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# Development of a miniaturized ring notched bend specimen for the assessment of the fracture toughness of ODS steel fuel claddings

Boualem Rais<sup>a,b, ID</sup>, Jérôme Garnier<sup>a</sup>, Elodie Pons<sup>a</sup>, Bernard Marini<sup>a</sup>, Jacques Besson<sup>b, ID, *</sup>

## ARTICLE INFO

**Keywords:**

Fracture toughness

Miniaturized ring bend specimen

Fracture resistance curve

ODS steel fuel cladding

## ABSTRACT

A miniaturized Ring Notched Bend test (mRNB) was developed as a means to assess the fracture toughness of thin-walled tubes. The fracture resistance curve (J-R curve) was derived from both the load-CMOD curve and the load-LLD curve, utilizing a single specimen technique. Following the procedures proposed in the ASTM E1820 standard [1], the elastic unloading methods were employed for the load-CMOD curve. Alternatively, the normalization method was utilized to analyze the load-LLD curve when CMOD measurements was not possible. However, for the application of this methodology to the mRNB test, the standard lacks the necessary functions, including the compliance function, the geometric stress intensity function, and the plastic factor, to process the test results. In this study, the finite element analysis was used to determine the geometric functions required for crack length evaluation during the test, as well as the elastic and plastic components of the J-integral, whether derived from the load-CMOD or load-LLD curve. This methodology was applied to experimentally investigate the crack growth resistance and damage of two types of ODS (Oxide Dispersion Strengthened) steel tubes, (containing specifically 9%Cr and 14%Cr), which are considered candidate materials for fuel claddings of future fast-neutron reactors.

## 1. Introduction

Future fast neutron reactors, often referred to as 4th generation reactors, offer several key improvements over current Pressurized Water Reactors (PWRs). These improvements primarily aim to enhance efficiency and reduce radioactive waste production. However, these advancements also introduce more severe thermo-mechanical loading conditions on various components of the 4th generation reactors, particularly fuel cladding. The fuel cladding serves as the initial containment barrier for radioactive fuel and is expected to operate at elevated service temperatures of approximately 650 °C.

Oxide Dispersion Strengthened (ODS) steels have emerged as promising candidate materials for the fuel cladding in these reactors due to their remarkable resistance to swelling under irradiation [3], and their high mechanical and creep resistance at high temperatures [4–6] It is worth noting that high-strength alloys such as ODS steels typically exhibit limited ductility. Consequently, ensuring the integrity of the fuel cladding and preventing contamination of the reactor’s primary circuit are paramount. It is essential to conduct comprehensive assessments

of the crack propagation resistance of these materials to achieve these goals.

These fuel claddings are thin-walled tubes with a wall thickness of 0.5 mm and an external diameter of 10.73 mm. They are manufactured through a cold rolling process, which modifies the material’s microstructure. Because of the thinness of the fuel cladding tubes and the change in microstructure during manufacture, it becomes imperative to evaluate the material’s fracture toughness in the final cladding geometry.

The ASTM E1820-22e1 standard [1] offers guidelines for conducting tests on specimens with cracks with the ultimate goal of obtaining a crack propagation resistance curve (J-R curve) from the force-displacement curve. When employing the unloading compliance technique or the normalization method, these procedures enable the determination of the complete curve using a single specimen.

However, it is essential to note that the ASTM E1820-22e1 standard describes the testing procedure and the data processing technique for a limited number of specimen types, such as C(T), DC(T), and SE(B). Therefore, applying this standard to thin-walled components like fuel claddings is impossible. Due to their geometry, the typical fracture spec-

<footnote><sup>a</sup> Université Paris-Saclay, CEA, Service de Recherches Matériaux et Procédés Avancés, 91191, Gif-sur-Yvette, France</footnote>

<footnote><sup>b</sup> MINES Paris, PSL Research University, Centre des Matériaux, CNRS UMR 7633, 91003 Evry, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: jacques.besson@minesparis.psl.eu (J. Besson).</footnote>
## Nomenclature

<table><tr><td>A<sub>pl</sub></td><td>Plastic area under load—displacement curve</td><td>J</td></tr><tr><td>a, a<sub>0</sub>, a<sub>i</sub>, a<sub>ic</sub>, a<sub>f</sub></td><td>crack length, initial crack length, crack length at unload point i, corrected crack length at unload point i, final crack length</td><td>m</td></tr><tr><td>B, b, b<sub>0</sub></td><td>Specimen thickness (two times the wall thickness for the mRNB specimen), ligament length, initial ligament length</td><td>m</td></tr><tr><td>CMOD</td><td>Crack Mouth Opening Displacement</td><td>m</td></tr><tr><td>C, C<sub>i</sub></td><td>Compliance, Compliance at unload point i</td><td>m N<sup>-1</sup></td></tr><tr><td>C(a/W)<sub>CMOD</sub>, C(a/W)<sub>LLD</sub></td><td>Compliance function for Load—CMOD curves, Compliance function for Load—LLD curves</td><td>m N<sup>-1</sup></td></tr><tr><td>E, E'</td><td>Young modulus, reduced Young modulus, E' = E/(1 - v<sup>2</sup>) (plane strain) or E' = E (plane stress)</td><td>Pa</td></tr><tr><td>e<sub>U</sub>, e<sub>R</sub></td><td>Uniform elongation, elongation at fracture</td><td>—</td></tr><tr><td>f(a/W)</td><td>Geometric function to determine the stress intensity factor</td><td>—</td></tr><tr><td>G(b/W), H(V<sub>p</sub>/W)</td><td>Functions used in the load separation theory [2]</td><td>—, N</td></tr></table>

<table><tr><td>J, J<sub>e</sub>, J<sub>p</sub>, J<sub>0.2</sub></td><td>J-integral, elastic part of the J-integral, plastic part of the J-integral, J-integral at 0.2 mm crack extension</td><td>J m<sup>-2</sup></td></tr><tr><td>K<sub>I</sub>, K<sub>J</sub></td><td>Stress intensity factor for mode I opening, Stress intensity factor at crack initiation (corresponding to J<sub>0.2</sub>)</td><td>Pa√m</td></tr><tr><td>LLD</td><td>Load Line Displacement</td><td>m</td></tr><tr><td>P, P<sub>max</sub>, P<sub>N</sub></td><td>Load, Maximum load, Normalized load</td><td>N</td></tr><tr><td>R<sub>int</sub>, R<sub>ext</sub></td><td>Internal radius, external radius of the mRNB specimen</td><td>m</td></tr><tr><td>S<sub>i</sub>j</td><td>Separation factor</td><td>—</td></tr><tr><td>t</td><td>Wall thickness of the mRNB specimen</td><td>m</td></tr><tr><td>u</td><td>Normalized compliance</td><td>—</td></tr><tr><td>V, V<sub>p</sub></td><td>Total displacement. Plastic displacement</td><td>m</td></tr><tr><td>W</td><td>mRNB specimen width</td><td>m</td></tr><tr><td>η<sub>p</sub>, γ<sub>p</sub></td><td>Factors used to compute J<sub>p</sub></td><td>—</td></tr><tr><td>Δa</td><td>Crack extension</td><td>m</td></tr><tr><td>ε<sub>p</sub></td><td>Accumulated plastic strain</td><td>—</td></tr><tr><td>ν</td><td>Poisson's ratio</td><td>—</td></tr><tr><td>σ<sub>u</sub>, σ<sub>YS</sub>, σ<sub>Y</sub></td><td>Ultimate tensile strength. Yield Stress. Average of σ<sub>u</sub> and σ<sub>YS</sub></td><td>Pa</td></tr><tr><td>σ<sub>F</sub></td><td>Flow stress</td><td>Pa</td></tr></table>

imens used in the standard testing procedures cannot be extracted from fuel claddings. Evaluating the toughness of fuel claddings in terms of Stress Intensity Factor (SIF) or J-integral requires the development and construction of specialized experimental setups. These setups are designed to be compatible with the unique characteristics of fuel claddings and must be accompanied by detailed procedures for processing the experimental load-displacement curves.

In the existing literature, various testing methods have been devised to assess the toughness of thin-walled tubes [7–10]. Sanyal et al. [11] introduced both multiple-specimen and single-specimen methodologies for assessing the fracture toughness of Zircaloy fuel cladding tubes, quantified in terms of J-integral, using the Pin Loading Tension test (PLT), a testing method initially proposed by Grigoriev et al. [7]. In their research, they applied the single-specimen approach based on the normalization method, as proposed in the ASTM E1820-22e1 standard, to construct the J-R curve. Similarly, Rais et al. [12] presented a single-specimen technique for determining the resistance curve of an ODS steel fuel cladding using the PLT test. Their method relied on the unloading compliance technique. The authors [12] showed that the increase in size of the contact zone between the pin and the specimen together with the bending of the specimen led to a decrease in the elastic unloading compliance during the test before eventually reaching a constant value. This effect complicates the estimation of crack extension, particularly in the initial loading stages or for low-toughness materials, as ODS steels. To address this issue, they incorporated optical measurement alongside the unloading compliance technique to accurately determine crack extension.

In this study, a new experimental procedure (specimen, setup, testing method, and data analysis) is proposed to assess the fracture toughness of fuel cladding thin-walled tubes. This new procedure uses a miniaturized version of the Pipe Ring Notched Bend (PRNB) specimen, which was initially proposed by Gubeljak et al. for the fracture toughness evaluation of pipelines [13,14]. This test was adapted to test fuel claddings used in nuclear reactors. This new design is referred to as a miniaturized Ring Notched Bend test (mRNB) in the following. Section 2 describes data processing methods used to construct the J-R curve from the Load-Displacement curve from a single mRNB specimen. These construction procedures are based on the ASTM E1820-22e1 standard for SE(B) specimens. Section 3 offers a detailed description of the materials and testing apparatus employed in this study. In section 4, the numerical validation procedure of the mRNB is described, and the geometric parameters

needed to analyze the experimental data for obtaining the J-R curve are determined using the Finite Element Method (FEM). Section 5 details the experimental results before providing, in the following two sections, a general discussion and conclusions about the results obtained in this study.

## 2. Procedures for the determination of the J-R curve

The ASTM E1820 standard [1] provides analytical procedures for the evaluation of the J-Δa curves from a single specimen. The use of the multiple technique was not investigated in this work, but it can be straightforwardly applied once the necessary functions for the single specimen technique are determined. It is required to both evaluate the crack advance and the J-integral. Two methods are proposed in the standard to determine the Δa: (i) the “Unloading Compliance Method”, (ii) the “Normalization Method”. Once the crack advance is determined, the J-integral can be evaluated using the same procedure. In the following, both methods are used.

**Unloading Compliance Method:** This method is the main method. It involves conducting elastic unloading cycles during the test. The instantaneous compliance at unloading i, denoted as C<sub>i</sub>, is determined from the Load-Displacement curve as shown in Fig. 1. It is then possible to estimate the instantaneous crack length a<sub>i</sub> using the measured value of C<sub>i</sub>. By analyzing the data obtained using this method, it becomes possible to construct the J-R curve, which describes the variation of the J-integral with crack extension.

**Normalization Method:** The normalization method offers an alternative approach to determining intermediate crack lengths. It does not require elastic unloading or any measuring of intermediate crack lengths. It relies on the normalized load, the initial crack length, and the final crack length. This method is useful when it is challenging or impossible to use the unloading compliance method or directly measure the intermediate crack lengths.

### 2.1. Evaluation of J for a non-propagating crack

For Mode I loading, the J-integral is the sum of an elastic part, J<sub>e</sub>, and a plastic part, J<sub>p</sub>, as follows:

$$J = J_e + J_p \quad (1)$$

where J<sub>e</sub> is expressed as:
<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic of a Load-Displacement curve with partial unloading and determination of elastic unloading compliance.</figcaption></figure>

$$J_e = K_I^2 / E'. \qquad (2)$$

In this work, $E' = E$ was used due to the thickness of the tube, which favors plane stress conditions. $E$ and $\nu$ are the Young's modulus and the Poisson's ratio, respectively. The Mode I stress intensity factor ($K_I$), defined for three-point bend specimen (SE(B)) as:

$$K_I = \frac{PS}{(B_N B)^{1/2} W^{3/2}} f\left(\frac{a}{W}\right) \qquad (3)$$

Where $P$ is the applied load, $S$ the support span, $B$ the specimen thickness, $B_N = B$ in the absence of Side Grooves (which is the case of the mRNB specimen), $W$ the specimen width, $a$ the crack length, and $f(a/W)$ a geometric function. $f(a/W)$ is given by the ASTM E1820 standard for standard SE(B) specimens and has to be determined for the mRNB specimen, which is a nonstandard specimen. According to Rice et al. [15], $J_p$ can be expressed as a function of the macroscopic plastic work as:

$$J_p = \eta_p \frac{A_p}{B b_0} \qquad (4)$$

Where $A_p$ is the plastic area under the load-displacement curve, $b_0$ the uncracked ligament length ($b_0 = A - a_0$). $\eta_p$ is a factor, which is geometry dependent and expressed as a function of $a/W$. $\eta_p$ needs to be determined for the proposed mRNB specimen.

## 2.2. Unloading compliance method

In the case of a growing crack, the area under the load-displacement curve differs from the corresponding one for a non-propagating crack, and eq. (4) cannot be directly applied [16]. The largely used procedure given in standards to evaluate the $J$-integral is an incremental procedure that evaluates $J_e$ and $J_p$ at every elastic unloading point (i) (see Fig. 1). To compute $J_{e(i)}$, eq. (2) is applied together with eq. (3) using the current crack length ($a_i$) and the corresponding load $P_i$. To compute $J_p$, the measured load-displacement records must be corrected for crack extension to obtain an accurate estimation. Therefore, for growing crack, at every elastic unloading point, $J_{p(i)}$ is then determined from load-displacement curve measurement as:

$$J_{p(i)} = \left[ J_{p(i-1)} + \frac{\eta_{p(i-1)}}{b_{i-1}} \frac{A_{p(i)} - A_{p(i-1)}}{B} \right] \left[ 1 - \frac{\gamma_{p(i-1)}}{b_{i-1}} (a_i - a_{i-1}) \right] \qquad (5)$$

where $\gamma_p$ is a factor given by:

$$\gamma_p = \eta_p - 1 + \frac{b}{W} \frac{\eta'_p}{\eta_p} \qquad (6)$$

where $\eta'_p$ is the derivative of $\eta_p$ with respect to $a/W$.

When the unloading compliance method is used, the crack length can be evaluated from the measured compliance using the $a/W(u)$ function, which is also provided by the ASTM E1820 standard for standard

specimens. It needs to be determined for mRNB specimen. The parameter $u$ used in the $a/W(u)$ function is the normalized compliance. For example, $u$ is given for the SE(B) specimen by:

$$u = \frac{1}{\sqrt{\frac{B_e W EC}{S/4} + 1}} \qquad (7)$$

where $B_e = B$ in the absence of side grooves.

## 2.3. Normalization method

The normalization method makes full use of the load separation principle [2,17], which states that the load can be expressed as the product of two functions:

$$P\left(\frac{a}{W}, \frac{V_p}{W}\right) = G\left(\frac{a}{W}\right) \times H\left(\frac{V_p}{W}\right) \qquad (8)$$

where $P$ is the applied load. It is worth remembering that load separation is required to be able to evaluate $J$ as the sum of $J_e$ and $J_p$ [2]. The plastic displacement is evaluated as:

$$V_p = V - CP \qquad (9)$$

where $V$ is the total displacement. Using eq. (8) a normalized load can be defined as:

$$P_N = \frac{P}{WB} \frac{1}{G(a/W)} = \frac{1}{WB} H(V_p/W) \qquad (10)$$

Sharobeam and Landes [18,19] have shown that, for all the fracture mechanics specimens usually used, including the SE(B) specimen, $G(a/W)$ can be written as follows:

$$G(a/W) = \left(\frac{W - a}{W}\right)^{\eta_p} \qquad (11)$$

This form is also valid for the PRNB specimen as shown in [14]. Therefore, $P_N$ can be expressed as:

$$P_N = \frac{P}{WB} \left( \frac{W - a_b}{W} \right)^{\eta_p} \qquad (12)$$

This equation accounts for crack blunting before crack initiation using the following estimate of the crack length:

$$a_b = a_0 + \frac{J}{2\sigma_Y} \qquad (13)$$

where $\sigma_Y$ corresponds to the average value of tensile yield strength $\sigma_{Y,S}$ and ultimate strength $\sigma_u$ of the material. It then becomes possible to compute the normalized load. Note that $\eta_p$ is computed using $a_b$ at this stage.

The function $H(V_p/W)$ depends on the material and the specimen geometry. The normalization method assumes a form of function for $H$ i.e. $P_N$ according to the following equation, which is valid for many materials, specifically those with low strain-hardening properties such as ODS steels [19,20]:

$$\frac{1}{WB} H(v_p) = P_N = \frac{(l + m v_p) v_p}{n + v_p} \qquad (14)$$

with $v_p = V_p/W$. The $H$ function is characterized by two linear asymptotes, one for $v_p \ll 1$ ($H \approx l/nv_p$) and another for $v_p \gg 1$ ($H \approx mv_p + l - mn$). The three parameters $l$, $m$, and $n$ are then fitted using data points for which the crack extension is known. This corresponds to data points with blunting only and the final point for which crack extension is measured using the 9-point method. The actual crack advance is assumed to start when $P_N$ reaches its maximum. For each data point not used to fit $P_N$, crack advance is searched so as to match eq. (14). Once all the values for $a$ are adjusted, the $J$-integral is determined according to the
<table><caption>**Table 1**
Mechanical properties of the studied ODS steels fuel claddings at room temperature and 650 °C.</caption><thead><tr><th>Material</th><th>T (°C)</th><th>E (GPa) [22]</th><th>σ<sub>YS</sub> (MPa)</th><th>σ<sub>UTS</sub> (MPa)</th><th>σ<sub>Y</sub> (MPa)</th><th>r (%)</th><th>e<sub>U</sub> (%)</th><th>e<sub>R</sub> (%)</th></tr></thead><tbody><tr><td>ODS9Cr</td><td>25</td><td>225</td><td>910</td><td>1026</td><td>968</td><td>0.89</td><td>3.5</td><td>11.2</td></tr><tr><td>ODS14Cr</td><td>25</td><td>229</td><td>1002</td><td>1082</td><td>1042</td><td>0.92</td><td>1.5</td><td>1.8</td></tr><tr><td>ODS9Cr</td><td>650</td><td>170</td><td>298</td><td>359</td><td>329</td><td>0.83</td><td>3.2</td><td>9.7</td></tr><tr><td>ODS14Cr</td><td>650</td><td>165</td><td>170</td><td>256</td><td>213</td><td>0.64</td><td>15.2</td><td>24.5</td></tr></tbody><tfoot><tr><td colspan="9">σ<sub>Y</sub> = 1/2(σ<sub>YS</sub> + σ<sub>UTS</sub>)</td></tr><tr><td colspan="9">Poisson's ratio v = 0.3 [22]</td></tr><tr><td colspan="9">r = σ<sub>YS</sub>/σ<sub>UTS</sub></td></tr></tfoot></table>

procedure detailed in section 2.2. More details about the procedure can be found in the ASTM E1820 standard.

### 3. Materials and experimental procedures

In this section, the materials studied in this paper are firstly presented in terms of engineering tensile and microstructural properties. The specimen and setup configuration are then described before presenting the used pre-cracking, testing, and crack lengths measurement procedures.

#### 3.1. Materials

Fuel claddings studied in this work are 1000 mm long and 9.73 mm inner diameter tubes with a wall thickness of 0.5 mm. They were manufactured by the French Atomic Energy Agency (CEA) at the Materials and Physico-Chemistry Research Department. The studied materials are a 9Cr-1W-0.3Ti-0.25Y₂O₃ (wt.%) martensitic ODS steel tube referenced as ODS9Cr and a 14Cr-1W-0.3Ti-0.25Y₂O₃ (wt.%) ferritic ODS steel tube referenced as ODS14Cr. When optimized, both ferritic and martensitic ODS steels exhibit good resistance to creep and swelling under irradiation. Martensitic ODS are easier to manufacture. During the tube manufacturing process, the high-temperature transition from the α (BCC) to the γ (FCC) phase leads to an isotropic microstructure with a low dislocation density. Ferritic ODS, on the other hand, offer better corrosion resistance but are more challenging to produce, requiring precise control over the material's recrystallisation. Their mechanical properties, particularly toughness, must be well understood due to the formation of the α' phase, which can weaken the material, especially under irradiation. A strongly textured ferritic material, like the one presented in this study, is not ideal. However, it allows for an analytical comparison between two very different microstructures (see below): one isotropic (Fe-9Cr) and the other highly textured (Fe-14Cr). The latter is theoretically less favorable, enabling us to define a lower bound for toughness. The engineering tensile properties of these materials at room temperature and 650 °C are given in Table 1. These properties were obtained from ring specimens tested in the hoop direction of the cladding [21]. The gauge length is then 4 mm.

The tubes were manufactured using powder metallurgy. Mechanical alloying was used to produce the ODS powder using a mixture of steel alloying powders and yttria powder. This powder was then consolidated by hot extrusion. Cold pilger rolling allowed the final shape to be obtained. Inter-pass heat treatments were used to reduce material hardness and relieve internal stresses. For the martensitic ODS steel tube (ODS9Cr), this involves total austenitization of the tube, followed by slow cooling to eliminate all work-hardening defects and obtain a ferritic state with optimum hardness and ductility for cold working. A final heat treatment at 1050 °C for 30 minutes, followed by rapid cooling and tempering at 750 °C for 1 h, provided a good compromise between mechanical strength and ductility. In the case of the ferritic ODS steel (ODS14Cr), an interpass stress-relieving annealing at 1200 °C during 1 h is used. The final heat treatment of the tube consists in a slow cooling followed by tempering (750 °C for 1 h), as austenitization is impossible for this material due to its higher chromium content.

In the following ND represents the direction of the tube thickness and RD to the tube axis or rolling direction. Back-scattered electrons SEM

micrographs observed in the RD-ND plane reveal that the microstructure of both ODS steel tubes is homogeneous throughout the thickness of the tube (Fig. 2). The ODS9Cr steel tube has a martensitic morphology containing sub-micron grains. Some columnar arrangements along RD can be observed (see Fig. 2-a). This microstructure is characteristic of tubular ODS steels at the end of the post-quenching forming stage, which inherits a martensitic grain morphology from the final quenching heat treatment followed by tempering. Quenching was performed under flowing Argon. Recrystallization during final austenitization before quenching can be selective, not necessarily leading to an isotropic microstructure and texture. The ODS14Cr has a microstructure consisting mainly of columnar grains oriented in the rolling direction (see Fig. 2-b). This microstructure is characteristic of ferritic steels produced by hot-extrusion followed by cold-rolling with inter-pass heat treatments, which recrystallize partially or not at all.

In both materials, submicron-sized precipitates (500 nm in diameter at most) are present (see black particles in Fig. 2). These precipitates are homogeneously distributed throughout the thickness. Chemical composition analyses using Energy Dispersive X-ray spectroscopy (EDX) revealed that these precipitates are mainly oxides with a high concentration of titanium (Ti). These oxides are considered undesirable because they are responsible for the embrittlement of the material.

According to the EBSD observations in the RD-ND plane, the microstructure of the ODS9Cr steel tube showed equiaxed grains with an average diameter of approximately 1.2 µm (see Fig. 3-a). The presence of some columnar grains can also be noticed, a feature inherited from the cold rolling process employed to achieve the final geometry. When examining the pole figures and orientation maps in both sections (refer to Fig. 3), it becomes evident that the microstructure is slightly textured, with some preferred crystallographic orientations. Specifically, the ⟨110⟩ axes of the grains align parallel to the axial direction (RD), while the ⟨111⟩ axes align parallel to the normal direction (ND). Inter-pass heat treatments and the final tempering diminish this texture, resulting from the cold-rolling process. Indeed, unlike ferritic ODS steels, recrystallization is possible in martensitic ODS steels thanks to the presence of an austenitic domain at temperatures above 850 °C. However, EBSD observations and pole figures of the ODS14Cr steel tube show a highly textured microstructure, with grain axes ⟨110⟩ predominantly oriented in the rolling direction (RD) (see Fig. 3-b). This is mainly due to the fact that the material does not undergo phase transformation after initial solidification during the entire manufacturing process, and remains ferritic due to its high chromium content. Similar microstructures have previously been observed in martensitic and ferritic ODS steels produced at CEA [4–6,22,23].

#### 3.2. Specimen and setup configuration

A specific test specimen and setup have been designed to adapt the PRNB test to fuel cladding geometries. The mRNB specimen is a ring with a total height of 4 mm taken from the fuel cladding. Two parallel, diametrically opposed and symmetrical notches were machined using electro-discharge machining (EDM) as in the case of the PLT specimen [12]. In addition to the fact that its diameter is much smaller than that of the PRNB specimen [14], two modifications to the original design of the specimen have been proposed (see Fig. 4-a). These modifications were designed to facilitate specimen positioning and test
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Scanning Electron Micrographs (Backscattered Electrons) of microstructures in the longitudinal RD—ND plane of the tubes: (a) ODS 9%Cr steel (ODS9Cr) and (b) ODS 14%Cr steel (ODS14Cr).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** EBSD mapping and pole figures in the longitudinal RD-ND plane of the tubes: (a) ODS 9%Cr steel (ODS9Cr) and (b) ODS 14%Cr steel (ODS14Cr). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)</figcaption></figure>

instrumentation. The first modification involves machining two diametrically opposed grooves with a radius of 2 mm and a depth of 0.5 mm on the upper part of the specimen (see Fig. 4-a). These grooves make it possible to guarantee and repeat the centering of the specimen with respect to the upper support and the center-to-center distance (S) of the two rollers. The effective height W of the specimen is then the distance between the specimen's top and bottom contact points with the roller supports and the loading upper support (7 and 6 in Fig. 4-b) (3 mm in the present case). The crack length a is taken to be the distance between the contact line between lower support and the specimen and the crack tip (see Fig. 5-a). The second modification entails machining extensions specifically designed for an easy attachment of the clip gauge. They are also diametrically opposed, machined on the lower part with an extension of 0.5 mm from the base of the specimen in contact with the lateral supports (see Fig. 4-a). The CMOD was measured using a single gauge wide enough to be clipped onto the machined extensions (see Fig. 4-a) on both sides of the specimens. Due to the small size of the specimens, the procedure was first tested using a moc=kup three times larger than the final one (copper tubes were used).

An inverted tensile setup was also designed and machined to allow the loading of the mRNB specimen (see the 3D schematic in Fig. 4-b). The assembly consists of an upper and a lower plate connected by columns to the lower and upper loading lines plates, respectively, to enable inverted traction. The two plates serve distinct roles: the “upper” plate functions as the loading support (or bending hammer denoted as

06 in Fig. 4-b), while the “lower” plate serves as the base for the roller supports (refer to 7 in Fig. 4-b), the separation of which is adjustable. The assembly is designed so that an extensometer can be inserted by machining long columns in the part connected to the bottom loading line plate and attached to the upper plate (bottom columns referenced 4 in Fig. 5-b) and machining a hole on the lower plate (see 3 in Fig. 5-b).

### 3.3. Pre-cracking, testing, and crack extension measurement procedures

Before testing, some mRNB specimens were fatigue pre-cracked using a RUMUL resonant fatigue testing machine. A 3-point bending setup was used for that purpose. Fatigue cycles were applied to the mRNB specimen at a maximum force $P_m$ corresponding to the one specified in ASTM E1820 for the SE(B) specimen. The corresponding maximum stress intensity factor ($K_{max}$) was calculated using eq. (3) using the expression for the $f(a/W)$ function proposed below (eq. (23)). It was verified that the radius of the plastic zone developed at the crack tip $r_p$ (with $r_p = 1/2\pi(K_{max}/\sigma_{YS})^2$) was significantly smaller than the tube wall thickness (i.e. $r_p < t/10$). Additionally, $K_{max}$ was confirmed to be significantly lower than the material toughness $K_{Ic}$. A load ratio equal to 0.25 was used. The resonant machine automatically adjusts the frequency decreases when the specimen stiffness decreases due to crack extension. The specimens were first fatigued using a stress intensity amplitude equal to $10 \text{ MPa}\sqrt{\text{m}}$. Fatigue loading was stopped after a drop of 0.4 Hz
<figure><img src="image_6.png" /><figcaption>Fig. 4. (a) Design of mPRNB specimen (b) 3D design of the mPRNB loading fixture (c) Loading fixture articles description table.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. (a) mRNB test specimen and loading setup (b) View of an mRNB test at 20 °C.</figcaption></figure>

<table><caption>**Table 2**

List of tests with details regarding the specimen dimensions and precracking. C: cracked specimen. N: notched specimen.</caption><thead><tr><th rowspan="2">Specimen</th><th rowspan="2">W (mm)</th><th rowspan="2">t (mm)</th><th rowspan="2">Temp. (°C)</th><th rowspan="2">C/N</th><th colspan="3">a<sub>0</sub> after pre-cracking (mm)</th><th rowspan="2">a<sub>0</sub>/W</th><th colspan="3">Final crack length a<sub>f</sub> (mm)</th></tr><tr><th>Front side</th><th>Back side</th><th>Average</th><th>Front side</th><th>Back side</th><th>Average</th></tr></thead><tbody><tr><td>ODS9Cr-1</td><td>3.00</td><td>0.515</td><td>20</td><td>C</td><td>1.91</td><td>1.88</td><td>1.89</td><td>0.63</td><td>2.12</td><td>2.18</td><td>2.15</td></tr><tr><td>ODS9Cr-2</td><td>3.03</td><td>0.515</td><td>20</td><td>C</td><td>1.97</td><td>1.97</td><td>1.97</td><td>0.65</td><td>2.15</td><td>2.20</td><td>2.17</td></tr><tr><td>ODS9Cr-3</td><td>3.04</td><td>0.51</td><td>20</td><td>C</td><td>1.82</td><td>2.00</td><td>1.91</td><td>0.63</td><td>2.14</td><td>2.33</td><td>2.23</td></tr><tr><td>ODS9Cr-6</td><td>3.04</td><td>0.53</td><td>20</td><td>N</td><td>1.50</td><td>1.52</td><td>1.51</td><td>0.50</td><td>2.14</td><td>1.92</td><td>2.03</td></tr><tr><td>ODS9Cr-7</td><td>3.00</td><td>0.52</td><td>20</td><td>N</td><td>1.48</td><td>1.49</td><td>1.49</td><td>0.49</td><td>1.98</td><td>1.91</td><td>1.94</td></tr><tr><td>ODS9Cr-4</td><td>3.01</td><td>0.52</td><td>650</td><td>N</td><td>1.49</td><td>1.49</td><td>1.49</td><td>0.50</td><td>2.21</td><td>2.16</td><td>2.18</td></tr><tr><td>ODS9Cr-5</td><td>3.01</td><td>0.54</td><td>650</td><td>N</td><td>1.49</td><td>1.50</td><td>1.50</td><td>0.50</td><td>1.99</td><td>1.94</td><td>1.96</td></tr><tr><td>ODS14Cr-1</td><td>3.02</td><td>0.535</td><td>20</td><td>C</td><td>1.93</td><td>1.60</td><td>1.77</td><td>0.59</td><td>2.22</td><td>2.21</td><td>2.22</td></tr><tr><td>ODS14Cr-2</td><td>3.05</td><td>0.535</td><td>20</td><td>C</td><td>2.19</td><td>1.72</td><td>1.96</td><td>0.64</td><td>2.40</td><td>2.07</td><td>2.24</td></tr><tr><td>ODS14Cr-3</td><td>3.03</td><td>0.545</td><td>20</td><td>C</td><td>1.68</td><td>1.92</td><td>1.80</td><td>0.59</td><td>2.08</td><td>2.24</td><td>2.16</td></tr><tr><td>ODS14Cr-6</td><td>3.02</td><td>0.55</td><td>20</td><td>N</td><td>1.49</td><td>1.51</td><td>1.50</td><td>0.50</td><td>1.88</td><td>1.88</td><td>1.88</td></tr><tr><td>ODS14Cr-7</td><td>3.02</td><td>0.54</td><td>20</td><td>N</td><td>1.48</td><td>1.49</td><td>1.49</td><td>0.49</td><td>2.06</td><td>2.06</td><td>2.06</td></tr><tr><td>ODS14Cr-4</td><td>3.03</td><td>0.54</td><td>650</td><td>N</td><td>1.49</td><td>1.50</td><td>1.50</td><td>0.50</td><td>2.15</td><td>1.97</td><td>2.06</td></tr><tr><td>ODS14Cr-5</td><td>3.02</td><td>0.535</td><td>650</td><td>N</td><td>1.50</td><td>1.49</td><td>1.49</td><td>0.50</td><td>1.92</td><td>1.66</td><td>1.79</td></tr></tbody></table>

in the loading frequency to check crack advance. The procedure was repeated until reaching a crack extension of approximately 300 µm on both sides of the specimen. The loading amplitude was then progressively reduced to 5 MPa√m until reaching a total crack advance equal to 500 µm. After pre-cracking, the normalized initial crack length (a₀/W) was about 0.6. The dimensions of all tested specimens are gathered in Table 2. This includes width and thickness as well as the initial and final crack lengths on both sides of the specimens. Precracking was difficult in the case of the ODS14Cr material so that crack advances on both sides significantly differed (see Table 2 for tests ODS14Cr-1, 2, 3). This diffi-

culty was attributed to the material and not to the setup as symmetric precracking could be obtained for the ODS9Cr material.

Pre-cracking, in the case of the ODS14Cr steel, was more challenging than it was for the ODS9Cr steel. In some cases, the crack extension on one side of the specimen was significantly larger than on the other side despite rotating the specimen by 180° to try to correct this effect. Fatigue crack propagation was also less stable in ODS14Cr mRNB specimens. This may be attributed to the fibrous and textured microstructure of the ODS14Cr steel tube, which also has a lower tearing modulus compared to the ODS9Cr steel. For that reason, uncracked specimens (referred to as “notched”) were also tested.
<figure><img src="image_8.png" /><figcaption>Fig. 6. 3D view of the crack surface and measurement of fatigue pre-crack and final crack lengths.</figcaption></figure>

Tests performed at room temperature were conducted on a 10 kN electro-mechanical testing machine equipped with a 1 kN load cell. The CMOD was measured using a clip gauge. The loading program consisted of the following cycles: (i) loading at 0.1 mm/min until the CMOD increases by 0.01 mm, (ii) holding the CMOD for 30 s to allow for relaxation, (iii) unloading at 0.1 mm/min until 60% of load reached at the end of the holding period, and (iv) reloading to this load at the same speed. The use of a relaxation period of 30 s, an important unloading amplitude, and a low unloading speed allow for a better estimation of the unloading/loading compliance.

High temperature (650 °C) tests were performed on the same testing machine with the same load cell and using a lamp oven. A K-type thermocouple in contact with the specimen was used to measure its temperature. Loading was initiated when the specimen temperature stabilized at 650±1 °C for 15 min. Performing elastic unloading during high-temperature tests was impossible, as the material undergoes considerable stress relaxation at this temperature at constant LLD. A monotonic loading at 0.1 mm/min was then used for mRNB tests performed at 650 °C. It must be noted that the CMOD measurement was also not possible during these tests. Therefore, the normalization method has to be used to analyze these tests. Note that less accuracy can be expected in this case because the load line displacement measurement is generally less sensitive and reliable than the crack mouth opening displacement measurement.

After testing, specimens were fully broken by propagating the crack in liquid nitrogen to view the different crack extension zones. Optical microscopy was used to observe the failed specimens and analyze fracture surfaces. A KEYENCE VHX-700F optical microscope was employed for this purpose. This microscope enables a 3D reconstruction from a series of scans. It was utilized for measuring crack advancements and analyzing the topography of fracture surfaces, as illustrated in Fig. 6. The fatigue pre-crack length and the final crack extension were then measured using the 9-point method proposed by the ASTM E1820 standard. The measurements were done on both sides of the specimens, and the average values were used to analyze the tests. After testing, the crack front of all specimens had a V-shape with a higher crack advance at the center of the tube wall (see Fig. 6). As seen in Fig. 6, it was difficult to obtain a straight crack front. The ASTM standard requires that the difference between the longest and shortest crack length should be less than $0.1\sqrt{Bb_0}$ (section 8.5.4 in the standard) which is about 90 µm in the present case. Measured differences after precracking were between 120 and 200 µm in this study. For this reason, tests were also performed keeping the EDM notches. The tests are nevertheless believed to be representative as cracks are not expected to be straight in actual loading conditions. The V-shape crack front is an indication that the local state is neither fully plane stress nor fully plane strain.

## 4. Determination of the geometric functions $C(a/W)$, $a/W(u)$, $f(a/W)$ and $\eta_p(a/W)$

### 4.1. Numerical procedures

All simulations were performed using Cast3M-21,<fnref n="1" /> a multi-physics finite element software developed by CEA. Four functions had to be determined to determine the $J$-R curve using the procedures presented in section 2.2: (i) the evolution of compliance as a function of normalized crack length $a/W$, (ii) the function providing the crack length from the normalized compliance $a/W(u)$ (iii) the function $f(a/W)$ used to evaluate the stress intensity factor, and (iv) the $\eta(a/W)$ function used to compute $J_p$ from $A_p$.

Following ASTM E1820, a polynomial regression was used to fit all functions in this study. The order of each function was set to obtain a coefficient of determination $R^2 \ge 0.99$ ensuring a high level of accuracy. The procedure followed in this section was also checked against results given for SE(B) and C(T) specimens in the ASTM E1820 standard.

The finite element mesh (FE) with boundary conditions and loading is shown in Fig. 7. Due to symmetries, a quarter of the assembly was meshed. Quasi-incompressible linear hexahedral elements were used [24]. Otherwise stated, the simulations were performed using the small strain assumption. Mesh convergence was checked with respect to the quantities of interest: i.e. the geometric functions to be determined. A frictionless contact condition was used to represent the free rotation of the rollers.

Simulations were carried out using a Young's modulus equal to $E = 225$ GPa and a Poisson's ratio equal to $\nu = 0.3$. In cases where plasticity was considered, the following hardening law was used:

$$ \sigma_F(p) = 910 + 89.7(1 - \exp(-492.9p)) + 180.2(1 - \exp(-28.3p)) \quad (\text{MPa}) \qquad (15) $$

where $\sigma_F$ is the flow stress and $p$ the accumulated plastic strain. This relationship corresponds to the hardening behavior of the ODS9Cr material at room temperature. Von Mises plasticity was used. The hardening law was obtained using tile and ring tensile tests as described in [6,25,26]. Experimental findings indicate that the ODS14Cr material exhibits a similar behavior along both the hoop and longitudinal tube directions [27]. This indicates that, despite its anisotropic microstructure, the material can be described as isotropic. In addition, the geometric functions to be determined are assumed to be largely material-independent.

<footnote n="1">¹ http://www-cast3m.cea.fr/.</footnote>
<figure><img src="image_9.png" /><figcaption>Fig. 7. Finite element model with boundary conditions for the mRNB test.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Simulation of the mRNB test: (a) Experimental and simulated Load—CMOD curves. (b) Crack opening stress (c) Von Mises stress. Views are in the crack plane.</figcaption></figure>

## 4.2. Analysis of stress fields in the mRNB specimen

The stress distributions (von Mises stress $\sigma_{eq}$ and opening stress $\sigma_{22}$) in the mRNB specimen were studied to verify that loading conditions allow the assessment of mode I fracture toughness. The von Mises stress and the opening stress are shown in Fig. 8-(b). The fields are displayed on the symmetry plane. The results correspond to crack initiation as experimentally determined in a test (large red circle in Fig. 8-(a)). The corresponding Load—CMOD curve is shown in Fig. 8-(a) and compared with a test. Note that the model does not account for damage and crack growth. In that case, the load decrease cannot be described. The von Mises stress is nearly symmetrically distributed along the thickness in the symmetry plane (see Fig. 8-(b)), but it tends to be higher near the inner surface. The same conclusions also hold for the opening stress, which is maximum at the center of the ligament. The corresponding maximum stress triaxiality is 1.6, which is a moderate level. The slight bending of the specimen (which can be observed in Fig. 8-(b)) causes this slight asymmetry. It is also interesting to note that the stress profile is much more symmetric compared to the stress profile obtained for the PLT test by Rais et al. [12]. The mRNB, therefore, appears to be more suitable for characterizing toughness.

Load separation must also be checked as it is a necessary condition to apply the ASTM E1820 analysis procedure. The approach followed by Sharobeam and Landes [18,19] is used here. Based on eq. (8), they introduced a separation parameter, referred to as $S_{iI}$. This parameter represents the ratio between the loads applied to specimens with different crack lengths but the same plastic displacement:

$$S_{iI} = \frac{P_i}{P_I} = \frac{G(a_i/W)H(V_p/W)}{G(a_I/W)H(V_p/W)} = \frac{G(a_i/W)}{G(a_I/W)} \quad (16)$$

In the scenario of a static crack, for two specimens with crack lengths $a_i$ and $a_I$, $S_{iI}$ remains constant, whatever the plastic displacement, if the specimen verifies the load separation theory. $S_{iI}$ were determined for the specimen for both the Load—CMOD and Load—LLD curves. Results are plotted in Fig. 9 with $a_I/W = 0.6$ and show that $S_{iI}$ can be considered as constant in both cases for all values of $a_i$. This further validates the use of mRNB specimens to measure toughness.

## 4.3. Determination of the C(a) and a/W(u) functions

As outlined in section 2, it is necessary to determine the compliance of the specimen to process the mechanical tests. Several simulations (assuming linear elasticity and small deformation) were performed for crack lengths between $a/W = 0.467$ and $a/W = 0.8$. Compliances were determined for both CMOD and LLD. The resulting compliances were fitted as:

$$C_{\text{CMOD}}(\bar{a}) = \frac{6S}{EBW} \bar{a} \left[ 238.51 - 1654.66\bar{a} + 4358.75\bar{a}^2 - 5101.52\bar{a}^3 + 2260.61\bar{a}^4 \right] \quad (17)$$

and

$$C_{\text{LLD}}(\bar{a}) = \frac{S^2}{EBW^2} \frac{1}{(1-\bar{a})^2} [5.764 - 10.325\bar{a} + 5.425\bar{a}^2] \quad (18)$$
<figure><img src="image_11.png" /><figcaption>Fig. 9. Load separation parameter $S_{iI}$ vs plastic displacement $V_p$ determined for (a) the Load—CMOD curves, (b) the Load—LLD curves. The numbers close to each curve represent $a_i/W$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. (a) Compliance as a function of CMOD/W for various values for $a/W$. (b) Compliance as a function of LLD/W for various values for $a/W$. Symbols: FE estimates. Lines Proposed relations. The numbers close to each curve represent $a/W$.</figcaption></figure>

with $\bar{a} = a/W$. These specific forms for the compliance functions are inspired by the ASTM E1820 standard for the SE(B) specimen with different expression of polynomials $P_{CMOD}$ and $P_{LLD}$.

In the case of large displacements, it was numerically observed that compliance tends to decrease as the displacement is increased. The change in the geometry of the specimen causes this. The following formula is proposed to account for this effect:

$$C_{\text{TYPE}}^{c} = C_{\text{TYPE}}(\bar{a}) - c_{\text{TYPE}} \frac{S}{EBW} \frac{\text{TYPE}}{W} \qquad (19)$$

In this equation, TYPE is either CMOD or LLD with $c_{CMOD} = 37.125$ and $c_{LLD} = 101.25$. Computed and estimated compliances are plotted as a function of the displacement in Fig. 10 for the CMOD (a) and LLD (b) cases.

It now becomes possible to determine the function $a/W(u)$. In the case where the compliance is based on the CMOD, $u$ can be defined following eq. (7). $u$ must be evaluated from the experimental compliance corrected for large displacements: i.e. $C_c = C_m + c_{CMOD}S/(EBW)CMOD/W$, where $C_m$ is the measured compliance. The normalized crack length can then be obtained using the following polynomial:

$$a/W_{\text{CMOD}}(u) = 0.975 - 3.127u - 7.15u^2 \qquad (20)$$

In the case where the compliance is determined based on the LLD, $u$ can be defined as:

$$u = \frac{1}{\sqrt{BEW^2/S^2C_c + 1}}, \qquad (21)$$

where $C_c$ is now computed as: $C_c = C_m + c_{LLD}S/(EBW)CMOD/W$. Using this definition of $u$, the normalized crack length can then be obtained using the following polynomial:

$$a/W_{\text{LLD}}(u) = 2.978 - 31.1u + 153.52u^2 - 270.27u^3. \qquad (22)$$

### 4.4. $f(a/W)$ function

The geometric function $f(a/W)$, necessary for evaluating the stress intensity factor $K_I$ and deducing $J_e$ (eq. (2)), was then determined. Here, the material was assumed to remain elastic. The method used by the authors in [12] was employed for that purpose. The method relies on the evaluation of the energy release rate based on the $G_\theta$ method implemented in Cast3M 21.<fnref n="2" /> Several contours were used for the calculation, and it was checked that the volume integral is contour-independent. The $f(a/W)$ is then directly deduced from eq. (3) using $K_I$ values deduced from the energy release rate (eq. (2)). The procedure was repeated for various values of $a/W$. A polynomial expression was then adjusted based on these simulations as:

<footnote n="2">² Cast3M21, http://www-cast3m.cea.fr/.</footnote>
<figure><img src="image_13.png" /><figcaption>Fig. 11. $f(a/W)$ for the mRNB specimen and comparison with the function for the SE(B) specimen (ASTM E1820).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. $\eta(a/W)$ for the mRNB specimen. Comparison with the function for the SE(B) specimen (ASTM E1820). Comparison with the results by Likeb et al. [14].</figcaption></figure>

$$f(a/W) = f(\bar{a}) = 89.872 - 628.307\bar{a} + 1664.927\bar{a}^2 - 1951.772\bar{a}^3 + 869.309\bar{a}^4 \quad (23)$$

This equation for $f(a/W)$ is plotted in Fig. 11 and compared to the similar function for the SE(B) specimens proposed in the ASTM E1820 standard. The difference between both functions is small. This strengthens the conclusion that the loading condition of the mRNB specimen is close to that of the standard SE(B) specimen.

### 4.5. $\eta_p(a/W)$ function

To determine the $\eta_p(a/W)$ function, the $G_\theta$ method was also used. Indeed, the elastoplastic behavior of the material must be considered. In that case, contour-independence may not be reached if the integration domain is too close to the crack tip. Different element layers were used to vary the integration domain size. The most stable values were obtained for contours intersecting the symmetry plane in the zone where the opening stresses change sign and where plastic strains are minimum at about $(W-a)/2$ from the crack tip. These converged $J$-integral value was then considered in the analysis. $J_p$ is then obtained as $J - J_e$ where $J_e$ is evaluated using the $f(a/w)$ fitted above. $A_p$ is computed from the load-displacement curve, where the compliance is used to determine the elastic part of the displacement. Plotting $J_p$ as a function of $A_p/[B(W-a)]$ results in a line whose slope corresponds to $\eta_p$. The pro-

<caption>Table 3</caption>

Value of $J_{0.2}$ (kJ/m²), $J_c/J$ (%), $K_J$ (MPa√m) and $T$ (MPa) for both materials at 25 °C and 650 °C.

<table><thead><tr><th></th><th colspan="2">9Cr<br>Cracked 25 °C</th><th colspan="2">14Cr<br>Cracked 25 °C</th></tr><tr><th></th><th>Comp.</th><th>Norm.</th><th>Comp.</th><th>Norm.</th></tr></thead><tbody><tr><td>J<sub>0.2</sub></td><td>90.2 ± 1.3</td><td>81.7 ± 1.3</td><td>70.8 ± 5.5</td><td>84.5 ± 13.4</td></tr><tr><td>J<sub>c</sub>/J</td><td>6.9 ± 1.1</td><td>8.0 ± 0.1</td><td>8.3 ± 2.5</td><td>7.6 ± 2.1</td></tr><tr><td>K<sub>J</sub></td><td>142.5 ± 1.0</td><td>135.6 ± 1.1</td><td>127.2 ± 5.0</td><td>138.6 ± 11.4</td></tr><tr><td>T</td><td>250.0 ± 31.0</td><td>180.4 ± 15.4</td><td>166.0 ± 24.8</td><td>145.2 ± 30.5</td></tr><tr><td></td><td colspan="2">Notched 25 °C</td><td colspan="2">Notched 25 °C</td></tr><tr><td></td><td>Comp.</td><td>Norm.</td><td>Comp.</td><td>Norm.</td></tr><tr><td>J<sub>0.2</sub></td><td>141.1 ± 12.0</td><td>139.8 ± 16.2</td><td>77.7 ± 3.0</td><td>80.5 ± 3.3</td></tr><tr><td>J<sub>c</sub>/J</td><td>9.8 ± 0.45</td><td>11.9 ± 0.7</td><td>16.1 ± 0.5</td><td>19.4 ± 0.6</td></tr><tr><td>K<sub>J</sub></td><td>178.0 ± 7.6</td><td>177.1 ± 10.3</td><td>133.3 ± 2.6</td><td>135.7 ± 2.8</td></tr><tr><td>T</td><td>407.1 ± 10.3</td><td>344.1 ± 52.3</td><td>153.8 ± 4.8</td><td>185.3 ± 12.2</td></tr><tr><td></td><td colspan="2">Notched 650 °C</td><td colspan="2">Notched 650 °C</td></tr><tr><td></td><td>Norm.</td><td></td><td>Norm.</td><td></td></tr><tr><td>J<sub>0.2</sub></td><td colspan="2">41.4 ± 2.8</td><td colspan="2">51.5 ± 9.9</td></tr><tr><td>J<sub>c</sub>/J</td><td colspan="2">6.6 ± 1.</td><td colspan="2">2.4 ± 0.8</td></tr><tr><td>K<sub>J</sub></td><td colspan="2">83.8 ± 2.9</td><td colspan="2">91.7 ± 8.9</td></tr><tr><td>T</td><td colspan="2">378.7 ± 1.7</td><td colspan="2">434.2 ± 13.2</td></tr></tbody></table>

cedure is repeated for various values of $a/W$. Depending on whether a Load-CMOD or Load-LLD curve is considered, $A_p$ and consequently $\eta_p$ will differ. The results are then fitted by polynomials, leading to the following expressions:

$$\eta_p^{\text{CMOD}} = 0.82 + 1.816\bar{a} - 1.972\bar{a}^2 \quad (24)$$

$$\eta_p^{\text{LLD}} = -0.125 + 5.845\bar{a} - 4.394\bar{a}^2 \quad (25)$$

$\eta_p$ can then be directly evaluated using eq. (6). $\eta_p^{\text{CMOD}}$ and $\eta_p^{\text{LLD}}$ are plotted as functions of $a/W$ in Fig. 12 and compared with values proposed in the ASTM E1820 standard for SE(B) specimens and in [14] for the PRNB specimens. In the case where $\eta_p$ is calculated for the LLD, the proposed function is relatively close to the ASTM E1820 estimate and larger than the one proposed in [14]. The function for $\eta_p$ in the case where the CMOD is used is significantly smaller than both other estimates. This is to be expected, as the CMOD is measured at points below (5 mm, see Fig. 4) the contact line between the rollers and the specimens where displacements are larger than on the contact line. Note that the values for $\eta_p$ proposed in [14] for both configurations are constant as their determination is based on the load separation parameter $S_{iI}$ (eq. (16)) which is fitted as $S_{iI} = (1 - a/W)^{\eta_p}$ (see [12,14,18,19]).

## 5. Experimental results

### 5.1. Determination of the $J-\Delta a$ curves and $J_{0.2}$

Tests listed in Table 2 were processed using the above-determined functions to obtain the $J-\Delta a$ curves. At 25 °C, the normalization and unloading compliance methods were used. At 650 °C, the signal of the clip used to measure the CMOD was too noisy, so that the compliance could not be accurately determined. In that case, the analysis was uniquely based on the normalization method. Tests at 25 °C were carried out with precracked and notched specimens, whereas notched specimens only were used at 650 °C.

The obtained $J-\Delta a$ curves were then fitted using the following function:

$$J = c_1 + c_2(\Delta a)^{c_3} \quad \text{with} \quad c_1 \ge 0, c_2 > 0, 0 < c_3 < 1 \quad (26)$$

where $c_1$, $c_2$ and $c_3$ are the coefficients to be fitted. The fit was performed for $J$ values less than $J_{\text{limit}} = B\sigma_Y/7.5$ following the ASTM E1820 standard. The standard also recommends using data points between the
<figure><img src="image_15.png" /><figcaption>Fig. 13. Examples of $J-\Delta a$ curves obtained for various testing conditions. Solid lines correspond to experiments. Dashed lines correspond to the fit using eq. (26). Black lines correspond to measurements using the unloading compliance method. Red lines correspond to measurements using the normalization method. On those curves, the red dots indicate the measured final crack extension.</figcaption></figure>

0.15 and 1.5 mm exclusion lines. Due to the small size of the specimens, these limits cannot be used in the present case. First, the upper limit is larger than the initial ligament (see Table 2). Second, the lower limit corresponds to a crack advance corresponding to about 5% of the specimen width (10 µm was used in this study). $J_{0.2}$ was then determined as the intersection of the fitted $J-\Delta a$ curve and the 0.2 mm offset line corresponding to the following equation (0.2 mm offset line): $J = 2\sigma_Y(\Delta a - 0.2)$. The toughness is then computed as $K_J = \sqrt{J_{0.2}E}$ where the Young's modulus is used as plane stress conditions prevail due to the thickness of the specimen. The resulting value is considered as "valid" if it is smaller than $J_{max} = B\sigma_Y/10$ for a crack advance less than $\Delta a_{max} = 0.25(W - a_0)$. After each test, the specimens were fully opened by breaking them in liquid nitrogen, which induces brittle fracture so that the different crack propagation zones can be easily distinguished on fracture surfaces. Thermal etching was used for specimens tested at

20 °C before opening to mark the final crack front. This made it possible to determine the final crack advance, which could then be compared to the advances determined using the unloading compliance. The normalization method uses the final crack length determined by fractography.

The dimensionless tearing modulus, $T$, can also be determined as [2]:

$$T = \frac{E}{\sigma_Y^2} \frac{dJ}{da} = \frac{E}{\sigma_Y^2} \frac{J_{0.2} - J_{0.1}}{0.1 \text{ mm}} \quad (27)$$

Due to the reduced size of the specimens, $T$ was computed for a crack advance between 0.1 and 0.2 mm ($\Delta a = 0.1$ mm). The corresponding values of $J$ ($J_{0.1}$ and $J_{0.2}$) correspond to the intersections of the 0.1 and 0.2 mm offset lines with the fitted $J-\Delta a$ curve (eq. (26)). Note that in the ASTM E1820, the tearing modulus is computed for crack advances between 0.2 and 1. mm.
<figure><img src="image_16.png" /><figcaption>Fig. 14. Macroscopic view of the fracture surface for both materials and both testing temperatures. 'Brittle' zones correspond to fracture in liquid nitrogen.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 15. Microscopic view of the fracture surface for both materials at 20 °C and 650 °C. Boxes show aligned dimples.</figcaption></figure>

Examples of the determined *J*—Δ*a* curves are given in Fig. 13 for the various testing configurations using both data analysis methods. Black curves correspond to the unloading compliance method and red curves to the normalization method. The red symbol corresponds to the measured final crack extension. Table 3 gathers mean values and scatter for *J*<sub>0,2</sub>, *K*<sub>J</sub> and *T*. One can observe that the unloading compliance and normalization methods provide very similar results. The largest differ-

ence is obtained for the ODS14Cr material for precracked specimens (Fig. 13-(b) and Table 3). This is attributed to asymmetric precracks. In the case of notched specimens, which are symmetric, for the same material (Fig. 13-(d)) the agreement is very good.

At room temperature, a valid value for *J*<sub>0,2</sub> is obtained except in the case of the ODS9Cr material when using notches (see Fig. 13-(a)). Using notches for the ODS14Cr material leads to toughness values close
to those obtained using pre-cracked specimens. At 650 °C, the decrease in σ<sub>Y</sub> leads to a strong reduction in the value of J<sub>max</sub> and the slope of the offset line so that it becomes difficult to obtain valid values. For the ODS9Cr material, an almost linear J—Δa curve (c<sub>3</sub> = 0.7) is obtained with a relatively high normalized tearing modulus. For the ODS14Cr material, a similar behavior is observed up to Δa = 0.25 mm (see Fig. 13(f)). Some J—Δa curves present an irregular shape (e.g., Fig. 13(f)) which could be attributed to crack deflection or non-symmetric propagation. J—Δa curves are always adjusted before irregularities appear. Since irregularities appear late, this has never prevented adjustment.

Comparing results for both materials (Table 3) indicates that the ODS9Cr material shows better toughness properties (J<sub>0.2</sub> and T) than the OCS14Cr material. The opposite trend is observed at 650 °C. This is, however, mainly due the much lower flow stress of the OCS14Cr material at this temperature. Table 3 also displays the J<sub>e</sub>/J ratio for J = J<sub>0.2</sub>. In all case, this ratio is less than 20%. This implies that the difference between an evaluation of J<sub>0.2</sub> assuming a plane stress state or a plane strain state is less than 2%.

## 5.2. Fractography

Fig. 14 shows the macroscopic fracture surfaces for both materials and test temperatures equal to 20 and 650 °C. As mentioned above, a straight fatigue crack is difficult to obtain (Fig. 14-(a) and (b)). This is due to the slightly non-symmetric stress field along the specimen thickness as shown in Fig. 8-(b). At room temperature, the crack begins as a flat triangular region perpendicular to the loading direction. The base of the triangle is slightly less than the tube thickness. The remaining fracture surface is slanted making an angle of about 45° with respect to the loading direction. This mode of rupture is characteristic of thin sheet fracture [28,29] in materials having a low or moderate work-hardening. For high work-hardening, the fracture surface tends to remain flat [30,31]. At high temperature, the fracture surface is always flat.

Fig. 15 shows the microscopic fracture surfaces for both materials at room temperature. Very small dimples can be observed for both materials which correspond to strengthening particles. Alignments of dimples are observed for both materials. They are frequent in the case of the ODS14Cr material whereas the fracture surface of the ODS9Cr material exhibits large areas with isotropically distributed small dimples. These alignments are along the tube axis. For the 14Cr material, some larger dimples can be observed that could correspond to clusters of particles.

In the case of the ODS9Cr material, fracture surfaces for specimens tested at 650 °C are highly oxidized and cannot be observed at high magnification. Fracture surfaces for ODS14Cr materials tested at 650 °C differ from those obtained at room temperature. Dimples are smaller and very large dimples (clusters) are not observed.

## 6. Conclusions

A miniaturized version of three points bending test on ring specimens proposed to test pipe sections was developed and used to assess the fracture toughness of ODS steels fuel claddings. The single specimen method based on the use of the unloading compliance method and the normalization method was developed. This required the use of FE simulation to analyze the test and derive equations for the geometric functions C(a/W), a/W(u), f(a/W) and η<sub>p</sub>(a/W) needed to apply the ASTM E1820 testing procedure. Formulas were developed to analyze test data using either the Load—CMOD curve or the Load—LLD curves. The procedure allows the determination of J—R curves.

The proposed test procedure was applied to two cladding ODS materials containing 9 or 14% Cr. Both materials were tested at 20 °C and 650 °C. The main findings are as follows at room temperature. The ODS9Cr steel has a higher toughness (+11%) and tearing modulus than the ODS14Cr material. However, it was difficult to pre-fatigue

the ODS14Cr material as different fatigue crack lengths were often obtained on both sides of the specimen. The ODS9Cr material showed a more stable crack propagation, while ODS14Cr material exhibited less stable crack propagation. This behavior was attributed to its fibrous microstructure. Fractographic observations showed chevron-shaped crack patterns and dimple ductile damage for both materials. Data analysis using the unloading compliance and the normalization method were compared showing good agreement. At 650 °C, it was not possible to apply the unloading compliance method due to noisy signals and stress relaxation during unloading. The normalization method was then used as an alternative. The ODS14Cr material showed higher toughness and tearing modulus. This is mainly attributed to the lower flow stress of this material at 650 °C. Fracture surfaces showed a chevron shape and remained flat.

The proposed technique was applied to ODS materials suitable for Gen IV nuclear reactors. It could also be used to characterize the fracture toughness of zirconium fuel claddings [32] currently used in Pressurized Water Reactor. The test method should therefore be further developed to test irradiated fuel claddings in “hot” cells facilities. It could be used as an alternative method to the Pin Loading Tension (PLT) test which is also used to characterize crack propagation in nuclear fuel claddings [7, 11, 12]. The main advantage of the mRNB test is that it uses less material and is much less affected by the friction between the specimen and the loading device which consists in two half-cylinder mandrels inserted in the specimen in the case of the PLT test.

## CRediT authorship contribution statement

**Boualem Rais:** Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Jérôme Garnier:** Writing – review & editing, Writing – original draft, Validation, Supervision, Investigation, Funding acquisition, Conceptualization. **Elodie Pons:** Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Investigation, Data curation. **Bernard Marini:** Writing – review & editing, Writing – original draft, Validation, Supervision, Funding acquisition, Formal analysis, Data curation. **Jacques Besson:** Writing – review & editing, Writing – original draft, Validation, Supervision, Investigation, Data curation, 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.

## Acknowledgements

The authors would like to thank: D. Sornin (CEA/DES/ISAS/DRMP/SRMA/LTMEX) and the LTMEX laboratory from the CEA for the manufacture of the ODS steel fuel claddings studied in this paper. E. Rouesne and F. Salliot (CEA/DES/ISAS/DRMP/SRMA/LA2M) for the production of the EBSD maps. R. Cluzet (Centre des Matériaux / MinesParis) for the manufacture of mRNB specimens.

## Data availability

Data will be made available on request.

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