A Finite Line Crack in a Pressurized Spherical Shell-folias
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Transcript of A Finite Line Crack in a Pressurized Spherical Shell-folias
8/3/2019 A Finite Line Crack in a Pressurized Spherical Shell-folias
http://slidepdf.com/reader/full/a-finite-line-crack-in-a-pressurized-spherical-shell-folias 1/27
A F I N I T E L I N E C R A C K I N A P R E S S U R I Z E D S P H E R I C A L S H E L L
E f t h y m i o s S . F o l i a s *
A B S T R A C T
T h e d e f o r m a t i o n o f a t h i n s h e e t h a v i n g i n i t i a l s p h e r i c a l c u r v a t u re i s s h o w n t o b e a s s o c i a t e dw i t h t h a t o f a n i n i t i a l l y f l a t p l a t e r e s t in g u p o n a n e l a s t i c f o u n d a t i o n . U s i n g a n i n t e g r a lf o r m u l a t i o n t h e c o u p l e d R e i s s n e r e q u a t i o n s f o r a s h e l l w i t h a c r a c k o f l e n g t h 2 c a r e s o l v e df or t h e i n - p l a n e a n d K i r c h h o f f b e n d i n g s t re s se s , a n d , a m o n g o t h e r t h in g s , i t is f o u n d t h a tt h e e x p l i c i t n a t u r e o f t h e s t re s se s n e a r t h e c r a c k p o i n t d e p e n d s u p o n t h e i n v e r s e h a l f p o w e ro f t h e n o n - d i m e n s i o n a l d i s ta n c e f ro m t h e p o i n t z . T h e c h a r a c te r o f t h e c o m b i n e d e x t e n s i o n -b e n d i n g s t re s s f i e l d n e a r t h e c r a c k t i p i s i n v e s t i g a t e d in d e t a i l f o r : h e s p e c i a l c a s e o f ar ad ~. al c r a c k i n a s p h e r i c a l c a p w h i c h i s s u b j e c t e d t o a u n i f o r m i n t e r n a l p r e s s u r e q o a n d i sc l a m p e d a t t h e b o u n d a r y R = Ro . P e n d i n g a c o m p l e t e s t ud y o f t h e s o l ut i o n , a p p r o x i m a t er e s u l ts f o r t h e c o m b i n e d s u r f a c e s t r es s e s n e a r t h e c r a c k t i p n o r m a l a n d a l o n g t h e l i n e o fc r a c k p r o l o n g a t i o n a r e r e s p e c t i v e l y o f t h e f o r m
% RW y (C ' ° )[ ' , ~ = 1 / 3 = 0 . 4 5 V l , ' ~- - i f - - + . . .
k=0.98
c = 0 . 2 8 i n
Ro = 4 . 25 in
a n d s i m i l a r l yqoR
,~x(,.o)[ ~=zls= o..~'V'z/, -- U +. . .
k = 0 . 9 8c = 0 . 2 3 i nR o = 4 . 2 5 i n
I t i s i n t e r e s t i n g t o n o t e t h a t t h e s t re s s erx a n d ~ y, a l o n g t h e l i n e o f c r a c k p r o l o n g a t i o n , f o r th i sg e o m e t r y a r e e q u a l , I n g e n e r a l , t h e y w i l l b e o f t h e s a m e s i g n a n d w i l l d i f f e r o n l y s l ig h t l yi n m a g n i t u d e d u e t o t h e b e n d i n g c o m p o n e n t . F i n a ll y , t h e e x p e r i m e n t a l a n d t h e o r e ti c a l r e -s u lt s f o r C y, a l o n g t h e l i n e o f c r a c k p r o l o n g a t i o n , c o m p a r e v e r y w e l l .
I N T R O D U C T I O N
I n t h e f i e l d o f f r a c t u r e m e c h a n i c s , c o n s i d e r a b l e w o r k h a s b e e n
c a r r i e d o u t o n i n i t i a l l y f l a t s h e e t s s u b j e c t e d t o e i t h e r e x t e n s i o n a l
o r b e n d i n g s t r e s s e s , a n d f o r s m a l l d e f o r m a t i o n s t h e s u p e r p o s i t i o n
o f t h e s e s e p a r a t e e f f e c t s [ i ] i s p e r m i s s i b l e . O n t h e o t h e r h a n d ,
i f a t h i n s h e e t p o s s e s s e s i n i t i a l c u r v a t u r e , a b e n d i n g l o a d w i l l
g e n e r a l l y p r o d u c e b o t h b e n d i n g a n d e x t e n s i o n a l s t r e s s e s , r a n d
v i c e v e r s a . T h e s u b j e c t o f e v e n t u a l c o n c e r n t h e r e f o r e i s t h a t
o f t h e s i m u l t a n e o u s s t r e s s f i e l d s p r o d u c e d i n a n i n i t i a l l y c u r v e ds h e e t c o n t a i n i n g a c r a c k .
I t i s o f s o m e p r a c t i c a l v a l u e t o b e a b l e t o c o r r e l a t e f l a t s h e e t
b e h a v i o r w i t h t h a t o f i n i t i a l l y c u r v e d s p e c i m e n s . I n e x p e r i m e n t a l
w o r k , f o r e x a m p l e , c o n s i d e r a b l e t i m e c o u l d b e s a v e d i f a r e l i a b l e
p r e d i c t i o n o f c u r v e d s h e e t r e s p o n s e b e h a v i o r c o u l d b e m a d e f r o m
f l a t s h e e t t e s t s . F o r t h i s r e a s o n a n e x p l o r a t o r y s t u d y w a s u n d e r -
t a k e n t o a s s e s s a n a l y t i c a l l y t h e m a n n e r i n w h i c h t h e t w o p r o b l e m s
m i g h t b e r e l a t e d . A l t h o u g h i t i s r e c o g n i z e d t h a t ' e l a s t i c a n a l y s i s
i s n o t c o m p l e t e l y s a t i s f a c t o r y d u e t o p l a s t i c f l o w n e a r t h e c r a c k
t i p , c o n s i d e r a b l e i n f o r m a t i o n c a n b e o b t a i n e d .
T w o i n i t i a l l y c u r v e d g e o m e t r i e s i m m e d i a t e l y c o m e t o m i n d :
a s p h e r i c a l s h e l l a n d a c y l i n d r i c a l s h e l l . I n t h e l a t t e r c a s e o n e
* F o r m e r l y R e s e a r c h P e l l o w , C a L i f o r n ia I n s t i t u t e o f Te c h n o l o g y, D r. F o l i a s i s c u r r e n t l y I nT h e s s a l o n i k i , G r e e c e . M a i l I n g a dd r e ss c / o t h e e d i to r i a l o f f i c e , P a s a d e n a , C a l i f o r n i a .
8/3/2019 A Finite Line Crack in a Pressurized Spherical Shell-folias
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8/3/2019 A Finite Line Crack in a Pressurized Spherical Shell-folias
http://slidepdf.com/reader/full/a-finite-line-crack-in-a-pressurized-spherical-shell-folias 3/27
Y
22 Efthymios S. Folias
TOP Vl EW
y / lZ
~ X
SIDE VIEW
R
=- X
CROSS
SECTION
Fig. 1 . Geom etry and Coo rdinates .
o n e h a s a l r e a d y f o u n d * a p a r t i c u l a r s o l u t i o n s a t i s f y i n g 4 a n d5 , b u t p r o d u c i n g a r e s i d u a l n o r m a l m o m e n t M 0 e q u i v a l e n tv e r t i c a l s h e a r Vy , n o r m a l i n - p l a n e s t r e s s N y , a n d a n i n - p l a n e
* See the next sect ion for an example .
8/3/2019 A Finite Line Crack in a Pressurized Spherical Shell-folias
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EfthymiosS. Folias 23
t a n g e n t i a l s t r e s s N x y , a l o n g t h e r e a l a x i s Ix < 1 , o f t h e f o r m :
M y ( P ) _ D m ( x ) (6 )c 2
DVy ( P ) = - v ( x ) ( 7 )
c 2
N y ( P ) = - n(x___J.) (8 )C 2
N x ~ P ) = _ t ( x )c 2 ( 9 )
T h u s w e n e e d t o f i n d t w o f u n c t i o n s o f t he d i m e n s i o n l e s s c o o r -d i n a t e s ( x , y ) , W ( x , y ) an d F ( x , y ) , s u c h th a t t h e y s a t i s f y th ep a r t i a l d i f f e r e n t i a l e q u a t i o n s 4 a n d 5 an d t h e f o l l o w i n g b o u n d a r y
c o n d i t i o n s . A t y = 0 a n d ]xJ < 1
M y ( X , O ) - D [ 8 2 W + v - -82W] = D, m (x ) (10 )e 2 8y 2 0x 2 c 2
D v ( x ) ( 11 )y ( x , 0 ) - D [ 0 3 W + ( 2 - V ) 8 3 W ] = C 3c 3 L S y 3 8 x 2 d y
Ny(X, . 0) - 1 O2F = n( x) ( 1 2 )c 2 8 x 2 c 2
Nxy ( x , 0 ) - 1 8 2 F _ * ( x ) ( 1 3 )c 2 8 x S y c 2
A t y = 0 a n d Ix l > 1, w e m u s t s a t i s f y t h e c o n t i n u i t y r e q u i r e m e n t s ,n a m e l y :
8 n 8 n
l i r a [ - - ( W + ) - ( W - ) ] = 0[Y] -- '*0 8 y n 8 7
( 1 4 )
8 n 8 n
l i m [ - - ( F +)[y ] - - -,0 8yn aYn
- - - ( F - ) ] -- 0 ( 1 5 )
( n = 0 , 1 , 2 , 3 ) .
F u r t h e r m o r e , b e c a u s e w e a r e l i mi t in g o u r s e l v e s to a l a r ger a d i u s o f c u r v a t u r e f o r t hi s s h a l l o w s he ll , i .e ., s m a l l d e v i a t i o n sf r o m a f l a t s h ee t , w e c a n a p p l y c e r t a i n b o u n d a r y c o n d i t io n s a ti nf in it y e v e n t h o u g h w e k n o w p h y s i c a l l y t h a t t h e s t r e s s e s a n dd i s p l a c e m e n t s f a r a w a y f r o m t he c r a c k a r e fi ni te . T h e r e f o r e ,t o a v o i d i n f i n i t e s t r e s s e s a n d i n f i n i t e d i s p l a c e m e n t s w e m u s tr e q u i r e t ha t t h e d i s p l a c e m e n t f u n c t i o n W a n d t h e s t r e s s f u n c t i o nF w i t h t h e ir f i rs t d e r i v a t i v e s t o b e f i n i te f a r a w a y f r o m t he
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2 4 Efthymios S. Folias
c r a c k . T h e s e r e s t r i c t i o n s s i m p l i f y t he m a t h e m a t i c a l c o m p l e x i t i e so f t he p r o b l e m c o n s i d e r a b l y , a n d c o r r e s p o n d t o t he u s u a l e x -p e c t a t i o n s o f t h e S t . Ve n a n t P r i n c i p l e . I t s h o u l d b e p o i n t e d o u tt h a t th e b o u n d a r y c o n d i t i o n s a t i n f i n i ty a r e n o t g e o m e t r i c a l l yf e a s i b l e . H o w e v e r i f t he c r a c k i s s m a l l c o m p a r e d to t he d i -m e n s i o n s o f t he s h e l l , t he a p p r o x i m a t i o n i s g o o d .
Method of Solution
R e i s s n e r [4 ] h a s s h o w n t h a t t he s o l u t i o n o f t h e s y s t e m 4 ,5 c a n b e w r i t t e n i n t h e f o r m
w = z + • ( 1 6 )
F = - R D V 2 ; ( + ~ ( 1 7 )c 2
w h e r e ¢ a n d ~ a r e h a r m o n i c f u n c t i o n s a n d X s a t i s f i e s t h e s a m ed i f f e r e n t i a l e q u a t i o n a s t h e d e f l e c t i o n of a p l a t e o n a n e l a s t i c
f o u n d a t i o n , i . e .
( V4 + X4) X = 0 (18 )
w h e r e
E h c 4 _ 1 2 ( 1 - v 2 ) ( ~ _ ) 4k 4 - - R 2 D - ( _ ~ ) 2 (19)
T h e f u n c t i o n JJ r e p r e s e n t s t h e i n e x t e n s i o n a l b e n d i n g p a r t o f t h es o l u t i o n , a n d ¢ r e p r e s e n t s t h e m e m b r a n e p a r t o f t he s o l u t i o n .
We n e x t c o n s t r u c t t he i n t e g r a l r e p r e s e n t a t i o n s
f ? e -~ ] s2-iX21 _ ~f 2+iX"-~lYl _ sly IW ( x , y + ) = { P 1 Y+I P 2 e + P 3 e } c o s x s d s ( 2 0 )
f ; - ~ /s2- ix21Yl - ~ k 2 1 y E - s ly IP ( x , y ± ) - - - iA 2 R D { P 1 e - P 2 e + P 4 e } e o s x s d s ( 21 )
c 2
w h e r e t h e P i ( i = 1 , 2 , 3 , 4 ) a r e a r b i t r a r y f u n c t i o n s o f s t o b ed e t e r m i n e d f r o m th e b o u n d a r y c o n d i t i o n s , a n d t he f s i g n s r e f e rt o y > 0 a n d y < 0 r e s p e c t i v e l y . I n r e f e r e n c e [ 5] , i t i s s h o w nt h a t th e p r o b l e m i s r e d u c e d t o t h e s o l u t i o n o f t w o c o u p l e d s i n g u l a ri n t e g r a l e q u a t i o n s w h i c h , b e c a u s e o f t he c o m p l i c a t e d n a t u r e o ft he k e r n e l s , a r e s o l v e d f o r s m a l l v a l u e s o f t he p a r a m e t e r k .T h i s i s c e r t a i n l y p e r m i s s i b l e s i n c e f o r m o s t p r a c t i c a l a p p l i c a t i o n st he p a r a m e t e r ~ a t t a i n s s m a l l v a l u e s a s fo l l o w s f r o m th e d e f i n i t i o no f ~ , n a m e l y
( 2 2 )
I t i s c l e a r t h a t k i s s m a l l f o r l a r g e r a t i o s o f R / h a n d s m a l lc r a c k l e n g t h s . A s a p r a c t i c a l m a t t e r , i f w e c o n s i d e r c r a c k
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Efthyrnios S. Folias 25
2~rRl e n g t h s l e s s t h a n o n e t e n t h o f t h e p e r i p h e r y , i . e . 2 c < " T O - ' a n d
f o r R / h < 1 0 3 a c o r r e s p o n d i n g u p p e r b o u n d f o r k c a n b e o b t a i n e d ,n a m e l y k < 2 0 . T h u s t h e r a n g e o f k b e c o m e s 0 < k < 2 0 a n d f o rm o s t p r a c t i c a l c a s e s i s b e t w e e n 0 a n d 2 .
F i n a l l y , w i t h o u t g o i n g i n t o t h e d e t a i l s w e o b t a i n t h e d i s p l a c e -m e n t f u n c t i o n W a n d t h e s t r e s s f u n c t i o n F ( s e e a p p e n d i x ) f o rm = m o , v = 0 , n = n o a n d t = 0 . * T h u s t h e s t r e s s e s a r e f o u n dt o b e :
Bending stresses:** O n t h e s u r f a c e Z = + h2
_ E h P l o 3 0 1 5 0O ~ X b 2 ( 1 + ~ ) c 2 V ~'~(-4 c ° s 2 + 4 c ° s - ~ ) + 0 ( ¢ ° )
PE h 1o
Cry = 2 ( 1 _ u 2 ) c 2 ~ -
TxYb
( 2 3 )
1 1 + 5 u 0 1-u
- - ( ~ c o s -~ + "-4 -- c o s - - ) + O ( c ° ) ( 2 4 )
G h P 1 o . 7 + y . 0 1 - u( ~ s l n ~ + s in -- ) + O ( e °)
( l - y ) c 2 ~( 2 5 )
w h e r e
A 1 "B 1 n o k 2 8 -7u o 4-3v o
P l o - -- ~ h 2 - - : ( ~ + -----if--- ~' +2 ~ f 2 ~ f 2 E h D ( 4 - ' o )
4 - 3 y o
1 6
m o 4 - 3 v ok 2( i + ~ n ~ ) > + < I + ~ - k
324 - Vo ) 4 _U o } + 0 ( ~ . 4 i n k )
S i m i l a r l y
Extensional s t resses:
( 2 6 )
20 3 0 1 5 0(-z ~ T - ~ - ( ) ( 2 7 )- X =e h a 4 ~ - - ~ C O S ~ + ' ~ C O S a ) + O - c ° -
P 2 0 5 0 1Cry e h c 4 ~ _e ( ~ - c o s 7 ~ ~ - c o s ) + 0 ( ¢ ° )
P 2 0 1 0 1 5 0X Y e he 4 ~ (~- s in g ~ s in -~-) + O ( z °)
(28)
(29)
* Reference [5] also discusses the me thod of solution for the case tha t m and n are functionsof x. The case of v # 0 and t # 0 is discussed in reference [6].# Note because of the Kirchhoff boundary conditions, the bending shear stress does not vanishin the free edge. For the flat sheet this problem was discussed by Knowles and Wang[ 7 ] .
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2 6 Efthymios S. Folias
w h e r e
~ 0 ~ m
X 4 R D
( A 1 + B I )
n c 2 m X 2 ~ - ~ c 2 3 .~. k 2
- V~" { i + ~ " %/2" (4 -U o )
+ 0 ( k 4 ~ n k ) .
(30)
I t i s a p p a r e n t f r o m t h e a b o v e e x p r e s s i o n s t h a t t h e r e e x i s t s a n
i n t e r a c t i o n b e t w e e n b e n d i n g a n d s t r e t c h i n g , e x c e p t t h a t i n t h e
l i m i t a s X - ~ 0 t h e s t r e s s e s o f a f l a t s h e e t a r e r e c o v e r e d a n d
c o i n c i d e w i t h t h o s e o b t a i n e d p r e v i o u s l y f o r b e n d i n g [ 8 ] a n d
e x t e n s i o n [ 9 ] . T h u s t h e l o c a l s t r e s s e s i n a s h e l l a r e e x p r e s s e d
i n t e r m s o f t h e l o c a l s t r e s s e s i n a f l a t s h e e t .
Combined s t r e s se s :
I n g e n e r a l , t h e c o m b i n e d s t r e s s e s w i l l d e p e n d u p o n t h e c o n -
t r i b u t i o n s o f t h e p a r t i c u l a r s o l u t i o n s r e f l e c t i n g t h e m a g n i t u d ea n d d i s t r i b u t i o n o f t h e a p p l i e d n o r m a l p r e s s u r e . O n t h e o t h e r
h a n d , t h e s i n g u l a r p a r t o f t h e s o l u t i o n , t h a t i s t h e t e r m s p r o -
d u c i n g ( m a t h e m a t i c a l l y ) i n f i n i t e e l a s t i c s t r e s s e s a t t h e c r a c k
t i p , w i l l d e p e n d o n l y u p o n t h e l o c a l s t r e s s e s e x i s t i n g a l o n g t h e
l o c u s o f t h e c r a c k b e f o r e i t i s c u t , w h i c h o f c o u r s e a r e p r e c i s e l y
t h e s t r e s s e s w h i c h m u s t b e r e m o v e d o r c a n c e l l e d b y t h e p a r t i c u l a r
s o l u t i o n s d e s c r i b e d a b o v e i n o r d e r t o o b t a i n t h e s t r e s s f r e e
e d g e s a s r e q u i r e d p h y s i c a l l y . H e n c e t h e d i s t r i b u t i o n q ( x , y ) d o e s
n o t - - t o t h e f i r s t o r d e r - - a f f e c t t h e l o c a l c h a r a c t e r o f t h es t r e s s e s a t t h e c r a c k p o i n t .
A P A R T I C U L A R S O L U T I O N
A s a n i l l u s t r a t i o n o f h o w t h e l o c a l s o l u t i o n m a y b e c o m b i n e di n a p a r ti c u la r c a se , c o n s i d e r a c l a m p e d s e g m e n t o f a s h a l l o ws p h e r i c a l s h el l o f b a s e r a d i u s R o a n d c o n t a i n i n g a t t h e a p e x af in it e r a d i a l c r a c k o f l e n g t h 2 c i n t h e d i r e c t i o n o f t h e X - a x i s( s ee f i g . 2). T h e s h el l i s s u b j e c t e d t o u n i f o r m i n te r na l p r e s s u r e
h
I
!
. . . - . - . - R o . - -_ . -. ~
Fig . 2 . Clam ped spher ica l cap: A par t icu lar case .
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Efthymios S. Folias 27
1q o w i t h r a d i a l e x t e n s i o n a l s t r e s s N r = ~ q o R , a n d b e c a u s e i t
i s c l a m p e d w e r e q u i r e t h a t t he d i s p l a c e m e n t a n d s l o p e v a n i s ha t R = R o . F o r t h i s p r o b l e m , R e i s s n e r [1 0] g i v e s t h e s o l u t i o no f t he c o u p l e d - e x t e n t i o n - b e n d i n g e q u a t i o n s f o r t h e u n c r a c k e ds h e l l a s :
W p ( r ) = C 1 b e r ( k r ) + C 2 b e i ( k r ) + C a ( 3 1 )
1F p ( r) = E h 2 { C 1 b e i ( k r) - C 2 b e r ( k r ) } - ~ q o R r 2 ( 3 2)
1 ~ ( 1 -U 2 )
w h e r e
C 1 =
C 2 q o R ~ 1 2 ( l - v 2 ) r o b e i ' ( ) fr o )
E h 2 k [ b e i , ( A r o ) ] 2 + [ b e r , ( k r o ) ] 2
b e t ' ( k r o )
C 2 = - C 1 { b e i ' ( k r o ) }
b e r ( X r ° ) b e i ' ( A r o ) - b e i ( ) t r o ) b e r '( k r o )
C 3 = - C 1 {' b e i ' ( l r ) }
A l o n g a n y r a d i a l r a y , a n d i n p a r t i c u l a r a l o n g 0 = 0 , ~r, t h eb e n d i n g a n d e x t e n s i o n a l s h e a r v a n i s h b y s y m m e t r y i a n d t hec i r c u m f e r e n t i a l b e n d i n g a n d s t r e t c h i n g s t r e s s e s a r e
M(P)0 ( r ) =D X 2 { C l [ v b e i ( k r ) - ( l - v ) b e r ' ( ) t r ) .c s r l
- C 2 [ y b e r ( k r ) + ( 1 - v ) ~ ] } ( 33 )
IkT(P) Eh2(k2/c2)-- C 1 bei"(kr) - C 2 ber"(kr)} -½qoR (34)" ' c o ( r ) = V 1 2 ( 1 : v ~ )
N r 0 ( r ) = 0 ( 3 5 )
Vr O ( r ) = 0 ( 3 6 )
T h e r e f o r e , t he h o m o g e n e o u s s o l u t i o n m u s t n e g a t e t h e s e v a l u e sf r o m th e p a r t i c u l a r s o l u t i o n . B u t s i n c e w e a l r e a d y h a v e o b t a i n e da s o l u t i o n f o r u n i f o r m l o a d i n g a l o n g t he c r a c k , n a m e l y m o a n dn o , t h e r e f o r e w e w i l l m a k e u s e o f t h e s e r e s u l t s i n o r d e r t o o b -t a i n a n e s t i m a t e o f t he s t r e s s e s i n t h e v i c i n i t y o f t h e c r a c k .A s a n e n g i n e e r i n g a p p r o x i m a t i o n , b y c o n s i d e r i n g a n u p p e r a n dl o w e r b o u n d o n m o a n d n o , w e m a y e s t i m a t e a n u p p e r a n d l o w e rb o u n d f o r th e s t r e s s e s i n t h e n e i g h b o r h o o d o f t h e c r a c k p o i n t .O n th e o t h e r h a n d , i f w e a r e i n t e r e s t e d i n t h e s t r e s s e s a w a yf r o m t he c r a c k , s a y t h r e e t i m e s t he c r a c k l e n g t h , t h e n b y S a i n tVe n a n t ' s p r i n c i p l e w e n e e d t o t a k e o n ly th e a v e r a g e v a l u e s .T h u s w e d e f i n e
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28 Effhymios S. F o l i a s
D m o ( l)
2C
- - - r a i n
Ixl<l,O=0M(P)0 0 < - -
m m t r H ¢
m a xC 2 Ixl<l, 0 =0
M(P),@0= D - -
m o ( u )
C 2
a n d s i m i l a r l y
n ( t )oC 2
- r a i n (P) n t r~ e lkT P) n° ( U )Ix l<z ,o=0N00 < c 2 ..< m a x = - -IxL<l,0 = o ~ ' 0 0 c 2
N e x t , l e t u s c o n s i d e r a s p h e r i c a l s h e l l w i t h t h eg e o m e t r i c a l d i m e n s i o n s :
K o = 4 . 2 5
R = 2 0 i n .
h = 0 . 0 0 9 i n .
v = i / 3
2 c = 0 . 4 6 i n .
E = 1 6 x 1 0 6 p s i
f o l l o w i n g
f r o m w h i c h w e c a n c a l c u l a t e t he p a r a m e t e r s
X = 0 . 9 8
r o = 1 8 . 5 0
T h e f o l l o w i n g t a b l e s h o w s t he v a r i a t i o n o f t he r e s i d u a l m o m e n t
r 'T (P ) a l o n g t h e c r a c k .(P ) a n d m e m b r a n e f o r c e ~ '0 0* " 0 0
XX X x = X - -
c N(P)o0 M(O~
- 0 . 5 0 q oR 0 .89 × 10 4 qoRh-0.50 qo R 0.8 7 x 10 -4 qoRh-0.50 qo R 0. 92 x 10 -4 qoRh-0.50 qo R 0. 94 x 10 4 qoRh-0.50 qo R 0.97 x 10 -4 qoRh
diff. 0% diff. 9%
0 00. 074 0.300. 117 0.500. 164 0.700. 235 1.00
I t i s c l e a r f r o m t h e t a b l e a b o v e t h a t M (~P) a n d N ( ~ a r e
u n i f o r m a l o n g t he c r a c k . T h e r e f o r e w e m a y c h o o s e
n (u )o
0 2= 0 . 5 0 q o R
a l m o s t
D m ( u)
c 2= - 0 . 9 7 × 1 0 - 4 q o R h
R e t u r n i n g n o w t o t h e s t r e s s e s a l o n g t he l i n e o f c r a c k p r o l o n g a t i o n ,f o r e x a m p l e t h e n o r m a l s t r e s s ~Y total ( X , 0 ) , o n e f i n d s u s i n g 2 4a n d 2 8 t h a t :
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E f t h y m i o s S . F o l ia s 2 9
CrYtota1 ( X , O ) ] v = l /3~b
~ / 2 ( X - 1 ) / c
~ e
2{1 +(0.16 +0. 03 In l~)k 2}
( 3 7 )
+~/2(X-1)/c{ 1 - ( 0 , 3 8 + 0 , 2 3 I n )t 2 }
w h i c h f o r X = O. 9 8 r e d u c e s to :
O'Ytota1 ( X , 0 ) - - (1.07} +q 2(X-l)/c q 2(X-l)/c
{ 1 . 2 7 } (38 )
w h e r e
~b6 D m o
h 2 c 2- " a p p l i e d b e n d i n g "
n o
- - "applied stretching"e c2 h
A n d f o r o u r p a r t i c u l a r e x a m p l e , w e c a n a s s o c i a t e
qo R qo R~ b u ) -- - 0 . 5 8 x 1 0 - 3 T = - 0 . 0 0 1 - - h - - ( 3 9 )
q o R~(u) _.. O. 50 (40)e h
F r o m e q u a t i o n 3 8 w e s e e t h at t he i n i t i a l c u r v a t u r e w i l l i n c r e a s et h e a p p l i e d b e n d i n g s t r e s s i n t h i s c a s e b y a p p r o x i m a t e l y 7% a n dt he a p p l i ed s t r e t c h i n g s t r e s s b y 2 7% . O n e d e d u c e s t h e r e f o r e t ha tt he c r i t i c a l c r a c k l e n g t h o f a sh e l l d e c r e a s e s w i th an i n c r e a s eo f )t o r a d e c r e a s e o f r a d i u s o f c u r v a t u r e . I n t h i s p a r t i c u l a rc a s e i t i s f o u n d t h a t t he d i r e c t b e n d i n g s t r e s s e s a r e n e g l i g i b l ec o m p a r e d t o t he e x t e n s io n a l o n e s an d t he c o m b i n e d s t r e s s i s
Crytota ( X , 0 ) v = l / 3 = ~ 0 "4 q°R--h---"= ~f2(x-1)0"4 ~ qo h (41)
cr ( X , 0 ) I v = l / 8 ~ 0 . 6 4 ~ q o Rx total a]'2( X - 1 ) h
(42)
w h e r e , b a s e d o n t he K i r c h h o f f t h e o r y , t h e t w o - d i m e n s i o n a l" h y d r o s t a t i c t e n s i o n " n a t u r e o f t he ax a n d a y s t r e s s e s p r e d i c t e df o r f la t p l a t e s i s p r e s e r v e d . F i n a l l y , t he c o r r e s p o n d i n g s t r a i n sa r e
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30 E f f h y r n i o s S . F o l i a s
X ~ _ I q ° Ry(X, 0) = 0.3 0 Eh
q° Rex(X, 0) = 0.3 0 Eh
(43)
(44)
GRIF FITH 'S THEORY OF FRACT URE FOR CURVED SHEETS
As i s we l l known in f r a c t u re mec han ics , t he p re d ic t io n o ff a i l u r e i n t he p r e s e n c e o f s h a r p d i s c o n t i n u i t i e s i s a v e r y c o m -p l i ca ted p rob l em. Some work has been done on f i a t shee t s ,b a s e d o n t he b r i t t l e f r ac tu re the ory o f A . A . Gr i ff i th [11].Hi s hypo the s i s i s t hat t he to t a l ene r gy o f a c ra ck ed sys tems u b j e c t e d t o l o a d i ng r e m a i n s c o n s t a n t a s t he c r a c k e x t e n d s a ni n f i n i t e s i m a l d i s t a n c e . It s h o ul d o f c o u r s e b e r e c o g n i z e d t h atth i s i s a nec ess a ry cond i t ion fo r f a i lu re bu t no t su ff i c i en t .
Gr i ff i th app l i ed hi s c r i t e r i on to an in f in i t e , i so t r op i c p la t e
con ta in ing a f i a t , s ha r p -e nde d c ra ck o f l eng th 2c . We sha l l nowp r o c e e d t o o b t ai n a s i m i l a r, b ut a p p r o x i m a t e , c r i t e r i o n fo ri n i t i a l l y c u r v e d s h e e t s b a s e d u p on o nl y t he s i n g u l a r t e r m s o ft h e s t r e s s e s .
T h e b a s i c c o n c e p t f o r c r a c k i n s t a b i l i t y i s
8Us y s t e m8c
= o (45)
w h e r e t he s y s t e m e n e r g y i s d e f i ne d b y
U s y s t e m = U l o a d i n g + U s t r a i n + U s u r f a c e ( 4 6 )
T h e a p p l i e d s t r e s s e s a r e h e l d c o n s t a n t s o t h a t
U l o a d i n g = U o - h f v ( ~ u n + " r ~ u ~ ) dP (47)
where Uo i s a r e f e re nce cons ta n t ene r gy, and the in t eg ra t ion i so v e r t h a t p o r t i o n o f t he o u t e r p e r i m e t e r w h e r e f o r c e s a c t to c a u sed i s p l a c e m e n t s . T he s t r a i n e n e r g y i s
U s t r a i n = ½ fvol (cr~e~ + ~'~n T~r~ + or7 6~) d vol . (48)
U s u r f a c e =T(2S + Ph + 4ch) ( 4 9 )
I t m a y b e s h o w n t h a t t h e s u r f a c e a r e a S o f t h e s h e l l f a c e s a n d
t h e o u t e r p e r i m e t e r P o f t h e s h e l l a r e i n d e p e n d e n t o f t h e c r a c k
l e n g t h .
T h e r e f o r e 4 6 m a y b e w r i t t e n a s
° f+ 4 T ' h c - h ( ~ u ~ + u r / T ~ r T ) d PU s y s t e m = U o P
+ ½ f (~ e ~ + ~~n ~Y~n + %7c ~ ) d v o l .v o l
( 5 0 )
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Ef thymios S . Fol ias 3 1
o r
Usystem = U " + 4 7 % -
Z 2 A* 7r
X I ¢ -It
h* ~w h e r e U 'o = U o + 3 ' *( 2 A + P h ) , Z 1 = Z o - ~ - - - , Z 2 = Z o + a n d A *
a r a d i u s t o b e d e t e r m i n e d ( s e e f i g . 3 ) . I n t h e a b o v e w e h a v ed e f i n e d
2 - R 1 - ( R - h / 2 ) ~ ~ R - _ ~ _ /2 1 -R R 2 R '2
Z o = ~ R 2 . ~ , 2 _ ( R - a )
w i t h 6 t he h e i g h t o f t he s h e l l s e g m e n t .
C R A C K - - ~ _ . .~ A~
, - / /R
Fig . 3 . Ge om et r ic de ta i l s fo r f rac ture c r i t e r ion .
I n t e g r a t i o n o f 5 0 w i t h r e s p e c t t o 0 g i v e s :
Z 2 A*
U
\ Z o
\ Z I
system c f ; I l d R *U ' + 4 T * c h - l i r a ~ - ~ d Z
0 ~ - - ~ 0 Z ~ :
(51)
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3 2 EfthymiosS . F o l i a s
w h e r e
1 - / 2
1 = ~% '-~ "" " - - " ' ' " " - - " "~ K o + K 2 1 2 " ~ K I + K a ) ~ I + [ tK o - K 2 ) ~ " ~ K I " K 3 ) 21( 5 2 )
w i t h
K 0 _ K 2 = E Z 7+_._5_.v --~ 1 P202 c 2 l _ v 2 P 1 0 2 h c 4 ( 5 3 )
E z 2
K 0 + K 2 = + - - - - ( 5 4 )c 2 ( l - u ) h e 4
K I + K 3 = 0 ( 5 5 )
E Z P ~ 0 1 P ~ 0= 4-
K 1 - K 3 - (' T '/ '~ ) 2 c 2 ~ c 4 (56)
A n d s i n c e w e a r e r e s t r i c t in g o u r s e l v e s t o t h e s i n g u l a r s t r e s s e s ,i t i s o n l y f a i r t o d e r i v e a G r i f f i t h c r i t e r i o n a p p l i c a b l e i n t hei m m e d i a t e n e i g h b o r h o o d o f t h e c r a c k t ip , w h e r e b e c a u s e l o c a l l yt h e s h e l l i s a l m o s t f l a t , w e c a n r e p l a c e w i t h o u t m u c h e r r o r t h e
h hl i m i t s o f in t e g r a t i o n Z 1 a n d Z 2 b y - 2 a n d s . T h u s 5 1 m a y b e
a p p r o x i m a t e d b y --
h/2 A*
C $ f f i l d R #sy ste m = U " + 4 T * c h - l i r a ~ d Z ( 5 7 )z-- .0
- h / 2 •
a n d a f t e r i n t e g r a t i o n
Usystem = U o ' + 4"y *ch
- 8G {1-'g~ c 2 ( l_ u ) ~ (h - -" -4 )c h (58 )
+
3 ( i _ p 2 ) 2 C 4 2 h2c-------8-h) A; :"+
w h i c h i n t h e l i m i t , a s R ---* oo a n d ~ b - " 0 , i n e q u a l b i a x i a l t e n s i o n~ e a l o n g t h e p e r i p h e r y o f a f la t s h e e t , m u s t b e e q u a l t o ( s e e r e f .[12])
S h ( k - l ) B e 2 r c 2 h ( 3 - k ) ( 59 )U sys tem = U " + 4 T ~ c h - ~ 8 G ~ -e 2
T h u s w e h a v e d e t e r m i n e d t h e v a l u e f o r A ;'~, n a m e l y
1 6 v c 1 6 ( 1 - u ) S ( 6 0)A * -- ~ + 9-7 'u ~--~
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Effhymios S. Folias 33
S u b s t i t u t i n g i n 5 8 f o r A * a n d a p p l y i n g t h e b a s i c c o n d i t i o n 4 5 , t h ef r a c t u r e c r i t e r i o n i s s e e n to b e:
w h e r e
_ ~rc {33+6u-7v 24"y':"h ~ -( 1 -u 2 )2 24
a n d
2 " 2 3E P 1 0 h
4c
9 -7u P20 16u+ 2(l+u) h e 8 } 9 - 7 u 0
1 6 - 1 3 u 4 - 3 v o4 h ~-e o
= - { 3 2 + TP I 0 % ~- R D ( 4 _ V o )
h 2 c 2 ~ -
+ { i +6 "V'2 (4 -U o)D
e 4 h ; e ( 1 + ~ '~ A
+
F i n a l l y i n t r o d u c i n g 62r e a r r a n g e m e n t at :
(61)
( 33+6y-7y 2)4u
3 ( 9 _ 7 u ) ( 4 _ U o ) 2 ( I +
4 - 3 u~r o ~ 232 4-u o } + 0 (X41 nX)
(62)
h2c2k4R ~b 13 3
6~/2 (4 -y ° )
( 6 3 )
a n d 6 3 i n t o 6 2 w e a r r i v e a f t e r s o m e
2 + 4 u 37rh2 --2x2)}b ~(i+i-6-,~) %
4 - 3 Vo
16 4-uO
16-3v 4-3v 4-3v+ [ _ ~ ( 3 3 + 6 U - 7 y 2 ) U ( . __ o + o T + - - ~ n 4 l-) +
3 ~ 2 " - ~ ( 9 - 7v ) ( 4 -Vo ) 2 8
+ 8 _ ~ _ ~ % / i - u 1 ~ 3 3 ~ 2 - - _ 1 6 G T * ( 6 4 )l+u (4-Yo)( + ~ T + ~ - f n ) ] k O'eO- ~'C
w h i c h , f o r t h e c a s e o f a f l a t s h e e t , i . e . )t = 0 , r e d u c e s t o t h ef o l l o w i n g s i m p l e f o r m :
( 3 3 + 6 v - 7 y 2 ) 4 v ~ 2 + 4 v ~ 2 _ 1 6 G 7 ° _ ( e ~ ) 2 (6 5)
1F o r v = "g e q u a t i o n 6 4 b e c o m e s
0.21 (I + 0.12X 2) --2Crb+ (i + 0.59X 2 ) ~2e
- ( 0 . 2 4 + 0 . 0 7 l n - ~ ) = ~'c
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3 4 Efthymios S. Folias
w h i c h c l e a r l y r e p r e s e n t s a f a m i l y o f e l l i p s e s . I n v i e w o f 6 6 w e
c a n o b t a i n a r e l a t i o n b e t w e e n t h e c r i t i c a l c r a c k l e n g t h i n a s h e l l
1a n d t h e c r i t i c a l c r a c k l e n g t h i n a p l a t e , i . e . f o r v = - ~
[ O . 2 1 ~ 2 + ~ 2 ] (~ )b e Pla te cr- Pla te
= X 2 - - - -('tcr)Shell [0.21(i+0. 12X2)~+(I+0.59~t2)~2e- (0. 24+0. 07*n~)X OeOb] Shel
( 6 7 )
f o r e x a m p l e i f k = 1 a n d ( ~- b) P = ( ~ b ) s = ( ~ e ) s = ( ~ ' e ) P ' t h e n 6 7r e d u c e s t o :
( ~ c r ) S h e l l ~ O . 7 6 ( l c r ) P l a t e ( 6 8 )
T h i s c l e a r l y s h o w s t h a t t h e c r i t i c a l c r a c k l e n g t h f o r a s p h e r i c a ls h e l l i s l e s s t h a n t h a t o f a f l a t s h e e t , a n d a s i s s e e n b y 6 7t h e r a t i o d e p e n d s u p o n t h e c u r v a t u r e . T h i s a g r e e s w i t h t h es t a t e m e n t f o l l o w i n g e q u a t i o n s 3 9 a n d 4 0 .
F o r t h e s p e c i a l c a s e w h e r e ( ~ b ) P = ( ~ b ) S = ( ~- e) S = ( L ) P w e
o b t a i n t h e f o l l o w i n g e x p r e s s i o n
i=
CrP i + (0.31 - O.061n~)~. 2(69)
1.0
0 . 8
,_, , . , 0 . 6
t!
~1 ~ 0 .4
0 .2
% p
o - % = o -e p
I I l I I t0 1 .0 2 . 0 5 . 0X
Fig. 4 . Square root ra t io of cr i t ic a l crack lengths in a spher ical cap and a f la t p la te , for
v = 1/3; X = ~J12 (1 -v 2) C
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E f t h ym ios S . F o l i a s 3 5
T h i s r a t i o i s l e s s t h a n 1 f o r a l l h . < 7 . W e c o n j e c t u r e t h a t f o r
k > 7 t h e s a m e c h a r a c t e r w i l l p o s s i b l y b e p r e s e r v e d , b u t m o r e
t e r m s i n t h e b a s i c s o l u t i o n , s a y u p t o k 4 w o u l d b e r e q u i r e d t o
v e r i f y t h i s p o i n t . A s w e h a v e i n d i c a t e d , h o w e v e r , f o r m o s t
p r a c t i c a l c a s e s k i s l e s s t h a n 2 h e n c e 6 9 g i v e s a g o o d a p p r o x -i m a t i o n . A p l o t o f e q u a t i o n 6 9 g i v e n i n f i g . 4 . T h i s t y p e o f
b e h a v i o r w a s a l s o o b t a i n e d e x p e r i m e n t a l l y , f o r c r a c k e d c y l i n d r i c a l
s h e l l s , b y S e c h l e r a n d W i l l i a m s [ 2 ] .R e t u r n i n g t o e q u a t i o n 6 6 w e n o t e t h a t i t c a n a l s o b e w r i t t e n
i n t h e f o r m
( 1 + O . 5 9 X 2 ) ( e_.;_)2 _ ( 0 . 2 4 + 0 . 0 7 l n ~ - ) X 2 ( - - ~ -- ) - (- - - ~ -) +(3- (3- o-
s s s
+ 0 . 2 1 ( 1 + O . 1 2 X 2 ) (~ _~ b,)2 = 1(7
S
(70)
T h i s o b v i o u s l y r e p r e s e n t s a f a m i l y o f e l l i p s e s , w h i c h a r e p l o t t e df o r d i f f e r e n t v a l u e s o f t h e p a r a m e t e r ~ , s e e f i g . 5 . F o r k g r e a t e r
1.0
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% & r *
F i g . 5 . E x t e n s i o n - b e n d i n g i n t e r a c t i o n c u r v e f or a s p h e ri c a l c a p , f or v = ! / 3 ;
X = ~ / 1 2 ( 1 - v 2 ) C4-ff
t h a n 1 . 5 w e w i l l n e e d h i g h e r o r d e r s o f X f o r t h e d e t e r m i n a t i o n
o f t h e e l l i p s e s ; t h e r e f o r e f o r X = 5 , i 0 w e s h o w j u s t t h e i n -t e r c e p t s . * I t i s a l s o c l e a r f r o m f i g . 5 t h a t t h e a p p l i e d s a f e
l o a d i n a c r a c k e d s p h e r i c a l s h e l l d e c r e a s e s w i t h a d e c r e a s e i n
r a d i u s o f c u r v a t u r e . F o r e x a m p l e , i f a l o n g t h e c r a c k t h e r e i s
a r e s i d u a l l o a d o f e q u a l b e n d i n g a n d s t r e t c h i n g a f l a t s h e e t
c a n c a r r y , b e f o r e f a i l u r e o c c u r s , u p t o a l o a d o f 0 . 8 8 ( - - y ) w h i l e
a s p h e r i c a l s h e l l c h a r a c t e r i z e d w i t h t h e p a r a m e t e r X = 1 c a n
c a r r y o n l y u p t o 0 . 7 6 ( _ _ ~ e ) , i . e . a p p r o x i m a t e l y 1 4 % l e s s l o a d( i *
t h a n a f l a t s h e e t .
* Cur v e fo r k = 2 fol l ows t he an t i c i p a t e d t r end .
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F i g . 6 . S p h e r i c a l c a p u s e d f or s t r ai n m e a s u r e m e n t s .
F i g . 7 . C l o s e - u p o f s t ra i n g a u g e l o c a t i o n s .
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Efthymios S. Folias 3 7
E X P E R I M E N T A L V E R I F I C A T I O N
Description of Experiment
T o c o m p a r e t h e o r e t i c a l a n d a c t u a l b e h a v i o r o f a n i n i t i a l l y
c u r v e d s p e c i m e n , a p r e l i m i n a r y e x p e r i m e n t w a s c o n d u c t e d . W e
h a v e c o n s i d e r e d a c l a m p e d s e g m e n t o f a s h a l l o w s p h e r i c a l s h e l l ,
c o n t a i n i n g a t t h e a p e x a r a d i a l c u t o f l e n g t h 0 . 4 6 i n . T h e s h e l lw a s s u b j e c t e d t o a u n i f o r m i n t e r n a l p r e s s u r e q o , a n d t h e s t r a i n
e y a t t h r e e d i f f e r e n t p o s i t i o n s a l o n g t h e d i r e c t i o n o f c r a c k
p r o p a g a t i o n w a s r e c o r d e d a s a f u n c t i o n o f q o " T h e d e s i g n o f
t h e e x p e r i m e n t d i d n o t p e r m i t a d e t e r m i n a t i o n o f c r i t i c a l c r a c k
l e n g t h , f u r t h e r m o r e t h e c o p p e r m a t e r i a l i s t o o d u c t i l e f o r
b r i t t l e f r a c t u r e t h e o r y t o a p p l y .
Preparations
T h e s h a l l o w s h e l l s eg ,m , e n t w a s c o n s t r u c t e d b y t h e m e t h o d o f" c o p p e r e l e c t r o f o r m i n g . I t s c h a r a c t e r i s t i c s w e r e h = 0 . 0 0 9
= 4 . 2 5 i n . , v 1n . , R = 2 0 i n . , 5 = 0 . 4 i n . , S o = ~ , E =
1 6 x 1 0 6 l b s / i n 2 . A h o l e o f 0 . 0 1 2 i n . d i a m e t e r w a s d r i l l e d a tt h e a p e x o f t h e s h e l l s e g m e n t , a n d a c r a c k w a s s a w e d w i t h aj e w e l e r ' s s a w o f 0 . 0 0 7 i n . t h i c k n e s s . F i n a l l y , t h e e n d s o f t h ec r a c k w e r e s m o o t h e d b y t h e " d i a m o n d t h r e a d m e t h o d , " * *( d i a m e t e r o f d i a m o n d t h r e a d l e s s t h a n 0 . 0 0 5 i n . ) . I n t h e p r o c e s so f d r i l l i n g a n d o f s a w i n g , a w a x b a c k i n g w a s u s e d i n o r d e r t oa v o i d d a m a g i n g o f t h e s h e l l . N e x t , a l o n g t h e l i n e o f c r a c k p r o -l o n g a t i o n , t h r e e s t r a i n g a g e s w e r e a t t a c h e d o n t h e s h e l l t o m e a s -u r e t h e s t r a i n s i n t h e Y d i r e c t i o n ( s e e fig s._ _ 6 , 7 ) . T h e s h e l l w a sc e m e n t e d b e t w e e n t w o c i r c u l a r r i n g s , w i t h R o = 4 i n . a s t h e i n s i d er a d i u s ( s e e f i g . 8 ). N e x t t h e c r a c k w a s s e a l e d i n t e r n a l ly w i t h t w ol a y e r s o f a c e t a t e f i b r e t a p e . T h e f i r s t l a y e r w a s a s q u a r e o f 2 x 2
r - C E M E N T C R A C K/ Z ~ r ' ~S H E L L
VALVE
Fig. 8 . Schem at ic of tes t ing apparatus .
i n c h e s , a n d t h e s e c o n d l a y e r w a s a r e c t a n g u l a r o n e o f 2 x ½ i n c h e s .T h e f o l l o w i n g t a b l e g i v e s t h e g a g e f a c to r s a n d p o s i t i o n s o f t h eg a g e s f r o m t h e c r a c k t i p .
Gage no . G .F .=G age Fac to r X - 1 - 1
1 2 . i 0 O . 0 7 i . 8 1
2 2 . 0 9 O . 2 9 O . 8 9
3 2 . 0 9 0 . 4 8 0 . 6 9
- See tef . [131.*~ A cot ton thread im pregnated wi th 6 m icron diamondp a s t e .
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3 8 Efthymlos S. Folias
I n f i g. 9 w e h a v e p l o t t e d v o l t a g e v s . g a g e p r e s s u r e , a n d b e c a u s et he c u r v e s f o r s m a l l p r e s s u r e s w e r e n o t q u it e s t r a i g h t l i n e s , as e c o n d r u n w a s c o n d u c t e d a f e w h o u r s l a t e r. I t g a v e b e t t e r r e s u l t s( fi g. 1 0) . T h e c h a n g e b e t w e e n f i r s t a n d s e c o n d r u n s i s a t t r i b u t e dt o w a r m i n g u p o f t h e r e s i s t a n c e g a g e s i n t he e l e c t r o n i c e q u i p m e n t
14
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a n d " s e t t i n g " o f t he s t r a i n g a g e s . E v e n f o r t he s e c o n d r u n , t hec u r v e s a r e s l i g h t l y c u r v e d a t t he o r i g i n . I t i s p o s s i b l e t h a t t het ape ca r r i e s a sma l l pa r t o f t he l oad . I n any ca s e , we con s id e rt he s l ope o f t he cu rv e wh ich i s g iven by
A Ev° l t ag e (71)C = sl op e = A. F. x Aqo
w h e r e A . F . = a m p l i f i e r f a c t o r = 5. I n v i e w o f t his , w e c a nc o m p u t e t h e s t r a i n s f r o m
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Ef thymiosS. Fo l i a s 39
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Fig . lO . T es t no .2 . Da ta for a c l amp ed sphe r i ca l cap con ta in ing a c r ack .
4 C q o
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1 70 x I O: o 5 4 x i 0 , % I ~ x I O - ~ 0 ~ x i 0 - 4 %
1 . 3 q x l0 - 4 0 . 4 4 x 1 0 - 4 q o i . 1 5 x 1 0 - 4 0 . 3 6 x 1 0 " 4 q
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4 0 Ef thymios S . Fo l ias
w h e r e P. S . V. = P o w e r s u p p l y v o l t a g e = 6 v o l t s ( m e a s u r e d ) .T h e t h e o r e t i c a l s t r a i n s w e r e c a l c u l a t e d f r o m e q u a t i o n 4 3 a n d
t he c o m p a r i s o n w i th e x p e r i m e n t a l l y d e t e r m i n e d v a l u e s f ol l o w s .
C o n c l u s i o n s
I n f ig . 11 , w e c o m p a r e t he t h e o r e t i c a l p r e d i c t e d s t r a i n s w i tht he e x p e r i m e n t a l o n e s . I t i s e a s y t o s ee t h a t c l o s e t o t h e c r a c kt ip th e t h e o r e t i c a l r e s u l t s a r e s l i g h t l y l o w e r t h a n t h e e x p e r i m e n t a lo n e s , e . g . t h e r e e x i s t s a n e r r o r o f a b o u t 3% f o r t h e f i r s t g a g e .
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F i g . i i . C o m p a r i s o n o f e x p e r i m e n t a l a n d t h e o r e t i c a l s t r a i n s a h e a d o f t he c r a c k .
We r e c a l l t h a t i n t he t h e o r e t i c a l f o r m u l a w e n e g l e c t e d t e r m so f O(X4) . Th i s f a c t cou l d con t r i bu t e t o t he d i f f e r e nce , a s we l la s t he a v e r a g i n g e f f e c t o f t h e f i ni t e g a g e t h i c k n e s s . A s wem o v e f u r t h e r a w a y f r o m t he c r a c k t ip th e t h e o r e t i c a l v a l u e sb e c o m e s m a l l e r t h a n t he e x p e r i m e n t a l o n e s . T h i s i s t o b ee x p e c t e d , s i n c e n o w ~ i s l a r g e a n d t he n o n - s i n g u l a r t e r m sb e c o m e s i gn i fi c an t . O u r t he or et ic al re s ul t s w e r e c o m p u t e d o nt he b a s i s o f o n l y t he s i n g u l a r t e r m a n d f u r t h e r m o r e o n l y u p t ot e r m s o f O ( k2 ) .
W h i l e it s h o u l d b e p o i n t e d o ut t ha t t h e b e n d i n g s t r e s s e s a r ep r a c t i c a l l y n e g l i gi b l e f o r t h i s p a r t i c u l a r l e s t c o n fi g u ra t i on , itw a s f o u n d t h at o n t h e w h o l e t he e x p e r i m e n t a l a n d t h eo r e ti c alr e su l ts c o m p a r e v e r y we ll .
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Ef thym ios S . Fo l ias
C O N C L U S I O N S
4 1
T h e l o ca l s t r e s s e s n e a r t he c r a c k p o in t a r e f o u n d t o b e p r o -p o r ti o n a l t o i / ~ - w h i c h is c h a r a ct e r i s ti c f or c r a c k p r o b l e m s .F u r t h e r m o r e , t he a n g u l a r d is tr ib ut io n a r o u n d t he c r a c k tip i se x a c t l y t h e s a m e a s t h a t o f a f l a t s he e t, a n d t h e c u r v a t u r e a p -p e a r s o n l y i n t h e in te n si ty f a c t o r s a n d i n s u c h a w a y t h at f o r
R - - * ~ w e r e c o v e r t h e f l at s h e e t b e h a v i o r . A t yp ic al t e r m i s
f fs he ll " ~ O 'p la t e 1 + ( c o n s t + c o n s t ~ n - c ) ~ - - ~ + 0 ( )
( 7 3 )
w h e r e t h e e x p r e s s i o n i n t h e p a r e n t h e s e s i s a p o s i t i v e q u a n t i t y .
T h e g e n e r a l e f f e c t o f i n i t i a l c u r v a t u r e , i n r e f e r e n c e t o t h a t o f
a f l a t s h e e t , i s t o i n c r e a s e t h e s t r e s s i n t h e n e i g h b o r h o o d o f
t h e c r a c k p o i n t . F u r t h e r m o r e , i t i s o f s o m e p r a c t i c a l v a l u e
t o b e a b l e t o c o r r e l a t e f l a t s h e e t b e h a v i o r w i t h t h a t o f i n i t i a l l y
c u r v e d s p e c i m e n s . I n e x p e r i m e n t a l w o r k o nr i t t l e f r a c t u r e f o re x a m p l e , c o n s i d e r a b l e t i m e m i g h t b e s a v e d s i n c e b y3 w e w o u l d
e x p e c t t o p r e d i c t t h e r e s p o n s e b e h a v i o r o f c u r v e d s h e e t s f r o m
f i a t s h e e t t e s t s .
T h e s t r e s s e s a l s o i n d i c a t e t h a t t h e r e e x i s t s a n i n t e r a c t i o n
b e t w e e n b e n d i n g a n d t r e t c h i n g , i . e . b e n d i n g l o a d i n g s w i l l
g e n e r a l l y p r o d u c e b o t h b e n d i n g a n d s t r e t c h i n g s t r e s s e s , a n d
v i c e v e r s a .
I t i s w e l l k n o w n t h a t l a r g e , t h i n - w a l l e d p r e s s u r e v e s s e l s
r e s e m b l e b a l l o o n s a n d i k e b a l l o o n s a r e s u b j e c t t o p u n c t u r e a n d
e x p l o s i v e l o s s . F o r a n y g i v e n m a t e r i a l , u n d e r s p e c i f i e d s t r e s s
f i e l d d u e t o i n t e r n a l p r e s s u r e , t h e r e w i l l b e a c r a c k l e n g t h i n
t h e m a t e r i a l w h i c h w i l l b e s e l f p r o p a g a t i n g . C r a c k l e n g t h s e s s
t h a n t h e c r i t i c a l v a l u e w i l l c a u s e l e a k a g e b u t n o t d e s t r u c t i o n .
H o w e v e r , i f t h e c r i t i c a l l e n g t h i s e v e r r e a c h e d , e i t h e r b y
p e n e t r a t i o n o r b y t h e g r o w t h o f a s m a l l f a t i g u e c r a c k , t h e e x -
p l o s i o n a n d c o m p l e t e l o s s f t h e s t r u c t u r e o c c u r s . T h i s r i t i c a l
c r a c k l e n g t h , u s i n g G r i f f i t h ' s c r i t e r i o n , w a s s h o w n t o d e p e n d
u p o n t h e s t r e s s f i e l d , t h e r a d i u s a n d t h i c k n e s s o f t h e v e s s e l ,
a s w e l l a s t h e m a t e r i a l i t s e l f ( s e e 6 4 ) . W e w e r e a l s o a b l e t o
o b t a i n a r e l a t i o n f o r t h e r a t i o c r i t i c a l c r a c k l e n g t h o f a s p h e r i c a l
s h e l l o v e r c r i t i c a l c r a c k l e n g t h o f a f l a t s h e e t ( s e e e q . 6 7 ) . I n
g e n e r a l t h i s r a t i o i s l e s s t h a n u n i t y , w h i c h a g a i n i n d i c a t e s
c l e a r l y t h a t a c r a c k e d i n i t i a l l y c u r v e d s h e l l i s w e a k e r t h a n a
c r a c k e d f l a t s h e e t s u b j e c t e d t o t h e s a m e l o a d i n g .
I n c o n c l u s i o n i t m u s t b e e m p h a s i z e d t h a t t h e c l a s s i c a l b e n d i n g
t h e o r y h a s b e e n u s e d n d e d u c i n g t h e f o r e g o i n g r e s u l t s . H e n c e
i t i s i n h e r e n t t h a t o n l y t h e K i r c h h o f f e q u i v a l e n t s h e a r r e e
c o n d i t i o n i s s a t i s f i e d a l o n g t h e c r a c k , a n d n o t t h e v a n i s h i n g o f
b o t h i n d i v i d u a l s h e a r i n g s t r e s s e s . W h i l e o u t s i d e t h e l o c a l r e g i o n
t h e s t r e s s d i s t r i b u t i o n s h o u l d b e a c c u r a t e , o n e m i g h t e x p e c t h e
s a m e t y p e o f d i s c r e p a n c y t o e x i s t n e a r t h e c r a c k p o i n t a s t h a t
f o u n d b y K n o w l e s a n d W a n g7 ] i n c o m p a r i n g K i r c h h o f f a n dR e i s s n e r b e n d i n g r e s u l t s o r t h e f l a t p l a t e c a s e [ 8 , 9 ] . I n t h i s
c a s e t h e o r d e r o f t h e s t r e s s s i n g u l a r i t y r e m a i n e d u n c h a n g e d b u t
t h e c i r c u m f e r e n t i a l d i s t r i b u t i o n a r o u n d t h e c r a c k c h a n g e d s o s
t o b e p r e c i s e l y t h e s a m e a s t h a t d u e t o s o l e l y e x t e n s i o n a l l o a d i n g .
P e n d i n g f u r t h e r i n v e s t i g a t i o n o f t h i s e f f e c t f o r i n i t i a l l y c u r v e dp l a t e s , o n e i s t e m p t e d t o c o n j e c t u r e t h a t t h e b e n d i n g a m p l i t u d e
a n d a n g u l a r d i s t r i b u t i o n w o u l d b e t h e s a m e a s t h a t o f s t r e t c h i n g .
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4 2 Ef thymios S. F o l ias
F i n a l l y f o r a c l a m p e d s p h e r i c a l s h e l l , t h e e x p e r i m e n t a l a n dtheo re t i ca l s t r a ins Cy, a t th ree d i ffe re n t loca t ions a long thec r a c k pr o l o n g at i o n , c o m p a r e v e r y w el l .
ACKNOWLEDGMENTS
T h e a u t h o r a c k n o w l e d g e s s e v e r a l u se fu l d i s c u s s io n s of t hi sp r o b l e m w i th P r o f e s s o r M . L . W i l l i am s a n d M r . J . L . S w e d l o wo f t he C a l if o rn i a In st it ut e o f Te c h n o l o g y . T h e r e s e a r c h r e p o r t e dh e r e in w a s s p o n s o r e d i n p a r t b y t he A e r o s p a c e R e s e a r c hL a b o r a t o r i e s , O f f i c e o f A e r o s p a c e R e s e a r c h o f t he U n i t e d S ta t esA i r F o r c e .
R e c e i v e d J u n e 1 9 6 4 .
REFERENCES
I . A n g , D . D . a n d Wi l l i am s , M . L . , " C o m b i n e d S tr e ss es i n a n O r t h o t ro p i c P l a t e H a v i n g
a F i n i t e C r a c k , " J o u r n a l o f A p p l i e d M e c h a n i c s , v o l . 28 , 1 9 6 1 , p p. 3 7 2 - 8 7 8 .
2 . S e c h l e r, E . E. a n d Wi l l i a m s , M . L . , " T h e C r i t i c a l C r a c k L e n g t h i n P r e s su r i ze d , M o n o -c o q u e C y l i n d e r s, " F i n a l R ep o rt o n C o n t r a c t N A w - 6 5 2 5 , C a l i f o rn i a I n st i t u t e o f Te c h -n o l o g y, S e p t e m b e r 1 9 5 9 .
3 . R e i ss n e r, E . , " O n S o m e P r o b l e m s i n S h e l l T h e o r y, " S t r u c t u r a l M e c h a n i c s , P r o c e e d i n g so f t h e F i r st S y m p o s i u m o n N a v a l S t r u c t u r a l M e c h a n i c s , P e rg a m o n P re ss , 1 96 0, p p ;7 4 - I 1 3 .
4 . R e i ss n e r, g . , " A N o t e o n M e m b r a n e a n d B e n d i n g S t re s se s i n S p h e r i c a l S h e l l s , " J . S o c .I n d u s tr. A p p l . M a t h . , v o l . 4, 1 9 5 6 , p p . 2 3 0 - 2 4 0 .
5 . F o l i a s , E f t h y m i o s S . , " T h e S t r es s es i n a C r a c k e d S p h e r i c a l S h e l l , " ( S u b m i t t e d t o t h eJ o u r n a l o f M a t h e m a t i c s a n d P h y s ic s , 1 9 6 4 ) .
6 . F o l i as , E f t h y m i o s S . , " T h e S t re s se s i n a S p h e r i c a l S h e l l C o n t a i n i n g a C r a c k , " A R L6 4 - 2 8 , A e r o s p a c e R e s e a r c h L a b o r a t o r ie s , O ff i c e o f A e r o s p a c e R e s e a rc h , U . S . A i rForce , Jan uary 1964 .
7 . K n o w l e s , J . K . a n d Wa n g , N . M . , " O n t h e B e n d i n g o f a n E l a st i c P l a t e C o n t a i n i n g aC r a c k , " J o u r n a l o f M a t h e m a t i c s a n d P h y s ic s , v o l . 8 9 , 1 9 6 0 , p p . 2 2 3 - 2 8 6 .
8 . W i l l i a m s , M . L . , " T h e B e n d i n g S t re s s D i s t r i b u t i o n a t t h e B a s e o f a S t a t i o n a r y C r a c k , "J o u r n a l o f A p p l i e d M e c h a n i c s , v o l . 2 8 , Tr a n s . A S M E , v o l . 8 8 , S e r i es E , 1 9 6 1 ,p p . 7 8 - 8 2 .
9 . W i l l i a m s , M . L . . " O n t h e S t re s s D i s t r i b u t i o n a t t h e B a se o f a S t a t i o n a r y C r a c k , " J o u r-n a l o f A p p l i e d M e c h a n i c s , v o l . 2 4 , M a r c h 19 5" /, p p . 1 0 9 - 1 1 4 .
1 0 . R e i s sn e r, E . , " S t r es s e s a n d S m a l l D i s p l a c e m e n t s o f S h a l l o w S p h e r i c a l S h e l l s I , I I , "J o u r n a l o f M a t h e m a t i c s a n d P h y s i cs , v o l . 2 5 , 1 9 4 6 , p p . 8 0 - 8 5 , 2 7 9 - 3 0 0 .
11 . G r i f f i t h , A . A . , " T h e T h e o r y o f R u p t u r e , " P r o c e e d i n g s o f t h e F i rs t I n t e r n a t i o n a l C o n -g r es s o f A p p l i e d M e c h a n i c s , D e l f t , 1 92 4, p p . 5 5 - 6 3 .
1 2 . S w e d l o w, J . L . , " O n G r i f f i t h ' s T h e o r y o f F r a c t u r e , " G A L C IT S M 6 3 - 8 , C a l i f o r n i a l n -s t i t u t e o f Te c h n o l o g y, M a r c h 1 9 6 8 .
1 3 . P a r m e r t e r, R . R . , " T h e B u c k l i n g o f C l a m p e d S h a U o w S p h e r i c a l S h e l ls U n d e r U n i fo r mP r e s s u re , " P h . D . D i s s e r t a t i o n , C a l i f o r n i a I n s t i t u t e o f Te c h n o l o g y, S e p t e m b e r 1 9 6 3 .
1 4 . M i k h l i n , S . G . , I n t e g r a l B q u a t i 0 n s, E n g l i sh t r a n s N t i o n b y A . H . A r m s t r o n g , p u b -l i shed by Perga mo n Press , 1957 .
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N O T AT I O N 4 3
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= Yo u n g ' s m o d u l m o f e l a s t i c i t y
- - s t r e s s f u n c t i o n
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= c r i t i c a l c r a c k l e n g t h o f p l a t e
-- c o n s t a n t a s d e f i n e d i n t e x t
-- m o m e n t c o m p o n e n t s
= c o n s t a n t a s d e f i n e d i n t e x t
= m e m b r a n e f o r ce s
= p e r i p h e r y
= i n t e r n a l p r e s s u r e
= u n i f o r m i n t e r n a l p r e s s u r e
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= c o n s t a n t a s d e f i n e d i n t e x t
= e q u i v a l e n t s h e a r
= d i s p l a c e m e n t f u n c ti o n
= d i m e n s i o n l e s s c o o r d i n a t e s w i t h r e s p e c t t o t he c r a c k l e n g t h
-- r e c t a n g u l a r c a r t e s i a n c o o r d i n a t e s
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4 4
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s t r a i n c o m p o n e n t s
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+ = +
b e n d i n g s t r e s s c o m p o n e n t s
s t r e t c h i n g s t r e s s c o m p o n e n t s
a p p l i e d s t r es s c o m p o n e n t s a t t h e c r a c k
c r i t i c a l ( f r a c t u r e ) s t re s s f o r s h e i l
c r i t i c a l ( f r a c t u r e ) s t r es s f o r p l a t e
h a r m o n i c f u n c t i o n s
d e f l e c t i o n o f a p l a t e o n a n e l a s t i c f o u n d a t i o n .
A P P E N D I X
T h e d e f l e c t i o n f u n c t i o n W a n d t h e s t r e s s f u n c t i o n F a r e :
W ( x , y * ) = P 1 e + P 2 e + P 3 e c o s x s d s
F ( x , y ± ) - i X2 RD c2 P i e - ~24~-~-ix21Yl P 2 e + P 4 e c o s x s d s
w h e r eX 2 A 2 J 2 ( s )
S {AIJ I(S ) + + O(14 }P ~ ( s) - ~ 3 s
~ t 2 B 2 J 2 ( s )s
P 2 ( s ) - ~ { B I J I ( S ) + 3 s + O ( X 4 ) )
X ~ ( A 2 + B 2 ) J 2 ( s )P 3 ( s ) = - ( A z + B z ) J I ( s ) 3 s
i X 2 ( A z _ B I ) J l ( S ) + 0 ( ; %/2 S 2
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w h e r e
E f t h y m i o s S . F o l ia s
X2 (A2-B2) J2 (s)P 4 ( S ) = - ( A 1 - B 1 ) J ~ ( s ) 3 s
) t 4 R D
2 ~ (AI + B 1 )
+ O ( X 4 )
45
n c 2 m t . 2 ' ~ E h D c 2o { 1 + 3 r o
+ O ( X 4 I n ) t )
~ 2 X 2 12 ) t 2 8 - 7 / /-- 0 { O
2~r2 (AI-B1) = ~/2EhD (4-v o) 32 +
{13 a_y 3 X2gg+ 8 + ~ lni-g}
m ° { 1 + ~ ) t2 4 - 3 u ° } + O ( / 4 1 p n / )~ / ~ ( 4 _ U o ) 3 2 4 - u o
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o t
A 2 + B 2 = _ -if- (AI-B I)
( l + ~ n ~ ) }
4-3//0 ~2A 2 - B 2 = _ (4-~-2~vo) ~ (At + B I)
R E S UM { - O n m o n t r e c lu e l a d g f o r m a t i o n d ' u n e f e u i l l e m i n c e a y a n t u n e c o u r b u r e s p h g r i q u ei n i t ia l s , e s t a s s o c i g e a v e e c e l l e d ' u n e p l a q u e m i n c e r e p o s a n t s u r u n e f o n d a ti o n g la s t iq u e .
g n u t i l is a n t u n e f o r m e i n t g g r a l e , o n rg s o u d l e s g q u a t i o n s c o u p l g e s d e R e i s sn e r , p o u r u n ec o q u e a v e c u n e f r a c t u r e d e lo n g u e u r 2 c , p o u r l e p la n i n t e r n e e t le s c o a t r a i u t e s d e f l e x i o nd e K i r c h h o f f . D ' a u t r e p a r t o n t r o u v e q u e l a f o r m e e x p l i c i t e d e s c o n t r a % t e s p r gs d u e p o i n td e f r a c tu r e , d g p e a d d e l 'i n v e r s e d e l a d e m i p u i s s a n c e d e l a d i s t a n c e , s a n s d i m e n s i o n , a up o i nt e . L e c a r a c t ~ r e d u c h a m p s d e c o n t r a i n te c o m b i n g e , f l e x io n t e n s io n , p r ~t d e l ' e x t r g m i t gd e l a f r a c t u r e , e s t g t u d i g e n d g t a i t d u n s l e c a s p a r t i c u l ie r d ' u n e f r a c t u r e r a d i a l e d u n s u n ec u l o t t e s p h g r i q u e s o u m i s e ~I u n e p r e ss i o n i n t e r n e u n i f o r m e q o e t q u i e s t e n c a s t r g e ~t l a l i m i t e
- - R o " g n a t t e n d a n t u n e & u d e c o m p t g t e d e l a s o l u t i o n , d e s r g s u lt a t s a p p r o c h g s p o u r l e s
c o n t r a i n t e s c o m b i n g e s d e s u r f a c e , p r o s d e l ' e x t r g m i t g d e l a f r a c t u r e n o r m a l e , e t l e l o n g d el a l i g n e d e p r o lo n g a t i o n d e l a f r a c t u r e , o a t d o n n g s r e s p e c t i v e m e n t
~ry (~,o) -- '~x (~,o) , , 0.45 q ' ~ (%R/h)
o u v = 1 / 3 , c = 0 . 2 3 i n c h , R o = 4 . 2 5 i n c h e t X = 0 . 9 8 .
I1 e s t i n t e r r e s s a n t d e n o t e r q u e le s c o n t r a i n t e s ~ x e t a y ! e l o n g d e l a l i g n e d e p r o l o n g a t i c ~d e f r a c t u r e s o n t S g a t e s p o u r c e t t e g g o m g t r i e . E n g g h g r a l , e l l e s o a t l e m ~ m e s ig n e e t n ed i f fe r e n t l d g ~ r e m e n t q u ' e n m o d u l e , d u f a i t d e l a c o m p o s a n t e d e f l e x io n . F i n a l e m e n t t e s r g -s u l t a ts e x p ~ r i m e n t a u x e t t h ~ o r i q u e s p o u r Cy l e l o n g d e l a l i g n e d e p r o l o n g a t i o n d e l a f r a c -t u re c o ~ c i d e n t b i e n .
Z U S A M M E N FA SS U N G - E s w i rd g e z e i g t , d a s s d i e Ve r f o r m u n g e i a e r d u e n n e n , a n f a e n g l i c h s p h a e -r is c h g e k r u e m m t e n P l a t te m i t d e r Ve r f o r m u n g e i n e r a n f a e n g l i c h e b e n e n P l a t t e , d i e e l a s ti s e h
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4 6 E f t h y m i os S . F o l ia s
g e b e t t e t i s t , v e r b u n d e n i s t. M i t H i l f e e in e r I n t e g r a lf o r m u l a t i o n w e r d e n d i e g e k u p p e i t e nR e i s s n e r - G l e i c h u n g e n f u e r e i n e S c h a l e , d i e e i n e n R is s d e r L a e n g e 2 c e n t h a e l t , f u e r d i e e b e n e nu n d d i e K i r c h h o f f B i e g e s p a n n u n g e n g e l o e s t. U n t e r a n d e r e m w u r d e f e s t g e s t e l l t, d a s s d i e e x -p l i z i t e F o r m d e r S p a n n u n g e n i n d e r N a e h e d e s B m c h p u n k t e s u m g e k e h r t p r o p o r t io n a l d e r Wu r z e ld e s d i m e n s i n n s l o s e n A b s t a n d e s y o r e P u n k t e i s t. D e r C h a r a k t e r d e s D e h n m a g s - B i e g e s p a n n u n g s-f e l d e s i n d e r U m g e b u n g d e r R i s s p it z e w i r d i m D e t a i l u n t e r s u c h t f u e r d e n s p e z i e l l e n F a l l e i n e sr a d i a l e n R i ss es in e i n e r s p h a e r i s c h e n K a p p e u n t e r g l e i c h f o e r m i g e n I r m e n d ru c k q o , d i e a ni h r er A u f l a g e R = Ro e i n g e k l e m m t i s t . D i e a n g e n a e h e r t e n E r g e b n is s e - a l as a u s f u e h r li c h eS t u d i n m d e r L o e s u n ge n i s t n o c h n i c h t ab g e s c h l o s s e n - f u e r d i e k o m b i n i e r t e n O b e r f l a e c h e n -s p a n n u n g e n n a h e d e r R i s s p i t z e s e n k r e c h t z u m R is s u n d i n R i c h t u n g d e s R i ss e s s i n d y o n d e rF o r m :
~ ;y (¢ ,0 ) ~ :~X (¢ ,0 ) "~ 0 .4 5 ~ (qoR /h)
e t v = 1 / 3 , c = 0 . 2 3 i n c h , R o = 4 . 2 5 i n c h u n d k = 0 . 9 8 .
E s i s t i n t e r e s s a n t , d a s s d i e S p a n n u n g e n ~ x u n d g y e n t l a n g d e r L i n i e d e r R i s s f o r tp f l a n z u n gg l e i c h s in d f u e r d i e ss e n F a l l . I m a l l g e m e i n e n w e r d e n s i e d a s s e l b e Vo r z e i c h e n h a b e n un ds i c h n u t w e n i g i n d e r a b s o l u t e n G r o e s s e u n t e rs e h e i d e n i n f o l g e d e r B i e g e k o m p o n e n t e . D i e e x -p e r i m e n t e l l e n u n d t h e o r e t i s c h e n E rg e b n i s s e f u e r C y e n t l a n g d e r L i n i e d e r R i s s f o r t p f la n z u n g
s t i m m e n s e h r gu t u e b e r e i n .