Department of Mechanical & Materials Engineering

The University of Western Australia

A PHENOMENOLOGICAL AND
MECHANISTIC STUDY OF FATIGUE
UNDER COMPLEX LOADING
HISTORIES

Yat Khin Wong

BEng (Hons.) – University of Western Australia
BSc – University of Western Australia

This thesis is submitted in fulfillment of the requirement for

the degree of Doctor of Philosophy (Ph.D.)

JUNE 2003
 Acknowledgements

I would like to thank my supervisors Associate Professor Xiao-Zhi Hu and Professor

Michael Peter Norton for their guidance, patience and support throughout the course
of my PhD studies. To my principle supervisor, Associate Professor Xiao-Zhi Hu, I
would like to thank him for being an excellent mentor, constantly encouraging and
supportive especially in times of need.

To Professor Michael Peter Norton, I dedicate my thesis in memory of him. His

departure is a great loss to us all.

Special thanks to Mr. Ian Hamilton, head of the Mechanical and Materials
Engineering workshop, for his assistance with the machining and preparation of test
specimens. I would also like to thank Mr. Nigel Laxton for providing training and
technical support in operating the Instron 8501 testing machine.

Thanks is also due to the Commonwealth Government of Australia for providing

financial support and funding through the Australian Postgraduate Award. Additional
financial support in the form of the William Lambden Owen scholarship provided by
the department of Mechanical and Materials Engineering is also greatly appreciated.

Finally, I would like to thank my beloved wife, Lyn, for her support and sacrifice over
the years of study. Without her constant motivation and encouragement, the
accomplishment of my studies would not have been possible.

i
 Synopsis

Over the years much work has been done on studying sequence effects under
multilevel loading. Yet, the underlying fatigue mechanisms responsible for such
interactions are not fully understood. The study of fatigue under complex loading
histories begins by investigating strain interaction effects arising from simple 2-step
loading sequences. Fatigue for all investigations were conducted under uniaxial push-
pull mode in strain-control. Fatigue is traditionally classified as either low or high
cycle fatigue (LCF and HCF respectively). The boundary for LCF and HCF is not
well-defined even though the fatigue life of LCF is typically dominated by crack
“initiation”, while for HCF, fatigue life is usually dominated by stable crack growth.
The terms LCF and HCF, apart from referring to the low and high number of fatigue
cycles required for failure, also bear little physical meaning in terms of describing the
state of fatigue imposed. As a result, conventional definitions of the two distinct
regimes of fatigue are challenged and a new method of classifying the boundary
between the two regimes of fatigue is proposed. New definitions are proposed and the
terms plastically dominant fatigue (PDF) and elastically dominant fatigue (EDF) are
introduced as suitable replacements for LCF and HCF respectively. PDF refers to the
condition of a material undergoing significant reverse plasticity during cyclic loading,
while for EDF, minimal reverse plasticity is experienced. Systematic testing of three
materials, 316 L stainless steel, 6061-T6 aluminium alloy and 4340 high strength
steel, was performed to fully investigate the cycle ratio trends and “damage”
accumulation behaviour which resulted from a variety of loading conditions. Results
from this study were carried over to investigate more complex multilevel loading
sequences and possible mechanisms for interaction effects observed both under 2-step
and multi-step sequences were proposed. Results showed that atypical cycle ratio
trends could result from loading sequences which involve combinations of strain
amplitudes from different fatigue regimes (i.e. PDF or EDF). Mean strain effects on
fatigue life were also studied. The objective of this study was to identify regimes of
fatigue which are significantly influenced by mean strains. Results indicated that
mean strains affected EDF but not PDF. 2-step tests, similar to those performed in
earlier studies were conducted to investigate the effects of mean strain on variable
amplitude loading. Again, atypical cycle ratio trends were observed for loading

ii
sequences involving combinations of PDF and EDF. It is understood that fatigue
crack growth interaction behaviour and mean stress effects are two dominant
mechanisms which can be used to explain cycle ratio trends observed. The
significance and importance of proper PDF/EDF definition and specification are also
stressed.

The study of fracture mechanics is an important component of any fatigue research.

Fatigue crack growth in 4140 high strength steel CT specimens, under conditions of
plane stress and plane strain were studied. In this investigation, the effects of R and
overload ratios were also studied for both plane stress and plane strain conditions.
Results indicate that differences in the point of crack “initiation” under both plane
stress and plane strain conditions decrease with increasing load range, while the
extent of crack retardation as a result of overloading, is greater under plane stress than
plane strain conditions. The extent of crack growth retardation increases with
decreasing R ratios and increasing overload ratios.

The final phase of this project involves the proposal of two practical models used to
predict cumulative “damage” and fatigue crack propagation in metals. The
cumulative “damage” model proposed takes the form of a power law and the
exponent which governs “damage” accumulation can easily be calculated by knowing
the failure life, Nf, for a given strain or load level. Predictions for the “damage” model
performed better when compared to other popular cumulative “damage” models. The
second model proposed predicts fatigue crack growth behaviour from known
monotonic and smooth specimen fatigue data. There are several benefits of having a
model that can predict fatigue crack growth from monotonic and smooth specimen
fatigue data: a) traditionally, engineers had to rely on expensive and time-consuming
crack propagation tests to evaluate and select materials for maximum fatigue
resistance, and b) monotonic and smooth specimen fatigue data are readily available.
The crack propagation model is proposed to alleviate the material selection process by
providing engineers a means to rapidly eliminate and narrow down selections for
possible material candidates. As a consequence, the number of materials required for
proper crack propagation studies is dramatically reduced, saving both time and
money. The proposition of such a model also bridges stress / strain-based approaches
and fracture mechanics fatigue analysis and in so, highlights the importance of

iii
smooth specimen testing. Crack growth predictions made by the proposed model
showed reasonably good agreement with experimental results and the model looks
like a promising tool for estimating fatigue crack growth behaviour in metals.

iv
Table of Contents
Acknowledgements...............................................................................................................i
Synopsis .............................................................................................................................. ii
Table of Contents.................................................................................................................v
Nomenclature......................................................................................................................iv
Chapter 1: Introduction ........................................................................................................1
1.1 Historical overview....................................................................................................1
1.2 Fatigue and factors affecting fatigue life ...................................................................3
1.3 Cyclic deformation in metallic materials...................................................................7
1.3.1 Cyclic stress – strain response ............................................................................7
1.3.2 Microstructural aspects of cyclic loading (dislocations) ....................................9
1.3.2.1 Slip character & Stacking Fault energies...................................................10
1.3.2.2 Dislocation structures (fcc metals with wavy slip systems - single and
polycrystals)...........................................................................................................11
1.3.2.3 Dislocation structures (fcc metals with planar slip systems - single and
polycrystals)...........................................................................................................14
1.3.3 Cyclic strain hardening .....................................................................................14
1.3.4 Cyclic strain softening ......................................................................................16
1.3.5 Cyclic saturation ...............................................................................................18
1.3.6 Strain hardening ................................................................................................18
1.3.7 Bauschinger effect ............................................................................................19
1.4 Fatigue crack “initiation” and propagation ..............................................................21
1.4.1 Stage I crack growth .........................................................................................24
1.4.2 Stage II crack growth ........................................................................................25
1.4.3 Crack “initiation” (micro-mechanisms involving dislocations) .......................28
1.5 Approaches to fatigue analysis ................................................................................31
1.5.1 Stress-Life approach .........................................................................................32
1.5.2 Strain-Life approach .........................................................................................34
1.5.3 Fracture mechanics approach............................................................................38
1.5.3.1 Linear Elastic Fracture Mechanics (LEFM) ..............................................40
1.5.3.2 Elastic-Plastic Fracture Mechanics (EPFM)..............................................47
Chapter 2: Literature Review.............................................................................................51
2.1 Effects of overloading and underloading.................................................................51
2.2 Short and long fatigue crack growth........................................................................62
2.2.1 Fatigue limit concepts .......................................................................................69
2.3 Mean stress and mean strain effects on fatigue life .................................................72
2.4 Cumulative fatigue and “damage” models ..............................................................76
Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing ..................................101
3.1 Introduction............................................................................................................101
3.2 Materials and specimen specifications...................................................................102
3.2.1 Materials .........................................................................................................102
3.2.2 Specimen specifications..................................................................................103
3.3 Tensile Tests ..........................................................................................................104
3.4 Fatigue Tests ..........................................................................................................106
Chapter 4: 2-Level Loading .............................................................................................119
4.1 Introduction............................................................................................................119
4.2 Materials and testing program ...............................................................................120
4.3 Experimental results...............................................................................................123
4.4 Discussion ..............................................................................................................143

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 4.5 Significance of proper PDF-EDF definition and classification .............................162
4.6 Conclusions............................................................................................................166
Chapter 5: Multilevel Loading.........................................................................................168
5.1 Introduction............................................................................................................168
5.2 Material and testing program .................................................................................168
5.3 Experimental results...............................................................................................170
5.4 Discussion ..............................................................................................................173
5.5 Conclusion .............................................................................................................176
Chapter 6: Mean Strain Effects........................................................................................177
6.1 Introduction............................................................................................................177
6.2 Materials and testing program ...............................................................................177
6.3 Experimental results...............................................................................................179
6.4 Discussion ..............................................................................................................182
6.5 Conclusion .............................................................................................................189
Chapter 7: Fatigue Crack Growth in CT Specimens .......................................................190
7.1 Introduction............................................................................................................190
7.2 Materials and testing program ...............................................................................192
7.2.1 Determination of KIC.......................................................................................194
7.2.2 Plane strain / stress crack growth tests with overloading ...............................195
7.3 Experimental results...............................................................................................199
7.3.1 Fracture toughness testing...............................................................................199
7.3.2 Plane strain fatigue crack growth without overloading ..................................200
7.3.3 Plane stress fatigue crack growth....................................................................201
7.3.4 Plane strain crack growth with overloading....................................................202
7.3.5 Plane stress crack growth with overloading....................................................205
7.4 Discussion ..............................................................................................................208
7.4.1 Effect of R ratio and OLR on fatigue crack growth........................................208
7.4.2 Plane strain vs plane stress crack growth........................................................211
7.5 Conclusion .............................................................................................................215
Chapter 8: Cumulative “Damage” & Fatigue Crack Propagation Modelling..................216
8.1 Introduction............................................................................................................216
8.2 An alternative cumulative fatigue “damage” model..............................................218
8.3 Crack propagation model based on smooth specimen data ...................................232
8.4 Conclusions............................................................................................................237
Chapter 9: Conclusions ....................................................................................................238
9.1 2-level loading .......................................................................................................239
9.2 Multilevel loading..................................................................................................240
9.3 Mean strain effects.................................................................................................241
9.4 Fatigue crack growth in CT specimens..................................................................241
9.5 Cumulative fatigue “damage” and fatigue crack propagation modelling..............242
9.6 Recommendations for future work ........................................................................243
Publications......................................................................................................................244
References........................................................................................................................245
Appendices – Published Papers .......................................................................................254

vi
Nomenclature
a = Crack length

b = Fatigue strength exponent

c = Fatigue ductility exponent

da = Change in crack length

dN = Change in number of cycles

D = Fatigue “damage”

E = Young’s modulus

E’ = Young’s Modulus corrected for plane strain

GI = Mode I driving force for fracture or energy release rate

GIC = Critical mode I driving force for fracture or energy release rate

J = J integral

KI = Mode I fracture toughness

KIC = Mode I critical fracture toughness

∆K = Fracture toughness range

∆Keff = Effective fracture toughness range

Nf = Number of cycles to failure

Nt = Transition life defining the boundary between low and high cycle fatigue

n = Strain hardening exponent

n’ = Cyclic strain hardening exponent

p = Damage exponent

rp = Monotonic plastic zone size ahead of crack tip

rpc = Cyclic plastic zone size ahead of crack tip

R = Stress ratio

U = Total system energy

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Wp = Potential energy

Ws = Surface energy

Y = Geometry factor

∆εb = Bauschinger strain

εa = Total strain amplitude

εea = Elastic strain amplitude

εpa = Plastic strain amplitude

∆ε = Strain range

εf’ = Fatigue ductility coefficient

σf’ = Fatigue strength coefficient

σa = Total stress amplitude

σar = Total stress amplitude adjusted for mean stress

∆σ = Stress range

σm = Mean stress

σUTS = Ultimate tensile strength

σYS = Yield strength

σCYS = Cyclic yield strength

σYSH = Cyclic hardened yield strength

σYSS = Cyclic softened yield strength

γs = Surface energy per unit area

γp = Plastic shear strain

τ = Shear stress

υ = Poisson’s ratio

v
Chapter 1: Introduction

Chapter 1: Introduction

1.1 Historical overview

Fatigue and fracture has been a problem for engineers since the earliest days of
civilisation. However, it was not until the 19th century, the time of early industrial
revolution, that studies on fatigue and fracture were conducted. The first study of
metal fatigue was conducted by a German mining engineer, W.A.J. Albert, who
performed repeated proof load tests on iron mine-hoist chains around 1829. Since
then, significant progress has been made in the study of fatigue and fracture.

The word “fatigue” originates from the Latin expression “ fati ga re”, which means
“to tire”. Over the years, the term fatigue has been a widely accepted term in
engineering to describe damage and failure of materials under repeated cyclic loading.
There are seven main types of fatigue: mechanical, creep, thermomechanical, sliding,
rolling, fretting and corrosion fatigue. Mechanical fatigue results from fluctuations in
applied stresses or strains. Cyclic loading in association with high temperatures,
results in creep-fatigue while a combination of cyclic loading and thermal fluctuation
causes thermomechanical fatigue. Repeated applications of load in conjunction with
sliding and rolling contact between material surfaces produces sliding and rolling
contact fatigue respectively, while fretting fatigue occurs as a result of pulsating
stresses along with oscillatory relative motion and frictional sliding between surfaces.
Corrosion fatigue is caused by fluctuating load in the presence of corrosive
environments.

An overview of the entire history, covering historical developments and major

advances in all areas of fatigue is beyond the scope of this section. Only contributions
pertaining to mechanical fatigue and those relevant to the contents of this thesis are
mentioned. It would be impossible to name and mention all researchers who have
made significant contributions to the understanding of mechanical fatigue. Those
whose work and contribution that are not discussed in this historical overview will be
mentioned in the relevant chapters to follow.

1
Chapter 1: Introduction

The first detailed research into fatigue was initiated in 1842 following a railway
accident in France. An early explanation for fatigue was the ‘crystallisation theory’,
which postulated that the cause of fatigue failure in materials resulted from
microstructural crystallisation. This theory remained unchallenged for several decades
until the work of Ewing and Humfrey in 1903 showed the development of slip bands
and subsequent fatigue cracks in polycrystalline materials. Work by A. Wöhler from
1852-1869 was one of the earliest systematic investigations of fatigue failure, where
full scale fatigue testing in torsion, bending and axial loading of railway axles were
conducted. His work led to the characterisation of fatigue behaviour in terms of stress
amplitude-life (S-N) curves and also to the introduction of the concept involving
fatigue endurance limit. In the later half of the nineteenth century, engineers such as
Gerber (1874) and Goodman (1899) began developing methods for fatigue design and
formulated ways to account for mean stresses. Differences in the elastic limit for
materials subjected to reversed loading were first observed by Bauschinger (1886),
and later in 1910, Bairstow investigated changes in stress-strain response during
cycling, subsequently identifying cyclic softening and hardening behaviour in metals.

Equations for the characterisation of S-N curves were first proposed by Basqin in
1910 and later refined by Coffin and Manson (1954) for strain-based characterisation
of fatigue. Investigations into damage accumulation models for fatigue were
performed by Palmgren (1924) and Miner (1945). Neuber, in 1946, studied the effects
of notch on monotonic and cyclic deformation, while Langer (1937) pioneered work
in variable amplitude fatigue.

Fracture mechanics, which involves the study of fracture and crack propagation, came
about after stress analyses by Inglis (1913) and energy concepts of Griffith (1921)
provided the mathematical tools for quantitative treatments of fracture in brittle
solids. However, these ideas could not be applied directly to characterise fatigue
failure of many metallic materials. In 1957, work by Irwin showed that stress
singularity in front of a crack tip can be defined by what he termed stress intensity
factor, K. This, in years to come, formed the basis of linear elastic fracture mechanics
(LEFM). In 1961, Paris, Gomez and Anderson proposed a power law which
characterises crack growth rate in terms of stress intensity factor, K. This later became

2
Chapter 1: Introduction

known as Paris Law. The next important contribution came in 1970, when Elber
showed that fatigue cracks could remain closed even when subjected to cyclic tensile
loads. From this, the concept of crack closure was born. Since then, Ritchie, and
Suresh, amongst many others, have intensified research in crack closure and its effects
on fatigue crack growth. Today, crack closure and its associated effects form the basis
for any mechanistic study of fatigue crack growth.

More recently, significant interest in fatigue research has centred around on short or
small fatigue crack growth. This problem was first identified by Pearson in 1975,
who observed that crack growth rates for short cracks are higher than those observed
for long cracks at the same stress intensity range. Crack growth, even at stress
intensity ranges below the threshold for long cracks has also been detected. Such
anomalous behaviour contradicts conventional LEFM theory and thus far, significant
progress has been made in characterising the types of short cracks and explaining this
unique behaviour. Miller, Brown, Suresh, Jono, Song, are just a few of the many
others who have made significant contributions to small crack studies over the past
two decades.

1.2 Fatigue and factors affecting fatigue life

Fatigue is one of the most common forms of engineering failure. Failure from fatigue
results from crack growth in a component subjected to cyclic loading. Figure 1.1
illustrates a typical constant amplitude loading with a sinusoidal waveform. Common
terminologies associated with fatigue waveforms include:

i. stress range, ∆σ = σ r = σ max − σ min (1.1)

∆σ
ii. stress amplitude, σ a = (1.2)
2

σ max + σ min
iii. mean stress, σ m = (1.3)
2

3
Chapter 1: Introduction

σ min
iv. stress ratio, R = (1.4)
σ max

Fig. 1.1 – Typical constant stress amplitude fatigue loading waveform.

Fatigue crack growth is commonly classified into three main stages (Fig. 1.2): 1)
crack “initiation”, 2) stable crack growth and propagation, 3) unstable crack growth
leading to complete fracture. These three stages are often studied using fracture
mechanics.

Stage (I) in Figure 1.2 refers to the crack “initiation” stage. If ∆K (fracture toughness
range) is less than some threshold fracture toughness, ∆Kth, (which is often assumed a
material constant), a crack cannot “initiate”. Therefore, one can consider ∆Kth as the
threshold value where if ∆K is > ∆Kth, a crack “initiates”, allowing for steady crack
growth to follow in the second stage. ∆Kth can thus be viewed as fracture mechanic’s
equivalent to the stress (∆σo) or strain (∆εo)-based fatigue limit, while ∆K is
equivalent to ∆σ or ∆ε depending on the fatigue approach used. The three approaches
to fatigue analysis, namely stress, strain and fracture mechanics will be discussed in
greater detail in the ensuing sections.

4
Chapter 1: Introduction

III

II
Log (da/dN)

∆Kth Log(∆K)

Fig. 1.2 – Three main stages of fatigue crack growth.

The second stage of fatigue crack growth is perhaps the most studied stage of all
fatigue research. This is when fatigue crack growth is steady and the growth rate
increases in a linear fashion, which is best described by the Paris Law,

da
= A(∆K)
n
(1.5)
dN

where A and n are material constants.

Stage (III) crack growth occurs when Kmax approaches the material critical fracture
toughness KIC. During this stage, crack growth is unstable and occurs at a rapid rate
leading to ultimate fracture.

Fatigue cracks typically “initiate” from regions of high stress concentrations. Grain
boundaries, triple points and inclusions are examples of typical crack initiation sites,
while macroscopic stress concentration areas include material discontinuities such as
changes in cross-sections, keyways, holes, internal cracks and irregularities caused by
machining. Factors that influence fatigue life are those which affect crack “initiation”
or propagation. The most obvious factors affecting fatigue lives are the loading
amplitudes and frequencies. Stress concentration regions such as those mentioned
previously also affect fatigue lives. The more severe the stress concentration factor

5
Chapter 1: Introduction

and the greater the number of stress concentration sites, the lower the fatigue life.
Exposure of a material to potentially harsh environments will also encourage crack
“initiation”. Environmental conditions such as temperature, acidity and humidity all
play significant roles in influencing fatigue lives. Exposure of metals to hydrogen for
example, causes metals to become brittle, which greatly increases the chance for
cracks to be “initiated”. It was discovered that, with time, small hydrogen atoms are
able to diffuse into the crystallographic structures of metals to reduce bonding
between the metallic ions, thus, reducing the strength of the material and its resistance
to crack “initiation” and propagation. Fatigue loading at elevated temperatures results
in a reduction in yield and tensile strength. Creep is introduced in conjunction with
the fatigue waveform to further decrease overall life. Acidic and corrosive
environments decrease fatigue life by eroding protective oxide layers, causing
material pitting and weakening of interatomic bonds, which introduces more crack
initiation sites and encourages crack initiation.

Fatigue lives have also been found to be highly dependent on grain size. Short crack
studies have shown that materials with small grain sizes are more resistant to crack
“initiation”. However, in the case of long cracks, materials with larger grain sizes
exhibit slower crack propagation rates as larger grains provide a more uneven and
heavily faceted crack surface. As a result, life enhancing factors such as roughness
induced closure become more effective. Thus, for optimal fatigue resistance, materials
are often heat treated to yield small surface grain and large internal grain structures.

The final factor which influences fatigue life is the processing techniques used in
manufacturing. Components produced from hot rolling, forging and extrusion have
directional properties resulting from grain orientation caused in the process of
manufacturing. Fatigue life is enhanced if a component is loaded in the grain-oriented
direction but reduced if loaded in the transverse direction. Manufacturing techniques
also control the surface finish and the amount of air pockets and shrinkage cavities
present in the final product. For example, quench cracks will result, if the material is
quenched too quickly. Component surfaces are usually heat treated to reduce the
presence of harmful residual stresses. Compressive residual stress enhances fatigue
life while tensile residual stress is detrimental to fatigue life. Welding also produces a
heat affected zone, which if untreated may be detrimental to fatigue life. Cold

6
Chapter 1: Introduction

working, shot peening, cold rolling are processes specifically designed to induce
compressive residual stress with the effect of enhancing fatigue life. Figure 1.3 below,
summarises the factors affecting fatigue life of a component.

Load Spectrum
(Sign, magnitude,
rate, history)

Microstructure Environment
(Grain size, (Temperature,
texture, defects) corrosive
medium)

Fatigue
Life

Geometry of Processing
component (Deformation
(Surface finish, history,
notches, manufacturing)
defects)

Fig. 1.3 - Factors affecting fatigue life of a component.

1.3 Cyclic deformation in metallic materials

1.3.1 Cyclic stress – strain response

When a metal is subjected to cyclic deformation, characteristic stress-strain response

curves are observed depending on the material tested, its initial condition (annealed or
work hardened) and the manner in which fatigue is controlled (stress or strain). Figure
1.4 summarises the four most common stress-strain response curves that one would
observe for constant amplitude strain and stress-controlled fatigue.

7
Chapter 1: Introduction

Fig. 1.4 – Cyclic stress-strain response curves for constant amplitude strain and stress-
controlled fatigue.

Under strain-controlled fatigue, two possibilities might occur: cyclic strain hardening
(Fig. 1.4a); or cyclic strain softening (Fig. 1.4b). Cyclic strain hardening occurs when
stress amplitudes increases with increasing numbers of cycles. In general, the rate of
hardening progressively diminishes and a quasi-steady state of deformation (cyclic
saturation) is reached with continued cyclic straining. The opposite occurs for cyclic
strain softening, where the stress amplitudes for hysteresis loops decrease with
increasing numbers of cycling. It is common for cyclic saturation to be attained within
half-life in most cases of fatigue loading. As a result, the calculation of fatigue
parameters (such as fatigue strength coefficient, σ’f, fatigue ductility coefficient ε’f,
etc) is usually based on mid-life hysteresis loop data. Fatigue parameters for both
stress and strain-based approaches are discussed in sections 1.5.1 and 1.5.2
respectively.

8
Chapter 1: Introduction

1.3.2 Microstructural aspects of cyclic loading (dislocations)

Explaining the stress-strain response of fatigue loading on the basis of microstructure

is important since microstructural processes underlie and influence the examination,
interpretation, and understanding of macroscopic behaviour. The sequence of
processes during fatigue of metallic materials can be summarised as follows:

Cyclic deformation Fatigue “damage”

Hardening / Softening Strain localisation

Saturation Crack “initiation” (surface)

(microstructural changes in
the bulk)
Crack propagation
(localisation trans-,
Strain localisation intergranular processes)

Basic dislocation Fatigue failure mechanism

mechanisms

The basic mechanism for cyclic deformation involves irreversible dislocation

movement with the microstructure. Dislocation is defined as a flaw in the lattice
structure of a material which causes slip to occur in crystallographic planes when
stress is applied to the lattice. Figure 1.5 illustrates two dislocation types (screw and
edge) amongst other microstructural features commonly found in polycrystalline
metals.

Fig. 1.5 – Structure of a polycrystalline metal. Source: [1].

9
Chapter 1: Introduction

1.3.2.1 Slip character & Stacking Fault energies

Slip character defines the type of dislocation arrangement formed during cyclic
loading. There are two primary types of slip character, namely: wavy and planar slip
character. Under the situation where cross slip occurs, atoms in ABCABC stacking
sequence is disturbed as the gliding of dislocations leads to a shift of a plane of atoms
in the <110> direction. Resistance to slip results in a dissociation of the primary
dislocation into two partial dislocations and the elastic interaction of the partial
dislocations leads to a repulsion that is balanced by the energy of the stacking fault
existing between the partial dislocations (stacking fault refers to the disruption of the
orderly ABCABC stacking arrangement e.g. B atoms in C positions). This
disturbance to the perfect lattice structure yields an energy increase per unit area
which is defined as stacking fault energy.

There is a link between the stacking fault energy and the slip character of a material.
Stacking fault energy is affected by the width of splitting up (or the distance between
corresponding partial dislocations) in that the smaller the width, the higher the
stacking fault energy (i.e. the smaller the distance between partial dislocations, the
higher the repulsion and the higher the energy of the stacking fault required to balance
this repulsion). In order for cross slip (screw dislocation) to occur, partial dislocations
must recombine to form a complete dislocation and the process of recombination is
made easier if stacking fault energy is high or the dissociation distance is small.
Hence, high stacking fault energy is typically associated with wavy slip character
(cross slip easy) while low stacking fault energy is associated with planar slip
character. The general idea is that friction effects impede the recombination of partial
dislocations causing planar slip (difficult cross slip) to occur. This correlation works
well for pure metals but not for alloys.

Figure 1.6 illustrates the dislocation arrangement in cyclically deformed fcc metals as
a function of slip character and number of cycles to fracture.

10
Chapter 1: Introduction

Fig. 1.6 – Dislocation structures in cyclically deformed fcc metals. Source: [2].

1.3.2.2 Dislocation structures (fcc metals with wavy slip systems - single
and polycrystals)

Single crystals: The dislocation arrangement for wavy slip systems during cyclic
saturation is highly dependent on the loading amplitude. Cyclic loading with small
plastic amplitudes yields arrangements of edge dislocation dipoles caused by single
slip. Dislocations often agglomerate into bundles or veins which are separated by
regions of low dislocation density (channels) (see Figure 1.7 or region A in Figure
1.8). At low plastic amplitudes, cyclic hardening is due almost entirely to the
accumulation of primary edge dislocations. Cyclic saturation occurs when a dynamic
equilibrium is achieved between bundles of edge dislocations and the surrounding
matrix piled by screw dislocations. During the initial phases of cycling, dislocations
are produced which accumulate on the primary glide plane. Fully reversed cyclic
strain produces approximately equal amounts of positive and negative edge
dislocations which are attracted over small distances to form dislocation dipoles. This
process of positive and negative dislocation attraction can also be termed trapping.
Only edge dislocations are likely to form dipoles since screw dislocations have a
tendency of cross slipping, which is promoted by materials with high stacking fault
energies, mutually annihilating each other in the process. After prolonged cycling, the
accumulation of dislocations occurs predominantly in the form of mutually trapped
edge dislocations, typically called veins, bundles or loop patches, making up
approximately 50% by volume of the matrix shown in Figure 1.7a. Dislocation veins
contribute to the cyclic hardening during the initial stages of fatigue by partially
impeding dislocation motion on the primary slip system. Matrix veins are known to
accommodate only small plastic strains of the order of 10-4 and undergo

11
Chapter 1: Introduction

microyielding. Cyclic saturation and deformation within the matrix occurs by a ‘flip-
flop’ mechanism involving dislocation loops moving back and forth produced by jogs
during cross slip of screw dislocations during cyclic strain [3, 4]. More detailed
discussions on cyclic response and its associated dislocation structures and
mechanisms can be found in [5-9].

Fig. 1.7 Schematic representation of the dislocation arrangements in (a) a matrix

structure and (b) a persistent slip band. Source: [10].

Fig. 1.8 - Schematic diagram showing different regimes of the saturation stress-strain
curve. Source: [10].

Persistent slip bands otherwise known as PSBs are composed of a large number of
slip planes formed during fatigue loading involving large plastic strain amplitudes. i.e.
region B in Figure 1.8. The dislocation structures in PSBs are considerably different
to that observed in a matrix. It consists of dislocation veins that only occupy 10% by
volume and are arranged into wall-like configurations (Fig. 1.7b).

12
Chapter 1: Introduction

PSBs can support high plastic shear strains of the order of 0.01 and undergo
macroyielding. A dynamic equilibrium between dislocation multiplication and
annihilation has been identified [11] as the saturation mechanism for fatigue involving
intermediate degrees of plastic straining and PSB formation. Dislocation walls (veins)
and dislocation poor channels take part in plastic deformation by marcoyielding of
PSB. Edge dislocations bow out from walls, traverse the channels, and penetrate
partially into the opposite wall leading to the existence of screw dislocation segments
that glide along the channels. See Figure 1.7b. During cyclic deformation, edge
dislocations will traverse to and fro between walls with constant renewal of
dislocations caused by a dynamic process of dislocation formation and annihilation.
Annihilation occurs by climbing of edge dislocations of opposite signs in the wall
structures of PSBs or by cross slip between screw dislocations. Therefore, bundles
and walls consist primarily of edge dislocations.

At higher loading amplitudes, secondary slip occurs and the multiple slip contributes
to the formation of labyrinth and cell-like dislocation structures (region C in Figure
1.8). Cell size has been reported [6, 8] to decrease with increasing plastic strain
amplitude and saturation cell size is independent of the initial condition of the
material (i.e. cell structures during cyclic saturation the same for annealed or cold
worked materials). Cell size in wavy slip systems also increases as the test
temperature increases.

Polycrystals: The microstructural processes that take place in polycrystals is basically

the same as that discussed for single crystals in that at low amplitudes, loose veins are
formed; in an intermediate range, veins with PSBs are formed and at high amplitudes,
cells are formed. Even when cyclic saturation is reached, minor changes to dislocation
structures have been found.

It is important to realise that dislocation arrangement and cell size is determined

primarily by the plastic strain amplitude and not the flow stress. Note that the above
microstructural processes described results in cyclic hardening which will be
discussed in the next section.

13
Chapter 1: Introduction

1.3.2.3 Dislocation structures (fcc metals with planar slip systems -

single and polycrystals)

In planar slip materials, dislocation band structures are formed with dislocations
staying in a planar array with difficult cross slip. Planar slip systems are promoted by
high friction stress, high shear modulus and a high concentration of solved foreign
atoms (increased resistance to partial dislocation recombination). As a generalisation,
cyclic deformation of alloys in fcc metals with planar slip is similar to that of the base
metal. In planar slip materials, dislocations are localised in discrete bands between
which relatively dislocation-poor regions exist. The bands consist of mainly edge
dislocations of the primary slip system parallel to the primary slip plane. At higher
amplitudes of loading, dislocation density increases and the distance between
dislocation bands increase.

Li and Laird [7, 8] have reported that for planar slip materials, dislocation structures
for annealed and cold worked materials differ significantly after cycling to 20% of
fatigue life. In annealed condition, dislocations are contained in discrete bands with
low interband dislocation density while for cold worked materials, fewer discrete
bands and high interband dislocation (tangled) density can be found. Similar to wavy
slip cell structures, band spacing increases and interband dislocation density increases
as plastic strain amplitude increases. Repeated loading of a metal between fixed limits
of stress or strain will produce either hardening or softening, depending on the initial
condition of the metal and the load amplitude.

1.3.3 Cyclic strain hardening

In the rapid hardening stage, the dominant features are bundles of dislocations which
are separated by regions relatively free of dislocations. Through repeated straining,
dislocations are accumulated into bundles via a mutual trapping mechanism whereby
screw dislocations are largely annihilated by cross-slip. Consequently, the majority of
dislocations within these bundles have been identified as primary edge dislocations
which have been trapped by other primaries over part of their length to form dipoles
and multi-poles. These bundles extend normal to the primary glide plane to form

14
Chapter 1: Introduction

walls of dislocation, which becomes fragmented and discontinue with fatigue. As

hardening progresses, dislocations are removed from active participation in the
straining by absorption into bundles with new dislocations generated and bundles
started by chance pinning in the previously clear regions. Slip distance is
subsequently diminished and with the number of obstacles increasing (i.e. the density
of bundles increases and the spacing between the bundles decreases), peak stress
required to produce the applied strain increases giving rise to cyclic hardening. As the
stress increases, slip on secondary systems may occur, resulting in the fragmentation
of the walls of primary dislocations, eventually developing into a three-dimensional
cell structure. Hence, hardening at higher strain amplitudes causes dislocations
bundles to become fragmented into shorter and shorter lengths, causing the density of
dislocations on secondary slip systems to increase.

It is not likely at early stages of hardening that point defects and their clusters will
have accumulated sufficiently to contribute greatly to the flow stress. Therefore, in the
early stages, the theory of flow stress will rely heavily on long-range interactions.
Point defects and clusters will only become significant in increasing flow stress as
concentrations build up through fatigue.

In summary, higher strain amplitudes intensify secondary slip, causing a cellular

dislocation structure to form, while low amplitudes simply yield dislocation structures
consisting of a collection of bundles with spacing characteristic of the strain
amplitude used. Cyclic hardening also occurs when it is easy to generate dislocations
i.e. dislocation sources are only moderately blocked by solute atoms and can be
activated once these barriers are overcome, producing a sudden strain burst. This
means that only low stress levels are required to generate the required number of
dislocations to carry the applied strain.

Cyclic hardening of well-annealed single crystals is caused mainly from the

accumulation of primary dislocations. In wavy slip systems, the formation of
dislocation bundles is a consequence of mutual trapping of primary edge dislocations.
The bundles and veins are obstacles to further plastic deformation because they
partially impede dislocation motion on the primary slip system. Dislocation density
within veins increase and the number of veins per unit volume increase with cyclic

15
Chapter 1: Introduction

numbers, giving rise to pronounced hardening. Therefore, an increase in dislocation

density increases tangling which in turn increases friction force and hence the flow
stress required to overcome these barriers. High amplitude loading increases
dislocation density, which causes rapid hardening as dislocations become tangled. In
the case of wavy slip materials, these dislocations tangle up and eventually link to
form cell structures. Hardening is due to the complex interaction of primary
dislocations and the tangling associated with secondary slip.

Solute atoms are known to have a hardening effect by blocking moving dislocations.
Saturation stress amplitude increases with increased strain amplitude and the rate at
which saturation is reached also increases with increasing strain amplitude. Increasing
test temperatures also increases the rate of hardening /softening and reduces final
saturation hardness for a given strain amplitude [12, 13].

Stacking fault energies are known to affect the hardening rates observed for Cu-Al
alloys. It has been observed that the rate of hardening decreases as stacking fault
energies decrease. Hardening rates are also increased when crystals are oriented for
multiple slip.

1.3.4 Cyclic strain softening

There are two ways to explain cyclic softening. Firstly, cyclic softening can result
from a decrease in dislocation density through dislocation annihilation and change in
dislocation structure. On the other hand, cyclic softening can also be explained by
using the concept of ‘dislocation starvation’ proposed by Li and Laird [7, 8].

According to Li and Laird, cyclic softening occurs for polycrystals due to ‘dislocation
starvation’ – i.e. the inability or difficulty in generating dislocations to carry the
applied strain in the first few cycles. The initial stress applied is high in order to
generate enough dislocations to carry the strain imposed. Dislocations are required to
carry strain – the more mobile the dislocations, the better they can carry the applied
strain. One cause of dislocation starvation is the prevention of a dislocation source
from operating i.e. obstruction by solute atoms. Dragging the dislocations from their

16
Chapter 1: Introduction

associated solute atoms can activate a source. During cycling, a portion of solute
atoms lying in the dislocation path is carried to two destinations between which the
mobile dislocations glide back and forth. Dislocations with low mobility will get
trapped in these destinations while the rest of the remaining dislocations will continue
gliding in the region with lower solute content and new dislocations will be generated
in their place i.e. more strain is carried with more dislocations and there are less
solutes to obstruct movement, resulting in softening. Essentially, there will be more
mobile dislocations as the less mobile dislocations are weeded out.

At high amplitude loading for 316L stainless steel [7, 8], changes in dislocation
structures were observed, where regularly arranged dislocations are cut by cross slip
into shorter loops to form dislocation tangles with reduced interband distance, but
overall planar slip was still preserved.

The main principle in cyclic softening is the reversion of initial deformation (e.g. cold
worked) structure to one which is characteristic of the cyclic test conditions.
Softening may not be complete if the reversed plastic strain amplitude is small and
when cross slip is difficult (i.e. planar slip mode). Incomplete softening at low
amplitudes in wavy slip mode materials is due to a required change in the type of
dislocation structure (i.e. wavy slip mode materials work hardened by high strain
fatigue or unidirectional deformation formed cell structures which when cycled at low
amplitudes must be reverted back to a dipole structure for complete softening to take
place). Softening for wavy slip materials at high amplitude requires only a change in
the “degree” or “intensity” of the structure and not of its kind. Cell structures formed
by unidirectional deformation or by high amplitude cycling need only undergo
rearrangement and slight changes in dislocation density to achieve complete softening
when cycled at lower amplitudes. Wavy slip mode materials also allow easy cross slip
and thus easy dislocation rearrangement (motion is in three dimensions). Little or
partial softening occurs in planar slip mode materials (even at high amplitudes)
because cross slip is difficult and infrequent. Partial softening occurs only due to
reversed plastic strain providing some chance for dislocations to find annihilation or
low energy sites, even though motion is constrained to two dimensions. Some
experiments show that planar slip mode materials may have higher strain softening
rates than wavy slip mode materials even though wavy slip allows for easy cross slip

17
Chapter 1: Introduction

and dislocation motion. This is because in some wavy slip mode materials,
dislocations are heavily jogged and tangled which produces severe drags in cross
slipping screws inhibiting the annihilation of dislocations tremendously.

As a result, cyclic softening can occur through dislocation generation (i.e. when more
dislocations are required to carry the applied strain but material finds difficulty in
generating dislocations due to ‘dislocation starvation’, or through dislocation
annihilation where reduced dislocation density reduces the resistance to dislocation
movement.

1.3.5 Cyclic saturation

Several models have been proposed to describe the micro-structural aspects during the
saturation-hardened fatigue state. The most important feature of any model is that it is
able to provide a mechanism whereby non-hardening reversible plastic strain can
occur and that saturation flow stress must be maintained. Of the numerous models
presented, the most realistic model appears to be one in which dipole flipping, cell-
shuttling, and point defect hardening all play a role. At saturation, the applied strain
amplitude is accommodated by dipole flipping and the shuttling of dislocations
between bundles at low amplitudes, and between cell walls at higher amplitudes.
Flow stress is provided by some friction stress encountered by the dislocations as they
traverse the cell volume. Point defects and their clusters also produce friction stress.
Therefore, saturation flow stress is comprised of the stress required to flip the average
dipole and the long- and short-range stresses encountered by the shuttling
dislocations.

1.3.6 Strain hardening

Strain hardening works on the same principle as that described for cyclic hardening in
section 1.3.3. Strain hardening during monotonic tensile tests occurs when all
ductility in the material is exhausted, increasing the need for more dislocations to be
generated to carry the increasing strain. This causes a rapid build up of dislocations,

18
Chapter 1: Introduction

thereby increasing dislocation density to such an extent that severe dislocation

tangling/jogging occurs, which in turn causes flow stress to increase, hence
hardening.

1.3.7 Bauschinger effect

The Bauschinger effect refers to a phenomenon of reduced yield stress in the phase of
loading when the direction of initial plastic straining is reversed. This phenomenon,
observed by Bauschinger in 1886, forms the basis of many strengthening theories and
its understanding is essential for the development of constitutive models for complex
cyclic deformation. Fundamentals of the Bauschinger effect are also used to
rationalise work hardening phenomena, mean stress relaxation, and cyclic creep.
Recent studies [14, 15] have also shown that the Bauschinger effect has a strong
influence on the level of plasticity-induced crack closure, implying that it too has a
prominent role to play in defining the mechanistic behaviour of fatigue crack
propagation in metallic materials.

A common method of quantifying the Bauschinger effect is by evaluating reverse

strain. Reverse strain or Bauschinger strain (∆εb), is defined as the magnitude of
additional strain after load reversal required to make the reverse yield stress equal in
magnitude to the maximum flow stress in the forward phase of deformation (Fig.
1.9b). From Figure 1.9b, cyclic softening can be seen as characteristic of the
Bauschinger effect since the forward hardening segment ABC, when extrapolated to
point C’, having an accumulated strain equal in magnitude to F for the reverse loading
phase, shows that the magnitude of flow stress at point C’ is larger than that observed
at point F. The difference in flow stress between C’ and F is thus defined as the
Bauschinger stress, ∆σb.

Another method of quantifying the Bauschinger effect is the modified Bauschinger

Effect Factor, BEF, which is defined as the ratio of yield stress upon load reversal to
the maximum yield flow stress in the direction of forward straining [16]. This index,
with values ranging from 0 – 1, is regarded as a direct measure of the yield lowering
effect, which is a consequence of the Bauschinger phenomenon.

19
Chapter 1: Introduction

Fig. 1.9 – (a) Stress-strain curve for fully reversed loading, (b) Stress vs. accumulated
strain plot [10].

Mechanistic models describing the Bauschinger effect have been oriented around
dislocation-based arguments. In polycrystalline materials, dissolution of cell walls or
sub-boundaries upon stress reversals has been deemed responsible for the
Bauschinger effect [17]. In addition, long-range internal stresses arising from strain
incompatibility between PSB walls and channels can also lead to easier reverse flow
in materials which form well defined PSBs. For particle-hardened alloys, Orowan
[18] suggested that the interaction of dislocations with strengthening particles could
explain the Bauschinger effect. In the case of alloys containing precipitates which are
coherent with the matrix and easily sheared by dislocations, the magnitude of flow
stress in reverse is comparable to that for the forward direction, implying that there is
little contribution to the flow stress from the internal stresses. On the other hand,
hardened alloys containing precipitates which are incoherent with the matrix and
impenetrable by dislocation shear, impede dislocation motion and thus act to raise the
yield stress during the forward phase of straining. Subsequent reduction in reverse
yield stress upon load reversal has been attributed to plastic relaxation arising from
either the removal of Orowan loops by the formation of secondary dislocations and
prismatic loops of primary Burger vectors [19], or by shrinkage of Orowan loops by
climb via pipe diffusion [20]. Several authors have also suggested [10, 21] that the
Bauschinger effect is caused by the back stress associated with dislocation pileups at
barriers such as precipitates, grain boundaries and other dislocations which facilitates
deformation in the direction opposite to initial pre-strain.

20
Chapter 1: Introduction

1.4 Fatigue crack “initiation” and propagation

The development of a fatigue crack is traditionally classified into three main phases:
i) crack initiation ii) stable crack growth and propagation iii) unstable crack growth
leading to fracture. The definition of crack “initiation” has evolved with the
advancement of crack detection methods. This limit of crack detection has improved
over the years from several millimetres to the order of 0.1 µm or less. Since a fatigue
crack must be of a certain length before it can be observed, some microcrack growth
will always occur before the measured cycles to crack “initiation” is detected. Miller
[22, 23] amongst others [24-26] studying the growth of short fatigue cracks have
shown that the “initiation” phase commonly associated with the first phase of fatigue
does not exist in polycrystalline materials and structures, since microscopic cracks as
small as several microns in size have been found to propagate during the onset of
fatigue loading. It has been shown that irregardless of how well prepared a specimen
surface is, micro irregularities on a highly polished surface still exists. In certain cases
of highly polished surface, fatigue “initiation” have been reported [27] to occur from
transverse surface grooves, inclusions or inherent defects which were all greater in
depth than the maximum peak-valley heights of the surface condition measured by a
Talysurf scan. As a result, it is obvious that surface micro-notches, inclusions or
inherent microscopic defects, all provide avenues for immediate crack growth.
Henceforth, the term fatigue crack “initiation” will be quoted in inverted commas to
remind the reader that such a phase though often quoted in textbooks and in literature,
does not actually exist in reality. References to the term “initiation” are still made
throughout the context of this thesis to facilitate in the understanding of general
concepts of fatigue crack growth. The reader should be reminded that the word
“initiation” is a loosely coined termed used simply to refer to the onset of a detectable
crack and not the commencement of crack growth. Details involving short crack
growth will be discussed later.

Fatigue undergoes microscopic changes, which can be described as follows. Micro-

changes in the bulk material occur as a result of cyclic deformation leading to strain
localisation. This strain localisation is caused by irreversible dislocation movement
(slip caused along crystallographic planes when stress is applied) and with time, the

21
Chapter 1: Introduction

dislocations agglomerate into bundles at critical stress or strain levels causing strain
localisation to occur, thus forming a thin lamellae of persistent slip bands also known
as PSBs (Fig. 1.10).

Fig. 1.10 – Schematic illustration of PSBs with extrusion and intrusions.

A free surface is usually the preferred area for the “initiation” of fatigue cracks. The
formation of extrusions and intrusions (within slip bands) as a result of surface
deformation during fatigue allows for cracks to propagate at an angle 450 (plane of
maximum shear) to the tensile stress and the surface. This initial crack propagation is
identified as stage I crack propagation and usually penetrates only a few tenths of a
millimetre into the specimen. The lower the fatigue stress/strain amplitude, the more
deeply stage I cracks tend to penetrate. Fatigue crack “initiation” and crack growth
during stage I, occurs by slip plane cracking. Fracture features of stage I fatigue in
general resemble those observed for cleavage. Stage I crack propagation is observed
in high cycle fatigue while not commonly observed in low cycle fatigue and is
strongly affected by environments and microstructures that tend to concentrate slip
and influence slip reversibility. Figure 1.11 shows the extrusion and intrusion marks
formed on the surface of a specimen after 50,000 cycles while Figure 1.12 shows the
fracture surface with slip bands formed during stage I of fatigue fracture. Stage I
crack growth in polycrystalline materials involve many individual slip band cracks
which eventually link up to form a dominant single crack at about the same time stage
II crack growth begins. In general, the higher the amplitude of the fatigue load, the
greater the multiplicity of crack “initiation”. Many such cracks become isolated and
stop propagating after the development of the main fatigue crack has occurred.

Following stage I crack “initiation”, a fatigue crack will propagate at right angles to
the direction of the tensile stress and this propagation phase is known as stage II crack
growth. During this phase, a fatigue crack advances by a certain distance with each
load cycle, leaving characteristic features such as fatigue striations. On a macroscopic

22
Chapter 1: Introduction

level, beach marks, which are parallel to striations are also characteristic features of
Stage II propagation. Stage II crack growth, which often occurs in a transgranular
fashion is influenced more so by stress amplitudes than mean stress or microstructural
features.

Fig. 1.11 – Surface morphology Fig. 1.12 – Fracture surface showing slip
showing extrusions and intrusions. band formation during stage I crack growth.
Source: [1]. Source: [1]

The transition between Stage I and Stage II crack propagation during fatigue is shown
in Figure 1.13. Stage I or microstructurally short cracks as they are commonly termed,
propagate by shear mode and are typically of the order of one or a few grains in
length. Stage II cracks on the other hand propagate by mode I fracture. The crack
length at the end of transition between stage I and stage II crack propagation is often
termed as transition crack length, at.

at

Fig. 1.13 – Transition from stage I to stage II crack propagation.

23
Chapter 1: Introduction

1.4.1 Stage I crack growth

Fractographic examination of stage I cracks is limited due to complications involved

when observing cracks on such minute scales. Studies conducted in compressive
loading amplitudes showed that stage I cracks failed to “initiate” despite having
amplitudes comparable to those producing failure in a tension-compression mode.
This lead to conclusions that stage I crack growth under normal fatigue stressing
modes is a consequence of reversed plastic deformation at the crack tip. Models for
crack extension are the plastic blunting model (applicable to materials with high
stacking fault energies and easy cross slip) and the unslipping model (applicable to
materials with low stacking fault energies and difficult cross slip). This latter model
would also apply to materials subjected to fatigue in torsion where no tensile stresses
act across the primary slip planes. Both models are illustrated in Figure 1.14.

Fig. 1.14 – Possible models for stage I crack growth. Left: plastic blunting model;
Right: unslipping model. Source [13].

The main locations for crack nucleation remain to be at either slip steps between
extrusions/matrix interface or at intrusion sites (Fig. 1.10), all of which are regions of
high stress concentration. Figure 1.15 illustrates various possible crack “initiation”
sites commonly found in fatigue cracking.

24
Chapter 1: Introduction

i. transgranular PSB induced surface

ii. intergranular cracks
iii. surface inclusion or pore

Defects
iv. inclusions
v. grain boundary voids
vi. triple point grain boundary Commonly observed at
intersections high temperatures

Fig. 1.15 – Possible sites for crack “initiation” due to defects.

The mechanism commonly adopted for environmentally assisted crack nucleation and
growth (Fig. 1.16) is that proposed by Thompson et al. [28]. Tension causes slip steps
to form and the exposure of these fresh surfaces to oxygen causes the formation of
oxide layers. During the compression phase, the oxide layer is destroyed thus,
absorbing oxygen into the body of the crystal in the form of dissolved atoms. As the
cycle repeats more oxygen is absorbed and diffuses along the slip bands causing a
weakening effect in the PSBs which lead to eventual cracking. It is important to note
that oxygen or any environmental factors assist rather than directly cause crack
nucleation.

Fig. 1.16 – Environmentally assisted crack nucleation.

1.4.2 Stage II crack growth

Fatigue cracks propagate predominantly in a ductile manner, involving extensive

plastic deformation in the vicinity of the crack tip. Striations (Figs. 1.17 and 1.18) are

25
Chapter 1: Introduction

a common feature of stage II fatigue crack growth with cracks often propagating on a
plateau on different elevations with respect to each other. Fatigue crack propagation
in polycrystalline materials occur with the crack front being locally subdivided onto
several separate planes resulting in the formation of microscopically visible striations
consisting of many parallel crack planes as shown in Figure 1.17.

Fig. 1.17 – Schematic illustration of fatigue striation orientation to crack growth

direction. Source: [1].

An important conclusion to be drawn from this is that stage II crack growth is a

repetitive process. At maximum stress, yielding occurs to the extent of the plastic
zone, rp. Reversal of the strain forces the crack faces together, and the new surface
created during tension is not completely rehealed by reversed slip (in an oxidising
environment, reversed slip may be completely suppressed). Much of the slip during
compression occurs on new planes and the crack tip takes on a folded appearance with
“ears” causing the formation of upsetting folds which stand out subsequently as
striation patterns. After completion of the compressive half-cycle, the crack tip is re-
sharpened and the sequence begins again. This model of stage II crack growth is
called the plastic blunting model proposed by Laird and Smith [29] and is illustrated
in the Figure 1.19.

Not to be confused with striations which result from small crack advancement during
fatigue, beach marks are formed due to changes in loading and/or due to changes in
the crack tip environment and the surface layers which subsequently form.

26
Chapter 1: Introduction

a. unstressed state
Fatigue striations b. small tensile stress
c. maximum tensile
stress
d. small compressive
stress
e. maximum
compressive stress

Fig. 1.18 – Fatigue striations. Fig. 1.19 – Plastic blunting model

Source: [2]. for stage II crack propagation.

The intensity of loading and the ease of slip deformation affects the variation in
profile. This leads to the important conclusion that maximum load is more important
in determining the crack propagation velocity than the amplitude of loading. Apart
from the plastic blunting model proposed by Laird, Neumann [30] also proposed a
sliding-off mechanism to describe crack growth and extension. With this model,
Neumann sees the crack tip as a ‘V’ groove with a constant angle. Crack progression
occurs as the crack tip is widened during the crack opening phase and crack closure
occurring by slip reversal. This sliding-off mechanism illustrating crack extension and
propagation is seen in Figure 1.20.

Fig. 1.20 – Sliding-off mechanism for stage II crack propagation.

Transition from stage I to stage II crack in uniaxial loading is illustrated in Figure

1.15. Short crack studies have shown that the difficulty in translating between stages
is what provides the material fatigue resistance. In high cycle fatigue (usually > 105),

27
Chapter 1: Introduction

stage I shear-type crack dominates lifetime whereas in low cycle fatigue stage II crack
growth dominate. Cracks can either grow away from the surface (CASE B type
cracks) or parallel to the surface (CASE A type cracks) which is illustrated in Figure
1.21.

Case A Case B

Identical stress-strain field but different crack growth directions with respect
to the surface (shown shaded)

Fig. 1.21 – Schematic illustration of Case A and Case B crack growth. Source: [31].

In many engineering situations, e.g. torsion of shafts or non-proportional loading, it is

the shear-type crack that can dominate over the entire lifetime of the component. Type
A cracks are commonly found in torsion while type B cracks can be found in uniaxial,
plane strain and equibiaxial loading. All fatigue cracks grow either as stage I or stage
II types [32].

1.4.3 Crack “initiation” (micro-mechanisms involving dislocations)

In section 1.4.2, the plastic blunting and unslipping models were discussed,
illustrating the possible mechanisms for slip band crack “initiation” or stage I crack
growth. In this section, crack “initiation” micromechanisms involving dislocations
[13] are discussed. As mentioned previously, there are four main crack nucleation
sites: slip bands, grain boundaries, inclusions and twin boundaries. Fatigue cracks
have been observed on fatigue induced slip bands on the surface of a number of
materials including most of f.c.c, b.c.c and h.c.p metals. The “initiation” mechanism
is highly dependant on the shear strain, and cracks always start on the slip planes with
the highest resolved shear stress. Slip bands are known to have cellular dislocation
structures characteristic of high plastic strain amplitudes and tend to initiate at the

28
Chapter 1: Introduction

surface, growing inwards during fatigue. One important finding is that the dislocation
cell structure in slip bands is extremely stable and is difficult to remove even after
annealing. Dislocation structures present in the matrix are however, removed
relatively easily by annealing. This observation is consistent with those that have
shown that annealing is incapable of altering fatigue life once saturation hardening
and slip band production is achieved.

While stacking fault energies may have a significant effect on the appearance of slip
bands in single crystals and on the rate in which cracks “initiate”, its effects in
polycrystals are less profound.

The mechanisms for slip band crack “initiation” [13] are summarised below:

1. Production of slip bands requires first the establishment of a background

dislocation structure (rapid hardening), after which irreversible displacements
occur at the surface at rates as high as 1500 Burgers vectors/cycle to produce
the notch-peak geometry which develops into a fatigue crack. The mechanism
proposed by Partridge and Watt in which dipoles are swept to the surface by
edge dislocations is illustrated in Figure 1.22. Essentially, the emergence of
intrinsic diploes will produce inclusions while extrinsic dipoles will produce
extrusions. An alternative mechanism involves local increases in strain to
trigger the release of stored dislocations to the surface, with the multipole
bundles acting as the background dislocation structure mentioned previously.

2. This highly localised plasticity originates at the surface and propagates into
the material along adjacent primary glide planes producing a dislocation cell
structure containing twist boundaries. This cell structure is extremely stable
against heat treatments.

3. The production of slip bands requires two highly stressed slip systems on the
primary glide plane. Less highly cross-slip planes for these systems are also
required. Cyclic reversal of the strain on these systems seems to be necessary.

29
Chapter 1: Introduction

Fig. 1.22 – Schematic illustration of dipole sweeping to the surface, forming

extrusions and inclusions. Source: [13].

Environment and temperature also have significant influence on the appearance of

slip bands and the rate of crack “initiation”. It has been determined that as
temperature approaches T/Tm = 0.5, the site for crack “initiation” changes to the grain
boundaries and a new mechanism appears in which voids are generated along the
boundaries with vacancy formation and diffusion playing a part in this mode of crack
nucleation. Experiments have also shown that the effect of environment on crack
growth rate far outweighed the effect on crack “initiation” rate.

“Initiation” of cracks at grain boundaries only become important at high strain

amplitudes and at high temperatures. In the case of twin boundary initiation, the
mechanism is used to account for the extensive fatigue damage which occurs along
twin boundaries for a variety of materials. In many cases, the “initiation” is a
consequence of persistent slip bands forming adjacent to the boundary such that the
mechanism is not different from that for the usual slip band initiation.

The “initiation” of fatigue cracks in materials containing more than one phase may,
apart from the other “initiation” mechanisms that have already discussed, involve two
other important mechanisms that are unique to multiphase alloys. The first involves
resolution of the second-phase precipitate, which normally provides strength, and the
second depends on the presence of undissolved second phases or impurities which
provide a local stress concentration for the initiation of cracks.

In the first mechanism, it is possible that dislocations are able to shear the second-
phase precipitates and to-and-fro motion of dislocations in fatigue will reduce
precipitates to points at which the second phase reverts into solid solution. The
process of reversion of the precipitate softens the material which aids the development
of slip bands and shortening the period required to nucleate fatigue cracks.

30
Chapter 1: Introduction

The second mechanism relies on the presence of undissolved second phases or

impurities inherited from improper fabrication techniques which provide a local stress
concentration for the “initiation” of cracks. The stress concentration around inclusions
causes concentrated slip bands to develop. Experiments have shown that fatigue
cracks seemingly “initiated” at the matrix-inclusion interface of inclusions around
where no visible slip occurred, contradicting what has been observed for single-phase
materials where fatigue cracks have been found to occur at slip bands. The reasoning
for such behaviour is that the boundaries between particles are probably weak and
provide suitable nuclei for crack “initiation”. Continued cycling opens voids (at the
interface between matrix and inclusion, and inclusion clusters) until they become
propagating cracks. The mechanism of nucleation therefore seems to be one in which
debonding of the interface occurs without the aid of detectable plastic deformation.
Larger inclusions and inclusion clusters are favoured sites for nucleation, presumably
because the interface is weaker.

1.5 Approaches to fatigue analysis

As mentioned previously, there are three main approaches (stress, strain and
“damage” tolerant or fracture mechanics) to fatigue analysis. Each approach is
discussed in the sections to follow and several references [10, 33] are recommended
for more in depth discussions on the relevant approaches and derivations pertaining to
the equations quoted.

Early fatigue investigations were carried out using predominantly stress-based

approaches. However, since the introduction of servo-hydraulic testing machines,
strain-based approaches have become increasingly popular in their use to analyse
fatigue problems. Both approaches represent extremes of completely unconstrained or
stress-cycling conditions and completely constrained or strain-cycling conditions.
However, fatigue under conditions of controlled strain is more favourable since most
practical engineering components are subjected to a certain degree of structural
constraint, reminiscent of strain-control.

31
Chapter 1: Introduction

Fracture mechanics on the other hand, has proved invaluable in facilitating the
understanding of crack growth mechanisms and characterising crack growth rates.
While, stress or strain-based approaches may provide total life data for smooth
samples subjected to complex loading histories, it is fracture mechanics that provides
researchers with the answers to fatigue crack growth mechanisms responsible for the
trends in total life data observed.

It should be noted that discussions on the various approaches to fatigue analysis are
general and it is advised that relevant papers and textbooks be consulted for more in
depth discussions on the derivation of equations quoted.

1.5.1 Stress-Life approach

S-N curves (Fig. 1.23) were first introduced in the 1860s by Wohler, who worked on
studying fatigue lives of railroad axles. The curves are used primarily for fatigue life
prediction under stress-based approach. A main feature commonly found in S-N
curves is the fatigue or endurance limit, σe, which defines the maximum stress
amplitude possible for a defect-free specimen to have infinite life. The endurance
limit for metallic materials is typically listed at 107 cycles.

σTS
Stress amplitude, σa

σe

Number of cycles to failure, N

Fig. 1.23 – Typical S-N curve showing fatigue limit, σe, and tensile strength σTS.
Source: [34].

32
Chapter 1: Introduction

Stress-based approach to fatigue is used primarily for long life, high cycle fatigue
(HCF), elastic and unconstrained deformation problems. In this regime, stress levels
are relatively low and within the yield limit. As a result, elastic strain contribution to
total strain is typically larger than that for plastic strain and fatigue life is controlled
by material strength. Under predominantly elastic conditions, large stress ranges
occur over small strain changes and difficulties in measuring small strain variations
due to sensitivity limitations meant that most HCF experimental data in the past were
obtained by stress-controlled fatigue. Through advances in electronic technology,
highly responsive and sensitive strain measurements are now possible with present
day strain gauges enabling accurate strain-controlled fatigue to be performed even at
very small strain amplitudes. At very high loading frequencies (> 20 Hz) however,
fatigue performed under conditions of controlled stress is still preferred since there is
a possibility of slippage for clip on type strain gauges.

When a S-N curve is replotted on a logarithmic scale, a linear relationship is

observed. This led Basqin, in 1910 to propose an equation (Eq. 1.6) to characterise the
linear trend observed.
∆σ
σa = = σ f (2N f )
' b
(1.6)
2

where σa is the stress amplitude, σ’f is the strength coefficient, b is the fatigue strength
exponent or Basqin exponent and Nf is the number of cycles to failure. The Basqin
equation is only applicable under zero mean stress conditions. For cases where mean
stress is non-zero, the actual stress amplitude applied has to be adjusted to an
equivalent stress amplitude σar, which is applied at zero mean stress. Appropriate
mean stress correction is crucial in achieving accurate life prediction. Mean stress
correction is performed using the following equations:

⎡ σ m ⎤⎥
Soderberg (1939) σ ar = σ a ⎢1− (1.7)
⎢⎣ σ YS ⎥⎦
⎡ σ m ⎤⎥
Modified Goodman (1899) σ ar = σ a ⎢1− (1.8)
⎢⎣ σ UTS ⎦⎥

33
Chapter 1: Introduction

⎡ ⎛ ⎞2 ⎤
⎢ σ ⎟⎟ ⎥
Gerber (1874) σ ar = σ a ⎢1− ⎜⎜ m
⎥ (1.9)
⎣ ⎝ σUTS ⎠ ⎦

Hence, fatigue life predictions for non-zero mean stress cases need to use either,

σ ar = σ 'f (2 N f ) b
f
(1.10)

Morrow (1968) σ a = (σ ' f − σ m )(2 N f )b (1.11)

or

SWT (Smith, Watson, Topper) σ maxσ a = σ 'f (2N f ) b (1.12)

Mean stress correction equations (Eq. 1.7-1.9) each have their respective merits and
disadvantages. Soderberg’s equation is generally conservative in estimating fatigue
life. The modified Goodman equation is relatively accurate when applied to brittle
materials but agreement with experimental results is poor when applied to ductile
materials. Fatigue life predictions using the Goodman equation (Eq. 1.8) are also non-
conservative for compressive mean stresses. The Gerber equation yields good fatigue
life estimation for ductile materials and positive mean stress conditions. However, the
equation does not distinguish the effects of positive or negative mean stresses.

1.5.2 Strain-Life approach

Strain-based approach to fatigue analysis is rather similar to the stress-based approach

mentioned in section 1.5.1. In this case, strain is held constant during the cyclic
loading. The advantage of using the strain-based approach over the stress-based
approach is that it handles local plasticity effects with greater detail than the latter and
takes into account both elastic and inelastic deformation under constrained conditions.
Contrary to the stress-based approach, load levels are high and usually exceed the
yield limit of the material. The result of this is considerable plastic deformation and
hence, plastic strain is much larger than elastic strain and fatigue life is controlled by

34
Chapter 1: Introduction

ductility. When used at long fatigue lives where elastic strain dominates, it essentially
becomes equivalent to the stress-based approach. Strain-control is used when
gathering experimental data for low-cycle fatigue since in the non-linear part of the
stress-strain curve, a small disturbance in the stress level can cause a large change in
strain. As a result, strain-based approach is commonly preferred in the analysis of
short fatigue lives (low cycle fatigue, LCF).

Justification for the use of controlled strain instead of controlled stress while
observing cyclic stress-strain response curves under high stress is illustrated in
Figures 1.24 and 1.25.

Fig. 1.24 – Cyclic softening of steel under controlled stress cycling. Source: [35].

Fig. 1.25 – Cyclic softening of steel under controlled strain cycling. Source: [35].

In Figure 1.24, plastic deformation occurs and the response of the steel is an accrual
of ever-increasing amounts of plastic strain, resulting in a ‘runaway’ effect as further
cyclic softening takes place. As a comparison, the cyclic response of steel under
controlled strain (Fig. 1.25) differs significantly in appearance from that observed

35
Chapter 1: Introduction

under stress-controlled conditions (Fig. 1.24). While a decrease in stress limits occur
with increasing cycles, no instability in the cyclic response is observed.

Many researchers also favour the use of strain-controlled fatigue tests as the standard
method for generating hardening and softening data. Firstly, experimental tests
performed under strain-controlled fatigue give the flow stress versus cumulative
plastic strain directly. Secondly, cyclic creep commonly found in stress-controlled
tests (especially at larger stress amplitudes) is avoided. Thirdly, a full strain amplitude
can be applied on the very first cycle, whereas a pre-hardening period is often
required before full a stress amplitude can be applied.

To characterise strain-life curves obtained from strain-controlled fatigue loading,

Coffin and Manson (1954) proposed an equation (Eq. 1.13) similar in form to the
Basqin equation.

∆ε '
εa = = ε f (2N f )
c
(1.13)
2

where εa is the strain amplitude, ε’f is the fatigue ductility coefficient, c is the fatigue
ductility exponent and Nf is the number of cycles to failure.

Total strain amplitude can be expressed in terms of elastic and plastic components as
shown in Eq. 1.14 below.

ε a = ε ea + ε pa (1.14)

Elastic strain and plastic strain amplitudes can further be defined as

σa σ 'f
ε ea = = (2 N f ) b (1.15)
E E
and

ε pa = ε 'f (2 N f ) c (1.16)

36
Chapter 1: Introduction

respectively.

The equation relating total strain amplitude εa to life is given in Eq. 1.17 and this
equation is often termed the Coffin-Manson relationship.

σ 'f
εa = (2N f ) + ε f (2N f )
b ' c
(1.17)
E

As mentioned previously, total strain amplitude is composed of both elastic and

plastic strains. Figure 1.26 illustrates a typical total strain-life curve with components
of elastic and plastic strain-life curves. From Figure 1.26, it is clear that at some point,
the elastic and plastic strain-life curves intersect one another in a unique situation
where elastic strain is equal to plastic strain. Fatigue life at which this occurs is
termed transition life, Nt, and is defined in equation (1.18).

1 ⎛⎜ σ f ⎞⎟ c −b
'

Nt = (1.18)
2 ⎜⎝ ε 'f E ⎟⎠

Nt is often used to locate the boundary between low and high cycle fatigue. Under
LCF, substantial plasticity occurs while HCF involves little plasticity.

Fig. 1.26 – Elastic and plastic strain components in a typical logarithmic strain
amplitude vs cycles of failure curve. Source: [35].

37
Chapter 1: Introduction

Equations for fatigue life prediction accounting for mean stress effects have been
proposed by Morrow. These equations are:

c
σ 'f ⎛
⎞ ⎛ ⎞b
εa = ⎜1 − σ m ⎟(2 N f ) b + ε 'f ⎜1 − σ m ⎟ (2 N f ) c (1.19)
E ⎜⎝ σ 'f ⎟⎠ ⎜ σ' ⎟
⎝ f ⎠

,where Nf has been adjusted to include mean stress effects, and

σ 'f ⎛ ⎞
εa = ⎜1 − σ m ⎟(2 N f ) b + ε 'f (2 N f ) c (1.20)
E ⎜⎝ σ 'f ⎟⎠

(modified Morrow where the mean stress dependence on the plastic strain term as
been removed).

SWT also proposed an equation (Eq. 1.21) to evaluate strain-based fatigue life
subjected to mean stress conditions.
c
2
σ 'f ⎛ σ m ⎞b
σ max ε a = (2 N f ) 2b + σ f ε f 1 − ' ⎟ (2 N f ) b + c
' ' ⎜
(1.21)
E ⎜ σ ⎟
⎝ f ⎠

1.5.3 Fracture mechanics approach

The fundamental difference between fracture mechanics or damage tolerant approach

and that of stress-strain based approaches is the incorporation of crack and defect
sizes in fracture approach to fatigue analysis while stress-strain based approaches
assume defect free materials. The origins of modern day fracture mechanics can be
traced back to the pioneering work of Griffith and Inglis. Adopting Inglis’s stress
analysis (1913) of elliptical holes in infinitely elastic plates, Griffith (1921) postulated
that for a unit crack extension, a decrease in the potential energy of the system must
be equal to a corresponding increase in surface energy due to crack growth. Hence,

38
Chapter 1: Introduction

the critical condition for crack growth is when the change in total system energy,
dU
equals zero (Eq. 1.23).
dA

dU
=0 (1.23)
dA
The system energy U is defined as

U = Wp + Ws (1.24)
,the potential energy Wp as

−π a2σ 2 B
Wp = (1.25)
E'

where, for plane strain and plane stress respectively,

' E '
E = 2 and E = E (1.26)
1− υ

and the surface energy Ws as

Ws = 4aB γ s (1.27)

In 1956, Irwin proposed an approach for characterising the driving force for fracture
or energy release rate, G, in cracked elastic bodies (Eqs. 1.28 and 1.29).

dW p
G =− (1.28)
dA

πσ 2 a
G= = 2γ s (1.29)
E'

Two main approaches of fracture mechanics (linear elastic and elastic plastic) have
evolved since Griffith’s energy based fracture theory. The general concepts of each of
the stress intensity based approaches are discussed in the sections to follow. However,

39
Chapter 1: Introduction

before considering the two types of fracture approach to fatigue analysis, it is

important to first discuss the possible modes of fracture. In all forms of failure,
fracture can be attributed to each or a combination of three principle modes, all of
which are illustrated in Figure 1.27. Mode I is tensile opening mode, mode II is
sliding mode and mode III is anti-plane shear mode.

Mode I Mode II Mode III

Fig. 1.27 – Three principle modes of fracture.

1.5.3.1 Linear Elastic Fracture Mechanics (LEFM)

Near-tip stress fields for linear elastic cracks, simplified from elastic-plastic stress
solutions (Eqs. 1.30-1.32) by omitting the second (T) and third terms (V(r)), were first
quantified by Irwin (1957) in terms of stress intensity factor, K (Eqs. 1.33-1.35).

σ πa θ⎛ θ 3θ ⎞
σx = cos ⎜1 − sin sin ⎟ + Tδixδ jx + V (r ) (1.30)
2π a 2⎝ 2 2⎠

σ πa θ⎛ θ 3θ ⎞
σy = cos ⎜1 + sin sin ⎟ + Tδ ixδ jx + V ( r) (1.31)
2π a 2⎝ 2 2⎠

σ πa θ θ 3θ
σx = sin cos cos + Tδ ixδ jx + V (r) (1.32)
2π a 2 2 2

KI θ⎛ θ 3θ ⎞
σx = cos ⎜1− sin sin ⎟ (1.33)
2π a 2⎝ 2 2⎠

40
Chapter 1: Introduction

KI θ⎛ θ 3θ ⎞
σy = cos ⎜1+ sin sin ⎟ (1.34)
2π a 2⎝ 2 2⎠

KI θ θ 3θ
τx = sin cos sin (1.35)
2πa 2 2 2

Considering the near-tip stress field around a sharp crack under plane strain, linear
elastic conditions and using polar coordinates (Fig. 1.28), the stress intensity factor
for each of the three fracture modes, when shear stress is zero (i.e. θ=0) is derived
(Eqs. 1.36-1.38).

r→0
(
K I = lim 2π rσ yy )θ=0
(1.36)

r→0
(
K II = lim 2πrσ xy )θ=0
(1.37)

r →0
(
K III = lim 2πrσ yz ) θ=0
(1.38)

Fig. 1.28 – Polar coordinate representation of stress-fields ahead of crack tip.

For mode I fracture, the general equation for fracture toughness is reduced to,

K I = Yσ π a (1.39)

41
Chapter 1: Introduction

where, Y is the geometry factor and a is the size of the defect. KIC is the critical
fracture toughness, defining the condition required for fracture and is a material
constant that is independent of the size and geometry of the cracked body.
Equivalence of the stress intensity and energy based theories have been found,
relating K to G (Eq. 1.40) where,

KI2
G= ' (1.40)
E

Under cyclic loading, Paris (1961) and his colleagues postulated that fatigue crack
growth under linear elastic conditions, can be characterised by the following relation:

da
= C∆K
m
(1.41)
dN

where, C and m are material constants and

∆Κ = Kmax - Kmin (1.42)

It is imperative to point out that the forgoing discussion does not consider plasticity or
other types of non-linear material behaviour. Stress singularity equations yielding K
values defined by Eqs. 1.36 to 1.38, are valid only when the plastic zone in front of a
crack tip is very small and within a singularity dominated zone, known as the K-
dominance field. Only then, do asymptotic results provide a reasonable approximation
to the full solution. Outside of the singularity dominated zone, higher order terms
(such as the T and V(r) terms in Eqs. 1.30-1.32 become significant and the stress
fields are no longer uniquely characterised by K. This condition for mode I loading is
illustrated in Figure 1.29.

42
Chapter 1: Introduction

Fig. 1.29 – Mode I loading illustrating stress singularity dominated zone.

Limits to the validity of LEFM are detailed in ASTM standard E-399, specifying
conditions to be met for obtaining valid KIC values.

Stress singularity equations such as those specified in Eqs. 1.33-1.35 imply that near
tip stresses approach infinity. However, in reality, stress distribution mirrors more
closely to that of the elastic-plastic curve since stresses cannot be greater than the
yield criterion (Fig. 1.30), defined either by the von Mises (Eq. 1.43) or Tresca (Eq.
1.44) yield criterion.

1
[(σ 1 − σ 2 )2 + (σ 2 − σ 3 ) 2 + (σ 3 − σ1 ) 2 ]
1/ 2
σ YS = (1.43)
2

σ1 − σ 3 = σ YS (1.44)

From Irwin’s stress intensity approach, Eqs. 1.33-1.35, with θ = 0, σxx = σyy =
KI/√(2πr), a first approximation of an elastic crack tip plastic zone size is determined.

2
1 ⎛ KI ⎞
ry = ⎜⎜ ⎟⎟ (elastic-perfectly plastic material: plane stress) (1.45)
2π ⎝ σ YS ⎠

2
1 ⎛ KI ⎞
ry = ⎜⎜ ⎟⎟ (elastic elastic-perfectly material: plane strain) (1.46)
6π ⎝ σ YS ⎠

43
Chapter 1: Introduction

For elastic-plastic materials, Irwin’s second estimate of plastic zone size is determined
according to Eq. 1.47.

2
1⎛K ⎞
r p = ⎜⎜ I ⎟⎟ (plane stress elastic-plastic material) (1.47)
π ⎝ σ YS ⎠

σyy
Elastic

σYS ry
Elastic-Plastic

rp

Fig. 1.30 – Elastic and elastic-plastic plastic zone size and stress distribution curves.

In Dugdales’s plastic zone analysis, a strip yield model assuming total concentration
of plastic deformation at the crack tip, yields the following plastic zone size
approximation (Eq. 1.48).

π⎛K ⎞
2

rp = ⎜⎜ I ⎟⎟ (1.48)
8 ⎝ σ YS ⎠

Dugdale’s yield strip model assumes that a plastic zone of length rp is subjected to
closure stress equalling yield stress applied at each crack tip. The resulting
mathematical derivations yield equation (Eq. 1.48), which is similar to Irwin’s second
plastic zone size approximation (Eq. 1.47), implying that both approaches predict
similar plastic zone sizes.

Two other approximations for plastic zone sizes have also been proposed. Kujawaski
and Ellyin proposed an expression for approximating plastic zone sizes for materials
having stress-strain relationships obeying the power law [36].

44
Chapter 1: Introduction

2 ⎛ K ⎞2
rp = ⎜⎜ I ⎟⎟ (1.49)
(1+ n)π ⎝ σ YS ⎠

On the other hand, for materials having stress-strain relationships of the Ramberg-
Osgood type, plastic zone size approximation can be determined using Eq. 1.50 [37].

1 ⎛ K ⎞2
rp = ⎜⎜ I ⎟⎟ (1.50)
(1+ n')π ⎝ σ YS ⎠

where
1+ n(W op /W oe )
n'= (1.51)
1+ (Wo p /Woe )

1
Wo p = σ YS εYSp (1.52)
(1+ n)

and
σ YS
2
Woe = (1.53)
2E

The plastic zone size approximation equations discussed thus far are only used under
monotonic linear elastic conditions with an assumed circular disc shaped plastic zone.
However, experimental work by Huhn and Rosenfield [38, 39], have shown that
actual plastic zone shape resembles more of a “figure of eight” configuration instead
of the simple circular disc shape. By substituting appropriate stress equations into the
yield criterion, equations defining the plastic zone size and shape are derived.
However, actual plastic zone shapes obtained from finite element analysis utilising
elastic-plastic crack tip solution are slightly different to that predicted from Eqs. 1.54-
1.57. Nevertheless, the estimates of plastic zone size and shape appear reasonable.

45
Chapter 1: Introduction

Von Mises’s yield criteria

K 2 ⎡3 2 ⎤
rp (θ ) = ⎢ sin θ + (1− 2υ ) 2
(1+ cosθ )⎥ (plane strain) (1.54)
4 πY 2 ⎣ 2 ⎦

K 2 ⎡3 2 ⎤
rp (θ ) = ⎢ sin θ + cosθ ⎥ (plane stress) (1.55)
4 πY 2 ⎣ 2 ⎦

Tresca’s yield criteria

K 2⎡ 2 θ⎛ θ ⎞⎤
2

rp (θ ) = ⎢cos ⎜1+ sin ⎟⎥ (plane strain) (1.56)

2πY 2 ⎣ 2⎝ 2 ⎠⎦

K 2⎡ θ⎤
2

rp (θ ) = ⎢1− 2υ + sin ⎥ (plane stress) (1.57)

2πY 2 ⎣ 2⎦

Figure 1.31 illustrates the effect of yield criteria on plastic zone shape for plane stress
and plane strain conditions.

a) b)
Fig. 1.31 – Plastic zone shape: a) von Mises yield criteria b) Tresca’s yield criteria.

Plastic zone sizes for fatigue cracks have been found to be markedly different from
zone sizes predicted using Irwin’s and Dugdale’s approximations due to the presence
of reversed plastic flow ahead of the crack tip. Rice [40] found that the plastic zone
size for fatigue loading is only a quarter of that observed for monotonic loading. Park
[41] modified Dugale’s equation such that for small scale yielding under plane strain

46
Chapter 1: Introduction

conditions, cyclic plastic zone size, rpc is defined by Eq. 1.58, while plastic zone size
under plane stress is defined by Eq.1.59.

π ⎜⎛ ∆K ⎟⎞
2

rpc = ⎜ ⎟ (1.58)
144 ⎝ σ YS ⎠

⎡ ⎤
π ⎛ ∆K ⎞⎟ ⎢ 1 σ 3 ⎛⎜ σ ⎞⎟ ⎥
2 2

rpc = ⎜⎜ ⎢ + 1− ⎥ (1.59)
8 ⎝ 2σ YS ⎟⎠ ⎢ 2 σ YS 4 ⎜⎝ σ YS ⎟⎠ ⎥
⎣ ⎦

1.5.3.2 Elastic-Plastic Fracture Mechanics (EPFM)

While the stress intensity factor, K, has found significant success in characterising
near-tip fields under small scale yielding, its application in cases of large-scale
yielding and non-linear deformation materials has been less successful. In essence,
LEFM is valid only if non-linear material deformation is confined to a very small
region surrounding the crack tip. In 1968, Rice proposed the J-integral concept to
characterise crack tip conditions for non-linear deformation (Fig. 1.32). The line
integral J is defined as

⎛ ⎞
∫ ⎜⎝wdy − T . dx ds⎟⎠
du
J= (1.60)
Γ

where, Γ is the contour encircling the crack tip, y is the distance along the direction
normal to plane of the crack, u is the displacement vector, T is the traction vector and
w is the strain energy density of the material and s the arc length around the contour.

Fig. 1.32 – J contour around a crack tip.

47
Chapter 1: Introduction

dU
Rice showed that J = , which reduces to the energy release rate for linear elastic
dA
materials, where J = G.

A principle assumption for the J-integral is that it assumes a non-linear elastic

material behaviour to simulate elastic-plastic behaviour providing no unloading takes
place. Differences between the two types of material behaviour are illustrated in
Figure 1.33. For a non-linear elastic material, both the loading and unloading paths
are the same, whilst in the case of an elastic-plastic material, the loading path is non-
linear while unloading is elastic, following a path with a slope equal to the Young’s
Modulus, E.

Fig. 1.33 – Non-linear elastic and elastic-plastic stress-strain behaviour.

There are several ways to estimate the values of J experimentally. One can use several
strain gauges placed in a contour around the crack tip and calculate J according to Eq.
1.60. However, this method involves cumbersome instrumentation and calculation.
Finite element analysis has also been used in the estimation of J but like the
aforementioned method, it is both complicated and time consuming. It is hence, more
dΠ
practical to deal with J = − , where load versus displacement plots (Fig. 1.34a) are
dA
recorded for various specimens with varying crack size and on obtaining U versus a
plots, J can be estimated from the slope of the tangent to the curves in Figure 1.34b.

48
Chapter 1: Introduction

Fig. 1.34 – J estimation using compact-tension (CT) specimens.

Another common method of determining J involves measuring the crack tip opening
displacement (CTOD) to characterise fracture toughness. In summary, by measuring
CTOD, δ, G or J can be estimated by using

4 G
δ= (1.61)
π σ YS

,where δ = 2 uy which is illustrated in Figure 1.35.

Fig. 1.35 – Crack opening displacement, 2 uy and effective crack extension ry.

49
Chapter 1: Introduction

The reader is advised to consult with ASTM standard E-813 for details and
recommended experimental procedures pertaining to the determination of the J-
integral. Limitations on the validity of EPFM are again detailed in the specified
standard which summaries to infer that EPFM is not valid when there is excessive
plasticity or large scale yielding such that there is no longer a region uniquely
characterised by J.

50
Chapter 2: Literature Review

Chapter 2: Literature Review

In order to understand the underlying mechanisms that relate to fatigue under

complex loading histories, it is important to appreciate several key topics. This
chapter is divided into four sections, all of which will cover, in depth, the present
understanding and theories dealing with fatigue. It is hoped that these will aid in the
interpretation and analysis of experimental results to come.

2.1 Effects of overloading and underloading

Overloading a cracked specimen can increase its fatigue crack growth life. Many
mechanisms have been proposed to explain the retardation effects after overloading;
of which the six frequently used mechanisms are: plasticity induced crack closure,
irregular crack front or crack deflection and associated roughness induced closure,
crack-tip strain hardening, crack-tip blunting, compressive residual stress ahead of
crack tip and change in crack front shape (Fig. 2.1). Conventional knowledge in crack
growth behaviour would lead us to expect a decrease in crack growth rate
immediately after overloading. This idea, stemming from Elber’s model, was the first
to employ the concept of crack closure to explain retardation in fatigue crack growth
following overloading. Elber [42] discovered that overloading produced compressive
residual stresses, which acted on fatigue crack surfaces causing crack closure. As a
result, subsequent tensile loading will see a portion of this tensile load consumed in
overcoming the crack closure caused by residual compressive stresses ahead of the
crack tip and crack growth retardation is achieved assuming that crack growth would
not take place when a crack is closed. It has been suggested that the primary
mechanisms of closure are reasoned to be associated with the wedging crack face
asperities (roughness induced closure) and more importantly closure induced by
cyclic plasticity in the crack wake [43].

Plasticity induced crack closure results from the enlarged region of compressive
residual stress generated ahead of a crack tip. The residual stress effectively reduces
the load ratio and promotes plasticity-induced closure in the post overload region as

51
Chapter 2: Literature Review

the crack enters the plastically deformed zone caused by the overload. In addition,
residual stresses have also been deemed to enhance plasticity-induced closure in the
wake of an advancing crack. Suresh [44] attributes the retardation effects of
overloading to micro–roughness and crack branching mechanisms. Crack branching
arising from a single overload have been found to reduce effective stress intensity of
up to 19%.

Wheeler [45] proposed a model emphasising more on the compressive stresses ahead
of the crack tip caused by an overload to explain the observed decrease in crack
growth rates. It was suggested that the crack tip must grow beyond the overload
plastic zone for fatigue crack growth rates to return to values seen prior to
overloading. However, the model proposed by Wheeler requires crack retardation to
occur immediately following an overload and does not allow for delayed retardation
as observed experimentally by several other workers [43, 46-50].

Fig. 2.1– Six primary mechanisms used to explain the retardation effects of
overloading. Source: [43].

52
Chapter 2: Literature Review

Crack tip blunting has often been observed after an overload. As a result, there have
been suggestions that retardation after an overload is caused by crack tip blunting as
the blunted crack behaves like a notch and a finite number of cycles are required to
“re-initiate” and propagate the arrested crack. Others have also argued that the crack
tip blunting mechanism may not necessarily be responsible for crack arrest unless
blunting is severe enough. Conflicting evidence has also been presented to argue that
temporary crack arrest is instead associated with a sharp crack tip [48]. In this case,
blunting is deemed responsible for the initial acceleration in crack growth rates after
an overload. If cracks blunted by an overload remained opened at zero load for some
distance behind the crack front, crack growth acceleration is caused by the removal or
reduced crack closure along the crack length by the overload.

Tensile overloads are known to induce compressive residual stresses, which act to
decrease the local tensile stress in front of the crack tip, thereby retarding crack
growth. It has been pointed out however, that mean stress relaxation occurs rapidly
such as to suggest that residual compressive stress ahead of an overloaded crack tip
accounts for only the initial phase of retardation. Strain hardening in front of crack
tips caused by excessively high plastic straining due to overloads has also been
dismissed as a governing mechanism to explain retardation effects [48].

Among the six mechanisms proposed (Fig. 2.1), predictions by crack tip strain
hardening, crack-tip blunting and compressive residual stress models predict
retardation to occur immediately after an overload while irregular crack front models
require certain degrees of crack growth for irregularities and closure to develop for
retardation to take effect. As seen in later discussions, several workers have reported
an acceleration in crack growth rates immediately following overloading, further
suggesting that crack tip strain hardening, crack-tip blunting and compressive residual
stress models cannot fully explain the behaviour of transient crack growth after
overload. However, there is no evidence disputing these models playing a secondary
role in retardation.

Experimental studies [43, 46-50] have shown that initial crack tip acceleration was
detected immediately after overloading followed by the expected decrease in crack
growth rate (Fig. 2.2). Possible influencing factors such as the crack closure effect

53
Chapter 2: Literature Review

have been ruled out as compliance measurements immediately after overloading and
crack acceleration showed no evidence of crack closure. This lead to the hypothesis
that changes in crack growth rates are not related to crack closure. Crack closure was
however, observed prior to overloading and after the crack has reached a length of
3mm after overloading. Additional experiments [43] on the other hand, indicated that
largest post overload accelerations were observed at the lowest baseline K levels
where crack growth during the overload cycle was very small and subsequent closure
in the immediate post overload region is less apparent. The mechanistic crack growth
sequence during and after overloads can be seen in Figure 2.3.

Fig. 2.2 – Transient crack growth behaviour showing initial acceleration and
subsequent retardation following an overload, for two overload ratios (OLR).
Source: [49].

Fig. 2.3 – Mechanistic crack growth sequence during and after an overload.
Source: [43].

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Chapter 2: Literature Review

Workers [43] have likened immediate post overload crack growth response to short
crack behaviour, since short cracks have been known to have initially high growth
rates due to minimal development of closure along their limited wake (see section
2.2). Crack growth rates decrease to approach those characteristic of long cracks as
closure levels build up. There are also reports [51] suggesting that crack closure stress
is approximately a constant, independent of crack length. Acceleration of post
overload crack growth rates have also been attributed to crack tip blunting. Transient
crack growth response seen in Fig. 2.2 is best summarised in the following paragraph
[43].

Normal crack growth – under constant amplitude loading, crack formation is followed
by a period in which crack closure builds up to a steady state level after which crack
behaviour is typical of that of long cracks [52].

Overload cycle – crack tip blunting and associated increased crack opening
displacements removes near tip crack closure and reduces (far-field) closure along the
length of the crack. At higher ∆K levels, overloading results in an increment of
ductile crack growth.

Crack growth rate acceleration – on return to baseline cycling, if the ductile crack
advance at the overload is small, the absence of near-tip crack closure leads to an
initial crack growth rate one or two orders of magnitude higher than would be normal
for steady state cycling conditions. With larger overload cycles, the removal of
closure behind the pre-overload crack tip may be offset by closure generated by the
ductile crack growth increment formed by the overload cycle, in which case the
acceleration may be reduced or be absent.

Delayed retardation – as the fatigue crack extends into the overload plastic zone, it
encounters and enlarged zone of residual compressive stresses which act to promote
plasticity induced crack closure in the wake of the crack tip; the effective stress-
intensity range is thus reduced, and fatigue crack growth rates are retarded.
Mechanisms such as crack deflection and consequent enhanced roughness induced

55
Chapter 2: Literature Review

closure may prolong the retarded region over distances far larger than the overload
plastic zone.

Apart from those mechanisms already discussed, another model for fatigue crack
growth has been proposed for describing the effects after a single overload [53]. This
model deals with several co-existing mutually competitive processes in crack tip
blunting, compressive residual stress, crack closure and strain cyclic hardening.
Blunting of the crack tip is said to increase the fatigue crack growth rate as it reduces
the crack closure effect due to an increase in crack tip opening displacement. Cyclic
strain hardening is also said to cause an increase in fatigue crack growth rate as strain
hardening increases yield stress and hence reduces the plastic zone size and
compressive stress level. On the other hand, compressive residual stress and crack
closure are responsible for reducing fatigue crack growth rate. Hence, this model
explains the transient increase in crack growth rate immediately after overloading by
suggesting that the mechanisms which accelerate fatigue crack growth rates (crack tip
blunting, cyclic strain hardening) are most effective only immediately after
overloading. Upon the emergence of a new sharp crack tip, blunted crack tip and
strain hardening effect are overwhelmed by compressive stresses. These mechanisms
are responsible for the deceleration in fatigue crack growth rates and retardation
effects seen after overloading. The decelerating mechanisms become less effective as
crack grows beyond the overload plastic zone, returning fatigue crack growth rates to
that seen prior to overloading.

The contradicting aspect of what has been discussed is that, by common deduction,
crack tip blunting reduces the stress concentration at the crack tip. Hence, one would
expect fatigue crack growth rates to decrease instead of increase as a consequence of
crack-tip blunting. Strain hardening occurs as increasing deformation produces more
dislocations and lattice defects which hinder crack propagation. Again, with this in
mind, one would also expect a reduction in fatigue crack propagation rates due to
cyclic strain hardening.

Wheatley [50] on the other hand, suggested that the main mechanism responsible for
what has been observed is related to “damage” zone formation. It should be noted that
the term “damage” in this context refers to material degradation, in the form of plastic

56
Chapter 2: Literature Review

deformation and micro cracking. During fatigue crack growth, stress concentration at
the crack tip exceeding the yield strength of the material will cause plastic
deformation. As these local stresses are repeatedly applied, a local “damaged” zone
ahead of the crack tip is formed which accompanies the fatigue crack as it grows.
During overload, the “damaged” zone is further weakened and when fatigue is
resumed, the crack travels faster through the “damaged” zone than it would have
without overload. Thus, the acceleration in crack growth rate is observed. Crack
growth slows down as the crack tip encounters a strain hardened (undamaged) zone
and new “damage” zones must be rebuilt over a period of time for further crack
propagation. It is noted that the newly established fatigue “damage” zone is smaller
than the fatigue “damage” zone prior to overloading as residual compression and local
strain hardening tend to suppress its development. This delay accounts for the
retardation in crack growth rate observed. Other observations made in the study are:

1. The level of overload has a stronger effect on crack growth retardation than
overload duration.

2. Level of overload does not seem to affect the size of the acceleration zone (i.e.
the crack length over which crack growth rate acceleration is observed)

Other results [48] also show that the delay between the application of an overload to
the occurrence of maximum retardation increases with the overload ratio. Increasing
the overload ratio also causes a larger extent of overload affect zone and thus, an
increase in the degree of retardation. The amount and extent of retardation is also seen
to decrease with increasing R ratio. Essentially, overloading experiments using 304
stainless steel have shown that for R ratios smaller then 0.6 momentary acceleration
and delayed retardation were observed which precludes crack-tip blunting,
compressive residual stress and crack tip strain hardening as major retardation
mechanisms. The main retardation mechanism in this case is plasticity induced crack
closure. However, with high R ratios (>0.6), crack-tip blunting and residual
compressive stress ahead of the crack tip may be responsible for retardation as crack
growth behaviour showed immediate retardation instead of the characteristic delayed
retardation seen in low R ratios. Crack-tip strain hardening mechanism has been ruled
out in both cases using microhardness measurements.

57
Chapter 2: Literature Review

Verification of the above discussions on crack growth acceleration following overload

was made by analysing the fracture surfaces with Field Emission Scanning Electron
Microscope (FESEM). FESEM observations suggest that crack growth acceleration
immediately following overloading stems from void and quasi-cleavage fracture
within the fatigue “damage” zone in the vicinity of the crack tip. Shin and Hsu [48]
mentioned that observations of the zone showing voids and quasi-cleavage fracture is
much smaller than the distance over which acceleration of crack growth occurred.
This, however, is explained by Damri et al [54] that the magnitude of voids and
quasi-cleavage fracture increases markedly in the interior of the specimen.
Overloading is also thought to cause the formation of stretch zones within the central
or plane strain region of the specimens.

The transient growth rate responses with block overloads are similar to those seen in
single overloads. While retardation is preceded by an initial acceleration caused by
the removal of near tip closure in the case of single overloads, the step-down from a
high to low block causes near tip closure to be re-established during crack growth at
the higher block, resulting in a more immediate and severe retardation than what is
observed for an equivalent single overload.

Effects of specimen thickness, stress intensity levels and R ratio following single,
multiple overloads and overload/underload events have been investigated by Shuter
and Geary [55]. Typical plots showing retardation effects of overloading can again be
seen in Figure 2.4 below.

Fig. 2.4 – Fatigue crack growth plots showing retardation effects of overloading.
Source: [55].

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Chapter 2: Literature Review

It has been found that crack growth retardation increases with decreasing specimen
thickness. Previous work by various researchers suggest that crack closure is
predominantly a near surface plane stress phenomenon. It has been pointed out that in
thinner materials, because a greater proportion of the material is under plane stress
condition, crack closure has proportionately greater influence and hence, the increase
in crack growth retardation. This observation is supported by the fact that the
minimum effective driving force (U) for crack propagation is found to be significantly
smaller in thin specimens as compared to thicker specimens (Fig. 2.5).

Fig. 2.5 – Variation of retardation effect with material thickness. Source: [55].

Lower R ratios also seem to exhibit greater crack growth retardation than higher R
ratios. At higher R ratios (0.5) back-face and near tip strain measurements provide
little evidence of crack closure suggesting that other mechanisms might be
responsible for the retardation observed. Crack tip blunting or residual stress
formation ahead of the crack tip has been suggested as likely mechanisms. At low R
ratios (0.1), closure resulting from compressive residual stresses formed in the plastic
zone ahead of the crack tip and plastically-induced crack closure (which is essentially
a near crack tip phenomenon and dependant on the amount of crack flank material)
have been sighted as mechanisms responsible for retardation.

The transient behaviour of fatigue crack growth after overloading has also been
observed by Tsukuda [49]. Tsukuda also observed a distinctly different transient
behaviour of fatigue crack propagation for both high and low R ratios after overload.
For low R ratios, crack growth behaviour follows that shown in Figure 2.2, where
there is an initial acceleration in crack growth followed by retardation after

59
Chapter 2: Literature Review

overloading. Baseline steady state conditions similar to those prior to overloading is

reached only when the crack grows out of the overload plastic zone.

Fig. 2.6 – Transient crack growth behaviour for high R ratios. Source: [49].

At high R-ratios (Fig. 2.6), it is clear to see that there is no acceleration in crack
growth rate immediately following an overload. Retardation is observed to occur
immediately after overloading. However, following retardation, there exists an
acceleration phase where crack grows at a rate greater than steady state. This
acceleration in crack growth rate becomes more prominent as the R-ratio and overload
ratio is increased. The length of the transient region is also significantly smaller than
that of small R-ratio loading.

The effect of baseline stress intensity levels have also been investigated with results
showing that a reduction in baseline stress intensity correspond to an increase in crack
growth retardation. This is explained by observing a larger increase in minimum crack
driving force than a corresponding increase in closure with higher baseline stress
intensity values. Other workers [47] have also found that the magnitude of retardation
(delay distance/cycles) increases with either decreasing ∆K toward the threshold or
with increasing ∆K toward instability. The duration of overload retardation effects
also increases with increasing overload ratios.

Rest time after overloading is also known to affect fatigue life [56]. Findings in this
area conclude that an increase in rest periods after overloading seems in general to
increase the crack growth rate and hence, result in a lesser increase in fatigue life.
Larger overloads have also been found to give a lesser increase in life with a given
rest period. An explanation for this observation is that there is more time available for
relaxation of stresses in the larger plastic zones for longer rest times. Hence,

60
Chapter 2: Literature Review

maximum retardation in crack propagation occurs when rest time after overload
equals zero.

Multiple overloads, overload/underload block and overload block-underload block

type loadings were also studied [55]. Essentially, for multiple overloads, as the block
size increases, retardation of crack propagation increases. This may be attributed to
increased fracture surface roughness and plasticity induced crack closure with
increasing block size. However, the extent of increase in retardation reaches a
saturation point for increasing number of overloads. For an overload/underload block
sequence [55], a reduction in retardation was observed with an even further reduction
in retardation for the case of overload block-underload block sequence. Underloading
compresses additional material on the crack flanks such that the influence of
plastically induced crack closure is dramatically reduced [51]. In the case of an
overload block followed by an underload block sequence, it is thought that
preferential compression of near tip crack flank material reduces the influence of the
plastically induced crack closure to an even greater extent. Overloads cause an
increase in crack closure stress (i.e. greater overall crack closure even with the
reduction in closure immediately in front of blunted tip), while compressive
underloads decrease crack closure stress. All experimental results [51] show that
compressive stress (cyclic/underload) decreases closure stress and increases crack
growth rate. In addition to what has been presented, it has been shown that both
threshold stress intensity in cracked specimens and fatigue limit in smooth specimens
decrease when the frequency of application of application of a periodic underload is
increased [49]. Essentially, underloads are known to result in crack growth
acceleration. This occurs because compression induced crushing of fracture asperities
will reduce (roughness induced) closure levels [47]. The application of an underload
prior to a single overload does not have any effect on subsequent retardation whereas
an underload following an overload tends to minimise retardation. As gathered from
previous discussions, overload severely blunts the crack tip (and since crack tip
remains blunted even after overloading from peak overload stress), subsequent
underload cycling causes little closure in the wake and hence could generate tensile
stresses at the crack tip which nullifies some of the beneficial retardation effects of
overload [44].

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Chapter 2: Literature Review

2.2 Short and long fatigue crack growth

Over the past few decades, investigations into fatigue crack growth behaviour in
metallic materials have yielded results manifesting differences between short and long
crack growth behaviour. Work by Schijve [57] also showed that short crack formation
could consume 60-80% of overall fatigue life for several types of commercial alloys,
highlighting its importance in influencing fatigue lives. Research in short crack
growth, driven by the need to understand its mechanistic behaviour and
characteristics, has been extensive over the past two decades. Through this time
significant progress has been made to classify the various types of small cracks and
more importantly explain the anomalous behaviour of short fatigue cracks. i.e. short
cracks grow faster than long cracks at the same nominal driving force even at stress
intensity factor ranges well below the long crack threshold.

Fatigue cracks are generally categorised into short and long cracks or microcracks and
macrocracks respectively. Short or small cracks are further classified as
microstructurally, mechanically and physically short cracks depending on their
relative sizes. Microstructurally short fatigue cracks are typically of lengths
comparable to the microstructural scale, such as grain size. On the other hand,
mechanically short fatigue cracks are of lengths comparable to local plasticity (i.e.
small cracks embedded in the plastic zone of a notch or of a length comparable with
their own crack tip plastic zones, typically ≤ 10-2 mm in ultrahigh strength materials
and ≤ 0.1-1 mm in low strength materials [26]), while physically short fatigue cracks
are of lengths less than 1mm. Transition from microstructurally to physically short
crack occurs when the crack size exceeds a dominant microstructural barrier, such as
a grain boundary [58], while the transition from physically short cracks to long cracks
is considered to occur when the crack size exceeds crack tip plasticity [58, 59].
Detailed overviews on studies made pertaining to short crack propagation can be
found in [23, 26, 60, 61].

Three important equations quantifying crack growth for the various crack type
regimes are listed below [22, 31, 32]. Eqs 2.1 and 2.2 are derived based on a non-
continuum approach which considers micro-structural influences on crack growth

62
Chapter 2: Literature Review

rates. Eq. 2.3, on the other hand, is based on the more familiar continuum mechanics
approach accounting for the effects of material non-linearity on the crack tip driving
force and crack closure transients. Derivations of equations similar in form to those
specified can be found in [62].

1. Micro structurally short crack growth regime:

da
= A∆γ αp (d − a) (2.1)
dN

2. Physically small crack regime:

da
= B∆γ pβ a − C (2.2)
dN

3. Long crack, low-stress regime:

da
= D∆K n (2.3)
dN

A, α, B, β, n, and D are material constants, while γp is the applied plastic shear strain
range. d refers to the largest possible distance a crack can grow until it reaches the
strongest barrier and C represents the longer crack threshold value.

With these equations, lifetime can be assessed by the integration and coupling of the
appropriate crack propagation equations. Figure 2.7 illustrates the effect of cyclic
stress level and barrier strength on the propagation and non-propagation of a fatigue
crack. Essentially, the separation distance between crack limits defined by
microstructurally short crack growth Eq. 2.1, and physically small crack growth Eq.
2.2 represents the fatigue resistance to crack propagation and the fatigue limit.
However, when stress levels are large enough such that the curves (defined by Eqs.
2.1 and 2.2) overlap, fatigue failure is imminent. In general, a material’s fatigue limit
is associated with crack arrest in the microstructurally short crack regime, while crack
propagation in the physically short crack regime determines fatigue life at stresses
above the fatigue limit.

63
Chapter 2: Literature Review

Fig. 2.7 – Effect of cyclic stress level and barrier strength on the propagation or non-
propagation of a fatigue crack. Source: [31].

Grain size plays an important role in controlling the fatigue limit of a material.
Decreasing grain size is thought to increase the fatigue limit of a metal. However,
research has also shown that increasing grain size can increase the fatigue limit of
structures containing long cracks. The latter is explained by observing a reduction in
crack opening displacement and an increase in crack face contact (crack closure)
resulting from mismatching fatigue crack faces during crack closure. For plain
specimens, it can be seen from Eq. 2.1 that if d is smaller, crack growth rates will be
reduced, thus corresponding to an increase in fatigue resistance. Therefore, it is fair to
say that small surface grains and large interior grains will give rise to maximum
fatigue resistance. Other microstructural features such as grain orientation, grain
boundary geometry, precipitates and second phase particles have also been known to
have a influence on short crack propagation. It is imperative to understand that the
conventional deformation approach (stress/strain) including cyclic softening /
hardening cannot be used to understand key fracture processes.

Since Pearson [24], many studies have been conducted to explain and verify the
anomalous crack growth behaviour of short fatigue cracks [25, 63-65]. Most have
found that short fatigue cracks grow faster than long fatigue cracks at the same ∆K
level, and grow at ∆K levels below the threshold value, ∆Kth, for long cracks. This
behaviour is illustrated in Figure 2.8.

64
Chapter 2: Literature Review

Crack growth rate da/dN

Short cracks

Long cracks

∆Kth

Stress intensity factor, ∆K

Fig. 2.8 – Typical crack growth behaviour of short and long fatigue cracks.

In some instances, the “initiation” of a short fatigue crack is accompanied by a

relatively sharp decrease in crack growth rate from initially high growth rates to zero
level creating a non-propagating crack. Such behaviour can be rationalised by
understanding the blocking effect of grain boundaries or microstructural inclusions. If
stress applied is not able to overcome such obstacles, the crack remains dormant and
crack propagation ceases resulting in the oft-observed fatigue limit. In other cases,
short crack propagation may accelerate and decelerate as microstructural obstacles
(such as grain boundaries) are overcome, with the influence of microstructural
features decreasing with progressive crack growth.

Suresh and Ritchie [26] suggested that crack growth transients (i.e.
acceleration/deceleration) and the cessation of crack growth for short cracks could be
interpreted by considering crack tip interaction with grain boundaries and the effect
crack deflection has on the propagation of short fatigue cracks. When a crack tip
encounters a grain boundary, crack deflection occurs and causes a reorientation in
crack growth direction that is defined by the most favourable slip systems in adjoining
grains. The general effect of crack deflection is a reduction in crack driving force and
depending on the degree of crack deflection, a short crack will either advance into an
adjacent grain or stop propagating at the grain boundary. This concept is explained by
examining the following equations approximating local near-tip mode I and II stress
intensity factors.

65
Chapter 2: Literature Review

1 1 1
K 1 / K I = cos 2 θ 0 cos 3 ( θ1 ) + 3 sin θ 0 cos θ 0 sin( θ1 ) cos 2 ( θ1 ) (2.4)
2 2 2

1 1
K 2 / K I = cos 2 θ 0 sin( θ 1 ) cos 2 ( θ 1 ) …
2 2
1 1 1
… sinθ 0 cosθ 0 cos( θ 1 )cos2 ( θ 1 )[1− 3sin 2 ( θ 1 )] (2.5)
2 2 2

K1 and K2 are near-tip mode I and II stress intensity factors, θ0 is the angle of
incidence of initial crack to the mode I plane, θ1 is the angle of crack deflection, and
KI is the nominal far field stress intensity. Examination of the above equations will
reveal that effective driving force Keff (approximated as the summation of squares of
K1 and K2) would decrease with an increasing angle of deflection (θ1). Thus, if Keff is
larger than Kth for short cracks, the crack will advance with only a temporary
deceleration in crack growth rate.

Some reasons have been put forward to explain the anomalous behaviour of short
crack growth. In mechanically and physically short cracks, large scale near-crack-tip
plasticity or a lesser extent of crack closure were suggested as reasons why short
cracks propagate appreciably faster than long cracks subjected to the same nominal
value of stress intensity factor. In the experimental work by Jono and Sugeta [66],
crack opening stress intensity, Kop, was found to be below or nearly equal to zero
immediately after a short crack was initiated, indicating that the incipient crack opens
even during a part of the compressive portion of the loading cycle. Crack opening
ratio, U, which is defined as,

K max − K op ∆K eff
= (2.6)
K max − K min ∆K

has been found to decrease from initially high values to much lower values and either
increased or remained steady with progressive crack growth [25, 64, 66]. It is widely
accepted that crack closure level of short cracks increases with crack growth, and
eventually approaches long crack closure levels. The absence or limited crack closure

66
Chapter 2: Literature Review

observed for short cracks, explains both the anomalous behaviour of short fatigue
crack growth and explains why short cracks are able to grow at stress intensity ranges
that are less than threshold conditions for long cracks. As crack length increases,
crack closure (plasticity induced, roughness induced and oxide induced) builds up,
progressively reducing ∆K and crack growth rates, at times even causing complete
crack arrest. Plastically induced crack closure results from permanent residual plastic
strains left in the wake of an advancing crack tip while roughness and oxide induced
closure occurs as a result of premature contact between asperities and incompatible
mating crack surfaces. Tanaka [67], McEvily [68], Nakai and Ohji [69] studied the
development of crack closure for short cracks subjected to different loading
conditions for different materials.

Conventional fracture mechanics implies that threshold stress intensity range (∆Kth)
for materials should be independent of crack length and is a material constant.
However, work by Kitagawa and Takahashi [60] first showed that ∆Kth for small
cracks decreased with decreasing crack length and threshold stress ∆σth approached
that of smooth bar fatigue limit, ∆σe, at very short crack lengths. Therefore, a general
conclusion for no crack growth is one of constant ∆Kth for long cracks and constant
∆σe for short cracks. Tanaka [67] derived a model, which characterises short crack
threshold conditions by considering whether crack tip slip bands are blocked or can
traverse the grain boundary to adjacent grain. The following two equations were
derived, defining threshold conditions for both long and short cracks respectively:

Kth = Kcm + 2(2/π)1/2σfr*wo1/2 (2.7)

σe = σfr* + Kcm/(πwo)1/2 (2.8)

where, Kcm is the microscopic stress intensity factor at the tip of the slip band, σfr is
the friction stress for dislocation motion and wo is the width of the blocked slip band.

From the above equations, it is clear to see that long crack threshold Kth increases
with grain size, while short crack threshold, or fatigue limit, σe, decreases.

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Chapter 2: Literature Review

It has long been recognised from earlier works that the characterisation of short crack
growth using LEFM yields anomalous results, since many experimental results have
shown that short cracks propagate faster than corresponding long cracks under the
same nominal driving force. This behaviour results not from any physical difference
between the behaviour of long and short cracks but rather with the inappropriate use
of LEFM when analysing short crack problems. As a result, asymptotic analyses have
been developed to define crack tip stress and strain fields in the presence of more
extensive local plasticity. While it may appear sensible to characterise short crack
growth using elastic-plastic fracture mechanics (EPFM), since short cracks are
comparable in size to the extent of self-induced near tip plasticity, it is still often
apparent that short cracks propagate at somewhat faster rates.

The validity of using the ∆J approach is also often questioned [26] since it appears to
contradict a basic assumption in the definition of J. i.e. stress is proportional to the
current plastic strain. As quoted by Suresh and Ritchie [26], the above contradiction
occurs because J is defined from the deformation theory of plasticity, which does not
allow for the elastic unloading and non-proportional loading effects accompanying
crack advance. The short crack problem also highlights a breakdown, not only in
LEFM, but in also in fracture mechanics similitude concept which implies that for
two cracks of different sizes subjected to the same stress intensity factor (under small
scale yielding) in a given material, crack tip plastic zones are equal in size and the
stress / strain distributions along the borders of these zones (ahead of the crack) are
identical. A lack of similitude is apparent for short cracks since local microstructural
features, which do not affect the growth of long cracks, can interact strongly to
influence crack growth behaviour for short cracks. In general, the similitude concept
cannot be applied when crack size approaches those of microstructural dimensions or
crack tip plasticity (i.e. short cracks) and in cases where crack closure effects are
strongly influential.

Various attempts have been made over the years to characterise and correlate short to
long crack growth as well as to model the crack growth trends (acceleration and
deceleration of crack propagation rates) observed for short fatigue cracks [31, 61-63,
70-74]. Models derived to characterise crack propagation trends for short cracks were

68
Chapter 2: Literature Review

based on a variety of factors ranging from, slip band formation, grain boundary
effects, and plastic zone interaction to the modification of LEFM equations, with each
having its own advantages and limitations.

2.2.1 Fatigue limit concepts

The process of fatigue has been long been separated into three stages: “initiation”,
propagation and fracture. Miller [22, 31] suggests that the crack “initiation” stage,
which we often refer to, simply does not exist. In fact, engineers have interpreted
crack “initiation” in the past as the establishment of a crack of a given length, usually
limited by current crack detection resolution. The drive to abolish the “initiation”
stage is supported by experimental results showing that cracks grow on the onset of
the very first cycle during fatigue and this growth occurs from surface scratches,
precipitates, inclusion and triple-point boundaries, all of which can act as stress
concentrations and plasticity “initiation” sites.

The concept of fatigue limit is reinvestigated in which misunderstandings of fatigue

limits and its implications are addressed. A common understanding is that fatigue
limit refers to the inability of a material to initiate a crack. However, experiments
conducted showed that some form of crack growth still occurs even though
conventional fatigue limit conditions are met. Figures 2.9 and 2.10 illustrate this idea.

Fig. 2.9 – Long (low stress) and short (high stress) cracks and their associated zones
of characterisation by linear elastic and elastic-plastic fracture mechanics.

69
Chapter 2: Literature Review

Source: [31].

Fig. 2.10 – Short crack growth behaviour as influence by microstructural barriers of

different strengths and spacing. Source: [31].

According to Miller [31, 32, 75], fatigue cracks will often grow from the very first
cycle of loading. These cracks can either propagate to final fracture or become
dormant and stop growing. Initial crack growth is usually arrested by “barriers” in the
form of twin boundaries, grain boundaries or pearlite zone in a ferrite-pearlite
microstructure. Consequently, a fatigue limit refers to the stress level required to
overcome the strongest barrier to propagation that will be represented by a
microstructural distance (EPFM, MFM regime) and is not the limit of a material to
“initiate” a crack but rather a limit below which a crack cannot continue to propagate.
If the cracks present are large, stress levels will have to be extremely low to approach
infinite life. Another form of fatigue limit is hence formed, where the fatigue limit
stress level must be associated with a given crack size (LEFM regime) as illustrated in
the Kitagawa-Takahashi curve shown in Figure 2.11.

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Chapter 2: Literature Review

Fig. 2.11 - Kitagawa-Takahashi crack classification curve. Source: [31].

From Figure 2.11 it is important to realise that the three fundamental fatigue limits of
(A) single crystal-microstructurally short crack, (B) metal component-physically
small crack, (C) structure-LEFM/EPFM, are different, whilst Figure 2.9 illustrates
two very important characteristics:

1. Cracks of a given size will not grow unless a sufficiently high stress is applied,
which implies that a fatigue limit is not only stress range dependent but also
defect size dependent.

2. Microstructurally small cracks can and do grow, but eventually may slow
down and arrest at a microstructural barrier to produce a different form of
fatigue limit.

In reality, a true fatigue limit will not exist as other degrading factors such as
environment, random loading history and manufacturing defects will act to yield a
finite life regardless of however low the loading stress range is. Numerous studies
have shown that surface finish has a significant influence on the fatigue limit of
materials. Specimens with rougher surfaces will tend to have a lower fatigue as
opposed to those with highly polished surfaces. However, as suggested [23],
preparation of exceedingly smooth surfaces does not necessarily provide a
corresponding increase in fatigue lifetime as it has been found that for extremely fine
ground or highly polished surfaces, fatigue initiation occurred from transverse surface

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Chapter 2: Literature Review

grooves, inclusions or inherent defects which were all greater in depth than the
maximum peak-valley heights of the surface condition measured by a Talysurf scan.

2.3 Mean stress and mean strain effects on fatigue life

In section 1.5.1 of Chapter 1, the subject of mean stress and its effects on fatigue life
was broached upon. As quoted in section 1.5.1, the Soderberg (Eq. 1.7), Goodman
(Eq. 1.8), and Gerber (Eq. 1.9) equations are three of the more widely known
equations for mean stress correction, each having its respective merits and limitations.
In the presence of a positive mean stress, all three equations yield higher effective
stress amplitudes, predicting a reduction in overall fatigue life. On the other hand,
when a negative mean stress is present, both the Soderberg, and Goodman equations
predict an increase in fatigue life while Gerber’s equation still predicts a decrease in
fatigue life. Consequently, Gerber’s equation is seldom used when compressive mean
stresses are known to be present, since it makes no distinction between tensile and
compressive mean stresses in its account of mean stress effect on overall fatigue life.
Figure 2.12 illustrates the typical effect of mean stress on S-N data, while Figure 2.13
shows a constant life trends for fatigue loading under non-zero mean stress based on
the above-mentioned mean stress correction equations. Apart from the three well-
known mean stress correction equations, two other equations, namely, the Morrow
(Eq. 1.11) and Smith, Watson, Topper (Eq. 1.12) equations have also found wide use.

Fig. 2.12 – Typical mean stress effect on S-N curves. Source: [33].

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Chapter 2: Literature Review

Fig. 2.13 – Constant life diagram based on mean stress equations.

Based on Morrow’s mean stress correction equation, Figure 2.14 illustrates the effect
tensile mean stress has on elastic and plastic strain components. As stated in an article
by Mitchell [76], self-imposed mean stresses resulting from preceding strain cycles,
affect overall fatigue lives much in the same way as seen under stress-controlled
fatigue. However, mean stress affects the fatigue strength coefficient, σ’f, which is
relevant only to the elastic portion of total strain (hence, the absence of mean stress
effect on the plastic strain curve). This explains a common observation that mean
stress effects are only apparent in HCF, since elastic strain dominates only when
applied strain amplitudes are low, usually corresponding to high numbers of cyclic
life. When high levels of plastic deformation occurs, mean stress relaxation occurs
and the rate of relaxation depends on both the applied strain amplitude and the
hardness of the material. For example, a ductile material loaded at the same strain
amplitude as a more brittle material will encounter mean stress relaxation at a higher
rate since increased ductility results in a greater degree of plastic deformation.

Total (Tensile mean stress)

(Zero mean stress)

(Zero mean stress)

σm
σ’f-σm/E Elastic (Tensile mean stress)

Fig. 2.14 – Mean stress effect on elastic and plastic strain components.

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Chapter 2: Literature Review

Over the years, extensive studies have been conducted on mean stress effects on
fatigue lives, and a quick literature survey will reveal a multitude of publications
relevant to this topic. A general consensus is that positive mean stress decreases
fatigue life since it has a ‘prying’ effect on cracks and thus, a tendency to promote and
encourage crack propagation, while, a negative mean stress decreases fatigue life due
to its ‘closing’ effect on cracks. This said, there are certain instances, where negative
mean stresses have been observed to decrease overall fatigue life. Such observations
are particularly common when high levels of compressive mean stresses are present,
which promotes buckling.

Mean strain effects on fatigue life are, however, studied to a much lesser extent, as
revealed by the comparative numbers of literature published on mean stress to mean
strain investigations. Since strain-based fatigue analyses have found increasing use in
early design stages of engineering components, the importance of understanding mean
strain effect on fatigue life is obvious. Even though the mean stress correction
equations (Eqs. 1.19-1.21) used under strain approach are similar in form to those
seen in stress-based approach, the manner of application is fundamentally different.
The mean stress used in this case is one that occurs locally, and its value is obtained
specifically by analysing the local plastic deformation or hysteresis loops.

Step loadings involving combinations of relatively large strain amplitudes do not

usually yield mean stresses, as any mean stress initially present would relax to zero as
a result of large plastic straining. On the other hand, low strain amplitudes preceded
by large strain amplitudes tended to yield mean stresses. Figures 2.15 and 2.16 show
self-imposed mean stresses resulting from particular transfer sequences.

It has been reported that step strain tests [77-79] conducted in the LCF range showed
that cycle ratio summations based on strain versus life relationships yielded
summation values close to unity. Studies by Topper and his associates [80], found that
effect of mean stress on cumulative “damage” cannot be discounted as it causes
significant deviations from unity. It is suggested, that “damage” summations based on
completely reversed strain versus life data are reduced if the mean stress is tensile and
are generally increased if the mean stress is compressive. Cumulative fatigue and
“damage” models are discussed in section 2.4.

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Chapter 2: Literature Review

Fig. 2.15 – Development of tensile mean stress on large to small strain amplitude
transfer from compressive peak - No. 4. Source: [35].

Fig. 2.16 – Development of compressive mean stress on large to small strain

amplitude transfer from tensile peak - No. 3. [35]

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Chapter 2: Literature Review

2.4 Cumulative fatigue and “damage” models

As mentioned previously, almost all engineering components will experience non-

uniform load histories during their service life. So far, the problem of predicting
service lives of engineering components subjected to fatigue with any degree of
accuracy has proved quite intractable. The main problem encountered in random
fatigue is quantifying “damage” and accounting for “damage” caused by previous
load histories and its influence on overall fatigue lives. “Damage” a term often used
by researchers to describe material degradation resulting from fatigue, but without a
true appreciation of its meaning, is in this case, unambiguously defined as crack
length. The definition of damage has been associated with a broad range of concepts,
ranging from dislocation generation, changes in endurance limits, void formation,
micro-cracking, entropy changes, reduction in fracture ductility, to crack “initiation”,
and growth. Where references to the term are made, “damage” in inverted comas
refers to a very general definition often used by workers as a convenient measure of
the state of the material during fatigue. Whereas, damage, as a term without inverted
comas refers to crack length or normalised crack length ratio.

Since its introduction, the linear cumulative “damage” rule, often known as the
Palmgren-Miner rule, has been the subject of much discussion, comparison, and
criticism due to its inherent deficiencies, which has motivated and inspired numerous
modifications and the development of alternative “damage” theories. Yet, to this very
day, it remains the most popular and widely used method of life prediction.
Numerous “damage” theories have since been proposed. However, only a select few
will be reviewed here. Most of the models discussed are empirical with very little
theoretical basis. The reader is advised to consult with several published reviews on
cumulative theories [81-84] for more comprehensive analyses on “damage” theories
proposed over the years.

Linear “Damage” Rule; Palmgren, Langer, Miner [84-86]

Palmgren [85] was the first to suggest a linear “damage “ rule in 1924, which was
later independently proposed by both Langer [84] and Miner [86]. Langer though,

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Chapter 2: Literature Review

was recognised more for separating the fatigue process into “initiation” and
propagation stages, which are represented by separate linear “damage” rules. Known
widely as the Palmgren-Miner rule, the rule simply states that fatigue failure is
expected when life fractions consumed my each level of loading, sum to unity; that is,
when 100% of the life is exhausted. “Damage” in this case, is assumed to be
proportional to the cycle ratio applied at a given stress/strain amplitude [84-86].

N1 N N3 Nj
+ 2 + + ... = ∑ =1 (2.9)
N f1 N f 2 N f 3 N fj

Loading history can be repeated more than once such that:

Nj
B f [∑ ] one = 1 (2.10)
N fj

where, Bf is the number of repetitions to failure.

The Palmgren-Miner rule is the most practical fatigue life assessment tool since it is
simple to apply and requires only constant amplitude S-N curves for full computation
(Fig. 2.17).

Fig. 2.17 – a) Application of the Palmgren-Miner rule by summation of life fractions

to unity, b) Estimating failure life at various stress amplitudes using a S-N curve.

However, it does have its limitations in that chronological load effects (sequence
effects), non-linear propertied of “damage” accumulation, and possible “damage”
caused by loading at stress levels below the fatigue limit, are all not taken into

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Chapter 2: Literature Review

consideration during analysis. The rule simply assumes that equal cycle ratios will
produce equal “damage” regardless of the stress amplitude imposed.

Accumulation of Plastic Strain

It is well established that plastic strains are necessary for inducing fatigue fracture. As
a result, it is common to conduct fatigue under strain-control with fixed plastic strain
amplitudes and measure “damage” accumulation in terms of cumulative plastic strain.

N
Γ = 4∑ γ pl ,i (2.11)
i

Γ is an approximate measure of global fatigue “damage”, N is the number of cycles

and γpl,i is the plastic strain for the ith cycle. Note again, that this rule is linear in
nature and does not accurately capture the extent of permanent fatigue “damage” at
low plastic strain amplitudes, where a significant fraction of dislocation motion can be
reversible.

Rainflow Counting Method; Matsuishi and Endo [87]

A more complex case of random loading occurs when the load histories are irregular
and a special cycle counting technique is required. Consensus has been reached by a
number of researchers that the best approach for simplifying complex loading
histories is to use the Rainflow cycle counting procedure.

The procedure starts by shifting the load history to begin and end with the peak or
valley having the highest absolute value of stress. The criteria for counting a cycle is
illustrated below (Fig. 2.18):

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Chapter 2: Literature Review

Fig. 2. – Rainflow counting method.

When a cycle is counted, the stress range and mean stress are evaluated as shown
above and the points counted are removed from future calculations. Stress amplitude
must then be evaluated for that cycle using the equation representative of the S-N
curve or using a supplied S-N curve. This counting process is continued until the
entire load history is exhausted and Palmgren-Miner rule can then be applied to
evaluate the overall fatigue life of a component subjected to that particular loading
pattern.

Similar concepts of cycle counting can be applied for fracture mechanics and strain-
based approach to fatigue life prediction. In the fracture mechanics approach, for
each cycle counted, R, Smax and ∆S j are evaluated and the equivalent ∆Se is evaluated
from

1/ m
⎡ NB m ⎤
⎢ ∑ (∆S j ) ⎥
∆K e
=⎢ ⎥
j =1
∆S e = (2.12)
F πa ⎢ NB ⎥
⎢ ⎥
⎣ ⎦

and substituted into

a 1f− m / 2 − ai1− m / 2
N if = (2.13)
C ( F∆S e π ) m (1 − m / 2)

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Chapter 2: Literature Review

to evaluate Nif, the life consumed by the history. NB is the number of cycles cycled at
∆S j, a is the crack length, F is and m is the material constant. Bif, the number of
repeatable loading history, can then be evaluated using the Palmgren-Miner rule to
work out the number of times the history can be repeated before failure. Note that for
a given material, C1 is constant at R=0. Hence, for the evaluation of life at R = 0, C1 =
C and ∆S = ∆S = S max (1 − R ) γ are to be used in the equation stated above. On the

other hand, when fatigue life is evaluated at R≠0, C, which is a material constant, is
used and ∆S = S max (1 − R) . Sequence effects have thus far been ignored but will be

discussed later. For strain-based approach to fatigue life prediction, the technique
used is similar to that mentioned for the stress-based approach. i.e. Nif is evaluated for
each cycle counted and using the Palmgren-Miner rule, the number of possible
repetitions of history can be evaluated.

Previous discussions made on the Palmgren-Miner Rule have ignored sequence

effects in fatigue loading. In reality, sequence effects play a significant role in
affecting overall fatigue life. The “damage” sum ∑ (n i / N i ) usually does not add to

1. A “damage” sum > 1 means that the actual component has a larger fatigue life than
that predicted by the linear cumulative “damage” rule. On the other hand, a “damage”
sum < 1 means that the actual component has a fatigue life less than that predicted by
the linear cumulative “damage” rule. It is has been suggested that the “damage” sum
at failure depends on the loading sequence and that High→Low and Low→High
sequences yield very different “damage” sums. Tests have shown that High→Low
sequences tend to decrease the “damage” sum, yielding values < 1, while Low→High
sequence loading tends to increase the “damage” sum and yield values > 1 (Fig. 2.19).
It can be reasoned that for a High-Low sequence, the higher load may have “initiated”
cracks, thus permitting crack growth even at subsequent lower loads.

As seen from the criteria for cycle counting, the Rainflow method involves shifting
the load history to begin and end with the peak or valley having the highest absolute
value of stress. This shift in load history violates the load sequence effect, which has
been found to have significant effects on fatigue life. As a result, a modified random
fatigue load counting method in keeping with the main principles of traditional

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Chapter 2: Literature Review

Rainflow counting method has been suggested [88]. This alternative counting
treatment allows for the assessment of “damage” caused by varying load cycles in the
order corresponding to the entire load history. This method of cycle counting is
explained using a random load spectrum as an example. Reference should be made to
[88] for the conditions and methodology of applying the proposed counting technique.

Fraction of high strain/strain cycles

Low→High

High→Low

Fraction of low strain/strain cycles

Fig. 2.19 – Typical cycle ratio trends for both High→Low and Low→High loading
sequences.

Many experiments utilising the Palmgren-Miner rule in accounting for cumulative

“damage” has met with limited success. As mentioned before, more often than not,
cycle ratio summation does not add to one. In recognising that non-linearity, stress
dependency, and sequence effects of cumulative fatigue are not taken into account
with linear cumulative “damage” rules, many non-linear cumulative “damage” rules
have been proposed for better representation of the “damage” caused by various
sequences and stresses of loads.

“Damage” association with dislocation generation; Machlin [89]

Machlin’s hypothesis featured a mechanism based on associating “damage” with the

number of dislocations generated at the tips of propagating fatigue cracks. It was
postulated that on loading, crack tips act as sources of dislocations and the number of
dislocations per source required to cause failure is constant for all materials. As a
result, for varying stress conditions, the failure criterion is defined by:

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Chapter 2: Literature Review

m P/2
M = ∑ ni ∫R gt dt (2.14)
i 0

where Rgt is the time rate of dislocation generation, P is the period for a sinusoidally
varying stress, m is the number of varying stress stages and ni is the number of
applied cycles. The fundamental flaw with this model is by assuming that dislocations
exist in short discrete lengths and that work absorption per cycle is constant. As
pointed out by O’Neill [81], dislocation lines can only end by joining itself to other
dislocations or by meeting a free surface. The generation of dislocations would also
result in an increased resistance to movements of dislocations such that work
absorption per cycle is not constant but rather an ever evolving phenomenon. Finally,
it is questionable if Machlin’s theory could be applied practically as the determination
of Rgt , seems to be an arduous task.

Crack initiation and propagation; Shanley [90]

Shanley proposed a fatigue mechanism that is entirely qualitative in explaining the

formation of fatigue cracks. Unbonding of atoms caused by the action of shear stress
through tensile loading, results in the formation of slip planes and the subsequent
development of a fatigue crack. The crack rate functions suggested by Shanley are
listed below:

h = Aσ x n (constant crack growth per cycle) (2.15)

h = Ae Cσ
x
n
(exponential crack growth rate) (2.16)

where h is crack depth, σ is the maximum stress, n is the number of cycles applied
and A,C, x are empirical constants which are stress independent. O’Neill [81] argues
that the above functions suggested by Shanely are actually linear “damage” rules of
the Palmgren-Miner form. In the case of the first function proposed, there is no
mistaking that it is a linear “damage” rule since crack growth or damage occurs over a
constant rate. The second function as shown by O’Neill, is also a linear function. In
addition, it has been discovered that a constant exponent for σ is valid only for a

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restricted range of h. Hence, it is clear that no advantage is gained by using the more
complicated crack growth functions as they obviously yield the same results.
Therefore, it was suggested that the Palmgren-Miner rule would be the preferred rule
over the proposed crack growth formulas since it is simpler to apply. Interestingly,
one observes that theories which lead to linear “damage” rules make similar
assumptions, in that they all assume a constant “damage” criterion.

Crack growth, Valluri [91, 92]

Valluri’s mechanism of fatigue failure applies specifically to high stress level fatigue
where hardening mechanisms operate and deformation is by coarse slip. The basis for
the mechanism proposed is merely hypothetical as it was assumed that rapid cycling
causes a crack to form and propagation of the crack occurs intermittently by a process
of strain hardening of the plastic zone at the crack tip over a number of cycles,
followed by rapid growth through the hardened area in the course of a single cycle.

The proposed crack growth equation takes the following form:

dl
= Cf (σ )l (2.17)
dN

where l is the length of the crack, N is the number of cycles applied, f(σ) is a function
which involves both maximum and minimum stresses in the load cycle and the
endurance limit stress, while C includes terms for mean grain size and Burger’s vector
of the material. Even though the above equation is similar in form to that proposed by
Shanely [90], the formula was developed from fundamental considerations and is
quite complex in detail. The failure criterion for Valluri’s model is fracture and this
occurs when a crack reaches a critical length. This marks the difference between the
Valluri’s and Shanley’s crack growth equations. Since the failure criterion (critical
crack length) is not a characteristic constant, the ordering effect which arises from
Valluri’s equation is a consequence of using varying values of critical crack length
which is stress dependent. Although this model is predictive in principle, some testing
is required to establish C and the stress concentration factor at the crack tip for the
specimen involved for successful application.

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Chapter 2: Literature Review

Non-linear accumualtion of “damage”, Macro and Starkey [93]

Marco and Starky [93] proposed a “damage” function (Eq. 1.10) on the basis of non-
linear “damage” accumulation.

xi
⎛n ⎞
Di = ⎜⎜ i ⎟⎟ (2.18)
⎝ Ni ⎠

At first, it may seem that the above function is similar to the exponential crack growth
equation proposed by Shanley [90]. However, exponent xi, is not an empirical
constant like x defined in Shanley’s function, but rather it is a function of stress.
Application of the Marco and Starky function will require the knowledge of xi for the
various levels of stress concerned. Apart from knowing the values of xi, life
calculations are more involving than the linear rule. For multilevel loading, it is
assumed that “damage” accumulated from previous levels of loading is continued at
the new stress level. That is to say, an equivalent cycle ratio will have to be calculated
at the present stress level for the “damage” incurred thus far by previous loadings
before adding the amount of cycle ratio and damage done at the present stress level.
This additional computation results in a more accurate solution than the linear
accumulation rule.

Corten and Dolan “damage” model [94]

Corten and Dolan developed a model which is more complex than the
aforementioned theories. “Damage”, in this case is defined by:

Di = mi ri ni
ai
(2.19)

where m is the number of “damage” nuclei, which is assumed to the determined by

the higher stress level; r, the “damage” rate coefficient; a, the “damage” rate exponent
and n, the number of cycles applied. The relation derived for 2-level loading is as
follows:

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Chapter 2: Literature Review

1 / ai (1− β ) N gA
N1 = N gα + R (2.20)

where Ng is total life under programme loading, N1 is life at the first and highest stress
level, S1 applied, α is the proportion of programme at S1, R = r2/r1, and A = a2/a1. In
his analysis report, O’Neill [81], commented that in the derivation of the “damage”
theory proposed by Corten and Dolan, a mistake arising from confusion over the
number of cycles actually applied at stress level S2 with their equivalent at S1, was
made. As a result, the above equation which is supposedly valid for all values of A, is
in fact true only if A =1. As evidenced in experimental results, this flaw has lead to
some poor predictions using the Corten and Dolan model.

Scharton and Crandall crack growth model [95]

Scharton and Crandal proposed a crack growth rate of the following form based on
the assumption that fatigue “damage” is the growth of a pre-existing crack to some
arbitrary size which defines failure.

dC
= C m +1 f ( S i , j ) (2.21)
dn

where C is the non-dimensional crack length, Si,j is a sinusoidal stress cycle of

amplitude Ai, and m is a constant. Assuming that the above formula can be applied to
individual cycles, stress interactions and transient effects on crack growth rate due to
changes in stress levels are neglected, the following equations were derived:

C −fijm − C oij
−m
nij
if m≠0, 1 = ∑ −m −m
⋅ (2.22)
ij C f −C o N ij

ln C fij − ln C oij nij

or, if m=0, 1 = ∑ ⋅ (2.23)
ij ln C f − ln C o N ij

where Cfij, Coij, are the final and initial crack lengths respectively and nij is the number
of cycles at a stress level, Sij. This model posed problems due to its oversimplification

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Chapter 2: Literature Review

of the crack growth formula and residual stresses at the crack tips and interaction
effects on crack growth are all neglected. As a consequence, when the final crack
length is very much greater than the initial crack length, Eqs. 2.22 and 2.23 yielded
life predictions similar to those seen using a linear rule. As commented by O’Neill
[81], predictions made using this model are both inaccurate and non-conservative,
presumably a consequence of the simple form of crack rate function used.

Palmgren-Miner modified rule; Swanson [96]

Swanson proposed modifications to the Palmgren-Miner rule to correct its main

deficiencies. Modification of the linear rule involved dividing the conventional S-N
curve into three distinct regimes as shown in Figure 2.20 below.

1) High stress- non-linear “damage”

2) Intermediate stress- linear “damage”

3) Low stress- non-linear “damage”

Fig. 2.20 – S-N curve divided into three distinct regimes based on Swanson’s model.

Swanson hypothesised that “damage” accumulation in regions 1 and 3 will be non-

linear whilst linear interactions will be observed in region 2. On this basis, a
“damage” relation encompassing all three regions of loading was proposed:

ΣR1x + ΣR2 + Σ( R3 -R3y )= 1 (2.24)

where, R refers to the cycle ratios at the respective regions of loading, x is the
exponent varying with stress level, and y = (u/uH-1)2; u = stress level applied and uH =
stress level giving maximum increase in life. While the proposal of such a model is
interesting, it must be noted that in reality, “damage” in each regime cannot be
considered separately and summations must be done sequentially. The summations
described by Swanson’s equation implies that the “damage” in each regime starts

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Chapter 2: Literature Review

from the origin regardless of any previous damage, which obviously is not the case in
reality.

Strain Rate Dependency of Cumulative Fatigue, Miller [97]

Two very important papers involving the study of cumulative “damage” laws have
been presented [97, 98]. In both papers, results support the general notion that when
for low→high loading sequences, the cycle ratio summation is usually greater than
unity but is less than unity if the sequence order is reversed. Several deficiencies in
the Palmgren-Miner rule have also been addressed. These include not taking into
account the effects of residual stresses, stress cycling below the fatigue limit and
coaxing effects in strain-ageing materials. As discussed, there have been many
alternative cumulative “damage” laws proposed to curb with the over-conservatism of
the Palmgren-Miner hypothesis. Miller [97], provides a quick overview on some of
the early hypothesis proposed highlighting their respective limitations and at the same
time proposes a new hypothesis which deals with the sequential effect during variable
amplitude loading. High strain fatigue life is deemed to be both dependant on the
degree of plastic deformation ∆εp and the cyclic strain rate. Given that two different
strain amplitudes may be cycled at the same frequency, the cyclic strain rates are
different. Miller [97] suggests that the load-sequence effect noted in all previous
studies in cumulative “damage” is a frequency effect which may be eradicated if the
tests are performed at a constant strain rate.

An alternative hypothesis has been proposed to describe and account for the effect of
strain rate variations on cumulative “damage” under a constant frequency of cycling.
Figure 2.21 illustrates this hypothesis by taking the example of a two step loading
sequence from low to high level strain ranges. Let a-a’ and b-b’ represent endurance
curves for a material deformed at low and high strain rates respectively. In this case,
the low and high strain ranges are demarcated by levels 2 and 1 respectively.
Assuming that a low level strain range is applied initially to account for 50% life, i.e.
2-4a’, when the second step and higher level of strain is applied, the 50% proportion
life already used is equal to the distance 1-3a at the same rate. If the second half of the
test is carried out at the same frequency as the first half, however, it is often assumed

87
Chapter 2: Literature Review

that 50% damage is equivalent to point 3b, not 3a. Therefore, there is an apparent
extension in life 3a-3b which shows that Low-High strain/stress sequences usually
yield an increase in life. High-Low strain/stress will typically yield an opposite trend
and “damage” summation is less than unity suggesting a reduction in life. A
“damage” summation of less than unity observed for High-Low Strain/Stress
sequences is said to be caused by a higher “damage” rate during the high strain/stress
block as compared to the low strain/stress block.

Fig. 2.21 – Effect of strain rate variations on cumulative “damage” at a constant

frequency of cycling. Source: [97].

It is important to realise that the crack growth rates differ from stage I to stage II and
so the rates of “damage” accumulation in the separate phases are likely to be different.
An important assumption is that when a fatigue failure crack emerges, all other
defects may be considered to have stopped growing and the now dominant stage I
crack can accelerate since the overall strain is now totally accommodated by plasticity
at the tip of this one defect.

Miller and Zachariah crack propagation model [98]

Miller and Zachariah [98] proposed a simple yet effective model for explaining
sequence effects witnessed during multi-step loading based on the fact that stage I and
stage II crack growth rates differ from one another. Hence, damage accumulation
occurs at different rates in the respective stages. Figure 2.22 shows simplified crack
growth curves, which have been separated into the various propagation stages. At
low stress levels, the rate of crack growth during the “initiation” phase is very much
reduced and crack length at cessation of the initiation period is also decreased.

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Chapter 2: Literature Review

Consider the case where an initial stress amplitude ∆σ1 is applied for 0.2Nf1 and then
the load level is changed to ∆σ2. i.e. Hi-Lo step down sequence. The damage caused
by the first loading is a crack of length, a1. When loading is changed to ∆σ2, the crack
resumes from its present length and continues to grow at the new stress amplitude.

However, the figure shows that for the lower stress amplitude to cause an equivalent
damage as the previous higher stress amplitude, more than 0.2Nf2 of life must be
exhausted. This implies that the fraction of life left will invariably be less than 0.8Nf2
predicted using the Palmgren-Miner rule. Thus, providing the rational for
High→Low sequences yielding “damage” summations < 1 and summations values to
be > 1 when Low→High sequences are applied. From this model, one is able to
deduce the typical “damage” curve plots for High→Low and Low→High loading
sequences presented earlier (Fig. 2.19).

Stage II
Crack Length

∆σ1 ∆σ2

a1

Ci
Stage I Nf1 Nf2

0.2Nf1 Life

Fig. 2.22 – Crack growth model in two stages as presented by Zachariah and Miller.

There are four main assumptions associated with this model:

1. There are 2 stages of crack growth for low stress/strain loading and a single
crack growth stage for high stress/strain loading.

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Chapter 2: Literature Review

2. The crack immediately assumes a speed corresponding to the new stress state.

3. Crack length at cessation of stage I is small in comparison with the final crack
length.

4. Linear crack growth behaviour occurs in all stages.

It is also important to appreciate the advantages and disadvantages of such a model.

Two advantages come to mind:

1. It takes into account the separate stages in crack propagation.

2. The model is easy to apply and provides a simple explanation for sequence
effects observed for multi-level loading. The main limitation of such a model
is that it neglects the effects of residual stress and crack closure induced when
changing from one loading level to another.

Thang, Dubuc, Bazergui and Biron model [99, 100]

Thang and his associates proposed a “damage” model based on the reduction of
endurance limit during fatigue. Previous studies have showed that the endurance limit
changes with the static tensile strength of the material. During fatigue, static
properties of a material changes as the fracture ductility of the material is exhausted.
Two papers discussing cumulative “damage” based on this concept have been
published [99, 100]; one under stress-controlled conditions and the other under strain-
controlled conditions. Both utilise the same concept that “damage” is influenced by
changes in endurance limit and has a stress or strain dependent relationship with cycle
ratio β. The theory is based on two assumptions:

1. Damage process is continuous.

2. At low stress levels, the proportional “damage” caused solely by fatigue is

more pronounced than at higher stress levels (i.e. yielding, creep etc can play a
part).

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Chapter 2: Literature Review

The reader is advised to consult the two papers mentioned for damage equations
proposed and their derivation. The authors also proposed a methodology for the
prediction of approximated fatigue curves from static properties. It is from these
fatigue curves where parameters required to evaluate the “damage” functions are
determined. Under multilevel loading, the cycle ratio remaining in the last load level
can be determined by first finding an equivalent cycle ratio at the present load level
which would have caused the same “damage” by previous stress levels.

Cycle ratio trends as seen from this approach (Fig. 2.23) is similar to actual crack
growth behaviour. The main disadvantage of using such a model is that even though it
explains sequence effects during multilevel loading, the fact that it does not take into
account the various stages of crack growth probably results in the model being
slightly conservative for life predictions involving High→Low type loading [98]. In
addition, it also does not take into account the effects of crack closure and residual
stress. However, the fact that fatigue curves can be approximated from static
properties and multi-level loading can be predicted based on parameters determined
from the fatigue curves, presents an attractive approximation life prediction tool albeit
its limitations.

Fig. 2.23 – Cycle ratio trend based on Thang, Dubuc, Bazergui and Biron’s change in
endurance limit model. Source: [99].

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Chapter 2: Literature Review

Fatigue Curve Convergence and Rotation; Manson [101]

One of the earlier cumulative “damage” analysis models proposed, involved using a
fatigue curve convergence and rotation approach as illustrated by Figure 2.24.

Fig. 2.24 – General location of S-log N lines for original and material subjected to a
single pre-stress condition. Source: [101].

It was found by several workers [101-103], that the endurance limit of materials
changed accordingly with varying degrees of pre-stressing and the S-N curves seem
to rotate about a single convergence point. The convergence concept is simple and
readily applicable, as once the rotation point is known, the effect of any loading
sequence can supposedly be determined by successive application of the concept. We
take the following as an illustrative example (Fig. 2.25).

P (1000 cycles) Original fatigue line

Damage line after pre-

S, STRESS AMPLITUDE

stress at stress level A

2 B

A’’ A’
1

C
2’
O’’ O’ O

N, CYCLES TO FAILURE

Fig. 2.25 – Convergence concept to fatigue “damage” accumulation.

Suppose a specimen is loaded at stress level 1 for x% of life at this stress level. Life
remaining as a result of the initial pre-stress would be A’ on the rotated S-N curve,

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Chapter 2: Literature Review

O’P. If the second loading is carried out at a higher stress level than the first i.e. at 2,
life remaining using this approach would be B. On the other hand, if the second
loading is carried out at a lower stress level 2’, life remaining is C.

However, further review of this concept by Manson and Halford [104] revealed
several limitations warranting alternative cumulative “damage” approaches to be
devised. One limitation was the contradiction that would occur if prior cycling
produced a remaining life less than at the convergence point, P. It is shown that if
prior loading at stress level 1 were applied to a degree that the remaining life were
less than 1000 cycles, such as A’’, then the damage line after initial cycling would
have a positive slope, O’’P, which means that a higher stress would produce longer
life, which is absurd. In addition, more tests on other materials revealed that the
“damaging” effect of multilevel loading is not always represented by the convergence
approach. In fact, S-N curves for various materials were found to both rotate and
translate upon exposure to pre-loading. As stated, it is often found that for
High→Low stress/strain sequences, cycle ratio summations are < 1, whereas for
Low→High stress/strain loading sequences, cycle ratio summations tend to be > 1. As
a result, using the above example as a point for discussion, experimental results
obtained under High→Low or Low→High stress/strain loading showed that life
remaining at the second level can either be greater or smaller than that predicted using
this approach. This serves to further highlight the inadequacies associated with the
convergence approach and its failure to properly account for sequence effects
observed under multilevel loading.

Damage Curve Approach (DCA); Manson and Halford [105] – similar to that
proposed by Macro and Starkey [93]

Richart and Newmark [106] first introduced the idea of using damage curves to
explain the sequence effects observed under multilevel loading in 1948. However,
definitive formulas representing damage curves were not provided for the qualitative
prediction of sequence effects observed under multilevel loading. Such an approach
was also proposed by Marco and Starkey [93]. Mason and Halford [105] proposed a
general damage curve equation (Eq. 2.25) based on analogy to early crack growth.

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Chapter 2: Literature Review

⎡ 1 ⎤⎢⎡ ⎛ n ⎞ (2/ 3)(N ) ⎤⎥

0. 4

D =⎢ ⎥ a + (0.18 − ao )⎜ ⎟ (2.25)
⎣ 0.18⎦⎢⎣ o ⎝N⎠ ⎥
⎦

ao is taken as the characteristic defect size of the specimen, while 0.18 (inches) is
chosen as the fracture crack length of a quarter inch diameter specimen. It was argued
that the crack growth rate is so rapid near the fracture condition that almost any
number close to 0.25 could be used as af. The reader is advised to consult [105] for
the derivation of Eq. 2.25.

Figure 2.26 shows a schematic of the damage accumulation curves based on Eq. 2.25,
while Figure 2.27 shows a schematic of the DCA of summing cumulative damage for
normalized damage curves in multilevel loading.

Fig. 2.26 – Damage accumulation based on Damage Curve Approach (DCA).

Source: [104].

Sequence effects commonly observed for High→Low and Low→High loading can be
accounted for by using DCA. In Figure 2.27, it is clear that High→Low sequences in
which current fatigue loading is preceded by higher stress or strain levels, amount to
cycle ratio summations < 1. Conversely, using the DCA, Low→High sequences
understandably yield cycle ratio summations > 1. Based on the DCA, a general
equation for the prediction of remaining life at the final phase of K loadings is
proposed [105].

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Chapter 2: Literature Review

Fig. 2.27 – Cycle ratio trends based on Damage Curve Approach (DCA).
Source: [105].

⎧⎪⎧⎪ ⎫⎪
{ }
( N 2 / N 3 )0 .4
⎨⎨ (n1 / N1 )( N1 / N 2 ) + n2 / N 2
0 .4
+ n3 / N 3 ⎬⎪...
⎪⎩⎪⎩ ⎭
0 .4
⎫⎪( NK −1 / N K )
⎬...+ n K−1 / N K−1 ⎬ + n K /N K = 1 (2.26)
⎪⎭

While the DCA accounts for the trends in cycle ratio summations observed for
High→Low and Low→High stress/strain sequences, the damage trends observed
experimentally did not always correspond closely with that predicted using DCA. As
a result, modifications in the form of a double-damage curve approach (DDCA) [104]
with double term expressions have been proposed to refine and yield more accurate
predictive damage curve trends.

Double-Linear Damage Rule (DLDR); Manson and Halford [104]

It has long been recognized that fatigue is at least a 2-stage process. The idea of a 2-
stage fatigue process was originally suggested by Langer [84]. Later, Grover [107]
refined the linear rules and distinguished the two stages by identifying them as
“initiation” and propagation stages respectively. Grover considered the propagation
stage to begin with the appearance of a “significant’ crack, not necessarily associated

95
Chapter 2: Literature Review

with physical observation of cracking but fitted empirically to satisfy test results. In
Grover’s rule,

ni mi
∑a N = 1 and ∑ N (1 − a ) = 1 (2.27)
i i i i

ai refers to the fraction of life for “initiation”, and ni, mi are cycles applied in the
“initiation” and propagation stages respectively. It is assumed that ai decreases as
stress increases. The main problem associated with Grover’s approach was that ai had
to be estimated empirically.

The principle behind the DLDR proposed by Manson and Halford [104] involves
separating “damage” into 2 distinct phases, (much like the one proposed by Zachariah
and Miller [98]) and under each phase all damage lines are reduced to a single
“damage” curve by normalization. The benefit of such an approach is that within each
phase, loading order and sequence effects become unimportant (which may not be
true in reality) as damage is accumulated via the linear “damage” rule and the entire
process of “damage” analysis is greatly simplified. Figure 2.28 illustrates this process.
The two stages initially believed to be “initiation” and propagation were subsequently
termed as phase I and phase II respectively after failed attempts to correlate crack
growth behaviour observed with the aforementioned definitions.

It was observed that damage curves for two-level loadings depended only on the ratio
N1/N2 and not on the individual values of N1 and N2. Figure 2.29 shows a comparison
between various “damage” rules. Kneepoints for the “damage” curves are determined
by using the following equations:

For High→Low stress/strain sequences

(n1 / N1)knee = 0.35(N )

0.25
;(n 2 / N 2 )knee = 0.65( N1 / N 2 )
0.25
/ N2 (2.28)

For Low→High stress/strain sequences

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Chapter 2: Literature Review

(n1 / N1)upperknee = 1− 0.65(N )

−0.25
;(n2 / N 2 )upperknee = 1− 0.35( N1 / N 2 )
−0.25
/ N2 (2.29)

Fig. 2.28 – Elements of double-linear damage accumulation (DLDR). a) Damage

accumulation curves. b) DLDR rule leading to non-linear life fraction summations. c)
Linear life fraction rule for phase I damage accumulation d) Linear life fraction rule
for phase II damage accumulation. Source: [104].

Fig. 2.29 – Comparison of the various “damage” rules proposed by Manson and
Halford. Source: [104].

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Chapter 2: Literature Review

Again derivations for the above equations can be found in [105]. Similarly, the
methodology in “damage” accumulation analysis for multilevel loadings involving
more than 2 load levels is detailed in [104, 105].

Manson and Halford [104, 105] refrained from using terms such as “initiation” and
propagation to separate the two stages of “damage” accumulation thus avoiding the
confusion associated with the observation that “initiation”, defined as such, would
mean that it depended not only upon that life but also the later life level at which that
fatigue process is continued. This conclusion was made when changes in kneepoint
coordinates were observed resulting from a change in N1/N2 ratio, with N1 unchanged.
This in itself did not make sense, since the number of cycles for crack “initiation”
during fatigue at the first amplitude should be characteristic of that amplitude and
independent of what is to come later. Simple tests on the DLDR yielded a cycle ratio
plot shown below. The first loading level with a life of N1 was kept constant while the
second loading level with life N2 was varied to investigate claims made by the
authors. It is obvious kneepoint changes with N1/N2 ratio, however, it is also clear that
n/N1 remains unchanged if the first loading level remains the same and thus, the
initiation point for N1 is unaffected by subsequent loading N2. Hence, the physical
model of “initiation” and propagation still holds. In addition, interaction based on the
DLDR for equal ratios of N1/N2 do not yield a uniform kneepoint as claimed by
Manson and Halford. The coordinates of the kneepoint are still controlled by
respective n/N1 and n/N2 of the first and second loading.

10 L1>L2>L3>L4
9
8
7
Crack Length (units)

6
5
4
3
2
1
0
0 20 40 60 80
Life (units)

L1 L2 L3 L4

Fig. 2.30 – Simplified two phase linear crack growth behaviour Manson and Halford.

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Chapter 2: Literature Review

1
L1>L2>L3>L4
0.9
0.8
0.7

n/Nf L2,L4
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1
n/Nf L1, L2

Palmgren-Miner L1-L2 L2-L1

L1-L3 L3-L1 L3-L4
L4-L3

Fig. 2.31 – Cycle ratio curves showing changes in kneepoints for various
combinations of loading sequences using the DLDR.

Using previously established equations, Manson and Halford [105] derived two
equations defining Phase I and Phase II lives.

N I = N f exp( ZN Φf ) (2.30)

N II = N f − N I = N f [1 − exp( ZN Φf )] (2.31)

where,

1 ⎧ Ln[0.35( N 1 / N 2 ) 0.25 ] ⎫
Φ= Ln ⎨ ⎬ (2.32)
Ln( N 1 / N 2 ) ⎩ Ln[1 − 0.65( N 1 / N 2 ) 0.25 ] ⎭

Ln[0.35( N 1 / N 2 ) 0.25 ]
Z= (2.33)
N IΦ

An important implication of the DLDR is that cycle ratio summations within lifetimes
of the two separate phases are independent of the sequence of loading which

99
Chapter 2: Literature Review

simplifies damage accumulation calculations significantly particularly when very

complex loading histories are involved.

O’Neill [81] points out that earlier work by Manson and his associates showed that
while the double linear rule appears to give good results in step tests, it does not seem
significantly better than the linear rule for programme tests. However, fatigue life
predictions based on the refined DLDR seems to correlate well with experimental
results obtained from a host of loading conditions and materials. Nevertheless, the
two-stage rule provides a qualitative description of the observed phenomena
involving multilevel loading.

Modified Dugdale Damage Accumulation ; Wu, Mai and Cotterell [120]

Wu et al. and his colleagues addressed the problem of fatigue damage accumulation
by utilising modifications to the Dugdale model and the Coffin-Manson Law for
strain-controlled fatigue. The authors assumed that the plastic zone ahead of the crack
tip is represented by a series of material elements of a finite width and the successive
failure of each element constitutes incremental crack propagation. A dimensionless
fatigue crack growth model was thus, derived based on the fact that fatigue damage
accumulates only when the plastic stretch range on the material element exceeds some
threshold value, below which, a crack incubation or “initiation” period exists.
Comparisons of theoretical predictions versus experimental results showed
encouraging correlation for a variety of materials. The model predicts the decreasing
trend of the fatigue threshold with increasing stress ratios but underestimates the
effect of the stress ratios. A plausible reason for such a discrepancy is that the model
was essentially developed for plane stress conditions where plasticity induced-crack
closure plays a major role. However, fatigue crack growth in the near-threshold region
is in plane strain, where other sources instead of cyclic plasticity of crack closure
become of importance.

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Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

Chapter 3: Tensile & Baseline Strain-Controlled

Fatigue Testing

3.1 Introduction

Prior to more complex fatigue testing programmes, baseline fatigue parameters as

well as important monotonic mechanical properties, such as ultimate tensile strength,
σUTS, yield strength, σYS, and Young’s Modulus, E, are required. This section details
the materials used, and the experimental procedures and conditions for obtaining both
monotonic and fatigue parameters. In addition, a comparison of results obtained from
monotonic and fatigue loading for all materials tested are discussed. All fatigue and
monotonic tests for smooth specimens were performed on a closed-loop servo-
hydraulic INSTRON™ 8501 testing machine (Figs. 3.1 and 3.2) under total strain-
control. The sample size for each test set is five, unless stated otherwise. Specimens
were sampled randomly and polished before being subjected to testing of any kind.
Load and strain were recorded using a data acquisition system. Strain was measured
over the gauge length of the specimens using a strain extensometer (25mm strain
gauge length) and failure is defined as the complete fracture of the specimen.

Fig. 3.1 – INSTRON™ 8501 servo-hydraulic testing machine.

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Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

Load cell

Specimen

Strain gauge

Specimen Actuator
grips

Fig. 3.2 – Specimen mounted with custom grips.

3.2 Materials and specimen specifications

3.2.1 Materials

Three materials, 316L stainless steel, 6061-T6 aluminium alloy and 4340 high
strength steel were used over the course of this study. Apart from their commonality
and extensive use in engineering components, these materials were selected as a result
of their unique characteristics, suited to the types of experiments conducted. As
mentioned, all three materials are widely used in engineering applications based on
their respective qualities. 316L stainless steel is widely used for processing equipment
in the oil, chemical, food, paper and pharmaceutical industries due to its excellent
corrosion resistant and mechanical properties, while 6061-T6 aluminium alloy has
found considerable use in aircraft fittings, couplings, marine fittings and hardware,
hinge pins, magneto parts, brake pistons, hydraulic pistons, valves, and valve parts,
just to name a few. 4340 high strength steel is considered the standard by which other
high strength steels are compared to. It is often used where severe service conditions
exist and where high strength in heavy sections is required. Typical applications
include bolts, screws, gears, crankshafts, piston rods for engines, landing gear, and
other critical structural members for aircraft.

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 Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

Materials were supplied in round bars and machined according to specifications

stipulated in section 3.2.2. The chemical composition for each material is tabulated in
Tables 3.1 to 3.3 respectively.

Table 3.1 – Chemical composition of 316L stainless steel (% Weight).

C 0.024 Cr 16.700 Mn 1.520 Mo 2.020 N 0.045
Ni 11.100 P 0.027 S 0.024 Si 0.310 Co 0.100
Cu 0.440 Fe remaining

Table 3.2 – Chemical composition of 6061-T6 aluminium alloy (% Weight).

Si 0.40-0.80 Fe 0.70 max Cu 0.15-0.40 Mn 0.15 max Mg 0.80-1.20
Cr 0.04-0.35 Zn 0.25 max Ti 0.15 Others 0.05 max each, 0.15 max
Al remaining total

Table 3.3 – Chemical composition of 4340 high strength steel (% Weight).

Si 0.15-0.35 P 0.025 max Cu 0.35 max Mn 0.6-0.8 Mo 0.20-0.30
C 0.38-0.43 Cr 0.7-0.9 Ni 1.65-2.00 S 0.025 max
Fe remaining

3.2.2 Specimen specifications

Hourglass shaped specimens were machined from round bars in as-received condition
according to the general specifications detailed in Figure 3.2.

31.75
M12 size
6.35 thread

R 50.8 20.0
110.0

Fig. 3.2 – Specimen specifications (dimensions in mm).

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Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

Prior to mechanical testing, specimens were progressively polished with No.400 and
No.1000 emery paper followed by finer grade EPA 1200 and EPA 2400 silicon
carbide paper until visible machining marks were removed to give a smooth mirror-
like surface finish. A low magnification comparison of the surface condition before
and after mechanical polishing is shown in Figure 3.3.

(a) (b)

Fig. 3.3a (left) – Unpolished specimen surface showing machining marks - horizontal
grooves, Fig. 3.3b (right) – Mechanically polished specimen surface.

3.3 Tensile Tests

Testing procedures are in accordance to AS 1391-1991, Methods for Tensile Testing

of Materials. The selection of strain rates used for testing are also in accordance to the
above standard, which specifies that for the measurement of yield, proof and
permanent set stress for metals sensitive to strain rate, the strain rate used should
range between 2.5 X 10-4 s-1 to 2.5 X 10-3 s-1 aimed at a target value of 8 X 10-4 s-1.
For this reason, all monotonic tests (tensile or compressive) were performed using a
uniform strain rate of 6%/min or 1.0 X 10-3 s-1 based on a 25 mm strain gauge length.
Stress-strain curves derived from experimental measurements were then used to
determine the E, σYS, and elastic limit of the material. Stress-strain plots (Figs. 3.4-
3.6) as well as tensile mechanical properties for each material (Table 3.4) are listed
below. It should be noted that the complete stress-strain plot for 316L stainless steel
(Fig. 3.4) was obtained under position-control since the extent of material elongation
in tension far exceeded the range capacity of the strain gauge. Strain measurements
were correlated to position readings by way of using a pre-calibrated conversion
factor. It was found that position and strain readings exhibited a linear relationship,

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Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

which meant that a conversion factor could easily be ascertained by relating position
and strain over a series of measurements during calibration. Mechanical properties of
316L stainless steel were, however, obtained using stress-strain curves obtained under
strain-control, as a result of the greater accuracy associated with strain measurements.

700

600

500
Stress (MPa)

400

300

200

100

0
0 10 20 30 40 50 60
Strain (%)

Tensile Compressive

Fig. 3.4 – Monotonic stress-strain curves for 316L stainless steel.

400

350

300

250
Stress (MPa)

200

150

100

50

0
0 2 4 6 8 10 12 14 16 18 20
Strain (%)

Tensile Compressive

Fig. 3.5 – Monotonic stress-strain curves for 6061-T6 aluminium alloy.

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Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

1200

1000

800

Stress (MPa)
600

400

200

0
0 5 10 15 20
Strain (%)

Tensile Compressive

Fig. 3.6 – Monotonic stress-strain curves for 4340 high strength steel.

Table 3.4 – Tensile mechanical properties*.

Material σY, 0.2% offset σUTS E Elongation

316L stainless steel 508 Mpa 642 MPa 175 GPa 51.5%
6061-T6 aluminium alloy 295 Mpa 324 MPa 68 GPa 19.0 %
4340 high strength steel 880MPa 1063 MPa 179 GPa 15.9 %
* Properties were obtained at ε&= 0.001s −1 or 6%/min.

3.4 Fatigue Tests

All fatigue tests were conducted under total strain-control in uniaxial push-pull mode
using an INSTRON 8501 servo hydraulic testing machine. Testing frequencies varied
from 1 Hz to 10 Hz depending on the strain amplitude used. Although the stress-strain
response of austenitic stainless steels are known to be strain rate sensitive, it has been
determined from trial experiments that any effects on fatigue life resulting from
variations in loading frequencies within the range specified were negligible. This
observation is also true for both 4340 high strength steel and 6061-T6 aluminium
alloy. (NB – strain refers to total strain unless stated otherwise).

The definition of LCF and high cycle fatigue HCF has been a subject of much
ambiguity. In general engineering practice, LCF is associated with fatigue lives

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Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

ranging from 1 to 1000 cycles and HCF is concerned with failure corresponding to
life cycles of greater than 103. There has also been suggestion that LCF, should be
associated with fatigue loading at stress levels above static yield stress, σy, while
fatigue loading at stress levels below σy be classified as HCF. However, a single
regime of either LCF or HCF is difficult to administer under such a definition under
strain-controlled conditions, since HCF, with initial load levels < σy may cyclically
harden, resulting in load levels > σy at later stages of fatigue i.e. initial HCF
conditions will gradually change to LCF. Conversely, what may initially be
considered as LCF, may revert to HCF conditions through cyclic softening. The
problem of using σy to determine the boundary between two distinct regimes of
fatigue is illustrated in Fig. 3.7. So far, the most reliable method for defining the
boundary between LCF and HCF is to determine transition life, Nt, from the fatigue
parameters obtained by fitting stress/strain life trends to the Coffin-Manson equation.
Nt refers to the transition point in which plastic and elastic strains become equivalent
for a specific total strain applied. Fatigue lives less than Nt are classified as LCF while
fatigue lives greater than Nt are considered as HCF. The terms LCF and HCF, apart
from referring to the low and high number of fatigue cycles required for failure, bear
little physical meaning in terms of describing the state of fatigue being imposed.
Even as a reference for the number of fatigue cycles to failure, the terms yield much
ambiguity as it has been found that Nt varies significantly between materials (Nt
ranges from 102-104 cycles depending on the type of material investigated). As a
result, the terms LCF and HCF are abandoned in favour of more descriptive terms
namely Plastically Dominant Fatigue (PDF) and Elastically Dominant Fatigue (EDF)
respectively, since for fatigue lives less than Nt (i.e. PDF), plastic strain contribution
to total strain is larger than elastic strain while for fatigue lives greater than Nt (i.e.
EDF), elastic strain contribution to total strain is larger than plastic strain.

Fatigue parameters were determined in order to define plastically dominant and

elastically dominant fatigue strain amplitudes. Cyclic response tests were carried out
over predetermined strain amplitudes at 0 mean strain to determine the fatigue
parameters (Table 3.8) and cyclic stress-strain curves at various strain amplitudes for
all materials concerned (Figs. 3.17-3.19). Tables 3.5-3.8 summarise the average
fatigue lives after cycling at various strain amplitudes under 0 mean strain condition

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 Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

while strain amplitude-life plots are illustrated in Figures 3.8-3.10. Fatigue parameters
were determined by fitting the cyclic response test results to the Coffin-Manson
relationship:
σ 'f
εa = (2 N f ) b + ε 'f (2 N f ) c (3.1)
E

i.e. ε a = ε ea + ε pa where elastic strain and plastic strain can be defined as follows:

σa σ 'f
ε ea = = (2 N f ) b (3.2)
E E

ε pa = ε 'f (2 N f ) c (3.3)

Table 3.5 – Fatigue lives (cycles) at various strain amplitudes, 0 mean strain for 316L
stainless steel.
Strain amplitude (%) Test 1 Test 2 Test 3 Test 4 Test 5 Avg. Cycles to failure
0.3 28244 27163 26181 27380 33514 28496
0.4 11927 8680 10942 11723 11570 10968
0.6 4270 4617 3529 4796 4053 4253
0.8 984 1404 1481 1509 1046 1285
1.0 591 726 855 676 409 651

σYSH Cyclic hardened σ-ε curve

Monotonic σ-ε curve

σCYS
Stress, σ

σYSS Cyclic softened σ-ε curve

Strain, ε

Fig. 3.7 – PDF/EDF classification based on strain at σCYS,σYSS and σYSH.

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 Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

1.2

0.8

Strain amp (%)

0.6

0.4

0.2

0
0 5000 10000 15000 20000 25000 30000 35000 40000
Fatigue life (cycles)

Avg. fatigue life Experimental fatigue life

Fig. 3.8 – Stress amplitude vs Fatigue life curve for 316L stainless steel.

Table 3.6 – Fatigue lives (cycles) at various strain amplitudes, 0 mean strain for 6061-
T6 aluminium alloy.
Avg. Cycles to
Strain amplitude (%) Test 1 Test 2 Test 3 Test 4 Test 5 Test 6
failure
0.3 42647 42611 46519 33128 32227 - 39426
0.4 4388 3258 3616 5259 5416 - 4387
0.5 1908 1524 1344 889 934 1079 1280
0.6 911 741 749 947 - - 837
1.0 159 199 189 158 - - 176

1.2

1
Strain amp (%)

0.8

0.6

0.4

0.2

0
0 10000 20000 30000 40000 50000
Fatigue life (cycles)

Avg. fatigue life Experimental fatigue life

Fig. 3.9 – Stress amplitude vs Fatigue life curve for 6061-T6 aluminium alloy.

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 Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

Table 3.7 – Fatigue lives (cycles) at various strain amplitudes, 0 mean strain for 4340
high strength steel.
Strain amplitude (%) Test 1 Test 2 Test 3 Test 4 Test 5 Avg. Cycles to failure
0.3 35188 43154 35750 42136 44154 40076
0.4 11497 9858 9071 6128 8870 9085
0.6 1886 1893 2555 3780 3020 2627
0.8 798 940 791 738 943 842
1.0 557 434 426 618 636 534

Fig. 3.10 – Stress amplitude vs Fatigue life curve for 4340 high strength steel.

1.2

0.8
Strain amp (%)

0.6

0.4

0.2

0
0 10000 20000 30000 40000 50000
Fatigue life (cycles)

Avg. fatigue life Experimental fatigue life

Table 3.8 – Fatigue parameters of 316L stainless steel, 6061-T6 aluminium alloy and
4340 high strength steel.
Material σ’f (MPa) b ε’f c Nt
316L stainless steel 779.1 -0.098 0.2089 -0.449 28232
6061-T6 aluminium alloy 573.8 -0.085 2.912 -1.063 197
4340 high strength steel 1038.9 -0.061 0.3086 -0.565 1332

From these parameters, transition life or the boundary between plastically dominant
and elastically dominant fatigue for each material was determined from Eq. 3.4,
yielding 28232 cycles for 316L stainless steel, 197 cycles for 6061-T6 aluminium
alloy and 1332 cycles for 4340 high strength steel respectively.

110
Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

1 ⎛⎜ σ f ⎞ c − b
'

Nt = ⎟ (3.4)
2 ⎝ ε 'f E ⎠

Therefore, in the case of 316L stainless steel, strain amplitudes yielding fatigue lives
greater than 28232 cycles were classified as EDF while amplitudes yielding lower
lives were classified as PDF. Similarly for 6061-T6 aluminium alloy and 4340 high
strength steel, strain amplitudes yielding fatigue lives greater than 197 cycles and
1332 cycles respectively were classified as EDF while amplitudes yielding lower lives
were classified as PDF. However, it was later found, that using the Coffin-Manson
approach to define PDF and EDF is inadequate, since EDF close to Nt still involves
considerable plastic strain; enough to induce mean stress effects during the secondary
stage of a 2-step EDF(Hi)→EDF(Lo) multilevel loading (Chapter 4), a characteristic
which is supposedly inherent only in PDF(Hi)→EDF(Lo) type loading sequences. It
is known that under certain instances of multilevel loading, depending on how large
the preceding strain amplitude is and how small the follow up amplitude is; mean
stress could be induced during the follow up stage, thereby, significantly influencing
fatigue life trends. The question is under what conditions and under what
combinations of fatigue amplitudes do mean stress get induced and having it playing a
prominent role in influencing fatigue life trends? Hence, an alternative method for
classifying PDF and EDF is required; one which meets both criteria of: i) terms used
to associate the various fatigue regimes should correlate to physical characteristics ii)
provide indication on the influence of mean stress on fatigue life. It is proposed that
the boundary between PDF and EDF be represented by the cyclic yield strain
determined by implementing a 0.05% offset to the cyclic stress-strain curve, similar to
the 0.2% strain offset method used to determine the yield point for monotonic
loading. A 0.2% strain offset was not used in this case as it was discovered that it too
yielded too much plastic strain for elastic conditions to prevail. For the purposes of
the current analyses, a 0.05% offset is deemed suitable to determine the cyclic yield
strains for the three materials from their respective cyclic stress-strain curves (Figs.
3.11-3.13). Results are compared to PDF/EDF (Nt based) boundary strain amplitudes
obtained via the Coffin-Manson relationship (Table 3.9). It should be noted that the
percentage strain offset should be selected to ‘best’ reflect the transition point
between EDF and PDF (i.e. if the offset boundary is appropriately selected,

111
Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

PDF→EDF interactions would clearly manifest mean stress effects upon step-down,
while EDF→EDF interactions will not exhibit any obvious mean stress effects.
Details and results pertaining to step loading are discussed in Chapter 4).

Table 3.9 – PDF/EDF boundary for the three materials investigated.

PDF/EDF boundary PDF/EDF boundary
Material Nt
(based on Nt)* (cyclic yield strain)*
316L stainless steel 28232 0.3% 0.17%
6061-T6 aluminium alloy 197 0.98% 0.46%
4340 high strength steel 1332 0.73% 0.38%
* Values refer to strain amplitude.

Note that the cyclic yield strain amplitude for 316L stainless steel was determined by
estimating the elastic portion of the cyclic stress-strain curve using the Ramberg-
Osgood relationship (Eq. 3.5),
1
σ a ⎛σ a ⎞n
εa = +⎜ ⎟ (3.5)
E ⎝H⎠

where n =b/c and H = σ’f/(ε’fn). It is clear to see from Figures 3.11-3.13 that the
predicted cyclic stress-strain curves correspond well with actual experimental results
for all materials concerned.

450

400
0.05% strain offset
350

300
Stress amp (MPa)

250

200

150

100

50

0
0 0.002 0.004 0.006 0.008 0.01 0.012
Strain

Ramberg-Osgood Actual cyclic stress amplitude at half life

Fig. 3.11 – Cyclic stress-strain curve for 316L stainless steel.

112
Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

400
0.05% strain offset
350

300

Stress amp (MPa)

250

200

150

100

50

0
0 0.002 0.004 0.006 0.008 0.01 0.012
Strain

Ramberg-Osgood Actual cyclic stress amplitude at half life

Fig. 3.12 – Cyclic stress-strain curve for 6061-T6 aluminium alloy.

800

700
0.05% strain offset

600

500
Stress amp (MPa)

400

300

200

100

0
0 0.002 0.004 0.006 0.008 0.01 0.012
Strain

Ramberg-Osgood Actual cyclic stress amplitude at half life

Fig. 3.13 – Cyclic stress-strain curve for 4340 high strength steel.

Henceforth, all PDF/EDF classification will be based on cyclic yield strain amplitudes
determined from cyclic stress-strain curves. The inadequacies of using Nt to classify
PDF/EDF boundary will be highlighted and discussed in Chapter 4. Figures 3.14-3.16
illustrate strain amplitude vs fatigue life curves for both fitted and experimental
results of the three materials concerned. Fitted curves were obtained by substituting
the relevant parameters tabulated in Table 3.8 into the Coffin-Manson relationship
(3.1). Results in all three plots (Figs. 3.14-3.16) show that experimental and fitted
curves based on the Coffin-Manson relationship, correspond well with each other.

113
Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

1.8
1.6
1.4

Strain amp (%)

1.2
1
0.8
0.6
0.4
0.2
0
100 1000 10000 100000 1000000
Total life to failure (cycles)

Total strain (%) Elastic strain (%) Plastic strain (%)

Fitted total strain Fitted elastic strain Fitted plastic strain

Fig. 3.14 – Stress amplitude vs Fatigue life (experimental and fitted) curves for 316L
stainless steel.

1.2

1
Strain amp (%)

0.8

0.6

0.4

0.2

0
100 1000 10000 100000 1000000
Total life to failure (cycles)

Total strain (%) Elastic strain (%) Plastic strain (%)

Fitted total strain Fitted elastic strain Fitted plastic strain

Fig. 3.15 – Stress amplitude vs Fatigue life (experimental and fitted) curves for 6061-
T6 aluminium alloy.

114
Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

1.8
1.6
1.4
1.2
Strain amp (%)

1
0.8
0.6
0.4
0.2
0
100 1000 10000 100000 1000000
Total life to failure (cycles)

Total strain (%) Elastic strain (%) Plastic strain (%)

Fitted total strain Fitted elastic strain Fitted plastic strain

Fig. 3.16 – Stress amplitude vs Fatigue life (experimental and fitted) curves for 4340
high strength steel.

15

10

5
Load (kN)

0
-1.5 -1 -0.5 0 0.5 1 1.5

-5

-10

-15
Strain (%)

0.3% amp 0.4% amp 0.6% amp 0.8% amp 1.0% amp

Fig. 3.17 – Cyclic stress-strain response of 316L stainless steel at half-life.

115
Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

15

10

5
Load (kN)

0
-1.5 -1 -0.5 0 0.5 1 1.5

-5

-10

-15
Strain (%)

0.3% amp 0.4% amp 0.5% amp 0.6% amp 1.0% amp

Fig. 3.18 – Cyclic stress-strain response of 6061-T6 aluminium alloy at half-life.

25

20

15

10

5
Load (kN)

0
-1.5 -1 -0.5 0 0.5 1 1.5
-5

-10

-15

-20

-25
Strain (%)

0.3% amp 0.4% amp 0.6% amp 0.8% amp 1.0% amp

Fig. 3.19 – Cyclic stress-strain response of 4340 high strength steel at half-life.

116
Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

Comparisons between cyclic stress-strain response curves and monotonic stress-strain

curves (Figs. 3.4-3.6) are shown in Figures 3.20-3.22. The cyclic stress-strain curves
obtained by joining the tips of stable hysteresis loops (Figs. 3.17-3.19) of various
strain amplitudes show lower stress-strain curves than that observed for monotonic
loading conditions, indicating a cyclic strain softening behaviour for both 316L
stainless steel (Fig. 3.20) and 4340 high strength steel (Fig. 3.22) when subjected to
fatigue. On the other hand, the cyclic stress-strain curve for 6061-T6 aluminium alloy
(Fig. 3.21) is higher than stress-strain curves obtained for monotonic conditions
indicating a cyclic hardening behaviour. Note that hysteresis loops are plotted at half-
life since it is typically expected that stability (i.e. no further hardening or softening)
is reached at half-life.

700

600

500
Stress (MPa)

400

300

200

100

0
0 10 20 30 40 50 60
Strain (%)

Tensile Cyclic

Fig. 3.20 – Monotonic / cyclic stress-strain curves for 316L stainless steel.

350

300

250
Stress (MPa)

200

150

100

50

0
0 2 4 6 8 10 12 14 16 18 20
Strain (%)

Tensile Cyclic

Fig. 3.21 – Monotonic / cyclic stress-strain curves for 6061-T6 aluminium alloy.

117
Chapter 3: Tensile & Baseline Strain-Controlled Fatigue Testing

1200

1000

Stress (MPa)
800

600

400

200

0
0 5 10 15 20
Strain (%)

Tensile Cyclic

Fig. 3.22 – Monotonic /cyclic stress-strain curves for 4340 high strength steel.

118
Chapter 4: 2-Level Loading

Chapter 4: 2-Level Loading

4.1 Introduction

Many engineering components will experience non-uniform loading histories during

the course of their service life. The unpredictable nature of random fatigue has been
the basis for motivating research on studying the effects of multilevel loading and the
continual development of more reliable and accurate life prediction models.

The study of complex loading histories and its associated mechanisms commences
with the investigation of multilevel loading in its simplest form i.e. a 2-level or 2-step
loading sequence. Results pertaining to the interaction between High→Low
(stress/strain) and Low→High (stress/strain) loading have been reported extensively
[46, 66, 80, 99, 100, 104, 108-111]. Yet, the underlying fatigue mechanisms
responsible for such interactions are not fully understood. As mentioned in section
2.4, High→Low sequences generally yield cycle ratio summations that are < 1, while
summation values are typically >1 for Low→High sequences. It can be appreciated
that the mechanisms governing fatigue behaviour under elastically dominant
conditions differ from those observed under predominantly plastic conditions.
Although there does not seem to be any literature reporting specifically on plastically
and elastically dominant fatigue interaction, examples of recent work dealing with
LCF and HCF interaction can be found in [112-115] while a rather detailed two part
literature review has been published by Skorupa [46, 109], discussing the reported
trends in fatigue crack growth observed under variable amplitude loading on metallic
materials and the mechanisms underlying such interactions.

This chapter presents results on the interaction between PDF and EDF in 316L
stainless steel, 6061-T6 aluminium alloy and 4340 high strength steel. Possible
underlying mechanisms responsible for the trends observed under different loading
regimes are discussed.

119
Chapter 4: 2-Level Loading

4.2 Materials and testing program

Materials and specimens used in this investigation are identical to those specified in
Chapter 3. Two basic loading sequences were investigated, High→Low, and
Low→High. Schematic illustrations of the various loading spectrums tested are
shown in Figures 4.1-4.10. Specimens were introduced to varying degrees of phase I
loading (PDF or EDF) before subsequent phase II (PDF or EDF) loading to complete
failure. Test programs investigated for each material and the respective strain
amplitudes used are tabulated in Table 4.1. Regimes of PDF and EDF for each
material are classified according to cyclic yield strains (0.05% offset) tabulated in
Table 3.9 of Chapter 3. All tests were performed in total strain-control, under zero
mean strain conditions. (NB – strain refers to total strain unless stated otherwise).

PDF

EDF
εa

Failure

Transition between PDF/EDF

Fig. 4.1 – (Type A) High→Low loading sequence going from PDF to EDF.

PDF(Hi)
PDF(Lo)
εa

Failure

Transition between PDF/PDF

Fig. 4.2 – (Type B) High→Low loading sequence going from PDF to PDF.

120
Chapter 4: 2-Level Loading

EDF(Hi)
EDF(Lo)

εa
Failure

Transition between EDF/EDF

Fig. 4.3 – (Type C) High→Low loading sequence going from EDF to EDF.
PDF
EDF
εa

Failure

Transition between EDF/PDF

Fig. 4.4 – (Type D) Low→High loading sequence going from EDF to PDF.

EDF(Lo) EDF(Hi)
εa

Failure

Transition between EDF/EDF

Fig. 4.5 – (Type E) Low→High loading sequence going from EDF to EDF.

PDF
EDF
εa

Failure

Transition between PDF/EDF (after PDF tensile phase)

Fig. 4.6 – (Type F) High→Low loading sequence going from PDF to EDF.

121
Chapter 4: 2-Level Loading

PDF(Hi)
PDF(Lo)

εa
Failure

Transition between PDF/PDF (after 1st PDF tensile phase)

Fig. 4.7 – (Type G) High→Low loading sequence going from PDF to PDF.

PDF(Hi)
PDF(Lo)
εa

Failure

Transition between PDF/PDF

Fig. 4.8 – (Type H) Low→High loading sequence going from PDF to PDF.

EDF(Hi)
EDF(Lo)
εa

Failure

Transition between EDF/EDF (after 1st EDF tensile phase)

Fig. 4.9 – (Type I) High→Low loading sequence going from EDF to EDF.

PDF
EDF
εa

Failure

Transition between EDF/PDF (after 1st EDF tensile phase)

Fig. 4.10 – (Type J) Low→High loading sequence going from EDF to PDF.

122
 Chapter 4: 2-Level Loading

Table 4.1 – Loading spectrums with corresponding strain amplitudes used.

Material Loading spectrum type Strain amplitudes used
316L stainless steel B, H PDF(0.8%), PDF(0.3%)1
6061-T6 aluminium alloy A, B, C, D, E, F, G, I PDF(1.0%), PDF(0.6%)1,
EDF(0.4%)2, EDF(0.3%)
4340 high strength steel A, B, C, D, F, G, I,J PDF(1.0%), PDF(0.6%)1,
EDF(0.3%)2, EDF(0.25%)
1
Lower amplitude PDF for types B,G and H loading spectrums.
2
Higher amplitude EDF for types C,E and I loading spectrums.

4.3 Experimental results

Figures 4.11-4.28 show the fatigue life trends, while Figures 4.29-4.35 show cycle
ratio trends after various loading regimes. The average total fatigue lives for similar
regimes are tabulated in Tables 4.2-4.19.

30000

Palmgren-Miner PDF (0.8% strain amp)

25000 →PDF (0.3% strain amp)

PDF (0.8% strain amp)→PDF (0.3% strain

Total Life (Cycles)

20000
amp) with error bar

15000

10000

5000

0
0 20 40 60 80 100
% initial PDF intro

Fig. 4.11 – Type B PDF(Hi)→PDF(Lo) fatigue life trend for 316L stainless steel.

123
Chapter 4: 2-Level Loading

30000

25000

Total Life (Cycles)

20000

15000

Palmgren-Miner PDF (0.3% strain amp)

10000 →PDF (0.8% strain amp)

5000 PDF (0.3% strain amp)→PDF (0.8% strain

amp) with error bar

0
0 20 40 60 80 100
% initial PDF intro

Fig. 4.12 – Type H PDF(Lo)→PDF(Hi) fatigue life trend for 316L stainless steel.

Table 4.2 – Average fatigue lives (cycles) after PDF (0.8% strain amp)→PDF (0.3%
strain amp) loading for 316L stainless steel (Type B).
% PDF (0.8% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 28496 28496
27 13447 21147
43 11353 16742
65 6225 10854
100 1285 1285

124
Chapter 4: 2-Level Loading

Table 4.3 – Average fatigue lives (cycles) after PDF (0.3% strain amp)→PDF (0.8%
strain amp) loading for 316L stainless steel (Type H).
% PDF (0.3% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 1285 1285
13 4848 4838
33 10267 10167
52 15535 15496
78 22590 22602
100 28496 28496

45000 Palmgren-Miner PDF (1.0% strain amp)

→EDF (0.3% strain amp)
40000

PDF (1.0% strain amp)→PDF (0.3% strain

35000
amp) with error bar
Total Life (Cycles)

30000

25000

20000

15000

10000

5000

0
0 20 40 60 80 100
% initial PDF intro

Fig. 4.13 – Type A PDF→EDF fatigue life trend for 6061-T6 aluminium alloy.

125
Chapter 4: 2-Level Loading

45000

40000

35000

30000

Total Life (Cycles)

25000

20000

15000 Palmgren-Miner EDF (0.3% strain amp)

→PDF (1.0% strain amp)
10000
EDF (0.3% strain amp)→PDF (1.0%
5000 strain amp) with error bar
0
0 20 40 60 80 100
% initial EDF intro

Fig. 4.14 – Type D EDF→PDF fatigue life trend for for 6061-T6 aluminium alloy.

Table 4.4 – Average fatigue lives (cycles) after PDF (1.0% strain amp)→EDF (0.3%
strain amp) loading for 6061-T6 aluminium alloy (Type A).
% PDF (1.0% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 39426 39426
17 14653 32746
34 13028 26065
51 9931 19384
68 7476 12703
100 176 176

126
Chapter 4: 2-Level Loading

Table 4.5 – Average fatigue lives (cycles) after EDF (0.3% strain amp)→PDF (1.0%
strain amp) loading for 6061-T6 aluminium alloy (Type D).
% EDF (0.3% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 176 176
5 2187 2184
13 5234 5236
26 10298 10216
41 16398 16240
63 25146 25065
100 39426 39426

100000
Palmgren-Miner PDF (1.0% strain amp)
→EDF (0.3% strain amp)
90000
PDF (1.0% strain amp)→PDF (0.3% strain
80000
amp) with error bar
70000
Total Life (Cycles)

60000

50000

40000

30000

20000

10000

0
0 20 40 60 80 100
% of initial PDF intro

Fig. 4.15 – Type F PDF→EDF fatigue life trend for for 6061-T6 aluminium alloy.

127
Chapter 4: 2-Level Loading

Table 4.6 – Average fatigue lives (cycles) after PDF (1.0% strain amp)→EDF (0.3%
strain amp) loading for 6061-T6 aluminium alloy (Type F).
% PDF (1.0% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 39426 39426
17 72733 32746
51 39972 19384
68 31815 12703
100 176 176

45000
Palmgren-Miner EDF (0.4% strain amp)
→EDF (0.3% strain amp)
40000

35000 EDF (0.4% strain amp)→EDF (0.3% strain

amp) with error bar
Total Life (Cycles)

30000

25000

20000

15000

10000

5000

0
0 20 40 60 80 100
% initial EDF intro

Fig. 4.16 – Type C EDF(Hi)→EDF(Lo) fatigue life trend for 6061-T6 aluminium
alloy.

128
Chapter 4: 2-Level Loading

45000

40000

35000

Total Life (Cycles)

30000

25000

20000

15000
Palmgren-Miner EDF (0.3% strain amp)
→EDF (0.4% strain amp)
10000

5000
EDF (0.3% strain amp)→EDF (0.4%
strain amp) with error bar
0
0 20 40 60 80 100
% initial EDF intro

Fig. 4.17– Type E EDF(Lo)→EDF(Hi) fatigue life trend for 6061-T6 aluminium
alloy.

Table 4.7 – Average fatigue lives (cycles) after EDF (0.4% strain amp)→EDF (0.3%
strain amp) loading for 6061-T6 aluminium alloy (Type C).
% EDF (0.4% strain Avg. total fatigue life Palmgren-Miner
amp) intro (cycles) predictions (cycles)
0 39426 39426
10 32489 35922
60 17173 18402
100 4387 4387

129
Chapter 4: 2-Level Loading

Table 4.8 – Average fatigue lives (cycles) after EDF (0.3% strain amp)→EDF (0.4%
strain amp) loading for 6061-T6 aluminium alloy (Type E).
% EDF (0.3% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 4387 4387
5 7670 6181
26 13967 13349
41 20789 18729
100 39426 39426

45000
Palmgren-Miner EDF (0.4% strain amp)
40000 →EDF (0.3% strain amp)

35000 EDF (0.4% strain amp)→EDF (0.3%

strain amp) with error bar
30000
Total Life (Cycles)

25000

20000

15000

10000

5000

0
0 20 40 60 80 100
% initial EDF intro

Fig. 4.18 – Type I EDF(Hi)→EDF(Lo) fatigue life trend for 6061-T6 aluminium
alloy.

130
Chapter 4: 2-Level Loading

Table 4.9 – Average fatigue lives (cycles) after EDF (0.4% strain amp)→EDF (0.3%
strain amp) loading for 6061-T6 aluminium alloy (Type I).
% EDF (0.4% strain Avg. total fatigue life Palmgren-Miner
amp) intro (cycles) predictions (cycles)
0 39426 39426
10 29392 35922
60 17191 18402
100 4387 4387

900 Palmgren-Miner PDF (1.0% strain amp)

→PDF (0.6% strain amp)
800
PDF (1.0% strain amp)→PDF (0.6% strain
700 amp) with error bar
Total Life (Cycles)

600

500

400

300

200

100

0
0 20 40 60 80 100
% initial PDF intro

Fig. 4.19 – Type B PDF(Hi)→PDF(Lo) fatigue life trend for 6061-T6 aluminium
alloy.

131
Chapter 4: 2-Level Loading

900
Palmgren-Miner PDF (1.0% strain amp)
→PDF (0.6% strain amp)
800
PDF (1.0% strain amp)→PDF (0.6% strain
700 amp) with error bar

600

Total Life (Cycles)

500

400

300

200

100

0
0 20 40 60 80 100
% initial PDF intro

Fig. 4.20 – Type G PDF(Hi)→PDF(Lo) fatigue life trend for 6061-T6 aluminium
alloy.

Table 4.10 – Average fatigue lives (cycles) after PDF (1.0% strain amp)→PDF (0.6%
strain amp) loading for 6061-T6 aluminium alloy (Type B).
% PDF (1.0% strain Avg. total fatigue life Palmgren-Miner
amp) intro (cycles) predictions (cycles)
0 835 835
51 545 499
100 176 176

Table 4.11 – Average fatigue lives (cycles) after PDF (1.0% strain amp)→PDF (0.6%
strain amp) loading for 6061-T6 aluminium alloy (Type G).
% PDF (1.0% strain Avg. total fatigue life Palmgren-Miner
amp) intro (cycles) predictions (cycles)
0 835 835
51 541 499
100 176 176

132
Chapter 4: 2-Level Loading

45000
Palmgren-Miner PDF (1.0% strain
amp)→EDF (0.3% strain amp)
40000

PDF (1.0% strain amp)→EDF (0.3%

35000
strain amp) with error bar
30000
Total Life (Cycles)

25000

20000

15000

10000

5000

0
0 20 40 60 80 100
% initial PDF intro

Fig. 4.21 – Type A PDF→EDF fatigue life trend for 4340 high strength steel.

45000

40000

35000
Total Life (Cycles)

30000

25000

20000

15000 Palmgren-Miner EDF (0.3% strain

amp)→PDF (1.0% strain amp)
10000
EDF (0.3% strain amp)→PDF (1.0%
5000 strain amp) with error bar

0
0 20 40 60 80 100
% initial EDF intro

Fig. 4.22 – Type D EDF→PDF fatigue life trend for 4340 high strength steel.

133
Chapter 4: 2-Level Loading

Table 4.12 – Average fatigue lives (cycles) after PDF (1.0% strain amp)→EDF (0.3%
strain amp) loading for 4340 high strength steel (Type A).
% PDF (1.0% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 40076 40076
2 16611 39336
4 12838 38596
9 14634 36375
37 7240 25272
100 534 534

Table 4.13 – Average fatigue lives (cycles) after EDF (0.3% strain amp)→PDF (1.0%
strain amp) loading for 4340 high strength steel (Type D).
% EDF (0.3% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 534 534
12 5540 5468
62 25457 25201
100 40076 40076

134
Chapter 4: 2-Level Loading

45000

Palmgren-Miner PDF (1.0% strain

40000
amp)→EDF (0.3% strain amp)
35000
PDF(1.0% strain amp)→EDF (0.3%

Total Life (Cycles)

30000 strain amp) with error bar

25000

20000

15000

10000

5000

0
0 20 40 60 80 100
% initial PDF intro

Fig. 4.23 – Type F PDF→EDF fatigue life trend for 4340 high strength steel.

Table 4.14 – Average fatigue lives (cycles) after PDF (1.0% strain amp)→EDF (0.3%
strain amp) loading for 4340 high strength steel (Type F).
% PDF (1.0% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 40076 40076
2 22616 39399
38 5978 25235
100 534 534

135
Chapter 4: 2-Level Loading

45000
Palmgren-Miner EDF (0.3% strain
40000
amp)→PDF (1.0% strain amp)

35000 EDF(0.3% strain amp)→PDF (1.0%

strain amp) with error bar

Total Life (Cycles)

30000

25000

20000

15000

10000

5000

0
0 20 40 60 80 100
% initial EDF intro

Fig. 4.24 – Type J EDF→PDF fatigue life trend for 4340 high strength steel.

Table 4.15 – Average fatigue lives (cycles) after EDF (0.3% strain amp)→PDF (1.0%
strain amp) loading for 4340 high strength steel (Type J).
% EDF (0.3% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 40076 534
62 25506 25201
100 534 40076

136
Chapter 4: 2-Level Loading

3000
Palmgren-Miner PDF (1.0% strain
amp)→PDF (0.6% strain amp)
2500
PDF(1.0% strain amp)→PDF (0.6%
strain amp) with error bar

Total Life (Cycles)

2000

1500

1000

500

0
0 20 40 60 80 100
% initial PDF intro

Fig. 4.25 – Type B PDF(Hi)→PDF(Lo) fatigue life trend for 4340 high strength steel.

3000
Palmgren-Miner PDF (1.0% strain
amp)→PDF (0.6% strain amp)
2500
PDF(1.0% strain amp)→PDF (0.6%
strain amp) with error bar
2000
Total Life (Cycles)

1500

1000

500

0
0 20 40 60 80 100
% initial PDF intro

Fig. 4.26 – Type G PDF(Hi)→PDF(Lo) fatigue life trend for 4340 high strength steel.

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Chapter 4: 2-Level Loading

Table 4.16 – Average fatigue lives (cycles) after PDF (1.0% strain amp)→PDF (0.6%
strain amp) loading for 4340 high strength steel (Type B).
% PDF (1.0% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 2627 2627
9 2167 2431
100 534 534

Table 4.17 – Average fatigue lives (cycles) after PDF (1.0% strain amp)→PDF (0.6%
strain amp) loading for 4340 high strength steel (Type G).
% PDF (1.0% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 2627 2627
9 2085 2429
100 534 534

900000 Palmgren-Miner EDF (0.3% strain

amp)→EDF (0.25% strain amp)
800000

700000
EDF(0.3% strain amp)→EDF (0.25%
strain amp) with error bar
Total Life (Cycles)

600000

500000

400000

300000

200000

100000

0
0 20 40 60 80 100
% initial EDF intro

Fig. 4.27 – Type C EDF(Hi)→EDF(Lo) fatigue life trend for 4340 high strength steel.

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Chapter 4: 2-Level Loading

900000
Palmgren-Miner EDF (0.3% strain
800000 amp)→EDF (0.25% strain amp)

700000 EDF(0.3% strain amp)→EDF (0.25%

strain amp) with error bar
600000

Total Life (Cycles)

500000

400000

300000

200000

100000

0
0 20 40 60 80 100
% initial EDF intro

Fig. 4.28 – Type I EDF(Hi)→EDF(Lo) fatigue life trend for 4340 high strength steel.

Table 4.18 – Average fatigue lives (cycles) after EDF (0.3% strain amp)→EDF
(0.25% strain amp) loading for 4340 high strength steel (Type C).
% EDF (0.3% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 850000 850000
80 83025 203296
100 40076 40076

Table 4.19 – Average fatigue lives (cycles) after EDF (0.3% strain amp)→EDF
(0.25% strain amp) loading for 4340 high strength steel (Type I).
% EDF (0.3% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 850000 850000
80 71586 203297
100 40076 40076

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Chapter 4: 2-Level Loading

1 Palmgren-Miner
PDF(1.0% amp)-PDF(0.3% amp) Type B
0.9 PDF(0.3% amp)-PDF(1.0% amp)% amp) Type H
0.8

n/Nf, PDF (0.3% amp) = 28496 cycles

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf, PDF(1.0% amp) = 651 cycles

Fig. 4.29 – Cycle ratio trends for 316L stainless steel.

2 Palmgren-Miner
1.8 PDF(1.0% amp)-EDF(0.3% amp) Type A

1.6 PDF(1.0% amp)-EDF(0.3% amp) Type F

n/Nf, EDF = 39426 cycles

EDF(0.3% amp)-PDF(1.0% amp) Type D

1.4

1.2

0.8

0.6

0.4

0.2

0
0 0.5 1 1.5
n/Nf, PDF = 176 cycles

Fig. 4.30 – PDF/EDF cycle ratio trends for 6061-T6 aluminium alloy.

140
Chapter 4: 2-Level Loading

1 Palmgren-Miner
EDF(0.4% amp)-EDF(0.3% amp) Type C
0.9 EDF(0.4% amp)-EDF(0.3% amp) Type I

n/Nf, EDF (0.3% amp) = 39426 cycles

EDF(0.3% amp)-EDF(0.4% amp) Type E
0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.5 1 1.5
n/Nf, EDF(0.4% amp) = 4387 cycles

Fig. 4.31 – EDF/EDF cycle ratio trends for 6061-T6 aluminium alloy.

Palmgren-Miner
1 PDF(1.0% amp)-PDF(0.6% amp) Type B
PDF(1.0% amp)-PDF(0.6% amp) Type G
n/Nf, PDF (0.6% amp) = 835 cycles

0.8

0.6

0.4

0.2

0
0 0.2 0.4 0.6 0.8 1
n/Nf, PDF(1.0% amp) = 176 cycles

Fig. 4.32 – PDF/PDF cycle ratio trends for 6061-T6 aluminium alloy.

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Chapter 4: 2-Level Loading

Palmgren-Miner
PDF(1.0% amp)-EDF(0.3% amp) Type A
PDF(1.0% amp)-EDF(0.3% amp) Type F
EDF(0.3% amp)-PDF(1.0% amp) Type D
1
EDF(0.3% amp)-PDF(1.0% amp) Type J
0.9

0.8
n/Nf, EDF = 40096 cycles

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf, PDF = 534 cycles

Fig. 4.33 – PDF/EDF cycle ratio trends for 4340 high strength steel.

1
Palmgren-Miner
0.9
PDF(1.0% amp)-PDF(0.6% amp) Type B
n/Nf, PDF (0.6% amp) = 835 cycles

0.8 PDF(1.0% amp)-PDF(0.6% amp) Type G

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf, PDF(1.0% amp) = 534 cycles

Fig. 4.34 – PDF/PDF cycle ratio trends for 4340 high strength steel.

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Chapter 4: 2-Level Loading

Palmgren-Miner
1
EDF(0.3% amp)-EDF(0.25% amp) Type C
0.9

n/Nf, EDF (0.25% amp) = 850000 cycles

EDF(0.3% amp)-EDF(0.25% amp) Type I
0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf, EDF(0.3% amp) = 40076 cycles

Fig. 4.35 – EDF/EDF cycle ratio trends for 4340 high strength steel.

4.4 Discussion

In general, fatigue life trends curves observed for High→Low sequences (Figs. 4.11,
4.13, 4.16, 4.18, 4.21, 4.23, 4.25-4.28) deviate from, and are consistently less than
classical Palmgren-Miner predictions. The only notable exceptions are shown in
Figs. 4.15, 4.19-4.20, where fatigue life data do not conform to typical trends
expected for High→Low sequences. Fatigue life trends for Low→High sequences
(Figs. 4.12, 4.14, 4.17, 4.22, 4.24), on the other hand, yielded overall fatigue lives
consistently greater than Palmgren-Miner predictions. However, the beneficial effect
of such a loading sequence is perhaps not well represented in total life plots, since the
increase in PDF cycles remaining is relatively small in comparison to overall fatigue
lives. As a result, fatigue life analysis based on total life plots can be misleading,
since overall fatigue life data points shown in total life plots often appear to mirror
Palmgren-Miner predictions. Cycle ratio plots shown in Figures 4.29-4.35, better
illustrate the effects of sequence loading on overall fatigue lives in comparison with
Palmgren-Miner predictions.

Before discussing the mechanisms responsible for the cycle ratio trends observed, it is
important to emphasise that the only way of realistically appreciating the observed

143
Chapter 4: 2-Level Loading

fatigue behaviour, is by adopting a fracture mechanics approach in understanding

crack growth behaviour under different loading regimes and conditions. As a result,
all mechanisms discussed will involve crack growth and appreciating how changes in
loading conditions could affect crack growth behaviour and subsequently, overall
fatigue life trends. Mentioned in section 2.4, the term “damage”, is a loosely coined
term often used to describe material degradation during fatigue. However, in this
thesis, damage is referred to as crack length, and the term “damage sum” will,
henceforth, be termed as cycle ratio summation to avoid confusion.

Even though overall fatigue lives decrease with increasing degrees of initial PDF
exposure, it is obvious that the proportionate decrease (i.e. deviation from linear
“damage” rule predictions) in overall fatigue lives is greater at lower degrees of initial
PDF exposure (Fig. 4.36). Similar trends were also observed for EDF(Hi)→EDF(Lo)
sequences. This observation can be explained using crack closure concepts. Short
cracks have minimal development of closure along their limited wake, hence crack
closure and subsequent crack growth retardation during EDF, following transition at
low degrees of initial PDF, would be limited as well. Consequently, a limitation in
crack closure contributes to less than expected fatigue lives and also larger
proportionate decreases in overall fatigue lives at lower degrees of initial PDF
exposure. On the other hand, long cracks have greater crack closure resulting from a
fully developed wake. Thus, the extent of crack closure reaches a saturation state
where no significant increases in closure develops with continued crack growth. As a
result, fatigue lives at higher degrees of initial PDF exposure can be seen decreasing
in a more consistent fashion with increasing PDF exposure due to a uniform crack
closure effect.

The above theory is substantiated by similar arguments presented to explain crack

growth behaviour during High→Low load sequences [66]. It is argued that for cracks
shorter than 0.1mm, closure is insufficient to control the crack growth behaviour and
the degradation (“damage”) zone introduced by high level loading, giving rise to an
accelerated effect to crack growth behaviour. Conversely, fatigue cracks longer than
0.1 mm usually have a permanent residual plastic deformation in the wake and behave
like long cracks in showing retardation after load reduction. It can be deduced from

144
Chapter 4: 2-Level Loading

these arguments that a temporary acceleration in crack growth rate would contribute
to less than expected lives and hence, a larger proportionate decrease in overall
fatigue life observed after low degrees of initial PDF exposure. To avoid confusion, it
is important to reiterate that total fatigue life increases with less exposure to PDF
since the overall damage by a lower percentage of initial PDF exposure is less. This
also applies to EDF(Hi)→EDF(Lo) sequences, where total fatigue life increases with
decreasing degrees of initial EDF(Hi) exposure. While the crack closure theory is able
to explain the larger proportionate decrease in overall fatigue life observed after low
degrees of initial PDF exposure, two other important factors must be considered to
adequately account for the observed trends: 1) Fatigue crack growth behaviour at
different strain amplitudes, 2) Introduction of mean stresses in step loading. These
two factors will be discussed in the ensuing paragraphs.

D1>D2

D1
Total life

D2

% initial PDF intro

Fig. 4.36 – Larger proportionate decrease (deviation from Palmgren-Miner prediction)

in overall fatigue life at low degrees of PDF intro.

Cycle ratio plots for 2-step loading sequences involving combinations of PDF and
EDF are presented in Figures 4.29-4.35. Total strain amplitudes used for each
respective regime of fatigue are specified in brackets. In Figures 4.29-4.35, it is
obvious that a High→Low sequence results in cycle ratio summations (commonly
referred to as “damage” sum) less than predictions made using the Palmgren-Miner
rule, while, a Low→High sequence consistently yielded cycle ratio summations
greater than unity (e.g. Fig. 4.42). Such results are not surprising and the general cycle
ratio trends conform relatively well to typical trends expected for both High→Low

145
Chapter 4: 2-Level Loading

and Low→High sequences, where cycle ratio summations often take on values <1 and
>1 respectively. This sequence effect can be explained by simply recognising the
subtle differences in crack growth behaviour between high (stress/strain) and low
(stress/strain) fatigue loading. Figure 1.13 in section 1.4 illustrates a typical fatigue
crack maturing from a stage I to stage II crack.

Differences in crack growth behaviour between high (stress/strain) and low

(stress/strain) fatigue loading are illustrated in Figure 4.37. Stage II crack growth is
known to dominate over a large fraction of fatigue life for high (stress/strain)
amplitudes while for low (stress/strain) amplitudes, the majority of fatigue life is
expended during stage I or stage I to II transition crack growth. An independently
derived qualitative model is, hereby, presented to account for sequence effects.
Similar models have also been proposed by Miller and Zarchariah [98], and Manson
and Halford [104]. The main difference between models proposed previously and the
present model is that the current model is based on realistic crack growth behaviour
while the previous two models assume linear crack growth behaviour in two stages.

Consider the fatigue crack growth behaviour for two different stress amplitudes, ∆σ1
and ∆σ2 presented in Figure 4.37, where ao is the initial defect size, at is the transition
crack size and af is the final crack length at complete fracture. Based on the Paris Law
(1.41), it is worth noting that unique crack growth curves are expected for different
stress amplitudes. It can also be seen that both at and the fraction of total fatigue life
spent on stage II crack propagation, increases with increasing stress amplitudes. As an
example, in a High→Low sequence, assuming that a fraction (X1) of fatigue life at
∆σ1 is initially applied before changing to ∆σ2, Palmgren-Miner rule predicts that the
fraction of fatigue life remaining at ∆σ2 is 1-X1. According to the fatigue crack growth
curve for ∆σ1, cyclic fatigue for X1 fraction of life generates a fatigue crack of length,
a1. Assuming that the crack continues to grow from a1 when stress amplitude is
changed from ∆σ1 to ∆σ2, one finds that a crack length of a1 corresponds to a fraction
life, X2 at ∆σ2, which is >X1, meaning that the actual fatigue life remaining is 1-X2
instead of 1-X1 and the cycle ratio summation value is <1. Conversely, in a
Low→High loading sequence, if Y2 fraction of fatigue life at ∆σ2 is applied, a crack
corresponding to a length a2 is generated. At ∆σ1, a2 corresponds to life fraction Y1

146
Chapter 4: 2-Level Loading

which is < Y2, suggesting that the actual fatigue life remaining 1-Y1, is > 1-Y2 and
cycle ratio summation >1. Figure 4.38 is a schematic illustration of typical cycle ratio
trends for both High→Low and Low→High loading based on the above model. The
general cycle ratio trends correspond relatively well to the results presented (Figs.
4.11, 4.13, 4.16, 4.18, 4.21, 4.23, 4.25-4.28), as well as those published in other
literature [99, 111]. This qualitative model provides a simple yet effective way for
rationalising sequence effects during multilevel fatigue loading.

While the above model may help explain sequence effects and fatigue trends in most
multilevel loading cases, certain limitations have to be highlighted to ensure that this
model is used discriminately. Firstly, the model does not take into account important
crack growth factors such as crack closure, residual stress and mean stress effects.
Hence, one would be expect discrepancies between fatigue accumulation trends
rationalised using the above model and trends obtained experimentally to occur,
especially in cases where mean stress, residual stress, and closure effects are
significant.

af

∆σ1 > ∆σ2

a2 ∆σ1

∆σ2
Crack length, a

High→Low
Stage II Low→High

1-Y1
a1

Stage I 1-X2
at1
at2
ao
N/Nf X1 X2 Y1 Y2
0 1
1-X1

1-Y2

Fig. 4.37– Fatigue crack growth behaviour at different stress amplitudes.

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Chapter 4: 2-Level Loading

If fatigue life trends do not conform to initial expectations, crack growth factors such
as residual stress, crack closure and mean stress should be carefully integrated with
the crack model (Fig. 4.37) to give a more realistic interpretation.

The fatigue crack growth trends illustrated in Figure 4.37 are supported by crack
growth measurements performed on 4340 high strength alloy steel. Figure 4.38 shows
the fatigue crack growth behaviour for 4340 high strength alloy steel subjected to total
strain amplitudes of 0.3%, 0.6% and 1.0%, while Figure 4.39-4.41 illustrates the
sequential stages of crack propagation. Crack measurements were made by analysing
surface replicas taken at intermittent stages during fatigue, and images were captured
digitally via a microscope mounted camera. Surface replicas were obtained by wetting
one side of an acetate film with acetone and applying the dissolved side to the
specimen surface to obtain a negative of the microscopic features. The origin of each
dominant failure crack was sequentially traced back from the point of failure. From
Fig. 4.38, it is clear to see that fatigue crack “initiation” begins on the onset of fatigue
loading and fully mature fatigue cracks are formed at very infant stages of fatigue. It
has been suggested by Miller [22, 31] that in polycrystalline metals, one can be safely
assume that crack initiation phase does not exist due to a variety of stress
concentration features such as grain boundaries, machining marks, surface breaking
inclusions, and grain boundaries inherent in engineering components, which can
immediately initiate a crack. Surface texture also roughens with increasing cycles as
extrusions/intrusions of slip bands are formed on the specimen surface.

Note that cracks lengths described in both Figures 4.38a and 4.38b refer to the
circumferential crack lengths determined from surface replicas.

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Chapter 4: 2-Level Loading

7000
0.3% amp

6000 0.6% amp

Circumferential crack length (microns)

1.0% amp
5000

4000

3000

2000

1000

0
0 0.2 0.4 0.6 0.8 1
n/Nf

Fig. 4.38a – Crack growth behaviour for 4340 high strength steel.

1000

900
0.3% amp
Circumferential crack length (microns)

800 0.6% amp

700 1.0% amp

600

500

400

300

200

100

0
0 0.1 0.2 0.3 0.4 0.5 0.6
n/Nf

Fig. 4.38b – Early stage crack growth behaviour for 4340 high strength steel.

149
Chapter 4: 2-Level Loading

Fig. 4.39 – Sequential crack imaging at 0.3% strain amplitude, 0 mean strain for 4340
high strength steel.

150
Chapter 4: 2-Level Loading

Fig. 4.40 – Sequential crack imaging at 0.6% strain amplitude, 0 mean strain for 4340
high strength steel.

151
Chapter 4: 2-Level Loading

Fig. 4.41 – Sequential crack imaging at 1.0% strain amplitude, 0 mean strain for 4340
high strength steel.

152
Chapter 4: 2-Level Loading

Fig. 4.42 – Schematic illustration of typical cycle ratio curves for 2-step loading
sequences based on the crack growth model (Fig. 4.37).

Figure 4.4.2 illustrates the typical cycle ratio trends for 2-step loading sequences. The
cycle ratio trends presented in Figures 4.29-4.35 can also be explained by analysing
the cyclic stress/strain states from hysteresis loops and considering mean stress
effects. Many authors have published experimental results suggesting that
compressive mean stress increases fatigue life while tensile mean stress decreases
fatigue life. Step loadings involving combinations of relatively large strain amplitudes
do not usually yield mean stresses as any mean stress initially present would relax to
zero state as a result of significant plastic straining. On the other hand, under certain
instances of multilevel loading, depending on how large the preceding strain
amplitude is and how small the follow up amplitude is, mean stress could be induced
during the follow up stage. The influence of mean stress induced by such loading
sequences under strain-controlled fatigue has significant bearing on the outcome of
overall fatigue life and cannot be neglected in any analysis. In general, mean stress is
induced when elastic fatigue loading is preceded by fatigue straining involving
significant degrees of plastic deformation. Consequently, based on our definition of
PDF and EDF, one would expect mean stress to be induced only during a PDF→EDF
type loading. The basic mechanics of how tensile mean stress is induced under
controlled-strain is illustrated in Figures 4.43-4.44.

153
Chapter 4: 2-Level Loading

PDF PDF PDF

EDF
εa

εa

εa
σ σ σ
o p
n

ε ε σm ε

Fig. 4.43 – Tensile mean stress induced by prior loading.

A tensile mean stress is induced upon transition from PDF→EDF because PDF ends
after the compressive cycle (Type A) and to get strain back to zero level for the
transition to EDF, a tensile force has to be implemented to force the specimen back to
zero strain state n. On the other hand, a compressive force will be implemented to
force the plastically deformed material to zero strain condition if PDF ends after the
tensile phase (Type F. e.g. PDF→EDF for 6061-T6 aluminium alloy in Fig. 4.15). In
addition to the hysteresis analysis presented in Figure 4.43, it is also useful to
consider a simple spring deformation analogy to help illustrate this concept (Fig.
4.44).

During PDF, the spring suffers sufficient plastic deformation, such that after its last
compressive phase before transition to EDF, the spring relaxes from its maximum
compressed state but never returns to its original length and remains permanently
deformed (compressed) even at zero stress. As a result, a tensile force is needed to
return the spring to zero strain conditions. During EDF, this residual tensile stress is
greater than the EDF stress amplitude imposed, such that overall, a tensile mean stress
is induced.

154
Chapter 4: 2-Level Loading

PDF Transition between PDF/EDF

EDF
q

εa
Failure
p

o r
n

n o p q r

σ=min

σ=0 but Tensile Reduction in

spring does stress tensile force
not return to imposed to applied at
original return Tensile force min. strain
length due to spring to applied at amplitude
plastic original max. EDF
deformation length strain
(stored amplitude
potential
energy)

Fig. 4.44 – Spring deformation analogy illustrating tensile mean stress induced going
from PDF→EDF.

The principles presented above can also be adopted to explain compressive mean
stresses induced by type F PDF→EDF sequences (i.e. transition from PDF→EDF
occurring after PDF tensile phase).

Tensile mean stresses were observed during the EDF phase for both 6061-T6
aluminium alloy and 4340 high strength steel after prior PDF exposure (Figs. 4.45-
4.46). PDF→PDF, EDF→EDF and EDF→PDF sequences yielded dramatically
different mean stress responses (Figs 4.47-4.53). Mean stresses either relaxed to zero
state or were absent during the second stage of loading. For PDF→PDF sequences,
mean stress relaxation to 0 stress state occurred on transition to second phase PDF
loading. On the other hand, sequences which begin with EDF did not induce mean
stress during the second phase of loading.

155
Chapter 4: 2-Level Loading

70

60

50

Mean Stress (MPa)

40

30 17% PDF intro

34% PDF intro
20
51% PDF intro
10 68% PDF intro
0
0 0.2 0.4 0.6 0.8 1
-10
N/Nf

Fig. 4.45– Type A PDF→EDF mean stress response for 6061-T6 aluminium alloy.

250
1.9% PDF intro
3.7% PDF intro
200 9% PDF intro
37% PDF intro
Mean Stress (MPa)

150

100

50

0
0 0.2 0.4 0.6 0.8 1
N/Nf

Fig. 4.46 – Type A PDF→EDF mean stress response for 4340 high strength steel.

156
Chapter 4: 2-Level Loading

50 25% PDF intro

40% PDF intro
40 60% PDF intro

30

Mean Stress (MPa) 20

10

0
0 0.2 0.4 0.6 0.8 1
-10
N/Nf

Fig. 4.47 – Type B PDF(Hi)→PDF(Lo) mean stress response for 316L stainless steel.

100

80
Mean Stress (MPa)

60 5% EDF intro
12.9% EDF intro
40 25.6% EDF intro
40.9% EDF intro
20

0
0 0.2 0.4 0.6 0.8 1
-20
N/Nf

Fig. 4.48 – Type D EDF∏PDF mean stress response for 6061-T6 aluminium alloy.

157
a)
Chapter 4: 2-Level Loading

100
80 PDF(1.0%)-PDF(0.6%) Type B
60 PDF(1.0%)-PDF(0.6%) Type G
40
20
0
-20 0 0.2 0.4 0.6 0.8 1

-40
-60
-80
-100
N/Nf

Fig. 4.49 – Type B and G PDF(Hi)→PDF(Lo) mean stress response for 6061-T6
aluminium alloy.

100 EDF(0.4%)-EDF(0.3%) Type C

80 EDF(0.4%)-EDF(0.3%) Type I
60
40
Mean Stress (MPa)

20
0
0 0.2 0.4 0.6 0.8 1
-20
-40
-60
-80
-100
N/Nf

Fig. 4.50 – Type C and I EDF(Hi)→EDF(Lo) mean stress response for 6061-T6
aluminium alloy.

158
Pa)
Chapter 4: 2-Level Loading

100
12% EDF intro
80 62% EDF intro

60

40

20

0
0 0.2 0.4 0.6 0.8 1
-20

-40
N/Nf

Fig. 4.51 – Type D EDF→PDF mean stress response for 4340 high strength steel.

100 PDF(1.0%)-PDF(0.6%) Type B

90 PDF(1.0%)-PDF(0.6%) Type G

80

70
Mean Stress (MPa)

60

50

40

30

20

10

0
0 0.2 0.4 0.6 0.8 1
N/Nf

Fig. 4.52 – Type B and G PDF(Hi)→PDF(Lo) mean stress response for 4340 high
strength steel.

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Chapter 4: 2-Level Loading

EDF(0.3%)-EDF(0.25%) Type C
90 EDF(0.3%)-EDF(0.25%) Type I

70

Mean Stress (MPa)

50

30

10

-10 0 0.2 0.4 0.6 0.8 1

-30

-50
N/Nf

Fig. 4.53 – Type C and I EDF(Hi)→EDF(Lo) mean stress response for 4340 high
strength steel.

Deviations from Palmgren-Miner predictions, apart from differences in crack growth

behaviour at different strain levels and crack closure, is also attributed to mean stress
effects, particular in situations of PDF→EDF type loading. Tensile mean stress
induced during the EDF phase as a result of preceding PDF is detrimental to overall
fatigue lives, evidenced (severe deviations from Palmgen-Miner predictions) in the
cycle ratio plots for type A PDF→EDF sequences. PDF→EDF experiments were not
conducted for 316L stainless steel since the boundary for PDF/EDF was estimated to
be in the order of 107 cycles. Hence, the number of cycles required to implement EDF
is far too large, rendering it impractical to investigate PDF→EDF loadings for 316L
stainless steel.

An interesting trend (undulating cycle ratio trend seen in Figure 4.33) is observed for
4340 high strength steel subjected to type A loading. Such a trend implies that
introducing very small degrees of initial PDF could prove to be more detrimental than
if greater degrees of initial PDF were introduced, which contradicts conventional
expectations. Monotonic and cyclic stress-strain curves, in section 3.4 showed that
6061-T6 aluminium alloy and 4340 high strength steel tested are cyclic hardening and
softening materials respectively. The anomalous trend seen in Figure 4.33 can be

160
Chapter 4: 2-Level Loading

explained by observing the mean stress response for type A PDF→EDF loading (Fig.
4.46). Figure 4.46 illustrates that the magnitude of tensile mean stress induced
increases with decreasing degrees of PDF exposure. This phenomena acts in
competition with the crack growth approach used to explain sequence effects,
resulting in the undulating cycle ratio trend observed. Mean stress response in Figure
4.46 is rationalised by analysing the stress/strain states from hysteresis loops (Fig.
4.54).

n Low degree of PDF o High degree of PDF

intro. i.e. early stages of intro. i.e. mature stages of
cyclic softening. cyclic softening.

σ σ

σm1
σm2 ε
ε

σm1>σ m2

Fig. 4.54– Magnitude of tensile mean stress induced depending on the state of cyclic
softening.

One would expect the opposite (magnitude of tensile mean stress induced increases
with increasing degrees of PDF exposure) to occur for a cyclic hardening material
based on the same principle (Fig. 4.55). However, mean stress behaviour in 6061-T6
aluminium alloy did not exhibit the gradual increase in tensile mean stress induced
during the EDF phase with increasing degrees of initial PDF exposure. Complete
cyclic hardening was observed to occur almost immediately and cyclic stability state
was reached within the first couple of cycles. Hence, the mean stress induced is quite
consistent and did not vary depending on the degrees of initial PDF exposure. As a
result, no anomaly in cycle ratio trend is evident.

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Chapter 4: 2-Level Loading

n Low degree of PDF o High degree of PDF

intro. i.e. early stages of intro. i.e. mature stages of
cyclic hardening. cyclic hardening.

σ σ

σm2
σm1 ε ε

σm1<σ m2

Fig. 4.55 – Magnitude of tensile mean stress induced depending on the state of cyclic
hardening.

4.5 Significance of proper PDF-EDF definition and

classification

Accurate boundary classification and the specification of the fatigue regimes involved
are important when investigating any form of multilevel loading conducted under
strain-control. As mentioned previously, a great majority of results presented in past
literature on multilevel loading failed to recognise this importance. Specification of
the type of fatigue (PDF or EDF) involved for various levels of loading, provides a
great deal of information regarding the possible mechanisms which may be
responsible for the interaction effects observed, particularly on the possibility of mean
stress effects coming into play. Classifying the two regimes of fatigue according to Nt
was found to be of little benefit, in terms of aiding the understanding of fatigue
interaction trends involving two different fatigue regimes. One of the main problems
of using Nt to define the two regimes is that it offers no indication as to whether mean
stresses induced by prior loading will completely relax to zero levels or remain. 316L
stainless steel for instance, yielded a PDF/EDF strain boundary of 0.3% based on Nt.
However, as seen in Figure 4.47, even if 0.3% strain amplitude was considered as
EDF, mean stress relaxed to zero state, thus, eliminating any benefit associated with
the proposed PDF/EDF classification. As a result, for effective and useful boundary
classification, PDF/EDF boundary strain amplitude has to be defined by cyclic yield
stress based on a 0.05% strain offset. Table 4.20 compares the PDF/EDF boundary

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 Chapter 4: 2-Level Loading

strain amplitudes, εaB, (defined by Nt and cyclic yield stress) for all three materials,
showing the significant differences in boundary criteria based on the two methods of
classification.

Table 4.20 – PDF/EDF boundary criteria for 4340 high strength steel and 6061-T6
aluminium alloy.

Material εaB based on σCYC YS εaB based on Nt

316L stainless steel 0.17% 0.3%

6061-T6 aluminium alloy 0.46% 0.98%
4340 high strength steel 0.38% 0.73%

The nature of mean stresses induced by prior high plastic strain levels can have a
significant role in affecting overall fatigue lives. Analyses of mean stress responses
for type A loading (Figs. 4.45-4.46) show that substantial tensile mean stresses are
induced during the EDF phase by prior PDF straining. By nature of our definition of
PDF and EDF, mean stress relaxation during EDF is marginal and tensile mean stress
persists over the remaining life of the specimen as expected.

In contrast to what is observed for type A spectrums, type B sequences yielded

dramatically different cycle ratio trends. Figures 4.56-4.57 show the changes in mean
stress when transition from PDF to EDF occurs after the tensile phase of initial PDF
(Type B). In this case, the strong effects of tensile mean stress seen in type A loading
are diminished and for 6061-T6 aluminium alloy, compressive mean stresses are
induced on transition, yielding a cycle ratio trend which is somewhat characteristic of
a Low∏High sequence instead (Fig. 4.30).

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Chapter 4: 2-Level Loading

0
0 0.2 0.4 0.6 0.8 1

-50

Mean Stress (MPa)

-100

-150 17% PDF intro

51% PDF intro
68% PDF intro
-200
N/Nf

Fig. 4.56 – Type F PDF→EDF mean stress response for 6061-T6 aluminium alloy.

100

80
1.9% PDF intro
60
3.7% PDF intro
Mean Stress (MPa)

40

20

0
0 0.2 0.4 0.6 0.8 1
-20

-40

-60
N/Nf

Fig. 4.57 – Type F PDF→EDF mean stress response for 4340 high strength steel.

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Chapter 4: 2-Level Loading

100

80 12% EDF intro

60

Mean Stress (MPa) 40

20

0
0 0.2 0.4 0.6 0.8 1
-20
N/Nf

Fig. 4.58 – Type J EDF→PDF mean stress response for 4340 high strength steel.

Type J EDF→PDF (Fig. 4.58) did not yield any mean stress during the second stage
of loading and cycle ratio trends did not vary significantly between type D and type J
EDF→PDF loading. As seen in Figures 4.48, 4.51 and 4.58, transition from
EDF∏PDF does not induce any significant mean stress during the secondary PDF
stage. The large extent of plastic strain relaxes any mean stress, which may be left
over from prior loading histories. Since the influence of mean stress is negligible, it is
not surprising that cycle ratio trends are not affected by the way in which transition
from EDF→PDF occurs. i.e. either after compressive or tensile EDF phase.
Therefore, the dominant mechanism governing cycle ratio trends and sequence effects
associated with EDF→PDF type loading can be attributed to crack growth mechanics
discussed previously. Similar High→Low loading programs were implemented for
fatigue interactions involving similar regimes (i.e. PDF∏PDF or EDF∏EDF) to
observe the effectiveness of the PDF/EDF classification method presented in this
paper. Results indicated that overall fatigue lives remained relatively unaffected
regardless if initial strain loading ended after a compressive or tensile phase. Zero
mean stress is observed (Fig. 4.47-4.53) throughout the second stage of loading for all
cases studied, further emphasising that mean stress plays no part in the interaction
effects observed for multilevel loading involving similar regimes.

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Chapter 4: 2-Level Loading

It is important to note that that PDF/EDF interaction under strain-controlled

conditions, depending on cyclic hardening or softening materials, will produce cycle
ratio trends that are very different to those we are commonly accustomed to (Fig.
4.59-4.60) and it is only through effective PDF and EDF classification that we can
appreciate and interpret the trends observed.

4.6 Conclusions

1. PDF, EDF conveys more meaning than LCF, HCF classification.

2. Initial PDF is detrimental to subsequent EDF life.

3. Total fatigue life decreases with increasing degrees of PDF introduction due to
the greater extent of damage (crack size) caused. However, the proportionate
decrease (i.e. deviation from Palmgren-Miner predictions) in overall fatigue
life increases with lower degrees of initial PDF exposure.

4. Cycle ratio and fatigue life trends can be explained by crack closure, crack
growth mechanics and mean stress effects.

5. The crack growth model does not take into account mean stress, residual stress
and crack closure effects. Its application is limited to cases where the
aforementioned factors are less significant. Nevertheless, it serves as a useful
first step in rationalising fatigue life trends in multilevel loading.

6. PDF, EDF descriptions better reflect mean stress effect on cycle ratio and
overall fatigue life trends.

7. Tensile or compressive mean stresses induced depend on the way, which

PDF→EDF transition occurs.

8. Higher degrees of PDF exposure in:

a) cyclic softening materials results in lower mean stresses induced during

166
Chapter 4: 2-Level Loading

EDF phase.

b) cyclic hardening materials results in higher mean stresses induced during

EDF phase.

9. Multilevel fatigue loading under controlled-strain yields atypical cycle ratio

trends (Figs. 4.59-4.60).

Conventional cycle ratio trends

PDF→EDF loading in
cyclic softening materials

N/Nlow

Higher tensile mean stress

induced at low degrees of PDF
intro

N/Nhigh

Fig. 4.59 – Example of an atypical cycle ratio trend expected for PDF→EDF loading
in cyclic softening materials.

Conventional cycle ratio trends

PDF→EDF loading in cyclic

hardening materials
N/Nlow

Higher mean stress induced at

higher degrees of PDF intro

N/Nhigh

Fig. 4.60 – Example of an atypical cycle ratio trend expected for PDF→EDF loading
in cyclic hardening materials.

10. PDF, EDF specification assists interpretation of observed cycle ratio trends.

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Chapter 5: Multilevel Loading

Chapter 5: Multilevel Loading

5.1 Introduction

Previously in Chapter 4, results for 2-level loading sequences involving various

combinations of fatigue regimes were presented. In this Chapter, results obtained
from multilevel loading sequences involving three or more strain amplitude variations
over total life are discussed. The main objectives of studying multilevel loading
sequences involving PDF/EDF regimes are summarised as follows:

1) Investigate sequence effects in multilevel loading.

2) Evaluate the effectiveness of PDF/EDF classification in describing mean

stress response and cycle ratio trends in loading sequences involving more
than 2 strain amplitude variations.

3) Investigate the effect of block size on overall fatigue life.

5.2 Material and testing program

Material used for investigating multilevel loading is 4340 high strength steel. Smooth
hourglass shaped specimens, machined according to specifications detailed in Chapter
3 were used. In total, five types of multilevel sequences were investigated. Schematic
illustrations of the various loading spectrums tested are shown in Figures 5.1-5.5.
Total strain amplitudes used for each phase of loading is shown in brackets.

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Chapter 5: Multilevel Loading

PDF(1.0%)
PDF(0.6%)
EDF(0.3%)
Failure

100 cycles 8000 cycles to failure

Fig. 5.1 – (Type K) High→Low→High loading sequence.

PDF(1.0%)
PDF(0.6%)
EDF(0.3%)
Failure

600 cycles 100 cycles to failure

Fig. 5.2 – (Type L) Low→High→Low loading sequence.

PDF(1.0%) PDF(0.8%)
PDF(0.6%)

Failure

100 cycles 600 cycles to failure

Fig. 5.3 – (Type M) High→Low→High loading sequence.

PDF(1.0%)
PDF(0.6%)
EDF(0.3%)
repeat

10 cycles 800 cycles 60 cycles

1 block

Fig. 5.4 – (Type N) High→Low→High loading sequence, repeat block to failure.

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Chapter 5: Multilevel Loading

PDF(1.0%) PDF(0.8%)
PDF(0.6%)

repeat

10 cycles 60 cycles 30 cycles

1 block

Fig. 5.5 – (Type O) High→Low→High loading sequence, repeat block to failure.

5.3 Experimental results

Results for the five cases of multilevel sequences investigated are tabulated in Table
5.1. From Table 5.1, it is clear to see that all five sequences yielded cycle ratio
summations less than unity. Cycle ratio trends for the first three cases of loading (Fig.
5.6) are reminiscent of type A High→Low cycle ratio trends presented in Chapter 4. It
can be inferred from such trends, that the overall cycle ratio behaviour is governed
essentially by the High→Low phase for both High→Low→High and
Low→High→Low sequences. As discussed in Chapter 4, significant deviations from
the Palmgren-Miner rule can occur, particularly in situations where High→Low
sequences involve going from PDF to EDF. Deviations from Palmgren-Miner
predictions are firstly caused by differences in crack growth behaviour occurring at
varying strain amplitudes. Such deviations can, however, be further aggravated by
tensile mean stresses induced during the EDF stage, for PDF→EDF sequences.
Figures 5.7-5.9 compare the overall cycle ratio sums obtained under the various cases
of multilevel loading history.

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Chapter 5: Multilevel Loading

Table 5.1 – Results for multilevel loading.

Loading Avg. total Life Cycle ratio
N/Nf1 N/Nf2 N/Nf3
type (cycles) sum
Type K 0.1872 0.1996 8789 0.2624 0.6493
Type L 0.2284 0.1872 11416 0.2674 0.683
Type M 0.1872 0.2284 1021 0.3808 0.7964
Loading Cycle ratio sum for 1 Avg. total Life Avg. number of Cycle ratio
type block (cycles) blocks sum
Type N 0.0615 8512 9.8 0.5892
Type O 0.0772 1149 11.5 0.8846

1
Palmgren-Miner
0.9
Type K
0.8 Type L
Type M
0.7

0.6
n/Nf3

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf1 + n/Nf2

Fig. 5.6 – Cycle ratio trends for types K,L,M multilevel loading sequences.

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Chapter 5: Multilevel Loading

0.9

0.8

Cycle ratio summation

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
Type K Type L Type M Type N Type O
Loading type

Fig. 5.7 – A comparison of cycle ratio summation for the five types of multilevel
sequences investigated.

0.7

0.68

0.66
Cycle ratio summation

0.64

0.62

0.6

0.58

0.56

0.54
Type K Type L Type N
Loading type

Fig. 5.8 – Cycle ratio summation for type K,L and N sequences.

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Chapter 5: Multilevel Loading

0.9

0.88

0.86

Cycle ratio summation 0.84

0.82

0.8

0.78

0.76

0.74
Type M Type O
Loading type

Fig. 5.9 – Cycle ratio summation for type M and O sequences.

5.4 Discussion

Cycle ratio trends observed in Figure 5.6 can again be explained by analysing the
cyclic stress/strain states from hysteresis loops and considering mean stress effects on
fatigue life. Figures 5.10 and 5.11 illustrate changes in stress amplitudes and mean
stress responses for the various scenarios of multilevel loading. Tensile mean stresses
are observed at all stages of loading for type K, L, M sequences. Table 5.1 shows that
the average cycle ratio summation for a type K sequence is less than a than type L
sequence. A comparison of overall fatigue lives also reveals that a type K sequence is
more detrimental than a type L sequence even though life fractions of PDF imposed
are similar. From Figure 5.11, it is clear that tensile mean stresses induced during
EDF are higher for type K sequences, explaining the lower overall fatigue lives
observed. This observation can again be rationalised based on Figure 4.50 presented
in Chapter 4. In type L multilevel loading, EDF is introduced after a greater number
of collective PDF cycles. Therefore, significant cyclic softening would have taken
place by the time EDF is introduced such that lower mean stresses are induced. This
argument is supported by observing significant reductions in stress amplitudes as
cyclic softening occurs (Fig. 5.10).

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Chapter 5: Multilevel Loading

The first two cases of multilevel loading involved combinations of two different
fatigue regimes (PDF and EDF). In type M multilevel loading, only PDF strain
amplitudes were used. This test was designed to investigate if mean stress effects
were absent for multilevel loading sequences involving only a single fatigue regime
(i.e. either PDF or EDF). However, as seen in Figures 5.10 and 5.11, failure occurred
before cyclic stability is reached and total mean stress relaxation could not be
completed. As a result, under such circumstances, mean stress effects might have to
be taken into consideration when analysing fatigue results, even though mean stress
effects are not expected for single regime multilevel loadings based on PDF and EDF
definition. Nevertheless, compared with cycle ratio summations for type K and L
sequences, the average cycle ratio sum for a type M sequence is much closer to unity.
This implies a significant reduction in the influence of mean stress for single regime
multilevel sequences.

1000

900
Stress amplitude (MPa)

800

700

600

500

400
0 0.2 0.4 0.6 0.8 1
N/Nf

Type K Type L Type M Type N Type O

Fig. 5.10 – Stress amplitude changes over the course of multilevel loading.

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Chapter 5: Multilevel Loading

200

180

160

140
Stress amplitude (MPa)

120

100

80

60

40

20

0
0 0.2 0.4 0.6 0.8 1
N/Nf

Type K Type L Type M Type N Type O

Fig. 5.11 – Mean stress responses over the course of multilevel loading.

Type N and O multilevel loading sequences are essentially type K and M sequences
broken down into smaller blocks. The number of cycles required to complete each
stage of loading in type K and M sequences were reduced by a factor of 10 in type N
and O sequences, such that smaller blocks of type K or type M sequences were
repeated until complete failure. This test was designed to investigate the influence of
block size on overall fatigue lives.

The average cycle ratio sum for a type N sequence is significantly lower than cycle
ratio summations observed for type K and L sequences (Table 5.1 and Fig. 5.8).
Analysing and comparing the various mean stress responses associated with each type
of multilevel loading sequence can explain this observation. Figure 5.11 shows that,
in general, tensile mean stresses experienced over the course of a type N sequence, are
significantly larger than the mean stresses experienced in both type K and L
sequences. PDF to EDF transition occurs after only 10 PDF (1.0% amp) cycles (as
opposed to 100 PDF (1.0% amp) cycles in type K multilevel loading sequence) and
because a small number of PDF cycles are introduced, only in a small extent of cyclic
softening takes place. The cyclic softening process, accelerated by PDF, is regularly

175
Chapter 5: Multilevel Loading

interrupted by transitions to EDF and mean stresses are induced before any significant
amount of softening takes place. Consequently, relatively large tensile mean stresses
are induced during the EDF phase of each block, decreasing as block loading
progresses.

Type O multilevel loading sequence yields an average cycle ratio sum higher than that
observed for a type M sequence (Table 5.1 and Fig. 5.9). An analysis of mean stress
responses in Figure 5.11, reveals that mean stress relaxation is more complete when
there are more stages involved in single regime multilevel loading. A further
reduction in the influence of mean stress for multiple block, single regime multilevel
sequences, results in an average cycle ratio sum that is closest to Palmgren-Miner
predictions, amongst all cases of multilevel loading sequences investigated.

5.5 Conclusion

1. Overall cycle ratio trends are governed by the High→Low phase for both
High→Low→High and Low→High→Low sequences.

2. PDF, EDF descriptions still reflect mean stress effect on cycle ratio and
overall fatigue life trends.

3. A type K sequence is more detrimental than a type L sequence, even though

life fractions of PDF imposed are similar.

4. Block size influences overall fatigue lives and cycle ratio sums.

5. Repeated block loading of smaller type K sequence is more detrimental than

a single continuous type K block.

6. A type O sequence yields is less “damaging” than a type O sequence.

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Chapter 6: Mean Strain Effects

Chapter 6: Mean Strain Effects

6.1 Introduction

Historically, mean strain effects on fatigue life have been studied to a much lesser
extent as compared to studies performed on mean stress effects. Nevertheless, the
importance of understanding mean strain effect on fatigue life cannot be ignored as
strain-based fatigue analyses have found increasing applicability in early design
stages of engineering components. Results for 2-level loading sequences involving
various combinations of fatigue regimes under 0 mean strain conditions were
previously discussed in Chapter 4. In this Chapter, 2-level loading sequences under
tensile mean strain conditions are investigated. The objectives of studying 2-level
loading sequences under tensile mean strain conditions are summarised as follows:

1) Investigate tensile mean strain effect on overall fatigue lives.

2) Evaluate the effectiveness of PDF/EDF classification in describing mean

stress response and cycle ratio trends in 2-level loading sequences under
plastic tensile mean strain conditions.

3) Identify differences and investigate the influence of mean strain effects during
PDF and EDF phases.

6.2 Materials and testing program

Specimens made from 4340 high strength steel, machined according to specifications
detailed in Chapter 3, were used for this investigation. In total, three types of 2-level
loading sequences under a 1.0% tensile mean strain were investigated. A 1.0% mean
strain was chosen to see if PDF/EDF definitions are still consistent even when peak
strain during EDF is brought to a plasticity dominated region. Schematic illustrations

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Chapter 6: Mean Strain Effects

of the various loading spectrums tested are shown in Figures 6.1-6.3. PDF and EDF
were conducted using strain amplitudes of 1.0% and 0.3% respectively.

PDF

EDF

εm=1.0% εa
Failure

Transition between PDF/EDF

Fig. 6.1 – (Type P) High→Low loading sequence going from PDF to EDF at 1.0%
mean strain.

PDF
EDF
εm=1.0% εa

Failure

Transition between EDF/PDF

Fig. 6.2 – (Type Q) Low→High loading sequence going from EDF to PDF at 1.0%
mean strain.

PDF
EDF
εm=1.0% εa

Failure

Transition between PDF/EDF (after PDF tensile phase)

Fig. 6.3 – (Type R) High→Low loading sequence going from PDF to EDF at 1.0%
mean strain.

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 Chapter 6: Mean Strain Effects

6.3 Experimental results

Mean strain tests were first conducted on single regime, PDF and EDF loading, to
observe its effects on fatigue lives. Figure 6.4 compares the fatigue life trends of
specimens subjected to mean strain and non-mean strain conditions, while the average
total lives for both scenarios are tabulated and compared in Table 6.1.

1.2

0.8
Strain amp (%)

0.6

0.4

0.2

0
0 10000 20000 30000 40000 50000
Fatigue life (cycles)

0 mean strain 1.0% mean strain

Fig. 6.4 – Strain amplitude vs Fatigue life curve for 4340 high strength steel.

Table 6.1 – Fatigue lives (cycles) at various strain amplitudes, for 4340 high strength
steel.
Strain amplitude (%) Test 1 Test 2 Test 3 Test 4 Test 5 Avg. Cycles to failure
0.3 (0 mean strain) 20472 18402 27514 20033 17588 20802
1.0 (0 mean strain) 557 666 503 361 670 551
0.3 (1% mean strain) 35188 43154 35750 42136 44154 40076
1.0 (1% mean strain) 557 434 426 618 636 534

Figures 6.5 and 6.6 show fatigue life trends after various plastically/elastically
dominant fatigue loading spectrums under mean strain and non-mean strain
conditions, while cycle ratio trends are shown in Figure 6.7. The average total fatigue

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Chapter 6: Mean Strain Effects

lives after various plastically/elastically dominant fatigue loading regimes are

tabulated in Tables 6.2-6.4.

40000
Palmgren-Miner PDF (1.0% strain amp)
35000
→EDF (0.3% strain amp)

30000
PDF (1.0% strain amp)→EDF (0.3%
strain amp) with error bar
Total Life (Cycles)

25000
PDF (1.0% strain amp)→EDF (0.3% strain
amp) after tensile phase with error bar
20000

15000

10000

5000

0
0 20 40 60 80 100
% initial PDF intro

Fig. 6.5 – Type P and R PDF→EDF fatigue life trends for 4340 high strength steel.

25000

20000
Total Life (Cycles)

15000

10000
Palmgren-Miner EDF (0.3% strain amp)
→PDF (1.0% strain amp)

5000
EDF (0.3% strain amp)→PDF (1.0%
strain amp) with error bar

0
0 20 40 60 80 100
% initial EDF intro

Fig. 6.6 – Type Q EDF→PDF fatigue life trend for 4340 high strength steel.

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Chapter 6: Mean Strain Effects

Table 6.2 – Average fatigue lives (cycles) after PDF (1.0% strain amp)→EDF (0.3%
strain amp) loading for 4340 high strength steel (Type P).
% PDF (1.0% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 20802 20802
0.9 26076 20618
3.6 20948 20067
9 20880 18966
36 9210 13457
100 551 551

Table 6.3 – Average fatigue lives (cycles) after EDF (0.3% strain amp)→PDF (1.0%
strain amp) loading for 4340 high strength steel (Type Q).
% PDF (1.0% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 551 551
24 5479 5419
58 12478 12233
100 20802 20802

Table 6.4 – Average fatigue lives (cycles) after PDF (1.0% strain amp)→EDF (0.3%
strain amp) loading for 4340 high strength steel (Type R).
% PDF (1.0% strain Avg. total fatigue Palmgren-Miner
amp) intro life (cycles) predictions (cycles)
0 20802 20802
3.7 25378 20049
100 551 551

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Chapter 6: Mean Strain Effects

1.4
Palmgren-Miner prediction
PDF(1.0%)-EDF(0.3%) Type P; @1.0% mean strain
EDF(0.3%)-PDF(1.0%) Type Q; @1.0% mean strain
1.2 PDF(1.0%)-EDF(0.3%) Type R; @1.0% mean strain
n/Nf EDF (0.3% amp, 1.0% mean) = 20802 cycles

0.8

0.6

0.4

0.2

0
0 0.2 0.4 0.6 0.8 1
n/Nf, PDF (1.0% amp, 1.0% mean)=551 cycles

Fig. 6.7 – PDF/EDF cycle ratio trends for 4340 high strength steel.

6.4 Discussion

It can be seen in both Figure 6.4 and Table 6.1 that a 1.0% tensile mean strain has
negligible effect on PDF, yet it has a significant influence on EDF, reducing overall
fatigue lives by almost 50%. This behaviour can be explained by analysing the stress
amplitude and mean stress responses obtained experimentally (Figs. 6.8 and 6.9
respectively).

In Figures 6.8 and 6.9, it is observed that mean strain influence on stress amplitude
and mean stress responses are negligible for PDF. As previously discussed, the large
extent of plastic deformation and straining during PDF causes complete mean stress
relaxation to take place. This is evident in Figure 6.9 where the character of mean
stress relaxation curves for both 0 and 1.0% mean strain conditions are similar. Stress
amplitude responses for both 0 and 1.0% mean strain conditions are also similar since
only a small degree of stress change is required for a large increase in strain beyond
the point of yielding. As a result, a 1.0% tensile mean strain does not affect PDF.

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Chapter 6: Mean Strain Effects

1000
900
800
Stress amp (MPa) 700
600
500
400 0.3% amp

300 1.0% amp

200 0.3% amp, 1.0% mean

100 1.0% amp, 1.0% mean

0
0 0.2 0.4 0.6 0.8 1
N/Nf

Fig. 6.8 – Stress amplitude response for 4340 high strength steel.

600

500

400
Mean Stress (MPa)

300
0.3% amp @ 0 mean strain
0.3% amp @ 1.0% mean strain
200
1.0% amp @ 0 mean strain
1.0% amp @ 1.0% mean strain
100

0
0 0.2 0.4 0.6 0.8 1
-100
N/Nf

Fig. 6.9 – Mean stress response for 4340 high strength steel.

On the other hand, the stress amplitude and mean stress responses for EDF are
severely affected in the presence of a 1.0% tensile mean strain. The stress amplitude
curve is higher than that observed for 0 mean strain conditions and the application of
a 1.0% pre-strain induces large tensile mean stresses, synonymous to a PDF→EDF
type loading sequence. The existence of large tensile mean stresses is the main reason

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Chapter 6: Mean Strain Effects

for the significant reduction in overall fatigue lives observed for EDF in the presence
of a 1.0% mean strain. Therefore, tensile mean strains affect EDF and the definitions
of PDF/EDF are still consistent (i.e. significant mean stress relaxation occurred for
PDF while mean stress relaxation is minimal and is present during EDF) despite the
fact that EDF peak strain is now in a plasticity dominant region. This observation
concurs with previous statements suggesting that mean stress effects are only apparent
when elastic strain dominates (EDF). i.e. when applied strain amplitudes are low. This
is because mean stress affects fatigue strength coefficient, σ’f, which is relevant only
to the elastic portion of total strain.

The influence of mean strain on PDF/EDF interactions was also studied. In Figure
6.7, one observes an unusual trend for type P PDF→EDF loading. On the other hand,
type R EDF→PDF sequences produced a cycle ratio trend similar to typical trends
expected of Low→High sequences. At first sight, it appears that cycle ratio trends
observed for type A (Fig. 4.33) and type R loadings show dissimilar trend behaviours.
However, a plot of overall fatigue life versus the number of PDF cycles introduced
reveals that both type A and type R loadings have similar overall fatigue life trends.
This is illustrated in Figure 6.10.

30000
PDF-EDF (1.0% mean strain)
PDF-EDF (no mean strain)
25000
Overall fatigue life

20000

15000

10000

5000

0
0 50 100 150 200 250
PDF cycles introduced

Fig. 6.10 – Overall fatigue life vs PDF cycles introduced plots for 4340 high strength
steel.

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Chapter 6: Mean Strain Effects

The unusual cycle ratio trend observed for type P loading in Figure 6.7 encompasses
features akin to type A loading in Figure 4.33. Type P cycle ratio trend also shows
that introducing very small degrees of initial PDF could prove to be more detrimental
than if greater degrees of initial PDF were introduced, which is consistent with the
type A cycle ratio trend observed. In addition, small degrees of PDF introduction also
yielded overall fatigue lives which were greater than Palmgren-Miner predictions.
This observation would seem absurd given the fact that prior PDF ending (as
discussed in Chapter 4) after the tensile phase induces tensile mean stresses during
EDF and High→Low sequences typically give cycle ratio summations < 1. The
anomalous trend seen in Figure 6.7 can be explained by observing the mean stress
response for type P PDF→EDF loading (Fig. 6.11).

600

500

400
0.3% amp, 1.0% mean
Mean Stress (MPa)

5 cycles PDF intro

300 20 cycles PDF intro
50 cycles PDF intro

200 200 cycles PDF intro

20.5 cycles PDF intro

100

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-100
N/Nf

Fig. 6.11 – Type P PDF→ EDF mean stress response for 4340 high strength steel.

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Chapter 6: Mean Strain Effects

20 cycles PDF intro

200
20 cycles PDF intro, 1.0% mean
180 50 cycles PDF intro
50 cycles PDF intro, 1.0% mean
160 200 cycles PDF intro
200 cycles PDF intro, 1.0% mean
140
Mean Stress (MPa)

120

100

80

60

40

20

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
N/Nf

Fig. 6.12 – Comparing Type A and P PDF→EDF mean stress responses for 4340 high
strength steel.

In Figure 6.11, it is clear that the magnitudes of tensile mean stresses induced during
the EDF stage (for type P sequences) are significantly lower than the mean stress
observed for baseline single-amplitude EDF loading with 1.0% mean strain. Cyclic
softening during initial PDF exposure was the essential cause for the lower mean
stresses observed during EDF and again, the magnitude of tensile mean stress induced
decreased with increasing degrees of PDF exposure. Since the tensile mean stresses
induced during EDF are much lower than baseline single-amplitude EDF, fatigue life
remaining as a result of prior PDF loading is actually much higher than anticipated,
thus, giving overall fatigue lives much larger than Palmgren-Miner predictions. These
factors, again acting in competition with the crack growth approach used to explain
sequence effects, results in the undulating cycle ratio trend observed for type P
loading. A comparison, between mean stress responses for corresponding PDF→ EDF
loading under 0 and 1.0% mean strain is shown in Figure 6.12. It can be seen that
tensile mean stress responses for PDF→EDF loading under 0 mean strain conditions
are consistently higher than those observed under 1.0% mean strain, thereby,
explaining the higher overall fatigue lives observed for type P loading in Figure 6.10.
It is also important to observe that differences in mean stress responses for 0 and

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Chapter 6: Mean Strain Effects

1.0% mean strain conditions progressively diminish with increasing degrees of PDF
introduction (i.e. at 200 cycles PDF introduction, the mean stress responses for 0 and
1.0% mean strain conditions are similar). The reason for this is that PDF hysteresis
peak stresses become indifferent for 0 and 1.0% mean strain conditions once cyclic
stability or saturation is reached.

Figure 6.13 compares the cycle ratio trends for both PDF→EDF and EDF→PDF
cycle ratio trends at 0 and 1.0% mean strain. The red bold arrow shows the relative
shift in cycle ratio trend as a result of 0 and 1.0% mean strain conditions. For type Q
EDF→PDF loading, it is observed (Fig. 6.13) that the cycle ratio trend corresponds
relatively well to those observed under 0 mean strain conditions (type D and J). This
can again be explained by considering the mean stress response of the material during
EDF→PDF, where although mean stress responses for type Q, D and J loading appear
dissimilar, mean stresses are not induced during PDF. In the case (type Q) where
significant tensile mean stress is present during initial EDF, the magnitude of mean
stress is similar to that observed for baseline single-amplitude loading, thereby
eliminating the possibility of additional mean stress effects arising from step loading
involving different fatigue regimes.

Palmgren-Miner prediction
1.4 PDF(1.0%)-EDF(0.3%) Type P; @1.0% mean strain
EDF(0.3%)-PDF(1.0%) Type Q; @1.0% mean strain
PDF(1.0%)-EDF(0.3%) Type R; @1.0% mean strain
1.2
PDF(1.0%)-EDF(0.3%) Type A
PDF(1.0%)-EDF(0.3%) Type F
1 EDF(0.3%)-PDF(1.0%) Type D
EDF(0.3%)-PDF(1.0%) Type J

0.8
n/Nf, EDF

0.6

0.4

0.2

0
0 0.2 0.4 0.6 0.8 1
n/Nf, PDF

Fig. 6.13 – Comparing PDF→EDF, EDF→PDF cycle ratio trends at 0 and 1.0% mean
strain for 4340 high strength steel.

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Chapter 6: Mean Strain Effects

600

500

Mean Stress (MPa) 400

300

0.3% amp 1.0% mean

200 EDF-PDF, 5000 cycles EDFintro, 1.0 mean
EDF-PDF, 12000 cycles EDFintro, 1.0 mean
EDF-PDF, 5000 cycles EDF intro
100 EDF-PDF, 25000 cycles EDF intro
EDF-PDF, 25000.5 cycles EDF intro

0
0 0.2 0.4 0.6 0.8 1

-100
N/Nf

Fig. 6.14 – Comparing Type Q, D and J EDF→PDF mean stress responses for 4340
high strength steel.

This implies that crack growth mechanics is the dominant mechanism governing cycle
ratio trend behaviours for EDF→PDF loading under both mean and non-mean strain
conditions.

The final five specimens in the batch of 4340 high strength steel were subjected to
type R loading to observe what we can of the loading sequence. From Figures 6.7 and
6.13, it is clear that for the same degree of initial PDF exposure, cycle ratio
summation is greater than the corresponding type P data point. This observation is
consistent with descriptions of earlier type F behaviour discussed in Chapter 4. From
Figure 6.11, one can see that the high levels of tensile mean stresses previously
observed in type P loading are removed by PDF→EDF transition occurring after PDF
tensile phase. It is regretful that due to insufficient specimen quantity, only a single
data point in Figure 6.7 is obtained and a complete cycle ratio trend curve could not
be obtained. Nevertheless, based on knowledge gained from previous experiments,
one can expect a cycle ratio trend fitting the character displayed in Figure 6.15.
However, more experiments are required to verify this hypothetical trend.

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Chapter 6: Mean Strain Effects

Type R PDF→EDF loading in

cyclic softening materials

N/Nlow
Higher compressive mean
stress induced at low degrees of
PDF intro

N/Nhigh

Fig. 6.15 – Expected cycle ratio trend for Type R PDF→EDF loading sequence.

6.5 Conclusion

1. Mean strain affects EDF but not PDF.

2. PDF/EDF definitions are still valid in the presence of plasticity dominant

tensile mean strains.

3. Tensile mean strains could enhance overall fatigue life for PDF→EDF
sequences by enhancing early stage cyclic softening such that mean stresses
induced during EDF are lower than those induced under 0 mean strain
conditions.

4. Mean strains do not affect EDF→PDF sequences.

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Chapter 7: Fatigue Crack Growth in CT Specimens

Chapter 7: Fatigue Crack Growth in CT

Specimens

7.1 Introduction

It is important to include the study of fracture mechanics as a component of any

fatigue research. In previous chapters (Chapters 3-6), results pertaining to smooth
specimen testing have been presented and discussed. In this chapter, fatigue crack
growth in compact tension (CT) specimens are studied and fracture mechanics is
employed to explain crack growth behaviour. An attempt to use philosophies derived
from smooth sample testing to supplement conventional fracture mechanics principles
in explaining fatigue crack growth under various loading conditions will be made.

In general, many classical solutions in fracture mechanics are reduced to two-

dimensional analysis and assumes that at least one of the principal stresses or strains
equals 0 i.e. plane stress and plane strain condition respectively. Plane stress and
plane strain conditions can occur in front of a crack tip depending on component
thickness. Conventional stress-strain analysis shows that high levels of stress normal
to the crack plane tend to cause material surrounding a crack tip to contract in the z
direction. For components with low cross-sectional thickness, free-contraction occurs,
where σz = 0 and εz ≠ 0 (plane stress - Fig. 7.1a). On the other hand, for components
with high cross-sectional thickness, contraction in the z direction is restricted by the
surrounding material giving rise to a traverse stress, σz, such that σz ≠ 0 and εz = 0
(plane strain - Fig. 7.1b).

As discussed in Chapter 1, the freedom for unconstrained deformation around a crack

tip under plane stress conditions for a given stress intensity value gives rise to a larger
plastic zone size when compared to similar, but plane strain conditions. In
components under plane strain conditions, plastic zone size and shape may vary
through the thickness of the member as a result of triaxial stress condition. This
situation is accentuated for components with intermediate thicknesses as shown in

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Chapter 7: Fatigue Crack Growth in CT Specimens

Figure 7.2. In Figure 7.2, the free surface on the side allows for unconstrained
material contraction in the thickness direction, which gives rise to plane stress
condition, while material constraint around the crack tip gives rise to plane strain
conditions in the mid-section.

Fig. 7.1 – Plastic zone, stress state and fracture mode a) plane stress and b) plane
strain.

Fig. 7.2 – Plastic zone size and shape variation across components with intermediate
thicknesses.

Crack growth rates can be affected by component thickness as a result of different

crack growth behaviour in plane stress and plane strain. Therefore, the main
objectives of this chapter are:

1. To investigate differences in crack growth behaviour under plane stress and

plane strain conditions.

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Chapter 7: Fatigue Crack Growth in CT Specimens

2. To investigate the effect of overloading under plane stress and plane stress
conditions for different R and overload ratios.

The reader is referred to Section 2.1 of Chapter 2 for a literature review of pervious
work on overloading.

7.2 Materials and testing program

The material used for testing is 4140 high strength steel. CT specimens were
machined according to ASTM E-399-90 standard (Standard Test Method for Plane-
Strain Fracture Toughness of Metallic Materials) standard specifications based on a
L-T configuration (Fig. 7.3).

Fig. 7.3 – Orientation configurations for CT specimens.

The chemical composition and mechanical properties for 4140 high strength steel are
tabulated in Tables 1 and 2 respectively.

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Chapter 7: Fatigue Crack Growth in CT Specimens

Table 7.1: Chemical composition of 4140 high strength steel (% Weight).

C 0.4 Cr 1.03 Mn 0.82 Mo 0.18 N 0.015

Ni 0.22 P 0.009 S 0.023 Si 0.23 Al 0.031
Cu 0.18 Sn 0.01 Fe remaining

Table 7.2: Tensile mechanical properties for 4140 high strength steel.

σY, 0.2% offset σUTS E Elongation Reduction Area

1340 MPa 1450 MPa 205 GPa 15.0% 50.0 %

Dimensional specifications for CT specimens used are detailed in Figure 7.4, while
Figure 7.5 illustrates the experimental set-up.

∅ 11.7

48.0 3.0 10.0

11.5

40.0
10.35 19.0

50.0 5.0 25.0

(a) (b)

Fig. 7.4 – CT dimensional specifications (dimensions in mm); a) side b) front-face.

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Chapter 7: Fatigue Crack Growth in CT Specimens

Specimen
CT specimen grips
Microscope
Transverse-
travel stage

Micrometer

Fig. 7.5 – Illustration of experimental set-up.

Crack growth was monitored and measured intermittently via a micrometer coupled
with a transverse-travelling microscope.

All tests were conducted under load-control in uniaxial push-pull mode using an
INSTRON 8501 servo hydraulic machine. Testing frequencies varied from 15 Hz to
25 Hz depending on the loading conditions. No discernable differences in overall
fatigue lives were observed for variations in loading frequencies within the range
specified.

7.2.1 Determination of KIC

KIC for 4140 high strength steel was determined according to testing conditions and
procedures suggested in ASTM standard E-399-90 (Standard Test Method for Plane-
Strain Fracture Toughness of Metallic Materials). Fatigue pre-cracking (to a fatigue
crack length of 9.155 mm) was performed under load-control with limits ranging from
2-20 kN at 25 Hz for the first 8 mm of pre-cracking. These limits were subsequently
reduced (1.5-15 kN at 25 Hz) for the final 1.155 mm of pre-cracking to ensure that a
sharp fatigue crack is obtained. Toughness testing was conducted by applying a
monotonic tensile load, increasing at a rate of 0.5kN/s, to complete failure.

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Chapter 7: Fatigue Crack Growth in CT Specimens

7.2.2 Plane strain / stress crack growth tests with overloading

All plane strain / stress crack growth tests were conducted under load-control. The
various types of loading conditions investigated are illustrated in Figures 7.6-7.21.
Load levels were selected such that stress at the crack tip is the same for both plane
strain and plane stress conditions. Overload was performed using a double-ramp
sequence at a rate of 0.5 kN/s. Loading regimes were designed to investigate the
effects R ratio (loadmin/loadmax), overload ratio (OLR), plane strain / stress conditions
and load baseline level have on fatigue crack propagation.

20 kN

Failure
11 kN Type 1, R=0.1 (plane strain)

2 kN

Fig. 7.6 – Type 1 loading for 25 mm specimen (plane strain), where R = 0.1.

4 kN
Failure 1.8 kN Type 2, R=0.1 (plane stress)
0.4 kN

Fig. 7.7 – Type 2 loading for 5 mm specimen (plane stress), where R = 0.1.

20 kN

Failure
17 kN Type 3, R=0.7 (plane strain)

14 kN

Fig. 7.8 – Type 3 loading for 25 mm specimen (plane strain), where R = 0.7.

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Chapter 7: Fatigue Crack Growth in CT Specimens

40 kN

20 kN

Failure Type 4, R=0.1 (plane strain),

11 kN
OLR = 2.0, at ac = 8 mm

2 kN

Fig. 7.9 – Type 4 loading for 25 mm specimen (plane strain), where R = 0.1 & OLR
=2.0.

8 kN
4 kN Type 5, R=0.1 (plane stress),
Failure OLR = 2.0, at ac = 8 mm
1.8 kN
0.4 kN

Fig. 7.10 – Type 5 loading for 5 mm specimen (plane stress), where R = 0.1 & OLR
=2.0.

26 kN

20 kN

Failure Type 6, R=0.1 (plane strain),

11 kN OLR = 1.3, at ac = 8 mm

2 kN

Fig. 7.11 – Type 6 loading for 25 mm specimen (plane strain), where R = 0.1 & OLR
=1.3.

5.2 kN Type 7, R=0.1 (plane stress),

4 kN
Failure
OLR = 1.3, at ac = 8 mm
1.8 kN
0.4 kN

Fig. 7.12 – Type 7 loading for 5 mm specimen (plane stress), where R = 0.1 & OLR
=1.3.

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Chapter 7: Fatigue Crack Growth in CT Specimens

45 kN

Failure
24.75 kN Type 8, R=0.1 (plane strain)

4.5 kN

Fig. 7.13 – Type 8 loading for 25 mm specimen (plane strain), where R = 0.1.

9 kN
Failure 4.95 kN Type 9, R=0.1 (plane stress)
0.9 kN

Fig. 7.14 – Type 9 loading for 5 mm specimen (plane stress), where R = 0.1.

90 kN

45 kN

Failure Type 10, R=0.1 (plane strain),

24.75 kN
OLR = 2.0, at ac = 3.5 mm

4.5 kN

Fig. 7.15 – Type 10 loading for 25 mm specimen (plane strain), where R = 0.1 &
OLR = 2.0.

18 kN
9 kN Type 11, R=0.1 (plane stress),
Failure OLR = 2.0, at ac = 3.5 mm
4.95 kN
0.9 kN

Fig. 7.16 – Type 11 loading for 5 mm specimen (plane stress), where R = 0.1 & OLR
= 2.0.
58.5 kN

45 kN

Failure Type 12, R=0.1 (plane strain),

24.75 kN
OLR = 1.3, at ac = 3.5 mm

4.5 kN
Fig. 7.17 – Type 12 loading for 25 mm specimen (plane strain), where R = 0.1 &
OLR = 1.3.

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Chapter 7: Fatigue Crack Growth in CT Specimens

11.7 kN Type 13, R=0.1 (plane stress),

9 kN
Failure
OLR = 1.3, at ac = 3.5 mm
4.95 kN
0.9 kN

Fig. 7.18 – Type 13 loading for 5 mm specimen (plane stress), where R = 0.1 & OLR
= 1.3.

18 kN Type 14, R=0.65 (plane stress),

9 kN
Failure OLR = 2.0, at ac = 3.5 mm
7.425 kN
5.85 kN

Fig. 7.19 – Type 14 loading for 5 mm specimen (plane stress), where R = 0.65 &
OLR = 2.0.

58.5 kN
45 kN
Failure Type 15, R=0.65 (plane strain),
37.125 kN
OLR = 1.3, at ac = 3.5 mm

29.25 kN

Fig. 7.20 – Type 15 loading for 25 mm specimen (plane strain), where R = 0.65 &
OLR = 1.3.

11.7 kN 9 kN Type 16, R=0.65 (plane stress),

Failure OLR = 1.3, at ac = 3.5 mm
7.425 kN
5.85 kN

Fig. 7.21 – Type 16 loading for 5 mm specimen (plane stress), where R = 0.65 &
OLR = 1.3.

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Chapter 7: Fatigue Crack Growth in CT Specimens

7.3 Experimental results

7.3.1 Fracture toughness testing

Figure 7.22 below illustrates the load vs load-line displacement response of a 4140
high strength CT specimen loaded in tension to failure. The ultimate tensile load with
a fatigue pre-crack length of 9.155 mm is 49.3 kN and based on equations stipulated
in ASTM E-399-90, KIC was determine to be 100 MPa√m, with a plane strain / stress
boundary critical thickness of 14 mm. Therefore, the 25 mm thick specimen used to
conduct KIC testing is subjected to plane strain condition while the 5 mm thick
specimens is under plane stress condition.

60

50

40
Load ( kN)

30

20

10

0
0 0.5 1 1.5 2
Load-line displacement (mm)

Fig. 7.22 – Load vs Load-line displacement plot for 4140 high strength steel with a
9.155 mm pre-crack length.

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Chapter 7: Fatigue Crack Growth in CT Specimens

7.3.2 Plane strain fatigue crack growth without overloading

Figures 7.23 and 7.24 illustrate the crack growth behaviours for type 1 and type 8
loading.

0.0008
Type 1 - (plane strain) R=0.1
0.0007

0.0006
da/dN

0.0005

0.0004

0.0003

0.0002

0.0001

0
0 20000 40000 60000 80000 100000 120000 140000 160000
Cycles

Fig. 7.23 – Type 1 (plane strain) fatigue crack growth behaviour, R=0.1.

0.002
Type 8 - (plane strain) R=0.1
0.0018

0.0016

0.0014
da/dN

0.0012

0.001

0.0008

0.0006

0.0004

0.0002

0
0 5000 10000 15000 20000
N

Fig. 7.24 – Type 8 (plane strain) fatigue crack growth behaviour, R=0.1.

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Chapter 7: Fatigue Crack Growth in CT Specimens

7.3.3 Plane stress fatigue crack growth

Figures 7.25 and 7.26 illustrates the crack growth behaviours for type 1 and type 8
loading.

0.0008
Type 2 - (plane stress) R=0.1
0.0007

0.0006

0.0005
da/dN

0.0004

0.0003

0.0002

0.0001

0
0 20000 40000 60000 80000 100000 120000
Cycles

Fig. 7.25 – Type 2 (plane stress) fatigue crack growth behaviour, R=0.1.

0.002
Type 9 - (plane stress) R=0.1
0.0018

0.0016

0.0014
da/dN

0.0012

0.001

0.0008

0.0006

0.0004

0.0002

0
0 5000 10000 15000 20000
N

Fig. 7.26 – Type 9 (plane stress) fatigue crack growth behaviour, R=0.1.

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Chapter 7: Fatigue Crack Growth in CT Specimens

7.3.4 Plane strain crack growth with overloading

Figures 7.27-7.32 illustrate the plane strain crack growth behaviours for type 4, 6, 12,
10 and 15 loading regimes.

0.0008 Type 1 - (plane strain) R=0.1

Type 4 - (plane strain) OLR=2.0, R=0.1
0.0007
Type 6 - (plane strain) OLR=1.3, R=0.1
0.0006

0.0005
da/dN

0.0004

0.0003

0.0002

0.0001

0
0 20000 40000 60000 80000 100000 120000 140000 160000
Cycles

Fig. 7.27 – Type 1, 4 and 6 (plane strain) fatigue crack growth behaviour with
overloading.

0.0008
Type 1 - (plane strain) R=0.1

0.0007 Type 4 - (plane strain) OLR=2.0, R=0.1

Type 6 - (plane strain) OLR=1.3, R=0.1

0.0006

0.0005
da/dN

0.0004

0.0003

0.0002

0.0001

0
15 25 35 45 55
∆K

Fig. 7.28 – Type 1, 4 and 6 da/dN vs ∆K plots.

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Chapter 7: Fatigue Crack Growth in CT Specimens

0.003
Type 8 - (plane strain) R=0.1
Type 10 - (plane strain) OLR=2.0, R=0.1
0.0025 Type 12 - (plane strain) OLR=1.3, R=0.1

0.002
da/dN

0.0015

0.001

0.0005

0
0 5000 10000 15000 20000
Cycles

Fig. 7.29 – Type 8, 10 and 12 (plane strain) fatigue crack growth behaviour with
overloading.

0.002
Type 8 - (plane strain) R=0.1
0.0018
Type 10 - (plane strain) OLR=2.0, R=0.1
0.0016
Type 12 - (plane strain) OLR=1.3, R=0.1
0.0014

0.0012
da/dN

0.001

0.0008

0.0006

0.0004

0.0002

0
30 40 50 60 70
∆K

Fig. 7.30 – Type 8, 10 and 12 da/dN vs ∆K plots.

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Chapter 7: Fatigue Crack Growth in CT Specimens

0.0004
Type 15 - (plane strain), OLR=1.3, R=0.65
0.00035

0.0003

0.00025
da/dN

0.0002

0.00015

0.0001

0.00005

0
0 20000 40000 60000 80000 100000
Cycles

Fig. 7.31 – Type 15 (plane strain) fatigue crack growth behaviour with overloading.

0.0004
Type 15 - (plane strain), OLR=1.3, R=0.65
0.00035

0.0003

0.00025
da/dN

0.0002

0.00015

0.0001

0.00005

0
15 17 19 21 23 25
∆K

Fig. 7.32 – Type 32 da/dN vs ∆K plot.

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Chapter 7: Fatigue Crack Growth in CT Specimens

7.3.5 Plane stress crack growth with overloading

Figures 7.33-7.38 illustrate the plane stress crack growth behaviours for type 5, 7, 11,
13, 14 and 16 loading regimes.

0.0008
Type 2 - (plane stress) R=0.1

0.0007 Type 5 -(plane stress) OLR 2.0, R=0.1

Type 7 - (plane stress) OLR 1.3, R=0.1

0.0006

0.0005
da/dN

0.0004

0.0003

0.0002

0.0001

0
0 20000 40000 60000 80000 100000 120000
Cycles

Fig. 7.33 – Type 2, 5 and 7 (plane stress) fatigue crack growth behaviour with
overloading.

0.0008
Type 2 - (plane stress) R=0.1

0.0007 Type 5 - (plane stress) OLR 2.0, R=0.1

Type 7 - (plane stress) OLR 1.3, R=0.1

0.0006

0.0005
da/dN

0.0004

0.0003

0.0002

0.0001

0
15 25 35 45 55
∆K

Fig. 7.34 – Type 2, 5 and 7 da/dN vs ∆K plot.

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Chapter 7: Fatigue Crack Growth in CT Specimens

0.003
Type 9 - (plane stress) R=0.1
Type 11 - (plane stress) OLR=2.0, R=0.1
0.0025
Type 13 - (plane stress) OLR=1.3, R=0.1

0.002
da/dN

0.0015

0.001

0.0005

0
0 5000 10000 15000 20000 25000
Cycles

Fig. 7.35 – Type 9, 11 and 13 (plane stress) fatigue crack growth behaviour with
overloading.

0.002 Type 9 - (plane stress) R=0.1

0.0018 Type 11 - (plane stress) OLR=2.0, R=0.1

0.0016 Type 13 - (plane stress) OLR=1.3, R=0.1

0.0014

0.0012
da/dN

0.001

0.0008

0.0006

0.0004

0.0002

0
35 45 55 65 75 85
∆K

Fig. 7.36 – Type 9, 11 and 13 da/dN vs ∆K plot.

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Chapter 7: Fatigue Crack Growth in CT Specimens

0.0004
Type 14 - (plane stress) OLR=2.0, R=0.65
0.00035 Type 16 - (plane stress), OLR=1.3, R=0.65

0.0003

0.00025
da/dN

0.0002

0.00015

0.0001

0.00005

0
0 50000 100000 150000 200000 250000
Cycles

Fig. 7.37– Type 14 and 16 (plane stress) fatigue crack growth behaviour with
overloading.

0.0004
Type 14 - (plane stress) OLR=2.0, R=0.65
0.00035 Type 16 - (plane stress), OLR=1.3, R=0.65

0.0003

0.00025
da/dN

0.0002

0.00015

0.0001

0.00005

0
15 20 25 30 35
∆K

Fig. 7.38 – Type 14 and 16 da/dN vs ∆K plot.

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Chapter 7: Fatigue Crack Growth in CT Specimens

7.4 Discussion

The effects of overloading on fatigue crack growth have been studied extensively in
the past [42-56]. The aim of the testing regimes is not to replicate what we already
know about overloading (i.e. overloading tends to enhance overall fatigue life) and its
associated influence on fatigue crack growth. Rather, it is an exercise to observe and
study overloading effects under plane strain and plane stress conditions to see if there
are any discernable differences under the two distinct conditions. The influence of R
ratios and overload ratios (OLR) have also been investigated (under plane stress and
plane strain) to a give a more systematic and complete appreciation for fatigue crack
growth under these two distinct conditions. Discussions of results presented in section
7.3 are separated into the following sections.

7.4.1 Effect of R ratio and OLR on fatigue crack growth

As stated in section 2.1, overloading in most situations produces an enhancement in

overall fatigue life. Crack growth rates typically decrease immediately following an
overload and in certain instances, complete crack arrest is observed. Retardation,
however, is temporary as crack growth rates gradually recover to pre-overload levels.
In some cases, acceleration in crack growth rates is observed prior to characteristic
retardation. It is observed in results presented, that R ratios do influence fatigue crack
growth behaviours. In general, it is observed that the higher the R ratio, the lower the
fatigue crack growth rate is. Comparing overall fatigue life cycles in Figures 7.35 and
7.37 (i.e. R=0.1 versus R=0.65), it is clear to see that overall fatigue lives are much
larger at R=0.65 and crack growth rates are almost three times smaller than crack
growth rates observed at R=0.1. For type 3 loading at R=0.7, fatigue crack growth
could not be detected even after 2000000 cycles. Therefore, ∆K had to be increased to
(type 10-16) study the effect of R ratio on fatigue crack propagation. From the results
presented, it also seems that at R=0.1, the extent of crack growth retardation is higher
than that observed at R=0.65 (i.e. comparing type 16 to type 13 in Figures 7.35-7.38
and type 15 to type 12 in Figures 7.29-7.32 respectively) correlating well with
previous observations by Shin [48]. A type 16 loading regime at R=0.65, showed

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Chapter 7: Fatigue Crack Growth in CT Specimens

negligible crack retardation after overloading. As suggested in Chapter 2, previous

studies have show that at higher R ratios, back-face and near tip strain measurements
provided little evidence of crack closure, suggesting other mechanisms might be
responsible for the retardation observed. Crack tip blunting or residual stress
formation ahead of the crack tip are likely mechanisms. On the other hand, at low R
ratios, crack closure resulting from compressive residual stresses formed in the plastic
zone ahead of the crack tip and plastically-induce crack closure have been sighted as
possible mechanisms responsible for retardation. One notable exception is the
complete retardation observed for a type 14 loading regime at R=0.65. It is thought
that reduced ∆K and reverse plasticity at a high R ratio coupled with a high OLR,
would generate a crack driving force that is insufficient to overcome the large
compressive residual zone generated by the overload, thus resulting in complete crack
arrest and a significant enhancement in overall fatigue life.

The dependence of transient crack growth behaviour on R ratios as suggested by

Tsukuda [49] was not observed. Tsukuda observed distinctly different transient crack
propagation behaviour for both high and low R ratios following overloading. For low
R ratios, there is usually an initial acceleration in crack growth followed by
retardation after overloading. For high R ratios, however, the acceleration phase is
absent and crack growth retardation ensues immediately after overloading. In the
results presented, it is clear to see that transient crack growth behaviour is dependent
more on OLR than on R ratio. For a high OLR (i.e. OLR=2.0), initial acceleration in
crack growth followed by retardation is observed after overloading. On the other
hand, immediate crack retardation is observed for a lower OLR (i.e. OLR=1.3)
following overloading. Apart from the mechanisms discussed in Chapter 2, initial
acceleration in crack growth rates for OLR=2.0 can be attributed to crack shearing
caused by excessive near-tip stresses induced by overloading. Figure 7.39 for type 14
loading clearly shows the extent of crack shearing after extensive overloading. Crack
shearing is more prominent when the maximum load used for baseline fatigue is high
since the maximum overload level for a given OLR will be correspondingly higher as
well (i.e. type 10-16).

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Chapter 7: Fatigue Crack Growth in CT Specimens

(a) (b) (c) (d)

Fig. 7.39 – Type 14 overloading sequence showing crack shearing a) crack before
overload b) crack at beginning of overload c) crack shearing at maximum load d)
extension of crack after overload.

The fracture surface for a specimen after type 14 loading (Fig. 7.40) shows a distinct
“finger-print” marking induced as a result of overloading.

Crack growth direction

Crack extension resulting “Finger-print” showing crack

from crack shearing extension as a result of overloading

Fig. 7.40 – Fracture surface of specimen after type 14 loading.

As a comparison, Figure 7.41 shows the crack profiles before, during and after
overloading for a specimen undergoing a type 16 loading regime, while Figure 7.42
illustrates the fracture surface. Both figures show that for OLR=1.3, extensive crack
shearing is not observed and as a result, crack retardation occurs immediately after
overloading. The extent of overloading is also governed by OLR, where the higher the
OLR the greater the extent of crack retardation.

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Chapter 7: Fatigue Crack Growth in CT Specimens

(a) (b) (c)

Fig. 7.41 – Type 16 overloading sequence showing a) crack before overload b) crack
at maximum overload c) crack after overload.

Crack growth direction

Faint line indicating point of overload.

Crack extension not observed.

Fig. 7.42 – Fracture surface of specimen after type 16 loading.

It should be noted that an OLR of 2.0 could not be sustained for a type 10 loading
regime since the specimen fractured before maximum overload could be reached. As
a result, a corresponding loading regime with R=0.65 was not tested.

7.4.2 Plane strain vs plane stress crack growth

Figure 7.43 shows that the “initiation” point for crack growth in plane strain and plane
stress specimens are quite different. For type 1 loading, a crack is detected only after
90000 cycles as opposed to 20000 cycles for type 2 loading. Such a discrepancy can
be attributed to the higher driving force, ∆G (1.40), experienced under plane stress
conditions. i.e. even though ∆K is similar for both conditions, E’ for plane strain has
to be corrected to a larger value using Eq. 26 such that ∆Gplane strain < ∆Gplane stress.
However, the earlier detection of a crack under plane stress can also be attributed to

211
Chapter 7: Fatigue Crack Growth in CT Specimens

the fact that fatigue cracks initiating from the centre of the specimen thickness will
surface earlier in plane stress than in plane strain specimens.

0.0008
Type 1 - (plane strain) R=0.1
0.0007 Type 2 - (plane stress) R=0.1

0.0006

0.0005
da/dN

0.0004

0.0003

0.0002

0.0001

0
0 20000 40000 60000 80000 100000 120000 140000 160000
Cycles

Fig. 7.43 – Type 1 and 2 fatigue crack growth behaviour.

0.002
Type 8 - (plane strain) R=0.1
0.0018
Type 9 - (plane stress) R=0.1
0.0016

0.0014

0.0012
da/dN

0.001

0.0008

0.0006

0.0004

0.0002

0
0 5000 10000 15000 20000
Cycles

Fig. 7.44 – Type 8 and 9 fatigue crack growth behaviour.

When the load range is increased dramatically (as in case of type 8 and 9 loading)
such that plane strain conditions are beginning to approach plane stress conditions due
to increased deformation in the crack plane, discrepancies between the crack

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Chapter 7: Fatigue Crack Growth in CT Specimens

“initiation” point for plane stress and plane strain conditions are significantly less
obvious (Fig. 7.44).

A comparison of transient crack growth behaviours after overloading reveals that for
moderate (20-2 kN for plane strain and 4-0.4 kN for plane stress) to perhaps low load
ranges, differences in plane strain and plane stress crack growth and the extent of
crack retardation are negligible (Fig. 7.45)

0.0008
Type 4 - (plane strain) OLR=2.0, R=0.1

0.0007 Type 5 - (plane stress) OLR 2.0, R=0.1

Type 6 - (plane strain) OLR=1.3, R=0.1

0.0006
Type 7 - (plane stress) OLR 1.3, R=0.1
0.0005
da/dN

0.0004

0.0003

0.0002

0.0001

0
15 20 25 30 35 40 45 50
∆K

Fig. 7.45 – A comparison of da/dN vs ∆K plots for plane stress an plane strain
conditions (load range 20-2 kN for plane strain and 4-0.4 kN for plane stress).

However, as the load range is increased to much higher levels (load range 45-4.5 kN
for plane strain and 9-0.9 kN for plane stress), the extent of crack retardation under
plane stress is clearly greater than plane strain conditions for given OLR and R ratios
(Figs. 7.46 and 7.47).

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Chapter 7: Fatigue Crack Growth in CT Specimens

0.002
Type 10 - (plane strain) OLR=2.0, R=0.1
0.0018
Type 11 - (plane stress) OLR=2.0, R=0.1
0.0016 Type 12 - (plane strain) OLR=1.3, R=0.1

0.0014 Type 13 - (plane stress) OLR=1.3, R=0.1

0.0012
da/dN

0.001

0.0008

0.0006

0.0004

0.0002

0
40 50 60 70 80 90 100
∆K

Fig. 7.46 – A comparison of da/dN vs ∆K plots for plane stress and plane strain
conditions (load range 45-4.5 kN for plane strain and 9-0.9 kN for plane stress).

0.0004
Type 14 - (plane stress) OLR=2.0, R=0.65
Type 15 - (plane strain), OLR=1.3, R=0.65
0.00035
Type 16 - (plane stress), OLR=1.3, R=0.65

0.0003

0.00025
da/dN

0.0002

0.00015

0.0001

0.00005

0
15 17 19 21 23 25 27
∆K

Fig. 7.47 – A comparison of da/dN vs ∆K plots for plane stress and plane strain
conditions (load range 45-29.25 kN for plane strain and 9-5.85 kN for plane stress).

This observation can be explained by understanding that the monotonic plastic zone
size generated by overloading is larger under plane stress conditions. A larger plastic

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Chapter 7: Fatigue Crack Growth in CT Specimens

zone would mean that the overload effected zone, consisting of compressive residual
stresses is also larger, hence posing a greater barrier to crack propagation. Therefore,
the extent of crack retardation under plane stress is greater than under plane strain
conditions.

7.5 Conclusion

1. Increasing R ratios increase fatigue crack growth rates.

2. In general, the extent of retardation decreases with increasing R ratios. An

exception occurs when both R ratio and OLR are high. In this case, large
compressive residual stress fields ahead of the crack induced by extensive
overloading serves as an effective barrier to the relatively weak crack driving
force due to a reduction in ∆K as a result of a higher R ratio.

3. Acceleration followed by retardation in crack growth rates are observed

immediately after overloading for OLR=2.0. Immediate crack growth
retardation is observed after overloading for OLR=1.3. Crack growth
acceleration for OLR=2.0 is caused by crack shearing, particularly when the
maximum load used for baseline fatigue is high.

4. Transient acceleration in crack growth rates immediately after overloading for

4140 high strength steel does not seem dependent on R ratio.

5. Differences in the point of crack “initiation” under both plane stress and plane
strain conditions decrease with increasing load range.

6. The extent of crack retardation is greater under plane stress than plane strain
conditions.

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Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

Chapter 8: Cumulative “Damage” & Fatigue

Crack Propagation Modelling

8.1 Introduction

As seen in section 2.4, a multitude of cumulative fatigue “damage” models have been
proposed over the years. Most of the models require the determination of various
parameters, which are valid only for certain types of materials and loading conditions,
thus, limiting their practical use. The laboriously tedious computational requirements
of certain models also render them impractical for general applications. In section 8.2,
an alternative “damage” accumulation model, similar in form to the non-linear
“damage” accumulation rule by Marco and Starkey [93] is proposed. Predictions
based on the proposed model are compared with experimental results to evaluate the
model’s effectiveness. Comparisons with the Damage Curve Approach (DCA) [105]
and Double Linear Damage Rule (DLDR) [104] (Manson and Halford) are also
performed to test and see how the model performs against other established
cumulative fatigue models.

At present, stress / strain-based approaches and fracture mechanics fatigue analysis

methods are often applied separately. Most fatigue analysis involves either one or the
other but seldom in conjunction. To enhance and facilitate the understanding of
fatigue and crack growth mechanics, however, it is felt that the two approaches should
be studied and used concurrently in a manner such that they compliment each other.

Prediction of fatigue crack growth by way of the Paris Law requires experimentally
determined constants. This is can be a time consuming and costly exercise especially
for material engineers attempting to select materials with appropriate fatigue
resistance. A number of “damage” models, employing low cycle fatigue data to
quantify “damage” at the crack tip have been developed [116-118]. Liu and Iino [116]
proposed a model by assuming that the points ahead of a crack tip to be made up of a
set of uniaxial fatigue specimens and crack propagation occurs as a result of

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Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

successive failure of each imaginary specimen. By experimentally measuring the

plastic zone size and strain distribution ahead of a crack tip, Liu and Iino derived a
crack propagation equation similar in form to the Paris Law (1.41) with an exponent,
n = 2. Majumdar [117] modified Liu and Iino’s approach by introducing a
microstructural parameter, p*, as the lower bound limit for his fatigue crack growth
equation, where for any distance ahead of a crack tip ≤ p*, strain amplitude effects are
ignored. Rice’s model [119] for antiplane shear (mode III) under small scale yielding
was adapted to estimate stress and strain distributions ahead of the crack tip.
Charkrabortty [118], on the other hand, envisioned fatigue crack growth as a
consequence of the exhaustion of ductility due to the cyclic plastic strain in reverse
yield zone ahead of the crack tip and a crack growth model with no adjustable
parameters is obtained.

In section 8.3 a crack propagation model based on monotonic properties is proposed.

The crack propagation model is proposed to alleviate the material selection process by
providing engineers a means to rapidly eliminate and narrow down selections for
appropriate fatigue resistance.

The objectives of this chapter are therefore to:

1. Develop a practical fatigue “damage” model for smooth specimens.

2. Develop a crack growth model for cracked specimens based on fatigue data
obtained from smooth specimen testing.

3. Justify the importance and significance of smooth specimen testing.

4. Combine smooth specimen testing and fracture mechanics principles in the

understanding of key fatigue processes and crack growth mechanics.

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Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

8.2 An alternative cumulative fatigue “damage” model

The model proposed in this section takes the form of Macro and Starkey’s [93] non-
linear “damage” accumulation model:

D = (n /N f i )
pi
(8.1)

In Macro and Starkey’s “damage” model, the exponent pi, is related to the stress
imposed and the application of such a function requires prior knowledge of pi for
various levels of stress concerned. In this model however, prior knowledge of pi for
various stress levels is not required and pi can easily be determined from known
values of fatigue life, Nfi.

The concept of this model is explained as follows. It is firstly assumed that for Nf = 0,
the “damage” curve takes the profile of a linear line stretching from the origin (0,0) to
point (1,1) (Fig. 8.2). From previous discussions, it is known that as Nf increases, the
fraction of life spent on stage I crack growth increases. Consequently, the curvature of
“damage” curves becomes sharper with increasing values of Nf (Fig. 8.1). As a
result, if one assumes that the fatigue limit corresponds to Nf = 1000000 cycles, it is
reasonable to suggest a corresponding 90 degree “damage” curve profile illustrated in
Figure 8.2. This implies that for Nf values close to 106 cycles, the fraction of fatigue
life spent in stage I crack growth is so much greater than the fraction of life spent at
stage II crack growth, that near 90 degree “damage” curves are expected even though
in reality, a perfect 90 degree “damage” curve would not be possible. With the upper
and lower bound limits identified, the x, y points of “damage” curves can be
calculated based on the following relationships (Fig. 8.2). Note that Eqn. 8.2 has been
derived from basic trigonometry calculations based on Fig. 8.2. The angle φ is
calculated based on the assumption that the angle 450 at Nf = 0, decreases by
105 6 N f (0.7854)
with increasing Nf until φ =0 when Nf = 1000000.
106

10 5 6 N f (0.7854)
φ = 0.7854 − (8.2)
10 6

218
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

2
H= 2 (8.3)
⎛ 10 5 6 N f (0.7854) ⎞
cos⎜ ⎟
⎜ 10 6 ⎟
⎝ ⎠

⎛ 105 6 N (H ) ⎞
⎜
h = H −⎜
f ⎟ (8.4)
10 6 ⎟
⎝ ⎠

x = 1− hcos φ (8.5)

y = h sinφ (8.6)
1

N1 > N2 > N3 > N4

“Damage”, D

N1

N2
N3
N4
0

0 N/Nf 1

Fig. 8.1 – “Damage” curve profiles with increasing Nf values.

Once points x and y are determined from Eqs. 8.5 and 8.6 respectively, the
exponential p is evaluated using:

log D log y
p= = (8.7)
log(n /N f ) log x

Note that the above model approximates “damage” accumulation and predicts
“damage” curves for single level fatigue loading only. The exponent p will always be
> 1 due to the way coordinates x and y are determined (Fig. 8.2). As a result,

219
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

“damage” curves predicted using the proposed model will adopt the general trends
indicated in Figure 8.1. Figure 8.4 illustrates a series of damage curve plotted based
on x,y positions calculated using the proposed methodology presented in Eqns. 8.2-
8.7.

1
“Damage”, D
Nf = 0

√2/2
H
y h
Ø
0

0 N/Nf Nf = 1 1

Fig. 8.2 – Calculating points x, y for a “damage” curve.

An example of such a calculation is shown below:

For Nf = 15000 cycles,

10 5 6 15000 (0.785398)
φ = 0.785398 − = 0.39536
10 6

2
H= 2 = 0.7645
⎛ 10 5 6 15000 (0.785398) ⎞
cos⎜⎜ ⎟
⎟
⎝ 106 ⎠

⎛ 105 6 15000 (0.7645) ⎞

h = H − ⎜⎜ ⎟⎟ = 0.384855
⎝ 10 6 ⎠

x = 1− 0.384855cos(0.39536) = 0.644834

y = 0.384855 sin(0.39536) = 0.14822

220
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

log 0.14822
p= = 4.35
log0.644834

Therefore, the “damage” curve obeys the following relationship:

D = (n /N f )
4.35

Figure 8.3 illustrates the “damage” curve profile for p = 4.35 and Figure 8.4 illustrates
a series of “damage” curves obtained based on the model proposed. It can be seen that
the “damage” curve trends reflect what is typically observed.

0.9

0.8

0.7

0.6

0.5
D

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf

Fig. 8.3 – “Damage” curve for exponent p = 4.35 based on Nf =15000.

There are many advantages for using this model. Firstly, the exponent p can be
calculated with relative ease with a known Nf value and p is independent of loading
conditions, material etc. Even in the presence of mean stresses, p is altered according
to the way Nf changes. The exponent p is solely a function of Nf which means that
the above model can be applied to a wide variety of loading conditions and materials
without consequences. For multilevel loading, it is just a simple case of calculating an
equivalent cycle ratio for the accumulated damage incurred for all previous levels of
loading and adding the amount of cycle ratio done at the present stress/strain level,
such that D = 1.

221
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

To check the validity of the proposed model, “damage” curve predictions have been
compared with those experimentally taken for 4340 high strength steel. Experimental
results showed very good agreement with predicted curves (Fig. 8.5). Comparisons
between predicted (based on the proposed model) and experimentally derived cycle
ratio plots for a variety of loading conditions are illustrated in Figures 8.6-8.11. (Note
that where mean stress effects are known to be influential, predictions with and
without mean stress correction have been performed to give a better comparison.)
Results in general show good agreement between predicted and experimental cycle
ratio trends. Figures 8.12-8.17 compare cycle ratios trends predicted using the
proposed model, DCA, DLDR and the Palmgren-Miner rule. In most cases, the
proposed model presents a much better fit to experimental results compared to the
other models.

0.9

0.8

0.7
“Damage”, D

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf

Fig. 8.4 – Series of “damage” curves based on model proposed. Circle points
indicated x, y positions calculated for each curve.

222
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

1
1.0% 0.6%

0.9 0.3% Predicted (D,n/Nf)

1.0% Prediction 0.6% Prediction
0.8 0.3% Prediction

0.7

0.6
“Damage”, D

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf

Fig. 8.5 – Comparison of predicted and experimental “damage” curves for 4340 high
strength steel. Figure shows good correlation between experimental and predicted
curve trends.

223
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

0.8
n/Nf, EDF = 40076 cycles

0.6

0.4

0.2

0
0 0.5 1 1.5
n/Nf, PDF = 534 cycles

Palmgren-Miner prediction
PDF(1.0% amp)-EDF(0.3% amp) Type A
PDF(1.0% amp)-EDF(0.3% amp) Type F
EDF(0.3% amp)-PDF(1.0% amp) Type D
EDF(0.3% amp)-PDF(1.0% amp) Type J
PDF-EDF prediction w/o mean stress
PDF-EDF prediction with mean stress
EDF-PDF prediction w/o mean stress

Fig. 8.6 – Comparison of predicted and experimental cycle ratio trends for PDF/EDF
interaction in 4340 high strength steel.

224
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

1 Palmgren-Miner prediction
PDF(1.0% amp)-PDF(0.6% amp) Type B
0.9
PDF(1.0% amp)-PDF(0.6% amp) Type G
0.8 PDF(1.0%)-PDF(0.65%) prediction

n/Nf, PDF (0.6% amp) = 835 cycles 0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf, PDF(1.0% amp) = 534 cycles

Fig. 8.7 – Comparison of predicted and experimental cycle ratio trends for PDF/PDF
interaction in 4340 high strength steel.

Palmgren-Miner prediction
1
EDF(0.3% amp)-EDF(0.25% amp) Type C
EDF(0.3% amp)-EDF(0.25% amp) Type I
n/Nf, EDF (0.25% amp) = 850000 cycles

0.9
EDF(0.3%)-EDF(0.25%) prediction
0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf, EDF(0.3% amp) = 40076 cycles

Fig. 8.8 – Comparison of predicted and experimental cycle ratio trends for EDF/EDF
interaction in 4340 high strength steel.

225
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

1.8

1.6
n/Nf, Nf (EDF) = 39426 cycles
1.4

1.2

0.8

0.6

0.4

0.2

0
0 0.5 1 1.5 2
n/Nf, Nf (PDF) = 176 cycles

Palmgren-Miner
PDF(1.0% amp)-EDF(0.3% amp) Type A
PDF(1.0% amp)-EDF(0.3% amp) Type F
EDF(0.3%)-PDF(1.0%) Type D
PDF-EDF prediction w/o mean stress
PDF-EDF prediction with mean stress
PDF-EDF prediction with mean stress Type F
EDF-PDF prediction

Fig. 8.9 – Comparison of predicted and experimental cycle ratio trends for PDF/EDF
interaction in 6061-T6 aluminium alloy.

226
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

Palmgren-Miner prediction
1 PDF(1.0% amp)-PDF(0.6% amp) Type B
PDF(1.0% amp)-PDF(0.6% amp) Type G
PDF(1.0%)-PDF(0.6%) prediction
0.8

n/Nf, PDF (0.6% amp) = 835 cycles

0.6

0.4

0.2

0
0 0.2 0.4 0.6 0.8 1
n/Nf, PDF(1.0% amp) = 176 cycles

Fig. 8.10 – Comparison of predicted and experimental cycle ratio trends for PDF/PDF
interaction in 6061-T6 aluminium alloy.

0.9
n/Nf, EDF (0.3% amp) = 39426 cycles

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.5 1 1.5
n/Nf, EDF(0.4% amp) = 4387 cycles

Palmgren-Miner prediction

EDF(0.4% amp)-EDF(0.3% amp) Type C

EDF(0.4% amp)-EDF(0.3% amp) Type I

EDF(0.3% amp)-EDF(0.4% amp) Type E

EDF(0.4%)-EDF(0.3%) prediction

EDF(0.3%)-EDF(0.4%) prediction

Fig. 8.11 – Comparison of predicted and experimental cycle ratio trends for PDF/PDF
interaction in 6061-T6 aluminium alloy.

227
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

n/Nf, EDF = 40076 cycles

0.8

0.6

0.4

0.2

0
0 0.5 1 1.5
n/Nf, PDF = 534 cycles

Palmgren-Miner prediction
PDF(1.0% amp)-EDF(0.3% amp) Type A
PDF(1.0% amp)-EDF(0.3% amp) Type F
EDF(0.3% amp)-PDF(1.0% amp) Type D
EDF(0.3% amp)-PDF(1.0% amp) Type J
Proposed model w/o mean stress
Proposed model with mean stress
DCA
DLDR

Fig. 8.12 – Comparison of cumulative “damage” models for PDF/EDF interaction in

4340 high strength steel.

228
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

Palmgren-Miner prediction
1
PDF(1.0% amp)-PDF(0.6% amp) Type
0.9 B
PDF(1.0% amp)-PDF(0.6% amp) Type
G

n/Nf, PDF (0.6% amp) = 835 cycles

0.8 Proposed model
0.7 DCA

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf, PDF(1.0% amp) = 534 cycles

Fig. 8.13 – Comparison of cumulative “damage” models for PDF/PDF interaction in

4340 high strength steel.

Palmgren-Miner prediction
1 EDF(0.3% amp)-EDF(0.25% amp) Type C
EDF(0.3% amp)-EDF(0.25% amp) Type I
0.9 Proposed model
n/Nf, EDF (0.25% amp) = 850000 cycles

DCA
0.8
DLDR
0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
n/Nf, EDF(0.3% amp) = 40076 cycles

Fig. 8.14 – Comparison of cumulative “damage” models for EDF/EDF interaction in

4340 high strength steel.

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Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

1.8

1.6
n/Nf, Nf (EDF) = 39426 cycles

1.4

1.2

0.8

0.6

0.4

0.2

0
0 0.5 1 1.5 2
n/Nf, Nf (PDF) = 176 cycles

Palmgren-Miner
PDF(1.0% amp)-EDF(0.3% amp) Type A
PDF(1.0% amp)-EDF(0.3% amp) Type F
EDF(0.3%)-PDF(1.0%) Type D
Proposed model w/o mean stress
Proposed model with mean stress
Proposed model with mean stress Type F
DCA
DLDR

Fig. 8.15 – Comparison of cumulative “damage” models for PDF/EDF interaction in

6061-T6 aluminium alloy.

230
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

Palmgren-Miner prediction
1 PDF(1.0% amp)-PDF(0.6% amp) Type B
PDF(1.0% amp)-PDF(0.6% amp) Type G
Proposed model

n/Nf, PDF (0.6% amp) = 835 cycles

0.8 DCA
DLDR

0.6

0.4

0.2

0
0 0.2 0.4 0.6 0.8 1
n/Nf, PDF(1.0% amp) = 176 cycles

Fig. 8.16 – Comparison of cumulative “damage” models for PDF/PDF interaction in

6061-T6 aluminium alloy.

Palmgren-Miner prediction
EDF(0.4% amp)-EDF(0.3% amp) Type C
EDF(0.4% amp)-EDF(0.3% amp) Type I
1
EDF(0.3% amp)-EDF(0.4% amp) Type E

0.9 Proposed model

DCA
n/Nf, EDF(0.3% amp) = 39426 cycles

0.8
DLDR

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.5 1 1.5
n/Nf, EDF(0.4% amp) = 4387 cycles

Fig. 8.17 – Comparison of cumulative “damage” models for EDF/EDF interaction in

6061-T6 aluminium alloy.

231
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

8.3 Crack propagation model based on smooth specimen data

The present model adopts Liu and Iino’s uniaxial strip model. However, instead of
experimentally determining the plastic zone size and strain distribution ahead of the
crack, analytical results obtained by Rice [119] and Majumdar [117] are employed.
The motivation behind developing such a model, is so that an estimation of fatigue
crack growth behaviour can be made from readily available monotonic and smooth
specimen fatigue data during the initial phase of selecting materials with maximum
fatigue resistance. With the objective of developing a simple and easy to apply crack
propagation model, Liu and Iino’s uniaxial strip model has in this case been
simplified to comprise of only a single element ahead of the crack tip. As a result,
fatigue “damage” on subsequent elements by present loading has been assumed to be
negligible. It is also reasonable to apply this model without consideration for
microscopic properties because its influence under conditions of extensive plastic
deformation is probably limited. The width of this element corresponds to the size of
the cyclic plastic zone and it is assumed that crack advance occurs only when the
fatigue life of this imaginary element is completely exhausted. The basic principle of
this model is illustrated in Figure 8.18.

1 2 3 4

……
rc

Advancing crack with increasing cyclic

plastic zone size.

Fig. 8.18 – Crack propagation model with single uniaxial element in front of fatigue
crack tip.

The process of computing and estimating a material’s fatigue crack growth behaviour
is a straightforward and simple process. In this case, monotonic properties for 4140
high strength steel will be used to estimate smooth specimen fatigue parameters and

232
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

using the crack propagation model proposed, fatigue crack growth behaviours for
both predicted experimental results will be compared.

The first stage of analysis is to determine smooth specimen fatigue parameters in the
absence of such data. From ASM Handbook 1 [35], it is recommended that fatigue
parameters can be estimated by:

σ’f = σUTS + 345 MPa (8.8)

⎛ 100 ⎞
ε'f = ln⎜ ⎟ (8.9)
⎝ 100 − %RA ⎠

n’=b/c (8.10)

where, %RA is the reduction in area and b and c are approximated as –0.085 and –0.6
respectively.

Using monotonic properties for 4140 high strength steel used in Chapter 7, the
following fatigue parameters are estimated (Table 8.1).

Table 8.1 – Estimated fatigue parameters for 4140 high strength steel.
σ’f (MPa) b ε’f c n’
1795 -0.085 0.693 -0.6 0.1467

The next stage of analysis is to determine the cyclic plastic zone size, rc, and the
fatigue life, Nf of the element in front of the crack tip. Using analytical results
obtained by Rice [119] and Majumdar [117], the following equations were used to
determine rpc, Nf.

∆K 2
rpc = (8.11)
4(1+ n')πσ y
2

233
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

∆σ = 2σ 'f (2N f )b (8.12)

∆ε = 2ε'f (2N f )c (8.13)

∆K 2
∆σ ∆ε = (8.14)
(1+ n')πEx

where, E is the Young’s Modulus and x is the distance ahead of the crack tip and it is
defined as rpc/2 in this case. For the model to work effectively, one assumes that x =
rpc1/2 of the first element. Since rpc increases with each advancement of crack as a
result of an increase in ∆K, the distance rpc/2 increases as well and there could be
instances where stress and strain levels used to calculate Nf might be lower than those
used when both rpc and crack length were smaller. This is contravenes logic, therefore,
an assumption of fixed rpc/2 values based on the width of the first element is used
such that stress and strain ranges experienced by elements increase as crack advances.

Once Nf is determined from numerical analysis, the number of cycles to failure for
that element is said to equal Nf and thus, dN = Nf and the incremental crack advance
length, da = rpc, which is the width for the relevant element. This process is repeated
until a fatigue crack growth trend is attained.

Figures 8.19-8.22 show and compare predicted and experimental crack growth
behaviours of 4140 high strength steel loaded under Type 1,2, 8 and 9 regimes (see
Chapter 7). Results appear promising as the predicted trends reflect the experimental
crack growth behaviours in general and could be improved if actual fatigue
parameters were obtained. The favourable results also suggest that PDF (or smooth
sample) studies are relevant to fatigue crack growth studies based on fracture
mechanics and that the collection of PDF data using smooth specimens is both an
important and useful exercise.

234
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

0.0008
Type 1 - (plane strain) R=0.1
0.0007
Predicted crack growth behaviour
0.0006

0.0005
da/dN

0.0004

0.0003

0.0002

0.0001

0
15 25 35 45 55
∆K

Fig. 8.19 – Comparison of predicted and experimentally observed crack growth

behaviour for Type 1 loading.

0.0008
Type 2 - (plane stress) R=0.1
0.0007
Predicted crack growth behaviour
0.0006

0.0005
da/dN

0.0004

0.0003

0.0002

0.0001

0
15 25 35 45 55
∆K

Fig. 8.20 – Comparison of predicted and experimentally observed crack growth

behaviour for Type 2 loading.

235
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

0.002
Type 8 - (plane strain) R=0.1
0.0018

0.0016 Predicted crack growth behaviour

0.0014

0.0012
da/dN

0.001

0.0008

0.0006

0.0004

0.0002

0
30 40 50 60 70 80
∆K

Fig. 8.21 – Comparison of predicted and experimentally observed crack growth

behaviour for Type 8 loading.

0.002
Type 9 - (plane stress) R=0.1
0.0018

0.0016 Predicted crack growth behaviour

0.0014

0.0012
da/dN

0.001

0.0008

0.0006

0.0004

0.0002

0
35 45 55 65 75 85
∆K

Fig. 8.22 – Comparison of predicted and experimentally observed crack growth

behaviour for Type 9 loading.

236
Chapter 8: Cumulative “Damage” and Fatigue Crack Propagation Modelling

8.4 Conclusions

1. A practical “damage” model that can be applied to any material and loading
condition is proposed.

2. Predicted cycle ratio trends correlate better to experimental results and other
cumulative “damage” models.

3. Predications of crack growth behaviour from the model presented reflect

experimental results. It appears as an effective model for estimating fatigue
crack growth behaviour from just monotonic material properties.

4. Smooth specimen data can be used to determine actual fatigue parameters and
thus, improve growth behaviour predictions.

5. Smooth specimen testing and fracture mechanics principles can be combined

to understand key fatigue processes and crack growth mechanics.

237
Chapter 9: Conclusions

Chapter 9: Conclusions

Investigations into the interaction between PDF and EDF have been performed.
Experiments have been carried out systematically to study cycle ratio trends and
fatigue mechanisms arising from variable 2-step and multi-step loading sequences.
Results indicate the importance and advantages of PDF / EDF classification and
specification over conventional LCF / HCF classification. Atypical cycle ratio trends
could be expected when combinations of strain amplitudes from different fatigue
regimes are involved. It has been shown that proper boundary classification between
the two distinct regimes of fatigue provides a great deal of information regarding the
possible mechanisms which may be responsible for the interaction effects observed,
particularly on the possibility of mean stress effects coming into play. In addition, a
qualitative crack growth model has been presented to explain and rationalise cycle
ratios trends and sequence effects observed for multilevel loading. Crack growth
profiles in the model presented are supported by intermittent surface crack
measurements made using acetate replicas. Mean strain effects on fatigue life and 2-
step loading were also studied and results showed that mean strain affects primarily
EDF with little or no influence on PDF life. The two primary mechanisms responsible
for cycle ratio trends observed are the crack growth interaction behaviour and mean
stress effect.

In Chapter 7, studies related to crack growth in notched CT specimens were

discussed. Crack growth and overloading were studied under conditions of plane
stress and plane strain. Results indicate that differences in the point of crack
“initiation” under both plane stress and plane strain conditions decrease with
increasing load range, while the extent of crack retardation is greater under plane
stress than plane strain conditions. The effects of R ratios and OLR have also been
studied. The extent of crack growth retardation increases with increasing OLR and
decreasing R ratios. Two “damage” models were presented in Chapter 8. The first
model is a simple “damage” model proposed to predict “damage” accumulation under
multilevel loading sequences. Predictions are in good agreement with experimental
results and in general preforms better as compared to other popular cumulative
“damage” rules. The second model is a fatigue crack propagation model proposed to

238
Chapter 9: Conclusions

predict and estimate fatigue crack growth behaviour based on monotonic and smooth
specimen fatigue data. It is shown from results presented that the model proposed can
serve as an effective means for estimating fatigue crack growth behaviour of metals
and enhancing the process of selecting materials for maximum fatigue resistance. It is
important to note that model proposed does not function to completely replace
conventional crack propagation studies using CT specimens. The model, however,
serves as a good first estimator for expected fatigue crack growth behaviour in metals.
This model also shows that a close link between the two different fatigue test methods
(smooth specimen and fracture mechanics) can be established. Listed below is a
summary of the conclusions drawn based on the results presented.

9.1 2-level loading

1. PDF, EDF conveys more meaning than LCF, HCF classification.

2. Initial PDF is detrimental to subsequent EDF life.

3. Total fatigue life decreases with increasing degrees of PDF introduction due
to the greater extent of damage (crack size) caused. However, the
proportionate decrease (i.e. deviation from Palmgren-Miner predictions) in
overall fatigue life increases with lower degrees of initial PDF exposure.

4. Cycle ratio and fatigue life trends can be explained by crack closure, crack
growth mechanics and mean stress effects.

5. The crack growth model does not take into account mean stress, residual
stress and crack closure effects. Its application is limited to cases where the
aforementioned factors are less significant. Nevertheless, it serves as a useful
first step in rationalising fatigue life trends in multilevel loading.

6. PDF, EDF descriptions better reflect mean stress effect on cycle ratio and
overall fatigue life trends.

239
Chapter 9: Conclusions

7. Tensile or compressive mean stresses induced depend on the way, which

PDF→EDF transition occurs.

8. Higher degrees of PDF exposure in:

9. cyclic softening materials results in lower mean stresses induced during EDF
phase.
10. cyclic hardening materials results in higher mean stresses induced during
EDF phase.

11. Multilevel fatigue loading under controlled-strain yields atypical cycle ratio
trends (Figs. 59-60).

12. PDF, EDF specification assists interpretation of observed cycle ratio trends.

9.2 Multilevel loading

1. Overall cycle ratio trends are governed by the High→Low phase for both
High→Low→High and Low→High→Low sequences.

2. PDF, EDF descriptions still reflect mean stress effect on cycle ratio and
overall fatigue life trends.

3. A type 1 sequence is more detrimental than a type 2 sequence, even though

life fractions of PDF imposed are similar.

4. Block size influences overall fatigue lives and cycle ratio sums.

5. Repeated block loading of smaller type 1 sequence is more detrimental than a

single continuous type 1 block.

6. A type 5 sequence yields is less “damaging” than a type 3 sequence.

240
Chapter 9: Conclusions

9.3 Mean strain effects

1. Mean strain affects EDF but not PDF.

2. PDF/EDF definitions are still valid in the presence plasticity dominant tensile
mean strains.

3. Tensile mean strains could enhance overall fatigue life for PDF→EDF
sequences by enhancing early stage cyclic softening such that mean stresses
induced during EDF are lower than those induced under 0 mean strain
conditions.

4. Mean strains do not affect EDF→PDF sequences.

9.4 Fatigue crack growth in CT specimens

1. Increasing R ratios increase fatigue crack growth rates.

2. In general, the extent of retardation decreases with increasing R ratios. An

exception occurs when both R ratio and OLR are high. In this case, large
compressive residual stress fields ahead of the crack induced by extensive
overloading serves as an effective barrier to the relatively weak crack driving
force due to a reduction in ∆K as a result of a higher R ratio.

3. Acceleration followed by retardation in crack growth rates are observed

immediately after overloading for OLR=2.0. Immediate crack growth
retardation is observed after overloading for OLR=1.3. Crack growth
acceleration for OLR=2.0 is caused by crack shearing, particularly when the
maximum load used for baseline fatigue is high.

4. Transient acceleration in crack growth rates immediately after overloading

for 4140 high strength steel does not seem dependent on R ratio.

241
Chapter 9: Conclusions

5. Differences in the point of crack “initiation” under both plane stress and
plane strain conditions decrease with increasing load range.

6. The extent of crack retardation is greater under plane stress than plane strain
conditions.

9.5 Cumulative fatigue “damage” and fatigue crack

propagation modelling

1. A practical “damage” model that can be applied to any material and loading
condition is proposed.

2. Predicted cycle ratio trends correlate better to experimental results and other
cumulative “damage” models.

3. Predications of crack growth behaviour from the model presented reflect

experimental results. It appears as an effective model for estimating fatigue
crack growth behaviour from just monotonic material properties.

4. Smooth specimen data can be used to determine actual fatigue parameters

and thus, improve growth behaviour predictions.

5. Smooth specimen testing and fracture mechanics principles can be combined

to understand key fatigue processes and crack growth mechanics.

242
Chapter 9: Conclusions

9.6 Recommendations for future work

Thus far, only three materials, namely 316L stainless steel, 6061-T6 aluminium alloy,
and 4340 high strength steel, are used to study 2-step and multi-step loading
sequences. A more rigorous and complete study can be made by executing similar
tests on more materials and building up on existing fatigue results. It is recommended
that materials with strong cyclic softening and hardening characteristics be used as
they will more interesting results particular when in loading regimes which involve
both PDF and EDF. The 0.05% offset rule for evaluating the boundary between PDF
and EDF is based on the cyclic stress-strain responses observed for the three materials
tested. Further tests conducted over a broader range of materials will test this
boundary definition rule and confirm present findings that such PDF/EDF
classification does indeed provide indications as to whether mean stress effects play
an important role in the interaction behaviour or not.

The boundary between PDF and EDF is not as distinct as its definition might have us
believe. The transition from PDF to EDF or vice versa is actually a continuum.
Fatigue studies can be made using strain amplitudes close to the boundary to
investigate the transitional profile between PDF and EDF and the consequences of
such interactions. Variations to current testing programs can also be studied. For
example studying PDF and EDF interactions with single or multiple overstrains.
Similar test programs can also be performed under different test conditions, e.g.
elevated temperatures or corrosive environments to see if present mechanisms and
results still hold. Finite elemental modelling of PDF and EDF interaction could also
be performed to add a new dimension to the fatigue analysis.

Finally, the models proposed could be tested against more materials and cumulative
“damage” models to provide a more complete evaluation of their effectiveness.

243
 Publications

The following technical publications and presentations have been delivered during the
course of this investigation:

1. Wong, Y.K., Hu, X.Z., Norton, M.P., “Low Cycle / High Cycle Fatigue
Interaction and Overstraining Behaviour in 316L Stainless Steel”, Structural
Integrity and Fracture Conference 2000, University Technology of Sydney,
pp. 88-98, 2000. (Won the best student paper award)

2. Wong, Y.K., Hu, X.Z., Norton, M.P., “Low and High Cycle Fatigue
Interaction in 316L Stainless Steel”, Journal of Testing and Evaluation,
JTEVA, Vol. 29, No. 2, pp. 138-145, March 2001.

3. Wong, Y.K., Hu, X.Z., Norton, M.P., “Plastically / Elastically Dominant

Fatigue Interaction”, 10th International Conference on Fracture, Hawaii,
Elsevier, 2001.

4. Wong, Y.K., Hu, X.Z., Norton, M.P., “Plastically Dominant / Elastically

Dominant Fatigue Interaction in 316L Stainless Steel and 6061-T6 Aluminium
Alloy”, Fatigue Fracture Engineering Materials Structure, Vol. 25, No. 2, pp.
201-214, 2002.

5. Wong, Y.K., Hu, X.Z., Norton, M.P., “An alternative definition for two
distinct regimes of fatigue”, Structural Integrity and Fracture Conference 2002
(accepted for publication).

244
References

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253
 Appendices – Published Papers

1. Wong, Y.K., Hu, X.Z., Norton, M.P., “Low Cycle / High Cycle Fatigue
Interaction and Overstraining Behaviour in 316L Stainless Steel”, Structural
Integrity and Fracture Conference 2000, University Technology of Sydney,
pp. 88-98, 2000. (Won the best student paper award)

2. Wong, Y.K., Hu, X.Z., Norton, M.P., “Low and High Cycle Fatigue
Interaction in 316L Stainless Steel”, Journal of Testing and Evaluation,
JTEVA, Vol. 29, No. 2, pp. 138-145, March 2001.

3. Wong, Y.K., Hu, X.Z., Norton, M.P., “Plastically / Elastically Dominant

Fatigue Interaction”, 10th International Conference on Fracture, Hawaii,
Elsevier, 2001.

4. Wong, Y.K., Hu, X.Z., Norton, M.P., “Plastically Dominant / Elastically

Dominant Fatigue Interaction in 316L Stainless Steel and 6061-T6 Aluminium
Alloy”, Fatigue Fracture Engineering Materials Structure, Vol. 25, No. 2, pp.
201-214, 2002.

5. Wong, Y.K., Hu, X.Z., Norton, M.P., “An alternative definition for two
distinct regimes of fatigue”, Structural Integrity and Fracture Conference 2002
(accepted for publication).

254