MacDonald Fracture and fatigue of welded joints and structures

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Fracture and fatigue of welded joints and structures
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© Woodhead Publishing Limited, 2011
Fracture and 
fatigue of welded 
joints and 
structures
Edited by 
Kenneth A. Macdonald
Oxford Cambridge Philadelphia New Delhi
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© Woodhead Publishing Limited, 2011
Published by Woodhead Publishing Limited, 
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 Contributor contact details ix
 Preface xiii
 Introduction 1
 K. A. Macdonald, University of Stavanger, Norway 
Part I Analysing fracture of welded joints and structures 
1 Constraint-based fracture mechanics in predicting 
the failure of welded joints 17
 n. o’dowd, University of Limerick, Ireland 
1.1 Introduction to constraint-based elastic-plastic fracture 
mechanics 17
1.2 Constraint parameters 18
1.3 Tabulation of Q-solutions 22
1.4 Development of a failure assessment diagram (FAD) 
approach to incorporate constraint 25
1.5 Effect of weld mismatch on crack tip constraint 27
1.6 Full field (local approach) analysis for fracture assessment 28
1.7 Conclusion 28
1.8 References 28
2 Constraint fracture mechanics: test methods 31
 K. a. Macdonald, University of Stavanger, Norway, 
E. Østby and B. Nyhus, SINTEF Materials and Chemistry, 
Norway 
2.1 Introduction 31
2.2 High strains 32
2.3 Two-parameter fracture mechanics 35
2.4 Development of the single edge notch tension (SENT) test 36
Contents
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vi Contentsvi
© Woodhead Publishing Limited, 2011
2.5 Standardising the single edge notch tension (SENT) test 51
2.6 Conclusions 54
2.7 References 55
2.8 Appendix: Codes and standards 57
2.9 Nomenclature 58
3 Fracture assessment methods for welded structures 60 
I. hadlEy, TWI, UK 
3.1 Introduction 60
3.2 Development of engineering critical assessment (ECA) 
methods 63
3.3 The failure assessment diagram (FAD) concept 64
3.4 Specific engineering critical assessment (ECA) methods: R6 67
3.5 Specific engineering critical assessment (ECA) methods: 
BS 7910/PD6493 72
3.6 Specific engineering critical assessment (ECA) methods: 
Structural Integrity Procedures for European Industry 
(SINTAP)/European Fitness-for-Service Network (FITNET) 81
3.7 Specific engineering critical assessment (ECA) methods: 
American Petroleum Institute (API)/American Society of 
Mechanical Engineers (ASME) 85
3.8 Future trends 87
3.9 References 88
4 The use of fracture mechanics in the fatigue 
analysis of welded joints 91
 a. hobbachEr, University of Applied Sciences 
Wilhelmshaven, Germany 
4.1 Introduction to fracture mechanics 91
4.2 Technical applications of fracture mechanics 93
4.3 Fatigue assessment of welded joints using fracture 
mechanics 97
4.4 Examples of practical application 107
4.5 Conclusions110
4.6 References 111
Part II Analysing fatigue of welded joints and structures 
5 Fatigue strength assessment of local stresses in 
welded joints 115
 w. FrIcKE, Hamburg University of Technology, Germany 
5.1 Introduction 115
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viiContents vii
© Woodhead Publishing Limited, 2011
5.2 Types of stress 117
5.3 Factors affecting the fatigue strength 124
5.4 Fatigue strength assessment 129
5.5 Conclusions 137
5.6 References 137
6 Improving weld class systems in assessing the 
fatigue life of different welded joint designs 139
 b. Jonsson, Volvo Construction Equipment, Sweden 
6.1 Introduction 139
6.2 Historic view 140
6.3 Weld class system ISO 5817 142
6.4 Weld class systems at Volvo 143
6.5 A consistent and objective weld class system 144
6.6 Discussion 162
6.7 Conclusions 163
6.8 Future trends 164
6.9 Source of further information and advice 166
6.10 References 166
7 Fatigue design rules for welded structures 168
 s. J. Maddox, formerly at TWI, UK 
7.1 Introduction 168
7.2 Key features of welded joints influencing fatigue 170
7.3 Fatigue crack propagation 175
7.4 Design rules 177
7.5 Future developments in the application of fatigue rules 189
7.6 Conclusions 202
7.7 References 203
7.8 Appendix: fatigue design codes and standards 206
8 Fatigue assessment methods for variable amplitude 
loading of welded structures 208
 G. b. MarquIs, Aalto University, Finland 
8.1 Introduction 208
8.2 Fatigue damage and assessment for variable amplitude 
loading 214
8.3 Variable amplitude fatigue testing 226
8.4 Future trends 233
8.5 Sources of further information and advice 234
8.6 References and further reading 235
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viii Contentsviii
© Woodhead Publishing Limited, 2011
9 Reliability apects in fatigue design of welded 
structures using selected local approaches: the 
example of k-nodes for offshore constructions 239
 c. M. sonsIno, Fraunhofer Institute for Structural Durability 
and System Reliability LBF, Germany 
9.1 Introduction 239
9.2 Selected decisive design parameters 239
9.3 Selected design concepts by the example of K-nodes 261
9.4 Conclusions 273
9.5 References 274
10 Assessing residual stresses in predicting the service 
life of welded structures 276
 M. n. JaMEs, University of Plymouth, UK, d. G. hattInGh 
and w. h. rall, Nelson Mandela Metropolitan University, 
South Africa and a. stEuwEr, ESS Scandinavia, Sweden 
10.1 Introduction 276
10.2 Origins and types of stress 278
10.3 Modification of stresses after welding 283
10.4 Measurement 285
10.5 Conclusions 292
10.6 Acknowledgements 293
10.7 References 293
11 Fatigue strength improvement methods 297
 P. J. haaGEnsEn, Norwegian University of Science and 
Technology (NTNU), Norway 
11.1 Introduction 297
11.2 Fatigue strength of welded joints 298
11.3 Increasing the fatigue strength by improved design 301
11.4 Improvements obtained by special plate, filler materials 
and welding methods 305
11.5 Special welding methods 307
11.6 Post-weld improvement methods 307
11.7 Future trends 324
11.8 Conclusions 327
11.9 References and further reading 327
 Index 331
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© Woodhead Publishing Limited, 2011
Editor
K. A. Macdonald
University of Stavanger
Department of Mechanical and 
Structural Engineering and 
Materials Science
N-4036 Stavanger
Norway
E-mail: kenneth.macdonald@uis.no
Chapter 1
Professor Noel O’Dowd
Department of Mechanical and 
Aeronautical Engineering
Materials and Surface Science 
Institute
University of Limerick
Ireland
E-mail: noel.odowd@ul.ie 
Contributor contact details
Chapter 2
K. A. Macdonald*
University of Stavanger
Department of Mechanical and 
Structural Engineering and 
Materials Science
N-4036 Stavanger
Norway 
E-mail: kenneth.macdonald@uis.no 
E. Østby and B. Nyhus
SINTEF Materials and Chemistry
Department of Applied Mechanics 
and Corrosion
N-7465 Trondheim
Norway
Chapter 3
I. Hadley
TWI
Abington Hall
Granta Park
Great Abington
Cambridge CB21 6AL
UK
E-mail: isabel.hadley@twi.co.uk 
(* = main contact)
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x Contributor contact detailsx
© Woodhead Publishing Limited, 2011
Chapter 4
A. Hobbacher
University of Applied Sciences 
Wilhelmshaven
Germany
E-mail: hobbacher@t-online.de
Chapter 5
W. Fricke
Ship Structural Design and 
Analysis
Hamburg University of Technology 
(TUHH)
Schwarzenbergstr. 95c
21073 Hamburg
Germany
E-mail: w.fricke@tu-harburg.de
Chapter 6
B. Jonsson 
Volvo Construction Equipment
HL Division
360 42 Braås
Sweden
E-mail: bertil.bj.jonsson@volvo.com
Chapter 7
S. J. Maddox
TWI
Granta Park
Great Abington
Cambridge CB21 6AL
UK
E-mail: stephen.maddox@twi.co.uk
Chapter 8
Professor G. B. Marquis
Aalto University
Department of Applied Mechanics
P.O. Box 14300
FI-00076 Aalto
Finland
E-mail: gary.marquis@tkk.fi
Chapter 9
C. M. Sonsino
Fraunhofer Institute for Structural 
Durability and System 
Reliability LBF
Bartningstr. 47
D-64289 Darmstadt
Germany
E-mail: c.m.sonsino@lbf.fraunhofer.de
Chapter 10
M. N. James*
School of Engineering
University of Plymouth
Drake Circus
Plymouth PL4 8AA
UK
E-mail: m.james@plymouth.ac.uk
D. G. Hattingh and W. H. Rall
Mechanical Engineering
Nelson Mandela Metropolitan 
University
Gardham Avenue
Box 77000
Port Elizabeth 6031
South Africa
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xiContributor contact details xi
© Woodhead Publishing Limited, 2011
A. Steuwer
ESS Scandinavia
Stora Algatan 4
22350 Lund
Sweden
Chapter 11 
P. J. Haagensen
Norwegian University of Science 
and Technology (NTNU) 
7491 Trondheim
Norway
E-mail: per.haagensen@ntnu.no
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© Woodhead Publishing Limited, 2011
The motivation for writing this book is primarily to convey those aspects of 
current fracture and fatigue research that are important to general concepts of 
designing welded structures to avoid failure; and the ongoing assessment of 
the condition of structures and plant in service. Collectively termed structural 
integrity, these concepts often embrace the use of fracture mechanics – a 
branch of solid mechanics concerned with characterising the conditions 
surrounding stable or unstable growth of cracks.
 Although some academic circles are experiencing difficulty in attracting 
research interest and funding, especially from national sources who increasingly 
view fatigue and fracture as a mature subject area, societies around the world 
continue to experience failure of components and structures in this day and 
age. Quite an alarming state of affairs recalling that Wöhler’s experimental 
investigations of fatigue in train axles date from 1871 (in terms of eventual 
publication) and the birth of modern fracture mechanics can be traced back 
to the late 1940s following the Second World War’s Liberty ship failures 
that first arose in 1943. Our unfolding understanding of fracture mechanics 
and development of new characterising parameters to keep apace with greater 
levels of plastic strain capacity evident in modern steels continues to this 
day. The development of fatigue design guidance for welds was prompted by 
the rapid post-war adoption of welding as a dominant fabrication method for 
almost all types of metallic structure and process plant. The broader scope 
for encountering problems with fatigue and fracture problems in structures 
thus became truly immense. Countering this, design guidance has improved, 
becoming less uncertain, and fracture mechanics has blossomed to become a 
useful tool for examining influential factors – principally the deleterious effect 
of welding flaws related to both normal and poor quality fabrication. The 
net effect of all this is that there now appears to be evidence of a stabilised 
rate of in-service failures, at least in contrast to the galloping scale of the 
problemsexperienced in the second half of the 20th century.
 The content of this book naturally separates into the general subject areas 
of fracture and fatigue, natural, that is, in the context of welded joints. The 
Preface
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© Woodhead Publishing Limited, 2011
nature and depth of the subject matter ranges from rigorous treatment of 
fundamental fracture mechanics parameters, to descriptive chapters covering 
topics of more general interest. While the fracture segment of the book is 
comprised of contributions from key researchers working on important 
developments in modern applied fracture mechanics, the remaining section 
of the book concerned with fatigue is largely drawn from a cohesive group 
of researchers and investigators from industry and academia who are 
all active members of Commission XIII of the International Institute of 
Welding. It is anticipated that this book will have relevance for researchers 
and post-graduate students of fatigue and fracture, as well as designers and 
materials specialists in an industrial setting responsible for issues of design 
and structural integrity of weldments.
 The intent of this book is that, by providing a collection of the important 
advances in fatigue design and fracture mechanics, it may encourage more 
robust design of new structures and improve the standard of care for structures 
in operation; and that it also initiates interest and further work on integrity 
of welded joints.
 In the preparation of this book, I am indebted to the contributing authors 
for their detailed and comprehensive treatment of their individual specialist 
subject areas and the resulting breadth of coverage achieved in the book. 
Kenneth A. Macdonald
Hafrsfjord
xiv Prefacexiv
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© Woodhead Publishing Limited, 2011
17
1
Constraint-based fracture mechanics in 
predicting the failure of welded joints
N. O’DOwD, University of Limerick, Ireland
Abstract: This chapter discusses constraint-based approaches, which have 
been introduced to reduce the level of conservatism inherent in a single 
parameter K- or J-based approach to fracture. The constraint-based approach 
incorporates additional information about the crack tip deformation to 
quantify the deviation from a high constraint stress field with the amplitude 
given by J. The theory behind constraint-based fracture is discussed 
and the different parameters used in the theory are outlined briefly. The 
incorporation of the constraint-based approach within the commonly used 
failure assessment diagram (FAD) approach is also described.
Key words: fracture mechanics, constraint, failure assessment diagram, 
numerical models, finite element analysis, fracture parameters, non-linear 
fracture mechanics, Q-stress, T-stress.
1.1 Introduction to constraint-based elastic-plastic 
fracture mechanics
Non-linear fracture mechanics (NLFM) is applied to elastic-plastic materials 
when the extent of plastic deformation is such that the concept of small-
scale yielding no longer holds. NLFM, using the J-integral, is based on the 
concept of J dominance, whereby the near tip stress and strain states are 
characterised by the J-integral (Rice, 1968) and for power law materials an 
associated Hutchinson, Rice, Rosengren (HRR) field (Rice and Rosengren, 
1968; Hutchinson, 1968). The region where the crack tip fields are closely 
represented by the HRR field is known as the J dominance zone.
 For elastic-plastic materials, the applicability of the J approach is limited 
to so-called high constraint crack geometries. For example, when moderately 
sized laboratory crack geometries are loaded to general yield under tensile 
stress states, the J dominance zone is smaller than physically relevant length 
scales and the zone of finite strains (see e.g. Hancock et al., 1993; Shih et 
al.,1993). Under such conditions the near-tip stress distribution at physically 
significant distances from the crack tip can be significantly lower than the 
high constraint J dominant state.
 A typical result is shown in Fig. 1.1. Here the solid line shows the stress 
field ahead of a sharp crack (determined by finite-element (FE) analysis) and 
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18 Fracture and fatigue of welded joints and structures
© Woodhead Publishing Limited, 2011
the symbols are the values for the HRR field at the corresponding J value. 
Distances are normalised by J/sy, where sy is the material yield strength, so 
the x-axis represents about 10 crack tip openings (assuming that the crack tip 
opening displacement, dt µ 0.5J/sy). It is clear that if the FE stress field is 
considered to represent the ‘actual’ stress field ahead of a sharp crack, then 
the HRR field significantly overestimates the crack tip stress field. While 
it is thus conservative to use the high constraint HRR field to represent the 
stress field ahead of a crack, in many cases it will be overly conservative. 
Indeed, fracture toughness values well above the critical mode I fracture 
(JIC) toughness have been measured in centre cracked tension specimens 
(e.g. Sumpter and Forbes, 1992).
 The concept of crack tip ‘constraint’ was thus developed to quantify this 
deviation from the stress state predicted by the use of the J integral and the 
HRR field alone. Under conditions of high crack tip constraint, such as those 
experienced in deeply cracked specimens under bend loading, the stress field 
will be close to the HRR distribution (considered to be the upper bound stress 
field for a power law hardening material) and under conditions of low crack 
tip constraint, such as those experienced in specimens under tension loading 
conditions the stress field will be below the HRR distribution (Fig. 1.1).
1.2 Constraint parameters
Two parameter approaches have been developed to analyse situations 
where J dominance does not hold, (see e.g. McMeeking and Parks, 1979; 
J/asy = 0.004
FE stress field
HRR
0.0 1.0 2.0 3.0 4.0 5.0
r/(J/sy)
s y
y/
s y
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.1 Finite-element stress field for a sharp crack with an applied 
tensile stress field compared with the analytical HRR field. Here J is 
the J integral, r measures distance from the crack tip and sy is the 
material yield stress. 
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19Constraint-based fracture mechanics
© Woodhead Publishing Limited, 2011
Sumpter and Forbes, 1992; Hancock et al., 1993; Shih et al., 1993). The 
second parameter (the fi rst being J) quantifi es directly or indirectly the loss 
of constraint. This second parameter, will, in general, depend on geometry 
and material properties. The most common parameters used for this purpose 
are (i) the elastic T stress (williams, 1957), (ii) the Q stress (O’Dowd and 
Shih, 1991) and (iii) the A2 parameter (Yang et al., 1993a). The latter two 
parameters aim to quantify directly the stress fi eld in the elastic-plastic 
material ahead of the crack tip, while the former aims to rank different 
geometries by their T values. 
1.2.1 T stress
The T stress is the amplitude of the second term in the williams crack tip 
fi eld for a linear elastic material. Using the convention of Fig. 1.2, the stress 
fi eld for a linear elastic material may be represented as:
 
s s
s s p
11s s11s s12
12s s12s s 22
12
22
 =
2
 + 
Ê
Ë
Á
Ê
Á
Ê
Ë
Á
Ë
ˆ
¯
˜
ˆ
˜
ˆ
¯
˜
¯
Ê
Ë
Á Á 
Ê
Á
Ê
Ë
Á
Ë
ˆ
¯
˜ ˜ 
ˆ
˜
ˆ
¯
˜
¯
K
r
f f11f f11 12f f12
f f12f f12 22f f22
TTT 0
0 0
Ê
ËÁ
Ê
Á
Ê
ËÁË
ˆ
¯̃
ˆ
˜
ˆ
¯̄̃
 
1.1
In Equation 1.1 K is the linear elastic stress intensity factor and r measures 
distance from the crack tip (see Fig. 1.2). The stress s11 is the stress parallel 
to the crack face, s22 is normal to the crack faces and s12 is the shear stress. 
The quantities f11, f12, etc. are dimensionless functions which depend only 
on angle q and are tabulated in most textbooks on linear elasticfracture 
mechanics.
 The parameter T has dimensions of stress and as for K is obtained by 
consideration of the remote boundary conditions applied to a cracked 
specimen. It may be noted that T is a stress parallel to the crack faces and 
under linear elastic conditions is therefore not expected to have a strong 
effect on the driving force for crack growth. However, it has been shown 
that T can act as a characterising parameter for crack tip constraint. That 
is, for a given material, geometries which have the same or similar level of 
T stress have similar near-tip distributions when distance is normalised by 
J/s0. This approach may be considered to be an extension of the concept 
X2
X1
r
q s12
s11
s22
1.2 Convention for defi nition of crack tip fi elds.
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20 Fracture and fatigue of welded joints and structures
© Woodhead Publishing Limited, 2011
of small-scale yielding. Under small-scale yielding conditions K continues 
to describe the crack tip fi elds in an elastic-plastic material, although the 
deformation near the crack tip is represented by the HRR fi eld and J.
 The T stress may be obtained from FE solutions using a line or contour 
integral similar to that used for J (see e.g. Sham, 1991) and tables of T stress 
solutions are available for a number of cracked geometries (e.g. Sherry et 
al., 1995). 
1.2.2 A2 parameter
Using an asymptotic mathematical analysis, a three-term solution was 
developed to describe the stress fi elds for a crack in an elastic-plastic material. 
A Ramberg–Osgood (power law) description was used:
 
e
e
s
s
s
s0 0s0 0s 0
 = + ÊË
Ê
Ë
ÊÊ
Á
Ê
ËÁË
Ê
Ë
Ê
Á
Ê
Ë
Ê ˆ
¯
ˆ
¯
ˆ̂
˜
ˆ
¯̄̃
ˆ
¯
ˆ
˜
ˆ
¯
ˆ n
 
1.2
where n is the strain hardening exponent and s0, e0 are material parameters. 
(In Yang et al., 1993a, an additional parameter a was included, but it has 
been shown by Harkegard and Sorbo, 1998, and Kamel et al., 2009, that only 
three independent parameters, n, s0, e0 are required to represent uniquely 
a Ramberg–Osgood material response.) For such a material Chao and co-
workers (Chao and Zhang, 1997; Yang et al., 1993b) showed that the stress 
fi eld in the vicinity of the crack is given by
 
s
s e s s e s
ij
0 0s e0 0s e 0 n ij
HRR
2
0 0e s0 0e s n
 = s s ij ij + 
1
+1J
I r0 nI r0 n
 
I r
 A J
I LnI Ln
nÊ
Ës eËs e0 0Ë0 0s e0 0s eËs e0 0s e
 Ë 
Ê
Ë
ÊÊ
Á
Ê
ËÁËs eËs eÁs eËs e Ë Á Ë 
Ê
Ë
Ê
Á
Ê
Ë
Ê ˆ
¯ ¯ 
ˆ
¯
ˆ̂
˜
ˆ
¯̄̃ ¯ ˜ ¯ 
ˆ
¯
ˆ
˜
ˆ
¯
ˆ Ê

ËËË
Ê
Ë
Ê
Ë
Ê
Ë
ÊÊ
Á
Ê
ËËËÁËËË
Ê
Ë
Ê
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
Ë
Ê
Ë
Ê ˆ
¯
ˆ
¯
ˆ̂
˜
ˆ
¯̄̃
ˆ
¯
ˆ
˜
ˆ
¯
ˆ Ê
ËÁ
Ê
Á
Ê
ËÁË
ˆ
¯̃
ˆ
˜
ˆ
¯̄̃
1
+1
ij
(1)
2
0 0 n
 +
n r
L
A J
I
s
s
e s0 0e s0 0
2
L nLL nL
r
L
t
Ê
Ë
Ê
Ë
ÊÊ
Á
Ê
ËÁË
Ê
Ë
Ê
Á
Ê
Ë
Ê ˆ
¯L n¯L n
ˆ
¯
ˆ̂
˜
ˆ
¯̄̃L n¯L n˜L n¯L n
ˆ
¯
ˆ
˜
ˆ
¯
ˆ Ê
ËÁ
Ê
Á
Ê
ËÁË
ˆ
¯̃
ˆ
˜
ˆ
¯̄̃
1
 + 1
 
L
 
L
 
Ë
 
ËËÁË
 
ËÁË ¯
 
¯̄̃̄
 
¯̄̃ ij
(2)
s
 
1.3
 It may be noted that Equation 1.3 reduces to the HRR fi eld when A2 = 0. 
Thus, J describes the amplitude of the HRR fi eld and A2 characterises the 
‘loss of constraint’, which results in a reduction in stress magnitude relative 
to the HRR fi eld. The exponents s and t, the angular functions s̃ij and the 
constant In in Equation 1.3 depend only on strain hardening exponent n. 
The parameter L is a characteristic, normalising length parameter which has 
generally been chosen as the crack length. 
 For a given stress distribution, the value obtained for A2 will depend on 
the choice of characteristic length, L, but the overall amplitude of the stress 
fi eld is unaffected. The functions s̃ij and the exponents s and t have been 
tabulated in Chao and Zhang (1997) for a range of n values. The value of A2 
may be obtained from FE analysis and will depend on specimen geometry, 
material properties and, to a lesser extent, the load magnitude. 
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21Constraint-based fracture mechanics
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1.2.3 Q parameter
It was shown in O’Dowd and Shih (1991) that the near tip elastic-plastic 
stress fi elds may be represented in the following form which provides ‘an 
approximate, but robust, description of the near tip fi elds over physically 
signifi cant distances’ (O’Dowd, 1992):
 sij = (sij)ref + Qs0dij 1.4
The fi rst term in the above expression is a high constraint reference distribution 
and the second term is the difference fi eld which quantifi es the deviation 
from this high triaxiality fi eld. The stress s0 is a normalising stress, typically 
representative of the material yield stress. The Kronecker delta term dij in the 
difference fi eld indicates that the fi eld represents a uniform hydrostatic stress 
ahead of the crack tip and the (dimensionless) parameter Q is the parameter 
which quantifi es the magnitude of the difference term (typically Q < 0).
 The choice of reference fi eld, sref, in Equation 1.4 will depend on the 
material. For a power law material a possible choice is the HRR distribution 
as in the J–A2 representation of Equation 1.3. However, numerical studies 
have shown that the uniformity of the hydrostatic stress is better satisfi ed 
when the difference was taken with respect to the stress fi eld from an FE 
small-scale yielding solution with a remotely applied K-fi eld (see Fig. 1.3). 
This stress distribution will show some deviation from the HRR fi eld due 
to the contribution of the linear elastic deformation in the crack tip region. 
Note that, although not shown explicitly in Equation 1.4, the amplitude of 
the fi rst term in the above expression will depend on the magnitude of the 
applied load and thus will depend on J. 
 The second term in Equation 1.4 has no dependence on distance from 
s
p
=
2
K
r
r
q
Crack tip 
plastic zone
1.3 Determination of reference stress fi eld from a ‘small-scale 
yielding’ analysis.
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22 Fracture and fatigue of welded joints and structures
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the crack tip, r, but will depend on the magnitude of the applied load. In 
practice, the magnitude of Q is evaluated from FE solutions at a characteristic 
normalised distance ahead of the crack tip, typically at r/(J/s0) = 2 as proposed 
in O’Dowd and Shih (1991), which ensures that Q is evaluated at a physically 
signifi cant distance (a multiple of the crack tip opening displacement). More 
recently, it has been proposed by Kamel et al. (2009) that Q be evaluated 
at a characteristic distance of r/(J/e0s0) = 0.004. This provides a value for 
Q which is less sensitive to the material description, but the characteristic 
distance is no longer a fi xed multiple of crack tip opening displacement.
1.2.4 Modifi cations to the two parameter approach
Equations 1.3 and 1.4 have been found to provide a close representation of 
the stress fi eld for tension geometries, shallow crack bend geometries and 
deep cracked bend geometries under low deformation. For deep cracked 
bend geometries under high levels of deformation the agreement between 
FE solutions and Equations 1.3 and 1.4 is less good. Therefore a three 
parameter approach has been proposed by Chao et al. (2004) to extend the 
application of constraint-based approaches to bend dominated geometries 
under extensive yielding. An additional parameter Dsb was defi ned to extend 
the applicability of the A2 approach (Equation 1.3) to account for the infl uence 
of the bending fi eld:
 
Ds = 3C C 
M
b
r
 
1.5
where M is the global bending moment per unit length, b the ligament length, 
r distance from the crack tip and C a constant which may depend on applied 
load. A similar approach has also been suggested by Zhu and Leis (2006) 
to adjust Equation 1.4 to account for the bending term.
1.3 Tabulation of Q-solutions
As the J–Q description of the crack tip stress fi elds, described in Section 
1.2.3, provides a relatively simple descriptionof the crack tip fi elds, efforts 
have been made to tabulate Q solutions for a range of geometries from FE 
solutions. A typical result obtained from a 2D FE analysis is shown in Fig. 
1.4.
 It may be seen that at low levels of deformation the value of Q is close 
to zero, indicating that high constraint conditions prevail while at larger 
levels of deformation when plasticity has spread throughout the specimen 
(large-scale yielding) the value of Q is signifi cantly negative. A range of 
such solutions are provided in Sherry et al. (2005), for example. It may be 
noted that the Q value depends on material (in particular the value of the 
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23Constraint-based fracture mechanics
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hardening exponent n), specimen geometry and load level. Thus methods have 
been developed to simplify the representation of Q in cracked geometries 
and components as discussed in the next section.
1.3.1 Use of T stress to evaluate Q
It was shown by Betegón and Hancock (1991) that although the T stress 
does not provide a direct description of the crack tip stresses for an elastic-
plastic material, T can be used as a characterising parameter to rank levels of 
constraint or to match the constraint in two geometries. In O’Dowd and Shih 
(1991) it was shown, furthermore, that, provided deformations are sufficiently 
low (outside the small-scale yielding regime but before conditions of large 
scale plasticity when the plastic zone has spread to the specimen boundary), 
there is a one-to-one relationship between T and Q (for a given material). 
Such a relation is shown in Fig. 1.5. The advantage of such an approach is 
that the T stress may be obtained from a linear elastic FE analysis (or from 
handbook solutions), avoiding the necessity for a non-linear (elastic-plastic) 
analysis which can be expensive in terms of computing resources.
 As discussed in Section 1.1, when constraint is considered, the crack 
tip stress fields no longer depend on a single parameter, J, but on J and Q. 
Thus, the fracture toughness is no longer expressed as a single number but 
as a toughness curve, Jc(Q), with the high constraint toughness JIC being a 
single point on this curve. By carrying out a range of tests on geometries of 
Centre-cracked tension, a/W = 0.1
Power law material, n = 10
–2.0 –1.0 0.0 1.0 2.0
log [J/(ae0s0)]
Q
0.5
0.0
–0.5
–1.0
–1.5
1.4 Typical value of Q versus normalised J curve, for a shallow 
cracked centre cracked tension geometry and a power law material 
with n = 10. 
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24 Fracture and fatigue of welded joints and structures
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different constraint (ranging from deeply cracked bend geometries to shallow 
crack tension) the toughness curve may be constructed.
 A typical curve for a high strength, high toughness steel is illustrated in 
Fig. 1.6. The toughness curve is phrased here, with no loss of generality, in 
terms of KC rather than JC with KC obtained from the small-scale yielding 
relation, J = (1 – n2)K2/E.
Q
n = 10
n = 5
–1.0 –0.5 0.0 0.5 1.0
T/sy
0.25
0.00
–0.25
–0.50
–0.75
–1.00
–1.25
–1.50
1.5 Relationship between T stress and Q for a power law hardening 
material.
K
c(
M
P
a 
m
1/
2 )
800.0
700.0
600.0
500.0
400.0
300.0
200.0
–2.0 –1.8 –1.6 –1.4 –1.2 –1.0 –0.8 –0.6 –0.4 –0.2 0.0
Q
Da = 1 mm
Da = 0.5 mm
Loading path for low 
constraint structure
Loading path for high 
constraint structure
1.6 Schematic of a toughness–constraint relation for a high strength, 
high toughness steel (adapted from O’Dowd and MacGillivray, 2004).
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25Constraint-based fracture mechanics
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 In order to carry out a fracture assessment therefore, the toughness–
constraint curve must be known and in addition the crack driving force, in 
terms of J–Q (or K–Q) must be known. Two representative loading curves 
are illustrated in Fig. 1.6, representing a component under low and high 
constraint conditions, respectively. Fracture occurs when the loading path 
intersects the fracture curve, with the associated extent of crack growth Da 
determined by the relevant toughness curve.
 Examples of the application of constraint-based fracture mechanics to 
high pressure pipeline steels is provided in the studies of Ruggieri and co-
workers, e.g. Ruggieri et al. (2006).
1.4 Development of a failure assessment diagram 
(FAD) approach to incorporate constraint
Structural integrity assessments are generally based on the lower bound 
fracture toughness, determined from a high constraint fracture toughness 
specimen, using e.g. deeply cracked single edge notch bend, SEN(B), or 
compact tension, C(T), specimens. This is the approach recommended in 
the British Standard BS 7910 (BS 7910:99, 1999) the UK Nuclear R6 (R6 
Rev. 4, 2001) and the ASTM testing procedures (ASTM E 1820–01, 2001). 
In some cases, however, low constraint specimens, e.g. single edge notch 
tension SEN(T) can be used to obtain the fracture toughness value, provided 
it can be demonstrated that the constraint conditions of the component are 
matched by those of the test specimen. This approach is known as constraint 
matching and is adopted for example in Recommended Practice DNV 
RP–F108 (DNV-RP-F108, 2006) for the fracture assessment of offshore 
pipelines. A more general approach to incorporate the effect of constraint 
into structural integrity procedures is through modifi cation of the failure 
assessment diagram (FAD). 
 The discussion here is based on the British Standard, BS 7910, level 2B 
FAD, which relies on measured uniaxial material properties. The failure 
assessment curve (FAC) for a level 2B BS 7910 analysis is defi ned as 
follows:
 
K f L J
Jr r
K fr rK f Lr rL
e
–1/2
K f =K fK fr rK f =K fr rK f (K f (K f L (Lr r (r rK fr rK f (K fr rK f Lr rL (Lr rL ) = 
Ê
Ë
Ê
Ë
ÊÊ
Á
Ê
ËÁË
Ê
Ë
Ê
Á
Ê
Ë
Ê ˆ
¯
ˆ
¯
ˆ̂
˜
ˆ
¯̄̃
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
 
1.6
In Equation 1.6 Lr is the ratio between the applied load and the theoretical 
limit load and Kr is the ratio between the applied K and the fracture toughness 
Kmat. The ratio between the elastic-plastic J and the elastic J, J/Je in Equation 
1.6 is given by
 
J
J
E
L
L
Ee
ref
y rLy rL
r
3L3L y
ref
 = 
L
 
L
+ 1 L Lr r
eEeE
s
s
eEeE2
Ê
ËÁ
Ê
Á
Ê
ËÁË
ˆ
¯̃
ˆ
˜
ˆ
¯̄̃ 
1.7
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26 Fracture and fatigue of welded joints and structures
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In Equation 1.7, eref is the strain defi ned from the uniaxial stress–strain curve, 
using the corresponding reference stress, sref, obtained from Lr via,
 sref = syLr 1.8
where in Equations 1.7 and 1.8, sy is a measure of the material yield 
stress.
 A constraint-based FAD was developed in Ainsworth and O’Dowd (1995) 
and has been incorporated into the British Energy R6 failure assessment 
procedure. A constraint-dependent FAD is both geometry and material 
dependent. A number of simplifi cations are introduced in the R6 procedure 
to allow the constraint-dependent FAD to be constructed using only two 
additional parameters, a and b. A linear dependence of toughness, Kc, on 
constraint, Q, is assumed, such that
 Kc = Kc0 (1 – aQ) 1.9
where Kc0 is the toughness value corresponding to Q = 0. For the majority of 
materials 0 < a < 1. The dependence of the constraint parameter, Q, is also 
assumed to have a linear dependence on applied load, represented by:
 Q = bLr 1.10
The parameter b will depend on geometry and (more weakly) on the tensile 
response of the material. For most specimens, b < 0, so constraint decreases 
with increasing load. A modifi ed FAD may then be constructed, with
 
K J
J
Lr
e
–1/2
r = [ [
J [J
J
 [
J
1 – ]Ê [Ê [Ë [Ë [ [
Ê [Ë [
Ê [ÊÁ
Ê [Ê [Á [
Ê [ËÁË [Ë [Á [Ë [ [
Ê [Ë [
Ê [Á [
Ê [Ë [
Ê [ˆ [ˆ [¯ [¯ [ [ˆ [¯ [
ˆ [ˆ˜
ˆ [ˆ [˜ [
ˆ [¯̄̃ [¯ [˜ [¯ [ [
ˆ [¯ [
ˆ [˜ [
ˆ [¯ [
ˆ [ abLabL
 
1.11
where the term in square brackets accounts for the effect of constraint on 
the FAD. Note that the value of Kmat used in the defi nition of Kr is taken 
to be Kc0. The modifi ed FAD thus depends on material through a and on 
geometry through b. 
 A typical FAD for a high strength steel pipeline with a shallow crack 
loaded in tension is shown in Fig. 1.7. The value of a has been determined 
from fracture toughness testing and b from 3D FE analysis of a cracked 
pipe. It may be seen that if the effect of constraint is incorporated, the 
FAD in Figure 1.7 is expanded signifi cantly, providing an increased safety 
margin for a given applied loading. Note also that at high values of Lr (near 
to plastic collapse) the constraint modifi ed FAD and the original FAD 
are almost coincident. Thus there is little or no benefi t from constraint in 
this region. The largest effect of constraint is seen in the region 0.4 < Lr 
< 1.0.
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27Constraint-based fracture mechanics
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1.5 Effect of weld mismatch on crack tip 
constraint
If a crack lies within a weld or on the fusion line, the crack tip constraint 
will depend on the material mismatch. Typically, for a crack lying within 
an overmatched weld, the constraint will be reduced compared with that 
obtained for a homogeneous material at the same level of applied loading, as 
the plastic zone can easily extend into the parent material. For a crack lying 
within an under-matched weld, the reverse is the case – the deformation in 
the crack tip zone will be constrained by the higher strength weld material. 
Note that an increase or decrease in constraint due to over- or under-matching 
does not necessarily imply an increase or decrease in crack driving force 
as the crack driving force also depends on J, which is sensitive to weld 
over- or under-match.
 If a crack is located on the fusion line, the crack behaves as an interface 
crack and the HRR field distribution discussed in Section 1.1 no longer holds. 
This case was examined by Zhang et al. (1996) and the authors concluded 
that the effect of geometrical constraint was independent of mismatch and 
that the effect of mismatch could be incorporated through an additional 
parameter M, which depends on mismatch and material properties of the 
parent/weld material.
Constraint modified level 2B FAD 
a = 0.6, b = –0.8
Level 2B FAD
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Lr
K
r
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1.7 FAD for a high strength steel pipeline with a shallow crack loaded 
in tension (adapted from O’Dowd and MacGillivray, 2004). 
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28 Fracture and fatigue of welded joints and structures
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1.6 Full field (local approach) analysis for fracture 
assessment 
Constraint-based fracture mechanics approaches generally rely on numerical 
simulations to determine the crack tip constraint via the Q stress or T stress. 
An alternative approach to account for constraint effects in fracture mechanics 
is to undertake a full-field FE approach. This approach, often known as 
the local approach, requires a physical failure mechanism to be introduced 
within the FE analysis, e.g. the Gurson or Rouselier model (Gurson, 1977; 
Chaboche and Rousselier, 1983) to model ductile fracture via void growth. 
The crack tip damage mechanisms are then accounted for explicitly within 
the FE analysis and crack growth determined directly as an output from 
the analysis. However, among the drawbacks of these approaches are the 
complexity of the numerical modelling, the need for careful calibration and 
validation of the models and the fact that the models often provide results 
which are dependent on the FE mesh resolution. 
 The constraint-based approach using parameters such as T, Q or A2 may 
thus be considered as a compromise between a conservative approach, based 
on a single parameter high constraint fracture toughness, and a more accurate 
complex approach, based on a full field FE solution. The local-based approach 
to fracture and consideration of constraint effects is discussed in detail by 
Dolby et al. (2005).
1.7 Conclusion
The accuracy of one parameter fracture mechanics approaches can be improved 
through the introduction of a constraint parameter or parameters. However, 
the improvement in accuracy needs to be balanced with the additional effort 
required to obtain the relevant information to carry out a constraint-based 
assessment. In general, a finite-element analysis will be required to obtain 
the constraint parameter, e.g. Q or A2, while additional fracture testing will 
be needed to obtain the dependence of fracture toughness on constraint for 
the material of interest.
1.8 References
Ainsworth R A and O’Dowd N P (1995), ‘Constraint in the failure assessment diagram 
approach for fracture assessment’, J Pressure Vessel Tech, 117, 260–267.
ASTM E 1820–01 (2001), Standard test method for measurement of fracture toughness, 
ASTM E1820, Annual Book of ASTM Standards.
Betegón C and Hancock J w (1991), ‘Two-parameter characterization of elastic-plastic 
crack-tip fields’, J Appl Mech, 58, 104–110.
BS 7910:99 (1999), Guide on methods for assessing the acceptability of flaws in metallic 
structures, British Standards Institute, BS 7910:99.
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29Constraint-based fracture mechanics
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Chaboche J L and Rousselier G (1983), ‘On the plastic and viscoplastic constitutive 
equations. II. Application of internal variable concepts to the 316 stainless steel’, J 
Pressure Vessel Tech, 105, 159–164.
Chao Y J and Zhang L (1997), Tables of plane strain crack tip fields: HRR and higher 
order terms, ME-Report 97-1, Department of Mechanical Engineering, University 
of South Carolina.
Chao Y J, Zhu X K, Kim Y, Lar P S, Pechersky M J and Morgan MJ (2004), ‘Characterization 
of crack-tip field and constraint for bending specimens under large-scale yielding’, 
Int J Fracture, 127, 283–302.
DNV-RP-F108 (2006), Recommended Practice DNV-RP-F108: Fracture Control 
for Pipeline Installation Methods Introducing Cyclic Plastic Strain, Det Norske 
Veritas.
Dolby R E, Wiesner C S, Ainsworth R A, Burdekin F M, Hancock J, Milne I and O’Dowd 
N P (2005), ‘Review of a procedure for performing constraint and attenuation – corrected 
fracture mechanics safety case calculations for magnox reactor steel pressure vessels’, 
Int J Pressure Vessels Piping, 82, 496–508.
Gurson, A L (1977), ‘Continuum theory of ductile rupture by void nucleation and 
growth. I. Yield criteria and flow rules for porous ductile media’, J Eng Matls Tech, 
99, 2–15.
Hancock J W, Reuter G and Parks D M (1993), ‘Constraint and toughness parameterized 
by T’ in Constraint Effects in Fracture, ASTM STP 1171, Hackett E M, Schwalbe 
K-H, and Dodds R H, Eds., American Society for Testing and Materials: Philadelphia, 
21–40. 
Harkegard G and Sorbo S (1998), ‘Applicability of Neuber’s rule to the analysis of stress 
and strain concentration under creep conditions’, J Eng Mater Tech, 120, 224–229.
Hutchinson J w (1968), ‘Singular behaviour at the end of a tensile crack in a hardening 
material’, J Mech Phys Solids, 16, 13–31.
Kamel S, O’Dowd N P and Nikbin K M (2009), ‘Evaluation of two-parameter approaches 
to describe crack-tip fields in engineering structures’, J Press Vess Tech, 131, 031406 
(8 pages). 
McMeeking R M and Parks D M (1979), ‘On criteria for J-dominance of crack tip fields 
in large scale yielding’ in Elastic-Plastic Fracture, ASTM STP 668, Landes J D, 
Begley J A and Clark G A, Eds., American Society for Testing and Materials, West 
Conshohocken, PA, 175–194.
O’Dowd N P (1992), ‘Applications of two parameter approaches in elastic-plastic fracture 
mechanics’, Eng FractureMech, 52, 445–465.
O’Dowd N P and MacGillivray H J (2004), Study of Girth Welds at High Strains, Imperial 
College Consultants report, ME025/3. 
O’Dowd N P and Shih C F (1991), ‘Family of crack-tip fields characterized by a triaxiality 
parameter – 1: Structure of fields’, J Mech Phys Solids, 39, 989–1015.
R6, Rev. 4 (2001), Assessment of the Integrity of Structures Containing Defects, R6 Rev. 
4, British Energy Generation Ltd, UK.
Rice J R (1968), ‘A path independent integral and the approximate analysis of strain 
concentration by notches and cracks’, J Appl Mech, 35, 379–386.
Rice J R and Rosengren G F (1968), ‘Plane-strain deformation near a crack tip in a 
power-law hardening material’, J Mech Phys Solids, 16, 1–12. 
Ruggieri C, Silva L A L and Cravero S (2006), ‘Correlation of fracture behavior in high 
pressure pipelines with axial flaws using constraint designed test specimens. Part II: 
3-D effects on constraint’, Eng Fracture Mech, 73, 2123–2138.
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30 Fracture and fatigue of welded joints and structures
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Sham, T L (1991), ‘Determination of the elastic T-term using higher order weight 
functions’, Int J Fracture, 48, 81–102.
Sherry A H, France C C and Goldthorpe M R (1995), ‘Compendium of T-stress solutions 
for two and three dimensional cracked geometries’, Fatigue Fracture Eng Mats 
Struct, 18, 141–155.
Sherry A H, Wilkes M A, Beardsmore D W and Lidbury D P G (2005), ‘Material constraint 
parameters for the assessment of shallow defects in structural components – Part I: 
parameter solutions’ Eng Fracture Mech, 72, 2373–2395.
Shih C F, O’Dowd N P and Kirk M T (1993), ‘A framework for quantifying crack 
tip constraint’, in Constraint Effects in Fracture, ASTM STP 1171, Hackett E M, 
Schwalbe K-H and Dodds R H, Eds., American Society for Testing and Materials, 
Philadelphia, 134–159.
Sumpter J D G and Forbes A T (1992), ‘Constraint based analysis of shallow cracks 
in mild steel’, in Proceedings of TWI/EWI/IS Int. Conf. Shallow Crack Fracture 
Mechanics Test and Applications, Dawes M G, Ed., Cambridge, UK. 
williams M L (1957), ‘On the stress distribution at the base of a stationary crack’, J 
Appl Mech, 24, 109–114. 
Yang S, Chao Y J and Sutton M A (1993a), ‘Higher order asymptotic crack tip fields in 
a power-law hardening material’, Eng Fracture Mech, 45, 1–20.
Yang S Chao Y and Sutton M (1993b), ‘Complete theoretical analysis for higher order 
asymptotic terms and the HRR zone at a crack tip for mode I and mode II loading of 
a hardening material’, Acta Mechanica, 98, 79–98.
Zhang Z L, Hauge M and Thaulow C (1996), ‘Two-parameter characterization of the 
near tip stress fields for a bi-material elastic-plastic interface crack’, Int J Fracture, 
79, 65–83.
Zhu X-K and Leis B N (2006), ‘Bending modified J–Q theory and crack-tip constraint 
quantification’, Int J Fracture, 141, 115–134.
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31
2
Constraint fracture mechanics: test methods
K. A. MAcdonAld, University of Stavanger, norway, 
E. ØStby and b. nyhUS, SintEf Materials and 
chemistry, norway
Abstract: Approaches to fracture assessment taking account of geometry 
constraint were first developed for offshore pipelines where predictions 
had typically been very conservative if conventional, deeply notched bend 
specimens were used. the single edge notch tension (SEnt) specimen 
provides a lower level of crack-tip constraint that more closely matches 
that of the flaw in the pipe. This chapter outlines the basis of the current 
guidance for the use of SEnt testing and how it is applied in practice, 
including consideration of the development of the SEnt test for use in 
fracture control of pipelines. Areas requiring further research are highlighted, 
including limitations and aspects of specimen preparation, testing and 
analysis procedures that need to be addressed in order to standardise the test.
Key words: fracture, welds, steel, constraint, testing, pipeline, girthweld, 
SEnt.
2.1 Introduction
Engineering critical assessments (EcAs) are now commonly conducted during 
the design of structures to calculate tolerable sizes for flaws in welds. An 
ECA is a method for assessing the acceptability of a flaw in a structure, i.e. 
to demonstrate fitness-for-purpose.
 Pipeline welding codes and standards, e.g. bS 4515, APi 1104 and dnV-
oS-f101, specify workmanship acceptance levels for welding defects in 
pipeline girth welds. these acceptance levels represent what a ‘good’ welder 
should be able to achieve. They are not fitness-for-purpose defect limits, 
nor do they always apply to all welded structures, or even all pipelines. 
fortunately, fracture mechanics forms a rational basis for reaching informed 
decisions with regard to structural integrity. The benefits of ECA lie in 
avoiding unnecessary repairs and in determining if workmanship acceptance 
levels are themselves fit-for-purpose for the intended application. The latter 
point is particularly relevant to the design of pipelines subject to high static 
or cyclic strains because the partly historical safe limits for flaws promoted 
by the standards may have little bearing on the complex or severely loaded 
situations that are often prominent features of modern pipeline designs.
 An EcA is not required in all cases. the majority of existing onshore and 
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32 Fracture and fatigue of welded joints and structures
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offshore pipelines have been designed without EcAs, and have been installed 
and operated without incident. the reasons for the rise in popularity of the 
EcA can be partly attributed to the increased complexity of more recent 
pipeline designs, e.g. high temperatures and high pressures, deep water, 
aggressive internal conditions; installation methods involving plastic strain; 
and the use during construction of automatic ultrasonic testing (AUt) – rather 
than radiography – as the main inspection method.
 bS 7910 describes in detail how to conduct an engineering critical 
assessment. codes and standards such as APi 579 and R6 also give guidance, 
but are less commonly used in the pipeline industry. these methods are 
primarily stress-based and it is not straightforward to apply them to strain-
based situations. it is interesting to note that Pd 6493: 1980, the precursor 
to bS 7910, included the strain-based crack tip opening displacement 
(ctod) design curve (dawes, 1974). these generic codes and standards are 
supplemented by additional guidance in pipeline design codes and standards. 
dnV-RP-f108 was developed to provide additional guidance for EcAs of 
girth welds subject to cyclic plastic strains during installation. dnV-oS-f101 
has since extended this guidance to consider both installation and operation. 
both are intended to supplement the guidance given in bS 7910.
 in summary, EcAs often have a reputation of being over-conservative. 
Assessment of pipelines subject to high strains may indicate that only very 
small flaws would be acceptable, whereas practical experience has shown 
that the girth welds are highly tolerant of the presence of flaws. It was 
important to understand why EcA predictions could be overly conservative, 
or perhaps even non-conservative. Wide-ranging international research efforts 
examined a number of the issues surrounding pipeline EcAs including 
fracture toughness, tensile properties, misalignment (wall thickness tolerances 
and ovality); but it is the work on experimental measurement of fracture 
toughness and the importance of geometry constraint that is the focus of 
this chapter.
2.2 High strains
Axial plastic strain has an impact on girth weld flaw tolerance – particularly 
if cyclic in nature – which is in general lowered in comparison with elastic 
loading. the consequences of axial plasticstrain during operation may be 
more severe than during installation because the pipeline is pressurised, 
further reducing flaw tolerance. Current procedures for assessing these 
conditions are either inadequate or inadequately validated (cosham and 
Macdonald, 2008).
 A number of factors will affect flaw tolerance in addition to axial strain. 
for instance, the crack driving force is substantially greater when the pipeline 
is already at pressure when large axial strains are applied. on the other hand, 
Welded-Mcdonald-02.indd 32 3/23/11 1:43:01 PM
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33Constraint fracture mechanics: test methods
© Woodhead Publishing Limited, 2011
material resistance is thought to be unchanged by the presence of biaxial strains 
from internal pressure. Early experimental work on wide plates for the UK 
nuclear power industry first revealed the deleterious effects on CTOD-based 
assessments of, in that case, equi-biaxial conditions (Garwood et al., 1989; 
Phaal et al., 1995). Recently published test results in pipes show that axial 
straining capacity is significantly reduced under biaxial load (Minaar et al., 
2007; Østby and hellesvik 2007). current assessment procedures based on 
codified methods such BS 7910 experience difficulties in dealing with these 
conditions since they are essentially stress-based procedures and uncertainty 
surrounds their validity and safety. to illustrate this, comparison of the crack 
driving force (phrased in terms of J) for a simple surface cracked plate model 
computed directly from finite element analysis with that predicted using the 
reference stress formulation in bS 7910 (at level 2b) typically reveals a 
pronounced divergence at relatively low applied strains, in this case beyond 
approximately 1.7% (fig. 2.1). the material’s stress–strain behaviour and 
geometry both have a significant bearing on this type of assessment. Similar 
comparisons in pipe geometries, but using the Kastner plastic collapse solution 
(Kastner et al., 1981) to define the reference stress, show a dependency of 
crack driving force slope on defect geometry, with both conservative and 
non-conservative results in different areas (tkaczyk et al., 2007).
 Although embedded flaws are more likely to arise during fabrication than 
surface ones, easing the analysis by treating embedded flaws as surface flaws 
of equivalent dimensions (as in dnV-RP-f108) is simplistic, has not been 
fully validated and may not be conservative; especially if joint misalignment 
J 
(N
 m
m
–1
)
5000
4000
3000
2000
1000
0
0 2 4 6 8 10
Strain (%)
BS 7910
Finite elements
2.1 Crack driving force for a surface cracked plate.
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34 Fracture and fatigue of welded joints and structures
© Woodhead Publishing Limited, 2011
is present as this may cause increased plastic straining in the remaining 
ligament (Macdonald and cheaitani, 2010).
 in more accurately replicating the levels of crack tip constraint of pipeline 
girth weld flaws, the single edge notch tension (SENT) specimen (Fig. 2.2)
has still to be fully validated for fracture assessment of combined axial and 
pressure loading. however, there is growing experimental evidence that 
biaxial loading may not significantly influence ductile tearing resistance 
in plates (Garwood et al., 1989) and pipes (Minaar et al., 2007) loaded in 
tension; and that crack growth resistance measured in pipes under combined 
bending and internal pressure is similar to the R-curve obtained from SEnt 
testing (Phaal et al., 1995; Østby and hellesvik, 2007). the latter result is 
also supported by numerical simulation (tkaczyk et al., 2007; tyson et al., 
2007), giving some cause for optimism that the SEnt specimen geometry 
may also be applicable under such conditions.
 How residual stresses transverse to the girth weld relax with significant 
applied plastic axial strain is not well documented and the strain level at 
which they can safely be ignored remains to be defined.
 the failure assessment diagram (fAd) has proved to be a useful means 
B
H
a
P
P
W
Gripped area
Gripped area
2.2 SENT specimen geometry.
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35Constraint fracture mechanics: test methods
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of evaluating the significance of flaws (witness the widespread use of BS 
7910). A strain-based methodology would allow the assessment of elastic-
plastic fracture (by comparison of the crack driving force with fracture 
toughness) and failure by yielding or excessive straining (e.g. by comparison 
of a reference strain with a parameter based on material elongation). in a 
strain-based fAd the vertical axis would be based on J or ctod and the 
horizontal axis would be phrased in terms of a reference strain rather than a 
reference stress. A general framework for such a procedure has been proposed 
where the form of the fAd is somewhat different compared with those 
for existing stress-based treatments (budden, 2006) (fig. 2.3). Research is 
ongoing to address the limitations of the current assessment methods, both 
in-house, e.g. ExxonMobil; and in joint industry projects, e.g. tWi, PRci 
and SintEf (Garwood et al., 1989; Wang et al., 2006; Østby, 2007).
2.3 Two-parameter fracture mechanics
the experimental approach of matching the constraint levels of test specimens 
to those of actual flaws in structures was facilitated by theoretical progress 
in two-parameter descriptions of crack-tip stress fields. A number of two-
parameter approaches have been developed to analyse situations where the 
dominance of a single parameter breaks down, e.g. the J integral, and to 
quantify the deviation of the actual stress field (normally taken from numerical 
simulation) from the stress field predicted using J and the hutchinson, Rice, 
Rosengren (HRR) field alone. This loss of constraint is readily quantified 
either directly or indirectly by a second single parameter (the first being J) 
which in general is dependent upon both geometry and material properties. 
the parameters that have so far found the most widespread acceptance are: 
0 0.25 0.50 0.75 1.00 1.25 1.50
Lr = sref/sY
0 10 20 30 40
Lr = eref/eY
J
r0.
5
J
r0.
5
1.2
1.0
0.8
0.6
0.4
0.2
0
1.2
1.0
0.8
0.6
0.4
0.2
0
Stress-based FAD
2.3 Strain-based FAD (Budden, 2006).
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36 Fracture and fatigue of welded joints and structures
© Woodhead Publishing Limited, 2011
the elastic t stress (Williams, 1957; betegón and hancock, 1991); the elastic-
plastic Q stress (o’dowd and Shih, 1991, 1992); and the A2 parameter (yang 
et al., 1993). The first of these orders different geometries by ranking their 
T stress values, while the remaining two parameters aim to directly quantify 
the stress field in the elastic-plastic material ahead of the crack tip. The 
general problem of crack tip constraint and research aimed at understanding 
its effects are considered in more detail in chapter 1.
 dissatisfaction with the general state of over-conservatism in pipeline 
weld flaw assessment, and the growing awareness that geometry constraint 
was important (fig. 2.4) (Pisarski and Wignal, 2002), drove efforts leading 
to the development of a methodology for design against fracture and plastic 
collapse of offshore pipelines (bruschi et al., 2005; Østby, 2005; Sandvik 
et al., 2005; thaulow et al., 2005). the need for guidance was great and 
development of the EcA methodology and improvements in fracture testing 
were consequently quickly introduced in dnV RP-f108 (Wästberg et al., 
2004) for general use by industry.
2.4 Development of the single edge notch tension 
(SENT) test
2.4.1 Fracture control project
central to the development of the SEnt specimen as a constraint-matched 
fracture mechanics test for pipeline girth welds was work performed at 
SENB, a/W = 0.50
SENT, a/W = 0.50
Pipe 16 in OD, bending loading,a/t = 0.50
Pipe 16 in OD, tensile loading, a/t = 0.50
Increasing constraint
SENB SENT Pipe
–0.2 0.0 0.2 0.4 0.6 0.8 1.0
–Q
J/
b
s y
0.05
0.04
0.03
0.02
0.01
0
2.4 Effect of specimen geometry on crack tip constraint (Q) as a 
function of applied load expressed in terms of J (Pisarski and Wignal, 
2002).
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37Constraint fracture mechanics: test methods
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ntnU1 and SintEf2 in norway, with the backdrop of the challenges facing 
conventional EcA in situations where over-conservatism was already a 
recognised problem. the academic and theoretical springboard for the work 
was the interest that advances elsewhere in two-parameter fracture mechanics 
were attracting. The objectives of the first fracture control project for offshore 
pipelines were to develop a methodology for design against fracture and plastic 
collapse of offshore pipelines (bruschi et al., 2005; Østby, 2005; Sandvik 
et al., 2005; thaulow et al., 2005), fracture control in this context being 
the design of pipelines to address the implications of high static and cyclic 
strains during installation/construction and operation. the methodology had 
to be suitable for including calibrated partial safety factors; and compatible 
with current design standards and other failure modes. A second project is 
underway aiming to address many of the obstacles to wider acceptance.
 it is important to complement analytical equations established for crack 
driving force based on strain (strain-based design) (Østby, 2005) with accurate 
measures of fracture resistance in order to develop a design guideline with 
calibrated safety factors. the majority of fracture toughness data in existence 
are derived from standardised deeply notched single edge notch bend (SEnb) 
specimens (crack depth a/W = 0.5). Such specimens have a much higher 
geometry constraint ahead of the crack tip than circumferential cracks in tubes, 
inevitably leading to conservative results (nyhus et al., 2002, 2003). indeed, 
fracture toughness values well above the Jic toughness have been measured 
in low constraint centre cracked tension specimens (Sumpter and forbes, 
1992). these differences in crack tip constraint, combined with the differences 
in crack depth relative to the specimen width and in crack orientation with 
respect to the welding axis and pipe result in a significantly higher fracture 
toughness being obtained with SEnt compared with SEnb specimens. 
furthermore, establishing the true level of conservatism is complicated by 
the high material dependence of increases in fracture toughness estimated 
from specimens with lower geometry constraint.
 comparison of the fracture toughness and geometry constraint of equal 
crack depths in SEnt specimens and pipes (nyhus et al., 2003) concluded 
that shallow notched SEnt specimens possess a level of geometry constraint 
similar to that of circumferential cracks in pipe (fig. 2.5). this allows more 
accurate and relevant fracture toughness data to be established. this accounts 
in part for the rising level of interest shown in SEnt specimens as the design 
has increased in popularity for estimating fracture toughness in pipes. the 
majority of fracture toughness data estimated from SEnt specimens are 
used in accordance with guidelines established in the fracture control project 
1the norwegian University of Science and technology, trondheim
2An independent research and technology organisation operating in partnership with 
ntnU
Welded-Mcdonald-02.indd 37 3/23/11 1:43:02 PM
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38 Fracture and fatigue of welded joints and structures
© Woodhead Publishing Limited, 2011
(Wästberg et al., 2004). this guideline, dnV RP-f108, requires the crack 
depth in the SEnt specimen to be deeper than the crack in the pipe to ensure 
safe use of this design of specimen. crack depths in the SEnt specimens are 
commonly chosen somewhat deeper than the typical weld bead height. the 
size of the weld bead is often taken as a maximum limit on weld flaw height 
for embedded fabrication flaws (Macdonald and Hopkins, 1995a,b) – such as 
porosity, slag and lack of fusion – as they will be contained entirely within 
the weld run in which they were formed. the argument does not hold for 
cracks which may form quite independently of the weld beads. flaws larger 
than the notch depth used in the SEnt specimens would then be repaired 
independently of the EcA.
 numerical simulations using fE show that SEnt specimens have higher 
levels of geometry constraint than pipes containing equal crack depths. these 
findings are independent of: crack length; location on the internal or external 
pipe surface; and tension or bending loading (nyhus et al., 2002; Wästberg et 
al., 2004). Assuming that geometry constraint increases with crack depth, it 
is therefore a requirement to introduce a crack depth in the SEnt specimen 
that is deeper than the assessed crack in the pipe. the dnV guideline 
(Wästberg et al., 2004) is intended for pipeline installation and therefore 
the later interest in the effects of internal pressure and biaxial stress were 
not addressed in the original qualification program. Subsequent introduction 
(a/W = 0.2)
SENB (a/W = 0.2)
SENB (a/W = 0.5)
SENT
Constraint (Q, T, M)
R
es
is
ta
n
ce
 (
J,
 C
T
O
D
)
2.5 Geometry constraint in SENB, SENT and circumferential flaws in 
pipes.
Welded-Mcdonald-02.indd 38 3/23/11 1:43:02 PM
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39Constraint fracture mechanics: test methods
© Woodhead Publishing Limited, 2011
of the SEnt specimen design into the main pipeline design standard, dnV 
oS-f101, had therefore to retain high constraint SEnb specimens in its 
requirements for EcA for operating pipelines (at pressure).
 Results from other projects indicate that the fracture toughness estimated 
from SEnt specimens is almost independent of the crack depth. in order to 
verify the wider dataset, and to confi rm that fracture toughness estimated 
from SEnt specimens is also valid beyond the original limits of applicability 
(Wästberg et al., 2004), a programme of testing and fi nite element (FE) 
simulation examined the infl uence on fracture toughness of a wide range of 
crack sizes and internal pressure. the effects of misalignment and dissimilar 
thickness (on either side of the crack) were also assessed in order to determine 
if any correction to SEnt-based fracture toughness is needed for girth welds 
with these quite normal (intrinsic) geometric discontinuities.
2.4.2 J-integral in SENT specimens
the guidance recommends that fracture toughness (crack growth resistance) 
be characterised by J–R curves. the J-integral values from SEnt specimens 
are calculated for clamped conditions from equations 2.1–2.3 in Si units:
 J = Jel + Jpl 2.1
 
J
K
Eel
2 2
 =
(2 2(2 21 – 2 21 – 2 2)u2 2u2 2
 
2.2
 
J
A
B W apl
plAplA
0
 =
(B W(B W – )
hAhA
 
2.3
where:
a total crack length after test (mm)
a0 original crack length (mm)
Apl plastic area under the load – crack mouth opening displacement (cMod) 
curve at z = 0 (n mm2)
B specimen width (mm)
E elastic modulus (n/mm2)
F load (n)
J J integral (n/mm)
Jel elastic J (n/mm)
Jpl plastic J (n/mm)
K stress intensity factor (n/mm1.5)
W specimen height (mm)
h eta factor
v Poisson’s ratio
and
Welded-Mcdonald-02.indd 39 3/23/11 1:43:03 PM
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40 Fracture and fatigue of welded joints and structures
© Woodhead Publishing Limited, 2011
 
K F
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2.4.3 Fracture mechanics testing
fracture mechanics testing was performed for SEnt specimens and SEnb 
specimens with a range of crack depths (fig. 2.6). the SEnt programme 
2.3 2.3
4.0 4.0
5.7 5.7
11.5
(a)
(c)
(e)
(b)
(d)
(f)
11.5
11.5 11.5
11.5 11.5
2.6 Geometry of the standard (equal thickness and aligned) SENT 
and SENB specimens.
Welded-Mcdonald-02.indd 40 3/23/11 1:43:04 PM
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41Constraint fracture mechanics: test methods
© Woodhead Publishing Limited, 2011
also included misalignment and dissimilar wall thickness (fig. 2.7), the aim 
being to study the effects of geometry constraint and asymmetric pipes/
specimens. the unstrained test material was 304 mm (12 inch) diameter 
grade X-65 linepipe with 14.9 mm wall thickness. the cross-section of the 
specimens was 2B ¥ B (23 ¥ 11.5 mm) (fig. 2.8). Pipe curvature restricts the 
specimen thickness to some degree. Specimens were notched from the internal 
pipe surface to a range of depths. the clamping separation distance for the 
SEnt specimens was 115 mm (based on 10W). All testing was performed 
at room temperature using a double clip gauge arrangement to calculate the 
ctod values. ctod–R curves were constructed using a multiple specimen 
technique whereby specimens were unloaded at different displacements 
(fig. 2.9–2.11). the target crack depth was 2.3 mm. After testing, the actual 
initial crack depths were measured, which varied from 2.36 to 2.64 mm. it 
is clear from the driving force curves, figs 2.12 and 2.13, that the strain 
capacity in the test was strongly dependent on this difference in the initial 
crack depth. figures 2.12 and 2.13 show the crack driving force (ctod) 
for SEnt specimens under test. ctod is plotted as a function of the strain 
in the specimen measured between the clamped region and the crack. the 
strain is in this area unaffected by both the crack and the clamps. the loading 
curve has several important features (fig. 2.12, curve for a0 = 2.36 mm):
(a)
(c)
(b)
(d)
11.5
14.0
14.0
14.011.5
11.5
11.5
11.5
2.3
2.3
2.3
4.0
2.5 2.5
2.52.5
2.7 Geometry of the SENT specimens with dissimilar wall thickness 
and misalignment. 
a
B
2B
2.8 Cross-section of SENT specimen and placement in the pipe wall.
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42 Fracture and fatigue of welded joints and structures
© Woodhead Publishing Limited, 2011
∑ initially the specimen and the ligament remain in the linear elastic 
region.
∑ the remaining ligament then begins to yield, and ctod increases with 
almost no corresponding increase of global strain in the specimen.
∑ once ctod reaches approximately 0.5 mm, the ligament has passed the 
lüders plateau and the material recommences strain hardening. yield is 
eventually reached in the bulk of the specimen away from the crack.
SENT (a/W = 0.2)
SENT (a/W = 0.35)
SENT (a/W = 0.5)
SENB (a/W = 0.5)
0.0 0.5 1.0 1.5 2.0 2.5
Da (mm)
d 
(m
m
)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
2.9. CTOD–R curves from SENT specimens with different crack 
depths compared with a conventional SENB specimen. 
SENT (a/W = 0.2)
SENT (a/W = 0.35)
SENT (a/W = 0.2) fusion line
SENT (a/W = 0.35) fusion line
0.0 0.5 1.0 1.5 2.0 2.5
Da (mm)
d 
(m
m
)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
2.10 Comparison of the CTOD–R curves for parent material and 
fusion line for SENT specimens with different crack depths. 
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43Constraint fracture mechanics: test methods
© Woodhead Publishing Limited, 2011
∑ the specimen continues deforming plastically where strain increases 
with increasing ctod.
∑ finally, the specimen reaches a maximum strain capacity dictated by 
necking and crack extension.
Other fracture mechanics test results
fracture toughness data from other sources were used to widen the basis 
of qualification. CTOD–R curves for SEnt specimens with crack depth 
SENT (a/W = 0.35), Fig. 2.6(c)
SENT (dif. wall thickness), Fig. 2.7(c)
SENT (misal.), Fig. 2.7(a)
SENT (dif. wall thickness), Fig. 2.7(b)
SENT (dif. wall thickness), Fig. 2.7(d)
0.0 0.5 1.0 1.5 2.0 2.5
Da (mm)
d 
(m
m
)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
2.11 CTOD–R curves for SENT specimens with different wall 
thickness and misalignment (refer to Figs 2.6 and 2.7).
0.00 0.01 0.02 0.03 0.04
Strain (mm/mm)
C
T
O
D
 (
m
m
)
3.0
2.5
2.0
1.5
1.0
0.5
0.0
a0 = 2.64 mm
a0 = 2.36 mm
Necking
Yielding of the specimen
Yielding in the ligament
Linear elastic loading
2.12 CTOD as a function of strain for SENT specimens with crack 
depth a/W = 0.2 (refer to Fig. 2.6).
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44 Fracture and fatigue of welded joints and structures
© Woodhead Publishing Limited, 2011
a/W equal 0.35 and 0.55 and SEnb specimens with a/W equal to 0.55 are 
compared in fig. 2.14 (nyhus et al., 2003). the specimens were taken 
from a pipe with outer diameter of 325 mm, and wall thickness of 12 mm. 
base material was a high grade supermartensitic stainless steel, S13% cr 
steel (2.5 Mo). the girth welds were made with a gas tungsten arc welding 
(GTAW) process and the filler material used was Thermanit 13/06 Mo. All 
specimens were notched in weld metal and testing was performed at room 
temperature. the cross-section of the SEnt and SEnb specimens was 20 
¥ 10 mm2 (fig. 2.8).
a0 = 2.63 mm
0.00 0.01 0.02 0.03 0.04
Strain (mm/mm)
C
T
O
D
 (
m
m
)
3.0
2.5
2.0
1.5
1.0
0.5
0.0
2.13 CTOD as a function of strain for SENT specimens with dissimilar 
wall thickness (refer to Fig. 2.7). 
SENB (a/W = 0.35)
SENT (a/W = 0.55)
SENT (a/W = 0.35)
0.0 0.5 1.0 1.5 2.0 2.5
Da (mm)
d 
(m
m
)
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
2.14 CTOD–R curves for 13% Cr filler material (Nyhus et al., 2003).
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45Constraint fracture mechanics: test methods
© Woodhead Publishing Limited, 2011
 ctod–R curves for SEnt specimens with crack depth a/W equal to 
0.20, 0.35 and 0.50 and SEnb specimens with a/W equal to 0.50 are also 
compared in fig. 2.15. the specimens were taken from a 13% cr pipeline 
with outer diameter of 340 mm, and wall thickness equal to 15.3 mm. All 
specimens were notched in parent material and testing was performed at 
room temperature. the cross-section of the SEnt and SEnb specimens 
was 25.4 ¥ 12.7 mm2 (fig. 2.8).
2.4.4 Numerical simulation
ductile tearing analyses formed the basis