## Abstract

Ferritic steels, which are typically used for critical reactor components, including reactor pressure vessels (RPV), exhibit a temperature-dependent probability of cleavage fracture, termed ductile-to-brittle transition. The fracture process has been linked to the interaction between matrix plasticity and second-phase particles. Under high-enough loads, a competition exists between cleavage and ductile fracture, which results from particles rupturing to form microcracks or particles decohering to form microvoids, respectively. Currently, there is no sufficiently adequate model that can predict accurately the reduced probability of cleavage with increasing temperature and the associated increase of plastic deformation. In this work, failure probability has been estimated using a local approach to cleavage fracture incorporating the statistics of microcracks. It is shown that changes in the deformation material properties are not enough to capture the significant changes in fracture toughness. Instead, a correction to the fraction of particles converted to eligible for cleavage microcracks, with an exponential dependence on the plastic strains, is proposed. The proposed method is compared with previous corrections that incorporate the plastic strains, and its advantages are demonstrated. The method is developed for the RPV steel 22NiMoCr37 and using experimental data for a standard compact tension C(T) specimen. The proposed approach offers more accurate calculations of cleavage fracture toughness in the ductile-to-brittle transition regime using only a decoupled model, which is attractive for engineering practice.

## Introduction

Ferritic steels are widely used for reactor pressure vessels and other pressure boundary components in nuclear reactors due to their high strength and good fracture toughness properties. However, like other body-centered cubic metals, they exhibit a ductile-to-brittle transition (DBT), which leads to an increased probability of cleavage fracture with decreasing temperature. At sufficiently low temperatures, termed the lower shelf, pure cleavage occurs. As temperature increases some ductile crack growth precedes cleavage in the transition regime, where the amount of plastic deformation increases with temperature. The transition temperature is the temperature at which ductile growth is observed first. At high enough temperatures, termed the upper shelf, the material fails at higher toughness in a ductile manner. In unirradiated material, the transition usually occurs at approximately—100 °C [1,2]. However, environmental factors such as irradiation or presence of embrittling species, due to material ageing, can alter the steel properties in service and progressively shift the brittle-to-ductile transition temperature to higher values, e.g., see Ref. [3]. In addition to this, a large scatter in the measured fracture toughness is observed in the transition regime—a consequence of the stochastic nature of the cleavage fracture process determined by the material's microstructure. The random distribution of second-phase particles, such as carbide and sulfide inclusions, has been shown to have a critical role [4,5]. Under high-enough loads, a competition between particles rupturing to form microcracks and particles decohering to form microvoids takes place. These processes are prerequisites for, respectively, cleavage and ductile fracture. Current fitness-for-service assessments of engineering structures are highly conservative. If we were able to predict more accurately the fracture toughness behavior, this would improve the economics of current reactor operation by, for example, enabling component lifetime extensions, more refined safety margins and more efficient future designs.

Local approaches to fracture (LAF) consider cleavage as a probabilistic event based on local fracture criteria for an unstable critical Griffith-like microcrack in a volume element and consider the probability density distribution of these microcracks [6]. Thus, these approaches were developed as a mechanistic micromechanical method for modeling cleavage as an alternative to, but also to complement, highly conservative global approaches, based on a single parameter such as $KIC$, $JIC$, or crack tip opening displacement (CTOD). The majority of LAF cleavage models are based on the Beremin model [7], which was initially developed to model brittle fracture using Weibull statistics of microcracks. The Weibull stress was introduced as a probabilistic fracture parameter, which relates the fracture processes at the microscale and the macroscopic fracture parameters such as the J-integral. The existing local approaches to cleavage usually correctly predict the lower shelf toughness, but with increasing temperature toughness is increasingly underestimated.

Many studies [811] have applied LAF to predict fracture in ferritic steels, but currently there is no sufficiently adequate model, which correctly reproduces the competition between cleavage and ductile damage in the DBT regime. Among some of the encountered issues are temperature and geometry-dependent model parameters, and the need for cross-calibration between different geometries [6,12]. Furthermore, the inherent locality of the cleavage process has made the task of correctly coupling the statistical variability of resistance to cleavage of the material with the plastic deformation very difficult.

The current work utilizes a cleavage fracture model, first introduced by the Beremin group [7], to estimate brittle failure probability. Based on the mathematical theory behind the Beremin model, which represents cleavage as a Poisson point process, we present a new temperature-dependent correction to the cleavage initiators density utilizing the effects of plastic strain. The new correction is compared with previous models for the plastic strain effects. We show that the mechanical fields as calculated using the finite element method are not sufficient to capture the large temperature dependence of the fracture toughness in the DBT regime. Furthermore, micromechanical processes must be affecting the fraction of eligible microcracks as temperature changes.

## Statistical Model of Cleavage Fracture

### Theoretical Basis of Local Approaches to Cleavage.

Cleavage fracture in ferritic steels initiates from elastic-brittle second-phase particles in the material, for example, carbides or sulfide inclusions, which rupture under high tensile stress of the plastically deforming surrounding matrix—a process referred to as plastic overload. Experimental observations by Gurland [4] showed that the number of broken particles increased linearly with increasing plastic strain . Other studies also showed a growth of microcracks with an increase of plastic strain. The statistical nature of cleavage originates from the spatial random distribution of second-phase particles of variable sizes, which are converted into microcracks of variable sizes. Depending on the local mechanical fields, a particle of a given size may: remain intact, i.e., continue deforming with the matrix; or detach from the matrix forming a void, which grows with further plastic straining; or break, forming a microcrack, which blunts due to local conditions and continues as in the second scenario; or break, forming a microcrack, which can propagate to cleave the component. Hence, it is important to distinguish between particles, voids/microcracks not eligible for cleavage initiation, and microcracks eligible for cleavage and include only the latter into the local approach calculation. Additional factors that might affect the behavior of a particle include the shape of the particle, and its orientation with respect to the load, the strength of the particle–matrix interface and the relative misorientation between the crystal structures of the particle and the matrix [13].

The existing probabilistic models assume cleavage to be approximated by an inhomogeneous spatial Poisson point process, which is a model of events that happen completely at random in mathematical space, for example, see Ref. [14]. In the case of the cleavage process, points correspond to microcracks eligible for cleavage. There are three main characteristics that define a Poisson process. First, failures of nonoverlapping volumes are completely independent; second, the probability of failure of a volume element $δV$ is proportional to its volume; and third, the probability of more than one failure in the volume element $δV$ is assumed to be zero. Based on the first assumption, weakest link theory states that the survival probability of the solid is equal to the product of survival probabilities of all volume elements. If the fracture process zone ahead of the crack tip is divided into statistically independent and uniformly stressed unit volumes, $δV$, which contain uniformly distributed microcracks with probability density, $fc(a)$, the probability of failure of a volume element $δPf$, and that of the solid $Pf$ are expressed as
$δPf=μcδV=(1V0∫ac∞fc(a)da)δV, Pf=1−exp(−∫VμcdV)$
(1)

where $μc$ is the number density of unstable microcracks and $ac$ represent a critical microflaw size. The reference volume $V0$ was approximated as the volume of a spherical grain with a $10 μm$ radius, which corresponds to the approximate grain size in reactor pressure vessel (RPV) steels [13]. Here, we included volume elements with a plastic strain larger than $0.2%$.

Calculations of a component's probability of cleavage fracture by this equation will be as realistic as is the assessment of the number density of cleavage initiators, $μc$, i.e., how particular mechanical fields determine the critical microcrack size, $ac$, and the probability density of eligible microcrack sizes, $fc(a)$. This is a challenging task due to the limited experimental data available. Experimental works [4,5] demonstrate a clear link between the level of plastic strain and the number of microcracks formed by rupturing particles, but do not provide sufficient evidence for determining $fc(a)$ and particularly its dependence on temperature. This lack of knowledge makes it necessary to test different possibilities for the functional form of $fc(a)$. Since the microcracks originate at second-phase particles, it is logical to relate the probability density of eligible microcrack sizes to the probability density of particles. Taking a step back to the mathematical theory of the Poisson point process, one possible random operation that acts upon a Poisson process is called thinning, which removes or retains points based on some probabilistic rule [15]. Both the removed and the retained points form two new Poisson processes. In the case of the cleavage process, the retained points represent the particles that have formed microcracks eligible for cleavage. And so, we can assign a spatially dependent thinning function, $θ(x)$, which represents the fraction of particles converted into eligible microcracks for an infinitesimal volume located at position $x$.

Next, considering that $fc(a)$ should be zero in the absence of plastic strains, the Beremin group's original suggestion was that within the plastic zone ahead of a macroscopic crack $fc(a)$ was approximated by a power law $(a0/a)β$ representing the tail of the particle size distribution. Here, $a0$ and $β$ are the scale and shape parameters, respectively. This is equivalent to the assumption that all particles in the tail are converted into eligible microcracks with the onset of plasticity. If it is assumed that the size distribution of eligible microcracks has the same shape as the size distribution of particles, i.e., the same $β$, then
$fc(a)=θ(x)(a0/a)β$
(2)
The critical microcrack size is calculated from a criterion for unstable growth, e.g., for a penny-shaped crack formed by rupture of a particle the critical size is
$ac=π E γ2(1−ν2)σ12$
(3)
where $E$ and $ν$ are, respectively, the elastic modulus and Poisson's ratio of the steel, $σ1$ is the maximum principal stress, and $γ$ is a measure of fracture energy. The free surface energy of the steel, $γs$, is assumed to be a good approximation for the fracture energy in the case of limited plasticity. Using Eqs. (3)(5) and integrating leads to an expression for the number density of cleavage initiators as a power law of the effective stress
$μc=1V0θ(x)(σ1σu)m$
(4)
where the shape parameter is $m=2β−2$, and the scale parameter $σu$ collects the elastic properties, the constant surface energy, and the scale of the power law. Then, substitution of Eq. (4) into Eq. (1) yields a two-parameter Weibull distribution, Eq. (5), of an integral stress parameter, $σw$, known as Weibull stress, Eq. (6). The Weibull stress is considered as a crack driving force for cleavage fracture [16], where at a critical value of the Weibull stress, a cleavage crack becomes unstable and propagates catastrophically
$Pf(σw)=1−exp [−(σwσu)m]$
(5)
$σw=[1V0∫Vθ(x) σ1mdV]1/m$
(6)

### Plastic Strain Effects.

One of the main disadvantages of the Beremin model is the treatment of the effect of plastic strains, as pointed out by Ruggieri and Dodds [17]. In the original Beremin model, the fraction of particles converted into eligible microcracks, i.e., the thinning function $θ(x)$, is expressed as a function of the plastic strain: $θ(εp)=H(εp)$, where $H(εp)$ is the Heaviside step function. In other words, yielding of the material causes all eligible microcracks to nucleate. However, as previously mentioned, experimental evidence [4,5,18,19] about the physical processes controlling cleavage show that there is a more complicated dependence between the plastic strain and the generation of microcracks eligible for cleavage. For instance, Gurland observed a linear dependence between the two parameters, although there was limited data at a single temperature (room temperature), whereas Lindley et al. [18] suggested a power law relationship. This possibility was explored using the local approach by Gao et al. [20], Kroon and Faleskog [21], and Ruggieri and Dodds [17]. The thinning function takes the form
$θ(εp)=εpη$
(7)

where $η=0$ recovers the original Beremin model and $η=1$leads to a linear dependence on the plastic strain. Ruggieri and Dodds [17] explored the effect of the $η$ on the Weibull stress, which decreased with increasing the exponent value due to the decrease of number of microcracks eligible to propagate.

Another type of correction that takes into account the plastic strains effect that was first proposed by Ruggieri and Dodds [16,17] by the example of an earlier study by Bordet et al. [22] represented an exponential dependence of the eligible microcracks on the plastic strain
$θ(εp)=1−exp[−λcεp]$
(8)
where the thinning function is represented by a Poisson distribution with a constant material parameter $λc$ and represents the average rate of fractured particles eligible for cleavage for a small strain increment. Consequently, values between 0.1 and 1 were considered for the constant $λc$, which were found to have a minor effect on the corrected Weibull stress. In this study, we extend this model by introducing a temperature dependence of the $λ$ parameter. We show that this temperature dependence is required in order to correctly represent the temperature effect on cleavage fracture. Moreover, the required values of the $λ$ parameter exhibit an exponential dependence on temperature. The thinning function is then represented as
$θ(εp)=1−exp[−λ(T)εp]$
(9)

## Methodology

### Material Properties.

The pressure vessel steel 22NiMoCr37, also known as euro reference material A, was considered for the development of the model. The chemical composition of the material is given in Table 1. The fracture toughness properties within the lower shelf and DBT are available from the Euro fracture dataset [23]. Data for a standard 1T compact tension, C(T) specimen with width $W$ = $50$mm, thickness $B=25$mm, and crack length $a=25$mm, at four temperatures: $−154$ °C, $−91$ °C, $−60$ °C, $−40$ °C, are used for the analysis in this work. The experimental fracture toughness values, $Jc$, have been sorted in ascending order, so that $Jck≤Jck+1$ for $k=1,…,N−1$, where $k$ is the ranking number and $N$ is the total number of measurements. The cumulative probability of cleavage fracture at each data point is provided by the median ranking $F(Jci)=(i−0.3)/(N+0.4)$. The cumulative probabilities for the four temperatures are shown in Figs. 5(a) and 5(b). These are compared with a Weibull fit using the maximum likelihood (ML) method.

Table 1

Chemical composition of 22NiMoCr37 [23]

CSiPSCrMnNiCuMo
0.210.240.0030.0040.0030.820.790.0490.56
CSiPSCrMnNiCuMo
0.210.240.0030.0040.0030.820.790.0490.56
Deformation properties of the material at the considered temperatures are given in Table 2, where $E$ is the Young's modulus, $σ0$ is the proportionality stress, $σUTS$ is the ultimate tensile strength, and $n$ is the power law hardening exponent [24]. $σu$ and $εu$ correspond to the true stress and strain, at which $σUTS$ occurs. The Poisson's ratio $ν$ was found to be independent of temperature with a value of $0.3$. Finally, a flow stress, $σf$, is defined as a weighted average between $σ0$ and $σu$ by the following computation ($ε0=σ0/E$):
$σf=1εu−ε0∫σ0σuσdε$
(10)
Table 2

Deformation properties of 22NiMoCr37 at the considered temperatures

Ti (ºC)E (MPa)σ0 (MPa)σUTS (MPa)nσut (MPa)σf (MPa)
−154219,860768.3856.316.812908.6848.5
−91214,190594.9727.111.978790.5723.6
−60211,400544.7686.511.030751.5683.5
−40209,600520.4665.810.596731.2663.0
Ti (ºC)E (MPa)σ0 (MPa)σUTS (MPa)nσut (MPa)σf (MPa)
−154219,860768.3856.316.812908.6848.5
−91214,190594.9727.111.978790.5723.6
−60211,400544.7686.511.030751.5683.5
−40209,600520.4665.810.596731.2663.0

### Finite Element Analysis.

Three-dimensional models of a quarter standard compact tension specimen, 1 T C(T), with finite crack tip radii were used in the large strain finite element analyses, as implemented in abaqus (version 2017). The crack tip radii $ri=1, 10, 15$, and $30$μm for the corresponding $Ti$, were chosen proportional to the maximum crack tip opening displacement obtained using a boundary layer model with sharp crack tips [25]. Boundary layer model calculations with models with smaller (half) and larger (double) radii showed a negligible effect of selection on the cleavage crack driving forces. Figure 1(a) shows as an example the quarter-C(T) model with a $30$-μm radius, where the applied symmetry conditions are denoted. To speed up convergence, the model was loaded by applying displacement increments on the edge of a purely elastic wedge, which was used as a model for the experimental loading pins, typically made of a high yield strength maraging steel [26]. Figure 1(b) shows the focused mesh near the crack tip—the 90-deg segment was divided in six equal-angle segments, and the element thickness ahead of the crack front increases with the thickness of the first element set to between two and three times the crack tip radius. Furthermore, a higher density mesh was used near the outer free surface of the specimen with $20$ variable thickness layers defined over the half-thickness in order to capture stronger variations in the stress and strain distribution. The $J$ integral is obtained at a contour in the same position of the specimen for all temperature cases. Computationally, the material is modeled as elastic-plastic and the flow properties were provided in Abaqus in a tabular form. Figure 2 shows the normalized maximum principal stress and the equivalent plastic strain fields ahead of the crack tip.

Fig. 1
Fig. 1
Close modal
Fig. 2
Fig. 2
Close modal

## Results and Discussion

First, we compare the behavior of the considered thinning functions as shown in Fig. 3. All thinning functions dependent on the plastic strain agree with the expected behavior of values between 0 and 1. These two values correspond to two opposite conditions—no particles and all particles converted into eligible microcracks, respectively. However, Eqs. (4) and (5) produce functions, which are dependent on the temperature only through the difference in the plastic strain fields between the corresponding temperature finite element analyses models, and so there is not a great difference between the thinning functions ahead of the crack, shown in Figs. 3(a) and 3(b). In contrast, by introducing a temperature-dependent material parameter, we observe a much greater difference exhibited by the thinning function in Eq. (9), as seen in Fig. 3(c).

Fig. 3
Fig. 3
Close modal

Figure 4 shows how the Weibull stress, scaled with the flow stress at the corresponding temperature, varies for a range of values of the Weibull shape parameter. In plot (a), the original Beremin model with the Heaviside step function is applied—the Weibull stress increases with decreasing temperature. And, although as the shape parameter is increased, that difference decreases, there is no shape parameter within reasonable limits, for which the Weibull stress at different temperatures can be made equal. This observation suggests that the mechanical fields alone cannot provide sufficient information for a model to be able to predict the change of toughness with temperature. As seen in Figs. 2 and 3, there is no sufficiently great difference in the mechanical fields ahead of the crack tip with changing temperature. The biggest differences are in the material volumes over which these stresses and strains act (note, that distances are normalized). These two fields play a central role in different existing formulations of the Weibull stress. Other crack tip fields, such as the stress triaxiality and the Lode parameter, which could be incorporated in such formulations, similarly show small differences between different temperature cases. It can be noted at this point that a formulation based solely on the mechanical fields might not be able to account for the measured differences in cleavage fracture toughness.

Fig. 4
Fig. 4
Close modal

In Fig. 4(b), we observe the Weibull stress calculated with different $V0$ at each temperature, specifically scaling with the square of the initial radius. Although assuming different scaling volumes is not physical, the results make a good discussion point. Scaling the volume with the square of the initial radius, which scales with the CTOD at J0, makes the Weibull stress with given shape (approximately) independent of temperature. The result at $T=−40 °C$ appears to differ slightly, possibly due to the proportionality of the initial crack tip radius to the CTOD being slightly smaller than the corresponding proportionality for the models at the other temperatures. The good coincidence of the Weibull curves suggests that the biggest effect of temperature is not so much on the functional form of the effective stress, but on the functional form of the density of eligible microcracks. Furthermore, for the same percentile $Jp$, the number of cleavage initiators is equal across different temperatures, but they are spread across a different plastic zone volume, i.e., the volume density of cleavage initiators changes.

Figures 4(c) and 4(e) show the Weibull stress normalized by the flow stress, at the corresponding temperature, calculated using a constant volume $V0$ and Eqs. (4) and (5), respectively. Again, the curves do not intersect so it is not possible to obtain a Weibull shape parameter that is valid for all temperatures. The normalized Weibull stress difference compared to that at the lowest temperature are shown in Figs. 4(d) and 4(f), which result in a significant difference of up to 70% between the different curves. These analyses show that no cross-calibration could be applied using the existing models based on a plastic strain correction in order to account for the temperature dependence on fracture toughness, as it is done, for instance, when correcting for the geometry constraint effects [12].

Figure 4(g) shows the Weibull stress calculated using a constant volume $V0$ and a correction to the fraction of particles converted into cleavage microcracks described by Eq. (9), where the $λ$ parameter is assumed to be a function of temperature. The corresponding values of $λ$, which make the scaled Weibull stress independent of the shape parameter are also denoted. The scaled Weibull stress curves as a function of $m$ appear to be very close to coinciding. The $λ$ parameter is dependent on temperature through the micromechanisms that are active at a given temperature. There is a significant drop of four orders of magnitude between the fitted value of $λ$ at the lowest and at the highest temperature, which corresponds to a large decrease in the fraction of particles converted into eligible microcracks as the temperature increases. The number of microcracks appears constant across temperatures, however, spread across a different volume of material. In Fig. 4(h), the normalized Weibull stress difference is shown to be within 6% of the curve at the lowest temperature, which is an excellent agreement between the curves and within the expected experimental error of the material mechanical properties.

Next, it is demonstrated how the probability of fracture can be predicted at any temperature, given a series of fracture toughness tests at a certain temperature. Figure 5 shows the Weibull stress as a function of $J$ for the four temperatures. The Weibull shape parameter was set to $5.4$, which corresponds to the experimentally measured distribution of particle sizes. The dotted line denotes the value of the Weibull stress at the characteristic $J0$, at which the probability of cleavage fracture is $F(Jc≤J0)=0.632$, using the experimental data at $T=−91 °C$. The points where this line crosses the Weibull curves correspond to the $J0$ value, since the scaled Weibull stress is set to be independent of temperature at $J0$. The corresponding characteristic $J0$ at the other temperatures as calculated based on the experimental data available and the prediction from Fig. 5 are shown in Table 3. For all three temperatures, the predicted values were found to be in good agreement with a slight underestimation of the experimental values. Therefore, the proposed calibration method only requires fracture toughness tests at a selected temperature, combined with deformation properties for any other temperature of interest. Weibull stresses can then be computed and $J0$ estimated using an FE model and numerical analysis.

Fig. 5
Fig. 5
Close modal
Table 3

Comparison between characteristic fracture toughness J0 calculated based on a Weibull distribution fit to the experimental data using the maximum likelihood method, and predicted using the proposed method based on the fracture toughness experimental data at T = −91 °C

T (ºC)$J0 EXP$ (Nmm−1)$J0SIM$ (Nmm−1)
1547.76.6
−9157.6
−60112.8107.4
−40235.3216.9
T (ºC)$J0 EXP$ (Nmm−1)$J0SIM$ (Nmm−1)
1547.76.6
−9157.6
−60112.8107.4
−40235.3216.9

The data at $T=−91$ °C were further used to fit a two-parameter Weibull distribution and calibrate a value for $σu=2.97σf$. It was then possible to estimate the cumulative probability functions of the experimentally measured $Jc$ for each of the other temperatures, as seen in Figs. 6(a)6(d). The experimentally measured cumulative probabilities are given with solid circles, the simulated results are represented by dashed curves, and the Weibull fits using the maximum likelihood method are shown using solid lines. In all temperature cases, Weibull distributions have not been rejected with the ML method, which suggests that there is a sufficiently large experimental dataset available. Since all fracture toughness data used have been derived with high-constraint geometries, the Weibull shape parameter is expected to be close to 2 (theoretically $α=2$). Maximum likelihood with fixed $α=2$ has also been used to determine the corresponding characteristic toughness. The obtained values of J0 as well as the Weibull fits with $α=2$ are also given in Fig. 6. Notably, the experimental data at temperatures −154 °C and –60 °C are not represented well with the theoretically expected shape parameter. This suggests, in the first instance, certain skewness of the experimental data points toward lower fracture toughness values at these temperatures. The curves represented by the current model using an exponential dependence on the equivalent plastic strain provides a good estimation of the fracture toughness probability distribution and coincide with the ML estimates, particularly at high fracture toughness values and at temperatures above $T0$. Across all temperatures, there is a small underestimation of the experimental results, which is preferable to an overestimation. At the very low temperature, there is a possibility for cleavage to initiate from pre-existing defects in the material, which are not captured by the model. Furthermore, temperature dependence of the fracture toughness was plotted together with a comparison of prediction by the Master Curve and the developed model, as seen in Fig. 7. The model predicted values follow better the experimental data compared with the Master Curve. In particular, in the DBT region, the Master Curve overpredicts the probability of cleavage.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

## Conclusions

A local approach to cleavage fracture utilizing the statistics of microcracks combined with finite element analysis were applied in calculating the cleavage failure probability of a ferritic steel in the ductile-to-brittle regime. The findings of the study can be summarized as follows:

• It was demonstrated that the change in the material deformation properties with change in temperature is insufficient to capture the change in fracture toughness across temperatures.

• An exponential correction to the density of microcracks eligible for cleavage, which is dependent on a material property, that changes with temperature, was proposed. It represents a thinning function of the Poisson process, which creates a new Poisson process with a subset of the points of the original Poisson process.

• The proposed correction was compared with plastic strain corrections in the literature, and it is clearly shown that the new correction possesses the correct physical behavior as a function of temperature.

• The developed procedure for probability of cleavage estimation would only require experimental fracture toughness tests at a given temperature, and deformation properties at any other temperature of interest.

• Very good agreement was shown between the predicted and experimentally measured characteristic fracture toughness as well as the predicted and experimentally fitted probability distributions.

• Future work will focus on investigating the reason behind the observed exponential temperature dependence of eligible microcracks density. This finding would be critical in order to introduce the new procedure for industrial purposes, which would be possible when the relationship with the physical processes responsible for cleavage is revealed.

## Acknowledgment

The authors gratefully acknowledge funding from the National Nuclear Laboratory and Électricité de-France. Furthermore, A. P. Jivkov acknowledges the financial support of the Engineering and Physical Sciences Research Council UK (EPSRC) via grant EP/N026136/1.

## Funding Data

• Engineering and Physical Sciences Research Council (Grant No. EP/N026136/1; Funder ID: 10.13039/501100000266).

## Nomenclature

• $a$ =

microcrack size

•
• $ac$ =

critical microcrack size

•
• $a0$ =

scale parameter of microcrack size distribution

•
• $B$ =

specimen thickness

•
• $E$ =

elastic modulus

•
• $fc$ =

probability density of microcracks

•
• $H$ =

Heaviside step function

•
• $Jc$ =

critical J-value (fracture toughness)

•
• $JIC$ =

critical value of the $J$-integral

•
• $J0EXP$ =

experimentally measured characteristic toughness of the Weibull distribution

•
• $J0SIM$ =

simulated measured characteristic toughness of the Weibull distribution

•
• $k$ =

ranking number

•
• $KIC =$ =

critical value of mode I linear elastic stress intensity factor

•
• $m$ =

Weibull modulus of Weibull stress distribution

•
• $n$ =

power law hardening exponent

•
• $N$ =

number of experimental fracture toughness measurements

•
• $Pf$ =

cleavage probability of failure

•
• $T$ =

temperature

•
• $V0$ =

reference volume

•
• $W$ =

specimen width

•
• $x$ =

spatial position of volume element

•
• $β$ =

shape parameter of microcrack size distribution

•
• $γ$ =

fracture energy

•
• $δPf$ =

elemental probability of failure

•
• $δV$ =

volume element

•
• $εp$ =

equivalent plastic strain

•
• $εut$ =

true strain, at which $σUTS$ occurs

•
• $ε0$ =

proportionality strain

•
• $η$ =

constant scaling exponent of plastic strain power law form of thinning function

•
• $θ$ =

thinning function

•
• $λ$ =

temperature-dependent constant parameter of Poisson distribution form of thinning function

•
• $λc$ =

temperature-independent constant parameter of Poisson distribution form of thinning function

•
• $μc$ =

number density of unstable microcracks

•
• $ν$ =

Poisson's ratio

•
• $σ0$ =

proportionality stress

•
• $σ1$ =

maximum principal stress

•
• $σf$ =

flow stress

•
• $σu$ =

scale parameter of Weibull stress distribution

•
• $σut$ =

true stress, at which $σUTS$ occurs

•
• $σUTS$ =

ultimate tensile stress

•
• $σw$ =

Weibull stress

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