AEAT/R/NT/0381
Issue 2
The Shape Of The
Ductile-To-Brittle
A report produced for HSE
S.R.Ortner
May 2001
AEAT/R/NT/0381
Issue 2
The Shape Of The
Ductile-To-Brittle
A report produced for HSE
S.R.Ortner
May 2001
AEAT/R/NT/0381
Title
The Shape Of The Ductile-To-Brittle Transition
Customer
HSE
Customer
reference
NUC 56/68/4/1
Confidentiality,
copyright and
reproduction
Issue 2
This document has been prepared by AEA Technology plc in
connection with a contract to supply goods and/or services
and is submitted only on the basis of strict confidentiality.
The contents must not be disclosed to third parties other than
in accordance with the terms of the contract.
File reference
VD 60121
Report number
AEAT/R/NT/0381
Report status
Issue 2
AEA Technology Nuclear Science,
220 Harwell,
Didcot,
Oxon. OX11 0RA
Telephone 01235 434180
Facsimile 01235 435947
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AEA Technology is certificated to BS EN ISO9001:(1994)
Name
Author
S.R.Ortner
Reviewed by
C.A.English
Approved by
S.G.Druce
Signature
Date
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Executive Summary
Ferritic steels fail by cleavage at low temperatures. The fracture toughness of the
steels is low, and does not change much either with temperature, or from specimen
to specimen. At high temperatures, the steels are ductile, failing at much higher
toughness levels which, again, do not vary significantly from specimen to specimen.
Between these two extremes lies the transition region. In the transition region, the
average fracture toughness increases rapidly with temperature, and there is marked
specimen-to-specimen variation. The wide range of possible toughnesses which
may be measured in the transition region makes it experimentally difficult to
determine the mean level of toughness as a function of temperature - the “shape” of
the ductile-to-brittle transition (the DBT).
Clearly, if either the shape or mean position of the transition could be fixed by
theoretical considerations, then all the available data could be used to determine just
the remaining parameter. The Mastercurve hypothesis of Wallin and co-workers
holds out the promise of just this simplification of data analysis. The Mastercurve
hypothesis suggests that the mean fracture toughness versus temperature curve will
have the same shape for all ferritic steels – the only difference between steels being
the position of the curve on the temperature axis (the transition temperature). If true,
this would render the analysis of scattered data much simpler, reduce the number of
tests required to define the transition temperature, and make the assessment of
relationships between transition temperature shifts and e.g. irradiation or mechanical
processing, far more reliable.
This report examines suggestions that the constant Mastercurve shape is due to a
unique material property common to all ferritic steels. It then uses a statistical model
of cleavage fracture (previously published by Ortner and Hippsley, OH) to examine
the effect of various material parameters on the mean toughnesses in the transition
region. The OH model is assessed and compared with other statistical models of
fracture, and found to be sufficiently plausible for this sensitivity study to be reliable.
The results of this sensitivity study are summarised below:
Parameter increased
Stress at fracture nucleation site, sc
(experimentally found to be a material
constant, invariant with temperature)
Effective surface energy, g+wp (also
known as work to fracture)
Yield stress, sy
Work-hardening exponent, n
Mean size of particles causing fracture,
r
Number of particles per unit area, Na
Effect on temperature
at which cleavage
becomes impossible
Decrease
Effect on slope of
transition curve
-
Increase
Increase
Increase
-
Decrease
Decrease
Decrease
-
Slight increase
Increase
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a-parameter describing shape of
distribution (when a increases, the
fraction of particles in the high tail of the
distribution decreases)
-
Issue 2
Increase
Various metallurgical processes are then considered, and comparisons made
between OH predictions and data. On the basis of these comparisons, the report
finds that metallurgical processes generally change more than one of the material
parameters affecting the transition shape. Thus some processes, by changing a
suitable combination of parameters, can leave the transition curve unchanged when
the transition temperature changes. If the Mastercurve describes the material
behaviour before these processes, then it will still do so after. This is not true for all
metallurgical processes, however, as summarised below:
Metallurgical Process
Irradiation hardening
Plastic strain
Tempering
Segregation and intergranular
failure
Warm prestressing
Mastercurve
Applicable ?
Possible
Possible
Possible
No
No
Explanation
sy increases while n decreases
sy increases while n decreases
r increases, Na decreases, sc
decreases, n increases, while sy
decreases
scrit and g+wp decrease
a and sc increase, r and Na
decrease slightly
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Contents
1
2
3
Introduction
1
1.1 MEASURING THE SHAPE OF THE DUCTILE-TO-BRITTLE
TRANSITION
1
The Mastercurve Hypothesis
3
2.1 MODELLING BACKGROUND
2.1.1 Crack Extension From A Flaw
2.1.2 The RKR Model Of Cleavage In The Transition
2.1.3 The WST Models
2.1.4 The Beremin Approach
2.1.5 Summary
2.2 DEVELOPMENT OF THE MASTERCURVE
2.2.1 Scatter In K1C Measurements
2.2.2 Temperature-Dependence Of K1C
2.2.3 Interpretation Of Mastercurve Behaviour
2.2.4 Overview
3
3
4
5
9
11
12
12
13
18
19
The Two-Component Model
20
3.1 DETAILS OF THE MODEL
20
3.2 CALCULATED EFFECTS OF CHANGING PARAMETERS IN OH
22
3.2.1 The Critical Stress
23
3.2.2 Effective Surface Energy
24
3.2.3 Flow Properties: Yield Stress And Work Hardening Exponent
24
3.2.4 The Particle Size Distribution
30
3.2.5 Summary
32
3.3 COMPARISON WITH DATA AND PARAMETER COMBINATIONS
32
3.3.1 Changing The Crack Path
32
3.3.2 The Effect Of Heat Treatment
36
3.3.2.1
Summary Of Heat Treatment Effects
3.3.3 Effects Of Radiation Damage
44
3.3.4 Comparisons Between Plates, Forgings And Welds
46
3.3.5 Warm Prestressing
49
3.4 OH ASSESSMENT OF MASTERCURVE APPLICABILITY
49
3.5 ASSESSMENT OF THE OH MODEL
51
4
Suggestions For Further Work
55
5
Summary And Conclusions
56
Appendix
Details Of Calculations For JSW And JRQ
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List of Figures
Figure 1. Impact energies measured in the non-recrystallised regions of C-Mn multipass submerged-arc welds [5].
2
Figure 2. Fracture toughness measurements made on 22NiMoCr37 forging steel,
from the Euro Fracture Toughness Data Base. Open symbols refer to samples
which cleaved; closed symbols to samples which showed ductility.
3
Figure 3. WST predictions of the DBT in a ferritic tool steel, and the effect allowing
the work to fracture to be temperature-dependent [18].
6
Figure 4. WST predictions of the DBT in a bainitic Cr-Mo-V steel, using a constant
work to fracture, or a work to fracture with a different exponential dependence to
that required for the ferritic tool steel [18 above].
7
Figure 5. Comparison between calculated DBT and fracture toughness data from the
HSST program [24].
11
Figure 6. Fit of exponential trend line through fracture toughness data from
unirradiated rpv weld 72W [1].
14
Figure 7. Fit of exponential trendline through fracture toughness data from
unirradiated rpv weld 73W [1].
15
Figure 8. Fit of exponential trend line through fracture toughness data from rpv weld
72W after irradiation [1].
16
Figure 9. Fit of exponential trend line through fracture toughness data from rpv weld
73W after irradiation [1].
16
Figure 10. Fit of exponential trend line through fracture toughness data from A533B
Class 1 steel HSST02 before and after irradiation [1].
16
Figure 11. Median K values for a range of rpv steels and welds plotted in terms of
adjusted temperature (T-T0). The T0 values are given in the key. The trend line
through the median values is the Mastercurve [1].
17
Figure 12. Crack opening stress and plastic strain distribution ahead of a sharp
crack. The hatched area is the area within which the probability of failure is
calculated in OH.
21
Figure 13. Effect on the DBT of changing the critical stress, Sc.
23
Figure 14. Effect on the DBT of changing the work to fracture, E
24
Figure 15. Effect of different yield stress (S0):Young’s modulus ratios (1:300 and
1:100) and work hardening exponents (0, 0.1, 0.2) on the distribution of the
principal tensile stress (S11) in the plastic zone ahead of a sharp crack in Mode I
loading and small scale yielding (according to McMeeking [34]).
25
Figure 16. Yield stresses used for calculations.
26
Figure 17. The effect on changing the yield stress on the DBT.
27
Figure 18. Effect of increasing the yield stress by 50MPa at all temperatures when
the effective surface energy is exponentially temperature-dependent ( using the
WST expression E=2.15+1.77exp[0.0104T]).
28
Figure 19. Effect of changing the work hardening rate, n, on the DBT.
28
Figure 20. Effect on the DBT of increasing the yield stress, S, while decreasing the
work-hardening exponent, n.
29
Figure 21. A selection of particle size distributions.
30
Figure 22. Effect on the calculated DBT of changing the particle size distribution.
31
Figure 23. Relation between critical stress measured at crack initiation sites in
A533B after different ageing times at 450°C, and values of g +wp derived from
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shape of calculated transition curves [4]. The straight lines have a slope 0.5 (cf.
Equation 2).
33
Figure 24. Effect on the DBT of changing the crack path (E and scrit).
34
Figure 25. Comparison between fracture toughness data from unaged A533B and
the Mastercurve
34
Figure 26. Comparison between fracture toughness data from aged (segregated)
A533B, OH trendlines (20, 50 and 80% probability) and Mastercurve (50%).
35
Figure 27. Effect of heat treatments on JSW particle sizes, and comparison with OH
distribution.
37
Figure 28. Effect of heat treatment on the JSW 0.2% offset yield stress.
39
Figure 29. Effect of heat treatment on the work hardening exponent in JSW material. 39
Figure 30. Effect of heat treatment on the fracture toughness of JSW materials [35] 40
Figure 31. Mastercurve (MC) fits to JSW fracture toughness data from [35].
40
Figure 32. Comparisons between individual JSW datasets and best fits from OH
calculations (see also over page).
41
Figure 33. Relation between the fitted critical stress, and the geometric mean of the
particle size distribution, or the 99th percentile carbide size in JSW material.
43
Figure 34. Fracture Toughness data for unirradiated JRQ plate, and trend line
calculated to fit to data.
45
Figure 35. Fracture toughness data for irradiated JRQ plate, and trend line
calculated using parameters derived from unirradiated material.
46
Figure 36. Effect of changing particle distribution parameters on the shift in the
transition temperature caused by a yield stress increment of 50MPa at each
temperature.
48
Figure 37. Effect on the DBT of increasing the yield stress by 50MPa at each
temperature in materials with different critical stresses.
48
List of Tables
Table 1. Summary of parameters investigated with OH model, and their effects
on the fracture toughness transition.
Table 2. Range of tempering treatments used in [35].
Table 3. Comparison of curve positions for JSW material according to
Mastercurve and trend line used in [35].
Table 4. Applicability of the Mastercurve description of the fracture toughness
transition.
Table 5. Comparison between different RKR-type models of cleavage fracture.
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38
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1
Issue 2
Introduction
This report aims to investigate factors affecting the shape of the ductile-to-brittle
transition (DBT) in reactor pressure vessel (rpv) steels, i.e. whether the curvature of
the line that best describes the temperature-dependence of the fracture toughness in
the transition region is variable. First, the report will show why it can be difficult to
determine the shape of the fracture toughness transition experimentally, even though
it is an mechanically and economically important attribute of any steel (Section 1.1).
The report will describe the Mastercurve hypothesis [1, 2, 3], which suggests that the
shape of the transition in fracture toughness is constant for all ferritic steels (Section
2). It will then use the description of the DBT derived by Ortner and Hippsley [4],
which does not assume shape-invariance, to assess explicitly how factors such as
yield stress, work-hardening rate, precipitate distribution etc. can affect the shape of
the fracture toughness transition (Section 3 to part 3.3). By comparing these
qualitative assessments with the predictions of the Mastercurve, the report aims to
assess the limits within which it is appropriate to use the Mastercurve for data
analysis (Section 3.4). Finally, the reliability of the Two-Component Model will be
assessed in the light of other approaches to modelling the DBT (Section 3.5).
1.1
MEASURING THE SHAPE OF THE DUCTILE-TO-BRITTLE
TRANSITION
Ferritic steels fail by cleavage at low temperatures, without widespread ductility. At
very low temperatures, both the fracture toughness and the Charpy impact energy
are low, and do not change much with temperature. These properties vary a little
from specimen to specimen of a given geometry. At high temperatures, the steels
are ductile, failing at much higher toughnesses or impact energies, which do not vary
significantly from specimen to specimen. Between these two extremes (the lower
shelf and upper shelf respectively) lies the transition region.
Within the transition region, the average toughness and impact energy increase
rapidly with temperature, accompanied by marked specimen-to-specimen variations.
Figure 1 shows the results of a very large number of Charpy tests on specimens of
one geometry and material [5], while Figure 2 shows fracture toughness data
measured on compact tension specimens of several sizes [6]. The Figures illustrate
the variability of measurements in the transition region.
The wide range of possible toughnesses which may be measured in the transition
region makes it difficult to determine the mean level of toughness or impact energy
as a function of temperature. The best fit line through a limited dataset will not
reliably represent the true mean trend. This makes it difficult to assess the
significance of alterations in best-fit lines required to describe other, limited sets of
data taken from steels of (say) different compositions, thermomechanical histories or
environmental exposures and, thence, to determine accurate relations between shifts
in transition temperature and changes in composition, exposure etc.
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In some situations, it is possible to offset the variability of measurements by carrying
out large numbers of tests. In many cases, however, the amount of material
available is small, and this is not feasible. This is a particular problem for the nuclear
industry, in which surveillance material is limited, and mechanical testing expensive.
Figure 1. Impact energies measured in the non-recrystallised regions of C-Mn
multi-pass submerged-arc welds [5].
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Figure 2. Fracture toughness measurements made on 22NiMoCr37 forging
steel, from the Euro Fracture Toughness Data Base. Open symbols refer to
samples which cleaved; closed symbols to samples which showed ductility.
Clearly, if either the shape or mean position of the transition curve could be fixed by
theoretical considerations, then all the available data could be used to determine the
other attribute of the curve. The derivation of a ductile-to-brittle transition
temperature (DBTT) from a limited dataset would then be made less uncertain, and
the relations between e.g. radiation flux or fluence and the DBTT could be
determined with increased accuracy. The Mastercurve hypothesis of Wallin and coworkers holds out the promise of just this simplification of data analysis.
2
The Mastercurve Hypothesis
The Mastercurve hypothesis is based on experimental observations, and has been
rationalised with reference to various models of fracture. The relevant models
consider the effect of temperature on the ability of the applied loading to induce
cleavage and, hence, the effect of temperature on fracture toughness. This section
will describe the historical development of the family of models generally used, to
show the framework within which the justifications of the Mastercurve fit. It will then
describe the experimental observations on which the Mastercurve hypothesis is
based, followed by some suggested explanations of these observations. The
explanations can then be judged against the details of the models.
2.1
MODELLING BACKGROUND
2.1.1
Crack Extension From A Flaw
In ideally brittle material, such as glass, the control of fracture is described by the
Griffith energy balance [7]:
s 2pr = 2E 'g
Equation 1
where E’=E in plane stress and
E
in plane strain
(1 -n 2 )
E = Young’s modulus
n = Poisson’s ratio
s = applied normal stress
r = crack size
g = surface energy
In this description, the material is assumed to contain a series of atomically-sharp
flaws. A flaw of length 2r will propagate as a crack under the load s, determined by
the material properties n, E and g. The brittle material will then fail catastrophically.
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Even on the lower shelf, however, a ferritic steel is not perfectly brittle, and does not
intrinsically contain a set of atomically-sharp flaws. Such flaws are only introduced
by localised plasticity causing the failure of brittle second-phase particles. For steels
in the transition region at least, this initial stage is not the critical step in inducing
ultimate fracture. Failure of a second-phase particle, however, exposes the ferrite
matrix to the required, sharp, fast-running crack. Experiment has shown that
propagation of this microcrack into the ferrite is the critical step [e.g. 8] in causing
fracture.
A straightforward adaptation of Equation 1 allows the criterion for the subsequent
cleavage of the ferrite to be given as [9]:
s 2pr = 2 E ' (g + w p )
Equation 2
In this relation (the Griffith-Orowan relation), any dislocation motion in the matrix
accompanying crack propagation is considered as a process which (like the
formation of new surface area) can absorb the elastic energy released by fracture.
wp in Equation 2 is thus half the plastic work done per unit area in extending the
microcrack through the ferrite. g +wp can be determined by comparing r (from direct
observations of crack initiators [10, 11], maximum observed carbide sizes, or carbide
size distributions [12, 13, 14, 15, 4]) with the fracture stress, or the shape of the
transition region. In this way, values of g +wp have been found between 5 and 14Jm2
, as compared with the surface energy of iron which is around 2Jm-2 [16]. (g +wp is
sometimes described as the effective surface energy, or as the work to fracture.)
2.1.2
The RKR Model Of Cleavage In The Transition
Ritchie, Knott and Rice (RKR) [17] found that a critical stress was required in order to
initiate cleavage in a mild steel, but that this criterion alone was not sufficient to
explain failure. They suggested that the critical stress had to be exceeded over a
critical distance if failure were to occur. By fixing the stress and the distance as
temperature-independent parameters, they found that they could predict a
dependence of fracture toughness on temperature caused by the temperaturedependence of the flow stress. Increasing the temperature decreased the flow
stress, and so changed the relation between stress and distance within the plastic
zone. Their predictions fitted their experimental data on 4-point bend specimens, at
least in the lower transition.
Curry and Knott [13] later suggested that the critical distance was dependent on the
distribution of the second-phase particles which could be cracked to produce
cleavage initiators. The critical distance thus indicated the probability of
encountering particles of suitable sizes to fulfil the Griffith-Orowan criterion (Equation
2) for crack propagation, under the locally-prevailing stress conditions.
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2.1.3
Issue 2
The WST Models
Wallin, Saario and Törrönen (WST) [18] treated Curry and Knott’s explanation of the
RKR model explicitly. They expressed the probability of fracture (Pf) of a precracked
specimen as the probability of encountering a particle of radius r ≥r0 (i.e.p(r ≥r0) )
where r0 is the size of particle which will cause cracking under the locally-prevailing
level of stress, i.e. :
Xp
Pf = 1 - Õ [1 - p (r ³ r0 )]
X =0
Equation 3
The product (P) is over volume increments in the entire plastic zone (Xp) which is,
itself, temperature-dependent. The main factors affecting the probability of
encountering a critical particle within the plastic zone (at a given load and
temperature) are:
B = specimen thickness
N = number of carbides per unit area
F = fraction of carbides taking part in the fracture process.
By solving Equation 3 for different levels of probability, they found that they could
determine an expectance value for the fracture toughness, K, or probability limits.
As for the original RKR description, at this stage, the temperature-dependence of the
predicted toughness is based solely on the temperature-dependence of the flow
stress. This is reflected in the stress distribution in the sample (i.e. the value of r0 as
a function of distance from the crack tip), and the size of the area involved in the
summation (Xp).
WST found, however, that they could not adequately predict the temperaturedependence of the fracture toughness of a ferritic carbon tool steel with this
formulation. The dashed line in Figure 3 shows the fracture toughnesses predicted
by this formulation (and using (g+wp) = 14Jm-2). As can be seen, the toughnesses at
low temperature are overestimated, while those at high temperature are
underestimated. The predicted temperature-dependence is not as strong as the
measured temperature-dependence.
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Figure 3. WST predictions of the DBT in a ferritic tool steel, and the effect
allowing the work to fracture to be temperature-dependent [18].
WST found that a suitable temperature-dependence could be achieved if g +wp was
allowed to have an exponential temperature-dependence. A value of
g +wp = 9.17 + 0.19exp[0.0183T]
Equation 4
is used in Figure 3. For a bainitic Cr-Mo-V steel, a slightly different temperature
dependence was required, as shown in Figure 4 and Equation 5. The work to
fracture in the bainitic steel is lower than that in the spheroidised ferritic carbon steel.
g +wp = 2.15 + 1.77exp[0.0104T]
Equation 5
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Figure 4. WST predictions of the DBT in a bainitic Cr-Mo-V steel, using a
constant work to fracture, or a work to fracture with a different exponential
dependence to that required for the ferritic tool steel [18 above].
It is worth noting that an exponential temperature-dependence of g +wp does not
produce an exponential temperature-dependence of the fracture toughness. It
merely increases the curvature of the predicted values of Kc vs. T.
WST considered that, for a crack to propagate, a critical dislocation density was
required around the crack tip. wp was required predominantly to create the
necessary number of dislocations. With increasing temperature, the dislocation
mobility increases, making it more difficult to maintain the critical density around the
tip. Thus the increase of wp with temperature was related to the increase in
dislocation mobility or, inversely, to the decrease in lattice friction - the PeierlsNabarro force. Since the Peierls-Nabarro force has an exponential temperature
dependence, this rationalised the use of an exponential temperature-dependence for
wp. The authors did, however, note that, where measurements had been made (e.g.
Veistinen and Wallin’s [19]), there was no evidence that g +wp did increase
exponentially with temperature.
In this analysis, the probability of failure is equivalent to encountering a microcrack of
such a size that it will extend under the locally-prevailing stress. In effect, all the
carbides are assumed to have cracked. WST later considered the cracking of brittle
particles as a result of fibre loading by a ductile matrix [20], and found that it could be
described by Weibull statistics.
i.e.
probability of a particle cracking = Pfr
m
é æ d ö3 æ d ö3 æ s - s
ö ù
min
÷ ×ç
÷ ú
= 1 - exp ê - ç ÷ × çç
êë è d ø è d N ÷ø çè s 0 - s min ÷ø úû
Equation 6
where d, d are the particle diameter, and mean diameter of the particle population, s
is the tensile stress acting on the particle, smin is the minimum fracture stress of the
particle, m is the Weibull inhomogeneity factor, and s0 and dN are normalising
parameters. If the stress on the particle is assumed to be proportional to the flow
stress of the matrix (strictly the shear flow stress rather than the tensile flow stress),
then
é æ d ö3
ù
Pfr = 1 - exp ê- ç ÷ × f (s )ú
êë è d ø
úû
Equation 7
m
3
matrix
matrix
æ d ö æç s flow - s min ö÷
ç
÷
where f (s ) = ç
for spherical particles, and has a slightly
÷ ×
÷
s1
è d N ø çè
ø
different form for rod-shaped particles etc.
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matrix
s min
is the matrix stress corresponding to the minimum particle stress, effectively
the matrix yield stress, and s 1 is a constant.
Wallin and co-workers [21, 22, 23] were then able to generalise Equation 3 using
P * (r ³ r0 ) = probability of finding a broken carbide of size greater than critical value
¥
= ò Pfr P{r0 }dr
r0
Equation 8
where P{r0} is the carbide size distribution, and
Pfr is the probability of a carbide fracturing,
Wallin [22] then considered that the summation in Equation 3 was over the plastic
zone (size Xp) in small increments of volume dX.X.sinJ, where J is an effective angle
describing the volume which might affect the fracture process (a constant). This
allowed him to approximate Equation 3 to
Xp
é
ù
P f » 1 - exp ê - N v BF sin J ò p(r ³ r0 ) X .dX ú
X =0
êë
úû
Equation 9
Where Nv is now the number of particles per unit volume.
æK
By the relation X = U çç 1C
è sY
became
2
ö
÷÷ , where U is a proportionality constant, Equation 9
ø
é
ù
K 4 Up
P f » 1 - exp ê- N v BF sin J 14C ò p (r ³ r0 )U .dU ú
s Y U =0
êë
úû
Equation 10
The integral is now taken to be a constant (though, strictly, r0 is a function of X also)
so
[
P f » 1 - exp - const × K14C
]
Equation 11
or in the more generally-used form
4
é B ì K-K
ü ù
min
Pf = 1 - exp ê í
ý ú
êë B0 î K 0 - K min þ úû
Equation 12
Where B
= specimen thickness
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B0 = reference specimen thickness (a normalisation constant, generally
K = fracture toughness for which probability of failure is being calculated.
Kmin= limiting K value below which cleavage fracture is impossible (generally
~20MPaÖm)
K0 = normalisation fracture toughness (fracture toughness corresponding to
Pf=63.2%).
Interestingly, at this point, although this equation has the same form as a Weibull
distribution, it appears to refer only to microcrack extension. The Weibull description
of particle cracking is not involved in this expression. In effect, the probability of
particle cracking has returned to unity.
2.1.4
The Beremin Approach
The Beremin group has considered the probability of finding a microcrack of a
suitable length to cause failure in slightly different terms to those in section 2.1.3 [24].
In their analysis of notched or cracked specimens, they divide the stressed region
into small volumes, of size V0, in each of which the probability of finding a crack of
length between l0 (=2r in the terms of the previous equations) and l0+dl0 is:
a
P(l 0 )dl 0 = b dl 0 ,
where a,b are material constants
l0
Equation 13
The microcracks are assumed to form from cracked grain boundary carbides, but the
precise source is not important for the development of the model. V0 is chosen to be
small enough for the stress within each volume to be approximately constant, but
large enough that the probability of finding a microcrack of reasonable length is not
vanishingly small, and each volume is independent of its neighbours. In practice,
V0=(50µm)3 is often chosen.
If the principal tensile stress in each volume element is s, then the probability of
failure occurring in a volume is:
¥
p(s ) = ò P(l 0 )dl 0
crit
l0
Equation 14
Which, by the relation between l0 and s given in Equation 1, becomes
æs
p(s ) = çç
èsu
m
ö
÷÷ ,
ø
Equation 15
where m=2b-2, and su is a constant incorporating the material parameters, a, b, E, g
and n.
Summing over all volume units leads to:
PR = 1 - Õ [1 - p (s i )]
i
Equation 16
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In a homogeneously-stressed specimen, this is simply solved as:
é æ s öm ù
÷ ú
ln (1 - PR ) = 1 - exp ê - çç
êë è s u ÷ø úû
Equation 17
When s varies from one volume element (Vj) to another, it becomes:
é æ s öm ù
where sW = Weibull stress =
PR = 1 - exp ê - çç W ÷÷ ú
êë è s u ø úû
m
æ
æ V j öö
÷÷
÷÷
V
è 0 øø
m
å çç (s 1, j ) çç
j
è
Equation 18
For a sharply-cracked specimen in Mode I loading and small-scale yielding, s 1, j (the
principal tensile stress in each element) is calculated as a function of distance/angle
from the crack tip which is, itself, a function of sY and K. The Weibull stress then
becomes
4
üï
a (J ) ì B æ K ö
p
ï
m
m
m
÷÷ fn(u )udu ý
s W = s Y ò g JJ (J )dJ ò í çç
-p
0 ïV0 è s Y ø
ïþ
î
distance from crack tip in x - direction
where u =
, gqq(q) = angular dependence of
2
æK
ö
ç s ÷
Yø
è
distance from crack tip at angle, J
stress a (J ) =
2
æK
ö
ç s ÷
Y ø
è
Equation 19
This leads to the formulation:
a (J )
p
é s m-4 K14c BC m ù
m
where
=
PR = 1 - exp ê - Y
C
g
(
J
)
ò
ò fn(u )udu
ú
JJ
m
V0s um
-p
0
úû
êë
The effect of this calculation is shown in Figure 5
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Figure 5. Comparison between calculated DBT and fracture toughness data
from the HSST program [24].
Since Cm was found, experimentally, to have only a slight temperature-dependence,
and su was originally considered to be temperature-independent (though it must
incorporate the mild temperature-dependence of E and n), in the Local Approach, the
temperature-dependence of K is, again, dependent chiefly on sY. The Beremin group
specifically do not incorporate an exponential temperature-dependence of g+wp.
In some recent works [e.g. 25], it has been necessary to assign a temperaturedependence to su in order to reproduce the experimentally-determined upswing in
fracture toughness with temperature. It is not clear, however, if this is an intrinsic
requirement of the material/model or due (at least in part) to an inaccurate
description of the stress field when constraint is lost at higher temperatures.
In the current use of the Local Approach, m is generally taken to be constant for all
steels (22 if su is constant, 8-14 when it is temperature-dependent). From the
derivation described here, however, it is possible that a and b will be affected by the
particle size distribution and hence both m and su could vary with heat treatment and
composition.
2.1.5
Summary
The models described in this section have several features in common:
1) The critical stage in cleavage is taken to be the extension of a microcrack. Thus
describing cleavage requires only a description of microcrack extension. This is
in accordance with experimental observations of cracks in carbides which have
not led to overall failure [8]. The microstructural source of the microcracks, and
their availability, are treated explicitly only in [20], which is not incorporated into
later derivations.
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2) The probability of failure is described as the probability of locating a microcrack
which will extend under the locally-prevailing stress according to a Griffith-type
criterion (e.g. Equation 2 or Equation 3).
3) The probability is summed over the plastic zone, which expands as sY decreases
(i.e. as the temperature increases) and as the loading increases. The total
probability of failure increases as the volume considered increases.
4) The levels of stresses in the plastic zone decrease with decreasing sY (increasing
temperature) and work hardening rate, which decreases the probability of failure
in a given distance (/area / volume) segment.
5) a) In the RKR, Curry and Knott, and initial Beremin approaches, the
temperature-dependence of K is due to the temperature dependence of the
stresses in the plastic zone and the size of the plastic zone.
b) In the WST approach, wp is also considered to be temperature-dependent,
and this dominates the temperature-dependence of the fracture toughness (Kc)
over temperature ranges in which the yield stress varies only weakly with
temperature. (The increase in wp is thought to be related to the decrease in sy,
but the relation is not described explicitly, or determined by independent
measurement.)
c) In some applications of the Beremin approach, su is taken to be temperaturedependent. It is not yet clear if this is an intrinsic requirement of the model (due to
the effects of temperature on E or n, or even g), or due to loss of constraint.
6) Specimen-to-specimen variations in Kc at a given temperature are due to
variations in the probability of encountering suitable flaws in the stressed volume.
7) a) In general, these models describe the lower shelf and lower transition well.
b) Kc can be underestimated in the middle and upper transition. This is not
necessarily due to inaccuracies in the model formulations, but to a change in the
failure mechanism. In the upper transition region, cleavage is preceded by
increasing amounts of (energy-absorbing) ductile crack extension. The models
described here do not allow for this process, and can thus only define lower
bounds in the upper transition. (The Beremin model at least can be coupled with
a ductile crack growth model to describe the upper transition region more
effectively [26].)
c) Nonetheless, the models can predict cleavage at toughness levels below
those at which ductility occurs, even at temperatures at which ductile crack
extension is the dominant mode. It is to avoid this underestimation of the Kc
required to induce cleavage at higher temperatures that extra temperature
dependences (of su or wp) have been introduced.
(This summary is reproduced in part of Table 5, at the end of section 3.5.)
2.2
DEVELOPMENT OF THE MASTERCURVE
The Mastercurve has two aspects; a description of the distribution of K1C
measurements at a given temperature; and the temperature-dependence of the
mean (or median) toughness with temperature.
2.2.1
Scatter In K1C Measurements
Since Curry and Knott [13] pointed out that cleavage fracture could be dependent on
the probability of encountering a particular microstructural features, brittle fracture of
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steels has been observed in many ways as being statistical in nature. Several
authors have used a Weibull probability distribution to describe the scatter in fracture
toughness of steels failing in a brittle way ( e.g. [27, 28]). For a Weibull distribution,
the probability of failure at a given applied K is given by:
Pf(K1)=1-S(K1)
Where S(K1) = probability of surviving under K1 = exp - cB(K1 - K min )a
K ³ K min
1
K £ K min
[
]
c and a are positive constants; Kmin≥0 and B is the crack front length.
c is material- and temperature-dependent. In the Mastercurve analysis, a is always
equal to 4. The rationale is as given at the end of Section 2.1.3. If small-scale
yielding conditions do not apply (if the plastic zone is not well-contained, or there is a
loss of crack tip constraint), then a will change value. In order to use the
Mastercurve description, therefore, it is necessary to limit data to tests on specimens
which remain in small-scale yielding. The ASTM standard [2] defines the limits as:
Eb0s Y
K JC (max imum ) =
M
Da < 0.05b0
Equation 20
Where b0 is the length of the specimen ligament, M is a defining constant (30), and
Da is the increment of ductile crack extension preceding cleavage.
The UK nuclear industry’s Technical Advisory Group On Structural Integrity has
issued a paper assessing the applicability of this description, and considers that it
should hold for structures / specimens containing a moderate crack front length [29].
It considers that, at very long crack fronts, the appropriate description of the flaws
changes from a Weibull to a normal distribution. It also suggest that M may need to
be larger than 30 in some cases (possibly 50 in the case of irradiated steels), and
that Da should not exceed 0.2mm, even if this remains below 0.05b0.
2.2.2
Temperature-Dependence Of K1C
In 1993, Wallin [1] investigated the effect of radiation damage on the fracture
toughness of a pair of rpv steel submerged-arc welds for which a large amount of
data was available. He fitted exponential forms to the fracture toughness
temperature dependences in both welds in the transition region (c.f. the ASME
reference curves, rather than any particular theory of fracture). The fits are shown in
Figure 6 and Figure 7.
In both these Figures, the main trend line is
K0 = 31+77exp[0.019(T-T0)]
Equation 21
(where K0 is calculated from all the K- data measured at a particular temperature,
according to Equation 12.)
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i.e. for the two welds (which use different fillers), the temperature-dependence of K0
is the same. The welds differ only in the position of the K0 vs. T curve on the
temperature axis, described by the single parameter T0 – the temperature at which
K0=100MPaÖm.
Figure 6. Fit of exponential trend line through fracture toughness data from
unirradiated rpv weld 72W [1].
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Figure 7. Fit of exponential trendline through fracture toughness data from
unirradiated rpv weld 73W [1].
The same form of curve applied to the welds after irradiation, to an A533B Class 1
steel HSST 02, and to a further set of rpv welds using slightly different filler
compositions, as shown in the Figures below.
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Figure 8. Fit of exponential trend line through fracture toughness data from rpv
weld 72W after irradiation [1].
Figure 9. Fit of exponential trend line through fracture toughness data from rpv
weld 73W after irradiation [1].
Figure 10. Fit of exponential trend line through fracture toughness data from
A533B Class 1 steel HSST02 before and after irradiation [1].
Although the shape of Equation 21 differs slightly from the best trend line that might
be fitted to the individual datasets (especially in the irradiated material and at the
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higher toughnesses), it is, nonetheless a reasonable fit, and most of the datapoints
do lie between the 5% and 95% limits.
A summary curve is shown in Figure 11 in which K0 (calculated from the valid K data
at each temperature for each material) is plotted versus (T-T0) for several rpv steel
welds and plate. The agreement between the trend line and K0(T) is impressive.
Figure 11. Median K values for a range of rpv steels and welds plotted in terms
of adjusted temperature (T-T0). The T0 values are given in the key. The trend
line through the median values is the Mastercurve [1].
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2.2.3
Issue 2
Interpretation Of Mastercurve Behaviour
Since the steels in Figure 11 possess a range of yield strengths and compositions,
Wallin deduced that the temperature dependence of K0 had to be due to some
intrinsic property of the ferrite matrix common to all of the steels. The only such
property he considered appropriate was the dislocation mobility. Thus, as already
described in section 2.1.3, as the yield stress decreased with temperature, so the
effective surface energy must also increase (exponentially). As he reported,
however, “The drawback with the above argument is that experimental estimations of
the effective surface energy for ferrite show practically no temperature dependence
whatsoever.”
Merkle, Wallin and McCabe [30] considered that the yield stress might be the
dominant parameter governing the temperature-dependence of the fracture
toughness. They plotted . Log (fracture toughness) versus the yield stress for an
A533B Class 1 rpv steel before and after and found the slope of the lines to change
with irradiation. Since the curvature of the DBT was considered to be unchanged
with irradiation, the total yield stress could not be the feature controlling the shape of
the DBT. They then plotted Log(fracture toughness) versus Log( “the thermal
component of the yield stress”), this factor being derived from the mathematical form
fitted to the yield stress vs. temperature data. For this, second plot, the same slope
was found for both the unirradiated and irradiated steel. Despite the mathematical
non-equivalence of the two comparisons, the thermal component of the yield stress
was thereby considered to be the feature controlling the temperature-dependence of
the fracture toughness, while the athermal part of the yield stress controlled the
position of the K:T curve on the temperature axis. They further suggest that the
thermal part of the yield stress might be related to the activation energy for
dislocation mobility. No analytical relation between the yield stress and the fracture
toughness is given, however.
Natishan and co-workers have approached the rationalisation of the Mastercurve
from different points of view. In [31], they consider that the temperature dependence
of the fracture toughness required to induce cleavage is due to the temperature
dependence of the plastic work (per unit volume) required to cause ductile failure.
(Again, no specific relation between the two temperature-dependent phenomena is
derived.) In [32], they support the interpretation of Merkle et al [30], noting that the
temperature-dependent part of the yield stress must be related to the ability of
dislocation to overcome short-range obstacles. In effect, the temperaturedependence of the yield stress is related to the Peierls-Nabarro stress, or lattice
friction. Although long-range obstacles are specifically excluded from consideration,
it is worth noting that the grain-size dependence in the Hall-Petch equation (ky) is
temperature-dependent, and grain boundaries are long-range obstacles. In addition,
no consideration is given to the increasing waviness of slip between –196°C and
room temperature, which permits dislocations to bypass long-range obstacles such
as particles. Merkle et al and Natishan and co-workers are undoubtedly correct in
considering that the temperature-dependence of the yield stress is important, but the
mechanistic interpretations of their arguments are open to question.
Odette and He [33] have interpreted Mastercurve behaviour in terms of the RKR
model. They have used a finite element method to parameterise an expression
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describing the area (A) ahead of a deep crack which is stressed above a particular
level (s). It refers to a specimen in small-scale yielding. From this they derive
[
*
K Jc = s Y A 10
]
- P 0.25
és * ù
és * ù
where P = C 0 + C1 ê ú + C 2 ê ú
ës Y û
ës Y û
2
Equation 22
(the stars denote the critical values of A and s, C0-2 are constants).
This fits reasonably well to the Mastercurve near the lower shelf but, like the original
WST model, underestimates the K levels at higher temperatures. Odette and He
therefore derive a temperature-dependence for s*. This, in turn, is rationalised, in
terms of dislocation mobility. In essence, this explanation is similar to the WST
interpretation in that the fracture stress is proportional to g +wp via the Griffith-Orowan
relation (Equation 2). As with the WST interpretation, there is no independent
verification that s* varies with temperature in the required way.
2.2.4
Overview
Although no satisfactory theoretical basis for the Mastercurve has been derived, data
have accumulated in support of the common trend line. This accumulation of data
supporting the Mastercurve has led the ASTM to issue a standard describing its
application [2], and the method is used in many data analyses, particularly in the U.S.
The Mastercurve hypothesis is an extremely attractive one: Tests at a single
temperature (as few as six in some circumstances) permit K0 for that temperature to
be determined, and the single K0-T pair then permits the entire transition toughness
curve to be drawn. The convenience of the hypothesis can, however, lead to its use
in circumstances which do not appear appropriate.
The interpretations of Mastercurve behaviour described in section 2.2.3 treat the
shape of the transition curve as being due to a fundamental property intrinsic to all
steels. Each interpretation of Mastercurve behaviour is thus an attempt to identify
this single property. If the shape really is fundamental, then the Mastercurve can be
drawn through all steel datasets. As described above, this would be very convenient.
Alternatively, it is possible to explain the similar transition curves found in a set of
steels by assuming the shape of the transition curve to be affected by several
material properties, some of which act to offset each other. This would produce
ranges of steel conditions for which the curvature of the transition is similar. The
Mastercurve would then be appropriate to some of these conditions. Other steel
conditions would, however, have significant different transition curvatures, and it
would not be appropriate to use the Mastercurve in these situations. In order to take
advantage of the Mastercurve, it is necessary to determine the situations under
which it likely to be appropriate.
The aim of the next section is to distinguish between these two explanations of the
Mastercurve, and assess the range of situations under which it might be appropriate
to use the Mastercurve analysis.
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3
The Two-Component Model
3.1
DETAILS OF THE MODEL
Issue 2
Like the WST model, the Two-Component description of the DBT by Ortner and
Hippsley [4] (OH) is based on the Curry and Knott analysis of cleavage. OH
investigated fracture initiation sites in compact tension specimens of A533B over a
range of temperatures, and in different heat treatment conditions. They determined
the stresses and strains operating at the initiation sites under the failure load using
the McMeeking [34] solutions for the stress fields (appropriate to small-scale
yielding), and found that the crack opening stresses at the initiation sites were
constant for a given material condition. No other stresses or strains were found to be
temperature-invariant at the initiation sites. This value of this stress was described
as the critical stress, sc. The distance between the precrack tip and the initiation site
was also found not to be affected by temperature, so did not correspond to a
particular position in the plastic zone (i.e. it was not at the peak in the stress, or at the
elastic-plastic boundary etc.).
The importance of the crack opening stress alone suggested that the critical process
in cleavage initiation was the extension of a microcrack out of a cracked particle.
The probability of failure was therefore calculated in terms of the probability of
encountering a particle of sufficient size that, as a microcrack, it could extend under
the locally-prevailing stress, in a similar manner to Equation 3. In Equation 3,
however, the summation is over the plastic zone. In OH, the summation is between
the precrack tip and the point at which sc is achieved (under increasing local load), as
illustrated in Figure 12 for a sc value of 2500MPa.
3000
2500
Stress (S11, Pa) or
Strain (e11, %x1000)
Stress
2000
1500
1000
500
Strain
0
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
Distance Ahead Of Crack Tip (m)
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Figure 12. Crack opening stress and plastic strain distribution ahead of a
sharp crack. The hatched area is the area within which the probability of
failure is calculated in OH.
The probability calculations investigate the material directly ahead of the crack tip.
The distance is divided into small increments, in each of which the stress is
determined. The size of microcrack which would extend under that stress is then
calculated, and the probability of encountering a particle larger than this derived from
the particle size distribution. The probability of survival is then (1-this probability).
The survival probability up to a given distance is the product of the survival
probabilities in all increments up to that distance. The failure probability is then
calculated as a function of (1-survival) for a given specimen width (B), number of
particles per unit area (Na), fraction of particles taking part in fracture (F) according to
X s =s crit
Pfx = 1 - Õ [1 - p (r ³ r0 )]
for given N a , B , dX , F
X =0
where r0 =
(
2pE g s + w p
(1 -n )
2
)
2
s 11,
at given position
Equation 23
In accordance with the experimental observations, at low temperatures, the point at
which sc is achieved is calculated to be beyond the limit to the plastic zone (defined
by some small value of strain, e.g. 0.01%), while at high temperatures it is close to
the peak in the stress field. In absolute terms, the experimentally-measured
distances were between 80-500µm, and not clearly affected by temperature.
Since the stressed distance increases with loading at a given temperature, the
probability of failure increases with applied K. A requirement for a constant value of
this probability of failure, (calculated up to the site at which sc was achieved) was
found to be sufficient to fix a transition curve [4]. A probability of 50% gives the mean
position of the curve, while other probability levels allows confidence limits to be
defined.
(In a sharply-cracked specimen, the peak stress does not vary with loading, but the
position of the peak stress is pushed further from the crack tip as the load increases.
Thus, for a progressively-loaded sample under a particular load, such as described in
Figure 12, the material between the crack tip and the current peak in the stress field
has survived the peak stress. In determining the probability of encountering a
particle of a size capable of failing under the local stress, therefore, the stress close
to the crack tip is considered to be effectively the peak stress, as shown in Figure
12.)
To use this model, the stress field ahead of the crack tip must be defined at all
temperatures in the transition region. This, in turn requires that the material’s flow
properties (the yield stress, the Young’s modulus and the work hardening exponent)
have defined values. To calculate the size of microcrack (r0) which would extend
under different stresses, g +wp must be defined. Finally, to determine the probability
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of encountering a particle of size r0, the particle size distribution must be defined. All
these values were measured, or derived in [4]. It is possible that some of these
parameters are interrelated but, in the calculations required by the model, it is not
necessary that they should be. In the following section, these parameters will be
varied independently to show the effect of each parameter on the shape of the
transition curve calculated according to OH.
3.2
CALCULATED EFFECTS OF CHANGING PARAMETERS IN OH
In the following calculations, the materials parameters for A533B in the quenched,
tempered and stress-relieved condition will be used as the baseline properties.
These are the properties of the material described as “Unaged” in [4]. The
temperature-dependence of the flow properties have been parameterised as:
sY (MPa) = 633.8528exp[-0.0004016T]+18.8712exp[-0.016277T]-75.5424
E (GPa) = 186.2474-0.094699T
n = 0.120392-0.0002825T-0.000002834T2 = work hardening exponent (as in the
description s = Ae n of a tensile flow curve)
where T is the temperature in °C.
The particle size distribution is as described in [4], where
Probability of encountering a particle of size, r = p( r )
é
ù
a
-c ú
c ( a -1) æ r ö
ê
=
ç ÷ exp
ê r aú
(a - 1)! è r ø
ëê r ûú
( )
Equation 24
where c, a are constants (specifically, a=4, c=3.9) and r is the geometric mean of the
particle size distribution (26nm). The number of particles per unit area was 1010m-2.
The effective surface energy, g +wp has been taken as 6Jm-2.
The critical stresses measured in [4] ranged between 2110MPa and 2620MPa. The
baseline value of the critical stress in the following assessments will be 2500MPa.
The trend line shown in each assessment will be derived from the 50% probability of
failure.
As a result of this choice of parameters, the absolute values of the calculated fracture
toughnesses are high. This should not be considered mechanistically significant.
Whereas in real materials, some ductility would be expected at the highest
toughnesses calculated, within this sensitivity study no contributions from ductility will
be considered.
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The Critical Stress
Figure 13 shows the effect of reducing sc from its baseline value of 2500MPa to
2000MPa; all other parameters remaining constant. In this graph, and those
following, the values of K required to cause cleavage become infinite after the
highest datapoints plotted for each curve.
400
350
300
250
K (MPaÖm)
3.2.1
Issue 2
Sc=2500, Baseline
Sc=2000
200
150
100
50
0
-200
-150
-100
-50
0
50
100
Temperature (°C)
Figure 13. Effect on the DBT of changing the critical stress, Sc.
As would be expected from a simple Ludwig-Davidenkov diagram, decreasing the
critical stress (the cleavage fracture stress) shifts the DBT to higher temperatures.
Figure 13 also shows that, at toughnesses around those used to define the transition
temperature (~100MPaÖm), the slope of the K:T curve has decreased. This is not
surprising, as the only input parameters which vary with temperature in this model
are the flow properties. The flow properties vary most strongly at low temperature.
By shifting the DBT to higher temperatures, reducing sc shifts the DBT to
temperatures at which the effects of temperature are reduced. In this aspect, OH
differs from the WST models, in which the dominant temperature-dependence is that
of wp. In OH wp is a constant. The reduction in slope of sY:T or n:T with increasing T
thus affects the curvature of K:T less in WST calculations than in OH’s predictions.
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3.2.2
Issue 2
Effective Surface Energy
Figure 14 shows the effect of changing the effective surface energy, or work to
fracture, g +wp. The original dataset uses 6Jm-2, while the new datasets show the
effect of doubling, or halving this.
700
600
K (MPaÖm)
500
400
E=6, Baseline
E=12
E=3
300
200
100
0
-200
-150
-100
-50
0
Temperature (°C)
Figure 14. Effect on the DBT of changing the work to fracture, E
The temperature at which the fracture toughness required to produce cleavage
becomes infinite (T∞) is not affected by changing , g +wp (described as E in the
diagram). The approach to that temperature is, however, changed significantly.
Increasing g +wp increases the slope of the low-temperature region, producing a
much steeper slope at technologically interesting fracture toughnesses.
3.2.3
Flow Properties: Yield Stress And Work Hardening Exponent
The distributions of stress and strain in the region ahead of a loaded crack are
dependent on the flow properties of a material. This is illustrated by the McMeeking
descriptions of the stress field, as shown in the Figure below. The stress in Figure 15
is normalised to the yield stress (S0 in the Figure), and the distance, R, ahead of the
crack tip is normalised to the crack tip blunting (b, which is a function of the flow
properties and load). The Figure shows that the peak in the stress field increases as
the work hardening increases, while the decrease in stress beyond the peak is more
rapid as the ratio of the yield stress to the Young’s modulus increases. (For most
steels, the sY/E ratio is close to 1/300 and n around 0.1.) The McMeeking
descriptions are relevant to a deeply-cracked specimen in Mode I loading, exhibiting
small-scale yielding.
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Since the extension of a microcrack depends on the locally-achieved stress, and the
probability of encountering a suitable microcrack depends on the absolute distance
over which the stress is applied, fracture is clearly dependent on both the yield stress
and the work-hardening exponent, n. These parameters will now be assessed
individually, starting with the yield stress.
6.00
5.00
S11/S0
4.00
1/300, n0
1/300, n.1
1/300, n.2
1/100, n0
3.00
2.00
1.00
0.00
0.00
2.00
4.00
6.00
8.00
10.00
12.00
R/b
Figure 15. Effect of different yield stress (S0):Young’s modulus ratios (1:300
and 1:100) and work hardening exponents (0, 0.1, 0.2) on the distribution of the
principal tensile stress (S11) in the plastic zone ahead of a sharp crack in Mode
I loading and small scale yielding (according to McMeeking [34]).
When a new set of obstacles to slip are introduced (e.g. by heat treatment, strain or
irradiation) the yield stress is raised. If neither the obstacles, nor the ability of
dislocations to overcome them, are sensitive to temperature, then the yield stress
increment will be independent of temperature. If there is some effect of temperature,
then the increment will generally decrease with temperature. Both possibilities are
illustrated in Figure 16. The first curve is that seen in [4]; the second is raised above
the first by 50MPa at all temperatures; the third curve is raised above the first by
50MPa at temperatures below –100°C, but the increment decreases linearly to zero
by 0°C. It should be emphasised that these two curves do not reflect a particular
micromechanism of hardening. They are merely mathematical devices used for
illustration.
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1000
Yield Stress (MPa)
900
800
Original
700
Original+50MPa
600
Original +50MPa
Below -100°C,
dropping linearly
to Original At 0°C
500
400
-200
-150
-100
-50
0
50
Temperature (°C)
Figure 16. Yield stresses used for calculations.
The effects of the different yield curves on the DBT are shown in Figure 17. As can
be seen, increasing the yield stress pushes the transition to higher temperatures.
Since the shift in the yield stress:temperature curve has been on the stress axis
rather than the temperature axis, the slope of the sY:T curve at a given temperature is
unchanged by the shift. At low temperatures, the slope of the sY:T curve is higher
than at high temperatures. By pushing the DBT to higher temperatures, the increase
in the yield stress (like a decrease in critical stress) moves the DBT to a temperature
range over which the yield stress changes more slowly. The curvature of the
transition being dependent on the temperature-dependence of the yield stress,
Figure 17 shows a temperature-independent increase in the yield stress producing a
decrease in the curvature of the DBT.
By making the stress increment temperature-dependent, the slope of the sY:T curve
is kept high up to a higher temperature. The shift in the DBT becomes smaller, but
there is less of a decrease in its curvature. Clearly, to obtain a constant shape of
DBT, the yield stress increment must have a suitable temperature-dependence over
the temperature range of the DBT and its shift. For example, for the initial DBT
shown in Figure 17, the yield stress increment would need to be temperaturedependent in the range –150°C to 0°C to produce a shift of the DBT without changing
its slope. These are very low temperatures.
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300
250
K (MPaÖm)
200
Baseline
S+50MPa at all T
Temp-dep dS
150
100
50
0
-200
-150
-100
-50
0
Temperature (°C)
Figure 17. The effect on changing the yield stress on the DBT.
Within the WST model, the decrease in the temperature-dependence of the flow
properties is offset by an increase in gs+wp with temperature. This allows the fracture
toughness to increase rapidly with temperature even if the yield stress is decreasing
only slowly. It has been suggested that this increasing gs+wp will permit the DBT to
retain its shape even after a temperature-independent increase in yield stress.
Figure 18 shows the DBT calculated using the expression for gs+wp associated with
Figure 4. Again, the increment of 50MPa in the yield stress has shifted the curve to a
higher temperature, and decreased the slope of the curve at ~100MPaÖm.
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180
160
140
K (MPaÖm)
120
100
E=f(T), otherwise as base
E=f(T), and Sy+50MPa
80
60
40
20
0
-150
-100
-50
0
Temperature (°C)
Figure 18. Effect of increasing the yield stress by 50MPa at all temperatures
when the effective surface energy is exponentially temperature-dependent (
using the WST expression E=2.15+1.77exp[0.0104T]).
300
250
K (MPaÖm)
200
Baseline
n=0.75*Baseline
n=0.5*Baseline
150
100
50
0
-200
-150
-100
-50
0
Temperature (°C)
Figure 19. Effect of changing the work hardening rate, n, on the DBT.
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Figure 19 shows the effect on the DBT of changing n while keeping all other
parameters constant. The baseline curve uses the flow parameters described at the
beginning of section 3.2. The other curves show the effect of reducing n by 25% or
50% at all temperatures.
Figure 15 shows that decreasing the work hardening rate reduces the maximum
stress that can be achieved at a given temperature. To achieve a given probability of
failure therefore requires a higher yield stress, thus the DBT is shifted to lower
temperatures, and becomes steeper at the indexing toughnesses (~100MPaÖm).
The effect of reducing n is thus opposite to that produced by increasing the yield
stress.
A decrease in work hardening exponent frequently occurs when materials are
hardened, so it is worth assessing the effect of combining the two changes. This is
shown in Figure 20 for several combinations of yield stress increase (not varying with
temperature) and work hardening decrease. On the basis of the plots in Figure 20,
increases of 50-75MPa combined with 25-35% decreases in n could be described as
producing bodily shifts of the transition to higher temperatures. The shift is not
strictly constant at all toughnesses, but it is fairly constant over a wide range of
toughness levels. Other combinations of yield stress increase+work hardening
decrease will probably also produce temperature shifts constant over significant
toughness ranges.
300
250
K (MPaÖm)
200
Baseline
S+50MPa at all T
S+50MPa, 0.75n
S+75, 0.65n
150
100
50
0
-200
-150
-100
-50
0
Temperature (°C)
Figure 20. Effect on the DBT of increasing the yield stress, S, while decreasing
the work-hardening exponent, n.
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The Particle Size Distribution
Figure 21 shows the particle size distribution used in [4], which uses r =26nm, a=4,
c= 3.9 and Na=1010m-2. The Figure also shows the distributions described by
changing the mean particle size, and the shape parameter, a, from Equation 24. The
reduction in a increases the proportion of larger particles (the size of the tail of the
distribution). By the time the probability of encountering a particle of given size has
decreased to <0.5%, the decrease in a has produced a similar effect on particle
availability as doubling the mean particle size at constant a.
0.6
Original,
r=26nm,
a=4
0.5
Fraction Of Particles
3.2.4
Issue 2
0.4
r=52nm,
a=4
0.3
r=52, a=3
0.2
r=26, a=3
0.1
0
0.00E+00
1.00E-07
2.00E-07
3.00E-07
4.00E-07
5.00E-07
Particle Size (m)
Figure 21. A selection of particle size distributions.
Figure 22 now shows the effect these changes may have on the DBT. (In Figure 22,
the change in a is referred to the smaller mean particle size only, not to the larger.)
Increasing the mean particle size (effectively increasing the carbide content by
increasing the size of every particle while leaving constant the number of carbides
and form of size distribution) lowers the fracture toughness at every temperature,
although the temperature at which cleavage becomes impossible (T∞) remains the
same. The curvature of the DBT becomes much shallower. Reducing the number of
particles per unit area (to offset the increase in carbide content, and just coarsening
an aspect of the distribution) increases the fracture toughness, without changing the
curvature very much. Thus the third curve resembles the second very closely,
showing a similar shift to higher toughness at all temperatures.
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300
250
r=26nm, N=10^10, a=4,
Baseline
r=52nm, N=10^10, a=4
K (MPaÖm)
200
150
r=52nm, N=2.5x10^9, a=4
r=26nm, N=10^10, a=3
100
50
0
-200
-150
-100
-50
0
Temperature (°C)
Figure 22. Effect on the calculated DBT of changing the particle size
distribution.
During extended heat treatments, precipitate coarsening occurs. The amount of
precipitated material remains constant, but the particle size increases, producing a
reduction in the number density of particles. In the third curve of Figure 22, the area
fraction of particles has been kept roughly the same as in the baseline curve (a
constant value of Nr2). Although the lower shelf has clearly been raised by this
“coarsening treatment”, the toughness levels in the transition region are practically
unchanged. Thus some aspects of heat treatment may well leave the transition
region constant.
The dominant effect of the coarser particles on the fracture toughness is shown by
the fourth curve in Figure 22. In this case, particle coarsening is approximated by
changing the value of a. Reducing the a-value, which increases the tail of the particle
size distribution at the expense of the main part of the distribution, drastically reduces
the fracture toughness. The reduction is very similar to that caused by doubling the
mean particle size (at constant a), yet, as shown in Figure 21, the two distributions
are similar only when particles larger than the 99.5th percentile are involved. This is
in fair accordance with [35, 36], in which the DBTT was found to shift with heat
treatment in direct proportion to the increase in the 99th percentile carbide size.
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3.2.5
Issue 2
Summary
Table 1 summarises the effects on the DBT of the parameters investigated in this
section.
Parameter increased
Stress at fracture nucleation site, sc
(experimentally found to be a material
constant, invariant with temperature)
Effective surface energy, g+wp (also
known as work to fracture)
Yield stress, sy
Work-hardening exponent, n
Mean size of particles causing fracture,
r
Number of particles per unit area, Na
a-parameter describing shape of
distribution (when a increases, the
fraction of particles in the high tail of the
distribution decreases)
Effect on temperature
at which cleavage
becomes impossible
(T∞)
Decrease
Increase
-
Increase
Increase
Increase
-
Decrease
Decrease
Decrease
-
Slight increase
Increase
Effect on slope of
transition curve
Table 1. Summary of parameters investigated with OH model, and their effects
on the fracture toughness transition.
3.3
COMPARISON WITH DATA AND PARAMETER COMBINATIONS
3.3.1
Changing The Crack Path
In sections 3.2.1 and 3.2.2, sc and g +wp were changed independently. In [4],
however, sc and g +wp were observed to be related (as would be expected from
Equation 2). When the A533B steel was aged at 450°C to encourage the grain
boundary segregation of P, the fracture mode around the initiation site changed from
partly transgranular-partly intergranular to wholly intergranular as a result of grain
boundary embrittlement. At the same time, the proportion of intergranular failure on
the fracture surface as a whole also increased, though more slowly. This change in
the nature of the crack path was reflected in changes in both sc and g +wp. The
relation between sc and g +wp observed in [4 above] is shown in Figure 23.
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Figure 23. Relation between critical stress measured at crack initiation sites in
A533B after different ageing times at 450°C, and values of g +wp derived from
shape of calculated transition curves [4]. The straight lines have a slope 0.5
(cf. Equation 2).
In the light of Equation 2, the relation between sc and g +wp is likely to be dependent
on the details of the particle size distribution. Since the particle size distribution
measured in [4] is being used in this work, it is reasonable to use the relation
between sc and g +wp observed there. Thus for a decrease in sc from 2500MPa to
2000MPa, g +wp decreases from 6Jm-2 to 3.84Jm-2. The effect of the combined
change is shown in Figure 24.
As can be seen, the effect of shifting the crack to a lower energy path (such as might
be achieved by segregation) is to push the transition to higher temperatures, while
decreasing the slope of K:T at toughnesses around 100MPaÖm, more markedly than
when sc alone is changed. In this context, it is worth noting that the material in [4]
with the shallow transition curve, exhibited intergranular failure.
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400
350
K (MPaÖm)
300
250
Sc=2500,E=6,
Baseline
200
150
Sc=2000, E=3.84
100
50
0
-200
-150
-100
-50
0
50
100
Temperature (°C)
Figure 24. Effect on the DBT of changing the crack path (E and scrit).
As a further comparison, Mastercurves were calculated for the material used in [4],
(using the multi-temperature fitting method [6], and censoring the data which were
invalid according to Equation 20). Figure 25 shows the data for the unaged material.
The Mastercurve is an excellent descriptor of these data.
450
Unaged
400
350
K (MPaÖm)
300
250
Data
Mastercurve
200
150
100
50
0
-200
-150
-100
-50
0
Temperature (°C)
Figure 25. Comparison between fracture toughness data from unaged A533B
and the Mastercurve
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Figure 26 shows the data for the two aged conditions, in which intergranular failure
was increasingly important. For these materials, the Mastercurve overestimates the
fracture toughness at most temperatures – it has too great a curvature.
450
400
Aged 1000h
350
K (MPaÖm)
300
Measured K
20%
50%
80%
Mastercurve
250
200
150
100
50
0
-200
-150
-100
-50
0
Temperature (°C)
350
300
Aged 5000h
K (MPaÖm)
250
Measured K
20%
50%
80%
Mastercurve
200
150
100
50
0
-200
-150
-100
-50
0
50
100
Temperature (°C)
Figure 26. Comparison between fracture toughness data from aged
(segregated) A533B, OH trendlines (20, 50 and 80% probability) and
Mastercurve (50%).
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3.3.2
Issue 2
The Effect Of Heat Treatment
A heat treatment can have a variety of effects on steel microstructure. Section 3.3.1
illustrated the effects of an ageing treatment which left the particle size distribution
and flow properties unchanged, but caused the embrittling grain boundary
segregation of P [4]. More generally, steel heat treatments affect the particle size
distribution. In [35], the effects of a range of quench and tempering treatments on
the microstructure and mechanical properties of a clean, though commerciallyproduced A508 forging were examined. The heat treatments are shown in Table 2.
They bracket those likely to be seen by an rpv forging.
Initial
normalisation
Austenitisation
Material
40-610
40-675
6-610
6-675
1h 1050°C, furnace cool
880°C 10h
Quench rate
40°C to 500°C, then 25°C/min to 300°C
“
6°C/min to 520°C, then 2°C/min to 300°C
“
Temper
5h 610°C
20h 675°C
5h 610°C
20h 675°C
Table 2. Range of tempering treatments used in [35].
The quench rate was found to affect chiefly the grain size, and the tempering
treatment affected chiefly the particle size distribution. Both the grain size and the
particle size distribution affected the flow properties, and all together affected the
transition properties. Clearly it is possible to separate out the effects of different
material parameters on the DBT mathematically, but extremely difficult to do so
metallurgically.
Figure 27 shows the particle size distributions measured in [35] (“the JSW material”),
and compares them with the baseline OH distribution used in the DBT calculations.
The details of the trend lines and other material parameters are given in Appendix 1.
For each material, the volume fraction of particles predicted from the distribution
functions illustrated in Figure 27 was compared with the iron carbide volume fraction
estimated from the carbon content [35]. The authors found that the agreement was
good in all four cases (i.e. within 30%) considering the approximate nature of the
calculations. From the point of view of the current study, these comparisons suggest
that the main difference between the tempering treatments was in the degree of
coarsening not in the extent of total precipitation. As a result, (c.f. section 3.2.4) the
change in particle size distribution may not, in itself, have a significant effect on the
transition toughnesses.
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Fraction Of Particles Encountered
0.7
0.6
0.5
6-675
6-610
40-675
40-610
OH A533B
0.4
0.3
0.2
0.1
0
0.00E+00
2.00E-07
4.00E-07
6.00E-07
8.00E-07
1.00E-06
Carbide Size (m)
Figure 27. Effect of heat treatments on JSW particle sizes, and comparison
with OH distribution.
The effect of the heat treatments on the flow properties is also very interesting.
Figure 28 and Figure 29 compare the best-fit lines through the yield stresses and
work hardening exponents after the different heat treatments, as a function of
temperature. (The individual points are to help distinguish between the different
material conditions, they do not show particular measurements. Comparisons
between trend lines and data are given in Appendix 1.) The highest yield stresses
are associated with the lowest work hardening exponents, as described in section
3.2.3. This suggests that the effect of the different heat treatments on the flow
properties will lead to shifts the transition curves without large changes in curvature.
The actual transition toughness data for these material conditions are given in Figure
30. The trend lines through the data are those produced by the authors, and are of
the form:
ln (Kj-35MPaÖm)=5.58-0.761QR+1.76TC+0.0252T
Equation 25
where T=temperature
QR = -0.5 for the 6°C/min quench and +0.5 for the 40°C/min quench
TC = -0.5 if the temper was at 675°C and +0.5 if it was at 610°C
This converts to
KJ = 35+exp[0.0252(T-To)] where To=-(5.58 - 0.761QR +1.76TC)/0.0252
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Equation 26
i.e. although the constants in the equation differ from those in the Mastercurve, the
DBTs can still be described by an exponential form, and the effect of the heat
treatments can be incorporated into a simple shift in To – the position of the curve.
If the Mastercurve itself is fitted to the data, Figure 31 is produced. Overall, the
Mastercurves fit the data reasonably well. In detail, the correspondence with the 40675 data is very good; for the 6-675, the Mastercurve fits at –120°C, but curves
more steeply than the data with increasing temperature, lying higher than the data
would suggest between –80°C and 0°C, though the fit improves again at higher
temperatures; a similar effect is found for the 6-610 (Mastercurve overestimating
between –80°C and –40°C); the Mastercurve may overestimate all of the 40-610
data; the curves are a little higher than the data for the 610°C temper, but are still
reasonable approximations.
Table 3 compares the values of T0 and To from the two calculations. The factors do
not scale precisely with each other.
Material
40-610
40-675
6-610
6-675
Mastercurve T0
-131
-32
-99
-10
Ref. 35 To
-241
-171
-271
-201
Table 3. Comparison of curve positions for JSW material according to
Mastercurve and trend line used in [35].
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1000
0.2% Offset Yield Stress (MPa)
900
800
700
600
40-610
40-675
6-610
6-675
500
400
300
200
100
0
-250
-200
-150
-100
-50
0
50
100
Temperature (°C)
Figure 28. Effect of heat treatment on the JSW 0.2% offset yield stress.
0.18
Work-Hardening Exponent, n
0.16
0.14
0.12
40-610
40-675
6-610
6-675
0.1
0.08
0.06
0.04
0.02
0
-250
-200
-150
-100
-50
0
50
100
Temperature (°C)
Figure 29. Effect of heat treatment on the work hardening exponent in JSW
material.
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Figure 30. Effect of heat treatment on the fracture toughness of JSW materials
[35]
800
Fracture Toughness (MPaÖ m)
700
600
6-675 data
6-675MC
500
6-610 data
400
6-610MC
300
40-675 data
40-675MC
200
40-610 data
100
0
-150
40-610MC
-100
-50
0
50
100
Temperature (°C)
Figure 31. Mastercurve (MC) fits to JSW fracture toughness data from [35].
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Fitting the OH model to these data required several stages. First, the precipitate
distributions in Figure 27, and the flow data from Figure 28 and Figure 29 were
incorporated into the spreadsheet for calculating K at a given T. The variation of the
Young’s modulus with temperature was also taken from [35]. It was assumed that
the effective surface energy would be the same for all heat treatment conditions
(7Jm-2). The only parameters that could then be changed were the fraction of
particles taking part in fracture (F) and the critical stress. Slight variations in F were
considered acceptable as the uncertainty in the given carbide volume fractions could
have been up to 30%, and F appears in the calculations merely as a multiplier of Na
(the number of particles per unit area). The critical stress values required to fit a
trend line through the data were then found to vary significantly with heat treatment.
(See Appendix 1 for more details.) The results of the calculations are shown in
Figure 32. Beyond the upper end of each trend line, the calculated fracture
toughness becomes infinite.
The fit to the data is generally at least as good as with the Mastercurve, though 6-675
was hard to match.
800
J S W 4 0 -6 1 0
700
600
K (MPaÖm)
500
D a ta
C a lc .
400
300
200
100
0
-1 6 0
-1 4 0
-1 2 0
-1 0 0
-8 0
-6 0
-4 0
-2 0
0
T e m p e ra tu re (° C )
Figure 32. Comparisons between individual JSW datasets and best fits from
OH calculations (see also over page).
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700
600
JSW 40-675
K (MPaÖm)
500
400
Data
Calc.
300
200
100
0
-140
-90
-40
10
Temperature (°C)
800
700
JSW 6-610
K (MPaÖm)
600
500
Data
Calc.
400
300
200
100
0
-140
-120
-100
-80
-60
-40
-20
0
Temperature (°C)
700
600
JSW 6-675
K (MPaÖm)
500
400
Data
Calc.
300
200
100
0
-125
-75
-25
25
75
125
Temperature (°C)
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The requirement to change the critical stress when the particle size distribution
changed is not surprising in the light of the Griffith relation (Equation 2). The relation
between the critical stress and the particle sizes are given in Figure 33. Both the
mean particle size and the 99th percentile carbide size are plotted. The latter is more
representative of the carbides actually present at the fracture initiation sites. There is
significant variability, but the value of the critical stress does tend to increase with
decreasing particle size. (The relation does not appear to be described very closely
by Equation 2, as there is clearly an offset on the critical stress axis. Such offsets
have been observed in experimental data before e.g. [15].)
2500
Critical Stress (MPa)
2000
1500
m ean r
99% r
1000
500
0
0
0 .0 5
0 .1
0 .1 5
0 .2
0 .2 5
0 .3
1 /Ö (P a r tic le r a d iu s , n m )
Figure 33. Relation between the fitted critical stress, and the geometric mean
of the particle size distribution, or the 99th percentile carbide size in JSW
material.
3.3.2.1 Summary Of Heat Treatment Effects
The particle size distribution affects the slope of the DBT, with larger particles
(produced either by a large mean particle size, or a broad size distribution) making
the transition shallower at lower toughnesses. (If only toughnesses less than
K~150MPaÖm are considered, this could appear as a shift to a higher transition
temperature.) The number density of crack-initiating particles also affects the
toughness in the transition region but, within the OH formulation, neither aspect
affects the temperature at which cleavage becomes impossible – the temperature at
which the fracture toughness becomes infinite (T∞). Consideration of the JSW data
shows , however, that changes in the particle size distribution produce changes in sc,
and sc has the strongest effect on the transition temperature. Thus heat treatments
which increase the sizes of the crack-nucleating particles and decrease sc, both shift
the DBT to higher temperatures and decrease its slope.
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The JSW data also show that it is difficult to change a particle size distribution by
heat treating a rpv steel without simultaneously changing the steel’s flow properties.
A heat treatment which increased the particle size but decreased the number density
would probably decrease the yield stress. Decreasing the yield stress shifts the DBT
to lower temperatures, and increases its slope. Thus particle-coarsening treatments
combine factors causing shifts to higher temperatures and decreases in slope ( r
increases, N decreases, scrit decreases, n increases) with factors causing the
opposite effect (sy decreases). Since the shifts are more dramatic than the changes
in curvature, the sum of all these effects can be a shift with a negligible change in
slope.
3.3.3
Effects Of Radiation Damage
In general, irradiating an rpv steel increases its yield stress while decreasing its rate
of work hardening. For steels containing high levels of P, radiation-induced
segregation to grain boundaries can cause a shift from transgranular to intergranular
failure after an incubation dose. Considering each of these aspects in turn:
The damage put in by radiation at power reactor temperatures (~200-350°C)
will be stable at room temperature and below. Thus no temperaturedependence of the yield stress will occur over the range of the DBT due to
defect annealing. Thermal activation of dislocations to overcome radiation
damage has been observed between 0°C and 200°C, but most studies find
that yield stress increments due to neutron irradiation do not diminish
significantly until above about 150°C. The degree of thermal activation
required to retain the transition curve shape by increasing the temperaturedependence of the yield stress thus does not appear physically realistic.
Increases in the yield stress due to radiation damage would therefore cause
the transition curve to become shallower as it shifted to higher temperatures.
The defects introduced by irradiation (matrix damage and copper-enriched
solute clusters) are cut by dislocations, which causes a reduction in the work
hardening exponent. This aspect of radiation damage would therefore cause
the transition curve to become steeper as it shifted to lower temperatures.
As described in Figure 20, a suitable combination of decreasing n and
increasing sy can produce a shift in the DBT curve without an accompanying
change of slope. This may well have occurred in [1].
The introduction of intergranular failure will shift the transition to higher
temperatures and decrease its slope.
(As well as shifting the DBT, irradiation decreases the upper shelf toughness. This
affects the upper transition rather than the region of brittle failure described by these
calculations.)
An attempt was made to compare the results of OH calculations with data from an
irradiated material. Unfortunately the calculations require knowledge of many
parameters which were not reported. Appendix 1 describes in detail the many
approximations and estimates of suitable data which had to be made before K:T
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curves could be calculated. A good fit was thereby made to the fracture toughness
data from unirradiated A533B plate JRQ [37], as shown in Figure 34. The flow
parameters were then changed according to the data reported in [37] for JRQ after
irradiation to 14mdpa at 290°C. All the other parameters (known or fitted) were kept
constant, in order to calculate the trend curve for the irradiated material. This is
shown in Figure 35, where it is compared with the reported fracture toughness data
for the irradiated material.
450
400
350
K (MPaÖm)
300
250
Data
Trend
200
150
100
50
0
-200
-150
-100
-50
0
50
Temperature (°C)
Figure 34. Fracture Toughness data for unirradiated JRQ plate, and trend line
calculated to fit to data.
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450
400
350
K (MPaÖm)
300
250
Data
Trend
200
150
100
50
0
-150
-100
-50
0
50
100
150
Temperature (°C)
Figure 35. Fracture toughness data for irradiated JRQ plate, and trend line
calculated using parameters derived from unirradiated material.
The calculations predict a slight reduction in slope around 100MPaÖm. This is in
agreement with the data, though the reduction in slope might not have been
observed within the scatter had a constant curve shape been expected. The
predicted shift at 100MPaÖm is about 85°C, which is also in agreement with the data.
At toughnesses above ~150-200MPaÖm, the calculated curve underestimates the
irradiated material toughness, and the upswing in toughness is predicted to occur at
112°C. This is a little high. A value of 80-100°C would appear to be more
appropriate. Nonetheless, given the simplistic nature of the calculations, (most
significantly that they do not incorporate any effects of ductility) and the many
approximations which had to be made (reported in Appendix 1), the fit is quite
reasonable.
3.3.4
Comparisons Between Plates, Forgings And Welds
The JSW forging described in section 3.3.2 was very clean. All of the fractureinitiating particles were M23C6 iron carbides. In less clean, older forgings and plates
[e.g. 36] other carbides (e.g. TiC) or inclusions can also act as initiators. These
particles tend to be larger than the M23C6 particles, if less numerous. Because of
their method of manufacture, inclusions in older forgings can have larger radii in the
weakest material plane than plates. Modern, cored forgings, however, have the
regions of highest inclusion content removed, so may even be less affected by
inclusion content than plates. In welds, inclusions are far more important crack
initiators than iron carbides.
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The difference in crack nucleators has a significant effect on the fracture toughness
properties. The coarser inclusions produce a lower sc and, thus, a higher transition
temperature for given flow properties. In addition, the inclusion distribution will not be
affected by tempering treatments. According to OH, therefore, tempering and stress
relief treatments which, by a convolution of changes in flow properties and relevant
particle distributions, produce a shift in the DBT without a change in slope for plate or
modern forgings, will result in both a shift and a change in slope in older forgings and
welds.
The difference in crack nucleators will also cause the radiation response of older
forgings or welds to differ from that of modern forgings and plates. Looking at the
effects of the particle size distribution parameters alone, Figure 36 shows three
curves with fairly close K:T produced by different distributions: The baseline
distribution ( r =26nm, Na=1010m-2, a=4); a distribution with increased r (=52nm) and
decreased Na (=2.5x109m-2); and a distribution with reduced a(=3) and Na, but
constant r . Increasing the yield stress by 50MPa produces similar shifts in the
baseline and 2 r “materials” at 150-250MPaÖm, but a smaller shift in the a=3
“material”.
These are not very marked differences in the DBT shift caused by a given degree of
hardening. The changes in sc produced by changing the particle size distribution are
more significant. Figure 37 compares the effect of increasing the yield stress by
50MPa at all temperatures, in materials with different critical stresses. With the lower
critical stress, the slopes of the transition curves are lower at lower toughness, and
the difference in T∞ is reduced. This causes the shifts measured at low toughnesses
(100 or 150MPaÖm in this set of calculations) to be smaller in the higher scrit material;
at intermediate toughness (200MPaÖm) the shifts are similar; at high toughnesses
(>250MPaÖm) the shifts are smaller in the lower scrit material.
Combining these effects suggests that, for equal initial transition temperatures, the
material with the coarser initiator distribution should exhibit a smaller DT∞/Ds, at least
at fracture toughnesses below ~150MPaÖm.
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300
rgm=28nm,
N=10^10, a=4,
Baseline
250
Baseline+50MPa
K (MPaÖm)
200
rgm=52nm,
N=2.5x10^9, a=4
150
Coarser
(52nm)+50MPa
100
rgm=26,
N=2.5x10^9, a=3
50
0
-180
Issue 2
Coarser(a=3)
+50MPa
-130
-80
-30
20
Temperature (°C)
Figure 36. Effect of changing particle distribution parameters on the shift in
the transition temperature caused by a yield stress increment of 50MPa at each
temperature.
400
350
K (MPaÖm)
300
250
Sc=2500,
Baseline
200
Sc=2000
150
Sc=2500,
Sy+50MPa
100
Sc=2000,
Sy+50MPa
50
0
-180
-130
-80
-30
20
70
120
Temperature (°C)
Figure 37. Effect on the DBT of increasing the yield stress by 50MPa at each
temperature in materials with different critical stresses.
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3.3.5
Issue 2
Warm Prestressing
The results of this study may also be used to clarify some of the benefits of warm
prestressing. The most significant microstructural effect of warm prestressing is the
reduction in the size of particles initiating failure. The largest particles are
encouraged to form blunted microcracks, which play no further part in cleavage. On
the basis of Figure 21, this is equivalent to increasing a. Figure 22 shows that
increasing a increases the slope of the DBT.
Since the particles which cause failure are at the very upper end of the size
distribution, warm prestressing may not affect the mean value of r or Na very much.
Any changes which do occur will be decreases. Figure 22 shows that a decrease in r
would reduce the slope of the DBT; a decrease in Na would increase it slightly; both
effects will be small in comparison with that of increasing a.
If the effective particle size is reduced, then it is likely that sc will increase. This both
reduces the transition temperature and increases the slope of the transition.
Overall, then, the change in initiating particle distribution due to warm prestressing
should decrease the transition temperature, and increase the slope of the mean K:T
curve in the transition region.
This study has not considered the effects of warm prestressing on the stress
distributions in a cracked sample.
3.4
OH ASSESSMENT OF MASTERCURVE APPLICABILITY
These sensitivity studies show that the shape of the fracture toughness transition is
sensitive to all of the parameters studied, as summarised in Table 1. On this basis, it
would be difficult to understand how the Mastercurve has such wide applicability.
Comparison with experimental data shows, however, that these parameters do not
change independently in practice. The three parameters sc, g+wp and r at least have
a relation of the form:
sc µ
(g + w p )
r
so that changes in r and g+wp automatically produce changes in sc. Other
parameters are less obviously connected, but tend to be changed simultaneously by
metallurgical processes. Thus irradiation increases sy but may decrease n. These
changes have opposing effects on the shape of the DBT, so suitable combinations of
Dsy and Dn can produce a shift in the DBT with negligible effect on its shape (e.g. 5075MPa increases in sy, - ~10% original value - combined with 25%-35% decreases
in n produced shifts with negligible shape change when the particle distribution from
[4] was used.). Changes in yield stress with irradiation are commonly reported;
changes in n less so. Nonetheless it is likely that the raw data exist from which n
may be determined. Assessment of Dsy:Dn combinations would then show the
doses, dose rate etc. to which the DBT shape could be expected to remain
unchanged.
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Thermomechanical treatments can also produce an increase in yield stress coupled
with a decrease in work hardening. In deciding whether a particular treatment will
leave the DBT shape unchanged, however, additional attention must be paid to the
fracture initiators. A given heat treatment and Dsy:Dn, may leave the shape
unchanged in a material in which the fracture initiators are also left unchanged (e.g. a
weld). The same treatment may decrease the slope in a material in which the
number of fracture initiators is simultaneously increased, or the population coarsened
(e.g. iron carbides in a clean plate). As illustration, the yield stress increased by up
to ~30% (in 500MPa) while n decreased by ~30% (in 0.16) in the JSW materials.
These are higher Dsy:Dn ratios than those calculated to leave the shape unchanged
in the absence of particle changes. In association with changes in the particle size
distribution, however, they produced transition temperature shifts without evident
changes in slope. When the change in particle size dominates over the changes in
flow properties, as in warm prestressing, the curve shape can again be changed.
Only a change in fracture path has an unequivocal effect on the slope of the
transition. A change from transgranular to intergranular failure, or an increase in
embrittling segregation once failure has become intergranular, will decrease the
slope of the DBT. The Mastercurve will then not be applicable.
These conclusions are summarised in Table 4:
Metallurgical Process
Irradiation hardening
Plastic strain
Tempering
Segregation and intergranular
failure
Warm prestressing*
Mastercurve
Applicable ?
Possible
Possible
Possible
No
No
Explanation
sy increases while n decreases
sy increases while n decreases
r increases, N decreases, sc
decreases, n increases, while sy
decreases
scrit and g+wp decrease
a and sc increase, r and N
decrease slightly
Table 4. Applicability of the Mastercurve description of the fracture toughness
transition.
*
Initial assessment based on initiating particle size effects only.
Using the Mastercurve to describe the transition region when it is not appropriate can
be conservative or non-conservative. After warm prestressing, the transition is likely
to become steeper so (assuming it to be applicable to non-warm-prestressed
material) the Mastercurve will underpredict the fracture toughness of warmprestressed material. After a change from transgranular to intergranular failure, or an
increase in segregation, however, the transition curve becomes flatter. Using the
Mastercurve would then be non-conservative, as it would overpredict the true fracture
toughnesses (cf. Figure 26).
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3.5
Issue 2
ASSESSMENT OF THE OH MODEL
The OH description of the DBT is very similar to the models described in section 2.1.
It uses the same interpretations of the fracture process, and most of the same
formulations, and is thus equally soundly based. It differs from the rest of the family
of models in the region over which the probability of failure is calculated.
The distance used by OH (Xc), and illustrated in Figure 12, is only very slightly
affected by temperature. The choice of this distance is based on experimental
observations of the location of crack initiation sites, which were found at the position
of constant s11 (=sc) regardless of load and temperature. There is, as yet, no full
theoretical explanation of Xc/ sc. The other models calculate the probability of failure
within the plastic zone, which is theoretically plausible, as at least local plasticity is
required to nucleate microcracks [8]. The size of the plastic zone increases markedly
with increasing temperature, however, and increasing the calculation distance /
volume increases the calculated probability of failure. As the temperature increases,
the models in section 2.1 convolute an increasing distance with a decreasing stress
level producing a higher probability of failure at low K than OH. It may be this small
difference between OH and the other models which permits OH to predict a strong
upswing in the K required to cause cleavage, without needing to introduce an
additional temperature-dependence in wp, su, or any other parameter. Since there is
no independent experimental evidence to show that wp or su should change with
temperature, OH appears better supported experimentally than the other models
described in section 2.1.
One obvious drawback of the OH model is that it is 2-dimensional. Only the material
in the crack plane (Y=0) is considered. The plastic zone ahead of a crack tip is,
however, 3-dimensional, with elevated stresses being found out of this plane. If the
particles away from the crack plane can break, then the probability of finding a
suitable particle within a given stressed dX-Y area increment should be considered,
rather than that within a given dX distance increment. As a result of this 2dimensionality, the current OH model predicts a B0.5 effect of specimen size on
fracture toughness at a given temperature rather than the WST B0.25 relation. Apart
from this, allowing for the third dimension might affect the absolute values of K(T) or
T∞, but it should not affect the qualitative manner in which changes in parameters
such as sc, sY etc affect the DBT.
An aspect of the OH model which requires justification is the role of F, the fraction of
particles taking part in fracture. This factor incorporates several features. It may
reflect the orientation of the particles with respect to the stress axes: Not all particles
will have a suitably-oriented cleavage plane to fail under the applied loading. F may
also reflect the orientation of the particles with respect to the ferrite matrix: If the
cleavage plane of the particle is parallel to that of the surrounding matrix, then
passage of the microcrack from the particle to the matrix will be easier than if the two
planes have a large angular separation. These features will not change much
throughout the stressed region ahead of a crack tip, and it is justifiable to use a
constant value of F to describe them. Other features may, however, change with
position in the plastic zone. If F reflects the chance of a particle cracking or
microcrack blunting, then these processes need to be considered within the model.
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For example, the energy balance associated with particle cracking may be described
as
2 ì
4(1 - n 2) 2 ü
r0 ípE¢e 2p + 2pe ps 11 +
s 11 ý > 2pr02g
3 î
pE
þ
[38]
Equation 27
where the left hand side represents the strain energy of the carbide, so E’ and g refer
to the carbide.
Clearly, this has a higher value close to the precrack tip than further away, though in
the region beyond the peak in the stress field (where OH found cleavage to initiate) it
is a steady function of s11. Since the critical particle size required to satisfy this
energy condition is smaller than that required to satisfy equation 2, not incorporating
it into the analysis does not cause a major problem. Incorporating an equation such
as Equation 6 (using either the tensile stress or, more consistently, the shear stress),
which does not simply consider the overall energy balance, but a particular cracking
mechanism, might have more complicated effects.
If consideration of the particle cracking mechanism does produce a variation in F with
position, it might help interpret the parameters sc and Xc. It is not clear why the
probability of encountering a vulnerable particle should be calculated only out to a
particular stress contour, rather than over the specimen as a whole. It is likely that sc
itself is an average value for a material, derived from a convolution of the stresses
required to cause microcrack extension with the tail of the particle size distribution.
The limit to the calculation is derived from experimental observation, but has not
received a satisfactory explanation.
Like all the Curry and Knott-based models described in this report, OH considers
microcrack extension. Implicitly, it considers extension from (a population of )
stationary microcracks. Dislocation modelling of shielding around stationary
microcracks [39] suggests why the far-field stress required to extend a microcrack of
given size should be constant regardless of temperature (as is sc). It also suggests
that the relation between the microcrack size and the stress to extend it is not in
accordance with the Griffith-Orowan equation, Equation 2. As found experimentally,
(cf. Figure 33) it predicts a stress offset in addition to the inverse size relation. The
dislocation model does not, however, incorporate the effect of the dislocations in
blunting the microcracks. McMahon and Cohen [8] found that if microcracks did not
extend into the matrix immediately upon formation, then they blunted under
increased loading, and this blunting prevented microcracks acting as cleavage
initiators. In this case, the F parameter should incorporate the velocity with which a
crack in a particle can travel, which may be a function of both the local stress field
and the particle size, and the likelihood of blunting under the locally-prevailing stress
field.
It is worth mentioning that the WST model also uses an F-parameter to arrive at
reasonable predictions of the fracture toughness. The Beremin model does not use
an F-parameter explicitly, but incorporates a flaw distribution without any particular
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microstructural reference. The WST or Beremin models thus exhibit no advantage
over OH in this area.
Comparisons between the various models described in this report are summarised in
Table 5.
Despite the various flaws described here, the OH description of the DBT is basically
well-founded, and supported in its assumptions by experiment. It is sufficient to the
current task of assessing the sensitivity of the DBT shape to various parameters.
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Table 5. Comparison between different RKR-type models of cleavage fracture.
Model
Critical step
Failure
criterion
Source of
microcrack
WST
Propagation
of
microcrack
Probability of
locating a
microcrack of
particular size
at such a
position in the
stress field that
it will extend
under the
locallyprevailing
stress
according to a
Griffith-type
criterion
Identified as
cracked
carbide,
inclusion
etc.
Not
important
Beremin
OH
Mastercurve
interpretation
s
Identified
Not
important
Region over which
failure probability
calculated
Plastic zone (3D)
Factors causing rapid
increase of K in
transition region
Temperaturedependence of flow
parameters and wp
Plastic zone (3D)
Temperaturedependence of flow
parameters and su.
Flow parameters
From precrack tip
to distance Xc
(beyond peak in
stress field) at
which critical
stress achieved (2D)
Unspecified
Plastic zone (3D)
(justification of
scatter)
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Applicability
Small-scale
yielding
Depends on
censoring
criterion used,
but generally
SSY
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4
Issue 2
Suggestions For Further Work
This report has described the sensitivity of the DBT shape to various material
parameters, within the framework of a particular description of the DBT. It opens up
several areas for further investigation.
1.
Refinement of the OH model.
As outlined in section 3.5, the OH model as it stands is fairly simplistic. It could be
refined by incorporating the third dimension and, more particularly, by considering F
in greater detail as a nucleation-related parameter. Particle cracking by fibre loading,
microcrack velocity, and crack blunting could all be examined for their effect on F.
Further consideration of the model may also lead to justification for the
experimentally observed parameters sc and Xc.
In [4], the probabilities of failure at the initiation sites were initially found to be low
when ductile crack extension had occurred prior to cleavage initiation. The
calculated probabilities at the higher temperatures were brought into accordance with
those at lower temperatures when it was recognised that the area swept out by the
ductile crack should also be considered as having been sampled for suitably sized
microcracks. On this basis, it should be possible to incorporate an effect of ductile
crack extension prior to cleavage into the OH model. This approach would still not
include the effects of crack blunting, but would nonetheless enhance the applicability
of the model in the upper transition region.
All of these refinements may be at least started by desk studies.
2.
Comparison of current predictions/assumptions with data
Some comparisons with data have been made within this report. Closer
investigations of raw transition toughness data will show the extent to which the
predictions of the OH model are in accordance with experiment. It would, for
example, be informative to check whether irradiations which are known to have
induced intergranular failure do indeed produce clear changes in the DBT shape.
OH predicts that the Mastercurve could be applied to steels undergoing radiation
hardening without intergranular failure if the changes in yield stress and work
hardening are in a suitable relation. Work hardening is generally agreed to reduce
under irradiation, but data are rarely made available. It would be valuable first to
examine tensile traces to check that decreases are commonly found, and then to
compare any yield stress increments and work hardening decrements with the
associated effects on DBT slope/position due to irradiation
No studies have been reported comparing the features initiating cleavage before and
after irradiation. It is key to the use of the OH model in describing irradiation shifts
that the cleavage initiating population be unchanged by irradiation (or at least
changed only in a predictable way). This should be checked by a fractographic study
of unirradiated and irradiated.
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3.
Further investigations using OH
This study has examined the main features contributing to the shape of the mean
fracture toughness versus temperature curve in the transition region. It is possible to
use the OH model to look at K:T for levels of failure probability other than 50%. Thus
the effects of the various parameters on the range or scatter in K at given
temperatures can also be examined. This is equivalent to assessing whether the
Beremin parameter m is a constant. Both the Beremin and Mastercurve analyses
assume the scatter in K to be unaffected by steel type.
This study has made an initial assessment of warm prestressing. This has not
included the important effect of the compressive stress field about the precrack in a
stressed and unloaded specimen. It should be possible to approximate the change
in the crack tip stress field, and incorporate this into a fuller assessment of the
technically important process of warm prestressing.
4.
Implications for other models
When the Beremin model is used to describe fracture toughness, the population of
weak links capable of inducing cleavage is treated as though it is constant from
material to material. If the volume increments are large enough to include many
particles, it is possible that sensitivity to initiating particle distributions is only slight,
but this has not yet been shown experimentally. The present work suggests that the
fracture toughness curve will be affected by details of the particle size distribution,
and this may be reflected in changes in the Beremin parameters, m and su (or the
relation between them). The availability of the JSW data with their quantified particle
distributions, offers an opportunity to verify this assumption within the current
methodology.
5
Summary And Conclusions
1. This report has looked at the family of models based on the Curry and Knott
description of the fracture toughness transition. These models describe cleavage
as the extension of pre-existing microcracks. The probability of failure at a given
temperature and load is calculated as the probability of encountering a microcrack
which will extend under the locally-prevailing stresses. The Beremin and WST
models calculate the total probability over the plastic zone; the OH model
calculates it out to a critical distance which is only weakly temperature-dependent.
In OH, the temperature-dependence of K is determined by the temperaturedependence of the material’s flow properties. To offset the increasing size of the
plastic zone with temperature (and therefore the increasing probability of locating
a microcrack of a particular size), the Beremin and WST models introduce extra
temperature-dependences, in su or wp, respectively, in order to predict the
required temperature-dependence of K.
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2. The Mastercurve hypothesis suggests that the slope of the ductile-to-brittle
transition (i.e. the curvature of the mean K versus T line in the transition region) is
constant for a wide range of ferritic steels after different heat treatments or
radiation exposures. The report has looked at justifications for this hypothesis
based on the existence of a unique property, constant for all steels, and not found
them thoroughly convincing.
3. The OH model has been used to carry out a sensitivity study on the mean slope
of the transition region. Effects of the critical stress, the effective surface energy,
the flow properties (yield stress, work hardening exponent) and the particle size
distribution have been investigated. All of the parameters investigated affect the
slope.
4. The applicability of the Mastercurve is thus found to be due to metallurgical
processes causing changes in more than one parameter at a time.
5. On this basis, it is possible for irradiation hardening, plastic deformation and
tempering treatments to cause shifts in the transition without greatly affecting its
slope. If the Mastercurve applies before these processes, it can apply after them.
6. A change from transgranular to intergranular failure, or an increase in embrittling
segregation once failure has become intergranular, will decrease the slope of the
DBT. The Mastercurve will then not be applicable. If used, it will be nonconservative.
7. A preliminary assessment of warm prestressing suggests that it, too, will produce
changes in the slope of the DBT, and the Mastercurve analysis will not be strictly
appropriate, but could be conservative in its toughness predictions.
8. The differences between welds, plates and forgings in their response to irradiation
hardening or heat treatment have been considered in the light of the sensitivity
study. The important feature is the nature of the fracture-initiating flaws. In
plates, the distributions of these flaws may be affected by heat treatment. In
welds, they are not. (Forgings represent an intermediate case.) Plates will also
have a finer distribution of initiators than welds. This affects the material
response to changes in flow properties.
9. A series of suggestions has been made for further work to improve the OH model,
to examine additional data to check the results of the current calculations, to
examine warm prestressing, and to apply the insights gained from this study to
other models.
Acknowledgements
I would like to thank Profs. P.B.Hirsch and J.F.Knott, and Drs. R.Boothby,
S.G.Roberts and A.S.Sherry for several extremely helpful and illuminating
discussions.
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Appendix
Details Of Calculations For JSW
And JRQ
JSW Materials
The particle size distributions were taken from [35], and an expression of
the form
Probability of encountering a particle of size, r = p( r )
é
ù
a
-c ú
c ( a -1) æ r ö
ê
=
ç ÷ exp
ê r aú
(a - 1)! è r ø
êë r úû
fitted to the measured data.
( )
The probability of encountering a particle of size ≥ri was then calculated,
and an expression of the form:
é
ù
ê
b ú
ú
P(r ³ ri ) = 1 - exp ê ê æ ri ö d ú
ê çè r ÷ø ú
ë
û
fitted to the data.
Without the precise values of Na being available, an approximate value
(based on the data available from [36]) was used for all the heat treatment
conditions. This was considered acceptable, as Na appears only in
conjunction with F, which could be fitted to compensate at a later stage in
the calculations.
The values of the fitting parameters are given in Table A1.
Material
6-610
6-675
40-610
40-675
a
2.3
3.9
2.4
3.8
c
0.5
0.73
0.4
0.8
r (nm)
17.1
120
25.7
92.2
b
4.3
4.02
5.84
4.36
d
3.2
0.96
1.8
1.42
Na (m-2)
1012
1012
1012
1012
F
0.23
4.5x10-4
1
10-3
Table A1. Particle distribution parameters for JSW materials.
The value of g+wp was taken to be 7Jm-2 in all cases.
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Terms of the form
Aexp[BT]+Cexp[DT]+E
where T=temperature
were fitted to the yield stress and work hardening data for each material.
The values for each material are given in Tables A2 and A3.
Material
6-610
6-675
40-610
40-675
A
435
410
500
430
B
-0.0003
-0.00048
-0.0006
-0.00018
C
18
15
19
18
D
-0.0163
-0.017
-0.015
-0.0162
E
-50
-70
10
-40
Table A2. Fitting parameters for the temperature-dependence of the
yield stresses of JSW materials
Material
6-610
6-675
40-610
40-675
A
-0.0056
-0.0049
-0.0042
-0.0042
B
-0.0145
-0.0165
-0.0136
-0.0145
C
-0.026
-0.026
-0.0285
-0.0285
D
0.002
0.002
0.0022
0.0022
E
0.18
0.195
0.147
0.165
Table A3. Fitting parameters for the temperature-dependence of the
work-hardening parameters, n, of JSW materials
Comparisons between data and fit are shown in Figures A1 and A2.
The Young’s modulus was taken from [35] to be the same for all heat
treatments:
E = 204-0.056T(°C)
The particle size distribution and flow properties for each material were put
into a spreadsheet for calculating K.
An appropriate value for F was determined by fitting the calculated values
of K to the observed values at low temperature. The fitted values are
given in Table A1.
An appropriate value of the critical stress was then determined by defining
T∞ from the available data. (On the basis of Figure 33, the estimated
value of scrit=1750MPa may be a little low for JSW 6-610, but the data in
Fig 23 do not reach to high enough temperatures to make this clear.)
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900
800
700
Stress (MPa)
600
500
Data
Fit
400
300
200
100
JSW 6-610 0.2% Offset Yield
-250
-200
-150
0
-100
-50
0
50
100
Temperature (°C)
900
800
700
Stress (MPa)
600
500
Data
Fit
400
300
200
100
JSW 6-675 0.2% Offset Yield
0
-250
-200
-150
-100
-50
0
50
100
Temperature (°C)
900
800
700
Stress (MPa)
600
500
Data
Fit
400
300
200
100
JSW 40-675 0.2% Offset Yield
0
-250
-200
-150
-100
-50
0
50
100
Temperature (°C)
Figure A1 Yield data for JSW material
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1200
1000
Stress (MPa)
800
Data
Fit
600
400
200
JSW 40-610 0.2% Offset Yield
-250
-200
-150
-100
0
-50
0
50
100
Temperature (°C)
Figure A1 contd.
0.140
0.120
0.100
0.080
n
Data
Fit
0.060
0.040
0.020
Work Hardening Coefficients In JSW 40-610
-250
-200
-150
-100
0.000
-50
0
50
100
Temperature (°C)
0.160
0.140
0.120
0.100
Data
Fit
n
0.080
0.060
0.040
0.020
Work Hardening Coefficients In JSW 40-675
0.000
-250
-200
-150
-100
-50
0
50
100
Temperature (°C)
Figure A2. Work hardening data for JSW material
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0.160
0.140
0.120
0.100
Data
Fit
n
0.080
0.060
0.040
0.020
Work Hardening Coefficients In JSW 6-610
-250
-200
-150
-100
0.000
-50
0
50
100
Temperature (°C)
0.180
0.160
0.140
0.120
0.100
n
Data
Fit
0.080
0.060
0.040
0.020
Work Hardening Coefficients In JSW 6-675
-250
-200
-150
-100
-50
0.000
0
50
100
Temperature (°C)
Figure A2. Contd.
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JRQ Material
Data for the JRQ A533B plate were taken from [37]. This contains fracture
toughness, yield and UTS data acquired over significant temperature
ranges for both unirradiated material, and material irradiated at 290°C to
14mdpa. This is a fuller data set than is available for many materials, but
is still not sufficient for use with the OH calculations without
approximations.
There are no particle size distribution data available for JRQ. The heat
treatment was reported to be
Normalised
900°C
Water quenched from 880°C
Tempered
665°C 12h
Stress relieved
620°C 40h
The rapid quench and extended temper+stress relief made it plausible to
use the size distribution data appropriate to JSW 40-610.
The yield stress data are shown in Figure A3
900
800
Yield Stress (MPa)
700
600
Unirradiated
Irradiated 14mdpa
Unirradiated trend
Irradiated trend
500
400
300
200
100
0.2% Offset Yield In JRQ
0
-200
-150
-100
-50
0
50
100
150
Test Temperature (°C)
Figure A3. 0.2% offset yield stress measured in JRQ before and after
irradiation.
In Figure A3, exponential trend lines are fitted through all of the
unirradiated and irradiated data. It is worth noting, however, that
irradiation has caused hardening at temperatures above –100°C, but not
at –176°C. It is probable that the mechanism by which plastic deformation
occurs differs at these two temperatures. At temperatures above –100°C,
dislocation motion causes yield. This is affected by radiation damage. At
–176°C yield is caused by twinning, and this is not affected by radiation
damage. Since the DBT occurs at temperatures at which dislocation
motion is relevant, it seemed inappropriate to use a temperature-
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dependence of the irradiated yield stress that was dominated by the effect
of twinning. When the data at –100°C and above were considered
independently, it was found that the same temperature dependence
provided a good fit to both unirradiated and irradiated data sets. The
hardening observed in JRQ for T>-100°C was temperature-independent.
The temperature coefficients of the yield stress are given in Table A4
Material
Unirradiated
14mdpa
Unirradiated
14mdpa
Stress
Yield
Yield
UTS
UTS
A
500
500
500
500
B
-0.0006
-0.0006
-0.0006
-0.0006
C
19
19
17
14
D
-0.015
-0.015
-0.015
-0.015
E
-30
+40
+102
-180
Table A4. Fitting parameters for the temperature-dependence of the
yield stresses and UTSs of JRQ materials
No work hardening data were reported in [37], so these were
approximated. The work hardening behaviour of JSW 40-610 was
assumed to be appropriate for the unirradiated JRQ. The change in work
hardening due to irradiation was then approximated from the UTS data.
The difference between the UTS and the yield stress is affected by the
work hardening exponent (and the uniform strain, but there was no
information on this). Figure A4 shows the UTS data.
1000
900
800
700
UTS (MPa)
600
Unirr
14mdpa
500
400
300
200
100
0
-200
-150
-100
-50
0
50
100
Temperature (°C)
Figure A4. UTS measured in JRQ before and after irradiation.
The UTS data are more scattered than the yield stress data, but trend
lines were fitted to them as before, and the differences between the UTS
and yield stress trend lines calculated, as shown in Figure A5.
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160.00
140.00
UTS-Sy (MPa)
120.00
100.00
Unirr
Irr
80.00
60.00
40.00
20.00
0.00
-200
-150
-100
-50
0
50
100
Temperature (°C)
Figure A5. Difference between UTS and yield stress trend lines at
different temperatures in JRQ.
The work hardening exponents for the irradiated material were taken to be
those for the unirradiated material scaled according to the ratio of the two
lines in Figure A5.
The Young’s modulus data in [37] were scattered, and showed no clear
trend with either temperature or irradiation. The temperature-dependence
of the Young’s modulus found for the JSW material was therefore used for
JRQ, and the temperature-independent term was found by best fit to all
the JRQ data.
E= 207-0.056T
was used for both the unirradiated and irradiated material.
The flow data and precipitate distribution data for unirradiated JRQ were
put into the spreadsheet for the OH calculations. As for the JSW
materials, values for F and scrit were derived from fitting to the fracture
toughness data. The values derived were:
F = 6.7x10-3
scrit = 1970MPa
The curve for the irradiated material was then calculated leaving all
parameters constant other than those for the yield stress and work
hardening exponents. The results of the calculations are shown in the
main text. Given the crude nature of the model, and the many
uncertainties in the input parameters used for these calculations, the
agreement between the calculated trend line and the irradiated data is
reasonably good.
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