OF PROBLEM. We shall consider the problem of the propagation of plane waves arising in an anisotropic thermoelastic half-space as the result of "heat shock. ~.
4.
5.
6.
7. 8. 9.
M. S. Kornishin, Nonlinear P r o b l e m s in the T h e o r y of Laminas and Hollow Shells and Methods of Solution [in Russian], Izd. Nauka, Moscow (1964). M. S. Kornishin, N. N. Stolyarov, and N. I. Dedov, " L a r g e deflections of non/inearly e l a s t i c laminas and hollow shells with hinged b o u n d a r i e s , * in: P r o c e e d i n g s of a S e m i n a r on Shell T h e o r y [in Russian], Vol. 2, Izd. Kazansk. F i z . - T e k h . Inst. Akad. Nauk SSSR (1971), pp. 49-58. A. I. S t r e t ' b i t s k a y a , " E [ a s t o p t a s t i c s t r a i n s and the toad c a r r y i n g capacity of hollow shells (review),* P r t k l . Mekh., 9. No. 8, 3-21 (1973). A. I. S t r e l ' b t t s k a y a , " E t a s t o p l a s t i c p e r f o r m a n c e of hollow shells under uniform load," Prikl. Mekh., 1 I No. 10, 25-35 0-975). I. S. T s u r k o v , " E i a s t o p l a s t i c bending of m e t a l hollow shell panels f o r finite deflections," Inzh.-Fiz. Zh., 1 No. 1, 145-153 (1961). I. S. Chernyshenko, "Calculations of a x i s y m m e t r i c shells of rotation with variable thickness allowing f o r g e o m e t r i c a l and p h y s i c a l n o n l i n e a r i t i e s , " in: The T h e o r y of P l a t e s and Shells [in Russian], tzd. Nauka, Moscow (1971), pp. 279-284.
PROPAGATION THERMOE Yu.
OF
LASTIC
PLANE
WAVES
IN A N A N I S O T R O P I C
HA L F - ~ S P A C E
A. R o s s i k h i n
UDC 539.3
1.
STATEMENT
OF
PROBLEM
We shall c o n s i d e r the p r o b l e m of the propagation of plane waves a r i s i n g in an a n i s o t r o p i c t h e r m o e l a s t i c h a l f - s p a c e as the r e s u l t of "heat shock. ~ We shall a s s u m e that the plane xiv i = 0 with unit n o r m a l v e c t o r ui is suddenly heated to a t e m p e r a t u r e 00 and that this t e m p e r a t u r e is subsequently maintained at the s a m e level. We shall f u r t h e r suppose that this plane is load f r e e and that the initial conditions a r e homogeneous. The p r o b l e m r e d u c e s to the solution of the s y s t e m of d iffe rentia i equations [2] qk,~ + c0 + ToB//J..; = 0; qk = --xk~0.l;
0..1)
% , ; = p'u~; % = Z~ik~Uk,~- - ~;0
(1.2)
with the boundary conditions 0(0, t ) = O o f l ( t ) ;
0-.3)
~(0, t ) = ~ ; ( 0 , t ) v j = 0
and the initial conditions 0 (xh, 0) = 0;
~ (xk, 0) = 0;
~ (xh, 0) = 0.
(1.4)
H e r e qj a r e v e c t o r components of the heat flux; 0 = T - T o is the t e m p e r a t u r e of the body; T O is the t e m p e r a t u r e of the body in its natural state; ce is the specific heat at constant strain; flij = ;~ijk/akl; C~kl a r e the t h e r m a l expansion coefficients; ) q j k / a r e the i s o t h e r m a l rigidity coefficients of the m a t e r i a l ; ~-qj a r e the t h e r m a l conductivity coefficients; ui a r e components of the d i s p l a c e m e n t vector; aij is the s t r e s s tensor; p is the density; H(t) is the Heavi,~ide unit function; t is the time; dots o v e r quantities denote d e r i v a t i v e s with r e s p e c t to time; and the s u b s c r i p t following a c o m m a denotes a d e r i v a t i v e with r e s p e c t to the c o r r e s p o n d i n g coordinate. 2. S O L U T I O N
OF
THE
PROBLEM
BY THE
SMALL-PARAMETER
METHOD
Elimi~Lating the quantities qk f r o m the relations (1.1) and crij f r o m ( 1 . 2 ) , a n d t a k i n g t h e L a p l a c e t r a n s f o r m with r e s p e c t to t of the resulting s y s t e m of equations, we have Voronezh Polytechnical Institute. Trartslated f r o m P r i k l a d n a y a Mekhanlka, Vo[. 12, Nol 4, pp. 60-64, A p r i l , !976. O r i g i n a l a r t i c l e submitted July 10, 1974. This materia? is protected by copvright regisre,ed in the name oy Pi. . . . . . . Publishing Corporation 227 W e s t 1 7 t h Street N e w York. N.Y. lOO11. No part 1 o f this publication m a y be reproduced stored in a retrieval system or transmitted in any form" or by any means rico'ironic mechanical, photocopying, [microfilming . . . . . rding or oil . . . . is . . . . i'thout ~ritten perT, ission o f tlte publiskler. A ~'op), o f t tis arl~cle i. . . . ilable "from the publisher for $7;50.
371
(2.1) w h e r e r = T 0 c ~ ; Ykl = Uk/Cal; Uk, 0 a r e the Laplace t r a n s f o r m s of the quantities Uk, 0; and p is a c o m p l e x v a r i a b l e (the h o m o g e n e i t y p r o p e r t y of the initial conditions h a s a l r e a d y beer~ used in these equations). Since r is a s m a l l quantity [2], the solution of the s y s t e m (2.1) can be r e p r e s e n t e d in the f o r m of a p o w e r s e r i e s in r, i.e., ~ = ' ~ r,,~(,,); u ~ = ~ r . u ~ , O . n=O
(2.2)
n=O
Setting (2.2) in Eqs. (2.1) and taking the boundary conditions ~(0, p) =. 0 # < , ~ij (0, p)vj = 0. into account, we obtain with an a c c u r a c y to O(r2) 3
-~:~ox,(_~./zl+.oov. ,, ,,-~,/~{' - - exp
(/~)] --
x
x
__ -~ 2~(,,)(p) V py
exp
~ (/?)~ --
x
;
(2.3)
3
~,=Oo2 p x.~_oxd_x,/--ill o- '>+ ~.,>~,.,]/, 8
n=!
X 372
2V~,-2,
+ .exp
~(.)erict'2-~-'~-~)
(2.4)
r
1
v (0 =
(a~ + o s ~ Dtm
Tobib,,.)[%.~k)l - - Ft~k-i} (t);
t- - - ~ . ~ ,., + rob,b=) 2
-~
(.
+ s+m \
,,,
+P
DP
,~t
)
"
Taking into account 3 0') ; "tn)'t"> stm = X p~)lt
