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


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



(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


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


(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


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



KG

, KR

(MN

/m3/

2) KR

Graph 7: Specimen with initial crack length (2a) 10.9 cm

KG

Table 17


LOAD(N)CRACK-LENGTH (2a) 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


(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


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


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


(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


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


(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


KG

,KR

(MN

m3/

2)

KR


KG

Table 23


LOAD (N)CRACK LENGTH (2a) 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


(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


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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deformation, fracture, and fatigue.

4. Dressman. (2005). Theories of failure and the failure theories . 8-16.

5. G.E, D. (1986). Mechanical metallurgy.

6. Griffith, J. (1961). The theory of transition-metal ions.

7. Irwin, G. R. (1968). Engineering Fracture Mechanics.

8. Jilin , Y., & Jonesj, N. (1991). Further experimental investigations on the failure of clamped

beams under impact loads. International Journal of Solids and Structures 27.9, 1113-1137.

9. Jolly. R. H., &. S. (1997). A mohr circle construction for opening of pre-existing fracture.

Journal of Structural Geology, 887-892.

10. Knott. (1973). Fundamentals of fracture mechanics.

11. Kupfer, H. H. (1969). Behavior of materials under biaxial stresses. ACI journal proceedings .

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Chemistry of Solids , 985-1005.

13. Nyakamba, D. &. (2009). Fracture of thin sheets due to biaxial loading .

14. Peterson, R. (1953). Stress concentration design factors.

15. Rading, G. .. (1984). Fatigue crack growth on mild steel. Msc Thesis

16. Rice, J. &. (1968). Plane strain deformation near a crack tip in a power-law hardening

material. Journal of the Mechanics and Physics of Solids, 1-12.

17. Rice, J. (1988). Elastic Fracture Mechanics concept for interfacial cracks. Journal of applied

mechanics, 98-103.

18. Sasaki, N. &. (1996). Stress-strain curve and Young's modulus of a collagen molecule as

determined by the X-ray diffraction Technique.

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