DECLARATION - University of...
Transcript of DECLARATION - University of...
UNIVERSITY OF NAIROBI
DEPARTMENT OF MECHANICAL AND
MANUFACTURING ENGINEERING FINAL YEAR PROJECT
THIS PROJECT IS SUBMITTED IN PARTIAL FULFILMENT
FOR THE AWARD OF DEGREE BACHELOR OF SCIENCE IN
MECHANICAL ENGINEERING.
PROJECT NUMBER: J.K.M. 03/2015
FRACTURE OF THIN METAL SHEETS DUE TO
BIAXIAL LOADING
Authors: 1. YIAILE MILIA
F18/36662/2010
2. WILSON KIMUTAI TAIGET
F18/1854/1997
PROJECT SUPERVISOR: PROF.J.K. MUSUVA
DECLARATION
We (Yiaile Milia and Wilson Kimutai Taiget) declare that the project report that includes
research work, experimental findings, discussions and conclusions is entirely our typical
effort and to the best of our knowledge is original. The project details entailed in our report
has not been presented before to the best of our knowledge.
1. YIAILE MILIA,
Signature...........................................................
This day.................................of.........................
2. WILSON KIMUTAI TAIGET,
Signature...........................................................
This day.................................of.........................
3. Project Supervisor: PROF.J.K. MUSUVA
Signature...........................................................
This day.................................of.........................
ABSTRACT
For several years, thin metal sheets have been used for diverse applications i.e. car bodies,
pressure vessel and aircraft bodies and tanks. Consumers of these products have been
experiencing recurrent fracture and fatigue problems
Fracture mechanics principles are used to investigate the fracture of thin metal sheets due to
bi-axial loading and in our case the metal is mild steel of 1.3 mm thickness.
The object of this project is to investigate the fracture toughness of this kind of thin sheets
under bi-axial loading. Specimens were prepared according to the standards. Initial cracks of
various lengths were introduced in the specimen. This was done by first making saw cuts, and
then further the cut is extended by a newly filled saw on the specimen. The surface around
the crack was cleaned and polished using emery paper. The specimen was loaded on the
fatigue tester for further pre-cracking.
After pre-cracking the specimen was then loaded to a rig frame that enables bi-axial loading.
The crack growth was monitored till fracture took place. The load and the respective crack
length were recorded. From the data obtained, KG, KR was generated. R-curves were drawn
from the values of KG and KR against the half crack lengths (a) mm.
Fromm the R-curves, the fracture toughness was obtained for every specimen. The fracture
toughness ranges from 120 MN/m3/2 to 125 MN/m3/2.
Since fracture toughness is dependent on the geometry and the specimen thickness, the results
obtained here are only valid for mild steel of 1.3 mm thickness. The initial crack extension
was assumed to have started under plane strain condition, and hence the fracture toughness
generated in this experiment is plain strain fracture toughness.
From the experiment is has been found that K0 (KR at the onset of crack extension) is
independent of specimen thickness, and it was found that the initial crack extension varied
from 78 MN/m3/2 to 80 MN/m3/2 for most of the specimen.
APRECIATION
We are grateful to our supervisor Professor J.K. Musuva whose expertise, guidance and
support are our project foundation.
We also appreciate the support of Chief Technologist Mr Adoul and the other staff of the
Mechanical Engineering Workshops.
GLOSSARY OF TERMS
a- Half crack length of centre crack
B- Specimen thickness
A-cross-sectional area
E- Elastic modulus
e- Eccentricity
G- Strain energy release rate
I- Second moment of area
J-Contour integral
K-Stress intensity factor
KIC- Plane strain fracture toughness
KC- Plane stress fracture toughness
U- Strain energy
x, y, z – Cartesian coordinates
- Crack opening displacement
s- Surface energy per unit area
p- Surface energy of plastic distortion
-direct strain
- Poisson’s ratio
- Crack tip radius
- Shear stress
- Normal stress
Y- Yield stress
uts –ultimate tensile strength
Subscripts
c- Critical
I, II, III – crack opening modes
i. initiation
o- Original
Abbreviations
ASTM- American Society of Testing Materials.
CCTS- centre crack tension Specimen
COD- Crack Opening Displacement
CS- Compact Specimen
FM-Fracture Mechanics
LEFM-Linear Elastic Fracture Mechanics
LBB-leak before-burst
DEDICATION
We dedicate our work to the Almighty God for his guidance in the five years programme, and
to our families and many friends. A special feeling of gratitude to our loving parents whose
words of encouragement and push for tenacity rings in our ears. We also appreciate those
who supported us throughout the project period.
TABLE OF CONTENTDECLARATION....................................................................................................................ii
ABSTRACT............................................................................................................................................................ iii
APRECIATION....................................................................................................................................................... iv
GLOSSARY OF TERMS............................................................................................................................................v
DEDICATION.....................................................................................................................vii
1. CHAPTER ONE..........................................................................................................1
1.1 INTRODUCTION.......................................................................................................................................1
1.2 PREVIOUS WORK.....................................................................................................................................4
1.2.1 2006 REPORT.................................................................................................................................4
2004 REPORT..................................................................................................................................................4
1.2.2 2003 REPORT.................................................................................................................................5
1.2.3 2000 REPORT.................................................................................................................................5
2. CHAPTER TWO.........................................................................................................7
2.1 LITERATURE REVIEW........................................................................................................................7
2.1.1 STRUCTURAL ANALYSIS AND FAILURE ANALYSIS...........................................................7
2.1.2 THEORIES OF FAILURE..............................................................................................................8
2.2 2-D STRESS/ BIAXIAL STRESS.................................................................................................................10
2.2.1 Stresses on inclined sections..........................................................................................................13
2.2.2 Transformation Equations..............................................................................................................13
2.3 THE MOHR CIRCLE.................................................................................................................................14
2.4 FRACTURE MECHANICS.........................................................................................................................15
2.4.1 Introduction...................................................................................................................................15
2.4.2 CRACK TIP PLASTICITY...........................................................................................................17
2.4.3 LINEAR ELASTIC FRACTURE MECHANICS..........................................................................19
2.4.4 POST- YIELD FRACTURE MECHANICS..................................................................................28
2.4.5 R-CURVES ANALYSIS...............................................................................................................32
2.4.6 DETERMINATION OF R-CURVES............................................................................................34
3. CHAPTER THREE...............................................................................................38
3.1 METHODOLOGY AND EXPERIMENTAL RESULTS ANALYSIS...................................................................38
3.1.1 APPARATUS................................................................................................................................38
3.1.2 REPAIR OF THE FATIGUE TESTING MACHINE....................................................................39
3.1.3 EXPERIMENTAL PROCEDURE................................................................................................40
3.2 RESULTS AND ANALYSIS.................................................................................................................42
3.2.1 LOAD CELLS CALIBRATION...................................................................................................42
3.2.2 SAMPLE CALCULATIONS........................................................................................................44
3.2.3 MEASURED AND CALCULATED RESULTS...........................................................................48
4 CHAPTER FOUR.................................................................................................67
4.0 DISCUSSION...........................................................................................................................................67
4.1 CONCLUSION AND RECOMMENDATIONS...................................................................................69
4.1.1 CONCLUSIONS...........................................................................................................................69
4.1.2 RECOMMENDATION.................................................................................................................70
REFERENCES.....................................................................................................................71
1. CHAPTER ONE
1.1 INTRODUCTION
For many years engineers have been pressured to develop high strength alloys for different
applications like aircrafts with a need to conserve energy. High strength alloys have been on
high demand for cars, railroad equipment and moving machine parts. In the nineteenth
century, industrial revolution resulted in the enormous increase in the use of metals for
structural applications (Jilin & Jones, 1991).
Due to the failure of these structures, many accidents occurred, and lives were lost. Some of
the accidents were due to poor design, but it was also discovered that the material had
numerous deficiencies in the form of pre-existing flaws that initiated cracking and fracture.
These defects can be reduced through the use of better production methods, and thus
structural failure is reduced drastically. In some cases, structures failed under very low
stresses. This anomaly led to extensive investigations that revealed that the fractures were
brittle and flaw and stress concentration where responsible. It was also discovered that low
temperatures promoted brittle fracture in the type of steel used.
Current manufacturing and design procedures can prevent the intrinsically brittle fracture of
welded steel structures by ensuring that the material has a suitable low transition temperature
and that the welding process does not raise it. Nonetheless, service induced embrittlement
remains a cause of concern. The objective of fracture mechanic thus is to provide quantitative
answers to particular problems concerning cracks in structures. Fracture mechanics is
fundamentally concern with fracture dominant failure. (Broek, 1982)
Design procedures consider materials as homogeneous isotropic and perfectly elastic-plastic,
but in practice materials do not meet these conditions. Materials may contain cracks and other
imperfections introduced by the factors discussed
Fracture mechanics is the branch of mechanics that deals with the behaviour of structures and
materials in the presence of cracks. Using fracture mechanics, we can put in quantitative
terms the combination of stress level, crack size and material resistance necessary for the
crack extension.
Motivation of fracture mechanics was a series of failures in the 1940s and 1950s ships,
airplanes and bridges in circumstances where the operating stresses were lower than the
specified stresses by the design codes in use at that time. Investigation into the above failures
established that the major causes could be traced:
a. to poor design resulting in high-stress concentration,
b. Cold weather causing the materials to have inadequate toughness,
c. Poor fabrication techniques that caused crack-like defects.
d. A sub-critical growth of cracks due to fatigue, stress corrosion and corrosion fatigue.
To eliminate the above failures, a science was required which would quantify the various
combinations of stress, crack size and material properties to prevent crack growth. This is the
science of fracture mechanics
Historically, the first major step in the direction of quantification of the effects of crack-like
defects was undertaken by C. E. Inglis (a professor of Naval Architecture). In 1913, Inglis
made a publication on stress analysis for an elliptical hole in an infinite linear elastic plate
loaded at its outer boundaries. (Knott, 1973)
Figure 1: Stresses on the edge of an infinite plate
For the above semi-axes we may have
c cosh∝o=a , c sinh∝o=b
And hence the equation of the ellipse is obtained as
x1 2a 2
+ x22b 2
=1
In the limit, as o 0, the ellipse becomes a crack of length 2c=2a. When a=b, the ellipse
becomes a circle.
A. A. Griffith, who was studying the effects of scratches and similar flaws in aircraft engine
components, transformed the Inglis analysis by calculating the effect of the crack on the
strain energy stored in an infinite cracked plate. Griffith also carried out tests on cracked
glass spheres and showed that the simple elastic analysis could be applied to describe the
propagation of different size cracks at different stress levels.
The mechanics of fracture progressed from being a scientific curiosity to an engineering
discipline, primarily because of the major failures that occurred in the Liberty ships during
World War II. The Liberty ships had an all-welded hull, as opposed to the riveted
construction of traditional ship designs. Of the roughly 2700 Liberty ships build during
World War II, approximately 400 sustained fractures. (Griffith, 1961).
After World War II, the fracture mechanics research group at the Naval Research, Dr. G.R.
Irwin led laboratory. Having studied the early work of Inglis, Griffith, and others, Irwin
found out that the basic tools needed to analyse fracture were already available. Irwin’s first
significant contribution was to extend the Griffith approach to metals by including the energy
dissipated by local plastic flow.
In 1956, Irwin developed the energy release rate concept, which is related to the Griffith
theory, but it is in a form that is more useful for solving engineering problems;
G=−dπdA
≥ R
Next, He used the Westergaard approach to show that the stresses and displacements near the
crack tip could be described by a single parameter that was related to the energy release rate.
This crack tip characterizing parameter later became known as the stress intensity factor.
In practice, all this work was largely ignored by engineers as it seemed too mathematical, and
it was only in the 1970's that fracture mechanics came to be accepted as a useful and an
essential tool. There were many reasons for this, for example, the development of non-
destructive examination methods which revealed hidden cracks in the structures, the demand
of space industry for high-strength high integrity pressure vessels, the increasing use of
welding and the severe conditions of offshore structures, etc. Hence, most of the practical
development of fracture mechanics has occurred in the last thirty years. (Sih, 1962)
1.2 PREVIOUS WORK
Previous works have shown that fracture toughness varies with the specimen thickness and
usually relatively high for the thin specimen, and it increases until a limiting value is reached.
The fracture toughness of different metals was analysed. Standard specimens were prepared,
tested, data collected and the results were analysed and conclusion made. These were the
results obtained from the previous experiments
1.2.1 2006 REPORT
The report was based on a mild steel thin sheet of thickness 1.18 mm which was being tested
to determine fracture toughness due to bi-axial loading. This can be summarized in the table
below of how fracture toughness varies with the initial crack length.[Kennedy Ruto,2006]
Table 1
Crack length2a (mm) KC
horizontal stress
σ x
Fracture stressσ C
Biaxiallity parameter λ
44 206 29.4 336.28 0.08742714445 184 30.2 367.72 0.082127706106 162 28.6 196.03 0.14589603659 164 32.01 279.39 0.1145710372 149 30.6 240.53 0.127219058103 174 27.5 219.83 0.125096666106 162 25.6 196.03 0.130592256107 118 31.3 146.97 0.21296863393 150 34.5 234.27 0.147265975101 141 35.7 190.57 0.187332739
2004 REPORT
Specimen material: mild steel.
Specimen thickness: 1.5 mm.
Specimen width: 190 mm.
Young’s modulus: 210 GPa.
Table 2
crack length, a,(mm)
Kc(Mpa/ m^(1/2)) σx (Mpa) σc (Mpa)
Biaxiallity parameter
48.5 61.6 0 151.9 042.8 62.2 2.73 157.68 0.01731354643.5 68.8 8.17 169.62 0.0481664944.6 71.6 15.43 173 0.08919075145.4 74.4 20.71 183.31 0.112978015
1.2.2 2003 REPORT
This report was based on a mild steel specimen that was 1.5 mm in thickness. They analysed
three specimens that have shown that KC decreases with increase in crack length as biaxiallity
decreases
Table 3
a (mm) Kchorizontal stress σx
Fracture stress σc
Biaxiallity parameter λ
43 71.54 34.53 180.46 0.191344342 44 64.74 13.05 134.14 0.097286417
45 62.24 0 145.61 0
1.2.3 2000 REPORT
Specimen material: mild steel.
Specimen thickness: 1.5 mm.
Specimen width: 190 mm.
Young’s modulus: 210 GPa.
crack length, a,(cm)
Kc(MPam^(1/2)) σx(Mpa) σc (Mpa)
Biaxiallity parameter
5.05 60.6 1.363 152.14 0.0089588545.31 66.55 5.2316 162.92 0.0321114664.96 70.12 10.73 177.62 0.0604098645.22 73.63 13.923 181.82 0.0765757345.16 71.98 23.702 178.86 0.132517052
From the above table, it is evident that the magnitude of the fracture toughness, KC, is
influenced by the biaxiallity ratio. In fact, a direct proportionality is found for the mild steel
specimen
2. CHAPTER TWO
2.1 LITERATURE REVIEW
2.1.1 STRUCTURAL ANALYSIS AND FAILURE ANALYSIS
The ultimate goal in the field of applied solid mechanics is to be able to design structures or
components that are capable of safely withstanding static or dynamic service loads for a
particular period. Most of the engineering decisions are based on the semi-empirical design
rules, which rely on phenomenological failure criteria calibrated by means of standard tests.
The failure criteria are derived based on extensive observations of failure mechanisms,
together with theoretical models that have been developed to describe these mechanisms.
In general failure mechanisms can be classified into two broad fields of deformation and
fracture. The above can be narrowed down to the following more detailed failure mechanism:
Excessive Elastic Deformation, Unstable Elastic Deformation (Buckling), Plastic
Deformation, Fracture, Fatigue, Creep and Stress Corrosion Cracking
Failure mode occurrence is dependent on factors such as environment, temperature, type of
load and the time the load is applied.
For many applications, it is sufficient to determine the maximum static or dynamic stresses
that the materials can withstand and then design the structure to ensure that the stresses
remain below acceptable limits. This involves fairly routine constitutive modelling and
numerical or analytical solution of appropriate boundary value problems. More critical
applications require some defect tolerance analysis. In these cases, the material or structure is
considered to contain flaws and we must decide whether to replace the part or leave it in
service under a more tolerable loading for a particular period. This kind of decision is usually
made using the disciplines of Fracture Mechanics.
There are three different design philosophies are used:
a. Safe life
The component is considered to be free of defects after fabrication and is designed to remain
defect-free during service, and withstand the maximum static or dynamic working stresses for
a particular period. If flaws, cracks, or similar damages are visited during service, the
component should be discarded immediately.
b. Fail safe
The component is designed to withstand the maximum static or dynamic working stresses for
a particular period in such a way that it’s probable failure would leak before burst (LBB)
condition should show leakage as a result of crack propagation. The aim is to prevent
catastrophic failure by detecting the crack at its early stages of growth and also reducing the
internal pressure.
c. Damage tolerance
The component is designed to withstand the maximum static or dynamic working stresses for
a certain period even in the presence of flaws, cracks, or similar damages of precise geometry
and size.
2.1.2 THEORIES OF FAILURE
In uniaxial tension yielding takes place whenσ=σ Y . If a structure to design will be subjected
to a multi-axial state of stress yielding will occur at a certain stress or a combination of
stresses/strains. We can use results from uniaxial test and apply those to multi-axial loading.
Theories that relate yielding in a multi-axial state of stress to yield strength (σ Y ) are termed
theories of failure or Yield Criteria. These theories are semi-empirical. Some of these theories
are briefly described below. (Dressman, 2005)
I. Maximum (direct) stress theory (Rankine Criterion)This theory predicts that yielding in a multi-axial state of stress takes place when σ1 reaches a
critical value. In uni-axial loading σ 1=σ Y
The drawbacks of the above theory is that it predicts yielding for hydrostatic state of stress,
and this contradicts the experimental results as it tends to deduce that hydrostatic state does
not cause yielding no matter how high it is.
II. Maximum shear stress theory (Tresca Criterion)This predicts that in multi-axial state of stress yielding occurs when max reaches a critical
value. But τ max=12
(σ 1−σ 3 )(1)
Now in uni-axial test at point of yielding σ 1=σ Y and σ 3=0
Therefore,
12
(σ 1−σ 3 )=σ Y2
(2) This theory correctly predicts that
hydrostatic stress state cannot cause yielding.
Maximum shear stress theory therefore agrees with experimental results and hence can be
used to predict failure in metals.
III. Maximum normal strain theory (St.Venant’s Criterion)
This predicts that when ε 1 reaches a critical value yielding occurs.
Recall that:
ε 1= 1E [σ 1−υ ( σ 2+σ 3 ) ](3)
In simple tension at yield
σ 1=σ Y Andσ 2=σ 3=0, Therefore:
ε 1=σ YE
(4)
Yielding will occur when:
1E [σ 1−υ (σ 2+σ 3 ) ]= σ Y
E(5)
This theory does not agree with experiments for metals but it has some agreement with
experiments for composite materials.
IV. Total strain energy theory (Bertram)This predicts that yielding in a multi-axial state of stress occurs when the total strain energy
stored in a material reaches a critical value.
In uni-axial tension at yield:
U = 12 E
[σ Y 2 ]=σ Y 22 E
(6)
In multi-axial stress, yielding occurs when:
12E
¿
Or
[σ 12+σ 22+σ 3 2−2 υ (σ 1 σ 2+σ 2 σ 3+σ 1σ 3 ) ]=σ Y 2 (8)
This only predicts yielding for hydrostatic state of stress
v. Distortion Energy Theory(Von Misses, Henky, Huber)
This was first proposed by Huber.it is generally termed Von-Misses criterion. This predicts
that yielding depend on the deviatoric strain energy, Ud. Yielding occurs when Ud reaches a
critical value.
At yielding in uni-axial tests:
U d=1+υ6 E
[ σ Y 2+σ Y 2 ]=1+υ6 E
[ 2σ Y 2 ](9)
For a multi-axial case, yielding occurs when
(σ 1−σ 2 )2+( σ 2−σ 3 ) 2+(σ 3−σ 1 )2=2 σ Y 2 (10)
This theory presents the best agreement with experiment for metals.
When Ud reaches a critical value τ oct also reaches a critical value. Theory may be recast as
octahedral shear stress theory. Note that τ oct is a geometric mean of the three maximum
shear stresses. At yielding,
τ oct=√23
σ Y .
Recall
τ oct=13 √ {[ (σ x−σ y ) 2+ (σ y−σ z ) 2+(σ z−σ x ) 2+6 (τ xy2+ τ yz 2+τ xz 2 ) ] }
Moreover recall
I ' 2=−16 [ (σ 1−σ 2 ) 2+ (σ 2−σ 3 )2+(σ 3−σ 1 ) 2 ](11)
Conversely when Ud reaches a critical value I’2 also reach a critical value.
2.2 2-D STRESS/ BIAXIAL STRESS
This is when system has a stress state in two directions and shear stress. When biaxial stress
occurs in metal sheets, all the stresses are in plane of the material. Such stresses are called
plane stresses.
Figure 2: A plate loaded biaxially (Kupfer, 1969)
If only the x and y faces of the element are subjected to stresses it is called plane stress in 2-D
as shown above.
Plane stresses can be seen in pressure vessels, aircrafts skins, car bodies, shafts and many
other structures.the ratio of the stress in the X- direction to the stress in the Y-direction
experienced by the bi-axially loaded specimen is called a biaxial parameter expressed as
shown below.
λ=σ xσ y (12)
ʎ=0 for uniaxial stress condition, ʎ=1 for equi-biaxial stress condition, ʎ=-1 for shearing
mode condition (Nyakamba, 2009)
Figure 3 crack tip (Rice J. &., 1968)
From the above figure it can be seen that the stress field in any linear elastic cracked body is
given by:
σ ij= k√r
f ij (θ )+other terms(13)
Where σij is the stress tensor, k is a constant, r and θ are as shown in the diagram below and
fij is a dimensionless function of θ. it’s apparent from the above equation that the stress near
the crack tip varies with1/√r, regardless of the configuration of the cracked bod. Note that as
r0 the stress approaches to. I.e. when a body contains a crack, a strong concentration
develops around a crack tip. However, for linear elastic material this stress concentration has
the same distribution close to the crack tip regardless of the size shape and specific boundary
conditions of the body. Only the intensity of the stress concentration varies. For the same
intensity, the stresses around the crack tip are identical. (Peterson, 1953)
There are three types of loading that a crack can experience:
MODE I: The principal load is applied normal to the crack plane, tends to open the crack.
MODE II: In-plane shear loading and tends to glide one crack face with respect to the other.
MODE III: Refers to out of plane shear.
A cracked body can be loaded in any one of these modes, or a combination of two or three
modes shown below
Figure 4: Modes of failure in materials
Each loading mode produces the 1/√r singularity at the crack tip, but the proportionality
constant, k and fij depend on mode. Therefore we replace k with the stress intensity factor K
where K= k √ (2π).
The opening mode is characterized by local displacements that are symmetric with respect to
x-y and x-z plane. This two facture surfaces are displaced perpendicular to each other in
opposite directions. Local displacements in the sliding or shearing mode, mode II, are
symmetric with respect to the x-y plane.
The two fracture surfaces slide over each other in a direction perpendicular to the line of the
crack tip. The tearing mode, mode III, is associated with local displacement that are skew
symmetric with both x-y and x-z planes. The fracture surfaces slide over each other in the
direction that is parallel to the line of the crack front.
In any problem the deformations at the crack tip can be treated as one or a combination of
these local displacement modes. Moreover, the stress field at the crack tip can be treated as
one or a combination of the three basic types of stress fields
2.2.1 Stresses on inclined sections
Figure 5: Stress on inclined Section
The stress system is known in terms of coordinate system x-y. We want to find the stresses in
terms of the rotated coordinate system x1y1. This is important because a material may yield or
fail at the maximum value of σ or τ. This value may occur at some angle other than θ= 0. (For
uni-axial tension the maximum shear stress occurred when θ= 45 degrees.
2.2.2 Transformation Equations
Figure 6: Stress on an inclined plane
The sum forces in the x1 direction
σ x 1 A secθ−σ x cosθ−τ xy A tan θ sin θ−σ y A tan θ sinθ−τ xy A tan θ cosθ=0 (14)
Sum of forces in the y-direction:
τ x1 y 1 A secθ+σ x sin θ−τ xy A cosθ−σ y A tan θ cosθ−τ xy A tan θ sin θ=0 (15)
Using τ xy=τ yx and simplifying gives
σ x=σ x cos2θ+σ y sin 2θ+2 τ xy sin θ cosθ (16)
τ x1 y 1=−(σ x−σ y )sin θ cosθ+τ xy cos2θ sin 2θ (17)
Using trigonometric functions gives the transformation equations for planes stresses as:
σ x 1=σ x+σ y2
− σ x−σ y2
cos2 θ+τ xy sin 2θ
(18)
τ x1 y 1=−(σ x−σ y )sin 2 θ+τ xy cos2θ (19)
To find the principal stresses, we must differentiate the transformation equations. This will
yield the two principal stresses as:
σ 1=σ x+σ y2
+√[(σ x−σ y2 )2+ τ xy 2] (20)
τ xy=σ x+σ y−2sin 2θ+2 τ xy cos2 θ (21)
And the principal stresses are obtained as follows:
σ 1=σ x−σ y2
+√ [( σ x−σ y2 )2+τ xy2] (22)
σ 2=σ x−σ y
2 −√[( (σ x−σ y )2 )2+τ xy 2] (23)
2.3 THE MOHR CIRCLE
The Mohr Circle was named after the German Civil Engineer Otto Mohr who developed the
graphical technique for drawing the circle in 1882.
The transformation equations for plane stress can be represented in graphical form by a plot
known as Mohr’s Circle.
This graphical representation is extremely useful because it enables you to visualize the
relationships between the normal and shear stresses acting on various inclined planes at a
point in a stressed body.
Using Mohr’s Circle you can also calculate principal stresses, maximum shear stresses and
stresses on inclined planes.
Figure 7: Mohr Circle for the Principal stresses (Jolly. R. H., 1997)
2.4 FRACTURE MECHANICS
2.4.1 IntroductionInvestigation into fracture mechanisms depend largely upon electron microscopy. This study
entails use of electron microscope in describing and explaining fractures. This is known as
electron fractrography. There are two essential fracture mechanisms:
I. Cleavage fracture
Since cleavage fracture usually undergoes little plastic deformation, it is also referred as
brittle fracture. Cleavage fracture is the most brittle form of fracture the can occur in
crystalline materials. Cleavage fracture occurs at lower temperatures and higher strain rates.
Below transition temperature, fracture requires only little energy and the metal behaves in a
brittle manner
Figure 8: brittle-ductile transition of steel (Anderson, 2005)
Cleavage in metals occurs by direct separation of crystallographic planes due to a simple
breaking of atomic bonds.
Figure 9: Cleavage spreading through grains (Rice J. , 1988)
Cleavage along a cube plane (100) of its unit cell causes relative flatness of a cleavage crack
within one grain as shown in figure above. Since neighbouring grains will have slightly
different orientations, the cleavage crack takes another direction at the boundary to continue
propagation on the preferred cleavage plane. The highly reflective flat cleavage facets
through the drains give the cleavage fracture a bright shiny appearance.
Under normal circumstances face-centred- cubic (FCC) crystal structures do not exhibit
cleavage fracture: extensive plastic deformation will always occur in these materials before
the cleavage stress is reached. Conversely, cleavage fracture does occur in body-centred-
cubic (bcc) and many hexagonal-closed-packed (hcp) structures.
II. Ductile fracture
Ductile materials undergo appreciable deformation before fracture. There is a permanent
deformation at the tip of the advancing crack that leaves distinct patterns in SEM images. The
surface of a ductile fracture tends to be perpendicular to the principal tensile stress, although
other components of stress can be factors. In ductile fracture, crystalline metals and ceramics
it is microscopically resolved shear stress that is operating to expand the tip of the crack.
Fracture surface is dull and fibrous. There has to be a lot of energy available to extend the
crack
2.4.2 CRACK TIP PLASTICITY
Recall that the LEFM applies to sharp cracks. The assumption of sharp cracks, however,
leads to the prediction of infinite stresses at the crack tip. On the other hand, stresses in real
materials are finite because the crack tip radius is finite. In addition, inelastic deformation,
e.g., plasticity in metals results in further reduction of crack tip stresses which is a
modification of the LEFM to account for the crack tip yielding.
The formation of the plastic zone depends on specimen or structural element configuration,
material properties and loading conditions. Most materials develop plastic strains when the
yield strength is exceeded in the region near a crack tip. Therefore, the amount of plastic
deformation is restricted by the surrounding material, which remains elastic during loading.
Theoretically, linear elastic stress analysis of sharp cracks predicts infinite stresses at the
crack tip.in inelastic deformation, such as plasticity in metals and crazing in polymers, leads
to relaxation of crack tip stresses caused by the yielding phenomenon at the crack tip. As a
result, a plastic zone is formed containing microstructural defects such as dislocations and
voids. Consequently, the local stresses are limited to the yield strength of the material. This
implies that the elastic stress analysis becomes sufficiently large and linear elastic fracture
mechanics (LEFM) is no longer useful for predicting the field equations.
The size of the plastic zone can be estimated when moderate crack tip yielding occurs. Thus,
the introduction of the plastic zone size as a correction parameter that accounts for plasticity
effects adjacent to the crack tip is vital in determining the effective stress intensity factor or a
corrected stress intensity factor.
Plastic develops is most common in materials subjected to an increase in the tensile stress
that causes local yielding at the crack tip.
Crack tip stresses reach infinite values (stress singularity) as the plastic zone size (r)
approaches zero: that is ij as r0. However, most engineering metallic materials are
subjected to an irreversible plastic deformation. If plastic deformation occurs, then the elastic
stresses are limited by yielding since stress singularity cannot occur, but stress relaxation
takes place within the plastic zone. This plastic deformation occurs in a small region and it is
called crack-tip plastic zone. A small plastic zone, (r < < a) is termed small-scale yielding.
Conversely, a large-scale yielding correspond to a large plastic zone, which occurs in ductile
materials in which r >> a. these suggest that the stress intensity factor within and outside the
boundary of the plastic zone are different in magnitude so that KI(plastic) > KI (elastic) must
be defined in terms of plastic stresses and displacement in order to characterize crack growth,
and subsequently ductile fracture.as a consequence of plastic deformation ahead of a crack
tip, the linear elastic fracture mechanics (LEFM) theory is limited to r >> a ; otherwise,
EPFM theory controls the fracture process due to large plastic zone size (r a).
Figure 10: Stress concentration at the crack tip (Mataga, 1987)
Fracture mechanics is divided into two major parts:
1. Linear elastic fracture – assumes the material to be elastic
2. Post- yield fracture mechanics – used in materials in which appreciable yielding has
taken place
2.4.3 LINEAR ELASTIC FRACTURE MECHANICS
a) Energy balance approach
This approach is based on Griffith’s study of an infinite plate of unit thickness crack of length
(2a). The plate is subjected to uniform tensile stress, .
Figure 11: infinite plate with uniaxial stress (Irwin, 1968)
Total energy U of the cracked plate may be written as:
U=U o+U a+U γ−F (24)
U o – Elastic energy of the loaded un-cracked plate.
U γ - Change in elastic surface energy caused by the formation of the
crack surfaces.
U a – Change in elastic energy caused by introducing the crack in the
plate in the plate.
F – Work performed by external forces.
Griffith used Ingli’s solution to show that the absolute value of U a is given by:
|U a|=πσ 2 a2 BE (25)
U γ=2 (2 aγ e )=4 aγ e (26)
For the case where no work is done i.e. Fix grip condition, F=0 and the change in elastic
energy U a is negative; there is a decrease in the elastic strain energy of the plate because it
losses stiffness and the load applied by the fixed grips with therefore drop.
Thus:
U=U o+U a+U γ (27)
Becomes
U=U o− πσ 2a2 BE
+4 aγ e (28
The equilibrium condition for crack extension is obtained by setting duda equal zero.
( dda )(−πσ 2 a2 B
E+4 aγe)=0 (29)
Figure 12: Variation of energy with crack length (Knott J. F., 1973)
From equilibrium condition
2 πσ 2 aBE
=4 γ e (30)
σ c=√ 2 Eγ eπaB
(31)
For a unit thickness plate
σ f =√ 2 Eγ sπa
(¿ plane stress) (32)
σ f =√ 2 Eγ sπ (1−υ2)a
(¿ plane strain) (33)
Griffith results may only be applicable to materials where non-linear effects, prior to fracture
are absent i.e. to ideally brittle materials.
Griffith’s experiments indicate that crack extension in brittle materials occurs when the
product σ √ a attains a constant critical value.
In-order to be applicable to engineering materials modifications were necessary to make
allowance for small plastic deformation at the crack tip. Irwin and Orowan independently
suggested that the Griffith theory could be modified and applied to both brittle materials and
metals that exhibit plastic deformation.
They suggested that energy of plastic distortion, p absorbed by the fracturing be added to the
surface energy.
Therefore:
2 πσ 2 aBE
=2(γ p+γ e) (34)
For relatively ductile materials γ p >>γ e and therefore equation (34) could be written as:
σ c=√ 2 Eγ pπa
(35)
Irwin considered the rate of strain energy release at the point of fracture. Fracture would
occur when the strain energy release rate reached a critical value. This critical value can be
regarded as a material property. The strain energy in an elastic body may be represented by:
U=P 2C2 (36)
Where C- is the compliance and P- force.
The strain energy release rate in relation to crack extension may then be represented by:
G= δUδa
=12
P2 δCδa (37)
Fracture occurred when G reached a critical value GC
Gc=πσ c2aE (38)
Experiments later showed that GC was a function of crack size, thickness, plate width and
dimensions of the plastic zone. The great drawback of the energy balance approach is that it’s
still limited to defining the conditions required for instability of an ideally sharp crack. And it
may not be used for analysis of sub-critical crack extension such as those due to fatigue,
stress corrosion, corrosion fatigue or creep. For these, a crack tip describing parameter is
required. This led to the stress intensity factor.
b) Stress intensity factor (K)
From the theory of elasticity, the determination of the stress and strain at a point in an elastic
body reduces to the solution of a stress function (Airy’s stress function). This stress function
was proposed by Westergaard in terms of complex variables for the determination of stresses
in the vicinity of the cracks.
Irwin used Westergaard solution to determine the stresses ahead of the crack. He showed that
the stresses in the vicinity of the crack tip take the form:
σ ij= K√2 πr
f ij (θ )+…
r and θ are polar co-ordinates of a point with respect to crack tip.
Crack surfaces moves relative to each other. These relative movements are called
displacement modes. Mostly sub-critical crack growth occurs in mode I displacement. Irwin
expressed the stress ahead of a sharp mode I crack as:
⌊σ xσ yσ z
⌋=K cos θ2
1√2 πr
⌊
1−sin θ2
sin 3 θ2
1+sin θ2 sin 3θ
2
sin θ2 cos 3 θ
2
⌋=terms of order rθ (39)
KI – stress intensity factor for mode I loading
We can now see the stresses at the crack tip can be characterized by a single parameter, K
and cracked components can be loaded to various levels of K just as un-cracked to various
stress level. The cracked component fails when K reaches a given level.
The stress intensity factor approach and the energy balance approach have been shown to be
equivalent.
Thus:
G c= K c2E
∈ plane stress (40)
G c= K c 2E (1−υ2 )
∈plane strain (41)
But
G c= K c2E
plane stress
Therefore
K c=σ c√πa (42)
The above indicate that the stress intensity factor must be linearly related to stress and
directly related to√ a. It also indicates that crack extension occurs when the product √ a
attains a constant critical value. The value of this constant can be determined experimentally
by measuring the fracture stress for a large plate that contains a through-thickness crack of
known length. This value can also be measured by using other specimen geometries or else
can be used to predict critical combination of stress and crack length in other geometries.
K for different specimen geometries can be determined from conventional elastic stress
analysis.
There are now several handbooks giving relationship between K and many types of cracked
bodies with different crack sizes, orientations and shapes and loading conditions.
As an unflawed component will fail by yielding or breaking when the stress applied to it
reaches a certain level (yield stress, ultimate yield strength), crack extension also occur in a
cracked member when the stress intensity factor reaches a critical value. As with yield stress,
this value is a material property and can be determine in the laboratory.
Irwin being the first to suggest that fracture will occur when the strain energy release rate
reaches a critical value GC. He also suggested the method for its measuring. In his
experiments he observed that dependent on strain rate, temperature and thickness
Figure 13: variation of KIC with specimen thickness
From the above figure, it’s clear that as B increases and plain strain conditions are
approached, KC approaches a constant value which is independent of geometry and hence is a
material property at the particular temperature and loading rate. KC is designated KIC and is
called the plane strain fracture toughness.
Furthermore, K is applicable to stable crack extension and does to some extend characterize
processes of sub-critical cracking like fatigue and stress corrosion.
σ F=√ E × constπa
(43)
The constant was found to be very much greater than the surface energy of the material.
The results led to Orowan and Irwin to suggest independently that the energy release rate in
the specimen was to large extent dissipated by producing plastic flow around the crack tip,
thus the critical fracture value was apparently much greater than 2.
It appeared that the amount of plastic work in the crack tip region which preceded unstable
crack propagation was independent of the initial crack length and was thus as a characteristic
a measure of the materials resistance to fracture as would be its surface energy if it were
breaking completely in an elastic manner.
Orowan re-wrote the Griffith relationship to give
σ c=√ [ E (2 γ+γ p )πa ]
Where p represents the energy expended in the plastic work necessary to produce unstable
crack propagation. Since it was found experimentally that p>>2 the equation can be
rewritten as:
σ c=√ Eγ pπa
Values of p could be determined directly from the fracture stresses of specimens containing
cracks of known lengths.
Irwin’s approach was similar to Orowan, but he took more pains to justify the use of a linear
elastic approach to relate fracture stress to crack length, even though crack-tip plasticity was
preceding fracture. He expressed his results in terms of the critical value of strain or potential
energy release rate at which unstable propagation occurred. This value, Gcrit, provided a
convenient parameter to include all supplementary energy-dissipating terms, such as plastic
flow, which could in turn produce heat or sound, in addition to the work required to fracture
the lattice. The constancy of Gcrit, and hence its use as a measure of a materials resistance to
fracture, depend on critically on experimental testing conditions, but, for situations where
small amounts of local plastic flow precede crack extension, which we shall call quasi-brittle
behaviour, Irwin’s parameter, Gcrit, became known as a material’s fracture toughness
The results available in Westergaard paper demonstrated that, the characteristic distribution
of elastic field quantities in the vicinity of a crack tip always resulted.
⌊σ xxσ yyσ zz
⌋=¿ (44)
The relationship between K and G may be generalized to cover the three basic loading
conditions
G I = k+18μ
K I 2
G II= ν+18 μ
K II 2
G III= K III 22 μ
However, under skew-symmetry and anti-plane loading conditions cracks tend to extend in
non-planar fashion. Hence, a criterion for fracture based on the attainment of critical values
of GII and GIII becomes difficult to justify.
Most practical cases are concerned with loading that is symmetric with respect to the crack
plane, in these cases only the variables with subscript I apply.
By means of tests on suitable shaped and loaded specimens it was possible to determine the
material property KIC or GIC by defining it as the value of KI or GI operative at the point of
fracture .it’s then possible to establish what flaws were tolerable in an engineering structure
under given conditions or to compare materials as to their utility in situations where fracture
is possible.
i. KIC Testing Two standard specimens are used i.e. single edge notched bend (SENB) and compact test
specimens. This method was first published in 1970 by the ASTM
a. ASTM standard notched bend specimen
Figure 14: ASTM Standard Notch bend specimen
b) ASTM standard compact tension specimen
Figure 15: ASTM Standard Compact tension specimen
The specimens must be fatigue pre-crack. The specimens contain starter notches to ensure
that the cracking occurs correctly. The purpose of pre-cracking and notching is to simulate an
ideal plane crack with essentially zero tip radius. There are several ASTM standards, but the
most frequently used is a chevron notch. The chevron notch forces fatigue cracking to initiate
at the centre of the specimen thickness and thereby increases chances of a symmetric crack
front.
s
Figure 16: Chevron notch crack starter
ii. Specimen size requirement
The accuracy of KIC depends on how well the stress intensity factor characterizes the
conditions of stress and strain immediately ahead of the tip of the fatigue pre-crack. After a
number of experiments works it has been found that the following dimensions should reach
some specified size requirement for nominal plane strain behaviour
a ≥ 2.5( K Icσ ys )2 (47)
B ≥2.5( K ICσ ys )2 (48)
W ≥ 5.0( K ICσ ys )2 (49)
Specification of a, B and W requires that the KIC value to be obtained must already be
known or at least estimated
2.4.4 POST- YIELD FRACTURE MECHANICS
This describes the fracture response of a material of limited ductility in the presence of a
defect. The most widely used parameters for fracture assessment after yielding are crack
opening displacement (COD), R-curves determination and J-integral concept.
a. Crack Opening Displacement
The COD approach was introduced by Wells [1961]. In regimes of fracture-dominant, the
stresses and strains in the vicinity of the crack or defect are responsible for failure. At the
crack tip, the stresses will always exceed the yield strength and plastic deformation will
occur. Hence failure is brought about by stresses and hence plastic strains exceeding certain
respective limits.
Wells argue that the stress at the crack tip always reaches the critical value (in purely elastic
case). If it’s so then it is the plastic strain in the crack tip region that controls fracture. Hence
it might be expected that at the onset of fracture this COD or t, has a characteristic critical
value for a particular material and therefore could be used as a fracture criterion.
Burdekin and Stone [1966] came up with an improved basis for the COD concept. They
evaluated the displacement at the crack tip as:
δt=8 σysaπE
ln sec ( πσ2 σys
)
(50)
δt=8 σysaπE [ 1
2 ( πσ2 σys )+ 1
12 ( πσ2σys )4 ] (51)
Taking only the 1st term and using the relation
EG=σ 2 πa (52)
δt=πσ 2 aEσys
= Gσys (53)
This is only valid forσ ≪σys, And for Irwin circular plastic zone analysis
δt=4π
K I 2Eσ ys (54)
This shows that in the elastic regime the COD approach is compatible with LEFM concepts,
however, the COD approach is not limited to LEFM range of applicability since occurrence
of crack tip plasticity is inherent to it.
The major drawbacks of the COD are the difficulty involved in measuring the COD and the
uncertainty as to whether the critical value of the COD corresponds to crack initiation or to
unstable crack growth.
The first drawback is overcome by inferring the COD from measurement taken at the mouth
of the notch. The second has not been resolved and in the standard available, it’s left to the
individual to specify which particular value is used.
i. The COD design curveThe basis of COD design curve was that critical COD values to provide measures of the
maximum permissible strains in the crack vicinity. The starting point for the approach of
obtaining COD design curve was the expression for δt in an infinite centre cracked plate.
∂ t=( K I 2Eσ ys )=( πσ 2a
Eσ ys) (55)
ii. The standard COD specimenThe standard COD test specimen conforms to the notched bend (SENB). The thickness of the
specimen B is specified to be 0.5W. B=W may also be used in special cases.
It’s impossible to measure the crack displacement directly at the crack tip. Instead a clip
gauge is used to measure the crack opening vg at the specimen surface. It is assumed that the
ligament b=W+ a act as a plastic hinge.
This shows that δt can be expressed as
∂ t= v g ×r × br× b+a+z
b. J-Integral
It is based on energy balance approach. It was first introduced by rice. In post-yield fracture
mechanics, a single parameter is sought which quantify the stress and strain fields ahead of a
sharp crack, such a parameter , the J-contour Integral has been developed and describes the
flow of energy into the tip region. By definition J may be expressed as:
J=∫❑
Wdy−∫❑
Ti ∂ ui∂ x
ds (56)
Where W =∫ σijdεij is the strain energy density,Ti are the components of surface fractions
over an arbitrary part of the surface of the body and Ui are the components of displacement s
is the distance along contour transverse counter-clockwise from the lower to the upper face
of the crack
Figure 17: Sketch of contour drawn around a crack
T ( dudx )ds= is the rate of work input form the stress field into area encircled by Г.
W=is the strain energy per unit volume due to loading.
U= displacement vector.
T=is the outward traction (stress) vector acting on the contour around the crack.
Г=path of the integral which encloses the crack
Since J is path independent, it can consequently be determined from a stress analysis where σ
and ε are established by finite element analysis around the contour enclosing the crack.
J integral can be interpreted as the potential energy difference between two identically loaded
specimens having different crack length.
Figure 18: Potential energy difference between loads
The equation that defines J is only applicable to non-linear elastic rather than to elastic-plastic
materials
J=G (57)
It is also demonstrated that for non-linear elastic material, J is path independent i.e it has the
same value irrespective of the contour chosen for its evaluation. From the laws of incremental
plasticity J is path dependent for materials, however numerical evidence suggest that the
value of the path dependence is small for mechanically applied forces.
For J-integral approach the assumption of non-linear elasticity is compatible with actual
deformation behaviour only if no unloading occurs in any part of the material. However at the
crack tip the material is unloaded when crack growth occurs.
Therefore, J is only applicable up to the beginning of crack extension and not for crack
growth.
By definition J=G for the linear elastic case. Thus the J integral concept is compatible with
LEFM.
Begley and Landes demonstrated the existence of a critical value of JC at which fracture will
occur with geometric independence.
Obtaining solutions for the J integral in actual specimens or components turns out to be
difficult. Thus necessitate use of finite element techniques. However, some simple
expressions have been developed for standard specimens.
i. The JIC specimens
The notched bend (SENB) and compact tension specimens are used. The J integral formulae
for these specimens are as follows.
Notched bend specimen (SENB)
J=2 U tBb
= 2U tB (W −a )
Compact tension specimen (CT)
J= 2UtB (W−a )
× f ( aW )
Where f ( aW )= 1+a
1+a2
And a=2√(ab )2+ a
b+ 1
2−2( a
b+ 1
2)
2.4.5 R-CURVES ANALYSIS
R-curve is a graphical representation of the resistance to the crack propagation R, versus the
crack length, a, in a material as a function of the actual or effective crack extension (∆a).
Here R represents the energy absorbed (dus) per increment of crack growth (da). It is
therefore given by the value of dus/das prior to unstable crack growth at the critical point R as
the same units as K which is the stress intensity factor.
For larger proportions of plane stress failure R is no longer independent of crack length.
A.S.T.M E561 provides specific instructions for specific instructions for measurement of R.
if curves of G (corrected to allow for size of the plastic zone ) are superimposed on the R-
curves then the point of instability occurs at the point of tangency between the two types of
curves. Irwin and Orowan suggested that Griffith theory can be modified to apply for both
ductile and brittle materials. This modification suggest that the R is the sum of the elastic
surface energy γe and plastic strain work,γp accompanying crack extension. (Popelar, 1985)
G= πσ 2aE
=2 ( γ e+γ p )=R=Gc (58)
For ductile materials γ p>> γethe surface energy can be neglected for plane strain only.
For plane stress and intermediate plane strain it is found that R is no longer constant
Figure 19: Slow stable crack growth in plane stress
As depicted in the above graph, there is constant crack growth; increase in stress increases the
crack extension. This process continues until a critical combination of stress and crack length
ac is reached at which point instability occurs.
Figure 20: The rising curve
In terms of energy balance approach the value of R is depicted as a rising curve with a
vertical segment corresponding to no crack growth at low stress and G levels.
If we consider the crack resistance for a thin sheet it is assumed that a slow stable crack
growth occurs under plane strain in the middle of the specimen thickness and is thus
independent of crack length. Many tests have shown that the form of the rising part of the R-
curve is also independent of crack length. (Hertzberg, 1989)
Thus we may expect the R-curves to be independent of the initial crack length a o i.e. an
invariant R-curve may be placed anywhere along the horizontal axis of a (G, R) crack length
diagram as in figure below.
Figure 21: Invariant R-curves and the points of instability
Crack initiation is independent of initial crack length. Instability depends on ao, a longer ao
results in more stable crack growth and a higher G value at instability. In comparison with the
plane strain situation it may thus be stated that instability definitely depends on total crack
length a (=ao+Δa)
Although no definitive analysis of the rising R-curves exists, a working hypothesis has been
given by Kraft, Sullivan and Boyle. The hypothesis models the R-curve behaviour under-
intermediate plane stress-plane-strain conditions. Kraft et al presuppose that in plane strain
the plastic deformation energy necessary for crack extension is related to the area of the
newly created crack surface, but in plane stress the plastic energy is related to the volume
contained by the plane stress(45degree) crack surfaces and their mirror images.
For a crack growth increment da the total energy consumption: (Anderson T.L., 1991)
dw=( dW sdA )B (1−S ) da+(dW p
dV )( B 2 S 22 )da (59)
Where dWsdA – energy consumption rate
dWpdV - Energy consumption rate for plane stress per unit volume
The crack resistance, R is given by:
R= 1B
dWda
=dW sdA
(1−S )+ dW pdV
( BS 22
) (60)
Experimentally it has been shown that dW pdV
≫ dW sdA so that from the above it’s evident that
as soon as shear lips (slant fracture) start to form the value of R will show a sharp increase.
This explains the generality of the rising R-curve.
In the literature and in practice R-curve are not considered in terms of G and R but instead
the stress intensity factors KG and KR are used. This because the stress intensity factor
concept has found widespread application and energy balance parameters G and R may be
simply converted to stress intensities via relation K I=√E ' G
2.4.6 DETERMINATION OF R-CURVES
R-curves can be determined by either two experimental techniques:
a) Load control
This involves rising load test with crack driving force (KG) curves. Under rising loading
conditions the crack extension gradually to the maximum of ∆a when unstable crack growth
occurs at KC which is determined by the tangency point between the KR-curves and one of the
lines representing a crack driving force curve, K G=f ( p ,√a , aW
). This testing method is
capable only of obtaining that portion of the R-curve up to KR=KC when instability occurs.
b) Displacement control
This results in negatively sloped crack driving force curves. The specimen is loaded by a
wedge, which must be progressively further inserted in order to obtain greater displacement
and further crack growth. For each displacement the crack arrests when the crack driving
force intersects the R-curve. because there can be no tangency to the developing crack growth
resistance, KR, the crack tends to remain stable up to a plateau level i.e. the entire R-curve
can be obtained.
i. Recommended specimens for R- curve testing
The ASTM recommended three types of specimens:
a. The centre cracked tension specimen(CCT)
Figure 22: Centre crack tension specimen
b. The compact specimen(CS)
c. Crack line wedge-loaded specimens(CLWL)
CLAMPING
CLAMPING
2a
W
L
Figure 23: Crack line wedge-loaded specimen
The first two types of specimens are tested under load control (rising KG- curves) while the
crack line wedge loaded specimen may be used for displacement control tests. The specimens
must be fatigue pre-cracked unless it can be shown the machined root radius effectively
simulates the sharpness of a fatigue pre-crack. For CCT specimen the machined notch must
be 0.3-0.5 of W with fatigue crack not less than 1.3 mm in length. For CS and CLWL
specimens the starter notch configuration is basically similar to that required for KIC testing ,
but owing to lesser thickness a chevron notch crack starter may not be necessary to obtain a
symmetrical crack front i.e. a straight through electric discharge machined (EDM) slot will
often suffice .The initial crack length must be between 0.35-0.45 of W
ii. Specimen Size
The specimen size is based the requirement that the un-cracked ligaments (W-2a) or (W-a)
must be predominantly elastic at all values of applied load. For CCT specimen the net section
stress based on the effective crack size 2(ao+∆a+ry) must be less than the yield stress, ry is the
radius of the plastic zone
0.1576W 0.303W
D
a
W
0.6W
1.2W
SAW CUT
EDM NOTCH TIP
FATIGUE CRACK
For CS and CLWL specimens the condition that the un-cracked ligaments must be
predominantly elastic is given by the more or less empirical relation as shown below
W −(ao+∆ a+r y)≥ 4π ( K max
σ ys )2 Where Kmax – maximum stress
3. CHAPTER THREE
3.1 METHODOLOGY AND EXPERIMENTAL RESULTS ANALYSIS
3.1.1 APPARATUS
Fatigue tester:
Travelling microscope
Strain gauge indicators
Tensile and Compressive Testing Machine
Load cells:
Figure 24: Tensile and Compressive testing machine
Figure 25: Standard Specimen
3.1.2 REPAIR OF THE FATIGUE TESTING MACHINE
The fatigue testing machine consisted of a motor with an eccentric arm, running at 1420 rpm.
The eccentricity of the arm can be varied from zero to 45 mm. The eccentric converts the
motors rotatory motion into oscillatory motion of the force. The motor was fixed to the base
that was bolted to the floor using rawl bolts. The motor was aligned by tightening the bolts
that were attached it to the base. The shaft that connected the motor to the fly wheel was
tapered and connected with the key. This shaft was removed, filled and machined several
times.
A new key was also machined. The plate that was attached to the eccentricity arm was also
loose. This plate was welded on the eccentricity arm. Its surface was then grinded to ensure it
was flat. The bearing that connects the shaft and the fly wheel was replaced twice. To ensure
that the bearing does not cess easily, a large load should be avoided.
There was also a transmitting lever pivoted at a pivot. The lever amplifies the force from the
motor. The amplification can be varied in steps by changing the position of the pivot. A
universal joint is provided to reduce the horizontal components of the motion so that only
axial forces are transmitted to the specimen. The universal joint is connected to the lower
clevis. The universal joint and the clevis were oiled to reduce friction. A multi pin gripping
assembly was used and grips are joined to the clevis by a pin. Spaces are used to fill the gap
of the clevis. The upper clevis is threaded at its top end to allow fastening by means of a nut.
The load was measured by a strain gauge based load cell in line with the specimen. The load
cell was fast set to zero. This was done by first removing the load cell and ensuring that it
was clean. (Rading, 1984)
The test rig components were also prepared. With all this done the machine was able to pre-
crack 10 specimens.
3.1.3 EXPERIMENTAL PROCEDURE
1. The standard specimen where prepared from the material selected for testing(mild
steel of thickness 1.3 mm )
2. A hole was drilled at the centre of the specimen
3. Specimen development started by cutting a square plate of 300 mm by 300 mm and
slots cut on the specimen as indicated in the specimen drawing above. A 10mm
diameter hole was drilled at the centre of the plate and an initial crack of 50 mm
length was cut with a junior saw.
4. Using various grades of the emery cloth, the area around the crack was polished to
enhance visibility when monitoring the crack growth.
5. The specimen is loaded onto a fatigue tester which is for pre-cracking of the specimen
by fatigue.
6. After pre-cracking the specimen were then mounted onto a rig that facilitates biaxial
loading.
7. Horizontal stress was applied by tightening a lock nut against the load cell from which
the strain was read using the digital strain indicator. From the calibration curve of the
horizontal load cell, the load corresponding to the strain was read from which the
stress could be determined. The horizontal stress was kept constant.
8. The specimen was then loaded onto a and compression testing machine
9. The cracks on the specimens were made ranging from a small size (40 mm) to larger
size (109 mm). (This is for this case but the size of the hole can be varied so long as it
meets the ASTM specimen recommended standards)
10. The load was then applied on the specimen and the crack length and the load applied
were measured and recorded
11. The load that caused the initial crack extension was noted and recorded and the crack
extension was measured using a travelling microscope.
12. All the readings of load and crack length were taken for all the test specimens with
varying initial crack lengths
13. From the above data, the values of KG and KR were calculated.
14. The KG and KR values were plotted to develop the R- curves from which the critical
stress intensity factors were determined for the test specimens.
15. The R-curves plotted gave us an estimate of the plane strain fracture toughness KIC of
the material since initial crack extension can be assumed to take place under plane
stress conditions. Hence the K value at the point of crack extension could be an
estimate of the KIC of the material.
16. From the R-curves, the variation of the fracture toughness KC (in plane strain) with
initial crack length was determine
17. From the above experiment the fracture toughness of thin steel was determined and
can be used to design and evaluate the safety of structures made from the same
material with the same thickness and near similar loading conditions.
3.2 RESULTS AND ANALYSIS
3.2.1 LOAD CELLS CALIBRATION
The load against strain were measured and recorded for horizontal load cell in table
Table 4
Load (N) Strain 0 0 342 6 1700 11 2832 27 4680 51 6550 73 8540 93 11230 116 13210 135 15750 158 18250 180 20650 202 22950 221 25060 241 27936 266 31850 301 35340 332 37200 349 38320 358 40600 380
0 5000 10000 15000 20000 25000 30000 35000 40000 450000
50
100
150
200
250
300
350
400
Horizontal Calibration Curve
Strain
Loa
d (N
)
Graph 1: Horizontal Calibration Curves
The load versus strain was measured and recorded for the vertical load cell and is given
in table below
Table 5
Load Strain 2200 0 3850 4 5660 12 7098 20 8270 27 10440 40 13036 55 15390 69 18962 90 22280 108 25180 124 30067 150 33580 168 36572 183 40450 203 44230 222 47215 237 50870 256 53120 267 61185 309 66880 338 73140 370 8000 403
0 10000 20000 30000 40000 50000 60000 70000 80000 900000
50
100
150
200
250
300
350
400
450
Vertical calibration curve
Strain
Loa
d (N
)
Graph 2: Vertical calibration curve
3.2.2 SAMPLE CALCULATIONS
a) Horizontal stress calculation:
The specimen thickness = 1.3 mm
Specimen width = 190 mm Specimencross−sectional area=2.47× 10−4 m 2
From the strain gauge at 75 the load is 6620 N
Horizontal Stress= ForceArea
= 66201.3 ×190 ×10−6
=26.8 MPa
f ( aw )=√sec( πa
w )For example when a=20mm
¿1.028244
The various values of a, f (a/w) were calculated
and tabulated as shown the table below
Table 6
a) Calculation of crack driving force, KG
at fracture
K G= f ( aw )σ c√πa
Half crack length
(a) cm
f(a/w)
0 1
0.1 1.000068
0.2 1.000273
0.3 1.000616
0.4 1.001095
0.5 1.001712
0.6 1.002468
0.7 1.003362
0.8 1.004397
0.9 1.005572
1.0 1.006890
1.1 1.008351
1.2 1.009957
1.3 1.011709
1.4 1.013610
1.5 1.015660
1.6 1.078630
1.7 1.020220
1.8 1.022735
1.9 1.025408
2.0 1.028244
2.1 1.031246
2.2 1.034416
2.3 1.037759
2.4 1.041277
2.5 1.044975
2.6 1.048858
2.7 1.052928
2.8 1.057193
2.9 1.062656
3.0 1.066322
3.1 1.071199
3.2 1.076292
3.3 1.081607
3.4 1.087152
3.5 1.092934
3.6 1.098962
3.7 1.105242
3.8 1.111686
For example at crack length a=51 mm
The stress at fracture is given as:
σ c= 439661.3 ×190 ×10−6 =178 MPa
Using this value of c, the crack driving force, KG was calculated for the various values of the
crack length until fracture, i.e. for a=20 mm, f (a/w) = 1.028244
K G=1.028244 × 178× 106√ π × 2−2=45 MPa
Using the same steps tables of KG were generated for various values of crack length for all
specimens.
a) Sample calculation for crack resisting force, KR
For a load of 45637 N the crack length is 110 mm f (a/w) corresponding to this crack length
is 1.27597
σ=45637Area
= 456371.3 ×190 ×10−6 =184.7 MPa
K R=1.27597 ×184.76 × 106×√π ×5.5 ×1 0−2=98 MPa/m32
3.2.3 MEASURED AND CALCULATED RESULTS
The tables that follow show measured results and calculated results
Table 7
Specimen with initial crack length of 4.0cm and a Horizontal stress of 27.6 MPa
LOAD (N) CRACK -LENGTH (2a) cm
HALF CRACK LENGTH a(cm)
CRACK DRIVING FORCE,KR (MN/m3/2)
0 4.0 2.0 0 75908 4.0 2.0 60 78934 4.4 2.2 75 79979 5.0 2.5 87 81065 5.4 2.7 94 82030 6.0 3.0 106 81535 6.8 3.4 118 81017 7.2 3.6 122 80466 7.6 3.8 126 79557 8.0 4.0 130 78886 8.6 4.3 136 82688 9.4 4.7 143 75482 10.2 5.1 150
Table 8
HALF CRACK LENGTH(a) Cm
CRACK DRIVING FORCE, KG
(MN/m3/2) 0 0
2.0 403.0 604.0 805.0 986.0 112
6.1 166
0 1 2 3 4 5 60
20
40
60
80
100
120
140
160
A GRAPH OF KG, KR AGAINST HALF CRACK LENGTH (a) cm
Half Crack Length (a) cm
KG
, KR
(MN
/m3/
2)
KG
KR
Graph 3: specimen with initial crack length (a) 4cm
Table 9
Specimen with initial crack length of 10cm and a Horizontal stress of 26.8 MPa
LOAD (N) CRACK-LENGTH (2a) cm
HALF-CRACK LENGTH (a) cm
CRACK RESTISTING FORCE, KR
(MN/m3/2) 0 10 5.0 0 45417 10 5.0 80 47491 10.4 5.2 94 48394 10.8 5.4 100 50263 11.2 5.6 106 50489 12.0 6.0 114 48220 12.8 6.4 123 45942 13.4 6.7 128
Table 10
HALF CRACK LENGTH
(a) cm
CRACK DRIVING FORCE,KG
(MN/m3/2)
0 01.0 202.0 403.0 574.0 755.0 956.0 1176.7 128
0 1 2 3 4 5 6 7 80
20
40
60
80
100
120
140
A GRAPH OF KG,KR AGAINST HALF CRACK LENGTH (a) cm
Half Crack Length (a) cm
KG
,KR
(MN
/m3/
2)
KR
Graph 4: Specimen with initial crack length (a) 5cm
KG
Table 11
Specimen with initial crack length 5.8 cm and a Horizontal stress of 29.6 MPa
LOAD (N) CRACK LENGTH (2a) cm
HALF CRACK LENGTH (a) cm
CRACK DRIVING FORCE ,KR
(MN/m3/2) 0 5.8 2.9 0 60356 5.8 2.9 79 62342 6.1 3.05 90 66668 6.6 3.3 102 74288 7.0 3.5 109 77159 7.6 3.8 120 76530 8.2 4.1 127 76932 8.8 4.4 134 74898 9.4 4.7 138
Table 12
HALF CRACK LENGTH (a) cm
CRACK DRIVING FORCE,KG
(MN/m3/2)
0 02.0 354.0 656.0 978.0 1269.4 138
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
20
40
60
80
100
120
140
160
A GRAPH KG,KR AGAINST HALF CRACK LENGTH (a) cm
Half Crack Length (a) cm
KG
,KR
(MN
/m3/
2)
KG
KR
Graph 5: Specimen with initial crack length (2a) 5.8cm
Table 13
Specimen with initial crack length 7.6cm and a Horizontal stress of 31.8 MPa
LOAD (N) CRACK-
LENGTH (2a) cm
HALF CRACK LENGTH(a) cm
CRACK DRIVING FORCE, KR
(MN/m3/2) 0 7.6 3.8 0 59567 7.6 3.8 80 60659 8.0 4.0 98 59606 8.4 4.2 106 62005 8.8 4.4 114 66362 9.2 4.6 120 65859 9.8 4.9 126 66162 10.0 5.0 129 66443 10.2 5.1 132
Table 14
HALF CRACK LENGTH
(a) cm
CRACK DRIVING FORCE,KG
(MN/m3/2)
0 01.0 252.0 563.0 824.0 1075.0 1295.1 132
0 1 2 3 4 5 60
20
40
60
80
100
120
140
A GRAPH OF KG,KR AGAINST HALF CRACK LENGTH (a)cm
Half Crack Length (a)cm
KG
,KR
(MN
/m3/
2)
KR
Graph 6: Specimen with initial crack (2a) 7.6 cm
Table 15
KG
Specimen with initial crack length of 10.9cm and a Horizontal stress of 28.8 MPa
LOAD (N)
CRACK LENGTH (2a) cm
HALF-CRACK LENGTH(a) cm
CRACK DRIVING FORCE,KR (MN/m3/2)
0 10.9 5.45 0 39263 10.9 5.45 80 41241 11.3 5.65 90 42074 11.8 5.90 96 41605 12.2 6.10 102 39347 12.8 6.40 111 45333 13.4 6.70 122 43472 14.0 7.00 130
Table 16
CRACK- LENGTH (2a) cm
CRACK DRIVING FORCE, KG
(MN/m3/2)0 02 204 386 548 7010 8612 10514 130
0 1 2 3 4 5 6 7 80
20
40
60
80
100
120
140
A GRAPH OF KG,KR AGAINST HALF CRACK LENGTH (a) cm
Half Crack Length (a) cm
KG
, KR
(MN
/m3/
2) KR
Graph 7: Specimen with initial crack length (2a) 10.9 cm
KG
Table 17
Specimen with initial crack length of 9.0cm and a Horizontal stress of 33.6 MPa
LOAD(N)CRACK-LENGTH (2a) cm
HALF CRACK LENGTH (a) cm
CRACK DRIVING FORCE, KR
(MN/m3/2) 0 9.0 4.5 0 48655 9.0 4.5 80 49932 9.4 4.7 100 55405 9.8 4.9 112 59882 10.2 5.1 121 60046 10.6 5.3 124 60540 11.0 5.5 128 58539 11.6 5.8 134
Table 18
HALF CRACK LENGTH
(a) cm
CRACK DRIVING FORCE,KG
(MN/m3/2)
0 01.0 292.0 553.0 774.0 985.0 1205.8 134
0 1 2 3 4 5 6 70
20
40
60
80
100
120
140
160
A GRAPH OF KG, KR AGAINST HALF CRACK LENGTH (a)cm
Half Crack Length (a)cm
KG
,KR
(MN
/m3/
2)
KR
Graph 8: Specimen with initial crack length (2a) 9.0 cm
KG
Table 19
Specimen with initial crack length of 10.2 cm and a Horizontal stress of 27.5 MPa
LOAD(N) CRACK-LENGTH (2a) cm
HALF CRACK LENGTH (a) cm
CRACK RESISTING FORCE KR (MN/m3/2)
0 10.2 5.1 0 42678 10.2 5.1 80 43097 10.6 5.3 91 45637 11.0 5.5 98 46092 11.4 5.7 103 45712 11.8 5.9 108 46282 12.0 6.0 110 45756 12.6 6.3 116.5 44212 13.2 6.6 123.3 44595 13.6 6.8 127 43966 14.0 7.0 132
Table 20 CRACK LENGTH (2a) cm
CRACK DRIVING FORCE,KG
(MN/m3/2)0 02 214 406 588 7510 9312 11214 132
0 1 2 3 4 5 6 7 80
20
40
60
80
100
120
140
GRAPH OF KR, KG AGAINST CRACK-LENGTH (a) cm
Half Crack length (a) cm
KR
,KG
(MN
/m3/
2)
KR
Graph 9: specimen with initial crack length (2a) 10.2cm
KG
Table 21
Specimen with initial crack length of 4.6 cm and a Horizontal stress of 32.4 MPa
LOAD (N)CRACK LENGTH (2a) cm
HALF CRACK LENGTH (a) cm
CRACK RESISTING FORCE KR (MN/m3/2)
0 4.6 2.3 0 65890 4.6 2.3 65 66572 5.0 2.5 77 67473 5.6 2.8 85 68534 6.0 3.0 90 67907 6.4 3.2 94 68036 6.8 3.4 98 66736 7.0 3.5 100 68154 7.6 3.8 105 66871 8.2 4.1 110 66812 9.6 4.8 120 65208 10.2 5.1 125
Table 22
CRACK LENGTH (2a) cm
CRACK DRIVING FORCE,KG
(MN/m3/2)0 0
1.0 352.0 653.0 924.0 1105.0 1235.1 125
0 1 2 3 4 5 60
20
40
60
80
100
120
140
A GRAPH OF KG,KR AGAIST HALF CRACK LENGTH (a) cm
Half Crack Length (a) cm
KG
,KR
(MN
m3/
2)
KR
Graph 10: Specimen with initial crack length (2a) 4.6 cm
KG
Table 23
Specimen with initial crack length of 8.0cm and a Horizontal stress of 31.3 MPa
LOAD (N)CRACK LENGTH (2a) cm
HALF CRACK LENGTH (a) cm
CRACK RESISTING FORCE, KR (MN/m3/2)
0 8.0 4.0 048590 8.0 4.0 7453271 8.4 4.2 9054878 9.0 4.5 9956445 9.4 4.7 10456417 10.0 5.0 11057266 10.4 5.2 11455883 11.0 5.5 11854814 11.6 5.8 12453352 12.2 6.1 130
Table 24
CRACK LENGTH (2a)cm
CRACK DRIVING FORCE,KG
(MN/m3/2)0 0
1.0 282.0 503.0 704.0 905.0 1126.0 1286.1 130
0 1 2 3 4 5 6 70
20
40
60
80
100
120
140
A GRAPH OF KG,KR AGAINST HALF CRACK LENGTH (a) cm
Half crack length (a) cm
KG
,KR
(MN
/m3/
2)
KR
Graph 11: Specimen with initial crack length (2a) 8.0cm
KG
Biaxiallity
Table 25
crack length, (2a) cm
Kc MPam^(1/2)
)
σx(Mpa) σC (Mpa) Biaxiallity parameter
4.0 120 27.6 305 0.0904924.6 121 32.4 256 0.12656255.8 120 29.6 269 0.11003717.6 120 31.8 216 0.14722228.0 125 31.3 178 0.1758427
9.0 122 33.6 237 0.1417722 10.0 124 26.8 186 0.1440860 10.2 124 27.5 178 0.1544944 10.9 122 28.8 176 0.1636363
4 CHAPTER FOUR
IV.0 DISCUSSION
From the data obtained in the experiment, it was possible to draw the R-curve for the each
tested specimen. This curves were obtained after calculating the values for crack driving
force (KG) and crack resisting force (KR).The crack growth was stable on the specimen as the
load was increased. This enables us to measure and record the crack length at any load.
The maximum load was that cause fracture was noted, and this load was latter used in
calculating the crack driving force (KG) and the crack resisting force (KR). When the values
for KG and KR were calculated as shown in section [3.1.4], R-curves were then drawn. These
curves were obtained by drawing a graph of KG and KR against Half crack length.
The R-curves are as shown in Graph 3 to Graph 11.
The initial crack was constant as the stress is increased. It begins to extend at a certain stress.
However when this stress is maintained the crack does not increase further. But when the
stress was increased, then it results in additional crack extension. The process of increasing
the stress accompanied by a stable crack growth continues until a critical combination of
stress,C, and crack length, ac, is reached, at which point of instability occurs.
Instability is thus preceded by a slow stable crack growth in the specimens. At the stresses
lower than the fracture stress, crack extension begins but the KR remains equal to KG since the
crack is still stable. This is indicated by the fact that from the R-curves KG intersects the R-
curve at G=R. the stable crack condition is maintained until fracture stress C and critical
initial crack length, aC are reached. Beyond this point, KG > KR, as shown in the Graphs and
instability occurs
The point where the KG and KR are equal is termed the fracture toughness of the material, KC.
From the graphs the fracture toughness of each specimen is shown below
Table 26
As shown in the table above the fracture toughness of the material was found to range from
120 MN/m3/2 to 125 MN/m3/2. The variation was due to the variance in the initial crack
lengths. Some of the cracks were not purely horizontal.
Therefore, it is approximated that the average value of 122.5 MN/m3/2 can be used for design
purposes.
The point of the initial crack extension was found to range from 78 MN/m3/2 to 82 MN/m3/2,
with most values taking a value of 80 MN/m3/2. However this value is supposed to be constant
and independent of specimen thickness for a particular material. This inconsistency was
brought by the fact the some cracks where not purely horizontal. Though some initial saw
cuts were horizontal, after loading the specimen into the fatigue tester, some of the sharp
cracks did not propagate purely horizontally.
It is evident from the fracture toughness that for this specimen (Mild steel 1.3 mm thickness)
is dependent on the applied horizontal stress, x. as the fracture toughness increases with
increase in the horizontal stress.
Initial Half Crack Length(a) cm
Fracture Toughness (KIC) (MN/m3/2)
4.0 120 4.6 121 5.8 120 7.6 120 8.0 125 9.0 122 10 124 10.2 124 10.9 122
From the previous works it has been found that the fracture toughness is a function of
specimen thickness. Thinner specimens have higher values of fracture toughness and can
consequently exhibit slower crack growth since the point of crack extension is constant.
From the most specimens with small initial crack lengths (40 to 76) mm it was imperative
that the load at initial stages of the crack extension was high, the load keeps increasing until a
point where it starts decreasing. This is because at the initial stages of crack propagation the
crack is still stable i.e. the material still has a high crack resisting force and this calls for a
larger load to propagate the crack. The load starts decreasing at the point where the crack is
no longer stable; implying that the force required propagating the crack is smaller as the
crack propagates further.
Specimens with relatively longer crack length (80 to 109) mm have the load decreasing right
after the initial extension, this is because at this crack length the crack is already unstable, and
hence the material has a small crack resisting force. A small force is therefore required to
propagate the crack. However the force that initiates the crack was independent of the initial
crack length. All specimens almost had equal initial crack initiation force.
The crack instability was also found to depend on the initial crack length. A longer initial
crack length, results in more stable crack growth and a higher value of KG at instability.
It was also found that the rate of crack propagation at the initial stages of the crack extension
was quite low, this as a result of high crack resisting force. However, after instability has
been attained the crack growth rate increases until the specimen fractures completely.
Some the R-curves obtained from the data from our experiment did not meet the required
standards. This was due to the errors encountered in the experiment. These errors were:
i. Poor crack propagation motoring method: the travelling microscope could
not view the crack clearly.
ii. There was too much vibration on the fatigue tester machine, this also
hindered appropriate monitoring of the crack growth.
iii. In some specimen the crack was not perfectly horizontal.
The biaxiallity was also determined as shown in table 23. The fracture toughness increases
with increase in in the biaxiallity. This is so because with an increase in biaxiallity, the
fracture load increases and consequently the fracture stress.
4.1 CONCLUSION AND RECOMMENDATIONS
4.1.1 CONCLUSIONS
The average fracture toughness of the material tested [mild steel of 1.3 mm thickness] was
found to be 122.5 MN/m3/2. This is reasonable compared to the previous results where the
fracture toughness of mild steel of 1.18 mm thickness was found to be approximately 150
MN/m3/2. This further shows that the fracture toughness of mild steel varies with material
thickness. Thinner specimens manifest higher values of K1C and consequently exhibit slow
stable crack growth. From the results we can conclude that fracture toughness is high for the
small initial crack length of the specimen and decreases as the initial crack length increases.
The fracture toughness also found too dependent on the biaxiallity.
4.1.2 RECOMMENDATION
This experiment can generate more accurate results when the difficulties encountered [3.4.0]
are eliminated. This can be done by:
i. Putting dampers on the fatigue testing machine to reduce vibrations
ii. Using and electronic travelling microscope to monitor the crack length
iii. A new standard testing rig that the specimen is mounted on to enable biaxial
loading, should be prepared to enhance equal stress distribution on the specimen.
To get sufficient data, data collection should start at the mid of the first semester and one
should first ensure that the fatigue testing machine is in good condition and that it has the
capacity to pre-crack the specimen.
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