-
a r t i c l e i n f o
Article history:Received 18 September 2007Received in revised
form 13 February 2008Accepted 24 February 2008
ical energy release rate (GC) has been widely used to
characterize the crack growth resistance. GC can be measured using
var-ious testing methods such as the double-cantilever beam (DCB)
test [1] for tension mode (mode I), the end-notched exure(ENF) test
[2] for sliding mode (mode II), and the mixed-mode exure (MMF) test
[3] for mixed-mode of I and II.
0013-7944/$ - see front matter 2008 Elsevier Ltd. All rights
reserved.
* Corresponding author. Tel.: +1 780 9900964; fax: +1 780
4922200.E-mail address: [email protected] (C. Fan).
Engineering Fracture Mechanics 75 (2008) 38663880
Contents lists available at ScienceDirect
Engineering Fracture Mechanics
journal homepage: www.elsevier .com/locate
/engfracmechdoi:10.1016/j.engfracmech.2008.02.010material strength
on the damage initiation for delamination. 2008 Elsevier Ltd. All
rights reserved.
1. Introduction
Delamination in bre reinforced polymers (FRP) and fracture of
adhesive joints are two types of failure that have longbeen the
centre of materials research. Their occurrence can result in
signicant loss in the structural stiffness, and is danger-ous
because they often occur inside the components, thus difcult to
detect from the surface until catastrophic failure isimminent. The
failure is known due to separation at an interphase region, caused
by cyclic loading, low velocity out-of-planeimpact, or defects
introduced during manufacture.
In experimental studies, this type of failure is generally known
within the framework of fracture mechanics, in which crit-Available
online 10 March 2008
Keywords:Damage material modelCohesive zoneDelaminationFEMa b s
t r a c t
A new approach is developed to implement the cohesive zone
concept for the simulation ofdelamination in bre composites or
crack growth in adhesive joints in tension or shearmode of
fracture. The model adopts a bilinear damage evolution law, and
uses criticalenergy release rate as the energy required for
generating fully damaged unit area. Multi-axial-stress criterion is
used to govern the damage initiation so that the model is able
toshow the hydrostatic stress effect on the damage development. The
damage materialmodel is implemented in a nite element model
consisting of continuum solid elementsto mimic the damage
development. The validity of the model was rstly examined by
sim-ulating delamination growth in pre-cracked coupon specimens of
bre composites: thedouble-cantilever beam test, the end-notched
exure test and the end-loaded split test,with either stable or
unstable crack growth. The model was then used to simulate
damageinitiation in a composite specimen for delamination without a
starting defect (or a pre-crack). The results were compared with
those from the same nite element model (FEM)but based on a
traditional damage initiation criterion and those from the
experimentalstudies, for the physical locations of the delamination
initiation and the nal crack sizedeveloped. The paper also presents
a parametric study that investigates the inuence ofCohesive zone
with continuum damage properties for simulationof delamination
development in bre composites and failureof adhesive joints
Chengye Fan a,*, P.-Y. Ben Jar a, J.J. Roger Cheng b
aDepartment of Mechanical Engineering, University of Alberta,
4-9 MECE Building, Edmonton, AB, Canada T6G 2G8bDepartment of Civil
and Environmental Engineering, University of Alberta, Edmonton, AB,
Canada T6G 2G7
-
For numerical studies of this type of failure, there are two
basic categories according to the objectives. The rst is to
pre-dict the energy release rate (G) for a given path of crack
growth and loading condition, and the second is to simulate
crackpropagation based on experimentally determined critical energy
release rate for the crack growth. In the rst category,numerical
techniques in fracture mechanics, such as virtual crack closure
technique (VCCT) [4], the J-integral [5,6], the com-pliance
derivative technique (CDT) [7] and the energy derivative technique
(EDT) [8], have been shown to be able to predictthe G value using
experimentally determined loading conditions. In the second
category, provided that the variation of G isdetermined from the
methods in the rst category, crack growth could be predicted based
on the Grifths theory of criticalenergy release rate [9]. However,
difculties exist for these techniques to achieve these objectives.
For example, the CDT andthe EDT are inherently impossible to
predict the crack growth, because the crack growth path and rate
are the prerequisitesfor applying these techniques. VCCT and the
J-integral are hindered by the lack of topological information of
the crack prolethat is needed to calculate G for the crack growth
[10]. As a result, these techniques are only used to predict the
onset of thecrack growth.
A different approach, known as cohesive zone model that was
rstly proposed by Barenblatt [11], provides an alternativesolution
to avoid the above difculties. Use of the cohesive zone model to
simulate crack growth is depicted in Fig. 1 in whicha cohesive zone
is bounded by upper and lower cohesive surfaces. A damage zone is
developed in the cohesive layer at thecrack front of which
properties degrade with deformation due to material damage or
plastic softening. A stress limit is set for
C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
38663880 3867the cohesive zone based on the material strength,
which serves as a criterion for the damage initiation. That is,
when thestress limit is reached, the damage starts to develop, and
the stress decreases with the increase of the relative
displacement(d) between the two cohesive surfaces. Eventually, the
stress is reduced to zero, leading to the formation of a new crack
area.Coupling between stress (r) and d is governed by the cohesive
constitutive law, with the area underneath the rd curve
rep-resenting the critical energy release rate, GC.
In the past, the cohesive zone model was implemented in FEM
using nonlinear spring [1115] or interface elements [1636] that
have all the three key properties mentioned above, namely, damage
initiation criterion, constitutive law (rd curve)and GC. In most
approaches, GC is treated as a material constant and can be
determined experimentally. However, these ap-proaches have some
deciency as discussed below.
Most of the constitutive laws in the literature were developed
in a phenomenological way, in which the rd curve wasexpressed in
several functions, such as an exponential function [36,37], a
trapezoidal function [38], or most commonly a bi-linear function
[10,19,23,32]. Mathematical and physical limitations of these
approaches have been discussed by Jin and Sun[39,40]. To our
knowledge, there is no preferred function for the rd curve, as
these functions show similar results in sim-ulation, especially in
terms of a load-displacement curve for the crack growth. This is
because the constitutive laws mainlygovern the behavior within the
cohesive damage zone that is too small to affect the global
load-displacement curve.
FEMwork in the literature based on the cohesive zone model has
shown that damage initiation in a pre-cracked FRP is notsensitive
to material strength [18,19], except when an extremely weak
material is used. The use of weak material results ina very large
cohesive damage zone, thus signicantly reducing the global
stiffness of the structures. In the past, the choice ofcohesive
zone strength was mainly to improve the computational efciency
without losing the simulation accuracy for thedamage initiation,
such as the work by Alfano and Criseld [18] and Blackman et al.
[19]. However, all the cases used in theirstudies have an initial
crack, thus the results are only meaningful for supporting damage
growth scenarios that are governedby GC values.
For damage initiation, such as delamination growth from matrix
cracking in FRP [4143], accurate simulation using thecohesive zone
model requires a proper choice of the cohesive zone strength as the
damage initiation criterion. Stress-basedcriteria, such as those
suggested by Choi et al. [42] and Hou et al. [44,45], have shown
good predictions. However, these cri-teria were never implemented
in any cohesive zone models. Instead, the cohesive zone models
presented in the literaturemainly use interface elements [1636]
based on displacement jump vector and the corresponding
energy-conjugated trac-tion vector that does not include in-plane
normal stress/strain components. Other approaches in the literature
[1215] use
cohesive zone (undamaged)
upper/lowercohesivesurfaces
cohesive damage zone
crack front 1 3 2
Gc
Fig. 1. Schematic cohesive zone model.
-
develo[43].
2 2 2 2(
3868 C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
38663880e rii=T r12 r13=S for rii > 0r212 r213 arii2=S2 for rii
6 0
2
where r13 and r23 are shear stresses, a a non-dimensional
parameter that is used to quantify the effect of the
compressivehydrostatic stress on the suppression of matrix
cracking, T the transverse tensile strength and S the shear
strength of the FRPmatrix or adhesive. The only adjustable
parameter a is determined using the critical load for the damage
initiation, as to bediscussed in Section 4. The value of e
determines whether the critical condition for the damage initiation
is satised. That is,the damage is initiated when e is equal to or
larger than 1.
The dependence of the damage initiation criterion on the
hydrostatic stress is to reect the common phenomenon thatthe
hydrostatic compression may slow down or suppress the damage
initiation [50]. Note that when subjected to pure ten-sile or shear
stresses, the above criterion is identical to the maximum tensile
or shear stress criterion, respectively. Section 4will discuss the
damage development in a stress state that is a combination of
normal and shear stresses, to evaluate validityof the criterion for
the damage initiation.
2.3. Damage evolution law
The damage evolution law adopted in this study, which governs
the development of the damage parameter d, is base on abilinear
stressstrain relationship as depicted in Fig. 2.2.1. Constitutive
relationship
The cohesive damage material adopts the isotropic damage elastic
constitutive relationship
rij 1 dEijklekl i; j; k; l 1;2;3 1where rij and eij are stress
and strain components, respectively, d a scalar damage variable in
the range from 0 to 1 whichcontrols the degradation of the initial
elastic stiffness Eijkl.
2.2. Damage initiation criterion
The previous experimental study on FRP [4143] has shown that
specimens without a starting defect are expected to ini-tiate
delamination from matrix cracking that usually occurs in the
resin-rich region between two adjacent layers. For thenew damage
material model, the criterion for the damage initiation is
expressed in two functions, depending on the hydro-static stress
component (rii) being positive or negative2. Damage material
model
The cohesive zone used here to simulate crack evolvement in FRP
delamination and adhesive failure is represented by athin layer of
continuum solid elements, of which the property change follows the
new damage material model. The cohesivezone is inserted between two
potentially separable surfaces with local coordinates that have
1-axis perpendicular to thecrack surfaces, as shown in Fig. 1, in
order to facilitate determination of the fracture mode. For
example, when subjectedto mode I loading, the cohesive zone layer
is expected to extend in the 1-direction. Thus, by examining
deformation behaviorof the cohesive zone layer, the mode for
deformation and fracture can be determined. In this paper, all
stresses are expressedaccording to the local coordinates.pment in
FRP without any pre-cracks, and comparison of the results with the
experimental data reported previouslyspring elements which can only
adopt the simple maximum stress criterion, since axial stress is
the only stress componentthat can be used as the cohesive zone
strength.
To overcome the above problems, we propose a new approach that
uses solid elements for the cohesive zone so thatdamage initiation
criteria based on multi-axial stresses can be implemented for the
simulation. Note that a similar ap-proach has been used in
ply-based damage models for the simulation of in-plane failure in
composite [4648]. However,to the authors knowledge, no damage model
has ever considered the critical energy release rate in the
simulation of thedamage evolution for delamination. Some examples
that have been considered are: a prescribed function for the
damagedevelopment [44,45], the immediate drop of material modulus
to zero after the damage initiation [42], and an intuitivematerial
softening law without considering the physical behaviour [49]. The
proposed new approach is different in thatits damage initiation
criterion includes multi-axial stress components, with all the
parameters having physical meaningsand being experimentally
obtainable. Using this approach, the paper will show that mode II
delamination in FRP without apre-crack can be predicted by a FEM
model, in terms of the critical load for the delamination
initiation, its location and thenal delamination size.
This paper will present topics in the following order: (1) the
new damage material model and its criteria; (2) simulation ofDCB,
ELS and ENF tests, and comparison of the results with the
analytical solutions; and (3) simulation of delamination
-
exponIn
As
K
I/shear
C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
38663880 3869depending on the dominant mode for the damage
development. In addition, positions for points A and B in Fig. 2
also varywith the fracture mode, in which point A refers to the
state where the damage initiation criterion is met. Beyond point A
thestress reduces to zero as a linear function of the separation
distance between the two surfaces (d) which is equivalent to
theproduct of strain and cohesive zone thickness. When the damage
evolvement is dominated by the shear mode, i.e. b valuebeing
greater than 1.0, shear separation (dshear) is used in Fig. 2 as
the governing parameter. For the opening mode of thedamage
development, i.e. b < 1, separation in opening mode (dI) is
employed. The nal separation (dfI=shear) for a bilinear func-tion
of r and d, i.e. point B in Fig. 2, is calculated based on the
following formula
dfI=shear 2GI=shearrmaxI=shear
4
where rmaxI=shear represents the normal or shear stress for
damage initiation (at point A), and GI/shear the energy release
rate forthe tensile or shear mode required for completion of the
fracture process.
Note that based on the BK criterion [52] for a mixed-mode damage
development, gThe co
The dad in Fi
whereTh
whening thshown in Fig. 2, the damage development is determined
by the surface separation in either shear or opening mode,ratio
that is equivalent to the ratio of Gshear to GI, in which Gshear
gives no distinction between the sliding mode (mode II) andthe
tearing mode (mode III) of fracture, following the suggestion by
Camanho et al. [22]. The total energy release rate due tomodes II
and III of fracture is simply regarded as Gshear.
In principle, the b value should be determined after the damage
has been fully developed, which is impractical for theFEM
simulation based on the damage material model. Therefore, b value
is assumed to be constant in each element duringthe property
degradation. This allowed us to determine b before the damage is
initiated, based on the ratio of energy that isresponsible for the
shear deformation to that for the tensile deformation. That is,
b r223=G23 r213=G13=lt; r11gt;2=E11 3where the stress values are
those in the element that rstly satisfy Eq. (2), and G23, G13 and
E11 are the corresponding moduliin the undamaged stage. The
denotation h i means that the value enclosed is unchanged if it is
positive, otherwise 0.ential, or linear/bilinear functions, or
Benzeggagh and Kenanes function [52] (also known as BK
criterion).order to allow the possibility of damage development in
a mixed-mode, a parameter b is dened as the mode-mixingIn a pure
mode of fracture, area of the triangle OAB in Fig. 2 represents the
critical energy release rate in the correspond-ing fracture mode.
In the mixed-mode fracture, the total critical energy release rate
GC is assumed to be a function of GIC andGIIC which are the
fracture toughness in pure opening and shearing modes,
respectively. There have been several approachessuggested to
determine GC for the mixed-mode of fracture. As summarized by
Reeder [51], these approaches use power-law,Fig. 2. Constitutive
law of the damage material. Area of OAB is equivalent to G .O
fshearI /o shear /I max
B
(1-d)KAG I/ shear
max
/shearIGC GIC GIIC GIC b1 b 5
rresponding energy release rates for tensile mode (GI) and shear
mode (Gshear) are
GI GC=b 1 and Gshear GCb=b 1 6mage variable dwas determined
using the following expression, based on the geometrical
relationship between r andg. 2.
d df1=sheardmax1=shear do1=shear
dmax1=sheardf1=shear do1=shear7
dmax1=shear is the maximum separation in history generated by
the loading in tension or shear.e FEM model used in this study does
not allow stiffness degradation in the out-of-plane direction
(1-direction in Fig. 1)the out-of-plane compressive strain is
present. This is to avoid penetration of the two delamination crack
surfaces dur-e damage development process.
-
3. Verication of the cohesive zone model for crack growth
The cohesive zone model with the damage material characteristics
was implemented in the commercial FEM code ABA-QUS/Explicit through
the user subroutine (VUMAT). The DCB, the ENF and the ELS tests
were chosen to verify validity of thenew model for simulation of
the crack growth. Congurations of the three tests are shown in Fig.
3, with parameters of thespecimen dimensions and test set-up listed
in Table 1. Note that the conditions in Table 1 were chosen to
generate stablecrack growth for the DCB test and unstable for the
ENF and ELS tests.
3.1. FEM model
FEM model for the DCB, the ENF and the ELS tests contains three
parts: upper beam, lower beam and a layer of cohesivezone in
between. The upper and lower beams were modeled based on transverse
orthotropic elasticity, as the specimen con-tains only 0 bre.
Values used in Ref. [37] were adopted as the material constants for
the upper and lower beams, and arelisted in the upper row of Table
2. The cohesive zone layer has thickness of 0.02 mm to minimize its
role on the global stiff-ness of the specimen. Material constants
for the cohesive zone and parameters for the damage material model
are listed inthe middle row of Table 3. Note that a in Eq. (2) is
set to be zero for these cases. This is because the three tests use
specimenswith a starting defect. Thus, the simulation is mainly for
the crack growth and should not be sensitive to the a value for
Eq.(2), as this is a criterion for the damage initiation for
delamination.
Mesh pattern of the FEM model, as shown in Fig. 3, consists of
200 4-node plane-strain continuum elements (CPE4R) foreach of the
upper and lower beams. To mimic the experimental condition, loading
was applied by specifying a reasonablyconstant displacement rate
that is computationally efcient but without introducing dynamic
effect such as that by the iner-tial force.
The element length in the cohesive zone was chosen to be 0.05
mm. According to Turon et al. [35], element length in thecohesive
zone layer should be carefully chosen to capture the continuum
stress eld in the cohesive damage zone. Amongmany theories used to
estimate the cohesive zone length (lcz), the most conservative
estimate for mode I test is [35]:
3870 C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
38663880Fig. 3. FEM model of the specimen with a starting
defect.
Table 1Geometrical parameters of the DCB, ENF and ELS tests
Span length L (mm) Initial crack length a (mm) Half thickness h
(mm) Loading rate w (mm/sec)
DCB 50 1.5 0.8ENF 100 30 1.5 0.5ELS 100 50 1.5 2.0
Table 2Material properties for unidirectional FRP used in the
study
Test method E11 (GPa) E22 = E33 (GPa) G12 = G13 (GPa) v23 v12 =
v13 Material damping (kg/sec)
DCB, ENF and ELS 150 11.0 6.0 0.3 0.3 100a
Beam test 34.3 6.0 6.0 0.3 0.3 100
a Material damping is not used in the DCB model.
Table 3Properties for the cohesive zone layer and parameters for
the damage material model
Test Method E (GPa) v GIC (J/m2) GIIC (J/m2) g Tensile strength
T (MPa) Shear strength S (MPa) a
DCB, ENF and ELS 11.0 0 300 300 2.0 20.0 20.0 0Beam test 6.0 0
500 2500 2.0 47 40 0.3
-
lcz 0:21 EGCT28
Based on the constants given in the middle row of Table 3, the
above expression yields lcz of 0.77 mm which is more than 15times
of the element size selected for the cohesive layer (0.05 mm).
Therefore, mesh size for the cohesive zone is deemed tobe small
enough to provide good resolution for the stress distribution. Note
that the same element size was used for simu-lation of all the
three tests.
3.2. DCB test
The DCB test, with the conguration shown in Fig. 4a, is to
measure fracture toughness of FRP for mode I delamination.
Itsspecimen consists of 0 ber with an insert lm to initiate the
delamination, placed at the mid-thickness of one end of
thespecimen. Analytical solution for the specimen compliance (C)
and energy release rate for the delamination growth (GDCB),based on
classical beam theory and linear elastic fracture mechanics,
are
CDCB wP 4a
3
E11h3 9
GDCB 12P2a2
E11h3 10
where E11 is the longitudinal Youngs modulus, a crack length, h
half specimen thickness,w the displacement at the specimenend where
the insert lm is placed, and P the reaction force at the point
where w is measured.
The load-displacement curve generated by the FEM model is
compared with the analytical solution in Fig. 5a, which in-
by elimbeforeimen
C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
38663880 3871Fig. 4. Congurations of (a) DCB, (b) ELS and (c) ENF
tests.ELS
L
a
w
ENF
L
aw
2h
b
ca
w
2h3.3. ENF test
The ENF test, with the conguration shown in Fig. 4b, is a common
test to measure mode II delamination resistance. Itsanalytical
expressions for C and G, based on the simple beam theory, are
CENF L3 12a332E11h
3 11
GENF 9P2a2
16E11h3 12
where L is span length between the supports.
wDCBainating a and assuming constant GDCB. The two curves in
Fig. 5a show good agreement, with discrepancy exists onlythe
delamination growth. Such discrepancy is mainly caused by the
underestimate of the compliance of the DCB spec-from the classical
beam theory [7], not by the FEM model.cludes an initial linear
loading section to represent the response before the commencement
of the delamination propagation.The nonlinear, descending section
that follows the onset of delamination growth was determined based
on Eqs. (9) and (10)
-
3872 C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
386638802
2.5aDifferent from the DCB test, the ENF test usually generates
unstable crack growth. This is because energy released by thecrack
propagation in the ENF test is more than that required for forming
new crack surfaces. The unstable crack growthcauses specimen
vibration that is eventually damped out. Therefore, its FEM model
has a material damping function todissipate the extra energy. Fig.
5b shows the load-displacement curves generated for the ENF test.
The dashed line representsanalytical solution of Eqs. (11) and
(12), which shows that after the maximum load is passed, a fast
load drop occurs.
0 2 4 6 80
0.5
1
1.5
Displacement (mm)
Load
(N)
FEMAnalytical
0 0.5 1 1.50
5
10
15
20
Displacement (mm)
Load
(N)
FEMAnalytical
0 2 4 6 8 100
1
2
3
4
5
6
Displacement (mm)
Load
(N)
FEM, triple SFEM, doulbe SFEMAnalytical
b
c
Fig. 5. Load-displacement curves for (a) DCB test, (b) ENF test,
and (c) ELS test.
-
The phenomenon of fast load drop is consistent with the FEM
solution (presented by a solid line with d), though theanalytical
solution suggests a decrease of the displacement with the load drop
while the FEM solution simply shows theload drop.
3.4. ELS test
The ELS test, with the conguration shown in Fig. 4c, is used to
generate relatively stable crack growth for the measure-ment of the
mode II delamination resistance. Its analytical expressions for
compliance and energy release rate are
CELS L3 3a32E h3
13
of the load for delamination on the strength of the damage
material model diminishes when a relatively large value was
used
ferentin thespecim
C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
38663880 38730.5mm offset
0-degree 90-degree z
x
Fig. 6. Beam test conguration.positions along the thickness. The
FEM study considered delamination in three types of specimens that
were differentposition of the 90 layer, placed at a distance of
either 1/4, 1/2, or 3/4 of the thickness from the bottom surface.
Theseens are to be named 1/4-beam, 1/2-beam, and 3/4-beam
specimens, respectively. Detailed information for the exper-for the
material strength. The above results suggest that with an existing
crack, the main role of material strength in thecohesive zone model
is to change the size of the damage zone.
4. Simulation of the beam test
Experimental results from the beam test [54] that applies
bending to FRP coupon specimens to generate delaminationinitiation
and propagation between bre layers of different orientations were
used to further assess validity of the new dam-age material model
and its criteria for the damage initiation. The main difference
between the beam test and the otherdelamination tests that have
been considered for the FEM simulation [1636] is that the former
does not have a startingdefect to initiate the delamination.
4.1. Experimental results from the beam test
Experimental results for the beam test were taken from a
previous study [54] that used coupon specimens of mainly
uni-directional bre along the specimen length direction, except one
layer that in the transverse direction (90 layer). Withoutany
starting defect, the beam test initiates delamination from the 90
layer to grow in the adjacent upper or lower interlam-inar region.
By placing the 90 layer in different positions of the bre stack,
the beam tests generated delamination at dif-11
GELS 9P2a2
4E11h3 14
where L is the distance between the loading point and the
support.Similar to the ENF test, the above expressions generate a
curve that indicates an unstable delamination growth. However,
the FEM solution from the same test conguration shows a
progressive drop of the force from the maximum load, as shownby the
thick solid line in Fig. 5c. The inconsistent crack growth behavior
generated by Eqs. (13), (14) and by the FEM is be-cause the initial
crack length used (0.5 L) is very close to, but shorter than the
minimum value for the stable crack growth,0.55 L [53]. Due to the
presence of a damage zone at the crack tip, which based on the
FEMmodel has a length of 9.5 mm (orabout 0.1 L), the effective
crack length for delamination in the FEM model should exceed the
minimum value for the stablecrack growth. Fig. 5c also shows that
by doubling and tripling the shear strength S used in the FEM
model, thus decreasingthe cohesive damage zone length to 3.9 mm
(0.04 L) and 2.4 mm (0.03 L), respectively, the delamination growth
becomesunstable. Since the two curves are very close to each other,
they are presented in Fig. 5c as two thin lines without
anydistinction from each other.
The curves from the FEM simulation in Fig. 5c suggest that even
by tripling the shear strength of the damage materialmodel, the
maximum load only increases slightly. Therefore, for the test
conguration used in this study, the load for delam-ination is not
sensitive to the material strength. This agrees with the conclusion
drawn in Refs. [18,19] that the dependency
-
3874 C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
386638807000aiments is given in Ref. [43]. Note that the
delamination was initiated only on one side of the loading pin, and
its growth wascompleted at constant deection.
0 0.5 1 1.50
1000
2000
3000
4000
5000
6000
Displacement (mm)
Load
(N)
FEM, 1/2-beam Test, 1/2-beam
0 0.5 1 1.50
1000
2000
3000
4000
5000
6000
7000
Displacement (mm)
Load
(N)
FEM, 1/4-beamTest, 1/4-beam
0 0.5 1 1.50
2000
4000
6000
8000
10000
Displacement (mm)
Load
(N)
FEM, 3/4-beamTest, 3/4-beam
b
c
Fig. 7. Comparisons of the load-displacement curves from the
beam tests for (a) 1/2-beam, (b) 1/4-beam, and (c) 3/4-beam.
-
C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
38663880 38754.2. FEM simulation of the beam test
4.2.1. FEM modelThe simulation used a 2D plane-strain model, as
shown in Fig. 6, with restriction from any vertical movement for
nodes at
the supports. The rigid body motion was excluded by restricting
the node at the middle of the bottom line from any hori-zontal
movement. The model contains three layers, with the top and bottom
layers being the 0 layers and having orthotro-pic and elastic
properties given in the bottom row of Table 2. Properties in the
middle layer, where delamination occurs, arebased on the cohesive
damage material model, with properties and parameters given in the
bottom row of Table 3. The load-ing pin had an offset to the right
by 0.5 mm in order to generate crack growth only on one side of the
loading pin. Constant
Fig. 8. Comparisons of the delamination onset location in the
beam test for (a) 1/2-beam, (b) 1/4-beam, and (c) 3/4-beam.
-
loading speed of 10 mm/s was applied until unstable delamination
occurred. The specimen was then unloaded at the samespeed.
Except Youngs modulus E and Poissons ratio m that were based on
the previous experimental study [43], values in thelower row of
Table 3 were chosen in the following way. Tensile strength T was
based on the tensile strength of the polyesterresin [55], and the
shear strength S based on the maximum shear stress of the 1/2-beam
test generated at the maximum load.The latter choice was because
for the 1/2-beam specimens, shear stress is the only non-zero
stress component at the mid-thickness and is uniformly distributed
between the loading pin and the support. GIIC was set to be equal
to 2500 J/m2 accord-ing to the value reported before [43], and GIC
500 J/m2 [56]. The parameter g, as before, was set to be equal to
2, following thesuggestion by Benzeggagh and Kenane for brittle
fracture [52]. Value of a was chosen to be 0.3 by matching the
predictedmaximum load for the 3/4-beam test with the experimental
results [54].
4.2.2. ResultsFig. 7 compares load-displacement curves for
1/4-beam, 1/2-beam, and 3/4-beam specimens from the experiments
with
those from the FEM simulation. All the curves show that the load
increased initially with displacement in a linear fashion butthen
dropped quickly to a lower level after a critical loading level was
reached. The Fig. suggests that the load-displacementcurves
generated by the FEM simulation are consistent with those from the
experiments.
Note that the 3/4-beam, Fig. 7c, has the maximum load over 30%
higher than that for the 1/2-beam or 1/4-beam. As men-tioned
earlier, since delamination in the 3/4-beam was initiated under
compressive hydrostatic stress, its maximum loadwas used to
determine a value in Eq. (2).
Fig. 8 compares delamination initiation locations in the three
types of Beam specimens observed experimentally (pointedout by an
arrow in the top photographs of all 3 gures) with those from the
FEM simulation. These comparisons suggest thatthe delamination was
initiated at a location within the range predicted by the FEMmodels
which is represented by the maxi-mum damage factor d (the brightest
region) in the FEM contour plots. Note that the model of 1/2-beam,
Fig. 8a, has almostconstant d values between the loading pin and
the support. This is because the 90 layer lies on the neutral plane
where theshear stress is distributed uniformly between the loading
pin and the support, and is the only stress component to
initiatethe delamination.
3876 C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
38663880Fig. 9. Comparisons of the nal delamination area generated
by the beam test for (a) 1/2-beam, (b) 1/4-beam, and (c)
3/4-beam.
-
It should be mentioned that two signicant features from the beam
testing were used to evaluate the FEM models andvalidity of the a
value in Eq. (2) for the damage initiation. The rst feature is the
location for delamination initiation thatas shown by the
photographs in Fig. 8, has occurred closer to the loading pin and
the support for the 1/4-beam and 3/4-beamspecimens, respectively,
than that in the -beam specimen. This can be explained by the
effect of hydrostatic stress. For the1/4-beam specimen,
delamination occurred in the interlaminar region that is subjected
to hydrostatic tension. Since thehydrostatic tension encourages the
damage initiation, the delamination is expected to occur close to
the loading point wherethe hydrostatic tension is high. For the
3/4beam specimen, on the other hand, delamination occurred in the
region that issubjected to hydrostatic compression which is known
to suppress the damage initiation. Therefore, delamination
initiationin the 3/4beam specimens is expected to occur in a region
close to the support where the hydrostatic compression is low.Fig.
8 shows that the FEM model successfully predicted the trend of the
locations for the delamination initiation.
The second signicant feature from the experimental results is
the nal delamination size generated by the beam tests. Asshown by
the top photographs in each of Fig. 9ac, taken from post-tested
specimens of 1/2-beam, 1/4-beam and 3/4-beamspecimens,
respectively, the delamination area can be detected by a bright
region appearing on the surface of the post-testedspecimens. The
photographs suggest that the delamination size is in the order of
1/4beam < 1/2-beam < 3/4-beam. Thetrend has been correctly
predicted by the FEM models, as shown by the contour plots of d in
Fig. 9ac that were taken rightafter the drop from the peak load.
The pair of white dashed lines in Fig. 9 indicates where the
loading pin (left line) and thesupport (right line) were.
The main difference in Fig. 9 between the photographs and the
contour plots of d is that the delamination area in the post-tested
specimens barely went beyond the left dashed line where the loading
pin was, while that in the FEM contour plot wasslightly over. This
was probably because friction between the fracture surfaces was not
considered in the FEM simulation,but should exist in the Beam
specimens due to the out-of-plane compression generated by the
loading pin. The friction musthave prohibited further propagation
of the delamination over the line of the loading pin.
4.2.3. Parametric studySensitivity of the simulation results to
the longitudinal normal stress r33 (the normal stress in
2-direction is always zero
since Poissons ratio is zero) was examined by changing the term
rii in Eq. (2) to r11 and setting a value equal to 0, followingthe
criteria proposed by Camanho et al. [22]. A study was also carried
out using the FEMmodel to examine the effect of mate-rial tensile
and shear strengths on the damage initiation for delamination, by
increasing rmaxI=shear in Fig. 2 by 50% but maintain-ing the same
values for the other properties.
C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
38663880 38770 0.5 1 1.5 20
2000
4000
6000
8000
Displacement (mm)
Load
(N)
FEM, with ii in Eq. (1)FEM, ii --> 11Test
1/4-Beamwith 33
1/4-Beamwithout 33
a
b
Fig. 10. Parametric study of the effect of r33 in the 1/4-beam
test: (a) load-displacement curves, and (b) the resulting
delamination area with and withoutthe consideration of r33.
-
3878 C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
386638800 0.5 1 1.5 20
2000
4000
6000
8000
10000
Displacement (mm)
Load
(N)
T and S1.5 T1.5 S
T and S
a
bUsing 1/4-beam as an example, results from the two parametric
studies are presented in Figs. 10 and 11. Fig. 10a showsthat
without the consideration of r33 (thin dashed line) the critical
load for the delamination initiation becomes much higherthan that
determined experimentally. Therefore, r33 should have played a
signicant role on the maximum load allowed be-fore the delamination
initiation. The corresponding delamination areas are compared in
Fig. 10b, which also suggests that r33should be considered in the
criterion for the delamination initiation in order to reduce the
delamination area to the sizesimilar to that observed
experimentally, Fig. 9b.
In addition to the above differences, it should be mentioned
that ignoring r33 for the simulation of delamination in the1/4-beam
also resulted in signicant difference in the location for the
delamination initiation, at a distance of 8 and19 mm away from the
loading pin for the simulation with and without the consideration
of r33, respectively.
Fig. 11 summarizes the effect of material tensile (T) and shear
(S) strengths used in the damage material model on thedamage
development in the Beam specimens. By increasing T or S by 50%,
Fig. 11a suggests that a considerable increaseof the maximum load
is required to initiate the delamination. Fig. 11b shows that the
resulted delamination area also in-creases, due to the increased
energy available for the fracture surface formation.
5. Conclusions
This paper presents a new approach that uses damage material
model and continuum elements in FEM to simulate dam-age initiation
and propagation for delamination in FRP and crack growth in the
adhesive layer. The damage evolution law isbased on the cohesive
zone model; and the damage onset criterion takes into account the
effect of in-plane normal stress onthe damage development. The new
approach has a major advantage over the existing interface
element-based approach inthat the former can easily adopt any
stress- or strain-based damage initiation criteria, beneted from
the use of solid con-tinuum elements.
The paper proposes a new damage initiation criterion that
considers both shear and hydrostatic stresses for the
damageinitiation. The paper shows that with a simple bi-linear
function for the damage evolution, the new damage material
model
1.5 T
1.5 S
Fig. 11. Parametric study of the effect of material strength (T
and S) in the 1/4-beam test: (a) load-displacement curves, and (b)
the resulting delaminationarea.
-
C. Fan et al. / Engineering Fracture Mechanics 75 (2008)
38663880 3879can accurately predict the delamination development in
either mode I or mode II fracture of FRP. The new model was
rstlyveried using benchmark problems such as the DCB, the ENF and
the ELS tests, by comparing the simulation results with
theanalytical solutions. The new model was then veried using the
delamination development in the FRP Beam specimens thatdid not
contain any pre-crack. The study shows that the new approach
successfully predicts the location for the delaminationinitiation
and the nal delamination size with good accuracy.
Using ELS test as an example, the new model generates consistent
results with those obtained previously, suggesting thatthe material
strength for the cohesive zone plays a very minor role on the
critical load for the onset of delamination growthfrom a starting
defect (or a pre-crack). However, using the beam test, the study
shows that the material strength for thecohesive zone has a
signicant effect on the critical load required for the delamination
initiation.
The study concludes that the new damage material model, with
only one adjustable parameter a, can accurately simulatethe damage
development process, from crack initiation, propagation, to
arrest.
Acknowledgements
The work was sponsored by Natural Sciences and Engineering
Research Council of Canada (NSERC) and Intelligent Sensingfor
Innovative Structures (ISIS Canada). Fan also acknowledges the
nancial support from Izaak Walton Killam MemorialScholarships and
Petro-Canada Graduate Scholarship during the course of the
study.
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Cohesive zone with continuum damage properties for simulation of
delamination development in fibre composites and failure of
adhesive jointsIntroductionDamage material modelConstitutive
relationshipDamage initiation criterionDamage evolution law
Verification of the cohesive zone model for crack growthFEM
modelDCB testENF testELS test
Simulation of the beam testExperimental results from the beam
testFEM simulation of the beam testFEM modelResultsParametric
study
ConclusionsAcknowledgementsReferences