Stress intensity factor and shape of edge crack in plate under tensio

Orange, T. W.

English

NASA Technical Reports Server

N 1 1 -  3 2 2 2 9  
NASA TM %-67891 
CRACK SHAPES AND STRESS INTENSITY FACTORS 
FOR EDGE-CRACKED SPECIMENS 
by Thomas W. Orange 
Lewis Research Center 
Clevelmd, Ohio 
TECBWBCAE PAPER proposed fo r  presentation at 
Fgth National Symposium on Fracture Mechanics 
University of Illinois, Champaign, Illinois, 
August 31 - September 2, 1971 
https://ntrs.nasa.gov/search.jsp?R=19710022753 2020-03-23T15:11:11+00:00Z
NA* 1 l n a l  A \ e ~ ~ n  391-  3 ( 5  and Spare AdrnLnlstrat ioa 
i w > .  R e s e a r ~ h  Csnret 
C l t r - ~ J a r s d ,  Ohlo  
A simple stress Lnrcnszsy factor e x p r e s s r o n  rs glven  for 3 deep  edge 
crack I n  a p l a t e  - n  pensrfsn The shapes of c r a c k s  opened by t ea~c io r i  3r 
b e n d ~ n g  are appe-oximaked by cornrc sec t - rons ,  and the  c o n i c  s e c t  r sn  coe f f r -  
s s e n t s  related r r :  pJ a t <  geni-rjetry hj r  v e r y  sample e m p ~ r i c a l  e q u a t r o n s ,  The 
magn-stucle o f  th c v a r k  dr_cplal-sment x i  a flsnee ron of applied l o a d ,  p la te  
geometcy, and r h e  - - - l a s * r / -  , - , t , s ,dn t s  ---I-? - sf t h e  p l a t e  rnaeeraal, The shhpe sf 
a loaded crdf -k  1-n c, sem: -3xrf rnxtc-  plate a s ,  approx imate ly ,  ci por t i o r :  of 
an ellipse whose :rrnlmajor d x ~ s  as abou t  three t u n e s  the crack Length, 
As t h e  crack 1-engtb i r e i d t r v e  t o  the p l a t e  wid th )  i n c r e a s e s ,  the crack 
shape becomes parabo%rr  , then  h y p t r b o l i c ,  t h e  a e u l t p  of the hyperbola  i n -  
c r e a s i n g  w7 th 1 he r e 1 a f  ~ V F  c r a c k  I ~ n g s h ,  
The problem of a crack ha the edge of a s s r l p  o r  a p l a t e  under rormal 
t e n s i o n  o r  b e n d ~ n g  i s  of interest t o  m a t e r i a l s  engineers as w e l l  as t o  
s t r u c t u r a l  analysts Fox maximum u t i l i t y  i n  both f i e l d s ,  expresslors for 
t h e  s t r e s s  ~ n l e n s i t y  f a c r o r  are needed over t h e  w i d e s t  p o s s i b l e  range of 
t h e  ratio of r r a - k  Lengch t o  p l a t e  wldth The e f f e c t s  of loading a n d  
g e o r n e t s ~ s ~ . ?  p a x n a + t F r s  t-17 ?he shape   if tb.2 o p c n ~ d  crack axe  of ' ~ ~ d a -  
mental  -I n l  er es t 
STRESS TNTENSTTP FACTOR FOR DEEP EDGE CRACK IN A TENSION SPECIMEN 
Wilson [ l j  has S ~ O I ~ ~ P  t ha t  c i i l P o c a t ~ ~ n  resukcs  t o r  bend and a ~ T P ~ S L  L 
tension specxmerls "TI bc c+xt capo la red  t o  a/W - 1-0 by expressing them 
i n  the Porn  ok. a p p r ? p r s a s  e drmens  onl less paramete rs ,  A s r m z  l.aa tori;: rcc- 
t f o n  is a l s o  p c r s s ~ h i e  f o r  the edge c r a c k  I n  p u r e  t ensaon ,  Then r b e  at-- 
rnensl o n l e s s  par ame t s r  
i s  computed From ro43a-arion results [ 2 , 3 ]  and p l o t t e d  ( F i g *  PI, ~t i c  
seen to s a p i d l ;  approavch a ~/all_ue c s  0,5 w x t h  increasqng a / ~ .  Phz ex- 
p r e s s  i o n  
i s  w i t h i n  1 p e r c e n t  of t h e  r e f e r e n c e d  c o l l o c a t i o n  r e s u l t s  f o r  d/W - 0 , 3  
I t  rs of i n t e r e s t  t o  n o t e  t h a t  (1) f o r  t h e  t e n s l o n  spec inwn a n d  
Wilson's expxessrons  f o r  the bend and compact t e n s l o n  sperrmens caixvergc. 
as a + W .  Zn t-hese equations a common d imens ion less  parametes 1s 
(bend specimen) 
(compact t e n s  ion  specimen) 
where M i s  t h e  nominal bendlng moment ( t a k e n  t o  b e  P(W -t a)  J 2  f a r  t h e  
compact specimen o r  P a / 2  f o r  t h e  p u r e  t e n s i o n  specimen) .  Th i s  parm- 
e t e r ,  computed from c o l l o c a t i o n  r e s u l t s  and from t h e  e x t r a p o l a t l o r !  -.qua- 
t i o n s ,  i s  p l o t t e d  i n  F i g .  2 where t h e  convergency can b e  r e a d r l y  sccn 
This  i n d i c a t e s  t h a t  t h e  s t r e s s  s t a t e  f o r  a  v e r y  deep c r a c k  approache i  one 
of pure  bending r e g a r d l e s s  of t h e  manner of l o a d i n g .  
THE SHAPE OF AN EDGE CRACK OPENED BY PURE TENSION OR BENDING 
T t  i s  g e n e r a l l y  w e l l  known t h a t  t h e  opening-mode e l a s t i c  s t r c s s  
f i e l d  n e a r  t h e  t i p  of a  c r a c k  may b e  complete ly  d e s c r i b e d  by the  - r r e s s  
i n t e n s i t y  f a c t o r  and  a p p r o p r i a t e  c o o r d i n a t e  f u n c t i o n s .  It i s  lcs ,ell 
known t h a t  t h e  s t r e s s  i n t e n s i t y  f a c t o r  a l s o  describes -the dn.-pi,i rT-.ntr, 
of t h e  c r a c k  s u r f a c e s  n e a r  t h e  c r a c k  t i p .  Tha t  i s ,  n e a r  t h e  r r a t r  c r p  
t h e  s t r e s s  z n t e n s i t y  f a c t o r  d e s c r i b e s  n o t  on ly  t h e  stresses I n  r ; i -  un- 
cracked r e g i o n  b u t  a l s o  t h e  deflections i n  t h e  cracked regLon, r i  o n l y  
t h e  f i r s t  term of t h e  displacemerlt  f u n c t i o n  641 i s  c o n s i d e r e d ,  ~3 f l a t  
c rack  opens under load i n t o  a  p a r a b o l a  whose t i p  r a d i u s  1s (eucl i i y l  
where E i s  Young's modulus and v i s  ~ o i s s o n ' s  r a t s o ,  S ~ n c e  i f  L i p  
r a d i u s  i s  a significant f e a t u r e  of t h e  e n t i r e  c rack  prof t i e ,  in::p.id- 
e r a t i o n  of c rack  shape  may prove u s e f u l .  
No a n a l y t i c a l  e x p r e s s i o n s  a r e  a v a i l a b l e  f o r  t h e  shape  &gk 
c r a c k  i n  a  f i n i t e - w i d t h  p l a t e  under  normal t e n s i o n  o r  b ~ n f j  il< !he shape 
of an  edge crack i n  a s e m i - s n f i n l t e  p l a t e  under remote n s r a t - . '  i *  r lclon is 
given by Wiggleswor~h  [ 5 ]  i n  t h e  form of an  i n f i n i t e  s i '  ~ r p g  
boundary c o P l o s a t i o n  , Gross [3!  computed opening d i sp3  a~ me-r.: 1,r:dc r 
normal t e n s i o n  o r  bending f o r  edge c r a c k s  30 ,  50, and 70 p e r c e n t  of the 
p l a t e  w i d t h .  The c r a c k  shapes  f o r  t h e s e  c a s e s  could  b e  d e s c r i b e d  by a 
polynomial of s u f f i c i e n t l y  h i g h  o r d e r  i n  two v a r i a b l e s  ( d i s t a n c e  from 
c r a c k  t i p  and r e l a t i v e  c r a c k  l e n g t h ) .  However, such a f u n c t i o n  would  
probably  b e  cumbersome t o  u s e ,  and t h e  i n f l u e n c e  of t h e  i n d i v i d u a l  coef- 
f i c i e n t s  on t h e  c r a c k  s h a p e  would b e  d i f f i c u l t  t o  v i s u a l i z e .  
Although t h e s e  c r a c k  shapes  may n o t  b e  e x a c t  c o n i c  s e c t i o n s ,  i t  i s  
e x p e d i e n t  t o  model them as such .  Then on ly  a s i n g l e  c o n i c  s e c t i o n  coef-  
f i c i e n t  needs t o  b e  c o r r e l a t e d  w i t h  t h e  r e l a t i v e  c r a c k  l e n g t h .  The gen- 
e r a l  e q u a t i o n  of a c o n i c  s e c t i o n  w i t h  t h e  o r i g i n  a t  t h e  v e r t e x  can b e  
w r i t t e n  a s  
where t h e  n o t a t i o n  i s  g i v e n  i n  F i g .  3  and m is  t h e  c o n i c  s e c t i o n  coef- 
f i c i e n t .  The p h y s i c a l  i n t e r p r e t a t i o n  of t h e  c o e f f i c i e n t  m i s  a s  
f o l l o w s  : 
m = -1 2 an e l l i p s e ,  (n/no) = 1 - ( 1  - y / a )  2  
- 1 x  m 0 a p o r t i o n  of an  e l l i p s e  (semimajor a x i s ,  -a/m) 
0  c m an  h y p e r b o l a  ( o r i g i n  a t  y  = -a/m) 
a = a a p a i r  of s t r a i g h t  l i n e s ,  (n/qo) = f y / a  
T h i s  can b e  s e e n  Tn ( F i g ,  3 ) ,  where 43) i s  p l o t t e d  f o r  t h r e e  values of m 
and t h e  l i m i t i n g  c a s e  m = 
A f t e r  s u i t a b l e  d i f f e r e n t i a t i o n  of (31 ,  t h e  e x p r e s s i o n  
i s  o b t a i n e d  f o r  t h e  c o n i c  s e c t i o n  model,  where r is  t h e  c r a c k  t i p  
r a d i u s  ( r a d i u s  of c u r v a t u r e  a t  y = 0).  This  e q u a t i o n  shows t h e  i n t e r -  
r e l a t i o n s h i p  between mouth d i s p l a c e m e n t ,  t i p  r a d i u s ,  and t h e  c o n i c  sec- 
t i o n  c o e f f i c i e n t .  Any two of t h e s e  t h r e e  terms a r e  s u f f i c i e n t  t o  deEine 
t h e  c o n i c  s e c t i o n .  Equa t ion  (4)  may b e  p u t  i n  a n o t h e r  u s e f u l  form using 
(2) and t h e  form e q u a t i o n  
where cr i s  P / B W  i n  t e n s i o n  o r  ~ M / B W ~  i n  bending,  M i s  a  bending 
moment opening t h e  c r a c k ,  and Y i s  t h e  d imens ion less  s t r e s s  i n t e n s i t y  
f a c t o r  ( c a l i b r a t i o n  f a c t o r ) .  Thus (4)  becomes 
where t h e  d imens ion less  mouth d i sp lacement  is  g iven  i n  terms of  the ca l i -  
b r a t i o n  f a c t o r  and t h e  c o n i c  s e c t i o n  c o e f f i c i e n t ,  
The c o n i c  s e c t i o n  model ( 3 )  was compared w i t h  t h e  a v a i l a b l e  crack 
p r o f i l e s  [ 3 , 5 ]  a s  f o l l o w s ,  Using c o l l o c a t i o n  v a l u e s  [ 6 ]  o r  t h e  e x a c t  
s o l u t i o n  151 f o r  t h e  d imens ion less  mou&h d i sp lacements  and t h e  c a l r b r a -  
t i o n  f a c t o r  e x p r e s s i o n s  of Brown and Srawley [ a ] ,  Wilson E l ] ,  and ( I ) ,  
c o n i c  s e c t i o n  c o e f f i c i e n t s  were  computed from ( 5 ) .  These a r e  p l o t t e d  
a g a i n s t  t h e  r e l a t i v e  c r a c k  l e n g t h  i n  F i g .  4  ( t h e  curves  shown w i l l  b e  
d i s c u s s e d  l a t e r ) ,  and may b e  s e e n  t o  i n c r e a s e  w i t h  i n c r e a s i n g  r e l a t i v e  
c r a c k  l e n g t h .  Using t h e s e  c a l c u l a t e d  c o e f f i c i e n t s  i n  ( 3 ) ,  r e l a e i v e  
c r a c k  d i sp lacements  (n/qo)  were  computed a t  y / a  = 0 . 1 ,  0 . 2 ,  0 , 3 ,  . . ., 
0.9 .  These were w i t h i n  3 , 2  p e r c e n t  of cor responding  d i sp lacements  d e t e r -  
mined from Wiggleworth 's  s o l u t i o n  6.53, G r o s s ' s  c o l l o c a t i o n  r e s u l t s  [ J I ,  
and unpubl i shed  c o l l o c a t i o n  r e s u l t s  (B. Gross ,  NASA Lewis Research 
Cente r ;  a/W = 0 . 2 ,  0 . 4 ,  and 0 . 6 ) .  Thus t h e  s h a p e  of edge c r a c k s  opened 
by bending o r  t e n s i o n  can b e  v e r y  c l o s e l y  approximated by t h e  c o n i c  sec -  
t i o n  model. 
The e f f e c t  of r e l a t i v e  c r a c k  l e n g t h  on c r a c k  shape  can b e  s e e n  i n  
F ig .  5 .  I n  a s e m i - i n f i n i t e  p l a t e  t h e  c r a c k  s h a p e  i s  (approx imate ly )  a  
p o r t i o n  of an e l l i p s e  whose semimajor a x i s  is  about  t h r e e  t imes  t h e  c r a c k  
l e n g t h .  The Wigglesworth s o l u t i o n  i s  a l s o  shown h e r e  f o r  comparison,  
As t h e  r e l a t i v e  c r a c k  l e n g t h  i n c r e a s e s ,  t h e  c r a c k  s h a p e  becomes p a r a b o l i c ,  
t h e n  h y p e r b o l i c ,  t h e  a c u i t y  of t h e  h y p e r b o l a  i n c r e a s i n g  w i t h  t h e  r e l a t i v e  
c r a c k  l e n g t h .  A s  would b e  expec ted ,  t h i s  t r e n d  i s  more pronounced f o r  
t h e  c a s e  of p u r e  bending t h a n  f o r  p u r e  t e n s i o n .  
SOME SIMPLE APPROXIMATE EXPRESSIONS 
As can b e  s e e n  i n  F i g .  4 ,  t h e  s i m p l e  e x p r e s s i o n  
(where n  is  2 - 3  f o r  bending o r  3 .3  f o r  t e n s i o n )  i s  a f a i r l y  good 
approximat ion o v e r  t h e  range  of t h e  a v a i l a b l e  s o l u f i o n s  (0  L a/W ( Q a 7 ) ,  
E x t r a p o l a t i o n  of (6) beyond a/W = 0 , 7  is  somewhat q u e s t i o n a b l e  a t  
p r e s e n t ,  s i n c e  c o l l o c a t i o n  r e s u l t s  a r e  n o t  a v a i l a b l e  f o r  comparison,  
Crack d i sp lacements  a r e  l i k e l y  t o  b e  v e r y  s e n s i t i v e  t o  geometry i n  this 
r a n g e ,  and (6)  may b e  o v e r l y  s imple .  It was shown e a r l i e r  t h a t  t h e  
s t r e s s  s t a t e  f o r  v e r y  deep c r a c k s  ( a  + W) approaches  one of p u r e  bending 
r e g a r d l e s s  of t h e  manner of l o a d i n g ,  I f  t h e  stress s t a t e  is  u n i q u e ,  t h e  
c r a c k  p r o f i l e  s h o u l d  a l s o  b e  un ique ,  Thus t h e  c o n i c  s e c t i o n  c o e f f i c i e n t s  
f o r  t e n s i o n  and bending shou ld  converge as a + W ,  En t h i s  r e s p e c t  t h e  
form of (6) i s  p r o p e r ,  
Equation (3) g ives  only t h e  r e l a t i v e  crack displacements ,  An ex- 
pres s ion  f o r  t h e  mouth displacement ,  no = f(a/W), i s  needed t o  determine 
abso lu t e  va lues .  This  could be  obtained by f i t t i n g  a polynomial i n  a/W 
bo a v a i l a b l e  c o l l o c a t i o n  s o l u t i o n s  [ 3 , 6 ] ,  However, a  simple approxima- 
t i o n  can b e  made by s u b s t i t u t i n g  (6) i n t o  (5) .  The dimensionless mouth 
displacement s o  ca l cu la t ed  i s  compared w i t h  c o l l o c a t i o n  va lues  161 i n  
Fig.  6 ,  and t h e  maximum e r r o r  is 3.7 percent  (bending) o r  2.9 percent 
( tens ion)  f o r  a/W 2 0.7. Ext rapola t ion  beyond a/W = 0.7 i s  not  ree- 
ommended . 
SUMMARY 
A simple s t r e s s  i n t e n s i t y  f a c t o r  express ion  is given f o r  an edge 
crack deeper  than  30 percent  of t h e  width of a p l a t e  i n  t ens ion ,  The 
shapes of edge cracks opened by t ens ion  o r  bending can be  c l o s e l y  approx- 
imated by conic  s e c t i o n s .  The conic s e c t i o n  c o e f f i c i e n t  can be  related 
t o  t h e  r e l a t i v e  crack length  by very  s imple empi r i ca l  equat ions ,  The 
magnitude of t h e  crack displacements  is a func t ion  of appl ied  lad, pla te  
geometry, and t h e  e l a s t i c  cons tan ts  of t h e  p l a t e  m a t e r i a l .  
The shape of a  loaded crack i n  a  s emi - in f in i t e  p l a t e  i s ,  appruxi-  
mately,  a  p o r t i o n  of a n  e l l i p s e  whose semimajor a x i s  is  about three times 
t h e  crack l eng th .  A s  t h e  crack l e n g t h  ( r e l a t i v e  t o  t h e  p l a t e  wid th )  in-  
c r eases ,  t h e  crack shape becomes pa rabo l i c ,  then  hyperbol ic ,  t h e  acui ty  
a£ t h e  hyperbola i nc reas ing  wi th  t h e  r e l a t i v e  c rack  length .  This  tread 
is  more pronounced f o r  t h e  c a s e  of pure bending than  f o r  pu re  tension, 
REFERENCES 
[l] W. K. Wilson, S t r e s s  I n t e n s i t y  Fac to r s  f o r  Deep Cracks i n  Bending 
and Compact Tension Specimens. Engng. F rac tu re  Mech. 2, 169-971 
(1970) . 
[ 2 ]  B.  Gross,  J. E. Srawley and W. F. Brown, Jr. ,  S t r e s s - I n t e n s i t y  Fac- 
t o r s  f o r  a Single-Edge-Notch Tension Specimen by Boundary ColLocation 
of a S t r e s s  Function. NASA Tech. Note No. 0-2395 (1964). 
[3] B ,  Gross,  Some Plane Problem E l a s t o s t a t i c  Solu t ions  f o r  P l a t e s  Having 
a V-Notch. Ph.D. Thes is ,  Case Western Reserve Univ. (19701, 
a41 G, R ,  I rwin ,  Analysis  of S t r e s s e s  and S t r a i n s  Near t h e  End sf a Crack 
Traversing a P l a t e .  -. 24, 361-364 (1957), (Disc.l;assion 
by M, L, Wil l i ams ,  -,Mech, 25, 300 (1958). 
653 L. A .  Wigglesworth, S t r e s s  D i s t r i b u t i o n  i n  a  Notched P l a t e ,  
&ithematika_ 6, 76-96 (1957) . 
[6] B.  Gross,  E. Roberts ,  Jr. and J, E. Srawley, E l a s t i c  Displaceaents  
f o r  Various Edge-Cracked P l a t e  Specimens. I n t .  J .  F rac tu re  Meeh, 4, 
267-276 (1968). 
[ 7 ]  W. F. Brown, Jr. and J. E. Srawley,  P l a i n  S t r a i n  Crack Toughness 
T e s t i n g  of High S t r e n g t h  M e t a l l i c  Materials. ASTM Spec. Tech,  
Pub l .  No. 410, pp.  12-14 (1966).  
-
.55 0 Collocation resu l t s  (ref. 2) 
Collocation resu l t s  (ref. 3) 
F igure  1. - Dimensionless stress in tens i t y  factor f r o m  
boundary collocation results. 
Collocation resu l t s  
0 Tension, M = Pal2, Gross (ref. 3) 
0 Compact tension, M = P(W + a)/?, 
m Wilson (ref. 1) 
i O Bending, Gross (ref. 3) 
3 y 5.5 
E 
.- 
a 3. I I I 
. 4  .6  . 8  1.0 
Relative crack length, alW 
Figure 2. - Dimensionless stress in tens i t y  factor. 
0 .2 .4 .6 . S  1.0 
Normalized distance i ror i i  crack tip, yia 
Figure 3. - Conic sections, eq. (3). 
0 Tension 
0 Bending)~ollocation (ref. 31 I3 
E Exact solution (ref. 5) / 
Shaded symbols: Unpublished 
-2 1 I I I I 
0 
I 
.1 . 2  . 3  .4 .5 .6  .8 
Relative crack length, alW 
Figure 4. -Effect of relative crack length on  the conic section coeffi- 
cient. 
Exact solution (ref. 5) 
Equations (3) and 15) 
Relative distance from crack tip, yla 
(a) Tension. (b) Bending. 
Figure 5. - Normalized crack shapes from equations (3) and (5) and a n  exact solution. 
, I --ua 
0 1 . 2  . 3  .4 . 5  . 6  . 7  
Relative crack length, aiW 
Figure 6. - Effect of relative crack length on the  plane-strain crack 
mouth displacernenl. 


Crack shapes and stress intensity factors for edge-cracked specimens

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server