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Barbero1991-ModelingDelamination.pdf
MODELING OF DELAMINATION
IN COMPOSITE
LAMINATES USING A LAYER-WISE PLATE THEORY
E. J. BARBERO~ and J. N.
REDDY:
Department of Engineering Science and Mechanics. Virginia Polytechnic Institute and
State University. Blacksburg. VA 24061-0219. U.S.A.
Abstract-The layer-wise laminate theory ot’ Ruddy is extended to account for multiple delaminations between layers. and the associated comput~ltional model is developed. ~laminat~ons
between layers of composite plates are modeled by jump discontinuity conditions at the interfaces.
Geometric nonlinearity is included to capture layer buckling. The strain energy release rate distrlhution along the boundary ofdelaminations iscomputed by a novel algorithm. Thecomputational
model presented is validated through several numerical examples.
I. l~TRODU~l0~
The
of this study is to characterize
objective
dclaminations
in laminated
composite piates
using 3 layer-wise theory. We wish to raise the quality of the analysis beyond that provided
by conventional. equivalent single-layer laminate theories without
resorting to a full three-
dimensional analysis. A computational model based on the layer-wise theory of Reddy
(1987) is presented. and the model is used in the analysis of plates with d~Iilmin~~tions_
The ;tdv;tnt;tgcs of an equiv:tlCttt singlc-layer theory over :I 3-D analysis are many. In
the application ol’3-D
must
linitc clemcnts to bending of plates. the itspoct ratio of the elcmcnts
bc kept to ;I rcasonahlc viiluc in ordsr
mod&d
to avoid shear locking. If the laminate is
with 3-D slomcnts, ;In cxccssivcly rclinod mesh in the plane of the plilte needs to
IX used bccausc the thickness of an individual lamina dictates the sspcct ratio of i)n clement.
On the other hand. ii finite clement mod4 has+.~lon it lamimitc theory ~OCS not have the
same aspect ratio limitation
through
bcc;~use the thickness dimension
the lamin:~tc thickness.
vcntional
the hypothesis
(ic., both cli~ssi~:~l and shear doformation)
rcprescntution
dissimillir
f Iowcvcr,
of strains
is ~lirttini~t~d by integrating
commonly
used in the con-
laminate theories Icnds to a poor
in c:lscs of intcrcst. namely. in thick composite laminates with
material layas.
A 2-D l;mGn:rtc theory that provides it compr~~mise between the 3-D theory and conventional plate theories is the layer-wise laminutr
continuous
comput:Aonally
intcrl;iminar
theory of Reddy (1987). with layer-wise
represcntntion ot’displaccmcnts through the thickness. Although
more exprnsivc
stressa
than the cbnventional
very accurately (Rrddy
CI &..
this theory is
laminnte theories. it predicts the
1989;
Ertrbcro et al., 1990rl,b). Fur-
thermore. it has the advantage of all platr theories in the sense that it is a two-dimensional
theory, and JOCS not sufTcr from aspect ratio limitations
mod&
tlssociated with 3-D finite element
The layer-wise rcprescntation of the displacements through thr thickness has proven
to be successful. Yu ( 1959) and Durochcr and Solecki ( 1975) considered the case of a threelayer pliltc. MLIU (1973). Srinivas (1973). Sun and Whitney (1972). and Seidc (1980) derived
theories for layer-wise linear displ~i~~nt~nts. Reissncr’s mixed v~lriiltion~l principle was used
by Murakami (1986) and Tolcditno and Murakami (1957) to include the intcriaminar
strcsscs as primary variables. Roth continuous functions and piccc-wise linear functions
were used. Rcddy’s theory is chosen in this work because of the generality it offers in
modeling delaminations.
Dclrtminntions
between laminac arc common defects in laminates, usually developed
t Presently Assistant Professor. Dcpwtmcnt of Mechanical and Aerospace Engineering, West Virginia University. Morgantown. WV. U.S.A.
: Clifton C. Garvin Professor.
373
374
E. J. BAKHI-.RO
.ind J ?. Riw))
either during manufacturing
Drlaminations
or during operational
life of thr
kninarc
(e.g..
may buckle and grow in panels subjected to in-pisnc
Delaminated
pan&
have reduced load-carrying
regimes. However.
capacir)
under certain circumstances.
in
both
the srou-th
arrested. An efficient use of laminated composite structures
fatigue.
i
imp;icr
cotnprcbslis
IO:I~~.
the prc- and po>t-buckling
of detaminations
requires
an
c;tn be
underst;tnding
of
the
onset and grotvth. An analysis methodology is nccessar! to mod4 composite
delamination
laminates in the presence of delaminations.
Self-similar
slorq an interface between layers is suggested
growth of the delamination
by the laminated nature of the panel. It was noted by Obreimoff
(19~)
and
Inow
md
Kobatake ( 1959) that axial compressive load applied in the direction of the d~l~i~ljn~~tj~~n
promotes
further
in the same direction.
growth
Kachanov
(1976).
One-dimensional
and two-dimensional
problem were proposed by Chai (1982). Simitscs (‘l trl. ( I%?).
models for the delamination
Ashizawa
(l951),
and Satlam and Simitses { 198s). According to these
models. the delamination
can grow only after the debonded portion of the laminate buckles.
However. the del~Imin~~tion can also grow due to shear modes II and III.
The spontaneous growth of a delamination
“unstable growth”.
while the applied load is constant is called
If the toad has to be increased to promote further
growth is said to be “stable growth”.
The onset of delamination
del~Inlin~ltiun. the
growth can he followed
by stable growth, or unstable indefinite growth or even unstable growth followed by arrest
and subsequent stable growth.
In most studies the buckling load of the debondcd laminate is calculated using bifurcation analysis (see Chai, I (tIY 2: Simitses ft <I/.. 1985; IVebster, 1981 : Ruttoga and Macwnl.
19X3). Bifurcation
analysis is not appropriate for dcbonclcd Iaminatcs that have bending
extension coupling. as not4
by Simitscs
symmetric, once dClillllill~ltCd,
nations arc iiasytnttl~tricaliy
t*l rrf. ( IOS_i). Even Iatrrinatos that arc c~ripinalty
cxpcricncc bending cstcnsion coupling. In gcncral, ticlaniiIocatctl with rcspcct to the rt~i~ipi~lii~ and the resulting dcl;imi-
natcd layers hcconic unsymnictric.
Thcrcforc.
in-plane coniprcssivc load produ0.3 IiltCMl
dcllcution and the primary cyuilihrium
path is not tri\iai (0. # Of. t:urthcr.mt)rc,
analysis tiocs not permit computation
of the strain cncrgy rclcase rate.
Nonlinear
piatc thcorics have been used to an;~ly~
~l~b~)~i~~~~l
laminates.
the post-buckling
type of
behavior ot
Uottcga and Macwal f 19X.5). \‘in (IYSS], :iml f%i and t’iti
analyLctl the problem of a circular plate \tith concentric, circular
Kirmb
hil’urc:ttion
d~l~1lllil~illi~~l~.
(I 9S5)
The
vu11
has been used in thcsc stutlics.
~lolliin~~lrity
Must of the analysts pcrformccl have been rcstrictcd to rclativcly simple models. The
matori:rI H’i1Sassttrmxl
the possibility
tc) he isotropic in most CziL’S itd
~~rtf~~~tr~)pi~
in ;I fc\v, thus pr~~jl~~ijt~~
of analyzing the inllucncc of the stackin g scqucncc and bending cstcnsion
coupling.
The RayleigbRitz
method has been used by Chai (1987).
Chai c’t (I/. (19SI).
and
Shivakumar
and Whitcomb
(1985) to obtain approximate solutions
to simple problems.
Orthotropic
laminates were considered by Chai and Rabcock (i9YS)
and circuinr delami-
nations by Wcbstcr (19x1).
The finite rtemcnt method was used by ~Vhit~omb (1% I ) IO analyze throlI~h-wi~iti~
dclaminatcd coupons. Plane-strain cicmcnts w3zre used 10 model sections of beams, or
in
cylindrical bending. The analysis of dclaminations
the use of three-dimension4
dimensional.
of arbitrary
pi:ltCS
shnpc in panels rcyuircs
clcmonts, with ;I considerable computational
COST. A thrcc-
fully nonlinear
tinitc c:lcment analysis was used by Whitcomb (1988), where
it was noted that *‘. . . pl:tts analysis is porentialiy aitractivc bccausc it is inhcrcntly much
Iess expensive than 3D analysis.” Plate elements and multi-point constraints (MPC) have
been used by Whitcomb
and Shivakumar
(1987) to study d~ia~iin~~tion buckling and by
Wilt cl ~1. (1988) to study free-edge dclaminations.
situations.
First,
the MPC
This approach is inconvenient in many
approach requires a large number of
nodes to simulate actual
contact between Iaminae. Second, a new plate element is added for each delamination. The
MPC approach becomes too complex for the practical situation of multiplc delaminations
through the thickness. Third, all plate elements have their middle surface in rhe same plane.
which is unrealistic for the case of delaminated laminac that have their middle surface at
different locations through the thickness of the plate.
Cklamination in cumpasikz laminates
3x
The layer-wise theory of Reddy (1987) is extended here to model the kinematics of
multiple delaminations. The theory is applied to embedded delaminations that are entirely
separated from the base laminate after buckling. Numerical results are presented for a
number of problems and the results are compared to existing solutions.
2. THE LAYER-WISE LAMINATE
PLATE THEORY
Increased use of iaminated composite plates has motivated the development of refined
plate theories to overcome certain shortcomings of the classical faminate theory. The firstorder and higher-order shear deformation theories (see Reddy, 1981, 1989, 1990) yield
improved global response. such as maximum deflections. natural frequencies and critical
buckling loads. Conventional theories based on a single continuous and smooth displacement field through the thickness of a composite laminate give poor estimation of the
interlaminar stresses.Since important modes of failure are related to interlaminar stresses,
refined plate theories that can model the local behavior of the plate more accurately are
required, The layer-wise plate theory is shown to provide excellent predictions of the local
response. i.e., interlnminar stresses. in-plane displacements and stresses, etc. (Barbero,
19X9). This is due to the refined representation of the laminated nature of composite plates
provided by the theory and to the consideration of shear deformation effects. Before we
present the theory for dctarnination modeling, a review of the basi\sic
elements of the theory
is first pwswtcrf.
Consider a iatninntcd plate composed of A’ orthotr~~~;c Iaminae. each king oriented
arbitrarily with respect to the ktmimttc (.u.r) coordinates. which are taken to bc in the
midplane of the lamin;ttc. The displacements (11,.H:.t+) at it point (.u, _r,:) in the laminate
:irc il!GlltIlL!tlt0 hC Of tIlC fOWl (SC0 Retldy, I9K7).
where (it’, W) are the dis~I~i~enl~nts of a point (x. ~~0) on the refcrcnce piane of the
laminate, and U and Fare functions which vanish on the reference plane :
U(.L
_I’, 0)
=
w.
)‘,
0) = 0.
(21
In order to reduce the three-dimensional theory to a two-dimensjonal one, Reddy
(1987) suggested layer-wise ~~p~r~xirn~~tionof the variation of U and V with respect to the
thickness coordinate, z :
where ~‘and Pare undctcrmined cocl?icicntsand #‘are any piece-wise continuous functions
that satisfy the condition
$‘fOt ==O forall
j-
1.2 . . . . . fl.
(4)
The approximation in eqn (3) can also be viewed as the global semi-discrete hniteefemont approxim~~tions (Reddy, t%Mt, through the thickness. In that case (6’ denote the
global interpolation functions. and lti and t” are the gtobaf nodal values of U and fl (and
possibly their derivatives) at the nodes through the thickness of the laminate.
E. J
376
3. A MODEL
Delamination
B.MUIEXOand 1. ‘1. REDUY
FOR THE STUDY
buckling in laminated
well recognized as a limiting
Modeling
kinematical
importance
of composite structures. While the
to the correct evaluation
the cost of analysis precludes the use of three-dimensional
section deals \vith the formulation
deiaminations
plates subjected to in-plane compressive toads is
factor on the performance
accuracy of the analysis is of paramount
in composites.
OF DELAMIN.ATIONS
of a laminated
of damage
models. This
plate theory th:lt can handle multiple
in composite plates.
of delaminations
in laminated
composite
plates rcquircs an appropriate
description to allow for separation and slipping. This can be incorporated
the layer-wise
theory by proper modi~c~lt~on of the expansion
of the displacements
into
(I)
through the thickness. The layer-wise theory can be extended to model the kinematics of a
layered plate, with provision
for delamin;~tions.
by using the following
espansion of the
displacements through the thickness of the plate:
whcrc the step functions
/I’ arc cornputcd in tcrrns of the f fcavisitlc step functions
In eqn (5) C/J’(:)arc iinoar Lagrange interpolation
functions,
iV is the nunibcr
ii as:
oflayers
used
to model the iaminatc and D is the number ol’dcl;~minatiolls. Thcjumps in the displacements
at thcjth delaminated interface arc given by L)I, li’ and II”. Using the step functions H’(z).
wecan modri any number ofdciamin;~tio~~s through the thickness ; the number of~Iddition~l1
variables is cyuat to the number of ci~larltifl;ltiOrls
the displaxmrnts
considercci. At ~I~l~lnlin~Ite~ intcrfxcs,
allowing for separation ant1
on adjacent Inyers remain indepdent,
slipping.
Although nonlinear etTccts arc important,
to be so large as to require a full nonlinear
rotations and displaccmcntsarc not expected
analysis. Only the von Kirmlin
in the kinematic equations needs to bc considcrcd.
The linear strains of the theory itrt’
nonlinearity
377
Delaminationin composite laminates
(8)
where the underlined terms are due to the introduction of the delamination variables Ui,
v’, w’.
The nonlinear portion of the strains are :
The virtual strain energy is now given by
(IOa)
where
HJ,,
cW,.,~
SC/,,,
6UDN,.
is the contribution
.
.
IS the contrtbutton
is the contribution
is the contribution
of
of
of
of
the classical linear terms
the von KHrmrin classical nonlinear terms
the new linear terms [underlined in eqn (S)]
the new nonlinear terms (underlined in eqn (9)].
(lob)
The contribution of the conventional displacements to linear terms is :
dA.
The contribution of the von Krkmin nonlinear terms is
(I I)
378
E. J. BARBERO and J. N. REDUY
The contribution
of the new linear terms is
(l3a)
where
h I
up&)
=
Mcr:.M\,,
=
-h :
(ur_. a,.,)H’(=) d:
h 2
(Iv:.
The contribution
of the new nonlinear
(13b)
(a,. u,.. o.,r)H’(z) d:.
-h
2
terms is
( I4a)
whcrc
It. ?
(Ml’. Iv:.,. M’,‘,.) =
The boundary conditions
The laminate constitutivc
(a,,a,,(T,,.)H’(=)H’(=)d=.
(l4b)
Force
u
N,n, + N,,,.n,
1
N,,n,+
11’
Qk~~,+Q,nI.
I(’
N!.n,+
1”
N&n, -I-Nin,
U’
Min,+
V’
M:, n, + M,i.n,
W’
Q:,n,+
N,.n,
N’_n,
Mi,n,
&.:nv.
equations can be obtained
(15)
in the usual manner (see Reddy.
1989).
The deformation
ticld obtained from the layer-wise theory is used to compute the strain
energy release rates along the boundary
planar growth.
h?
of the theory are given below :
Geometric
1988; Barbero.
I
of the delamination.
Delminations
i.e., the crack grows in its original plane. However,
usually exhibit
the shape of the crack
D&mmation
in composite laminates
379
may vary with time. For example. a crack shape initially elliptic usually grows with variable
Therefore. the crack growth is not self-similar. However. a self-similar virtual
crack extension will be assumed in order to compute the distribution of the strain energy
release rate G(S) along the boundary of the delamination.
The virtual crack extension method postulates that the strain energy release rate can
be computed from the strain energy Li and the representative crack length a as:
aspect ratio.
G(s) = dCi/da.
(16)
In the actual implementation of the virtual crack extension method, however. eqn (16) is
approximated by a quotient AC:I’A~that in the limit approximates the value of the strain
energy release rate G. In the Jacobian derivative method (Barber0 and Reddy, 1990). the
G(s) is computed without approximating the derivative, so it does not require that we
choose the magnitude of the virtual crack extension Au.
The boundary of a delamination is modeled as boundary conditions on the delamination variables C?. b”. and It”. by setting them to zero. A self-similar virtual crack
extension of the crack (delamination) is specified for the nodes on the boundary of the
delamin~~tion, i.e.. the two components of the normal to the delamination bound~lry are
specified for each node on the boundary. The JDM is then used at each configuration (or
load stop) to compute G(s) from the displacement field.
The gcncrali~ed displacements (it, I’,IL’,[I’,f!‘, Ii’, I/‘, IV) are expressed over each
clement iISii linear ~~)tIlbitl~lti~)n
of the two)-diInension;ll interpolation functions $, and the
nodal values (II,,I’,,S(‘,.f/i, C’i.I”, I”. CC”)as follows :
clement. Using eqn (17) in eqn (IOa), we obtain the
I
(184
A
‘
where
[A’;‘= ftt:,~f....,tZ m.c’)
nr
and the submiltri~es [X-“1. [A-,“], [X-j’]. fk&‘], [k:‘t, [k)/], fkj:j with i,j= I,...,#
and
f-,s = I, . , , , D arc given in Barber0 (1989). The load vectors {q}, (4’) _. . fq”), and
:fl’; . . . . , (4”; are analogous to {A;, (A’). . . ., {A’] and {A’], . . . , {AD}, respectively [see
eqn (IXb)]. The nonlinear algebraic system is solved by the Newton-Raphson algorithm.
The components of the Jacobian matrix arc also given in Barber0 (1989).
The nonlinear equations are linearized to formulate the eigenvalue problem associated
with bifurcation (buckling) analysis:
380
E. J
B~~tmo
and J. \
= 0
([K,;,]-i.[k;;])[email protected]
where [&,I
matrix.
is the linear part of the direct stiffness
-I
REUUI
SL’%lERIC.-\L
ilY)
matrix and [KG] is the geometric stiffness
RESCLTS
Several examples are presented in order to validate the proposed formulation.
lytical solutions
Ana-
can be developed for simple cases and they are used for comparison with
the more general approach presented here.
A thin layer delaminated from an isotropic
square plate (Rexural
concentrated load P at its center is considered. The delamination
square with side Za. The
rigidity.
D) by a
is also assumed to be
base laminate is assumed to be rigid with respect to the thin
delaminated layer. An analytical solution
rate can be derived assuming
for the linear detlection and strain energy release
that the delaminated layer is clamped to the rigid
laminate. Due to the biaxial symmetry,
a quadrant of the square delamination
base
is analyzed
using 2 x 2 and 5 x 5 meshes of nine-node elements. Either the clamped boundary condition
is imposed on the boundary of the delamination
closed delamination
or an additional band of elements with a
is placed around the delaminated arca to simulate the nondclaminatcd
region. Roth models product consistent results for transverse dctlcctions and accragc strain
cncrgy rclcase rates. A tint mesh is ncccssary to obtain ;I smooth distribution
cncrgy rclcasc raw 6 along the houd:~ry
P = IO cornparcs well with the ~~n;~lytic~~lsolutiw
mixirnurn
dclarnination
of the applied Iwd
of the strain
of :I scluilrc rlcl:~mination. The linear solution
(WC Fig.
for
I). The linear and nonlinear
opening Cl’antl avcragc strain cncrgy rclcasc rate G,,, as ;i function
i’ arc show3 in I.‘ig. 2. It is cvitlcrit that the nwnbranc
1
_
wO/a’P
G.“O/P’
wO/o’P
---___---
_
-
-
Nonlinear
Linear
strcsscs caused
Delamination in composite laminates
381
by the geometric nonlinearity reduce the magnitude of the average strain energy release
rate considerably (Whitcomb and Shivakumar. 1987).
In this example we consider an isotropic thin layer delaminated From a thick plate in
its entire width. Due to symmetry, only one half of the length of the plate strip is modeled
with the cylindrical bending assumptions and a nonuniform mesh of seven elements. The
cylindrical bending assumption is satisfied by restraining alI degrees of Freedom in the _V
direction. The base laminate is considered to be much more rigid than the thin delaminated
layer so that it will not buckle or deflect during the postbuckling of the delaminated layer.
First. an eigenvalue (buckling) analysis is performed to obtain the buckling load and
corresponding mode shape. Then a nonlinear solution For the postbuckling configuration
is sought. Excellent agreement is found in the jump discontinuity displacements CJ’and W
across the delaminiltion. The values of delamination opening W and strain energy release
rate G are shown in Figs 3 and 4 as a Function of the applied load. where E,, is the critical
strain at which buckling occurs For a delamination length of 7,~(see Yin. 1988).
Axisymmetri~ buckling of a circular, isotropic, thin-~lrn delamination can be reduced
to a ono-dimensional boundary-value problem by means of the classical plate theory (CPT).
One quadrant OFa square plate of total width 2h with a circular delamination of radius a
i
0.2
5
I-
0.0
l
x
1.4
Closed-form
Present
study
$3
1.8
2.0
N./b
l:ig. 3. M;~nimum drl:unin:ttion opening IV liar ;Lthin tilm buckled delamination.
N, 1.6
Strain
energy release
rate
Fig. 4. Strain cncrgy release rate for buckled thin Mm delamination
382
E.
J.
BAABEHO and f.
S.
R~uuv
is modeled. The layer-wise elements are capable of representing discontinuities
of the
displacements at the interface between layers. The symmetry boundary conditions used are :
tI(o.Jg = tr’(O,_r) = c:‘(o,J.)
r(x.0)
= 0
= z~‘f.r.0) = C”(.u.o)
zz
0
bvith I’ = 1. , . . .V and
of delaminations
j = I.. . . D; where :c’ is the number of layers and D is the number
through the thickness ; in this example. we have D = I. The boundary of
the delamination
is specified by setting the jump discontinuity
zero on the boundary
boundary
of the delamination
of the plate is subjected to the following
state of axisymmetric
boundary conditions.
tt’( h. J’) = C”(k,r)
= 0; N,(h,_v) =
r’(.r.h) = r*‘(.U,h) = 0; N,(.\..h)
distributed
The
which produce a
:
stress on the circular delamination
where fi is iI uniformly
variables tTJ. V’ and W’ to
and wherever the pfate is not delaminated.
-,G
= -g
compressive force por unit length. The same material
propcrtics are used for the delaminated
layer of thickness t and for the substrate of thickness
(Ir - I). To simulate the thin-film assumptions, a ratio I/,/r = 100 is used. First an eigenvalue
analysis is pcrfbrmcd to obtain the blic~lin~ load and corrcspondin~ mode shape. Then :I
Nowton-Raphson
solution for the post-buckling
is sought. The Jacobian derivative method
(see Rarbcro iInt1 Reddy, 1990) is used at cquilihrium
of the strain cntrgy
cx;~nlplc, C;(S) is a constant.
Itt this cxanlplc
analytical
hccausc rwrlt~
in the litcraturc. Compared
Thcrdorc,
A quasi-isotropic
n:ttion
ol’ Jiarnrtcr
tlistributcd
ol’;~ thrcL.-dirncnsic,nal
in-plane loacl A’, , The cxamplc
analysis arc available (Whitcomb.
198X)
G(x)
varies aiong the boundary
of the dclatnination.
[+45iO/90]
01’total thickness h = 4 mm and a circular delumi-
3tr located at : = 0.4 mm is considered.
arc those ofASJ/PEEK
in a square plate
IWittd
to the last c.uamplc. this problem dots not admit a11axisym~lcfric
the distributiotl
laminate
For this
solution (see Yin. 1085).
wc analyze a circular ~~c~~iflli~?;lti~)n,
centrally
of total width 2~ a11d subjcctccl to ;I uniformly
solution.
of the dcfamination.
Its value, shown in f:ig. 5 as ;I function of the appficd load, is
in cxccllcnt agrccnicnt with the appronimatc
is cotlsiclcrcd
solution to compute the distribution
rclcasc rate Ci(.c) along the boundary
The material
propertics
usrd
: E, = 134 GPa, El = 10.2 GPa, (21, = 5.52 GPa, G:> = 3.43 GPa,
“I .’ = 0.3. As is well known,
the quasi-isotropic
laminate
exhibits
equivalent
isotropic
Delamination in composite laminates
383
behavior when loaded in its plane. The bending behavior, however, depends on the orientation. To avoid complications in the interpretation of the results introduced by the nonisotropic bending behavior. an equivalent isotropic material is used by Whitcomb (1988).
where the equivalent stiffness components of the 3-D elasticity are found from the relation :
c,,=g’t, wk.
Due to the transverse incompressibility used in this work. it is more convenient and
customary to work with the reduced stiffness. Equivalent material properties can be found
directly from the A-matrix of the quasi-isotropic laminate as follows. First. compute the
equivalent reduced stiffness coefficients
where h is the total thickness of the plate and A,, are the extensional stiffnesses. Next, the
equivalent material properties can be found as :
An eigcnvalue analysis reveals that the deluminatcd portion of the plate buckles at
N, = 286,816 N m ’ for u 1: IS mm and at N, = 73,666 N m . ’ for 18= 30 mm. The
maximum transverse opening of the delamination as it function of the applied in-plane
strain c,~is shown in Fig. 6. The ditierences observed with the results of Whitcomb (1988)
are due to the fact that in the latter an arti~c~ally zero transverse dc~lection is imposed on
the base laminate to reduce the computational cost of the three-dimensional finite element
solution. The differences are more important for N = 30 mm, as indicated by the dashed
line in Fig. 6, which represents the transverse dellection w*of the midplane of the plate. The
square symbols denote the total opening (or gap) of the delamination, while the solid line
represents the opening reported by Whitcomb (1988) with H*= 0. The ditkrences in the
Fig. 6. Maximum transverse opening Wof a circular delamination of diameter 20 in a square plate
subjected to in-plane load N, as a function of the in-plane uniform strain.
total opening
IC’ and transverse deflection )I’ have an intlucnce on the distribution
strain energy release rata. as can be seen in Fig. 7. Both solutions
(i.c., solutions
of the
of 3-D
elements 2nd the present 2-D elements) coincide for rhc small delamination of radius r~ = I.5
mm. as shown in Fig. 8. For
iongcr valid. and difkrcnocs
the ktrgcr radius (I = 30 mm, the xsumption
coincide and the shapes of the distrib~~t~~~r~~
of C; are yuitc similnr.
at kttst fwo efcnWnts &SC
to the ~j~l~~~~~jl~~ltj~~n
hOttntfi1~~.
sudden changes in dcflcrtions
and stqe
in ;l narrow
lllllst
br:
Mesh r&ncmcnt,
ttszA
rvgiort &xc
to
:tcwitrlt
of the ciCl:tltlillilti~~I1
with
hbf
lhc
to the ~~~l;~~~?ir~~lti~~ll
boundary, similar to the pbcnomcna dcscrihcd by fktdncr ( I 954). Distributit)ns
cncrgy rclcaso rates G(s) along the bound:iry
X for the dilkrcnt
II’ = 0 is no
can bo obscrvcd in Fig. 8, ;tlthough the nxiximunx values of G
:tI’c sllon
4th~
tl it1 i:igS
strain
7 ilnd
delamination
radii, (I = I.5 mm antI LI = .X1 mm. rcspcctivcly. Thu value
s = 0 corresponds to (x = II, 1’ = 0) and .c = u~T,‘:!to ( v = 0, JS= u). Ncpli\c v;tIucs ofG(s)
indicate that energy should hc provided to :rdvancc the dcl;lmin;ltion in that direction.
Negative values ilre obtained ;IS ;L rustlIt of’d&~~in;~ttiti
eliminating the contribution
of mode I ot’ fixture
analysis does not include Wntitct
by ~hit~on?b
(iY88).
not ~~~n~~i~nnt~y&ctcd
constraints
the strain energy r&xc
surl’x~s
that CO~W in COIIIX~,
111~
hut not ol‘ motlcs I I and I I I. The present
Tinti therefore luycrs may ovcrlnp. AS noted
rate G(s) in the rqion
by imposing conlttct constraints
without
overlap is
on the smaft overfap W-2.
Delamination
385
in composite laminates
7.0
6.0
5.04....,....
-0.5
-1.0
load
4.0
ratio:
r =
. . . .. . .
..I
0.5
1 .o
bW’Jy)/(Nx+Ny)
Fig. 9. Buckling load versus the ratio of in-plane loads M, and N, for a circular delamination.
in.. and the delamination
is located at a distance
I = 0.005
eigenvalue analysis is used to obtain the magnitude
thin delaminated
in. from
the surface. An
of the in-plane load under which the
layer buckles. Due to the orthotropic
nature of the material
example. it is interesting to study the elTcct of diRerent combinations
used in this
of loads N, and IV,.
Let US denote the load ratio as
N,-N,
r=N,+N,.’
The magnitude of the buckling load as a function of the load ratio r, with - I < r < I,
is s,hown in Fig. 9. The distribution
of the strain cncrgy rclcasc rate G(s) along the boundary
s = (14. 0 < C/J< n/2 is plotted in Figs IO I2 for thr load ratios of r = - I. 0. and I. and
for several values of the applied in-plane load. in multiples i. of the buckling load N,.,. For
r = I (i.e., N, = N and NJ = 0) it is clear that (see Fig. IO) the delamination
propagate in (I the direction approximately
of load in the direction
perpendicular
perpendicular
is likely to
to the load direction. Introduction
to tibers causes the maximum
value of G to align
closer to the .r-axis. Note that both the magnitude and the shape of the distribution
of G(S)
change as the load ratio changes (see Fig. 13). The plots suggest that the propagation
arrest of delaminations
dominant
is greatly intlurnced
by the anisotropy
of the material
load.
10.0
‘1
I
X N/N,
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 6.0
-5.0-v.
s
Fig. IO. Distribution
1
I
-
.
orp
.
,
tmml
.
,
,
.
,
(Oqwr/Z)
of the strain energy release rate along the boundary
for various values of applied load N.
or
and the
of a circular delamination
E. J BARHRO 3rd J. N. Rmn
386
10.0
‘“1
5
CT
0.0
#“-.
\tf:
Q
-10.0
-20.0
0.0
-
x =
I
1.0
5 =
Fig.
Ii. Distribution
afthe
N/N,
t
,
2.0
I
3.0
a9
*
4.0
ifflm~
3
5.0
7
6.0
j
7.0
_
a
3
(0<9~n/z>
strain energy release rate ~lonp the bounJar~
for the ratio I = 0.
ofa circular
-10.0
-15.0
-20.0
0,O
1.0
5
s =
2.C
=
3.0
oy,
a9
4.0
6.0
[mm1
(Ocpn/2)
Cmml
i 0 8 3
(CH9<n/2)
Jvlamination
Delamination tn composite lemirtafes
5. SUMMERY
AND
387
CONCLUSfONS
A layer-wise theory and associated finite-element mod4 for the study of detaminations
in laminated composite plates is developed. The same displacement distribution in the
individual layers is capable of representing displacement discontinuity conditions at interfaces between layers. The finite element model predicts accurate distributions of strain
energy release rates along the boundary of delaminations of arbitrary shape. The model
can be used to study multiple delaminations through the thickness of the plate.
The layer-wise laminate plate theory provides an tidequnte framework for the analysis
of laminated composite plates. Particularly, the layer-wise linear approximation of the
displacements through the thickness and the use of Heaviside step functions to model
delaminations prove to be an effective approach for an accurate analysis of local effects in
laminated composite plates. lit must be noted, however. that the computational cost of the
pro~sed analysis makes it unattractive for prediction of global behavior when compared
with conventional theories. For the prediction of local ef%cts (i.e., delaminations, interlaminar stresses. etc.), the theory and formulation presented in this study shows its potential
as an alternative to three-dimensional analysis. The model can also be used in a globallocal analysis scheme wherein the local regions are modeled using the layer-wise theory and
global regions are modeled using less refined theories, say the first-order laminate theory.
Transition elements must be developed to join regions modeled by the laycr-wise theory to
regions modeled by less expensive theories in global-local analysis procedures.
ft is expected that accurate stress distributions obtained with this type of analysis,
along with meaningful failure theories. will enable rcnlistic prediction of failure initi;lti~~n
and prop~ig~~t~onin composite Laminates. Also, the layor-wise plate theory can be used as
a pos[-processor to enrich the stress prcdicti~?n of the lirst-or&r &car dcfl?rn~;lti~?lltheory.
Barbcro. E. 1.1 *Reddy. J. N, and Ti$y. J. L. (1990~).A gtxrr:tl two-dimensional theory of Iaminatcd cylindrical
shells. /ll.J,.t J. t&3). 544 ,552.
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using a generalized plate theory. ltrr. J. ,Vunlrr. .f/crh. ErI,yrrq29, I .I4
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in Meclrmtic.r,
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REDRY
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