<figure><img src="image_1.png" /></figure>

HAL
open science

# Spatial Persistence of High Strain Events During Brittle Failure

Jessica Mcbeck, Benoît Cordonnier, Wenlu Zhu, François Renard

► To cite this version:

Jessica Mcbeck, Benoît Cordonnier, Wenlu Zhu, François Renard. Spatial Persistence of High Strain Events During Brittle Failure. Geophysical Research Letters, 2024, 51 (21), pp.e2024GL110668. 10.1029/2024GL110668. hal-05099490

HAL Id: hal-05099490

https://cnrs.hal.science/hal-05099490v1

Submitted on 5 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Geophysical Research Letters®

## RESEARCH LETTER

10.1029/2024GL110668

**Key Points:**

* X-ray tomography quantifies the evolving spatial evolution of high strain events during brittle failure

* At the onset of strain localization, on average 65% of the volume of the rock cores undergo dilation

* Localized zones of high strain events persist in space from the onset of loading

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

J. McBeck,
j.a.mcbeck@geo.uio.no

**Citation:**

McBeck, J., Cordonnier, B., Zhu, W., & Renard, F. (2024). Spatial persistence of high strain events during brittle failure.

*Geophysical Research Letters*, *51*, e2024GL110668. https://doi.org/10.1029/2024GL110668

Received 5 JUN 2024

Accepted 15 OCT 2024

**Author Contributions:**

**Conceptualization:** Jessica McBeck, Wenlu Zhu, François Renard

**Data curation:** Benoît Cordonnier

**Funding acquisition:** Jessica McBeck, François Renard

**Investigation:** Benoît Cordonnier, Wenlu Zhu

**Methodology:** Jessica McBeck, Benoît Cordonnier

**Project administration:** François Renard

**Resources:** François Renard

**Supervision:** François Renard

**Writing – original draft:** Jessica McBeck

**Writing – review & editing:** Wenlu Zhu, François Renard

## Spatial Persistence of High Strain Events During Brittle Failure

<figure><img src="image_3.png" /></figure>

Jessica McBeck¹,² ID, Benoît Cordonnier³, Wenlu Zhu⁴ ID, and François Renard¹,⁵ ID

¹Departments of Physics and Geosciences, Njord Centre, University of Oslo, Oslo, Norway, ²Now at Department of Geoscience, NTNU, Trondheim, Norway, ³European Synchrotron Radiation Facility, Grenoble, France, ⁴Department of Geology, University of Maryland, College Park, MD, USA, ⁵Université Grenoble Alpes, Université Savoie Mont Blanc, Université Gustave Eiffel, CNRS, IRD, Grenoble, France

**Abstract** The onset of brittle failure in rocks includes dilatancy and strain localization. To better understand this nucleation process, we analyze the evolution of the local three-dimensional strain tensor using X-ray tomograms acquired during triaxial compression experiments on granite and sandstone. The onset of the localization of the compaction, dilation, and shear strain occurs when ~65% of the rock volume experiences dilation. Tracking the locations of the high strains throughout loading suggests that the deformation that occurs early in loading influences the location of the system-sized fracture network that produces macroscopic failure. This influence is larger in the sandstone experiments than the granite experiments, likely due to the microstructure of the sandstone. These results have important implications for detecting precursors to catastrophic failure.

**Plain Language Summary** We investigate the fundamental processes that lead to brittle failure in rocks. We deform two common types of crustal rocks, granite and sandstone, under upper crustal stress conditions. As the stress applied to the rock increases, the rock tends to expand (dilate) more than compact, particularly as it approaches catastrophic, macroscopic failure. A larger portion of the rock undergoes dilation when the strain field starts to localize, indicating that accelerating dilation is a precursor to macroscopic failure. We observe different localization patterns in the rocks: in sandstone, strain localization progresses monotonically with increasing stress, whereas phases of delocalization can occur in the granite. Two competing models describe the development of the system-sized fracture network that produces macroscopic failure: the network develops from (a) the coalescence of fractures that form early in loading, or from (b) the propagation of a process zone of interacting fractures through relatively intact rock. We find that the high strain events persist at the same location throughout the experiments more than expected by chance, particularly in experiments on sandstone. The results provide perhaps the most robust experimental confirmation yet that the fracture network that causes macroscopic failure evolves from the deformation that occurs earlier in loading.

## 1. Introduction

Analytical formulations from linear elastic fracture mechanics struggle to describe the transition from distributed, isolated fracture propagation to linkage and coalescence in rocks under compression in the brittle regime (e.g., Jaeger et al., 2009). This transition controls the timing of catastrophic failure of rocks and other heterogeneous materials, and the production of precursors to catastrophic failure (e.g., Cartwright-Taylor et al., 2020). One of the most consistent precursors is the macroscopic dilation of the rock (Paterson & Wong, 2005). Early laboratory measurements on granite suggest that the opening of mode I fractures aligned parallel to the maximum compression direction drives the increase of dilatancy (e.g., Brace et al., 1966; Mogi, 1977; Peng & Johnson, 1972; Tapponnier & Brace, 1976). Following macroscopic failure, the identified system-sized fracture network is often oriented oblique to the maximum compression direction, and thus in the direction expected to accommodate shear, under a few megapascals of confining stress (e.g., McCormick & Rutter, 2022). Previous analyses suggest that the linkage of en échelon mode I fractures produces this network (Brace et al., 1966; Lockner et al., 1991; Peng & Johnson, 1972; Petit & Barquins, 1988; Stanchits et al., 2006). In contrast, such networks may develop from a process zone of interacting fractures that propagates into relatively unfractured regions of the rock following the peak stress (Reches & Lockner, 1994). According to this idea, the deformation that precedes the peak stress has only a limited influence on the location and geometry of the fracture network that produces catastrophic failure.

© 2024. The Author(s).
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.
In addition to dilatancy, deformation localization has been recognized as a critical component of catastrophic failure, including large crustal earthquakes and macroscopic failure during triaxial compression (e.g., Ben-Zion & Zaliapin, 2020; Lockner et al., 1991; Rizzo et al., 2018; Rudnicki & Rice, 1975). Recent analyses indicate that localization can occur preceding the peak stress in triaxial compression experiments on rock cores that lack through-going faults (McBeck, Ben-Zion, & Renard, 2021; McBeck et al., 2022), and in experiments with through-going faults (Bolton et al., 2023; Dresen et al., 2020). However, fractures, strain events, and low-magnitude earthquakes can experience phases of reduced localization or delocalization (Ben-Zion & Zaliapin, 2020; McBeck, Ben-Zion, & Renard, 2021; McBeck et al., 2022), in contrast to expectations from linear elastic fracture mechanics (e.g., Isida, 1971). These phases of delocalization complicate efforts to forecast the timing of catastrophic failure using localization statistics.

Previous analyses suggest that the amount of preexisting fractures, or heterogeneity, influences localization (e.g., Tang et al., 2000). The greater amount of preexisting fractures in heat-treated granite cores promotes delocalization of the strain events (McBeck, Cordonnier, Ben-Zion et al., 2023). This type of damage influences localization by inhibiting propagation into stress shadows created by neighboring fractures, or promoting propagation via coalescence with neighboring fractures. Another type of heterogeneity that influences fracture development arises from the cemented grain structure of sandstone. The stress concentrations that develop between neighboring grains can nucleate fractures on the edges of grains (intergranular) and within grains (intragranular) (e.g., Baud et al., 2014; Zhang et al., 1990). Differences in the mechanical properties of inter- and intragranular material in sandstone can inhibit fracture growth. Consequently, the contrasting microstructures of cemented granular aggregates such as sandstone, and of low porosity, crystalline rocks with interlocking minerals such as granite, have a profound influence on fracture development in these rocks.

We conduct triaxial deformation experiments on water-saturated intact and heat-treated Westerly granite and Darley Dale sandstone. We compare deformation among these rock types in order to test the influence of pre-existing damage and heterogeneities on deformation. We introduce fluids (water) in these experiments because water is pervasive in the upper crust and can influence fracture network development through processes such as subcritical crack growth (Atkinson, 1984). Dynamic synchrotron X-ray microtomography experiments reveal the evolving three-dimensional microstructure of the deformed samples. Using Digital Volume Correlation (DVC), we calculate the local, internal strain fields from pairs of tomograms. The results provide one of the most comprehensive comparisons of the stress conditions at the onset of strain localization, and the onset of dilation dominance, when more than 50% of the rock volume undergoes dilation. In these experiments, strain begins to localize after the onset of dilation dominance, and before the peak stress. To test the idea that the system-sized fracture network that leads to catastrophic failure evolves from the coalescence of fractures that develop early in loading (Reches & Lockner, 1994), we quantify the degree to which the high strain events overlap in space throughout loading in each experiment. The results suggest that the deformation that occurs early in loading influences the location of the final, system-sized fracture network that produces macroscopic failure, particularly in Darley Dale sandstone, and to a lesser extent, in Westerly granite.

## 2. Methods

To characterize rock deformation in both granite and sandstone, we perform 12 triaxial compression experiments while acquiring X-ray tomograms on beamline ID19 at the European Synchrotron Radiation Facility (ESRF) using the Hades deformation apparatus (Renard et al., 2016). We deform heat-treated granite, intact granite, and sandstone cylinders to test the influence of heterogeneities, such as porosity, on localization. The samples are 10 mm in height and 4 mm (granite) or 5 mm (sandstone) in diameter. The smaller diameter of the granite cores allows these cores to fail within the force limit of the Hades apparatus. Text S1 in Supporting Information S1 describes the experimental protocol, reconstruction methods, and postprocessing steps. In each experiment, we impose a constant confining stress and fluid pressure, and then increase the axial stress in steps until the rock fails in a sudden stress drop. Table S1 and Table S2 in Supporting Information S1 describe the loading conditions of each experiment. The experiments on Darley Dale sandstone have the same effective pressure, the confining pressure minus the pore fluid pressure, $P_e = 10$ MPa. Five of the six experiments on Westerly granite have $P_e = 5$ MPa, and one has $P_e = 10$ MPa. Figure S1 in Supporting Information S1 shows the differential stress and macroscopic axial strain, $\epsilon_{zz}$, when each tomogram was acquired. The initial porosity of the cores of Darley Dale
sandstone used here are about 11%, while Westerly granite has initial porosities of <1% (Text S1 in Supporting Information S1).

We use DVC of pairs of tomograms to quantify the three-dimensional strain fields with the software TomoWarp2 (Tudisco et al., 2017). For each experiment, we identify the tomograms for the calculations by separating each experiment into 20 equal increments of macroscopic axial strain, $ε_{zz}$, following our previous analyses (e.g., McBeck et al., 2018). Text S2 in Supporting Information S1 describes additional details of these calculations, including the spatial resolution (0.13 mm). The resulting data are time series of the three-dimensional incremental strain tensor field within the entire volume of the rock sample, from which we calculate the incremental volumetric strain (the first invariant of the strain tensor, $I_1$) and the incremental shear strain (the second invariant of the deviatoric strain tensor, $J_2$). In the adopted sign convention, a positive $I_1$ represents a volumetric strain increase (dilation), and a negative $I_1$ represents a volumetric strain decrease (compaction). The strain calculations use an engineering sign convention while the stress measurements use the geologic sign convention (compressive stress is positive). Figure S1 in Supporting Information S1 shows the macroscopic stress and strain conditions of the tomograms used in the DVC calculations.

# 3. Results

## 3.1. Strain Localization and Dilation Dominance

The local, three-dimensional strain fields captured in triaxial compression X-ray tomography experiments reveal fundamental characteristics of strain localization (Figure 1). Under lower differential stress, the local incremental strain fields in experiments on sandstone and granite appear relatively distributed or delocalized, with high values (the mean plus two standard deviations) spread throughout the rock core (Figure 1). With increasing differential stress, the high values of the dilation, $I_1 > 0$, and shear strain, $J_2$, cluster together, or localize. The high strain values are more localized immediately preceding macroscopic failure in the sandstone experiment than the granite experiment shown in Figure 1 (Animation S1, Animation S2). All but one (DD04) sandstone experiments show progressively increasing localization of the dilation and shear strains (Figure S2 in Supporting Information S1). The exception, DD04, produces a concentrated dilation and shear zone early in loading that evolves into a larger structure with increasing differential stress. In contrast, the majority of the granite experiments do not produce monotonic increases in localization throughout loading (Figure S3 in Supporting Information S1). Instead, phases of localization are often followed by various degrees of delocalization before localization occurs again.

To quantify localization, we extract the locations of the highest magnitudes of the incremental strain populations ($>80$th percentile of the compaction, dilation and shear strains at a given stress state) in each DVC calculation, identify the polyhedron that surrounds these high strains, and calculate the volume of this polyhedron (Figure 2d inset), following the approach of McBeck, Cordonnier, Ben-Zion et al. (2023). We then report this volume as the volume of the polyhedron divided by the total volume of the rock core used in the calculation, $v_{loc}$. When $v_{loc}$ is close to one, the high strains are diffusely distributed throughout the rock, and when $v_{loc}$ is lower, the high strains are more localized (Figure 2d inset). Later, we show that using different thresholds to identify the high strains does not change the key conclusions of the analysis.

In the sandstone experiment DD05, the shear strain and dilation localize with increasing stress, producing smaller $v_{loc}$, while the compaction does not localize by similar amounts, producing $v_{loc}$ with similar values earlier and later in loading (Figure 2c). In the granite experiment WG14, all of the strains localize, producing smaller $v_{loc}$ closer to macroscopic failure (Figure 2d).

In addition to the onset of strain localization, Figure 2 highlights another fundamental behavior of strain accumulation. At the beginning of each experiment, compaction ($I_1 < 0$) dominates the volumetric strain field (Figures 2a and 2b). Then, the proportion of the total volume of rock occupied by compaction, $v_{vol}$, decreases, while the $v_{vol}$ of the dilation increases with loading. The $v_{vol}$ of the dilation at the end of each experiment is on average (± one standard deviation) 0.78 (± 0.12). This result indicates that some portion of the rock cores undergoes compaction immediately preceding catastrophic failure.

Figure 2 and the supporting figures (Figures S4 and S5 in Supporting Information S1) show that the onset of dilation dominance, when more than 50% of the rock undergoes dilation, generally occurs early in loading, and in particular, in the linear phase of the macroscopic differential stress and axial strain. The one exception to this trend
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Slices of tomograms and the local, incremental strain field in experiments on sandstone (a) and granite (b). The images in the first row show two-dimensional slices of portions of tomograms acquired immediately preceding macroscopic failure. The plots in the second row show the macroscopic axial strain and differential stress when each tomogram was acquired (black dots), and the conditions of the tomograms used in each Digital Volume Correlation calculation (blue dashed lines). The red and gray rectangles and numbers indicate the mechanical conditions of the strain slices shown in the figure. The slices show the compaction, ($I1 < 0$), the dilative strain ($I1 > 0$), and the shear strain, $J2$. When $I1$ is positive (dilative), it is shown as zero in the slice of the compaction, and vice versa for the slices of dilation. The range of the color bars are set by the mean, $\mu$, and standard deviation, $\sigma$, of the incremental local strain calculated for the pair of tomograms.</figcaption></figure>

is experiment WG05 (Figure S4 in Supporting Information S1), which only experiences dilation dominance late in loading, and close to macroscopic failure.

To compare when the onset of localization occurs among the different experiments, we identify the differential stress, $\sigma_D$, when $v_{\text{loc}}$ becomes less than 0.95, divided by the $\sigma_D$ at macroscopic failure, $\sigma_D^F$, $\bar{\sigma}_D = \sigma_D/\sigma_D^F$. If $v_{\text{loc}}$ drops below 0.95, but is above this threshold at the end of the experiment, then we classify that strain component as not localizing. Varying this threshold to 0.9 does not change the key conclusions of the analysis.

To identify when the dilation begins to dominate the system, we identify $\bar{\sigma}_D$ when the $v_{\text{vol}}$ of the dilation is greater than the $v_{\text{vol}}$ of the compaction. To identify the $\bar{\sigma}_D$ at the onset of dilation dominance, and the $\bar{\sigma}_D$ at the onset of
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Evolution of the volume of rock occupied by the volumetric strain components, $v_{vol}$, and evolution of the localization of the strain fields, $v_{loc}$, for representative experiments on sandstone (a, c) and granite (b, d). (e–f) Normalized differential stress, $\overline{\sigma}_D = \sigma_D/\sigma_D^F$, of key events for the sandstone (e) and granite (f) experiments. The inset in (d) shows $v_{loc}$ calculated at two different loading steps for the shear strain >80th percentile (black dots) and the corresponding polyhedron (red). Both the localization and volumetric strain evolutions are reported as the proportion of the volume out of the total volume of the rock used in the Digital Volume Correlation analysis. In the evolutions of $v_{vol}$ (a–b), the light and dark blue lines show the volume occupied by the dilation and compaction, respectively. The pink lines and right axis show the differential stress, $\sigma_D$. The triangle shows when the $v_{vol}$ of the dilation begins to exceed the $v_{vol}$ of the compaction, indicating the onset of dilation dominance. In the evolutions of $v_{loc}$ (c–d), the dark blue, light blue, and red lines show the localization of the compaction, dilation, and shear strain, respectively. The circles show when $v_{loc}$ first becomes less than 0.95, indicating when the strain fields begin to localize. When $v_{loc}$ is greater than 0.95 at the end of loading, then we do not classify that strain component as localizing, and no circle is shown. (e–f) Timing of key events in terms of $\overline{\sigma}_D = \sigma_D/\sigma_D^F$ for sandstone (e) and granite (f) experiments. The triangles show when the volume of the rock, $v_{vol}$, dilating exceeds the volume of rock compacting. The circles show when the compaction (dark blue), dilation (light blue), and shear strain (red), begin to localize, that is, when $v_{loc}$ of the high strains (>80th percentile) becomes below 0.95. When no circles are shown, for example, for granite experiment WG05 (f), then the strain components did not localize using the chosen thresholds of high strains and localization onset.</figcaption></figure>
strain localization, we linearly interpolate between the data provided by the discrete time steps of the tomogram calculations. Consequently, the symbols shown in Figure 2 that specify this timing may fall on a line between data points.

We now compare the $\tilde{\sigma}_D$ at the onset of dilation dominance and onset of strain localization among all the experiments (Figures 2e and 2f). Some of the experiments do not produce localization for all or some of the three strain components using the chosen criteria of localization (WG05, WG18). For the strain components that experience localization, the onset of this localization always occurs following the onset of dilation dominance. The timings of these events are similar for the experiment DD04, but in all the other experiments, the rock accumulates several tens of megapascals of differential stress between the onset of dilation dominance and the onset of localization (Table S3 in Supporting Information S1). At the onset of localization of the three strain components, on average 60% ± 6% of the rock volume undergoes dilation for granite and 73% ± 12% for sandstone (Table S4 in Supporting Information S1). Figures S6 and S7 in Supporting Information S1 show the results using the 80th and 70th percentiles as thresholds of the high strains, and that varying this threshold does not change the central results.

These data reveal key insights into the monotonicity of localization. In particular, the experiments on damaged (heat-treated) granite experience maximum localization earlier in loading than the intact granite or sandstone, on average at 0.78 $\tilde{\sigma}_D$, rather than 0.92 $\tilde{\sigma}_D$ (intact granite), and 0.99 $\tilde{\sigma}_D$ (sandstone) (Text S3, Figure S8 in Supporting Information S1). Here, we consider maximum localization to occur when the $v_{loc}$ is the smallest. Consequently, the monotonicity of the strain localization process increases from the damaged granite to the intact granite and sandstone. In other words, the damaged granite is more likely to experience phases of delocalization than the intact granite or sandstone. These results are consistent with previous work that linked the amount of heterogeneities with the production of precursors (Cartwright-Taylor et al., 2020; McBeck, Ben-Zion, et al., 2021; McBeck, Cordonnier, Ben-Zion et al., 2023; Tang et al., 2000; Vasseur et al., 2015).

## 3.2. Spatial Persistence of the High Strains

A key question about rock deformation leading to catastrophic failure is whether the deformation that develops early in loading and preceding the peak stress controls the location of the system-sized fracture network that produces macroscopic failure. Early analyses on granite suggested that the location of fractures that form early in loading do not strongly influence the eventual location of the fracture network that produces macroscopic failure (Lockner et al., 1991; Moore et al., 1990). These observations led to the idea that faults develop by the formation of a process zone of interacting fractures (Reches & Lockner, 1994). In this framework, the interaction of a few mode I fractures leads to the development of a process zone that extends into relatively pristine rock via fracture growth at the tip of the zone. This process zone begins to propagate and undergo shear only after the peak stress.

Here, we test this idea by tracking the position of the high strain events throughout loading. Our experiments only include deformation preceding the peak stress, and so according to Reches and Lockner (1994), the positions of the high strain events are expected to shift in space randomly. We thus track the positions of the high (>80th and 70th percentile threshold) values of the compaction, dilation and shear strain. We then calculate the volume fraction of the high strain events that overlap from one DVC calculation to the next, $f_{over}$, using the high strains identified in the current, or $t_n$, step, $v_{high}$, and the volume that the high strains overlap in step $t_n$ and the previous step, $t_{n-1}$, $v_{over}$, as $f_{over} = v_{over}/v_{high}$ (Figure 3a). $f_{over}$ is thus the proportion of the high strain events that overlap in space from one step to the next. We report the difference between this value, and the $f_{over}$ calculated from synthetic data sets created by spatially shuffled strain values, $f_{over}^{rand}$, as $\Delta f_{over} = f_{over} - f_{over}^{rand}$. The value of $f_{over}^{rand}$ changes with loading and between experiments, but on average, it is 0.1 for the dilation and compaction, and 0.2 for the shear strain using a threshold of the high strains of the 80th percentile. $\Delta f_{over}$ represents the proportion of the high strain events that overlap above the proportion that overlaps due to randomness (Text S4 in Supporting Information S1).

According to Reches and Lockner (1994), $\Delta f_{over}$ should be close to zero throughout loading. Examining the evolution of $\Delta f_{over}$ throughout each experiment reveals that it never falls below zero in the sandstone experiments, and is negative for only 8% of the calculations in the granite experiments. Moreover, $\Delta f_{over}$ is greater than the average $f_{over}^{rand}$ for the volumetric strains (0.10), and the shear strains (0.20) for 60% (granite) and 85% (sandstone) of the calculations. Interestingly, $\Delta f_{over}$ increases with loading for the vast majority of the strain components and
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Persistence of the location of the high strain events measured with $\Delta f_{\text{over}}$. (a) Method of calculating $\Delta f_{\text{over}}$ using the high strains identified in the previous stress step (or digital volume correlation calculation, $t_{n-1}$) (dark blue region in #1), the high strains identified in the current stress step, $t_n$ (light blue region in #2), and the volume that these strains overlap (outlined region in #3). (b) Example of high shear strains (>80th percentile) in subsequent steps: $t_{n-1}$ (black) and $t_n$ (blue), and resulting overlapping region (pink) in experiment DD05. (c–d) The average $\pm$ one standard deviation of $\Delta f_{\text{over}}$ throughout all the experiments on sandstone (c) and granite (d) using a threshold of the high strains of the 80th percentile.</figcaption></figure>

experiments. In particular, the $\Delta f_{\text{over}}$ of the final strain field immediately preceding failure is greater than the $\Delta f_{\text{over}}$ at the onset of loading in 11 of the 18 calculations of the granite experiments (three strain components and six experiments), and 17 of the 18 calculations of the sandstone experiments (Table S5 in Supporting Information S1).

The mean $\Delta f_{\text{over}}$ throughout each experiment is greater in the sandstone experiments than the granite experiments (Figure 3). In the sandstone experiments, the mean $\Delta f_{\text{over}}$ for the compaction, dilation, and shear strain are 0.22, 0.31 and 0.42, while in the granite experiments, they are 0.14, 0.11 and 0.25 (Figure 3). The key results do not change when we vary the threshold used to identify the high strains (Figure S9 in Supporting Information S1). In summary, (a) the high strain events do not randomly fluctuate in space, and instead produce $\Delta f_{\text{over}}$ greater than the average $f_{\text{over}}^{\text{rand}}$ for more than 60% (granite) and 85% (sandstone) of the loading steps, (b) the high strain events increase in spatial persistence with loading in the majority of the granite experiments, and in all but one of the sandstone experiments, and (c) the high strain events are more persistent in the sandstone experiments than the granite experiment.

# 4. Discussion

## 4.1. Dilation Dominance Precedes Strain Localization Onset

In all of the experiments, the volumetric strain field evolves from compaction-to dilation-dominated (Figure 2, Figures S4 and S5 in Supporting Information S1). The onset of dilation dominance precedes the onset of the localization of the compaction, dilation, and shear strain. This timing suggests that the dilation of the rock is a phase required for strain localization in these experiments. However, rock types that produce compaction bands may host localized strain without significant dilation. Previous work has recognized the importance of mode I fractures during triaxial compression of low porosity crystalline rock, such as granite (Brace et al., 1966; McBeck, Cordonnier, Cooke, et al., 2023; Peng & Johnson, 1972; Tapponnier & Brace, 1976). More generally, the onset of dilatancy recognized by the macroscopic volumetric strain is a key phase of rock deformation leading to catastrophic failure (Paterson & Wong, 2005). The present work identifies a precursory phase of rock deformation (dilation dominance of the incremental strain field) that occurs preceding macroscopic yielding and preceding the onset of strain localization in Westerly granite and Darley Dale sandstone.

We may expect that dilation dominance is required for the localization of shear strain. Because fractures are rough preceding slip (Brodsky et al., 2011; Candela et al., 2011; McBeck, Mair, & Renard, 2021; Sagy et al., 2007), we would expect that dilation of the rough fracture surfaces are required for significant shear. Here, we show that in order for shear strain to localize in laboratory experiments, more than 50% of the rock volume must be dilating. Furthermore, dilation must dominate the volumetric strain field in order for the high magnitudes of the compaction and dilation to localize.
## 4.2. Persistence of the High Strain Events

To test the idea that a process zone of interacting fractures produces the system-sized network that causes failure (Reches & Lockner, 1994), we characterize the degree to which the high strain events persist at the same location, or fluctuate randomly in space. The DVC technique likely captures deformation and strains that are smaller than those detectable from acoustic emissions, particularly at lower differential stresses. Consequently, the present analysis tracks deformation that analyses using acoustic emissions, such as Reches and Lockner (1994), could not reveal.

Our approach cannot reveal the structure of the deformation that occurs following the peak stress. However, assessing the persistence of the high strain events in space can help indicate if the locations of these events are indeed random preceding the peak stress, or are influenced by the location of the high strain events that previously developed.

The results show that the high strain events do not randomly fluctuate in space, and instead produce $\Delta f_{over}$ above zero in more than 92% of the stress steps of the granite and sandstone experiments, in contrast to the expectations of Reches and Lockner (1994). Moreover, the high strain events increase in spatial persistence with loading in the majority of the granite experiments, and in all but one of the sandstone experiments (Table S5 in Supporting Information S1). This result has important implications for detecting precursors to catastrophic failure: they suggest that monitoring the characteristics of the fractures and strain fields early in loading can help determine the geometry of the fracture network that produces failure.

The persistence of the high strain events is greater in the sandstone experiments (the mean $\Delta f_{over}$ ranges from 0.2 to 0.6) than in the granite experiments (the mean $\Delta f_{over}$ ranges from 0.1 to 0.3) (Figure 3). The deformation that occurs earlier in loading in sandstone appears to have a significant influence on the system-sized fracture network that produces macroscopic failure, while this early deformation does not exert as strong of a control on granite. This result may occur because of the varying degree to which the rock microstructure controls fracture propagation in sandstone and granite. In Darley Dale sandstone, fracture development is impeded by pervasive pores and cement. In Westerly granite, in contrast, the heterogeneity introduced by minerals of varying stiffness does not exert as strong of an influence on fracture development (e.g., Wu et al., 2000). This relative homogeneity of Westerly granite, compared to Darley Dale sandstone, allows fracture growth to accelerate rapidly from several smaller fractures to a few system-sized fractures. The impedance of fracture growth in Darley Dale sandstone allows many smaller fractures and high strain zones to grow and coalesce gradually with increasing loading, rather than quickly produce macroscopic failure. The varying microstructure thus produces more persistent high strain zones in the sandstone than the granite. In summary, the idea proposed by Reches and Lockner (1994) that fracture development is relatively random earlier in loading may apply to a limited extent in Westerly granite, but does not accurately describe the spatial persistence of the high strain zones in Darley Dale sandstone.

## Data Availability Statement

The experimental data (time series of X-ray tomograms of 12 new experiments) are available on Norstore (J. A. McBeck, 2024).

## References

* Atkinson, B. K. (1984). Subcritical crack growth in geological materials. *Journal of Geophysical Research*, 89(B6), 4077–4114. https://doi.org/10.1029/jb089ib06p04077

* Baud, P., Wong, T. F., & Zhu, W. (2014). Effects of porosity and crack density on the compressive strength of rocks. *International Journal of Rock Mechanics and Mining Sciences*, 67, 202–211. https://doi.org/10.1016/j.ijrmms.2013.08.031

* Ben-Zion, Y., & Zaliapin, I. (2020). Localization and coalescence of seismicity before large earthquakes. *Geophysical Journal International*, 223(1), 561–583. https://doi.org/10.1093/gji/ggaa315

* Bolton, D. C., Marone, C., Saffer, D., & Trugman, D. T. (2023). Foreshock properties illuminate nucleation processes of slow and fast laboratory earthquakes. *Nature Communications*, 14(1), 3859. https://doi.org/10.1038/s41467-023-39399-0

* Brace, W. F., Paulding Jr, B. W., & Scholz, C. H. (1966). Dilatancy in the fracture of crystalline rocks. *Journal of Geophysical Research*, 71(16), 3939–3953. https://doi.org/10.1029/jz071i016p03939

* Brodsky, E. E., Gilchrist, J. J., Sagy, A., & Collettini, C. (2011). Faults smooth gradually as a function of slip. *Earth and Planetary Science Letters*, 302(1–2), 185–193. https://doi.org/10.1016/j.epsl.2010.12.010

* Candela, T., Renard, F., Schmittbuhl, J., Bouchon, M., & Brodsky, E. E. (2011). Fault slip distribution and fault roughness. *Geophysical Journal International*, 187(2), 959–968. https://doi.org/10.1111/j.1365-246x.2011.05189.x

* Cartwright-Taylor, A., Main, I. G., Butler, I. B., Fusseis, F., Flynn, M., & King, A. (2020). Catastrophic failure: How and when? Insights from 4-D in situ x-ray microtomography. *Journal of Geophysical Research: Solid Earth*, 125(8), e2020JB019642. https://doi.org/10.1029/2020jb019642

## Acknowledgments

This study was funded by the Research Council of Norway (project no. 300435 to JM), and the European Research Council under the European Union's Horizon 2020 research and innovation program (Grant 101019628 BREAK to FR). We thank the ESRF for provision of synchrotron radiation facilities on beamline ID19 (Long Term Proposal ES-1190). We thank Clémence Muzelle for technical support on beamline ID19 at ESRF. Sigma2 (the National Infrastructure for High-Performance Computing and Data Storage in Norway) provided data storage (Grant NS9806K to JM). FR and JM acknowledge support from the FricFrac project funded by the Center for Advanced Study at the Norwegian Academy of Science and Letters during the academic year 2023–2024. WZ acknowledges support from the US National Science Foundation under Grant EAR-2218314. We thank Editor Germán Prieto, an Associate Editor, Georg Dresen and an anonymous reviewer that provided suggestions that helped improve this manuscript.
Dresen, G., Kwiatek, G., Goebel, T., & Ben-Zion, Y. (2020). Seismic and aseismic preparatory processes before large stick-slip failure. *Pure and Applied Geophysics*, **177**(12), 5741-5760. https://doi.org/10.1007/s00024-020-02605-x

Isida, M. (1971). Effect of width and length on stress intensity factors of internally cracked plates under various boundary conditions. *International Journal of Fracture Mechanics*, **7**(3), 301-316. https://doi.org/10.1007/bf00184306

Jaeger, J. C., Cook, N. G., & Zimmerman, R. (2009). *Fundamentals of rock mechanics*. John Wiley and Sons.

Lockner, D., Byerlee, J. D., Kuksenko, V., Ponomarev, A., & Sidorin, A. (1991). Quasi-static fault growth and shear fracture energy in granite. *Nature*, **350**(6313), 39-42. https://doi.org/10.1038/350039a0

McBeck, J., Ben-Zion, Y., & Renard, F. (2021). Fracture network localization preceding catastrophic failure in triaxial compression experiments on rocks. *Frontiers in Earth Science*, **9**, 778811. https://doi.org/10.3389/feart.2021.778811

McBeck, J., Ben-Zion, Y., & Renard, F. (2022). Volumetric and shear strain localization throughout triaxial compression experiments on rocks. *Tectonophysics*, **822**, 229181. https://doi.org/10.1016/j.tecto.2021.229181

McBeck, J., Ben-Zion, Y., Zhou, X., & Renard, F. (2021). The influence of preexisting rock damage on fault network localization. *Journal of Structural Geology*, **153**, 104471. https://doi.org/10.1016/j.jsg.2021.104471

McBeck, J., Cordonnier, B., Ben-Zion, Y., & Renard, F. (2023). The influence of confining pressure and preexisting damage on strain localization in fluid-saturated crystalline rocks in the upper crust. *Journal of Geophysical Research: Solid Earth*, **128**(8), e2023JB026987. https://doi.org/10.1029/2023jb026987

McBeck, J., Cordonnier, B., Cooke, M., Fattaruso, L., & Renard, F. (2023). Deformation evolves from shear to extensile in rocks due to energy optimization. *Communications Earth & Environment*, **4**(1), 352. https://doi.org/10.1038/s43247-023-01023-w

McBeck, J., Kobchenko, M., Hall, S. A., Tudisco, E., Cordonnier, B., Meakin, P., & Renard, F. (2018). Investigating the onset of strain localization within anisotropic shale using digital volume correlation of time-resolved X-ray microtomography images. *Journal of Geophysical Research: Solid Earth*, **123**(9), 7509-7528. https://doi.org/10.1029/2018jb015676

McBeck, J., Mair, K., & Renard, F. (2021). Decrypting healed fault zones: How gouge production reduces the influence of fault roughness. *Geophysical Journal International*, **225**(2), 759-774. https://doi.org/10.1093/gji/ggab003

McBeck, J. A. (2024). Filtered tomograms of sandstone and granite [Dataset]. Norstore. https://doi.org/10.11582/2024.00029

McCormick, C. A., & Rutter, E. H. (2022). An experimental study of the transition from tensile failure to shear failure in Carrara marble and Solnhofen limestone: Does "hybrid failure" exist? *Tectonophysics*, **844**, 229623. https://doi.org/10.1016/j.tecto.2022.229623

Mogi, K. (1977). Dilatancy of rocks under general triaxial stress states with special reference to earthquake precursors. *Journal of Physics of the Earth*, **25**(Supplement), S203-S217. https://doi.org/10.4294/jpe1952.25.supplement_s203

Moore, D. E., Summers, R., & Byerlee, J. D. (1990). *Faults, fractures, and other deformation features produced during loading of granite in triaxial equipment* (pp. 90-349). US Geological Survey.

Paterson, M. S., & Wong, T. F. (2005). *Experimental rock deformation: The brittle field*. Springer.

Peng, S., & Johnson, A. M. (1972). Crack growth and faulting in cylindrical specimens of Chelmsford granite. *International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts*, **9**(1), 37-86. https://doi.org/10.1016/0148-9062(72)90050-2

Petit, J. P., & Barquins, M. (1988). Can natural faults propagate under mode II conditions? *Tectonics*, **7**(6), 1243-1256. https://doi.org/10.1029/tc007i006p01243

Reches, Z. E., & Lockner, D. A. (1994). Nucleation and growth of faults in brittle rocks. *Journal of Geophysical Research*, **99**(B9), 18159-18173. https://doi.org/10.1029/94jb00115

Renard, F., Cordonnier, B., Dysthe, D. K., Boller, E., Tafforeau, P., & Rack, A. (2016). A deformation rig for synchrotron microtomography studies of geomaterials under conditions down to 10 km depth in the Earth. *Journal of Synchrotron Radiation*, **23**(4), 1030-1034. https://doi.org/10.1107/s1600577516008730

Rizzo, R. E., Healy, D., Heap, M. J., & Farrell, N. J. (2018). Detecting the onset of strain localization using two-dimensional wavelet analysis on sandstone deformed at different effective pressures. *Journal of Geophysical Research: Solid Earth*, **123**(12), 10460-10478. https://doi.org/10.1029/2018jb015898

Rudnicki, J. W., & Rice, J. R. (1975). Conditions for the localization of deformation in pressure-sensitive dilatant materials. *Journal of the Mechanics and Physics of Solids*, **23**(6), 371-394. https://doi.org/10.1016/0022-5096(75)90001-0

Sagy, A., Brodsky, E. E., & Axen, G. J. (2007). Evolution of fault-surface roughness with slip. *Geology*, **35**(3), 283-286. https://doi.org/10.1130/g23235a.1

Stanchits, S., Vinciguerra, S., & Dresen, G. (2006). Ultrasonic velocities, acoustic emission characteristics and crack damage of basalt and granite. *Pure and Applied Geophysics*, **163**, 975-994. https://doi.org/10.1007/3-7643-7712-7_4

Tang, C. A., Liu, H., Lee, P. K. K., Tsui, Y., & Tham, L. (2000). Numerical studies of the influence of microstructure on rock failure in uniaxial compression—Part I: Effect of heterogeneity. *International Journal of Rock Mechanics and Mining Sciences*, **37**(4), 555-569. https://doi.org/10.1016/s1365-1609(99)00121-5

Tapponnier, P., & Brace, W. F. (1976). Development of stress-induced microcracks in Westerly granite. *International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts*, **13**(4), 103-112. https://doi.org/10.1016/0148-9062(76)91937-9

Tudisco, E., Andò, E., Cailletaud, R., & Hall, S. A. (2017). TomoWarp2: A local digital volume correlation code. *SoftwareX*, **6**, 267-270. https://doi.org/10.1016/j.softx.2017.10.002

Vasseur, J., Wadsworth, F. B., Lavallée, Y., Bell, A. F., Main, I. G., & Dingwell, D. B. (2015). Heterogeneity: The key to failure forecasting. *Scientific Reports*, **5**(1), 13259. https://doi.org/10.1038/srep13259

Wu, X. Y., Baud, P., & Wong, T. F. (2000). Micromechanics of compressive failure and spatial evolution of anisotropic damage in Darley Dale sandstone. *International Journal of Rock Mechanics and Mining Sciences*, **37**(1-2), 143-160. https://doi.org/10.1016/s1365-1609(99)00093-3

Zhang, J., Wong, T. F., & Davis, D. M. (1990). Micromechanics of pressure-induced grain crushing in porous rocks. *Journal of Geophysical Research*, **95**(B1), 341-352. https://doi.org/10.1029/jb095ib01p00341