This document analyzes the accuracy of mathematical expressions used to calculate the critical strain energy release rate (Gc) for delamination in fiber composites. It investigates how Gc values change with specimen thickness by using finite element models of double cantilever beam (DCB) and end-notched flexure (ENF) specimens. Three beam theory expressions are considered: simple beam theory, transverse shear deformation theory, and corrected beam theory accounting for transverse shear and crack tip effects. The study finds the corrected beam theory provides Gc values that are least sensitive to specimen thickness changes in both opening and shear deformation modes.

1. Analysis of specimen thickness effect on interlaminar fracture
toughness of fibre composites using finite element models
Arun Agrawal, P.-Y. Ben Jar*
Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G8
Received 19 March 2002; received in revised form 22 January 2003; accepted 22 January 2003
Abstract
This work investigated accuracy of various mathematical expressions used to calculate the critical strain energy release rate (Gc)
for delamination in fibre composites. Three mathematical expressions were considered here, based on (i) a simple beam theory, (ii) a
transverse shear deformation theory, or (iii) a corrected beam theory with consideration of transverse shear deformation and crack
tip singularity. Variable selected to examine accuracy of these expressions was specimen thickness. Since Gc is a material property,
change of specimen thickness should not affect its value. The study used 2-dimensional finite element models with a blunt starting
defect, which have length and geometry simulating the test coupons used for the delamination tests. For delamination in the shear
mode (Mode II), we assumed that contact surfaces along the starting defect were free from friction, in order to be consistent with
the beam theory expressions used for the calculation of Gc. As the finite element analysis used is static in nature, only the strain
energy release rate for crack initiation was examined. The study firstly assigned a constant load of 1 N for the 10 mm-thick models,
and then calculated the corresponding loads for models of other thickness based on constant strain energy release rates, GI and GII
for Mode I (tension mode) and Mode II respectively, using the three beam theory expressions. For each model under the given load,
stresses in the vicinity of the starting defect were then examined to determine whether the specimen thickness affects the stress
values. Stresses used were the maximum principle stress and the von Mises stress along the contour of the starting defect, and the
normal stress and shear stress along the boundary of the interlaminar resin-rich region, which were treated as the stress criteria
for fracture initiation. The study concludes that the corrected beam theory provides Gc expressions that are least sensitive to the
specimen thickness in both deformation modes.
# 2003 Elsevier Science Ltd. All rights reserved.
Keywords: A. Polymer-matrix composites; B. Fracture toughness; C. Delamination
1. Introduction
Laminated fibre reinforced polymer composites
(named fibre composites hereafter) have attracted a
wide range of uses in civil, marine, automotive, aero-
space and sports applications on account of their
superior tailor-made properties that are not attainable
from conventional material. However, due to low inter-
laminar strength fibre composites are susceptible to
delamination damage during processing or in service.
By far, delamination is known to be the most critical
damage mode that limits fibre composite’s load-carrying
capability. The presence and growth of delamination
may cause severe stiffness reduction in a structure,
leading to a catastrophic failure. Hence, reliable
measure of delamination resistance is essential in selec-
tion and design of fibre composites.
The resistance to delamination is usually character-
ized by interlaminar fracture toughness, often char-
acterised in terms of critical strain energy release rate
(Gc). A popular approach to development of an expres-
sion for Gc has been through the application of energy-
based linear elastic fracture mechanics. Gc for delami-
nation in an opening mode (Mode I) is known as GIc
while that for a sliding shear mode (Mode II) is GIIc.
Expressions for GIc and GIIc have been under investi-
gation by experimental, theoretical and numerical
simulation in the last two decades. Studies for Mode I
delamination have yielded a standard test method that
uses Double Cantilever Beam (DCB) specimen with
unidirectional fibres [1–3]. On the other hand, studies
0266-3538/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0266-3538(03)00088-5
Composites Science and Technology 63 (2003) 1393–1402
www.elsevier.com/locate/compscitech
* Corresponding author. Fax: +1-780-492-2200.
E-mail address: ben.jar@ualberta.ca (P.-Y. Ben Jar).

2. for GIIc measurement have proven to be a more compli-
cated exercise. The end-notched flexure (ENF) test [4] is
most widely used due to its simple fixture, and is adopted
by Japan Industrial Standards Group (JIS) as a standard
test [3]. However, ENF test does not provide resistance
curve (GIIc as a function of crack growth length, com-
monly known as R-curve). Therefore, only initial crack
length is available for the GIIc calculation, at the critical
load for the on-set of crack growth. This requires experi-
ence and careful specimen preparation to yield consistent
results. The measurement is further complicated by
uncertainty of pure shear loading at the crack tip [5–9].
As a result, some groups prefer different test configur-
ations for the GIIc measurement [10–12]. At this point of
time when the manuscript is prepared, there is no com-
monly accepted ASTM (American Society for Testing
and Materials) standard for the measurement of GIIc.
This study used 2-dimensional finite element models
with no contact frictional force to examine sensitivity of
beam theory expressions to specimen thickness change
for GI and GII calculation. Based on the results, accuracy
of the beam theories for the calculation of GI and GII
using DCB and ENF specimens, respectively, was deter-
mined. The beam theories examined were (i) simple beam
theory, (ii) transverse shear deformation theory, and (iii)
corrected beam theory that considers the transverse shear
deformation and the crack tip singularity.
2. Expressions for GI and GII
Because of the requirement from the finite element
analysis, as detailed in the next section, expressions of
GI and GII from the three beam theories have to exclude
vertical deflection d. That is, expressing GI and GII as
functions of load P and specimen parameters only. The
expressions used in the study are as follows.
2.1. The simple beam theory
The expression for GI, GBT
I , is [1,2]
GBT
I ¼
12P2
E1B2h3
a2
ð1Þ
and for GII, GBT
II , is [13]
GBT
II ¼
9P2
a2
16E1B2h3
ð2Þ
2.2. The transverse shear deformation theory
The expression for GI, GSH
I , is [14]
GSH
I ¼
12P2
E1B2
a
h
2
þ
1
10
E1
G13
ð3Þ
and for GII, GSH
II , is [15]
GSH
II ¼ GBT
II 1 þ 0:2
E1h2
G13a2
ð4Þ
2.3. The corrected beam theory with consideration of
transverse shear deformation and crack tip singularity
The expression for GI, G
I , is [16–21]
G
I ¼ 12
P2
a þ Ihð Þ2
B2E1h3
ð5Þ
in which the expression for the correction factor I is [18]:
I ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a66
18K a11ð Þ
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 À 2
G
ðG þ 1Þ
s 2
ð6Þ
Nomenclature
a11, a33, a66 Elastic compliances
a Crack length
B Specimen width
C Specimen compliance
E1 Flexure modulus along the fibre
direction
G13 shear modulus
GI Mode I strain energy release rate
GII Mode II strain energy release rate
h Thickness of each arm of DCB or
ENF specimen
L Half span length for ENF specimen or
full span length of DCB specimen
P Load for a given GI or GII value.
13 Major Poisson’s ratio, À 3
1
;
31=minor Poisson’s ratio, À 1
3
I Correction to crack length at elastic
singular root of a DCB specimen
II Correction to crack length at elastic
singular root of an ENF specimen
Vertical deflection (in loading
direction) at the loading point
S1 Major principle stress
SEQV von Mises stress
GBT
Strain energy release rate based on the
simple beam theory, Eqs. (1) and (2)
GSH
Strain energy release rate based on the
Transverse Shear Deformation
Theory, Eqs. (3) and (4)
G
Strain energy release rate based on
the Corrected Beam Theory that
takes into account of the transverse
shear deformation and the crack tip
singularity, Eqs. (5) and (7)
1394 A. Agrawal, P.-Y. Ben Jar / Composites Science and Technology 63 (2003) 1393–1402

3. where G ¼ 1:18 a66ffiffiffiffiffiffiffiffiffia11a33
p , a11 ¼ 1
E1
, a33 ¼ 1
E3
, a66 ¼ 1
G13
, and
K is a function of Poisson’s ratio.
It should be noted that empirical observations [18–20]
have suggested that the best results of G
I for DCB spe-
cimens are obtained when the value of ‘‘18K’’ in Eq. (6)
is equal to 11.
It should also be noted that value of I can be deter-
mined from a plot of C1/3
versus crack length ‘‘a’’ in which
the intercept with the x-axis is equal toIh. The current
ASTM Standard [1] adopts this modified beam theory for
GIc calculation, but the correction factor for the crack
length ‘‘a’’ is not explicitly expressed as a function of h.
The expression for GII, G
II, is [17–19]
G
II ¼
9P2
a þ IIhð Þ2
16E1B2h3
¼ GBT
II 1 þ II
h
a
2
ð7Þ
The correction factor II in the above expression
accounts for both intense local shear deformation at the
crack tip and global transverse shear deformation of
the beam. Its value for the finite element models used in the
study was determined based on the expression below
[17,20,22] with the parameter ‘‘18K’’ being 63 [20]:
II ¼
ffiffiffiffiffiffiffiffiffi
11
18K
r
I ð8Þ
3. Review of past experimental studies
Several studies in the past were devoted to under-
standing the effect of specimen thickness on the inter-
laminar fracture toughness of fibre composites.
Hashemi et al. [21], using DCB specimens and based on
the corrected beam theory, i.e. Eq. (5), measured GIc for
initiation and propagation of interlaminar cracks in
carbon fibre composites that have thickness variation
from 1 to 6 mm. The results showed no thickness
dependence of GIc, which was supported by Davies et al.
[23] who used specimens in the same thickness range.
Despite the independence of GIc on specimen thick-
ness, GIIc from ENF test was found in many studies to
be dependent on the specimen thickness, using load at
either the first non-linear point of the load-displacement
curve [24] (the initiation of crack growth) or the point of
the maximum load [23–25]. The thickness dependence of
GIIc was attributed to friction between surfaces of the
starting defect [24] or fibre bridging in the pre-crack
generated in Mode I [23] or Mode II [25] pre-cracking
processes. The conclusion of the frictional force affect-
ing the measured GIIc [24] was consistent with that
reported by Hashemi et al. [21] using the end-loaded
split (ELS) test, with the former based on the simple
beam theory and the latter on the corrected beam the-
ory. Unfortunately, even after excluding the frictional
energy, the measured GIIc values still could not be used
to determine accuracy of the beam theories for the GIIc
calculation, due to significant scattering of the experi-
mental results [21].
Using finite element modelling, the work presented
here has avoided data scattering and excluded the con-
tact frictional force, thus enabling us to investigate
effect of the specimen thickness on GIIc calculated from
different beam theory expressions.
4. Finite element analysis
Two-dimensional linear elastic finite element models
were developed using ANSYS finite element code version
5.7 [26]. Schematic diagrams of DCB and ENF specimen
models are shown in Figs. 1 and 2, respectively. The two
models are similar except loading and boundary condi-
tions. The dimensions and boundary conditions of the
models correspond to full-scale test coupons with varia-
tion of the overall thickness 2h from 5 to 15 mm.
4.1. Material properties of the finite element models
Each model has three layers. The top and the bottom
layers have orthotropic properties that simulate unidirec-
tional fibre composites with fibre in the specimen length
direction. Two sets of material properties were used: one
for glass fibre/epoxy composite of medium fibre volume
fraction (around 40%), and the other carbon fibre/epoxy
composites of high fibre volume fraction (around 60%).
The middle layer of 26 mm thick has isotropic properties
that represent the thin, interlaminar resin-rich region.
Values of the material properties are given below.
For the two orthotropic outer layers:
Glass fibre/epoxy [27,28]:
E1 ¼ 26:6 GPa; E3 ¼ 4:7 GPa; 31 ¼ 0:09;
G13 ¼ 2:8 GPa
Carbon fibre/epoxy [22,29,30]:
E1 ¼ 115:1 GPa; E3 ¼ 9:7 GPa; 31 ¼ 0:09;
G13 ¼ 4:478 GPa
For the middle layer of the interlaminar resin-rich
region:
E ¼ 3:1 GPa; ¼ 0:35
A starting defect of 13 mm thick was created at the centre
of the interlaminar region. Length of the starting defect ‘‘a’’
was 50 mm for the DCB model, and 25 mm for the ENF
model with a/L ratio of 0.5. The following expression for
an ellipse with an aspect ratio of 2 was used to represent the
crack tip geometry of the starting defect, as shown in Fig. 3.
A. Agrawal, P.-Y. Ben Jar / Composites Science and Technology 63 (2003) 1393–1402 1395

4. x
xo
2
þ
z
zo
2
¼ 1
0 4 x 4 a
Àb 4 z 4 b
ð9Þ
where 2xo ¼ 6:5 m and 2zo ¼ 13 m.
The elliptical contour of the starting defect represents
the blunt tip of the insert film, which has been shown to
truly represent contour of the starting defect in many
test coupons that we used in the past [27,28]. However,
this approach is different from most of finite element
Fig. 1. Finite element model of DCB specimen, thickness=10 mm.
Fig. 2. Finite element model of ENF Specimen, thickness=10 mm.
1396 A. Agrawal, P.-Y. Ben Jar / Composites Science and Technology 63 (2003) 1393–1402

5. works reported in the past, in which a sharp crack was
used to model the starting defect in DCB or ENF speci-
mens [29–32]. As to be discussed in Criteria for Fracture
Initiation, the approach adopted in this study requires
stress analysis in the vicinity of the starting defect to
determine when crack growth is initiated from the
starting defect. Therefore, it is necessary to use a blunt
tip to represent the realistic contour of a starting defect.
It should be noted that results presented here may still
be applicable to test coupons that use a delamination
crack as the starting defect, even though this type of
starting defect may have different crack tip contours or
bluntness. However, inconsistency of the crack tip con-
tour may have caused significant variation of the mea-
sured Gc values [23,25], nullifying the difference caused
by the beam theories.
4.2. Meshing of the models
Eight-node plane strain elements, PLANE82, were
used to generate mesh in the models. The mesh near the
crack tip is shown in Fig. 4, of which size has been
selected following that used in the previous studies
[27,28], to ensure that the critical stress values are not
sensitive to the change of the mesh size. In addition, the
same mesh lay-out was used in the vicinity of the start-
ing defect in all models used in the study, to ensure that
stress value changes were not caused by the change of
the mesh size and the lay-up. For the ENF model, fol-
lowing the previous approach [28,29], bar elements (or
non-linear truss elements) were used to resist the com-
pressive force between surfaces of the starting defect, as
shown in Fig. 5. Material properties for the bar ele-
ments are 3.1 GPa for the Young’s modulus and 0.35
for the Poisson’s ratio, which are the same as those for
the interlaminar resin-rich region.
Another approach to simulate the starting defect in
the ENF specimen is the use of contact elements, which
was reported to be most rigorous and allow for con-
sideration of friction [29,31,32]. However, determina-
tion of contact pressure is computationally demanding,
Fig. 3. Contour of the starting defect. Arc ABC has an elliptical
shape.
Fig. 4. Mesh at the crack tip for DCB and ENF models.
A. Agrawal, P.-Y. Ben Jar / Composites Science and Technology 63 (2003) 1393–1402 1397

6. thus not selected in the current study. Besides, friction
between the contact surfaces was not considered in the
above expressions for GII. Therefore, it was not neces-
sary to use contact elements in this study.
4.3. Determination of loading for models of various
thickness
Variation of the model thickness, from 5 to 15
mm, was achieved by varying thickness of the two
outer orthotropic layers, without changing dimensions
for the middle interlaminar region and the starting
defect. To determine the load for each model, a load
of 1 N was first selected as a reference load and
applied to the models of 10 mm thick specimen, as
shown in Figs. 1 and 2. Appropriate loading for
other models of different thickness was then deter-
mined using Eqs. (1), (3) and (5) for DCB models and
(2), (4) and (7) for ENF models, to obtain the same
values of GI and GII as for the 10 mm-thick models. The
loads for the models are listed in Table 1 in which values
from Eqs. (5) and (7) were determined with I and II
being equal to:
For glass fibre/epoxy I=1.19; II=0.50
For carbon fibre/epoxy I=1.80; II=0.76
4.4. Criteria for fracture initiation
As shown in the previous work [27,28], fracture was
expected to start along contour of the starting defect, pro-
vided that sufficient bonding existed along the interface
between the inter-laminar resin-rich region and the ortho-
tropic layers (abbreviated as ‘‘interface’’ hereafter). Other-
wise, the fracture might start along the interface, nearby
the tip of the starting defect. Criterion for fracture initia-
tion from the starting defect was based on values of the
maximum principal stress (S1) and the von Mises stress
(SEQV), and criterion for fracture initiation from the
interface was based on normal stress (SZ) and shear stress
(SXZ), with SXZ only considered for the ENF models.
As material properties, GIc and GIIc are expected to be
independent of specimen thickness. Main concept used
in this study is that an appropriate beam theory for
calculation of GI and GII should provide a relationship
between load P and specimen thickness 2h so that the
same critical stress values are generated in the vicinity of
the starting defect in models of different thickness.
Gillespie et al. [29] have also used finite element
models to predict GII, GFE
II , from the ENF specimen
based on virtual crack closure [33] and compliance
techniques [29]. The study examined accuracy of
expressions from simple beam theory and transverse
shear deformation theory by comparing changes of the
ratio of GFE
II to GII that was calculated from the beam
theories. Their techniques allowed for the use of a sharp
crack tip in the finite element models, thus, greatly sim-
plifying the analysis process. However, the loads for the
finite element models were arbitrarily selected, bearing no
correlation among models of different specimen thickness.
Therefore, the study could not clearly show whether the
GFE
II was indeed independent of the specimen thickness.
Consequently, the results did not clarify which beam the-
ory is more appropriate for the GII calculation.
Fig. 5. Bar elements that provide constraint between surfaces of the
starting defect. (Ni,u, Nj,u) and (Ni,l, Nj,l) are the nodes on the upper
and lower defect surfaces, respectively.
Table 1
Critical loads used for the finite element models
Thickness
2h(mm)
Glass fibre/epoxy Carbon fibre/epoxy
PBT
I Eq. (1) [1] PSH
I Eq (3) [14] P
I Eq. (5) [18] PBT
I Eq. (1) [1] PSH
I Eq. (3) [14] P
I Eq. (5) [18]
DCB 5.0 0.3536 0.3548 0.3734 0.3536 0.3569 0.3827
10.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
15.0 1.8371 1.8264 1.7444 1.8371 1.8090 1.7069
ENF PBT
II Eq. (2) [13] PSH
II Eq. (4) [15] P
II Eq. (7) [20] PBT
II Eq. (2) [13] PSH
II Eq. (4) [15] P
II Eq. (7) [20]
5.0 0.3536 0.3633 0.3704 0.3536 0.3786 0.3784
10.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
15.0 1.8371 1.7610 1.7572 1.8371 1.6679 1.7239
1398 A. Agrawal, P.-Y. Ben Jar / Composites Science and Technology 63 (2003) 1393–1402

7. 5. Results and discussion
5.1. DCB Model
Typical stress plots from the 10-mm-thick model with
glass fibre/epoxy properties are presented in Figs. 6 and
7. The former shows stress variation along contour of
the starting defect and the latter along the interface. As
shown in Fig. 6, maximum values of S1 and SEQV
occurred at the same position, and were much higher
than values of SZ in Fig. 7. This suggests that fracture is
expected to start from the corner of the starting defect,
located near point A in Fig. 3, instead of from the fibre/
matrix interface, provided that the fibre/matrix interface
has sufficient bonding strength. This conclusion is con-
sistent with that reported previously [27], using both 2-
dimensional and 3-dimensional FEM analysis.
Maximum values of S1 and SEQV along the starting
defect and SZ along the interface are presented Table 2.
Percentage given in the parentheses under S1 column
indicates variation of the stress values due to the thick-
ness change, compared to the value from the 10-mm-
thick model. It should be noted that percentage varia-
tion for SEQV and SZ is the same as that for S1, on
account of linear elastic behaviour of the model, thus
not listed in Table 2.
The percentage shown in Table 2 suggests that the
load predicted by Eq. (5), P
I , generates stresses that
have the minimum variation with the change of the
specimen thickness. For example, by increasing speci-
men thickness from 10 to 15 mm, stress variation by the
corresponding load P
I is 0.5%, compared to 4.9 and
4.2% by PBT
I and PSH
I , respectively.
When percentage change of S1 is compared between
models of glass fibre/epoxy and carbon fibre/epoxy,
Table 2 suggests that difference of the percentage
change is also minimum using the load predicted by Eq.
(5), P
I . For example, for the 15-mm-thick model the
change is from 0.5 to 0.2% by the load P
I , compared to
the change of 4.9 to 7.4% by PBT
I and 4.2 to 5.7% by
PSH
I . The above conclusions are applicable to SEQV
and SZ, as the percentage change for these stresses are
the same as that for S1.
5.2. ENF model
In addition to the above stresses, SXZ along the
interface was also considered as a critical stress for the
ENF model. Typical variations of these stresses are
presented in Figs. 8 and 9 for the 10mm-thick model
with properties of glass fibre/epoxy. Positions of the
maximum stress values support the previous conclusion
[28] that with sufficient interfacial bonding, crack is
expected to grow from the tip of the starting defect
towards the interface.
Maximum values of S1 and SEQV along the starting
defect and SZ and SXZ along the interface from the
ENF models are summarised in Table 3, using loads
determined by Eqs. (2), (4) and (7) for constant GII
(with P=1N for the 10 mm-thick model). Again, values
given in the parentheses of the S1 column represent the
percentage changes of stress values compared to that
from the 10 mm-thick model. The results suggest that
S1 values generated by PSH
II and P
II, predicted from Eqs.
(4) and (7) respectively, show a much smaller variation
with thickness than that generated by PBT
II , from Eq. (2).
Fig. 6. Variation of the maximum principal stress (S1) and von Mises
stress (SEQV) for the path along contour of the starting defect in the
10-mm-thick DCB model with properties of glass fibre/epoxy.
Fig. 7. Variation of the normal stress (SZ) for the path along the
interface in the 10 mm-thick DCB model with properties of glass fibre/
epoxy.
A. Agrawal, P.-Y. Ben Jar / Composites Science and Technology 63 (2003) 1393–1402 1399

8. The same conclusion can also be applied to SEQV, S1
and SXZ, as the percentage changes for these stresses
are the same as that for S1.
Comparing the percentage values in Tables 2 and 3,
we notice that while the load PSH
I generates stresses that
vary quite significantly with the change of specimen
thickness, the load PSH
II does not. Therefore, it is
believed that consideration of the transverse shear
deformation in the beam theory provides a reasonable
prediction for GIIc of interlaminar fracture in fibre
composites, but not for GIc.
5.3. Discussion
As mentioned in the previous section, various experi-
mental studies [21,23] that consider crack tip singularity
for GIc calculation confirmed the thickness-indepen-
dence of the measured GIc values. Results from our
Fig. 8. Variation of the maximum principal stress (S1), shear stress
(SXZ) (based on the coordinates defined in Fig. 2) and von Mises stress
(SEQV) for the path along the contour of the starting defect in the 10
mm-thick ENF model. Properties used were based on glass fibre/epoxy.
Fig. 9. Variation of the interlaminar normal stress (SZ) and shear
stress (SXZ) for the path along the interface in the 10 mm-thick ENF
model. Properties used were based on glass fibre/epoxy.
Table 2
Values of the critical maximum principle stress (S1), von Mises stress (SEQV) and normal stress along the interface (SZ) predicted from the DCB
models of constant GI, using the critical loads provided in Table 1
Thickness
2h (mm)
S1 (MPa) SEQV (MPa)
PBT
I PSH
I P
I PBT
I PSH
I P
I
Glass fibre/epoxy 5.0 94.89 (À4.2%) 95.21 (À3.9%) 100.21 (+1.1%) 82.57 82.85 87.19
10.0 99.07 99.07 99.07 86.20 86.20 86.20
15.0 103.89 (+4.9%) 103.24 (+4.2%) 98.61 (À0.5%) 90.36 89.83 85.80
SZ (MPa)
5.0 31.063 31.168 32.809
10.0 32.866 32.866 32.866
15.0 34.654 34.452 32.906
Carbon Fibre/Epoxy S1 (MPa) SEQV (MPa)
5.0 45.39 (À6.9%) 45.81 (À6.0%) 49.13 (+0.8%) 39.49 39.86 42.74
10.0 48.76 48.76 48.76 42.42 42.42 42.42
15.0 52.36 (+7.4%) 51.56 (+5.7%) 48.65 (À0.2%) 45.56 44.86 42.33
SZ (MPa)
5.0 18.30 18.47 19.80
10.0 19.82 19.82 19.82
15.0 21.36 21.03 19.85
Number in the parentheses indicates percentage of increase or decrease from the values in the 10-mm-thick model. The percentage variation for
SEQV and SZ is similar to the percentage variation for S1, as expected on account of linear elastic behaviour.
1400 A. Agrawal, P.-Y. Ben Jar / Composites Science and Technology 63 (2003) 1393–1402

9. finite element models support those conclusions, and
suggest that Eqs. (1) and (3) are expected to produce GI
values that show thickness dependence.
It is interesting to note that since SZ at the interface
shows the same percentage variation as that for S1 and
SEQV along the starting defect, the thickness dependence
of GI, as determined by Eqs. (1), (3), and (5), can also be
applied when fracture starts at the interface, provided that
the interfacial bonding strength remains constant and is
independent of the specimen thickness.
Earlier experimental studies on ENF test have sug-
gested that frictional force between contact surfaces of
the starting defect may contribute to the thickness
dependence of GIIc [24]. Due to the frictional force, the
calculated GII value is expected to increase with
increased specimen thickness. Further finite element
analysis will be carried out to investigate such an effect,
based on the expression including the term for correc-
tion of friction [15].
6. Conclusions
Accuracy of beam theory expressions for GI and GII
has been investigated, using finite element models with
thickness as the variable. It was found that the corrected
beam theory with the consideration of transverse shear
deformation and crack tip singularity provides the
expression with the least sensitivity to specimen thick-
ness in both modes of deformation. On the other hand,
the expression based on the simple beam theory man-
ifests the specimen thickness effect by about 9% for GI
and 7% for GII, in the thickness range of 5 mm to 15
mm. It is therefore recommended that expressions based
on the corrected beam theory be used for the calculation
of GIc and GIIc of fibre composites.
Acknowledgements
The work was sponsored by NSERC, Research
Grants scheme. The first author also acknowledges
some financial support from Department of Mechanical
Engineering, University of Alberta for his scholarship
during the study for Mater of Science degree.
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