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Engineering Fracture Mechanics 67 (2000) 87±100

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Prediction of cleavage failure probabilities using the Weibull stress N.P. O'Dowd *, Y. Lei, E.P. Busso Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, Exhibition Road, London SW7 2BX, UK

Abstract In this work, the use of the Weibull stress as a measure of the failure probability of a cracked body is examined. Closed form expressions for the Weibull stress are presented for linear elastic and power law materials. These expressions allow Weibull stress values and failure probabilities to be estimated without the need for ®nite element analyses and provide insight into the use of the Weibull stress as a parameter for the prediction of cleavage failure of cracked bodies. Application of the Weibull stress to the prediction of transition region and upper shelf cleavage failure probabilities of ferritic steel welds is also discussed. The e€ect of ductile tearing, described by a Gurson-type material model, is examined. Numerical analyses of compact tension fracture tests are carried out and used to interpret experimental fracture toughness data for ferritic steel welds. A comparison between the ®nite element results and the experimental data indicates that the scatter in the cleavage fracture toughness values in the transition region are quite well captured by the Weibull stress. However, cleavage failure probabilities close to the upper shelf using the two parameter Weibull model are considerably higher than those observed in practice. A number of reasons for this discrepancy and suggestions to improve the predictions are proposed. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Weibull stress; Cleavage fracture; Ductile tearing; Finite element analysis

1. Introduction Weibull statistics have been widely used to describe the scatter in strength and fracture toughness of brittle materials [1,2]. The Weibull stress, rw , was de®ned in Ref. [2] as " #1=m ne X ÿ i
m Vi r1 ; …1† rw ˆ Vo iˆ1 where Vo is a reference volume, Vi is the volume of the ith material unit in the crack tip plastic zone experiencing a maximum principal stress ri1 and ne is the number of material units in the plastic zone. *

Corresponding author. Tel.: +44-0171-594-7059; fax: +44-0171-823-8845. E-mail address: [email protected] (N.P. O'Dowd).

0013-7944/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 0 ) 0 0 0 5 1 - 5

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The cumulative probability of failure, in terms of the modulus m and scaling parameter, ru associated with the Weibull stress, is then given by   m  rw : …2† Pf ˆ 1 ÿ exp ÿ ru The Weibull approach has been extended in Ref. [3], wherein a three parameter Weibull stress distribution has been examined which includes a threshold stress below which the probability of failure is negligible. The use of the three parameter Weibull distribution will be discussed later in this work, when the application of the model to data for a pressure vessel weld steel is examined. Analytical and semi-analytical solutions for the Weibull stress are ®rst presented. These solutions allow direct calculation of the Weibull stress for cracked-elastic and elastic±plastic geometries under small scale yielding conditions and beyond. Cleavage failure probabilities in terms of K and J are then presented and connection made with the recent work of Ruggieri and Dodds [4] and Faleskog [5]. In Ref. [2], the reference volume Vo was identi®ed as a material volume related to the likelihood of ®nding a cracked carbide. As discussed in Ref. [6], however, Vo can be any value provided it is kept constant for the material under study. In a ®nite element analysis, the computed Weibull stress value will then be independent of the ®nite element mesh used (provided the stress ®elds are non-singular). As will be seen later, when the stress ®elds are singular, as is the case for a sharp crack, the Weibull stress obtained from a ®nite element analysis may be mesh dependent. In this case, the choice of a physical length scale for the mesh size may be appropriate. An alternative, more physically meaningful approach, is to assign a ®nite notch root radius to the crack tip. The latter approach is adopted here. Replacing the sum in Eq. (1) by an integral and considering plane stress or plane strain conditions, Eq. (1) becomes  Z 1=m B m r dA ; …3† rw ˆ Vo A 1 where B is the out-of-plane thickness of the cracked body and A denotes the area of material inside the fracture process zone, de®ned as the region where r1 P kro : Here, k is a parameter P1 which scales the yield stress ro . Introducing a length scaling parameter, L, Eq. (3) may be written as  Z 1=m 1 m r dA ; rw ˆ L2 A 1

…4†

…5†

where

r Vo : Lˆ B

…6†

The Weibull stress for two-dimensional problems can be evaluated by integrating Eq. (5), when the stress ®eld near the crack tip is known. 2. Weibull stress solutions A range of analytical and semi-analytical Weibull stress solutions have been provided in Ref. [7]. The results are summarised in this section.

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is

89

For an in®nitely sharp crack in a linear elastic material under K dominant conditions, the Weibull stress 2 rw Ce ˆ4 kro …4 ÿ m†

K2 …kro †2 L

!2 31=m 5

for m < 4;

where L is the length parameter de®ned in Eq. (6). The constant Ce is given by  4 Z p  1 h h cos 1 ‡ sin dh ˆ 0:456: Ce ˆ 2 p 0 2 2

…7†

…8†

For an elastic±plastic analysis, r0 plays the role of an yield stress. In Eq. (7) (the solution for a linear elastic material), it is simply a normalising stress and the Weibull stress value is obtained from that region of the body where the principal stress is greater than or equal to this value multiplied by k. Note that if a ®nite region, over which the Weibull stress is determined, is not speci®ed, asymptotic (e.g., K ®eld) solutions will always lead to unbounded Weibull stress values, since no remote boundary exists. From Eq. (7), the Weibull stress for the elastic stress ®eld remains ®nite as long as m < 4 in spite of the stress singularity at the crack tip. However, if m P 4, the Weibull stress tends to in®nity. The singularity in the Weibull stress may be removed by having a notch of ®nite root radius, q, at the crack tip. Based on the solution for the stress ®eld near a notch in an elastic material [8], the result obtained for the Weibull stress is  mÿ4 L 2m K ^ p ; for m P 4: …9† rw =ro ˆ Cen …m† q Lro The function C^en …m† has been determined numerically and is well approximated by the relation C^en ˆ 0:181=m ÿ 0:02. For values of m < 4, it has been found that Eq. (7) (i.e. the sharp crack solution) gives a good estimate of the Weibull stress for the notch case. The solution for an elastic±plastic (or non-linear elastic) material with a Ramberg±Osgood material law has also been obtained. For such a material, the uniaxial constitutive behaviour is given by  n  r r ˆ ‡a ; …10† o ro ro where a is a material constant, n is the strain hardening exponent, and ro and o are reference stress and strain, respectively. The crack tip stress ®eld for this material is given by the HRR ®eld [9,10] and the following expression for the normalised Weibull stress is obtained: "  2 #1=m ^ rw J U…n† ˆ for m < 2…n ‡ 1†; …11† m kro 1 ÿ 2…n‡1† ao ro Lkn‡1 ^ where U…n† is a function which depends only on the hardening exponent n, #2…n‡1† Z p" ~ 1 f … n; h † ^ dh: U…n† ˆ 2 In 0 ro

…12†

^ ^ The function U…n† can be evaluated numerically and depends strongly on n. For instance, U…5† ˆ 764:54 ^ and U…10† ˆ 3:4671  107 , with In the usual HRR n-dependent constant. From Eq. (11), the Weibull stress for the elastic±plastic HRR ®eld is ®nite as long as m < 2…n ‡ 1† despite the stress singularity at the crack tip. Following a similar method to that used to obtain Eq. (9), an approximate Weibull stress relation can be obtained for a notch in an elastic±plastic material.

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N.P. O'Dowd et al. / Engineering Fracture Mechanics 67 (2000) 87±100 1  mÿ2…n‡1†  n‡1 rw L m…n‡1† J ^ ˆ Cpn …m; n† q ao ro L ro

for m P 2…n ‡ 1†:

…13†

The function C^pn …m; n† in Eq. (13) depends on m and n and has been determined numerically. A close approximation to C^pn …m; n† for the range of n and m values examined in this work is: 1 !2…n‡1†   ^ 10…n ‡ 1† U…n† 1‡ ; for n P 3: …14† C^pn …m; n† ˆ m2 n‡1 For m < 2…n ‡ 1†; the sharp crack solution provides a good approximation to the Weibull stress around the notch. 3. Finite element comparisons Small strain ®nite element analyses have been performed to verify the analytical and semi-analytical Weibull stress expressions under small scale yielding conditions and to examine their applicability to real crack geometries. Further details of these validations are provided in Ref. [7]. A comparison is provided here between the Weibull stress solution and the results from a ®nite element analysis of a CT specimen with a=W ˆ 0:5. Results are presented for the case, a=L ˆ 25, q=L ˆ 0:01 (i.e., a ®nite notch at the crack tip) and n ˆ 5. Fig. 1 shows the elastic±plastic ®nite element results of a notch for three di€erent values of m. The close correspondence with the semi-analytical (small scale yielding) solution is clear. At very low loads, J =a0 r0 L < 20, the component behaves elastically (no sharp crack tip) and the approximate notch plasticity solution (Eq. (13)) overestimates the Weibull stress in this region. In order to predict Weibull stresses accurately over the full range of J levels an interpolation scheme between the elastic and the elastic±plastic regime may therefore be appropriate. At higher loads, the e€ect of specimen geometry may also become signi®cant, i.e. the near tip ®elds may no longer be well represented by the small scale yielding notch distributions from which Eq. (13) is derived.

Fig. 1. Weibull stress for a CT specimen obtained numerically and using Eq. (13).

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However, for the CT specimen this e€ect does not appear to be signi®cant for the load levels examined. For low constraint specimens such close agreement with the small scale yielding solutions may not hold under large scale yielding conditions. In addition to comparing the Weibull stress evaluated from a numerical J value, the dashed line in Fig. 1 also provides a comparison between the estimate of Weibull stress, based on the EPRI estimation method for J [11] in conjunction with Eq. (13), i.e. this line could be obtained directly without the need for an FE solution. Again the good agreement is clear. 4. Discussion of Weibull stress solutions 4.1. E€ect of mesh size on Weibull stress As discussed earlier, under certain conditions, the analytical solutions can predict an in®nite value for the Weibull stress. In practice, this will lead to mesh dependence of Weibull stress ®nite element solutions for sharp cracks. Fig. 2 shows the normalised Weibull stress, obtained from a small scale yielding ®nite element analysis, plotted against crack tip element size for an elastic±plastic material with hardening exponent n ˆ 5. Here the mesh size, i.e. the size of the crack tip element, is normalised by the ®xed outer mesh radius. It may be seen that for m ˆ 30 the calculated Weibull stress does not converge to a constant value as the mesh size is decreased. However, for m ˆ 11 the Weibull stress values are independent of the mesh size when the crack tip element size is less than 10ÿ4 of the outer mesh radius. This mesh dependence, for m > 2…n ‡ 1†, may be eliminated by having a ®nite notch root radius, as discussed earlier, or a ®xed mesh size, both of which limit the maximum stress at the crack tip, and introduce a length scale into the problem. 4.2. Cleavage failure probabilities based on the Weibull stress The Weibull stress is used to determine the probability of cleavage failure. Rather than presenting the results in terms of the Weibull stress, one could equivalently rewrite them as cleavage probability predictions. Substituting the Weibull stress solutions for a power law hardening material into Eq. (2) yields

Fig. 2. Mesh dependence of Weibull stress for a sharp crack.

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(

2 ) J Pf …J † ˆ 1 ÿ exp ÿ m < 2…n ‡ 1† ao ro Lkn‡1 ) (    mÿ2…n‡1†  m n‡1 m n‡1 ro L J C^pn m P 2…n ‡ 1†: ˆ 1 ÿ exp ÿ ru q ao ro L 

kro ru

m

^ U m 1 ÿ 2…n‡1†



…15†

The notch solution has been used for m P 2…n ‡ 1† as the crack solution predicts Pf ˆ 1 in this case. The ®rst term of Eq. (15) is consistent with a similar equation presented in Ref. [2]. The advantage of Eq. (15) is that all quantities (apart of course from m and ru ) can be obtained from uniaxial tensile data. It may also be seen that in the ®rst equation, J is raised to the power of 2 which is consistent with the ASTM E-1921 [12] procedure and a threshold value Kmin ˆ 0. Further modi®cation to the second term of Eq. (15) also leads to a similar result. While the e€ect of crack blunting has not been explicitly accounted for in this analysis, the notch solutions provided, with q taken to be the current crack tip radius, are representative of the solutions which would be obtained from a ®nite strain analysis. This is illustrated in Fig. 3. Here, the Weibull stress solutions obtained from ®nite strain analyses are compared with small strain analyses with the same current notch radius. The close agreement between the two solutions is noted. We can therefore account for notch blunting through the use of the second of Eq. (15). If, under J dominant conditions, the notch radius, q, is replaced by q ˆ rn …J =r0 † where rn is a constant relating the current notch root radius to J [13], the equation reduces to, (    m=n‡1  2 ) m r0 1 J C^pn ‰mÿ2…n‡1†Š=m …16† Pf …J † ˆ 1 ÿ exp ÿ Lr0 ru a0 rn which again is consistent with ASTM E-1921. Note also that if cleavage data is presented in this way, the values of m and ru cannot be obtained uniquely, as discussed in Refs. [4,5], since they both appear only as a scaling parameter for J. Indeed, this conclusion can be obtained immediately from a dimensional analysis. However, if cleavage probability data is available over a range of temperature and if one or both Weibull parameters are assumed temperature

Fig. 3. Comparison of small strain and ®nite strain Weibull stress ®nite element solutions.

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independent, then Eq. (15) or Eq. (16) may be used to determine m and ru . In this work, the values of both m and ru are assumed constant over the temperature of interest, implying no change in cleavage failure mechanism over these temperatures. Recently, it has been argued that such an assumption leads to poor predictions of Weibull stress values over the full transition range [14]. However, in the absence of any strong evidence to the contrary, in this work the temperature independence of m and ru for this material is assumed. Having obtained m and ru , solutions for Weibull stress or failure probabilities can be obtained from the equations presented here.

5. Analysis of failure probabilities for a weld steel An analysis of cleavage failure probabilities for a low alloy ferritic weld steel is next discussed. The failure mode of such steels changes from ductile to unstable cleavage over a relatively narrow temperature range. At low temperatures, the majority of failures are by cleavage while at high temperatures (the upper shelf region) crack growth is primarily due to the growth and coalescence of micro-voids from non-metallic inclusions. Modelling of local fracture processes using continuum mechanics-based approaches [6] provides a mechanistic interpretation of the macroscopic observations and allows the competition between ductile and cleavage mode in the transition region to be examined in a quantitative manner. In what follows, the probability of cleavage failure is determined using the Weibull stress, as discussed. As well as the two parameter Weibull stress, Eq. (1), a three parameter model is also examined, whereby a threshold stress rth is introduced, below which the cleavage failure probability is zero. This modi®ed Weibull stress is de®ned as " rw ˆ

ne X ÿ iˆ1

ri1

ÿ rth


m Vi

#1=m :

Vo

…17†

Ductile tearing is treated in a deterministic way using the Gurson model [15] which prescribes material softening linked to the growth and coalescence of voids, and a ®nal critical void volume fraction, beyond which all stress carrying capacity of the material is lost. Within the ®nite element analysis, this loss of stress carrying capacity corresponds to an increment of crack growth. The yield condition for the voided material is given as  Uˆ

re r0



2 ‡ 2q1 f cosh

ÿ q2

3p 2r0



ÿ
ÿ 1 ‡ q21 f 2 ˆ 0;

…18†

where f is the current void volume fraction, re is the von Mises stress and p, the hydrostatic pressure. The material parameters q1 and q2 depend on the hardening exponent n and on the ratio E=r0 [16], where E is the Young's modulus. In the numerical analysis, the local mesh consists of square elements ahead of the growing crack tip, thus allowing the crack to grow through a region of constant sized elements. The size of the elements ahead of the crack tip, D, implicitly introduces a characteristic length scale into the analysis. The Weibull stress will also depend on a characteristic length scale which as discussed earlier can be considered to be the (current) notch root radius. Since the mesh size controls the notch size in the ®nite element analysis beyond the initiation of ductile tearing, it therefore constitutes the only characteristic length for both fracture modes after the onset of ductile crack growth. The implications of the use of a single length scale to characterise both the cleavage and ductile fracture is discussed in more detail in Ref. [17].

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6. Weld material properties 6.1. Uniaxial tensile properties The weld material data presented in this work were obtained from tests conducted on weld samples [18]. The dependence of the weld material yield stress on temperature was obtained from data for both plate and weld steel and takes the following form [17], r0 ˆ 420 ‡ 48eÿ0:015T ;

…19†

with r0 in MPa and temperature in degrees Celsius. The hardening behaviour of the steel is well represented by power law hardening with the hardening exponent relatively constant over the temperature range, n  7:2. 6.2. Calibration of local fracture parameters Fracture toughness tests were carried out on CT specimens over a range of temperatures. The cleavage parameters, m and ru , are calibrated from the low temperature fracture tests where ductile tearing processes are expected to be inactive and the ductile fracture properties are determined from the high temperatures fracture resistance curves. The cleavage parameter calibration was based on fracture toughness data measured over a range of temperatures, ÿ130°C 6 T 6 ÿ170°C. The maximum likelihood method, based on Eq. (2), was used to calibrate the Weibull parameters, m and ru . A notch root radius of 10 lm was used, thus avoiding any mesh sensitivity in the analysis. Furthermore, no ductile tearing was modelled in the analysis ± at these temperatures, the amount of crack growth before cleavage failure is expected to be very small. For a reference volume of V0 ˆ 1 mm3 and with rth ˆ 0, the calibrated cleavage parameters were m ˆ 30 and ru ˆ 2022 MPa. Fig. 4 shows the data points used in the calibration as well as the ®nal failure probability function

Fig. 4. Calibration of Weibull parameters from cleavage fracture probability data at ÿ131 C 6 T 6 ÿ171 C:

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Fig. 5. Near tip ®nite element mesh used in ductile tearing analysis and de®nition of mesh size D.

Pf …rw †. These values were obtained directly from ®nite element calculations rather than through Eq. (15), though it is expected that similar results would be obtained using either method. It should be pointed out that the value of m obtained is considerably higher than recent values reported for ferritic steels, e.g., see Ref. [4]. Values for the void growth parameters in the Gurson model, q1 and q2 , were obtained from [16] giving q1 ˆ 1:44 and q2 ˆ 0:94. The magnitude of the critical void volume fraction at coalescence, fe , was found to have a relatively small e€ect on the results (cf., [6]) and was taken to be fe ˆ 0:16. The length scale D was associated with the crack tip opening displacement, dt , at the initiation of ductile crack growth (cf., [6]). Such correlation provides a convenient initial estimate for D from fracture toughness data associated with ductile tearing, either directly from dt measurements or via the dt ±J relation [13]. Experimental J-resistance curves were used to calibrate the values of D and f0 . The calibration procedure is done iteratively ®nite element analyses are performed for various values of f0 and D until the experimental J-resistance curve is well approximated. A B A Q U S explicit [19] was used for the analyses, with the standard porous plasticity model, without void nucleation, to model the e€ect of ductile tearing. When the evolving void volume fraction, f, reaches fe the element is removed from the analysis. The characteristic length scale, D, and initial void volume fraction, f0 , obtained were 0:1 mm and 0.003, respectively. The size of the smallest square element in the ®nite element mesh of the CT geometries used in all the ductile crack growth analyses is based on this value of D (Fig. 5). The result of the ®tting procedure is shown in Fig. 6 ± the solid line indicates the experimental, target J-resistance curve [20,21] and the computational ®t obtained with di€erent values of f0 ˆ 0:003 and D ˆ 0:1 mm ®xed are shown by the symbols. It is clear that these values of f0 and D capture the experimental data quite accurately.

7. Analysis of fracture in the transition regime 7.1. Weibull stress results Having determined the material parameters required in the modelling of the ductile tearing and cleavage fracture processes, analyses of fracture in the transition temperature regime, where both failure modes are active, were carried out. The computed evolution of the Weibull stress is shown in Fig. 7. (As the e€ect of ductile tearing is included in these calculations, the analytical expressions in Section 2 are not applicable to these analyses.)

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Fig. 6. Comparison between experimental and numerically generated J resistance curve.

Fig. 7. Normalised Weibull stress obtained from ®nite element analysis.

Also included in Fig. 7 are lines corresponding to constant cumulative cleavage probabilities of Pf …rw † ˆ 0:05, 0.50 and 0.95. (The Weibull stresses corresponding to these probabilities can be obtained from Eq. (2)). The fracture toughness corresponding to a given failure probability is found by the intersection of these Pf lines with each of the rw ±J curves. As expected these values increase with increasing temperature.

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For temperatures above 50°C, it can be seen from Fig. 7 that the rw ±J curves does not intersect the Pf ˆ 0.95 line. Above this temperature, a cleavage failure probability of 95% is not reached irrespective of the applied load. The analysis instead predicts that a ductile instability will occur and the crack grows in an unstable fashion with no further increase in load. 7.2. Cleavage fracture probabilities In Fig. 8, the experimental KC ±KJ =Da data, [18], versus temperature are compared with the numerical predictions over the full temperature range. The notation KJ =Da implies that there is some (unspeci®ed) amount of ductile tearing associated with the data points. Predicted values of fracture toughness corresponding to cleavage failure probabilities of Pf ˆ 0:05, 0:5 and 0.95 are shown as solid lines. These results reveal that the majority of the experimental data falls between the 0.05 and 0.95 probability lines. Also included on Fig. 8 is a curve showing the computed K values corresponding to crack growth of 0.2 mm at various temperatures, labelled KJ …Da ˆ 0:2†. The value of Da ˆ 0:2 mm is chosen based on a typical minimum detectable amount of crack growth and therefore indicates the K value for initiation of ductile tearing. This curve is essentially temperature independent in the upper-shelf region. The curve labelled as KJ …P ˆ Pmax † on Fig. 8 represents the K value at which ductile instability is predicted to occur at each temperature. Failure which takes place above this line would be expected to always occur by unstable ductile tearing. In the region between the initiation and unstable tearing lines, ductile tearing will begin if cleavage instability has not already intervened. Therefore, if cleavage occurs within this region, it will be accompanied by prior tearing. The cleavage fracture probability, Pf , corresponding to a 0.2 mm crack extension in the CT specimens is ˆ 0). At low temperatures Pf  1:0 and at approximately ÿ30°C, plotted in Fig. 9 (the curve labelled r=r100 0 it can be seen that the cleavage failure probability starts to decrease rapidly as temperature increases. It should however be noted that even at temperatures of 200°C, the cleavage fracture probability, at J levels

Fig. 8. Experimental and predicted fracture toughness of weld material from CT tests.

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Fig. 9. Predicted cleavage failure probabilities for a range of threshold stress values.

corresponding to the initiation of ductile tearing, is predicted to be on the order of 12% though no cleavage failures were reported in the fracture toughness tests at this temperature [18]. Diculties in predicting cleavage failure probabilities over a wide temperature range have been encountered by many authors, e.g., Ref. [14]. In some cases, this has been explained by a change in cleavage mechanism at high temperatures ± large inclusions which can act as cleavage initiators at low temperatures, debond from the ferrite matrix at high temperatures and thus are no longer available as cleavage sites. This results in a bi-modal probability distribution (see e.g., Ref. [22], where di€erent failure mechanisms are used to explain a similar trend). For the weld material examined here, however, the non-metallic inclusions consist mostly of de-oxidation products such as silicates or oxides, in contrast with the (generally larger) MnS-type inclusions found in plates of the same material. As argued in Ref. [17] because of their relatively small size, these inclusions do not act as cleavage initiators at any temperatures and thus the cleavage mechanism is not expected to change over the full temperature range. The use of a threshold stress term, rth in Eq. (17) can also result in reduced cleavage probabilities at high ˆ 2, 3 and 4, temperatures. Analyses have been performed for three di€erent values of rth , namely rth =r100 0 is the value of the yield stress at 100°C (Eq. (19)). Note that a new calibration of the cleavage where r100 0 parameters m and ru for each value of rth is required. The increase in rth has a pronounced e€ect on the values of the cleavage failure probabilities Pf at higher temperatures as shown in Fig. 9. The much sharper transition from high to low cleavage probabilities may also be noted. In principle, a value of rth could be selected from this type of analysis so that the resulting cut-o€ temperature would agree with experimental observations of only ductile fracture modes. However, such judgement could not be made in this study from the limited data available. Scatter in the ductile properties can also a€ect cleavage failure probabilities in the transition and upper shelf region. This issue has been examined computationally in Ref. [17], where it was found that when such variability is taken into account, cleavage failure probabilities on the upper shelf can alter signi®cantly. Two such cases are shown in Fig. 10, namely when the initial void volume fraction, f0 is taken to be zero (corresponding to an inclusion free weld) and to be 0:3% (close to the mean value for the weld data). Note that the probability for f0 ˆ 0 could have been obtained directly from Eq. (15). It is clear that an increase in f0 leads to a reduction in the cleavage failure probability in the transition and upper-shelf temperature region.

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Fig. 10. Predicted cleavage failure probabilities for two di€erent values of initial void volume fraction, f0 .

8. Conclusions Analytical and semi-analytical Weibull stress expressions have been presented for sharp cracks and notches in terms of the remote loading parameter, J or K, and material properties. It has been shown that Beremin type cleavage failure probability expressions in terms of J or K and material properties can be obtained by the use of the above closed form Weibull stress solutions. The limit of applicability of the proposed relations has been discussed. The Weibull stress has been used to quantify the cleavage failure probability fracture of a ferritic steel weld throughout the brittle-to-ductile temperature transition region. The Weibull parameters were assumed to be temperature independent over the temperature range examined. Good correlation is found between the experimental data and the numerical results throughout the transition regime with the use of a threshold stress leading to decreased cleavage failure probabilities at high temperatures. Acknowledgements Support for this work by the IMC (Industry Management Committee of the United Kingdom) is gratefully acknowledged. The authors would also like to thank Dr. R. Moskovic of BNFL MAGNOX Generation and Dr. R. Ainsworth of British Energy for their assistance with this work. Useful discussions with Prof. R.H. Dodds of University of Illinois, Urbana and Prof. G.A. Webster of Imperial College are also acknowledged. The A B A Q U S programme was provided under academic licence by Hibbitt, Karlsson, and Sorensen Inc., Providence, Rhode Island. References [1] Freudenthal, A.M. Statistical approach to brittle fracture. In: H. Liebowitz, editor. Fracture: an advanced treatise, vol. 2. Academic Press: New York, 1968. p. 591±619.

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