Development of a numerical evaluation strategy to predict crack

Master of Science Thesis
Development of a numerical evaluation strategy to
predict crack branching and crack turning in the
latest generation aluminium alloys
H.P.A. Dijkers, B.Sc.
March 27, 2015
Faculty of Aerospace Engineering
Development of a numerical
evaluation strategy for predicting
crack branching and crack turning in
the latest generation aluminium alloys
Master of Science Thesis
For the degree of Master of Science in Aerospace Engineering at Delft
University of Technology
H.P.A. Dijkers
March 27, 2015
Faculty of Aerospace Engineering (AE) · Delft University of Technology
c H.P.A. Dijkers
Copyright All rights reserved.
Delft University of Technology
Department of
Aerospace Structures and Materials (ASM)
The undersigned hereby certify that they have read and recommend to the Faculty of
Aerospace Engineering (AE) for acceptance a thesis entitled
Development of a numerical evaluation strategy for predicting crack
branching and crack turning in the latest generation aluminium
alloys
by
H.P.A. Dijkers
in partial fulfillment of the requirements for the degree of
Master of Science Aerospace Engineering
Dated: March 27, 2015
Supervisor(s):
Dr. ir. R.C. Alderliesten
Ir. S. van der Veen, Airbus France
Reader(s):
Dr. S. Teixeira De Freitas
Dr. S.J. Garcia Espallargas
Preface
TV-shows, like Air Crash Investigation, and the lectures of Professor Stoop on forensic engineering got me
interested in fatigue of metals. I found an opportunity to do a thesis in collaboration with Airbus, which
would allow me to explore the fatigue behaviour of some advanced aluminium alloys. Also it would allow
me to perform fatigue tests, with the goal to support the development of a predictive numerical method
for the particular fatigue behaviour of these alloys. The connection with Airbus and the opportunity to
combine theory and praxis were more than enough reasons for me to choose this thesis as the final part
of my study at the Delft University of Technology.
Hererby, I want to express my sincere appreciation to my supervisors dr. ir. René Alderliesten and ir.
Sjoerd van der Veen, who were able to provide me with the right amount and right kind of information
to really support the progress on my thesis. René and Sjoerd both have proven to be worthy sparring
partners to discuss ideas with, if I was unsure if my approach was correct or not.
I would like to extend my gratitude to ir. Sjoerd van der Veen and many of his colleagues at Airbus
for their industry-input into this thesis. I would like to acknowledge both Airbus and Constellium for
providing the funding and material for the manufacturing of the fatigue test coupons. Also, I am very
grateful towards the technical staff of the DASML laboratory, especially Berthil Grashof, Bob de Vogel,
Gertjan Mulder and Frans Oostrum, for helping me with many of my questions and providing assistance
with my experiments.
Finally, I want to thank my family, friends, and Sanne for their support, love, and at times much needed
relaxation. Also my fellow students at the department of Structural Integrity and Composites deserve
my appreciation. Lastly I want to thank Robert Smit for helping me designing the cover for this report.
Without all the help I received throughout my thesis I would not have produced this good result.
Delft University of Technology
March 27, 2015
H.P.A. Dijkers
i
ii
Summary
Artificially aged aluminium-lithium alloys have recently seen renewed interest in them in the aviation
industry. They have a decreased density, due to the addition of lithium, while keeping or improving the
fatigue properties compared to other aluminium alloys. Recent and past research has shown that these
aluminium-lithium and other advanced aluminium alloys exhibit crack turning and crack branching in a
direction parallel to the principal loading direction. It has been observed that these alloys exhibit fracture
toughness anisotropy. Currently no predictive capabilities exist for such phenomena, therefore designers
cannot use the full potential of these alloys.
This report contains a thorough explanation of the development of a numerical strategy that can predict
in-plane crack turning and accounts for anisotropy in fatigue fracture resistance. The models have been
calibrated with the use of fatigue tests, where crack turning is observed. The numerical strategy has been
developed to be as general as possible, and can deal with multiple material and structure geometries.
Additionally, fatigue tests on C(T) and DEN(T) coupons have been performed in this study to gain
insight into the effects of crack turning and crack branching on the fatigue crack life, crack paths and
FCGR. Two different aluminium alloys are used in these tests, Al 2050-T84 and Al 7010-T7451, both of
which are expected to exhibit these phenomena. Lastly, the main drivers are established for both in-plane
crack turning and crack branching in a crack arrestor configuration.
The fatigue tests in this study have been performed with C(T) and DEN(T) coupons in seven different
orientations in the ST-L material plane, where the L-axis has been rotated increasingly w.r.t the
expected mode-I crack growth direction. The results of the fatigue tests show that the orientation of
the microstructure w.r.t the primary crack and loading direction affects the crack life and FCGR. The
ST90-L orientation, also known as L-ST, has a significant higher fatigue crack life and lower FCGR.
For orientations close to ST00-L the difference in crack life and FCGR is rather small. Each coupon
orientation produced different crack paths, and the investigated alloys exhibit crack turning and crack
branching. Crack branches and kinks in the crack path cause the FCGR to stay constant or even decrease
with increasing ΔK*.
Microscopy revealed that both Al 2050-T84 and Al 7010-T7451 show a banded microstructure typical
for rolled artificially aged aluminium plate products. The grains have large dimensions in the rolling
direction (L) and small dimensions in the ST-direction. The grain size of Al 2050 is larger than that of
Al 7010, which is likely the cause for the larger observed difference in FCGR in Al 2050 for the different
coupon orientations.
Two separate numerical methods are developed in this study, the k2-method and the Pettit-method. Both
methods are based on a LEFM framework. The key hypotheses of the k2-method have been judged to be
invalid, so this method should not be considered for future use. On the other hand, the key assumptions
iii
iv
for the Pettit method have proven to be very reasonable. This method assumes that a crack likely grows
in a direction which maximizes the ratio of available crack driving force over crack growth resistance.
This notion is also applied in conventional mixed-mode criteria, however these do not account for fatigue
fracture resistance anisotropy.
To account for fatigue fracture resistance anisotropy, the modified strain energy release rate approach is
used. The desired crack propagation direction can be found by maximizing the sum of the cubes of the
ratio of crack driving force over fracture toughness for each individual crack opening mode. Additionally,
to account for fatigue fracture resistance anisotropy, that is dependent on the orientation of the
microstructure w.r.t the primary crack, a polar interpolation function is used. This function determines
the relative fatigue fracture resistance of a crack with an arbitrary orientation in the ST-L plane, based on
the relative crack growth resistance of the ST- and L-direction of the material. Next to the dependency
of the fracture resistance on the crack orientation, it is made dependent on Kmax as well. The value of Kmax
determines the shape factor of the polar interpolation function and the value of the relative crack growth
resistance of the L-direction of the alloy.
The values of the parameters used in the Pettit method have been calibrated with the use of fatigue tests
that were performed by Airbus. With these calibrated values, predictions are made for the crack paths
of the seven coupon orientations and two coupon types used in the fatigue tests. A comparison of the
predictions with the actual crack paths, shows that the predictions of the Pettit-method are accurate for
coupon orientations in between ST00-L and ST45-L. For ST60-L a comparison of the actual crack path
with the predicted path showed some larger deviations. Recalibration of the model parameters, with the
crack paths observed in the fatigue tests, may improve the predictions for orientations close to ST90-L.
Fractography revealed that the fracture surface of a branch in a DEN(T) ST90-L coupon of Al 2050 is
formed by void growth and coalescence. No indications of intergranular crack growth or shearing of the
dimples have been observed. A recent study confirms that void growth and coalescence is the primary
fracture process for crack branches in the crack arrestor configuration. A combination of shielding and
amplification effects on a microstructure level is responsible for this void growth, which concentrates on
grain boundaries between favourable stiff/soft grain pairings. The main drivers for crack branching are
determined to be an elevated stress level on the grain boundary between a stiff/soft grain pair, along with
increased void growth due to the higher stress and the damaging effects of the PFZ’s and δ’ precipitates.
These observations make it unlikely that crack branching in a crack arrestor configuration can be
predicted with macroscopic crack growth criteria.
The main driver for the observed crack turning and fatigue fracture resistance anisotropy, observed
in the fatigue tests, likely is the orientation of the microstructure w.r.t the primary crack and loading
direction. It is shown that the orientation of the microstructure determines the FCGR. Therefore likely it
determines the fatigue fracture resistance as well. Numerical simulations reveal that turning cracks grow
in near mode-I conditions, which is confirmed by fractography of such a turning crack path. The fact
that such cracks grow in near mode-I conditions, makes them well predictable with macroscopic crack
growth criteria, as proven by the predictions of the Pettit-method.
Table of contents
Prefacei
Summaryiii
Glossaryvii
1 Introduction1
2 Basic theory
2-1
2-2
2-3
2-4
2-5
2-6
Fatigue in metallic alloys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear elastic fracture mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Crack kink theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decimal mark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12-1 2-22-2 2-32-3 2-42-4 2-52-5 2-62-6 3 Literature review
3-1
3-2
3-3
3-4
3-5
3
4
6
7
7
8
9
Microstructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Fracture anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Branch forming and crack turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Mixed-mode fatigue fracture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3-13-1 3-23-2 3-33-3 3-43-4 3-53-5 4 Requirements numerical method
4-1
4-2
4-3
4-4
3
17
Requirements on numerical method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Current available methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Required additions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4-14-1 4-24-2 4-34-3 4-44-4 v
vi
1 1
5 k2-method21
5-1
5-2
5-3
5-4
Model origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Method process flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Details on modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5-15-1 5-25-2 5-35-3 5-45-4 6 Pettit-method29
6-1
6-2
6-3
6-4
Model origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Method process flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Details on numerical modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6-16-1 6-26-2 6-36-3 6-46-4 7 Experiments37
7-1 Setup of fatigue tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7-2 Data processing of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7-3 Reasoning for design of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7-17-1 7-27-2 7-37-3 8 Results49
8-1
8-2
8-3
8-4
Microstructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Fatigue tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Fractography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
8-18-1 8-28-2 8-38-3 8-48-4 9 Discussion65
9-1
9-2
9-3
9-4
Discussion on literature and fatigue tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Discussion on numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Drivers for crack branching and turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9-19-1 9-29-2 9-39-3 9-49-4 10 Conclusions and recommendations
75
10-1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
10-2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10-110-1 10-210-2 11 References79
A Fatigue test data
A1
A-1 Fatigue test data Al 2050-T84. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1
A-2 Fatigue test data Al 7010-T7451. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A4
A-3 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A6
A-1A-1 A-2A-2 A-3A-3 B Photographs of specimens
B1
B-1 Photographs of Al 2050-T84 specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B1
B-2 Photographs of Al 7010-T7451 specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . .B4
B-3 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B6
B-1B-1 B-2B-2 B-3B-3 C Certificates Al 7010-T7451
C1
Glossary
List of acronyms
Acronym Definition
AR
Adaptive remeshing
ASTM
American Society for Testing and Materials
C(T)
Compact tension coupon
CC(T)
Center crack tension coupon
CZM
Cohesive zone modelling
DEN(T)
Double edge notched tension coupon
FCG
Fatigue crack growth
FCGR
Fatigue crack growth rate
FE
Finite element
FEA
Finite element analysis
FEM
Finite element model
LEFM
Linear elastic fracture mechanics
M(T)
Middle crack tension coupon
MSERR
Modified strain energy release rate
MSS
Maximum shear stress
MTS
Maximum tangential stress
PFZ
Precipitate free zone
SEM
Scanning electron microscope
SERR
Strain energy release rate
SIF
Stress intensity factor
SSY
Small scale yielding
TEM
Transmission electron microscope
XFEM
Extended finite element method
vii
viii
List of symbols
Greek symbols
Symbol
Definition
α
Angle to reference line
β
Geometry factor
Δap
Predetermined crack extension length
ΔK
Difference between maximum and minimum stress intensity factor of a fatigue cycle
ΔK
Equivalent stress intensity factor
Δθ
Crack kink angle of virtual crack extension
Δθmixed
Mixed-mode crack growth direction
θ
Main crack angle
σnom
Nominal applied stress
σr
Radial stress
σrθ
In-plane shear stress
σ0
Far-field stress
σθ
Tangential stress
τmax
Intergranular shear strength
*
Upper case symbols
Symbol
Definition
a/W
Relative crack length
a/W
Equivalent relative crack length
An
Data point from reduced data set
B
Thickness of a fatigue test coupon
Ccg
Material constant in Paris-law
Ci
Coefficient in crack length polynomial for compact tension coupon
Cij
Matrix coefficient virtual stress intensity factor
D
Hole diameter in compact tension coupon
*
da/dN
Crack growth speed
da/dN
Equivalent crack growth speed
E
Young’s modulus
Ki
stress intensity factor
KI
Mode-I stress intensity factor
KI,C
Mode-I fracture toughness
KII
Mode-II stress intensity factor
KII,C
Mode-II fracture toughness
Kmax
Maximum stress intensity factor of a fatigue cycle
*
ix
Symbol
Definition
Kmin
Minimum stress intensity factor of a fatigue cycle
KP
Fracture resistance
Kp,1
Relative fracture resistance in the L-direction
Kp,2
Relative fracture resistance in the ST-direction
Kt
Stress concentration factor
M
Mixed-mode ratio
N
Number of fatigue cycles
P
Applied force
Pmax
Maximum applied force in a fatigue cycle
R
Stress ratio of a fatigue cycle
Sa
Stress amplitude in a fatigue cycle
Sm
Mean stress in a fatigue cycle
Smax
Maximum stress in a fatigue cycle
Smin
Minimum stress in a fatigue cycle
W
Width of a fatigue test coupon
Lower case symbols
Symbol
Definition
a
Crack length
ao
Notch depth in a DEN(T) fatigue test coupon
c
Material parameter for branch length determination
h
Height of a DEN(T) fatigue test coupon
k1
Mode-I virtual stress intensity factor
k1,eq
Equivalent mode-I virtual stress intensity factor
k2
Mode-II virtual stress intensity factor
k2,thres
Threshold of branch initiation
k3
Mode-III virtual stress intensity factor
n
Shape parameter of Polar interpolation
ncg
Material constant in Paris-law
t
Thickness of a fatigue test coupon
ux
Compliance parameter
v
Deflection of a fatigue test coupon
x
Chapter 1
Introduction
To maximize the profitability and performance of aircraft they are designed to be as light as possible.
Decreasing the density of a material, while keeping or even improving strength and fatigue properties,
represents a major opportunity to save weight in an aircraft structure. For this reason there is renewed
interest in aluminium-lithium based alloys in the aviation industry, as these alloys have a lower density
and improved fatigue life compared to other alloys. Also other advanced aluminium alloys are being
developed with an improved fatigue life, however without the decreased density, as no lithium is used
in those alloys. Airbus for example is using these aluminium-lithium and other advanced alloys in their
A380 and A350XWB.
Past and recent investigations on these Al-Li alloys revealed aspects that have not been considered up
until today. Where fatigue cracks in aluminium alloys are predominantly propagating as single cracks
perpendicular to the principal load, several of these new alloys exhibit crack branching or crack turning
in the direction of principal loading. Also an apparent anisotropy in the fracture toughness is observed.
As currently no predictive capabilities exist for such behaviour, designers cannot use the full benefits
of aluminium-lithium alloys. An aircraft manufacturer should be able to use predictive capabilities to
design for damage tolerance and durability, and in case cracks are detected, to develop sustaining repair
solutions.
Although these phenomena have been observed in the past on few alloys and are qualitatively understood,
the absence of prediction capabilities limits the application of these new alloys in aircraft structures. Due
to the crack branching phenomena and likely the influence of individual grains and grain boundaries,
thorough analysis would require detailed FEM at a scale that becomes impractical for industry wide use,
and the use on a wide range of materials and structures. For that reason an efficient numerical evaluation
strategy should be developed that can assess the effects of crack branching and crack turning in advanced
aluminium-lithium alloys and other newly developed aluminium alloys, which exhibit these phenomena.
1
2
1 Introduction
Therefore, the main goal of this thesis is:
Develop a numerical method for the prediction of crack branching and crack turning in advanced aluminium
alloys, by identifying the main drivers behind crack branching and crack turing in these aluminium alloys,
implementing this into a FE-model, and calibrating the numerical method using coupon fatigue tests.
Current widely used mixed-mode fatigue criteria are not capable of predicting crack branching or crack
turning, so new criteria are needed to determine the crack growth direction. This study focuses on crack
branching in the crack-arrestor configuration and in-plane crack turning. The numerical method is
implemented in a FE-environment, where it is required that the method can handle different materials
and geometries of a structure. Coupon tests are performed to identify the effects of crack branching and
crack turning on the fatigue behavior of these alloys. Additionally, the fatigue tests aid in calibrating and
validating the numerical method.
This report is structured as follows. An overview of some basic theory and the definition of the material
coordinate system is presented in Chapter 2. A review of the available literature on the topic of this
study is given in Chapter 3. The requirements of the numerical strategy are set out in Chapter 4. The
two developed numerical methods are the topics of Chapter 5 and 6 respectively. Chapter 7 discusses
the setup of the fatigue tests, the data processing and the reasoning for the design of the experiments.
Next, Chapter 8 presents the results of the fatigue tests and both numerical methods. In Chapter 9 both
numerical methods are evaluated, and the drivers for crack branching and crack turning are discussed.
Finally, some conclusions and recommendations for future research are presented in Chapter 10.
Chapter 2
Basic theory
This chapter describes some basic concepts used throughout this thesis. A general overview is given of
fatigue in metallic alloys and of analytical models that are available to predict fatigue crack growth. An
introduction into linear elastic fracture mechanics is given, as this serves as the theoretical basis for the
crack growth models presented in this thesis. A crack kinking theory, originally developed by Cotterell
and Rice, is explained for a quasi-3D stress field [Cotterell & Rice, 1980]. The material coordinate system
is defined, along with a standardized notation for specimen orientations. Lastly, some definitions are
given in this chapter for key terminology used in this study.
2-1 Fatigue in metallic alloys
2-1 The first scientific description of fatigue is attributed to Wöhler in 1870. In his work he explained how a
repetition of a load far below the static strength of a structure could lead to catastrophic failure of that
structure [Schijve, 2009]. Hereafter, fatigue has been a subject of study for many researchers in a wide
variety of scientific disciplines, like the railroad and maritime industry [Schütz, 1996]. In the aviation
industry, fatigue has more than once led to catastrophic failures, an infamous example being the accident
with Aloha airlines in 1988, where fatigue damage caused inflight separation of part of the fuselage
[NTSB, 1989].
A (fatigue) stress cycle has several characteristic properties, see Figure 2-1 . These properties are the main
stress Sm, the stress amplitude Sa, the maximum stress Smax, the minimum stress Smin and the stress ratio R:
R=
Smax
Smin
(2-1) From these quantities only two are needed to fully define a stress cycle.
3
4
2 Basic theory
Figure 2-1: Characteristics of a load cycle.
Crack initiation, due to fatigue loading, is the result of a process of atomic slipping in the crystal structure
of a stressed material. Very small defects in the crystal structure or on the surface of the material can
have a significant effects on the duration of this process, which makes it difficult to predict. From an
engineering approach, a way to deal with these uncertainties is to introduce a stress concentration factor
Kt. Here Kt is defined as the quotient of the peak stress to the nominal stress, where the latter is defined as
the stress without any stress concentration. Every geometry or defect has a different stress concentration,
and its respective Kt-value is usually obtained using handbooks like Peterson’s Stress Concentration Factors
[Pilkey et al., 2008]. A larger Kt-value results in a shorter period to crack initiation, because locally,
around the defect, notch, etc., the stress is expected to be higher, easing the process of atomic slipping.
In order to deal with fatigue crack growth in structures, it may be assumed that a certain crack of a
certain length is already present, where a fixed number of load cycles is assumed for the crack initiation
phase [Alderliesten, 2009]. The predictability of crack growth has become increasingly important, as
current design philosophies are aimed at controlled crack growth and structural integrity. This is a
logical consequence when one looks at an aircraft structure, where numerous bolts, rivets, windows and
inspection holes are present. All of these features introduce stress concentrations in the structure, making
crack initiation and growth a near certainty.
2-2 Linear elastic fracture mechanics
2-2 The stress field around a growing crack is influenced by many factors, such as material properties,
environmental conditions etc. Within linear elastic fracture mechanics (LEFM) two assumptions must
hold. Firstly, the plasticity around a crack tip is assumed to be confined to a small area, i.e. small scale
yielding (SSY). For many materials, like aerospace grade aluminum alloys, ductility is limited, validating
this first assumption [Dursun & Soutis, 2014]. Secondly, the applied loading is assumed to be far below
the static yield limit, or limit load of the structure, which is mostly the case for fatigue loading. With
the LEFM framework in place, the full 3-D stress field around a crack tip can be reduced to three single
parameters, the stress intensity factors (SIF’s) [Tada et al., 2000]. There exist three distinct fracture modes,
so each fracture mode has its own SIF. The first crack extension mode is in-plane opening, or mode-I,
mode-II crack growth is in-plane shearing, mode-III is defined as out-of-plane shearing. For clarity all
crack growth modes are shown in Figure 2-2.
2-2 Linear elastic fracture mechanics
5
Figure 2-2: The three crack extension modes and associated loadings.
The value of the SIF is given in units of MPa√m, which is a measure of the crack driving force. Each mode
has its own crack growth resistance, where tests can be employed to find the exact value for each mode.
A general relation for the stress intensity factor is given by:
K = βσ 0 π a
(2-2) Where β is a dimensionless factor depending on the geometry of the structure, σθ is the far-field stress a a
is the crack length. As the SIF is proportional to the stress, the stress ratio R can also be written as Kmin/‍Kmax.
Kmax and Kmin are the maximum and minimum stress intensity factor of a stress cycle respectively. When
a crack is loaded in more than one mode, one speaks of mixed-mode loading, or mode-mixity. The stress
fields for various configurations have been published in books like The Stress Analysis of Cracks Handbook
[Tada et al., 2000]. In contrast to Kt, which just depends on the geometry, the stress intensity factor Ki
depends on both the geometry and the applied stress.
In real-life aerospace structures, a crack is rarely loaded purely in one crack extension mode only. Rather
the crack is loaded in an arbitrary mix of the three crack extension modes. In a 2-D case, i.e. ignoring
mode-III, one can define the mixed-mode ratio as:
M=
K II
KI
(2-3) Here M is the mixed-mode ratio, KI and KII are the SIF’s of mode-I and -II respectively.
This thesis focuses on cyclic fatigue loading, so in that case the SIF will also vary cyclically. As this cyclic
variation further complicates determination of crack growth in a structure or material, a different quantity
is used for the SIF, namely ΔK. This quantity is the difference between the maximum and minimum
SIF within one load cycle. Fatigue crack propagation in materials, especially aluminium alloys, tends to
behave according to a power law, called the Paris relation, when the stress intensity factor ΔK is not too
small or too large [Paris & Erdogan, 1963]:
n
da
= Ccg ( ∆K ) cg
dN
(2-4) Here Ccg and ncg are material constants. This relation was later extended by for example Newman and
NASGRO [Newman, 1984; Forman & Mettu, 1992].
6
2 Basic theory
2-3 Crack kink theory
2-3 Based on the original work of Cotterell and Rice, multiple methods have been developed to predict the
fatigue crack growth direction in planar and 3-D conditions for isotropic solids [Cotterell & Rice, 1980].
Commonly known methods are the maximum tangential stress criterion (MTS) and the maximum
strain energy release rate (SERR) [Bouchard et al., 2003]. To be able to predict the crack driving force
in any direction with respect to the crack tip can be key to predicting the crack growth direction. To
illustrate this process, Figure 2-3 gives an overview of the cylindrical coordinate system around an
arbitrarily orientated infinitesimal crack tip extension of an existing crack. Δθ is defined as positive in the
anticlockwise direction, i.e. from the X- to the Y-axis.
Figure 2-3: Definition of coordinate system and stress tensors around a crack tip [Pettit et al., 2013].
Ki are the stress intensity factors of the main crack, θ is the main crack orientation, σθ, σr and σrθ are the
tangential, radial and in-plane shear stress respectively. As the loading of the main crack is arbitrary,
the crack is assumed to be loaded in a mixed-mode condition. The virtual mixed-mode stress intensity
factors, k1, k2 and k3, at the newly formed crack tip can be expressed as functions of the stress tensors or
of the main crack stress intensity factors:
=
k1
σ θθ
∆θ
∆θ
∆θ
=
cos3
K I − 3sin
cos 2
K II
2
2
2
2π r
σ rθ
∆θ
∆θ
∆θ 
∆θ
k2 =
=
sin
cos 2
K I + cos
1 − 3sin 2

2π r
2
2
2 
2
=
k3
σ zθ
∆θ
= cos
K III
2π r
2
(2-5) 
 K II

(2-6) (2-7) This can be more easily expressed in matrix format, where the influence coefficients Cij are a function of
Δθ only:
C11 C12   K I  k`1 
C
⋅  =
 
 21 C22   K II   k2 
(2-8) Here it is assumed that KIII << KII and KIII << KI. For a 2D case KIII is ignored altogether, both cases result
in the 2x2 matrix with only mode-I and –II SIF’s.
2-4 Material coordinate system
7
2-4 Material coordinate system
2-4 To have one clear definition of the various material planes and test coupon orientations, a standard
notation is introduced here. Figure 2-4 shows the material coordinate system and the three principal
materials planes. The three principal material planes are: L-LT, LT-ST and ST-L. In the notation of the
test coupon orientation the first material axis indicates the loading direction, the second material axis
indicates the expected crack growth direction. So a L-LT coupon is loaded in the L-direction, and crack
growth is expected in the LT-direction.
The orientations of the principal material axes does not necessarily have to align with the expected crack
growth and loading direction. To clearly indicate the rotation of the microstructure w.r.t. to the loading
and expected mode-I crack growth direction, an angle is added to the coupon notation. This angle
indicates how large this rotation is. So, a coupon in the ST45-L orientation has the ST and L axes rotated
45 degrees w.r.t the loading and expected crack growth direction. A ST90-L coupon, commonly known
as L-ST, has a 90 degree rotated microstructure, so the expected crack growth direction is aligned with
the ST-axis and the loading direction with the L-axis.
ion
ing
ll
Ro
ect
dir
L45-LT
LT-ST
T
0-L
L9
L-L
T
L
ST-
ST
LT
9
0-S
T
LT
L
Figure 2-4: Definition of material coordinate system and various crack growth coupon orientations.
2-5 Definitions
2-5 The definitions for the main crack, crack branching, - turning, - deviation and - deflection are given here
to explain their use and meaning to the reader.
From an analysis perspective for a multi-crack scenario, the main crack would be the crack path with the
highest overall driving force, i.e. the largest total energy release rate. The main crack does not necessarily
have to follow the intended direction. It might require a subjective judgment to decide what is the main
crack and what is a “lesser” branch.
8
2 Basic theory
Crack branching is defined as crack growth in multiple directions at once, aside from the inherent
symmetric crack growth found in symmetric specimens, such as a CC(T). This study focuses on
crack branching in the crack arrestor configuration. Figure 2-5 presents a branch in the crack arrestor
configuration, along with the two other types of branches generally identified in literature.
Crack turning is a gradual change in direction of the main crack. The crack growth direction changes
under influence of various factors, such as stiffness anisotropy or fracture toughness anisotropy. Usually,
this cannot be predicted with common mixed-mode crack growth criteria. In this study, the focus is laid
on in-plane crack turning.
Crack deviation is crack growth in a direction, which deviates from the intended/expected direction.
Crack kinking is an abrupt deviation from the inteded/expected crack growth direction, not to be
mistaken with crack branching. The expected direction for mode-I loading is perpendicular to the
principal loading direction, possibly with left/right symmetry for a symmetric crack growth specimen.
For mixed-mode loading, the crack path can be predicted with mixed-mode criteria, such as maximum
tangential stress (MTS). Crack deviation and crack kinking cannot be predicted with common mixedmode fracture mechanics.
Crack deflection is the point where a “lesser” branch becomes the main crack, and vice versa. This switch
is defined by which continuous crack path has the highest overall crack driving force. Again this may
sometimes require a subjective judgment to determine the exact point where the crack is deflected from
it’s original main crack path.
Figure 2-5: Standard crack branching configurations. This study focuses on the arrestor configation, particulary in the L-S
configuration [Messner, 2014].
2-6 Decimal mark
2-6 In this report a full stop is used as decimal mark.
Chapter 3
Literature review
In this chapter an outline of the research performed on advanced Al-Li alloys is presented. The first
part, targeted on the microstructure, highlights all the important characteristics hereof. Usually, the
microstructure largely determines the crack growth behaviour and crack growth resistance of a material.
Hereafter, in section 3-2, the macrostructural effects of the anisotropy in the grain structure are reviewed.
Both static and fatigue fracture anisotropy are discussed. Next, section 3-3 treats macroscopic crack
branching and crack turning. Finally, some observations from mixed-mode fatigue tests are presented
in section 3-4.
3-1 Microstructure
3-1 Grains
The demand for aluminium alloys with higher static strengths has led to the development of more highly
crystallographic textured alloys such as Al 7085-T7651 and various artificially aged Al-Li alloys, in
particular the Al 2050-T84 alloy [Rioja & Liu, 2012; Dursun & Soutis, 2014]. The observed microstructure
in these artificially aged Al 7xxx or Al-Li alloys is quite similar [Sinclair & Gregson, 1994; Gregson &
Sinclair, 1996]. The grain structure is anisotropic in nature, with long pancake like grains, which have
large dimensions in the rolling direction (L) of the plate product and small dimensions in the thickness
direction (ST) [Suresh, 1985; Suresh et al., 1987]. In Figure 3-1 one can see the grain structure of Al
2099‑T86, a 3rd generation Al-Li alloy, in a rolled plate product.
9
10
3 Literature review
Figure 3-1: Etch of the grain structure of an Al 2099-T86 rolled plate product [Rioja & Liu, 2012].
Precipitates
Not only the grain structure, but also the precipitates in the grains influence the fracture toughness of
artificially aged alloys. The precipitates found in artificially aged Al-Li alloys actually promote intergranular
failure, as they act as stress raisers [Suresh, 1985; Suresh et al., 1987; Rao et al., 1988; Hernquist, 2010;
De et al., 2011]. The metastable spherical δ’ (Al3Li) precipitate increases the tendency for intergranular
failure. It has a small matrix misfit, which results in localized slip. This localized slip generates stress
concentrations on the grain boundaries, thus increasing the tendency for intergranular fracture.
The T2 (Al6CuLi) precipitate, which is mostly found on the grain boundaries, also promotes intergranular
fracture. Especially in peak- or over-aged alloys these T-type precipitates have been linked to intergranular
failure and a decrease in fracture resistance [Hernquist, 2010; Rioja & Liu, 2012]. Precipitates found on
the grain boundaries of Al 7050-T7451 have similar effects [Sinclair & Gregson, 1997].
Next, it is known that bending stress increases the multi-axial stress state or stress triaxiality, which helps
precipitates act as stress raisers and increases the likelihood of intergranular failure [De et al., 2011]. In
contrast one can look towards the well-known Al-2024 alloys. In these alloys, no such precipitates or
associated intergranular failure have been observed in the past [Hahn & Rosenfield, 1975; Kung & Fine,
1979; Kamp et al., 2007b].
Precipitate free zones
The δ’ (Al3Li) precipitate additionally promotes grain boundary failure by formation of precipitate free
zones (PFZ’s). The δ’ precipitates nucleate during the quenching process of the heat treatment, concurrently
δ (AlLi) grows in the same regions. Because this generates a competition for lithium, it leaves less lithium
to form δ’ precipitates on the grain boundaries. As δ’ are the primary strengthening precipitates, little
in these grain boundaries can inhibit further dislocation motion [Jha et al., 1987; Prasad et al., 2003;
Hernquist, 2010].
Recent research shows that the quench rate influences the formation of PFZ’s. It is shown that a slower
cooling rate, which can be found in the centre of thick material billet, promotes the formation of PFZ’s.
The width of the PFZ’s was significantly larger when a slower cooling rate was applied. In addition, the
grain boundary precipitates are observed to be larger as well [Reese & Wood, 2013]. Figure 3-2 shows a
transmission electron microscopy (TEM) image of the PFZ’s in Al 7085-T7651, microstructural similar
to Al 2050-T84. In the image the PFZ’s can be reconignized by their lighter color and relative large size.
3-2 Fracture anisotropy
11
Figure 3-2: TEM image of the grain boundary and precipitate free zones in Al-7085 T7651 [Reese & Wood, 2013].
Figure 3-2
Micro-voids and slip planarity
Contrary to conventional aluminium alloys, the fracture toughness of Al‐Li alloys may increase at
cryogenic temperatures [Rao et al., 1989; Chen et al., 1998]. McDonald found that Al‐Li 2099 developed
larger and more frequent crack branches at room temperature than at cryogenic temperatures regardless
of the location within the plate from which the sample was obtained [McDonald, 2009]. At room
temperatures micro-voids, which act as stress concentrations, can be found near the grain boundaries,
due to the liquid phase of some metallic impurities [Rao et al., 1989].
Another microscopic characteristic found in Al-Li alloys is slip planarity [Suresh et al., 1987; Rao et al.,
1988; De et al., 2011]. This promotes crack growth along slip bands, which may manifest as macroscopic
crack deflection or a zig-zag pattern. The plane of maximum tensile stress however still remains the
governing crack growth direction.
3-2 Fracture anisotropy
3-2 Static fracture anisotropy
Rolled plate products from high yield strength aluminium alloys often show a grain structure as shown
in Figure 3-1. This grain structure is a result of the production process of the material billet, for example
hot rolling or extrusion. Static fracture anisotropy is observed is many materials which are used in these
types of production processes [Ertürk et al., 1974; Benzerga et al., 2004; Mir et al., 2005; Beese et al., 2010;
Luo et al., 2012].
Hot rolled medium carbon steel exhibits significant anisotropic deformability. The deformability in the
ST- or LT-direction can be up to 50% higher compared to the L-, or rolling-direction [Ertürk et al., 1974].
The observed zig-zag crack paths in static fracture tests are found to be a result of the anisotropic plastic
behaviour of the involved steel alloy and are linked to anisotropic grain structure [Benzerga et al., 2004].
In the automotive industry aluminium extrusions are often used for parts of the primary load carrying
structure. Due to the extrusion process an anisotropic grain structure is formed in these sheet materials.
This leads to various forms of anisotropy with regards to fracture initiation and plastic response [Hooputra
et al., 2004; Beese et al., 2010; Luo et al., 2012].
12
3 Literature review
It remains a key challenge how to model the anisotropy in the mentioned material characteristics. Most
fracture models therefore focus on either ductile normal or ductile shear failure. The crachFEM model
couples the stress state of the material, in terms of triaxiality, to the allowable plastic strain for both ductile
normal and ductile shear failure [Hooputra et al., 2004; Dell et al., 2007; Gese et al., 2013]. An orthotropic
knockdown factor has been suggested, in the form of a series of sine, to allow for the anisotropic fracture
resistance. A downside of this model is that one needs a large number of tests to calibrate the model for
a specific material.
As a crack turns or branches, the actual fatigue crack growth resistance may be different compared to the
resistance in the principal material directions, e.g. L-LT or ST-L. Therefore, to fully characterize the inplane anisotropy so called off-axis tests are performed [Metzmacher et al., 2007; Beese et al., 2010; Rioja
& Liu, 2012]. The in-plane anisotropy can be related to both the plastic properties and fatigue properties
of a material. The static fracture mode may vary with in-plane test angle, i.e. either ductile shear fracture
or ductile normal fracture [Beese et al., 2010]. A large series of off-axis tests showed that the allowable
plastic strain to ductile failure, for in-plane variations of the orientation of the microstructure, can be up
to a factor 10x smaller compared to the reference value, the L-direction in the L-LT plane [Metzmacher
et al., 2007]. Furthermore, some orientations, e.g. L30-LT, showed an increase in allowable plastic strain
compared to the reference value.
Fatigue fracture anisotropy.
The demand for aluminium alloys with higher static strength has led to the development of alloys such as
Al 7085-T7651 and various artificially aged Al-Li alloys, e.g. Al 2050-T84. Early tests on first generation
Al-Li alloys seemed promising, as an increase in static strength was observed, accompanied by a decrease
in density and an increase in fatigue life, due to frequent crack branching and turning [Suresh et al.,
1987; Rao et al., 1988]. This branching and turning decreases the available crack driving force, effectively
slowing crack growth. However, this increased fatigue life is not present in all material orientations. On
the contrary, in some material orientations, the fatigue strength is actually reduced [Rao et al., 1988;
Dursun & Soutis, 2014]. Two orientations which show a significant decrease in fracture toughness and
fatigue strength are the ST-L and LT-ST orientation, for reference the orientations can be found in Figure
2-4. Furthermore, in-plane fatigue resistance anisotropy can lead to crack deviation, as was observed in
tests with Al-8090 [Rioja & Liu, 2012].
Next, this type of aluminium alloy is more likely to show intergranular crack paths and delamination
mechanisms [Sinclair & Gregson, 1994; McDonald, 2009; Hernquist, 2010]. Associated with an
intergranular crack path is an increase in fatigue crack growth rate (FCGR) as the grain boundaries are
weakened due to several aspects, see section 3-1. An example of delaminations in a component can be
seen in Figure 3-3. Delaminations are a fairly unique crack growth mechanism in aluminium alloys and
occur mainly on planes perpendicular to the ST-direction [Gregson & Sinclair, 1996]. Precipitates can
contribute to delamination tendency, on delamination surfaces T1 or T2 precipitates are found together
with the presence of PFZ’s [McDonald, 2009; Hernquist, 2010]. Slip banding is also found to contribute,
supported by strain-rate insensitivity or negative strain-rate sensitivity [Hamel, 2010]. This promotes
local instabilities rather than damping them, causing premature fracture.
3-3 Branch forming and crack turning
13
Figure 3-3: Crack arrestor delaminations during fatigue test of integrally stiffened structure in a 3rd generation Al-Li alloy
[Messner, 2014].
Figure 3-3
3-3 Branch forming and crack turning
3-3 Especially in the late 1980’s and 1990’s research on crack branching and turning in Al-Li alloys, but also
other Al-alloys, has been extensive [Suresh, 1985; Suresh et al., 1987; Rao et al., 1988; Rao & Ritchie,
1992; Sinclair & Gregson, 1994; Wanhill, 1994; Gregson & Sinclair, 1996]. The first aim was to establish a
(microscopic) reason for the observed crack paths in Al-Li alloys. As mentioned in Section 3-1 and 3-2,
the anisotropic grain structure and slip planarity are linked to the observed zig-zag crack paths. Kmax is
identified as a key parameter in explaining crack turning in different test specimens [Sinclair & Gregson,
1997; Crill et al., 2006].
Delaminations play a major role in determining the FCGR and crack path, as they influence the stress
field surrounding the crack [Kalyanam et al., 2009; De et al., 2011]. The exact origin of delaminations is
not known yet, however researchers have been able to identify some parameters influencing this process.
The local shear stress, both in a branched and non-branch configuration, and the weakened intergranular
strength play a role in the formation of delaminations [Kalyanam et al., 2009; Hamel, 2010].
In the Al 7050-T7451 alloy it is found that branches are formed frequently, when the material is loaded in
the ST90-L configuration, an example can be seen in Figure 3-4. Although the apparent FCGR is slowed,
it is noted that this is due to the crack splitting branches [Bao et al., 2013]. Also it is observed that the size
of the branches increases with increasing main crack length and higher ΔK values.
In the research of Schubbe fractography revealed indications of crack branching at the internal centerline
of the specimen, i.e. the mid-thickness plane, almost immediately after the test start. An intergranular,
delamination-type failure mechanism dominated these branches, which formed along the roll plane
[Schubbe, 2009]. Crack branching is also observed in AA2324-T39, the higher strength version of Al
2024-T351 [Zhang et al., 2014]. One of the reasons this Al-Cu-Mg alloy shows crack branches, is the
relative large grain size. A higher loading level seemingly increased the tendency for crack branching.
14
3 Literature review
Figure 3-4: Macroscopic crack deflection in thick AA7050-T651 plate, ST90-L fatigue specimen [Joyce & Sinclair, 2014].
Figure 3-4
3-4 Mixed-mode fatigue fracture
3-4 The tendency for intergranular failure in Al-Li alloys leads to a change in mixed-mode fatigue crack
growth behavior, compared to isotropic alloys, such as Al 2024-T3. A characterization of this mixedmode behaviour is done by several researchers [Bush et al., 1993; Kfouri & Brown, 1995; Sinclair &
Gregson, 1997; Forth et al., 2004; Kamp et al., 2007a] [Joyce & Sinclair, 2014].
In the research of Sinclair and Gregson, three-point bending tests are used to characterize the initial
deflection angle of a straight crack in Al 7050‑T7451 under mixed-mode conditions. The use of LEFM
for predicting the initial turning angle worked quite well, for low Kmax values. Hereafter, intergranular
failure started to influence the crack path, and the deflection angle started to decrease. A plot of the
initial deflection angle can be seen in Figure 3-5. With the use of fractography, it is concluded that the
fracture surfaces at high Kmax values are more sheared in nature and show more intergranular facets
[Joyce & Sinclair, 2014]. The fact that sustained coplanar crack growth is favoured at high KII/KI values, is
found to be consistent with intergranular failure resulting from shear strain concentration in the PFZ’s.
For isotropic materials the initial deflection angle of a straight crack in mixed-mode conditions is not
expected to show a similar relation with respect to Kmax. The angle is more dominated by the mixed-mode
ratio. At a certain ratio, which is different for each material, a switch is made between the maximum
tangential stress (MTS) and maximum shear stress (MSS) regime for the crack growth direction [Pettit et
al., 2013]. In the MSS regime the deflection angles are negative for a positive shear load, while for MTS
the opposite is true.
3-5 Conclusion
15
Figure 3-5: Initial crack propagation angle, θ, in Al 2297 / 7050 ST-L SEN(B) coupons for various mixed-mode ratios and Keq,max
levels. Also shown are the LEFM predictions [Joyce & Sinclair, 2014].
Figure 3-5
3-5 Conclusion
3-5 In this chapter, a literature review is presented on the microstructure and on the macrostructural
observations in experiments on artificially aged aluminium(-lithium) alloys, such as Al 2050-T84. In
rolled plate products and extruded alloys often an anisotropic grain structure is observed, with large
dimensions in the rolling-, or L-direction, and small dimensions in the ST-direction. Precipitates
found in these alloys contribute to the fracture toughness anisotropy, as they increase the tendency for
intergranular fracture. The formation of precipitate free zones during the heat treatment increases this
tendency as well, as the primary strengthening precipitates are absent in these zones. Slip planarity and
micro-voids, a result of the liquid phase of some impurities, both contribute to the increased tendency
for intergranular fracture as well.
Static and fatigue fracture anisotropy are observed in a wide variety of aluminium and steel alloys. The
anisotropy in allowable plastic strain before fracture is often linked to the microstructure of these alloys.
The material plane which shows the most prominent fracture anisotropy is the ST-L plane. The crachFEM
model allows to account for this anisotropic fracture resistance, together with a way to deal with two
competing failure mechanisms, ductile shear and ductile normal fracture.
Crack turning and crack branching in advanced aluminium alloys is reported by many researchers.
Factors determined to have an influence on the tendency for crack branching and/or turning are: the load
level, often represented by Kmax, the intergranular fracture strength of the alloy and the microstructure.
Delaminations play a key role in fatigue fracture anisotropy and the initiation of branches. A delamination,
which mainly occurs on planes perpendicular to the ST-direction, influences the local stress field, and
can affect the crack growth rate and possibly the crack path.
In mixed-mode loading conditions the increased tendency for intergranular fracture changes the crack
path, compared to isotropic alloys, such as Al 2024-T3. It was shown that for increasing values of Kmax,
intergranular fracture started to dominate the crack path, the mixed-mode ratio influences this as well.
Where for low loading levels LEFM predictions for the crack path are reasonably accurate, at higher levels
this does not hold anymore. With fractography it is concluded that at higher loading levels the fracture
surfaces become more sheared in nature and show more intergranular facets.
16
3 Literature review
Chapter 4
Requirements numerical method
It is important to establish the requirements of the numerical method before a solution is sought and
implemented. Section 4-1 presents the requirements for a numerical method to predict crack turning and
branching in advanced aluminium alloys. In section 4-2 an overview is provided of the methods which
currently exist and which are possibly best suited to meeting parts of these requirements. This chapter is
concluded with a discussion the available methods found in literature and the required additions to the
available methods.
4-1 Requirements on numerical method
4-1 Requirements connected to the research goal
The main goal of this study is to develop a numerical method capable of predicting crack branching and
crack turning in the latest generation aluminium alloys. The numerical method can be calibrated with
the aid of the fatigue coupon tests.
From this goal some requirements flow down to the numerical method. First, the method should be able
to predict fatigue crack growth (FCG) with a LEFM framework. Furthermore, it should be able to predict
both crack turning and crack branching, which result from the anisotropy in fracture toughness and
microstructure in these alloys. This study focuses on in-plane crack turning and crack branching in the
crack-arrestor configuration. The fatigue experiments conducted in this study can be used to calibrate the
numerical method. Last, the method needs to be as generally applicable as possible, to facilitate the use in
multiple crack growth scenarios, and make it suitable for multiple materials and structures.
17
18
4 Requirements numerical method
Requirements related to the implementation
To make the method as user friendly as possible some requirements are put on the implementation
of the numerical method. First, the crack propagation should be done automatically, this significantly
reduces the required interaction between the user and the tool. Next, the method should be as general as
possible, in order to maximize its usefulness for predicting crack growth in advanced aluminium alloys.
The model generation within the numerical domain should be as easy as possible and changes to a certain
standardized specimen geometry should be easily made. Lastly, the computational time needed for the
analyses should be minimized, while retaining accuracy within crack growth and crack driving force
predictions.
4-2 Current available methods
4-2 Cracks within a structure represent a very strong discontinuity in stiffness and various other material
properties. Analytical methods show that a very steep stress gradient is to be expected in the vicinity
of the crack tip. This poses a modelling challenge, as this dictates the use of a very fine mesh around
the crack tip in order to accurately capture the stress gradients. Commonly three methods are used to
represent cracks within a structure: the extended finite element method, cohesive zone modelling and
adaptive remeshing. Below each of these is discussed and the most suitable method is chosen.
Extended finite element method
The extended finite element method (XFEM) is a relatively new method of dealing with cracks, or other
discontinuities, in a FE-environment. The basic idea of XFEM is to model the crack as a discontinuity,
where the nodes around the crack and crack-tip are enriched to model the additional degrees of freedom.
By using a Heaviside step function, along with a crack tip function and element subdivision, the crack can
be modelled in an accurate manner. The advantage of XFEM is that it inherits the characteristics of the
basic mesh and there is no need to change the mesh in order to incorporate a strong discontinuity, such
as a crack. XFEM does not provide a way to model damage in the material, a suitable material model still
has to be implemented [Moës et al., 1999; Moës & Belytschko, 2002; Giner et al., 2009]. A large number
of researchers has used XFEM to investigate cracks and damage mechanics in various solid materials,
e.g. concrete, aluminium and composites [Mohammadi, 2008]. With LEFM and XFEM a very reasonable
approximation can be made of the crack growth in mixed-mode conditions [Dumstorff & Meschke,
2007; Giner et al., 2009]. Currently, the XFEM method is incorporated within various FE-packages, e.g.
Abaqus/CAE. However, to obtain accurate results and allow for fatigue damage modelling, often the use
of user-subroutines is required [Giner et al., 2009; Xu & Yuan, 2009; Shi et al., 2010]. This makes the
XFEM method rather time consuming and requires detailed knowledge of the mathematical method
behind XFEM.
Cohesive zone modelling
With the use of cohesive elements, crack growth is simulated as the degradation of cohesion in an element,
causing the gradual reduction of shear and normal stress transfer across an element [Roy & Dodds, 2001].
In cohesive zone modelling (CZM) special-purpose cohesive elements provide the traction between the
regular elements. With the use of a traction-separation law and appropriate traction degradation models,
crack growth can be simulated. The use of CZM frequently requires a priori knowledge of the crack path,
as cohesive elements have to be placed in the mesh to allow the numerical model to simulate a cohesive
crack. With proper calibration of the traction-separation models, accurate results can be obtained for
4-3 Required additions
19
mode-I and mixed-mode conditions [Roy & Dodds, 2001; Maiti et al., 2009]. However, to incorporate
fatigue crack growth in CZM, the use of cyclic dependent traction-separation models is required
[Ural et al., 2009]. This cyclic-dependent material behaviour is quite difficult to accurately model and
measure. Also there are difficulties associated with non-isotropic materials [Zhang & Paulino, 2005].
Adaptive remeshing
In order to capture the effects of crack tip stress singularity and to calculate the strain energy release rate,
a refined mesh in needed around the crack tip [Bouchard et al., 2003]. XFEM partially solves this issue
by introducing extra degrees of freedom in the elements around the crack tip, however still a relative
fine mesh is needed in order to reflect the singularity [Vethe, 2012]. One can also make use of adaptive
remeshing (AR) to have a refined mesh around the crack tip, while keeping numerical efficiency [Khoei et
al., 2008; Maligno et al., 2010; Khoei et al., 2012]. However, AR of a numerical model can be become
very demanding in complex geometric structures or structures incorporating discontinuities. A blocked
approach can bring an outcome, as the region with the crack is captured in a different block than the
remaining structure [Maligno et al., 2010]. One of the main issues with AR is to determine if a mesh is
refined enough in a certain region, and vice versa if the mesh can be coarsened. By comparing the FE
results to known analytical solutions the FE error can be estimated. The mesh is then adapted, such that
the error around the crack tip is reduced, while computational efficiency is maintained by using a coarser
mesh where possible [Khoei et al., 2008; Khoei et al., 2012].
Method implemented
The aforementioned methods, i.e. XFEM, CZM and AR, each provide their own benefits and come at their
own costs. The extended finite element method could provide very accurate results, without requiring a
very detailed mesh. Yet it requires significant effort to implement into a FE-package. The cohesive zone
modelling approach also can provide very accurate results and allows for modelling of complex crack
growth phenomena, such as crack retardation. But the required a priori knowledge of the crack path and
associated difficulty with calibration pose a challenge. Finally, adaptive remeshing provides the benefit of
a computationally efficient mesh, while keeping the needed refinement at the crack tip. It may however
be difficult to determine if the mesh is refined enough, or too refined for that matter. Because of the ease
of implementation and the relative high accuracy provided, adaptive remeshing is chosen as the most
suitable method for this thesis.
4-3 Required additions
4-3 Adaptive remeshing only covers the part how to incorporate an arbitrary crack within a FE-model, a
suitable material model is still needed. AR in this case does not require a special cyclic damage model,
because crack propagation is realized as a series of static analyses, where the stress distribution provides
the crack propagation direction and crack driving forces. So the required additions are solely focused
on the LEFM framework. Current LEFM methods do not predict the initiation of branches and do not
accurately predict crack turning in anisotropic solids [Xie et al., 2011; Pettit et al., 2013]. Therefore, the
required additions are: to allow for the initiation of branches, to allow for modelling of fatigue fracture
anisotropy and to predict crack turning within anisotropic solids.
20
4 Requirements numerical method
4-4 Conclusion
4-4 The requirements on the numerical method have been presented in this chapter. It is identified that
current linear elastic fracture mechanics methods do not allow for prediction of branch initiation or
crack turning in anisotropic solids. Furthermore, three methods have been investigated, which can be
used to represent a crack within in numerical model: the extended finite element method, cohesive
zone modelling and adaptive remeshing. Because of the ease of implementation and the accuracy of
the results, the method best suited for this study is adaptive remeshing. The required additions to the
LEFM framework are the modelling of fatigue fracture anisotropy, and the prediction of the initiation of
branches in the crack-arrestor configuration and in-plane crack turning within anisotropic solids.
Chapter 5
k2-method
This chapter presents the first numerical method that can be used to predict crack turning and branching.
The k2-method is implemented into a FE-environment, namely in Abaqus/CAE. In the first section an
overview is given of the model origin. Hereafter, the method’s process flow is explained in section 5-2.
Next, details are given on each specific module, explaining all involved steps. Lastly, this method is
calibrated and some remarks are given on its use.
5-1 Model origin
5-1 The formation of branches in advanced aluminium alloys is reported by many researchers, as mentioned
in section 3-3. Characteristics of the microstructure have been linked to the initiation and subsequent
growth of these branches. The nature of the branches seems to be mostly intergranular, as discussed in
section 3-2 and 3-3. With the use of a crystal plasticity model, McDonald determined that a branch in the
crack-arrest configuration, i.e. in-plane branching, initiates due to both the shear and normal stress around
the crack tip. Also, it is concluded that the grain boundary shear interface strength was approximately
1/3 of the axial yield stress. This supports the notion that a weakened material plane exists along the
elongated grain boundaries, that may contribute to branching or delamination [McDonald, 2009].
In mixed-mode conditions it is observed that at higher Kmax values fracture surfaces of alloys such as
Al 7050-T7451 appear to be more sheared in nature and show more intergranular facets. This is linked
to the intergranular failure from shear strain concentrations in PFZ’s [Joyce & Sinclair, 2014]. Other
research shows a correlation between Kmax, or ΔK, and the length of branches [Kalyanam et al., 2009;
Schubbe, 2009]. Branches can also lead to an apparent decrease in the FCGR, especially crack dividing
branches have been observed to have this effect [Bao et al., 2013].
As just presented, many researchers have observed crack branching in artificially aged aluminium alloys,
such as Al 2050-T84 or Al 7050-T7451. The formation of branches is linked to shear strain concentration,
shear stress and the values of Kmax, or ΔK. In particular the formation of delamination type branches
seems to be largely influenced by shear stress and the weakened intergranular strength [Kalyanam et al.,
2009; Hamel, 2010]. Therefore, it is hypothesized that the intergranular shear strength and the local shear
stress are the two main parameters which define both the initiation and the length of a branch. Once
the local shear stress surpasses the intergranular shear strength a branch is formed of a certain length.
21
22
5 k2-method
This length should be sufficient to lower the local shear stress below the intergranular shear strength,
otherwise fracture will continue to occur. It is known that a delamination influences the local shear
stress field, therefore it is difficult to determine the exact length of branch, as also small variations in the
microstructure may influence this process, as presented in Section 3-1 and 3-3.
In a numerical framework the procedure of finding the appropriate branch length can therefore be an
iterative procedure. First, an initial guess is taken for the branch length. Hereafter, a new analysis provides
the local shear stress field, which then allows to evaluate if the branch continues to grow or is arrested.
Subsequently, if the branch has not reached it’s final length, another guess is taken for the additional
extension length of the branch. Otherwise, normal mixed-mode crack growth can be resumed.
The initiation and growth of branches is only one of the requirements of the numerical method, it should
also be able to predict crack turning. Several researchers have observed crack turning in fatigue tests
[Wu et al., 1994; Sinclair & Gregson, 1997; Forth et al., 2004; Crill et al., 2006; Joyce & Sinclair, 2014]. In
C(T) fatigue tests crack path orientations in between the grain direction and symmetry axis are found
[Wu et al., 1994; Forth et al., 2004]. These tests are so called off-axis tests, i.e. the material axes are rotated
in-plane with respect to the loading direction. From these tests it is concluded that the orientation of the
material axis primarily determines the crack growth direction in such off-axis conditions.
Another key parameter for the crack growth direction, especially in mixed-mode conditions, is Kmax
[Sinclair & Gregson, 1997; Joyce & Sinclair, 2014]. As seen in Figure 3-5, an increase in Kmax leads to a
lower initial crack propagation angle of a straight crack in mixed-mode loading. The coupons used for
these tests are loaded in the ST-L configuration. So, with increasing Kmax the cracks become more aligned
with the L-direction in the ST-plane. In alloys like Al 7050-T7451 this direction is known to have lower
crack growth resistance, see Section 3-2.
Figure 3-5 also shows that mode-mixity ratio (M) has influence on the crack growth behaviour of
the tested materials. A higher KII/KI ratio promotes sustained coplanar crack growth. The fact that
coplanar growth is favoured at a higher M is consistent with shear strain failure of the grain boundaries
[Joyce & Sinclair, 2014]. With the use of a LEFM simulation, Joyce and Sinclair investigated the local
crack tip loading of a macroscopically deflected crack and compared those results with experiments. They
found that a broad correlation exists between the predicted and the experimental values for the crack
length at crack deflection. The predicted length was calculated using a Kmax and mode-mixity criterion.
Their results show that as an initial approximation the onset of macroscopic crack deflections appears to
be controlled by macroscopic mixed-mode crack growth criteria.
Based on the result that macroscopic mixed-mode crack growth criteria seem to govern the initiation
of crack deflections, it is hypothesized that the mechanism, which governs the growth of branches, also
affects the tendency for coplanar crack growth. The increased tendency, associated with higher M and
Kmax values, could be explained by the local shear stress. Increasing either Kmax or M leads to a higher shear
stress, which increases the tendency for grain boundary failure, especially in the weaker L-direction, see
section 3-2.
Essentially, it is hypothesized that observed macroscopic crack deviation is governed by the formation of
intergranular crack extensions in the weak plane of the alloy, governed by a local shear stress criterion.
Macroscopic crack deviation then applies to both branch initiation and off-axis crack growth. In a
numerical model the iterative nature of the crack growth process, as described in this section, would not
lead to the actual crack path, as observed in experiments. Rather an equivalent crack path is constructed,
which if the crack growth increments are kept reasonably small, can approximate the actual crack path.
In Figure 5-1 an example is given of such an equivalent and off-axis crack path.
5-2 Method process flow
23
Y position
Equivalent crack path
Off−axis crack
Equivalent crack
X position
Figure 5-1: Straight off-axis and equivalent crack paths.
Figure 5-1
Concluding, in the model crack growth is determined with the use of LEFM mixed-mode equations. If
and where a crack branch initiates is determined using a criterion based on the local shear stress and
intergranular shear strength. To calculate the correct length of a crack branch is an iterative procedure, as
a branch extension alters the local stress field.
5-2 Method process flow
5-2 In this section a clear overview is given of the numerical implementation in a FE-framework of the
k2‑method. The flowchart of one cycle of the numerical process is shown in Figure 5-3. Hereafter, a short
description is presented of each of the modules in the flowchart, the equations that are used can be found
in section 5-3. The FEM program that is used is Abaqus/CAE, use of Python is made to ease model
generation, perform intermediate calculations and perform part of the post-processing. Also Python is
used to create a loop for the analysis cycles and loops until the desired number of cycles has been reached.
As mentioned in section 4-3, no actual crack growth is simulated in the numerical models, rather a series
of static analysis provides the stress state around one particular crack configuration. The stress field can
then be used to determine the crack extension and growth direction. So, the analysis cycle shown in
Figure 5-3, is repeated until the desired number of analysis cycles is reached
The first step is to generate a FE-model, based on the known geometry of the coupon and known crack
tip coordinates within the coupon. For the first analysis cycle a predetermined crack is provided. The
geometry and the crack are parameterized, and a FE-model is generated using Python. A very refined
mesh is used in a small region around the crack tip, to ensure that the expected stress gradients can be
captured. By using a refined mesh around the crack tip and coarsening the mesh in the rest of the model,
the computation time for one cycle can be kep low, while the results are expected to be quite accurate.
The second module, FE-simulation and SIF-extraction, provides the FEA of the previously generated
model and extracts the 2-D stress intensity factors, KI and KII . The SIF’s are then used as input for the
next step in the analysis cycle.
Crack kinking theory by Cotterell and Rice, see section 2-3, provides a field of virtual stress intensity
factors for all possible crack growth directions. This is used in the third module to calculate k1 and k2 in
the weak plane of the material, in this case the L-direction. k1 and k2 are the virtual stress intensity factors
of an infinitesimal crack extension. Basically this is a measure for the available crack driving force in that
particular direction. Figure 5-2 shows how to determine the crack kink angle, Δθ, needed to calculate
24
5 k2-method
k1 and k2, using equation (2-5) and (2-6). As can be seen in Figure 5-2 the symmetry axis of the coupon
serves as the reference line. If no symmetry line exists, then a line perpendicular to the principal loading
direction would also suffice.
P
k1(Δθ), k2(Δθ)
Δθ
ST
Ref. line
L
P
Figure 5-2: Infinitesimal crack extension in the L-direction, with associated k1 and k2. Also shown is the crack kink angle Δθ and
the reference line.
Figure 5-2
Based on the values of k1 and k2 in the L-direction of the material, it is determined if a branch is formed.
The criterion for branch initiation is formulated such that a branch initiates, if the local shear stress field,
generated by an infinitesimal crack extension, is high enough to surpass the intergranular shear strength.
If this is not the case, then normal mixed-mode crack extension will occur.
In the case of mixed-mode crack extension, LEFM equations give the crack growth direction, and a
predetermined crack extension length is used. If a branch should initiate, then the branch length
is calculated such, that its length is a reasonable guess for the local shear stress to decrease below the
intergranular shear strength.
In the module 6, based on the input of the crack extension length and direction, new crack tip coordinates
are determined. These new coordinates are then added to the series of points describing the old crack
geometry. In the next analysis cycle this is used to generate the new crack geometry.
start
1. Model
generation
2. FEA and SIF
extraction
3. Calculate k1 and k2 in
weak plane material
4. Determine if a
branch is formed
5A. Crack extension
mixed-mode
5B. Calculate
branch length
6. Determine new
crack tip coordinates
Figure 5-3: Flowchart of one analysis cycle in the k2-method.
Figure 5-3
5-3 Details on modules
25
5-3 Details on modules
5-3 In module 1 and 2, Abaqus/CAE is used to generate the models and perform the FEA. The models are
generated as 2-D plane strain shell structures, which gives representative conditions for the mid-thickness
of the coupon. As the mid-thickness plane of the coupon gives the most constrained stress field, it is
expected that branches initiate there first [Schubbe, 2009; Bao et al., 2013]. The loading is introduced into
the coupons via couplings, which ensure proper distribution of the forces. Relevant boundary conditions
are placed to prevent rigid body motions.
The SIF’s are calculated using contour integrals readily available in Abaqus/CAE. A total of 15 contour
integrals are requested, where an average over the last five contours is used to find the SIF’s. The values of
the SIF’s are stored for post-processing. Also, the calculated displacement of the upper loading reference
point is stored, which can be used for comparison with the fatigue tests.
The description of the numerical model starts at the module 4, as as the equations to calculate k1 and k2
are given in section 2-3. The initiation of a branch occurs when the local crack driving force in the weak
plane of the material exceeds the intergranular crack resistance. The local crack driving force is expressed
in terms k1 and k2. The intergranular crack resistance can also be represented by a SIF. It must be noted
that the simplifications of SSY-conditions and a K-dominated stress field merely are abstractions of the
actual mechanisms on a grain level to a macroscale. Clearly, this does not model the effects of a microbranch on the stress field, yet it allows for the analysis of a macroscale branch. Basically, the branch
initiation criterion checks if k2 exceed the permissible value:
k 2 ≥ k 2,allow
(5-1) The length of a branch is determined in module 5B. In SSY-conditions, the K-field is dominant for the
stress field surrounding a crack tip. Hence, the local shear stress can be expressed in terms of the SIF’s of a
virtual crack extension. As explained in section 5-1, the length of a branch is a guess for the length needed
to decrease the local crack driving force below the permissible values. This is done using a quadratic
relation between k2 and the intergranular strength τmax:
Lbranch =
c
k 22
τ max 2
(5-2) Here, c and τmax are material parameters. Potentially one could measure the intergranular strength τmax.
The coefficient c is used to more adequately determine the branch length. This coefficient can prevent an
overshoot, or undershoot, in the estimated branch length. The coefficient c is dimensionless. A value of
1.56 for c is used in this study, and τmax can be used to adjust the predicted crack branch length. For τmax a
value of 170 MPa seems reasonable and is used as reference value for the calibration in section 5-4.
The mixed-mode crack direction is determined in module 5A. The preferred crack growth direction is
assumed to be the direction which maximizes KI, or k1 in case of a virtual crack extension. This direction
is called the maximum tangential stress direction, MTS. Using equation (2-5), after differentiating and
rewriting one gets:
∆θmixed
1 − 1 + 8 K K
( II I
=
2 arctan 

4 (K II K I )

)
2




(5-3) Here, KI and KII are the stress intensity factors of the crack, and Δθmixed is the crack growth direction. The
length of the crack extension is predetermined and kept sufficiently small to obtain an accurate crack
path prediction.
26
5 k2-method
5-4 Calibration
5-4 In this section the calibration of the k2-method is discussed. The coupon geometry chosen to calibrate
the FEA results is a C(T) coupon, because of the ease of modelling with the presence of only one crack.
The ST30-L orientation serves as reference for the simulations in this section. First, the parameter is
identified which best can serve as steering parameter for the average crack direction. Here, the average
crack direction is referred to as the average crack path angle of the equivalent crack path generated by
the k2-method.
By systematically varying the three dominant parameters of the model k2,thres, τmax and Δap, it is found that
τmax provides the best steering capability, see Figure 5-4. The baseline value of τmax is 170 MPa, this value
is used in the remainder of this study, unless mentioned otherwise. The predictions for the crack path
angle are relatively insensitive to variation in Δap, the predetermined crack extension length, see Figure
5-5. However, the effects of variations in k2,thres are not well predictable, as can be seen in Figure 5-6. For a
too low value of k2,thres crack extension is exclusively by branching, while for a high value a slightly lower
average angle is achieved.
It must be said that the value of k2,thres is dependent on the initial value of Kmax, i.e. at the analysis start.
This makes the choice for k2,thres rather arbitrary and a suitable value seems to be one slightly higher than
k2 calculated for the first analysis cycle. Although this variation of the branching threshold likely does not
represent reality anymore, the reasoning for the presence of such a threshold may still be valid.
Effect of τmax on crack path angle θavg
25
Original τmax
τmax −25%
τmax +25%
θavg [deg]
22.5
20
17.5
1
2
3
4
5
6
7
8
9
10
11
12
Iteration [−]
Figure 5-4: Plot of the effect of τmax on the predicted crack path angle.
Figure 5-4
Effect of Δap on crack path angle θavg
25
Original Δap
Δap = 0.5 mm
Δap = 1.5 mm
θavg [deg]
22.5
20
17.5
1
2
3
4
5
6
7
8
9
10
11
12
Iteration [−]
Figure 5-5: Plot of the effect of Δap on the predicted crack path angle.
Figure 5-5
5-4 Calibration
27
Effect of k2,thres on crack path angle θavg
32.5
Original k2,thres
k2,thres −25%
k2,thres +25%
30
θavg [deg]
27.5
25
22.5
20
17.5
15
1
2
3
4
5
6
7
8
9
Iteration [−]
Figure 5-6: Plot of the effect of k2,thres on the predicted crack path angle.
Figure 5-6
In order to calibrate the k2-model, the FEA results are compared with experimental results obtained from
an internal Airbus report. In Figure 5-7 the differences between the predictions and the experiments are
presented. Again as coupon geometry a C(T) is used, with an initial Kmax around 8.1 MPa√m. One can see
that the errors made in the FEA predictions are in the order of a couple of degrees. It must be noted that
all crack path angles have been overpredicted, i.e. the observed crack paths have a lower angle. The reason
that ST00-L provides a perfect match to the report, is the fact that the crack extension is pure mode-I
and the expected crack path is aligned with the ST-axis. In Chapter 9 a comparison is made between the
predicted crack paths with the k2-method and the crack paths observed in the fatigue coupons tested in
this study. This comparison is based on the actual crack paths, not the average crack path angle.
During the calibration of the k2-method, it is observed that the equation used to calculate the branch
length results in very small lengths at low Kmax values. Furthermore, many consecutive branches are
needed to decrease the local k2 below the threshold value and switch back to mixed-mode crack growth
criteria. For this reason, a minimum branch length is suggested to speed up the numerical process and
overcome the modelling issues associated with very small branch lengths. A value of 0.5mm proved to be
effective, and is used in the FEA to calibrate the k2-method.
Error of predicted crack path angles
4
ε [deg]
3
2
1
0
ST00−L
ST15−L
ST30−L
Orientation
ST45−L
Figure 5-7: Error in degrees of the predicted crack path angles with the k2-method for four coupon orientations.
Figure 5-7
28
5 k2-method
Overall, one can say that the predictions for the off-axis crack angle of the k2-method are reasonably
accurate and therefore are considered calibrated. However, the arbitrary value of k2,thres is a point of
concern, that could show this macroscopic method does not represent the physical mechanisms on the
microscopic scale correctly. The concept of a branch formation threshold, in terms of crack driving force,
can still be true. Yet it likely does not explain why off-axis crack growth is possible in a range of Kmax
values.
Chapter 6
Pettit-method
This chapter presents the second numerical method that can be used to account for crack turning and
fatigue fracture anisotropy. This method is based on the works of Pettit, and is implemented into a FE‑
environment, namely in Abaqus/CAE. In section 6-1 an overview is given of the model origin and some
adjustments made to the model based on observations from literature. Next, the general process flow is
explained to give an overview of the numerical implementation. Thirdly, details on each specific module
are presented and the model parameters are established. In the last section the method is calibrated and
some remarks on its use are given.
6-1 Model origin
6-1 The second numerical method to predict crack turning and branching is based on the studies of Pettit
[Pettit, 2000; Pettit et al., 2013]. In those two studies, a set of crack propagation equations is developed
that can handle a wide variety of loading conditions, and can hadle materials with fracture resistance
anisotropy and fracture mode asymmetry.
Within the aerospace industry a trend can be seen towards more integrated structures, and the use of
artificially aged aluminium alloys is increasing. In integrated structures, crack turning can serve as a
mechanism to slow crack growth or even arrest it completely. An example of crack turning is the observed
deviation of a crack growing in a fuselage panel close to a stiffener. Under influence of a stiffener the crack
will turn in the direction of that stiffener. This considerably decreases the available crack driving force,
leading to slower crack growth or even arrest.With integrated structures the fuselage panel and stiffeners
are made from the same bill of material, creating a seamlessly connected component. A very similar
effect on the crack path is expected of a stiffener in an integrated and non-integrated structure. As shown
in Chapter 3, artificially aged aluminium alloys often exhibit fracture toughness anisotropy, both in
static and fatigue conditions. So, the need is apparent for a crack propagation criterion that can handle
crack turning and fracture toughness anisotropy.
The starting point for this method is the theory that a crack most likely grows in the direction where the
relative crack resistance is minimal. This concept is aligned with the principle of minimum total potential
energy. Formally, the direction in which a crack in an arbitrary structure or material will grow can be
found by maximizing the ratio of available crack driving force over crack resistance, or:
29
30
6 Pettit-method
crack driving force (θ )
crack resistance (θ )
(6-1) max
Both quantities are given as a function of θ, the crack orientation. Although more parameters may
influence the crack driving force or crack resistance, the basic idea remains that the maximum of this
ratio will provide the preferred crack direction. A good approach to evaluate both quantities is to use a
modified strain energy release rate (MSERR) [Kfouri & Brown, 1995]. MSERR uses the sum of the cubes
of the ratio of crack driving force over fracture toughness for each individual mode. The crack is assumed
to be critical if the sum of those cubes is equal to 1. In a sub-critical loading state, the desired crack
propagation direction can be obtained by maximizing sum of the cubes of the relative crack driving force.
Not all materials show a gradual transition between tensile dominated and shear dominated crack growth.
Rather a sharp transition is exhibited between the tensile and shear dominated regime. This leads to a
so called modal criterion, where the crack grows in a direction which either maximizes tensile or shear
stress. In this case the crack is said to be critical, when either the tensile or shear crack driving force
exceeds the fracture toughness for that particular mode. In section 6-4 it is investigated if the artificially
aged alloys, used in this thesis, show either modal or MSERR crack growth behaviour.
The modal and MSERR criteria account for fracture mode asymmetry, however not yet for fracture
resistance anisotropy dependent on the orientation of a crack. The term fracture mode asymmetry is first
described by Kfouri and Brown [Kfouri & Brown, 1995]. It describes the difference in fracture resistance
of the three different crack growth modes within a material. In order to incorporate anisotropy dependent
on the orientation of a crack, regardless of the fracture mode, an equation is needed for the fracture
resistance dependent on the orientation of the crack. With such an equation, the fracture resistance
can be scaled to the appropriate value, for an arbitrary orientation of a virtual crack extension. A polar
interpolation function is suggested to accomplish this scaling:


1
K P (α ) = 
 cos 2 α sin2 α
+

n
K p ,2n
 K p ,1






1
n
(6-2) KP is the fracture resistance for a particular direction α, which is the angle to the reference material
orientation, e.g. the ST-direction. Kp,1 and Kp,2 are the relative fracture resistance values of the two in-plane
principal material orientations. The coefficient n is used to adjust the shape of the interpolation function.
In Figure 6-1 one can see the polar interpolation function for various values of n. For large values of n, Kp
almost instantly drops to the value of Kp,2. On the contrary, for negative values of n, Kp remains close to
the value of Kp,1 for most angles. With this interpolation function the fracture resistance can be scaled to
the value belonging to that particular orientation of the (virtual) crack extension.
With the scaled fracture resistance the crack propagation direction is found when either the modal or
MSERR criterion is maximized. In this thesis, the fracture toughness anisotropy that is accounted for
is in-plane anisotropy, where the material is loaded in a direction that lies in a principal material plane.
This could be for example the observed anisotropy in a L45-LT coupon. A full 3-D fracture anisotropy
criterion is given in the research of Pettit, with a vectorial addition to determine the fracture resistance
for any arbitrary direction [Pettit et al., 2013].
6-1 Model origin
31
Polar interpolation KP
1
n = -100
n = -10
n = -5
n = -2
n = -1
n=1
n=2
n=5
n = 10
n = 100
0.95
0.9
0.85
KP [−]
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
10
20
30
40
50
60
70
80
90
θ [deg]
Figure 6-1: Plot of the polar interpolation KP for different values of n, with Kp,1 = 1 and Kp,2 = 0.5.
Figure 6-1
As explained in section 3-4, materials can also show a dependency on Kmax and M for the crack propagation
direction. The effect of the Kmax on the crack deflection angle is most pronounced, see Figure 3-5. M rather
seems to determine if the material has MSERR or modal crack growth behaviour. At a low mode-mixity
(M ≤ 0.35) the observed behaviour of Al 7050 seems to follow the MSERR criterion. In this thesis the
following equation for Kmax is used:
=
K max
K I 2 + K II 2
(6-3) The stress intensity factors of the existing crack, KI and KII, are used here without any scaling parameters.
In order to account for the effect of Kmax on KP(θ), two parameters in that function are made dependent
on Kmax, namely n and Kp,1. The results of SEN(B) tests on Al 7050 / 2297 loaded in the ST-L orientation,
see Figure 3-5, are used to find the optimal values of n and Kp,1. The optimal values minimize the error
between the predictions and the observations. Next, a function can be fitted on these optimal values, to
introduce the dependency on Kmax in these two coefficients. In Figure 6-2 and 6-3 plots are shown with
the optimal values, and the functions that have been fitted through these points.
The function that has been fitted on the results of the optimization for n is a linear relation. For Kp,1
an exponential fit is used. The fitted functions for these two coefficients are calibrated and adjusted, if
needed, in section 6-4.
In this section the general approach is shown for a numerical method that predicts crack turning, and
accounts for fracture mode asymmetry and fracture resistance anisotropy, dependent on both crack
orientation and Kmax. Two criteria have been developed, MSERR and modal, which each characterize
a certain set of materials in certain loading conditions. Although it is hypothesized that a similar
mechanism causes crack branching and crack turning, see section 5-1, the nature of the polar function
for KP prevents this. If one principle material direction has a significantly lower crack resistance, both the
MSERR and modal criterion would lead to a crack turning towards the weak direction. Therefore, this
method is not able to satisfy all criteria as set out in Chapter 4. Specifically, this method does not allow for
the initiation and growth of branches. Yet the results are expected to quite accurately reflect crack turning
in anisotropic alloys.
32
6 Pettit-method
n depence on Kmax
Optimized values
Linear fit
1
0
−1
n [−]
−2
−3
−4
−5
−6
−7
2.5
5
7.5
10
12.5
15
17.5
20
Kmax [MPa√m]
Figure 6-2: Relation for n as function of Kmax as determined with least squares approximation, R2 = 0.995.
KP depence on Kmax
Optimized values
Exponential fit
1
KP,1 [−]
0.9
0.8
0.7
0.6
0
5
10
15
20
Kmax [MPa√m]
Figure 6-3: Relation for Kp,1 as function of Kmax as determined with least squares approximation, R2 = 0.998.
6-2 Method process flow
6-2 A clear overview of the numerical implementation in a FE-framework of the Pettit-method is given in
this section. The numerical process is analogue to the k2-method described in Chapter 5. Crack growth
is simulated via a series of analyses, where in each cycle the crack geometry is updated. The flowchart of
one analysis cycle is shown in Figure 6-4. Next, a short description of each module is presented, equations
are presented in section 6-3. Again Abaqus/CAE is used for the FE-models and Python to aid in model
generation, calculations, and controlling the entire process.
The first two modules are similar to the first two modules of the k2-method. Python is used to generate
a parameterized model, and Abaqus/CAE calculates the stress field and SIF’s of the current crack
configuration. With KI and KII known, module 3 generates the SIF’s of an infinitesimal crack extension
in a number of discrete directions around the crack tip, with a step size of 0.05 degrees. The values for k1
6-3 Details on numerical modules
33
and k2 in all these directions are given as input to module 5. Module 4 calculates the values of Kp(α) for
those same n directions. Module 2 provides KI and KII, to take into account the effect of Kmax. The values
of Kp are then passed on to module 5. There the direction which maximizes k1,eq is calculated. This is
done according to the MSERR or modal criterion, depending on the expected material behaviour. Which
criterion is used, has to be determined on forehand, but can be adjusted per analysis cycle. With a fixed
crack extension length and the crack growth direction from step 5, module 6 generates the new crack tip
coordinates. These new coordinates are then used to generate a new model and start a new analysis cycle.
This process is repeated for a predetermined number of analysis cycles.
start
1. Model
generation
2. FEA and SIF
extraction
3. Calculate k1 and k2
field around crack tip
4. Calculate Kp(α)
6. Determine new
crack tip coordinates
5. Find α that
maximizes k1,eq
Figure 6-4: Flowchart of one analysis cycle of the Pettit-method.
Figure 6-4
6-3 Details on numerical modules
6-3 The first two modules use Python and Abaqus/CAE to model the coupon and crack geometry, and
to perform the FEA. The coupons are modelled as 2-D shell structures with plane strain elements,
representative for mid-thickness conditions. The loading is introduced into the model via couplings to
two loading points, ensuring proper distribution of the forces. Similar to the k2-method, contour integrals
are used to extract the SIF’s. Relevant boundary conditions prevent rigid body motions. The SIF’s and
displacements of the loading points are stored for post-processing.
Crack kinking theory is used in module 3 to find all k1 and k2 values for the all directions that are evaluated.
module 4 finds KP belonging to those same directions using equation (6-2). The value of Kp,2, belonging to
the ST-direction, gets a relative crack toughness value of 1. The value of Kp,1, belonging to the L-direction,
is determined with the exponential fit obtained in section 6-1. The values of the coefficients are given in
Table 6-1, for clarity the function for Kp,1 is presented here:
K p ,1 =
a ⋅ e b ⋅K
max
+c
(6-4) The coefficient n in equation (6-2) is determined using a linear relation for Kmax of the existing crack. The
relation for the parameter n is given below, the values of d and f can be found in Table 6-1:
n =⋅
d K max + f
(6-5) With equation (6-2), (6-4) and (6-5) one can calculate the value of KP belonging to a respective virtual
crack extension direction. These values are passed on to the next module together with the values for k1
and k2 for those same directions.
34
6 Pettit-method
Table 6-1: Values of the coefficients in the relations for n and Kp,1.
Table 6-1
Parameter
Value
a
-6.0E-3
b
2.1E-7
c
1.0
d
-4.7E-7
f
3.6196
In module 5, the desired crack propagation direction is calculated. This is done by finding the virtual
crack extension direction which maximizes the crack driving force according to the MSERR or modal
crack growth behaviour. For MSERR the equivalent crack driving force can be expressed as:
k 1,eq =
K
2
k 1 ( ∆θ )  +  I ,C
K
 II ,C

2
 k 2 ( ∆θ ) 

(6-6) Again k1 and k2 are obtained in module 3, Δθ is the crack kinking angle of the virtual crack extension. KI,C
and KII,C are the fracture toughness values of mode I and II respectively. By maximizing k1,eq one finds the
desired crack propagation direction. Fracture mode asymmetry is taken into account with the ratio of
the fracture toughness values of mode-I and -II, a value of 1.4 is chosen for KI,C/KII,C. Now the integration
with the fracture toughness scaling of KP is straightforward:
k 1,eq=
(α , θ )
1
K p (α )
K
2
k 1 ( ∆θ )  +  I ,C
K
 II ,C

2
 k 2 ( ∆θ ) 

(6-7) max
As one notices, KP, and the infinitesimal SIF’s k1 and k2 are not dependent on the same angle. Rather, α is
the angle of the virtual crack extension w.r.t. the reference material orientation, the L-direction. Δθ is the
angle of the virtual crack extension w.r.t. the existing crack.
For the modal criterion the equivalent crack driving force is not obtained through addition of the
contributions of each mode. Rather it is determined which mode is most critical, by finding the maximum
of the crack driving force scaled by the fracture toughness of its respective mode. This can be expressed
as:
  k ( ∆θ ) 
1

K
α

  p ( )  max
k 1,eq (α , θ ) = max  
 K
,  I ,C
 K II ,C
 k 2 ( ∆θ )  
 

 K p (α )  max 
(6-8) Similar to the MSERR criterion, KP is dependent on Kmax, using again equations (6-2), (6-4) and (6-5). The
choice for the modal or MSERR criterion is predetermined for each analysis cycle, and generally is fixed
throughout all analyses of that particular coupon.
With the desired crack propagation direction known, module 6 determines the new crack tip coordinates.
A fixed crack growth increment is assumed, in this case 1 millimetre. This ensures a reasonably accurate
path prediction and limits computation time.
6-4 Calibration
35
6-4 Calibration
6-4 The calibration of the Pettit-method is presented here. The same geometry is chosen for the calibration
as for the k2-method, a C(T) coupon. Four coupon orientations are used for the calibration: ST00-L,
ST15-L, ST30-L and ST45-L. The MSERR approach is chosen to investigate if the model parameters
have appropriate values and if the crack extension length has influence. With parameter values, as shown
in section 6-3, the difference proved to be too large between the predictions and the crack path angles
observed in an internal Airbus report. Therefore, the coefficients determined in section 6-3 are adjusted
to provide a better fit. Also, a minimum value of 0.9 is introduced for Kp,1, the fracture resistance in
ST-direction. This minimum is chosen to prevent the crack from always turning to the ST-direction at
higher Kmax values, as this is not observed in the internal report. With the parameter values, as shown
in section 6-3, the average difference is more than 5 degrees between the experimenal values and the
predicted average crack path angle. The new values give an average difference of only 1.7 degrees and less
variation in the local crack path angle, as can be seen in Figure 6-5. Table 6-2 presents the new values of
the model parameters.
Table 6-2: Adjusted values of the coefficients in the relations for n and Kp,1.
Table 6-2
Parameter
Value
a
-6.0E-3
b
1.8E-7
c
0.96
d
-4.7E-7
f
3.6196
Kp,1
≥ 0.9
Equivalent crack path
−3
6
x 10
Y position [m]
5
4
3
2
1
adjusted parameters
original parameters
linear fit on adj. parameters
0
0
0.005
0.01
0.015
0.02
X position [m]
Figure 6-5: Predicted crack paths with original and adjusted model parameters. Linear fit for adjusted parameters is also given.
Figure 6-5
Next,.it is investigated if variations in the crack extension length have a noticeable effect on the predicted
crack path angle. The chosen extension lengths are 0.5, 1.0 and 2.0 millimetre. Taking the predicted crack
path angle of 1.0 mm as reference, the other two extension lengths result in a difference of only 0.2 and
0.4 degrees, for 0.5 and 2.0 mm respectively. Therefore, it can be concluded that variations in the crack
extension length do not introduce noticeable differences in the predicted crack path angle, so 1.0mm is
chosen to be used in all subsequent numerical simulations.
36
6 Pettit-method
To calibrate the Pettit-method, the FEA results are compared with experimental results obtained from an
internal Airbus report. In Figure 6-6 the difference in degrees in average crack path angle is presented for
various coupon orientations. Again the coupon geometry is a C(T), with an initial Kmax about 8.1 MPa√m.
For ST00-L, ST15-L and ST30-L coupons the difference with respect to the observed angles is very small,
however for ST45-L the error is significantly larger. This larger error is mostly due to a more curved crack
path, with rather low local angles in the first few cycles. This results in a lower average crack path angle
than expected, causing the large difference to the observed angle. For a ST60-L coupon the crack path
becomes even more curved, so this trend is expected to continue on orientations close to ST90-L as well.
In Chapter 9 a comparison is made between the predicted crack paths with the Pettit-method and the
crack paths observed in the fatigue coupons tested in this study. This comparison is based on the actual
crack paths, not the average crack path angle. This allows for a more fair assessment of curved crack paths.
Error of predicted crack path angles
6
5
ε [deg]
4
3
2
1
0
ST00−L
ST15−L
ST30−L
Orientation
ST45−L
Figure 6-6: Error in degrees of the predicted crack path angles with the Pettit-method for four coupon orientations.
Figure 6-6
For the calibration, all crack path angle predictions are made with the MSERR criterion. Using the modal
criterion to predict the crack path, leads to a very different result. The difference with respect to the
observed experimental value is 8.5 degrees, much larger than MSERR prediction for ST30-L. Therefore,
it is decided that for the type of materials focused on in this thesis, e.g. Al 2050-T84, the MSERR criterion
gives the most accurate crack path predictions.
For orientations close to ST00-L the predictions for the crack path of the Pettit method are quite accurate.
The results for a ST45-L coupon show that the predicted crack path becomes more curved for orientations
closer to ST90-L. The Pettit-method is calibrated, however for orientations closer ST90-L some larger
errors on the average crack path angle may show.
Chapter 7
Experiments
An overview is given in this chapter of the experimental setup and how the test data is processed. Section
7-1 discusses the coupon specifics, the used test standards, specifics of the fatigue test machines and
the loading parameters. Next, the data processing procedure is explained, and details are given on the
compliance method and the noise reduction process. Also, section 7-2 gives the polynomials used to
determine the stress intensity factors. Lastly, reasoning for trends in the fatigue tests is disussed in
section 7-3.
7-1 Setup of fatigue tests
7-1 First, the used test standards and measurement setup is described. The loading parameters, compliance
method and camera setup are discussed as well. Hereafter, the seven coupon orientations that are used
in the fatigue tests are presented. Lastly, the chemical composition of the two alloys used in this study is
given.
Test standards
The American Society for Testing and Materials (ASTM) provides test standards for both the C(T) and
DEN(T) coupons [ASTM International, 2003, 2013] . These standards define the appropriate specimen
geometry, and outline the testing procedure and data processing. The reason to select two coupon
geometries is to check if the geometry influences the fatigue crack growth behaviour of these aluminium
alloys. So both a symmetrical and asymmetrical coupon geometry is chosen. A C(T) has the advantage that
it can be relatively small to still provide accurate results. Also this test standard is widely used throughout
the industry and sufficient reference material is therefore available. Lastly, C(T) coupons are relatively
easy to manufacture. A DEN(T) provides symmetrical crack growth and is easier to manufacture than
the comparable CC(T) or M(T) coupons. To give a clear overview of the coupons used in the fatigue tests,
Table 7-1 and 7-2 give the characteristic dimensions, and Figure 7-1 shows a diagram of both specimens.
37
38
7 Experiments
The loading of coupons is chosen such that SIF’s of the growing cracks during the fatigue tests fall within
the range of 5-45 MPa√m. This range represents loading conditions within general aviation components,
and should let crack growth speeds fall within the Paris-regime. Prior to the start of the fatigue tests, a
crack of a certain size must be initiated, in accordance with the ASTM standards. This crack initiation
is performed with a lower force than the actual fatigue test, in accordance with ASTM standards. The
initiation frequency is between 15-25 Hz, depending on the coupon, as this reduces the time it takes
for the crack to initiate. The complete set of loading parameters is given in Table 7-3. The environment,
in which the specimens are tested, is normal laboratory air with an average temperature of 20 degrees
Celsius and an average humidity level of 75%.
Table 7-1: Characteristic dimensions of the C(T) coupons.
Table 7-1
Parameter Dimension [mm]
W
50.8
ao
10.16 (0.2*W)
t
6.35
D
10
Table 7-2: Characteristic dimensions of the DEN(T) coupons, values for ST90-L variants are given between brackets.
Table 7-2
Parameter Dimension [mm]
W
38.1 (84)
ao
3.175 (12.7)
t
6.35
h
92 (200)
0.25W
D = 10mm
0.325W
a0
h
1.2W
C(T)
a0
DEN(T)
1.25W
W
Thickness t for both coupons
Figure 7-1: Diagram of C(T) and DEN(T) coupon geometries.
Figure 7-1
7-1 Setup of fatigue tests
39
Table 7-3: Loading parameters of the fatigue tests for both coupon types.
Table 7-3
C(T)
Value
DEN(T)
DEN(T) ST90-L
R
0.1
0.1
0.1
finitiation [Hz]
20-25
20-25
20-25
ffatigue [Hz]
10
10
10
Pinitiation [kN]
2
14
17-25
Pfatigue [kN]
2.5
15
17
Parameter
The compliance method to determine the crack length is based on the method shown in the ASTM 647‑13
standard [ASTM International, 2013]. By measuring the displacement of the loading actuators at given
intervals at a given force, the compliance of the coupon can be determined. The interval used in all fatigue
tests is 1,000 cycles, for the initiation phase this is 2,000 cycles. Compliance is simply a measurement of
the stiffness of the coupon, i.e. the ratio of displacement over force applied, units m/N. The displacement
measurement must be corrected for the deformation of the fatigue test machine itself, to ensure only the
compliance of the coupon is used in the crack length calculation. Other corrections, e.g. misfits in the
loading pins, must be made as well. The details of the compliance method are discussed in section 7-2.
Next to the compliance method, a camera setup is used to determine the crack length. At the same
intervals as the displacement measurements a picture is taken, i.e. every 1,000th cycle in the fatigue tests.
In this way the crack growth can be monitored, and it provides a backup in case the compliance method
does not produce useful results. The chosen value for the length of the measurement intervales ensures
the required accuracy in the crack growth increments, damin = 0.01*W, can be achieved by both the
compliance method and the pictures.
Coupon orientation
Two types of standard fatigue tests are used in this study, compact tension (C(T)) and double edge notched
(DEN(T)) coupons. The crack length, at any time during the test, can be determined with a compliance
method. A camera setup records the crack as well, this allows for the capture of specific events, such as
branch initiation, and provides means for calibration and redundancy. Some samples are selected to be
examined with optical and electron microscopy. The coupons are machined in seven orientations in the
ST-L material plane. The orientations have increasing rotation in steps of 15 degrees with respect to the
ST-axis: ST00-L, ST15-L, ST30-L, ST45-L, ST60-L, ST75-L and ST90-L (also known as L-S). Figure 7-2
shows how these seven coupon orientations are situated within their material bill, the C(T) coupon is used
as example, DEN(T) coupons are similary situated. For each orientation two coupons are manufactured.
Also two different materials are used, Al 7010-T7451 and Al 2050-T84. Of Al 2050-T84 DEN(T) ST90-L
three specimens have been machined, bringing the total number of specimens to 57.
t/2
t/2
ST
ST00-L
ST15-L
ST30-L
ST45-L
ST60-L
ST75-L
ST90-L
L
Figure 7-2: The seven C(T) coupon orientations and their respective placement within the material bill.
Figure 7-2
40
7 Experiments
Fatigue test machines
The fatigue test machines used to perform the experiments are the MTS 10kN elastomer for the C(T)
coupons and the MTS 100kN for the DEN(T) coupons. Both machines have sufficient accuracy on the
load actuators to comply with the respective ASTM standards. The C(T) is fixed in the fatigue bench using
two pins, ensuring the coupon can rotate freely around these pins. The DEN(T) is clamped to prevent
rotation of the coupon, contrary to the ASTM norm. This is done to promote symmetrical crack growth,
as it known that this coupon can exhibit asymmetrical crack growth. It is expected that the clamping will
affect the stress field around the notches and growing cracks. However, calibration of the calculated crack
length, with pictures of the cracks, can ensure the deviations are sufficiently small.
Chemical composition alloys
The chemical composition of the plates used for the manufacturing of the specimens are given in Table
7-4 and 7-5, for Al 2050-T84 and Al 7010-T7451 respectively. The production certificates of the two plates
of Al 7010 are included in Appendix C for reference. Unfortunately it is not known which specimens
are manufactured from which plate. For Al 2050 a production certificate of the plate is not available,
as this plate is an unique experimentally produced and rolled plate. The densities of Al 2050-T84 and
Al 7010‑T7451 are 2.70 g/cm3 and 2.82 g/cm3 respectively.
Table 7-4: Typical chemical composition of Al 2050-T84 [Hafley et al., 2011].
Wt. %
Cu
Li
Mg
Mn
Ag
Zr
Si
Fe
Zn
Al
Min.
3.20
0.70
0.20
0.20
0.20
0.06
-
-
-
-
Max.
3.90
1.30
0.60
0.50
0.70
0.14
0.08
0.10
0.25
rest
Table 7-5: Typical chemical composition of Al 7010-T7451, according to ASN-A 3098.
Wt. %
Cr
Cu
Fe
Mg
Mn
Ni
Si
Ti
Zi
Zr
Other
Al
Min.
-
1.50
-
2.10
-
-
-
-
5.70
0.10
-
-
Max.
0.05
2.00
0.15
2.60
0.10
0.05
0.12
0.06
6.70
0.16
0.15
rest
7-2 Data processing of experiments
7-2 This section discusses how the data obtained in the fatigue experiments is processed to generate useful
quantities like the crack length and FCGR. First, the method to obtain the compliance of the fatigue
test coupons is explained. Hereafter, the equations needed to determine the crack length are presented.
Finally, the noise reduction process is discussed.
Compliance of coupon
As mentioned, compliance is the ratio of force applied versus deflection observed. Compliance is a measure
of the crack length, as a longer crack leads to a less stiff coupon, resulting in a larger displacement, and
vice versa. To accurately determine the crack length, one only needs the compliance of the coupon itself.
To determine this, one must seperate the contributions to the total compliance of the various components
in the test setup.
7-2 Data processing of experiments
41
To separate the contribution of the fatigue bench itself, calibration tests are performed using sufficiently
thick steel dummy coupons. The results of these tests are shown in Figure 7-3 and 7-4. The compliance
in the linear force regime of the MTS 100 kN fatige test machine is 5.7E-3 mm/kN and for the MTS
10 kN elastomer fatigue fatigue test machine it is 4.2E-2 mm/kN. By simply substracting the expected
deformation of the fatigue test machine from the measured displacement, the contribution of the machine
itself is excluded.
For the C(T) coupons the contribution to the total compliance of the misfit in the loading pins must
be excluded as well. To ensure free rotation of the coupon around the loading pins, the pins have a
slightly smaller diameter than the holes in the coupons. Also the hole diameter may vary slightly, which
is significant enough to introduce an error into the compliance estimates. Therefore, a measurement of
the deflection is done at a set force level prior to the initiation of crack, so these contributions can be
identified and subtracted from the total compliance.
The DEN(T) coupons are clamped, therefore only the correction of the deformation of the fatigue test
machine itself must be applied to the measured displacement to arrive at the compliance of the coupon
itself.
Compliance calibration 10 kN MTS
Displacement [mm]
0.4
0.3
0.2
0.1
Test data
Linear fit
0
0
1000
2000
3000
4000
5000
6000
7000
8000
Force [N]
Figure 7-3: Calibration of the compliance of the 10 kN fatigue test bench used for C(T) coupons.
Figure 7-3
Compliance calibration 100 kN MTS
0.6
Displacement [mm]
0.5
0.4
0.3
0.2
0.1
Test data
Linear fit
0
0
10
20
30
40
50
60
70
80
90
100
110
Force [kN]
Figure 7-4: Calibration of the compliance of the 100 kN fatigue test bench used for DEN(T) coupons.
Figure 7-4
42
7 Experiments
Crack length
Polynomial fits for the crack length in terms of compliance have been obtained by the ASTM for both
DEN(T) and C(T) configurations [ASTM International, 2003, 2013]. So after correction of the measured
displacements in the fatigue tests, to exclude the contributions of the misfits and fatigue tests machine
itself, one can calculate the corresponding crack length.
The compliance method for both coupons assumes a straight pure mode-I crack. So, in the case of
asymmetrical crack growth, off-axis crack growth or crack branching, this method yields an projected
straight crack length, which has the same effect on the coupon compliance as the actual crack. Figure 7-5
shows how the projected crack length is determined. This means that also the corresponding stress intensity
factors are of a projected mode-I crack. The actual mixed-mode SIF’s occurring during the fatigue tests
are not known, yet the projected KI is expected to give a good indication of the actual mixed‑mode KI.
Δa
θ
Δa*
Figure 7-5: Projected crack, length Δa*, as determined with the compliance method. Actual crack has length Δa at an angle θ to
the expected crack growth direction.
Figure 7-5
The polynomial equations used to calculate the crack lengths for the C(T) coupons are shown first:
a
*
W
C0 + C1u x + C2u x 2 + C3u x 3 + C4u x 4 + C5u x 5
=
ux =
(7-1) 1
(7-2) EvB
+1
P
Here, E is the Young’s modulus, P is the applied force, B is the thickness of the coupon, v is the deflection
of the coupon only and ux is the compliance parameter of the coupon. The values of the coefficients
C0 through C5 are given in Table 7-6. These values differ from the values given in the corresponding
ASTM standard, as modification of these values with the help of FEA proved to be more accurate in the
prediction of the crack length, see Figure 7-6. The range of applicable a/W values is from 0.2 up to 0.95,
more than sufficient for the expected crack lengths in the fatigue tests.
7-2 Data processing of experiments
43
Table 7-6: Values of the constants C0 through C5 for the C(T) coupons.
Table 7-6
Constant
Value
C0
C1
C2
C3
C4
C5
1.0113
-4.4625
21.898
-207.80
879.81
-1336.3
Compliance polynomial C(T)
0.8
FEM results
Original function
Improved fit
0.7
a/W [−]
0.6
0.5
0.4
0.3
0.2
0.05
0.1
0.15
0.2
0.25
Ux [-]
Figure 7-6: Original and improved polynomials relating the compliance to the crack length in a C(T) coupon.
Figure 7-6
For the DEN(T) a similar procedure is used to determine the crack length using the compliance. However
there is no readily available equation which directly relates the compliance of the entire coupon to the
crack length. Instead an inverse method is available, i.e. relating a known crack length to the expected
increase in overall displacement of the coupon [Tada et al., 2000]. As this equation cannot be analytically
inversed, a numerical solver is used to find the crack length corresponding to the measured displacement
of the coupon. An assumption of the initial crack length has to be made, in order to calculate the remaining
crack lengths. For the numerical solver only the extra displacement w.r.t. a baseline value is used as input.
The crack length used to determine this baseline displacement value is 1mm. This crack is chosen, because
the respecitve ASTM standard assumes a crack is fully initiated at this length in a DEN(T) coupon. The
extra displacement is already corrected for the deformation of the fatigue test machine itself.
Stress intesity factor
To determine KI in a C(T) coupon a sixth order polynomial is used, where the loading, geometry and
crack length is taken into account. The polynomial used is this thesis is developed by Newman and is
valid for a/W values between 0.2 and 0.8 [Newman, 1974]. For clarity the equations are given here:
KI =
4.55 40.32 a + 414.7 a
( W ) =−
W
W
f a
2
P
B W
( W)
f a
− 1698 a
3
W
+ 3781 a
(7-3) 4
W
− 4287 a
5
W
+ 2017 a
6
W
(7-4) 44
7 Experiments
For DEN(T) coupons the stress intensity factor can be determined using a handbook such as The stress
analysis of cracks handbook [Tada et al., 2000]. With the use of FEA, the polynomial to determine the SIF
is adjusted, because the size of the DEN(T) coupons is deviant to the respective ASTM standard. Both
the original and improved polynomial can be seen in Figure 7-7. Any 2a/W value may be used to obtain
a valid KI:
(
K I σ nom 2π a ⋅ f 2a
=
(
f 2a
W
)=
(
1.0974 − 0.7026 2a
W
)
(7-5) + 0.3249 ( 2a ) − 0.0489 ( 2a ) − 0.0319 ( 2a )
W)
W
W
W
1 − ( 2a )
W
2
3
4
(7-6) Here σnom is the nominal stress applied to the coupon. It must be mentioned that these equations and the
compliance method assume two symmetrical cracks with half crack length a to be present in the coupon,
hence 2a/W instead of a/W. In practice the two cracks emanating from the notches in the DEN(T) will
never be of equal length. So the effects of both cracks are combined to give one representative symmetrical
half crack length.
Stress intensity factor polynomial DEN(T)
1
FEM results
Original function
Improved fit
f(2a/W) [−]
0.9
0.8
0.7
0.6
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2a/W [−]
Figure 7-7: Original and improved polynomials to determine the stress intensity factor in a DEN(T) coupon.
Figure 7-7
Noise reduction
One other important aspect is the noise level in the measurements of the displacements and forces, and
the resulting crack length and FCGR. Two methods are commonly used to filter out the noise in crack
length and FCGR measurements: the incremental polynomial method, outlined in ASTM E647-13
[ASTM International, 2013], and the point-to-point method. By using either method on the raw data, the
noise level on the FCGR still remained too high, i.e. of a similar order of magnitude as the actual FCGR.
For this reason the raw data sets are reduced, which means only every nth point is included in the
analysis, the reduction factor can be adjusted for each coupon. This ensures that the calculated crack
growth increments, between two consecutive data points, become significantly larger than the noise
level, making it possible to filter out the noise. With the reduction of the raw data and the use of the
incremental polynomial method, most issues are solved, however in some instances the calculated FCGR
is still negative.
7-2 Data processing of experiments
45
Therefore, the incremental polynomial method is only used to filter out the noise in the calculations of
the crack length. To determine the FCGR a point-to-point method is used on crack lengths calculated
with the polynomial method. For clarity, the process of calculating the crack length with the incremental
polynomial method and the point-to-point determination of the FCGR are both explained hereafter.
As mentioned, the incremental polynomial method is outlined in ASTM E647-13. Here a set of
2m+1 successive points is taken from the reduced data set around point An, so this set resembles
[An‑m … An … An+m]. Next, a second order polynomial is fitted through this set of points. The averaged
value of the crack length is taken at the number of cycles corresponding to point An in this polynomial.
By repeating this process for all points in the reduced data set, as far as possible, the noise is filtered out.
The incremental polynomial method is illustrated in Figure 7-8.
The FCGR, da/dN, can be now be determined in a point-to-point fashion with use of the crack lengths ,
calculated with the polynomial method, and the number of fatigue cycles. This is done using the following
equation, here a is the crack length and N the respective number of cycles:
ai − ai −1
N i − N i −1
da
=
Ni ) =
, with N i
(
dN
N i − N i −1
2
(7-7) For each coupon the processed crack length is compared with the pictures at 5 instances. If the discrepancy
in crack length proves to be too big, the data set may be discarded or corrected. If for example the noise
level still is too high or negative crack growth still is predicted, the pictures can provide the needed
crack lengths. Pictures can also be used to identify interesting events, such as crack deviation or crack
branching. In addition, they provide information about the crack path angle, which can be compared
with the predictions of the numerical methods.
Crack length
Incremental polynomial method
Polynomial set 1
Polynomial set 2
Averaged point 1
Averaged point 2
Fatigue cycles
Figure 7-8: Incremental polynomial method as outlined in ASTM E647-13.
Figure 7-8
46
7 Experiments
7-3 Reasoning for design of experiments
7-3 This section aims at explaining the reasoning behind the design of the experiments. To be able to
investigate the effects of different orientations of the microstructure w.r.t. the loading direction, the
coupons have been manufactured in seven different orientations in the ST-L plane, as seen in Figure
7-2. From observations made in literature, it is clear that the fatigue crack life is affected by the coupon
orientation. Orientations closer to ST90-L are expected to have a larger crack life than orientations closer
to ST00-L. This increase in crack life can be attributed to crack deflection, a more tortuous crack path, and
a higher fatigue fracture resistance, as predicted with the Pettit-method. Also branch forming contributes
to the decrease of the apparent crack growth rate. The effect of the orientation of the microstructure on
the fatigue crack life likely is non-linear. So the decrease in crack growth speeds for orientations close to
ST00-L can be rather small, even though the crack path may be different.
In terms of the crack paths, orientations close to ST00-L likely show approximately straight crack paths.
For these orientations the crack likely grows in a direction in between the L-axis and a line perpendicular
to the principal loading direction. With an increased tendency for intergranular failure, the crack paths
can be quite tortuous and can show frequent branching or crack deviation. In orientations closer to
ST90-L branches are expected to initiate and they can possibly cause crack deflection. At higher Kmax
values the crack is more likely to turn towards the weak material direction, where final fracture is expected
to occur. So, by testing the alloys used in this study in seven different coupon orientations, the effects of
the microstructure on the crack life and crack paths can be identified.
Also two different coupons types are used in the fatigue tests. The main reason to do this is that one can
ensure that the observed effects of the microstructure orientation are not related to the geometry of the
coupon. If similar effects are observed in both coupon types, one can exclude the influence of the coupon
geometry.
The coupon geometry will have some effects on the crack growth behaviour. The DEN(T) coupons will
show a higher crack growth speed at similar ΔK levels. This has a couple of causes, mainly stress related.
Due to the higher nominal stress present in the DEN(T) coupons, crack closure effects are expected
to occur less. Due to the crack tip plasticity, at the minimum load a zone of compressive stress may
exist adjacent to the crack tip. This compressive zone effectively slows down crack growth. Due to the
reduction of this compressive zone, the expected FCGR for the DEN(T) coupons is higher compared to
C(T) coupons of similar orientations.
Next, the clamping used on a DEN(T) will constrain the Poisson’s contraction of the coupon. This causes
a larger contribution of the Poisson’s effect on the stress at the mid-section, where the cracks are present.
This larger contribution results in a slightly more plane strain state, compared to the plane stress state
of the C(T). It is known that crack growth resistance decreases in more plane strain like conditions, this
again leads to an increase in FCGR for DEN(T) coupons.
In terms of accuracy of the compliance method, likely the C(T) coupons perform better, because the
deflection can be measured directly at the loading pins. For the DEN(T) the deflection of the entire
coupon is measured, which inherently introduces more spread. Also the DEN(T) coupons are known to
have an increased tendency for asymmetrical crack growth, again introducing errors in the crack length
prediction.
Artificially aged aluminium alloys, e.g. Al 2050-T84, are known to exhibit fracture anisotropy, see Chapter
3. If one compares the microstructure of the alloys used in this study, both show a banded structure,
typical for rolled artificially aged aluminium alloys, see Figure 8-2 and 8-3. The grain size however is
very different, Al 2050-T84 has much larger grains than Al 7010-T7451. Also Al 7010 does not contain
any lithium, which in literature has often been linked to fracture anisotropy, see Chapter 3. The fatigue
7-3 Reasoning for design of experiments
47
tests of both these alloys allows to identify the effect of the grain size and to see if the well-known alloy
Al 7010-T7451 also exhibits similar fatigue behaviour as Al 2050-T84.
Artificially aged aluminium alloys, especially those with a relative large grain size, may show an increased
tendency for intergranular fracture. This can show in different ways, e.g. a zig-zag like crack path or branch
forming. The tendency for intergranular fracture can be increased by PFZ’s. Most likely Al 2050‑T84 will
show more frequent crack branching than Al 7010, due to these PFZ’s and the larger grain size.
48
7 Experiments
Chapter 8
Results
This chapter presents the results from the fatigue tests and the numerical simulations. In section 8-1 the
investigation of the microstructure is discussed. Hereafter, the results of the fatigue tests are presented in
section 8-2. This section is split up into three sub-sections: first fatigue crack life, secondly crack length,
crack growth speed and Kmax, lastly crack paths. Section 8-3 presents the findings of fractography on three
specimens, C(T) ST00-L 1 and ST45-L 1, and DEN(T) ST90-L 2, all Al 2050-T84. Finally, section 8-4
gives the predictions of the crack paths with both numerical methods, for both specimen types. The
discussion and interpretation of the results can be found in Chapter 9.
8-1 Microstructure
8-1 To gather detailed insight into the causes of the expected fatigue fracture anisotropy in Al 7010-T7451 and
Al 2050-T84, the microstructure is characterized. Two DEN(T) coupons were selected to be sectioned in
order to obtain samples of the three principal material planes (L-LT, LT-ST, ST-L). The selected coupons
are Al 7010 ST90L 2 and Al 2050 ST90L 1, where these specimens allow for decent size samples to be
extracted in the vicinity of the fracture surface. Figure 8-1 shows a schematic of the sample extraction
locations in these two coupons. Sample 1 gives the LT-ST plane, 2 the L-LT plane and 3 the ST-L plane.
1
3
2
L
ST
= cutting-lines
Figure 8-1: Schematic of sample extraction for optical microscopy.
49
50
8 Results
The coupons are cut using a water-cooled diamond-coated rotating saw blade. The samples are then
casted into wear-resistant epoxy to allow for grinding and proper handling. The samples are grinded
and polished with an incrementally finer grain size, up to 1μm, to ensure a mirror finish. Hereafter, the
samples are etched with Keller’s reagent to bring forward the grain structure. The required etch time for
Al 7010 is between 60-80 s, while for Al 2050 this is around 100 s. The etched samples are then inspected
with a Leica optical bright field microscope, grains with different orientations will show as different
shades of grey. Copper rich areas and precipitates will show as black spots. Figure 8-2 and 8-3 show the
microstructure of Al 7010-T7451 and Al 2050-T84 respectively.
m
0μ
10
ST
L
LT
Figure 8-2: Grain structure of Al 7010-T7451.
m
0μ
50
ST
L
LT
Figure 8-3: Grain structure of Al 2050-T84.
8-2 Fatigue tests
51
The microscopy of Al 7010-T7451 reveals a partially recrystallized structure, consistent with findings in
literature [Hafley et al., 2011]. The recrystallized grains are elongated in the rolling-direction (L). Some
inclusions can be found in the recrystallized grains, with an average size of 5-10 μm. The L-LT plane
shows relative big grains, compared to the other two material planes, the grains are somewhat elongated
in the L-direction. It must be noted that not all grain boundaries have been revealed.
In the LT-ST plane the typical banded structure is visible, with the largest grain sizes found in the LTdirection. In the ST-L plane a similar banded structure is found, with grains elongated in the L-direction
and small dimensions in the ST-direction.
As one can see, the etching agent used, Keller’s reagent, produced little contrast in the L-LT plane. In
general it can also be observed that not all grain boundaries have been revealed, yet some areas are
already very dark, usually indicative of over-etching. Overall it may be said that in Al 7010-T7451 a
banded microstructure is observed, typical for rolled aluminium plate products.
It must be noted that, during the investigation of the microstructure, it is found that the investigated
Al 7010 DEN(T) ST90-L coupon is milled in the wrong orientation, namely the L-LT configuration.
This conclusion is made on the observation that the grain structure had basically switched the ST-L
and L-LT planes and had a 90-degree rotated LT-ST plane. Inspection of the material bill certificates
showed the thickness of one of the material bills was not sufficient to cover the width of 84 mm used for
this coupon. Because of the large similarities in the fracture surfaces between the two Al 7010 DEN(T)
ST90-L coupons it is likely both have been milled in the L-LT configuration, rather than the intended
ST-L configuration. This does not invalidate their crack growth data, however it does not make them
suitable to compare other results to.
The grain morphology observed in Al 2050-T84 is elongated and pancake shaped, typical for rolled Al-Li
plates. The grains are largely unrecrystallized, with some inclusions and some copper-rich grain boundary
particles. Approximate dimensions found in literature for the grains are 500 μm in the L-direction, 200 μm
in the LT-direction and 50 μm in the ST-direction [Hafley et al., 2011]. It is observed that some grains
are somewhat larger, especially in the L- and LT-direction. The inclusions and grain-boundary particles
are clearly visible in the L-LT plane. Also some scratches are still visible, yet they are not detrimental to
the observations. Similar to Al 7010-T7451, not all grain boundaries have been revealed, yet some areas
already tend to be rather dark.
In the LT-ST plane the grain size is slightly more uniform, still the grain dimensions remain anisotropic.
Some inclusions can be observed in the grains there as well. The larger dark spots are a result of an
imperfect mirror finish and the etching process. The grain size anisotropy is most prominent in the ST-L
plane. Clearly the grains are highly elongated in the L-direction, and also here some inclusions and grain
boundary particles can be observed. It is concluded that the microstructure of Al 2050-T84 is highly
anisotropic, and shows the presence of some inclusions and grain boundary particles.
8-2 Fatigue tests
8-2 In this section the results from the fatigue tests are presented. To gain insight into the effects of the
microstructure on the fatigue crack behaviour, different aspects of the results are looked at. First the
overall fatigue life of the initiated cracks is shown. Hereafter some exemplary da/dN* vs ΔK* graphs are
given. Lastly some pictures of broken specimens, as well branch initiation are presented. The full data set
for all tested coupons, in terms of da/dN*, ΔK*, etc., can be found in Appendix A.
52
8 Results
Fatigue crack life
Looking at the fatigue crack life, one can observe a few interesting things. Figure 8-4 and 8-5 give the
fatigue crack life of the C(T) and DEN(T) coupons respectively, for both materials. Overall it is observed
that the spread is quite acceptable in the results between two coupons of the same orientation and
material. It must be noted that a few coupons are not shown here, because they fractured prematurely,
these coupons are the DEN(T) 7010 ST00-L 1 and C(T) 7010 ST75-L 1. Also the fatigue crack lives of the
ST90-L DEN(T) coupons are not shown, due to the difference in coupon size.
The fatigue life of the C(T) coupons is significantly higher compared to the DEN(T) coupons. This due to
the higher nominal stress in the DEN(T), increasing the damaging effect of a load cycle, and the shorter
available coupon width to be cracked. For both specimen types Al 2050 shows an increase in crack life,
for increasing in-plane rotation of the ST-axis. In Figure 8-4 it can be observed that for Al 7010-T7451
no significant change in fatigue life occurs, for increasing rotation of the ST-axis. On the other hand Al
2050‑T84 shows a gradual increase in crack life up to ST75-L and a large jump for ST90-L. This jump
could be partially explained by the differences in crack paths found in these coupons, see Appendix B.
In Figure 8-5 one can see that Al 7010-T7451 shows a gradual increase in fatigue life up to ST45-L,
hereafter the crack life starts to slowly decrease. This decrease may be explained by the observed crack
paths, which are presented in the sub-section on crack paths. A remarkable decrease in fatigue life is
observed for Al 2050-T84 for ST15-L. For other orientations a large increase in fatigue life is seen, up to a
factor 6x larger than ST00-L. For both the ST45-L and ST75-L the spread in fatigue life is larger than for
the other orientations, yet remains acceptable.
Fatigue crack life C(T) specimens
5
10
x 10
Al 7010−T7451
Al 2050−T84
2.5
Cycles to fracture [−]
8
Cycles to fracture [−]
Fatigue crack life DEN(T) specimens
5
3
6
4
2
x 10
Al 7010−T7451
Al 2050−T84
2
1.5
1
0.5
0
0
ST00−L
ST15−L
ST30−L
ST45−L
ST60−L
ST75−L
ST90−L
Orientation
Figure 8-4: Fatigue crack life of all C(T) specimens for
all tested orientations, for both Al 7010-T7451 and Al
2050‑T84.
ST00−L
ST15−L
ST30−L
ST45−L
ST60−L
ST75−L
Orientation
Figure 8-5: Fatigue crack life of DEN(T) specimens for
orientations up to ST75-L, for both Al 7010-T7451 and Al
2050-T84.
Crack length, FCGR and Kmax
With the data reduction and compliance methods described in Section 7-2 the crack lengths, crack growth
rates and stress intensity factors have been obtained throughout the duration of the fatigue tests. All a/W*
– N and da/dN* – ΔK* graphs can be seen in Appendix A. Typical examples of crack length and FCGR
curves can be seen in Figure 8-6 and 8-7 respectively. The crack length can be approximately represented
by an exponential function, while the FCGR is typical for the Paris-regime. In this regime the FCGR can
be represented by a power law. If crack turning or branching occurs in a coupon, both the a/W* – N and
da/dN* – ΔK* graphs show evidence of thereof. If a crack turns or branches the apparent crack growth rate
is slowed, which is can be seen in Figure 8-8, where the results for 2050-T84 ST75-L are shown.
8-2 Fatigue tests
53
Crack length for Al 2050−T84 ST00−L
0.8
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
0.7
Crack growth rates for Al 2050−T84 ST00−L
−2
10
C(T) ST00−L 1
C(T) ST00−L 2
DEN(T) ST00−L 1
DEN(T) ST00−L 2
0.6
0.5
0.4
−3
10
−4
10
C(T) ST00−L 1
C(T) ST00−L 2
DEN(T) ST00−L 1
DEN(T) ST00−L 2
0.3
0.2
−5
0
0.5
1
1.5
10
2
N [cycles]
x 10
7
5
Figure 8-6: Crack lengths for all Al 2050-T84 ST00-L
specimens.
9
10
12
ΔK* [MPa√m]
15
20
Figure 8-7: Crack growth rates of all Al 2050-T84 ST00-L
specimens
Crack growth rates for Al 2050−T84 ST75−L
−3
10
da/dN* [mm/cycle]
8
−4
10
−5
10
C(T) ST75−L 1
C(T) ST75−L 2
DEN(T) ST75−L 1
DEN(T) ST75−L 2
−6
10
6
7
8
9
10
12
ΔK* [MPa√m]
15
20
Figure 8-8: Crack growth rates of all Al 2050-T84 ST75-L specimens.
One can notice that generally the FCGR for DEN(T) coupons is significantly higher than for C(T)
coupons. The resulting fatigue crack life for DEN(T) coupons is also less w.r.t. C(T) coupons, about
a factor 4x for ST00-L. The net section limit for both coupons is a/W = 0.734 and 2a/W = 0.862 for
C(T) and DEN(T) respectively, assuming a yield strength of 450 MPa. As the thickness of all coupons,
6.35 mm, will result in a plane stress state, the actual KI,c value is likely much higher than range Kmax values
for all coupons, i.e. Kmax = 7-45 MPa√m.
Figure 8-6 and 8-7 show that Al 2050-T84 follows the expected trends for the applied stress levels. The
spread in the results is somewhat larger for the DEN(T) coupons, which is also representative for all
results. The crack growth rate approximately shows a linear trend in the log-log plot of Figure 8-7, in
accordance with the Paris-regime.
In Figure 8-8 one can see the effects of crack turning and branching. The crack paths of the respective
C(T) and DEN(T) specimens are shown in Appendix B. The C(T) shows a gradually turning crack, while
the DEN(T) shows crack branches in the weak direction of the material, along with approximate mode-I
crack growth. The local dips in the crack growth speed for both ST75-L DEN(T) coupons correspond to
the formation of branches. The two areas, around ΔK* = 12 MPa√m, with decreasing da/dN* for the C(T)
coupons correspond to the points where the crack path is highly curved. Due to the crack turning and
54
8 Results
crack branching the spread in the FCGR has become larger, but still shows a general trend. Again it must
be noted that the crack growth speeds and crack lengths calculated are for an equivalent straight mode-I
crack, so the results have not been corrected to take into account the off-axis crack growth direction.
To reveal differences in FCGR between the two used materials, Figure 8-9 compares the FCGR for C(T)
ST30-L coupons. These coupons have been chosen, because the last recorded crack length was very
similar for both materials. As one can see the FCGR of the two materials is quite similar. Yet, for ΔK* in
between 7-9 MPa√m the FCGR for Al 2050-T84 is lower, up to a factor to a factor 10x. For most coupon
orientations, a similar observation is made. This explains why the overall crack life for Al-Li coupons is
generally higher than for Al 7010.
Crack growth rates for C(T) ST30−L
−3
da/dN* [mm/cycle]
10
−4
10
2050 ST30−L 1
2050 ST30−L 2
7010 ST30−L 1
7010 ST30−L 2
−5
10
7
8
9
10
12
ΔK* [MPa√m]
15
20
Figure 8-9: Crack growth rates for Al 2050-T84 and Al 7010-T7451 C(T) ST30-L specimens.
To see the effect of coupon orientation, the FCGR of three coupon orientations are compared to that of
ST00-L. For the comparison ST30-L ST45-L and ST90-L Al 2050-T84 C(T) coupons are used, because
of the relative low spread in the FCGR and the similarities in the crack paths for coupon of the same
orientation.
If one would compare the FCGR of different coupon orientations, without correcting the FCGR for the
observed crack path, the comparison would not be fair. Because the crack paths are not aligned with the
expected mode-I direction, except for ST00-L, the actual FCGR is larger than the calculated equivalent
FCGR da/dN*. A possible way of accounting for the effect of the crack path, is to apply a correction factor,
based on the average crack path angle, on both the FCGR and ΔK. da/dN* should then be multiplied
1‍/‍cos(θ), where θ is the average crack path angle, and to obtain more realistic values for ΔK, one should
multiply ΔK* with cos(0.5*θ) [Forth et al., 2004].
This correction method is not suitable for turning cracks or cracks with branches, as it assumes an
approximately straight crack path without branches. So for the ST90-L coupons this method cannot be
applied. Also, for large in-plane rotations (more than 40 degrees) of a crack, w.r.t the expected mode-I
direction, this method tends to overcorrect. Figure 8-10 shows the comparison for the three different
orientations, where the FCGR of ST30-L and ST45-L have been corrected for their crack path.
With use of the correction procedure, one can see that the FCGR for the ST00-L, ST30-L and ST45-L
coupons are very similar. The average crack path angles were easily determined, as the crack paths are
fairly straight. For ST30-L an average angle of 20.5 degrees is found, for ST45-L this is 31.8 degrees.
Apparently the crack paths of these coupon orientations do not seem to significantly affect the FCGR. For
ΔK* = 7-9 MPa√m, the FCGR of the ST30-L and ST45-L coupon shows somewhat lower values than for
ST00-L. This explains the higher fatigue crack life for ST30-L and ST45-L, as seen in Figure 8-4.
8-2 Fatigue tests
55
For ST90-L, a different situation arises. As mentioned, the correction method to account for the effect
of the crack path cannot be used. The first portions of the crack paths for both ST90-L coupon are
approximately aligned with the expected mode-I direction, see Figure B-8. So, for ΔK* = 7-10 MPa√m,
the calculated values of the equivalent FCGR and ΔK* should be very similar to the actual values. If one
compares the FCGR for those values of ΔK*, it clear that the FCGR of ST90-L is considerably lower than
that of ST00-L. Because the crack paths are not very different, for ΔK* = 7-10 MPa√m, the lower FCGR
is attributed to the 90-degree in-plane rotation of the microstructure. This provides clear evidence of
fatigue fracture resistance anisotropy.
For ΔK* > 10 MPa√m, the values of the actual FCGR and SIF’s likely differ significantly from the calculated
values shown in Figure 8-10 C. At this point, no methods are available to obtain the actual FCGR using
the values of the equivalent FCGR obtained with the compliance method. Therefore, it is recommended
that research is done how one can determine the actual FCGR of turning cracks with a large in-plane
rotation (>40 degrees) or branched cracks. Pictures taken during the fatigue tests in this study can serve
a basis for finding the actual FCGR of such crack paths.
A
−4
10
C(T) ST00−L 1
C(T) ST00−L 2
C(T) ST30−L 1
C(T) ST30−L 2
−5
10
7
8
9
10
12
15
ΔK* [MPa√m]
C
−4
10
−5
10
C(T) ST00−L 1
C(T) ST00−L 2
C(T) ST45−L 1
C(T) ST45−L 2
−6
10
20
7
8
9
10
12
ΔK* [MPa√m]
15
20
Crack growth rates for C(T) Al 2050−T84
−3
10
da/dN* [mm/cycle]
Crack growth rates for C(T) Al 2050−T84
−3
10
da/dN [mm/cycle]
da/dN [mm/cycle]
B
Crack growth rates for C(T) Al 2050−T84
−3
10
−4
10
−5
10
C(T) ST00−L 1
C(T) ST00−L 2
C(T) ST90−L 1
C(T) ST90−L 2
−6
10
7
8
9
10
12
15
ΔK* [MPa√m]
20
25
30
35
Figure 8-10: Comparison of corrected and uncorrected FCGR for three orientations Al 2050-T84 C(T) coupons to the
baseline FCGR of the ST00-L orientation. A) Comparison of the corrected FCGR of the ST30-L orientation. B) Comparison
of the corrected FCGR of the ST45-L orientation. C) Comparison of the uncorrected FCGR of the ST90-L orientation.
56
8 Results
Crack paths
In terms of observed crack paths during the fatigue tests, some very characteristic features are present.
Pictures of all broken specimens can be seen in Appendix B. In this sub-section only some crack paths are
shown, to highlight what type of characteristic features were observed. Figure 8-11 shows 4 specimens of
the Al-Li alloy of various orientations and from both specimen types.
Immediately one can notice the large difference in appearance of the crack path with a different material
orientation. Branches can clearly be seen in Figure 8-11 A and B, with material orientations ST75-L
and ST90-L respectively. The crack path of ST75-L is fairly complex, and shows various branches, crack
kinking and crack deflection. The main crack initiated at the right notch, where it eventually deflected
into the large upwards branch and was arrested. Hereafter, a crack initiated at the left notch, forming
various branches, whereatfter final failure occurred. Such a crack path shows that analysis of the equivalent
FCGR or SIF alone does not suffice in such a case, as the expected crack growth direction does not match
the observed crack path.
Figure 8-8 shows the FCGR of this coupon, and the steep drop in da/dN* at ΔK* =11 MPa√m matches the
initiation of the large branch. Hereafter the reliability of the calculated SIF’s decreases, as the crack path
clear is not a simple straight crack path anymore.
Figure 8-11 B shows the formation of branches in a ST90-L specimen, where braches initiate both
upwards and downwards at the same time. Some branches are only apparently formed in one direciton,
yet detailed analysis of the fracture surface might reveal branches that do not show on the surface yet.
In Figure 8-11 C a straight off-axis crack path can be observed in a ST30-L coupon. Both fatigue crack
growth and final failure align in the same direction, which is rotated 21.5 degrees w.r.t. the expected crack
growth direction. As the materials weak direction is rotated 30 degrees, the crack path lies in between the
weak material direction and the expected 0 degree direction. This type of crack path is mostly observed
for specimens of ST15-L, ST30-L and ST45-L.
For orientations closer to ST90-L a crack path is found similar to the one shown in Figure 8-11 D. This
is especially true for C(T) specimens, as crack paths in DEN(T) specimens tend more to the one shown
in Figure 8-11 A. A slowly turning crack path, with ultimate failure aligned with L-direction is mostly
observed in C(T) coupons of ST60-L and ST75-L. In some DEN(T) coupons such a turning crack path is
observed as well, but more often branches are formed and the subsequent crack paths are complex.
In general three types of crack paths were observed in the specimens: approximately straight (off-axis)
crack paths, turning crack paths with ultimate failure aligned with L-direction and crack paths with
(frequent) branches. The last type makes the analysis of the SIF’s unrealiable, when the first branch is
initiated, due to the complexity crack path. Furthermore, the exact influence of a branch on the SIF’s of
the main crack is not known, but it is expected that a branch lowers the availble crack driving force of the
main crack, or forces the main crack to reinitiate along the length of a branch.
8-3 Fractography
A
C
57
B
D
Figure 8-11: Fatigue tested specimens showing various characteric crack paths. A) Al 2050-T84 DEN(T) ST75-L 1 showing
various branches and complex crack path. B) Al 2050-T84 DEN(T) ST90L 3 showing various branches and reinitation of the
main crack along branches. C) Al 2050-T84 C(T) ST30-L 1 showing a straight off-axis crack path. D) Al 2050-T84 C(T) ST60-L 1
showing a turning crack path, with ultimate failure approximately aligned with L-direction.
8-3 Fractography
8-3 To gain insight into the actual failure mechanisms in the fatigue tests, one can use a scanning electron
microscope (SEM). For this thesis three samples are analysed using a SEM, C(T) ST00-L 1 and ST45-L
1 and DEN(T) ST90-L 2, all Al 2050-T84. The fracture surfaces are extracted from the specimens by
making a straight cut just below the fracture surface. These three specimens are chosen, because they
represent the identified crack path types, where ST00-L can serve a baseline comparison. The SEM used is
a JEOL JSM-7500F. In all scans the main crack growth direction is from left to right and is indicated with
an arrow in the scans. The scans are made at approximately the mid-thickness of the samples.
The scans of the ST00-L sample are displayed in Figure 8-12, 8-13 and 8-14, the used magnification factors
and relevant scale bars are included in these figures. Clear striations are visible in Figure 8-12, with some
inclusions present on the fracture surface. The FCGR determined in Figure 8-12 is 3.39E‑4 mm/‍cycle,
which is close to the calculated value of 3.17E-4 mm/cycle with the compliance method for an a/W* = 0.47.
Figure 8-13 and 8-14 show the region of unstable crack growth, where Figure 8-14 is a zoomed in view
of typical features found. In this region no striations are visible, and macroscopic the surface looks very
facetted. On the plateaus visible in Figure 8-13, locally dimples and intergranular facets are present, see
Figure 8-14. Initially the region of unstable fracture shows more dimples, and with increasing crack
length more intergranular facets show. Also little to none oxide product is observed here.
58
8 Results
Growth direction
Figure 8-12: SEM image of C(T) Al 2050-T84 ST00L 1 fracture surface showing striations, a/W = 0.47, ΔK* = 13.9 MPa√m.
Growth direction
Figure 8-13: SEM image of C(T) Al 2050-T84 ST00-L fracture surface showing unstable crack growth region, a/W = 0.58,
ΔK* = 20.1 MPa√m.
Growth direction
Figure 8-14: SEM image of C(T) Al 2050-T84 ST00-L fracture surface showing dimples and intergranular facets, a/W = 0.74,
ΔK* = 44.4 MPa√m.
8-3 Fractography
59
In Figure 8-16, 8-16 and 8-17 the scans of ST45-L are presented, again crack growth direction is indicated
with an arrow. Shortly after crack initiation a facetted fracture surface is visible, with some lines visible,
possibly due to slip planarity, see section 3-1. On this facetted surface little to none fatigue striations are
visible, likely the striations have been flattened by crack closure effects, due to the low loading and R-ratio
used in C(T) specimens. An SEM scan of such a surface can be seen in Figure 8-15. The ST45-L and
ST90-L samples both showed possible indications of slip bands on the fracture surface.
For somewhat larger crack lengths a fracture surface as in Figure 8-16 can be observed. This folded fan
like structure might indicate a fatigue fracture mechanism, yet classical striations do not seem to be
present. With larger crack length striations become clearly visible, and the fracture surface resembles
Figure 8-12. A very distinct rectangular inclusion was found on a region with clear striations, and similar
inclusions were found on the ST00-L sample. A comparison of the FCGR obtained from the SEM scans
and the compliance method shows a larger difference than for ST00-L, 1.22E-4 and 7.01E-5 mm/cycle
respectively, for an a/W* = 0.37.
In the area of unstable fracture a fracture surface as in Figure 8-17 shows. Here, dimples are present with
a clear orientation, most likely due to shearing of material during dimple formation. For crack lengths
close to ultimate failure, the fracture surface resembles Figure 8-13 again, where the dimples do not have
an orientation.
Growth direction
Figure 8-15: SEM image of C(T) Al 2050-T84 ST45-L fracture surface showing a facetted surface with possible slip bands,
a/‍W = 0.26, ΔK* = 8.95 MPa√m.
Growth direction
Figure 8-16: SEM image of C(T) Al 2050-T84 ST45-L fracture surface showing a folded fan like structure, a/W = 0.35,
ΔK* = 10.1 MPa√m.
60
8 Results
Growth direction
Figure 8-17: SEM image of C(T) Al 2050-T84 ST45-L fracture surface showing shear dimples, a/W = 0.42, ΔK* = 12.2 MPa√m.
The investigation of the ST90-L sample is split into two parts, one part concerning the regular fracture
surface and one part concentrates on the fracture surface of a branch. Figure 8-18 and 8-18 show the
scans of the regular fracture surface of the ST90-L sample. It must be noted that the crack growth in this
coupon is asymmetrical, so the presented 2a/W and ΔK* values are only an indication. Macroscopically
a banded structure can be seen, of what appears to be grains and grain boundaries, with some lines
present. Again these lines may be caused by slip planarity. For small crack lengths the fracture surface is
flatter compared to the surfaces of ST00-L and ST45-L. Some folded fan like surfaces are present here as
well. For larger crack lengths striations become more clearly visible, with some inclusions on the fracture
surface. Some secondary cracking also becomes visible for larger crack lengths, as can be seen in Figure
8-18 and 8-18. In Figure 8-18 one can see that up to the edge of the secondary crack striations are present.
After the secondary crack a rougher surface emerges, with no clear striations or facets. Macroscopically
the secondary cracking seems to be on grain boundaries.
Growth direction
Figure 8-18: SEM image of DEN(T) Al 2050-T84 ST90-L fracture surface showing striations and secondary cracking,
2a/W = 0.88, ΔK* = 19.4 MPa√m.
8-3 Fractography
61
Growth direction
Figure 8-19: SEM image of DEN(T) Al 2050-T84 ST90-L fracture surface showing a banded structure and secondary cracking,
2a/W = 0.89, ΔK* = 19.7 MPa√m.
Scans of the investigated fracture surface of a branch in the Al-Li DEN(T) ST90-L 2 specimen are
presented in Figure 8-20 and 8-21. The branch extension is from left to right. Close to the main crack the
fracture surface of the branch looks somewhat rough, with no clear facets, lines or striations. Dimples
are present on the surface close to the main crack. As the branch extends the fracture surface flattens, as
can be seen in Figure 8-20. On the branch fracture surface lots of black oxide product is present. Figure
8-21 is a zoomed in view and shows clear dimples without a clear orientation. No other distinct features
or indications of fatigue damage are found on the surface of this branch. It must be mentioned that not
the entire length of the branch could be investigated. A part of the branch still remains in the original
specimen, but it is expected that the fracture surface looks similar. It is recommended to investigate more
branch fracture surfaces, including the part of branch arrest.
Growth direction
Figure 8-20: SEM image of DEN(T) Al 2050-T84 ST90-L showing an overview of the branch fracture surface where light grey
particles are oxide product, branch length = 2.4 mm.
62
8 Results
Growth direction
Figure 8-21: SEM image of DEN(T) Al 2050-T84 ST90-L showing non-orientated dimples on the branch fracture surface,
branch length = 2.4 mm.
8-4 Numerical simulations
8-4 The results of both numerical methods, introduced in Chapter 5 and 6 are presented in this section. The
predicted crack paths are given for the seven orientations used in the fatigue tests as well. This allows for
a comparison to be made, with the crack paths observed in the fatigue tests. The results are split up, such
that each coupon and each method is presented in a different figure. Figure 8-22 through 8-24 give the
predicted crack paths for the Pettit- and k2-method, for both the C(T) and DEN(T) specimens. In these
figures the crack paths are plotted on a millimetre scale.
The predicted crack paths for ST90-L are not included in Figure 8-22, 8-23 and 8-25. In Figure 8-22 and
8-23 the predicted crack path coincides with that of ST00-L, so for convenience these have been left out
of the figures. In Figure 8-25 the crack path of ST90-L has been left out for two reasons, first the trend
seen in the other orientations makes the predicted crack path of ST90-L very unrealistic, secondly various
modelling issues were encountered.
Comparing the predicted crack paths of the Pettit-method, the average crack path angle of C(T) specimens
is higher than for DEN(T) specimens. For orientations up to ST45-L the curvature of the predicted paths
is very small, where for ST60-L and ST75-L the predicted paths are curved. The predicted crack path for
ST75-L does not seem to follow the trend of increasing crack paths angles with increasing rotation of the
ST-axis. Rather its path is very similar to that of ST60-L, especially in Figure 8-22.
The k2-method, especially in DEN(T) coupons, does not seem to act as expected. Figure 8-24 show
unrealistic crack paths, due to the fact the value of k2 only increases if a branch is formed. So the k2‑method
chooses to extend the crack with another branch, this again increases k2, resulting in a crack path solely
in the weak direction of the material. In C(T) specimens the method behaves more as expected, for
orientations up to ST45-L. Hereafter the value of k2 seldomly drops below the initial threshold, again
producing unrealistic crack paths.
8-4 Numerical simulations
Crack paths in C(T) specimens with Pettit-method
12
ST00−L
ST15−L
ST30−L
ST45−L
ST60−L
ST75−L
3
y position [mm]
8
Crack paths in DEN(T) specimens with Pettit−method
4
ST00−L
ST15−L
ST30−L
ST45−L
ST60−L
ST75−L
10
y position [mm]
63
6
4
2
1
2
0
0
0
2
4
6
8
10
12
x position [mm]
14
16
18
Figure 8-22: Plot of the crack paths in C(T) specimens
predicted with the Pettit-method, for six orientations.
6
x position [mm]
8
10
6
4
2
12
ST00−L
ST15−L
ST30−L
ST45−L
ST60−L
ST75−L
10
y position [mm]
8
4
Crack paths in DEN(T) specimens with k2−method
12
ST00−L
ST15−L
ST30−L
ST45−L
ST60−L
ST75−L
ST90−L
10
2
Figure 8-23: Plot of the crack paths in DEN(T) specimens
predicted with the Pettit-method, for six orientations.
Crack paths in C(T) specimens with k2−method
12
y position [mm]
0
20
8
6
4
2
0
0
2
4
6
8
10
12
14
x position [mm]
Figure 8-24: Plot of the crack paths in C(T) specimens
predicted with the k2-method, for six orientations.
0
0
2
4
6
x position [mm]
8
10
12
Figure 8-25: Plot of the crack paths in DEN(T) specimens
predicted with the k2-method, for six orientations.
Finally, the mode-mixity values of the predicted crack paths are looked at. The k2-and Pettit-method
both have been calibrated for low mode mixity, i.e. M ≤ 0.4. Figure 8-26 presents M for both methods,
where the models of C(T) specimens have been used to obtain M. One can clearly see the large difference
between the two methods, as was to be expected looking at the crack paths. The k2-method shows
alternations between low mode-mixity, i.e. the parts of the crack path were normal mixed mode criteria
apply, and high mode-mixity, i.e. branch forming, see Figure 8-26 A. With increasing rotation of the STaxis, the peak values of M increase for the k2-method, this is a logical consequence of the predicted crack
path. However, one can also notice the values of the mode-mixity gradually decrease as the crack length
becomes longer (higher analysis number). For orientations up to ST45-L the values for M are below 0.4.
For the Pettit-method, the variations in mode-mixity are much lower, and the values themselves as well,
see Figure 8-26 B. The mode-mixity shows a slow increasing trend up to a certain point, whereafter the
mode-mixity gradually decreases again. The peak is found to coincide with the point where the minimum
value of Kp,1 has been reached. The values of M for the Pettit-method are within the low mode-mixity
regime.
64
8 Results
A
Mode−mixity C(T) specimens with k2−method
0.7
ST15−L
ST30−L
ST45−L
ST60−L
ST75−L
0.2
ST15−L
ST30−L
ST45−L
ST60−L
ST75−L
0.6
0.5
0.15
M [−]
M [−]
B
Mode−mixity C(T) specimens with Pettit−method
0.25
0.1
0.4
0.3
0.2
0.05
0.1
0
0
1
3
5
7
9
11
13
Analysis number [−]
15
17
19
21
1
3
5
7
9
11
13
15
17
19
21
Analysis number [−]
Figure 8-26: Mode-mixity in C(T) specimens as predicted with both numerical methods. A) Pettit-method. B) k2-method.
Chapter 9
Discussion
In Chapter 8 the results of the fatigue tests, fractography and the investigation of the microstructure
are given. This chapter discusses these results and compares the results to the expected outcomes and
trends, as presented in Chapter 3 and 8. Also some literature presented in Chapter 5 and 6 is used in
this discussion and the hypotheses given in those chapters are judged for validity. The judging of these
hypotheses aids in assessing the results of the numerical methods. Last, the drivers for crack branching
and turning are established and the conclusions are presented.
9-1 Discussion on literature and fatigue tests
9-1 Microstructure
As observed in literature, the grain structure of Al 2050-T84 and of Al 7010-T7451 is anisotropic in
nature, with large dimension in the L-direction and small dimensions particularly in the ST-direction.
Copper rich inclusions and grain boundary particles have been found. Although the exact composition
of these particles has not been determined, section 3-1 supports the notion that these particles may lead
to an increased tendency for intergranular fracture. The large grain size found in Al-Li may contribute
to a decreased fatigue strength in some directions, as generally alloys with smaller grain sizes have better
fatigue properties.
Crack paths and fracture surfaces
Crack branches, crack turning and crack deviation found in the tested Al-Li alloy indicates a strong
influence of the microstructure on the macroscopic fatigue crack growth behaviour, and likely also the
static fracture behaviour, see Figure B-14 C. Crack branches initiate frequently, especially in ST75-L and
ST90-L coupons, and their length differs greatly, as Figure 8-11 A and B show.
Fractography revealed secondary cracking on what appear to be grain boundaries in the Al-Li ST90-L
sample, see Figure 8-17 and 8-18. Some secondary cracking is observed on the investigated ST45-L sample
as well, though not as often as on the ST90-L sample. Essentially these secondary cracks can be considered
65
66
9 Discussion
as branches, regardless of the actual fracture mechanism. Literature shows that crack branching has a
major influence on the stress field surrounding the crack, possibly explaining the change in fracture
surface observed in Figure 8-17. It may be concluded that Al 2050 shows an increased tendency for crack
branching, as is suggested in section 3-3 as well.
In terms of mixed-mode fatigue behaviour it may be said that both tested materials are likely to show
crack turning towards the materials weak direction for higher Kmax values (>25 MPa√m). The specimens
show approximately straight off-axis crack paths for orientations up to ST45-L, where the crack is clearly
growing in low mixed-mode conditions, M below 0.2, see Figure 8-26 A. The curved crack paths that are
seen in C(T) specimens of Al 2050, for orientations close to ST90-L, seem to grow in low mixed-mode
conditions as well, according to the numerical analyses performed in section 8-4. This indicates that main
crack driving force still is mode-I, even for the (highly) curved crack paths in the C(T) specimens.
Fractography has revealed that Al 2050 shows a very facetted surface for short crack lengths, i.e. shortly
after crack initiation, see Figure 8-15. The investigated DEN(T) sample showed more distinct striations
for shorter crack lengths than C(T) samples, again indicating crack closure affects the striations. The
ST45-L and ST90-L samples both showed possible indications of slip bands on the fracture surface.
Crack life and FCGR
The fatigue fracture anisotropy, in terms of fatigue crack life and FCGR, found in literature is confirmed
by the fatigue tests, where the anisotropy is most prominent in Al 2050, see Figure 8-4 and 8-5. There is
a large difference found in fatigue crack life between the ST00-L and ST90-L orientation. The crack life
of C(T) ST90-L Al-Li specimens is about 4.5x larger than the ST00-L orientation. Also crack deviation is
found in specimens of both materials, as was found in tests with Al 8090 as well, see section 3-2. All this
indicates that the fatigue fracture resistance of Al 2050, and Al 7010 for that matter, is dependent on the
orientation of the crack path with respect to the material axes.
Not all of the increased crack life may be attributed to the increased fatigue resistance of the material,
as the crack path also has an effect, especially if branches or kinks are present. With the use of Figure
8-10, it can be determined that an Al 2050 ST90-L C(T) specimen has a FCGR about 5-10 lower than a
comparable ST00-L specimen, for ΔK* below 10 MPa√m. For those SIF’s, the crack paths in the specimens
are more or less aligned with the 0-degree direction, i.e. the expected mode-I direction. Looking at the
crack paths in the ST90-L specimens for higher ΔK* values, one can see that they turn or kink towards the
materials weak direction. The effect on the FCGR of this sharp turn or kink in the crack path is a constant
or even lower value with increasing ΔK*. The effect of branches on the FCGR seems to be similar, see
Figure A-13 and A-14.
So, although the crack turns or branches in the material weak direction, associated with a lower fatigue
crack growth resistance, the FCGR stays constant or even drops. Clearly the crack geometry has a large
influence on the available crack driving force, causing the actual crack driving force to be lower than
the calculated value using the compliance method. This is supported by FEA of specimens with kinking
cracks, such as seen in Figure 8-24. When a kink is formed in the crack path the available crack driving
force actually decreases, and M is increased significantly, resulting in a much lower mode-I SIF than
before the kink. As mentioned, the compliance method does not account for the actual crack geometry,
it determines an equivalent crack length, which is assumed to be pure mode-I. The exact influence of
a branch on the actual crack driving force is not known, and it is recommended to perform additional
research on this topic.
Figure 8-10 shows that a rotation of the microstructure w.r.t. the orientation of the crack does not always
lead to a significant effect on the FCGR. It is observed that the FCGR for the Al-Li C(T) ST00-L, ST30-L
9-1 Discussion on literature and fatigue tests
67
and ST45-L coupons are very similar. The ST30-L and ST45-L coupons showed an average crack path
angle of 20.5 and 31.8 respectively. To help explain why the difference in FCGR is minimal, although the
crack paths are different, one has to look towards the orientation of the crack w.r.t. the microstructure.
The angle of the crack paths to the L-axis of the material is 9.5 and 13.2 degrees for ST30-L and ST45-L
respectively. So apparently a rotation of about 13 degrees of a crack w.r.t. the L-axis of the material does
not change the fatigue crack growth resistance.
The data of the ST90-L coupon shown in Figure 8-10, shows that when the rotation of the crack w.r.t. the
L-axis of the material is about 90 degrees, the fatigue crack growth resistance does change significantly.
This indicates that in the case of an approximately straight crack of gradually turning crack the FCGR is
mainly determined by the orientation of the crack w.r.t. the microstructure.
Additional research on the effects of the orientation of the microstructure on the crack growth resistance
is recommended. The aim of this research is to find a relation between the crack growth resistance and the
orientation of the crack w.r.t. the microstructure. In this research one should investigate the effects of the
number of grains and grain boundaries a crack encounters in its path. In the case of the ST90-L coupons,
for ΔK* = 7-10 MPa√m, the crack grows in a direction where the grain dimensions are much smaller
compared to the ST00-L case, see Figure 8-2 and 8-3. For ST90-L the crack than grows the ST-direction,
where for ST00-L the crack grows in the L-direction.
The anisotropy in fatigue crack growth resistance is much less pronounced in Al 7010-T7451, as it only
shows a small increase in fatigue crack life for DEN(T) specimens and virtually none for C(T) specimens.
The crack paths however are very different from one another, indicating that a relative small amount of
anisotropy in crack growth resistance, can result in rather different crack paths than one would expect.
As much fewer sharp turns, kinks or branches are present in Al 7010 specimens, likely the majority of
the cracks grows predominantly mode-I. Therefore a large difference in crack life is not expected. The
kinks that do show in the crack path, are usually in the unstable crack growth regime, quickly leading to
ultimate fracture of the specimen. So their effect on the crack life will be minimal.
Looking at Figure 8-5, one can note that the crack life of ST-45L is the highest for Al 7010 DEN(T)
specimens. Both the ST60-L and ST-75-L specimens have lower crack lives, which is counterintuitive.
The crack length curve of the ST75-L specimens increases almost linear, with quite a high FCGR even
for small crack lengths, see Figure A-25. To find the reason for this unexpected trend in FCGR more
work is needed. The FCGR of the ST60-L specimens shows much more spread compared to ST45-L, the
decrease in crack life therefore may be due to inherent material variation. Concluding, it may be said that
Al 7010‑T7451 likely has a relative small amount of anisotropy in fatigue crack growth resistance.
Coupon types
Comparing the two specimen types for Al 2050, one can see that the effect of material axis orientation
on the fatigue crack life is more pronounced in DEN(T) specimens. For Al 2050 an increase in fatigue
crack life of about 4.5x is found between DEN(T) ST00-L and ST75-L, compared to 2.5x for the same
orientation C(T) specimens. It is unknown if this difference would change when comparing ST90-L and
ST00-L specimens, as the DEN(T) ST90-L specimens were of a deviant size. Therefore it is recommended
to repeat the fatigue tests of the ST90-L orientation with DEN(T) specimens of similar size as the other
DEN(T) specimens. This would allow to compare the two principal material directions present in the
ST-L plane, making the assessment about the difference between the two specimen types more fair.
From the fatigue tests, it is evident that the C(T) coupons have a lower FCGR than the DEN(T) coupons.
The reasons for this are explained in section 7-3, the most important being the higher nominal stress,
and the influence of the clamping for the DEN(T) coupons. The accuracy of the compliance method
68
9 Discussion
for the C(T) coupons is found to be better than for the DEN(T) coupons, based on the comparison of
the pictures of the coupons with the compliance data. Also many of the DEN(T) coupons exhibited
asymmetrical crack growth, which reduces the reliability of the SIF’s determined for those coupons.
9-2 Discussion on numerical methods
9-2 To evaluate the performance of the numerical methods presented in Chapter 5 and 6, the results
are compared with data from the actual fatigue tests and the stated hypotheses in these chapters are
investigated. And it is assessed to what extents the requirements set out in Chapter 4 are met. Table 9-1
gives the predicted average crack path angles for both numerical methods for C(T) specimen orientations
ST15-L, ST30-L and ST45-L. The prediction for ST00-L is a simple straight mode-I crack path, which is
also seen on the actual fatigue test specimens. The average crack path angles found on the fatigue test
specimens for those same orientations are also given in Table 9-1. It seems that the predictions of the
k2-method are reasonably accurate, while the differences for the Pettit-method are much larger.
However, looking at the predicted crack paths in C(T) specimens with the Pettit-method, an example can
be seen in Figure 9-1, the curvature of the predicted crack path is very small for larger crack lengths. The
first few analysis cycles predict a relatively low in-plane rotation of the crack path. This is because he first
analysis cycle has a straight 0-degree crack path, which is not shown in any of the predicted crack paths.
Hereafter the crack rotates and tends to grow an approximately steady angle w.r.t. x-axis (the symmetry
axis of the coupon). To give a more fair assessment of the average crack path angle with the Pettit-method,
one has to determine the crack path angle of the approximately straight part of the predicted crack path,
as is done in Figure 9-1. For the four orientations presented in Table 9-1, that straight part is determined
to start at the 11th analysis cycle, and continues to last completed analysis cycle. The value of the crack
path angle of that portion of the crack path is given in Table 9-1 as well in the right most column.
Table 9-1: Average crack path angles for C(T) Al 2050-T84 specimens.
Table 7-1
Specimen
Fatigue test
Pettit-method
k2-method
Adjusted Pettit-method
ST15-L
11.5
8.7
10.1
10.1
ST30-L
20.5
17.1
21.6
19.9
ST45-L
31.8
24.9
32.9
30.1
Adjusted fit on crack path
10
ST45−L
Adjusted fit
y position [mm]
8
6
4
2
0
0
2
4
6
8
10
12
x position [mm]
14
16
18
20
Figure 9-1: Adjusted average crack path angle, prediction with Pettit-method shown for a C(T) ST45-L coupon.
9-2 Discussion on numerical methods
69
Finally, one can overlay the actual crack path with the predicted crack path from either numerical method.
Such an overlay is presented in Figure 9-2 for a ST45-L and ST60-L coupon. The pictures obtained during
the fatigue tests have been scaled appropriately to match the scale of the axes of Figure 9-2. For the
ST45-L coupon the predicted crack path very accurately matches the actual crack path as observed for the
Al 2050 C(T) ST45-L 1 coupon. For the ST15-L and ST30-L configurations, such an overlay has not been
performed, however one may expect that similar correlations with the actual crack path as for ST45-L are
obtained, based on the prediction shown in Table 9-1.
Comparing the actual crack path of the ST60-L coupon to the predicted path, one sees that these do not
seem to match. The Pettit-method predicts that the crack immediately starts to rotate and after a certain
point becomes approximately straight. The actual crack shows much smaller crack path angles for smaller
crack lengths, where for larger crack lengths the crack rotates towards the material’s weak plane, i.e.
rotated 60 degrees w.r.t. the x-axis.
Recalibration of the model parameters of the Pettit-method may improve the predictions made for ST60-L
and likely also ST75-L coupons which show crack turning. Therefore, it is recommended to perform a
recalibration of the model parameters of the Pettit-method, where the actual crack paths can be used to
compare the predicted crack paths to. Overlaying the actual crack path with the predicted crack path
immediately shows how large the deviation is, and gives more information than just an average crack
path angle.
A
Predicted and actual crack paths C(T) ST60−L coupon
12
Pettit−method
k2−method
10
Pettit−method
k2−method
10
8
y position [mm]
y position [mm]
B
Predicted and actual crack paths C(T) ST45−L coupon
12
6
4
2
8
6
4
2
0
0
0
2
4
6
8
10
12
x position [mm]
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
x position [mm]
Figure 9-2: Comparison of the crack paths observed in the fatigue tests and the predictions with the Pettit-method. A) Crack
paths for an Al 2050-T84 C(T) ST45-L coupon. B) Crack paths for an Al 2050-T84 C(T) ST60-L coupon.
k2-method
To fully justify the use of either numerical method, one also has to judge the hypotheses that form the
basis of that particular method, even though the predictions may be accurate. In Chapter 5 a couple of
hypotheses were stated, which are now investigated and judged for validity.
First, it is hypothesized that the intergranular shear strength and local shear stress are the two main
parameters which define both the formation and length of a branch. Fractography of a branch has
revealed no signs of shear influence on the fracture surface, as non-orientated dimples were found, see
Figure 8-21. Also no intergranular facets or fatigue markings are observed, indicating the fracture surface
of the branch was formed by void growth and coalescence. It must be noted that only a portion of the
branch has been investigated, namely the part of the branch is directly adjacent to the main crack.
70
9 Discussion
A recent study by Messner confirms that branches in the crack arrestor configuration, as observed on
specimens in this study, are formed by void growth and coalescence [Messner, 2014]. To explain how this
fracture process can occur in a direction aligned with the loading direction, one must look towards the
stress distribution on a microstructure level. It is found that branches or delaminations tend to initiate on
the grain boundaries between specific pairs of grains. These pairs of grains have a stiff/soft configuration,
or vice versa, where the soft grain shows large plastic deformation at low stress levels, leading to elevated
stress on grain boundary with stiff grain.
Numerical simulations show that in the crack arrestor configuration, such a stiff/soft grain pair causes
elevated normal stress on the grain boundary, also in symmetrical coupons like a M(T) [Messner 2014].
Figure 9-3 shows the stress distribution on a grain level, for a stiff/soft grain pair. Also presented in the
same figure is the value of Rice-Tracey parameter, along a line parallel to the primary crack. Larger values
of the Rice-Tracy parameter indicate regions more prone to ductile, void-driven fracture.
Figure 9-3: Stress distribution and Rice-Tracey damage parameter on the mid-thickness plane around a crack in ST90-L
configuration in SSY-conditions with KI = 45 MPa√m. A) Equivalent stress distribution σe, dashed line marks location of the line
plot shown in B. B) Values of Rice-Tracey parameter along a line in the ST-direction [Messner et al., 2014].
Furthermore, it is found that the PFZ’s cause damage localization on the grain boundaries and around the
δ’ precipitates slip localization is found. Both these factors combined with the elevated normal stress on
the grain boundary increase the void growth on the grain boundary. Shear stress plays a role in creating
the necessary conditions for a crack arrestor delamination [Messner 2014].
Additionally a soft/stiff grain pair shields the primary crack front while not affecting the normal stresses
acting on the adjacent grain boundary for crack arrestor delamination. Shielding of the primary crack
increases with the application of a compressive T-stress. These results indicate a somewhat higher
tendency for crack arrestor delaminations in M(T) specimens (negative T-stress) than in C(T) specimens
(positive T-stress) consistent with recent experiments [Messner 2014].
So, the combination of a region of sustained/increased void growth on the nearby grain boundaries,
significant shielding of material ahead of the primary crack, and continued high stresses just above the
primary crack plane and normal to the grain boundary apparently favour the initiation of delaminations.
These shielding and amplification effects remain highly localized to the grain pairs that generate them.
9-2 Discussion on numerical methods
71
At this point it is not known how large the tensile stress in ST-direction, i.e. normal to the grain boundary
was, in order to initiate the branches observed in the Al 2050-T84 DEN(T) ST90-L 2 specimen. Therefore,
it is recommended to investigate the stress state at time of branch initiation both a macrostructural
and microstructural level. One could use digital image correlation to find the macrostructural stress
distribution in a test coupon at the time of branch initiation. For the investigation of the microstructure
one could use a crystal plasticity approach, as is done in the research of Messner [Messner 2014].
The second hypothesis that is stated in Chapter 5 is that macroscopic crack deflection is governed by
the formation of intergranular crack extensions in the weak plane of the material, governed by a shear
stress criterion. As fractography of the Al 2050-T84 C(T) ST45-L sample showed, no indications of
intergranular crack extension in the direction of the weak plane are found. The fracture surface shows
clear fatigue striations and influence of shear stress in the region of unstable fracture, see Figure 8-17.
Only in the region of unstable fracture intergranular facets are observed, so this hypothesis is judged to
be invalid.
Furthermore, it is stated in Chapter 5 that there is a correlation between the branch length and Kmax
or ΔK. However Figure 8-11 B and Figure B-7 show that the branch length in the DEN(T) specimens
seems rather uncorrelated to the apparent SIF of the main crack. So to see if any relation exists, one
should focus only on the first branch formed, as the stress state is less complicated and more comparable
between the specimens. Interrupting the fatigue test after the first branch has formed allows for detailed
investigation of the fracture surfaces. Considering the sample set in this study is limited, it seems that
there is a correlation in branch length in the Al 2050-T84 DEN(T) specimens for the first branch formed.
The fractography on the samples of Al 2050-T84 revealed a macroscopically very rough fracture surface
in the unstable fracture regime, as can be seen in Figure 8-13. In the area of unstable fracture increasingly
more intergranular facets became visible with increasing crack length, alongside of dimples. In a majority
of the dimples particles with various shapes are present, likely these particles are the void initiators which
eventually result in the dimples. Also no sign of cleavage has been found, so at this point it unclear how
such a rough fracture surface is formed, if the dominant fracture mechanisms seems to be of tensile and
intergranular nature. Therefore, it is recommended to investigate which fracture processes would cause a
fracture surface as seen in Figure 8-13 and 8-14.
It may be concluded that branch formation seems to be determined by a combination of shielding and
amplification effects on a microstructural level, leading to void growth and coalescence. This is confirmed
by the finding on the fracture surface of the investigated branch. This invalidates the first hypothesis stated
in Chapter 5, that branch formation would be primarily determined by intergranular shear strength and
shear stress. The second assumption, that crack deviation is caused by intergranular crack extensions is
also invalidated, as no intergranular markings have been found on a deviated crack surface, such as the
C(T) ST45-L investigated in section 8-3. So, although the predicted crack path angles with the k2-method
are reasonably accurate, it is concluded that this method does not explain this physical mechanisms
causing macroscopic crack deviation and/or crack branching.
Considering the requirements set out in Chapter 4, the k2-method does not satisfy any of three required
additions to the LEFM framework. It does not accurately model fatigue fracture anisotropy, it does not
accurately predict the initiation of branches or in-plane crack turning.Therefore, it recommended to
not use this method for future crack path predictions on artificially aged aluminium alloys, such as
Al 2050‑T84.
72
9 Discussion
Pettit-method
The underlying key concepts of the Pettit-method are discussed here and judged for validity. The basic
concept of the Pettit-method is that a crack grows in the direction that maximizes the ratio of crack
driving force over crack resistance, see equation (6-1). Next, it assumes that the fracture resistance of
the material is dependent on the orientation of the crack. Also fracture mode asymmetry is taken into
account, i.e. the difference in fracture resistance for the three different crack modes. Furthermore, it
is assumed that the fracture resistance of the material is dependent on Kmax. It must be noted that the
parameters of this method have been validated for low mixed-mode conditions.
As FEA shows, both an approximately straight off-axis and turning crack path are in low mixed-mode
conditions, i.e M < 0.2, see Figure 8-25 A. The crack paths observed in the fatigue tests and those predicted
with the Pettit-method show similarities, indicating that fracture resistance anisotropy may indeed be the
cause of such crack paths. As the calibrated values for the parameters of this model show, only a small
amount of anisotropy is needed, the ratio between the lowest and highest relative fracture resistance is at
most 0.9, see Table 6-2. Furthermore, the predictions of the Pettit-method are reasonably accurate, see
Figure 9-1, 9-2 and Table 9-1. So, one may conclude that the Pettit-method predicts turning crack paths
rather well.
One downside of the Pettit-method is that it does not allow for the initiation of branches. It always
predicts one continuous crack, no new cracks are initiated. The model would rather predict a sharply
turning crack, in the case of a significant difference in fracture resistance. As earlier in this section was
shown, branches seem to be formed by void growth and coalescence. So, to expect a numerical method
that is calibrated to predict fatigue crack paths to also predict such branches would not be reasonable.
However, the capability of the numerical method to predict branches is one of the required additions to
the LEFM framework described in Chapter 4. Therefore, it concluded that the Pettit method incorporates
the effects of fatigue fracture anisotropy and accurately model in-plane turning crack paths. It is not
capable of predicting branches or delaminations.
Intersecting cracks
As was presented during this study is that most commonly available FE packages do not allow the
modelling of complex crack paths, as for example discussed in Figure 8-11 A and B. Usually it is allowed
to model multiple crack in one model, however it is not allowed to have multiple intersecting cracks,
which would be needed if branches are present. Even more sophisticated crack modelling methods, such
as XFEM, do not allow for intersecting cracks, while this is essential to accurately model complex crack
geometries, such as branches, forked cracks, etc. Therefore, it is recommended that intersecting cracks
can be (more easily) modelled in future FE-packages.
9-3 Drivers for crack branching and turning
9-3 This section summarizes the finding on the drivers for crack branching, -turning and off-axis crack
growth as found in the artificially aged alloys Al 2050-T84 and Al 7010-T7451. The drivers for crack
branching are discussed first.
Crack branching in the crack arrestor configuration manifests itself as delaminations in alloys such as Al
2050-T84. In this case a delamination means crack growth along a grain boundary. Recent studies and
fractography performed in this study show that delaminations are formed by void growth and coalescence.
Delaminations initiate on favourable stiff/soft grain pairs. The effect of this stiff/soft pairing, along with
9-4 Conclusions
73
damage localization in PFZ’s along the grain boundaries and slip localization, is an elevated normal stress
and increased void growth on the grain boundary between a stiff/soft grain pair. The elevated stress, crack
shielding of the main crack, due to the stiff/soft grain pair, and the increased void growth are the main
drivers for the initiation of delaminations in a crack arrestor configuration. Shear stress plays a role in
creating the necessary conditions for a crack arrestor delamination.
As a result of the plate rolling process, the microstructure becomes anisotropic, with large dimensions in
the L- or rolling-direction, and small dimension in the ST-direction. The main driver for crack turning
seems to be the orientation of that microstructure w.r.t. the primary crack and the loading direction.
Numerical simulations show that the mode-mixity for turning crack paths is low, M < 0.2. In other words
such crack grow in near mode-I conditions. This is confirmed by fractography of the fracture surface of
a C(T) ST45-L coupon, which exhibited a turning crack path.
In terms of fatigue fracture resistance anisotropy, comparison of the FCGR of ST00-L and ST90-L coupons
showed that such anisotropy is present in these alloys, see Figure 8-10. However a similar comparison
of the FCGR between ST00-L, ST30-L and ST45-L coupons, showed little to no difference, see Figure
8-10. This is likely explained by the fact that the difference in orientation of the primary crack w.r.t. the
microstructure was small in those three coupon configuration. So, the orientation of the microstructure
is not only the main driver for crack turning but also for fracture resistance anisotropy.
9-4 Conclusions
9-4 The fatigue tests on Al 2050-T84 and Al 7010-T7451 showed that the fatigue fracture anisotropy is
present in both these alloys, where it is most prominent in Al 2050-T84. The effects of this anisotropy
are noticeable on the crack life, fatigue crack growth rate and crack paths. The fracture resistance is
dependent on the orientation of the crack with respect to the principal material axes. The crack paths
observed also play a major role in determining the FCGR. Crack branches, sharp turns or kinks in the
crack path cause the FCGR to stay constant or even decrease with increasing ΔK*. It must be noted that all
ΔK* values calculated in this thesis are for an equivalent straight mode-I crack, not the actual crack in the
fatigue specimen. Branches also seem to have an influence on the surrounding stress field, as changes in
the fracture surface morphology were observed around secondary cracking. Furthermore, the specimen
geometry seems to influence the size of the effect of fracture resistance anisotropy, as the anisotropy in
crack life was observed to be most prominent in DEN(T) specimens. The difference in stress distribution
in the specimens seems to cause this.
As far as the numerical methods go, the k2-method does prove to be useful, although the predictions were
reasonably accurate. The key hypotheses of this method have been invalidated, with the results of the
fatigue tests and fractography. The k2-method does not satisfy the requirements as set out in Chapter 4,
so it is recommended to not use this method in the future.
The Pettit-method seems to function properly, predicting the crack paths reasonably accurate, as is shown
in section 9-2. The key concepts of the Pettit-method are judged to be true or at least plausible. The
simulations have shown that only a small amount of anisotropy in fracture resistance for two principal
material orientations is sufficient to create the crack paths as observed in the fatigue specimens. For
the simulations the minimum ratio between the fracture resistances of the weakest over the strongest
direction is 0.9. The Pettit-method does not satisfy all the requirements described in Chapter 4, as it does
not allow for the initiation of branches. For crack turning this method delivers promising results and its
use can be continued in the future.
74
9 Discussion
Finally, a part of the goal of this thesis is to establish the drivers for crack branching and turning, and
off-axis crack growth. The main driver for crack turning seems to be the orientation of the microstructure
w.r.t. the primary crack and loading direction. The fatigue tests have revealed that anisotropy exists in the
fatigue fracture resistance for the principal material orientations, see section 8-xx. Also indications have
been found that the alloys investigated in this study exhibit static fracture anisotropy. The Pettit-method
has shown that crack turning and off-axis crack growth can be predicted with an modified strain energy
release rate (MSERR) criteria.
Crack branching in the crack arrestor configuration seems to be caused by a combination of shielding and
amplification effects on a microstructure level. Recent research shows that delaminations tend to initiate on
favourable grain pairings, which cause an elevated stress to be present on the grain boundary. Additionally
damage localization in PFZ’s on the grain boundaries and slip localization due to δ’ precipitates, and the
elevated stress lead to increased void growth on the grain boundary. The favourable grain pairing also
shields the primary crack, which all combined leads to the formation of delaminations along these grain
boundaries. So, the fracture process that forms a delamination is void growth and coalescence. This is
confirmed by findings on the fracture surface of a branch in DEN(T) coupon investigated in this study.
Chapter 10
Conclusions and recommendations
10-1 Conclusions
10-1 Recently renewed interest in aluminium-lithium alloys is seen in the aviation industry. Aluminiumlithium and other advanced aluminium alloys present a major opportunity to save weight in an aircraft
structure. The addition of lithium in the alloy decreases the density, while keeping or increasing the
fatigue life, compared to other aluminium alloys. Where fatigue cracks in aluminium alloys are expected
to grow in a direction perpendicular to the principal loading, these aluminium-lithium alloys can exhibit
crack branching or crack turning in a direction parallel to the principal loading. Furthermore, anisotropy
in both fracture toughness and fatigue crack life is observed in these alloys. These phenomena have been
observed in the past on other alloys, and are qualitatively understood.
An investigation of the microstructure of Al 2050-T84 and Al 7010-T7451 confirms the findings in
literature, that these alloys have a banded microstructure typical for rolled artificially aged aluminium
plate products. The grains have large dimension in the rolling- or L-direction and small dimension in
the ST-direction. Literature has shown that Al 2050-T84 has precipitate free zones around the grain
boundaries, caused by a competition for lithium.
The fatigue tests in this study showed that fatigue fracture anisotropy is present in both Al 2050-T84
and Al 7010-T7451, affecting the fatigue crack life, the crack path and fatigue crack growth rate. The
crack paths observed play a major role in determining the fatigue crack growth rate. Crack branches,
sharp turns or kinks in the crack path cause the crack growth rate to stay constant or even decrease with
increasing ΔK*. Branches also seem to have an influence on the stress field surrounding the primary crack,
as changes in the fracture surface morphology were observed around secondary cracking. Findings in
literature also confirm this effect of branches. Furthermore, the specimen geometry seems to influence
the magnitude of the effect of fracture resistance anisotropy, differences in the stress distribution are the
cause of this.
The investigation of the fracture surface of a branch in a DEN(T) ST90-L coupon of Al 2050-T84 showed
that the branch seemed to be formed by void growth and coalescence. No indications of fatigue markings
or intergranular facets have been observed on this surface. The dimples that are observed do not seem
to be sheared in nature, indicating these dimples are formed by tensile void growth. Macroscopically the
fracture surface of the branch looks very flat and lots of aluminium oxide product is present.
75
76
10 Conclusions and recommendations
A recent study confirms the findings that a branch in the crack arrestor configuration is formed by void
growth and coalescence. In this configuration branches tend to initiate on favourable grain pairings, i.e.
stiff/soft grain pairs. These grain pairs cause an elevated stress to be present on the grain boundaries
between these grain pairs. Additionally damage localization in PFZ’s on the grain boundaries, slip
localization due to δ’ precipitates and the elevated stress lead to increased void growth on these grain
boundaries. The favourable grain pairing also shields the primary crack, which all combined leads to
the formation of delaminations along these grain boundaries. So, the stiff/soft grain pairing along with
damage localization in the PFZ’s on the grain boundaries and slip localization due to δ’ precipitates are
determined to be the main drivers for crack branching in the arrestor configuration.
On the other hand, the main driver for crack turning and fatigue fracture resistance anisotropy seems to
be the orientation of the microstructure w.r.t. the primary crack and loading direction. Comparison of
the fatigue crack growth rate for various coupon orientations shows that orientation of the crack w.r.t.
the microstructure seems to determine the fatigue fracture resistance. Numerical simulations show
that turning cracks grow in near mode-I conditions, and seem to be well predictable with a MSERR
approach. Fractography on an Al 2050 C(T) ST45-L coupon, which exhibits crack turning, reveals that
normal striations are found on the fracture surface. No indications of intergranular crack growth or
macrostructural branches have been observed outside of the region of unstable fracture.
For the numerical strategy two methods have been developed, the k2-method and the Pettit-method. The
first method hypothesizes that crack deviation is caused by intergranular crack extensions, governed by a
shear stress criterion. The latter method assumes that cracks in these advanced alloys grow in a direction
which maximizes the ratio of available crack driving force over fracture resistance in that particular
direction. The results of the fatigue tests and fractography invalidated the assumptions of the k2-method,
leaving it unsuitable for future use.
The key concepts of the Pettit-method have proved to be valid or at least plausible. The predictions of
the crack paths in various specimens with this method are reasonably accurate. This method shows
it is possible to predict crack turning with the use of macroscopic crack growth criteria. Additionally,
simulations have shown that only a small amount of anisotropy in fatigue fracture resistance is sufficient
to create the crack paths observed in the fatigue specimens. A downside of the Pettit-method is that is
does not allow for the initiation of branches.
Summarizing, the main driver for crack turning and the observed fatigue fracture resistance anisotropy
seems to be the orientation of the microstructure w.r.t. the primary crack and loading direction. The
FCGR, the crack paths and crack life are all affected by the fatigue fracture resistance anisotropy. The
main drivers for crack branching in an arrestor configuration seem to be stiff/soft grain pairs, damage
localization in the PFZ’s and slip localization around δ’ precipitates. The fracture process of a branch
in this configuration seems to be void growth and coalescence. The numerical strategy has shown that
crack turning can be predicted by macroscopic crack growth criteria, such as MSERR. However, crack
branching likely cannot be predicted by such macroscopic criteria.
10-2 Recommendations
77
10-2 Recommendations
10-2 Several recommendations are considered in this section to improve the work presented in this thesis,
and to extend the knowledge on crack branching and crack turning. First, it is recommended to perform
a recalibration of the model parameters of the Pettit-method, where the actual crack paths can be used
to compare the predicted crack paths to. Overlaying the actual crack path with the predicted crack path
immediately shows how large the deviation is, and gives more information than just an average crack
path angle.
Secondly, additional research on the effects of the orientation of the microstructure on the crack growth
resistance is recommended. The aim of this research is to find a relation between the crack growth
resistance and the orientation of the crack w.r.t. the microstructure. In this research one should investigate
the effects of the number of grains and grain boundaries a crack encounters in its path.
Thirdly, it is recommended to investigate the stress state at time of branch initiation both a macrostructural
and microstructural level. One could use digital image correlation to find the macrostructural stress
distribution in a test coupon, at the time of branch initiation and shortly thereafter. For the investigation
of the microstructure one could use a crystal plasticity approach. Suitable test coupons likely are C(T)
and M(T) coupons in the ST90-L configuration. Additional fatigue tests allow to investigate more branch
fracture surfaces, including the part of branch arrest. It is recommended to interrupt the fatigue test after
the initiation of the first branch, to preserve as much of the original fracture surface as possible. The use
of digital image correlation aids in the understanding of the influence of branch in the crack arrestor
configuration on the stress distribution around the primary crack.
Fourthly, most widely used FE-packages do not allow for the modelling of intersecting cracks, while
this is a necessary condition to be able to accurately model the effects of one or multiple branches in
a structure. So, it is recommended that an efficient method is developed to allow for the modelling of
intersecting cracks in FEM.
Crack turning and off-axis crack growth affect the crack length calculations if a compliance method is
used. In literature methods are available to determine the crack length and FCGR of the actual crack,
however they are limited to approximately straight crack paths and up to 40 degrees in-plane rotation
w.r.t. the expected mode-I crack growth direction. Therefore it is recommended to develop a method
that can determine the actual FCGR of a turning crack and allows for up to 90 degrees in-plane rotation
of the crack w.r.t. the expected mode-I direction. As a basis for this research one can use the crack paths
observed in the fatigue tests performed in this study, along with the compliance measurements. The
numerical models from this study can be used to determine the compliance an arbitrary crack path,
which can help to improve the accuracy of the method.
The fatigue tests in this thesis have shown that the considered aluminium alloys exhibit fatigue fracture
resistance anisotropy. The magnitude of this anisotropy has not been quantified accurately and consistently
yet. Partly this is due to the choice of coupon geometries. The DEN(T) specimens proved to significantly
promote asymmetrical crack growth, leading to inaccuracies in the calculations of the FCGR and SIF’s.
Therefore, it is recommended to repeat similar fatigue tests with a symmetrical coupon, which does not
promote asymmetrical crack growth, such as a CC(T) specimen. These additional tests allow to quantify
the anisotropy more accurately.
Additionally, fractography revealed fracture surface morphology in the regime of unstable fracture that
needs further explaining. In this regime some intergranular facets along with dimples are observed, and
macroscopically the surface looks very rough. Additional work, to explain how this fracture surface
is formed, may aid in the understanding of the special crack growth properties of these advanced
aluminium-lithium alloys.
78
10 Conclusions and recommendations
Finally, the DEN(T) ST90-L specimens used in the fatigue tests in this thesis have a different geometry
than the other DEN(T) specimens. The reason a different geometry is chosen, is that a wider and longer
specimen would allow for longer crack lengths, likely increasing the tendency for crack branching. Yet the
choice for a different geometry for the ST90-L orientation does not allow for a full comparison between
the two coupon geometries used in the fatigue tests. Therefore, it is recommended to use similar sized
specimens for all orientations in future experiments.
Chapter 11
References
Alderliesten, R.C. (2009). Fatigue & Damage Tolerance of Hybrid Materials & Structures – Some Myths,
Facts & Fairytales. In ICAF 2009, Bridging the Gap between Theory and Operational Practice, pp.
1245-1260.
ASTM International. (2003). ASTM E338-03. Standard Test Method of Sharp-Notch Tension Testing of
High-Strength Sheet Materials.
ASTM International. (2013). ASTM E647-13ae1. Standard test method for measurement of fatigue crack
growth rates.
Bao, R., Zhao, X.C., Zhang, T. & Zhang, J.Y. (2013). Fatigue Crack Growth of Alloy 7050-T7451 Plate in
L-S Orientation. Key Engineering Materials, 525-526, pp. 221-224.
Beese, A.M., Luo, M., Li, Y., Bai, Y. & Wierzbicki, T. (2010). Partially coupled anisotropic fracture model
for aluminum sheets. Engineering Fracture Mechanics, 77, pp. 1128-1152.
Benzerga, A.A., Besson, J. & Pineau, A. (2004). Anisotropic ductile fracture: Part I: experiments. Acta
Materialia, 52, pp. 4623-4638.
Bouchard, P.O., Bay, F. & Chastel, Y. (2003). Numerical modelling of crack propagation: Automatic
remeshing and comparison of different criteria. Computer Methods in Applied Mechanics and
Engineering, 192, pp. 3887-3908.
Bush, R.W., Bucci, R.J., Magnusen, P.E. & Kuhlman, G.W. (1993). Fatigue crack growth measurements
in aluminum alloys forgings: effects of residual stress and grain flow. In Fracture Mechanics: TwentyThird Symposium, pp. 568-589.
Chen, P.S., Kuruvilla, A.K., Malone, T.W. & Stanton, W.P. (1998). The effects of artificial aging on the
microstructure and fracture toughness of Al-Cu-Li alloy 2195. Journal of Materials Engineering and
Performance, 7, pp. 682-690.
Cotterell, B. & Rice, J.R. (1980). Slightly curved or kinked cracks. International Journal of Fracture, 16,
pp. 155-169.
79
80
11 References
Crill, M., Chellman, D., Balmuth, E., Philbrook, M., Smith, P., Cho, A., . . . Feiger, J. (2006). Evaluation of
AA 2050-T87 Al-Li Alloy Crack Turning Behaviour. In 10th International Conference on Aluminum
Alloys, Vancouver, Canada, pp. 1323-1328.
De, P.S., Mishra, R.S. & Baumann, J.A. (2011). Characterization of high cycle fatigue behavior of a new
generation aluminum lithium alloy. Acta Materialia, 59, pp. 5946-5960.
Dell, H., Gese, H. & Oberhofer, G. (2007). CrachFEM - A comprehensive approach for the prediction of
sheet metal failure. pp. 165-170.
Dumstorff, P. & Meschke, G. (2007). Crack propagation criteria in the framework of X-FEM-based
structural analyses. International Journal for Numerical and Analytical Methods in Geomechanics,
31, pp. 239-259.
Dursun, T. & Soutis, C. (2014). Recent developments in advanced aircraft aluminium alloys. Materials &
Design, 56, pp. 862-871.
Ertürk, T., Otto, W.L. & Kuhn, H.A. (1974). Anisotropy of ductile fracture in hot-rolled steel plates-an
application of the upset test. Metallurgical Transactions, 5, pp. 1883-1886.
Forman, R.G. & Mettu, S.R. (1992). Behavior of surface and corner cracks subjected to tensile and bending
loads in Ti–6Al–4V alloy. In Fracture Mechanics: 22nd Symposium, ASTM STP 1131, pp. 519-546.
Forth, S.C., Herman, D.J. & James, M.A. (2004). Fatigue crack growth rate and stress-intensity factor
corrections for out-of-plane crack growth. Fatigue and Fracture Mechanics: 34th Volume, ASTM STP
1461, pp.
Gese, H., Dell, H., Oberhofer, G., Oehm, M. & Heath, A. (2013). CRACHFEM – A comprehensive
approach for the prediction of sheet failure in multi-step forming and subsequent forming and crash
simulations. In Proceedings of Forming Technology Forum 2013, Germany.
Giner, E., Sukumar, N., Tarancón, J.E. & Fuenmayor, F.J. (2009). An Abaqus implementation of the
extended finite element method. Engineering Fracture Mechanics, 76, pp. 347-368.
Gregson, P.J. & Sinclair, I. (1996). Deviant crack path behaviour of aluminium-lithium alloy AA8090 plate.
Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering,
210, pp. 117-121.
Hafley, R.A., Domack, M.S., Hales, S.J. & Shenoy, R.N. (2011). Evaluation of Aluminum Alloy 2050T84 Microstructure Mechanical Properties at Ambient and Cryogenic Temperatures. NASA Langley
Research Center; Hampton, VA, United States. NASA/TM-2011-217163.
Hahn, G.T. & Rosenfield, A.R. (1975). Metallurgical factors affecting fracture toughness of aluminum
alloys. Metallurgical Transactions A, 6, pp. 653-668.
Hamel, S.F. (2010). A Parametric Study of Delaminations in an Aluminum-Lithium Alloy. M.Sc. thesis,
University of Illinois at Urbana-Champaign.
Hernquist, M. (2010). Effects of crack arresting delaminations in aluminum-lithium alloys. M.Sc thesis,
University of Illinois at Urbana-Champaign.
Hooputra, H., Gese, H., Dell, H. & Werner, H. (2004). A comprehensive failure model for crashworthiness
simulation of aluminium extrusions. International Journal of Crashworthiness, 9, pp. 449-463.
Jha, S.C., Sanders Jr, T.H. & Dayananda, M.A. (1987). Grain boundary precipitate free zones in Al-Li
alloys. Acta Metallurgica, 35, pp. 473-482.

81
Joyce, M.R. & Sinclair, I. (2014). Experimental assessment of mixed mode loading effects on macroscopic
fatigue crack paths in thick section Al-Cu-Li alloy plate.
Kalyanam, S., Beaudoin, A.J., Dodds Jr, R.H. & Barlat, F. (2009). Delamination cracking in advanced
aluminum–lithium alloys – Experimental and computational studies. Engineering Fracture
Mechanics, 76, pp. 2174-2191.
Kamp, N., Gao, N., Starink, M.J., Parry, M.R. & Sinclair, I. (2007a). Analytical modelling of the influence
of local mixed mode displacements on roughness induced crack closure. International Journal of
Fatigue, 29, pp. 897-908.
Kamp, N., Gao, N., Starink, M.J., Parry, M.R. & Sinclair, I. (2007b). Influence of grain structure and slip
planarity on fatigue crack growth in low alloying artificially aged 2xxx aluminium alloys. International
Journal of Fatigue, 29, pp. 869-878.
Kfouri, A.P. & Brown, M.W. (1995). Fracture criterion for cracks under mixed-mode loading. Fatigue and
Fracture of Engineering Materials and Structures, 18, pp. 959-969.
Khoei, A.R., Azadi, H. & Moslemi, H. (2008). Modeling of crack propagation via an automatic adaptive
mesh refinement based on modified superconvergent patch recovery technique. Engineering Fracture
Mechanics, 75, pp. 2921-2945.
Khoei, A.R., Moslemi, H. & Sharifi, M. (2012). Three-dimensional cohesive fracture modeling of nonplanar crack growth using adaptive FE technique. International Journal of Solids and Structures, 49,
pp. 2334-2348.
Kung, C.Y. & Fine, M.E. (1979). Fatigue Crack initiation and microcrack growth in 2024-T4 and 2124-T4
aluminum alloys. Metallurgical Transactions A, 10, pp. 603-610.
Luo, M., Dunand, M. & Mohr, D. (2012). Experiments and modeling of anisotropic aluminum extrusions
under multi-axial loading – Part II: Ductile fracture. International Journal of Plasticity, 32–33, pp.
36-58.
Maiti, S., Ghosh, D. & Subhash, G. (2009). A generalized cohesive element technique for arbitrary crack
motion. Finite Elements in Analysis and Design, 45, pp. 501-510.
Maligno, A.R., Rajaratnam, S., Leen, S.B. & Williams, E.J. (2010). A three-dimensional (3D) numerical
study of fatigue crack growth using remeshing techniques. Engineering Fracture Mechanics, 77, pp.
94-111.
McDonald, R. (2009). Characterization of Delamination in 2099-T861 Aluminum-Lithium. Ph.D.
dissertation, University of Illinois at Urbana-Champaign.
Messner, M. (2014). Micromechanical models of delamination in aluminum-lithium alloys. Ph.D.
dissertation, University of Illinois at Urbana-Champaign.
Messner, M.C., Beaudoin, A.J. & Dodds, R.H., Jr. (2014). Mesoscopic modeling of crack arrestor
delamination in Al–Li: primary crack shielding and T-stress effect. International Journal of Fracture,
188, pp. 229-249.
Metzmacher, G., Dell, H., Oberhofer, G., Gese, H. & Reese, E.D. (2007). Numerical simulation of the cold
expansion process - development and validation of an enhanced material model. Technical report.
EADS Innovations Works Germany.
Mir, A.A., Barton, D.C., Andrews, T.D. & Church, P. (2005). Anisotropic ductile failure in free machining
steel at quasi-static and high strain rates. International Journal of Fracture, 133, pp. 289-302.
82
11 References
Moës, N. & Belytschko, T. (2002). Extended finite element method for cohesive crack growth. Engineering
Fracture Mechanics, 69, pp. 813-833.
Moës, N., Dolbow, J. & Belytschko, T. (1999). A finite element method for crack growth without remeshing.
International Journal for Numerical Methods in Engineering, 46, pp. 131-150.
Mohammadi, S. (2008). Extended finite element method for fracture analysis of structures. Blackwell
Publishing Ltd.
National Transportation Safety Board. (1989). Aircraft Accident Report Aloha Airlines Flight 243, 28 april
1988. NTSB/AAR-89/03.
Newman, J. (1974). Stress analysis of the compact specimen including the effects of pin loading. In Fracture
Analysis: Proceedings of the National Symposium on Fracture Mechanics, pp. 105-121.
Newman, J.C., Jr. (1984). A crack opening stress equation for fatigue crack growth. International Journal of
Fracture, 24, pp. 131-135.
Paris, P. & Erdogan, F. (1963). A Critical Analysis of Crack Propagation Laws. Journal of Fluids Engineering,
85, pp. 528-533.
Pettit, R. (2000). Crack turning in integrally stiffened aircraft structures. Ph.D. dissertation, Cornell
University.
Pettit, R., Annigeri, B., Owen, W. & Wawrzynek, P. (2013). Next generation 3D mixed mode fracture
propagation theory including HCF-LCF interaction. Engineering Fracture Mechanics, 102, pp. 1-14.
Pilkey, W.D., Pilkey, D.F. & Peterson, R.E. (2008). Peterson’s stress concentration factors (third edition).
John Wiley & Sons.
Prasad, N.E., Gokhale, A.A. & Rao, P.R. (2003). Mechanical behaviour of aluminium-lithium alloys.
Sadhana, 28, pp. 209-246.
Rao, K.T.V. & Ritchie, R.O. (1992). Fatigue of aluminium-lithium alloys. International Materials Reviews,
37, pp. 153-186.
Rao, K.T.V., Yu, W. & Ritchie, R.O. (1988). Fatigue crack propagation in aluminum- lithium alloy 2090:
Part I. long crack behavior. Metallurgical Transactions A, 19, pp. 549-561.
Rao, K.T.V., Yu, W. & Ritchie, R.O. (1989). Cryogenic toughness of commercial aluminum-lithium alloys:
Role of delamination toughening. Metallurgical Transactions A, 20, pp. 485-497.
Reese, E. & Wood, P. (2013). Summary on findings of TEM investigation on Al 7085-T7651. EADS
Innovation Works, Germany.
Rioja, R. & Liu, J. (2012). The Evolution of Al-Li Base Products for Aerospace and Space Applications.
Metallurgical and Materials Transactions A, 43, pp. 3325-3337.
Roy, Y.A. & Dodds, R.H., Jr. (2001). Simulation of ductile crack growth in thin aluminum panels using 3-D
surface cohesive elements. International Journal of Fracture, 110, pp. 21-45.
Schijve, J. (2009). Fatigue of structures and materials. Springer.
Schubbe, J.J. (2009). Evaluation of fatigue life and crack growth rates in 7050-T7451 aluminum plate for
T-L and L-S oriented failure under truncated spectra loading. Engineering Failure Analysis, 16, pp.
340-349.
Schütz, W. (1996). A history of fatigue. Engineering Fracture Mechanics, 54, pp. 263-300.

83
Shi, J., Chopp, D., Lua, J., Sukumar, N. & Belytschko, T. (2010). Abaqus implementation of extended finite
element method using a level set representation for three-dimensional fatigue crack growth and life
predictions. Engineering Fracture Mechanics, 77, pp. 2840-2863.
Sinclair, I. & Gregson, P.J. (1994). Mixed mode fatigue crack growth in the commercial AlLi alloy, 8090.
Scripta Metallurgica et Materiala, 30, pp. 1287-1292.
Sinclair, I. & Gregson, P.J. (1997). The effects of mixed mode loading on intergranular failure in
AA7050-T7651. In 5th International Conference on Aluminum Alloys, pp. 175-180.
Suresh, S. (1985). Fatigue crack deflection and fracture surface contact: Micromechanical models.
Metallurgical Transactions A, 16, pp. 249-260.
Suresh, S., Vasudevan, A.K., Tosten, M. & Howell, P.R. (1987). Microscopic and macroscopic aspects of
fracture in lithium-containing aluminum alloys. Acta Metallurgica, 35, pp. 25-46.
Tada, H., Paris, P. & Irwin, G. (2000). The Stress Analysis of Cracks Handbook (Third edition edition).
ASME Press.
Ural, A., Krishnan, V.R. & Papoulia, K.D. (2009). A cohesive zone model for fatigue crack growth allowing
for crack retardation. International Journal of Solids and Structures, 46, pp. 2453-2462.
Vethe, S. (2012). Numerical Simulation of Fatigue Crack Growth. M.Sc. thesis, Norwegian University of
Science and Technology.
Wanhill, R.J.H. (1994). Status and prospects for aluminium-lithium alloys in aircraft structures. International
Journal of Fatigue, 16, pp. 3-20.
Wu, X.J., Wallace, W., Raizenne, M.D. & Koul, A.K. (1994). The orientation dependence of fatigue-crack
growth in 8090 aluminum-lithium plate. International Journal of Fatigue, 16, pp. 575-588.
Xie, Y.J., Hu, X.Z., Wang, X.H., Chen, J. & Lee, K.Y. (2011). A theoretical note on mode-I crack branching
and kinking. Engineering Fracture Mechanics, 78, pp. 919-929.
Xu, Y. & Yuan, H. (2009). Computational analysis of mixed-mode fatigue crack growth in quasi-brittle
materials using extended finite element methods. Engineering Fracture Mechanics, 76, pp. 165-181.
Zhang, T., Bao, R. & Fei, B. (2014). Load effects on macroscopic scale fatigue crack growth path in 2324T39 aluminium alloy thin plates. International Journal of Fatigue, 58, pp. 193-201.
Zhang, Z. & Paulino, G.H. (2005). Cohesive zone modeling of dynamic failure in homogeneous and
functionally graded materials. International Journal of Plasticity, 21, pp. 1195-1254.
84
11 References
Appendix A
Fatigue test data
All data from the fatigue tests on Al 205--T84 and Al 7010-T7451 C(T) and DEN(T) specimens are
presented in this appendix. For all specimens of one specific orientation a graph is given of both the
recorded crack length and fatigue crack growth rate obtained with the compliance method. The crack
length for C(T) specimens is given a/W, for DEN(T) this is given as 2a/W. The fatigue crack growth rate
is calculated as the derivative of the recorded crack length.
A-1 Fatigue test data Al 2050-T84
A-1 ST00-L
Crack length for Al 2050−T84 ST00−L
0.8
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
0.7
Crack growth rates for Al 2050−T84 ST00−L
−2
10
C(T) ST00−L 1
C(T) ST00−L 2
DEN(T) ST00−L 1
DEN(T) ST00−L 2
0.6
0.5
0.4
−3
10
−4
10
C(T) ST00−L 1
C(T) ST00−L 2
DEN(T) ST00−L 1
DEN(T) ST00−L 2
0.3
0.2
−5
0
0.5
1
N [cycles]
1.5
10
2
x 10
7
5
Figure A-1: Crack length of Al 2050-T84 ST00-L
specimens.
8
9
10
12
ΔK* [MPa√m]
15
20
Figure A-2: Crack growth speeds of Al 2050-T84 ST00-L
specimens.
A1
A2
A Fatigue test data
ST15-L
Crack length for Al 2050−T84 ST15−L
0.8
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
0.7
Crack growth rates for Al 2050−T84 ST15−L
−2
10
C(T) ST15−L 1
C(T) ST15−L 2
DEN(T) ST15−L 1
DEN(T) ST15−L 2
0.6
0.5
0.4
−3
10
−4
10
C(T) ST15−L 1
C(T) ST15−L 2
DEN(T) ST15−L 1
DEN(T) ST15−L 2
0.3
−5
0.2
0
0.5
1
1.5
2
10
2.5
N [cycles]
7
5
x 10
Figure A-3: Crack length of Al 2050-T84 ST15-L
specimens.
8
9
10
12
ΔK* [MPa√m]
15
20
Figure A-4: Crack growth speeds of Al 2050-T84 ST15-L
specimens.
ST30-L
Crack growth rates for Al 2050−T84 ST30−L
Crack length for Al 2050−T84 ST30−L
0.9
0.8
0.7
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
−3
10
C(T) ST30−L 1
C(T) ST30−L 2
DEN(T) ST30−L 1
DEN(T) ST30−L 2
0.6
0.5
0.4
0.3
−4
10
C(T) ST30−L 1
C(T) ST30−L 2
DEN(T) ST30−L 1
DEN(T) ST30−L 2
−5
10
0.2
0
0.5
1
1.5
2
2.5
3
N [cycles]
x 10
7
5
Figure A-5: Crack length of Al 2050-T84 ST30-L
specimens.
8
9
10
12
ΔK* [MPa√m]
15
20
Figure A-6: Crack growth speeds of Al 2050-T84 ST30-L
specimens.
ST45-L
Crack length for Al 2050−T84 ST45−L
0.9
0.7
−3
10
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
0.8
Crack growth rates for Al 2050−T84 ST45−L
−2
10
C(T) ST45−L 1
C(T) ST45−L 2
DEN(T) ST45−L 1
DEN(T) ST45−L 2
0.6
0.5
0.4
−4
10
−5
10
C(T) ST45−L 1
C(T) ST45−L 2
DEN(T) ST45−L 1
DEN(T) ST45−L 2
0.3
−6
0.2
0
0.5
1
1.5
2
2.5
N [cycles]
3
3.5
4
4.5
x 10
5
Figure A-7: Crack length of Al 2050-T84 ST45-L
specimens.
10
6
7
8
9
10
12
ΔK* [MPa√m]
15
20
Figure A-8: Crack growth speeds of Al 2050-T84 ST45-L
specimens.
A-1 Fatigue test data Al 2050-T84
A3
ST60-L
Crack length for Al 2050−T84 ST60−L
1
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
0.8
Crack growth rates for Al 2050−T84 ST60−L
−3
10
C(T) ST60−L 1
C(T) ST60−L 2
DEN(T) ST60−L 1
DEN(T) ST60−L 2
0.6
0.4
−4
10
−5
10
C(T) ST60−L 1
C(T) ST60−L 2
DEN(T) ST60−L 1
DEN(T) ST60−L 2
−6
0.2
0
1
2
3
4
5
10
6
N [cycles]
x 10
6
5
Figure A-9: Crack length of Al 2050-T84 ST60-L
specimens.
7
8
9
10
12
ΔK* [MPa√m]
15
20
25
Figure A-10: Crack growth speeds of Al 2050-T84
ST60-L specimens.
ST75-L
Crack length for Al 2050−T84 ST75−L
0.9
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
0.7
Crack growth rates for Al 2050−T84 ST75−L
−3
10
C(T) ST75−L 1
C(T) ST75−L 2
DEN(T) ST75−L 1
DEN(T) ST75−L 2
0.5
0.3
−4
10
−5
10
C(T) ST75−L 1
C(T) ST75−L 2
DEN(T) ST75−L 1
DEN(T) ST75−L 2
−6
0.1
0
1
2
3
4
5
10
6
N [cycles]
x 10
6
5
Figure A-11: Crack length of Al 2050-T84 ST75-L
specimens.
7
8
9
10
12
15
ΔK* [MPa√m]
20
Figure A-12: Crack growth speeds of Al 2050-T84
ST75-L specimens.
ST90-L
Crack length for Al 2050−T84 ST90−L
1
Crack growth rates for Al 2050−T84 ST90−L
−3
a/W* or 2a/W* [−]
0.8
0.6
C(T) ST90−L 1
C(T) ST90−L 2
DEN(T) ST90−L 1
DEN(T) ST90−L 2
DEN(T) ST90−L 3
0.4
0.2
0
2
4
6
8
N [cycles]
10
12
14
16
x 10
5
Figure A-13: Crack length of Al 2050-T84 ST90-L
specimens.
da/dN* [mm/cycle]
10
−4
10
−5
10
C(T) ST90−L 1
C(T) ST90−L 2
DEN(T) ST90−L 1
DEN(T) ST90−L 2
DEN(T) ST90−L 3
−6
10
6
7
8
9 10
12
15
20
25
30
35
40
ΔK* [MPa√m]
Figure A-14: Crack growth speeds of Al 2050-T84
ST90-L specimens.
A4
A Fatigue test data
A-2 Fatigue test data Al 7010-T7451
A-2 ST00-L
C(T) ST00−L 1
C(T) ST00−L 2
DEN(T) ST00−L 2
0.8
−3
10
0.7
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
Crack growth rates for Al 7010−T7451 ST00−L
Crack length for Al 7010−T7451 ST00−L
0.9
0.6
0.5
0.4
−4
10
C(T) ST00−L 1
C(T) ST00−L 2
DEN(T) ST00−L 2
0.3
−5
0.2
0
0.5
1
1.5
2
2.5
3
N [cycles]
10
3.5
7
5
x 10
Figure A-15: Crack length of Al 7010-T7451 ST00-L
specimens.
8
9
10
12
ΔK* [MPa√m]
15
20
Figure A-16: Crack growth speeds of Al 7010-T7451
ST00-L specimens.
ST15-L
C(T) ST15−L 1
C(T) ST15−L 2
DEN(T) ST15−L 1
DEN(T) ST15−L 2
0.8
0.7
−3
10
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
Crack growth rates for Al 7010−T7451 ST15−L
Crack length for Al 7010−T7451 ST15−L
0.9
0.6
0.5
0.4
−4
10
C(T) ST15−L 1
C(T) ST15−L 2
DEN(T) ST15−L 1
DEN(T) ST15−L 2
0.3
−5
0.2
0
2
4
6
8
N [cycles]
10
12
14
16
x 10
4
Figure A-17: Crack length of Al 7010-T7451 ST15-L
specimens.
10
7
8
9
10
12
15
ΔK* [MPa√m]
20
25
Figure A-18: Crack growth speeds of Al 7010-T7451
ST15-L specimens.
A-2 Fatigue test data Al 7010-T7451
A5
ST30-L
C(T) ST30−L 1
C(T) ST30−L 2
DEN(T) ST30−L 1
DEN(T) ST30−L 2
0.8
0.7
−3
10
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
Crack growth rates for Al 7010−T7451 ST30−L
Crack length for Al 7010−T7451 ST30−L
0.9
0.6
0.5
0.4
−4
10
C(T) ST30−L 1
C(T) ST30−L 2
DEN(T) ST30−L 1
DEN(T) ST30−L 2
0.3
−5
0.2
0
0.25
0.5
0.75
1
1.25
1.5
1.75
N [cycles]
10
2
x 10
7
5
Figure A-19: Crack length of Al 7010-T7451 ST30-L
specimens.
8
9
10
12
15
ΔK* [MPa√m]
20
25
Figure A-20: Crack growth speeds of Al 7010-T7451
ST30-L specimens.
ST45-L
C(T) ST45−L 1
C(T) ST45−L 2
DEN(T) ST45−L 1
DEN(T) ST45−L 2
0.8
0.7
−3
10
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
Crack growth rates for Al 7010−T7451 ST45−L
Crack length for Al 7010−T7451 ST45−L
0.9
0.6
0.5
0.4
−4
10
−5
10
C(T) ST45−L 1
C(T) ST45−L 2
DEN(T) ST45−L 1
DEN(T) ST45−L 2
0.3
−6
0.2
0
0.4
0.8
1.2
1.6
N [cycles]
10
2
x 10
7
5
Figure A-21: Crack length of Al 7010-T7451 ST45-L
specimens.
8
9
10
12
15
ΔK* [MPa√m]
20
25
Figure A-22: Crack growth speeds of Al 7010-T7451
ST45-L specimens.
ST60-L
C(T) ST60−L 1
C(T) ST60−L 2
DEN(T) ST60−L 1
DEN(T) ST60−L 2
0.8
0.7
−3
10
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
Crack growth rates for Al 7010−T7451 ST60−L
Crack length for Al 7010−T7451 ST60−L
0.9
0.6
0.5
0.4
−4
10
C(T) ST60−L 1
C(T) ST60−L 2
DEN(T) ST60−L 1
DEN(T) ST60−L 2
0.3
−5
0.2
0
0.4
0.8
N [cycles]
1.2
1.6
2
x 10
5
Figure A-23: Crack length of Al 7010-T7451 ST60-L
specimens.
10
7
8
9
10
12
15
ΔK* [MPa√m]
20
25
Figure A-24: Crack growth speeds of Al 7010-T7451
ST60-L specimens.
A6
A Fatigue test data
ST75-L
Crack length for Al 7010−T7451 ST75−L
1
Crack growth rates for Al 7010−T7451 ST75−L
−3
10
C(T) ST75−L 2
DEN(T) ST75−L 1
DEN(T) ST75−L 2
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
0.8
0.6
0.4
0.2
−4
10
C(T) ST75−L 2
DEN(T) ST75−L 1
DEN(T) ST75−L 2
−5
0
2.5
5
7.5
10
12.5
N [cycles]
10
15
x 10
7
4
Figure A-25: Crack length of Al 7010-T7451 ST75-L
specimens.
8
9 10
12
15
20
25
ΔK* [MPa√m]
30
35
40
Figure A-26: Crack growth speeds of Al 7010-T7451
ST75-L specimens.
ST90-L
Crack length for Al 7010−T7451 ST90−L
1
Crack growth rates for Al 7010−T7451 ST90−L
−2
10
da/dN* [mm/cycle]
a/W* or 2a/W* [−]
0.8
0.6
0.4
C(T) ST90−L 1
C(T) ST90−L 2
DEN(T) ST90−L 1
DEN(T) ST90−L 2
0.2
0
0.3
0.6
0.9
1.2
N [cycles]
1.5
1.8
−4
10
C(T) ST90−L 1
C(T) ST90−L 2
DEN(T) ST90−L 1
DEN(T) ST90−L 2
−5
2.1
x 10
−3
10
5
Figure A-27: Crack length of Al 7010-T7451 ST90-L
specimens.
10
6
7
8
9 10
12
15
20
ΔK* [MPa√m]
25
30
35 40 45 50
Figure A-28: Crack growth speeds of Al 7010-T7451
ST90-L specimens.
A-3 Remarks
A-3 Some specimens have not been included in the fatigue test data, because the specimens failed prematurely
due to overloading. The prematurely failed specimens are: Al 7010-T7451 C(T) ST75-L 1 and DEN(T)
ST00-L 1.
After the fatigue test of specimen Al 7010-T7451 DEN(T) ST90-L 1 it was determined that applied stress
was too high, therefore the loading for the remaing DEN(T) ST90-L coupons is adjusted to a lower level.
Appendix B
Photographs of specimens
All photographs of the specimens, after the fatigue testing, are presented in this appendix. A millimetre
scale is present on most specimens for reference.
B-1 Photographs of Al 2050-T84 specimens
B-1 ST00-L
A
B
C
D
Figure B-1: Post fatigue test Al 2050-T84 specimens. A) DEN(T) ST00-L 1. B) DEN(T) ST00-L 2. C) C(T) ST00-L 1.
D) C(T) ST00-L 2.
B1
B2
B Photographs of specimens
ST15-L
A
B
C
D
Figure B-2: Post fatigue test Al 2050-T84 specimens. A) DEN(T) ST15-L 1. B) DEN(T) ST15-L 2. C) C(T) ST15-L 1.
D) C(T) ST15-L 2.
ST30-L
A
B
C
D
Figure B-3: Post fatigue test Al 2050-T84 specimens. A) DEN(T) ST30-L 1. B) DEN(T) ST30-L 2. C) C(T) ST30-L 1.
D) C(T) ST30-L 2.
ST45-L
A
B
C
D
Figure B-4: Post fatigue test Al 2050-T84 specimens. A) DEN(T) ST45-L 1. B) DEN(T) ST45-L 2. C) C(T) ST45-L 1.
D) C(T) ST45-L 2.
B-1 Photographs of Al 2050-T84 specimens
B3
ST60-L
A
B
C
D
Figure B-5: Post fatigue test Al 2050-T84 specimens. A) DEN(T) ST60-L 1. B) DEN(T) ST60-L 2. C) C(T) ST60-L 1.
D) C(T) ST60-L 2.
ST75-L
A
B
C
D
Figure B-6: Post fatigue test Al 2050-T84 specimens. A) DEN(T) ST75-L 1. B) DEN(T) ST75-L 2. C) C(T) ST75-L 1.
D) C(T) ST75-L 2.
ST90-L
A
B
C
Figure B-7: Post fatigue test Al 2050-T84 specimens. A) DEN(T) ST90-L 1. B) DEN(T) ST90-L 2. C) DEN(T) ST90-L 3.
B4
B Photographs of specimens
A
B
Figure B-8: Post fatigue test Al 2050-T84 specimens. A) C(T) ST90-L 1. B) C(T) ST90L 2.
B-2 Photographs of Al 7010-T7451 specimens
B-2 ST00-L
A
B
C
D
Figure B-9: Post fatigue test Al 7010-T7451 specimens. A) DEN(T) ST00-L 1. B) DEN(T) ST00-L 2. C) C(T) ST00-L 1.
D) C(T) ST00-L 2.
ST15-L
A
B
C
D
Figure B-10: Post fatigue test Al 7010-T7451 specimens. A) DEN(T) ST15-L 1. B) DEN(T) ST15-L 2. C) C(T) ST15-L 1.
D) C(T) ST15-L 2.
B-2 Photographs of Al 7010-T7451 specimens
B5
ST30-L
A
B
C
D
Figure B-11: Post fatigue test Al 7010-T7451 specimens. A) DEN(T) ST30-L 1. B) DEN(T) ST30-L 2. C) C(T) ST30-L 1.
D) C(T) ST30-L 2.
ST45-L
A
B
C
D
Figure B-12: Post fatigue test Al 7010-T7451 specimens. A) DEN(T) ST45-L 1. B) DEN(T) ST45-L 2. C) C(T) ST45-L 1.
D) C(T) ST45-L 2.
ST60-L
A
B
C
D
Figure B-13: Post fatigue test Al 7010-T7451 specimens. A) DEN(T) ST60-L 1. B) DEN(T) ST60-L 2. C) C(T) ST60-L 1.
D) C(T) ST60-L 2.
B6
B Photographs of specimens
ST75-L
A
B
C
D
Figure B-14: Post fatigue test Al 7010-T7451 specimens. A) DEN(T) ST75-L 1. B) DEN(T) ST75-L 2. C) C(T) ST75-L 1.
D) C(T) ST75-L 2.
ST90-L
A
B
C
D
Figure B-15: Post fatigue test Al 7010-T7451 specimens. A) DEN(T) ST90-L 1. B) DEN(T) ST90-L 2. C) C(T) ST90-L 1.
D) C(T) ST90-L 2.
B-3 Remarks
B-3 Some specimens have failed prematurely due to overloading, these specimens are: Al 7010-T7451 C(T)
ST75-L 1 and DEN(T) ST00-L 1.
Appendix C
Certificates Al 7010-T7451
This appendix contains the production certificates with the chemical composition and the tensile
properties of the Al 7010-T7451 plates used for manufacturing the fatigue specimens.
C1
C2
C Certificates Al 7010-T7451

C3
C4
C Certificates Al 7010-T7451

C5
C6
C Certificates Al 7010-T7451