TEMPERATURE PLATE

WITH

TRANSFER Yu.

STRESSES

IN AN ANNULAR

TEMPERATURE-DEPENDENT

ANISOTROPIC HEAT

COEFFICIENT M. Kolyano

and

V.

L.

Lozben'

UDC 678.5:539.377

T h e nonsteady t e m p e r a t u r e s t r e s s e s a r e d e t e r m i n e d for an annular plate, an infinite plate with a c i r c u l a r opening, and a c i r c u l a r plate, whose coefficients of heat t r a n s f e r f r o m the l a t e r a l s u r f a c e s a r e functions of t e m p e r a t u r e .

Consider a thin plate p o s s e s s i n g t h e r m a l and elastic cylindrical a n i s o t r o p y and having at each point a plane of t h e r m a l and elastic s y m m e t r y to which the z axis is n o r m a l . Through the l a t e r a l s u r f a c e s of the plate heat t r a n s f e r with the e x t e r n a l m e d i u m at constant t e m p e r a t u r e t o t a k e s place in a c c o r d a n c e with Newton's law, the heat t r a n s f e r coefficient depending on t e m p e r a t u r e e ( T ) . At the initial m o m e n t the t e m p e r a t u r e of the plate is a s s u m e d to be constant, i.e., Tlx=o=Tn"

(1)

If the t h e r m a l conductivity of the plate in the middle s u r f a c e is v e r y s m a l l as c o m p a r e d with that in the d i r e c t i o n of the z axis, the heat conduction equation for d e t e r m i n i n g the g e n e r a l i z e d t w o - d i m e n s i o n a l t e m p e r a t u r e field h a s the following f o r m [1]: OT

-~(r) (r-t0)=ey(,

(2)

w h e r e c = Cv5; Cv is the v o l u m e specific heat: 25 is the t h i c k n e s s of the plate: ~" is t i m e . Any t e m p e r a t u r e dependence of the heat t r a n s f e r coefficient can be s a t i s f a c t o r i l y a p p r o x i m a t e d by a p i e c e w i s e - c o n t i n u o u s function of t e m p e r a t u r e [11. E x p e r i m e n t a l data on the t e m p e r a t u r e dependence of the heat t r a n s f e r coefficient obtained in the c o u r s e of Investigating s h e e t g l a s s fluid hardening p r o c e s s e s a r e p r e s e n t e d in [2]. T h i s dependence can be a p p r o x i m a t e d by a step function (Fig. 1), i.e.,

~f

~.(T) =~xoq- (~zl-ao) [ S + ( T - T I ) - S + ( T - T o ) ]

ao r Fig. 1. H e a t t r a n s f e r coefficient as a function of the s u r f a c e t e m p e r a t u r e of the plate.

(~)

,

where ~l > c~0:~1 = eonst: ce0 = eonst: T O > T I. T l = const, T O= const: S + ( T - T i ) i s the a s y m m e t r i c unit function. B e a r i n g in mind the dependence S + ( T - T i ) = S+(Ti--~) , instead of (3) we obtain ~(~) =~o+ (a~-ao) [S+(~-x0) -S+(~-TI)] .

(4)

P h y s i c o - M e e h a n i c a l Institute, A c a d e m y of Sciences of the Ukrainian SSR, L ' v o v . T r a n s l a t e d f r o m Mekhanika P o l i m e r o v , No. 5, pp. 949-950, S e p t e m b e r - O c t o b e r , 1971. Original a r t i c l e s u b m i t t e d M a r c h 16, 1971. © t974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g'est 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced~ stored in a retrieva~ system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the pu~blisher. A copy of this article is available from thee publisher for $15.00.

846

05

I

-

-

* 2

5

0

3

f

F i g , 2. V a r i a t i o n of the t e m p e r a t u r e s t r e s s e s : a) as a ftmction of B = 0 (1), 1 (2); 2 (3), and 3 (4) at B = 2, Mi 0 = 0.5: b) as a function of B = 2 (1), 1.5 (2), and 4 {4) at B = 0.5, Mi 0 =0.5; and B = 1.5 (3) at B = 2, Mi 0 = t . T h e solution o f Eq. (2), in w h i c h the h e a t t r a n s f e r c o e f f i c i e n t h a s the f o r m (4) with initial condition (1), is w r i t t e n in the f o r m O= exp {-ivii-B[(Yii-[~{io)S+(34i-Mio)

-(Mi-,~3Mio)S+(Mi-f'}Mio)]},

(5)

w h e r e ® = ( T - t 0 ) / ( T n - t 0 ) ; Mi = ~ 0 r / c is the Mikheev n u m b e r : Mi 0 = o%%/e: Mi i = (~0ri/e: B = (oq/~ 0) - t: B = M i i / M i 0. We now d e t e r m i n e the t e m p e r a t u r e s t r e s s e s in an a n n u l a r plate f r e e of e x t e r n a l load, keeping in mind t h a t the i n s i d e r = d and outside r = D c o n c e n t r i c s u r f a c e s a r e t h e r m a l l y insulated. F r o m the e x p r e s s i o n s g i v e n in [3] we find the n o n d i m e n s i o n a l t e m p e r a t u r e s t r e s s e s f o r this c a s e : Ur

(7~

E,n U+(m) - 11

G,* = . . . . . . . . . . . . . . . . . . . . . .

to

t0

EoN[kf-(m) 0 ;

- II

* =~

O,

w h e r e E . : E / ( t - Vr~o yrs,) ; N : [( at* - at*)/( k 2 - 1 ) ] ( 1 + Vr~) + at* ; m : d / D ; f (m) ~ : [ ( 1 - m k+ t)/( l _ m 2 k ) ] ( r / D ) k + 1[(1 - m k - 1 ) / ( 1 ~mak) ] ( d / r ) k + 1; k 2 = E ~ / E r ; Cr t + Vr~0a~ot; a* t = C~rt Vr~0 + k 2 ~ot; E r , v~r, a r t, and Eq~ Vr~ ~ o t a r e the YoungVs moduli, P o i s s o n ' s r a t i o s , and l i n e a r expansion coefficients in the r a d i a l and tangential d i r e c t i o n s . T h e n o n d i m e n s i o n a l t e m p e r a t u r e s t r e s s e s f o r a c i r c u l a r plate and an infinite plate with a c i r c u l a r opening c a n be o b t a i n e d b y s u b s t i t u t i n g the c o r r e s p o n d i n g v a l u e s of d and D in E q s . (6). T h e g r a p h s in F i g . 2 show the v a r i a t i o n of the n o n d i m e n s i o n a l t e m p e r a t u r e s t r e s s e s (6) with the n o n d i m e n s i o n a l p a r a m e t e r s B and B. C l e a r l y , the n o n d i m e n s i o n a l n o n s t e a d y t e m p e r a t u r e s t r e s s e s d e c r e a s e c o n s i d e r a b l y as the p a r a m e t e r B ( r e l a t i v e height of step) i n c r e a s e s , a f t e r the m o m e n t at which the heat t r a n s f e r c o e f f i c i e n t s t a r t s to i n c r e a s e . A c h a n g e in the width o f the step £ at fixed B does not have m u c h effect on the n o n d i m e n s i o n a l t e m p e r a t u r e s t r e s s d i s t r i b u t i o n

LITERATURE 1o

2. 3.

CITED

Yu. M. Kolyano, Dokl. A k a d . Nauk BSSR, 14, 1000 (t970). I. A. B o g u s l a v s k i i , Dokl. Akad. Nauk SSSR, 1.73, 1298 (1967). Yu. M. K o l y a n o and E. A. Pakula, Mekhan. P o l i m . , No. 4, 725 (1970).

847