Global-local Finite Element Fracture Analysis of Curvilinearly Stiffened Panels and Adhesive Joints
Mohammad Majharul Islam
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 Mechanics
Rakesh K. Kapania, Chair Mark S. Cramer Scott W. Case
Surot Thangjitham Mayuresh J. Patil
June 26, 2012 Blacksburg, VA
Keywords: Global-local finite element analysis, nonlinear finite element analysis, VCCT,
J-integral, cohesive zone finite element analysis
Copyright©2012, Mohammad Majharul Islam
Global-local Finite Element Fracture Analysis of Curvilinearly Stiffened Panels and Adhesive Joints
Mohammad Majharul Islam
Abstract
Global-local finite element analyses were used to study the damage tolerance of
curvilinearly stiffened panels; fabricated using the modern additive manufacturing process, the so-called unitized structures, and that of adhesive joints. A damage tolerance study of the unitized structures requires cracks to be defined in the vicinity of the critical stress zone. With the damage tolerance study of unitized structures as the focus, responses of curvilinearly stiffened panels to the combined shear and compression loadings were studied for different stiffeners’ height. It was observed that the magnitude of the minimum principal stress in the panel was larger than the magnitudes of the maximum principal and von Mises stresses. It was also observed that the critical buckling load factor increased significantly with the increase of stiffeners’ height.
To study the damage tolerance of curvilinearly stiffened panels, in the first step, buckling analysis of panels was performed to determine whether panels satisfied the buckling constraint. In the second step, stress distributions of the panel were analyzed to determine the location of the critical stress under the combined shear and compression loadings. Then, the fracture analysis of the curvilinearly stiffened panel with a crack of size 1.45 mm defined at the location of the critical stress, which was the common location with the maximum magnitude of the principal stresses and von Mises stress, was performed under combined shear and tensile loadings. This crack size was used because of the requirement of a sufficiently small crack, if the crack is in the vicinity of any stress raiser. A mesh sensitivity analysis was performed to validate the choice of the mesh density near the crack tip. All analyses were performed using global-local finite element method using MSC. Marc, and global finite element methods using MSC. Marc and ABAQUS. Negligible difference in results and 94% saving in the CPU time was achieved using the global-local finite element method over the global finite element method by using a mesh density of 8.4 element/mm ahead of the crack tip. To study the influence of different loads on basic modes of fracture, the shear and normal (tensile) loads were varied differently. It was observed that the case with the fixed shear load but variable normal loads and the case with the fixed normal load but variable shear loads were Mode-I. Under the maximum combined loading condition, the largest effective stress intensity factor was very smaller than the critical stress intensity factor. Therefore, considering the critical stress intensity factor of the panel with the crack of size 1.45 mm, the design of the stiffened panel was an optimum design satisfying damage tolerance constraints.
To acquire the trends in stress intensity factors for different crack lengths under different loadings, fracture analyses of curvilinearly stiffened panels with different crack lengths were performed by using a global-local finite element method under three different load cases: a) a shear load, b) a normal load, and c) a combined shear and normal loads. It was observed that 85% data storage space and the same amount in CPU
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time requirement could be saved using global-local finite element method compared to the standard global finite element analysis. It was also observed that the fracture mode in panels with different crack lengths was essentially Mode-I under the normal load case; Mode-II under the shear load case; and again Mode-I under the combined load case. Under the combined loading condition, the largest effective stress intensity factor of the panel with a crack of recommended size, if the crack is not in the vicinity of any stress raiser, was very smaller than the critical stress intensity factor.
This work also includes the performance evaluation of adhesive joints of two different materials. This research was motivated by our experience of an adhesive joint failure on a test-fixture that we used to experimentally validate the design of stiffened panels under a compression-shear load. In the test-fixture, steel tabs were adhesively bonded to an aluminum panel and this adhesive joint debonded before design loads on the test panel were fully applied. Therefore, the requirement of studying behavior of adhesive joints for assembling dissimilar materials was found to be necessary. To determine the failure load responsible for debonding of adhesive joints of two dissimilar materials, stress distributions in adhesive joints of the nonlinear finite element model of the test-fixture were studied under a gradually increasing compression-shear load. Since the design of the combined load test fixture was for transferring the in-plane shear and compression loads to the panel, in-plane loads might have been responsible for the debonding of the steel tabs, which was similar to the results obtained from the nonlinear finite element analysis of the combined load test fixture.
Then, fundamental studies were performed on the three-dimensional finite element models of adhesive lap joints and the Asymmetric Double Cantilever Beam (ADCB) joints for shear and peel deformations subjected to a loading similar to the in-plane loading conditions in the test-fixtures. The analysis was performed using ABAQUS, and the cohesive zone modeling was used to study the debonding growth. It was observed that the stronger adhesive joints could be obtained using the tougher adhesive and thicker adherends. The effect of end constraints on the fracture resistance of the ADCB specimen under compression was also investigated. The numerical observations showed that the delamination for the fixed end ADCB joints was more gradual than for the free end ADCB joints.
Finally, both the crack propagation and the characteristics of adhesive joints were studied using a global-local finite element method. Three cases were studied using the proposed global-local finite element method: a) adhesively bonded Double Cantilever Beam (DCB), b) an adhesive lap joint, and c) a three-point bending test specimen. Using global-local methods, in a crack propagation problem of an adhesively bonded DCB, more than 80% data storage space and more than 65% CPU time requirement could be saved. In the adhesive lap joints, around 70% data storage space and 70% CPU time requirement could be saved using the global-local method. For the three-point bending test specimen case, more than 90% for both data storage space and CPU time requirement could be saved using the global-local method.
This dissertation is dedicated to my parents, Mongal Miah and Sufia Begum, and my
fiancée, Samina Islam, without whose continuous support, encouragement, and
inspiration it would never have been accomplished.
Acknowledgements
Foremost, I would like to express my deepest gratitude to Dr. Rakesh K. Kapania who
took me under his wings and whose invigorating charisma prompted me to continue my
academic education. His professionalism, genuineness, and trust together with a dazzling
sense of humor are a true inspiration to all his students. I am most grateful for PhD
duration under his guidance, knowing that without his support I would not be where I am
now. It is my sincere hope that we can continue working together beyond the boundaries
of this appointment. My thanks also go to Dr. Mark S. Cramer, Dr. Scott W. Case, Dr.
Surot Thangjitham, and Dr. Mayuresh J. Patil, my committee members, who have
provided educational guidance and incentive through course work or through valuable
advice regarding my research.
Most importantly, I want to thank my fiancée, Samina Islam, who I met two years ago.
The love we have for each other is the meaning of my life and without her all my
accomplishments fade into obscurity. Thank you, Samina, for the beautiful smiles I could
always count on to make today a better day.
To this extent, I would like to express my deepest appreciation to Dr. Ali Yeilaghi
Tamijani and Dr. Sameer B. Mulani, who I hold in highest regards, and who have been
exceptionally helpful in catalyzing my thoughts and ideas. I would like to express my
sincere appreciation to Ms. Lisa Smith and Mr. Steve Edwards for their exceptional
administrative and technical support. I would also like to extend my deepest gratitude to
Ms. Madhu Kapania for her brilliant comments and suggestions. I would like to thank my
best friends Ahsan Zaman and Todd White for their continuous support and for the
numerous debates and discussions that I have had with them. My cooking experiments
would go in vein if I would not have such cheerful and spontaneous friends to share
gourmet meal with. I would also like to thank my friends Giovanni Sansavini, Yasser
Aboelkassem, Arnab Gupta, Ganesh Balasubramanian, Faiysal Ahmed, Patrick Poitras,
Ryan Poitras, Pankaj Kumar, Pankaj Joshi, Souvick Chatterjee, Chialiang Tsai, Sandeep
Shiyekar, Mehdi Ghommem, Alireza Karimi, Naseem, Kabir, Lincoln, Likhon, Dipu, and
Shahriar Khandaker for spending memorable time with. Special thanks go to the Sharir
v i A c k n o w l e d g e m e n t s
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Narachara Soccer team and VT Badminton Club for letting me to play for them and have
the priceless memories during my life in Blacksburg.
I want to thank my parents, Mohammad Mongal Miah and Sufia Begum, who saw
their son traveling to USA promising to visit them after one year but still not returning in
four years. Your love and support always guided me in finding my little spot in the world.
Thank you to all of my family members, Mukul, Mohsin, Mostofa, Asma, Meem, Tuli,
Srabon, Mayameen, Monir, Shabuj, Alamgir, Gianna, Ethan, Ayra, Mrs Mukul, and Mrs
Mostofa, your support, love, and kind encouragement provided the essential foundation
to thrive and succeed.
I want to send a warm thank-you across the ocean to the many friends that have kept
in touch with me. In particular, I would like to thank my friends, Ibrahim Khan, Farida
Khan, Tushar, Kibria, Zahid, Sharif, Asif, Sazedur Rahman, and Lokman Hossain, who
constantly contacted me and encouraged me over the phone.
Finally, parts of the work presented here were funded under NASA Subsonic Fixed
Wing Hybrid Body Technologies NASA Research Announcements (NRA) (NASA NN
L08AA02C) with Karen M. Brown Taminger as the application programming interface
and the Contracting Officer’s Technical Representative, and R. K. Bird as the Technical
Monitor. I would like to thank both of them for their suggestions. I would also like to
thank our partners in the NRA project, Lockheed Martin Aeronautics Company of
Marietta, GA, for technical discussions.
Table of Contents
Abstract .............................................................................................................................. ii
Acknowledgements ........................................................................................................... v
Table of Contents ............................................................................................................ vii
List of Figures .................................................................................................................... x
List of Tables .................................................................................................................. xiv
1 Introduction .................................................................................................................. 1
1.1 Damage Tolerance .................................................................................................. 1
1.2 Global-Local Finite Element Methods.................................................................... 4
1.3 Adhesive Joints ....................................................................................................... 8
1.4 Global-Local Finite Element Methods to Study the Characteristics of Adhesive Joints and Crack Propagation................................................................................ 11
1.5 Objectives and the Scope of the Work .................................................................. 12
1.6 Organizations of the Dissertation.......................................................................... 14
2 Global-local Finite Element Methods to Curvilinearly Stiffened Panels .............. 15
2.1 Modeling Curvilinearly Stiffened Panels for Global-local Finite Element Analyses 15 2.1.1 Formulations of stiffened panels ................................................................. 16 2.1.2 Finite element model formulations ............................................................. 17 2.1.3 Numerical solution of the buckling problem ............................................... 19
2.2 Finite Element Analyses of a Curvilinearly Stiffened Panel under Shear and Compression Loadings for Different Heights of the Stiffeners ............................ 19
2.3 Formulations of Global-local Finite Element Methods ........................................ 26 2.3.1 Continuity of the kinematic conditions between global and local method . 28 2.3.2 Application of the global-local finite element method on a plate under
tension ......................................................................................................... 28 2.3.3 Global finite element analyses of a curvilinearly stiffened panel under shear
and compression loads ................................................................................ 30 2.3.4 Global-local finite element analyses of a curvilinearly stiffened panel under
shear and compression loads ....................................................................... 31 2.3.5 Analyses of refined global models and refined local models under shear and
compression loads ....................................................................................... 32
2.4 Conclusions ........................................................................................................... 35
3 Multi-load Case Damage Tolerance Study of Curvilinearly Stiffened Panels Using Global-local Finite Element Analyses ............................................................ 37
3.1 Fracture Mechanics Approaches ........................................................................... 37 3.1.1 J-integral procedure ..................................................................................... 37
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3.1.2 Extraction of stress intensity factors from the domain integral for a mixed mode loading case ....................................................................................... 38
3.1.3 Virtual Crack Closure Technique ................................................................ 39
3.2 Framework of the Global-local Finite Element Method for the Fracture Analysis 42 3.2.1 Global-local finite element analyses of the center crack and side crack
tension plates ............................................................................................... 43 3.2.2 Fracture analyses of a center crack tension plate ........................................ 44 3.2.3 Fracture analyses of a side crack tension plate ........................................... 46
3.3 Fracture Analyses of Curvilinearly Stiffened Panels Using Global and Global-local Finite Element Methods ............................................................................... 48 3.3.1 Determining the location of the critical stress in the panel ......................... 51 3.3.2 A mesh sensitivity study for fracture analyses of a curvilinearly stiffened
panel under the combined shear and normal loads ..................................... 52 3.3.3 Fracture analysis of a curvilinearly stiffened panel under a fixed shear but
different normal load for a crack tip mesh density of 8.4 element/mm ...... 57 3.3.4 Fracture analyses of a curvilinearly stiffened panel under a fixed normal but
a different shear load for a crack tip mesh density of 8.4 element/mm ...... 61
3.4 Fracture Toughness of the Curvilinearly Stiffened Panel with a Crack ............... 66
3.5 Conclusions ........................................................................................................... 68
4 Global-local Finite Element Methods for Fracture Analyses of Curvilinearly Stiffened Panels for Different Crack Sizes .............................................................. 70
4.1 Framework for the Fracture Analysis of Curvilinearly Stiffened Panels for Different Crack Lengths with a Crack Tip Mesh Density of 8.4 element/mm ..... 70
4.2 Fracture Analyses of a Curvilinearly Stiffened Panel for Different Crack Lengths under a Fixed Normal Load .................................................................................. 74
4.3 Fracture Analyses of a Curvilinearly Stiffened Panel for Different Crack Lengths under a Pure Shear Load ....................................................................................... 78
4.4 Fracture Analyses of a Curvilinearly Stiffened Panel for Different Crack Lengths under the Combined Shear and Normal Loads ..................................................... 82
4.5 Fracture Toughness of the Panel ........................................................................... 86
4.6 Conclusions ........................................................................................................... 87
5 Static Stress and Fracture Analyses of Adhesive Joints ......................................... 89
5.1 Formulations of Static Stress and Fracture Analyses of Adhesive Joints ............. 89 5.1.1 Formulation of the boundary value problem for the single lap adhesive joint
89 5.1.2 Modeling the cohesive zone for the crack propagation ............................... 90
5.2 Delamination Analyses of the Compression-Shear Test-Fixture .......................... 96
5.3 Finite Element Simulation of the Adhesive Lap Joint ........................................ 102
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5.4 Stress Analysis across the Thickness of the Adhesive Joint under the Combined Loadings .............................................................................................................. 105
5.5 Finite Element Simulation of the Cohesive Zone Interface ................................ 107
5.6 Modeling the Asymmetric Double Cantilever Beam (ADCB) ........................... 109 5.6.1 Influence of adhesive material property to fracture resistance of the ADCB
joint ............................................................................................................ 109 5.6.2 Influence of adherends geometric asymmetry to fracture resistance of the
ADCB joint ............................................................................................... 111 5.6.3 Influence of adherend material asymmetry to fracture resistance of the
ADCB joint ............................................................................................... 113
5.7 Compression Delamination under Different Constrained End Conditions ......... 113 5.7.1 The modified Riks method ........................................................................ 114 5.7.2 Compression delamination study on the adhesive joints .......................... 115
5.8 Conclusions ......................................................................................................... 119
6 Global-Local Finite Element Analyses of Crack Propagation and Adhesive Joints .................................................................................................................................... 122
6.1 Global-Local Analyses of an Adhesively Bonded Double Cantilever Beam Specimen ............................................................................................................. 122
6.2 Global-Local Analyses of a Single Lap Adhesive Joint ..................................... 125
6.3 Global-local Analyses of a Three-point Bending Test Specimen ....................... 129
6.4 Conclusions ......................................................................................................... 133
7 Conclusions and Future Directions ........................................................................ 135
7.1 Conclusions ......................................................................................................... 135 7.1.1 Global-local finite element methods to curvilinearly stiffened panels...... 135 7.1.2 Multi-load case damage tolerance study of curvilinearly stiffened panels
using global-local finite element analyses................................................. 136 7.1.3 Global-local finite element methods for fracture analyses of curvilinearly
stiffened panels for different crack sizes ................................................... 138 7.1.4 Static stress and fracture analyses of adhesive joints ................................ 139 7.1.5 Global-local finite element analyses of crack propagation and adhesive
joints .......................................................................................................... 141
7.2 Future Directions ................................................................................................ 142 7.2.1 Optimization of the curvilinearly stiffened panels using the global-local
finite element method ................................................................................ 142 7.2.2 Global-local finite element methods to study the complex 3D structural
adhesive joints ........................................................................................... 143 7.2.3 Global-local finite element methods to study the systems that can be
disintegrated .............................................................................................. 143
Bibliography .................................................................................................................. 144
List of Figures
Fig. 2-1 A stiffened panel under combined loadings ........................................................ 15 Fig. 2-2 Bending and twisting moments due to stiffeners ................................................ 17 Fig. 2-3 Triangular shell element ...................................................................................... 18 Fig. 2-4 Geometry, loading, and boundary conditions of the curvilinearly stiffened panel
............................................................................................................................. 20 Fig. 2-5 Finite element model of the curvilinearly stiffened panel ................................... 21 Fig. 2-6 Deformations of the panel for different heights of stiffeners .............................. 22 Fig. 2-7 Distribution of stresses in the panel for different heights of stiffeners ............... 24 Fig. 2-8 Variation of buckling load factors of the panel for different heights of stiffeners
............................................................................................................................. 25 Fig. 2-9 Global-local interface definition ......................................................................... 26 Fig. 2-10 Schematic of the global-local solution strategy ................................................ 27 Fig. 2-11 Global-local finite element analyses of a simple plate under tension ............... 29 Fig. 2-12 Global finite element analyses of the stiffened panel under shear and
compression loads ................................................................................................ 31 Fig. 2-13 Global-local analyses of the panel under shear and compression loads ........... 32 Fig. 2-14 Comparison of results obtained using global and global-local finite element
analyses ................................................................................................................ 34 Fig. 3-1 An arbitrary path on which the line integral (J integral) is to be calculated [120]
............................................................................................................................. 37 Fig. 3-2 Normal stress ( yσ ) distribution ahead of the crack tip ....................................... 39
Fig. 3-3 Virtual crack closure technique ........................................................................... 39 Fig. 3-4 Procedure to calculate SIFs using global-local finite element method ............... 43 Fig. 3-5 Plates with center and side cracks ....................................................................... 43 Fig. 3-6 Finite element model of the plate problem with a center crack .......................... 45 Fig. 3-7 Results of the plate problem with center cracks .................................................. 46 Fig. 3-8 Finite element model of the side crack plate ....................................................... 47 Fig. 3-9 Result of the side crack plate under tension ........................................................ 48 Fig. 3-10 Schematic of the panel under combined shear and compression loadings ....... 49 Fig. 3-11 Finite element model of the panel under shear and compression loadings ....... 49 Fig. 3-12 Buckling analysis of the panel under combined shear and compression loads . 50 Fig. 3-13 Critical stress location in the curvilinearly stiffened panel ............................... 51 Fig. 3-14 Procedure for fracture analyses of the curvilinearly stiffened panels ............... 53 Fig. 3-15 Different mesh densities .................................................................................... 54 Fig. 3-16 Global and global-local analyses of the panel for mesh sensitivity analyses
under shear and normal loads .............................................................................. 55 Fig. 3-17 Path independence of J-integral estimation ...................................................... 56 Fig. 3-18 Crack tip stresses for different normal loads ..................................................... 58 Fig. 3-19 Percentage of savings obtained using global-local analyses over a global
analysis for different normal loading cases ......................................................... 58 Fig. 3-20 Energy release rates for different normal loads ................................................ 59 Fig. 3-21 Stress intensity factors for different normal loads ............................................ 60
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Fig. 3-22 Contribution of individual stress intensity factor to the effective stress intensity factor for different normal loads .......................................................................... 61
Fig. 3-23 Crack tip stresses for different shear loads ........................................................ 62 Fig. 3-24 Percentage of savings obtained using global-local analyses over a global
analysis for different shear loading cases ............................................................ 62 Fig. 3-25 Energy release rates for different shear loads ................................................... 63 Fig. 3-26 Stress intensity factors for different shear loads ............................................... 64 Fig. 3-27 Contribution of individual stress intensity factor to the effective stress intensity
factor for different shear loads ............................................................................. 65 Fig. 3-28 A curvilinearly stiffened panel with a crack of length 2a ................................. 66 Fig. 4-1 Framework for fracture analyses of curvilinearly stiffened panels ..................... 71 Fig. 4-2 Procedure for the fracture analysis of the curvilinearly stiffened panel for
different “a” ......................................................................................................... 72 Fig. 4-3 Global and local models for different crack lengths ........................................... 73 Fig. 4-4 Path independence of the J-integral estimation ................................................... 74 Fig. 4-5 Percentage of savings obtained using global-local analyses over a global analysis
for the normal loading case .................................................................................. 75 Fig. 4-6 Energy release rates for different crack lengths under the normal loading ......... 76 Fig. 4-7 Contribution of individual mode to the total energy release rate for different
crack lengths under the normal loading ............................................................... 76 Fig. 4-8 Stress intensity factors for different crack lengths under the normal loading ..... 77 Fig. 4-9 Percentage of savings obtained using global-local analyses over a global analysis
for the pure shear loading case ............................................................................ 79 Fig. 4-10 Energy release rates for different crack lengths under the pure shear load ....... 79 Fig. 4-11 Contribution of individual mode to the total energy release rate for different
crack lengths under the pure shear load ............................................................... 80 Fig. 4-12 Stress intensity factors for different crack lengths under the pure shear load ... 81 Fig. 4-13 Percentage of savings obtained using global-local analyses over a global
analysis for the combined shear and normal loading case ................................... 82 Fig. 4-14 Energy release rates for different crack lengths under the combined shear and
normal loads ......................................................................................................... 83 Fig. 4-15 Contribution of individual mode to the total energy release rate for different
crack lengths under the combined shear and normal loads ................................. 84 Fig. 4-16 Stress intensity factors for different crack lengths under the combined shear and
normal loads ......................................................................................................... 85 Fig. 5-1 Single lap adhesive joint ..................................................................................... 90 Fig. 5-2 Symmetric double cantilever beam ..................................................................... 91 Fig. 5-3 Asymmetric double cantilever beam ................................................................... 91 Fig. 5-4 Cohesive elements: a) 3D cohesive element, b) Bilinear cohesive material model
............................................................................................................................. 92 Fig. 5-5 a) Schematic of the test, b) Steel tabs bonded onto the Aluminum panel, c) Steel
tabs are debonded from the panel ........................................................................ 96 Fig. 5-6 Shell elements in the test fixture model .............................................................. 97 Fig. 5-7 Elastic-plastic material model for steel tabs and the aluminum panel ................ 98 Fig. 5-8 Panel with adhesively bonded steel tabs where adhesive is modeled using linear
spring elements .................................................................................................... 98
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Fig. 5-9 Finite Element Model for the compression-shear test fixture with the panel ..... 99 Fig. 5-10 Stress analysis along the interface between tabs and the panel ......................... 99 Fig. 5-11 The node list along the interface between steel tabs and the panel at the top left
corner of the system ........................................................................................... 100 Fig. 5-12 von Mises stresses along the node list on the interface between steel tabs and
the panel ............................................................................................................. 101 Fig. 5-13 In-plane tangential stresses along the node list on the interface between tabs
and the panel ...................................................................................................... 101 Fig. 5-14 In-plane normal stresses along the node list on the interface between tabs and
the panel ............................................................................................................. 102 Fig. 5-15 Detail model with boundary and loading conditions ...................................... 103 Fig. 5-16 von Mises stress distribution along the adhesive of the lap joint specimen .... 104 Fig. 5-17 von Mises stress profiles across the thickness of the adhesive ....................... 105 Fig. 5-18 Schematic of the 3D lap joint .......................................................................... 106 Fig. 5-19 Finite element model with boundary and loading conditions ......................... 106 Fig. 5-20 von Mises stress distribution along the adhesive layer at the three different
locations across the thickness of the joint .......................................................... 107 Fig. 5-21 Finite element model of the DCB configuration ............................................. 108 Fig. 5-22 Load–displacement curves using the nominal interface strength 40 MPa for a
DCB test ............................................................................................................. 108 Fig. 5-23 Load-displacement profiles of ADCB specimens for different critical energy
release rates for the adhesive (in N/mm) ........................................................... 110 Fig. 5-24 Change in reaction force due to the change in the critical energy release rates of
the adhesive ....................................................................................................... 111 Fig. 5-25 Load-displacement profiles of ADCB specimens for different adherend
thicknesses ......................................................................................................... 112 Fig. 5-26 Change in the reaction force due to the change in the top adherend thickness 112 Fig. 5-27 Load-displacement profiles of ADCB joints for different adherend stiffness 113 Fig. 5-28 Riks method (a) unstable static response (b) modified Riks method [120, 139]
........................................................................................................................... 115 Fig. 5-29 ADCB configuration for a fixed end ............................................................... 116 Fig. 5-30 Configurations for the top adherend with a free end ....................................... 116 Fig. 5-31 Load displacement profile for fixed end ADCB specimen with different mesh
densities of the cohesive interface ..................................................................... 117 Fig. 5-32 Deformation profiles of the fixed end ADCB specimen with t1 = 2 mm and t2 =
4 mm under compression ................................................................................... 118 Fig. 5-33 Load displacement profile for free end ADCB specimen with different mesh
densities of the cohesive interface ..................................................................... 119 Fig. 6-1 Global-local models of a DCB specimen .......................................................... 122 Fig. 6-2 Superimposed deformations of the DCB specimen for the global and global-local
methods .............................................................................................................. 123 Fig. 6-3 Comparisons of the end reaction force profiles obtained using the global-local
method ............................................................................................................... 124 Fig. 6-4 Global-local models for a single lap adhesive joint .......................................... 126 Fig. 6-5 Superimposed deformations of the lap joint for the global and global-local
methods .............................................................................................................. 127
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Fig. 6-6 von Mises stress along the adhesive joint using the global-local method ......... 128 Fig. 6-7 von Mises stress along the adhesive thickness using the global-local method . 128 Fig. 6-8 A three-point bending test ................................................................................. 129 Fig. 6-9 Global-local results for a three-point bending test specimen ............................ 130 Fig. 6-10 Load-displacement profiles in the global analysis .......................................... 131 Fig. 6-11 Comparison of results using the global-local method ..................................... 132
List of Tables Table 2-1 Heights and thickness of stiffeners and the panel ............................................ 21 Table 2-2 Summary of global and global-local finite element analyses ........................... 33 Table 3-1 Empirical formulas for the dimensionless constant F(a/b) of the center crack
specimen .............................................................................................................. 44 Table 3-2 Empirical formulas for the dimensionless constant F(a/b) of the side crack
specimen .............................................................................................................. 47 Table 5-1 Material properties used in the test-fixture finite element model ..................... 97 Table 6-1 Comparison of global-local results ................................................................. 125 Table 6-2 Comparison of global-local results ................................................................. 129 Table 6-3 Comparison of global-local results ................................................................. 132
1 Introduction The current trend in the aircraft design is driven by the strength and stiffness
requirement considering the weight as a major design constraint. The manufacturers of
large commercial aircrafts are, therefore, looking for suitable light weight metallic
structures with directional strengths and adhesively bonded joints [1].
1.1 Damage Tolerance
To make it possible for the metallic structure to resist directional loads, stiffeners are
used in the major loading directions. The biggest advantage of the stiffeners is the
increased bending stiffness of the panel with a minimum of additional material, which
makes these structures highly desirable for supporting out-of-plane loads and
destabilizing compressive loads [2].
According to the demand of the Federal Aviation Regulations, the design of all
primary structure airframe components must satisfy the principles of damage tolerance
[3]. Researchers at Virginia Tech have been developing a computer environment,
EBF3PanelOpt [4-7], which will help the aerospace industry to optimally design
stiffened panels fabricated using the modern additive manufacturing process, the so-
called unitized structures. This environment allows researchers to perform the
optimization of unitized structures with multiple constraints including damage tolerance
constraints for panels that may be stiffened using curvilinear stiffeners. A number of
studies have recently been conducted dealing with the performance of structural
optimization in the presence of constraints related to the damage tolerance.
For example, the weight optimization of a vessel structure with instability yield stress
and static fracture strength constraints was performed by Elabdi et al. [8]. The shape
2 Globa l - loca l f in i t e e leme nt f rac tu re ana lys i s o f curv i l inear ly s t i f fened panel s and adhes ive jo in t s
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optimization of structures with fatigue life and static fracture strength constraints have
been considered by some researchers [9-11]. Kale et al. [12] developed an efficient
technique to carry out a reliability-based optimization of a structural design and an
inspection schedule for a fatigue crack growth. Nees and Canfield [13] proposed a
methodology for implementing fracture mechanics in a global structural design of
aircraft, in which they performed safe-life structural optimization of F-16 wing panels to
obtain the minimum structural weight for fatigue crack growth under a service load
spectrum. Dang et al. [14] performed the optimization of stiffened panels with cutouts
and curvilinear stiffeners for mass minimization under multiple load cases considering
multiple failure modes, for example, damage tolerance, buckling, stress, and crippling.
Arsene et al. [15] developed a software package in Alcan’s research center for damage
tolerance analysis. This tool called PAnel was based on the commercially available
software MSC. Marc Mentat and MSC. Marc and acted as a large macro for meshing,
calculations and post-treatment operations. The user only needed to provide a text file
containing some parameters describing the desired configuration. The tool would then
obtain the stress intensity factor vs crack length curve needed for the damage tolerance
analysis.
Some research efforts have been made on the study of the damage tolerance behavior
of integrally stiffened panels subjected to uniaxial and biaxial loadings [16 - 18]. Isida
[16] performed a theoretical analysis of the stress intensity factor for the tension of a
centrally cracked strip with stringers along edges based on the Laurent series expansions
of the complex stress potentials. He used a perturbation technique to determine constants
of the series from the boundary conditions. The influence of biaxial loadings on the crack
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growth in stiffened panels was reported by Joshi and Shewchuk [17] for the fatigue crack
growth, and by Ratwani and Wilhem [18] for the biaxial tension. Fracture analysis of the
FAA/NASA wide stiffened panels was performed to predict loads against the crack
extension for wide panels with a single crack and multiple site damage cracking at many
adjacent rivet holes [19]. A considerable amount of research on the analysis of crack tip
stress intensity factors for stiffened panels has been conducted: Yeh [20] performed such
an analysis on orthotropic plates under tension, Nishimura [21] on isotropic plate with
cracked stiffeners under tension, Lee and Kim [22] on remote normal stress on the plate
reinforced with a sheet by spot weld, Yeh and Kulak [23] on orthotropic skin panels with
riveted stiffeners under tension, Horst and Hausler [24] on fatigue crack growth behavior
in welded panels influenced by residual stress caused by welding, Swift [25] on a cracked
sheet under tension considering plasticity effect of the stiffener and its adhesive
attachment to the cracked sheet, Penmesta et al. [26] for risk based design plots for flat
and stiffened panels under tension, and Moreira et al. [27] performed research on tensile
and fatigue growth tests on the stiffened panels.
Buckling stability is an important characteristic for structures under combined biaxial
loadings [28-35]. Some research has been conducted on the compression behavior of
stiffened panels with cracks. For example, Magaritis and Toulios [36] evaluated ultimate
strength and collapse characteristics of stiffened panels with cracks of different length
and configuration without considering the crack propagation; Brighenti and Carpinteri
[37] studied the crack sensitivity for the collapse load of unstiffened panels; Paik [38]
examined ultimate strength characteristics of stiffened plates with cracking damages
under compression; Alinia et al. [39] investigated the buckling and post-buckling
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behavior of the shear panels that had edge cracks; and Brighenti [40] studied the effects
of the crack length and orientation on the buckling load of rectangular unstiffened panels.
However, crack analyses require a very fine mesh of finite elements. To analyze a very
fine mesh finite element model, a high CPU time is required. Therefore, to save CPU
time in the fracture analysis, instead of, using a very fine mesh of finite elements, a
global-local finite element method can be used.
1.2 Global-Local Finite Element Methods
Analyzing a complex stiffened structure in sufficient detail to obtain accurate results
everywhere is difficult. One possible alternative is to use the global-local finite element
analysis [41]. In the global-local finite element analysis, first, the global finite element
analysis is performed with a coarse mesh. The mesh is then refined in the neighborhood
of the location of interest by isolating that area from the remaining system, using
appropriate boundary conditions for the localized area. For such a procedure, kinematic
boundary conditions can be applied to the boundary of isolated local domain modeled
using finer mesh. In such a case, the boundary values of the local model are the
displacement and rotations obtained from the global analysis of the complete model using
the coarse mesh.
Since the 1971 pioneering work of Mote, the literature on global-local modeling
techniques has grown at an astonishing rate [42-56]. For instance, Zeghal and Abdel-
Ghaffar [57] used a global-local finite element method for studying nonlinear seismic
behavior of earth dams. They found this method was very effective in saving
computational time. Han and Atluri [58] used the Schwartz-Neumaan alternating method
for analyzing three-dimensional arbitrary surface cracks by modeling the cracks in a local
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finite-sized subdomain using the symmetric Galerkin boundary element method. They
found that the alternating procedure converged unconditionally by imposing prescribed
displacements and tractions in the alternating approach.
Hirai et al. [59] presented an efficient and exact formulation of the global-local
method for finding the stress concentration factors without introducing any new
approximation. They used the global-local method performing simulation only in the
local area of interest without considering the region outside the local zone. Noor [60]
found the global-local method as a potential future direction of research in predicting
nonlinear and post-buckling responses of structures. Whitcomb [61] developed an
iterative algorithm for the global-local analysis. He used two distinct meshes, one for a
global and another for a local model, and obtained same level of accuracy as one would
obtain from the single refined global model.
A considerable research has been performed on the global-local methods on composite
materials. Ransom and Knight [62] employed interpolation functions satisfying the linear
plate bending equations to determine the displacements and rotations from a global
model. Those displacements and rotations from the global model used as boundary
conditions for the local model for analyzing the local model independently from the
global model. Voleti et al. [63] addressed the solution to large scale periodic structures
made up of multi-material composite systems. They found that both specified boundary
method and multi-point constraint method offered the potential choice because of their
feasibility in use. Srirengan [64] developed a global-local method based on the modal
analysis facilitating the three dimensional stress analysis of plain weave composite
structures. Fish et al. [65] developed a composite grid method to solve symmetric
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indefinite linear systems enforcing compatibility and traction continuity conditions
between independently modeled substructures. They found out that for a thin shell, the
direct solution of the local problem seemed to be a better choice compared to the iterative
solution.
A considerable research has been performed on the global-local analysis of the
damage mechanics of structures. Xu et al. [66] employed a bottom-up global-local
strategy to determine local stresses in a multi-phase and multi-layer plain weave
composite structure. They found that the global-local method provided a basis for the
prediction of the damage and strength of the multi-phase and multi-layer composite
structures. Haryadi, Kapania, and Haftka [67] presented a simple and accurate global-
local method to calculate the static response of simply supported composite plates with a
small crack. They used displacement boundary conditions in the local model boundary
obtaining from the global analysis using Ritz analysis. They found that this method gave
the accurate stress result saving 85% of the CPU time. Guidault et al. [68] proposed a
multiscale strategy for crack propagation enabling researchers to use refined mesh only in
the required crack’s vicinity, using micro-macro approach based on a mixed domain
decomposition method. The accuracy was retained by using an iterative procedure for
enforcing equilibrium between the global and local domains. Haider et al. [69] performed
analysis of a simplified computational technique based on a refined global-local method
applied to the failure analysis of concrete structures. Their finite element solution was
divided into two parts: a linear elastic analysis on a coarse mesh over the whole model
and a non-linear analysis over a small region of the structure. A non-local damage model
was implemented in the non-linear calculation. These two models were coupled with the
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help of an iterative scheme. Gendre et al. [70] developed one computational strategy to
solve structural problems with localized nonlinear phenomena defining two finite element
models: one for global linear model for the whole model, and another for a local
nonlinear model replacing the global model for the area of interest. Sun et al. [71]
proposed a combined micromechanics analysis and global-local finite element method to
study the interaction of particles and matrix at the nano-scale near the crack tip. They
used a global-local multiscale finite element model with homogenous material properties
to study the fracture of a compact tension sample. They concluded that their proposed
combined method could be used to study the toughness mechanism.
The effectiveness of the global-local analysis capability was demonstrated by Knight
Jr. et al. [72] by obtaining the detailed stress states of a blade-stiffened graphite epoxy
panel with a discontinuous stiffener. In their study, the accurately detail stress state was
found in the locally refined model from the region of interest. Alaimo et al. [73]
presented a Hierarchical approach for the analysis of advanced aerospace structures. They
used two kind of numerical methods for their global-local models. The first step of the
Hierarchical procedure was performed by the finite element method using a coarse mesh
to study the global structure without cracks. Then, the local region with a crack was
analyzed by using a boundary element method based on the multi-domain anisotropic
technique. Their global-local model predicted stress concentrations at crack tips with a
reduction of the modeling efforts and of the computational time. However, displacement
values defined in the boundary of the local finite element model calculated from the
global finite element analysis without cracks would be different than the displacement
values calculated from the global finite element analysis with cracks, especially when the
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crack size is large and/or the local model is small. To the author’s knowledge, no study
found so far on the stress intensity factors and fracture analyses of the curvilinearly
stiffened panels using global-local finite element methods defining cracks in both global
and local models.
1.3 Adhesive Joints
This research was primarily motivated by an adhesive joint failure that we faced on a
test-fixture that we used to experimentally validate the design of curvilinearly stiffened
panels under compression-shear load. In the test fixture, steel tabs were adhesively
bonded to an aluminum panel and this adhesive joint debonded before all the design loads
on the panel were fully applied. Therefore, the requirement of a stronger adhesive joint
for the current test fixture and a detailed understanding of the adhesive joints between
two dissimilar materials are both deemed necessary.
Adhesive bonding has been used in the fabrication of primary aircraft fuselage and
wing structures for many years [74]. For example, joining stringers to skins of fuselage
and wing structures, metallic honeycomb to skins of elevators, ailerons, tabs, and spoilers
constitute the main uses of adhesives in aircraft structures. Adhesively bonded aircraft
structures are stable and durable, and, hence, this joining method has a good potential for
future lightweight structures. However, application of a new technology needs
corresponding development of design and assessment methods. In industry, earlier
analytical design procedures are being replaced by the Finite Element Method (FEM) so
that the complex structure including adhesive bonds can be simulated and their
performances assessed. The detailed FEM computations are based on detailed
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understanding and modeling of all relevant material behaviors, including the adhesives
used in the joined or layered materials.
To predict the strength of adhesive joints, knowledge of the stress distribution must be
coupled with a knowledge of fracture propagation through the interface of adherends.
Often, a lap joint is used to gain such knowledge. Considerable research has been
performed on the adhesive lap joint. Kuczmaszewski and Wlodarczyk [75] studied the
effect of types of finite elements on stress distribution in adhesive joints of metallic
structures. They found that the division of the adhesive into more than three layers had no
significant effect on the stress distribution over the lap length in a lap joint. Goncalves et
al. [76], da Silva et al. [77], Pearson and Mottram [78], and Diaz et al. [79] performed
three dimensional stress analyses on single lap adhesive joints. Since the peel stress is
associated with the shear lap joint because of the eccentricity between the adhesive and
the loading axes, accurately capturing the stress distribution in the adhesive requires
discretizing the adhesive layer with a very fine mesh, resulting in models with a very high
number of degrees of freedom.
In layered materials, delamination is one of the most common failure modes that may
result from joint imperfections, edge effects, and complex loadings. The presence of
delamination can cause significant reduction in both the stiffness and the strength of a
joined structure, which leads to a failure. A clear understanding of the failure behavior of
the joined structure under shear/ tension/ compression is, therefore, extremely important.
Joined structures can be of symmetric or asymmetric configurations. The asymmetry in
the joined structures can arise from the asymmetry in material properties, geometric
properties, and/or loadings, in the joint elements. Asymmetric configurations lead to the
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mixed mode fracture of the joint interface. A number of studies have been performed on
the Asymmetric Double Cantilever Beam (ADCB) to study the delamination in the
bonded interface for different fracture modes. Sundararaman and Davidson [80]
evaluated interfacial fracture toughness from an ADCB using the analytical and finite
element methods. Bennati et al. [81] proposed a mechanical model of the ADCB test, in
which case the specimen was an assembly of two sublaminates that were partly
connected by an elastic–brittle interface. They obtained a complete explicit solution to
the problem including analytical expressions for the interfacial stresses, internal forces,
and displacements.
Park and Dillard [82] proposed a hybrid Asymmetric Tapered Double Cantilever
Beam (ATDCB) specimen. They found a limited range of mode mixities with a single
specimen by combining a constant thickness and tapered adherends in the asymmetric
TDCB configuration. da Silva et al. [83] performed research on the determination of the
fracture toughness of steel/adhesive/steel joints under mixed mode loadings for
Asymmetric Tapered Double Cantilever Beam (ATDCB), Single Leg Bending (SLB),
and Asymmetric Double Cantilever Beam (ADCB). They concluded that the
introduction of a small amount of Mode II loading (shear) in the joint resulted in a
decrease of the total fracture energy, GT = GI + GII, when compared to the pure Mode I
fracture energy.
Alfredsson and Hogberg [84] studied the fracture behavior of adhesive joints under
mixed mode loading for the ADCB specimen, and they found that the mode-mixity of
their model was strongly dependent on the relative stiffness of the adherends and the
adhesive layer. Szekrenyes [85] developed mixed mode specimen for the interlaminar
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fracture testing of laminated transparent composite materials. They determined that the
mode ratio changed along the specimen width for mixed-mode cases. Research has also
been performed on the Mode-I fracture [86-90] and also the mixed-mode fracture [91-97]
of adhesive joints. However, due to the increasing use of bonded structures in aircraft and
other applications, adhesive joints are increasingly being used for bonding two different
materials, complicated geometries, and loading combinations.
Adhesively bonded structures can be used for the compression loading case, as well. A
considerable work has been done in the compression delamination of composite
structures [98-105]. However, very little work appears to have been done on compressive
delamination of adhesive joints of the ADCB joint [106]. These fracture and
delamination analyses of the ADCB joints require a very fine mesh of finite elements,
which is associated with the requirement of a high CPU time. Therefore, to save CPU
time requirement in the fracture and delamination analyses, global-local finite methods
can be used.
1.4 Global-Local Finite Element Methods to Study the Characteristics of Adhesive Joints and Crack Propagation
Since the peel stress is associated with the shear lap joint because of the eccentricity
between the adhesive and the loading axes, accurately capturing the stress distribution in
the adhesive requires discretizing the adhesive layer with a very fine mesh, resulting in
finite element models with a very high number of degrees of freedom. Since simulation
of a finite element model with a fine mesh requires a high computational time,
researchers are looking for alternative methods to simulate adhesive joints saving CPU
time. Another set of studies that require high CPU time are the evaluation of the
delamination growth in adhesive joints and crack propagation in brittle material both
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using cohesive zone modeling along with a detailed finite element mesh to model the
bulk material.
The concept of cohesive zone models assumes fracture as a gradual phenomenon in
which a crack opening takes place across an extended crack tip region (cohesive zone)
and is resisted by cohesive tractions [107]. Cohesive zone elements are, therefore, placed
between continuum elements as an interface for crack propagation. These cohesive zone
elements open when damage growth occurs in order to simulate crack growth. As the
crack path can only follow these elements, the direction of crack propagation strongly
depends on the presence of cohesive zone elements. This implies that the crack path is
very much cohesive element mesh dependent. In addition, the initial stiffness of the
cohesive zone elements has a large influence on the overall elastic deformation, and
should be very high in order to obtain realistic results [107, 108]. In short, although one
of the best advantages of the cohesive zone modeling using finite element methods is that
it can predict the propagation of delamination, the simulation of progressive delamination
using cohesive elements poses numerical difficulties due to the requirement of accurate
definition of the stiffness of the cohesive layer and extremely refined meshes [108 - 111].
A very fine mesh is required for the cohesive zone modeling using the finite element
method which makes analyses very time consuming and computationally expensive.
Since the global-local finite element method is a less expensive method in terms of the
CPU time requirement, this method can be used to study the characteristics of adhesive
joints and crack propagation in brittle materials.
1.5 Objectives and the Scope of the Work
The objectives and scope of the present work are:
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i. Motivated by a need for understanding the damage tolerance of unitized structures
fabricated using the modern additive manufacturing process, a global-local finite
element method was used to study the stress and fracture of curvilinearly stiffened
panels, unitized structures, by defining a crack in the location of the critical stress.
ii. Motivated by a need for acquiring the trends of stress intensity factors for
different crack lengths under different loadings, fracture analyses of curvilinearly
stiffened panels with different crack lengths were performed by using a global-
local finite element method under three different load cases: a) a shear load, b) a
normal load, and c) a combined shear and normal loads.
iii. Motivated by an adhesive joint failure that we faced on a test-fixture while
experimentally validating the design of curvilinearly stiffened panels, stress
analyses of adhesive joints on the test-fixture using a nonlinear finite element
model of the compression-shear test fixture were performed under a gradually
increasing compression-shear load.
iv. To better understand the physics of the adhesive joint employed in the test-fixture,
analyses of three-dimensional finite element models of the adhesive lap and
ADCB joints were performed for shear and peel deformations under loadings that
were similar to the loading observed in the test-fixture analysis.
v. Motivated by a need for reducing the computational time in the simulation of
adhesive joints and crack propagation, three cases were studied by using a global-
local finite element method: a) an adhesively bonded DCB, b) an adhesive lap
joint, and c) a three-point bending test specimen.
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1.6 Organizations of the Dissertation
The rest of the dissertation is organized as follows: Chapter 2 focuses on using the
global-local finite element method to the curvilinearly stiffened panels for the static stress
analysis; Chapter 3 focuses on the global-local finite element method for stress, buckling,
and fracture analyses of a curvilinearly stiffened panel under combined shear and
compression/tension loadings; Chapter 4 focuses on the global-local finite element
analysis for acquiring the profiles of stress intensity factors of the curvilinearly stiffened
panels for different crack lengths under the combined shear and normal loadings, and on
the fracture analyses of the curvilinearly stiffened panels with the presence of a crack of
the recommended size [112]; Chapter 5 focuses on the stress analyses of adhesive joints
on the compression-shear test-fixture using a nonlinear finite element model of the
compression-shear test-fixture under a gradually increasing compression-shear load, on
the static stress and fracture analyses of adhesive lap joints and ADCB joints, and on the
compression delamination of the ADCB joints, and Chapter 6 focuses on the global-local
finite element method to study the characteristics of an adhesive lap joint and the crack
propagation on a DCB joint and three-point bending test specimen. Major conclusions
and contributions from this work and future directions of the research are summarized in
Chapter 7.
2 Global-local Finite Element Methods to Curvilinearly Stiffened Panels
This chapter describes the influence of the biaxial loadings on the response of the
curvilinearly stiffened panel. Since the global-local finite element method will be applied
to the curvilinearly stiffened panel for the fracture analysis, before starting the fracture
analysis of the curvilinearly stiffened panels using the global-local finite element method,
it is important to perform a comparative analysis of the curvilinearly stiffened panels for
studying the feasibility of the proposed global-local finite element method.
2.1 Modeling Curvilinearly Stiffened Panels for Global-local Finite Element Analyses
Suppose the panel is subjected to the biaxial normal and shear loads (Fig. 2-1). When
the panel is reinforced with stiffeners located at an offset from the panel middle plane, the
stiffener bends at the same time producing a bending displacement to the panel, as well.
In such a case, in addition to the in-plane displacements, bending should also be
considered [113, 114].
Fig. 2-1 A stiffened panel under combined loadings
Ny
Nxy y
x
2h
2b t
Nx
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2.1.1 Formulations of stiffened panels The governing differential equations for the stiffened panels are based on the
following assumptions:
a. Both the stiffener and the panel are the linear elastic materials
b. The deflection in the z-direction depends on x and y only
c. The deflection of any points of the panel is small compared to the thickness of the
panel
d. The bending deformation follows Mindlin’s hypothesis (linear section
perpendicular to the middle plane remains straight)
e. The common normal to the panel and the stiffener before bending remains straight
after bending
The governing differential equations for the stiffened panels are [113, 114]:
2 2 2
2 2
2 2 2
2 2
1 10
2 2
1 10
2 2
u v u
x x y y
v u v
y x y x
(2.1)
2 22 2 2 2
2 ( 2 )2 2 2 2
M MM w w wxy yx N N Nx xy yx y x yx y x y
(2.2)
where Mx and My are bending moments at y and x edges, and Mxy and Myx are twisting
moments along y and x edges (Fig. 2-2).
2 2
2 2x
w wM D
x y
2 2
2 2y
w wM D
y x
2
1xy yx
wM M D
x y
(2.3)
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where, v is the Poisson’s ratio and D is the flexural rigidity of the plate,
3
212 1
EtD
,
E is the Young’s modulus and t is the thickness of the panel.
Fig. 2-2 Bending and twisting moments due to stiffeners
Substituting the moment equations, the final governing differential equation for
buckling is [113, 114]:
4 4 4 2 2 2
4 2 2 4 2 2
12 ( 2 )x xy y
w w w w w wN N N
x x y y D x x y y
(2.4)
2.1.2 Finite element model formulations
The finite element formulation is developed based on the Mindlin’s plate theory
(points of the plate originally on the normal to the undeformed middle surface remain on
a straight line, but the line is not necessarily normal to the deformed middle surface)
[115-117]. This technique allows for considering shear deformation. This theory is
basically the generalization of the Kirchhoff hypothesis (normal line to the mid-surface
remains normal after deformation).
x
y
My Myx
Mxy
Mx
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Fig. 2-3 Triangular shell element
Three node triangular shell elements were used for the finite element model
formulation based on the discrete Kirchhoff theory (Fig. 2-3). Discrete Kirchhoff
elements were originally presented by Wempner et al. [115]. Three node triangular shell
elements were used for the finite element model formulation based on the discrete
Kirchhoff theory. where,
e e ef sK K K (2.5)
where eK is the total stiffness matrix of a element, efK is the bending part of the
stiffness matrix, esK is the transverse shear part of the stiffness matrix.
bending stiffnesse
e Tf f f f
A
K B D B dA
shear stiffnesse
e Ts s s s
A
K B D B dA (2.6)
where, B’s and D’s are strain and elasticity matrices respectively.
The solution of the stiffness matrices, formation of the global matrix, and application
of the boundary conditions were done using MSC. Marc [118, 119] and ABAQUS [120]
ζ
η
1
2
3
x’
y’
z’ x
y
z
xyz
, , , , ,x y zu v w
Node 1 Node 2
Node 3
, , , , ,x y zu v w , , , , ,x y zu v w
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using bilinear interpolation function and using Gauss quadrature rule with three Gauss
integration points.
2.1.3 Numerical solution of the buckling problem
In an eigenvalue buckling problem, we look for the loads for which the model tangent
stiffness matrix becomes singular, so that the problem [118-120],
0K v (2.7)
has nontrivial solutions. K is the tangent stiffness matrix when the loads are
applied, and the v are nontrivial displacement solutions. An incremental loading
pattern, Q , is defined in the eigenvalue buckling prediction step. The magnitude of this
loading is not important; it will be scaled by the load multipliers, i , found in the
eigenvalue problem [120]:
0 0i iK K v (2.8)
where, 0K is the stiffness matrix corresponding to the base state, K is the
differential initial stress and load stiffness matrix due to the incremental loading pattern,
Q , i are the eigenvalues, iv are the buckling mode shapes (eigenvectors), and
refer to the degrees of freedom, and I refers to the ith buckling mode. The critical
buckling loads are then iP Q . Normally, the lowest value of i is of interest.
2.2 Finite Element Analyses of a Curvilinearly Stiffened Panel under Shear and Compression Loadings for Different Heights of the Stiffeners
To study the influence of the curvilinear stiffeners on the structural response of the
panel, static stress analyses of the curvilinearly stiffened panels were performed for
different stiffener heights. The geometry, loading, and boundary conditions of the
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curvilinearly stiffened panel under shear and compression loadings are shown in Fig. 2-4.
The geometric properties of the model were: length = 1067 mm, height = 1067 mm, and
thickness = 3.5 mm. The material properties were: Young’s modulus = 73 GPa, Poisson’s
ratio = 0.33, mass density = 2700 kg/m3, and shear modulus = 27.5 GPa.
Fig. 2-4 Geometry, loading, and boundary conditions of the curvilinearly stiffened panel
Top point
Left point Center point
Right point
Bottom point
2h
2b
t1 t2
t0
h1 h2
h = (h1+h2)/2
X, u
Z, w
Y, v
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s 2 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 2-5 Finite element model of the curvilinearly stiffened panel
The finite element mesh, with three-node isoparametric triangular elements, of the
curvilinearly stiffened panel with boundary and loading conditions is shown in Fig. 2-5.
Global static stress analysis of the panel for different stiffener heights was performed to
study the influence of these stiffeners on the structural response of the panel under
combined shear and compression loadings. Table 2-1 shows the list of values of the
stiffeners height and thickness.
Table 2-1 Heights and thickness of stiffeners and the panel
h(mm) h1(mm) h
2(mm) t
1(mm) t
2(mm) t
0(mm)
0 0 0 3 2.5 3.5
19.3 18.5 20 3 2.5 3.5
38.5 37 40 3 2.5 3.5
57.8 55.5 60 3 2.5 3.5
22 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0 20 40 600
0.5
1
1.5
2
Average stiffener height, h (mm)
Max
dis
pla
cem
ent,
wm
ax (
mm
)
a) 0 10 20 30 40 50 60
0
0.5
1
1.5
2
Average stiffener height, h (mm)
Dis
pla
cem
ent,
w (
mm
)
Top pointBottom pointLeft pointRight pointCenter point
b)
0 20 40 60
-0.585
-0.58
-0.575
-0.57
-0.565
Average stiffener height, h (mm)
Max
dis
pla
cem
ent,
vm
ax (
mm
)
c) 0 10 20 30 40 50 60
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
Average stiffener height, h (mm)
Dis
pla
cem
ent,
v (
mm
)
Top pointBottom pointLeft pointRight pointCenter point
d)
0 20 40 60-0.165
-0.16
-0.155
-0.15
-0.145
Average stiffener height, h (mm)
Max
dis
pla
cem
ent,
um
ax (
mm
)
e) 0 20 40 60
-0.08
-0.06
-0.04
-0.02
0
0.02
Average stiffener height, h (mm)
Dis
pla
cem
ent,
u (
mm
)
Top pointBottom pointLeft pointRight pointCenter point
f)
Fig. 2-6 Deformations of the panel for different heights of stiffeners
The panel was divided into five zones to study the deformation and stress response on
those zones of the panel under in-plane compression and shear loadings, as shown in Fig.
2-4. The deformation response of the panel for different stiffener heights can be seen in
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s 2 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 2-6. The panel had negligible out-of-plane deflection under in-plane loading when
there were no stiffeners. The maximum out-of-plane deflection was observed at the
center of the panel under in-plane loading for different stiffeners height. This out-of-
plane deflection of the panel under in-plane loading is due to the bending moment exerted
on the panel by the eccentric stiffeners attached to the panel. In addition, the out-of-plane
deflection increased with stiffener heights until the average stiffener height was 38.5 mm.
After this value of the stiffener height, the out-of-plane deflection decreased (Fig. 2-6 a,
b).
On the other hand, the change in in-plane deflections was very small, as shown in Fig.
2-6 c, d, e, and f. In short, although the out-of-plane deflection of the stiffened panel
increased with the stiffener height, it decreased when the optimum stiffener height was
reached. In the current case, the optimum average stiffener height was 57.8 mm. This
observation suggests that the bending rigidity of the panel can be improved if the stiffener
height is optimum.
The stress distribution in the panel for different stiffener heights can be seen in Fig.
2-7. Although the change is marginal, the maximum von Mises stress, maximum
principal stress, and minimum principal stress for the complete panel (Fig. 2-7 a, c, and e)
decreased with stiffener heights. The magnitude of the minimum principal stress was
larger than the magnitudes of the maximum principal stress and the von Mises stress.
Unlike other stress variations in the five zones, the magnitude of minimum principal
stress increased with stiffener heights until the average stiffener height was 38.5 mm, as
shown in Fig. 2-7 b, d, and f.
24 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0 20 40 60122.2
122.4
122.6
122.8
123
Average stiffener height, h (mm)
Max
vo
n M
ises
str
ess,
vo
n (
MP
a)
a) 0 10 20 30 40 50 60
33
34
35
36
37
38
39
40
41
Average stiffener height, h (mm)
von
Mis
es s
tres
s,
von (
MP
a)
Top pointBottom pointLeft pointRight pointCenter point
b)
0 20 40 6074.1
74.2
74.3
74.4
74.5
74.6
74.7
Average stiffener height, h (mm)
Max
pri
nci
pal
str
ess,
m
ax (
MP
a)
c) 0 10 20 30 40 50 60
6
8
10
12
14
16
18
20
Average stiffener height, h (mm)
Max
pri
nci
pal
str
ess,
m
ax (
MP
a)
Top pointBottom pointLeft pointRight pointCenter point
d)
0 20 40 60-202.5
-202
-201.5
-201
Average stiffener height, h (mm)
Min
pri
nci
pal
str
ess,
m
in (
MP
a)
e)
0 10 20 30 40 50 60-50
-48
-46
-44
-42
-40
-38
-36
-34
-32
Average stiffener height, h (mm)
Min
pri
nci
pal
str
ess,
m
in (
MP
a)
Top pointBottom pointLeft pointRight pointCenter point
f)
Fig. 2-7 Distribution of stresses in the panel for different heights of stiffeners
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s 2 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
With a further increase in the ratio of the stiffener heights, the minimum principal
stress (compressive) began to decrease. It can, therefore, be concluded that the minimum
principal stress is the critical stress in the stiffened panel under combined loadings, and
the magnitude of this stress can be reduced by adding stiffeners of an optimum height to
the panel. In the present case, the optimum average height of the stiffeners was found to
be 57.8 mm. Buckling analysis of the curvilinearly stiffened panels was performed to
study the variation of the first five buckling load factors of the panel for different
stiffener heights. The variation of the first five buckling load factors with respect to the
average stiffener height is shown in Fig. 2-8. All buckling load factors increased almost
linearly with the stiffener heights. This result suggests that the buckling load of the
stiffened panel increases with the increase of stiffener heights.
0 10 20 30 40 50 600.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Average stiffener height, h (mm)
Bu
cklin
g lo
ad f
acto
r
Mode IMode IIMode IIIMode IVMode V
Fig. 2-8 Variation of buckling load factors of the panel for different heights of stiffeners
26 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
2.3 Formulations of Global-local Finite Element Methods
Suppose we have a problem of domain g with essential boundary condition at one
end, and with force field f(t) at the other end (Fig. 2-9), where, l is the local domain
with the common kinematic interface u between the local and semi-global model
'g (domain excluding the local domain.) Suppose the displacement vector at any point x,
y, z is x y zu u v w .
Fig. 2-9 Global-local interface definition
Statement: For any x, y, and z location inside the global domain g containing the
local domain l , the displacement at the common boundary ui is always same for any
load case when the entire problem is in equilibrium [59, 68]. Then, if
, on
'' ' ', on
' 'and ;
lu u q ul l l ii i
gu u Qg g g ui i i
g g gu l l
(2.9)
uli is displacements of the local model, 'ugi
is displacements in the semi-global model,
is real, Q is the displacement vector for the semi-global model 'g at the boundary
'gu , and q is the displacement vector at the local boundary l
u for the local model, then
gl()f t
u
Global model Semi-global
model Local model
= +
x
y 'g
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s 2 7
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
oni i u gQ q x (2.10)
i.e., displacements at the common boundaries are equal. The schematic of the global-
local method can be shown in Fig. 2-10.
Fig. 2-10 Schematic of the global-local solution strategy
Perform Global Analysis (Creating geometry file, generating mesh, applying boundary and loading conditions, and performing static stress analysis)
Global Modeling and Analysis
Identify Critical Region (In the analysis of the global
static stress)
Specify Interpolation Region
(In the original global mesh)
Define Local Model Boundary (Generating local interpolated
displacements)
Global-Local Interface Definition
Generate Local Model (Impose kinematic boundary
conditions)
Perform Local Analysis (Stress analysis, define crack,
and fracture analysis)
Local Modeling
and Analysis
28 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
2.3.1 Continuity of the kinematic conditions between global and local method Let us start with the system mentioned in the schematic Fig. 2-9. The global model can
be represented in the stiffness matrix form:
K u fgg g
(2.11)
where g
K is the stiffness matrix, g
u is the deformation vector, and g
f is the force
vector for the nodes g
N respectively for the global representation; and l
K is the
stiffness matrix, l
u is the deformation vector, and l
f is the force vector for the nodes
lN respectively for the local representation. After isolating certain portion from the
whole system, the system becomes [59, 68, 69]:
'
g g l (2.12)
So, the solution of the equation g ggK u f provides the displacement vectors
required to define kinematic boundary conditions in local model to solve our current
equation of interest l llK u f for the local model.
2.3.2 Application of the global-local finite element method on a plate under tension
To validate the global-local finite element method, a simple plate with one end fixed
and another end with a static traction of 400 MPa was chosen. The geometric, material,
and loading conditions of the plate model are shown in Fig. 2-11 a. In the first step, the
stress distribution in the complete model was studied under the prescribed loading
conditions. Since the plate was subjected to a uniaxial tension, the component of the
Cauchy stress along the loading direction would be equal to the applied traction. The
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s 2 9
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
finite element mesh, with four-node isoparametric quadrilateral elements for plane stress
applications, of the panel with boundary and loading conditions are shown in Fig. 2-11 b.
a. A simple plate under uniaxial tension
b. Finite element mesh of the entire plate
c. Normal stress along the loading direction in the global model
d. Local model
e. Local model with kinematic BCs
f. Normal stress along the loading direction
in the local model
Fig. 2-11 Global-local finite element analyses of a simple plate under tension
The component of the Cauchy stress along the loading direction was exactly equal to
the applied traction (Fig. 2-11 c). In the second step, an area within the panel was
30 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
zoomed, which was the top right portion in the current case. The remaining elements and
the corresponding nodes were deleted subsequently (Fig. 2-11 d). The kinematic
boundary conditions obtained from the results of analyses of the complete model were
then applied to the zoomed (local) model (Fig. 2-11 e). Simulation was done for the
zoomed (local) model, and then the stress distribution was studied. The component of the
Cauchy stress along the loading direction of the zoomed (local) model was exactly equal
to the applied traction (Fig. 2-11 f). It can be concluded that the global and global-local
finite element methods produced similar results.
2.3.3 Global finite element analyses of a curvilinearly stiffened panel under shear and compression loads
To perform global-local finite element analyses, initial global results with a coarse
mesh of the finite element are necessary for developing the local model. Global finite
element analysis was, therefore, performed on the curvilinearly stiffened panel. The
complete schematic and finite element models of the curvilinearly stiffened panel for the
global analysis are shown in Fig. 2-4 and Fig. 2-5. The geometric properties of the panel
were: length = 1067 mm, height = 1067 mm, and thickness = 3.5 mm. The material
properties were: Young’s modulus = 73 GPa, Poisson’s ratio = 0.33, mass density = 2700
kg/m3, and shear modulus = 27.5 GPa. Using three nodes triangular elements, shell finite
element was used for the panel model. The finite element results of the global analysis
are shown in Fig. 2-12. Maximum von Mises stress was 122 MPa. The critical zone for
the major principal stress, von Mises stress, and strain energy density was at the lower
right corner of the panel. The shear stress variation was not significantly visible in the
global analysis. The CPU time required for the global finite element model analysis was
1.72 s.
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s 3 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Displacement, w
von Mises stress
Max principal stress
Min principal stress
Shear stress
Strain energy density
Fig. 2-12 Global finite element analyses of the stiffened panel under shear and compression loads
2.3.4 Global-local finite element analyses of a curvilinearly stiffened panel under shear and compression loads
Global-local analyses were performed on the same model defining kinematic
boundary conditions in the local model. To perform the global-local analysis, the critical
zone within the panel was zoomed, which was the lower right portion in the current case.
The remaining elements and then nodes were deleted subsequently. The kinematic
boundary conditions obtained from the results of analyses of the complete model were
then applied to the zoomed (local) model. Simulation was done for the zoomed (local)
model for further analyses.
32 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Displacement, w
von Mises stress
Max principal stress
Min principal stress
Shear stress
Strain energy density
Fig. 2-13 Global-local analyses of the panel under shear and compression loads
The finite element results of the global-local analysis are shown in Fig. 2-13.
Maximum von Mises stress in the local model was 122 MPa, which was equal to the
result obtained in the global analysis. The critical zone for major principal stress, von
Mises stress, and strain energy density in the local model were at the lower right corner
of the panel. It can be concluded that the global and global-local finite element methods
produced similar results for the curvilinearly stiffened panel. Most importantly, a better
variation of shear stress was observed in the global-local analysis. The CPU time required
for the local finite element model analysis was 0.17 s.
2.3.5 Analyses of refined global models and refined local models under shear and compression loads
Static analysis was performed on the global and global-local finite element models
with similar finite element mesh refinement under similar boundary and loading
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s 3 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
conditions. For the global analysis, the finite element mesh was refined for the complete
model; whereas for the global-local analysis, the finite element mesh was refined only for
the local model. Analysis was performed for three element lengths: 5 mm, 10 mm, and 20
mm.
Table 2-2 Summary of global and global-local finite element analyses
Element length (mm)
Global/local Total degrees of freedom
(#)
CPU time (s)
Size of data library (megabytes)
20 Global 21156 1.72 5.5
Local 1224 0.17 0.30
10 Global 83238 7.44 21.9
Local 4560 0.39 1.15
5 Global 330198 35.25 87.5
Local 17586 1.41 4.55
The summary of results of the global and global-local finite element analyses is given
in Table 2-2. Considerable improvement in the CPU time required to simulate the
models and the data library size required to save the output file was achieved with global-
local analyses. The percentage of the improvement in local analyses was calculated by
using the following equation:
(%) 100Global result Local result
improvementGlobal result
(2.13)
34 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
5 10 15 200
20
40
60
80
100
Element length (mm)
Siz
e o
f d
ata
libra
ry (
meg
abyt
es)
Global analysisLocal analysis
a) 5 10 15 20
0
1
2
3
4x 105
Element length (mm)
Deg
rees
of
free
do
m (
#)
Global analysisLocal analysis
b)
5 10 15 2090
91
92
93
94
95
96
Element length (mm)
Per
cen
tag
e im
pro
vem
ent
(%)
CPU time
c)
Fig. 2-14 Comparison of results obtained using global and global-local finite element analyses
Comparison of the size of the data library required for global and local analyses with
refined finite element meshes is shown in Fig. 2-14 a. Refined local analysis with element
length 5 mm could save almost 95% of the data library space. The number of degrees of
freedom required for global and local analyses is shown in Fig. 2-14 b. Refined global
model with 5 mm element length required 95% more degrees of freedom than refined
local model with same mesh density. Total CPU time required for local analysis with a
mesh density was compared with the CPU time required for global analysis with same
mesh density. Local analysis with the element length 5 mm could save around 95% CPU
time compared to the global analysis with the same mesh density (Fig. 2-14 c).
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s 3 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
2.4 Conclusions
Global finite element analyses of the curvilinearly stiffened panels for different
heights of the stiffeners were performed to study the influence of stiffeners on the
structural response of the panel under combined shear and compression loadings. The
panel had negligible out-of-plane deflection under the in-plane loading when there was
no stiffener. The maximum out-of-plane deflection was observed at the center of the plate
under the in-plane loading for different stiffener heights. The out-of-plane deformation
increased with stiffeners height until the stiffeners height was 38.5 mm. After this value
of stiffeners’ height, the out-of-plane deflection decreased. In short, although the out-of-
plane deflection of the stiffened panel increased with the stiffeners height, it decreased
when the optimum heights of the stiffeners were reached. The height of the stiffeners to
yield maximum stress was found to be 57.8 mm.
The magnitude of the minimum principal stress was larger than the magnitudes of the
maximum principal stress and the von Mises stress. In addition, unlike other stress
variation, the minimum principal stress increased with stiffeners height until the average
stiffeners height was 38.5 mm. With a further increase in the ratio of the stiffeners’
height, the minimum principal stress (compressive) began to decrease. It can, therefore,
be concluded that the minimum principal stress is the critical stress in the panel under
combined loading, and the magnitude of this stress can be reduced by adding stiffeners of
an optimum height to the panel.
Buckling analysis was performed to study the variation of the first five buckling load
factors for different stiffener heights. Critical buckling load factor increased significantly
with the increase of stiffeners’ height. This result suggests that the buckling stability of
the stiffened panel increases with the increase of stiffeners’ height.
36 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
To considerably reduce both the CPU time and the data storage space, a global-local
finite element method can be employed for studying the structural response of
curvilinearly stiffened panels. Global-local finite element analyses with a mesh
refinement were performed on a curvilinearly stiffened panel under combined shear and
compression loadings for three element-lengths: 5 mm, 10 mm, and 20 mm. The refined
global model with the 5 mm element length required 95% more degrees of freedom than
the refined local model with the same mesh density. The refined local analysis, with
elements of dimension 5 mm, can save 95% CPU time as compared to the global analysis
with the same mesh density.
3 Multi-load Case Damage Tolerance Study of Curvilinearly Stiffened Panels Using Global-local Finite Element Analyses
This chapter describes the methodology to use a global-local finite element method to
study fracture behavior of curvilinearly stiffened panels with a crack defined at the
location of the critical stress under combined shear and tensile loadings.
3.1 Fracture Mechanics Approaches
The fracture analysis was performed using J-integral and VCCT methods.
3.1.1 J-integral procedure The J-contour integral is a popular choice for characterizing fracture for linear and
nonlinear materials. Rice [121] provided the basis for fracture mechanics methodology by
idealizing elastic-plastic deformation as nonlinear elastic behavior.
Fig. 3-1 An arbitrary path on which the line integral (J integral) is to be calculated [120]
The J-integral for two dimensional crack problems in linear and nonlinear elastic
materials is a line integral surrounding a two dimensional crack tip, and can be written as
[122]:
, 1, 2uiwdy f dsi xJ i
a
(3.1)
b)
1
nh
2
0
0
m
n
A a)
x
y
Ѳ r
dS
Crack 1m;m
h
38 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Where is the total potential energy, w is the elastic strain energy density, f is the
traction vector on the contour , u is the displacement vector, ds is the length increment
along the contour (Fig. 3-1 a), and
0
eijw deij ij
f mi ij j
(3.2)
Where m is the outward unit normal vector to .
According to Shih and Asaro [123],
1 0 2 0 0 0
J d d
u
m M g S gx (3.3)
where M is written as,
w
u
M I σx
(3.4)
where I is the identity matrix, σ is the stress tensor, g is the weighting function within
the region ‘A’ (Fig. 3-1 b) having the value g = h on 2 and g = 0 on 1 , and m is the
outward normal to the region ‘A’. S is the surface traction on the crack surfaces 0 and
0 .
After simplification [123]:
A
J dA g
Mx
(3.5)
This is the integral used for numerical integration of J-integral. For a linear elastic
material, J is the general version of the strain energy release rate (G).
3.1.2 Extraction of stress intensity factors from the domain integral for a mixed mode loading case
Individual stress intensity factors IK , IIK , and IIIK play an important role in the linear
elastic fracture mechanics. The stress intensity factors can be related to the J-integral for
a linear elastic material by [120, 123],
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 3 9
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
2 2 211I II IIIJ K K K
EE
(3.6)
where E E for plane stress and 21
EE
for plane strain, axisymmetry, and three
dimensions.
3.1.3 Virtual Crack Closure Technique The Virtual Crack Closure Technique (VCCT) offers a simpler but more general way
for obtaining the energy release rate [124, 125].
Fig. 3-2 Normal stress ( yσ ) distribution ahead of the crack tip
The normal stress distribution ahead of a crack tip in an infinite isotropic plate
subjected to a remote Mode-I type loading can be seen in Fig. 3-2.
Fig. 3-3 Virtual crack closure technique
If the initial half crack size a extends to a a , for an infinitesimal value a , the
crack opening displacement behind the crack tip will be approximately the same as those
behind the original crack tip. Then the energy necessary to extend the crack from a to
b)
Δai
j
k’
I
J
K L
Ѳ x
y
k
jxF
ixF
jyFiyF
tn
inFit
Fa)
......
F
δ
Δa
Δa
x
y
i j
k
k’
I J
K L
Δa x
y
a
a+Δa
r yσ
40 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
a a is the same as that necessary to close the crack tip from a a to a. The work
can be computed as [126]:
1
( ) ( )2 0
aW v r a r dry
(3.7)
where v(r) is the crack opening displacement at a distance r behind the crack tip at
a a . The strain energy release rate can be obtained as,
1
lim ( ) ( )0 2 0
aWG v r a r drya a
(3.8)
According to Rybicki and Kanninen [127], in finite element analysis, the strain energy
release rate can be calculated as (Fig. 3-3 a):
'
'
1
2
1
2
i
i
I y k k
II x k k
G F v va
G F u ua
(3.9)
where Fxiand Fyi
are the nodal forces at node i in the x- and y-directions,
respectively, and uk and vk are the displacements at node k in the x- and y-directions,
respectively. This technique of calculating energy release rate is very attractive because
of the values of G can be computed from a single finite element analysis very accurately.
In order to evaluate individual energy release rate in a mixed mode problem (Fig. 3-3 b),
the following procedure is used: the forces at nodes i are calculated in the global
coordinate system; these forces are then transferred to the tangential (t) and normal (n)
coordinate system at the crack tip as follows (Fig. 3-3 b) [125]:
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 4 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
cos sin
sin cos
F F Fx yt i iiF F Fn x yi i i
(3.10)
The displacement components at node i, k, and k’ are calculated in the global
coordinate system. Then the relative displacement components are calculated as follows:
u u uirelative kkv v virelative kk
''
''
u u uirelative kkv v virelative kk
(3.11)
These displacements are then transferred to the tangential (t) and normal (n)
coordinate system at the crack tip as follows:
cos sin
sin cos
cos sin' ' '
sin cos' ' '
u u ut relative relativek k kv v vn relative relativek k ku u ut relative relative
k k kv v vn relative relativek k k
(3.12)
Using equations (3.13) and (3.15), individual energy release rate for a for 3-D case
[127],
42 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
2; *
2; *
*1 2 ;1
2 2 21 * for planestress
22 2 for plane strain
1
KIG K G EI I IE
KIIG K G EII II IIE
G EIIIG K KIII III IIIE
K K K KI II IIIeff
KIIIK K KI IIeff
(3.13)
where E E for plane stress and 21
EE
for plane strain, axisymmetry, and three
dimensions.
3.2 Framework of the Global-local Finite Element Method for the Fracture Analysis
This section describes the framework for global-local finite element methods in
fracture analysis. The details of this method for fracture analysis can be seen in Fig. 3-4.
At first, simulations are run either for the symmetric part of the panel or for the complete
panel with a coarse mesh (global analysis). Then, a neighborhood of a local area is
isolated from the global model to analyze further. Next, calculating the displacements and
rotations form the results of the global analysis, the kinematic boundary conditions are
applied to the boundary of the local model. Finally, the local model with kinematic
boundary conditions is analyzed for fracture analyses.
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 4 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 3-4 Procedure to calculate SIFs using global-local finite element method
3.2.1 Global-local finite element analyses of the center crack and side crack tension
plates To validate the proposed global-local finite element method for fracture analyses, two
studies on the fracture analysis of plates with the center crack and the side crack were
performed using the global-local finite element method, as shown in Fig. 3-5.
a. Center crack plate
b. Side crack plate
Fig. 3-5 Plates with center and side cracks
These problems have through the thickness cracks in finite plates. In such a case, one
technique is to approximate KI with the appropriate correction factor [122]:
.
2.
*
y yI
I I
F vG
a
K E G
44 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
I
aK a F
b
(3.14)
Where F(a/b) is the dimensionless constant that depends on the geometry and the
mode of the loading. Several empirical formulas have been developed for this
dimensionless constant. The difference in results of energy release rate was calculated
comparing with J-integral result using the formula:
Difference in (%) 100II
G JG
J
(3.15)
The difference in results of the stress intensity factor was calculated using the formula:
,73
,73Difference in (%) 100
IsidaI I
I IsidaI
K KK
K
(3.16)
3.2.2 Fracture analyses of a center crack tension plate
The summary of some empirical formulas for F(a/b) of the center crack plate are
presented in Table 3-1.
Table 3-1 Empirical formulas for the dimensionless constant F(a/b) of the center crack specimen
Formula Accuracy Method of Derivation (Reference)
Exact values for up to a/b = 0.9
Laurent series expansion of a complex stress potential [16]
2tan
2
a b aF
b a b
Error is less than 5% for a/b ≤ 0.5
Approximation by periodic crack solution [126]
2 4
1 0.025 0.06 sec2
a a a aF
b b b b
Error is 0.1% for any a/b
Modification to the Isida’s result [128]
2 3
1 0.128 0.288 1.525a a a a
Fb b b b
Error is 0.5% for a/b ≤ 0.7
Least square fitting to the Isida’s result [129]
The finite element model for one quarter of the center cracked specimen was made
with 2500 four-node quadrilateral elements.
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 4 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
a. Global model
b. Local model
c. Typical von Mises stress near crack tip (Pa)
Fig. 3-6 Finite element model of the plate problem with a center crack
The symmetric quarter plate finite element model, the local model with kinematic
boundary conditions applied at the boundaries, and the typical crack tip von Mises stress
for the center plate crack problem can be seen in Fig. 3-6. The geometric properties of the
model were: b = 1m, height = 2m. The material properties were: E = 200 GPa, ν = 0.3.
The plate was subjected to a normal traction of magnitude 400 MPa acting on the top
edge. The results for the plate problem with center cracks under tension can be seen in
Fig. 3-7. The energy release rate obtained using the global-local method matched very
well with the results of the global analysis obtained using both the J-integral and the
VCCT methods, as shown in Fig. 3-7 a.
46 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
a/b
10-6
X G
I (N
/m)
Global-LocalJ-integralVCCT
a.
0.1 0.2 0.3 0.4 0.53.43
3.44
3.45
3.46
3.47
3.48
a/b
Dif
fere
nce
in G
I (%
)
Global-LocalVCCT
b.
0.1 0.2 0.3 0.4 0.5200
300
400
500
600
700
a/b
KI (
MP
a m
)
Isida, 73Brown, 66Tada, 73Global-LocalJ-integralVCCT
c.
0.1 0.2 0.3 0.4 0.50
1
2
3
4
a/b
Dif
fere
nce
in K
I (%
)
Global-LocalJ-integralVCCT
d.
Fig. 3-7 Results of the plate problem with center cracks
The error for the energy release rate using the global-local finite element method was
very small (Fig. 3-7 b). Mode-I stress intensity factors were calculated and compared
with the available results in the literature. The stress intensity factors were calculated
using the global-local method and were found to be in good agreement with the results
available in the literature (Fig. 3-7 c).
3.2.3 Fracture analyses of a side crack tension plate The summary of some empirical formulas for F(a/b) of the side crack plate is
presented in Table 3-2. The finite element model for one half of the side crack specimen
was made with 2500 four-node quadrilateral elements.
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 4 7
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
a. Global model
b. Local model
c. Typical von Mises stress near crack tip (Pa)
Fig. 3-8 Finite element model of the side crack plate
The symmetric half plate finite element model, the local model defining kinematic
boundary conditions at boundaries, and the typical crack tip von Mises stress for the side
crack plate problem can be seen in Fig. 3-8. The geometric properties of the model were:
b = 1m, height = 2m. The material properties were: E = 200 GPa, ν = 0.3. The plate was
subjected to a normal traction of magnitude 400 MPa acting on the top edge.
Table 3-2 Empirical formulas for the dimensionless constant F(a/b) of the side crack specimen
Formula Accuracy Method of Derivation (Reference)
2 3 4
1.122 0.231 10.550 21.710 30.382a a a a a
Fb b b b b
Error is 0.5% for a/b ≤ 0.6
Least square fitting [129]
3
0.752 2.02 0.37 1 sin22
tan2cos
2
a aa b ab b
Fab a bb
Error is less than 0.5% for any a/b
[128]
48 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
The results for the plate problem with side cracks under tension are shown in Fig. 3-9.
The energy release rate obtained using the global-local finite element method matched
very well with the results of the global analysis obtained using both the J-integral and the
VCCT methods, as shown in Fig. 3-9 a. The error for the energy release rate using the
global-local method was very small (Fig. 3-9 b). The stress intensity factors were
calculated using the global-local method and were found to be in good agreement with
the results available in the literature (Fig. 3-9 c).
0.1 0.2 0.3 0.4 0.50
2
4
6
8
10
12
a/b
10-6
X G
I (N
/m)
Global-LocalJ-integralVCCT
0.1 0.2 0.3 0.4 0.53.2
3.4
3.6
3.8
4
a/b
Dif
fere
nce
in G
I (%
)
Global-LocalVCCT
0.1 0.2 0.3 0.4 0.50
500
1000
1500
a/b
KI (
MP
a m
)
Tada, 73Brown, 66Global-LocalJ-integralVCCT
0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
a/b
Dif
fere
nce
in K
I (%
)
Global-LocalJ-integralVCCT
Fig. 3-9 Result of the side crack plate under tension
3.3 Fracture Analyses of Curvilinearly Stiffened Panels Using Global and Global-local Finite Element Methods
This section describes the fracture analysis of the curvilinearly stiffened panels. In the
first step, buckling analysis of the panel was performed under shear and compression
loads to determine whether the thickness of the panel satisfied the buckling constraint.
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 4 9
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 3-10 Schematic of the panel under combined shear and compression loadings
Fig. 3-11 Finite element model of the panel under shear and compression loadings
The geometry, loading, and boundary conditions of the curvilinearly stiffened panel
under shear and compression loading can be seen in Fig. 3-10. The geometric properties
of the model were: length = 812 mm and height = 1016 mm. The material properties
were: Young’s modulus = 73 GPa, Poisson’s ratio = 0.33, mass density = 2700 kg/m3, and
shear modulus = 27.5 GPa. The complete finite element model of the stiffened panel
with three-node isoparametric triangular shell elements can be seen in Fig. 3-11. All
2h
2b2b
2h
50 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
analyses were performed using both MSC. Marc and ABAQUS. In both cases, the critical
buckling load factor was observed to be slightly less than one for the thickness of the
panel with a value of 3.25 mm, as shown in Fig. 3-12.
Minimum buckling load factor in ABAQUS = 0.937 for plate thickness 3.25 mm
Minimum buckling load factor in MSC. Marc = 0.934 for plate thickness 3.25 mm
Minimum buckling load factor in ABAQUS = 1.009 for plate thickness 3.27 mm
Minimum buckling load factor in MSC. Marc = 1.006 for plate thickness 3.27 mm
Fig. 3-12 Buckling analysis of the panel under combined shear and compression loads
The thickness was then gradually increased until the panel critical buckling load factor
was observed to be greater than one. The required thickness of the panel was determined
to be 3.27 mm for satisfying the buckling constraint.
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 5 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
3.3.1 Determining the location of the critical stress in the panel The global stress analysis of the panel of thickness 3.27 mm was performed to
determine the locations of the critical stress under combined shear and compression
loads. The critical stress zone was the common location with the maximum magnitude of
the principal and von Mises stresses in both MSC. Marc and ABAQUS, and the location
of the critical stress was observed at 153 mm along the horizontal and 183 mm along the
vertical directions respectively, from the bottom-left corner of the panel (Fig. 3-13).
Fig. 3-13 Critical stress location in the curvilinearly stiffened panel
52 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
The global-local finite element fracture analysis of the panel was then performed by
defining a crack of half crack length 0.725 mm in the location of the earlier obtained
critical stress [14] using MSC Patran [130] and MSC Nastran [131]. This crack size was
used because of the requirement of a sufficiently small crack, if the crack is in the vicinity
of any stress raiser [14, 112]. A simple configuration of the crack was selected because of
the simplicity of embedding a very small crack in the model [23, 132, 133]. Since the
crack generally opens in tension and shear loading, the fracture analysis was performed
under tension and shear loadings.
3.3.2 A mesh sensitivity study for fracture analyses of a curvilinearly stiffened panel under the combined shear and normal loads
The schematic of the complete panel under shear and normal loads, the global finite
element model with a refined mesh near the crack tip, the global-local model, and a
typical von Mises stress profile near the crack tip are shown in Fig. 3-14. To compare the
results of the fracture analysis obtained using global-local method, all analyses were
performed using the global-local method using MSC. Marc, and global methods using
MSC. Marc and ABAQUS. However, the results of fracture analysis are generally finite
element mesh dependent. It is, therefore, necessary to use a fine mesh near the crack tip
to obtain accurate results of fracture analysis. Mesh sensitivity analysis was, therefore,
performed on the panel with a crack. Two typical mesh densities used for the mesh
sensitivity analysis are shown in Fig. 3-15. A mesh sensitivity analysis was performed
under the combined shear and tensile loads for different mesh densities, i.e., different
number of elements per unit length, along the distance ahead of the crack tip.
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 5 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Combined load system
Refined mesh near crack tip
Defining crack (half crack length
= 0.725 mm)
Typical von Mises stress near
crack tip (MSC. Marc) Local model in MSC. Marc
Imported in ABAQUS for comparison
Fig. 3-14 Procedure for fracture analyses of the curvilinearly stiffened panels
The comparison of the results of the data library size of the output file, number of
nodes in the model, and the CPU time required to simulate a model between the global
analysis and the global-local analysis can be seen Fig. 3-16 a, b, and c, respectively.
Since the mesh refinement was performed only near the crack tip, the overall variation of
these results was small. However, the global-local finite element analysis showed a
2a
54 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
significant improvement in CPU time over the global finite element method for larger
mesh densities.
4.2 element/mm
8.4 element/mm
Fig. 3-15 Different mesh densities
The percentage improvement was calculated using:
_
(%) 100Global local result Global result
ImprovementGlobal result
(3.17)
The percentage improvement in data library size and CPU time requirement for the
global-local finite element analysis can be seen in Fig. 3-16 d. In higher mesh densities,
94% CPU time could be saved using the global-local finite element method over the
global finite element method. The energy release rate for different mesh densities can be
seen in Fig. 3-16 e. Although the energy release rate increased with the mesh density, it
remained almost constant after the mesh density of 8.4 element/mm ahead of the crack
tip. The percentage difference in the change of energy release rates for the corresponding
change of mesh densities can be seen in Fig. 3-16 f. The difference became almost
negligible when the mesh density of 8.4 element/mm ahead of the crack tip was used.
This mesh density was, therefore, used in the further analyses.
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 5 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0 5 10 15 200
5
10
15
20
25
30
Number of element per unit dimension (#/mm)
Dat
a lib
rary
siz
e (m
egab
ytes
)
GlobalLocal
a.
0 5 10 15 200
0.5
1
1.5
2x 104
Number of element per unit dimension (#/mm)
Nu
mb
er o
f n
od
es (
#)
GlobalLocal
b.
0 5 10 15 200
2
4
6
8
10
Number of element per unit dimension (#/mm)
CP
U t
ime
(s)
GlobalLocal
c.
0 5 10 15 2092
92.5
93
93.5
94
94.5
95
Number of element per unit dimension (#/mm)
Per
cen
tag
e (%
)
CPU time change (local)Data size change (local)
d.
0 5 10 15 20540
550
560
570
580
590
600
Number of element per unit dimension (#/mm)
En
erg
y re
leas
e ra
te, G
(N
/m)
GlobalLocal
e.
0 5 10 15 200
2
4
6
8
10
Number of element per unit dimension (#/mm)
Dif
fere
nce
(%
)
GlobalLocal
f.
Fig. 3-16 Global and global-local analyses of the panel for mesh sensitivity analyses under shear and normal loads
Fracture analyses of a curvilinearly stiffened panel with a half crack size of 0.725 mm
were performed under combined shear and tensile loads using the crack tip mesh density
of 8.4 element/mm. To study the influence of the individual load on the basic modes of
56 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
fracture, the shear and normal loads were varied differently. When the shear load was
varied, the normal load was held constant. Likewise, when the normal load was varied,
the shear load was held constant. All analyses were performed using the global-local
finite element method using MSC. Marc, and global finite element methods using MSC.
Marc and ABAQUS.
Fracture analyses in ABAQUS were performed using J-integral approach in which the
energy release rate would be independent of the path considered. The energy release rate
for the J-integral approach was calculated along different contours, considering the radius
of first contour equals to the length of three elements and of the subsequent contour
includes one additional length of the element.
0 20 40 60 80 1000
100
200
300
400
500
600
Percentage of load applied (%)
To
tal e
ner
gy
rele
ase
rate
, G (
N/m
)
Contour2Contour3Contour4
Crack length 1.45 mm and mesh density 8.4 element/mm
under combined loading
Fig. 3-17 Path independence of J-integral estimation
Typical energy release rates at different percentages of load along different contours
using J-integral approach are shown in Fig. 3-17. Although the energy release rate was
quite independent of the path considered, the energy release rates at 100% load along the
contour 4 were used for further analyses.
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 5 7
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
3.3.3 Fracture analysis of a curvilinearly stiffened panel under a fixed shear but different normal load for a crack tip mesh density of 8.4 element/mm
In this case, the shear load was maintained fixed while the edge tensile load was
introduced along the direction of the stiffeners and varied from zero to the final design
load. Defining:
0 0 00
; ( ) ; where ( ) 462, 200 / ;and ( ) 106,100 /( )
and ζ is varied from 0.0 to 1.0.
ns n s
n
NN fixed N N m N N m
N
where Nn and Ns are variable normal and shear loads, respectively.
The variation of in-plane normal and shear stresses at the crack tip for different normal
loads can be seen in Fig. 3-18. Results obtained using the global-local finite element
method using MSC. Marc matched very well with the results obtained using the global
finite element method using MSC. Marc and ABAQUS. While the crack tip shear stress
decreased with increasing normal load, the normal stress increased. Besides, considering
the magnitude, the increased rate of the crack tip normal stress was considerably larger
than the decreased rate of the shear stress. Comparative analysis of the results of the CPU
time and the data library size obtained in those three approaches can be seen in Fig. 3-19.
Considering the size of the data library of the output file and the CPU time required to
perform the simulations, the global-local method had considerable advantages over other
two methods for all load cases.
58 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0 0.2 0.4 0.6 0.8 135
40
45
50
Normalized tension,
12
(M
Pa)
GlobalLocalAbaqus
0 0.2 0.4 0.6 0.8 10
100
200
300
400
500
Normalized tension,
22
(P
a)
GlobalLocalAbaqus
Fig. 3-18 Crack tip stresses for different normal loads
0 0.5 10
20
40
60
80
100
Normalized tension,
Dat
a lib
rary
siz
e (m
egab
ytes
)
GlobalLocalAbaqus
0 0.5 10
20
40
60
80
Normalized tension,
CP
U t
ime
(s)
GlobalLocalAbaqus
Fig. 3-19 Percentage of savings obtained using global-local analyses over a global analysis for different normal loading cases
Energy release rates for different normal loads can be seen in Fig. 3-20 a. The energy
release rate increased significantly with the increase of normal loads. The difference in
the energy release rate was calculated by comparing the results obtained using the global-
local finite element method with the J-integral result obtained using ABAQUS as:
Difference in (%) 100G Jglobal local
GJ
(3.18)
The difference in energy release rates between the results obtained using J-integral
and the results obtained using global-local finite element method and global finite
element method using VCCT for all normal loading cases was very small, as shown in
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 5 9
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 3-20 b. Comparison of energy release rates among three approaches can be seen in
Fig. 3-20 c. The results matched very well with each other. Not only did the Mode-I
energy release rate increase with the normal load, but it was also the dominant mode of
fracture over the entire load range. The influence of the individual mode to the total
energy release rate can be seen in Fig. 3-20 d. The first data point was for pure shear
load. After the introduction of the normal load, most of the data was dense at the right-
bottom corner. The contribution of the Mode-I energy release rate to the total energy
release rate for the cases with different normal loads was notable.
0 0.2 0.4 0.6 0.8 10
100
200
300
400
500
600
Normalized tension,
To
tal e
ner
gy
rele
ase
rate
, G (
N/m
)
GlobalLocalAbaqus
a.
0 0.5 10
1
2
3
4
Normalized tension,
Dif
fere
nce
(%
)
Global GLocal G
b.
0 0.2 0.4 0.6 0.8 10
100
200
300
400
500
600
Normalized tension,
En
erg
y re
leas
e ra
te, G
(N
/m)
Global GLocal GGlobal G
I
Local GI
Global GII
Local GII
c.
0 0.5 10
0.5
1
GI/G
GII/G
GlobalLocal
d.
Fig. 3-20 Energy release rates for different normal loads
The effective stress intensity factor for different normal loads can be seen in Fig. 3-21
a. Stress intensity factor increased significantly with the increase of the normal load.
60 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
While the Mode-I stress intensity factor increased significantly with the increase of the
normal load (Fig. 3-21 b), Mode-II stress intensity factor decreased at a considerably
smaller rate over the load range (Fig. 3-21 c). The difference in the stress intensity factor
was calculated by comparing the results obtained using the global-local finite element
method with the J-integral result obtained using ABAQUS as:
Difference in (%) 100global local AbaqusK K
KAbaqusK
(3.19)
0 0.5 11
2
3
4
5
6
7
Normalized tension,
Kef
f (M
Pa m
)
GlobalLocalAbaqus
a.
0 0.2 0.4 0.6 0.8 10
2
4
6
8
Normalized tension,
KI (
MP
a m
)
GlobalLocalAbaqus
b.
0 0.2 0.4 0.6 0.8 10.8
0.9
1
1.1
1.2
1.3
Normalized tension,
KII (
MP
a m
)
GlobalLocalAbaqus
c. 0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
Normalized tension,
Dif
fere
nce
(%
)
KI global
KI local
KII global
KII local
Keff
global
Keff
local
d.
Fig. 3-21 Stress intensity factors for different normal loads
The difference in the stress intensity factors between the results obtained using J-
integral and the results obtained using global-local finite element method and global
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 6 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
finite element method using VCCT for all normal loading cases was very small as shown
in Fig. 3-21 d. The influence of individual mode to the effective mode of fracture can be
seen in Fig. 3-22. The first data point was for pure shear load. After the introduction of
the normal load, like the energy release rate, most of the data was dense at the right-
bottom corner. Similarly, the contribution of the Mode-I stress intensity factor to the
effective stress intensity factor for the cases with different normal loads was notable.
Considering the magnitude of the stress intensity factors, the case with the fixed shear but
different normal loads was, therefore, Mode-I case.
0.2 0.4 0.6 0.8 10
0.5
1
KI/K
eff
KII/K
eff
GlobalLocal
Fig. 3-22 Contribution of individual stress intensity factor to the effective stress intensity factor for different normal loads
3.3.4 Fracture analyses of a curvilinearly stiffened panel under a fixed normal but a
different shear load for a crack tip mesh density of 8.4 element/mm In this case, the edge tensile load was maintained fixed while the shear load was
introduced and varied from zero to the final design load. Defining,
0 0 00
; ( ) ; where ( ) 462, 200 / ;and ( ) 106,100 /( )
and is varied from 0.0 to 1.0.
sn n s
s
NN fixed N N m N N m
N
62 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
The in-plane normal and shear stresses at the crack tip for different shear loads can be
seen in Fig. 3-23. Considering the magnitude, the crack tip shear stress decreased
initially, then increased after 0.2 . On the other hand, the crack tip normal stress
increased at the same rate with the increase of the shear load.
0 0.5 1-20
-10
0
10
20
30
40
Normalized shear,
12
(M
Pa)
GlobalLocalAbaqus
0 0.5 1370
380
390
400
410
Normalized shear,
22
(M
Pa)
GlobalLocalAbaqus
Fig. 3-23 Crack tip stresses for different shear loads
0 0.5 10
20
40
60
80
100
Normalized shear,
Dat
a lib
rary
siz
e (m
egab
ytes
)
GlobalLocalAbaqus
0 0.5 1
0
20
40
60
80
Normalized shear,
CP
U t
ime
(s)
GlobalLocalAbaqus
Fig. 3-24 Percentage of savings obtained using global-local analyses over a global analysis for different shear loading cases
Comparative analysis of the results of the CPU time and data library size obtained in
those three approaches can be seen Fig. 3-24. Considering the size of the data library of
the output file and the CPU time required to perform the simulations, the global-local
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 6 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
finite element analysis showed considerable advantages over the other two finite element
methods for all shear load cases.
0 0.2 0.4 0.6 0.8 1500
520
540
560
580
600
Normalized shear,
To
tal e
ner
gy
rele
ase
rate
, G (
N/m
)
GlobalLocalAbaqus
a. 0 0.5 10
1
2
3
4
5
Normalized shear, D
iffe
ren
ce (
%)
Global GLocal G
b.
0 0.2 0.4 0.6 0.8 10
100
200
300
400
500
600
Normalized shear,
En
erg
y re
leas
e ra
te, G
(N
/m)
Global GLocal GGlobal G
I
Local GI
Global GII
Local GII
c.
0.98 0.99 10
0.005
0.01
0.015
0.02
GI/G
GII/G
GlobalLocal
d.
Fig. 3-25 Energy release rates for different shear loads
Energy release rates for different shear loads can be seen in Fig. 3-25 a. The difference
in energy release rates between the results obtained using J-integral and the results
obtained using global-local finite element method and global finite element method using
VCCT for all shear loading cases was very small, as shown in Fig. 3-25 b. Comparison of
energy release rates among three approaches can be seen in Fig. 3-25 c. The results
matched very well with each other. Not only did the Mode-I energy release rate increase
with the shear load, but it was also the dominant mode of fracture over the shear load
64 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
range. The influence of the individual mode to the total energy release rate can be seen in
Fig. 3-25 d. As the most of the data was dense at the right-bottom side, the contribution
of the Mode-I energy release rate to the total energy release rate for the cases with
different shear loads was notable.
0 0.5 14
5
6
7
8
Normalized shear,
Kef
f (M
Pa m
)
GlobalLocalAbaqus
a.
0 0.5 15
6
7
8
Normalized shear, K
I (M
Pa m
)
GlobalLocalAbaqus
b.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Normalized shear,
KII (
MP
a m
)
GlobalLocalAbaqus
c. 0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
Normalized shear,
Dif
fere
nce
(%
)
KI global
KI local
KII global
KII local
Keff
global
Keff
local
d.
Fig. 3-26 Stress intensity factors for different shear loads
The effective stress intensity factor for different shear loads can be seen in Fig. 3-26 a.
The Mode-I stress intensity factor increased at a very slow rate with an increase in the
shear load (Fig. 3-26 b). The Mode-II stress intensity factor decreased first, and then
increased after 0.2 (Fig. 3-26 c). This is due to a decrease in the magnitude of the
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 6 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
crack tip shear stress with the increase of the shear load until 0.2 (Fig. 3-23), and the
increase of the magnitude of the crack tip shear stress with the increase of the shear load
after 0.2 (Fig. 3-23). Since the crack tip shear stress is mainly responsible for the
Mode-II fracture, the first decreasing and then the increasing profile of the Mode-II stress
intensity factor might be due to the variation of the magnitude of the shear stress at the
crack tip in the similar nature.
0.99 0.995 10
0.05
0.1
0.15
0.2
KI/K
eff
KII/K
eff
GlobalLocal
Fig. 3-27 Contribution of individual stress intensity factor to the effective stress intensity factor for different shear loads
The difference in the stress intensity factors between the results obtained using J-
integral and the results obtained using global-local finite element method and global
finite element method using VCCT for all shear loading cases can be seen in Fig. 3-26 d.
The difference in all cases was very small. The influence of the individual mode on the
total mode of fracture can be seen in Fig. 3-27. Like the energy release rate, most of the
data were dense at the right-bottom side. Similarly, the contribution of the Mode-I stress
intensity factor to the effective stress intensity factor for the cases with different shear
66 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
loads was notable. Considering the magnitude of the stress intensity factors, the case with
the fixed normal but different shear loads was, therefore, Mode-I dominant case, as well.
However, the magnitude of the effective stress intensity factors for the case with the fixed
shear but different normal loads was varied in a much wider range than the case with the
fixed normal but different shear loads. The influence of the normal load to the effective
stress intensity factor was, therefore, greater than the influence of the shear load.
3.4 Fracture Toughness of the Curvilinearly Stiffened Panel with a Crack
Fracture toughness may be defined as the ability of a part with a crack or defect to
sustain a load without catastrophic failure. The critical stress intensity factor is a measure
of the fracture toughness of the material in the plane strain condition. Fracture toughness
of the curvilinearly stiffened panel with a crack was analyzed under combined shear and
normal loadings.
Fig. 3-28 A curvilinearly stiffened panel with a crack of length 2a
For a through the thickness crack in the panel, the stress intensity factor (Fig. 3-28) is:
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 6 7
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
IK a (3.20)
where, is the dimensional dependent correction factor, and for a stiffened panel,
1 2 1 2 1 2 1 2, , , , , , , , , , with two curved stiffeners
, with no stiffeners
stiff
no stiff
f t t h h a b h b b
ag h
b
(3.21)
where, no stiff stiff
The critical stress intensity factor, KIC, for a panel with the material Al7050-T7451
[134] was given by,
27.47 Unstiffened panelICK MPa m (3.22)
Analogous to the equation for the stress intensity factor, we can write:
stiffI stiff
no stiffI no stiff
K a
K a
(3.23)
For our current problem,
33 0.4064 0.725*10
0.725*10 ; 0.2032 ;So, 0.00357,2 0.2032
, 1.00 [16]no stiff
aa m b m and
ba
g h Isidab
In this case,
no stiffI no stiffK a a
(3.24)
Since no stiff stiff , so it should be stiffIK a
Therefore, if the critical stress intensity factor of the un-stiffened panel is considered
as the design standard, the design is in the reasonably safe side for the stiffened panel. In
this analysis, the half crack size was 0.725 mm. Under the maximum combined loading
condition (Nn=462,200 N/m, and Ns=106,100 N/m), the largest effective stress intensity
factor was,
68 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
6.614combined loadeffK MPa m (3.25)
which was very small compared to the KIC:
0.24combined loadeff
IC
K
K
Therefore, considering the critical stress intensity factor, the design of the stiffened
panel was indeed an optimum design.
3.5 Conclusions
A global-local finite element method was used to study the damage tolerance of
curvilinearly stiffened panels. All analyses were performed using global-local finite
element method using MSC. Marc, and global finite element methods using MSC. Marc
and ABAQUS. Before starting the damage tolerance study, stress distributions on the
panel were analyzed to find the location of the critical stress, which was the common
location with the maximum magnitude of the principal stresses and von Mises stress,
under the combined shear and compression loadings.
To perform the damage analyses of the curvilinearly stiffened panels, a half crack size
of 0.725 mm was defined, using MSC Patran and MSC Nastran, in the earlier obtained
critical stress zone. This crack size was used because of the requirement of a sufficiently
small crack, if the crack is in the vicinity of any stress raiser. A mesh sensitivity analysis
was then performed under the combined shear and tensile loads to validate the choice of
the mesh density near the crack tip. The difference between the subsequent results
became almost negligible when the mesh density of 8.4 element/mm ahead of the crack
tip was used. In addition, 94% saving in the CPU time was achieved using the global-
G l o b a l - l o c a l m e t h o d s t o s t i f f e n e d p a n e l s f o r f r a c t u r e a n a l y s e s 6 9
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
local finite element method over the global finite element method using this mesh density
near the crack tip. This mesh density was, therefore, used in the further analyses.
Fracture analyses of a curvilinearly stiffened panel with a half crack size of 0.725 mm
were performed under combined shear and tensile loads using the crack tip mesh density
of 8.4 element/mm. To study the influence of the individual loads on the basic modes of
fracture, shear and normal loads were varied differently. When the shear load was varied,
the normal load was held constant. Likewise, when the normal load was varied, the shear
load was held constant. Considering the size of the data library of the output file and the
CPU time required to perform the simulations, the global-local finite element method had
considerable advantages over the other two finite element methods for all load cases.
Considering the magnitude of the stress intensity factors, the case with the fixed shear
but different normal loads was a Mode-I dominant. Considering the magnitude of the
stress intensity factors, the case with the fixed normal but different shear loads was a
Mode-I dominant, as well. However, the magnitude of the effective stress intensity
factors for the case with the fixed shear but different normal loads was varied in a much
wider range than the case with the fixed normal but different shear loads. In short, the
influence of the normal load to the effective stress intensity factor was, therefore, greater
than the influence of the shear load.
The critical stress intensity factor of the curvilinearly stiffened panel with the half
crack size of 0.725 mm was analyzed. The largest effective stress intensity factor of the
panel under the maximum combined shear and tensile loads was very smaller than the
critical stress intensity factor. Therefore, considering the critical stress intensity factor,
the design of the panel was an optimum design.
4 Global-local Finite Element Methods for Fracture Analyses of Curvilinearly Stiffened Panels for Different Crack Sizes
This chapter deals with the fracture analysis on the curvilinearly stiffened panels for
different crack sizes, with predefined half crack lengths of 5, 10, 15, 20, and 25 mm. The
starting half crack size of 5 mm was chosen since the recommended half crack size in the
critical zone [14] should be more than 3 mm, if the crack is not in the vicinity of any
stress raiser [112]. To be on the conservative side, a moderately larger crack size than the
required minimum crack length was chosen. The geometric properties of the complete
model were: length = 812 mm, height = 1016 mm, and thickness = 3.27 mm. The
material properties were: Young’s modulus = 73 GPa, Poisson’s ratio = 0.33, mass
density = 2700 kg/m3, and shear modulus = 27.5 GPa.
4.1 Framework for the Fracture Analysis of Curvilinearly Stiffened Panels for Different Crack Lengths with a Crack Tip Mesh Density of 8.4 element/mm
Fracture analyses of the curvilinearly stiffened panel were performed for different
crack lengths under three load cases: a) a pure shear load, b) a normal load, and c) a
combined shear and normal loads. The overall procedure for fracture analyses of the
curvilinearly stiffened panel using global-local finite element methods can be seen in Fig.
4-1. The initial stiffened panel was designed in the MSC Patran [130], and then one bulk
data file was created for MSC Nastran [131]. This bulk data file was then imported in
MSC. Marc for defining BCs and performing buckling analysis to determine whether the
thickness of the panel satisfied the buckling constraint. Global stress analysis on the panel
was then performed to find the location of the critical stress under combined shear and
compression loadings. The location of the critical stress was the common location with
the maximum magnitude of the principal stresses and von Mises stress.
G l o b a l - l o c a l m e t h o d s t o p a n e l s w i t h d i f f e r e n t c r a c k s i z e s 7 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 4-1 Framework for fracture analyses of curvilinearly stiffened panels
The same model was then exported to the ABAQUS for comparison of the results, and
was also exported to MSC Patran to edit geometry for defining cracks in the critical stress
zone and refining mesh near the crack tips. After defining cracks, another bulk data file
was created for the stiffened panels with cracks using MSC Patran, and then the model
was imported back to the MSC. Marc for global-local fracture analyses. The schematic of
the complete model, the global finite element model with a refined mesh near the crack
Design Variables Uncracked Geometry and Meshing in MSC. Patran
Creating BDF File for MSC. Nastran
Importing in MSC. Marc and Defining BCs
Static Stress Analyses under Combined Loadings in MSC. Marc for Locating Critical Stress Zone
Importing in MSC. Patran, Editing Geometry for Future Crack Modeling and Refining Mesh in
the Critical Stress Zone
Importing in MSC. Marc, Defining BCS for Global Models
Defining Crack, Local Modeling, Global-Local Methods for Fracture
Analyses, Calculating Effective Stress Intensity Factor under Combined
Loads in MSC. Marc
Responses for Optimization: • Effective Stress Intensity Factor, Critical Buckling Load Factor, and Maximum von
Mises Stress. • Comparing with the Results Obtained in
Abaqus
Optimizer: Mass Minimization Satisfying
Constraints
Optimal Design
Global Buckling Analyses in MSC. Marc
72 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
tip, the definition of a crack, the global-local model, and a typical von Mises stress profile
near the crack tip are shown in Fig. 4-2. The fracture analyses are performed in the two
crack tips of the through the thickness cracks, as shown in Fig. 4-2.
Combined load system
Panel with a crack
Refined mesh near the crack (half
crack length, a = 20 mm)
Local model in MSC. Marc
Imported in ABAQUS for
comparison
von Mises stress (MSC. Marc)
Fig. 4-2 Procedure for the fracture analysis of the curvilinearly stiffened panel for different “a”
a a
b1 b2
u = v = 0
v = 0
462200 N/m
106100 N/m
Edge 1: w = 0
Ed
ge
2:
w =
0
Edge 3: w = 0
X
Y
2b
106100 N/m
462200 N/m
Tip 1 Tip
2h
Tip 1 Tip 2
G l o b a l - l o c a l m e t h o d s t o p a n e l s w i t h d i f f e r e n t c r a c k s i z e s 7 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Global model for a crack length 10 mm
Local model for a crack length 10 mm
Global model for a crack length 20 mm
Local model for a crack length 20 mm
Global model for a crack length 30 mm Local model for a crack length 30 mm
Global model for a crack length 40 mm
Local model for a crack length 40 mm
Global model for a crack length 50 mm
Local model for a crack length 50 mm
Fig. 4-3 Global and local models for different crack lengths
To develop a local model, a small neighborhood of the area containing the whole crack
within the panel was, first, zoomed. The remaining elements and nodes were, then,
deleted subsequently. In the final step, the kinematic boundary conditions, calculating
Tip 1 Tip 2
74 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
from the global analysis, were applied to the boundaries of the local model (Fig. 4-2).
The global crack models along with the corresponding local crack models for different
crack lengths can be seen in Fig. 4-3.
Fracture analyses in ABAQUS were performed using the J-integral approach. The
energy release rate for the J-integral approach was calculated along different contours,
considering the radius of first contour equals to the length of three elements and of
subsequent contour includes one additional length of the element. Typical energy release
rates at different percentages of load along different contours using J-integral approach
for crack lengths 10 mm under combined loading can be seen in Fig. 4-4. Although the
energy release rate was quite independent of the path considered, the energy release rates
at 100% load along the contour 4 were used for further analyses.
0 20 40 60 80 1000
1000
2000
3000
4000
Percentage of load applied (%)
To
tal e
ner
gy
rele
ase
rate
, G (
N/m
)
Tip1:Contour2Tip1:Contour3Tip1:Contour4Tip2:Contour2Tip2:Contour3Tip2:Contour4
Crack length 10 mm and mesh density 8.4 element/mm under combined loading
Fig. 4-4 Path independence of the J-integral estimation
4.2 Fracture Analyses of a Curvilinearly Stiffened Panel for Different Crack Lengths under a Fixed Normal Load
In this case, the shear load was zero while a constant edge tensile load of 462,200 N/m
was applied along the direction of the stiffeners. Results obtained using the global-local
finite element method were compared with the results obtained using the global finite
G l o b a l - l o c a l m e t h o d s t o p a n e l s w i t h d i f f e r e n t c r a c k s i z e s 7 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
element method. Comparative analyses between the performances of the global-local
finite element method and the global finite element method can be seen in Fig. 4-5.
Considering the size of the data library of the output file and the CPU time required to
perform the simulations, the global-local finite element method had considerable
advantages over the global finite element method for cases with different crack lengths.
More than 85% data storage space and 85 % CPU time requirement could be saved using
the global-local finite element method (Fig. 4-5).
5 10 15 20 2574
76
78
80
82
84
86
88
Half crack length, a (mm)
Per
cen
tag
e sa
vin
g (
%)
CPU timeDegrees of freedomData library size
Fig. 4-5 Percentage of savings obtained using global-local analyses over a global analysis for the normal loading case
Energy release rates for different crack lengths under the tensile loading can be seen in
Fig. 4-6 a-c. The difference in energy release rate for the cases with different crack
lengths was calculated by comparing the results obtained using the global-local analysis
with the J-integral result obtained using ABAQUS. The difference in all cases with
different crack lengths was very small as shown in Fig. 4-6 d. In different crack length
cases, results obtained using the global-local finite element method matched very well
with the results obtained using the global finite element method. In addition, compared to
76 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
the Mode-II energy release rate, the Mode-I energy release rate increased significantly
with the increase of crack lengths.
5 10 15 20 250
0.5
1
1.5
2
2.5x 104
Half crack length, a (mm)
To
tal e
ner
gy
rele
ase
rate
, G (
N/m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
a. 5 10 15 20 25
0
0.5
1
1.5
2
2.5x 104
Half crack length, a (mm)
Mo
de-
I en
erg
y re
leas
e ra
te, G
I (N
/m)
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
b.
5 10 15 20 250
10
20
30
40
50
Half crack length, a (mm)
Mo
de-
II en
erg
y re
leas
e ra
te, G
II (N
/m)
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
c.
5 10 15 20 250
1
2
3
4
5
Half crack length, a (mm)
Dif
fere
nce
(%
)
Tip1:Local GTip2:Local G
d.
Fig. 4-6 Energy release rates for different crack lengths under the normal loading
0.997 0.998 0.999 10
0.5
1
1.5
2
2.5x 10-3
GI/G
GII/G
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
Fig. 4-7 Contribution of individual mode to the total energy release rate for different crack lengths under the normal loading
G l o b a l - l o c a l m e t h o d s t o p a n e l s w i t h d i f f e r e n t c r a c k s i z e s 7 7
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0 10 20 3010
15
20
25
30
35
40
45
Half crack length, a (mm)
Kef
f (M
Pa m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
a.
0 10 20 3010
15
20
25
30
35
40
45
Half crack length, a (mm)
KI (
MP
a m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
b.
5 10 15 20 250
0.5
1
1.5
2
Half crack length, a (mm)
KII (
MP
a m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
c. 5 10 15 20 25
0
1
2
3
4
5
Half crack length, a (mm)
Dif
fere
nce
(%
)
Tip1:Local K
I
Tip1:Local KII
Tip1:Local Keff
Tip2:Local KI
Tip2:Local KII
Tip2:Local Keff
d.
0.9985 0.999 0.9995 10
0.01
0.02
0.03
0.04
0.05
KI/K
eff
KII/K
eff
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
e.
Fig. 4-8 Stress intensity factors for different crack lengths under the normal loading
The magnitudes of the total energy release rates with the increasing crack lengths were
larger at the crack Tip 1 than at the crack Tip 2 (Fig. 4-6 a). The reason could be that the
crack Tip 1 is at a farther distance from the stiffener location than the crack Tip 2, and
that is why the crack experiences less resistance at the crack Tip 1 to propagate than at
78 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
the crack Tip 2. The influence of the individual mode to the total energy release rate for
different crack lengths under the normal loading case is shown in Fig. 4-7. The
contribution of the Mode-I energy release rate to the total energy release rate was
significant.
The stress intensity factors for different crack lengths under the tensile loading can be
seen in Fig. 4-8 a-c. The difference in the stress intensity factors was calculated by
comparing the results obtained using the global-local finite element analysis with the J-
integral results obtained using ABAQUS. The difference in all cases for different crack
lengths was very small as shown in Fig. 4-8 d. Contribution of the Mode-I stress intensity
factor to the effective stress intensity factor was significant as shown in Fig. 4-8 e.
Considering the magnitude of the stress intensity factors, and considering the change of
the stress intensity factors with the change in the crack lengths, the farther crack tip from
the stiffener was critical under the normal load case for different crack lengths. The
difference in the values of the stress intensity factors for the crack Tip 1 and the crack Tip
2 can thus be considered to be the crack growth arresting capability of the stiffeners.
Furthermore, every case for different crack lengths under the normal loading was Mode-I
dominant, and it was the Mode-I stress intensity factor that increased significantly with
the increase of the crack length.
4.3 Fracture Analyses of a Curvilinearly Stiffened Panel for Different Crack Lengths under a Pure Shear Load
In this case, the normal load was zero while a constant edge shear load of 106,100
N/m was applied in the four edges of the panel. Results obtained using the global-local
finite element method were compared with the results obtained using the global finite
element method.
G l o b a l - l o c a l m e t h o d s t o p a n e l s w i t h d i f f e r e n t c r a c k s i z e s 7 9
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
5 10 15 20 2575
80
85
90
Half crack length, a (mm)
Per
cen
tag
e sa
vin
g (
%)
CPU timeDegrees of freedomData library size
Fig. 4-9 Percentage of savings obtained using global-local analyses over a global analysis for the pure shear loading case
5 10 15 20 250
200
400
600
800
1000
1200
Half crack length, a (mm)
To
tal e
ner
gy
rele
ase
rate
, G (
N/m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
a. 5 10 15 20 25
0
0.5
1
1.5
2
Half crack length, a (mm)
Mo
de-
I en
erg
y re
leas
e ra
te, G
I (N
/m)
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
b.
5 10 15 20 250
200
400
600
800
1000
1200
Half crack length, a (mm)
Mo
de-
II en
erg
y re
leas
e ra
te, G
II (N
/m)
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
c.
5 10 15 20 250
1
2
3
4
5
Half crack length, a (mm)
Dif
fere
nce
(%
)
Tip1:Local GTip2:Local G
d.
Fig. 4-10 Energy release rates for different crack lengths under the pure shear load
80 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Comparative analyses between the performances of the global-local finite element
method and the global finite element method can be seen in Fig. 4-9. Considering the size
of the data library of the output file and the CPU time required to perform the
simulations, the global-local finite element method had considerable advantages over the
global finite element method for the cases with different crack lengths. More than 85%
data storage space and 85% CPU time could be saved using the global-local finite
element method (Fig. 4-9).
Energy release rates for the cases with different crack lengths under the shear loading
can be seen in Fig. 4-10 a-c. The difference in energy release rates in all cases for
different crack lengths was very small as shown in Fig. 4-10 d. In different crack length
cases, results obtained using the global-local finite element method matched very well
with the results obtained using the global finite element method. The influence of the
individual mode to the total energy release rate for different crack lengths under the pure
shear loading case is shown in Fig. 4-11. The contribution of the Mode-II energy release
rate to the total energy release was significant.
0 0.5 1 1.5 2x 10
-3
0.998
0.9985
0.999
0.9995
1
1.0005
GI/G
GII/G
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
Fig. 4-11 Contribution of individual mode to the total energy release rate for different crack lengths under the pure shear load
G l o b a l - l o c a l m e t h o d s t o p a n e l s w i t h d i f f e r e n t c r a c k s i z e s 8 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
5 10 15 20 253
4
5
6
7
8
9
Half crack length, a (mm)
Kef
f (M
Pa m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
a. 5 10 15 20 25
0
0.1
0.2
0.3
0.4
Half crack length, a (mm)
KI (
MP
a m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
b.
5 10 15 20 253
4
5
6
7
8
9
Half crack length, a (mm)
KII (
MP
a m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
c.
5 10 15 20 250
1
2
3
4
5
Half crack length, a (mm)
Dif
fere
nce
(%
)
Tip1:Local K
I
Tip1:Local KII
Tip1:Local Keff
Tip2:Local KI
Tip2:Local KII
Tip2:Local Keff
d.
0 0.02 0.040.9985
0.999
0.9995
1
1.0005
1.001
KI/K
eff
KII/K
eff
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
e.
Fig. 4-12 Stress intensity factors for different crack lengths under the pure shear load
The stress intensity factors for different crack lengths under the shear loading can be
seen in Fig. 4-12 a-c. The difference in the values of the stress intensity factors in all
82 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
cases for different crack lengths under the pure shear loading was considerably small as
shown in Fig. 4-12 d. The contribution of the Mode-II stress intensity factor to the
effective stress intensity factor was much more significant as shown in Fig. 4-12 e.
Considering the magnitude of the stress intensity factors, and considering the change of
the stress intensity factors with the change in the crack lengths, every case for different
crack lengths under the pure shear loading was seen to be Mode-II dominant, and the
Mode-II stress intensity factor increased significantly with the increase in the crack
lengths.
4.4 Fracture Analyses of a Curvilinearly Stiffened Panel for Different Crack Lengths under the Combined Shear and Normal Loads
Fracture analyses were performed on a curvilinearly stiffened panel under combined
shear and normal loads. In this case, the shear load of magnitude 106,100 N/m and the
edge normal (tensile) load of magnitude 462,200 N/m were maintained constant.
5 10 15 20 2575
80
85
90
95
Half crack length, a (mm)
Per
cen
tag
e sa
vin
g (
%)
CPU timeDegrees of freedomData library size
Fig. 4-13 Percentage of savings obtained using global-local analyses over a global analysis for the combined shear and normal loading case
Comparative analyses between the performances of the global-local finite element
method and the global finite element method can be seen in Fig. 4-13. Considering the
size of the data library of the output file and the CPU time required to perform the
G l o b a l - l o c a l m e t h o d s t o p a n e l s w i t h d i f f e r e n t c r a c k s i z e s 8 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
simulations, the global-local finite element method had considerable advantages over the
global finite element method for all cases with different crack lengths. In most of the
cases for different crack lengths, more than 85% data storage space and 85% CPU time
could be saved using the global-local finite element method (Fig. 4-13).
5 10 15 20 250
0.5
1
1.5
2
2.5x 104
Half crack length, a (mm)
To
tal e
ner
gy
rele
ase
rate
, G (
N/m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
a. 5 10 15 20 25
0
0.5
1
1.5
2
2.5x 104
Half crack length, a (mm)M
od
e-I e
ner
gy
rele
ase
rate
, GI (
N/m
)
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
b.
5 10 15 20 250
500
1000
1500
2000
Half crack length, a (mm)
Mo
de-
II en
erg
y re
leas
e ra
te, G
II (N
/m)
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
c.
5 10 15 20 250
1
2
3
4
Half crack length, a (mm)
Dif
fere
nce
(%
)
Tip1:Local GTip2:Local G
d.
Fig. 4-14 Energy release rates for different crack lengths under the combined shear and normal loads
Energy release rates for different crack lengths under the combined shear and normal
loadings can be seen in Fig. 4-14 a-c. The difference in energy release rates for the cases
with different crack lengths under the combined shear and normal loadings was very
small as shown in Fig. 4-14 d. Results obtained using the global-local finite element
method matched very well with the results obtained using the global finite element
84 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
method. Both the Mode-I and the Mode-II energy release rates increased significantly
with the increase of crack lengths. In addition, the magnitudes of the total energy release
rates for most of the cases for different crack lengths were larger at the crack Tip 1 than
at the crack Tip 2 (Fig. 4-14 a).
0.92 0.94 0.96 0.980.03
0.04
0.05
0.06
0.07
0.08
GI/G
GII/G
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
Fig. 4-15 Contribution of individual mode to the total energy release rate for different crack lengths under the combined shear and normal loads
The influence of the individual mode to the total energy release rate for different crack
lengths under the combined shear and normal loadings is shown in Fig. 4-15. The
contribution of the Mode-I energy release rate to the total energy release rate was
significant. The stress intensity factors for different crack lengths under the combined
shear and normal loadings can be seen in Fig. 4-16 a-c. The difference in the values of
the stress intensity factors for all cases with different crack lengths was considerably
small as shown in Fig. 4-16 d. The influence of individual mode to the effective mode of
fracture can be seen in Fig. 4-16 e. The contribution of the Mode-II stress intensity factor
to the effective stress intensity factor was significant (Fig. 4-16 e).
G l o b a l - l o c a l m e t h o d s t o p a n e l s w i t h d i f f e r e n t c r a c k s i z e s 8 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0 10 20 3010
20
30
40
50
Half crack length, a (mm)
Kef
f (M
Pa m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
a.
0 10 20 3010
15
20
25
30
35
40
45
Half crack length, a (mm)
KI (
MP
a m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
b.
5 10 15 20 252
4
6
8
10
12
Half crack length, a (mm)
KII (
MP
a m
)
Tip1:GlobalTip1:LocalTip1:AbaqusTip2:GlobalTip2:LocalTip2:Abaqus
c.
5 10 15 20 250
0.5
1
1.5
2
2.5
3
Half crack length, a (mm)
Dif
fere
nce
(%
)
Tip1:Local KI
Tip1:Local KII
Tip1:Local Keff
Tip2:Local KI
Tip2:Local KII
Tip2:Local Keff
d.
0.95 0.96 0.97 0.98 0.99
0.2
0.25
0.3
0.35
KI/K
eff
KII/K
eff
Tip1:GlobalTip1:LocalTip2:GlobalTip2:Local
e.
Fig. 4-16 Stress intensity factors for different crack lengths under the combined shear and normal loads
86 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Considering the magnitude of the stress intensity factors, and considering the change
in the stress intensity factors with the change in the crack lengths, the farther crack tip
from the stiffener was more critical than the tip near to the stiffener under the combined
shear and normal loads for different crack lengths. In addition, although for this
combined load condition all cases for different crack lengths were Mode-I dominant, and
both the Mode-I and Mode-II stress intensity factors increased significantly with the
increase in the crack lengths, the contribution of the Mode-II stress intensity factor to the
effective stress intensity factor was significant.
4.5 Fracture Toughness of the Panel
The fracture toughness of the curvilinearly stiffened panel under the combined shear
and normal loadings for a recommended crack length of 10 mm [112] was analyzed and
compared with the effective stress intensity factors for the same loadings. The critical
stress intensity factor of a panel of the material Al7050-T7451 [134] is given by:
27.47 Unstiffened plateICK MPa m (4.1)
Under the maximum combined loading condition ( 0( ) 106,100 /sN N m and
0( ) 462, 200 /nN N m ), the largest effective stress intensity factor was observed,
16.65combined loadeffK MPa m (4.2)
which was small compared to the KIC;
0.6combined loadeff
IC
K
K
Therefore, considering the critical stress intensity factor, the curvilinearly stiffened
panel was indeed an optimum design.
G l o b a l - l o c a l m e t h o d s t o p a n e l s w i t h d i f f e r e n t c r a c k s i z e s 8 7
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
4.6 Conclusions
Fracture analyses of a curvilinearly stiffened panel were performed for different crack
lengths under three load cases: a) a shear load, b) a normal load, and c) a combined shear
and normal loads. Half crack lengths of 5, 10, 15, 20, and 25 mm were defined in the
common location with the maximum magnitude of the principal stresses and von Mises
stress, and the finite element mesh was refined near the crack tips with the mesh density
of 8.4 element/mm. All analyses were performed using the global-local finite element
method using MSC. Marc, and using global finite element methods using MSC. Marc and
ABAQUS. Comparative analyses between the performances of the global-local finite
element method and other two global finite element methods showed that considering the
size of the data library of the output file and the CPU time required to perform the
simulations, the global-local finite element method had considerable advantages over the
global finite element method for all cases. More than 85% data storage space and 85%
CPU time could be saved using the global-local finite element method.
Fracture analyses of the curvilinearly stiffened panels with different crack lengths
showed that the farther located crack tip from the stiffener was critical under the
combined load case. It was also observed that the panel with different crack lengths was
Mode-I dominant case under the normal load, Mode-II dominant case under the shear
load, and Mode-I dominant case under the combined load. However, under the combined
loading, both Mode-I and Mode-II stress intensity factors increased significantly with the
increase in the crack lengths, and the contribution of Mode-II stress intensity factor to the
effective stress intensity factor was significant.
The critical stress intensity factor of the curvilinearly stiffened panel with the half
crack size of 5 mm was analyzed under the maximum combined loadings. The largest
88 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
effective stress intensity factor of the panel was smaller than the critical stress intensity
factor. Therefore, considering the critical stress intensity factor, the curvilinearly
stiffened panel was an optimum design.
5 Static Stress and Fracture Analyses of Adhesive Joints This chapter describes the stress analysis in adhesive joints of the three dimensional
nonlinear finite element model of the compression-shear test fixture performed under a
gradually increasing compression-shear load. To determine the failure load responsible
for debonding of adhesive joints, stress distributions in adhesive joints of the finite
element model of the test-fixture were studied under a gradually increasing compression-
shear load. This chapter also focuses on the static stress and fracture analysis of adhesive
lap and ADCB joints, and on the compression delamination of the ADCB joints.
Adhesive lap joints and ADCB joints were studied under loadings similar to the loadings
found in the analysis of the test-fixture under compression-shear loads.
5.1 Formulations of Static Stress and Fracture Analyses of Adhesive Joints
Formulations for the static stress and fracture analyses of adhesive lap and ADCB
joints and for the compression delamination of the ADCB joints are presented here.
Single lap adhesive joints were studied for determining static stresses using the finite
element method. Crack propagation and delamination growth under compression were
studied using cohesive zone modeling and finite element methods using ABAQUS,
commercially available software.
5.1.1 Formulation of the boundary value problem for the single lap adhesive joint The finite element formulation of the adhesive for performing static stress analysis is
performed by considering the adhesive as an undamaged continuum. The geometric
model for the adhesive lap joint used in the finite element analysis can be seen in Fig.
5-1.
90 Globa l - loca l f in i t e e leme nt f rac tu re ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 5-1 Single lap adhesive joint
In the static stress analysis, the static equilibrium equations are solved using the finite
element analysis [120, 135]:
Given b : , : , and : , find : ,Such that
0 in (equilibriumequation)
on
on
i i
i
i
i i u i t i
i i u
ij j i t
q t u
b
u q
n t
(5.1)
where, is the domain, is the smooth boundary of , and it is formed with the
union of the closed set of locations with essential boundary conditions, u , and with
natural, t , boundary conditions, and n is the outward unit normal at . Let iw are the
arbitrarily slow varying virtual displacements. Let i denote the trial solution space and
i the variation space. Each member i iu satisfies onii i uu q , whereas each
i iw satisfies 0 onii uw .
However, to study crack propagation in an adhesive joint, the fracture zone is required
to be modeled using a different approach so that a crack can be introduced in the
continuum body.
5.1.2 Modeling the cohesive zone for the crack propagation A crack propagation system has three major areas for modeling: the undamaged
continuum, the cohesive zone constitutive relationship formulation, and modeling the
P l
t
t1
t2
b
t1
t2
t
E1
E2
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 9 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
damage evolution. If the system is symmetric in loading, geometry, and material
properties between the adherends, then the system is the classical symmetric Double
Cantilever Beam (DCB), as shown in Fig. 5-2. If the system has any asymmetry between
loading, material properties, or geometric properties then the system becomes ADCB, as
shown in Fig. 5-3.
Fig. 5-2 Symmetric double cantilever beam
Fig. 5-3 Asymmetric double cantilever beam
a0
F
F L
δ
E1
E2 t2
t1
ta
Ѳ
t1
t2
b
P
a0 F
F L
δ
E1
E2 t2
t1
ta
t1
t2
b
P
92 Globa l - loca l f in i t e e leme nt f rac tu re ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
The idea of the cohesive zone model goes back to the strip yield models of Dugdale
[136] and Barenblatt [137]. Instead of letting stresses become infinite as predicted by the
linear theory of elasticity, finite stresses are introduced in a cohesive zone ahead of the
crack tip. The finite value of the stress is assumed to be same as the yield stress [136] or
as some function of the distance to the crack tip [137]. The cohesive stresses or tractions
have been introduced as functions of the local separation, u u u , of the
material (Fig. 5-4 a). This local separation is a vector having three components in
mutually perpendicular directions for the selected orthogonal coordinate system. The
cohesive model for crack propagation analysis of ductile materials was introduced by
Needleman [110].
Fig. 5-4 Cohesive elements: a) 3D cohesive element, b) Bilinear cohesive material model
The interface element for cohesive finite element is an isoparametric element, as
shown in Fig. 5-4 a. The thickness of the element is about 1/50th of the adherend
thickness, and it is inserted as a numerical layer between the adherend layers. The strain
vector, ε, and the traction vector, t, are calculated in a local coordinate system (s,t,n),
which is located at the element midplane. The local separation, δ, is calculated as the
relative movement of the two surfaces from this midplane by [110, 138]
b)
Separation
Traction
δc δm
a)
x,
y,z
2
6
3
7
4
8
1
5 n
Top face
Bottom face
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 9 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
1 1
2 2
3 3
top bottom
t
top bottoms
top bottomn
u u
u u
u u
(5.2)
A bilinear constitutive law is chosen which is based on 2-D cohesive model (Fig. 5-4
b). This constitutive law relates the effective traction, teff, to effective opening
displacement, eff . The effective traction and effective separation are positive continuous
quantities, and when there are no compressive normal traction and separation, they are
equal to the norm of relative traction and displacement vectors; respectively, and are
defined by [120]
2 2 2eff n s tt t t t
2 2 2
eff n s t (5.3)
where < > means if the inside parameter value is negative, it becomes zero. This
constitutive law consists of three different parts, as shown in Fig. 5-4 b:
i) If the effective opening displacement is less than the critical value c , the interface
material behaves as linear elastic, and no damage is present in the element.
ii) If the effective opening displacement reaches c , the interlaminar damage initiates.
After this point, the interface traction decreases linearly.
iii) m refers to the complete decohesion.
The nominal traction stress vector, t, consists of three components (two components in
two-dimensional problems): tn, ts, and (in three-dimensional problems) tt. The
corresponding separations are denoted by n , s , and t . Denoting by T0 the original
thickness of the cohesive element, the nominal strains can be defined as [110, 138]
94 Globa l - loca l f in i t e e leme nt f rac tu re ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0 0 0
, ,n s tn s tT T T
(5.4)
The elastic behavior can be written as:
n nn ns nt n
s ns ss st s
t nt st tt t
t K K K
t t K K K K
t K K K
(5.5)
This elasticity matrix provides a fully-coupled behavior among all the components of
the traction vector t and the separation vector . We can set the off-diagonal terms in
the elasticity matrix to zero, if an uncoupled behavior between the normal and any of the
two shear components is desired.
The stability criterion for uncoupled behavior requires that 0nnK , 0ssK ,
and 0ttK . For coupled behavior the stability criterion requires that [110, 138]:
0, 0, 0;nn ss ttK K K
ns nn ssK K K
st ss ttK K K
nt nn ttK K K
det 0nn ns nt
ns ss st
nt st tt
K K K
K K K
K K K
(5.6)
The delamination initiation is predicted by the quadratic failure criterion. Damage is
assumed to initiate when a quadratic interaction function involving the nominal stress
ratios reaches a prescribed value. This criterion can be represented as [110, 120, and
138],
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 9 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
2 2 2
0 0 01n s t
n s t
t t t
t t t
(5.7)
where t0n, t0
s, t0t are the Mode-I, Mode-II, and Mode-III directional strengths,
respectively. For linear softening, an evolution of the damage variable, D, is defined as
[120]:
max
max
eff eff effm c
eff eff effm c
D
(5.8)
where2eff c
meff
G
t with teff as the effective traction at damage initiation. max
eff refers to
the maximum value of the effective displacement attained during the loading history. The
damage evolution law describes the rate at which the material stiffness is degraded once
the corresponding initiation criterion is reached. A scalar damage variable, D, represents
the overall damage in the material and captures the combined effects of all the active
mechanisms. It initially has a value of 0. If the damage evolution is modeled, D
monotonically evolves from 0 to 1 upon further loading after the initiation of the damage.
The stress components of the traction-separation model are affected by the evolution of
the damage according to [120]:
1 , 0
, Otherwise (nodamagein compressivestiffness);
1 ,
1
n nn
n
s s
t t
D t tt
t
t D t
t D t
(5.9)
where nt , st and tt are the stress components predicted by the elastic traction-
separation behavior for the current strains without a damage.
96 Globa l - loca l f in i t e e leme nt f rac tu re ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
5.2 Delamination Analyses of the Compression-Shear Test-Fixture
As a first step to determine the failure load responsible for the debonding of the
adhesive joints from the test fixture, a stepwise combined shear and compression load
was applied to the three-dimensional materially nonlinear finite element model of the test
fixture, and the stress distribution in the adhesive layer between the steel tabs and
aluminum panel was studied until the joint debonded (Fig. 5-5).
a)
b)
c)
Fig. 5-5 a) Schematic of the test, b) Steel tabs bonded onto the Aluminum panel, c) Steel tabs are debonded from the panel
The test-fixture nonlinear finite element model was developed by the Lockheed
Martin Aeronautics Company using a total of 44186 shell, beam, and gap elements for
the fixture and connections. Shell elements were used to model the steel test fixture upper
and lower L-arm as well as the picture frame secured to the panel (Fig. 5-6). The elastic-
x
y
Panel
Steel tabs
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 9 7
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
plastic material properties used for the test-fixture and aluminum panels are given in
Table 5-1and Fig. 5-7. The adhesive in the adhesively bonded steel tabs was modeled
using linear spring elements (Fig. 5-8). The complete finite element model of the
compression-shear test fixture with the loading system can be seen in Fig. 5-9.
Fig. 5-6 Shell elements in the test fixture model
Table 5-1 Material properties used in the test-fixture finite element model
Steel (H-1025) Tabs Panel (Al-7050)
Young’s Modulus, E (ksi) 29,400 10,600
Poisson’s Ration, ν 0.32 0.33
Yield Stress, (ksi) 152.9 59.4
98 Globa l - loca l f in i t e e leme nt f rac tu re ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0 0.005 0.01 0.0150
50
100
150
200
Strain, e (in/in)
Str
ess,
(
ksi)
SteelAluminum
Fig. 5-7 Elastic-plastic material model for steel tabs and the aluminum panel
Fig. 5-8 Panel with adhesively bonded steel tabs where adhesive is modeled using linear spring elements
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 9 9
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 5-9 Finite Element Model for the compression-shear test fixture with the panel
In order to perform simulation for fracture analysis on the test fixture, we needed to
define crack at the location with maximum stress. Stress analysis was performed by
applying 15400 lb compression and 4900 lb shear on the shear test fixture to determine
the location of the maximum stress (Fig. 5-9.) The maximum von Mises stress was found
to be on the top left and bottom right corners along the interface between tabs and the
panel (Fig. 5-10 a).
a. The von Mises stress distribution
b. Node list along the interface between tabs
and the panel in the debonding zone
Fig. 5-10 Stress analysis along the interface between tabs and the panel
100 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Since the debonding took place at the top left corner of the frame, a list of nodes,
considered to have significantly high stresses, was developed along the interface between
the steel tabs and the panel at the top left corner of the shear test fixture having the
maximum von Mises stress (Fig. 5-10-b and Fig. 5-11).
Fig. 5-11 The node list along the interface between steel tabs and the panel at the top left corner of the system
The von Mises stress distribution along the node list is shown in Fig. 5-12. The von
Mises stress showed some mild nonlinear behavior at each node in the list because of the
material nonlinearities of steel tabs and the aluminum panel. Although the maximum von
Mises stress was found to exist at the node near the load transfer zone, the profiles of the
von Mises stress were almost similar in those three locations. In order to characterize the
stresses responsible for the steel tabs debonding, it was important to study the influence
of the individual stress components. The profiles of the in-plane tangential (x-direction is
considered along the direction perpendicular to the compressive loading direction) stress
components on the node list along the interface are shown in Fig. 5-13. The maximum
tangential stress was in the vicinity of the zone having the maximum von Mises stress
(Fig. 5-13).
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 0 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 5-12 von Mises stresses along the node list on the interface between steel tabs and the panel
Fig. 5-13 In-plane tangential stresses along the node list on the interface between tabs and the panel
The maximum normal (y-directional) stress was found to be at the top left corner node,
the node near the zone of the load transfer from tabs to the panel (Fig. 5-14.) The stress
was compressive with a magnitude larger than that of the tangential stress. Since the
design of the combined load test fixture was for transferring the in-plane shear and
compression loads to the panel, in-plane loads might have been responsible for the
102 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
debonding of the steel tabs, which was similar to the results obtained from the nonlinear
finite element analysis of the combined load test fixture. Although the in-plane loads
were responsible for debonding the tabs, the adhesive could have experienced both the
peel and shear deformations caused by the in-plane loads.
Fig. 5-14 In-plane normal stresses along the node list on the interface between tabs and the panel
Therefore, the following sections describe the three-dimensional finite element models
of adhesive lap joints and the ADCB joints for shear and peel deformations subjected to a
loading similar to the in-plane loading conditions in the test-fixture. These analyses were
performed to understand the physics of the adhesive joints in the test-fixture.
5.3 Finite Element Simulation of the Adhesive Lap Joint
Using fine cohesive elements, for a lap joint under in-plane loading conditions,
analyses of the adhesive joints for determining the fracture behavior of bonded joints
were performed in ABAQUS, and the obtained numerical results were validated with the
available experimental results. In addition, the influence of both the adherend materials
and geometric asymmetries on the fracture resistance of the adhesive joint was analyzed.
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 0 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
To determine the location of the crack across the thickness of the adhesive joint, we
needed to investigate stress distribution in the adhesive joints. For the validation of the
stress analysis, a three-dimensional adhesive lap joint was studied first, and the results
were compared with the available results in the literature. The boundary and the loading
conditions of the 3D lap joint configuration are shown in Fig. 5-15. The geometrical
properties were: adherend length L=100 mm, the arm thickness h = 1.5 mm, adherend
width B = 25 mm, adhesive length = 12.5 mm, and adhesive width B = 25 mm. The
material properties were: adhesive E = 2.5 GPa, adhesive ν = 0.34, adherends E = 70
GPa, and adherend ν = 0.34 [75].
Boundary conditions and external loads were defined within the model global
coordinate system, which was described by x,y,z coordinates (Fig. 5-15). The left end of
the top plate was fixed; i.e., no displacement in any direction was allowed. The right end
of the bottom plate was assumed to be free along the loading direction only. External load
acting on the joint lap model is a static load of value 3125 N and is applied to the lower
lap free end in the x direction.
Fig. 5-15 Detail model with boundary and loading conditions
104 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
The profile of the von Mises stress along the lap joint using 3D, 8-node linear
hexahedral C3D8R elements for the adhesive is shown in Fig. 5-16. The von Mises stress
profile along the adhesive joint was almost close to the result of the Kuczmaszewski and
Wlodarczyk [75], as shown in Fig. 5-16. The use of 3D, 8-node brick elements is
necessary to investigate stress distributions along the thickness of the adhesive. The von
Mises stress distribution along the thickness of the adhesive joint is shown in Fig. 5-17.
The von Mises stress profiles were found to be similar to each other for all the three
locations; the mid, top, and the bottom points located along the thickness of the adhesive
joint.
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20
40
60
80
250 elements
20 elements
Kuczmaszewski 2006
Fig. 5-16 von Mises stress distribution along the adhesive of the lap joint specimen
σvo
n(M
Pa)
Distance (mm)
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 0 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 5-17 von Mises stress profiles across the thickness of the adhesive
5.4 Stress Analysis across the Thickness of the Adhesive Joint under the Combined Loadings
A three dimensional adhesive lap joint was modeled and studied under combined
shear and compression loadings to study the stress distribution across the thickness of the
adhesive joints under multi-load conditions. The schematic of the 3D lap joint
configuration with boundary and the loading conditions is shown in Fig. 5-18. The
geometrical properties were: joint length = 100 mm, arm thickness = 1.5 mm, and width
= 25 mm. The material properties were: adhesive Young’s modulus = 2.5 GPa, adhesive
Poisson’s ratio = 0.34, adherends’ Young’s modulus = 70 GPa, and adherend’s Poisson’s
ratio = 0.34.
106 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 5-18 Schematic of the 3D lap joint
The finite element model of the 3D lap joint configuration with boundary and the
loading conditions is shown in Fig. 5-19. All the edges of one of the plates were fixed,
restraining displacements in all directions. Static tractions of value 100 MPa in the
normal direction and 100 MPa in the shear direction were applied to the upper lap.
Fig. 5-19 Finite element model with boundary and loading conditions
The variations of the von Mises stress along the adhesive joint at three different
locations (top, middle, and bottom) across the thickness of the adhesive, using fine 3D, 8-
node brick elements, are shown in Fig. 5-20. The von Mises stress profiles were found to
be similar to each other at all three locations across the thickness of the adhesive joint.
Furthermore, the maximum stress was found to exist near the loading zone due to the
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 0 7
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
moments from the eccentric loadings caused a significant stress concentration at the
loading end of the lap zone. From this analysis, it can be concluded that the fracture
mechanics should be performed by adding a crack near the loading zone, and by adding
interface elements in the mid section of the adhesive joint.
0 20 40 60 80 100 1200
10
20
30
40
50
60
70
80
90
Distance, d (mm)
von
Mis
es,
v (M
Pa)
Top surface of the adhesiveMiddle layer of the adhesiveBottom surface of the adhesive
Fig. 5-20 von Mises stress distribution along the adhesive layer at the three different locations across the thickness of the joint
5.5 Finite Element Simulation of the Cohesive Zone Interface
The DCB test specimen is generally used for the characterization of Mode-I fracture.
To validate the cohesive zone finite element model, the geometry and the loading
conditions of the DCB configuration and the finite element mesh used for the analysis are
shown in Fig. 5-21. One layer of linear quadrilateral elements, CPS4R (continuum plane
stress 4 nodes reduced integration), was used for adherends, and one layer of linear
COH2D4 elements (cohesive, 2-dimensional, 4 node) was used for modeling the
adhesive. The simulation consisted of a load that was applied at the end blocks attached
to the DCB specimen. The geometrical properties were the length L=203mm, the arm
108 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
thickness h = 6.35mm, and width B = 25.4mm. The initial crack length a0 was about
55mm. The mechanical properties of the DCB specimen were E = 69 GPa, ν = 0.3, GC =
1.6 N/mm [109].
Fig. 5-21 Finite element model of the DCB configuration
The load history from the finite element simulation was compared to the results
available in the literature. The corresponding load–displacement curves of the DCB tests
are shown in Fig. 5-22. For the quasi-static loading, the finite element result was in
excellent agreement with the experimental result [109].
Fig. 5-22 Load–displacement curves using the nominal interface strength 40 MPa for a DCB test
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 0 9
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
5.6 Modeling the Asymmetric Double Cantilever Beam (ADCB)
When asymmetries are introduced in the adherends through material asymmetry,
ADCB-specimens become materially asymmetric, but geometrically balanced joints.
Likewise, when asymmetries are introduced to the adherends through a geometric
asymmetry, ADCB-specimens are materially balanced, but geometrically asymmetric
joints [84]. To perform fracture analysis, on the compression delamination test fixture
with dissimilar adherends, the Asymmetric Double Cantilever Beam (ADCB) was
studied by initiating cracks on the adhesive. The effect of the geometrical and material
asymmetries was analyzed for the ADCB. The schematic of the ADCB configuration is
shown in Fig. 5-3. The geometrical properties of the interested ADCB were: the length
L=100 mm; adherend thicknesses t1 = 2 mm, t2 = 4 mm, width B = 4 mm, and the initial
crack length a = 50 mm. The material properties for adherends were E1= E2=200 GPa, ν
= 0.3, and those for the adhesive were: ultimate strength = 35 MPa, and the critical
energy release rate = 0.7 N/mm.
5.6.1 Influence of adhesive material property to fracture resistance of the ADCB joint
The influence of adhesive material properties on the fracture resistance of the adhesive
joint is shown in Fig. 5-23. Simulations were performed for the case where the top
adherend stiffness was equal to the stiffness of the bottom adherend. However, the
bottom adherend thickness was twice that of the top adherend. The critical energy release
rate of the adhesive was varied from 0.7 to 1.05 N/mm. The fracture resistance of the
adhesive joint slightly increased with the critical energy release rate of the adhesive (Fig.
5-23).
110 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
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0.0 0.1 0.2 0.3 0.4 0.5 0.60
5
10
15
20
25
30
35
G=1.05
G=0.95
G=0.85
G = 0.7
Fig. 5-23 Load-displacement profiles of ADCB specimens for different critical energy release rates for the adhesive (in N/mm)
The change in the reaction force with respect to the change in the critical energy
release rate of the adhesive is shown in Fig. 5-24. The reaction force increased linearly
with the critical energy release rate of the adhesive. The change in the reaction force due
to the change in the critical energy release rate of the adhesive was considerably large
(Fig. 5-24), about 16 N/(N/mm.) The reason could be that the higher its critical energy
release rate, the tougher the adhesive is; the adhesive thus can absorb more energy before
the crack starts to propagate.
d (mm)
P (N)
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 1 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0.7 0.8 0.9 1 1.127
28
29
30
31
32
33
34
GC
(N/mm)
Rea
ctio
n f
orc
e, R
(N
)
y = 16.4*x + 15.5
t1/t
2 = 0.5
Linear
Fig. 5-24 Change in reaction force due to the change in the critical energy release rates of the adhesive
5.6.2 Influence of adherends geometric asymmetry to fracture resistance of the ADCB joint
The influence of adherend geometrical asymmetry on fracture resistance of the
adhesive joint is shown in Fig. 5-25. Simulations were performed for different ratios of
top to bottom adherend thicknesses. The ratio is defined as, r = t1/t2, where t1 and t2 are
the top and bottom adherend thicknesses. We kept t2 constant (4 mm), and varied t1 from
2 to 6 mm. The fracture resistance of the adhesive joint significantly increased with the
thickness of the top adherend (Fig. 5-25). This is due to the fact that the overall stiffness
of the joint increases as the adherend thickness increases. In this case, the bending
rigidity of the adherend increases with the increased stiffness, which results in a stronger
joint.
112 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
20
40
60
80
r=1.50
r=1.25
r=1.0
r=0.75
r = 0.5
Fig. 5-25 Load-displacement profiles of ADCB specimens for different adherend thicknesses
The change of the reaction force with respect to the change of the top adherend
thickness is shown in Fig. 5-26. The reaction force increased linearly with the thickness
of the top adherend.
2 3 4 5 620
30
40
50
60
70
80
90
Top adherend, t1 (mm)
Rea
ctio
n f
orc
e, R
(N
)
y = 14.6*x - 0.601
GC
= 0.7 N/mm
Linear
Fig. 5-26 Change in the reaction force due to the change in the top adherend thickness
d (mm)
P (N)
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 1 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
The change in the reaction force due to unit change in the thickness of the top
adherend was quite significant (14 N/mm) as shown in (Fig. 5-26). Therefore, it can be
concluded that larger the thickness of the adherend, stronger will be the adhesive joint.
5.6.3 Influence of adherend material asymmetry to fracture resistance of the ADCB joint
The influence of adherend material properties on the fracture resistance of the
adhesive joint is shown in Fig. 5-27. Two simulations were performed for different
stiffness values of the top and the bottom adherends: one with similar stiffness, and
another with top adherend being 1.5 times stiffer than the bottom adherend. The fracture
resistance of the adhesive joint slightly increased with the stiffness of the top adherend
(Fig. 5-27).
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0.0 0.1 0.2 0.3 0.4 0.5 0.60
5
10
15
20
25
30
35
E1=1.5*E2
E1=E2
Fig. 5-27 Load-displacement profiles of ADCB joints for different adherend stiffness
5.7 Compression Delamination under Different Constrained End Conditions
This section describes the ADCB joint under compression loading similar to the
compressive loading found in the test-fixture analysis. This analysis was performed to
d (mm)
P (N)
114 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
understand the compression behavior of the ADCB joint in the test-fixture. Under
compression loading, the load-displacement response can exhibit a type of unstable
behavior (Fig. 5-28 a), it is often necessary to obtain nonlinear static equilibrium
solutions for unstable problems. During some parts of the loading, the load and/or the
displacement may decrease as the solution evolves. The modified Riks method is an
algorithm that allows effective solution of such cases.
5.7.1 The modified Riks method For unstable problems, the modified Riks method assumes that all load magnitudes
vary with a single scalar parameter and the response is reasonably smooth [120, 139].
The solution in this method is viewed as the discovery of a single equilibrium path in a
space defined by the nodal variables and the loading parameter. The basic algorithm is
such that at any time there will be a finite radius of convergence, which may result in the
path-dependent response. It is, therefore, essential to limit the increment size. The
increment size is limited by moving a given distance along the tangent line to the current
solution point. The next solution point is determined by searching for equilibrium along
the plane that passes through the obtained point, and in a plane that is normal to the same
tangent line (Fig. 5-28 b).
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 1 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Fig. 5-28 Riks method (a) unstable static response (b) modified Riks method [120, 139]
For instance, PM (M is the degrees of freedom) is the loading pattern, µ is the load
magnitude parameter, and uM is the displacement vector. The load magnitude at any time
is µ*PM. The solution space is scaled by measuring the maximum absolute value of all
displacement variables, u, in the linear iteration. It is also defined, 1
2M MP P P . The
scaled space is then spanned by [120, 139], load ,M
M M PP P
P ; and displacements
MM u
uu
. The solution path is now the continuous set of equilibrium points described
by the vector ;Mu in this scaled space. The schematic of the algorithm is shown in
(Fig. 5-28 b) and the detail algorithm is explained in the references [120, 139].
5.7.2 Compression delamination study on the adhesive joints The ADCB-specimen was studied under compression. The adhesive layer was
modeled using cohesive elements. The effect of the compression and different end
constraints on the fracture was studied for different mesh densities of the cohesive
interface. The geometry and the loading conditions of the ADCB configurations are
b)
µ
Displacement
B0
B1
B2 ρ1
1Mv
0Mv
a)
Load
Displacement
Maximum displacement Maximum load
Minimum load
Minimum displacement
116 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
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shown in Fig. 5-29 for the fixed end and Fig. 5-30 for the top adherend with free end. The
geometrical properties were the length L=100mm; the arm thickness t1 = 2 mm, t2 = 4
mm, width B = 4 mm, and the initial crack length a0 = 50mm. The material properties for
adherends were: E1= E2=200 GPa, ν = 0.3, and those for the adhesive were the ultimate
strength = 35 MPa, and the critical energy release rate = 0.7 N/mm.
Fig. 5-29 ADCB configuration for a fixed end
Fig. 5-30 Configurations for the top adherend with a free end
The results for the delamination under compression for one end constraint (fixed) and
loadings at the other end, for different mesh densities of the cohesive interface are shown
in Fig. 5-31. The delamination was gradual after it started to propagate. This behavior can
be explained from the deformation profile. The deformation profile for the fixed one end
case is shown in Fig. 5-32. The bending deflection of the top adherend was higher before
the load reached 42% of the total displacement (Fig. 5-32), which could have resulted in
an increase of the reaction force before delamination started. When the delamination
started to propagate (Fig. 5-32), around 66% of the total displacement, the bending
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 1 7
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
deflection decreased and so did the axial deformation. After the delamination started to
propagate, the system responded as in the case of the peel deformation, as shown in Fig.
5-32, after the load reached the value of 66% of the final displacement.
0 50 100 1500
100
200
300
400
500
600
End displacement, d (m)
Rea
ctio
n f
orc
e, P
(N
)
300 elements12000 elements16000 elements20000 elements
Fig. 5-31 Load displacement profile for fixed end ADCB specimen with different mesh densities of the cohesive interface
The bending deflection of the top adherend was higher before the load reached 42% of
the total displacement (Fig. 5-32), which might result in an increase of the reaction force
before delamination started. When the delamination started to propagate (Fig. 5-32),
around 66% of the total displacement, the bending deflection decreased and so did the
axial deformation. After the delamination started to propagate, the system responded as in
the case of the peel deformation, as shown in Fig. 5-32, after the load reached at 66% of
the total displacement.
118 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
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0% displacement
18% displacement
42% displacement
66% displacement
85% displacement
95% displacement
100% displacement
Fig. 5-32 Deformation profiles of the fixed end ADCB specimen with t1 = 2 mm and t2 = 4 mm under compression
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 1 9
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0 500 1000 15000
500
1000
1500
2000
End displacement, d (m)
Rea
ctio
n f
orc
e, P
(N
)
2000 elements4000 elements8000 elements
Fig. 5-33 Load displacement profile for free end ADCB specimen with different mesh densities of the cohesive interface
The delamination characteristic of the ADCB configuration with one end free and the
other end loaded is shown in Fig. 5-33 for different mesh densities of the cohesive
interface. The reaction force had increased to a certain peak value before the
delamination started to propagate rapidly. Therefore, the ADCB joint with one end free
and the other end loaded would not be safe to use in an adhesive joint subjected to
compression-shear load. In short, since for the one end constrained ADCB case, the
delamination was gradual, this configuration should be preferred for an adhesive joint
that is subjected to a combined compression-shear load.
5.8 Conclusions
To determine the failure load responsible for the debonding of the adhesive joints from
the test fixture, a gradually increasing combined shear and compression load was applied
120 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
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to the three-dimensional nonlinear finite element model of the test fixture. The stress
distribution in the interface between steel tabs and the panel was studied. Since the design
of the combined load test fixture was for transferring the in-plane shear and compression
loads to the panel, in-plane loads might have been responsible for the debonding of the
steel tabs, which was similar to the results obtained from the nonlinear finite element
analysis of the combined load test fixture. Although the in-plane loads were responsible
for debonding the tabs, the adhesive could have experienced both the peel and shear
deformations caused by the in-plane loads. Therefore, further fundamental studies were
performed on the three-dimensional finite element models of adhesive lap joints and the
ADCB joints for shear and peel deformations subjected to a loading similar to the in-
plane loading conditions in the test-fixture. These studies were performed to determine
configurations that would lead to stronger adhesive joints using the knowledge gained
from these analyses.
The analyses of three dimensional finite element models of adhesive lap joints and
ADCB joints under the loads similar to the loads found in the test-fixture analyses were
performed to understand the physics of the adhesive joints in the test-fixture, and to
determine means to acquire stronger different adhesive joints representing the test-fixture
design and loadings. To determine the location of the crack across the thickness of the
adhesive joint, a three-dimensional adhesive lap joint was modeled and analyzed under
combined loadings. The profiles of the von Mises stress along the adhesive joint at
different locations (the top, mid, and bottom sections of the adhesive joint) along the
thickness were found to be similar. Furthermore, the maximum von Mises stress was
observed near the loading zone due to the moments from the eccentric loadings caused a
S t a t i c s t r e s s a n d f r a c t u r e a n a l y s e s o f a d h e s i v e j o i n t s 1 2 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
significant stress concentration at the loading end of the lap zone. In addition, the
influence of adhesive material properties and the adherend material and geometric
properties on the fracture resistance of the adhesive joint of ADCB configurations was
studied considering the crack propagation path through the mid section of the adhesive
interface. It was observed that the reaction force linearly increased with the critical
energy release rate of the adhesive. The reason could be that higher the critical energy
release rate, the tougher is the adhesive; the adhesive can thus absorb more energy before
the crack starts to propagate. It was also observed that the reaction force for the adhesive
joint significantly increased with the thickness of the top adherend. This is due to the fact
that the overall stiffness of the joint increases as the adherend thickness increases.
To understand the compression behavior of the adhesive joints in the test-fixture, an
ADCB was studied under compression loading similar to the compressive loading found
in the test-fixture analysis. The numerical observations showed that for the one end fixed
and the other end loaded ADCB configuration, the delamination was gradual. Therefore,
this configuration would be safer to use in an adhesive joint for structures subjected to a
combined compression-shear load, the case for the test fixture.
6 Global-Local Finite Element Analyses of Crack Propagation and Adhesive Joints
This chapter describes the procedure for studying crack propagation and adhesive
joints using global-local finite element analysis. Three cases – a double cantilever beam
specimen, a single lap adhesive joint, and a three-point bending test specimen – for the
global-local finite element analysis are described in the following sections.
6.1 Global-Local Analyses of an Adhesively Bonded Double Cantilever Beam Specimen
If the joint is symmetric in loading, geometry, and material properties between the
adherends, then the system is the classical symmetric Double Cantilever Beam (DCB), as
shown in Fig. 5-2.
Global model Local model
Global deformed model
Local deformed model
Fig. 6-1 Global-local models of a DCB specimen
To study the crack propagation in the adhesive joints using the global-local method, an
adhesively bonded Double Cantilever Beam (DCB) was studied. Crack propagation
studies were performed considering the crack propagation path in the mid section of the
adhesive. The geometrical properties of the DCB specimen were: the length L=203mm,
G l o b a l - l o c a l a n a l y s e s o f c r a c k p r o p a g a t i o n a n d a d h e s i v e j o i n t s 1 2 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
the arm thickness h = 6.35mm, and width B = 25.4mm. The initial crack length a0 was
about 55mm. The mechanical properties of the DCB specimen were: E = 69 GPa, ν = 0.3,
and GC = 1.6 N/mm [109]. The global finite element model, local finite element model,
and their corresponding deformed configurations are shown in Fig. 6-1.
0% displacement
20% displacement
60% displacement
80% displacement
90% displacement
100% displacement
Fig. 6-2 Superimposed deformations of the DCB specimen for the global and global-local methods
The reaction force and the corresponding displacement at the end of the beam were
studied for the analysis and comparison. In the first step, a global analysis was performed
to obtain an output file containing global results using the entire model. The local
Common locations between the global and the global-local models
124 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
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analysis was, then, performed by applying properly defined kinematic boundary
conditions to the local boundaries connecting to the global model. The plots of the
deformations of the global and global-local models are shown in Fig. 6-2. It could be
seen from these plots that both the global and the global-local models produced very
similar deformations in the area of the interest at every step of the total deformation. The
profiles of the end point reaction forces of the global model and the global-local model
with respect to the corresponding displacements are shown in Fig. 6-3. The profile
obtained using the global-local method matched exactly with the global solution, and
matched very well with the available solution in the literature [109].
Fig. 6-3 Comparisons of the end reaction force profiles obtained using the global-local method
Comparisons of the global and global-local results are shown in Table 6-1.
Considering the total degrees of freedom associated with every model, data library size
required to store the output file, and the CPU time required to simulate the model, the
global-local method showed a considerable improvement over the global analysis. More
G l o b a l - l o c a l a n a l y s e s o f c r a c k p r o p a g a t i o n a n d a d h e s i v e j o i n t s 1 2 5
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
than 80% data storage space and more than 65% CPU time requirement could be saved
using the global-local method (Table 6-1).
Table 6-1 Comparison of global-local results
Global/local Total degrees of freedom (#)
Size of data library (megabytes)
CPU time (S)
Global 18300 65.4 53.5
Local 6924 11.7 18.6
% Difference 63 83 65
6.2 Global-Local Analyses of a Single Lap Adhesive Joint
To study the stress analysis on the adhesive lap joint using the global-local method, a
single lap adhesive joint was studied. The geometrical properties of the lap joint were:
adherend length L = 100 mm, the arm thickness h = 1.5 mm, adherend width B = 25 mm,
adhesive length = 12.5 mm, and adhesive width B = 25 mm. The material properties
were: adhesive E = 2.5 GPa, adhesive ν = 0.34, adherends E = 70 GPa, and adherends ν =
0.34 [75].
In the first step, global analysis was performed to obtain an output file containing
global results using the entire model. The local analysis was performed by applying the
properly defined kinematic boundary conditions to the local boundaries connecting to the
global model. The global finite element model, local finite element model, and their
corresponding deformed configurations are shown in Fig. 6-4. A small area shown in the
red circle in Fig. 6-4 was selected for local analyses. The von Mises stress profiles along
the adhesive length and along the thickness of the adhesive joint were studied for the
analyses and comparisons.
126 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
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Global model
Local model
Global deformed model
Local deformed model
Fig. 6-4 Global-local models for a single lap adhesive joint
The plots of the deformations of the global and global-local models are shown in Fig.
6-5. It could be seen from these plots that both the global and the global-local models
produced very similar deformations in the area of the interest at every step of the total
deformation. The von Mises stress profiles of the global model and the global-local
model along the adhesive length are shown in Fig. 6-6. The profiles obtained using the
global-local method matched very well with the global solution as well as with the
available solution in the literature [75].
G l o b a l - l o c a l a n a l y s e s o f c r a c k p r o p a g a t i o n a n d a d h e s i v e j o i n t s 1 2 7
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
0% displacement
20% displacement
60% displacement
80% displacement
90% displacement 100% displacement
Fig. 6-5 Superimposed deformations of the lap joint for the global and global-local methods
The von Mises stresses obtained using the global-local method along the adhesive
thickness are shown in Fig. 6-7. The profiles were found to be similar to each other in all
the three locations, in the mid, top, and bottom, across the thickness of the adhesive joint.
Common locations between the global and the global-local models
128 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
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Fig. 6-6 von Mises stress along the adhesive joint using the global-local method
Fig. 6-7 von Mises stress along the adhesive thickness using the global-local method
G l o b a l - l o c a l a n a l y s e s o f c r a c k p r o p a g a t i o n a n d a d h e s i v e j o i n t s 1 2 9
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
Comparisons of the global and global-local results are shown in Table 6-2.
Considering the total degrees of freedom associated with every model, data library size
required to store the output file, and the CPU time required to simulate the model, the
global-local method showed a considerable improvement over the global analysis.
Around 70% data storage space and CPU time requirement could be saved using the
global-local method (Table 6-2).
Table 6-2 Comparison of global-local results
Global/local Total degrees of freedom (#)
Size of data library (megabytes)
CPU time (S)
Global 171054 383.5 870
Local 42476 104.3 267.7
% Difference 75 73 70
6.3 Global-local Analyses of a Three-point Bending Test Specimen
To study the fracture propagation in a brittle material using the global-local finite
element method, a specimen of a very common bending test - three-points bending test -
has been selected (Fig. 6-8).
Fig. 6-8 A three-point bending test
The geometric, material, and loading conditions of the system for the three-point
bending test are shown in Fig. 6-8. The geometric properties of the model were: length
2c
b
2c
P
L
130 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
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between the supports = 560 mm, total length = 600 mm, and width = 150 mm. The
material properties were: Young’s modulus = 20 GPa, and Poisson’s ratio = 0.2 [140].
The reaction force and the corresponding displacement at the center of the beam were
studied for the analysis and comparison.
a. Global deformed model
b. Local deformed model
c. Deformations of the global and global-local models
Fig. 6-9 Global-local results for a three-point bending test specimen
Since the main objective here was to apply the global-local finite method for studying
the fracture propagation, in the first step, a global analysis was performed to obtain an
output file containing global results using the entire model. The local analysis was, then,
performed by applying the existing loads or/and boundary conditions in the local model
along with properly defined kinematic conditions to the local boundaries connecting to
the global model.
Common locations between the global and the global-local models
G l o b a l - l o c a l a n a l y s e s o f c r a c k p r o p a g a t i o n a n d a d h e s i v e j o i n t s 1 3 1
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
The deformations of the global model, deformation of the global-local model, and the
plot of the deformations of the global and global-local models, are shown in Fig. 6-9 a, b,
and c, respectively. It could be seen from these plots (Fig. 6-9 c) that both the global and
the global-local models produced very similar deformations in the area of the interest.
The center point reaction force profile of the global model with respect to the
corresponding displacement is shown in Fig. 6-10. The profile matched very well with
the available solution in the literature [140].
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æ
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æ æ æ
ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
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æ
æææ
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æææææ
æææææææææææææææææææææææææææææææææ
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350
20
40
60
80
Global
Patzak - 2003
Fig. 6-10 Load-displacement profiles in the global analysis
The center point reaction force profile of the global-local model with respect to the
corresponding displacement is shown in Fig. 6-11. The profile matched exactly with the
result of the global analysis, and matched very well with the available solution in the
literature [140].
d (mm)
P (N)
132 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
æ
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æææææææææææææææææææææææææææææ
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350
20
40
60
80
Local
Global
Patzak - 2003
Fig. 6-11 Comparison of results using the global-local method
Table 6-3 Comparison of global-local results
Global/local Total degrees of freedom (#)
Size of data library (megabytes)
CPU time (s)
Global 66496 120.3 146.4
Local 6652 9.7 13.3
% Difference 90 92 91
Comparison of the global and global-local results is shown in Table 6-3. Considering
the total degrees of freedom associated with every model, data library size required to
store the output file, and the CPU time required to simulate the model, the global-local
method showed a considerable improvement over the global analysis. More than 90%
data storage space and CPU time requirement could be saved using the global-local
method (Table 6-3).
d (mm)
P (N)
G l o b a l - l o c a l a n a l y s e s o f c r a c k p r o p a g a t i o n a n d a d h e s i v e j o i n t s 1 3 3
C o p y r i g h t © 2 0 1 2 , M o h a m m a d M a j h a r u l I s l a m
6.4 Conclusions
Global-local finite element methods were applied to study the fracture propagation and
the characteristics of adhesive joints. Three cases were studied using the global-local
finite element method: the double cantilever beam joint, the lap joint, and three-point
bending test specimen.
In the study of the adhesively bonded Double Cantilever Beam (DCB) using the
global-local finite element method, it was observed that both the global and the global-
local models produced very similar deformations in the area of the interest for every step
of the total deformation. Considering the total degrees of freedom associated with every
model, data library size required to store the output file, and the CPU time required to
simulate the model, the global-local method showed a considerable improvement over the
global analysis. More than 80% data storage space and more than 65% CPU time
requirement could be saved using the global-local method.
To study the stress distribution in the adhesive lap joints using the global-local
method, a single lap adhesive joint was studied. The von Mises profiles obtained using
the global-local method matched very well with the global solution as well as with the
available solution in the literature. The von Mises stress profiles using the global-local
method were found to be similar to each other at all the three locations, in the mid, top,
and bottom, across the thickness of the adhesive joint. Furthermore, around 70% data
storage space and CPU time requirement could be saved using the global-local method.
To study the fracture propagation in the brittle materials using the global-local finite
element method, a three-point bending test specimen was studied. The simulation of
crack propagation was performed using the cohesive zone finite element method. The
center point reaction force profile of the global-local model with respect to the
134 Globa l - loca l f in i t e e le ment f rac ture ana lys i s o f curv i l inear ly s t i f fened pane l s and adhes ive jo in t s
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corresponding displacement matched exactly with the result of the global analysis, and
matched very well with the available solution in the literature. In addition, more than
90% data storage space and CPU time requirement can be saved using the global-local
method.
7 Conclusions and Future Directions Major conclusions and contributions from this work, along with the future work, are
summarized in this Chapter.
7.1 Conclusions
The major conclusions and contributions from different chapters are presented in the
following subsections.
7.1.1 Global-local finite element methods to curvilinearly stiffened panels Global finite element analyses of the curvilinearly stiffened panels for different
heights of the stiffeners were performed to study the influence of stiffeners on the
structural response of the panel under combined shear and compression loadings. The
panel had negligible out-of-plane deflection under the in-plane loading when there was
no stiffener. The maximum out-of-plane deflection was observed at the center of the plate
under the in-plane loading for different stiffener heights. The out-of-plane deformation
increased with stiffeners height until the stiffeners height was 38.5 mm. After this value
of stiffeners’ height, the out-of-plane deflection decreased. In short, although the out-of-
plane deflection of the stiffened panel increased with the stiffeners height, it decreased
when the optimum heights of the stiffeners were reached. The height of the stiffeners to
yield maximum stress was found to be 57.8 mm.
The magnitude of the minimum principal stress was larger than the magnitudes of the
maximum principal stress and the von Mises stress. In addition, unlike other stress
variation, the minimum principal stress increased with stiffeners height until the average
stiffeners height was 38.5 mm. With a further increase in the ratio of the stiffeners’
height, the minimum principal stress (compressive) began to decrease. It can, therefore,
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be concluded that the minimum principal stress is the critical stress in the panel under
combined loading, and the magnitude of this stress can be reduced by adding stiffeners of
an optimum height to the panel.
Buckling analysis was performed to study the variation of the first five buckling load
factors for different stiffener heights. Critical buckling load factor increased significantly
with the increase of stiffeners’ height. This result suggests that the buckling stability of
the stiffened panel increases with the increase of stiffeners’ height.
To considerably reduce both the CPU time and the data storage space, a global-local
finite element method can be employed for studying the structural response of
curvilinearly stiffened panels. Global-local finite element analyses with a mesh
refinement were performed on a curvilinearly stiffened panel under combined shear and
compression loadings for three element-lengths: 5 mm, 10 mm, and 20 mm. The refined
global model with the 5 mm element length required 95% more degrees of freedom than
the refined local model with the same mesh density. The refined local analysis, with
elements of dimension 5 mm, can save 95% CPU time as compared to the global analysis
with the same mesh density.
7.1.2 Multi-load case damage tolerance study of curvilinearly stiffened panels using global-local finite element analyses
A global-local finite element method was used to study the damage tolerance of
curvilinearly stiffened panels. All analyses were performed using global-local finite
element method using MSC. Marc, and global finite element methods using MSC. Marc
and ABAQUS. Before starting the damage tolerance study, stress distributions on the
panel were analyzed to find the location of the critical stress under the combined shear
and compression loadings.
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To perform the damage analyses of the curvilinearly stiffened panels, a half crack size
of 0.725 mm was defined, using MSC Patran and MSC Nastran, in the location of the
critical stress, which was the common location with the maximum magnitude of principal
stresses and von Mises stress. This crack size was used because of the requirement of a
sufficiently small crack, if the crack is in the vicinity of any stress raiser. A mesh
sensitivity analysis was then performed under the combined shear and tensile loads to
validate the choice of the mesh density near the crack tip. The difference between the
subsequent results became almost negligible when the mesh density of 8.4 element/mm
ahead of the crack tip was used. In addition, 94% saving in the CPU time was achieved
using the global-local finite element method over the global finite element method using
this mesh density near the crack tip. This mesh density was, therefore, used in the further
analyses.
Fracture analyses of a curvilinearly stiffened panel with a half crack size of 0.725 mm
were performed under combined shear and tensile loads using the crack tip mesh density
of 8.4 element/mm. To study the influence of the individual loads on the basic modes of
fracture, shear and normal loads were varied differently. When the shear load was varied,
the normal load was held constant. Likewise, when the normal load was varied, the shear
load was held constant. Considering the size of the data library of the output file and the
CPU time required to perform the simulations, the global-local finite element method had
considerable advantages over the other two finite element methods for all load cases.
Considering the magnitude of the stress intensity factors, the case with the fixed shear
but different normal loads was a Mode-I dominant. Considering the magnitude of the
stress intensity factors, the case with the fixed normal but different shear loads was a
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Mode-I dominant, as well. However, the magnitude of the effective stress intensity
factors for the case with the fixed shear but different normal loads was varied in a much
wider range than the case with the fixed normal but different shear loads. In short, the
influence of the normal load to the effective stress intensity factor was, therefore, greater
than the influence of the shear load.
The critical stress intensity factor of the curvilinearly stiffened panel with the half
crack size of 0.725 mm was analyzed. The largest effective stress intensity factor of the
panel under the maximum combined shear and tensile loads was very smaller than the
critical stress intensity factor. Therefore, considering the critical stress intensity factor,
the design of the panel was an optimum design.
7.1.3 Global-local finite element methods for fracture analyses of curvilinearly
stiffened panels for different crack sizes Fracture analyses of a curvilinearly stiffened panel were performed for different crack
lengths under three load cases: a) shear load, b) normal load, and c) combined load. Half
crack lengths of 5, 10, 15, 20, and 25 mm were defined in the location of the critical
stress, which was the common location with the maximum magnitude of principal
stresses and von Mises stress, and the finite element mesh was refined near the crack tips
with the mesh density of 8.4 element/mm. All analyses were performed using the global-
local finite element method using MSC. Marc, and using global finite element methods
using MSC. Marc and ABAQUS. Comparative analyses between the performances of the
global-local finite element method and other two global finite element methods showed
that considering the size of the data library of the output file and the CPU time required
to perform the simulations, the global-local finite element method had considerable
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advantages over the global finite element method for all cases. More than 85% data
storage space and 85% CPU time could be saved using the global-local finite element
method.
Fracture analyses of the curvilinearly stiffened panels with different crack lengths
showed that the farther located crack tip from the stiffener was critical under the
combined load case. It was also observed that the panel with different crack lengths was
Mode-I dominant case under the normal load, Mode-II dominant case under the shear
load, and Mode-I dominant case under the combined load. However, under the combined
loading, both Mode-I and Mode-II stress intensity factors increased significantly with the
increase in the crack lengths, and the contribution of Mode-II stress intensity factor to the
effective stress intensity factor was significant.
The critical stress intensity factor of the curvilinearly stiffened panel with the half
crack size of 5 mm was analyzed under the maximum combined loadings. The largest
effective stress intensity factor of the panel was smaller than the critical stress intensity
factor. Therefore, considering the critical stress intensity factor, the curvilinearly
stiffened panel was an optimum design.
7.1.4 Static stress and fracture analyses of adhesive joints
To determine the failure load responsible for the debonding of the adhesive joints from
the test fixture, a gradually increasing combined shear and compression load was applied
to the three-dimensional nonlinear finite element model of the test fixture. The stress
distribution in the interface between steel tabs and the panel was studied. Since the design
of the combined load test fixture was for transferring the in-plane shear and compression
loads to the panel, in-plane loads might have been responsible for the debonding of the
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steel tabs, which was similar to the results obtained from the nonlinear finite element
analysis of the combined load test fixture. Although the in-plane loads were responsible
for debonding the tabs, the adhesive could have experienced both the peel and shear
deformations caused by the in-plane loads. Therefore, further fundamental studies were
performed on the three-dimensional finite element models of adhesive lap joints and the
ADCB joints for shear and peel deformations subjected to a loading similar to the in-
plane loading conditions in the test-fixture. These studies were performed to determine
configurations that would lead to stronger adhesive joints using the knowledge gained
from these analyses.
The analyses of three dimensional finite element models of adhesive lap joints and
ADCB joints under the loads similar to the loads found in the test-fixture analyses were
performed to understand the physics of the adhesive joints in the test-fixture, and to
determine means to acquire stronger different adhesive joints representing the test-fixture
design and loadings. To determine the location of the crack across the thickness of the
adhesive joint, a three-dimensional adhesive lap joint was modeled and analyzed under
combined loadings. The profiles of the von Mises stress along the adhesive joint at
different locations (the top, mid, and bottom sections of the adhesive joint) along the
thickness were found to be similar. Furthermore, the maximum von Mises stress was
observed near the loading zone due to the moments from the eccentric loadings caused a
significant stress concentration at the loading end of the lap zone. In addition, the
influence of adhesive material properties and the adherend material and geometric
properties on the fracture resistance of the adhesive joint of ADCB configurations was
studied considering the crack propagation path through the mid section of the adhesive
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interface. It was observed that the reaction force linearly increased with the critical
energy release rate of the adhesive. The reason could be that higher the critical energy
release rate, the tougher is the adhesive; the adhesive can thus absorb more energy before
the crack starts to propagate. It was also observed that the reaction force for the adhesive
joint significantly increased with the thickness of the top adherend. This is due to the fact
that the overall stiffness of the joint increases as the adherend thickness increases.
To understand the compression behavior of the adhesive joints in the test-fixture, an
ADCB was studied under compression loading similar to the compressive loading found
in the test-fixture analysis. The numerical observations showed that for the one end fixed
and the other end loaded ADCB configuration, the delamination was gradual. Therefore,
this configuration would be safer to use in an adhesive joint for structures subjected to a
combined compression-shear load, the case for the test fixture.
7.1.5 Global-local finite element analyses of crack propagation and adhesive joints
Global-local finite element methods were applied to study the fracture propagation and
the characteristics of adhesive joints. Three cases were studied using the global-local
finite element method: the double cantilever beam joint, the lap joint, and three-point
bending test specimen.
In the study of the adhesively bonded Double Cantilever Beam (DCB) using the
global-local finite element method, it was observed that both the global and the global-
local models produced very similar deformations in the area of the interest for every step
of the total deformation. Considering the total degrees of freedom associated with every
model, data library size required to store the output file, and the CPU time required to
simulate the model, the global-local method showed a considerable improvement over the
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global analysis. More than 80% data storage space and more than 65% CPU time
requirement could be saved using the global-local method.
To study the stress distribution in the adhesive lap joints using the global-local
method, a single lap adhesive joint was studied. The von Mises profiles obtained using
the global-local method matched very well with the global solution as well as with the
available solution in the literature. The von Mises stress profiles using the global-local
method were found to be similar to each other at all the three locations, in the mid, top,
and bottom, across the thickness of the adhesive joint. Furthermore, around 70% data
storage space and CPU time requirement could be saved using the global-local method.
To study the fracture propagation in the brittle materials using the global-local finite
element method, a three-point bending test specimen was studied. The simulation of
crack propagation was performed using the cohesive zone finite element method. The
center point reaction force profile of the global-local model with respect to the
corresponding displacement matched exactly with the result of the global analysis, and
matched very well with the available solution in the literature. In addition, more than
90% data storage space and CPU time requirement can be saved using the global-local
method.
7.2 Future Directions
The probable future work is summarized in the following subsections:
7.2.1 Optimization of the curvilinearly stiffened panels using the global-local finite element method
Optimization of the curvilinearly stiffened panels can be performed using the global-
local finite element methods as well as using the available resources. Damage tolerance
constraint would be evaluated in every iteration of the design optimization process by
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calculating stress intensity factors using the global-local finite element methods. In such a
case, not only will the process facilitate the fracture criteria in the optimization scheme,
but it will also reduce significant time in the overall optimization process.
7.2.2 Global-local finite element methods to study the complex 3D structural adhesive joints
When a structural design is very complex containing 3D multilayered adhesive joints,
it is almost impossible to perform detailed fracture analyses of the adhesive joints of the
entire structure using available resources. In such a case, detailed fracture analyses of the
adhesive joint in the specific area interest can be performed using the global-local finite
element method. Following the appropriate steps discussed in the earlier chapters, not
only will the global-local method make it possible to study the fracture propagation in the
complex 3D structural adhesive joints, but it will also save significant amount of
computational time.
7.2.3 Global-local finite element methods to study the systems that can be disintegrated When a structure consists of multiple components, and each component requires very
accurate analyses, global-local finite element methods can be applied to the individual
components for detailed analyses with very fine finite element meshes using the available
resources. In such a case, by saving considerable amount of computational time, the
accuracy of the results obtained using finer meshes for the individual components can be
improved significantly using the global-local finite element methods.
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