Special 2-D and 3-D Geometrically Nonlinear Finite Elements for Analysis of Adhesively Bonded ...

164 Pages·1998·0.89 MB·English

Special 2-D and 3-D Geometrically Nonlinear Finite
Elements for Analysis of Adhesively Bonded Joints
By
Raul H. Andruet
Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State
University in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Engineering Science and Mechanics
APPROVED:
__________________________ ____________________________
D. A. Dillard, Chairman S.M. Holzer, Co-chairman
__________________________ ____________________________
E. R. Johnson B.J. Love
__________________________
R. D. Kriz
April, 1998
Blacksburg, Virginia
Special 2-D and 3-D Geometrically Nonlinear Finite
Elements for Analysis of Adhesively Bonded Joints
By
Raul H. Andruet
David A. Dillard, Chairman
Siegfried M. Holzer, Co-chairman
Engineering Science and Mechanics
(ABSTRACT)
Finite element models have been successfully used to analyze adhesive bonds in actual
structures, but this takes a considerable amount of time and a high computational cost.
The objective of this study is to develop a simple and cost-effective finite element
model for adhesively bonded joints which could be used in industry. Stress and
durability analyses of crack patch geometries are possible applications of this finite
element model. For example, the lifetime of aging aircraft can be economically extended
by the application of patches bonded over the flaws located in the wings or the fuselage.
Special two and three- dimensional adhesive elements have been developed for stress and
displacement analyses in adhesively bonded joints. Both the 2-D and 3-D elements are
used to model the whole adhesive system: adherends and adhesive layer. In the 2-D
elements, adherends are represented by Bernoulli beam elements with axial deformation
and the adhesive layer by plane stress or plane strain elements. The nodes of the plane
stress-strain elements that lie in the adherend-adhesive interface are rigidly linked with
the nodes of the beam elements. The 3-D elements consist of shell elements that represent
the adherends and solid brick elements to model the adhesive. This technique results in
smaller models with faster convergence than ordinary finite element models. The
resulting mesh can represent arbitrary geometries of the adhesive layer and include
cracks. Since large displacements are often observed in adhesively bonded joints,
geometric nonlinearity is modeled.
ii
2-D and 3-D stress analyses of single lap joints are presented. Important 3-D effects can
be appreciated. Fracture mechanics parameters are computed for both cases. A stress
analysis of a crack patch geometry is presented. A numerical simulation of the debonding
of the patch is also included.
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Acknowledgements
I would like to express my sincere appreciation and gratitude to Dr. S.M. Holzer for his
teaching, guidance, patience, friendship, and support. I admire him as a teacher, as a
leader, and, above all, as a friend. I also want to express my gratitude to Dr. D.A. Dillard
for providing me expert advice in the mechanics of adhesives, and for his support and
understanding. I want to extend my gratitude to Dr. E.R. Johnson, Dr. B.J. Love, and
R.D. Kriz for serving on the dissertation committee, and to Dr R.H. Plaut for serving on
my examining committee.
Aida Mendez-Delgado for her smile, support, care, and for being always there when I
needed her.
My infinite gratitude goes to Marcela Ruiz-Funez, Gustavo Maldonado, Daniela
Verthelyi and Eduardo Romano for their friendship and hospitality. To them I am forever
grateful.
I am grateful to all my friends who made my stay in Virginia an unforgettable experience,
to Virgilio Centeno, Tatiana Boluarte, Paul Prato, Clara Diaz, Eduardo Miles, Alberto
and Patricia Perez Rigau, Juan Sabate, Margarita, Cristina y Mario Sanchez, Diana
Rubio, Mercedes Pastor, Luis Moreschi , Guillermo Aberboch, Daniel Quinones, and
many other exceptional friends.
Very special thanks to Jacem Tissaoui, Samruam Tongtoe, and Amara Loulizi . I learned
a great deal and was honored to work with them.
My gratitude goes forever to Gail, Siegfried, Mathew, and Michael Holzer for their
generosity, friendship, and hospitality, for making me feel at home.
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I would like to thank Gustavo Maldonado and Sergio Preidikman for their friendship and
the long conversations we had about this work. Special thanks to Azar Alizadeh for her
dedication and patient in helping me with my writing, and for being an unconditional
friend.
This work was partially supported by CASS, Center for Adhesive and Sealant Sciences.
Finally, I would like to dedicate this work to my family; to my parents, Titi and Nene, to
my brothers, Carlos, Luis y Mario, to my sisters in law, Ines, Raquel, and Viviana, and to
my nieces and nephews, Florencia, Maria Belen, Tomas, Maria Jose, Agustin, Virginia,
and Maria Ines.
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Table of Contents
1 Introduction
1.1 Background 1
1.2 Objectives 2
1.3 Significance 2
2 Literature Review
2.1 Adhesively Bonded Joints 4
2.1.1 Closed-Form Solutions 5
2.1.2 Numerical Solutions 7
2.1.3 Finite Element Solutions 8
2.1.4 Two-dimensional Solutions 9
2.1.5 Three-dimensional Solutions 11
2.1.6 Nonlinear Finite Element Methods 12
2.2 Crack Analysis Using Finite Elements 17
2.2.1 Two-dimensional Crack Analysis 21
2.2.2 Three-dimensional Crack Analysis 29
2.3 Fatigue Life Prediction of Adhesively-Bonded Joints 32
3 Geometric Nonlinear Finite Element Formulation
3.1 Introduction 37
3.2 Nonlinear Finite Element Analysis 37
3.3 Isoparametric Finite Element Discretization 43
3.4 Solution Technique for Nonlinear Finite Element 45
Analysis
4 Two-Dimensional Adhesive Finite Element
4.1 Introduction 47
4.2 Formulation 47
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4.2.1 Adherends 48
4.2.1 Adhesive 50
4.2.3 Thermal and Moisture Effects 54
4.2.5 Single Lap Joint Example 55
4.3 Fracture Mechanics Analysis 60
4.5 Conclusions 37
5 Three-Dimensional Adhesive Finite Element
5.1 Introduction 68
5.2 Adherends 69
5.3 Adhesive 74
5.4 Thermal and Moisture Effects 79
5.5 Examples 80
5.5.1 Single Lap Joint 80
5.5.2 Crack-Patch Geometry 89
5.5.3 Geometrically Nonlinear Single Lap Joint 97
5.6 Conclusions 104
6 Analysis of Fatigue-Crack Growth in Adhesively
Bonded Joints
6.1 Introduction 106
6.2 Determination of Load Characteristics 107
6.3 Fracture Mechanics Parameters 108
6.4 Fatigue-Crack-Growth Law 113
6.5 Determination of Crack Extensions 114
6.6 Algorithm to Simulate Debond 115
6.7 Examples 116
6.7.1 Crack Lap Shear Geometry 117
6.7.2 Crack-Patch Repair Geometry 121
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6.8 Summary and Conclusions 126
7 Conclusions and Recommendations
7.1 Introduction 128
7.2 Conclusions 130
7.3 Future Work and Recommendations 131
References 133
Appendix A 139
Vita 150

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List of Figures
Figure 2.1. Single lap joint configuration. 5
Figure 2.2. Geometric nonlinearities in a finite element: (a) Large displacements,
small strains; (b) Large displacements, large strains. 14
Figure 2.3. Graphical representation of the Newton-Raphson method. 15
Figure 2.4. Graphical representation of the Riks/Wemper method. 16
Figure 2.5. Three fractures modes. 18
Figure 2.6. J-integral notation and parameters. 21
Figure 2.7. Quarter-point two-dimensional continuum finite element. 22
Figure 2.8. Collapsed quarter-point two-dimensional continuum finite element. 23
Figure 2.9. Arbitrary body with a region with stress concentration. 23
Figure 2.10. Modified crack closure integral method parameters and notation. 28
Figure 2.11. Virtual crack extension method (from Nikishkov and Atluri 1987). 31
Figure 2.12. Fatigue crack extension law. 33
Figure 2.13. Crack-Patch repair configuration. 35
Figure 3.1 Large displacements of a body in stationary Cartesian coordinate
system. 38
Figure 3.2 Modified Newton-Raphson method. 46
Figure 4.1 Two dimensional adhesive finite element. 48
Figure 4.2 Nonlinear beam, generalized displacements. 49
Figure 4.3 Nonlinear beam, generalized displacements. 50
Figure 4.4 Single lap joint specimen. 55
Figure 4.5 Finite element model of a simple lap joint 56
Figure 4.6. Peel stress distribution for a single lap joint. 57
Figure 4.7 Shear stress distribution for a single lap joint. 57
Figure 4.8 Finite element model of a simple lap joint for a Geometric
Nonlinear Analysis. 58
Figure 4.9. Peel stress distribution for a single lap joint. Geometric nonlinearity
is included. 58
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Figure 4.10. Shear stress distribution for a single lap joint. Geometric
nonlinearity included. 59
Figure 4.11. Modified Crack Closure Integral Method. 60
Figure 4.12. Double Cantilever Beam (DCB) Specimen. 61
Figure 4.13. Crack Lap Shear (CLS) Specimen. 62
Figure 4.14. Strain energy release rate vs. applied load. 63
Figure 4.15. Total strain energy release rate results of ASTM round-robin
crack lap shear specimen including new results from ADH2D
approach. 64
Figure 4.16. Mode mix results of ASTM round-robin crack lap shear specimen
including new results from ADH2D approach. 65
Figure 4.17. Comparison between different analyses of strain energy release
rate values vs. debond length for the equal adherend
thickness CLS specimen of the round-robin study (Johnson 1989). 66
Figure 5.1. Configuration of the adhesive element. 68
Figure 5.2. Adherend element configuration. 69
Figure 5.3. Degrees of freedom per node. 70
Figure 5.4. 3-D adhesive finite element configuration. 74
Figure 5.5. Finite and parent elements. 75
Figure 5.6. Adhesive element finite element configuration. 75
Figure 5.7. Three-dimensional single lap joint geometry. 81
Figure 5.8. Single lap joint: symmetry condition. 81
Figure 5.9. Finite element mesh of the single lap joint. 82
Figure 5.10. Displaced finite element mesh of the single lap joint. 83
Figure 5.11. Peel stress (s ) distribution in the midplane of the adhesive layer
zz
of the single lap joint example. 84
Figure 5.12. Shear stress (t ) distribution in the midplane of the adhesive
xz
layer of the single lap joint example. 85
Figure 5.13. Shear stress (t ) distribution in the midplane of the adhesive
xy
layer of the single lap joint example. 86
Figure 5.14. Shear stress (t ) distribution in the midplane of the adhesive
yz
layer of the single lap joint example. 87
Figure 5.15. Axial stress (s ) distribution in the midplane of the adhesive
zz
layer of the single lap joint example. 88
x