A new statistical local criterion for cleavage fracture in steel. Part I - Model presentation
S R Bordet a , A D Karstensen a , D M Knowles b , C S Wiesner a ,*
a TWI, Granta Park, Great Abington, Cambridge, CB1 6AL, U.K.
b Gracefield Research Centre, Gracefield Road, PO Box 31-310, Lower Hutt 6009, New Zealand
* Corresponding author.
Paper published in Engineering Fracture Mechanics, vol.72. issue 3. February 2005. pp.435 - 452
Since it was first proposed in the early eighties, the Local Approach to cleavage fracture, or Beremin model, has been applied to a wide range of engineering problems. However, several results have cast doubts on both the predictive capabilities and fundamental assumptions of the Beremin model. This three-part paper shows that most problems encountered with the original Beremin model result from an unnecessarily oversimplified description of the local cleavage event. A new statistical local criterion for cleavage is proposed, which expresses the necessity of maintaining a critical dynamic connectivity between microcrack nucleation and unstable propagation in order for cleavage fracture to occur.
Following the pioneering works of Ritchie, Knott and Rice (RKR)  on cleavage toughness modelling and McClintock on fatigue,  Beremin  suggested that cleavage fracture could be characterised in terms of a local fracture criterion. [3,4] Such characterisation, in addition to providing a better insight into the physics of fracture, enables fracture analysis in virtually any kind of geometry or loading situation, given that the local fracture criterion is valid and correctly modelled, and that the mechanical environment in which it operates is accurately described. Despite important developments in the field of micromechanical modelling of cleavage fracture, notably in the last five years, which saw the emergence of the first major alternatives to the Beremin model, [5-8] this model, proposed two decades ago, remains the most widely applied micromechanical cleavage model. It serves as the starting point for the new statistical local criterion and cleavage model proposed below.
The Beremin model is based on assumptions shared by many other cleavage models:
- Cleavage is stress-controlled, but is always preceded by plastic flow, which is necessary for the nucleation of microcracks (typically at hard brittle particles such as carbides or non-metallic inclusions);
- The failure event follows a weakest-link-in-a-chain principle, meaning that the instability of a single microcrack leads to complete fracture.
Assuming that the microcrack size distribution follows an inverse power law and that the microcracks can be modelled as Griffith flaws, the global fracture probability can be expressed as a Weibull distribution [3,4] :
where m and σ u are respectively the shape and scaling factors of the Weibull distribution, V p is the fracture process zone (FPZ) which is some fraction of the plastic zone, σ 1 is the maximum principal stress and σ w is the so-called Weibull stress. Eq. (1) can be modified to account for the existence of a critical threshold stress, σ th , below which cleavage propagation is not possible.
The principle of the original Local Approach method is to assume that m and σ u (usually called Weibull parameters) are material parameters independent of temperature, loading rate and constraint, which logically derives from the acceptance of the critical stress criterion. [9-11] Once calibrated in a given geometry, the Weibull parameters can then be transferred to another to calculate the risk of cleavage fracture.
Although reported applications of the Beremin model look generally promising, a certain number of difficulties have been unveiled with time, among them [12-15] :
- Different shape factors m are obtained for the same material depending on crack geometry (i.e. constraint), temperature or loading rate, in contradiction with the critical stress criterion.
- The iterative calibration procedure for the shape parameter m diverges or produces very high m-values, which do not relate well with the recorded scatter in experimental data.
- The model sometimes gives poor cleavage fracture predictions.
This three-part paper is organised as follows. The first part explores the above difficulties experienced with the Beremin model. It re-assesses previous cleavage fracture work and introduces a new local fracture criterion for cleavage fracture in steel. The latter is then implemented into a probabilistic framework based on weakest-link statistics so as to establish a new expression for cleavage fracture risk evaluation. In the second part of the paper, the assumptions and predictive capabilities of the model are validated using a modern offshore structural steel. The third part links the new model with the Beremin model in the particular situation of J-dominated crack tips, and establishes an alternative, material-dependent, Master Curve method.
Explanations for difficulties with the Beremin model
The integration over the process zone in Eq. (1) indicates the need for plastic flow to nucleate cleavage microcracks. However, adopting the usual Weibull expression in Eq. (1) of the product of a stress function and a volume imposes the assumption of a microcrack population created at the onset of plasticity, which then remains active over the entire loading history. As the plastic strain component only appears through V p , no difference is made as to whether the material has just yielded or is work hardening. Work which demonstrated the importance of slip in carbide cracking also showed the paramount importance of plastic strain level, temperature and strain rate in this process. [16-20] Furthermore, they clearly established that arrested microcracks blunted with further straining, making them very unlikely to ever propagate again, even if they could turn into local stress raisers of their own. In other words, only freshly nucleated carbide microcracks are truly expected to lead to global macroscopic cleavage fracture. On this basis, Eq. (1) should be reserved for the treatment of pre-existing, constant active microcrack populations, which might be acceptable for very brittle materials such as ceramics, but not generally for steels.
The non-recognition of the combined influence of plastic strain, temperature and strain rate on microcrack nucleation in Eq. (1) offers sound materialistic reasons for the difficulties stated in the introduction. Assume that the actual cleavage fracture probability, P f , increases with increasing plastic strain as well as with stress. If Eq. (1) is applied to describe the fracture stress scatter at some given temperature and strain rate, the left-hand side of Eq. (1), determined by the experimental results, would rise more quickly than predicted by the right-hand side, which scales with stress only. The data set of stresses would therefore wrongly appear to be less scattered relative to the evolution of P f and this would be 'corrected' during the iterative calibration process of the Weibull parameters through an m-value larger than its true value. Thus, according to the fracture geometry employed, different local stress vs. plastic strain evolutions could lead to different Weibull parameters if one uses the current definition of Eq. (1). The divergence of the calibration procedure sometimes observed is therefore the result of the impossibility to explain the variation of P f with that of the stress field only.
It must be emphasised here that the divergence during calibration is often caused by mistakenly applying the classical Minami et al.  iterative calibration procedure to crack geometries approximating small-scale yielding (SSY) conditions while alternative methods (e.g. Gao et al.  ) should be used in such cases.  Indeed, under SSY conditions, the stress field self-similarity means that the evolution of P f (but not its absolute magnitude) is independent of m, so that any computed m at the beginning of the Minami et al. iterative calibration process should be found as converged. As pointed out by Gao et al.  , one would then find an infinite number of valid sets ( m, σu). This does not happen in practice, however, because these ideal conditions are never met. Either convergence or divergence is achieved instead.
Divergence is most likely when σ th is underestimated, which is the case when the two-parameter Weibull distribution is used. Suppose the iterative procedure is initialised at the value, m e,0 . The use of the two-parameter distribution means that σ th is forced to zero, while it is actually finite: the 'true' m-value corresponding to Eq. (1) will thus systematically be overestimated, since the estimates for σ th and m- are negatively correlated.  Because the stress field is quasi self-similar, one should get an m-estimate at the end of the first iteration little different from m e,0 . However, because of the use of the two-parameter distribution, the new 'best' m-estimate, m e,1 , will be higher than m e,0 . Since
, a new iteration is performed with the actualised m-value, m e,1 . The same situation arises, and the new estimate, m e,2 , is found larger than m e,1 . Depending on how close the stress field is to self-similarity, and how scattered is the experimental data set, further iterations will result in either converging to an 'optimised' m (realistic or not) or diverging until the computer resolution is exceeded. The same thing is true when overestimating σ th: in this case, the iterative process converges towards zero, unless an 'optimised' m is found before reaching zero.
The poor cleavage predictions that are sometimes reported in the literature are thus a result of the unrealistic treatment of the microcrack population in Eq. (1) and the approximations inherent to the calibration process of the Weibull parameters, whether purely statistical or simply methodological. For example, the ignorance of the existence of a threshold critical stress, σth, in cleavage predictive models, if tolerable in a high-constraint geometry, is unlikely to be acceptable in a low-constraint geometry, where the range of critical stress values will typically be much closer from σth.
The relative success of Eq. (1), despite these insufficiencies, owes to the general validity of the critical stress criterion and weakest-link assumptions for describing cleavage fracture in steel,  which dictate the form and main trends of Eq. (1). For a given geometry, it is quite conceivable to consider that the plastic strain component can be implicitly integrated in the model through the stress. In a notched tensile (NT) geometry for example, no stress unloading is observed, unless the initial notch tip is sharp and gets blunted with loading. Thus, one can assume that a Ramberg-Osgood type relation exists between the stress and the plastic strain for each material point of the FPZ. The effect of plastic strain is to increase the number of microcracks: it plays the same role as an extension of the FPZ volume, and should therefore appear as a multiplicative term of the stress power law inside the exponential function in Eq. (1). Whatever the actual form of the plastic strain function (linear, power law etc.), the weakest-link nature of cleavage guaranties the exponential form of Eq. (1).  It is then reasonable to assume that the influence of the plastic strain variable will be implicitly accommodated through an adjustment of m. In sharp cracked geometries, the situation is made more complex by the presence of stress unloading regions at the crack-tip. However, if one considers the overall probability of fracture, it is also quite likely that the great flexibility offered by m in Eq. (1) would be enough to artificially, rather than effectively, account for the plastic strain effects. That is, Eq. (1) may be perfectly capable of correctly fitting fracture data for a given configuration, without the calibrated parameters, especiallym, matching the true values of the physical features they are supposed to represent (e.g. m theoretically relates to the shape of the end-tail microcrack size distribution). Yet discrepancies in the model might not be observed, unless the parameters affecting microcrack formation, i.e. plastic strain level, temperature and strain rate, are sensibly different between the geometry that serves calibrating the Weibull parameters and the one for which predictions are made.
Reassessing experimental data on cleavage
Plastic strain effects on carbide cracking
There is experimental evidence showing that the number of cracked carbides in steel increases with plastic strain and decreasing temperature [16-20] . The number of cracked carbides in steels at room temperature and below was found to increase proportionally with plastic strain by Brindley  and Gurland  ( Fig.1), the same was observed by Lindley et al.  up to a certain strain level, after which the nucleation rate decreased continuously. Similar linear trends have been reported in the literature on damage of Metal Matrix Composites (MMCs). [26-29]
Fig.1. Percentage of broken cementite particles vs. tensile strain (after Gurland )
In addition to these effects, it was noted that cracking occurred preferentially in carbides oriented with their longest axis parallel to the direction of maximum tensile strain, [19-20] and that the probability of fracturing increased as the centre of the carbide was approached.  Furthermore, it was found that the proportion of the largest ferrite microcracks tended to increase with straining.  All these observations are consistent with a fibre-loading type mechanism, as argued by Lindley et al. 
The fibre-loading mechanism has not attracted as much attention as dislocation pile-up based models for carbide cracking and microcrack formation. One possible reason is that it led Lindley et al.  to the conclusion that thinner carbides cracked more easily than larger carbides of same length. However, Gurland  soon after established experimentally that the average size of the fractured carbides was larger than the average size of all particles at constant shape factor, thus apparently invalidating the fibre-loading model. The discrepancy came simply from Lindley et al.'s implicit assumption that carbides fail at a constant 'deterministic' value, whereas carbides, as virtually all materials, especially brittle ones, present strength variability and therefore a size effect. For a given length, thinner carbides will indeed develop larger stresses by the fibre-loading mechanism, but this will be done at the expense of the volume, so that, depending on the carbide flaw size distribution, this will either lead to a decrease or an increase of the cracking risk. Wallin  has confirmed that the fibre-loading model, when combined with weakest-link type statistics, was able to predict correctly, both qualitatively and quantitatively, particle cracking in metals.
The constancy of the proportionality between particle cracking probability and plastic deformation for very different matrix-particle systems suggests a general underlying law rather than a simple coincidence. Because carbides are brittle, it can be considered that their fracture is fully characterised by the attainment of a critical stress within the carbide 1 (whereas, as indicated earlier, a stress-only criterion is insufficient at fully describing cleavage in steel). A first requirement for explaining carbide cracking in terms of plastic strain, ε p , is then obviously that the evolution of the carbide internal stress can suitably be described by that of the plastic strain only, which is usually the case if a fibre-loading mechanism is operating. Under this condition, it was shown using asymptotic theory  that such linear behaviour will be observed at a relatively low damage level 2 , given that cracking is a weakest-link process involving a large number of particles, statistically independent with regard to fracture, and whose strength distribution can be expressed as a left-truncated distribution.
If carbide failure originates from surface flaws, then the interfacial stress appears a more relevant parameter. However, static equilibrium of the carbide prior to its cracking imposes that the interfacial stress globally scales with the internal stress, so that both parameters are interchangeable.
Low damage level here means that the number of cracked carbides remains small compared with that of the uncracked population.
Plastic strain and temperature effects on microcrack nucleation
Fig.2 represents ferrite microcrack density measurements performed by Kaechele and Tetelman,  plotted as a function of uniform tensile plastic strain. These authors selected a ferritic steel inside which arrested microcracks, typically one or two grain diameters in length, could be produced before final fracture. As found for carbide cracking in the previous section, Fig.2 shows that the number of nucleated microcracks globally scales in proportion with plastic strain. At first sight, this linearity seems logical, since one might expect this number to scale in proportion to the number of cracked carbides, which, according to the previous section, is predicted to be a linear function of plastic strain at low damage level. However, because larger strains are accompanied by larger flow stresses (all above experiments correspond to tensile tests), one would also expect that the probability that a carbide crack propagates into the ferrite increase and the lines curve upwards in Fig.2 as εp is raised 3 . The persistence of the linearity thus suggests that the level of local matrix stress has little influence on microcrack nucleation and early propagation in the matrix. Conversely, this suggests that the parameter controlling ferrite microcrack formation is somewhat related to the one controlling carbide cracking, i.e. to the internal carbide stress.
Fig.2. Ferrite microcrack density per cm2 vs. ε p (after Kaechele and Tetelman )
The effect of temperature on microcrack nucleation can be inferred from the relative evolution of the data points in Fig.2 with temperature. In this purpose, the data points at each test temperature were linearly interpolated. The slope of each interpolation line is proportional to the microcrack nucleation rate with respect to plastic strain in a given volume of material. Fig.3 represents a plot of these slopes vs. yield strength at the corresponding temperature: a linear relationship is found, meaning that, for a given plastic strain increment, the probability of propagating a carbide microcrack into the matrix is proportional to yield strength. Flow curves in steel at different temperatures are typically found to lie essentially parallel, so that they scale with yield strength. As a first approximation, it seems reasonable to assume that the internal particle stress will also globally scale in proportion to the yield strength. On this basis, the observed proportionality between the microcrack nucleation rate and the yield strength indicates that the same proportion of cracking carbides leads to ferrite microcrack nucleation at the three tested temperatures. In other words, the likelihood of crossing the carbide/ferrite interface is the same whatever the test temperature, i.e. temperature independent. This is in complete agreement with the temperature independence of the crack arrest toughness, K Ia , for ferrite crystals 4 , theoretically valid for temperatures up to 100°C  , extended here to the carbide/ferrite interface.
It is doubtful that the reduction in the uncracked carbide population matches exactly the greater likelihood in forming a ferrite microcrack due to the increase in flow stress, so as the microcrack nucleation rate remains overall constant.
This explains the existence of a critical stress for cleavage propagation independent of temperature.
Fig.3. Relationship between yield strength and ferrite microcrack nucleation rate
Cleavage nucleation mechanism
The proposed mechanism for cleavage nucleation is the following. Considering the carbides as reinforcement particles, the cracking of the carbide is accompanied by a release of elastic strain energy, which, in some cases, is sufficient to cause the carbide crack to propagate into the matrix for a short distance and form a ferrite microcrack.  If long enough, the ferrite microcrack will continue propagating under the action of the local matrix stress, if not, it will arrest and then blunt. The probability of crossing the carbide/ferrite interface is typically quite small  and only a small fraction of eligible carbides, i.e. the largest, most elongated, best-orientated and toughest carbides 5 , will be capable of achieving this. The lower the temperature, the higher the yield stress and the more effective the clamping of the carbide, and thus the greater is the load transferred to the particle by the deforming matrix. For brittle particles such as carbides, the critical stress (or more correctly the distribution of critical stress) can be taken as almost temperature independent, at least for temperatures below 300°C.  Consider a carbide in a specimen at low temperature that cracks at a given plastic strain. The same carbide in the same specimen, tested at higher temperature, will crack at a larger strain, due to the reduction in yield stress. The internal carbide critical stress, however, will remain the same, and so will the elastic strain energy released by the carbide. Thus, on average, the cracking carbides will consistently present the same crack features (length, tip acuity, speed, orientation relative to the ferrite cleavage planes, elastic energy remaining in the carbide etc.) to the carbide/matrix interface, which may or may not be sufficient for the carbide crack to cross the interface. To be successful in crossing, the carbide crack must develop very high strain rates at its front, hindering dislocation activation, so as to force a rigid (cleavage) response in the ferrite matrix,  which explains why the likelihood of crossing is essentially temperature independent 6 .
Although the model follows a fibre-loading type approach, the role played by local stress raisers, such as dislocation pile-ups  , in assisting in the cracking of carbides and the formation of microcracks, is not ruled out. However, these additional effects are not explicitly modelled thereafter. By way of simplification, it is considered that their effects are randomly distributed throughout the process zone and equivalent to a general weakening of the carbides and of the matrix resistance, which is implicitly accommodated through the model variables and the statistical treatment.
The largest carbides offer a larger Griffith crack on cracking which helps propagation. Elongated carbides build more stress by the fibre loading mechanism: although it makes these carbides particularly susceptible to cracking, they may sometimes be the only carbides able to raise the stress sufficiently high to trigger cleavage, if their fracture is sufficiently delayed. Elongated particles also tend to fracture, whereas more round ones tend to separate from the matrix by interfacial decohesion. The best-orientated carbides are carbides whose fracture plane has a low angle relative to ferrite cleavage planes. Tough carbides are carbides containing small critical flaws which crack at a higher applied load than average, which allow them to store more elastic strain energy.
It is clear however that this temperature independence is increasingly challenged as the test temperature approaches the ductile/brittle fracture transition temperature for ferrite (see discussion of Section 0).
Local statistical criterion for cleavage fracture
The philosophy of the new local criterion for cleavage is to effectively recognise that there are two distinct and necessary steps in order for cleavage fracture to take place: initiation and propagation. The previous sections showed that the first step is not correctly 'quantified' in the Beremin model, since not only is it assumed that all microcracks are created at the onset of plasticity, but also that they remain active over the entire loading history. The new model, in contrast, aims at accounting for both the activation of cleavage nuclei and their de-activation, until global instability ensues.
The new model for cleavage fracture is based on the same fundamental assumptions as the original Beremin model,  restated here for clarity:
- Microcracks nucleate at some brittle particles (grain-boundary carbides, non-metallic inclusions) under the action of plastic flow;
- These microcracks propagate unstably into the steel matrix if the local stress exceeds a critical stress value.
- Cleavage obeys a weakest-link principle.
It is recognised that cleavage actually involves a cascade of critical events characterised by run-arrest bursts of cleavage,  which suggest fibre-bundle type modelling, but the weakest-link assumption remains acceptable considering that this cascade originates from one initial critical event. [24,33] No distinction is therefore made whether the critical event is the propagation of carbide microcracks into the matrix or that of grain-sized microcracks into neighbouring grains. The adopted point of view is that cleavage is a dynamic process requiring continuity in a chain of events, consisting of initiation in a carbide, propagation into the immediately adjacent grains, then into the neighbouring grains, up to reaching macroscopic fracture. Thus, the critical stress is viewed as a general local quantity high enough to trigger global fracture.
The major departure from the Beremin model concerns the treatment of the slip-nucleated microcrack population, which is not viewed as fixed, but rather as evolving as a function of plastic deformation, temperature and strain rate. Any freshly nucleated microcrack that does not immediately propagate into the matrix, or that is rapidly stopped (e.g. at a grain-boundary), so that the macroscopic integrity of the specimen is preserved, will blunt and is very unlikely to propagate subsequently. The damage effect that such arrested microcracks represent (in terms of potential sites for void growth and/or stress concentration for example) is not explicitly accounted for in the present analysis.
Based on the experimental results of the previous section, the ferrite microcrack nucleation rate with respect to plastic strain is assumed to be a constant independent of the local matrix stress, but proportional to the matrix yield strength. Strain rate has similar effects to temperature on the flow stress, by making more or less difficult the generation and movement of dislocations, so that the higher the strain rate, the larger the flow stress. It is assumed that both variations in temperature and strain rate shift the whole stress-strain curve in proportion to the yield strength. Temperature and strain rate effects on the ferrite microcrack nucleation rate are thus simply integrated through the changes incurred by the yield strength.
Local cleavage fracture probability
The new local criterion is defined as the statistical event of simultaneously fulfilling the conditions for microcrack nucleation and propagation. In the model, microcrack nucleation and propagation are treated as two statistically independent events. Rigorously, one should speak of asymptotic (with respect to distance) statistical independence, since the continuity of the critical chain of events leading to unstable cleavage will necessarily impose a transition between the nucleation and propagation stages, where both descriptive parameters intervene. Statistically, this physical transition stage is ignored, since sustained propagation implies that the transition stage was necessarily passed (i.e. the probability attached to the 'transition event' is equal to 1). In effect, the 'nucleation' event designates 'carbide cracking leading to microcrack formation', while the 'propagation' event refers to 'local cleavage instability driven by the local stress only'.
The local cleavage fracture probability is thus generally expressed as:
P cleav = P nucl x P propag
where P nucl is the probability of nucleating a ferrite microcrack and P propag is the probability of propagating the nucleated microcrack.
Definition of local cleavage nucleation probability, P nucl
In accordance with the above assumptions, the probability of nucleating a ferrite microcrack over an equivalent plastic strain increment d εp is:
is the yield strength at temperature T and plastic strain rate ε p . This expression is only valid if the cracked carbide population, or more generally, the number of cracked sites within the carbides (in case of multiple carbide cracking), remains negligibly small relative to the number of potential uncracked sites, N unc . Rigorously, the probability of cleavage nucleation is proportional to the number of uncracked carbides, N unc :
Since the carbide cracking rate with respect to ε p is constant, N unc ( ε p) varies as an exponential function of ε p. This is easily deduced by noting that P nucl is proportional to the diminution in uncracked carbide number, dN unc , over the plastic increment d ε p:
Let σ ys,0 be the yield strength at a reference temperature T 0 and ε p,0 a characteristic plastic strain value such as:
Assuming that microcrack nucleation is possible as soon as plasticity occurs:
where N 0 is the initial number of uncracked carbides before yielding ( εp = 0). Finally, P nucl can generally be rewritten as:
If carbide cracking remains limited (i.e. if σ ys,0 . ε p,0 / σ ys, >> ε p ), then Eq. (4) reduces to Eq. (2).
Definition of local cleavage propagation probability, P propag
As in the Beremin model, the nucleated carbide microcracks are treated as Griffith cracks distributed as an inverse power law:
where f is the size frequency density distribution and α and β are positive material constants independent of temperature, constraint and strain rate, in accordance with the critical stress criterion. There is no strict reason for choosing a power law, except for the fact that power laws, and in particular fractal laws, appear to underlie many natural phenomena. [34,35] If the growth of carbide precipitates obeys a power law, then it seems logical to assume that the length of the microcracks originating from them should also be distributed as a power law.  The form of the chosen carbide size distribution does not change the general format of the model, which can therefore be altered for any better description of the actual crack size distribution 7 . However, the link with fractal geometry offered by choosing a power law allows some valuable a priori inference about the shape parameter m  .
A general recommendation, in the context of extreme value phenomena, is that one should concentrate on correcty describing the tail of the size distribution of active particles. 
If the local matrix stress was not present when the carbide crack bursts into the ferrite, it is obvious that the ferrite microcrack would stop at some point, after the initial dynamic effects, generated by the sudden release of carbide strain energy, have dissipated. An upper bound for nucleated ferrite microcrack size, l max , is consequently introduced, which represents the minimum local stress required to propagate the largest ferrite microcrack that could ever be nucleated. Hence,
where σ 1 is the maximum principal stress and l c is the critical ferrite microcrack size given by the Griffith's relationship for penny-shaped cracks 8 :
Beremin assumed through-thickness cracks, so that their expressions for l c and σ u differ slightly.
E, ν and γ p are respectively the Young's modulus, Poisson's ratio and effective surface energy.
Replacing l c in Eq. (5):
are material constants.
Similarly to Beremin's treatment  , the effect of crack orientation relative to the maximum principal stress direction is therefore assumed to be implicitly accommodated through σ u .
Definition of local cleavage fracture probability, P cleav
Over an infinitesimally small plastic strain increment, d ε p , the acting stress, can be considered as constant. Hence, combining Eqs. (4) and (6), the probability that a ferrite microcrack is nucleated and propagated over d ε p is:
Note that this equation does not mean that the nucleated microcracks remain active over d ε p , but that the condition for propagation for each of these microcracks is determined by the value of the local stress acting at the precise moment of its creation. If the local stress is above σ th when the ferrite microcrack is nucleated, then there is a finite probability that it will propagate ( P cleav > 0), and the greater the local stress σ 1 , the higher this probability (since m > 0). If the local stress is below σ th , then the nucleated microcrack does not propagate and its contribution to P cleav is nil.
The probability that a ferrite microcrack is nucleated and propagated over a given plastic strain range [0, ε p,u ] is:
If σ l < σ th over the whole plastic strain range [0, ε p,u ], then:
Eq. (7) therefore indicates that plastic straining under insufficient stress for cleavage propagation leads to a 'pre-strain effect' through a reduction of the remaining number of uncracked carbide sites, as expressed by Eq. (3).
Global cleavage fracture probability, Pf
Eq. (7) pertains to one particular statistical material point. Suppose the FPZ volume contains n c potential cleavage initiation sites. By virtue of the weakest-link principle,  the global probability of cleavage fracture, P f , is equal to:
where σ 1, i and ε p,u,i are respectively the maximum principal stress and the equivalent plastic strain acting at site i and k is a scaling constant. Since the number of initiation sites is proportional to volume, the previous equation can be rewritten in terms of the FPZ volume, V p . Be dV an infinitesimal volume of the FPZ, sufficiently small so that σ l and ε p are taken as constant over dV, then:
where V 0 and Σ are scaling constants ( V 0 is only introduced in order that Σ 1/m has the dimension of stress). To ease comparison with the Beremin model, the modified Weibull stress, σ w*, is defined as:
so that, with σ u * = Σ 1/m ,
If the ratio σ ys,0 • εp, 0 / σ ys is large relative to the actual critical ε p -values, i.e. if the number of cracked carbide sites is small relative to the uncracked potential population, Eq. (8) can be simplified as:
and σ w * is approximately equal to
The benefits of Eq. (8) are as follows. Consider the case of cracked specimens where cleavage starts in the region between the peak stress location and the crack tip, characterised by unloading stress and increasing strain, which is often observed in tough steels. [13,37] The original Beremin expression is unable to deal with such a situation. The use of the maximum stress ever experienced during loading instead of the current stress σ 1 avoids the cumulative fracture probability decreasing,  which is not physically possible, but this substitution ignores the fact that the microcrack actually propagated under the current stress. Eq. (8) addresses these difficulties by acknowledging that fracture initiation actually resulted from the nucleation of a microcrack of sufficient size through plastic straining. The local stress was of 'higher criticality' before, as it is now lower (and decreasing further), but the potential criticality was not 'activated' because no microcrack of critical dimensions was nucleated. Thus, Eq. (8) rightly predicts that regions experiencing higher tensile stresses may be less critical than regions with lower stresses but higher plastic strain levels. Similarly, Eq. (8) adapts well to situations involving crack growth prior to fracture 9 , whereas application of the original stress-only Weibull expression (Eq. (1)) forced some researchers (e.g. Ruggieri and Dodds  ) to ignore earlier contributions by unloaded parts of the process zone to the fracture probability, which is incorrect. Lastly, Eq. (8) predicts that carbide cracking under insufficient tensile stress leads to a decrease of the uncracked carbides population, and consequently to a decrease of the microcrack nucleation rate, which is equivalent to a pre-strain. Indeed, for the same cleavage fracture probability, the average stress in Eq. (8) will have to increase in order to counterbalance the diminution in microcrack nucleation rate 10 .
Obviously, Eq. (8) will need to be modified to account for any significant effects affecting the cleavage criterion as a result of crack growth (e.g. reduction of contributing carbides due to debonding, stress localisation due to void growth etc.).
Note that the cleavage fracture stress also tends to increase due to microstructural changes (e.g. higher density of dislocations provide additional barriers to cleavage crack propagation). This effect is not accounted for in Eq. (8).
Eq. (8) has similarities with a set of expressions recently proposed by Stöckl et al.,  who shared the same concerns about the Beremin model and its treatment of the microcrack population. They also assumed a linear relationship between the number of nucleated microcracks and plastic strain, but did not model the effect of temperature and strain rate on the microcrack nucleation rate. Prior to their paper, Bernauer et al.  proposed modifications to the Beremin model for cleavage fracture in the transition region to account for both the continuous generation of microcracks with plastic deformation and the decreasing number of contributing carbides due to ductile void growth around them. The microcrack nucleation rate with respect to ε p was modelled as varying linearly with plastic strain (whereas here it is constant under fixed temperature and strain rate). Unlike the present and Stöckl et al.'s models, nucleated microcracks that did not propagate unstably were not 'deactivated': their number was cumulated over time, only reduced by the number of cracked particles around which a void had formed. This delayed deactivation of the microcracks, in contradiction with the experimental observations reported earlier (e.g. those of  ), certainly explains some of the difficulties reported in their paper.
Validity domain of the model
One has to be aware of the validity domain of the model and its inherent limitations. The first limitation that must be acknowledged concerns the fundamental assumption of Local Approach methods, which consists in equating P f in Eq. (8) with the macroscopic fracture probability. Eq. (8) expresses the cumulative macroscopic probability of generating local cleavage instabilities. It does not, however, give any indication whether global instability will follow or not. One can envisage an infinite crack-front, leading to an infinite σ w * because some of the stress field is above σ th , yet with a global crack-driving force insufficient to reach global instability. Equating P f with the macroscopic failure probability means that the macroscopic mechanical environment ensuring the continuity of the chain of events, from microcrack nucleation to complete failure, is necessarily met. Therefore, in addition to have σ 1 > σ th , the condition for propagation should rigorously be conditioned by the prior attainment of sufficient elastic strain energy to drive the main cleavage crack and allow for the local cleavage area to link up with the macroscopic crack-tip. In other words, σ w * should only be integrated once K > K min or J > J min , where K min and J min represent the minimum (macroscopic) cleavage fracture toughness of the cracked body. It is worth noting that σ w *, like σ w , is a scalar quantity independent on how the stress is distributed in space, whereas this spatial repartition (e.g. expressed in terms of strain energy density  ) plays a determinant role in global instability. Ideally, one would like to introduce the RKR concept  , extended to 3-D, of a critical stress over a critical area into the local criterion  11 , which could be linked with some real measurable microstructural quantities, such as the grain size for example. 12
Kroon and Faleskog  have just proposed a new cleavage model precisely based on this idea.
V 0 in Eq. (1), although meant to represent some microstructural entity (a few grains in  ), actually plays no role in the derivation of Eq. (1): the latter is a consequence of the weakest link assumption and the power law form assumed for the microcrack size distribution only. 
Nevertheless, it can be argued that Eq. (8) will in general be applicable even without conditioning it on K > K min . This is especially true with tough structural steels, where the main difficulty is in generating a critical local ferrite microcrack, as proved by the observation of a unique primary site at the origin of macroscopic cleavage fracture. [24,33] Although the stress criterion ultimately applies, initiation is generally the critical event in causing cleavage fracture in most modern steels, because enough elastic strain energy for propagation is usually available at the crack-tip long before macroscopic fracture actually takes place.  .
A second limitation is the simplifying assumption that the probability of microcrack nucleation is a linear function of plastic strain. This assumption is applied to the whole carbide population without distinction of size or shape, whereas it is clear that there will be a tendency for the largest and most elongated carbides to crack preferentially, firstly because of the size effect, and secondly due to the fibre loading mechanism. Fig.2 suggests that the linearity is preserved for plastic strain values in excess of 10%. However, the microcrack counts in Fig.2 do not distinguish between the carbide categories from which they originate, so that the early disappearance of the largest and most elongated carbides may pass unnoticed because these categories are typically much less numerous than carbides of average size and shape. Thus, even if the probabilities of carbide cracking and microcrack nucleation can overall be approximated by a linear law up to a given strain ε p( T) depending on temperature, some carbide categories will display some non-linearity 13 beforehand due to their early cracking (cf. Eq. (3)). Because the largest and most elongated carbides play a disproportionate role in the difficult triggering of global fracture, the linear approximation used for P nucl increasingly overestimates the generation of potentially critical microcracks. Fortunately, this overestimation is delayed by the tendency in forming larger ferrite microcracks as ε p is raised,  because the 'natural selection' operating during straining means that increasingly tougher carbides are left to crack, which, on average, will have stored more strain energy.
Here non-linearity means that the microcrack nucleation rate decreases. Note that an acceleration of carbide cracking with respect to ε p is sometimes observed,  often due to stress concentration caused by extensive damage (impinging stress field of cracked particles, void growth etc.), which is not appropriately quantified by the evolution of ε p alone.
Although the model does not consider other temperature effects than those affecting the yield strength, it is acknowledged that temperature will somehow alter the microcrack formation process, notably through its role played in the relaxation of the carbide internal stress. If stress relaxation is mainly the result of plastic flow, then it seems reasonable to think that the relaxed stress will approximately scale with yield strength, as is implicitly assumed here. However, if a significant part of stress relaxation occurs through diffusional rearrangement, which is strongly temperature dependent, then it is unlikely that the microcrack nucleation rate with respect to ε p will still be a linear function of yield strength. The test temperatures in Kaechele and Tetelman's experiments  are certainly too low, and their range too small (~30°C), to make apparent these effects. Nevertheless, one expects the straight line in Fig.3 to display some non-linearity, i.e. the line falling down more sharply, if the temperature is raised and the range of yield strength extended to lower enough values. A compensatory consequence is that, as long as σ u * is calibrated at a test temperature lower than the temperature range for which predictions are made, the model will give over-conservative results, which is always preferable than underestimating the fracture risk.
The possible debonding of the carbide-matrix interface is also not explicitly addressed in the model: whether it promotes cleavage fracture by acting as a competitive cleavage nucleation mechanism, or diminishes its risk of occurrence by preventing subsequent carbide cracking, debonding is likely to be temperature dependent as well. It is also clear that temperature will promote ferrite microcrack as well as carbide microcrack arrests 14 , by favouring dislocation activity, responsible for an 'overall increase' in the work of fracture. The latter is to distinguish from the local fracture work involved right at the fast moving crack tip. The very high strains developed there leaves no time for dislocation emission, hence a relatively temperature independent and constant local fracture work, in accordance with the critical stress criterion. However, the 'macroscopic' work of fracture does increase with temperature, essentially due to plastic deformation taking place in regions of small and moderate strain rates and to ductile rupturing of ligaments left behind the moving crack tip. This leads to an increased likelihood, but not certainty, as temperature is raised, of slowing down the crack tip speed sufficiently to allow enough time for dislocations to blunt the crack and cause its arrest. This explains in part why the statistical mean toughness value increases significantly with temperature, while the minimum cleavage toughness increases very little in comparison.
Linaza et al.  found that the energy for crossing the particle/matrix interface in micro-alloyed steels was essentially temperature independent, whereas the crossing of grain-boundaries was much more temperature dependent.
Lastly, as mentioned in Section 0, the present model, although recognising carbide cracking as a trigger of cleavage fracture, does not explicitly account for the weakening (damage) effect of cracked and debonded carbides and, to a lesser extent, arrested microcracks. If the proposed local criterion for cleavage is not altered by limited damage, the success of Eqs. (8) and (9), independently of the model's own limitations, will rely up on the correct characterisation of the stress and strain fields in the process zone (involving precise elastic-plastic FE analyses in most cases). However, void growth and coalescence prior to cleavage, as well as any other change to the local cleavage criterion induced by extensive damage, must specifically be modelled. The present model can nonetheless serve as a framework for modelling the final cleavage fracture stage, after inserting the adequate local criterion, based e.g. on GTN (Gurson-Tveegard-Needleman) or Rousselier models. [43-45]
Such temperature and damage effects on cleavage fracture are complex to model and express in a simple and synthetic form. An immediate amendment to Eq. (8) could be to make ε p, 0 temperature, strain rate and/or damage dependent, to reflect changes that are not accounted for by the sole evolution of the yield strength, σ ys ( T, ε), with the non-negligible advantage that ε p,0 is easily interpretable, and theoretically measurable.
This paper argued that most difficulties in the application of the original Local Approach to cleavage fracture (the Beremin model) comes from an unnecessarily oversimplified treatment of the nucleation stage of cleavage. The use of the classical stress-only Weibull expression, albeit resolved over the plastic process zone V p , amounts to assuming that microcracks are created at the onset of yielding and remains active with further straining, contradicting experimental evidence and undoubtedly affecting the predictive performances of the model.
The new model is based on the same fundamental assumptions as the Beremin model, with the difference that the local criterion accounts for both the nucleation and propagation stages of cleavage. The new local criterion expresses the necessity of maintaining a critical dynamic connectivity between microcrack nucleation and unstable propagation and corresponds to the statistical event of simultaneously fulfilling the conditions for both stages. Microcrack nucleation and propagation are treated as statistically independent, so that, locally, the probability of cleavage is simply the product of the probabilities attached to each. To simplify, the probability of microcrack nucleation was assumed to be a linear function of plastic strain only, whose slope scales with yield strength, while the probability of propagation was modelled as a lower-truncated power law of the maximum tensile stress.
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| α, β ||positive material constants in f( l) = α / l β |
| Δa ||average stable crack extension |
| δ ||crack tip opening displacement |
| δ 0 ||reference CTOD (usually designates the 63.2% fracture probability toughness) |
| δ c ||critical CTOD with Δ a < 0.2 mm |
| δ u ||critical CTOD with Δ a 0.2 mm |
| δ min ||minimum critical CTOD |
| ε p ||equivalent plastic strain |
| ε p,0 ||reference plastic strain |
| ε p,u ||value of ε p in dV at the loading history point for which σ w * is calculated |
| ε p ||equivalent plastic strain rate |
| γ p ||plastic work of fracture |
| ν |
|Poisson's ratio |
| σ 1 ||maximum principal stress |
| σ flow ||flow stress |
| σ th ||threshold critical stress |
| σ u , σ u * ||Weibull scaling parameters for conventional and modified Weibull stress model |
| σ w , σ w * ||conventional and modified Weibull stress |
| σ ys ||yield strength |
| σ ys,0 ||reference yield strength |
| a 0 ||nominal crack depth of fracture specimen |
| a c ||critical Griffith crack size |
| B ||thickness of fracture specimen |
| dV ||infinitesimal volume used in the integration of σ w * over V p |
| E ||Young's modulus |
| f ||size frequency density distribution of microcracks in material |
| J ||nonlinear elastic strain energy release rate |
| J c ||critical J-value |
| J 0 ||reference J-value (usually designates the 63.2% fracture probability toughness) |
| J min ||minimum critical J |
| K ||stress intensity factor |
| K 0 ||reference toughness (usually designates the 63.2% fracture probability toughness) |
| K Ic ||plane strain fracture toughness |
| K Ia ||crack arrest toughness |
| K Jc ||elastic-plastic equivalent stress intensity factor |
| K min ||minimum critical K |
| l ||microcrack size |
| l c ||critical Griffith crack size |
| m ||Weibull shape parameter |
| n ||size of data set |
| ||strain hardening exponent |
| N ||size of data set |
| N 0 ||initial number of carbides |
| N unc ||number of uncracked carbides |
| P cleav ||local cleavage probability |
| P f ||global cleavage probability |
| P carbide cracking ||local probability of carbide cracking |
| P nucl ||local probability of microcrack nucleation |
| P propag ||local probability of microcrack propagation |
| T 0 ||reference temperature |
|V 0 ||elementary volume of V p |
|V p ||volume of the fracture process zone |
|CMOD ||Crack Mouth Opening displacement |
|CT ||Compact tensile |
|CTOD ||Crack Tip Opening Displacement |
|CVN ||Charpy V-notched |
|DBT ||Ductile to brittle transition |
|FE ||Finite element |
|FEA ||Finite Element Analysis |
|FPZ ||Fracture Process Zone |
|GRD ||Gao-Dodds-Ruggieri |
|GTN ||Gurson-Tvergaard-Needleman |
|LLD ||Load line displacement |
|LLV ||Load line velocity |
|MC ||Master Curve |
|NT ||Notched Tensile |
|RKR ||Ritchie-Knott-Rice |
|SEM ||Scanning Electron Microscope |
|SENB ||Single Edge Notched Bend |
|SSY ||Small-Scale Yielding |
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