ComputationalMechanics16(1995)17o-1899 Springer-Verlag1995
Strength reduction and delamination growth in thin and
thick composite plates under compressive loading
B. P. Naganarayana, S. N. Atluri
17o
Abstract In this paper, the coupled local-global buckling
behavior in laminated composite plates with elliptic
delaminations and the associated mechanisms of delamination
growth under compressive loads are critically examined. The
J-integral technique is used for delamination growth prediction
in terms of pointwise energy release rate distribution along
the delamination edge. A Multi-plate model, in conjunction with
a 3-noded quasi-conforming shell element, is used to model
the delaminated plates. The incremental equilibrium equations
are set up based on total Lagrangian formulation. The solution
strategy incorporates Gauss elimination in a cycle of
Newton-Raphson iterations and is augmented with automated
arc-length controled load incrementation and equilibrium
iterations; and with automated post-buckling path tracing based
on a linearised asymptotic solution. The effects of structural
parameters such as delamination thickness, size and shape, on
the post-buckling behavior and on the delamination growth are
critically examined.
1
Introduction
It is known that delaminations are the most frequent causes
of failure of laminated structures, particularly under
compressive loads. The presence of delaminations leads to
a reduction in the overall buckling strength of the structure.
In addition, the delaminations tend to grow rapidly under
post-buckling loads, causing a further reduction in structural
strength, and finally, leading to fatal structural failure. In this
paper, in an effort towards gaining an understanding of the
possible strength reduction due to the presence of the
delaminations, and the mechanisms of their growth,
a parametric study of the influence of the thickness, shape and
size of elliptic delaminations in plates is undertaken.
The Reissner-Mindlin theory of plate flexure is used to model
the undelaminated plate and the delaminated sublaminates
Communicated by S. N. Atluri, 10 January 1995
B. P. Naganarayana, S. N. Afluri
Computational Modeling Center, Georgia Institute of Technology,
Atlanta GA 30332-0356, USA
This work was supported by the FAA to the Center of Excellence for
Computational Modeling of Aircraft Structures at Georgia Institute
of Technology, and in part by a grant from ONR. These supports
are gratefully acknowledged. The first author also acknowledges the
support by the INDO-US Science and Technology Fellowship under the
auspices of United States Agency for International Development and
Department of Science and Technology, India
Correspondence to: B. P. Naganarayana
independently; and the junction between the delaminated
sublaminates and the undelaminated plate is modelled by using
a two-dimensional generalisation of the so called Multi-Domain
Model, presented earlier for one-dimensional problems by Chai,
Babcock and Knauss (1981) [Naganarayana and Atluri (1994)].
A quasi-conforming 3-noded plate element [Huang, Shenoy
and Atluri (1994)] is used to model the delaminated structure.
An automated incremental post-buckling solution strategy,
incorporating an arc-length controlled load incrementation,
and branch switching based on a linearised asymptotic solution
[Huang and Atluri (1994)], is utilised while using the
displacement type finite element model. The stresses are
post-processed to obtain pointwise energy release rate
distribution along the delamination front, by using the adapted
J-integral approach [Naganarayana and Atluri (1994)] for each
load increment.
In this paper, the influence of the structural parameters such as the location of the delamination with reference to the
plate mid-surface, the shape and the size of the elliptic
&lamination, and the structural thickness, material orthotropy,
fiber orientation and lay-up sequence in general laminates - on
the coupled local-global buckling phenomena, reduction in
structural strength, and mechanisms of delamination growth
is critically examined as the delaminated plate is subjected to
compressive inplane loads.
2
Multi-Plate model
In this paper, we shall consider a general laminated composite
plate with a single delamination of an arbitrary shape and
location subjected to arbitrary buckling loads (Fig. 1). The
structure is modelled using the multi-plate model
[Naganarayana and Atluri (1994)] where the delaminated plate
is assumed to be assembled with three distinct plates-(1)
Laminate: the undelaminated zone 12~ (2) Delaminate: the
thinner side of the delaminated zone 12(2) and (3) Base: the
thicker side of the delaminated zone 12(3/. The three plates have
midsurface areas sr thicknesses t(~ boundaries 812{i);and
midsurface boundaries 8 d (~ for i = 1, 2, 3 respectively (Fig. 1).
The delamination edge is denoted by F. The Reissner-Mindlin
theory of plate bending is used for modeling each of the plates
and the interface between them. Thus, for each plate, the
3-dimensional displacement field (U {U1U2U3}) can be
expressed in terms of the corresponding midsurface
displacement (u - {u~u2u3}) and rotation (0 = {01020}) fields
=
as,
u (') (x~, x3) = u {') (x~) - x~i) 0 (~)(x~)
(1)
#
DeLaminote fl~2~
Delaminotion"~F
I
X2
L
;
Ondetominote f~m
Iw
N
Bose
Assumed deformotion at pLote-junction
(U I~l = UI~I= Ul~l)r
Xl
a,
IX
_
(2>~
L~ ~ •
.
T
/
3
171
I
/
"
- -
"1
iI
I
I
i
I
Iq)A t
-
X3 T/
XF, - - - x ,
I
(3)
h'
-l/._
Ft:l.0N/mm
t
t
t
/
,
Fig. la-r
.
.
.
I
r
I
e
},1
.
iI
!
L
I*[
I
I
I
XF~
b
- -
t!
//U:31
t"
_
/I
/ Um
I
Multi-plate model, a Delamination in an arbitrary plate, b Clamped square plate with central elliptic delamination, c Section x-x
where x~~ (~ = 1, 2) are the inplane coordinates and x~~ is the
thickness coordinate for the ?h (i = 1, 2, 3) plate. The structural
continuity at the delamination front F is maintained by
assuming the deformation to be unique at the junction of the
three plates i.e. U ~ = U (2) = U (3) on F.
For each plate, the inplane components (en, e=, q2) and the
transverse shear components (e~3, e23) of the 3-dimensional
Green-Lagrange strain tensor (including large deformations)
can be expressed in terms of the membrane, flexural and
transverse shear (engineering) strain c o m p o n e n t s - e , ~,
7 respectively - as,
where the material constitutive parameters E!!) are functions
of the thickness coordinate of each plate x~i/~JGenerally, for
a laminate with orthotropic layers, E!!
1 are assumed to be
U
piecewise constants over the laminate thickness.
Now, for each plate, the strain energy density (per unit
volume) can be calculated using Eqs. (2), (3) and (4) as,
1
,•(i)
~a=~[u
~+
~(u) =
Ufl,.
+ u~y~,e,
1(i)
W(~)=
[o-. ( e +
and the total strain energy stored in the i ~hplate is given by,
~(1
= l [ u .,fl -F uB,. -I- u3,=u3,fl] (i) - x(i) !2 tr~ o:,fl -I- O fl,~.](i)
~
_ ~ [u3,~
~[ 3,~+u~,3]~'~-~
- 0J0
(3)
respectively where, again, ~, fl = 1, 2 and i = 1, 2, 3.
Assuming orthotropic material behaviour for each lamina,
the inplane stresses ~r/~)= {G~ cr22cr12}l~/and the transverse shear
stresses r (~ = {z13z23}(~ are related to the corresponding strain
components as,
f:t
i) _~
"En
EI2
El6
0
0'(i)
El2
E22
E26
0
0
El6
E26
E66
0
0
0
0
0
E44
%
0
0
0
E45 Ess
/
(6)
i= 1 \~r
(2)
1 v(i)
= ~.~
-(2i
89 N - e + M - r + Q. r)U/d~r
=
~(i) __
_ _1 U
Q)
~ ~ W~Od
i=1 \
~_ ~(i) 4 - "v(i)~"(i)
(5)
x3x)](~) + [r. r] (i)
where N - {N u N= N12} are the inplane stress resultants; M -{M11M22M~2} are the bending moment resultants; and Q =
{Q13Q23}are the transverse shear stress resultants given, for each
plate, by,
(i)
.(i)
(i)
(7)
and
(4)
(Akl;Bkl;DkS I= f E~i)(x3)(1;x3;x2)dx3
and
G~
tl
as k,l= 1,2,3 correspond to
correspond to i, j = 4, 5.
=
~E(o
mk(x3)dx3
ti
i,j=
1,2,6 and
re, n=1,2
172
If f(0= {f~ f2 f3} (~ are reference body forces acting on D(~
F = {F~ F2 F3}/~>are the traction forces acting on ~d(~ and 2 is the
current load factor, the potential of the current external forces
is given by,
(hAW(U) - A2bB (U) - (62W(U) -- A2c3B(U)) = 0).t(u,~)
(14)
2 B ( U ) = 2 ~ =( ~1 f .
3q- [ K t ] . A q - A23q.F = 0
i)
UdD@
~.Q(i)
F.Ud~)
where, assuming that f and F3 do not vary over the plate
thickness,
"f(i)= ~ f(Odx3; ~(i) = ~ F(i)dx3; ~(i) = ~ F(i)x3dx3.
ti
ti
For a typical finite element application, incremental
equilibrium Eq. (14) can be written as,
(15)
where Aq are the incremental nodal displacements; F are the
discretised reference nodal forces (typically as specified in the
input for the problem); and [Kt] is the tangent stiffness of the
system. The incremental equilibrium equations thus become,
[Kt]-Aq - A2F = 0
(16)
On the other hand, the total equilibrium for the system
(Eq. (12)) can be written, in finite element context, as,
ti
Applying the minimum total potential energy principle
(cSYi = c~( W (U) - )3 (U)) = 0), we obtain internal equilibrium
equations:
[Ks] .q + 2F = 0
(17)
where [Ks] is the secant stiffness (or simply stiffness) matrix
for the system.
{N~/~,~+ j ~ = 0 ; M~,~+T~=O; T~,~+f3=0}~(~/
(9)
For each incremented external load vector, the incremental
equilibrium equations (Eq. (16)) are solved in a cycle of Newton
the boundary conditions on the external boundary:
type iterations and using the total equilibrium condition
(Eq. (17)) for computing the residual load vector which in turn
{ G = 0 or N~.n~=F~; 0 ~ = 0 or M~.n~=lfI~;
is used for convergence check. The load increment is optimised
for each iteration using the constant arc-length criterion to
u3=O or T..n.=F3}o:g(~,
(10)
assure faster and definite convergence of the solution even while
and the equilibrium condition on the delamination front:
traversing limit points on the solution path. The presence and
location of an instability point is determined by the singular
{ : ~ ( N ) = 0 ; : ~ ( M ' ) = O; : ~ ( T ) = 0}r
(11)
nature of the tangent stiffness matrix at the instability point;
and the nature of the instability point - whether limit point or
where T~ = O~3 + N~u3 ~; (M')(~) = M(i) + h(~)N(~ ~ n (*) = (,)(1) __ bifurcation p o i n t - is determined in a heuristic sense by
(,)(2) _ (,)(3) with (,)(~)'corresponding to the value of (,) at the comparing the load increment and increment in the generalised
specified point on the delamination boundary of the i~hplate;
deflection across the instability point. If the instability point
and n is the unit vector normal to delamination front.
identified is a bifurcation point, a cost-effective branch
However, it should be noted here that the equilibrium Eq. (11) switching technique is used to follow the desired postbuckling
may be violated when the delamination edge intersects the
branch - the nonlinear fundamental state between two points
structural boundary and when the delaminated plies come into in the neighborhood of the bifurcation point is linearised to
contact after global buckling.
obtain an asymptotic solution on the desired postbuckling
solution branch. Refer to Huang and Atluri (1994) and
3
Naganarayana (1994) for the automated post-buckling solution
Incremental solution
algorithms.
Here, we shall derive the incremental equilibrium equations
for an elastic conservative system D under linear loading. The 4
generalised deflection B (U) is assumed to be a linear function J-integral adaptation for pointwise energy release rate along
in U; and the load factor 2 is assumed to be an arbitrary constant. a planar delamination front
In an incremental solution strategy, the solution is known at
In the case of a delamination, the growth is assumed to be along
the initial point (U, 2) of the current increment. Then by the
the interlaminar zone parallel to the midsurface of the plate (i.e.
principle of minimum total potential energy, we have,
the crack cannot shear into the neighboring laminae). In other
words, the crack is assumed to grow in a self-similar sense
(3W(U) -- 2c3B (U) = 0
(12)
[Naganarayana and Atluri (1994) ].
The pointwise energy release rate for self-similar
Now, the equilibrium at the unknown point (U + AU, 2 + A2)
3-dimensional crack growth (~(_F)) is defined as [Atluri
requires,
(1986)1,
3W(U + AU)
- (2 + A2) cSB(U + AU) = 0
(13)
Expanding the incremented functions W and B and neglecting
the higher order incremental terms; and by using the
equilibrium conditions at the initial point (U, 2) (Eq. (12)) we
get,
N ( r ) A F = L i m i t ~ L[ W e l - G ,_ n p OU~q
~JdA
e~O
Ae
A1
-- ~ 6,2~
A2
~0~ dA
+ i G2 ff~x~
891
(18)
x2
r
aX3
7
73
,n
/ ~ - ~ ~ 1
~
xl
,n
F
A2
where A i (j = 1-9) are the segments forming the surface area
A e of the rectangular tube (Fig. 2c). Since the Reissner-Mindlin
assumptions (#13,#23 are constant over plate thickness; and
#33 = 0) are used for the element formulations as well as to
achieve displacement continuity at the delamination edge; and
since fi= {00 +_1} on the segments Aj, j = 4 - 9 (see Fig. 2c),
the integral in Eq. (19) vanishes over the segments Aj, j -- 4-9.
Now, since fi= { + 1 0 0} on the segment A~ and fi = { - 1 0 0}
on the segments A2 and Ar Eq. (19) becomes,
(20)
AF-~0
Now, carrying out the integration through the thickness for each
plate, we get,
b
As
§ ....
A~
6~(F)AF'= j" ffg[lk7,r- (]k[s:fi. 1+MI~O~, 1 q-(7)13U3,1)1dr' (21)
AF
,._.2k_.a:
x~l
11
{1)L.~. . . . *'-"
. ._3.
X ~,
A~
Therefore, as A F ~ 0, the pointwise energy release rate at any
p o i n t on the delamination front Ng' computed from the
Gauss-point variables, in a finite element model using constant
strain elements, is given by,
i ....
. _ .9.t-
. -t
xL!'A.I
Xg'A31
. -
Aa
13
Ng(F) = fig [ ~/-- (/q, d2~,~+ M~O~,l +
Av
Fig. 2a-c. J-Integral for delamination growth in a plate model
where, c~, fi=1,2,3; A e is the area of the tube of radius
e enclosing the crack front; A1and A2 are the areas covering the
ends of the tube (Fig. 2a); and ri, #, 0 and o7are defined in the
crack tip coordinate system 2 (Fig. 2b).
For the present problem of delamination progression, it is
assumed that the material is homogeneous along the :~1, such
that (8W/8,~)e~p~, = 0, in other words,
8W
0XI
8G~
[~W~
O-aft~1AV ~ 1
(22)
where, Yg (,) = ( * )g(1) - - ( * )g(2) - - ( * )g(3)and (*)g(,)corresponds
to the quantities (,) evaluated at specified points on the annular
surface. For example, in a finite dement analysis, these specified
points would be preferably the optimal stress recovery
points - normally the Gauss points corresponding to reduced
integration [Barlow (1976); Naganarayana (1991)] - i n the
adjoining element of the i'h plate nearest to the delamination
front F. The local stress resultants (N, M, Q) and displacement
gradients (~.~) can be obtained from their global Cartesian
counterparts (N, M, Q; Urn), by applying the regular tensorial
transformations between the reference coordinate system x and
the crack tip coordinate system ~.
5
Numerical experiments
8G~
explicit
which case, the path-independence of the ]-integral
is maintained [Atluri (1986)] for opening mode of crack
growth and hence the tube can have a cross section of any
shape.
Consider a rectangular tube enclosing the delamination front
and passing through the nearest stress recovery points (S(~ of
the adjoining elements. Note that, the integrals over the areas
A1 and A2 nearly cancel each other in a constant strain/stress
element model, since the quantities like 6"and 80~/0~ do not
vary in the neighbourhood of a point in an element domain.
Then, Eq. (18) for the opening mode of delamination growth
becomes,
in
(19)
AFj=I Aj
(~)13/~3,1)]
In this section, we shall analyse a number of delaminated
structures for their post-buckling behavior and the associated
delamination growth.
5.1
Problem description
An isotropic square plate of edge length L with a central elliptic
delamination (Fig. lb), is considered for the following
numerical experiments. The plate is subjected to biaxial
compressive loads and its boundary is assumed to be clamped
against out-of-plane deformations. One quarter of the plate is
modelled by imposing appropriate symmetry conditions. 264
shell elements are used for the undelaminate plate and 192
elements each are used for the delaminate and base plates
respectively. The reference applied biaxial compressive loads
are assumed to be of unit intensity (Fs = 1.0) and the equilibrium
equations are solved at each load step for an applied load
F = 2Fs, where 2 is the corresponding load factor (Fig. lb).
173
200"
strength of the delaminate plate (G(
[] 'AnaLytical' solution /
-
"~l~rFx/t,))is given by
(23)
IOO
The local buckling strength of the delaminate plate obtained
from the finite element analysis compares very accurately with
the analytical estimate (Eq. (23)) as shown in Fig. (3a).
Further, assuming that the post-buckling deformation is
axisymmetric and nearly linear in the neighbourhood of local
buckling point, the pointwise energy release rate is given by
[Evans and Hutchinson (1984)],
.i.r
0"
0
174
0.1
0.2
0.3
t2/tl
0.~
0.5
2-'o1
(ffrb(V)
A,*: 3./.55
-
-
( 1 - v2) t 2
(1.8-28-5~v)E (r _ 0-2)
(24)
where rr0( = 2FJq) is the actual stress level at which the energy
release is being computed.
The ratio, ffFe/~qrb, is plotted along the delamination
1.60
periphery, in Fig. (3b) for a case of very thin delaminate
1./.0
~
1.084
configuration (tE/t~ = 0.01). It can be observed that, fife is close
,%*= 2.105
to Nrb when the post-buckling loads are in the close vicinity of
1.20'
x*= x/;%
local buckling point (i.e. 2* = M2tcr-- 1.0). However, ffrb is
1.00
under-estimated when compared to ffF~ even when 2 -~ )]cr"This
0o
3bo
6'0~
9'00
8
is because, in the present problem, though the &laminate plate
is very thin when compared to the total laminate thickness
(t2/t 1 = 0.01), the base plate is flexible as opposed to the rigid
base as considered in Evans and Hutchinson (1984).
Laminate is also thin when compared to its edge length,
L
(tJL
= 0.05). Hence, the finite element model represents
[ ] Represents'analyticaL'
a reasonably flexible laminate and base plates. The deviation
Energy Release
increases as the buckling load increases beyond its critical value
[ Evans and
Hutchinson (198l,) ]
for local buckling of the delaminate plate. This is because, the
analytical solution (Eq. (24)) is based on the assumption of
[ ] Represents 'actual'
E
o
Energy Release
a quasi-linear post-buckling behavior for the delaminate plate.
Z
But, in practice, particularly when the laminate is thin,
post-buckling behavior of the delaminate plate is highly
'Normalised' displacement (A~)
nonlinear. Accordingly, much higher energy-release-rates are
Fig. 3a-c. Clamped square plate with central circular delamination
expected when compared to Nrb as shown in Fig. (3c). Note that,
(Validation of the model) a Criticallocal buckling strength. Comparison in Fig. (3c), actual stress ((r) and displacement (A) are
of F. E. solution with 'analytical' solution, b Pointwise energy release
'normalised' by the critical stress G and the associated critical
rate distribution Comparison of F. E. solution with 'analytical'
inward radial displacement A c respectively.
solution (t21t1=0.01; 2/r=0.125). C Effect of 'quasi-linear'
post-buckling behavior for delaminate on energy-release-rate
5.3
[] Represents 'analytical' energy release [Evans and Hutchinson
Effects of delaminate-plate-thickness
(1984)] [] Represents 'actual' energy release
The numerical experiment is repeated for varying delaminate
thicknesses, for a fixed laminate thickness (tJg = 0.05),
delamination size (a/L = 0.3), and aspect ratio of delamination
(a/b = 1.5). The mechanisms are examined for both
supplementary and complementary modes of post-buckling
5.2
deformations of the delaminated plate.
Validation of the model
The structure is assumed to be isotropic with Young's modulus
5.3.1
E = 6500 and Poisson's ratio v = 0.3. The laminate thickness
is chosen as tl = 0.05 L. The numerical experiment is conducted Supplementary modes of post-buckling deformation
For very thin delamination (t2/tl = 0.02), the delaminate plate
for a near-surface circular delamination with tJtl = 0.01,
undergoes a second mode of buckling before reaching the global
a/b = 1.0 and all = 0.3. Assuming that the base plate and the
undelaminated plates are infinitely stiff when compared to the buckling load level. Typical post-local-buckling modes of
deformation are presented in Fig. (4a) and (b) for first and
delaminated plate, the delaminate plate can be considered as
second mode respectively. The deflection at the centroids of
a clamped circular plate under the same radial compressive
the delaminate and base plates are traced in Fig. (5). For very
stress [Evans and Hutchinson (1984)]. Then, the buckling
= 1.80
X * : 2.793
0
-0.5
-1.0
5O
lo
2o
~u
30
~
a,
10
u
50
0
0
-0.2
-0.4
-0.6
-0.8
0
b
~
50
z,O
1.5
1.0
0.5
0
50
0
'-~
30
,
~
~
10
Fig. 4a-c. Post-local and post-global buckling deformation a/b = 1.50;
a/L = 0.30; tJL = 0.05. a Typical post-local-buckling deformation
(First Mode) t2/t1 = 0.02; )~= 58.5. b Typical-local-buckling deformation
(Second Mode) tE/t~= 0.02; )o= -48.8. c Typical post-global-buckling
deformation tE/t~ =0.20; )0 = 218.0
thin delaminate (t2/t ~ = 0.02), there exists a load l i m i t p o i n t at
2 = 92.3 as the delaminate plate undergoes second mode of
buckling (see Fig. (4b) and Fig. (5a)). However, for larger
delaminate thickness, the delaminate plate does not undergo
a second mode of buckling. On the other hand, the locally
buckled delaminate plate acts as a geometric imperfection for
the base plate and leads to premature global buckling of the
structure. As expected, the local buckling strength of the
delaminate plate increases as its thickness increases (see Fig. (5)
and Fig. (6)). However, the global buckling strength of the
structure decreases as the delaminate thickness increases (see
Fig. (5) and Fig. (6)). The reduction in the global buckling
strength due to the presence of delamination is nearly 50% (see
Table 1). In fact, for t 2 =< 0.2t~, the global buckling strength of
the structure is less than that of the plate with the delaminate
plate completely removed. This is understandable since the
locally buckled delaminate plate acts as a geometric imperfection
and can lead to a highly premature structural failure.
The energy release rate distribution is nearly uniform along
the delamination edge for very thin elliptic delaminate plate
in the vicinity of the local buckling point (Fig. (7)). However,
the energy release distribution pattern changes as the load factor
increases, and as the thickness increases. The maximum energy
release rate is observed to be in the vicinity of the direction
of maximum local-buckling-stiffness. In this case, the
delaminat e plate exhibits maximum stiffness against buckling
in the minor axis direction and hence the point of maximum
pointwise energy release rate is found to be at an angle of 75 ~
with reference to the major axis. As the delaminate plate
thickness increases the energy release rate becomes negative at
certain zones. This indicates, in a qualitative sense, that the
delaminate and base plates are in contact. This happens because,
for higher delaminate thickness, the local buckling point is very
close to the global buckling point (see Fig. (5f)).
For very thin delaminates, the energy release rate increases
upto the load limit point ()~ = 92.3); then decreases as the
delaminate enters the second buckling mode whence the
delamination starts closing; and takes a non-zero positive value
for )~ = 0.0. Then as the traction forces become tensile in nature,
the buckled delaminate plate starts unfolding from its second
mode of deformation, and tends to open the delamination
at its edge and hence the energy release rate increases from that
point onward. The variation of the maximum and the average
pointwise energy release rates with reference to the load factor
are presented in Fig. (8). Since, for very thin delaminates, the
energy release rate distribution is nearly even, the average
energy release rate is very close to the maximum energy release
rate.
The variation of the maximum and the average pointwise
energy release rates with reference to the load factor
are presented for varying thicknesses in Fig. (9) and
Fig. (10) respectively. It can be observed from Fig. (9) that,
for thin delaminates, the energy release rate is predominantly
due to the local flexural deformation of the delaminate plate
after the load level crosses its buckling strength. In fact the
energy release rate is very much negligible for 5[ < Z~crfor thin
delaminates (see Fig. (9a)). However, for thick delaminates, the
maximum energy release rate is positive - and comparable with
that with thin delaminate in post-local-buckling
conditions- even when the load level is lower than
corresponding local buckling strengths. In this case, the growth
mechanisms are predominantly due to the membrane action.
However, the maximum pointwise energy release rate increases
steeply as the global buckling of the structure approaches
(see Fig. (gb)), irrespective of the delaminate thickness; and
delamination growth is relatively more rapid for thinner
delaminate configurations.
For thin delaminates, the variation of the average pointwise
energy release rate with the load factor is similar to that of the
maximum pointwise energy release rate (see Fig. (9)). However,
for a very thick delaminate, the delaminate and base plates are
in contact for lower load factor on the average (i.e. the
average energy release rate is negative) as depicted in the
corresponding energy release rate distributions (see Fig. (7)).
This is because, the local buckling strength becomes very close
to the global buckling strength as delaminate thickness
increases (see Fig. (5)). In fact, for t J t 1 = 0.40, local and global
buckling occur almost simultaneously (see Fig. (50).
Accordingly, the pointwise energy release rate increases steeply
175
1.5
a~ t2/t ~: 0.02
'.C
u
7.5
I
~.o
!
/
b t2/t 1=0.05
w3 /
e
t2/t 1=0.10
/ /
6.0
5.0
://
-~ o.s
8.084
/
//
VO
W2
2.5
g
2.0
W3
(..2
- 100
176
- 50
0
50
100
5.0 84
d t2/tl:0.20
?
I
2.0
200
30O
6.0-
/
54.0
o=
3.0
100
e t2/tl=0.30
/
,
Z,.O-
/
5.0 84
260
300
f t2/tl= 0.40
4.0 84
2.0
[...
100
20
~.o
0
0
100
200
Load footer ,L
300
0 I ----~0
16o
260
Load factor L
360
Ok
10o
260
Load factor X
360
Fig. 5. Effect of delaminate thickness on load-deflection response a/b = 1.50; a/L = 0.30; tt/L = 0.05
as the load factor crosses beyond the corresponding local critical
value.
It is observed that, the thicker delaminates reduce the overall
global buckling strength of the structure (Fig. (6)), while the
thinner delaminations are more critical from the energy release
rate point of view (Fig. (9) and Fig. (10)).
5.3.2
Complementary modes of post-buckling deformation
Normally the locally buckled delaminate plate acts as
a geometric imperfection and causes a premature global
buckling in a supplementary (signifying that the deflections of
the delaminate and the base plates are in the same transverse
direction) sense (Fig. (4c)). However, when the delaminate plate
is thick, the eccentricity of the geometric imperfection is small;
and, as a result, the delaminate and the base plates may
snap into complementary (signifying that the delaminate and
the base plates deflect in opposite directions) modes of
post-global-buckling deformations (Fig. ( 11 ) and Fig. (12) ). The
load-deflection curves are plotted for the centroids of the
delaminate and the base plates in Fig. (12). The centroid of the
delaminate plate undergoes a displacement limit point as the
plate enters global buckling mode. Then both the base and the
delaminate plates, with comparable thicknesses, enter higher
order buckling modes. As they enter second buckling mode, the
centroids of both the sublaminates pass a load limitpoint as
shown in Fig. 12. It is also interesting to note that the critical
load factors, triggering local and global buckling, remain the
same whether the two modes are complementary or
supplementary to each other.
It is also interesting to study the energy release patterns in
the complementarymodes. For thick delaminate configurations,
the delaminate and base plates close the crack at many parts
of the delamination periphery, in supplementary modes
(Fig. (7)). However, they tend to open the crack at all points
in the complementary modes. All-positive energy release rate
distribution shown in Fig. (13) signify this observation. As
a result, though the maximum energy release rate is slightly
lower in case of the complementary mode (Fig. (14)), the crack
tends to open at all points on the average (Fig. (15)). Note that,
the crack tends to close on the average in case of supplementary
modes for higher delaminate thicknesses in post-globalbuckling range of loads (Fig. (10) and Fig. (15)). Thus, in case
of thicker delaminate plate configurations, the complementary
modes of buckling pose a serious threat of rapid delamination
growth and fatal failure. In case of thin delaminate plate
configurations, the residual strength of the base plate is still
comparable with the original strength; while both the base and
the delaminate plates may face failure almost simultaneously
(in case of complementary local and global buckling modes)
leading to catastrophic failure of the whole structure.
5.4
Effects of aspect ratio of delamination
The experiment is repeated for varying aspect ratios (a/b) of
the elliptic delamination with specified thickness (t2/t ~= 0.1)
and major axis length (a/L = 0.3) and fixed total plate thickness
(tl/L = 0.05).
A typical solution path for the transverse deflections at center
of delaminate (w2) and base (w3) is shown in Fig. (5c) for an
aspect ratio a/b = 1.5, depicting local and global buckling. It
is observed that the local buckling load increases as aspect ratio
increases (Fig. (16)). The global buckling strength also increases
as the aspect ratio increases. However, global buckling strength
Table 1. Reduction in global buckling strength of plate with elliptic
delamination: Effects of the delaminate thickness
200
(;~~ (t = t~))o~- (;~)~.
.... * 100
(;.~T(t = tl))oo~,
%lc~
'< 100
Rb -
,100
(,t~
= t3))ooo~
(2~r(t = t))~aIF~= 5.30n2Et3/12 (1 --v 2) (Timoshenko (1940)]
0.1
0.2
t2/tl
0'.3
177
0.4
Fig. 6. Effect of delaminate thickness on local and global buckling
strengths
becomes saturated asymptotically as aspect ratio increases
beyond 2.0 as seen in Fig. (16). The global buckling strength
of the delaminated plate is compared with that of the
undelaminated plate of full thickness and of base thickness in
Table 2. It can be noted here, again, that the delaminate buckles
locally prior to the global buckling; acts as a geometric
eccentricity to the global behavior of the plate; and hence leads
to premature failure of the plate. For the structure considered,
the global buckling strength reduces to nearly half its original
strength. In fact, the delaminated structure is nearly 30% weaker
than the structure with the delaminate plate being completely
peeled off. Thus, the reduction in the buckling strength of
the delaminated plate is very high due to the presence of the
delamination irrespective of its shape.
The variations of the maximum and the average point-wise
energy release rates with reference to the applied load are
presented in Fig. (17) and Fig. (18) for different aspect ratios.
As the aspect ratio increases, local buckling strength increases
and hence the energy release rate is lower at lower load factors.
But, as the external load increases, the energy release rate
increases sharply for higher aspect ratios. Thus, for
a delamination with higher aspect ratio, the onset of
delamination growth may be delayed, but the delamination
growth rate is much higher once the growth is set-in.
The location of the point on the delamination edge where
the point-wise energy release rate is maximum is traced
in Fig. (19). It is observed that the point of maximum
energy release rate occurs at an angle of 0m = 45 ~ with
reference to the major axis when a/b = 1.0. It is observed that
as the aspect ratio increases, maximum energy release rate is
experienced at 0~ = 75 ~ This is, again, due to the larger
crack opening at minor-axis-tips than at major-axis-tips for
a given locally buckled configuration.
5.5
Effects of size of delamination
Now, the numerical experiment is repeated for varying sizes
of the elliptic delamination with specified delaminate thickness
(tJt 1 = 0.1), aspect ratio of the delamination (a/b = 1.5) and
plate thickness (q/L = 0.05). The major axis length of the elliptic
delamination, a, is varied. As the delamination size increases,
both the local and global buckling strengths decrease as
t2/t 1
0.02
0.05
0.10
0.20
0.30
0.40
Rb
34.39
25.72
1.39
-37.08
- 108.16
Rf
43.75
45.48
49.50
52.98
55.04
presented in Fig. (20). The percentage reduction in the
global buckling strength of the structure is given in Table 3
for varying delamination size. Again the delaminated structure
is weaker in buckling when compared to undelaminated
laminates.
For a smaller delamination, the local buckling begins at
a higher load level; and hence, the energy release rate becomes
significant at higher load levels. However, the energy release
rate increases steeply once the local buckling sets in. In the
pre-global-buckling stages, both the average energy release rate
(Fig. (21)) and the maximum energy release rate (Fig. (22)) are
lower for smaller delaminations since the local buckling
strength is higher. However, in post-global buckling stages, the
smaller delaminations tend to grow more rapidly. Thus the
smaller the delamination, the later the delamination growth
sets-in; but once it sets in, the delamination growth will be much
more rapid than with the larger delaminations.
5.6
Effects of total plate thickness
Here, the numerical experiment is repeated for varying
thicknesses ratio of the plate (tl/L), for a fixed delamination
thickness ratio (t2/t 1 =0.1), delamination aspect ratio
(a/b = 1.5), and delamination size (a/L = 0.3). The critical
buckling load increases as the cube of the plate thickness
(Fig. 23). Both local and global buckling strengths increase as
the plate thickness increases. However, the increase in the
critical load factors is slightly lower than that expected from the
expected variation proportional to cubic variation of the
thickness (Fig. (23)). This is because, the actual delaminate
thickness also increases as the plate thickness increases for
a fixed delaminate thickness ratio. Hence, the eccentricity of
the delaminate plate with reference to the plate midsurface
increases as the plate thickness ratio increases with fixed
delaminate thickness ratio. Accordingly, the rate of reduction
in the global buckling strength of the structure increases as the
plate thickness increases. In case of delaminated plate, 30-60%
strength reduction is observed with reference to undelaminated
0.3(
2.0
t2/t 1 : 0,02
0.2!
,?:o;o
1.8
. . . . .
1.6
0.20
1.4
0.15
1.0
1.2
0.8
0.10
0.6
0.4
0.2
0.05
t78
0
1;
20
3'o ~'o 5o
6o
io
8o 9o
0.25
0 ,
%
25
t2/t 1: 0.20
0.20
15
0.10
10
0.05
5
0
0
-0.05
-5
i
0
1;~ 2o
='297.2
2O
0.15
-0.10
%
3o
i
~o
5'0
i
~o
~o 8o
~o
-10
t 2/tl : 0.40
xQg.9
0
~
10
'
20'
30
'
~0
60
'
5'0
70
'
80
'
90
Fig. 7. Pointwise energy release rate distribution along delamination edge a/b = 1.50; a/L = 0.30; t,/L = 0.05
o,o6.
0.050.3"
0.04
i 0.03"
"
0.2
0.02
1
2
3
4
t2/t~=O.02
tJt~:O.05
tJt~:O.lO
t2/tl=0.20
5 t2/t1:0.30
5 t2/tl:0.40
i
1;/7
//
//
0.01
0
0.1
0
a
10
20
30 z,O 50 60
Load factor X
70
80
30
J
- 100
- 50
i
i
0
50
100
;k
Fig. 8. Average and maximum pointwise energy release rate for very
thin delamination a/b = 1.50; a/L = 0.30; tilL = 0.05; tJtl = 0.02
20
t2/tl=O.05
tjfl=0.10
t2/t1:0.20
2"///
3////
&//4//L
t2/tl: 0.30
IIII l
tJtl:O.L
10
plate of thickness tl, while 1 0 - 4 5 % strength reduction is
observed with reference to u n d e l a m i n a t e d plate of thickness
t l - t2 (see Table 4).
The n o r m a l i s e d average and m a x i m u m pointwise energy
release rates - ~q~J)/cr and ~max/)Jcr -- are plotted in Fig. (24) and
Fig. (25) respectively with reference to n o r m a l i s e d load factor
2/)/cr. As the plate thickness increases, even though the
d e l a m i n a t i o n growth m a y start at m u c h higher load factors, the
0
0
16o
260
360
Load factor ~,
Fig. 9a, b. Influence of delamination thickness on maximum energy
release rate a/b = 1.50; a/L = 0.30; tall = 0.05. a Pre-global-buckling
regime, b Post-global-buckling regime
0.1
300
0
/t21q:
O.~'~~.3-
g
-0.1 '
200
100
t79
-0.2
10
20
30
40
50
Load factor k
60
70
80
0
-6
-i
0
2
w
/*
,.
Fig. 12. Load-deflection paths at the centroids of delaminate and
base plates. (In complementary mode), t2/t~ = 0.20, q/L = 0.05,
a/b = 1.50, a/L = 0.30
9
0.6
0.5
0.4
90.3
0.2
-2
2;o
o
b
3oo
Load factor: ;L
9
0.1
Fig. lOa, b. Influence of delaminate thickness on average energy
release rate a/b = 1.50; a/L = 0.30; tilL = 0.05. a Pre-global-buckling
regime, b Post-global-buckling regime
10
20
30
40
50
60
70
80
90
8
Fig. 13. Pointwise energy release rate distribution. (In complementary
mode) t2/q = 0.20, till = 0.05, a/b = 1.50, a/L = 0.30
6.0
1.0
(1) In complementary mode
(2) ]n supptementary mode
/..5
/
50
3.0
Lu
30 ~
/
/ ~
10
1.5
Fig. 11. A typical complementary post-buckling deformation mode.
fi = 205.8, tz/t ~ = 0.20, tJL = 0.05, a/b = 1.50, a/L = 0.30
-1.5
e n e r g y release rate i n c r e a s e s v e r y steeply. Thus, t h e t h i c k e r the
l a m i n a t e plate, the later t h e d e l a m i n a t i o n g r o w t h begins; a n d
the m o r e d a n g e r o u s the d e l a m i n a t i o n g r o w t h will be.
5.7
Effects of the material orthotropy
A n o r t h o t r o p i c s q u a r e plate of edge l e n g t h L a n d t h i c k n e s s
q = 0.05L, w i t h a c e n t r a l elliptic d e l a m i n a t i o n (Fig. l b ) , is
c o n s i d e r e d for the following n u m e r i c a l e x p e r i m e n t s . The
0
. . . . . .
100
9
, 9
200
,
X
Fig. 14. Average pointwise energy release rate distribution. (In
complementary mode) t2/t ~= 0.20, t~/L = 0.05, a/b = 1.50, all = 0.30
d e l a m i n a t i o n c o n f i g u r a t i o n is fixed as: a/L = 0.3; a/b = 1.5; a n d
t2/t 1 = 0.1. The l o a d i n g a n d b o u n d a r y c o n d i t i o n s , a n d t h e finite
e l e m e n t m o d e l i n g are t h e s a m e as in the p r e v i o u s p r o b l e m s .
The m a t e r i a l p r o p e r t i e s are t a k e n as: E 2 = 26000; vi2 = v~3 =
/
15
(1)
10'
In
comptementarymode
supplementary mode
(2) In
0.03
[
(2 t/
1
2
3
t,
0.02
•
u
E
5
j ~ ( 1 ]
a/b:1.00
a/b=1:50
a/b=2.00
a/b =3.00
0.01
180
0
o
1;o
x
200
so
Fig. 15-. Maximum pointwise energy release rate distribution. (In
complementary mode) tg/t~ = 0.20, t~/L = 0.05, a/b = 1.50, a/L = 0.30
80
220
16o
1so
2so
L
Fig. 17. Influence of aspect ratio on maximum energy release rate
a/L = 0.30; tJL = 0.05; t2/t 1 = 0.10
i:iill lab100
III
III
2 o/b:150
3 a/b~2100
60
I
/
2t0
x? 40
200
20
0
1.0
,
1.5
,
2.0
a/b
2.5
190
3.0
Fig. 16. Effect of aspect ratio of delamination on local and global
buckling strengths a/L = 0.30; tJL = 0.05; t2/t ~= 0.10
Olo
50
100
2,.
150
200"
250
Fig. 18. Influence of aspect ratio on average energy release rate
a/L = 0.30; tJL = 0.05; t2/t 1 ~ 0.10
900
Table 2. Reduction in
global buckling strength
of plate with elliptic
delamination: Effects of
the aspect ratio
a/b
R~
Rf
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
31.86
27.40
25.70
23.52
22.81
22.46
22.25
22.14
22.11
50.33
47.07
45.49
44.24
43.73
43.47
43.32
43.24
43.22
800
70 o
600
500
o/b =1.0
z,0o
v23 = 0.16; G12 = G13 = G 2 3 = 7500. T h e n u m e r i c a l e x p e r i m e n t s
are c o n d u c t e d for v a r y i n g r a t i o o f Y o u n g ' s m o d u l i i , E1/E 2 =
1.0-10.0; while fiber o r i e n t a t i o n is fixed a l o n g t h e m a j o r axis
o f t h e elliptic d e l a m i n a t i o n .
30 o
200
10 ~
0o
1
50
100
150
200
250
X
Fig. 19. Location of point of maximum pointwise energy release rate
a/L = 0.30; tJL = 0.05; tg/tt = 0.10
B o t h the local a n d global b u c k l i n g s t r e n g t h s i n c r e a s e as E1/E 2
increases (Fig. (26)). The local b u c k l i n g s t r e n g t h (2~c,) o f the
d e l a m i n a t e plate, as expected, increases linearly as E1 increases.
However, the global b u c k l i n g s t r e n g t h (2~r) of the d e l a m i n a t e d
50
230
0.035
III
2 o:30
0.030
220
40
0.025
g
>
210
-xu30
s,~ 0.020
0.015
20O
0.010
20
4~ ~
19o
0.O05
0
10
20
2;
3o
35
~0
0
180
a
Fig. 20. Effect of delaminate size on local and global buckling strengths
a/b = 1.50; tJL = 0.05; t i t 1 = 0.10
50
100
450
x
200
250
Fig. 22. Effect of delamination size on maximum energy release rate
a/b = 1.50; tJL = 0.05; t2/t 1 = 0.10
100000
Table 3. Reduction in
global buckling strength
of plate with elliptic
delamination: Effects of
the delamination size
10 000
1 000
100
a/L
Rb
Rf
0.40
0.50
0.60
0.70
0.80
22.26
23.32
25.69
31.86
31.86
43.45
44.22
45.51
48.07
52.70
10
g o
(~,gcr)FE
1
(2~)FE
0.1
0101
.
.
.
0.01
0.010
0.008
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.1
L/L
Fig. 23. Influence of the plate thickness (t~) on buckling strengths
of the delaminated plate a/b = 1.50; a/L = 0.30; t2/t I = 0.10
II I
I/I
1 Q--2o
2~
Table 4. Reduction in
global buckling strength
of plate with elliptic
delamination: Effects of
the laminate thickness
0.006
>
a
0.00~,
0.002
50
100
150
200
250
k
Fig. 21. Effect of delamination size on average energy release rate
a/b = 1.50; tl/L = 0.05; t2/t ~ = 0.10
structure increases initially a n d reaches a saturation point as
E 1 increases. Finite element solutions are obtained for the
critical buckling strength of equivalent u n d e l a m i n a t e d plates
of original laminate thickness tl a n d of base plate thickness t3:
~ r (t = tl) and 2~r(t = t3) respectively. As it can be observed from
Fig. (26), the critical buckling strength of the actual delaminated
plate is m u c h lower t h a n that of the original u n d e l a m i n a t e d
tJL
Rb
Rf
0.01
0.05
0.10
0.15
0.20
0.25
7.05
25.71
33.85
45.12
42.11
37.91
31.93
45.50
51.77
59.99
57.80
54.73
plate. It can be observed that, the delaminated plate is actually
weaker than the plate with the delaminated layers completely
peeled off.
Typical energy release rate distributions along the
d e l a m i n a t i o n front are shown in Fig. (27) for E~/E 2 = 8.0. Again,
the m a x i m u m energy release rate is observed to be in the vicinity
of the direction of m a x i m u m local-buckling-stiffness. For the
109
40
t2/t 1=0.10
o/b =1.50
a/L:0.30
8
7
7<./
/
~82
1
t~/L:O.05
0
0
10
20
30
40
tl/L =0.01
;
1;
50
60
70
~0
90
X
15
Fig. 27. Typical p o i n t w i s e e n e r g y release rate d i s t r i b u t i o n for
d e l a m i n a t e d o r t h o t r o p i c plate E1/E 2 = 8 (a/b = 1.50; a/L = 0.30;
tl/L = 0,05; tE/t ~ = 0.10)
Fig. 24. Influence of laminate t h i c k n e s s o n average energy release
rate a/b = 1.50; a/L = 0.30; t2/t ~ = 0.10
//
50
,oi
/
t2/t1=0.10
o/,:o.3o
8
/
~/
/
/
~-/
5
E
/
..;
~X-
~x //
40
/
?/~
~o 30
El/E2:
10
60
s~
~/
~.-I
,,'/.~ /
4
20
o
2
0
.
0
5
10
15
,~,~
1250
1000
~< 750, ~ . ~ 500 1
250 1
]
1
•Lcr
"
i
2
9
i
3
,
i
4
9
J
5
600
800
t i t I = 0.i0
aspect ratio of the delamination (a/b = 1.5) and the material
orthotropy (EllE 2 = 8.0) considered, the delaminate plate
exhibits maximum stiffness against buckling in the fiber
direction and hence the point of maximum pointwise energy
release rate is found to be at an angle of 0 ~ with reference to
the major axis. As E1 increases, the local buckling strength
increases and hence the energy release rate become significant
at higher compressive loads. However, the average and
maximum pointwise energy release rates are higher for higher
material orthotropy at local-post-buckling range of loads
(Fig. (28-29)).
1750
0
,
400
X
Fig. 28. Effect of orthotropic material properties on maximum
pointwise energy release rate a/b = 1.50; a/L = 0.30; tl/L = 0.05;
Fig. 25. Influence of l a m i n a t e thickness o n m a x i m u m e n e r g y release
rate a/b = 1.50; a/L = 0.30; talt ~ = 0.10
1500
200
9
i
9
i
6
7
El/E2
9
i
8
9
i
9
,
10
Fig. 26. Influence of material o r t h o t r o p y o n b u c k l i n g s t r e n g t h s
a/b = 1.50; a/L = 0.30; tJL = 0.05; ta/t ~ = 0.10
5.8
Effects of fiber orientation
Now the numerical experiments are repeated for the problem
considered in the previous section for varying fiber orientation
(~b = O, 15, 30, 45, 60, 75, 90); and the material orthotropyis fixed
at EllE 2 = 8.0.
The post-buckling deformation is plotted in Fig. (30) for
typical fiber orientations. When the fiber is along the major axis
of the delamination, the delaminate plate stiffness against
buckling along major axis is comparable with that along the
El/E2:
10
12.5
S
10.0
B
6
2
4
7.5
-1.0
. . . .
40
5.0
0
zu
a
2.5
n
200
400
>,
600
30 ~
0
50
0
10
183
S00
Fig. 29. Effect of orthotropic material properties on average pointwise
energy release rate a/b = 1.50; a/L = 0.30; tt/L= 0.05; t2/t ~= 0.10
-0.2
5-0.4
--0
0..8
6~
C
minor axis for the aspect ratio of delamination (a/b = 1.50) and
5O
LO
material orthotropy (E~/E 2 = 8.0) considered. Hence, the
0
lO 2o
20
postbuckling deformation conforms with the fundamental
Lu 30 540~ ~ 0 0 10
b
buckling mode in both x 1- and x2-directions. However, the
delaminate plate becomes much weaker in the major axis
direction against compressive loads as the fiber orientation
deviates from the major axis direction. Accordingly, the
delaminate plate deforms at a higher mode in the x~-direction
0
while it is undergoing postbuckling deformation in the
-0.2
f u n d a m e n t a l mode in the x2-direction. Initially, the local
buckling strength (21cr)slightly decreases as the fiber orientation ):-0.4
-0.6
(qb) increases; however, 2~crincreases for q5 > 30 ~ (Fig. (31)). The
finite element solutions for the critical buckling strengths of
50
4C
the equivalent undelaminated plates of thicknesses t~ and
0
t 3 -A~r(t = ti) and 2~ = t3) respectively- are found to be
~u 30 5L0~ ~ 0 0 10
maximum for fiber orientation ~b = 45 ~ The global buckling
o
strength of the actual delaminated plate (2gr) is, again, highly
underestimated when compared to both the undelaminated
configurations as shown in Fig. (31).
In the supplementary modes of buckling (shown in Fig. (30)),
the base plate comes in contact with the delaminate plate and
-0.1
accordingly the energy release rate becomes partially negative
-0.2
-0.3
along the delamination front. However, when the local and
global post-buckling deformations are complementary, the
50
pointwise energy release rate is positive all along the
30
delamination front (Fig. (32)). Accordingly, the location of
,v 30 ~
10
maximum energy release rate changes from 0 = 0 ~ to 0 = 75 ~
as the fiber direction changes from ~b = 0 ~to 0 = 90 ~ as shown in d
Fig. (33). The average and the maximum energy release rates
E1/E2:8.0; t2/t 1:0.10; a/b:1.50; tl/L=0.05; o/L:0.60
are plotted in Fig. (33) and Fig. (34) for typical fiber orientations
Fig. 30a-d. Influenceoifiber orientation on postbuckling deformation
and for varying load levels. For a typical post-buckling load
EI/E 2 = 8.0; a/b = 1.50; a/L = 0.30; tJL = 0.05; t21t~= 0.10. a ~b=0~
level (Z = 740.0), the average and the maximum energy release
2 = 465.3. b ~b= 30~ ). = 259.8. c ~b= 60~ 2 = 287.5. d ~b= 90~
rates vary as fiber orientation varies as shown in Fig. (35). The ~. = 299.6
energy release rate is again found to be maximum in the vicinity
of the direction of maximum-stiffness against compressive
loads (Fig. (36)).
an antisymmetric sense. The plate thickness is assumed as
5.9
t 1 = 0.05 L. The delamination configuration is fixed as: a/L = 0.3;
Antisymmetric laminate with central elliptic delamination
a / b = 1.50; and t2/t1 =0.10. The loading and boundary
In the following numerical experiments, the plate is assumed
conditions, and the finite element modeling are the same as in
to be a laminated composite with orthotropic layers stacked in the previous numerical experiments. The material properties
-
-
3o'~
70
X2
60
50
40
30
20
1 800
10
1 600
0
1 400
184
-10
25
1 200
,4
b ~ = 30 ~
1 000
20
800
15
600
400
0
10
x~r
200
o
/s
6'o
r
9o
E~/E2:8.0, t 2 / t ~ = 0 . 1 0 , { a / b : l . 5 0 : t J L - - 0 . 0 5 : a / L : 0 . 6 0
0
25
Fig. 31. Influence of fiber orientation on local and global buckling
strengths E~/E 2 = 8.0; a/b = 1.50; alL = 0.30; t J L = 0.05; t2/t ~ = 0.10
20
15
10
for each layer are taken as: E~ = 208000; E z = 26000; v12 = v~3 =
v23 = 0.16; Ga2 = G13 = Gz3 = 7500. All layers are assumed to be
of equal thickness. The following three lay-up sequences are
considered: (i) the laminate is composed of 20 laminae stacked
0
in the recursive sequence 0~176 (ii) the laminate is composed
14
of 100 laminae stacked in the recursive sequence 0~176 and
(iii) the laminate is composed of 20 laminae stacked in the
12
d 0:90 ~
recursive sequence - 450/45 ~ The numerical experiments are
10
conducted for symmetric and antisymmetric configurations for
8
the delaminate plate.
The local and global buckling strengths of the delaminated
6
plates are plotted for varying delaminate thicknesses in
4
Fig. (37). It can be noted that local buckling strength increases as
2
the delaminate thickness increases and becomes nearly equal
to or greater than the global buckling. The finite element
0
solution for the equivalent undelaminated plate with
-2
base-plate-thickness is also plotted for comparison. The global
10' 20' 30' 4o" 5'o 60' 7b 8'0 90
@
strength of the actual structure is less than the equivalent
undelaminated-base-plate-strength. The behavior is nearly
E l / E 2 = 8.0: t2/t 1 =0.10; o / b =1.50, t l / L = 0 . 0 5 r a / L = 0.60
identical for case-i and case-ii. The threshold delaminate
thickness for case-i and case-ii is found to be t2/t ~ = .175 while Fig. 32. Influence of fiber orientation on pointwise energy release
rate distribution E1/E 2 = 8.0; t2/t 1 = 0.10; a/b = 1.50; tt/L = 0.05;
that for case-iii is t2/tt = .08 below which critical buckling
a/L = 0.30
strength of the actual structure is less than that of the base-plate.
The residual drop of buckling strength is found to be
considerably less in case-iii when compared to the other two
cases.
Even though, case-i and case-ii show identical results with
symmetric delaminate configuration (tE/t 1 -- 0.15). It is found
reference to their buckling strengths, the energy release rate
that the energy release rate drops considerably when the number
distribution is found to be quite different. The energy release
of layers (for the same total laminate thickness) increases
rates are plotted along the delamination front for the three cases (compare case-i and case-ii). Since the angle-ply configuration
with antisymmetric delaminate configuration (t2/t ~ = 0.10) and (case-iii) makes the plate stiffer (note the rise in buckling
90
~=0 ~
10
60o
30~
75'
---.60
j
6
45-
90 o
4
3015-
2 84
0
'
0
i
01
0o
,
100
200
300
400
X
500
600
700
800
.
,"
. . . .
150
300
9
450
. . . .
600
75 ~
90~
*
Fig. 33. Influence of fiber orientation on average pointwise energy
release rate E J E 2 = 8.0; t21q = 0.10; a/b = 1.50; tJL = 0.05; a/L = 0.30
Fig. 36. Angle of m a x i m u m energy release rate for different fiber
orientations E J E 2 = 8.0; t2/t ~ = 0.10; a/b = 1.50; tJL = 0.05; a/L = 0.30
0=0 ~
20
18 5
1600 J ' N
o
60"
30~
12001
90~
~ 1 5 84
600
o
S
o11
a
i
1
i
200
400
600
BOO
Fig. 34. Influence of fiber orientation o n m a x i m u m pointwise energy
release rate E1/E2 = 8.0; t2/t ~ = 0.10; a/b = 1.50; tJL = 0.05; a/L = 0.30
~.0
L
X~cr
0'.2
ola
o.a
tJtl
2 000
.1200! ~
8001
///
"".
4oo
~%=740.0
30
0
0'.1
b
012
t2/tl
0.3
0.4
~'~20
j
$
Fig. 37a, b. A n t i s y m m e t r i c laminate with central elliptic delamination.
Influence of delaminate thickness on local a n d global buckling
strengths, tJL = 0.05; a/L = 0.30; a/b = 1.50; a n d E1/E2 = 8 . 0 for each
lamina, a 10 x (0o/90 ~ b 10 x ( - 45~ ~
-----...
10
9
0o
i
150
.
'
i
300
,
i
45 ~
,
J
60 ~
9
i
75~
90 ~
Fig. 35. Variation of average a n d m a x i m u m pointwise energy release
rates with fiber orientation E1/E 2 = 8.0; t2/t 1 = 0.10; a/b = 1.50;
tJL = 0.05; a/L = 0.30
s t r e n g t h s , Fig. (37b) w h e n c o m p a r e d to Fig. (37a), t h e e n e r g y
r e l e a s e r a t e is c o n s i d e r a b l y l o w e r w h e n c o m p a r e d to c a s e - i f o r
t h e s a m e l o a d level. T h e a v e r a g e a n d t h e m a x i m u m e n e r g y
release rates are plotted for different cases considered here in
Fig. (39).
16
18/
,
~"o/,~
1/-.
14[
12
,>'i,
10
8
6
6
qqS.'~
4
2
0
]86
I
-20o
I
10~
I
20 ~
I
I
30 o
40 ~
a,
50 o
I
600
I
I
70 o 50 ~
-2
e
0.010 /
.
.
.
i
90 ~
0~
i
100
i
i
20 ~
i
300
I
/*0~
d
i
50 o
i
600
i
70 ~ 500 90"
e
2.5
.
0.009 k
0.008 t ~ =87/..2
0.007 ~ "
2.0
0.006 Ix. 787.1
1.5
~ 0.005 k~69~. 8
0.00/,1 \
~\
0.003 612./,
1.0
0.5
o
o. ooi
0
~ 0 ~O01
I
0~
--
10 0
I
20"
I
I ~
30 ~
50 ~ 60 ~ 70 ~ 50 ~ 90 ~ -0.5 0 o
4"0~
b
e
10 ~
I
I
A
20 ~ 30 ~
I
400
,('~
10
30
I
60 ~ 70 o
I
500 900
@
e
35
I
500
.
.
.
.
.
.
.
8
25
6
20
~
~.~15
10
4
2
5
0
0
-5
i
0~
100
i
i
i
I
20 ~ 30 o 40"
500
I
i
-2
i
600 70 ~ 80 ~ 900
8
i
O"
f
10 ~
I
20"
i
300
i
i
40 ~ 500
e
i
~
i
600 70 ~ 50 ~ 90"
Fig. 38a-f. Antisymmetric laminate with central elliptic delamination. Influence of delaminate thickness on pointwise energy release rate
distribution a/b = 1.50; tl/L = 0.05; a l l = 0.30; E J E z = 8.0 for each lamina, a 10 x (00/90~ laminate tJt~ = 0.10. b 50 x (0o/90~ laminate
t2/t I =0.10. r 10 x ( - 450/45 ~ laminate tz/t ~ = 0.10. d 10 • (0o/90~ laminate t2/t ~ = 0.15. e 50 x (00/90~ laminate t2/t ~ = 0.15. f 10 x
( - 45o/45 ~ laminate t2/t ~ = 0.15.
5.10
Symmetric laminate with central elliptic delamination
In this section, the l a m i n a t e d plate considered in the previous
example is assumed to be symmetrically constituted with 32
orthotropic laminae in the following fashion: (i) ( 0 1 9 0 / 4 5 1 - 45),
a n d (ii) ( 4 5 / 9 0 / - 4 5 / 0 ) , . Material properties, laminate a n d
delaminate configurations, loading a n d b o u n d a r y conditions,
a n d finite element m o d e l i n g r e m a i n the same as in the previous
example. The d e l a m i n a t i o n is assumed to be between the n th
a n d the (n + 1) th layer from the top surface where n takes values
from 1 to 15.
The characteristic buckling strengths for the delaminated
structure are plotted in Fig. (40) for both case-i a n d case-ii. It
is found that the local buckling strength becomes nearly equal
for n = 8 or t J t l = 0.25. The threshold delaminate thickness for
which equivalent u n d e l a m i n a t e d base plate is stronger than the
actual structure in buckling is found to be 0.125 and 0.15 for
case-i a n d case-ii respectively. It is interesting to note that the
global buckling strength does not vary m u c h with the
delaminate thickness for case-ii.
The average a n d the m a x i m u m energy release rates are
plotted for case-i and case-ii for typical delaminate thickness
(n < 8) in Fig. (41) and Fig. (42). It should be noted that local
buckling does not occur when n > 8 a n d hence no delamination
growth is expected because of buckling action. Since the
delaminate alternates between symmetric a n d antisymmetric
2 000
1 800
1 600
1 4O0
200
/
x~
1 000
800
2a
6O0
4O0
~--...~,~ .
187
200
,
i
9
,
,
,
9
500
,
,
,
,
,
1000
1500
a
0
,
,
,
,
k
2 000
1 800
15
t 600
1400
1 200
~10
~
'~ I 000
E
J
800
6O0
5
400
2000
,
i
,
i
,
i
,
500
b
i
1000
,
i
,
i
1 10,(0~ ~ [erninote
2 50,(0~
~ tominute
3 lO,,(-/-,5~176
E~./E::8.0
(o)
(o)
(o)
for
Fig. 39a, b. Antisymmetric laminate with central elliptic delamination
a/b = 1.50; tJL = 0.05; a/L = 0.30. a Average pointwise energy release
rate. b Maximum pointwise energy release rate
configurations as n increases, the energy release rate variation
also changes for different delaminate thicknesses as shown
in the figures. Variation of the average and the maximum
energy release rates with the delaminate thickness is plotted
for typical load levels in Fig. (41c-d) and Fig. (42c-d)
respectively. It is observed that the average energy release rate
is lower for case-i while the maximum energy release rate is
lower for case-ii.
6
Concluding
0.05
b
t 2 / t l : 0 . 1 0 (b) t2/tl=0.15
t2/tl=0.10 (b) t2/t~=0.15
t2/t~=0.10 (b) t2/tl=O.15
each Lomino
remarks
In this paper, the reduction in buckling strength of a plate with
an elliptic delamination and the mechanisms of delamination
growth under post-buckling conditions, were critically
examined using a finite element method. A quasi-conforming
3-noded plate element based on Reissner-Mindlin plate theory
and the multi-plate modelling technique are used to model the
delaminated plate. Gauss elimination solution algorithm is used
to solve the incremental equations in a cycle of NewtonRaphson iterations. Arc-length controled load incrementation
and a linearised asymptotic solution for branch switching
are used for effectively traversing both limit and bifurcation
points.
,.
0
,
1500
X
Xt
0.10
0.15
0.20
0.25
t2/tl
Fig. 40a, b. Symmetric laminate plate with central elliptic delamination
Buckling strength reduction for different delaminate thickness, a Case-i.
b Case-ii a/b = 1.50; tt/L = 0.05; all = 0.30
The post-buckling solution and the energy release rate
models are validated using a simple classical p r o b l e m isotropic square plate with central near-surface delamination.
Influence of the following structural parameters on the local and
global buckling strengths of the structure, and on the pointwise
energy release rates along the delamination front, are critically
studied:
delaminate plate thickness (tJtl),
aspect ratio of the delamination (a/b),
size of delamination (a/L),
total laminate thickness (tJL),
material orthotropy (EJE2),
fiber orientation (~b),
antisymmetric laminate with
symmetric/antisymmetric delaminate plates
(viii) symmetric laminate with symmetric/asymmetric
delaminate plates
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
The influence of these parameters are in particular examined
regarding the reduction of the structural strength against
buckling, coupling of local and global post-buckling solutions,
and pointwise energy release rate at the delamination front.
Several observations are made regarding the post-buckling
failure mechanisms.
15
!5
I
.
"1
aC
4
bC
10
15"
>
TO "
5
54
~88
O]
400
800
1200
.,
. . . . . . . . . . .
0
400
~00
X
1200
x
0.4
25
e-]l
Case -11
20
0.3
15
02
~10
0.1
Fig. 41a-d. Symmetric
composite plate with central
elliptic delamination Average
pointwise energy release rate
a/b = 1.50; tJL = 0.05;
a/L = 0.30
5
0
0
0
c At X- z,00
0.1
0.2
0
t2/tl
0.1
d At ;L =1300
Z,0
0.2
t2/tl
40
a Case-I
[/ 5
30
/
b Case-]I
30'
20
~-~ 20'
10
10.
0
0
0
400
800
5
1200
9
i
0
9
i
,
r
9
i
/,00
'
i
9
800
i
'
1200
x
4O
0.6
30
~
-
I
0.4
~ 20
0.2
10
0.1
c At,L : 400
0.2
t~/tl
00.1
d At ;~ =1300
0.2
t,~/tl
Fig. 42a-d. Symmetric
composite plate with central
elliptic delamination Maximum
pointwise energy release rate
a/b = 1.50; tl/L = 0.05;
a/L = 0.30
of Excellence of Computational Modelling of Aircraft Structures,
References
Atluri, S. N. 1986: Energetic approaches and path independent integrals Georgia Tech., Atlanta
Naganarayana, B. P. 1991: Consistency and correctness principles in
in fracture mechanics. In: Atluri, S. N. (ed): Computational methods
quadratic displacement type finite elements. Ph.D. Dissertation, Indian
in mechanics of fracture, Amsterdam: North-Holland
Barlow, ]. 1976: Optimal stress locations in finite element models. Int. Institute of Science, India
Naganarayana, B. P.; Atluri, S. N. 1994: Energy release rate evaluation
J. Numer. Meths. Engg. 10:243-251
for delamination growth prediction in a multi-plate model of a laminate
Chai, H.; Babcock, C. D.; Knauss, W. G. 1981: One-dimensional
modelling of failure in laminated plates by delamination buckling. Int. composite. Comput. Mech. (to appear)
Naganarayana, B. P. 1994: Incremental iterative strategies for
J. Solids and Struct. 17(11): 1069-1083
automated post-buckling analysis. (To be published)
Huang, B-Z.; Shenoy, V. B.; Atluri, S. N. 1994: A quasi-conforming
triangular laminated composite shell element based on a refined first Timoshenko, S. P. 1940: Theory of plates and shells. 1st Ed., McGraw
Hill, New York, 1940
order theory. Comput. Mech. 13:295-314
Huang, B.-Z.; Atluri, S. N. 1994: A simple method to follow postbuckling paths in finite element analysis. Internal Report, FAA Center
189