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International Journal of Adhesion & Adhesives 28 (2008) 256–265

www.elsevier.com/locate/ijadhadh

On using a penalty-based cohesive-zone finite element approach,Part II: Inelastic peeling of an epoxy-bonded aluminum strip

Ted Diehl�

DuPont Engineering Technology, Chestnut Run, Building 722, Wilmington, DE 19880-0722, USA

Accepted 18 June 2007

Abstract

Previous work has demonstrated the use of a penalty methodology for enhancing the use of cohesive-zone elements to simulate double

cantilever beam (DCB) and flexible-arm peeling problems, both involving elastic arm deformation. This approach is extended to the

general case of inelastic peel arm deformations. Refinements to the original penalty selection methodology are included to account for

the influence of plasticity in the peel arms. Accuracy of the method is demonstrated by comparing simulation results to experimental data

of epoxy-bonded aluminum arms being peeled at different angles from a rigid substrate. This work addresses significant complexities in

the analysis that arise due to the inelastic deformation of the aluminum peel arms.

r 2007 DuPont. Published by Elsevier Ltd. All rights reserved.

Keywords: Finite element stress analysis; Peel; Fracture mechanics; Cohesive elements

1. Introduction

Part I of this paper demonstrated a penalty methodologyfor effectively utilizing cohesive elements to simulate failureonset and failure propagation in bonded joints. Thepenalty framework set forth offered an alternative methodto analyze such problems with a Cohesive Zone Method(CZM) where only the critical energy release rate, Gc, isknown. The more classical CZM approach uses aformulation requiring the specification of multiple para-meters to define the bond, namely the critical energy releaserate, critical limiting maximum stress, and the shape of thetraction–separation law ([1–5]). Validity of this new penaltyframework was demonstrated for two classes of problems:(1) double cantilever beam analysis and (2) flexible-armpeeling (see Fig. 1). In both the cases, only elasticdeformations in the arms were considered.

The current work extends the penalty-CZM approach toinclude the important effects of inelastic arm deformation.Compared to the deformations occurring in a DCBspecimen, it is far more likely that inelastic arm deforma-tions will occur in flexible-arm peeling problems, and thus

e front matter r 2007 DuPont. Published by Elsevier Ltd. Al

dhadh.2007.06.004

999 6775; fax: +1 302 999 6273.

ess: ted.diehl@usa.dupont.com

we concentrate on this case only. Consider commonexamples of peeling masking tape from a surface or peelinga thin metallized lid sealed to a drink container. Afterpeeling, the peel arm (masking tape or metallized lid) istypically curled—evidence of inelastic arm deformation.This inelastic arm deformation can contribute significantlyto the overall seal strength of the peeling system ([1–3, 5,6])and therefore it is important to properly include its effectsfor a valid analysis.

2. Idealized flexible-arm peeling

The energy analysis that was applicable for a DCBspecimen (Fig. 1a; analysis presented in Part I of thispaper) can be employed to the peeling of a flexible arm asdepicted in Fig. 1b. If the peel arm is idealized as aninfinitely rigid string (IRS), defined by infinite membranestiffness and zero bending stiffness, and if the bondingagent is assumed to have no compliance, then there will beno change in internal energy as the arm peels. The onlyenergy change will be derived from the external load andenergy dissipated by the crack itself, namely

1

b

dUext

da

� �¼

1

b

dU int

da

� �þ Gc. (1)

l rights reserved.

ARTICLE IN PRESS

P

P

b

a

h

y

L

P

P

P

P

P

P

P

Time 2 Time 3

P

P

Time 1

peel arm

Peel front

A1A2

A3

P + ΔP

P + ΔP

y + Δy

State 2

State 1 current bond length

DCB arm

Δa

θ

Fig. 1. Idealized crack growth in a DCB specimen and the deformation history during single arm peeling. (a) Double cantilever beam (DCB), and (b)

following segment ‘‘A’’ demonstrates arm bending and then unbending during peeling.

1It is noted that the original publishing of the analysis method used in

ICPeel had a typographical error in a term related to inelastic deformation

(f1ep(k0), Georgiou [5, p. 264]). The error, in the third line of this term, was

published as

�4

ð1þ 2NÞð1þNÞk2N0

1�N

1þN

� �.

This was corrected by Kinloch [7] to be

�4

ð1þ 2NÞð1þNÞk2N0 �

1�N

1þN

� �.

All the ICPeel calculations in this current work utilize the corrected

version of this formulation.

T. Diehl / International Journal of Adhesion & Adhesives 28 (2008) 256–265 257

Eq. (1) and simple geometry calculations indicate thatthe peel load, PIRS ¼ P, is related to the critical releaseenergy of the bond via

PIRS ¼Gcb

1� cosðyÞ. (2)

For most physically realizable systems, this idealizationwill not hold true and there will be changes in the internalenergy as the peel advances. Sources of internal energychanges are elastic and inelastic stretching of the peel arm(left of the peel front in Fig. 1b), inelastic bending of thepeel arm as portions continuously transition from bondedto unbonded status, shear across the arm’s cross-section inthe vicinity near the crack front, and highly localizedinelastic deformations in the peel arm very near the crackfront. The inclusion of these additional energy-consumingmechanisms into Eq. (1) leads to very complex analyticalformulations that do not lend themselves to simplisticclosed-form formulae. Analytical solutions based on non-linear shear-flexible beam theory that include somerepresentations of these inelastic terms have been publishedby the Adhesion, Adhesives and Composites research group

at Imperial College, London ([2–5]). These analyses alsoinclude improved estimates of the ‘‘clamped boundarycondition’’ at the root of the crack front, allowing for rootrotation and compliance based on the elastic stiffnessof a finite thickness bond and peel arm. The resultingformulation becomes quite complex with several highlynon-linear, transcendental equations needing to be solved.The method has been coded into a computer programcalled ICPeel [10].1

ARTICLE IN PRESST. Diehl / International Journal of Adhesion & Adhesives 28 (2008) 256–265258

An important energy contribution that ICPeel includes isthe influence of inelastic bending and unbending of the peelarm. Following material segment ‘‘A’’ in Fig. 1b, we seethat an initially straight segment in the bonded regionbecomes bent as it begins to peel upward and thenstraightens again as the segment continues into theunbonded peel arm far away from the peel front. If thepeel arm were to deform only elastically, then this set of deformations would have no net energy change becausethe energy to bend the segment would be returned whenthe segment was unbent. However, if the arm deformsinelastically, then energy will be required to bend thesegment (A2) and further energy will be required to unbendthe segment (A3). There is also the possibility that for amild inelastic deformation case only the initial bending willbe inelastic, but that the unbending may remain elastic([3,5]). Relative to the energy balance defined in Eq. (1),energy contributions from inelastic bending and unbendingcan be very large.

3. Peeling of an epoxy-bonded aluminum strip

Fig. 2 depicts the peeling of an epoxy-bonded aluminumstrip from a rigid substrate. Physical experimental data ofpeel load vs. peel angle for this problem was measured bythe Adhesion, Adhesives and Composites research group atImperial College, London [8]. This set of data forms thebasis of our benchmark.

P x

20

θ aluminum peel

arm, thickness h

epoxy

thickness ha

epoxy

(1/2 layer

each)

rigid

substratedepth (width) = b

Alum. arm thickness, h (mm)

Epoxy thickness, ha (mm)

Depth, b (mm)

0.307 ± 0.157

1.03

Peel angle, θ θ (deg) 45

Physically Measu

Peel force per depth, P/b (N/mm) 16.69

Calculated Critical Re

“Infinitely rigid string”formula (J/m2) 4888

1333ICPeel program (J/m2)

Fig. 2. Description of single arm peeling experiment of an aluminum substrat

problem, (b) measured peeling force vs. peel angle, and (c) estimates of critica

3.1. Experimental data—benchmark

The physical descriptions of the aluminum peel arm andepoxy bond are provided in Fig. 2a and the relevantcharacterizations of these materials are displayed in Fig. 3.The experiments of Kawashita [8] indicated that the epoxyfractured cohesively, roughly down the middle of the epoxybond. All experiments were performed at quasi-staticloading rates. The resulting peel forces displayed in Fig. 2were steady state peel forces experimentally measured forthree different peel angles of 451, 901, and 1351.Also displayed in Fig. 2c are two analytically based

estimates of the critical release energy for the bondinterface during steady-state peeling. The first set ofestimates was based on the IRS formula (Eq. (2)). Thesecond set of estimates was based on the ICPeel program.In both cases, the input data was peel angle and peel force.The ICPeel calculations also utilized material data for thealuminum and epoxy. Kawashita [8] modeled the alumi-num with a power-law plasticity model [6] using a yieldstress of 85MPa and a power-law hardening coefficient of0.22. This power-law yield stress is slightly lower than theactual yield stress of 100.3MPa (Fig. 3a) because a best-fitcompromise was required for the overall stress/strain curvewhen modeled with a power-law. (Only minor differencesin Gc estimates were found when a yield stress of 100.3MPawas utilized in the ICPeel power-law model.) The epoxywas simply modeled as an elastic foundation [8].

00

5

10

15

20

Pe

el fo

rce

pe

r d

ep

th,

P/b

(N

/mm

)

Peel angle (deg)

45 90 135 180

Experimental data

(Kawashita, 2006)

90 135

red Response

6.05 4.11

lease Energy, Gc

6050 7016

9931147

e adhered to a rigid support via an epoxy bond. (a) Definition of peeling

l release energy computed from experimental peel force data.

ARTICLE IN PRESS

0 5 10 15 200

100

200

300

1

True stress vs true strain, entire range *Plastic data for ABAQUS FE models

Tru

e s

tress,

σ (M

Pa)

True strain, εln (%)

Elastic modulus (GPa)

Poisson’s ratio

Density (kg/m3)

66.0

0.35

2750

Elastic modulus (GPa)

Poisson’s ratio

Density (kg/m3)

4.0

0.35

1496

pointTrue Stress

(MPa)

True plastic

strain (%)

2

3

4

5

6

7

8

9

10

100.3

106.6

110.0

122.4

136.7

170.1

207.3

234.2

275.0

286.0

0

0.031

0.150

0.612

1.13

3.00

5.48

8.95

15.3

19.6

Fig. 3. Material data of aluminum and epoxy utilized in finite element simulation of peeling experiment. (a) Data for Aluminum, ISO 5754-O, 1.0mm

thickness, and (b) data for ESP110 Epoxy.

T. Diehl / International Journal of Adhesion & Adhesives 28 (2008) 256–265 259

For both analytical approaches (presented in Fig), avalue of Gc was calculated separately for each peel angle.The analytical approaches estimated very different valuesof critical release energy. The IRS method estimated valuesranging from 4888 J/m2 at 451 to 7016 J/m2 at 1351. TheICPeel program calculated much lower values that rangedfrom 1333 J/m2 at 451 to 993 J/m2 at 1351.

Theoretically, a material characteristic such as Gc shouldbe independent of peel angle for isotropic materials that arepeeled at quasi-static rates and have the same mode ofcrack propagation for all peel angles tested (mode I).However, both methods resulted in angular dependent

values of Gc (variations of approximately715% from their901 estimates). This angular dependence implies that thesemethods insufficiently represent some portion of a physicalmechanism(s).

Relative to the two classes of estimates for Gc shown inFig. 2c, it is deemed that the ICPeel results are likely to becloser to the actual values since these were computed withformulae that included more physics of the problem,specifically the inclusion of inelastic deformations forbending and unbending. Since energy was continuouslyconsumed during peeling for the inelastic bending andunbending of aluminum at it passed from the bondedsection to the peel arm section, the Gc value computedwith this consideration is lower than the rigid stringmethod.

3.2. Finite element models

Fig. 4a lists four finite element modeling approachesutilized to study the epoxy-bonded aluminum strip bench-mark. The models ranged from highly detailed analysesthat utilized solid element representations of the aluminumand epoxy (Fig. 4c) to lower fidelity simulations that onlyutilized beam elements for the aluminum with no epoxycompliance (picture not shown).

3.2.1. Model setup

The modeling approach essentially follows that de-scribed in Part I of this paper. The cohesive failure ofthe epoxy bond was modeled with a zero-thickness layerof cohesive elements using a cohesive over-meshingfactor of 5. The traction–separation characterization ofcohesive elements is depicted in Fig. 5. As in Part I of thispaper, the penalty-CZM approach considers Gc as the onlymaterial parameter to represent the bond. The otherclassical CZM parameters (critical limiting maximum stressand the shape of the traction–separation law) are taken tobe related to a single penalty parameter, the cohesiveductility (failure separation), df. In a penalty-CZManalysis, we desire to make the cohesive ductility (a penaltyparameter) as small as possible to approach a zero-compliance bond. As previously demonstrated, there is aminimum value that this penalty can be defined as, below

ARTICLE IN PRESS

Fig. 4. Various finite element modeling approaches utilized in peeling study. (a) Finite element modeling approaches analyzed, (b) explicit transient FE

model of 135 (degree symbol here) peel at various stages during solution, (c) highly zoomed view showing model detail near crack front (modeling

approach #1).

0 �o �f

effK

0

Gc = lim Gc

Keff

Tra

ction

, T(n

om

inal str

ess)

Tult

0

Tra

ction

, T(n

om

inal str

ess)

Tult

Separation, δ0 �f

�f → 0

Separation, δ

(area under entire curve)

Gc

Elastic limit.

Damage occurs

for δ > δo.

Gcˆ

area under

entire curve

Taking the limit as δfapproaches zero

equates to an

impulse, which

implies a non-

compliant cohesive

material

ˆ

Fig. 5. Traction–separation law utilized in finite element cohesive model and an interpretation of the classical Griffith energy release approach cast in a

cohesive material law form. (a) ABAQUS traction-separation cohesive material law, and (b) Effective cohesive material law for classical energy release

approach.

T. Diehl / International Journal of Adhesion & Adhesives 28 (2008) 256–265260

which the numerical solution becomes ill-behaved.The absolute value of this minimum penalty is dependenton the CZM mesh density, but the results are consistent for

a common value of the mesh-relative cohesive ductilityratio, df=DLCOH2D4 (a non-dimensional measure of thepenalty intensity). Table 1 lists 10 cases of cohesive

ARTICLE IN PRESS

Table 1

Cohesive element modeling parameters

(a) Parameters common to all FE models studied

Cohesive over-meshing factor 5

COH2D4 element length DLCOH2D4(mm) 0.080

Damage initiation ratio, dratio 0.50

(b) Cohesive ductility values utilized for each FE modeling approach

Case Mesh-relative

cohesive ductility,

df=DLCOH2D4

Cohesive ductility

relative to Aluminium

thickness, df/h

Cohesive ductility

relative to epoxy

thickness, df/ha

Ultimate nominal stress

for cohesive law, Tult

(MPa)

Additional cohesive

density scale-up

factor

1 6.00 0.466 1.564 4.17 19.36

2 4.50 0.350 1.174 5.56 10.89

3 3.00 0.233 0.782 8.33 4.84

4 2.50 0.194 0.651 10.0 4.84

5 2.00 0.155 0.520 12.5 4.84

6 1.50 0.117 0.393 16.7 4.84

7 1.00 0.078 0.261 25.0 2.25

8 0.50 0.039 0.130 50.0 2.25

9 0.10 0.0078 0.0261 250.0 1.0

10 0.05 0.0039 0.0130 500.0 1.0

T. Diehl / International Journal of Adhesion & Adhesives 28 (2008) 256–265 261

ductility used with each of the modeling methods fromFig. 4a.

3.2.2. FE model results

For the analytical methods discussed in Sections 2.1 and3.1, the value of Gc was a calculated output derived fromthe peel angle, peel force, geometries, and materialbehaviors. With cohesive elements in a finite elementmodel, the value of Gc is typically a model input and thepeel force is a model output. As a starting point for the FEmodel, we utilized the ICPeel values of Gc derived from theexperimental data shown in Fig. 2. Ultimately, all the FEmodels were analyzed for two values of Gc; namely 1.0 and1.32 kJ/m2. Results using these two values demonstratethe sensitivity of the different FE modeling approaches tothe value of Gc.

The FE models were intended to be quasi-static analyses,but to improve solution robustness (convergence), theywere solved as transient dynamic models using ABAQUS/Explicit [9], a commercial FEA code utilizing an explicitdynamics solution algorithm. Fig. 4b depicts several stagesof a typical solution for peeling at 1351. The solution wasdriven by an imposed velocity boundary condition on theleft end of the peel arm. For cases with solid elements in thesolution (as depicted in Fig. 4c), a multipoint constrainttechnique was employed to apply the boundary conditionon a single node, but have its effect smeared across the endcross-section of the arm. The velocity was imposed in arotated coordinate system. To minimize transient behavior,a ramped velocity method was used which increased thevelocity from zero up to a constant value during the first10% of the solution and then was constant thereafter.Solution reaction forces from this imposed boundarycondition were then utilized to obtain the applied peelforce.

It is noted that Fig. 4b shows the peel arm having apermanent bend at its ‘‘pulled end’’ (through-out thesolution). Because there was an initially unbonded cracklength at the beginning of the solution, this small portion ofmaterial did not endure the same deformation history asthe material which was initially bonded and then peeled. Asthe peel arm increased in length (as more was peeled free),the influence of this permanent bend was lessened andultimately became negligible by the end of the solution. It isnoted that once this end region of the peel arm wassufficiently away from the active peel front it exhibited nofurther straining, just rigid body translation. Analyzingthese models with zero initial crack length (in an attempt toavoid this issue) created other problems due to excessivelyhigh stresses around the crack front that generallyprevented any valid solutions.Since the explicit dynamics method utilizes a time-

marching algorithm, it is desirable to artificially increasethe imposed peeling rate of the problem for increasedsolution efficiency. The trick is to increase the imposeddisplacement rate up to a point where the dynamicinfluences on the solution are still negligible, but thecomputational duration is as short as possible. Fig. 6demonstrates how this concept was pushed further thantypically done, while still obtaining acceptable quasi-static results. In Fig. 6a, the kinetic energy from a typicalpeeling solution is plotted relative to the total externalwork. For a quasi-static solution, this level of kineticenergy relative to the external work would be consideredtoo high because it significantly distorts the peel loadspredicted by the model. However, notice how the kineticenergy rises at a constant slope after about 1/3 of thesolution has developed. This implies that the kinetic energyis not causing excessive vibrations. Instead, as thealuminum is peeled free from the bond (Fig. 4b), the peel

ARTICLE IN PRESS

0 100 150 200 250 30050

Peel Distance ux (mm)450400350

0 100 150 200 250 30050

Peel Distance ux (mm)450400350

Pe

el fo

rce

pe

r d

ep

th,

P/b

(N

/mm

)

8.0

6.0

4.0

2.0

0.0

Pke

P = Pqs

Ptotal

Pqs

Ptotal

= 0.576

(ux = 350mm)

En

erg

y,

U (

J)

2.5

2.0

1.5

1.0

0.5

0.0

kinetic energy

external work

Fig. 6. Improving calculation of peel load by compensating for excessive

kinetic energy in Explicit model. Results shown for modeling approach #1

(solid aluminum mesh and solid epoxy mesh), Case 7 (df=DLCOH2D4 ¼

1:0), peel angle ¼ 1351 and Gc ¼ 1.32 kJ/m2. (a) External work and total

kinetic energy calculated by model, and (b) peel force per depth.

0

10

20

30

40

50

Peel fo

rce p

er

depth

, P

/b (

N/m

m)

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Mesh-relative cohesive ductility δf / ΔLCOH2D4

Mesh-relative cohesive ductility δf / ΔLCOH2D4

Boundary belowwhich model beginsto behaves poorly

45 deg

90 deg

135 deg

Symbols are FEA calculated results.

Lines are linear connections between FEA data points.

Peel fo

rce p

er

depth

, P

/b (

N/m

m)

20.0

15.0

10.0

5.0

0.00.0 1.0 2.0 3.0 4.0 5.0

bestdirectresults

45 deg

90 deg

135 deg

improvedestimates

Symbols are FEA calculated results.

Lines are least squares quadratic fit to FEA results.

Fig. 7. Influence of mesh-relative cohesive ductility on FEA predicted peel

force and an extrapolation method to obtain ‘‘improved estimates.’’

Results are for modeling approach # 1 with Gc ¼ 1.32 kJ/m2. (a) Influence

of mesh-relative cohesive ductility on predicted peel force, and (b)

Extrapolating to obtain ‘‘improved estimates’’ of predicted peel force.

T. Diehl / International Journal of Adhesion & Adhesives 28 (2008) 256–265262

arm’s ever-increasing mass (traveling at constant velocity)is simply causing the kinetic energy to rise. Fig. 6bdemonstrates how the ‘‘kinetic force’’, Pke, can beeffectively removed from the model’s raw calculation ofthe peel force, Ptotal (obtained directly from the FE model’sreaction forces at the imposed velocity boundary location),to get an improved estimate of the quasi-static peel force,Pqs. The kinetic force is simply computed by taking thederivative of the kinetic energy relative to the peel distance(same as crack growth assuming negligibly small peel armstretching).

For the example depicted in Fig. 6, the ratio of theresulting quasi-static peel force, Pqs, relative to the model’sraw calculation of peel force, Ptotal, was 0.576; a significantadjustment. This technique was validated on several casesby running the models at half the loading rate shown here(such that the kinetic energy was 1/4 its previous values).The estimates of quasi-static peel force from the two speedswere nearly identical for each case. The benefit of thiskinetic force compensation technique yields computationalspeed-ups of two times to four times. For individual model

computations that range between 20min and 2 h, such asavings is significant.Fig. 7a shows the influence of the mesh-relative cohesive

ductility penalty on the resulting quasi-static peel forcepredictions for modeling approach # 1 (Fig. 4a and c).Results are shown for all the three peel angles. Similar tobehavior reported previously in Part I of this paper, thefinite element model begins to behave poorly when themesh-relative cohesive ductility penalty is set too stiff.Due to a transitioning effect, the exact cut-off of this poorbehavior is not precisely defined, but clearly occurssomewhere below df=DLCOH2D4 ¼ 1:0 for the case shownhere. For all the FE models in this current study, thetransition range was below 1.0, but it could be different forother classes of problems (see Part I of this paper forexamples). Fig. 7b demonstrates how the ‘‘good’’ data(to the right of the boundary defined in Fig. 7a) is furtherutilized to extrapolate to an ‘‘improved estimate’’ of peelforce. This extrapolation is based on the concept that themesh-relative cohesive ductility penalty should be 0.0 inthe ideal case (for a penalty-CZM approach) if numerical

ARTICLE IN PRESST. Diehl / International Journal of Adhesion & Adhesives 28 (2008) 256–265 263

ill-behavior could be avoided. Since the epoxy complianceis separately modeled using solid continuum elements (inmodeling approach #1 and #3), driving this cohesivestiffness to infinity is theoretically desirable (ignoringnumerical problems) to mimic a Griffith Criterion repre-sentation of the bond based solely on Gc.

It is noted that the curvature of peel force vs. mesh-relative cohesive ductility shown in Fig. 7b was caused bythe increasing clamping stiffness of the cohesive elementsaround the crack front as the penalty was stiffened. Thisinfluenced the resulting inelastic deformation imposed onthe peel arm (mainly the bending plasticity). If the peel armhad behaved purely elastically, then there would be nodependency on peel force as a function of mesh-relativecohesive ductility (until the numerical ill-behavior of anexcessively stiff penalty influenced the solution). Thisconcept was demonstrated in Part I of this paper, whereonly elastic deformations were considered.

Table 2 provides a summary of all the FE modelresults. Accuracy of the models is judged based onnormalizing the FE-predicted quasi-static peel forcesrelative to the actual experimental values of peel forcemeasured by Kawashita [8]. For each modeling approachlisted, two results are provided: (1) ‘‘improved estimates’’based on extrapolation (df=DLCOH2D4 ¼ 0:0) of peel forcesvia the method depicted in Figs. 7b, and 2) ‘‘bestdirect results’’ using the stiffest penalty viable beforenumerical ill-behavior (df=DLCOH2D4 ¼ 1:0). The FE model

Table 2

Assessment of predictions of peel force per depth using various finite element

Modeling

approach #

Aluminum peel

arm

Epoxy

compliance

Relative cohes

ductility, df=D

FEA predictions using Gc ¼ 1.00 kJ/m2

1 Solid elements Solid elements 0.0a

1.0

2 Solid elements None 0.0a

1.0

3 Beam elements Solid elements 0.0a

1.0

4 Beam elements None 0.0a

1.0

FEA predictions using Gc ¼ 1.32 kJ/m2

1 Solid elements Solid elements 0.0a

1.0

2 Solid elements None 0.0a

1.0

3 Beam elements Solid elements 0.0a

1.0

4 Beam elements None 0.0a

1.0

aMesh-relative cohesive ductility of df=DLCOH2D4 ¼ 0:0 represents extrapola

predictions are further divided into two sections, results forGc=1.0 and 1.32 kJ/m2.Ideally, the most accurate FE model should be the one

with the most physics included—modeling approach #1(all solid elements for both aluminum and epoxy). Indeed,this model demonstrated that using the ‘‘improved esti-mates’’ (df=DLCOH2D4 ¼ 0:0) predicted nearly identical peelforce values to the experiment for all three peel angles whenGc=1.32 kJ/m2. Most importantly, this FE model showednegligible angular dependence of Gc. Even with the ‘‘bestdirect results’’ (df=DLCOH2D4 ¼ 1:0), this model showedvery little angular dependence of Gc. This also held true forthis model when a value of Gc=1.0 kJ/m2 was utilized,albeit the predicted peel forces were about 25% lower thanthe experiment (as expected since Gc was lower). Modelingapproach #2 indicates that removing the epoxy compliancecaused the model to exhibit noticeable angular dependenceas well as a slightly higher overall peel force. Because theepoxy compliance was removed, inaccurate clampingstiffness is imposed to the aluminum peeling arm, causingincorrect amounts of inelastic deformation to occur.It is noted that others [3,5] have rationalized the initial

stiffness of a cohesive element material law, Keff (Eq. (9)),to represent components such as the epoxy stiffness. Thisleads to a model based on a linear elastic foundationapproach which does not sufficiently capture the complexstiffness interactions that occur at the peel front. The FEmodels here demonstrate that more accurate results can be

modeling approaches

ive

LCOH2D4

Peel force per depth, P/b, assessment

FEA calculations relative to Kawashita [8] experiment

(FEA/experiment)

451 peel 901 peel 1351 peel

0.77 0.77 0.75

0.68 0.70 0.71

0.95 1.03 1.07

0.82 0.90 0.96

0.95 1.00 1.05

0.83 0.90 0.97

0.96 1.02 1.07

0.84 0.91 0.99

0.98 0.98 0.97

0.88 0.90 0.91

1.19 1.32 1.39

1.03 1.15 1.24

1.16 1.28 1.33

1.03 1.14 1.23

1.18 1.31 1.36

1.04 1.16 1.26

ted values.

ARTICLE IN PRESST. Diehl / International Journal of Adhesion & Adhesives 28 (2008) 256–265264

obtained when the individual materials are analyzed ascontinuums (solid elements) and that the cohesive elementis strictly used to model the actual fracture interface via apenalty-CZM approach.

When the aluminum peel arm was modeled with beamelements (modeling approach #3 and #4), a value ofGc=1.0 kJ/m2 best fit the experimental peel force data.However, in all of these cases, the model exhibitednoticeable angular dependence (although not as much asthe ICPeel model; Fig. 2c). It is also noted that the epoxycompliance had much less influence on the results for thebeam models as compared to the solid models (modelingapproach #1 and #2). This is likely because the beammodels are not able to sufficiently capture the complexinelastic stress state in the peel arm that is occurring in thegeneral neighborhood of the peel front.

3.2.3. Benchmark summary

Three different types of models were evaluated: theidealistic IRS model, the ICPeel model, and various FEmodels. For this study, only the continuum-based FEmodel using cohesive elements defined in a penaltyframework was able to accurately model an inelasticpeeling problem with negligible angular dependence ofcritical release energy Gc. Direct modeling of all thephysical components in the system and the proper penaltyapproach for the cohesive elements were the primaryreason for this success. While not analyzed here, it wouldlikely be true that accurate results could also be obtainedusing the full CZM method (where critical energy releaserate, critical limiting maximum stress, and the shape of thetraction–separation law are all physically meaningfulmaterial parameters of the bond).

Of the two analytical models studied, the calculatedcritical release energy values from the ICPeel model weregenerally similar to those of the FE models (as compared tothe predictions from the IRS model). However the ICPeelmodel exhibited angular dependence of Gc. It is believedthat the primary deficiency in this model is related to itsbeam-based formulation’s inability to fully analyze thecomplex continuum stress state in the neighborhood of thepeel front, including the 2-D stiffness interactions betweenthe epoxy and aluminum. Despite this limitation, theICPeel model has significant value in that it computesextremely fast (1–2 s) and it predicts Gc values that aresimilar (on average) to the much more computationallyintense FE models.

4. Practical modeling considerations

The methodology utilized for the FE models was quitedetailed and computationally intensive in both computa-tion time, number of models run, and extrapolationmethods to obtain ‘‘improved estimates.’’ In manypractical cases, such rigor may not be required. As theanalyst repeatedly works with a similar class of problems,the need for constantly running mesh-relative cohesive

ductility studies will lessen. They may be able to simplyrun various analyses using a single setting for the penalty(‘‘best direct results’’).Another issue to further consider is the influence and

meaning of the effective critical limiting maximum stress,Tult, of the cohesive material (Fig. 5a). As stated previouslyfor a penalty-CZM approach, this is just a dependentpenalty parameter that is directly related to the physicalvalue of Gc and the penalty value of cohesive ductility, df.As the cohesive ductility penalty is stiffened, the value ofTult will increase. Since this stress will be the stress actingnormal to the cohesive element at the time that damageinitiates and the crack begins to propagate, its value canhave an influence on the solution. In particular, if its valueis set higher than the maximum allowable stress that theadjacent material connected to the cohesive element canendure, then a solution will not be possible and thesimulation will terminate. In this case, the cohesiveductility penalty will be clearly too stiff for the physics ofthe problem defined. This result might be viewed asphysically reasonable. Adhering a weak material (relativelylow ultimate strength) with a high strength bond (Gc ishigh) is likely to cause bulk material failure in the weakmaterial, not a failure in the bond interface. This isconsistent with the general behavior predicted by a penalty-CZM model. This also points out that the value of stressnormal to the cohesive element near the crack front is ill-defined with this approach in that it is driven by a penaltyparameter. This is consistent with the view that in using anenergy-based approach to analyze the crack, we areimplicitly taking a global or smeared approach to theproblem, as opposed to a highly local or detailed analysisthat is utilized with classical fracture mechanics methodsderived around stress intensity factors. The user needs to bemindful of these inter-relationships when analyzing suchproblems with this approach. The alternative is to use amore traditional CZM approach, but this requires the userto obtain physically accurate values for the additionalCZM material parameters.

5. Conclusions

This study has presented accurate simulations of thepeeling of an epoxy-bonded aluminum strip from a rigidsubstrate. All the models were compared to experimentalresults. Relative to the various analytical and FE modelsevaluated, it has been shown that only the continuum-based FE model, using cohesive elements defined in apenalty framework, was able to accurately model theinelastic peeling problem with negligible angular depen-dence of critical release energy Gc. The ability of thisparticular FE model to fully analyze the complex stressstate in the peel arm near the peel front, including the 2-Dstiffness interactions between the epoxy and aluminum,was the primary reason for its accuracy. This study furtherdemonstrated how to rationally determine the cohesiveductility penalty required to convert the traditional

ARTICLE IN PRESST. Diehl / International Journal of Adhesion & Adhesives 28 (2008) 256–265 265

three-parameter cohesive element approach into a single-parameter law dependent on the physically meaningfulvalue of Gc. The work also confirmed the usefulness of thehighly efficient ICPeel model and provided a potentialexplanation of why the ICPeel method exhibits mildangular dependence of Gc.

Acknowledgments

The author is grateful to Professor A.J. Kinloch,Imperial College, London, for his initial suggestion of thisbenchmark problem (aka The Kinloch Challenge) and hismany interactions throughout this project. The continuedcommunications with ABAQUS developers Harry Hark-ness, Kingshuk Bose, and David Fox are acknowledged.The support of DuPont colleagues Mark A. Lamontia,Clifford Deakyne, Delisia Dickerson, Leo Carbajal, JaySloan, James Henderson, David Roberts, Patrick Young,and James Addison is also recognized.

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