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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
thic kness,
[1] Adams, R.D. and Peppiatt, N.A., "Stress analysis of adhesive bonded tubular lap
joint", Journal of Adhesion, 9, 1 977, 1 - 1 8.
[2] Adams, R.D., "Stress analysis: A finite element anaiysis approach, Develo~ments
in Adhesives-2, Edited by A.J. Kinloch, Applied Science Publishea, 198 1.
[3] Adams, R.D., Atkins, R.W., Harris, J.A. and Kinloch, A.J., "Stress analysis and
failure properties of carbon-fibre reinforced-plastic/steel double-lap joints".
Journal of Adhesion, Vol. 20, 1986,2962.
[4] Anderson, G.P., Devnes, K.L. rnd Williams, M.L., "Peel test in experirnental
adhesive- fracture mechanics", Experimental hlechanics, 24, 1 976, 1 1 .
[SI ASTM, "Standard test method for climbing d m peel for adhesives", Amencan
Society of Testing Materials (ASTM), ASTM D 1 78 1-98.
[6] ASTM. "Standard test method for floating roller peel resistance of adhesives':
Arnerican Society of Testing Materials (ASTM), ASTM D3 167-97.
[7] ASTM, "Standard test methods of tension testing wrought and cast alurninum and
magnesium alloy products [metric]", Amencan Society of Testing Materials
(ASTM), ASTM D557M-94.
[8] ASTM, "Standard test method for tende properties of plastics", American Society
of Testing Materials (ASTM), ASTM D638-99.
[9] Barenblatt, G.I., "The mathematical theory of equilibriurn cracks in bnnle
fracture", Advances of Appiied Mechanics, Vol. 7, 1 962,SS- 1 29.
[10] Bigwood, D.A. and Crocombe, A.D., "Elastic anaiysis and engineering design
formulae for bonded joints", International Journal of Adhesion and Adhesives,
Vol. 9, 1989, 229-24 1.
[Il] Bigwood, D.A. and Crocombe, A.D., "Non-linear adhesive bonded joint design
analyses", I~ternational Journal of Adhesion and Adhesives, Vol. 10, 1990,3 1-4 1.
[12] Bikerman, J.J., "Theory of peeling through a hookean solid", Journal ofApplied
Physics. 28, 1957, 1484- 1485.
[ 1 31 Cao, H.C. and Evans, A.G.. "An expenmental study of the fracture tesistance of
bimaterial interfaces", Mechanics of Materials, 7, 1 989,295-304.
[14] Chang, MD., Devries, K.L. and Williams, M.L., "The effects of plasticity in
adhesive fracture", Journal of Adhesion, Vol. 4., 1 972,22 1 .
[ l S] Charalambides, M.A., Kinloch, A.J., Wang, Y and Williams, J.G., "On the
anal y sis of mixed-mode fracture", lnternutional Journal of Fracîi4re. Vol. 54.
1993,269-29 1.
[16] Chen, W.T. and Falvin, T.F., "Mechanics of tilm adhesion: elastic and elastic-
plastic behavior." IBM Journal cf Research and Development, Vol. 1 6, 1 972, 203-
213.
[ 171 Chiang, M.Y .M. and Chai, H., "Plastic deformation analysis of cracked adhesive
bonds loaded in shear", International Journul of Solids and Structures. 3 1, 1 994.
2477-2490,
[la] Cooper, P.S. and Sawyer, J. W., NASA Technical Report, 1979, 1 507- 1 5 1 8.
[19] Comell, R. W., Journal of Applied Mechanics, Transactions of ASME, 75, 1953,
3 55.
[20] Crocombe, AD. and Adams, R.D., "An elasto-plastic investigation of the peel
test", Journui of Adhesion, Vol. 13, 1982,24 1-267.
[ZI] Crocombe, A.D. and Adams, R.D., "Peel analysis using the finite element
method", Journui MAdhesion, Vol. 12, 198 1, 1 2% 1 39.
[22] Crocombe, AD. and Bigwood, D.A., "Development of a full elastic-plastic
adhesive joint design analysis", Journal ofSfmin Analysis, Vol. 27. No. 4. 1992,
21 1-218.
[23] Delale, F. and Erdogen, F., NASA NGR 39-007-01 1, Lehigh University,
Bethlehem, Pennsylvania, 1 980.
[24] Delale. F. and Erdogen, F., NASA NOR 39-007-0 1 1, Lehigh University.
Bethlehem. Pe~sylvania, 198 1.
[î 51 Donne, C. D., "The crack tip displacement vector approach to mixed-mode
fracture", ASTM STP 1359. 1999,21-40.
[26] Dowling, NE., Mechanical Behavior of Materials, Prentice-Hall Inc.. New
Jersey, USA, 1999.
[27] Femlund, G., "Quasi-static failure of structural adhesive joints", M.A.Sc. Thesis,
University of Toronto. 199 1.
[28] Femlund. G. and Spelt, J.K., "Analytical method for calculating adhesive joint
fracture parameters", Engineering Fracture Mechanics, Vol. 40, 1 99 1, 1 19- 1 32.
1291 Fernlund, G. and Spelt, J.K., "Mixed-mode fracture characterization of adhesive
joints", Composites Science and Technology, 50, 1 994,44 1 -449.
[30] Femlund, G. and Spelt, J.K., "Mixed-mode energy release rates for adhesively
bonded beam specimens", Journul of Composites Technolugy & Research, Vol. 1 6,
1 994,2340243.
[3 11 Gent, A.N. and Hamed, G.R., "Peel mechanics for an elastic-plastic adherend",
Journal of Applied Poiymer Science, 2 1, 1 977,28 1 7-283 1.
[32] Gent, A.N. and Hamed, G.R., "Peel mechanics of adhesive joints". Polymer
Engineering and Science, 1 7, 1 977,462-466.
[33] Gledhill, R.A., Kinloch, A.J., Yarnini, S., Young, R.J., "Relationship between
mechanical properties of and crack propagation in epoxy resin adhesives".
Polymer, 19, 1978,574.
[34] Goland, M. and Reissner, E., "The stresses in cemented joints". Journal of
Applied Mechanies. Vol. 66, 1 944, 1 7-27.
[35] Grant, P. and Taig, 1. C., "Strength and stress anaiysis of bonded joints", BAC
Report No. SOR (P), 1979. 10% 1 14.
[36] Greenwood, L., Boag, T.R. and Mclaren, AS.. Adhesion: Fundamentals and
Practice. Ministry of Technology, Maclaren, London, 1969.
[37) Griflïth. A.A., 'The phenornena of rupture and flaw in solids", Philosophical
Transaction Royal Society London, Series A, 22 1, 1 920, 163- 198.
[38] Hartsmith, L.J., **Designhg to minimize peel stresses in adhesive bonded joints",
ASTM STP 876, Edited by W.S. Johnson, American Society of Testing and
Materials, Philadelphia, PA, 1985,238-266.
[39] Hartsmi th, L.J., "Stress analysis: a continuum mechanics approach".
Develovments in Adhesives-2, Edited by A.J. Kinloch, Applied Science Publishers,
1981, 1-45.
[40] Invin, G.R., "Analysis of stresses and strains near the end of a crack traversing a
plate", Journal of Applied Mechanics, 24, 1957,36 1-364.
[41] Ishai, O., Peretz, 0. and Gali, S., "Direct determination of interlaminar stresses in
pol y meric adhesive Iayer", Experimentaf Mechanics, Vol. 1 7, 1 977,265.
1421 Kaelble, D.H. and Ho, C.L., "Biaxial bond stress analysis in peeling".
Transactions of Society of Rheology, 1 8, 1974,2 19-23 5.
[43] Kaelble, D.H., "Peel adhesion", Adhesives Age, 3, 1960,37042.
[44] Kaelble, D.H., Transactions of Society of Rheology, 4, 1960,4549.
[45] Kanninen, M.F., "An augmented double cantilever beam mode1 for studying
propagation and arrest", lnternational Journal of Fracture, 9, 1 973,8399 1.
[46] Kenball, K.. "Thin-film peeling - the elastic terni", Journal of Applied Physics,
Vol. 8, 1973, 1449-1452,
[47] Kim, J. and Kim, K.S. and Kim, Y.H.. "Mechanical effects in peel adhesion test".
Journui of Adhesion Science and Techno fogy, 3, 1 989, 1 75- 1 87.
[48] Kim, K.S. and Avaras, N., "Elastic-plastic analysis of the peel test", Infernarional
Journal ofloiids and Structures, 24, 1 988,2 1 4-23 5.
[49] Kim. K.S. and Kim. J., "Elastic-plastic anaiysis of the peel test for thin film
adhesion", Journal ofEngineering Materials und Technology, 1 1 0, 1988,266-273.
[SOI Kinloch, A.J. and Hadavinia , H., " The peel behavior of adhesive joints".
Proceedings of the ~ 3 ' ~ Annual Meeting ojAdhesive Society, 2000.
[51] Kinloch, A.J. and Shaw, S.I., "The fracture resistance of a toughened epoxy
adhesive", Journal ofAdhesion, Vol. 12, 1981.59-77.
[52] Kinloch, A.J., Adhesion and Adhesives-Science and Technolopy, Chapman and
Hail Ltd., NY, USA, 1987.
[53] Kinloch, A.J. and Young, R.J., Fracture Behavior of Polvmers, Applied Science
Publishers, London, 1983, 1 16- t 17.
[54] Kinloch, A.J., Lau, C.C. and Williams, G.J., "The peeling of flexible laminates".
lnrernnrional JournaI of Fracture, 66, 1 994,45 -70.
[55] Kody, R.S., and Lesser, A.J., "Deformation and yield of epoxy networks in
constrained states of stress", Journal of Materials Science, Vol. 32, 1997, 5637-
5643.
1563 Livey, D.T. and Mccarthy, J.C., "Composite-metal joining technology for vehicle
weight reduction", EC Report EUR 10551, EN, 1986.
[57] Lesser, A.S. and Kody, R.S., "A generalized mode1 for the yield behavior of
epoxy networks in multiaxial stress states", Journal of Polymer Science. 1997,
161 1-1619.
[58] McClintock, F.A. and Invin, G.R., "Plasticity aspects of fracture mechanics".
ASTM STP 381, 1965,96,
1591 Moidu, A.K., Sinclair, A.N. and Spelt, J.K., "Analysis of the peel test: prediction
of adherend plastic dissipation and extraction of fracture energy in metal-metal
adhesive joints", Journul of Tesring and Evaluation, Vol. 23, No. 4, 1995,2419253.
[60] Moidu, A.K., Sinclair, A.N. and Spelt, J.K., "On the determination of fracture
energy using the peel test", Journal of Testing and Evuluafion Vol. 26, No. 3,
1998,247-254.
[6 11 Needleman, A., "An analysis of tensile decohesion dong an interface", Journal of
the Mechanics and Physics of Solids, Vol. 38, 1990,289-324.
[62] Orowan, E.. "Fracture and strength of solids", R e m s on Progress in Physics, 1 2 ,
1948, 185-20 1.
[63] Packham, D.E., Handbook of Adhesion, Longman Scienti fic & Technical,
Harlow, England, 1 992.
[64] Papini. M., Fernlund, G. and Spelt, J.K., "The effect of geometry on the fracture
of adhesive joints", International Journal of Adhesion and Adhesives, 14, 1 994, 5-
13,
[65] Post, D., Czarnek, R., Wood, J.D. and Joh, D., "Deformations and strains in a
thick adherend Iap joint", ASTM STP 981, Edited by Johnson. W.S., American
Society of Testing Materials, 1988, 107- 1 18.
(661 Rice, J.R., Fracture - an Advanced Treatise, Academic Press, New York. 1968.
[67] Ripling, E.J., Mostovoy. S. and Patrick, R.L., "Application of fracture mechanics
to adhesive joints", ASTM STP 360, 1963.
[68] Ripling, E L , Mostovoy, S. and Patrick, R.L., "Measuting fracture toughness of
adhesive joints", Materials Research and Standards, ASTM 64, 1964.
1691 Ripling, E.J., Mostovoy, S. and Cortem, K., "Fracture mechanics: a tool for
evaluating structurai adhesivrs," Journal of Adhesion, Vol. 3,1971, 107-1 23.
[70] Shirani, A. and Liechti , KM., "A calibrated fiacture process zone mode1 for thin
film blistering", International Journal of F ~ u r e , Vol. 93, 1 998,28 1 -3 14.
[7 1 ] Suo, Z.G. and Hutchinson, J. W., "Interface crack between two elastic Iayers",
Intemutional Journal of Fracture, 43, 1 990, 1 - 1 8.
[72] Spies, G.J., "The peeling test on redux-bonded joints," Journal of Aircrafr
Engineering, Vol. 25, 1953,64-70.
[73] Tvergaard, V. and Hutchinson, J. W., "On the toughness of ductile adhesive
joints", Journal of the Mechanics and Physics of Solids, Vol. 44, 1996, 789-800.
[74] Tvergaard, V. and Hutchinson, LW., "The influence of plasticity on mixed-mode
interface toughness", Journal of the Mechanics and Physics of Solids. Vol. 4 1,
1993, 11 19-1 135.
[75] Tvergaard. V. and Hutchinson, LW., "The relation between crack growth
resistance and fracture process parameters in elastic-plastic solids", Journal qf the
Mechunics and Physics of Solids, Vol. 40, 1 992, 1 377- 1 3 97.
[76] Tvergaard, V. and Hutchinson, LW., "Toughness of an interface along a thin
ductile joining elastic solids", Philosophical Magazine A, 70, 1994.64 1-656.
[77] Varias, A.G., Zuo, 2. and Shih, C F . , "Ductile fracture of a constrained metal
FoiI", Journal of the Mechanics and Physics of Solids, 7 , 1 99 1 ,963-986.
(781 Vincent, RI., Poiymer, Vol. 12, 197 1,534.
(791 Volkersen, O., "Research on the theory of cemented joints'', Construction
Metalligue, No. 4, 1965,343.
[80] Volkersen, O., "Rivet strength distribution in tensile-stressed rivet joints with
constant cross-section", Luifahhrjorschung, Vol. 1 5,1938,4 1 -47.
[81] Wang, R.X., "Adhesive failure with aâherend yielding: experiments and
analytical models", M.A.Sc. Thesis, University of Toronto, 200 1.
[82] Wei, Y.G. and Hutchinson, J.W., "Interface strength, work of adhesion and
plasticity in the peel test", Inîernational Journal of Fracture, Vol. 93, 1998, 3 15-
333.
[83] Wei, Y.G. and Hutchinson, J. W., "Models of interface separaiion accompanied by
plastic dissipation at multiple scales", Internaiional Journal of Fracture, Vol. 95.
1999, 1-17.
[84] Westergaard, H.M., "Bearing pressures and cracks", Transactions oj' the
A rnerican Society of Mechanical Engineers, 6 1 . 1 939,4995 3.
[85] Williams, G.J.. Fracture Mechanics of Polvmers, John Wiley and Sons, New
York, 1984.
[86] Williams, G.J., "Root rotation and plastic work effects in the peel test". Journal of
Adhesion, 4 1. 1993,225-239.
1871 Wooley, G.R. and Carver, D.R., Journal ofAircru#, 8, 1971,817-823.
[88] Yamada, S.E., "Elastic-plastic fracture anal ysis for bonded joints", Engineering
Fracture Mechanics, Vol. 27, l987,3 15-328.
1891 Yang, Q.D., "Fracture analysis of plastically-defoming adhesive joints", Ph.D.
thesis, University of Michigan, 2000.
1901 Yang, Q.D., Thouless, M.D. and Ward, S.M., "Numerical simulations of
adhesively-bonded beams failing with extensive plastic deformation", Journal of
the hlechanics and Physics of Solids, Vol. 47, 1999, 1337- 1353.
[9l] Qum, S.C., "Mechanical properties of srnicnual adhesives", B.A.Sc. thesis,
University of Toronto, 1 99 1.
[92] ANSYS, "Finite element code and manual, version 5.5", ANSYS Inc. U.S.A.
[93] Chiu, 1, "Adhesive bonding of alurninurn sheet structures for automotive
applications", Sumrner Proiect Remrt, University of Toronto, 1999.
[94] ASTM. "Standard specification for alurninum and aluminum-alloy sheet and
plate", American Society of Testing Materials (ASTM), ASTM B209M-95.
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)
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