CRACK ARREST CAPABILITIES OF AN ADHESIVELY BONDED SKIN AND
STIFFENER
A Thesis By
Shamsuzuha Habeeb
B. E., Bangalore Institute of Technology, India 2005
Submitted to the Department of Aerospace Engineering
and the faculty of the Graduate school of
Wichita State University
in partial fulfillment of
the requirements for the degree of
Master of Science
Dec 2012
© Copyright 2012 by Shamsuzuha Habeeb
All Rights Reserved
CRACK ARREST CAPABILITIES OF AN ADHESIVELY BONDED SKIN AND
STIFFENER
The following faculty members have examined the final copy of the Thesis for form and
content, and recommend that it be accepted in partial fulfillment of the requirement for the
degree of Master of Science, with a major in Aerospace Engineering.
________________________________
Keshavanarayana Suresh Raju, Committee Chair
________________________________
Charles Yang, Committee Member
________________________________
Krishna Krishnan, Committee Member
iii
DEDICATION
To my parents and my brother
iv
ACKNOWLEDGEMENTS
I thank Allah for providing me the opportunity to come to US for my studies.
I would like to express my sincere gratitude to my guru Dr. K. Suresh Raju, for his
continuous support and guidance in helping me towards a successful completion of this
research work. He has helped me gain understanding of fundamental details and apply it in
my thesis. For his valuable time, help, support and patience I am greatly indebted to him. I
am greatly obliged to him for providing me a stage to work on something I am passionate
about.
I would also like to thank Dr. Charles Yang and Dr. Krishna Krishnan my committee
members for their valuable time spent reviewing my thesis and giving feedback.
Lastly and importantly I would like to thank my family and friends for supporting
and encouraging me throughout my lifetime and studies.
v
ABSTRACT
The crack arrest capabilities and the load bearing characteristic of a stiffened and
unstiffened panel subjected to remote tension loading is examined in this work.
We make use of cohesive interface elements to predict the behavior of the center crack
stiffened and unstiffened panels using a nonlinear traction-separation relationship. This
thesis describes an application of traction-separation model on a four stringer stiffened wide
panel with and without a disbond, and also an unstiffened panel with a center crack and
compare their residual strengths. Aluminum 2024-T3 alloy panels were tested as a part of
another study and the measured values were used to compare the numerical results. The
predicted crack growth and residual predictions shows fair convergence with experimental
values. The strength of the stiffened panel was observed to be much higher (55%) than that
of that unstiffened panel. Moreover the presence of the disbond of 1” caused a 9% drop in
the residual strength and disbond lesser than 1” does not lead to a drop in strength. The
stiffened panel readily supported the transfer of load from the plate to the stiffener, thus
aiding in the crack growth arrest.
vi
TABLE OF CONTENT
1 INTRODUCTION ............................................................................................................... 1
1.1
1.2
1.3
1.4
Overview ......................................................................................................... 1
1.1.1
Stiffened Panel ................................................................................. 2
1.1.2
Type of Fractures ............................................................................. 2
CTOD / CTOA ............................................................................................... 3
Cohesive Zone Model (CZM) ........................................................................ 4
1.3.1
Importance of CZM ......................................................................... 6
Organization of the thesis document .............................................................. 6
2 LITERATURE REVIEW .................................................................................................... 8
2.1
2.1
Use of CZM to simulate crack propagation in Aluminum ............................. 8
Use of CZM to simulate behavior of Adhesive ............................................ 15
3 STATEMENT OF PROBLEM ......................................................................................... 20
3.1
3.2
Statement of Problem ................................................................................... 20
Objectives ..................................................................................................... 25
4 COHESIVE ZONE FORMULATION .............................................................................. 26
4.1
Cohesive Law ............................................................................................... 29
5 FINITE ELEMENT MODEL ............................................................................................ 31
5.1
5.2
5.3
5.4
Model and Boundary Conditions .................................................................. 31
5.1.1
Modeling of plate and stiffener ..................................................... 32
5.1.2
Modeling of adhesive zone ............................................................ 36
Material Properties ........................................................................................ 37
5.2.1
Material Properties of plate and stiffener [57] ............................... 37
5.2.2
Material Properties of adhesive zone [58] ..................................... 37
Modeling of cohesive zone ........................................................................... 38
5.3.1
Cohesive parameters for the stiffener region ................................. 40
5.3.2
Cohesive parameters for the adhesive region ................................ 40
Mesh ............................................................................................................. 41
vii
TABLE OF CONTENT CONTD
5.5
5.6
Solution Procedure ........................................................................................ 44
Data reduction Procedure ............................................................................. 45
5.6.1
Crack Extension ............................................................................. 45
5.6.2
CMOD ........................................................................................... 46
5.6.3
Radius of Plastic Zone ................................................................... 47
5.6.4
Height of Plastic Zone in the Nearest Stiffener ............................. 49
6 VERIFICATION AND VALIDATION ............................................................................ 51
6.1
6.2
Verification of CZM with 3 Element Model [14] ........................................ 51
Validation ..................................................................................................... 54
7 RESULTS AND DISCUSSION ........................................................................................ 57
7.1
7.2
7.3
7.4
Effect on Residual Strength with different types of constraint ..................... 58
7.1.1
Unstiffened panel ........................................................................... 58
7.1.2
Stiffened Panel ............................................................................... 64
Comparison of the residual strengths of stiffened panel with and without a
disbond .......................................................................................................... 70
Effect of presence of stiffeners on the residual strength of a panel .............. 74
Shear stress in the adhesive region ............................................................... 77
8 CONCLUSION AND FUTURE WORK .......................................................................... 81
8.1
8.2
Conclusions ................................................................................................... 81
Future Work .................................................................................................. 82
9 MARC ELEMENTS .......................................................................................................... 90
9.1
9.2
MARC element # 7: Three-dimensional Arbitrarily Distorted Brick ........... 90
Marc element # 188: Eight-node Three-dimensional Interface Element ...... 91
10 STIFFENED FILE WITH GUIDE PLATE CONSTRAINT INPUT FILE .................... 92
REFERENCES ....………………………………………………………………………… .83
APPENDIX ……………………………………………………………………………….. 89
viii
LIST OF TABLES
Table
1.
Page
Comparison of cohesive parameters used previously…………………………….…14
ix
LIST OF FIGURES
Figure
Page
1. CTOD and CTOA definitions [18] .................................................................................... 3
2. Representation of CZM [21] .............................................................................................. 5
3. Traction separation law [43] .............................................................................................. 9
4. Common problems seen in riveted and adhesive joints................................................... 20
5. Geometry of unstiffened panel ........................................................................................ 22
6. Geometry of a stiffened panel.......................................................................................... 23
7. Illustration of a disbond ................................................................................................... 24
8. Normal traction Tn across the cohesive surface as a function of ∆n[13] ......................... 28
9. Shear traction Tt across the cohesive surface as a function of ∆t [13] ............................. 29
10. Boundary condition of the unstiffened panel................................................................... 33
11. Illustration of three different types of anti-buckling supports for unstiffened panel ....... 34
12. Boundary condition of the stiffened panel....................................................................... 35
13. Illustration of two different types support for buckling for stiffened panel .................... 36
14. Strain hardening for Aluminum[55] ................................................................................ 37
15. Strain hardening for Adhesive ......................................................................................... 38
16. Traction separation law used in the study........................................................................ 39
17. 1/8th symmetry FE model of unstiffened panel, with close of through thickness
mesh refinement............................................................................................................... 42
th
18. 1/4 symmetry FE model of stiffened panel, with close up of through thickness
mesh refinement............................................................................................................... 43
19. Illustration of a crack ....................................................................................................... 46
20. CMOD measurements ..................................................................................................... 47
x
LIST OF FIGURES CONTD
21. Illustration of forces around the crack tip [18] ................................................................ 48
22. Plane stress and plane strain plastic zone shape for mode-I cracking [18]. .................... 48
23. Shape of plastic zone at different crack lengths in unstiffened and stiffened
panels ............................................................................................................................... 49
24. Height of plastic zone measurement in the nearest stiffener ........................................... 50
25. Three element model with boundary conditions applied ................................................. 52
26. Cohesive law followed by the element ............................................................................ 53
27. Stress comparison of three element verification model with and without CZM ............. 53
28. Comparison of the predicted values for a 3” wide M(T) specimen with the values
reported [21] .................................................................................................................... 55
29. Comparison of the predicted values for a 12" wide M(T) specimen with the
values reported [1] ........................................................................................................... 55
30. Comparison of the predicted values for a 24" M(T) specimen with the values
reported [1] ...................................................................................................................... 56
31. Load vs. crack extension response for the unstiffened M(T) panel ................................. 59
32. Load vs. CMOD comparison for the unstiffened M(T) panel ......................................... 60
33. CMOD vs. physical crack length prediction.................................................................... 61
34. KR vs. effective crack length comparison (microscope reading) ..................................... 62
35. KR vs. effective crack length comparison (CMOD reading) ........................................... 62
36. KR and KG curves ............................................................................................................. 63
37. Load vs. Physical crack length response for stiffened M(T) specimen ........................... 65
38. Load vs. CMOD comparison for the stiffened M(T) specimen....................................... 66
39. CMOD vs. physical crack length comparison ................................................................. 67
40. Radius of plastic zone size vs. effective crack length ..................................................... 68
xi
LIST OF FIGURES CONTD
41. Height of plastic yielding in the inner stiffener vs. physical crack length ...................... 69
42. Residual strength prediction in stiffened panel out of plane displacement
constrained using guide plates with and without a disbond ............................................ 71
43. Load vs. CMOD comparison of the stiffened panel constrained with guide plates ........ 72
44. Radius of plastic zone at different effective crack lengths .............................................. 73
45. Change in the amount of yielding in the inner stiffener with crack extension ................ 73
46. Difference in residual strength between unstiffened and stiffened panel ........................ 74
47. Load vs. CMOD comparison ........................................................................................... 75
48. COMD vs. physical crack length comparison ................................................................. 76
49. Radius of plastic zone as the crack propagates ................................................................ 76
50. Shear stress in adhesive region at a crack length of 0.001" ............................................. 78
51. Shear stress in adhesive region at a crack length of 0.1" ................................................. 78
52. Shear stress in adhesive region at a crack length of 1.1" ................................................. 79
53. Shear stress in the adhesive region in plate with large lengths........................................ 80
54. MARC element # 7 [59] .................................................................................................. 90
55. MARC element # 188 [59] .............................................................................................. 91
xii
CHAPTER 1
1INTRODUCTION
1.1
Overview
The continual need for light weight, large scale metallic structures has brought about
a new set of problems related to fracture. Safety assessments of aging airframes frequently
requires predictions of the residual strength of aluminum components that have accumulated
damage due to repeated pressurization cycles during normal in-service operation. Such
components must maintain suitable safety margins in case a catastrophic failure like largescale crack propagations or coalescence of multiple macroscopic cracks may occur [1].
According to Jiang [2], “failure in an engineering structure can occur due to various
reasons, voids in materials, environment, building processes, loading, inadequate design or
improper constructions. Often fracture is a result of cracks in the structure either during
production or service and can lead to loss in human lives and economy”. Defects and cracks
are imminent in a structure and various approaches, both at the microscopic and
macroscopic level have been reported [1-3] to deal with problems like these. Fracture
mechanics is a study of the driving force of a macroscopic crack extension [4] . During
crack extension, the crack tip experiences a high level of stress triaxiality[1], and reliable
methods are required to adequately describe this field. Thin panels when subjected to
repeated pressurization cycles are prone to failure very quickly; one way to improve the
strength of the skin (fuselage) is to bond stiffeners to the skin either by adhesive bonding or
using rivets.
1
1.1.1
Stiffened Panel
In adhesively bonded stiffened panels, the interface (adhesive) between the skin and
the stiffener helps the transfer of load as the crack approaches the stiffener boundary. The
crack tip stress in the skin also decreases as the load starts to get transferred to the stiffener
from the skin, thus facilitating crack arrest. This helps to achieve a high strength in the panel
while keeping the weight to a minimum. Stiffeners are also attached to the skin using rivet.
The need of making small rivet holes in the skin and the stiffener which introduces stress
concentration around the holes is a major issue with riveted joints. On the other hand while
the adhesive overcomes the problems experienced with riveted joints, it is easily affected by
the environment such as moisture and the constant contraction and expansion cycles
experienced by aircraft structures. In addition, process induced disbonds due to the improper
tooling and surface preparation could be detrimental to the crack arrest capability of the
stiffened panel.
1.1.2
Type of Fractures
Fracture can be of two types brittle and ductile [5]. Fracture in brittle material is
predominantly linear elastic, i.e. the formation of process zone ahead of the crack
propagation is small. Linear Elastic Fracture Mechanics (LEFM)[6] adequately explains the
behavior of brittle crack. Ductile materials experience a large plastic region, where the
material yields (substantial amount) before fracture occurs. Prediction of growth of crack
with high crack tip plasticity is difficult [1, 7]. To overcome these difficulties several
approaches have been proposed as a part Elastic Plastic Fracture Mechanics (EPFM).
2
These include Crack Tip Opening Angle (CTOA) [7-9], J- integral [2, 10], various
forms of traction-separation models Cohesive Zone Model (CZM) [1, 3, 11-16]. J-integral
can be related to stress intensity factor only in the linear region, it means that J-integral can
predict residual loads and the crack tip fields prior to crack extension but fails to predict
large amount of crack propagation[17]. CTOA reliably predicts peak loads but fails to
estimate crack front tunneling effect [2].
1.2
CTOD / CTOA
This methodology [18] states that stable crack growth occurs when the local
displacement at the original crack tip or the angle made by the upper crack plane, crack tip
and the lower crack plane reaches a critical value as shown in Figure 1.
Figure 1 CTOD and CTOA definitions [18]
CTOD was proposed by Wells [19] during the study of structural steels, since LEFM
could not characterize crack growth in high toughness material. He noted that the crack
surfaces had moved apart before the fracture occurred, and the initial sharp crack had
blunted due to plastic deformation. The degree of the crack blunting increased
proportionally to the toughness of the material, which led to the proposal of CTOD to
measure fracture toughness.
3
While CTOD was adequate to predict large amount of crack propagation, it fails to
consider the crack tip triaxiality and as a result fails to predict crack front tunneling effect.
1.3
Cohesive Zone Model (CZM)
CZM overcomes some of the aforementioned deficiencies and gives a very good
estimate of the peak stress [1, 13, 14], the residual stresses and the crack tunneling effect.
CZM was first applied by Dugdale [20]. CZM uses the traction-separation model to simulate
failure. It idealizes the fracture process instead of an exact representation of fracture zone,
and this also helps in simulating the complex features of void nucleation, growth and
coalescence which are not provided by the other approaches[1].
Figure 2 shows the representation of CZM. Process zone is represented by the
extended region of crack i.e. between the fictitious crack and the real crack tip region[2, 21].
The presence of a process zone eliminates the problem caused by triaxial stresses developed
at the crack tip (the exponential cohesive law provides for gradual unloading of the
elements, simulating a crack). Various cohesive laws or traction-separation models can be
used to define the behavior of the process zone namely exponential, linear, bilinear and
trapezoidal.
4
Figure 2 Representation of CZM [21]
The traction-separation behavior characterizing the cohesive law can be determined
either experimentally or by assumption. Various cohesive laws have been defined and
successfully applied to predict crack behavior. The cohesive laws are defined by three
parameters, cohesive strength (critical traction), the cohesive energy (area underneath the
curve), and the separation value at peak load, among which two parameters are independent.
All the cohesive laws follow a common trend; as the separation between the top and the
bottom surfaces increases the magnitude of traction increases till the peak (critical) value is
reached, unloading process begins while the separation continues to increase. The
descending part of the curve after peak traction indicates the unloading process of the
5
material. The cohesive strength sets the maximum allowable traction that can be applied
before failure. The cohesive energy represents the energy dissipated at complete separation.
1.3.1
Importance of CZM
CZM predicts the behavior of crack (entire life) from crack initiation to complete
failure [13]. The presence of process zone helps the prediction of crack front tunneling and
eliminates the difficulties due to crack tip triaxiality.
CZM has been successfully applied in simulating crack growth in metals, concrete,
polymers, ceramics and in composites [1, 14, 22-26]. CZM effectively simulates
macroscopic crack propagation. CZM also has been successful in simulating crack behavior
in thick specimens which is difficult due to crack tip stress triaxiality developed, which leads
to tunneling[1, 13, 14].
1.4
Organization of the thesis document
The objective of this study is to investigate the crack arresting capabilities of an
adhesively bonded stiffened panel. That is to find the increase in the strength of a stiffened
panel over an un-stiffened panel and compare the results to the experimental results reported
in [27] and discuss the applicability of CZM to simulate similar failures.
This report is organized as follows: chapter 2 is dedicated to the background study
where in, an overview of various form of cohesive laws employed to study failure in
different types of material by other researchers is presented. Description of the problem and
the objective of this study is presented in chapter 3. Chapter 4 describes the cohesive
element formulation that governs the progressive crack growth. The description of model,
the material properties, the cohesive parameters and the solution scheme used are described
6
in chapter 5. Validation of CZM is tackled in this chapter 6. Chapter 7 closes with results
and discussion.
7
CHAPTER 2
2LITERATURE REVIEW
2.1
Use of CZM to simulate crack propagation in Aluminum
The key component in the assessment of the structural integrity of an aircraft focuses
on the reliable prediction of the residual strength. The development of reliable strategies to
predict and simulate the behavior of the crack growth has been a challenge in fracture
mechanics[1]. For typical aluminum alloys used in airframe structures, ductile extension of a
macroscopic crack occurs by nucleation [28], subsequent crack growth and coalescence of
voids along the path of the crack extension. Void growth under the relatively low crack front
triaxiality that is seen in these panels are not adequately defined by the Gurson type damage
model [28], which provides a realistic prediction of crack extension in thick panels (eg.
pressure vessel) where a high level of stress triaxiality is present at the crack tip.
Two approaches proposed to overcome these difficulties for engineering simulation of large
scale crack extension include crack tip opening angle (CTOA) [7-9, 29] and traction
separation models [1-3, 11, 12, 14-17, 21, 23, 30-42]. Newman and coworkers [7-9] have
performed extensive computational and experimental studies using CTOA methodology,
while Roy and coworkers [1, 14] have studied traction separation models to predict crack
propagation. During the crack propagation, much of the life of the crack is spent during
initiation process. Since initiation takes place on a smaller scale than that of the overall
structure, the factors that influence the initiation also occur at smaller length scale [13].
8
Figure 3 Traction separation law [43]
Cohesive law is characterized by a relationship between the closure traction and
separation illustrated in Figure 3. In the cohesive zone approach, the fracture process occurs
between thin layers confined by adjacent surfaces. Earlier studies [7, 28, 40, 43] of fracture
mechanics yielded expressions for stresses at the crack tip for linear elastic bodies which
were valid for small scale yielding. However the stress singularity at the crack tip does not
represent the real behavior since the material is always undergo yielding at the crack tip.
Tractions are applied to the process zone, eliminating the stress singularity at the crack tip.
A cohesive model can be defined by three parameters, traction (local strength of the
material), separation and the cohesive energy needed for opening the crack (area under the
curve) with many variations.
In recent times the cohesive zone approach to model crack propagation has gained
popularity[1, 11, 12, 14, 15, 21, 30, 34, 35, 44-46]. However the concept was introduced
9
when Dugdale [20] and Barenblatt [47] proposed a model in 1960’s that incorporated a
process zone in front of the crack tip. Dugdale [20] used a cohesive zone to model yielding
in a plastic strip in front of a crack tip to study the ductile fracture in a thin sheet of mild
steel. Upper and lower surfaces of the cohesive zone were acted upon by a constant traction
equal to the yield stress of the material. Barenblatt [47] defined a cohesive zone model to
eliminate crack tip stress singularity. He proposed that fracture occurs due to decohesion of
upper and lower surfaces of the cohesive zone behind the crack tip. According to his
definition, the tractions are zero when the distance between two atomic planes is equal to
normal intermolecular distance and the traction increases with distance until the traction
reaches the value equal to the atomic bond strength and then it is decreased to zero.
Some of the first examples of the use of interface elements for modeling cohesive
behavior were in the field of civil engineering. Ngo and Scordelis [22] simulated the bond
slip between concrete and steel reinforcement which was modeled by a zero thickness
element. Similar kind of idea was applied to rock fracture by Goodman [48] to model the
behavior of joints in rock. In 1976 Hillerborg [23] introduced the concept of fictitious crack
model to study failure in concrete. Cohesive zone model has also been employed to study
fracture processes in other quasi – brittle materials like, ceramics (eg. Camacho and Ortiz
[16]) polymers (eg. Knauss [30]) and ductile metals (eg. Needleman [2, 3])
The work of Needleman [3] was one of the first studies on the use of cohesive zone
in fracture simulations. He carried out interface cracking simulations for isotropic hardening
elastic viscous plastic matrix (axisymmetric model containing an array of spherical voids)
using FEM to simulate applications such as inclusion of debonding and decohesion. In this
10
study the interface was described through a constitutive relation dependent on both normal
and tangential displacements and is a polynomial. Since then CZM has emerged as a popular
tool for simulating fracture in materials and structures due to the computational
convenience. In the following work, Needleman [2] analyzed tensile decohesion along an
interface using cohesive model.
Tvergaard and Hutchinson [49] employed a special Rayleigh-Ritz finite element
method to numerically calculate the K resistance curve for a plane strain mode-I growth for
small scale yielding in a semi-circular elastic-plastic solid using trapezoidal cohesive model
and concluded that the cohesive traction and the energy play a far more important role than
the actual shape the curve follows in predicting the final fracture behavior. This model was
extended to include the tangential crack face displacement and traction [50]. This was
achieved by defining a non-dimensional crack separation term that combined both normal
and tangential separation. Further, the same authors also worked on determining the effects
of including crack tip plastic straining on the peak separation stress, assumed to arise from
either strain-induced void nucleation or accelerated void growth [11, 12]. Xu and
Needleman [31, 32, 51] used cohesive model to study dynamic interfacial crack growth in
brittle solids. They studied the behavior of crack propagation of a plane strain isotropic
hyper-elastic solid with an initial center crack subjected to impact tensile loading with
accommodation for branching. They found that crack branching emerges as a natural
outcome instead of predefining the crack path, and crack speed reaches 45% of the Rayliegh
wave speed. In the work that followed, they investigated crack growth in bi-material block
PMMA/Al both away from and along the bondline and found that the calculations carried
11
out provide a unified description of a variety of observed characteristic of crack growth in
brittle solids. They also introduced mixed mode traction separation relationship.
Camacho and Ortiz [16] investigated dynamic failure and fragmentation in brittle
materials. The study involved two test cases, spall and double cantilever beam, and the data
from field experiments was used to validate the law used to describe fracture. They found
that parameters such as impact velocity, mesh size, stress field, fracture patterns and energy
distribution affect crack growth.
Roy and Dodds [14] and coworkers compared the experimental study conducted on a
2.3mm thick aluminum2024-T3 alloy as a part of NASA-Langley Aging Aircraft Program
[21] to the 3D crack growth simulation with two thin panels( 12” and 24” wide) using finite
element (FE) code WARP3D with the help of cohesive zone modeling. They found that the
predicted crack growth response shows a rapid convergence with through-thickness mesh
refinement. He extended his work [1] to include the effects of large displacement. The large
displacement formulation helped describe the modeling of mode-I crack extension in thin
panels which experience significant out of plane displacement under remote tensile loads. It
was found that the interface of cohesive model accurately captures the behavior over the full
loading history. The same test was analyzed previously using CTOA methodology and was
found that the CTOA was able to predict peak loads measured in the various specimens but
it overestimates crack extensions early in the loading process.
Siegmund [34] used CZM to simulate crack growth in thin sheet metal with the help
of finite element code Abaqus. He made the first attempt to extend the CZM to shell
elements. In this work, crack propagation for M(T) specimen and a panel with multi-site
12
damage under mode-I/III conditions was studied in Aluminum panels. He also simulated
crack propagation in three panels of widths 3” 12” and 24”. Further, the far field tension and
the crack extension computations were used to determine the CTOA and it was found that
they were nearly identical for all the simulations. Brocks [43] compared the capabilities of
CZM with CTOD, and CTOA, in the structural integrity assessment of aluminum 5083 C(T)
sheets and found that CZM proved to be formidable in predicting mode-I crack propagation
as well as behaviors like mix mode fractures, crack kinking or branching (e.g., simultaneous
cracking of skin and stiffener) compared to the rest of the approaches, because unlike
cohesive models where the crack growth is controlled by two parameters and can predict
crack growth in 3D, CTOA is limited to 2D simulations and uses an experimentally
determined resistance curve to control crack growth.
Sun and Jin [16] focused on the development of CZM based on a physical
mechanism and the mathematical and physical restrictions on the functional form of
cohesive laws. Chen and Kolednik [36] investigated the inter-relations between the cohesive
zone parameters and crack tip triaxiality for a 10mm thick CT specimen made of a pressure
vessel steel 20MnMoNi55. It was found that stress triaxiality depends on the position of the
crack; and it increases dramatically during crack initiation but becomes stable during crack
growth. It was also found that the decrease of cohesive strength results in decrease in stress
triaxiality, while the decrease in separation energy results in increase in triaxiality.
Zavattieri [21] extended the cohesive interface model to handle crack for 3D shell
elements. In addition to a traction displacement law, a bending moment-rotation relationship
was included and the behavior of the crack was studied. A triangular law introduced by
13
Espinosa and Zavattieri [21] was used to predict the behavior of mode-I and mode-II for the
5” wide 0.23” thick steel specimen. He also simulated 3D effects of the ductile cracking in
thin walled specimens using a 3” wide and 0.1” thick Aluminum sheets. Predictions of
behavior were made using both shell elements and solid elements using a finite element
model in LS-Dyna [21] and compared to the experimental results. His analysis revealed that
the 2D nature of shell elements together with cohesive model is not appropriate to accurately
predict the 3D deformation state of crack front that evidently affects the overall behavior of
the material.
CZM’s are increasingly becoming popular in the finite element simulations of crack
propagation, adhesively bonded joints, interface cracks in bimaterials, delamination [1, 13,
14, 24, 25, 45] in composites and so on. The behavior of the cohesive elements is expressed
in terms of traction, separation and energy. These three terms can be expressed in various
forms such as Bilinear, Exponential, Triangular and Trapezoidal. Volokh [3] made an effort
to compare some cohesive zone models in an effort to show the specific shape of the
traction-separation curve can significantly affect results of the fracture analysis. He
compared four different cohesive zone models, bilinear, parabolic, sinusoidal, and
exponential considering a block-peel test between two rigid blocks. He concluded that the
CZMs with different shapes will always behave differently (crack initiation loads) in
fracture simulation. The different cohesive laws employed by past researchers and the
critical values of traction employed by them is summarized in table 1. As seen from the
table, the critical traction values reported by different investigators are not the same even for
the same material
14
Table 1 COMPARISON OF COHESIVE PARAMETERS USED PREVIOUSLY
Paper
Material
E(GPa)/
Cohesive law
and thick
nu
KIc
σy(MPa)
T
G
38.5
MPa√m
345
2σy
19
kJ /m^2
35
Al2024-T3
Exponential 71.4 /0.3
2.3mm
MPa√m
345
33
MPa√m
390
38.5
Roychowdhury Al2024-T3
Exponential 71.3 /0.3
[29]
2.3mm
MPa√m
345
Al2024-T3
46.6
Exponential 71.3 /0.3
2.3mm
MPa√m
345
P. Zavattieri
[48]
Siegmund [27]
Haynes [21]
Roy [24]
2.1
Al2024-T3
Linear
2.3mm
71.3 /0.3
Al2024-T3
Exponential 68.9/0.3
3.2mm
2σy
2σy
17
kJ /m^2
15
kJ /m^2
2.7σy
19
kJ /m^2
2σy
27
kJ/m2
Use of CZM to simulate behavior of Adhesive
Adhesively bonded and riveted joints are widely used in Aerospace, Automotive and
other industries. Riveted joints have been extensively studied and the lack of reliable failure
criteria for adhesive joints is limiting a more widespread application [20]. Accurate strength
predictions of crack propagation in adhesive bonded models are essential to decrease the
amount of expensive testing at the design stage.
For elastic adhesive joints in which failure process is predominantly brittle fracture
and can be predicted using LEFM. Hutchinson and Suo [52] have done substantial amount
15
of work in this area. However, adhesive joints deform plastically during fracture process,
complication due to not being able to accommodate plastic behavior prevented the
development of reliable strategies. Inclusions of plasticity increases the amount of fracture
energy and the total fracture energy increases until the R-curve (plastic zone) is fully
developed and the steady state fracture toughness is obtained. A number of groups [2, 3, 11,
12, 35, 49, 50, 53] successfully demonstrated the use of traction-separation relation to
predict coupling between plasticity and fracture process.
A plane strain mode-I crack growth in small scale yielding of an elastic plastic
material was studied with a trapezoidal cohesive law by Tvergaard and Hutchinson [49]
which shows that plastic dissipation becomes comparable to cohesive energy when plastic
zone fully develops and for a fixed cohesive energy Г and maximum traction T, strain
hardening is inversely proportional to KR i.e., in the absence of strain hardening or sudden
step strain hardening R-curve increases dramatically and thus steady state fracture toughness
is low compared to slow hardening which allows the plastic zone to reach a higher stress
thus increasing the toughness. This was also validated by Li and Chandra [35] increase in
strain hardening increases the steady state resistance.
As a continuation of the previous study, Tvergaard and Hutchinson [50] found the
influence of mode mixity on plasticity and found that steady state fracture toughness
(plasticity) increases with an increasing in proportion between mode 2 and mode 1.
This study [54] was continued further to find the toughness of ductile adhesive joints
between 2 elastic substrates. They showed that the plastic zone has an influence on the
toughness of the overall structure. Also the ratio of the substrate elastic modulus to that of
16
the adhesive layer has been shown to have a fairly significant effect such that the joints
having relatively stiff substrates have higher toughness.
Chowdury and Narasimhan [13] used trapezoidal cohesive law to investigate quasistatic crack growth in a pressure sensitive adhesive layer between two semi-circular
aluminum alloys substrates under plane strain condition and found that for a fixed level of
pressure sensitivity, the steady state fracture toughness increases more due to mode-II than
mode-I and also that steady state fracture toughness depends on the thickness of the
adhesive layer (saturates as the thickness increases). This was validated by Tvergaard (2001)
[33].
Li and Chandra [35] studied crack propagation in aluminum 2024-T3 CT specimen
with an exponential law and showed that for values lower than T < 1.5σ y plasticity is never
reached and the R-curve develops rapidly at much lower tractions resulting in lower
toughness, and a higher value of strain hardening results in higher toughness. They also used
a bilinear cohesive law to show that the effect of CZM on the R-curve; as the critical
separation of cohesive element increases the energy required for the crack initiation
increases and the transition segment of R-curve shortens and the size of plastic zone
decreases.
In 2003 Loh [55] simulated failure behavior of bonded joints using linear elastic
material properties with some success. In CZM approach the crack path is predefined and
the crack propagation is simulated using a traction separation law between the initially
coincident nodes on either side of the crack. The parameters that govern the cohesive law
17
are maximum traction T and the cohesive energy Γ. Many researchers have used a rather
rigorous approach to determine the CZM parameters.
Sorensen and Jacobsen [17] determined the cohesive laws in a double cantilever
beam specimen bonded with adhesive using J-integral approach. An extensometer was
mounted with it pins at the neutral axis of the beam and the end opening displacement was
recorded. By differentiating the J-integral with respect to the end opening displacement it
was possible to derive a meaningful cohesive law.
Andersson and Stigh [10] also employed a similar approach; elongation of the
adhesive was measured by illuminated lasers focused on indentation on each adherent and
using J-integral approach to find the cohesive parameters.
Others have compared numerical and experimental methods to predict the cohesive
parameters. Yang [56] presented a numerical study of mode-II elastic-plastic fracture in a
adhesively bonded end notched flexural specimen loaded in 3pt bending and found that the
local strain in crack tip region was atleast 40%. Therefore a value of stress corresponding to
the 40% strain value was chosen and cohesive energy is chosen to best fit the experimental
data.
Li and co [24, 25] found the cohesive parameters using mode-I fracture of adhesively
bonded polymer matrix composite by matching numerical to experimental observations. 2
cohesive parameters the cohesive energy Γ and the max traction T were obtained from the
plots of applied load vs. displacement for a double cantilever beam. A third parameter has to
be found for the intrinsic strength of the interface and becomes significant when the
characteristic dimension of the specimen would be small.
18
Liljedahl [5] investigated damage modeling of a mixed mode flexural joint
and single lap joint and found the cohesive parameters that best fit the experimental loaddisplacement and load-crack length plots and CZM parameters found from the mixed mode
flexural specimen predicted the failure well.
19
CHAPTER 3
3STATEMENT OF PROBLEM
3.1
Statement of Problem
Joints used for structural applications can be broadly classified into two types,
riveted and adhesive. One of the biggest worries in riveted joints is that the holes can cause
small cracks to initiate due to stress concentration, which in turn can cause failure of the
panel/structure by coalescence [14](Multiple Site Damage). On the other hand the presence
of voids is the main concern in adhesive bonding, which can lead to disbonding between
materials. Figure 4 shows the illustration of the above mentioned problems.
Figure 4 Common problems seen in riveted and adhesive joints
20
In this study an attempt is made to compare the crack propagation behavior and the residual
strength of a adhesively bonded stiffened panel to a unstiffened panel (M(T) specimen),
under tensile loading. Figure 5 and Figure 6 show the overall geometry of the stiffened and
unstiffened panel. Further the crack arrest capability of the stiffened panel in the presence of
a disbond is also simulated, in an effort to find the loss in strength due to voids in the
adhesive layer. Figure 7 shows an enlarged view of the disbond region.
The study provides a comparison of the residual strength between the unstiffened
panel and the stiffened panel with the help of a load (lbs) vs. crack extension (inches) plot.
Further the CMOD (Crack Mouth Opening Displacement) is studied in the stiffened and the
unstiffened panel to help us compare the effects of presence of a stiffener on the crack
propagation. The study of the radius of plastic zone size compares the amount of yielding
and the amount of stress transferred from the panel to the stiffener at various crack lengths.
21
Figure 5 Geometry of unstiffened panel
22
Figure 6 Geometry of a stiffened panel
23
Figure 7 Illustration of a disbond
24
3.2
Objectives
•
Evaluate the capability of CZM to predict the crack propagation in stiffened panels
with adhesively bonded stiffeners.
•
Study the residual strength and crack propagation behavior in an unstiffened panel
using a FE model. The effects of out-of-plane deformation constraints will also be
addressed.
•
Study the crack arrest capabilities of a stiffened panel using a FE model, compared to
unstiffened panel.
•
Predict the change in residual strength in a stiffened panel in the presence of a
disbond.
•
Compare the FE model results with experimental data.
25
CHAPTER 4
4COHESIVE ZONE FORMULATION
The cohesive law relating the traction between cohesive surfaces may be written in
terms of an energy potential as [31].
∂φ
T =− r
∂∆
(4.1)
An exponential form of the potential function reported by Xu and Needleman [32] given by
the following equation is chosen for the current study
 ∆ n   
 ∆ t2  
∆ n  1 − q   r − q  ∆n 
−
+
−
+
1
r
q
exp
 
 − 2 



δ n  r − 1   r − 1  δ n 
 δ n   
 δ t  
φ ( ∆ ) = φn + φn exp  −
(4.2)
In 3-dimensions, ∆n and ∆t , are the normal and tangential separations, Tn and Tt are the
corresponding tractions , δn and δ t represent critical normal and tangential separations
corresponding to the normal and tangential cohesive strengths σ max and τ max respectively. In
the above equation, q and r are given by q = φt φn and r = ∆ *n δ n where φn and φt are the
area under the normal and shear traction-displacement curve respectively. ∆ *n is the value of
∆ n after complete shear separation without resulting in normal traction. φn and φt can be
written as φn = eσ maxδ n , φt = e 2τ maxδ t , and represent the areas under the corresponding
traction-separation curves. The equation for normal and shear traction can be obtained from
equations (4.1) and (4.2) as [31]
26
Tn =
φn
e
δn
 ∆n 
−

 δn 
 ∆2 
 2



− t 
 − ∆t  
 δ2   

∆
 ∆ n  δt2   1 − q  
n 
 t 
+
r
−
 e


 1 − e

δn 
 r −1 

 δn




φ ∆
Tt = 2  n 2 t
 δt
(4.3)
 2
 ∆   ∆t 
   r − q  ∆ n   − δ nn   − δt2 
e
 q + 
 e
δ
r
−
1


n 

(4.4)
Figure 8 and Figure 9 represent the normalized traction-separation curves in the uncoupled
state. In these figures, the dimensionless normal and shear tractions are plotted against
dimensionless normal and shear separations respectively. Normal traction separation curve
shows the traction increases and causes the amount of separation to increase along with
traction until the maximum traction corresponding to the cohesive strength is reached, after
which the stiffness decays to zero. When the separation is given in the negative direction the
traction becomes negative. However, under compression, the traction increases (in
magnitude) unboundedly to prevent penetration of the cohesive surfaces. In the case of shear
traction-separation curve separation in the negative direction leads to shear stress in the
negative direction, thus resulting in anti-symmetric traction-separation curves as shown in
figure 9.
27
2
Tn/σmax
0
-2
-4
-6
-8
-2
0
2
4
6
∆n/δn
8
10
12
Figure 8 Normal traction Tn across the cohesive surface as a function of ∆n[13]
28
1.5
1
Tt τmax
0.5
0
-0.5
-1
-1.5
-3
-2
-1
0
∆t/δt
1
2
3
Figure 9 Shear traction Tt across the cohesive surface as a function of ∆t [13]
4.1
Cohesive Law
Following the work of Roy and Dodds [13], the energy potential may also be written
in a more computationally convenient form given by




φ = eσ cδ c 1 −  1 +
δ
δc

 δ 
 exp  −  

 δc 
(4.5)
Under pure normal loading condition the relation between the separation δ and traction t is
given by [14]
t=
 δ 
∂φ
δ
= eσ c exp  − 
∂δ
δc
 δc 
29
(4.6)
The work of separation per unit deformed area of the cohesive surface is given by[13]
∞
Γ c = ∫ tdδ
(4.7)
0
The cohesive energy Γc is the area enclosed by the traction separation curves and represents
the energy dissipated when the material separates or when new surfaces are formed[13].
Combining equations (4.6) and (4.7) we get the cohesive energy as [14]
Γc = eσ cδ c
(4.8)
Per Roy and Dodds [13], the cohesive energy is related to the fracture toughness KIC at the
onset of stable ductile tearing and is given by
Γc =
2
K IC
E
(4.9)
In the present study, the exponential traction-separation law was chosen to simulate
the crack growth in stiffened and unstiffened panels. The present study is aimed at analyzing
the effects of bonded stiffeners on the crack-arrest capability in the presence of disbonds.
30
CHAPTER 5
5FINITE ELEMENT MODEL
5.1
Model and Boundary Conditions
In this work cohesive interface elements are modeled to handle cracks in 3D solid
elements using a 1/4th symmetry model. Here we describe and apply a 3D computational
framework to simulate ductile tearing in thin aluminum panels using interface elements. The
fracture process occurs between layers which is governed by an interface tractions and
displacement separation to provide a phenomenological description for the complex features
of void nucleation, growth and coalescence to form new traction free crack faces as a result
of loss of cohesion, leads to progressive decay of otherwise tension and shear stresses
subjected surface.
The experiments on cracked panels provide a great insight in the behavior of the
cracked panel under load. 3D interface cohesive elements and traction separation model are
used to simulate mode-I crack growth and validate with the experimental results obtained
from tests carried out on thin sheet 2024-T3 aluminum alloy. A non-linear FE model was
assembled using the MSC Marc FEM code to study the behavior of crack propagation in an
unstiffened and a stiffened panel. The effects of the presence of an initial crack of 5” in a
panel 40” wide and 0.032” thick subjected to uniaxial tensile stress with and without the
stiffener were assessed by comparing the load-displacement response. The test uses
displacement control to propagate the crack from an initial saw cut notch, sharpened by
fatigue cycles. The values of remote stress and crack extension obtained from experimental
31
data were compared with numerical studies. The experiment shows the transition of an
initial flat crack to a slant (45°) crack in the 2024-T3 sheet. The crack thus extends into
mixed mode configuration locally while the bulk geometry maintains mode-I features.
Modeling the transition of crack from flat to slant is beyond the realm of this study. The
numerical models restrict crack fronts to a mode-I type crack propagation throughout.
Comparisons are mainly made for the amount of crack extension on the outer surface and
the overall capability to predict the behavior of load crack extension. The development of
FE model, their implementation, limitation and comparison with the experimental data are
presented in the following sections.
5.1.1
Modeling of plate and stiffener
Eight noded isoparametric brick elements are used to model the bulk material of
plate and stiffener. For an unstiffened panel, a 1/4th symmetry model was assembled for the
purpose of this study. The model and the associated boundary conditions are illustrated in
Figure 10 and the three different configurations for buckling constraint are shown in Figure
11.
32
Figure 10 Boundary condition of the unstiffened panel
33
Figure 11 Illustration of three different types of anti-buckling supports for unstiffened panel
The remote load is applied along the far edge, while the portion Y=0 and X=0 for
the uncracked ligament were constrained to impose symmetric boundary conditions to have
similar loading as the experimental setup as shown in Figure 10.
Three different buckling constraints illustrated in Figure 11 (one without any
constraint, another with partial constraint to simulate guide plates and finally, a completely
constrained panel) were applied on the plate and the behavior of the panel was studied and
compared.
34
Similarly for a stiffened panel, a 1/4th symmetry model was assembled with and
without a disbond and supported with two different out-of-plane displacement constraints as
shown in the Figure 12 and Figure 13.
Figure 12 Boundary condition of the stiffened panel
Remote displacement is applied on the far edge (Y= Ymax) along Y direction, while
Y=0 was constrained to simulate symmetry. A partial constraint was applied at X=0 to
simulate the presence of a crack Figure 12 also the out of plane displacement was
constrained with guide plates and the results were compared to a panel without any
constraints Figure 13.
35
Figure 13 Illustration of two different types support for buckling for stiffened panel
5.1.2
Modeling of adhesive zone
Eight noded isoparametric solid elements are used to model the adhesive layer
between the plate and the stiffener. The element has 8 integration points and 3 translational
degrees of freedom per node.
36
5.2
Material Properties
5.2.1
Material Properties of plate and stiffener [57]
The plate (bulk material) is made of a 2024-T3 bare aluminum alloy having a yield
strength of 50 Ksi, Young’s Modulus of 10.6 Msi and a Poisson’s ratio of 0.3. The behavior
of the bulk material is governed by a piece wise linear curve shown in Figure 14.
True stress (ksi)
80
60
40
Aluminum strain hardening
20
0
0.05
0.1
0.15
0.2
Plastic strain
Figure 14 Strain hardening for Aluminum[55]
5.2.2
Material Properties of adhesive zone [58]
The adhesive (FM 73 Cytec Industries) has a young’s modulus, E = 348.5 Ksi a yield
stress, σ y = 8.1 Ksi and a Poisson’s ratio of 0.3. The strain hardening behavior of the
37
adhesive is shown in Figure 15. At the onset of yielding, shear yield stress in pure shear is
1/√3 times the tensile yield stress in the case of simple tension(von-Mises yield criterion)
11
True stress (Ksi)
10
9
8
Adhesive strain hardening
7
6
0
0.2
0.4
0.6
0.8
1
Effective Plastic strain
Figure 15 Strain hardening for Adhesive
5.3
Modeling of cohesive zone
The cohesive zone was modeled using 3-dimensional, 8 noded interface elements.
The element has four integration points and each node has 3 translational degrees of
freedom. The behavior of these elements is expressed in terms of traction versus relative
displacement between top and bottom surfaces. The relative displacements include one
normal and two shear components. The element is allowed to be infinitely thin in which case
the top and the bottom surfaces coincide, this property enables this element to be employed
to model interface between two elements or the uncracked portion in a material.
38
The interface element behavior is defined by two key independent parameters:
cohesive energy Γc (area under the curve), effective traction t, and the effective separation v.
For the purpose of this study, we consider an exponential variation of the cohesive law as
shown in Figure 16 in which the effective traction is introduced as a function of Gc and an
effective opening displacement which is characterized by an initial reversible response
followed by an irreversible response when a critical opening displacement vc is reached
given by the formula [59]
−v
v
t = Gc 2 e vc
vc
(5.1)
Figure 16 Traction separation law used in the study
The calibration study conducted by Roychowdhury and Dodds [1] to match the
measured load vs. crack extension on the outside surface provided an estimate of Gc of
108.59 lb-in/in2 and due to the fact that the radius of the plastic zone are relatively small
39
0.2” we neglect any elevation of Gc over Γc and use Γc = 108.59 lb-in/in2. Subsequent study
by varying the peak stress performed to match the overall curve and the peak stress of the
experimental data gave a value of t = 2σy where yield stress σy = 50 Ksi. As previously
mentioned, for the purpose of consistency in the calibration process and comparison with
experimental crack growth measurement the effective crack is defined where the cohesive
stress decreases to 0.05t as shown in Figure 16. Siegmund and Brocks [34, 43] have also
reported a calibrated value of t = 2σy for the crack growth simulation of MSD (multiple site
damage)in 0.394” thick, 2024-T3 panels using 2D model with plane stress condition.
5.3.1
Cohesive parameters for the stiffener region
Considering the fact that failure in stiffener region is based on net section yielding
and not ductile crack propagation, the stress at the failure region is much lesser compared to
critical stress as crack tip during propagation. Thus the use of same cohesive parameters as
the plates will over predict the strength of the stiffener. The stiffener behaves more like a
tension (dog bone) specimen. Accordingly, when the stress at the fracture region reaches
ultimate stress σu the specimen has to fracture, thus T = σu = 1.65σy was selected for this
study. Adopting a value of Gc = 108.59 lb-in/in2 , the specimen follows the traction
separation curve more closely.
5.3.2
Cohesive parameters for the adhesive region
It was found that for adhesive failures (with the crack running at an interface) for
mode-I failures, a peak traction of T = 8.7 Ksi and a cohesive energy of 5.7 lb-in/in2
provided the best fit to experimental data and will be discussed in a later section.
40
5.4
Mesh
The 3D FEM models must have adequate mesh refinement to successfully resolve
interactions between the boundary conditions and the loads applied. Earlier studies [7-9] of
crack growth have demonstrated the presence of high triaxial stress even in thin panels
which leads to crack front tunneling from an initial straight fatigue crack front, consistent
with the experimental studies[7]. Studies conducted by Roy and Dodds [14] on a 5.9 inches
wide C(T) specimen of thickness 0.1”, with 2 to 4 elements across half thickness with
various combinations of interface mesh refinement and interface element sizes Le,
1.
Large Le values fail to resolve the stress-strain fields in the bulk material subjected to
plastic deformation.
2.
Large (Le / vc) ratios fail to resolve the decohesion response.
They also found that an element length of 0.01” and through thickness mesh refinement 4
layers gave good comparison with experimental data.
41
Figure 17 1/8th symmetry FE model of unstiffened panel, with close of through thickness
mesh refinement
The current unstiffened model has an element length of 0.01” for 7 layers
immediately above the cohesive zone with 2 layers of uniformly sized elements over half
thickness as shown in Figure 17. This provides satisfactory resolution to capture load versus
crack extension behavior and crack front tunneling. The ratio Le/vc was set to 0.3 following
the work of Roychowdhury [1]. A typical model for crack growth simulation contains 57900
volumetric and 3550 interface elements.
42
Figure 18 1/4th symmetry FE model of stiffened panel, with close up of through thickness
mesh refinement
The stiffened model has an element length of 0.008” for 7 layers immediately above
the cohesive zone with four layers of uniformly sized elements over the plate thickness, and
two layers over the stiffener thickness as shown in Figure 18 a typical model for crack
growth simulation contains 119784 volumetric and 10424 interface elements.
43
5.5
Solution Procedure
The computations and the model reported here were generated using non-linear finite
element code MSC Marc [59]. Eight noded interface elements governed by traction
separation law, surrounded by conventional eight node solid elements which constitute the
bulk material were used to study crack propagation in unstiffened and stiffened panels. Full
Newton-Raphson iteration [60] is used to solve the non-linear equations for each increment
of applied displacement. Large displacement increments generally lead to satisfactory
convergence, however the response of the decohesion behavior accumulates significant
errors[13]. The interface elements may pass from pre- to post-peak without enforcing the
peak stress level in the traction separation law. To overcome this, small displacement
increments are used along with adaptive load control in MSC Marc. The adaptive load
control following each step, locates the interface element which undergoes the largest
change in effective displacement (opening) and finds the residual forces. The code
continually adjusts the sizes of subsequent displacement increments to maintain an absolute
normalized value ∆/vc and relative residual force tolerance || Fresidual || / || Freaction || < TOL is
within limit. Earlier studies [14] have shown that values of ∆/vc upto 0.3 and relative force
tolerance (TOL) of 0.1 show close prediction of load versus crack extension curves when
compared to experimental data.
The adopted form of cohesive model tends exponentially towards zero with
increasing effective opening displacement. To provide a consistent definition on the position
of the advancing crack front, it was assumed to be at the point where the opening stress is
decreased to 0.05tmax. Once the interface element reaches this level of deformation, the
44
element extinction criterion removes the element, causing a free surface (crack) over the
next few increments. This occurs simultaneously over several interface elements ahead of
the crack.
5.6
Data reduction Procedure
Tensile loading is simulated on the panel with a displacement control loading, which
provides for crack propagation. The correlation between the amount of crack propagation
and the far field stress or between CMOD and crack extension is important to present the
results and in order to understand the behavior of the panel under different conditions. Thus
the data reduction procedure becomes crucial. The details of the analysis of the data from
simulations are described in the following paragraphs.
5.6.1
Crack Extension
The value of crack extension can be obtained by finding the relative displacement of
the cohesive nodes in the crack plane, i.e., the amount of separation between the top and the
bottom nodes of the cohesive elements. Figure 19 shows the real and fictitious crack tips.
The fictitious portion of the crack is representative of the process zone. To achieve
consistency with respect to the measurement of the position of the crack front, the real crack
tip is located at a point where the traction between the top and the bottom surface decreases
to 0.05t and the corresponding X co-ordinate value is taken as real crack tip. A Matlab
program was written1 to calculate the relative displacement of the crack plane (obtained
from the path plot data in MSC Marc). The corresponding X value or the crack tip value
where the traction reduces to 0.05t generated by the program.
1
In collaboration with Farzad Ghods (gradute student)
45
Figure 19 Illustration of a crack
5.6.2
CMOD
CMOD is a measure of the amount of crack mouth opening displacement. Increase in
CMOD is directly related to the crack length and thus could be used as a measure of the
same. It is measured 0.2” (to get a realistic value compared to the test data) away from the
crack plane along the center line of the panel, where the crack opening is the maximum as
shown in Figure 20. This distance was chosen to facilitate comparisons with experiments
which used the same dimensions.
46
Figure 20 CMOD measurements
5.6.3
Radius of Plastic Zone
Stresses near the crack tip are significantly high, exceeding the yield strength of the
material and thereby creating a plastic zone as shown in Figure 21. However the material
around the plastic zone is elastic and tends to get back to its original position causing
retardation in crack growth. One of the early explanations of the plastic zone size was given
by G.R.Irwin[18]. He theorized that this region extends by a distance
1
r=
2π
 KI 


 σ ys 
2
ahead of the crack tip and termed it as radius of plastic zone size ‘r’ shown in Figure 23.
47
(4.10)
Figure 21 Illustration of forces around the crack tip [18]
Figure 22 Plane stress and plane strain plastic zone shape for mode-I cracking [18].
The shape of the plastic zone is not circular. Figure 22 shows a typical shape of
plastic zone size for mode-I crack for plane stress and plane strain conditions. The shape and
48
size of the plastic zones in the stiffened and unstiffened panels were studied using the FE
models. To identify the shape and size of the plastic region, a binary contour plot of the
effective plastic strain was used. Figure 23 shows the growth of plastic zone size for the
unstiffened and the stiffened panels at different crack extension.
Figure 23 Shape of plastic zone at different crack lengths in unstiffened and stiffened panels
5.6.4
Height of Plastic Zone in the Nearest Stiffener
This is a measure of the extent of plastic deformation the stiffener undergoes for a
given crack extension. As the crack extends towards the stiffener, a bigger fraction of the
49
load gets transferred from the panel to the stiffener, thus facilitating crack arrest. Due to the
increase in stiffener load, yielding occurs over a certain length of the stiffener as shown in
Figure 24 (depiction of yielding in the skin is suppressed to provide better visualization).
Figure 24 Height of plastic zone measurement in the nearest stiffener
50
CHAPTER 6
6VERIFICATION AND VALIDATION
6.1
Verification of CZM with 3 Element Model [14]
The stiffness in the cohesive element is provided in only one direction and thus the
orientation is vital for correct analysis. When the load is applied both the surfaces begin to
separate following the constitutive laws defined. To verify that the cohesive elements follow
the defined cohesive law, a simple three element model is generated with one cohesive
element sandwiched between two volumetric elements of unit volume as shown in Figure
25.
Boundary conditions on the model are applied as shown in the Figure 25. Bottom
surface in constrained in all directions and a displacement in applied on the top surface to
simulate tension loading. Relative displacement between the top and bottom surfaces is
measured at every step and the relation between the traction and separation is derived and
compared with the cohesive law applied as shown in Figure 26. Further, the load carrying
capacity of the three element model with and without the cohesive element were compared
to study the change in behavior. Figure 27 shows the plot for stress in Y-dir (σyy). While
both the models follow the strain hardening criteria employed, only the model with the
cohesive element is able to simulate the load drop associated with necking and the fracture
behavior effectively, whereas the stress keeps increasing beyond the ultimate strength value
of the material in the absence of the cohesive elements.
51
Figure 25 Three element model with boundary conditions applied
52
80000
Traction (psi)
60000
Cohesive law
40000
20000
0
0
0.001
0.002
0.003
0.004
Separation (inches)
Figure 26 Cohesive law followed by the element
80000
Stress in y dir (psi)
60000
40000
with CZM
without CZM
20000
0
0
0.1
0.2
0.3
0.4
0.5
Strain
Figure 27 Stress comparison of three element verification model with and without CZM
53
6.2
Validation
The validation part describes the use of interface-cohesive elements to simulate crack
propagation in M(T) specimen. The calibrated values of cohesive parameters are employed
to validate the crack growth behavior in 3”, 12” and 24” wide 2024-T3 M(T) specimens
(thickness of 0.1” and a/W ratio of 0.3) with the experimental and numerical results reported
by Zavattieri [21] and Roychowdhury and Dodds [1] to validate the modeling approach used
in the current study. They conducted an experiment of 2.3mm thick aluminum panel with
unconstrained out of plane displacement. Zavattieri analyzed the specimens using dynamic
tension test on both top and bottom boundaries with unconstrained out of plane displacement
using shell elements. All the properties for both the aluminum and the interface elements
were taken as mentioned in sections 5.2.1 and 5.2.2, and the residual strength and the crack
extension were compared for a maximum crack extension of about 0.8” as shown in the
following figures.
Figure 28 shows the comparison of load vs. change in crack extension for the 3”
wide panel to the results presented in .The numerical results presented by the current
cohesive model matches the trend observed in the experimental data reported by
Roychowdhury [1]. The present model tends to predict higher loads when compared to the
predictions of Roy and Dodds [14].
54
12000
10000
Load (lbs)
8000
6000
4000
Dodds experiment data
Dodds analysis data
2000
Current CZM analysis
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Change in physical crack length ap (inches)
0.9
1
Figure 28 Comparison of the predicted values for a 3” wide M(T) specimen with the values
reported [21]
40000
35000
Load (lbs)
30000
25000
20000
15000
Dodds experiment data
10000
Dodds analysis data
5000
Current CZM validation
0
0
0.2
0.4
0.6
0.8
1
Change in physical crack length ap (inches)
Figure 29 Comparison of the predicted values for a 12" wide M(T) specimen with the values
reported [1]
55
Figure 29 provides the comparison of predicted load versus change in physical crack
(δ) length to the experimental results and also the numerical result
results reported in [1]. The
present CZM numerical predictions overestimate the load carried by about 2% after the
crack reachess 0.6” and the rest of the points match closely to the reported values
values.
70000
60000
Load (lbs)
50000
40000
30000
Dodds experiment data
20000
Dodds analysis data
10000
Current CZM validation
0
0
0.2
0.4
0.6
0.8
1
Change in physical crack length ap (inches)
Figure 30 Comparison of the predicted values for a 24"" M(T) specimen with the values
reported [1]
The numerical results for a 24” wide panel is compared to the numerical and
experimental presented in [1] is shown in Figure 30.. The computed response using CZM
shows an excellent match till the crack extends to about 0.6” after which the response
underestimates
derestimates the predicted
predicte load.
The current model predicts well for the 3” wide specimen. However under predicts
(during initial stage) the loads for wider specimens.
56
CHAPTER 7
7RESULTS AND DISCUSSION
This study employs a 3D finite element model along with traction-separation law to
simulate crack propagation and predict residual strength. It also attempts to investigate the
crack arrest capabilities of the stiffened panel in the presence of a disbond and compares the
results with the residual strength predictions for the unstiffened panel. Furthermore the
numerical results are compared and validated against the experimental results conducted
[27].
The experimental work was performed a part of NIS (NIAR - Industry – State
research program) [27]. This report includes test conducted on M(T) unstiffened and
stiffened specimen.
The results and discussion are focused on the following:
1.
Effects of applying different types of constraints on unstiffened and stiffened panel
2.
Residual strength in stiffened panel with and without a disbond
3.
Difference in strength between the unstiffened and stiffened panels
4.
Shear stress distribution in the adhesive region for stiffened panel with different
constraints
57
7.1
Effect on Residual Strength with different types of constraint
7.1.1
Unstiffened panel
The numerical predictions for the unstiffened panels are compared to the experimental
results. Experiments were conducted on a thin, wide center crack panel with the out of plane
displacement being constrained by guide plates. The experiments were conducted with two
different configurations of guide plate constraint. That being type A and Type B, both
supports consisted of 12” deep web and 4” wide flange with thickness of the flange being
0.25” and that of the web at 0.4” and they were clamped at the ends with fasteners. The only
difference between type A and type B support was that the web in the type B support was
reinforced with a 0.5” thick flange. Both the supports had a 1” window to measure the crack
and it was 13” in length in type A and it was increased to 21” in type B support [27].
Numerical simulations were conducted on an unstiffened panel with 3 different
configurations
1.
Out of plane displacement was allowed
2.
Out of plane displacement was completely constrained
3.
Out of plane displacement was partially constrained (guide plates)
A displacement control loading was applied to simulate crack extension. Numerical
predictions were compared to the experimental values and the results are given below.
Figure 31 and Figure 32 show the comparison of load with crack extension and
CMOD for the unstiffened panel M(T) specimen with experimental results. For the case
with no buckling constraint, the predicted value of residual strength is lower than the
measured value. This is consistent with the fact that during crack extension the plate
58
undergoes buckling thus the state of stress is higher compared to the panel having
constrained out of plane displacement. Furthermore the panel with completely constrained
out-of-plane displacement experiences a lower state of stress and has a higher residual
strength. Figure 31 and Figure 32 also show that the residual strength prediction for the
partially constrained panel is in good agreement with the measured value.
60000
50000
Load (lbf)
40000
30000
Experiment, type A support
20000
Experiment, type B support
Analysis, out of plane displacement allowed
10000
Analysis, out of plane displacement constraint
Analysis, guide plates constraint
0
4
5
6
7
8
9
10
11
12
13
Physical crack length 2*ap (inches)
Figure 31 Load vs. crack extension response for the unstiffened M(T) panel
59
14
60000
50000
Load (lbf)
40000
30000
Experiment, type A support
20000
Experiment, type B support
Analysis, out of plane displacement allowed
10000
Analysis, out of plane displacement constraint
Analysis, guide plates constraint
0
0
0.05
0.1
0.15
0.2
CMOD (inches)
Figure 32 Load vs. CMOD comparison for the unstiffened M(T) panel
60
0.25
The CMOD vs. physical crack length plot in Figure 33 also shows that the numerical
predictions using CZM closely match the experimental values.
0.25
CMOD (inches)
0.2
0.15
Experiment, type A support
0.1
Experiment, type B support
Analysis, out of plane displacement allowed
0.05
Analysis, out of plane displacement constraint
Analysis, guide plate constraint
0
4
6
8
10
12
14
16
18
Physical crack length 2*ap (inches)
Figure 33 CMOD vs. physical crack length prediction
Figure 34 shows the plot of crack growth resistance curve against effective crack
length. The case with no buckling constraint underestimates the KR curve, while the panel
with completely constrained out of plane displacement overestimates it. For the case with
guide plate constraints we get closely matching results.
The experiment predictions obtained from the CMOD gauge are not entirely reliable,
due to the fact that the plate undergoes localized buckling exactly where the CMOD gauge is
fixed. This is clearly seen in Figure 35 there is a large variation in the CMOD prediction
obtained from the experimental results of plate with type A and type B support.
61
300000
250000
KR (psi√in)
200000
150000
Experiment, type A support (microscope reading)
100000
Experiment, type B support (microscope reading)
Analysis, out of plane displacement allowed
50000
Analysis, out of plane displacement constraint
Analysis, guide plates constraint
0
0
1
2
3
4
5
6
7
8
9
10
Effective crack length ∆ae (inches)
Figure 34 KR vs. effective crack length comparison (microscope reading)
300000
250000
KR (psi√in)
200000
150000
Experiment, type A support (CMOD gauge)
Experiment, type B support (CMOD gauge)
Analysis, out of plane displacement allowed
Analysis, out of plane displacement constraint
Analysis, guide plates constraint
100000
50000
0
0
2
4
6
8
Effective crack length ∆ae (inches)
10
Figure 35 KR vs. effective crack length comparison (CMOD reading)
62
12
Crack resistance KR
Stress Intensity Factor, K (psi√in)
250000
200000
150000
KR curve
100000
P1 = 17000 lbs
P2 = 25000 lbs
50000
P3 = 35000 lbs
P4 = 48800 lbs
0
0
2
4
6
8
10
12
Effective Crack Length ae (inches)
Figure 36 KR and KG curves
Figure 36 shows the crack growth resistance (KR) curve for the unstiffened panel.
Also plotted are a series of K-curves with constant load called the KG curves. KR is related to
crack resistance and KG to an energy term that drives the crack. For stable crack extension
KG = KR and dKG da < dK R da , and for unstable crack growth KG = KR and
dKG da > dK R da . The value of K corresponding to the point of tangency between the KR
and the KG curve is the critical fracture toughness; from the plot we get 48800lbs as the
critical load which also the peak load obtained from the load vs. crack length curve and
beyond which the crack becomes unstable.
63
7.1.2
Stiffened Panel
Experiment was conducted on wide stiffened panel with a center crack and the out of
plane displacement constrained with 2 pairs of C sections 10” apart [27]. The problem
encountered during experiment (C-sections being inadequate to resist the buckling of the
plate) was addressed in the numerical simulation and the results are presented below.
Numerical simulation were conducted on stiffened panel having 2 different configuration
1.
Out of plane displacement allowed
2.
Out of plane displacement constrained partially
64
Figure 37 shows the residual load plot. In the case of unconstrained specimen the
predicted value closely follows the experimental this is consistent with the problem
encountered during the experiment related to buckling as a result in subjected to a higher
state of stress and thus having a much lower residual strength. While proper application of
guide plate constraint will yield a result with higher residual strength as in the case of the
plate with guide plate constraint shown in Figure 37.
80000
Load (lbs)
60000
40000
Analysis, out of plane displacement allowed
20000
Analysis, guide plate constraint
Experimental
0
-0.5
0.5
1.5
2.5
Change in physical crack length ap (inches)
Figure 37 Load vs. Physical crack length response for stiffened M(T) specimen
65
Figure 38 also shows a similar behavior, the numerical prediction for the stiffened
panel with no constraint closely matches the measured values whereas the numerical
predictions with the guide plate constraint are much higher.
80000
Load (lbs)
60000
40000
Analysis, out of plane displacement allowed
20000
Analysis, guide plate constraint
Experimental
0
0
0.1
0.2
0.3
CMOD (inches)
Figure 38 Load vs. CMOD comparison for the stiffened M(T) specimen
66
0.4
Figure 39 provides a comparison of numerical and experimental results for CMOD.
The experimental predictions are far lower compared to the numerical owing to the fact that
the readings obtained from the CMOD gauge are not accurate due to the presence of
localized buckling in the crack region.
0.4
Analysis, out of plane displacement allowed
Analysis, guide plate constraint
Experimental
CMOD (inches)
0.3
0.2
0.1
0
1
2
3
4
5
6
Physical crack length ap (inches)
Figure 39 CMOD vs. physical crack length comparison
67
7
8
Figure 40 shows a comparison between the plastic zone size ahead of the crack vs.
the length of the effective crack extension. The increase in the radius of plastic zone size in
the plate with guide plate constraint can be attributed to the difference in residual strength of
each plate as radius of plastic zone is proportional to remote load applied. As thee crack tip
comes close to the stiffeners the load gets transferred and as the stiffeners extends the skin
tends to move with it which in turn helps the radius of plastic zone to spread in the plate
more in stiffened panels than unstiffened panels. The radius of plastic zone starts decreasing
after a certain point, because of the finite wide of the panel. At a crack extension of about 4”
the plastic zone ahead of the crack would have reached the outer edge of the plate, after
which any increase in the crack length will reduce the plastic zone size.
16
Analysis, out of plane displacement allowed
Radius plastic zone r (inches)
Analysis, guide plate constraint
12
8
4
0
0
5
10
15
Effective crack length ae (inches)
Figure 40 Radius of plastic zone size vs. effective crack length
68
20
Figure 41 shows the amount by which the inner stiffener yields as crack extends. The
panel constrained with guide plate can carry more load thus the amount of yielding in the
stiffener attached to panel with guide plate constraint yields more than the other.
7
Height of plastic zone size (inches)
6
5
4
3
2
Analysis, out of plane displacement allowed
Analysis, guide plate constraint
1
0
2
3
4
5
Physical crack length ap (inches)
Figure 41 Height of plastic yielding in the inner stiffener vs. physical crack length
69
6
7.2
Comparison of the residual strengths of stiffened panel with and without a
disbond
One of the main disadvantages of using adhesives to bond the stiffener to the plate is,
having to deal with inclusion of voids, which can cause disbond. The presence of disbonds
in the adhesive layer will affect the effective transfer of load from the plate to the stiffener.
This section will compare the loss in residual strength of the stiffened panel in the presence
of a disbond. Three different configurations of stiffened panel with guide plates have been
studied and compared along with the experimental results
1.
Stiffened panel with no disbond
2.
Stiffened panel with 0.5” disbond
3.
Stiffened panel with 1” disbond
Figure 42 shows the residual strength predictions in a stiffened panel constrained
using guide plates, and the effect of presence of a disbond on the load carrying capacity of
the panel. The load carrying behavior of the panel with 0.5” disbond is similar to that with
no disbond, but the plate experiences a 9% loss in strength with a disbond of size 1”. The
change in amount of the total crack extension is due to the fact that the presence of a
disbond allows for localized yielding the material immediately next to the crack plane,
which in turn allows for crack propagation. The fact that the out of plane displacement of the
stiffened panel was not effectively constrained during the experiments prevents active
comparison with experimental values.
70
80000
Load (lbs)
60000
40000
Analysis, no disbond
Analysis, 0.5" disbond
20000
Analysis, 1" disbond
Experimental, no disbond
0
-1
0
1
2
3
4
5
Change in physical crack length ap (inches)
Figure 42 Residual strength prediction in stiffened panel out of plane displacement
constrained using guide plates with and without a disbond
Figure 43 shows a comparison of load vs. CMOD for the stiffened panel with guide
plate constraint. The change in the size of the disbond does not affect the crack mouth
opening displacement at any other point except towards the end. This can be attributed to the
presence of disbond as explained earlier.
71
80000
Load (lbs)
60000
40000
Analysis, no disbond
Analysis, 0.5" disbond
20000
Analysis, 1" disbond
Experimental, no disbond
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
CMOD (inches)
Figure 43 Load vs. CMOD comparison of the stiffened panel constrained with guide plates
Figure 44 and Figure 45 compares the change in radius of plastic zone size and the
height of plastic yielding as the crack propagates. The variation is very similar in all the
cases, thus we can say that radius of plastic zone in the skin and height of plastic yielding in
stiffener is not affected with the existence of disbonds.
72
Radius of plastic zone r (inches)
16
12
8
Analysis, no disbond
4
Analysis, 0.5" disbond
Analysis, 1" disbond
0
0
5
10
15
20
25
Effective crack length ae (inches)
Figure 44 Radius of plastic zone at different effective crack lengths
Height of plastic yielding (inches)
8
6
4
Analysis, no disbond
2
Analysis, 0.5" disbond
Analysis, 1" disbond
0
2
3
4
5
6
7
Physical crack length ap(inches)
Figure 45 Change in the amount of yielding in the inner stiffener with crack extension
73
8
7.3
Effect of presence of stiffeners on the residual strength of a panel
The increase in residual strength of a stiffened panel over unstiffened panel is
achieved by adding stiffeners to the panel. As the crack tip comes closer to the stiffener the
stiffener takes up most of the load, the load is transferred from the plate to the stiffener by
the adhesive. Figure 46 shows a gain in load carrying capacity between an unstiffened and a
stiffened panel of more than 55%. In the presence of a disbond of 1” also the load carrying
capacity is much higher to that of unstiffened panel.
80000
Load (lbs)
60000
40000
20000
Analysis, unstiffened panel
Analysis, stiffened panel (no disbond)
Analysis, stiffened panel (1" disbond)
0
-1
0
1
2
3
4
5
Change in physical crack length ∆ap (inches)
Figure 46 Difference in residual strength between unstiffened and stiffened panel
74
6
80000
Load (lbs)
60000
40000
Analysis, unstiffened panel
20000
Analysis, stiffened panel (no disbond)
Analysis, stiffened panel (1" disbond)
0
0
0.1
0.2
0.3
CMOD (inches)
Figure 47 Load vs. CMOD comparison
Figure 47 and Figure 48 show the amount of crack mouth opening displacement at
different remote load and crack lengths respectively. From the above figure it is clear that
for loads up to 40000 lbs the crack mouth opening is double in the unstiffened panel
compared to the stiffened at the same loads, after which CMOD value increases drastically
until the unstiffened panel can no longer support any more load. However the stiffened panel
is capable of supporting much heavier loads. Figure 48 shows that for the same amount of
crack extension in the stiffened panel the crack mouth opening is twice as that of a
unstiffened panel, thus stiffened panel has a higher residual strength than that of an
unstiffened panel.
75
0.4
0.5
Analysis, unstiffened panel
Analysis, stiffened panel (no disbond)
CMOD (inches)
0.4
Analysis, stiffened panel (1" disbond)
0.3
0.2
0.1
0
2
3
4
5
6
7
8
9
Physical crack length ap (inches)
Figure 48 COMD vs. physical crack length comparison
Radius of plastic zone size r (inches)
16
Analysis, unstiffened panel
Analysis, stiffened panel (no disbond)
12
Analysis, stiffened panel (1" disbond)
8
4
0
0
5
10
15
Effective crack length ae (inches)
Figure 49 Radius of plastic zone as the crack propagates
76
20
Figure 49 shows the relation of the radius of plastic zone to the effective crack length. Due
to the fact that the stiffeners in the stiffened panel aide crack arrest more than the unstiffened
panel by itself, the size of the plastic zone is larger in stiffened panel than in unstiffened
panel. All these results show that the stiffened panel has much larger residual strength than
the unstiffened panel.
7.4
Shear stress in the adhesive region
As the crack length increases, the plate experiences a larger displacement in the Y-
direction as compared to the stiffener. As a result of this the adhesive is subjected to shear
stress. Graphs below show the distribution of shear stress in stiffened panel with different
constraints. We can see in Figure 50, Figure 51 and Figure 52 that the presence of a disbond
does not cause any change in shear stress distribution in the stiffened panel with guide plate
constraint for different crack lengths. Whereas the shear stress in the panel with completely
constrained Z displacement and the panel with no Z displacement constraint is lower than
the panel with guide plate constraint for all the crack lengths.
77
Length of the adhesive region (in)
25
20
completely constrained Z displacement
15
Z displacement allowed
guide plate constrained
10
guide plate with 0.5" disbond
guide plate with 1" disbond
5
0
-200
-100
0
100
Shear stress σ23 (psi)
200
300
400
Figure 50 Shear stress in adhesive region at a crack length of 0.001"
Length of the adhesive region (in)
25
20
15
completely constrained Z displacement
Z displacement allowed
10
guide plate constrained
guide plate with 0.5" disbond
5
guide plate with 1" disbond
0
-1500
-1000
-500
0
Shear stress σ23 (psi)
500
1000
Figure 51 Shear stress in adhesive region at a crack length of 0.1"
78
1500
25
Length of the adhesive region (in)
completely constrained Z displacement
20
Z displacement allowed
guide plate constrained
15
guide plate with 0.5" disbond
guide plate with 1" disbond
10
5
0
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
Shear stress σ23 (psi)
Figure 52 Shear stress in adhesive region at a crack length of 1.1"
Furthermore the shear stress in the adhesive in the region away from the crack tip
decays to zero as the length of the panel is increased as shown in Figure 53
79
Length of the adhesive region (in)
60
δa = 0.001" sp no z displacement
δa = 0.001" sp guide plate const
50
δa = 0.1" sp no z displacement
δa = 0.1" sp guide plate const
40
δa = 1.1" sp no z displacement
δa = 1.1" sp guide plate const
30
20
10
0
-4000
-3000
-2000
-1000
0
1000
2000
Shear stress σ23 (psi)
Figure 53 Shear stress in the adhesive region in plate with large lengths
80
3000
CHAPTER 8
8CONCLUSION AND FUTURE WORK
8.1
Conclusions
This study describes the use three dimensional interface-cohesive elements for the
FE analysis of crack propagation in thin stiffened panel, with the use of cohesive element
formulation and compared to the experimental results. Based on this study, the following
conclusions can be drawn
1.
The calibrated cohesive parameters were adequate in accurately predicting the
residual loads and the crack propagation behavior in the unstiffened and the
stiffened panels.
2.
The residual strength is much higher in panels constrained with guide plates
compared to panels without constraint.
3.
A 55% increase in residual strength is achieved with the stiffened panels over the
unstiffened panels.
4.
Inclusion of stiffeners provides a mechanism for crack arrest when compared to
unstiffened panels.
5.
The radius of plastic zone is much larger in stiffened panels compared to
unstiffened panels.
6.
The presence of disbond of 1” causes a 9% loss in residual strength compared to a
plate without disbond. A disbond of size less than 1” in size does not cause any
loss in strength.
81
7.
The crack extends farther in a stiffened panel with disbond before catastrophic
failure of the panel, compared to the one without due to free localized yielding near
the crack tip.
8.2
Future Work
Based on the observations from the current study, following issues may be addressed
in the future.
1.
Sensitivity of the simulations to the choice of traction-separation law must be
studied.
2.
Cohesive elements have to be employed to obtain a better visualization of the
stresses in the adhesive region.
3.
Mode-II and Mode-III parameters have to be used to simulate real life conditions
as the adhesive region experiences these types of stresses.
4.
Yield stress and cohesive parameters for the adhesive were derived from the shear
stress and shear strain curve, shear stress and shear strain curve may not be
appropriate for arriving at values of yield stress and cohesive parameters.
5.
The out-of-plane constraints used in the simulations represent an extreme case of
rigid constraints.
Explicit modeling of the anti-buckling supports must be
donealong with appropriate contact conditions between the panel and the supports.
82
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83
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[2]
A. Needleman, "An analysis of tensile decohesion along an interface", Journal of the
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A. Needleman, "A Continuum Model for Void Nucleation by Inclusion Debonding",
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J. C. Newman, M. A. James and U. Zerbst, "A review of the CTOA/CTOD fracture
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V. Tvergaard and J. W. Hutchinson, "Effect of Strain Dependent Cohesive Zone
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88
APPENDIX
89
9MARC ELEMENTS
9.1
MARC element # 7: Three-dimensional Arbitrarily Distorted Brick
This is a three dimensional eight node isoparametric arbitrary hexahedral element
shown in Figure 54. The element has 8 nodes and each node has 3 translational degrees of
freedom u, v and w. The stiffness of this element is formed using eight-point Gaussian
integration.
Figure 54 MARC element # 7 [59]
90
9.2
Marc element # 188: Eight-node Three-dimensional Interface Element
This is an eight-node three-dimensional interface element, which can be used to
simulate the onset and progress of delamination shown in Figure 55. The stress components
of the element are one normal and two shear. The element stiffness matrix is integrated
numerically using a four-point integration scheme. The deformations are the relative
displacements between the top and the bottom face of the element which allows the element
to be infinitely thin.
Figure 55 MARC element # 188 [59]
91
10STIFFENED FILE WITH GUIDE PLATE CONSTRAINT INPUT FILE
title
job1
$....MARC input file produced by Marc Mentat 2008 r1 (64bit)
$...................................
$....input file using extended precision
extended
$...................................
sizing
0
140208
178232
44406
alloc
25
elements
7
elements
188
version
11
table
0
0
2
1
processor
1
1
1
0
$no list
update
0
2
1
constant d
assumed st
all points
no echo
1
2
3
elsto
40960
ibooc
setname
1717
end
$...................
solver
8
0
0
0
0
0
optimize
11
connectivity
0
0
1
49394
7
4
3
13
6
49395
7
120606
120601
120641
120611
49396
7
6
13
14
7
1
0
0
0
0
120606
120608
120611
120601
120603
120641
120641
120643
120646
120611
120613
120616
431924
431926
431922
431924
0
0
.
.
.
202532
188
2570
2571
489731
489729
7647
202533
188
2571
5110
489733
489731
7648
coordinates
3
178232
0
1
3 2.000000000000000-1 0.000000000000000+0 4.064000000000000-1
4 0.000000000000000+0 0.000000000000000+0 4.064000000000000-1
6 0.000000000000000+0 2.000000000000000-1 4.064000000000000-1
.
.
.
489731 5.077999999999798+2 0.000000000000000+0 2.032000000000000-1
489733 5.079999999999798+2 0.000000000000000+0 2.032000000000000-1
define
element
set
stiffener
123740
to
135099
define
element
set
plate
49394
49395
49396
49397
49398
49399
49404
49405
49406
c
49407
54486
54487
54488
54489
54490
54495
54496
54497
c
54498
54499
54500
54501
54502
54503
54508
54509
54510
c
.
.
.
141695
141696
141697
141698
141699
141700
141705
141706
141707
c
141708
141709
141710
141711
141712
141713
fixed disp
1
0
0
0
1
0x
0.000000000000000+0
0
1
2
x_nodes
1
0
0
0
1
0y
0.000000000000000+0
0
2
2
y_nodes
1
0
0
0
1
0z
0.000000000000000+0
0
2
z_nodes
1
0
0
0
1
0x_const
92
7648
7649
49400
49401
49402
49403
54491
54492
54493
54494
54504
54505
54506
54507
141701
141714
141702
141715
141703
141716
141704
141717
0.000000000000000+0
0
1
2
x_const_nodes
1
0
8.000000000000000+0
1
2
2
ydisp_nodes
1
0
0.000000000000000+0
0
3
2
guide_pla_nodes
loadcase
job1
6
x
y
z
x_const
ydisp
guide_pla
tying
2
100961
280723
2
100963
280723
2
100965
280723
2
100974
0
0
1
0ydisp
0
0
1
0guide_pla
0
0
0
0
0
0
0
0
.
.
.
280723
2
480323
0
0
280723
2
480325
0
0
280723
2
480327
0
0
280723
no print
post
4
16
17
0
0
19
20
0
2
0
0
0
0
0
0
0
7
0
17
0
311
0
411
0
parameters
1.000000000000000+0 1.000000000000000+9 1.000000000000000+2 1.000000000000000+6 2.500000000000000-1 5.000000000000000-1
1.500000000000000+0-5.000000000000000-1
8.625000000000000+0 2.000000000000000+1 1.000000000000000-4 1.000000000000000-6 1.000000000000000+0 1.000000000000000-4
8.314000000000000+0 2.731500000000000+2 5.000000000000000-1 0.000000000000000+0 5.670510000000000-8 1.438769000000000-2
2.997900000000000+8 1.00000000000000+30
0.000000000000000+0 0.000000000000000+0 1.000000000000000+2 0.000000000000000+0 1.000000000000000+0-2.000000000000000+0
1.000000000000000+6
end option
$...................
$....start of loadcase lcase1
title
lcase1
loadcase
lcase1
6
x
y
z
x_const
ydisp
guide_pla
control
99999
10
0
5
2
1
0
0
1
0
1
0
1.000000000000000-1 0.000000000000000+0 0.000000000000000+0 0.000000000000000+0 0.000000000000000+0 0.000000000000000+0
0.000000000000000+0 0.000000000000000+0
0.000000000000000+0 0.000000000000000+0 0.000000000000000+0 0.000000000000000+0 5.000000000000000-3 0.000000000000000+0
0.000000000000000+0 0.000000000000000+0
parameters
1.000000000000000+0 1.000000000000000+9 1.000000000000000+2 1.000000000000000+6 2.500000000000000-1 5.000000000000000-1
1.500000000000000+0-5.000000000000000-1
8.625000000000000+0 2.000000000000000+1 1.000000000000000-4 1.000000000000000-6 1.000000000000000+0 1.000000000000000-4
8.314000000000000+0 2.731500000000000+2 5.000000000000000-1 0.000000000000000+0 5.670510000000000-8 1.438769000000000-2
auto increment
1.000000000000000-5
2000
10 1.000000000000000-3 1.000000000000000+1 1.000000000000000+0 1.000000000000000-5
1
20
2
0
1
2
0 3.000000000000000+0
2
continue
$....end of loadcase lcase1
93