# Effect of temperature and geometry on the brittle failure of an 18MND5 steel in the lower part of the ductile to brittle transition

Emmanuelle Catel, Aboubakr Amzil, Eric Lorentz, Anna Dahl, Jacques Besson

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## To cite this version:

Emmanuelle Catel, Aboubakr Amzil, Eric Lorentz, Anna Dahl, Jacques Besson. Effect of temperature and geometry on the brittle failure of an 18MND5 steel in the lower part of the ductile to brittle transition. Engineering Fracture Mechanics, 2025, 314, pp.110745. 10.1016/j.engfracmech.2024.110745. hal-04926386

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# Effect of temperature and geometry on the brittle failure of an 18MND5 steel in the lower part of the ductile to brittle transition

Emmanuelle Catel<sup>a,b</sup>, Aboubakr Amzil<sup>a,b</sup>, Eric Lorentz<sup>c,d</sup>, Anna Dahl<sup>b,d</sup>, Jacques Besson<sup>a,d,*</sup>

## ARTICLE INFO

### Keywords:

Nuclear pressure vessel steel

CT and SENT specimens

Beremin model

Master curve

## ABSTRACT

Tests on cracked CT and SENT specimens (152 tests) extracted from a nuclear pressure vessel steel (18MND5) were conducted between -150 °C and -50 °C together with tests on smooth and notched bars. Tests on tensile bars were used to calibrate a temperature-dependent hardening law. Tests on cracked specimens that exhibited brittle failure only were analyzed using the Beremin model. Special care was taken to evaluate local stresses using enhanced mixed finite elements. The Beremin model was able to represent the entire database provided the reference failure stress $\sigma_u$ increases with temperature. Finally, the results were also interpreted using the Master Curve approach, which was adapted to account for different stress states using the Q-parameter.

## 1. Introduction

In order to assess the integrity of Nuclear Pressure Water Reactor (NPWR) in the Ductile to Brittle transition (DBT) regime [1], the global approach to failure is often used. This method, originally based on a single parameter (for example, toughness) for the whole structure, can be very conservative as it does not consider the potentially beneficial effects of geometry or thickness of a structure. However, the effect of the stress state can be accounted using either the *T*-stress or the *Q*-parameter [2–4]. The thickness effect can be accounted for using a correction as proposed by the Master Curve approach [5]. An alternative approach is based on the local approach to fracture [6], which can be used to characterize brittle fracture in the lower part of the transition. The local approach to rupture focuses on the detailed study of stress and strain fields in regions close to the crack front; this was why the approach was described as “local”. This approach was also proposed as a better way to ensure the transferability from laboratory-size specimens to structures. In the case of brittle cleavage fracture, an overview of the approach and the different models was presented in [7].

One of the most popular and oldest local approach models for brittle failure is the seminal Beremin model [8]. The model is based on the statistical description of the microcrack distribution and the Griffith cleavage criterion [9] to describe subsequent propagation in the stressed material. The model is characterized by a Weibull law describing the failure of the material introducing two parameters known as *m*, a shape factor, and $\sigma_u$, a characteristic cleavage stress. *m* can be linked to the size distribution of particles at the origin of cleavage [6]. The approach also incorporates the effect of plasticity, assuming that particle cracking and subsequent cleavage crack propagation is only possible when plasticity is active. This hypothesis is based on the assumption that microcracks are formed due to an inhomogeneous distribution of plastic deformation within the grains following a Stoh and/or Cottrel mechanism

https://doi.org/10.1016/j.engfracmech.2024.110745

Received 24 July 2024; Received in revised form 13 December 2024; Accepted 15 December 2024

Available online 21 December 2024

0013-7944/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/).

<footnote><sup>a</sup> Mines Paris, PSL Research University, Centre des Matériaux, UMR CNRS, 7633, 10 rue Henri Desbruères, Evry 91000, France</footnote>

<footnote><sup>b</sup> EDF R&D, MMC, Lab Renardières, Avenue des renardières, Écuelles 77250, France</footnote>

<footnote><sup>c</sup> IMSIA (UMR EDF-ENSTA-CNRS-CEA 9219), 7 Boulevard Gaspard Monge, Palaiseau 91120, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: tojacques.besson@minesparis.psl.eu (J. Besson).</footnote>
<table><thead><tr><th colspan="2">Nomenclature</th></tr></thead><tbody><tr><td>CT</td><td>Compact Tension specimen</td></tr><tr><td>MC</td><td>Master Curve</td></tr><tr><td>NT</td><td>Notched Tensile specimen</td></tr><tr><td>SENT</td><td>Single Edge Notch Tension specimen</td></tr><tr><td>ST</td><td>Smooth Tensile specimen</td></tr><tr><td>b<sub>1</sub>, b<sub>2</sub></td><td>Parameters of the hardening model</td></tr><tr><td>B, B<sub>N</sub></td><td>Specimen thickness, specimen net thickness</td></tr><tr><td>E</td><td>Young's modulus</td></tr><tr><td>E&#773;</td><td>Hencky (logarithmic) strain tensor</td></tr><tr><td>E&#773;</td><td>Corrected strain tensor</td></tr><tr><td>J<sub>F</sub></td><td>Determinant of F</td></tr><tr><td>J</td><td>J-integral</td></tr><tr><td>K<sub>J</sub></td><td>Toughness</td></tr><tr><td>F</td><td>Force</td></tr><tr><td>F</td><td>Transformation gradient tensor</td></tr><tr><td>L, L<sub>m</sub>, L<sub>inco</sub></td><td>Helmholtz free energy (total, mechanical, quasi-incompressibility)</td></tr><tr><td>h</td><td>Mesh size ahead of the crack tip</td></tr><tr><td>l<sub>0</sub></td><td>gage length</td></tr><tr><td>L</td><td>Mesh size along the crack front</td></tr><tr><td>m</td><td>Shape parameter for the Beremin model</td></tr><tr><td>n</td><td>Parameter of the Φ(ε<sub>p</sub>) function</td></tr><tr><td>n<sub>x</sub>, n<sub>y</sub></td><td>Parameters determining the size of the refined mesh</td></tr><tr><td>N<sub>brit</sub></td><td>Number of fully brittle specimens</td></tr><tr><td>N<sub>tot</sub></td><td>Total number of specimens</td></tr><tr><td>P</td><td>Nodal pressure</td></tr><tr><td>P<sub>exp</sub></td><td>Experimental failure probability</td></tr><tr><td>P<sub>f</sub></td><td>Theoretical failure probability</td></tr><tr><td>Q</td><td>Q constraint factor</td></tr><tr><td>Q<sub>1</sub>, Q<sub>2</sub></td><td>Parameters of the hardening model</td></tr><tr><td>r<sub>c</sub></td><td>characteristic length (J/σ<sub>Y</sub>)</td></tr><tr><td>r<sub>inco</sub></td><td>Penalty term (quasi-incompressibility)</td></tr><tr><td>R<sub>0</sub></td><td>Yield stress (model)</td></tr><tr><td>R<sub>m</sub></td><td>Ultimate tensile stress</td></tr><tr><td>R<sub>p0.2</sub></td><td>Conventional yield stress</td></tr><tr><td>R(ε<sub>p</sub>)</td><td>Flow stress</td></tr><tr><td>S<sub>0</sub></td><td>Initial cross section</td></tr><tr><td>t</td><td>Time</td></tr><tr><td>T</td><td>Temperature</td></tr><tr><td>T</td><td>Stress tensor (work conjugate of E)</td></tr><tr><td>T<sub>I</sub></td><td>Maximum principal stress of T</td></tr><tr><td>u</td><td>Displacement</td></tr><tr><td>V<sub>0</sub></td><td>Reference volume for the Beremin model</td></tr><tr><td>W</td><td>Specimen width</td></tr></tbody></table>

(see [8] and references herein). A model focusing on the importance of active plasticity as an initiator of cleavage has been proposed by Bordet et al. [10]. Other works, such as [11–15], study the defects statistics to incorporate them directly into the models. They also investigate the role of temperature on fracture.

The completeness of the database is a key point for accurately calibrating the Beremin model and predicting size and constraint effects on brittle fracture. A commonly used database for such a study is given by [16] for a pressure vessel steel commonly used in nuclear power plants. This European experimental program consists of over 750 toughness tests on CT specimens with thicknesses of 12.5 mm, 25 mm, 50 mm, and 100 mm, covering the entire ductile-to-brittle transition range from –154 °C to 20 °C. The Beremin model is, for example, used in [13,17] to represent this database. As many similar works have been conducted on these experimental data, they appear as a reference case study of brittle fracture of ferritic steel. The present work aims to present a new comprehensive database established for a ferritic steel with the originality of presenting a variety of specimen geometries and sizes. The database covers the complete ductile-to-brittle transition from purely ductile failure (future work) to purely brittle failure (present work).
<table><caption>**Table 1**</caption><tr><td>α</td><td>Parameter of the modified MC model</td></tr><tr><td>Δa</td><td>Ductile crack advance</td></tr><tr><td>Δl</td><td>Elongation</td></tr><tr><td>ΔΦ</td><td>Variation of the minimum diameter</td></tr><tr><td>ε<sub>p</sub></td><td>Accumulated plastic strain</td></tr><tr><td>λ</td><td>Parameter of the Φ(ε<sub>p</sub>) function</td></tr><tr><td>ν</td><td>Poisson's ratio</td></tr><tr><td>Φ(ε<sub>p</sub>)</td><td>Function affecting the Weibull stress</td></tr><tr><td>Φ<sub>0</sub></td><td>Initial minimum diameter</td></tr><tr><td>σ<sub>eff</sub></td><td>Effective stress (Beremin model)</td></tr><tr><td>σ<sub>u</sub></td><td>Reference fracture stress (Beremin model)</td></tr><tr><td>σ<sub>Y</sub></td><td>Effective yield stress 1/2 (R<sub>p0.2</sub> + R<sub>m</sub>)</td></tr><tr><td>σ<sub>w</sub></td><td>Weibull stress</td></tr><tr><td>θ</td><td>Nodal volume variation</td></tr></table>

Chemical composition (wt. %) of the 18MND5 ferritic steel.

<table><tr><td>C</td><td>Mn</td><td>Si</td><td>Ni</td><td>Cr</td><td>Mo</td><td>Cu</td><td>S</td><td>P</td><td>Al</td><td>V</td><td>Ti</td><td>Fe</td></tr><tr><td>0.19</td><td>1.5</td><td>0.23</td><td>0.66</td><td>0.17</td><td>0.485</td><td>0.084</td><td>&lt; 0.001</td><td>0.004</td><td>0.011</td><td>0.004</td><td>0.006</td><td>bal.</td></tr></table>

This paper uses the experimental database considering temperatures from -150 °C to -20 °C on different specimen geometries on a ferritic steel. Firstly, the material and tests are presented. Then experimental results including tensile tests on smooth and notched bars, toughness results on CT and SENT specimens and fractographic observations, are summarized. The elastoplastic behavior is then identified and validated thanks to finite element computations on all geometries. Finally, the failure tests in the lower shelf of the ductile-to-brittle transition (i.e. specimens for which brittle fracture only is observed) are analyzed and modeled thanks to the Beremin local approach model. The methodology to obtain an accurate evaluation of the Weibull stress is described. In the final discussion, the obtained results are compared to the literature, focusing particularly on the Master Curve approach [5].

## 2. Materials and experimental procedures

### 2.1. Material

Tests were performed on an A533 (18MND5) ferritic steel representative of nuclear pressurized water reactors. The chemical analysis of the material was carried out during the study. Results are shown in Table 1. The microstructure of the material, shown in Fig. 1, exhibits a mixed ferrite/bainite structure. The sulfur content is low, and few manganese sulfides (MnS) can be observed as shown in Fig. 1. Titanium nitrides (TiN) can also be observed.

### 2.2. Material characterization procedures

To characterize the elastoplastic behavior, smooth tensile bars (ST) and notched axisymmetric bars (NT<sub>r</sub> where r is the notch radius) were used. The various specimens are displayed in Fig. 2. An edge tracing technique [18,19] was used to track the variation of the minimum diameter of NT<sub>r</sub> specimens. This technique allows for determining the hardening law well beyond the strain at the onset of necking in smooth bars. The material was tested at four different temperatures: -20, -50, -100, and -150 °C.

SEM observations were performed on NT specimens broken at -100 °C to check damage mechanisms. On NT2 and NT4 specimens, ductile initiation was observed confirming the observations presented in [20] on the same steel. An example is presented on Fig. 3. As illustrated in [21], there is a transition in the strain at failure for notched specimens. The current microscopic observations permit linking high deformation at failure (up to 30%) to a ductile initiation. Based on [20], very low strain at failure are reached for NT specimens on the same material at -188 °C and microscopic observations show pure cleavage. There is therefore a change in the mechanism of failure initiation around -150 °C. These observations lead to the exclusion of NT specimens in the fitting of the Beremin model as this work deals only with pure cleavage.

Compact Tension (CT) and Single Edge Notch Tension (SENT) specimens were tested to evaluate brittle fracture in cracked specimens. The list of tested specimens is given in Table 2. CT specimens were machined according to the ASTM E1820 standard with thicknesses (B) equal to 12.5, 25, and 50 mm. Side grooves were machined in some cases, as indicated by * in Table 2. In that case, the net thickness (B<sub>n</sub>) was 80% of the thickness for CT specimens and 90% for SENT specimens. All specimens were fatigue pre-cracked according to the recommendations of the ASTM E1820 standard to obtain a crack length about equal to half of the specimens' thickness (a<sub>0</sub>/W = 0.5). The SENT specimens were pre-cracked using a 3-point bending setup following the recommendations for Single Edge Bend (SEB) specimens. Side grooves were machined after pre-cracking. This process aimed to eliminate any potential curved crack near the free sides.
<figure><img src="image_5.png" /><figcaption>Fig. 1. Microstructure of the 18MND5 material. MnS and TiN inclusions can be observed.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. Geometries of the smooth tensile specimen ST6, Notch tensile specimens NT₂, NT₄, NT₁₀ and NT₂₀ (unit: mm).</figcaption></figure>

<caption>Table 2</caption>

Number of tested cracked specimens at the different temperatures.

<table><thead><tr><th></th><th>-150 °C</th><th>-100 °C</th><th>-50 °C</th></tr></thead><tbody><tr><td>CT12.5</td><td>11</td><td>20</td><td>21<sup>a</sup></td></tr><tr><td>CT25</td><td>11<sup>a</sup></td><td>15</td><td>14<sup>a</sup></td></tr><tr><td>CT50</td><td>-</td><td>15</td><td>15</td></tr><tr><td>SENT</td><td>10<sup>a</sup></td><td>10<sup>a</sup></td><td>10<sup>a</sup></td></tr></tbody></table>

<footnote>a Indicates that side grooves were used with $B_n = 0.8B$ for CT specimens and $B_n = 0.9B$ for SENT specimens.</footnote>
<figure><img src="image_7.png" /><figcaption>Fig. 3. SEM micrograph observation of ductile initiation in a NT2 specimen tested at -100 °C.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4. (a) Geometry of SENT specimens (unit: mm). (b) View of the SENT specimen together with the bolted extensions.</figcaption></figure>

Hydraulic grips were used to test the SENT specimens. Extensions were bolted to the SENT specimens as these grips could not be used at low temperatures. For that purpose, five holes were machined on both sides of the specimens (see Fig. 4-a). A photograph of the bolted specimen can be seen in Fig. 4-b. The load (F) and the Crack Mouth Opening Displacement (CMOD), measured using a clip gage, were continuously recorded during the test. The tests were conducted at -50, -100, and -150 °C.
<figure><img src="image_9.png" /><figcaption>Fig. 5. Stress-strain curves for different test temperatures.</figcaption></figure>

<table><caption>Table 3
Standard mechanical properties for different temperatures.</caption><thead><tr><th>T<br>(°C)</th><th>R<sub>ρ0.2</sub><br>(MPa)</th><th>R<sub>m</sub><br>(MPa)</th><th>σ<sub>Y</sub><br>(MPa)</th><th>E<br>(GPa)</th></tr></thead><tbody><tr><td>-20</td><td>537</td><td>674</td><td>605</td><td>205</td></tr><tr><td>-50</td><td>562</td><td>709</td><td>635</td><td>207</td></tr><tr><td>-100</td><td>612</td><td>700</td><td>691</td><td>209</td></tr><tr><td>-150</td><td>761</td><td>872</td><td>816</td><td>211</td></tr></tbody></table>

# 3. Experimental results

## 3.1. Results for smooth and notched bars

Fig. 5 displays the stress-strain curves for different test temperatures obtained on smooth tensile bars ($F/S_0$: nominal stress, $\Delta l/l_0$: nominal strain where $S_0$ is the initial minimal cross section and $l_0$ the gage length of the extensometer). The results are plotted up to the onset of necking. A Lüders plateau is always observed. Its length increases as temperature decreases. The 0.2% proof stress ($R_{\rho0.2}$) and the ultimate tensile stress ($R_m$) are assessed for each temperature and gathered in Table 3 together with the Young's modulus ($E$). As expected, both values increase with decreasing temperature. In all cases, the 0.2% proof stress corresponds to the level of the Lüders plateau. Poisson's coefficient is fixed to the usual value for such steel, $\nu = 0.3$. The effective yield strength, $\sigma_Y$, is computed as follows:

$$ \sigma_Y = \frac{1}{2} (R_{\rho0.2} + R_m) \qquad (1) $$

Fig. 6 shows the test results for $NT_{10}$ (Fig. 6-(a)) and $NT_{20}$ (Fig. 6-(c)) specimens at various temperatures. The nominal stress ($F/S_0$) is plotted against the relative diameter reduction ($\Delta\Phi/\Phi_0$, where $\Delta\Phi$ is the change in the minimum diameter and $\Phi_0$ its initial value). The Lüders plateau cannot be seen on the tests, except for the $NT_{20}$ at -196 °C. In that case, the plateau is strongly reduced compared to results obtained on smooth bars. At a given temperature, the maximum load increases with increasing notch severity, as often observed in the literature [22,23]. This well-understood evolution is caused by increased stress triaxiality at the center of the specimens as the notch radius decreases. All specimens failed in an unstable manner.

Figs. 6-(b) and (d) show the normalized force, $F/S_0\sigma_Y$, as a function of $\Delta\Phi/\Phi_0$ for $NT_{10}$ and $NT_{20}$ specimens. Upon normalization by $\sigma_Y$, all curves collapse to a single one for a given specimen geometry. Results for $NT_4$ and $NT_2$ are shown in Fig. 10 where they are compared with simulations.

## 3.2. Results for CT and SENT specimens

Tests were conducted on several specimen sizes and geometries from -50 °C to -150 °C. At -150 °C, cleavage fracture occurs in the linear part of the force-opening curve for specimens in which small-scale yielding prevails. At -100 °C, cleavage fracture occurs at the beginning of the non-linear part for CT50 and CT25. Plasticity is more extended for CT12.5 and SENT specimens. At -50 °C,
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** NT₁₀ tests: (a) Force–diameter reduction curves at various temperatures. (b) Normalized force–diameter reduction curves. NT20 tests: (c) Force–diameter reduction curves at various temperatures. (d) Normalized force–diameter reduction curves.</figcaption></figure>

the force-opening curves are highly non-linear for all specimens, corresponding to large scale plasticity and ductile crack extension for some of the specimens.

Toughness values for CT specimens were obtained following the recommendations of ASTM standards E1820 [24] and E1921 [5]. In the case of SENT specimens, the values of toughness were computed using the finite element method to account for specific boundary conditions as detailed below in part 4 as the BS8571 [25] could not be used. Experimental results corresponding to Fig. 7 are gathered in Appendix.

The comparison of toughnesses for different specimen sizes and geometries at several temperatures illustrates geometry and temperature effects on toughness. Fig. 7 and Table 4 show that for all specimens, toughness values increase with temperature and decrease with increasing thickness as expected. The same trend is observed for scatter. In Fig. 7 closed symbols represent tests for which ductile crack extension before cleavage fracture was observed. This only occurs at –50 °C for CT25 (2 out of 14), CT12.5 (12 out of 21) and all SENT specimens, for which only the lowest measured toughness is indicated.
<table><caption>**Table 4**
Mean and standard deviation of toughness values by geometry and temperature.</caption><thead><tr><th>Temperature (°C)</th><th>Geometry</th><th>Mean value (MPa√m)</th><th>Standard deviation (MPa√m)</th></tr></thead><tbody><tr><td rowspan="3">-150</td><td>CT25</td><td>37.8</td><td>7.7</td></tr><tr><td>CT12.5</td><td>48</td><td>6</td></tr><tr><td>SENT</td><td>50.4</td><td>12.9</td></tr><tr><td rowspan="4">-100</td><td>CT50</td><td>66</td><td>15.1</td></tr><tr><td>CT25</td><td>85</td><td>22.5</td></tr><tr><td>CT12.5</td><td>100</td><td>26.3</td></tr><tr><td>SENT</td><td>139.6</td><td>51.5</td></tr><tr><td rowspan="3">-50</td><td>CT50</td><td>136</td><td>26.1</td></tr><tr><td>CT25</td><td>218</td><td>82.9</td></tr><tr><td>CT12.5</td><td>281</td><td>94.9</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Values of $K_J$ evaluated for the different specimens and temperatures. Open symbols: brittle fracture. Closed symbols: mixed ductile/brittle. All SENT specimens above the arrow presented ductile failure and were not considered in the study.</figcaption></figure>

SENT specimens have higher toughness values and a larger scatter at each temperature. At a given temperature, the comparison between CT12.5, CT25 and CT50 illustrates the “Size effect” [26]: a higher thickness results in a lower toughness because of the increase in the volume of the specimen. The higher toughnesses observed for SENT specimens illustrate the “Constraint effect” [27]. Due to the specimen geometry, stress triaxiality is lower and large-scale yielding can occur at lower values of $J$.

### 3.3. Fractographic examination of CT and SENT specimens

The objective of fractographic examinations is to confirm the origin of failure in various specimens and illustrate the different failure mechanisms in the transition region. Representative fracture surfaces are shown in Fig. 8.

At -150 °C, Fig. 8-(a) shows the fracture surface of a CT12.5 specimen which failed for a value of $K_J$ equal to 32 MPa√m. In that case, the fracture surface is fully brittle without any significant stretch zone. This is consistent with low toughness and failure in the small-scale yielding regime.

At -100 °C, Fig. 8-(b) shows the fracture surface of a SENT specimen which failed for a value of $K_J$ equal to 175 MPa√m. In that case, a stretch zone (SZ) is clearly visible after the pre-fatigue area (F). The stretch zone, as described in [28], corresponds to the propagation of the crack due to blunting. A small shear zone (S) is observed before the brittle failure area (B). The shear zone is assumed to be created after the initiation of brittle failure ahead of the crack tip in a highly stressed volume due to crack blunting [29]. This interpretation of the fracture surface was initially proposed in [30,31].

At -50 °C, Fig. 8-(c) shows the fracture surface of a CT12.5 specimen which failed for a value of $K_J$ equal to 240 MPa√m. In that case, the fracture surface is more complex. Four zones can be distinguished: the stretch zone (SZ), the sheared zone (S), the ductile zone (D), where dimples can be seen, and finally, the brittle fracture zone (B). In that case, ductile fracture clearly occurred before cleavage, as evidenced by the presence of dimples on the fracture surface. All specimens presenting such a fracture surface
<figure><img src="image_12.png" /><figcaption>Fig. 8. Fractographic examination of cracked specimen: (a) CT12.5 at -150 °C, (b) SENT at -100 °C, (c) CT12.5 at -50 °C. Labels indicate the different zones: F fatigue precrack, SZ Stretch Zone, S Shear zone, D Ductile tearing zone, B Brittle zone. Arrows also indicate the zones.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 9. Blunted crack for an applied K<sub>J</sub> equal to 200 MPa√m showing no ductile crack advance.</figcaption></figure>

are shown by closed symbols in Fig. 7. Ductile crack initiation was never observed at -150 and -100 °C and in CT50 at -50 °C. On the other hand, it was always observed for SENT specimens tested at -50 °C.

To check the absence of ductile tearing in specimens for which only a shear zone was observed (case (b) in Fig. 8), a CT12.5 specimen was loaded up to a value of K<sub>J</sub> equal to 200 MPa√m. The test was performed at -20 °C to avoid the possibility of brittle fracture before reaching the desired load. The specimen was then sliced using electro-discharge machining. The slice is cut at the center of the specimen where cracks initiate. The slice was then polished. The resulting SEM micrograph is shown in Fig. 9 where no significant ductile damage can be seen neither close to the crack tip nor at a distance of about 2 × CTOD where stresses are maximum. In particular, shearing (S) is not observed.

## 4. Model for the elasto-plastic behavior

### 4.1. Identification on TC and NT specimens

Based on the observation that normalized forces are similar for a given geometry whatever the testing temperature, it is proposed to express the flow stress as:

$$R(\epsilon_p) = \sigma_Y(T) (R_0 + Q_1(1 - \exp(-b_1 \epsilon_p)) + Q_2(1 - \exp(-b_2 \epsilon_p))) \quad (2)$$
<table><caption>**Table 5**
Non-dimensional parameters for the flow stress.</caption><thead><tr><th>R<sub>0</sub></th><th>Q<sub>1</sub></th><th>b<sub>1</sub></th><th>Q<sub>2</sub></th><th>b<sub>2</sub></th></tr></thead><tbody><tr><td>0.772</td><td>0.475</td><td>3.888</td><td>0.307</td><td>29.0</td></tr></tbody></table>

<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Comparison of simulated (dashed lines) and experimental (thick lines) Force-Diameter reduction for (a) NT₄ and (b) NT₂ specimens at -100 °C, -50 °C and -20 °C.</figcaption></figure>

where $\sigma_Y(T)$ is given in Table 3 for the various test temperatures. The non-dimensional parameters $R_0$, $Q_1$, $b_1$, $Q_2$ and $b_2$ are assumed temperature independent. They were fitted using test results on NT₁₀ and NT₂₀ specimens at -20 °C. The fit of the hardening function was done assuming von Mises plasticity. The fitting was performed by considering the difference between the experimental load—diameter reduction curves and the simulation of the same curves as an objective function and minimizing it thanks to the 3D-VAR algorithm [32]. The Lüders plateau was not taken into account. The adjusted parameters are gathered in Table 5. Fig. 10 presents the corresponding results for NT₄ and NT₂ specimens at various temperatures where a very good agreement is found.

## 4.2. Simulation of the CT and SENT specimens

Finite elements calculations were conducted to estimate the elastoplastic behavior of the different specimens. Due to symmetry, one-quarter of the specimens was modeled and usual boundary conditions were applied on the symmetry planes. FE calculations conducted with the optimized hardening law (eq.(2)) compare well with the experimental force-opening curves for CT specimens at all temperatures as exemplified in Fig. 11 for a testing temperature of -100 °C. The calculations on CT specimens were performed for an initial crack length $a_0$ corresponding to the experimental mean value. This is sufficient to capture the force-opening curves for all specimens due to the relatively small scatter in initial crack lengths.

At -100 °C and -50 °C, tests on SENT specimens were not repeatable (two specimens with the same initial crack length may present different force-opening curves). Furthermore, usual boundary conditions (i.e. clamping at the end of the specimen) appeared insufficient to catch the SENT specimens' behavior well. The problem was attributed to the clearance between the specimen and the bolts, which were not tightened at a constant torque. A specific test was conducted at room temperature and monitored using Digital Image Correlation (DIC). This test revealed that while the intended translation was applied to the specimen's top surface, it also experienced unexpected rotation. A FE simulation of the test was then conducted by applying the measured boundary conditions. A good agreement between the experimental and simulated force-opening curves was obtained in that case. This validated that poor control of the boundary conditions caused the problem. Using the test conducted at room temperature, it was demonstrated that comparable outcomes can be achieved by loading the upper surface of the specimen via a tension and torsion spring, with the stiffnesses adjusted accordingly for every test. As DIC measurements were not available at low temperatures, these stiffnesses were adapted by comparing experimental and simulated force-opening curves. The phenomenon is emphasized at low temperatures because of differential thermal dilatation between the specimen, the pins, the bolts, and the loading device. The last test campaign was conducted at -150 °C using a controlled and higher torque to tighten the setup. This procedure led to a considerable reduction in the difference between the simulations (assuming perfect clamping) and the tests. This further validated the proposed interpretation of the initially observed discrepancy.

As the geometry of SENT specimens differed from the standard, the experimental toughness values presented in Fig. 7 were not calculated directly from the experimental force-opening curves using the BS8571 standard but from corresponding finite element calculation which allows for the calculation of the value of J using the G(Θ) method [33] implemented as a post-processing routine.
<figure><img src="image_15.png" /><figcaption>Fig. 11. Elastoplastic simulation of the force-opening curves at -100 °C for CT and SENT (black thick lines) compared to experimental testings for various initial crack length $a_0$ (gray thin lines). To model CT specimens, the mean of experimental values was used for $a_0$. In the case of SENT specimens, simulation corresponding to test 2141-SENT-1 is presented (experiment presented with the red line).</figcaption></figure>

# 5. Modeling of brittle fracture

## 5.1. Finite strain formulation

The elastoplastic behavior described above was used to evaluate the stress fields for each specimen using the Finite Element method. Even for low values of the applied $J$, the crack tip undergoes blunting so that a finite strain formulation is always needed to model the specimens. In the following, finite strains are modeled using the logarithmic strain formulation [34,35]. The logarithmic strain tensor $E$ is defined as:

$$E = \frac{1}{2} \log(F^T \cdot F) \qquad (3)$$

where $F$ is the transformation gradient. The corresponding dual stress tensor ($T$) is the work conjugate of $E$ so that:

$$T : \dot{E} = S : \frac{1}{2} \dot{C} \qquad (4)$$

where $S$ is the second Piola-Kirchhoff stress tensor. $C = F^T \cdot F$ is the Cauchy-Green strain tensor. This formulation allows using an additive decomposition of $E$ into an elastic and a plastic part.

## 5.2. Finite element technique

Because of the quasi-incompressibility of the elastoplastic behavior, standard finite element formulation based on displacement results in “volumetric locking” and stress oscillations. This leads to a poor evaluation of the criteria for assessing brittle fracture. The phenomenon is illustrated in Fig. 12 in the case of 2D plane strain simulations of a CT specimen loaded up to $K_J = 200$ MPa$\sqrt{m}$ using an element size of 9 µm around the crack tip. This corresponds to a severely deformed crack tip. The conclusions drawn from the 2D case are also valid for 3D elements (hexahedra, tetrahedra or pentahedra). Fig. 12 shows the values of the trace of the Cauchy stress tensor at Gauss points close to the crack tip in the undeformed configuration. Bi-linear squares with full integration (C2D4) lead to a very strong checkerboard pattern. The phenomenon is slightly attenuated using quadratic square or triangular elements with full integration (C2D8/C2D6), and further reduced using reduced integration (C2D8R/C2D6R). A well-known solution relies on the use of $\overline{B}/\overline{F}$-methods [36,37] and leads to the suppression of oscillations (C2D4 $\overline{F}$ and C2D6R $\overline{F}$ in Fig. 12). This technique is often used to evaluate brittle fracture (see e.g. [38]). The $\overline{F}$-method relies on a multiplicative decomposition of the transformation
<figure><img src="image_16.png" /><figcaption>Fig. 12. Stress fields obtained for a CT12.5 at -100 °C for different types of square elements. C2D4: linear standard element. C2D8/C2D6: quadratic standard element. C2D4 F: linear element with the F-method, C2D8R/C2D6R: quadratic reduced standard element, C2D8/C2D6 P-θ: quadratic element with mixed formulation, C2D8R/C2D6R P-θ quadratic reduced element with mixed formulation. C2D4, C2D8 and C2D8R elements are quadrilateral elements. C2D6 and C2D6R elements are triangular elements. Simulations are carried out up to $K_J = 200$ MPa√m. The mesh size around the crack tip is 9 µm. $r_c = J/σ_y = 265$ µm.</figcaption></figure>

gradient tensor into a deviatoric and a dilatational part. The dilatational part corresponds to the average of the dilatation over the element.

In this work, a mixed formulation developed by [39–41] is used. These elements use additional nodal unknowns to control the stress field better: the pressure P and the volume variation θ. In the following, u represents the nodal displacements. Displacements are interpolated using quadratic shape functions. P and θ are interpolated using linear shape functions associated with the vertices of the elements. Therefore, the mixed formulation can only be used with quadratic elements. These so-called P-θ elements can be described by their Helmholtz free energy L:

$$ \mathcal{L}(u, \theta, P, \text{i.v...}) = \mathcal{L}_m(\tilde{E}(u, \theta), \text{i.v...}) + \mathcal{L}_{\text{inco}}(u, \theta, P) \quad (5) $$

where i.v... represents the internal state variables, which do not need to be detailed here. $\mathcal{L}_m$ corresponds to the usual Helmholtz free energy evaluated using a corrected strain $\tilde{E}$ such that:

$$ \tilde{E} = E + \frac{1}{3} (\theta - \log(J_F)) \mathbf{1} \quad (6) $$

where E is evaluated from the nodal displacements (u). $J_F$ is the determinant of F. The additional part ($\mathcal{L}_{\text{inco}}$) is added to deal with quasi-incompressibility:

$$ \mathcal{L}_{\text{inco}}(u, \theta, P) = \int_{\Omega_0} \left( P(\log(J_F) - \theta) + \frac{1}{2} r_{\text{inco}} (\log(J_F) - \theta)^2 \right) d\Omega_0 \quad (7) $$

The pressure field P acts as a Lagrange multiplier used to weakly enforce the condition: $\theta = \log(J_F)$. The use of an augmentation term ($\frac{1}{2}r_{\text{inco}}(\log(J_F) - \theta)^2$) was proposed in [35] to ensure a better convergence of the iterative numerical scheme. The value of the $r_{\text{inco}}$ parameter is set to $r_{\text{inco}} = 10\sigma_y$ as proposed in [42].

Fig. 12 compares the stress field obtained using the mixed formulation for full (C2D8/C2D6-P-θ) and reduced integration (C2D8R/C2D6R-P-θ) with results obtained with standard displacement-based elements. Pressure fluctuations are considerably reduced. Residual fluctuations are caused by the severe deformation of the elements close to the crack tip. They, however, remain
sufficiently limited not to affect the evaluation of the Weibull stress. F̄-elements also avoid stress field oscillations. They were not selected for this study as the proposed mixed elements can also be used to represent ductile fracture using a "nonlocal" framework [43]. They, therefore, can potentially be applied to represent the entire (lower and upper shelf) ductile-to-brittle transition within a unified framework. 3D quadratic elements (hexahedra/20 nodes and pentahedra/15 nodes) using the mixed formulation with reduced integration will be used in the following.

### 5.3. Beremin model for brittle fracture

The Beremin model [8] is used in this work. It is based on the use of a Weibull stress defined as:

$$ \sigma_w = \left( \int_{\Omega_0} \sigma_{\text{eff}}^m \frac{d\Omega_0}{V_0} \right)^{\frac{1}{m}} \qquad (8) $$

where

$$ \sigma_{\text{eff}} = \max_{0 \to t} (T_I |\epsilon_p| > 0, T_I > 0) \qquad (9) $$

$T_I$ is the maximal principal value of the stress tensor $T$. $\dot{\epsilon}_p$ is the accumulated plastic strain rate and $\Omega_0$ the total volume of the considered structure in its initial configuration. The condition $\dot{\epsilon}_p > 0$ expresses the fact that cleavage fracture only occurs if plasticity is active following [10,44]. At a given time $t$, the maximum value of $T_I$ over the time interval $[0, t]$ is considered to account for a potential local stress drop due to crack blunting or ductile crack advance [44,45]. This phenomenon is exemplified in Fig. 13 where the stress profile ahead of the crack tip is plotted for increasing values of $K_J$ ($50 \to 150 \text{ MPa}\sqrt{\text{m}}$). The maximum stress is approximately constant, but the position where the maximum is reached moves away from the initial crack tip. Consequently, the stress decreases in the elements between the crack tip and the current position of the stress peak. The maximum of the principal stress over the load history is shown by the red dashed curve. As the product $\sigma_u^m V_0$ is a constant, only the Weibull parameters $m$ and $\sigma_u$ need to be identified (see below). In the following, the reference volume is taken equal to $V_0 = (50 \text{ µm})^3$. This model requires an accurate computation of stress fields. The Weibull exponent $m$ is usually high (about 20), so relatively small errors on $T_I$ are strongly amplified. In particular, the stress peak ahead of the blunted crack front must be adequately described. The proposed FE elements fit that purpose. Finally, the failure probability is obtained as:

$$ P_f = 1 - \exp\left(-\left(\frac{\sigma_w}{\sigma_u}\right)^m\right) \qquad (10) $$

The model used in this work is the simplest version of the Beremin approach. In particular, the model does not account for possible dependence on the accumulated plastic strain. Such a dependence was considered in the models proposed in [10,20,46] by considering continuous strain-controlled nucleation of micro-cracks and their potential propagation and arrest. So-called multi-barrier models [12,47,48] describe the consecutive rupture of the particles, the particle/matrix interface and the grain boundaries. As such, they may indirectly induce a deformation effect on fracture. They are appealing as they can be microstructure-informed [48] and therefore directly use experimentally determined particle and grain size distributions. They, however, still introduce parameters that need to be fitted. A simpler way to account for the effect of plastic strain is based on the following definition of the Weibull stress [7]:

$$ \sigma_w = \left( \int_{\Omega_0} \Phi(\epsilon_p) \sigma_{\text{eff}}^m \frac{d\Omega_0}{V_0} \right)^{\frac{1}{m}} \qquad (11) $$

Following [7], $\Phi$ represents the fraction of brittle inclusions that nucleate microcracks eligible for unstable propagation. A possible form for $\Phi$ is then: $\Phi(\epsilon_p) = 1 - \exp(-\lambda\epsilon_p)$ where $\lambda$ needs to be calibrated [49]. In [50], $\Phi = \epsilon_p$ based on the approximation of a constant crack nucleation rate as observed in [51,52]. These data were also used to motivate the model proposed in [10]. The model proposed in [49] is equivalent provided that $\lambda\epsilon_p \ll 1$. $\Phi$ is expressed as a power law ($\Phi(\epsilon_p) \propto \epsilon_p^n$) in [53] but using a different form of the stress dependence. It is worth remembering that the seminal Beremin paper [8] also introduced a plastic strain dependence with $\Phi(\epsilon_p) = \exp(-m\epsilon_p/4)$. The correction factor accounts for grain size reduction in directions perpendicular to the loading direction. In that case, the effect of deformation is beneficial, whereas all other models represent a deleterious effect. However, the effect is important only if strains are large enough. The model also implies that fracture is controlled by the resistance of grain boundaries to cracking. Bernauer et al. [54] also proposed the description of a possible beneficial effect of plasticity. In that case, carbides on which voids have nucleated (ductile damage mechanism) are assumed to no longer act as potential initiation sites for cleavage. All the models for brittle fracture mentioned above assume that scatter is due to the size distributions of particles and grains. They, therefore, neglect that local stresses may vary because of the local crystallography, as outlined in [55–57]. Following these papers, crystal plasticity is used to evaluate the scatter in the principal stress within grain aggregates. This effect is likely to increase scatter.

### 5.4. Mesh design

In order to accurately compute the Weibull stress, a methodology has been developed to design meshes and ensure mesh convergence with respect to $\sigma_w$. For the convergence study, a fixed value of $m = 20$ is considered.
<figure><img src="image_17.png" /><figcaption>Fig. 13. (a) Stress profile ahead of the crack tip for different load levels (CT12.5 at -100 °C). The dashed red line represents the maximum value of $T_i$ over the load history for an applied $K_J$ equal to 150 MPa√m. (b) Value of the Weibull stress as a function of the relative mesh size for different load levels.</figcaption></figure>

The crack tip is often represented as a notch with a small radius, $\rho_0$ (see e.g. [58]). The Weibull stress depends on the initial notch radius if the CTOD is less than 4 times the initial radius. Above this value, the Weibull stress is independent of $\rho_0$. In this study, a sharp crack was used. In that case, blunting appears due to finite strains. The computed Weibull stress becomes mesh-independent when the CTOD is about 4 times the initial mesh size. It was checked that the two different meshing strategies lead to similar Weibull stresses once the respective validity criteria are met. Using a sharp crack instead of a notch simplifies meshing, particularly for complex geometries or structures.

The mesh size ahead of the crack front is then considered. The mesh size must be small enough to capture the stress peak well ahead of the crack tip. As noted above, a poor evaluation of the local stresses may greatly impact the Weibull stress due to the high value of $m$. The mesh size must be adapted to the load level to be represented. Fig. 13-(a) presents the stress profile ahead of the crack for applied stress intensity factor between the minimum (50 MPa√m) and the maximum (150 MPa√m) values reached during tests at -100 °C on CT12.5 specimens. As expected, the stress peak width increases with increasing load. A low load level requires a small mesh size to enable accurate discretization of the stress peak and guarantee capturing of the maximum stress, whereas, at a high load level, larger mesh sizes may suffice. A convergence study on the Weibull stress is required for each type of specimen over the entire load range to be represented to determine the adequate mesh size. This convergence study was performed for 2D calculations. The methodology is summarized here with the example of CT12.5 tests at -100 °C. The computed Weibull stress for different load levels is plotted in Fig. 13-(b) as a function of the normalized mesh size $h/r_c$ where $r_c = J/\sigma_y$. For a ratio $h/r_c$ lower than 0.35 (represented by the red line on the figure), the Weibull stress reaches a constant value for all load levels. For CT12.5 at -100 °C, this ratio at the lower load ($K_J = 47$ MPa√m) corresponds to a mesh size around 5 µm while at the maximal load (corresponding to $K_J = 142$ MPa√m), a mesh size of 44 µm is sufficient.

The previous convergence study was performed assuming a large refine zone ahead of the crack tip. It is, however, important to limit the extent of this zone in the case of 3D simulations. Calculations highlight that the refined zone could be rather small as far as the peak of maximal principal stress is well described. Thanks to the enlargement of the stress peak with the load presented on Fig. 13, larger mesh sizes far from the crack peak are not prejudicial to the Weibull stress computation and give similar results than a fine mesh over a long distance. Based on simulations, the following rule can be adopted. First, the minimum mesh size must be selected as $h = 0.35r_c^{\min}$ where $r_c^{\min}$ is the value of $r_c$ corresponding to the minimum value of the measured $K_J$. The size of the finely meshed zone can then be chosen as $(n_x h) \times (n_y h)$ with $n_x = 50$ and $n_y = 10$ which is sufficient to have a converged Weibull stress. The mesh size can then be gradually increased around this zone. It is not necessary to use a large number of elements in the loading direction, as high stresses only develop close to the symmetry plane.

In the case of 3D calculations, it is necessary to check the convergence of the Weibull stress with respect to the mesh size in the direction parallel to the crack front. In this study, a constant mesh size was used. It was shown that the mesh size can be relatively coarse. This is due to two factors: (i) At the center of the specimens, the stress gradient is moderate, so few elements are needed to describe it. (ii) At the free surface of the specimens, stresses drop. Therefore, this zone does not significantly contribute to the volume integral defining the Weibull stress (eq. (8)). The convergence study on mesh size along the thickness direction (L) for CT12.5 is presented on Fig. 14-(b). There is little dependence of the Weibull stress on L. In this case, a size of L = 2.1 mm gives a similar value of the Weibull stress than a size equal to 0.2 mm. An example of used mesh is presented in Fig. 14-(a).

## 5.5. Fitting and application of the model

The fitting of Weibull parameters ($m, \sigma_u$) is presented in this section. At each temperature, an experimental probability of failure is associated with each specimen by ordering by increasing Weibull stress all the specimens and giving to the specimen $i = 1... N_{tot}$ the probability:

$$P_{\text{exp}} = \frac{i - 0.3}{N_{\text{tot}} - 0.4} \qquad (12)$$
<figure><img src="image_18.png" /><figcaption>Fig. 14. (a) Example of mesh for a CT50. (b) Weibull stress for different mesh sizes in the thickness direction.</figcaption></figure>

<table><caption>Table 6
Values of the identified reference stress $\sigma_u(T)$ for the different temperatures ($m = 18$).</caption><thead><tr><th></th><th>-150 °C</th><th>-100 °C</th><th>-50 °C</th></tr></thead><tbody><tr><th>σ<sub>u</sub>(T) (MPa)</th><td>3030</td><td>3152</td><td>3455</td></tr></tbody></table>

where $N_{tot}$ is the total number of specimens. The objective function for the identification is then the least square error on the probabilities of failure defined by:

$$ \text{Error}(m, \sigma_u) = \frac{1}{N_{\text{brit}}} \sum_{i=1}^{N_{\text{brit}}} \left| P_{\text{exp}} - P_f(\sigma_u(m, \sigma_u)) \right|^2 \quad (13) $$

where $N_{brit}$ is the total number of specimens in which brittle failure only occurred (i.e. brittle failure without initial ductile propagation). As the present model does not include ductile tearing, it is not possible to evaluate the Weibull stress and, consequently, the failure probability in the case of non-purely brittle specimens. However, these tests are accounted as a part of the population of the considered specimens ($N_{tot} - N_{brit}$ specimens) so that it remains possible to assign a failure probability to fully brittle specimens.

The identification of the optimal values $m^{opt}$ and $\sigma_u^{opt}$ Beremin parameters is then done by minimizing the error on $\sigma_u$ and $m$ in two steps. The first step involves optimizing $\sigma_u$ for a series of fixed values for $m$ between 10 and 30.

$$ \sigma_u^{\text{opt}}(m) = \underset{\sigma_u}{\operatorname{argmin}} \text{Error}(m, \sigma_u) \quad \text{for } m \in [10, 30] \quad (14) $$

In the second step, the value of $m$ corresponding to the smallest error is selected together with the corresponding value for $\sigma_u$:

$$ m^{\text{opt}} = \underset{m \in [10,30]}{\operatorname{argmin}} \text{Error}(m, \sigma_u^{\text{opt}}(m)) \quad (15) $$

The identification at -100 °C was done using all specimens. Fig. 15-(a) presents the optimal $\sigma_u^{opt}(m)$ and the corresponding Error($m, \sigma_u^{opt}$) as a function of $m$. The retained couple corresponds to the minimum of the error function: $m = 18$ and $\sigma_u = 3152$ MPa. The obtained theoretical probabilities of failure compared to experimental values are presented on Fig. 15-(b).

At -150 °C and -50 °C, the same shape factor $m = 18$ was used following [6,50] in which $m$ was considered to be temperature independent. This assumption is consistent with the fact that $m$ essentially represents some scatter in the microstructure (particle and grain size distribution) [6,8]. $\sigma_u$ is then identified using the methodology described above (eq. (14)). At -150 °C, all specimens were considered for the optimization. At -50 °C, only CT50 and CT25 were considered for the identification due to the low number of purely brittle CT12.5 and SENT specimens. This procedure leads to a $\sigma_u$ parameter, which increases with temperature. Optimized values are reported in Table 6.

The predicted and experimental failure probabilities applied for each temperature and each specimen type are presented in Fig. 16. Because of the low number of specimens in each group, differences between theoretical and experimental data are more important than on the identified model in Fig. 15-(b). At -50 °C, the dotted part of predicted probabilities corresponds to the domain where ductile failure occurs before brittle failure. The model does not represent this domain, as its description also requires modeling ductile crack growth. The results are discussed in the next part.
<figure><img src="image_19.png" /><figcaption>Fig. 15. Identification and application of Beremin parameters on all specimens at -100 °C.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 16. Comparison of experimental and simulated failure probabilities for all specimens and all temperatures.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Fig. 17.** (a) Error function as a function of *m* for CT specimens only (dashed black curve) and for CT and SENT specimens (red curve). (b) Comparison of the normalized principal stress ($T_i/\sigma_y$) as a function of the normalized distance to the blunted crack $x/r_c$ for CT and SENT specimens for an applied stress intensity factor equal to 100 MPa$\sqrt{m}$.</figcaption></figure>

# 6. Discussion

## 6.1. Identification procedure

The identification of the Beremin model at –100 °C is shown in Fig. 15. The fit was performed based on the computed Weibull stresses for all tested specimens. They are sorted to compute the experimental failure probabilities. The ordering of specimens in terms of Weibull stress permits the comparison of all geometries and specimen sizes. The main advantage of this methodology is that it focuses on the stresses that generate the initiation of cleavage on defects. Experimental probabilities of failure agree correctly with theoretical probabilities. For each geometry, it appears that probabilities of failure are well distributed between 0 and 1: as an example for CT50, $P_{exp}$ are between 0.09 and 0.87 while for SENT specimens, they are between 0.04 and 0.9. Checking the even distribution of each specimen type is important as one can imagine situations where the CT50 specimens could correspond to low failure probabilities while SENT specimens could correspond to high probabilities. One good overall fit could also be obtained in that case.

As explained in [58], different specimen geometries are primordial to guarantee the uniqueness of ($m, \sigma_u$). This necessity is exemplified in this study by trying to identify the Beremin parameters using only results for CT specimens (at –100 °C). The error plotted as a function of $m$ is shown in Fig. 17-(a) (dashed black curve). It can be noticed that no clear minimum can be found for $m$ between 20 and 30. Adding results on SENT specimens allows defining a unique minimum associated with the optimum values for $m$ and $\sigma_u$ (red curve in Fig. 17-(a)). Plotting the stress profile ahead of the blunted crack for an applied stress intensity factor equal to 100 MPa$\sqrt{m}$ allows us to interpret that difficulty. Stress profiles for all CT specimens are similar, so the information brought by the tests on various CT geometries is the same. The stress profile in the case of the SENT specimen is different, exhibiting a lower stress maximum at a smaller distance from the crack tip. This additional data allows the determination of a unique minimum for the error function. As less data was available at –150 °C and –50 °C, $m$ was considered constant to fit $\sigma_u$ as a function of temperature.

The identified parameters were applied for each specimen type to directly compare predicted and experimental toughness values. Fig. 16 presents the results at all temperatures. The fit between experimental results and theoretical probabilities of failure is good at –100 °C, with similar errors for all specimen types. At –150 °C, the lower number of specimens for each geometry leads to a large discrepancy, particularly for CT12.5 where experimental data does not fit well with the Beremin model. At –50 °C, the identification was made using only CT50 and CT25 specimens as all these specimens exhibited pure brittle fracture. Applying the fitted model to the brittle CT12.5 specimens also provides results consistent with experiments. In the case of SENT specimens, only one showed full brittle failure (out of ten); however, the model remains consistent with these tests as it predicts a failure probability of about 10% in that case.

This presentation of results in Fig. 16 permits reproducing the sample and constraints effects previously observed experimentally and well-known in the literature [3,4]. In the case of CT specimens, the model well represents the size effect, which leads to a higher failure probability at a given stress intensity factor for thick specimens, as the probability of finding a defect is larger for large specimens. SENT specimens, which exhibit lower stress triaxialities than CT specimens, present a lower probability of failure for a given stress intensity factor.

## 6.2. Comparison with literature results

The identification method and chosen Error functions appear to have some influence on the identified parameters. In fact, the usual linear regression method, used by [59] gives an identification of the shape parameter independent of $\sigma_u$ while the chosen least
<caption>**Table 7**</caption>

Synthesis of comparable results for *m* and *σ*<sub>u</sub> identification in the literature. *σ*<sub>u</sub> value is corrected by the ratio $\frac{V_0}{V_0^{ref}}$ with $V_0^{ref} = (50 \ \mu m)^3$ to take into the account the dependence to the chosen volume $V_0$.

<table><thead><tr><th>Reference</th><th>Steel reference</th><th>m</th><th>Temperature (°C)</th><th>σ<sub>u</sub>(V<sub>0</sub>/V<sub>ref</sub>)<sup>1/m</sup> (MPa)</th></tr></thead><tbody><tr><td>[8]</td><td>A508</td><td>22</td><td>-196</td><td>2560</td></tr><tr><td rowspan="3">[45]</td><td rowspan="3">A508</td><td rowspan="3">17.8</td><td>&lt; -80</td><td>3287</td></tr><tr><td>-50</td><td>≈ 3480</td></tr><tr><td>&gt; -50</td><td>increasing σ<sub>u</sub></td></tr><tr><td>[61]</td><td>A533B C1.1</td><td>20.7</td><td>-150</td><td>2127</td></tr><tr><td>[62]</td><td>A508</td><td>20</td><td>-20</td><td>3384</td></tr><tr><td rowspan="2">[63]</td><td rowspan="2">A508</td><td rowspan="2">21</td><td>-90</td><td>2500</td></tr><tr><td>-30</td><td>2980</td></tr><tr><td rowspan="3">[17]</td><td rowspan="3">DIN 22NiMoCr37</td><td rowspan="3">24</td><td>-154</td><td>2515</td></tr><tr><td>-91</td><td>2561</td></tr><tr><td>-40</td><td>2609</td></tr></tbody></table>

square error simultaneously provides *m* and *σ*<sub>u</sub>. The choice of method could explain the difference in the shape factor *m* compared to [60], which reported *m* = 16 and different values for *σ*<sub>u</sub> on the same database.

The results obtained are in accordance with studies on similar ferritic steels in the literature. Concerning the identified parameters at -100 °C, the shape factor *m* = 18 is around usual values. Table 7 proposes a synthesis of some literature results on similar materials for the same 2-parameter Beremin model (eq. (10)). It appears that the values of *m* are in agreement with each other and close to 20. The values of *σ*<sub>u</sub> tend to differ more depending on temperature, material and possibly testing conditions. For instance, the lowest value (2127 MPa) is obtained at -150 °C in the case of Small Punch Tests [61] in which the strain to failure is high even at low temperatures.

Lei [64] proposes a synthesis of the dependence of the parameters on the temperature based on previously cited papers. The results of this study confirm that *σ*<sub>u</sub> (Table 6) increases with temperature as already presented in [45,63] on a similar material. According to Tanguy et al. [45], even if the cleavage initiation sites are the same, they are more difficult to trigger when the temperature increases. Haušild et al. [63] explained the increase in *σ*<sub>u</sub> with temperature by an evolution in the defect population distribution and barriers to cleavage such as carbide/ferrite interfaces or ferrite grain boundaries. In the study by Bernauer et al. [54], various specimens and temperatures were used to characterize brittle fracture. However, the authors could not evidence any clear dependence of *σ*<sub>u</sub> on temperature. Gao et al. [50] used a Beremin-type model in which a threshold stress is introduced. This stress corresponds to the value of the Weibull stress for which $K_{min} = 20 \text{ MPa}\sqrt{m}$ is reached. It is, therefore, predetermined and decreases with temperature. The resulting *σ*<sub>u</sub> is, in that case, an increasing function of temperature. An interesting study on 22Ni-MoCr37 pressure vessel steel was done by Wasiluk et al. [65] with a similar error function as eq. (13). They highlight the fact that an identification of the Beremin model thanks to the Master Curve approach (E1921 standard [5]), as presented in [38], permits finding such a dependence of *σ*<sub>u</sub> on temperature.

## 6.3. Comparison with the master curve (MC) methodology

Finally, a comparison with the Master Curve (MC) methodology is performed. For this comparison, the identified model on all specimens at -100 °C is considered (*m* = 18), and the retained values for *σ*<sub>u</sub> are those presented in Table 6. The Master Curve global approach was developed by [66] and is fully described in the standard E1921 [5]. It permits plotting the transition curve and the tolerance bounds for the chosen cumulative probability level. The median toughness $K_{Jc(med)}$ is expressed as

$$K_{Jc(med)} = 30 + 70 \exp[0.019(T - T_0)], \quad (\text{MPa}\sqrt{m}) \qquad (16)$$

where $T_0$ is the reference temperature such that the median toughness is equal to 100 MPa√m. The approach uses a reference specimen geometry corresponding to a CT25. Thickness corrections, as described in the standard, are used to compare specimens of different thicknesses. The MC approach was developed on CT and SE(B) specimens. Therefore, $T_0$ was identified on all CT specimens. SENT specimens were not used. The procedure detailed in the standard was followed. This led to a value of $T_{0CT} = -92.8$ °C. As stress triaxiality is smaller in SENT specimens, higher toughnesses are obtained, and results do not match the master curve well. It was proposed in [67] to correct measured value by a factor depending on the *T*-stress. This correction was introduced in the SINTAP procedure [68]. However, using the *T*-stress in the case of large-scale yielding (observed in this study for SENT) is not appropriate. It requires the use of the *Q* parameter [2] or some similar method capable of describing large-scale yielding effects as already outlined in [67]. Such a correction of the measured toughness has been proposed in [69]. For a given toughness of the reference specimen $K_{Jc}^{ref}$, the corrected toughness of the considered specimen $K_{Jc}^{spe}$ can be expressed as :

$$K_{Jc}^{spe} = K_{min} + (K_{Jc}^{ref} - K_{min}) \exp [0.019 (\alpha (Q^{spe} - Q^{ref}))] \qquad (17)$$

with $K_{min} = 20 \text{ MPa}\sqrt{m}$. The thickness correction is applied to the measured toughness before applying the *Q*-factor correction. The effect is almost negligible in the present case, as the thickness of the SENT specimen is 30 mm, i.e. close to the reference thickness
<figure><img src="image_22.png" /><figcaption>Fig. 18. Comparison of Master Curve approach to Beremin model considering CT25 as the reference specimen and identifying the Master Curve on all CT specimens. Symbols marked with an ‘x’ are censored data due to excessive plasticity or excessive ductile crack advance. A small temperature shift has been applied to separate data for different specimen types. The shift is 0 for CT25 specimens.</figcaption></figure>

of 25 mm. This method is applied for the present database based on the field stress described in Fig. 17-b considering the CT25 as the reference specimen. Therefore, there is no need to compute the absolute Q value by considering the HRR solution [2]. The difference of Q parameter is equal to $\Delta Q = (T_{yy}^{spe} - T_{yy}^{ref})/\sigma_Y$ for a given load value J and at a relative position equal to $x/r_c = 2$ (see Fig. 17-(b)). One notes that $\Delta Q = 0$ for CT12.5 and CT50 specimens based on results shown in Fig. 17-(b) The parameter $\alpha$ has been identified ($\alpha = 48.62$ K) in [69] for SENB considering the linear relation between $T_0$ and Q:

$$T_0^{spe} = T_0^{ref} + \alpha(Q^{spe} - Q^{ref}) \quad (18)$$

Considering SENT specimens in the present database, $T_0^{SENT} = -123.8$ °C and $\Delta Q = -0.45$ at -100 °C for a load of 100MPa$\sqrt{m}$. The resulting $\alpha$ is equal to 68.9 K.

Based on eq. (17), an equivalent CT25 toughness $K_c^{ref}$ is associated with each toughness value for SENT specimens. For CT12.5 and CT50, the standard length correction is applied to toughness values. This correction permits the comparison of all specimens on the identified Master Curve as presented on Fig. 18.

As geometries are different, the comparison of toughnesses evaluated using the Weibull-Beremin model with experimentally measured ones is not direct. To compare the resulting probabilities of failure to those predicted by the Master Curve, the Weibull stress is converted into a toughness value based on finite element calculations. The obtained toughness is converted into an "equivalent CT25" toughness with the same correction as the one used for the Master Curve. In Fig. 18, the 5% and 95% probabilities of failure obtained thanks to the Beremin model are presented for each temperature and geometry and compared with the same bounds provided by the MC approach. Firstly, comparing computed toughnesses for CT specimens to the master curve gives consistent results. The lower probabilities of failure predicted by the master curve are overestimated, while the current Beremin model better fits experimental data, giving more conservative results and providing a lower envelope curve for toughnesses and probabilities of failure. At -50 °C, the 5% curves are slightly underestimated, leading to an excessive conservatism. The Beremin model also permits catching the higher toughnesses for CT12.5 and SENT specimens, whereas the MC approach underestimates the probability of failure. Furthermore, the current approach, by taking into account all geometries, seems more appropriate in order to predict the failure of a whole structure. It also allows for avoiding censoring of a part of the data. Similar comparisons were made in [14] for a similar statistical model and confirm the advantage of local approaches compared to a global approach.
## 7. Conclusions

This work aimed to present the application of the Beremin model on a comprehensive experimental database consisting of various fracture tests on an 18MND5 steel. Tests were conducted from -150 °C to -50 °C to cover the lower part of the Ductile-To-Brittle transition. The experimental results on Compact Tension (CT) specimens having thicknesses ranging from 12.5 mm to 50 mm thicknesses and Single Edge Notched Tension (SENT) specimens presenting a different stress triaxiality illustrate the well-known size and constraint effects, with a decrease in toughness when thickness increases and more important toughness values and scatter for SENT than for CT specimens due to a lower stress triaxiality. Fractographic observations were used to determine the nature of failure i.e. fully brittle and brittle after some ductile crack extension. For sufficiently high levels of toughness (above $100\text{MPa}\sqrt{\text{m}}$), a clear stretch zone due to blunting is observed. At moderate values of toughness ($100...200\text{MPa}\sqrt{\text{m}}$), a small shear fracture zone is observed between the initial crack tip and the brittle area. Based on the observation of a CT specimen unloaded after reaching a value of $K_J$ equal to $200\text{MPa}\sqrt{\text{m}}$, it can be concluded that this shear zone is formed after brittle crack initiation to link the initiation site to the blunted crack tip. Based on this conclusion, specimens for which brittle failure occurred before any ductile crack advance could be determined.

The second part of the paper aims to model the experimental results in the fully brittle regime. The elastoplastic behavior of the material was identified by using tests on smooth and notched bars. The method uses the difference between experimental and finite element results as an objective function to be minimized. This allows to identify the hardening law well beyond necking accurately. It is also shown that this law can be expressed as the product of $\sigma_Y$ (temperature dependent) and a function that only depends on the accumulated plastic strain. The simulation of cracked specimens used mixed finite elements to evaluate the stresses and, therefore, the Weibull stress better. A convergence study with respect to the Weibull stress was performed. The optimal mesh size in the plane perpendicular to the crack front is chosen depending on the load level (represented by $K_J$) to be reached. The mesh size along the crack front direction can be coarser as the stress gradient is not pronounced in that direction.

The simplest form of the Beremin model was then considered for interpreting fracture test results. It was shown that using different specimen geometries (in particular CT and SENT) ensured the uniqueness of the Weibull parameters. The procedure led to a value of the m parameter equal to 18 corresponding to a value of $\sigma_u = 3152 \text{ MPa}$ at -100 °C for a reference volume equal to $(50 \ \mu\text{m})^3$. These values compare well with literature data. The already reported increase of the characteristic cleavage stress $\sigma_u$ with temperature was confirmed and highlights the material's increased resistance to cleavage as temperature increases. Finally, it is proposed to adapt the master curve approach to represent the SENT and CT samples in a unified framework. Future work will include the description of the upper shelf of the ductile-to-brittle transition in cases where significant ductile crack extension occurs prior to brittle failure.

### CRediT authorship contribution statement

**Emmanuelle Catel:** Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Aboubakr Amzil:** Investigation, Data curation.
**Eric Lorentz:** Writing – review & editing, Supervision, Software, Funding acquisition, Formal analysis, Conceptualization.
**Anna Dahl:** Writing – review & editing, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Data curation.
**Jacques Besson:** Writing – review & editing, Supervision, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix. Experimental data

See Table A.1.

### Data availability

Data will be made available on request.
<table><caption>**Table A.1**
Experimental data.</caption><thead><tr><th>Type</th><th>B (mm)</th><th>B<sub>N</sub> (mm)</th><th>W (mm)</th><th>a<sub>0</sub> (mm)</th><th>Δa (mm)</th><th>T (°C)</th><th>Toughness (MPa√m)</th></tr></thead><tbody><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.33</td><td>0</td><td>-100</td><td>52</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.38</td><td>0</td><td>-100</td><td>77</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.58</td><td>0</td><td>-100</td><td>59</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>49.84</td><td>0</td><td>-100</td><td>85</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>50.26</td><td>0</td><td>-100</td><td>91</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>50.01</td><td>0</td><td>-100</td><td>61</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>49.74</td><td>0</td><td>-100</td><td>60</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>50.59</td><td>0</td><td>-100</td><td>78</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>50.03</td><td>0</td><td>-100</td><td>56</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>49.59</td><td>0</td><td>-100</td><td>55</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.53</td><td>0</td><td>-100</td><td>51</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.60</td><td>0</td><td>-100</td><td>59</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.19</td><td>0</td><td>-100</td><td>49</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.31</td><td>0</td><td>-100</td><td>69</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.35</td><td>0</td><td>-100</td><td>94</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>50.77</td><td>0</td><td>-50</td><td>131</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>50.82</td><td>0</td><td>-50</td><td>145</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.85</td><td>0</td><td>-50</td><td>191</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.57</td><td>0</td><td>-50</td><td>152</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>50.31</td><td>0</td><td>-50</td><td>108</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>50.67</td><td>0</td><td>-50</td><td>150</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>49.89</td><td>0</td><td>-50</td><td>108</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>50.13</td><td>0</td><td>-50</td><td>148</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>50.61</td><td>0</td><td>-50</td><td>135</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>50.45</td><td>0</td><td>-50</td><td>119</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.1</td><td>50.27</td><td>0</td><td>-50</td><td>99</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.38</td><td>0</td><td>-50</td><td>176</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.71</td><td>0</td><td>-50</td><td>126</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>50.92</td><td>0</td><td>-50</td><td>140</td></tr><tr><td>CT50</td><td>50.1</td><td>50.1</td><td>100.0</td><td>49.30</td><td>0</td><td>-50</td><td>107</td></tr><tr><td>CT25</td><td>25.0</td><td>20.0</td><td>50.0</td><td>27.50</td><td>0</td><td>-150</td><td>31.6</td></tr><tr><td>CT25</td><td>25.0</td><td>20.0</td><td>50.0</td><td>27.50</td><td>0</td><td>-150</td><td>28.9</td></tr><tr><td>CT25</td><td>25.0</td><td>20.0</td><td>50.0</td><td>27.50</td><td>0</td><td>-150</td><td>44</td></tr><tr><td>CT25</td><td>25.0</td><td>20.0</td><td>50.0</td><td>27.50</td><td>0</td><td>-150</td><td>42.7</td></tr><tr><td>CT25</td><td>25.0</td><td>20.0</td><td>50.0</td><td>28.00</td><td>0</td><td>-150</td><td>43</td></tr><tr><td>CT25</td><td>25.0</td><td>20.0</td><td>50.0</td><td>28.00</td><td>0</td><td>-150</td><td>39</td></tr><tr><td>CT25</td><td>25.0</td><td>20.0</td><td>50.0</td><td>27.50</td><td>0</td><td>-150</td><td>30.4</td></tr><tr><td>CT25</td><td>25.0</td><td>20.0</td><td>50.0</td><td>28.00</td><td>0</td><td>-150</td><td>50.1</td></tr><tr><td>CT25</td><td>25.0</td><td>20.0</td><td>50.0</td><td>27.50</td><td>0</td><td>-150</td><td>30.4</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>24.94</td><td>0</td><td>-100</td><td>139</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.20</td><td>0</td><td>-100</td><td>123</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.17</td><td>0</td><td>-100</td><td>91</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>24.98</td><td>0</td><td>-100</td><td>68</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.15</td><td>0</td><td>-100</td><td>88</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.05</td><td>0</td><td>-100</td><td>84</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.17</td><td>0</td><td>-100</td><td>96</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.31</td><td>0</td><td>-100</td><td>73</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.04</td><td>0</td><td>-100</td><td>93</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.08</td><td>0</td><td>-100</td><td>76</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>24.93</td><td>0</td><td>-100</td><td>100</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>24.97</td><td>0</td><td>-100</td><td>62</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.09</td><td>0</td><td>-100</td><td>81</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.05</td><td>0</td><td>-100</td><td>73</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.21</td><td>0</td><td>-100</td><td>58</td></tr><tr><td>CT25</td><td>25.1</td><td>25.1</td><td>50.0</td><td>25.06</td><td>0</td><td>-100</td><td>54</td></tr><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.0</td><td>25.17</td><td>0</td><td>-50</td><td>129</td></tr><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.1</td><td>25.24</td><td>0.14</td><td>-50</td><td>228</td></tr><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.1</td><td>25.21</td><td>0.23</td><td>-50</td><td>283</td></tr><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.0</td><td>25.10</td><td>0.22</td><td>-50</td><td>279</td></tr><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.0</td><td>23.64</td><td>0.16</td><td>-50</td><td>229</td></tr><tr><td>CT25</td><td>25.0</td><td>20.2</td><td>50.0</td><td>25.33</td><td>0</td><td>-50</td><td>121</td></tr><tr><td>CT25</td><td>25.0</td><td>19.9</td><td>50.1</td><td>25.16</td><td>0</td><td>-50</td><td>155</td></tr><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.0</td><td>25.20</td><td>0.15</td><td>-50</td><td>223</td></tr><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.0</td><td>25.28</td><td>0.00</td><td>-50</td><td>128</td></tr><tr><td>CT25</td><td>25.0</td><td>20.2</td><td>50.0</td><td>25.26</td><td>0.69</td><td>-50</td><td>401</td></tr><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.0</td><td>25.12</td><td>0.00</td><td>-50</td><td>123</td></tr></tbody></table>

(continued on next page)
<table><caption>**Table A.1 (continued).**</caption><thead><tr><th>Type</th><th>B (mm)</th><th>B<sub>N</sub> (mm)</th><th>W (mm)</th><th>a<sub>0</sub> (mm)</th><th>Δa (mm)</th><th>T (°C)</th><th>Toughness (MPa√m)</th></tr></thead><tbody><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.0</td><td>25.17</td><td>0.00</td><td>-50</td><td>204</td></tr><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.0</td><td>25.22</td><td>0.12</td><td>-50</td><td>233</td></tr><tr><td>CT25</td><td>25.0</td><td>20.1</td><td>50.0</td><td>25.21</td><td>0.33</td><td>-50</td><td>312</td></tr><tr><td>CT12.5</td><td>12.2</td><td>12.2</td><td>25.0</td><td>12.55</td><td>0</td><td>-150</td><td>48</td></tr><tr><td>CT12.5</td><td>12.2</td><td>12.2</td><td>25.0</td><td>12.69</td><td>0</td><td>-150</td><td>52</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.64</td><td>0</td><td>-150</td><td>35</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.39</td><td>0</td><td>-150</td><td>53</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.02</td><td>0</td><td>-150</td><td>53</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.40</td><td>0</td><td>-150</td><td>49</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.61</td><td>0</td><td>-150</td><td>54</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.69</td><td>0</td><td>-150</td><td>54</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.71</td><td>0</td><td>-150</td><td>41</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.63</td><td>0</td><td>-150</td><td>46</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.64</td><td>0</td><td>-150</td><td>44</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.59</td><td>0</td><td>-150</td><td>51</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.61</td><td>0</td><td>-100</td><td>75</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.69</td><td>0</td><td>-100</td><td>120</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.64</td><td>0</td><td>-100</td><td>125</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.59</td><td>0</td><td>-100</td><td>55</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.64</td><td>0</td><td>-100</td><td>106</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.63</td><td>0</td><td>-100</td><td>117</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.67</td><td>0</td><td>-100</td><td>77</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.76</td><td>0</td><td>-100</td><td>122</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.74</td><td>0</td><td>-100</td><td>126</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.61</td><td>0</td><td>-100</td><td>113</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.86</td><td>0</td><td>-100</td><td>107</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.61</td><td>0</td><td>-100</td><td>88</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.68</td><td>0</td><td>-100</td><td>94</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.74</td><td>0</td><td>-100</td><td>142</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.60</td><td>0</td><td>-100</td><td>115</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.61</td><td>0</td><td>-100</td><td>47</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.64</td><td>0</td><td>-100</td><td>84</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.61</td><td>0</td><td>-100</td><td>61</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.67</td><td>0</td><td>-100</td><td>102</td></tr><tr><td>CT12.5</td><td>12.5</td><td>12.5</td><td>25.0</td><td>12.74</td><td>0</td><td>-100</td><td>122</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.58</td><td>1.58</td><td>-50</td><td>476</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.1</td><td>25.0</td><td>13.43</td><td>0.08</td><td>-50</td><td>206</td></tr><tr><td>CT12.5</td><td>12.5</td><td>9.8</td><td>25.1</td><td>13.30</td><td>0.3</td><td>-50</td><td>287</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.46</td><td>0.05</td><td>-50</td><td>198</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.1</td><td>25.0</td><td>13.61</td><td>0.51</td><td>-50</td><td>372</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.56</td><td>0.25</td><td>-50</td><td>289</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.1</td><td>25.0</td><td>13.37</td><td>0</td><td>-50</td><td>152</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.1</td><td>25.0</td><td>13.50</td><td>0.21</td><td>-50</td><td>271</td></tr><tr><td>CT12.5</td><td>12.5</td><td>9.7</td><td>25.0</td><td>13.59</td><td>0.64</td><td>-50</td><td>381</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.1</td><td>25.0</td><td>13.51</td><td>0.71</td><td>-50</td><td>387</td></tr><tr><td>CT12.5</td><td>12.5</td><td>9.6</td><td>25.0</td><td>13.51</td><td>0.63</td><td>-50</td><td>377</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.52</td><td>0.19</td><td>-50</td><td>251</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.1</td><td>25.0</td><td>13.38</td><td>0.73</td><td>-50</td><td>397</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.52</td><td>0</td><td>-50</td><td>133</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.50</td><td>0.29</td><td>-50</td><td>292</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.47</td><td>0.54</td><td>-50</td><td>350</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.45</td><td>0</td><td>-50</td><td>142</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.45</td><td>0.06</td><td>-50</td><td>189</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.46</td><td>0.28</td><td>-50</td><td>290</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.42</td><td>0.07</td><td>-50</td><td>215</td></tr><tr><td>CT12.5</td><td>12.5</td><td>10.0</td><td>25.0</td><td>13.45</td><td>0.15</td><td>-50</td><td>255</td></tr><tr><td>SENT</td><td>30.0</td><td>27.1</td><td>15.0</td><td>7.81</td><td>0</td><td>-150</td><td>60.1</td></tr><tr><td>SENT</td><td>30.0</td><td>27.0</td><td>15.0</td><td>7.8</td><td>0</td><td>-150</td><td>80.8</td></tr><tr><td>SENT</td><td>30.0</td><td>27.1</td><td>15.0</td><td>7.8</td><td>0</td><td>-150</td><td>42.3</td></tr><tr><td>SENT</td><td>30.0</td><td>27.1</td><td>15.0</td><td>7.9</td><td>0</td><td>-150</td><td>38.6</td></tr><tr><td>SENT</td><td>30.0</td><td>27.0</td><td>15.0</td><td>7.85</td><td>0</td><td>-150</td><td>53.3</td></tr><tr><td>SENT</td><td>30.0</td><td>27.1</td><td>15.0</td><td>7.84</td><td>0</td><td>-150</td><td>50.4</td></tr><tr><td>SENT</td><td>30.0</td><td>27.1</td><td>15.0</td><td>7.87</td><td>0</td><td>-150</td><td>35.2</td></tr><tr><td>SENT</td><td>30.0</td><td>27.1</td><td>15.0</td><td>7.91</td><td>0</td><td>-150</td><td>46.4</td></tr><tr><td>SENT</td><td>30.0</td><td>27.0</td><td>15.0</td><td>7.95</td><td>0</td><td>-150</td><td>51.4</td></tr></tbody></table>

(continued on next page)
<table><caption>Table A.1 (continued).</caption><thead><tr><th>Type</th><th>B (mm)</th><th>B<sub>N</sub> (mm)</th><th>W (mm)</th><th>a<sub>0</sub> (mm)</th><th>Δa (mm)</th><th>T (°C)</th><th>Toughness (MPa√m)</th></tr></thead><tbody><tr><td>SENT</td><td>30.0</td><td>27.0</td><td>15.0</td><td>7.81</td><td>0</td><td>-150</td><td>45.8</td></tr><tr><td>SENT</td><td>30.2</td><td>27.0</td><td>15.0</td><td>7.82</td><td>0.05</td><td>-100</td><td>159.4</td></tr><tr><td>SENT</td><td>30.2</td><td>27.0</td><td>15.0</td><td>7.82</td><td>0.08</td><td>-100</td><td>181.6</td></tr><tr><td>SENT</td><td>30.2</td><td>27.0</td><td>15.0</td><td>7.79</td><td>0</td><td>-100</td><td>58.5</td></tr><tr><td>SENT</td><td>30.2</td><td>27.0</td><td>15.0</td><td>7.81</td><td>0.06</td><td>-100</td><td>202</td></tr><tr><td>SENT</td><td>30.2</td><td>27.0</td><td>15.0</td><td>7.83</td><td>0.02</td><td>-100</td><td>117.9</td></tr><tr><td>SENT</td><td>30.1</td><td>27.0</td><td>15.0</td><td>7.80</td><td>0</td><td>-100</td><td>108.5</td></tr><tr><td>SENT</td><td>30.2</td><td>27.0</td><td>15.0</td><td>7.77</td><td>0</td><td>-100</td><td>82.4</td></tr><tr><td>SENT</td><td>30.2</td><td>27.0</td><td>15.0</td><td>7.77</td><td>0</td><td>-100</td><td>104.2</td></tr><tr><td>SENT</td><td>30.2</td><td>27.0</td><td>15.0</td><td>7.82</td><td>0.08</td><td>-100</td><td>189.3</td></tr><tr><td>SENT</td><td>30.2</td><td>27.0</td><td>15.0</td><td>7.86</td><td>0.06</td><td>-100</td><td>192.5</td></tr><tr><td>SENT</td><td>30.2</td><td>27.0</td><td>14.9</td><td>7.68</td><td>0.28</td><td>-50</td><td>264</td></tr></tbody></table>

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