AN ELASTIC-PLASTIC ANALYSIS OF FATIGUE CRACK CLOSURE
IN MODES I AND 11.
**
*
and S a t y a N . Acluri
School o f Engineering S c i e n c e and Mechanics
Georgia I n s t i t u t e o f Technology
A t l a n t a , Georgia 30332.
M. Nakagaki
Abstract
I n t h i s p a p e r , an e f f i c i e n t e l a s t i c - p l a s t i c
f i n i t e element procedure t o a n a l y z e c r a c k - c l o s u r e
and, i t s e f f e c t s on f a t i g u e
c r a c k growth under
g e n e r a l spectrum l o a d i n g , is p r e s e n t e d . A hybriddisplacement f i n i t e element procedure i s used t o
p r o p e r l y t r e a t t h e s t r e s s and s t r a i n s i n g u l a r i t i e s
n e a r t h e c r a c k - t i p ; and crack-growth under c y c l i c
l o a d i n g i s s i m u l a t e d by t h e t r a n s l a t i o n of c e r t a i n
"core" elements, n e a r t h e c r a c k - t i p , i n which prop e r s t r e s s and s t r a i n s i n g u l a r i t i e s were embedded.
Both pure Mode I and Mode I1 t y p e s of c y c l i c loadi n g a r e c o n s i d e r e d . I n t h e mode I c a s e , f o ~ r rtypes
o f c y c l i c l o a d i n g , v i z . , c o n s t a n t a m p l i t u d e block
l o a d i n g , high-to-low block l o a d i n g , low-to-high
b l o c k , and a s i n g l e o v e r l o a d i n an o t h e r w i s e cons t a n t amplitude b l o c k loading a r e c o n s i d e r e d ;
whereas i n Mode 11, o n l y a c o n s t a n t amplitude
b l o c k loading i s c o n s i d e r e d . D e t a i l e d r e s u l t s a r e
p r e s e n t e d f o r c r a c k - c l o s u r e and o p e n i n g - s t r e s s e s ,
c r a c k - s u r f a c e d e f o r m a t i o n p r o f i l e s , e t c . , i n each
c a s e . C e r t a i n o b s e r v a t i o n s , based on t h e p r e s e n t
numerical r e s u l t s , concerning v a r i o u s f a c t o r s t h a t
c a u s e krack growth a c c e l e r a t i o n o r r e t a r d a t i o n und e r g e n e r a l spectrum l o a d i n g , a r e p r e s e n t e d and
discussed.
Introduction:
A s d i s c u s s e d r e c e n t l y i n an e x p o s i t o r y a r t i c l e
by S c h i jve [ 11, t h e mechanism of c r a c k - c l o s u r e , as
f i r s t observed e x p e r i m e n t a l l y by E l b e r 11 23 , i s gene r a l l y c o n s i d e r e d t o be a predominant mechanism
t h a t c o n t r i b u t e s t o " i n t e r a c t i o n e f f e c t s " , which
c a u s e crack-growth r e t a r d a t i o n o r a c c o l e i a t i o n ,
under v a r i a b l e - a m p l i t u d e f a t i g u e l o a d i n g . I t i s
a l s o g e n e r a l l y understood [l] t h a t t h e c r a c k c l o s u r e phenomenon i s cuased by r e s i d u a l p l a s t i c
deformations remaining i n t h e wake o f t h e advanci n g c r a c k - t i p , a s i n i t i a l l y p o s t u l a t e d by E l b e r .
A n a l y t i c a l models t h a t lend t h e o r e t i c a l s u p p o r t t o
t h e e x i s t e n c e o f c r a c k - c l o s u r e phenomenon i n f a t i g u e crack-growth, and provide some r a t i o n a l i t y
f o r t h e adoption of an e f f e c t i v e s t r e s s - i n t e n s i t y
r a n g e , based on c l o s u r e e f f e c t s , f o r t h e c o r r e l a t i o n of f a t i g u e crack-growth r a t e s , have a l s o been
As for
proposed by Budiansky end Hutchinson [3]
a more g e n e r a l a n a l y s i s of e x t c n d i n g c r a c k s under
g e n e r a l block c y c l i c l o a d i n g , t o o b t a i n c r a c k c l o s u r e s t r e s s e s , c r a c k opening s t r e s s e s , d e t a i l s
o f c r a c k - s u r f a c e d e f o r m a t i o n s , and r e s i d u a l s t r e s s e s i n tkie c r a c k - t i p r e g i o n , e t c . , e l a s t i c - p l a s t i c
f i n i t e element a n a l y s i s were f i r s t performed by
Newman an1 h i s c o l l e a g u e s [4,5]. Apart from t h e s e
a n a l y s e s , t h e a u t h o r s a r e aware of s i m i l a r a t t e m p t s
The s t u d o n l y bv Oh51 and h i s co-workers [6,7].
L C ; ' ~ Ln L4- / J L U ~ L G L U ~ Z ~ U
r i ~ d eI case u r l ~ y . ALs o , s i n c e t h e crack-growth was s i m u l a t e d i n [4-71
K~esearch
Engineer,
P r o f e s s o r , Member, A I A A
.
C
-
-**
Cop~righl@AmericanInstilute of Aeronautic.) and
Astronnrlics. l n c .
W Q A l l riuhto *:*.;:-r.. ;?
by s h i f t i n g a f i n i t e element node ( t h e c u r r e n t
c r a c k t i p ) t o an immediately a d j a c e n t node, and
s i n c e c o n s t a n t s t r a i n - t r i a n g l e type f i n i t e e l e ments were used t o model t h e cracked s t r u c t u r e ,
a very f i n e f i n i t e element mesh (with t h e smalle s t element o f t e n b e i n g of t h e o r d e r of 10-3
t i m e s t h e crack l e 2 g t h ) i s necessary i n t h e mode l i n g procedures of 1 4 - 7 1 . Thus t h e f i n i t e e l e ment computations o f t h e type given i n [4-71 can
be very expensive, e s p e c i a l l y when c y c l i c loadi n g o f a r b i t r a r y spectrum a r e considered.
I n t h e p r e s e n t paper, an a l t e r n a t e c o s t e f f e c t i v e and a c c u r a t e e l ; s t i c - p l a s t i c f i n i t e
element procedure t o analyze f a t i g u e c r a c k c l o s u r e , and i t s e f f e c t s , under g e n e r a l spectrum
I n t h e p r e s e n t procedure,
l o a d i n g , is p r e s e n t e d .
t h e well-known Hutchinson-Rice-Rosengren [ 8 , 9 ]
t y p e s t r a i n and s t r e s s s i n g u l a r i t i e s , f o r s t r a i n hardening m a t e r i a l s , a r e embedded i n s p e c i a l l y
T h i s elimdeveloped elements n e a r t h e c r a c k - t i p .
i n a t e s t h e need f o r a very f i n e mesh n e a r t h e t i p .
For i n s t a n c e , t h e c r a c k - t i p elements i n t h e pres e n t procedure a r e o f t h e order of 10-I of t h e
c r a c k - l e n g t h , a s compared o c o n s t n t s t r a i n t r i a n g l e s of t h e o r d e r o f lo-' t o lo-' times t h e
c r a c k - l e n g t h g e n e r a l l y used i n t h e procedures of
A hybrid-displacement f i n i t e element me[4-71
thod [ 10,11] i s used i n developing t h e s e s p e c i a l
elements. Also i n t h e present procedure, crack( i ) the t r a n s l a t i o n of
growth i s simulated by:
a c o r e of t h e forementioned s p e c i a l elements by
an a r b i t r a r y amount i n t h e d e s i r e d d i r e c t i o n ,
( i i ) r e i n t e r p o l a t i o n of r e q u i s i t e d a t a i n t h e new
f i n i t e element mesh, and ( i i i ) p r o p o r t i o n a l r e l a x a t i o n of t r a c t i o n s i n o r d e r t o c r e a t e a new crack
s u r f ace. S i n c e t h e forementioned s p e c i a l elements
n e a r t h e c r a c k - t i p a r e of c i r c u l a r - s e c t o r shape,
c e n t e r e d a t t h e c r a c k - t i p , crack growth i n an
a r b i t r a r y d i r e c t i o n , from t h e i n i t i a l - c r a c k axis,
under g e n e r a l m i ~ e dmode c y c l i c l o a d i n g , can be
modeled. The p r e s e n t l y considered c a s e s , of
crack-geometries and f a r - f i e l d a p p l i e d loading,
r e s u l t i n "small-scale" y i e l d i n g c o n d i t i o n s near
In t h e s e c a s e s , a s t a t i c - c o n d e n the crack-tip.
s a t i o n procedure i s employed, wherein, t h e
p l a s t i c p o r t i o n of t h e s t r u c t u r e is i s o l a t e d from
t h e e l a s t i c ; t h e s t i f f n e s s of t h e former keeps
changing whereas t h a t of t h e l a t t e r remain f i x e d .
T h i s condensat i o n procedure r e s u l t s i n a conside r a b l e saving of computer time.
.
I n t h e p r e s e n t paper, both Modes I and I1
t y p e c y c l i c l o a d i n g s a r e considered.
I n the
mode I c a s e , f o u r d i f f e r e n t types of c y c l i c block
loading, v i z . , constant-amplitude, high-to-low,
low-to-hieh. and
s i n p l ~n . r o r l - = A i n
nth~vw l s e c o n s t a n t a n p l l t u d e block, a r e c o n s i d e r e d .
D e t a i l e d r e s u l t s a r e presented f o r crack-opening
I ~ I
9
~ A -dL L~V ~ ~c L b L L e b b t 4 b ,
c ~ t a a e s ,L
~ldcK-biiKLiiCe
deformation p r o f i l e s , and e f f e c t i v e s t r e s s - i n t e n s i t y f a c t o r ranges f o r f a t i g u e c r a c k growth, i n
each c a s e o f block loading. Also p r e s e n t e d h e r e i n a r e d e t a i l e d d i s c u s s i o n s , based on t h e o b t a i n e d
numerical r e s u l t s , concerning v a r i o u s f a c t o r s
t h a t c a u s e c r a c k growth a c c e l e r a t i o n o r r e t a r d a t i o n and d e l a y e f f e c t s under d i f f e r e n t t y p e s of
c y c l i c loading. F i n a l l y , t h e r e s u l t s o f an i n v e s t i g a t i o n of a c e n t e r - c r a c k e d panel under ext e r n a l pure s h e a r (Mode 11) c y c l i c loading, o f
conscz
~ ~ p l i t u d ea r, c p r e s e n t e d .
B r i e f Discussion o f Mathematical
D e t a i l s of Analysis Procedure:
Ln t h e p r e s e n t paper an incremental, updated
Lagrangean f i n i t e element formulation, f o r f i n i t e deformation e l a s t o - p l a s t i c i t y , i s used. The flow
t h e o r y o f p l a s t i c i t y t h a t i s used i s c h a r a c t e r i z e d
by t h e w e l l known ~ u b e r - ~ i s e s - H e n c k yi n i t i a l y i e l d
c r i t e r i o n , and a P r a g e r - Z i e g l e r type k i n e m a t i c
hardening r u l e .
I n t h e p r e s e n t work, a s ment i o n e d e a r l i e r , c i r c u l a r - s e c t o r shaped s i n g u l a r i t y
elements ( i n which, a displacement f i e l d t h a t c o r responds t o t h e forementioned Hutchinson-Rice
Rosengren s i n g u l a r f t i e s i n s t r a i n s and s t r e s s e s ,
for strain-hardening e l a s t o - p l a s t i c m a t e r i a l s ,
a r e embedded) a r e used n e a r t h e c r a c k - t i p , a s
shown i n Fig. 1. The above s i n g u l a r elements a r e
surrounded by "regular" eight-noded i s o p a r a m e t r i c
q u a d r i l a t e r a l elements, a s a l s o shown i n Fig. 1.
Compatibi l i t y of d i s p l a c e m e n t s , ane r e c i p r o c i t y
of t r a c t i o n s , between t h e s e " s i n g u l a r " and "regul a r " elements a r e enforced through a Lagrangean
M u l t i p l i e r technique ( t h u s l e a d i n g t o a ' h y b r i d '
element formulation), a s shown below.
be-the current (say, i n a s t a t e C )
Let y
i
C a r t e s i a n S p a t i a l c o o r d i n a t e s of a p a r t i c l e , l t o
be used a s a r e f e r e n c e system f o r t h e c u r r e n t i n crement, i e . fromC t o C
Let sYj be t h e t r u e
N
w1..
Let S i j
(Cauchy) s t r e s s i n C
pa'
(-
s~' ijW
T~
h o f f (P-K)
ij
)
be t h e r a t e of Second P i o l a - K i r c h -
s t r e s s r e f e r r e d t o CN.
that
sty1
Cwl
a s r e f e r r e d t o CN.
is the Second P-K
t N)
I t is noted
Stress i n State
-
In t h e above,
F is t h e f u n c t i o n a l f o r t h e group of s i n g u l a r
8
e h m e n t a , w h i l e F is t h a t f o r t h
oup of r e g u l a r elements. ~ h & , i n t h e a i n p f s r :few. t one
assumes: ( i ) an a r b i t r a r y displacement f i e l d ui
w i t h i n t h e element nm t o i n c l u d e t h e proper s t r a i n /
stress s f n g u l a r i t i e s as w e l l a s non-singular polynomial v a r i a t i o n s ; ( i i ) an independent displacement
f i e l d vi a t t h e s i n g u l a r - e l e m e n t boundary, anm,and
( i i i ) Lagrange M u l t i p l i e r TLi a t t h e boundary t o
e n f o r c e t h e c o n s t r a i n t ;=+
a t t h e boundaries of
s i n g u l a r elements. Whoriasf i n t h e r e g u l a r elements,
developed through t h e u s u a l c o m p a t i b l e - d i s p l a c e rnent method, one assumes a s i n g l e compatible d i s Thus, i t is seen t h a t i f vi
placement f i e l d Gi.
a t t h e boundary o f s i ~ g l u a relement i s s o chosen
t h a t i t matches w i t h vi o f r e g u l a r elements a t t h e
i n t e r f a c e s of s i n g u l a r and r e g u l a r elements, t o t a l
displacement c o m p a t i b i l i t y i s enforced throughout
the structure.
~ l s o ,i n E q . ( I ) , W i s tbe r a t e p o t e n t i a l f o r
t h e r a t e - o f second P-K s t r e s s Sij, such t h a t ,
aW/ai
where 6
i s t h e updated Lagrangean
i j7
i j.
s t r a i n r a t e , E i j = ( f ) (ui, + u
) . T h i s r a t e poj, i
t e n t i a l h a s been conabwantl:' d e r i v e d from a pos t u l a t e d r a t e p o t e n t i a l f o r t h e c o r o t a t i o n a l (Jaumanx$
r a t e o f Kirchhoff s t r e s s , u s i n g t h e well-known
c l a s s i c a l r a t e t h e o r i e s o f ( f i n i t e deformation),
r a t e - i n d e p e n d e n t e l a s t o - p l a s t i c i t y , as d i s c u s s e d
i n d e t a i l e l s e w h e r e [12, 131. These d e t a i l s a r e
omitted h e r e due t o space r e a s o n s .
The d e t a i l s o f assumed v a r i a b l e s hi,
, and
for t h e s i n g u l a r elements and a r e ornltted h e r e .
However, f o r t h e p r e s e n t p u r p o s e s , i t i s worth n o t i n g t h a t each sector-shaped s i n g u l a r element has
3 nodes a l o n g t h e c i r c u m f e r e n c e , and 4 nodes along
each of t h e r a d i a l boundary l i n e s . Along t h e c i r c u m f e r e n t i a l boundary, t h e f i e l d v . i s assumed i n
t h e form,
TLi
L e t Gi r e p r e s e n t t h e r a t e
o f deformation from CN, and l e t u
N
3bi/aYj.
i,j
Also, l e t elements rn = 1, 2 , . . p be t h e s e c t o r shaped s i n g u l a r i t y elements, and m = ptl
M be
the surrounding r e g u l a r e l e m e n t s . I t can be shown
t h a t t h e v a r i a t i o n a l p r i n c i p l e governing: ( i ) t h e
c o n d i t i o n s of e q u i l i b r i u m i n each element ( s i n g u l a r
a s well' a s r e g u l a r ) , ( i i ) t r a c t i o n boundarv condit i o n s f o r each element, where such e x i s t ,
and ( i i i ) c o n d i t i o n s of c o m p a t i b i l i t y o f d i s p l a c e ments and r e c i p r o c i t y of t r a c t i o n s between s i n g u l a r and r e g u l a r elements, can be s t a t e d a s t h e
s t a t i o n a r y condrtion of t h e f u n c t i c n a l :
..
where 9 i s t h e c i r c u m f e r e n t i a l a n g l e ; and %i..a3i
a r e e x p r e s s e d i n terms of t h e r e s p e c t i v e d i s p l a c e ments a t t h e 3 nodes a l o n g t h e circumference. I t
$an b e s e e n t h a t t h e above boundary displacement,
v
i s compatible with t h a t o f t h e surrounding 8
i'
noded i s o p a r a m e t r i c q u a d r i l a t e r 3 l s . Along-the r a d i a l boundaries of the s i n g u l a r elements, vi i s
assumed a s :
dl.-re r i s t h e r a d i a l d i s t a n c e from t h e c r a c k - t i p ;
n i s t h e exponent i n t h e m a t e r i a l hardening law
(ie , , from t h e u n i a x i a l s t r e s s - s t r a i n curve,
6
0 " ; BP t h e 2 i a s t i c - s t r a i n , and a t h e s t r e s s ) ;
aRd t h e c ~ cf fi c i e n t s bli
bli a r e expressed i n
terms o f tire r e s p e c t i v e nodal d ~ s p l a c e m e n t sa t t h e
forementioned 4 nodes.
-
...
n..
L:ISL
REPRODUCIBILITY OF THE
-
r
nc?
L
-
ORIGINAL PAGE IS POOR
.,*
.*I
r
".$k.@.-
- -1y i t
IS
- . . , -...
I!u,--
LL:SL
.
.
-L
.
.
- - - - .S L
p r i n c i p l e i n Eq. (1) assumes t h a t t h e c u r r e n t
s t a t e CN i s e x a c t l y e q u i l i b r a t e d . I n g e n e r a l ,
such i s not t h e c a s e ; and hence, e q u i l i b r i u m
, t i L L P : 3 t 10118 t l f ' I L ,.rewton~
k a p t ~ s o ~ tby p e
have t o be performed a t t h e end of e a c h i n c r e m e n t a l s o l u t i o n . The d e t a i l s of these i t e r a t i o n s have
a l r e a d y been p r e s e n t e d e l s e w h e r e [ 1 1 , 1 2 1 .
corr-ec.t i
F i n i t e Element M o d e l l i n g o f C r a c k Growth
The s t e p s i-I t h e f i n i t e e l e m e n t s i m u l a t i o n o f
c r a c k growth i n t h e p r e s e n t p r o c e d u r e may be de( i ) g e o m e t r i c a l change i n t h e crack
scribed as:
s u r f a c e boundary; ( i i ) t r a n s l a t i o n o f t h e c r a c k - t i p
s i n g u l a r i t i e s t o t h e advanced crack- t i p ; and ( i i i )
r e l e a s e o f s u r f a c e t r a c t i o n s on t h e n e w l y c r e a t e d
crack surface.
The change i n t h e c r a c k s u r f a c e boundary is
made by t r a n s l a t i h g t h e whole s e t o f c r a c k - t i p c o r e
e l e m e n t s , a s shown i n F i g . 1 , by a r b i t r a r y d i s l ~ n c e
ba i n t h e d i r e c t i o n o f i n t e n d e d c r a c k e x t e n s i o n ;
t h u s t h e new c r a c k - t i p node whlch i s d e s i g n a t e d by
t h e c e n t e r o f t h e s e c t o r - s h a p e d c o r e e l e m e n t s need
n o t b e c o i n c i d e n t w i t h any p r e v i o u s l y e x i s t i n g f i r
n i t e e l e m e n t node b e f o r e e x t e n s i o n . Thus e v e n
though t h e f i x e d b o u n d a r y i n t h e u n c r a c ked 1igament
o f t h e s t r u c t u r e is changed, t h e cons t r a i n i n g condit i o n o f t h e n o d e s n e e d n o t be a l t e r e d . Elements
i m m e d i a t e l y a d j a c e n t t o t h e c o r e nius t b e r e a d j u s t e d
t o f i t t o t h e t r a n s l a t e d c o r e . T h i s p r o c e s s of
t r a n s l a t i n g t h e c o r e mesh a l s o moves t h e embedded
s i n g u l a r i t y i n t h e e l e m e n t s t o t h e new c r a c k t i p
a r e a , l e a v i n g n o s i n g u l a r i t i e s b u t l a r g e deformat i o n s and s t r a i n s i n t h e wake o f advanced c r a c k - t i p .
A l l t h e 5x5 Gaussian d a t a p o i n t s i n e a c h o f the
t r a n s l a t e d c o r e e l e m e n t s (and a l s o t h e 5x5 p o i n t s f o r
t h e c o n v e n t i o n a 1 e l e m e n t s ) may g e n e r a l l y n o t c o i n c i d e w i t h t h o s e b e f o r e t r a n s l a t i o n , a t which p l a s t i c
h i s t o r y data such a s current s t r e s s e s , p l a s t i c
s t r a i n s , p l a s t i c a l l y d i s s i p a t e d work, y i e l d s u r f a c e
t r a n s l a t i o n , e t c . , a r e a v a i l a b l e . Therefore the
d a t a a t p o i n t s i n t h e new mesh a r e e s t i m a t e d by
l i n e a r l y i n t e r p o l a t i n g d a t a on f o u r Gaussi-an p o i n t s
i n t h e o l d mesh t h a t a r e n e a r e s t t o t h e p o i n t u n d e r
q u e s t i o n i n t h e new mesh. The s i m p l e b u t cumbersome
m a t h e m a t i c a l d e t a i l s of t h i s i n t e r p o l a t i o n and
s m o o t h i n g p r o c e s s a r e o m i t t e d he;e f o r t h e s a k e of
b r e v i t y . W i t h t h e f i t t e d p l a s t i c d a t a and t h e new
element geometry, element s t i f f n e s s m a t r i c e s a r e
r e c a l c u l a t e d f o r the c o r e elements a s w e l l a s f o r
t h e surrounding r e a r r a n g e d elements and t h e globa 1
s t i f f n e s s i s a p p r o p r i a t e l y modified. Subsequent
e q u i l i b r i u m c h e c k i n e r a t i o n s u s i n g t h e new s t i f f n e s s
o f t h e s t r u c t u r e c o r r e c t f i t t i n g e r r o r s , i f ar , i n
t h e p l a s t i c i t y d a t a i n t h e new mesh. A t t h e sdme
t i m e , t h e t r a c t i o n s o v e r t h e d i s t a n c e AB (ha s s
shown i n F i g . 1) a r e i n c r e m e n t a l l y removed, w i t h
e q u i l i b r i u m check i t e r a t i o n s being used a t each s t e p ,
t o c r e a t e a new t r a c t i o n - f r e e c r a c k s u r f a c e o f
l e n g t h Aa. Tht f i n i t e e l e m e n t s i m u l a t i o n o f c r a c k
e x t e n s i o n by t h e d e s i r e d amount, L a , i s now completed.
A n a l y s i s o f F ? t i ~ u eCrsck Growth Under
Mode I C y c l i c L o a d i n 2 D e s c r i p t i o n o f t h e Problem
Throughout
t h e s e r i e s of t h e p r e s e n t e l a s t i c p l a s t i c f i n i t e element analyses of f , t i g u e crack
g r o w t h under Mode I t y p e c y c l i c l o a d i n g , a t h i n
r e c t a n g u l a r p l a t e w i t h a c e n t r a l c r a c k and under
u n i f o r m t e n s i l e s t r e s s e s , i n a d i r e c t i o n normal t o
t h e c r a c k - a x i s . a t t h e e d p p ? of t h e p l a t e , i s cons i a ~ r e a (See k ~ g .2 ) .
lnca u l m e n s l o n s of the p l a t e
a r e : h a l f w i d t h w = 230mm; and ' l a l f - c r a c k l e n g t h
a = 27.3mm, r e s p e c t i v e l y . The m a t e r i a l is cons y d e r e d t o be a 2024-T3 Aluminum a l l o y , whose
~ ~ r r ~ ~ ~p r~o p~e ~r t l el ac qar el c l r a r a c t e r l , r a o i l : y l e l a
stress,^
= 350MNImL; and young's modulus, E =
70,000 MN%'.
The m a t e r i a l i s assumed t o be e l a s t ic-perfectly-plast ic.
I t is noted t h a t t h e above
problem d e f i n i t i o n i s i d e n t i c a l t o t h a t used b y
Newman [ 5 ] . The p l a t e i s assumed t o be i n a s t a t e
of p l a n e s t r e s s .
Because of t h e s y m e r r i e s of g e o m e t r y , a p r l i e d
l o a d i n g , and m a t e r i a l homogeneity, o n l y a q u a r t e r
of t h e c r a c k e d p l a t e i s a n a l y s e d . F i g . 2 shows t h e
f i n i t e elemen& breakdowr t h a t is used. A t o t a l of
6 sector-shaped s i n g u l a r i t y elements near t h e c r a c k t i p , and 43 c o n v e n t i o n a l quad^ tt it isoparame t r i c
e l e m e n t s a r e employed. Some o f t h e s e isoparame t r i c
e l e m e n t s a r e 6 noded t r i a n g l e s , w h i l e t h e m a j o r i t y
a r e 8 noded q u a d r i l a t e r a l s (See F i g . 2 ) .
It is
s e e n t h a t t h e t o t a l number of nodes i n t h e f i n i t e
e l e m e n t mesh f o r t h e q u a r t e r - p l a t e is 171, w i t h a
t o t a l number o f d e g r e e s o f f n i e d o m of 311.
The r a d i u s of t h e s e c t o r - s h a p e d s i n g u l a r i t y
e l e m e n t s is c h o s e n a s D = 2.;tmn;
i e . , p / a = -103.
While t h e c r a c k - e x t e n s i o n per c y c l e 9f l o s d i n g , Aa,
can be a r b i t r a r y ( i e . , not r e l a t e d t o the f i n i t e
e l e m e n t mesh s i z e ) i n t h e p r e s e n t a n a l y s i s procedu r e , i t i s chosen t o be h a = 0 . 1 4 m i n t h e p r e s e n t
s e r i e s of c o m p u t a t i o n s .
T e c h n i q u e s t o Minimize
Computational( CPU) Time
F i r s t l y , we n o t e t h a t t h e n e a r - t i p e l e m e n t s
used p r e s e n t l y a r e of t h e o r d e r 1 0 - I t i m e s t h e sernic r a c k l e n g t h ; and t h e t o t a l number r f a l g e b r a i c equat i o n s f o r t h e above problem a r e only 311.
I n t h e p r e s e n t p r o c e d u r e , a t a n g e n t modulus
( s t i f f n e s s ) approach is used i n e a c h increment and
i n each i t e r a t i o n i n the respective increment.
Thus a f a s t e r c o n v e r g e n c e is o b t a i n e d i n t h e p r o c e s s of i t e r a t i o n s of e q u i l i b r i u m c o r r e c t i o n , e t c .
However, i t is n o t e d t h a t c n l y t h e s t i f f n e s s m a t r i c e s of t h e p l a s t i c p o r t i o n of t h e s t r u c t u r e need
t o b e changed i n t h e p r e s e n t t a n g e n t modulus a p p r o a c h , w h e r e a s , t h o s e o f t h e e l a s t i c p o r t ion r e main f i x e d
.
For t h e p r e s e n t l y c o n s i d e r e d l e v e l s of a p p l i e d
f a r - f i e l d t e n s i o n on t h e specimen, o n l y " s m a l l s c a l e " yielding conditions prevail near the crackt i p . A typical plastic-zone s i z e near t h e crackt i p f o r t h e p r e s e n t l y c o n s i d e r e d c l a s s o f Mode I
probiems i s shown i n F i g . 3 , b e i n g superimposed on
t h e f i n i t e e l e m e n t mesh.
Tt i s s e e n t h a t t h e p l a s t i c p o r t i o n of t h e s t r u c t u r e ( d e s i g n a t e d "Region P")
is c o n s i d e r a b l y s m a l l e r t h a n t h e e l a s t i c p o r t i o n
Thus o n l y t h e s t i f f n e s s
( d e s i g n a t e d "Region F").
m a t r i x of r e g i o n P need t o be changed i n e a c h loads t e p and each i t e r a t i o n i n t h e p r e s e n t p r o c e d u r e . '
However, t h e t o t a l number of t i m e s , s a y N , t h a t
t h e combined s t i f f n e s s m a t r i x ( f o r Regions P + E )
must be i n v e r t e d i n t h e c o u r s e of a n a l y z i n g a t y p i c a l problem, s a y t h e c a s e a growing c r . ? c k u n d e r
c o g s t a n t s m p l i t u d e c y c l i c l o a d i n g , is h = [ ( n o . of
*Even though t h e g e n e r a l f o r m u l a t i o n p r e s e n t e d i n
(1) i s f o r f i n i t e d e f o r m a t i o n s , t h e f i n i t e def o r m a t i o n e f f e c t s i n t h e p r e s e n t c l a s s o f problems
- -- .,.I "-,.. c . L - .."-A?....;".,"t
,-,!.I..
4"
the . . i r : - . : c . -
Eq.
..--
of t h e c r a c k - t i p .
Thus, changes t o t h e s t i f f n e s s
o f r e g i o n E due t o f i n i t e d e f o r m a t i o n e f f e c t s , if
any, a r e ignored h e n c e f o r t h .
i c e r a t i o n s / e v c l e ) X (number o f l o a d i n c r e m e n t s p e r
For a t y p i c a l p r o , y c i e ) XCno. o r l o a d cycles)].
b l e m , s a y F! c y c l e s o f c o n s t a n t a m p l i t u d e l o a d i n g ,
a t y p i c a l v a l u e f o r N c a n be N = 4 x 28 x 8 = 8 9 6 ,
( i t . , 4 i t e r s t i o n s p e r e ; c l e , w i t h 28 l o a d i n c r e m e n t s / c y c l e , e t c . ) . Th - is - a r a t h e r enormous a mount of c o m p u t i n g ; and h e n c e a more e c o n o m i c a l way
of s o l v i n g the s t i f f n e s s e q u a t i o n s i s mandatory.
S i n c e t h e p l a s t i c - z o n e i s a s m a l l - s i z e , by a n
a p p r o p r i a t e node n u m b e r i n g s c h e m e , t h e s t i f f n e s s
m a t r i x o f t h e p l a n t i c z o n e c a n be a r r a n g e d , a s i n
F i g . 3 . b , s o t h a t i t is a small s u b - s e t of t h e g l o b a l s t i f f n e s s m a t r i x of t h e s t r u c t u r e (even though
t h e p l a s t LC zone s i z e keeps c h a n g i n g w i t h l o a d , a n
a p p r o x i m a t e p r e l i m i n a r y a n a l y s i s can be made t o
d e t e r m i n e i t s s i z e a t maximum l o a d ) . Then we use a
sta t i c condensat ion procedure t o f i r s t e l i m i n a t e
t h e e q u a t i ~ - n sc o r r e s p o n d i n g t o t h e n o d e s i n t h e e l a s t i c portion, in the very f i r s t load-increment.
T h e r e a f t e r o n l y t h e e q u a t i o n s f o r t h e n o d e s i n Reg i o n P need t o b e o p e r a t e d u p o n , i n a l l s u b s e q u e n t
l o a d s r e p s and i n t e r a t i o n s . T h u s , i n t h e e x a m p l e
c i t e d a b o v e , (N = 8 9 6 ) , i n a l l b u t one o f t h e 8 9 6
s o l u t i o n s , t h e number o f e q u a t i o n s b e i n g s o l v e d is
r a t h e r v e r y s m a l l , and c o r r e s p o n d t o t h e t o t a l
number o f u n c o n s t r a i n e d n o d a l d i s p l a c e m e n t s i n Region p. T h i d enables t h e present computations t o
be e c o n o m i c a l l y f e a s i b l e .
A l s o , t h e computer p r o g r a m i s s o a r r a n g e d
t h a t t h e d a t a obtained from computation u p t o t h e
end of a g i v e n s p e c t r u m o f l o a d i n g c a n b e u s e d as
i n - p u t d a t a a t the beginning of a d i f f e r e n t s p e c trum o f l o a d i n g . For i n s t a n c e , a t t h e e n d o f a
c o n s t a n t Hi -amplitude c y c l i c loading, t h e d a t a is
s t o r e d on a d i r e c t a c c e s s permanent d i s c f i l e a n d
used a s i n i t i a l c o n d i t i o n s f o r a low a m p l i t u d e c y c l i c load spectrum; i n t h i s process not o n l y t h e
c a s e of Hi-amplitude l o a d i n g but a l s o t h e c a s e of
H i t o - L o b l o c k l ~ a d i n gi s s o l v e d .
M o n i t o r i n n of C r a c k - C l o s u r e and O p e n i n g
i n t h e F i n i t e E l e m e n t Mqdel
-
Let u s assume, t h a t a t a g i v e n i n s t a n t o f t i m e
(at a given point i n t h e loading h i s t o r y ) , t h e loc a t i o n of t h e c r a c k - t i p , t h e l o c a t i o n o f t h e 4
n o d e s on t h e r a d i a l l i n e (which c o i n c i d e s w i t h t h e
c r a c k s u r f a c e ) of the s i n g u l a r s e c t o r e l e m e n t , t h e
l o c a t i o n s o f a l l o t h e r n o d e s on t + e c r a c k a x i s ,
as w e l l t h e c u r r e n t ( d e f o r m e d ) p r o f i l e o f t h e c r a c k
e d e n o t e t h e c u r r e n t no&
s u r f a c e , a r e a l l known. W
on t h e c r a c k s u r f a c e as " u p d a t e d L a g r a n g e a n Nodes".
L e t u s now assume t h a t t h e c r a c k t i p i s now f u r t h e r
e x t e n d e d b y ,an a r b i t r a r y a m m n t (Aa) and t h e s t r u c t u r e is t h e n s u b j e c t e d t o f u r t h e r l o a d i n g . W
e
f i r s t n o t e t h a t , i n t h e p r e s e n t d e v e l o p m e n t , the
b o u n d a r ) ~d i s p l a c e m e n t ( i n t h e d i r e c t i o n o f t h e app l i e d normal s t r e s s ) a l o n g a r a d i a l l i n e o f a
" s i f i g u l a r " s e c t o r e l e m e n t f s of the form g i v e n i n E q .
(3).
Using t h e above e q u a t i o n , and k n o w i n g , a ~ r i o r i . the r a d i a l c o o r d i n a ' e s , ( i e . , r , a s measi : * i
from the c u r r e n t c r a c k - t i p ) , o f t h e r e s p e c t i v e n o d e s
on t h e r a d i a l l i n e of the s e c t o r e l e m e n t i n i t s
i m m e d i a t e l y p r e v i o u s l o c a t i o n , one c a n compute t h e
v a l u e s o f Av a t t h e a b o v e m e n t i o n e d "Updated Lag r a n g e a n ~ o d e g . " By a d d i n g ( o r s u b s t r a c t i n g , a s t h e
c a s e mav b e , ) t h e s e i n c r e m e n t a l d i s p l a c e m e n t s t o
t h e rev i n l ~ lsv known
l x r c ZT. - - ., .,,
. -.
D L :he c u r r e n t c r a c k s u r f a c e d e f o r m a t i o n p r o f i l e
i s made.
-7-
.
.,
. .
1
-
p r e s e n t c y c l i c l o a d i n g c a s e , a t t t , e i n e t a n ~t h e
d is p l a c e m e n t ( i n t h e d Erect i o n of a p p l i e d t e n s i o n )
a t one o r more n o d e s o n t h e c r a c k s u r f a c e becomes
n e g a t i v e , f u r t h e r u n l o a d i n g bebopped. The c o m p u t a t i o n a l p r o c e d u r e is t h e n s w i t c h e d t o a d i s p l a c e ment c o n t r o l t y p e , and t h e a b o v e n e g a t i v e d i s p l a c e ments a r e p r e c i s e l y e n f o r c e d t o be z e r o ; t h u s f i n d Eng t h e p r e c i s e s t r e s s L e v e l a t which t h e c l o s u r e
c o n s t r a i n t on L h e r e e p e c t i v e node must be c n f o r e a d .
A f t e r t h e c r a c k - c l o s u r e i s d e t e c t e d , t h e respective node(s1 a r e c o n s t r a i n e d t h e r e a f t e r , u n t i l
t h e r e s t r a i n i n g f o r c e ( e ) a t t h e node(s) just becomes zero and begins t o b e t e n s i l e i n n a t u r e .
The c o r r e s p o n d i n g a p p l i e d s tress leve 1 d e f inee t h e
c r a c k - o p e n i n g streas. The d e t a i l s o f t h e a b o v e
p r o c e s s a r e d i s c u s s e d and g r a p h i c a l l y i l l u s t r a t e d e l s e w h e r e [ 12,143.
F i n a l l y , some comments o n t h e p r e s e n t l y observed p a t t e r n s c f c r a c k - c l o s u r e a r e g i v e n , bef o r e p r o c e e d i n g t o a d i s c u s s i o n of s p e c i f i c c a s e s .
In g e n e r a l , c l o s u r e was n o t i c e d t o o c c u r a t t h e
node c l o s e s t t o t h e c u r r e n t c r a c k - t i p , as i n d i c a ted i n tt,e sequence of unloading s t e p s i n F i g s .
4 a - c . However, if t h e c u r r e n t c r a c k - s u r f a c e p r o f i l e i s i r r e g u l a r , a s ~n t h e c a s e o f Hi-to-Low
block l o a d i n g t o be d i s c u s s e d l a t e r , c r a c k - c l o s u r e
rnqy f i r s t o c c u r a t t h e node c l o s e s t t o t h e c r a c k t i p ; however, i n t h e s u b s e q u e n t unloading s t e p ,
c l o s u r e may o c c u r a t a n o d e f a r - r e m o v e d from t h e
c r a c k - t i p , a s i n d i c a t e d i n F i g . 4 d . From t h e r e s u l t s t o be d i s c u s s e d l a t e r , t h i s p a t t e r n o f
crack-closure appears t o c o n t r i b u t e s i g n i f i c a n t l y
t o g r o w t h r e t a r d a t i o n and d e l a y e f f e c t s .
C r i t e r i o n For Crack-Extension
S t r e s s Level
I n t h e p r e s e n t w o r k , a s t u d y is made t o a r r i v e
a t a c r i t e r i o n f o r t h e s t r e s s l e v e l , g , a t which
f a t i g u e c r a c k growth o c c u r s .
In p r i o r e f i t e r a t u r e ,
t h i s c r a c k - e x t e n s i o n s t r e s s l e v e l was c h o s e n a r b i t r a r i l y . For i n s t a n c e , i n [ 5 j the crack is ext e n d e d a t t h e maximum a p p l i e d s t r e s s i n e a c h c v c l e
even in a g e n e r a l spectrum ( f o r i n s t a n c e , h i g h t o - l o w , l o w - t o - h i g h , e t c . , ) l o a d i n g , w h e r e as i n
[ 7 ] t h e c r a c k was e x t e n d e d a t t h e a p p l i e d s t r e s s
l e v e l a t which the r e s t r a i n i n g nodal f o r c e at t h e
new c r a c k - t i p becomes z e r o .
In t h e p r e s e n t s t u d y ,
for i n s t a n c e i n a c o n s t a n t - a m p l i t u d e ( z e r o - t o - t e n s i o n ) c y c l i c l o a d i n g , i t was found t h a t t h e c r a c k
respecto p e n i n g and c l o s u r e s t r e s s e s , oo and o
i v e l y , w e r e s e n s i t i v e t o t h e choBen o F' I n t h e
p r e s e n t w o r k , a c r i t e r i o n , oex = 0
qX p ( o max-"od
OP
w h e r e p i s a c o n s t a n t o f p r o p o r t i o n a l i t y , is p o s t u l a t e d ; and p is o b t a i n e d b y c a l i b r a t i o n s u c h t h a t
c o r r e l a t e d with that abserved
the calculated a
i n e x p e r i m e n t a l ''studies
s u c h a s i n [ 2 2 . However,
i t i s noted t h a t this c o n s t a n t of p r o p o r t i o n a l i t y
p m a y , t o some e x t e n t , be d e p e n d e n t on t h e n u m e r i c a l m e t h o d o l o g y employed i n f a t i g u e c r a c k m o d e l i n g
i t s e l f . T h u s t h e above d e s c r i b e d c a l i b r a t i o n may
be c o n s i d e r e d a s v a l i d o l l l y i n t h e c o n t e x t o f t h e
p a r t i c u l a r m e t h o d o l o g y employed i n t h e p r e s e n t
work. Three d i f f e r e n t t e s t c a s e s , each w i t h a
d i f f e r e n t magnitude of c o n s t a n t amplitude ( z e r o
t o t e n s i o n ) c y c l i c l o a d i n g , were s t u d i e d w i t h
. . ., , . .
.
c.
-,
C d l - 1 6 ( kIYe LOX
t h c above mentioned c o n s t a n t of propor t i o n a l i t y ,
p . The i d e a was t o s e l e c t a I p ' t h a t y i e l d s r e a u l t s , i n e a c h c a s e , f o r (oo /arna ) t h a t a r e i n
b e s t a g r e e m e n t v i t h t h e expe!imenfal
r e s u l t s [21
. --
9 . .
\
h i
.
L
L
V
~
~
L1
I
for 2024-T3 Aluminum a l l o y , which i s t h e m a t e r i a l
s i m u l a t e d i n a n a l y s i s . The r e s u l t s , f o r i n s . a n c e .
f o r t h e c a s e (omax/u
.40) and (I? = umin/umaxrOj
YS
a r e summarized ae f o 1 lows :
p
Levelled - o f f
1.0
.85
,62
.4q
[o
OP
/omax] a t s t e a d y s t a t e
115
MP
.82
94
II
.67
79
II
.56
58
11
( i ) The v a r i a t i o n of c r a c k - o p e n i n g s t r e s s ,
.4 1
,
+
C o n s t a n t Amplitude Z e r o - t o - T e n s i o n
C y c l i c Loading
f o r t h e c a s e of
( i ) The r e s u l t s f o r a
) = 0 , a r e shown
(omax/BS) = 0 . 4 and R =
m
(' i n lamax
i n Fig. 5 , f o r 8 c y c l e s of loading.
I t i s observed
r e a c h e s a " s t e a d y s t a t e " v a l u e of 0.56~
that o
a f t e r OP the 4 t h o r 5 t h c y c l e .
I t is a l s o notedmaX
t h a t t h i s v a l u e f o r o /amax(=O. 56) i s i n r e a s o n a b l e
r e s u l t s [2] f o r t h e
accord w i t h experimenP%l
same m a t e r i a l , a 2024 T3 Aluminum a l l o y . Knowing
0
i n e a c h c y c l e , we d e f i n e t h e e f f e c t i v e s t r e s s i8Pens i t f a c t o r a s :
+
NAa)
(amax-
uop) where C 1
is the f i n i t e s i z e c o r r e c t i o n f a c t o r f o r the pres e n t c r a c k geometry (which was found t o b e C - =
1
1 . 0 1 7 from a f i n i t e e l e m e n t l i n e a r a n a l y s i s of t h e
c r a c k w i t h a = a ; and t h e r e a f t e r assumed t o he
c o n s t a n t ) ; N is e h e number of c y c l e s and Aa is t h e
I t is t h u s s e e n t h a t bKeff
c r a c k growth p e r c y c l e .
l e v e l s o d i a f t e r a few c y c l e s t o a s t e a d y s t a t e
value.
.
A two l e v e l b l o c k l o a d i n g , from low t o h i g h ,
w i t h omex i n t h e h i g h e r l e v e l b e i n g 1.273 t i m e s t h e
i n t h e lower l e v e l , is c o n s i d e r e d . The
z%fmum s t r e s s i n t h e lower l e v e l is t a k e n such t h a
(O
( u s ) = -314. A s m e n t i o n e d e a r l i e r , tte
d a ~ P a p Y h ee n 8 o f 4 c y c l e s o f low l e v e l b l o c k loadi n g (See F i g . 7) i s r e c o v e r e d from a c o n s t a n t - a m p l i t u d e t e s t c a s e , w i t h t h e c o r r e s p o n d i n g stress l e v e l .
The f o l l o w i n g r e s u l t s w e r e o b t a i n e d :
a , a s t h e c y c l i c l o a d i n g p r o g r e s s e s , is shown i n
F@.
(7).
I t can be s e e n t h a t immediately a f t e r
t h e s t e p u p i n t h e l e v e l o f a p p l i e d stress, a
Similar results
/a ) = .229 and
w e r e o b t a i n e d W f o r t h e c a s e s , (a
/o
= .314.
From t h e s e ? % $ e e Y ~ e t s of res(urnax
it
wag o b s e r v e d t h a t ~ 4 . 6 2v i e l d a r e s u l t s f o r
(3 /a 1 t h a t a r e i n b e e t agreement w i t h e x p r i m e R t a T a & s e r v a t i o n s , which i n d i c a t e t h a t [o /bma
a t s t e a d y s t a t e i s a b o u t 0.56.
I t is hypotf$size3
t h a t t h e above c o n s t a n t ~ 1 0 . 6 2may b e used t h r o u g h o u t t h e r e s t of t h e a n a l y s i s , i e . , f o r c a s e s of
g e n e r a l s p e c t r u m l o a d i n g . We a l s o n o t e t h a t when
t h e load l e v e l o
d u r i n g any c y c l e i s f i r s t d e t e r m i n e d , t h e nurnbegQ(and s i z e ) o f l o a d s t e p s between
ando
, i n t h e r e e p e c t i v e c y c l e , is s o
thisa
a d j u s t g f l t h a t TI% p r e - c h o s e n l e v e l of o [ = a
) ] c o i n c i d e s w i t h one o f t h g x l o a d - o P
!ik%?nt:o~n the cycle. W
e now d i s c u s s t h e r e s u l t s of a n a l y s i s o f e a c h o f t h e l o a d i n g c a s e s .
bKeff = C1 & ( a o
Low-'To-ti i n h Block Load i n q
( i i ) F i g . 6 shows t h e c r a c k s u r f a c e d e f o r m a t i o n p r o f i l e s , f o r i n s t a n c e , a t v a r i o u s s t a g e s of
u n l o a d i n g d u r i n g t h e 8 t h c y c l e of t ! ~ e p r e s e n t cons t a n t - a m p l i t u d e ( R = o ) c y c l i c l o a d i n g . The l a r g e
b l u n t i n g a t the i n i t i a l c r a c k - t i p l o c a t i o n ( a = ao)
i s o b s e r v e d t o r e m a i n p e r m a n e n t l y . The s u r f a c e
o f t h e e x t e n d e d c r a c k is o b s e r v e d t o b e f a i r l y
smooth. During t h e 8 t h c y c l e , i t is o b s e r v e d t h a t
p r e c i s e crack-closure occurs only over the area h a
( i e . , o n l y a t t h e previnlts I n r a f i n n - F +-hn
L A P n e a e ) e v e n upon f u i i u n i o a d ~ n g ; however, 1 t . i s
a l s o n o t e d t h a t a t t h i s s t a g e , t h e o p e n i n ) of t h e
c r a c k f a c e s between t h e p o i n t s a and t h . p r e c i s e l y
c l o s e d node is v e r y s m a l l (See F P 6~ ) .
--cl--
4
d e c r e a s e s by a b o u t 33% o f i t s s t e a d y s t a t e vaP8e
c o r r e s p o n d i n g t o t h e lower Level of b l o c k l o a d i n g .
i n c r e a s e s mono t o n i c a ? l y t o
S u b s e q u e n t t o t h i s , 0,
a s t e a d y s t a t e value cgrresponding t o t h e higher
l e v e l o f b l o c k l o a d i n g , w i t h :n about 5 c y c l e s .
P r i o r t o t h i s s t a b i l i z a t i o n , ~ K ~ f d d e f i n eads bef o r e ) i n t h e h i g h e r l e v e l of block l o a d i n g remains
considerably higher than the steady s t a t e value
corresponding to t h i s s t r e s s level; thus indicating
g r o w t h a c c e l e r a t i o n f o l l o w i n g the l o a d s t e p - u p .
( i i ) The r e p r e s e n ' a t i v e c r a c k s u r f a c e p r ~ f i l e s
f o r i n s t a n c e , a t v a r i m s s t a g e s of u n l o a d i n g a t
t h e end of t h e 8 t h c y c l e ( h i g h s t r e s s ) o f t h e c u r r e n t low- t o - h i g h l e v e l b l o c k l o a d i n g , a r e shown i n
F i g . ( 8 ) . From t h i s F i g u r e , i t c a n be s e e n t h a t
t h e s t e p - u p i n t h e l e v e l o f l o a d i n g c a u s e 8 a. b l u n t i n g o f t h e c r a c k - t i p (ie., a t t h e l o c a t i o n x / a =
1 . 0 2 i n Fig. 8 , when t h e s t e p - u p i n l o a d i n g o c c u r s
i n t h e p r e s e n t f i n i t e e l e m e n t s i m u l a t i o n ) . Even
d u r i n g t h e u n l o a d i n g a t t h e end o f t h e p r e s e n t two
l e v e l b l o c k l o a d i n g , a s s e e n from F i g . (8), t h e
c r a c k - c l o s u r e o c c u r s only over t h e a r e a La ( i e . ,
o n l y a t t h e p r e v i o u s l o c a t i o n of t h e c r a c k - t i p
node).
Hi~h-To-Low Block Loading
A f t e r 8 c o n s e c u t i v e c y c l e s of a h i g h l e v e l
b l o c k l o a d i n g ( t h e d a t a a t which p o i n t is r e c o v e r e d from t h e c v r r e s p o n d i n g c o n s t a n t a m p l i t u d e t e s t
is r e d u c e d by 21.4% and 8 more cyc a s e ) , the a
c l e s o f thismFgduced l e v e 1 b l o c k l o a d i n g were cons i d e r e d . The magnitude o f t h e a p p l i e d s t r e s s i n
t h e h i g h - l e v e l b l o c k was such t h a t (urnax)
' b o
= 0 . 4 0 . The f o l l o w i n g r e s u l t s w e r e o b k ~ l n e d .
ys
-
( i ) The v a r i a t i o n o f t h e c r a c k - o p e n i n g s t r e s s
a s t h e l o a d i n g p r o g r e s s e s , is shown i n F i g . ( 9 )
gpIt is s e e n t h a t i n m e d i a t e l y a f t e r t h e step-down
a s was
i n t h e load l e v e l , n o a b r u p t d e c r e a s e iw
t h e c a s e i n Low- to-High l o a d i n g , o c c u r s igp;he
p r e s e n t High- to-Low b l o c k l o a d i n g c a s e . A f t e r t h e
load-levelstepdown,
tayedatabout0.70(omax)br
witF.in
t h e number o cyclar of low- l e v e l
Load c o n s i d e r e d p r e s e n t i y , I t may be p o s s i b l e t h a t ,
as f u r t h e r number of l o w - l e v e l load c y c l e s a r e cons i d e r e d a n d t h e c r a c k - t i p grows f u r t h e r and event u a l l y s u r p a s s e s t h e p l a s t i c zone c r e a t e d d u r i n g t h e
decreases t o a
high l e v e l block loading, the o
b a s e l i n e v a l u e c o r r e s p o n d i n g t g P t h e lower l e v e l
cornouter
b l o c k l o a d i n g . However, l i n i i 4 a t i o n s o f ..
.
f:!nds F r P c l ~ d 9 d th" nt n- ~- =-*--h i i : t y n f l a r g e r number of l o a d c y c l e s a t the low l e v e l . I t
i s a l s o s e e n t h a t a f t e r t h e l o a d - l e v e l s t e p down,
AK
remains r e m a r k a b l y lower than its b a s e l i n e
v a f u e c o r r e s p o n d tng t o the low- l c v e 1 b l o c k l o a d i n g ;
.
,
Y
L,
f
f
.,, ,, L C .. r . r C j
preseuce o t a considerable r e
+ a r c l a ; L;,I o f gr3wtt:. but n c d e l a j .
( i i ) F i g . ( 1 0 ) shows t h e c r a c k s u r f a c e p r o f i l m
d u r i n g v a r i o u s s t a g e s of u n l o a d i n g i n one of t h e
low l e v e l c y c l e s of t h e p r e s e n t High to-Low b l o c k
loading.
I t i s seen t h a t a t t h e stage o f u n l o a d i n g
i n d i c a t e d by p o i n t ' R ' i n F i g . ( l o ) , t h e c r a c k c l o s e s o n l y a t the previous c r a c k - t i p ( c l n s u r e a r e a =
a
F u r t h e r u n l o a d i n g , r e p r e s e n t e d by p o i n t C ,
c a u s e s a n o t h e r nade away from rhe c u r r e n t c r a c k - t i p
t o c l o s e , a s seen i n F i g . 10.
The a r e a o f c r a c k c l o s u r e thus i n c r e a s e s as t h e u n l o a d i n g p r o g r e s s e s .
To u n d e r s t a n d t h e e f f e c t s o f t h e f e a t u r e s o f
c r a c k - c l o s u r e a s i n the p r e s e n t c a s e , t h e p r o b l e m
was s e a n a l y s e d w i t h the c o n s t r a i n t of c l o s u r e b e i n g removed on t h e node ( a s d i s c u s s e d a b o v e ) f a r
away from t h e c r a c k - t i p , b u t l e a v i n g t h e c l o s u r e c o n s t r a i n t on the node c l o s e s i t o t h e c r a c k - t i p .
The c o r r e s p o n d i n g c h a n g e s i n o
a r e i n d i c a t e d by
a broken l i n e i n F i g . 9. ~ h e s g ' r e s u l t s i n d i c a t e
t h e i n f l u e n c e of p r o p e r l y imposing t h e c l o s u r e c o n s t r a i n t s on nodes e v e n f a r away from t h e c r a c k t i p : when t h e c o n s i d e r e d l o a d i n g , a s t h e p r e s e n t
H i g h - to-Low c a s e , c a u s e s s u c h a t y p e of c r a c k c losure.
S i n n l e Over Load
cf
I
-
of
::ya8%6
s?~bk~:)ac
%$PZ&I
u s i n g t h e above d is c u s s e d 3 d a t a p o i n t s , is shown i n F i g . 12. By ext r a p o l a t i o n , t h e t h r e s h o l d v a l u e of t h e o v e r l o a d
r a t i o , a t which r e t a r d a t i o n e f f e c t s come i n t o
In c o n t r a s t ,
p l a y , is s e e n t o be a b o u t 1.10.
Bernard e t . a 1 161 r e p o r t a t h r e s h o l d r ~ e r l o a drat i o of 1.3
1 . 4 based on a s e r i e s o f e x p e r i m e n t s
0,: t h e m a t e r i a l Ducol W30B whose y i e l d s t r e n g t h is
366 MP (, comparable t o t h e p r e s e n t l y c o n s i d e r e d
o
J~OMP,)
It i s n o t e d however, t h a t t h e prarex!! a n a l y s i s is b a s e d on a p l a n e - s t r e s c t a s s u m p t i o n ,
w h i l e B e r n a r d e t . a l [ 1 6 ] n o t e t h e dependence o f t h e
e x p e r i m e n t a l l y d e t e r m i n e d t h r e s h o l d v a l u e on t h e
specimen t h i c k n e s s .
-
I
-
OPrnax
Oop. b a s q
max- base- = o p . b a s e
.
e-
The curve depicting the
the r a t i o ( cro
) vi!~?%e o v e r load
(gii)
-
"
( i f ) It is seen t h a t hKeff e x p e r i e n c e s a s u d den jump innmediately a f t e r t h e o v e r l o a d , and t h e n d e creases below i t s b a s e l i n e l e v e l c o r r e s p o n d i n g :o
level;
a constant-amplitude cycling a t a
t h u s i n d i c a t i n g t h e p r e s e n c e o f r%%&@?on
and d e l a y e f f e c t a ( t h e s e terms are used here in t h e same
sense a s d e f i n e d by B a r n a r d , e t . a 1 [16])in t h i s b i n It is also n o t e d t h a t t h e quang l e overload case.
t i t a t i v e e f f e c t s of r e t a r d a t i o n and d e l a y depend on
t h e overload r a t i o .
variation
The c a s e of a s i n g l e o v e r l o a d a f t e r 4 c y c l e s of
a c o n s t a n t a m p l i t u d e b l o c k l o a d i n g , f o l l o w e d by f u r t h e r c y c l e s o f c o n s t a n t a m p l i t u d e ( e q u a l i n magnitude
The f o l l o w t o t h a t b e f o r e o v e r l o a d ) was c o n s i d e r e d .
i n g t h r e @ c a a e s were c o n s i d e y d .
_
r a t i o 2 . 0 , ~ is s t i l l i n c r e a s i n g (Fig. l l c ) . T h i s
implies t h a f P t h e higher t h e overload r a t i o is, t h 0 . m
r e m a r k a b l e b o t h t h e r e t a r d a t i o n and d e l a y e f f e c t s
are, This c o n c l u s i o n is i n a c c o r d w i t h t h e obrervat i o n o f Schijve [ I ] , bared on t h e e x p e r i m e d t a l res u l t ~of A r k e m [15!,
I n t h e above, a
i s he maximum a p p l i e d
max. b a s e
stress prior to o r a f t e r overload;
is t h e
'over l o a d
overload s t r e s s ;
is t h e maximup c a l c u l a t e d
O o p . max
v a l u e f o r c r a c k - o p e n i n g s t r e s s a f t e r o v e r l o a d ; and
o
i s t h e base l i n e o p e n i n g s t r e s s f o r a n 0 t h o base
e & i s e c o n s t a n t - a m p l i t u d e c y c l i c load a t l e v e l
and a l l t h e s e s t r e s s e s a r e i l l u s t r a t e d
'max.baseY
i n F i g . ( 1 1 ) . The o b t a i n e d r e s u l t s a r e d i s c u s s e d
.
I
.
( i v ) The c r a c k - l i n e d e f orrnat i o n p r o f i l e s a t
v a r i o u s s t a g e s o f u n l o a d i n g a t the end of t h e c o n s i d e r e d number o f c y c l e 8 a r e shown i n F i g . 13 f o r
t h e case of o v e r l o a d r a t i o o f 2 , w h i l e s i m i l a r res u l t # were n o t e d f o r t h e o t h e r two o v e r l o a d - r a t i o
I t i s seen t h a t t h e a p p l i c a t i o n of the
cases also.
s i n g l e o v e r l o a d t o t h e s p e c i m e n ( a t t h e i n s t a n t when
a/no = 1.02 i n Fig. 13)causes a l a r g e ( p l a s t i c )
b l u n t i n g which is r e t a i n e d i n t h e c r a c k - s u r f a c e prof i l e even a s t h e c r a c k a d v a n c e s i n f u r t h e r c y c l i c
l o a d i n g . When t h e specimen is f u l l y u n l o a d e d , a t
t h e end of t h e c y c l e i l l u s t r a t e d i n F i g . 1 3 , a l m o s t
t h e whole s u r f a c e a r e a ahead o f t h e p r e v i o u s l y mrnt i o n e d L o c a t i o n of b l u n t i n g is n o t i c e d t o c l o s e ,
w h i l e t h e c r a c k s u r f a c e a r e a behind t h i s b l u n t i n g
l o c a t i o n is s e e n n e v e r t o c l o a e .
/
Analvs is of a C e n t e r - C r a c k e d Specimen
u n d e r Pure Mode 11 C y c l i c Load'ing
below:
( i ) T h e v a r i a t i o n s of o
d u r i n g the l o a d c y c l i n g , for the three d i f f e r & ? r a t i o s of s t r e s 3 o v e r l o a d , a r e i n d i c a t e d i n F i g s . ( I ].a, b , c ) r e s p e c t ively.
I n a l l t h r e e o v e r l o a d c a s e s an a b r u p t d e cl ease i n o
(which r e l a t i v e d e c r e a s e becomes more
predominantoPs t h e oyer l o a d s t r e s s - r a t i o i n c r e a s e s )
is noticed irmediately a f t e r the s i n g l e overload
a p p l i c a t i o n . After t h i s , i n a l l the three c a s e s .
3
i n c r e a s e s a g a i ~t o r c a c h a peak v a l u e a
"P
op-max
before levelling off a steady-state value.
This
r e l a t i v e v a l u e s of 0
increases as t h e averload
opmax
s t r e s s r a t i 3 i n c r e a s e s . Also as t h e o v e r l o a d s t r e s r a t i o increases, the l a t t e r is the oczurance o f
r-- . -.
t h i s .r . .
. . - - * . -- - .
.. st r e s s - i a t Lo
occurs i n the 4 t h cyc l e a f t e r o v eOr lf o a1d' 2 7( 3F' l g*. O Y T ~ f; o r o v e r l o a d r a t i o
of 1.455 t h i s occurs i n the 8 t h c y c l e a f t e r overl o a d (Fig. l l b ) ; w h i l e f o r t h e c a s e of o v e r l o a d -
-
1.4
C,,
I
'
4..
*.
A c e n t e r c r a c k e d s q u a r e p l a t e under a c o n s t a n t
a m p l i t u d e c y c l i c l o a d i n g o f p u r e s h e a r , which i s
u n i f o r m l y d i s t r i b u t e d a t t h e e d g e s of t h e p l a t e , i s
analysed.
P l a n e s t r e s s c o n d i t i o n s a r e assumed.
The m a t e r i a l i s c o n s i d e r e d t o b e 2024-T3 Aluminum
a l l o y , ( ~ a m ea s i n t h e Pure Mode I c a s e d i s c u s s e d
e a r l i e r ) . The d i m e n s i o n s of t h e p l a t e a r e : L=W=
140mm; a = 40mm. The maximum a m p l i t u d e of t h e
u n i f o r m l ~ d i s t r i b u t e d s h e a r is t a k e n t o b e r
=
80MP ( T x
~ /o,
= .23).
In t h e p r e s e n t probTg%,
t h e &omef!ry
b f t h e p l a t e w i t h t h e c r a c k i s symm e t r i c a l a b o u t b o t h t h e x and y a x i s (See F i g . 1 4 ) ,
and t h e e x t e r n a l l o a d i n g i s a n t i - s y m m e t r i c w i t h
.
C-.l.-
- .
,
.
. , . . I .
L A
II
d i i ~j
d.t
ia
.
A s e a r l i e r , t h e p r e s e n t m a t e r i a l is modeled a s
an e l a s t i c - p e r f e c t - p l a s t i c m a t e r i a l . We n o t e a l s o
t h a t the p r e s e n t l y considered m a t e r i a l has the
same p r s p e r t i e s i n t e n s i o n as i n c o m p r e s s i o n . T I ~ ~ I S ,
t h e d i s p l a c e m e n t f i e l d h a s t!le f o l l o w i n g a n i i s y m metric properties:
where u f a the displacement i n x c l i r e c t i o n , e t c .
~ u r t h e r li t is n o t e d t h a t t h e s e d i s p l a c e m e n t s may
be d i s c o n t i n u o u s a t t h e crack s u r f a c e , - a < x < a
0'
T h u s , ir. t h e f i n i t e e l e m e n t m o d e l i n g , o n l y a q u a r t e r
o f the p l a t e is modeled ( a s show1 i n F i g , 1 4 ) w i t h
t h e d i s ~ l a c m e n tb o u n d a r y c o n d i l i o r s : u = 0 a l o n g
y = 0, i n t h e u n c r a c k e d l i g a m e n t o n l y ; a8d u = 0
a t n o d e s a l o n g x = 0. The L i n e a r e l a s t i c r e Ys u l t s ,
b a s e d on t h e f i r s t l o a d i n c r e n t e n t , 'rom t h e p r e s e n t
f i n i t e element a n a l y s i s i n d i c a t e :
KI = 0.075;
K I I = 3.777
6
w h i c h compare f a v o r a b l y w i t h t h e f o l l owing r e s u l t s
( o b t a i n e d by u s i n g t h e f i n i t e - s i z e c o r r e c t i o n f a c t o r s ) o f Bowie a n d P e a l [ I 7 2 f o r a n I d e n t i c a l p r o blem:
KI = 0.0
K I I = 3.899
7
The f a c t t h a t K f O i n t h e p r e s e n t f i n i t e e l e m e n t
I
a n a l y s i s i s t h e r e s u l t o f i n h e r e n t n u l r ~ e r i c a le r r o r s ,
s u c h a s r o u n d - o f f and t r u n c a t i o n , i n t h e f i n i t e
element analysis.
In F i g . ( 1 5 a ) t h e r e s u l t s f o r t h e d i s p l a c e m e n t s
a t t h e u p p e r and l o w e r c r a c k s u r f a c e s , uU and uL
Y
Y
Y
respectively, are plotted for the linear e l a s t i c
c a s e . These n u m e r i c a l r e s u l t s f o r uU and u a a r e i d e n t i c a l i n the l i n e a r e l a s t i c case.' T h i s 'equality
o f uU and u 4 i s n o t i c e d a s t h e l o a d i n g c o n t i n u e s i n
t h e F i r s t c y c l e ( c r a c k b e i n g s t a t i o n a r y ) and when
t h e p l a s t i c z o n e s i z e i s s i g n i f i c a n t a t r = 70 MPa
( s e e F i g . 1 6 ) f o r t h e s h a p e o f t h e p l a s t i c i t y zone
a t 7 = 70 MPa . F o r l a c k o f any o t h e r c r i t e r i a ,
t h e c r a c k was e x t e n d e d , i n t h e p r e s e n t p r o c e d u r e ,
a t 70MPa , ( t h e maximum a p p l i e d s t r e s s b e i n g 80 HPaX
i n the d i r e c t i o n of t h e i n i t i a l c r a c k i e . , i n the
I t i s s e e n from F i g . ( 1 5 a ) t h a t s i g x-direction.
n i f i c a n t i n c r e a s e i n u i s b r o u g h t a b o u t by t h e proc e s s o f c r a c k e x t c n s i o X and h e n c e t h e a t t e n d a n t pie
t i c u n l o a d i n g ; h o w e v e r , a g a i n , uU and ue a r e a l m o s t
i d e n t i c a l ( t o t h e 4 t h s i g n i f i c a n t d i g i t r . Thus i t
i s s e e n t h a t t h r o u g h t h e a l l s t a g e s of l o a d i n g ,
c r a c k e x t e n s i o n and p l a s t i c - u n l o a d i n g , and f u r t h e r
l o a d i n g ( t o 8 MPa i n t h i s c a s e ) a f t e r c r a c k e x t e n s i o n , t h e u p p e r and l o w e r c r a c k f a c e s e x p e r i e n c e id e n t i c a l displacements i n the y d i r e c t i o n , i e . ,
p e r p e n d i c u l a r t o t h e i n i t i a l c r a c k a x i s . On t h e
o t h e r h a n d , t h e d i s p l a c e m e n t s u a t t h e u p p e r and
l o w e r s u r f a c e s o f t h e c r a c k , u L ' znd o a a r e p l o t t e d
i n F i g . ( 1 5 b ) f o r t h e c a s e s *of l o a B i n g when l i n e a r - e l a s t i c c o n d i t i o n s p r e v i a l (7 = 3 1 . 1 MPa); when
appreciable p l a s t i c i t y develops a t the c r a c k - t i p
(T = 7 0 MPa), when t h e c r a c k i s e x t e n d e d (and h e n c e
t h e r e i s p l a s t i c l o a d i n g ) a t T = 7 0 MPa, and when
t h e l o a d i s f u r t h e r i n c r e a s e d r o 80 MPa a f t e r c r a c k
extension.
I t i s s e e n t h a t t h e m a g n i t u d e s o f uU
and u A a r e n e a r l y i d e n t i c a l ( t o t h e 4 t h s i g n i f i ? s n t
d i g i t x ) , b u t w i t h o p p o s i t e s i g n , i n a l l t h e above
c a s e s . However i t is i n t e r e s t i n g t o o b s e r v e t h a t
t h e change ( a s a r a t i o of the r e s p e c t i v e value'
. , ,. y. - : --. ,
,L - - .,.e..
,. - T q L h > t
u
A,,
- 1
%
<
_ ' . C C - . . " L " . . ,
4
C
.L
C
^-A
.
#
about by t h e p r o c e s s of r r a c k e x t e n s i o n (and hence
p l a s t i c u n l o a d i n g ) i n u i s much more p r o n o u n c e d
Y
than i n u .
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 i n the l i n e a r
e l a s t i r c a s e the i r n r l - - s t i r f a c e d i s ~ l a r e r sn ~.I: ~ ~- ~ d
uk ( s e e Fig. 15a) e x h i b i t almost a l i n e a r v a r i a r ion
fYom t h e c r a c k - t i p ; t h u s i n d i c a t i n g a l a c k o f any
r a ( i n p a r t i c u l l r , ,,F t y p e f o r l i n e a r e l a s t i c i t y i
component i n u f o r t h e l i n e a r e l a s t i c c a s e . On
tile o t h e r hand:
for the linear e l a s t i c c a s e , ti,?
t a n g e n t i a l d i s p l a c e m e n t s uU and l l a
(See F i g . 1 5 5 )
do e x h i b i t a f i behavior naar thexcrack- t i p , f o r
t h e p r e s e n t Mode I 1 p r o b l e m . A l s o , i t i s s e e n from
F i g . ( 1 5 ) t h a t , a s p l a s t i c i t y d e v e l o p s , t h e tangen
r i a l d i s p l a c e m e n t s uU and ue e x h i b i t a P (a
$j
v a r i a t i o n n e a r t h e c:ack-t
However, f o r t i l e
p u r e Mode 11 c a s e , e v e n i n che p r e s e n c e o f p l a s t i c i t y , t h e a n a l y s e s o f H u t c h i n s o n [8] and R i c e and
R a s e n g r e n [ 9 : , i n d i c a t e t h a t t h e r e may n o t be a
P (a <
t y p e v a r i a t i o n i n uv n e a r t h e c r a c k - t i p .
g u t t h e p r e s e n t r e s u l t s f o r u i n t h e p r e s e n c e of
p l a s t i c i t y , Fig. 1 6 , a r e s e e n Y t o c o n t a i n such an
P(CI<
1 / 2 ) t v p e v a r i a t i o n n e a r t h e c r a c k - t i p . Howe v e r , i t s h o u l d be n o t e d t h a t t h e a n g u l a r v a r i a t i o r ,
o f t h e s i n g u l a r i t y f u n c t i o n s , G i ( d ) , a s embedded i n
t h e p r e s e n t s e c t o r e l e m e n t s a r e being approximated
a s q u a d r a t i c polynomials i n each s e c t o r element.
< 1 / 2 1 t y p e v a r i a t i o n s i n rl a r e
The f a c t t h a t
n u m e r i c a l l y o b t a i n e d a l o n g t h e r a d i a l l i n e of Y t h e
s e c t o r e l e m e n t l y i n g on t h e c r a c k s u r f a c e , aa i n
Fig. 15a s u g g e s t s t h a t t h e above a n g u l a r v a r i a t i o n s
a r e n o t being s o l v e d h i g h l y a c c u r a t e l y i n t h e pres e n t n u m e r i c a l method. However, i t a p p e a r s t h a t
t h e s e numerical e r r o r s a r e i d e n t i c a l a t 6 = + n
a s w e l l a s a t d = - n , s o t h a t u$ = u$ a s i n F - , 1 5 1 ~
F i n a l l y t h e c r a c k s u r f ace d i s p l a c e m e n t s (uu,uy) and
( u u d j a t t h e end o f t h e f i r s t c y c l e a f lo%diXg ( i e .
X' X
when t h e a p p l i e d stress is b r o u g h t b a c k t o z e r o ) a r e
a l s o i n d i c a t e d i n F i g s . ( l S a ) and (15b) r e e p e c t i v e l v .
Once a g a i n i t i s s e e n t h a t e v e n a f t e r c o m p l e t e unl o a d i n g , uU and ua a r e i d e n t i c a l i n m a g n i t u d e and
x a r e ident i c a l i n magnitude
d i r e c t ion,'where 8 s uU
b u t o p p o s i t e i n d i r e c t i o n . Thus f o r t h e p r e s e n t
m a t e r i a l , with i d e n t i c a l properties i n tension as
i n compression, i t is seen t h a t i n a l l c a s e s of
p u r e - s h e a r type e x t e r n a l load'ng, t h e upper and
l o w e r s u r f a c e s of t h e c r a c k move t o g e t h e r i n the
d i r e c t i o n perpendicular t o the i n i t i a l c r . 4 ~ ~ - a x i s ,
whereas they s l i d e past one another i n t h e d i r e c t i o n of t h e c r a c k - a x i s .
rhus, i t appears f o r these
t y p e s o f m a t e r i a l s t h e phenomenon of c r a c k - c l o s u r e ,
a s o b s e r v e d e x p e r i m e n t a l l y and a s a n a l y s e d p r e s e n t l y i n Mode I t y p e l o a d i n g c o n d i t i o n s , d o e s n o t occur
i n p u r e Mode 11 t y p e c y c l i c l o a d i n g . However t h e
p r e s e n t e x p e r i e n c e i n d i c a t e s t h a t c r a c k - c l o s u r e may
o c c u r e v e n i n p u r e Mode I 1 c y c l i c l o a d i n g i f t h e
m a t e r i a l has d i f f e r e n t p r o p e r t i e s i n u n i a x i a l t e n s i o n and c o m p r e s s i o n . C o n s i d e r a t i o n of s u c h m a t e r i a l s is n o t p u r s u e d i n tile p r e s e n t work. F i n a l l y ,
t h e computed s h a p e s and s i z e s o f t h e p l a s t i c zone
near the c r a c k - t i p a t various stages of pure shear
l o a d i n g a r e i n d i c a t e d i n F i g . 16.
i.5.
%,
Summary and C o n c l u s i o n s :
I f t h e c r a c k - g r o w t h r a t e , d a , d , i n Mode 1
f a t i g ~ l el o a d i n g , c a n be assumed t o r e r e l a t e d t h e
the pree f f e c t i v e s t r e s s - i n t e n s i t y range, A K e f f ,
s e n t r e s u l t s i n d i c a t e t h a t : ( i ) g r o w t h r e t a r d a t ion
o c c u r s i n h i g h t o low m d s i n g l e o v e r l o a d c a s e s ,
and (ii) s i g n i f i c a n t f l e l a y e f f e c t s p r i o r t o r e t a r d a t i o n o c c u r i n t h e c a s e of a s i-n-g l e - o v e r l o a-d -and
.,,=LC
n P '
..\
1
d IU
~ C C c l L u dA u u
CLLCLLI
V G C .
-
L.L..L-
I
\
d o m i n a n t a s t h e o v e r l o a d r a t i o i n c r e a s e s . From t h e
c r a c k s u r f a c e d e f o r m a t i o n p r o f i l e s shown i n F i g s .
8 a n d 13, i t i s s e e n t h a t a c o n s i d e r a b l e c r a c k s u r f a c e b l u n t i n g o c c u r s a t the i n s t a n t when t h e a p p l i d
l o a d is s t e p p e d up. T h i s b l u n t i n g a t t h e ~ n s t a n t
of l o a d - s t e p up a l t e r s t h e p o s s i b l e p a t t e r n of
c r a c k c l o s u r e d u r i n g s u b s e q u e n t l o a d c y c l e s , and i s
s e e n t o b e r e s p o n s i b l e f o r the "c'elay" e f f e c t 8 s u c h
as the delayed r e t a r d a t ion i n t h e s i n g l e overload
c a s e , and t h e d e l a y e d t r a n s i t i o n o f o p e n i n g stress
l e v e l s f r o m the b a s e l i n e v a l u e s f o r l o w e r a m p l i tude block loading t o the h i g h e r b a s e - l i n e v a l u e
f o r t h e h i g h e r a m p l i t u d e b l o c k l o e d i n g , i n t h e Lowt o - h i g h c a s e . The p h y s i c a l mechanism b e h i n d t h e s e
e f f e c t s is f u r t h e r d e t a i l e d i n [ 1 2 , 1 4 ] .
Finally,
t h e phenomenon of c r a c k . c l o s u r e was n o t o b s e r v e d
i n t h e p r e s e n t n u m e r i c a l m o d e l i n g of a t h i n c e n t e r c r a c k e d s h e e t ( o f an e l a s t i c - p l a s t i c m a t e r i a l w i t h
i d e n t i c a l ~ r n p e r t i e ei n u n f a x f a l t e n s i o n a s i n c o r n
p r e s s i o n ) , s u b j e c t a p u r e s h e a r (Mode 11) c y c l i c
Loading o f c o n s t a n t a m p l i t u d e . T h u s , i t a p p e a r s
t h a t i n t h e s t u d y of t h e inore g e n e r a l p r o b l e m o f
f a t i g u e c r a c k - g r o w t h u n d e r mixed-mode l o a d i n g , t h a t
causes small scale yielding9 near the c r a c k - t i p ,
c r a c k - c l o s u r e e f f e c t s need b e c o n s i d e r e d i n t h e c a s e
mode I component o n l y . However, t h e p r e s e n t e x p e r i e n c e i n d i c a t e s t h a t c r a c k - c l o s u r e may o c c u r e v e n
i n p u r e Mode I 1 i f t h e m a t e r i a l h a s d i f f e r e n t p r o p e r t ies i n u n i a x i a l t e n s i o n and c o m p r e s s i o n .
e n i n g b l a t e r i a l , " J o u r n a l o f Mechanics and Phva&
o f S o l i d s , V o l . 1 6 , 1 9 6 8 , pp. 1 - 1 2 .
[ l o ) A t l u r i , S. N . m d Nakagaki, M., " J - I n t e g r a l
Estimates f o r Strain-Hardening Materials i n
Duct i n e F r a c t u r e P r o b l e m s , I t AIAA J o u r n a l , Vol.
1 5 , No. 7 , 1977. pp. 9 2 3 - 9 3 1 .
[ l l ] A t l u r i , S . N. and N a k a g a k i , M. and Chen, W . t i . ,
"Fiac t u r e Ana l y s is Under L a r g e S c a l e Y i e l d Fng :
A F i n i t e Defarmat i o n Embedded S i n g u l a r i t y ,
E l a s t i c - p l a s t i c I n c r e m e n t a l Finite Element
S o l u t i o n i n Flaw Growth and F r a c t u r e " .
STP 631, American S o c i e t y f o r T e s t i n g and
M a t e r i a l s , 1 9 7 7 , pp. 43-61.
[I21 N a k a g a k i , M . ,
and A t l u r i , S . N . , " E l a s t i c P l a s t i c F i n i t e Element A n a l y s e s of F a t i g u e
Crack Growth i n Mode I and Mode I 1 C o n d i t i o n s " ,
NASA -CR- 1 5 8 9 8 7 , Nov. 7 8 , 82 p e g e s .
!3]
Acknowledntmen t s : T h i s work was s u p p o r t e d i n p a r t s
by N A S A g r a n t No. NSG-1351 and by AFOSR c o n t r a c t
No. F49620 78-C-0085. T h e s e s u p p o r t s a r e g r a t e
f u l l y acknowledged.
At l u r i , S . N . , "On Some New G e n e r a l and Complem e n t a r y E n e r g y Theorems f o r t h e Rate Problems
i n Finite Strain, Claesical Elas to-Plast icity",
G e o r g i a I n s t i t u t e of T e c h n o l o g y , R e p o r t GITESM-SA-78 1 0 , Aug. 1978. ( I n r e v i e w f o r publ i c a t i o n i n J . Mechanics and P h y s i c s of S o l i d s ) .
f 141 N a k a g a k ~ , M . , and A t l u r i , S .
N . , "Fatigue
Crack C l o s u r e and D e l a y E f f e c t s Under Mode I
9 p e c t r u m l o a d i n g : An E f f i c i e n t E l a s t i c - P l a s t i c P r o c e d u r e " t o a p p e a r i n P r o c e d . of 3 r d I n t .
Conf. on Mech. Beh. o f M a t e r i a l s , U n i v . o f Qamtri%e
U. K . , S e p t . 1979 ( 9 p a g e s ) .
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S c h i j v e , J . , "Four L e c t u r e s on F a t i g u e C r a c k
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E l b e r , W . , " F a t i g u e Crack C l o s u r e Under C y c l i c
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C o n f e r e n c e , Las Vega, N e v . . 17-19 A p r i l 1 9 7 4 .
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Growth. ASTM S T p 590, American S o c i e t y f o r
T e s t i n g and M a t e r i a l s , 1 9 7 6 , pp. 2 8 1 - 3 0 1 .
-
O h j i , K . , O g u r a , K . , and Okubo, Y . , " C y c l i c
A n a l y s i s o f a P r o p a g a t i n g Crack and i t s C o r r e lat ion with ~ a 'i tg u e C r a c k Growth ," E n g i n e e r i n g .
F r a c t u r e Mechanics, 1 9 7 5 , V o l . 7 , pp. 4 5 7 - 4 6 4 .
O g u r a , K . a n d O h j i , K . , "FEM A n a l y s i s o f Crack
C l o s u r e and Delay E f f e c t i A l F a t i g u e C r a c k
Growth Under V a r i a b l e Amplitude L o a d i n g . "
E n n i n e e r i n p F r a c t u r e M e c h a n i c s , 1977, V a l . 9 ,
pp. 4 7 3 - 4 8 0 .
H u t c h i n s o n , J . W . , " S i n g u l a r Behaviour a t t h e
-6
-,.. 4 ~ i a i C i E i ? L i ~ ig. ; ; l ; C i - i d i " .
A
J o u r n a l of Mechanics and P h y s i c s o f S o l i d s ,
1968, Vol. 16 pp. 1 3 31.
.V.."..
3
LL:-
u
I G L I U L ~ F
LLQI-P.
R i c e , J . ?. and R o s e n g r e n , G . F . , " P l a n e S t r a i n
Dcformatio? Near a Crack T i p i n a Power Low Hard-
[ 1 5 ] Arkema, W. J . , r e s u l t s q u o t e d i n J . S c h i j v e ,
" O b e e r v a t i o n s on t h e P r e d i c t i o n of F a t i g u e
Crack Growth P r o p a g a t i o n Under V a r i a b l e - Amp1 i t u d e L o a d i n g " , ASTM STP 5 9 5 , p. 3 , 1976.
L16:
B e r n a r d , P. J . , L i n d l e y , T. C . , and R i c : l a r d s ,
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C.
[17] Bowie, 0. L. and N e a l , P. M . , "A Note on t h e
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Vol. 2 , Nov. 1 9 7 0 , p p . 1 8 1 - 1 8 2 .
P:AII. 'N OF DETAIL 'A' IN FIG. 2
MESH ANL NODES BEFORE
MESH AND NODES AFTER
TRANSIATION
LP
PIASTIC ZONE
FIG 30. REPRESENTATIVE SIZE OF YIELD ZONE AT Omx.
FIG 1. SCHEMATIC REPRESENTAT ION OF TRANSIATION OF
SINGLUR ELPIENTS.
FIG 3b. S C U M T I C REPRESENTATION OF INCRPIENTAL
EQUATIONS IN THE PRESENCE OF YIELDING.
/---- DETAIL 'A'
CLOSED
FIG 4. REPRESENTATIVE FATTERNS OF CRACK CLOSTJRE ;
( 4 8 c) : CRACK CLOSES ONLY AT NODES W S E S T
TO CRACK TTP. bd. PQACY 17.OSTmF n r r r m c r r r n
AT NODES AWAY FROM CRACK TIP.
-
F T C . 2 . FINTTE
nmwr M o n n w
4 C E N T F ~ruArYLn SPECTMEN ~
E
UNIAXIAL CYCLIC LOADING ( S XNG IJLAR SECTOR ELEMENTS SHOWN
WITHIN DETAIL ' A ' ) .
R
8 C#A1;7( CLOSURE
O CRACK OPENTNG
C1
.96
1.
.98
1.02
1.04
X/ao
FIG 5
CRACK CLOSURE AND CRACK OPENING STRESSES IN [XIH!?I'AVT AKPLITUnE
(LO) m n I c LUADING.
FIG 8. CRACK LINE PROFILE DURING UNLOADING IN
LOW- TO-HIGH BLOCK LOADING.
4
$3
0
V)
8 *
;
< 0
LYCLE
FIG 9 . CHhLX CLOSURE
LOADING.
(b)
CRACK OPENING STRESSES IN HIGH-m-UU BLOCK
X I ro
F I G 6 a . C R A a LINF PROFILE DURlNG UNLOADING I N CONSTAKI.
AWPLITLRIE LOADING: 6b ZNSET 'A' I N F I G 6. I S
K 4 G N I F I E D AND SHOW.
0 CRACK OPENING
CRACK CLOSUP-E
FIG 7
'-7(ACN
L'YCLt
CLOSURE AND CRACK OPENTNG STRESSES IN W - T O - H I G H BMO(
LCHDINC.
AI
e
a: CRACK L I N E PROFILE DURING UNLOADING I N HIGH- TO-LOW
BLOCK LOADING; b: INSET ' A ' I N F I G 10a. I S MAGNIFIED
AND SHOWN.
FIG 11. CRACK CLOSURE AM) CRACK OPENING STRESS FOR
THREE DIFFERENT CASES OF A SINGLE OVER 1A3AI)
FIG 15, a NORMAL D I S I ~ A C P I E N TFROFILE5 OF THE L!P% ANP LOh'Ffi
SURFACES OF THE CRACK: b: TANGENTIAL DISPIACEKK'I PROFILES
OF UPPER AND UlWER C IFACES OF THE CRACK.
'op.mx
Omex. bare
'
'op.bars
- 'op. bare
FIG 12. EFFECT OF OVERLOAD-STRESS RATIO ON uop'mx' oap'baee
*mx. bare- 'op,bare'