**
T o s h i h i s a Nishioka* and Satya N. A t l u r i
School of Engineering Science and Mechanics
Georgia I n s t i t u t e o f Technology
A t l a n t a , Georgia 30332
Abstract
I n t h e p r e s e n t paper a new assumed s t r e s s
f i n i t e element method, based on a complementary
energy method, i s developed, f o r t h e a n a l y s i s of
cracks and h o l e s i n angle-ply laminates. I n t h i s
procedure, t h e f u l l y three-dimensional s t r e s s s t a t e
( i n c l u d i n g t r a n s v e r s e normal and s h e a r s t r e s s e s ) i s
accounted f o r ; t h e mixed-mode s t r e s s and s t r a i n
s u n g u l a r i t i e s , whose i n t e n s i t i e s vary within each
layer, near the crack front, are b u i l t into the
formulation a p r i o r i ; t h e i n t e r - l a y e r t r a c t i o n
r e c i p r o c i t y c o n d i t i o n s a r e s a t i s f i e d a p r i o r i ; and
t h e i n d i v i d u a l c r o s s - s e c t i o n a l r o t a t i o n s of each
l a y e r a r e allowed; thus r e s u l t i n g i n a highly
e f f i c i e n t and c o s t - e f f e c t i v e computational scheme
f o r p r a c t i c a l a p p l i c a t i o n t o f r a c t u r e s t u d i e s of
laminates. R e s u l t s obtained from t h e present procedure, f o r t h e c a s e of an uncracked laminate und e r bending and f o r t h e case o f a laminate with a
through-thickness crack under f a r - f i e l d tension;
t h e i r comparison with o t h e r a v a i l a b l e d a t a ; and
pertinent discussion, a r e presented.
Introduction
An a c c u r a t e three-dimensional s t r e s s a n a l y s i s
of angle-ply laminates with c r a c k s and/or h o l e s , a s
opposed t o t h e u s e of simpler " c l a s s i c a l laminated
p l a t e t h e o r i e s , " i s o f t e n times mandatory t o understand Q i ) t h e complicated f e a t u r e o f t h e oftenobserved n o n - s e l f - s i m i l a r crack growth i n symmetric
angle-ply l a m i n a t e s ; ( i i ) t h e s u b c r i t i c a l damage i n
t h e form of m a t r i x crazing, s p l i t t i n g , and delamina t i o n t h a t i s observed t o precede f i n a l f a i l u r e i n
a laminate; and ( i i i ) t o more c l e a r l y understand
t h e h o l e - s i z e e f f e c t s i n laminates. S p e c i f i c a l l y ,
consider a symnetric angle-ply laminate of the form
I 2 . . . .1 Bnhn)s where Bn and hn
(k Blhi/
B
+
+
2h2
a r e , r e s p e c t i v e l y , t h e o r i e n t a t i o n and thickness of
t h e n t h angle-ply component o f t h e laminate; and
consider t h e c a s e of a through-the-thickness crack
i n t h e laminate. ALSO, c o n s i d e r t h e crack-axis t o
be p a r a l l e l t o t h e f i b e r - o r i e n t a t i o n of t h e K-th
p l y . Thus, even though t h e crack i s located symmetrically with respect t o the preferred material
d i r e c t i o n s of t h e k t h p l y , i t i s , i n general,
o r i e n t e d unsynunetrically with r e s p e c t t o t h e p r i n c i p a l m a t e r i a l d r i e c t i o n s of t h e p l i e s , n
k, i n
which l o c a l (layerwise) mixed-mode c r a c k - t i p condit i o n s can be seen t o e x i s t . This may, f o r i n s t a n c e ,
lead t o a model f o r n o n - s e l f - s i m i l a r crack growth.
R e s t r i c t e d quasi-three-dimensional analyses of
angle-ply l a m i n a t e s , t r e a t i n g each p l y a s a homogeneous a n i s o t r o p i c medium, and w i t h o t h e r assumpt i o n s of p e r f e c t bonding between l a y e r s and zero
t r a n s v e r s e n o r m a l - s t r e s s were r e c e n t l y reported by
Wang, e t a 1 i n [I!.
However t h e procedures i n [ 11
when applied t o t h e a n a l y s i s o f c r a c k s , do not
account , a p r i o r i , f o r t h e above mentioned mixed
mode 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 the crack
f r o n t , and hence involve expensive computations
using very f i n e f i n i t e element meshes ( i n f a c t , t h e
s o l u t i o n s i n [ I ] were obtained i n two stages: one
w i t h a f i n i t e element mesh f o r t h e e n t i r e s t r u c t u r q
and t h e second, a much f i n e r mesh f o r t h e nearcrack zone) of conventional, polynomial-based, e l e ments. Even though from t h e above t h e procedures,
i n t h e l i m i t as t h e mesh s i z e becomes very small,
one may o b t a i n high s t r e s s - g r a d i e n t s o l u t i o n s , i t
i s o f t e n inconvenient t o e x t r a c t parameters, such
a s t h e mixed mode s t r e s s - i n t e n s i t y f a c t o r s t h a t
vary through t h e thickness of t h e laminate, which
a r e needed f o r a r a t i o n a l a p p l i c a t i o n o f t h e o r i e s
o f f r a c t u r e i n i t i a t i o n a t cracks and h o l e s . Moreover, t h e most v e r s a t i l e f i n i t e element procedure
developed i n t h e l i t e r a t u r e so f a r , f o r the analys i s even of u y r a c k e d laminates, and used i n the
s t u d i e s of [ l i , i s t h e assumed s t r e s s multi-layer
hybrid element developed by Mau, e t a 1 [ 2 ] . I n t h e
procedure of [2), a s t r e s s f i e l d i s assumed independently i n each l a y e r and i n t e r - l a y e r s t r e s s cont i n u i t y ( o r more g e n e r a l l y , t r a c t i o n r e c i p r o c i t y )
c o n d i t i o n s a r e enforced through Lagrange Multip l i e r s , which n e c e s s a r i l y complicates t h e a l g e b r a i c
formulation and r e s u l t s i n expensive computations.
F u r t h e r , t h e e f f e c t s of t r a n s v e r s e normal s t r e s s e s ,
which a r e important i n delamination s t u d i e s and
s t u d i e s of s t a c k i n g sequence, a r e ignored i n [ 1 , 2 j .
Also, s i n c e t h e s t r e s s e s (and hence the undetermined parameters i n them) a r e assumed independentlyf o r each l a y e r , t h e computational procedure i n [ I , 21
becomes p r o h i b i t i v e l y expensive when t h e number of
l a y e r s i n c r e a s e s . However, t h e assumed boundary
displacements i n [2] a r e such t h a t each l a y e r can
undergo independent c r o s s - s e c t i o n a l r o t a t i o n s ; an
assumption i n l i n e with t h e s o l u t i o n s a n t i c i p a t e d
from a f u l l y three-dimensional t h e o r y , and an i m provement over t h e c l a s s i c a l laminated p l a t e theory.
L a t e r , t o study t h e e f f e c t s o f t h e t r a n s v e r s e
normal s t r e s s (aZZ), Wang e t a1 [3] employed a
f u l l y three-dimensional h y b r i d - s t r e s s f i n i t e e l e ment method a s o r i g i n a l l y proposed by Pian [ 4 ] .
Thus, i n t h e procedure of [3], each l a y e r of the
laminate was modelled by t h e usual three-dimensiona l f i n i t e elements whose s t i f f n e s s formulation i s
based on t h e h y b r i d s t r e s s model [4]. Since t h e
elements used i n [ 31 a r e f u l l y three-dimensional,
eachnode of each element has t h r e e displacement
degrees of freedom. F y r t h e r , i n t h e a p p l i c a t i o n
o f t h e procedure i n [ 3 J t o t h e a n a l y s i s of cracks
i n laminates, s i n c e t h e general mixed-mode s t r e s s
s i n g u l a r i t i e s ne.w t h e crack-front a r e n o t accounted f o r a p r i o r i , a very f i n e mesh of 3D f i n i t e e l e ments i s necessary. rhus, f o r i n s t a n c e , i n t h e
a n a l y s i s of a through-thickness edge c r a c k i n a
t h i n 9 0 ~ / 0 ~ / 0 0 / 9 laminate
0~
under f a r - f i e l d tensiorq
t h e s t r e s s s o l u t i o n was obtained i n [3] using a
two-stage s o l u t i o n technique i. e. , t h e f i r s t with
a r e l a t i v e l y c o a r s e mesh f o r t h e e n t i r e p l a t e and
t h e second with a f i n e i n n e r mesh t o model t h e subThe f i r s t solus t r u c t u r e around t h e crack-front
t i o n was obtained [3] from a c o a r s e 3D f i n i t e e l e -
.
rnr.-:&
* Post -Doctoral Fellow
** Professor, and Member,
AIM.
Copkright ';/American In,titutc oC Aeronautic* and
Astronaufic\. I nc... I'i79. All righi\ reserved.
Z E
0 V
lQ
l
!
9
(3
h-J-CI
<-
CL
L1-'r.vr.
.
d i r e c t i o n ) w i t h 1710 degrees ofwfreedom, while the
i n n e r mesh c o n s i s t e d o f 11 X 19 X 4 3~ elements
31 5
-
, ,c
~ w c c I
LL,..L>LUACLLLA.Y
T
+
i ~ 1 5 1 ,
'llif-
1
12-
ment solution method f o r cracks i n laminates, which
st the same should y i e l d information, d i r e c t l y ,
cmcerning the mixed-ede s e t a s t,?rtensity f a c t o r s
ne:r t h e crack f r o n t . The development of such a
method i a , thus, one of t h e primary o b j e c t i v e s of
s h e presently reported research.
I n the present paper, a new assumed s t r e s s
f i n i t e element method'is developed, f o r t h e analys i s of cracks and holes i n angle-ply l w i n a t e s ,
which circumvents t h e above c i t e d d i f f i c u l t i e s and,
more over, r e s u l t s fn a highly c o s t - e f f e c t i v e and
accurate procedure. I n t h i s procedcre, t h e f u l l y
three-dimensianaL s t r e s s s t a t e , including t r a n s verse normal s t r e s s , is accounted f o r ; t h e mixedmode 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 , whose intens i t i e s vary within each l a y e r , near the crack front
a r e b u i l t - i n t o the formulation a p r i o r i ; and the
t n t e r l a y e r s t r e s s - c o n t i n u i t y conditions a r e s a t i s f i e d a p r i o r i i n t h e formulation; thus r e s u l t i n g a
highly e f f i c i e n t computational scheme f o r p r a c t i c a l
a p p l i c a t i o n t o f r a c t u r e s t u d i e s of laminates. Res u l t s , based on the present analysis procedure, are
presented f o r two problems: ( i ) bending of a simply
supported three-layer ( 0 ~ / 9 0 ~ / 0rectangular
~)
lamina r e under a sinusoidal transverse load, f o r which
an exact 3-D s o l u t i o n i s available [ 5 ] , and ( i i )
through-thickness edge crack i n a t h i n 90°/00/00/9(P
laminate under unlform f a r - f i e l d tension normal t o
the crack d i r e c t i o n , which problem was a l s o solved,
Comparing t h e present
as mentioned before, i n [3].
r e s u l t s with those a v a i l a b l e i n p r i o r l i t e r a t u r e
t 2 , 3 , 4 1 , the possible advantages of t h e p r e s e n t
method a r e discussed i n d e t a i l .
Description of the Present Analysis Procedure
Let the laminate c o n s i s t of-K l a y e r s , i = 1,2,
...K; and l e t the planar domain of t h e laminate be
divided i n t o M f i n i t e elements, n = 1,. . .M.
We
consider here t h a t each f i n i t e element c o n s i s t s of
the e n t i r e stack of l a y e r s i n the laminate. Let
V; be the volume of t h e i t h layer within t h e n t h
element;
be t h e boundary of
s&, be t h e p a r t
of
where t r a c t i o n s a r e prescribed ( a s , f o r
instance, zero t r a c t i o n s on t h e crack f a c e ) . Furt h e r , we use the n o t a t i o n t h a t (m) under a symbol
denotes a vector and (m) under a symbol denotes a
matrix. Let zZ denote t h e vector (6 X 1) of 3dimensional s t r e s s i n t h e i t h l a y e r , and let &i
be the compliance matrix of t h e i t h Layer ( t r e a t e d
h e r e as general a n i s o t r o p i c ) i n the element coordin a t e s . (The compliance p r o p e r t i e s of each l a y e r ,
i n element coordinates, a r e assumed t o be obtained
from those i n t h e layer-principat-material-direct ions through appropriate tensor t r a n s f armat ions. )
For the moment, l e t us assume t h a t a candidate
s t r e s s f i e l d i s chosen such t h a t i t s a t i s f i e s the
tSree-dimensional stress-equilibrium equations,
a p r i o r i , everywhere within each l a y e r i n each
f i n i t ? element. For t h e sake of g e n e r a l i t y , l e t
us assume that t h i s s t r e s s f i e l d does n o t s a t i s f y
t h e t r a c t i o n r e c i p r o c i t y conditions e i t h e r a t t h e
i n t e r l a y e r i n t e r f a c e s within each elemem, o r a t
t h e interelement boundaries of adjoining f i n i t e
elements. ( b t e r , i n t h e d e t a i l s of t h e chosen
s t r e s s f i e l d i n the p r e s e n t formulation, i t w i l l be
s e w t h a t the i n t e r l a y e r t r a c t i o n r e c i p r o c i t y cond i t i o n i s , however, s a t i s f i e d a p r i o r i . ) Let us
a l s o agO*mp. for the FtPs-nt* +hqt
zk-:cz
stress f i e l d does not s a t i s f y t h e t r a c t i o n boundary
conditions a t s&, a p r i o r i . (Again, i t w i l l be
~vA
&I.-&
This sunnests t h e
g i t h 3522 Gegrees of freedom.
1-
L*
*
--
'
'
"'.">
L . ' b - " L 1
.>L
L <
-
-
LiePd, the condition of vanishing t r a c t i o n s on t h e
crack f a c e a r e , f o r the most p a r t , s a t i s f i e d a
priori.)
Under these assumptions, i t can be shown,
following t h e basic theory o f hybrid s t r e s s f i n i t e
clezzer?te presented i n [ 4 , 6 , 7 ] and elsewhere, t h a t
t h e conditions of compatibility of s t r a i n s corresponding t o the assumed s t r e s s e s , the interelement/
and i n t e r l a y e r t r a c t i o n r e c i p r o c i t y conditions, and
t r a c t t o n boundary conditions follow from t h e v a r i a t i o n a l p r i n c i p l e , which i s s t a t e d as t h e s t a t i o n a r y
condition of the following (modified) complementary
energy f unc t i o n a l :
M
K
ui
where
a r e Lagrange M u l t i p l i e r s that a r e i n t r o duced t o enforce the t r a c t i o n r e c i p r o c i t y condition
a t t h e i n t e r e l e m e n t l i n t e r l a y e r i n t e r f a c e s ;&t a r e
another s e t of Lagrange M u l t i p l i e r s _to enforce t h e
a r e pret r a c t i o n boundary condifions ( b - c ) ;
s c r i b e d t r a c t i o n s ; and TIT i n d i c a t e s t h e transpose
of t h e vector
e t c . The idea of introducing i n dependent Lagrange M u l t i p l i e r s
t o enforce t r a c t i o n b - c , a s accurately a s desired, i s d e t a i l e d i n
[73. For purposes of conceptual c l a r i t y , imagine
t h e domain of a cracked laminate t o be d e s c r i t i z e d
i n t o f i n i t e elements a s shown i n Fig. 1, where a
type 1 element i s a ' r e g u l a r ' element; type 2 i s
an element with the crack f r o n t as one of i t s edgeg
and hence has vanishing t r a c t i o n s on t h e crack face;
and type 3 i s an element which does not have t h e
crack f a c e as a p a r t of i t s boundary, but may have
t h e crack f r o n t as one of i t s edges. I n general,
t h e t h r e e f i e l d v a r i a b l e s i n the functional of Eq.
(1) can be assumed as:
zi
zi
&
vA;
From Eq. ( 2 ) , the boundary t r a c t i o n s can be derived
as
i
We n o t e t h a t i n E q . ( 4 ) , _
% a r e point-wise Lagrange
M u l t i p l i e r s t o enforce t r a c t i o n boundary condition4
a s accurately as d e s i r e d , using the c o l l a t i o n technique a s d e t a i l e d i n [ 7 ] .
I n Eq. (2) , & a r e undetermined parameters i n t h e assumed e q u i l i b r a t e d
s t r e s s f i e l d of a regular polynomial nature; while
a r e undetermined parameters i n the assumed equil i b r a t e d s t r e s s f i e l d of s i n g u l a r (inverse square
rpot from the crack f r o n t ) n a t u r e . We draw a t t e n t i o n here t o the f a c t t h a t t h e parameters & a r e
conanon t o a l l layers ( i = 1,. k) within an e l e tuent; t h i s i s due t o t h e f a c t t h a t as seen from t h e
d e t a i l s t o follow, t h e r e g u l a r s t r e s s f i e l d i n an
element s a ~ , ~ , , , s c& l ~ t = i L ~ ~= ~r a c t l o
r enc i p r ~ c i ~ y
conditionapriori, butnot the interelwent trfction
are
r e c i p r o c i t y condition. On t h e other hand,
1:
..
ES
57
*
I t a
' 1 -.--12,ci LU L u J L A
$,L-L,.
within an element; and the i n t e r l a y e r reciprocity
condition f o r t r a c t i o n 6 corresponding t o the singul a r s t r e s s f i e l d i s then s t i s f l e d by exactly
matching t h e parameters
a t the interlayer interf c e s , afi shown i n t h e following. I n equation (3)
a r e i n t e r p o l a t i o n functione a t t h e boundaries of
av; such t h a t t h e boundary displacement f i e l d
is
uniquely i n t e r p o l a t e d i n terms of generalized nodal
displacements CJ~. F i n a l l y i n Eq. (4) , t h e Lagrange
which a r e used t o enforce the tracMultipliers
t i o n boundary conditions (such as on t h e crack
f a c e ) , at3 a c c u r a t e l y as desired, contain displacement v a r i a b l e t h a t a r e not only i n t e r p o l a n t s from
t h e respective nodal d i splacement v a r i a b l e s , but
a l s o a d d i t i o n a l p o i n t wise Lagrange M u l t i p l i e r s ,
f o r reasons discussed i n d e t a i l i n [7].
r,
{,I,
r
,
.&bJ
2
&,B
~
Q
L
~
U
~
LP ~
In
L S
the t i n l c e element comprising of
a l l t h e layers.
Further, --e assume t h a t cS3 11 0.
The inplane s t r e s s e s G&i
a r e d e r i v e d from e R
QB as:
within t h e i t h layer
ui
$
Even though Eqs. (2-5) were w r i t t e n i n t h e i r
most general form, we now note c e r t a i n s p e c i f i c
s i m p l i f i c a t i o n s : (i) t h e type 1 (Fig. 1) ltregularll
elements, away from the crack-front, t h e s t r e s s
f i e l d can be expected t o be f a i r l y smooth, and t h a t
t r a c t i o n boundary conditions do n o t play f c r i t i c a l
r o l e ; hence f o r t h e s e elements, we t a k e Bs = 0 and
$I = 0; ( i i ) f o r type 2 (Fig. 1) t*singularllelements, t h e most general assumptions a s i n Eqs. (2-5)
a r e u s ~ d ,and s p e c i f i c a l l y f o r a s t r e s s - f r e e crack
= 0; ( i i f ) f o r type 3 (Fig. 1) 11singular48
face,
elements which do not share the s t r e s s - f r e e crack
face, the a d d i t i o n a l Lagrange m u l t i p l i e r s +JI. i as i n
Eq. (4) a r e removed, i . e . , $I = 0 .
i
where E,gyb i s the general anisotropic e l a s t i c i t y
t e n s o r , i n element coordinates, corresponding t o
inplane s t r e s s e s , f o r t h e i t h 1z;er.
It i s clear
from Eq. (8) t h a t , i n general, 0% i s discontinuous a t an i n t e r l a y e r i n t e r f a c e , and t h i s i s perm i s s i b l e i n s p i r e of the requirement of i n t e r l a y e r
t r a c t i o n r e c i p r o c i t y . The transverse shear and
normal s t r e s s e s , ua3 and a33, respectively, are
obtained by i n t e g r a t i n g t h e equilibrium equations
(ignoring body f o r c e s , f o r the present) as:
and
S u b s t i t u t i n g Eq. (8) i n t o Eqs. (9,10), one obtains
xi
With t h e above assumptions, t h e development of
a "multi-layer f i n i t e element" s t i f f n e s s matrix
follows the f a i r l y standard procedure so d e t a i l e d
f o r instance i n ( 4 , 6 , 7 , 8 1 , and these d e t a i l s a r e
omitted here f o r b r e v i t y . However, s i n c e the crux
of t h e present problem, of a p p l i c a t i o n of the
hybrid-stress f i n i t e element method t o cracked
laminates, l i e s i n a judicious choice (with a view
towards computational economy) of t h e f i e l d variables, a s symbolically expressed i n Eqs. (2-5), the
d e t a i l s of t h e s p e c i f i c choices made i n the present
work are given below:
-Field
Variables f o r Regular (Type 1) Elements
Consider Xa (a = 1,2) t o be t h e coordinates i n
t h e plane of t h e laminate and Xg be t h e thickness
coordinate. For t h e so-called r e g u l a r elements, we
s t a r t with t h e assumption f o r t h e inplane s t r a i n s
B&
(wherein t h e s u p e r s c r i p t R denotes "regular") i n
t h e e n t i r e s t a c k of l a y e r s i n each f i n i t e element,
as:
Thus, t h e inplane s t r a i n s vary c u b i c a l l y i n the
thickness d i r e c t i o n of the Laminate, which variat i o n i s an extension much beyond t h e c l a s s i c a l
laminated p l a t e theory, and i s considered t o be an
adequate approximation f o r t h i c k p l a t e - l i k e s t r u c t u r e s . Further, each of the q u a n t i t i e s r (m) (m=O,
Q@
1. .3) i s assumed as :
.
s3 c6
where 5 = 54 = s7 = 1; S2 = 55 = xl;
=
f ~ 2 ;
~f a r e )unctions of X3 i n t h e i t h l a y e r ; and C? are
i n t e g r a t i o n constants. I t can be seen e a s i l y $hat
and 043 have a q u a r t i c v a r i a t i o n i n t h e thickn e s s a d i r e c t i o n . The constants of i n t e g r a t i o n ,
7;i=l,
k) can be so chosen, a p r i o r i ,
(j=1,
t h a t t h e t r a c t i o n r e c i p r o c i t y condition a t the
i n t e r l a y e r i n t e r f a c e s i s s a t i s f i e d exactly. Assumi n g t h a t the lamina a r e of constant thickness, and
t h a t a l l the i n t e r l a y e r i n t e r f a c e s a r e perpendicul a r t o t h e X3 coordinate, t h i s t r a c t i o n r e c i p r o c i t y
(i=1,2,3) where +
condition reduces t o o+ =
and
denote, a r b i t r a r f ? y , e i t h e r s i d e of t h e i n t e r l a y e r i n t e r f a c e . W e assume t h a t t h e applied tract i o n s on the bottom surface of the laminate, within
each f i n i t e element, can be expressed as:
...
C3
...
0i3
-
where A? ( j = l , . .7) a r e known constants. Thus, the
constanis of i n t e g r a t i o n i n the bottom-must layer
can be adjusted t o r e f l e c t the above known t r a c t i o n s on the bottom s u r f a c e of t h e laminate. Thus,
t h r o ~ g hs t r a i g h t -forward algebra, t h e s t r e s s f i e l d ,
which s a t i s f i e s t h e conditions of i n t e r l a y e r t r a c t i o n r e c i p r o c i t y a s well a s the boundary conditions
on t h e bottom surfaces a p r i o r i , can be w r i t t e n ,
f o r each layer within each element, as:
(ml
vhere ijdy, a c b i i i ~ ~ ~ ~ e n i , i pl ;ae~dh r l l e i e r s Lor each
m = 0,1..3, and cq = 11, 22, and 12. As seen from
E q s . (6,7), t h e r e a r e a t o t a l of 72 undetermined
[ REPRODUC l B l L ITY."* - OF THE., OR l G INAL
-*C'Anhw.&A
*bc
;.pi"G"
&;rr*
PAGE I S- POOR
"-
T + .L,,..lf
hr - .
cllllC
LLe a t r C L V t ) m ~of
~~a
t
s t r e s s f i e l d for a layered element, analogous t o the
above, was a l s o used i n [9,101; however, s i n c e only
a l i n e a r v a r i e t ton of e a ~i n the x c o o r d i n a t e is
assumed i n [9,10], t h e r e s u l t s of 29,101 e s s e n t i a l l y reduce t o a c l a s s i c a l laminated p l a t e t h e o r y i n
c o n t s a s t t o the p r e s e n t development. Also, u n l i k e
i n [9,101, independent c r o s s - s e c t i o n a l r o t a t i o n s of
each lamina a r e allawed i n t h e present formulation.
Moreover, i n c o n t r a s t t o t h e formulatfon i n [ 2 3 ,
the p r e s e n t method s z t i s f i e s i n t e r l a y e r r e c i p r o c i t y
c o n d i t i o n s a p r i o r i , thus r e s u l t i n g i n computational
economy. S ince the s t r e s s e s a r e assumed independentl y i n each layer, t h e computer core-storage and exe c u t i o n - t ime requirements f o r the procedure i n 121
grow r a p i d l y with t h e i n c r e a s i n g number of l a y e r s
i n t h e laminate; i n c o n t r a s t , i n the p r e s e n t proced u r e , the s t r e s s parameters a r e common f o r the e n t i r e
laminate
9
L L .A
\
and x ( ~ a) r e the thickness co3
o r d i n a t e s of the bottom and top s u r f a c e s , r e s p e c t i v e l y , of the f t h l a y e r (See Fig. 3). I t is seen from
Eqs. (15, 9, 10, and 16-19) t h a t t h e undetermined
parameters involved i n the three-dimensional s t r e s s
f i e l d of each l a y e r a r e ~ $ 1 of Eq. ( 7 ) , which, a s
mentioned before, a r e common t o a l l l a y e r s w i t h i n
a f i n i t e element. Thus, f i n a l l y , t h e s t q e s s f i e l d
f o r r e g u l a r elements can be w r i t t e n a s 2' = pi&.
I n the above,
('-')
x3
fi:
The topology of a type 1 "regular" element is
shown in Fig. 2 . The boundary displacement f i e l d
f o r t h i s r e g u l a r element i s assumed t o be r e g u l a r
pol ynomia 1s , i n the boundary coordinates ; and these
boundary displacements a r e expressed unique l y interms of the r e s p e c t i v e nodal displacements. Thus,
f o r instance, along t h e s i d e A-B-C i n F i g . 2 , the
boundary displacements a r e assumed a s :
.
F i n a l l y , the embedding of s t r a i n and s t r a i n
s i n g u l a r i t i e s near t h e c r a c k f r o n t , i n each a n i s o t r o p i c lamina, i n c l u d i n g the transvcr s e normal s t r e s s
s i n g u l a r i t i e s , a s discussed below, have been
considered here f o r t h e f i r s t time.
Field Variables f o r S i n g u l a r
(Types 2 and 3, Fig. 1) Elements
In the f i r s t p l a c e , we note t h a t the f i e l d
v a r i a b l e s f o r the "singular" elements a r e assumed
i n t h e i r most genera 1 form a s given i n Eqs ( 2 - 5 ) ,
i e . , t h e assumed s t r e s s f i e l d f o r these elementsc o n s i s t s o f both r e g u l a r gi&)
and s i n g u l a r ( %')
terms. The regular v a r i a t i o n s
are i d e n k 8
c a l to those given i n Eqs. (15-lq)
Thus, i t is t o
the assumed e q u i l i b r a t e d s i n g u l a r (l/fi)
type s t r e s s
f i e l d i n s i n g u l a r elements, t h a t a t t e n t i o n is focused i n the following. We a l s o note t h a t e a r l i e r
s t u d i e s ell] of the cracked laminate problem sugges ted t h a t a 1/JF s t r e s s s i n g u l a r i t y w i l l be maint a i n e d f o r the inplane s t r e s s e s a& while t h e s t r e s s
a i 3 , i n regions away from f r e e s u r f a c e s (such a s
upper and lower s u r f a c e s of the laminate which a r e
normal t o the crack f r o n t ) and i n t e r l a y e r i n t e r f a c e s ,
i s given by a p l a n e - s t r a i n condition.
(However, the
s t u d y i n [ll] is l i m i t e d t o laminates wherein each
l a y e r i s a n i s o t r o p i c medium). F u r t h e r , t h e inf luence of the f r e e s u r f a c e s on the s t r e s s - i n t e n s i t y
factors for
f u r t h e r complicates t h e problem.
To circumvent t h i s problem, and t o f a c i l i t a t e t h e
a c c u r a t e enforcement of s t r e s s - f r e e c o n d i t i o n s (u3i
= 0 , i = 1,...3) a t t h e bottom and top s u r f a c e s of
t h e laminate a t the p o i n t of t h e i r i n t e r - s e c t i o n
w i t h t h e crack-front i n types 2 and 3 f i n i t e elements
( s e e F i g . I ) , we p r e s e n t h e r e a simple approach
which ignores, a p r i o r i , t h e p l a v e - s t r a i n n a t u r e of
t h e dependence of the s i n g u l a r
on the s i n g u l a r
a,
= 1 , 2 ) . I n t h i s process, we g e n e r a l i z e t h e
f a m i l i a r metals-based concepts of modes I , 11 and
I11 s t r e s s - i n t e n s i t y f a c t o r s , and introduce s e v e r a l
such i n t e n s i t y f a c t o r s i n each of the s t r e s s compone n t s . For conceptual c l a r i t y , l e t the s i n g u l a r
s t r e s s f i e l d i n types 2 and 3 elements be represented
by 9
.
(zig
.
i
where, % (01=1,2) a r e i n p l a n e displacements, uj t h e
t r a n s v e r s e displacement, a t the boundary of the i
t h l a y e r ; the s u p e r s c r i p t s ( i - 1 ) and ( i ) denote the
bottom and top s u r f a c e s of the l a y e r and - I 5 5 , 5
5 1 a r e non-dimensional coordinates a t t h e boundary
segment A-B-C, a s i n d i c a t e d i n Fig. 2 . S i m i l a r d i s placement f i e l d s a r e assumed a t the o t h e r boundary
segments. I t is s e e n from Eq. ( 2 0 ) t h a t t h e inplane
displacements vary l i n e a r l y i n the t h i c k n e s s coord i n a t e (5) of each l a y e r , thus allowing independent
c r o s s - s e c t i o n a l r o t a t i o n s of each l a -y e r ;- whereas
the transverse displacement u3 i s cons t a n t throughout each layer a s w e l l a s through the t h i c k n e s s of
the e n t i r e laminate. This assumption f o r u3 i s thus
consis t e n t w i t h t h e assumption t h a t 6 3 3 2 0. Den o t i n g by a , b, and c , the number of parameters a,
t h e number o f element nodal displacements q , and
t h e number of r i g i d body modes of the element, res p e c t i v e l y , i t is w e l l known from t h e theory of
hybrid-s t r e s s f i n i t e elements [ 7 , 8 j t h a t these parameters must obey the c o n s t r a i n t , b I a + c , i n order
t o avoid spurious kinema t i c modes of t h e element.
The number of nodal displacements corresponding t o
assumptions of the type given i n Eq. (20) and (21)
can be seen t o be, b = 8(k+l) + 4 f o r a 4-noded ( i n
t h e plan form) element , whereas, b = 16 (k+l) + 8 For
a n 8-noded ( i n the plan form) element, where k =
number of layers i n the element. I f t h e number of
A I s - i~ , - - 1
: l..-p
---s.
CJ
aficu a
ez :I: L y . \ t j , j. L
i s seen t h a t the above i n e q u a l i t y can be s a t i s f i e d
f o r z 4-noded element cons i s t ing of upto 8 l a y e r s .
J
013
can be c a l l e d the vector of " s t r e s s - i n t e n for the i th layer.
.
. , %-?$ - - j , kc s c y i ~ n a~r ~ polar-coordlnace
d
i
system centered a t t h e c r a c k - f r o n t , a s shown i n F i g .
I. F i r s t , we note t h a t t h e e q u i l i b r a t e d s i n g u l a r
- ,,
-
I L.rC.34
-,
,
-...IAYfy t h e equiLfLi-tuz
e q u a t i o n s , i n t h e absence of body f o r c e s , as :
a
A
- &
(a::) ,, = 0 and
.ILC.L.
02
..,k)
is
(m,na1 ,2,3 ;is1 ,
= gm.
(23a ,b)
..k)
is
We s t a r t by assuming u33 i n each l a y e r (i.1,.
as:
3
~ t ~ rcos, )( 8 1 2 )
is = a33 ,/K
'.
+
(~4
fi
K;
The p a r t i c u l a r s o l u t i o n s of Eqs. (31 and 32), respect i v e l y , can be obtained b y s e t t i n g ,
s i n '(8/2)
,
(24)
where Z
XI + i X 2 = reie.
Note t h a t K g and Kg a r e
func r i m s of X3. The t h i r d of the e q u i l i b r i u m equations, viz.,
is
= ai s h =i;p where,
3a!
3~
3a
the a d d i t i o n a l s u p e r s c r i p t s h and p i n d i c a t e t h e
homogeneous and p a r t i c u l a r s o l u t i o n s , r e s p e c t i v e l y .
The p a r t i c u l a r s o l u t i o n of Eq. (25) can be obtained
be s e t t i n g ,
is then solved by s e t t i n g 0
+
and
oisp = 0
12
we n o t e t h a t on the crack face, (which l i e s along x l
axis)
(and hence, i n the present procedure,
= + n ) . Upon u s i n g t h i s
o.$fP) must vanish ( a t
boundary c o n d i t i o n , and upon s u b s t i t u t i o n f o r a i s
from Eqs ( 2 8 and 30) i n t o Eq. ( 3 4 ) , we o b t a i n 2 3
aiz
.
In the p r e s e n t c a s e , the crack is assumed t o be pres e n t along t h e x l a x i s , w i t h t h c~r a c k f a c e perpend i c u l a r t o t h e x2 a x i s . Thus
must vanish a t 0
(5)i n the type 2 element (Fig. 1 ) . Noting t h i s
f a c t , the s o l u t i o n s t o Eq. (26) can be w r i t t e n as:
~ $ 9
where,
=
s t i t u t i n g from Eqs.
obtain,
and
,./2no:ip=~
4 ,3
=
and
-
1
i
(cos8uj,sine)
- K ~ ( X)
%j==f
i Reii:~
+ s 3iX 2 = r ( c o s g + si3 s i n 8 ) ; and S3i i s a
1
complex number, depending on the a n i s o t r o p i c e l a s t i c
compliance m a t r i x components C i j a s given i n [12].
Combinjng Eqs. (27 and 29), and 4q. (24,30), respeclSP
+ a$sh,
t i v e l y , t h e s i n g u l a r s o l u t ion. ats = =3Q4
rv
-,I
where, Z 3 = Y
....
cau~lsReC.
-,
i
etc.
A l s o , upon sub-
( 2 7 and 29) i n t o E q . (33), w e
I( r ) l i ( - s i n ~ / 2 ) + ~xlll )+Kg , 3 ( r ) f (-cos8 /2)
= a [ ~ 4 ( ~ 3 ) ] / a ~ e3t,c . F o r the homogewhere, K
4,3
neous s o l u t i o n o f Eq. (25), we take the asymptotic
two-dimensional a n t i-plane s h e a r s o l u t i o n of a
cracked a n i s o t r o p i c s o l i d , given i n S i h and Liebowitz l121, a s :
...
~ 2 b,2 ,L K~(x3)l/ax3,
~ 2
We now c o n s i d e r tire f i r s t two e q u i l i b r i u m equations,
For t h e homogeneous s o l u t i o n of E q s . (31,321 w e take
asymptotic two-dimensional s o l u t i o n of a cracked
as :
a n i s o t r o p i c s o l i d LIZ],
r . l c c s ox laminate, tor
:''
are satisfied exactly,
a p r i o r i , as described below.
Each of $he s t r e s s i n t e n s i t y f a c t o r s w i t h i n
each l a y e r , K ( i a 1,. .K; p " 1 , 2 , , .5) is i n t e r po l a ted us i n $ ~ e r m i t i a o po lynomia 1s as be low :
.
.
I t is t o be noted t h a t t h e above homogeneous solut i o n s i d e n t i c a l l y s a t i s f the crack f a c e t r a c t i o n f r e e conditions, v i z . , orsh = a i s h = 0 , Upon comb i n i n g Eqs (37 and 38) ,2&s.
and 3 9 ) , and Eqs
(35 and 4O), r e s p e c t i v e l y we o b t a i n t h e r e q u i r e d
solutions,
.
(4g
.
Thus the r e q u i r e d , s e l f equi!ibra t e d , s i n g u l a r s t r e s s
f i e l d is now e s t a b l i s h e d : 01 a s i n Eq. (24) ;
a s t h e sum of
a s t h e rum of Eqs. (27 ?nd 247;
= (a,g= , 2 ) a s through Eq.
Eqs. (28 and 3 0 ) ;
(41). We f u r t h e r
a t the s t r e s s - i n t e n s i t y
parameters K
!,
. ~ af r e , a t the moment, independent
f o r each layer, and, more over, vary w i t h the thickness coordinate, X3, i n each l a y e r .
..
Also, i t is important to note t h a t t h e above
K; m,n = 1
3) identically
1
derived o i g ( i
s a t i s f i e s , a p r i o r i , t h e t r a c t i o n bvundary condi1S
0 ( i = 1,.
t ions on the c r a c k s u r f a c e , v i z , 02,
K; rn = 1,2,3).. A s mentioned e a r l i e r , t h i s s i n g u l a r
s t r e s s f i e l d gls i s augmented, i n type 2 and type 3
elements (as shown jn Fig. l ) , by t h e r e g u l a r polyThis qegular polynomial
nomial functions,
s t r e s s f i e l d does Got, i n g e n e r a l , s a t i s f y t h e
= 0,
crack-face s t r e s s f r e e conditions ( t h a t
i = 1
K; m = 1,..3), i n type 2 elements. S i n c e
an a c c u r a t e s a t i s f a c t i o n o f s t r e s s - f r e e c o n d i t i o n s
on the crack f a c e is important t o o b t a i n a c c u r a t e
e s t i m a t e s of s t r e s s ; i n t e n s i t y f a c t o r s , t h e r e g u l a r
( i n t h e term
of Eq. (2)).
polynomial s t r e s s
a r e enforced t o vanish on the c r a c k 2 G c e i n type 2
e lemen ts through a co 1l o c a t i o n technique a s d e t a i l e d
i n Ref. [71. The Lagrange M u l t i p l i e r s introduced,
i n t h i s p o i n t - c o l l o c a t i o n technique, (which a r e
h ~ n c ec a l l e d point-wise Lagrange M u l t i p l i e r s ) , a r e
a s i n Eq. (4). By i n c r e a s i n g t h e number of coll o c a t i o n pojnts on t h e c r a c k face (and hence the
number of -1) i n type 2 elements a t which s t r e s s "e a r e enforced e x a c t l y , a high-degree
f r e e conditions
of accuracy can be achieved, i n g e n e r a l , a s shown
fran the r e s u l t s of the present paper a s w e l l a s i n
Ref. [73.
,...
.
,...
..
xl&.
02,
,...
~$2
zip
&
I
I t remains t o enforce: ( i ) i n t e r l a y e r t r a c t i o n
r e c i p r o c i t y conditions f o r the s i n g u l a r p a r t , b ls,
of t h e s t r e s s f i e l d ; ( i i ) t h e t r a c t i o n boundary
conditions a t the top and bottom s u r f a c e s of the
laminate f o r the s i n g y l a r p a r t of t h e s t r e s s f i e l d
( i . e . conditions on ffiS
i = 1 and K; m = l , . . 3 ,
including zero c o n d i t 3m'
r o n s ) ; and (iil) t h e conditions
a t the to^ s u r f a c e of t h e l f ~ i n a t efor t h e r e g u l a r
part of the s t r e s s f i e l d ( ~ 7 i~ =~ K' , and m = 1 , 2 , 3 ) .
The conditions u l r a t the top s u r f a c e ( i = K) of the
laminate a r e allawed t o follow a s natured boundary
c o n d i t i o n s from t h e varga t i o n a l p r i n r i n l ~ ZC;;;,;,,,.urias LU v a r r a r I o n s tp,
cf Eq. (L)
Uowever the ucr
c o n d i t i o n s of
interlayer tracrion reciprocity, as
w e l l a s boundary conditions a t t o p and bottom s u r 9.'
.
w h y K ( ~ end
)
K (i'l)are,
respcc t i v e l y , t l u values
of K otPtheitop akdibot tom s u r f a c e s of t h e i t h
The well-known
lay&, and K , = a [ ~ ]/ax,, e t c .
Hermite int&#mlatespare given h e r e , f o r conven ience, a s :
r
(iwhere t i s defined a s t ' L x 3 - ' 3
0 5 t 5 1. These values a r e shown i n
c l a r i t y , where hi is t h e thickness of
layer.
i / his
;
F i g . 3, f o r
the i t h
Now i t is c l e a r , assuming t h a t the lamina
a r e of constant thickness and t h a t X i s perpend i c u l a r t o the i n t e r f a c e s , t h a t the ? n t e r l a y e r
t r a c t i o n r e c i p r o c i t y c o n d i t i o n ( i .e , c o n t i n u i t y
of
a t the i n t e r f a c e ) f o r t h e s i n g u l a r ( l / r j r )
.
component of the s t r e s s f i e l d canibe ~ a s i l ys a t i s e s (K
K and )'K
f i e d ~ f q ; ~ $ t n g K ~ Q ~ ) v a l uof
and ( 3 s
4 s
tj
a t t h e corn& i h t e r f a c a .
Thus, one has unique v a l u e s of s t r e s s - i n t e n s i t y
, K4, and K a t i n t e r l a y e r i n t e r f a c e s ,
factors t e s e
whereas
vary q u a d r a t i c a l l y w i t h i n
each l a y e r . However, t h e s t r e s s - i n t e n s i t y f a c t o r s
K1 and K2, while a l s o v a r y i n g q u a d r i t i c a l l y w i t h i n
each l a y e r , are a 1lowed t o be discontinuous a t t h e
interfaces.
%
quantities
Now, assuming t h a t one of s u r f a c e s , say the
bottom s u r f a c e , which t h e crack f r o n t i n t e r s e c t s ,
i s s t r e s s f r e e (s eFigs. 1 and 3 ) , t h i s s t r e s s
f r e e c o n d i t i o n a(')'=
0 (m = 1 2 3) can be e a s i l y
( 0 ) = 0.0.
= Kg,)
s a t i s f i e d by s e t t E g K(!)=
K(:J='Kiyi
Thus, s p e c i f i c a l l y , the vanishing of a
a t the
33
bottom s u r f a c e has n o t a f f e c t e d the s t r e s s - i n t e m i t y
f a c t o r s K and K i n all and u *, s i n c e , i n t h e
1
2
p r e s e n t procedure, t h e assumptton of plane-s t r a i n
f o r s i n g u l a r s t r e s s f i e l d s has been removed; and
i n t h i s process, d i f f e r e n t s t r e s s - i n t e n s i t y f a c t o r s
K1.. .K f o r d i f f e r e n t components of s t r e s s have
5
have introduced.
F i n a l l y , we comment t h a t the s i n g u l a r a s w e l l
a s r e g u l a r s t r e s s - f i e l d s introduced i n t h e above,
w h i l e being s e l f - e q u i l i b r a t e d , do not a p r i o r i ,
This
correspond t o compatible s t r a i n f i e l d s
c o m p a t i b i l i t y , however, i s an a p o s t e r i o r i condit i o n of t h e v a r i a t i o n a l p r i n c i p l e i n E q . (1).
M reover, 3 ince the stress-iiitensiiy parameters
K'z'and
t ( i 3 1~ = 1,..5;
i = 0
K] which form
S
t h e v e c t o r & , a r e d i r e c t v a r i a b l e s i n t h e complementary energy f u n c t i o n a l of Eq .' (1) : t h e s e
.
,...
1
REPRODUCIBILITY OF THE ORIGINAL PAGE I S
POOR
t
c a n oe computed d i r e c t l y from t h e f i n i t e element
solution.
Thus t h e p r e s e n t p r o c e d u r e , w h i l e not o n l y
y i e l d i n g a c c u r a t e s o l u t i o n s t o cracked laminates,
a l s o l e a d s t o a d L r e c t e v a l u a t i o n o f v a r i o u s stressi n t e n s f t y f a c t o r s and t h e i r v a r i a t i o n through t h e
t h i c k n e s s , i . e , K~ (Xj). (a = 1.. .5;
i = 0 , . .K)
i n the v a r i o u s strgss components. A l s o , i t is w o r t h
emphas i z i n g t h a t t h e p r e s c k i b e d ( f n c l u d i n g z e r o )
c o n d i t i o n s for t r a c t i o n s n o t o n l y on t h e c r a c k f a c e ,
b u t a l s o on s u r f a c e s w i t h which t h e c r a c k - f r o n t
i n t e r s e c t s , p a r t i c u l a r l y i n the v i c i n i t y of t h e
crack-front, a r e s a t i s f i e d exactly.
.
R e f e r r i n g t o Fig. 4
d a r y s u r f a c e s of types 2
dary displacement f i e l d ,
a s follows. For faces 1
and f o r f a c e s 1 and 4 of
f i e l d is:
.
f o r t h e numbering o f bounand 3 e l e m e n t s , the bouno f Eg. ( 3 ) , i s assumed
and 2 of- t y p e 2 element
type 3 e l e m e n t , the assumed
LIIEDC
~ ~ i r e g r a ni sl g h l y a c c u r a t e l y . Tnc .
d e t a i l s of t h e s e r u l e s a r c g i v e n i n [ 137, wherein
f u r t h e r mathematical d e t a i l s of t h e p r e s e n t proc e d u r e a r e more f u l l y e l a b o r a t e d upon,
evaluate
R e s u l t s and D i s c u s s i o n
(i)
Bending o f a ( 0 ~ / 9 0 ~ / 0Laminate:
~)
The problem c o n s i d e r e d is t h a t o f bending of
a simply suppor t c d , t h r e e - l a y e r (0°/900/0 ) , rect a n g u l a r (2W x 2L) l a m i n a t e , under a t r a n s v e r s e
s i n (nX 1 2 ~ )s i n (nX2/2L) w i t h t h e
l o a d [q =
o r i g i n , X1 q0 X2 = 0 & e i n g l o c a t e d a t t h e lower l e f t
c o r n e r of t h e r e c t a n g l e 1 a p p l i e d a t the t o p s u r f a c e
of t h e p l a t e , f o r which an e x a c t 3-D s o l u t i o n is
a v a i l a b l e [5]
The p r o p e r t i e s o f each lamina a r c
( i n t h e laming p r i n c i p a 1 m a t e r i a l d i r e c t i o n s ) :
E l l = 25 x 10 p s i ; E2* = lo6 p s i ; E
= lo6 p s i ;
G12 =
x 105 p s i ; GZ3 = 2 x lo5 ps2? and G31 =
5 x 10 p s i ; a n d y
= p 1 3 = U 2 3 = 0.25.
The
problem was s o l v e d l % o r v a r i o u s v a l u e s o f t h e lamin a t e tf - c k n e s s p a r a m e t e r , S = (2W/h), where h i s
t h e laminate- t h i c k n e s s . Because o f t h e geometric ,
m a t r i c a l and l o a d i n g symmetries, o n l y a q u a r t e r of
t h e p l a t e was modeled, w i t h (N X M ) f i n i t e elements
( i . e . , N and M e l e m e n t s , r e s p e c t i v e l y a l o n g t h e
l e n g t h and w i d t h d i r e c t i o n s )
.
:
.
In t h e above, r is the d i s t a n c e perpendicular
t o the c r a c k f r o n t , i . e . , r = I x ( on f a c e 1 and
r a 1x21 on f a c e 2 o f type 2 e d e n t . On f a c e s 5
and 6 of t y p e 2 e lemen t , which a r e perpendicu l a r
t o t h e c r a c k f r o n t , t h e d i s p l a c e m e n t f i e l d i s assumed
as:
The convergence s t u d i e s f o r v a r i o u s values o f
S , N and M a r e i n c l u d e d i n [13], b u t a r e n o t given
h e r e due t o s p a c e r e a s o n s ; however, t h e s e r e s u l t s
i n d i c a t e t h a t t h e p r e s e n t method converges e x c e l l e n t l y
f o r both t h i c k (Sc 4 ) a s w e l l a s t h i n S 2 10) lamin a t e s . A l s o , t h e r a t e s of convergence were observed
t o be f a s t e r t h a n t h o s e i n d i c a t e d i n [2]. W e pres e n t h e r e o n l y t h e r e s u l t s f o r a t h i c k , s q u a r e lamin a t e w i t h S = (2W/h) = 4.0; L = W ; and when the l a m i n a t e was modeled by a (4 x 4 ) f i n i t e element mesh
(we n o t e once a g a i n t h a n each f i n i t e element cons i s t s
o f t h e e n t i r e s t a c k o f lamina).
The v a r i a t i o n o f 0
w i t h t h e t h i c k n e s s coordin a t e (X3) a t t h e edge o i l t h e p l a t e (X1=O; XZ=L)
i s shown i n F i g . 5 a l o n g w i t h t h e e x a c t s o l u t i o n .
The v a r i a t i o n o f U13 w i t h X 3 a t t h e l o c a t i o n (X1 =
L; X = L) is shown i n F i g . 6 a l o n g w i t h t h e e x a c t
I n Eq. (46) r = [ ( x ~ )+~ (X,)*lf: and 9 i s t h e
w i t h Xj
s o l u t ~ o n . Likewise t h e v a r i a t i o n of G
a n g l e from c r a c k - a x i s , a s i n ~ i 4.~ Noting
t
that
= X2 = ( 7 / 8 ) ~ j i s shown i n F i g . 7?3 The v a r l a at
f o r t h e p r e s e n t f i n i t e element, t h e r e a r e 8 nodes
a t t h e t o p s u r f a c e of t h e
t i o n o f t h e computed Q
a t each i n t e r l a y e r s u r f a c e , t h e above c o n s t a n t s
p l a t e , a l o n g t h e X1 coo!ainate,
is shown i n F i g . 8
alcu"a6cu; a 7. .ag, and b lm..b8m a r e e x p r e s s e d i n terma
f o r X 2' ( 7 / 8 ) L , and t h i s v a r i a t i o n is s e e n t o a g r e e
o f t h e y e t unknciwn n o d e l d i s p l a c e m e n t s . L a s t l y ,
e x c e l l e n t l y w i t h t h e applied s t r e s s q ; thus indic a t i n g t h a t t h e s a t i s f a c t i o n of t r a c t i o n boundary
t h e d i s p l a c e m e n t s a t f a c e s 3 and 4 o f t h e type 2
c o n d i t i o n s is b e i n g accompl i s h e d e x c e 1l e n t l y i n the
element a r e a s s u n e d t o be o f t h e same form a s i n
p r e s e n t assumed s t r e s s f i n i t e element p r o c e d u r e .
Eqs (20,21) s u c h t h a t they a r e c o m p a t i b l e w i t h t h e
F i n a l l y , the thickness-variation of inplane d i s b o u n d a r y ' d i s p l a c e m e n t s o f t h e n e i g h b o u r i n g elements
placements a t (X = 0 and X 2 = L) i s shown i n F i g .
9, from which
c a n be s e e n t h a t t h e p r e s e n t bounF i n a l l y , we n o t e t h a t i n t h e p r o c e s s of development of t h e e l e m e n t s t i f f n e s s m a t r i x based on Eq. ( I ) , d a r y - d i s p l a c e m e n t a s s u m p t i o n s , which a l l o w f o r t h e
1s
independent c r o s s - s e c t i o n a l r o t a t i o n s o f each l a y e r ,
s i n c e t h e assumed s i n g u l a r 5 i s o n l y e q u i l i b r a t e d
y i e l d r e s u l t s i n e x c e l l e n t accord w i t h t h e e x a c t
b u t doesn't correspond t o compatible s t r a i n f i e l d ,
3-D e l a s t i c i t y t h e o r y . The above r e s u l t s may i l l u s a p r i o r i , i t becomes n e c e s s a r y t o numer;.cally e v a l t r a t e the a c c u r a c y and e f f i c i e n c y o f t h e p r e s e n t
u a t e i n t e g r a l s o f t h e type:
method f o r an a n a l y s i s of t h e t h r e e - d i m e n s i o n a l
s t r e s s s t a t e i n uncracked l a m i n a t e s under g e n e r a l
1oad i n g
.
.
if
.
i n a r e c t a n g u l a r domain 0 iX,<a;OsX,Sb; where
w e FO l a r coordinate<; ccvtcrcd I t X - X, = [ ) >
and where f (r,Q) c o n t a - h s a n r-" t y p e s i n g u f a r i t y
(0
cu % 1). S p e c i a l q u x d r a t u r e r u l e s have been
developed i n t h e c o u r s e of t h e p r e s e n t work t o
r ,a
The geometry of t h e cracked l a m i n a t e i s shown
i n F i g . 10 ; w i t h (L/w) = 1.0; ( a h ) = 0.2; ( h , / ~ ) '
f117nn?: h
,,.
,
*Lo; z
*
c
l
"
,.,
PLY
P'
C
,,,,,,,
pr r n c t p a l material d i r e c t i o n s ) which
a r e t y p i c a l of a medi urn modulus gr p h i t e / (epoxy,
a r e chosen t o be: Ell r 18.25 x 10 p s i ; E 2 p "
1 . 5 x lo6 p s i ; E33 = 1.5 X lo6 p s i ; G12
G13
G23
= 0.95 x lo6 psf; ~ 1 =2 v23
v y j = 0.24. Note t h e
coordinate system (XI, X 2 , X j ) , as used i n t h e pres e n t paper, which i s shown i n Fig. LO.
-
t
-
The 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
A uniformly d i s t r i b u t e d s t r e s s sp
t h e one i n [3].
i s applied a t the b m n d a r i e s , XL = +_ L; and t h e top
and bottom surfaces of t h e laminate (Xg = 0 , and
4h0) as well a s t h e crack-face, a r e assumed t o be
t rac t ion- f ree.
Because of t h e appropriate geometric, m a t e r i a l ,
and loading symmetries only a quarter of t h e p l a t e
bounded by -a S X i S (W
a) ; 0 4 Xg S L ; and 2h0
S X3 S 4h0, need t o be modeled.
It i s noted t h a t
i n Ref. [3], the s o l u t i o n i s obtained i n two stages;
one with a coarse mesh and the second with a very
mesh f o r a substructure near t h e crack f r o n t . T ~ U S ,
i n Ref. 121, the coarse mesh consisted of (9 x 18
x 2) elements ( i . e . , 9 , 18, an? 2 elements i n X2,
XI, and X3 d i r e c t i o n s , respectively) i n t h e quarterp l a t e with 570 nodes and 1710 degrees of freedom;
while the inner mesh consisted of a (11 x 19 x 4)
f i n i t e element mesh ( i n X2, X i , X3 d i r e c t i o n s , respectively) with 3600 degrees of freedom. I n cont r a s t , t h e persent s o l u t i o n i s obtained i n a s i n g l e
s t a g e using a (11 x 7 x 1 ) f i n i t e element mesh t i n
XI, Xz, and X3 d i r e c t i o n s r e s p e c t i v e l y ) , i n t h e
q u a r t e r p l a t e a s shown i n Fig. 11, with a t o t a l of
1562 degrees of freedom.
-
The values of t h e normalized s t r e s s i n t e n s i t y
f a c t o r s ~1 (which a r e d i r e c t l y computed i n t h e present procedure) and i t s v a r i a t i o n with X3 i s shown
i n Fig. 12. As can be expected, t h e r e i s a d i s c o n t i n u i t y i n K l value a t the i n t e r f a c e between the
O0 and 90° p l i e s , and moreover, the K 1 value i s
much higher i n the 90° p l y than i n t h e O0 ply. The
s t r e s s i n t e n s i t y f a c t o r Kq i n the t r a n s v e r s e normal
s t r e s s ojg, and i t s v a r i a t i o n with X3, i s shown i n
Fig. 13. Note t h a t K4 i s zero ( a c t u a l l y s e t t a
zero) a t the Xg = 0 , i s continuous a t t h e i n t e r f a c e
between o0 and 90° p l i e s , and i t ' s magnitude i s
much smaller than t h a t of K1. It is a l s o i n o t e d t h a t
i n t h e present example, t h e f a c t o r s K$, K3, and ICf
were found t o be n e a r l y zero, a s can be expected.
I f plane-strain conditions a r e assumed t o p r e v a i l i n
egch l a e r , i t can be shown t h a t i n each i t h l a y e r ,
= (K4/ci) h e r e Ci i s a material constant f o r the
general anisotropic medium. This m a t e r i a l constant
C i can be derived i n a s t r a i g h t forward manner f o r
an anisotropic medium (and i s equal t o 2v i n t h e
i s o t r o p i c case, v = Pois on r a t i ), and is given i n
(131. A comparfson of K1, and &Ci
i s shown f o r
each of the 0' and 90° l a y e r s i n Fig. 14, which sugg e s t s t h a t for t h i s t h i n laminate (W/h = 75; a/h =
15) , plane s t r a i n conditions a r e not a t t a i n e d even
a t the i n t e r i o r of t h e laminate, away from f r e e
s u r f a c e s , or away from t h e lamina i n t e r f a c e s . However, r e s u l t s f o r t h i c k i s o t r o i c and a n i s o t r o p i c
laminates (which a r e given i n fl31, but hor here,
due t o space reasons) do i n d i c a t e t h a t Kq and K1
meet the plane s t r a i n requirement a t t h e i n t e r i o r
of t h e laminate, away from f r e e surfaces. However,
the results for stress-intensity factor variations
a r e not given i n [3], because t h e nrocedu- *la-J
~i
r
B
-
-.
. *"bI .".. r r ~ u y l a n estre.8
(normal
t o crack-axis) , at t h e mid-plane of each p l y , along
x1 (near the crack f r o n t ) is shown i n Fig. 15, .low
w i t h comparison r e s u l t s i n [31, which ate noted t o
c o r r e l a t e excellently with t h e ptesent r e s u l t s .
Similar c o r r e l a t i o n between present r e s u l t s , f o r t h e
o t h e r inplane s t r e s s e s (not cthown here, f o r w a n t of
space) both a s functions of X1 and X q n e a r t h e
crack- f ront , and those i n [ 31 , was noted. The d i s t r i b u t i o n of interlaminar shear s t r e s s e s 013 (X3)
and oZ3(X3) along t h e thickness coordinate a t t h e
0.8 and Xl
01 a r e shown i n
l o c a t i o n [x2/hO)
Figs. 16 and 17 r e s p e c t i v e l y ; once again, good
c o r r e l a t i o n i s obtained between preaent r e s u l t s and
those i n [33. Fig. 18 shows t h e v a r i a t i o n of 033
a s a function of (x1/ h,) a t (Xg/ho)
1.0 ( i n t e r
f a c e ) ; while Fig. 19 shows 033 as a function of
t h e r a d i a l distance ( r / h ) along a d i r e c t i o n a t 45'
t o t h e crack-axis a t (x39hO) = 2.0 (midplane of
laminate). It should be remarked here t h a t i n t h e
present procedure, c.& has a (I/Jr) s i n u l a r i t y
b u i l t i n t o it, while t h e s o l u t i o n i n 137 i s based
on t h e use of regular polynomial s t r e s s f i e l d s .
F i n a l l y , the v a r i a t i o n of 033 with the t h i c k n e s s coordinate X
a t the l o c a t i o n along the crack a x i s ,
(xl/h0) = 8:8 a r e shown i n Fig. 20 along with comp a r i s o n r e s u l t s from [33, except a t (xl/h0)
0.75.
The present r e s u l t s were observed t o p r e d i c t a
higher value of 033, i n general, as seen f r m t h e
s p e c i f i c case i n Fig. 20; f u r t h e r , a s seen from
Fig. 18 the present r e s u l t s i n d i c a t e no s i g n r e v e r s a l i n a33 a t the i n t e r f a c e , a t short d i s t a n c e s
from t h e crack-tip, as i n [3].
-
-
-
-
-
Closure
Considering t h e f e a t u r e s : ( i ) t h a t t h e present
s o l u t i o n method leads t o a d i r e z t evaluation of
s t r e s s - i n t e n s i t y f a c t o r s (and t h e i r v a r i a t i o n i n
t h e laminate thickness d i r e c t i o n ) i n t h e t h r e e dimensional stres. f i e l d , a&, i n each i t h p l y and
(ii) i n t h e s p e c i f i c example t r e a t e d here, t h a t
accurate r e s u l t s f o r d e t a i l s of s t r e s s - f i e l d s a r e
obtained more economically ( i . e . , i n a one s t e p
s o l u t i a n Kith 1562 degrees of freedom ( d . 0 . f . ) i n
t h e present work, a s contrasted t o a two-step solut i o n with (1710 + 3600) d o. f i n previously reported [3] procedures); it appears t h a t t h e pres e n t l y reported I1multi-layer hybrid crack element1'
procedure o f f e r s a v i a b l e t o o l for s t r e s s a s w e l l
a s f r a c t u r e analyses of laminates. Moreover, t h e
procedure o f f e r s new and convenient ways f o r accounting f o r s t r e s s - s i n g u l a r i t i e s i n a l l t h e 6
s t r e s s components o&, i n each layer, t h e i r v a r i a t i o n through t h e thickness of each p l y , and t h e
e f f e c t s of f r e e surface (normal t o t h e crack-front)
on t h e s e s t r e s s - i n t e n s i t y f a c t o r variations. The
implication of the p r e s e n t r e s u l t s i n formulating
mechanisms f o r i n i t i a t i o n of f r a c t u r e , and subsequent s u b c r i t i c a l damage i n cracked laminates, i s
t h e o b j e c t of our work i n progress.
. .
Acknowledgement
This work was supported by the U. S. AFOSR
under c o n t r a c t No. F49620-78-C-0085 with t h e Georg i a I n s t i t u t e of Technology (G.I.T.) and by supplemental funds from G.I.T.
The authors g r a t e f u l l y
acknowledge t h i s support. The authors a l s o wish t o
thank D r . J. D. Morgan, 111 f o r h i s timely encour-
-___ .
-
-
-
--.o
-.-"
b U C L . Y L
W&
LLlLLL
WOfK.
-References
Wang, S. S., Mandell, J. F., and McGarry, F. J.,
Engineering Fracture Mechanics, Vol. 9, 1977,
p. 217.
Mau, S. T . , Tong, P., and P i a n , T. H. H,, J.
Composite M a t e r i A , Vol. 6, 1972, p. 304.
Wang, S. S., Mandell, J. F., and McGarry, F.,
J. Fracture Mechanics of Composites, AS% STP
593, 1975, p. 36.
Pian, T. H. H., AIAA Journal, Vol. 7, 1964,
p. 1333.
Pagano, N. J., and Mo., S. L., 3. Composite
Materials, Vol. 4, 1970, p. 330.
Atluri, S. N., in Advanced in Computer Methods
for Partial Differential Equations, (R.
Vishnevetsky, Ed.) AICP,, Rutgers University,
1975, p. 346.
Atluri, S. N., and Rhee, H. C., AIAA Journal,
Vol. 21, 1978, p. 529.
Pian, T. H. H. and Tong, P. in Advances in
_Applied Mechanics, Vol. 12 (C. S. Yih, Ed.)
Acad. Press, 1972, p. 2.
Deak, A. L. , and Atluri, S. N., in Computational Methods in Nonlinear Mechanics, ( J . T. Oden,
Ed.) University of Texas at Austin, TICOM,
1974, p. 79.
Spilker, R. L., Chou, S. C., and Orringer, O.,
J. Composf te Materials, Vol. 11, 1977, p. 5;,
Hilton, P. D., and Sih, G. C., in Fracture
Mechanics of Composites ASTM STP 593, 1975,
p. 3.
Sih, G. C . , and Libowitz, H., Chapter 2 in
Fracture, Vol. I1 (H. Liebowitz, Ed.) Academic
Press, N. P., 1968, p. 108.
Nishioka, T., and Atluri, S. N., "An Efficient
Assumed Stress Finite Element Procedure for
Fracture Analysis of Multilayer Anisotrop -c
Laminates ," Georgia Institute of Technology,
Technical Report (in preparation) 1979.
Fig.3
Nomenclature for Element Variables.
Fig,4 Singular Element Topology.
--OI
I
Fig. 1 Types of Finite Elements.
I
8
4r4 Mesh
-
4x4 Mesh
---'(I
Fig. 2
Element
Topology.
Fig.6
Transverse Shear Stress UI3 at (xl,x2)=(0,L)
F i g . 10 Geometry of Cracked Laminate.
Fig.7
Transverse Normal S t r e s s d
33
at (xI/w,x2/L)=(7/8,7/8).
Fig.11
F i n i t e Element Model of Cracked Laminate.
-4Calculated Stre8.g
A p p l ~ e dStresa
0
1.o
C.5
X'/w
Fig.8
Comparison of Calculated 0
( a t x3-h
33
and x2/L=7/8) at Various x L ~ c a t i o n sw i t h
1
Applied Transverse S t r e s s q ,
K, (X,) /(WiZl
Fig.12
V a r i a t i o n of K1 through the Thickness of
Laminate.
..-
CQT
1.0
WZwd.25
L-w
1
-1.0
4x4 Mesh
:
C-
1.0
-
u1
F i g . 13
Fig.9
Inplaoe Displacement u (x ) a t (xl,x2)=(0,L),
1 3
V a r i a t i o n of K 4 through t h e Thickness of
Laminate.
Fig. 14 Comparison of Actual and Nominal
Plane Strain Conditions.
m.0
(0.0
3
\
Fig.18
bd
Y
Variation of 033with x
1.o
0.1
Fig.15
Variation of CTZ2 (with x ) Near the Crack
Front
1
.
Fig. 16
Variation of 013 with xj a t (xl=O,x2-0. ah0).
Fig. li'
Variation of CT23 with x
3
-
a t ( x =0,x2=0.ah0)
1
--
I
.
! REPRODUCIBILITY
- ,. --r;7Pt-w--*,aap-
-s
I.
. ,-
-I?BV-"-..
w-
-- -
OF THE ORIGINAL PAGE I S POOR
1
a t (xfO,x
3
=h ) .
0