Analysis of specimen thickness effect on interlaminar fracturetoughness 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) atransverse shear deformation theory, or (iii) a corrected beam theory with consideration of transverse shear deformation and cracktip 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 shearmode (Mode II), we assumed that contact surfaces along the starting defect were free from friction, in order to be consistent withthe 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 stressvalues. Stresses used were the maximum principle stress and the von Mises stress along the contour of the starting defect, and thenormal 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 thespecimen 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 awide range of uses in civil, marine, automotive, aero-space and sports applications on account of theirsuperior tailor-made properties that are not attainablefrom conventional material. However, due to low inter-laminar strength fibre composites are susceptible todelamination damage during processing or in service.By far, delamination is known to be the most criticaldamage mode that limits fibre composite’s load-carryingcapability. The presence and growth of delaminationmay cause severe stiffness reduction in a structure,

leading to a catastrophic failure. Hence, reliablemeasure 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 numericalsimulation in the last two decades. Studies for Mode Idelamination have yielded a standard test method thatuses Double Cantilever Beam (DCB) specimen withunidirectional 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).

for GIIc measurement have proven to be a more compli-cated exercise. The end-notched flexure (ENF) test [4] ismost widely used due to its simple fixture, and is adoptedby Japan Industrial Standards Group (JIS) as a standardtest [3]. However, ENF test does not provide resistancecurve (GIIc as a function of crack growth length, com-monly known as R-curve). Therefore, only initial cracklength is available for the GIIc calculation, at the criticalload for the on-set of crack growth. This requires experi-ence and careful specimen preparation to yield consistentresults. The measurement is further complicated byuncertainty 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 oftime when the manuscript is prepared, there is no com-monly accepted ASTM (American Society for Testingand Materials) standard for the measurement of GIIc.

This study used 2-dimensional finite element modelswith no contact frictional force to examine sensitivity ofbeam theory expressions to specimen thickness changefor GI and GII calculation. Based on the results, accuracyof 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 beamtheory, (ii) transverse shear deformation theory, and (iii)corrected beam theory that considers the transverse sheardeformation and the crack tip singularity.

2. Expressions for GI and GII

Because of the requirement from the finite elementanalysis, as detailed in the next section, expressions ofGI and GII from the three beam theories have to excludevertical deflection d. That is, expressing GI and GII asfunctions of load P and specimen parameters only. Theexpressions used in the study are as follows.

2.1. The simple beam theory

The expression for GI, GBTI , is [1,2]

GBTI ¼

12P2

E1B2h3a2 ð1Þ

and for GII, GBTII , is [13]

GBTII ¼

9P2a2

16E1B2h3ð2Þ

2.2. The transverse shear deformation theory

The expression for GI, GSHI , is [14]

GSHI ¼

12P2

E1B2

a

h

� �2þ

1

10

E1

G13

� �� �ð3Þ

and for GII, GSHII , is [15]

GSHII ¼ GBT

II 1þ 0:2E1h

2

G13a2

� �ð4Þ

2.3. The corrected beam theory with consideration oftransverse shear deformation and crack tip singularity

The expression for GI, G�I , is [16–21]

G�I ¼ 12

P2 aþ �Ihð Þ2

B2E1h3ð5Þ

in which the expression for the correction factor �I is [18]:

�I ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia66

18K a11ð Þ

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3� 2

GðG þ 1Þ

� �s 2

ð6Þ

Nomenclature

a11, a33, a66 Elastic compliancesa Crack lengthB Specimen widthC Specimen complianceE1 Flexure modulus along the fibre

directionG13 shear modulusGI Mode I strain energy release rateGII Mode II strain energy release rateh Thickness of each arm of DCB or

ENF specimenL Half span length for ENF specimen or

full span length of DCB specimenP Load for a given GI or GII value.�13 Major Poisson’s ratio, � "3

"1;

�31=minor Poisson’s ratio, � "1"3

�I Correction to crack length at elasticsingular root of a DCB specimen

�II Correction to crack length at elasticsingular root of an ENF specimen

� Vertical deflection (in loadingdirection) at the loading point

S1 Major principle stressSEQV von Mises stressGBT Strain energy release rate based on the

simple beam theory, Eqs. (1) and (2)GSH Strain energy release rate based on the

Transverse Shear DeformationTheory, Eqs. (3) and (4)

G� Strain energy release rate based onthe Corrected Beam Theory thattakes into account of the transverseshear deformation and the crack tipsingularity, Eqs. (5) and (7)

1394 A. Agrawal, P.-Y. Ben Jar /Composites Science and Technology 63 (2003) 1393–1402

where G ¼ 1:18 a66ffiffiffiffiffiffiffiffiffia11a33

p , a11 ¼1E1, a33 ¼

1E3, a66 ¼

1G13

, andK 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 ofC1/3 versus crack length ‘‘a’’ in whichthe intercept with the x-axis is equal to�Ih. The currentASTM Standard [1] adopts this modified beam theory forGIc calculation, but the correction factor for the cracklength ‘‘a’’ is not explicitly expressed as a function of h.The expression for GII, G

�II, is [17–19]

G�II ¼

9P2 aþ �IIhð Þ2

16E1B2h3¼ GBT

II 1þ �IIh

a

� �2

ð7Þ

The correction factor �II in the above expressionaccounts for both intense local shear deformation at thecrack tip and global transverse shear deformation ofthe beam. Its value for the finite element models used in thestudy was determined based on the expression below[17,20,22] with the parameter ‘‘18K’’ being 63 [20]:

�II ¼

ffiffiffiffiffiffiffiffiffi11

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 onthe corrected beam theory, i.e. Eq. (5), measured GIc forinitiation and propagation of interlaminar cracks incarbon fibre composites that have thickness variationfrom 1 to 6 mm. The results showed no thicknessdependence 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 tobe dependent on the specimen thickness, using load ateither the first non-linear point of the load-displacementcurve [24] (the initiation of crack growth) or the point ofthe maximum load [23–25]. The thickness dependence ofGIIc was attributed to friction between surfaces of thestarting defect [24] or fibre bridging in the pre-crackgenerated in Mode I [23] or Mode II [25] pre-crackingprocesses. The conclusion of the frictional force affect-ing the measured GIIc [24] was consistent with thatreported by Hashemi et al. [21] using the end-loadedsplit (ELS) test, with the former based on the simplebeam theory and the latter on the corrected beam the-ory. Unfortunately, even after excluding the frictionalenergy, the measured GIIc values still could not be usedto 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 investigateeffect of the specimen thickness on GIIc calculated fromdifferent beam theory expressions.

4. Finite element analysis

Two-dimensional linear elastic finite element modelswere developed using ANSYS finite element code version5.7 [26]. Schematic diagrams of DCB and ENF specimenmodels are shown in Figs. 1 and 2, respectively. The twomodels are similar except loading and boundary condi-tions. The dimensions and boundary conditions of themodels 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 bottomlayers have orthotropic properties that simulate unidirec-tional fibre composites with fibre in the specimen lengthdirection. Two sets of material properties were used: onefor glass fibre/epoxy composite of medium fibre volumefraction (around 40%), and the other carbon fibre/epoxycomposites of high fibre volume fraction (around 60%).The middle layer of 26 mm thick has isotropic propertiesthat 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-richregion:

E ¼ 3:1 GPa; � ¼ 0:35

A starting defect of 13 mm thick was created at the centreof the interlaminar region. Length of the starting defect ‘‘a’’was 50 mm for the DCB model, and 25 mm for the ENFmodel with a/L ratio of 0.5. The following expression foran ellipse with an aspect ratio of 2 was used to represent thecrack 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

x

xo

� �2

þz

zo

� �2

¼ 104 x4 a�b4 z4 b

ð9Þ

where 2xo ¼ 6:5 �m and 2zo ¼ 13 �m.

The elliptical contour of the starting defect representsthe blunt tip of the insert film, which has been shown totruly represent contour of the starting defect in manytest 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

works reported in the past, in which a sharp crack wasused to model the starting defect in DCB or ENF speci-mens [29–32]. As to be discussed in Criteria for FractureInitiation, the approach adopted in this study requiresstress analysis in the vicinity of the starting defect todetermine when crack growth is initiated from thestarting defect. Therefore, it is necessary to use a blunttip 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 ofstarting defect may have different crack tip contours orbluntness. However, inconsistency of the crack tip con-tour may have caused significant variation of the mea-sured Gc values [23,25], nullifying the difference causedby the beam theories.

4.2. Meshing of the models

Eight-node plane strain elements, PLANE82, wereused to generate mesh in the models. The mesh near thecrack tip is shown in Fig. 4, of which size has beenselected following that used in the previous studies[27,28], to ensure that the critical stress values are notsensitive to the change of the mesh size. In addition, thesame mesh lay-out was used in the vicinity of the start-ing defect in all models used in the study, to ensure thatstress value changes were not caused by the change ofthe mesh size and the lay-up. For the ENF model, fol-lowing the previous approach [28,29], bar elements (ornon-linear truss elements) were used to resist the com-pressive force between surfaces of the starting defect, asshown in Fig. 5. Material properties for the bar ele-ments are 3.1 GPa for the Young’s modulus and 0.35for the Poisson’s ratio, which are the same as those forthe interlaminar resin-rich region.Another approach to simulate the starting defect in

the ENF specimen is the use of contact elements, whichwas 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

thus not selected in the current study. Besides, frictionbetween the contact surfaces was not considered in theabove expressions for GII. Therefore, it was not neces-sary to use contact elements in this study.

4.3. Determination of loading for models of variousthickness

Variation of the model thickness, from 5 to 15mm, was achieved by varying thickness of the twoouter orthotropic layers, without changing dimensionsfor the middle interlaminar region and the startingdefect. To determine the load for each model, a loadof 1 N was first selected as a reference load andapplied to the models of 10 mm thick specimen, asshown in Figs. 1 and 2. Appropriate loading forother 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 samevalues of GI and GII as for the 10 mm-thick models. Theloads for the models are listed in Table 1 in which valuesfrom 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 wasexpected to start along contour of the starting defect, pro-vided that sufficient bonding existed along the interfacebetween the inter-laminar resin-rich region and the ortho-tropic layers (abbreviated as ‘‘interface’’ hereafter). Other-wise, the fracture might start along the interface, nearbythe tip of the starting defect. Criterion for fracture initia-tion from the starting defect was based on values of themaximum principal stress (S1) and the von Mises stress(SEQV), and criterion for fracture initiation from theinterface 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 usedin this study is that an appropriate beam theory forcalculation of GI and GII should provide a relationshipbetween load P and specimen thickness 2h so that thesame critical stress values are generated in the vicinity ofthe starting defect in models of different thickness.Gillespie et al. [29] have also used finite element

models to predict GII, GFEII , from the ENF specimen

based on virtual crack closure [33] and compliancetechniques [29]. The study examined accuracy ofexpressions from simple beam theory and transverseshear deformation theory by comparing changes of theratio of GFE

II to GII that was calculated from the beamtheories. Their techniques allowed for the use of a sharpcrack tip in the finite element models, thus, greatly sim-plifying the analysis process. However, the loads for thefinite element models were arbitrarily selected, bearing nocorrelation among models of different specimen thickness.Therefore, the study could not clearly show whether theGFE

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

PBTI Eq. (1) [1]

PSH

I Eq (3) [14]

P�I Eq. (5) [18] PBT

I Eq. (1) [1]

PSHI 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

PBTII Eq. (2) [13] PSH

II Eq. (4) [15]

P�II Eq. (7) [20] PBT

II Eq. (2) [13]

PSHII 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

5. Results and discussion

5.1. DCB Model

Typical stress plots from the 10-mm-thick model withglass fibre/epoxy properties are presented in Figs. 6 and7. The former shows stress variation along contour ofthe starting defect and the latter along the interface. Asshown in Fig. 6, maximum values of S1 and SEQVoccurred at the same position, and were much higherthan values of SZ in Fig. 7. This suggests that fracture isexpected 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 interfacehas 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 columnindicates 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, onaccount of linear elastic behaviour of the model, thusnot 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 thespecimen thickness. For example, by increasing speci-men thickness from 10 to 15 mm, stress variation by thecorresponding load P�

I is 0.5%, compared to 4.9 and4.2% by PBT

I and PSHI , respectively.

When percentage change of S1 is compared betweenmodels of glass fibre/epoxy and carbon fibre/epoxy,Table 2 suggests that difference of the percentagechange is also minimum using the load predicted by Eq.(5), P�

I . For example, for the 15-mm-thick model thechange is from 0.5 to 0.2% by the load P�

I , compared tothe change of 4.9 to 7.4% by PBT

I and 4.2 to 5.7% byPSHI . The above conclusions are applicable to SEQV

and SZ, as the percentage change for these stresses arethe same as that for S1.

5.2. ENF model

In addition to the above stresses, SXZ along theinterface was also considered as a critical stress for theENF model. Typical variations of these stresses arepresented in Figs. 8 and 9 for the 10mm-thick modelwith properties of glass fibre/epoxy. Positions of themaximum stress values support the previous conclusion[28] that with sufficient interfacial bonding, crack isexpected to grow from the tip of the starting defecttowards the interface.Maximum values of S1 and SEQV along the starting

defect and SZ and SXZ along the interface from theENF models are summarised in Table 3, using loadsdetermined by Eqs. (2), (4) and (7) for constant GII

(with P=1N for the 10 mm-thick model). Again, valuesgiven in the parentheses of the S1 column represent thepercentage changes of stress values compared to thatfrom the 10 mm-thick model. The results suggest thatS1 values generated by PSH

II and P�II, predicted from Eqs.

(4) and (7) respectively, show a much smaller variationwith 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

The same conclusion can also be applied to SEQV, S1and SXZ, as the percentage changes for these stressesare the same as that for S1.Comparing the percentage values in Tables 2 and 3,

we notice that while the load PSHI generates stresses that

vary quite significantly with the change of specimenthickness, the load PSH

II does not. Therefore, it isbelieved that consideration of the transverse sheardeformation in the beam theory provides a reasonable

prediction for GIIc of interlaminar fracture in fibrecomposites, but not for GIc.

5.3. Discussion

As mentioned in the previous section, various experi-mental studies [21,23] that consider crack tip singularityfor 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)

PBTI

PSH

I

P�I PBT

I

PSHI P�

I

Glass fibre/epoxy

5.0 94.89 (�4.2%) 95.21 (�3.9%) 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 (�0.5%) 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 (�6.9%) 45.81 (�6.0%) 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 (�0.2%) 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

finite element models support those conclusions, andsuggest 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 andSEQV along the starting defect, the thickness dependenceof GI, as determined by Eqs. (1), (3), and (5), can also beapplied when fracture starts at the interface, provided thatthe interfacial bonding strength remains constant and isindependent of the specimen thickness.Earlier experimental studies on ENF test have sug-

gested that frictional force between contact surfaces ofthe starting defect may contribute to the thicknessdependence of GIIc [24]. Due to the frictional force, thecalculated GII value is expected to increase withincreased specimen thickness. Further finite elementanalysis 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 withthickness as the variable. It was found that the correctedbeam theory with the consideration of transverse sheardeformation and crack tip singularity provides theexpression 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 15mm. It is therefore recommended that expressions basedon the corrected beam theory be used for the calculationof GIc and GIIc of fibre composites.

Acknowledgements

The work was sponsored by NSERC, ResearchGrants scheme. The first author also acknowledgessome financial support from Department of MechanicalEngineering, University of Alberta for his scholarshipduring the study for Mater of Science degree.

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Composites 1995;26:257–67.

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Table 3

Values of the critical maximum principle stress (S1), von Mises stress (SEQV), normal stress along the interface (SZ) and shear stress along the

interface (SXZ) predicted from the ENF models of constant GII, using the critical loads provided in Table 1

Thickness

2h (mm)

S1 (MPa)

SEQV (MPa)

PBTII

PSH

II

P�II PBT

II

PSHII P�

II

Glass Fibre/Epoxy

5.0 14.95 (�3.3%) 15.37 (�0.6%) 15.67 (+1.3%) 12.96 13.32 13.57

10.0

15.46 15.46 15.46 13.40 13.40 13.40

15.0

16.02 (+3.6%) 15.37 (�0.7%) 15.32 (�0.9%) 13.88 13.30 13.27

SZ (MPa)

SXZ (MPa)

5.0

2.33 2.39 2.44 1.80 1.85 1.89

10.0

2.41 2.41 2.41 1.87 1.87 1.87

15.0

2.50 2.40 2.39 1.94 1.86 1.85

Carbon Fibre/Epoxy

S1 (MPa) SEQV (MPa)

5.0

7.32 (�5.4%) 7.84 (+1.3%) 7.84 (+1.2%) 6.32 6.77 6.77

10.0

7.74 7.74 7.74 6.68 6.68 6.68

15.0

8.14 (+5.1%) 7.39 (�4.6%) 7.63 (�1.4%) 7.02 6.38 6.59

SZ (MPa)

SXZ (MPa)

5.0

1.33 1.43 1.43 1.3 1.39 1.39

10.0

1.41 1.41 1.41 1.37 1.37 1.37

15.0

1.48 1.35 1.39 1.44 1.31 1.36

Number in the parentheses indicates percentage of increase or decrease from the values in the 10-mm-thick model. The percentage variation for

SEQV, SZ and SXZ is similar to the percentage variation for S1, as expected on account of linear elastic behaviour.

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