# Explanation of edge defect influence on sapphire bending strength scatter using the coupled criterion

Aurélien Doitrand, Ronan Henry, Sylvain Meille

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Aurélien Doitrand, Ronan Henry, Sylvain Meille. Explanation of edge defect influence on sapphire bending strength scatter using the coupled criterion. Journal of the European Ceramic Society, 2024, pp.117052. 10.1016/j.jeurceramsoc.2024.117052. hal-04779228

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Submitted on 13 Nov 2024

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# Explanation of edge defect influence on sapphire bending strength scatter using the coupled criterion

Aurelien Doitrand<fnref n="1" />, Ronan Henry<fnref n="2" />, and Sylvain Meille<fnref n="1" />

Sapphire fracture is studied by means of four-point bending tests on mirror-polished millimetric specimens having their cristallographic **a**-axis oriented along the specimen length. The scattering of bending strengths between 500 MPa and 750 MPa is mainly due to edge defects of some tens of microns resulting from the specimen cutting process. Crack initiation occurs from an edge defect along a a-plane, perpendicular to the direction of maximum tensile stress, and further deviates along weaker m-planes. Numerical simulations of edge defect-induced crack initiation based on the coupled criterion reveal that the material sensitivity to edge defect-induced crack initiation mainly depends on Irwin's length. For Irwin's lengths larger than twice the defect depth, the bending strength is the same as the one obtained without defect. By retrieving the bending strength variation as a function of the defect depth measured experimentally, the proposed approach enables the identification of sapphire a-plane 850 ± 90 MPa tensile strength for a $G_c = 15 \text{ J m}^{-2}$ critical energy release rate.

**Keywords** Sapphire, failure, defects, coupled criterion, tensile strength

## 1 Introduction

Brittle fracture is common in ceramics, it is usually explained by the presence of defects within the microstructure, which locally weaken the material and lead to its fracture. One way to estimate the critical defect size from the measured fracture stress $\sigma_R$ is to use the energy criterion defined in the framework of linear elastic fracture mechanics (LEFM) (Griffith 1921; Griffith 1924). Irwin's approach (Irwin 1957; Irwin 1958; McClintock et al. 1965) gives the critical crack size $a_c = \frac{K_{\text{Ic}}^2}{\pi\sigma_R^2}$ that propagates under such a remote tensile stress $\sigma_R$ as a function of the material critical stress intensity factor $K_{\text{Ic}}$. However, it should be noted that this approach is only applicable under the assumption of a crack subjected to uniform tension in an infinite medium. It thus cannot assess configurations other than a sufficiently long pre-existing crack. The approach is valid provided the pre-existing crack length is larger than the Irwin's length, which is a characteristic of the material defined as $\ell_{\text{mat}} = \frac{EG_c}{(1-\nu^2)\sigma_c^2}$, where E is the material Young's modulus, $\nu$ the material Poisson's ratio, $G_c$ its critical energy release rate and $\sigma_c$ its tensile strength. The Irwin's length is often compared to the size of the fracture process zone, which represents a region near the crack tip where nonlinear phenomena, such as damage or plasticity, occurs. The fracture process zone is present even in brittle materials (Irwin 1948; Orowan 1949), which explains why the true critical energy release rate is several orders of magnitude larger than the surface energy density originally proposed by Griffith. Irwin's length is crucial in fracture mechanics as it is related to the initiation length in the coupled criterion (Leguillon 2002; Doitrand et al. 2020c), it defines the regularization length in phase-field fracture models (Molnár et al. 2020), and it is related to the critical displacement jump and the process zone extent in cohesive zone models (Barenblatt 1959; Dugdale 1960). It is also used as an input for non-local stress evaluation in the theory of critical distances (Taylor 2007). The application of Irwin's approach to evaluate a critical defect size in a ceramic neglects both the location of the crack within the specimen and the actual shape of the defect (a porosity would then be assimilated to a crack if this approach is used).

The criticality of a defect can be evaluated by studying the phenomenon of crack initiation from the defect. However, when studying a more general defect shape than a crack such as, e.g., a V-notch (Leguillon et al. 1987; Dunn et al. 1997; Yosibash et al. 2006), a U-notch (Nishida et al.

<footnote n="1">¹ Univ Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS UMR5510, MATEIS, F-69621 Villeurbanne, France, aurelien.doitrand@insa-lyon.fr</footnote>

<footnote n="2">² CNRS, Univ Rouen Normandie, INSA Rouen Normandie, Normandie Univ, GPM UMR 6634, F-76000 Rouen, France</footnote>
1994; Picard et al. 2006; Sapora et al. 2021), or a pore (Li et al. 2006; Leguillon et al. 2006; Martin et al. 2012; Romani et al. 2015), the sole use of an energy criterion is not sufficient as the energy release rate tends towards zero for vanishing crack lengths (Leguillon et al. 1987; Labossiere et al. 1999). The limitation of LEFM in studying crack initiation can be overcome using the coupled criterion (CC) (Leguillon 2001; Leguillon 2002). The underlying idea of the CC is to combine the energy criterion together with a stress criterion (Leguillon 2002; Cornetti et al. 2006) to address crack initiation. Among various applications of the CC (Weißgraeber et al. 2016a; Doitrand et al. 2024), different defect configurations were studied such as circular (Rosendahl et al. 2017; Felger et al. 2017; Sapora et al. 2018; Cornetti et al. 2019; Carrère et al. 2021), spherical (Chao Correas et al. 2021; Ferrian et al. 2022) or elliptical holes (Carpinteri et al. 2012; Weißgraeber et al. 2016b). These approaches somehow generalize Irwin's approach to study the influence of the size of other defect shapes on fracture. It was also shown that not only the defect size but also its location within the specimen has a first order influence on crack initiation, both under bending (Doitrand et al. 2021a; Uhl et al. 2022) or tensile loading (Doitrand et al. 2021b).

The input fracture parameters of the CC are the material critical energy release rate $G_c$ and the tensile strength $\sigma_c$. From a modeling perspective, $\sigma_c$ corresponds to the strength of a sufficiently long specimen loaded under tension (Leguillon 2002; Doitrand et al. 2020a; Jiménez-Alfaro et al. 2021). Translating this definition into experimental terms is not trivial when dealing with brittle ceramics. Tensile tests are generally not performed on monolithic ceramics as they are not easy to set up and because the resulting homogeneous stress state exacerbates the sensitivity of fracture to the presence of a defect. Instead, flexural tests are typically preferred, as they induce a stress gradient and create a preferential zone for crack initiation on the tensile side of the specimen. The definition of the tensile strength for polycrystalline ceramics was established by studying the fracture of specimens with a surface defect (Leguillon et al. 2018; Martin et al. 2018). For surface defects larger than the ceramic average grain size, the fracture stress increased as the defect size decreased, following Irwin's approach provided that the defect that can be assimilated to a crack. However, for surface defects smaller than the grain size, the fracture stress reached a plateau that corresponded to the tensile strength of the ceramic. However, the tensile strength measurement were still scattered due to possible heterogeneities (such as grain size variation or internal pores) in the polycrystalline ceramic microstructure.

The objective of this work is to study sapphire fracture in order to consider the influence of edge defects and determine its tensile strength. Mechanical tests of millimetric specimens and observations of related defects are presented in Section 2. The modeling of crack initiation from a defect based on the CC is then presented in Section 3. The influence of the defect size on sapphire fracture and its tensile strength determination is finally presented in Section 4.

## 2 Experiments

The material under investigation is sapphire (Saint-Gobain Lumilog), which is a crystal with a hexagonal compact structure. A sapphire wafer was cut with a diamond wire saw to prepare 2.5 × 0.8 × 30 mm³ bars. One face of the sample is mirror polished. The specimen are cut so that the $\bar{a}$ direction corresponds to the bar main dimension, which means that a $\bar{a}$-plane (1120 type) is perpendicular to the sample neutral axis. Twenty-three specimens, whose dimensions are summarized in Table 1, are tested under four-point bending using a Instron 8862 (USA) machine equipped with a 5 kN load cell at 0.1 mm/min displacement rate. A specimen before test is shown in Figure 1(a). The specimen length is 30 mm, the distance between the supports is 12 mm and the distance between the loading spans is 3 mm. The mirror-polished specimen face is the one that it is loaded under tension, so the possible presence of surface defects is mainly due to the specimen cutting step from the original disk, i.e., these defects are not purposely induced in the specimens. For all specimens, a linear elastic behavior until brittle failure is recorded. The measured failure force $F_c$ enables the specimen bending strength $\sigma_f$ to be determined based on Euler-Bernoulli's beam theory. Figure 1(b) shows a specimen after testing which, at first sight, exhibits a V-shaped failure pattern. The specimen actually breaks into three parts, following two fracture paths oriented at 30 deg. from the bar cross section plane. This failure pattern is observed for all the tested specimen and the fracture planes oriented at 30 deg. correspond to sapphire m-planes
<figure><img src="image_2.png" /><figcaption>**Figure 1:** Optical observation of a bending specimen a) before and b) after failure. Scanning electron microscope observations of the specimen after failure c) top view, d) fracture surface view and e) focus on an edge defect located at the fracture surface (two examples of edge defect are shown).</figcaption></figure>

<table><caption>**Table 1:** Dimensions (width *W* and thickness *B*), failure force (*F*<sub>c</sub>), bending strength (*σ*<sub>f</sub>) and critical edge defect depth (*d*) measured for the tested sapphire specimens.</caption><thead><tr><td>Specimen #</td><td>B (mm)</td><td>W (mm)</td><td>F<sub>c</sub> (N)</td><td>σ<sub>f</sub> (MPa)</td><td>d (μm)</td></tr></thead><tbody><tr><td>1</td><td>2.45</td><td>0.78</td><td>59.3</td><td>537.1</td><td>20</td></tr><tr><td>2</td><td>2.45</td><td>0.78</td><td>63.4</td><td>574.2</td><td>25</td></tr><tr><td>3</td><td>2.45</td><td>0.79</td><td>81.9</td><td>723.1</td><td>20</td></tr><tr><td>4</td><td>2.43</td><td>0.79</td><td>72.0</td><td>640.9</td><td>15</td></tr><tr><td>5</td><td>2.43</td><td>0.78</td><td>53.9</td><td>492.2</td><td>40</td></tr><tr><td>6</td><td>2.44</td><td>0.78</td><td>60.1</td><td>546.6</td><td>25</td></tr><tr><td>7</td><td>2.44</td><td>0.78</td><td>64.8</td><td>589.3</td><td>30</td></tr><tr><td>8</td><td>2.44</td><td>0.78</td><td>81.0</td><td>736.6</td><td>15</td></tr><tr><td>9</td><td>2.44</td><td>0.78</td><td>78.0</td><td>709.3</td><td>10</td></tr><tr><td>10</td><td>2.43</td><td>0.78</td><td>72.9</td><td>665.7</td><td>10</td></tr><tr><td>11</td><td>2.44</td><td>0.79</td><td>62.3</td><td>552.3</td><td>30</td></tr><tr><td>12</td><td>2.45</td><td>0.78</td><td>75.2</td><td>681.1</td><td>15</td></tr><tr><td>13</td><td>2.44</td><td>0.78</td><td>62.4</td><td>567.5</td><td>20</td></tr><tr><td>14</td><td>2.45</td><td>0.78</td><td>74.5</td><td>674.7</td><td>10</td></tr><tr><td>15</td><td>2.45</td><td>0.78</td><td>76.3</td><td>691.0</td><td>5</td></tr><tr><td>16</td><td>2.45</td><td>0.79</td><td>79.4</td><td>701.0</td><td>5</td></tr><tr><td>17</td><td>2.45</td><td>0.78</td><td>82.3</td><td>745.4</td><td>10</td></tr><tr><td>18</td><td>2.44</td><td>0.79</td><td>85.0</td><td>753.5</td><td>5</td></tr><tr><td>19</td><td>2.45</td><td>0.79</td><td>75.3</td><td>664.8</td><td>15</td></tr><tr><td>20</td><td>2.44</td><td>0.78</td><td>76.6</td><td>696.6</td><td>20</td></tr><tr><td>21</td><td>2.45</td><td>0.79</td><td>65.1</td><td>574.8</td><td>25</td></tr><tr><td>22</td><td>2.45</td><td>0.79</td><td>78.0</td><td>688.3</td><td>15</td></tr><tr><td>23</td><td>2.45</td><td>0.79</td><td>79.7</td><td>703.4</td><td>5</td></tr></tbody></table>

which are among their weakest plane (Wiederhorn 1969). Figure 1(c) shows a Scanning Electron Microscope (SEM) top view of the specimen (i.e., the apparent face is the mirror-polished face). Most of the fracture surface makes a 30 deg. angle with respect to the specimen cross section, but it is observed that a part of the fracture surface is actually perpendicular to the specimen neutral
axis and thus coincide with a a-plane. As a consequence, two scenarios are possible to explain the observed fracture pattern: either crack initiation on a a-plane perpendicular to the specimen neutral axis, or crack initiation along a m-plane oriented at 30 deg.

A first possible scenario would be crack initiation along a m-plane. However, since its orientation is 30 deg. from the specimen cross section plane, it means that the local opening stress is smaller than the one acting on the cross section plane. In addition, this scenario would not explain the V-shaped fracture pattern, that is more likely to originate from crack branching after initiation. We thus conclude that crack initiation rather occurs on a a-plane perpendicular to the specimen neutral axis, on which the opening stress is the highest. This conclusion is supported by having a closer look at the fracture surfaces, that evidence the presence of defects located on the edge of the mirror-polished surface under tension (Figure 1(d-e)). A flat-like surface is observed just around the defect (Figure 1(e)), indicating a dynamic crack initiation from the defect at the specimen surface under tension. The fracture surface becomes rougher at the end of this flat zone, indicating that the crack initiated on the specimen a-plane tends to deviate to propagate along a plane with a weaker fracture energy, i.e., the m-plane. This crack branching also occurs from the specimen face under tension, as evidence by a second flat-like zone indicating a dynamic crack propagation along the m-plane stemming from the surface under tension (Figure 1(d)). Once initiated from the edge defect and deviated along the m-plane from the face under tension, the end of the crack propagation is indicated by fracture surface that becomes rougher when approaching the face initially located under compression. This roughening is typical of the final ceramic specimen failure (Quinn 2020) (Figure 1(d)). The observations of all the tested specimen fracture surfaces reveal that the shape of the critical defect slightly differ depending on the specimen as shown in Figure 1(e). The defect shapes are mainly wedge-like and the defect size was measured as the largest depth either along (Õy) or (Õz) axis. The defect sizes were evaluated for all the specimens based on SEM observations and are reported in Table 1, ranging between 5 µm and 40 µm. An overall trend is observed that the lower the measured bending strengths, the deeper the edge defect.

# 3 Edge defect-induced fracture

## 3.1 The coupled criterion

Surface defect-induced crack initiation is assessed using the CC (Leguillon 2001; Leguillon 2002) in the framework of Finite Fracture Mechanics (Hashin 1996; Nairn 2000) under the assumption of small deformations and considering linear elastic material behavior. The configuration under investigation of a four-point bending specimen including an edge defect is depicted in Figure 2(a), assuming a defect shape close to the ones observed experimentally, but involving a single parameter, its depth d. Note that influence of the defect shape on the failure loading depends on the Irwin's length, since the defect morphology has a limited influence on the failure loading if it size ranges between $l_{mat}/10$ and $l_{mat}/2$ (Martin et al. 2018), and that differences up to 20 % may be obtained depending on the shape of defects larger than $l_{mat}/2$ (Leguillon et al. 2007; Martin et al. 2018). The objective of the CC is to determine, for a given defect depth, the imposed displacement at which a crack initiates from the defect. The underlying idea of the CC is that crack initiation occurs provided the fulfillment of both a stress criterion and an energy criterion. The energy criterion derives from the energy conservation principle (Leguillon 2002; Doitrand et al. 2020c) and writes:

$$ \mathcal{G}_{\text{inc}}(S) \geq \mathcal{G}_{c}, \qquad (1) $$

where $\mathcal{G}_{\text{inc}}$ is the incremental energy release rate (IERR) defined as $\mathcal{G}_{\text{inc}} = -\Delta W_{el}/S$, i.e., the ratio between the elastic strain energy ($W_{el}$) variation due to crack initiation and the crack surface ($S$). This criterion reverts to Griffith's criterion when it is applied to study the propagation of a semi-infinite crack embedded in an infinite medium. The stress criterion is a non-local condition that must be fulfilled at all points $x$ over the new crack path $\Gamma$ before initiation:

$$ \sigma_{nn}(x) \geq \sigma_{c}, \forall x \in \Gamma \qquad (2) $$
<figure><img src="image_3.png" /><figcaption>**Figure 2:** a) Geometry of the bending specimens under investigation including an edge defect of depth *d*. b) Stress field under 4-point bending and emphasis on the singular stress field around the surface flaw, based on which the potential crack surfaces are defined to further calculate the incremental energy release rate.</figcaption></figure>

where $\sigma_{nn}$ is the stress component normal to the crack path $\Gamma$ and $\sigma_c$ is the material tensile strength. The main difficulty when applying the CC in 3D is that the crack path $\Gamma$ is a priori unknown. A first way to address this issue is to describe the crack shape by several parameters (García et al. 2016; Yosibash et al. 2016; Doitrand et al. 2017; García et al. 2018; Papšik et al. 2023; Vereecke et al. 2024; Burhan et al. 2024). Previous works showed that the alternative choice of potential crack shapes based on the stress isocontours is a reasonable assumption to accurately determine the initiation loading (Leguillon 2014; Doitrand et al. 2018; Doitrand et al. 2020b; Sevecek et al. 2019; Girard et al. 2024). The advantage of this method is that it provides a unique description $\Gamma(S)$ of the crack topology for a given crack surface. Moreover, the definition of the stress isocontours ensure that an increasing crack surface corresponds to a decreasing stress level. As a consequence, if the stress criterion is fulfilled on the crack front of $\Gamma(S)$ before initiation, it is also fulfilled anywhere inside $\Gamma(S)$. Thus, the stress criterion can unambiguously be written as a function of the crack surface:

$$ \sigma_{nn}(S) \geq \sigma_c. \qquad (3) $$

Under the assumptions of linear elasticity and small deformations, the stress components are proportional to the applied displacement $U$ and the IERR is proportional to the square applied displacement, which writes:

$$ \left\{ \begin{array}{l} \sigma_{nn}(S) = k(S) \frac{E}{1-\nu^2} \frac{U}{W} \geq \sigma_c, \\ \mathcal{G}_{inc}(S) = A(S) \frac{E}{1-\nu^2} W \left(\frac{U}{W}\right)^2 \geq G_c. \end{array} \right. \qquad (4) $$

The dimensionless functions $k$ and $A$ are obtained by finite element (FE) calculations for different crack surfaces, as detailed in next section, $W$ is the specimen width, $E$ and $\nu$ are the $\vec{a}$-axis direction Young's modulus and Poisson's ratio. The coupling of both conditions reverts in determining the minimum imposed displacement $U_c$ for which both conditions are simultaneously fulfilled. For monotonic variations of the stress and of the IERR, the crack surface at initiation $S_c$ is determined by combining the two equalities given in Equation (4) and solving the following equation:

$$ \frac{A(S_c)}{k(S_c)^2} = \frac{\ell_{mat}}{W} \qquad (5) $$

Then, the initiation imposed displacement $U_c$ is deduced from Equation (4) as:

$$ U_c = \frac{W \sigma_c}{E k(S_c)} = \sqrt{\frac{W \mathcal{G}_c}{E A(S_c)}} \qquad (6) $$

The CC thus takes as input the material elastic and fracture properties, and yields as outputs the initiation imposed displacement and crack surface.
## 3.2 Numerical simulation

The CC is implemented through FE calculations (Abaqus quasi-static implicit solver) of the stress and of the IERR. The FE calculations are performed under linear elasticity and assuming small deformations. The mesh consists of quadratic tetrahedra with 10 nodes and 4 integration points. It is refined near the defect on the crack plane (minimum mesh size of 0.3 µm) and typically has around 1.5 M degrees of freedom, which results in differences in the initiation displacement smaller than 1 % for a finer mesh. Dirichlet boundary conditions are imposed at the span and support locations to replicate the bending loading. The anisotropic elastic properties, taken from the literature, are $C_{11}=466$ GPa, $C_{12}=117$ GPa, $C_{13}=127$ GPa, $C_{22}=506$ GPa, $C_{36}=94$ GPa, $C_{44}=235$ GPa, where C is the stiffness matrix in Voigt's notation. The stiffness matrix is defined in the material local axis system that corresponds to the global axis system ($O, \vec{x}, \vec{y}, \vec{z}$), i.e., the $\vec{a}$-axis is along ($O\vec{x}$) and the $\vec{c}$-axis is along ($O\vec{y}$). A first calculation without crack is performed in order to determine the stress fields on the crack plane around the defect. This step enables computing the stress isocontours and thus the function k (see Section 3.1). An example of crack shapes based on the stress field isocontours is depicted in Figure 2(b). Once the crack shapes are defined, the lines corresponding to their crack fronts are included in the specimen geometry so that the mesh follows the crack front shapes. This is of practical importance for the calculation of the elastic strain energy variation as a function of the crack surface. Indeed, the elastic strain energy is calculated by successively unbuttoning the nodes lying on the surfaces delimited by the stress isocontour lines. This step allows computing the IERR and thus the function A as a function of the crack surface.

The IERR variation as a function of the normalized crack surface is shown in Figure 3 for different edge defect depths up to 40 µm (symbols in Figure 3) and compared to the case without defect (solid line in Figure 3). The overall increasing trend as a function of the crack

<figure><img src="image_4.png" /><figcaption>**Figure 3:** a) IERR as a function of the normalized crack surface obtained for several edge defect depths. b) Focus on small normalized crack surfaces for which the IERR increases with increasing defect size compared to the case without defect.</figcaption></figure>

surface is retrieved whatever the edge defect depth, so that the IERR variation is similar for large enough normalized crack surfaces whatever the defect depth (Figure 3(a)). However, significant differences arise for small crack surfaces as the larger the defect depth, the larger the IERR (Figure 3(b)). It can also be noted that the larger the defect depth, the larger the zone over which the IERR in presence of a defect significantly increases compared to the case without defect. After the initial increase corresponding to the influence of the edge defect, the IERR slightly decreases as the crack surface spans the whole specimen width to retrieve a similar value as in the case without defect. As a consequence, it can be expected that the initiation loading significantly decreases if the initiation surface is small enough whereas if the crack surface at initiation is sufficiently large, the IERR remains the same as in the case without defect.

The variation of the stress normal to the crack plane before initiation is shown in Figure 4. Whatever the defect depth, there is a decreasing stress variation as a function of the normalized
<figure><img src="image_5.png" /><figcaption>**Figure 4:** a) Opening stress as a function of the normalized crack surface obtained for several edge defect depths. b) Focus on small normalized crack surfaces for which the stress increases with increasing defect size compared to the case without defect.</figcaption></figure>

crack surface, with similar values for sufficiently large crack surfaces whatever the defect depth Figure 4(a). However, when approaching the defect location, the stress becomes significantly larger than in the case without defect. Indeed, without defect, the stress variation has a linear trend up to the maximum value corresponding to the bending strength for the imposed initiation displacement. The presence of a defect induces a singularity which makes the stress theoretically tend towards infinity when approaching the defect location (Figure 4(b)) (Leguillon et al. 1987; Carpinteri 1987). The larger the defect depth, the larger the stress locally and the larger the zone over which the stress is higher than in the case without defect. It means that similarly to the analysis about the IERR, it can be expected that the initiation loading remains similar to the case without defect if the crack surface at initiation is sufficiently large, and that a smaller initiation loading is obtained for sufficiently small crack surfaces at initiation.

## 4 Sapphire fracture under bending

We now study the influence of the edge defect depth on the initiation imposed displacement and resulting bending strength. It is observed from Equation (5) that for a given specimen geometry and defect depth, the crack surface at initiation only depends on the Irwin's length. The imposed displacement required for crack initiation is shown as a function of the normalized Irwin's length in Figure 5(a). As expected from the analysis of the stress and of the IERR in the previous section,

<figure><img src="image_6.png" /><figcaption>**Figure 5:** a) Imposed displacement at crack initiation ($U_c$) and b) ratio between the imposed displacements at crack initiation with ($U_c$) and without defect ($U_0$) as a function of Irwin's length to specimen width ratio for different defect depths.</figcaption></figure>
for a given defect depth, the imposed displacement at crack initiation is the same as the one without defect provided that Irwin's length is large enough. It means that materials exhibiting larger Irwin's length are more tolerant to the presence of large defects, and vice-versa. For smaller Irwin's length, the influence of the defect is significant as it induces a decrease in the imposed displacement at initiation, which is smaller for larger defect depths. The transition between the two regimes can be evidenced when studying the variation of the ratio between the imposed displacement at initiation in presence of a defect or without it (Figure 5(b)). The same imposed displacement at initiation is obtained for sufficiently large Irwin's length, whereas the ratio of the initiation displacement decreases with decreasing Irwin's length. The transition between both situations happens for $\ell_{mat} \approx 2d$. It means that if the Irwin's length is larger than $2d$ (which may happen, e.g., if the tensile strength is sufficiently small), it is like there was no edge defect in the material, i.e., the defect size is too small to induce a decrease in the bending strength. We finally analyze the bending strength decrease with increasing defect depth obtained using the CC and compare it to the experimental measurements presented in Section 2. Figure 6 shows the bending strength to tensile strength variation as a function of the defect depth for $\sigma_c = 750$ MPa (Figure 6(a)) or $\sigma_c = 850$ MPa (Figure 6(b)) and different Irwin's lengths. If the

<figure><img src="image_7.png" /><figcaption>**Figure 6:** a-b) Bending strength ($\sigma_f$) to tensile strength ($\sigma_c$) ratio as a function of the defect depth ($d$) for two Irwin's lengths and a) $\sigma_c = 750$ MPa, b) $\sigma_c = 850$ MPa. Tensile strengths and c) Critical ERR or d) Irwin's length couples that well represent the bending strength decrease as a function of the defect depth.</figcaption></figure>

Irwin's length is smaller than 10 µm (for $\sigma_c = 750$ MPa, Figure 6(a)) or than 5 µm (for $\sigma_c = 850$ MPa, Figure 6(b)), the decrease in bending strength predicted by the CC is too severe compared to the one measured experimentally. On the contrary, if the Irwin's length is larger than 20 µm (for $\sigma_c = 750$ MPa, Figure 6(a)) or 11 µm (for $\sigma_c = 850$ MPa, Figure 6(b)), the obtained bending strength variation overestimates the experimental measurements. For Irwin's length between 10 µm and 20 µm (Figure 6(a)) or between 5 µm and 11 µm (Figure 6(b)), the model provides a reasonably good representation of the bending strength variation measured experimentally. For a given tensile strength, it is thus possible to determine the critical ERR range for which the
failure stresses obtained with the model lies within the range measured experimentally. It yields $15 \text{ J m}^{-2} \le \mathcal{G}_c \le 30 \text{ J m}^{-2}$ for $\sigma_c = 750 \text{ MPa}$ and $10 \text{ J m}^{-2} \le \mathcal{G}_c \le 22 \text{ J m}^{-2}$ for $\sigma_c = 850 \text{ MPa}$.

This procedure can be repeated for other values of $\sigma_c$, which yields the admissible couples of tensile strengths and critical ERR (or equivalently, Irwin's length) for the studied material. They are represented in Figures 6(c) and 6(d). It can be noted that for $\sigma_c \le 650 \text{ MPa}$, the bending strength variation is not well captured. In general, the ceramic critical ERR can be determined based on notched bending tests (Ozcoban et al. 2010; García-Prieto et al. 2012; Schlacher et al. 2023) or indentation tests (Quinn et al. 1997; Haney et al. 2011; Henry et al. 2022) for instance. The proposed approach thus enables determining the corresponding tensile strength. For instance, considering a critical ERR $\mathcal{G}_c = 20 \text{ J m}^{-2}$ (Doitrand et al. 2020a) yields a corresponding tensile strength of $\sigma_c = 805 \pm 90 \text{ MPa}$. Considering a $2.32 \text{ MPa} \sqrt{\text{m}}$ a-plane critical stress intensity factor (Huang et al. 2021), which corresponds to $\mathcal{G}_c = 15 \text{ J m}^{-2}$, it yields a sapphire a-plane tensile strength of $\sigma_c = 850 \pm 90 \text{ MPa}$.

Considering the edge defect-induced crack initiation, the proposed approach is able to explain the overall scattering level measured experimentally on this batch of specimens. For a given Irwin's length and tensile strength, the proposed model provides a single value of the bending strength whereas there is also a certain scattering in the experimental measurements for a given defect depth. This may be due to the fact that the defect shape slightly varies depending on the specimen, whereas a single shape is considered in the proposed model. The defect shape may thus also have an influence on the bending strength (Leguillon et al. 2018; Martin et al. 2018; Doitrand et al. 2020a), nevertheless it seems to be a second order parameter when compared to the defect size.

# 5 Conclusion

Bending tests on mirror-polished millimetric sapphire specimens with $\bar{a}$-axis oriented along the specimen length reveal that the scattering in the measured bending strength is mainly due to edge defects of some tens of microns resulting from the specimen cutting process. The larger the defect, the smaller the bending strength, which may also slightly depend on the defect shape. Crack initiation occurs from an edge defect along a specimen a-plane before deviating along weaker m-planes that favour further crack propagation. The coupled criterion shows that the sensitivity of the material to edge defect-induced crack initiation mainly depends on Irwin's length. For Irwin's lengths larger than twice the defect depth, the bending strength is the same as the one obtained without defect. For smaller Irwin's lengths, the decrease in the bending strength compared to the case without defect is more pronounced for larger defects. In the present experiments, the sapphire Irwin's length is sufficiently small so that the material is sensitive to the defects, which explains the bending strength decrease with increasing defect size. For a given tensile strength, the comparison between the bending strength decrease with increasing defect depth obtained experimentally and numerically enables determining a range of material critical ERR. Conversely, based on the knowledge of the material critical ERR, the proposed approach yield an indirect way of determining the ceramic a-plane tensile strength, which turns to be evaluated as $850 \pm 90 \text{ MPa}$ for the sapphire under investigation.
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**Authors' contributions** SM performed the mechanical tests, AD performed the fracture surface observations, numerical simulations and drafted the manuscript. SM, RH and AD developed the methodology, conceived of the study, and participated in its design, coordination, and critical review of the manuscript. All authors read and approved the final manuscript.