Knott2004-ProbabilisticFracture-PressureVesselSte+

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Probabilistic Aspects Of Brittle Fracture In Pressure-Vessel Steels
J F Knott
The University of Birmingham, U.K.
ABSTRACT: This paper addresses issues relating to the treatment of distributions of tensile strength, local
cleavage fracture stress, and fracture toughness in pressure-vessel steels and weld deposits. Attention is drawn to
the differences in behaviour between steels which have “quasi-homogeneous” microstructures and those which
exhibit spatial heterogeneity. The differences are of greatest significance when statistical analysis is applied to
distributions to derive “lower-bound” values. The findings from model systems are used to re-assess an earlier
analysis of Weibull fracture stress and to comment on the use of a Master Curve methodology in the analysis of
the “Euro” fracture toughness data-set.
1 INTRODUCTION
The scatter associated with the measurement of a mechanical property, such as flow strength or
toughness, is of importance because it forms part of the overall probabilistic failure assessment of a
structure. Fracture toughness distributions may be fitted by a variety of statistical formulations, but, if
these are not related to a physically-based model, there is concern that the tail of a distribution (to
which the overall failure probability is highly sensitive) does not have the appropriate form. The
modelling behind the “local approach” or Weibull methodology [Beremin 1983] is based on a “test
volume” concept. The linear dimension of a “process zone” is proportional to K2 and so the process
zone area is proportional to K4. This is also a part of the Curry/Knott [1979] treatment of cleavage
fracture. The test volume is then taken to be proportional to the length of the crack front (in test-pieces,
the test-piece thickness, B) multiplied by K4. This enables values of Weibull parameters to be
compared for different test volumes. For low probability events, a lower bound needs to be set. This
may be at some level of probability appropriate to a particular safety case (e.g. 10-4 = 0.01%) or
expressed as a number of standard deviations (s.d.) below the mean (3.09 s.d. = 0.1%). Both an
assumed Gaussian distribution or a two-parameter Weibull (with a cut-off value of zero) require that
this lower bound be set independently. A three-parameter Weibull includes a “cut-off” value in its
formulation, but a robust methodology needs to be established to establish its value. [Neville and Knott
1986]. It may be of significance to note that the value chosen for the “cut-off” affects the value of
Weibull modulus, so that the usual interpretation of a Weibull modulus as representing the degree of
scatter in a data-set becomes somewhat ambiguous. The “Master Curve” method [ASTM 2002]
includes an independent cut-off of 20 MPa m0.5. This value is taken as constant for a range of pressurevessel steels (once the reference temperature, To, corresponding to a median fracture toughness of
100 MPa m0.5 has been determined) and is also constant throughout the whole transition range for
which the Master Curve applies (a range of over 150K). Results given later in this paper suggest
however, that, for a wider range of steels, this single value underestimates the lower bound for some
steels/heat-treatments and overestimates it for others. It might also be expected that the “cut-off” value
would vary, if weakly, with temperature, reflecting the decrease of yield strength with increase in
temperature. In an application of Master Curve methodology to the “Euro” data-set, Wallin [2002] used
both the standard value, and also “fitted” values for individual data sets. Interestingly, some of these
“fitted” values were negative. As will be seen below, this is a strong indicator of heterogeneity in
material, but the raw findings highlight the dangers inherent in the simple fitting of a standard
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statistical form to a distribution, without paying due recognition to microstructures and micromechanisms of fracture.
Even for wrought products, common structural steels are unlikely to possess spatially uniform
properties, because chemical segregation in the original casting may persist throughout the working
processes. There may be a spatial distribution of non-metallic inclusions, which affect the critical value
of Ji the J-Integral, or δi, the crack-tip opening displacement (CTOD), at the initiation of ductile
fracture; a spatial distribution of major alloying elements, which, through their effects on hardenability,
influence transformations and the toughnesses of transformed microstructures; and a spatial distribution
of trace impurity elements, which can induce inter-granular embrittlement. Similar factors, on a coarser
scale, affect multi-pass weldments [Todinov et al. 2000]. It may be noted, in terms of flow properties,
that the 0.2% proof strength is likely to show more variability in spatially heterogeneous material than
is the UTS, because the proof strength reflects the resistance to the production of a macroscopic plastic
strain of just 0.2%: this could well be accomplished by plastic deformation in only the softest
microstructure present. The UTS is a property reflecting the total behaviour of all microstructures
present, deforming as a continuum. The point has been clearly illustrated for an as-deposited C-Mn
weld metal [Tweed and Knott, 1987]. Here, it was shown that the softer grain-boundary ferrite strained
by up to ~7% until it had hardened to the same level as that of the acicular ferrite. From then on, both
microstructures deformed equally.
Recent work has attempted to distinguish between the fracture toughness properties of quasihomogeneous and heterogeneous materials. No engineering material is truly homogeneous, but, if a
steel contains a high volume fraction of relatively small carbides, distributed in a smooth “wellbehaved fashion, a large number of potential crack nuclei will be sampled in the process-zone. Whether
the “critical” size is taken as the 95th or 98th percentile, the size of the particle, co, in the “wellbehaved” distribution will not vary by any large amount from sample to sample and the critical fracture
stress, which depends on co-0.5 will be, to all intents and purposes, single-valued. This implies that
values of the local fracture stress, σF, in notched bars fractured at low temperatures should be singlevalued (within the limits of random experimental errors) and it further follows that the size of the
process zone and the value of K at fracture in fracture toughness tests should be the same for whichever
sample in a batch is tested. The following sections discuss the extent to which these concepts are
validated and how working definitions of “quasi-homogeneous” and “heterogeneous” behaviour may
be established.
2 VARIABILITY IN ULTIMATE TENSILE STRENGTH
The aim of this section is to show how use of a basic model can be used to predict the form of a
statistical distribution, which can then be compared with experimental values. The model for UTS uses
Considère’s criterion, which balances the rate of strain-hardening against the loss of cross-sectional
area as a specimen undergoes plastic deformation (as a uniform continuum). If the material’s true stress
(σt) – true strain (εt) curve is represented by the form, σt = Kεtn, the criterion for necking is given by εt =
n, giving the true stress at the UTS as Knn and the UTS, which equates to the nominal stress (σnom) = σt
(A/A0), where A is the area at the UTS and A0 is the original area, as:
UTS = σnom = Knn/(exp n)
(1)
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Equation (1) predicts that the value of the UTS is determined by the parameters K and n in the
relationship between true stress and true strain. These may be taken as constant for uniform,
homogeneous material of constant grain size and heat-treated/cold-worked condition. The prediction of
the model is that the UTS is a single-valued parameter.
There is value in presenting distributions in graphical form. The probability density function,
pdf, is the histogram of the number of occurrences of a particular value as ordinate, plotted versus the
range of values on the abscissa. For a single-valued parameter, the pdf is a delta (spike) function,
located, in this case, at the value of the UTS. The cumulative distribution function, CDF, is the integral
of the pdf, representing the number of occurrences of the range of values up to a point of interest: it is
usually scaled as a fraction or percentage of the total number of values. For a single-valued parameter,
the CDF is a step function, having a value of zero up to the UTS, and unity (or 100%) above the UTS.
In any set of experimental results, there are experimental errors. Here, these are assumed to be small
and randomly distributed about the mean. The central limit theorem [Sokolnikov and Redheffer 1966]
shows that random errors are distributed in a normal manner about the mean. For a single-valued
parameter, the pdf is then a Gaussian distribution (with a standard deviation s.d. reflecting the degree
of variation in random variables) and the CDF is the error function, erf, which is the area under the
Gaussian, but scaled by a factor of 2/π0.5 to provide a value of unity for the total range. For a second
steel/heat-treatment, having a different UTS, the effect is to shift the value of the mean, but not to alter
the s.d. because the random experimental errors have not changed. (Noting that, in the absence of any
such errors, equation (1) predicts both UTS values to be delta functions). The ratios of s.d. to mean
will, however, be different, because the means have shifted. It is useful to represent the CDF on normal
probability paper, which has its ordinate scaled about 50% (or 0.5) so that the erf plots as a straight
line: ±one s.d. corresponds to frequencies of 16% and 84%. Median ranking is employed, with the
median rank of order n, Fn, in a set of total number N, derived from the close approximation [BompasSmith 1973]:
Fn = (n - 0.3) / (N + 0.4)
(2)
Fig. 1 shows the CDFs for UTS values in PWR pressure-vessel steels [Marshall 1982].
Attention is drawn first to the “French” data (69 samples) for A533B Class1, which conform to linear
behaviour, with a mean of 617 MPa and s.d.±19 MPa, approx. 3% of the mean. Similar values
(614±15 MPa) were obtained by the Japanese for A533B Class1. Although the s.d. values may reflect
some genuine material variability from sample to sample, arguably they simply reflect random
experimental test variables (variations in load measurement, measurement of cross-sectional area,
precision of identifying the position of the load maximum etc.). If so, they are predicted, by the Central
Limit Theorem, to be distributed about the mean in a Gaussian manner. For the purposes of this paper,
this is what is assumed, and with this assumption, it becomes clear that the basic prediction of the UTS
model is supported: in homogeneous material, the UTS is a single-valued function. A second set of
“French” data (61 samples), for A508 Class 3, have a similar mean, 612 MPa, but a somewhat larger
s.d. ±29 MPa, 4.8% of the mean. Since the care taken in tensile testing may be assumed to be of similar
quality for both sets of French material, the increase in s.d. is now more likely to reflect a degree of
material variability from sample to sample. The level of variability is, however, unlikely to give rise to
concern in general structural integrity assessments. A -2 s.d lower bound would be 554 MPa, derived
from the mean of 612 MPa (contrasted with 579 MPa, 617 MPa). Since safety factors of 2 are applied
to the yield strength for the initial design of PWR steels, this decrease (and the difference between the
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two batches of steel) is insignificant. Similar changes to the flow strength, affecting the position of an
assessment point on the Lr abscissa on the R6 Failure Analysis Diagram, are, again, unlikely to cause
problems, except in extreme circumstances, where the assessment point (Kr,Lr) is already close to the
near-vertical part of the failure locus. Results for American PWR steels gave s.d. values of about
30 MPa (5% of the means).
99.99
PWR Pressure-Vessel Steels
French A533B1
99
French A5083
(data bars
omitted for
clarity)
90
Mean
612 MPa
s.d.
29 MPa
70
50
Probability %
30
10
s.d.
19
MPa
1
Mean
617 MPa
0.01
500
550
600
650
700
U.T.S. MPa
Figure 1 Cumulative distribution functions for UTS values in PWR steels
Data from Marshall (1982)
In lower strength steels, there are significant changes, particularly of yield strength, but also of
flow strength, with grain size. A large block of material, cooled from high temperature, may contain a
range of grain sizes throughout its cross-section, as a result of the different cooling-rates experienced
by near-surface and central regions. This is recognised in specifications for steel plate, where thicker
plates do not have as high a guaranteed lower-bound yield strength as that stipulated for the same
composition in thinner section. Tensile test samples from surface layers of thick sections will exhibit a
high mean yield strength, representative of homogeneous fine-grained material with a low s.d.
(reflecting random experimental errors). Samples from the centre will exhibit a lower mean,
representative of homogeneous coarse-grained material, again associated with a low s.d. A range of
samples from all positions throughout the section will have an intermediate mean value, but with a
higher s.d. attributable to the fact that not all samples belong to the same “metallurgical” population.
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3 VARIABILITY IN LOCAL FRACTURE STRESS IN NOTCHED BARS
The sequence of events leading to cleavage fracture ahead of a notch in steel at low test temperatures is
that plastic deformation nucleates a micro-crack in a brittle second-phase particle, which then
propagates under the influence of a critical tensile stress, σF. It is found experimentally that the value of
σF is independent of temperature over a range of low temperatures [Knott 2000]. The brittle particle
may be a carbide, an inclusion (such as an oxide or silicate), or titanium carbo-nitride. The values of σF
are determined by measuring the fracture load in a notched bar, and using an elastic/plastic finite
element stress analysis to determine, either the maximum value of local tensile stress ahead of the notch
[Griffiths and Owen 1971] or a characteristic stress, which represents an averaged stress across the
notched cross-section [Beremin 1983]. The treatment of scatter in the latter case is to fit values of the
characteristic stress to a Weibull distribution. Drawing on the results of a number of studies [Curry and
Knott 1979, Tweed and Knott 1987, Knott 2000] it is possible to develop a micro-mechanical model for
cleavage fracture as the Griffith-like propagation of a disc-shaped micro-crack nucleus from a brittle
particle:
σF = {πEγp/ 2(1 – ν2)c0}0.5
(3)
where c0 is the radius of the micro-crack, E is Young’s modulus, ν is Poisson’s Ratio and γp is the local
work of fracture, found experimentally to be of order 9-14 J m-2. [Knott 1994]. The scatter in σF values,
from test-piece to test-piece, is then related to the probability of finding a given size of particle in the
high stress region ahead of a notch, at a given applied load. For the failure loads and notched bar
geometry commonly used, the region ahead of the notch in which the local stress rises from 0.95 σmax
to σmax and then falls to 0.95σmax is of order one root radius (0.25mm). The lateral extent is of similar
dimension. Although the size distribution of brittle particles is not normal: there being many small, and
fewer large, particles; it is argued that, if a “typically large” (say, 95th percentile) particle is
consistently sampled, from test-piece to test-piece, closely similar values of σF will be produced. The
fracture stress varies as the inverse square root of particle size.
The predictions of the model may be tested, using results [Wu and Knott 2004] for a
reproduction weld metal, both in an as-received condition and after an embrittling treatment, which
coarsened the grain size, and caused the material to become susceptible to inter-granular embrittlement,
by promoting the segregation of impurity elements to grain boundaries. The material was further
subjected to 8% cold prestrain. The brittle particles acting as cleavage initiation sites were non-metallic
inclusions, typically some 2 µm in size, representing the 95th percentile. For a similar weld metal, 309
inclusions (following a log-normal distribution) were found [Widgery and Knott 1978] in an area of
14450 m2, i.e. approx. 0.12 mm x 0.12 mm; of order 1200 are then expected in the 0.25 x 0.25 mm2
high stress region ahead of a notch – hence 60 of the 95th percentile size. A “weakest link” argument is
not strictly sustainable, but the figures show that, if a series of micro-fractures is triggered–off in an
“avalanche” once a crack starts to spread from the most susceptible particle, so many particles are
present in the high stress region that all test-pieces are expected to behave in near-identical manner.
The distribution of σF is predicted to be single-valued. With random errors, the CDF, on normal
probability paper, is predicted to be linear, with small s.d., reflecting the random experimental errors.
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Fig. 2 shows the experimental CDF distributions for σF for the two conditions. The distributions
are seen to be linear, with different means: 1555 MPa and 1400 MPa, but with similar values of s.d.
35 MPa: 2.2% and 2.5% of the mean respectively. The prediction is therefore confirmed: the CDFs are
straight lines with s.d. values which are small, noting that they are obtained from a combination of load
and finite-element stress analysis. Wu and Knott estimated experimental and numerical errors to be of
order ±80 MPa (±2.3 s.d.) and all data points for either condition were found to lie within these bounds.
(Note that the s.d. for the UTS distribution in Fig.1 for material assumed to be homogeneous is ~3% of
the mean). The embrittlement procedure has decreased the local fracture stress of the weld metal,
effectively by lowering the work-of-fracture, γp, in Eqn (3). It has not, however, altered the distribution
of crack-initiating particles in any given test-piece or the random experimental errors which have been
identified as the primary cause of scatter in the results. The absolute values of s.d. are identical,
although they vary as a proportion of the mean. The results obtained here for “reproduction” (carefully
fabricated) weld metal are more typical of those for wrought products than of those for welds in
general. These may contain a much wider size-range of inclusions: both as the expected de-oxidation
products (up to ~4 µm in size) and “exogenous” inclusions, which may enter the weld pool from an
unexpected source e.g. from the binder for the coating on a welding rod. An isolated inclusion of this
sort of size 10 µm has been observed [Tweed and Knott 1987]. This would produce a clearly
observable “outlier” to any predicted CDF, as an unexpectedly low value of σF or, perhaps, as a “popin” at low applied load. Such an anomalous point provides an incentive for further investigation,
identification of its cause and subsequent remedial action.
The data in Fig. 2 have been drawn to 1% and 99% limits (approx. ±2.3 s.d.(80 MPa) - the
estimated maximum experimental error) and show a small overlap between the lowest “as-received”
condition and the highest “degraded” condition. This is a slight extrapolation beyond the range of the
experimental results, for which the highest “degraded” value was 1462 MPa (at the 97.5% level) and
the lowest “as-received” value was 1482 MPa (at the 97% level). Consider now the analysis if it were
not initially known which test-pieces were “as-received” and which were “degraded”, such that the
whole set was treated as a single population. The resultant re-plotting of data is shown in Fig. 3. There
are now two branches, reflecting the original distributions at the lowest and highest values, joined by a
central portion, reflecting the “uneasy marriage” of the two separate distributions (forced by the
assumed lack of knowledge). The separation is more clearly seen for the normal CDF than for a
Weibull representation, and provides a “template” for the diagnosis and analysis of distributions.
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99.99
99.99
1462 MPa
99
99
Degraded
90
Cumulative
Probability %
Cumulative
Probability %
50
30
1
70
50
30
35
MPa
35
MPa
mean
1400 MPa
10
extrapolation
of upper branch
1
mean
1555 MPa
0.01
1300
As-received
90
70
10
1482 MPa
1400 MPa
1555 MPa
0.01
1400
1500
Local Fracture Stress σ F MPa
1600
1300
1400
1500
1600
Local Fracture Stress σF MPa
Figure 2 CDF distributions of local cleavage fracture Figure 3 The data from Fig. 2 re-plotted as if for a
stress for as-received and “degraded” weld metal – single distribution
data from Wu and Knott (2004)
The presence of sharp changes in slope (not predicted by the physical model), demands thorough
investigation. Given the results in Fig.3, it would be necessary to undertake fractographic and
metallographic examination, ideally on all specimens, but particularly on those whose fracture stresses
spanned the central region. It would then be found that those which had failed at stresses at and above
1482 MPa would reveal trans-granular failure and relatively fine grain size; those failing at and below
1462 MPa would reveal inter-granular fractures and large grain sizes. These observations then provide
justification for re-analysing the two sets as different populations, with the results shown in Fig. 2.
These data are clear-cut, with overlap occurring only at the 1%/99% level. More overlap could confuse
the details, but, once a change (or changes) in slope has revealed the presence of two (or more)
populations, the degree of confidence in their independence can be assessed by conventional techniques
such as Students t-Test. The detection of slope changes (based on the visual “template” in Fig.3), not
predicted by the underlying model, is, however, a critically important first step in the statistical
analysis. Slope changes may be seen but are less easy to interpret in the Weibull presentation. The
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“ln.ln.vs. ln” scales tend to provide too close a grouping of points and the predictive model is less clear.
To clarify the “change in slope”, consider equal numbers of samples of two steels with different “ideal”
UTS values. From equation (1), each pdf is a delta function, each CDF a step function. The CDF for
the combined set is a double-step function: two vertical “risers” below and above 50% probability and
no values on the horizontal “tread”. The incorporation of random errors transforms the “risers” to two
steeply sloping lines and the “tread” to a shallow slope: see Fig. 3.
The Wu and Knott paper [2004] used the approach to re-assess some of the Beremin [1983] analyses of
fracture stress results In the original paper, results for A508 steel were analysed by means of a twoparameter Weibull. The results for steel Heat A were stated to give a Weibull modulus m=22, although
subsequent regression analysis of the published data gives m=13.3. This is still a high value and the
distribution could therefore be equally well plotted as a normal CDF, since the Weibull and Normal
distributions are virtually identical for m>4 [Bompas-Smith 1973]. This was done by Wu and Knott to
derive mean and s.d. values of 2493 MPa and 221 MPa, 9% of the mean. It is of interest to compare
these results with the values for the weld metals treated as a single, or two separate, distributions (Figs.
3 and 2). The distribution in Fig.3 as a single population of the results from the two different conditions
gives a value of m = 19, corresponding to a mean of 1477 MPa and a “standard deviation” of 89 MPa
(6% of the mean). The value of “standard deviation” for the Beremin results, 221 MPa (9% of the
mean), is significantly larger than this figure (which is, of course, already for the “combined”
distribution). Since there is no reason to suppose that random experimental errors in the Beremin work
were any greater than for the present experiments (approx. 2.4% of the mean for the separate
distributions), it may be conjectured that the Beremin results reflect spatial heterogeneity in the
material. Bowen et al [1986] have measured σF values for a number of different simulated
microstructures in A533B (a steel very similar to A508), obtaining values in the range of 20003500 MPa, which spans those for Beremin’s Heat A. Generally, this range splits into two bands. The
lower values correspond to coarse ferrite/carbide distributions (pearlite, coarse upper bainites); the
higher values to fine distributions (tempered martensites, lower bainites). Wu and Knott carried out a
sensitivity study on 10 or more data points in the “lower branch” of the Beremin distribution to try to
detect sub-sets. Points ranked 4 to 13 gave a mean of 2319 MPa with a standard deviation of 56 MPa
(2.4% of the mean) and a total error range of approximately ±85 MPa. The Weibull modulus for these
10 points was 46, similar to those for the separate distributions in Fig.2. It was inferred that this set of
points could represent an independent subset. If generic to French A508 steel, the presence of spatial
heterogeneity is consistent with the finding that the s.d. for the A508 UTS distribution was higher than
that for the UTS distribution (Fig.1).
4 THE FRACTURE TOUGHNESS OF “QUASI-HOMOGENEOUS” STEEL
The model adopted to treat the fracture toughness of steel follows the RKR model [Ritchie, Knott and
Rice 1973] for cleavage fracture in mild steel. This identified a critical distance, X, needed to reconcile
the dimensions of local fracture stress, σF (stress) and fracture toughness, K1c (stress x length0.5). In a
study of cleavage fracture in spheroidised microstructures [Curry and Knott 1979], X was defined in
terms of a statistically-averaged distance, with the microstructure characterised in terms of the
histogram of number density (per unit volume, or per unit area in 2-D) of carbides as a function of
carbide radius, co. The local value of σF (in notched bars) was calculated for each radius. Reference
was then made to the tensile stress distribution in the process zone ahead of a crack loaded to a given
stress-intensity factor, K, and the stress “available” at a given position r, θ was compared with the
probability that, at this position, there would be a carbide of radius sufficiently large for a newly-
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initiated (disc-shaped) microcrack to propagate at this level of local stress. If this were not the case for
any r, θ values at a given applied K, the value of K was increased until the condition was met
(at K = KIC). A similar approach has been adopted by Lin, Evans and Ritchie [1986, 1986a]. Any
variation in KIC is related to the distribution of carbide sizes. For the spheroidised steels (uniformly
austenitised, quenched and tempered) the distribution is smooth and continuous and there is a high
probability that a “typically coarse” (95th or 98th percentile) carbide will be found consistently in an
equivalent σ (r, θ) position for a given applied K value, so that systematic variability from test-piece to
test-piece should be small. This conclusion is supported by measurement [Neville and Knott 1986] of a
number of fracture toughness values at –115oC in samples of En8 steel, quenched from 810oC and
tempered at 220oC for 1 hour. A normal distribution was obtained, with a mean of 32.4 MPa m0.5 and
s.d. of 1.03 MPa m0.5 i.e. 3.2% of the mean. (All results fell within 6% of the mean). Random
experimental errors were estimated as only ±2% and it was shown that, even within the narrow K1c
distribution, there was sufficient variability in the hardness of different samples (592-609 VDH) to
produce a small systematic variation in the K1c values. Fracture toughness distributions for other finescale steel microstructures are discussed below.
In the Beremin [1983] “local approach”, the (averaged) local fracture stresses are assigned to a
Weibull distribution. Stress distributions around a crack tip at a given K-level are analysed and the
probability that the stress at a given r, θ position in the plastic zone exceeds the “Weibull stress” is
assessed. If the maximum local stress in a cylindrically notched test-piece were used, this would equate
to the input fed into the original RKR model. In more recent work, particularly on weld metals, it is
possible to determine the value of stress at the initiation site in the notched bar or the pre-cracked
fracture toughness specimen. In the “Local Approach” the concept of a “critical distance” is still
necessary (although it is defined differently) and is, in one example, ~0.25mm (in RKR, using σmax, X
= 0.12mm). The Beremin use of the “Weibull stress” does not relate it to any micro--structural feature,
such as the carbide distribution, and the Wu and Knott [2004] re-analysis of the Beremin results,
discussed above, allows the possibility that this lack of correlation could lead to inappropriate analysis,
e.g. using an average Weibull stress derived from a distribution containing results for both fine and
coarse microstructures to treat a crack tip located predominantly in either a fine, or a coarse,
microstructure.
The tensile stress model relates to transgranular cleavage or intergranular fracture, in which
the critical event is the propagation of a brittle micro-crack. In low strength steel, the ductile fracture
process is dominated by the internal necking between, and coalescence of, voids initiated on nonmetallic inclusions. In a microstructure containing ferrite grains, grain-boundary carbides and nonmetallic inclusions, there is a clear separation between the value of fracture toughness for brittle
fracture and that for ductile fracture. In high-strength steels, the transition is less distinct. High strength
is produced by microstructural refinement and a high volume-fraction of closely-spaced particles,
combined with a high density of transformation dislocations. Proof strengths of over 1500 MPa can be
generated in low-alloy steels, quenched and tempered in the range 300-450˚C. The cleavage fracture
process, below the transition temperature, is similar to that for En8 steel [Neville and Knott 1986].
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99.99
300M and Maraging (G150, G125) Steels
99
320oC
Temper
77K o
90
o
x
o
70
o
o Temper
R.T.
o
x
x
x
o
x
x
33 x 41x 51o
o
x
x
21 o
50
Probability
30
320oC
Temper
R.T.
o
x
450oC
G150
R.T.
x
o
o
o
x
x
o
o
x
x
o
x
x
o
o
o
o
o
76o
o
o
o
o
10
2.5
G125
R.T.
o
2.5
o
2.5
1.2
1
0.01
0
1.5
20
40
60
80
100
Fracture Toughness MPam0.5
Figure 4 CDF plots for the distributions of fracture toughness in 300M and
maraging steels – data courtesy Dr J E King and Dr B Wiltshire
Above the transition temperature, ductile fracture initiates by the formation of voids on nonmetallic inclusions, but linkage does not occur by the “internal necking” process. Instead, the
accumulating plastic strain between expanding voids, comprising dislocations looped around the
carbide hardening particles, causes carbide/matrix interfaces to decohere, leading to a “fast-shear”
linkage. The overall strain/toughness associated with this process can be quite low. In the “cleanest”
(lowest inclusion content) high-strength steels, the ductile fracture is dominated by fast shear, and since
this operates on the scale of the tempered carbides, the material is expected to behave in a “quasihomogeneous” manner. It is predicted that once a high-strength steel is sufficiently clean (in terms of
inclusions), no improvement in fracture toughness will be obtained by a further decrease in inclusion
content (increase in spacing) because the fracture process is dominated by carbide/matrix de-cohesion
[Smith, Cook and Rau 1977, Slatcher and Knott 1982]. The effects of these features on fracture
toughness distributions are illustrated in Figs. 4 and 5 which show CDFs, plotted on normal probability
scales, for 300M and G125, G150 maraging steels, used in aerospace applications, and for a QT steel
used in gun barrels. In Fig. 4, the data for 300M [King and Knott 1980] indicate “quasi-homogeneous”
behaviour, at both -196˚C (cleavage fracture) and room-temperature (cleavage and micro-voids). The
values of s.d. are ±2.5, ±1.5 and ±2.5 MPa m0.5. The values for G150 and G125 [Wiltshire and Knott
1980] are ±1.2 and ±2.5 MPa m0.5. For 300M, the room-temperature fracture toughness is greater for
the 450˚C temper than for the 320˚C temper. This is due to “350˚C embrittlement”, arising from a
balance between inter-lath cementite development and matrix softening. The 320˚C temper has a proof
strength of 1900 MPa, compared with 1500 MPa for the 450˚C temper.
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99.99
Air-melted and ESR Gun-barrel Steel
99
o
x
Bend Testpieces
CTS Testpieces
o
x
90
70
50
Probability
30
o
o x
o
10
o
1
o
o x
ox
ox
o
ox
ox
xo
o
oxx
o
x
x
5.5 MPa m
0.5
0.5
Mean = 85.5 MPa m
0.01
70
80
90
Fracture Toughness MPa m
100
0.5
Figure 5 Cumulative distributions of fracture toughness for a gun-barrel steel
Data courtesy Dr S Slatcher
Fig. 5 shows fracture toughness results [Slatcher and Knott 1982] for a gun-barrel steel (0.34C
3.2Ni 0.8-1.15Cr 0.7Mo), initially ignoring small variations in the material’s chemistry, processing
route and heat-treatment. The s.d. is ±5.5 MPa m0.5. The values obtained from SEN Bend and CTS testpieces were agreeably similar. Some test-pieces were cut from air-melted (AM) stock (with 1.15Cr);
others, from electro-slag-refined (ESR) stock (with 0.8Cr). The CDFs for the two conditions were
found effectively to superimpose. This is as predicted, because both AM and ESR material had low
inclusion contents and the fast shear process is dominated by de-cohesion of the carbide/matrix
interfaces. An effect of tempering temperature could be deduced: as for 300M, the 300-350˚C tempers
were less tough than the 400-450˚C tempers (although the yield stress had decreased only slightly, from
~1400 MPa to ~1300 MPa). The s.d. values did not drop to the ± 2-3 MPa m0.5 level, suggesting that
further variables, e.g. the spatial location of a test-piece in a forging would be needed to be explored to
produce more convincing “quasi-homogeneous” behaviour. In general more variability is to be
expected here than in high-quality aerospace alloys and, for service application, the s.d. of ±5 MPa m0.5
may be acceptable. A review of other data-sets for steels of this type [May 1965] suggests that a
pragmatic value for the s.d. of material to be defined as “quasi-homogeneous” is ±5-6 MPa m0.5.
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836
5 THE FRACTURE TOUGHNESS OF “HETEROGENEOUS” STEEL
The behaviour of quasi-homogeneous material contrasts sharply with that for spatially heterogeneous
material [Zhang and Knott 1999, 2000]. Cleavage fracture-toughness values were measured at -80˚C
for A533B pressure-vessel steel, heat-treated to generate three microstructures: 100% (autotempered)
martensite, α′; 100% (coarse) upper bainite, β; 30% β, 70% α′; all with a prior austenite grain size of
250µm. Fig. 6 shows that the CDFs for 100% α′ and 100% β plot as “quasi-homogeneous” functions,
with s.d. values less than 6 MPa m0.5. Interest lies in the behaviour of the 30% β, 70% α′ mixture. Due
to carbon and alloy segregation in the melt and/or during processing, the hardenability of the material is
not spatially uniform and the microstructure consists of a “patchy” mixture of “brittle” bainite (β) and
“tough” martensite (α′). There is also a systematic spatial variation of the proportion of bainite (from
15% to 45%) over a “wavelength” of some 250-500µm. In a set of fracture-toughness tests, the critical
region ahead of the crack tip may, in some specimens, be located in a region with a high amount of
“brittle” bainite and, in others, a low amount. It is, therefore, expected that the CDF for the mixed
microstructure lies between “tramlines”, bounded by the CDF for β and the CDF for α′. Fig.6 shows
that this is what is observed. A “best-fit” straight line drawn through the data-points exhibits a s.d. of
20MPa m1/2, or 33% of the mean (over three times that for α′ or β). A single straight line is, of course,
not appropriate, and, although there are rather few data points, there are suggestions, following the
“template” of Fig. 3 that the distribution is (at least) bi-modal. The material is not “quasihomogeneous” and the physical reasons for this are clear. Fitting the data empirically to (twoparameter) Weibull distributions gives m = 14.5 for 100% β and m = 3.2 for 30% β, 70% α′.
This metallurgical system was designed to demonstrate a point of general applicability to
“heterogeneous” microstructures. These include not only micro/meso-scale variations in heat-treated
forging steels, but also meso/macro-scale variations in multi-pass welds and nano/micro/meso-scale
variations in the segregation of “embrittling” species (such as P, Sn, Sb) to grain boundaries in
quenched-and-tempered steels [Islam, Bowen and Knott 2001]. The spatial extent of a “coarse” or a
“fine” region in a multi-pass weld is a few mm. The root radius of a Charpy notch is 0.25 mm and the
extent of uniform high stress is similar. It is found that wide scatter in toughness is obtained for weld
deposits, not only in sharply-cracked testpieces but also in Charpy tests [Newmann, Benois and Hibbert
1968, Todinov et al. 2000].
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99.99
99.9
A533B Pressure-Vessel Steel
(as-quenched condition)
Tested at -80o C
99
Bainite
90
70
Cumulative 50
Probability %
30
10
Martensite
x
o
Mixed Bainite/
Martensite
(30%/70%) x
x
x
o
o
o
o
o
o
o
o
o
x
o
o
o
o
o
o
x
x
x
o
o
o
x
x
o
1
0.1
0.01
20
40
60
100
80
0.5
Fracture Toughness MPa m
Figure 6 CDF plots for the fracture toughness at –80oC of A533B, austenitised
at 1250oC, grain size ~200 µm – after Zhang and Knott (2000)
A final point relates to the use of statistical analysis to derive “lower-bound” values for fracture
toughness, typically at a level of 10-4 (0.01%). Such values may be required for the establishment of an
acceptably low probability of failure for critical engineering plant. For quasi-homogeneous material, it
is reasonable to extrapolate the CDF to the required level, recognising that, if an out-lier is found, its
cause must be investigated. For 300M (Fig. 4), the lower-bounds, at the 0.01% level, are 12, 36 and
39.5 MPa m0.5, whereas, for the 100%martensite and 100% bainite CDF distributions in Fig. 6 they are
65 MPa m0.5 and 22 MPa m0.5. For the mixed microstructure, extrapolation to 0.01% probability gives a
value which is not just much lower than the lower-bound for the more brittle constituent, but is
negative! This is equivalent to the negative values obtained by Wallin’s “fitting” of cut-off values to
the Euro data base, using Master Curve methodology. The physical reasons for the negative value are
clear when Fig. 6 is examined, because the limiting “tramlines” are clear, but consider a situation for
which the only data-set available were that for 30%β 70%α′ material. In the absence of a model, these
data would be fitted empirically to a Weibull distribution, as has been done by Wallin, and the negative
value would be obtained. The “common sense” lower bound can be established by setting the value for
the mixed microstructure equal to that for the more brittle bainite phase, but the clarity of Figs. 4-6 is
obscured by the rather convoluted Weibull presentation. The present Master Curve recommendation of
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836
a cut-off value of 20 MPa m0.5 has some merit for a defined set of steels when no other information is
available. It is quite close to the “common-sense” value for bainite in Fig. 6 but it would be extremely
conservative for the martensitic condition. It would be unsafe for the 300M steel tested at 77K (Fig. 4).
Any physically-based lower bound might be expected to increase with test temperature, following
perhaps the RKR prediction, which suggests a weak increase, because the yield strength, and hence the
maximum stress generated in the process zone ahead of the crack-tip, decreases with increase in
temperature.
6 FINAL REMARKS
Whether or not extrapolation to low failure probabilities is viable or not requires a clear distinction
between what is “quasi-homogeneous” and what is “heterogeneous”. Judgement is involved. The
following procedure is proposed:
i) Plot fracture toughness results as a CDF on normal probability paper;
ii) If the line is straight and the s.d. equates to that for random experimental errors (or a “pragmatic”
value of <6 MPa m0.5) class the material as “quasi-homogeneous” and extrapolate with some
confidence (noting the need to identify, and investigate the causes of, outliers). Note that any
exceptionally low, or negative, “lower bound” value is indicative of spatially heterogeneous material.
There are strong suggestions of this, not only in the microstructures deliberately produced to generate
Fig. 6, but also in the French A508 tensile results in Fig. 1, in the Beremin σF results, and in the Euro
data-base.
iii) If ii) is not satisfied, class the material as “heterogeneous”. “Forensic fractography” must then be
employed to determine the more brittle constituent. If the ability of the tougher phase to arrest cracks
formed in the more brittle phase were understood to the extent that it could be quantified, it might
prove possible to use a weighted distribution directly to estimate the lower-bound, but, at present, the
safest approach is to model the more brittle phase microstructurally and to carry-out fracture toughness
tests on this (quasi-homogeneous) microstructure. These data can then be used for extrapolation
purposes to determine the lower bound. In many cases, it will not be possible to have access to
materials or to have sufficient time and resource to perform the necessary fractography and testing to
follow such a detailed sequence. It is, however, important to appreciate the general effects of spatial
heterogeneity, because the understanding gained leads to more intelligent “expert judgment”, to more
confidence in the “data-mining” of other existing data-bases and to more precision in identifying what
is “like-with-like”. Particularly to treat low-probability events, it is critically important to underpin
analysis of fracture results with good metallurgical knowledge and appropriate micro-mechanical
models.
ACKNOWLEDGEMENTS
The results analysed have appeared (in other forms) in referenced open literature, but thanks are due to
Prof.P.Bowen, Dr.J.E.King, Dr.D.J.Neville, Dr.M.Novovic, Dr.S.Slatcher, Dr.B.Wiltshire, Dr.S.Wu
and Dr.X.Zhang for access to their original experimental results.
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