DELAMINATION
CRACK ORIGINATING
FROM
TRANSVERSE CRACKING IN CROSS-PLY
COMPOSITE LAMINATES UNDER EXTENSION
T.
Department
W. KIN
H. J. K~ac and S. lm
of Mechanical Engineering. Korea Advanced Institute of Science and Technology.
P.O. BOI 150. CheongRyang. Seoul. Korea
Abstract-Based
upon rhe Swoh formalism for anisotropic elastic materials and upon the method
of eipnfunction
expansion. the stress redistribution
due to delamination cracks originating from
transverse cracking is examined from [90,0]. and [O,YOj, lammates under extension. The structure
of the solution. in the form of a series expansion. is determined from the eigenvulue equation
resulting from appropriate near-field conditions. Tocomplete the solution. use is made ofa houndury
collocation tcchniquc in conjunction with the eipenfunction series that includes ;I large number of
terms. enough lo represent the elastic slate throughout the appropriate domain concerned. The
fracture mechanics parameters. such ;LS stress mtcnsity factcars and cncrgy relcasc rates. arc calculated
and the major characteristics of stress di\tnhution arc discusrcd. The stability of dcl;miinaticw
cracks is examined for varying ratios of ply thlckncv in terms of the cncrgy rcIc;wc rate.
I. INTRODUCIION
Fracture
analysis
of an anisotropic
conipositc
laminntc
has
gained
substantial
in
ittttXlti0n
association with the understanding of its fundamcntnl fracture behavior. which is of critical
importance for the design of composite structures.
delamination
In particular.
the last two dccadcs. For example. stress singularities
delamination
changes
works
composite laminates.
along this
delamination
trclttd
problems concerning transvcrsc
damage. including crack initiation
of structural
contributions
Various
and growth. stilfncss
reduction and other
properties. have been discussed by many rcscarchers;
line is too great to give individual
citation
cracks in
cracking and
herein.
the number of
Many
important
may be found in the thesis by Lim (1988). for example.
Recently. the asymptotic stress field near transvcrsc cracks occurring in the 90
cross-ply
during
wcrc lirst rcportcd by Ting and Hoang
(19X4) for transverse cracks, and Wang and Choi (IW3a)
anisotropic
transvcrsc cracking and
fracture have been among the subjects under intcnsivc investigation
laminates
was examined by Im (1989)
increases, the transverse cracks in the 90
and Im and Kim
ply of a cross-ply
(1989).
ply of
As the load
laminate that arc arrested at
the interface tend to kink into the delamination cracks along the ply interface (Lim,
1988).
The
in the
purpose of this work
is to obtain the stress
presence of such interfacial cracks originating
the fracture behavior, including the stability
redistribution
from transvcrsc
of delamination
under extension
cracking. and to examine
cracks and the inlluence of
geometric parameters. It is assumed that these interface cracks run through the width of a
laminate, with a uniform spacing and crack length. The asymptotic structure of the solution
is obtained. under the assumption
anisotropic
elasticity
of plane strain elasticity.
and the eigenfunction
expansion
from the Stroh
[see Ting
(1986)
formalism
of
or rcfcrences
thcrcin]; appropriate near-field conditions are imposed to Icad to the eigcnvalue equations,
which dctcrminc the structure
of the asymptotic solution,
including the stress singularity.
To complctc the solution, use is then made of a boundary collocation technique in conjunction with the eigenfunction series that includes a large number of terms. enough to
cover the elastic state of the far field as well as of the near field. The convergence behavior
of the solution. including the influence of the number of cigcnvalucs truncated in the
scrics solution and the number of collocation stations, is dcmonstratcd through numerical
cxamplcs and the role of the two translational planar rigid body modes, in association with
the solution accuracy, is critically examined. The characteristics of the singular stress field
near the crack tip arc brietly discussed for [90/O]. and [O/90],. respectively; under the given
T U
IV26
loading
of extension.
the delamination
while [0 901, laminates
delamination
intensity
crack
are found
proposed
crack
by Rice (1988).
the energy
crack
turns out to be opened
for [OO O]. laminates.
to ha\e ;1 full> clo~cd delamination
in [90 01, luminatcs
for an interfacial
KIV ,‘I d
uhich
hits an o\cillatorq
using the analog
is computed.
crack.
of linear
elastic
fracture
The elTect of ply thickness
of delamination
release rate. and the stability
For the opened
singularity.
the stress
mechanics.
as
is assessed in terms of
cracks is examined
under ;L fixed
load condition.
2. STATEXIEST
Consider
the t\vo types of cross-ply
to extension.
parallel
As the load
increases.
of the laminates.
transverse
of a laminate.
These cracks.
further.
cracking
Figure
(Lim.
cracks
;IS ShO\Vfl
IOildillg
tlircctiun.
transverse
into delamination
I is typical
spacing along
to the ply interface
cracks along the interface
cracks originating
that the cracks run through
loadin_c.
the width
in the li~mini~tc due to the presence
To simplify
rlic c~vcr~~ll arrangcmcnt
coordin;ltc
cracks running
uniform
of such dclnminution
the problem.
ilrri~ngetl with the configuration
rcprcscntativc
unil cell.
WC take iI rectangular (‘;lrtcsiall
tips. We choose
[90 01, and [0 901.. subjected
perpendicular
the stress redistribution
2, so that
iI1 Fig.
laminate.
terminating
under extensional
thal the cracks ;lre uniformly
plilIlL!.
HASIC’ EQL’ATIOSS
occur numerous
1988). It is noticed
We will examine
of such interfacial
composite
there will
0 ply, tend to develop
because of the stronger
as the load increases
r\SD
of the 90 ply. tvith an approximately
to the fiber orientation
the length
from
OF Tt1E PROULEM
sys[cm
is ohtaincd
with
wx assume
symmetric ;tbout ihe midhy rcpctition
the origin
of the
at OIIC of the crack
lillllill~ltCs
or along the
lhc .v, axis I0 bc 3lollg Lhc lcllgth of the composi~c
\vhilc
tllc v2 axis ih LilhCll to hc ;llong Lhc tlirccIion of Ihc lamini~tc
thickness, illld then Lhc V, axis i\ illogic IIIC laminate witl~h. which is p:~ri~IIcl 10 the cracks
(xc
I:ig. 2). Each ply of the composite
and the ply oriciitatioii
0 is tlcliiiccl
Lh;lI the lihcr tlircction
makes
(lanlin;tlc
Ihe .v, Jirccrion
illld
thaf
I:imin;llc
tllc
The plant
stress distribution
i~rc)ll~~tl composite
( 1979). For a11 anisotropic
nonirro
the normal
strain
ilpprO;hchCs
used
by
trcaul
and
Choi
iind
their asociatcs
the I;lllliniltc
above.
011 thC _\‘,~-.\‘J pl;lnC.
by milny
rcscarchcrs.
including
e.g. Zilk
and Sinclitir
Jisplaccmcnt
relation.
is in gcncral
This is SO cvcn when
to bc Lcro (which is the USC
courclinatcs
on
Lckhnitskii’s
[c.g. U’arlg
in
thickness,
miltcri:lls.
is ilssunlcd
the two
to caaminc the boundary-layer
from
isotropic
the l>ut-of-plitnc
dirwtion
only
vicwcd
( 1972) ilnd &mp~~y
in the slrcss -str;lill
in i~n:~lyAng such elastic ddormationr.
Wang
Yuitn (l983)j
upon
wcrc
ilntl Enlogan
body.
coupling
in the out-of-plant
that clcpcnd
with
comprising
wcdgcs.
composite
due M the associittcd
for deformations
bdy
( I97 I ). COOk
( I YOJ), tl~sy
I;rrgc comp;Ircd
10 bc iII LhC S;L;lLC01’ plallc slraiii
ol’ iI composik
to lllc .v, .\‘, plilllc.
ailglc.
Wc ;IssunIc that the Iarninatc tlimcnsion
is sutlicicntly
is ilSSulllCll
problem
ilnd Williams
wilh fhc .v, ais.
width)
lies in ;I pl;lllc pilrilllcl
I;imiii;~lcs
to hc llic ~ourl[cr-clo~kwisc
and
ttlc
plitnc).
cornpl~x
Choi
Among
polcntials
( 1982). Wang
[he
were
and
elrcct in the free-c&c problem. Ting and his
associi~tes. on the other hand. usd
Stroh’s Ltpproitch (Stroh.
1963) 10 treat the stress
singularities near various anisolropic composite ucdgcs and cracks [Ting ( 1986) and
references cited fhcrcinj. This
;lpprl)ilch
has the advantage. as shoivn by Ting
( IYYb),
that
it leads to il simple ilntl nml Ibrniul;~tion of problciiis in anisotropic ClilStiCity. Due 10 such
iln adwntagc. bvcuse the Stroh forn~alisni to dclcrniinc the struclurc ol’thc solution for the
present problem.
Let II,. E,, ilnd (T,, denote the Ci~rtcsian components ofdisplxcm~nt,
respectively. For deformations
the governing cquiltions :
Equilibrium
equation,
Strain-displacement
Stress-strain
strain and stress.
that dcpcntl upon two coordinates .v, and x2 only, wx hllvc
o,,,, +a,:,:
= 0 (no body force).
relation. E,, = (II,,, + II,,,) 2.
relation. 6,, = C,,AIcI,.
Cj,k, = C,,A, = CL,,,.
(lil)
(lb)
(Ic)
Transverse
cracking in cross-ply composite hmioztcs
unit
ce!l
---w--J
whcrc C’,,4,arc the fourth-o&r
still’ncss tensors. and the conlm;i inclic;ltcs partial dilyerwith rcspcct to .Y,. Introducing the “coil;lpsccl” rcprcscntation, WC may write the
stress strain relation as
rntiation
0, = c’,,i:, .
(Lb)
t:, = S,,b, ,
where
(2cd)
and C,, nnd S,, arc the sti(Tncss and the compliance matrix.
[90/O], i\nd [O/90], laminates.
c I4
--
WC
For orthotropic
materials like
hilvc
c15= CI6 = c’:, = c-11,=
C’, =
c,, = c,, =
c3,
=
c,, = c,, = c,, =
0.
(3)
For such materials. the displacement componsnt 11~is decoupled from II,. 11:. and the plane
strain assumption can bc made for extension along the s, axis. 3s in the present deformation.
Following Ting and Chou (198 1). WCc;ln show (hilt the general solution
the present problem.
takes the form for
1930
T.
w.
KIM
PI
ul.
4
11, =
1
I=I
l-x*/A (A ).
(Jab)
(k = 1.2.3.4).
-_A = XI +j&.r,
where the Greek letters 2. 8. ;’ take the value 1 or 7. and summation
is implied on repeated
Greek indices onl! ; pi are the eigenvalues to be determined, and /;(zi)
Substituting
D,,,(/IA k,,i =
For the existence of nontrivial
0.
solutions,
we have
det[&APL)I = ID,,,( jr* )I
and the solutions
values (Stroll.
o.
=
of this quartic equation yield the two pairs of complex conjugate cigen-
(r = I. 2).
(8)
The associated eigcnvcctor v,,~ can now bc obtained from (64
cqn
Using
(7)
IY61).
ji, , : = jix
ization.
is a function of z~.
eqns (5a.b) into eqn (la), we obtain the equations:
the collapsed symmetry
relation
through ;t proper normal-
(3). we obtain the explicit cxprcssion
for
(7)
The four eigcnvalucs jlr appear as two pairs of complex conjugates. and they are purely
imaginary if the inequality
:c,l/C’,,,-C'i2/(C,,C~~)-2C,~/Cz~)'-4(C,,jC2~)
2 0
holds among the still’ncss components C,,; otherwise they have ;L nonzero real part. For the
advnnced fibrous composite materials such as graphite epoxy and boron epoxy. the above
inequality
holds and therefore the values of 11A are assumed to be purely imaginary hrre-
after. It is worthwhile to note that these t’our cigenvrrlues 11~are the constants that make eqn
(Jb) the complex ch;alctcrIstIcs
Lekhnitskii’s
ol‘ the governing elliptic
complex stress potential (Lekhnitskii.
3. ASYMPTOTIC
EIGENFUNCTION
IWUNDARY
partial differential
equation t’or
1963).
COLLOCATION
EXPANSION
AND
SOLUTION
In this section, the asymptotic eigenfunction expansion for the function /i(zL) is examcrack tip. and use is made of this expansion to obtain the
ined near the delamination
complete numerical solution with the aid of the boundary collocation method. We assume
;I power-type eigcnfunction for/;(z,)
ils given by (Wang and Choi, 1982; Ting. 1986)
/L(G) =
which leads to the expression
2 c,A”/(s,+
n= I
I).
for the displacement and <tress field
(10)
where the superscript tfz = 1.2 indicutes the upper and the lower ply. respectively. Here the
eigenvalucs ri, are to be determined
from the so-called “near-field”
crack tip. including the traction or displacement
ply interface. The coefficient Ci, is dependent
be determined
conditions around the
conditions on the crack surface and on the
upon the associated eieenvalues tin, and can
within an ;Lrhitrary constant when 5, is obtained.
We assume that there is no friction on the delamination
crack surface and that the two
plies are rigidly bonded alonp the rest of the ply interface. Then the nettr-field conditions
for the present case may be written ;ts
rr':~(.~,.O~)
= a'1',1(.s,.O-)
= 0.
or
[a,,(s,.O)]
= [u:(s,.O)]
= 0
(on the crack surface. s, g 0).
(r’,!(.v,. 0 i, = n::,‘(.v,. 0 -) = 0,
[fl::Cs,.O)l = [fi,:(.s,. O)] = 0,
(12)
{on the crack surface. .vi G 0).
[II, (.s,,O)] = [11:(.r,.0)] = 0
(i4a.b)
fan the intcrfnce, .Y{ 2 0).
whcrc the bntckct [o] dcnotcs thr discontinuity
c.g.. [rrJ:(.u,,O)l = a’~i(.u,.O ‘)--~‘,<(.v,.O
(13
of the quantity in it itcross the pty interface,
). The first condition
in cqn (12) is valid for the
opcnctl region of the crnck surf;tcc antI the luttcr is for the closed region. Substituting
rxprcssions for tlisplaccmcnt
;md stress ( I I ) into the above near-field conditions,
the
we obtain
;I system of X x X homogcncous linc:tr equations :
4,bj,,w,,,=
0
(iJ = I X),
For the existence of nuntrivi:ll
u,,, = cl;‘.
f&+&N = Ck2
(k = l-4).
(I 5.1)
solutions, we h;ive
IA,,Wl = 0,
(1Sb)
which determines the eigenvalurs S,,. When the eigenvalues S, are known, within unknown
constants the eigenvectors C,,, are computed from eqn (I Su) and the asymptotic form of the
solutions for the stress and displacement
A,,(&).
is given by eqns (I 1a.b). From the structure of
wt‘ can show that if 6, is 3 root of the ~h~i~~~t~risti~ equation.
conjugate &,, and the expressions (I ta,b) for the stress and displacement
so is its complex
become real. To
take only the real part of eqns (I li1.b) for convenience, we may introduce
where h::’ is the solution of cqn (1.51). computed by a proper normalization,
and 7,“. yL
and ;‘?. arc constirnts to be dctcrmincd to complete the solution. The expressions for the
stress and displacsmcnt (I 1;l.b) are then written as
(I 7a.b)
where Qy;::“’
and Pz$ are given by
if 3, is complex. or
and
The unknown constants ;‘,,, can bc dctcrrnincd by matching the asymptotic expressions with
the far-field conditions
(the conditions
collocation technique: (Wails
at the remote boundary)
through
the boundary
2nd C’hoi. I YXZ), or through special linitc: clcmcnt nicthods in
which the prcscnt ~~synlpt~?ti~sofutions arc cmbcddcd in 2 singular crack tip cicnicnt (W;ins
and Y u;In. I983). or in which the asymptotic solutions arc introduced over thcctltirctioIllilin
(Stolarski
;ind
C’hiang, I9XO). Exlctisivc
lists of the litcrnturc
on the boundary collocation
methods hind the special finite clcntont tfchniqucs may he found in C’hung ( I%SX~ami Atluri
and Nakagaki ( 1986). WC use the hountlary collocation tcchnic~uc for the prcscnt problem
hcrc bccausc of its simplicity.
A brieftlcscription
of this tcohniquc is given in Appendix A
for self-sullicicncy of’ the prcscnt dcvelopmcnt. and for details the rcadcrs may bc rcfcrrcd
to Wang and Choi ( 19X3).
The power-typt: expansion (10) fails to he camplcte when the algebraic multiplicity
greater than the gcomctric multiplicity,
eigenvectors associated with thcsc multiple eigenvnlues [see Dcmpscy and Sinclair
Ting and Chou
(19% 1) and the rcl’crrncss cited therein].
resolved thisditlicutty
the problem of anisotropic
enough to span the solution.
(1979).
Dempsey and Sinclair
by introducing logarithmiceigenfunctions,
of the sets ol‘eigenvrctors,
is
that is, there arc not enough sets of power-type
(1979)
which ensure theexistence
Subsequently
this was extended to
composite laminates by Ting and Chou (1981). The existence
of the logarithmic ~i~enfun~tjons
can be examined by calculating the algebraic mui~ipli~ity
of the eigenvalues and the rank of the associated coeflicient matrices A,,(&) in (I 5).
An essential prerequisite for the success of the aforementioned
technique is that the asymptotic representation
boundary collocation
(17rl,b) should be complete in the sense that
it can represent the elastic field over the entire domain concerned, including the far-field
region as well LLS the near-tield region. For
example, the present boundary collocation
method is ditlicult to apply if there is another singular region or u strong boundary layer
in the far ticld. which the ei~~nfLin~~ion expansion (17a.b) fails to reprcsat: for the single
expression for the elastic state valid throughout the entire domain is not available. For such
;I case. we may USC‘the hybrid F.E.M.
whcrcin each singular or boundary-layer
region is
mod&d as a special elcmcnt into which the rtssociatcd asymptotic solution is incorporated,
or the enriched F.E.M. wherein the asymptotic solutions are defined over the entire domain.
Roughly speaking. WC can approximate the far field by retaining ;I sufticient number of
terms in the ncur-ticld cigonfunction scrics. 3s long as the asymptotic oi$enfunction series
is complete in the near field and there arc no strong boundary layers other than the ncarfield singular region. For such crtses, we in gcncral have no difficulties in applying the
boundary collocation mcthod in conjunction with the asymptotic rcprcsontation in the near
tield.
Trwwrrw
Another
important
IV33
crackmp in cross-ply composite lammates
issue related to the completeness
is the treatment
modes. If only traction conditions are involved in the remote boundary,
of rigid body
such’ as in the free-
edge problems (Wang and Choi. 1982). we exclude rigid body modes from the eigenfunction
representation
because they are indeterminate
are. however. any displacement
boundary
which is the case for the present problem.
displacements
in the asymptotic
for displacement
If there
in the remote boundary.
we have to incorporate
all relevant rigid body
(I la) ; otherwise the asymptotic
representation
solution
and this will lead to an inaccurate
expansion (I la). however. excludes the two displacement
to rigid translation
placements are both compatible
incompleteness
involved
(I la) or (17a) fails to be complete
solution. The power-type
corresponding
under traction boundary conditions.
conditions
in the .K, and X? directions,
with the near-field conditions
is due to the inherent
introduced in the denominator
limitation
(IO) wherein
and therefore the constant translational
where II:’ (I = I. 2) is the constant translational
solution (l7a)
rigid displacement
rigid displacements
as
or the crack tip dis-
and Q!:z’ (N >, I) is the same as before. The significance of
In this section. WC dotcrminc
collocation
and
For the numsrical
epoxy T300/520X
the solution.
DISCUSSION
free constants 7,“. y:,, and yr,, through
WC will confirm
the convcrgcncc of the
of the near-tip
singular fields in terms
and the fracture behavior of delamination cracks, including
the elrccts of ply thickness upon the energy release rates.
computation,
(Whitcomh.
E,. = I 3-I C 1’;~
AND
solution, and discuss the nature
of the stress intensity,
of cracks
RESULTS
the unknown
to complctc
boundary collocation
stability
en’ect
through numerical examples.
4. NtJMliRICAI.
boundary
Such
(B,+ I) is
including these terms is apparent in the light of completeness. and their quantitative
will bc illustrated
modes
these rigid dis-
(12). (13) and (l4a.b).
of assumption
are excluded. To rectify this, we modify the asymptotic
placement in the _v, direction,
although
E,. = E,
WC:11s~’the following
material
the
data for the graphite
1987) :
= 10.2GPa.
GIeI. = Cl./ = 5.52GPu.
\‘,_I. = \‘L/ = 0.3,
Gr/
= 3.43GPa.
vrz = 0.49.
To determine the free constants, we match the near-field solution (18) with the remote
boundary conditions
through boundary collocations.
crack length. the displacement
the 90
Let c. 6, II,,,. /I,, and b denote half the
prescribed at the right end of the unit cell, the thickness of
and 0’ ply, and half the spacing of the transverse cracks, respectively (see Fig. 2).
The remote boundary conditions
for the case of [90/O], laminates may then be written as
a~~(-~~-~:)=~,~(-~,.K~)=O
O<_K~ ~h,,ontheleftendofthe90’ply;
~21(.~,./IY,,)
= a,:(.\-,.IJ,,,) = 0
u,(h-c,.K~) = 17,
L’II,(.‘c,,
----.-___
ss,
-/I,,)
=
‘L(huJ
--c < .rI < b-con
=
L7.\-,
ll?b,
,,, ( _ c, S?) = _I____:
(‘s ,
9 -A,,)
= 0
=
0
-It0
0
-/I,,
-C
the top surface;
,< 12 < h,, on the right end;
< X, < b-c
on the mid-plane;
d sz < 0 on the left end of the 0” ply,
(19~)
(l9b)
(l9c)
(19d)
(l9e)
For [O/90], laminates. WC have similar boundary conditions. Note that the boundary conditions involve the prescribed displacements for U, and u:, and this requires the inclusion
T
IO:_1
of the rigid
blxik
modes
Lb’ li IV ,‘I <I/
in the eigenfunction
solution
as in eqn
I IS). LC’cnow truncate
the
srrio
Aution
(IS) to retain a finite number OF tcrmb. and determine the free constants
.,
..
such that the resulting
solution
ma)
, 13,., ?,I and ;s,,, through the bounder\ collocations
approltimate
procedure
the above remote
iq outlined
The
resultins
opened u hilt
Thus.
surface.
dcl:~mination
that
[O YO]. I;lminates.
ith the crack
to ;iyi;il
towrd
wherein
is rclati\cly
untl:~niaccd
w~ion.
c
Tllc eigen~alue~
that the i1n;rgin;lry
cracks
lilcralurc.
binguI;ir
ahic
lsriii
alItI
of the lirst
sclu;irc
/I,()‘. n) -I/,(/..
the 90
the
the crack to
constr;lintxi
ply is brought
contraction
region
into
of the 90
cnriiparcd
pl>
with
the
cracks
of
are gi!-cn as :
‘/ =
. . .).
is not zero
cigcnv;lluc
is oscilli~lory.
wliilc
root singul:irily.
(lY77)
0.03
( I7a.b),
In ,.)+I],,,
ficltl 1ic;ir the closctl
and C’hoi (IW3h).
Wang
cspnsion
cos(q
lhc opcncd
for
lhc litrcss
‘I’his agrees will1 the results
and
tip and the crack tip opening
= [:I,,,
cawt’s
on the
around
ti)r the closed cracks iI1 [O!W],
in lhc cigcnl~unctioii
fr,,,(l..O)
cracks
hand.
the .v,
constraints
ply
:
uniform
along
ply is symmetrically
t,icc 011
1ic;ir the dclaniin:ition
c.g. C’omninou
licld ahcad 01’ lhc crack
00
the crack
observations
ply. on the other
ot‘ the YO
hcca~~sc the thickness
sm;~ll
siligularily
~~l‘[O~~A~J,
IUS tlic iiIvcrsc
in the c;lrlicr
stress
part
the htrcss
det;lilcd
almost
the damaged zone
the YO ply art‘ free from
the intcrinr
(!I = 0, I. 2, 3,
Notice
remains
of the 90
from
tier the two types of inttxfacial
[UO OJ, ;iricl tl1crcliIr.c
0,.
the free surface
ply.
ply
II - !, arid II
the thllceving
the undamaged region and this
~;ICC of the 0
cxtcnsion
uith
contraction
anay
ntxr
and In\ver t’ices hy the 0
1%
sense. The
crack for [90 01, is full)
the delamination
the axial stress
to
I+ herein
elements
are pulled
in the least square
is consistent
proceed
for [YO 01. lamin;ltes.
rcpion
the upper
dirt
show
the thicknec
as we
the material
rend to ~~pcn. Fur
c<jnt;lct
ivhilst
due to shear lag
coordin~~tc.
cvtcrior
solutions
fulls closed for [0 YO],, This
contraction
of the 0 ply dut:
the .v, coordinatr.
increases
on
numtxical
conditions
,A.
it is
the thickness
along
boundary
in Appendi\
wc can write
- n) = [tl, cc)?;(q 111r)+h,
’ ‘+
the
the asymptotic
diqkiccmcnts
sin (11 In r)jr
rcportcd
Taking
2s
(‘Oa)
‘.
sin (11 In r)],-’ ‘+
for the openccl cracks in [%J,‘O],
(‘Oh)
;lnd
a,,,(r.O)
’ ‘f
= (‘,,,r
u,(r.n)--u,(r.
..‘,
fi)r the closal
whcrc
/I,
“q” is the imaginary
and
C’,.
(‘,/I ‘,,, =
normaliad
xro.
r goes
10
crack
model
li,r
in l‘rictiunlcss
clnscd
To
crack
arc almost
:lll~l
ll I,,,
=
I in Table
arid thcrclix
rcctiry
such ;I shortcoming.
dissimilar
inod4
isotropic
nc;Ir
the tips.
both
: moreover.
Willlg
Slllilll in ;I filly
of’ the fully
the partially
singularity.
(2Oc.d)
paramctcrs
The
I:,, = ii,,/h.
strain
I. ‘fhc stress
whcrcin
and
Choi
anisotropic
:I,,,.
II,,,.
C’,,,. II,.
arc t:ibulatcd
for
the case
singularity
inadmissible
Comninou
materials.
and
‘f ..
in [O;YOJ,.
physically
il
to the GISC of dissimilar
is negligibly
the same Ibr
(-‘hc>i. IOS3h)
II,,
normal
cracks
contact
iiitcrpcnctratioli
part of the stress
by the applied
I. I7 I/%,,,= 4 and
inlcrl~icial
opcnctl
cracks
= c,r’
-7r)
( 1977)
is oscillatory
intcrpcnctration
proposed
the partially
closed crack
tHowcvcr.
opcncd ;ind
model
the partially
;ilW
closed
ICICCSarc
:I parti:illy
the zone
opened crack and the asymptotic
closed
a5
;I new intcrl’ricial
(I 983h) cxtcntlcd such
materials.
I’or the
occurs
stress
mod4s
has an inconsistency
01
fields
(Wallg
that it is
Transverce
I.
Table
Asymptotic
1935
cracking in cross-ply composite lammares
stress held ahcad of the crwk
tip and crack tip opening
displacements
E,, = u‘,h. h,,. h,,,
= I. h,,:h
= 0.15.
c h = 0.25
PO
I I
B!,
.4 :1
aJr.0)
B ..
I.4407J
I.06751
-
8,:
C\,
C::
C,,
[ll,(r.I[)--l(,(r.
-n)j,r:,,
h:
(‘I
(‘2
n,,,(r. n),‘~:,, = [, t,,, cos Or In r) + B.,,
1
0.
0.
- I .7.+41?
0.67692
0.424I2
-0.7V’?I
0.5x705
-
(1I
h,
‘12
Crack tip opening displacement
-
- I .~‘YXV
_..
.-r;;
I:,,
lo w.
0.68’446
-0.93OSI
0.9105-l
.q
Stress held ahead of the
crack tip
01.
sin (q In r)]r
[u,(r. xl -u,(r. - n)j,‘x,,= [tt, cos (q In
-O.XOSIJ
0.
’ ’+
r)+b,,
sin (0 In r)lr’
:+
..
for the opened cracks ill [VO.O~..
;tnJ
nJr.0)
I:,, = C‘.,,r
’ !-t ” ‘. [u,(r. Jr)-tr,(r. -n)j:c.,, = c,r’ :+ “’
for the clo~cd cracks in [O,VO~..
not rduccd
to the
solution
for ;I homogcncous body when the two plies bccomc itlcntical.
l:or such reasons. WC do not seek the solution
To
frxturc
ch;lr:lctcrizc
for the partially closed moclcl hcrc.
the stress licld near the prcscnt delamination
mechanics paramctcrs such as tho stress intensity
cracks. WC may use
factors K,, A’,, and the cncrgy
rclcasc rates G. l:or closed cracks in [O/YOl,, thcrc arc no dillicultics
introducing
the classic;~l
stress intensity factors for this purpose. In fully opcncd intcrfacc cracks in [YO’O],, howcvcr,
modes I and II arc inhcrcntly coupled, and the stress intensity
factor is not dctincd in the
s;~me way as in the classical fracture mechanics. One possible dctinition,
as
suggcs~otl
by
Rice (198X), is to dclins the stress intensity factor by obtaining the stress value at ;I specific
small distance from the crack tip, that is. for ;I very smull value of r’,
h’, =
J~nJa,,(i,O),A;,
=
JZnir_+(i.O).
For the prcsrnt USC, wc take i to bc c/SO. Because the imaginary part of singularity
very small (~1= 0.03 . . . ). changing the value of i within
3 factor of IO or Icss, dots not aftict the vulucs of K, and A’,, signiticantly
To contirm the convcrgcncr of the solution.
“11” is
a modest range. say. an order ot
(Rice. 1988).
we computed the stress intensity
and the maximum mismatch of the rcmotc boundary conditions,
factors
varying the number of
the eigenvalues and the number of the collocation stations (Table 2). WC xc that the present
solution has a very stable convcrgcncc characteristic. The accuracy of the solution
has been
confirmed through ;I comparison with the result obtained from the singular hybrid clcmcnt
tcchniquc (Jang. 1990):
the stress intensity
factors A’, and A’,, from
the two numsrical
solutions agree up to the first two digits and the rclativc diflcrcrrnceis Icss than two per cent.
when the boundary collocation solution with the 63 terms truncated in the eigcnfunction
series is compared with the solution from the F.E.M. mod4 comprising 51 regular elcmcnts
and one singular element. Such an cxccllcnt agreement bctwcen the solutions from the
diffcrcnt approaches has been rcportcd for a stress ticld near cracks normal to the ply
interface (Im and Kim,
IYYY) and for ;I stress ticld near the free edge (Stolarski
1989).
As discussed earlier. the rigid translational
field conditions
(II)
through
(I4a.b).
and Chiang.
modes. which apparently mctt the ncar-
are not included in eqn (1721) due to the inhcrcnt
1936
T. W.
Tltble 2. Solutmn
convergence
KIM YI al.
versus the number of eigenvalues
collocation stations
E. = li,h. c’h,,, = I. h h.,, = 4. h,,/h.,, =
and the number
I
PO01.
unit
No. of
of
: GPa (mm)’
’
No. of
eigenvalues
COk3Ci#tiOfl
51
K, ‘E,,
stations
IO0
I20
I40
100
120
I40
IO0
I?0
I40
51
51
63
63
63
I10
II0
I10
Max. mismatch
2.6514
2.6514
2.6512
2.7016
2.6862
2.6606
2.7207
2.11%
1.58%
1.57%
1.50%
1.07%
I .27%
1.16%
1.15%
I. 14%
3.0656
3.9660
3.0662
3.05 I I
3.0844
3.1 I40
3.029 I
3.059il
3.0708
2.6652
2.6867
(n = 0. I. 2.3..
1 = 0.03..
No. of
cigcnwlucz
No. of
colloc;ilion skltions
51
51
51
62
63
63
I IO
I IO
II0
Eiycnvalucs:
-32
-
5 =
-
’
I
‘25
b
I
a-
Max. mismiltch
who
100
I20
I40
I IO
I20
I40
I tn,
I20
I40
.).
I .70%
I .06%
I .020/u
1.26%
I 47 %
I :ow,
0.53%
I. IO%
1.21%
I .X063
I .7732
I.7751
IX105
I .76X2
I .x090
I .7706
I .7870
I .x070
,I
1
(a = 0. I.23
.25
. . . . ).
5
75
x,/b
Fig. 3. Stress distribution
limitation
of the functional
for [9O/OJ..
from (10) or (I la). Thus we can obtain an accurate solution in
the light of completeness only when we add these terms to the expression for displacements,
as in eqn (18). Indeed. numerical results show that the maximum mismatch of the remote
boundary conditions increases by a factor of IO and the stress intensity factors deviate more
than 30% when the rigid body displacements are omitted, although the solution shows a
good convergent behavior depending upon the numbers of eigenvalues and collocation
stations.
Figures 3 through 6 show the stress distribution along the .K, axis at x2/hue = 0 (the
Transverv
cracking in cross-ply composite
Fig. 4. Stress distribution
’
-30
-
‘2s
iaminatcs
for [90/O]..
1
6
.i5
1937
.i
i
.75
x,/b
Fig. 6. Stress distribution
for [O/90],.
ply interface) and at xr/hpn = +0.5. For the closed cracks in [O/90].. the normal stress is
compressive, and the shear mode is responsible for delamination growth. It is also noticed
for the opened cracks in [90/O], that the interlaminar shear stress component cl I is greater
than the normal stress component uz2. and this suggests that the high shear stress is also an
important driving force for the growth of opened delamination cracks. The’characteristics
of the stress distributions uz2 and CJ,?.near the closed crack tip of [O/90],. agree with the
T
results for ;I fWt’tiitlly closed
th;tt 0, .‘
is sitlgtdar
at
illtcrfilCC
.x1 = O+,
W
crack
tiiw
4.1
in isotropic bi-m~lt~ri~tls (C~~~~~~~irlol~.
197X). in
while rr,,
,x2 = 0.
rrl
is singular at .s, = 0
and .\‘: = 0 and
bountlctl ;I t .Y, = 0 + ;~rld .v: = 0. It is noticed that both [OO,Oj,and [O/90], have peculiar i~Xidl
stress
distributions
w
the clack f;lcc of the 90 ply due to the complexity of loading and
gcomctry. For [OO/OJ,.the stress cc~nlponcnt CT,, at ,Y: = 0, goss to nqitivc
infinity
first as
tlrc crack tip is ;tpproachcd from the nugitivc s , axis. It then undcrgocs 3 violent oscillation
until it critis up at positive inlinity 3s wt approach the crack tip front the positive .Y( axis.
I:or [O,OOj,. the stress coniponcnt fr, , at x1 = 0 goes to ncgativc infinity with no oscillations
this time as the crack tip is ;ippro;tchcd from the ncgativc s , axis but it is hounded iIS WC
approach the cr;~ck tip from the positive .v , axis. “I‘hc large ncgativc values of ths axial stress
on the crack I&X of the 00
ply ;IS the crack tip is approached from thr: negative .Y, itxis
;tppcar to hc contrary to our illttiiti~n.
Howtzvcr, such negative: axial stress may be explained
whtn wvccxatninc ths stress distribution
through the thickness
(.v, = 0) and the global forcc equilibrium
of the 90 ply itt the crack tip
of thr part ofthc 90 ply comprising
-c ,< .r, < 0.
IIL’~;ILISC thcrc is no axial force through the thickness on the left rnd (.u, = -l*) of the part.
the net axi;il force through the thickness on the right end (_Y, = 0) must bc zero. However,
the axial stress on the 90
ply, which continuously
dccrcascs as the crack tip is approached
from the right end (.Y~ = h-c) does not disappear tzactly at the cross-section of.r,
= 0
where the crack tip is loc:ttcd. That is, the axial stress 6, t &xs not decrease to zero but still
remains positive on the greater part
of thr
cross-section at _v, = 0. To maintain the overall
force quilibrium
of the part considrred above in the _v, direction, there must be ;I portion
of cross-section
under large compressive iIxi:tl stress (see Fig. 7). For the [90/O], Iaminntes,
the stress ~~istribtIti~~n through
overnll
I~l~itl~Ilt
the thickness should be such that it alone may meet the
~~~~ilibri~~rn in addition
to the force ~4uilibrium.
complic;~tcd than the cast of [0,‘90], wherein
and hcncc it is more
the moment cyuilibrium
is automatically
satislicd due to syntmctry with rcspcct to the mid-plane.
For both oponcd und closed delamination cri\cks. the cncrgy rclcilsc rates G can be
calculated front Irwin’s virtual crack cxtcnsion concept. For an opcncd delamination crack,
modes I znd II ;irc strongly coupled and the calcttlntion of the cncrgy rttc;tsc: rates involves
;L s~ibst~iltti~il amount of algcbrri including com~ut~ition of a complex contour integral
(Willis.
I971 ). For self-suficicncy of development. ths expressions for the energy rcleasc
rata ilrc outlined in Appendix B. Figures 8 ilnd 9 show the stability of delamination cracks
in terms ofencrgy rclcasc rates under a fixed load condition. Thcsc figures reveal important
fatturrs rcgrtrding the growth of delamination cracks originating from transverse cracking
in cross-ply laninatcs. When we assumt‘ a failure criterion based upon the critical energy
rciertse rate. G = G, (G, is a material constant). we see that any delamination crack will
Trans~crw
cracking
m cross-ply cumpoGtr
I9.W
lamtn;ltc\
undergo unstable growth until the crack length rauzhes ;I critical vnluc associated with the
maximum G. The crack will continue to grow under ;I tixcd loading until the energy rrlcasr
rate drcreascs to the critical value C,. Thus.
mechanism for thr present delamination
there exists an inhcrsntly
built-in
crack arrest
cracks. The critical crack length is drprndsnt
on
the rrlativc
crack spacing hi/r ‘,,, its well as on the relative ply thickness lz,,,‘h,,,,. As the
of the 90 ply incrcrscs relative to the 0’ ply. the energy rclciisc rutc notabl)
thickness
incrclscs
and thcrcforc the criick growth after the onset of the dclnmination
more unstable. This
crack will bc
is consistent with thr observation that a larger portion of ths load uill
bc carried by the 90
layer i1s the thickness
of the 90
ply increases. and the cncrgy
rcIcrtsc rate will accordingly incrcnsc under it fixed load condition. On the other hand, it is
worthwhile
to note that the cncrsy rcicasr rate tads
to bccomc indepcndcnt of the crack
Icngth. sxccpt uhcn the crack is of the order of the thickness of the 90 ply. as the thickness
of the 90 ply dccrcascs.
The cncrgy rclcasc rates arc zero at the onset of the delamination cracks. as shown in
Figs s and 9. as the stress singularity of the transvcrsc cracks in ;I cross-ply Iaminatc under
cxtcnsion is wcakcr than the invcrsc square root singularity and thcrcfore the stress intensity
for a vanishing delamination crack approaches zero under cxtcnsion (Im and Kim.
I9S9).
.,(c,P,tr,~~/c,~k~~,nrcrr,.r-Ths authors grarcfully acknowlcdpc Ihe tinancial suppurl fur this work. prnvdcd by the
Korw Scicncc and Engineering Foundation (KOSEF).
They thank professor T. C .T. Tinp of The Univcrsiry of
Illinois 31 Chwago fur the fruitful discussion.
I YJO
T. W
KIM <f (I/
REFERENCES
Atluri. S. N. and Nakag;lkl. M. ( IYXh). Computational
methods for pl.mc problcmh of fracture. In C~qwr~rronui
.L/crhodv ;II .Wrchtmnic.r
o/~Frcwture
(Edited by S. N. A~lun), pp. 16Y 227 EIx\wr. .Amsterdam
Bogy. D. B. (lY71 J. On the plane clastostatic problem oi .I loaded crxk tcrmln.ltlng at d m;ttcrlal Interface. J.
.4ppl. .tlt,ch.
38. Y I I -9 I Y.
Chung. S. K. (IYXX). Anulysls of crack in tuo-dimcnslonal
anwtroptc
hud~cs Hlth \arIous shawd boundarw.
Ph.D. Thesis. Korea Advanced Institute of Science and Tcchnolog>
Comninou. M. ( 1977). The mterfxe crack. .4S.UEJ.
.l,+.
.1/w/1 44. h! I 636.
Comnmou. hf. (197X). The intrrfxr
crack in ;L shear tirld. .-IS.t/EJ.
-lypi. II<,ch. -Is. 2X7 2uO.
Cook. T. S. and Erdogan. F. (1972). Stresses In bonded materials with rl crack peqwndlcular to the interface. Inr.
J. &nqnq Sci. IO, 677-6Y7.
Dempse). J. P. and Slnclalr. G. B. (lY79). On the stress singulurlttes m plane el.~rt~c~ty of the composite wedge.
1. &ltr.r,rU~r 9, ?73--39 I.
Im. S. t IYXY). Asymptotic stress field around ;I crack normal IO the plk-Interrace of an xnsotroplc
composite
laminate. Int. J. Sr~litb Swrrcrctre.~ 26. I I I -I 27.
Im. S. and Kim. T. W. (IYXY). Stress tield near trunsversc cracks under e\tcnsion or in-plane shear in crusx-ply
composite laminates. A’S,VE /I 3, I? I - 119.
Jang. J. S. (1090).
Stress analysts around delamination cracks orlglnatmg from transverse crac~rng in ;Lcross-ply
laminate using a hybrid F.E.M. MS. Thesis, Korea Advanced Instirule ol’Science and Technology.
Lskhnilskii. S. G. ( 1963). Tlreor~ o/‘Eltrsfiri/~ in (m &~i.wrrr~pic BIU<!.. Holden-Day.
San Francisco.
Lim. S. K. (IVXH). Transverse failure in cross-ply laminated composites. Ph.D. Thesis. Korea Ad\anccd Institute
of Science and Technology.
R~cc. J. R. (198X). El,lrtic frxcturc mechanics concepts (or intcrfacc crxkr. J. :lp/‘l .\l<x,h. 55. YX 103.
Stolarrkl. H. K. nnd Chianp. M. y. M. (IYXY). On the slgniticancc of the logarithmic term in the free edge strcsb
sinpul;irily of composite I;lminalcs. Inr. /. S&l\
Srrrcc’lrtrc~.r2.5. 75 Y!.
SIroh. A. N. ( IYh?).
Steady SI;IIC problems in anisotropic clmt~c~t~ J. .Jltrtlr. f’hb~ 41. 77 IO!.
Ting. T. C. T. (19X6). Exploit solulion and invariance of the ~~ngllaritlc~ at .III Intcrl’wc crack III anisotropic
coiiip~~~ille~,.Inf. J. Sditlr
S/rwrwc*v
22, 965 YH?.
Tins. T. C. T. and Chou. S. C. ( IYHI ). Edge singularllws
in ;Inl\otrcqxc CCUU~O~IIC~.fort .I. .Srddv .S!rucfurcv
17,
1057 II)hX.
APPENDIX
A
The free con.s~;m~r ;‘,,, (k = I. 2. 3) in cqn (16) are determined through
cigcnfunction solu&m (17b) and (IX) mert the rernolc boundary conditions
For convenience, we write oqns (I 7h) and ( IX) as
boundary collocatwn so [hat the
( IYa c) in [he lwzt square xmsc.
where /f. dcnole one of lhe 7,” or II:. and y:. /I,”represent the correspondmg cigcnl’unctions in the capressions for
w and P,$. Lcl 5,. 17,. &,/?s, and &,jS.r, (i = I 6. j = I. 3) dcno1c the prescribed wlucs t’or the stresses.
QM
dispkicemcnls and displitccmcnl derivalrves on lhc remote boundaries. The boundary conditions ( IV;1 c) can then
be written in the form
where .\t,. .?: denote the dimensionless coordinates of the remO[e houndaries. and /I;,. 11;: are the partial derivatives
of h; with respect IO the s,. s: coordinates. respectively.
The functional to be minimized in the boundary collocation is willen ac
Transverse
IWI
cracking in cross-ply composite laminates
Here CB, denotes the whole remote boundaries. and .4,. B,. C, and D, are equal to I if the terms correspond to
the prescribed boundup conditions. und otherwise become zero. For example. eqns Il9a-e)
indicate. for the
[UU UJ, lammates. that .A, = A6 = I and the other terms are zero on the let’1 end ol’the YU pll. .4: = A, = I and
the others are zero on the lop surface of the lammates. 8, = C: = I and the others are zero on the right end. etc.
Minimization
of the above residual leads to the folIowIng system of slmultaneuus hnear equations for the free
constants j. :
ds
(nr=
1.?.3.4...).
(A3)
The system of the above infinite number of linear equations can be approximated by an appropriate truncation.
say nt. n = I-N. and in general. 3 proper scaling procedure is required for successful numerial
computation.
APPENDIX
According to Irwin’s virtual
deformation may be written as
I
.*I,.o :nr
G = G, + G,, = lim
Fl
crack cxtcnsion concept.
the energy rclcasc rate for the
[n::(r.O)(lc:(cjr-r.R)-Ic:(;Sr-r.
C;ISC
of plant
-K); +n,:(r.O)ltc,(,sr-r.x)-~,,(,ir-r.
strain
-n);]dr.
(AJ)
Substitutmg
the form :
cqns (lOa d) into ISIS cxprcssiljn.
(; = ,,,.,,2:r~[.‘,c~)“{,,ls(‘V;r)~
lini
;III~
performing
Icngthy hut str;tightfurw;lrd
+,,,:~ii,{,,l.(dr;r)~]Jlir;rd~
dr = :
KU,
(
;llgchr;l. wc obtain
for;lriopcllc‘lcr;,ck
fur
;I
closed crxk.
(A.0
(Ah)
where
A,,. B,,. c’.,. (1,. h, and C, are the coetiicients appearing
Willis (1’171) showed that the intcgrdl (AS) lads to
and
in eqns (2Oa d). With the aid of contour
c = : n ‘!l!_G!!Y!
cash (f/n)
which is rcduccd to (Ab) whrn the oscillation disappears (,I = 0).
’
integration.