continuum mechanics approaches to the study of …
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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
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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
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
Figure 2.2.23. Load-displacement curve, V-notch (4340 steel): (a) 15,9; (b)
15,10.5; (c) 12,6. 114
Figure 2.2.24. Load-displacement curve, V-notch (4340 steel): (a) 10,5; (b)
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
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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
Figure 2.2.32. V-notch stress-strain state, free-cutting brass 15,9: (a) σ -r; (b)
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
Figure 2.2.35. V-notch stress-strain state, free-cutting brass 8,4: (a) σ -r; (b)
σh-r; (c) pε -r. 127
Figure 2.2.36. Equivalent plastic strain contour, V-notch (free-cutting brass)
15,4.5. 129
Figure 2.2.37. Equivalent plastic strain contour, V-notch (free-cutting brass)
15,6. 130
Figure 2.2.38. Equivalent plastic strain contour, V-notch (free-cutting brass)
15,9. 130
Figure 2.2.39. Equivalent plastic strain contour, V-notch (free-cutting brass)
15,10.5. 131
Figure 2.2.40. Equivalent plastic strain contour, V-notch (free-cutting brass)
10,5. 131
Figure 2.2.41. Equivalent plastic strain contour, V-notch (free-cutting brass)
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
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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
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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 3.2.4. Stress-strain curve, E.30.2. 186
Figure 3.2.5. Stress-strain curve, E.28.1. 186
Figure 3.2.6. Stress-strain curve, E.28.2. 187
Figure 3.2.7. Stress-strain curve, E.26.1. 187
Figure 3.2.8. Stress-strain curve, E.26.2. 188
Figure 3.2.9. Stress-plastic strain curve, E.30.1. 189
Figure 3.2.10. Stress-plastic strain curve, E.30.2. 189
Figure 3.2.11. Stress-plastic strain curve, E.28.1. 190
Figure 3.2.12. Stress-plastic strain curve, E.28.2. 190
Figure 3.2.13. Stress-plastic strain curve, E.26.1. 191
Figure 3.2.14. Stress-plastic strain curve, E.26.2. 191
Figure 3.2.15. Voltage-time curves, T.26.1. 194
Figure 3.2.16. Temperature-time curves, T.26.1. 194
Figure 3.2.17. Temperature-time curves, T.26.1. 195
Figure 3.2.18. Temperature-time curves, T.26.2. 196
Figure 3.2.19. Temperature-time curves, T.26.3. 196
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
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List of Figures
XXIII
Figure 3.3.1 Linear form of ∆Wp-Nf curve plotted on log-log axes. 208
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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
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1
1. INTRODUCTION
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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
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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].
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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.
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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
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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)
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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
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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
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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
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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
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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)
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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
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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)
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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
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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
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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
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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
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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].
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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
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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
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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
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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
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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)
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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].
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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)
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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)
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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
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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)
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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)
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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)
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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
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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)
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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′
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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].
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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
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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)
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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
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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.
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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.
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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.
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41
2. FAILURE DUE TO MONOTONIC LOADING
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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
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Failure due to Monotonic Loading
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)
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Failure due to Monotonic Loading
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
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Failure due to Monotonic Loading
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.
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Failure due to Monotonic Loading
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.
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Failure due to Monotonic Loading
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)
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Failure due to Monotonic Loading
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=
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Failure due to Monotonic Loading
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
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Failure due to Monotonic Loading
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 +−=
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Failure due to Monotonic Loading
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.
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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Figure 2.1.15. Clamped test specimen with extensometer.
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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 =
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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).
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Failure due to Monotonic Loading
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∫=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
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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)
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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)
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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
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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∆
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Failure due to Monotonic Loading
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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.
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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.
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Failure due to Monotonic Loading
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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.
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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)
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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
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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)
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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).
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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
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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
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Failure due to Monotonic Loading
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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.
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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)
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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
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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−=
−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
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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).
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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Figure 2.2.3. Specimen images and dimensions for free-cutting brass: (a) undeformed; (b) deformed
(necked) immediately prior to failure.
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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( )
++
=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.
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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-
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Failure due to Monotonic Loading
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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).
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Failure due to Monotonic Loading
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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).
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Failure due to Monotonic Loading
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Figure 2.2.14. Equivalent stress contour, uniform section (4340 steel).
Figure 2.2.15. Hydrostatic stress contour, uniform section (4340 steel).
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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)
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Failure due to Monotonic Loading
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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)
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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)
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Failure due to Monotonic Loading
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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)
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Failure due to Monotonic Loading
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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)
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Failure due to Monotonic Loading
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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)
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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)
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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)
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Failure due to Monotonic Loading
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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)
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Failure due to Monotonic Loading
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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)
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Failure due to Monotonic Loading
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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
Figure 2.2.31. V-notch stress-strain state, free-cutting brass 15,7.5: (a) σ -r; (b) σh-r; (c) pε -r.
0
100
200
300
400
500
600
0 1 2 3 4 5
r (mm)
o (M
Pa)
(b)
(c)
(a)
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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
Figure 2.2.32. V-notch stress-strain state, free-cutting brass 15,9: (a) σ -r; (b) σh-r; (c) pε -r.
0
100
200
300
400
500
600
700
0 1 2 3 4 5 6
r (mm)
o (M
Pa)
(b)
(c)
(a)
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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
Figure 2.2.33. V-notch stress-strain state, free-cutting brass 15,10.5: (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 3
r (mm)
o (M
Pa)
(b)
(c)
(a)
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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)
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Failure due to Monotonic Loading
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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
Figure 2.2.35. V-notch stress-strain state, free-cutting brass 8,4: (a) σ -r; (b) σh-r; (c) pε -r.
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)
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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
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Failure due to Monotonic Loading
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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.
Figure 2.2.36. Equivalent plastic strain contour, V-notch (free-cutting brass) 15,4.5.
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Failure due to Monotonic Loading
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Figure 2.2.37. Equivalent plastic strain contour, V-notch (free-cutting brass) 15,6.
Figure 2.2.38. Equivalent plastic strain contour, V-notch (free-cutting brass) 15,9.
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Failure due to Monotonic Loading
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Figure 2.2.39. Equivalent plastic strain contour, V-notch (free-cutting brass) 15,10.5.
Figure 2.2.40. Equivalent plastic strain contour, V-notch (free-cutting brass) 10,5.
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Failure due to Monotonic Loading
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Figure 2.2.41. Equivalent plastic strain contour, V-notch (free-cutting brass) 8,4.
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.
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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.
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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.
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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).
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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
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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.
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Failure due to Monotonic Loading
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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
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Failure due to Monotonic Loading
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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].
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Failure due to Monotonic Loading
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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.
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146
3. FATIGUE FAILURE DUE TO
CYCLIC LOADING
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Fatigue Failure due to Cyclic Loading
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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.
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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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.
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Fatigue Failure due to Cyclic Loading
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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.
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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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.
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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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−
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Fatigue Failure due to Cyclic Loading
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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.
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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)
Uniform Test Section
PP z
r
r0
r
z
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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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)
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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.
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Fatigue Failure due to Cyclic Loading
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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
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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.
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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
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Fatigue Failure due to Cyclic Loading
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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
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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
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Fatigue Failure due to Cyclic Loading
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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)
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( ) ( )( )
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.
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Fatigue Failure due to Cyclic Loading
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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
Uniform Test Section
P
P
z
r
r0
z0
z0
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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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.
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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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
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
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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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.
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Fatigue Failure due to Cyclic Loading
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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.
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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-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
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Fatigue Failure due to Cyclic Loading
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-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 3.2.4. Stress-strain curve, E.30.2.
-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
)
Figure 3.2.5. Stress-strain curve, E.28.1.
σ (MPa)σ (MPa)
ε
σ (MPa)σ (MPa)
ε
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Fatigue Failure due to Cyclic Loading
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-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
)
Figure 3.2.6. Stress-strain curve, E.28.2.
-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
)
Figure 3.2.7. Stress-strain curve, E.26.1.
σ (MPa)
σ (MPa)
εp
ε
σ (MPa)
σ (MPa)
ε
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Fatigue Failure due to Cyclic Loading
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-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
)
Figure 3.2.8. Stress-strain curve, E.26.2.
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
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Fatigue Failure due to Cyclic Loading
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-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
)
Figure 3.2.10. Stress-plastic strain curve, E.30.2.
σ (MPa)
εp
σ (MPa)
εp
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Fatigue Failure due to Cyclic Loading
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-400
-300
-200
-100
0
100
200
300
400
-0.003 -0.002 -0.001 0 0.001 0.002 0.003
εp
(MPa
)
Figure 3.2.11. Stress-plastic strain curve, E.28.1.
-400
-300
-200
-100
0
100
200
300
400
-0.003 -0.002 -0.001 0 0.001 0.002 0.003
εp
(MPa
)
Figure 3.2.12. Stress-plastic strain curve, E.28.2.
σ (MPa)
εp
σ (MPa)
εp
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Fatigue Failure due to Cyclic Loading
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-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
)
Figure 3.2.13. Stress-plastic strain curve, E.26.1.
-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
)
Figure 3.2.14. Stress-plastic strain curve, E.26.2.
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
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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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.
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Fatigue Failure due to Cyclic Loading
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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
Figure 3.2.17. Temperature-time curves, T.26.1.
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Fatigue Failure due to Cyclic Loading
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
)
Exp eriment- Symmetry PlaneExp eriment- 5 mmExp eriment- 10 mmAnalys is - Symmetry PlaneAnalys is - 5 mmAnalys is - 10 mm
Figure 3.2.18. Temperature-time curves, T.26.2.
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
)
Exp eriment- Symmetry PlaneExp eriment- 5 mmExp eriment- 10 mmAnalys is - Symmetry PlaneAnalys is - 5 mmAnalys is - 10 mm
Figure 3.2.19. Temperature-time curves, T.26.3.
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Fatigue Failure due to Cyclic Loading
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).
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Fatigue Failure due to Cyclic Loading
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20
22
24
26
28
3032
34
36
38
40
42
44
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Exp eriment- Symmetry PlaneExp eriment- 5 mmExp eriment- 10 mmAnalys is - Symmetry PlaneAnalys is - 5 mmAnalys is - 10 mm
20222426283032343638404244
7000 8000 9000 10000 11000 12000 13000 14000 15000 16000
t (s)
T (o C
)
Exp eriment- Symmetry PlaneExp eriment- 5 mmExp eriment- 10 mmAnalys is - Symmetry PlaneAnalys is - 5 mmAnalys is - 10 mm
Figure 3.2.21. Temperature-time curves, T.24.2: (a) transient and steady state; (b) crack propagation
and failure.
(b)
(a)
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171921232527293133353739414345
0 1000 2000 3000 4000 5000 6000 7000 8000
t (s)
T (o C
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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
)
Exp eriment- Symmetry PlaneExp eriment- 5 mmExp eriment- 10 mmAnalys is - Symmetry PlaneAnalys is - 5 mmAnalys is - 10 mm
Figure 3.2.22. Temperature-time curves, T.24.3: (a) transient and steady state; (b) crack propagation
and failure.
(b)
(a)
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Fatigue Failure due to Cyclic Loading
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t (s)
T (o C
)
Exp eriment- Symmetry PlaneExp eriment- 5 mmExp eriment- 10 mmAnalys is - Symmetry PlaneAnalys is - 5 mmAnalys is - 10 mm
Figure 3.2.23. Temperature-time curves, T.24.4 (transient and steady state).
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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)
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Fatigue Failure due to Cyclic Loading
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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
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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.
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Fatigue Failure due to Cyclic Loading
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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.
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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.
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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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)
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Fatigue Failure due to Cyclic Loading
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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.
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Fatigue Failure due to Cyclic Loading
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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
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Fatigue Failure due to Cyclic Loading
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.
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Fatigue Failure due to Cyclic Loading
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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.
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4. SUMMARY OF CONCLUSIONS
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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
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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.
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215
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