# Mixed-mode deformation in a rock bridge between two fault segments

Erina Prastyani, Benoît Cordonnier, Jessica Mcbeck, Jérôme Weiss, François Renard

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Erina Prastyani, Benoît Cordonnier, Jessica Mcbeck, Jérôme Weiss, François Renard. Mixed-mode deformation in a rock bridge between two fault segments. Tectonophysics, 2025, 897, 10.1016/j.tecto.2025.230624. hal-04933191

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# Mixed-mode deformation in a rock bridge between two fault segments

Erina Prastyani<sup>a,*</sup>, Benoît Cordonnier<sup>b</sup>, Jessica McBeck<sup>c</sup>, Jérôme Weiss<sup>d</sup>, François Renard<sup>a,d</sup>

## ARTICLE INFO

**Keywords:**

Time-lapse X-ray tomography

Fault growth

Westerly granite

Rock damage

## ABSTRACT

Analyzing how fracture networks develop from preexisting isolated fault segments in immature faults may provide knowledge of the preparation process leading to fault growth and dynamic rupture. We reproduce this process experimentally using triaxial compression experiments coupled with time-lapse in-situ synchrotron X-ray tomography in Westerly granite core samples (10 mm diameter, 20 mm height) that contain two parallel notches oriented at 30° with respect to the axis of the cylinder. We conducted the experiments at room temperature with a constant confining pressure of 20 MPa. We image microfracture development, characterize the microphysical processes of damage and fault growth in an intact rock bridge between the notches, and analyze the evolution of fractures oriented at 0°–17° (extensile) and 17°–32° (shear) as the rock approached failure. We also elucidate the strain localization process in the deforming rock using digital volume correlation. Our study offers a detailed comparison of microfracture and strain field development during fault growth.

Results indicate that the rock bridge between the notches becomes damaged with a damage rate and a fracture rate diverging as the rock approaches macroscopic failure. Immediately preceding failure, shear fractures dominate the fracture networks. DVC analysis showed that the deformation process is mixed-mode, accompanied by dilation and shear strain, with the shear strains being more localized than the dilation. Furthermore, regions of high dilative strain host both extensile and shear fractures. These findings provide valuable insights into the fault growth process within an intact rock bridge between two fault segments in nature, demonstrating how mixed-mode deformation facilitates fault maturation.

## 1. Introduction

Immature faults grow when isolated fault segments progressively link and coalesce (Tchalenko, 1970), breaking rock bridges that separate them. Understanding the evolution of faults is crucial to understanding the process of earthquake nucleation (Lockner et al., 1991; Lockner and Byerlee, 1992; Harbord et al., 2017). However, field observations aimed at elucidating long-term fault growth are currently constrained by time limitations and instrumentation capabilities. Consequently, laboratory studies of fault growth are vital for addressing this research gap.

Several studies using analog experiments have shown how fracture segment linkage can lead to the growth of normal faults (An, 1998; Mansfield and Cartwright, 2001; Schreurs, 2003; Schlagenhauf et al., 2008). Other studies have demonstrated how shear faults grow in laboratory experiments (Cox and Scholz (1988), Lockner et al. (1991), Otsuki and Dilov (2005)). These experiments typically measure new

faults on the 2D surface in analog systems or record acoustic emissions in 3D volumes to characterize the fault growth process. However, analog experiments must be scaled to natural systems, which involves assumptions about material properties, stress levels, and boundary conditions.

Numerous studies have also been conducted in the laboratory to mimic the fault growth process in intact rocks (Lockner and Byerlee, 1992; Reches and Lockner, 1994; Vermilye and Scholz, 1998; Lockner and Beeler, 2002; Paterson and Wong, 2005; Goebel et al., 2012; Kwiatek et al., 2014; Renard et al., 2017, 2018; Kandula et al., 2019). Fractures in the rock occur due to microcrack coalescence (Kranz, 1983). These microcracks often nucleate from the tips of preexisting flaws in rocks (Paterson and Wong, 2005), such as pores and grain boundaries. The type of preexisting flaws significantly affects how the fractures initiate, propagate, and coalesce (Zhao et al., 2019; Gómez et al., 2022). In the laboratory, several studies have created an artificially preexisting

<footnote><sup>a</sup> Njord Centre, Departments of Physics and Geosciences, University of Oslo, Boks 1072 Blindern, Oslo NO-0316, Norway</footnote>

<footnote><sup>b</sup> European Synchrotron Radiation Facility, 71 Av. des Martyrs, Grenoble 38000, France</footnote>

<footnote><sup>c</sup> Norwegian University of Science and Technology, Department of Geoscience, Trondheim NO-7491, Norway</footnote>

<footnote><sup>d</sup> ISTerre, Université Grenoble Alpes, Grenoble INP, Université Savoie Mont Blanc, CNRS, IRD, Université Gustave Eiffel, Grenoble 38000, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: erina.prastyani@mn.uio.no, prastyani.erina@gmail.com (E. Prastyani).</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Schematic model of how a rock core that contains two preexisting notches breaks during triaxial compression experiment. The present study investigates how these notches promote shear localization which leads to fault growth and system-size failure. The black-filled shape inside the notch is a piece of Teflon inserted to avoid the jacket becomes pinched into the notch during experiments. $σ_1$ and $σ_3$ are the axial stress and confining pressure. Two experiments were performed under constant confining pressure and ambient temperature, and the axial stress was increased until the sample reached failure.</figcaption></figure>

weakness in crystalline rock, such as a notch, to guide fault growth (Goebel et al., 2012; Aben and Brantut, 2023; Patton et al., 2023). Goebel et al. (2012) used acoustic emission monitoring to examine the relationship between microseismicity and fault heterogeneities as well as the occurrence of dynamic slip events. Instrumenting rock cores with in-situ fluid pressure sensors and acoustic emission monitoring, Aben and Brantut (2021, 2023) demonstrated that dynamic rupture propagation and fault slip in water-saturated notched Westerly granite can be stabilized by a dilatancy strengthening effect due to a drop in local pore pressure, resulting in quasi-static behavior during the rupture phase. Additionally, Patton et al. (2023) showed through acoustic emission monitoring that large-scale heterogeneities induced by saw-cut notches affect the predictability of failure behavior in Westerly granite samples. However, acoustics emission experiments have limitations as mode I fracture, or small-scale high frequency often not detected (Cartwright-Taylor et al., 2022), and recorded acoustic emissions only capture a tiny fraction of the total strain. Numerical modeling has also been used to mimic and predict the evolution of strike-slip faults in nature (Wang et al., 2017; McBeck et al., 2021b). However, numerical modeling may not fully represent the complex structures and processes observed in natural faults.

Furthermore, the relative importance of tensile and shear fractures during fault growth remains unclear. This is partly because most previous laboratory studies have relied on post-mortem microstructural analyses of deformed samples (Katz and Reches, 2004; Tal et al., 2016), which do not provide real-time insights into the dynamic evolution of fractures. Additionally, many of these studies have focused on axisymmetric loading of intact rock without preexisting notches, where shear localization typically occurs only at or after peak stress (Katz and Reches, 2004; Tal et al., 2016; Cartwright-Taylor et al., 2020).

To address these limitations, we propose experiments in which rock samples are imaged in three dimensions over time, under conditions analogous to the shallow crust, with continuous strain measurements until a well-developed fault forms. Here, we conducted two experiments to image how faults grow in Westerly granite core samples with parallel notches oriented at 30° relative to the maximum compressive axis. The two notches separate an intact rock bridge that becomes damaged with loading and in which fault growth occurs. Such geometry is relevant to study the weakening and failure of rock bridges in landslides or asperities in fault zones. We used time-lapse three-dimensional X-ray tomography imaging in triaxial compression experiments (Fig. 1). We provide in situ observations of the tensile and shear processes leading to failure, measuring the amount of damage that develops and tracking the spatial distribution of local high strain events using digital volume correlation (DVC) following previous studies by McBeck et al. (2019, 2020, 2021a). Unlike acoustic emission monitoring, which only captures

detectable seismic events, DVC provides both the seismic and aseismic components of the deformation (deformation that does not produce detectable seismic events).

Our results indicate that the compressive failure behavior of the notched Westerly granite occurs by progressive damage of the rock bridge, in which the fault propagates. This damage follows an acceleration toward failure, with the damage rate diverging more sharply than the fracture rate as approaching macroscopic failure, and is reminiscent of what physicists call a critical phenomenon, which has been observed previously in the system-size failure of intact rocks (Renard et al., 2018; Kandula et al., 2019). As failure approaches, the overall deformation of the notched Westerly granite is predominantly accommodated by fractures oriented at 17°–32°, characterized by a combination of high dilation and shear strain, exhibiting mixed-mode behavior with both fracture opening and shearing.

## 2. Materials and methods

### 2.1. Sample preparation

We drilled two core samples with a diameter of 10 mm and a height of 20 mm in a single block of Westerly granite. This low porosity crystalline rock was extracted from a quarry in Westerly, Rhode Island, USA, and contains 27.8 % quartz, 32.6 % potassium feldspar, 32.8 % plagioclase, 3.5 % biotite and 1.9 % muscovite (Chayes, 1950), with the mean grain size of approximately 0.75 mm. The initial porosity is lower than 1 % (Brace, 1965). Westerly granite has been widely studied in the rock mechanics community (Tapponnier and Brace, 1976; Hadley, 1976; Tullis and Yund, 1977; Soga et al., 1978; Wong, 1982; Kawakata et al., 1999; Lockner, 1998; Aben and Brantut, 2021, 2023; Patton et al., 2023) and is a model rock for the continental and seismogenic crust. Each core sample contains two parallel notches oriented at 30° from the maximum principal stress ($σ_1$). The length of the notches is close to 5 mm (longer side) and 3 mm (shorter side), and their aperture is 2 mm (SI 1, Fig. SI 1). They are made to guide the location and orientation of the final fault during the brittle failure process. We used Viton® fluoropolymer elastomer as a jacket surrounding the sample before inserting it into the deformation apparatus. The jacket aims to prevent the confining oil from entering the pore space of the rock. Furthermore, to prevent the jacket from entering the notch during the experiment, we inserted a Teflon piece inside each notch (Fig. 1).

### 2.2. In-situ time-lapse triaxial compression experiments

We used time-lapse synchrotron X-ray tomography, also called 4D X-ray imaging, to study the in-situ and *opercando* evolution of fractures in
<caption>**Table 1**</caption>

Experimental conditions. At room temperature, we deformed two rock samples with preexisting notches under a constant confining pressure $P_c$. $\sigma_D^f$ and $\epsilon_{zz}^M$ indicate the differential stress at failure and macroscopic axial strain, respectively.

<table><thead><tr><th>Sample name</th><th>P<sub>c</sub> (MPa)</th><th>σ<sub>D</sub><sup>f</sup> (MPa)<sup>a</sup></th><th>Total ε<sub>zz</sub><sup>M</sup></th><th>Rate of increase in axial stress</th><th>Number of 3D scans</th></tr></thead><tbody><tr><td>WGN-01</td><td>20</td><td>315.4</td><td>0.0089</td><td>10 MPa → 5 MPa /step</td><td>36</td></tr><tr><td>WGN-02</td><td>20</td><td>317.2</td><td>0.0098</td><td>5 MPa → 2 MPa → 1 MPa /step</td><td>80</td></tr></tbody></table>

<sup>a</sup> The samples failed in a brittle manner immediately after the last loading phase.

the two rock samples deformed under controlled conditions. We performed the experiments using the newly built KORE triaxial deformation apparatus. This apparatus can be installed on a rotating stage on the ID19 and BM18 beamlines at the European Synchrotron Radiation Facility. KORE apparatus is almost similar to the HADES apparatus (Renard et al., 2016); the main differences are that the maximum axial stress is 400 MPa (200 MPa for the HADES apparatus) and the sample dimensions are 20x10mm (10x5mm for the HADES apparatus). These specifications of the KORE apparatus allowed us to perform experiments using notched granite samples.

The experimental protocol for the time-lapse imaging of rocks followed previous studies (Renard et al., 2017, 2018; Kandula et al., 2019, 2022). We used the full synchrotron beam with 200 mA and a pink spectrum, with a 5.6 mm Al filter. Due to the filtering of the energy by the KORE rig, the energy reaching the sample was in the range of 40–60 keV. The X-ray beams then pass through the rock sample and hits the scintillator, producing a 2D image (radiograph) that measures how the X-rays were absorbed by the sample from a specific angle. We performed a 180° rotation, capturing 4500 projections, which resulted in an angular resolution of 0.04°. During the experiments, we acquired a series of three-dimensional X-ray microtomography data (tomograms)

with a spatial sampling of 6.5 µm/voxel side length. The axial stress was increased incrementally, and each tomograph scan was performed during a step of constant stress to prevent deformation from blurring the sample until the rock ultimately failed. The schematic diagram of the KORE deformation apparatus and how the entire sample was scanned into three subvolumes are shown in the SI 1, Fig. SI 1.

With the specified X-ray beam height of 10.53 mm used at beamline ID19, we acquired three subvolumes of each sample. The lower subvolume contains a part of the lower piston and the lower part of the sample. The middle subvolume contains the middle part of the sample, between the two notches. The upper subvolume contains a part of the upper piston and the upper part of the sample. An overlap of 2 mm between these three subvolumes was used so that they could be stitched later into a single volume. Each acquisition of the three subvolumes lasted for 3 min in total, including the time to offset the sample vertically by 8 mm between scans. Each reconstructed subvolume contains 1600 radiographs acquired at a 32-bytes grayscale resolution of X-ray absorption and has dimensions of 2560 × 2560 × 1600 voxels.

## 2.3. Experimental conditions

Table 1 shows the experimental conditions. We deformed the two samples, labeled as WGN-01 and WGN-02, under constant confining pressure of $P_c = 20$ MPa, at room temperature of 24–25 °C, and under dry conditions (without pore fluid pressure, and under atmospheric humidity). A confining pressure of 20 MPa represents the pressure condition in the upper crust at about 0.8 km depth if one assumes a rock density of 2600 kg/m³ and a gravitational acceleration of 9.8 m/s². We increased axial stress in successive steps until the core sample failed, and we reported the values of differential stress, $\sigma_D$, in the present study. The steps of axial stress increase were 10 MPa to 5 MPa for sample WGN-01 and 5 MPa to 2 MPa to 1 MPa for sample WGN-02. The WGN-01 sample served as a “scope” test, while the WGN-02 sample was used for more detailed experiment. Although we used slightly different rates of increase in axial stress when acquiring tomogram scans in the two

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Axial strain versus differential stress for all the acquired tomograms (black dots), and the tomograms shown here (red circles) for experiments WGN-01 (a) and WGN-02 (b) in a domain located between the two notches (inset area). Blue colors in the tomograms show the segmented pores and fractures, respectively with volumes larger than 1000 voxels. $\epsilon_{zz}^M$ and $\sigma_D/\sigma_D^f$ are the macroscopic axial strain where the tomogram is captured and the normalized proximity to failure, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
samples, the strain rate used in both experiments is nearly the same, falling within the range of 10⁻⁷/s (SI 3, Fig. SI 5). Furthermore, even if we had used different strain rates, Paterson and Wong (2005) noted that brittle fracture in dry rocks is relatively insensitive to small changes in strain rate compared to the behavior of ductile rocks. We acquired a total of 36 and 80 tomograms for samples WGN-01 and WGN-02, respectively. The differential stress at failure, $\sigma'_D$, at which both samples broke, was similar (315.4 MPa for WGN-01 and 317.2 MPa for WGN-02). We deformed the two core samples under similar conditions to ensure reproducible results.

We measured mechanical data during the experiment, to track the macroscopic deformation of the rock. Mechanical data includes axial and radial strains, from which we calculated the volumetric strain. We use the geological convention to represent shortening strains and compressive stresses with positive values. We measured the axial strain by tracking the movement of the upper piston in the tomograms during the deformation process and verifying that the lower piston remained immobile, following the procedure described in Renard et al. (2018, 2019a, 2020a, 2020b). Following the same procedure, we measured the radial strain and volumetric strain by tracking the change in diameter and volume of the rock sample in the tomograms (SI 3, Movie S1 and Movie S2). Fig. 2 shows the plot of differential stress, $\sigma_D$, versus macroscopic axial strain ($e''_{zz}$) from both samples. Radial and volumetric strains are shown in SI 3, Fig. SI 4.

## 2.4. X-ray tomograms reconstruction and segmentation

We reconstructed the 3D tomograms from the radiographs using both X-ray absorption and phase contrast imaging (Mirone et al., 2014). Then, we stitched each group of three subvolumes into a single volume to get the entire sample in a single tomogram. We processed the data using the software Avizo3D™ with the following workflow to extract information on cracks from the tomograms. First, we removed the beam hardening artifact that occurs in the image, making the center of the image appear brighter (SI 2, Fig. SI 2a and Fig. SI 3). To remove this artifact, we subtracted the mean gray value of each horizontal slice with the background image, which was calculated by averaging the gray values of xy slices across all z-axis slices (depth), thereby representing the entire rock volume (SI 2, Fig. SI 2a and Fig. SI 3). Second, we cropped the volumes to 2557 × 2557 × 3451 voxels. This cropping does not affect the data inside the rock sample because it only affects the area where the sample jacket and confining oil are located. Third, we segmented each volume using a global thresholding technique. For each volume scan, we could distinguish the microfracture from the rock because of their different range of densities and gray levels. The global thresholding technique separates the light-density materials (pores and cracks) from the high-density materials (rocks and minerals) (SI 2, Fig. SI 2b). We used global thresholding technique to segment the tomograms because it provides low computational cost. Additionally, varying the threshold value for ± 100 does not produce systematic change in the calculated fracture properties (volume, surface area, length, and thickness of the fracture) (McBeck et al., 2021b). Fourth, after segmenting the microfractures, we applied a cylinder mask to remove the jacket, piston, and confining oil around the rock sample, and a mask to remove the part of the rock volume that corresponds to the notches.

From the segmented image, we isolated groups of pore and fracture voxels that are directly connected in 3D space, based on the maximum possible number of neighbors (26) a voxel can have. The software also computed the covariance matrix and corresponding eigenvectors and eigenvalues of each of these pores and fractures, allowing to quantify geometrical properties. We utilized the module Label Analysis in the software Avizo3D™ to obtain statistics on the geometrical properties of segmented objects, such as volume, length of the main axes (e.g. eigenvalues of the covariance matrix), anisotropy, and orientation. Given

that the spatial resolution of the technique is two to three times the voxel size, we only selected the segmented objects with a volume larger than 30 voxels.

As the gray values of the fracture and pore are similar, our segmentation result still contains both of them. Therefore, we isolated segmented fractures from pores by filtering the anisotropy values of the segmented objects (one minus the aperture divided by the greatest length), following McBeck et al. (2020). We selected fractures with anisotropy values larger than the peak of the distribution (0.93 for both experiments), while values below this threshold were considered as pores (SI 5, Fig. SI 10).

## 2.5. Digital volume correlation analyses

We measured the spatial distribution of strain increments using the Digital Volume Correlation (DVC) technique that we applied on pairs of tomograms. DVC finds the best mapping of a subset in the reference tomogram [im1] with the subset from the subsequent tomogram [im2]. We obtained displacement vectors based on the intensity of correlation between two subsets. These displacement vectors are then used to calculate the incremental strain between the two tomograms. We used TomoWarp2, an open source to compute DVC (Tudisco et al., 2017).

Following the DVC calculation protocol that has been performed in McBeck et al. (2018) and Renard et al. (2019a, 2019b), we used a similar correlation window size of 10 voxels (65 µm), node spacing of 20 voxels (130 µm), and search window of 10 voxels (65 µm) as the input parameters of TomoWarp2. We used these parameters to make sure that each window contains a subvolume of at least one grain and maximizes the contrast for volume correlation. The output of TomoWarp2 consists of the list of spatial coordinates, including the displacement, rotation component, and correlation coefficient. Using the incremental displacements between pairs of tomograms, we calculated the first invariant ($I_1$) of the incremental strain tensor, and the von Mises strain ($J_2$). $I_1$ and $J_2$ represent incremental volumetric and shear strains, respectively. The positive and negative volumetric incremental strains, $I_1 > 0$ and $I_1 < 0$, show dilation and contraction, respectively.

We applied the DVC calculations to a series of pairs of tomograms, separated by almost constant increments of macroscopic axial strain. For samples WGN-01 and WGN-02, we divided the macroscopic axial strain into eleven and twenty equal increments and performed DVC calculations on the resulting tomogram pairs, yielding a total of ten and nineteen DVC calculations, respectively. We acquired more DVC calculations on sample WGN-02 because more scans were acquired compared to sample WGN-01 (Table 1).

The main outcome of the DVC calculations are time series of incremental volumetric and shear strains. To quantify how strain localization occurs within the rock sample, we present the evolution of high-magnitude dilation and shear strain components with magnitudes above the 95th percentile of the data population in a given tomogram pair.

Finally, due to the stitching step used to produce a single volume from three subvolumes, the overlapping regions between the subvolumes may contain artifacts in the form of some spatially correlated strain increments in the DVC data, with strain values below 10⁻³, which is close to the strain resolution of the DVC technique (Renard et al., 2019a). These artifacts appear as horizontal bands on the DVC volumes and are ignored when interpreting the DVC data.

# 3. Results

## 3.1. Imaging damage close to system-size failure

Fig. 2 illustrates the evolution of fracture growth in both experiments. To image damage, tomograms of both experiments were selected at a differential stress, $\sigma_D > 90\%$ of the differential stress at failure, $\sigma'_D$.
<figure><img src="image_5.png" /><figcaption>Fig. 3. Panels a) and c) show the evolution of the damage index, $D_{\phi}$, and the total number of fractures, respectively, as a function of the proximity to failure ($\sigma_D/\sigma_D^f$) in a domain located between the two notches. Panels b) and d) show the rate of increase of the damage index and the total number of fractures in semilog plots as a function of one minus the proximity to failure. $\sigma_D$ and $\sigma_D^f$ are the differential stress and the differential stress when failure occurs, respectively.</figcaption></figure>

In the experiment WGN-01, accumulation of fractures was observed mainly near the upper notch (Fig. 2a), even in the last tomogram acquired before failure when $\sigma_D$ is equal to 0.98 $\sigma_D^f$. In contrast, the accumulation of fractures was starting to localize within the rock bridge between the notches in the experiment WGN-02 (Fig. 2b). As failure approached in experiment WGN-02, fracture linkage became apparent, particularly in the last scan acquired at a differential stress level equal to 0.99 $\sigma_D^f$. Moreover, rotating the image by 180° along a vertical axis parallel to the axis of the rock cylinder shows that the fractures in the WGN-02 experiment are located parallel between the notches, but have not yet been linked to form a fault (Fig. 9).

## 3.2. Evolution of damage approaching failure

We tracked the evolution of several quantities as the samples approached failure, such as the dimensionless damage index (in linear and semi-log plots) and the total number of fractures (Fig. 3). The

damage index is defined as $D_{\phi} = \frac{\phi - \phi_i}{1 - \phi_i}$, where $\phi$ is the volume fraction of the void space (pores and fractures), at a given stress step, and $\phi_i$ is the initial volume fraction of the void space (initial porosity) before loading. Fig. 3c displays the total number of segmented fractures (after removing the pores).

Both samples show a significant increase in the damage index and the total number of fractures when $\sigma_D > 0.8 - 0.9 \sigma_D^f$ (Fig. 3a and c). Both samples exhibit similar values in terms of the evolution of the damage index (Fig. 3a) and the total number of fractures (Fig. 3c). However, when $\sigma_D > 0.96 \sigma_D^f$, the sample WGN-01 contains 1000–2000 less fractures than sample WGN-02.

For both samples, we found that the equation fitted in the semilog plot of the evolution of the damage index toward failure follows a linear trend (Fig. 3b, as a function of $1 - \frac{\sigma_D^f}{\sigma_D^f}$). The slope of the fitting equation is positive, indicative of a growth of damage, and exhibits almost similar values between the two samples, indicating a good reproducibility

<figure><img src="image_6.png" /><figcaption>Fig. 4. The damage and fracture rates from WGN-02 sample. a) The damage rate, $\frac{dD_{\phi}}{d\Delta}$, with $\beta_D$ value of 1. b) The fracture rate, $\frac{dN}{d\Delta}$, with $\beta_N$ values of 0.75 and 1. Note that the $\beta_N$ values of 0.75 and 1 are not derived by fitting the fracture number data to Eq. (1); instead, they represent approximations for $\beta_N$, and we select the value of $\beta_N$ that shows the largest $R^2$.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 5. The number of fractures (N - N₀) of the WGN-02, considers N = N₀ for Δ = 1, as a function of 1/βₙ⁻¹ [Δ⁻βₙ⁺¹ - 1] with different values of βₙ and R².</figcaption></figure>

between the two experiments. Meanwhile, the semilog plot of the evolution of the number of fractures to failure (Fig. 3d) does not follow a similar trend as the evolution of the damage index.

Following the analysis of Renard et al. (2018) who proposed that rock failure in crystalline monzonite rock is a critical phenomenon, we make the assumption that the damage and fracture rates diverge when approaching macroscopic failure and follow Eq. (1), i.e.:

$$ \frac{dX}{d\Delta} \sim -\Delta^{-\beta}, \qquad (1) $$

where X is the damage (D_φ) or the number of fractures (N), Δ = σ_D^f - σ_D, σ_D^f is a normalized distance to failure and σ_Y is a yield stress corresponding to the onset of microfracturing and damage, i.e. the mechanical response is essentially elastic below σ_Y (we found that σ_Y ≈ 0.85 σ_D^f), β is a critical exponent, and the negative sign on the right hand-side of Eq. (1) comes from the definition of Δ (decreasing as approaching faulting). Here we have chosen to normalize the stress by σ_Y because the number of fractures does not vary significantly between σ_D = 0 and σ_D = σ_Y (Fig. 3c). For the damage rate, we call the exponent β_D. We can test the

<figure><img src="image_8.png" /><figcaption>Fig. 6. The evolution of the spatial distribution of high incremental strain values (> 95th percentile) of dilation (I₁ > 0) and shear strain (J₂) in a domain located between the two notches (inset area) at four stages of macroscopic axial strain for both experiments in three dimensions. The thresholds of high incremental values (> 95th percentile) for dilation and shear are shown as I₁⁹⁵ and J₂⁹⁵, respectively. The colors indicate the location of the rock matrix that is experiencing high dilation (blue) and shear (red). Both dilation and shear strains may appear in different shades of color, indicating areas of concentrated strain regions. The high incremental values located horizontally in some of the plots correspond to the stitching artifact that occurs during the reconstruction process of the tomogram (shown by red arrows). eM and σ_D/σ_D^f are the macroscopic axial strain and the normalized proximity to failure, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 7. The evolution of the spatial distribution of high incremental strain values (> 95th percentile) of dilation ($I_1 > 0$) and von Mises strain ($I_2$) in a domain located between the two notches (inset area) at three stages of macroscopic axial strain for both experiments in two dimensions. $\sigma_D/\sigma_D^f$ is the ratio between the macroscopic axial strain and the normalized proximity to failure. The color bar labels of $\mu + \sigma$ refers to the mean plus one standard deviation. Approaching failure, the shear strain becomes more localized than the dilative strain.</figcaption></figure>

divergence of the rates (Eq. (1)), at least for the WGN-02 sample, by differentiating the damage data (Fig. 4a). We see that Eq. (1) is a reasonable model, over almost two orders of magnitude in $\Delta$ and with a $\beta_D$ exponent close to 1. If $\beta_D = 1$, Eq. (1) for the damage rate becomes $\frac{dD\phi}{d\Delta} \sim \frac{1}{\Delta}$, which implies:

$$D_{\phi} \sim -\ln(\Delta), \qquad (2)$$

which is consistent with the result of shown on Fig. 3b.

For the rate of the number of fractures, we call the exponent $\beta_N$. Calculating a fracture rate, $\frac{dN}{d\Delta}$, as shown in Fig. 4b suggests that $\beta_N < 1$ with $\beta_N \approx 3/4$ as a reasonable value. If $\frac{dN}{d\Delta} \sim -\Delta^{-\beta_N}$, with $\beta_N \neq 1$, this implies that:

$$N - N_0 \sim \frac{1}{\beta_N - 1} [\Delta^{-\beta_N + 1} - 1], \qquad (3)$$

where $N_0 \approx 7540$ is the plateau value observed for $\sigma_D < 0.85$ in sample WGN-02 (Fig. 3c). The integration considers that $N = N_0$ for $\Delta = 1$. This $N_0$ value is meaningless if one analyzes the rate, but has to be taken into account if analyzing the integrated number of fractures. As in Renard et al. (2018), we plot $N - N_0$ of the WGN-02 data as a function of $\frac{1}{\beta_N - 1} [\Delta^{-\beta_N + 1} - 1]$ for different values of $\beta_N$ on Fig. 5, and then search for the best fit. We find that $\beta_N = 0.8$ best fits the data for fracture rate with $R^2$ value larger than 0.99 (Fig. 5), while $\beta_D$ value for the damage rate is 1.0 (Fig. 4a).

### 3.3. Spatial distribution of local high strain events

We used DVC data to analyze the spatial distribution of dilation and shear strains in both experiments. Fig. 6 shows the evolution of the spatial distribution of high incremental values (above the 95th percentile) of dilation and shear strains in three dimensions across four DVC

calculations at different macroscopic axial strains. The full evolution of this spatial distribution is also available in SI (Movie S3 and Movie S4) for samples WGN-01 and WGN-02, respectively. Fig. 7 illustrates the same data in two dimensions. These figures highlight the strain localization process that leads to the growth and failure of the fault.

The spatial distributions of the incremental dilation and shear strains in sample WGN-01, shown in Figs. 6 and 7, indicate that high strain is concentrated around the upper notch. As the rock approaches failure, the high-strain regions extend downward from the upper notch. Figs. 6 and 7 also show the strain localization process in sample WGN-02. When $\sigma_D \approx 0.62 \sigma_D^f$, the rock exhibited high magnitudes of dilation and shear strain evenly spread out between the notches (Fig. 6a). The high incremental strain values (dilation and shear) in the notches, shown in Figs. 6 and 7, indicate that the Teflon pieces moved during deformation. Consequently, we excluded these values from interpretation and focused on the high strain events between the notches. As the stress increases, the high strain values of dilation and shear indicate that the future fault is already initiated at $\sigma_D \approx 0.83 \sigma_D^f$. As the rock approaches failure ($\sigma_D > 0.97 \sigma_D^f$), these high values of dilation and shear strain become progressively localized, facilitating the link between the two notches and the growth of the fault within the rock bridge. Comparing the spatial distribution of dilation and shear strain in sample WGN-02 just before failure (Fig. 7d), the dilation strain is more distributed, while the shear strain is more localized.

### 3.4. Evolution of extensile and shear fractures

To characterize how the shear and extensile deformation components of the fractures evolve throughout deformation, we follow the procedure proposed in McBeck et al. (2023), where each fracture is categorized as dominantly extensile or shear depending on its orientation with respect to the main compressive stress. In this study, we
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Panels a) and b) show the evolution of total volume of extensile ($v_{\theta}^{e}$) and shear ($v_{\theta}^{s}$) fractures in samples WGN-01 and WGN-02, respectively. Panel c) displays the ratio of these volumes ($r = v_{\theta}^{e}/v_{\theta}^{s}$). Panels d) and e) present the evolution of corrected extensile ($\bar{v}_{\theta}^{e}$) and shear ($\bar{v}_{\theta}^{s}$) fracture volumes in both samples. Panel f) illustrates the ratio between the corrected volume fractures in both samples ($r = \bar{v}_{\theta}^{e}/\bar{v}_{\theta}^{s}$). All panels are plotted as a function of the proximity to failure ($\sigma_{D}/\sigma_{D}^{f}$). The plots on the right with gray background shows the data at $\sigma_{D}/\sigma_{D}^{f} > 0.93$.</figcaption></figure>

assumed that the potential failure orientation of rock follows the Mohr-Coulomb criterion for intact rock under compression, given by the angle $\frac{\pi}{4} - \frac{\varphi}{2}$, where $\varphi$ is the angle of internal friction. The internal friction coefficient, which defines $\varphi$, ranges from 0.5 to 1.5 (Paterson and Wong, 2005). Thus, we define extensile fractures as those with an orientation less than 17° with respect to the vertical loading direction ($\sigma_{1}$), and shear fractures as those with orientation ranging from 17° to 32°.

Fig. 8a and b illustrate the total volume evolution of extensile ($v_{\theta}^{e}$) and shear ($v_{\theta}^{s}$) fractures for both samples. Shear fractures consistently dominate until failure (as also shown in SI 6, Fig. SI 11). To account for preexisting fractures, we correct the volumes by subtracting initial values: $\bar{v}_{\theta}^{e} = v_{\theta}^{e}(i) - v_{\theta}^{e}(1)$ and $\bar{v}_{\theta}^{s} = v_{\theta}^{s}(i) - v_{\theta}^{s}(1)$. Fig. 8d and e show the corrected volume evolution, highlighting shear fracture dominance before failure. Fig. 8c tracks the relative growth ratio $r = v_{\theta}^{e}/v_{\theta}^{s}$, and show that the ratio increases when $\sigma_{D}$ exceeds 0.8 $\sigma_{D}^{f}$ and 0.87 $\sigma_{D}^{f}$ for samples WGN-01 and WGN-02, respectively, but drops near failure ($\sigma_{D} \approx 0.98 \sigma_{D}^{f}$ and 0.99 $\sigma_{D}^{f}$), indicating shear-dominated failure. Lastly, Fig. 8f shows that the ratio $r = \bar{v}_{\theta}^{e}/\bar{v}_{\theta}^{s}$ goes above and below 1, reflecting mixed-mode deformation, yet shear dominance is evident just before failure. The trends indicating shear dominance just before failure, as highlighted in the combination where extensile-dominated fractures have orientations of 0°–17° and shear-dominated fractures have orientations of 17°–32°, are similarly observed across other orientation combinations (SI 6, Fig. SI 12).

### 3.5. Spatial correlation of the high strain events and fractures preceding system-size failure

Fig. 10 shows a three-dimensional rendering of the last tomogram before failure for both experiments. The image is superimposed with segmented extensile and shear fractures and the three-dimensional strain distribution map of high-incremental strain values (> 95th percentile) obtained from the DVC analysis.

In the experiment WGN-02, extensile and shear fractures are co-located in the area near the upper and lower notches (Fig. 10b(i) and (v)). In the experiment WGN-01, the co-location between extensile and

shear fractures could only be seen in the area close to the upper notch (Fig. 10a(i)). In both experiments, the total volume of the shear fractures was still higher than that of the extensile fractures (Fig. 8a and b).

The location where shear fractures were found in both experiments is the same as where shear strain was detected. This result can be observed in Fig. 10b(ii), (vi) for experiment WGN-02 and Fig. 10a(ii) for experiment WGN-01. The location of the shear fractures also coincides with the location of the high dilation strain, indicated by strong blue color shades in Fig. 10b(iv), (viii) for sample WGN-02 and Fig. 10a(iv) for sample WGN-01.

## 4. Discussion

### 4.1. Effect of the geometry of the notch tip and presence of pores

The notches serve as macroscopic defects within the rock, acting as focal points for stress concentration (Pluvinage and Gilbert, 2003). Our analysis, as illustrated in both segmentation results (Fig. 2) and DVC analysis (Figs. 6 and 7), captures the evolution of fracture growth and strain localization process in the rock bridge between the notches. In the experiment WGN-01, the propagation of fractures was mainly observed around the upper notch. We did not observe a continuous development of fractures and high-strain events coalescence between the two notches, probably because of the limited number of tomograms acquired close to failure. However, we observed a continuous development of fractures and coalescence of strain events between the two notches before system-size failure in experiment WGN-02.

The comparison between the notch geometry of sample WGN-01 and WGN-02 in Fig. 11 shows that both notches in sample WGN-01 have slightly sharper tips than in sample WGN-02. A sharper notch tip is expected to concentrate stresses to a larger extent than a blunt notch tip (Wang, 2006). Furthermore, we also observe a presence of pores just below the upper notch and above the lower notch of sample WGN-01 (Fig. 11a and c). The presence of heterogeneities, such as this pore, might add stress shielding and block the propagation of the rupture. This could prevent us from observing the fracture and coalescence of strain events before rock failure occurs in sample WGN-01.
Despite similar peak stress and road to failure between the two samples, the fracture propagation paths differ (Fig. 2). In WGN-01, the connection between the notches occurs much later than in WGN-02, potentially leading to a more sudden rupture over a larger surface. This delay is presumably due to strong, resistant heterogeneities within the sample (as suggested in Fig. 11a and c). Only time-lapse X-ray tomography imaging reveals these differences, offering valuable insights into how sample heterogeneities influence failure, rupture, and potentially total energy release.

## 4.2. Evolution of damage before failure

The theory of brittle compressive failure as a critical transition, proposed by Weiss et al. (2014), involves long-range elastic interactions and local disorder, resulting in finite-size effects. In intact quartz monzonite and Carrara marble rocks, Renard et al. (2018) and Kandula et al. (2019) demonstrated that damage evolution during compressive failure follows a power law divergence (Eq. (1)) toward faulting, arguing for such a critical transition. This power law divergence is also observed in Vu et al. (2019) and Patton et al. (2023), who measured the acoustic emission energy rate and acoustic emission event rate respectively in concrete, and in intact and notched samples of Westerly granite.

Our study, which examines samples with preexisting notches, also shows that the damage and fracture rates follow as well this type of evolution as the rock approaches failure (Eq. (1)), consistent with these previous studies. The model proposed in Eq. (1) is consistent with the data of sample WGN-02, for both the damage rate $\frac{dD_\kappa}{d\Delta}$ and the fracture rate $\frac{dN}{d\Delta}$. If the damage rate exponent, $\beta_D = 1$ in sample WGN-02, is smaller than the values measured in intact samples of monzonite ($\beta_D$ in the range 1.4–1.8 (Renard et al., 2018)) and Carrara marble ($\beta_D = 2.6$, (Kandula et al., 2019)), the existence of such divergence at $\sigma_D = \sigma_D^f$ still indicates that damage evolves as a critical transition between an unfractured state to a failed stage.

While both damage and fracture rates exhibit power law behavior in sample WGN-02, the damage rate shows a higher power law exponent ($\beta_D = 1$) than the fracture rate ($\beta_N = 0.8$), resulting in a sharper divergence of the former compared to the latter. Furthermore, the value of $\beta_N$ is similar to the exponent in the acoustic emission energy and event rate reported by Patton et al. (2023), who found values in the range 0.6–0.8.

There are two possible reasons for the faster damage rate compared to the fracture rate. First, as the stress increases during loading, the propagation of existing fractures may increase the damage volumetrically faster than the nucleation and coalescence of the fractures. Second, newly forming fractures, on average, may become larger as the rock approaches failure. Although we did not track the competition between the volumes of propagating, nucleating, and coalescing of the fractures in this study, McBeck et al. (2021c) demonstrated that the growth of preexisting fractures in intact low-porosity crystalline rock (monzonite) dominates over the nucleation of new fractures throughout loading. Therefore, the faster damage rate is more reliably attributed to the volumetric increase of propagating fractures rather than the nucleation of new fractures as the rock approaches failure.

## 4.3. Mechanisms of fracture and fault growth

The mechanisms of fault growth in notched Westerly granite are summarized in Fig. 12. At the beginning of rock loading, dilation strain is widely distributed across the rock matrix (Fig. 6a). According to Brace et al. (1966), this dilation is caused by pervasive microcracks aligned with the maximum compressive stress, $\sigma_1$. When the differential stress exceeds 80 % and 87 % of the differential stress at failure for WGN-01 and WGN-02, respectively, the growth rate of the total volume of extensile fractures increases relative to their earlier stages (Fig. 8c). As differential stress continues to rise, the total volume of stress-induced shear fractures increases substantially before failure (Fig. 8d and e).

<figure><img src="image_11.png" /><figcaption>Fig. 9. Segmented fractures (extensile and shear) in the final scan before failure for both experiments as seen from three different angles (left: front view, middle view, right: back view).</figcaption></figure>

Wong (1982) showed that some shear fractures in Westerly granite may involve shear modes influenced by the rotation of the stress tensor caused by grain-scale anisotropy. This rotation can also result from many individual extensile fractures linking together in an *en echelon* array, inclining the faults’ macroscopic trend in the direction of maximum compression (Peng and Johnson, 1972; Tapponnier and Brace, 1976; Lockner et al., 1991; Lockner and Byerlee, 1992; Reches and Lockner, 1994; Menéndez et al., 1996; Renard et al., 2019a; Scholz, 2019). Our observations are also consistent with the “classical scenario” described by Ashby and Sammis (1990), who demonstrated that the final large shear fault results from the coalescence of *en-echelon* arrays formed from wing cracks that nucleate from frictional sliding along preexisting defects, such as joints and boundaries. In the brittle regime, transgranular fractures, which account for the majority of extensile cracks in crystalline rock (Renard et al., 2018), promote the quick linkage of mesoscale shear fractures, which in turn leads to fault nucleation, a process known as shear localization (Lockner et al., 1991; Brace et al., 1966; Paterson and Wong, 2005). As loading continues and differential stress exceeds 97 % of the failure stress, dilation and shear strains have become progressively localized (Fig. 6c).

As the rock approaches system-size failure, we observed a mixture of dilation and shear strain (Fig. 10 and SI 7, Fig. SI 13), where shear strain
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Three-dimensional rendering of the last tomogram before failure for samples WGN-01 (a) and WGN-02 (b), overlaid with segmented fractures (with volume exceeding 30 voxels) and strain distribution map of high incremental values (> 95th percentile) obtained from DVC. The yellow and gray colors indicate segmented extensile and shear fractures respectively, while the blue and pink color shades represent the incremental dilation and shear strain, respectively. The spatial distribution overlap from the areas near the upper and lower notches are shown in panels (i-iv) and (v-viii), respectively. Specifically, panels (i, v) show the overlap between the shear and extensile fractures; panels (ii, vi) show the overlap between the shear fractures and incremental shear strain DVC; panels (iii, vii) show the overlap between extensile fractures and incremental dilation strain DVC; and panels (iv, viii) show the overlap between the shear fractures and incremental dilation strain. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

is more localized than the dilation strain (Fig. 7d). Our finding about the mixture of dilation and shear strain is consistent with the previous study by McBeck et al. (2020) who demonstrated that the most frequent deformation mode in intact crystalline rock (monzonite) under higher differential stress and moderate confining pressure is a paired increase in dilation and shear strains. As shown in Fig. 10b(iii), (vii), (iv), (viii), and a(iii), (iv), the mixture of dilation and shear strain allows fractures to accommodate both opening and shearing, similar to what has been proposed for the formation of wing cracks (Ashby and Sammis, 1990; Wong and Chau, 1998).

Immediately preceding failure, the fracture networks are dominated by shear fractures (Fig. 8d and e). Large shear fractures accommodate deformation in the last scan before failure (Figs. 9 and 10). This observation is consistent with theoretical and laboratory observations of

triaxial compression under moderate confining pressure, which suggests a shift to shear-dominated deformation (Peng and Johnson, 1972; Wong, 1982; Lockner et al., 1991; Reches and Lockner, 1994; McCormick and Rutter, 2022). Our observation differs from McBeck et al. (2023), who demonstrated extensile failure as the final failure mode of intact Westerly granite. Two factors might explain the difference between our study and that of McBeck et al. (2023). First, the conditions that we used to deform the rock differ from those of McBeck et al. (2023). Specifically, McBeck et al. (2023) used an effective pressure of 5 MPa (with and without pore fluid pressure), while we used 20 MPa. The 20 MPa confining pressure, corresponding to a greater depth than 5 MPa, promotes a shear failure rather than only extensile. A transition from axial splitting and extensile, mode I fracture at uniaxial conditions and low confining stress, to more shear-dominated deformation at higher
<figure><img src="image_13.png" /><figcaption>**Fig. 11** Geometry of the upper and lower notches of samples WGN-01 and WGN-02. a) Upper notch of sample WGN-01, with a sharper tip (red circle) and a large pore below the notch (yellow circle). b) Upper notch of sample WGN-02. c) Lower notch of sample WGN-01, showing a large pore above the notch (yellow circle). d) Lower notch of sample WGN-02. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

confining stress are also observed in other previous studies (Tapponnier and Brace, 1976; Jaeger and Cook, 2014; McCormick and Rutter, 2022). Second, the notch itself plays a role in facilitating the growth of damage in the rock bridge between the notches, aligning with a preferred failure plane based on the Coulomb criterion. In this scenario, fractures propagate along this failure plane guided by the notches, transitioning to shear failure. This shear failure behavior has also been demonstrated in previous studies using notches (Goebel et al., 2012; Aben and Brantut, 2021, 2023; Patton et al., 2023).

### 4.4. Implication to fault growth at larger scales

Our laboratory experiments provide valuable information on the fault growth process between two fault segments in nature. This geometry is often encountered in immature fault zones (Tchalenko, 1970). We studied specifically how dilation and shear strains are localized as the rock bridge approaches failure (Figs. 2 and 6). This localization mirrors the natural process in which the transition from distributed microcracking to localized shear zones facilitates the growth and maturation of faults.

Field studies indicate that many natural faults originate from pre-existing structures such as joints, veins, or pressure solution surfaces (Crider and Peacock, 2004). Before faulting initiates, an array of *en echelon* joints often develop parallel to the maximum compressive stress, $\sigma_1$. These joints later act as weaknesses that facilitate the initiation and growth of faults. As stress accumulates around these preexisting structures, slip occurs across the joints, potentially leading to the formation of wing cracks between the *en echelon* joints. This process is crucial because

<figure><img src="image_14.png" /><figcaption>**Fig. 12.** Road to failure of notched granite. a) The future fault between the notches is already initiated, indicated by the high dilation strain value. The accumulation of fractures is mainly concentrated in the upper notch. The deformation mode is mixed-mode, indicated by the ratio of extensile to shear that fluctuates around 1. b) Both dilation and shear strain begin to localize progressively. c) Highly localized dilation and shear strain. Shear fractures accommodate the overall deformation before system-size failure of the rock bridge located between the notches.</figcaption></figure>

linking existing fault segments is a more efficient mechanism for fault growth than propagation from fault terminations. This is due to the tendency of individual faults to lengthen in significant jumps, optimizing the overall evolution of the fault network and the localization of the strain (Mansfield and Cartwright, 2001; Cooke and Madden, 2014; Wang et al., 2017). Thus, as faults mature, wing cracks propagate, the shear zones narrow, and fault slip may become more efficient with less off-fault deformation (Hatem et al., 2017; McBeck et al., 2021b).

## 5. Conclusion

To capture detailed in-situ and operando fracture development and strain localization process in Westerly granite with 30° notches orientation relative to the maximum compressive axis, we conducted time-lapse three-dimensional X-ray tomography imaging in triaxial compression experiments combined with digital volume correlation analyses. Our results indicate that as stress increases, fractures grow and localize parallel to the notch direction. These fractures damage the rock bridge between the notches, with accelerating trends in damage and fracture rates, reminiscent of a critical phase transition. Prior to system-size failure, the overall deformation is mainly accommodated by shear fractures with both dilation and shear strains. This observed mixed-mode strain behavior, involving fracture opening and shearing,
provides insight into the complex interplay between different deformation modes during faulting. Consequently, the fault propagates by first damaging the rock bridge between the notches and then localizing tensile and shear fractures in the damaged bridge before the system-size failure.

Future studies may consider varying the angle of the notches to further explore the onset and relative contribution of shear and dilation strains as the rock approaches failure. Such research may help elucidate the mechanisms driving fault growth and earthquake nucleation, offering deeper insights into fault dynamics across different geological settings.

Supplementary data to this article can be found online at https://doi.org/10.1016/j.tecto.2025.230624.

### CRediT authorship contribution statement

**Erina Prastyani:** Writing – original draft, Visualization, Investigation, Formal analysis, Conceptualization. **Benoît Cordonnier:** Writing – review & editing, Validation, Supervision, Data curation. **Jessica McBeck:** Writing – review & editing, Supervision, Software, Methodology. **Jérôme Weiss:** Writing – review & editing, Supervision, Formal analysis. **François Renard:** Writing – review & editing, Validation, Supervision, Investigation, Funding acquisition, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

Data will be made available on request.

### Acknowledgments

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 101019628 BREAK). FR, JM, and EP acknowledge support from the project FricFrac funded by the Center for Advanced Study (CAS) at the Norwegian Academy of Science and Letters during the academic year 2023-2024.

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