# A simple criterion to exclude the risk of brittle fracture in the brittle-to-ductile transition temperature range

Déborah Clement, C. Jacquemoud, S. Chapuliot, S. Marie

► To cite this version:

Déborah Clement, C. Jacquemoud, S. Chapuliot, S. Marie. A simple criterion to exclude the risk of brittle fracture in the brittle-to-ductile transition temperature range. Engineering Fracture Mechanics, 2025, 314, pp.110739. 10.1016/j.engfracmech.2024.110739. hal-04850478

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Submitted on 20 Dec 2024

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Distributed under a Creative Commons Attribution 4.0 International License
A simple criterion to exclude the risk of brittle
fracture in the brittle to ductile transition
temperature range

D. Clement<sup>a,*</sup>, C. Jacquemoud<sup>a</sup>, S. Chapuliot<sup>b</sup>, S. Marie<sup>c</sup>

<sup>a</sup> Université Paris-Saclay, CEA, Service d'Etudes Mécaniques et Thermiques, 91191, Gif-sur-Yvette, France

<sup>b</sup> EDF/R&D Les Renardières, France

<sup>c</sup> FRAMATOME, France

**Abstract**

This paper presents the use of a simple threshold stress criterion to exclude the risk of brittle fracture in the brittle-to-ductile transition temperature range of carbon-manganese ferritic steel. In order to compare its predictions for laboratory specimens as well as for structures, fracture tests on CT specimens and three pipe structures representative of the in-service auxiliary piping system of French PWR, were analysed. In association with Finite Element analyses, this criterion allowed to predict a lower bound of the non-fracture threshold.
## Nomenclature

<table><tr><td>a</td><td>crack depth</td></tr><tr><td>B</td><td>CT specimen thickness</td></tr><tr><td>c</td><td>crack half-length</td></tr><tr><td>Δa</td><td>crack extension</td></tr><tr><td>E, ν</td><td>Young and Poisson's modulii</td></tr><tr><td>F</td><td>slope linking the probability of failure to threshold volumes V<sub>th</sub> or V<sub>th_intg</sub></td></tr><tr><td>J</td><td>non-linear strain energy release rate</td></tr><tr><td>J<sub>i</sub></td><td>J at crack initiation</td></tr><tr><td>J<sub>max</sub></td><td>maximal value of J at the deepest point of the crack</td></tr><tr><td>K<sub>min</sub>(T)</td><td>minimum stress intensity factor required</td></tr><tr><td>m</td><td>Weibull modulus</td></tr><tr><td>n<sub>T</sub></td><td>number of specimens at T temperature</td></tr><tr><td>P<sub>r</sub></td><td>Total probability of failure</td></tr><tr><td>σ<sub>I</sub></td><td>maximum principal stress</td></tr><tr><td>σ<sub>th</sub></td><td>threshold stress (MPa)</td></tr><tr><td>σ<sub>u_CT</sub></td><td>normalization stress</td></tr><tr><td>σ<sub>y</sub></td><td>yield stress</td></tr><tr><td>σ<sub>y0</sub></td><td>threshold yield stress</td></tr><tr><td>t</td><td>pipe thickness</td></tr><tr><td>V<sub>th</sub></td><td>threshold volume</td></tr><tr><td>V<sub>th_intg</sub></td><td>integral formulation of the threshold volume</td></tr><tr><td>V<sub>th_min</sub></td><td>threshold volume corresponding to K<sub>min</sub>(T)</td></tr><tr><td>V<sub>0</sub></td><td>elementary volume</td></tr><tr><td>W</td><td>CT specimen width</td></tr></table>

## Principal abbreviations

<table><tr><td>CMOD</td><td>Crack Mouth Opening Displacement</td></tr><tr><td>CT</td><td>Compact Tension specimen</td></tr><tr><td>CT20-15</td><td>CT specimen with a nominal thickness of 20 mm reduced to 15 mm</td></tr><tr><td>FE</td><td>Finite Element</td></tr><tr><td>NT</td><td>Notched Tensile specimen</td></tr><tr><td>PWR</td><td>Pressurized Water Reactor</td></tr><tr><td>RPV</td><td>Reactor Pressure Vessel</td></tr><tr><td>SE(B)</td><td>Single Edge-crack Bend specimen</td></tr><tr><td>SE(T)</td><td>Single Edge-crack Tension specimen</td></tr></table>

# 1. Introduction

The auxiliary loop of PWR contains thin pipes made of carbon-manganese ferritic steel, in which temperatures during incidental situations can come close to the brittle-to-ductile transition temperature range where the fracture process changes from ductile tearing to cleavage fracture. Therefore, in-service inspection rules for mechanical components of PWR nuclear islands [1] impose to carry out a fracture mechanical analysis to demonstrate the integrity of those pipes for both brittle and ductile fracture modes.

Regarding brittle fracture, which is a brutal fracture mode with serious consequences, this analysis aims at defining a critical defect size under which cleavage cannot occur. However, in some cases, the low fracture toughness of ferritic steel can lead to critical defect sizes smaller than detection capacities of non-destructive examinations, which complicates defect tolerance demonstration. One can note that conventional fracture mechanics analysis leans on results from tests performed on laboratory specimens, such as standard CT specimens, to define materials' transition temperature and toughness. Crack-tip constraint in those deep-cracked specimens (a/W ~ 0.5) is generally higher than in components such as thin pipes, inducing a lower toughness. Hence, critical defect sizes ascertained with standard laboratory specimens are not representative of the dimensions of a real defect found in thin pipe structures.

An extensive work was done in order to understand and quantify the constraint-effect [2]. As in components, a lower crack-tip constraint was also observed in laboratory specimens with shallow cracks (a/W ~ 0.1). This loss of constraint is due to interaction between the elastic-plastic crack tip stress field and the specimen surface, resulting in a reduction of the hydrostatic stress close to the crack tip. In order to derive accurate safety margins on critical defect sizes, VOCALIST project aimed, in part, at quantifying the constraint effects on cleavage fracture toughness of ferritic steels used in the fabrication of nuclear reactor pressure vessels [3]. It illustrated a shift in the brittle to ductile transition curve towards lower temperatures for SE(B) specimens with shallow cracks (a/W ~ 0.1), compared with standard CT, SE(B) and SE(T) specimens (a/W ~ 0.5), showing a decrease of the Master Curve reference temperature T<sub>0</sub> as defined in [4].

With the same objective of evaluating safety margins on critical defect sizes, further work was done to compare mechanical behavior of pipe structures representative of the auxiliary loop piping, and standard laboratory specimens [5, 6]. The authors used a simple stress-based cleavage criterion to size circumferential surface defects before performing structure tests in the brittle-to-ductile transition temperature range. This recent criterion allows to predict simply the absence of cleavage risk by defining non-fracture thresholds [5, 7, 8, 9]. One structural test was performed at the highest temperature at which cleavage becomes possible on CT specimens (-50°C) [5]. The results showed an important difference of ductile behavior between the standard CT specimens and the pipe structure, which developed a much larger tearing length.
Following the same experimental approach, three structural tests were performed at the highest temperature at which CT specimens fracture by cleavage without any ductile initiation (-70°C). The test on the first pipe, and the results, are precisely described in [6]. The present study was dedicated to the validation of the threshold stress model for structures, applying it to analyze the results of the tests on those three pipes. Moreover, it was used to compare the mechanical behavior of the CT specimens and the pipes.

## 2. Threshold stress model

### 2.1 Basic assumptions of the model

The authors used a threshold stress model to dimension the defect and analyze the test results. This model was first proposed in [7] with the objective to simply assess the risk of brittle fracture of structures within the brittle-to-ductile transition temperature range. It is based on a minimal threshold stress $\sigma_{th}$ below which cleavage cannot occur. $\sigma_{th}$ is assumed to be a material property. As initially proposed in [10], the threshold stress criterion is based on the hypothesis that the probability of brittle failure is directly related to the volume at the crack tip, submitted to a maximum principal stress higher than $\sigma_{th}$. This volume, referred to as $V_{th}$, is the volume susceptible to cleavage.

As a first approximation, and postulating a weakest link assumption, according to which the failure of an elementary volume induces the failure of the whole specimen, the probability of failure $P_r$ is simply calculated with equation (1):

$$P_r = 1 - \exp[-F(T) \cdot V_{th}] \quad (1)$$

In this equation, $F(T)$ is a sensitivity function to cleavage, depending on the yield stress $\sigma_y$, according to equation (2):

$$F(\sigma_y) = \left( \frac{\sigma_y - \sigma_{y0}}{\sigma_{u\_CT} \cdot V_0^{1/m}} \right)^m \quad (2)$$

where $\sigma_{y0}$ is the threshold yield stress, representative of the temperature above which cleavage cannot occur; $m$ is the Weibull modulus; $V_0$ is an elementary volume; $\sigma_{u\_CT}$ is a normalization stress. This definition of $F(\sigma_y)$ was validated with results of tests performed on CT specimens with different sizes. One should note that, in this first approximation, triaxiality variation with the specimen geometry is not taken into account. One can also write equation (1) as follows:

$$-\ln(1 - P_r) = F(\sigma_y) \cdot V_{th} \quad (3)$$

Therefore, the parameter $-\ln(1-P_r)$ should evolve linearly with $V_{th}$. In order to validate this linear assumption, an analysis of fracture tests performed on CT25 specimens was proposed in [7]. The tests were performed over a wide temperature range, extending from very low temperatures (-253°C) to the onset of ductile tearing (-50°C). The probability of failure of the $i^{th}$ specimen is expressed as follows:

$$P_{ri} = \frac{i-0.5}{n_T} \quad (4)$$

$n_T$ was the number of tests carried out at a given temperature. For each test on CT25 specimen, $V_{th}$ was calculated from a FE analysis, corresponding to the volume of material subjected, at fracture, to a principal stress higher than $\sigma_{th}$. For the lowest probabilities, the authors showed that the parameter $-\ln(1-P_r)$ evolved linearly with $V_{th}$, thus validating equation (3). They showed that the slope $F(\sigma_y)$ depended strongly on test temperature. Moreover, for a given $T$ temperature, the linear evolution does not pass through the origin, which means that a minimum $V_{th}$ volume is required to complete specimen fracture. This minimum volume, referred to as $V_{th\_min}(T)$, is associated to a minimum stress intensity factor depending on temperature, referred to as $K_{min}(T)$. From these results, the authors proposed a new formulation of the fracture criterion, rewriting equation (3) as follows:

$$-\ln(1 - P_r) = F(T) \cdot (V_{th} - V_{th\_min}(T)) \quad (5)$$
with $F(T) = \left( \frac{\sigma_y}{\sigma_{u\_CT} \cdot V_0^{1/m}} \right)^m$.

One can transform equation (1) to:

$$P_r = 1 - \exp[-F(T) \cdot (V_{th} - V_{th\_min}(T))] \quad (6)$$

Equation (6) means that, for a $V_{th}$ volume lower than $V_{th\_min}$, corresponding to a toughness lower than $K_{min}(T)$, probability of failure is equal to zero.

In [9], the authors compared the sensitivity functions $F$ determined for different specimen geometries, as a function of the temperature. They showed that it was necessary to normalize $V_{th}$ by the crack front length, to obtain a good consistency between the configurations. Moreover, large loading variations along the crack front, such as those seen in cracked pipes, can influence the global probability of failure. In order to transfer the threshold stress criterion to real structures with real defects, it is important to take into account this loading variation along the crack front. The authors proposed a new normalized integral formulation of the criterion:

$$V_{th\_intg} = \int_{front} \left[ \max \left( 0; \frac{dV_{th}(S)}{ds} - \frac{dV_{th}(S)}{ds} \right) \right]_{min} ds \quad (7)$$
In equation (7), $dV_{th}/ds$ is the $V_{th}$ volume calculated on an incremental part $ds$ of the curvilinear abscise of the crack front. $dV_{th}/ds_{min}$ is an incremental threshold volume calculated from equation (8).

$$ \left. \frac{dV_{th}(S)}{ds} \right|_{min} = \frac{V_{th\_min(CT)}}{B} \qquad (8) $$

$V_{th\_min(CT)}$ is the minimal threshold volume determined at a given temperature on a CT specimen, and $B$ is the CT specimen thickness. This integral formulation allows coming back to a local 2D configuration by determining stresses distribution around each point of the crack front in the normal plan. Replacing ($V_{th}-V_{th\_min}$) by $V_{th\_intg}$ in equation (5) gives:

$$ P_r = 1 - \exp[-F(T) \cdot V_{th\_intg}(T)] \qquad (9) $$

Therefore, equation (9) allows one to predict simply the absence of brittle fracture risk if the $V_{th\_intg}$ volume is null.

## 2.2 Calibration of the model

The threshold stress $\sigma_{th}$ can be determined with series of tests performed at very low temperature on NT specimens with a reduced notch radius. In these conditions, plastic strain is limited and cleavage is mainly driven by the stress level, generally initiating on a second-phase particle. $\sigma_{th}$ is set as the lower bound of the maximum principal stress reached at fracture for each NT specimen.

The minimal threshold volume $V_{th\_min(CT)}$ used in Equation (8), can be determined with a FE analysis of tests performed on CT specimens at the temperature of interest (-70°C in the present study).

For the material studied in this paper, the model was calibrated in [6], with a $\sigma_{th}$ value of 1250MPa and a $V_{th\_min}(-70°C)$ value of 0.8mm$^3$.

# 3. Experimental results

## 3.1 Description of the material

The material considered in this study is a TU42C (French denomination) carbon-manganese steel, representative of C-Mn steels constituting the in-service auxiliary piping system of French PWR. The dimensions of the pipe structures studied were also representative of this piping system, with an external diameter of 406.4mm and a thickness $t$ of 23mm. Three pipes, with a length of 7.5m, were tested in 4-point bending. All the laboratory specimens dedicated to chemical and mechanical characterization of the material, were sampled from a 4th pipe. Table 3-1 gives its chemical composition, characterized by a particularly high sulfur level.

<table><caption>Table 3-1: Chemical composition of Tu42C carbon-manganese steel [6]</caption><thead><tr><th>C</th><th>Mn</th><th>Si</th><th>P</th><th>S</th><th>Cu</th><th>Sn</th></tr></thead><tbody><tr><td>0.153</td><td>0.7</td><td>0.29</td><td>0.019</td><td>0.012</td><td>0.19</td><td>0.01</td></tr></tbody></table>

<table><caption>Table 3-2: Mechanical properties of TU42C carbon-manganese steel at 20°C [6]</caption><thead><tr><th>R<sub>p0.2</sub> (MPa)</th><th>R<sub>m</sub> (MPa)</th><th>Elongation (%)</th><th>K<sub>v</sub></th></tr></thead><tbody><tr><td>322</td><td>485</td><td>30.3</td><td>144</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>Figure 3-1: True stress-strain curve at -70°C</figcaption></figure>

Table 3-2 presents the material mechanical properties at 20°C. The true stress-strain curve of the material was previously determined at -70°C with tensile tests performed on smooth cylindrical specimens. Figure 3-1 shows the stress-strain curve used in this study.

## 3.2 Tests on compact tension specimens

Series of tests were carried out on 7 compact tension specimens at -70°C. The tested specimens were CT20 specimens with a nominal thickness reduced to 15mm, noted hereafter as CT20-15. Presence of side grooves reduced the effective thickness to 12mm. Five specimens fractured by cleavage without any ductile initiation and two specimens fractured by cleavage after small ductile propagation of less than 0.2mm.

## 3.3 Four-point bending test on pipe structures

Mechanical behavior at fracture of the cracked pipes was studied by performing four-point bending tests at -70°C with a servo-hydraulic testing system Instron associated with a 2500kN load cell. Figure 3-2 shows the testing device internally developed. The inner and outer spans were respectively set to 1200mm and 5000mm. An extensometer fixed at the surface of the external crack measured the CMOD during the tests. A phase of fatigue pre-cracking at ambient temperature preceded the fracture tests. A cooling system allowed maintaining the central part of the cracked pipe at -70°C.
<figure><img src="image_3.png" /><figcaption>Figure 3-2: Testing device internally developed to perform a 4-point bending test on a 7.5m long pipe with a servo-hydraulic system Instron associated with a 2500kN load cell</figcaption></figure>

The bending test was controlled with imposed displacement to avoid any resonance phenomenon. For the first pipe, hereafter noted Pipe 1, the operators ended the test as the pipe structure entered into a global plastic collapse. The tests on the two other pipes, hereafter noted Pipe 2 and Pipe 3, were conducted up to fracture.

After fracture, the fracture surfaces were observed with a digital microscope VHX-1000 (Keyence) and a FE-SEM Sigma 500 (Zeiss). Each pipe presented a different crack geometry, characterized by the ratios $a/t$ and $c/a$. As presented in Table 3-3, Pipe 1, already presented in [6], included a semi-circular defect; Pipe 2 included a long semi-elliptical defect; Pipe 3 included a shallow semi-elliptical defect. Figure 3-3 shows a macrography of the fracture surface of Pipe 3. One can see, from top to bottom, the notch zone, the pre-cracking zone and the cleavage zone. Figure 3-4 present MEB observations of the zone close to the deepest point of the crack. They show the existence of ductile tearing initiation between the pre-cracking and the cleavage zone. As presented in Table 3-3, the maximum propagation $\Delta a$, observed at the deepest crack point, stands between 0.3 and 0.45mm.

<table><caption>Table 3-3: Initial crack geometry of each pipe and ductile propagation measured after testing</caption><thead><tr><th></th><th>a (mm)</th><th>c/a</th><th>a/t</th><th>Δa (mm)</th></tr></thead><tbody><tr><td>Pipe 1</td><td>9.2</td><td>1.2</td><td>0.41</td><td>0.3</td></tr><tr><td>Pipe 2</td><td>10.9</td><td>4.1</td><td>0.45</td><td>0.3</td></tr><tr><td>Pipe 3</td><td>5.5</td><td>3.3</td><td>0.25</td><td>0.45</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Figure 3-3: Macrography of one of the fracture surfaces of Pipe 3</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3-4: SEM observation of one of the fracture surfaces of Pipe 3 (a) with a zoom on the ductile tearing zone (b)</figcaption></figure>

## 4. Finite Element modelling

3D numerical analyses were carried out using the FE software CAST3M [12] to analyze the experimental results from the tests performed at -70°C on the CT specimens and the pipes.
## 4.1 CT specimens

Owing to two symmetry planes, the model was composed of only one quarter of specimen as shown on Figure 4-1. The crack (in red) was meshed with refined elements of 50µm size along the crack front. A quarter of cylinder (represented in blue) was modeled as a rigid body to simulate the pin through which the specimen was loaded.

The symmetry conditions were set along the crack plan, and along the plan perpendicular to the latter and passing through the center of the crack. An additional locking condition was applied on one point of the model to prevent any rigid body motion. A vertical displacement was imposed on the pin. The elastic-plastic material was assumed to follow a von Mises model with isotropic hardening, defined with the tensile curve obtained at -70°C (Figure 3-1).

A first simulation was carried out with the small displacements assumption to calculate precisely the *J* parameter along the crack front. A second simulation was performed with the large strains and large displacements formulation to obtain precise stress fields for *V*<sub>th</sub> calculation. Figure 4-2 shows the comparison between load-CMOD curves obtained experimentally on a CT specimen and with simulation.

<figure><img src="image_6.png" /><figcaption>Figure 4-1: Meshing of one quarter of a CT20-15 specimen with the pre-crack in red, and the pin for loading in blue modeled as a quarter of cylinder</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 4-2: Comparison between load-CMOD curves obtained experimentally on a CT specimen and with simulation</figcaption></figure>

Small and large displacements assumptions gave the same load-CMOD curves. Simulation fits well the experimental curve up to the beginning of the elasto-plastic domain. Afterwards, simulation overestimates experimental results up to 8%.

## 4.2 Pipe structures

Owing to two planned symmetries, the model was composed of only one quarter of pipe. Moreover, during a four-point bending test, the bending moment imposed to the pipe part located between the inner supporting rolls is constant. Therefore, the rolls were not modeled, and a pure bending moment was directly applied to a 1m long pipe, representing the center part of the pipes. Figure 4-3 shows the mesh used for the defect sizing. As shown on the zoom, the crack was meshed with refined cubic elements around the front, with a side size of 50µm.

The symmetry conditions were set along the crack plan (in blue), and along the plan perpendicular to the latter and passing through the center of the crack (in pink). The crack lip (in red) was free of symmetry conditions. An additional locking condition was applied on one point of the model to prevent any rigid body motion (pointed with the black triangle). A rotation was imposed at the opposite section of the pipe regarding to the crack, by applying an opposite axial displacement at the two extreme points of the section. The section planarity was ensured by kinematic conditions applied to the other nodes of the section. The elastic-plastic material was assumed to follow the same von Mises model at -70°C (Figure 3-1).

For each pipe, containing a specific defect geometry, two simulations were performed. A first simulation was carried out with the small displacements assumption to calculate precisely the *J* parameter along the crack front. A second simulation was performed with the large strains and large displacements formulation to obtain precise stress fields for *V*<sub>th</sub> calculation.
<figure><img src="image_8.png" /><figcaption>Figure 4-3: Meshing of one quarter of pipe showing the symmetry plans in blue and pink, and the crack lip in red, with refined elements of 50µm size around the crack front</figcaption></figure>

As ductile tearing initiated in the pipes, the authors integrated a node release approach in this second simulation to simulate the crack extension with increasing loading. For given crack mouth openings, the symmetry condition imposed on each node of the crack front was released. This node release was controlled according to the curve da-CMOD presented on Figure 4-4.

<figure><img src="image_9.png" /><figcaption>Figure 4-4: Δa-CMOD curves controlling release of the crack nodes to simulate ductile tearing for the three pipes</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 4-5: Comparison between the experimental and simulated load-CMOD curves obtained for Pipe 3</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 4-6: Evolution along the crack front of J, normalized by $J_{max}$, for the loading corresponding to a J value at the deepest crack point of 200kJ/m² for the pipes and the CT specimen</figcaption></figure>

Those curves were constructed so that ductile tearing initiated at a J value of 180 kJ/m², and the crack length corresponding to the crack mouth opening at pipe fracture was equal to the experimental final crack length.

Figure 4-5 compares, for Pipe 3, the experimental load-CMOD curve with the simulation integrating the node release approach. The curves are in very good agreement, which allows validating the FE model.
Figure 4-6 presents the evolution along the crack front of $J$, normalized by its maximum value noted $J_{max}$, for the loading corresponding to a $J$ value at the deepest crack point of $200\text{kJ/m}^2$. It highlights an influence of the specimen and the defect geometry on the loading variation along the crack front. It also shows that $J_{max}$ is obtained at the deepest crack point for all the geometries of specimen and defects. The authors assumed $J_{max}$ to be more representative of the fracture probability. Therefore, the evolutions of the important parameters are represented as a function of $J_{max}$ in the following analysis.

### 4.3 Prediction of the risk of brittle fracture

Figure 4-7 presents the $V_{th}$ evolution obtained by simulation for a CT specimen tested at $-70^\circ\text{C}$. Symbols stand for experimental fractures. According to the initial stress-based criterion, expressed according equation 1, there is no risk of brittle fracture up to a $J$ of $40\text{kJ/m}^2$. However, all CT specimens fractured for $V_{th}$ values higher than $0.8\text{mm}^3$. As proposed in [7], these results illustrate the existence of a minimal volume $V_{th\_min}$ necessary to complete specimens fracture. In accordance with the minimal volume identified in [6], $V_{th\_min}$ was set to $0.8\text{mm}^3$ to calculate the integral $V_{th\_intg}$ volume according equation 7. Moreover, two CT specimens reached a $J_{max}$ value higher than $200\text{kJ/m}^2$, for which small ductile bands less than $0.2\text{mm}$ thick developed. Therefore, the $J_i$ value, corresponding to the $J$ value at ductile tearing initiation, was estimated at $200\text{kJ/m}^2$ [6].

Figure 4-8 compares the $V_{th\_intg}$ evolutions obtained by simulation for CT specimens and for the three pipes. This integral formulation predicts no risk of brittle fracture, corresponding to a $V_{th\_intg}$ null, up to a $J$ of $50\text{kJ/m}^2$ for the CT specimen, and $120\text{kJ/m}^2$ for the pipes. Afterward, the risk of brittle fracture exists, in particular before ductile tearing initiation ($J<J_i$), for all the geometries. This is in agreement with experimental fracture of the CT specimens, for which a large majority fractured before ductile tearing initiation, for $V_{th\_intg}$ volumes lower than $2.5\text{mm}^3$. As for the pipes, they fractured for $J$ values from $265$ to $350\text{kJ/m}^2$, which are much higher than the predicted non-fracture threshold of $120\text{kJ/m}^2$. These values are also higher than the estimation of $J$ at ductile initiation, which is in accordance with the development of a short ductile tearing observed experimentally in the three pipes. Moreover, all pipes fractured for $V_{th\_intg}$ volumes higher than for the last CT fracture. In particular, Pipe 2 fractured for a much higher $V_{th\_intg}$ volume of $10\text{mm}^3$.

<figure><img src="image_12.png" /><figcaption>Figure 4-7: $V_{th}$ evolution obtained by simulation for a CT specimen; symbols stand for the experimental fractures</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 4-8: Comparison of the $V_{th\_intg}$ evolutions obtained by simulation for a CT specimen and for the pipes; symbols stand for the experimental fractures</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 4-9: Comparison between the maximal principal stress $\sigma_i$ obtained from the deepest crack point along the ligament, at a loading corresponding to a $J_{max}$ of $120\text{kJ/m}^2$, for the pipes and a CT specimen</figcaption></figure>

Figure 4-9 compares the evolutions of $\sigma_i$ obtained from the deepest crack point along the ligament for the pipes, and from the center crack point along the ligament for a CT specimen, for a $J_{max}$ of $120\text{kJ/m}^2$ (corresponding to the non-fracture threshold predicted for the pipes). $\sigma_i$ is higher than $\sigma_{th}$ ($1250\text{MPa}$) from the deepest crack point up to $0.1\text{mm}$ for the CT specimen, and up to $0.3\text{mm}$ for the pipes. Therefore, the volume susceptible to cleavage extends twice more for the pipes than for CT specimens.
<figure><img src="image_15.png" /><figcaption>Figure 4-10: Comparison between the triaxiality factor obtained from the deepest crack point along the ligament, at a loading corresponding to a $J_{max}$ of 120kJ/m², for the pipes and a CT specimen</figcaption></figure>

Figure 4-10 compares the evolutions of the triaxiality factor obtained from the deepest crack point along the ligament, for the pipes and a CT specimen, for a $J_{max}$ of 120kJ/m². It corresponds to the ratio between the hydrostatic pressure $p$ and the von Mises equivalent stress $q$. For the Cauchy stress tensor $\tilde{\sigma}$, it is defined by:

$$ \tilde{\sigma} = \tilde{s} - p\tilde{I} \qquad (12) $$

$$ p = -tr(\tilde{\sigma})/3 \qquad (13) $$

$$ q = \sqrt{\frac{3}{2}tr(\tilde{s}^2)} \qquad (14) $$

$\tilde{s}$ is the deviatoric stress tensor, and $\tilde{I}$ is the identity matrix. The triaxiality factor is minimum at the deepest crack point and maximum around 0.2mm for the pipes, and 0.35mm for the CT specimen. It is much higher for the CT specimen at the deepest crack point, and it becomes similar for the pipes at 0.2mm from the crack front. Therefore, for the pipes, the zone around 0.2mm from the deepest crack point is more favorable for microcracks creation than at the deepest crack point.

## 5. Discussion

In this study, the threshold stress criterion has been applied to compare non-fracture thresholds for laboratory specimens and pipe structures. As proposed in [7], the results confirmed the importance to introduce in the criterion formulation, a minimal threshold volume $V_{th_min}$, necessary to complete specimens fracture. With this minimal volume, the integral formulation defined by equation (7), allows to predict a non-fracture threshold in accordance with experimental fractures of the CT specimens, which mostly fractured before ductile initiation. However, for pipe structures, this formulation seems severe as it predicts a low non-fracture threshold compared to experimental fractures.

The criterion predicts an existing risk of brittle fracture before ductile initiation, whereas the pipes fractured for a high $V_{th_intg}$ volume, especially for pipe 2 which contains a long defect, after a small ductile tearing development.

Analysis of the maximum principal stress showed that the volume susceptible to cleavage extends from the deepest crack point up to 0.2mm in the ligament. However, in this volume, the triaxiality factor is very low at the deepest crack point, and maximum at 0.2mm in the ligament. In the literature, the authors agree that brittle fracture mechanism occurs in two steps [12, 13, 14]:

- microcracks nucleation at some brittle particles under the action of plastic flow;

- unstable propagation of one of the microcracks if the local stress exceeds the threshold stress.

However, all the created microcracks are not susceptible to provoke unstable propagation as part of them blunt with further straining [13]. According to Chen statistical model for cleavage fracture [12], specimens need to reach a minimum triaxiality rate to prevent the nucleated microcracks from blunting and allow brittle fracture. In the present case, although the stress in the volume susceptible to cleavage was higher than the threshold stress, the triaxiality factor near the crack front was too low to prevent the microcracks from blunting and induce brittle fracture. The only sharp microcracks susceptible to provoke brittle fracture were the one created around 0.2mm from the crack front, where the triaxiality factor was high enough. Therefore, ductile tearing could develop up to 0.2mm, and brittle fracture occurred once the propagated crack encountered the sharp microcracks. In comparison, for the CT specimens, the triaxiality factor was high enough all along the ligament to prevent all the microcracks created from blunting. Therefore, the microcracks created near the crack front could induce brittle fracture before any ductile tearing.

This analysis seems to indicate the important role of the triaxiality factor in the microcracks propagation mechanism. However, this important parameter does not appear in the present formulation of the threshold stress criterion. This might explain why it only predicts a lower bound of the real non-fracture threshold, in particular for the pipe structures. The introduction of the triaxiality factor within the sensitivity function to cleavage $F(\sigma_y)$ might improve the non-fracture threshold predictions.
# 6. Conclusion and perspectives

This study presents the use of a simple threshold stress criterion to predict non-fracture thresholds in the brittle-to-ductile transition temperature range for a C-Mn steel. In order to assess the precision of its predictions for structures compared to laboratory specimens, an extensive experimental work was done to develop and perform fracture tests on pipe structures representative of the in-service auxiliary piping system of French PWR. This work presents a synthesis of the previous fracture tests on pipe structures performed in [6] and a new test performed by the present authors.

The deviation of the predicted non-fracture threshold from the loading at experimental fracture of the CT specimens and the pipes show that the present formulation of the threshold stress criterion only allows to predict a lower bound of this non-fracture threshold. This deviation is particularly accentuated for the pipes which exhibit a low triaxiality factor near the crack front, compared with CT specimens.

As initiated in [5], further tests on laboratory specimens with various geometries will be performed in order to gain more insight into the triaxiality factor influence. The final goal would be to integrate this parameter within the formulation of the sensitivity function to cleavage $F(\sigma_y)$, in order to improve the threshold stress criterion predictions.

# Acknowledgments

The authors gratefully acknowledge T. Le Grasse and J. Lecour for performing the pipe structure test at CEA (Saclay, France). The authors also acknowledge EDF for financial support.

# References

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