# Synchrotron X-ray 3D characterisation of fatigue crack initiation during in-situ torsion cyclic tests

Viet-Duc Le, Franck Morel, Nicolas Saintier, Pierre Osmond, Daniel Bellett,

Wolfgang Ludwig, Marta Majkut, Jean-Yves Buffiere

► To cite this version:

Viet-Duc Le, Franck Morel, Nicolas Saintier, Pierre Osmond, Daniel Bellett, et al. Synchrotron X-ray 3D characterisation of fatigue crack initiation during in-situ torsion cyclic tests. International Journal of Fatigue, 2025, 193, pp.108762. 10.1016/j.ijfatigue.2024.108762. hal-05122137

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# Synchrotron X-ray 3D characterisation of fatigue crack initiation during in-situ torsion cyclic tests

Viet-Duc Le<sup>aID*</sup>, Franck Morel<sup>a</sup>, Nicolas Saintier<sup>b</sup>, Pierre Osmond<sup>d</sup>, Daniel Bellett<sup>aID</sup>, Wolfgang Ludwig<sup>c</sup>, Marta Majkut<sup>e</sup>, Jean-Yves Buffiere<sup>c</sup>

## ARTICLE INFO

### Keywords:

Fatigue crack initiation

Cast aluminium alloy

Synchrotron

Tomography

Torsion

## ABSTRACT

This paper focuses on the characterisation of fatigue crack initiation mechanisms under fully reversed torsional loads for the porosity-containing cast AlSi7Mg0.3 aluminium alloy by synchrotron X-ray tomographic imaging and Diffraction Contrast Tomography (DCT). The aim is to analyse the relation between crack initiation and the microstructure, in the fatigue regime close to the fatigue limit of the material (fatigue lives of approximately $2 \times 10^6$ cycles). In-situ torsion fatigue tests have been conducted at the European Synchrotron Radiation Facility. A large number of cracks (approximately 80 cracks) were analysed, at the surface and in the bulk, highlighting the role of the surrounding microstructure on the crack initiation mechanisms.

Contrary to uniaxial loads in which fatigue crack initiation from casting pores in an opening mode (mode I) is generally only observed for this material, it is shown in this work that multiple crack initiations mechanisms can appear simultaneously under torsional loads. This is very interesting because it is possible to analyse different mechanisms (and the effect of the surrounding microstructure) using the same specimen, that is under identical loading conditions. More precisely, two different crack initiation mechanisms were observed. The first mechanism involves intra-granular fatigue crack initiation from Persistent Slip Bands (PSBs) formed on the slip system with the highest Schmid factor. For this mechanism, the grain orientation is the key parameter. A shear stress threshold of approximately 80 MPa is observed to trigger this mechanism. The presence of a pore, in the crack initiation zone reduces this threshold. The second mechanism concerns crack initiation from pores, in mode I, on planes of maximum principal stress. No link to the local grain orientation was found in that case. The pore size is the key factor governing this crack initiation mechanism. Regarding the competition between these mechanisms in the formation of the final crack, it was observed that at high applied stress, the cracks initiated from PSBs are dominant due to their high density and high growth rates resulting from coalescence. At lower stress, cracks initiated from pores are dominant, mainly because of the little effect of grain boundaries and eutectic zones, while most of cracks initiated from PSBs are arrested within the grain where they have initiated.

## 1. Introduction

Cast Al-Si alloys have been used for automotive applications for many years due to their high strength-to-weight ratio and their good castability. In the literature, the fatigue behaviour, including the damage mechanisms, of the cast Al-Si alloys under uniaxial loads has been intensively investigated [1–6]. It is widely admitted, for this loading condition, that the pore size is the first order factor controlling fatigue crack initiation. The influence of the other microstructural factors

including the orientation and size of alpha grains, the Dendrite Arms Spacing (DAS), the Secondary Dendrite Arms Spacing (SDAS), or the presence of inclusions, is much less pronounced. However, for torsional loads, the roles of the different microstructural heterogeneities are not fully understood. In torsion, the crack initiation mechanisms are more complex to analyse when compared to uniaxial loading conditions. The brief state-of-the-art given below summarises some observations reported in the literature.

<footnote><sup>a</sup>Arts et Metiers Institute of Technology, LAMPA, Angers, France</footnote>

<footnote><sup>b</sup>Arts et Metiers Institute of Technology, I2M, Bordeaux, France</footnote>

<footnote><sup>c</sup>INSA de Lyon, MATEIS, Lyon, France</footnote>

<footnote><sup>d</sup>CETIM, Nantes, France</footnote>

<footnote><sup>e</sup>ESRF, Grenoble, France</footnote>

<footnote>* Corresponding author.
E-mail address: viet-duc.le@ensam.eu (V.-D. Le).</footnote>
As previously mentioned, for uniaxial loads, cracks initiate principally from pores in an opening mode (or mode I) [7–9]. For torsional loads however, it was shown that two initiation mechanisms co-exist: cracks initiate from pores and from the aluminium matrix [1,8,10] either in an opening mode or in shearing modes (modes II and/or III) [11].

The influence of porosity on the torsional fatigue strength is less pronounced than for uniaxial loads [11–14]. The critical pore size $\sqrt{area_c}$ in the Kitagawa-Takahashi diagram (i.e. the pore size marking the decrease of the fatigue strength from the horizontal plateau corresponding to a defect-free material) is larger for torsional loads ($\sqrt{area_c} \approx 700$ µm, value obtained at a loading ratio R=-1) than for uniaxial loads ($\sqrt{area_c} \approx 100$ µm, obtained at R=-1). For uniaxial loading conditions, the effect of pore morphology is most of time considered to be negligible when compared to the effect of the pore size [8]. For torsional loads, no work was found in the literature addressing this aspect.

In cast hypo-eutectic aluminium alloys, the aluminium matrix is formed of alpha grains, containing at a lower scale secondary dendrite arms [11,15]. Serrano-Munoz et al. [16] investigated the effect of grain orientation on fatigue crack initiation under torsion loads for a cast aluminium alloy containing natural pores (with a maximum equivalent diameter of 57 µm). The authors observed that cracks systematically initiate in a shearing mode in the aluminium matrix without the presence of neighbouring pores. In most cases, cracks are formed on the slip system plane with the highest Taylor factor. This means therefore that grain orientation seems to be the first order factor that governs crack initiation from Persistent Slip Bands (PSBs). The same conclusion was also reported in our previous work [17] using the Schmid factor. However, the grain size effect was not addressed in these studies.

Regarding the role of the SDAS, in the work of Houria et al. [8], the authors observed that the fatigue strength decreases when the SDAS increases from 30 µm to 80 µm. The authors argued that, under torsional loads, cracks grow through the eutectic zones and silicium particles can play a role as microstructural barriers that tend to slow the crack growth. This argument seems to contradict the observation that eutectic particles accelerate crack initiation in grains with a high Schmid factor, as discussed by Le [17]. These conclusions remain, however, qualitative.

Serrano et al. [18] observed that the fracture surface shape under torsional loads depends on the load level. For high stress levels ($\tau_{max} > 105$ MPa), fracture occurs on the planes perpendicular to the specimen axis with crystallographic initiation sites, leading to macroscopically flat fracture surfaces. For low stress levels ($\tau_{max} < 85$ MPa), cracks propagate on planes at 45° with respect to the specimen axis in mode I and crystallographic initiation sites exhibit thick murky debris. The intermediate stress levels produce fracture surfaces with tortuous morphologies and the propagation is chaotic as cracks seem to indistinctly evolve in either opening or shearing modes.

Even though the presence of the two fatigue crack initiation mechanisms (from pores or PSBs) seems to be clearly established for torsional loads, the nature of the most efficient barrier for each of these mechanisms is still unclear. This issue is difficult to address experimentally. While surface crack initiation can be studied fairly reliably (by SEM or Optical microscopy), uncertainties concerning the sub-surface microstructure (shape of the grain, presence of a pore for example) prevent an exhaustive analysis. As mentioned above, fatigue crack initiation has been studied in 3D using tomography [19], but mainly for tension/compression tests. Because of the very small crack opening values obtained in fatigue, synchrotron tomography is compulsory in order to capture the cracks during the initiation stage. A drawback of this type of experiment is the small number of cracks that can be studied during one experiment (typically between 1 to 5 cracks). However, when torsional loads are used, the number of fatigue cracks that can be investigated is much larger because initiation in a shear mode occurs very rapidly in the fatigue life, when compared to cracks

initiated in tension [20]. Therefore, torsional cyclic tests would potentially make it possible to obtain good statistical data to study the conditions of crack initiation in relation to the microstructure. Fatigue cracks initiated in torsion have been studied in 3D using tomography in a very limited number of papers [16,21,22]. However, in those studies the interaction between crack initiation and the microstructure was either not addressed [21,22] or did not provide an exhaustive investigation of the crystallographic orientation in 3D [16].

The present work is focused principally on the mechanisms of crack initiation of a cast AlSi7Mg03 alloy under torsional loads. Some main characteristics of crack propagation mechanisms are also addressed in the discussion of the formation of the main crack causing the failure of the specimen. A large effort has been undertaken to characterise in-situ and in 3D the fatigue cracks together with their surrounding 3D microstructure. Diffraction Contrast Tomography [23] has been used to characterise the shape and orientation of the grains present in the gauge length of the fatigue samples. The large coherence of the X-ray photons enabled the visualisation of crystallographic cracks with subvoxel openings as well as Si particles which cannot be detected with laboratory X-ray sources. All this provided a nearly complete “picture” of the microstructure around the initiated fatigue cracks, making it possible to shed new light on the initiation and early stages of growth of small fatigue cracks in this class of material and loading mode.

## 2. Experimental conditions and material properties

### 2.1. Diffraction Contrast Tomography (DCT)

Before mechanical cycling, the shape and orientation of the grains present in the fatigue specimen gauge length were analysed by Diffraction Contrast Tomography (DCT) [23–25]. The DCT images were recorded using a pixel size of 10.3 µm and an angular resolution of 0.1° with 3600 recorded images covering 360°. Two slightly overlapping scans were recorded in the central region of each specimen, covering a length of approximately 5 mm along the specimen axis. The identification and reconstruction of the orientation, the position and the grain morphology were obtained thanks to the software developed by Ludwig et al. [25,26]

### 2.2. In-situ fatigue tests under X-ray micro-tomography

The in-situ torsional fatigue tests were conducted at ID19 beamline using a dedicated fatigue machine that was designed and built by INSA Lyon. Torsional loads were applied to cylindrical samples (nominal diameter of 3 mm, Fig. 1(a)) under angular displacement with a constant angular amplitude. The resulting torque is measured by a ±1000 N mm sensor. During the tomography scans that last several minutes, the fatigue tests were interrupted. The test frequency was set between 10 and 15 Hz. Fig. 1(b) shows the set-up of the in-situ fatigue tests in the synchrotron beamline.

Even though the machine is constant angular displacement controlled, the machine was set at the beginning of the fatigue tests so that the obtained load ratio R was approximately -1 (with an error less than 3%). During the fatigue tests, the variation of the torque amplitude is generally lower than 1% if there is no initiated crack. The shear stress on the specimen surface is then calculated from the applied torque, supposing a linear elastic behaviour of the material.

Three specimens were tested and monitored. The experimental conditions, including the applied stress, number of cycles, number of scans as well as the microstructural properties, mentioned in the previous sections, are given in Table 1. It is worth noticing that because the aim of this study is to investigate crack initiation in the regime close to the fatigue limit of the material, the applied stresses are relatively low compared to the work of Serrano et al. [16] and therefore the investigated number of cycles are much higher. Therefore, to optimise the beam time, all specimens were pre-cycled, on the same fatigue
<figure><img src="image_4.png" /><figcaption>Fig. 1. (a) Set-up of in-situ fatigue tests in the synchrotron beamline; (b) Geometry of fatigue specimens; (c) Schematic of in-situ fatigue test protocol.</figcaption></figure>

machine, before going to ESRF (see the schematic shown in Fig. 1(c)). The number of cycles in the pre-cycling step is given as the number of cycles at the first scan in Table 1. Two specimens (nC-2A-3 and nB-1B-2) were tested at relatively high stress levels (higher than 90 MPa). At this shear stress amplitude, the average fatigue life is expected to be slightly less than $10^6$ cycles [11]. The last specimen (nB-3A-3) was cycled at a lower shear stress amplitude ($\tau_a$ = 75.6 MPa). This specimen was pre-cycled for $N$ = 2.59×$10^6$ cycles and only one scan was performed after pre-cycling. Among the three tested specimens, the final failure (with a final crack of several millimetres in length) was only observed for specimen nC-2A-3. For the other specimens, no long crack, indicating final failure, was observed.

The tomography scans were performed using a (2560 × 2160 pixel) camera with a pixel size of 1.3 µm and a beam energy of 26.5 keV. 1800 images were captured for each volume covering 180°. For each specimen, two slightly overlapping volumes with a height of approximately 5 mm were scanned. The duration of each scan was approximately

5 min. During the scans, the samples were maintained at either the maximum ($\tau_{max}$) or minimum ($\tau_{min}$) torsion loads in order to maximise contrast around crack during CT scans.

## 2.3. Crack detection methodology and correlation between DCT - Tomographic images

One major advantage of synchrotron X-rays compared to laboratory X-ray sources is the high coherence of the beam that makes it possible to obtain phase contrast in combination with the classical attenuation contrast [27]. This is of particular importance here as small cracks formed in torsion are hard to visualise for two reasons. First, as explained in the introduction, some of the cracks form from slip bands and are therefore very flat and not opened. Second, in contrast to what happens in uniaxial tension/compression cyclic loads, there is no tensile stress acting perpendicular to the planes of maximum shear [28], which could help to open the cracks under load and facilitate their visualisation. In their early stages, fatigue cracks formed in
<table><caption>**Table 1**
Experimental conditions of the in-situ tests.</caption><thead><tr><th>Specimen</th><th>Applied shear stress amplitude τ<sub>a</sub> (MPa)</th><th>Nb of cycles at the 1st scan</th><th>Nb of cycles at the last scan</th><th>Nb of scans during fatigue</th></tr></thead><tbody><tr><td>nC-2A-3</td><td>91.7</td><td>2.5×10<sup>5</sup></td><td>5.33×10<sup>5a</sup></td><td>5</td></tr><tr><td>nB-1B-2</td><td>93.4</td><td>2.6×10<sup>5</sup></td><td>4.76×10<sup>5</sup></td><td>8</td></tr><tr><td>nB-3A-3</td><td>75.6</td><td>2.59×10<sup>6</sup></td><td>–</td><td>1</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Orthogonal reconstructed slices of specimen nC-2A-3 captured at 3.18×10⁵ and 5×10⁵ cycles. In their early propagation stage, torsion cracks lie along crystallographic planes and have a very low opening which results in a very thin white line indicated by white arrows (b). The presence of the crack is confirmed by looking at the same slice after a larger number of fatigue cycles (c).</figcaption></figure>

torsion are therefore small with a low opening (typically 1 µm or less). Thanks to diffraction effects, such cracks with subvoxel opening can nevertheless be imaged as shown in [27]. The origin and interpretation of the contrast of flat crystallographic cracks in tomographic images is however not trivial and has been discussed in detail in [29].

Fig. 2-(a) shows an example of such cracks within a reconstructed slice perpendicular to the rotation axis. In this image, artefacts caused by the presence of a small crack (specimen cycled for $N = 3.18×10^5$ cycles at $τ_a = 91.7$ MPa) can be seen. The crack, located on a plane perpendicular to the specimen axis, is difficult to detect. Fig. 2-(b) shows a reconstructed slice located at the red dashed line given in Fig. 2-(a). In this figure, the crack can be observed more easily as a thin straight white line (indicated by the white arrows). However, as explained in the work of Xiao et al. [29], at the early stage of the crack initiation, the contrast on the reconstructed images does not allow a reliable determination of the crack boundaries (and hence the crack size). Only the crack plane can be determined, as shown in Fig. 2-(b). For this reason, the presence of the crack has to be confirmed (traced back) by analysing the same region at a higher number of cycles, when the crack opening is larger as shown in Fig. 2-(c).

In order to study the influence of the local crystallographic orientation on crack initiation, the 3D images of the cracks need to be correlated with the DCT volume. To do so, the following post-analyses were carried out:

* Schmid factor and resolved shear stress calculations: the objective is to estimate which slip systems are activated within each grain. This is done by estimating the resolved shear stress on each of the 12 slip systems of the FCC structure based on the value of the Schmid factors $f_i$ obtained from the 3D orientation of the grains (for the case of torsion $0<f_i<1$). The plastic activity within the aluminium matrix is assumed to occur on the classical {111}{110} slip system of the FCC structure. However, the Schmid factor may be not completely relevant in the case of torsional loads because of the stress gradient between the grains intersecting the surface and those fully in-bulk. The resolved shear stress computed from the stress tensor has to be corrected by considering the stress gradient. In the analysis of the present work, two values of resolved shear stress are considered. The first one is computed at the centroid of the grain (so it can be considered as an “average” value). The second value of resolved shear stress is calculated at the point of the grain which is the furthest from the specimen centre. For a grain intersecting the specimen surface, this resolved shear stress value is simply the multiplication of the Schmid factor by the surface macroscopic applied stress. An example of resolved shear stress (calculated at the grain centroid) of grains intersecting with a slice perpendicular to the specimen axis (specimen nB-1B-2) is shown in Appendix B, Fig. B.1. The effect of the macroscopic stress gradient under torsional loads on

<footnote>a Final failure observed with a crack of several millimetres in length.</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 3. (a) Schematic representation of the transformation of the slip system trace vectors from the DCT coordinate system to the cutting plane coordinate system: (b) Superposition of a micro-tomography image showing a crack with the corresponding DCT grains map.</figcaption></figure>

the resolved shear stress in the grains, can clearly be seen in this figure.

* Slip system trace plots: for visualisation of slip system traces of a grain on any cutting plane (perpendicular or parallel to the specimen axis). An algorithm was developed (using the Mtex toolbox [30]) to transform the slip systems traces vector from the DCT coordinates and show it on the cutting plane.

Fig. 3(a) shows a schematic representation of the transformation of slip system traces from the DCT coordinate into the cutting plane coordinate system.

* Superposition of the DCT and tomography images: the absorption image obtained when recording the DCT scan (in parallel with the diffraction images) make it possible to register the DCT grain maps with the high-resolution tomography image captured during the in-situ fatigue test. In practice, the large pores, visible in the DCT absorption images despite the relatively large voxel size used, were used as natural fiducial marks to correlate with the in-situ tomography images. Fig. 3(b) illustrates the superposition of the tomography image of the zone shown in Fig. 2 and the corresponding DCT grain map with projected slip system traces.

## 2.4. Specimen fabrication and material properties

The material investigated in the present work is an AlSi7Mg0.3 alloy, cast by the lost foam process within moulds in the form of sheets, 200 mm × 150 mm × 18 mm in size. After casting, the material was heat treated (T7 condition). The monotonic tensile properties are given in Table 2. Specimens used for the in-situ fatigue tests were machined from the cast sheets with the geometry shown in Fig. 1(a). The aluminium matrix has an average SDAS of 77 µm and a mean micro-hardness of 99 HV025. More details related to the microstructural properties can be found in [11].

### 2.4.1. Grain map and grain size distribution

Thanks to the DCT observations, the grain maps of the investigated specimens were reconstructed in 3D. Fig. 4(a) shows an example of a reconstructed 3D grain map and Fig. 4(b) shows the distribution of the grain size (in terms of $V^{1/3}$) for the three specimens, investigated here. The distributions have been truncated at a grain size of 100 µm (corresponding to approximately 10 voxels). This was done in order to discard grains that are too small and could be artefacts or noise. In this figure, the Y-axis corresponds to the empirical cumulative
<table><caption>**Table 2**
Monotonic tensile properties of the investigated alloy (mean (μ) and standard deviation (s) values)</caption><thead><tr><th>Young's modulus<br>E (μ ± s) (GPa)</th><th>0.2% Yield stress<br>σ<sub>y</sub> (μ ± s)(MPa)</th><th>Ultimate tensile strength<br>σ<sub>uts</sub> (μ ± s) (MPa)</th><th>Elongation<br>(μ ± s)(%)</th></tr></thead><tbody><tr><td>68±5</td><td>240±5</td><td>251±6</td><td>0.8±1.2</td></tr></tbody></table>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 4.** (a) Reconstructed 3D grain maps of a sample; (b) Grain size distributions of the investigated specimens; (c) Comparison of an orthogonal slice of the grain map obtained by the DCT technique and the corresponding experimental EBSD grain map.</figcaption></figure>

density function. The numerical details of the grain size distributions are summarised in Table 3.

For cast Al-Si alloys, because of the presence of large eutectic zones and the large range in the grain size (from several tens of micrometers to several hundred micrometers), the grain reconstruction is more difficult than in materials with a more homogeneous grain size [25,31]. In order to validate the obtained results, a DCT scanned sample was sectioned and observed by the EBSD technique. The 2D grain map obtained by the latter was compared with the same grain map reconstructed by the DCT technique. Fig. 4(c) shows the comparison of the orientation and morphology of the grains in the slice. It can be seen that, for the large grains, the results obtained by the DCT are in very good agreement with the experimental EBSD grain map, except for one grain located on the right of the map. The average misorientation between the two maps ranges between 1° to 3°.

The accurate result of DCT reconstructed grains compared to the EBSD observation reassures the analyses, presented below in Section 3, that use information from the DCT reconstructed grains as essential data.

### 2.4.2. Microstructure and pore size distribution

Thanks to the small voxel size and to the presence of phase contrast within the reconstructed images, the microstructure of the material, including the alpha phase with dendrite arms, eutectic particles as well as porosities can be identified clearly as shown in Fig. 5.

Regarding the pore population, Fig. 6(a) shows a 3D rendering of the pores in the gauge length of a fatigue sample, highlighting the presence of relatively large pores. The pore size distributions of the three analysed specimens are given in Fig. 6(b). The distributions are truncated at a pore size of 40 µm (corresponding to a volume of approximately 30 voxels) in order to discard small pores that could be artefact or noise. In this figure, the Y-axis corresponds to the reduced variable $Y_j$, calculated from the cumulative function $F_j$ by following equation $Y_j = -\ln(-\ln(F_j))$. The latter makes it easier to visualise the distribution of the largest pores in the volume (i.e. the distribution tail). It is often these pores that are critical in terms of fatigue crack initiation. Table 3 summarises some key parameters related to the microstructure of the investigated specimens characterised by DCT and tomography.
<figure><img src="image_13.png" /><figcaption>Fig. 5. A reconstructed micro-tomography slice of the specimen nC-2A-3 scanned at N=3.18×10⁵ cycles. The eutectic zones containing Si particles (highlighted in blue) surrounding the α dendrites (highlighted in red) are made visible by the phase contrast.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 6. (a) 3D rendering of porosities observed by tomography in a sample. The small spots can be either real pores or artefacts (eutectic particles, numerical artefacts); (b) Pore size distribution.</figcaption></figure>

<table><caption>**Table 3**

Microstructural properties of in-situ tested specimens characterised by DCT and Tomography observations.</caption><thead><tr><th>Specimen</th><th>Scan. vol.<br>(mm³)</th><th>Nb of grains<br>V¹/³ >100 µm</th><th>Gr. size V¹/³ (µm)<br>Avg. (Min-Max)</th><th>Nb of pores<br>V¹/³ >40 µm</th><th>Pore size V¹/³ (µm)<br>Avg. (Min-Max)</th></tr></thead><tbody><tr><td>nC-2A-3</td><td>≈33</td><td>≈1600</td><td>240(100-809)</td><td>131</td><td>69(40-150)</td></tr><tr><td>nB-1B-2</td><td>≈33</td><td>≈1500</td><td>243(100-1029)</td><td>154</td><td>74(40-210)</td></tr><tr><td>nB-3A-3</td><td>≈33</td><td>≈2000</td><td>227(100-792)</td><td>115</td><td>71(40-186)</td></tr></tbody></table>
<figure><img src="image_15.png" /><figcaption>**Fig. 7.** Two crack initiation mechanisms observed from a single pore on specimen nB-1B-2, captured at *N* = 310000 cycles (figures (a) and (c)) and *N* = 476000 cycles (figures (b) and (d)); (a) and (b) reconstructed images parallel to the specimen axis; (c) and (d) projected grey levels on a plane perpendicular to the specimen axis. The specimen is loaded statically at *τ*max. The red star marks the position of a crack that propagates very slowly.</figcaption></figure>

**3. Results and discussion of the crack initiation mechanisms**

Three reconstructed volumes with an average size of approximately 30 mm³ were examined. Approximately 80 cracks were observed and analysed. From these data, the observed crack initiation mechanisms as well as the effect of surrounding microstructure is discussed in the following sections.

**3.1. Two coexisting crack initiation mechanisms**

As explained in the introduction section, two coexisting crack initiation mechanisms are observed for the experimental conditions investigated in this work. Fig. 7 illustrates a typical example in which the two crack initiation mechanisms were observed from the same crack initiation site (i.e. a pore). The reconstructed tomography images of the surface of specimen nB-1B-2 are shown in Figs. 7(a) and 7(b), captured respectively at *N* = 3.1×10⁵ cycles and *N* = 4.76×10⁵ cycles. Figs. 7(c) and 7(d) show a projection of the minimum grey level volumes within a reconstructed volume around the crack on a plane perpendicular to the specimen axis. It is possible to visualise the original pore as well as the projected crack shape. This image was recorded when the specimen was loaded statically at *τ*max in order to make the crack more visible, in particular for the cracks initiated in opening mode.

It can be seen that the two cracks initiate from the same pore (highlighted in yellow in Fig. 7(c) and (d)). The crack on the left side, is on a plane at an angle of approximately 49° with respect to the specimen axis. This plane is close to the maximum normal stress planes for torsional loads (45°), which indicates that the crack, on the left, is growing in an opening mode (or mode I). This crack initiates from a “corner” of the pore intersecting with the specimen surface (with a crack length on the sample surface of 122 µm at N=310000 cycles) and propagates to reach a length of approximately 330 µm at N=476000 cycles. The second crack, on the right side, was initiated on a plane at an angle of approximately 81° with respect to specimen axis. That is, on a plane close to the maximum shear stress planes (90°). It can therefore be assumed that this crack is developing in a shear opening

mode. Compared to the first crack, whose length is roughly tripled between the two scans, the propagation length of the second crack, for the same number of cycles, is much lower. Another difference between the two cracks is the crack opening distance. It can be seen that the mode I crack (on the left) has a larger opening distance on the surface that makes it visible, compared to the crack in mode II (on the right) which is less visible.

In the following sections, the two crack initiation mechanisms will be investigated in depth in order to characterise the role of different microstructural factors on each mechanism.

**3.2. Crack initiation in a shear mode from PSBs**

Cracks initiated from PSBs were observed to occur in grains either connected or not connected to pores. For all of the cracks analysed (66 cracks), **the crack planes within these grains were always coincident with the crystal slip plane with the highest Schmid factor.** In most cases, the crack planes were close to the maximum shear stress planes. As an example, Fig. 8 shows a crack observed in specimen nC-2A-3 at 5×10⁵ cycles. In Fig. 8-(a), the crack marked with black arrows can be seen on a plane tangent to the surface. The crack appeared within grains g586 and g293 and propagated to grain g119. In grains g586 and g293, the crack plane appears to be parallel to the trace of the (11̄1) slip plane. A second observation, shown in Fig. 8-(b), on a plane perpendicular to the plane in Fig. 8-(a) confirms that: in grain g293, the crack lies parallel to the trace of the (11̄1) plane. Note that this double check was systematically done in all of the observations in this work to confirm the coincidence of the crack plane with a slip plane. The resolved shear stresses calculated at the specimen surface for the these three grains g320, g293 and g586, showed that the slip plane (11̄1) corresponding to the crack plane, contains the direction that has the maximum resolved shear stress, among the twelve values (τ<sub>rss</sub> > 80 MPa). Within grain g119, the crack grows on a different slip plane (11̄1) compared to the three previously discussed grains. For this grain, the resolved shear stress (τ<sub>rss</sub> > 80 MPa) is also the highest on this plane.
<figure><img src="image_16.png" /><figcaption>**Fig. 8.** Specimen nC-2A-3 at $5 \times 10^5$ cycles: a crack (crack-tips marked with black arrows, white arrows added along the crack path) initiated from PSBs in grains without any neighbouring pores, observed on (a) a plane tangential to the specimen surface and (b) on a plane perpendicular to the surface tangent corresponding to the black dashed line shown in figure (a). The resolved shear stress values are also given for the grains in which the crack initiates and grows. Figure (c) shows the same region as Figure (b) at a higher number of cycles ($5.33 \times 10^5$ cycles). When entering grain g10, the crack path changes the direction and, at this scale of observation, shows less pronounced crystallographic facetting.</figcaption></figure>

In Fig. 8-(b), it can be seen that, in the volume, the crack does not seem to grow into grain g10 (right arrow in Fig. 8-(b)). An analysis at a greater number of cycles ($5.33 \times 10^5$ cycles) shows that, at this grain boundary, the crack changes its propagation mode going from a crystallographic mode in grain g293 to a more chaotic path in grain g10 (Fig. 8-c). It is worth noticing that the propagation plane in grain 10 is closer to the macroscopic maximum shear stress plane, as opposed to the crystallographic (111̄1) plane in grain g293.

Beside the effect of the stress gradient, the large misorientation between the two grains is one explanation for this change of crack propagation mode. For cracks propagating along crystallographic planes, it has been suggested [32,33] that the misorientation between activated slip planes can provide an explanation for the efficiency (or the lack of efficiency) of grain boundaries to arrest cracks. Zhai et al. [32] proposed to measure this misorientation using tilt and twist angles. Schematically, large twist and (to a less extent) tilt angles increase the

efficiency of the grain boundary to arrest a crack. In the case studied here, the values of the tilt and twist angles between the two most activated slip planes (i.e. the (11̄1) plane in both grains) are equal to 71.8° and 31.7° respectively. These relatively large values could explain the change in crack path observed in grain g10 in Fig. 8-(c).

Another illustration of crack initiation from PSBs is given in Fig. 9. In this case, two cracks were initiated close to a sub-surface pore. Fig. 9-(a) shows a plane tangential to the surface and Fig. 9-(b) shows a slice perpendicular to the specimen axis. It can be seen that the crack which initiated in grain g342 occurs on the (11̄1) plane i.e. the plane with the trace close to the vertical direction. The crack which initiated in grain g203 occurs on the (11̄1̄1) plane with the trace close to the horizontal direction. Even though these two cracks initiated in the two different slip planes ((11̄1) vs (11̄1̄1)) that have a trace with very different orientation (vertical vs horizontal), these planes always correspondent to the one with highest $\tau_{rss}$ value. It means that
<figure><img src="image_17.png" /><figcaption>**Fig. 9.** Specimen nB-1B-2 at $4.76 \times 10^5$ cycles: two cracks initiated due to PSBs from a pore on the specimen sub-surface (crack ends marked with black arrows, white arrows added along the crack path), observed on (a) a plane tangential to the specimen surface and (b) on a plane perpendicular to the specimen surface. The resolved shear stress values are also given for the grains in which the cracks initiate.</figcaption></figure>

even though a pore is present, the cracks always occur on the most activated slip plane. Besides, this case highlights the importance of conducting the investigation in 3D because the pore is located beneath the specimen surface and cannot be detected on the surface.

The effect of grain orientation and grain size on crack initiation from PSBs has been investigated statistically. Fig. 10 shows the relationship between the highest resolved shear stress of each grain (among the twelve values computed for the twelve slip systems) and the grain size. The data from specimens nB-1B-2 and nC-2A-3 have been merged on the same graph (Figs. 10(a) and 10(b)) since these two specimens were loaded at roughly the same stress amplitude ($\tau_a \approx 92$ MPa) and the observations were made at approximately the same number of cycles ($N \approx 5 \times 10^5$ cycles). The data from the third specimen (nB-3A-3), loaded at a lower stress amplitude (75.6 MPa) and corresponding to a much greater number of cycles ($2.59 \times 10^6$ cycles) is plotted in Figs. 10(c) and 10(d).

In these diagrams, three categories of grains are presented: (i) all grains intersecting the specimen surface (1487 grains in Figs. 10(a) and 10(b); 1069 grains in Figs. 10(c) and 10(d)) (ii) grains connected to at least one pore in which cracks from PSBs are observed and (iii) grains not connected to pores in which cracks from PSBs are observed. Only grains located on the specimen surface are plotted in these figures because all initiated cracks were observed on the surface. The resolved shear stress is computed at the specimen surface without taking into account the stress raising effect of the pores (if present). Regarding the grain size, two definitions are considered. The first one is the total size (in terms of cubic root of the 3D volume) of the whole grain. For the second definition, only the part of the grains located in a 50 µm thick subsurface “ring” is taken into account in the calculation of the volume. This thickness results from a compromise. It was chosen so that only the

grains intersecting with the specimen surface are considered. In fact, a larger ring thickness would result in subsurface grains being included in analysis and if a smaller ring thickness were to be used, some small surface grains would have been excluded. To do this, the core of the 3D grains map was removed and the grain size calculation was done on the subsurface “ring” volume (an example is shown in Fig. A.1). The reason for using the second definition of grain size is that, again, cracks always initiate at the sample surface because of the torsional stress gradient at the early stage of crack propagation.

It can be seen that the grains without a connected pore, which contain cracks initiated from PSBs (i.e. the blue points in Fig. 10(a)) are submitted, on average, to a higher maximum resolved stress ($\tau_{rss} > 80$ MPa), when compared to grains that are connected to pores. The scatter in terms of stress is also larger for the latter. It can be explained by the fact that the presence of a pore increases the shear stress amplitude at the local point, making it easier to activate crack initiation from PSBs. The effect of the pore size was also analysed, but has not been included in this paper, as no clear influence of this parameter on the resolved shear stress, triggering crack initiation, was seen.

Regarding the effect of grain size, no clear trend can be drawn from Figs. 10(a) and 10(b) between the maximum resolved shear stress leading to crack initiation and the grain size, for grains with or without pores. Therefore, this observation does not support models in the literature which predict a decrease in the stress necessary to initiate fatigue cracks when increasing the grain size [34], at least for torsional loads. When the stress level is not considered, however, it seems that the probability of development of cracks is higher on average for the large grains than for the small grains. This trend is more clear in Fig. 10(b) in which the size of partial grain close to the
<figure><img src="image_18.png" /><figcaption>**Fig. 10.** Maximum resolved shear stress vs size of grains for (i) all grains located on the surface in the observation volumes (grey points); (ii) grains connected to a pore containing cracks initiated from PSBs (orange points) and (iii) grains without any pore in the neighbourhood containing cracks initiated from PSBs (blue points), plotted for the specimens submitted to a relatively high stress level (nB-1B-2 at 4.76×10⁵ cycles at 93.4 MPa and nC-2A-3 at 5.33×10⁵ cycles at 91.7 MPa) (a) and (b); specimen nB-3A-3 ($\tau_a$=75.6 MPa at N = 2.59×10⁶ cycles(c) and d) (lower stress level).</figcaption></figure>

surface is presented. A similar trend was also observed in the work of Weidner et al. [35]. These authors showed, for polycrystalline nickel, that the occurrence probability of PSBs in small grains (grain diameter less than 50 µm) is lower (48% of grains having PSBs) than for larger grains (grain diameter greater than 80 µm, 74% of grains having PSBs). A same result was also observed in the work of Ou and Liu [36]. The authors show that the maximum PSB width of annealed copper (grain size = 104.2 µm) is statistically larger than that of the stress-relieved copper (grain size = 14.2 µm). They explained that in large grains, there is more space for the growth and motion of dislocation within the grain. This explanation seems reasonable to us. It seems logical that an increase in the grain size, would result in increased PSB length and width. This, in turn would result in increased slip irreversibility and hence, increase the probability of crack initiation in larger grains. Nevertheless, this effect observed in Figs. 10 remains relatively slight. However, note that the microstructure of the material investigated in

this work is more heterogeneous than those in the literature discussed above. In particular, the eutectic zones can play a role as an barrier in the dislocation movement process and in PSBs formation, and hence, attenuate the effect of the grain size.

For specimen nB-3A-3, loaded at a lower shear stress amplitude (75.6 MPa) and at a greater number of cycles (2.59×10⁶ cycles), the same trends can be seen in Fig. 10(d). **A notable difference between the high and low loads is the crack density, especially for the cracks initiated in grains without connected pores: it can be seen that the crack density in grains without a connected pore in the specimen loaded at $\tau_a$ ≈ 76 MPa at N ≈ 2.6×10⁶ cycles (2 cracks in 1 specimen) is remarkably lower than in the specimens loaded at $\tau_a$ ≈ 92 MPa at N ≈ 5 × 10⁵ cycles (10 cracks in 2 specimens).** This tends to show that, if the applied stress was to be further decreased, then crack initiations from PSBs would unlikely occur in grains without a connecting pore. They only occur in grains connected to pores due to the local stress concentration effect.
<figure><img src="image_19.png" /><figcaption>**Fig. 11.** Specimen nC-2A-3: coalescences of cracks initiated from PSBs observed at the sample surface, recorded at different numbers of cycles (4 × 10⁵, 5 × 10⁵ and 5.33 × 10⁵ cycles). The black arrows indicate the ends of the cracks. The black circles indicate the coalescence of cracks.</figcaption></figure>

Another point that is worth discussing for crack initiated in PSBs is the crack arrest and crack propagation mechanisms. It is observed that among the 66 cracks initiated in the three investigated specimens, **only 4 cracks grow beyond the boundary of the initiating grain (i.e. less than 10%)**. The rest are arrested inside the initiation grain (or significantly slowed down). An example is shown in Fig. 7. In this example, the growth of the crack on the right side of the pore, initiated in the green grain, seems to be significantly slowed down at the boundary between the initiating grain (g69) and the adjacent grain (g32). As previously discussed, it can be assumed that the main crack arrest mechanism is a high degree of misorientation between the initiation grain and the adjacent ones in terms of tilt and twist angles. For the case shown in Fig. 7, the value of these two angles, for the planes with the highest $\tau_{RSS}$ values, between grain g69 (plane 11̄1) and grain g32 (plane 11̄1) are 8° and 92° respectively. In this case the twist angle seems to be the main cause of the crack arrest.

When the number of cycles increases, the coalescence of multiple cracks that “escaped” their initiation grain was observed. An example is given in Fig. 11. At $4 \times 10^5$ cycles, cracks that initiated in grains g586, g293 and g320 coalesce at $5 \times 10^5$ cycles to form the main crack. In addition, a crack initiated in grain g36 at $5 \times 10^5$ cycles coalesces to the main crack at $5.33 \times 10^5$ cycles.

### 3.3. Crack initiation in an opening mode from pores

The other crack formation mechanism that coexists with the first mechanism observed in this work is the formation of cracks by mode I opening.

Fig. 12-(a) and (b) show a plane tangential to the specimen surface in which a crack is growing from a pore in mode I. The crack can be observed to occur on a plane, at approximately 45° to the specimen axis (i.e. a plane of maximum normal stress). The projected images of the crack (Fig. 12-(c) and (d)) show that the crack initiates at the

intersection between the sample surface and the pore and seems to grow preferentially along the specimen surface. In order to investigate a possible effect of grain orientation on this crack initiation mechanism, Fig. 13 shows the trace of the slip planes in the grains adjacent to the pore, observed on the planes tangential (Fig. 13-(a)) and orthogonal (Fig. 13-(b) and (c)) to the specimen surface.

When the cracks start to propagate away from the pore, a corrugated crack path (≈100 µm long) can be observed (arrows shown in Fig. 13-(c)) before becoming flat. Despite the limited resolution of the tomographic images, the crack seems to be growing by alternating between the (111) plane (green) and another plane. By looking at Fig. 13-(b), the latter seems to correspond to the (11̄1) plane (blue) but this is not confirmed by Fig. 13-(a). Regarding the resolved shear stresses, the maximum value, found for the (111) plane, is lower than 40 MPa. This is much lower than the shear stress levels evaluated above for the activation of the PSB crack initiation mechanism (see Fig. 10). After this crystallographic-like stage, the crack plane becomes flat making an angle of 45° with respect to the specimen axis (maximum normal stress plane).

The planes of the mode I crack have been compared with the traces of the low index crystallographic planes (Fig. 13-d and -e). This is motivated by experimental evidences, in the literature, for aluminium alloys, of fatigue cracks propagating on low index planes, (mainly {100}) under uniaxial loads [37,38]. The authors assumed that this crack propagation mechanism may be related to environmental hydrogen-assisted cracking. However, in the present work, no correlation could be observed.

In the present work, because the resolution of the tomographic images is limited (typically lower than the SEM images used in [38]), small deviations (such as the one shown in Fig. 13-c) of the cracks cannot be accurately linked to the local crystallography. However, one sure result is that the crystallographic nature of the cracks is no longer observed at a larger scale (≈ 10–50 µm). The principal crack follows
<figure><img src="image_20.png" /><figcaption>Fig. 12. Specimen nB-1B-2 at 2.60 × 10⁵ cycles (figures (a) and (c)) and at 4.76 × 10⁵ cycles (figures (b) and (d); (a) and (c)) cracks initiating and growing from a pore observed on a plane tangential to the specimen surface; (b) and (d) images of projected grey level on a plane perpendicular to the specimen axis. Images obtained with the specimen under load τmax.</figcaption></figure>

a path that is on the plane with the maximum normal stress. This confirms the governing role of the opening mode.

Fig. 14(a) shows the relationship between the pore size and the crack length. The length of each crack is measured on the specimen surface, not including the initial pore size. The cracks of the same specimen are measured at N=4.76 × 10⁵ cycles for specimen nB-1B-2; at N=5.33 × 10⁵ cycles for specimen nC-2A-3 and at N=2.59 × 10⁶ cycles for specimen nB-3A-3. The pore size is defined as the square root of the maximum cross section of the equivalent sphere (with radius r<sub>eq</sub>) having the same volume as the pore under consideration (Eq. (1)) where V is the volume of the pore.

$$ \text{Pore size in } \sqrt{area} = \sqrt{\pi \times r_{eq}^2} = \sqrt{\pi \times \sqrt[3]{\frac{3 \times V}{4 \times \pi}}} \quad (1) $$

It can be seen that, globally, the crack length depends on the size of the connected pore. For specimens nB-1B2 and nC-2A-3, the relationship between the crack length and the pore size is similar. For specimen nB-3A-3, it can be seen that, for a same pore size, the crack length is much larger. This is likely to be due to the larger number of cycles at which the measurements were made. This trend implies that, for cracks initiated from a pore in an opening mode, the size of the initiating pore seems to be the governing factor while, as discussed previously, the crystalline orientation of the adjacent grains does not seem to be a first order parameter.

Fig. 14(a) highlights the cracks that are considered to be growing cracks (open diamond markers). The criterion for determining if a crack is growing or arrested is based on the fact the crack has propagated beyond the boundary of the influence zone created by the initiating pore. The influence zone can be defined as the maximum lateral dimension of the pore below the surface following the work of Buffiere et al. [7] on a cast AlSi7Mg0.3. In their work, these authors showed that a crack that has a length measured on the specimen surface larger than the maximum lateral size of the pore is propagating under uniaxial loads. It can be seen in Fig. 14(a) that only cracks initiating from a relatively large pore can grow beyond the influence zone of the pore (as already observed in [7]). The influence of the pore size can be clearly confirmed by using the stress intensity factor K<sub>I</sub>, computed as per the Murakami approach with the formula given in Eq. (2) for surface pores [39]. The pore size (√area) in this equation corresponds to the size of the initiating pore, without taking into account the crack size. The stress σ<sub>I,max</sub> is the first principal stress calculated from the macroscopic applied stress, computed at the centroid of the pore. The max subscript indicates that it is the maximum value that occurs during

the fatigue cycle. The data for specimens nB-1B-2 and nC-2A-3 have been plotted in the same diagram while the data corresponding to specimen nB-3A-3, loaded at the lower stress, is plotted separately.

$$ K_{I,max} = 0.65 \times \sigma_{I,max} \times \sqrt{\pi \times \sqrt{area}} \quad (2) $$

In these graphs, the K<sub>I</sub> values of some pores, located at the surface but without crack, have been included. It can be seen that for specimens nB-1B-2 and nC-2A-3 (τ<sub>a</sub> > 90 MPa and N ≈ 5 × 10⁵ cycles), there is a clear threshold of K<sub>I</sub> ≈ 1.1 MPa√m distinguishing the pores from which cracks initiate and propagate and the pores from which cracks (if initiated) are arrested. This value is in a good agreement with the one found in the work of Buffiere et al. [7]. In this work, the authors obtained a threshold between 1 and 1.5 MPa√m for the same type of material under uniaxial fatigue loads (R=0.1).

Concerning the crack arrest mechanism, no clear evidence was observed in the present work relating to the role of the DAS/SDAS, the eutectic zones or the grain boundaries as the first crack barriers. In some cases, a slight change in the crack path direction was observed when a crack crossed a grain boundary, such as the crack shown in Fig. 13-(a) (crack path between the grains g10 and g912). However, the 3D shapes of the mode I cracks observed do not suggest a clear arrest mechanism, like the one observed for crack initiation from PSBs. Regarding the statistical aspect, the number of cracks initiated from pores in an opening mode (14) is much lower than the number of cracks initiated from PSBs (66). Among these 14 cracks, 8 cracks grew beyond the influence zone of the initiating pore and were considered as growing at the last scans.

### 3.4. Competition between the two crack initiation mechanisms

In this section, the competition between the two crack initiation mechanisms leading to the formation of the final crack that causes the specimen failures is addressed. Table 4 summarizes the number of cracks in each mode for the three specimens investigated. Note that, for the specimen nB-3A-3 only one tomography scan was realised. Hence, the criterion to define a growing or arrested crack is based on the crack size compared to the size of the originating pore (for cracks initiated from pores in the opening mode) and the size of the initiating grain (for cracks initiated from PSBs).

Based on these data and on the analysis given in the previous sections, a schematic view of four crack initiation and propagation scenarios can be suggested at high and low shear stress levels (Fig. 15). Even though further works need to be done to clearly understand
<figure><img src="image_21.png" /><figcaption>Fig. 13. Specimen nB-1B-2 at $4.76 \times 10^5$ cycles: cracks initiating and growing in mode I from a pore, observed on (a) a plane tangential to the specimen surface and (b) on a plane perpendicular to the surface tangent corresponding to the black dashed line on figure (a); (c) a zoom of the crack shown in figure (b). The resolved shear stress values are also given for the grains crossed by the crack. (d) and (e) comparison of the crack planes with low index systems traces. The black arrows shown in figure (c) indicate crystallographic-like crack growth.</figcaption></figure>

<table><caption>**Table 4**
Summary of all of cracks detected in the three specimens under investigation.</caption><thead><tr><th rowspan="2">Specimen</th><th colspan="3">Nb of cracks from PSBs<br>in a shear mode</th><th colspan="2">Nb of cracks from pores<br>in an opening mode</th></tr><tr><th>Total</th><th>w/-w/o pore</th><th>Growing-Arrested</th><th>Total</th><th>Growing-Arrested</th></tr></thead><tbody><tr><td>nC-2A-3</td><td>25</td><td>19-6</td><td>3-22</td><td>3</td><td>1-2</td></tr><tr><td>nB-1B-2</td><td>24</td><td>18-6</td><td>0-24</td><td>9</td><td>5-4</td></tr><tr><td>nB-3A-3</td><td>17</td><td>15-2</td><td>1-16</td><td>2</td><td>2-0</td></tr></tbody></table>

and statistically evaluate the effect of the microstructure on the crack propagation stage, the results related to this stage, presented earlier in the article, are discussed in this section.

**Schematic (a) - Crack initiation under relatively high stress levels from pores and PSBs:** cracks initiate from PSBs, with a relatively

high density, in grains with or without a connected pore. Cracks initiate in an opening mode from pores with a lower density than PSBs induced cracks.

**Schematic (b) - Crack propagation under relatively high stress levels:** many PSB induced cracks are arrested due to the misorientation
<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /><figcaption>**Fig. 14.** Relationship between the pore size and (a) length of the crack growing from the originating pore and (b) the stress intensity factor $K_{I,max}$ calculated from the size of the originating pore. The fatigue life at which the measurements were performed of each specimen is given in Figure (a).</figcaption></figure>

between the initiating grain and the adjacent ones. However, due to the high crack density, certain PSB induced cracks coalesce to form the final crack. The cracks that initiated in an opening mode propagate but their growth rate is not sufficiently high to compete with the PSB induced cracks, which have a much higher growth rate, (at least for a fatigue life lower than $2 \times 10^{-6}$ cycles). A study of the crack propagation kinetics, conducted in this project, showed that the growth rate of cracks initiated from PSBs can be up to $1 \times 10^{-8}$ (m/cycle) while for cracks growing in an opening mode from pores, the crack growth rate is approximately $1 \times 10^{-9}$ (m/cycle). This will be the subject of another publication.

**Schematic (c) - Crack initiation under relatively low stress levels:** the cracks density is lower than for the high applied stress scenario. Only the grains connected with pores can initiate PSB induced cracks. Regarding the second mechanism, opening-mode cracks are

initiated only from large pores (pore size calculated from pore volume as per Eq. (1)).

**Schematic (d) - Crack propagation under relatively low stress levels:** most of the PSB induced cracks are arrested. The probability of coalescence of PSB induced cracks to form the final crack is much lower than at higher stresses. Due to the higher fatigue life (fatigue life > $2 \times 10^6$ cycles) and also the less pronounced effect of microstructural barriers, opening-mode cracks, initiated from pores, have enough time to grow and lead to the final crack.

The four proposed schematics highlight the complexity of the competition between the two mechanisms for the formation of the final crack. Based on our observations, at least three factors can play a role: the initiated crack density, the microstructural barriers and the crack growth rate of the different mechanisms. The mechanism leading to the formation of the final crack, proposed with these scenarios is in good agreement with the observations made in the work of Serrano et al. [16]. These authors observed that, at high stress levels (corresponding to a fatigue life less than $2 \times 10^5$ cycles), the final fracture surface is relatively flat and perpendicular to the specimen axis. This implies that the shearing mode is the dominant mechanism of the final crack causing the failure of the specimen. At low stress levels (corresponding to a fatigue life higher than $3 \times 10^6$ cycles), the fracture surfaces are on planes inclined at 45° with respect to the specimen axis, meaning that the opening mode is dominant.

## 4. Conclusion

In this study, the crack initiation mechanisms of a cast AlSi7Mg0.3 alloy have been investigated by using DCT and high resolution 3D tomography. In-situ torsional fatigue tests have been conducted on three specimens and the gauge volume of each specimen were monitored to track crack initiation. In total, 80 cracks were detected and analysed. Two crack initiation mechanisms have been observed: crack initiation from PSBs in a shearing mode and crack initiation from pores in an opening mode. These two crack initiation mechanisms coexist (in the investigated fatigue regime corresponding to a fatigue life of approximately $N = 2 \times 10^6$ cycles) and compete to form the final crack. Based on this large dataset, the effects of the grain size and orientation and of the pore size on the two coexisting crack initiation mechanisms have been studied. The principal results are given as below.

**For crack initiation in a shear mode from PSBs:**

* The local grain orientation and the maximum resolved shear stress are the governing factors. All the cracks formed by this mechanism initiate in the slip system plane with the highest resolved shear stress. A threshold value of the maximum resolved stress of approximately 80 MPa has been measured for crack initiation at $N \approx 5 \times 10^5$ cycles. This value does not depend on the grain size. However, crack initiation tends to be more frequent within large grains.

* The presence of a pore increases the shear stress amplitude at the local point, making it easier to activate crack initiation from PSBs.

* A decrease in the applied stress significantly reduces the crack initiation density, in particular for the grains not connected to pores.

* A large number of cracks are arrested when the first grain boundary is reached (>90%).

**For crack initiation in an opening mode from pores:**

* The role of the orientation of the surrounding grains is not significant, even though corrugated crack propagations on the slip systems were observed for a few cracks at the beginning of propagation.
<figure><img src="image_24.png" /><figcaption>Fig. 15. Schematic diagrams of crack initiation and propagation for high and low stress levels (see the text for a detailed description).</figcaption></figure>

* The pore size is the first order factor controlling this mechanism. Clear relationships between the pore size and the initiated crack length were observed. A stress intensity factor threshold $K_{I,th} \approx 1$ MPa$\sqrt{m}$ in amplitude must be achieved for a crack to grow beyond the influence zone of the initiating pore. In contrast to the first mechanism, a much larger number (>50%) of these cracks continue to propagate.

Thanks to these analysis, four crack initiation and propagation scenarios have been proposed in order to explain the competition between these two mechanisms in terms of the formation of the final crack. At high stress levels, crack initiation from PSBs governs the formation of the final crack, due to (a) the high number of initiated cracks, which allows a greater probability of crack coalescence and (b) a higher crack growth rate compared to the cracks initiated in an opening mode. At low stress levels, a much lower number of PSB initiated cracks significantly reduces the probability of coalescence. Also, most of these cracks are arrested at the boundary with adjacent grains. On the contrary, cracks initiated from pores in an opening mode do not have any clear microstructural barriers for the investigated stress levels and grow steadily to form the final crack.

### CRediT authorship contribution statement

**Viet-Duc Le:** Writing – review & editing, Writing – original draft, Software, Resources, Methodology, Investigation, Data curation, Conceptualization. **Franck Morel:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition. **Nicolas Saintier:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition. **Pierre Osmond:** Writing – review & editing, Validation, Supervision, Methodology, Investigation, Funding acquisition. **Daniel Bellett:** Writing – review & editing, Supervision, Project administration, Methodology, Investigation. **Wolfgang Ludwig:** Writing – review & editing, Validation, Software, Investigation. **Marta Majkut:** Writing – review & editing, Software, Investigation. **Jean-Yves Buffiere:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition.

### 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

This study is part of the FANTOM project, financially supported by the CARNOT ARTS institute, in France. The DCT observations and in-situ fatigue tests were conducted at the European Synchrotron Radiation Facility - ESRF (Grenoble, France) (proposal number of MA4486 [40,41]). Thanks to STELLANTIS for supplying the specimens in this project but also for financial and technical supports during numerous collaborative projects in the last two decades.

### Appendix A. 3D rendering of subsurface grains truncated by a subsurface ring

See Fig. A.1.

### Appendix B. Resolved shear stress of grains intersecting a cross section plane

See Fig. B.1.

### Data availability

Data will be made available on request.
<figure><img src="image_25.png" /><figcaption>**Fig. A.1.** 3D Rendering of subsurface grains, truncated by a subsurface ring (50 µm thick), used to calculate the partial size of grains located on the specimen surface. The holes observed in the sub-surface ring are due to the non indexed zones or non reconstructed grains.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Fig. B.1.** A resolved shear stress map of grains intersecting a cross section plane (specimen nB-1B-2) showing the effect of macroscopic stress gradient on the maximum resolved shear stress under torsion loads.</figcaption></figure>

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