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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

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c t
http://dx.doi.org/10.1016/j.compstruct.2010.12.006mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.compstruct.2010.12.006http://www.sciencedirect.com/science/journal/02638223http://www.elsevier.com/locate/compstructhttp://www.elsevier.com/locate/compstructhttp://www.sciencedirect.com/science/journal/02638223http://dx.doi.org/10.1016/j.compstruct.2010.12.006mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.compstruct.2010.12.006

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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


Wang's result [42]

Local

buckling

Compressiveload(N)


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


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


Wang's result [42]

Case-C


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)


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)


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)


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)


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)


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)


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.

References

[1] Cox BN, Yang QD. In quest of virtual tests for structural composites. Science

2006;314:11027.

[2] Lu K. The future of metals. Science 2010;16:31920.

[3] Chang FK, Chang KY. A progressive damage model for laminated composites

containing stress concentrations. J Compos Mater 1987;21:83455.

[4] Tay TE, Liu G, Yudhanto A, Tan VBC. A micro-macro approach to modeling

progressive damage in composite structures. Inter J Damage Mech

2008;17:528.

[5] Liu PF, Zheng JY. Progressive failure analysis of carbon fiber/epoxy composite

laminates using continuum damage mechanics. Mater Sci Eng A-Struct

2008;485:7117.

[6] Liu PF, Zheng JY. Review on methodologies of progressive failure analysis of

composite laminates. In: Koppel A, Oja J, editors. Continuum mechanics. New

York: Nova Science Publishers; 2009 [Chapter 11].

[7] Liu PF,Zheng JY. Mechanical strength andoptimal design of complex structure.

In: Covali A, editor. Mechanical strength and optimal design of complex

structure. Saarbrcken: Lap Lambert Academic Publishing; 2010.

[8] Kim RY, Soni SR. Experimental and analytical studies on the onset of

delamination in laminated composites. J Compos Mater 1984;18:7080.

[9] Raju IS, Crews Jr JH, Aminpour MA. Convergence of strain energy release rate

components for edge-delaminated composite laminates. Eng Fract Mech

1988;30:38396.

[10] Camanho PP, Davila CG, de Moura MF. Numerical simulation of mixed-mode

progressive delamination in composite materials. J Compos Mater

2003;37:141538.

[11] Foo CC, Chai GB, Seah LK. A model to predict low-velocity impact response and

damage in sandwich composites. Compos Sci Technol 2008;68:134856.

[12] Tita V, de Carvalho J, Vandepitte D. Failure analysis of low velocity impact on

thin composite laminates: experimental and numerical approaches. Compos

Struct 2008;83:41328.

[13] Faggiani A, Falzon BG. Predicting low-velocity impact damage on a stiffened

composite panel. Compos Part A-Appl Sci 2010;41:73749.

[14] Sleight DW. Progressive failure analysis methodology for laminated compositestructures. NASA Report 1999:4863.

Table 3

Local buckling load, initial delamination load and global buckling load for five cases

(for multiple delaminations, the initial delamination load denotes one at which the

upper sub-laminate starts growing).

Case Lay-ups Local buckling

load (N)

Initial delamination

load (N)

Global buckling

load (N)

A 04/012//04 1627 2160 3216

B 04/012//04 474 1816 3331

C 04//012//04 493 1971 1661D 04/(45)6//04 716 1125 1407

E 04//(45)6//04 384 1350 445

Table 4

Mode-I energy release rate at which the delamination starts growing (for multiple

delaminations, the Mode-I ERR denotes one at which the upper sub-laminate starts

growing).

Case Lay-ups Mode-I ERR (N/mm)

A 04/012//04 0.0812

B 04/012//04 0.0467

C 04//012//04 0.0307

D 04/(45)6//04 0.0632

E 04//(45)6//04 0.2093


8/11/2019 Liu VCCT

12/12

[15] Tay TE, Liu G, Tan VBC, Sun XS, Pham DC. Progressive failure analysis of

composites. J Compos Mater 2008;42:192166.

[16] Garnich MR, Akula Venkata MK. Review of degradation models for progressive

failure analysis of fiber reinforced polymer composites. Appl Mech Rev

2009;62:135.

[17] LiuPF, Zheng JY. Recent developments on damage modeling andfiniteelement

analysis for composite laminates: a review. Mater Des 2010;31:382534.

[18] Kim H, Kedward KT. A method for modeling the local and global buckling of

delaminated composite plates. Compos Struct 1999;44:4353.

[19] Hu H, Belouettar S, Potier-Ferry M, Makradi A. A novel finite element for local

and global buckling analysis of sandwich structures. Compos Struct2009;90:2708.

[20] Chai H, Babcock C, Knauss W. One dimensional modelling of failure in

laminated plates by delamination buckling. Int J Solids Struct 1981;17:

106983.

[21] Chai H, Babcock C. Two-dimensional modelling of compressive failure in

delaminated laminates. J Compos Mater 1985;19:6798.

[22] Bolotin VV. Delamination in composite structures: its origin, buckling, growth

and stability. Compos Part B-Eng 1996;27:12945.

[23] Hwang SF, Mao CP. Failure of delaminated carbon/epoxy composite plates

under compression. J Compos Mater 2001;35:163453.

[24] Kutlu Z, Chang FK. Modeling compression failure of laminated composites

containing multiple through-the-width delamination. J Compos Mater

1992;26:35086.

[25] Cappello F, Tumino D. Numerical analysis of composite plates with multiple

delamination subjected to uniaxial buckling load. Compos Sci Technol

2006;66:26472.

[26] Suemasu H, Irie T, Ishikawa T. Buckling and post-buckling behavior if

composite plates containing multiple delamination. J Compos Mater 2009;

43:191202.

[27] Rybicki EF,KanninenMF. A finite element calculation of stressintensityfactors

by a modified crack closure integral. Eng Fract Mech 1977;9:9318.

[28] Rybicki EF.An energyrelease rate approach for stablecrack growthin thefree-

edge delamination problem. J Compos Mater 1977;11:4704 87.

[29] Irwin GR. Fracture dynamics, fracturing of metals. Cleveland: Am Soc Metals

1948:14766.

[30] Krueger R. Virtual crackclosure technique: history, approach, and applications.

Appl Mech Rev 2004;57:10943.

[31] Rice JR. A path independent integral and the approximate analysis of strain

concentration by notches and cracks. J Appl Mech 1968;35:37986.

[32] Xie D, Biggers SB. Progressive crack growth analysis using interface element

based on the virtual crack closure technique. Finite Elem Anal Des

2006;42:97784.

[33] Leski A. Implementation of the virtual crack closure technique in engineering

FE calculations. Finite Elem Anal Des 2007;43:2618.

[34] Orifici AC, Thomson RS, Degenhardt R, Bisagni C, Bayandor J. A finite element

methodology for analysing degradation and collapse in postbuckling

composite aerospace structures. J Compos Mater 2009;43:323963.

[35] Gaudenzi P, Perugini P, RiccioA. Post-buckling behavior of compositepanels inthe presence of unstable delamination. Compos Struct 2001;51:3019.

[36] Krueger R, Goetze D. Influence of finite element software on energy release

rates computed using the virtual crack closure technique. NASA/CR-2006-

214523; 2006.

[37] Krueger R. An approach to assess delamination propagation simulation

capabilities in commercial finite element codes. NASA TM/2008-215123;

2008.

[38] Pietropaoli E, Riccio A. On the robustness of finite element procedures based

on virtual crack closure technique and fail release approach for delamination

growth phenomena definition and assessment of a novel methodology.

Compos Sci Technol 2010;70:1288300.

[39] Benzeggagh M, Kenane M. Measurement of mixed-mode delamination

fracture toughness of unidirectional glass/epoxy composites with mixed-

mode bending apparatus. Compos Sci Technol 1996;56:43949.

[40] Wu EM, Reuter Jr RC. Crack extension in fiber-glass reinforced plastics. Univ Ill

TAM Report 1965:275.

[41] Reeder J, Kyongchan S, Chunchu PB, Ambur DR. Postbuckling and growth of

delamination in composite plates subjected to axial compression. In: 43rd

AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials

conference. Denver, Colorado; 2002.

[42] Wang S, Zhang Y. Buckling, post-buckling and delamination propagation in

debonded composite laminates part 2: numerical applications. Compos Struct

2009;88:13146.

[43] Kutlu Z, Chang FK. Composite panels containing multiple through-the-width

delamination and subjected to compression part II: experiments and

verification. Compos Struct 1995;31:297314.


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