The applied fracture model12 defines a conditional probability of fracture assuming that the distribution of flaws in the material follows a statistical distribu­tion. The conditional probability term takes into account the possibility of void formation (blunting of a crack initiator) and the crack arrest-propagation event (Figure 4). The fracture event is controlled in the model by assessing the criticality of a single crack initiator from the weakest link principle. The basic elements of the methodology — the scatter definition and the specimen size correction6,7 — are based on this cleavage fracture model. It is assumed that the material has uniform macroscopic properties and

Pr{I} = Probability of cleavage initiation Pr{V} = Probability of void initiation Pr{O} = Probability of ‘no event’

Pr{I/O} = Conditional probability of cleavage initiation (no prior void initiation)

Pr{V/O} = Conditional probability of void initiation (no prior cleavage initiation)

Pr{P/I} = Conditional probability of propagation (in the event of cleavage initiation)

Pr{A/I} = Conditional probability of arrest (in the event of cleavage initiation)

Figure 4 Definition of the conditional probability of cleavage fracture. Reproduced from Wallin, K.; Laukkanen, A. Eng. Fract. Mech. 2008, 75(11), 3367-3377.

that no global interaction exists between the crack initiators. An overview (basic equations) of the cleav­age fracture model (also called the Wallin, Saario, Torronen (WST) model) is described next.

The conditional probability of cleavage initiation, Pr{I/O}, is expressed as a product of the probability of having a cleavage initiation and that of not having a void initiation as follows:

Pr{I/O} = Pr{I}(1 — Pr{V/O}) [7]

Equation [7] can be approximated as:

Pr{I/O} « Pr{I}(1 — Pr{I}) [8]

The WST model expresses Pr{I/O} as the product of the particle fracture and nonfracture probabilities so as to take into account the previously broken parti­cles which do not contribute to the cleavage process as follows:

The probability of particle fracture is described by a Weibull-type dependence accounting for particle size (d) and particle stress (spart) as follows:

Pfr = 1 — exp

where dN and s0 are scaling factors.

The particle size distribution, P{d}, is given by eqn [12], which describes the size distribution with two parameters, the average particle size (d) and the shape factor n. For pressure vessel steels, the shape factor appears generally to be in the range 4-6.

P{d}

A detailed description of the WST model is given in Wallin et at}2 and Wallin and Laukkanen.1 The recent numerical validation ofthe model is presented in Wallin and Laukkanen.13

define the curves for the mean and lower bound fracture toughness. If the dataset covers several test temperatures, T0 is found as a solution of equations giving the maximum likelihood estimate to this value. Other than the LEFM testing standards for fracture characterization, there is no specified limit for the minimum specimen size. However, the number of specimens to be tested has to be at least six and generally increases when the specimen size is decreased in order to produce a statistically accept­able confidence level for the estimate. The method also includes a censoring procedure that allows the use of adjusted invalid test data, which contain statis­tically useful information.

The probability of cleavage fracture initiation is described as a three-parameter Weibull distribution, which defines the relationship between the cumula­tive failure probability (Pf) and the fracture toughness level before or at KI as follows:

Pf (KJc < K) = 1 — exp where Kmin is the theoretical lower bound fracture toughness, set normally to 20 MPa Vm for steels with yield strength from 275 to 825 MPa, and K0 is the scale parameter corresponding to Pf = 63.2%. The Weibull exponent is assumed to have a constant value equal to 4 based on theoretical and experimental arguments.7

The measuring capacity (maximum KJc) of a spec­imen depends on its dimensions (ligament size) and the material yield strength as follows:

Eb0s ys

Kjc(limit) = M(1 — n2)

where E is the modulus of elasticity, n is Poisson’s coefficient, b0 is the initial specimen ligament length (specimen width minus the initial crack depth, W — a0), M is the constraint value (usually set equal
to 30), and sys is the material yield strength at the test temperature. Toughness values exceeding this mea­suring capacity require censoring and are set at the level of maximum Kjc.

The expression for predicting the specimen size effect is based on the cleavage fracture model. The fracture toughness (Кед) corresponding to the desired specimen thickness or crack front length (Bx) is obtained from the values of Kc(o) and B0, respectively, as follows:

Kjc(x) = Kmin + (Kjc(0) — Kmin) [15]

When analyzing measured data according to ASTM E 1921, the size adjustment is normally made to 1 in. (Bx = 1 in. or 25.4 mm). Note that the influence of side grooves on the specimen thickness is ignored. After the size adjustment is made for each Kjc mea­surement, the data for different size specimens can be described as one population following the same Master Curve form and scatter as shown in Figure 5. The aforementioned formula applies in the transi­tion range, and it is not necessary to perform the size conversion at temperatures below (T0 — 50 °C), because the size effect diminishes (see the fracture toughness curve 50 MPa Vm shown in Figure 6).

The procedure for estimating the maximum like­lihood solution for T0 from data measured at various temperatures was published in 199514 and added to the second and later revisions of ASTM E 1921. The value of T0 is solved iteratively from the fol­lowing equation, which includes a factor d for data censoring:

di exp{0.019[T, — T0]}

11 + 77exp{0.019[Tj — T0]}

ys (J) — Kmin)4exp{0.019[Ti — T0]} 1=1 (11 + 77exp{0.019[Ti — T0]})5

B (mm)

where is the size-adjusted fracture toughness measured at temperature T) and d, is the censoring coefficient: d, = 1 if the KjcW datum is valid (less than the limit determined from eqn [14]) or d= 0 if the KJc(i) datum is not valid but may be included as censored data.

When the value of T0 has been solved, the fracture toughness curves for specific levels of fracture prob­ability are obtained from the following equations for probability level 0.xx

KJc(0.xx) = 20 +

{11 + 77 exp[0.019(T — T>)]} [17]

The mean curve (50% failure probability) for

25.4 mm specimen thickness is obtained as a function of temperature from

KJc(mean)1 T = 30 + 70 exp[°.°19(T — T0)] [18]