CONTINUUM MECHANICS APPROACHES TO THE STUDY OF …

CONTINUUM MECHANICS

APPROACHES TO THE STUDY

OF FRACTURE AND FATIGUE

IN METALS

A thesis submitted in fulfilment of the

requirements for the award of the degree

DOCTOR OF PHILOSOPHY

From

UNIVERSITY OF WOLLONGONG

By

Bradley Smyth Glass

B.E. (Mechanical)

FACULTY OF ENGINEERING

2004

I

CERTIFICATION

I, Bradley Smyth Glass, declare that this thesis, submitted in fulfilment of the

requirements for the award of Doctor of Philosophy, in the Faculty of Engineering,

University of Wollongong, is wholly my own work unless otherwise referenced or

acknowledged. The document has not been submitted for qualification at any other

academic institution.

Bradley Smyth Glass

10 October 2004

II

ABSTRACT

This thesis investigates continuum mechanics based means of metal failure

assessment. A basic science approach was employed throughout the study to examine

the fundamental relationships responsible for metal failure. The extension of

previously existing continuum mechanics based theories to encompass a wider range

of application was considered in this thesis. Research was conducted as two separate

studies which examine specific aspects of the metal failure spectrum, namely failure

due to monotonic loading, and fatigue failure due to cyclic loading.

The failure due to monotonic loading research was conducted to examine the

influence of hydrostatic stress on metal ductility. A fundamental relationship in the

form of a monotonic failure criterion was proposed based on a relationship between

equivalent plastic fracture strain and hydrostatic stress. An experimental program

incorporating uniaxial tensile testing of notched specimens was conducted to examine

the proposed relationship for the hydrostatic tensile stress range. Finite element

analyses were produced to confirm the mechanical properties and obtain the stress-

strain state present at specimen failure. A good correlation was established between

the load-displacement results obtained from experiment and finite element analysis,

providing confirmation of the stress-strain data. The stress-strain results confirmed the

existence of a relationship between hydrostatic stress and ductility in the form of a

monotonically decreasing value of equivalent plastic fracture strain with increasing

hydrostatic tensile stress. The relationship determined was in accordance with the

trend indicated by various researchers for the hydrostatic compressive stress range.

The potential application of such a criterion to finite element methods was amply

demonstrated from this research.

The fatigue failure due to cyclic loading research examined the application of energy

based methods to fatigue life characterisation. Based on the hypothesis that

irreversible damage may be attributed entirely to plastic deformation, the application

of the plastic strain energy approach to the entire fatigue life spectrum was pursued.

For application to high cycle fatigue, a thermodynamic approach was developed to

Abstract

III

allow plastic strain energy determination beyond the range of application of

conventional mechanical measurement. Thermodynamic models consisting of varying

degrees of free surface contribution to heat dissipation were developed as possible

representations of the high cycle fatigue damage process. An experimental program

was conducted incorporating mechanical and thermodynamic means of measurement.

Thermodynamic measurement was achieved via an experimental apparatus

incorporating precision temperature measurement and thermal isolation at the

specimen surface. Assuming an appropriate thermodynamic model, a finite difference

analysis allowed a quantitative determination of plastic strain energy. Close

agreement was indicated from comparison of the low cycle fatigue plastic strain

energy results obtained from mechanical and thermodynamic measurement. A

qualitative determination of plastic strain energy for high cycle fatigue was achieved,

subject to confirmation of the thermodynamic model. The qualitative assessment

verified the existence of measurable plastic strain energy during high cycle fatigue.

IV

ACKNOWLEDGEMENTS

As the author of this thesis, I would like to personally thank the following people for

their contribution, support and assistance throughout the course of my postgraduate

research.

Firstly, I would like to thank Professor Michael P West for his assistance and support

as supervisor of my postgraduate research project. His knowledge, experience and

creative input have been a major contributing factor to the direction of this research.

Secondly, I would like to thank Professor Richard E Collins of the Department of

Physics, University of Sydney, for his technical advice and loan of the Julabo water

bath, along with Paul Stathers and Ken Short of the Materials and Engineering

Science Division, Australian Nuclear Science and Technology Organisation, the

Cooperative Research Centre for Welded Structures, and all of the academic and

technical staff of the Faculty of Engineering, University of Wollongong who have

rendered their technical assistance throughout the course of my studies.

Thirdly, I would like to thank fellow postgraduate students Peter Sorrenson, Benjamin

Lake and Geoffrey Slater for their friendship and support over these years.

Finally, I would like to thank my family and my beautiful wife Parisa. Their love,

patience and support throughout this ordeal have been a major contributing factor in

the success of my postgraduate research. In time, I hope I can give back to them all

they have given me.

V

ABBREVIATIONS

1-D One dimensional

2-D Two dimensional

3-D Three dimensional

CTOD Crack tip opening displacement

EPFM Elastic-plastic fracture mechanics

FEA Finite element analysis

LEFM Linear-elastic fracture mechanics

RKR Ritchie-Knott-Rice

SED Strain energy density

SWT Smith-Watson-Topper

VI

CONTENTS

CERTIFICATION I

ABSTRACT II

ACKNOWLEDGEMENTS IV

ABBREVIATIONS V

NOMENCLATURE X

LIST OF FIGURES XVI

LIST OF TABLES XXIV

1. INTRODUCTION 1

1.1 Background 2

1.2 Motivation for the Present Study 4

1.2.1 Monotonic Failure 4

1.2.1.1 Multiaxial Stress-Strain Relationships 4

1.2.1.2 Yield Criteria 10

1.2.1.3 Hydrostatic Stress Influence 12

1.2.1.4 Failure Criteria Incorporating Hydrostatic Stress Effects 13

1.2.1.5 Porous Metal Plasticity 14

1.2.1.6 Fracture Mechanics Approach to Modelling of Cracks 15

1.2.2 Cyclic Failure 21

1.2.2.1 Fatigue Failure Phenomenon 21

1.2.2.2 Stress Based Approach 22

1.2.2.3 Strain Based Approach 24

Contents

VII

1.2.2.4 Energy Based Approach 26

1.2.2.5 Hydrostatic Stress Influence 31

1.2.2.6 Non-Proportional Loading Influence 34

1.2.2.7 Damage Accumulation 35

1.3 Objectives 38

1.4 Approach 40

2. FAILURE DUE TO MONOTONIC LOADING 41

2.1 Research Methodology 42

2.1.1 Concept Development 42

2.1.1.1 Effects of Hydrostatic Stress on Ductility 42

2.1.1.2 Proposed Fracture Criterion 47

2.1.2 Experimental Program 51

2.1.3 Analytical Program – Equivalent Stress-Strain Curve 56

2.1.3.1 Equivalent Stress-Strain Curve Determination to Point of Necking 56

2.1.3.2 Bridgman Approximation of Stress State in Necked Region 58

2.1.3.3 General Form of Equivalent Stress-Strain Curve 70

2.1.3.4 Video Imaging Technique for Bridgman Approximation 71

2.1.3.5 Finite Element Modelling 74

2.1.3.6 Comparison of Experimental Results with Finite Element Analysis 78

2.1.4 Analytical Program – Fracture Curve 82

2.1.4.1 Finite Element Modelling 82

2.1.4.2 Comparison of Experimental Results with Finite Element Analysis 85

2.1.4.3 Comparison of Stress-Strain State with Fracture Mechanics Theory87

2.1.4.4 Fracture Curve Determination 90

2.2 Experiments and Results 93

2.2.1 Establishment of Equivalent Stress-Strain Curve 93

2.2.1.1 Experimental Derivation of Equivalent Stress-Strain Curve 93

2.2.1.2 Analytical Confirmation of Equivalent Stress-Strain Relationship 101

2.2.2 Establishment of Fracture Curve 110

2.2.2.1 Correlation of V-Notch Specimen Load-Displacement Curves 110

2.2.2.2 Stress-Strain State at Fracture Cross-Section 116

Contents

VIII

2.2.2.3 Comparison of Stress-Strain State with Fracture Mechanics 127

2.2.2.4 Fracture Curve Determination 132

2.3 Analysis and Discussion 137

2.3.1 Determination of Equivalent Stress-Strain Curve 137

2.3.2 Determination of Fracture Curve 140

2.4 Conclusion 143

2.4.1 Research Outcomes 143

2.4.2 Recommendations for Future Work 145

3. FATIGUE FAILURE DUE TO CYCLIC LOADING 146

3.1 Research Methodology 147

3.1.1 Theory Development 147

3.1.1.1 Plastic Strain Energy Approach to Fatigue Life Characterisation 147

3.1.1.2 Thermodynamic Approach to High Cycle Fatigue 150

3.1.2 Experimental Program 158

3.1.2.1 Materials Selection and Specimen Design 158

3.1.2.2 Temperature Measuring Equipment 160

3.1.2.3 Temperature Calibration Facility 163

3.1.2.4 Achievement of Thermal Isolation at Specimen Surface 165

3.1.2.5 Fatigue Testing Program 168

3.1.3 Analytical Program 171

3.1.3.1 Determination of Plastic Strain Energy Density - Mechanical

Measurement 171

3.1.3.2 Determination of Plastic Strain Energy Density - Thermodynamic

Measurement 173

3.1.3.3 Comparison of Mechanical and Thermodynamic Measurement 181

3.2 Experiments and Results 183

3.2.1 Thermistor Calibration 183

3.2.2 Experiments Conducted using Mechanical Measurement 184

3.2.2.1 Low Cycle Fatigue Experiments 184

3.2.2.2 Determination of Plastic Strain Energy Density 188

3.2.3 Experiments Conducted using Thermodynamic Measurement 193

Contents

IX

3.2.3.1 High Cycle Fatigue Experiments 193

3.2.3.2 Determination of Plastic Strain Energy Density 195

3.2.4 Comparison of Mechanical and Thermodynamic Results 203

3.3 Analysis and Discussion 206

3.4 Conclusion 209

3.4.1 Research Outcomes 209

3.4.2 Recommendations for Future Work 211

4. SUMMARY OF CONCLUSIONS 212

REFERENCES 215

X

NOMENCLATURE

A Cross-section area

A2 Necked section area

A2i Initial section area

As Surface area

a Crack length

B Integral fracture criteria material coefficient

b Stress-life equation exponent

C Integral fracture criteria material constant

c Strain-life equation exponent

cp Specific heat

D2 Necked section diameter

D2i Initial section diameter

Di Inner diameter

Do Outer diameter

Dσ Deviatoric stress tensor

df Normalised accumulated damage

dn Shih relationship parameter

da Crack growth

ds Contour path increment

dλ Associative flow rule incremental constant

E Young’s modulus of elasticity

E′ Equivalent modulus of elasticity

Ep Plastic modulus

F Factor, view factor, stability factor

Fr Force component in the r-direction

f Frequency

fo Yield stress function about stress origin

fN Volume fraction of nucleated voids

fv Void volume fraction

vf& Void volume fraction rate

Nomenclature

XI

grvf& Void growth rate

nuclvf& Void nucleation rate

G Strain energy release rate

g Non-linear damage rule exponent function

H Strength coefficient

h Height

h′ Differential height

I1, I2, I3 First, second and third stress invariants

I Current

J J-integral strain energy release rate

J1, J2 First and second deviatoric stress invariants

K Stress intensity factor

KI, KII, KIII Stress intensity factors for modes I, II and III fracture

KIc Critical mode I stress intensity factor

Kc Critical stress intensity factor

k Thermal conductivity

L Length

l, m, n Principal axis direction cosines

M Thermistor resistance decay constant

N Number of fully reversed cycles

Nf Number of fully reversed cycles to failure

n Strain hardening exponent

P Load (force)

Pa Load amplitude

Pmax Failure load

PQ Offset load

p Pressure

Q1, Q2, Q3, Finite difference scheme variables

Q4, Q5, Q6

gQ& Heat generation rate

inQ& Heat flux

stQ& Transient heat rate

Nomenclature

XII

q Yield surface growth function

q& Internal heat generation rate per unit volume

q′ Heat flux per unit area

qo Constant yield surface

q1, q2, q3 Porous metal plasticity void material constants

R Resistance

r Radius

r1 Uniform section radius

r2 Necked section radius

ir2 Initial section radius

ri Inner radius

ro Outer radius

rs Vacuum chamber surface radius

Sξ Yield surface history dependent parameter

sN Standard deviation of nucleation strain

T Temperature

Ts Vacuum chamber surface temperature

oo zz TT −+ , Boundary condition temperatures

Tε Strain tensor

Tσ Stress tensor

t Time

U Strain energy

u, v, w Displacement components in x (r), y (θ) and z-directions

V Voltage

W Strain energy in crack vicinity

X Normal contour stress vector

x, y, z Cartesian coordinate system axes

z1 Temperature measurement point distance

zo Boundary temperature measurement distance from symmetry plane

α Pressure equation constant

β Pressure coefficient of ductility

∆ Finite change

Nomenclature

XIII

maxt∆ Maximum stable time increment

∆We Elastic strain energy density per cycle

∆Wp Plastic strain energy density per cycle

∆Wt Total strain energy density per cycle

δ Displacement (deflection)

δCTOD Crack tip opening displacement

ξij Origin offset stress tensor components

ε Strain, engineering strain

ε Equivalent strain

ε′ Emissivity

ε1, ε2, ε3 Principal normal strain components

aε Total strain amplitude

eaε Elastic strain amplitude

paε Plastic strain amplitude

fε Equivalent fracture strain

fε ′ Strain-life equation coefficient

εf Fracture strain

ofε Uniaxial fracture strain

εij Strain tensor components

pMε Equivalent plastic strain of matrix material

pMε& Equivalent plastic strain rate of matrix material

εN Mean nucleation strain

εp Plastic strain

pε Equivalent plastic strain

1pε , 2pε ,

3pε Principal plastic strain components

fpε Equivalent plastic fracture strain

ijpε Plastic strain tensor components

opε Equivalent plastic fracture strain at zero hydrostatic stress

Nomenclature

XIV

rpε ,θ

ε p ,zpε Plastic strain components in the r-θ-z cylindrical coordinate system

εr, εθ, εz Normal strain components in the r-θ-z cylindrical coordinate system

εv Volumetric strain

pvε& Volumetric plastic strain rate

εx, εy, εz Normal strain components in the x-y-z Cartesian coordinate system

εxx, εyy, εzz, Strain tensor components

εxy, εyz, εzx

Φ Yield function

φ Monotonic failure criterion function

Γ Enclosed line integral contour path

γrz Shear strain component in the r-θ-z cylindrical coordinate system

γxy, γyz, γzx Shear strain components in the x-y-z Cartesian coordinate system

ηp Plastic strain energy-life equation coefficient

ηt Total strain energy-life equation coefficient

ϕ Neck angle

ϕ′ Oscillating sphere angle

κ Density

µ Secant slope

ν Poisson’s ratio

θ Angle

ρ Neck radius

ρ′ Oscillating sphere radius

σ Stress, engineering stress

σ Equivalent (von Mises) stress

σ′ Boltzmann constant (5.67 × 10-8 W/m2.K4)

σ1, σ2, σ3 Principal normal stress components

σa Cyclic stress amplitude

σar Equivalent stress amplitude

σh Hydrostatic stress

σhc Critical hydrostatic stress

σij Stress tensor components

Nomenclature

XV

σM Yield stress of fully dense material

σm Cyclic mean stress

σmax Maximum normal stress

σn Nominal stress

σo Yield stress

oσ ′ Yield stress at point of unloading

σr, σθ, σz Normal stress components in the r-θ-z cylindrical coordinate system

σUTS Ultimate tensile stress

σu Fracture stress

σx, σy, σz Normal stress components in the x-y-z Cartesian coordinate system

σxx, σyy, σzz, Stress tensor components

σxy, σyz, σzx

maxzσ Maximum normal stress in the z-direction

2rzσ Normal stress component in z-direction at free surface

τ Shear stress

τ1, τ2, τ3 Maximum shear stress components

τxy, τyz, τzx Shear stress components in the x-y-z Cartesian coordinate system

τrz Shear stress component in the r-θ-z cylindrical coordinate system

ωp Plastic strain energy-life equation exponent

ωt Total strain energy-life equation exponent

ξij Stress tensor centre components

ψ Tangent angle

XVI

LIST OF FIGURES

Figure 1.2.1. Stress-strain curve. 5

Figure 1.2.2. Mohr’s circle representation of stress state. 7

Figure 1.2.3. Hydrostatic axis and deviatoric plane in the σ1-σ2-σ3 principal axis

system. 8

Figure 1.2.4. Metal microstructure: (a) grain crystallographic orientation; (b)

dislocation movement. 10

Figure 1.2.5. Comparison of Tresca and von Mises yield criteria in the σ1-σ2

plane. 12

Figure 1.2.6. Strain energy release rate: (a) crack growth da due to applied load

P; (b) strain energy release dU with crack growth da. 16

Figure 1.2.7. Modes of fracture; Mode I (normal), Mode II (forward shear),

Mode III (parallel shear). 17

Figure 1.2.8. Crack tip opening displacement (CTOD) model. 19

Figure 1.2.9. Application of J-integral approach to crack growth. 20

Figure 1.2.10. Cyclic loading, indicating stress amplitude and mean stress. 21

Figure 1.2.11. Comparison of proportional and non-proportional loading in the

σ1-σ2 plane. 22

Figure 1.2.12. Typical σa-Nf curve plotted on log-log axes. 23

Figure 1.2.13. Typical apε -Nf curve plotted on log-log axes. 25

Figure 1.2.14. Comparison of aeε -Nf, apε -Nf and aε -Nf curves. 26

Figure 1.2.15. Stress-strain hysteresis loop indicating elastic SED and plastic

SED. 27

Figure 1.2.16. Typical ∆Wp-Nf curve plotted on log-log axes. 28

Figure 1.2.17. Typical ∆Wt-Nf curve plotted on log-log axes. 29

Figure 1.2.18. Comparison of isotropic hardening and kinematic hardening. 31

Figure 1.2.19. Comparison of Goodman and Gerber equations. 33

Figure 1.2.20. Strain hardening due to non-proportional loading. 35

Figure 1.2.21. Typical damage accumulation curves derived from Palmgren-

Miner and non-linear damage accumulation rules. 37

List of Figures

XVII

Figure 2.1.1. Uniform section specimen geometry. 43

Figure 2.1.2. Notched specimen geometry: (a) transverse hole; (b) 90°

circumferential V-notch. 43

Figure 2.1.3. Free-cutting brass stress-strain state at failure: (a) hydrostatic

stress vs. radius; (b) equivalent plastic strain vs. radius. 45

Figure 2.1.4. Fracture cross-sections for 4340 steel: (a) uniform section specimen;

(b) 90° circumferential V-notch specimen. 46

Figure 2.1.5. Effect of hydrostatic tension on crack geometry. 47

Figure 2.1.6. Effect of hydrostatic compression on crack geometry. 47

Figure 2.1.7. General form of predicted equivalent plastic fracture strain vs.

hydrostatic stress curve. 48

Figure 2.1.8. Typical equivalent stress-strain curve depicting equivalent plastic

fracture strain fpε . 49

Figure 2.1.9. Possible linear form of equivalent plastic fracture strain vs.

hydrostatic stress curve. 50

Figure 2.1.10. Uniform section specimen geometry. 51

Figure 2.1.11. Uniform section specimen. 52

Figure 2.1.12. 90° circumferential V-notch specimen geometry. 53

Figure 2.1.13. 90° circumferential V-notch specimen. 53

Figure 2.1.14. Instron servohydraulic uniaxial testing machinery. 54

Figure 2.1.15. Clamped test specimen with extensometer. 55

Figure 2.1.16. Engineering stress-strain curve. 56

Figure 2.1.17. Range of application of true stress and true strain formulae. 57

Figure 2.1.18. Necked region axisymmetric geometry and representative element.

58

Figure 2.1.19. Geometry and representative element of necked region expressed

in terms of cross-section radius r, neck radius ρ, and angles θ and ϕ. 64

Figure 2.1.20. Range of application of true stress, true strain and Bridgman

approximation formulae. 69

Figure 2.1.21. Equivalent stress-strain curve. 70

Figure 2.1.22. Video imaging equipment. 72

Figure 2.1.23. Necked specimen image. 73

List of Figures

XVIII

Figure 2.1.24. Dimensioned specimen geometry: (a) undeformed; (b) deformed

(necked). 74

Figure 2.1.25. Uniform section specimen finite element model, geometry, loads

and constraints. 75

Figure 2.1.26. Femcad 2000 finite element mesh model of uniform section

specimen. 78

Figure 2.1.27. σ -r curve, uniform section specimen. 79

Figure 2.1.28. σh-r curve, uniform section specimen. 80

Figure 2.1.29. pε -r curve, uniform section specimen. 80

Figure 2.1.30. V-notch specimen finite element model geometry, loads and

constraints. 83

Figure 2.1.31. Femcad 2000 finite element mesh model of 15,7.5 circumferential

V-notch specimen. 84

Figure 2.1.32. V-notch specimen stress-strain state: (a) σ -r; (b) σh-r; (c) pε -r. 86

Figure 2.1.33. Circumferential V-notch. 88

Figure 2.1.34. Load-displacement curve for determination of KIc validity. 89

Figure 2.1.35. Equivalent plastic strain-hydrostatic stress curve for uniform

section specimen. 90

Figure 2.1.36. Equivalent plastic strain-hydrostatic stress curve for

circumferential V-notch specimen. 91

Figure 2.1.37. Superposition of uniform section and V-notch equivalent plastic

strain-hydrostatic stress curves. 92

Figure 2.2.1. Load-displacement curve, uniform section (free-cutting brass). 93

Figure 2.2.2. Load-displacement curve, uniform section (4340 steel). 94

Figure 2.2.3. Specimen images and dimensions for free-cutting brass: (a)

undeformed; (b) deformed (necked) immediately prior to failure. 95

Figure 2.2.4. Specimen images and dimensions for 4340 steel: (a) undeformed;

(b) deformed (necked) immediately prior to failure. 96

Figure 2.2.5. True stress-true strain curve, free-cutting brass. 99

Figure 2.2.6. True stress-true strain curve, 4340 steel. 99

Figure 2.2.7. Equivalent stress-strain curve, free-cutting brass. 100

Figure 2.2.8. Equivalent stress-strain curve, 4340 steel. 101

List of Figures

XIX

Figure 2.2.9. Load-displacement curve comparison, uniform section (free-cutting

brass). 102

Figure 2.2.10. Load-displacement curve comparison, uniform section (4340 steel).

102

Figure 2.2.11. Equivalent stress contour, uniform section (free-cutting brass). 103

Figure 2.2.12. Hydrostatic stress contour, uniform section (free-cutting brass).

104

Figure 2.2.13. Equivalent plastic strain contour, uniform section (free-cutting

brass). 104

Figure 2.2.14. Equivalent stress contour, uniform section (4340 steel). 105

Figure 2.2.15. Hydrostatic stress contour, uniform section (4340 steel). 105

Figure 2.2.16. Equivalent plastic strain contour, uniform section (4340 steel). 106

Figure 2.2.17. Uniform section stress-strain state (free-cutting brass): (a) σ -r;

(b) σh-r; (c) pε -r. 107

Figure 2.2.18. Uniform section stress-strain state (4340 steel): (a) σ -r; (b) σh-r;

(c) pε -r. 108

Figure 2.2.19. Load-displacement curve, V-notch (free cutting brass) 15,4. 110

Figure 2.2.20. Load-displacement curve, V-notch (free cutting brass): (a) 15,6;

(b) 15,7.5; (c) 15,9. 111

Figure 2.2.21. Load-displacement curve, V-notch (free cutting brass): (a)

15,10.5; (b) 10,5; (c) 8,4. 112

Figure 2.2.22. Load-displacement curve, V-notch (4340 steel): (a) 15,4; (b)

15,6; (c) 15,7.5. 113


15,10.5; (c) 12,6. 114


8,4. 115

Figure 2.2.25. Equivalent stress contour, V-notch (free-cutting brass) 15,7.5.

117

Figure 2.2.26. Hydrostatic stress contour, V-notch (free-cutting brass) 15,7.5.

118

Figure 2.2.27. Equivalent plastic strain contour, V-notch (free-cutting brass)

15,7.5. 118

List of Figures

XX

Figure 2.2.28. V-notch specimen fracture cross-section for free-cutting brass. 119

Figure 2.2.29. V-notch stress-strain state, free-cutting brass 15,4.5: (a) σ -r; (b)

σh-r; (c) pε -r. 121

Figure 2.2.30. V-notch stress-strain state, free-cutting brass 15,6: (a) σ -r; (b)

σh-r; (c) pε -r. 122

Figure 2.2.31. V-notch stress-strain state, free-cutting brass 15,7.5: (a) σ -r; (b)

σh-r; (c) pε -r. 123


σh-r; (c) pε -r. 124

Figure 2.2.33. V-notch stress-strain state, free-cutting brass 15,10.5: (a) σ -r;

(b) σh-r; (c) pε -r. 125

Figure 2.2.34. V-notch stress-strain state, free-cutting brass 10, 5: (a) σ -r; (b)

σh-r; (c) pε -r. 126


σh-r; (c) pε -r. 127


15,4.5. 129


15,6. 130


15,9. 130


15,10.5. 131


10,5. 131


8,4. 132

Figure 2.2.42. Equivalent plastic strain-hydrostatic stress, uniform section

specimen (free-cutting brass). 133

Figure 2.2.43. Evolution of equivalent plastic strain-hydrostatic stress curves for

V-notch specimen, free-cutting brass 15,7.5. 134

List of Figures

XXI

Figure 2.2.44. Combined equivalent plastic strain-hydrostatic stress curves (free-

cutting brass). 135

Figure 2.2.45. Combined (σh, fpε ) failure points obtained from uniform section

and V-notch specimens (free-cutting brass). 136

Figure 2.3.1. Equivalent plastic fracture strain vs hydrostatic stress, possible

linear form (free-cutting brass). 142

Figure 3.1.1. Stress-strain curve hysteresis loops. 148

Figure 3.1.2. Uniform test section, internal heat generation and temperature

distribution. 151

Figure 3.1.3. Internal heat generation models: (a) uniform model and

temperature distribution; (b) free surface model and temperature

distribution. 153

Figure 3.1.4. Time-varying temperature boundary conditions. 154

Figure 3.1.5. Uniform test section model. 156

Figure 3.1.6. Fatigue specimen dimensions. 160

Figure 3.1.7. Fatigue specimen indicating 5 mm interval markings about plane of

symmetry. 160

Figure 3.1.8. Miniature glass bead NTC thermistor dimensions. 161

Figure 3.1.9. Wheatstone bridge circuit. 162

Figure 3.1.10. Temperature probe. 162

Figure 3.1.11. Temperature calibration facility: (a) water bath and temperature

measuring equipment; (b) facility overview. 164

Figure 3.1.12. Vacuum pump. 166

Figure 3.1.13. Vacuum chamber design: (a) cylinder and bottom flange; (b) top

flange. 167

Figure 3.1.14. Thermodynamic method consisting of clamped specimen enclosed

by vacuum chamber. 169

Figure 3.1.15. Typical stress-strain hysteresis loop. 171

Figure 3.1.16. Trapezoidal rule application to stress-plastic strain curve. 172

Figure 3.1.17. Typical temperature-time curve. 173

Figure 3.1.18. Assumed thermodynamic model (uniform internal heat

generation). 176

Figure 3.1.19. Surface radiation exchange model. 177

List of Figures

XXII

Figure 3.1.20. Finite difference model incorporating heat flux, internal heat

generation and transient effects. 179

Figure 3.2.1. Thermistor calibration curves. 183

Figure 3.2.2. Load-displacement curve, E.26.1. 185

Figure 3.2.3. Stress-strain curve, E.30.1. 185






Figure 3.2.9. Stress-plastic strain curve, E.30.1. 189






Figure 3.2.15. Voltage-time curves, T.26.1. 194

Figure 3.2.16. Temperature-time curves, T.26.1. 194




Figure 3.2.20. Temperature-time curves, T.24.1 (transient and steady state). 197

Figure 3.2.21. Temperature-time curves, T.24.2: (a) transient and steady state;

(b) crack propagation and failure. 198

Figure 3.2.22. Temperature-time curves, T.24.3: (a) transient and steady state;

(b) crack propagation and failure. 199

Figure 3.2.23. Temperature-time curves, T.24.4 (transient and steady state). 200

Figure 3.2.24. Temperature-time curves, T.22.1: (a) transient; (b) steady state;

(c) crack propagation and failure. 201

Figure 3.2.25. Combined ∆Wp-Nf low cycle fatigue data obtained from

mechanical and thermodynamic measurement. 204

Figure 3.2.26. Combined ∆Wp-Nf low cycle and high cycle fatigue data obtained

from mechanical and thermodynamic measurement. 205

List of Figures

XXIII

Figure 3.3.1 Linear form of ∆Wp-Nf curve plotted on log-log axes. 208

XXIV

LIST OF TABLES

Table 2.1.1. Failure load, nominal stress and deflection data. 44

Table 2.1.2. Typical nominal mechanical properties. 51

Table 2.1.3. V-notch specimen configurations with reference to Figure 2.1.12. 53

Table 2.1.4. Circumferential V-notch specimen model summary. 84

Table 2.2.1. Bridgman approximation data for free-cutting brass. 97

Table 2.2.2. Bridgman approximation data for 4340 steel. 97

Table 2.2.3. Bridgman correction sample data. 98

Table 2.2.4. Mechanical properties. 100

Table 2.2.5. Material porous metal plasticity parameters. 101

Table 2.2.6. Comparison of cross-section radii obtained from experiment and

finite element analysis. 106

Table 2.2.7. Location of maximum normal stress. 119

Table 2.2.8. Determination of KIc validity. 128

Table 2.2.9. KIc calculations for V-notch specimen configurations (free-cutting

brass). 128

Table 3.1.1. Temperature distribution derived from Equation (3.3). 157

Table 3.1.2. Nominal mechanical properties. 158

Table 3.1.3. Chemical composition comparison. 158

Table 3.1.4. Material properties and thermodynamic constants. 159

Table 3.1.5. Miniature glass bead NTC thermistor characteristics based on

Equation (3.4). 161

Table 3.1.6. Data acquisition precision. 165

Table 3.1.7. Typical emissivity values for selected materials. 168

Table 3.1.8. Radius and emissivity values for finite difference calculation. 180

Table 3.2.1. Low cycle fatigue results. 184

Table 3.2.2. Low cycle fatigue plastic SED results. 192

Table 3.2.3. High cycle fatigue results. 193

Table 3.2.4. High cycle fatigue plastic SED results. 202

Table 3.2.5. Combined low and high cycle fatigue plastic SED results. 203

Table 3.2.6. Comparison of low cycle fatigue plastic SED results. 204

1

1. INTRODUCTION

Introduction

2

1.1 Background

The characterisation of failure in metals through the development of failure criteria is

of major importance in the design and assessment of structures and components. The

study of the metal failure phenomena and the development of methodology and

criteria for performing accurate failure prediction has been a major focus of research

over the past 150 years, with many of the methods currently in use due to significant

developments and advancements which have occurred over the last fifty years.

Associated with failures are the potentially disastrous consequences of major

structural failures, particularly in regards to human safety and financial cost.

Numerous examples can be found which graphically illustrate the consequences of

failure in metal structures and emphasise the importance of failure research, including

bridges, ships, aircraft and railway infrastructure. The range of sources and the

potential outcome of failures outline the importance and requirement for the continual

development of accurate means of failure assessment.

The phenomena of metal failure may be attributed to the accumulation of damage due

to the application of a load. The applied load may be monotonically increasing until

such load is reached where the damage results in failure, known as fracture, or may be

repetitive or cyclic such that the total damage accumulated from each load application

results in failure, commonly referred to as fatigue. Such failures may be induced by

mechanical or thermal loading, and are directly influenced by a variety of factors

including material properties, state of stress, surface finish, temperature and

environmental effects.

In regards to fracture, the amount of damage accumulated as a result of monotonic

loading may be expressed in the form of plastic deformation or ductility. The ductility

of a metal is a measure of its ability to undergo plastic deformation prior to failure.

Metal alloys are commonly classified in terms of their ductility as either ductile or

brittle subject to a uniaxially applied tensile load. Ductile behaviour results in metal

failure due to a significant or large amount of plastic deformation, whereas brittle

behaviour results in metal failure with little or no associated plastic deformation. As

Introduction

3

ductile failure is preceded by significant amounts of plastic deformation, fracture

occurs due to gradual crack formation and growth within the material. In the case of

brittle failure, fracture occurs due to rapid unstable crack growth, and as such there is

little warning prior to failure. There are situations where brittle failure may occur in

metal alloys considered to be normally ductile, such as that seen in the case of

notched components or sudden impact of large objects with structures. The complex

loading present in such situations may impose states of multiaxial stress quite

different to that of the uniaxial stress state, and as such the conditions from which

ductility is normally determined for a metal alloy may not be applicable. The presence

of complex multiaxial states of stress and the effects of such stress states on metal

ductility highlight the importance and requirement for accurate fracture prediction

criteria.

The characterisation and prediction of fatigue failure is of great importance in

engineering analysis. It has been estimated that as great as ninety percent of all

engineering structural and component failures may be attributed to fatigue failure [1],

the occurrence of which has become more prevalent with increased use of high speed

machinery and resulting structural vibration. Fatigue damage is accumulated as a

result of cyclic loading, where each load cycle causes an amount of irreversible

damage within the metal alloy. The accumulated damage from each loading cycle

leads to the formation of cracks followed by crack growth resulting in failure of the

metal. The complex nature of fatigue failure has led to the establishment of numerous

criteria and incorporation of these criteria into standards or codes which formalise the

life assessment procedure. The fatigue codes employed in fatigue life assessment are

usually applied to loading situations involving uniaxial or weak biaxial stress states,

with resulting life predictions varying markedly from conservative to highly non-

conservative depending on the code applied and interpretation of the code [2].

Although the criteria used by many of these codes may be applied accurately to a

limited range of stress states, they are in widespread use and have become accepted as

industry standard. Despite widespread acceptance of current fatigue methodology, the

discrepancy in life prediction between the various codes highlights the continual need

for the development of accurate fatigue failure criteria applicable to the general

multiaxial stress state [2].

Introduction

4

1.2 Motivation for the Present Study

1.2.1 Monotonic Failure

1.2.1.1 Multiaxial Stress-Strain Relationships

Failure due to monotonic loading may be related to the material properties which are

characteristic of a given metal alloy. The majority of continuum mechanics based

failure criteria relate the imposed state of stress to the ductility of a metal alloy. The

relationship between stress and ductility is normally represented by the material

stress-strain curve illustrated by Figure 1.2.1, obtained from uniaxial tensile loading

of specimens consisting of uniform geometry. The resulting curve is expressed in

terms of normal stress σ and normal strain ε defined by a distinct elastic range and

plastic range. The stress-strain relationship within the elastic range is related by

Hooke’s law, whereby the stress σ and strain ε are related by the modulus of elasticity

constant E. The yield stress σo represents the point of transition from elastic material

behaviour to plastic material behaviour, where permanent plastic deformation is

sustained by the material with constant or increasing stress. For metal alloys where an

increasing stress is required to increase plastic deformation, the process is referred to

as strain hardening. The fracture strain εf represents the strain at which fracture

occurs, and is representative of the ductility of a material.

Introduction

5

Figure 1.2.1. Stress-strain curve.

The general multiaxial state of stress may be represented by the stress tensor Tσ,

illustrated by Equation (1.1). The corresponding multiaxial strain state may be

represented by the strain tensor Tε, illustrated by Equation (1.2). The orientation of an

element on which the stress tensor is resolved may be such that there is a state of

normal stress with no associated shear stress. These resulting normal stresses are

referred to as principal normal stresses, and may be found from solving the

determinant of Equation (1.3).

=

==

zyzzx

yzyxy

zxxyx

zzzyzx

yzyyyx

xzxyxx

ijTστττστττσ

σσσσσσσσσ

σσ (i = 1, 2, 3) (j = 1, 2, 3) (1.1)

=

==

zyzzx

yzy

xy

zxxyx

zzzyzx

yzyyyx

xzxyxx

ijT

εγγ

γε

γ

γγε

εεεεεεεεε

εε

22

22

22

(i = 1, 2, 3) (j = 1, 2, 3) (1.2)

σ

ε

E

σo

∆σ

∆ε

Elastic Range Plastic Range εf

Introduction

6

=

−−

−

000

i

i

i

izyzzx

yziyxy

zxxyix

nml

σστττσστττσσ

(i = 1,2,3) (1.3)

1222 =++ iii nml (1.4)

From solving the determinant of Equation (1.3) subject to the condition imposed by

Equation (1.4), the cubic polynomial of Equation (1.5) is obtained, consisting of

constants I1, I2 and I3 referred to as the stress invariants. The stress invariants I1, I2 and

I3 are illustrated by Equations (1.6)-(1.8) respectively, expressed in terms of the

normal and shear stress components of the stress tensor. As implied by the cubic

polynomial form of the equation, the principal normal stresses are defined by three

orthogonal stress components, denoted σ1, σ2 and σ3.

0322

13 =−+− IσIσIσ iii (σ3 ≤ σ2 ≤ σ1) (1.5)

zyx σσσI ++=1 (1.6)

2222 zxyzxyxzzyyx τττσσσσσσI −−−++= (1.7)

2223 2 xyzzxyyzxzxyzxyzyx τστστστττσσσI −−−+= (1.8)

For a given multiaxial state of stress, the principal normal stresses represent the

maximum and minimum normal stresses for any resolved orientation within an

element of material. From the principal normal stresses, the maximum shear stresses

τ1, τ2 and τ3 may be obtained according to Equations (1.9)-(1.11) respectively.

Expressed in this form, the multiaxial state of stress may be represented in the σ-τ

plane in terms of Mohr’s circles, as illustrated by Figure 1.2.2.

232

1σστ −

= (1.9)

213

2σστ −

= (1.10)

Introduction

7

221

3σστ −

= (1.11)

Figure 1.2.2. Mohr’s circle representation of stress state.

When a given state of stress is expressed in terms of principal normal stresses, a stress

point (σ1,σ2,σ3) may be found in the σ1-σ2-σ3 principal axis system as illustrated by

Figure 1.2.3. The state of stress in this coordinate system may be resolved into

hydrostatic stress and deviatoric stress components, according to the hydrostatic axis

and the deviatoric plane.

σ1 σ2 σ3

τ1

τ2

τ3

σ

τ

Introduction

8

Figure 1.2.3. Hydrostatic axis and deviatoric plane in the σ1-σ2-σ3 principal axis system.

The hydrostatic stress component is directly related to the first stress invariant I1, and

may be expressed in the form illustrated by Equation (1.12), denoted σh. The

deviatoric stress tensor Dσ may be obtained from the stress tensor Tσ and hydrostatic

stress component σh, as illustrated by Equation (1.13). The corresponding deviatoric

stress invariants J1 and J2 are indicated by Equations (1.14)-(1.15) respectively. The

hydrostatic stress is a measure of the average normal stress acting on an element of

material, and is related to volumetric expansion and contraction. Hydrostatic stress

may be directly related to the volumetric strain for elastic behaviour through the

generalised form of Hooke’s law. The general form of Hooke’s law is indicated by

Equations (1.16), expressed in terms of modulus E, normal stress components σ1, σ2

and σ3, and Poisson’s ratio ν. Volumetric strain εv is expressed by Equation (1.17),

defined in terms of the principal normal strain components ε1, ε2 and ε3. The

relationship between hydrostatic stress σh and volumetric strain εv is illustrated by

Equation (1.17). The deviatoric stress component is independent of the hydrostatic

stress component, and may be viewed as a measure of pure shear stress acting in an

element of material.

σ1

σ2

σ3

Hydrostatic Axis

Total Stress Vector

Deviatoric Plane

Deviatoric Stress

Introduction

9

3331321 Izyx

h =++

=++

=σσσσσσσ (1.12)

−−

−=

hzyzzx

yzhyxy

zxxyhx

Dσσττ

τσστττσσ

σ (1.13)

01 =J (1.14)

( ) ( ) ( ) ( )[ ]2222222 6

61

zxyzxyxzzyyx τττσσσσσσJ +++−+−+−= (1.15)

( )[ ]32111 σσνσε +−=E

( )[ ]13221 σσνσε +−=E

(1.16)

( )[ ]21331 σσνσε +−=E

( ) hv EEσνσσσνεεεε 21321

321321−=++−=++= (1.17)

From experimental evidence, plastic deformation within metal alloys has been shown

to be a shear dominated event as a result of dislocation movement within the metal

matrix. On a microstructural level, a metal matrix consists of a series of crystals or

grains. A grain consists of a crystalline lattice of atoms, with each grain possessing an

individual crystallographic orientation as illustrated by Figure 1.2.4 (a). The regions

between grains, where the crystallographic structures of individual grains meet, are

known as grain boundaries. Each grain consists of defects throughout the crystalline

structure, known as dislocations, as illustrated by Figure 1.2.4 (b). The dislocations,

under a certain magnitude of applied shear stress, move along favourably orientated

crystallographic planes known as slip planes [3]. When the orientation of such grains

is random on a statistical basis, the resulting material properties on a macroscopic

scale are relatively uniform in all orientations. A metal alloy with uniform material

properties in all orientations is referred to as isotropic, whilst a metal alloy that

Introduction

10

possesses an initially un-voided and uniform grain structure throughout the metal

matrix is referred to as homogeneous.

Figure 1.2.4. Metal microstructure: (a) grain crystallographic orientation; (b) dislocation movement.

1.2.1.2 Yield Criteria

From both a theoretical and experimental basis, plastic deformation or flow is

hypothesised to occur within a metal alloy once a certain value of stress in the

deviatoric plane away from the hydrostatic axis has been reached, known as the yield

surface. Various yield criteria have been developed to determine yield surface

(a)

(b)

Dislocation

Grain A

Grain B

Grain boundary

Slip plane

Slip plane orientation

Grains

Grain boundaries

Introduction

11

expansion and subsequent plastic deformation within predominantly ductile metal

alloys. The two most commonly used yield criteria for homogeneous, isotropic metals

are the maximum shear stress criterion and the octahedral shear stress criterion. The

maximum shear stress criterion or Tresca criterion, originally formulated by Tresca

and later applied by Saint-Venant [4], is related to the multiaxial state of stress as

represented by Mohr’s circle. From calculation of the three principal normal stresses

σ1, σ2 and σ3, the maximum shear stresses are determined, the largest of these shear

stress values representing the maximum shear stress. By formulating the equation in

terms of the uniaxial normal stress, the maximum shear stress is expressed in terms of

an equivalent normal stress σ which may be related directly to the uniaxial stress-

strain curve, as illustrated by Equation (1.18).

( )133221 ,,max σσσσσσσ −−−= (1.18)

The octahedral shear stress criterion, also known as the distortion energy criterion or

von Mises criterion, was originally proposed by von Mises and later applied by

Hencky [4]. The criterion may be formulated from the second deviatoric stress

invariant, J2, obtained from the deviatoric stress tensor Dσ, as illustrated by Equation

(1.15). The von Mises yield criterion is expressed in the form of an equivalent normal

stress σ related to the uniaxial stress-strain curve, obtained from formulating the

equation in terms of the uniaxial normal stress as illustrated by Equations (1.19)-

(1.20). Although slightly less conservative than the Tresca criterion, the von Mises

criterion is commonly used as a yield criterion due to the fact that the formulation

employed is continuous and statistically provides a more effective correlation with

experimental data [5]. Both criteria generally predict yield of ductile metal alloys

quite accurately. A comparison of the Tresca and von Mises yield criteria in the σ1-σ2

plane is illustrated by Figure 1.2.5.

23J=σ (1.19)

( ) ( ) ( ) ( )222222 62

1zxyzxyxzzyyx τττσσσσσσ +++−+−+−=

( ) ( ) ( )213

232

2212

1 σσσσσσσ −+−+−= (1.20)

Introduction

12

Figure 1.2.5. Comparison of Tresca and von Mises yield criteria in the σ1-σ2 plane.

1.2.1.3 Hydrostatic Stress Influence

Experimental evidence suggests that the hydrostatic stress component has as strong

influence on the ductility of metal alloys. The pioneering work of Bridgman [6]

clearly revealed that a strong relationship exists in metals between hydrostatic stress

and ductility. The experimental work of authors Brownrigg et al. [7] and

Lewandowski and Lowhaphandu [8] further support this notion. The experiments

conducted consisted of a uniaxial tensile load with a superimposed pressure load

applied to the surfaces of cylindrical specimens. The experimental work of these

researchers in regards to the effects of hydrostatic pressure on homogeneous, isotropic

metal alloys has clearly shown that hydrostatic pressure has the effect of increasing

the fracture strain. The experimental work of Bridgman on various grades of steel

clearly revealed a strong linear relationship between hydrostatic pressure and fracture

strain [6]. The tests performed on spheroidised steel by Brownrigg et al. indicate the

relationship between superimposed pressure and fracture strain to be a linear

relationship [7]. An excellent compilation is provided by Lewandowski and

Lowhaphandu [8] of experimental work performed by various authors in investigating

the effects of hydrostatic pressure on a wide range of metal alloys.

σ1

σ2

σo

− σo

− σo

von Mises

Tresca

σo

Introduction

13

The research of Lewandowski and Lowhaphandu [8] has also revealed that, for

homogeneous, isotropic metal alloys, hydrostatic stress has no measurable influence

on yield and subsequent plastic flow. The implications of these findings support the

notion that plastic flow is accompanied by zero volume change within metals, the

assumption of which forms the basis of continuum plasticity theory to justify the

deviatoric stress plane concept and subsequent derivation of the Tresca and von Mises

yield criteria.

The existence of a relationship between hydrostatic stress and ductility was confirmed

by an experimental program conducted by Glass and West [9]. A series of monotonic

loading experiments were conducted on plain and notched cylindrical tensile

specimens from a variety of brass, steel and aluminium metal alloys. It was shown

from these experiments that material considered normally ductile may exhibit brittle

behaviour depending on the notch geometry. From close correlation with

experimental results, elastic-plastic finite element analyses concluded that the

apparent notch toughening effect for particular notch geometry could be attributed to

the hydrostatic stress state and associated plastic strain present at the fracture cross-

section. The degree of ductility was also found to be related to material

characteristics, including yield strength, rate of strain hardening and fracture ductility

obtained from the uniaxial stress case.

1.2.1.4 Failure Criteria Incorporating Hydrostatic Stress Effects

From the linear correlation of experimental data obtained over a large range of

hydrostatic stress values for various grades of steel, Bridgman proposed a relationship

in the form of an equation of ductility [6]. Indicated by Equation (1.21), the linear

expression defines hydrostatic pressure p in terms of a pressure coefficient of ductility

β and true fracture strain εf. The pressure coefficient of ductility β was determined to

be a function of the material, remaining constant throughout the applied hydrostatic

pressure range for a given grade of steel.

fp βεα += (1.21)

Introduction

14

Following the work of Bridgman, numerous criteria have been proposed over the past

thirty years in relation to ductile fracture of metal alloys which incorporate the

hydrostatic stress influence on ductility. Two such criteria are the modified

McClintock criterion [10] and Oyane criterion [11]. The criteria, expressed in integral

form, relate the process parameters to a corresponding equivalent strain range ε . The

modified McClintock criterion relates equivalent stress σ and hydrostatic stress σh

over a specified equivalent fracture strain range fε to a material constant C, as

indicated by Equation (1.22). The Oyane criterion is expressed in a similar form to the

modified McClintock criterion, with the incorporation of a material dependent

coefficient B, as indicated by Equation (1.23).

Cdf

h =

∫ ε

σσε

0

(1.22)

CdB

f

h =

+∫ ε

σσε

0

1 (1.23)

Based on the findings of Brownrigg et al. [7], a ductile failure criterion was recently

proposed by Oh [12] which relates the hydrostatic stress σh to the fracture strain fε as

illustrated by Equation (1.24). The general form of the equation is similar to that of

the equation of ductility proposed by Bridgman, defined as a linear function between

hydrostatic stress and strain for any given hydrostatic stress value.

hoff σθ

εε ∆

−+=

tan1 (1.24)

1.2.1.5 Porous Metal Plasticity

The yield and failure criteria discussed thus far are generally applicable to

homogeneous, isotropic metal alloys. For metal alloys that undergo large amounts of

plastic deformation, dislocation movement may result in the formation of voids within

the material [1,3]. The nucleation, growth and subsequent coalescence of these voids

Introduction

15

results in inhomogeneous material behaviour. Criteria associated with modelling yield

and failure within voided metal alloys are categorised as porous metal plasticity

criteria. A commonly used yield criterion is that proposed by Gurson [13] , which was

later modified by Tvergaard [14] to provide a yield criterion expressed as a function

of the equivalent (von Mises) stress σ , yield stress of fully dense material σM,

hydrostatic stress σh, void volume fraction fv and material constants q1, q2 and q3 as

indicated by yield function Φ of Equation (1.25). The criterion accounts for the

influence of hydrostatic stress on void growth, with constants q1, q2 and q3 accounting

for the void geometry. For a homogeneous or fully dense material, the form of the

Tvergaard criterion is identical to that of the von Mises criterion.

( ) 012

3cosh2 2

32

1

2

=+−

+

=Φ v

M

hv

M

fqq

fqσ

σσσ (1.25)

1.2.1.6 Fracture Mechanics Approach to Modelling of Cracks

For the modelling of crack growth within metal alloys, the fracture mechanics

approach was developed. First proposed by Griffith, and later modified by Orowan

and Irwin [1], the fracture mechanics concept is based on modelling of the stress field

in the vicinity of a crack tip. Due to the geometry of the formed cracks and applied

loads, stresses are raised locally at the crack tip to levels that approach the theoretical

cohesive strength of the metal matrix. Upon reaching a critical value of applied load,

the crack will propagate to complete fracture [1]. The fracture mechanics theory based

on brittle fracture of metals, where the material exhibits linear-elastic behaviour with

little or no plastic deformation prior to fracture, is denoted linear-elastic fracture

mechanics [5].

Linear-elastic fracture mechanics (LEFM) is based on the concept of strain energy

release rate. With the growth of a crack, elastic strain energy U is released according

to Equation (1.26) and Figure 1.2.6. According to the theory, once the rate of strain

energy release G reaches a critical value, unstable crack growth da will take place

resulting in complete fracture. The elastic stress state in the vicinity of a crack tip may

Introduction

16

be represented as a function of radius r and angle θ . Equations (1.27)-(1.29) indicate

the stress components σx, σy and τxy for plane stress conditions, expressed in terms of

radius r, angle θ and constant K referred to as the stress intensity factor [15].

dadU

LG 1−= (1.26)

Figure 1.2.6. Strain energy release rate: (a) crack growth da due to applied load P; (b) strain energy

release dU with crack growth da.

+=

23sin

2sin1

2cos

2θθθ

πσ

rK

x (1.27)

−=

23sin

2sin1

2cos

2θθθ

πσ

rK

y (1.28)

23cos

2sin

2cos

2θθθ

πτ

rK

xy = (1.29)

Viewing the normal stress component σy normal to the crack plane (θ = 0°), the stress

state in the vicinity of a crack tip may be conveniently represented by the stress

intensity factor K of Equation (1.30), expressed in terms of factor F, nominal stress σn

and crack length a. The critical stress intensity factor, Kc, is representative of the

fracture toughness of a material [5]. The stress intensity factor K may be related to the

P

a da

P

∆L

P

dU

U - dU

a

a + da

L

Introduction

17

strain energy release rate G by Equation (1.31) [16], indicated here for plane stress

and plane strain, expressed in terms of modulus of elasticity E and Poisson’s ratio ν.

aFK n πσ= (1.30)

EGK ′=2 EE =′ (Plane Stress) (1.31)

21 ν−=′ EE (Plane Strain)

The value of Kc differs depending on the mode of fracture, whether it be normal

(mode I), forward shear (mode II) or parallel shear (mode III) in relation to the crack

geometry, as displayed by Figure 1.2.7. For mode I, mode II or mode III fracture, the

stress intensity factors are denoted KI, KII and KIII respectively. For a given material,

Kc has been determined to be a material parameter independent of load, crack

geometry and crack length [1]. Materials with higher values of Kc require higher

levels of stress intensity in the vicinity of a crack to instigate fracture, and hence have

a higher resistance to fracture or fracture toughness. The stress intensity factor

formula has been derived for application to a wide range of commonly encountered

components and associated crack geometry. The LEFM approach has been used

successfully for the prediction of fracture in highly brittle metals with known crack

size and geometry.

Figure 1.2.7. Modes of fracture; Mode I (normal), Mode II (forward shear), Mode III (parallel shear).

The linear-elastic fracture mechanics concept assumes that the size of the plastic zone

surrounding the crack tip is negligible. When the plastic zone is not small enough to

be ignored, an adjustment to account for the plastic zone size is required to allow

accurate determination of K [17]. The plastic zone correction is applied in the

Mode I

Mode II

Mode III

x

y

z

Introduction

18

determination of K by assuming an effective crack length a′ , consisting of the

summation of the actual crack length a and the plastic zone radius rp, as indicated by

Equation (1.32) [17]. The plastic zone radius rp for plane stress and plane strain

conditions may be approximated in terms of the stress intensity factor K and yield

stress σo by Equations (1.33)-(1.34) respectively [17,18].

praa +=′ (1.32)

2

21

=

op

Krσπ

(Plane Stress) (1.33)

2

61

=

op

Krσπ

(Plane Strain) (1.34)

The linear-elastic fracture mechanics approach is mostly applicable to higher strength

materials which exhibit brittle failure characteristics. In such failures, the plastic zone

radius rp is relatively small compared to the crack length a. An accurate determination

of K is not possible once the plastic zone size becomes an appreciable fraction of the

crack length [1]. For lower strength ductile materials where the plastic zone size at the

crack tip is relatively large, elastic-plastic fracture mechanics (EPFM) approaches

such as the crack tip opening displacement (CTOD) method or J-integral approach

may be used.

The CTOD method may be described in terms of a hypothetical series of miniature

tensile specimens ahead of the crack tip within the plastic zone [19]. Crack growth

occurs once the specimen adjacent to the crack has failed. Unstable crack propagation

occurs if failure of the specimen adjacent to the crack is immediately followed by

failure of the next adjacent specimen without increase in load. Stable crack growth

occurs if an increasing load is required to continue crack growth. The CTOD, denoted

δCTOD, is expressed for unstable crack growth in terms of nominal stress σn, crack

length a, modulus of elasticity E and yield stress σo as indicated by Equation (1.35)

and Figure 1.2.8 [1]. For linear-elastic conditions, the CTOD equation may be related

to K in accordance with Equation (1.36) [1].

Introduction

19

o

nCTOD E

aσ

πσδ2

= (1.35)

oCTOD E

Kσ

δ2

= (1.36)

Figure 1.2.8. Crack tip opening displacement (CTOD) model.

The J-integral, originally proposed by Rice [20], is defined by a line integral

inscribing an enclosed path Γ representing the strain energy W in the vicinity of a

crack as indicated for 2-D plane stress by Equation (1.37), with crack propagation

occurring once a critical value of J is reached. Figure 1.2.9 illustrates the application

of the equation, with X representative of the normal stress vector acting on the contour

and ds representing the increment along the contour path. The integral has been

shown to be path independent, allowing application to any convenient path that

encloses the crack [5]. The value obtained for J is numerically equivalent to the strain

energy release rate G as applied to LEFM, and for plane stress and plane strain

conditions may be related to K in accordance with Equation (1.38) [1].

∫Γ

∂∂−= ds

xuXWdyJ (1.37)

EKJ

′=

2 EE =′ (Plane Stress) (1.38)

21 ν−=′ EE (Plane Strain)

δCTODρ

2 rp

Plastic Zone

Introduction

20

Figure 1.2.9. Application of J-integral approach to crack growth.

A generalised relationship between δCTOD and J for EPFM was derived by Shih [21]

for plane stress and plane strain conditions. The relationship may be expressed in

terms of yield stress σo and correlation parameter dn as indicated by Equation (1.39),

where dn is a function of yield stress σo, modulus of elasticity E and strain hardening

exponent n [21].

onCTOD

Jdσ

δ = (1.39)

y

x

Γ

ds

X

Introduction

21

1.2.2 Cyclic Failure

1.2.2.1 Fatigue Failure Phenomenon

Fatigue failure may be attributed to the accumulation of damage within a metal alloy

as a result of repetitive or cyclic load application. The accumulation of fatigue

damage may be measured in terms of the fatigue life or life fraction of a metal. The

vast majority of fatigue failure methodology and life assessment criteria relate the

accumulation of fatigue damage in terms of fatigue life, usually in the form of a

relationship between fatigue life and the stress or strain state present due to the

applied cyclic load. The fatigue life is normally expressed as the number of load

cycles at a given load level required to cause failure of the metal. A load cycle is a

repetitively applied load unit, and may be categorised in terms of an amplitude and

mean value. Illustrated by Figure 1.2.10 for an equivalent uniaxial stress, a cyclic

stress loading may be represented by a stress amplitude σa and a mean stress σm. A

cycle of load categorised in these terms with a mean stress σm of zero may be referred

to as a fully reversed load cycle.

Figure 1.2.10. Cyclic loading, indicating stress amplitude and mean stress.

σ

Cycle Cycle

t

σa

σm

σa 0

Introduction

22

An applied load may be classified as either proportional or non-proportional. A

proportional or in-phase load exists when the stress or strain components are

increasing or decreasing proportionally, resulting in a constant direction of equivalent

stress or strain. A non-proportional or out-of-phase load exists when the stress or

strain components are not applied proportionally, producing an equivalent stress or

strain which changes direction throughout the load cycle. A comparison between

proportional and non-proportional loading is illustrated by the tension-torsion

example of Figure 1.2.11 displayed in the σ1-σ2 plane.

Figure 1.2.11. Comparison of proportional and non-proportional loading in the σ1-σ2 plane.

Depending on the number of fully reversed cycles to failure, fatigue life may be

categorised as either low cycle fatigue or high cycle fatigue. Low cycle fatigue refers

to the cyclic range within 104 cycles, whereas high cycle fatigue refers to the cyclic

range beyond 105 cycles. For ductile metal alloys, low cycle fatigue is normally

associated with significant amounts of plastic deformation per cycle, whilst high cycle

fatigue usually occurs within the elastic range of the metal.

1.2.2.2 Stress Based Approach

The majority of methods devised to characterise fatigue life have been in the form of

stress based methods. The stress based approach has been used over the past 150 years

σ1

σ2

Axial

Torsion

Proportional

Non-proportional

Combined

Introduction

23

for the study of fatigue failure, and can be traced back to the pioneering work of

Wöhler in the 1850’s in the study of metal alloy fatigue failure due to axial, bending

and torsion loading cases [5]. Most stress based methods are usually expressed in the

form of stress amplitude σa versus number of fully reversed cycles to failure Nf

curves, or σa-Nf curves. For engineering metal alloys, the σa-Nf curve normally takes

the form of a line of negative slope with increasing number of cycles plotted on log-

log axes, until the region of 106 to 107 cycles where the curve decreases in slope or

becomes constant. The region of zero slope where a constant value of stress amplitude

is assumed is commonly referred to as infinite life or the endurance limit. The general

form of the σa-Nf curve for typical metal alloys is illustrated by Figure 1.2.12. The

equation that is most commonly used to define the region of negative slope is

expressed by Equation (1.40), defined in terms of stress amplitude σa, number of fully

reversed cycles to failure Nf, material dependent coefficient fσ ′ and exponent b.

Stress based methods have been shown to be mostly applicable to high cycle fatigue

life characterisation where the strains are essentially elastic [22].

Figure 1.2.12. Typical σa-Nf curve plotted on log-log axes.

( )bffa N2σσ ′= (1.40)

104 101 1

Nf

102 105 103 106 107

σa

(log10)

(log10)

Introduction

24

1.2.2.3 Strain Based Approach

The strain based approach was developed to provide accurate means of fatigue life

assessment for low cycle fatigue. The strain based approach has undergone

widespread development over the past fifty years, mostly as a result of the

development of closed-loop testing facilities enabling strain controlled testing [22].

Numerous approaches have been developed using either elastic strain, plastic strain or

total strain to characterise fatigue life. The strain based approach is expressed along

similar lines to that of the stress based approach, usually in the form of strain

amplitude εa versus number of fully reversed cycles to failure Nf curves, or εa-Nf

curves. As with the σa-Nf curve, the εa-Nf curve normally takes the form of a line of

negative slope with increasing number of cycles plotted on log-log axes.

For characterisation of low cycle fatigue, the plastic strain approach is usually

employed. The research of Coffin and Manson showed independently that a linear

relationship exists between plastic strain amplitude and the number of fully reversed

cycles to failure, as plotted on log-log axes [22,23]. The general form of the plastic

strain amplitude apε versus number of fully reversed cycles to failure Nf curve (

apε -

Nf curve) for typical metal alloys is illustrated by Figure 1.2.13. The subsequent

relationship used to define the region of negative slope became known as the Coffin-

Manson equation, expressed by Equation (1.41), defining plastic strain amplitude apε

in terms of number of fully reversed cycles to failure Nf and material dependent

fatigue fracture constants fε ′ and c [22].

Introduction

25

Figure 1.2.13. Typical apε -Nf curve plotted on log-log axes.

( )cffap N2εε ′= (1.41)

The total strain approach unifies the low cycle and high cycle fatigue regimes by the

addition of components of the equations used by the elastic strain approach and plastic

strain approach. The resulting equation used to define the total strain approach is

indicated by Equation (1.42), expressing total strain amplitude aε in terms of elastic

strain and plastic strain amplitude components [5]. The general form of the total strain

amplitude versus number of fully reversed cycles to failure curve (εa-Nf curve) for

typical metal alloys is indicated by Figure 1.2.14, in comparison to the elastic strain

amplitude curve ( aeε -Nf curve) and plastic strain amplitude curve (apε -Nf curve) [22].

The εa-Nf curve is very similar in form to the σa-Nf curve, depicting a linear region of

negative slope until the region of 106 to 107 cycles where the curve decreases in slope

or becomes constant.

( ) ( )cff

bf

fa NN

E22 ε

σε ′+

′= (1.42)

104 101 1

Nf

102 105 103 106 107

apε

(log10)

(log10)

Introduction

26

Figure 1.2.14. Comparison of aeε -Nf, apε -Nf and aε -Nf curves.

1.2.2.4 Energy Based Approach

The stress and strain components are load path dependant as defined by the stress-

strain curve of a given metal alloy. The amount of work done on a metal may be

measured by the area underneath the curve defined by the stress-strain relationship,

and is referred to as strain energy. Fully reversed cyclic loading of a metal alloy

results in the formation of a stress-strain path which is commonly referred to as a

hysteresis loop, as illustrated by Figure 1.2.15 on σ-ε axes. The strain energy density

(SED) per fully reversed cycle forms the basis for the energy based approach to

fatigue life characterisation. The energy based approach has undergone significant

development over the past thirty years, mostly due to the pioneering research of

Garud [24] and Ellyin [25]. The energy based approach may be categorised in terms

of elastic strain energy, plastic strain energy or total strain energy. Illustrated by

Figure 1.2.15, the area within the hysteresis loop represents plastic SED, while the

area underneath the elastic portion of the stress-strain curve represents elastic SED.

Total SED represents the addition of the elastic and plastic SED values as indicated.

104 101 1

Nf

102 105 103 106 107

εa

Elastic Strain Plastic Strain

Total Strain

(log10)

(log10)

Introduction

27

Figure 1.2.15. Stress-strain hysteresis loop indicating elastic SED and plastic SED.

The plastic strain energy approach was developed specifically to relate the fatigue

damage process to the energy attributed to plastic deformation. The plastic strain

energy method, as proposed by Garud [24] and later by Ellyin [25], relates the plastic

SED per cycle ∆Wp to the number of cycles to failure Nf. The general form of the

plastic strain energy equation is indicated by Equation (1.43), defining plastic SED

per cycle ∆Wp as the integral of the corresponding stress and plastic strain components

σij and ijpε respectively.

ijpcycle

ijp dεσ∆W ∫= (i = 1, 2, 3) (j = 1, 2, 3) (1.43)

From the research of Garud [24] and Ellyin [25], the use of plastic SED for fatigue

life characterisation has clearly shown that a linear log-log relationship exists between

plastic work per cycle ∆Wp and number of fully reversed cycles to failure Nf, as

illustrated by Figure 1.2.16. The corresponding equation used to describe the line of

ε

σ

Plastic SED

Hysteresis Loop

Elastic SED

Introduction

28

negative slope for plastic SED per cycle ∆Wp is indicated by Equation (1.44),

expressed in terms of number of cycles to failure Nf and material dependent constants

ηp and ωp.

Figure 1.2.16. Typical ∆Wp-Nf curve plotted on log-log axes.

( ) pfpp N∆W ωη 2= (1.44)

The research of Ellyin [22] has revealed that the cyclic plastic strain energy may be

related to the uniaxial plastic strain energy by calculating ∆Wp using equivalent

uniaxial stress and strain components. A comparison between experiments and

analysis assuming a von Mises equivalence between the uniaxial and multiaxial

conditions resulted in a close correlation of results. From this comparison, the

calculation of the plastic strain energy may be related to the area within an equivalent

uniaxial hysteresis loop, expressed in terms of von Mises stress and equivalent plastic

strain.

The plastic strain energy approach is generally applicable to low cycle fatigue where

the state of stress usually results in significant amounts of plastic deformation. The

accurate determination of plastic deformation by mechanical means is difficult during

high cycle fatigue, as the state of stress present is usually within the elastic range of

104 101 1

Nf

102 105 103 106 107

∆Wp

(log10)

(log10)

Introduction

29

the material. The total strain energy approach was proposed by Ellyin and Golos [26]

as a means of unifying the low cycle and high cycle fatigue regimes. The total strain

energy approach is essentially the summation of components of the elastic strain

energy and plastic strain energy. Elastic SED per cycle ∆We may be calculated from

the corresponding components of stress and strain by use of the elastic Hooke’s law

relationship from Equations (1.16). The stress and strain components may be

expressed in terms of equivalent stress σ and the first stress invariant I1 to obtain

∆We as indicated by Equation (1.45) [22]. The total SED per cycle ∆Wt versus number

of cycles Nf curve exhibits a linear log-log relationship of negative slope, until the

region of 106 to 107 cycles where the curve decreases in slope or becomes constant, as

illustrated by Figure 1.2.17. The corresponding ∆Wt relationships defining the region

of negative slope are indicated by Equations (1.46)-(1.47), expressed in terms of

elastic and plastic strain energy components.

21

2

621

31 ∆I

Eνσ∆

Eν∆We

−++= (1.45)

Figure 1.2.17. Typical ∆Wt-Nf curve plotted on log-log axes.

pet ∆W∆W∆W += (1.46)

( ) tftt N∆W ωη 2= (1.47)

104 101 1

Nf

102 105 103 106 107

∆Wt

(log10)

(log10)

Introduction

30

The hysteresis loop formed from the stress-strain path during cyclic loading has been

shown to result in a constant plastic strain energy value for a given cyclic load. The

formation of a stable hysteresis loop allows the stress-strain path to be modelled in

terms of a yield surface, hardening rule and associative flow rule. The yield surface

expansion may be determined from the hardening rule. Typical hardening rules

include the isotropic and kinematic hardening rules. In accordance with Equation

(1.48), the isotropic hardening rule assumes that the yield surface q expands

uniformly about a fixed centre, according to the monotonically increasing hardening

parameter Sξ, when the stress surface fo is coincident with the yield surface [22]. The

kinematic hardening rule assumes that the size of the yield surface qo remains constant

and translates by an amount ξij when the stress surface fo is coincident with the yield

surface, as indicated by Equation (1.49) [22]. The isotropic and kinematic hardening

rules result in the equivalent stress-strain paths illustrated by Figure 1.2.18. The

kinematic hardening rule accounts for the yield and subsequent plastic deformation

behaviour observed in metal alloys, referred to as the Bauschinger effect [1]. The

associative flow rule, indicated by Equation (1.50) [22], is used to determine the

incremental plasticity relationship between the stress and plastic strain components

according to the yield function Φ and incremental constant dλ.

( ) ( ) 02 =−=Φ ξSqσf ijo (i = 1, 2, 3) (j = 1, 2, 3) (1.48)

( ) 02 =−−=Φ oijijo qσf ξ (i = 1, 2, 3) (j = 1, 2, 3) (1.49)

( )ij

ij

ijp σσ

dλdε∂Φ∂

= (i = 1, 2, 3) (j = 1, 2, 3) (1.50)

Introduction

31

Figure 1.2.18. Comparison of isotropic hardening and kinematic hardening.

1.2.2.5 Hydrostatic Stress Influence

In terms of cyclic loading, hydrostatic stress has been shown to have a strong

influence on the fatigue life of metal alloys. In fatigue terminology, variation of the

mean stress σm is equivalent to variation of hydrostatic stress. The mean stress is the

mean value of normal stress present due to the applied cyclic state of stress. For a

given stress amplitude σa, an increase in mean stress σm results in a decrease in life Nf,

whereas a decrease in mean stress σm results in an increase in life Nf [1,5,22]. The

relationship between mean stress and fatigue life for cyclic loading in this sense is

analogous to the hydrostatic stress effect present in monotonic loading situations.

oσ2 ′

σo

oσ′

o2σ

ε

σ

E

∆ε

∆σ

Introduction

32

Numerous expressions have been proposed to incorporate the mean stress effect in

fatigue life assessment. The relationships for the stress based approach proposed by

Goodman and Gerber relate a given stress amplitude and mean stress to an equivalent

stress amplitude with zero mean stress [5]. Expressed in terms of actual stress

amplitude σa, equivalent stress amplitude σar, mean stress σm and true fracture stress

σu from a uniaxial tensile test, the Goodman and Gerber expressions are indicated by

Equations (1.51)-(1.52) respectively. A graphical comparison of the two equations,

plotted on σm-σa axes, is illustrated by Figure 1.2.19. The Goodman equation may be

applied to tensile and compressive mean stress situations, whereas the Gerber

equation is specifically intended for the tensile mean stress region. Given a σa-Nf

curve determined for zero mean stress, the Goodman and Gerber formulae allow the

determination of a stress amplitude with zero mean stress equivalent in terms of

fatigue life for a given stress amplitude and mean stress. A general expression

incorporating a modified form of the Goodman equation and the σa-Nf curve

expression is indicated by Equation (1.53), where fσ ′ is substituted for σu.

1=+u

m

ar

a

σσ

σσ

(1.51)

12

=

+

u

m

ar

a

σσ

σσ

( )0≥mσ (1.52)

Introduction

33

Figure 1.2.19. Comparison of Goodman and Gerber equations.

( )( )bfmfa Nσσσ 2−′= (1.53)

The Goodman equation may also be used to determine an equivalent elastic strain in

terms of the total strain amplitude. A combined form of the Goodman equation and εa-

Nf curve equation is indicated by Equation (1.54) [5]. A method of accounting for

means stress effects called the Smith-Watson-Topper (SWT) parameter was proposed

by Smith, Watson and Topper which relates the mean stress effect on fatigue life to a

product of maximum normal stress σmax and strain amplitude εa [5]. The SWT

parameter may be formulated for low cycle fatigue in terms of strain, or for high cycle

fatigue in terms of stress. The general form of the SWT parameter is indicated by

Equation (1.55) for low cycle fatigue, expressed in terms of the total strain equation.

Various forms incorporating mean stress effects have also been proposed for energy

based criteria by Ellyin and Kujawski [27], and Golos [28], to incorporate mean stress

effects with the total strain energy approach subject to general multiaxial stress

conditions.

σm

σa

σar

σu

Goodman

Gerber

fσ′

Introduction

34

( ) ( )cf

bc

f

mf

bf

f

mfa N

σσ

εNσσ

Eσ

ε 2121

′−′+

′−

′= (1.54)

( ) ( ) ( )

′+

′′= c

ffb

ffb

ffa NεNEσ

Nσεσ 222max (1.55)

1.2.2.6 Non-Proportional Loading Influence

Non-proportional loading has been shown to have a significant effect on the fatigue

life of metal alloys, particularly due to low cycle fatigue where there is significant

plastic deformation. Experimental data obtained from various researchers has revealed

that, with increase in the degree of non-proportionality, there is an associated increase

in strain hardening within a metal alloy, and hence non-proportional loading has a

direct bearing on the fatigue life [22]. Under proportional loading, the equivalent

stress orientation remains constant throughout the fatigue life, resulting in strain

hardening due to slip along favourably orientated grains. When there is out-of-phase

loading, the equivalent stress orientation changes in accordance with the degree of

non-proportionality, resulting in additional strain hardening due to the occurrence of

slip in different directions. A comparison of proportional and non-proportional

loading situations and subsequent material stress-strain behaviour is illustrated by

Figure 1.2.20 [22]. The strain hardening phenomena was characterised by the research

of Lamba and Sidebottom [29], where it was revealed from experimental data that the

stable material behaviour from additional strain hardening due to non-proportional

loading would remain regardless of preceding loading history of equal or lower strain

magnitudes, providing the subsequent strain paths remains enclosed by the previous

strain path. This phenomena was termed “erasure of memory property” from the

research of Lamba and Sidebottom [29].

Introduction

35

Figure 1.2.20. Strain hardening due to non-proportional loading.

1.2.2.7 Damage Accumulation

The concept of damage accumulation forms an integral part of fatigue life assessment.

For a constant cyclic stress state, the fatigue life may be determined directly from the

predicted number of fully reversed cycles until failure. For the accumulation of

damage due to varying cyclic stress states, a damage accumulation law is required to

determine the remaining life by summation of the irreversible damage attributed to

each cyclic stress state. Several forms of damage accumulation law have been

proposed. The linear damage accumulation law, first proposed by Palmgren and later

adopted by Miner, is the most widely used form of damage accumulation and is

commonly referred to as the Palmgren-Miner rule [5]. The law is defined as a

summation of the ratio of the number of accumulated cycles N to number of cycles to

failure Nf for each cyclic stress state, with the summation equal to unity for fracture as

indicated by Equation (1.56). The Palmgren-Miner rule assumes that the accumulation

of damage for any given cyclic stress state is linear.

Proportional

Non-Proportional

σ

ε

Introduction

36

11

=

∑ =

ji

ifNN (1.56)

Experimental evidence from various researchers has since revealed that the damage

accumulation process in metal alloys is a highly non-linear process. The research of

Halford [30] in particular has shown that damage accumulation becomes increasingly

non-linear with increased life, and that a linear damage accumulation law may result

in highly non-conservative life estimates by factors ranging from five to ten. Various

forms of non-linear damage accumulation laws have been proposed which account for

accumulation of damage for a given cyclic stress state. The general form of these

damage accumulation laws is indicated by Equation (1.57) [22,30], expressed as the

summation of damage due to cyclic loading at each cyclic stress state, with the

summation equal to unity for failure. A comparison of the general forms of the

Palmgren-Miner rule and non-linear damage rule is illustrated by Figure 1.2.21 [30],

displaying the progressively non-linear form of the non-linear damage rule with

increasing life in terms of the normalised accumulated damage df.

1

1

3

2

2

1

1

1

2

2

1

1 =+

++

+

−

−

−

jf

j

NN

g

jf

j

NN

g

f

NN

g

f NN

NN

...NN

NN

fj

fj

f

f

f

f

(1.57)

Introduction

37

Figure 1.2.21. Typical damage accumulation curves derived from Palmgren-Miner and non-linear

damage accumulation rules.

0.5 0 1

0

0.5

1

fNN

df

Palmgren-Miner Rule

Non-Linear Rule

103 104

105 106

Introduction

38

1.3 Objectives

The primary objective of this thesis is to advance means of failure assessment of

metal alloys through the development of continuum mechanics based relationships. A

basic science approach is to be followed throughout this study to conduct exploratory

research into the fundamental relationships responsible for metal failure. The

development of alternative failure theories or the extension of previously existing

failure theories is to be conducted as part of this research.

In terms of monotonic loading until failure, the fundamental relationship between

hydrostatic stress and ductility is to be investigated. From the literature a relationship

between hydrostatic stress and plastic strain was clearly evident, with this relationship

determined to be linear for hydrostatic compression. A research program is envisaged

to verify the existence of this relationship for hydrostatic tension for a selection of

metal alloys based on previous research conducted by Glass and West [9]. The

applicability of a fracture criterion based on a relationship between hydrostatic stress

and plastic strain is to be investigated, in particular the linear form proposed by

Bridgman [6]. The effects of hydrostatic tensile stress on the resulting fracture mode

of a metal alloy are to be considered.

For the case of cyclic loading until failure, the energy based approach is to be

considered for further investigation, in particular the plastic strain energy approach of

the form proposed by Garud [24] and Ellyin [25]. As was demonstrated from the

literature, the energy based method provides a scalar approach to fatigue life

characterisation invariant of the applied cyclic stress state, allowing application to

complex multiaxial states of stress. Given that plastic strain energy is responsible for

plastic deformation, the hypothesis that irreversible damage may be attributed entirely

to plastic deformation is to be considered. Means of applying the plastic strain energy

approach to high cycle fatigue through the determination of the existence of plastic

strain energy are to be explored.

Introduction

39

A desirable outcome of the research is the development of continuum mechanics

based failure theories which allow application of numerical analysis methods, in

particular finite element analysis (FEA). The finite element method is continuum

mechanics based, essentially modelling a continuous object in terms of a mesh

consisting of discrete, finite objects or elements. Non-linear elastic-plastic analysis

incorporating non-linear geometry and non-linear material behaviour allows the finite

element technique to be applied to situations where significant amounts of plastic

deformation may be encountered, subject to the applied loads and imposed boundary

conditions. The incorporation of numerical analysis techniques in verification of

experimental results would demonstrate the potential application, and hence is

considered an essential component of this thesis.

Introduction

40

1.4 Approach

Research throughout this thesis is to be conducted as two separate studies which

explore specific aspects of the metal failure spectrum, namely failure due to

monotonic loading, and fatigue failure due to cyclic loading. The study of failure due

to monotonic loading will specifically explore the relationship between hydrostatic

stress and ductility for hydrostatic tensile stresses. The fatigue failure due to cyclic

loading research will examine the fatigue phenomena associated with high cycle

fatigue to verify the existence of plastic strain energy, and hence validate the

application of the plastic strain energy approach to the high cycle fatigue regime.

The failure due to monotonic loading and fatigue failure due to cyclic loading

research will be presented as two separate studies. Each study will consist of concept

development followed by an experimental program, analytical program, and

concluding with analysis and discussion. The concept development will extend further

on the literature presented, and develop specific hypotheses subject to verification.

The experimental program will be presented and conducted to obtain specific

experimental results relevant to the concept development. The analytical program will

be conducted incorporating numerical techniques to verify the results obtained from

experiments. Finally, an analysis and discussion will follow to critically examine the

results, verify the proposed theories and present findings and observations.

41

2. FAILURE DUE TO MONOTONIC LOADING

Failure due to Monotonic Loading

42

2.1 Research Methodology

2.1.1 Concept Development

2.1.1.1 Effects of Hydrostatic Stress on Ductility

The existence of a relationship between hydrostatic stress and ductility was amply

demonstrated from the literature. The work of Bridgman [6], Brownrigg et al. [7] and

Lewandowski and Lowhaphandu [8] clearly illustrated the existence of a linear

relationship between hydrostatic stress and fracture strain for a variety of metal alloys.

The linear relationship exhibited formed the basis for the equation of ductility

proposed by Bridgman [6], indicated by Equation (1.21). Numerous failure criteria

have since been proposed which incorporate the observed hydrostatic stress effects on

ductility.

Given that an increase in hydrostatic compression has the effect of increasing

ductility, it is logical to conclude that an increase in hydrostatic tension would have

the effect of decreasing ductility. This notion was presented by Bridgman [6] in

relation to research conducted to investigate the effects of non-uniformities of stress at

the neck of tensile specimens. Experiments and analyses were performed on steel

cylindrical tensile specimens to investigate the state of stress associated with the

observed necking phenomena present in ductile metal alloys. An analytical

approximation of the state of stress present at the fracture cross-section revealed that

the von Mises stress and associated equivalent plastic strain were uniform across the

fracture cross-section, whereas the hydrostatic tensile stress reached a peak value at

the axis of symmetry. From experiments performed by numerous researchers

including Bridgman [6], it was concluded that fracture first occurs at the axis of

symmetry of a necked tensile specimen. The combination of maximum hydrostatic

tensile stress and occurrence of fracture at the axis of symmetry is in agreement with

the notion of reduced ductility with increased hydrostatic tension. The assumption of


43

strain uniformity across the necked region used in the derivation of the analytical

formulae was confirmed from experiments conducted by Bridgman [6] and

independently by Davidenkov and Spridonova [31] investigating the uniformity of the

cross-section area reduction.

The monotonic loading experiments conducted on notched specimens by Glass and

West [9] revealed a strong relationship between hydrostatic stress and ductility. The

specimen materials selected for the experiments were free-cutting brass, 6061-T651

aluminium, 4340 steel, 1080-O (high purity) aluminium and gray cast iron. The

materials chosen provided tensile properties ranging from strong and tough (4340

steel) to soft and ductile (1080 aluminium), as well as inherently brittle (gray cast

iron). From the uniform test section geometry of the original tensile specimens

illustrated by Figure 2.1.1, two types of notched specimens were produced, namely a

90 degree circumferential V-notch specimen and a transverse hole specimen, as

illustrated by Figures 2.1.2.

Figure 2.1.1. Uniform section specimen geometry.

Figure 2.1.2. Notched specimen geometry: (a) transverse hole; (b) 90° circumferential V-notch.

φ 12.85

63.5 25.4 25.4

90°

φ 6.35

R 0.1

φ 6.35

(a) (b)


44

From the experimental results obtained from monotonic tensile loading, the 1080

aluminium specimens exhibited ductile behaviour, whereas the gray cast iron

specimens exhibited brittle behaviour. The remaining materials, namely free-cutting

brass, 6061-T651 aluminium and 4340 steel, exhibited ductile behaviour for the

uniform and transverse hole specimens, but brittle behaviour for the V-notch

specimen. When the engineering stress calculated at the smallest cross section was

plotted against displacement for these materials, the V-notch specimens displayed a

significantly greater nominal stress σn at fracture compared to the other specimen

types, as outlined by Table 2.1.1.

Table 2.1.1. Failure load, nominal stress and deflection data.

Specimen Type Plain V-Notch Transverse Hole Material P (N) σn (MPa) δ (mm) P (N) σn (MPa) δ (mm) P (N) σn (MPa) δ (mm)

Free-Cutting Brass 53000 408.677 14.3427 23350 718.441 0.2056 26200 391.91 0.970436061 Aluminium 46900 361.64 2.69336 21250 653.828 0.38242 25200 376.952 0.55923

4340 Steel 11300 871.329 0.39064 54800 1686.11 0.34952 49000 732.962 0.41121080 Aluminium 9750 75.1811 17.2704 3950 121.535 2.42608 - - - Gray Cast Iron 27100 208.965 0.3084 11000 338.452 0.07813 - - -

Non-linear elastic-plastic finite element analyses were conducted to determine the

state of true tress present at the fracture cross-section. From close correlation of load-

displacement curves between the experiments and analysis, the triaxial state of stress

was obtained for the fracture cross-section of the V-notch specimens. The stress state

at the fracture cross-section consisted of a high hydrostatic tension value σh present

from a small distance inward from the free surface to the axis of symmetry, as

illustrated by Figure 2.1.3 (a) for free-cutting brass. The corresponding equivalent

uniaxial plastic strain pε was greatest at the free surface, followed by a sharp drop to

a smaller value for the remainder of the cross-section, as indicated by Figure 2.1.3 (b)

for free-cutting brass. In these figures, the dashed line indicates the location of the

free surface. The equivalent plastic strain value for the majority of the cross-section

was well below the fracture strain for a uniform section specimen tensile test. In

comparison to the uniform section and transverse hole specimens, the V-notch

specimen exhibited a significantly greater hydrostatic tensile stress value and lower

equivalent plastic strain for the majority of the fracture cross-section. The triaxial state


45

of stress present in the V-notch specimen also accounts for the significantly higher

value of nominal engineering stress σn obtained from the experimental results.

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5 3 3.5

r (mm)

h (M

Pa)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.5 1 1.5 2 2.5 3 3.5

r (mm)

op

Figure 2.1.3. Free-cutting brass stress-strain state at failure: (a) hydrostatic stress vs. radius; (b)

equivalent plastic strain vs. radius.

Further evidence of the hydrostatic stress influence on ductility was obtained from the

fracture surface appearances exhibited by the free-cutting brass, 6061 aluminium and

4340 steel specimens. The uniform section and transverse hole specimens displayed a

typical cup-cone like fracture appearance normally associated with ductile behaviour.

The V-notch specimens exhibited a flat, shiny fracture surface normally associated

with brittle behaviour. Figure 2.1.4 illustrates the observed fracture surfaces for the

uniform and V-notch specimens of 4340 steel.


46

Figure 2.1.4. Fracture cross-sections for 4340 steel: (a) uniform section specimen; (b) 90°

circumferential V-notch specimen.

In terms of void growth and coalescence in metal alloys, hydrostatic stress may be

deemed to have a significant influence on the opening and closing of voids. The

observed influence of hydrostatic stress on the effectiveness of voids is directly

incorporated in numerous porous metal plasticity criteria, including the modified yield

criterion proposed by Tvergaard [14], as indicated by Equation (1.25). Hydrostatic

stress may be directly related to volumetric strain as indicated by Equation (1.17),

where the volumetric strain is a measure of volume deformation within a material. As

illustrated by Figure 2.1.5, hydrostatic tension would have the effect of opening a

crack, hence increasing the effectiveness of the crack geometry and promoting

subsequent crack growth. Conversely, hydrostatic compression would have the effect

of closing a crack, hence decreasing the effectiveness of the crack geometry and

retarding crack growth, as illustrated by Figure 2.1.6. Given the demonstrated

independence of the hydrostatic stress component from plastic flow within metal

alloys, hydrostatic stress may be viewed as a stress component which determines the

fracture strain of a given stress-strain curve, and hence would provide a logical

continuum mechanics based parameter for a monotonic failure criterion.


47

Figure 2.1.5. Effect of hydrostatic tension on crack geometry.

Figure 2.1.6. Effect of hydrostatic compression on crack geometry.

2.1.1.2 Proposed Fracture Criterion

Based on the observed relationship between hydrostatic stress and fracture strain, a

monotonic failure criterion is proposed which relates hydrostatic stress σh to the

equivalent plastic fracture strain fpε . The proposed form of the criterion is indicated

by Equation (2.1), where the equivalent uniaxial plastic fracture strain fpε is

expressed as a function φ of the hydrostatic stress σh.

( )hp fσφε = (2.1)


48

The equivalent plastic fracture strain fpε is the equivalent uniaxial plastic strain pε

at fracture, taking the von Mises criterion form and expressed in terms of the principal

plastic strain components 1pε ,

2pε and 3pε as indicated by Equation (2.2).

( )2223213

2pppp εεεε ++= (2.2)

In terms of the hydrostatic stress component σh, hydrostatic tension may be

represented by a positive value of σh, whilst hydrostatic compression may be

represented by a negative value of σh. From the experimental evidence gathered from

the literature, the general form of the fracture criterion would be one which depicts a

monotonically decreasing value of equivalent plastic fracture strain fpε with

increasing hydrostatic stress σh. A general form of the fracture criterion is illustrated

by the fracture curve of Figure 2.1.7, plotted on fpε -σh axes.

Figure 2.1.7. General form of predicted equivalent plastic fracture strain vs. hydrostatic stress curve.

Given a material stress-strain relationship determined from the equivalent uniaxial

stress-strain curve, the proposed fracture criterion would determine the fracture strain

according to the imposed state of hydrostatic stress. The demonstrated independence

σ h

fpε

( )hp σεf

φ=


49

of hydrostatic stress from plastic flow in homogeneous, isotropic metal alloys allows

the stress-strain behaviour to be expressed in terms of the equivalent (von Mises)

stress σ and equivalent strain ε , indicated by Equations (2.3)-(2.4) respectively. For

a typical equivalent stress-strain curve illustrated by Figure 2.1.8, the equivalent

plastic strain at fracture fpε would be determined directly from the fracture criterion

of Equation (2.1).

( ) ( ) ( )213

232

2212

1 σσσσσσσ −+−+−= (2.3)

pEε εσ += (2.4)

Figure 2.1.8. Typical equivalent stress-strain curve depicting equivalent plastic fracture strain

fpε .

Given the invariant form of the hydrostatic stress, equivalent stress and equivalent

strain parameters, the proposed fracture criterion in combination with the equivalent

stress-strain curve would allow fracture determination for the general multiaxial state

of stress. Assuming the linear form of the equation of ductility proposed by Bridgman

[6], a possible expression for the fracture criterion is depicted by Equation (2.5) and

illustrated by Figure 2.1.9, expressing equivalent plastic fracture strain fpε in terms

σ

ε

E

σo

σ∆

ε∆

fpε


50

of hydrostatic stress σh and constants opε and chσ .

opε would represent the

equivalent plastic fracture strain at zero hydrostatic stress, and chσ the critical

hydrostatic stress value required to cause a purely brittle failure without any

associated plastic deformation. The possibility of fracture occurring at zero strain due

to a hydrostatic tensile stress resulting in a purely brittle failure was discussed by

Bridgman [6], the notion of which forms a fundamental component of the failure

criterion proposed by Oh [12].

ophch

opp f

εσσε

ε +−= (2.5)

Figure 2.1.9. Possible linear form of equivalent plastic fracture strain vs. hydrostatic stress curve.

chσ σ h

fpε

opε

oc

o

f phh

pp εσ

σε

ε +−=


51

2.1.2 Experimental Program

Inspired by the experimental work conducted by Glass and West [9], an experimental

program consisting of uniaxial tensile testing of notched specimens was undertaken to

determine the relationship between hydrostatic tensile stress and equivalent plastic

fracture strain. The program focused on the testing of two normally ductile metal

alloys which exhibited notch-dependent brittle behaviour in earlier experimental

work, namely free-cutting brass (C36000) and 4340-HR steel. Typical nominal

mechanical properties characteristic of these metal alloys, including modulus of

elasticity E, Poisson’s ratio ν, yield stress σo, ultimate tensile stress σUTS and ductility

εf are presented in Table 2.1.2 [5,32,33].

Table 2.1.2. Typical nominal mechanical properties.

Material E (GPa) ν σo (MPa) σUTS (MPa) εf Free-cutting brass 97 0.35 124-310 338-469 0.18

4340 steel 207 0.293 825-1670 965-1875 0.1-0.19

Two cylindrical specimen configurations were produced for testing, namely a uniform

section specimen and a 90 degree circumferential V-notch specimen. The geometry of

the uniform specimens is illustrated by Figure 2.1.10, manufactured from 15 mm

diameter cylindrical bars of 120 mm length. The test section consists of a 55 mm

length of 10 mm uniform diameter. A typical uniform section specimen is illustrated

by Figure 2.1.11 for 4340 steel.

Figure 2.1.10. Uniform section specimen geometry.


52

Figure 2.1.11. Uniform section specimen.

The 90 degree circumferential V-notch specimens were produced consisting of

various inner and outer diameters forming the notch geometry. Figure 2.1.12 outlines

the general configuration of the V-notch specimens, depicting the outer diameter,

inner diameter and notch root radius. The V-notch specimens were manufactured

from 15 mm diameter cylindrical bars of 120 mm length, with finished specimen

outer diameters ranging from 8 mm to 15 mm. A typical V-notch specimen is depicted

by Figure 2.1.13 for 4340 steel with an outer diameter of 15 mm and inner diameter of

7.5 mm. The outer diameters Do and corresponding inner diameters Di of the tested

specimens are outlined in Table 2.1.3, with each specimen indicated by the notation

Do,Di. Each circumferential V-notch specimen consisted of a notch root radius in

the vicinity of 0.1 mm as confirmed by measurement using a Nikon V-12 profile

projector with 20 × zoom magnification.


53

Figure 2.1.12. 90° circumferential V-notch specimen geometry.

Figure 2.1.13. 90° circumferential V-notch specimen.

Table 2.1.3. V-notch specimen configurations with reference to Figure 2.1.12.

Do (mm) Di (mm) iDoD ,

15 4.5 15,4.5 15 6 15,6 15 7.5 15,7.5 15 9 15,9 15 10.5 15,10.5 12 6 12,6 10 5 10,5 8 4 8,4


54

For the purpose of specimen testing, 100 kN and 500 kN Instron servohydraulic

uniaxial testing machines were used as typically illustrated by Figure 2.1.14.

Monotonic, uniaxial tensile loading was applied to each specimen until failure

occurred. The specimens were held in place at each end by friction clamp type grips.

Testing was carried out under load control with the lowest achievable ramp loading

rate of 0.74 kN/s in an attempt to minimise load rate effects on the material behaviour.

For measurement of deflection a 25 mm gauge length strain extensometer was used.

Load-displacement data was acquired via a PC-based National Instruments PCI-

6021E 12-bit analog-to-digital data acquisition card with data acquisition software

developed specifically for the application. The load-displacement data was recorded at

a sampling rate per channel of 25 Hz. Figure 2.1.15 depicts a clamped 4340 steel V-

notch specimen with the extensometer attached prior to testing.

Figure 2.1.14. Instron servohydraulic uniaxial testing machinery.


55

Figure 2.1.15. Clamped test specimen with extensometer.


56

2.1.3 Analytical Program – Equivalent Stress-Strain Curve

2.1.3.1 Equivalent Stress-Strain Curve Determination to Point of Necking

Using the load-displacement data obtained from the uniform section specimen testing,

equivalent stress-strain curves were analytically derived and verified for each metal

alloy. The derivation of the equivalent stress-strain curves from the load-displacement

curves firstly required the determination of the engineering stress-engineering strain

curves. Engineering stress is calculated from the applied load P and the original cross-

section area A, while engineering strain is determined from the original length L and

change in length ∆L, as indicated by Equations (2.6)-(2.7) respectively. The general

form of the resulting engineering stress-engineering strain curve is illustrated by

Figure 2.1.16.

AP=σ (2.6)

LLL δε =∆= (2.7)

Figure 2.1.16. Engineering stress-strain curve.

σ

ε

E

σo

∆σ

∆ε

fε

σUTS


57

From the engineering stress-engineering strain curves, true stress-true strain curves

were determined to the point of ultimate tensile stress σUTS, beyond which necking of

the specimen begins and the assumption of uniaxial stress is invalid. Based on the

constant volume assumption, the true stress σ accounts for the change in cross-

section area, and may be determined from the engineering stress and engineering

strain according to Equation (2.8). True strain ε accounts for the instantaneous

changes in length, and may be evaluated from the engineering strain as indicated by

Equation (2.9). The range of application of the true stress and true strain formulae to

the engineering stress-engineering strain curve is illustrated by Figure 2.1.17. For

stress prior to yield, the true stress may be assumed to be equal to the engineering

stress, as the difference between engineering stress and true stress for small strains is

negligible.

( )εσσ += 1 (2.8)

( )εε += 1ln (2.9)

Figure 2.1.17. Range of application of true stress and true strain formulae.

σ

ε

Yield Necking Begins

( )ε1σσ +=

( )ε1lnε +=

Failure

σσ =


58

2.1.3.2 Bridgman Approximation of Stress State in Necked Region

To determine the true stress-true strain curve beyond ultimate tensile stress, an

analytical approximation of the stress state in the necked region is required. The

analytical approximation of Bridgman [6] is employed here, based on the assumptions

of rotational symmetry of geometry and strain uniformity in the necked region. Figure

2.1.18 represents the axisymmetric geometry of a necked region, defined in terms of

cylindrical coordinates r, θ and z. The stress state may be represented by the normal

stress components σr, σθ and σz, and the shear stress component τrz. The stress

components are related according to the stress equations of equilibrium, indicated by

Equations (2.10)-(2.11).

Figure 2.1.18. Necked region axisymmetric geometry and representative element.

σz

τrz

σr

r1

r2

z

O

r

ψ


59

0=−

+∂

∂+

∂∂

rσσ

zτ

rσ θrrzr (2.10)

0=+∂

∂+

∂∂

rτ

zσ

rτ rzzrz (2.11)

The corresponding normal strain components εr, εθ and εz, and the shear strain

component γrz, may be related to the displacements u and w according to the strain-

displacement relationships of Equations (2.12). Here, u represents r-direction

displacement, and w represents z-direction displacement.

ruεr ∂

∂= ruεθ =

zwε z ∂

∂= (2.12)

rw

zuγrz ∂

∂+∂∂=

From Figure 2.1.18, at a location O on the free surface we may evaluate the boundary

conditions given the radius r and tangent angle ψ. Given the zero stress component

normal to the free surface, the stress equations required to satisfy equilibrium are

outlined by Equations (2.13)-(2.14). Solving these equations simultaneously yields

expressions for σr and τrz in terms of σz and angle ψ, indicated by Equations (2.15)-

(2.16) respectively.

0sincos =− ψψ rzr τσ (2.13)

0sincos =− ψψ zrz στ (2.14)

ψtanzrz στ = (2.15)

0sintancos =− ψψψ zr σσ

ψ2tanzr σσ = (2.16)

The total force P over the cross-sectional area defined by radius r may be determined

from the area integral of σz according to Equation (2.17).


60

∫=r

z drrσπP0

2 (2.17)

At the plane of symmetry representing the smallest cross-section of the necked region,

defined by radius r2, the boundary conditions may be represented according to

Equations (2.18). The conditions σz and τrz are obtained from z-symmetry about the

symmetry plan whereas the σr conditions are defined by the free surface condition and

stress continuity at the axis of symmetry.

0=∂

∂zσ z (0 ≤ r ≤ r2)

0=rzτ (0 ≤ r ≤ r2) (2.18)

0=rσ (r = r2)

θσσ =r (r = 0)

From the presented boundary conditions we may eliminate τrz from Equations (2.10)

and (2.11). Rearranging and solving of the equations yields an expression which

relates the three normal stress components σr, σθ and σz at the plane of symmetry,

indicated by Equation (2.19).

zrσσ

rσ rzθrr

∂∂

−=−

+∂

∂ τ

zσ

rτ zrz

∂∂

−=∂

∂ rzστ z

rz ∂

∂∂

−=∂ drzσ

τr

zrz ∫

∂∂

−=2

0

drzσ

rσσrσ

rr

zθr

r ∫

∂∂

+=+∂

∂ 2

02

2

( )rσ

rσrσr

rrr ∂

∂+=

∂∂

( ) drzσrσrσ

r

rz

θr ∫

∂∂

+=∂∂ 2

02

2

(2.19)

To resolve the state of stress present at the smallest cross-section, we may relate the

normal stress components in terms of the von Mises function to obtain the equivalent


61

stress σ , expressed by Equation (2.20). An equivalent plastic strain pε may also be

determined from the plastic strain components rpε ,

θε p and

zpε using the von Mises

function as indicated by Equation (2.21). The assumption is made here that the

variation in strain across the plane of symmetry is negligible, implying that the

equivalent stress is independent of the radius r.

( ) ( ) ( )222

21

rzzθθr σσσσσσσ −+−+−= (2.20)

( )222

32

zpθprpp εεεε ++= (2.21)

The stress and strain states due to strain hardening may be related by deformation

plasticity theory, where Equations (2.22) analogous to Hooke’s law may be used

defined in terms of plastic modulus Ep. The equations assume a Poisson’s ratio of 0.5

to maintain the constant volume condition. The plastic modulus Ep relates the

equivalent stress σ and equivalent plastic strain pε as indicated by Equation (2.23).

( )

+−= zθr

prp σσσ

Eε

211

( )

+−= rzθ

pθp σσσ

Eε

211 (2.22)

( )

+−= θrz

pzp σσσ

Eε

211

pp ε

σE = (2.23)

Given the constant volume condition present in the necked region according to

Equation (2.24), if we assume that zpε is constant across the plane of symmetry, from

Equations (2.12) we may obtain a function for the radial displacement u. Indicated by

Equation (2.25), the radial displacement u may be determined if we assume the

integration constant c1 is zero to avoid an infinite value at the axis of symmetry.


62

0=++zpθprp εεε (2.24)

0=++∂∂

zpεru

ru

zp-εru

ru =+

∂∂

( )drdRu

drduRRu

drd +=

zpRεruR

drduR −=+

rR

drdR =

∫∫ =rdr

RdR

r R lnln =

R = r

( )zprεru

drd −=

1

2

2c

rεdrrεru zp

zp +−=−= ∫

rcrε

u zp 1

2+−= c1 = 0 (r = 0)

2rε

u zp−= (2.25)

From substitution we may obtain functions which relate rpε and

θε p to

zpε , as

indicated by Equations (2.26)-(2.27).

22zpzp

rpεrε

rε −=

−

∂∂= (2.26)

22 zpzp

θpε

r

rε

ε −=−

= (2.27)


63

Satisfying the conditions imposed by Equations (2.26)-(2.27) implies that σr is equal

to σθ along the plane of symmetry, which in turn satisfies σz. From these conditions,

the strain functions from Equations (2.22) may be defined according to Equations

(2.28).

σr = σθ (0 ≤ r ≤ r2)

( )rzp

zrr

prp σσ

Eσσσ

Eε −−=

−−=

21

221

( )rzp

θp σσE

ε −−=2

1 (2.28)

( )rzp

rrz

pzp σσ

Eσσσ

Eε −=

−−= 1

221

In addition, if we assume the function according to Equation (2.29) which relates σz in

terms of 2rzσ and σr, by substitution into Equation (2.20) we obtain Equation (2.30)

which satisfies the von Mises function by relating 2rzσ equal to the equivalent

uniaxial stress value σ . The stress state implied here consists of a uniform tensile

stress across the plane of symmetry with a superimposed hydrostatic tension which

has a maximum value at the axis of symmetry and is zero at the free surface.

rrzz σσσ += 2 (2.29)

( ) ( )[ ] ( )[ ]222

222

21

rzrrrzrrzrrr σσσσσσσσσσ =−+++−+−= (2.30)

From Equations (2.17) and (2.19), a condition may be imposed on the contour of the

neck given that Equation (2.17) holds for the necked region. According to the diagram

and element of Figure 2.1.19, if we define the necked region in terms of a circle and

approximate one of the principal stress surfaces in terms of a sphere, we may define

the radius of the circle or neck radius ρ as an independent parameter to be determined

from experiment.


64

ϕϕ ′==′ rrρ 2

ϕϕ

=′

2rr

Figure 2.1.19. Geometry and representative element of necked region expressed in terms of cross-

section radius r, neck radius ρ, and angles θ and ϕ.

From a force summation in the radial direction given the element depicted in Figure

2.1.19, we obtain the relationship according to Equation (2.31). From the geometry,

dimensions h and h′ may be determined in linearised form as indicated by Equations

(2.32)-(2.33), assuming angles θ and ϕ are small.

r2 dr r

h h′

ϕ

ρ

ϕ

ρ′

dϕ′ ϕ′

(r + dr) θ r θ

dr

dr

h′ h


65

0=∑ rF

( )

0θsinθ

θsinθ2

=−−

′+

∂∂

++′

+

∂∂

+

h dr σhrσ

h drrrσdrσφdr drr

zσhσ

θr

rr

zz (2.31)

( )φφρρφh coscos −′′+=

( )[ ]φφdφρρφh coscos −′+′′+=′

θθsin ≈

φrrφφ

=′≈′

2sin dr

rφ

ρdrφd

2=

′=′

( )

−+=

−+=

−+=

′−′

+=

+−

′−′+=

2

222

2

22

22

2

22

2222

2

22

2

221

21

rrrρφ

rrφr

ρφ

φrrφ

φrρφ

φφρρφφφρρφh

(2.32)

( ) ( )

( )

+−+=

−−−+=

−−−+=

−

−

−+=

+−′′−′−′

+=

+−

′+′−′+=′

2

222

2

2

22

22

22

2

2

22

2

22

2222

22

222

22

2

22

22222

2

222

22

22

222

12

1

rdrrrρφ

rdr

rrdr

rrrρφ

drrφdr

rrφ

rφrφ

φrρφ

drrφdr

rφφ

rrφ

rrφ

φrρφ

φdφφdφφρρφφφdφρρφh

(2.33)


66

From substitution of Equations (2.32)-(2.33) into the radial force summation of

Equation (2.31), and given the zσ z

∂∂ boundary condition from Equations (2.18), we

obtain the expression indicated by Equation (2.34).

0=∂

∂zσ z

0θ

θθθθθ2

θθ 22

22

=−

−′∂∂+′

∂∂+′+′++

hdrσ

hrσhrσdrhr

rσdrhdrσhrσdrφ

rrσdrφ

rrrσ

θ

rrr

rrzz

0θ2

θ2

θ2

θ2

θ2

θ

2

222

2

222

2

222

2

222

22

222

2

2

=

−+−

−+−

−+

∂∂

+

−++

−

−++

drφr

rrρσrφ

rrr

ρσdrφr

rrρ

rσ

r

drφr

rrρσrφr

rdrr

rrρσdrφrrσ

θrr

rrz

0222 2

222

2

222

2

222

2

2

2

2=

−+−

−++

−+∂∂+

−

r

rrρσr

rrρσr

rrρrσr

rrσ

rrσ θr

rrz

0222

3

2

222

2

222

2

222

2

2=

−+−

−+

∂∂

+

−++

rrr

ρσr

rrρ

rσ

rr

rrρσ

rrσ θ

rrz (2.34)

If we substitute the expressions obtained previously for σθ and σz in terms of σr and

2rzσ into Equation (2.34), we obtain Equation (2.35) which defines σr as a function of

2rzσ . The constant of integration c1 may be resolved from the zero normal stress

condition present at the free surface, producing Equation (2.36). This equation may

then be substituted into the relationship which defines σz in terms of σr and 2rzσ ,

given by Equation (2.29), to obtain σz in terms of 2rzσ as indicated by Equation

(2.37).

θr σσ = rrzz σσσ +=2


67

0222

3

2

222

2

222

2

222

2

2

2

2

2=

−+−

−+∂∂+

−++

+

r

rrρσr

rrρrσr

rrrρσ

rrσ

rrσ r

rrrrz

02 22

222

2=

+

−+rrσ

rrrρ

drdσ

rzr

12

222

2

222

2

2ln

2

22c

rrr

ρσdr

rrr

ρ

rr

σσ rzrzr +

−+=

−+

= ∫ (2.35)

r = r2

02

ln 12

222

2=+

−+ c

rrrρσ rz ρσc rz ln

21 −=

−+=

−

−+=2

22

22

2

222

22lnln

2ln

22 ρrrρrrσρ

rrrρσσ rzrzr (2.36)

−++=

−++=2

22

22

2

22

22

22ln1

22ln

222 ρrrρrrσ

ρrrρrrσσσ rzrzrzz (2.37)

By substituting Equation (2.37) into Equation (2.17), we obtain the following integral

expression for applied load P in terms of 2rzσ . By parametric substitution and

integration by parts, the expression for applied load P is obtained according to

Equation (2.38).

−++

=

−++=

−++=

∫

∫

∫

22

2

2

2

2

2

0 2

22

22

0

2

0 2

22

22

0 2

22

22

22ln

22

22ln2

22ln12

rr

rz

r

rz

r

rz

drρr

rρrrr rπσ

drρr

rρrrr rπσ

drρr

rρrrrσπP

Let q = r2 dq = 2 r dr


68

dqρr

qρrrπσrπσP

r

rzrz ∫

−++=

2

220 2

22

222 2

2ln

[ ] ∫∫ ′−=′2

22

00

0

rr

r

tsstts

Let s′ = 1 s = v

−+=

2

22

2

22

lnρr

sρrrt

sρrrt

−+−=′

22

2 21

[ ] ( ) ( )[ ] 22

2

22

22

2

022

222

20

02

22

2

0 22

2

22

02

22

2

0 22

202

22

2

0 2

22

2

2ln222ln

22

122

ln

222ln

22ln

rrr

rr

rrr

qρrrρrrqρr

qρrrq

dqqρrr

ρrrρr

qρrrq

dqqρrr

vρr

qρrrq dqρr

qρrr

−++−+−

−+=

−++

−−

−+=

−++

−+=

−+

∫

∫∫

[ ] ( ) ( )[ ]

( ) ( ) ( )[ ] ( )

++=

+−−++−=

−++−+−

−++

=

2

22

22

22

22

22

222

222

2

02

22

222

202

02

22

222

22

22ln2

2ln2ln2

2ln222ln

2

2

22

2

2

2

ρrρrrρrrπσ

ρrrrρrrρrrπσ

rρrrρrrrρr

rρrr rπσ

rπσP

rz

rz

rrr

rz

rz

( )

++= 1

2ln2 2

22

22 ρrρrrπσ rz (2.38)

From Equation (2.38), noting that 2rzσ is equal to the equivalent stress σ from the

von Mises function, we may define the equivalent uniaxial stress in terms of the

applied load P, necked radius of curvature ρ and cross-section radius at the plane of

symmetry r2, as indicated by Equation (2.39). Subsequently, if we know the initial


69

radius ir2 , we may obtain the expression for the equivalent strain ε according to

Equation (2.40).

( )

++

=1

2ln2 2

22

2 ρrρrrπ

Pσ (2.39)

=

=

=

2

2

2

2

2

2 ln2ln2lnrr

DD

AA

ε iii (2.40)

From the equivalent stress and equivalent strain expressions obtained by Bridgman at

the plane of symmetry of a necked region, the true stress-true strain curve may be

determined from ultimate tensile stress to failure, as illustrated by Figure 2.1.20.

Figure 2.1.20. Range of application of true stress, true strain and Bridgman approximation formulae.

σ

ε

Yield Necking Begins

( )ε1σσ +=

( )ε1lnε +=

Failure

σσ =

( )

++

=1

2rlnr2rπ

Pσ2

ρρ

=

rr2lnε i


70

2.1.3.3 General Form of Equivalent Stress-Strain Curve

From application of the equations depicted in Figure 2.1.20 to the load-displacement

curve, a true stress-true strain or equivalent stress-strain curve may be obtained which

describes the material stress-strain behaviour to the point of fracture. The general

form of the equivalent stress-strain curve is depicted by Figure 2.1.21, defined in

terms of the modulus of elasticity E, yield stress oσ and true fracture strain fε .

Figure 2.1.21. Equivalent stress-strain curve.

For many metal alloys, the stress-strain relationship beyond yield is linear when

viewed on log-log axes. As a matter of mathematical convenience, the strain

hardening region of the equivalent stress-strain curve beyond yield stress oσ may be

approximated in terms of a power law relationship. The general form of the power law

relationship is depicted by Equation (2.41), where equivalent stress σ is defined as a

function of the equivalent strain ε , strength coefficient H and strain hardening

exponent n.

nHεσ = (2.41)

σ

ε

E

oσ

σ∆

fε

ε∆


71

The power law relationship approximation of the stress-strain behaviour beyond yield

allows the stress-strain relationship to be predicted or approximated beyond fracture

strain obtained from a uniaxial tensile test. For a homogeneous, isotropic metal alloy,

the equivalent stress-strain curve may be completely expressed in terms of Hooke’s

law for the elastic region prior to yield, and by the power law relationship for the

plastic region beyond yield, as indicated by Equations (2.42)-(2.43) respectively.

εσ E= (σ ≤ oσ ) (2.42)

nHεσ = (σ > oσ ) (2.43)

2.1.3.4 Video Imaging Technique for Bridgman Approximation

To obtain the necked geometry of the specimen during uniaxial loading, a video

imaging technique was developed. Recording of the deforming specimen test section

was accomplished via a tripod-mounted VHS video camera as illustrated by Figure

2.1.22. The camera was positioned level with the specimen test section with a

horizontal distance between the lens and specimen of approximately 200 mm,

enabling a 6 × zoom magnification to be used. A spirit level was used to ensure

horizontal positioning of the camera. A white background was placed behind the

testing machine to reduce background interference and to improve image clarity. A

high intensity light was used to intensify the image of the specimen.


72

Figure 2.1.22. Video imaging equipment.

Synchronised recording of video image and load-displacement data was achieved by

triggering the recording mechanisms of the VHS video camera and data acquisition

software simultaneously. The video image was converted to a PC-based Windows

AVI video file using a Pinnacle DC-10 video capture card and Pinnacle Studio

Version 7 movie editing software. The digitised video image was captured with 720 ×

576 resolution at a sample rate of 25 Hz, allowing a direct correlation between the

video images and load-displacement data. The individual video frames representing

the undeformed specimen geometry at zero load and the deformed specimen test

section beyond ultimate tensile stress were converted to 1500 × 1125 resolution

Windows BMP image files, as illustrated by Figure 2.1.23.


73

Figure 2.1.23. Necked specimen image.

The necked specimen geometry was determined from the images using AutoCAD

software. From the BMP image files, each image depicting the necked geometry was

scaled and dimensioned according to the known geometry of the initial undeformed

specimen image. Using this procedure, the neck radius of curvature and cross-section

radius were determined corresponding to the load-displacement data at each time

interval. The general procedure illustrating the comparison between the undeformed

and necked specimen images is outlined by Figure 2.1.24.


74

Figure 2.1.24. Dimensioned specimen geometry: (a) undeformed; (b) deformed (necked).

2.1.3.5 Finite Element Modelling

For comparison with the experimental results and to verify the stress-strain behaviour

at fracture, finite element mesh models of the uniform section specimen for each

material were constructed for detailed finite element analysis. The mesh modelling

was conducted using Femcad 2000 finite element mesh modelling software [34],

taking into account specimen geometry, material properties, applied loads and

(b)

(a)


75

constraints. The finite element analysis was conducted using Abaqus finite element

analysis software [35].

For modelling of the uniform section specimen, an axisymmetric model was

generated using 8-node axisymmetric solid elements as illustrated by Figure 2.1.25.

To induce necking at the point of instability where ultimate tensile strength is reached,

a small geometric imperfection was incorporated at the free surface along the plane of

symmetry of the model. The geometric imperfection was in the form of a

circumferential notch, with geometry and dimensions as displayed by Inset A of

Figure 2.1.25. The dimensions were obtained via an iterative mesh generation and

analysis process such that the notch geometry required to induce necking was

minimised.

Figure 2.1.25. Uniform section specimen finite element model, geometry, loads and constraints.

A displacement δ was applied to the specimen end and constraint boundary conditions

were applied to the plane of symmetry, as illustrated by Figure 2.1.25. The effects of

the applied displacement and the notch geometry on the necked region in terms of the

R 5 mm

12.5 mm

δ

0.2 mm

0.05 mm Inset A Inset A


76

model dimensions were accounted for in accordance with Saint-Venant’s principle

[36], which implies that the localised effects of an applied load, boundary condition or

geometric feature are negligible outside the region enclosed by a characteristic

dimension. In the case of the depicted finite element mesh model, the specimen

diameter can be considered a dimension characteristic of the geometry where the

displacement is applied, while the notch dimensions are characteristic of the region

which encloses the localised stress concentration. The finite element mesh model

length of 12.5 mm was specified in accordance with these localised effects and to

coincide with the extensometer clip gauge half length.

Material properties for the finite element model were defined in accordance with

experimentally derived data. The stress-strain behaviour of each material was

specified in terms of modulus of elasticity E, Poisson’s ratio ν and an equivalent

stress-equivalent plastic strain curve defined beyond yield to a true strain of unity. For

ductile metal alloys, large plastic strains are generally present in the necked region,

associated with a raised hydrostatic tension near the axis of symmetry. Significant

void nucleation, growth and coalescence would be expected under such conditions,

and as such the porous metal plasticity model of Tvergaard [14] was incorporated to

simulate the resulting inhomogeneous material behaviour. Indicated by Equation

(2.44), the porous metal plasticity criterion essentially modifies the yield surface of

the fully dense material in terms of the hydrostatic stress σh, void volume fraction fv

and material constants q1, q2 and q3. Typical values for q1, q2 and q3 range between 1

to 1.5 for q1, 1 for q2 and 1 to 2.25 for q3 as indicated by the literature [13,14], where a

value of 1 for q1, q2 and q3 indicates spherical void geometry and recovers the original

form of the criterion proposed by Gurson [13].

( ) 012

3cosh2 2

32

1

2

=+−

+

=Φ v

M

hv

M

fqq

fqσ

σσσ (2.44)

In addition to the material constants, the porous metal plasticity algorithm of Abaqus

requires the specification of additional parameters which statistically determine void

initiation and growth in accordance with the criterion. Assuming a normal distribution

of the nucleation strain, the void initiation, growth and coalescence is determined


77

from the mean nucleation strain εN, standard deviation of the nucleation strain sN, and

volume fraction of nucleated voids fN according to the void fraction nucleation rate

nuclvf& relationship of Equation (2.45) [35]. Void growth fraction rate grvf& is

determined in accordance with the current void fraction fv and volumetric plastic

strain rate pvε& as indicated by Equation (2.46) [35]. The total change in void volume

fraction vf& in accordance with void nucleation nuclvf& and void growth grvf& is

indicated by Equation (2.47) [35]. In these expressions, the matrix equivalent plastic

strain pMε represents an equivalent plastic strain in the matrix material according to

equivalent plastic work relationship of Equation (2.48) [35]. The growth of voids due

to hydrostatic tension results in a net softening of the material.

2

21

2

−−

= N

NpM

s

N

Nnuclv e

sf

f

εε

π& (2.45)

( ) pvvgrv ff ε&& −= 1 (2.46)

grvnuclvv fff &&& += (2.47)

( ) ∑=

=−3

11

iipipMMvf εσεσ && (2.48)

The resulting mesh model produced by Femcad 2000 software is illustrated by Figure

2.1.26, displaying significant detail of the geometric imperfection or notch. The finite

element mesh model consisted of 6427 nodes and 2060 8-node axisymmetric solid

elements.


78

Figure 2.1.26. Femcad 2000 finite element mesh model of uniform section specimen.

For the model analysis, quasi-static non-linear, elastic-plastic analyses using Abaqus

Standard were conducted with the inclusion of non-linear geometry. Reduced

integration quadratic axisymmetric elements were used in the analysis due to the

ability of these elements to handle near-incompressible (perfectly plastic) behaviour

as a result of large plastic strains. Automatic load incrementation was used throughout

the analyses, with load incrementally increased until a converged result was achieved

with the total specified displacement δ applied to the model.

2.1.3.6 Comparison of Experimental Results with Finite Element Analysis

The accuracy of the finite element analyses were verified by comparison of the load-

displacement curves obtained from experiment and analysis. An iterative technique

was adopted here whereby the porous metal plasticity parameters and equivalent

stress-strain curve beyond necking were adjusted in the finite element model until the

load-displacement curve obtained from the finite element analysis closely matched the

Notch


79

load-displacement curve derived from experiment. Verification of the equivalent

stress-strain curve was obtained once a close correlation existed between the load-

displacement curves and necked geometry obtained from experiment and analysis.

From a close correlation between results obtained from experiment and analysis, the

stress-strain state at the plane of symmetry of the necked region was determined

corresponding to failure of the specimen. The stress-strain state present at the fracture

cross-section was presented in terms of curves which illustrate variation of the stress

or strain state with respect to radius r, namely equivalent stress σ , hydrostatic stress

σh and equivalent plastic strain pε . The expected general form of the σ -r, σh-r and

pε -r curves for the uniform section specimen based on the Bridgman approximation

are illustrated by Figures 2.1.27-2.1.29 respectively, with zero radius indicating the

specimen axis of symmetry and the dashed line indicating the free surface. The pε -r

curve depicted here accounts for the slightly increased value of pε expected towards

the axis of symmetry due to void growth and coalescence.

Figure 2.1.27. σ -r curve, uniform section specimen.

σ

r


80

Figure 2.1.28. σh-r curve, uniform section specimen.

Figure 2.1.29. pε -r curve, uniform section specimen.

The σ -r, σh-r and pε -r curves allow determination of the stress-strain state at

fracture, assuming occurrence of failure at the axis of symmetry. To confirm the

Bridgman approximation and the assumptions of constant von Mises stress and plastic

strain conditions at the plane of symmetry, a comparison between the σ -r, σh-r and

pε -r curves obtained from the Bridgman approximation and finite element analysis

was to be conducted. Assuming constant σ and ε distributions in accordance with

σh

r

pε

r


81

Equations (2.39)-(2.40), distributions for σh and pε were obtained respectively

according to Equations (2.49)-(2.50). Noting that 2rzσ is equal to σ , and assuming

the equivalence of normal stress components σr and σθ, the normal stress components

σr, σθ and σz may be obtained from Equations (2.36)-(2.37), resulting in Equation

(2.51) for σr and σθ, and Equation (2.52) for σz.

3zr

hσσσσ θ ++

= (2.49)

Epσεε −= (2.50)

−+==

2

22

22

22

lnρr

rρrrσσ r θσ (2.51)

−++=

2

22

22

22ln1ρr

rρrrσσ z (2.52)


82

2.1.4 Analytical Program – Fracture Curve

2.1.4.1 Finite Element Modelling

To verify the experimental results and to determine the stress-strain state at fracture,

finite element mesh models of the 90 degree circumferential V-notch specimens for

each material were produced for detailed finite element analysis. For modelling of the

V-notch specimens, an axisymmetric mesh model was generated in Femcad 2000

using 8-node axisymmetric solid elements as illustrated by Figure 2.1.30. The notch

root radius was modelled with significant detail, with successively finer element sub-

divisions approaching the notch root as displayed by Inset A of Figure 2.1.30.

A displacement δ was applied to the specimen end and constraint boundary conditions

were applied to the plane of symmetry, as illustrated by Figure 2.1.30. The effects of

the applied displacement and the notch geometry in terms of the model dimensions

were accounted for in accordance with Saint-Venant’s principle, whereby the

specimen diameter and notch geometry were considered characteristic of localised

effects. The finite element mesh model length of 12.5 mm was specified in

accordance with these localised effects and to coincide with the extensometer clip

gauge half length.

Material properties obtained from the uniform section specimen testing were

incorporated into the V-notch specimen finite element model. The stress-strain

behaviour of each material was specified in terms of modulus of elasticity E,

Poisson’s ratio ν and an equivalent stress-equivalent plastic strain curve defined

beyond yield to a true strain of unity. The porous metal plasticity model of Tvergaard

[14] was incorporated to simulate the resulting inhomogeneous material behaviour,

defined by the material constants q1, q2 and q3, mean nucleation strain εN, standard

deviation of the nucleation strain sN, and volume fraction of nucleated voids fN in

accordance with Equations (2.44)-(2.48).


83

Figure 2.1.30. V-notch specimen finite element model geometry, loads and constraints.

oo r 2D =

ii r 2D =

ri

ro

Inset A

R 0.1 mm Inset A

δ

12.5 mm


84

The general form of the resulting mesh models produced by Femcad 2000 software

are illustrated by Figure 2.1.31 for a 15 mm outer diameter and 7.5 mm inner

diameter, displaying significant detail of the notch geometry and a progressively finer

mesh towards the notch root. The total number of nodes and 8-node axisymmetric

solid elements for each V-notch specimen mesh model is summarised by Table 2.1.4.

Figure 2.1.31. Femcad 2000 finite element mesh model of 15,7.5 circumferential V-notch specimen.

Table 2.1.4. Circumferential V-notch specimen model summary.

Model Do (mm) Di (mm)

Nodes Elements (8-node axisymmetric)

15 4.5 8741 2800 15 6 9233 2960 15 7.5 9041 2900 15 9 9317 2990 15 10.5 8973 2880 12 6 9041 2900 10 5 9041 2900 8 4 8421 2700

For the model analysis, quasi-static non-linear, elastic-plastic analyses using Abaqus

Standard were conducted with the inclusion of non-linear geometry. Reduced


85

integration or reduced integration hybrid quadratic axisymmetric elements were used

in the analysis due to the ability of these elements to handle near-incompressible

(perfectly plastic) behaviour as a result of large plastic strains encountered in the

notch root vicinity. Automatic load incrementation was used throughout the analyses,

with load incrementally increased until a converged result was achieved with the total

specified displacement δ applied to the model.

2.1.4.2 Comparison of Experimental Results with Finite Element Analysis

Assuming the material properties obtained from the uniform section finite element

models, the accuracy of the V-notch specimen finite element analyses were

determined by comparison of the load-displacement curves obtained from experiment

and analysis. Provided that a close correlation was obtained between the experimental

and finite element load-displacement results, the stress-strain behaviour corresponding

to the fracture load at the plane of symmetry was assumed to be representative of the

stress-strain state required to cause failure of the specimen. Following the convention

adopted in the uniform section specimen analyses, the stress-strain state present at the

fracture cross-section was presented in terms of curves which illustrate variation of

the stress or strain state with respect to radius r, namely equivalent stress σ ,

hydrostatic stress σh and equivalent plastic strain pε . The expected general form of

the σ -r, σh-r and pε -r curves for the circumferential V-notch specimen according to

the experimental work of Glass and West [9] are illustrated by Figures 2.1.32, with

zero radius indicating the specimen axis of symmetry and the dashed line indicating

the free surface.


86

Figure 2.1.32. V-notch specimen stress-strain state: (a) σ -r; (b) σh-r; (c) pε -r.

σ

r σh

r

pε

r

(a)

(b)

(c)


87

2.1.4.3 Comparison of Stress-Strain State with Fracture Mechanics Theory

Following previous experimental work, the circumferential V-notch specimens of the

selected metal alloys were expected to exhibit notch-dependent brittle behaviour

according to the stress-strain state presented by the σ -r, σh-r and pε -r curves of

Figures 2.1.32. For a V-notch specimen exhibiting brittle behaviour, the peak

magnitudes of equivalent stress σ and equivalent plastic strain pε exist at the free

surface, rapidly decreasing towards the axis of symmetry. Hydrostatic tension σh

greatly increases and peaks at a small distance away from the free surface, gradually

decreasing to a steady state value towards the axis of symmetry. The peak magnitude

of σh is greatly increased relative to the magnitude present at the free surface. As was

illustrated by Figures 2.1.32, the comparison between the σ -r, σh-r and pε -r curves

depicts a competing effect in the region between fracture occurring due to high σh and

small pε away from the free surface, or fracture occurring due to lower σh and high

pε at the free surface. For brittle failure, crack initiation and propagation is unlikely

to first occur at the free surface, as brittle failure is associated with small pε . From

the comparison between the σ -r, σh-r and pε -r curves, brittle failure in the case of a

circumferential V-notch specimen would most likely occur at a small distance inward

from the free surface, where the magnitude of σh is greatly increased and the

associated pε magnitude is small.

Given the confined nature of plastic deformation surrounding the notch root, fracture

mechanics theory in the form of LEFM would be considered applicable in the analysis

of this situation. Fracture toughness research conducted by Tetelman and McEvily

[19] has revealed that metal alloys with V-notch geometry subjected to uniaxial

loading and plane-strain conditions may exhibit brittle failure due to the plastic

constraint present in the notch region. Detailed analyses concluded that the

requirements for brittle cleavage fracture are met once a state of stress is reached at a

small distance inward from the free surface at the boundary of the confined plastic

zone. According to Tetelman and McEvily, this location corresponds to the maximum

magnitude of the axial normal stress σz. The initiated crack propagates unstably in


88

opposite directions along the fracture surface, propagating towards the free surface

and towards the axis of symmetry. Similar conditions for brittle failure could be

expected in the circumferential V-notch due to the triaxial constraint present as a

result of the rotational symmetry.

Given the invariant representations of the equivalent stress σ , hydrostatic stress σh

and equivalent plastic strain pε , the stress-strain state present at the crack initiation

site would represent the conditions required for fracture in an invariant form similar to

the critical stress intensity factor KIc. Assuming conditions present in the notch region

of the circumferential V-notch specimen are applicable to LEFM, the critical stress

intensity factor KIc for a V-notch specimen subjected to plane strain conditions may

be represented by the Ritchie-Knott-Rice (RKR) relationship, in terms of the radius ρ,

maximum normal stress maxzσ and yield stress σo, by Equation (2.53) and Figure

2.1.33 [18]. From Equation (2.53), the critical stress intensity KIc derived from each

specimen test would indicate that, for brittle failure conditions, a constant KIc value is

obtained. Verification of the stress-strain results as representative of the fracture stress

state may be determined from obtaining a consistent stress-strain state present at the

crack initiation site, approximated by the location of the maximum axial normal stress

σz at the plane of symmetry.

Figure 2.1.33. Circumferential V-notch.

Do

Di

ρ

Inset A

Inset A


89

−=

−19.2

1max

o

z

eK oIcσ

σ

ρσ (2.53)

Prior to KIc determination, the applicability of LEFM may be determined from

analysis of the load-displacement curve as illustrated by Figure 2.1.34. In accordance

with ASTM standard E399-90 [37], a line tangent to the initial linear region of the

load-displacement curve is produced, indicated by slope µ. A second line is produced

of slope 0.95 µ which intersects the curve to obtain load PQ. The maximum load is

designated Pmax as indicated. According to the standard, a valid KIc is determined

provided that the ratio of Pmax to PQ does not exceed 1.1, as indicated by Equation

(2.54).

Figure 2.1.34. Load-displacement curve for determination of KIc validity.

QPPmax ≤ 1.1 (Valid KIc) (2.54)

P

δ

PQ

Pmax µ 0.95 µ


90

2.1.4.4 Fracture Curve Determination

The general trend of the proposed equivalent plastic fracture strain-hydrostatic stress

curve, or fracture curve, may be determined from the stress-strain data obtained at the

fracture cross-section. The hydrostatic stress and equivalent plastic strain values

obtained at a crack initiation site represent a point on the fracture curve, expressed in

terms of hydrostatic stress σh and equivalent plastic fracture strain fpε as indicated

by Equation (2.1) and Figure 2.1.7. The uniform section specimen and 90 degree

circumferential V-notch specimen configurations represent two possible fracture

points along the curve in the hydrostatic tensile stress region.

Assuming initiation of fracture at the axis of symmetry, the equivalent plastic strain-

hydrostatic stress state along the plane of symmetry for the uniform section specimen

may be represented by the pε -σh curve of Figure 2.1.35. The curve illustrates the pε

-σh relationship in terms of an approximately constant pε value with the lowest

magnitude of σh present at the free surface and highest σh magnitude present at the

axis of symmetry. The curve ends correspond to the free surface and axis of symmetry

as indicated. The σh and pε values obtained at the axis of symmetry represent a

(σh, fpε ) point on the fracture curve.

Figure 2.1.35. Equivalent plastic strain-hydrostatic stress curve for uniform section specimen.

σ h

pε

Free Surface

Axis of Symmetry

),(fph εσ


91

The general form of the equivalent plastic strain-hydrostatic stress state at the plane of

symmetry for the circumferential V-notch specimen may be represented by the pε -σh

curve of Figure 2.1.36. The relationship of the curve to the specimen geometry is

indicated by sections I, II and III. Section I represents the pε -σh relationship present

within the plastic zone near the notch root, indicated by the high magnitude of pε and

low value of σh, with the curve end representing the free surface. Section II represents

the pε -σh relationship at a small distance inward from the free surface, where the

magnitude of pε is lower with a significantly higher σh magnitude which reaches a

peak value as clearly shown. Section III of the curve represents the relationship

present away from the plastic zone towards the axis of symmetry, typified by the low

magnitude of pε and gradually decreasing value of σh, with the curve end

representing the axis of symmetry. Assuming brittle failure, the (σh, fpε ) point on the

fracture curve may be approximately located at a location where maxzσ occurs.

Figure 2.1.36. Equivalent plastic strain-hydrostatic stress curve for circumferential V-notch specimen.

From superposition of the pε -σh data, the (σh, fpε ) points obtained from the uniform

section and circumferential V-notch specimens may be illustrated according to Figure

2.1.37. The relative positioning of the pε -σh curves is indicated here, where brittle

Free Surface

Axis of Symmetry ),(fph εσ

pε

σh

I

II

III maxzσ


92

failure in the V-notch specimen would result in a fracture point significantly different

to that obtained from ductile failure of the uniform section. The expected

monotonically decreasing trend of fpε with increasing σh for the hydrostatic tensile

stress range is depicted here.

Figure 2.1.37. Superposition of uniform section and V-notch equivalent plastic strain-hydrostatic stress

curves.

fpε

σh

Uniform Section

V-Notch


93

2.2 Experiments and Results

2.2.1 Establishment of Equivalent Stress-Strain Curve

2.2.1.1 Experimental Derivation of Equivalent Stress-Strain Curve

From monotonic loading experiments performed on the uniform section specimens,

load-displacement curves were obtained for free-cutting brass and 4340 steel in terms

of the applied load P and extensometer deflection δ. Video images of the deforming

specimen test sections were captured corresponding to the load-displacement data.

The load-displacement curves obtained for free-cutting brass and 4340 steel are

illustrated by Figures 2.2.1-2.2.2 respectively.

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8

δ (mm)

P (k

N)

Figure 2.2.1. Load-displacement curve, uniform section (free-cutting brass).


94

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6

δ

P (k

N)

Figure 2.2.2. Load-displacement curve, uniform section (4340 steel).

From the load-displacement curves, engineering stress-engineering strain curves were

derived according to Equations (2.6)-(2.7). The engineering stress σ was based on the

original cross-section area A according to the undeformed specimen diameter of 10

mm, while engineering strain ε was obtained according to the initial extensometer

length L of 25 mm. True stress-true strain curves were derived from the engineering

stress-strain curves according to Equations (2.8)-(2.9) and Figure 2.1.17 to the point

of ultimate tensile stress, prior to the onset of necking. Beyond the onset of necking,

the Bridgman stress approximation equations were applied to determine the true stress

and true strain at the plane of symmetry of the necked region. Video image frames

captured simultaneously with the load-displacement data were scaled and

dimensioned to obtain the neck radius of curvature ρ and cross-section radius r2.

Images scaled and dimensioned using the AutoCAD software procedure are illustrated

by Figures 2.2.3 for free-cutting brass, and by Figures 2.2.4 for 4340 steel. The

illustrated images depict the initial undeformed specimen geometry and final necked

geometry image immediately prior to specimen failure, indicating cross-section

diameter and radius of curvature dimensions.


95

Figure 2.2.3. Specimen images and dimensions for free-cutting brass: (a) undeformed; (b) deformed

(necked) immediately prior to failure.


96

Figure 2.2.4. Specimen images and dimensions for 4340 steel: (a) undeformed; (b) deformed (necked)

immediately prior to failure.

The applied load P, neck radius of curvature ρ and cross-section radius r2

corresponding to the test section images are displayed by Table 2.2.1 for free-cutting

brass, and Table 2.2.2 for 4340 steel. Extremely large values of ρ are assigned a

value of infinity, indicating that a uniform section may be assumed.


97

Table 2.2.1. Bridgman approximation data for free-cutting brass.

P (N) D2 (mm) ρ (mm) r2 (mm) 0.0 10 ∞ 5

37.7441 8.8441 ∞ 4.42205 37.6465 8.8224 ∞ 4.4112 37.5000 8.7664 ∞ 4.3832 37.5000 8.6729 197.0931 4.33645 37.3535 8.5607 74.2942 4.28035 37.2070 8.5402 58.4527 4.2701 37.0605 8.4673 53.0284 4.23365 36.7188 8.2991 49.0559 4.14955 36.2305 8.2056 40.6584 4.1028 35.5469 8 24.7348 4

Table 2.2.2. Bridgman approximation data for 4340 steel.

P (N) D2 (mm) ρ (mm) r2 (mm) 0.0 10 ∞ 5

81.2988 9.5656 ∞ 4.7828 81.2988 9.5509 ∞ 4.7754 81.1523 9.5362 ∞ 4.7681 81.2500 9.5067 ∞ 4.7534 81.0059 9.4926 ∞ 4.7463 81.0547 9.4797 ∞ 4.7398 81.0059 9.4667 ∞ 4.7334 80.8105 9.4538 ∞ 4.7269 80.7617 9.4408 ∞ 4.7204 80.6152 9.4149 ∞ 4.7075 80.3223 9.3449 ∞ 4.6724 80.1758 9.2271 ∞ 4.6136 79.8340 9.1977 ∞ 4.5988 79.1992 9.1830 ∞ 4.5915 78.6133 9.0211 70.4631 4.5105 77.5391 8.9475 65.9302 4.4738 76.2695 8.8483 49.7190 4.4242 74.2677 8.5648 32.5829 4.2824 70.8496 8.2651 23.3592 4.1325 65.2832 7.6261 13.6023 3.8130

True stress σ and true strain ε were determined from the tabulated data of Tables

2.2.1-2.2.2 according to the Bridgman stress approximation and true strain

expressions of Equations (2.54)-(2.55). For large ρ indicated by infinity, the true

stress formula of Equation (2.56) was applied, assuming a relatively uniform section.

A sample calculation based on the free-cutting brass data of Table 2.2.3 illustrates

application of the Bridgman approximation.


98

( )

++

=1

2ln2 2

22

2 ρrρrrπ

Pσ (2.54)

=

2

2ln2rr

ε i (2.55)

22r

Pπ

σ = (2.56)

Table 2.2.3. Bridgman correction sample data.

P (kN) r2i (mm) D (mm) ρ (mm) r2 (mm) 36.2304688 5 8.2056 49.0559 4.1028

σ = ( ) ( )( )( ) ( )

++ 149.055924.1028ln 4.102849.055924.1028π

36230.46882

= 671.271 MPa

ε = 2 ln

4.10285 = 0.39553649

From the Bridgman approximation data, complete true stress-true strain curves or

equivalent stress-strain curves were obtained for free-cutting brass and 4340 steel.

The resulting equivalent stress-strain curves for free-cutting brass and 4340 steel are

illustrated by Figures 2.2.5-2.2.6 respectively.


99

0

100

200

300

400

500

600

700

800

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

εο

ο (M

Pa)

Figure 2.2.5. True stress-true strain curve, free-cutting brass.

0

200

400

600

800

1000

1200

1400

0 0.1 0.2 0.3 0.4 0.5 0.6

εο

ο(M

Pa)

Figure 2.2.6. True stress-true strain curve, 4340 steel.

To allow approximation of the stress-strain behaviour beyond fracture strain, a power

law curve was fitted to the equivalent stress-strain data beyond yield according to the

method of least squares. Table 2.2.4 outlines the mechanical properties obtained for

each material, including modulus of elasticity E, Poisson’s ratio ν, yield stress σo,

strength coefficient H and strain hardening exponent n, which define the equivalent

stress-strain curve according to Hooke’s law and the power law relationship of

Equations (2.57)-(2.58) respectively.


100

Table 2.2.4. Mechanical properties.

Material E (GPa) ν σo (MPa) H (MPa) n Free-cutting brass 93.026 0.35 379.385 805.95 0.1983

4340 steel 206.022 0.293 954.044 1407.1 0.0908

εσ E= (σ ≤ oσ ) (2.57)

nHεσ = (σ > oσ ) (2.58)

The equivalent stress-strain curves for free-cutting brass and 4340 steel, approximated

to a true strain of unity, are illustrated respectively by Figures 2.2.7-2.2.8.

0

100

200

300

400

500

600

700

800

900

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

εο

ο (M

Pa)

Figure 2.2.7. Equivalent stress-strain curve, free-cutting brass.


101

0

200

400

600

800

1000

1200

1400

1600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

εο

ο(M

Pa)

Figure 2.2.8. Equivalent stress-strain curve, 4340 steel.

2.2.1.2 Analytical Confirmation of Equivalent Stress-Strain Relationship

Finite element analyses were performed on the uniform section specimen finite

element mesh models using Abaqus software. The equivalent stress-strain curves

derived from experiment were incorporated into the finite element models in

accordance with the mechanical properties outlined in Table 2.2.4 and Equations

(2.57)-(2.58). From an iterative procedure, porous metal plasticity parameters were

determined such that a close correlation between the load-displacement curves from

experiment and analysis was obtained. Porous metal plasticity parameters required for

the analysis included the constants q1, q2 and q3, mean nucleation strain εN, standard

deviation of the nucleation strain sN, and volume fraction of nucleated voids fN in

accordance with Equations (2.44)-(2.48). The porous metal plasticity constants

obtained for free-cutting brass and 4340 steel are outlined in Table 2.2.5.

Table 2.2.5. Material porous metal plasticity parameters.

Material q1 q2 q3 εN sN fN Free-cutting brass 1.5 1 2.25 0.32 0.1 0.04

4340 steel 1.5 1 2.25 0.2 0.08 0.04


102

A comparison of the load-displacement curves obtained from experiment and finite

element analysis is illustrated by Figure 2.2.9 for free-cutting brass, and Figure 2.2.10

for 4340 steel. The close correlation illustrated by the comparisons indicate excellent

agreement between the results obtained from experiment and finite element analysis.

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8

δ (mm)

P (k

N)

FEA

Experiment

Figure 2.2.9. Load-displacement curve comparison, uniform section (free-cutting brass).

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6

δ (mm)

P (k

N)

FEA

Experiment

Figure 2.2.10. Load-displacement curve comparison, uniform section (4340 steel).

Deformed mesh contour plots were produced using Femcad 2000 Post post-

processing software [38]. Contour plots for equivalent stress σ , hydrostatic stress σh

and equivalent plastic strain pε are illustrated by Figures 2.2.11-2.2.13 for free-


103

cutting brass, and Figures 2.2.14-2.2.16 for 4340 steel, corresponding to the true

stress-true strain state and necked geometry immediately prior to failure. The

relatively constant σ and pε distributions at the plane of symmetry illustrated by

these figures appear consistent with the assumptions of Bridgman. The σh distribution

depicting a peak value towards the axis of symmetry is also consistent with the

analytical approximation. A comparison of the plane of symmetry radii r2 obtained

from image measurement and finite element analysis for free-cutting brass and 4340

steel, incorporating the initial notch depth of 0.05 mm, is illustrated by Table 2.2.6,

indicating a close correlation between experiment and analysis.

Figure 2.2.11. Equivalent stress contour, uniform section (free-cutting brass).


104

Figure 2.2.12. Hydrostatic stress contour, uniform section (free-cutting brass).

Figure 2.2.13. Equivalent plastic strain contour, uniform section (free-cutting brass).


105

Figure 2.2.14. Equivalent stress contour, uniform section (4340 steel).

Figure 2.2.15. Hydrostatic stress contour, uniform section (4340 steel).


106

Figure 2.2.16. Equivalent plastic strain contour, uniform section (4340 steel).

Table 2.2.6. Comparison of cross-section radii obtained from experiment and finite element analysis.

r2 (mm) Specimen Experiment FEA

Free-cutting brass 4 4.0813 4340 steel 3.8130 3.7287

From the finite element results, the stress-strain state present at the fracture cross-

section was obtained in the form of σ -r, σh-r and pε -r curves. For comparison,

variation of σ , σh and pε with respect to radius r was determined according to the

Bridgman approximation formulae of Equations (2.54)-(2.56), based on the applied

load P, neck radius of curvature ρ and cross-section radius r2 at specimen failure

obtained from Tables 2.2.1-2.2.2. The resulting σ -r, σh-r and pε -r curves depicting

the comparison are illustrated by Figure 2.2.17 for free-cutting brass, and Figure

2.2.18 for 4340 steel, with zero radius indicating the specimen axis of symmetry and

the dashed line indicating the free surface.


107

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

0 1 2 3 4 5

r (mm)

o (M

Pa)

FE A

Bridgm an

0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

0 1 2 3 4 5

r (mm)

h (M

Pa)

FE A

Bridgm an

0

0 .0 5

0 .1

0 .1 5

0 .2

0 .2 5

0 .3

0 .3 5

0 .4

0 .4 5

0 1 2 3 4 5

r (mm)

op

FE A

Bridgm an

Figure 2.2.17. Uniform section stress-strain state (free-cutting brass): (a) σ -r; (b) σh-r; (c) pε -r.

(a)

(b)

(c)


108

0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

0 1 2 3 4 5

r (mm)

o (M

Pa) FE A

Bridgm an

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5

r (mm)

h (M

Pa)

FEA

Bridgm an

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5

r (mm)

op

FEA

Bridgm an

Figure 2.2.18. Uniform section stress-strain state (4340 steel): (a) σ -r; (b) σh-r; (c) pε -r.

(a)

(c)

(b)


109

It is evident from the comparison that discrepancies exist between the finite element

data and the Bridgman approximation data. The comparison of σ -r curves for free-

cutting brass indicated a good correlation, whilst for 4340 steel the Bridgman

approximation predicted a significantly higher σ distribution. The σh-r curve

comparison for free-cutting brass and 4340 steel displays an underprediction of peak

σh at the axis of symmetry and an overprediction of σh at the free surface, indicated by

the smaller curvature of the Bridgman σh-r curves in comparison with finite element

results. The pε -r curves obtained from finite element analysis for free-cutting brass

and 4340 steel display a decreasing value with increasing r compared to a constant

value predicted from the Bridgman approximation, producing a large discrepancy in

results towards the free-surface particularly for 4340 steel. The differences indicated,

although present in both materials, are distinctively more pronounced in the case of

4340 steel.

The differences between the Bridgman approximation and finite element results may

be attributed to a number of assumptions. The Bridgman approximation appears to

decrease in accuracy with decreasing neck radius of curvature ρ, which is consistent

with the small angle theory assumed in the derivation and subsequent linearisation of

the trigonometric functions. In addition, the Bridgman approximation assumes an

unvoided material, whilst the finite element model incorporated the porous metal

plasticity yield surface with associated void nucleation, growth and coalescence.

These differences aside, the comparison indicates that the constant von Mises stress

distribution at the plane of symmetry is a valid assumption. The assumption of

constant plastic strain appears valid also, providing that the plastic strains are small

and the material remains unvoided.


110

2.2.2 Establishment of Fracture Curve

2.2.2.1 Correlation of V-Notch Specimen Load-Displacement Curves

From monotonic loading experiments performed on the circumferential V-notch

specimens, load-displacement curves were obtained for free-cutting brass and 4340

steel in terms of the applied load P and extensometer deflection δ. Corresponding

finite element analyses were performed on the V-notch specimen finite element mesh

models using Abaqus software, assuming the mechanical properties derived from the

uniform section specimen experiments. The equivalent stress-strain curves were

defined in accordance with the mechanical properties outlined in Table 2.2.4 and

Equations (2.57)-(2.58). Porous metal plasticity parameters were included in the

material models as specified by Table 2.2.5 and Equations (2.44)-(2.48). A

comparison of the load-displacement curves obtained from experiment and finite

element analysis is illustrated by Figures 2.2.19-2.2.21 for free-cutting brass, and

Figures 2.2.22-2.2.24 for 4340 steel. The configuration of each specimen is indicated

by the outer diameter Do and inner diameter Di, identified by the notation Do,Di, in

accordance with the specimen geometry of Figure 2.1.12.

0

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

Figure 2.2.19. Load-displacement curve, V-notch (free cutting brass) 15,4.


111

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

Figure 2.2.20. Load-displacement curve, V-notch (free cutting brass): (a) 15,6; (b) 15,7.5; (c)

15,9.

(a)

(b)

(c)


112

0

5

10

15

20

25

30

35

40

45

50

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

0

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

Figure 2.2.21. Load-displacement curve, V-notch (free cutting brass): (a) 15,10.5; (b) 10,5; (c)

8,4.

(a)

(b)

(c)


113

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

0

10

20

30

40

50

60

70

80

90

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

Figure 2.2.22. Load-displacement curve, V-notch (4340 steel): (a) 15,4; (b) 15,6; (c) 15,7.5.

(b)

(a)

(c)


114

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

0

20

40

60

80

100

120

140

160

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

Figure 2.2.23. Load-displacement curve, V-notch (4340 steel): (a) 15,9; (b) 15,10.5; (c) 12,6.

(b)

(a)

(c)


115

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1

δ (mm)

P (k

N)

FEA

Experiment

Figure 2.2.24. Load-displacement curve, V-notch (4340 steel): (a) 10,5; (b) 8,4.

A comparison of the load-displacement curves of Figures 2.2.19-2.2.21 for free-

cutting brass, with the exception of Figure 2.2.20 (a) for the 15,6 specimen, indicate

good agreement between data obtained from experiment and finite element analysis.

Figure 2.2.20 (a) for the 15,6 specimen illustrates a significant offset between the

load-displacement curves towards failure load, indicating that the ductility of the

experimental data was underpredicted by the finite element model. A comparison of

the load-displacement curves of Figures 2.2.22-2.2.24 for 4340 steel illustrate a

consistent overprediction by the finite element model of the failure load in correlation

with the experimental results. Illustrated by Figures 2.2.22 (a), 2.2.23 (a) and 2.2.23

(c) respectively, the 15,4.5, 15,9 and 12,6 specimens indicate a large

(b)

(a)


116

discrepancy between load-displacement results obtained by experiment and finite

element analysis, the difference of which is particularly evident from comparison of

the predicted failure loads.

Overall, the free-cutting brass load-displacement curves were found to be in good

agreement, indicating a close correlation between results obtained from experiment

and finite element analysis. The 4340 steel load-displacement data illustrated large

discrepancies between results obtained from experiment and finite element analysis,

indicating that a close agreement had not been found. The correlation discrepancies

may be attributed to small sample size and possible bending effects, although in the

case of 4340 steel an additional reason would be due to excessive element

deformation in the region immediately surrounding the notch root within the finite

element model. Although hybrid formulation elements were used in conjunction with

large deformation plasticity theory, the presence of high strains and absence of

adaptive meshing resulted in excessively distorted elements prior to the achievement

of the fracture load. These differences between the free-cutting brass and 4340 steel

finite element models may be attributed to the comparatively higher ductility of 4340

steel.

On the basis of these comparisons, the stress-strain behaviour at the plane of

symmetry corresponding to the fracture load of the free-cutting brass specimens was

assumed to be representative of the stress-strain state required to cause failure of the

specimen. As a close correlation could not be obtained for the 4340 steel specimens,

analysis of these specimens was not pursued further.

2.2.2.2 Stress-Strain State at Fracture Cross-Section

The true stress-true strain state present in the free-cutting brass V-notch specimens

immediately prior to failure was obtained using Femcad 2000 Post post-processing

software [38]. Deformed mesh contour plots from each finite element model were

produced for equivalent stress σ , hydrostatic stress σh and equivalent plastic strain

pε . The resulting σ , σh and pε contour plots for free-cutting brass are typified for the


117

15,7.5 specimen by Figures 2.2.25-2.2.27 respectively. The σ and pε contour

plots depicted here illustrate the constrained plastic zone immediately surrounding the

notch root region typical of these specimens, with an associated σh contour depicting

a dramatic increase in hydrostatic stress at a small distance inward from the free

surface, gradually decreasing towards the axis of symmetry.

Figure 2.2.25. Equivalent stress contour, V-notch (free-cutting brass) 15,7.5.


118

Figure 2.2.26. Hydrostatic stress contour, V-notch (free-cutting brass) 15,7.5.

Figure 2.2.27. Equivalent plastic strain contour, V-notch (free-cutting brass) 15,7.5.


119

From the finite element results, the stress-strain state present at the fracture cross-

section was obtained for each specimen configuration in the form of σ -r, σh-r and

pε -r curves. The location of the fracture site was approximated to coincide with the

maximum axial normal stress maxzσ , in accordance with the plane strain brittle

fracture analysis of Tetelman and McEvily [19]. Table 2.2.7 outlines for each

specimen configuration the maxzσ value and approximate fracture site location along

the plane of symmetry, indicated by radius r.

Table 2.2.7. Location of maximum normal stress.

Do (mm) Di (mm) maxzσ (MPa) r (mm)

15 4.5 742.3445 2.155368 15 6 764.0158 2.908359 15 7.5 769.8022 3.661091 15 9 764.4943 4.398855 15 10.5 775.742 5.17685 10 5 726.9948 2.422292 8 4 736.0161 1.913833

In all instances, the free-cutting brass V-notch specimens exhibited a flat, shiny

fracture surface normally associated with brittle behaviour. Figure 2.2.28 for the

15,9 specimen illustrates the observed fracture surface typical of the free-cutting

brass specimens.

Figure 2.2.28. V-notch specimen fracture cross-section for free-cutting brass.


120

The resulting σ -r, σh-r and pε -r curves for free-cutting brass are illustrated by

Figures 2.2.29-2.2.35, with zero radius indicating the specimen axis of symmetry,

dashed line indicating the free surface and dotted line indicating the approximate

fracture initiation point. It may be clearly seen from these curves that the approximate

fracture initiation point, determined according to the location of maxzσ , coincides

with the maximum σh value as indicated by the σh-r curves. A comparison of the σ -

r, σh-r and pε -r curves indicate similar stress-strain profiles at the plane of symmetry

for all of the specimen configurations. The σ -r and pε -r curves illustrate peak σ

and pε values at the free surface which rapidly decrease towards the axis of

symmetry. The pε distribution illustrates a confinement of the plastic zone to a region

surrounding the notch root, indicative of the triaxial constraint present due to the

circular notch geometry. The σh-r curves illustrate a peak σh magnitude at a small

distance inward from the notch root significantly higher than the σh magnitude at the

free surface, depicting a gradual decrease from the peak value towards the axis of

symmetry. The plastic zone size indicated by the pε -r curves appears consistent for all

specimen configurations in accordance with the consistent notch geometry.

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5

r (mm)

o (M

Pa)

(a)


121

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5

r (mm)

h (M

Pa)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.5 1 1.5 2 2.5

r (mm)

o

Figure 2.2.29. V-notch stress-strain state, free-cutting brass 15,4.5: (a) σ -r; (b) σh-r; (c) pε -r.

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5 3 3.5

r (mm)

o (M

Pa)

(c)

(b)

(a)


122

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5 3 3.5

r (mm)

h (M

Pa)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.5 1 1.5 2 2.5 3 3.5

r (mm)

o

Figure 2.2.30. V-notch stress-strain state, free-cutting brass 15,6: (a) σ -r; (b) σh-r; (c) pε -r.

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5 3 3.5 4

r (mm)

o (M

Pa)

(c)

(b)

(a)


123

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5 3 3.5 4

r (mm)

h (M

Pa)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.5 1 1.5 2 2.5 3 3.5 4

r (mm)

o


0

100

200

300

400

500

600

0 1 2 3 4 5

r (mm)

o (M

Pa)

(b)

(c)

(a)


124

0

100

200

300

400

500

600

0 1 2 3 4 5

r (mm)

h (M

Pa)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 1 2 3 4 5

r (mm)

o


0

100

200

300

400

500

600

700

0 1 2 3 4 5 6

r (mm)

o (M

Pa)

(b)

(c)

(a)


125

0

100

200

300

400

500

600

0 1 2 3 4 5 6

r (mm)

h (M

Pa)

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6

r (mm)

o


0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5 3

r (mm)

o (M

Pa)

(b)

(c)

(a)


126

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5 3

r (mm)

h (M

Pa)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.5 1 1.5 2 2.5 3

r (mm)

o

Figure 2.2.34. V-notch stress-strain state, free-cutting brass 10, 5: (a) σ -r; (b) σh-r; (c) pε -r.

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5

r (mm)

o (M

Pa)

(b)

(c)

(a)


127

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5

r (mm)

h (M

Pa)

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.5 1 1.5 2 2.5

r (mm)

o


2.2.2.3 Comparison of Stress-Strain State with Fracture Mechanics

Determination of KIc values for each free-cutting brass specimen configuration was

conducted as a means of verifying brittle fracture behaviour. Prior to KIc calculation,

the validity of LEFM was determined from the load-displacement curves in

accordance with ASTM standard E399-90 [37], as indicated by Figure 2.1.34. From

PQ and Pmax obtained for each free-cutting brass and 4340 steel specimen

configuration, validity of KIc was verified according to Equation (2.54). Illustrated by

Table 2.2.8, the ratio of Pmax to PQ indicates that, with the exception of the 15,6

specimen, a value less than or approximately equal to 1.1 was obtained in each case

(b)

(c)


128

for free-cutting brass, verifying the validity of KIc and confirming the observed brittle

behaviour. For comparison, the ratio of Pmax to PQ obtained for 4340 steel for all

specimen configurations was greater than 1.1, indicating ductile fracture behaviour.

Table 2.2.8. Determination of KIc validity.

Material Free-cutting brass 4340 steel

Do (mm) Di (mm) PQ (kN) Pmax (kN)

Q

maxP

P

PQ (kN) Pmax (kN)

Q

maxP

P

15 4.5 8.1543 8.8379 1.0838 23.5352 28.0273 1.1909 15 6 14.4531 16.3086 1.1284 40.9668 51.1719 1.2491 15 7.5 22.2168 24.4629 1.1011 65.4297 78.1250 1.1940 15 9 30.8105 31.4941 1.0222 73.2153 100.8539 1.3775 15 10.5 38.2812 42.4805 1.1097 90 129 1.4333 12 6 - - - 38.0371 46.1426 1.2131 10 5 8.0078 8.5938 1.0732 28.1250 32.2539 1.2535 8 4 5.5664 5.9570 1.0702 16.1621 22.9492 1.4120

The critical stress intensity factor KIc was determined from the finite element

maximum normal stress maxzσ value for each free-cutting brass specimen

configuration according to the RKR relationship of Equation (2.53). The resulting KIc

values are indicated by Table 2.2.9, based on a notch radius ρ of 0.1 mm and yield

stress σo value from Table 2.2.4.

Table 2.2.9. KIc calculations for V-notch specimen configurations (free-cutting brass).

Do (mm) Di (mm) maxzσ (MPa) KIc

21

MPa.m

15 4.5 742.3445 27.2574252 15 6 764.0158 29.3615442 15 7.5 769.8022 29.9385832 15 9 764.4943 29.4090152 15 10.5 775.742 30.5377266 10 5 726.9948 25.8201599 8 4 736.0161 26.6596247

Table 2.2.9 indicates that a range of similar KIc values was obtained from each

specimen configuration, resulting in a mean KIc value of 28.4263 and standard

deviation of 1.8196. The variability observed between the KIc values may be largely

accounted for by the small test sample size of two tests per specimen configuration. A


129

further contributing factor would be the possibility of slight load misalignment

introducing bending in the specimens, the presence of which would become further

pronounced with increasing load. Given the consistency of the KIc values and notch

geometry, a similar trend could be expected from the pε -σh data representing the

failure stress-strain state of the material.

The pε contour plots for free-cutting brass, illustrated by Figure 2.2.27 and Figures

2.2.36-2.2.41 for each specimen configuration, depict the confinement of the plastic

zone to the region immediately surrounding the notch root. Accounting for scale

differences between Figure 2.2.27 and Figures 2.2.36-2.2.41 due to specimen

dimensions, the plastic zone sizes indicated by the pε contour plots appear consistent

for each specimen configuration. The pε contour plots and the σ -r, σh-r and pε -r

curves presented indicate an invariant stress-strain state surrounding the notch root

immediately prior to failure.



130

Figure 2.2.37. Equivalent plastic strain contour, V-notch (free-cutting brass) 15,6.



131




132


2.2.2.4 Fracture Curve Determination

From the free-cutting brass σh-r and pε -r curves obtained from finite element

analysis, the equivalent plastic strain-hydrostatic stress state immediately prior to

failure was derived for each specimen type and configuration in the form of pε -σh

curves. The pε -σh curve corresponding to failure of the uniform section specimen is

illustrated by Figure 2.2.42, indicating the highest magnitude of σh at the axis of

symmetry and a gradually increasing pε from the free-surface to the axis of

symmetry in accordance with Figure 2.1.35.


133

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 50 100 150 200 250 300 350

σh (MPa)

ο

Figure 2.2.42. Equivalent plastic strain-hydrostatic stress, uniform section specimen (free-cutting

brass).

For the circumferential V-notch specimens, each pε -σh curve was obtained

corresponding to the applied uniaxial load P, resulting in a final pε -σh curve

representing the failure stress-strain state at load Pmax indicated by Table 2.2.8. An

evolution of the pε -σh curve with applied load P is illustrated by Figure 2.2.43 for the

15,7.5 specimen configuration. With reference to Figure 2.1.36, the evolution of the

pε -σh curve indicates a significant increase in pε towards the free-surface in section

I, accompanied by a smaller increase in pε near the assumed fracture point in section

II. The confinement of the plastic zone is clearly illustrated by the relative absence of

any significant pε in section III towards the axis of symmetry. The pε -σh curve

evolution is accompanied by a significant overall increase in σh, with a peak value

occurring in section II of the curve near the fracture point. Associated with the pε -σh

curve evolution, the relative increase in σh towards the free-surface in section I

compared to the peak value in section II illustrates the competing effect present

between fracture occurring due to a higher σh and smaller pε away from the free

surface, and fracture occurring due to a lower σh and higher pε at the free surface.


134

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 100 200 300 400 500 600

σh (MPa)

o

P = 7.276 1 kNP = 11.7358 kNP = 17.2 92 8 kNP = 23 .32 01 kNP = 24 .958 2 kN

Figure 2.2.43. Evolution of equivalent plastic strain-hydrostatic stress curves for V-notch specimen,

free-cutting brass 15,7.5.

The resulting pε -σh curves obtained for the uniform section specimen, and for each

V-notch specimen configuration at Pmax, are illustrated by Figure 2.2.44. As illustrated

by the V-notch specimen pε -σh curves, a higher pε -σh state corresponding to the

higher KIc range exhibited by the 15,6, 15,7.5, 15,9 and 15,10.5 specimens is

indicated, whilst a lower pε -σh state is indicated corresponding to the lower KIc

values obtained for the 15,4.5, 10,5 and 8,4 specimens.


135

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 100 200 300 400 500 600 700 800

σh (MPa)

op

Unifo rmV-Notch 15,4 .5V-Notch 15,6V-Notch 15,7.5V-Notch 15,9V-Notch 15,10 .5V-Notch 10 ,5V-Notch 8 ,4

Figure 2.2.44. Combined equivalent plastic strain-hydrostatic stress curves (free-cutting brass).

From the pε -σh curves obtained for the uniform section and V-notch specimens,

failure points were determined representing the fpε -σh curve in accordance with the

proposed fracture criterion. For the uniform section specimen, the (σh, fpε ) failure

point may be obtained at the axis of symmetry. From the experimental evidence and

analytical justification presented, brittle failure was assumed for the V-notch

specimens, allowing determination of the (σh, fpε ) failure points corresponding to the

location of maxzσ in accordance with Table 2.2.7. The (σh, fpε ) failure points

corresponding to the uniform section and V-notch specimens are illustrated by Figure

2.2.45. The two distinctly different stress-strain states represented here by the uniform

section and V-notch specimens depict a monotonically decreasing trend of fpε with

increasing σh in accordance with the proposed form of the fracture criterion indicated

by Figure 2.1.7.


136

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 100 200 300 400 500 600 700 800

σh (MPa)

opPlainV-No tch 15,4 .5V-No tch 15,6V-No tch 15,7.5V-No tch 15,9V-No tch 15,10 .5V-No tch 10 ,5V-No tch 8 ,4

Figure 2.2.45. Combined (σh, fpε ) failure points obtained from uniform section and V-notch

specimens (free-cutting brass).


137

2.3 Analysis and Discussion

2.3.1 Determination of Equivalent Stress-Strain Curve

The close correlation illustrated by the load-displacement curve and necked section

geometry comparisons for free-cutting brass and 4340 steel indicated excellent

agreement between the results obtained from experiment and finite element analysis.

The verification of the equivalent stress-strain curve by close comparison of the

experimental and finite element results allowed the equivalent stress-strain curve to be

assumed for subsequent analyses and determination of the stress-strain state at the

fracture cross-section corresponding to specimen failure.

The derivation of the equivalent stress-strain curve from the experimental load-

displacement curves illustrated application of the Bridgman approximation in

determining the stress-strain state at the necked region plane of symmetry. In

combination with the video imaging technique developed for determining the necked

geometry, the Bridgman approximation provided a good approximation of the

equivalent stress and equivalent strain to allow determination of the equivalent stress-

strain curve beyond the onset of necking to the point of fracture. The video imaging

technique, consisting of digitised images in combination with measurement using

AutoCAD software, allowed accurate determination of the necked radius of curvature

ρ and cross-section radius r2 by comparison of the deformed images with the original

specimen geometry. Although a high degree of measurement accuracy was obtained

from this procedure, the Bridgman approximation relied on the necked section

geometry of the specimen remaining concentric. Verification of measurement

accuracy was determined by calliper measurement of the fractured cross-sections for

the free-cutting brass and 4340 steel uniform section specimens which confirmed the

presence of concentric geometry throughout the necked region.

The Bridgman approximation allowed determination of the equivalent stress-strain

curve representing the homogeneous or unvoided material behaviour. Using Abaqus


138

finite element analysis software, the finite element analyses allowed the inclusion of

the porous metal plasticity criterion of Tvergaard [14] for modelling of

inhomogeneous material behaviour as a result of void initiation, growth and

coalescence. The iterative procedure adopted to obtain the material stress-strain

behaviour required incremental adjustment of the porous metal plasticity parameters

such that a close correlation of load-displacement curves and necked geometry

between experiment and finite element analysis was obtained. The procedure did not

require adjustment of the equivalent stress-strain curve to obtain the close correlation,

which tends to indicate that the Bridgman approximation adequately predicted the

equivalent stress-strain curve for the homogeneous or unvoided material. A high

degree of sensitivity was indicated by adjustment of the porous metal plasticity

parameters. Adjustment of the mean nucleation strain εN, standard deviation of the

nucleation strain sN, and volume fraction of nucleated voids fN greatly affected the

slope of the load-displacement curves and the displacement at which ultimate tensile

strength was reached, whilst the material constants q1, q2 and q3 influenced the

correlation between the experimental and finite element load-displacement curves.

Typical values for q1, q2 and q3 obtained by Tvergaard [14] of 1.5, 1 and 2.25

respectively were assumed for the analysis and proved adequate in this instance.

The introduction of a small geometric imperfection at the free surface in the form of a

notch proved adequate in inducing necking in the finite element model. The onset of

necking coincided with the ultimate tensile strength in accordance with the load-

displacement data and test section images obtained from experiment. From the

assumed equivalent stress-strain curves and optimised porous metal plasticity

parameters, the necked geometries exhibited by the deformed mesh were in close

correlation with the necked specimen images obtained from experiment.

Comparison of the cross-section radii at the plane of symmetry immediately prior to

failure indicated that a close correlation had been obtained between experiment and

finite element analysis.

A comparison between experiment and finite element analysis of the stress-strain state

at the fracture cross-section indicated the existence of discrepancies between the

results. As was indicated previously, the discrepancies illustrated by the σ -r, σh-r


139

and pε -r curves could be attributed to the assumptions made in derivation of the

Bridgman approximation equations, in particular the assumptions of small angle

theory and unvoided material. The discrepancies were more pronounced in the 4340

steel specimen results due to the smaller neck radius and larger amounts of plastic

deformation in the necked region. Given the discrepancies, a comparison of the σ -r

and σh-r curves obtained from experiment and finite element analysis verified the

assumption of constant σ through the cross-section and occurrence of peak σh at the

axis of symmetry, indicating initiation of failure at the axis of symmetry. The close

correlation between the experimental and finite element load-displacement curves and

necked geometry indicate that the Bridgman approximation provided an

approximation of the stress-strain state adequate for homogeneous material behaviour

prediction.

From analysis of the results, the procedure adopted to obtain and verify the equivalent

stress-strain curve proved successful for the metal alloys tested. The load-

displacement curves obtained from experiment allowed determination of the true

stress-true strain curve to the point of ultimate tensile stress. Application of the

Bridgman approximation beyond ultimate tensile stress to fracture strain allowed an

approximate determination of the true stress-true strain curve to the point of failure.

The true stress-true strain behaviour beyond yield was adequately modelled using a

power law equation which allowed approximation of the stress-strain state beyond

fracture strain. The resulting equivalent stress-strain curves allowed accurate

determination of the homogeneous or unvoided stress-strain behaviour, whilst the

inclusion of the porous metal plasticity criterion and associated material parameters

allowed accurate prediction of the inhomogeneous material behaviour due to the

nucleation, growth and coalescence of voids.


140

2.3.2 Determination of Fracture Curve

A comparison of the circumferential V-notch specimen load-displacement curves of

Figures 2.2.19-2.2.24 obtained from experiment and finite element analysis indicated

good agreement for free-cutting brass and fair to poor agreement for 4340 steel. As a

result of the discrepancies observed in the 4340 steel load-displacement data, further

analysis could not be conducted as the stress-strain state obtained from finite element

analysis could not be verified. As discussed previously, major causes of the

discrepancies may be attributed to the small sample size tested for each V-notch

specimen configuration and the possibility of bending due to load misalignment. The

correlation obtained for free-cutting brass allowed verification of the finite element

results and subsequent determination of the stress-strain state at the fracture cross-

section corresponding to failure.

The stress-strain state immediately prior to failure was obtained assuming the free-

cutting brass equivalent stress-strain curve and porous metal plasticity parameters

derived from the uniform section specimen. The σ , σh and pε contour plots

illustrated the triaxial constraint present due to rotational symmetry, typified by the

constrained plastic zone immediately surrounding the notch root. The σ -r, σh-r and

pε -r curves illustrated peak σ and pε values at the free surface which rapidly

decreased towards the axis of symmetry, associated with a peak σh magnitude present

at a small distance inward from the notch root significantly higher than the σh

magnitude at the free surface. The location of maximum σh coincided with the

predicted point of failure in accordance with the fracture toughness research of

Tetelman and McEvily [19], assuming the presence of brittle failure conditions.

The presence of brittle failure conditions and applicability of LEFM was confirmed

from the free-cutting brass V-notch experimental and finite element analysis results.

Analysis of the load-displacement curves in accordance with ASTM standard E399-

90 [37] determined that conditions were sufficiently brittle to allow application of

LEFM and determination of a valid KIc value. The pε contour plots illustrated the


141

confinement of the plastic zone to a small region surrounding the notch root,

providing a clear indication of the triaxial constraint present and consistency of the

plastic zone size at failure for each specimen configuration. Further indication was

provided by a flat, shiny fracture surface appearance for each specimen configuration

typically associated with brittle failure. The presence of brittle failure conditions

enabled verification of the point of failure in accordance with the fracture toughness

research of Tetelman and McEvily [19], assuming similarity between axisymmetric

and plane strain conditions for V-notch geometry.

The determination of valid KIc values for each V-notch specimen configuration

provided a basis for comparison of the stress-strain state at failure. The (σh, fpε )

fracture points determined from the pε -σh curves were consistent with the KIc values,

where similar (σh, fpε ) points were obtained for specimens with similar KIc values.

An increase in KIc corresponded to an increase in σh and pε at failure. The invariant

form of σh and pε , and the demonstrated consistency with KIc, indicates that the

(σh, fpε ) fracture points provide an invariant representation of the stress-strain state at

failure.

The uniform section and V-notch specimens allowed the determination of two

distinctly different (σh, fpε ) failure points for free-cutting brass which clearly

illustrate the trend in the fpε -σh curve for the hydrostatic tensile stress region. The

monotonically decreasing trend depicted by the (σh, fpε ) failure points are in

accordance with the trend demonstrated by Bridgman [6], Brownrigg et al. [7], and

Lewandowski and Lowhaphandu [8] for hydrostatic compression. For illustration

purposes, if a linear form of the fpε -σh curve is assumed to exist for hydrostatic

tension corresponding to the linear relationship demonstrated for hydrostatic

compression [6,7], an expression of the form presented by Equation (2.59) may be

obtained. From least squares analysis, the possible linear form of the fpε -σh curve

for free-cutting brass is indicated by Equation (2.60) and Figure 2.3.1.


142

ophch

opp f

εσσε

ε +−= (2.59)

0057.10019.0 +−= hp fσε (2.60)

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500 600 700 800

σh (MPa)

op

PlainV-Notch 15,4 .5V-Notch 15,6V-Notch 15,7.5V-Notch 15,9V-Notch 15,10 .5V-Notch 10 ,5V-Notch 8 ,4Line o f Bes t Fit

Figure 2.3.1. Equivalent plastic fracture strain vs hydrostatic stress, possible linear form (free-cutting

brass).

In accordance with the concept proposed by Bridgman [6] and the criterion of Oh

[12], a value chσ of 529.32 MPa may be obtained from Equation (2.60) which

represents a possible critical hydrostatic stress value required to cause brittle failure

with zero associated plastic deformation.


143

2.4 Conclusion

2.4.1 Research Outcomes

The influence of hydrostatic stress on the ductility of a metal alloy has been amply

demonstrated from this research. In particular, the existence of a relationship between

hydrostatic stress and ductility was clearly illustrated for the hydrostatic tensile stress

range. The proposed monotonic failure criterion form, consisting of a fundamental

relationship between hydrostatic stress and equivalent plastic strain, provided an

invariant representation of the stress-strain state at failure generally applicable to

multiaxial states of stress.

The concept of a fracture curve defined in terms of hydrostatic stress and equivalent

plastic strain was strongly supported by the research of Bridgman [6]. The

experiments and analyses performed on the uniform section and circumferential V-

notch specimens enabled the determination of two distinctly different fracture points,

expressed in terms of hydrostatic stress and equivalent plastic fracture strain. A

monotonically decreasing trend in equivalent plastic strain with increasing hydrostatic

stress was demonstrated for the hydrostatic tensile stress region, similar to the

monotonically decreasing trend for hydrostatic compression indicated by Bridgman

[6], Brownrigg et al. [7], and Lewandowski and Lowhaphandu [8]. The proposed

monotonic failure criterion, based on the invariant parameters of hydrostatic stress

and equivalent plastic strain, is similar in form to the equation of ductility proposed by

Bridgman [6].

As part of this research, a methodology was established for accurate determination of

the equivalent stress-strain curve which effectively incorporated experiment and finite

element analysis. The equivalent stress-strain curve corresponding to a homogeneous

or unvoided material was accurately determined from the experimental uniform

section specimen load-displacement curve, in combination with the video imaging

technique, by application of the analytical approximation of Bridgman [6]. The


144

inhomogeneous or voided material behaviour was adequately modelled by

determination of material parameters corresponding to the porous metal plasticity

criterion of Tvergaard [14], determined from finite element analysis via an iterative

procedure. The iterative approach adopted for the finite element analyses enabled a

close correlation between the uniform section specimen load-displacement curves and

necked geometry to be established, providing a means of verification of the

mechanical properties and stress-strain state present at failure.

Numerous questions may be presented from this research in relation to the form of the

proposed failure criterion. The monotonically decreasing trend of the equivalent

plastic fracture strain-hydrostatic stress data for hydrostatic tension introduces the

possibility of a linear relationship in accordance with the hydrostatic compression

research of Bridgman [6], Brownrigg et al. [7], and Lewandowski and Lowhaphandu

[8]. Corresponding to a linear relationship, the existence of a critical stress value

which would cause failure at zero equivalent plastic strain was proposed as a distinct

possibility. The notion of a purely elastic fracture was presented by Bridgman [6] in

relation to the equation of ductility, expressed as a linear relationship between

hydrostatic pressure and equivalent strain. The influence of hydrostatic stress on

ductility also presents the possibility of fracture mode dependence of a metal alloy on

the imposed state of stress, introducing the notion of a hydrostatic compressive stress

value where a maximum equivalent plastic fracture strain may be reached. The

concepts of fracture mode influence of hydrostatic stress and subsequent maximum

fracture strain were presented by the research of Lewandowski and Lowhaphandu [8].


145

2.4.2 Recommendations for Future Work

The determination of additional failure points corresponding to the proposed

monotonic failure criterion is a requirement for verification of the fracture curve in the

hydrostatic tensile stress region. A possible means of determining additional failure

points exists through the circumferential V-notch specimen by variation of the notch

root radius, where a smaller notch root radius would tend towards an increase in

brittleness in accordance with fracture mechanics theory [39,40]. In addition, a

substantially higher sample size for each metal alloy tested would be required to

increase the effectiveness of the correlation between experiment and finite element

analysis.

Correlation between failure points determined from hydrostatic tension and

hydrostatic compression would allow a more accurate determination of the form of

the fracture curve. The testing of uniform section specimens subjected to an imposed

hydrostatic pressure would provide additional failure points throughout the

hydrostatic compressive stress region. The combined failure point data would allow

verification of the existence of a linear equivalent plastic strain-hydrostatic stress

relationship, determination of a critical hydrostatic stress value and study of the

influence of hydrostatic stress on the fracture mode of a specific metal alloy.

146

3. FATIGUE FAILURE DUE TO

CYCLIC LOADING

Fatigue Failure due to Cyclic Loading

147

3.1 Research Methodology

3.1.1 Theory Development

3.1.1.1 Plastic Strain Energy Approach to Fatigue Life Characterisation

The application of the plastic strain energy approach to fatigue life characterisation

was clearly demonstrated by the work of Garud [24] and Ellyin [25]. The plastic SED

per cycle calculated from the stress-strain path or hysteresis loop was shown to

represent irreversible damage in a scalar form. Ellyin [22] demonstrated that the scalar

form of plastic SED is invariant, allowing multiaxial stress states to be resolved in

terms of equivalent stress and equivalent plastic strain components.

Based on the notion that irreversible damage may be attributed entirely to plastic

deformation, means of determining plastic SED throughout the entire fatigue life are

to be considered. During low cycle fatigue, the plastic SED per cycle may be

determined from the area within a stress-strain hysteresis loop resulting from

significant plastic deformation, as illustrated by Figure 3.1.1. Mechanical means of

measurement such as the strain gauge or extensometer may be used here to determine

the displacement corresponding to the applied load. As high cycle fatigue is

approached, the hysteresis loop becomes progressively smaller corresponding to

smaller amounts of plastic deformation as illustrated by Figure 3.1.1, until such point

is reached where the area within the hysteresis loop can no longer be accurately

determined by conventional mechanical means of measurement.


148

Figure 3.1.1. Stress-strain curve hysteresis loops.

Although mechanical means of measurement suggest that plastic deformation is not

existent within the elastic limit of a metal alloy, experimental evidence suggests that

plastic deformation does exist in the high cycle fatigue regime. The existence of

irreversible damage attributed to plastic deformation during high cycle fatigue is

supported by the research of Bathias [41], and Miller and O’Donnell [42]. Previously

it was assumed that a fatigue limit existed in the vicinity of 106-107 cycles for most

metal alloys. The research of Bathias, and Miller and O’Donnell, has revealed that

fatigue failures may occur well beyond 107 cycles. The experimental work of Bathias

[41] has demonstrated that fatigue failures may occur beyond 1010 cycles for a variety

of metal alloys. The research of Miller and O’Donnell [42] has explored the

possibility of fatigue failures in the 106-1012 cycles region, challenging the previously

held notion that fatigue cracks could not propagate beyond microstructural barriers at

such stress levels.

To quantify the presence of plastic deformation during high cycle fatigue, an

alternative means of measurement is required. It is well known that, due to the

ε

σ

Hysteresis Loop

Equivalent Stress-Strain Curve


149

conservation of energy principle, all energy must be conserved during a process.

During elastic deformation, the elastic strain energy that is put into the system through

an applied load is stored and fully recovered once the load is released. During plastic

deformation, the plastic strain energy that is put into the system results in permanent

deformation, and as such the majority of the energy must be released in other forms.

According to Callister [3], approximately 95 percent of plastic strain energy is

released in the form of heat, whilst the remaining plastic strain energy is stored

surrounding dislocations within the metal alloy in regions of tensile and compressive

strain. The research of Rosakis et al. [43,44], Briottet et al. [45] and Rittel [46]

investigating the partition of plastic work into heat and stored energy reach a similar

conclusion in regards to the heat dissipation fraction. The dissipation of heat as a

result of plastic deformation would therefore be expected to result in a temperature

rise within a metal alloy.

In recent years, numerous researchers have used heat dissipation associated with

plastic deformation as a means of investigating fatigue related phenomena. An

extensive amount of research has been conducted in this area by Mast, Badaliance and

co-workers [47-49] investigating strain induced damage in composite materials based

on the dissipated energy density. The thermal imaging technique of infrared

thermography has been adopted by many researchers to determine the surface

temperature profile of test specimens [50-54]. The existence of plastic deformation in

these experiments was determined from the temperature difference between the

specimen surface and the surrounding ambient air. Due to the use of open test

facilities for these experiments, the thermographic technique applied in this form

allowed determination of crack initiation, crack growth, and prediction of the fatigue

limit, but did not allow a quantitative determination of plastic SED due to the

thermally uncontrolled environment surrounding the specimen surface. Accurate

plastic SED measurement during high cycle fatigue would depend on the development

of a quantitative technique of thermodynamic measurement.


150

3.1.1.2 Thermodynamic Approach to High Cycle Fatigue

The development of a quantitative thermodynamic technique for determining plastic

SED was proposed, based on measurement of the surface temperature distribution of a

fatigue test specimen, subjected to clearly defined boundary conditions. Through the

application of heat transfer theory, the plastic strain energy would be determined by

equating the plastic SED equal to the heat dissipation. The proposed method to

experimentally determine heat dissipation required incorporation of specifically

designed fatigue specimens, achievement of thermal isolation at the specimen surface,

appropriate temperature measuring equipment and servohydraulic uniaxial testing

machinery.

The development of the thermodynamic method required an analysis of the heat

transfer processes. The heat dissipation due to cyclic loading is equivalent to internal

heat generation in terms of heat transfer terminology. Illustrated by Figure 3.1.2 for

the case of surface dominated internal heat generation, uniaxial loading of a specimen

with a uniform diameter test section of sufficient length was proposed such that the

section would experience a uniform stress amplitude, thereby creating an internal heat

generation rate q& distribution through the cross-section that remains constant in the z-

direction. Assuming thermal isolation of the free surface, a temperature distribution T

would exist through the cross-section at any location along the uniform section, as

illustrated by Figure 3.1.2.


151

Figure 3.1.2. Uniform test section, internal heat generation and temperature distribution.

By measuring the temperature distribution along the uniform test section surface, a

heat transfer analysis may be performed to calculate the internal heat generation rate

q& present within the section, providing the distribution of q& is known. In this

instance, the distribution of q& through the cross-section for low cycle fatigue may be

assumed uniform due to significant plastic deformation throughout the uniform

section. In the case of high cycle fatigue, the possibility of a non-uniform distribution

of q& exists due to the influence of surface effects. The degree to which surface effects

influence the fatigue damage process during high cycle fatigue would determine the

distribution of q&. The influence of surface effects on the fatigue life of metals has

been documented by numerous researchers including Manson [55] and McClintock

and Argon [56]. The notion was presented that fatigue failure during high cycle

q& r0

Uniform Test Section

P

P

z

r

r0

Adiabatic Surface

r0 r

T

0

r

z


152

fatigue in the vicinity of 106 cycles is primarily caused by the development of surface

cracks as a result of dislocation movement at the free surface. The relative freedom of

dislocation movement and surface flaws or notches present at the free surface further

justified this notion. The influence on fatigue life has led to the incorporation of

surface effects in industry standard fatigue codes such as BS 7608 [57]. Numerous

techniques have been developed in industry to reduce the influence of surface effects

on fatigue life, include surface polishing, nitriding and shot peening.

More recent experimental evidence from the research of Bathias [41] suggests that

cracks initiate at different locations within a fatigue test specimen of polished surface

depending on the number of cycles to failure. For low cycle fatigue, cracks have been

shown to initiate from multiple sites at the free surface, whilst for high cycle fatigue

in the vicinity of 106 cycles, a single crack initiation site at the free surface was

usually present. When the number of cycles to failure was beyond 106 cycles, the

crack initiation site was located at an internal zone. The experimental evidence of

Bathias appears to present the notion that the surface effects become less effective

with increased number of cycles, particularly beyond the vicinity of 106 cycles.

In terms of the internal heat generation rate q& distribution through the cross-section

for high cycle fatigue, the range of possibilities may be bounded by two distinct

thermodynamic models. The first model assumes that the difference between plastic

deformation at the free surface and that which occurs internally is negligible, and

hence the resulting q& distribution is uniform. The second model assumes that the

plastic deformation at the free surface is significantly higher than that which occurs

internally, resulting in the vast majority of q& occurring at the free surface. The two

proposed internal heat generation models and associated temperature distributions are

illustrated by Figure 3.1.3 (a) for the uniform internal heat generation model, and

Figure 3.1.3 (b) for the free surface internal heat generation model.


153

Figure 3.1.3. Internal heat generation models: (a) uniform model and temperature distribution; (b) free

surface model and temperature distribution.

Assuming thermal insulation at the test section surface, the internal heat generation

models would result in a 1-D heat conduction problem for the uniform q& model, or a

2-D axisymmetric heat conduction problem for the free surface q& model. Regardless

of a uniform or free surface q& model, the resulting heat dissipation would be uniform

along the test section surface, producing similar temperature distribution profiles at

the free surface. According to Figure 3.1.4, specified locations along the test section

would define the boundary conditions of the heat conduction problem in terms of time

varying temperatures, indicated by ( )tT oz+ and ( )tT oz− respectively.

r

T1 (t)

T

r

r0

r0

q&

T

T1 (r,t)

r r0

r

r0

q&

Axis of Symmetry

Axis of Symmetry

Plane of Symmetry

Plane of Symmetry

T1 T1

Free Surface (Adiabatic)

Free Surface (Adiabatic)

z z


154

Figure 3.1.4. Time-varying temperature boundary conditions.

A finite difference scheme may be applied here to calculate the internal heat

generation rate q& from the transient and steady state surface temperature profile T.

The generalised heat transfer equations to be used as the basis for development of a

finite difference scheme are indicated by Equation (3.1) for 1-D heat conduction, and

by Equation (3.2) for 2-D axisymmetric heat conduction, defined in terms of

temperature T, internal heat generation rate q&, thermal conductivity k, density κ,

specific heat cp, dimensions r and z, and time t.

tTc

kq

zT

p ∂∂=+

∂∂ κ&

2

2 (3.1)

tTc

kq

zT

rT

rrT

p ∂∂=+

∂∂+

∂∂+

∂∂ κ&

2

2

2

2 1 (3.2)

From the heat equations, specific material properties and thermodynamic constants

including density κ, specific heat cp and thermal conductivity k must be known for the

specimen metal alloy prior to undertaking a rigorous heat transfer analysis.

Temperature sensors located on the specimen surface would require a high level of

r0 Adiabatic Surface

r

z

Symmetry Plane

z0

z0

( )tT oz+

( )tT oz−


155

precision in order to measure minute temperature changes and allow determination of

the temperature distribution. The infrared thermography equipment used by numerous

researchers investigating fatigue life have temperature measurement precision in the

vicinity of 0.015 K [51,53], which indicates the level of precision required for

accurately undertaking such measurements.

After discussions with Professor Richard E Collins of the Department of Physics,

University of Sydney, the use of thermistors as a means of point temperature

measurement was proposed. A thermistor is essentially a variable resistor with a

highly non-linear relationship between resistance and temperature. The most

commonly used thermistors are negative temperature coefficient (NTC) thermistors,

which decrease in resistance with increase in temperature according to a non-linear

relationship. Miniature glass bead NTC thermistors were considered for use as a

temperature sensor due to high sensitivity and small bead dimensions (typically 1-2

mm diameter) which would allow accurate point temperature determination. In

association with appropriate Wheatstone bridge circuitry and a temperature calibration

procedure, miniature glass bead NTC thermistors offer the possibility of achieving

levels of precision in the order of 10-4 K and lower.

To determine the viability of performing such experiments using the proposed

thermodynamic approach, preliminary calculations were performed assuming the 1-D

uniform internal heat generation model, steady state conditions and a thermally

insulated free surface. Assuming the uniform specimen test section of Figure 3.1.5,

the solution to the steady state heat conduction problem is indicated by Equation (3.3),

where the temperature difference ∆T between two points separated by distance z1 may

be determined according to the internal heat generation rate q& and thermal

conductivity k.


156

Figure 3.1.5. Uniform test section model.

kzq

TTT2

21

12&

=−=∆ (3.3)

Possible values of internal heat generation rate q& were calculated from the plastic

SED per cycle value pW∆ based on representative values of stress σ, corresponding

plastic strain εp and fatigue test cyclic loading frequency f. The predicted q& values

were used to determine the temperature difference ∆T between possible measurement

points according to Equation (3.3). Displayed by Table 3.1.1, the results of these

calculations indicate that for the representative stress-strain values, the temperature

difference ∆T between possible measurement points would allow accurate

determination of the temperature distribution well within the achievable measurement

precision of miniature glass bead NTC thermistors.

σ = 100 MPa εp = 0.001 (predicted stress-strain values)

∫=∆cycle

pp dW εσ = 4 (100 × 106) (0.001) = 400000 J/m3 (calculated plastic work)

q&

r0

Adiabatic Surface

T1 (t)

z1

T2 (t)


PP z

r

r0

r

z


157

f = 5 Hz (cyclic frequency)

q&= ∆Wp f = (400000) (5) = 2 × 106 W/m3 (internal heat generation rate)

k = 60.5 W/mK (typical thermal conductivity value for steel)

Table 3.1.1. Temperature distribution derived from Equation (3.3).

∆T (K) q& (W/m3) ∆z1 = 0.001m ∆z1 = 0.002m ∆z1 = 0.003m ∆z1 = 0.004m ∆z1 = 0.005m

2 × 106 0.016529 0.066116 0.14876 0.264463 0.41322 2 × 105 0.0016529 0.0066116 0.014876 0.0264463 0.041322 2 × 104 0.0001653 0.0006617 0.001488 0.0026446 0.0041322 2 × 103 0.0000165 0.0000662 0.0001488 0.0002645 0.0004132


158

3.1.2 Experimental Program

3.1.2.1 Materials Selection and Specimen Design

The design of the fatigue specimens incorporated various considerations regarding the

servohydraulic uniaxial testing machinery, measurement by thermodynamic and

mechanical means, and specimen buckling. The specimens were machined from

AS3679-300 steel and designed in accordance with the specimen design

recommendations of ASTM standard E466-96 [58]. AS3679 steel was chosen due to

its widespread use in structural applications in which cyclic fatigue loading situations

are encountered. Typical nominal mechanical properties characteristic of this metal

alloy, including modulus of elasticity E, Poisson’s ratio ν, yield stress σo, ultimate

tensile stress σUTS and ductility εf are presented in Table 3.1.2 [32,59].

Table 3.1.2. Nominal mechanical properties.

Material E (GPa) ν σo (MPa) σUTS (MPa) εf AS3679-300 210 0.293 350-400 450-490 0.32

The AS3679 grade of steel is a mild steel with chemical composition and heat

treatment similar to that of mild steel grades used under different naming conventions.

A comparison between the chemical composition of AS3679 steel [59] and that of

AISI-SAE1010 obtained from the ASM Metals Reference Book [28] is displayed in

Table 3.1.3. From the close comparison, values for density κ, specific heat cp and

thermal conductivity k according to the ASM Metals Reference Book [28] are

presented for AS3679 steel in Table 3.1.4.

Table 3.1.3. Chemical composition comparison.

Chemical Composition (%) Material C Mn P S

AS3679-300 steel 0.15 0.25 0.03 0.03 AISI-SAE1010 steel 0.08-0.13 0.3-0.6 0.008-0.04 0.028-0.05


159

Table 3.1.4. Material properties and thermodynamic constants.

Material κ (kg/m3) cp (J/kg.K) k (W/m.K) AS3679-300 Steel 7854 434 60.5

In accordance with the limitations imposed by the servohydraulic uniaxial testing

machinery in terms of stroke length and friction grip diameter, cylindrical specimens

of 210 mm length and 15 mm diameter were proposed. For mechanical

measurements, a minimum uniform test section length of 25 mm was required in

accordance with the 25 mm gauge length strain extensometer. From the preliminary

calculations of Table 3.1.1, a thermistor spacing of 5 mm was proposed for the

thermodynamic measurements, with a thermistor positioned at the plane of symmetry

and additional thermistors to be equi-spaced at 5 mm intervals. An arrangement of

five thermistors positioned symmetrically about the plane of symmetry was proposed,

resulting in a total test length of 20 mm. ASTM E466-96 [58] specifies that for a

cylindrical specimen with a uniform test section and tangentially blending fillets

between the test section and ends, the uniform test section should be approximately

two times longer than the test section diameter. The blending fillet radius should be at

least eight times the test section diameter in order to minimise the stress concentration

factor, and the grip cross-sectional area should be at least 1.5 times the test cross-

sectional area. The test section diameter should remain between 5.08 mm and 25.4

mm. Considering Saint-Venant’s principle [36] with a 20 mm total length required for

the thermodynamic measurements and a 25 mm test section length, a 10 mm diameter

and 102.5 mm fillet radius were proposed to ensure uniform stress in the test section

whilst preventing buckling from occurring during low cycle fatigue.

The resulting design consisted of a specimen of 210 mm length and 15 mm diameter,

with 70 mm ends and a 70 mm machined section as illustrated by Figures 3.1.6-3.1.7.

The 10 mm diameter section was marked with blue ink to prevent surface corrosion

and to allow thermistor positioning markings to be easily identified. All specimens

were machined and polished to the same specifications, and as such the influence of

initial defects were not considered in the scope of this research. The thermistor

positioning markings were placed along the circumference via a fine black felt-tip

permanent marker attached to the tool stock of a lathe, with the plane of symmetry


160

and 5 mm symmetric spacings marked accordingly. The ends of the specimens were

chamfered and polished to allow a tight seal to be formed with the vacuum chamber.

Figure 3.1.6. Fatigue specimen dimensions.

Figure 3.1.7. Fatigue specimen indicating 5 mm interval markings about plane of symmetry.

3.1.2.2 Temperature Measuring Equipment

Based on the conceptual development, thermistors were selected to be used as

temperature measurement probes for attachment to the specimen surface, with each

probe connected as a resistor to Wheatstone bridge circuitry powered by a regulated

voltage. The thermistors chosen for the application were RS miniature glass bead

NTC thermistors with a 220 kΩ resistance at 298 K. Thermistors with high resistance

R were specifically chosen as increased resistance results in decreased power for a

balanced bridge circuit of constant voltage, and hence decreased heat dissipation from

the sensor at the point of measurement. Characteristic data and dimensions of the

miniature glass bead NTC thermistor are displayed by Table 3.1.5, Equation (3.4) and

Figure 3.1.8.


161

Table 3.1.5. Miniature glass bead NTC thermistor characteristics based on Equation (3.4).

Characteristic Value Rbead +25°C (kΩ) 220 Rmin (Hot) (kΩ) 1.3 Rbead Tolerance (%) ±20 Tamax (Ambient Temperature Range Maximum Dissipation) (°C) -55 - +200 Maximum Dissipation (mW) 130 Derate to Zero at (°C) 200 Dissipation Constant (mW/K) 0.75 Thermal Time Constant (s) 5 M Constant (+25°C to +85°C) (K) 4145 M Tolerance (%) ±3

−

= 12

11

12TT

M

eRR (3.4)

Figure 3.1.8. Miniature glass bead NTC thermistor dimensions.

Each thermistor was soldered to insulated wiring commonly referred to as figure-8

wire, consisting of a positive and negative wire, with the exposed wire insulated by

heat shrink insulation up to the glass bead. The wire was broken by male and female

lug type connectors to allow the thermistor probes to be connected directly to the

bridge circuitry wiring or via an alternative connection. The Wheatstone bridge

circuitry was comprised of four resistors configured in a bridge arrangement as

illustrated by Figure 3.1.9, consisting of the miniature glass bead NTC thermistor, a

resistor and potentiometer arranged in series, and two resistors. The two resistors each

had a resistance of 220 kΩ, whilst the resistor and potentiometer in series had a

resistance of 150 kΩ and a variable resistance of 100 kΩ respectively. The bridge was

powered by a 12 V Ni-Cd battery, regulated to 5 V. The battery was used in order to

eliminate the 50 Hz frequency noise associated with most power supply units. The

output for each temperature sensor was in the form of the voltage Vo measured across

the bridge circuit, with connectors allowing the measurement of the output via banana

5 mm 25 mm

1.6 mm


162

type plug connectors or BNC coaxial cable. The assembled temperature probe is

illustrated by Figure 3.1.10.

Figure 3.1.9. Wheatstone bridge circuit.

Figure 3.1.10. Temperature probe.

The use of a potentiometer in the opposing arm of the bridge circuit to the thermistor

allowed the possibility of zeroing the circuit for each temperature sensor at the

V

Vo

RR R4 = RR + RP

RP

R1 R3

I1

I3

I2

R2

V – 5V voltage source Vo – Voltage output I1 – Total current I2 – Current arm 1 I3 – Current arm 2 R1 – Thermistor R2 – 220 kΩ resistor R3 – 220 kΩ resistor RR – 150 kΩ resistor RP – 100 kΩ potentiometer R4 – Combined resistance

Thermistor glass bead


163

beginning of an experiment after the initial temperature reading had been recorded.

This would increase the accuracy of measurement associated with small temperature

changes as the data acquisition equipment could be set to measure voltage for a

smaller voltage range with a higher precision.

3.1.2.3 Temperature Calibration Facility

To provide the means for accurate calibration of the temperature measuring

equipment, a purpose built temperature calibration facility was designed using

existing precision calibrated equipment. A temperature controlled water bath was used

as the basis for the calibration facility to enable the calibration of the temperature

measuring equipment over an expected wide temperature range. The water bath

equipment, a Julabo HC-10, incorporated both heating and refrigeration cooling

facilities and the ability to set temperature with 0.1 K increments with an accuracy of

0.01 K. From preliminary calculations, a calibration temperature range from 15°C to

40°C was proposed to incorporate the minimum expected initial temperature and a

reasonable expected maximum temperature at the specimen surface.

The calibration facility consisted of the water bath, tripod and clamp, large glass test

tube, temperature measuring equipment and data acquisition equipment. The base of

the glass test tube was filled with thermal conductive grease or heat sink compound.

The five thermistor probes were inserted into the heat sink compound. The remaining

space within the test tube was filled with fine grain sand to minimise air gaps and

provide a thermal insulator between the thermistors and ambient air. The test tube,

acting as a large temperature probe due to its thin Pyrex glass construction and heat

sink compound base, was two-thirds submerged through a cardboard opening into the

water bath and held in position by clamp and tripod. The cardboard was taped using

electrical insulation tape to the opening of the water bath to reduce heat transfer

effects arising from convection between the water surface and ambient air. The

temperature measuring equipment was configured to output a bridge voltage with the

potentiometers in a fixed position, set at the minimum series output of 100 kΩ. The


164

facility is illustrated by Figures 3.1.11, depicting the temperature calibration facility,

temperature measuring equipment and data acquisition equipment.

Figure 3.1.11. Temperature calibration facility: (a) water bath and temperature measuring equipment;

(b) facility overview.

Data was acquired through a PC-based National Instruments PCI-6021E 12-bit PC-

based analog-to-digital data acquisition card with associated data acquisition software.

Data was acquired through five channels simultaneously at a rate of 1 Hz in the form

of voltage V, with a precision according to the voltage range illustrated by Table

3.1.6. BNC coaxial cables were used to input the bridge voltages with a 40 Hz low

(a)

(b)

Water bath

Water

Temperature probes

Sand

Heat sink compound

Cardboard


165

pass filter installed near the BNC input board to remove excessive 50 Hz noise

present due to surrounding electrical equipment and power sources.

Table 3.1.6. Data acquisition precision.

Voltage Range Precision ± 10 V 4.88 mV ± 5 V 2.44 mV

± 500 mV 244.14 µV ± 50 mV 24.41 µV

The calibration program began at 15°C, incrementing at 1°C increments and

terminating at 40°C. For each temperature, a ten minute period was allowed for water

temperature equilibrium to be reached, followed by a 600 second data acquisition

period at a frequency of 1 Hz. The average value and standard deviation for each

temperature increment was calculated from the acquired data. The average value for

each thermistor was taken as a calibration point at each temperature, providing the

associated standard deviation value was negligible. The water bath calibration was

regularly checked by comparison of the indicated temperature with that of a precision

calibrated thermometer. The calibrated thermometer used for the experiment was an

AMA 9975-4-96 thermometer, calibrated to 0.1°C increments.

3.1.2.4 Achievement of Thermal Isolation at Specimen Surface

Thermal isolation of the specimen surface was achieved through the development of a

vacuum chamber. The vacuum chamber design permitted the fatigue specimen to be

gripped at either end by the friction grips of the servohydraulic uniaxial testing

machine and the temperature sensors to be attached to the specimen surface, whilst

providing a vacuum surrounding the test section of the fatigue specimen. A vacuum

pump was used in conjunction with the vacuum chamber to provide a constant

vacuum throughout the duration of a fatigue test.

The design of the vacuum chamber incorporated a thin-walled cylinder, blank flange

ends with concentric holes for the specimen (the top flange is welded while the

bottom flange is fixed by two tie rods), electrical wiring and lugs, and vacuum pump


166

hose attachment, as illustrated in Figures 3.1.12-3.1.13. A thin-walled steel cylinder of

90 mm diameter was used containing fittings for the electrical wiring and a copper

tube for attachment of the vacuum pump hose. The blank flanges each contain a

centred 15 mm diameter hole consisting of a rubber o-ring for forming a tight seal

around the specimen ends. A vacuum seal was formed around the electrical wiring by

formation of an Araldite plug. The vacuum pump used was of the belt-driven rotary

oil type, consisting of the pump, main and bleeder valves, and a vacuum gage,

connected to the vacuum chamber via reinforced hosing and clamped to the copper

tube. Dow Corning silicon high vacuum grease was used to coat the ends of the

specimens before placement into the chamber, forming a seal capable of achieving a

vacuum in the order of 2 torr (0.2666 kPa).

Figure 3.1.12. Vacuum pump.


167

Figure 3.1.13. Vacuum chamber design: (a) cylinder and bottom flange; (b) top flange.

The male and female connectors of the thermistors were attached to corresponding

connectors located in the vacuum chamber to allow thermistors attachment to the

specimen surface. The high vacuum created had the effect of reducing any convection

to a level that could be considered negligible, hence simplifying the heat transfer

(b)

(a)


168

problem to the 1-D or 2-D axisymmetric heat conduction solutions. The effects of

radiation were minimised by lining of the inside surface of the cylinder with highly

reflective aluminium foil, greatly reducing radiation absorption at the specimen

surface. Typical emissivity ε ′ values representative of the radiation absorption for

reflective aluminium foil and steel specimen surface are indicated by Table 3.1.7.

Table 3.1.7. Typical emissivity values for selected materials.

Material ε′ Steel (black body) 1

Aluminium foil 0.06

3.1.2.5 Fatigue Testing Program

Testing of the fatigue specimens was conducted using a 100 kN Instron

servohydraulic uniaxial testing machine. Fully reversed cyclic, uniaxial loading was

applied to each specimen under load control until failure occurred. The specimens

were held in place at each end by friction clamp type grips. Testing was carried out

under load control according to a specified load amplitude and cyclic frequency. For

measurement of deflection a 25 mm gauge length strain extensometer was used. Data

was acquired via a PC-based National Instruments PCI-6021E 12-bit analog-to-digital

data acquisition card with associated data acquisition software in the form of load-

displacement data for the mechanical method, and voltage-time data for the

thermodynamic method. The thermodynamic method is illustrated by Figure 3.1.14,

displaying the servohydraulic uniaxial testing machine, fatigue specimen, vacuum

chamber, vacuum hose and temperature measuring equipment.


169

Figure 3.1.14. Thermodynamic method consisting of clamped specimen enclosed by vacuum chamber.

Fatigue testing of the AS3679 specimens was conducted under sinusoidal cyclic

loading with zero mean load, implying fully reversed cyclic loading consisting of a

load amplitude equal in tension and compression. A maximum cyclic loading

frequency of 5 Hz could be applied by the uniaxial testing machinery. Monitoring of

the cyclic load amplitude was verified by oscilloscope measurement from the load

output channel.

For the mechanical method of measurement, an extensometer was placed along the 25

mm length test section prior to testing. Tests using the mechanical method were

carried out at a cyclic loading rate of 0.5 Hz, and two channels of data for each test

corresponding to displacement and load were recorded at a data acquisition rate of 50

Hz. For the thermodynamic method, the thermistors were placed vertically along the

specimen with the glass beads in contact with the circumferentially marked positions.

Each thermistor bead was coated with heat sink compound and fastened tightly to the

specimen test section with electrical insulation tape. Tests for the thermodynamic

method were conducted at cyclic loading rates ranging from 0.5-5 Hz in accordance

with the heat dissipation per cycle such that the upper bound of the calibrated

temperature range (40°C) was not exceeded. A minimum cyclic frequency of 0.5 Hz

was determined to minimise the influence of the thermoelastic effect on the recorded

data [60,61]. Five channels of voltage data were recorded for each test at a data


170

acquisition rate of 1 Hz. The number of cycles for each test were counted by the PC-

based software interface to the Instron servohydraulic uniaxial testing machine.

The fatigue testing program incorporated the low cycle fatigue and high cycle fatigue

regimes ranging from 102-106 cycles. The mechanical approach was applied to the

low cycle fatigue experiments conducted up to 104 cycles, where extensometer

measurement was used to determine load-displacement hysteresis loops. The

thermodynamic approach was applied to the high cycle fatigue experiments ranging

from 104-106 cycles, where voltage-time curves were determined corresponding to the

positioning of the thermistors. Prior to each thermodynamic test, specimens were pre-

cycled for 1000 cycles at the test load level and cyclic loading frequency to stabilise

the internal heat generation rate. An overlap of the mechanical and thermodynamic

measurement experiments was produced at 104 cycles to allow verification of the

thermodynamic method, assuming applicability of the uniform internal heat

generation model to low cycle fatigue.


171

3.1.3 Analytical Program

3.1.3.1 Determination of Plastic Strain Energy Density - Mechanical

Measurement

From the load-displacement hysteresis data obtained from mechanical measurement

of low cycle fatigue testing, stress-strain hysteresis curves were derived in terms of

engineering stress σ and engineering strain ε, as indicated by Equations (2.6)-(2.7)

respectively. The general form of the stress-strain hysteresis loop is illustrated by

Figure 3.1.15.

Figure 3.1.15. Typical stress-strain hysteresis loop.

From the stress-strain data, hysteresis curves were obtained in terms of stress σ and

plastic strain εp according to Equation (3.5). A numerical integration scheme was

applied to the stress-plastic strain data to obtain the plastic SED per fully reversed

cycle ∆Wp. The numerical scheme incorporated the trapezoidal rule to obtain the area

within the hysteresis loop, the area equivalent to plastic energy per unit volume. The

ε

σ


172

trapezoidal integration scheme and method of application to the stress-plastic strain

curve are indicated by Equation (3.6) and Figure 3.1.16, where i denotes the current

data point and j denotes the total number of data points.

Epσεε −= (3.5)

( )∑−

=+

+ +−

=∆1

11

1

2

j

iii

ipippW σσ

εε (3.6)

Figure 3.1.16. Trapezoidal rule application to stress-plastic strain curve.

For the purpose of calculating the area within the hysteresis loop of the stress-plastic

strain data, an Excel spreadsheet incorporating Visual Basic programming was

developed. For the low cycle fatigue testing using extensometer measurement, the

cyclic loading rate of 0.5 Hz and load-displacement data acquisition at a rate of 50 Hz

resulted in smooth curves consisting of 100 data points per fully enclosed hysteresis

loop. For mechanical measurement, failure was defined as crack initiation to a

specific size required for steady crack growth. The initiation of a crack of critical size

was assumed to coincide with the rapid increase in area within the hysteresis loop

following steady state behaviour.

εp

σ

σi

1ipε +

σi+1

∆Wp

ipε


173

3.1.3.2 Determination of Plastic Strain Energy Density - Thermodynamic

Measurement

From thermodynamic measurement of high cycle fatigue testing, voltage-time curves

were obtained representing the temperature profile of the test section surface. From

temperature calibration of the thermistors, temperature T was obtained from the

voltage data resulting in the formation of temperature-time curves. The general form

of the temperature-time curves obtained from the thermistor locations is illustrated by

Figure 3.1.17, indicating the initial transient temperature rise, a constant temperature

at steady state conditions, and rapid temperature rise in the final fatigue life stages due

to crack propagation. For the thermodynamic measurement, failure was defined as

crack initiation to a specific size required for steady crack growth. The initiation of a

crack of critical size was assumed to coincide with the beginning of the crack

propagation region of the temperature-time curve.

Figure 3.1.17. Typical temperature-time curve.

For analysis of the temperature-time data, a finite difference scheme based on the heat

transfer equation was developed. The finite difference scheme allowed the internal

Failure

T

t

Transient Temperature

Steady State Temperature

Crack Propagation


174

heat generation rate q& to be progressively determined through numerical analysis of

the transient and steady state regions of the temperature-time curves, accounting for

the experimentally determined time varying boundary conditions. By obtaining the

internal heat generation rate q& from the finite difference scheme, the plastic SED per

cycle ∆Wp may be determined according to the cyclic loading frequency f as indicated

by Equation (3.7).

fqWp&

=∆ (3.7)

For the finite difference scheme, the 1-D thermodynamic model was assumed for low

cycle and high cycle fatigue based on a uniform q& distribution. The assumption of

uniform q& appeared valid for low cycle fatigue due to significant uniform plastic

deformation present throughout the uniform test section. For high cycle fatigue, in the

absence of specific evidence regarding contribution of surface effects, the uniform q&

distribution model provided a qualitative assessment of the existence of plastic strain

energy. By substitution of the formulae for first order and second order numerical

differentiation, a finite difference scheme was obtained using a central difference

formulation for space and a forward difference formulation for time. The numerical

differentiation terms for second order central difference in space and first order

forward difference in time are indicated by Equations (3.8)-(3.9) respectively. Here, i

denotes an increment ∆z in space, and j denotes an increment ∆t in time. By

substitution of Equations (3.8)-(3.9) into the 1-D heat transfer differential expression

of Equation (3.1), an explicit finite difference scheme was obtained as indicated by

Equation (3.10).

( )211

2

2 2∆z

TTTzT j

ij

ij

i −+=

∂∂ −+ (3.8)

∆tTT

tT j

ij

i −=

∂∂ +1

(3.9)


175

( ) ( )( )

kc∆tqTT

∆zc∆t

∆zc∆tTT

p

ji

ji

pp

ji

ji κκκ

&+++

−= −+

+1122

1 21 (3.10)

The explicit finite difference scheme is incremental in terms of the calculations

performed, and was divided into equal divisions in terms of space and time. Given the

initial and boundary conditions, the finite difference scheme increments forward in

time, calculating new values of temperature in space from the previous temperature

values. The scheme accounts for internal heat generation rate q& throughout the

model, and requires specific material properties and thermodynamic constants in the

form of density κ, specific heat cp and thermal conductivity k (cp and k were assumed

constant within the experimental temperature range). The application of the finite

difference scheme to the thermodynamic model is illustrated by Figure 3.1.18 and

Equations (3.11)-(3.13) for a generalised situation, displaying the modelling of the

uniform test section and the finite difference scheme incorporating boundary

conditions.


176

Figure 3.1.18. Assumed thermodynamic model (uniform internal heat generation).

( ) ( )( )

kc∆tqTT

∆zc∆t

∆zc∆tTT

p

ji

ji

pp

ji

ji κκκ

&+++

−= −+

+1122

1 21 (3.11)

( ) ( ) kc∆tqT

∆zc∆t

∆zc∆tTT

p

ji

pp

ji

ji κκκ

&++

−= +

+122

1 221 (z = 0) (3.12)

ji

ji TT =+1 (z = zo) (3.13)

For heat dissipation at the free surface, the effects of radiation are considered. The

radiation exchange problem may be modelled in terms of two infinite concentric

cylinders, or an infinite cylinder enclosed within an infinite cylinder, as illustrated by

q& Axis of

Symmetry

Plane of Symmetry

Ti-1

Free Surface

(Adiabatic)

z0

Ti

Ti+1 ∆z

∆z

z

0 r


P

P

z

r

r0

z0

z0


177

Figure 3.1.19. The heat flux between the inner and outer surfaces 12q′ corresponding

to this situation may be determined according to Equation (3.14) [62]. The area ratio

and view factor F12 are applied according to Equations (3.15)-(3.16) [62]. The

resulting equation from substitution is indicated by Equation (3.17), expressing heat

flux 12q′ in terms of inner and outer surface temperatures T1 and T2, Boltzmann

constant σ ′ , inner and outer radii r1 and r2, emissivities of the inner and outer

surfaces 1ε ′ and 2ε ′ , and inner surface area A1.

Figure 3.1.19. Surface radiation exchange model.

22

2

12111

1

42

41

12 111Aεε

FAAεε

)T(Tσq

′′−

++′

′−−′

=′ (3.14)

2

2

1

2

1

=

rr

AA (3.15)

112 =F (3.16)

( )

′

′−+

′

−′=′

2

1

2

2

1

42

411

12 11rr

εε

ε

TTAσq (3.17)

r1 r2

Surface 1

Surface 2


178

To incorporate the effects of radiation at the specimen surface, an energy balance

model was devised in terms of the element illustrated by Figure 3.1.20, indicating the

heat flux q′ contribution due to conduction and radiation, and internal heat generation

q&. Convection at the specimen surface was assumed to be negligible due to the

presence of a constant high vacuum. Considering the element section area A and

surface area As of Equations (3.18), an energy balance calculation was performed

incorporating heat flux inQ& , internal heat generation gQ& and transient heat effects stQ&

according to Equation (3.19) [62]. By substitution, explicit terms were derived for

inQ& , gQ& and stQ& according to the element of Figure 3.1.20, indicated by Equations

(3.20)-(3.22) respectively.


179

Figure 3.1.20. Finite difference model incorporating heat flux, internal heat generation and transient

effects.

2 π orA = (3.18)

∆z r A os π2=

stgin QQQ &&& =+ (3.19)

( ) ( )

′

′−+

′

−′

−−+−= −+

s

o

sp

isj

ij

ij

ij

iin

rr

εε

ε

TTAσTT

∆zkATT

∆zkAQ

2

2

1

44

11 11& (3.20)

Ti-1

Ti

Ti+1

q& Ts

( )

′

′−+

′

−′=′

s

o

2

2

1

4s

4is

rr

εε1

ε1

TTAσ-q

( )∆z

TTkAq i1i −=′ +

( )∆z

TTkAq i1i −=′ − A

As

∆z

∆z

A

r0 rs

Specimen Surface

Chamber Surface


180

zAqQg ∆&& = (3.21)

∆tTT

zAκcQj

ij

ipst

−=

+1

∆& (3.22)

The resulting finite difference equation is indicated by Equation (3.23), incorporating

the effects of radiation at the free surface whilst maintaining a 1-D heat conduction

problem with internal heat generation. From theoretical calculations, the heat loss at

the free surface was assumed negligible compared to the transference of heat via

conduction, allowing the incorporation of radiation heat flux into the 1-D heat

conduction problem with minimal error, as opposed to specifically developing a 2-D

axisymmetric model. Constant radii values ro and rs, and emissivity values ε1 and ε2

for the finite difference calculation are listed in Table 3.1.8.

5

444

5

31

5

21

5

1

5

2

5

11 1QQ

TTQQ

TQQ

TQQ

QQ

QQ

TT sj

ij

ij

ij

ij

i +

−+++

−−= +−

+ (3.23)

∆zkr

Q o2

1 = ∆zkr

Q o2

2 =

−+

′=

s

o

o

rr

εε

ε

zrσQ

2

2

1

3 11∆2

∆zrqQ o2

4 &= ∆zrcQ op2

5 κ=

Table 3.1.8. Radius and emissivity values for finite difference calculation.

Quantity Value ro (mm) 5 rs (mm) 45

ε1 1

ε2 0.06

The finite difference calculations were performed using a purpose designed Excel

spreadsheet incorporating Visual Basic programming. The spreadsheet program

allowed direct graphical comparison between the experimentally obtained data and

that obtained from the finite difference scheme. Experimental data obtained at 10 mm

from the plane of symmetry was used to determine the time varying boundary

conditions, requiring the internal heat generation q& to be specified for each time

increment. From the time varying boundary conditions and internal heat generation


181

values, a temperature distribution along the uniform section was calculated with

respect to time. An iterative procedure was adopted to vary q& until a negligible

difference was obtained between the temperature profile of the experimental and finite

difference data. The spreadsheet program allowed the temperature-time data for the

transient and steady state periods of the experimental data to be closely matched by

the finite difference analysis, with the transient period of particular importance in

verifying the accuracy of the assumed thermodynamic model.

The number of increments in space ∆z was set to four to coincide with the thermistor

locations, whilst the incrementation in time was automatically calculated and applied

to produce data points matching the time incrementation of the experimental data. The

minimum allowable time increment ∆t for a convergent solution was calculated from

a stability equation derived from the finite difference scheme, as indicated by

Equation (3.24) [62]. In this equation, F represents the stability factor which must be

greater than or equal to one for unconditional stability of the finite difference scheme.

The largest time interval ∆tmax possible to achieve a stable solution is indicated by

Equation (3.25), assuming a critical stability factor F of unity.

( )zrc

tQQFop ∆

∆+−= 2

211κ

(3.24)

21

2

max QQzrc

t op

+∆

=∆κ

(F = 1) (3.25)

3.1.3.3 Comparison of Mechanical and Thermodynamic Measurement

To enable verification of the thermodynamic approach, low cycle fatigue results

obtained from mechanical and thermodynamic measurement were to be compared in

the vicinity of 104 cycles. A correlation of the plastic SED per cycle ∆Wp values

obtained from numerical integration of the stress-strain hysteresis loop and those

obtained via the finite difference scheme using temperature-time data would enable a

direct comparison of the two methods. Assuming validity of the uniform internal heat

generation model for low cycle fatigue and correctness of the thermodynamic


182

constants, a close correlation would determine the validity of the thermodynamic

approach, enabling a qualitative assessment of the high cycle fatigue data to be

performed.


183

3.2 Experiments and Results

3.2.1 Thermistor Calibration

Calibration of the glass bead NTC thermistors was conducted using the temperature

calibration facility in 1°C increments over a temperature range of 15-40°C. Mean

bridge voltage values V were obtained for the thermistors corresponding to each

temperature increment. The resulting thermistor voltage-temperature calibration

curves are illustrated by Figure 3.2.1.

-600

-400

-200

0

200

400

600

800

1000

1200

1400

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

T (oC)

V (m

V)

Thermistor 1

Thermistor 2

Thermistor 3

Thermistor 4

Thermistor 5

Figure 3.2.1. Thermistor calibration curves.

The voltage data acquisition rate of 1 Hz over a 600 second period produced a

negligible standard deviation for each steady state temperature setting. As illustrated

by Figure 3.2.1, the voltage-temperature curves for each 1°C interval are piecewise

linear, allowing intermediate values to be obtained via linear interpolation. With

reference to Table 3.1.6, a precision in the order of 10-3 °C was obtained from the

calibration procedure.


184

3.2.2 Experiments Conducted using Mechanical Measurement

3.2.2.1 Low Cycle Fatigue Experiments

From cyclic loading experiments performed on the AS3679 fatigue specimens using

extensometer measurement, load-displacement hysteresis curves in the fatigue life

range of 102-104 cycles were obtained in terms of the applied load P and extensometer

deflection δ. The low cycle fatigue tests conducted using extensometer measurement

are outlined by Table 3.2.1, indicating the cyclic loading frequency f, load amplitude

Pa and number of cycles to failure Nf. Each fatigue test was designated according to

the convention E.Pa.i, where E indicates extensometer measurement, Pa the applied

load, and i the test number. A typical load-displacement hysteresis loop is illustrated

for the E.26.1 fatigue test by Figure 3.2.2.

Table 3.2.1. Low cycle fatigue results.

Specimen f (Hz) Pa (kN) Nf E.30.1 0.5 30 690 E.30.2 0.5 30 981 E.28.1 0.5 28 2369 E.28.2 0.5 28 2670 E.26.1 0.5 26 6723 E.26.2 0.5 26 9556


185

-30

-20

-10

0

10

20

30

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

δ (mm)

P (k

N)

Figure 3.2.2. Load-displacement curve, E.26.1.

From the load-displacement hysteresis curves, stress-strain curves were obtained in

terms of engineering stress σ and engineering strain ε corresponding to the steady

state hysteresis loop formation prior to crack initiation. The steady state σ-ε hysteresis

curves obtained from each fatigue test are illustrated by Figures 3.2.3-3.2.8.

-500

-400

-300

-200

-100

0

100

200

300

400

500

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008

ε

(MPa

)

Figure 3.2.3. Stress-strain curve, E.30.1.

δ (mm)

P (kN)

σ (MPa)σ (MPa)

ε


186

-500

-400

-300

-200

-100

0

100

200

300

400

500

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008

ε

(MPa

)


-400

-300

-200

-100

0

100

200

300

400

-0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005

ε

(MPa

)


σ (MPa)σ (MPa)

ε

σ (MPa)σ (MPa)

ε


187

-400

-300

-200

-100

0

100

200

300

400

-0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005

ε

(MPa

)


-400

-300

-200

-100

0

100

200

300

400

-0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

ε

(MPa

)


σ (MPa)

σ (MPa)

εp

ε

σ (MPa)

σ (MPa)

ε


188

-400

-300

-200

-100

0

100

200

300

400

-0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

ε

(MPa

)


3.2.2.2 Determination of Plastic Strain Energy Density

From the steady state σ-ε hysteresis curves, σ-εp hysteresis curves were obtained for

plastic SED determination in accordance with Equation (3.5). The steady state σ-εp

curves for each fatigue test are illustrated by Figures 3.2.9-3.2.14, displaying vertical

curve sections indicating elastic loading and unloading of the specimen due to the

removal of the elastic strain component. In accordance with Figure 3.1.1, the σ-εp

hysteresis curves indicate a large decrease in εp with a small decrease in σ as the yield

stress is approached.

σ (MPa)σ (MPa)

ε


189

-500

-400

-300

-200

-100

0

100

200

300

400

500

-0.006 -0.004 -0.002 0 0.002 0.004 0.006

εp

(MPa

)

Figure 3.2.9. Stress-plastic strain curve, E.30.1.

-500

-400

-300

-200

-100

0

100

200

300

400

500

-0.006 -0.004 -0.002 0 0.002 0.004 0.006

εp

(MPa

)


σ (MPa)

εp

σ (MPa)

εp


190

-400

-300

-200

-100

0

100

200

300

400

-0.003 -0.002 -0.001 0 0.001 0.002 0.003

εp

(MPa

)


-400

-300

-200

-100

0

100

200

300

400

-0.003 -0.002 -0.001 0 0.001 0.002 0.003

εp

(MPa

)


σ (MPa)

εp

σ (MPa)

εp


191

-400

-300

-200

-100

0

100

200

300

400

-0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002

εp

(MPa

)


-400

-300

-200

-100

0

100

200

300

400

-0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002

εp

(MPa

)


From Excel spreadsheet analysis incorporating Visual Basic programming, numerical

integration of the σ-εp hysteresis data was performed to obtain the plastic SED per

fully reversed cycle ∆Wp in accordance with Equation (3.6) and Figure 3.1.16. The

calculated ∆Wp values for each fatigue test is displayed by Table 3.2.2, indicating a

decreasing trend in ∆Wp with increasing number of cycles to failure Nf.

σ (MPa)

εp

σ (MPa)

εp


192

Table 3.2.2. Low cycle fatigue plastic SED results.

Specimen f (Hz) ∆Wp (J/m3) Nf E.30.1 0.5 6352967.29 690 E.30.2 0.5 6022594.05 981 E.28.1 0.5 3249052.49 2369 E.28.2 0.5 3074141.77 2670 E.26.1 0.5 2027363.16 6723 E.26.2 0.5 1761466.55 9556


193

3.2.3 Experiments Conducted using Thermodynamic Measurement

3.2.3.1 High Cycle Fatigue Experiments

From cyclic loading experiments performed on the AS3679 fatigue specimens using

thermodynamic measurement, voltage-time curves were obtained for the fatigue life

range of 104-106 cycles in terms of voltage V and time t. The high cycle fatigue tests

conducted using thermodynamic measurement are displayed by Table 3.2.3,

indicating the cyclic loading frequency f, load amplitude Pa and number of cycles to

failure Nf. Following the convention adopted for the low cycle fatigue testing, each

fatigue test was designated according to the convention T.Pa.i, where T indicates

thermodynamic measurement, Pa the applied load, and i the test number. Typical

voltage-time curves are illustrated for the T.26.1 fatigue test by Figure 3.2.15,

indicating voltage measurement at 5 mm intervals about the plane of symmetry in

accordance with Figure 3.1.7.

Table 3.2.3. High cycle fatigue results.

Specimen f (Hz) Pa (kN) Nf T.26.1 0.5 26 6345 T.26.2 0.5 26 10201 T.26.3 0.5 26 10230 T.24.1 3 24 61153 T.24.2 3 24 87564 T.24.3 3 24 112693 T.24.4 3 24 124369 T.22.1 5 22 1751029


194

00.10.20.30.40.50.60.70.80.9

11.11.21.31.4

0 5000 10000 15000 20000 25000

t (s)

V (V

)

Experiment- Symmetry PlaneExperiment- 5 mmExperiment- 10 mm

Figure 3.2.15. Voltage-time curves, T.26.1.

From the voltage-time curves, temperature-time curves were obtained in accordance

with the thermistor calibration curves of Figure 3.2.1. A typical temperature-time

curve is illustrated for the T.26.1 fatigue test by Figure 3.2.16, indicating the initial

transient temperature, steady state temperature and crack propagation sections in

accordance with Figure 3.1.17.

22

24

26

28

30

32

34

36

38

40

42

0 5000 10000 15000 20000 25000

t (s)

T (o C

)

Exp eriment- Symmetry PlaneExp eriment- 5 mmExp eriment- 10 mm

Figure 3.2.16. Temperature-time curves, T.26.1.


195

3.2.3.2 Determination of Plastic Strain Energy Density

From Excel spreadsheet analysis incorporating Visual Basic programming, finite

difference analyses were performed to obtain comparable temperature-time curves in

accordance with the finite difference scheme of Equation (3.23). Assuming a 1-D heat

conduction model, an iterative procedure was adopted to obtain the constant internal

heat generation rate q& which produced a close correlation between the transient and

steady state sections of the experimental and analytical temperature-time curves. A

comparison of the temperature-time curves obtained from experiment and analysis for

each fatigue test is illustrated by Figures 3.2.17-3.2.24.

22

24

26

28

30

32

34

36

38

40

42

0 5000 10000 15000 20000 25000

t (s)

T (o C

)

Exp eriment- Symmetry PlaneExp eriment- 5 mmExp eriment- 10 mmAnalys is - Symmetry PlaneAnalys is - 5 mmAnalys is - 10 mm



196

16

18

20

22

24

26

28

30

32

34

36

38

0 5000 10000 15000 20000 25000 30000 35000 40000

t (s)

T (o C

)



18

20

22

24

26

28

30

32

34

36

38

40

0 5000 10000 15000 20000 25000 30000 35000 40000

t (s)

T (o C

)




197

20

22

24

26

28

30

32

34

36

38

40

0 1000 2000 3000 4000 5000 6000 7000 8000

t (s)

T (o C

)

Experiment- Symmetry PlaneExperiment- 5 mmExperiment- 10 mmAnalys is - Symmetry PlaneAnalys is - 5 mmAnalys is - 10 mm

Figure 3.2.20. Temperature-time curves, T.24.1 (transient and steady state).


198

20

22

24

26

28

3032

34

36

38

40

42

44

0 1000 2000 3000 4000 5000 6000 7000 8000

t (s)

T (o C

)


20222426283032343638404244

7000 8000 9000 10000 11000 12000 13000 14000 15000 16000

t (s)

T (o C

)


Figure 3.2.21. Temperature-time curves, T.24.2: (a) transient and steady state; (b) crack propagation

and failure.

(b)

(a)


199

171921232527293133353739414345

0 1000 2000 3000 4000 5000 6000 7000 8000

t (s)

T (o C

)

Experiment- Symmetry PlaneExperiment- 5 mmExperiment- 10 mmAnalys is - Symmetry PlaneAnalys is - 5 mmAnalys is - 10 mm

171921232527293133353739414345

7000 8000 9000 10000 11000 12000 13000 14000 15000

t (s)

T (o C

)


Figure 3.2.22. Temperature-time curves, T.24.3: (a) transient and steady state; (b) crack propagation

and failure.

(b)

(a)


200

20

22

24

26

28

30

32

34

36

38

0 1000 2000 3000 4000 5000 6000 7000 8000

t (s)

T (o C

)


Figure 3.2.23. Temperature-time curves, T.24.4 (transient and steady state).


201

20

22

24

26

28

30

32

34

36

38

40

42

0 1000 2000 3000 4000 5000

t (s )

T (o C

)

Exp eriment - S ymmet ry P laneExp eriment - 5 mmExp eriment - 10 mmAnalys is - S ymmet ry P laneAnalys is - 5 mmAnalys is - 10 mm

20

22

24

26

28

30

32

34

36

38

40

42

0 500 1000 1500 2000 2500 3000 3500 4000

t (s )

T (o C

)

Exp eriment - Symmet ry P laneExp eriment - 5 mmExp eriment - 10 mmAnalys is - Symmet ry P laneAnalys is - 5 mmAnalys is - 10 mm

20

22

24

26

28

30

32

34

36

38

40

42

0 1000 2000 3000 4000 5000

t (s )

T (o C

)

Exp eriment - Symmet ry P laneExp eriment - 5 mmExp eriment - 10 mmAnalys is - Symmet ry P laneAnalys is - 5 mmAnalys is - 10 mm

Figure 3.2.24. Temperature-time curves, T.22.1: (a) transient; (b) steady state; (c) crack propagation

and failure.

(a)

(b)

(c)


202

The close correlation illustrated by the temperature-time curve comparison of Figures

3.2.17-3.2.24 indicate good agreement between the results obtained from experiment

and finite difference analysis. The agreement is particularly evident for the transient

and steady state temperature regions where a constant q& is assumed. With the

exception of the T.26.3 and T.24.3 fatigue experiments, the crack propagation event is

clearly indicated by a rapid temperature rise approaching specimen failure. The plastic

SED per fully reversed cycle ∆Wp for each fatigue test was obtained from the internal

heat generation rate q& according to the cyclic loading frequency f as specified by

Equation (3.7). The calculated ∆Wp values for each fatigue test are displayed by Table

3.2.4, indicating a decreasing trend in ∆Wp with increasing number of cycles to failure

Nf similar to the low cycle fatigue extensometer measurement results.

Table 3.2.4. High cycle fatigue plastic SED results.

Specimen f (Hz) q&(W/m3) ∆Wp (J/m3) Nf T.26.1 0.5 1400000 2800000 6345 T.26.2 0.5 1200000 2400000 10201 T.26.3 0.5 1600000 3200000 10230 T.24.1 3 2500000 833333.333 61153 T.24.2 3 1800000 600000 87564 T.24.3 3 2700000 900000 112693 T.24.4 3 2200000 733333.33 124369 T.22.1 5 2000000 400000 1751029


203

3.2.4 Comparison of Mechanical and Thermodynamic Results

The cyclic loading experiments performed on the AS3679 fatigue specimens using

extensometer and thermodynamic measurement encompassed the fatigue life range of

102-106 cycles. The combined low cycle and high cycle fatigue experiments are

outlined by Table 3.2.5, indicating the cyclic loading frequency f, plastic SED per

cycle ∆Wp and number of cycles to failure Nf.

Table 3.2.5. Combined low and high cycle fatigue plastic SED results.

Specimen f (Hz) ∆Wp (J/m3) Nf E.30.1 0.5 6352967.29 690 E.30.2 0.5 6022594.05 981 E.28.1 0.5 3249052.49 2369 E.28.2 0.5 3074141.77 2670 T.26.1 0.5 2800000 6345 E.26.1 0.5 2027363.16 6723 E.26.2 0.5 1761466.55 9556 T.26.2 0.5 2400000 10201 T.26.3 0.5 3200000 10230 T.24.1 3 833333.333 61153 T.24.2 3 600000 87564 T.24.3 3 900000 112693 T.24.4 3 733333.33 124369 T.22.1 5 400000 1751029

Assuming applicability of the uniform internal heat generation model, a correlation of

the plastic SED per cycle ∆Wp values obtained from 102-104 cycles was conducted to

verify the accuracy of the thermodynamic approach. A comparison of the low cycle

fatigue extensometer and thermodynamic measurement results is illustrated by Table

3.2.6 and the ∆Wp–Nf curve of Figure 3.2.25. Assuming the linear trend of the ∆Wp–Nf

curve on log-log axes in accordance with the research of Garud [24] and Ellyin [25],

the comparison indicates the achievement of good agreement between the

extensometer and thermodynamic measurements. The agreement verifies the validity

of the 1-D heat conduction model for low cycle fatigue and the qualitative

determination of ∆Wp for high cycle fatigue.


204

Table 3.2.6. Comparison of low cycle fatigue plastic SED results.

Specimen f (Hz) ∆Wp (J/m3) Nf E.30.1 0.5 6352967.29 690 E.30.2 0.5 6022594.05 981 E.28.1 0.5 3249052.49 2369 E.28.2 0.5 3074141.77 2670 T.26.1 0.5 2800000 6345 E.26.1 0.5 2027363.16 6723 E.26.2 0.5 1761466.55 9556 T.26.2 0.5 2400000 10201 T.26.3 0.5 3200000 10230

1000

10000

100000

1000000

10000000

100000000

1.0E+02 1.0E+03 1.0E+04 1.0E+05

Nf

Wp

Jm3

Extensometer DataThermod ynamic Data

Figure 3.2.25. Combined ∆Wp-Nf low cycle fatigue data obtained from mechanical and thermodynamic

measurement.

In accordance with Table 3.2.5, the combined results obtained from extensometer and

thermodynamic measurement are illustrated by the ∆Wp–Nf data of Figure 3.2.26,

depicting a monotonically decreasing trend in the data.


205

1000

10000

100000

1000000

10000000

100000000

1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07

Nf

Wp

Jm3

Exp erimental Data

Figure 3.2.26. Combined ∆Wp-Nf low cycle and high cycle fatigue data obtained from mechanical and

thermodynamic measurement.


206

3.3 Analysis and Discussion

A good correlation was indicated by comparison of the low cycle fatigue plastic SED

per cycle ∆Wp values obtained from extensometer and thermodynamic measurement.

The comparison illustrated the successful application of the thermodynamic approach,

verifying the achievement of thermal isolation, accurate temperature measurement,

appropriate assumed thermodynamic constants and applicability of the uniform

internal heat generation model to low cycle fatigue. The comparison of the

temperature-time curves obtained from experiment and finite difference analysis

enabled an iterative approach to be adopted in determining the internal heat

generation rate, allowing direct calculation of ∆Wp in accordance with the cyclic

loading frequency.

Two distinct thermodynamic models were proposed as possible representations of the

heat dissipation due to high cycle fatigue. The 1-D heat conduction model based on

uniform internal heat generation assumed that plastic deformation at the free surface

was negligible compared to the internal plastic deformation, whilst the 2-D

axisymmetric heat conduction model assumed that the vast majority of the internal

heat generation occurred at the free surface. The applicability of the 1-D heat

conduction model to low cycle fatigue was verified from comparison of results

obtained from extensometer and thermodynamic measurement. A possible means for

determination of the appropriate internal heat generation model for high cycle fatigue

exists in surface hardening of the fatigue specimen test section. A hardened layer

present at the free surface would reduce the magnitude of plastic deformation

associated with a particular cyclic load in comparison to the remaining cross-section.

A method of surface hardening called plasma immersion ion implantation, or PI3, was

proposed by Ken Short of the Materials and Engineering Science Division, Australian

Nuclear Science and Technology Organisation (ANSTO). The surface hardening

process involves treatment of specimens at 380°C for five hours in pure N2 gas,

resulting in a compound layer on the specimen surface consisting of Fe4N and Fe3N,

commonly referred to as the “white layer”. For typical low carbon steels, a surface

compound layer of approximately 5 µm in thickness and hardness of approximately


207

10 GPa can be achieved via this process, compared to the untreated material with

hardness less than 2 GPa. From application of thermodynamic measurement, a

comparison between the temperature-time curves of the normal and surface hardened

fatigue specimens would provide a direct means of determining an appropriate

internal heat generation model for high cycle fatigue.

Assuming applicability of the 1-D heat conduction model for high cycle fatigue, a

correlation between the low and high cycle fatigue ∆Wp–Nf data indicates a linear

relationship on log-log axes. The linear relationship depicted is in accordance with the

form of the ∆Wp–Nf curve obtained from the research of Garud [24] and Ellyin [25],

the curve of Ellyin depicting a linear relationship from actual test data approaching 6

× 105 cycles. A least squares analysis was performed on the data to obtain a linearised

equation of the form indicated by Equations (3.26)-(3.27). For this linear model, a

log-normal fatigue life distribution with constant variance along the Nf interval was

assumed without the inclusion of suspended test data in accordance with ASTM

standard E739-91 and prior fatigue testing [63,64]. The test data was categorised as

preliminary and exploratory due to the small sample size and low replication. The

determination of 95 percent confidence bands indicate the degree of variance present

in the test data. The test data, ∆Wp–Nf curve of best fit and 95 percent confidence

bands are illustrated on log-log axes by Figure 3.3.1.

( ) pfpp N∆W ωη 2= (3.26)

( ) 3817.06 2108258.91 −×=∆ fp NW (3.27)


208

1000

10000

100000

1000000

10000000

1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07

Nf

Wp

Jm3

Exp erimental DataCurve o f Bes t FitUpp er 9 5% Co nfid ence BandLower 95% Confidence Band

Figure 3.3.1 Linear form of ∆Wp-Nf curve plotted on log-log axes.


209

3.4 Conclusion

3.4.1 Research Outcomes

The existence of plastic deformation in metal alloys during high cycle fatigue was

confirmed from this research. The successful application of the thermodynamic

approach to high cycle fatigue allowed determination of plastic deformation beyond

the range of application of conventional mechanical measurement. The validity of the

thermodynamic approach was confirmed from comparison between low cycle fatigue

results obtained from mechanical and thermodynamic measurement, indicated by the

attainment of a close correlation.

Assuming application of an appropriate thermodynamic model, application of the

finite difference method allowed an iterative procedure to be adopted in determining

plastic strain energy. Matching of the transient and steady state temperature regions of

the temperature-time curves obtained from experiment and finite difference analysis

enabled the determination of a constant internal heat generation value corresponding

to the fatigue life prior to crack initiation. The attainment of thermal isolation at the

specimen test section surface allowed a quantitative analysis of the heat transfer

problem, enabling a direct determination of the plastic strain energy generated per unit

of loading cycle.

Several questions may be presented in relation to the application of thermodynamic

measurement to high cycle fatigue. The determination of an appropriate

thermodynamic model for high cycle fatigue would depend on the degree to which

plastic deformation at the free surface dominates the fatigue failure event. The two

thermodynamic models presented from this research, namely the uniform and free

surface internal heat generation models, indicate two possible representations of this

situation. Determination of the magnitude of plastic deformation present at the free

surface in comparison to the magnitude present throughout the remaining cross-

section would determine the correct model for application to high cycle fatigue. The


210

influence of surface effects also introduces a possible size effect in relation to the

specimen geometry. As the specimen diameter is increased, the ratio of the free

surface area to the volume would decrease, indicating a substantial reduction in the

surface area compared to the volume of the internal material. A reduction in the

influence of surface effects could be expected as the specimen dimensions are

increased.


211

3.4.2 Recommendations for Future Work

Determination of the influence of surface effects is a requirement for accurate

application of the thermodynamic approach to high cycle fatigue. The fatigue testing

of surface hardened specimens could provide a means of determining an appropriate

thermodynamic model. The degree to which surface effects contribute to heat

dissipation could be determined by a direct comparison between normal and surface

hardened specimens under identical cyclic fatigue loading conditions. Further testing

could be conducted to explore a possible size effect in relation to the test section

surface area compared to the volume. The method of PI3 surface hardening, coupled

with specific hardness testing of the hardened surface layer, could be applied to these

situations.

Following verification of an appropriate thermodynamic model for application to high

cycle fatigue, further fatigue testing investigating the effects of mean stress and non-

proportional loading would enable the development of a generalised fatigue failure

criterion characterised in terms of plastic strain energy. Cyclic testing of fatigue

specimens with variation of the mean stress would allow the influence of hydrostatic

stress to be determined. The application of non-proportional loading via tension-

torsion testing machinery would enable the influence of loading non-proportionality

effects to be considered, particularly in relation to material strain hardening. Further

testing could also be performed on notched specimens incorporating mean stress and

non-proportional loading which, in conjunction with finite element analysis, would

allow plastic strain energy determination for complex multiaxial states of stress.

212

4. SUMMARY OF CONCLUSIONS

Summary of Conclusions

213

The two separate research studies conducted, namely failure due to monotonic loading

and fatigue failure due to cyclic loading, have fulfilled the primary objectives outlined

in this thesis. The basic science approach of this thesis has allowed the determination

of the fundamental relationships which directly influence metal failure. The research

conducted has allowed the extension of previously existing continuum mechanics

based failure theories to encompass a wider range of application to metal alloy failure

assessment.

For the case of monotonic loading until failure, a fundamental relationship between

hydrostatic stress and ductility was confirmed. In particular, a distinct relationship

between hydrostatic tensile stress and equivalent plastic fracture strain was verified

through experiment and finite element analysis. The basis for a monotonic failure

criterion incorporating the hydrostatic tensile stress range was proposed and

confirmed from analysis. A monotonically decreasing equivalent plastic fracture

strain with increasing hydrostatic stress was illustrated for the hydrostatic tensile

stress range in accordance with the trend indicated for hydrostatic compression. The

establishment of a procedure for accurate determination of the equivalent stress-strain

curve allowed finite element analyses to be conducted on the various specimen

geometries tested, enabling determination of fracture points corresponding to the

proposed equivalent plastic fracture strain-hydrostatic stress failure curve relationship.

A methodology for determination of hydrostatic tensile stress fracture points through

application of the uniaxial tensile test was confirmed.

In the case of cyclic loading until failure, the existence of plastic strain energy during

high cycle fatigue was verified. The development of a thermodynamic method of

measurement in conjunction with finite difference analysis provided a quantitative

means for determination of plastic SED beyond the range of conventional mechanical

measurement. Corresponding to a constant cyclic loading amplitude, the procedure,

via an iterative approach, allowed the determination of a constant plastic SED value

prior to crack initiation. The accuracy of the method was confirmed by close

correlation of low cycle fatigue plastic strain energy results obtained from mechanical

and thermodynamic measurement. Two distinct thermodynamic models were

proposed as possible representations of heat dissipation during high cycle fatigue,

subject to verification. Assuming applicability of the uniform heat dissipation

Summary of Conclusions

214

thermodynamic model to low and high cycle fatigue, a distinct monotonically

decreasing value of plastic SED with increasing number of cycles to failure was

illustrated.

Numerous original contributions have been made to the fields of fracture and fatigue

in metals in terms of basic and applied knowledge. For monotonic loading, a

relationship between hydrostatic stress and equivalent plastic fracture strain was

demonstrated which could potentially unify the brittle and ductile fracture regimes via

a single failure criterion. A methodology combining experiment, analytical techniques

and finite element analysis was outlined and demonstrated for establishing the

equivalent stress-strain curve for metal alloys. A technique combining experiment and

finite element analysis using circular V-notch specimens was utilised which is capable

of obtaining equivalent plastic fracture strain-hydrostatic stress data points that span

the brittle and ductile fracture regimes. For cyclic loading, the basis for quantitative

plastic strain energy density measurement was established for the high cycle fatigue

regime. A thermodynamic approach to plastic strain energy measurement was

demonstrated which, consisting of a fatigue test specimen, thermally isolated chamber

and thermistor temperature measurement, allowed accurate determination of heat

dissipation when coupled with an appropriate thermodynamic model and finite

difference scheme. A concise methodology was established which allowed the

characterisation of fatigue over the entire fatigue life spectrum in terms of plastic

strain energy.

The incorporation of numerical analysis techniques in verification of the experimental

results demonstrated the applicability of the proposed monotonic failure criterion and

thermodynamic approach to numerical methods of analysis. In particular, the failure

due to monotonic loading research demonstrated the potential application of a

equivalent plastic fracture strain-hydrostatic stress based monotonic failure criterion

to finite element analysis, incorporating non-linear geometry and non-linear elastic-

plastic material behaviour. The determination of plastic SED from thermodynamic

measurement illustrated the application of the finite difference method in determining

heat dissipation due to plastic deformation. The application of the continuum

mechanics based failure theories outlined in this thesis to numerical analysis

techniques was amply demonstrated from this research.

215

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