Investigation of sapphire strength across the scales

Ronan Henry, Aurélien Doitrand, Sylvain Meille

► To cite this version:

Ronan Henry, Aurélien Doitrand, Sylvain Meille. Investigation of sapphire strength across the scales. Materialia, 2025, pp.102439. 10.1016/j.mtla.2025.102439. hal-05115386

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Submitted on 16 Jun 2025

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# Investigation of sapphire strength across the scales

Ronan HENRY a*, Aurélien DOITRAND b, Sylvain MEILLE b

## Keywords

Micro-cantilever, bending test, alumina, size-effect, strength, multi-scale

## Abstract

The strength of sapphire across several length scales has been studied by micro-cantilever bending tests on specimens with the a-plane perpendicular to the main loading direction. The combined machining of two types of FIB (Ga and plasma Xe) is used to produce specimens with a wide range of dimensions, with specimen heights ranging from a few microns to several tens of microns. All specimens exhibit a fracture surface along m-planes. These micro-bending strength results are compared with macroscopic bending results obtained on the same sample with the same orientation. A marked size effect is observed on the measured strength, ranging from around 20 GPa at the micrometric scale down to 650 MPa for macroscopic specimens. A comparison with data from literature on sapphire samples of different size is made. Statistical models are found unable to capture the size effect on the entire range of sample sizes.

## 1. Introduction

Micromechanical tests are useful for measuring local mechanical properties [1]. Bending tests are particularly used because they are technically relatively easy to perform in terms of specimen fabrication and loading [2]. They are typically well suited for probing small objects, such as coatings [3][4], but are also used to test in tension individual elements of the microstructure in order to understand the overall behavior of the material [5][6][7]. One classic method used is to fabricate micro-cantilevers with a Focused Ion Beam (FIB) and to test them with a nano-indenter. Most FIBs use Ga ions to produce specimens a few microns wide. The less common use of a FIB with plasma technology (e.g., Xe ions) allows the fabrication of larger specimens, typically several tens of microns wide.

The evaluation of brittle material fracture toughness based on micromechanical testing of notched specimens may be representative of the material fracture toughness measured at the macroscopic scale [8]. However, the bending strength of plain specimens is clearly subject to a size effect, i.e. the measured flexural fracture stress is much higher on micro-specimens than the one measured on macroscopic samples of the same material [9]. As a result, this value cannot be used for dimensioning purposes, for example, because it is not representative of a macroscopic part. In many cases, the fracture size effect of brittle materials is studied using a statistical approach: a larger probed volume is statistically more likely to contain a larger critical defect, leading to premature failure. This statistical size effect, derived from Weibull statistics [10] and widely used in the brittle materials community, helps to explain the higher fracture stresses measured on smaller specimens [11]. However, an

alternative explanation may come from a deterministic size effect [12] and these two approaches may in fact complement each other [13]. This size effect can also be related to a lack of representativeness in the microstructure, for instance in polycrystals, for which millimeter scale specimens contain thousands of grains whereas sub-grain scale specimens can be tested by FIB. Moreover, only few experimental studies cover a wide range of dimensions in a given brittle and crystalline material to evaluate the fracture size effect and eventually the possible influence of defects on it.

Therefore, the objective of this work is to study the fracture of monocrystalline sapphire specimens from the micron scale (manufactured by FIB) to the millimeter scale (produced by conventional machining): the results of these macroscopic tests have already been published in the literature [14]. We choose to work on a single crystal in order to have the same microstructure whatever the scale studied and few defects induced by the shaping of the sample (e.g. sintering porosity). We also choose sapphire because it is available and extensively studied, including through micromechanical testing. Since sapphire is an anisotropic crystal, we systematically select the same crystalline orientation whatever the scale, and test it in bending.

## 2. Materials et methods

### 2.1. Sapphire sample

Sapphire is corundum, the most stable phase of alumina $Al_2O_3$, and is known for its great hardness. It crystallizes in a rhombohedral system and can be described by a hexagonal lattice. All the samples are obtained from a sapphire substrate in the shape of a 0.8 mm thick disc supplied by Saint-Gobain Lumilog (Vallauris, France). The

<footnote>a Normandie Univ, CNRS, Univ Rouen Normandie, INSA Rouen Normandie, GPM UMR 6634, F-76000 Rouen, France</footnote>

<footnote>b Univ Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS UMR5510, MATEIS, F-69621 Villeurbanne, France</footnote>

<footnote>* Corresponding author: ronan.henry@cnrs.fr</footnote>
crystal is oriented so that the c-plane (0001) corresponds to the surface of the single-crystal wafer. The disc is mirror-polished by successive mechanical polishing before cutting. All the bending specimens fabricated and tested in this work are oriented along the [1120] direction, so that an a-type plane is tested and is perpendicular to the specimen's neutral plane (Figure 1). The reason for this choice is the desire to use exactly the same conditions as for macroscopic tests, in order to study the effect of size on fracture over a very wide range of dimensions [14].

<figure><img src="image_2.png" /><figcaption>Figure 1: Descriptive schematic of the orientation of a micro-bending specimen showing the different parameters and the associated crystalline orientation. The beam is embedded in the surface on the left. An (x,y,z) axis system fixed to the specimen is shown for a better understanding of the orientation in the other figures.</figcaption></figure>

## 2.2. Specimen preparation

A wide range of specimen sizes are prepared and tested, from a few microns in height and width to several tens of microns (Figure 2 – (a,b)). These micro-specimens are fabricated using a dual-beam SEM/FIB microscope. Since sapphire is a poor electrical conductor, the sample is coated with approximately 10 nm of carbon using a PVD deposition system, and silver paint is used to improve charge evacuation towards the microscope stage. Specimens in the shape of cantilever beams with a rectangular cross-section are manufactured. They are therefore positioned on the edge of the sample so that the bottom cut can be made with the ion beam after the sample is positioned on a pre-tilted stub at 90°. The smallest specimens are made using a FIB with a Ga liquid metal source (Helios 5UX, ThermoFisher Scientific) and the largest ones using a plasma source and Xe ions (Helios5 PFIB CXe, ThermoFisher Scientific). The ion acceleration voltage is always 30 kV, but because the dimensions of each specimen vary considerably, the milling currents used to produce each sample are different. However, the overall procedure is the same, i.e. a maximum ion current is first used to clear the preform and make the cut underneath. The currents are then progressively reduced to refine and polish all the surfaces to produce a sample with smoother surfaces. To compensate for the gaussian shape of the ion beam, an over-tilt angle is used during the refining stage, depending on the type of FIB and current used.

## 2.3. Mechanical testing

Cantilever bending of FIB-prepared micro-specimens is performed with a nano-indenter (PI87R, Brucker) in situ

in a SEM (XB540, Zeiss). A cube corner tip is used to ensure precise positioning of the load applied to the specimens. As the tested material is rigid and brittle, the fact that the tip slightly penetrates the surface is not a problem since only the maximum load is used to calculate the strength. Tests are displacement-controlled, with a loading speed corresponding to a bending strain rate of approximately 0.003 s⁻¹. A wide range of specimen sizes are produced, from 1 µm to 47 µm in height, 0.75 µm to 39 µm in width and 4 µm to 320 µm in length. To calculate the flexural strength σ<sub>F</sub> in micro-bending, the maximum recorded load F<sub>c</sub>, which also corresponds to the fracture force, is used based on beam theory (equation 1).

$$ \sigma_F = F_c \frac{6(L - L_0)}{BW^2} \quad (1) $$

In this equation, W is the height of the beam, B is its width, L is the distance between the point of the force application and the embedment and L<sub>0</sub> is the minimum distance between the fracture plane and the embedment. All the dimensions, measurements and calculations of the specimens are shown in Table 1.

<table><caption>Table 1: Geometry parameters (as indicated in Figure 1), failure force, fracture stresses and calculated critical defect size from all the tested micro-cantilevers.</caption><thead><tr><th>Ion</th><th>N°</th><th>W (μm)</th><th>B (μm)</th><th>L (μm)</th><th>L<sub>0</sub> (μm)</th><th>F<sub>c</sub> (mN)</th><th>σ<sub>F</sub> (GPa)</th><th>a<sub>c</sub> (nm)</th></tr></thead><tbody><tr><td>Ga</td><td>1</td><td>1.0</td><td>0.8</td><td>4.1</td><td>0.4</td><td>0.72</td><td>20.0</td><td>11</td></tr><tr><td>Ga</td><td>2</td><td>1.5</td><td>1.1</td><td>5.1</td><td>0.5</td><td>1.4</td><td>15.6</td><td>19</td></tr><tr><td>Ga</td><td>3</td><td>1.6</td><td>0.9</td><td>4.7</td><td>0.2</td><td>1.4</td><td>16.4</td><td>17</td></tr><tr><td>Ga</td><td>4</td><td>2.1</td><td>1.9</td><td>21.5</td><td>3.5</td><td>0.90</td><td>11.6</td><td>34</td></tr><tr><td>Ga</td><td>5</td><td>2.4</td><td>2.3</td><td>8.8</td><td>1.1</td><td>4.4</td><td>15.3</td><td>19</td></tr><tr><td>Ga</td><td>6</td><td>2.5</td><td>1.1</td><td>10.1</td><td>0.3</td><td>0.90</td><td>7.7</td><td>77</td></tr><tr><td>Ga</td><td>7</td><td>2.6</td><td>2.2</td><td>12.5</td><td>0.4</td><td>2.6</td><td>12.7</td><td>28</td></tr><tr><td>Ga</td><td>8</td><td>2.6</td><td>2.4</td><td>9.6</td><td>0.0</td><td>5.3</td><td>18.8</td><td>13</td></tr><tr><td>Ga</td><td>9</td><td>2.7</td><td>2.9</td><td>11.7</td><td>0.0</td><td>4.9</td><td>16.3</td><td>17</td></tr><tr><td>Ga</td><td>10</td><td>2.7</td><td>3.6</td><td>21.6</td><td>0.4</td><td>3.6</td><td>17.4</td><td>15</td></tr><tr><td>Xe</td><td>11</td><td>3.2</td><td>3.7</td><td>22.8</td><td>1.3</td><td>5.5</td><td>18.7</td><td>13</td></tr><tr><td>Ga</td><td>12</td><td>4.3</td><td>4.0</td><td>29.0</td><td>2.0</td><td>8.3</td><td>18.2</td><td>14</td></tr><tr><td>Ga</td><td>13</td><td>4.3</td><td>4.4</td><td>27.0</td><td>0.0</td><td>6.1</td><td>12.1</td><td>31</td></tr><tr><td>Ga</td><td>14</td><td>4.5</td><td>3.2</td><td>17.4</td><td>3.8</td><td>6.8</td><td>8.6</td><td>62</td></tr><tr><td>Xe</td><td>15</td><td>6.6</td><td>10.3</td><td>77</td><td>0.0</td><td>14.7</td><td>15.1</td><td>20</td></tr><tr><td>Xe</td><td>16</td><td>12.7</td><td>26.0</td><td>172</td><td>18.0</td><td>27.5</td><td>6.1</td><td>124</td></tr><tr><td>Xe</td><td>17</td><td>15.4</td><td>16.5</td><td>134</td><td>3.0</td><td>27.5</td><td>5.5</td><td>150</td></tr><tr><td>Ga</td><td>18</td><td>12.8</td><td>5.6</td><td>53</td><td>27.3</td><td>16.7</td><td>2.8</td><td>580</td></tr><tr><td>Xe</td><td>19</td><td>17.0</td><td>16.1</td><td>47.0</td><td>0.0</td><td>49.3</td><td>3.0</td><td>512</td></tr><tr><td>Xe</td><td>20</td><td>19.2</td><td>22.3</td><td>145</td><td>20.0</td><td>62.0</td><td>5.7</td><td>143</td></tr><tr><td>Xe</td><td>21</td><td>21.4</td><td>10.2</td><td>206</td><td>15.7</td><td>19.2</td><td>4.7</td><td>207</td></tr><tr><td>Xe</td><td>22</td><td>21.5</td><td>15.5</td><td>44.0</td><td>0.0</td><td>75.0</td><td>2.8</td><td>598</td></tr><tr><td>Xe</td><td>23</td><td>32.0</td><td>39.0</td><td>198</td><td>32.0</td><td>192.0</td><td>4.8</td><td>199</td></tr><tr><td>Xe</td><td>24</td><td>35.2</td><td>37.3</td><td>316</td><td>0.0</td><td>81.0</td><td>3.3</td><td>414</td></tr><tr><td>Xe</td><td>25</td><td>47.0</td><td>15.0</td><td>175</td><td>0.0</td><td>95.0</td><td>3.0</td><td>504</td></tr></tbody></table>

Millimeter samples (Figure 2 - c) prepared by mechanical cutting using a diamond saw with were made from the same sapphire disc and then tested in four-point bending (the results have already been published in the literature [14]). In this paper, a fracture model is applied to a set of macroscopic specimens. The stresses at fracture measured ranged from 540 MPa to 703 MPa. The results obtained are compared with those of the present study.
<figure><img src="image_3.png" /><figcaption>Figure 2: Example of a bending test specimen (a) cantilevers prepared by FIB Ga, (b) cantilevers prepared by FIB Xe, (c) four-point bending specimen prepared by mechanical cutting (adapted from [14]).</figcaption></figure>

### 3. Fracture size effect

All the specimens showed brittle fracture, with no sign of plasticity. All the load curves, once corrected for the penetration of the indentation tip into the beam [15], seem linear up to failure (Figure 3).

<figure><img src="image_4.png" /><figcaption>Figure 3: Representative examples of stress-strain curves for three specimens of different sizes. The curves are correction for the penetration of the indent into the material and appears linear up to failure.</figcaption></figure>

Over the whole range of dimensions, the fracture planes are not perpendicular to the neutral axis of the beam, as might be expected in an isotropic material. The directions perpendicular to the fracture surfaces rather present an angle of 30° with this axis (Figure 4). This was also observed on macroscopic four-point bending specimens [14]. On several micro-cantilevers after fracture, a line is present at about 30 degrees from the axis of the beam, therefore with the same angle as the cracking plane (see red arrows on Figure 4 and Figure 5).

As the fracture plane is not straight and we do not know a priori where initiation took place, it is assumed that failure occurs at the point closest to the embedment, where the stress is at its greatest. $L_0$ is therefore measured for the side of the fracture plane closest to the embedment (Figure 1). For each dimension measurement ($W, B, L$ and $L_0$), we estimated the measurement uncertainty by considering manufacturing defects: the faces of the specimen may not be straight, may be asymmetrical or may show redeposition induced by the FIB. When these uncertainties are propagated to the stress calculation, the uncertainty in the fracture stress values is estimated at 20%, not least because sapphire is a difficult material to machine correctly with FIB.

<figure><img src="image_5.png" /><figcaption>Figure 4: Fracture planes of specimens (top view) showing a plane whose normal make a 30° angle with respect to the beam neutral axis. The sample's width is (a) 4.4 µm, (b) 16.1 µm and (c) 2.45 mm (from [14]). The red arrow in Fig 4b indicates the line visible after the beam fracture of this particular sample, presenting an angle with the beam axis close to that of the fracture plane (approximately 30°).</figcaption></figure>
For all the tested micro-bending specimens, it is not possible to identify defects that clearly have initiated the fracture. However, because the fracture surface is not straight and has some degree of roughness, it is not easy to obtain very good quality images of the fracture surface (Figure 5).

For macroscopic specimens [14], there are relatively large defects induced by the cutting process at the edges of the samples. The fracture initiating defect can be identified thanks to the characteristic facies around this defect, with a defect size of up to 50 µm.

<figure><img src="image_6.png" /><figcaption>Figure 5: SEM (front view) image of the fracture surface of a micro-bending specimen after fracture (a) small micro-cantilever prepared by Ga FIB, (b) small micro-cantilever prepared by Xe FIB. The red arrows indicate the lines visible after the beam fracture, presenting an angle with the beam axis close to that of the rupture plane (approximately 30°)</figcaption></figure>

Brittle material such as sapphire fails due to crack initiation and propagation mainly loaded under tension and from a flaw in the material, therefore the bending strength may depend on the volume of the material loaded under a tensile stress.

The size of the defect $a_c$ that caused the specimen fracture can be estimated from the value of the bending strength $\sigma_F$ and of the critical intensity factor $K_{IC}$. Using Irwin's relation [16] based on Griffith's criterion [17] for brittle fracture, with $Y$ the shape factor, taken here to be $0.66\sqrt{\pi}$ [18] and $K_{IC} = 2.5$ MPa.m$^{0.5}$ for alumina single crystal [18], $a_c$ can be estimated from (2):

$$ \sigma_F = \frac{K_{IC}}{Y} \frac{1}{\sqrt{a_c}} \qquad (2) $$

It should be noted that this approach has to be cautiously used as it is only theoretically valid for studying the propagation of a sufficiently large crack in an infinite medium. Thus, it does not consider the actual defect shape or location within the specimen, and does not provide an accurate estimate of the failure stress if the defect is smaller than the Irwin's length. We recall that the Irwin's length is defined as: $\ell_{mat} = \frac{K_{IC}^2}{\sigma_c^2}$, where $\sigma_c$ is the strength of a sufficiently large specimen without external defect. For instance, in the case of Feilden's work on sapphire platelets, $\sigma_c$ was found to be 1.1 GPa [34]. For each specimen, the theoretical defect sizes obtained using this approach are estimated and given in Table 1, ranging from 10 nm for the smallest specimens to around 500 nm for the largest one. For macroscopic samples c, these calculations give defect sizes ranging between 5 µm to 15 µm.

## 4. Discussions

The apparent fracture stress values measured in this work are subject to a significant size effect (Figure 6), i.e. the bending strength value measured depends on the dimensions of the specimen tested (here the characteristic dimension $W/2$, which corresponds to the height of the part in tension).

<figure><img src="image_7.png" /></figure>
Figure 6: Apparent strength (graph in log-log) of monocrystalline alumina as a function of the characteristic dimension W/2 of the specimen in the same sample of sapphire in the same crystallographic orientation (red), for the micro-cantilevers of this work (circles) and macroscopic 4PB (diamond) [14] and for those found in the literature (squares). The error bars represent the uncertainties resulting from a calculation and estimated at 20% of the stress value. F4P means Four-Point Bending and B3B means Ball-on-3-Balls.

## Plasticity at micro-scale

Several authors have shown that when brittle materials are tested by micromechanics, plasticity can occur, even at room temperature. In compression, it is easier to promote plasticity [19], and some examples also exist when the material is stressed in tension or bending [20][21]. When the material has no obvious defects (and therefore statistically when the size of the specimens is reduced) acting as stresses concentrators, the material is more likely to plasticize before fracture. It has been shown that for a brittle material such as monocrystalline silicon nitride, plasticity effects are clearly visible on the bending curves, but also by visualizing the dislocations by TEM, and that these effects are strongly dependent on the crystalline orientation tested [22].

However, in all the stress-strain curves obtained in this work over the whole range of micro-sample sizes tested (Figure 3), and despite the fact that the single-crystal material has no obvious defect visible by the SEM, no deviation from linearity is found. This result may, however, be influenced by the chosen loading speed, the orientation of the crystal chosen to correspond to the one used for macroscopic bending tests [14], or to the nature of the tested sample (sapphire), which may lead to unfavorable plasticity.

## Sapphire strength data

Sapphire is a fairly well-documented material in the literature, with a wide range of specimen dimensions tested at fracture.

Numerous micromechanical tests have been carried out on single crystal of alumina. Crystalline orientations can be different, so the fracture planes can be also different depending on the reference. As all the mechanical tests are not of the same type, we have extracted a characteristic length W/2 in order to be able to compare them (Figure 6). Stresses as high as 39.1 GPa have been measured by Wang *et al.* [23] by loop test for nano-whiskers with almost no defects, which is quite close to the theoretical strength of the crystal, estimated to be around $\frac{E}{10} \approx 46$ GPa. Pejchal *et al.* [18] used micrometric C-shape tests on alumina particles which gave values of around 10 GPa on alumina single crystal with no apparent defect, but fall to around 5 GPa when pores or grain boundaries are present. So, the strength values around 10 GPa are rather consistent with our results, with the same order of magnitude for the failure stress scattering. Another study by Feilden *et al.* [24] gave strengths between 3 GPa and 7 GPa, with an average of 5.3 GPa obtained by three-point micro-bending tests using micrometric specimens on alumina single crystal platelets. The measured failure stresses seem to be smaller than the ones obtained with our specimens, but the method used to produce these samples differs considerably from the processes used to grow single crystals (these platelets are processed from nanoparticles of an aluminum precursor mixed with diluent and heat treated to form alpha alumina platelets). Also, it may be explained by the presence of internal pores and edge defects in the tested alumina platelets [7][24]. Other authors have also shown that there is a significant size effect on the tensile strength of whiskers, with a power law dependency between the whiskers diameter and their tensile strength [25][26]. In the case of polished whiskers (flame or chemically), the exponent of the power law is reduced, showing that the variation in tensile strength is influenced by the presence or absence of surface defects on the whiskers tested.

Many results are also available at a macroscopic scale. Muzy *et al.* [27] tested two types of specimen: 4-point bending (4PB) and ball-on-three-balls (3B3) on small disks, the latter making it possible to largely decrease the influence of the sample edges (such as cutting defects, for example) on the fracture strength. For 2 mm thick 4PB specimens, they found strengths of 490 ± 94 MPa and 450 ± 83 MPa for two different manufacturing processes. Using 3B3 specimens, they found much higher values of 1840 ± 630 MPa and 1740 ± 720 MPa, showing that the apparent strength is much higher when the influence of edge defects is lowered. Graça *et al.* [28], for 4PB tests on 3x4x45 mm³ specimens, measured fracture stresses of 591 ± 156 MPa and 726 ± 254 MPa respectively on a-planes and m-planes. These macroscopic results are fairly consistent with 4-point bending results of [14], with a-plane strengths of the same order of magnitude, i.e. 650 ± 80 MPa and 840 ± 80 MPa for the two tested sizes [14]. The difference in strength compared with the literature may be due to the variations in dimensions, or to defects produced by the cutting, or even possibly by the sapphire manufacturing process.

## Fracture planes

The direction of the fracture plane appears to be influenced by the crystallography of the crystal. The plane under tension was an a-plane (11̅20) because it was perpendicular to the beam neutral plane, yet the fracture plane mainly follows a m-plane (1̅1̅100) oriented at 30° with respect to the a-plane (Figure 1). The expected cleavage plane is the a-plane [29][30], although the fracture energy are fairly close between a- and m-planes (around 6 - 7 J/m² [29][31]). One possibility is that a a-type plane initiates the fracture and then further propagation takes place on an m-type plane, as suggested by the small ligament visible on the macroscopic specimens (Figure 4 - c). In all cases, as all the specimens for all sizes were tested in the
same crystallographic direction, they are directly comparable with each other, and in any case highlight the size effect on fracture even if the underlying fracture mechanism potentially differs between the scales.

Csanàdi et al. [32] have also worked on microcantilever bending on a brittle material (ZrB₂) with a hexagonal crystal lattice. In some orientations of the tested grains, they have noted clear traces of plasticity on the bending curves, and also some stripes appearing at the base of the beam fracture surface. They attribute these lines to slip lines. These stripes strongly resemble the stripes seen on some cantilevers (Figure 4), although in our case these lines appear to be more like cracks. In the end, they could be slip lines that turn into cracks and then lead to fracture. Given the orientation of the fracture surfaces and these lines, they would a priori correspond to a prismatic plane, i.e. m-plans.

## Scatter in the experimental data

If we perform a Weibull analysis on the macro flexural data set with visible edge defects [14], we find a Weibull modulus *m* of 10. In the literature, statistical studies show that the Weibull's modulus for sapphire is around 6 for macroscopic bending tests [27], and in view of the typical standard deviation of this parameter [11], these results are fairly consistent.

By reducing the sample size, we avoid the largest flaws in the sample. One might therefore expect a smaller scattering of data at the micro-bending scale. However, the results of this work are still highly scattered (Table 1)(Figure 6). Such a dispersion for a given specimen size seems to correspond to a fracture driven by a defect distribution. Nevertheless, no surface defects or porosities are observed in the micro-cantilever beams, which raises the question whether the scattering could be explained by crystal defects such as different dislocation densities in the specimens. There are studies in the literature that show the presence of defects of nanometric dimensions difficult to observe with SEM: dislocations (whose density depends on the manufacturing process), or even local disorientations of the crystal that can form during crystal growth [28][27]. In any case, these defects are very small compared to our specimens, even for the smallest micro-bending test. One possible explanation is also that sapphire is not an easy material to mill with the FIB (charging effects), and so the dispersion observed may be due, at least in part, to manufacturing or testing imperfections (such as, e.g., asymmetry, misalignments).

## Micro-bending of FIB samples

As ions are used to prepare the micro-cantilevers with FIB machining, it could influence the local properties of the sapphire (e.g., amorphisation, ion implantation, residual stress). In fact, this is an issue that is frequently discussed in the literature, particularly for notched specimens, which are more likely to be influenced by the effects of FIB at the notch tip [31][34][35]. For plain specimens, FIB-milled surfaces are smooth and, in principle, the tensile surface of the specimens is not much affected by the ion beam. However, the ion beam may affect this surface to some extent. A priori, any modification is influenced by the ions used and their energy (and therefore the milling voltage applied). In this case, all the FIB machining is carried out at 30 kV, so we can imagine that the size and extent of the changes that can be caused by the FIB beam are the same for all the samples prepared. Specimens prepared with Xe ions and Ga ions appear to show some overlap in terms of sample size, with no real change in trend of strength versus size (Figure 6). In addition, other results in the literature (Figure 6) without FIB preparation also show this increase in strength with a decrease in specimen size [25][26]. Also the work of Pejchal et al. [18] shows that specimens with a surface obtained by FIB that initiates fracture give similar strength values to specimens without a surface obtained by FIB. In fact, even if the FIB slightly changes the local properties of the material, this should have a limited influence on our tests, and on the effect of specimen size on strength.

## Size effect

The decrease in strength with specimen height appears to follow roughly a power law evolving in $W^{-0.5}$ for our set of data, and this trend is approximately maintained for different results found in the literature taken as a whole (Figure 6). However, there are some discrepancies, probably because the distribution of defects in sapphire differs from one study to another. This size effect on strength is generally only explained by the defects in the material that initiate the fracture. In the absence of a defect in the single crystal, we should reach the theoretical fracture stress of the crystal at around 46 GPa (one tenth of Young's modulus), which is almost reached for nanometric sapphire fibers [23]. If we now only consider the tests with no apparent defects such as porosities or flakes, i.e. our micrometric tests, loop tests on alumina fibers [23], and the tests taken from [27] for macroscopic 3B3 bending tests, we still observe a size effect on strength, but the slope seems different and closer to $W^{-0.3}$.

## Origin of the size effect

The analysis in terms of linear elastic fracture mechanics (Equation (2)) gives fairly consistent results for macroscopic specimens, since this simple model gives a calculated critical defect size of about 15 µm; this type of surface defect is actually observed on the fracture surface using SEM [14]. At the macroscopic scale, the visible flaws that lead to fracture are identified as surface defects, generally induced by the preparation, such as scratches [27] or chips produced during cutting [14]. On the other hand, for micro-cantilevers, the range of calculated defect sizes is between 10 nm and 500 nm (Table 1), and this time these defects are not observed on the fracture surface. For the largest micro-cantilevers, a defect of 500 nm in diameter
should be easily visible with a SEM, but no defect initiating fracture is identified here. A similar conclusion has already been noted on plain micro-specimens prepared in another brittle material [9]. This is the reason why linear elastic fracture mechanics is not directly applicable since it is able to study the propagation of a sufficiently long crack, but fails to predict its initiation. However, these statements need to be tempered because the fracture surfaces in micro-bending tests are not perfectly straight and smooth (Figure 5), making it difficult to visualize small defects, and also because Equation (2) is valid for an isolated defect of negligible size compared to the specimen.

Yamaguchi et al. [20] has shown that plasticity can occur in micro-cantilevers in single grain of brittle materials, leading to areas with dislocations in the tensile region. These high-density dislocation zones can then behave like defects and initiate fracture. When the specimen size is reduced, the size of this zone decreases, resulting in a higher fracture stress. This conclusion is consistent with the results shown here; however, we were unable to demonstrate the presence of plasticity in the sapphire studied.

## Modelling of size effect

The Weibull theory [10] is commonly used to explain variations in specimen strength with variations in specimen size; a smaller volume of material means less chance of a defect initiating failure, and therefore a higher average strength.

Using a Weibull strength scaling model with effective volumes $V_{eff}$ calculated specifically for four-point bending and micro-cantilevers [33], the strengths expected $\sigma_F^2$ by this model can be calculated by equation (3) for any other effective volume. The values of the Weibull modulus $m$ and the reference value of $\sigma_F^1$ measured for an effective volume $V_{eff}^1$ were determined on at the macroscopic samples [14].

$$ \sigma_F^2 = \sigma_F^1 \left( \frac{V_{eff}^1}{V_{eff}^2} \right)^{\frac{1}{m}} \qquad (3) $$

This strength calculations are compared with the experimental results as a function of the effective volume of the tested specimens (Figure 7).

<figure><img src="image_8.png" /><figcaption>Figure 7: Comparison of experimental fracture stresses as a function of effective volume calculated by Weibull scaling model.</figcaption></figure>

In our case, this does not fit well with our experiments on microcantilevers (Figure 7). It should also be noted that the distribution of micro-bending results is very scattered. This approach is perhaps not really appropriate here, as it covers two very different dimensional ranges. Zagar et al. [36] also found similar results by carrying out fracture tests on polycrystalline alumina fibers on two scales of specimen dimensions, millimeter tensile and micrometer flexural, and showing that the Weibull scale model does not transfer well from one scale to the other in this case.

Another approach to analyze the size effect in strength is to use deterministic mechanical models [37][38][39][40][41]. These models describe a transition between an energy-controlled failure stress at small scale and a strength-controlled failure stress at large scale. They provide a decreasing strength with increasing specimen size up to an asymptotic value theoretically corresponding to $\sigma_c$. The transition between these two regimes occurs for characteristic specimen size around $\ell_{mat}$. For the experiments in this study, the Irwin's length ranges from a few microns to a few tens of microns [14]. This is precisely the size range for which experimental data are lacking. Additional experimental effort is needed to evaluate the ability of these models to reproduce the experimental size effect.
**5. Conclusions**

Brittle fracture under bending is observed for sapphire plain specimens over several length scales, from a few microns (FIB-prepared specimens) to nearly 1 mm in height (mechanically-cut specimens) [14]. Although the sapphire a-plane is perpendicular to the main loading direction, fracture mainly occurs along m-plane, which shows a lower strength. The measured flexural fracture stress shows a pronounced size effect, ranging from about 20 GPa for micrometer specimens to about 650 MPa for millimeter specimens [14], i.e. the strength decreases by around thirty times. Although clearly visible fracture initiating defects are present in millimeter specimens, SEM observations of the micro-cantilever fracture surfaces do not reveal any significant flaws. As the scatter in the micro-cantilever data is important, fracture seems to be driven by the presence of defects. However, a Weibull scaling model does not describe well the evolution of fracture strength for the entire range of sample size. A possible improvement could be obtained considering mechanical deterministic models. In this case, additional experiments in the intermediate size range of tens of microns would be required. Considering this data set, collected on a single-crystal sample with few intrinsic defects, it is clear that plain micro-bending tests should be used with caution, as the measured strength values are subject to a size effect and are therefore not directly representative of the material strength at another scale of investigation.

**Declaration of interest**

None

**CRediT authorship**

**Ronan Henry:** Writing – original draft, Investigation, Methodology, Data curation. **Aurélien Doitrand:** Writing – review & editing, formal analysis. **Meille Sylvain:** Writing – review & editing, Conceptualization.

**Acknowledgements**

SEM for in-situ mechanical experiments have been carried out on the GENESIS facility which is supported by the Region Normandie, the Metropole Rouen Normandie, the CNRS via LABEX EMC3 and the French National Research Agency as a part of the program "Investissements d'avenir" with the reference ANR-11-EQPX-0020.

**References**

[1] G. Dehm, B. N. Jaya, R. Raghavan, and C. Kirchlechner, "Overview on micro- and nanomechanical testing: New insights in interface plasticity and fracture at small length scales," *Acta Mater.*, 2017, doi: 10.1016/j.actamat.2017.06.019.

[2] D. Di Maio and S. G. Roberts, "Measuring fracture toughness of coatings using focused-ion-beam-machined microbeams," *J. Mater. Res.*, vol. 20, no. 02, pp. 299-302, 2005, doi: 10.1557/JMR.2005.0048.

[3] K. Matoy et al., "A comparative micro-cantilever study of the mechanical behavior of silicon based passivation films," *Thin Solid Films*, vol. 518, no. 1, pp. 247-256, 2009, doi: 10.1016/j.tsf.2009.07.143.

[4] J. Schaufler, C. Schmid, K. Durst, and M. Göken, "Determination of the interfacial strength and fracture toughness of a-C:H coatings by in-situ microcantilever bending," *Thin Solid Films*, vol. 522, pp. 480-484, 2012, doi: 10.1016/j.tsf.2012.08.031.

[5] J. Schlacher, T. Csanádi, M. Vojtko, R. Papšik, and R. Bermejo, "Micro-scale fracture toughness of textured alumina ceramics," *J. Eur. Ceram. Soc.*, vol. 43, no. 7, pp. 2943-2950, 2023, doi: 10.1016/j.jeurceramsoc.2022.06.028.

[6] J. Tatami et al., "Local fracture toughness of Si3N4 ceramics measured using single-edge notched microcantilever beam specimens," *J. Am. Ceram. Soc.*, vol. 98, no. 3, pp. 965-971, 2015, doi: 10.1111/jace.13391.

[7] R. Henry, H. Saad, A. Doitrand, S. Deville, and S. Meille, "Interface failure in nacre-like alumina," *J. Eur. Ceram. Soc.*, vol. 40, no. 13, pp. 4694-4699, 2020, doi: 10.1016/j.jeurceramsoc.2020.05.068.

[8] B. N. Jaya, C. Kirchlechner, and G. Dehm, "Can microscale fracture tests provide reliable fracture toughness values? A case study in silicon," *J. Mater. Res.*, vol. 30, no. 5, 2015, doi: 10.1557/jmr.2015.2.

[9] R. Henry et al., "Irradiation effects on the fracture properties of UO2 fuels studied by micro-mechanical testing," *J. Nucl. Mater.*, vol. 536, no. 152179, 2020, doi: 10.1016/j.jnucmat.2020.152179.

[10] W. Weibull, "A statistical theory of the strength of materials," *Proc. R. Swedish Inst. Eng. Res.*, vol. 151, 1939.

[11] R. Danzer, "On the relationship between ceramic strength and the requirements for mechanical design," *J. Eur. Ceram. Soc.*, vol. 34, no. 15, pp. 3435-3460, 2014, doi: 10.1016/j.jeurceramsoc.2014.04.026.

[12] Z. P. Bazant, "Size effect on structural strength: a review," *Arch. Appl. Mech.*, vol. 69, pp. 703-725, 1999, doi: 10.1007/s004190050252.

[13] D. Leguillon, É. Martin, and M. C. Lafarie-Frenot, "Flexural vs. tensile strength in brittle materials," *Comptes Rendus - Mec.*, vol. 343, no. 4, pp. 275-281, 2015, doi: 10.1016/j.crme.2015.02.003.

[14] A. Doitrand, R. Henry, and S. Meille, "Explanation of edge defect influence on sapphire bending strength scatter using the coupled criterion," *J. Eur. Ceram. Soc.*, vol. 45, no. 4, p. 117052, 2025, doi: 10.1016/j.jeurceramsoc.2024.117052.

[15] E. Camposilvan, O. Torrents, and M. Anglada, "Small-scale mechanical behavior of zirconia," *Acta Mater.*, vol. 80, pp. 239-249, 2014, doi: 10.1016/j.actamat.2014.07.053.
[16] G. R. Irwin, "Fracture," in *Flügge, S. (eds) Elasticity and Plasticity / Elastizität und Plastizität. Handbuch der Physik / Encyclopedia of Physics, vol 3 / 6*. Springer, Berlin, Heidelberg, 1958.

[17] A. A. Griffith, "The phenomena of rupture and flow in solids," *Philos. Trans. R. Soc. London*, vol. 221, pp. 163-198, 1921, doi: 10.1098/rsta.1921.0006.

[18] V. Pejchal, M. Fornabaio, G. Žagar, and A. Mortensen, "The local strength of individual alumina particles," *J. Mech. Phys. Solids*, vol. 109, pp. 34-49, 2017, doi: 10.1016/j.jmps.2017.08.005.

[19] H. Masuda, "Orientation-dependent plasticity in microcompression of oxide single crystals," no. January, pp. 1-16, 2025, doi: 10.1111/jace.20476.

[20] H. Yamaguchi et al., "Dislocation-controlled microscopic mechanical phenomena in single crystal silicon under bending stress at room temperature," *J. Mater. Sci.*, vol. 55, no. 17, pp. 7359-7372, 2020, doi: 10.1007/s10853-020-04528-3.

[21] T. Yahagi et al., "Deformation Behavior and Fracture Strength of Single-Crystal 4H-SiC Determined by Microcantilever Bending Tests," *Adv. Eng. Mater.*, vol. 2400095, pp. 1-9, 2024, doi: 10.1002/adem.20240095.

[22] M. Tanabe et al., "Deformation behaviors and fracture strength of β-Si3N4 single crystals," *J. Am. Ceram. Soc.*, no. April, pp. 1-9, 2023, doi: 10.1111/jace.19167.

[23] S. Wang et al., "An improved loop test for experimentally approaching the intrinsic strength of alumina nanoscale whiskers," *Nanotechnology*, vol. 24, no. 28, 2013, doi: 10.1088/0957-4484/24/28/285703.

[24] E. Feilden et al., "Micromechanical strength of individual Al2O3 platelets," *Scr. Mater.*, vol. 131, pp. 55-58, 2017, doi: 10.1016/j.scriptamat.2017.01.008.

[25] R. F. Regester, P. D. Gorsuch, and L. Girifalco, "Structure and properties of aluminium oxide whiskers," *Mater. Res. Stand.*, pp. 957-960, 1967.

[26] P. D. Bayer and R. E. Cooper, "The effects of chemical polishing on the strength of sapphire whiskers," *J. Mater. Sci.*, vol. 2, no. 4, pp. 347-353, 1967, doi: 10.1007/BF00572418.

[27] J. Muzy et al., "Influence of growth process and crystal defects on sapphire brittleness," *J. Cryst. Growth*, vol. 618, no. April, 2023, doi: 10.1016/j.jcrysgro.2023.127327.

[28] S. Graça et al., "Influence of mosaicity on the fracture behavior of sapphire," *Acta Mater.*, vol. 67, pp. 67-80, 2014, doi: 10.1016/j.actamat.2013.12.004.

[29] A. Azhdari, S. Nemat-Nasser, and J. Rome, "Experimental observations and computational modeling of fracturing in an anisotropic brittle crystal (Sapphire)," *Int. J. Fract.*, vol. 94, no. 3, pp. 251-266, 1998, doi: 10.1023/A:1007565624473.

[30] S. M. Wiederhorn, "Fracture of Sapphire," *J. Am. Ceram. Soc.*, vol. 52, no. 9, pp. 485-491, 1969, doi: 10.1111/j.1151-2916.1969.tb09199.x.

[31] A. D. Norton, S. Falco, N. Young, J. Severs, and R. I. Todd, "Microcantilever investigation of fracture toughness and subcritical crack growth on the scale of the microstructure in Al2O3," *J. Eur. Ceram. Soc.*, vol. 35, no. 16, pp. 4521-4533, 2015, doi: 10.1016/j.jeurceramsoc.2015.08.023.

[32] T. Csanádi, A. Azizpour, M. Vojtko, and W. G. Fahrenholtz, "The effect of crystal anisotropy on fracture toughness and strength of ZrB2 microcantilevers," *J. Am. Ceram. Soc.*, vol. 107, no. 3, pp. 1669-1681, 2024, doi: 10.1111/jace.19359.

[33] G. D. Quinn, "Weibull strength scaling for standardized rectangular flexure specimens," *J. Am. Ceram. Soc.*, vol. 86, no. 3, pp. 508-510, 2003, doi: 10.1111/j.1151-2916.2003.tb03329.x.

[34] J. P. Best, J. Zechner, J. M. Wheeler, R. Schoeppner, M. Morstein, and J. Michler, "Small-scale fracture toughness of ceramic thin films: the effects of specimen geometry, ion beam notching and high temperature on chromium nitride toughness evaluation," *Philos. Mag.*, vol. 96, no. 32-34, pp. 3552-3569, 2016, doi: 10.1080/14786435.2016.1223891.

[35] J. P. Best et al., "A comparison of three different notching ions for small-scale fracture toughness measurement," *Scr. Mater.*, vol. 112, pp. 71-74, 2016, doi: 10.1016/j.scriptamat.2015.09.014.

[36] G. Žagar, V. Pejchal, M. G. Mueller, A. Rossoll, M. Cantoni, and A. Mortensen, "The local strength of microscopic alumina reinforcements," *Acta Mater.*, vol. 100, pp. 215-223, 2015, doi: 10.1016/j.actamat.2015.08.026.

[37] D. Leguillon, "Strength or toughness ? A criterion for crack onset at a notch," *Eur. J. Mech. A/Solids*, vol. 21, pp. 61-72, 2002, doi: 10.1016/S0997-7538(01)01184-6.

[38] Z. P. Bažant, "Size Effect in Blunt Fracture: Concrete, Rock, Metal," *J. Eng. Mech.*, vol. 110, no. 4, pp. 518-535, 1984, doi: 10.1061/(asce)0733-9399(1984)110:4(518).

[39] A. Carpinteri, B. Chiaia, and G. Ferro, "Size effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder," *Mater. Struct.*, vol. 28, no. 6, pp. 311-317, 1995, doi: 10.1007/BF02473145.

[40] M. Baldassari, A. Monaco, A. Sapora, and P. Cornetti, "Size effect on flexural strength of notched and un-notched concrete and rock specimens by Finite Fracture Mechanics," *Theor. Appl. Fract. Mech.*, vol. 125, no. November 2022, p. 103787, 2023, doi: 10.1016/j.tafmec.2023.103787.

[41] A. Doitrand, R. Henry, and S. Meille, "Brittle material strength and fracture toughness estimation from four-point bending test," *J. Theor. Comput. Appl. Mech.*, pp. 0-17, 2021, doi: 10.46298/jtcam.6753.