RV C OLLEGE OF E NGINEERING

D EPARTMENT OF A EROSPACE E NGINEERING

Fundamentals of Fatigue and Fracture

Mechanics

Author:
Benjamin R OHIT

A compilation of notes available on fatigue and fracture mechanics. Only for use at
the department of Aerospace Engineering, RVCE

August 30, 2019

 iii

“Thanks to my solid academic training, today I can write hundreds of words on virtually any
topic without possessing a shred of information, which is how I got a good job in journalism.”

Dave Barry
 v

Acknowledgements
The acknowledgments and the people to thank go here, don’t forget to include your
project advisor. . .
 vii

Contents

Acknowledgements v

1 Fundamentals of Fracture Mechanics 1

1.1 Examples and Historical Background . . . . . . . . . . . . . . . . . . . 1
1.2 Need For Fracture Mechanics in Design . . . . . . . . . . . . . . . . . . 2
1.3 Micro Mechanics of Fracture . . . . . . . . . . . . . . . . . . . . . . . . . 2
Size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Micro structure and Defects . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Crack Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.3 Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Griffith’s and Orowan’s Theory of Brittle Fracture . . . . . . . . . . . . 5
Irwins Modification . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Different Types of Failures . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5.1 Brittle vs Ductile Fracture . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Ductile Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 Brittle Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 Ductile to Brittle Transition . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.9 Ductile to Brittle Transition Temperature . . . . . . . . . . . . . . . . . . 10
1.9.1 Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.10 Fracture mechanics Approach to Design . . . . . . . . . . . . . . . . . . 11
1.10.1 Energy Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.11 Fracture Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.12 Time dependent crack growth and damage tolerance . . . . . . . . . . 13
1.12.1 Time Dependant Crack Growth . . . . . . . . . . . . . . . . . . . 13
1.12.2 Crack Growth as a Function of ∆K . . . . . . . . . . . . . . . . . 14
Crack Growth in region I . . . . . . . . . . . . . . . . . . . . . . 14
Crack Growth in region II . . . . . . . . . . . . . . . . . . . . . . 14
Crack Growth in region III . . . . . . . . . . . . . . . . . . . . . . 15
1.12.3 Damage Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 LEFM - Linear Elastic Fracture Mechanics 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Conditions for Validity of LEFM . . . . . . . . . . . . . . . . . . 17
2.2 Griffith’s Energy balance criterion . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Energy Balance Approach . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Compliance Change . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Stability of Crack growth and the R Curve . . . . . . . . . . . . . . . . . 23
2.3.1 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Shape of the R Curve . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Stress Intensity Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Stress Intensity Factor vs Stress Concentration . . . . . . . . . . 25
2.4.2 Stress Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
viii

2.4.3 Stress Intensity factor(SIF) for different geometries . . . . . . . 26

2.4.4 Relationship between K and G . . . . . . . . . . . . . . . . . . . 26
2.4.5 Role of Material Thickness . . . . . . . . . . . . . . . . . . . . . . 26

3 • 29

4 Fatigue in Structures 31
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 S-N curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.1 Fatigue Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Stress Life Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3.1 Mean Stress Effects on Fatigue Life . . . . . . . . . . . . . . . . . 33
General Observations . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4 Strain Life Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Low cycle and high cycle fatigue lifes . . . . . . . . . . . . . . . 36
4.4.1 Cyclic Stress-Strain Behavior . . . . . . . . . . . . . . . . . . . . 37
Bauschinger Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Cyclic Strain Hardening and Softening . . . . . . . . . . . . . . 37
4.4.2 Mean Stress Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5 Stress Life vs Strain Life . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.6 Notched Stress - Neubers Stress Concentration . . . . . . . . . . . . . . 40

5 Statistical Aspects of Fatigue Behavior 41

5.1 Fatigue - HCF and LCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1.1 High cycle fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1.2 Low Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Coffin Manson Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Transition Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Cycle Counting Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.4.1 Procedures for Cycle Counting . . . . . . . . . . . . . . . . . . . 43
Level-Crossing Counting . . . . . . . . . . . . . . . . . . . . . . 43
Peak Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Simple-Range Counting . . . . . . . . . . . . . . . . . . . . . . . 44
5.5 Paris Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.6 Damage rule for irregular loads . . . . . . . . . . . . . . . . . . . . . . . 45
5.6.1 Linear Damage Rules(LDR) . . . . . . . . . . . . . . . . . . . . . 45
5.6.2 Marco-Starkey theory . . . . . . . . . . . . . . . . . . . . . . . . 46
5.7 Variable Amplitude Loading . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.8 Miner’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.8.1 Miner’s Rule for variable loading . . . . . . . . . . . . . . . . . . 47
Sequence Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Crack Length at failure . . . . . . . . . . . . . . . . . . . . . . . . 49
 ix

List of Figures

1.1 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 a) Transcrystalline crack, b) intercrystalline crack . . . . . . . . . . . . . 5
1.3 Stress-Strain curves for Brittle and Ductile materials . . . . . . . . . . . 7
1.4 Stress-Strain curves for Brittle and Ductile materials . . . . . . . . . . . 7
1.5 Stages of crack growth in Ductile failure. . . . . . . . . . . . . . . . . . 8
1.6 True stress-strain curves at room and elevated temperatures(Image
Courtesy-ICME) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Fracture behaviour of different materials with temperature, including
some that don’t have a transition zone.)) . . . . . . . . . . . . . . . . . . 10
1.8 Constant Displacement and Constant Load)) . . . . . . . . . . . . . . . 11
1.9 Load vs Displacement graphs for Constant Load and constant dis-
placement type crack growth)) . . . . . . . . . . . . . . . . . . . . . . . 11
1.10 Fracture Modes: Mode I is a normal-opening mode and is the one we
shall emphasize here, while modes II and III are shear sliding modes. . 12
1.11 crack growth rate as a function of stress intensity . . . . . . . . . . . . . 14
1.12 The damage tolerance approach to design. . . . . . . . . . . . . . . . . 15

2.1 Plate with centrally cracked hole and two crack edges . . . . . . . . . . 18
2.2 Griffiths energy graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Load vs Displacement graph for constant displacement and constant
load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Compliance as a function of crack length . . . . . . . . . . . . . . . . . 23
2.6 Driving force & R curve as a function of crack length . . . . . . . . . . 23
2.7 R-curve for various crack lengths . . . . . . . . . . . . . . . . . . . . . . 24
2.8 K is measured as a function of specimen thickness . . . . . . . . . . . . 27
2.9 K is measured as a function of specimen thickness . . . . . . . . . . . . 27

4.1 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.6 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.7 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.8 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.9 Strain Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.10 Strain Softening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
x

5.5 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.6 Time compressed representation of the load history . . . . . . . . . . . 47
5.7 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.8 Two different sequences of blocks in a simple VA load history (R = 0)
applied to a notched specimen . . . . . . . . . . . . . . . . . . . . . . . 48
 xi

List of Tables
 xiii

List of Abbreviations

LAH List Abbreviations Here

WSF What (it) Stands For
 xv

Physical Constants

Speed of Light c0 = 2.997 924 58 × 108 m s−1 (exact)

 xvii

List of Symbols

a distance m
P power W (J s−1 )
ω angular frequency rad
 xix

For/Dedicated to/To my. . .

 1

Chapter 1

Fundamentals of Fracture
Mechanics

Generally in engineering studies we consider that a structure is homogeneous, isotropic

without any flaws. But most of the times it is not so. Materials tend to have flaws
whether it be inherent or formed during a manufacturing process or may be during
a heat treatment process. When solids contain such flaws or also sometimes cracks,
the equations are not completely defined by the theory of elasticity(the mechanics
of materials approach) since it does not consider that there could be a stress singu-
larity phenomenon near a crack tip or the flaw. Fracture mechanics is the study of
cracked materials or materials with flaws when subjected to an applied load. Frac-
ture mechanics also deals with crack nucleation, crack propagation, crack growth
and rupture. Crack nucleation is mostly dependent on the micro-structure, envi-
ronment and loading. Example, ductile fracture is generally a high energy process
as large amounts of energy is absorbed with large plastic strains before crack attain
critical crack length and becomes instable. But in the case of a brittle fracture there
is very less absorption of energy and crack propagates at a rapid rate. One of the
most famous disaster of the last several centuries was the sinking of the transatlantic
ocean liner Titanic on April 15, 1912, it was recently discovered that brittle failure at
low temperatures was the main reason which was not accounted for during initial
design.

1.1 Examples and Historical Background

SS Schenectady, 16 days after being inducted into service fell victim to brittle failure
due to cold temperatures. The hull broke into two when docked in calm waters. The
failure was due to brittle fracture of low-grade steel components. Cold temperatures
reduced the toughness of steels making them brittle which led to the catastrophic
failure.
There are plenty of failures in aviation history due to fatigue failure and lack of
knowledge of fracture mechanics.

1. Comet: Three comet air planes crashed in a short period of 12 months, leading
to the aircraft being grounded. The crashes were found to be caused by cracks
growing from corners of the square fuselage windows which served as stress
concentration points leading to accelerating crack formation.

2. Aloha Airlines Flight 243: The famous failure of the upper fuselage of the air-
craft is one of the landmark fatigue failures in aviation history. The failure was
the result of fatigue cracking of the skin panel adjacent to rivet holes at a lap
joint which was accompanied by corrosion.
2 Chapter 1. Fundamentals of Fracture Mechanics

3. Unexpected failures of ships and aircraft called for an expansion in the field of
fracture mechanics with the early era of research being limited to linear elastic
mechanics. In the present day research has advanced to transient crack growth
and crack behaviour even in exotic materials.

1.2 Need For Fracture Mechanics in Design

Three main points must be considered in design.

1. The weight of the structure

2. The strength to meet the loading requirements

3. Prediction of the growth rate of cracks when they initiate from damage or fa-
tigue cycling.

If the structure and the operating environment could be perfect, then the struc-
ture would be designed to have no cracks during its operating life. However, given
the random variations in the real environment, it is necessary to deal with this three-
requirement problem using the concept of ’fail safe’ design, whose aim is not so
much to completely avoid the appearance of cracks due to defects in the material
itself, or manufacturing method, or to accidental damage in service., but to under-
stand the growth rate of the cracks. If this is achieved, the cracks can be found and
repaired before they become catastrophic.
The central difficulty in designing against fracture in high-strength materials is
that the presence of cracks can modify the local stresses to such an extent that the
elastic stress analyses done so carefully by the designers are insufficient. When a
crack reaches a certain critical length, it can propagate catastrophically through the
structure, even though the gross stress is much less than would normally cause yield
or failure in a tensile specimen . The term "fracture mechanics" refers to a vital spe-
cialization within solid mechanics in which the presence of a crack is assumed, and
we wish to find quantitative relations between the crack length, the material’s inher-
ent resistance to crack growth, and the stress at which the crack propagates at high
speed to cause structural failure.
The conventional approach to the design against fatigue failure (fatigue strength-
endurance limit approach) involves cyclic-life predictions based on nominal stress
(or strain) vs. elapsed-cycles data developed from endurance tests on smooth labora-
tory specimens. However, such data do not distinguish between the crack-initiation
and crack-growth phases of fatigue. Consequently, smooth-specimen endurance-
test fatigue data do not provide information regarding the effect of pre-existing flaws
on component life.

1.3 Micro Mechanics of Fracture

Fracture is the separation of a body into two or more parts. During this process the
bonds between the components of the material are broken. At the microscopic level,
these are for instance bonds between atoms, ions, molecules, etc. The bonding force
between two of those elements can be expressed by means of the relation

a b
F=− m
+ n (1.1)
r r
1.3. Micro Mechanics of Fracture 3

Here, the first term represents attractive forces, while the second term describes re-
pulsive ones. The parameters a, b, m, and n (m < n) are constants which depend on
the bond type.

F IGURE 1.1: •

For small displacements from the equilibrium configuration d0, the bonding
force F(r) can be approximated by a linear function. This is equivalent to a material
behaviour which macroscopically is described by Hooke’s law. During the release of
bonds, i.e., the separation of elements, a negative material specific work WB is done
by the bonding force.
we consider in a somewhat simplified manner the separation process of two
atomic lattice planes of a crystal. For the separation stress σ(x) we assume a de-
pendence on the separation displacement x similar to the bonding force.
Equating the latter with Hooke’s law
x
σ = E·e = E· (1.2)
d0
yields for the so-called theoretical strength, i.e., the cohesive stress that has to be
overcome during separation.
a
σc ≈ E · (1.3)
π · d0
If we further assume that the bonds are completely broken i.e

d0 ≈ a (1.4)

then we get,
E
σc ≈ (1.5)
π
This can only be true for defect free crystals which is an ideal condition in nature.
For real polycrystalline materials, the fracture strength is 2-3 orders of magni-
tude less. In parallel, the energy necessary for the creation of new fracture surfaces
exceeds the value defined (σc ≈ πE ) by several orders of magnitude. The reasons for
this can be found in the inhomogeneous structure of the material and, first of all, in
its defect structure.
4 Chapter 1. Fundamentals of Fracture Mechanics

Size effects
With respect to experiments the stress needed to fracture bulk glass is around 100
MPa but the theoretical stress needed for breaking atomic bonds of glass is approx-
imately 10,000MPa as per the equation 1.5. Also, experiments on glass fibres that
Griffith himself conducted suggested that the fracture stress increases as the fibre
diameter decreases. Hence the uni-axial tensile strength, which had been used exten-
sively to predict material failure before Griffith, could not be a specimen-independent
material property. Griffith suggested that the low fracture strength observed in ex-
periments, as well as the size-dependence of strength, was due to the presence of
microscopic flaws in the bulk material.

1.3.1 Micro structure and Defects

A polycrystalline material consists of crystals (grains) which are joined with one an-
other along grain boundaries. The individual grains have anisotropic properties and
the orientation of their crystallographic planes and axes differ from grain to grain.
Furthermore, e.g., due to segregation, the properties of grain boundaries differ sub-
stantially from those of the grains. In addition to these irregularities of the material’s
structure, a real material contains from the beginning on a number of defects of dif-
ferent size.
Usually they are classified according to their dimension as point imperfections
(e.g., vacancies, interstitials, impurity atoms), line imperfections (dislocations), and
area imperfections (e.g., stacking faults, phase boundaries, twin boundaries).
A particular role regarding the mechanical behaviour is played by dislocations.
Under the action of sufficiently high shear stresses the atoms in the vicinity of the
dislocation line rearrange their bonds which leads to a displacement of the disloca-
tion. Under the action of sufficiently high shear stresses the atoms in the vicinity
of the dislocation line rearrange their bonds which leads to a displacement of the
dislocation. The dislocation movement results in a relative ’slip’ of the lattice planes
and may lead to the formation of a new surface. This microscopic mechanism is the
origin of macroscopic plastic material behaviour.
In general, dislocations can not move unlimited. At obstacles such as grain
boundaries or inclusions they may stop and accumulate.Macroscopically such a dis-
location pile-up is observed as strain hardening.
In contrast to crystalline materials, the molecules and atoms are completely dis-
ordered in amorphous solids such as glasses or many polymers.While in these mate-
rials no disturbances of a regular lattice, e.g. by dislocations or grain boundaries can
be identified, the defects are essentially given by foreign particles and micro-voids.

1.3.2 Crack Formation

In initially crack-free polycrystalline materials there are different mechanisms of
micro-crack formation. A rather important mechanism of microcrack formation is
the dislocation pile-up at obstacles. It causes high stress concentrations which can
lead to bond breaking along preferred lattice planes and as a consequence to cleav-
age. If such a crack runs through several grains, the orientation of the separation
surface changes according to the local lattice planes and axes. Such a type of frac-
ture is called transcrystalline cleavage.
If the grain boundaries are sufficiently weak, the separation on account of dis-
location pile-up and (or) grain boundary sliding will take place along these bound-
aries. This is called intercrystalline fracture.
1.4. Griffith’s and Orowan’s Theory of Brittle Fracture 5

F IGURE 1.2: a) Transcrystalline crack, b) intercrystalline crack

A dislocation pile-up causes not only stress concentrations. It also can be the
responsible source for the formation of microscopic voids and cavities.

1.3.3 Crack Growth

A fracture process is always connected with crack growth. Both, fracture and crack
growth, can be classified from different phenomenological viewpoints. The typical
stages in the behavior of a loaded crack are characterized as follows. As long as the
crack does not change its length, the crack is called stationary. At a specific critical
load or deformation, respectively, crack initiation takes place, i.e., the crack starts to
propagate and becomes non-stationary.
One can distinguish different types of crack propagation. Crack growth is called
stable if an increase of crack length requires an increase of external load. In contrast,
crack growth is unstable if the crack, starting froma specific configuration, advances
spontaneously without any increase of the external load. It should be noted here
that stable or unstable crack growth is governed not solely by material properties.
In fact, the geometry and the type of loading have a significant influence on the crack
behaviour.
Very slow crack propagation under constant loading in a creeping manner (e.g.,
at a velocity of 1 mm/s or less) is called sub-critical. Under cyclic loading the crack
can propagate in small ’steps’ (e.g., of about 10−6 mm per cycle). This type of crack
propagation is called fatigue crack growth. If the crack propagates with a velocity
which approaches the order of the speed of sound in the solid material (e.g., 600 m/s
or more), the crack is called fast. If such a fast crack comes to rest, we call this crack
arrest.

1.4 Griffith’s and Orowan’s Theory of Brittle Fracture

In the 1920s, A.A. Griffith, while testing glass rods, observed that the longer the rod,
the lower the strength. This led to the idea that the strength variation in the glass
rods was due to defects, primarily surface defects.
As the rods became longer, there was a higher probability of encountering a flaw
large enough to cause failure. These flaws lower the fracture strength because they
amplify the stress at the crack tip. This led to an instability criterion that considered
the energy released in a solid at the time a flaw grows catastrophically under an
applied stress.
Griffith developed a criterion for the elliptical crack in a plate using an energy
balance approach. He equated the elastic strain energy that is stored in the material
as it is elastically deformed to the surface energy created when two new free sur-
faces form during crack propagation. He concluded that the crack will propagate
when the elastic energy released as a result of crack propagation exceeds the energy
6 Chapter 1. Fundamentals of Fracture Mechanics

required to propagate the crack.

His analysis showed that the critical stress required to propagate a crack in a brittle
material is: r
2Eγs
σc = (1.6)
πa
where E is the modulus of elasticity, γs is the surface energy, and a is one half the
length of an internal crack.
Orowan later modified the Griffith equation, replacing γs with γs + γ p where γ p is
the plastic deformation associated with crack extension. Griffith’s equation can then
be rewritten as:
s
2E(γs + γ p )
σc = (1.7)
πa
If the material is highly ductile, then γ p > γs and:
r
2Eγ p
σc = (1.8)
πa

Irwins Modification
In the 1950s, G.R. Irwin incorporated γ p &γs into a single term, Gc, known as the
critical strain energy release rate:

Gc = 2(γs − γ p ) (1.9)
The strain energy release rate or the crack extension force (G) is the change in
potential energy (U) of the system per unit increase in crack area (G = dU/da).
The original Griffith criterion can now be written for both brittle and ductile
materials as:
πσ2 a
Gc = (1.10)
E
πσ2 a
Therefore, crack extension occurs when exceeds the value of Gc for the
E
material in question.

1.5 Different Types of Failures

Conventional failure criteria has classified failure broadly as brittle and ductile fail-
ures which are the two extremes.

1.5.1 Brittle vs Ductile Fracture

In brittle failures the structure or the specimen fails or fractures after a large defor-
mation which may occur over a long time period and may be associated with yield-
ing(the phenomenon of increasing deformation at a constant load) or plastic flow.
Brittle failure is generally preceded by small deformations and its failure is sudden
and catastrophic. Defects have a major role to play during these types of failures.
It can also be noted that toughness of a material is the integrated area under the
entire stress-strain curve to the break point and resilience is the integrated area up
to yield. Hence it can be also concluded that ductile materials are tougher(or have
higher toughness) that brittle materials.
1.5. Different Types of Failures 7

F IGURE 1.3: Stress-Strain curves for Brittle and Ductile materials

Defects associated with ductile failure are:

1. Dislocations

2. Grain boundary Spacings

3. Precipitates

F IGURE 1.4: Stress-Strain curves for Brittle and Ductile materials

Defects associated with brittle failure are:

1. Sharp notches

2. Surface scratches

3. Cracks

It should also be noted that materials may have ductile or brittle behavior, depend-
ing on the temperature, rate of loading and other variables
8 Chapter 1. Fundamentals of Fracture Mechanics

1.6 Ductile Failure

Ductile fracture in tension is usually preceded by a localized reduction in cross-
sectional area called necking. Further it exhibits the following stages.
1. Necking
2. Formation of micro voids
3. Coalescence of micro voids to form a crack
4. Crack propagation by shear deformation
5. Fracture
After the onset of necking there is a rapid crack growth which leads to failure.
During necking cavities form later these cavities grow and further growth leads to
their coalesce resulting in formation of crack that grows outward in direction per-
pendicular to the application of stress and final failure involves rapid crack propa-
gation at 45 degrees to the tensile axis. This angle represents the direction of max-
imum shear stress which in turn causes shear slip. This shear slip causes the crack
to propagate at the periphery of the neck and failure occurs in the form of cup and
cone fracture.

F IGURE 1.5: Stages of crack growth in Ductile failure.

1.7 Brittle Fracture

In this case the direction of crack propagation is very nearly perpendicular to the di-
rection of applied load. This crack propagation corresponds to successive breaking
1.8. Ductile to Brittle Transition 9

of atomic bonds along crystallographic planes and hence also called cleavage frac-
ture. This fracture is also said to be transgranular as the crack propagates through
grains. Thus it has a grainy and faceted texture when the cross section of failure is
viewed. Crack propagation across grain boundaries is called transgranular, while
propagation along grain boundaries is termed intergranular. Think of the metal
micro-structure as a 3-D puzzle. Transgranular fracture cuts through the puzzle
pieces whereas intergranular fracture propagates along the puzzle pieces precut
edges. Another important point to notice is that the surface of the brittle fracture
tends to be almost perpendicular to the direction of load(uniaxial) or perpendicular
to the direction of principal tensile stress. Brittle fracture usually occurs at stress lev-
els well below those predicted theoretically.
Properties of a brittle fracture.

1. No appreciable plastic deformation

2. Crack propagation is very fast

3. Crack propagates nearly perpendicular to the direction of the applied stress

4. Crack often propagates by cleavage - breaking of atomic bonds along specific

crystallographic planes (cleavage planes).

1.8 Ductile to Brittle Transition

There are a few factors which cause brittle fractures to lean towards behaving like
ductile fractures. One of the most important factors is temperature. We should also
note that as the temperature increases the yield strength decreases and the ductility
increases. Which also has a role to play in the toughness of the material.

F IGURE 1.6: True stress-strain curves at room and elevated tempera-

tures(Image Courtesy-ICME)

Fig 1.6 shows the true stress-strain curves at room and elevated temperatures.
As temperature increases, the atoms within the material absorb energy and vi-
brate with greater frequency and amplitude. This additional energy allows the
atoms under stress to break bonds and form new bonds. This behaviour is seen
10 Chapter 1. Fundamentals of Fracture Mechanics

as plastic deformation. Similarly, When temperature decreases the exact opposite

happens. As the energy to vibrate is less the bonds between atoms do not break and
when stress increases these bonds just break leading to catastrophic failure.

1.9 Ductile to Brittle Transition Temperature

As the temperature begins to drop, some metals that would behave in a ductile man-
ner at room temperature tend to behave as brittle material. This is called as ductile
to brittle transition. The ductile to brittle transition temperature(DBTT) is very much
dependent on the composition of the metal. This term Ductile-to-Brittle transition
(DBT) is used in relation to the temperature dependence of the measured impact en-
ergy absorption.The temperature where DBT occurs is termed as Ductile-to-Brittle
Transition Temperature (DBTT).

F IGURE 1.7: Fracture behaviour of different materials with tempera-

ture, including some that don’t have a transition zone.))

From the figure 1.7 it can be concluded that there is no single criterion that de-
fines the DBTT. With decreasing temperature, the yield strength increases rapidly to
the point where it equals the tensile stress for brittle failure, and below this temper-
ature, fracture usually occurs in brittle/cleavage mode, hence we can say below the
DBTT,σy = σ f

1.9.1 Strain Rate

We know that as temperature declines, materials become stronger and stiffer but less
impact resistant, while increasing temperatures produce the opposite response. This
suggests that we will observe the same effect on material properties by varying strain
rate as we do when we change temperatures. So while keeping the temperature
constant, we can produce a change from ductile failure to brittle failure by simply
changing the velocity of the impact. AS the velocity of impact increases i.e the strain
rate increases the failure mechanism changes from ductile to brittle. o just as there is
a ductile-to-brittle transition temperature for plastic materials there is also a ductile-
to-brittle transition strain rate.
1.10. Fracture mechanics Approach to Design 11

1.10 Fracture mechanics Approach to Design

The central difficulty in designing against fracture in high-strength materials is that
the presence of cracks can modify the local stresses to such an extent that the elas-
tic stress analyses done so carefully by the designers are insufficient. When a crack
reaches a certain critical length, it can propagate catastrophically through the struc-
ture, even though the gross stress is much less than would normally cause yield or
failure in a tensile specimen. The term "fracture mechanics" refers to a vital special-
ization within solid mechanics in which the presence of a crack is assumed, and we
wish to find quantitative relations between the crack length, the material’s inher-
ent resistance to crack growth, and the stress at which the crack propagates at high
speed to cause structural failure.

1.10.1 Energy Approach

In this segment we will look at a brief introduction to the energy based approach to
fracture mechanics and how thee theory was derived by Griffith. To understand the
strain energy concept derived by Griffith. Let us consider a hole with radius "a" in
a plate. The elastic strain energy in that plate with the hole can be defined as the
total strain energy of that plate minus the strain energy of the hole. Note that when
a crack forms two new surfaces are formed and to form two new surfaces energy is
required.

F IGURE 1.8: Constant Displacement and Constant Load))

F IGURE 1.9: Load vs Displacement graphs for Constant Load and

constant displacement type crack growth))

Let us now consider the crack growth under two different conditions.
12 Chapter 1. Fundamentals of Fracture Mechanics

1. Constant Load: Under a constant load, half the work during fracture goes into
elastic strain, and half goes into fracture. So we can say that the strain energy
increases. Because some amount of external work is being done.

2. Constant Displacement: Under a constant displacement, all the fracture energy

comes from elastic strain energy stored in the body (it has to; there is no other
energy source!)Hence we can say that in the end the strain energy decreases.

By looking at the constant displacement graph in fig 1.9 we can say that the
strain energy decreases when the crack grows during constant displacement and the
strain energy increases when the grows under constant load. We can now see that
the elastic strain energy increases when work is done by increasing displacement at
constant load. So hence we can say that the work done can be the area AEFB, and the
amount by which strain energy has increased is AEA, by simple math we can note
that 2×OEA = AEFB, i.e, Under a constant load, half the work during fracture goes
into elastic strain, and half goes into fracture. A detailed explanation along with the
mathematics will be discussed in the next unit.

1.11 Fracture Modes

The literature treats three types of cracks, termed mode I, II, and III as illustrated
in Fig 1.10. Mode I is a normal-opening mode and is the one we shall emphasize
here, while modes II and III are shear sliding modes. Mode-I corresponds to fracture
where the crack surfaces are displaced normal to themselves. This is a typical tensile
type of fracture. In mode-II, crack surfaces are sheared relative to each other in a
direction normal to the edge of the crack. In mode-III, shearing action is parallel
to the edge of the crack. To indicate different modes, it is normal practice to add
the corresponding subscript. There are two extreme cases for mode-I loading. With
thin plate-type specimens the stress state is plane stress, while with thick specimens
there is a plane-strain condition. The plane-strain condition represents the more
severe stress state and the values of K are lower than for plane-stress specimens.

F IGURE 1.10: Fracture Modes: Mode I is a normal-opening mode and

is the one we shall emphasize here, while modes II and III are shear
sliding modes.

The stress intensity approach states that fracture occurs due to stress concentra-
tion at flaws, like surface scratches, voids, etc. If ’c’ is the length of the crack and
ρ the radius of curvature at crack tip, the enhanced stress (σm ) near the crack tip is
given by:
 0.5
c
σmax = 2 · σnom (1.11)
ρ
1.12. Time dependent crack growth and damage tolerance 13

The above equation states that smaller the radius, higher is the stress enhancement.
Another parameter, often used to describe fracture toughness is known as critical
stress concentration factor, K, and is defined as follows for an infinitely wide plate
subjected to tensile stress perpendicular to crack faces: In traditional engineering
design such stress concentrations are denoted by the stress concentration factor kt
σmax
Kt = (1.12)
σnom
Do not confuse the stress concentration factor kt with the stress intensity factor
K or KIc . Typical situations where stress concentrations arise are changes of section
or point loads. For such situations the use of appropriate kt values in design is
essential, particularly in fatigue design.
For a sharp crack the radius of the tip is extremely small and hence we can say
that as σmax reaches infinity as ρ equals zero. This suggests that for a sharp crack,
any applied stress will cause infinitely high stresses at the tip. Also for very sharp
cracks, this approach cannot distinguish between long and short cracks whereas we
know that failure stress depends on crack length. Hence we can conclude that the
concept of stress concentration factor breaks down as crack tip radius tends to zero.
Hence we introduce a stress intensity factor which is more finite and often used to
describe fracture toughness denoted by K and is defined as follows for an infinitely
wide plate subjected to tensile stress perpendicular to crack faces.
√
K = σ cρ (1.13)
This relation holds for specific conditions, and here it is assumed that the plate
is of infinite width having√a through-thickness crack.It is worth noting that K has
the unusual units of MPa m. It is a material property in the same sense that yield
strength is a material property. The stress intensity factor K is a convenient way
of describing the stress distribution around a flaw. For the general case the stress
intensity factor is given by, √
K = ασ cπ (1.14)
where α is a parameter that depends on the specimen and crack sizes and ge-
ometries, as well as the manner of load application.

1.12 Time dependent crack growth and damage tolerance

1.12.1 Time Dependant Crack Growth
Fracture mechanics often plays a role in life prediction of components that are sub-
ject to time- dependent crack growth mechanisms such as fatigue or stress corrosion
cracking. The rate of cracking can be correlated with fracture mechanics parameters
such as the stress-intensity factor, and the critical crack size for failure can be com-
puted if the fracture toughness is known.For example, the fatigue crack growth rate
in metals can usually be described by the following empirical relationship:

da
= C (∆K )m (1.15)
dN
where da/dN is the crack growth per cycle, ∆K is the stress-intensity range, and
C and m are material constants. In static loading, the stress intensity factor for a
small crack in a large specimen can be expressed as eqn 1.13. If the stress is kept
14 Chapter 1. Fundamentals of Fracture Mechanics

constant, we will get fracture for a certain crack length, a=ac , which will give KI =
KIc where ac is the critical crack length and KIc is the critical stress intensity factor
in mode I.
For a < ac (KI < KIc ) the crack will not propagate. In dynamic loading, however,
for (KI < KIc ) , the crack may still propagate. This means that a (and KI) will increase,
we will eventually obtain fracture when a = ac.

1.12.2 Crack Growth as a Function of ∆K

In experiments, crack propagation has been measured as a function of the stress
intensity factor There exists a threshold value of ∆K below which fatigue cracks will
not propagate. At the other extreme, Kmax will approach the fracture toughness
KC , and the material will fail. Note that ∆K depends on the crack size. This is not
shown in the plot(figure 1.11).

F IGURE 1.11: crack growth rate as a function of stress intensity

Crack Growth in region I

For small ∆K (region I), crack propagation is difficult to predict since it depends on
microstructure and flow properties of the material Here, the growth may even come
to an arrest. Crack growth rate is sensitive to the size of the grains.

Crack Growth in region II

For larger magnitudes of ∆K (region II), the crack growth rate will be governed by
a power law (such as Paris’ law) The crack growth rate is fairly insensitive to the
microstructure (however, the constants m and C are, of course, different for different
materials). On a log-log plot, the two Paris law model parameters, C and m, can be
determined graphically, the slope line when extended backwards will intersect the
Y-axis, this will give you the material constant c and the slope of the line will give
you m. In region II the fatigue life can be directly estimated by integrating Paris’ law
as it is a linear relationship between crack growth rate and stress intensity factor.
1.12. Time dependent crack growth and damage tolerance 15

Crack Growth in region III

If the stress intensity ratio is increased even further (region III), the crack growth rate
will accelerate and finally fracture will occur The behaviour of this fracture is rather
sensitive to the micro structure and flow properties of the material. Thus it can be
concluded that the rate of crack growth is a function of stress intensity range(Kmin-
Kmax ), stress intensity ratio(Kmax/Kmin) and stress history.

1.12.3 Damage Tolerance

Damage tolerance, as its name suggests, entails allowing subcritical flaws to remain
in a structure. Repairing flawed material or scrapping a flawed structure is expen-
sive and is often unnecessary. Fracture mechanics provides a rational basis for es-
tablishing flaw tolerance limits. The initial crack size is inferred from nondestructive
examination (NDE), and the critical crack size is computed from the applied stress
and fracture toughness. Normally, an allowable flaw size would be defined by di-
viding the critical size by a safety factor. The predicted service life of the structure
can then be inferred by calculating the time required for the flaw to grow from its
initial size to the maximum allowable size.

F IGURE 1.12: The damage tolerance approach to design.

 17

Chapter 2

LEFM - Linear Elastic Fracture

Mechanics

2.1 Introduction
Linear Elastic Fracture Mechanics (LEFM) first assumes that the material is isotropic
and linear elastic. Based on the assumption, the stress field near the crack tip is cal-
culated using the theory of elasticity. When the stresses near the crack tip exceed the
material fracture toughness, the crack will grow. The concepts provide an analytical
method based upon the stress intensity factor, which characterises the stress distri-
bution in the vicinity of the crack tip, and are valid in design applications provided
that gross yielding does not occur.
Linear elastic fracture mechanics can be used to describe ultimate static failure of
low toughness high strength materials used in aerospace and other specialised ap-
plications.
Under fatigue loading, crack growth rates can be correlated by the stress intensity
factor for a wide range of materials, because even in the more ductile materials the
amount of plastic flow that can occur under fatigue loading is restricted. It should
not be used in cases where the fatigue cycles involve extensive plastic deformation,
often referred to as "low cycle" fatigue. In Linear Elastic Fracture Mechanics, most
formulas are derived for either plane stresses or plane straines, associated with the
three basic modes of loadings on a cracked body: opening, sliding, and tearing.
Again, LEFM is valid only when the inelastic deformation is small compared to the
size of the crack, what we called small-scale yielding. If large zones of plastic defor-
mation develop before the crack grows, Elastic Plastic Fracture Mechanics (EPFM)
must be used.

2.1.1 Conditions for Validity of LEFM

Linear Elastic Fracture Mechanics can deal with only limited crack tip plasticity, i.e.
the plastic zone must remain small compared to the crack size and the cracked body
as a whole must still behave in an approximately elastic manner. If this is not the
case then the problem has to be treated elasto-plastically.
ASTM has specified that the following requirements must be met for LEFM to be
applicable (for metals).

KI
a, B, (W − a) ≥ 2.5 (2.1)
σy
Where a is half the crack length, B is the thickness of the specimen and W is
the width of the specimen. These requirements are to assure that the plastic zone is
18 Chapter 2. LEFM - Linear Elastic Fracture Mechanics

sufficiently small (i.e. nearly plane strain conditions (B) and elastic conditions (W-a))
for the assumption of LEFM to be valid

2.2 Griffith’s Energy balance criterion

When A.A.Griffith (1893–1963) began his pioneering studies of fracture in glass in
the years just prior to 1920, he was aware of Inglis work in calculating the stress con-
centrations around elliptical holes, and naturally considered how it might be used
in developing a fundamental approach to predicting fracture strengths. However,
the Inglis solution poses a mathematical difficulty: in the limit of a perfectly sharp
crack, the stresses approach infinity at the crack tip. This is obviously non physical
and using such a result would predict that materials would have near-zero strength:
even for very small applied loads, the stresses near crack tips would become infinite,
and the bonds there would rupture. Rather than focusing on the crack-tip stresses
directly, Griffith employed an energy-balance approach that has become one of the
most famous developments in materials science.

2.2.1 Energy Balance Approach

Now we know that the strain energy changes (decreases) when crack grows when no
external work is done during constant displacement and during constant load. But
how do we find out the strain energy in a cracked component. Let us now consider
a infinite plate with a central crack under uniaxial tension. This crack now has two
tips.

F IGURE 2.1: Plate with centrally cracked hole and two crack edges

Griffith made the important connection in recognising that the driving force for
crack extension is the energy which can be released and that this is used up as the
energy required to create the two new surfaces. This thermodynamic description of
the fracture process has the huge advantage of removing attention from the small
area at the crack tip and the precise micro mechanism of fracture. Let us now con-
sider the plate with hole. The strain energy in the presence of the crack is given
by

σ2
Ua = Volume of Triangles × (2.2)
2E
σ2
2E is the stain energy within elastic limit i.e the area under the stress strain curve
upto elastic limit(Area of the Triangle). The two surfaces formed are the triangles
whose height is 2a λ where 2a is the crack length and λ is the proportionality factor.
For thin plates, λ = π/2,
2.2. Griffith’s Energy balance criterion 19

σ2
Ua = 2 · (0.5 × 2a × 2aλ) × (2.3)
2E
hence we get,
σ 2 · a2 π
Ua = (2.4)
E
which is the strain energy in the presence of a crack with two crack edges per unit
thickness of the plate. Now because the new surfaces have been formed, the total
energy of the system can be thought of as being the sum of the potential energy term,
U plus the surface energy of the crack, S, because no work is being done externally,
The surface energy required for thickness B is given by,

Us = 4aν (2.5)
where ν is the surface energy (e.g., Joules/meter2 ) and the factor 4 is needed
since two free surfaces have been formed at each crack tip. But we know that Ua is
negative as no external work is being done, hence we get total energy as
hence we get,
σ 2 · a2 π
Ua + Us = − + 4aν (2.6)
E

F IGURE 2.2: Griffiths energy graph

Here we should note that the line which describes the total energy reaches a
peak and drops, if we draw a line perpendicular to the x-axis from the max point
that would indicate the critical crack length. After differentiating equation 2.6 and
equating it to zero to obtain maximum value, we get,

σ2 · aπ
= 4ν (2.7)
E
which is nothing but

σ2 · aπ
G= = R = 4ν (2.8)
E
20 Chapter 2. LEFM - Linear Elastic Fracture Mechanics

where G is the energy release rate and R is the surface energy per unit extension.
hence by rearranging we get the following for a plane stress condition,
r
2Eν
σf = (2.9)
πa
and for a plane strain condition we get,
s
2Eν
σf = (2.10)
πa(1 − µ2 )

In reality these relations are only the basis for further extrapolation since these two
equations are really only valid for truly brittle materials such as glass.Thus we need
to modify these relations before we can apply them to the problem of fracture in
materials, which are not classically brittle.

2.2.2 Energy Release Rate

The next step in the development of Griffith’s argument was consideration of the
rates of energy change with crack extension, because the critical condition corre-
sponds to the maximum point in the total energy curve.

F IGURE 2.3: Energy Release Rate

Fracture is deemed to occur when the potential energy release rate, G exceeds
the surface energy per unit crack extension which must be provided to the system if
crack growth is to occur. Where these two lines meet the crack length is the critical
or Griffith crack length, ac. The problem is to determine G either analytically or by
experiment such that we have an input value in order to predict the fracture stress of
a supposed cracked body. Again if we look at our elastic loading diagram for crack
lengths a and a+δa we can hopefully see an experimental method by which we could
determine G.

2.2.3 Compliance Change

To produce a load-displacement diagram for the condition where the crack length
is a and then superimpose the diagram for the condition when the crack length is
a+δa. let us consider fixed grip i.e constant displacement condition. As the crack
extends the stiffness of the plate will decrease such that because the grips are fixed
(equivalent to fixed displacement, i.e. constant U1 ) the load applied by the grips
2.2. Griffith’s Energy balance criterion 21

will decrease as the crack extends. As we know for crack length a the elastic strain
energy is given by
1
P1 U1 (2.11)
2
and this changes to
1
P2 U1 (2.12)
2

F IGURE 2.4: Load vs Displacement graph for constant displacement

and constant load

Hence under fixed grip conditions the extension of the crack from a to a+δa re-
sults in the release (decrease) of elastic strain energy to

1
( P1 − P2 )U1 (2.13)
2
and this loss of energy is because energy is being consumed in the work of frac-
ture required to create the two new crack surfaces.
Now we should consider what happens under constant load condition since this
represents the other end of the spectrum. The strain when the crack is of length a is
given by

1
P1 U1 (2.14)
2
and the strain energy when the crack is developed to a+δa is given by

1
P1 U2 (2.15)
2
which is greater, and from the graph we can say that the work done externally is

P1 (U2 − U1 ) (2.16)

from the graph we can note that the triangle O,P1(U1),P1(U2) is the overall change
in energy which can be expressed as,
22 Chapter 2. LEFM - Linear Elastic Fracture Mechanics

1 1
P1 (U2 − U1 ) + P1 U1 − P1 U2 (2.17)
2 2
simplifying it we get

1
P1 (U2 − U1 ) − P1 (U2 − U1 ) (2.18)
2
hence from the above equation we can say that half the work done goes into
creating new cracks and other half goes into stored strain energy. Now we can note
that energy released in constant load condition is

1
P1 (δU ) (2.19)
2
and energy released during a constant displacement condition is

1
(δP)U (2.20)
2

Also we need to consider the relationship between load and displacement in the
general case. As for any elastic system the displacement and load are related through
a simple linear equation such that for any given crack length we can write that

U = CP(Similar to P=KU where K is stiffness) (2.21)

where C is a constant referred to as the compliance of the system. (Note that C

has the inverse units to stiffness since compliance is in effect the inverse of stiffness)
also we can write

δU = C · δP (2.22)
and substituting this into Eqs. 2.19 and 2.20 we get,

1
P · C (δP) (2.23)
2

1
P · C (δP) (2.24)
2
hence we can conclude that There is no difference in the energy released when an
infinitesimally small increment of crack growth occurs under conditions of constant
load or constant displacement condition.
We know that the strain or potential energy release for an increment of crack growth
δa is given by G δa per unit thickness and if we define B as the thickness of the plate
we can say that:

1
G · δa · B = P(δU ) (2.25)
2
invoking the compliance relationship,

1 2
G · δa · B = P (δC ) (2.26)
2
rearranging,
2.3. Stability of Crack growth and the R Curve 23

P2 (δC )
G= (2.27)
2δa · B

F IGURE 2.5: Compliance as a function of crack length

2.3 Stability of Crack growth and the R Curve

Crack growth occurs when

σ2 · aπ
G= = R = 4ν (2.28)
E
as shown in figure 2.6. But crack growth may be stable or unstable, depending
on how G and R vary with crack size. A plot of R vs. crack extension is called a
resistance curve or R curve. The corresponding plot of G vs. crack extension is the
driving force curve. Consider a wide plate with a through crack of initial length 2a.
At a fixed remote stress σ , the energy release rate varies linearly with crack size.

F IGURE 2.6: Driving force & R curve as a function of crack length

The first case, shows a flat R curve, where the material resistance is constant
with crack growth. When the stress is σ1 , the crack is stable. Fracture occurs when
the stress reaches σ2 ; the crack propagation is unstable because the driving force
increases with crack growth, but the material resistance remains constant. In the
second case, the crack grows a small amount when the stress reaches σ2 , but cannot
24 Chapter 2. LEFM - Linear Elastic Fracture Mechanics

grow further unless the stress increases. When the stress is fixed at σ2 , the driving
force increases at a slower rate than R. Stable crack growth continues as the stress in-
creases to σ3 . Finally, when the stress reaches σ4 , the driving force curve is tangent
to the R curve. The plate is unstable with further crack growth because the rate of
change in the driving force exceeds the slope of the R curve. Imagine testing a series
of thin (plane stress conditions) centre-cracked panels to instability with different
initial crack lengths. The results of such test would appear schematically as shown
below.

F IGURE 2.7: R-curve for various crack lengths

The graph(figure 2.7) shows a locus of points of initial crack length and critical
crack length.

2.3.1 Stability Criteria

The conditions for stable crack growth can be expressed as,

G=R (2.29)

and

dG dR
≤ (2.30)
da da
unstable crack growth occurs when

dG dR
≥ (2.31)
da da

2.3.2 Shape of the R Curve

Some materials exhibit a rising R curve, while the R curve for other materials is
flat. The shape of the R curve depends on the material behaviour and, to a lesser
extent, on the configuration of the cracked structure. The R curve for an ideally
brittle material is flat because the surface energy is an invariant material property.
When non-linear material behaviour accompanies fracture, however, the R curve can
2.4. Stress Intensity Factor 25

take on a variety of shapes. For example, ductile fracture in metals usually results
in a rising R curve; a plastic zone at the tip of the crack increases in size as the
crack grows. The driving force must increase in such materials to maintain the crack
growth. If the cracked body is infinite (i.e., if the plastic zone is small compared to
the relevant dimensions of the body) the plastic zone size and R eventually reach
steady-state values, and the R curve becomes flat with further growth. The size and
geometry of the cracked structure can exert some influence on the shape of the R
curve. A crack in a thin sheet tends to produce a steeper R curve than a crack in a
thick plate because there is a low degree of stress tri-axiality at the crack tip in the
thin sheet, while the material near the tip of the crack in the thick plate may be in
plane strain. The R curve can also be affected if the growing crack approaches a
free boundary in the structure. Thus, a wide plate may exhibit a somewhat different
crack growth resistance than a narrow plate of the same material. Ideally, the R
curve, as well as other measures of fracture toughness, should be a property only
of the material and not depend on the size or shape of the cracked body. Much
of fracture mechanics is predicated on the assumption that fracture toughness is a
material property.

2.4 Stress Intensity Factor

2.4.1 Stress Intensity Factor vs Stress Concentration
Stress concentration factors are due to geometrical changes of cross sections and
regardless of the load condition such as bending,tension,compression or shearing.
Therefore, you can find these factors mentioned in tables or figures in handbooks.
No matter of what is the kind of load condition, you can choose the one correspond-
ing to your geometry during your design. The theoretical stress concentration factor
is
σmax
Kt = (2.32)
σ
However ,when there is a crack in your model, the stress intensity factor comes
into design which not only is dependent to geometry also extremely to load con-
dition and that’s why you can’t find these factors easily in handbooks. They are
determined experimentally according to each part geometry and load condition.

2.4.2 Stress Intensity

Look at the equation
πaσ2
G = 2ν = (2.33)
E
let us rearrange all material property terms,

G · E = πaσ2 (2.34)
√ √
G · E = σ πa (2.35)
This parameter is called the stress intensity factor(K) which is the crack driv-
ing force, and its critical value is a material property known as fracture toughness,
which, in turn, is the resistance force to crack extension. Hence we can say,
√ √
K= G · E = σ πa (2.36)
26 Chapter 2. LEFM - Linear Elastic Fracture Mechanics

Which is a material property and a function of the crack length and stress. The
stress intensity factor, K , is a single parameter which completely specifies the ampli-
tude of the stress field in the vicinity of the crack tip. In general, the stress intensity
factor depends on the geometry of the cracked body (including the crack length) and
it is usual to express it as
√
K = Yσ a (2.37)
where Y is called the shape factor and is a function of body geometry and crack
length.

2.4.3 Stress Intensity factor(SIF) for different geometries

2.4.4 Relationship between K and G
Stress intensity factors are used in design and analysis by arguing that the material
can withstand crack tip stresses up to a critical value of stress intensity, termed KIc
, beyond which the crack propagates rapidly. This critical stress intensity factor is
then a measure of material toughness. The failure stress is then related to the crack
length a and the fracture toughness.

K = G · E = πaσ2 (2.38)
√ √
K = G · E = σ πa (2.39)
K2 = G · E (2.40)
This equation applies only for a plane stress condition, extending it to a plane
strain condition we get,
K 2 = G · E (1 − µ2 ) (2.41)
we can note that µ is 0.3 for metals and hence(1-µ2 )=0.91 which is not a very
big change, however, the numerical values of G or KI are very different in plane
stress(thin specimens) or plane strain(thick specimens) situations. This is a direct
consequence of an effective stiffness increase experienced when an object is pulled
in tension, but with one lateral plane constrained from contracting under Poisson
effects.

2.4.5 Role of Material Thickness

Specimens having standard proportions but different absolute size produce differ-
ent values for KI . This results because the stress states adjacent to the flaw changes
with the specimen thickness (B) until the thickness exceeds some critical dimension.
Once the thickness exceeds the critical dimension, the value of KI becomes relatively
constant and this value, KIc , is a true material property which is called the plane-
strain fracture toughness. The relationship between stress intensity, KI , and fracture
toughness, KIc , is similar to the relationship between stress and tensile stress. The
stress intensity, KI , represents the level of “stress” at the tip of the crack and the frac-
ture toughness, KIc , is the highest value of stress intensity that a material under very
specific (plane-strain) conditions that a material can withstand without fracture. As
the stress intensity factor reaches the KIc value, unstable fracture occurs. As with a
material’s other mechanical properties, KIc is commonly reported in reference books
and other sources.
2.4. Stress Intensity Factor 27

F IGURE 2.8: K is measured as a function of specimen thickness

Above a certain thickness Bmin , the fracture toughness has a minimum value
which is independent of thickness. This minimum value of KIc is known as the
plane strain fracture toughness and is denoted KIc.

F IGURE 2.9: K is measured as a function of specimen thickness

Particular attention is paid to KIc because this is the minimum toughness that can
be achieved under the most severe conditions of loading. The changes in KIc with
thickness are accompanied by corresponding changes in fracture geometry. In the
plane strain regime the fracture surface is oriented at 90o to the direction of loading
(ie “square” fracture). As the thickness decreases, 45 deg “shear lips” appear on
either side of a flat central regime. At and below the thickness corresponding to the
maximum KIc position, the shear lips occupy the full thickness and one has a 45 deg
“shear” or plane stress fracture.
 29

Chapter 3

•
 31

Chapter 4

Fatigue in Structures

4.1 Introduction
Fatigue, as understood by materials technologists, is a process in which damage
accumulates due to the repetitive application of loads that may be well below the
yield point. The process is dangerous because a single application of the load would
not produce any ill effects, and a conventional stress analysis might lead to a as-
sumption of safety that does not exist. In one popular view of fatigue in metals, the
fatigue process is thought to begin at an internal or surface flaw where the stresses
are concentrated, and consists initially of shear flow along slip planes. Over a num-
ber of cycles, this slip generates intrusions and extrusions that begin to resemble a
crack. A true crack running inward from an intrusion region may propagate initially
along one of the original slip planes, but eventually turns to propagate transversely
to the principal normal stress. The modern study of fatigue is generally dated from
the work of A. Wöhler, a technologist in the German railroad system in the mid-
nineteenth century. Wohler was concerned by the failure of axles after various times
in service, at loads considerably less than expected. A railcar axle is essentially a
round beam in four-point bending, which produces a compressive stress along the
top surface and a tensile stress along the bottom. After the axle has rotated a half
turn, the bottom becomes the top and vice versa, so the stresses on a particular re-
gion of material at the surface varies sinusoidally from tension to compression and
back again. This is now known as fully reversed fatigue loading.

4.2 S-N curves

Well before a microstructural understanding of fatigue processes was developed,
engineers had developed empirical means of quantifying the fatigue process and
designing against it. Perhaps the most important concept is the S-N diagram, in
which a constant cyclic stress amplitude S is applied to a specimen and the number
of loading cycles N until the specimen fails is determined. Millions of cycles might
be required to cause failure at lower loading levels, so the abscissa in usually plotted
logarithmically.
In some materials, notably ferrous alloys, the S - N curve flattens out eventu-
ally, so that below a certain endurance limit σe failure does not occur no matter how
long the loads are cycled. Obviously, the designer will size the structure to keep the
stresses below σe by a suitable safety factor if cyclic loads are to be withstood. For
some other materials such as aluminum, no endurance limit exists and the designer
must arrange for the planned lifetime of the structure to be less than the failure point
on the S - N curve. Statistical variability is troublesome in fatigue testing; it is neces-
sary to measure the lifetimes of perhaps twenty specimens at each of ten or so load
32 Chapter 4. Fatigue in Structures

levels to define the S - N curve with statistical confidence. It is generally impossible

to cycle the specimen at more than approxi- mately 10Hz (inertia in components of
the testing machine and heating of the specimen often become problematic at higher
speeds) and at that speed it takes 11.6 days to reach 107 cycles of loading. Obtaining
a full S - N curve is obviously a tedious and expensive procedure.

F IGURE 4.1: •

The fatigue life is usually split into a crack initiation period and a crack growth
period. The initiation period is supposed to include some microcrack growth, but
the fatigue cracks are still too small to be visible. In the second period, the crack
is growing until complete failure. It is technically significant to consider the crack
initiation and crack growth periods separately because several practical conditions
have a large influence on the crack initiation period, but a limited influence or no
influence at all on the crack growth period.
From such an experiment, the stress amplitude σa for fully reversed loading
(equal to one-half of the stress range from the maximum tension to maximum com-
pression), is plotted against the number of fatigue cycles to failure. For materials
which harden by strain-ageing, under constant amplitude loading conditions, these
alloys exhibit a plateau in the stress-life plot typically beyond about 106 fatigue cy-
cles. Below this plateau level, the specimen may be cycled indefinitely without caus-
ing failure. This stress amplitude is known as the fatigue limit or endurance limit.
The value of at is 35% to 50% of the tensile strength UTS for most steels and copper
alloys.

4.2.1 Fatigue Ratio

Through many years of experience, empirical relations between fatigue and tensile
properties have been developed. Although these relationships are very general, they
remain useful for engineers in assessing preliminary fatigue performance. The ratio
4.3. Stress Life Approach 33

of the endurance limit Se to the ultimate strength Su of a material is called the fatigue
ratio. It has values that range from 0.25 to 0.60, depending on the material. For steel,
the endurance strength can be approximated by:

Se = 0.5Su (4.1)
In addition to this relationship, for wrought steels the stress level corresponding
to 1000 cycles, S1000 , can be approximated by:

S1000,steel = 0.9Su (4.2)

4.3 Stress Life Approach

The stress-life approach to fatigue was first introduced in the 1860s by Wohler. Out
of this work evolved the concept of an ’endurance limit’, which characterizes the
applied stress amplitude below which a (nominally defect-free) material is expected
to have an infinite fatigue life. This empirical method has found widespread use
in fatigue analysis, mostly in applications where low-amplitude cyclic stresses in-
duce primarily elastic deformation in a component which is designed for long life.
This chapter deals with the stress-life approach to fatigue where the effects of stress
concentrations, mean stresses and surface modifications are discussed. Methods
for characterizing the fatigue life in terms of nominal stress amplitudes using ex-
perimental data obtained from rotating bend tests on smooth specimens emerged
from the work of Wohler (1860) on fatigue of alloys used for railroad axles. In this
approach, smooth (unnotched) test specimens are typically machined to provide a
waisted (hour-glass) cylindrical gage length, which is fatigue-tested in plane bend-
ing, rotating bending, uniaxial compression-tension (push-pull) or tension-tension
cyclic loading.
Many high strength steels, aluminum alloys and other materials do not generally
exhibit a fatigue limit. For these materials, σa continues to decrease with increasing
number of cycles. An endurance limit for such cases is defined as the stress ampli-
tude which the specimen can support for at least 107 fatigue cycles. If the (true) stress
amplitude plotted as a function of the number of cycles or load reversals! to failure,
a linear relationship is observed. The resulting expression relating the stress ampli-
tude, σa = ∆σ/2, in a fully-reversed, constant-amplitude fatigue test to the number
of load reversals to failure, 2N f , is (Basquin, 1910)

∆σ
= σa = σ0f (2N f )b (4.3)
2
where σ0f is the fatigue strength coefficient (which, to a good approximation,
equals the true fracture strength σ, corrected for necking, in a monotonic tension
test for most metals) and b is known as the fatigue strength exponent or Basquin
exponent.

4.3.1 Mean Stress Effects on Fatigue Life

Most basic S-N fatigue data collected in the laboratory is generated using a fully-
reversed stress cycle. However, actual loading applications usually involve a mean
stress on which the oscillatory stress is superimposed. The mean level of the im-
posed fatigue cycle is known to play an important role in influencing the fatigue
34 Chapter 4. Fatigue in Structures

behavior of engineering materials. In this case, the stress range, the stress amplitude
and the mean stress, respectively, are defined as

F IGURE 4.2: •

The mean stress is also characterized in terms of the load ratio, R = σmin /σmax .
With this definition, R = -1 for fully reversed loading, R = 0 for zero-tension fatigue,
and R = 1 for a static load.
When the stress amplitude from a uniaxial fatigue test is plotted as a function of
the number of cycles to failure, the resultant S-N curve is generally a strong function
of the applied mean stress level.
Figure 4.5 shows the typical S-N plots for metallic materials as a function of four
different mean stress levels. One observes a decreasing fatigue life with increasing
mean stress value.
Mean stress effects in fatigue can also be represented in terms of constant-life
diagrams, as shown.
Here, different combinations of the stress amplitude and mean stress providing
a constant fatigue life are plotted. Most well known among these models are those
due to Gerber (1874), Goodman (1899), and Soderberg (1939).

1. Soderberg (USA, 1930)

σa σm
+ =1 (4.4)
σe σy

2. Goodman (England, 1899)

σa σm
+ =1 (4.5)
σe σu
4.4. Strain Life Approach 35

F IGURE 4.3: •

F IGURE 4.4: •

3. Gerber (Germany, 1874)

σa σm
+ ( )2 = 1 (4.6)
σe σu
4. Morrow (USA, 1960s)
σa σm
+ =1 (4.7)
σe σf

General Observations
• Most actual test data tend to fall between the Goodman and Gerber curves.
• For most fatigue situations R<1 ( i.e. small mean stress in relation to alternat-
ing stress), there is little difference in the theories
• In the range where the theories show large differences (i.e. R values approach-
ing 1) there is little experimental data. In this case the yield stress may set the
design limits.
• The Soderberg line is very conservative and seldom used

4.4 Strain Life Approach

The information derived from cyclic-stress-based continuum analysis mainly per-
tains to elastic and unconstrained deformation. In many practical applications, en-
gineering components generally undergo a certain degree of structural constraint
36 Chapter 4. Fatigue in Structures

F IGURE 4.5: •

and localized plastic flow, particularly at locations of stress concentrations. In these

situations, it is more appropriate to consider the strain-life approach to fatigue.
when the logarithm of the plastic strain amplitude, ∆e p /2, was plotted against
the logarithm of the number of load reversals to failure, 2N f , a linear relationship
resulted for metallic materials,(Coffin (1954) and Manson (1954) relation)

∆e p
= e0f (2N f )c (4.8)
2
where e0f is the fatigue ductility coefficient and c is the fatigue ductility exponent.
In general, e0f is approximately equal to the true fracture ductility e f in monotonic
tension.

Low cycle and high cycle fatigue lifes

Since the total strain amplitude in a constant strain amplitude test, ∆e/2, can be
written as the sum of elastic strain amplitude, ∆ee /2, and plastic strain amplitude,
∆e p /2.

∆e ∆ee ∆e p
= + (4.9)
2 2 2
from hookes law,

∆ee ∆σ σa
= + (4.10)
2 2E E
4.4. Strain Life Approach 37

F IGURE 4.6: •

from Basquin’s relation

∆ee σ0f
= (2N f )b (4.11)
2 E
combining Coffin manson’s relation and Basquin relation,

∆e σ0f
= (2N f )b + e0f (2N f )c (4.12)
2 E
The first and second terms on the right hand side, are the elastic and plastic
components, respectively, of the total strain amplitude. Equation forms the basis for
the strain-life approach to fatigue design and has found widespread application in
industrial practice.

4.4.1 Cyclic Stress-Strain Behavior

Cyclic Stress Strain curves are useful for assessing the durability of structures and
components subjected to repeated loading. The response of a material subjected to
cyclic inelastic loading is the form of a hysteresis loop. The total width being Deltae
and the total height being ∆σ.
This can be stated in terms of amplitude. Strain Amplitude is ea = ∆e a and stress
amplitude σa = ∆σ 2 . The total strain is ∆e = ∆e e + ∆e p

Bauschinger Effect
The stress-strain behavior obtained from a monotonic test can be quite different from
that obtained under cyclic loading. This was first observed by Bauschinger. His ex-
periments indicated the yield strength in tension or compression was reduced after
applying a load of the opposite sign that caused inelastic deformation. Thus, one
single reversal of inelastic strain can change the stress -strain behavior of metals.

Cyclic Strain Hardening and Softening

The stress-Strain behavior is often altered due to repeated loading. If maximum
stress increases with each cycle then this is called strain hardening and if the max-
imum stress decreases with each cycle is it called strain softening. It was observed
38 Chapter 4. Fatigue in Structures

F IGURE 4.7: •

F IGURE 4.8: •

by Manson that if
σult
> 1.4 (4.13)
σy
is satisfied then the material will cyclically harden. And if
σult
< 1.2 (4.14)
σy

then the material will cyclically soften.

4.4.2 Mean Stress Effect

Mean stress effects are seen predominantly seen at longer lives. They can either
increase with a nominal compressive force or decrease it with a nominal tensile load.
At high strain amplitudes where plastic strains are significant mean stress relaxation
4.5. Stress Life vs Strain Life 39

F IGURE 4.9: Strain Hardening

F IGURE 4.10: Strain Softening

occurs and mean stress tends to zero. (This is not cyclic softening.) Mean stress
relaxation occurs only in materials that are cyclically stable. Modifications to strain
life equation have been made to account for mean stress effects.

∆ee σ0f − σo
= (2N f )b (4.15)
2 E
where σo is the mean stress.

4.5 Stress Life vs Strain Life

Stress life methods are most useful at high cycle fatigue, where the applied stresses
are elastic, and no plastic strain occurs anywhere other than at the tips of fatigue
cracks. At low cycles, scatter in the fatigue data makes these methods increasingly
less reliable. On the other hand, strain life methods can be used for low cycle fatigue,
where there the loading is a combination of elastic and plastic on the macro scale.
For most stress life calculations, the math is relatively easy, since there is only one
stress component. In strain life calculations, the math is more difficult, as the elastic
and plastic components of the strain must be dealt with separately. Stress life and
strain life test data often do not correlate well to each other.
40 Chapter 4. Fatigue in Structures

4.6 Notched Stress - Neubers Stress Concentration

In theory, the peak stress near a stress raiser would be Kt times larger than the nom-
inal stress at the notched cross section. However, Kt is an ideal value based on lin-
ear elastic behavior and depends only on the proportions of the dimensions of the
stress raiser and the notched part. To deal with the various phenomena that influ-
ence stress concentration, the concepts of effective stress concentration factor and
notch sensitivity have been introduced. The effective stress concentration factor is
obtained experimentally. The effective stress concentration factor of a specimen is
defined as:
σmax
Kt = (4.16)
σnom
For fatigue loading, the definition of experimentally determined effective stress
concentration is
fatigue strength without notch
Kf = (4.17)
fatigue strength with notch
We can define notch sensitivity index q provides a means of estimating the effects of
stress concentration by the following relation.

K f = 1 + q ( K t − 1) (4.18)

Notch sensitivity in fatigue decreases as the notch radius decreases and as the
grain size increases. A larger part will generally have greater notch sensitivity than
a smaller part with proportionally similar dimensions. This variation is known as
the scale effect. Larger notch radii result in lower stress gradients near the notch,
and more material is subjected to higher stresses.
In almost all cases, the fatigue notch factor is less than the stress concentration
factor, and is less than 1. That is: 1≤ K f ≤ Kt
Neuber has developed the following approximate formula for the notch factor
for R= -1 loading, and the Neuber Equation for notch factor sensitivity is:

1
q= q (4.19)
ρ
1+ r

where r is the notch root radius and ρ is a material constant that is related to the
grain size of the material. The notch factor K f is usually used to correct the fatigue
strength for the notched member.
Neuber also established a rule that is useful beyond the elastic limit relating the
effective stress and strain concentration factors to the theoretical stress concentration
factor. Neuber’s rule contends that the formula

Kσ Ke = Kt2 (4.20)
which can be rewritten as,

σmax emax = Kt2 σnom enom (4.21)

 41

Chapter 5

Statistical Aspects of Fatigue

Behavior

5.1 Fatigue - HCF and LCF

By far the majority of engineering design projects involve machine parts subjected
to fluctuating or cyclic loads. Such loading induces fluctuating or cyclic stresses that
often results in failure by fatigue. There are three commonly recognized forms of
fatigue: high cycle fatigue (HCF), low cycle fatigue (LCF) and thermal mechanical
fatigue (TMF).
High-cycle fatigue is generally associated with low loads and long life (> 103
cycles), is commonly analyzed with a "stress-life" method (the S-N curve), which
predicts the number of cycles sustained before failure, or with a "total-life" method
(endurance limit), which puts a cap stress that allows the material to have infinite
life (> 106 cycles). On a conventional stress-life curve, commonly called a S–N curve
or a Wöhler diagram, HCF occurs at the right end of the curve where the number of
cycles is usually too large to be able to obtain sufficient statistically significant data to
be able to characterize the material behavior with a very high degree of confidence.
Generally HCF is associated with higher frequencies and purely elastic strain but
LCF is a combination of elastic and plastic strains.
Factors to Differentiate HCF and LCF:

1. Life

2. Frequency

3. Elastic and Plastic Strains

4. Stress Amplitude

5.1.1 High cycle fatigue

High-cycle fatigue involves a large number of cycles (N > 105 cycles) and an elas-
tically applied stress. High-cycle fatigue tests are usually carried out for 107 cycles
and sometimes 5 × 10 8 cycles for nonferrous metals. Although the applied stress is
low enough to be elastic, plastic deformation can take place at the crack tip. High-
cycle fatigue data are usually presented as a plot of stress (S) versus the number of
cycles to failure (N). A log scale is normally used for the number of cycles.
The fatigue life is the number of cycles to failure at a specified stress level, while
the fatigue strength is the stress at which failure does not occur at a predetermined
number of cycles. As the applied stress level is decreased, the number of cycles
to failure increases. Normally, the fatigue strength increases as the static tensile
42 Chapter 5. Statistical Aspects of Fatigue Behavior

strength increases. For example, high-strength steels heat treated to over 1380 MPa
(200 ksi) yield strengths have much higher fatigue strengths than aluminum alloys
with 480 MPa (70 ksi) yield strengths. For a large number of steels, there is a di-
rect correlation between tensile strength and fatigue strength; that is, higher-tensile-
strength steels have higher endurance limits. The endurance limit is normally in the
range of 0.35 to 0.60 of the tensile strength. This relationship holds up to a hard-
ness of approximately 40 HRC ( 1240 MPa, or 180 ksi, tensile strength), and then the
scatter becomes too great to be reliable.
This does not necessarily mean it is wise to use as high a strength steel as pos-
sible to maximize fatigue life because, as the tensile strength increases, the frac-
ture toughness decreases and environmental sensitivity increases. In addi- tion, the
endurance limit of high-strength steels is extremely sensitive to surface condition,
residual stress state, and the presence of inclusions that act as stress concentrations.
Fatigue cracking can occur quite early in the service life of the member by the for-
mation of a small crack, generally at some point on the external surface. The crack
then propagates slowly through the material in a direc- tion roughly perpendicular
to the main tensile axis.

5.1.2 Low Cycle Fatigue

During cyclic loading within the elastic regime, stress and strain are directly re-
lated through the elastic modulus. However, for cyclic loading that produces plastic
strains, the responses are more complex and form a hysteresis loop. In cyclic strain-
controlled fatigue, the strain amplitude is held constant during cycling. Because
plastic deformation is not completely reversible, the stress-strain response during
cycling can change, largely depending on the initial condition of the metal. The
metal can either undergo cyclic strain hardening, cyclic strain softening, or remain
stable. Cyclic hardening leads to an increasing peak strain with increasing cycles.
Strong metals tend to cyclically soften, and low-strength metals tend to cyclically
harden. However, the hysteresis loop tends to stabilize after a few hundred cycles,
when the material attains an equilibrium condition for the imposed strain level. This
has been discussed in detail in the previous chapter.

5.2 Coffin Manson Relation

The notion that plastic strains are responsible for cyclic damage was established by
Coffin (1954) and Manson (1954). Working independently on problems associated
with fatigue due to thermal and high stress amplitude loading, Coffin and Manson
proposed an empirical relationship between the number of load reversals to fatigue
failure and the plastic strain amplitude. This so-called Coffin-Manson relationship.

5.3 Transition Life

For ’short’ and ’long’ fatigue lives, it is useful to consider a transition life, which is
defined as the number of reversals to failure (2N f )t at which the elastic and plastic
strain amplitudes are equal. At short fatigue lives, plastic strain amplitude is more
dominant than the elastic strain amplitude and the fatigue life of the material is
controlled by ductility. At long fatigue lives, the elastic strain amplitude is more
significant than the plastic strain amplitude and the fatigue life is dictated by the
rupture strength.
5.4. Cycle Counting Techniques 43

Optimizing the overall fatigue properties thus inevitably requires a judicious bal-
ance between strength and ductility.
Mean stress effects have also been incorporated into the uniaxial strain-based
characterization of fatigue life in a simple manner. Assuming that a tensile mean
stress reduces fatigue strength.
Taking cyclic relaxation into account, Cyclic hardening reduces the plastic strain
range and increases the stress range for a fixed total strain.

5.4 Cycle Counting Techniques

These practices are a compilation of acceptable procedures for cycle-counting meth-
ods employed in fatigue analysis in accordance with ASTM E 1049 – 85. Cycle count-
ing is used to summarize (often lengthy) irregular load-versus-time histories by pro-
viding the number of times cycles of various sizes occur. The definition of a cycle
varies with the method of cycle counting.

5.4.1 Procedures for Cycle Counting

Level-Crossing Counting
One count is recorded each time the positive sloped portion of the load exceeds a
preset level above the reference load, and each time the negative sloped portion of
the load exceeds a preset level below the reference load. Reference load crossings are
counted on the positive sloped portion of the loading history. It makes no difference
whether positive or negative slope crossings are counted. The distinction is made
only to reduce the total number of events by a factor of two.

F IGURE 5.1: •

Peak Counting
Peak counting identifies the occurrence of a relative maximum or minimum load
value. Peaks above the reference load level are counted, and valleys below the ref-
erence load level are counted. Results for peaks and valleys are usually reported
separately. A variation of this method is to count all peaks and valleys without re-
gard to the reference load. To eliminate small amplitude loadings, mean-crossing
peak counting is often used. Instead of counting all peaks and valleys, only the
largest peak or valley between two successive mean crossings.
44 Chapter 5. Statistical Aspects of Fatigue Behavior

F IGURE 5.2: •

Simple-Range Counting
For this method, a range is defined as the difference between two successive rever-
sals, the range being positive when a valley is followed by a peak and negative when
a peak is followed by a valley. Positive ranges, negative ranges, or both, may be
counted with this method. If only positive or only negative ranges are counted, then
each is counted as one cycle. If both positive and negative ranges are counted, then
each is counted as one-half cycle. Ranges smaller than a chosen value are usually
elimi- nated before counting.

F IGURE 5.3: •

5.5 Paris Law

Certainly in aircraft, but also in other structures as well, it is vital that engineers
be able to predict the rate of crack growth during load cycling, so that the part in
question be replaced or repaired before the crack reaches a critical length. A great
deal of experimental evidence supports the view that the crack growth rate can be
correlated with the cyclic variation in the stress intensity factor.

da
= A∆K m (5.1)
dN
5.6. Damage rule for irregular loads 45

where da/dN is the fatigue crack growth rate per cycle, ∆K = Kmax - Kmin is
the stress intensity factor range during the cycle, and A and m are parameters that
depend the material, environment, frequency, temperature and stress ratio. This is
sometimes known as the “Paris law,”.
The crack growth rate (da/dN) can be determined from the slope of the curve.
Initially, the crack growth rate is slow but increases with increasing crack length.
Of course, the crack growth rate is also higher for higher applied stresses. If one
can characterize the crack growth, it is then possible to estimate the service life or
inspection intervals required under specific loading conditions and service environ-
ment. In the fracture mechanics approach to fatigue crack growth, the crack growth
rate, or the amount of crack extension per loading cycle, is correlated with the stress-
intensity parameter (K ). This approach makes it possible to estimate the useful safe
life and inspection intervals.

F IGURE 5.4: •

An idealized da/dn versus ∆K curve is shown below.

In region I, ∆Kth is the fatigue crack growth threshold, which is at the lower
end of the ∆K range, where crack growth rates approach zero. In region II, the
crack growth rate is stable and essentially linear and can be modeled by power-law
equations, such as the Paris equation.
The Paris equation does not account for the load ratio; however, there are other
expressions that do account for the sensitivity of crack growth rate to the load ratio.

5.6 Damage rule for irregular loads

Fatigue damage increases with applied cycles in a cumulative manner which may
lead to fracture. Cumulative fatigue damage is an old, but not yet resolved problem,
More than seventy years ago, Palmgren suggested the concept which is now known
as the ’linear rule’. In 1945, Miner first expressed this concept in a mathematical
form called as the Miner’s rule discussed in later sections.

5.6.1 Linear Damage Rules(LDR)

In the LDR, the measure of damage is simply the cycle ratio with basic assumptions
of constant work absorption per cycle, and characteristic amount of work absorbed
46 Chapter 5. Statistical Aspects of Fatigue Behavior

F IGURE 5.5: •

at failure. The energy accumulation, therefore, leads to a linear summation of cycle

ratio or damage.

5.6.2 Marco-Starkey theory

To remedy the deficiencies associated with the LDR, Richart and Newmark intro-
duced the concept of damage curve (or D-r diagram) in 1948 and speculated that the
D-r curves ought to be different at different stress-levels. Upon this concept and the
results of load sequence experiments, Marco and Starkey proposed the first nonlin-
ear load-dependent damage theory in 1954, represented by a power relationship, D
= ∑ rixi , where Xi is a variable quantity related to the ith loading level.
Life calculations based on Marco-Starkey theory would result in ∑ri > 1 for L-H
load sequence, and in ∑ri < 1 for H-L load sequence.

5.7 Variable Amplitude Loading

Constant-amplitude (CA) fatigue loading is defined as fatigue under cyclic loading
with a constant amplitude and a constant mean load. Sinusoidal loading is a classical
example of CA fatigue loads applied in many fatigue tests. But various structures in
service are subjected to variable-amplitude (VA) loading, which can be a rather com-
plex load-time history. For structures subjected to VA load cycles in service, it may
be desirable that fatigue failures should never occur. It implies that all load cycles of
the load spectra should not exceed the fatigue limit. However, this requirement can
lead to a heavy structure and it can be unnecessarily conservative, especially if the
number of more severe load cycles above the fatigue limit is relatively small.
5.8. Miner’s Rule 47

F IGURE 5.6: Time compressed representation of the load history

Moreover, a complete avoidance of fatigue is not always required. Failures after

a sufficiently long life can be acceptable from an economical point of view, the more
so if safety issues are not involved. The discussion starts with considerations on
the well-known Miner rule with its long lasting reputation. Reasons why and how
this rule can be misleading are discussed. Results of various fatigue tests under VA
loading are considered.

5.8 Miner’s Rule

When the cyclic load level varies during the fatigue process, a cumulative damage
model is often hypothesized. To illustrate, take the lifetime to be N 1 cycles at a stress
level σ1 and N2 at σ2 . If damage is assumed to accumulate at a constant rate during
fatigue and a number of cycles n1 is applied at stress σ1 , where n1 < N1, then the
fraction of lifetime consumed will be n1/N1. To determine how many additional
cycles the specimen will survive at stress σ2 , an additional fraction of life will be
available such that the sum of the two fractions equals one,

n1 n2
+ =1 (5.2)
N1 N2
Note that absolute cycles and not log cycles are used here.

5.8.1 Miner’s Rule for variable loading

A specimen is fatigue tested under CA loading until a certain percentage of its fa-
tigue life, say x%. Fatigue damage must then be present in the specimen, because
its original life (N) has been reduced to (100 - x)% of the fatigue life N. The damage
may still be invisible, but it is present in the material of the specimen.
The amplitude is changed only once. Obviously, such a load history is not related
to service load spectra, but it is considered here to discuss the basics of the Miner
rule. Moreover, this simple VA load sequence was widely used in many older test.
In the figure n1 cycles at a stress amplitude Sa1 are applied, followed by cycles with
an amplitude reduced to Sa2 . The test is continued until failure occurs after n2 cycles
at the lower amplitude. Two different blocks of load cycles are thus applied in this
test. The problem is to predict n2. According to this rule, applying n 1 cycles with a
stress amplitude Sa1 and a corresponding fatigue life endurance N 1 , is equivalent
48 Chapter 5. Statistical Aspects of Fatigue Behavior

F IGURE 5.7: •

to consuming n1 /N1 of the fatigue resistance. The same assumption applies to

any subsequent block of load cycles. Failure occurs if the fatigue resistance is fully
consumed.

n1
∑ N1 = 1 (5.3)

Sequence Effect

F IGURE 5.8: Two different sequences of blocks in a simple VA load

history (R = 0) applied to a notched specimen

Two simple VA load sequences are shown in Figure 10.3. These sequences are ap-
plied to a notched specimen. The same two amplitudes are used in both sequences,
but in the first sequence the test starts with the low amplitude (low-high sequence,
or LoHi), and in the second one with the high amplitude (high-low sequence, or
HiLo). The stress ratio is supposed to be zero (S min = 0, or R = 0).
The peak stress at the root notch exceeds the yield stress only in the block with
the high amplitude, whereas this does not occur in the block with the low amplitude.
It implies that notch root plasticity did not occur in the LoHi sequence during the
low-amplitude cycles of the first block of the LoHi sequence.
However, in the other sequence (HiLo), notch root plasticity occurs immediately
in the first block with the high amplitude. In this case, compressive residual stresses
at the root of the notch are present at the beginning of the second block with the
low-amplitude cycles. This is favorable for fatigue in the second block. Although
5.8. Miner’s Rule 49

the number of high-amplitude cycles (n2 ) as observed in the LoHi test is applied
in HiLo test, the fatigue life will be larger in this test due to the favorable residual
stress. In other words, the sequence of the two blocks is significant for the fatigue
life. This sequence effect is not predicted by the Miner rule because the rule ignores
any change of residual stresses induced by previous cycles.

Crack Length at failure

The two load sequences of Figure are considered again. Assume that a crack length a
= 2 mm (as an example) is reached at the end of the first low-amplitude block while
the crack at this low amplitude could grow until a = 20 mm until failure occurs.
However, a change to the second high-amplitude block could lead to immediate
failure, because a small crack is more critical at a higher stress level.