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Finite element analysis of postbuckling and delamination of composite
laminates using virtual crack closure technique
P.F. Liu a, S.J. Hou a, J.K. Chu a, X.Y. Hu b, C.L. Zhou a, Y.L. Liu a, J.Y. Zheng a,, A. Zhao a, L. Yan a
a Institute of Chemical Machinery and Process Equipment, Zhejiang University, Hangzhou 310027, Zhejiang Province, Chinab School of Engineering, Zhejiang A & F University, Linan 311300, Zhejiang Province, China
a r t i c l e i n f o
Article history:
Available online 28 December 2010
Keywords:
Buckling and postbuckling
Delamination
Virtual crack closure technique (VCCT)
Finite element analysis (FEA)
Composite laminates
a b s t r a c t
The two-dimensional and three-dimensional parametric finite element analysis (FEA) of composite flat
laminates with two through-the-width delamination types: 04/(h)6//04 and 04//(h)6//04 (h= 0, 45,
and // denotes the delaminated interface) under compressive load are performed to explore the effects
of multiple delaminations on the postbuckling properties. The virtual crack closure technique which is
employed to calculate the energy release rate (ERR) for crack propagation is used to deal with the delam-
ination growth. Three typical failure criteria: B-K law, Reeder law and Power law are comparatively stud-
ied for predicting the crack propagation. Effects of different mesh sizes and pre-existing crack length on
the delamination growth and postbuckling properties of composite laminates are discussed. Interaction
between the delamination growth mechanisms for multiple cracks for 04//(h)6//04composite laminates
is also investigated. Numerical results using FEA are also compared with those by existing models and
experiments.
2010 Elsevier Ltd. All rights reserved.
1. Introduction
Currently, carbon fiber reinforced polymer composites have
been increasingly used in areas of the aeronautics, astronautics,
fuel cell vehicle, new energy utilization, pressure vessel and piping,
electricity generation, construction, boats and sport equipments
due to their advantages such as high strength/stiffness-to-weight
ratio, excellent fatigue- and corrosion-resisting behavior as well
as satisfactory durability.
Generally, the carbon fiber composite laminates are manufac-
tured by designing fiber layup orientation for each layer. In this
case, these stacked angle-ply layers are expected to achieve high
stiffness and strength in different orientations. Since the stiffness
and strength of an individual layer are much higher in the fiber
direction than in the transverse direction, the mechanical proper-
ties of composites in the fiber principal orientation bear different
external loads[1].
The complex failure mechanisms of laminated composites un-
der various environment pose a big challenge to the design and
practical application of composites though they exhibit more
advantages than the traditional metal materials [2]. Firstly, the
intralaminar damage and failure of composites in the forms of fiber
breakage, matrix cracking and fiber/matrix interface debonding
according to the composite mesomechanics may appear which
leads to the stiffness degradation and strength loss due to variable
physical and mechanical properties of polymer materials [37].
Secondly, the interlaminar delamination may often occur due to
poor bonding strength between neighbouring layers depending
merely on the polymer matrix[810]. In addition, the instabilities
and imperfections arising from the manufacturing process are also
important factors leading to the interlaminar debonding. More-
over, the interaction between the intralaminar and interlaminar
failure modes in the presence of defects adds the difficulty for
studying the failure mechanisms of laminated composites
[1113]. Recently, Sleight [14], Tay et al. [15], Garnich and Akula
Venkata[16], and Liu and Zheng[17]gave comprehensive review
on the progressive failure analysis of composite laminates in terms
of the general methodologies on the damage constitutive modeling
by continuum damage mechanics and fracture mechanics, the
failure criteria, the damage evolution law simulating the stiffness
degradation, and the finite element implementation of progressive
failure analysis which predicts the mechanical properties of
composites in process of continuous failure.
Among the failure modes above, a special case is the buckling
and postbuckling of composite laminates with multiple interlami-
nar delaminations under compressive load. In general, this type of
failure mode can be divided into two categories: local buckling and
global buckling [18,19]. The sub-laminates under compressive load
may locally buckling and impose the additional bending stress on
the neighbouring sub-laminates, which may lead to the failure of
0263-8223/$ - see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruct.2010.12.006
Corresponding author. Tel.: +86 571 87953370; fax: +86 571 87953393.
E-mail addresses:[email protected](P.F. Liu), [email protected](J.Y. Zheng).
Composite Structures 93 (2011) 15491560
Contents lists available at ScienceDirect
Composite Structures
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remaining sub-laminates. The global buckling for composite lami-
nates with through-the-width delamination appears before the
immediate unstable delamination and evolves accompanied by rel-
atively long delamination process with increasing compressive
strains. Chai et al. [20,21] established the one-dimensional and
two-dimensional delamination buckling models by evaluating
the crack tip energy release rate (ERR) which was defined as the
drive force for crack propagation. Bolotin[22]studied the delami-
nation buckling mechanism by considering local buckling of
delamination and the interaction between the local buckling, dam-
age accumulation, crack growth and global buckling. Hwang and
Mao [23]studied the buckling loads, buckling modes, postbuckling
behavior, and critical collapse loads of delamination growth for
unidirectional composite laminates using FEA. Kutlu and Chang
[24], Cappello and Tumino [25], Suemasu et al. [26] investigated
the buckling and postbuckling properties of composite laminates
with multiple delaminations using FEA.
Nowadays, the ERR as a typical fracture parameter is widely
used to predict the delamination crack propagation. The ERR can
be efficiently solved by the virtual crack closure technique (VCCT),
which was proposed by Rybicki and Kanninen [27,28] based on the
Irwins crack tip energy analysis[29]. The sole assumption of VCCT
is that the energy required for the crack propagation length Da is
equal to that for closing two separate crack surface with crack
length Da. Krueger [30] gave a full-scale overview on the VCCT
in terms of the solid/shell element approach, the calculation for-
mula for the 2D and 3D problems, the modified VCCT with geomet-
rically nonlinear FEA and the delamination growth behavior for
dissimilar materials. Compared with the well-known J-integral
proposed by Rice[31], the VCCT exhibits strong calculation ability:r the VCCT can be applicable to 3D structure,s the mesh require-
ment for the VCCT is lower than that for theJ-integral in the FEA,t
the mixed-mode crack propagation can be evaluated. Xie and Big-
gers[32], Leski[33]and Orifici et al.[34]pointed out that the FEA
using the VCCT is simple since the assumption above can be easily
realized. Already, a lot of models had been proposed to study the
effect of delamination on the postbuckling properties using VCCT.Gaudenzi et al. [35] explored the non-linear behavior of delaminat-
ed composite panels under compressive load using an incremental
continuation method (Riks method) and modified VCCT. As the FEA
is used to implement the VCCT to deal with the delamination prob-
lems, Krueger and Goetze[36,37]analyzed effects of some param-
eters such as the element type, integration order, release tolerance
and damage factor. In order to reduce the dependency of the
delamination growth rate on the element size and load step using
the VCCT and a fail release approach, Pietropaoli and Riccio [38]
proposed a novel method which allows an automatic load step size
adjustment based on the ERR levels and on the shape of delaminat-
ed area computed at each load increment.
In this analysis, the buckling and postbuckling properties of
composite flat laminates with multiple through-the-width delam-inations are studied under compressive load using the VCCT. Influ-
ence of some parameters such as the element size, load step
number, symmetry boundary conditions, pre-crack size and failure
criteria on the delamination growth in the FEA are discussed. Spe-
cially, the interaction mechanism between the delamination and
postbuckling is studied. Numerical results using the VCCT are also
compared with those by experiments and other existing models.
2. Virtual crack closure technique (VCCT)
As the FEA is associated with the VCCT,Fig. 1 shows crack prop-
agation from the crack tip node i to j with the increment crack
lengthD
a. Nodei is separated into two nodes i1 andi2 after crackpropagation. For node i, the relative displacements in three direc-
tions (x, y, z) are Duix, Duiy and uizafter propagation and the node
forces before propagation are Fix, Fiy andFiz. The total ERR due to
crack propagation is expressed as
G GI GII GIII
limDa!0
1
2S
Z Da
0
FixaDuixada
Z Da
0
FiyaDuiyada
Z Da0
FizaDuizada 1
whereGI,GII andGIII are ERR for the mode-I, mode-II and mode-III,
andSis the new area generated due to a crack propagation length
Da.
Assume a two-step method is used based on the node force be-
fore crack propagation and node relative displacement after crack
propagation, the ERRG due to crack propagation on the crack clo-
sure surface is calculated as [30]
G 1
2SFixDuix FiyDuiy FizDuiz 2
Often, the two-step method can be approximately substituted by
the one-step method if the mesh sizes on the crack closure surface
are sufficiently small, where the relative displacement after crackpropagation can be substituted by the relative displacements be-
tween nearest node pairs before crack propagation[32]. For exam-
ple, the relative displacements in three directions for the node pair
i1 and i2 can be approximately substituted by those for the node pair
e1 ande2. In this analysis, the one-step method is used for the cal-
culation of ERR.
In the FEA, the node bonding technique is used to simulate
crack propagation which divides the node pair at the same position
into two nodes by releasing the coupling freedom degree if the fol-
lowing crack propagation criterion is satisfied
Gequ=GequC 1 3
where Gequ and GequC are the equivalent and critical ERR,
respectively.Currently, three typical crack propagation criteria are used
(1) B-K law[39]
GequC GIC GIIC GIC GII GIIIGI GII GIII
g4
(2) Power law[40]
Gequ=GequC GIGIC
am
GIIGIIC
an
GIIIGIIIC
ao5
(3) Reeder law[41]
GequC GIC GIIC GIC GII GIIIGI GII GIII
g
GIIIC GIIC GIIIGII GIII
GII GIIIGI GII GIII
g
6
where GIC, GIIC and GIIIC are critical ERR for mode-I, mode-II and
mode-III crack propagation. g,am, an andao are constants.
3. FEA of postbuckling and delamination for composite flat
laminates with through-the-width delamination
3.1. Geometry models and sizes for composite laminates with
delamination
The delamination buckling analysis concentrates mainly on the
interlaminar through-the-width delamination for composite flat
laminates regardless of intralaminar damage and failure. TheT300/976 composite materials are used and the material parame-
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ters are listed inTable 1.Fig. 2shows the geometry model of com-
posite laminates, in which the delamination configuration includes
two types: one through-width delamination with length a1 and
two through-width delamination with length a1 and a2. The delam-
ination locates at the center of composite laminate span. Four
layer-up types from the bottom to top of composite laminatesare 04/012//04, 04//012//04, 04/(45)6//04 and 04//(45)6//04 in turn.
Here, the symbol // denotes the delamination interface. The
geometry sizes of models for five cases are listed in Table 2. Each
layer has equal thickness.
3.2. Finite element analysis
For Case-A, B and C, 2D model can deal with the problem since
each sub-laminate behaves as an orthotropic material for zero-
angle layup. But, 3D model must be established for Case-D and
Case-E. The parametric finite element model accounts for a half
of the geometry model in the axial direction due to symmetry.
The finite element analysis is carried out using ABAQUS software.
The 2D model with 5.08 mm width is dealt with by introducing
section properties in ABAQUS. The four-node plane element CPS4
and eight-node solid element C3D8R in ABAQUS are used to mesh
the 2D and 3D model, respectively. The number and coordinates of
nodes and elements are specially designed (as detailed in what fol-
lows) so that the FEA can be effectively performed. For 2D prob-
lems, symmetry constraint is exerted on the line at y=A/2 andthe axial displacement is applied on the clamped edge. For 3D
problems, the symmetry constraint is exerted on the symmetry
section aty=A/2 and degrees of freedoms in the yandz directions
are constrained and the axial displacement is applied on clamped
edge.
The flow chart of the FEA is shown in Fig. 3. In the buckling anal-
ysis, the multi-point constraint is used to tie the nodes at the same
position on delaminated surface. The buckling analysis is per-
formed using the subspace iterative method, from which the eigen-
value of buckling modes provides an initial imperfection for the
Fig. 1. Schematic representation of crack propagation between composite layers: (a) before propagation and (b) after propagation.
Table 1
Material parameters [42,43].
Ply longitudinal modulus E1 139.3GPa
Ply transverse modulus E2 9.72GPa
Out-of-plane modulus E3 5.58GPa
Inplane shear modulus G12 5.58GPa
Out-of-plane shear modulus G13 5.58GPaG23 3.45GPa
Poissons ratio v12 0.29
v13 0.29
v23 0.40
Critical ERR for mode-I GIC 0.0876 N/mm
Critical ERR for mode-II GIIC 0.3152 N/mm
Critical ERR for mode-III GIIIC 0.3152 N/mm
Fig. 2. Configuration of delaminated composite laminates under uniform compressive strain.
Table 2
Geometry sizes for five cases.
Case Lay-ups a1 (mm) a2 (mm) h (mm) A (mm) B (mm)
A 04/012//04 19.05 0 2.54 50.8 5.08
B 04/012//04 38.1 0 2.5908 50.8 5.08
C 04//012//04 38.1 19.05 2.5654 50.8 5.08D 04/(45)6//04 25.4 0 2.54 50.8 5.08
E 04//(45)6//04 25.4 12.7 2.54 50.8 5.08
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postbuckling analysis. The postbuckling analysis employs the mas-
terslave node technique, in which the slave nodes on the slave
surface are to be debonded from those on the master surface.
The delamination of composite laminate is simulated by releasing
the tied node pairs at the crack tip front into two separate nodes if
a specified fracture criterion (such as the B-K law, Power law or
Reeder law) f reaches 1.0 within a given tolerance ftol. In general,
the B-K failure criterion for crack propagation is used.
Small time increments (the initial, maximum and minimum
time increments are 0.001 s, 1e8s and 0.01 s, respectively) and
1000 load steps are specified to ensure the calculation accuracy.
The initial imperfection in the postbuckling analysis is set to be
0.001 of the first mode (eigenvalue) from the buckling analysis.
In order to improve the convergence, the viscous regularization
technique is used, in which an appropriate damping factor is intro-
duced to cause the tangent stiffness matrix of the softening
material to be positive. The linear scaling technique is used for
3D problems to reduce the solution time to reach the onset of crack
growth. The line search technique is used to accelerate the conver-
gence velocity.Parallel calculations are implemented on the high-performance
computer and the main configurations are Intel Xeon Central
Processing Unit (CPU) with 8 processors (the main frequency of
each processor is 2.33 GHz) and 3.99 GB memory. Each calculation
lasts for about 0.5 h for 2D problems and about 5 h for 3D
problems.
3.3. Numerical results for five cases
3.3.1. Case-A and B: 04/012//04 composite laminate
The finite element model with boundary conditions is shown in
Fig. 4which includes 4000 elements. The 0.18 mm uniform axial
displacement is applied on the clamped edge. Figs. 5 and 6show
the compressive loadstrain curves for Case-A and B composite
laminates. The two curves using VCCT are basically consistent with
those obtained by Wang using finite strip method [42]. In Ref.[42],
the layer-wise finite strip method was developed to account for the
delamination kinematics and the interface spring model was used
to simulate the crack propagation. The finite strip method is a spe-
cial finite element method, which replaces the continuous displace-
ment shape function in the FEA with a piece-wise polynomial
function. It may efficiently reduce the order of stiffness matrix
and improves the calculation efficiency for some particular cases.
Figs. 7 and 8 show the compressive load-central deflection
curves for Case-A and B. Fig. 9 shows the whole delamination
growth process for Case-A composite laminate. For Case-A, the ini-
tial local buckling appears at the compressive strain 1.904 103
using VCCT, which approaches 2.024 103 using finite strip
method. The compressive load first increases linearly with strain,
but abruptly jumps at the 2.53 103 strain, which indicates the
delamination crack propagates initially from the initial length
a1 = 19.05 mm in an unstable manner, as shown in Fig. 9a. The
change reflects a large axial stiffness change of the whole laminate.
From then, the load continues to increase linearly with strain until
a stable delamination growth appears form the 4.6 103 strain to
7.086 103 strain representing the global buckling stage, as
shown in Fig. 9d. The unstable and stable delamination growth
stages as shown in Fig. 9b and c above are also verified by the
experimental results [43]. The global buckling load for Case-A is
3216 N using FEA, which approaches 3392 N obtained by Wang[42]and about 2900 N by experiment[43]. For Case-B, the initial
local buckling appears at the strain 5.54 104, which accounts
for only about one quarter of 2.024 103 strain for Case-A. In
contrast with evident delamination unstable stage for Case-A, no
abrupt jump appears in the compressive loadstrain curve for
Case-B, which indicates a stable local buckling of upper sub-lami-
nate and small change for axial stiffness. The global buckling load
for Case-B using FEA is 3331 N, slightly larger than 3216 N for
Case-A.
In terms of Case-A, the compressive load first increases and then
becomes constant with increasing deflection for the bottom sub-
laminate, but experiences a sudden drop with the central deflec-
tion for the upper sub-laminate. The unstable delamination growth
stage attributes mainly to the interaction between the postbuck-ling and delamination. The change tendency for the bottom
sub-laminate for Case-B is consistent with that for Case-A, but no
load drop and obvious unstable delamination appear, which may
Fig. 3. Flow chart of FEA using VCCT.
Fig. 4. Finite element model with boundary conditions for Case-A and B composite laminates.
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be due to a longer initial delamination length for Case-B. At the glo-
bal buckling stage, the loaddeflection curves enter a plateau stage
though the upper-laminate exhibits larger deflection than the bot-
tom sub-laminate. By comparison, the loaddeflection curves for
Case-A and B using FEA are basically consistent with those ob-
tained by Wang[42].
The energy cumulates near the crack tip front on the slave sur-
face with strain, and the mode-I ERR at the crack tip node increasesrapidly from 1.035 102 N/mm at the 2.153 103 strain to
8.122 102 N/mm at the 2.53 103 strain. After then, the crack
starts to propagate rapidly and the nodes on the crack closure sur-
face add. The value for the new generated crack tip decreases
slowly from 8.122 102 N/mm at the initial delamination stage
to 1.5 102 N/mm at the global buckling stage. The change ten-
dency for the ERR is also consistent with the results[42]. In addi-
tion, the predicted global buckling loads for Case-A and Case-B are
3206 N and 3319 N, respectively if the number of finite elements
increases to 8000. The calculation errors of global buckling load be-
tween two mesh sizes are within 5%.
If the initial time increment 0.0001 s, the minimum time incre-
ment 1e8s, minimum time increment 0.001 s and 10,000 load
steps in the FEA are specified, the predicted final global bucklingloads for Case-A and B are about 3422 N and 3615 N, respectively.
The errors for these two cases due to changed load step number
and time increment are 6.0% and 7.8%, respectively.
3.3.2. Case-C: 04//012//04composite laminate
Fig. 10 shows the finite element model with boundary condi-
tions, which includes 4000 elements. The 0.14 mm uniform axial
displacement is applied on the clamped edge. Fig. 11 shows the
compressive loadstrain curve. Fig. 12 shows the axial compressive
load-central deflection curve. Fig. 13 shows the delamination
growth process with increasing strain. The calculated local buck-ling strain is 5.426 104, which is smaller than those for Case-A
and B, is in good agreement with the result 5.621 104 by Wang
[42]. When the strain increases to 2.56 103 (2.6 103 by Wang
[42]), slightly later than the 2.53 103 strain for Case-A compos-
ite laminate, the upper sub-laminate starts to delaminate, as
shown inFig. 13a. When the strain increases to 2.77 103, the
delamination crack length for the upper sub-laminate adds from
the initial length a1= 38.1 mm to 38.6 mm, and at this time the
bottom sub-laminate also starts to debond, as shown in Fig. 13b.
After that, the bottom sub-laminate delaminates in a higher crack
growth rate than the upper sub-laminate, as shown in Fig. 13c.
Similar to Case-A and B, the compressive load takes on the ten-
dency to first increase and then rapidly decrease to a constant with
increasing strain, and the plateau represents the global bucklingstage form the strain 3.5 103 to 5.5 103, as shown in
0 1 2 3 4 5 60
500
1000
1500
2000
2500
3000
3500Delamination
starts growing
Case-A
Local buckling
Finite element result
Wang's result [42]Compressiveload(N)
Compressive strain (10-3)
Global buckling
Fig. 5. Axial compressive loadstrain curve for Case-A composite laminate.
0 1 2 3 4 5 60
500
1000
1500
2000
2500
3000
3500
Delamination
starts growing
Case-B
Finite element result
Wang's result [42]
Local
buckling
Compressiveload(N)
Compressive strain (10-3)
Global buckling
Fig. 6. Axial compressive loadstrain curve for Case-B composite laminate.
-1 0 1 2 30
500
1000
1500
2000
2500
3000
3500Case-A
Local
buckling
Global bucklingGlobal
buckling
Unstable
delamination
stage
Central deflection (mm)
Compressiv
eload(N)
(1)Finite element results Bottom sub-laminate
Upper sub-laminate
(2)Wang's results [42]
Bottom sub-laminate
Upper sub-laminate
Fig. 7. Axial compressive loadcentral deflection for Case-A composite laminate.
-1 0 1 2 3 40
500
1000
1500
2000
2500
3000
3500
Delamination
starts growing
Central deflection (mm)
Compressiveload(N)
Local
buckling
Global
bucklingGlobal
buckling
Case-B
(1)Finite element results
Bottom sub-laminate
Upper sub-laminate
(2) Wang's results [42]
Bottom sub-laminate
Upper sub-laminate
Fig. 8. Axial compressive loadcentral deflection for Case-B composite laminate.
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Fig. 9. Delamination growth process of Case-A composite laminate at the compressive strain: (a) 2.53 103, (b) 2.65 103, (c) 3.58 103, and (d) 7.086 103.
Fig. 10. Finite element model with boundary conditions for Case-C composite laminate.
0 1 2 3 4 50
250
500
750
1000
1250
1500
1750
2000
Unstable delamination
Delaminationstarts growing
Local
buckling
Finite element result
Wang's result [42]
Case-C
Compressive strain (10-3)
Compressiveload(N)
Global buckling
Fig. 11. Axial compressive loadstrain curve for Case-C composite laminate.
-2 -1 0 1 2 3 40
250
500
750
1000
1250
1500
1750
2000
Delamination
starts growing
Local
buckling
Global buckling
Case-C
(2)Wang's results [42]
Upper sub-laminate
Middle sub-laminate
Bottom sub-laminate
(1)Finite element results
Upper sub-laminate
Middle sub-laminate
Bottom sub-laminate
Compressiveload(N)
Central deflection (mm)
Global buckling
Fig. 12. Axial compressive loadcentral deflection for Case-C composite laminate.
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Fig. 13d. Generally, the loadstrain curve using FEA is in consistent
with that by Wang [42] except the decreasing stage which is due to
unstable delamination after the laminate enters into the global
buckling stage. Compared with Case-A and B, Case-C exhibits
weaker load-bearing ability since the postbuckling response with
two delaminations is different from that with one delamination be-
cause of the interaction between two delaminating sub-laminates.
The global buckling load 1661 N using VCCT is in good agreement
with 1673 N by Wang[42]and about 1620 N by experiments[43].
At the initial postbuckling stage, the upper sub-laminate moves
largely upward and the bottom and middle sub-laminates moveslightly downward. As the strain increases, the bottom sub-lami-
nate continues to move downward, but the middle sub-laminate
changes to move upward. The energy cumulates near the crack
tip front for the upper sub-laminate, and the mode-I ERR at the
crack tip node first increases rapidly from 1.852 102 N/mm at
the 1.788 103 strain to 3.068 102 N/mm at the 2.56 103
strain. At the strain larger than 2.56 103, the delamination crack
on the slave surface for the upper sub-laminate starts to propagate.
At the 2.77 103 strain, the value for the bottom sub-laminate is
3.4379 103 N/mm, which is smaller than the 1.852 102 N/
mm, driving the bottom sub-laminate to debond. After that, the
values for upper and bottom sub-laminates add rapidly. As the
strain increases to 3.14 103, the values for upper and bottom
sub-laminates are 4.307 102
N/mm and 8.648 102
N/mm,respectively. This indicates a stronger resistance to the delamina-
tion growth for the bottom sub-laminate than that for the upper
sub-laminate. From then, the value for the upper sub-laminate re-
mains basically constant while the value for the bottom sub-lami-
nate still slightly increases with increasing strain.
Fig. 13. Delamination growth process for Case-C composite laminate at the compressive strain: (a) 2.56 103, (b) 2.77 103, (c) 3.14 103, and (d) 5.5 103.
0 1 2 3 4 50
250
500
750
1000
1250
1500
1750
2000
Delamination
starts growing
Global buckling
B-K law or Reeder law
Power law:am=an=ao=1
Power law: am=an=ao=0.5
Power law: am=an=ao=2
Case-C
Compressiveload(N)
Compressive strain (10-3)
Local
buckling
Fig. 14. Axial compressive loadcentral deflection for Case-C composite laminateusing three failure criteria.
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Fig. 14 shows the compressive loadstrain curves using three
failure criteria. Since the critical mode-II and mode-III ERR is the
same, the Reeder law is reduced to the B-K law. The initial delam-
ination appears for the parameter set am= an= ao = 0.5 earlier than
for another two parameter sets am= an= ao= 1 andam= an= ao= 2
using the power law failure criterion. The final global buckling
loads are the same using three failure criteria though four load
strain curves take on slightly different evolvement tendency after
initial delamination.
3.3.3. Case-D: 04/(45)6//04composite laminate
The finite element model with boundary conditions is shown in
Fig. 15, which includes 40,000 elements. The uniform compressive
displacement 0.127 mm is applied on the clamped edge. In order to
validate the accuracy of symmetric model, the FEA with full geom-
etry model for Case-D is also performed. Fig. 16 shows the compres-
sive loadstrain curve and Fig. 17 shows the axial compressive
load-central deflection curve using symmetric model and full mod-
el. From Figs. 16 and 17, two models lead to consistent results.
Fig. 18 shows the buckling mode after buckling analysis and
Fig. 19shows the delamination growth process using full model.
The initial buckling for the upper sub-laminate appears at the
strain 1.068 103, which is smaller than those for Case-A and
B, approaching the value 1.172 103
by Wang[42]. At the strain1.6 103, the laminate starts to delaminate as shown in Fig. 19a
and the crack propagates from the initial length 25.4 mm to
36.6 mm at the strain 1.89 103 as shown in Fig. 19b, and to
44.2 mm at the strain 2.3 103 as shown in Fig. 19c. It can be
seen that the load experiences the change process of first increas-
ing and then decreasing with strain twice, indicating complex
unstable delamination growth. From the strain 3.9 103 as
shown in Fig. 19d until the strain 5.0 103 as shown in
Fig. 19e, the composite laminate enters the global buckling stage.
By comparing the unidirectional laminate, both the collapse load
and global buckling load for the angle-ply composite laminates de-
crease largely. The global buckling load 1407 N at the strain
5.0 103 by FEA approaches the experimental value 1334 N
[43]. It should be emphasized the delamination analysis is muchtime-consuming and an equilibrium between the finite element
Fig. 15. Finite element model with boundary conditions for Case-D composite laminate.
0 1 2 3 4 50
200
400
600
800
1000
1200
1400
1600
Delamination
starts
growing
Global bucklingLocal
buckling
Symmetric
model
Full model
Unstable
delamination
growth
Compressiveload(N)
Compressive strain (10-3)
Case-D
Fig. 16. Axial compressive loadstrain curve for Case-D composite laminate.
-1 0 1 20
200
400
600
800
1000
1200
1400
1600
Local buckling
Global
buckling
(2)Full model
Upper Sub-laminate
Bottom Sub-laminate
(1)Symmetric model
Upper Sub-laminate
Bottom Sub-laminate
Case-D
Unstable
delamination
growth
Compressiveload(N)
Central deflection (mm)
Global buckling
Fig. 17. Axial compressive loadcentral deflection curve for Case-D compositelaminate.
Fig. 18. Buckling mode for Case-D composite laminate with full geometry model.
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mesh and calculation efficiency should be considered. Krueger
[36,37] pointed out too dense mesh sizes are also disadvantage
to the calculation convergence using VCCT even if various robustconvergence technique is employed. In fact, the mode-I ERR is
found to remain almost constant if the number of nodes at a crack
increment length Dain Fig. 1 increases. Thus, an appropriate selec-
tion for the mesh sizes is important to ensure the calculation con-vergence and precision.
Fig. 19. Delamination growthprocess for Case-D compositelaminateat thecompressivestrain: (a)1.6 103, (b) 1.89 103, (c) 2.3 103, (d) 3.9 103, and (e)5 103.
Fig. 20. Finite element model with boundary conditions for Case-E composite laminate.
0 1 2 3 40
200
400
600
800
1000
1200
1400
Delaminationstarts
growing
Global
buckling
Unstabledelamination
growth
Compressiv
eload(N)
Compressive strain (10-3)
Case-E
Local
buckling
Fig. 21. Axial compressive loadstrain curve for Case-E composite laminate.
-2 -1 0 1 20
200
400
600
800
1000
1200
1400
Local
buckling
Global buckling
Unstable
delamination
growth
Upper Sub-laminate
Middle Sub-laminate
Bottom Sub-laminate
Compres
siveload(N)
Central deflection (mm)
Case-E
Global
buckling
Fig. 22. Axial compressive loadcentral deflection curve for Case-E composite
laminate.
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The loaddeflection curve for the upper sub-laminate is similar
to the loadstrain curve. For the bottom sub-laminate, a small
unstable delamination stage appears at the small deflection dis-
placement. At the strain smaller than 3.9 103, the delamination
gradually grows to the clamped edge. From then, the delamination
growth rate decreases and the bottom sub-laminate starts to move
downward slowly with increasing strain.
In terms of the mode-I ERR, the value rapidly increases to
6.7 102
N/mm at the strain 1.6 103
, and 0.2428 N/mm atthe strain 2.6 103, and then remains constant until the strain
5 103, which indicates a relatively long stable delamination
stage. By comparison, the delamination growth for angle-ply com-
posite laminates requires more energy to drive the crack propaga-
tion than that for unidirectional composite laminate at the same
crack increment length Da.
3.3.4. Case-E: 04//(45)6//04 composite laminate
Fig. 20 shows the finite element model with boundary condi-
tions, which includes 40,000 elements. The uniform compressivedisplacement 0.11 mm is applied on the clamped edge. The
Fig. 23. Delamination growth process for Case-E composite laminate at the compressive strain: (a) 2.09 103, (b) 3.51 103, (c) 3.64 103, (d) 3.75 103, (e)
4.11 103, and (f) 4.33 103.
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compressive loadstrain curve is shown in Fig. 21. The loaddeflec-
tion relationship is shown in Fig. 22. The whole delamination
growth process with increasing strain is shown inFig. 23. The ini-
tial buckling for the whole laminate appears at the strain
4.828 104, which is smaller than 5.426 104 for Case-C com-
posite laminate and 1.068 103 for Case-D composite laminate.
Due to the interaction of two crack growth, the loadstrain curve
for multiple delaminations takes on more complex delamination
behavior than that for one delamination. The global buckling load
at the strain 4.33 103 as shown inFig. 23f is 445.9 N using FEA,
approaching the experimental value 420 N [43]. By comparing
1661 N for Case-C composite laminate and 1407 N for Case-D com-
posite laminate, the global buckling load 445.9 N for Case-E com-posite laminate is largely decreased. If the initial time increment
0.0001 s, the minimum time increment 1e8s, minimum time
increment 0.001 s and 10,000 load steps in the FEA are specified,
the predicted final global buckling loads for Case-E is 445.7 N.
The upper sub-laminate starts to delaminate at the strain
2.22 103 as shown inFig. 23a and the delamination crack rap-
idly propagates from the initial length 25.4 mm to 47.2 mm at
the strain 3.51 103 as shown inFig. 23b. At this time, the bot-
tom sub-laminate starts to debond. From then, the crack propaga-
tion rate for the upper sub-laminate becomes small, but the
bottom sub-laminate delaminates very rapidly until two delamina-
tion cracks reach the clamped edge at the strain 4.11 103 as
shown in Fig. 23e. By comparing Case-C composite laminate,
Case-E exhibits different delamination behavior in terms of theinteraction between the propagation of two delamination cracks,
which may be attributed to the angle-ply effect and initial crack
length. The maximum mode-I ERR at the crack tip nodes for the
upper sub-laminate increases to 0.2093 N/mm at the strain
2.09 103, but the value for the bottom sub-laminate is only
2.724 103 N/mm. After then, the value for the upper sub-lami-
nate slowly increases to 0.2923 N/mm at the strain 3.51 103,
but the value for the bottom sub-laminate increases rapidly to
0.7583 N/mm. From the strain 4.11 103 to 5 103, the values
for two sub-laminates keep almost unchanged.
By comparison, the delamination growth for Case-E angle-ply
composite laminate with multiple delaminations requires more
energy to drive the crack propagation than for the Case-C unidirec-
tional composite laminate with two delaminations and the Case-Dangle-ply composite laminate with one delamination at the same
increment length Da. Besides, the slight increase for the tolerance
in the failure criterion can improve the calculation convergence to
some extent, but may affect the local calculation precision, which
is consistent with Kruegers conclusion [36,37]. However, almost
no effect on the global buckling behavior of composite laminate
occurs.
Finally, the local buckling load, the load at which the delamina-
tion growth starts (stable or unstable) and global buckling load for
five cases are summarized inTable 3. The mode-I ERR at which the
crack starts growing for five cases are listed inTable 4.
4. Conclusions
The parametric finite element model using ABAQUS is proposed
to study the multiple through-the-width delaminations and post-
buckling behavior of composite flat laminates. The VCCT is used
to calculate the energy release rate and predict the delamination
crack propagation. The finite element results using VCCT are in rel-
atively good agreement with those by existing model and experi-
ments. It can be concluded from finite element results multiple
delaminations largely decrease the collapse load and global buck-
ling load of composite laminate, but the initial delamination length
has relatively small effect on the global buckling load. Different
failure criteria such as the B-K law and Power law lead to the same
global buckling load. The number of load steps in the nonlinear FEA
has almost no effect on the load-compressive strain curve, but has
a little effect on the calculation efficiency to some extent. In addi-
tion, an appropriate selection for the mesh sizes is important to im-
prove the calculation precision and convergence.
Acknowledgements
This research is supported by the national natural science fund-
ing of China (Number: 50905197), the crossover research seed
funding for young teacher in Zhejiang University, the key project
Chinese universities scientific funding of Zhejiang University.
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Initial delamination
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