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: chengye@ualberta.ca (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.

References

[1] ASTM D5528-01. Standard test method for mode I interlaminar fracture toughness of unidirectional ber-reinforced polymer matrix composites.Annual Book of ASTM Standards, 15.03.2000.

[2] Russell AJ, Street KN. Moisture and temperature effects on the mixed-mode delamination fracture of unidirectional graphite/epoxy. In: Delaminationand debonding of materials. Pittsburgh, PA, USA: ASTM; 1985. p. 34970.

[3] ASTM D 6671/D 6671M-04e1. Standard test method for mixed mode Imode II interlaminar fracture toughness of unidirectional ber reinforcedpolymer matrix composites. Annual Book of ASTM Standards, 13.05.2005.

[4] Rybicki EF, Kanninen MF. A nite element calculation of stress intensity factors by a modied crack closure integral. Eng Fract Mech 1977;9(4):9318.[5] Shih CF, Needleman A. Fully plastic crack problems. I. Solutions by a penalty method. Trans ASME J Appl Mech 1984;51(1):4856.[6] Shih CF, Needleman A. Fully plastic crack problems. II. Application of consistency checks. Trans ASME J Appl Mech 1984;51(1):5764.[7] Wang Y, Williams JG. Corrections for mode II fracture toughness specimens of composites materials. Compos Sci Technol 1992;43(3):2516.[8] Fan C, Jar PYB, Cheng JJR. Prediction of energy release rates for crack growth using FEM-based energy derivative technique. Eng Fract Mech

2007;74(8):124354.[9] Grifth AA. The phenomena of rupture and ow in solids. Philos Trans R Soc London 1920;221(A 587):16398.[10] Turon A, Camanho PP, Costa J, Davila CG. A damage model for the simulation of delamination in advanced composites under variable-mode loading.

Mech Mater 2006;38(11):107289.[11] Barenblatt GI. The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 1962;7:55129.[12] Wisnom MR. Shear fracture of unidirectional composites without initial cracks. Compos Sci Technol 1994;52(1):917.[13] Wisnom MR. Modelling the effect of cracks on interlaminar shear strength. Compos - Part A: Appl Sci Manufact 1996;27(1):17.[14] Wisnom MR. Modelling of stable and unstable fracture of short beam shear specimens. Composites 1994;25(6):394400.[15] Weicheng Cui, Wisnom MR, Jones M. Effect of specimen size on interlaminar shear strength of unidirectional carbon bre-epoxy. Compos Eng

1994;4(3):299307.[16] Yan A, Marechal E, Nguyen-Dang H. A nite-element model of mixed-mode delamination in laminated composites with an R-curve effect. Compos Sci

Technol 2001;61(10):141327.[17] Alfano G, Criseld MA. Solution strategies for the delamination analysis based on a combination of local-control arc-length and line searches. Int J

Numer Meth Eng 2003;58(7):9991048.[18] Alfano G, Criseld MA. Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues.

Int J Numer Meth Eng 2001;50(7):170136.[19] Blackman BRK, Hadavinia H, Kinloch AJ, Williams JG. The use of a cohesive zone model to study the fracture of bre composites and adhesively-bonded

joints. Int J Fract 2003;119(1):2546.[20] Bui VQ, Marechal E, Nguyen-Dang H. Imperfect interlaminar interfaces in laminated composites: interlaminar stresses and strain-energy release rates.

Compos Sci Technol 2000;60(1):13143.[21] Camanho PP, Davila CG, Pinho SS. Fracture analysis of composite co-cured structural joints using decohesion elements. Fatigue Fract Eng Mater Struct

2004;27(9):74557.[22] Camanho PP, Davila CG, De Moura MF. Numerical simulation of mixed-mode progressive delamination in composite materials. J Compos Mater

2003;37(16):141538.[23] Qiu Y, Criseld MA, Alfano G. An interface element formulation for the simulation of delamination with buckling. Eng Fract Mech

2001;68(16):175576.[24] Corigliano A. Formulation, identication and use of interface models in the numerical analysis of composite delamination. Int J Solids Struct

1993;30(20):2779811.[25] Criseld MA, Alfano G. Adaptive hierarchical enrichment for delamination fracture using a decohesive zone model. Int J Numer Meth Eng

2002;54(9):136990.[26] Davila CG, Camanho PP. Analysis of the effects of residual strains and defects on skin/stiffener debonding using decohesion elements. In: 44th AIAA/

ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, April 710, 2003. Norfolk, VA, United States: American Inst.Aeronautics and Astronautics Inc; 2003. p. 6008.

[27] Davila CG, Camanho PP, De Moura MF. Mixed-mode decohesion elements for analyses of progressive delamination. In: 42nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference and exhibit technical papers, April 1619, 2001. Seattle, WA: American Inst Aeronauticsand Astronautics Inc; 2001. p. 227788.

[28] de Moura MFSF, Goncalves JPM, Marques AT, de Castro PMST. Prediction of compressive strength of carbon-epoxy laminates containing delaminationby using a mixed-mode damage model. Compos Struct 2000;50(2):1517.

[29] de Moura MFSF, Goncalves JPM, Marques AT, de Castro PMST. Modeling compression failure after low velocity impact on laminated composites usinginterface elements. J Compos Mater 1997;31(15):146279.

[30] El-Sayed S, Sridharan S. Predicting and tracking interlaminar crack growth in composites using a cohesive layer model. Compos Part B: Eng2001;32(6):54553.

[31] Liu D, Xu L, Lu X. Stress analysis of imperfect composite laminates with an interlaminar bonding theory. Int J Numer Meth Eng 1994;37(16):281939.[32] Mi Y, Criseld MA, Davies GAO, Hellweg H. Progressive delamination using interface elements. J Compos Mater 1998;32(14):124672.[33] Pantano A, Averill RC. A mesh-independent interface technology for simulation of mixed-mode delamination growth. Int J Solids Struct

2004;41(14):380931.[34] Turon A, Camanho PP, Costa J, Davila CG. Interface Damage model for the simulation of delamination under variable-mode ratio in composite

materials. NASA/TM-2004-213277 2004;132.[35] Turon A, Davila CG, Camanho PP, Costa J. Engineering Solution for Using Coarse Meshes in the Simulation of Delamination with Cohesive Zone Models.

NASA/TM-2005-213547 2005; 126.[36] Xu X, Needleman A. Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 1994;42(9):1397434.[37] Goyal VK, Johnson ER, Davila CG. Irreversible constitutive law for modeling the delamination process using interfacial surface discontinuities. Compos

Struct 2004;65(34):289305.[38] Tvergaard V, Hutchinson JW. The relation between crack growth resistance and fracture process parameters in elasticplastic solids. J Mech Phys Solids

1992;40(6):137797.[39] Jin Z, Sun CT. Cohesive zone modeling of interface fracture in elastic bi-materials. Eng Fract Mech 2005;72(12):180517.[40] Jin Z, Sun CT. Cohesive fracture model based on necking. Int J Fract 2005;134(2):91108.[41] Choi HY, Downs RJ, Chang F. New approach toward understanding damage mechanisms and mechanics of laminated composites due to low-velocity

impact. Part I. Experiments. J Compos Mater 1991;25(8):9921011.[42] Choi HY, Wu HT, Chang F. New approach toward understanding damage mechanisms and mechanics of laminated composites due to low-velocity

impact. Part II. Analysis. J Compos Mater 1991;25(8):101238.[43] Yuen S, Fan C, Kuboki T, Jar P-B, Forest TW, Cheng JJR. Delamination resistance of bre-reinforced polymers (FRP) under transverse loading - Criteria for

on-set of delamination. In: ANTEC 2004 - Annual Technical Conference Proceedings, May 16-20 2004, Chicago, IL., United States: Society of PlasticsEngineers; 2004. p. 13741378.

[44] Hou JP, Petrinic N, Ruiz C, Hallett SR. Prediction of impact damage in composite plates. Compos Sci Technol 2000;60(2):27381.[45] Hou JP, Petrinic N, Ruiz C. A delamination criterion for laminated composites under low-velocity impact. Compos Sci Technol 2001;61(14):206974.[46] Lapczyk I, Hurtado JA. Progressive damage modeling in ber-reinforced materials. Compos Part A 2007;38(11):233341.[47] Maimi P, Camanho PP, Mayugo JA, Davila CG. A continuum damage model for composite laminates. Part I. Constitutive model. Mech Mater

2007;39(10):897908.[48] Maimi P1, Camanho PP, Mayugo JA, Davila CG. A continuum damage model for composite laminates: Part II Computational implementation and

validation. Mech Mater 2007;39(10):90919.[49] Williams KV, Vaziri R, Poursartip A. A physically based continuum damage mechanics model for thin laminated composite structures. Int J Solids Struct

2003;40(9):2267300.[50] Hoppel CP, Bogetti TA, Gillespie JW. Report, Effects of hydrostatic pressure on the mechanical behavior of composite materials. United States: Army

Research Lab, Aberdeen Proving Ground, MD; 1995.[51] Reeder JR. Evaluation of Mixed-Mode Delamination Failure Criteria. Report, NASA Langley Research Center, Hampton, VA, United States: 1992.

3880 C. Fan et al. / Engineering Fracture Mechanics 75 (2008) 38663880[52] Benzeggagh ML, Kenane M. Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos Sci Technol 1996;56(4):43949.

[53] Davies P, Sims GD, Blackman BRK, Brunner AJ, Kageyama K, Hojo M, et al. Comparison of test congurations for determination of mode II interlaminarfracture toughness results from international collaborative test programme. Plast, Rubber Compos 1999;28(9):4327.

[54] Kuboki T, Gallagher E, Jar PYB, Cheng JJR. A New method to quantify delamination resistance of bre reinforced polymers (FRP) under transverseloading. Appl Compos Mater 2005;12(2):93108.

[55] Sediadi Y, Jar PYB, Cheng JJR. Evaluation of toughness transfer from matrix to random-bre-reinforced polymers (FRP) under cyclic loading.In: Proceedings of the Fifth Canadian-International Composites Conference. UBC, Vancouver; 2005. p. 112.

[56] Kuboki T, Jar PYB, Forest TW. Inuence of interlaminar fracture toughness on impact resistance of glass bre reinforced polymers. Compos Sci Technol2003;63(7):94353.

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