Boundary value problems for spherically isotropic elastic media solved in terms of infinite serie

Moghe, S. R.

Breton 

NASA Technical Reports Server

T e c h n i c a l  Report ,  No. 111 
ON SOME MIXED BOUNDARY VALUE PROBmMS 
FOR SPHERICALLY ISOTROPIC ELASTIC MEDIA 
S.  R .  Moghe 
U n i v e r s i t y  of Minnesota ,  Minneapo l i s ,  Minnesota  
A p r i l  1967 
https://ntrs.nasa.gov/search.jsp?R=19680006302 2020-03-12T12:03:29+00:00Z
On Some Mixed Boundary Value Problems 
F o r  S r s h e r i c a l l v  I s o t r o r s i c  E l a s t i c  Media 
S .  R .  Moghe 
U n i v e r s i t y  of  Minnesota ,  Minneapo l i s ,  Minnesota  
I n  t h i s  p a p e r ,  some mixed boundary  v a l u e  problems 
for s p h e r i c a l l y  i s o t r o p i c  e l a s t i c  medium a re  d i s c u s s e d .  
G e n e r a l  s o l u t i o n s  t o  t h e s e  problems a re  o b t a i n e d  i n  terms 
of  i n f i n i t e  s e r i e s .  Mi.xed boundary  v a l u e s  on t h e  s u r f a c e  
g i v e  r i s e  t o  t r i p l e  se r ies  e q u a t i o n s .  They are  s o l v e d  
b y  r e d u c i n g  t o  a s i n g l e  Fredholm i n t e g r a l  e q u a t i o n .  
INTRCDUCTI ON 
The boundary  v a l u e  problems f o r  a e l o t r o p i c  media 
have  been  c o n s i d e r e d  b y  many i n v e s t i g a t o r s  i n  t h e  
p a s t  [l, 21 *. 
i s o t r o p i c  mater ia ls .  For such  materials t h e  p h y s i c a l  
p r o p e r t i e s  are  t r a n s v e r s e l y  i s o t r o p i c  a b o u t  t h e  r a d i u s  
v e c t o r .  R e c e n t l y  Chen[3]  s o l v e d  some problems c o n c e r n i n g  
s p h e r i c a l l y  i s o t r o p i c  e l a s t i c  medium bounded by two 
c o n c e n t r i c  s p h e r i c a l  s u r f a c e s .  H i s  c o n s i d e r a t i o n s  
were p a r t i c u l a r l y  c e n t e r e d  on s e v e r a l  ax i - symmet r i c  
problems such  as a c o n c e n t r a t e d  f o r c e  i n  a n  i n f i n i t e  
medium, t he  s t r e s s  c o n c e n t r a t i o n  due to s p h e r i c a l  
c a v i t y  and  a s t e a d i l y  r o t a t i n g  s h e l l .  I n  h i s  p a p e r  
o n l y  one type of boundary v a l u e  problems where in  s t ress  
components a re  p r e s c r i b e d  on t h e  boundary  has been d e a l t  
w i t h .  The r e s u l t s  a r e  based e s s e n t i a l l y  on t h e  s o l u t i o n s  
s imi la r  to t h o s e  o b t a i n e d  b y  Hu[4] who has shown t h a t  
t h e  g e n e r a l  s o l u t i o n s  t o  t h e  e q u i l i b r i u m  e q u a t i o n s  may 
be  e x p r e s s e d  i n  t e r m s  of s p h e r i c a l  ha rmon ics .  
I n t e r e s t s  have been  f o c u s e d  on s p h e r i c a l l y  
I n  t h i s  r e p o r t  a n  a t t e m p t  i s  made t o  i n v e s t i g a t e  
g e n e r a l  mixed boundary  v a l u e  problems f o r  a s p h e r i c a l l y  
i s o t r o p i c  e l a s t i c  medium bounded by  s p h e r i c a l  s u r f a c e s .  
For these  t y p e s  of problems a s e t  of  t r i p l e  i n f i n i t e  
se r ies  e q u a t i o n s  i s  g e n e r a l l y  i n v o l v e d  and may be 
r educed  t o  a d o u b l e  s e r i e s  e q u a t i o n s  as s p e c i a l  c a s e s .  
A c o n s i d e r a b l e  amount of e f f o r t  has been  g i v e n  i n  s o l v i n g  
these  e q u a t i o n s  b y  C o l l i n s [ S ]  a n d  o t h e r s [ 6 ] .  R e s u l t s  
nave a l s o  been  a p p l i e d  
p rob lems .  The p u r p o s e  
to t h e  s t u d y  of boundary  v a l u e  
of t h i s  i n v e s t i g a t i o n  i s  to show 
*Numbers i n  t h e  s q u a r e  
t h e  e n d .  
b r a c k e t s  r e f e r  to r e f e r e n c e s  a t  
t h a t  c e r t i i n  e q u i l i b r i u m  problems w i t h  p r e s c r i b e d  
boundary  d i s p l a c e m e n t s  and s t r e s ses  g i v e  r i s e  t o  t h e  
t r i p l e  s e r i e s  e q u a t i o n s  which can  be s o l v e d  as a s i n g l e  
Fredholm i n t e g r a l  e q u a t i o n .  
I 
1 
I 
8 
U 
P 
S 
1 
8 
1 
8 
E 
1 
8 
8 
I 
f 
I 
I 
- 4 -  
EQUILIBRIUM PROBLEIvi 
The fundamen ta l  system of  f i e l d  e q u a t i o n s  i n  
e l a s t i c i t y  c o n s i s t s  of t h e  l i n e a r i z e d  s t r a i n - d i s p l a c e -  
ment r e l a t i o n s ,  s t r e s s - s t r a i n  r e l a t i o n s  aad e q u i l i b r i u m  
e q u a t i o n s  t o g e t h e r  w i t h  t h e  p r o p e r  boundary  c o n d i t i o n s .  
Us ing  s p h e r i c a l  c o o r d i n a t e  system ( cff ,  @ ) t h e  s t ress -  
s t r a i n  r e l a t i o n s  for a s p h e r i c a l l y  i s o t r o p i c  and  homo- 
geneous  medium, symmetr ic  a b o u t  t he  r a d i u s  v e c t o r ,  may 
b e  g i v e n  as f o l l o w s :  
%i = CII pee + c12 %p -+ c13 er-r 
%/J = 
0;r = c13 + c,, e$$ -f- c& e/-,-- 
T o  = C44 ere 
q$ = %4- 
Ti+ = 2e-I 2 - CI2) q+ (1. 1 
C,, eee +- G I  e++ + c,, e,, 
where t h e  e q u a n t i t i e s  r e p r e s e n t  t h e  s t r a i n s  i n  s p h e r i c a l  
c o o r d i n a t e s  and  t h e  e l a s t i c i t y  c o e f f i c i e n t s  a re  shown 
a s  t h e  v a r i o u s  C q u a n t i t i e s .  The s t r a i n - d i s p l a c e m e n t  
and  e q u i l i b r i u m  e q u a t i o n s  i n  s p h e r i c a l  c o o r d i n a t e s  may 
be found  i n  [ 2 ] .  
a l o n g  t h e  ( 5  d , )  d i r e c t i o n s  i n  terms of t h e  p o t e n t i a l  
f u n c t i o n s  &(< a n d  g ( c q @ )  as 
We e x p r e s s  t h e  d i s p l a c e m e n t s  ( ur3 Q'&, /+) 
- 5 -  
a n d  assume t h a t  
where (B, # ) i s  a s u r f a c e  harmonic of o r d e r  n . 
shown i n  [ 3 ]  t h e  e q u i l i b r i u m  e q u a t i o n s  i n  t h e  a b s e n c e  
o f  body f o r c e s  are  t h e n  s a t i s f i e d  b y  d$n and Qn if 
As 
where 
-t 2c44 ( = , I  + C,L  -- c,3q 
cc,, n ( n + O  - t x , + - t  C, , -C, /  ++ CJ3] 
- n ( ~ ~ l ~ [ 2 C q q + c , ~ - + c , , - t ( C 1 3 + c ~ ~ ~ _ l  
b, = ’ {p4 n<n+O -4- 2 (C, I  +c,z - CJ,>+$ S J X  
=33 c44 
’I (5) 
E q u a t i o n  ( 4 )  has fou;. r o o t s  qL‘ ( L ’ =  ;’,2,JJ4 ) . For 
conven ience ,  t h e y  are  assumed to b e  d i s t i n c t  and  r e a l .  
S imi l a r ly  h, has two v a l u e s  Ani  (i= 32) which s a t i s f i e s  
4 
( 6 )  
“I = O  G- c/2 ( n - i ) ( n - + 2 )  t 
One i m p o r t a n t  r e s u l t  f rom t h e  above  e q u a t i o n s  i s  t h a t  
f o r  large 77 e q u a t i o n  ( 4 )  becomes 
- 6 -  
and  (6) becomes 
c44 
T h e r e f o r e  f rom ( 7 )  and  ( 8 )  we can  e x p e c t  t h a t  and 
2, a re  p r o p o r t i o n a l  t o  Y) . 
Without  l o s i n g  g e n e r a l i t y ,  l e t  u s  s p e c i a l i z e  t o  t h e  
p rob lems  w i t h  p r e s c r i b e d  boundary  c o n d i t i o n s  symmet r i ca l  
w i t h  r e s p e c t  t o  t h e  Z-axis. A s  a consequence  t he  s u r f a c e  
harmonic  % ( BJ@ ) i n  ( 3 )  becomes a Legendre po lynomia l  
P ( C O s B ) .  Using t h e  n o t a t i o n  n 
; g == cos8  
t he  d i s p l a c e m e n t  components can be  e x p r e s s e d  as :  
( 9 )  
where 
- 7 -  
The s t r e s s  components are  e x p r e s s i b l e  as follows: 
where A n i a n d  Bn; are c o n s t a n t s  t o  be  de te rmined  from 
the  boundary  c o n d i t i o n s .  E q u a t i o n s  ( 1 0 )  and  ( 1 2 )  a re  
g e n e r a l  e x p r e s s i o n s ,  good f o r  a n y  domain. 
f o r  a s o l i d  s p h e r e  ( d 5 r_L R )  w i t h  the  p r e s c r i b e d  boundary  
c o n d i t i o n s  on T= q s u r f a c e  as :  
Now l e t  u s  c o n s i d e r  a g e n e r a l  boundary v a l u e  problem 
- 8 -  
Fay s ; q l i e i t y ,  . -  l e t  us aSS-dr,e t l y a t  n- 
w q  = G&$) 1 o ( G S  e 5 T )  
and  g(8) and  F(0) a r e  s u f f i c i e n t l y  smooth f u n c t i o n s  of 
8 .  Then s i n c e  t h e  s o l u t i o n  i s  r e q u i r e d  to be bounded 
/4n; everywhere  i n  t h e  r e g i o n  0 C T Z  R ,  we choose t h o s e  
and  Br)(. c o r r e s p o n d i n g  to terms which i n v o l v e  n e g a t i v e  
e x p o n e n t s  of r , t o  be zero. We assume also t h a t  t h e s e  
terms are  g i v e n  b y  
t h e n  s e t  
( 1 2 )  i n t o  (13) w e  o b t a i n  
Lo= 3,4 i n  u$i and i = 2 i n  h/7[* 
A, = An+ = 6 1 - , r = D .  S u b s t i t u t i n g  ( 1 0 )  and  
and  
F i n a l l y  we o b t a i n  w i t h  some a l g e b r a i c  m a n i p u l a t i o n s ,  
where 
cos 8, = , 
- 9 -  
and 
Here i t  c a n  be  shown e a s i l y  t h r o u g h  t h e  u s e  of ( 7 ) ,  (8 )  
a n d  (11) t h a t  f o r  l a r g e  l'7, kln+o( A) .  
e q u a t i o n s  (15 )  c o n s t i t u t e s  t r i p l e  s e r i e s  e q u a t i o n s  a n d  
have  been  i n v e s t i g a t e d  i n  d e t a i l s  by C o l l i n s [ 5 ] .  Fol low-  
i n g  [ 5 ]  t h e  s o l u t i o n  to (15) can be o b t a i n e d  and  shown 
t o  e x i s t  i f  H,O( A )  f o r  l a r g e  n. U n f o r t u n a t e l y  t h e  
f i n a l  form o f  t h e  s o l u t i o n  i s  q u i t e  c o m p l i c a t e d  and  
e v a l u a t i o n  may b e  v e r y  much i n v o l v e d .  However, f o l l o w i n g  
t h e  a p p r o a c h  of  [ 6 ]  t h e  s o l u t i o n  t o  (15 )  can  be  o b t a i n e d  
c o n v e n i e n t l y  i n  t h e  fo rm of a s i n g l e  Fredholm i n t e g r a l  
e q u a t i o n  and  t h u s  s i m p l i f y  t h e  s o l u t i o n .  To do t h i s ,  
l e t  u s  d e f i n e  a f u n c t i o n  f(r,) s u c h  t h a t  
The system of 
(17) 
M u l t i p l y  b o t h  s i d e s  of ( 1 7 )  by g(5.J 
o v e r  ( - f J  I ) w i t h  t h e  h e l p  of t h e  o r t h o g o n a l i t y  c o n d i t i o n s  
of  Legendre p o l y n o m i a l s  we g e t  
and  i n t e g r a t e  
(18) 
The u s e  of Mehler - D i r i c n l e t  fo rmula  
r e s u l t s  i n  a u s e f u l  r e l a t i o n  as  f o l l o w s :  
i' 
I -  7- cos ( n j  L 2) c o s - b y  = -1;. &/ 
I- 
- 10 - 
Now m i i l t i p l y  b o t h  s ides  of t h e  second e q u a t i o n  of (15)  
b y  (x-'6 1 -"2 and i n t e g r a t e  o v e r  (T3 1 ) .  Then th rough  
t h e  u s e  of ( 1 9 )  we o b t a i n  
00 
( 20 ) 
S u b s t i t u t i n g  for A, t h e  v a l u e s  shown i n  (18) and u s i n g  
a w e l l  known r e l a t i o n  r 
I 4 
0 1 
w e  can  r e d u c e  ( 2 0 )  t o  t h e  f o l l o w i n g  form: 
T h i s  i s  a n  Abe l  t y p e  s i n g u l a r  i n t e g r a l  e q u a t i o n  and can 
b e  i n v e r t e d  to d e t e r m i n e  r ( Z )  to g i v e  
P 4 
- ii - 
where 
I f  w e  d e n o t e  
P 
G(T) = J w d x  $. ~ 1 - y ~  T ;iJf.-r;. 
x- Y -1 ia-K 'd 
1 
-t- ( 1 -+ J',) ~ ( 3 x 1  Y W J X  (2s 1 
4' - 1  
Then ( 2 3 )  becomes 
o( 
(26) // 
T h i s  i s  a Fredholm i n t e g r a l  e q u a t i o n  of t h e  second k i n d  
a n d  can be  s o l v e d  e a s i l y .  It can be r e a d i l y  shown t h a t  
t h e  k e r n e l  s ( 5  
e q u a t i o n  has a un ique  s o l u t i o n .  Knowing ~ ( t )  i t  i s  a 
mat ter  of  s i m p l e  computat ion of A n a n d  a l l  t h e  r e q u i r e d  
q u a n t i t i e s .  
The e q u a t i o n  ( 2 3 )  or ( 2 6 )  can f u r t h e r  be s i m p l i f i e d  
) i s  r e g u l a r  if H,,,-O( 4 ) and t h e  
i f  some more i n f o r m a t i o n  about  t h e  v a l u e s  o f  4 and 4 
i s  known. F o r  i n s t a n c e ,  i f  0, j B2 T t h e n  d=- 
n P and 
I 
8 
8 
- 12 - 
I n  a d d i t i c n ,  i f  ? ( % ) i s  a n  even f u n c t i o n  i . e .  g(5) = gI-5) 
t h e n  the  l as t  t e r m  i n  
t h u s  ( 2 7 )  r e d u c e s  t o  t h e  f o l l o w i n g  s e t  of e q u a t i o n s :  
G(7) w i l l  have o n l y  even terms and  
G(T) = 
A -  zn - 
and  
where 
cl 'I J Y ;g 
2
I 
-1 I" ?(-GI 
* m + q  = o  
JCJ 2 
If 
w i l l  g e t  e q u a t i o n s  s imi la r  t o  ( 2 8 )  f o r  A It can 
g(5) i s  a n  odd f u n c t i o n ,  g ( t )  = - fj(-<), t h e n  w e  
2n-t.l' 
I 
I 
I 
I 
8 
I 
I 
1 
8 
I 
8 
B 
I 
8 
8 
8 
8 
8 
I 
- 13 - 
be s e e n  t h a t  ,A2,,,= 0 and &(a; 2 ) i s  t o  b e  r e p l a c e d  by 
s,(T3T ) d e f i n e d  as 
( 2 9 )  
Fur the rmore ,  l e t  u s  now c o n s i d e r  0, - 0 .  Then o(= 7 
and the  system of t r i p l e  s e r i e s  e q u a t i o n s  r e d u c e s  t o  a 
d u a l  s e r i e s  system. The s o l u t i o n  to t hese  e q u a t i o n s  can 
be  o b t a i n e d  by s u b s t i t u t i n g  i n  (25 )  and ( 2 6 )  as 
a s p e c i a l  c a s e .  
I f ,  i n s t e a d  of  (13), t he  boundary  c o n d i t i o n s  are  
p r e s c r i b e d  on r = R f o r  t h e  s h e a r  components as  t h e  f o l l o w i n g :  
( 3 0 )  
and  T r ( R J G )  = 0 ( o S e L 7 - r )  
where ?(e) and %(e)  a r e  s u f f i c i e n t l y  smooth f u n c t i o n s  
o f  B . 
For s i m p H c i t y ,  we c o n s i d e r  a l s o  t h a t  
S u b s t i t u t i n g  ( 1 0 )  and ( 1 2 )  i n t o  ( 3 G )  we o b t a i n ,  
L 
8 
I 
I 
I 
I 
I 
I 
I 
8 
1 
I 
I 
I 
1 
8 
8 
8 
1 
m 
- 13 - 
b e  seer, t h a t  0 and S,('B, 3 ) i s  t o  be r e p l a c e d  by  
s,(T3z ) d e f i n e d  as 
( 2 9 )  
Fur the rmore ,  l e t  u s  now c o n s i d e r  6, =o. Then o(= 7 
and t h e  system of t r i p l e  s e r i e s  e q u a t i o n s  r e d u c e s  to a 
d u a l  s e r i e s  sys t em.  The s o l u t i o n  to t h e s e  e q u a t i o n s  can 
be o b t a i n e d  by s u b s t i t u t i n g  a= ! i n  ( 2 5 )  and ( 2 6 )  as 
a s p e c i a l  c a s e .  
I f ,  i n s t e a d  of  ( l 3 ) ,  the boundary  c o n d i t i o n s  a r e  
p r e s c r i b e d  on r = R f o r  t h e  s h e a r  components as t h e  f o l l o w i n g :  
where F(B)  and % ( e )  a r e  s u f f i c i e n t l y  smooth f u n c t i o n s  
of B . 
For simplicity, we c o n s i d e r  a l s o  t h a t  
S u b s t i t u t i n g  ( 1 0 )  and  ( 1 2 )  i n t o  ( 3 G )  w e  o b t a i n ,  
8 
I 
I 
I 
8 
I 
8 
I 
8 
I 
I 
I 
I 
1 
8 
I 
8 
I 
I 
- 14 - 
I n t e g r a t i n g  b o t h  s i d e s  of t he  l a s t  two e q u a t i o n s  w i t h  
r e s p e c t  to 
l a t i o n s  
5 we can w r i t e  a f t e r  some a l g e b r a i c  manipu- 
I n  t he  above e q u a t i o n s ,  
-v 
where = h( I +  G). yo and m, are  a r b i t r a r y  
c o n s t a n t s  and  must be chosen c o n s i s t e n t l y .  One of  t he  
c o n s t a n t s  can  be chosen  to be  z e r o  s i n c e  o t h e r w i s e  i t  
c o n t r i b u t e s  o n l y  to t h e  r i g i d  d i s p l a c e m e n t  and d o e s  n o t  
i n f l u e n c e  t h e  s t ress  system. The o t h e r  c o n s t a n t  can be 
chosen  s u c h  that  t h e  d i sp lacemen t  Ue on the  s u r f a c e  i s  
c o n t i n u o u s .  
n-- 
- 
Here H n i s  0 (A) f o r  l a r g e  tl and i s  
- 15. - 
d e f i n e d  as 
I 
I 
8 
8 
1 
I 
1 
and 
E q u a t i o n s  i n  ( 3 2 )  are s imilar  t o  (15) and can be  s o l v e d  
e a s i l y  as b e f o r e .  
A s imple  b u t  i m p o r t a n t  problem when o n l y  qr 
r r o  are  p r e s c r i b e d  on t h e  s u r f a c e  can be c o n s i d e r e d  
a g a i n  as a s p e c i a l  c a s e  of (13) and ( 3 0 ) .  The s o l u t i o n  
i n  t h i s  c a s e  i s  d i r e c t l y  o b t a i n e d  i f  t he  p r e s c r i b e d  
s t resses  a re  expanded i n  terms of Legendre po lynomia l s .  
I n  t h i s  way, we o b t a i n  a system of s imple  a l g e b r a i c  
e q u a t i o n s  for t h e  unknown c o n s t a n t s  A,; which can be 
r e a d i l y  s o l v e d .  
and 
So far  we have d i s c u s s e d  the  mixed boundary  v a l u e  
problems for a s o l i d  sphe re  ( 0 5  r - R  ). 
i n f i n i t e  medium w i t h  s p h e r i c a l  c a v i t y  (R 5 r < d )  t h e  
same p r o c e d u r e  can be fo l lowed w i t h  t h e  o n l y  d i f f e r e n c e  
t h a t  we choose An, ,  An2, ra ther  t h a n  An3, A,,, and 
&; to be  z e r o .  
s o l u t i o n s  d;n i n  (3 )  such  t h a t  $n goes to z e r o  as r-00 . 
The problem u l t i m a t e l y  r educes  to s o l v i n g  a t r i p l e  s e r i e s  
I n  t h e  c a s e  of 
T h i s  co r re sponds  t o  assuming t h e  
I 
I 
I 
I 
I 
I 
I 
I 
1 
B 
I 
8 
E 
I 
I 
I 
1 
I 
1 
- 16 - 
e q u a t i o n s  s imilar  t o  (15) or ( 3 2 ) .  Again t h e  s c l u t i m  
may be o b t a i n e d  i n  terms of Fredholrn i n t e g r a l  e q u a t i o n  
o f  t h e  second k i n d  w i t h o u t  any d i f f i c u l t y .  
- 1 7  - 
EEFEEENC3S 
I 
I 
I 
I 
I 
II 
I 
I 
I 
I 
I 
1) S .  G .  L e k h n i t s k i i ,  Theory of  E l a s t i c i t y  of an  Aniso-  
t r o p i c  E l a s t i c  Body, T r a n s l a t i o n ,  Holden Day I n c . ,  
San F r a n c i s c o ,  1963. 
2 )  A. E .  H. Love, T r e a t i s e  on Mathemat i ca l  Theory of 
E l a s t i c i t y ,  F o u r t h  E d i t i o n ,  Dover P u b l i c a t i o n .  
11 3 )  W .  T .  Chen, On Some Problems i n  S p h e r i c a l l y  I s o t r o p i c  
E l a s t i c  Materials", Jour. Appl.  Mech., V o l .  33, No. 3, 
p .  539, 1966.  
4 )  Hai-Chang Hu, "On t h e  Genera l  Theory of E l a s t i c i t y  
f o r  a S p h e r i c a l l y  I s o t r o p i c  Medium", Acta  S c i e n t i a  
S i n i c a ,  V o l .  3, p .  247, 1954.  
5 )  W .  D.  C o l l i n s ,  "On Some T r i p l e  S e r i e s  E q u a t i o n s  and 
T h e i r  A p p l i c a t i o n s " ,  Arch. R a t i o n a l  Mech. A n a l y s i s ,  
V o l .  11, No. 2, p .  122, 1962.  
6 )  N .  Kh. A r u t i u n i a n  and B.  L. Abramian, On t h e  Impress ion  
of a R i g i d  D i e  i n t o  a n  E l a s t i c  Sphere" ,  PMM, V o l .  28, 
No. 6, p .  1101, 1964.  
11  
ACKNOWLEDGMENT 
It i s  a g r e a t  p l e a s u r e  to t hank  P r o f e s s o r  C .  C .  Hs iao  
for h i s  h e l p  and i n t e r e s t  d u r i n g  t h i s  i n v e s t i g a t i o n .  


On some mixed boundary value problems for spherically isotropic elastic media

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