Global-local Finite Element Fracture Analysis of …...Global-local Finite Element Fracture Analysis of Curvilinearly Stiffened Panels and Adhesive Joints Mohammad Majharul Islam Abstract

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


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

v i i i C o n t e n t s


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

C o n t e n t s i x


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

L i s t o f f i g u r e s x i


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

x i i L i s t o f f i g u r e s


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

L i s t o f f i g u r e s x i i i


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


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

I n t r o d u c t i o n 3


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



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



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



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



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



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



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

10 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


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

I n t r o d u c t i o n 1 1


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



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:

I n t r o d u c t i o n 1 3


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.



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



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)

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 1 7


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



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



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



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



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



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


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


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


Dis

pla

cem

ent,

v (

mm

)


d)

0 20 40 60-0.165

-0.16

-0.155

-0.15

-0.145


Max

dis

pla

cem

ent,

um

ax (

mm

)

e) 0 20 40 60

-0.08

-0.06

-0.04

-0.02

0

0.02


Dis

pla

cem

ent,

u (

mm

)


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



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.



0 20 40 60122.2

122.4

122.6

122.8

123


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


von

Mis

es s

tres

s,

von (

MP

a)


b)

0 20 40 6074.1

74.2

74.3

74.4

74.5

74.6

74.7


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


Max

pri

nci

pal

str

ess,

m

ax (

MP

a)


d)

0 20 40 60-202.5

-202

-201.5

-201


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


Min

pri

nci

pal

str

ess,

m

in (

MP

a)


f)

Fig. 2-7 Distribution of stresses in the panel for different heights of stiffeners



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


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



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



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



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



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



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.



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.



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



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)



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).



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.



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



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


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



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]:



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],



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.



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



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.



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.



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 (%

)


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.



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]



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)


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 (%

)


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.



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



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.



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



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.



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



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.



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


Nu

mb

er o

f n

od

es (

#)

GlobalLocal

b.

0 5 10 15 200

2

4

6

8

10


CP

U t

ime

(s)

GlobalLocal

c.

0 5 10 15 2092

92.5

93

93.5

94

94.5

95


Per

cen

tag

e (%

)

CPU time change (local)Data size change (local)

d.

0 5 10 15 20540

550

560

570

580

590

600


En

erg

y re

leas

e ra

te, G

(N

/m)

GlobalLocal

e.

0 5 10 15 200

2

4

6

8

10


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



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.



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.



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



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.



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



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



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



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



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



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



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:



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,



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-



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.



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


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



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



Global model for a crack length 10 mm

Local model for a crack length 10 mm



Global model for a crack length 30 mm Local model for a crack length 30 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



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



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



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


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


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


Mo

de-

II en

erg

y re

leas

e ra

te, G

II (N

/m)


c.

5 10 15 20 250

1

2

3

4

5


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


Fig. 4-7 Contribution of individual mode to the total energy release rate for different crack lengths under the normal loading



0 10 20 3010

15

20

25

30

35

40

45


Kef

f (M

Pa m

)


a.

0 10 20 3010

15

20

25

30

35

40

45


KI (

MP

a m

)


b.

5 10 15 20 250

0.5

1

1.5

2


KII (

MP

a m

)


c. 5 10 15 20 25

0

1

2

3

4

5


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


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



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.



5 10 15 20 2575

80

85

90


Per

cen

tag

e sa

vin

g (

%)


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


To

tal e

ner

gy

rele

ase

rate

, G (

N/m

)


a. 5 10 15 20 25

0

0.5

1

1.5

2


Mo

de-

I en

erg

y re

leas

e ra

te, G

I (N

/m)


b.

5 10 15 20 250

200

400

600

800

1000

1200


Mo

de-

II en

erg

y re

leas

e ra

te, G

II (N

/m)


c.

5 10 15 20 250

1

2

3

4

5


Dif

fere

nce

(%

)


d.

Fig. 4-10 Energy release rates for different crack lengths under the pure shear load



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


Fig. 4-11 Contribution of individual mode to the total energy release rate for different crack lengths under the pure shear load



5 10 15 20 253

4

5

6

7

8

9


Kef

f (M

Pa m

)


a. 5 10 15 20 25

0

0.1

0.2

0.3

0.4


KI (

MP

a m

)


b.

5 10 15 20 253

4

5

6

7

8

9


KII (

MP

a m

)


c.

5 10 15 20 250

1

2

3

4

5


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


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



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


Per

cen

tag

e sa

vin

g (

%)


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




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


To

tal e

ner

gy

rele

ase

rate

, G (

N/m

)


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

)


b.

5 10 15 20 250

500

1000

1500

2000


Mo

de-

II en

erg

y re

leas

e ra

te, G

II (N

/m)


c.

5 10 15 20 250

1

2

3

4


Dif

fere

nce

(%

)


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



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


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).



0 10 20 3010

20

30

40

50


Kef

f (M

Pa m

)


a.

0 10 20 3010

15

20

25

30

35

40

45


KI (

MP

a m

)


b.

5 10 15 20 252

4

6

8

10

12


KII (

MP

a m

)


c.

5 10 15 20 250

0.5

1

1.5

2

2.5

3


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


e.

Fig. 4-16 Stress intensity factors for different crack lengths under the combined shear and normal loads



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.



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


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



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.

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


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


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



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



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]



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],



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.



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



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



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



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



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


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



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.



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



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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0 2 4 6 8 10 120

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)



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.



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



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



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



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).



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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)



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.



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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)



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)



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).



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



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



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.



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



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



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



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


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



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



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 (#)


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.



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].



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.




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



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).




CPU time (S)

Global 171054 383.5 870

Local 42476 104.3 267.7


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



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.




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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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)



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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




CPU time (s)

Global 66496 120.3 146.4

Local 6652 9.7 13.3


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)



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



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,



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.

C o n c l u s i o n s a n d f u t u r e w o r k 1 3 7


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.



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.


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



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



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



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



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



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



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.

Bibliography

[1] Renton, W. J., Olcott, D., Roeseler, W., Batzer, R., Baron, W., and Velicki, A.,

“Future of Flight Vehicle Structures (2000-2023),” Journal of Aircraft, Vol. 41,

No.5, 2004, pp. 986-997.

[2] Haftka, R. T., and Gürdal, Z., “Elements of Structural Optimization.” Kluwer

Acadernic Publishcrs, P.O.Box 17, 3300 AA, Dordrecht, The Netherlands, 1992.

[3] Mohaghegh, M., “Evolution of Structures Design Philosophy and Criteria,”

Journal of Aircraft, Vol. 42, No. 4, 2005, pp. 814-831.

[4] Slemp, W. C. H., Bird, R. K., Kapania, R. K., Havens, D., Norris, A., and Olliffe,

R., “Design, Optimization, and Evaluation of Integrally Stiffened Al-7050 Panel

with Curved Stiffeners,” Journal of Aircraft, Vol. 48, No. 4, 2011, pp. 1163-1175.

[5] Mulani, S. B., Joshi, P., Li, J., Kapania, R. K., and Shin, Y. S., “Optimal Design

of Unitized Structures Using Response Surface Approaches,’’ Journal of

Aircraft, Vol. 47, No. 6, 2010, pp. 1898-1906.

[6] Joshi, P., Mulani, S. B., Slemp, W. C. H., and Kapania, R. K., “Vibro-Acoustic

Optimization of Turbulent Boundary Layer Excited Panel with Curvilinear

Stiffeners,” Journal of Aircraft, Vol. 49, No. 1, 2012, pp. 52-65.

[7] Gurav, S. P., and Kapania, R. K., “Development of Framework for the Design

Optimization of Unitized Structures,” 50th AIAA/ASME/ASCE/AHS/ASC

Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA,

May 4-7, 2009, AIAA-2009-2186.

B i b l i o g r a p h y 1 4 5


[8] Elabdi, R., Touratier, M., and Convert, P., “Optimal Design for Minimum Weight

in a Cracked Pressure Vessel of a Turboshaft,” Communications in Numerical

Methods in Engineering, Vol. 12, 1996, pp. 271-280.

[9] Chaperson, P., Jones, R., Heller, M., Pitt, S., and Rose, F., “A Methodology for

Structural Optimization with Damage Tolerance Constraints,” Engineering

Failure Analysis, Vol 7, 2000, pp. 281-300.

[10] El-Sayed, M. E. M., and Lund, E. H.,”Structural Optimization with Fatigue Life

Constraints,” Engineering Fracture Mechanics, Vol. 37, 1990, pp. 1149-1156.

[11] Jones, R., Peng, D., Chaperson, P., Pitt, S., Abramson, D., and Peachey, T.,

“Structural Optimisation with Damage Tolerance Constraints,” Theoretical and

Applied Fracture Mechanics, Vol 43, 2005, pp. 133-155.

[12] Kale, A. A., Haftka, R.T, and Sankar, B.V., “Efficient Reliability-Based Design

and Inspection of Stiffened Panels Against Fatigue,” Journal of Aircraft, Vol. 45,

2008, pp. 86-97.

[13] Nees, C. D., and Canfield, R. A., “Methodology for Implementing Fracture

Mechanics in Global Structural Design of Aircraft,” Journal of Aircraft, Vol. 35,

1998, pp. 131-138.

[14] Dang, T. D., Kapania, R. K., Slemp, W. C. H., Bhatia, M., and Gurav, S. P.,

“Optimization and Postbuckling Analysis of Curvilinear-Stiffened Panels Under

Multiple-Load Cases,” Journal of Aircraft, Vol. 47, No. 5, 2010, pp. 1656-1671.



[15] Arsene, S., Van-der-Veen, S., and Muzzolini, R., “A Finite Element Approach to

Optimizing Damage Tolerance of Airframe Panels” MSC VPD Conference,

Huntington Beach, CA, October 2004.

[16] Isida, M., “Analysis of Stress Intensity Factors for the Tension of a Centrally

Cracked Strip with Stiffened Edges,” Engineering Fracture Mechanics, Vol. 4,

1973, pp. 647-655.

[17] Joshi, S. R., and Shewchuk, J., “Fatigue-Crack Propagation in a Biaxial-Stress

Field” SESA Spring Meeting, Huntsville, Ala, May 19-22, 1970, pp. 529-533.

[18] Ratwani, M. M., and Wilhem, D. P., “Influence of Biaxial loading on Analysis of

Cracked Stiffened Panels.” Engineering Fracture Mechanics, Vol. 11, 1979, pp.

585-593.

[19] Seshadri, B. R., Newman, J. C. Jr., Dawicke, D. S., and Young, R. D., “Fracture

Analysis of the FAA/NASA Wide Stiffened Panels”, Langley research Center,

Hampton, Virginia, NASA/TM-1998-208976.

[20] Yeh, J. R., “Fracture Analysis of a Stiffened Orthotropic Sheet,” Engineering

Fracture Mechanics, Vol. 46, No. 5, 1993, pp. 857-866.

[21] Nishimura, T., “Stress Intensity Factors for a Cracked Stiffened Sheet with

Cracked Stiffeners,” Journal of Engineering Materials and Technology,

Transactions of the ASME, Vol 113, 1991, pp. 119-124.

[22] Lee, K. Y., and Kim, O. W., “Stress Intensity Factor for Sheet-Reinforced and

Cracked Plate Subjected to Remote Normal Stress,” Engineering Fracture

Mechanics, Vol. 61, 1998, pp. 461-468.



[23] Yeh, J. R., and Kulak, M., “Fracture Analysis of Cracked Orthotropic Skin Panels

with Riveted Stiffeners,” Solids and Structures, Vol. 37, 2000, pp. 2437-2487.

[24] Horst, P., and Hausler, S., “Residual Stress Effects in Stiffened Structures-An

Alternative Approach,” Key Engineering Materials, Vol. 417-418, 2010, pp. 757-

760.

[25] Swift, T., “Fracture Analysis of Adhesively Bonded Cracked Panels,” Journal of

Engineering Materials and Technology, Transactions of the ASME, Vol. 100,

1978, pp. 10-15.

[26] Penmetsa, R., Tuegel, E., and Shanmugam, V., “Rapid Risk Assessment Using

Probability of Fracture Nomographs,” Fatigue and Fracture of Engineering

Materials and Structures, Vol. 32, 2009, pp. 886-898.

[27] Moreira, P. M. G. P., Richter-Trummer, V., Tavares, S. M. O., and De-Castro, P.

M. S. T., “Characterization of Fatigue Crack Growth Rate of AA6056 T651 and

T6: Application to Predict Fatigue Behavior of Stiffened Panels,” Material

Science Forums, Vol. 636-637, 2010, pp. 1511-1517.

[28] Putra, I. S., Dirgantara, T., Firmansyah, and Mora, M., “Buckling Analysis of

Shells with a Circumferential Crack,” Key Engineering Materials, Vols. 306-308,

2006, pp. 55-60.

[29] Sih, G. C., and Lee, Y. D., “Tensile and Compressive Buckling of Plates

Weakened by Cracks,” Theoretical and Applied Fracture Mechanics, Vol. 6,

1986, pp. 129-138.



[30] Riks, E., Rankin, C. C., and Brogan, F. A., “The Buckling Behavior of a Central

Crack in a Plate under Tension,” Engineering Fracture Mechanics, Vol. 43, No.

4, 1992, pp. 529-548.

[31] Purbolaksono, J., and Aliabadi, M. H., “Buckling of Cracked Plates,” Key

Engineering Materials, Vols. 251-252, 2003, pp. 153-158.

[32] Markstorm, K., and Storakers, B., “Buckling of Cracked Members,” International

Journal of Solids and Structures, Vol. 16, 1980, pp. 217-229.

[33] Barut, A., Madenci, E., Britt, V. O., and Starnes Jr., J. H., “Buckling of a Thin,

Tension-loaded, Composite Plate with an Inclined Crack,” Engineering Fracture

Mechanics, Vol. 58, No. 3, 1997, pp. 233-248.

[34] Shaw, D., and Huang, Y. H., “Buckling Behavior of a Central Cracked Thin Plate

under Tension,” Engineering Fracture mechanics, Vol. 35, No. 6, 1990, pp. 1019-

1027.

[35] Brighenti, R., “Buckling of Cracked Thin-plates under Tension or Compression,”

Thin-Walled Structures, Vol. 43, 2005, pp. 209-224.

[36] Margaritis, Y., and Toulios, M., “The Ultimate and Collapse Response of Cracked

Stiffened Plates Subjected to Uniaxial Compression,” Thin-Walled Structures,

Vol. 50, 2012, pp. 157–173.

[37] Brighenti, R., and Carpinteri, A., “Buckling and Fracture behavior of cracked

Thin Plates under Shear Loading,” Materials and Design, Vol. 32, 2011, pp.

1347-1355.



[38] Paik, J. K., “Residual ultimate strength of steel plates with longitudinal cracks

under axial compression–experiments,” Ocean Engineering, Vol. 35, 2008, pp.

1775–1783.

[39] Alinia, M. M., Hosseinzadeh, S. A. A., and Habashi, H. R., “Buckling and Post-

Buckling Strength of Shear Panels Degraded by near Border Cracks,” Journal of

Constructional Steel Research, Vol. 64, 2008, pp. 1483-1494.

[40] Brighenti, R., “Buckling Sensitivity Analysis of Cracked Thin Plates under

Membrane Tension or Compression Loading,” Nuclear Engineering and Design,

Vol. 239, No. 6, 2009, pp. 965-980.

[41] Mote, C. D., “Global-Local Finite Element,” International Journal of Numerical

Methods for Engineering, Vol. 3, 1971, pp. 565-574.

[42] Saito, K., Araki, S., Kawakami, T., and Moriwaki, I., “Global-local Finite

Element Analysis of Stress Intensity Factor for a Crack along the Interface of

Two Phase Material,” Proceedings of 10th International Conference on Composite

Materials Society, Whistler, British Columbia, Canada, August, 1995, pp. 261-

268.

[43] Kapania, R. K., Haryadi, S. G., and Haftka, R. T., “Global/Local Analysis of

Composite Plates with Cutouts,” Computational Mechanics, Vol. 19, No. 5, 1997,

pp. 386-396.

[44] Jara-Almonte, C. C., and Knight, C. E., “The Specified Boundary Stiffness/Force

(SBSF) Method for Finite Element Subregion Analysis,” International Journal

for Numerical Methods in Engineering, Vol. 26, No. 7, 1988, pp. 1567-1578.



[45] Farris, T. N., and Doyle, J. F., “A Global/Local Approach to Lengthwise Cracked

Beams: Static Analysis,” International Journal of Fracture, Vol. 50, No. 2, 1991,

pp. 131-141.

[46] Farris, T. N., and Doyle, J. F, “A Global/Local Approach to Lengthwise Cracked

Beams: Dynamic Analysis,” International Journal of Fracture, Vol. 60, No. 2,

1993, pp. 147-156.

[47] Kim, D. -J., Pereira, J. P., and Duarte, C. A., “Analysis of Three-dimensional

Fracture Mechanics Problems: A Two-scale Approach Using Coarse-generalized

FEM Meshes,” International Journal for Numerical Methods in Engineering, Vol.

81, 2010, pp. 335-365.

[48] Gendre, L., Allix, O., and Gosselet, P., “A Two-scale Approximation of the Schur

Complement and Its Use for Non-intrusive Coupling,” International Journal for

Numerical Methods in Engineering, Vol. 87, 2011, pp. 889-905.

[49] Srinivasan, S., Biggers Jr., S. B., and Latour Jr., R. A., “Identifying Global/Local

Interface Boundaries Using an Objective Search Method,” International Journal

for Numerical Methods in Engineering, Vol. 39, 1996, pp. 805-828.

[50] Kerfriden, P., Allix, O., and Gosselet, P., “A Three-scale Domain Decomposition

Method for the 3D Analysis of Debonding in Laminates,” Computational

Mechanics, Vol. 44, 2009, pp. 343-362.

[51] Fish, J., Suvorov, A., and Belsky, V., “Hierarchical Composite Grid method for

Global-local Analysis of Laminated Composite Shells,” Applied Numerical

Mathematics, Vol. 23, 1997, pp. 241-258.



[52] Martinez-Esnaola, J. M., Bastero, J. M., and Miranda, I., “The Use of Global-

local Functions of Blended Type for Computer Simulation of Dynamic Crack

Propagation with Thermal Loading Generation and Prediction Study,”

Engineering Fracture Mechanics, Vol. 28, No. 4, 1987, pp. 387-401.

[53] Madenci, E., Shkarayev, S., and Sergeev, B., “Thermo-mechanical Stresses for a

Triple Junction of Dissimilar Materials: Global-local Finite Element Analysis,”

Theoretical and Applied Fracture Mechanics, Vol. 30, 1998, pp. 103-117.

[54] Leung, A. Y. T., and Tsang, K. L., “Mode III Two-dimensional Crack Problem by

Two-level Finite Element Method,” International Journal of Fracture, Vol. 102,

2000, pp. 245-258.

[55] Vemaganti, K., and Deshmukh, P., “An Adaptive Global-local Approach to

Modeling Functionally Graded Materials,” Computer Methods in Applied

Mechanics and Engineering, Vol. 195, 2006, pp. 4230-4243.

[56] Her, S. –C., “Fracture Analysis of Interfacial Crack by Global-local Finite

Element,” International Journal of Fracture, Vol. 106, 2000, pp. 177-193.

[57] Zeghal, M., and Abdel-Ghaffar, A. M., “Local-global Finite Element Analysis of

the Seismic Response of Earth Dams,” Computers and Structures, Vol. 42, No. 4,

1992, pp. 569-579.

[58] Han, Z. D., and Atluri, S. N., “SGBEM (for Cracked Local Subdomain) – FEM

(for Uncracked Global Structure) Alternating Method for Analyzing 3D Surface

Cracks and Their Fatigue-growth,” Computer Modeling in Engineering Science,

Vol. 3, No. 6, 2002, pp. 699-716.



[59] Hirai, I., Uchiyama, Y., Mizuta, Y., and Pilkey, W. D., “An Exact Zooming

Method,” Finite Element Analysis and Design, Vol. 1, No. 1, 1985, pp. 61-69.

[60] Noor, A. K., “Global-Local Methodologies and Their Application to Nonlinear

Analysis,” Finite Elements in Analysis and Design, Vol. 2, 1986, pp.333-346.

[61] Whitcomb, J. D., “Iterative Global/Local Finite Element Analysis,” Computers

and Structures, Vol. 40, No. 4, 1991, pp. 1027-1031.

[62] Ransom, J. B., and Knight, Jr. N. F., “Global/Local Stress Analysis of Composite

Panels,” Computers and Structures, Vol. 37, No. 4, 1990, pp. 375-395.

[63] Voleti, S. R., Chandra, N., and Miller, J. R., “Global-Local Analysis of Large-

Scale Composite Structures Using Finite Element Methods,” Computers and

Structures, Vol. 58, No. 3, 1996, pp. 453-464.

[64] Srirengan, K., Whitcomb, J., and Chapman, C., “Modal Technique for Three-

Dimensional Global/Local Stress Analysis of Plain Weave Composites,”

Composite Structures, Vol. 39, No. 1-2, 1997, pp. 145-156.

[65] Fish, J., Belsky, V., and Pandheeradi, M., “Composite Grid Method for Hybrid

Systems,” Computer Methods in Applied Mechanics and Engineering, Vol. 135,

1996, pp. 307-325.

[66] Xu, Y., Zhang, W., and Bassir, D., “Stress Analysis of Multi-Phase and Multi-

Layer Plain Weave Composite Structure Using Global/Local Approach,”

Composite Structures, Vol. 92, 2010, pp. 1143-1154.



[67] Haryadi, S. G., Kapania, R. K., and Haftka, R. T., “Global/Local Analysis of

Composite Plates with Cracks,” Composites Part B, Vol. 29 B, 1998, pp. 271-

276.

[68] Guidault, P. A., Allix, O., Champaney, L., and Cornuault, C., “A Multiscale

Extended Finite Element Method for Crack Propagation,” Computer Methods in

Applied Mechanics and Engineering, Vol. 197, 2008, pp. 381-399.

[69] Haidar, K., Dube, J. F., and Cabot, G. P., “Modeling Crack Propagation in

Concrete Structures with a Two Scale Approach,” International Journal for

Numerical and Analytical Methods in Geomechanics, Vol. 27, 2003, pp. 1187-

1205.

[70] Gendre, L., Allix, O., and Gosselet, P., “Non-Intrusive and Exact Global/Local

Techniques for Structural Problems with Local Plasticity,” Computational

Mechanics, Vol. 44, 2009, pp. 233-245.

[71] Sun, L., Gibson, R. F., and Gordaninejad, F., “Multiscale Analysis of Stiffness

and Fracture of Nanoparticle-Reinforced Composites Using Micromechanics and

Global-Local Finite Element Models,” Engineering Fracture Mechanics, Vol. 78,

2011, pp. 2645-2662.

[72] Knight Jr., N. F., Ransom, J. B., Griffin Jr., O. H., and Thompson, D. M.,

“Global/Local Methods Research Using a Common Structural Analysis

Framework,” Finite Elements in Analysis and Design, Vol. 9, 1991, pp. 91-112.



[73] Alaimo, A., Milazzo, A., and Orlando, C., “Global/Local FEM-BEM Stress

Analysis of Damaged Aircraft Structures,” Computer Modeling in Engineering

Science, Vol. 36, No. 1, 2008, pp. 23-41.

[74] Higgins, A.,” Adhesive Bonding of Aircraft Structures,” International Journal of

Adhesion and Adhesives, Vol. 20, 2000, pp. 367-376.

[75] Kuczmaszewski, J., and Wlodarczyk, M., “Numerical Analysis of Stress

Distributions in Adhesive Joints,” IUTAM Symposium on Multiscale Modelling of

Damage and Fracture Processes in Composite Materials, 2006, pp. 271-278.

[76] Goncalves, J. P. M., de Moura, M. F. S. F., and de Castro, P. M. S. T., “A Three-

Dimensional Finite Element Model for Stress Analysis of Adhesive Joints,”

International Journal of Adhesion and Adhesives, Vol. 22, 2002, pp. 357-365.

[77] da Silva, L. F. M., Carbas, R. J. C., Critchlow, G. W., Figueiredo, M. A. V., and

Brown, K., “Effect of Material, Geometry, Surface Treatment and Environment

on the Shear Strength of Single Lap Joints,” International Journal of Adhesion

and Adhesives, Vol. 29, 2009, pp. 621-632.

[78] Pearson, I. T., and Mottram, J. T., “A Finite Element Modelling Methodology for

The Non-Linear Stiffness Evaluation of Adhesively Bonded Single Lap-Joints:

Part 1. Evaluation of Key Parameters,” Computers and Structures, Vol. 90-91,

2012, pp. 76-88.

[79] Diaz, J., Romera, L., Hernandez, S., and Baldomir, A., “Benchmarking of Three-

Dimensional Finite Element Models of CFRP Single-Lap Bonded Joints,”

International Journal of Adhesion and Adhesives, Vol. 30, 2010, pp. 178-189.



[80] Sundararaman, V., and Davidson, B. D., “An Unsymmtric Double Cantilever

Beam Test for Interfacial Fracture Toughness Determination,” International

Journal of Solids and Structures, Vol. 34, No. 7, 1997, pp. 799-817.

[81] Bennati, S., Colleluori, M., Corigliano, D., and Valvo, P. S., “An Enhanced

Beam-Theory Model of the Asymmetric Double Cantilever Beam (ADCB) Test

for Composite Laminates,” Composites Science and Technology, Vol. 69, 2009,

pp. 1735-1745.

[82] Park, S., and Dillard, D. A., “Development of a Simple Mixed-Mode Fracture test

and the Resulting Fracture Energy Envelope for an Adhesive Bond,”

International Journal of Fracture, Vol. 148, 2007, pp. 261-271.

[83] da Silva, L. F. M., Esteves, V. H. C., and Chaves, F. J. P., “Fracture Toughness of

a Structural Adhesive under Mixed Mode Loadings,” Mat.-wiss.u.Werkstofftech,

Vol. 42, No. 5, 2011, pp. 460-470.

[84] Alfredsson, K. S., and Hogberg, J. L., “Energy Rate and Mode-Mixity of

Adhesive Joint Specimens,” International Journal of Fracture, Vol. 144, 2007,

pp. 267-283.

[85] Szekrenyes, A., “Prestressed Composite Specimen for Mixed-Mode I/II Cracking

in Laminated Materials,” Journal of Reinforced Plastics and Composites, Vol. 29,

2010, pp. 3309-3321.

[86] Yuan, H., and Xu, Y., “Computational Fracture Mechanics Assessment of

Adhesive Joints,” Computational Materials Science, Vol. 43, 2008, pp. 146–156.



[87] Plausinis, D., and Spelt, J. K., “Designing for Time-Dependent Crack Growth in

Adhesive Joints,” International Journal of Adhesion and Adhesives, Vol. 15,

1995, pp. 143-154.

[88] Tvergaard, V., and Hutchinson, J. W., “On the Toughness of Ductile Adhesive

Joints,” Journal of the Mechanics and Physics of Solids, Vol. 44, No. 5, 1996, pp.

789-800.

[89] Katnam, K. B., Sargent, J. P., Crocombe, A. D., Khoramishad, H., and Ashcroft,

I. A., “Characterization of Moisture-Dependent Cohesive Zone Properties for

Adhesive Joints,” Journal of Engineering Fracture Mechanics, Vol. 77, 2010, pp.

3105-3119.

[90] Ameli, A., Papini, M., Schroeder, J. A., and Spelt, J. K., “Fracture R-Curve

Characterization of Toughened Epoxy Adhesives,” Journal of Engineering

Fracture Mechanics, Vol. 77, 2010, pp. 521-534.

[91] Yang, Q. D., and Thouless, M. D., “Mixed-Mode Fracture Analysis of Plastically

Deforming Adhesive Joints,” International Journal of Fracture, Vol. 110, 2001,

pp. 175-187.

[92] Madhusudhana, K. S., and Narasimhan, R., “Experimental and Numerical

Investigations of Mixed Mode Crack Growth Resistance of a Ductile Adhesive

Joint,” Journal of Engineering Fracture Mechanics, Vol. 69, 2002, pp. 865-883.

[93] De-Morais, A. B., and Pereira, A. B., “Mixed Mode II+III Interlaminar Fracture

of Carbon/Epoxy Laminates,” Composites Science and Technology, Vol. 68,

2008, pp. 2022-2027.



[94] Szekrenyes, A., “Delamination Fracture Analysis in the GII-GIII Plane Using

Prestressed Transparent Composite Beams,” International Journal of Solids and

Structures, Vol. 44, 2007, pp. 3359-3378.

[95] Marannano, G. V., Mistretta, L., Cirello, A., and Pasta, S., “Crack Growth

Analysis at Adhesive-Adherent Interface in Bonded Joints Under Mixed Mode

I/II,” Journal of Engineering Fracture Mechanics, Vol. 75, 2008, pp. 5122-5133.

[96] Guo, Y. J., and Weitsman, Y. J., “A Modified DCB Specimen to Determine

Mixed Mode Fracture Toughness of Adhesives,” Journal of Engineering Fracture

Mechanics, Vol. 68, 2001, pp. 1647-1668.

[97] Sun, C., Thouless, M. D., Waas, A. M., Schroeder, J. A., and Zavattieri, P. D.,

“Rate Effects for Mixed-Mode Fracture of Plastically-Deforming Adhesively-

Bonded Structures,” International Journal of Adhesion and Adhesives, Vol. 29,

2009, pp. 434-443.

[98] Tafreshi, A., and Oswald, T., “Global Buckling Behavior and Local Damage

Propagation in Composite Plates With Embedded Delaminations,” International

Journal Pressure Vessels Piping, Vol. 80, 2003, pp. 9–20.

[99] Withcomb, J. D., and Shivakumar, K. N., “Strain Energy Release Rate Analysis

of Plates With Postbuckled Delaminations,” Journal of Composite Materials, Vol.

23, 1989, pp. 714–734.

[100] Riccio, A., Perugini, P., and Scaramuzzino, F., “Modelling Compression

Behaviour of Delaminated Composite Panels,” Composite Structures, Vol. 78,

2000, pp. 73-81.



[101] Chai, H., Babcock, C. D., and Knauss, W. G., “One Dimensional Modelling of

Failure in Laminated Plates by Delamination Buckling,” International Journal of

Solids and Structures, Vol. 17, No. 11, 1981, pp. 1069-1083.

[102] Gaudenzi, P., “On Delamination Buckling of Composite Laminates Under

Compressive Loading,” Composite Structures, Vol. 39, No. 1-2, 1997, pp. 21-

330.

[103] Hwang, S. F., and Mao, C. P., “Failure of Delaminated Carbon/Epoxy Composite

Plates under Compression,” Journal of Composite Materials, Vol. 35, No. 18,

2001, pp. 1634-1653.

[104] Kim, H., and Kedward, K. T., “A Method for Modeling the Local and Global

Buckling of Delaminated Composite Plates,” Composite Structures, Vol. 44,

1999, pp. 43-53.

[105] Kapania, R. K., and Wolfe, D. R., “Buckling of Axially Loaded Beam Plate With

Multiple Delaminations,” Journal of Pressure Vessel Technology, Vol. 111, 1989,

pp. 151-158.

[106] Chow, C. L., and Ngan, K. M., “Method Fracture Toughness Evaluation of

Adhesive Joints,” Journal of Strain Analysis, Vol. 15, No. 2, 1980, pp. 97-101.

[107] Ortiz, M., and Pandolfi, A., “Finite-Deformation Irreversible Cohesive Elements

for Three Dimensional Crack-Propagation Analysis,” International Journal for

Numerical Methods in Engineering, Vol. 44, 1999, pp. 1267-1282.



[108] Turon, A., Davila, C. G., Camanho, P. P., and Costa, J., “An Engineering Solution

for Mesh Size Effects in the Simulation of Delamination Using Cohesive Zone

Models,” Engineering Fracture Mechanics, Vol. 74, 2007, pp. 1665-1682.

[109] Makhecha, D. P., Kapania, R. K., Johnson, E. R., Dillard, D. A., Jacob, G. C.,

and Starbuck, M. J., “Rate-Dependent Cohesive Zone Modeling of Unstable

Crack Growth in an Epoxy Adhesive ,” Mechanics of Advanced Materials and

Structures, Vol. 16, 2009, pp. 12-19.

[110] Needleman, A., “An Analysis Decohesion Along an Imperfect Interface,”

International Journal of Fracture, Vol. 42, 1990, pp. 21-40.

[111] Tay, T. E., “Characterization and Analysis of Delamination Fracture in

Composites: An Overview of Developments from 1990 to 2001,” Applied

Mechanical Review, Vol. 56, No. 1, 2003, pp. 1-32.

[112] JSSG-2006, “Joint Service Specification Guide,” Department of Defense, 1998.

[113] Timoshenko, S., and Woinowsky-Krieger, S., “Theory of Plates and Shells,”

Second ed., McGraw-Hill Book Company, USA, 1959.

[114] Venstel, E., and Krauthammer, T., “Thin Plates and Shells-Theory, Analysis, and

Applications,” Marcel Dekker, Inc., NY, USA, 2001.

[115] Wempner, G. A., Oden, J.T., and Kross, D. A., “Finite Element Analysis of Thin

Shells,” Mechanical Engineering Division, Proceeding ASCE, 94, EM6, 1968, pp.

1273-1294.



[116] Hughes, T. J. R., and Hinton, E., “Finite Element Methods for Plate and Shell

Structures (Volume 1: Element Technology),” Pineridge Press Limited, Mumbles,

Swansea, U.K., 1986.

[117] Batoz, J. L., Bathe, K. J., and Ho, L. W., “A Study of Three-Node Triangular

Plate Bending Elements,” International Journal for Numerical Methods in

Engineering, Vol. 15, 1980, pp. 1771-1812.

[118] MSC Software, MSC.Marc Theory and User guide, 2010.

[119] MSC Software, MSC.Marc Mentat User’s Guide, 2010.

[120] ABAQUS Software, ABAQUS User Manual, Version 6.10, ABAQUS, Inc.,

Providence R.I.

[121] Rice, J. R., “A Path Independent Integral and the Approximate Analysis of Strain

Concentration by Notches and Cracks,” Journal of Applied Mechanics, Vol. 35,

1968, pp. 379-386.

[122] Anderson, T. L., “Fracture Mechanics Fundamentals and Applications,” Third ed.,

Taylor and Francis, Boca Raton, FL, 2005.

[123] Shih, C. F., and Asaro, R. J., “Elastic-Plastic Analysis of Cracks on Bimaterial

Interfaces: Part I—Small Scale Yielding,” Journal of Applied Mechanics, 1988,

pp. 299–316.

[124] Krueger, R., “Virtual Crack Closure Technique: History, Approach and

Applications,” Applied Mechanics Reviews, Vol. 57, No. 2, 2004, pp. 109–143.



[125] Raju, I. S., “Calculation of Strain-Energy Release Rates with Higher Order and

Singular Finite Elements,” Engineering Fracture Mechanics, Vol. 28, No. 3,

1987, pp. 251-274.

[126] Irwin, G. R., “Analysis of Stresses and Strains Near the End of a Crack

Transversing a Plate,” Journal of Applied Mechanics, Transactions ASME, Vol.

24, 1957, pp. 361.

[127] Rybicki, E. F., and Kanninen, M. F., “A Finite Element Calculation of Stress-

Intensity Factors by a Modified Crack Closure Integral,” Engineering Fracture

Mechanics, Vol. 9, 1977, pp. 931-938.

[128] Tada, H., Paris, P. C., and Irwin, G. R., “The Stress Analysis of Cracks

Handbook,” 1st ed., Del Research Corporation, Hellertown, PA, 1973.

[129] Brown, W. F. Jr., and Srawley, J. E., “Plane Strain Crack Toughness Testing of

High Strength Metallic Materials,” ASTM STP, 410, 1966.

[130] MSC Software, MSC.Patran User’s Guide, 2010.

[131] MSC Software, MSC.Nastran Theory and User Guide, 2010.

[132] Beom, H. G., and Jang, H. S., “A Wedge Crack in an Anisotropic Material under

Antiplane Shear,” International Journal of Engineering Science, Vol. 49, 2011,

pp. 867-880.

[133] Aggelopoulos, E. S., Righiniotis, T. D., and Chryssanthopoulos, M. K.,

“Debonding of Adhesively Bonded Composite Patch Repairs of Cracked Steel

Members,” Composites: Part B, Vol. 42, 2011, pp. 1262-1270.



[134] Niu, M. C. Y., “Airframe Structural Design,” Hong Kong, Conmilit Press Ltd.,

1988.

[135] Hughes, T. J. R., “The Finite Element Method: Linear Static and Dynamic Finite

Element Analysis,” Dover Publications Inc., N. Y., U. S. A., 2000.

[136] Dugdale, D. S., “Yielding of Steel Sheets Containing Slits,” Journal of the

Mechanics and Physics of Solids, Vol. 8, 1960, pp. 100-104.

[137] Barenblatt, G. I., “The Mathematical Theory of Equilibrium Crack in the Brittle

Fracture,” Advances in Applied Mechanics, Vol. 7, 1962, pp. 55-125.

[138] Daudeville, L., Allix, O., and Ladeveze, P., “Delamination Analysis by Damage

Mechanics: Some Applications,” Composites Engineering, Vol. 5, No. 1, 1995,

pp. 17-24.

[139] Crisfield, M. A., “A Fast Incremental/Iteration Solution Procedure that Handles

Snap-Through,” Computers and Structures, Vol. 13, 1981, pp. 55–62.

[140] Patzak, B., and Jirasek, M., “Process Zone Resolution by Extended Finite

Elements,” Engineering Fracture Mechanics, Vol. 70, 2003, pp. 957-977.

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Global-local Finite Element Fracture Analysis of …...Global-local Finite Element Fracture Analysis of Curvilinearly Stiffened Panels and Adhesive Joints Mohammad Majharul Islam Abstract