FINITE ELEMENT MODELING OF ADHESIVE

FAILURE WITH ADHEREND YIELDING

Jun Cui

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

O Copyright by Jun Cui 200 1

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FlNlTE ELEMENT MODELING OF ADHESIVE F A U R E

wim ADHEREND YIELDINO Master of Applied Science

2001

Jun Cui Graduate Department of Mechanical and Industrial Engineering

University of Toronto

Abstract

In this thesis, both stress-analysis and cohesive zone modeling (CZM ) üpproiic hcs

were used to deveiop peel finite element models. aimed at predicting the strenpths 01'

steady-state peel tests with top adiierend ranging from 1 mm to 3 mm and pcd :11iglc\

ranging from 30 to 90 degrees. For the stress-andysis approach. a critical Von-Miw~

strain failure criterion was investigated and found to be independent of the perl ünglc hiit

dependent on the peel arm thickness. For the CZM. an energy-based failure criterion wn\

used. It was observed that once the parameten of the traction-separation çur~-c\

characterizhg the CZM were calibrated using a 1 mm, 90" peel test, it could be usrd io

give reasonable predictions of peel strengths for the 1 mm and 2 mm peel tests. Howvci-.

the predicted peel strengths of the 3 mm peel tests were significantly lower than ttic

experimental results.

Acknowledgements

First and foremost, 1 thank my thesis supervisor, Dr. Jan K. Spelt for his constant

support and intensive guidance throughout the course of this work. There is no doubt that

the pst two years have been the greatest leaming experience of my life. Jan played a

significant role in that. I would also like to express my sincere gratitude to Professor

Anthony Sinclair. for his inspiration and insightful comments which 1 have received

during various stages of this research.

Special thanks to my dear wife, Wen, for her encouragement and invaluable

suggestions. 1 am also grateful to my extended family for their long time understanding

and encouragement in my pursuit of knowledge.

Thanks are also due to al1 my colleagues and friends who made my stay here fmitful

and enjoyable; each of thern helped me, one way or another.

iii

To Wen

List of Figures

Nomenclature

Table of Contents

Abstract

Acknowledgements

Table of Contents

List of Tabies

iii

Chapter 1 Introduction 1

................................................................................. 1 .1 Background.. 1

........................................................................... 1.2 Thesis objectives.. 4

1.3 Literature review.. ........................................................................ ..S

1.3.1 Stress analysis of adhesive joints.. ................................................ .5

1.3.2 Fracture mechanics analysis of adhesive joints.. ............................... 13

1.3.3 Cohesive zone modeling of adhesive joints.. .................................... 19

. . 1.4 Thesis organization.. .................................................................... - 2 3

Chapter 2 Numerical Study of the Peel Test Using Stress Analysis 24

............................................................................. 2.1 tntroduction.. . 2 4

.................................................................................. 2.2 Peel tests. - 2 8

2.2.1 Peel specimens.. ................................................................... - 2 8

2.2.2 Peel test resul ts.. ................................................................... -29

2.3 Development of a non-linear large displacement. steady-state peel finite element

mode1 ......................................................................................... 31

2.3.1 Material property tests .............................................................. 31

................... 2.3.2 Development of the steady state peel finite element mode1 37

................. 2.4 Numerical simulations and cornparisons with experimental results 52

2.4.1 The approach used in the numerical simulations ................................ 52

.......................................... 2.4.2 Cornparisons with expenmental results 52

.............................. 2.5 Cornparisons of the initiation state and the steady state 56

2.6 investigation of the mode ratio ........................................................... 59

2.6.1 The stress-based definition ........................................................ .6O

......................................................... 2.6.2 The strain-based definition 67

............................................................. 2.7 Discussions and conclusions 71

2.7.1 The peel a m thickness dependence of the critical Von-Mises

. . strain failure criterion ............................................................... 71

........................................................................... 2.7.2 Mode ratio 73

Chapter 3 Study of DCB Fracture Test Using

Cohesive Zone Modeling

3.1 Introduction ................................................................................ -75

3.1.1 Fracture characterization of elastic adhesive joints ............................. 75

3.1.2 Cohesive zone modeling ............................................................ 76

3.1.3 Objectives of this Chapter ......................................................... 77

3.2 DCB fracture test ........................................................................... 77

3.2.1 Adhesive system used ............................................................... 77

3.2.2 DCB specimen ....................................................................... 78

............................................................................. 3.2.3 Apparatus 81

.................................................................... 3.24 Fracture envelope 83

3.3 Prediction of the fracture envelope using cohesive zone modeling .................. 86

3.3.1 Establishment of a mixed-mode DCB finite element model for the

452368 adhesive system ......................................................... -37

3.3.2 Characterization of the parameters used to speci fy the traction-

................................................................... separation curves -88

3.3.3 Prediction of the fracture envelope and discussions ............................ 93

3.3.4 Prediction of the mode ratio ........................................................ 94

3.3.5 Predictive sensitiviiy caused by the shape parameters ......................... 96

3.3.6 Numerical analyses of the fracture envelope for Betamate 1044-3 /

.................................................... AM06 1 -T6 adhesive system 10 1

............................................................ 3.4 Discussion and Conclusions 104

Chapter 4 Numerical Study of the Peel Test Using Cohesive Zone

Modeling (CZM) 1 06

................................................................................ 4.1 Introduction 106

4.2 Establishment of the peel finite element mode1 ....................................... 107

. . ......................................................................... 4.3 Failure cntenon.. I O 8

4.4 Characterization of parameters used to specify the traction-separation curves ... 109

4.5 Numerical simulations and cornprisons with experimental results ............... I I I

4.6 Investigation of the mode ratio ......................................................... 114

4.6. t S tress-based definition ........................................................... -115

vii

4.6.2 Strain-based definition ......., .. .. .. . ., ..... . . ... ... ..... . . .. ... . ... ..... ...... . .. 1 18

Chapter 5 Conclusions and Recommendations 121

5.1 Conclusions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 2 1

5.2 Recornmendations.. . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 23

References 124

Appendix A Finite Element Program for the Peel Continuum

Model Based on the Critical Von-Mises Strain

Failure Criterion 134

Appendix B Finite Element Program for the Peel Model

Based on Cohesive Zone Modeling 143

Appendix C Data Files for DCB Fracture Tests of

Betamate 1044-3 Adhesive System

viii

List of Tables

Table 2.1 Peel test results ........................................................................... 31

Table 2.2 Mechanical properties of AA5754-0 and AA606 1 -T6 (Experimental data) ... 32

Table 2.3 Mechanical properties of AA5754-0 and AA606I-T6 (Published data) ........ 33

Table 2.4 Mechanical properties of Betamate 1044-3 .......................................... 36

Table 2.5 Cornparisons of the Clayer mode1 and the 8-layer model ........................ -49

Table 2.6 FEA prediction for peel loads based on the critical Von-Mises strain failure

. . cri terion .................................................................................. -54

Table 2.7 FEA prediction for root curvature based on the critical Von-Mises strain failure

. . cnfenon ................................................................................. -36

Table 2.8 The plastic zone length detemined based on the adhesive interfacial Von-

Mises stress distribiition ................................................................ 61

Table 2.9 The influence of the peel angle and the peel a m thickness on the tensile zone

lenpth ...................................................................................... 64

Table 3.1 Parameter combinations chosen for the traction-separation curves ............... 91

Table 3.2 Mode ratio predictions .................................................................. 94

Table 3.3 Shape parameter combinations (strategy one) ....................................... 97

Table 3.4 Shape parameter combinations (strategy two) ....................................... 99

Table 4.1 Different combinations of shape parametea. ô. i and corresponding

predicted peel forces using the cohesive zone mode1 ............................. 1 I O

Table 4.2 FEA prediction for peel loads based on the steady state peel finite element

model using the cohesive zone modeling approach calibrated at

1 mm. 90' peel.. ...................................................................... - 1 12

Table 4.3 The influence of the peel angle and the peel arm thickness on the tensile zone

length (obtained from the peel finite element model based on the cohesive zone

................................................................. modeling approach). . 1 15

List of Figures

...................................................... Figure I . 1 The traction-separation law.. .2 1

................................... Figure 2.1 Schematic representation of a typical peel test. 24

............................... Figure 2.2 Schematic representation of the peel specimen.. . .28

.................................................. Figure 2.3 Illustration of the peel test load jig 29

Figure 2.4 The uniaxial stress-strain curve for the top adherend material (AAj754-0).33

Figure 2.5 The uniaxial stress-strain curve for the bottom adherend material (AA6O6 1 - ................................................................................... TS).. -34

........................ Figure 2.6 The uniaxial stress-strain curve for Betarnate 1044-3.. .35

Figure 2.7 Definition of the plastic modulus as the secant modulus to a point on the

................................... effective stress versus effective strain curve.. .46

........... Figure 2.8 The peel finite element mode1 with 4 layers of adhesive elements.. 50

........... Figure 2.9 The peel finite element mode1 with 8 layers of adhesive elements.. 50

Figure 2.10 The interfacial adhesive Von-Mises stress cornparison of the Clayer model

............................................................ and the 8-layer model.. ..5 1

Figure 2.1 1 Photograph of the peel test with 2 mm peel m thickness and 60" peel angle

- - during steady state.. ................................................................ ..33

Figure 2.12 The predicted macroscopic defonned shape of the 1 mm. 90' peel test at the

. . . . initiation state ....................................................................... -57

Figure 2.13 The predicted macroscopic defomed shape of the 1 mm, 90" peel test at the

........................................................................ steady state.. -37

Figure 2.14 The predicted nodal Von-Mises stress distribution of the 1 mm. 90" peel test

. . . . at the initiation state.. .............................................................. ..58

Figure 2.15 The predicted nodal Von-Mises stress distribution of the 1 mm. 90' peel at

the steady state.. ...................................................................... 59

Figure 2.1 6 Average phase angle over the plastic zone as a function of the peel a m

thic kness (stress-based de finition). ............................................... .6 1

Figure 2.17 Average phase angle over the plastic zone as a function of the peel angle

(stress-based de fi nit ion). ............................................................ .62

Figure 2.18 Variation of the adhesive normal stress with distance from the peel front for

the 1 mm. 90' peel test ............................................................... 63

Figure 2.19 Average phase angle over the tensile zorir as a function of the peel am

thic knrss (stress-based defini tion). ............................................... .65

Figure 2.20 Average phase angle over the tensile zone as a function of the peel angle

* a . ............................................................ (stress-based definition). .65

Figure 2.2 1 Local phase angle on the adhesive node at the peel front as a function of the

peel arm thickness (stress-based de finition). .................................... .66

Figure 2.22 Local phase angle on the adhesive node at the peel front as a function of the

peel angle (stress-based definition). .............................................. -66

Figure 2.23 Average phase angle over the plastic zone as a function of the peel am

thickness (strain-based definition). ............................................... .67

Figure 2.24 Average phase angle over the plastic zone as a function of the peel angle

. ........................................................... (strain-based definition). ..68

xii

Figure 2.25 Average phase angle over the tensile zone as a function of the peel arm

................................................ thic kness (strain-based de finition) -69

Figure 2.26 Average phase angle over the tensile zone as a function of the peel angle

.............................................................. (strain-based definition) 69

Figure 2.27 Local phase angle on the adhesive node at the peel front as a function of the

...................................... peel ami thickness (strain-based definition) 70

Figure 2.28 Local phase angle on the adhesive node at the peel front as a function of the

................................................ peel angle (strain-based definition) 70

Figure 3.1 Geometry and dimensions of the DCB specirnen (dimension in mm unless

.................................................................................. stated) 78

............................................ Figure 3.2 illustration of the mixed-mode load jig 81

............................................ Figure 3.3 Photograph ofthe mixed-mode load jig 82

...... Figure 3.4 Fracture envelope for Betamate 1044-3 / AA606 1 -T6 adhesive system 85

...... Figure 3.5 Fracture envelope for Cybond 4523GB / AA7075-T6 adhesive system 86

................................................... Figure 3.6 Normal trac tion-separation curve 88

..................................................... Figure 3.7 Shear traction-separation curve 88

.................................................... Figure 3.8 Mode I traction-separation work 92

................................................... Figure 3.9 Mode II traction-separation work 93

Figure 3.10 The Frocture envelope prediction for the 452368 adhesive system using the

........................................................ cohesive mode) ing approach 95

Figure 3.1 1 Predictions of the fracture envelope based on the shape parameter

. . ......................................................... combination-strategy one 98

xiii

Figure 3.12 Predictions of the fracture envelope based on the shape parameter

. . ....................................................... combination-strategy two. 100

Figure 3.13 The fracture envelope prediction for Betarnate 1 044-3 adhesive system using

................................................ the cohesive modeling approach.. 103

Figure 4.1 Average phase angle over the tensile zone as a function of the peel a m

.............................................. thickness (stress-based detinition). . 1 16

Figure 4.2 Local phase angle on the springs at the peel front as a function of the peel

.......................................... a m thickness (stress-based definition). 1 17

Figure 4.3 Average phase angle over the tensile zone as a function of the peel a m

.............................................. thickness (strain-based definition). . I l8

Figure 4.4 Local phase angle based on the sprinp at the peel front as a function of the

.................................. peel a m thickness (strain-based definition). ,119

xiv

Nomenclature

LI

A

b

CZM

dl

DCB

dW'

E

4 1

E'

Crack length

Area of the crack surface

Width of the adherend

Cohesive zone modeling

Virtual crack advance

Double canti lever beam

Change of the elastic strain energy of the system

Plane stress Young's modulus

Plastic modulus

Plane strain Young's modulus

f ;, ( O ) A known function

The work done by the extemal forces

Eneqy release rate

Adhesive shear modulus

Critical energy release rate

Mode 1 traction-separation work absorbed by the fracture process

Mode 11 traction-separation work absorbed by the fracture process

Mode 1 critical energy release rate

Mode II critical energy release rate

Maximum value of adhesive fracture energy

J An elastic-plastic energy release rate calculated using the J-integral

J,. A critical value of elastic-plastic energy release rate calculated using the J-integral

K Stress intensity factor

K, . Critical value of the stress intensity factor

LEFM Linear elastic fracture mechanics

Total force

Plastic potential

Adhesive layer thickness

A specific bond thickness

The strain energy stored in en adhesive joint

Plastic deformation energy

A geometric function

The work of separation per unit area of crack advance

The work of separation per unit area of crack propagation corresponding to the

normal traction-separation curves

The work of separation per unit area of crack propagation corresponding to the

shear traction-separation curves

Shear strain

'Shape" parameter

"S hape" parameter

Critical displacement

xvi

Normal component of the relative displacement of the crack faces across the

interface in the zone where the fracture process occurs

Tangential component of the relative displacement of the crack faces across the

interface in the zone where the fracture process occurs

Critical normal displacement

Critical shear displacement

Critical normal displacement where fracture occurs

Critical shear displacement where fracture occurs

Plastic strain in the principal direction

Plastic strain in the principal direction

Plastic strain in the principal direction

Ultimate Strain

Effective total strain or Von-Mises strain

Effective plastic strain or Von-Mises stress

Plastic multiplier

Coe tficient of intemal friction

Poisson's Ratio

0.2% Y ielding Strength

Equivalent or Von-Mises stress

The stress at a point near the crack tip

Hydrostatic stress

xvii

Modified Von-Mises yield stress

Ultimate Strength

Yield stress of the peel adherend

Normal peak stress in the traction-separation curve

Effective stress

Shear stress

Shear peak stress in the traction-separation curve

Peel angle

Phase angle

( r , B ) Polar coordinates

[B] Strain-displacement matrix

[D] Material property ma~ix

[ K ] Stitrness matrix

{F} Extemal nodal force vector

{u} Nodal displacement vector

(E) Strain vector

(a} Stress vector

xviii

Chapter 1 Introduction

1.1 Background

Interest in the use of adhesives for structural bonding began in the 1940s. when

high-strength. polymer-modified and phenolic adhesives becarne available [go]. It was

soon realized that adhesives could be used in place of other traditional joining methods.

e.g., screws. nails, bolts and rivets. as well as welding. Indeed. the use of adhesives in the

construction of aircrafi resulted in lighter, more aerodynamic structures. Subsequentl y.

other resin types were introduced and structural bonding has been extended to many key

industries including automotive, construction. composite materials and semicoiiductor

industries.

Compared with conventional joining techniques. adhesive bonding offers a number

of advantages. Fint. the fact that an adhesive distributes applied loads over the entire

bonded area and avoids points of stress concentration. leads to joints exhibiting

outstanding fatigue resistance. Second, adhesives are particularly suitable for joining

dissimilar materials; if different metals are involved, then galvanic corrosion can be

prevented. Third. the use of mechanical fasteners generally means that holes have to be

drilled in the materials to be joined. This may weaken the material. provide stress

concentration points and introduce sites for corrosion. These problerns can be avoided by

the use of an adhesive, which thus ensures the integrity of the components. Last, for some

applications adhesive bonding is the only practical joining method. One of the best

examples, of particular importance in the aircrafi industry, is the bonding of thin metal

skins to honeycomb cores to provide lightweight, rîgid structures.

Although adhesives have so many advantages, they suffer from a number of

limitations, which must also be considered when assessing the suitability of an adhesive

for a particular application. Most adhesives are strong in shear and tensile loading, but

weak when peel or cleavage stresses are present. Low joint strength is oflen caused by

poor adhesion between the adhesive and the substmte surface. Moreover. the

performance of adhesives may degrade on exposure to hostile environments. However. in

contrast to more conventional joining techniques, the major concem about the use of

adhesives is that, there exist presently no generally accepted guidelines for the prediction

of joint strength, especially for the case when the adherends experience large plastic

deformation. The aerospace industry, which developed the technology. is still designing

joints mainly based on previous experience and rules of thumb. These qualitative design

techniques are very costly since they must be verified with full-scale experiments under

the realistic operating conditions to ensure satisfactory joint designs. Therefore, simple

and reliable methods for joint strength prediction. similar to those established for riveted.

bolted and welded joints, need to be developed before it is possible to use adhesives in

the design of structurally critical components.

The c u m t state-of-the-art designs of adhesive joints are primarily based on two

approaches: the stress analysis approach and the linear elastic fracture mechanics

(LEFM) approach. The former approach generally utilizes a maximum stress or

maximum strain as a failure criterion. Failure is assumed to occur when the maximum

stress or strain ai the end of the bonded overlap reaches a critical value. This theory

assumes that flaws are present in materials but that they are very small and unifomly

distributed. The stress analysis approach is more complicated than it seems for several

reasons. One complexity arises in the determination of local stresses in the adhesive joint.

In addition, stresses typically occur from the application of loads on a system; however,

deformation of adherends with respect to the adhesive and stress concentrations in the

joint can also produce large local stresses. Another reason for complications is that each

joint geometry or design can produce different types of stresses and in different locations.

Adhesive materials, as well as al1 polymea, inherently contain flaws such as porosity.

voids or microcracks. The realization that these voids actually govem the performance of

the material has led to the application of linear elastic fracture mechanics in thc study of

adhesive joints. LEFM has proven to be an effective tool for analyzing the behavior of

adhesive joints when the adherends deform elastically [45]. Moreover, a pseudo-LEFM

approach was used by Fernlund and Spelt [28. 291 and Papini and Spelt [64]. who

employed an energy-based fracture mechanics criterion to predict fai lure of el astic

adhesive joints which contain a substantial plastic zone occumng ahead of the

macroscopic crack tip of the adhesive layer. However. when the adherends defom

plasticall y, the application of the LEFM is inappropriate because the plastic de formation

of the adherends will, in general, affect the crack tip stress field and thus the fracture

process occumng in the adhesive layer. Although it has been well recognized that the

influences of adherend plasticity are significant [3 1, 32. 421, analytical tools are Far from

being established.

Recently. Tvergaard and Hutchiiison [73-761 made great progress in developing a

cohesive zone modeling (CZM) approach to anaiyze the interfacial failure of bi-material

systems. The CZM is characterized by the traction-separation relation that describes the

fracture process occumng ahead of the crack tip. It was shown that this modeling

approach is a promising tool for analyzing interfacial failure in the presence of extensive

plasticity in the surrounding materials, which is similar to the case of plastically

deforming adhesive joints. A detailed review of the stress analysis approach, the fracture

mechanics approach and the cohesive zone modeling approach will be given in Section

1.3.

1.2 Thesis objectives

So far. a very successfui method exists for predicting the fracture loads of a wide

range of adhesive joint geometries under combinations of mode-1. mode4 and mode-Ill

[29. 30, 641. The principal restriction to this approach, which provides the motivation of

the present research. is the assumption that the bonded members behave elastically. Le..

they do not yield. However. adhesively bonded thin sheet structures will yield under

impact situations, which obviously violates the small-scale deformation assumption

inherent in the previous approach of fracture load predictions. Therefore. if structural

adhesives are to be used extensively in the automotive industry. it is important to develop

a comprehensive engineering approach to predict the adhesive fracture strength in the

presence of the adherends' plastic deformation, which is the overall objective of this

M.A.Sc. thesis. The more specific objectives include: (1) to study the applicability of the

critical Von-Mises strain in the fracture analysis of plastically deforming adhesive joints:

(2) to develop a non-linear, large displacement steady state peel finite element mode1

based on the stress analysis and fracture mechanics approaches; (3) to analyze the

fracture of double cantilever bearn (WB) specimens and peel specimens using the

cohesive zone modeling approach in order to develop a generally applicable engineering

method for the fracture analysis of plastically defonning adhesive joints; (4) to

incorporate the cohesive zone modeling approach into the numerical study of the steady

state peel test and to develop the corresponding finite element model.

1.3 Litera ture ieview

1.3.1 Stress aaalysis of ad hesive joints

The stress analyses of elastic adhesive joints were studied earliest. The first to merit

special mention is that of Volkersen who, in 1938, studied the adhesive shear stress

distribution alonp the bond line in the single lap joint loaded in tension [go]. By assuming

that both the adherend and adhesive materials are elastic, the shear stress is calculated in

ternis of the differential stretching of the adherends. It was found that the shear stress is

not uniformly distributed because of the stress concentration o c c h n g nt the ends of the

overlap area. Volkersen's theory is incomplete because it does not account for the

bending of the adherends from the eccentricity of the loading path. Predictions based on

Volkenen's work would seem more valid for double lap joints where bending is

minimized. Goland and Reissener [34] were the first to take into account the bending of

the adherends in the stress analysis of the single lap joint. Their analysis makes use of

several assumptions: (1) the joint is in a plane strain or triaxial stress state; (2) the

adherends and adhesives behave as elastic materials; and (3) the tinite deflec tion theory

for the cylindrically bent plates cm be applied to calculate the deflection of the

adherends. They found that the stress concentration at the edge of the lap joint, which

accounts for the eccenincity of the load, is twice of that predicted by Volkersen [80].

Many researchen have performed subsequent analyses of lap shear joints. Greenwood et

al. [36] experirnentally proved the validity of the Goland and Reissner [34] theory for

specific combinations of adherends and adhesive matenals. Ishai et al. [4 11 also obtained

experimental results in agreement with the stress distribution predicted by Goland and

Reissner [34]. Comell [19] analyzed the lap shear joint and characterized the adherends

as simple beams while considering the adhesive to behave as a system of shear and

tension springs. Bigwood and Crocombe [IO] presented a full elastic analysis of the

adhesive joint which calculates the adhesive shear and tensile stresses in the overlap

region. and this analysis was validated for a range of load cases using a finite element

program. In their analyses, the adhesive joint is modeled as an adhesive-adherend

sandwich with any combination of tensile, shear and moment loading applied at the ends

of both adherends.

The stress analysis approach was fùrther used to explore the stresses in a variety of

adhesive joints such as double lap joints and p e l joints. Volkersen [79] considered the

double lap joint and found that bending of the adherends still occurs but to a lesser extent

and that the resulting normal stress in the joint is reduced. A number of resrarchers have

considered the analyses of the peel test. Most analyses have involved the peeling of a

flexible mernber from a rigid adherend. One of the first analyses was done by Bikerman

[12], who assumed that both the flexible and tigid substrates behave as perfectly elastic

materials. Also the shear stresses were eliminated because the adhesive could be modeled

as individual fibers extending between substrates with no interde pendence. B ikennan

concluded that the adhesives with low modulus should perform well in peel, but that low

modulus may hinder the performance in shear. Kaelble [42-441 also did work on the

analysis of peeling and found that the distribution of peel stress along the bond line is a

highly darnped harmonic fwiction involving altemating regions of tension and

compression. Moreover, Kaelble's analysis takes into account the peel angle, while in

Bikerman's [12] analysis, only 90' peel angle was considered.

So far al1 of the analyses discussed or mentioned above have assumed linear

elasticity of the adhesive. However, adhesives typicall y show either plastic or elastic-

plastic behavior. depending on the nature of the joint materials. Many investigators have

sought to include the nonlinear or time dependent behavior of the adhesive into their

studies. Delale and Erdogan (23, 241 performed an analysis where they considered the

adherends as linear elastic and the adhesive as viscoelastic. They found that with time the

stresses in the adhesive redistribute and that the normal stress is greater than the

corresponding shear stress. The work of Hart-Smith [38. 391 is probably the best known

in the area of modeling non-linear adhesive behavior. By using closed-fom analytical

niethods together with numerical iterative solving techniques. he presented a series of

non-linear analyses of several cornplicated joint configurations loaded main1 y in shear.

These studies showed that the inclusion of adhesive plasticity in an analysis might

decrease the stress concentration substantially and thus increase the joint toughness

significantly. Howevcr. the author did not couple the adhesive shear and peel stresses but

considered the shear to be elastic-perfectly plastic and the peel stress to be elastic.

Further. it was assurned that. by proper design of the geornetry of the adhesive joint. the

mode 1. or peel stresses could be reduced to the point where they do not contribute to the

failure of the joint. The bondline was thus asswned to be under pure shear. A closer look

at the most common geornetries reveals that they contain significant peel stresses.

Bcsides, the peel stresses in these joint geometries cannot be easily reduced by improved

design. The approach cannot, therefore, be considered generally applicable. Grant and

Taig [35] used a more realistic model of the adhesive non-linear stress-strain curve but

their analysis is based on the Volkersen's shear lag approach and thus neglects the effect

of the peel stress. Livey and McCarthy [56] coupled the adhesive shear and peel stresses

to predict the onset of adhesive yield but only analyzed a simple tension overlap.

Bigwood and Crocombe [ 1 11 investigated a general plane strain problem of adhesively

bonded structures that consist of two different adherends. In a similar way to the general

elastic analysis outlined in an earlier paper by the same authors [IO]. the adhesive joint

was modeled as an adherend-adhesive sandwich allowing the application of any

combination of tensile. shear and moment loading at the adherend ends. The adherends

were assumed to behave as linear elastic, cylindrically bent plates with the adhesive

forming a non-linear interlayer between them. The deformation theory of plasticity was

used to model the stress-strain characteristics of the adhesive, with the stress-strain curve

i tself being approximated by any continuous mathematical function. Unlike some other

approaches to this problem. here both the adhesive shear and peel stresses contribute to

the yield of the adhesive through the Von-Mises criterion, and the non-linear responses of

both are modeled,

While the plastic deformation in the adhesive layer of the joint has been studied

extensively. and the dependence of the bond strength on the plastic deformation in the

adhesive is relatively weil understood, it should be mted that most of these analyses are

subject to the limit that the adherends exhibit only elastic behavior before the adhesive

joint fails. It is, however, often found that the failure of an adhesive joint is accompanied

by extensive plastic deformation in the adherends when the bonding is reasonably strong

or the adherends are relatively thin. However, so far, there are no analytical tools that cm

provide a reliable prediction of the adhesive joint strength when the adherends experience

extensive plastic deformation.

Research on the stress analysis of plastically deforming adhesive joints is scarce.

One of these few studies is that of Crocombe and Bigwood [22], who extended their

previous adhesive joint analysis, which accommodated non-linear adhesive behavior. to

mode1 the elastic-plastic response of the adherends. The non-linear behavior of both the

adhesive shear and transverse direct stresses was modeled to predict the yield of the

adhesive, and the adherends were modeled as cylindrically bent plates that yield under

the action of combined tension and bending. The problem was reduced to a set of six

non-linear first order ordinary differential equations that were solved numerically using a

finite difference method.

As shown in the above discussion, owing to the complications introduced by the

adherend and adhesive plasticity, it is very difficult to obtain theoretical solutions of

plastic stresses in the adhesive of an adhesive joint. However. non-linear numerical

simulation techniques such as non-linear finite element analyses (FEA) have been shown

to be powerful alternatives for such studies. Numerical solutions or finite element

rnethods cm alleviate some of the limitations of analytical solutions while also

eliminating the need for simplifying assumptions. Finite element methods can also be

used to tackle problems that are impossible to compute by analytical methods. In pneral.

finite element analysis involves the representation of the adhesive joint by a network of

elements which contain positions called nodes. Each element is characterized by stresses

and displacements. At the nodes, either stresses or displacements are unknown,

depending on the method of the analysis. These unknowns are used to determine the

coefficients of the elemental stresses and displacements by potential energy

considerations. As a result, stresses and displacements are determined for each element.

Many researchers have applied finite element analysis to structural adhesive joints.

Wooley and Carver [87] used finite element analysis for the single lap joint and reported

agreement with Goland and Riessner [34] when cornparhg tearing stress concentrations.

Cooper and Sawyer [18] accounted for the non-elastic properties of the adherends. the

adhesives, or both. They found that the elastic-plastic behavior can have a pronounced

effect on joint stresses. Varias er al. [77] employed the finite element technique to study

the stress distribution in the plastic zone at the crack tip in an adhesive joint with elastic

adherends and with an elasto-plastic adhesive subject to remote mode-1 loading. It was

found that possible âilure mechanisms include near-tip vuid growth. high triaxiality

cavitation that may occur at several bond line thicknesses ahead of the crack tip. and

interfacial debonding at the site of highest interfacial normal traction. Chiang and Chai

[17] calculated the elasto-plastic stresses and strains in cracked adhesive joints subject to

shear loading using a large strain finite element technique. It was found that the overall

plastic zone size ahead of a crack tip might be up to forty times the bond thickness and

the predicted failure load increases as the plastic zone increases. Adams and Peppiatt [ l .

21 used the finite element technique to analyze the stress distribution of lap joints. When

an adhesive fillet was included, the highest stresses were found to be near the adherend

corner at an angle of approximately 45' to the surface of the adherend. It was also

observed that the direction of cracks in failed lap joints is perpendicular to the predicted

maximum tensile stresses. The authors concluded that the failure of the lap joint is

initiated by a tensile failure of the adhesive within the spew fillet. Crocombe and Adams

[2 11 developed an elastic large displacement finite element program for peel analysis. In

their analysis, both non-cracked and cracked configurations were presented, representing

initial and continuous failure of the peel test. Analysis of the former indicated that the

initial failure was caused by the adhesive principal stresses driving a crack towards the

interface with the flexible adherend. Investigation of the cracked configuration showed

that the amount of mode II loading at the crack tip is significant and is cssentially

independent of the peel angle, load and adhesive or adherend modulus. only decreasing

as the adhesive becomes incompressible. Further. the strength measured by the peel test

is not proportional to the actual strength of the adhesive. and a small increase in the

adhesive strength will cause a much larger increase in the applied peel load. In a

subsequent piece of work. Crocombe and Adams [20] presented an elasto-plastic

investigation of the peel test by extending their previous elastic, large displacement finite

element analysis [2 11 to include elasto-plastic material behavior. In the paper. two

common peel tests which used high and low yield strength aluminum adherends.

respectively. were analyzed. The Von-Mises yield citenon was used to mode1 the

yielding of the aluminum and a modified (parabolic) Von-Mises yield fùnction was used

for the adhesive. A failure criterion based on effective plastic strain of the adhesive was

employed to predict the relative strengths of the peel tests. The adhesive stresses near the

crack tip were s h o w to be finite while the corresponding strains remain singular.

Further. it was found that the effective plastic strain failure critenon could be used

successfully to predict the relative strengths of the same peel test. However, values of the

adhesive effective plastic strain fiom the peel test with the low yield strength adherend

were considerably lower than those fiom the peel test with the high yield strength

adherend. As another important aspect conceming the elasto-plastic response of the peel

test, the amount of energy dissipated in the plastic deformation of the peeling adherends

was also assessed by a series of tests and was show to be a considerable proportion,

about 50%, of the total energy supplied to the peeling system. Thus, the non-linear nature

of the peel test, established by the elastic analysis is significantly increased by plastic

deformation for the two systems considered. It was also observed that the energies

dissipated in plastic deformation was similar although the two alriminum alloys had

grossly di fferent yield strengths.

It should be noted that al1 these numerical simulations discussed so far are only valid

up to crack initiation. However, the situation duririg steady state crack propagation is

different from that of crack initiation. In the peel test, for example, the steady state

peeling load is several times the peeling load when the crack initiaies: There is more

energy dissipated in the peeling ami under steady state conditions; and the stresses and

strains in the adhesive layer are also changed. Therefore, an adhesive joint finite element

analysis with steady state crack propagation is more usehl and will capture more realities

than the analysis based on crack initiation. It is surprising then, no such models were

found during the extensive literature search of this study. Consequently. the steady state

finite element peel analysis based on the stress analysis approach became one of the core

parts of this M.A.Sc. thesis.

13.2 Fracture mecba nics analysis of adhesive joints

As discussed in Sections 1.1 and 1.3.1, it is always a challenging task to detemine

the stress distribution along the bond line because it is highly depndent on joint

geometry. Also, there exists another complexity in ternis of deciding the proper failure

criterion applicable to designing structural adhesive joints. These difficulties of the stress

analysis approach led to the application of fracture mechanics to adhesive joint failure

analysis.

Fracture mechanics studies the effect of stress concentrations that occur when a load

is applied to a body containing a void, independent of the geometry or material of the

body. By definition it would seem logical that the fracture toughness of a material. as

determined by specific fracture mechanics techniques, would be an appropriate design

cri terion. Since adhesive joints always fail by the initiation and propagation of flaws. the

application of fracture mechanics theory to analyze the failure of adliesive joints has

received considerable attention. The following discussion will begin with a reviaw of the

basic theories of fracture mechanics, focusing on the energy balance approach. the stress

intensity approach and the I-integral approach. The application of fracture mechanics to

the adhesive joint analysis will be given afierwards.

Fracture mechanics based on a consideration of energy balance cornes From the

work of Griffith [37], who stated that for an infinitely sharp crack in a brittle material. the

crack will propagate when the energy released is greater than the energy needed to create

a new surface. As a continuation of the GrifFith approach, Orowan [62] realized that the

energy necessary to propagate a crack is much greater than the material's surface energy.

He attributed this to the local plastic deformation that occua at the crack tip. The energy

release rate, G , is therefore defined as:

where F is the work done by the extemal forces, U, is the strain energy stored in an

adhesive joint. and A is the area of the crack surface. Failure occurs when G reaches a

critical value. G,. . which is termed as the critical energy release rate or the fracture

toughness of a specific adhesively jointed system. Another approach used in LEFM is

based on stress intensity factors. which were proposed by lnvin in the 1960s. Irwin [JO]

modified the stress function derived by Westergard 1841 to obtain the following equation:

where O,, is the stress at a point near the crack tip defined by polar coordinates ( r . 0 ) .

K is the stress intensity factor, and j, ( O ) is a known function. The critical value of the

stress intensity factor. K,. , is expressed in the following way:

where K,. is considered a matenal property and is refened to as the fracture toughness,

a is the crack length, Y is a geometric function which has been derived for many test

specimen geomctries and cm be found in an excellent book on fracture mechanics of

polymers by Williams [85]. Both the energy balance approach and the stress intensity

approach assume that energy dissipation occurs locally near the crack tip in an area

termed the plastic zone. Another theory which accounts for the non-linear behavior of

materials is the path independent J-contour integral developed by Rice [66]. The basic

assumption of this theory is that materials undergo non-linear elastic deformation such

that the unloading curve follows the same path as the loading curve. In other words. this

non-linear elastic behavior cm be used to mode1 plastic behavior of a material. which is

known as the deformation theory of plasticity. It has also been stated that under certain

restrictions the J-integral can be used as an elostic-plastic energy release rate. The path

independency of the J-integral expression allows calculation along a contour remote from

the crack tip. Such a contour cm be chosen to contain only elastic loads and

displacements. Thus an elastic-plastic energy release rate can be obtained from an elastic

calculation along a contour for which loads and displacements are known. Moreover.

because J may be considered as an elastic-plastic energy release rate it is to be expected

that there is a critical value, J,. , which predicts the onset of crack extension. This is by

analogy with G,. in the LEFM.

The energy release rate for an elastic adhesive joint can be calculated or

experimentally measured from the work done by extemal forces and this makes it

possible to predict the Fracture of adhesive joints without knowing the complicated stress

distribution in adhesives.

The application of the LEFM to the fracture analysis of elastic adhesive joints has

been studied extensively in the last four decades [13, 29-32,45, 5 1, 67,68. 7 11. Ripling,

Mostovoy and Patrick [67] first investigated the application of general Fracture mechanics

in the elastic adhesive joint analyses. Kaminen (451 presented an improved analytical

mode1 for the double cantilever beam fracture specimen by treating a finite length beam

which is partly fiee and partly supported by an elastic spnng foundation. Gent and

Hamed [31, 321 used the fracture mechanics approach to study peel joints and it was

found that the measured joint toughness is dependent upon the adherend thickness - it

reaches a maximum as the thickness is increased and then decreases as the thickness is

further increased. Explanations of the plasticity in adherends and adhesives were

postulated but without confirmation. In Kinloch and Shaw [5 11, the fracture resistance of

a rubber-modified epoxy adhesive was studied using a continuum fracture mechanics

analysis and it was observed that the adhesive fracture energy, G,Jjoint) of joints

consisting of steel adherends bonded with epoxy adhesive is a strong function of adhesive

bond thickness, r . A maximum value, G,c.A, (joint) was recorded at a specitic bond

thickness. r,,, . Further, the value of G,(,, (joint) was compared to the fracture energy.

G,. (bulk) of the e p x y material and under many conditions the f m e r parameter was

found to be greater in value. Cao and Evans [13] experimentally measured the fracture

resistance of bimaterial interfaces for a wide range of phase angles. These experiments

revealed that the critical energy release rate increases with the increase in phase angles,

especially when the crack opening displacement becomes srnail. Suo and Huchinson [71]

derived an analytical solution for a semi-infinite interface crack between two infinite

isotropic layers under general edge loading conditions. Femlund and Spelt 1291

developed closed-form solutions for the energy release rate and the mode ratio of an

equal adherend adhesively bonded beam specimen subject to a mixed mode bending load.

The developed expressions explicitly account for the thickness and the marerial

properties of the adhesive layer using a beam on an elastic foundation model. The

accuracy of the expressions was verified using dimensional analysis and cornparison with

finite element results, and it was show that they correlate well with experimental

fracture data frorn adhesively bonded beam specimens with varying crack lengths.

Fiwther, the J-integral technique was used to take into account the material and geometry

non-linearity when calculating the energy release rate in Femlund and Spelt [30].

The combination of theoretical work with experimentally measured critical

quantities has made LEFM an extraordinary tool in malyzing adhesively bonded elastic

materials. Based on the analytical approach, a variety of failure phenornena such as

interfacial debonding, delamination and elastic instability in brinle adhesive joints have

been studied. However, the LEFM approach is incapable of dealing with those joints with

large-scale plastic de formation occumng in the adherends. In such situations as

mentioned before. the energy absorbed by the fracture process is coupled with the energy

dissipated by the macroscopic plasticity in adherends. It is very diffkult to separate one

from the other and as a result, the measured joint toughness will depend on the joint

geometry and cannot be treated as a material property. The fracture analyses of plastically

defonning adhesive joints are relatively few in the Iiterature because of these difficulties.

The earliest and most extensive study on the fracture analyses of plastically deforming

adhesive joints might be the work of Kim, Aravas and their CO-workers [47-491. They

proposed a generalized elastic-plastic slender beam theory for the analysis of the

detached part of the adherend in a peel test. By taking account of the elastic unloading

and reverse plastic bending of the strip, they gave a closed-form solution for the

maximum curvature (root curvature), and hence the plastic dissipation attained by an

elastic-perfectly plastic adherend. Their expressions are in terms of the peel force, peel

angle, adherend properties, and the rotation at the root of the adherend. and it was shown

that the plastic dissipation strongly depends on the root rotation. Yamada [88] extended

the common approach of beam-on-elastic foundation for bonded joints to include the

elastic-plastic bond response by replacing the plastic zone by a unifomly distributed load

on the b e m . Williams [86] analyzed the role of root rotation due to the adherend

cornpliance in the peel test, following Kaminen's approach [45] for double cantilever

beam specimens. It was assumed that the adherend behaved elastically at the root.

although elastic-plastic behavior was taken into account for the detached part of the

adherend. Kinloch [54] used this approach to study the peeling of laminated materials and

found good agreement with experiments. Moidu. Sinclair and Spelt [59] presented an

analytical approach to predict the adherend plastic dissipation in the peel test for metal-

metal adhesive joints. thereby allowing the fracture energy to be extracted from the test

data using an energy balance approach. In this study, expressions were developed for the

deflection of an elastic-plastic beam on an elastic foundation, which was then combined

with known solutions for the defonnation of an elastic-plastic strip under large

displacements. In a subsequent paper by the same authon 1601, an improved model was

developed for the prediction of the adherend plasticity in the peel test, based again. on the

elastic foundation model but considering the adhesive shear stress.

Despite the fact that considerable efforts have been invested, a full understanding of

the facture analysis of the plastically deforming adhesive joints is still far from being

accomplished. As discussed above, the major dificulty cornes from the coupling between

the rnicroscopic fracture process and the macroscopic plastic deformation process, as well

frorn the physical and geometrical non-linearity that usually accompanies the plastically

defoming joints. Recently, the cohesive zone modeling approach was proposed and

developed to study this type of ptoblem and it was demonstrated to be a very effective

approach. A detailed review will be given in the following section.

1.33 Cohesive zone modeling of adhesive joints

Metallurgical research has predicted that nucleation and growth of voids play a key

role in the fracture process of ductile rnaterials. This cannot be described by classical

fracture theory based on conventional continuum rnechanics, which does not consider any

effects of rnicroscopic behavior. Therefore, it becomes necessary to introduce new

constitutive models into continuum mechanics. As a first step of simplification. the

effects of voids will be neglected. This means that the fracture process zone can be

represented by a thin micro-scaled strip ahead of the actual crack tip. which is

characterized by a cohesive zone mode1 (CZM) with its own constitutive requirements.

The core of the CZM is the traction-separation relation that mirnics the effect of the

fracture process. The interfacial tractions and separations providc the link between the

fracture process and the macroscopic deformation in the surrounding materials. The CZM

incorporates more details of the separation process than the modeling with continuum

mechanics. but does not contain effects of the order of atomic discreteness. The region

ahead of the growing crack tip is represented by a traction-separation strip joining the two

elastic-plastic continua. They intemct with each other in such a way that a traction-

versus-separation relation applies along the gradually separating ligament. In general. the

traction across the ligament is taken to be a function of the separation. which cm be

undeotood to be a collective reaction of accumulation of micro-cracking and void

growth.

The damage zone model was originally proposed by Barenblatt [9] to describe

nonlinear crack behavior by means of the cohesive forces in the so-called process zone".

A constant traction was used in Bareiiblatt 's traction-separaiion curve. which is the

sirnplest form of the CZM. Due to its simple formulation, the CZM can be implemented

into finite element codes based on conventional continuum mechanics. Needleman [61]

pioneered the analysis of interfacial crack problems in the presence of plastic deformation

by using a polynomial traction-separation cume in finite elernent analysis. The most

extensive study on the CZM bas done by Tvergaard and Hutchinson [73-761. They used

a trapezoidal traction-separation relation to study the crack growth resistance curve

behavior under small-scale yielding conditions in homogeneous materials [XI. The

model involves representing the bonding across a putative fracture plane by a layer of

special elements whose constitutive properties describe the traction-separation law for

bonding. The traction-separation law (see Figure 1.1) is characterized by r, . the work of

separation per unit area of crack advance (equal to the area under the traction-separation

curve), the peak stress supported by the bonding tractions, b , the critical displacement.

6, and the "shape" parameten 6, and 4 .

4 4 4 S

Figure 1.1 The traction-separation law

It was show that the first two parameten, r, and 6 , are the dominant ones in the

numerical analysis of fracture; the precise shape of the traction-separation law. as

represented by the two shape parameters, is less important. Tvergaard and Hutchinson

(74) deveioped a mode-independent potential function to derive the normal and shear

traction-separation curves for mixed-mode fracture at the interface of bimaterials. In the

subsequent work of the authors. the CZM was used to explore a variety of coupled

fracture and plasticity phenornena: the influence of plasticity on the mixed-mode

toughness of interfaces [74], the contribution of plastic deformation to the effective work

of fracture for an interface along a thin layer joining two elastic solids [73. 761. Shirani

and Liechti [70] investigated the feasibility of using the CZM for extracting the adhesive

fracture energy of thin films on a thick substrate from the circular blister experiments that

involve a substantial amount of inelastic deformation in the thin tilm. Non-linear spring

elements were used in their finite element study to simulate the traction-separation laws

in the directions normal and tangentid to the interface. In the work of Wei and

Hutchinson [83], the CZM was proposed to analyze the steady-state peeling of a thin

rate-independent, elastic-plastic film bonded to an elastic substrate. By embedding a

traction-separation description of the interface within continuum characterizations of the

film and substrate, the relationship of the peel force to the work of adhesion of the

interface and its strength was examined. Yang and Thouless [90] also investigated the

applicability of the CZM to the fracture of adhesive joints when the adherends experience

extensive plastic deformation. In this paper, r,, was detemined from wedge test; 6 was

obtained from the cornparison between the tinite element results and the wedge

experimental data. Then, they used this CZM in finite element prediction of the load-

displacement curve for the T-peel test and acquired excellent agreement with the

experimental curve. However, The T-peel test is still predominantly mode4 failure. In

Kinloch et al. [SOI. the CZM was used to investigate the peel test. They concluded that

for a given peel test configuration. a unique pair of values of T, and ô. which predict

the load versus displacement curve in agreement with the experimental results. does not

exist. In addition, they found that T, from the peel test was less than that from the TDCB

(tapered double cantilever beam) specimen. Further work is clearly needed to investigate

how to extract a characteristic, geometry independent value of r, .

In this thesis. the cohesive zone modeling approach has been used to study the

fracture of the double cantilever beam specimen and the peel specimen. which represent

elastic and plastic adhesive joints. respectively.

1.4 Thesis organiza tion

Chapter 2 discusses a numerical study of the peel test based on the traditional stress

analysis approach and the fnicture mechanics analysis approach. A non-linear. large

displacement peel finite element model, which utilizes the critical Von-Mises strain as

the failure criterion, is successfully developed. The predictive capability of this rnodel is

then examined by comparing the numerical predictions with the experimental results.

Chapter 3 presents a fracture envelope (critical energy release rate as a function of

the mode ratio) for the Betamate 1044-3 adhesive system measured using DCB

specimens. A CZM approach is used for the fracture analysis of the DCB under mixed-

mode load conditions. The numerical predictions for the fracture envelope are compared

with the associated DCB fracture test results.

In Chapter 4. the elastic-plastic mixed-mode fracture of adhesive joints is studied

using the cohesive zone modeling approach. The traction-separation relation is employed

to simulate the interfacial failure of adhesively bonded peel specimens. with extensive

plastic deformation occurring in the peeling arm. The fiacture parameten for the traction-

separation law are determined by comparing the numerical and experimental results for

one configuration of the peel samples. The parameters are then used without Further

modification to simulate the fracture of peel samples with different configurations.

Chapter 5 outlines the conclusions, and gives the limitations of the present analysis.

Recommendations for future work are also presented.

Chapter 2 Numerical Study of the Peel Test Using Stress Analysis

Adhesive

I Rigid adherend l

Figure 2.1 Schematic representation of a typical peel test

A peel test is illustrated schematically in Figure 2.1. It is one of the most frequently

used test methods for assessing adhesion strength. There are a variety of peel tests, and

the Arnerican Society of Testing Materials (ASTM) has issued sevenl different standards

such as ASTM D3 167-97 and Dl 78 1-98. In a typical peel test, a thin flexible adherend,

which is bonded to a rigid adherend by a layet of adhesive, is pulled apart at a specified

angle and rate from the underlying substrate. The test reaches a steady state d e r a

substantial amount of crack extension. The steady state peel force required to separate the

top adherend and the substrate is t ened the peel strength and has been widely used to

characterize adhesive bond strength. Because of its wide application in many key

industries such as the aerospace, automotive and microelectronics. the peel test has k e n

studied extensively for almost five decades. A large amount of experimental and

theoretical work exist in a variety of aspects. such as the effects of the peel angle, the

non-linear behavior of the materials, the degree of intrinsic adhesion acting between the

adherends, the effects of the test rate and temperature, etc.

Two types of analysis, the stress analysis approach and the fracture mechanics

analysis approach, have been developed in parallel to predict the strength of peel joints.

The former involves the stress analysis of the joint coupled with a stress or strain based

failure criterion. This approach dates back to the work of Bickerman [12] and has been

continuously developed and improved to accommodate such behaviors as non-linear

deformation in the adhesive layer [20], the effect of adherend plasticity [16,3 11 and non-

linear effects due to the large rotation of the adherends [ I I l . Based on the stress analysis

approach, a numerical solution of the elastic-plastic peel problem was presented by

Crocombe and Adams [20,21]. in which the finite element method was used to calculate

the stress distribution ahead of the interfacial crack. It was observed that the effective

plastic strain failure criterion could be used successfully to predict the relative strengths

of the same peel test. However, a unique value of the adhesive effective plastic strain was

not found for peel tests consisting of adherends with diftierent yield strengths. As

mentioned in Chapter 1, a limitation existing in Crocombe and Adams [20,21] is that the

numerical analysis is only valid up to crack initiation. A non-linear, large displacement,

steady-state peel finite element model, which is based on the critical Von-Mises strain

failure cnterion, will be developed to analyze the peel tests in this Chapter.

The fracture rnechanics approach uses an energy-based failure criterion such as a

critical energy release rate or a critical J-integral to predict failure of peel joints. An

energy balance is used to relate the experimentaily measured peel force to the specific

fracture energy. Following this approach, Spies [72] analyzed elastic peeling by

considering the still attached part of the adherend as an elastic bearn on an elastic

foundation and the detached part of the beam as an elastic barn under large

displacement. During elastic peeling, part of the work done by the peel force is stored in

the elastically deforming system and the rest is used to pmvide the work required to

break the adhesive bond and create the new fracture surface. Therefore, in steady state

elastic peeling the energy balance cm be written as [48]:

Pd1 = dW' +Gbdl , (2.1)

where P is the total force, dW' is the change of the elastic strain energy of the system.

b is the width of the adherend, dl is a virtual crack advance, and G is the fracture

toughness of the peel joint. During the steady state elastic peeling, the peel bend remains

constant in shape, and therefore the change of the elastic strain energy dWi' is due to the

extension of the adherend alone. In most cases, the quantity dW' is small and for an

inextensible adherend is exactly zero. Thus, the above equation can be written as:

where P is the peel force per unit width of the adherend. By taking the peel angle into

account, Kendall [46] derived the following expression:

where # is the peel angle. Equation (2.3) shows that, for elastic peeling, the peel force is

a direct mesure of the fracture energy G .

However, if severe plastic deformation occurs in the flexible strip during peeling,

the above energy balance is not valid anymore and we have to take into accowt the

plastic dissipation in the peeling adherend. Assuming again that the change of the elastic

strain energy d W e due to the extension of the adherend is small, we can write the energy

balance as:

dW' = !'dl(]-cos()=~bdl +dwl ' , (2.4)

where W' is the total applied energy per unit area, P is the steady state force and W " is

the plastic deformation energy. The above equation c m also be written as:

1 d ~ " where - - is the work expenditure per unit width of the adherend per unit advance of

b dl

the interfacial crack. W" in a peel test, is due io plastic deformation in the bending of the

flexible adherend. Equation (2.5) makes it clear that the experimental determination of

the peel force is not enough for the calculation of the interfacial fracture energy; one

needs. in addition, to calculate the plastic work consumed in the flexible strip during

elastic-plastic peeling. Chang et al. [14] considered the energy balance of end loaded

cantilever beams and provided an approximate method used for caiculating w " . The

systematic study for the calculation of W" was presented by Kim and Aravas 1481,

Williams 1861, Kincloch [54] and Moidu et al. [59,60].

2.2 Peel tests

2.2.1 Peel specimens

The following peel tests were conducted by Wang [8 11 as part of his M.A.Sc. thesis.

The peel specimens used for the current research are shown schematically in Figure

2.2. AA5754-0 (from Alcan International Ltd.) and AA6061-T6 (from Alcan

International Ltd.) were used for the flexible and ngid adherend, respectively. The

adhesive used was a one-part. heat-curing, rubber-toughened epoxy. Betamate 1044-3

(from Essex Specialty Products Inc). Betamate 1044-3 is designed for bonding

automotive aluminum structures.

The substnite had dimensions of 300 mrnx 20 mmx 12.7 mm. The peel arms had a

length of 410 mm, width of 20 mm, and thickness of 1 mm. 2 mm and 3 mm. The

adhesive thickness was fixed at 0.4 mm.

Extension length

tt T '

Figure 2.2 Schematic representation of the peel specimen

Prior to bonding, the aluminum adherends were degreased with acetone and then

were subjected to a Henkel pretreatment procedure, which consisted of a two-part

process. The Alumiprep 33 was used to clean and brighten the surface. and the Alodine

5200 was used to produce a titanium based conversion coating. Teflon spacers of

thickness 0.4 mm were laid along the edges of the bond-line in order to maintain a

uniform bond-line thickness. The final specimens were cured in a preheated oven at

1 70°C for 2 hours and then cooled to room temperature in this oven with door closed.

2,2,2 Peel test results

The peel experiments were performed in a specially designed peel test load jig

illustrated in Figure 2.3.

Crosshead

Load Cell ç7

Pee 1 specimen /

Trolley

I Test Machine l

Figure 2.3 Illustration of the peel test load jig

The peel specimens were bolted into the load jig attached to the base of the load

h e using a linear-bearing trolley, which allowed free movement as indicated in the

figure and hence enabled the jig to align itself during testing, thus maintaining a constant

peel angle. Different peel angles were obtained by adjusting the position of the specimen

in the load jig.

Peel tests at angles of 30°, 60'. and 90' were camed out. Crosshead speeds were

set to 5 mdmin. By peeling a specimen along only part of its length, it could be used for

a number of tests. On average, 4 measurements were conducted to evaluate the steady-

state peel force for each particular peel configuration, Le., thickness of the peel adherend

and the peel angle. Altogether there were 36 measurements were obtained out of 18 peel

specimens. A sumrnary of the experimental results is show in Table 2.1. The locus of

fai lure was visuall y assessed as cohesive in the adhesive layer, however extremel y close

to the adhesivehp adherend interface. From Table 2.1, it can be clearly seen that the peel

strengths increase with decreasing peel angles for the specimens with the same peel arm

thickness. The peel strengths of the 30" peel tests are almost 5 times higher than those of

the 90' peel tests. Furthemore, it cm be observed that increasing the peel a m thickness

increases the experimental peel loads

As discussed in Chapter 1, for the peel situation, it is believed that G,. will be a

funciion of the phase angle and the root curvature, the latter representing the degree of

stress concentration at the peel front. Consequently, it was worth checking the ability of

the finite element mode1 to predict the root curvature. Durhg the steady-state peeling for

each peel configuration, three photographs of the peel front were taken using a Kodak

120 digital carnera. The root curvature during the steady-state peeling for al1 of the peel

cases was then calculated based on these three picture files fiom the carnera. Table 2.1

lists the exprimental results of the root curvature.

2.3 Developmen t of a non-linear large displacemen t, steady-state peel

Table 2.1 Peel test resiilts

finite elemen t model

Peel arm thickuess

(mm)

1

2.3.1 Material property tests

The establishment of the peel finite element model requires information on the

elastic-plastic behavior of the materials involved in the peel tests. This was obtained by

performing uniaxial tensile tests. Details of the testing for the two types of aluminum

alloy and the adhesive are given below.

Peel

30"

60"

2

3

Experimental peel load +, Standard Deviation

(N/mm)

35,Sf 0.1

12.1 I 0.3

Root curvature f Standard Deviation

(l/m m)

0.070 k 0.005

0,082 f 0.007

90"

30"

60"

90"

30"

60"

90"

6.82 _+ 0.4

50.7 2.4

16.2 * 2.0

8.43 ' 0.8

68.4 f 1.8

20.8 I 1.7

12.2 f 2.0

0.093 + 0.009

0.022 f 0.003

0,034 f 0.004

0.035 +, 0.005

0.014I 0.001

0.017k 0.001

0.023 k 0.003

Uniaria1 tensife tests ofAAS7JI-O and AA6061- T6

Specimens for both types of alurninum alloy, AA5754-0 and AA606LT6, were

machined according to the Amencan Society of Testing Materials standard specification

(ASTM D557M-94). These were then tested under quasi-static conditions in an Instron

mode1 4400 universal testing machine with a loading ce11 of 5 W. The specimen

extension was measured using the Instron series 2630 strain gauge extensometer with a

gauge length of 25 mm. The test speed was 1 mm/min, which was also chosen according

to the ASTM D557M-94. The unimial stress-strain cwves of AA5754-0 and AA6061-

T6. obtained from the tensile tests, are shown in Figures 2.4 and 2.5, respectively. The

mechanical properties of these two aluminum alloys are summarized in Table 2.2. The

published data (ASTM B29M-95) for these two aluminum alloys are also Iisted in Table

2.3 for the sake of cornparison. It can be seen from Tables 2.2 and 2.3 that the measured

data agree well with the published ones.

Table 2.2 Mec hanical properties of AA5754-0 and AM061 -T6 (Experimental data)

Ultimate Poisson's Strength Ratio a. ( M W v

Ma terial Young's Modulus

0.2% Y ielding Shrngîh

Table 2.3 Mechanical properties of AA5 754-0 and AA6O6 1 -T6 [93.94]

Strain

Material

Figure 2.4 The uniaxial stress-strain curve for the top adherend material (AA5754-0)

Young's Modulus E (CPa)

0.2% Y ieldiog Streogth OU ( M W

Ultimate Stnngth m. ( M W

Poisson's Ratio

v

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Strain

Figure 2.5 The uniaxial stress-strain curve for the bottom adherend material (AA6061-T6)

Uniaxid tensile test of the Be-te 1044-3 bufk specimen

In addition to the mechanical properties of the adherend materiais. a knowledge of

the basic engineering properties of the adhesive is also necessary for analyzing the

stresses in the adhesive joint. Typically, the main properties required are the tensile, or

Young's modulus ( E , ) , the shear modulus (G, ) . the yielding stresses. the ultimate

stresses and the ultimate strains in tension and in pure shear. The preparation of bulk

specimens in order to measure the mechanicd properties of the adhesive has been an

approach adopted by many workers [3, 201. Following this approach, the tensile stress-

strain pmperties of Betarnate 1044-3 were measured by testing specimens that had been

cast and machined into "ciurnb-bell" shaped test pieces according to ASTM D638-99. The

parallel gauge length was carefùlly polished to avoid the premature failure due to sudace

scratches. The tests were then c h e d out in an Instron mode1 4400 universal testing

machine with a maximum capacity of Sm. An Instron series 2630 strain gauge

extensometer with a gauge Iength of 25 mm, was mounted ont0 the specimen to rneasure

the longitudinal displacement. It should be noted that the test specimen should be held in

such a way that slippage relative to the grips was prevented insofar as possible. In an

effort to overcome this problem, the grips with fine serration surfaces were chosen. In

addition, double-sided abrasive paper was also utilized to put between the tails of the

specimen and the grip surfaces, which has been proved to be another effective technique

to eliminate slippage. Both the loads and displacements were measured by a computer

acquisition system associated with this testing machine. Three bulk adhesive samples

were prepared to perfonn the tensile tests. nie mechanical properties obtained from these

three tests agree quite well with one another except the ultimate strain, the values of

which for two samples are much smaller than that of the third because of the existence of

bubbles in the fracture surface of these two samples. The stress-strain curve of Betamate

1044-3 for the third sarnple is shown in Figure 2.6 and the relevant material properties are

summarized in Table 2.4.

0.00 0.01 0.02 0.03 0.04

Strain

Figure 2.6 The uniaxial stress-strain curve for Betarnate 1 044-3

Table 2.4 Mechanical properties of Betamate 1044-3

It should be noted that the yielding strength for polymers, such as the adhesive used

for the current research, is usually detennined based on the 0.2% offset standard.

Young's modulus is obtained through linear regression of the linear segment in the stress-

strain curve.

One point to consider conceming the approach to determine the engineering

properties of the adhesive is whether the adhesive is the "same material" in bu1 k form as

when present as a thin adhesive layer between substrates. For example. the presence of

the substrates might alter the kinetics of any chernical reaction by which the adhesive

hardens, either by changing the local temperature or by removing the curing agent or

filler by preferential absorption [52]. Also, residual stresses are more likely to be present

in the adhesive when it is cured between substrates. These factors might obviously

influence the mechanical properties of the bulk adhesive compared to the in siru cured

adhesive. Little work has been reported on this topic, but the study by Post (651, using a

high-sensitivity Moire interferometry, has indicated that the strain distribution across the

adhesive layer in the thick-adherend lap-shear joints was vinually uniforrn. which

suggested that the adhesive's modulus was uniform across the thickness of the adhesive

layer. These aspects are undoubtedly worthy of further investigation. In this thesis, the

tensile properties obtained from the uniaxial tensile test of the adhesive bulk specimen

Young's Modulus E (GPa)

0.2./0 Y ielding S trengt b

O,, ( M W

Ultima te S treagt h 0, ( M W

Ultimate Strain

E u

were used to model the non-linear material behavior of the adhesive in the subsequent

finite element analysis of the peel test.

23.2 Development of the steady state peel finite element model

Characteristics of the steady stute peel test

An adhesive bond is inherently weak when it is subjected to peel loads. For this

reason. the peel test has been widely used to compare the perfomance of adhesion under

this type of loading. For a typical peel test, one of the important features is the geometric

non-linearity, i.e.. the large displacement experienced by the tlexible adherend when it is

pulled away from the rigid substrate. Furthemore. it has k e n generally realized that the

non-linear material behavior of both the top adherend and the adhesive should be taken

into account in order to accurately predict the stress distribution along the bond line and

give a sound prediction of the failure strength. This will further increase the non-lineanty

of the peel test. Another characteristic that needs to be considered when performing peel

analysis is the failure mode. Gledhill et al. [33] have tried to apply fracture mechanics

principles to the analysis of bi-material systems by considering cohesive fracture of the

adhesive. Other researchers, such as Anderson d al. 141 investigated interfacial failure of

bi-material systems using the concept of the interfacial hcture energy. As the locus of

fa i lw in the peel test is either interfacial between the adhesive and the flexible adherend

or cohesive, extremely close to the flexible adherend, the interfacial failure mode was

assumed in the finite element modeling of this study. This is also in agreement with the

visual observation of the peel experiments performed For the current research, where the

crack is propagating along a path that is quite close to the upper interface. Analysis for

interfacial failure is similar to that for cohesive fracture. This facture can be considered in

two parts [21]: the first, called mode 1 in fracture mechanics terminology, involves tensile

fracture and the second, mode II, involves shear fracture. Practical situations. such as peel

tests, usually consist of a combination of both modes.

All of the above characteristics associated with the peel test need to be accounted for

when developing the steady state peel finite element model. This will be further discussed

in the following sections.

The application of the non-lhear finite eieme~t method in the peel

analysis

A bief introduction of the non-lincar finire element method

It cannot be overemphasized that the finite element method is an approximate

method for solving differential equations, and the way an object is modeled, such as the

choice of the element or the representation of the loading and constraint conditions. is

vital. Furthemore, considerable engineering judgment may be required to analyze the

very detailed information that a tinite element model can produce. Consequently.

although sophisticated commercial finite element packages are available. the theory of

finite elements and the practical (modeling and analysis) aspects of the technique must be

clearly understood so as to find out whether the developed model is valid and accurate. It

is worthwhile, therefore, to discuss the basic theories of finite element analysis,

especially non-linear finite element analysis.

Within the field of adhesive technology, the finite element rnethod has k e n widely

employed to analyze a large number of adhesive joints including double lap shear, single

lap shear, scarf, peel, cracked lap shear, edge notched flexure, tubular lap, butt and many

others. As in many other numericd methods, solution is obtained by assuming an

arbitrary form of trial fùnction. The trial function consists of a nurnber of known

parameters that are selected to make the fûnction a best fit to the true solution. What sets

the finite element method apart fiom the other numerical methods is that the region of

interest is discretized into a number of small finite-sized elements. The trial function is

then defined in a piecewise marner over each of these elements sepmtely, with certain

continuity requirements enforced between trial funciions at element boundhes. In the

case of stress analysis, the unknown parameters which make up the trial function are

structural displacements defined at points called nodes. Having found the displacements,

the strains and hence the stresses cm easily be reconstituted. The following is a brief

symbolic synopsis of a linear finite element analysis. The stress vector {a} and the strain

vector { E } at an arbitrary point in an element can be expressed in tems of the nodal

displacement vector {LI ] as:

where [B] is the strain-displacement matrix, [D] is the material property matrix.

Virtual work principles are ofien used to find expressions for intemal nodal forces

that are equivalent to the stress distribution and these forces in tum are summed and

equated to the extemal nodal force vector (F} to establish the finite element equations

which involve a stiffhess matrix [ K I :

( F I = [KI(uL where

For the linear finite element analysis, the stiffiess matrix is simply a function of the

geometry of the structure and the material properties and can be calculated readily.

However, when it cornes to non-linear problems. the stifiess rnatrix [KI andfor the force

vector { F } are functions of the nodal displacement vector {u). This may occur as a

result of non-linearities in the material properties, the geometry, combined effects of the

material and geornetric non-linearities (this is the case for the peel test) or the contact

conditions of the problem. What follows is the application of the non-linear FEM in the

peel analysis.

Geometric non-Iineariiy analysis

Geometric non-Iinearities refer to the non-linearities of the structure or component

due to the changing geometry as it deflects. For a typical peel test, a large-displacement

analysis is required because the flexible adherend's deflection becomes so large that the

original matrix no longer adequatel y represents the structure. The large displacement

theory assumes that the rotations are large but the mechanical strains (those that cause

stresses) are small. The structure is assumed not to change shape except for ngid body

motions. In large-displacement problems due to applied loads, since the geometry

changes. its stiffness matrix needs to be adjusted accordingly. There are two ways in

which this cm be achieved. The first approximate method assumes that the size of the

individual elements is constant, so that a reorientation of the elemental stiffness matrix

due to the elements' rotation andor translation is al1 that is required. The second method

is more accurate and recalculates the stiffness matrix of the elements after adjusting the

nodal coordinates with the calculated displacements. The latter is the method chosen for

the analysis of the current research. In both cases, an incremental solution is perfonned,

usually by the Newton-Raphson method. If the spatial motions are noi large, then it is

possible to apply the load in a single step with several iterations, but for large deflection

such as the peel case, the extemal load must be applied in as small load step as possible

in order to achieve convergence.

Materid non-llneariw cinalysits

As mentioned in previous sections, the material non-linearity is another important

feature that needs to be considered when developing the steady-state peel finite element

model. This section presents a brief analysis on the material non-linearity of the peei test.

Material non-linearities are due to the non-linear relationship between the stress and

the strain. that is, the stress is a non-linear function of the strain. The relationship is also

path dependent. so that the stress depends on the strain history as well as the strain itself.

ANSYS, which is the finite element program used for this study, can account for 8 types

of material non-linemities: rate-independent plasticity, rate-dependent plasticity. creep.

non-linear elasticity, hyperelasticity, concrete and swelling. In this research. the option of

the rate-independent plasticity, which is characterized by the irreversible straining that

occurs in a materiai once a certain level of stress is reached, is chosen to model the non-

Iinear material behavior of the adhesive and adherend. Plasticity theory provides a

mathematical relationship that characterizes the elasto-plastic response of materials.

There are three ingredients in the rate-independent plasticity theory in the ANSYS

program: yield criterion, flow le and hardening nile. These are discussed below.

Yield criterion

The yield criterion determines the stress level at which yielding is initiated. For

multi-component stresses. this is represented as a function of the individual components,

/'({a)), which can be interpreted as an equivalent stress a, :

where {a} is the stress vector.

When the equivalent stress is equal to a material yield parameter O, :

the material will develop plastic strains. If a, is less than a,, the material is elastic and

the stresses will develop according to the elastic stress-strain relations. Note that the

equivalent stress can never exceed the material yield since in this case plastic strains

would develop instantaneousl y, thereby rcducing the stress to the mate rial yield. In this

thesis. the Von-Mises yield function has been employed to mode1 the yielding of the

adherend and the adhesive,

Flow rule

The flow rule determines the amount and direction of plastic straining and is given as:

where A is the plastic multiplier which determines the arnount of plastic straining, Q is

the function of stress termed the plastic potentiai and determines the direction of plastic

straining. If Q is the yield function (as is normally assumed), the flow nile is termed

associative and the plastic strains occur in a direction normal to the yield surface.

The hardening rule describes the change of the yield surface with progressive

yielding, so that the conditions (i.e., stress states) for subsequent yielding can be

established. Two hardening rules are available: work (isotropic) hardening and kinematic

hardening. In the work hardening, the yield surface remains centered about its initial

centerline and expands in size as the plastic strains develop. For materials with isotropic

plastic behavior this is termed isotropic hardening. The kinematic hardening assumes that

the yield surface remains constant in size as the surface translates in stress space with

progressive yielding. In the current research, the multilinear isotropic hardening was used

to mode1 the plasticity behavior of the adherend and adhesive material.

Characteristics of the steudy stage peeljinite eCcmcnt rnodel

The geometry and dimensions of the finite element model were in agreement with

the configuration s h o w in Figure 2.2. The stress state was assumed as plane strain for

both the adhesive layer and the adherend. The yielding of the adhesive and adherend was

modeled using the Von-Mises yield fùnction based on the uniaxial stress-strain curve

presented in Section 2.3.1. The finite element meshes used for the analysis are shown in

Figure 2.8, where the adhesive layer and the top adherend were divided to 4 layers and 5

layers through the thickness, respectively. Along the bond-line direction (x). unifonn

meshes with element width of 0.1 mm were taken. 4-node quadrilaterai isoparametric

elements. which are good general purpose elements and have been used successfully by

many workers, were employed to mesh the whole model. Another significant featuw of

the element selected for the current research is that it supports such behavior as large

displacement and material non-linearity. The flexible adherend extension lengths were

chosen according to the rule that the full development of the plastic region of deformation

in the fiee adherend adjacent to the bonded region should be able to be achieved. The

degrees of freedom of the bottom nodes in the substrate were constrained in al1

directions.

Owing to the incremental nature of the equations goveming the geometric and

material non-linearity, especially for the peel situation that is a combination of the two

cases, it is necessary to approach a full load solution in a series of small load steps.

Within each load step, the analysis was done incrementally and the modified Newton-

Raphson rnethod was utilized to solve the equilibRum equations. It should be mentioned

that, through the author's experience of running this steady state peel finite elernent

model, the traditional Newton-Raphson method cannot be used alone since the tangent

stiffness matrix rnay become singular, thus causing severe convergence problems. A

number of convergence-enhancement and recovery features such as line search.

automatic load step and bisection, were activated to help the solution converge.

Moreover, an alternative iteration scheme, the arc-length method, was used to help

stabilize the solution. The arc-length method causes the Newton-Raphson equilibrium

iterations to converge along an arc, thereby oflen preventing divergence even when the

dope of the tangent stiffness matrîx becomes zero or negative. The ANSYS finite

element software was used for the caiculations. Non-linear, large displacement finite

element programs, which enable the analysis of the steady state peel test, were developed

using the ANSYS Parametric Design Language (APDL). The program for the 1 mm, 90'

peel mode 1 was included in Appendix A. Because of the demanding non-linear features

of the peel situation, a considerable amount of computing time was required.

Critical Von-Mises stni in fuiiure criterion

Deformation ~lasticitv theorv

Before proceeding to present the critical Von-Mises strain failw criterion used in

this research, it is worthwhile giving a brief introduction of the deformation plasticity

theory.

Total strains during yielding c m be split into elastic and plastic cornponents as

indicated in the following expression:

E = &', +&,, . (2.1 3)

The elastic components of the stain E, cm be obtained from the stresses using the

generalized Hooke's law for elasticity:

where the elastic constants E and G are defined in the usual mmer.

Assuming that the plastic components of strain, E,, , cm be obtained from the

equations analogous to the generalized Hooke's law while relating the stresses and the

plastic strains:

- -- 7 p.~ - 'xy, YpJz = r y z t YF EP E P EP

Comparing these to the generalized Hooke's law (Eqn. 1.14-2.1 7). the elastic

modulus E is replaced by a plastic modulus E,, which is defined as:

-

where 5 and Et, represent the etrective or Von-Mises stress and the effective plastic

strain, respectively. Graphically. E,, corresponds to a secant modulus drawn to a point on

the a vs. Zl, curve as shown in Figure 2.7. Hence, E, is a variable that decreases as

plastic deformation progresses along the vs. F,. curve for the material.

Figure 2.7 Definition of the plastic modulus as the secant modulus to a point on the effective stress versus effective strain curve

Poisson's ratio in Hooke's law is replaced by 0.5 in Eqn. 2.18-2.21, which is

equivalent to the assumption that plastic strains do not contribute to volume change.

Further cornparhg Eqn. 2.14-2.17 and 2.18-2.2 1, the elastic shear modulus is replaced by

E,, 13 . The equations given above constitute stress-strain relationships that c m be used

beyond the point of yielding. Although the elastic and plastic strains are treated

separately, Eqn. 2.14-2.17 and 2.18-2.2 1 can be substituted into Eqn. 2.13 to obtain the

relationships between the effective stress and the effective total strain as follows:

where E is defined as the effective total strain or Von-Mises strain, 5 is Von-Mises

stress, 3,. is the effective plastic strain which can be derived from the following equation:

- 47 &, =- fi,,, - E p l y +(cl>? - g P J ) ? +(&pl - ~ p l ) l 9

3

where E,, . spi. E,, are plastic strains in the principal directions.

An important use of the above equations is in predicting the effects of state of stress

on stress-stmin curves. To do so for a particular material. it is necessary to have the

stress-strain curve for one state of stress, such as the uniaxial one which has been verified

to be the sarne as the curve relating the Von-Mises stress-strain [26] and has k e n

discussed in previous sections. The key feature of the de formation plasticity theory is its

prediction that a single curve relates a and B in al1 States of stress.

Critical Von-Mises strain failure critenon

The failure criterion used in the curent research is based on the approach of Vincent

[78], who suggested a failure critenon following an earlier idea by McClintock and Irwin

[ B I . The failure was assumed to occur when the strain at a certain distance fiom the

crack tip reached a cntical level. Subsequently, Crocombe and Adams [20] assessed the

peel strengths employing an effective plastic strain failure criterion, which was

determined at the Gauss point closest to the crack tip. The main reason that failure is not

determined at the crack tip is that the singularity of stress and strain at that point makes it

quite dificult to calculate the corresponding values. This approach is panicularly

appealing to the finite element analysis that can obtain values of strains at discrete points

in the material sumunding the crack tip. Which distance tu use in the analysis would

necessitate a separate study and so, as an attempt to use this type of failure criterion, the

strain value was determined at the third interfacial adhesive node from the peel front. The

term of the "peel front" is used here rather than the "crack tip" in order to indicate that no

sharp crack was introduced into the model. The Von-Mises strain discussed in earlier

sections was employed in this study as a failure criterion. Failure was assumed to occur

when the value of the Von-Mises strain at the third interfacial node reached a critical

value. In the meantirne, the interfacial adhesive elrment at the crack tip was "killed"

utilizing a function provided by the ANSYS finite element code in order to make the

crack propagate and the failure criterion was determined at the next corresponding third

interfacial node. AAer a substantial amount of crack growth, the peel m will reach a

geometrically stable state and the peel load will reach the plateau, a value that was used

to compare with the experimental peel load.

Meshhg effect

The meshing effect is always an interesting issue when analyzing the peel situation

using the finite element method. This issue was also investigated in this thesis by

comparing two peel steady state models consisting of different layers of adhesive

elements. These two models are shown in Figures 2.8 and 2.9, respectively and they have

the sarne dimensions: the peel a m thickness of 1 mm. the adhesive thickness of 0.4 mm

and the bottom adherend thickness of 12.7 mm. In Figure 2.8, the adhesive was meshed

as 4 layers dong the thickness direction and a double meshing density (i.e., 8 layers) was

chosen for the rnodel shown in Figure 2.9. The rneshes for the top and bottom adherends

were the same for both models and the peel angle was chosen as 90'. The value of the

critical Von-Mises strain. 1.07%. which was found to be able to predict the experimental

peel load - 6.8 N/mm for the 1 mm, 90' peel test using the Clayer model. was then used

as a failure cntenon in the 8-layer model to predict the peel strength. Four parameten

were chosen for the cornparison in order to check the meshing dependence: the peel

strength, the root cuwature. the crack propagation length and the interfacial Von-Mises

stress distribution. The former three parameters are listed in Table 2.5 and the last

parameter is shown in Figure 2.10.

Table 2.5 Cornparisons of the 4-layer model and the 8-layer model

Adhesive layers

4

8

Peel stnngth (N/mm)

6.8

6.82

Root cuwature (ifmrn)

0.093

0.094

Crack propagation length (mm)

4.9

4.9

Figure 2.8 The peel finite element model with 4 layen of adhesive elements

1 Top adherendL

Figure 2.9 The peel finite element model with 8 layea of adhesive elements

1

4.00 3.00 2.00 1 .O0 0.00

Distance from the peel front (mm)

Figure 2.10 The interfacial adhesive Von-Mises stress cornparison of the 4-layer model and the 8-layer model

From Table 2.5, it can be obviously seen that the prediction results of the peel

strength, the root curvature and the crack propagation length from both models are

extremely close. Figure 2.10 also shows that the Von-Mises stress distribution of the

adhesive nodes along the upper interface coincide with each other. Therefore, it can be

safely concluded that the 4-layer meshing of the adhesive is accurate enough from the

viewpoint of prediction. Increasing the meshing density will not influence the prediction

results, but will require more computation effort, so that the Clayer model was employed

for the following numerical simulations of different peel configurations.

2.4 Numerical simulations and comparisoas with experimental results

2.4.1 The approach used in the numerical simulations

The strategy of the numerical simulation employed in the curent analysis is that, for

each peel arm thickness, the critical Von-Mises strain was calibrated based on the 90'

peel case and then used to predict the 30' and 60' cases for the same peel arm thickness.

It should be mentioned that when the critical Von-Mises strain obtained from one peel

ami thickness was used to predict the peel tests with different peel arm thickness, the

predicted peel strengths were found to be much lower than the experimental results. This

will be discussed in more detail later.

2.4.2 Cornparisons with experimental results

Peel strength cornparisons

The measured peel strengths and the corresponding numerical simulation results

obtained from the steady state peel finite element models are summarized in Table 2.6. It

c m be observed that. for the peel configurations with the same peel am thickness. the

critical Von-Mises strains calibrated based on the 90' peel give a good prediction of peel

loads for the other two peel angles. The percent prediction errors are within or close io

10%. Therefore, it can be argued that there exists a characteristic critical Von-Mises

strain, which is independent of the peel angles for peel tests with the same peel am

thickness. However, when the peel arrn thickness increases from 1 mm to 3 mm. the

critical Von-Mises strain has to be increased accordingly fiom 1.07% to 1.8% in order to

simulate the experimental peel loads. Hence, another conclusion which can be drawn here

is that the critical Von-Mises strain failure criterion is dependent on the peel ami

thickness. The adhesive appears to be effectively stronger as the adherend becomes

thicker. A similar phenornena was reported in the work of Crocombe and Aravas [20],

who used a failure criterion based on the effective plastic strain to predict the relative

strengths of two common peel tests consisting of top adherends with different yield

strengths. They concluded that the effective plastic strain failure criterion could be used

successfully to predict the relative strengths of the same peel test, Le., this failure

criterion is independent of peel angles. However, values of the adhesive effective plastic

strain from the peel test with low yield strength adherend are considerably lower than

those from the peel test with high yield strength adherend. Here, the peel am material's

yield strength is a dependent parameter that is analogous to the peel a m thickness iactor

obsewed in our analyses. The dependence of the critical Von-Mises strain failure

criterion on the peel a m thickness may be related to hydrostatic stress effects on

adhesive yield and hence to varying degrees of adhesive constraint in the vicinity of the

peel front. This will be discussed Further in Section 2.7.

Root curvature comp~isons

It has been mentioned in earlier sections that photographs of the peel front were

taken using a Kodak 120 digital camera during the peel tests as illustrated in Figure 2.1 1.

Consequently, by aid of Windig, a data sampling and digitalization software. root

curvature during steady-state peeling could be calculated based on these picture files. The

two scales shown in Figure 2.1 1 formed a reference system which was used for

calibration between the pixels, a measurement unit in Windig, and the real distance.

Table 2.6 FEA prediction for peel loads based on the cntical Von-Mises strain failure cri terion

Peel arm thickaess

(mm)

1

7 -

3

Peel angle

30"

60"

90"

30"

60'

90"

30"

60"

90'

Critical Von-Mises Strain

(Calibrated based on 90" case)

t .07%

1.07%

1.07%

1.4%

1.4%

1.4Yo

1.8%

1.8%

1.8%

FEA predicted peel loads @/mm)

30.8

11.9

6.80

46.0

17.1

8.40

60.4

19.3

12.2

Expt. peel load * Standard Deviation (N/mm)

35.2 10.1

12.1 * 0.3

6.82 * 0.4

50.7 * 2.4

16.2 * 2.0

8.43 * 0.8

68.4 1 1.8

20.8 + 1.7

12.2 * 2.0

FE A prediction

Error

1 2.5%

1.67%

O

8.00%

5.62%

O

1 1.7%

7.2 1 %

O

Table 2.7 lists measured results of root curvature and the predicted values from finite

element models. It can be seen that the finite element prediction enors for al1 of the peel

cases are within or close to 5%. This indicates that the numerical prediction of root

curvature based on the critical Von-Mises strain failure criterion is even better than that

of the peel load. In other words, this failure critenon can accurately predict the degree of

stress concentration for the peel tests with the same peel arm thickness.

Figure 2.1 1 Photograph of the peel test with 2 mm peel a m thickness and 60" peel angle during steady state

Table 2.7 FEA prediction of root curvatw based on the critical Von-Mises strain failure cri terion

Peel arm thickaess (mm)

2.5 Cornparisons of the initiation state and the steady state

In Chapter 1, the significance of canying out the steady state peel finite element

analysis rather than the initiation one has ken discussed. Here, more details will be given

to verify the previous argument. For the convenience of discussion, only one peel

configuration was chosen: the peel ami thickness is 1 mm and the peel angle is 90". The

initiation state is defined as the state when the failure criterion is first reached, however

no element is broken yet. Figure 2.12 and 2.13 show the macroscopic deformed shapes

Expt. Root curvature f Standard Deviation Wmm)

FEA prediction

Error

Peel angle

Critical Von-Mises Strain

(Calibrnted bas& on 90" case)

FEA predicted

r00t curvature Wmm)

corresponding to the two states, i.e., the initiation state and the steady state. In both cases,

the load applied to the initially straight and horizontal peel strip was in the vertical

direction.

Figure 2.1 2 The predic ted macroscopic de fonned shape of the 1 mm, 90" peel test at the initiation state

Figure 2.13 The predicted macroscopic deformed shape of the 1 mm, 90" peel test at the steady state

Cornparhg Figures 2.12 and 2.13. it is obvious that the peel ami has been

straightened up during steady state and the root curvature is much bigger in the initiation

state. The root curvature is a representation of the degree of stress concentration; the

bigger the root curvatwe, the higher the degree of stress concentration, hence, in the

steady state situation, the stress in the root region is more concentrated Uian in the

initiation state. To hirther investigate this point, enlarged plots of the root regions for

both states are s h o w in Figure 2.14 and 2.15, where it can be observed that the stress

concentration zone of the peel front during the steady state is much smaller than in the

initiation state. Furthemore, the steady state peel load, i.e., the peel strength, is much

higher than that of the initiation state. Taking the 1 mm, 90' peel as an example. the load

ratio of the two states is close to 10. This can be easiiy understood when considering the

continuous energy contribution made by the increasing peel load in order to defonn the

peel am until the whole system reaches a stendy state.

Figure 2.14 The predicted nodal Von-Mises stress distribution of the 1 mm, 90" peel test at the initiation state

Stress concentration zone

Figure 2.15 The predicted nodal Von-Mises stress distribution of the 1 mm. 90" peel ai the steady state

2.6 Investigation of the mode ratio

The peel test may be considered as a mixed-mode fracture of some combination of

mode 1 (opening mode) and mode I I (shearing mode). Therefore, it is of interest to know

the mode ratio during steady state peel. Since there is no esiablished definition for the

mode ratio under such conditions of adherend yielding, two tentative defmitions were

used in this study: a stress-based definition and a strain-based definition:

where p is defined as the phase angle which characterizes the mode ratio, a, and r,,

are the normal stress and shear stress for the interfacial nodes of the adhesive,

respectively, E, and y, are the nomal and shear strain of the interfacial nodes of the

adhesive, respectively. For both definitions, three types of phase angles were studied: the

average phase angle based on the adhesive plastic zone, the average phase angle based on

the adhesive tensile zone and the local phase angle based on the interfacial adhesive node

at the peel front. The definitions of the adhesive plastic zone and the adhesive tensile

zone are the sarne for both crises.

2.6.1 The stress-based definition

Average phase angle based on the adhesive plastic zone

The adhesive plastic zone was determined according to the Von-Mises stress

distribution of the interfacial adhesive nodes by using the adhesive yielding stress as a

criterion. Figure 2.10 shows a typical adhesive Von-Mises stress distribution dong the

upper interface and the corresponding plastic zone lengths are listed in Table 2.8. It can

be seen that the plastic zone lengths of 3 mm peel tests are almost twice those of 1 mm

peel tests, and the sizes in 30' peel tests are slightly greater than those in 60' and 90"

peel tests for peel configurations with the same peel adherend thickness. In al1 cases. the

plastic zones are smaller than the thickness of the adhesive layer (0.4 mm). Phase angles

predicted by Eqn. 2.25 are plotted in Figures 2.16 and 2.17 as funciions of the peel ami

thickness ami the peel angle, respectively. It is interesting to note that the phase angle is

independent of the peel angle and the peel arm thickness and relatively small phase angle

is predicted, Le., mode 1 dominant fracture is predicted.

Table 2.8 The plastic zone length determined based on the adhesive interfacial Von- Mises stress distribution

1 Peel m n tbieknns (mm) 1 Peel angle

2

Ped arm thickness (mm)

Figure 2.16 Average phase angle over the plastic zone as a function of the peel arm thickness (stress-based definition)

O 1 mm peel O 2mmpeel

60

Peel angk (degree)

A 3 mm peel 30

25

Figure 2.17 Average phase angle over the plastic zone as a function of the peel angle (stress- based de finition)

20

15

Average phase angle based on l e adhesive tende zone

The adhesive tensile zone is defined as a zone where the adhesive normal stress

along the upper interface is positive. Figure 2.18 shows the variation of the adhesive

normal stress (a,.) with the distance fiom the peel front of the peel configuration with 1

-

-

mm peel am thickness and 90' peel angle. The stresses were obtained from the

interfacial adhesive nodes. Similar distributions were also found for other peel tests

analyzed. It can be seen that the shape of the stress distribution is a darnped, harmonic

function, which is similar to that in the elastic peel analysis. The oscillation exhibited by

the stress distribution in Figure 2.18 was also observed by Crocombe and Aravas (201 in

elastic-plastic peel finite element analysis. From their study, the oscillation was found to

be more pronounced in the higher stressed regions, at high peel angles and with the soft

aluminwn as the peel m. The effects of the peel angle and the adherend thickness were

also investigated here by looking into the tensile zone length discussed above. Table 2.9

lists the tensile zone length corresponding to the various peel angles and peel atm

thicknesses.

Distance from the peei front (mn)

Figure 2.18 Variation of the adhesive normal stress with distance from the peel Front for the 1 mm, 90" peel test

Table 2.9 The influence of the peel angle and the peel a m thickness

on the tensile zone length

1 Peel arm thiehess (mm) 1 Peel angle / Teosik zone Ggth 4

From Table 2.9, one cm find that the tensile zone length. which represents the

sharpness of the oscillation, decreases with an increase in the peel angle or a decrease in

the peel arm thickness. It indicates that the oscillation is more abrupt at higher peel angles

and smaller peel a m thickness, which is in agreement with the conclusion of Crocombe

and Aravas [20].

Figures 2.19 and 2.20 plot the average phase angles over the tensile zone predicted

by Eqn. 2.25 as functions of peel angte and peet arm thickness. It cm be seen that the

increase of the average phase angle is within 5 degrees as the peel am thickness

increases fiom 1 mm to 3 mm. Thecefore, the average phase angle is basically

independent of the peel ami thickness. However, it can also be observed that the average

phase angle predicted for the 30' peel angle is significantly higher than that for 60' and

90°, which are quite close to each other. In other words, the average phase angle defined

based on the tensile zone is more strongly dependent on the peel angle. This is different

fiom the conclusion drawn for the average phase angle over the plastic zone.

A 900 peel 600Pel l Peel arm thickness (mm)

Figure 2.19 Average phase angle over the tensile zone as a function of the peel a m thickness (stress-based definition)

O lmrn peel

A 3mm peel 10 I

60

Peel angle (degree)

Figure 2.20 Average phase angle over the tensile zone as a function of the peel angle (stress-based definition)

Local phase angle based on the adhesive mode at the peel front

Figures 2.21 and 2.22 show the local phase angle, which was calculated based on the

adhesive node at the peel front, as a function of the peel angle and the peel arm thickness.

Peel arm thickness (mn)

Figure 2.2 1 Local phase angle on the adhesive node at the peel front as a function of the peel ami thickness (stress-based definition)

O Imm peel 2mm peei

A 3mrnpee1

Peel angle (degree)

Figure 2.22 Local phase angle on the adhesive node at the peel front as a function of the peel angle (stress-based definition)

As predicted in Figures 2.16, 2.17, the local phase angles are quite insensitive to the

variation of the peel angle and the peel ami thickness. Again, mode 1 dominant fracture is

predicted according to the local phase angle definition.

2.6.2 The strain-based definition

For the strain-based definition, the same three types of phase angle were

investigated as those for the stress-based definition.

Average phase angle based on the adhesive plastic Gone

The plastic zone was the sarne as that detemincd for the stress-based definition.

Average phase angles predicted by Eqn. 2.26 are plotteci in Figures 2.23 aiid 2.24 as

functions of the peel arm thickness and the peel angle, respectively.

O Wpeel O Wpeel A 900 peel

Peel a m thickness (mm)

Figure 2.23 Average phase angle over the plastic zone as a function of the peel an thickness (strain-based definition)

50 1 O M m peel I

60

Peel angle (degree)

Figure 2.24 Average phase angle over the plastic zone as a function of the peel angle (strain-based detinition)

From Figures 2.23 and 2.24, it is interesting to note that, the average phase angle

over the plastic zone determined according to the strain-based de finit ion is independent

of the peel angle and the peel ami thickness, nonetheless, the mode I I dominant fracture

was predicted, compared with the mode 1 dominant fracture predicted from the stress-

based definition,

Average phase angle based on the adhesive t e d e zone

The tensile zone was the same as that detennined for the stress-based definition.

Figures 2.25 and 2.26 show the average phase angle as a hinction of the peel a m

thickness and the peel angle.

Peel arrn thickness (mn)

Figure 2.25 Average phase angle over the tende zone as a function of the peel arm thickness (strain-based definition)

O Imrn peel O 2mm peel n 3mm peel

Peel angle (degree)

Figure 2.26 Average phase angle over the tende zone as a fùnction of the peel angle (strain-based definition)

From Figures 2.25 and 2.26, one can find that the predicted phase angle over the

tensile zone is independent of the peel angle for the same peel arm thickness.

Nevertheless, there seems to exist peel ami thickness dependence, especially for the 60'

peel case since the predicted average phase angle for 1 mm peel test is significantly

higher than those of the 2 mm and 3 mm peel tests. This is different from the conclusion

drawn for the stress-based definition when investigating the average phase angle over the

tensile zone. where it was found that there existed a peel angle dependence rather than a

peel arm thickness dependence.

Local phase angle bosed on the adhesive node a# the peel front

Figures 2.27 and 2.28 show how the local phase angle varies with the peel angle and

the peel a m thickness. The same trend was found as that show in Figures 2.23 and 2.24.

Peel arm thickness (mm)

Figure 2.27 Local phase angle on the adhesive node at the peel Front as a function of the peel a m thickness (strain-based definition)

70 - o qmm peel I

n O 2mm peel A 3rnm peel

1 I 30 60 90

Peel angle (degree)

Figure 2.28 Local phase angle on the adhesive node ai the peel front as a function of the peel angle (strain-based definition)

2.7 Discussion and conclusions

2.7.1 The peel arm thickness dependence of the critical Von-Mises strain

failu re criterion

As discussed before. the critical Von-Mises strain failure criterion is independent of

the peel angle for peel configurations with the sarne peel am thickness, but it is

dependent on the peel ami thickness. The possible explanation is:

Hydrostatic stress effecf on adhesive yield

In general, different materials require différent yield fùnctions. The Von-Mises yield

function has been pmved to be a good criterion to mode1 the yielding of the aluminum

alloys. However, it has been demonstrated that the yield and facture responses of

polymeric materials, such as adhesives, are sensitive to hydrostatic pressure [20, 53. 551.

A modified Von-Mises criterion can be used to describe the yield behavior of the

adhesive as below:

where is the modified Von-Mises yield stress, a, is the Von-Mises yield stress

obtained from the uniaxial tende test, p is the coefficient of intemal friction, and a, is

the hydrostatic stress. For the present study, the hydrostatic stress obtained at the third

adhesive interfacial node away from the peel front is show in Figure 2.29 as a function

of the peel am thickness.

2

Peel arm thickness (mm)

Figure 2.29 The hydrostatic stress as a fùnction of the peel arm thickness

From Figure 2.29, it can be seen that the hydrostatic stresses are quite close to one

another for the peel tests with the same peel arm thickness while with different peel

angles. However, the hydrostatic stress is dependent on the peel m thickness, Le., the

former increases with an increase in the latter. By substituting the hydrostatic stress into

Eqn. 2.27. it can be easily found that the modified Von-Mises yield stress will be smaller

if the peel am thickness becomes bigger. Consequentl y, the corresponding cri tical Von-

Mises failure strain for peel tests with thicker adherends should decrease close to the

value for peel tests with thinner adherends. This would account for the adherend

thickness dependence observed in Section 2.4.2.

2.7.2 Mode ratio

Several conclusions cm be drawn corresponding to the two tentative definitions for

the mode ratio:

1) The average phase angles over the plastic zone and the local phase angle. obtained

according to the stress-based definition, predict that: the phase angle is independent

of the peel angle and the peel am thickness. Furthemore, the mode 1 dominant

fracture is predicted for the steady state peel according to this definition. However.

the average phase angle over the tensile zone predicts that: the phase angle is still

independent of the peel am thickness, but is dependent on the peel angle with the

value for 30' peel significantly greater than for 60' and 90' peel.

2) The average phase angle over the plastic zone and the local phase angle. obtained

according to the strain-based definition, predict the similar trend as that given by the

equivalent phase angle fiom the stress-based detinition, i.e., the phase angle is

independent of the peel angle and the peel arm thickness. Nevertheless, the average

phase angle over the tensile zone shows that the phase angle is dependent on the peel

arm thickness rather than the peel angle, as predicted by the stress-based definition.

Moreover, the mode II dominant fracture is predicted for al1 of the cases.

Similar phenornena were also found in the work of Crocombe and Aravas [2 11, where

they analyzed the cracked systems of the elastic peel configurations. By using the ratio of

the normal stress and the shear stress as the phase angle definition, it was observed that

the mode II loading at the crack tip was significant and essentially independent of the

peel angle. load and adhesive or adherend modulus. The current study agrees with

Crocombe and Aravas [2 1 ] that the phase angle is independent of the peel angle and the

material property. However, the mode II component predicted fiom the stress-based

definition in this analysis is relatively small which disagrees with the significant mode II

proportion stated by Crocombe and Aravas [2 11. The prediction from the strain-based

definition, which shows a mode Il dominant fracture with a phase angle between 50" and

70°, is cioser to their conclusions than the prediction results fkom the stress-based

de finition.

Chapter 3 Study of DCB Fracture Test Using Cohesive Zone Modeling

3.1 Introduction

3.1.1 Fracture characterizatioa of elastie adhesive joints

One of the main concerns when designing structml adhesive joints is the possibility

of the crack initiating and propagating in the bondline. and this hm led to the wide-spread

use of fracture mechanics to analyze the crack-growth behavior occumng in elastic

adhesive joints. In the study of cracks in btittle homogeneous materials, the main focus

has been on cracks subject to mode 1 loading (opening mode) because there is ample

experimental evidence that a crack subject to a mixed-mode loading will grow by kinking

in a direction such that the crack tip is in pure mode 1. However, a crack in the bondline

of an adhesive joint is constrained by the adherend, and thus, in general. will propagate

under mixed-mode conditions. Among the Iiterature of elastic fracture analysis of

adhesive joints, a large number of studies have made great progress in correlating the

energy release rate, G,. , with the »>-situ fracture of adhesive joints and it was found that

G,. is generally dependent on the mode of loading with G,,. (in-plane shear) typically

higher than G,(. (opening mode). The use of adhesively-bonded specimens for the

characterization of the in-situ fracture toughness of elastic adhesive joints dates back to

the work of Ripling et al. [69], who evaluated the symmetrically loaded double cantilever

beam (DCB) specimen. Because of the simple geometry of the DCB specimen, other

investigators studied the sarne specimen geometry subject to other loading conditions. In

addition to the DCB fraçture test, several other tests have been used for mode 1, mode II

and mixed-mode fracture characterization of adhesive joints and composite laminates,

such as the end-notch-flexure (ENF) test, the cracked-lap-shear (CLS) test. the mixed-

mode- flexure (MMF) test, the tapered-double cantilever beam (TDC B) test, the mixed-

mode-bending (MMB) test, etc. With these fracture tests, the characteriration of mode 1,

mode II or some mixed-mode with a specific phase angle cm be obtained. Nevertheless.

none of these tests allow the fracture testing in the entire mode ratio ranging from pure

mode 1 to pure mode II. In the present study, DCB fracture tests were perfonned using a

mixed-mode load jig, which was designed by Femlund and Spelt [29], to generate a

fracture envelope (critical energy release rate as a function of the mode ratio) for an

adhesive system consisting of AA6G61-T6 aluminurn adherends bonded with Betamate

1044-3 epoxy adhesive (from Essex Specialty Products Inc.). The fracture envelope was

then compared with the numerical predictions obtained from the DCB finite element

model, which was based on a cohesive zone modeling approach. This mixed-mode load

jig which enables fiacture testing over the entire range of mode ratios by using a single

DCB specirnen. will be briefly described in Section 3.2.3.

3.1.2 Co hesive zone modeling

The detailed description and literature survey of the cohesive zone modeling (CZM)

approach have been given in Chapter 1. The core of the present modeling approach was

to use the CO hesive zone model represented by traction-separation relations to mimic the

role that the adhesive layer plays during the deformation of adhesive joints. More

speci ficall y, DCB finite element models incorporating the CZM approach were

developed to predict the fracture envelope obtained h m the DCB fracture tests. This will

be discussed in more detail in the following sections.

3.1.3 Objectives of this Chapter

The intent of this chapter was to investigate the prediction ability of the CZM in the

fracture analysis of elastic DCB specimens. This was motivated by the attempt of this

M.A.Sc study to establish a general modeling approach to analyze plastically-deforming

adhesive joints, such as peel joints. To fulfill this objective, DCB fracture tests were

carried out in a specially designed mixed-mode load jig to obtain the fracture envelope

and the DCB finite element mode1 was accordingly developed to give numerical

predictions. The prediction ability of the CZM was studied by comparing the

experirnental results and the numerical results.

3.2 DCB fracture test

3.2.1 Adhesive system used

In this research, AA6061-T6 (fiom Alcan international Ltd.) with a half-inch

thickness was chosen for the adherend materials. Please refer to Table 2.2 and Figure 2.4

for this material's mechanical properties. The adhesive adopted here was the same as that

used in the peel tests presented in Chapter 2, Betamate 1044-3 (fiom Essex Specialty

Products Inc.). The uniaxial tende properties of bulk Betamate 1044-3 specimens were

show in Table 2.3 and Figure 2.6. The bondline thickness was 0.4 mm, which i s again,

the same value as that of the peel specimens. The pretreatment procedure will be given in

the following section.

3.2.2 DCB specimen

Specimen dimensions

/-- 0.25 inch diameter

Figure 3.1 Geometry and dimensions of the DCB specimen (dimension in mm unless stated)

The geometry and dimensions of the DCB specimen used for this study are show in

Figure 3.1, in which a is the length of a sharp mode I initial crack (about 100 mm)

obtained by driving a cold-chisel between the adherends.

Specimen fubricafion

Materlol prepPraio~

O Adherends: the alurninum plates (AA606LT6) were cut into two rectangle pieces

1 40 mm x 300 mm.

0 Adhesive: Betamate 1044-3 was used to bond the adherends together and it

should be taken out of a fndge about 2 hours before bonding.

Chemicals: there were three kinds of chemicals used for manufacturing the DCB

specimens, and they were: Acetone, Alumiprep 33 in an aqueous solution of 5%

by volume and Alodine 5200 in an aqueous solution of 7.5% by volume.

Other materials and equipment which were also necessary for the specimen

fabrication include: Teflon spacers, aluminum foil, latex gloves, a spatula, 2 inch

binder clamps, an oven with a thermostat, a drill press with % inch bit and a table

saw.

Decreased the bare aluminum plates (140 mmx 300 mm) using Kimwipe tissue

soaked with acetone until the bonding surface was clean enough.

Completely rinsed the surface with distilled water.

Sprayed Alumiprep 33 ont0 the surface of alwninum plates for 5 seconds. and

then left wet for 3 minutes.

Completely rinsed off the residual Alumiprep 33 with distilled wiiter.

Sprayed Alodine 5200 onto the bonding surface for 3 seconds. and then again left

wet for 3 minutes. This step aimed at forming a thin layer of titanium-based

conversion coating to improve the bonding strength of the joint.

Rinsed the residual Alodine 5200 off the surface irnmediately afier the 3-minute

wetting pdod in order to avoid an owrweight coating, and thus decrease the

bonding strength.

Dried the pretreated aluminum plates in an oven for approximately 10 minutes at

80°C.

Bonding and curing

The approximate weight of the adhesive required for bonding was obtained using

a standard balance.

Applied the adhesive in the center of the bonding surface and then spread it using

a small spatula.

0.4 mm thick Teflon shims were evenly laid dong the edges of the aluminum

plates in order to maintain a uniform bondline thickness.

Pressed the two plates togcther and clamped with 2 inch binder clamps. The edges

of the plates were scraped fiee of adhesive periodically so that excess adhesive

was able to flow out from between the plates. The joiat was left in this manner for

approximately one hour, or until no adhesive flow was evident.

The joint was put into an oven and increase the oven temperature to 170°C at

which the adhesive was cured for at least 2 hours. Then the oven was turned off

and the specimens were allowed to cool in the oven.

Final specimett

The joint was then cut into 20 mm wide specimens using an ordinary table saw

with a 10 inch, 80 tooth. carbide blade. The width of the adherends was chosen as

20 mm because a previous study has show that the critical energy release rate

remains constant for adhesive sandwiches over about 15 mm in width (Femlund,

1991).

The !4 inch diarneter loading pin holes were then drilled into the specimens.

The sides of the specimens at the bondline were bnished with a coating of diluted

typing correction fluid to improve crack visibility.

A sharp mode 1 initial crack was made by driving a cold-chisel between the

adherends.

3.2.3 Apparatus

The apparatus employed in perfonning the DCB fracture tests included: a compu iter

controlled ATM load frarne with a 2000 Ib capacity screw driven actuator, a mixed-mode

load jig, a light source and a crack propagation length measuring system consisting of a

traveling microscope mounted ont0 the frame, a Sony CCD video camera and a monitor.

Figure 3.2 schematically illustrates the mixed-mode load j ig designed by Femlund and

Spelt 1291 and it was reproduced here for convenience of discussion. Figure 3.3 is a

photograph of the load jig.

4 TO actuator

Pin Specimen

Figure 3.2 Illustration of the mixed-mode load jig

Figure 3.3 Photograph of the mixed-mode load jig

An important feature of this load jig is that a single equal adherend double cantilever

beam (DCB) specimen can be used for the entire mode ratio range, which avoids the

controversy over how the mode ratio should be calculated for unequal adherend

specimens [29]. This mixed-mode load jig (see Figure 3.2) consists of a link-arm systern,

which allows different loads to be applied to the upper and lower adherends of the DCB

specimens by altering the geometry of the load jig. Two equai and opposite forces cause

equal and opposite moments in each arm of the DCB specimens and correspond to pure

mode 1. Two equal bending moments in the sarne direction give pure mode 11. Many

other combinations of the forces applied to the top and bottom adherends can lead to the

full range of mode ratios.

In order to accurately determine the energy release rate, it was necessary to measure

the failure load, the specimen width, the adherend thickness and the crack length. The

failure load was recorded by the software, which was designed to control the load fiame,

at the onset of crack propagation. The width of the specimen was measured using a

micrometer at fivesentimeter intervais dong the length of the specimen, and the average

value was taken as the width for calculation purposeS. The adherend thickness was taken

to be the thickness of the bare aluminum plate. The crack length was measured accurately

by using a traveling microscope mounted on a vernier scale with a precision of f 0.1 mm.

A reference mark was made with a pend at an approximate distance of 14 cm away from

the center of the loading pin, the crosshair of the microscope was positioned on the mark

and a reference reading was taken on the microscope scale. The crosshair was then

positioned on the crack tip and a reading of the scale was taken. The real distance from

the loading pins to the reference mark was measured using a ruler. The difference

between the microscope reading at the crack tip and the reading at the reference mark

was added to the measured distance from the pins to the reference mark in order to obtain

the real distance fiom the pins to the crack tip. In this manner, the crack length could be

accurately measured as the specimen was tested.

33.4 Fracture envelope

DCB fracture tests were conducted to detennine the fracture envelope of the 1044-

3/6061-T6 adhesive system using the above procedure. G,. was detennined for steady-

state crack propagation at six different nominal phase angles, q , defined as:

Figure 2.4 shows the average value of G,. , plus minus one standard deviation, at

each phase angle. For each p, the number of data points was at least 30. The total

number of data points used to plot the fracture envelope was around 200 and specimens

from five different batches were used. Figure 3.4 shows that G,. is increasing with

increasing (D and that G,,(. (5305 ~ / r n ~ ) is approximately three times G,(. ( 1 68 1 .I/rn2) for

this system. It is also observed that the scatter of the data is greater at higher p. G(. is

virtually independent of (p for 9 c 30'. The same observation has been made previously

in the comprehensive Fracture study of adhesive joints by Femlund and Spelt [29j. who

also showed that G,. exhibits a linearly increasing trend with q for (D > 30'. However,

in Figure 3.4, G,,(. seems to be slightly lower than the expected value in order to form a

linear relation with the points with mode ratios greater than 30". This is attributed to the

lack of enough experimental data.

O 10 20 30 40 50 60 70 80 90

(degrees)

Figure 3.4 Fracture envelope for Betamate 1044-3 / AA606 1 -T6 adhesive system

3.3 Prediction of the fracture envelope usiag cohesive zone modeling

It should be mentioned that much work of the numerical predictions and analyses

conducted in this chapter was based on the fracture envelope of Cybond 452368 1 AA

7075-T6 adhesive system, which was obtained by Fenilund and Spelt [29]. The reason

was that the DCB fracture tests of the Betarnate 1044-3 / AA606LT6 adhesive system

had not ken finished yet. Nevertheless, the numerical prediction for the fracture

envelope of Betamate 1044-3 adhesive system was also given. Figure 3.5 shows the

fracture envelope for the Cybond 4523 GB / AA7075-T6 adhesive system.

Figure 3.5 Fracture envelope for Cybond 452368 1 AA7075-T6 adhesive system

3.3.1 Establishment of a mixed-mode DCB finite element model for the

452368 adhesive system

The geometry and dimensions of the DCB specirnen were the same as that s h o w in

Figure 3.1. The ANSYS finite element package (version 5.5) was employen to develop a

2-D model in this analysis. The stress state for the adherends was assumed to be plane

stress and linear elastic material properties were chosen for the adherend materials

(AA7075-T6), which have a Young's modulus E of 71.7 GPa and a Poisson's ratio o of

0.3. Quadrilateral isopararnetric plane finite elements were used to mesh the adherends. A

layer of non-linear spting elements, which were characterized by the traction-sepmat ion

laws. was used to represent the adhesive layer in the numencal aiiiilysis. Since the

adhesive layer, in general, is subjected to mixed-mode loading conditions, two springs. in

the normal and shear directions, respectively, were utilized to connect the corresponding

two nodes lying in the top and bottom adherends. The fiat two spring elements were

meshed at about 100 mm away from the center of the loading pins. which was the

distance close to the length of the initial crack made in the DCB specimens. According to

the study of Femlund and Spelt [29], the steady state energy release rate G,. . obtained

from the DCB fracture tests, was insensitive to the lengthof the initial crack. Therefore. it

can be assumed here that the numerical predictions will not be affected by this value.

Along the bondline direction, a uniforni meshing was taken with an element interval of

0.1 mm. Concentrated loads were applied at the upper and lower load pins, respectively,

with different combinations in order to achieve the full range of mode ratios.

3.3.2 Characterization of the parameters used to specify the traction-

separation curves

Figures 3.6 and 3.7 show the nonnal and shear traction-separation curves, which

could be transferred to the comsponding force-displacement curves used to describe the

mechanical properties of the two non-linear springs.

Figure 3.6 Normal traction-separation curve

Figure 3.7 S hear traction-separation curve

As shown in Figures 3.6 and 3.7, following the notations introduced by Tvergaard

and Hutchinson [74], T;,, and f,, are tenned the work of separation per unit area of

crack propagation (equal to the area under the traction-separation curve); 6 and P are

the normal and shear peak stresses, respectively, supported by the fracture process zone;

8, and 8, are the normal and tangential components of the relative displacement of the

crack faces across the interface in the zone where the fracture process occurs, and 4,

and 6, represent the critical values of these displacements. Moreover, two sets of shape

""' - for the normal traction-sepration curve parameters were defmed as follows: - , f i r w fim.

41 di2 and - , - for the shear traction-separation curve. S I C 4

The parameters goveming the normal and shear traction-separation curves are the

work of separation per unit area of crack advance c,, or c,,, , the peak stresses 6 or i

4,' & and the shape parameters -, 4, 4, or - -. Numerical study of Tvergaard and 4, 4 4 4

Hutchinson [74, 751 indicated that the details of the shape of the traction-separation laws

were relatively unimportant, and thus constant values of 0.15 and 0.5 were used for the

two shape parametea. In this thesis. the sensitivity of the predicted results to the shape

parameters was also investigated and will be discussed later. The two most important

parameters characterizing the fracture process in this cohesive zone model. according to

the study of Tvergaard and Hutchinson [74, 751, were T;,, , T;,, and &, i . Furthermore,

there exist two inherent relations among these parameters for these two traction-

separation laws:

and

and

These two expressions can be transfomed as:

In the present study, T;,, was equal to the pure mode I critical energy release rate,

G,. , obtained from pure mode 1 DCB fracture tests, and T;,, was equal to the pure mode

11 critical energy release rate. G,,(. , obtained from pure mode II DCB fracture tests. For

the Cybond 4523GB /AA7075-T6 adhesive system, the experimental results for G,(. and - were chosen to be 0.1 5 and G,. were 220 Pm2 and 570 ~ l m ' . respectively. -. 6, 4

- , - were chosen to be 0.5, the same values as those used in the studies of 4' 6,.

Tvergaard and Hutchinson [73-761. By substituting the values of shape parameters into

Eqn. 3.4 and 3.5. the following equations cm be obtained:

From Eqns. 3.6 and 3.7, it can be seen that, once the values of f,, , Tu, and the

shape parameters were specified, only one parameter was left to be detemined for each

traction-separation curve. Furthemore, in the work of Quan [91], uniaxial tensile tests

and shear tests were conducted for the bulk 452368 specimens. The maximum tensile

and shear strains were measured to be within 0.6% - 1 % and 3.7 - 6%, respectively.

Consequently, 6, and 6 , were determined based on the above experimental results. It

should be mentioned that the strains could be related to the displacements by multiplying

the original adhesive thickness, 0.4 mm, in both normal and shear directions. For

convenience of discussion, the displacements were stated as strains. In this research. six

combinations of 6, and 6,'. were chosen to determine those two traction-separation

curves. Four of them were within the experimental range found by Quan [91] and the

other two were out of the range. Afler 4, and 4, were specified, & and i could be

detemined accordingly based on Eqns. 3.6 and 3.7. The different parameter

combinations used in this study are listed in Table 3.1.

Table 3.1 Parameter combinations chosen for the traction-separation curves

Parameter combination 8, ô (MPa)

1

4 i (MPa)

3.33 Prediction of the fracture envelope and discussion

The failure criterion used in the present study was a simple mixed-mode fracture

criterion [15,52,89]:

Gl G, - + - = I , 4, r,,

where G, and G,, represent the mode 1 and mode II

(3.8)

traction-separation work absorbed

by the fracture process, respectively. These two ternis c m be calculated by integrating the

mode 1 and mode II traction-separation curves from zero displacement to the

displacements where fracture occurs:

G, = f 46, )d6, , (3.9)

G, = fr(6,)ds, . (3.10)

where 6; and 6: represent the critical normal and shear displacements where fracture

occurs. G, and G,, are also schematically show in Figures 3.8 and 3.9.

Figure 3.8 Mode I traction-separation work

Figure 3.9 Mode II traction-separation work

The total traction-sepmat ion work absorbed by the mixed-mode hcture process,

i.e., the critical energy release rate at different mode-mixedness, G,. , is the summation of

G, and G,, :

The fracture envelope predicted using the cohesive zone mode1 and the

conesponding experimental results are s h o w in Figure 3.10, where Gc upper limit and

Gc lower limit refer to the average Gc values plus minus one standard deviation at each

phase angle. [t can be seen that the numerical predictions for Gc and thereby the fracture

envelope based on the parameter combinations 1 to 4, lie well within the range of

experimental results. Especially for the results fiom parameter combinations 1 and 2. the

predicted fracture envelopes almost coalesce with the experimental fracture envelope.

However, the predictions fiom the parameter combinations 5 and 6 exceed the range of

experimental results. Consequently, the conclusions which cm be drawn here are that:

excellent numerical predictions for the fracture envelope cm be achieved provided that

the two components of critical displacements, 6, and 4, are taken io be within the

range of the maximum tensile and shear strains obtained from the tensile and shear tests

using bulk adhesive specimens; the predictions of the fracture envelope are not good if

6, or 6 , are outside the experimental range the maximum tensile and shear strains of

adhesive.

3.3.4 Prediction of the mode ratio

Parameter combination Set 2 (Figure 3.10), which gave the best prediction of the

fracture envelope, was used to examine the predictive ability of the mode ratio. Table 3.2

lists the input mode ratio calculated using the equation associated with the load jig [29]

and the predicted mode ratio calculated from the finite element mode1 using the same

equation.

Table 3.2 Mode ratio predictions

Table 3.2 shows that the FEA percentage prediction error is quite small. In addition

to the excellent prediction for the fracture envelope, a precise prediction for the mode

ratio was obtained as well.

Input mode ratio (degree)

FEA Prdicted mode ratio (degree) Prodiction errer

700 .. - - a - Gc-Upper-Limit ...*.. . Gc-Lower-Limit

. - - - - . Gc-Ave rag e-Va l ue 600 -*Set 1 ,

O 10 20 30 40 50 60 70 80 90

Phase angle (degrees)

Figure 3.10 The Fracture envelope prediction for the 452368 adhesive system using the cohesive modeling appmach

(Set refers to the different parameter combination shown in Table 3.1)

3.33 Sensitivity to the shape parameters

In the numerical predictions of the fracture envelope discussed above, the two shape

parameters were taken to be constant values of 0.15 and 0.5, respectively, the same

values as those used by Tvergaard and Hutchinson [75]. However, it is of interest to

investigate the sensitivity to the values of the shape parameters. The strategy used here is

4 4 l l 6 1 6, to keep - (including - and - ) constant at 0.15 and vary ; (including - and 4 &- 4 4 4,

g) from 0.5 to 0.9, and vice versa. It should be noted thai the same combination of

shape parameters was used for both the normal and shecv traction-separation curves. The

normal and shear critical displacements 6 , and 6 , were chosen to be 0.8% and 6%.

respectively, which gave the best prediction of the fracture envelope. I;,, and T;,, were

still equal to G,. and G,..

4 Strategy one: Keep - consfant al 0.15 and vury 5 f r o ~ 0.5 to 0.9 4 8,

Different combinations of shape parameters are summan'zed in Table 3.3 and the

corresponding prediction results are show in Figure 3.1 1.

Table 3.3 Shape parameter combinations (strategy one)

From Figure 3.1 1, it can be found that the numerical predictions of the fracture

envelope based on the shape parameter combinations, which were obtained using the

stratepy one, demonstrate excellent quantitative agreement with the experimental fracture

Shape panmeter combiaation

1

4 envelope. More specifically, when - varies from 0.2 to 0.5, the prediction results a 4 alrnost coalesce with the Gc average values. When - increases further to 0.7 and 0.9. 4

4 - 6,

0.15

the predicted Gc values are close to the Gc lower limit. however, still within the range of

experimental results. Consequently, the numerical predictions of the fracture envelope are

indeed not sensitive to the specific values of shape parameten. This is in agreement with

the conclusion of Tvergaard and Hutchinson [75] that the shape parameters would not

significantly affect the numerical prediction.

- - -- -

- 4 4 0.2

Phase an& (degrees)

Figure 3.1 1 Predictions of the fracture envelope based on the shape parameter combination-strategy one

Different combinations of shape parameters are summarized in Table 3.4 and the

corresponding prediction results are shown in Figure 3.12.

Table 3.4 S hape parametet combinations (stnitegy two)

Shape panmeter combination

1 ---- upper-li mi t

Phase angle (degrees)

Figure 3.12 Predictions of the frafture envelope based on the shape parameter corn bination-strategy two

The same trend is observed in Figure 3.12; different combinations of shape

panuneten obtained using the sirategy two, can quantitatively predict the shape of the

fracture envelope. In other words, the numerical predictions of the fracture envelope are

insensitive to the shape parameters.

3.3.6 Numerical analyses of the fracture envelope for Betamate 1044-3 /

AA6061-T6 ad hesive system

The numerical predictions for the fracture envelope of Betarnate 1044-3 / AA6061-

T6 adhesive system were also conducted afier obtaining the experimental fracture

envelope. which was show in Figure 3.4. Following the parameter specification strategy

employed for Cybond 4523GB adhesive system, T;,, and c,,, were again taken to be

equal to the measured G , (1681 .I/m2) and G,,(. (5305 ~/rn~), respectively. Shape

parameters were taken to be constant values of 0.15 and 0.5. It should be noted that

critical normal and shear displacements were not taken to be input parameters. since only

the uniavial tensile test of the bulk adhesive was done, i.e., only 6 , was available.

Rather, the dope of the linearly rising stage in both traction-separation curves was

speci fied, with Young's modulus for the normal traction-separation curve and shear

modulus for the shear traction-separation curve. By taking into account the plane strain

stress state of the adhesive layer in a peel specimen, plane strain Young's modulus

E' was used instead of plane stress Young's modulus E . E' was defined as:

where v is Poisson's ratio and a typical value of 0.4 for the epoxy resin was employed in

the current analysis. E was equal to 2.58 GPa, which was obtained from the uniaxial

tensile test of the bulk adhesive specimen s h o w in Chapter 2. G was defined as:

k2 '

Hence, 4. and 8,. could be derived as:

The peak stresses & and i could be detemined accordingly as:

B=O.ISx6,, x E' (3.16)

i = 0 . 1 5 ~ 6 , x G (3.17)

The cornparisons of numerical predictions and experimental results are shown in

Figure 3.1 1. Although the Betamate 1044-3 adhesive system has a much higher fracture

resistance (Figure 3.4) than the Cybond 4523GB adhesive system (Figure 3.5) (G, [ . is

approximately 7 times higher, and G,. is almost 9 times higher). the numerical

predictions of the fracture envelope still exhibit excellent quantitative agreement with the

experimentai results for the Betamate 1044-3 adhesive system.

- - - - Gc-Lower-Limit

--A-, Gc-Average-Value

+ FEM-predicted resuits

Phase angle (degrees)

Figure 3.13 The fiacture envelope prediction for Betarnate 1044-3 adhesive system using the cohesive modeling approach

3.4 Discussion and Conclusions

From the above analyses, it car. be seen that excellent numerical predictions of the

fracture envelopes for two different adhesive systems (Cybond 4523 and Betamate 1044-

3) could be obtained using cohesive zone modeling. The precise prediction of the mode

ratio was also demonstnited for the Cybond 4523 adhesive system.

The reason why the numerical prediction does such an excellent job might be

explained from two aspects: a proven energy-based failure criterion and the cohesive

zone modeling approach.

The failure criterion used in the current analysis is a simplification of a commonly

used. empirically suggested form [15,52,89]:

which has been found to work well for epoq adhesives. In this study, m and n were

taken to be 1 for the sake of simplification. Charalambides et al. [15] attempted to give

this physical meaning by assuming that a joint will fracture when the total mode1 1

component is equal to a constant, G , . That is, mode II component does not lead to failure

by itselc but only when comected to a mode I loading via a mechanism such as surface

roughness, which was thought to be able to transform shear displacement to an opening

displacement.

The cohesive zone modeling approach is the other reason for the excellent numerical

prediction. It has already k e n demonstnited that the prediction results are not sensitive to

the shape of the traction-separation c w e . Moreover, the yielding stress of the adherend

used to manufacture the DCB specimens was pretty high in order to make sure the

deformations occumng in the adherends remain elastic. According to the work of

Tvergaard and Hutchinson [75], B would not be an important parameter provided that it

is less than 3uy

DCB specimen,

where a, i s the yield stress of the adherend. In the case of the present

& - is well less than 3. Consequently, T;,, and T;,, were the only key O,,

parameters for the present analysis. Once these two parameters were calibrated by

experiments, it was expected to have gooà numerical predictions of the energy release

rates for the other phase angles using the appropriate energy combination rule and failure

criterion.

Chapter 4 Numerical Study of the Peel Test Using Cohesive Zone Modeling (CZM)

4.1 Introduction

In Chapter 3, the mixed-mode CZM model was established to predict the fracture of

elastic double cantilever beam (DCB) specimens, and it was demonstrated that this

modeling approach cm provide excellent quantitative predictions for the fracture

envelope of both Cybond 4523GB and Betamate 1044-3 adhesive systems. It is of

interest to further investigate the predictive ability of the CZM approach to mixed-mode

fracture of plastically deforming adhesive joints. As discussed in Chapter 2, the peel test

is one of the most commonly used mixed-mode joint confiigurations with adherend

yielding and thus it will be studied in this chapter using the CZM approach. The

numerical method adopted here was based on finite element analysis incorporating the

cohesive zone model (CZM), where cohesive elements represented by traction-separation

curves were introduced to replace the adhesive layer. A non-linear, large displacement

peel tïnite element model was developed accordingly to analyze the steady-state peel

behavior. Fracture parameten characterizing traction-separation curves were detennined

by comparing the numerical and expetimentai results for one configuration of the peel

samples. The parameters were then used without Further modification to predict the

fracture of peel samples with different configurations.

4.2 Establishment of the peel finite element model

The geomehy and dimensions of the peel specimen were the same as that shown in

Figure 2.2. A 2-D model was developed using the ANSYS finite element code (version

5.5). The stress state for the top adherend (AA5754-0) was assumed to be plane strain

and the corresponding material properties were already given in Chapter 2. Cnode

quadrilateral isoparametric finite elements were used to mesh the top adherend. The

bottom adherend in the numerical analysis was assumed to be an infinitely rigid substrate.

The whole adhesive layer in the peel specimen was replaced by a layer of non-linear

spring elements characterized by traction-separation curves. Since the adhesive layer in

the peel specimen is generally subjected to mixed-mode loading conditions, two spnngs

in the normal and shear directions, respectively, were utilized to connect the

corresponding two nodes lying in the top and the infinitely rigid substrates. This is

similar to the numerical analysis of DCB fracture tests. As show in Figure 2.2, the

flexible adherend extension Iengths were chosen according to the rule that the full

development of the plastic region of deformation in the fiee adherend adjacent to the

bonded region should be able to be achieved. Owing to the combined geornetric and

material non-linearity of the peel test, a considerable arnot.int of computing time was

required to reach the steady state. Non-linear, large displacement finite elernent

programs, which enable the analysis of the steady state peel test, were developed using

the ANSYS Parametric Design Language (APDL). The program for the 1 mm. 90" peel

model is included in Appendix B.

4.3 Failure criterion

The failure cnterion adopted here is an energy-based mixed-mode fracture criterion,

which is the sarne as that used in Chapter 3 for the numerical analyses of DCB fracture

tests. Spring elements in the normal and shear directions were deactivated continuously

when the failure criterion was satisfied, Le.:

It should be mentioned that, in Chapter 2, the interfacial adhesive element

(PLANE42) was "killed" using an "EKILL" command provided by ANSYS in order to

grow the crack. For the current analysis, the spring element (COMBIN39). which is the

only non-linear spnng eiemed available in ANSYS to descnbe the trapezoidal traction-

separation curves, was used to mode1 the adhesive layer. However, this element does not

support the "EKILL" function, Le., the spring element cannot be deactivated in the same

manner as that used for the continuum element (PLANE42). Therefore, an alternative

way was utilized to realize the same objective; namely, the degrees of freedom of the

bottom nodes of the spring element were released. Nonetheless, senous convergence

difficulties arose if the degrees of freedom of the springs were suddenly released due to

the non-linear properties of the springs. Consequently, three continuous steps were taken

to gradually deactivate the spnng elements. First, compressive loads were applied to the

bottom nodes of the normal and shear springs when the failure criterion was reached so

as to make the springs contract to around half of their critical displacements. Then.

additional compressive loads were applied to make the springs contract to 1% of their

critical displacements. Finally, the bottom nodes of the springs were released with least

convergence difficulties using a command "DDELE" provided by ANSYS since almost

al1 of the loads applied on the springs have been unloaded.

Steady-state peel was reached after a substantial amount of crack propagation.

4.4 Characterization of parameters used to speeify the traction-

separation curves

The strategies to determine the parameters used to characterize the traction-

separation curves were similar to those used in Chapter 3 for the numerical analyses of

DCB fracture tests. As shown in Figures 3.6 and 3.7, the two shape parameters 6, and

4 were still taken to be 0.1 5 and 0.5. These values were already demonstrated to have

no significant effect on the predicted fracture envelope for the 1044-3 DCB adhesive

system. Following the sarne approach used in Chapter 3, the dopes of the rising stages of

the two traction-separation curves were set as the plane strain Young's modulus and

shear modulus, respectively. At a first attempt, the area under the normal traction-

separation curve, i.e., the normal work of separation per unit area of crack advance. was

taken to be equal to the mode I critical energy release rate G,. (1681 .Vrn2). obtained

from mode I DCB fracture test for the 1044-3 adhesive system. Similady, G,/(. (5035

.I/m2), obtained h m mode II DCB fracture test, was taken as the area under the shear

traction-separation curve. These two traction-separation curves could be uniquely

determined after the above parameters were specified. Next, the numerical analysis of the

1 mm, 90' peel test was perfomed using the parameters specified this way. However. the

predicted peel load was 18 N h m , which was almost three times the value 6.8 N/mm

obtained fiom peel experiments. The two panuneters that were adjusted to lower the

predicted load to match the experimentai result were t? (nonnal peak stress) or i (shear

peak stress) and T;,, or T,, , which were pointed out to be the two most important

parameters in the application of traction-separation laws [73-76, 82, 83, 901. Fint, the

areas under the curves, i.e., the work of separation in mode 1 and II, were kept constant at

G,c. and G,,(. , respectively. Meanwhile, the peak stresses of the curves were modified to

investigate the dependence of the predicted peel loads on these parameters. Table 4.1

shows different combinations of shape parameters, 8 , î and the corresponding

predicted peel forces.

Table 4.1 Di fferent combinations of shape parameters, ci , i and corresponding predicted peel forces using the cohesive zone mode1

From Table 4.1. it cm be observed that the predicted peel forces are quite insensitive

.

& t? to â when - is well below I . Nonetheless. when - increases UD to 1.61. the

predicted value (25 N/mm) is significantly higher than the results fiom the other four

No.

1

2

3

4

5

Ptedicted peel force (N/mm)

16.5

17.5

18.0

18.7

6,

0.05

0.15

0.15

0.3

1 35.0

0.5

0.95

0.5

0.5

1 '

- f l .

0.63

0.98

1.1

1.7

3.4

6 (MP.1

29.9

46.5

53.7

80.5

161.1

t.v

(MPs)

50

50

50

50

50

I

(MPa)

31.67

41.23

56.84

85.26

170.53

=Y

(MPi)

1 O0

1 O0

IO0

100

t O0

- t5 by

0.30

0.47

0.54

0.8 1

1.61

predictions. In the work by Wei and Hutchinson [83], it was show that if the maximum

cohesive stress along the interface (ô) is greater than about three times the yield stress

(a,.) of the peel ann, the steady-state peel force per unit width increases dramatically

with &. However, the results of the current analysis demonstrate that the peel forces

were significantly increased even when ô was relatively small. Similar phenornena were

observed in the work of Yang [89], who found that a substantial elevation of the peel

force could be obtained for the T-peel test when the peak stress was small provided that

there was extensive plastic deformation in the amis of the laminate.

Since the variation of 6 could not lower the predicted peel force, the areas under

the two curves were then adjusted in order to simulate the experimental peel loads. To

minimize the number of arbitrary parameters used to specify the traction-separation

curves, a simple relation was assumed between the two variables cl, and T;,, with:

The reason to use the coefficient 3 was that the DCB fracture test showed that G,.

was about three times G,(. . It was assurned that the sarne relation holds for the peel

situation.

4.5 Numerical simulations and cornparisons with experimental results

AAer speïifying the simple relationship shown in Eqn. 4.2, there was only one

arbitrary parmeter remaining: c,, or cl;,, . In this study, Tl, and T;,,, were then

detemined to be 422.5 j/m2 and 1325 .I/m2, respectively, by perfoming a series of

numerical simulations of 1 mm, 90' peel tests using different values of I;,, and I;,, and

finding the best fit to the associated experimental results. The same traction-separation

curves were then used without any M e r modifications, to predict the fracture of al1

other peel specimens with different peel am thickness ranging from 1 mm to 3 mm and

peel angles ranging from 30" to 90". The numerical results and the corresponding

experimental results are shown in Table 4.2.

Table 4.2 FEA prediction for peel loads based on the steady state peel finite element model using the cohesive zone modeling approach calibrated at 1 mm, 90" peel

In Table 4.2, the FEA prediction error refers to the percentage error of the FEA

predicted load corresponding to the average experimental load for each peel

configuration. From this table, it can be seen that the steady-state peel finite element

Peel arm thickness

(mm)

Peel angle

FEA prodicted peel loads (Nlmm)

Expt. peel load Standard Deviation

(N/mm)

FEA prediction Error

model based on the cohesive zone modeling approach does a reasonable job in predicting

the dependence of the peel strength on the peel angle and the peel am thickness, i.e., the

peel strength increases with decreasing the peel angle or increasing the peel arm

thickness. The model gives better predictions of the peel force for p e l configurations

with peel a m thickness of 1 mm and 2 mm since the percentage prediction error is well

below or close to 1 0%. The anthmetic average error for al1 of the peel cases is 9.4%. It

was quite encouraging to see that one single set of traction-separation parameters can

capture so many realities. Nonetheless, from Table 4.2. it can also be observed that the

percentage prediction error for the peel test with 3 mm peel arn~ thickness is relatively

large, especially for the 30' and 60" cases, which are 15.9% and 19.3%. respectively. In

order to precisely mimic the experimental peel loads, the work of separation per unit area

of crack advance, i.e., the areas under the two traction-separation curves, namely T;,, and

/;,, , need to be increased. When the peel ami thickness increases to an extremely large

value, such as in the DCB case, T;,, and q,,, are expected to be much higher than the

values used for the prediction of 1 mm and 2 mm peel tests. This is in agreement with the

DCB numerical analyses presented in Chapter 3, where T;,, and c,,, are 1,680 ~/m' and

5.305 ~ / r n ~ (i.e., the values of Glc and Gffc, respectively). The thickness dependence of

the work of separation used in the traction-separation curves was attributed to the effect

of adhesive constraint, which will be increased with increasing adherend thickness. The

higher the adhesive constraint, the bigger the damage zone in the vicinity of the peel

front. Therefore, more energy is required to break new materials in order to extend the

crack. More research needs to be done in the future to further investigate whether there

exists an adhesive constraint related parameter, on which the cohesive zone model is

dependent.

Compared with the critical Von-Mises strain failure critenon presented in Chapter 2,

the cohesive zone modeling approach has the following advantages: it avoids the

complicated stress analysis in the adhesive layer; it does not have the shortcoming of

ignoring the hydrostatic stress; it is not dependent on the peel arm thickness, Le.. there is

no need to find a peel a m thickness dependent failure criterion provided that the peel

am thickness is less than 2 mm.

4.6 Investigation of the mode ratio

As discussed in Chapter 2, since the peel test is mixed-mode fracture of some

combination of mode 1 and mode 11, it is worthwhile to investigate the mode ratio

provided by the steady-state peel finite element model based on the cohesive zone

modeling approach. Following the same strategy used in Chapter 2, both the stress-based

definition and the strain-based definition were adopted here to define the mode ratio:

where a, and r,, correspond to the normal and the shear stresses occumng in the

normal and shear springs, respectively. Similady, E, and y, are the normal and shear

strains occurring in the two springs, respectively. For each definition, two types of phase

angle were studied: the average phase angle based on the adhesive tensile zone and the

local phase angle based on the springs at the peel fiont (mot).

4.6.1 Stress-based definition

Average phase angle based on the adhesive tensile zone

The definition of the adhesive tensile zone is the same as that used in Chapter 2; i.e.,

a zone where the adhesive normal stress in the y-direction is positive. Table 4.3 lists the

tensile zone length corresponding to the peel angles and the peel a m thickness.

Table 4.3 The influence of the peel angle and the peel a m thickness on the tensile zone length (obtained from the peel finite element modei based on the cohesive zune modeling approach)

From Table 4.3, it can be seen that the tensile zone length exhibits the sirnilar trend

as that obsewed in Table 2.9; namely, it increases with increasing the peel am thickness

or decreasing the peel angle. However, The magnitude of the predicted value in Table 4.3

Peel arm thickneso (mm)

1

2

Peel angle

30"

60"

90"

30"

60"

90'

Tensile zone length (mm)

1.6

1.5

1.3

2.4

2.3

2.1

is greater than that show in Table 2.9. Furthemore, within the peel tests with the same

peel am thickness, the predicted tensile zone length is quite close to one another. This

supports the above conjecture that varying adhesive constraint is responsible for the

observed increase of the fracture energy with increasing peel am thickness; Le. increased

constraint implies an increased tensile zone length.

Figure 4.1 shows the average phase angles over the tensile zone predicted by Eqn.

4.3 as a function of the peel a m thickness.

Peel arm lhickness (mm)

Figure 4.1 Average phase angle over the tensile zone as a function of the peel arm thickness (stress-based definition)

From Figure 4.1, one c m find the similar trend as that obsewed in Figure 2.19: the

predicted phase angle is independent of the peel arm thickness; but is strongly dependent

on the peel angle since the phase angle of 30" is significantly higher than those of 60" and

90'.

Local phase angle based on d e springs ut the peel fmd

Figure 4.2 shows the local phase angle, which was calculated based on the normal

and shear springs at the peel front, as a function of the peel arm thickness.

1 2

Peel arm thickness (mm)

Figure 4.2 Local phase angle on the springs at the peel front as a function of the peel arm thickness (stress-based definition)

As in Figure 2.21, the predicted phase angle in Figure 4.2 is insensitive to the peel

arm thickness and peel angle. Nevertheless, mode 11 dominant fracture with a mode ratio

around 53" is predicted rather than the mode 1 dominant prediction obtained in Chapter 2.

4.6.2 Strain-based definition

Average phase angle based on the udhesive tensile zone

The tensile zone length was the same as that used for the stress-based definition.

Figure 4.3 shows the average phase angle as a function of the peel arrn thickness.

Peel arm thickness (mm)

Figure 4.3 Average phase angle over the tensile zone as a fùnction of the pee1 a m thickness (strain-based definition)

The same trend is observed in Figure 4.3 as that in Figure 4.1 for the stress-based

definition. That is, the predicted phase angle is independent of the peel arm thickness and

dependent on the peel angle between 30' and 60". However, this is different fiom the

conclusion drawn in Chapter 2 for the average phase angle over the tensile zone based on

the strain-based definition obtained fiom the steady-state continuum peel model, which

was developed using the stress analysis approach. A peel a m thickness dependence was

found there instead of the peel angle dependence observed here.

Local phase angle hsed on the springs ut the peeI front

Figure 4.4 shows how the local phase angle varies with the peel angle and the peel

arm thickness.

Peel arm thickness (mm)

Figure 4.4 Local phase angle based on the spnngs at the peel Front as a function of the peel arm thickness (shain-based definition)

From Figure 4.4, it can be seen that the predicted phase angle is independent of the

peel arm thickness and the peel angle, which is the same trend as that found in Figure 4.2

for the local phase angle obtained fiom the stress-based definition.

Comparing the different mode ratio definitions used in this study, the average phase

angle over on the tensile zone based on the strain-based definition might be the best one

for the following reasons: (1) One assumption used in this study and also for most of the

related literature is that the aâhesive is a homogeneous material. However, the real

adhesive material is not exactly homogenous since it has many rnicroscopic voids near

the peel root darnage zone. Therefore, stresses calculated based on continuum mechanics

should be different from the real values for this material. The shah component, however,

would not be significantly affected by this homogeneous assumption. Therefore, the

strain-based de finition is more meaningful than the stress-based definition. (2) The local

phase angle is not as good a definition since it will be quite sensitive to errors caused by

the singularity existing at the peel front. (3) The tensile zone is the main area to generate

most of the peel strength and create the adhesive darnage zone, and thus it is more

meaningful to study the mode ratio in this zone.

Chapter 5 Conclusions and Recommendations

5.1 Conclusions

In this thesis, a steady-state peel finite element model based on the traditional stress

analysis approach, was first developed to analyze the elastic-plastic fracture behavior of

the peel test. Next, a DCB finite element model incorporating a cohesive zone model

which was intended to describe the fracture process occurring in the adhesive layer, was

established to predict the fracture envelope for two different adhesive systems. The

parametric study regarding the shape parameter sensitivity was aiso performed using this

model. In Chapter 4, the cohesive zone modeling approach was used to develop a steady-

state peel finite element model in order to investigate the predictive ability of this

promising rnodel for adhesive joints with plastically-deforming adherends. The fracture

parameters for the traction-separation law were determined by comparing the numerical

and experimental results for one configuration of the peel samples. The parameters were

then used without further modification to simulate the fracture of peel samples with

different configurations. Excellent predictions of the peel forces were obtained.

The overall objective of this work was to establish a failure criterion for adhesive

joints undergoing large scale adherend yielding.

The most important conclusions are summarized as follows:

(i) Good numerical predictions of the peel strength for peel tests with the sarne peel

a m thickness but different peel angles were obtained from the steady-state peel finite

element model, which was established using the critical Von-Mises strain failure

criterion. There existed a peel a m thickness dependence in the failure critenon; Le., the

critical Von-Mises strain failure criterion increased with increasing peel ami thickness.

This was attributed to the effect of hydrostatic stress on adhesive yield.

(ii) Accurate numerical predictions of the frafture envelopes for both Cybond 4523

and Betamate 1044-3 DCB adhesive systerns were obtained using the cohesive zone

modeling approach. The predicted fracture energies were demonstnited to be insensitive

to the specific values and combinations of the shape parameters provided that the critical

normal and shear displacements were within the range of experimental results obtained

fiom uniaxial tensile or pure shear test using bulk adhesive samples.

(iii) A numerical study of peel using the cohesive zone modeling approach was

presented in Chapter 4 and it was found that the traction-separation curves calibrated

based on the 1 mm, 90' peel test gave reasonable numerical predictions of the peel

strength for peel tests with the peel a m thickness of 1 mm and 2 mm and different peel

angles. However, for the 3 mm peel tests, the predicted peel forces were relatively lower

than the corresponding experimental results. The areas under the two traction-separation

curves (Le.. the works of separation) should increase relative to I and 2 mm values in

order to better match the experimental results for thicker adherends.

(iv) Both the stress-based definition and the strain-based definition were used to

investigate the mode ratio associated with steady-state peel for both the peel mode1 based

on the traditional stress anaiysis approach and that based on cohesive zone modeling. The

average strain-based phase angle over the tensile zone was concluded to be the best

definition in both cases.

5.2 Recornmendations

(i) For the peel model established by using the cohesive zone modeling approach, it

is worthwhile to obtain the experimental and numerical results for the peel cases with

thicker adherends to see whether the peel ami thickness dependence really exists. In other

words, whether the energy under the traction-separation curve is dependent on the peel

arm thickness. If the answer is positive, the adhesive constraint effect caused by the

adherend thickness could be used to account for this as discussed in Chapter 4. Then, it

will be of interest to seek an adhesive constraint related pararneter and an approach to

characterize the work of separation, which is the key pararneter to determine the traction-

separation curves. The tensile zone length mi@ be an appropriate pararneter sincc it

varies with the adherend thickness. Once the re!stion between the tensile zone length and

the work of sepamtion is established, the next interesting issue would be how to measure

or predict this length. The elastic sandwich model developed by Fernlund and Spelt (281

might be a useful tool for this job since it could estimate the stress distribution and thus

the tensile zone length using specimen geometry and the given reactions applied to the

sandwich.

(ii) The peel cohesive zone model should be further used to analyze adhesive joints

with a variety of geometries such as T-peel tests with the same or different adherend

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Finite Element Program for the Peel Continuum Mode1 Based on the Critical

Von-Mises Strain Failure Criterion

The following is a typical finite element program written using the ANSYS

Parametnc Design Language (APDL) in order to develop the elastic-plastic, large

displacement, steady-state peel finite element model based on the critical Von-Mises

strain failure criterion. For convenience of discussion, only the program for 1 mm, 90"

peel was given.

KOM Elastic-plastic, large displacement, steady-state peel finite element program based on the critical Von-Mises strain failure criterion

KLEAR ! Cornmand used to clear the database

PREP7 !Enter the mode1 creation processor.

/TITLE. 1 mm, 90" peel analysis using the stress analysis approach

!!Establish the mathematical model of the peel test.

!!The unit used in this program to develop the mathematical model is m.

TADT=I .OE-3 !Top adherend thickness TADL=100E-3 !Top adherend length

BADT=6E-3 ! Bottom adherend thickness BADT 1=0.4E-3 !Region 1 (including 2-layers of element) BADT2=0.3 E-3 !Region 2 (this is a transition layer.) BADT3=2.4E-3 !Region 3 (including 2-layers of element) BADT4=0.9E-3 !Region 4 (this is a transition layer.)

BADL=18E-3 !Bottom adherend length

AHT=0.4EW3 !Adhesive thickness

AHE W=O. 1 E-3 ! Adhesive element width

ENUM=7 1 8 !The third adhesive element along the upper interface TNNUM=188 !The third adhesive node used to calculate the failure criterion

KENUM=720 !The adhesive element at the peel front KTNNUM=182 !The interfacial adhesive node at the peel front

!Note: when the failure criterion was reached, the element KENUM was "killed". Then KENUM and KTNNUM were increased to the values of the next correspondhg element and node.

K, 1,0,0 !Specify a keypoint. K,2,BADL,O

!****** BOTTOM ADHEREND A, 1,2,20,2 1 !Al A,25,20, 1 8,24 !A2 A,19,18,16,17 !A3 A,22,16,3,23 !A4 A,4,3,5,6 !A5

!******ADHESIVE LAYER A.6,5,8,7

!*******TOP ADHEREND A,7,9,12,13 A,9,10,11,12 A,10,14,15,11

! !Finish establishing the mathematical model.

! !De fine material properties.

ET, 1 ,PLANE42,,,2 !Plain strain stress state for the adherend and adhesive MPTEMP, 1,O.O

MP,EX, 1,7 1 E9 !Top adherend Young's modulus MP,NUXY, 1 ,0.3 !Poisson's ratio

MP,EX,2,2.58 1 E9 !Adhesive Young's mudulus MP,NUXY ,2,0.4 !Poisson's ratio

MPTEMP, 1,O.O MP,EX,3,7 1 E9 ! Bottom adherend Young's modulus MP,NUXY ,3,0.3 !Poisson's ratio

!Top adherend property MAT, 1 TB,BISO, 1,1 TBDATA, 1,l OOE6,0.48272E9 !Dethe the yielding stress and tangent rnodulus. !TBPLOT,BISO, 1

! Adhesive property input MAT,2 TB,MIS0,2,1,88

!Bottom adherend property MAT,3 TB,BIS0,3,1 TBDATA, 1,24OE6,2E9 ! Define the yielding stress and tangent modulus TBPLOT,BIS0,3

!The following is to mesh the mathematical model.

!Mesh the adhesive layer TYPE, 1 MAT,2 MSHKEY,I !specify mapped meshing LESIZE,2 1 ,,,4 LESIZE, 19,AHEW AMESH,6

!Mesh the top adherend TYPE, 1 MAT, 1 LESIZE,24,AHE W LESIZE,27,,,5 AMESH.7

MSHKEY, 1 LESIZE,3 1,1 E-3 AMESH,9

!Mesh the bottom adherend TYPE, 1 MAT,3 LESIZE, l8,,,2 AMESH,S

LESIZE,9.,,60 LESIZE, 1 O,,J

LESIZE,3,,,20 LESIZE,S,,, 1 AMESH, 1

MSHKEY,O LESIZE, 14,,, 1 LESIZE, 1 6,,, 1 LESIZE, 13,,, 1 AMESH,4

!Finish the rneshing stage.

ALLSEL,ALL SAVE

!Below is to apply the boundary conditions. NSEL,S.LOC,Y,-0.005E-3,O.OOSE-3 D,ALL,AL L,O

ALLSEL,ALL SAVE

!Below is the solution program. /SOLU

ANTYPE,STATIC,NE W ! Speci fy analysis type and restart status. SOLCONTROL,ON !Specify whether to use optimized non-linear solution

defaults and some enhanced intemal solution algorithms.

NLGEOM,ON !Large de formation is twned on. SSTIF,ON AUTOTS,ON !Speci@ whether to use automatic time stepping or load

stepping.

NEQIT,30 !Specify the maximum equilibrium iteration for non-linear anal ysis.

DELTIM,O.l ,O. 1,0.5,ON !Specify the time step sizes to be used for his load step. PRED,ON !Turn on the prediction for the non-linear analysis. CNVTOL,F, 1 E5,0.00 1 !Convergence criterion for the force

!Output control OUTRES,STRS OUTRES,EPEL OUTRES,EPPL OUTPR,BASIC

TM-START= 1 TM_END=50 !In additional load steps, these two numbers should be changed to

the corresponding values. TM-MCR= 1

* DO,TM,TM-START,TM-END,TMMCR TIME,TM FCUM,ADD F,3 1 54,FY,4O ALLSEL,ALL SOLVE SAVE

*IF,ENUM,EQ.718,TWEN ESEL,S,ELEM,,ENUM,ENUM !Determine which material is chosen.

NSEL,S,NODE,,TNNUM !Choose the node for detemining the failure cntenon.

*GET,M WON,NODE,TNNUM,S.EQV !Get the nodal Von-Mises stress. *GET.MYPLS,NODE,TNNUM,NL,EPEQ !Get the nodal effective Von-Mises

plastic strain.

!Calculsite the total effective Von- Mises strain.

NSEL,S,NODE,,TNNUM *GET,MYVON,NODE,TNNUM,S,EQV *GET,MY PLS,NODE,TNNUM,NL,EPEQ ELS=MYVON/2.581 E9 TOTAL=ELS+MYPLS ALLSEL,ALL * IF,TOTAL,GE, 1.07E-2,THEN !The element KENUM was

deactivated when the failure criterion was reached.

EKILL,KENUM KENUM=KENUM- I KTNNUM=KTNNUM+ 1 ENUMsENUM-1 TNNUM=TNNUM+l

*ENDIF

ALLSEL,ALL SAVE

! End of the program

Appendix B

Finite Element Program for the Peel Mode1 Based on Cohesive Zone Modeling

The following is a typical finite element program written using the ANSYS

Parametric Design Language (APDL) in order to develop the elastic-plastic, large

displacement, steady-state peel finite element model based on the cohesive zone

modeling (CZM) approach. As in Appendix A, only the program for 1 mm, 90' peel was

given. Miich of the notes appeared within this prograrn is the same as that in Appendix A.

/COM Elastic-plastic, large displacement, steady-state peel finite element program based on the cohesive zone modeling approach

ICLEAR ! Command used to clear the database

PREP7 !Enter the mode1 creation processor.

/TITLE, 1 mm, 90' peel analysis using the cohesive zone modeling approach

!!Establish the mathematical model of the peel test.

!The unit used in this prograrn to develop the mathematical model is m.

TADT=I .OE-3 !Top adherend thickness TADL=l OOE-3 !Top adherend length

BADL=l8E-3 !Length of the adhesive layer AHT=0.4E-3 ! Adhesive thic kness AHE W=O. 1 E-3 ! Adhesive element width

NENUM= 1880 !First normal spring SENUM=206 1 ! First shear spring KNODE=23 12 !This is the peel front node based on which the DOF was released

when the failw criterion in order to grow the crack.

!Parameters of the normal traction-separation curve NDLTC=2.33004E-OS !& SIGMAP=26.84744342 ! 6 GAMMA 1 =422,25 ! <II

NDLTI =NDLTCSO. 15 ! 4, NDLT2=NDLTC*O.S ! ' n 2

! Parameters of the shear traction-separation curve SDLTG6.9 1 E-OS ! 4 TOWP=28.42 15023 ! f GAMMA2= 1 325 ! CUI SDLTI =SDLTC*O. 15 ! 4 SDLT2=SDLTCf 0.5 ! &,?

!!Finish establishing the mathematical model.

! !De fine material properties.

ET. 1 ,PLANE42,,,2 !Plain strain stress state for the top adherend

ET,2,COMBIN39,0,0,2,0,, 1 ! SpeciQ the nonnal spring element ET,3,COMBN39,0T0, 1 ,O,, 1 ! SpeciS, the normal spring eletnent

MPTEMP, 1,O.O MPTEX, t ,7 1 E9 MPTNUXY, 1 90.3

!Top adherend Young's modulus !Poisson's ratio

!Top adherend property MAT, I TB,BISO, 1, l TBDATA, 1,l 00E6,O048272E9 !Define the yielding stress and tangent modulus !TBPLOT,BISO, 1

!The following is to mesh the mathematical model.

! Mesh the top adherend. TYPE, 1 MAT, 1 MSHKEY, 1 LESIZE,l,AHEW LESIZE,2,,,5 AMESH, 1

MSHKEY, 1 LESIZET8, 1 E-3 AMESH,3

NKPT,, 1 NKPT,,2 FILL,23 1 1,23 12,179,23 1 3,1 !Create bottom nodes which were used to rnesh the

spring elements

!Mesh the adhesive layer.

!Below are the normal spnng properties. R92,0,0.8.74E-07,671 .19,1.758-06,1342.37 RMORE,2.62E-06,20 1 3 .56,3. JOE-06,2684.74,4.85E-06,2684.74 RMORE,6.2 1 E-06,2684.74,7.57E-06,2684.?4,8.93E-06,2684.74

RMORE, l.O3E-OS,2684.74,l,17E-O5,2684.74, t .3 1 E-05,2349.1s RMORE. 1 -46E-05.20 t 3 .56,1.60E-OS, 16'V.W, t .?SE-OS, 1 342.37 RMORE. 1.89E-05, IOO6.78,2.ME-OS,67l. 19,2.18E-OS,335.59 RMORE,2.33E-O5,O.OO

! Form the normal springs. TYPE,2 REAL,2 E,23 1 2,182 BOTTOM-N=249 1 *DO,TMN, 1 8 1,3,- 1 E,BOTTOM-N,TMN BOTTOM-N43OTTOM-N- 1 *ENDDO E,2311,1

!Below are the shear spring properties. R,3,0.0000E+00,0,2.5900E-06,710.54,5.1800E-06,1421 .O8 RMORE,7.7699E-06,2 1 3 1.6 1,1.0360E-05,2842.15,1.43898-05,2842.15 RMORE. 1 $4 1 SE-O5,2842.15,2.24468-05,2842.I 5,2.6475 E-O5,2842.15 RMORE,3.0504E-05,2842.15,3.4533E-05,2842.15,3.885OE-05,2486.88 RMORE,4.3 1668-05.2 13 1.6 1,4.74838-05,l776.34,5.l8OOE-O5,l42 1 .O8 RMORE,S.6 1 16E-O5,lO65.8 I,6.04338-05,7lO.54,6.4749E-O5,355.27 RMORE.6.9066E*OS,O.OO

!Fom the shear springs. TYPE,3 REAL,3 E,23 12,182 BOTTOMN=249 1 *DO,TMN,I81,3,-1 E,BOTTOM-N,TMN BOTTOMdN=BOTTOMN- I *ENDDO E,2311,l

!Finish meshing the mathematical model.

ALLSEL,ALL SAVE

! Below is to apply the boundary conditions. NSEL,S,LOC,Y,-O.005E-3,O.OOSE-3

ALLSEL,ALL SAVE

!Below is the solution program. BOLU

ANTYPE,STATIC,NEW !SpeciQ analysis type and restart status. SOLCONTROL,ON !Specify whether to use optimized non-linear solution

defaults and some enhancecl interna1 solution algorîthms.

NLGEOM,ON !Large deformation is tumed on. SSTIF,ON AUTOTS.ON ! Speci fy whether to use automatic time stepping or load

stepping. NEQIT,30 ! Speci fy the maximum equilibrium iteration for non-linear

analysis. DELTiM,O. 1 ,O. 1,OS.ON !Specify the time step sizes to be used for his load step. PRED.ON !Tum on the prediction for the non-linear analysis.

!Output control OUTRES,STRS OUTRES,EPEL OUTRES,EPPL OUTPR,BASIC

TM-START= 1 TMEND= 100 !In additional load steps, these two numbers should be

changed to the corresponding values.

*DO,TM,TM-START,TM-END,TM-iiUCR TIME,TM FCUM,ADD F,2250,FY,30 ALLSEL,ALL SOLVE SAVE ALLSEL,ALL

!************Calculation of the energy consumed in the normal spring2*********

*GET,NDIS,ELEM,NENUM,NMISC, 1 !Obtain the displacement of the normal spring

!The first part *IF,NDIS,LE,NDLTl ,THEN GI=O.S*NDIS*NDIS*SIGMAP/NDLTl* 1 E6

*ENDIF

!The second part *IF,NDIS,GT,NDLT 1 ,THEN

*IF, NDIS.LE.NDLT2,THEN GI=SIGMAP1(NDIS-NDLTI )' 1 E6+0.5*NDLT1 *SIGMAP* 1 E6 * ENDIF

*ENDIF

!Thc third part *IF,NDIS,GT,NDLT2,THEN *1F, NDIS,LE,NDLTC,THEN GIGAMMA 1 -OS*(NDLTC-NDIS)*(NDLTC-NDIS)I(NDLTC-

NDLT2)*SIGMAP1 1 E6 *ENDIF

*ENDIF

! ** *** * ** * * * *CaIculation of the energy consumed in the shear spring* * * ** ** ***

*GET.SDIS,ELEM,SENUM,NMISC, I !Obtain the displacement of the shear spring

!The first part * IF,SDIS,LE,SDLT I ,THEN GII=O.S*SDIS*SDlS*TOWP/SDLTl* 1 E6

*ENDIF

!The second part *IF,SDIS,GT,SDLTI ,THEN

*IF, SDIS,LE,SDLT2,THEN GII=TOWP*(SDIS-SDLTl)* 1 E6+0S*SDLTl *TOWP* 1 E6 *ENDIF

*ENDIF

!The third part *IF,SDIS,GT,SDLT2,THEN

LANMDA=GI/GAMMA 1 +GIVGAMMA2 !Calculate the failure criterion A .

*GET,BNODEUY,NODE,KNODE,U,Y !Get the displacement of the bottom node of the two springs. This displacement should not be zero after apply the compressive load to the springs.

*ENDIF * ENDIF *ENDIF

!The failure criterion was reached

!The fmt step to release the springs: apply compressive loads to the two springs in order to make the two springs contract to half of their cniical displacements.

!If there i s a non-zero displacement in the bottom node.

!The second step to release the springs: apply compressive loads to the two springs in order to make the two springs contract to 1% of their critical displacements.

!The last step to release the springs: spnngs were released completely.

!After the first two spnngs were "killed", the values of these two springs were changed accordingly .

!The corresponding first node in the bottom

!The following sentences were to realize automatic loading. FYAPP was determined according to the value of LANMDA, which was an output of a previous load step. * IF,LANMDA,LT, 1 ,THEN * IF,LANMDA,GE.O.999,THEN FYAPP=I *ENDIF *ENDIF

ALLSEL,ALL SAVE

FINISH

!End of the program

- Appendix C

Data Files for DCB Fracture Tests of Betamate 1044-3 Adhesive System

Specimn No, #1 Test t[m: Mar,25Vi,O1

Made Ratio

Specimen Dimensions:

Average

Load Jig Geornetry sl s2 s3 s4

Test Results

Fons (N) 354 752

1462 1552 1600 1622 1600 1585 1600 1558 1547 1543 1537 1 544 1 548 1537 1526 15% 1521 1526 1515 1530 1520 1 488 1474 1440 1423

Wdth (mm) Adbrenâ Thickness (mm) 19.89 12.7 19.9 19.8 BondIlne Thkkness (mm)

19.95 0.4 19.89

Measumd Crack Lenth (mm)

119.3 122.43

123 124.4

125 125.72 126.81 127.58 128.27 128.48 129.34 129.83 130.18 130.22 130.68 131.42 131.92 132.12 132.2

13241 132.33 132.82 134.2 134.4

i3s.n 136.58 137.54

Actual Crack kngth (mm)

103.48 106.61 107.18 108.58 109.18

109.9 110.99 111.76 1 12.45 1 12.66 1 13.52 1 14.01 1 t 4.36 114.4

1 14.86 115.6 t 16.1 116.3

1 16.38 1 16.59 1 16.91

117 118.38 118.58 1 19.95 120.76 121 -72

Adwl Crack Length (m)

0.10348 0.10661 0.10718 0.10858 0.10918 0.1099

0.1 1099 0.1 1 176 0.11245 0.11266 0-11352 0.11401 0.1 l a 6 0.1 144

0.11486 0.1 156 0.1 161 0.1 163

0.1 1638 0.1 1659 0.1 7691

0.1 17 0.4 1838 0.1 1858 O. 1 1995 0.12û76 0.12172

Nate: the values with ' adjacent will be distarded d u e Co i m s i n g crack lmgth 1 spaiiad measummsnts

Oc-Average 1672.80 J / mA2

crack length (m)

Spscimen No. #2 Test time: ~ p n l 5ttt,2001

Mode Ratio O

Wdth (mm) 19.8

19.57 19.78 19.5

Average 19.66

Load Jig Geornetry s i s2 s3 s4

Test Results

Farce (N) 400 790 854

1 O68 1204 1350 1388 1469 1455 1634 1600 1 700 1724 1700 1690 1692 1 708 1697 1688 1 700 1676 1680 1639 1633 1623 1619 1591 1590 lsal 1585 1578 1570

Measured Crack Lenth (mm)

104.55 105.92 106.35 106.8 107.9

108.66 1 O9

109.95 1 10.4

111.36 113.15 113.9

114.67 t 15.47 175.96 1 16.32 1 16.91 1 1 7.94 1 18.42 119.31 120.61 121.6

122.26 1 23

123.97 124.83 125.34 125.73 126.43 126.72 127.28 127.76

Adhemnd Thickness (mm) 12.7

Bondine Thickmss (mm) 0.4 NIA

MPa

Eh3/12 (Flexural rigidity)

Adual Crack Lengih (mm)

90.25 91.62 92.05 92.5 93.6

94.36 94.7

95.65 96.1

97.06 98.85 99.6

100.37 101.17 101.66 102.02 102.61 103.64 104.12 105.01 106.31 107.3

1 O?.% 108.7

109.67 110.53 111.04 111.43 11213 1 12-42 112.98 1 13.46

Actual Crack Lsngth (m)

0.09025 0.09162 0.09205 0.0925 0.0936

0.09436 0.0947

0.09565 0.0961

0,09706 0.09885 0.0996

0.10037 0.10117 0.10166 o. 10202 0.10261 0.1 0364 0.10412 0.10501 0.10631

O. 1073 0.10796 0.1087

0.10967 0.1 1053 0.1 1 104 0.1 1143 0.1 3213 0.1 $242 0.1 1298 0.1 1346

Analysis of Data

Note: the values with ' adjacent wilt k discarded due to lncrsrslng crack Iength 1 spoiîed masuremnts

Gc- Average 1706.01 J 1 mA2

0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12

crack length (m)

Specimn No. Ki Test Urne: April9lh.2001

Adherend: 6061-T6 Ad hesive: Essex 1044-3

Mode Ratio O

Average

Load Jig Geometry s l s2 s3 54

Test Uesuîts

Offsets Specirnem

Microscope

Force (N) 767

1664 1725 1 800 1883 1943 1929 1920 1933 1934 1925 1900 1971 1966 1966 1960 1982 1956 1918 1882 7 8% 1829 1824 18% 1808 1800 1 782 1 780 1 740 1 700 1690 1680 1662

Width (mm) Adharand lhkkness (mm) 19.16 12.7 19.46 19.26 Bondlins ïhickness (mm) 19.58 0.4 19.37 NIA

69000 MPa

1 1778.20 Eh311 2 (flexural rigidiîy)

Measursd Crack Lanth (mm)

89.5 90.62 91.28 92.22 93.08 93.9

94.57 95.1 95.4

%.OS 96.75 97.02 97.8

98.46 99.4

99.83 101.1

102.12 103.73 105.15 106.11 106.95 107.65 109.15

1 10.3 711.4

11 1.89 1 13.6

1 15.46 116.5

1 17.48 118.06 1 19.07

Aclual Actual Crack Lenplh (mm) Cmk Length (m)

0.07496

Analysis of Data

Note: the values with ' adjacent will be discarded due to imasing crack Iength I spolied measummnts

Gc-Average 1662.94 J I mA2

crack length (m)

Specimn No. #4 Test tim: April 1 lîh,2ûû1

Specirnen Dimensions:

Wdth (mm) 19.54 19.45 19.46 19.57

Average 19.51

Adhsmnd Material Proparties

Test Resulrs

offsets Speclmm 140

Microscope 154.15

Force (N) 760

1009 1600 1692 1790 1800 1883 1830 1810 1862 1800 1776 1730 1 700 1618 1650 1610 1600 1580 1560 1570 1550 1535 1510 lm 1476 1457 1460 1452 1448 1440 1443 1414

Masurad Crack Lenth (mm)

99.26 99.96

100.82 101 ,SB 102.64 104.87 106.94 107.75 109.01 110.04 112.07 114.37 1 15.37 1 16.68 119.5

120.86 121.66 122.77 123.68 124.85 125.4

125.91 127.07 128.75 t 29.79 131.13 131.72 132.04 132.52 133.57 134.07 135.6

136.38

Adherend Thlckness (mm) 12.7

Bandline Thickness (mm) 0.4 NIA

MPa

Eh311 2 (F lexural rigidiîy)

Actual Crack Lsngth (mm)

85.1 1 85.81 86.67 87.43 88.49 90.72 92-19 93.6

94.86 95.89 97.92

100.22 101.22 10253 105.35 106.71 lO7Sl 108.62 109.53 110.7

111.25 111.76 1 12.92 114.6

1 15.64 116.98 Il7.57 1 1 7.89 118.37 1 19.42 1 19-92 121.45 122.23

Adual Crack Lenglh (m)

0.0851 1 0.08581 0.00667 0.08743 0.08849 0.09072 0.09279 0.0936

0.09486 0.09589 0.09792 0.10022 0.10122 0.10253 0.10535 0.10671 0.10751 0.10862 0.10953 0.1 107

0.1 1125 0.1 1176 0.11292 0.1 146

0.11564 0.1 1698 0-1 1757 0.1 1789 0.1 1837 0.1 1942 0.1 1992 0.12145 0.12î23

Analysis of Data

Note: the values with 'adjacent will be discardod due 10 increasing crack imgth 1 spoiled measurements

crack length (m)

Spacimen No. #5 TM! ü m : April, 13th,01

Mode Ratio

Specimen Dimensions:

Average

Loed Jlg Gsametry s 1 s2 s3 s4

Test Resulls

mssts Specimem

Microscope

Width (mm) A d M Thickness (mm) 19.59 12.7 19.4

19.61 Bondline Thicknsss (mm) 19.65 0.4 19.56 N A

Measured Crack Lenth (mm)

122.3 125.6 127.7

129 129.88 131 .O3 132.1

133.09 134.2

135.56 136.1 138.4

140.34 141.2 142.3 143.1

144.45 145.37

146 146.8

147.67 148.9 149.8 150.3

151.56 152.23 153.2 153.8

154.12 155.09 156.3

157.44

Actual Crack Longth (mm)

107.52 110.82 1 12.92 114.22 115.1

116.25 117.32 198.31 1 19.42 120.78 121.32 123.62 125.56 126.42 127.52 128.32 t 29.67 130.59 131 22 132.02 13289 134.12 135.02 135.52 136.78 137.45 138.42 139.02 139.34 140.31 141.52 142.66

Adwl C m k Length (m)

O. 10752 0.1 1082 0.1 1292 0.1 1422 0.1 151

0.1 1625 O. 1 1732 0.1 1831 0.1 1942 0.12070 0.12132 0.12362 0.1 2556 o. 12642 0.12752 0.12832 O. 1 2967 0.1 3059 0.13122 O.t32O2 0.13289 0.13412 0.13502 0.1 3552 0.13678 0.13745 0.13842 O. 13902 0.13934 0.14031 0.14152 0.1 4266

Note: the values with ' adjacent wilt be discaidsd due to incmasing crack Iongai / spaikd mesuremonts

Gc-Average 1 735.72 J 1 mA2

crack lcngth (m)

Speclnwn No. iY6 Test ffm: April, 14ü1,Ol

Adbisnd: 6061-16 Ad hesive: Essbx 1044-3

Mode Ratio 16.1

Specimen Dimensions:

Wdth (mm) 19.4 19.5 19.7

19.56 Average 19.54

Load Jig Geornetry sl 92 s3 s4

Test Results

Force (N) 7 80

1254 1490 1720 1820 1910 1895 1888 1840 1 860 1875 1820 1815 la00 1 797 1 750 1715 1734 1 740 1680 1677 1650 1638 1670 1635 1592 1585 1573 1566 1540 1570 1557

Msasured Crack Lenth (mm)

124.3 126.3

129.22 130.4 1 32.3

133.54 134.9 135.5

137 137.56 138.5 140.3

14254 143.29 144.73 145.13 146.88

1 48 148.4

149 149.33 150.9 151.4

153.21 155.32 156.04 156.9 157.4

1 58 158.62

159 159.5

Adbmnd Thicknass (mm) 12.7

BondIlne Thkkness (mm) 0.4 NIA

Actual Crack Lenglh (mm)

110.3 1 12.3

115.22 1 16.4 118.3

1 19.54 120.9 121.5

123 123.56 124.5 126.3

128.54 129.29 130.73 131.13 132.88

134 134.4

1 35 135.33 136.9 137.4

139.21 141.32 142.04 142.9 143.4

1 44 144.62

1 45 145.5

Adwl Crack Length (m)

0.1103 0.1 123

0.11522 0.1 164 0.1 183

0.11954 0.1209 0.1215

O. 123 0.12356 0.1245 0.1263

O. 1 2854 O. 1 8 2 9 0.13073 0.131 13 0.1 3288

0.134 0.1344 0.135

0.13533 0.1369 0.1374

O. 1 3921 0.14132 0.14204 0.1429 0.1 434 0.144

0.14462 0.145

O.? 455

Analysis of Data

Note: the values with *adjacent will be discarâed dw to increasing crack bngth 1 spoiled maasuisments

Gc-Average 1 772.79 J 1 mA2

O. I OS 0.1 15 O. 1 25 O. 135 0.145 O. 155

crack length (m)

Spacimen No. #7 Test tlme: April, 17th,01

Adhmnd: 6061-T6 Adhssive: Essex 1044-3

Mode Ratio 27.5

Specimsn Dimensions:

Width (mm) 19.55 19.67 19.4 19.5

Average 19.53

Adherend Material Proparties

Load Jig Geometry sl s2 s3 s4

Test Results

offsels Specimem

Microscope

Force (N) 830

1100 1340 1650 1690 1687 1675 1630 1615 1610 1605 1595 1580 1575 1600 1560 1536 1522 lsoo 1532 1511 1495 1486 1450 1430 1422 1413 1410 1405 1390 1385 1400

Measured Crack Lenth (mm)

120 1 Z2.34 123.4 125.3

126.78 127.8

1 29 1 30.23 130.9

131.45 132.6

132.57 133.3 T 34.2

135.23 136.4 137.5

138.23 139.1

140.41 141.7

142.56 143.68 144.55 145.9

146.72 147.81

148 148.9

149.23 1 50.43 151.64

Adherend Thkknsss (mm) 12.7

Bondlina Thicknsss (mm) 0.4 NIA

MPa

Eh3112 (flexurai rigldity)

Adual Crack Lengîh (mm)

108.6 110.94

112 113.9

115.38 116.4 1 1 7.6

118.83 119.5

120.05 121.2

121.17 121.9 1228

123.83 125

126.1 126.83 127.7

129.01 130.3

131 -16 132.28 133.15 134.5

135.32 136.41 136.6 137.5

137.83 139.03 140.24

Actwl Crack Length (m)

0.1086 0.1 1094

0.1 12 0.1 139

0.1 1538 0.1 164 0.1176

0.1 4883 0.1 195

o . l m 5 0.1212

0.12117 0.1219 0.12î8

0.12383 0.125

0.1261 0.1 2683 0.1277

0.12901 0,1303

0.13116 0.13228 0.13315 0.1345

0.13532 0.13641 0.1366 0.1375

0.13783 0.13903 0.14024

Analysis of Data

Note: the values with ' adjacent will bs discarded dua ta incmasing crack Isngth 1 spollsd masuremen(s

Gc-Averaqe 1844.76 J 1 rnA2

0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 0.145

crack lcngth (m)

Spsclmn No. #8 Test t im: April, 201h,01

Mode Ratio

Speclmen Dimensions:

Average

Load Jig Geometry sl s2 93 s4

Test Results

cnk8ts Specimern

Microscope

Force (N) 810

1010 1430 1 580 1693 1677 1640 1620 1610 1590 1585 1578 1530 1545 1533 1505 1498 1486 1490 1506 1450 1477 1464 1438 1420 1417 1400 1421 1410 1408 1 385 t 365

Wldth (mm) Adhersnd Thickness (mm) 19.87 12.7 19.6

19.76 Bondlino Thickness (mm) 19.5 0.4

19.68 NIA

69000 MPa

1 1778.20 Eh311 2 (Fiaura1 rigidlty)

Measud Crack Lenth (mm)

1 23 125.4 127.8 128.4

129.65 130.6

133 135.67 136.1 137.3

138.03 139.1 140.4 141.1 142.3 143.1

144 144.5 145.2 146.3 147.2

148.13 149

150.5 150.9 151.4 152.2 153.1

19.08 155.1

t 56.53 157

Actual Crack Longth (mm)

1 t0.2 112.6

115 115.6

li6.85 117.8 120.2

122.87 123.3 124.5

125.23 126.3 127.6 128.3 129.5 130.3 131.2 131.7 1324 133.5 134-4

135.33 136.2 137.7 138.1 138.6 139.4 140.3

141.28 1423

143.73 144.2

Actual Crack L w t h (m)

0.1 102 0.1 126 0.115

0.1 156 0.11605 0.1 178 0.1202

0.12287 0.1233 O. 1245

0.1 2523 0.1 263 0.1276 0.1283 0.1295 0.1303 0.1312 0.1317 0.1324 0.1 335 O. 1344

0.1 3533 O. 1362 0.13731 0.1381 0.1 386 0.1394 0.1403

0.14128 0.1423

0. 14373 0.1442

Analysis of Data

Note: Vis values with ' adjacent will be discardad due to increasing crack !mgth I spdrsd msuraments

crack length (m)

Specimen No. # l -O degrea and 48 dagrsas Test tirne: April21 th,Ol

Adhemnd: 6061-T6 Adheslve: Essex 1044-3

Mode Ratio

Spscirnen Dirrtensions:

Average

Load Jig Gaometry sl s2 s3 !à4

Test Rssults

Microscope

Force (N) 365 669 797

t O98 1289 1319 1406 1498 1525 1597 1604 1635 1667 T m 0 1 730 im 1810 1 894 1 900 1867 1872 1861 1893 1830 1845 1840 1806 1800 1802 1812 1 798 1787

MdIh (mm) Adherend Thicknsss (mm) 19.89 12.7 19.9 19.8 BondIlne Thicknsss (mm)

19.95 0.4 19.89 NIA

69000 MPa

1 1 778.20 Eh311 2 (Flexural rlgiôity)

Measursd Crack Lenlh (mm)

131.2 131.95 134.27 137.01 138.12 139.12 140.05 140.75 141.8

142.33 143.45 145.57 146.48 1 47.08 147.83 148.85 149.7 150.3

150.43 152.27 152.66 153.79 155.05 155.73 155.98 156.36 156.78 158.34 158.6

159.33 t6O.05 161.31

Actual Crack Lenglh (mm)

128.12 128.87 131.19 133.93 135.04 136.04 136.97 1 37.67 138.72 139.25 140.37 142.49 143.4

1 44 144.75 145.?7 146.62 147.22 147.35 149.1 9 149.58 150.71 751.97 152.65 1529

153.28 153.7

155.S 155.52 156.25 156.97 158.23

SXL Prfmary 405 Secondary

Acîwl Crack Length (m)

0.12812 0.12887 0.131 19 0.13393 0.13504 0.13604 0.13697 0.13767 0.13872 0.1 392s 0.14037 0.14249 0.1434 0.144

0.14475 0.14577 O. 14662 0.14722 0.14735 0.14919 0.14958 0.15071 0.15197 0.1 5265 0.1529

0.1 5328 0.1537

0.1 5526 0.15552 0. 1 562s 0.15697 0.1 5823

0.16005 0.16146 0 . t m O.167f 2 0.16901 0.1 7019 0.17152 0.1 7322 O.? 7468 0.1 76û2 0.1 7652 0.17812 0.1 7922 0.18'142 0.18196 0. t 8342 O. 18526

Analysis of Data

Note: the values wilh ' aâjaeent will bs discardeci due fo incmaslng crack length 1 spaikd measurements

Gc- Average 3263.29 J I mA2

0.125 0.135 0.145 0.155 0.165 0.175 0.185

crack tength (m)

Specimn No. W Test Ume: April22,Ol

Spscimen Dimensions:

Width (mm) 19.54 19.45 19.46 19.57

Average 19.51

Load Jig Gsomtry sl 92 s3 s4

Test Results

offsets Specirnem

Microscope

F o m (N) 523 705 826 955

1100 1303 1432 1419 1516 1570 1640 1688 1694 1693 1730 1688 1690 1 680 1710 1667 1680 1650 1657 1640 1581 1 646 1620 1610 1607 1569 1560 1554

Measursd Crack Lenth (mm)

136.72 137.39 138.14 140.19 141.58 144.6

146.6 1 146.9

148.05 150.37 153.8

155.13 158.3

160.55 161.62 164.67 165.47 166.3

168.29 169.1

171 3 7 172.9

173.43 174.11 175.05 176.1 1 1 77-05 178.1

179.26 180.95 183.31 185.1

Adhafend Thkkness (mm) 12.7

Bondlim ThIckri8ss (mm) 0.4 NIA

Actwl C m k Longth (mm)

135.07 135.74 136.49 138.54 139.93 142.95 144.96 145.25 146.4

148.72 152.15 153.48 156.65 158.9

159.97 163.02 163.82 164.65 166.64 167.45 169.72 1 71.25 171.78 1 72.46 173.4

174.46 175.4

1 76.45 177.61 179.3

181.66 1 83-45

Actual Crack Length (m)

0.13507 0.13574 0.13649 0.1 3854 0.13993 0.14295 O. 14496 O. 14525 0.1464

O. 14872 0.15215 0.1 5348 0.15665 0.1589

0.1 5997 O. 16302 O. t 6382 O. 1 6465 O. 1 6664 0.16745 O. t 6972 0.17125 0.17178 O. 1 7246 0.1 734

0.1 7446 O. 1 754

O. 1 7645 o. t 7761

O. 1793 0.18166 O. 1 8345

Noie: Via values with 'àdjwnt will be discarded due to incrsasing crack langth I spoilsd mawmmsnts

Gc-Average 3331.53 J l mA2

crack length (m)

Specim No. #9 Test Ume: Apfil25.01

Adhemnd: 6061 -16 Adhesive: Esssx 1044-3

Mode Relio

Specimn Dimensions:

Average

Adherend Matenal Properties

Load Jig Geometry s 1 s2 s3 s4

Test Results

011sets Specimem

Microscope

Force (N) 646

121 7 1420 1509 1 608 1666 1 704 1760 1810 1854 1889 1906 1928 1932 1918 1940 1900 1910 1860 1850 1855 1816 1756 1 745 1732 1692 1659 1596 1630 1610 1566 1 583

Crack Lent!? (mm) 119.73 124.45 125.01 125.84 127.13 130.39 131.02 132.03 133.88 135.03 t38.51 141.3

142.26 144.35 145.09 146.9

147.81 149.05 150.58 f 51.43 152.5

156.73 159.4

161.42 16216 165.64 166.79 168.52 170.61 172.65 173.45 174.83

Adherend Thicknsss (mm) 12.7

Bondlino ïhickness (mm) 0.4 NIA

MPa

Eh311 2 (Fiexutal rl~idlty )

Adual Crack Lenglh (mm)

1 18.36 123.08 1 23.64 124.47 125.76 129.02 129.65 1 30.66 132.51 133.66 137.14 139.93 140.89 T 42-90 143.72 145.53 146.44 147.68 149.21 150.06 151.13 155.36 158.03 160.05 160.79 1ô4.27 165.42 167.15 169.24 177.28 17208 173.46

ActuA Crack Lsngth (m)

0.1 1836 0.12308 0.1 2369 O. 12447 O. t ZS76 0.12902 0.1 2965 0.13066 0.13251 O. 13366 0.13714 0.1 3993 0.14089 0.14298 0.14372 0.14553 O. 14644 0.14768 0.14921 0,15006 0.151 13 O. 15536 0.15803 0.16005 0.16079 0.16427 0.16542 O. 1671 5 0.16924 0.17128 O. 1 7208 O. 1 7346

Note: the values with ' adjacent will be discardeci due to inmashg crack iength 1 spdbd msuremnts

crack length (m)

Specimn No. #lO Test rime: May lst,Ol

Adherend: 6061-16 Adhasive: Esssx 1044-3

Mode Ratio 68.95

Sgaclmen Dimensions:

Width (mm) 19.51 19.83 19.59 19.32

Average 19.56

Adherend Materlai ProperUes

Load Jig Geometry SI s2 s3 s4

Test Results

Onsets Specimem

Microscope

Force (N) 947 1090 1316 1583 1743 1794 1987 2130 2247 2455 2463 2554 2552 2640 2683 2695 2701 2689 2710 2659 2684 2638 2645 2569 2598 2602 2490 2538 2513 241 3 2388 2410

Measwsd Crack Lenth (mm)

141.94 142.79 144.41 146.17 146.94 147.7 148.7 149.94 152.48 155.16 155.87 157.13 158.35 161.42 162.56 163.5 t64.67 165.12 166.34 167.55 169.15 169.71 170.1 171 .O5 172.36 174.03 175.09 176.68 178.6 180.65 181.34 182.6

Adherend Thkkms (mm) 12.7

ûondline Thlcknsss (mm) 0.4 NIA

MPa

Eh311 2 (Ftexural rigidlty)

Acîual Crack Length (mm)

126.61 t 27.46 129.08 130.84 131.61 132.37 133.37 134.61 137.15 139.83 140.54 141.8 143.02 146.09 147.23 148.17 149.34 149.79 151.01 152.22 153.82 154.38 154.77 155.72 157.03 158.7 159.76 161.35 163.27 165.32 166.01 167.27

Analpis of Data

Note: the values with adjacent will ôe discarâed due to incrsasing c m k iength / tgoikd measmmnts

Gc-Average 4357.89 J 1 mA2

crack tength (m)

Tsst Ume: May 4.01

Adhafend: 6061 -T6 Adhesive: WX 10443

Mode RaUo 68.95

Spscimen Dimensions:

Width (mm) 19.56 19.71 19.75 19.85

Average 19.72

Adheisnd Material Properties

Load Jig Geometry s 1 s2 s3 s4

Test Results

Farce (N) 1315 1 774 1 786 1936 Mg9 2164 2254 2335 2386 2414 2461 2577 2û13 2646 2658 2638 2643 2630 2603 2564 2498 2500 248 1 2478 2450 2431 2484 2389 2370 2345 2381 2313

Muasureci Crack Lenth (mm)

150.86 1S.58 155.55 156.92 157.3 158.05 160.37 161.59 162.1 163.85 165.34 166.85 167.82 769.32 171.83 174.02 174.75 176.02 177.8 178.33 180.02 182.35 163.46 185.69 186.23 186.98 187.3 188.45 189.1 1 19O.76 191.5 191.9

Bondlins Thicknsss (mm) 0.4

NIA

MPa

Eh311 2 (Rexurai rigidify)

Adual Crack Lingth (mm)

134.92 138.64 139.61 140.98 141.36 142.11 144.43 145.65 146.16 147.91 149.4 150.91 151.88 153.38 155.89 158.08 l58.8l 160.08 161 .û6 162.39 164.08 166.41 167.52 169.75 170.29 1 71 -04 171 36 172.51 173.17 174.82 175.56 175.96

Acml Crack Lingai (m)

0.13492 O. 13864 0.13961 0.14098 0.14136 0.1421 1 0.1 4443 0.14565 0.14616 0.14791 0.1494 0.15091 O. 151 88 0.15338 0.15589 0.15808 0.15881 0.16008 0.16186 0.16239 0.16408 0.16641 0.16752 0.16975 0,17029 0.17104 0.17136 0.1M51 0.17317 O. 17482 O. 1 7556 0.175%

Note: the values with 'adjacent will be discardeci due to i m s i n g crack length 1 spoilsd muasuremonts

Gc-Average 4540.85 J 1 mA2

Botamab 10113 Modo rr#o 8.95 dagnrr (DCB ml)

0.115 0.125 0.135 0.145 0.155 0.165 0.175 0.185

crack length (m)

Specimsn No. #12 -68.95 dagrees Test Um: May 6,Ol

Mode Ratk 68.95

Specimen Dimensions:

Wdth (mm) 19.43 19.58 19.6 1 19.63

Average 19.S

Load Jlg Gsometry sl s2 s3 s4

Test Results

Force (N) 1677 1 923 2100 2434 2510 263 1 2666 2674 2705 2700 2718 2670 2628 2550 2553 2593 2490 2477 2474 2470 2435 241 3 2352 2334 2364 2389 2316 2290 2315 2230 2286 2250

Maasured Crack Lenlh (mm)

148.2 150.34 153.65 156.32 157.89 161.33 162.71 164.3

165.28 167.3 168.4

170.73 171.2

172.56 174.5

176.19 t78.33 179.1 180.3 182.8 183.4

184.66 186.2

187.03 187.9

188.34 790.3

i 9 i . n 193.1 194.6 195.3 195.8

Bondfina Thickneas (mm) 0.4

NIA

MPa

Eh311 2 (Flexurat rigidity)

Acfual Crack Length (mm)

132.42 134.56 137.87 140.54 142.1 1 145.55 146.93 148.52 149.5

151.52 t 52.62 154.95 155.42 156.70 158.72 160.41 162.55 163.32 164.52 167.02 167.62 t 68.88 170.42 171 .25 172.12 172% 174.52 i 75.93 1 77.32 178.02 179.52 180.02

Actwl Crack Length (m)

0.13242 0.13456 O. 13787 0.14054 O.142t 1 0.14555 0.14693 0.14852 0.1495

0.15152 O. 1 5262 0.15495 0.15542 0.15678 0.15B72 0.16041 0.16255 O. 16332 0. 7 6452 0.16702 0.16762 0.16888 0.17042 0.17125 0.17212 0.1 7256 0.1 7452 0. 17599 O. 17332 O.lfôô2 0.17952 0.18002

Anaiysis of Data O

Note: the values with adjacent will be discarâeci dm to increaslng crack longth 1 spdled measummcnts

Gc-Average

so

crack length (m)

S~ecimen No. Hl3 -90 dagrecs Test tirne: May 26.01

Mode Ratio 90

Specimen Dimensions:

Wdth (mm) 12.5 12.5 12.5 12.5

Average 12.50

Load Jig Geometry sl s2 s3 s4

Test Results

msdts Specimem

Microscope

Force (N) 1677 2300 3513 4100 5200 6104 6346 671 0 721 0 7310 7243 6935 7233 7002 6898 6972 6883 6790 6773 681 7 6702 6659 ô491 631 0 6617 6423

Measured Crack Lenth (mm)

150 155.8 158.2

163.43 165.82 169.23 171.1 173.2

1 75.34 177.2

178.89 180.05 162.3

183.19 184.57 186.77 187.39 188.4 1

1 89 191 .25 19234 193.55 194.72 195.03 197.4

1 98

Adherend Thickness (mm) 33

BondIlne fhickness (mm) 0.8

NIA

Actual Crack LengVi (mm)

140 145.4 f 48.2

153.43 155.82 159.23 161.1 163.2

165.34 167.2

168.89 170.05 172.3

173.19 :74.57 176.77 177.39 178.41

1 79 181.25 182.34 183.55 184.72 185.03 187.4

1aa

Note: the values with ' adjacent wilf be discardsd due to incmasing crack mgfh f spoiiad msummants

Gc-Average 5528.57 J 1 mA2

crack length (m)

Specimen No. Test tim:

Mode Ratio

1114 -90 dagtess Jum 4,Ol

6061 -T6 Essex 1044-3

Specimn Dimensions:

Width (mm) Adbmnd Thlckness (mm) 12.6 34.1 12.6 12.6 Bondine Thicknsss (mm) 12.6 0.4

Average 12.60 NIA

E 69000 MPa

Load Jig Geomstry sl 3 s2 3 s3 6 s4 6

Test Results

Measured Acîual Actual Force (N) Crack Lenth (mm) Crack LsngUi (mm) Crack Lsngth (m) F I F2 G

Analysls of Data

Note: th vatws wiVi ' adjacent will be discarded dus to incrsosing crack I q t h 1 spoilad m s a w m t s

crack lcngth (m)