = ~2/~1), = Poisson's ratio (~ = p 2 / P l ) , = n o n d i m e n s i o n a l radius Co = r/a), = stress c o m p o n e n t s , = n o n d i m e n s i o n a l time (r = r I t/a2), = stress function, = positive root o f eq. (16).
Reference [1] Y. Takeuti et al., Nucl. Eng. Des. 41 (1977) 335.
159
ASYMMETRICAL TRANSIENT THERMOELASTIC PROBLEMS IN A COMPOSITE HOLLOW CIRCULAR CYLINDER Y. TAKEUTI and Y. TANIGAWA Department o f Mechanical Engineering, University o f Osaka Prefecture, Mozu, SakaL Osaka 591, Japan. Received 19 July 1977
The present paper is concerned with the transient thermal stresses in a bonded composite hollow circular cylinder under an arbitrary asymmetrical heat supply. Numerical work is carried out using Laplace transforms and is given for the composite cylinder made of the different materials under three, band heat sources on its outer surface.
1. Introduction In a previous paper [ 1], transient thermal stresses in a bonded, composite, hollow, circular cylinder made of different materials were obtained for the problem under an axisymmetrical temperature distribution. In cases of actual engineering problems, however, we often encounter cases of unsymmetrical heating, So it is necessary to consider the problems due to arbitrary heating in a circumferential direction. The present paper is concerned with the corresponding results for the composite, hollow, circular cylinder under an arbitrary temperature distribution due to a heat supply on the outer surface expressed by the form of arbitrary function of angle. Numerical work was carried out for the transient thermal stress distribution under three band heat sources on the outer surface.
2. Analysis 2.1. H e a t c o n d u c t i o n p r o b l e m
Consider a homogeneous, isotropic, composite, hollow, circular cylinder, as shown in fig. 1, having radii a and b. The temperature distribution on the outer surface of the composite cylinder can be expressed as an arbitrary function f ( O ) with respect to the coordinate 0, which is periodically distributed with an interval of 2 n / k . The inner boundary remains zero. The expression for the temperature condition on the outer boundary is given by f(0) =
(1)
~ a m cos m k O , m=O
in which ao =
(0) dO ,
0
am = -~- o o
f ( O ) cos m k O d O .
Throughout the paper, lower case indices i(1,2) are associated with inner and outer cylinder, respectively. The heat conduction equation of the transient state of the composite cylinder takes the form Ti, t = g i A T i ,
(i = 1,2),
(2)
160
Y. Takeuti and Y. Tanigawa i Thermoelastic problems in hollow cylinder /
~T=fl(e)
~2
~0
/ I Fig. 1. Composite hollow c~cular cylinder with different materials.
where, AT= T,r r +r-lT, r +r-2T, oo . Throughout the paper, a comma denotes a partial differentiation with respect to a variable. The initial and the boundary conditions are Ti = 0 ,
t=O,
(3)
TI = 0 ,
r=a,
(4)
T 1 = Tl , 3`1 Tl,r = 3`2T2,r
r =b ,
(5)
T 2 =f(O)
r = c.
(6)
In order to obtain the solution of eq. (2), we may introduce the Laplace transformation with respect to time. Performing the Laplace transforms on eq. (2) and eqs. (4)-(6) under the consideration of the initial condition (3), we obtain
T;,r+r-IT;r+r-2Ti~oo--q~r;=o,
(7)
in which the Laplace transform of a function is denoted by an asterisk (*) and qi =
•
(8)
Considering the symmetry for 0 = 0, the solution of the fundamental equation (7), with the aid of the separation of variables, is given by t~
T7 = ~
(-4niln(qir) + Bn,Kn(q:) ) cos nO.
(9)
n=O
After substitution of the boundary conditions, we can determine the above integral constants A'ni and Bni, and substituting these integral constants into eq. (9) again, we obtain the required temperature solution in the subsidiary domain as follows ao
T; = ~ am~k2 m=O P ~ ( l m k ( q l a ) K m k ( q l r ) -- I m k ( q l r ) K m k ( q l a ) ) COS mkO , oa
T~ = m~=0 plAI am cosmk0 {ink ~ (3`2 - 3"1)[Imk(q~b) Kink(q2 r) -- Imk(q2r) Kmk(q2b)] X [Imk(qla ) Kmk(ql b) - l m k ( q l b) Kmk(qla)]
(10)
Y. Takeutiand Y. Tanigawa/ Thermoelasticproblems in hollow cylinder
161
+ Xlq I [Imk(q2b ) Kmk(q2r) - Im~(q2r ) Kmk(q ~b)] [Imk(qla ) Kink+ 1(q 1b) + Imk+ 1 ( q l b ) Kmk(q la)]
+ k~qz [lm~+z (q~b) Km~(q~r) + Im~(q~r) Km~+ l(q~b)] [Im~(q~a) Km~(qlb) - Imk(q~b) Km~(q~a)l I , 1 ,(11)
in which IAI
mk =--b-(k2 -
kl )[Im~(qla ) Kmk(qlb ) - Imk(q 1b) Km~(qla)] [Imk(q~b ) Km~(q~c) - Imk(q~c ) Kmk(q2b)]
+ Xlq I [Imk(q2b) Kmk(q2c) -Imk(q2c) Kmk(q2b)] [[mk(qla) Kmk+l(ql b) +Im~+l(qlb) Kmk(qla)] + X2q2 [lmk(ql a) Kmk(ql b) - Imk(qlb) Kmk(qla)] [Imk+ l(q2 b) Kink(q2 c) +Imk(q~c) Kink+ 1(qzb)] " (12) Using the residue theorem, we may perform the inverse Laplace transformation on eqs. (10) and (11), and find the temperature solution in the dimensionless form ~o k l n p + 2X ~ a-m (p-ink -- pmk) cos mkO -- 4 X ~ ~ e-Wins 2 r ~nAms -- COS mkO T ~ ° ' 0' r) = (h - 1) In b + In cm=~Amk ~r m=o s=~ wmsCms (13)
~,2(,o,O,r)=a-o{(h-1)lnb+ln,p} "(X - 1) In b- + In c
~
a-m
+ m=l Amk--[i~mk(P/b)-mk + -~mk(p/b)mk] X cos mkO
e- t o 2ms~ gmBms cos mkO , m=O s= 1 b.)msCms
+2 55
(14)
in which the following dimensionless variables are introduced:
Ti = Tt/T' , -b = b/a ,-5 = c/a, t9 = r/a,
K=~,
~=~2[XI,Z=KIt/a2 ,
am =am/T"
(15)
and the eigenvalue cornsfor each value m are determined as the positive root of the following characteristicequation.
mk(x- 1)[:mk(~o)Ym~(~o)- :mk(~o) Ymk(~o)l[:,.k(K~CO)Ym~(~o)-:mk(K~) Ym~(~O)] + -b(~o[Jmk(~) Ymk+ 1(-b¢~) - Jmk+ 1(b-w) Ymk(¢O)] [Jmk(K-b~o) Ymk(KC-60)-- Jmk(K-C60) Ymk(K-b~)l - x~
[:mk(~) Ymk(b~) -- Jmk(b~) Ymk(O~)] [Jm~+ 1( K ~ ) Ymk(~o~) - J m k ( K ~ ) Ym~+ 1( K ~ ) ] = 0 , (16)
and ~mk, ~mk, 7trek and Ams, Bins, Cms are given by the following expressions;
~mk = (1 + X)-b -ink ÷ (1 - X)-b mk ,
(17)
-~mk = - ( 1 - ~) ~ - m k _ (1 -I-~,)-~mk,
(18)
Amk = ~mk(C /ff )-ink + -~mk(Cf6) mk,
(19)
Ams = Jmk(C.Oms) gmk(P6Oms) -- Jmk(PbJms) Ymk(~Oms),
(20)
Bm~ = m k ( X - 1)[Jmk(~ms) YmkQ~¢Oms)- Jm~(-t;~ms) Ym~(~Oms)] [Jm~(~-t~¢Oms) Ym~(~P6Oms) -
Jmk(KP6Oms) Ymk(Kbc~ms)]
Y. Takeuti and Y. Tanigawa/ Thermoelastic problems in hollow cylinder
162
+ btoms [Jmk(tom,) rink+ 1(btoms) - Jmk÷ l(btoms)Ymk (toms)] X [Jmk(Kbtoms) Ymk (KPtoms) --Jmk(KOtoms) Ymk(KF)torns)] - Xr-btoms [Jmk(toms)rmk(-btoms) -Jmk(btoms)Ymk(toms)]
X [Jmk+ 1(Kbtoms)Ymk(KPtoms)--Jmk(KPtoms)Ymk+ 1 (Kbtom•)] ,
(21)
Crns = -b=( 1 - )~g=)toms [Jm k ( toms)Ym k (IT"toms) -- Jm k (t)toms) Ym k(toms) ] X
[Jmk(Kbtoms)Ymk(r-(toms) --Jmk(Ke~,~rns) Ymk(Kbtorns)]
+ Kb(1 + X) mk [Jmk(toms) Ymk(btoms) - JmkQJtoms) Ymk(toms)]
X [Jmk+ I (gbtoms) Ymk(t~Cb.)ms) -- Jmk(KC~ms) rmk+ 1 (gbtoms)] + KF(1 - k) mk[Jmk(toms ) Ymk(-btoms) - Jmk(btorns) Ymk(toms)]
X [Jmk(K-btoms) Ymk+ l(liCtorns) -- Jmk+ 1(t~Ctoms) rmk(K-btoms)]
+ )~K2-b'(torns[Jmk(toms) Ymk(-btoms) -- Jmk~toms) Ymk(toms)] X [Jmk+ 1(K-btoms) Ymk+ 1(KCtorns) -- Jmk+ 1(KCtoms) Ymk+ I (t~b-toms)] - b(1 + X) mkiJmk(toms ) Ymk+ 1(btoms) - Jmk+ 1(btoms) Ymk(toms)]
x [Jmk(~btoms) Ymk(K~toms)- Jmk(K~ms) Ymk(~bto,.s)] + (1 - ~.) mk [Jmk+ 1(toms) Yrnk(-btoms) - Jmk(-btoms) rink+ 1(toms)]
x [Jmk(Kbtoms)rm~(K~toms)-Jm~(K~toms) rmk(Kbtoms)] -- -btoms [Jmk+ 1(toms) rmk+ 1(btoms) - Jmk+ 1(btoms) Ymk+ 1(tOms)]
x [Jmk(Kbtoms) Ymk(K~toms)- Jmk(~toms) Ymk(K-btom~)] -
-
Kb-2 (1
-- ~)
toms [Jmk(toms) Ymk+ 1(-btoms) - Jmk+ 1(-btoms) Ymk(toms)]
x [Jmk+, (Kt"toms) Ym,~(K~Oms) -- Jm,~(K~toms) Ymk+ 1(~btoms)] -
-
Kbc-toms [Jmk(toms) Ymk+ I (btoms) - Jmk+ 1(btoms) Ymk(torns) ]
X [Jmk (Kbtoms) rink+ 1(KCtoms) -- Jmk+ 1(K-(toms) Ymk(K-btoms)]
+ X~-btoms [Jmk+l (toms) Ymk(btoms) -- Jmk(bto.,s) Ym~+ 1(toms) ] X [Jmk+l (Kbtoms) Ymk(KC-toms) -- Jmk(KCtoms) Ymk+ l(Kbtoms) ] •
(22)
Thus, the solution of the temperature distribution is obtained entirely.
2.2. Transient thermoelastic problem The fundamental equation governing the two-dimensional asymmetrical thermal stress problems may be expressed in the known form,
AAXi = - K i A r i ,
(i = 1,2).
(23)
Now, the fundamental eq. (23) can be resolved into two differential equation systems,
AXi + KiTi = Pi ,
(24)
AP i = 0 .
(25)
Y. Takeuti and Y. Tanigawa/ Thermoelastic problems in hollow cylinder
163
On the other hand, Michell's conditions expressed in the following three integral relations are introduced so as to satisfy the single-valuedness of displacements and rotation in doubly-connected region, 2~r
2r¢
f rei, rdO = 0 , o
21r
f [(sin O)re~r-(cosO)?i,o] d0 = 0 , o
f [(cosO)rei, r+(sinO)?t,o] d 0 = 0 .
(26)
Using the method of separation of variable, the solution of Laplace's eq. (25) can easily be obtained, and the solution satisfying the conditions (26) can be expressed as follows,
Pi = Bot + Co~(O - ~r) In r + DoiO + A i f cos 0 + (Clir - Cot) sin 0 +~
[(Anirn + Bni r-n) cos nO + (Chit n + Ontr -n) sin nO] .
(27)
ri=2
On the other hand, Poisson's eq. (24) can be treated in the same manner as eq. (25), then the complementary solution Xci of eq. (24) can be expressed as Xa = aoi In r + boi + Coco In r + doiO + ~
[(anirn + bntr -n) cos nO + (Chitn + dni r-n) sin nO] ,
(28)
n=l
and taking into account of the solution (13), (14) and (27), the particular solution Xpt of eq. (24) is expressed as
×pt = ~Boi r2 + ~Col(O - n) r2(ln r - 1) + ¼Doir20 + ~A xir3 cos 0 + (~Clir _ 15Cot) r 2 sin 0
r2~--~[(AnirnBni n - 1 r-n ) cos n0 + (~I r n
+-~-n__/=2L\ n + 1
nDni Cir-
n)sinn Ol +~0 i,
(29)
in which
~l=-Kl'a
+
..~ 2 [- ~-oXP2(ln p _ 1) ~m~=l~raP2 ( p-ink pink) L4((--ff~-~ln~--+lnc) ~ n k \ink---l* + m k + l cosrnkO
= = e-t°ms r (omsCms COSink07_J'
~2 = - K I T ' a 2 [ ?tOp-z(Inp-- 1 " ( X - - 1 ) l n b ) 4 (-(~- -- ~ + In ~
-
(30) 1 ~mp21U~mk~/-b)-7 k " 4 m = l Amk [ mk-l*
~mk(P/b)m~ m--k-+l -jcosmkO
2 amBms e -t°msz ~ cos mk . ¢.OmsCms
(31)
Thus, the stress function Xl are determined entirely. The stress components for this problem are
°rri = r-2Xi, oo + r-lXi, r ,
Oooi =
Xt, rr,
OrOi= --(r--l~,O ),r •
(32)
Substituting eqs. (28) and (29) into eq. (32), and taking into account of the symmetrical property for 0 = O, stress components may be expressed by the following relations,
orrl = r-2aol - r -2 ~ n[(n - 1)an1 rn + (n + 1) bnl r-n] cos nO n=l
Y. Takeuti and Y. Tanigawa/ Thermoelasticproblems in hollow cylinder
164
oo
+ { B o l + ¼A11r cos 0 - ,{ ~
[(n - 2 ) A . i r n - (n + 2 ) B n l r - h I cos nO
n.=2
+ K _,1- ~-oX(1 - 2 In p) k x"x dm , 1 ) l n b + l n - c } - - 2 m ~ = l A m k [(mk + 2 ) p - m ~ + ( m k - 2 )
pmk]
llL4{(x-
+ -4Xm~Os~l -~r = =
e-tOmsr 2 ~~mp [ m k-2( m k ms
Oool = - r - 2 a o l + r -~ ~
¢osmkO
--, - 1)Ares +p~msAms] cosmk O],
(33)
r~s
n[(n - 1)an1 rn +(n + 1)bn~r -~] cosnO
~=1
+~B01 +lAllrCOSO +~ C [ ( # +2)An1 ~n - ( n - 2 ) B n l r - n l cosn0 n=2 ~m
~,[+ K11 I-
KoX(1 + 2 In P) L 4{(X 1) In + In
4X ff
~-~°2-'r ~mP-a =
=
Orol =r -2 ~
}
~ Km +-- ~ ------ [(ink* - 2) p -ink + (ink + 2) pink] cos mkO 2 m=l Amk ([ink(ink
1)-p
*)ms] Ams + PC°m~tms } c°s m k
(34)
(.Oms~ms
nI(n-
1)anlr n - (n + 1 ) b n , r -n] sinnO + ~ A l x r s i n O + ~ ~ n ( A n , r n + B n l r - n ) s i n n O
n=l
n=2
_
7r m=
= e-Wins r W3s~ms [ ( m k - 1)Ams - PWm~4ms] sinmk
,
(35)
1
n[(n - 1)an2r n + (n + 1) bn2r -n] cos nO
orr2 = r-2a02 - r - 2 ~ n=l
+ ~Bo2 + ~A 12r cos 0 - ~ ~
[(n - 2) An2r n - (n + 2) Bn2r - n ] cos nO
n=2
+K2T,t -
~o[2(X-1)lnK-l+Zlnpl
I ~
~ m k ( m k - 2)(o/b) mk ] cos mkO -
e-°ams r ~
m= 0002
=
- r - 2 ao2 + r -2 ~
~m
=
[ink(ink - 1) Bins + ~PwmsBms ] cos m k
Wms~ mS
n [(n - 1)an2r n + (n + 1)bn2r -n] cos nO
n=l
+ ~Bo= + ~]a,~r cos o + ~ ~C [(,, + 2) a . 2 e ' - (. - 2) Bn=r-"] cos,,0 n=2
(36) J
,
Y. Takeuti and Y. Tanigawa/ Thermoelasticproblems in hollow cylinder +
1o.1
_
oo
- ( m k + 2) ~mk(p~) rak] cosmkO + 2
165
--
oo
--
,O--2.
~_~ ~.~e_~Om: ~
~ 2 m =o s= I
C°TnsCrns
{[ink(ink- 1)
r2-2co 2 1~ P
msJ
ms
(37)
+ KpCOms~ms} cos mkO], ara2 = r -2 ~ n[(n - 1) an2rn - (n + 1) bn2r -n] sin nO + ~A 12r sin 0 + ~ ~ nOn2r n + Bn2r -n) sin nO n=l. n=2 + K2 T'
I-
~
= ~
l~mk(P/-b ) -ink• + ~mk@l-b ) mk] sin mkO
= ~o2 r -amp-2mk ~-] __2 ~ ~ e - ms ~ [ ( m k - 1)Bms--Kp~msBms ] sinmkvj + g 2 m=O s=1 ¢.omsCms
(38)
where, 74m~ = 1mk(~OmD Ymk +~(P~OmD -- Jmk* , (P~OmD Ymk( ~OmD ,
(39)
Bms = m k ( ~ - 1)[Jmk(COms) Ymk(bwms)-Jmk6~Oms) Ymk(COms)]
x [Jm~(KF~OmDYm~+ ~(K/X°m,). Jm~* 1(~:~Om.) Ym~(~F~OmD] + -b~°ms [Jmk(~°ma) Ymk+ 16Corns) -- Jm~+ ~(-booms) Ym~(COms)] X [:mk(K-b~Oms) Ymk+ 1(gPC.Oms) Jmk+ 1(KP('Oms) Ymk(K-bC'°ms)] -
-
-
~r-b~o,.s[Jm~(~Oms) Ym~6Wm~) - Jm~6~Oms) Ym~(COms)]
X [:mk* 1(gbc-°ms) Ymk+ 1(KPC.Oms)-- ]mk+ 1(KPC.Oma)Ymk* 1(Kb~ms)] •
(40)
In the above expressions (30), (31) and (33)-(38), it is necessary to be arranged in the case of mk equal unity. Then, we may rewrite these equations in such case as follows; 1) For eqs. (30) and (31),
1/(mk-1)*
-+1 - 2 1 n r ,
2) For eqs. (33), (34), (36) and (37),
mk* ~ O,
(41)
3) For eq. (35),
p-ink* ._>2/p , 4) For eq. (38),
~o/~)-m~* -, 2~o/~)-~. Thus, the expressions for stress components are obtained entirely, and the integral constants are determined so as to satisfy the boundary conditions. For the composite cylinder, the boundary conditions at the interface are given
by the stress components and the displacements. Then, we have to derive the expressions for displacements. The
166
Y. Takeuti and Y. Tanigawa / Thermoelastic problems in hollow cylinder
stress-strain relation sv for the plane strain problems are expressed in the known forms; 1 erri = ~ei (Orri - VeiO00i) + (XeiTi '
1
e°°i = ~
( ° ° ° i - Vei°rri) + °leiTi '
2 eroi = Eei (1 + Vei) oro i ,
(42)
in which
1/Eei = (1 - v~)/Ei , Vei = vi/(1 - vi) , Otei= (1 + vi) ~i "
(43)
On the other hand, strain components in terms of the displacements can be expressed as follows;
eooi = r - l ( u r i
erri = Uri, r ,
+ Uoi, o ) ,
eroi = UOi,r - r - l u o i + r - l u r i , O •
(44)
Substituting eq. (44) into eq. (42), and using the expressions for the stress components (33)-(38), displacements
Uri and Uoi can be given by the following relations, EelUrl = -(1 + Vel) r-laol + (1 + vel ) r-2bll cos 0 - (1 + Vel ) r -1 ~ n(anlr n - bnl r-n) cos nO n=2
+ (1 -- v e l ) B o t r + (1 --83re1 ) A l l r2 cos 0
r 5{ 1 1 } 4n=2 -fiT-1 [n - 2 + vel(n + 2)] Anxrn +n - l [n + 2 + vel(n - 2)] Bnl r-n cosn0 oo
+(I+Vel)KIT,[_
~'orX(l-21np) = 4(g7~
+
4La~ .
.
EelttOl
2 amp-' e -wrest . .
.
m=O
~n b + l n c " }
(mkAras
X ~ gmr ( m k - 2 * p-ink + _m_k + 2 pmk) cos mkO 2 m = 1 m----~--k m k - 1 mk+l - ' ) cos mk p(,Om#lms
(45)
+ C1 cos 0 ,
=
= (1 +/"el)
r-2bll sin 0 + (1
+ Pel) r - I
~
l'l(anl rn + bnl r-n) sin nO + ~ A l l
r2 sin 0
n=2
r
S,
1
. n=2
+ (1 +
1
(n +4 +nUel)Anlr n
-
Uel)KiT'[~m~=l -amrmk( ~
4La 5
5
/1' m=0 s=l
e
-¢°2 rYZ'nP-1 ms
3 -- t.OmsCms
n-1
1"
(n-
4 +nVel)Bnlr -n sinnO
#-rak +m~+lpmk
) sin mkO
mkAms sinmkO] - C l s i n 0
J
Ee2ur2 = - (1 + re2) r-Xa02 + (1 + re2) r-2b12 cos 0 - (1 + re2) r -l e~
X ~
n=2
n(an2 rn - bn2 r - n ) cos nO +
- ' 4r n =~ 2 [ ~ 1
12
re2 B02 r + -1 - - 3re2 A 12 r2 cos 0
8
{n-2+Ve2(n+2)}An2r n+ n -l 1 {n+2+Ve2(n - 2)}Bn2r- nl XcosnO
K T' V~-°r{2(X - 1) In b - 1 + 2 In p }
(46)
Y. Takeuti and Y. Tanigawa / Thermoelastic problems in hollow cylinder
167
oo
T' 1 ~ ~m r ( mk - 2" ~mkCO/~)._mk + mk + 2 -~mk(19/-b)mk} cos mkO -4 m~=a'--~--mk[ mk------~--I mk +-----~ 1
-- ~.--~- ~ 5 e -- C°ms'r 3"''''=~ (mk~ms_t~pCOms~ms) CosmkO] + ~ 2 c o s O 2 ffmp-l /¢ m=Os=l ~omsC ras
(47)
oo
Ee2uo2 =.(1 + re2 ) r-2 bl2 sin 0 + (1 + Ve2)r -1 ~ n(an2r n + bn2r -n) sin nO + 5 + re2 A12r 2 sin 0 n=2 8
+4n@2{n~(n+4+nVe2)An2rn-
1s - - -i~ ( nn- 4 +n n VOe 2 ) B n n 2-r - n1}
+ (1 + re2 ) K2T'
"amr mk Amk
,(p/-b)-mk
55
'
ol
+
K m =0 $= 1
2 amO e-Wmsr 3 -- - m k B m s s i n m k (.OrasCm$
~mk (p/-~)mk sin mkO ink+ 1
-C2sinO.
(48)
In the above expressions for displacements, we have to transform these expressions for mk equal unity. Then, we may rewrite in such case as follows; 1) For eq. (45),
{(ink - 2)~(ink - 1)}* -~ 2 In p , 2) For eq. (46),
1~(ink - 1)* -+ 2(1 - In a ) ,
(49)
3) For eq. (47),
{(mk - 2)/(mk - l) }* ~ 2 ln(p/b), 4) For eq. (48),
l/(mk-
1)* -~2{1 - In(p/b)}.
Thus, the expressions for displacements are obtained entirely. Next, we may consider the boundary conditions in order to determine the integral constants. For the composite cylinder, we can assume that there in no free axial displacement of the outer cylinder relative to the inner one. It follows from this assumption that at the interface, the stress normal to the interface and the shear stress along the one, and the displacements are continuous. Then, the boundary conditions available for the asymmetrical thermoelastic problems are given as follows,
Orr1 = 0 ,
OrO1 = 0 ,
r=a,
Orrl =Orr2,OrOl =Ore2, Url =Ur2, UOl =u02 ,
r=b ,
tIrr2 = 0 ,
r =c .
°rO 2 = 0 ,
(50)
The procedure to determine the unknown integral constants can be classified into three cases as shown in the following manner. 1)n=0 In the case of n = 0, four integral constants aoi and Boi can be determined from the following relations,
Y. Takeuti and Y. Tanigawa / Thermoelastic problems in hollow cylinder
168 Orr 1 = 0 ,
r=a,
Orr 1 = Orr 2 , Url = Ur2 ,
r=b,
Orr 2 = 0 ,
r=c,
(51)
and the remaining boundary conditions are satisfied identically. 2)n=mk
= 1,i.e., m = 1 a n d k = 1 .
In the case o f n = m k = 1, integral constants to be determined are b l i , A li and C'i. However, among those constants, Ci are included in the expressions for the displacements. Then, only the constants b 1,' and A li have to be determined in calculating the thermal stress components. First of all, we may find that in this case the boundary conditions for normal stress Orr is identically equivalent to the statement for the shear stress or0. On the other hand, integral constants Ci can be eliminated from the boundary conditions with respect to the radial and circumferential displacements. Thus, the integral constants b l i and A 1i can be determined from the boundary conditions shown as follows, Orr 1 = 0 ,
r=a,
Orr 1 = Orr 2 , (Url + UOl ) = (Ur2 + U 0 2 ) ,
r =b ,
Orr2 = 0 ,
r=c.
(52)
3)n = mk > 1 In the case of n = m k > 1, integral constants ani, bni, A n i and Bni can be determined from the boundary conditions (50). And in the other case n =~ m k , solutions become trivial. Thus, all the integral constants become determined values if the temperature solution is given. For the present problem, we consider only plane strain, then the corresponding axial thermal stress component becomes (53)
Ozzi ='l)i(Orri + O00i ) -- c~iEiTi .
3. Numerical results and conclusion
The foregoing analytical solution will now be applied to a composite cylinder made from alluminum alloy and steel in which the inner material is steel. For illustrative example, it will be assumed that the temperature distribution on the outer surface is given by the band heat source T' as shown in fig. 2. Then, the coefficients am denoted
T'
Fig. 2. Temperature distribution on the outer surface.
Y. Takeutiand Y. Tanigawa/ Thermoelasticproblems in hollow cylinder
169
in eq. ( I ) b e c o m e
k-ST' a°-
27r '
2 T' mk-~ am = - - s i n - m~r
(54)
2
In the above expression, k denotes the number of the periodically distributed band heat source on the outer surface, and ~ is the angle. The computation is carried out for the following data;
-b=b/a=2.0,
g=c/a=3.0,
~k=)k2/~k1 =0.54/0.11 = 4 . 9 0 9 ,
K=qKl/K2 =(0.116/0.94) U= = 0 . 3 5 1 ,
(5s) e = ct=/ei = 2 3 . 1 / 1 2 . 0 = 1.925,
v=v=/vl =0.34/0.3 = 1.133,
E=E=/EI = 0 . 7 2 / 2 . 2 = 0 . 3 2 7 3 .
The temperature and thermal stress distributions are shown in figures for several dimensionless times with the parameter k = 3 and several values of ~. In figs. 3 - 6 , the temperature and thermal stress distributions are shown against the ratio p for several dimensionless time r with respect to ff = 20 °. As an another numerical example, these results for ~ = 120 ° are shown in fig. 7. In the case of ~- = 120 °, the band heat source distributes uniformly on the outer surface, then the problem considered can be treated as an axi-symmetrical one. From the results, it is clearly seen that the discontinuity o f the hoop stress distribution arises at the interface of the composite cylinder, and also, the discontinuity of the gradient of the temperature and radial stress distributions appear at the interface. The hoop and axial stress distributions on the specific points of the composite cylinder are shown against the di-
1-01-
.
i# -
1.0L iV-
%p.,\
T,T"
UOO,F
0.5
°,,\~
~.o4 ~.~.o -,.o
~ -
a.o
I ..¢/
01.0 ~_
iliT, a%.ooo
%!o I
ÜIA
_
pc=
I k~ /~Y'/-Wo °
t?-o.~I-
~
i
~'.o' ~
II~;,~E \ \
~ L / / - ~u
oo
o LJ~ ) ' ' ~ - 2 ; °
128:
~o ~
..... 2 . ~
Le=40~60 °
I\\
J
--
1.o
~:~s~T°~ >
\\~
7
1!0 -
~'~\~
-<:o.o~,~:~o:
Fig. 3. Temperature and thermal stress distributions for ~ = 20° at r = 0 . 0 1 .
"
|
0"=,/k,T'
\
\
"t:O.04, ~=20 °
Fig. 4. Temperature and thermal stress distributions for ~ = 20 ° at r = 0.04.
Y. Takeuti and Y. Tanigawa I Thermoelastic problems in hollow cylinder
170
'° I
I~
""
/ o.~f-
Z.U / 0 6 0 @u
O~!o~ o
60'
~/
I
,,
~!o •
~"
.
°
t
o ~ ~ .
070
0.1-
01.0
°
%0°
,!o -"~.o~
O-oo/k,T"
\\ 2 0
0=-
o 3_0£
-o_-ooJ'~, e :20° i k
~.
~o_ o.,-o..;~:; ~ , ~
. ~,.
f :~t?i:o -o.~I
-o.,~;~~
O,.o_~_~.°
.o---e:,-"
~
2o~
,
\
~. <~
'x '-c= 0"16,, ~-= 20°
-0.II-
Fig. 5. Temperature and thermal stress distributions for ~ = 20 ° at r = 0.16.
o.2~-
/
:
,._ _:~o
-c:oo, ~. =20 o
Fig. 6. Temperature and thermal stress distributions for ~ = 20 ° atr=~.
~.o
o.~
//..-'~.~o.
o.~
y
"
i
-0.5~-
__40
,:>u~',
/
,.o~-
~0=20
"~ ,
~,L ~ ~ o o o.o2~-i ~ .-o:oo
j..~
o.-,^ i.u
O../k,T'
.-~f
,;I
• 03o
_o.,"
l--
--. i
f':
6
\\\Z ;;!
oo
/
T/T"
l,--
0.02
:%
04
~,,'~'" Fo:
,;'7~ ~ /..,#
o . ~ J ~ ~ I.U
'°!
/
~,.= 12 0 °
Fig. 7. Temperature and thermal stress distributions for ~ = 120 °.
1.0
O-ee2/kJ"
-
178 : ~ " Fig. 8. Hoop stress distribution for several values of ~.
Y. Takeuti and Y. Tanigawa / Thermoelastic problems in hollow cylinder
171
O'zz/k~T'
..~ 0.2
~
0 -0.2 -0.4
-0.6
0J -0-2
~ - ~ _ _
O'zz,l klT
"~.,.~
,2 = 20°
o~= 8 0 ° ~
= ov
t
o I
t -0.2 -0.4
~
z
O.C)1 ~ . . . ~ _~_ ~zz,/k,T
~ 0!1
~ = 20I!0 °
" ~ . ~ . _ ~
(0~=~ ~'=2) 120 ° J
Fig. 9. Axial stress distribution for several values of ~.
mensionless time r for several values o f ~ in figs. 8 and 9, respectively. From the results shown in these figures, it is concluded that the temperature and stress distributions of the composite cylinder become too complicated in accordance with the material property and the ratio of the radius. However, we may reduce the maximum stresses by taking into account the quality of materials and the ratio of the radius of the composite cylinder.
Nomenclature a, b, c -b,~
= radius of inner, interface and outer surface of cylinder. = nondimensional radius (b = b/a, ~ = c / a ) ,
am
= Fourier coefficients off(0) (am = a m / T ' ) ,
constants which appear in temperature distributions, ani, bni, Ani, Bni = constants which appear in stress and displacement components, el~ f(o)
Young's modulus (E = E 2 / E x ) , = strain components, = Fourier expansion of temperature on the outer surface,
In, Kn j,,,y,,
= modefied Bessel functions of first and second kind, = Bessel functions of first and second kind,
Ki
= material constant i.e~, K = aE for plane stress problem, K = aE/(1 - u) for plane strain problem,
k p
= number of the band heat source, = parameter of Laplace transforms,
qi
= x/p/Ki,
El
=
172 r, O, z T' Ti t ur, Uo
t~i ri hi ~i p ai/ r X/ Wms
Y. Takeuti and Y. Tanigawa / Thermoelastic problems in hollow cylinder -
= cylindrical coordinates, = temperature o f b a n d heat source o n the outer surface, = temperature change (Ti = T i l T ' ) , = time, = radial and circumferential displacement, = angle o f the b a n d heat source on the outer surface, = linear expansion coefficient (c~ = t~2/~ 1), = t h e r m a l diffusivity (K = Vrrl/K2), = thermal conductivity (~ = ~2/~1), = Poisson's ratio (~ = p 2 / P l ) , = n o n d i m e n s i o n a l radius Co = r/a), = stress c o m p o n e n t s , = n o n d i m e n s i o n a l time (r = r I t/a2), = stress function, = positive root o f eq. (16).
Reference [1] Y. Takeuti et al., Nucl. Eng. Des. 41 (1977) 335.