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
Paper published in Engineering Fracture Mechanics vol.72. issue 3. February 2005. pp.453 - 474
View part 1
In the first part of this paper, a new model for cleavage fracture in steel was presented, based on a new statistical local criterion, which expresses the necessity of simultaneously fulfilling the conditions for both cleavage microcrack nucleation and propagation. In this second part, the assumptions and predictive capabilities of the new model are assessed using a modern offshore structural steel plate (Grade 450EMZ). It is shown that the model assumptions are consistent with the cleavage fracture behaviour of the steel and that the new model has the potential of correctly quantifying the effects of size, constraint, temperature and strain rate on cleavage fracture risk.
In the first part of this paper , a new statistical local criterion for cleavage fracture in steel was presented, which allowed writing the cleavage fracture probability in steel as a Weibull distribution:
where m and σ u * are respectively the shape and scaling factors (also referred to as Weibull parameters), and σ w * is a modified Weibull stress defined as:
In Eq. (2), σ 1 is the maximum principal stress, σ th is the minimum threshold local stress for cleavage propagation, ε p is the equivalent plastic strain, ε p,u is the value of ε p in dV at the loading history point for which σ w * is calculated, σ ys is the yield strength at the temperature and strain rate under consideration, σ ys,0 is a reference yield strength, V p is the fracture process zone (FPZ) volume for which σ 1 > σ th and V 0 is a reference volume. If the ratio σ ys,0 . ε p,0 / σ ys is large relative to the actual critical ε p -values, σ w * can be simplified as:
Eqs. (2) and (3) account for the continuous generation of microcracks with plastic straining and their deactivation if propagation is not coincident with their nucleation. In addition, Eq. (2) takes into account the reduction in microcrack nucleation rate due to the diminution of the uncracked carbide population, as expressed by the exponential term in Eq. (2).
Fig.2 of  shows that the linearity between the probability of microcrack nucleation, P nucl , and ε p can be effective up to 10% strain and beyond, which suggests that ε p,0 in Eq. (2) has a large value, so that Eq. (3) is generally a good approximation of Eq. (2). Keeping in mind that the probability P nucl above was determined without distinguishing between carbide categories, meaning that the probability of nucleating critical microcracks rather than any microcrack is not necessarily a linear function of ε p , it was nonetheless assumed here that Eq. (2) could be approximated by Eq. (3), thus removing the need of calibrating ε p,0 . Previous application of Eq. (2) for a sensible value of ε p,0 = 0.5 resulted in quite similar results to Eq. (3) for the studied material (see [2,3] ). Furthermore, Eq. (3) has the advantage of being more conservative than Eq. (2) for the same material parameters m, σ u * and σ th , which is always more desirable than underestimating ε p,0 , and thus the risk.
Eqs. (2) and (3) contrast with the original Local Approach formulation  :
A threshold critical stress for cleavage, σth , can also be introduced in Eq. (4). To establish σw , Beremin  assumed an inverse power-law distribution of microcrack size in the upper tail. Introducing σth means that the microcrack size power-law distribution is upper-truncated at the length a c ( σth) (see ), so that:
Note that Eq. (5) differs from the form usually (and most often arbitrarily) adopted
Although one can find ways of justifying the form of Eq. (6), notably through its link with extreme value theory  (Eq. (6) leads to a three-parameter Weibull distribution for P f ), the rigorous expression deriving from Beremin's assumptions is nonetheless Eq. (5). 
The stress-only description of Eqs. (4) to (6) signifies that all microcracks are assumed to be created at the onset of plastic yielding and remain active over the entire load history, which was shown in  to be in contradiction with experiments. It was also argued that this oversimplification, acknowledged by Beremin but certainly overlooked thereafter, was to a great extent responsible for the difficulties encountered in the application of the Beremin model.
The aim of this paper is to assess the ability of the new model to predict size, constraint, temperature and strain rate effects on the cleavage fracture toughness of a modern structural steel.
The material investigated is a Grade 450EMZ offshore steel plate made by British Steel (now Corus). The plate is 50 mm thick and was produced to meet the requirements of British Standard BS 7191:1989.  The steel chemical composition is given in Table 1. The micrograph ( Fig.1) reveals a fine-grained mixed microstructure of bainite and ferrite.
Fig.1. Grade 450EMZ microstructure (2% nital etch)
Table 1: Chemical composition of Grade 450EMZ 50 mm plate (in weight %)
Determination of true stress - true strain curves
The quality of the elastic-plastic finite element (FE) analyses, used to calculate the Weibull stress and calibrate the Weibull parameters, depends strongly on the material's input data, given as a series of true stress-true strain data points. The method employed to obtain accurate stress versus strain data is therefore very important. For this purpose, waisted tensile samples were used ( Fig.2). This geometry was considered preferable to the classical straight-sided one because it ensured that necking took place at the position of the clip gauge, which measured the diametrical contraction of the specimens throughout the test.
Fig.2. Waisted tensile specimen geometry
The specimens were taken from the mid-thickness position transverse to the rolling direction, the same direction of the crack opening in fracture tests. The applied load and diametrical contraction at the minimum section were continuously recorded during the test. The Bridgman correction was used to calculate the true stress values after necking of the specimen.  The true stress-true plastic strain curves for quasi-static loading conditions are presented in Fig.3, and the evolution of the corresponding lower yield strength with temperature is shown in Fig.4.
Fig.3. True stress vs. true strain in Grade 450EMZ at various temperatures
Fig.4. Variation of lower yield stress vs. temperature of Grade 450EMZ
The study of high strain rate effects on cleavage toughness requires the knowledge of the evolution of the yield stress with strain rate at a given temperature. A series of tensile experiments at various strain rates were therefore conducted at -100°C.  The extension, time and load were recorded at a sampling rate of 50 kHz during the tests. The plastic strain rates corresponding to the displacement speed were calculated using the proposed guidance in  . The Bridgman method was applied to calculate the true stress after necking of the specimen. Fig.5 shows the true stress - true plastic strain at -100°C for different strain rates.
Fig.5. True stress vs. true strain in Grade 450EMZ at -100°C for various strain rates
Notched tensile testing
Notched tensile (NT) tests for determining the Weibull parameters were performed at -196°C under quasi-static loading conditions. The NT geometry ( Fig.6) had a notch radius of 2.5 mm, a minimum section diameter of 6 mm and a maximum section diameter of 11 mm. As for the waisted tensile specimens, the NT samples were taken from the mid-thickness position of the plate, transverse to the rolling direction. The diametrical contraction at failure was determined from two perpendicular measurements on the fracture surface of the broken specimens. The strains at failure are plotted in Fig.7.
Fig.6. Notched tensile specimen geometry
Fig.7. Plastic strain at fracture in notched tensile (NT) tests at -196°C
Two NT specimens (specimens no. 21 and no. 25) failed at significantly larger strains than the rest of the set ( Fig.7). The fracture surfaces of specimen nos. 21 and 25 were examined on the SEM, along with those of other specimens that failed at lower strains for comparison.  It was found in all cases that the cleavage initiation area was close to the centre of the specimen. No particular feature was detected to explain the very large strain values of specimen nos. 21 and 25.
A total of 61 full-thickness 50x50 mm 2 Single Edge Notch Bend (SENB) fracture toughness specimens were tested at different temperatures.  The specimens were cut transverse to the rolling direction and notched in the through thickness direction. As part of a study on the Master Curve method,  twenty small 10x10mm 2 small SENB specimens, machined from the broken halves of the full-thickness specimens in the same orientation, were also tested. All small-sized and full-thickness specimens had a nominal crack depth to width ratio a0/W of 0.5. Specimen preparation, testing and analysis were carried out according to BS 7448: Part 1. 
The CTOD results for the full-thickness specimens are shown in Fig.8, where the ductile-to-brittle transition is clearly identifiable. Some of the specimens tested at -120°C and above experienced limited ductile tearing, Δa, but the latter was small enough for the results to fall in the δ c c-category (i.e. Δa < 0.2 mm). Grey symbols designate tests that experienced ductile tearing, whatever its extent. Open symbols correspond to specimens that fractured without ductile tearing, but for which cleavage is likely to have been initiated by a mixed ductile/cleavage mechanism 1 (see Section 0). Solid symbols correspond to the main lower-shelf cleavage initiation mechanism.
Fig.8. CTOD results vs. temperature for full-thickness 50x50 mm 2 SENB specimens. Results of small-sized 10x10 mm 2 SENB specimens tested at 140°C are also plotted (for clarity, these results are plotted at -140±1°C)
1 Several specimens tested at -130°C (the toughest) wre examined by scanning electron microscopy confirming a mixed ductile/cleavage initiation mechanism.
The CTOD results for 10x10 mm 2 SENB specimens tested at -140°C are included in Fig.8. All small specimens failed without ductile tearing. Note that one of the specimens fractured at a very low load (δ c = 0.006 mm) compared with the rest of the data set, due to an unexpected intergranular fracture that changed to cleavage. Fractographic evidence of this distinctive behaviour and of the other cleavage initiation modes is presented in Section 0.
High strain rate testing
Several full-thickness 50x50 mm 2 SENB specimens, identical to those used in the quasi-static fracture toughness tests, were tested [9,12] in accordance with BS 6729:1987.  Crosshead displacement rates of 10 mm/s  and 120 mm/s  were used. The results are presented in Fig.9. As expected  , higher strain rates result in a sharp decrease in toughness.
Fig.9. Dynamic CTOD results of full-thickness SENB specimens tested at load line velocities of 10 and 120 mm/s. To avoid overlapping, CTOD results at - 100°C are plotted at -100±1°C
Finite element modelling
Notched tensile tests
The notched tensile specimens were modelled using large strain axisymmetric finite element (FE) analysis, employing the commercial FE package ABAQUS. The mesh was set up according to the recommendations of the European Structural Integrity Society.  The analysis used the measured true stress-true strain data. Fig.10 is a comparison of the true stress vs. true strain curve obtained from FEA with the experimental failure points: good agreement is found, although the FE results slightly overpredict the stress level.
Fig.10. Comparison between FE analysis and experimental failure points for the NT specimens
Quasi-static CTOD tests
The full-thickness SENB specimens were modelled using large strain three-dimensional FE analysis. The same mesh was used for all temperatures. Due to symmetry about the crack plane and the mid-thickness of the specimens, only one quarter of the SENB specimen was modelled. Eight nodes first order brick elements were used throughout the mesh. Only six layers of elements were used through the half thickness of the specimen, since the stress in the width direction is essentially constant in the interior of the material, where plane strain conditions exist. The width of the layer was decreased as the free surface was reached and the stress gradients in the width direction became larger due to the change from plane strain to plane stress dominance.
The mesh was refined near the crack tip ( Fig.11) to account for the steep stress gradients as the tip is approached and led to element section areas as small as 55µm 2 at the crack tip. The crack was treated as a very narrow notch with a semi-circular tip to accommodate blunting. Gao et al.  found that the Weibull stress vs. δ relationship becomes independent of the initial root radius for δ/ ρ 0 > 4. Since the smallest δ c -value, measured in a full-thickness specimen tested at -196°C, was found equal to 4 µm, ρ 0 was chosen equal to 1 µm for all temperatures.
Fig.11. Near tip configuration of quarter symmetric 3-D mesh for SENB specimen
The load was applied by means of nodal displacements. The displacement increment was set in such a way as to allow precise linear interpolation of the Weibull stress values at the corresponding CTOD. The CTOD was calculated according to BS 7448: Part 1, in the same way as for the actual tests. A row of brick elements was used to model the experimental knife-edge position, where the Crack Mouth Opening Displacement (CMOD) is determined.
The material properties used in computation were the elastic modulus, E, the Poisson's ratio, ν (fixed to 0.28) and the material's plastic behaviour description beyond the yield point 2 , given as a series of true stress - true plastic strain data points obtained from the tensile tests ( Fig.3) at the corresponding temperature. When an experimental flow curve was not available for the temperature of interest, it was obtained by linear interpolation between the existing curves.
2 Here, the yield point is taken as the lower yield strength σ ys . In the FE input file, σ ys is defined as the stress for which ε p = 0.
Fig.12 is a comparison of the load, normalised 3 by (W- a 0 ) 2 , vs. the clip opening obtained from FEA, together with the experimental data points at failure. It is seen that the FE outputs are in good agreement with the experimental data.
3 As it is difficult to control with complete accuracy the final fatigue crack length in the fracture specimens, there were unavoidably some small differences between the specimens, so that the elastic compliance slightly differed between the specimens, hence the normalisation performed here.
Fig.12. Comparison between FE analysis and experimental failure points for the full-thickness specimens
High strain rate CTOD tests
The FE models used to determine the effects of strain rate on cleavage toughness were the same as those employed to model quasi-static loading conditions. Dynamic loading was simulated by applying the same load line velocities as in the experiments. Only the effects of loading rate on the material's yielding properties were modelled, i.e. the material inertial effects, negligible here, were ignored. The material's input data were entered as tables of yield stress values versus equivalent plastic strain at different strain rates. Because the strain rates at the crack tip are much higher than those determined experimentally (the strain rate can be as high as 200 s -1 in the specimens tested at a displacement rate of 120 mm/s), it was necessary to extrapolate the experimental results. Lower yield strength versus strain rate curves typically display two regions  : a region of moderate increase with strain rate ( ε < 10 4 s -1 ) and a region of very steep increase with strain rate ( ε > 10 4 s -1 ). It was therefore assumed that all strain rates generated in the fracture specimens belonged to the region of mild sensitivity to strain rate. Examination of Figs.3 and 5 shows that the flow curves at different temperature and strain rates are approximately parallel to each other, so that they can be approximately constructed from the knowledge of the yield strength at the corresponding strain rate and temperature. The experimental lower yield strength values at -100°C were fitted by a logarithm function of the strain rate ( Fig.13), so that the evolution of the yield stress, σ flow , as a function of strain rate and temperature was expressed as:
Fig.13. Experimental lower yield strength values vs. ln( edot /0.0005) at -100°C
where σ ys (T) and σ ys ( T 0 ) are respectively the (quasi-static) lower yield strength at temperatures T and T 0 , and ε = 0.0005 s -1 corresponds to the experimental quasi-static strain rate. Because the yield stress at a given strain rate is linearly interpolated from the tables, a large number of the latter was generated from Eq. (7) for each temperature of interest to allow fine linear interpolation.
Fig.14 shows that the curves of the load, normalised by (W- a 0 ) 2 , vs. the clip opening obtained from FEA at -100°C for both load line velocities of 10 and 120 mm/s, agree well with the experimental data points at failure.
Fig.14. Comparison between FE analysis and experimental failure points for dynamic fracture tests at -100°C
Assessment of the new cleavage model
Validation of the model assumptions in grade 450 EMZ steel
Fractographic analyses were carried out on the small 10x10 mm 2 SENB fracture specimens tested at -140°C, to investigate the conditions of cleavage initiation. The main initiation point was determined using river markings on the cleavage facets, and its distance from the crack tip was measured. In almost all fractured specimens, a unique cleavage origin at a grain boundary could be traced, confirming the validity of the weakest-link assumption. In a few cases, a precise initiation point could not be unambiguously determined, but a unique local area, from where cleavage started, was always clearly observed.
Notable differences in the initiation conditions existed, however, for specimens with toughness lying at the extremes of the δ c -values range. Three different cleavage initiation mechanisms were identified:
- An unexpected early intergranular failure transforming to cleavage in one of the specimen ( Fig.15), which explains its very low toughness ( δ c = 0.006 mm), compared with the δ c -values of the rest of the set.
- A transgranular type with moderate plastic strain at initiation ( Fig.16), which corresponds to the main cleavage nucleation mechanism. Specimens falling in this category are later referred to as Group 1.
- A transgranular type with large plastic strain and microductility at initiation ( Fig.17), which corresponds to specimens with the highest δ c -values. For these specimens, the initiation points were found to be located closer to the crack tip than in most specimens with lower CTOD. This group of specimens is later referred to as Group 2.
Fig.15a) Scanning electron micrograph of the fracture initiation area in front of the fatigue crack-tip of specimen no. 2, corresponding to an unexpected early intergranular failure, which then changes to transgranular cleavage
Fig.15b) Magnification of the initiation area showing the change from intergranular failure to transgranular cleavage
Fig.16a) Scanning electron micrograph of the cleavage initiation area in front of the fatigue crack-tip of specimen no. 1, representative of the main cleavage initiation mechanism (Group 1), characterised by moderate local plastic strain
Fig.16b) Magnified view of the initiation area. The arrow indicates the likely initiation point at the grain-boundary
Fig.17a) Scanning electron micrograph of the cleavage initiation area in front of the fatigue crack-tip of specimen no. 9, representative of the toughest CTOD specimens (Group 2), characterised by large local plastic strain and microductility
Fig.17b) Magnified view of the initiation area. The arrow indicates the likely initiation point, found in the immediate vicinity of microductility occurring at the grain-boundary
The finite element (FE) models for the SENB geometry, presented in Section 0, were used to infer the original location of the initiation sites in the specimen before loading. For simplicity, the initiation points were assumed to lie in the plane of the fatigue crack (Brückner-Foit et al.'s measurements  showed that the fatigue crack tip plane is the most likely location for the cleavage initiation points, and that the latter tend to cluster around it). The distance from the crack tip to the peak stress occurring on the crack plane was also determined in the undeformed configuration at different CTOD values. Both distances are plotted against δ c in Fig.18. Only initiation point distances of specimens for which the distance could be determined unambiguously are reported in Fig.18. Also plotted are 6 initiation distances, measured on the full-thickness 50x50 mm 2 SENB specimens tested at -130°C, which had previously been measured. 
Fig.18. Initiation site and peak stress distances from fatigue crack-tip vs. CTOD
Fig.18 confirms that initiation points can be divided into two groups, corresponding to the two main cleavage initiation mechanisms already identified: one, containing most results (Group 1), the other one containing only the highest δ c -values of the two geometries (Group 2). An interesting feature is that the two different cleavage initiation mechanisms apparently took place in both small and large geometries. All initiation points lie below the 'peak stress lines', which indicates that all initiation points were located between the crack tip and the peak stress location. Group 1 lies above and to the left of Group 2, expressing the tendency that cleavage nucleates nearer to the crack tip for the toughest specimens. There is also a general trend for the distances of both groups to increase proportionally with CTOD, or equivalently with peak stress location distances, so that both groups remain at an approximately constant distance from the peak stress position. Chen and Wang  , Urabe  and Koide et al.  reported similar linear relationship between initiation point distance and CTOD or J c values.
Fig.18 confirms the model's fundamental assumptions: cleavage is stress-controlled, but sufficient plastic strain is needed to achieve cleavage initiation. The fact that cleavage is stress-controlled implies that nucleation sites are to be expected around the maximum stress value, which is the case, since the initiation point location follows the peak stress location as it moves away from the crack tip with increasing load in Fig.18. However, initiation points are not found at the peak stress location, but closer to the crack tip, which expresses the necessity for sufficient plastic strain to nucleate a microcrack, yet still enough tensile stress to propagate it. Further supporting evidence is the fact that all initiation points have 'experienced' the maximum stress at some time during the specimen loading, but this was not coincident with failure. Indeed, ahead of a blunting crack in SSY, the maximum tensile stress stays essentially constant as it spreads with increasing load ( Fig.19). Any point in the specimen which 'experienced' the peak stress will therefore have been subjected approximately to the maximum tensile stress value developing in the specimen. The δ c -value at which the peak stress position coincided with the later initiation point location can be inferred from Fig.18. It corresponds to the δ c -value at the intersection between the relevant peak stress line and a horizontal line passing through the datum point of interest. For datum point A ( δ c = 0.06 mm) in Fig.18, this gives a CTOD of 0.035 mm. Fig.18 links up with Fig.20, showing the maximum principal stress - equivalent plastic strain history up to failure for point A evaluated from FE calculations. Finally, it is important to note that most of the δ c -results scatter is caused by the varying location of the critical nucleation sites relative to the crack tip  . This observation implies both that the nucleation of a critical microcrack is rare, as discussed in , and that the nucleators capable of generating such critical microcracks are scarce in the present material.
Fig.19. Normal stress ahead of the crack-tip at four different CTOD levels in the full-thickness SENB geometry, showing approximate self-similar behaviour
Fig.20. Maximum principal stress - equivalent plastic strain history of datum point A in Fig.25
Identification of the cleavage nucleators
While it was always possible to identify the precise location or area of cleavage initiation on the fracture surfaces of the small specimens, it was not possible to observe any inclusion or carbide, which might have served as a nucleation site. This does not mean, however, that no particle was involved in cleavage initiation. Carbides in Grade 450EMZ are very fine, which render their observation difficult, especially as the fracture surfaces were not etched, so that the river patterns were not damaged in the process. Franklin  similarly failed to associate most of the cleavage initiation sites ahead of a blunted notch with a specific particle. Nevertheless, whenever he did succeed, a relatively large particle was observed. Franklin concluded that a dislocation mechanism, such as that proposed by Cottrell  was the most likely explanation for the absence of particles at the initiation sites. However, if one accepts that large particles tend to make cleavage initiation easier, it is unlikely that the main cleavage mechanism (that with no visible particle) corresponded to a dislocation-based mechanism, which is considered to occur with greater difficulty.  Thus, it is argued that brittle carbides, albeit not visible, are most likely to have been responsible for cleavage initiation in Franklin's tests, their smaller sizes being counterbalanced by a larger internal stress at failure (due to the size effect), providing more elastic energy per carbide unit volume available for microcrack propagation (see ).
Another strategy for identifying the cleavage nucleators has been to test double notch bend specimens, whose geometry is presented in Fig.21. The specimens were tested at the same temperature as the small CTOD specimens, i.e. -140°C. It was verified that the cleavage mechanism was the same as those in the small CTOD specimens. Transverse sections of the unpropagated notch were cut, polished and etched with 2% nital, and then observed on a field emission gun JEOL 6340F scanning electron microscope ( Fig.22). Although no grain-size arrested microcrack was observed, a large number of cracked grain-boundary carbides were visible ( Figs.22 and 23). A few microvoids were also detected, mostly linked with carbides (Region (3) in Fig.23). Although precise counts were not made, it was apparent that the closer the notch-tip, the higher the surface density of both cracked carbides and microvoids ( Fig.22). Cracked carbides were most often aligned with the direction of maximum principal strain (Regions (1) and (2) in Fig.23), as testified by the deformed shape of the grains near the notch tip ( Fig.22). A general tendency for the cracking of the largest and most elongated carbides was clearly noted, whereas void nucleation was more often associated with round shaped carbides ( Fig.23). Carbide cracking took place preferentially near the centre of the particle, but it could also be found near the end, in both narrow and broad regions of the carbide (Region 2 in Fig.23). It was sometimes possible to link the carbide crack with features likely to cause stress localisation, such as a carbide located astride a grain-boundary (Region 2 in Fig.23), or a carbide with an angular shape (Region 4 in Fig.23). All these observations are consistent with those of previous studies on steel. [24-27] Although these authors were not unanimous regarding the adequate fracture criterion, these observations are coherent with a fibre loading type mechanism, [25, 28-29] as discussed in .
Fig.21. Geometry of double-notched bend specimen
Fig.22. Scanning electron micrograph (2% nital etch) of the region below the unpropagated notch in the double-notched bend specimen. Cracked carbides are clearly visible in the large strain region immediately below the notch root (arrows indicate a few of them). The spatial density of cracked carbides diminishes with distance from the notch tip
Fig.23. Scanning electron micrograph of cracked carbides below the unpropagated notch root. (1) The carbide crack was stopped at the carbide/matrix interface, and was subsequently blunted. (2) The cracked carbides tend to be aligned with the direction of maximum tensile strain, in accordance with the fibre-loading model. (3) Evidence of carbide/matrix debonding and microvoid growth. (4) The cracking of the carbide on the right may have prevented the cracking of the elongated carbide on its left through local stress relaxation
There is no categorical proof that cracked carbides are responsible for cleavage fracture initiation in the CTOD specimens. However, whenever a crack or a void was observed in the transverse sections of the double notch specimens, it was almost always linked with a carbide, making it the most likely cleavage nucleator. The absence of observable ferrite microcracks at the moment of fracture of the double notch bend specimens, suggests that the propagation of carbide microcracks into the ferrite, i.e. the generation of a critical ferrite microcrack, is the controlling stage in cleavage fracture of Grade 450EMZ steel. Obviously, an increase in temperature may require the critical length to be larger than the mean grain-size, so that the crossing of grain-boundaries may become the ultimate controlling stage. In all cases, global unstable propagation will only be achieved if the initial carbide stress was large enough to 'launch' the ferrite microcrack sufficiently far to allow the local tensile stress to drive it further.
Calibration of the model parameters
The model parameters were calibrated using the fracture results of the NT specimens tested at -196°C. SEM examinations revealed similar cleavage fracture surfaces as those obtained at -140°C in the CTOD specimens of Group 1.  It was therefore assumed that the same local cleavage conditions applied for both the NT specimens and the SENB specimens of Group 1, so that toughness predictions based on the calibrated parameters could then be checked against the experimental toughness results of Group 1.
The modified Weibull stress, σ w *, given by Eq. (3), was calculated numerically from the FE results as follows:
where σ ys is the yield strength at temperature T and strain rate ε,
is the averaged maximum principal stress and
is the plastic strain increment at integration Gauss point (IP) i within element j between load step numbers l - 1 and l of the FE analysis (in the dynamic case, l represents the actual time). n e is the number of elements in the volume delimiting the fracture process zone, n ip,j is the number of IPs for element j, V i,j is the IP volume in element j and ε p,0 , σ ys,0 and V 0 are scaling constants. The maximum load increment allowed in the FE calculations was set in such a way to leave σ w * relatively unchanged when reduced to a smaller increment size. The reference volume V 0 was conveniently fixed to one unit volume for all calculations.
Because σ w * is a function of m, the latter was calibrated using an iterative procedure similar to that proposed by Minami et al.  Sets of σ w *-values at failure were calculated by linking each diametrical contraction recorded experimentally with that of the FE model. The Maximum Likelihood (ML) method was then employed to estimate m from the set of σ w *-values, until the m-estimate matched the value assumed in the calculation of σ w *. The two NT specimens, which failed at critical strains much larger than the rest of the data set (see Section 0), were right-censored, in order to focus on the typical fracture behaviour 4 . The Maximum Likelihood method for multi-censored data proposed by  was therefore employed, and the m-estimates were unbiased using the empirical relation derived by Ross. 
4 The choice is to favour statistical inference by avoiding introducing 'outliers', which potentially failed by a different mechanism.
The value of the m-estimate is very sensitive to the choice of the threshold stress σ th . The value for σ th was fixed so that the σ u *-values in Eq. (1) (representing 63.2% risk of fracture) were approximately the same in the NT and in the SENB geometry, tested at the same temperature T 0 = -196°C. The following estimates were obtained: m e,unb = 11.2, σ u * = 2632 MPa and σ th = 1570 MPa. The 95% confidence intervals for m e,unb and σ u * were [6.9, 18.1] and [2502, 2768] respectively. The value of 1570 MPa is a reasonable estimate for the minimum critical stress, as illustrated in Fig.24, which represents the critical stresses calculated in the centre of the specimen (where cleavage initiation points were found: see Section 0) from FE analysis. The results of a second set of NT tests, with notch radius r = 6.4 mm, are also presented. It can be verified that all critical stresses are above 1600 MPa. Fig.24 is a good illustration of cleavage being stress-controlled, since for the NT tests with r = 6.4 mm, it can be argued that there was by far enough strain to nucleate microcracks, but not enough stress to propagate them. To some extent, this is the reverse situation to that of the high constraint SENB geometry, where stress builds up rapidly to very high levels, yet the crack-tip needs to blunt before large enough plastic strains are available near the peak stress region.
Fig.24. Critical stresses at the centre of the NT specimens evaluated from FEA
Quasi-static lower-shelf toughness predictions
Fig.25 represents a comparison between toughness results obtained on 50x50 mm 2 SENB specimens and the predictions calculated from Eqs. (1) and (3) using the material parameters determined at -196°C: the only varying parameter is the yield strength, whose evolution with temperature was obtained experimentally ( Fig.3). One observes that the toughness predictions are in good agreement with measured values 5 . The 95% risk curve correctly predicts the change of fracture mode between the main lower shelf cleavage mechanism (Group 1) and the mixed ductile/cleavage mechanism (Group 2) occurring at -130°C. However, the same curve over predicts the results at -110°C as it captures regions corresponding to Group 2 initiation mechanism. The reason for this is that it becomes increasingly difficult to build tensile stresses above σ th when the upper-shelf region is approached, so that more deformation and work hardening are required to raise the stress. By the time enough tensile stress is available for propagation, strains are likely to have been large enough to have already triggered the ductile/cleavage mechanism beforehand. Hence the disappearance of Group 1 initiation mechanism for temperatures above -120°C and its gradual replacement by Group 2 mechanism. Further increase in temperature leads to the replacement of Group 2 mechanism by more and more ductile mechanisms, involving void growth, coalescence and crack growth. For these temperatures, the model is obviously irrelevant and cleavage toughness predictions would be characterised by curves abruptly 'jumping' towards very large toughness values as the upper-shelf is approached, expressing the difficulty of building stresses above σ th . Interestingly, the collapse of the model is proof of the physical relevance of σ th .
Fig.25. Lower shelf toughness predictions of Grade 450EMZ using Eqs. (1) and (3)
5 Recall that the model is only supposed to capture the solid data points corresponding to the main cleavage mechanism of Group 1.
At temperatures of -150 and -130°C, the number of measured δ c -values is sufficient to allow for acceptable statistical treatment. Fig.26 shows a comparison between the predicted evolution of P f with CTOD (solid line) and median rank probabilities, P i , for the measured δ c -values (symbols) at both temperatures (at -130°C, only toughness values belonging to Group 1 were considered). P i was calculated using the best (approximate) estimate of the median rank probability P i = ( i - 0.3)/( N + 0.4), where i denotes the rank number and N defines the total number of fracture tests. The dashed lines represent the 90% confidence limits for the estimates of the experimental rank probabilities. These confidence limits were computed assuming that P f , given by Eq. (1), provides the expected median rank probability for an experimental data set containing the number of measured values N. The numerical procedure to compute the 90% confidence limit values follows Wallin  and is briefly described in the Appendix of  . The calibrated model fits both data sets relatively well. As already visible in Fig.25, the experimental results at -150°C are slightly less dispersed than predicted by the model, hence the steeper slope recorded for the experimental median rank probabilities, P i , and a somewhat under-predicting model for the highest probabilities. At -130°C, the shape of the predicted median fracture probability curve, P f , is in better accordance with the experimental median rank probabilities. The latter are slightly shifted to the right relative to the predictions, meaning that the model is this time slightly over-predictive. The last result is in line with the expectedly increasing over-conservatism of the model with temperature, due to the over-estimation of the microcrack nucleation rate as temperature is raised, as explained in .
Fig.26. Comparison between predicted failure probabilities (solid lines) and median rank probabilities for the quasi-static delta c -results (symbols) at (a) -150°C and (b) -130°C. The dashed lines represent the 90% confidence limits for the estimates of the experimental rank probabilities
Rigorously, the calculations of σ w * should have been conditioned by the prior attainment of a minimum value of CTOD, δ min , as discussed in . This was not done here to avoid adding a fourth parameter into Eq. (3), and also because the accurate estimation of δ min is still an open issue. This omission is not too influential in the present case because the integration of σ w * was conditioned on σ 1 being larger than the minimum critical stress, σ th , estimated from the NT tests. The fracture stress values obtained from the NT geometry are paradoxically more conservative than in most other fracture geometries. Hahn  reports Kotilainen's estimates of the fracture stress for three different test geometries made of the same steel  : values of 1230, 2500 and 3015 MPa were found for tensile, Charpy V-notched (CVN) bend specimens and K Ic geometries respectively. This substantial change in fracture stress is explained by a much larger fracture process zone (FPZ) in the NT geometry relative to the other two geometries, but also by a much larger proportion of the FPZ subjected to high plastic strains.  As δ min represents the minimum fracture toughness, which necessarily implies that σ 1 > σ th in a substantial part of the process zone, conditioning the integration of the cleavage risk to σ th , rather than δ min , corresponds to an even more conservative approach 6 . It was nonetheless verified that fixing the minimum toughness to the recommended value in ASTM E1921, K min = 20 MPa √ m, had only minor effects on the overall predictions 7 . Obviously, K min is quite influential at very low fracture probabilities, since it sets the zero probability of macroscopic cleavage. Thus, to avoid being overly conservative, K min should be included in the analysis if the model is used for fracture assessment purposes.
6 This over-conservatism is somewhat illustrated in Fig.26, (as well as in Fig.29 presented in the next section), by the fact that the model tends to over-predict the experimental median rank probabilities at low probability, whatever the evolution of P f might be at higher fracture probabilities.
7 It did however improve statistical inference at low probability (cf. previous footnote).
High strain rate toughness predictions
Fig.27 shows a comparison between toughness results of the 50x50 mm 2 SENB specimens at load line velocity (LLV) of 10 mm/s and the predictions based on the set of material parameters determined from the quasi-static NT tests at -196°C. As for the quasi-static case, the only varying parameter is the yield strength, whose evolution with temperature and strain rate is given by Eq. (7). Fig.28 presents the same comparisons for LLV = 120 mm/s. For both strain rates, the predictions are in good general agreement. However, it is noticed that the results are increasingly underpredicted as the temperature is raised, although the targeted toughness values (filled symbols in Figs.27 and 28) are within the range where Group 1 cleavage type is expected to be the dominant initiation mechanism (cf. Fig.25). Comparison of the predictions at -100°C between Figs.27 and 28 shows that while this underprediction is pronounced for LLV = 10 mm/s, this is not the case for LLV = 120 mm/s. This is apparent on Fig.29, where, as for the quasi-static case, P f is compared with median rank probabilities for the dynamic δ c -results, belonging to Group 1 initiation mechanism (filled symbols in Figs.27 and 28). The model predicts the shape of the experimental toughness distribution well for both LLVs. However, whereas the model suitably predicts the position of the experimental toughness distribution with respect to CTOD for LLV = 120 mm/s, the predicted P f curve is noticeably shifted to the left 8 relative to the median rank probabilities for LLV = 10 mm/s.
8 Note that underpredictions of the model in toughness vs. temperature diagrams ( Figs.25, 27 and 28) appear as overpredictions in P f vs. CTOD diagrams ( Figs.26 and 29).
Fig.27. Cleavage toughness predictions for LLV = 10 mm/s using Eqs. (1) and (3)
Fig.28. Cleavage toughness predictions for LLV = 120 mm/s using Eqs. (1) and (3)
Fig.29. Comparison between predicted failure probabilities (solid lines) and median rank probabilities for the dynamic δ c -results (symbols) at -100°C with load line velocities (LLVs) of (a) 10 mm/s and (b) 120 mm/s. The dashed lines represent the 90% confidence limits for the estimates of the experimental rankprobabilities
These discrepancies reflect the effects of temperature on work of fracture, stress relaxation, carbide debonding etc. , which become stronger as temperature is increased and which are not presently modelled. The fact that the model underpredicts dynamic CTOD ranges, which were suitably predicted in the quasi-static loading case, reveals that a higher strain rate cannot fully compensate for an increase in temperature. For example, a temperature shift Δ T, causing the same change in yield strength as an imposed strain rate shift Δ ε p , will have a stronger effect on carbide stress relaxation due to diffusional processes than Δ ε p . Also, once a carbide crack or a ferrite microcrack is running, the strain rates generated at their tips are far superior to the strain rates imposed by the dynamic loading conditions, so that the temperature is likely to have stronger effects than Δ ε p for equivalent change in yield strength. The work of fracture behind the moving cleavage crack, involving the elongation and rupture of ductile ligaments, will also vary more as a function of Δ T than Δ ε p . The temperature shift observed between two toughness curves of the same steel at different strain rates is therefore smaller than what would be obtained if all temperature dependent parameters, with the exception of the yield strength, were kept constant. The model, only considering the effect of temperature on the yield strength, will therefore predict a transition temperature higher than the actual one. As a result, the model will predict a lower shelf toughness region extending to temperatures for which the actual toughness already belongs to the fast increasing ductile-brittle transition region, which explains why the model increasingly underpredicts cleavage toughness as the temperature is raised in Figs.27 and 28. This also explains why the model underpredicts experimental toughness data at -100°C for LLV = 10 mm/s ( Fig.27), but not for LLV = 120 mm/s ( Fig.28), if one considers that the model overestimation of the probability of cleavage fracture at small probabilities in Fig.29(b) is principally due to assuming δ min = 0. This arises from the fact that the higher the loading rate, the higher the ductile-brittle transition (DBT) temperature. For LLV = 120 mm/s, the experimental cleavage toughness values at -100°C still belong to the lower shelf, whereas those for LLV = 10 mm/s belong to the lower part of the DBT region, hence a greater gap with the model predictions in the second case.
Microscopic work has shown that the model assumptions are consistent with the cleavage fracture behaviour of a modern offshore structural steel plate (Grade 450EMZ). The model parameters were calibrated from notched tensile (NT) fracture specimens, tested at -196°C, and then used to predict the lower shelf toughness transition curve of the steel. Good predictions were obtained, validating the ability of the model to predict effects of size, constraint and temperature on cleavage toughness, and confirming that the critical stress criterion is a valid assumption for characterising cleavage fracture, given that the nucleation stage is accounted for in the local criterion. The new model also showed potential to correctly predict the effect of dynamic loading on toughness, although it appeared too conservative, due to temperature effects on material parameters not presently modelled.
Calibration of the shape parameter m must be done in a stress-varying geometry if a single testing geometry is used, i.e. a geometry where the stress field is not self-similar (as is the case in deeply cracked SENB and CT geometries). The NT geometry is recommended because it allows an accurate and conservative estimation of the lower critical stress, σ th , and because testing, modelling, and parameter calibration are made easier compared with other geometries, such as Charpy-type notch bend tests for example. In addition, there already exists an ESIS guidance document  for the calculation of the Local Approach parameters which employs this geometry, and which could easily be adapted to the new model. The fact that the stress field in the NT geometry is relatively homogeneous is also favourable in terms of statistical inference, especially if the number of tests is low. However, because σ th will usually be quite close to the lowest critical stress value recorded in the weakest NT test, the m-estimate will be strongly dependent on the value of σ th . It is therefore advisable, if not necessary, to use a second geometry tested at the same temperature and select the set of ( σ th , m) parameters that produce the same scaling parameter, σ u *, in both geometries. The approach is similar to the GRD (Gao-Ruggieri-Dodds) procedure  , with the difference that m is cross-checked through the classical calibration procedure on the NT fracture data.
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