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Fatigue crack growth experiments and analyses- from small scale to large scale yielding atconstant and variable amplitude loading
Par Ljustell
Doctoral thesis no. 81, 2013KTH School of Engineering Sciences
Department of Solid MechanicsRoyal Institute of TechnologySE-100 44 Stockholm Sweden
TRITA HFL-0531
ISSN 1654-1472
ISRN KTH/HFL/R-12/15-SE
Akademisk avhandling som med tillstand av Kungliga Tekniska Hogskolan i Stockholmframlagges till offentlig granskning for avlaggande av teknisk doktorsexamen fredagen den1 februari kl. 10.00 i sal F3, Kungliga Tekniska Hogskolan, Lindstedtsvagen 26, Stockholm.
Failure is simply the opportunity to begin again, this time more intelligently.
Henry Ford
Abstract
This thesis is on fatigue crack growth experiments and assessments of fatigue crack growth
rates. Both constant and variable amplitude loads in two different materials are considered;
a nickel based super-alloy Inconel 718 and a stainless steel 316L. The considered load levels
extend from small scale yielding (SSY) to large scale yielding (LSY) for both materials.
The effect of different load schemes on the fatigue crack growth rates is investigated on
Inconel 718 and compact tension specimens in Paper A. It is concluded that load decreasing
schemes give a too high Paris law exponent compared to constant or increasing load amplitude
schemes. Inconel 718 is further analyzed in Paper B where growth rates at variable amplitude
loading in notched tensile specimens are assessed. The predictions are based on the fatigue
crack growth parameters obtained in Paper A. The crack closure levels are taken into con-
sideration and it is concluded that linear elastic fracture mechanics is incapable of predicting
the growth rates in notches that experience large plastic cyclic strains. Even if crack closure
free fatigue parameters are used and residual stresses due to plasticity are included. It is also
concluded that crack closure free and nominal fatigue crack growth data predict the growth
rates equally well. However, if the crack closure free parameters are used, then it is possible
to make a statement in advance on the prediction in relation to the experimental outcome.
This is not possible with nominal fatigue crack growth parameters.
The last three papers consider fatigue crack growth in stainless steel 316L. Here the load is
defined as the crack tip opening displacement parameter. Paper C constitutes an investigation
on the effect of plastic deformation on the potential drop and consequently the measured crack
length. It is concluded that the nominal calibration equation obtained in the undeformed
geometry can be used at large plastic deformations. However, two conditions must be met: the
reference potential must be taken in the deformed geometry and the reference potential needs
to be adjusted at every major change of plastic deformation. The potential drop technique
is further used in Paper D and Paper E for crack length measurements at monotonic LSY.
Constant amplitude loads are considered in Paper D and two different variable amplitude
block loads are investigated in Paper E. The crack tip opening displacement is concluded in
Paper D to be an objective parameter able to characterize the load state in two different
geometries and at the present load levels. Furthermore, if the crack tip opening displacement
is controlled in an experiment and the local load ratio set to zero, then only monotonic LSY
will appear due to extensive isotropic hardening, i.e. elastic shake-down. This is also the
reason why the linear elastic stress-intensity factor successfully could merge all growth rates,
extending from SSY to monotonic LSY along a single line in a Paris law type of diagram, even
though the generally accepted criteria for SSY is never fulfilled. For the variable amplitude
loads investigated in Paper E, the effect of plastic deformation on measured potential drop
is more pronounced. However, also here both the crack tip opening displacement parameter
and the linear elastic stress-intensity factor successfully characterized the load state.
Preface
The research presented in this thesis was conducted at the Department of Solid Mechanics,
Royal Institute of Technology (KTH). The research was financed by the Consortium for
Materials and Thermal Energy Processes (KME) and the Swedish Radiation Safety Authority
(SSM). The financial support is greatly acknowledged.
Fatigue crack growth is a very interesting and challenging area of solid mechanics. From an
industrial viewpoint, fatigue crack growth is of particular interest due to its close connection to
quality and safety, two factors which require special attention since products are increasingly
optimized in terms of material consumption, etc. With this in mind, I hope the outcome of
this thesis will help engineers in making fatigue design decisions when the current limits of
linear elastic fracture mechanics theory are exceeded.
As a PhD graduate I have had the opportunity to meet a lot of people from the Universities
and the Swedish industry. I would like to acknowledge some of them. First and foremost is my
supervisor Prof. Bo Alfredsson, whose patience, encouragement and support made it possible
for me to finalize this research project and thesis. His guidance as a scientist, mentor and
friend is greatly appreciated. Thank you Bo! My former supervisor, late Prof. Fred Nilsson
is also acknowledged. He gave me the opportunity to start my PhD studies and supported
me during the first years. Special thanks also go to Mr. Hans Oberg for contributions to
this work through his extensive experience in the experimental area. I learned a lot from the
interesting discussions we have had! The people in the mechanical workshop, Messrs Bertil
Dolk and Kurt Lindqvist, are also acknowledged for immediate support on test specimens
and fixtures. This thesis includes a few extensive numerical computations for which updated
computer hardware was required; for that I thank Mr. Per Berg. I also thank all my colleagues
and friends at the department for creating a good and inspiring atmosphere at work.
My sincere thanks also goes to Prof. Soren Sjostrom, Lic. Bjorn Sjodin (Siemens Industrial
Turbomachinery AB) and Dr. Tomas Hansson, Dr. Tomas Mansson (GKN Aerospace Sweden
AB) for the cooperation during the KME project. I really appreciate the industrial perspective
that you brought into the research!
My family is the solid foundation of my life, with whom I so gladly share my joy and
happiness. But it’s also the solid foundation that supports and keeps me up when life’s hard.
Thank you my dear Josefin for your support, encouragement and patience during these years!
Twice during my time as a PhD graduate, I have had the good fortune to become a father,
obviously the best thing that has ever happened to me. Jonatan and Elvira, you always bring
out a smile on my face and help me to focus on the important things in life, thank you!
Thanks also to my sister Camilla and my parents Mats and Mona-Britt, who have always
been there for me. Better sister and parents cannot be found!
Stockholm, November 2012
List of appended papers
Paper A: Effects of different load schemes on the fatigue crack growth rate
P. Ljustell and F. Nilsson
Journal of Testing and Evaluation, 34(4), 2006, 333 − 341
Paper B: Variable amplitude crack growth in notched specimens
P. Ljustell and F. Nilsson
Engineering Fracture Mechanics, 72(18), 2005, 2703 − 2720
Paper C: The effect of large scale plastic deformation on fatigue crack length measurement
with the potential drop method
P. Ljustell
Journal of Testing and Evaluation, 39(6), 2011, 985 − 1002
Paper D: Fatigue crack growth experiments on specimens subjected to monotonic large scale
yielding
P. Ljustell
Submitted for international publication
Paper E: Variable amplitude fatigue crack growth at monotonic large scale yielding experi-
ments on stainless steel 316L
P. Ljustell and B. Alfredsson
Submitted for international publication
Individual contributions
Paper A: Effects of different load schemes on the fatigue crack growth rate
P. Ljustell: Experiments; Numerical Analyses; Manuscript
F. Nilsson: Supervision
Paper B: Variable amplitude crack growth in notched specimens
P. Ljustell: Experiments on CT-specimens; Numerical Analyses; Manuscript
F. Nilsson: Supervision; Concept
Siemens Industrial Turbomachinery AB: Experiments on NT-specimens
Paper C: The effect of large scale plastic deformation on fatigue crack length measurement
with the potential drop method
P. Ljustell: Experiments; Numerical Analyses; Manuscript
B. Alfredsson: Supervision
Paper D: Fatigue crack growth experiments on specimens subjected to monotonic large scale
yielding
P. Ljustell: Experiments; Numerical Analyses; Manuscript
B. Alfredsson: Supervision
Paper E: Variable amplitude fatigue crack growth at monotonic large scale yielding experi-
ments on stainless steel 316L
P. Ljustell: Experiments; Numerical Analyses; Manuscript−figures
B. Alfredsson: Supervision; Manuscript−text
Contents
Introduction 11
Crack growth and fatigue of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Engineering approach to the fatigue problem . . . . . . . . . . . . . . . . . . . . . 12
Project backgrounds and research motivation . . . . . . . . . . . . . . . . . . . . . 14
Crack closure 17
Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Origins to crack closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Crack closure effects on the fatigue growth parameters . . . . . . . . . . . . . . . . 19
Crack closure in growth rate assessments . . . . . . . . . . . . . . . . . . . . . . . . 21
Finite elements and crack closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
The applicability of linear elastic fracture mechanics 25
Intermediate scale yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Large scale yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
The potential drop method for crack length measurements 31
Characterization of fatigue crack growth outside the SSY limit 35
The ΔJ parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
The Δδ parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Conclusions 45
Summary of appended papers 49
Bibliography 53
Introduction
The subject for this thesis is fatigue crack propagation in metals. A description of the different
phases of the fatigue crack life will be given in this introductory part together with a short
overview of the two main approaches to fatigue design and a background description and
research motivation of the research project.
Crack growth and fatigue of metals
Crack growth is on the atomic level breakage and separation of the bonds linking the atoms
and/or movement and gathering of dislocations (imperfections in the atomic structure). Thus,
new surfaces are created in the solid as the crack nucleates and continues to grow. This can
be interpreted as an adaption of the material to an applied load of a critical level. Application
of a strictly increasing load to a flawed structure may not be critical at first since the load
needs to be at some critical level in order for crack growth to occur. At this point stable or
unstable crack growth occurs dependent on the characteristics of the material resistance to
continuous growth and the driving force dependence vs crack extension. However, at some
point, assuming a strictly increasing load, the component or structure will fail due to loss of
load bearing capacity or functionality.
In the case of fatigue, which is characterized by time variable loads, the amplitude of the
loads may be considerably lower than the level at which strictly increasing loading causes
fracture. This is due to local plasticity on the micro-scale, although the applied loads on a
structural level only give elastic stresses. The initiation position of a fatigue crack is often
found at some inclusion, impurity or surface flaw which acts as a local stress raiser giving
small scale plastic deformation. The inclusion may have sharp edges and very much act as
a crack right from the start or have a smooth surface giving a local stress concentration
where initiation of the crack starts. Another possibility for crack initiation is concentrated
plastic deformation on certain planes in the grains orientated such that shear deformation
occurs between them. These planes, or band of planes, are called persistent slip bands. As
the plastic deformation is locally accumulated with the number of load cycles local stress
concentrations are created and form initiations spots for fatigue cracks. At this stage the
nucleated fatigue crack size ranges from sub micrometer to a few tens of micrometers. It
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is not certain that the fatigue crack will continue to grow. Important parameters here are
the applied load level combined with the micro-structural local conditions around the fatigue
crack. Obstacles such as grain boundaries, inclusions and unfavorable orientation of the
grains locally around the crack tip may act as shielding mechanisms for further growth. Some
materials show a distinct load limit dependence. If the load amplitude is above the limit,
then the fatigue crack continuous to grow and if it is below the limit, then the fatigue crack
stops to grow. The limit is called the endurance limit or fatigue limit, usually defined as 50 %
probability of survival at 106 load cycles. If the fatigue crack nucleates from persistent slip
bands at the surface of the component, then the first part of the fatigue life is often shear
driven with growth in the maximum shear direction. As the crack continues to grow into
the component a shift in direction generally occurs. The new direction is on the continuum
level perpendicular to the principal stress axis. If the fatigue crack nucleates at an inclusion
within the material, then the initial growth direction can be either in the maximum shear
direction or perpendicular to the principal stress axis somewhat dependent on the size, shape
and orientation of the inclusion. Further growth of the fatigue crack, of submillimeter length
and above, may be driven by plastic deformation on slip bands emanating from the crack tip.
The crack growth direction at this stage is perpendicular to the principal stress axis and the
growth rate can be charaterized from continuum theory. Finally, the fatigue crack becomes
sufficiently long for unstable crack growth to take place and the component functionality will
be lost.
Although the damage process to some extent is qualitatively known, still many questions
remain to be answered, in particular with respect to life computation. Also, a big challenge lies
within finding an explanation of how a crack is created in a material initially free from flaws.
Such local phenomena are of great complexity, both from a theoretical and computational
point of view, compared to an approach on the continuum level. Hence, the traditional
engineering treatment of the the fatigue problems with elastic theory on a continuum level.
Engineering approach to the fatigue problem
From the engineering point of view there are traditionally two ways to approach the fatigue
design problem. The first and the oldest approach is the safe-life design methodology. This
methodology is based on the assumption of no existing flaws in the component/structure and
no distinction is made between initiation and propagation of the fatigue crack. Furthermore,
the stress amplitude and mean stress level are chosen such that the fatigue life (number of
cycles to failure) is either finite or infinite based on the knowledge of the material fatigue
properties and other significant factors. In order to maintain the design safety when the
design life is used, the component is replaced with a new one, even though a considerably
large part of the functional life may remain. Another assumption is that no inspections of
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the component/structure needs to made during the design life.
Traditionally, because of the observed large scatter in the fatigue life, safety factors are
included. These reduce the allowed load level, or instead of reducing the load level, increase
the dimensions of the component/structure. However, safety factors are hard to interpret
from reliability and a durability point of view. A better approach, allowing for an easier
interpretation of the component/structural reliability, is to use statistical methods to judge
the probability of survival. From a historical point of view the inherent scatter in the fatigue
life was observed as early as in 1860, which also is the time around which the safe-life design
methodology originates. However, the scatter was first treated on a scientific level around
1945. From the engineering point of view the use of statistical methods has still not penetrated
some branches of the mechanical industry. For a comprehensive historical review on fatigue
see for instance [1].
The second approach is called damage-tolerant design. Here it is assumed that initial
cracks exist in the component/structure and periodical inspections for the presence of these
needs to be performed in order to maintain the reliability. The size of the assumed initial
crack is connected to the largest crack size that may be missed during an inspection, which
in turn is set by the detection limit of the non-destructive technique used for the inspection.
The design life of the component/structure then becomes the number of in-service fatigue
load cycles required for the initial crack to grow to a critical size. The damage-tolerant
methodology consequently focus on predictions of the fatigue crack growth rate and the
remaining fatigue life whereas the safe-life design methodology focuses on estimating the
total life. Thus, estimations of the remaining fatigue life of a flawed component/structure
is only possible through use of the damage-tolerant approach. The strive for increasing the
accuracy and the precision of the estimated fatigue crack growth rate is driven by cutting
costs by increasing the inspection length interval since inspections are expensive (downtime
of the industrial process) while maintaining the safety requirements.
A theory suitable for the damage-tolerant design methodology is the fracture mechanics
theory. The fracture mechanics theory offers methods for estimating the driving force of the
applied load to the crack. The main part of the fracture mechanics theory was developed
during the 1950s and the 1960s. The required input information to the fracture mechanics
models for fatigue crack growth rate estimates are the in-service load, material properties and
geometry of the fatigue crack and component. Historically, the damage-tolerant approach
has been applied in areas where high demands on reliability and safety are required, often
combined with limitations on structural weight such as for the aviation and space industry.
The damage-tolerant methodology was first introduced in 1974 by the US Air Force in their
structural specifications. The cause to the change of structural specifications was that the
US Air Force did not manage to obtain sufficient structural integrity and durability of the
aircrafts [1].
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The papers that are appended to this thesis may be categorized under the damage-tolerant
methodology. The papers address the fatigue crack growth from an experimental, numerical
and theoretical point of view based on fracture mechanics with emphasis on the first two
categories. In particular, the work focused on extending the life prediction methods to larger
crack tip loads than is currently accepted and to include some aspects of variable amplitude
loads.
Project backgrounds and research motivation
The research presented may be divided into two parts based on the research financing. The
first two appended papers, Paper A and Paper B, were conducted and reported within the
project KME-101 entitled ”Fatigue crack growth and life predictions in gas turbine materials”.
This project was a collaboration between The Department of Solid Mechanics at the Royal
Institute of Technology (KTH) and the Swedish gas turbine industry represented by Siemens
Industrial Turbomachinery AB and GKN Aerospace Sweden AB (former Volvo Aero Corpo-
ration AB). The project plan spanned three years (2001−2003) with a total budget of 4.6
MSEK, equally financed by the Swedish Energy Authority (Energimyndigheten) and the the
Swedish gas turbine industry. The project leader was the now late Prof. Fred Nilsson (KTH).
Other participants from the academia were Dr. Tomas Mansson (PhD graduate), Lic.1 Par
Ljustell (PhD graduate, who entered the project in mid-year 2002), research engineer Jan
Skantz and senior research engineer Hans Oberg. The companies’ main representatives were
Prof. Soren Sjostrom, Lic. Bjorn Sjodin (Siemens Industrial Turbomachinery AB) and Dr.
Tomas Hansson (GKN Aerospace Sweden AB).
The outcome from the project was two scientific papers, Paper A and Paper B, published
in international scientific journals, Ljustell and Nilsson [2, 3]. In addition, three other papers
were published within the project, Mansson et al. [4, 5] and Nilsson et al. [6]. The research
in the current Paper A and Paper B were conducted in parallel but mainly after the work in
papers [4, 5, 6] and naturally the papers share similar thoughts and approaches to the fatigue
crack growth problem. Furthermore, three academic exams are connected to the project; one
Doctoral degree Tomas Mansson (2002); one Licentiate of Engineering degree Par Ljustell
(2005); one Master of Science degree Peter Bodin (2004).
In order to stay competitive and increase customer value the products are optimized
and the material properties must be used as efficiently as possible. The gas turbine industry
identified the need of an improved treatment of the fatigue phenomenon which in turn required
a thorough understanding of the fatigue crack growth process. Thus, current practice needed
to be enhanced. The ultimate goal was to safeguard against fatigue failure of gas turbines
and the project goal was stated as ”To develop methods whereby complex fatigue crack
1Abbreviation for: Licentiate of Engineering degree. Swedish translation: Teknologie licentiat
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growth situations can be analyzed and predicted using data from constant amplitude testing
of standard specimens”. The reason for developing methods based on data from constant
amplitude testing of standard specimens is that the majority of all existing fatigue propagation
data in the industry are tested accordingly. The following four topics were identified as
especially important for the gas turbine technology.
� Methodology development for testing under constant and variable amplitude loading.
� Growth of cracks initiated at stress concentrations under variable amplitude loading.
� Surface crack growth under variable amplitude loading.
� Mixed mode crack growth.
In all these areas, the goal has been to develop or verify engineering practice for treat-
ment of the fatigue crack growth phenomenon. The research presented in Paper A considers
methodology development for testing under constant amplitude loading while Paper B con-
siders predictions of growth rates of fatigue cracks initiated at stress concentrations under
variable amplitude loading.
The research in Paper C to Paper E constitutes the second part of the thesis. It was
conducted and reported within the project: ”Variable amplitude fatigue crack growth in
the low cycle life regime with reference to thermal fatigue”. This project was funded by the
Swedish Radiation Safety Authority (Stralsakerhetsmyndigheten2). The project plan spanned
three years (2004−2006) with a budget of 3.1 MSEK. The project ended in early 2007 but was
prolonged by SSM until mid 2012. In early 2009 Prof. Bo Alfredsson join the project and took
the project leadership as Prof. Fred Nilsson sadly passed away in mid 2009. Others attending
the project was senior research engineer Hans Oberg and Lic. Par Ljustell (PhD graduate).
The results from the project were published in three scientific papers currently submitted
to or published in international scientific journals; Ljustell [7, 8], Ljustell and Alfredsson [9].
Furthermore, van Eeten and Nilsson [10] established and published the constitutive model
that was later used in [7, 8, 9]. In addition, three academic exams are connected to the
project; one Doctoral dissertation Par Ljustell (this thesis); two Master of Science degrees,
Paul van Eeten (2005) and Christian Partel (2007).
The motivation for the research was that second only to stress corrosion cracking, thermal
cyclic and consequently thermal fatigue is the most common damaging mechanism in Swedish
nuclear power plants [11, 12]. The thermal fatigue mechanism is also identified as major life
limit issue worldwide. Three international projects have been finalized during the last decade,
THERFAT [13], NESC-V [14] and the Thermal Fatigue Benchmark (OECD/CSNI) [15]. The
THERFAT and NESC-V projects aimed at preparing the foundation for an European proce-
dure for handling of thermal fatigue of components in nuclear power plants. No comprehensive
2Abbreviation: SSM
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instruction for handling the thermal fatigue problem was presented due to the complex nature
of the subject. However, the projects surveys and discusses the vast experience and interdisci-
plinary aspects on thermal fatigue, such as fluid mechanics, fracture mechanics and material
mechanics. Thus, the reports provide valuable information and guidance of handling thermal
fatigue.
The project background is that pipe systems in nuclear power plants are vulnerable to
thermal fatigue, especially at mixing points where fluids of different temperatures are mixed.
The turbulent flow in these areas act as a relatively high frequency thermal cyclic load on
the inner surface of the pipe. This is a well-known phenomenon that has been studied in the
past. The damage at the inner surface is best described as an ”elephant skin” or a ”dried up
lake bed of clay” with arresting surface cracks in a network pattern. The depth of the surface
cracks are relatively small compared to the thickness of the structure and the operational
initiation time may be less than a year. Unfortunately, several cases (about 45 in the NESC-
V report) with deep or through the thickness fatigue cracks are reported internationally for
operational components. The propagation phase is often shorter than the initiation time. The
NESC-V report discusses possible conditions for deep fatigue cracking and it is argued that
three-dimensional thermal distributions are a major factor driving the fatigue crack through
the pipe wall. The three-dimensional thermal distributions occurs at large scale instabilities
in the fluid flow, i.e. stratification of the fluid flow in combination with a turbulent flow or
turbulent penetration of a nominally stagnant fluid in a pipe. The thermal load frequencies
for these three-dimensional temperature distributions are low and consequently generating
large temperature differences throughout the wall thickness as oppose to a high frequency
turbulent flow.
Due to the difficulty in performing well-controlled crack growth experiments under thermal
loading a pure mechanical load was used for Paper C to Paper E. Therefore, some features
of thermal fatigue such as strain gradients and possible material temperature dependence
were not included here. However, in the early and preliminary experimental series on current
material (stainless steel 316L) no major difference was observed between the fatigue crack
growth rate at room temperature and at +280�C.
The research presented in Paper C establishes the experimental setup and evaluates the
important factors needed to be taken into consideration for low cycle fatigue crack propaga-
tion experiments. Paper D and Paper E reports the numerical and experimental low cycle
fatigue results with constant amplitude loading and some variable block amplitude loading,
respectively.
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Crack closure
The fracture mechanics theory is commonly used to characterize the ”severity” of the applied
load on the fatigue crack tip. If the plastic deformation at the crack tip due to the applied
load is small relative to the characteristic dimensions of the geometry, then linear elastic
fracture mechanics (LEFM) is applicable. The parameter used for the linear description of
the load on the fatigue crack tip is called the stress-intensity factor, KI. Numerous researchers
have shown that the main factor controlling the propagation rate, in the case of fatigue crack
propagation, is the range of the stress-intensity factor, ΔKI. In 1963 Paris and Erdogan [16]
expressed the crack growth per cycle as a linear relationship with the stress-intensity factor
range in a double logarithmic diagram, cf. Fig. 1. The curve can be divided into three parts
where the linear part of the curve is described by,
da
dN= a0
(ΔKI
K0
)β
. (1)
The constant a0 is an arbitrary chosen reference growth rate and ΔKI is the entire stress-
intensity factor range. Equation 1 is a rewrite of the original definition by Paris in order to
get interpretable dimensions of the two parameters, K0 and β. The two parameters K0 and
β have shown experimentally to depend on the load ratio R = KI,min/KI,max.
Paris regime
Stage I Stage II Stage III
Stress-intensity factor range
Fatiguecrackgrow
thrate
Figure 1: Illustration of the fatigue crack growth rate vs the stress-intensity factor range.
17
Basic concepts
In the early 1970s Elber [17, 18] discovered that the fatigue crack can be partially closed even
if a global tensile load is applied. Elber observes a change of the compliance during loading
and interpreted this as a gradual change of the crack length due to closure of the crack faces
behind the crack tip. An effective stress-intensity factor range ΔKI,eff is suggested to be used
as the crack driving parameter instead of the nominal stress-intensity factor range ΔKI. The
effective stress-intensity factor range is defined as,
ΔKI,eff = KI,max −KI,closure, if KI,closure > KI,min (2)
where KI,closure is the stress-intensity factor level as the crack opens/closes. Elber finds that
the use of ΔKI,eff eliminates the R-ratio dependence in fatigue parameters K0 and β and all
fatigue data coincide to a common line in double-logarithmic diagram as long as ΔKI,eff is
correctly measured/calculated. Andersson et al. [19], Dougherty et al. [20] and many others
show the same result, eliminating the R-ratio dependence in fatigue parameters by use of an
effective stress-intensity factor range. Elber also states that the crack closure phenomenon
accounts for acceleration and retardation effects in variable amplitude crack growth. The
phenomenon has since then provided a rationale for different observed behaviours such as:
retardation of the fatigue crack growth rate due to overloads, the load ratio effect on the
growth rate, some of the observed fatigue threshold effects and small crack growth behaviour
to mention a few.
Another indirect definition of the crack closure level, with the aim of extending the appli-
cability of Eq. 1 below the load ratio where crack closure first is seen, is the one stating that
the difference between crack propagation rates at different R-ratios is due to crack closure.
This procedure requires the growth law, Eq. 1, to be first established by initial experiments
where the minimum nominal stress-intensity factor exceeds the closure level. These growth
law parameters, K0 and β, from ΔKI = ΔKI,eff are then independent of the R-ratio. The
effective stress-intensity factor range in other experiments with K0 (R) and β (R) dependent
of the R-ratio are then computed as
ΔKI,eff = K0
(ΔKI
K0 (R)
)β(R)β
. (3)
The crack closure level in R-dependent experiments is finally defined as the limit giving the
effective stress-intensity factor range from Eq. 3, i.e.
KI,limit = KI,max −ΔKI,eff, if KI,limit > KI,min. (4)
This idea is not new, cf. [21]. It eliminates the need for accurate crack closure measure-
ments during the actual test, but requires a fatigue crack growth rate experiment where no
18
crack closure is present. Also, with the two different definitions mentioned above it is possible
to assess the accuracy of the other crack closure measuring techniques, cf. Mansson et al.
[4]. This indirect definition of the crack closure level is used in Paper A and Paper B where
a nickel-based superalloy Inconel 718 is used at +400�C. Most of the existing fatigue crack
growth data in the industry today is of constant amplitude at different load ratios, since
such procedures are recommended in standards. By conducting one additional crack closure
free fatigue crack growth experiment it is possible to estimate the crack closure level in all
other constant amplitude experiments. This knowledge is of high value for the engineer when
predicting the fatigue crack propagation life of a component.
Origins to crack closure
Crack closure effects may have several different origins. The common base is some change of
length of the crack faces behind the crack tip or some obstacle between the crack faces. The
origin to this may be due to corrosive environment that causes the newly created crack faces
to oxidize/corrode. Another cause is the plastic deformation formed due to the growth of the
crack leaving a wake of deformed material behind the crack tip. This plastic stretch will vary
if the load amplitude is variable causing the closure level to change with crack length. This
type of crack closure is called plasticity induced crack closure. A third type may occur if a
Mode I growing fatigue crack is subjected to a Mode II overload. This mismatch between
the crack surfaces is called roughness induced crack closure. Due to the relative displacement
in the crack plane, retardation effects occur that may affect the crack growth rate several
millimeters ahead, Dahlin and Olsson [22, 23]. For a summary of proposed mechanisms, see
for instance Suresh [24].
Crack closure effects on the fatigue growth parameters
A study on the effect of different load schemes on the fatigue crack rate data, i.e. the exponent
β and the constant K0 in Eq. 1 is described in Paper A. It is found that ΔKI-reducing load
schemes effects the fatigue data and may result in a too high exponent β. Figure 2 shows the
evaluation of the exponent on the ordinate for three different experiments with constant load
ratios, two with load decreasing schemes and one with an load increasing scheme. Here the
stress-intensity factor range is kept constant at each load level for a crack extension sufficiently
long for attaining steady-state growth conditions before ΔKI is increased or decreased. Each
level is divided into five equally long growth increments Δa. The number on the abscissa tells
the number of increments included, counted from the last increment at constant ΔKI and
towards the point of load change, when determining the crack growth parameters.
Figure 2 shows that a shielding mechanism is present, reducing the fatigue crack growth
rate, in the two experiments with ΔKI-reducing load schemes. The shielding effect also
19
1 2 3 4 5
3
3.5
4
4.5
5
Number of crack length inc., Δa, incl. in the eval. of the fatigue exp.T
he e
xpon
ent,
β, in
Par
is la
w
Load decreasing test, R = 0.05, Δa = 0.4 mmLoad decreasing test, R = −1, Δa = 0.2 mmLoad increasing test, R = 0.5, Δa = 0.2 mm
Figure 2: The effect of different load schemes on the slope of the fatigue crack growth data. (Paper A).
diminish and finally disappears as the crack tip extends away from the point of load change,
observed as converging value of β when fewer increments Δa are included in the evaluation
of the fatigue parameters. Thus, plasticity induced crack closure is present. This transient
shielding effect is not present in the ΔKI-increasing experiment. The main recommendation
from Paper A is to use load increasing or constant ΔKI schemes at fatigue crack growth
testing. Decreasing load schemes introduce a risk of obtaining load history effected fatigue
crack growth data. This has implications not only on the fatigue crack growth parameters
but also on the fatigue crack growth threshold cf. Newman [25].
By arguments of self-similarity in LEFM the crack closure level should be proportional
to the maximum stress-intensity level. Thus, fatigue crack growth rates described by curves
with constant R-values should also be parallel in diagram such as Fig. 1, i.e. the curves
should have the same exponent β.
15 20 25 30 35
3
4
5
6
7
8x 10
−5
Crack length a / mm
Cra
ck g
row
th r
ate
/ mm
/cyc
R=
0.05
R=
0.1
R=
0.15
R=
0.2
R=
0.25
R=
0.3
R=
0.35
R=
0.4
R=
0.45
R=
0.5
R=
0.55
R=
0.05
R=
0.1
R=
0.15
R=
0.2
R=
0.25
R=
0.3
R=
0.35
R=
0.4
R=
0.45
R=
0.5
R=
0.55
Figure 3: Crack growth rate at constant ΔKI = 18 MPa√m and incremental increase of R. (Paper C).
One way of investigating the load ratio dependence of a material is to keep the ΔKI con-
stant while increasing R-value, starting the experiment with a low R-value. In the author’s
20
view, based on the research findings in Paper A and Paper C, the load ratio dependence ob-
served primarily consists of two components, crack closure and in some materials a maximum
load effect. Inconel 718 show no dependence of the maximum load level used at +400�C while
the stainless steel 316L used in Paper C may be sensitive to the maximum load level at room
temperature, Fig. 3. In the stainless steel 316L the fatigue crack growth rate increases rather
proportional to the increase of the R-ratio up to a R-ratio of approximately 0.2. Above this
limit the average crack growth rate is rather unaffected by the R-ratio until a R-ratio of 0.35
is applied. Here the crack growth rate starts to increase again indicating a dependence on
the KI,max-level.
Crack closure in growth rate assessments
Incorporating estimates of the crack closure level in predictions of the fatigue life can be
easy or very difficult dependent on the type of load history. If the component is exposed to
a constant amplitude load history a good estimate of the crack closure level would be the
experimentally obtained one, Eq. 3. For a totally random load history it is a lot harder to
estimate the crack closure level on a cycle by cycle basis. It would require complete knowledge
about the load history and the coupling between the load level and crack closure level since
plastic deformation is load history dependent. If complete knowledge is available, then it is
theoretically possible, although not practically possible to compute the crack closure level
with the finite element method (FEM). Consequently, empirical models such as the Wheeler
[26] and Willenborg et al. [27], which only account for retardation and not acceleration due
to underloads, and later Chang [28], which also includes acceleration are proposed. These
estimate the yield-zone size in which the crack growth is assumed to be affected. Empirical
crack closure models are also proposed, see for instance Bell and Schijve [29] and de Koning
[30]. Semi-analytical models such as the Dugdale or strip-yield models are also developed
and numerically implemented. These allow for numerical simulation of the crack closure
behaviour opposed to the empirical models which assume a specific behaviour. Since the
Dugdale and strip-yield models are based on plane stress a constraint parameter is introduced
in the numerical models. The constraint parameter allows in an artificial way to account for
geometrical constraints such as material thickness, crack location relative external boundaries
and type of applied load (tension/bending type of load). Furthermore, models assume elastic-
perfectly-plastic constitutive behaviour. Basically if none of the empirical or semi-analytical
models are at hand or unsuitable to the considered application, then the option for the
engineer is to make an average estimate of the crack closure level based on for instance
experimental data.
A third type of load history are those that are irregular but possible to describe with
repeatable load blocks. The typical application for such load histories may be start-stop type
21
of applications. In Paper B the crack growth rates are predicted in notched specimens for
a number of different variable amplitude load histories that are described by load blocks.
The predictions are based on constant amplitude experiments conducted in Paper A and on
a cycle weighted average of Eq. 1. For each load block the average crack growth rate are
computed as
(da
dN
)average
=n
n+m
da
dN(ΔKI,small) +
m
n+m
da
dN(ΔKI,large) , (5)
where the n and m are the number of small and large cycles in the variable amplitude block
sequence. The stress-intensity factor ranges are either chosen as nominal (the whole stress-
intensity factor range acts detrimentally to the crack tip) or as effective (the crack closure
level is taken into account, which reduces the stress-intensity factor range) ranges depending
on the evaluation method used. Here results from the effective method are presented in Fig.
4(b), i.e. Reduced data and the Closure computed with FEM for one type of load block, see
inset in Fig. 4(a). For the complete set of data see Fig. 6. In the Reduced data the crack
closure level is assumed to be constant and controlled by the largest load cycle in the load
block. This assumption is justified by the small distance traveled by the crack tip between two
large load cycles compared to the plastic zone size given by the large cycles. A steady state
crack closure level is thus expected, exclusively determined by the the large load cycle. Such
an approach are also used by Socie [31] and Mansson et al. [4, 5]. The effective stress-intensity
factor ranges are then computed according to Eq. 3.
0 1 2 3 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Crack length a / mm
Nor
mal
ized
clo
sure
val
ues
Pcl
/Pla
rge,
max
−1−0.5
00.5
1
P/P
larg
e,m
ax
(a)
0 1 2 3 4 5 610
−4
10−3
10−2
Crack length a / mm
da/d
N /
mm
/cyc
le
Closure computed with FEMReduced dataExperimental data, specimen no. 1Experimental data, specimen no. 2
(b)
Figure 4: (a) The crack closure level computed with the finite element method. Inset in the figure is also thetype of load cycle. (b) Predictions of experimental fatigue crack growth rates. (Paper B).
22
Finite elements and crack closure
Another way to obtain the crack closure level is to use the finite element method. The
development and implementation of the finite element models into computer codes in the
1970s gave further possibilities to analyze crack closure induced by plastic deformation at
the crack tip. The early models were very small and simple. But along with the immense
development of the computer capacity larger and more complex models can now be solved.
Solanki et al. [32] present an extensive review paper on simulations of plasticity induced
crack closure by the finite element method. The basic approach is to model the crack and
apply a cyclic load onto the component. Crack growth is then accomplished by releasing the
crack tip node at some stage in the load cycle. Combined with an elastic-plastic constitutive
model the plastically deformed material is built up in the wake and the crack closure level
can be studied as the first node behind the crack tip comes into contact (crack closure) or
leaves contact (crack opening) with the opposite crack surface. In Paper B, finite element
simulations of the crack closure levels are presented for two different types of load blocks. A
two-dimensional plane strain model is used and the plastic part of the constitutive model is
based on incremental, rate-independent classical plasticity with linear kinematic hardening.
Figure 4(a) shows the computed crack closure level averaged over two consecutive load blocks
vs crack length. Figure 4(b) shows the prediction of the fatigue crack growth rates (Eq. 5)
with the simulated crack closure level included (Closure computed with FEM ). One should
have in mind when interpreting such simulations that several researchers show a dependence
of the crack closure level on the finite element size, cf. McClung and Sehitoglu [33, 34],
Solanki et al. [35] and Gonzales and Zapatero [36]. Furthermore, the choice of constitutive
model may also influence the the results, cf. Jiang et al. [37] and Pommier and Bompard [38]
and Pommier [39].
23
24
The applicability of linear elastic
fracture mechanics
A prerequisite for accurate fatigue life predictions is to have reliable fatigue data at hand.
From the damage-tolerant point of view this means to have a proper description of the relation
between the applied load at the crack tip and the corresponding measured fatigue crack
growth rate. The purpose of conducting fatigue crack growth experiments is to establish
this relationship. The relationship needs only to be applicable to the specific geometry and
load from the fatigue design point of view. Thus, full-scale fatigue tests are an alternative
to establish the fatigue data and total life of the component. However, full-scale tests are
often very expensive in terms of time, technique and money and often practically impossible
to use as a stage in the iterative design process. The other choice is to rely on the concept of
similitude or transferability. The concept states that if there are enough similarities between
two situations one may be able to infer that the results would be the same or similar. This gives
the opportunity to carry out material tests on small specimens that are far cheaper than full-
scale tests and assume the same behaviour in both cases. The iterative design process would
then take place as simulations in the computer environment, with the material data from
the small test specimens, and drastically lowering the development costs. Fracture mechanics
theory provides the theoretical basis and measures for transfer of fatigue propagation data
from the small specimen in the laboratory to the full-scale field application.
The use of LEFM would be very limited if it only applies to the completely elastic ma-
terials. Fortunately LEFM can be used in more general situations under certain limitations.
These limitations are mainly connected to the size of the non-linearly deformed zone relative
the dimensions of the component. From a theoretical point of view considering the auton-
omy of the crack tip state, one can conclude that the only requirement needed for control of
the crack tip field by the stress-intensity factor is to have elastic boundaries and vanishing
contribution of higher order terms on the boundary between the linear and non-linear defor-
mation, cf. [40]. However, from a practical engineering point of view, a small scale yielding
(SSY) requirement is imposed on the non-linearly deformed zone in order for the similitude
concept to be applicable. The requirement states that the size of the plastic zone must be
significantly smaller than any dimension of the component. This limit is generally expressed
25
by the stress-intensity factor and yield strength, σY of the material,
min (crack length, uncracked ligament, thickness) > κ
(KI
σY
)2
. (6)
The constant κ is obtained by systematic experimentation or numerical computation by solv-
ing the full non-linear problem. In the linear elastic fracture toughness standard ASTM
E399 κ is conservatively set to 2.5. If Eq. 6 is fulfilled along with other requirements at
crack growth initiation, then the applied stress-intensity factor value is stated as the material
fracture toughness, KIc. This size conditions, i.e.
min (crack length, uncracked ligament, thickness) > 2.5
(KIc
σY
)2
, (7)
is often considered as a general limit of application of LEFM. However, the requirement on
the thickness is not connected to the applicability of the LEFM and the control of the crack
tip state by the stress-intensity factor. It is connected to the similitude concept since the
plastic zone size relative the component thickness determines the state of plane stress or
plane strain. The linear elastic fracture toughness, KIc depends on the component thickness
with a higher fracture toughness for plane stress states. This is not a problem as long as
the small experimental specimen is of the same thickness as the component to be used in the
application. However, the ASTM E399 standard has a requirement on the thickness implying
a plane strain state and assuring specimen stability against out of plane buckling.
In the case of fatigue crack growth the same kind of requirement as Eq. 6 is used. In the
Standard Test Method for Measurement of Fatigue Crack Growth Rates ASTM E647 κ is set
to 4/π,
min (crack length, uncracked ligament) >4
π
(KI
σY
)2
, (8)
primarily based on empirical results applicable to compact tension (CT)-specimens. In Eq.
8 the thickness requirement is relaxed in order for the standard to be applicable to thin
structures. However, one should be aware of the results may be dependent on specimen
thickness and the limitations connected to the concept of similitude.
Predictions of variable load amplitude fatigue crack growth rates in notched specimens,
Eqs 3 and 5 are presented in Fig. 6 and Paper B based on constant load amplitude parameters
from Paper A. For load block type, see inset in each subfigure, Fig. 6. The numbers on the
inset ordinates are normalized load levels based on the maximum load in each block. Two
different geometries are used, i.e. the concept similitude is tested. The fatigue parameters
obtained in Paper A fulfill Eq. 8 while the target fatigue data in Paper B do not. Figure 5
shows the maximum stress-intensity factor applied and the two measures on the right hand
side of the diagram reflects how well the conditions of LEFM are satisfied. According to ASTM
26
0 1 2 3 4 5 60
50
100
150
200
KI,m
ax /
MP
a(m
)½
0 1 2 3 4 5 60
1
2
3
4
5
6
7
8
9
10
(KI,m
ax/σ
Y)2 /B
0 1 2 3 4 5 60
2
4
6
8
10
12
14
16
18
20
(KI,m
ax/σ
Y)2 /a
Crack length a / mm
KI,max
(KI,max
/σY)2/a
(KI,max
/σY)2/B
SSY limit
Figure 5: The scale of the left axis gives the maximum stress-intensity factor. The two scales of the right axescorrespond to the two measures of the applicability of linear elastic fracture mechanics. (Paper B).
0 1 2 3 4 5 610
−4
10−3
10−2
Crack length a / mm
da/d
N /
mm
/cyc
le
Cycle type 1
0 1 2 3 4 5 610
−5
10−4
10−3
10−2
Crack length a / mm
da/d
N /
mm
/cyc
leCycle type 2
0 1 2 3 4 5 610
−4
10−3
10−2
Crack length a / mm
da/d
N /
mm
/cyc
le
Cycle type 3
0 1 2 3 4 5 610
−4
10−3
10−2
Crack length a / mm
da/d
N /
mm
/cyc
le
Cycle type 4
0 1 2 3 4 5 610
−4
10−3
10−2
Crack length a / mm
da/d
N /
mm
/cyc
le
Cycle type 5
0 1 2 3 4 5 610
−4
10−3
10−2
Crack length a / mm
da/d
N /
mm
/cyc
le
Cycle type 6
0
0.5
1
0
0.5
1
−1−0.5
00.5
1
0
0.5
1
−1−0.5
00.5
1
−1−0.5
00.5
1
Figure 6: Experimental fatigue crack growth rates (markers) from two specimens for each cycle type. Thepredictions (dashed dotted line) include consideration of crack closure according to Eqs 3 and 5, i.e.corresponding to Reduced data in Fig. 4(b). The numbers on the inset ordinates are normalizedload levels based on the maximum load in each block. (Paper B).
27
E647 they should fall below 0.785 for LEFM to be applicable in a CT-specimen. Reasonable
predictions of the fatigue crack growth rates are obtained as the fatigue crack becomes longer
than the notch affected zone, despite that the concept of SSY is never fulfilled. At crack
lengths a > 0.5 mm Fig. 5 gives κ = 1/15 ≈ 0.07, which is considerably smaller than
4/π ≈ 1.27 in Eq. 8. Note the growth rates from one experiment (cycle type 3 filled markers)
does not conform with the general trend. The specimen developed shear lips right from the
start which caused a decrease in the growth rates throughout the ligament.
The applicability of LEFM is further investigated for load levels above the commonly
accepted monotonic SSY limit given by Eq. 8, see Paper C to Paper E. A stainless steel
316L is used and the crack tip opening displacement (CTOD), denoted δ, is controlled in the
experiments. The applied load levels relative to the SSY limit given by Eq. 8 are presented
in Fig. 7(a). Each line represents a constant CTOD at the crack tip. The flow stress σFS
used in Eq. 8 is defined as the average of the yield strength and ultimate tensile strength,
since extensive hardening occurred. The maximum load levels applied in the experiments are
above the monotonic SSY limit. However, at cyclic loading following Rice’s scheme [41] by
replacing KI with ΔKI and σY with 2σY,cyc
min (crack length, uncracked ligament) >4
π
(ΔKI
2σY,cyc
)2
, (9)
the load levels fall below the cyclic SSY limit due to the extensive isotropic hardening and
elastic shake-down of the material, Fig. 7(b). Consequently, monotonic large scale yielding
(LSY) rather than cyclic LSY took place in the experiments.
15 20 25 30 35 400
0.5
1
1.5
2
2.5
3
3.5
4
Crack length a / mm
(KI,m
ax/σ
FS)2 /a
δmax
= 8 μm
12
16
20
24
28
SSY limit
Increasing δmax
(a)
15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Crack length a / mm
Δδ = 7.6 μm11.4
15.219
22.826.6
(ΔK
I/(2σ
Y,c
yc))
2 /a
SSY limit
Increasing Δδ
(b)
Figure 7: Comparison of load and crack length to the condition for LEFM stated in ASTM E647, (a) mono-tonic increase to the maximum load, and (b) the condition adapted to cyclic loading. (Papers Cto Paper E).
The experimentally obtained fatigue crack growth rates for Δδ = 7.6−26.6 μm in Paper D
are presented in Fig. 8. The stress-intensity factors are computed from measured crack
28
lengths and applied forces. Two different specimens are used; the CT-specimen and an in-
house developed edge crack tension panel, the (ECT)-specimen. Surprisingly, all data are
consolidated along the extrapolated line from the SSY crack closure free data, the blue circles
in Fig. 8(a), i.e. the stress-intensity factor is able to characterize and correlate the load state
at the crack tip throughout the entire load range from SSY to monotonic LSY. Two possible
explanations for the encouraging results may be: firstly, no crack closure is present in the
LSY fatigue data since it conforms with the SSY fatigue data obtained at crack closure free
conditions; secondly, elastic conditions is prevailing at steady state conditions due to cyclic
isotropic hardening and elastic shake-down.
10 20 30 40 50 60 708010
0
101
102
103
104
ΔK / MPa(m)1/2
da/d
N /
nm/c
ycle
Const. KI,max
, CT 10.0 mm,R
G = [0.5, 0.8]
CT 10.0 mm, RG
= 0.5
CT 10.0 mm, RL = 0.05
CT 19.5 mm, RL = 0.5
CT 19.5 mm, RL = 0.05
ECT 9.8 mm, RL = 0.5
ECT 9.8 mm, RL = 0.25
Lsq curve fit to const. KI,max
data
(a)
20 30 40 50 60 70 8010
2
103
104
ΔK / MPa(m)1/2
da/d
N /
nm/c
ycle
(b)
Figure 8: Linear evaluation of the fatigue crack growth data. (a) Summary view. (b) Enlarged view on themonotonic LSY fatigue data. (Paper D).
(a) (b)
Figure 9: Deformed specimens. (a) The CT-specimen and (b) the ECT-specimen. (Paper D).
Figure 9 show the deformed specimens after the experiments from which some of the
29
highest fatigue crack growth rates in Fig. 8 are obtained. The specimens show large rotation
and plastic deformation in the crack plane and also through the thickness.
The results in Figs 5−8 shows that the ASTM E647 criterion, Eq. 8 for applicability
of LEFM is conservative and may be relaxed in some situations. In Paper B a nickel-based
superalloy, Inconel 718, is used which shows limited hardening compared to the stainless
steel 316L used in Paper C to Paper E. The stress-intensity factor range characterized the
fatigue crack growth in both materials at loads outside the LEFM criterion limits. Thus, this
knowledge may be very valuable for the experienced engineer in the design situation. However,
for safety reasons a conservative criterion is necessary to insure the concept of similitude in
all situations.
Intermediate scale yielding
This load region is characterized by a plastic zone size at the crack tip that is about the same
as the characteristic dimensions of specimen; still the plastic zone should not interact with the
outer boundaries of the body. As stated above, the stress-intensity factor may be a possible
characterizing measure even in this region but will not always work. A better measure would
be the J -integral or the CTOD, which are able to account for the non-linear deformation.
The validity condition for elastic-plastic fracture mechanics (EPFM) is not equally well
established and developed as for LEFM. A similar criterion is suggested, cf. Hutchinson and
Paris [42] that any characteristic dimension l in the body should be larger than
l > κ2
(J
σY
), (10)
for the J -integral to be applicable.
The non-dimensional constant κ2 is usually set to 25−100 dependent on the type of
loading (pure bending or pure tension). In the ASTM E1820 standard for non-linear fracture
toughness testing κ2 is set to 25. If Eq. 10 is fulfilled, then the experimentally obtained
J-value may be interpreted as size independent fracture toughness value, JIc.
Large scale yielding
When extensive plasticity occurs and the size of the yield zone becomes large compared to
characteristic dimensions in the body the single parameter description breaks down. The
size of the J -dominated zone becomes strongly dependent on the configuration of the test
specimen, which means that the distributions of stresses and strains in the crack tip vicinity
are not unique.
The Paper B to Paper E make no distinction between ISY and LSY. Thus, data that falls
outside the monotonic SSY limit is termed LSY.
30
The potential drop method for
crack length measurements
The potential drop (PD) is one of two techniques recommended in ASTM E647 for measuring
the fatigue crack length at SSY conditions. The idea is to pass a constant electrical current
through the specimen and measure the potential difference across the crack mouth. The PD
across the crack increases with increasing crack length. Thus, the technique is well suited for
crack length measurements at SSY. Several researchers have also used the method for crack
closure measurements. The idea is that the current passes through the contact between the
crack surfaces as long as contact exists. Consequently, when the contact is released and the
crack opens, changes in the PD signal takes place until the crack is fully opened after which
the PD value remains constant. However, a number of studies indicates problems with the
PD technique for crack closure measurements, Bachmann and Munz [43], Pippan et al. [44],
while others find the technique useful, Shih and Wei [45], Spence et al. [46] and Andersson
et al. [47]. The problems are mainly connected to fatigue crack propagation in vacuum and
air in connection to the formation of an oxide layer on the crack surfaces which prevents
current flow. The PD technique was used in the present Paper C to Paper E for crack length
measurements. By keeping the local load ratio, RL = δmin/δmax above zero in all experiments
no crack closure is expected. Consequently, the PD technique is primarily used for crack
length measurements even though crack closure effects are searched for but never found.
Different factors that affect the measured PD values and consequently the calibration
equation that connects the normalized potential to the normalized crack length are inves-
tigated in Paper C. The material used is the stainless steel 316L. For SSY fatigue, several
studies have been conducted that investigate different techniques for establishing the calibra-
tion equation, cf. Clark and Knott [48], Wei and Brazill [49], Johnson [50]. However, when
using a normalized calibration equation a reference potential is needed giving the correct
crack length at an arbitrary but known calibration point. If a significantly higher load level is
applied relative to the load level at which the original reference potential was taken, a change
of the state may be expected due to geometry and resistivity changes. Thus, the original
reference potential and shape of the calibration curve may be affected by the plastic defor-
mation at the substantial load change, compared to the nominal calibration curve. Hence,
31
substantial load changes in the LSY range may give errors in measured fatigue crack length.
The error seems to be very small in the SSY region, cf. Ritchie and Bathe [51], Hicks and
Pickard [52] and Wilson [53].
The change in measured PD due to plastic deformation origins from a resistance change
which in turn consists of a geometrical part and resistivity part. Figure 10 shows the results
from a cyclic tension-tension test on a smooth cylindrical specimen where the PD is contin-
uously measured. The strain range is controlled and increased in steps up to a total strain
range of 3 % with 10−30 cycles at each strain range.
0 0.5 1 1.5 2 2.5 3−500
−400
−300
−200
−100
0
100
200
300
400
500
Plastic strain εpl
/ %
Tru
e st
ress
σ /
MP
a
(a)
0 0.5 1 1.5 2 2.5 3−1
0
1
2
3
4
5
6
7
Rel
ativ
e ch
ange
of r
esis
tanc
e ΔR
/R0 /
%
Plastic strain εpl
/ %
(b)
0 0.5 1 1.5 2 2.5 3−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Plastic strain εpl
/ %
Rel
ativ
e ch
ange
of r
esis
tivity
Δρ/
ρ 0 / %
(c)
Figure 10: (a) Cyclic stress-strain curve for Rε = 0. (b) Cyclic relative change of resistance for Rε = 0. (c)Cyclic relative change of resistivity for Rε = 0. (Paper C).
As shown in Fig. 10(a) the stainless steel 316L exhibits extensive strain hardening. The
Figs 10(b)−10(c) show the cyclic change of resistance and resistivity, respectively. The relative
change of resistance is on the same order as the plastic strain while the relative change of
resistivity is an order lower compared to the applied strain. The same relative change is
observed at a monotonic tension test, i.e. the resistivity contributed with <10 % of the total
32
resistance change. Assuming that the same relation holds in the PD measured crack length
situation, which may not be the case, the major part (>90 %) of the error in measured crack
length can be ascribed to a change in geometry while the effect of the material property
change will be relatively small (<10 %).
As stated, the plastic deformation will affect the resistance and consequently the crack
length. Figure 11(a) shows the ratio between the first and forthcoming PD measured crack
lengths for 15 different experiments on CT-specimens with two different thicknesses, 10 mm
and 19.5 mm. The maximum load level applied in each experiment corresponds to the load
level used later in the fatigue experiment. A shift of the PD measured crack length within
±10 % immediately occurred as the large load is applied. It means that an error of ±10 % in
measured crack length, or even more, can be expected if the calibration point is taken in the
nominal geometry. Expected outcome would be and an artificial increase in crack length but
surprisingly about half of experiments showed an artificial decrease in crack length at the first
load cycle. The reason for the diverging results is not known to the author. However, finite
element simulations of the geometrical effect on the PD signal all point towards an artificial
increase in crack length, see Paper C.
0 100 200 300 400 500 6000.94
0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
a PD
/aP
D,0
Cycle number
10 mm19.5 mm
11639
(a)
100 120 140 160 1801
3
5
7
CM
OD
/ m
m
Time / s
100 120 140 160 180
0.74
0.76
0.78
V1 /
mV
Time / s
100 120 140 160 1801.341.361.381.4
1.42
V2 /
mV
Time / s
(b)
Figure 11: (a) The ratio between the first and the forthcoming PD measured crack lengths for 15 exper-iments. (b) Measures at first application of the load level used during the fatigue experimenton specimen 11639. Top: The crack mouth opening displacement, middle: the potential in thecracked specimen, below: the potential in the reference specimen. (Paper C).
Figure 11(b) shows the crack mouth opening displacement (CMOD), denoted ν, the PD
signal over the fatigue crack, V1 and the PD signal in the reference specimen, V2 from exper-
iment 11639. The first large load is applied at ∼ 110 s and initial drop in V1 is measured.
Towards the end of the load increase, V1 also started to increase and finally a too long crack
length is measured. The PD in the reference specimen is temporarily affected by the initial
large load change. Changes in V2 is not desired since it ought to be unaffected by changes in
other parts of the system. However, no long term affect from changes in geometry, material
33
hardening/softening or crack extension is observed in V2, which is most importantly.
Figure 12 shows the measured crack lengths vs cycle number. In order to verify the
PD sampled crack lengths additional visual measurements are conducted on one side of the
specimen. The main observation is the overall conformity of the two measurements techniques.
However, in some experiments with a crack growth over at least 10 mm, a weak tendency
existed for the PD method to give a too short crack length estimate compared to the visually
measured crack length. That tendency is consistent with the ideas that Wilkowski and Maxey
[54] schematically present. In spite of the weak tendency, as the results show, it seems that
the calibration curve taken from the nominal geometry and applied in the monotonic LSY
case (calibrated in the deformed state) still yields a reliable measurement.
0 1 2 3 4 5
x 104
10
15
20
25
30
35
40
Cycle number
Cra
ck le
ngth
/ m
m
aPD
aVisual
1163811631
Disturbances in the PD signal
11639
Figure 12: Crack length measurements on 10 mm thick CT-specimen. (Paper C).
34
Characterization of fatigue crack
growth outside the SSY limit
The limits given by Eqs 8−9 define in a conservative manner the applicability of LEFM.
Outside these limits LEFM may still be applicable. However, the applicability of LEFM
beyond Eqs 8−9 can only be judged based on EPFM which takes the non-linear constitutive
behaviour into consideration. The two main EPFM parameters suggested in literature for
characterization of fatigue crack growth at non-linear conditions are the ΔJ and Δδ.
The ΔJ parameter
The J -integral and ΔJ -integral are two measures used in the EPFM field. This means that
the measures are capable of characterizing the state at the crack tip even if the plastic zone
is not confined to a very small region compared to the in-plane dimensions. Still the plastic
zone may not interact with any in-plane dimension for the theory to be applicable.
The J contour integral is presented in 1968 by Rice [55]. He idealizes the elastic-plastic
non-linear constitutive behaviour to a non-linear elastic behaviour. Rice then shows that the
non-linear energy release rate J, can be written as a path independent line integral,
J =
∫Γ
(Wdy − Ti
∂ui∂x
ds
). (11)
Here are W the strain energy density, Ti are the components of the traction vector, ui are
the displacement vector components and ds is the length increment along the a contour Γ
around the crack tip.
The following equation needs to be satisfied in order for the contour integral to be strictly
path independent,
σij =∂W
∂εij. (12)
The requirement of elastic material response, linear or non-linear, may be relaxed to a least
approximately include elastic-plastic materials under the condition of monotonic loading. The
35
response of the two different constitutive models is approximately identical (apart from a small
change of the Poisson’s ratio) as long as no unloading occurs. In three dimensions this may
not be true because unloading may occur due to stress redistribution in the elastic-plastic
material. Still, the assumption of non-linear elastic behaviour may be a good assumption in
cases where a high triaxiality exists at the crack tip and no global unloading occurs.
The J -value can also be calculated as a deformation-J denoted JD. The deformation
JD is basically calculated as an integration of the load-deformation path. In an elastic-
plastic material, however, history dependence exists that is removed when the integration is
performed. The J -value from the contour integral and the load-deformation path may differ
somewhat dependent on the amount of crack growth in the elastic-plastic material.
The situation becomes a lot more complicated in the case of fatigue. When unloading
occurs in an elastic-plastic material, deformation plasticity theory no longer describes the
actual behaviour of the material. Thus, highly path dependent values result in the situation
of unloading and are therefore questionable to use in the fatigue situation. Despite the loss
of generality several researchers have tried to use the ΔJ when correlating the crack growth
rate with different degrees of success. The definition of ΔJ is first presented by Lamba [56]
in 1975 in connection to stress concentration factors.
The material in front of the crack tip experiences a cyclic stress-strain range and conse-
quently the process is characterized by Δσij and Δεij. Figure 13 shows a cyclic stress-strain
loop where the initial value is indicated with number 1 and final value with number 2. The
range of the J -integral is then defined as [56], Dowling and Begley [57] and Lambert et al.
[58]
ΔJ =
∫Γ
(ψ (Δεij) dy −ΔTi
∂Δui∂x
ds
). (13)
Here Γ is the integration path around the crack tip. Further, ΔTi and Δui are the changes
in traction and displacement between the initial point 1 and the finial point 2. In addition,
ψ corresponds to the strain energy density and is defined in an analogous way by
ψ (Δεij) =
∫ ε2ij
ε1ij
Δσijd (Δεij) =
∫ ε2ij
ε1ij
(σij − σ1ij
)dεij. (14)
Usually only the loading part of the cyclic loop is included in the integration of the strain
energy density rather than the whole loop. If the initial point 1 is located at zero stress
and strain then ΔJ = J . Equation 13 is a generalization of Eq. 11 to include the situation
when the stress and strain at point 1 is not at zero. The requirements for ΔJ to be path
independent are analogous to those for the original J mentioned above, i.e. σij = ∂W/∂εij .
As in the case of monotonic loading it is also possible to calculate the ΔJD from the cyclic
load-displacement curve. The general form is written as
36
ΔJD =η
Bb
∫ ΔV
0ΔPd (ΔV ) =
η
Bb
∫ Vmax
Vmin
(P − Pmin) dV, (15)
where η is dimensionless constant, B the specimen thickness and b the uncracked ligament.
Pmax, Pmin, Vmax and Vmin are the maximum and minimum load and displacement, respec-
tively, during the specific load cycle.
(σ1ij , ε1ij)
(σ2ij, ε2ij)
Strain
Stress
Figure 13: Stress-strain curve.
If Eq. 10 is met, then Eqs 13 and 15 should give the same result. In general, Eqs 13 and
15 do not necessarily give the same value under equal conditions. This may happen when
excessive plastic deformation occurs in the geometry (LSY). Still, the state at the crack tip
may be characterized by Eq. 15 meaning that the path independence is lost in Eq. 13.
Several researches: Tanaka et al. [59], Mowbray [60] and Rolfe and Barsom [61] show
experimentally that ΔJD is an unambiguous parameter correlating the fatigue crack growth
rate at SSY and cyclic LSY conditions. The fatigue crack growth rates fall onto a single line
plotted in a double logarithmic diagram vs ΔJD. However, the parameter is not suitable for
all situations. For instance, fatigue crack growth at high load ratios, load control and LSY
is a situation shown to be difficult for ΔJD to correlate due to ratcheting effects, Tanaka et
al. [62] and Dowling [63]. While ΔJD has shown to correlate fatigue data at both SSY and
LSY one cannot expect ΔJ calculated as a contour integral to do the same. If the body is
plastically deformed out to the boundaries, i.e. LSY then, the behaviour locally at the crack
tip becomes dependent on the overall behaviour of the geometry and thereby result in a path
dependent ΔJ . However, in the SSY region [64] and in the ISY region both parameters are
expected to give similar results.
The advantage of the ΔJD measure, compared to both ΔJ and Δδ, lays with the possibility
to evaluate the parameter from the load-displacement data. Furthermore, the theoretical
base for use of ΔJ in elastic-plastic materials is not without doubts. The definition of the
path independence of the J -integral relies on linear elastic, non-linear elastic or deformations
37
plasticity. Another issue is the crack closure mechanism which is problematic to incorporate
into the contour integral, however, possible in the ΔJD concept, cf. [57]. Furthermore, the
physical meaning of ΔJ and ΔJD is also hard to interpret.
The Δδ parameter
An easily interpreted measure is the crack tip opening displacement (CTOD), denoted δ.
Originally this parameter was developed by Wells [65] who discovers that several structural
steels could not be characterized by LEFM, i.e. KIc is not applicable. He also discovers while
examining the fracture surfaces that the crack surfaces moves apart prior to fracture. Plastic
deformation precedes the fracture and the initially sharp crack tip is blunted. The plastic
deformation increases with increasing fracture toughness and Wells proposes the opening at
the crack tip as a fracture toughness parameter.
The CTOD has no unique definition. The most common one was defined by Rice [55].
It is measured as the distance between two lines separated by 90◦ lines extending from the
crack tip and intersecting the crack surfaces behind the tip, shown in Fig. 14. Others define
the CTOD at the elastic-plastic boundary or at a distance beyond the plastic zone where the
crack face is no longer deformed. The CTOD range is used in connection to fatigue, i.e. Δδ.
90◦ δ
Figure 14: A common definition of the CTOD.
The drawback with the CTOD measure, compared with the JD, is the rather expensive
experimental equipment that is required for the measurements. However, the CTOD measure
has several advantages. It has an obvious physical interpretation and crack closure can easily
be incorporated. Furthermore, it is a local parameter defined close to the crack tip and
possible to evaluate with finite elements, even though a fine mesh is required.
The definition of δ that Paper C to Paper E use is very similar to the one in Fig. 14.
However, instead of using the intersection between the crack surface and the two straight lines
emanating from the crack tip at 90◦ a straight line is fitted to node number 2 and 3 behind
the crack tip and that line is regarded as the crack surface, see Fig. 15. By excluding the
first node behind the crack tip less scatter is achieved in the parameter. Hence, node number
38
15.49 15.492 15.494 15.496 15.498 15.50
1
2
3
4
5x 10
−3
Crack length a / mm
Def
orm
atio
n / m
m
Crack tip
45°
Lsq fit to node 2 and 3
Node number 2 and 3 behind the crack tip
½ δ
Figure 15: Definition of δ. (Paper C to Paper E).
2 and 3 are used as a good compromise between capturing the local behaviour at the crack
tip while minimizing the scatter in the computed Δδ.
The mesh size dependence in δ is investigated in Paper D. No obvious convergence is
observed. However, in order to have a properly computed Δδ-value a minimum element size
is required. The required element size is found to be strongly dependent on the definition of
the CTOD parameter and the mesh structure in the crack tip region. No published results
are found on the convergence behaviour of the CTOD parameter in connection to fatigue
and decreasing element size including possible crack closure. A comparison with convergence
studies on the required element size for computation of the crack closure level show that
no convergence is achieved with decreasing element size, cf. Gonzalez-Herrera and Zapatero
[36]. The comparison is justified by fact that both measures (δ and the crack closure level)
are evaluated in the vicinity behind the crack tip.
The fatigue crack growth experiments in Paper C to Paper E are controlled through δ.
Based on finite element computations for the two geometries in Fig. 9 the coupling between
δ, applied external load P , and crack length a, is derived as
Δδ = 2a0
(ΔP
PL (a0)
)2
f1
( a
W
)(16)
and
δmax = 2a0
(Pmax
PL (a0)
)2
f2
( a
W
). (17)
Here, a0 is a constant reference crack length set to 12.5 mm for the CT-specimen and 6 mm
for the ECT-specimen; PL (a0) is the limit load at a0; ΔP and Pmax are the load range and the
maximum load; f1 and f2 dimensionless functions obtained by non-linear least square curve
fits to finite element results, see Fig. 16 for results on the ECT-specimen. At linear conditions
δ is proportional to the square of the applied load P . Hence, the square in Eqs 16−17. The
limit load and the reference crack length a0 are added in order to have dimensionless functions
f1 and f2.
39
Figure 16 displays the result from two-dimensional plane deformation finite element com-
putations on the ECT-specimen where a cyclic load with constant maximum and minimum
load is applied. The global load ratio, RG = Pmin/Pmax was 0.5 in all computations. Each
computation continued until gross plastic deformations occurred and finally stopped due to
plastic collapse of the ligament, i.e. a crack is propagated through the ligament. Each line
shows the development of δ as a function of crack length. Filled markers represent an applied
maximum load that is larger than the limit load at the evaluated crack length and opposite,
an unfilled marker represent a load level lower than the limit load. The black unmarked lines
in Figs 16(c)−16(d) are the dimensionless functions f1 and f2.
5 10 15 20 25 30 350
2
4
6
8
10
12
Crack length a / mm
Δδ/2
/ μm
10.26 kN/mm12.82 kN/mm15.39 kN/mm17.95 kN/mm20.51 kN/mm23.08 kN/mm25.64 kN/mm
Increasing load
(a)
5 10 15 20 25 30 350
2
4
6
8
10
12
14
16
18
Crack length a / mm
δ max
/2 /
μm
10.26 kN/mm12.82 kN/mm15.39 kN/mm17.95 kN/mm20.51 kN/mm23.08 kN/mm25.64 kN/mm
Increasing load
(b)
5 10 15 20 25 30 350
1
2
3
4
5
6
7x 10
−3
Crack length a / mm
Δδ/(
2a0(Δ
P/P
g(a0))
2 )
10.26 kN/mm12.82 kN/mm15.39 kN/mm17.95 kN/mm20.51 kN/mm23.08 kN/mm25.64 kN/mmNonlinear Curve Fit
f1
(c)
5 10 15 20 25 30 350
0.5
1
1.5
2
2.5
3
3.5x 10
−3
Crack length a / mm
δ max
/(2a
0(Pm
ax/P
g(a0))
2 )
10.26 kN/mm12.82 kN/mm15.39 kN/mm17.95 kN/mm20.51 kN/mm23.08 kN/mm25.64 kN/mmNonlinear Curve Fit
f2
(d)
Figure 16: The CTOD vs crack length for different load levels in the ECT-specimen. (a) Δδ/2. (b) δmax/2.(c) Normalized Δδ/2. (d) Normalized δmax/2. (Paper D).
In Paper D constant amplitude loading is investigated in terms of δ, i.e. the force is
continuously decreased with the fatigue crack growth. The applied load level is computed on
a cycle-by-cycle routine based on Eqs 16 and 17. Given the user specified Δδ and δmax along
with measured crack length (aPD) in the previous load cycle, ΔP and Pmax are computed
40
and applied in the next load cycle. The crack lengths are simultaneously measured using
direct current PD, compliance and visually by a low magnifying long distance microscope.
The compliance based measurements are excluded from the figures due to large scatter. The
microscope allowed for manual in-plane crack length measurements on one side of the speci-
men. Since LSY occurred and in the case of the CT-specimen large accumulated rotations,
see Figs 9 and 17(b), this is taken into account by correcting for the rotation according to
aVisual = anotch +
√a2meas +
(ymeas
2
)2, (18)
where ymeas is the vertical distance at the electro discharge notch root and ameas the horizontal
distance from the notch root to the crack tip, see Fig. 17(a). Here, anotch = 10 mm.
ameas
10 aVisual
ym
eas
Initial loadline
Current loadline
(a) (b)
Figure 17: (a) Deformed specimen that illustrates the rotational compensation of visual measured cracklength, aVisual. (b) Specimen 11901, after completed experiment. (Paper E).
Figure 18(a) shows the measured crack lengths vs cycle number from the constant am-
plitude experiments on the ECT-specimen. The results show a relatively linear relationship
between the crack length and cycle number. This supports the assumption that the CTOD
is able to characterize the crack tip load state at monotonic LSY fatigue crack growth. This
rather linear relation is also seen in the results from the CT-specimen, see Paper D. The
deviation from the linear trend towards the end in some experiments are connected to large
extrapolations, i.e. the validity range for the control functions are 5 ≤ a ≤ 35 mm for the
ECT-specimen.
41
0 2 4 6 8
x 104
15
20
25
30
35
40
45
50
55
12518−4
Cycle number
Cra
ck le
ngth
/ m
m
12518−2
12519
12520
aPD
aVisual
(a)
15 20 25 30 35 40 45 50 550
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Crack length / mm
Cra
ck g
row
th r
ate
/ μm
/cyc
le
12520
1251912518−4
12518−2
PD dataVisual dataPrediction(a
PD)
(b)
Figure 18: Fatigue crack growth data for the 9.8 mm thick ECT-specimen. (a) Crack length vs cycle number.(b) Fatigue crack growth rate vs crack length. (Paper D).
The corresponding fatigue crack growth rates vs crack lengths for the ECT-specimen are
shown in Fig. 18(b). As a consequence of the relatively linear relations in Fig. 18(a) the rates
are rather constant, except towards the end of the experiments due to extrapolations. Figure
18(b) includes predictions of the growth rates based on LEFM. These predictions correspond
to the data presented in Fig. 8.
0.1 1 1010
0
101
102
103
104
Δδ / μm
da/d
N /
nm/c
ycle
Const. KI,max
, CT 10.0R
G=[0.5,0.8]
CT 10.0, RG
= 0.5
CT 10.0, RL = 0.05
CT 19.5, RL = 0.5
CT 19.5, RL = 0.05
ECT 9.8, RL = 0.5
ECT 9.8, RL = 0.25
Lsq curve fit to constant KI,max
data
(a)
2 3 4 5 6 7 8 910 20 30 4010
2
103
104
Δδ / μm
da/d
N /
nm/c
ycle
(b)
Figure 19: The fatigue crack growth data vs Δδ. The black line is a least square curve fit to the con-stant KI,max-data. (a) Summary view. (b) Enlarged view on the monotonic LSY fatigue data.(Paper D).
Figure 19 summarizes all fatigue crack growth rates vs Δδ from both the ECT- and CT-
specimen. The blue circles are the constant KI,max-data obtained at SSY conditions. The
black line is a least squares fitted line, exclusively fitted to the constant KI,max-data. The
other markers are the fatigue crack growth data obtained at monotonic LSY conditions. All
42
fatigue crack growth rates are consolidated vs Δδ, cf. Tanaka et al. [59] and McClung and
Sehitoglu [66]. Hence, the objectivity of Δδ was checked up to monotonic LSY and δ can be
considered as a correlating parameter giving transferable results between geometries. This is
a prerequisite for predictive purposes in engineering fatigue design.
Furthermore, the fatigue crack growth at variable amplitude loading is described in Pa-
per E. Experiments similar to the ones in Paper D are conducted. However, here two different
load histories are used, see Fig. 20. The load blocks consist of one large cycle and eight small
cycles and are continuously repeated throughout the experiments. The load ratios for the
large cycles are RL = 0.05 and for the small cycles RL = 0.7 − 0.8 in sequence 1, Fig. 20(a).
In sequence 2 the load ratio is RL = 0.05 for all cycles, Fig. 20(b).
0 1 2 3 4 5 6 7 8 9
δ
Cycle number
(a)
0 1 2 3 4 5 6 7 8 9
δ
Cycle number
(b)
Figure 20: Block load: (a) sequence 1, large cycle followed by 8 small cycles at high maximum load, i.e.effect of underload, and (b) sequence 2, large cycle followed by 8 small cycles at low load, i.e.effect of overload. (Paper E).
Figures 21(a) and 21(b) display examples of measured and predicted fatigue crack growth
rates for an experiment with load sequence 1 and 2 respectively. The data are based on
three independent crack length measurement systems; the low magnifying long distance mi-
croscope (Visual data); the direct current PD technique (PD data and Prediction(aPD));
post-experiment measurement of the start and stop crack length according to the average
nine-point formula along the crack front recommended in ASTM E1820 (Prediction(aASTM)).
Note that all rates are plotted against aPD, which was selected for the abscissa because it was
used for controlling the experiments.
The predictions are based on crack closure free fatigue parameters from Paper D (the
KI,max-experiment) and on the cycle weighted average of the growth rates according to Eq.
5. No crack closure are observed in the PD data. This is also supported by the fact that
RL > 0 for all experiments and crack lengths. The respective ΔKI-values are determined
using force and crack lengths from the experiments. The predictions based on aPD are of the
43
same size but consistently lower than the measured crack growth rates. The difference are
due to a consistently shorter measured aPD relative to aVisual. The difference also increased
with crack extension due to accumulated plastic deformations and consequently increased
deflections. The reason is the effect of geometrical changes on the reference potential, which
could have been solved by taking a new reference potential. However, that is not conducted.
The aASTM-based predictions utilize the measured aPD-values but linearly interpolate the
intermediate crack lengths to coincide with the start and stop aASTM-values. The predictions
are consistently higher than those based on aPD and in most cases somewhat below the
measured rates but in some cases they over predict the crack growth rates.
27 28 29 30 31 32 33 34 35 360
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4
4.4
Specimen: 12475
Crack length / mm
Cra
ck g
row
th r
ate
/ μm
/cyc
le
PD dataVisual dataPrediction(a
PD)
Prediction(aASTM
)
(a)
19 20 21 22 23 24 25 26 270
0.6
1.2
1.8
2.4
3
3.6
4.2
4.8
5.4
6
Specimen: 11897
Crack length / mm
Cra
ck g
row
th r
ate
/ μm
/cyc
le
PD dataVisual dataPrediction(a
PD)
Prediction(aASTM
)
(b)
Figure 21: Crack growth rates at variable amplitude experiments for (a) sequence 1 with underloads and for(b) sequence 2 with overloads. (Paper E).
The present experiments are performed on CT-specimens and a directly comparable load
measure is not available in literature. McClung and Sehitoglu [66] quantify the cyclic LSY load
on centre cracked specimens in terms of strain over the crack. For variable load amplitudes
and R = −1, the strain ranges are Δεlarge = 0.5 % and Δεsmall = 0.2 % in their experiments.
Returning to Paper D the growth rates on both the ECT- and CT-specimens are transferable
between specimen types and the deformation in one ECT-experiment corresponded to Δε =
0.1 − 0.3 % with crack growth. The load in the actual experiment is Δδ = 5 μm and
δmax = 10 μm. For the variable load amplitude series in Paper E Δδlarge ≈ 20 − 28 μm and
Δδsmall ≈ 4− 11 μm, which suggests that the present loads correspond to Δεlarge ≈ 0.4− 1.2
% and Δεsmall ≈ 0.1 − 0.3 % for a set-up following that of McClung and Sehitoglu [66].
Hence, expressed in specimen Δε the variable amplitude loads in Paper E seemed to be in the
same range as the LSY fatigue loads that McClung and Sehitoglu use. However, the present
variable load amplitude experiments are closure free with RL = 0.05 or 0.5 (RG ≈ 0.4− 0.6)
and on a material with substantial monotonic hardening whereas McClung and Sehitoglu use
RG = −1 with crack closure and reversed plastic deformation at the crack tip.
44
Conclusions
During the present work a series of conclusions were made on fatigue crack growth experi-
ments, the possibilities to use linear elastic fracture mechanics beyond its established limits
and conditions for such extensions. The results are presented in detail in the appended papers.
The main conclusions are summarized here.
� The accuracy of fatigue crack growth rate assessments and life predictions are heavily
dependent on the fatigue crack growth rate data obtained from experiments. In order
to ensure generality of the fatigue growth data and successful transferability of results
between test specimen and applications no load history effects is desired in the fatigue
crack growth experiments.
– When establishing the fatigue crack growth data, the experimental ΔK-reducing
procedures should be avoided. The reason is to minimize the risk of introducing
transient crack closure effects into the data, which is not representative for the
load history of the assessed application.
– One method of producing fatigue crack growth data with minimal transient load
history effects is to propagate the crack under constant low ΔK for a couple of
millimeters and then do the same at a high ΔK. Thereby the ligament is used for
producing fatigue data with high accuracy at two different load levels.
– The constant KI,max-method gives a statistically significant lower slope of the fa-
tigue data compared to other load schemes in the nickel-based superalloy Inconel
718 at +400�C. The difference in fatigue crack growth rates is small in the actual
measuring range but extrapolations should be avoided as the errors may become
significant.
� Even though engineering structures in general exhibit variable amplitude loading his-
tories, most of the existing fatigue crack growth data in the industry are of constant
amplitude type. Ideally, it should be possible to predict variable amplitude crack growth
rates with experimental data from constant amplitude fatigue crack growth experiments.
– Mean fatigue crack growth rates are computed from variable amplitude loading
experiments described with repeated simple load blocks. Predictions are based on
45
constant amplitude fatigue crack growth data. Although experimental data showed
rather large a scatter it was concluded that variable amplitude crack growth rates
can be predicted with reasonable agreement using constant amplitude fatigue data.
– Due to the relative levels of large and small loads and the rather limited number
of small cycles within each load block in the present experiments, the large cycle
influenced the overall crack growth rate for a block.
� It is desired from an engineering point of view to use linear elastic fracture mechanics in
fatigue life assessments due to the increased complexity of the analyses if one substitutes
linear elastic fracture mechanics with elastic-plastic fracture mechanics. Examples of
such situations are fatigue crack growth in notch affected regions and fatigue crack
growth at load levels above the commonly used monotonic small scale yielding criterion
given by American Society for Testing and Materials (ASTM).
– The crack growth rates in a notch affected region could not be predicted by linear
elastic fracture mechanics despite consideration of crack closure and residual stress
effects. However, the fatigue crack growth rates are predicted as the length is
comparable to the plastic zone size due to the notch (0.5 mm).
– The use of crack closure free fatigue crack growth data in fatigue crack growth
rate assessments enables statements in advance about the prediction accuracy in
relation to the expected outcome from the application. This is not possible to do
with nominal fatigue crack growth data.
– For the nickel-based superalloy Inconel 718, crack closure free fatigue growth from
the small scale yielding load range is able to predict growth rates where the load
exceeded the monotonic small scale yielding criterion by a factor of about 15.
– For the stainless steel 316L, crack closure free fatigue growth data at small scale
yielding is able to predict growth rates where the load exceeded the monotonic
small scale yielding criterion with approximately a factor 4. The main reasons are
concluded to be the large isotropic hardening of the material that gave small scale
cyclic yielding and the absence of crack closure.
– For the stainless steel 316L it is noted that if the crack tip opening displacement
range Δδ is kept constant and the local load ratio is close to zero, then the corre-
sponding cyclic small scale yielding criterion will still be fulfilled due to the elastic
shake-down in the material. Also, in a fatigue experiment with maximum load
levels far beyond the monotonic small scale yielding limit.
� The crack tip opening displacement range Δδ, which is a non-linear elastic-plastic frac-
ture mechanics parameter, is investigated as measure for characterizing the load state
46
at the fatigue crack tip at monotonic large scale yielding. Both constant amplitude
and variable amplitude load histories constituting of repeated load blocks are consid-
ered. The crack tip opening displacement is controlled in the experiments and ideally
an applied constant Δδ would result in a constant fatigue crack growth rate.
– The experimental results show sufficiently constant fatigue crack growth rates to
conclude that Δδ is able to characterize the load state at monotonic large scale
yielding constant and variable amplitude fatigue crack growth.
– The fatigue crack growth rates from two different geometries based on Δδ, are
consolidated to a straight line in a Paris law type of diagram. Thus, Δδ is concluded
to be an objective parameter for the current geometries and load levels.
� The direct current potential drop method is a recommended technique for crack length
measurements in standard small scale yielding fatigue crack growth rate testing. The
technique is also applicable to crack length measurements at large scale yielding when
certain conditions are fulfilled.
– The nominal calibration curve obtained from the undeformed geometry can be
used at large scale yielding if the reference potential is adjusted at every major
change of (plastic) deformation. This applies in particular, at the first fatigue load
cycle.
– The major part of the difference between the nominal and the true calibration
curve in the large scale yielding case should be attributed to geometry change and
only a minor part to the material resistivity change.
– The fatigue crack length can be measured in situ with the potential drop technique,
i.e. the experiment does not need to be stopped.
47
48
Summary of appended papers
Paper A: Effects of different load schemes on the fatigue crack growth rate
This paper presents an experimental study on a nickel-based superalloy, Inconel 718, at
an elevated temperature of +400�C. Inconel 718 is preferably used in high temperature appli-
cations and extreme environments such as in gas turbines, chemical processing applications
and pressure vessels. The basic question was how to conduct fatigue crack growth experi-
ments based on linear elastic fracture mechanics in order to obtain growth data as free from
load history effects as possible in the stage II region. This was investigated by use of four
different load sequences on the compact tension geometry. An important question was to in-
vestigate if the so called constant KI,max-method provides an upper bound of the fatigue crack
growth rate in the stage II region. The results indicated that the constant KI,max-method
gave an upper bound in fatigue crack growth rate in the current measuring range but also
gave a statistically significant lower exponent of the crack growth equation. A more reliable
method to establish the fatigue crack growth parameters was to propagate the crack under
constant stress-intensity factor range. This should be done at two different stress-intensity
factor ranges for a few millimeters at each range. Also, ΔK-reducing (constant load ratio)
procedure should not be used, when performing fatigue crack growth testing in the stage II
region, in order to minimize the risk of transient crack closure effects. Transient crack closure
effects showed up as a too steep gradient of the fatigue growth rates vs ΔK plotted in a
double logarithmic diagram, i.e. too large an exponent in Paris law.
Paper B: Variable amplitude crack growth in notched specimens
The typical conditions for an engineer, in the fatigue life design situation, are to have
constant amplitude fatigue crack growth data from fracture mechanics specimens at specific
load ratios at his disposal. However, variable load amplitude and complex geometries are
rather the rule in applications. The load history can sometimes be divided into load blocks.
Especially if the product is intended for a repeated start-stop type of use under similar
conditions as for a gas turbine. In this paper, the constant amplitude fatigue crack growth
data from Paper A was used for predictions of variable amplitude fatigue crack growth in
49
notched tensile specimens. The variable load consisted of load blocks repeatedly applied to
the specimens until final failure. The notched specimens were manufactured from the nickel-
based superalloy, Inconel 718, and the experiments were conducted at an elevated temperature
of +400�C and relative high loads. Mean crack growth rates were computed over each load
block in the notched specimens. Crack closure levels were computed, from the compact
tension specimens and applied in the prediction of the fatigue crack growth rates in the
notched specimens, by comparison of crack closure free experimental data obtained with the
so called constant KI,max-method and nominal data, i.e. fatigue data at a constant load ratio.
Also, numerical simulations of the closure level throughout the notched tensile specimens were
performed. Although the experimental data from the notched specimens showed rather large
a scatter it was concluded that variable amplitude crack growth rates can be predicted with
reasonable agreement using constant amplitude fatigue data. This conclusion is valid despite
that the load levels applied to the notched specimens were considerably above the ASTM
criterion for linear elastic fracture mechanics to be applicable. Furthermore, crack growth
rates in the inelastic notch affected zone could not be predicted by linear elastic fracture
mechanics, even if absence of crack closure was assumed.
Paper C: The effect of large scale plastic deformation on fatigue crack length measurement
with the potential drop method
One of several difficulties encountered in fatigue crack growth testing, as the constitutive
response moves from linear elastic to non-linear elastic-plastic, is to ensure proper crack
length measurements under the conditions which the measurements technique is to be used.
In this Paper A combined experimental and numerical investigation was conducted on the
effects of plastic deformation and material resistance on the relationship between the potential
difference and crack size, denoted calibration curve, or equation. The stainless steel 316L was
used at room temperature for investigating the limitations of the calibration curve. It is
concluded that the nominal calibration equation, obtained from the undeformed geometry,
can be used for fatigue crack length measurements at large plastic deformation. However, the
reference potential must, for reliable crack length measurements, be measured at the deformed
state and later adjusted at every major change of (plastic) deformation. Furthermore, the
major part (∼ 90 %) of the change in reference potential was concluded to be attributed to
the geometry change and only a minor part (∼ 10 %) to the resistivity change in the current
material. The scatter in the potential drop measured crack length, measured on a cycle by
cycle basis, was about 30 times smaller here compared to the compliance measured crack
length. Also, in situ potential drop sampling was possible, i.e., the test did not need to be
stopped for crack length measurements. Thus, the constitutive time dependent effects were
minimized.
50
Paper D: Fatigue crack growth experiments on specimens subjected to monotonic large scale
yielding
The conclusions in Paper C formed guidelines for further experiments where the fatigue
crack growth behaviour was investigated for through thickness long cracks at monotonic large
scale yielding. The stainless steel 316L was used for two different specimens; the compact ten-
sion specimen and an in-house developed edge cracked tension specimen. Constant amplitude
loading defined as the crack tip opening displacement, δ, was investigated. Finite element
computations on the coupling between the applied load, the crack length and δ, were con-
ducted. These coupling equations were used to control Δδ in situ in the experiments based on
the potential drop measured crack length. The Δδ was able to characterize and correlate the
fatigue crack growth load state for the present geometries and loads. Also the stress-intensity
factor range, ΔKI, could predict the growth rates due to large isotropic hardening at cyclic
conditions and absence of crack closure.
Paper E: Variable amplitude fatigue crack growth at monotonic large scale yielding experi-
ments on stainless steel 316L
Fatigue crack growth in stainless steel 316L was investigated for two different variable
amplitude block loads in the monotonic large scale yielding load range. The experiments
were performed on compact tension specimens. The crack loads were defined as crack tip
opening displacement, δ. The crack length was continuously monitored by potential drop
with reference potential from the deformed configuration. The force for the next load cycle
was in situ controlled using the measured crack length from the previous cycle. The control
program used numerically determined control functions that combined the measured crack
length and target Δδ to the desired force. The average growth rates over the block sequences
were determined from potential drop and visually measured crack lengths. The experimental
growth rates were also predicted, based on small scale yielding fatigue crack growth data from
closure free experiments, using a straight forward average procedure. The possibility to use
small scale yielding material data was explained the by the large isotropic material hardening
and closure free crack growth.
51
52
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