# Grain-size effects during semi-brittle flow of calcite rocks

###### Abstract

We study the role of grain size in the rheological behaviour of calcite
aggregates in the semi-brittle regime. We conduct triaxial deformation tests
on three rocks, Solnhofen limestone, Carrara marble and Wombeyan marble, with
average grain sizes of 5–10 µm, $200$ µm and 2 mm, respectively, at pressures
in the range 200–800 MPa and temperatures in the range 20–400∘C. At all
conditions, both strength and hardening rate increase with decreasing grain
size. Flow stress scales with the inverse of grain size to a power between 1/3
and 2/3. Hardening rate decreases linearly with the logarithm of grain size.
In-situ ultrasonic monitoring reveals that P-wave speed tends to decrease with
increasing strain, and that this decrease is more marked at room temperature
than at 200 and 400∘C. The decrease in wave speed is consistent with
microcracking, which is more prevalent at low temperature and low pressure.
Microstructural observations reveal high twin densities in all deformed
samples. Twin density increases with stress, consistent with previous
datasets. Spatial distributions of intragranular misorientation indicate that
twins are sometimes obstacles to dislocation motion, but this effect is not
ubiquitous. Computed slip-transfer statistics indicate that that twins are
typically weaker barriers to dislocation glide than grain boundaries, so that
their effect on dislocation accumulation and hardening rates is likely smaller
than the effect of grain size. Indeed, our data reveal that grain size exerts
a first-order control on flow stress and hardening in calcite, whereas
twinning may only have a secondary impact on these behaviours.

JGR: Solid Earth

Department of Earth Sciences, University College London, London, UK Department
of Earth Sciences, University of Cambridge, Cambridge, UK

Christopher<EMAIL_ADDRESS>
Experimental investigation of semi-brittle flow of calcite rocks with variable
grain size

Grain size exerts a first-order control on the semi-brittle flow stress of
calcite rocks

Deformation twinning has limited influence on the strength of calcite rocks

## Plain Language Summary

Rocks are strongest when their failure mode is mixed between brittle (fracture
processes) and crystal plastic (individual grains deform by movement of
crystal defects). Unfortunately, we have a limited understanding of how these
two mechanisms accommodate deformation together. Yet, both failure modes are
sensitive to grain size. We selected three calcite rocks of varying grain size
and deformed them at elevated pressure and temperature to simulate coupled
brittle and crystal-plastic deformation. Calcite also forms twins (small
planar structures in the crystal) when deformed that may act to strengthen the
rock during deformation. By monitoring the speed at which sound waves pass
through the rock during experiments, we were able to track brittle behaviour.
We find that smaller grain size results in greater strength. We observe twins
in all deformed samples, but they do not seem to have a big effect on the
resulting strength. Microscopic observations of samples and wave-speed
measurements show that brittle behaviour is suppressed by increases in
pressure and temperature. We suggest a new approach to model these processes
that could be used to model coupled brittle and crystal-plastic behaviour on a
large scale.

##  1 Introduction

In the shallow lithosphere, rocks deform by localised brittle failure and
strength is described by a friction law Scholz (2002); Townend Zoback (2000).
In the lower lithosphere, high temperatures and pressures promote the onset of
crystal-plastic deformation mechanisms. Here, strength is described by flow
laws sensitive to temperature ($T$) and strain rate ($\dot{\varepsilon}$)
Goetze Brace (1972); Evans Kohlstedt (1995). At intermediate conditions,
deformation is ductile, i.e., remains macroscopically distributed (following
the terminology of Rutter1986), and is often termed ”semi-brittle”, which is
characterised by coupled cracking and crystal plasticity. Semi-brittle
deformation is likely to support the highest stresses in the lithosphere
Goetze Evans (1979); Brace Kohlstedt (1980), impacting geodynamic processes
Burov (2011) and the maximum depth of earthquake nucleation Scholz (1998).
Despite the significance of semi-brittle flow, there is a paucity of simple
models to describe the rheological behaviour of rock in this regime, and the
relative importance of key processes, such as friction, tensile cracking,
dislocation motion, twinning, or grain-boundary sliding is not well
constrained. Thus, to advance our understanding of semi-brittle flow, we must
first improve our understanding of interactions among microscale deformation
processes.

Calcite rock is an important constituent of tectonic terrains, and undergoes a
transition to semi-brittle flow at modest pressure and low temperature
<e.g.,¿[]Heard1960,Fredrich1989. This combination of characteristics has led
to numerous laboratory investigations into the rheological behaviour of
calcite rock across the brittle-ductile transition (see Rybacki2021 and
references therein). The main deformation behaviours of calcite rock can be
illustrated using the case of Carrara marble, a relatively isotropic, pure
calcite aggregate with little initial crack porosity and equant grain shapes
with sizes in the range 60–200 $\mu$m. At room temperature, Carrara marble is
dilatant and brittle at low pressure ($P<30$ MPa), and its strength is
controlled by friction. At intermediate pressures ($30<P<300$ MPa), calcite
rocks deform in a semi-brittle manner, and strength becomes decreasingly
pressure sensitive with increasing pressure Fredrich . (1989). When pressure
is high enough ($P>300$ MPa), strength becomes independent of confining
pressure and deformation is nondilatant Edmond Paterson (1972). Furthermore,
increases in temperature reduce the pressure required to promote semi-brittle
flow Rybacki . (2021). Edmond1972 and Fischer1989 report in-situ changes of
volumetric strain during deformation of calcite rocks, finding that dilatancy
is reduced by increasing pressure. Furthermore, sample volume remains constant
when strength is independent of pressure Edmond Paterson (1972). More
recently, Schubnel2005 and Rybacki2021 measured in-situ P-wave velocity
($V_{\textrm{p}}$) and demonstrated that wave-speed decreases during
deformation are suppressed by pressure increases. In-situ measurements
therefore support microstructural observations in implying that increasing
pressure suppresses cracking and frictional sliding whilst promoting crystal-
plastic processes.

Microstructural observations from calcite rocks deformed in the semi-brittle
regime reveal that distributed cracking, twinning, and dislocation motion act
together to accommodate strain Fredrich . (1989). Post-mortem crack density
decreases with increasing confining pressure Fredrich . (1989), approaching
zero when strength is independent of confining pressure. Additionally,
deformation twins are present in calcite rocks deformed at conditions below
800∘C Rybacki . (2021), and twin spacing depends on stress Rowe Rutter (1990).
Detailed strain measurements at the grain scale reveal that twinning occurs
readily in well oriented grains (those with high Schmid factor for twinning),
and that it is likely associated with a local backstress that causes hardening
of twinned grains Spiers (1979). In addition, microscale strain mapping also
indicates the existence of shear localised along grain boundaries at
temperatures $<800^{\circ}$C, potentially identifying grain-boundary sliding
as a possible deformation mechanism Quintanilla-Terminel . (2017).

Another common rheological behaviour of semi-brittle flow in calcite rocks is
strain hardening, which can persist to high temperatures <$<800^{\circ}$C,
see¿[]Rybacki2021. At low temperature, strain hardening can arise from
microscopic frictional slip across distributed defects, such as grain
boundaries <e.g.,¿[]David2020. In dislocation-mediated deformation regimes,
strain hardening occurs due to an increase in dislocation density caused by
inefficient recovery mechanisms (e.g., lack of dislocation climb) Mecking
Kocks (1981) and is sensitive to microstructure. In particular, grain
boundaries can act as barriers to dislocation motion, resulting in increases
in strength and sometimes increases in hardening rates with decreasing grain
size for most metals <grain-size strengthening, see ¿[]Cordero2016, and some
geological materials <e.g., olivine,¿[]Hansen2019. Alternatively, strain
hardening in calcite rocks has been proposed to be controlled by twinning
Rybacki . (2021). Twins may not contribute significantly to the total strain,
but they could indirectly control strength and hardening by providing
additional barriers to dislocations. Indeed, TEM observations show that
dislocation densities are elevated adjacent to twin boundaries Barber Wenk
(1979); Fredrich . (1989); Rybacki . (2021). In the metallurgy literature De
Cooman . (2018), the additional hardening provided by twins is commonly known
as twinning-induced plasticity, or TWIP. TWIP originates from observations of
high-manganese steels, where high hardening rates and ductility occur as a
result of the formation of deformation twins De Cooman . (2018). The hardening
rates in these metals are attributed to a ”dynamic Hall-Petch effect” in which
twinning refines the intracrystalline microstructure and thereby reduces the
dislocation mean free path. In this case, twin spacing largely controls
strength and hardening and thereby limits sensitivity to grain size.

The semi-brittle regime in calcite rock is thus characterised by many
interacting deformation mechanisms. One way to quantify the relative
contribution of each mechanism to the overall rheological behaviour is to test
the impact of independent variables beyond the usual pressure and temperature
conditions and imposed strain rate. One such variable is the grain size. At
room temperature, yield stress ($\sigma_{\textrm{y}}$) and the transition
pressure to semi-brittle deformation increase with decreasing grain size
Olsson (1974); Fredrich . (1990). Grain size also impacts the strength of
marble tested at elevated temperature in the dislocation-creep regime, which
could be consistent with a Hall-Petch effect Renner . (2002). In the high-
pressure, moderate-temperature range ($P>200$ MPa, $T<400^{\circ}$C) where
twinning is ubiquitous, models of twinning-induced plasticity suggest that
twin density rather than grain size is the main control on strength Rybacki .
(2021). To test this hypothesis, here we explore the role of grain size in
semi-brittle rheological behaviour and the brittle-plastic transition in
calcite rocks for comparison to the role of twin density.

We performed a series of experiments at a range of pressures ($P\leq$ 800 MPa)
and temperatures ($\leq$ 400∘C) using calcite rocks spanning two orders of
magnitude in grain size: Solnhofen limestone (grain size less than $10$ µm),
Carrara marble (grain size on the order of $100$ µm), and Wombeyan marble
(grain size on the order of $1$ mm). Experiments are supplemented by in-situ
measurements of axis-parallel P-wave speed and post-mortem microstructural
investigations to infer deformation mechanisms. We find that grain size has a
first-order impact on both strength and hardening rate, and that the influence
of twins is only indirect and likely smaller than anticipated from TWIP
models.

##  2 Methods

###  2.1 Sample materials

Three calcite rocks were selected for experiments, with grain size, $D$,
spanning three orders of magnitude. Solnhofen limestone is a lithographic
limestone with a grain size of 5–10 µm French . (2022), composed of $>99.9\%$
calcite, and with an initial porosity (measured with helium pycnometry) of 4%
Baud . (2000). Carrara Marble is a medium-grained marble with equant grains of
size 60–220 µm, comprised of $>99.9\%$ calcite with a porosity of $<$ 0.1%. In
the starting material, most grains exhibit at least one twin set. Wombeyan
marble is a coarse-grained marble with a grain size of 1–2 mm, comprised of 96
% calcite. Grains are equant and typically twinned in at least one plane, with
an initial twin density of 60 mm-1. This material was obtained thanks to Ian
Jackson at the Australian National University, and is the same rock that was
used by Paterson1958. See Table 1 for further petrographic details.

Table 1: Composition and physical properties of starting materials | Solnhofen limestone | Carrara marble | Wombeyan marble
---|---|---|---
Abbreviation | SL | CM | WM
Composition |  $>$99% CaCO3 |  $>$99% CaCO3 | 96% CaCO3
|  |  | 2.5% MgCO3
Porosity (%) | 4% |  $<$0.5% |  $<$0.5%
$d$ (mm) | 0.005–0.01 | 0.06–0.22 | 1–2
Initial $V_{\textrm{p}}$ (m s-1) | 5600 | 5900 | -
Figure 1: (a) Detail of sample geometry in the pressure vessel. Insets
(b)–(d): Polarised incident-light images of initial sample microstructure of
(b) Solnhofen limestone, (c) Carrara marble and (d) Wombeyan marble.

###  2.2 Mechanical testing

A total of 42 experiments were conducted in the ”Murrell” gas-medium triaxial
apparatus hosted at University College London Harbord . (2022). Rectified
cores, 10 mm in diameter and 22 mm in length, were dried in an oven at 70∘C
for at least 24 hours before testing. Samples were inserted into annealed
copper jackets of 0.1 mm wall thickness and were swaged onto the deformation
rams using a slip ring (Figure 1a). The jacketed samples were inserted into
the pressure vessel and pressurised to the target confining pressure using
argon gas. Here the confining pressure is provides the intermediate and
minimum principal stresses. Throughout the tests, the measured pressure
remained within 5% of the target pressure. An internal furnace was used to
heat samples to $200^{\circ}$C and $400^{\circ}$C. Axial load was applied
vertically, generating the differential stress, $\sigma$. Axial stress was
measured by an external load cell, and axial shortening was measured by a pair
of linear variable-displacement transducers. Axial deformation was then
applied by a piston that moved at a constant imposed shortening rate of 0.22
µm s-1, equivalent to a strain rate of $10^{-5}$ s-1. All samples were
deformed to a total strain of 7.5%. A detailed list of test conditions is
given in Table 3.

###  2.3 Data processing

Several corrections were made to the mechanical data. Seal friction and jacket
strength were subtracted from load measurements (as reported in Harbord2022)
and displacement due to machine compliance was subtracted from shortening
measurements. The differential stress supported by the sample was computed by
dividing the corrected force by the cross-sectional area of the sample, which
was assumed to linearly increase with deformation consistent with a constant
sample volume. Mechanical data were then further processed to obtain estimates
of the tangent modulus ($h=\mathrm{\delta}\sigma/\mathrm{\delta}\varepsilon$)
by numerical differentiation of the stress data over a moving window of 1%
strain. Uncertainties in seal friction during deformation result in error on
the hardening modulus that is $<$ 20 % of reported values. The yield stress
($\sigma_{\mathrm{y}}$) is defined as the stress at which $h$ falls to 90% of
the sample-specific Young’s modulus determined from elastic loading.

###  2.4 In-situ P-wave speed measurements

In-situ P-wave speed was measured during tests conducted on Solnhofen
limestone and Carrara marble. Measurements were made parallel to the sample
axis during deformation using the pulse-transmission method Birch (1961).
Every 10 s during the experiments, a 200 V pulse of $0.4$ µs duration and
$2.5$ MHz frequency was sent to a lead-titanate-zirconate ceramic disk mounted
centrally in the bottom ram, which was received by a transducer mounted
centrally at the top of the sample assembly and recorded digitally at 100 MHz
<see ¿[ for further details]Harbord2022. The signal-to-noise ratio was
improved by stacking 256 raw traces at each time interval. Changes in axial
P-wave speed were computed relative to a reference waveform using cross
correlation, and corrected for interfacial delays following the methods
outlined in Harbord2022. Measurements of P-wave speed are reported as the
change of wave speed ($\Delta V$) and are normalised relative to the wave
speed measured at the start of loading ($V_{0}$). Selected deformation tests
were subsequently repeated, and wave-speed changes were found to be
reproducible (see Supplementary Material Figure S1).

To gain insight into the microstructural state of samples after deformation
during the return to ambient pressure, we continued to perform wave-speed
surveys after removal of differential stress and throughout the staged
decompression. The change in wave speed during decompression is quantified as
the relative change in wave speed ($\Delta V$) normalised by the wave speed
measured at the end of deformation ($V_{\mathrm{final}}$), and when at
temperature, after cooling of the sample. These measurements were also
complemented by post-mortem measurements of wave speed at room pressure (0.1
MPa).

###  2.5 Microstructural analysis

####  2.5.1 Optical microscopy

To investigate deformation microstructures, selected samples were mounted in
epoxy, sectioned parallel to the deformation axis, and polished. A set of thin
sections was also made for visible-light microscope observations. Imaging
using transmitted and reflected light was performed using a Leica DM750P
microscope furnished with an ICC50 camera. To quantify the prevalence of
intragranular cracks in each sample, we followed the method outlined by
Fredrich1989. Samples were imaged using reflected light at $\times 4$
magnification, and we counted intersections between cracks (excluding grain
boundaries) and a square grid of 2 mm by 2 mm with a spacing of 0.2 mm. These
counts were used to determine the resulting crack surface area per volume,
$S_{\mathrm{v}}$ (mm2/mm3).

####  2.5.2 Electron microscopy

Scanning electron microscopy was performed using a Jeol JSM-6480LV scanning
electron microscope (SEM) hosted at University College London and a Zeiss
Gemini 300 field-emission gun SEM at the University of Cambridge. Electron
backscatter diffraction (EBSD) patterns and forescattered electron images were
collected using an Oxford Instruments Symmetry detector and AZtec 4.0
acquisition software. Table 2 lists the acquisition parameters for each EBSD
dataset. All diffraction patterns with acquired with low detector gain.
Datasets collected for conventional EBSD were acquired with a reduced number
of pixels in the diffraction patterns to increase the mapping speed. Datasets
collected for high-angular resolution electron backscatter diffraction (HR-
EBSD) were collected with the maximum number of pixels permitted by the
detector.

HR-EBSD is a postprocessing technique that analyses distortion of the
diffraction patterns to measure lattice rotations and intragranular
heterogeneity in elastic strain Wilkinson . (2006); Britton Wilkinson (2011,
2012). Full details of the technique are given by Wallis2019 and here we
provide a summary of the key points. One diffraction pattern from the host
grain within each mapped area was manually selected to be a reference pattern
based on the quality of the diffraction pattern and its position within the
map. 100 regions of interest, 256$\times$256 pixels in size, were extracted
from all diffraction patterns within the host grain. Each region of interest
from each diffraction pattern was cross-correlated with the corresponding
region of interest from the reference pattern to determine shifts in their
positions. Shifts in the diffraction pattern due to beam scanning were
corrected using a calibration determined on an undeformed Si single crystal
following Wilkinson2006 and the position of the pattern centre was calibrated
using diffraction patterns collected over a range of detector insertion
distances Maurice . (2011). A deformation-gradient tensor was fit to the field
of shifts in each diffraction pattern. The deformation-gradient tensor was
decomposed into its symmetric and antisymmetric parts, which respectively give
the elastic-strain and rotation tensors Wilkinson . (2006). We used the
pattern remapping approach of Britton2012, in which a first pass of cross-
correlation measures lattice rotations that are used to rotate each pattern
back into the orientation of the reference pattern before a second pass of
cross-correlation measures the elastic strain and a small correction to the
rotations. The elastic strains were converted to stresses using Hooke’s law.
The measured stresses are relative to the unknown stress state at the
reference point, giving maps of intragranular stress heterogeneity, rather
than absolute values. We subtracted the arithmetic mean value of each
component of the stress tensor within the map area from each measured value so
that the final maps provide stress heterogeneity relative to the unknown mean
stress state within each grain Mikami . (2015). Alongside, spatial gradients
in the lattice rotations were used to estimate densities of geometrically
necessary dislocations (GNDs). Densities of each dislocation type on the slip
systems summarised by DeBresser1997 were fit to the measurable components of
the lattice curvature following the approach applied to quartz by Wallis2019.
We emphasise that the stress heterogeneity and GND densities are determined
independently, being respectively derived from the distinct symmetric and
antisymmetric parts of the deformation-gradient tensor. Data points were
filtered out if they had a mean normalised peak height in the cross-
correlation function of $<$0.3 or a mean angular error in the fitted
deformation gradient tensor of $>$0.004 radians Britton Wilkinson (2011).

We analysed the probability distributions of the stress heterogeneity to
assess whether the stresses are imparted by dislocations. The probability
distribution of the stress field of population of dislocations has a
characteristic form with tails that depart from a normal distribution towards
higher stresses Jiang . (2013); Wilkinson . (2014). We assess for the presence
of these tails using a normal-probability plot, in which the cumulative-
probability axis is scaled such that a normal distribution falls on a straight
line. Importantly, the tails of the probability distribution, $P(\sigma)$,
have a specific form if the stresses, $\sigma$, are imparted by dislocations,
whereby $P(\sigma)\rightarrow C\rho|\sigma|^{-3}$, where $C$ is a constant
that depends on the material, type(s) of dislocation, and considered stress
component, and $\rho$ is the total dislocation density Groma Bakó (1998);
Wilkinson . (2014). To test whether the measured stress fields exhibit this
form, we compute the restricted second moment, $\nu_{2}$, which is a metric
that characterises the shape of a probability distribution based on the
integral over restricted ranges in stress, calculated as
$\nu_{2}(\sigma)=\int_{-\sigma}^{+\sigma}P(\sigma)\sigma^{2}\,d\sigma$
Wilkinson . (2014); Kalácska . (2017). A plot of $\nu_{2}$ versus ln($\sigma$)
becomes a straight line at high stresses if the tails of the probability
distribution of the stresses exhibit the form $P(\sigma)\propto|\sigma|^{-3}$
expected of a population of dislocations Wilkinson . (2014); Kalácska .
(2017). We apply this analysis to the $\sigma_{12}$ component of the stress
tensor as this component is the least modified by sectioning the sample and is
a shear stress capable of exerting glide forces on dislocations Wallis .
(2019). This approach has recently been applied to olivine by
Wallis2021,Wallis2022. We include in these plots data from an undeformed Si
wafer measured by Wallis2022 to provide an indication of the noise level of
the stress measurements.

Table 2: EBSD acquisition parameters for each map Figure | Expt. | Lithology | Step (µm) | Map (X$\times$Y) | EBSP pixels (X$\times$Y)
---|---|---|---|---|---
Fig. 11a | Run0119 | WM | 15 | 480$\times$110 | 622$\times$512
Fig. 11b | Run0093 | CM | 2 | 479$\times$359 | 622$\times$512
Fig. 11c | Run0167 | SL | 0.15 | 250$\times$191 | 622$\times$512
Fig. 12a | Run0093 | CM | 2 | 479$\times$359 | 622$\times$512
Fig. 12b | Run0093 | CM | 0.5 | 450$\times$470 | 622$\times$512
Fig. 12c | Run0093 | CM | 0.5 | 340$\times$285 | 622$\times$512
Fig. 12d | Run0093 | CM | 0.5 | 560$\times$560 | 622$\times$512
Fig. 13a | Run0093 | CM | 0.2 | 260$\times$175 | 1244$\times$1024
Fig. 13b | Run0093 | CM | 0.15 | 300$\times$130 | 1244$\times$1024

####  2.5.3 Twin density measurements

Twin density was measured using a combination of forescattered electron images
and EBSD maps. For each sample for which twin density was reported, we chose
between 20 and 60 grains in a representative forescattered electron image, and
measured twin spacing and twin width perpendicular to the selected twin set.
Using an EBSD map of the same area, we determined the orientation of each
grain and active twin set, and used the angle between the normal to the twin
plane and the normal to the section to correct the measured twin width and
spacing <the same procedure as used by¿[]Rutter2022.

##  3 Results

###  3.1 Mechanical data and in-situ P-wave speed

####  3.1.1 General characteristics

Table 3: Table of experiments conducted at a range of conditions. Tests denoted with * are accompanied by measurements of P-wave speed parallel to the specimen axis. Experiment | Lithology |  $P$ (MPa) |  $T$ (∘C) |  $\sigma_{5}$ (MPa) |  $h_{5}$ (GPa)
---|---|---|---|---|---
Run0147* | Solnhofen limestone | 207 | 20.1 | 466 | 0.34
Run0149* | Solnhofen limestone | 416 | 20.4 | 535 | 1.57
Run0150* | Solnhofen limestone | 628 | 19.8 | 517 | 1.85
Run0152* | Solnhofen limestone | 225 | 195 | 437 | 1.25
Run0153* | Solnhofen limestone | 440 | 180 | 474 | 2.22
Run0154* | Solnhofen limestone | 217 | 393 | 436 | 1.36
Run0162* | Solnhofen limestone | 435 | 394 | 456 | 1.56
Run0167* | Solnhofen limestone | 610 | 186 | 474 | 2.54
Run0168* | Solnhofen limestone | 617 | 398 | 416 | 1.84
Run0075 | Carrara marble | 200 | 19.8 | 344 | 1.22
Run0078 | Carrara marble | 405 | 19.8 | - | -
Run0084 | Carrara marble | 615 | 19.8 | 386 | 1.83
Run0086 | Carrara marble | 236 | 197 | 187 | 0.73
Run0089 | Carrara marble | 366 | 204 | 272 | 1.91
Run0090 | Carrara marble | 213 | 390 | 203 | 0.82
Run0091 | Carrara marble | 378 | 405 | 186 | 0.72
Run0093 | Carrara marble | 599 | 192 | 295 | 1.37
Run0094 | Carrara marble | 594 | 401 | 198 | 0.65
Run0095 | Carrara marble | 602 | 206 | - | -
Run0097 | Carrara marble | 597 | 204 | - | -
Run0098 | Carrara marble | 769 | 20.1 | 377 | 1.67
Run0129* | Carrara marble | 403 | 17.6 | 385 | 1.67
Run0131* | Carrara marble | 202 | 18.1 | 335 | 1.12
Run0132* | Carrara marble | 407 | 18.2 | 354 | 1.31
Run0137* | Carrara marble | 556 | 19.1 | 378 | 1.73
Run0138* | Carrara marble | 220 | 172 | 267 | 1.41
Run0141* | Carrara marble | 241 | 387 | 221 | 1.22
Run0143* | Carrara marble | 579 | 20.2 | 376 | 1.63
Run0145* | Carrara marble | 417 | 370 | 222 | 0.93
Run0163* | Carrara marble | 616 | 171 | 247 | 1.58
Run0164* | Carrara marble | 622 | 368 | 255 | 1.28
Run0165* | Carrara marble | 391 | 195 | 321 | 2.17
Run0166* | Carrara marble | 188 | 183 | 306 | 1.71
Run0169* | Carrara marble | 203 | 16.3 | 341 | 1.36
Run0104 | Wombeyan marble | 206 | 19.3 | 279 | 0.76
Run0106 | Wombeyan marble | 594 | 20.2 | 330 | 1.15
Run0110 | Wombeyan marble | 231 | 207 | 189 | 1.19
Run0119 | Wombeyan marble | 609 | 187 | 218 | 0.55
Run0120 | Wombeyan marble | 408 | 19.2 | 338 | 0.75
Run0121 | Wombeyan marble | 428 | 190 | 226 | 0.71
Run0124 | Wombeyan marble | 233 | 408 | 128 | 0.08
Run0126 | Wombeyan marble | 574 | 416 | 100 | –0.02
Run0127 | Wombeyan marble | 400 | 392 | - | -

The stress-strain behaviour and evolution of in-situ P-wave speed are
qualitatively similar across the range of conditions and tested materials
(Figure 2), and are typical of ductile behaviour. Taking the example of
Carrara marble at $P=600$ MPa and $T=20^{\circ}$C (Figure 2b, solid dark blue
curve), the mechanical data is characterised by a rapid linear increase in
stress at low strain ($\varepsilon<0.5$%), representing elastic loading,
during which P-wave speed remains relatively constant (Figure 2b, dashed dark
blue curve). Above a stress of around 250 MPa, the rate of increase in stress
with strain begins to decrease and the P-wave speed begins to decrease
concomitantly. Beyond approximately 1.5% strain, stress continues to increase
approximately linearly with strain, which is accompanied by a steady decrease
in wave speed. Similar qualitative behaviour occurs in all samples at all
conditions tested, with quantitative variations in the yield stress and degree
of hardening.

Strain hardening is observed at nearly all test conditions (Figure 2), with
the exception of Wombeyan marble deformed at $T=400^{\circ}$C (Figure 2i),
solid curves). Between $20^{\circ}$C and $200^{\circ}$C, hardening rates are
unaffected by temperature increases, however the hardening modulus is lower
for all lithologies at $T=400^{\circ}$C.

Figure 2: Differential stress ($\sigma$, solid lines) and normalised P-wave
speed ($\Delta V/V_{0}$, dashed lines) from constant-strain rate experiments
at $P$ = 200, 400, 600, and 800 MPa, indicated by increasing intensity of
blue, at 20, 200 and 400∘C increasing downward in the plot. Plots (a), (d) and
(g) correspond to experiments performed on Solnhofen limestone at 20, 200 and
400∘C respectively. Plots (b), (e) and (h) are tests performed on Carrara
marble at 20, 200 and 400∘C respectively. Plots (c), (f) and (i) are tests
performed on Wombeyan marble at 20, 200 and 400∘C respectively. Figure 3:
Differential stress at 5% strain ($\sigma_{5}$) as a function of confining
pressure in Solnhofen limestone (a), Carrara marble (b) and Wombeyan marble
(c), at temperatures of 20, 200 and 400∘C. All tests were conducted at a
strain rate of $10^{-5}$ s-1.

####  3.1.2 Effect of pressure and temperature

Figure 4: Evolution of yield stress ($\sigma_{\mathrm{y}}$), differential
stress at 2.5% strain ($\sigma_{2.5}$) and 5% strain ($\sigma_{5}$) as
functions of grain size. Circles correspond to data obtained in this work, and
other symbols are taken from the literature (see Table 4 for list of labels).

The absolute strength, i.e. the differential stress at a given strain, of all
rocks tested decreases with increasing temperature. The temperature
sensitivity is greater at elevated pressure (400 and 600 MPa) than at low
pressure, and is greater in Wombeyan marble than in Carrara marble and
Solnhofen limestone (Figure 3).

Both Carrara marble and Wombeyan marble have strengths and yield stresses that
tend to increase with increasing pressure (Figure 3). However, strength seems
to become pressure-independent beyond a pressure threshold. In Carrara marble,
there are no quantitative differences between the tests conducted at 600 MPa
and at 800 MPa, and only small differences can be detected between 400 MPa and
600 MPa. In Wombeyan marble, strength does not change appreciably with
pressure above 400 MPa. The pressure sensitivity of strength in Carrara marble
and Wombeyan marble is reduced by increasing temperature, and strength becomes
pressure independent at a lower pressure as temperature increases. Hardening
rates in Carrara marble and Wombeyan marble also increase with increasing
pressure, except for at a temperature of $400^{\circ}$C, at which hardening
rates are independent of pressure.

Solnhofen limestone exhibits some differences compared to the other
lithologies in terms of the pressure dependence of strength. At all
conditions, the yield stress of Solnhofen limestone decreases with increasing
pressure, in contrast with the other lithologies. In contrast, at 5% strain,
the strength of Solnhofen limestone increases slightly with pressure from 200
to 400 MPa, but decreases at 600 MPa (Figure 3a).

Wave-speed changes also depend on pressure and temperature. Decreases in axial
P-wave speed with strain are smaller at higher temperature in both Solnhofen
limestone and Carrara marble. For example, in Carrara marble at a pressure of
$200$ MPa and strain of $7.5\%$, the relative wave-speed decrease is 8% at a
temperature of $20^{\circ}$C (Figure 2b ) and is 2.5% at $400^{\circ}$C
(Figure 2h). In Carrara marble, at all temperatures, increasing pressure
results in smaller wave-speed decreases during deformation. In contrast, in
Solnhofen limestone at temperatures of $20^{\circ}$C and $200^{\circ}$C the
final wave-speed drop increases with increasing pressure (Figure 2a,d).

Figure 5: Hardening modulus at 5% strain as a function of grain size, at (a)
20∘C, (b) 200∘C and (c) 400∘C. Circles correspond to data obtained in this
work, and other symbols are taken from the literature (see Table 4 for list of
symbols).

####  3.1.3 Effect of grain size

Table 4: Summary of references for literature data presented in Figure 4 and 5. Reference | Lithology | Grain size | Legend
---|---|---|---
Paterson1958 | Wombeyan marble | 1–2 mm |
Heard1960 | Solnhofen limestone | 6–10 µm |
Edmond1972 | Solnhofen limestone | 6–10 µm |
Edmond1972 | Carrara marble | 60–120 µm |
Fredrich1989 | Carrara marble | 60–120 µm |
Rybacki2021 | Carrara marble | 60–120 µm |
Fredrich1990 | Wombeyan marble | 1–2 mm |
Donath1971a | Beldens marble | 0.5 mm |
Mogi1964 | Yamaguchi ’Fine marble’ | 0.4 mm |
Mogi1964 | Mito marble | 1 mm |
Mogi1964 | Yamaguchi ’Coarse marble’ | 3.5 mm |
Rutter1974 | Solnhofen limestone | 6–10 µm |

Grain-size has a marked effect on the mechanical behaviour. Solnhofen
limestone is always the strongest material and Wombeyan marble the weakest,
independent of test conditions (Figures 2 and 4). For example, at a
temperature of $20^{\circ}$C and pressure of 600 MPa, the stress at $7.5\%$
strain is 570 MPa for Solnhofen limestone, 400 MPa for Carrara marble, and 310
MPa for Wombeyan marble (Figure 4).

Grain-size strengthening is further promoted by increases in strain. For
example, at a temperature of $20^{\circ}$C the range of stresses across the
grain sizes tested is $\sigma_{\mathrm{y}}=$ 100–300 MPa at the yield stress,
$\sigma_{2.5}=$ 250–500 MPa at 2.5% strain, and $\sigma_{5}=$ 300–600 MPa at
5% strain (Figure 4). This observation is consistent with measurements of the
hardening modulus, which increases with decreasing grain size at all tested
temperatures (Figure 5).

Wave-speed evolution is also sensitive to grain size, as highlighted by
comparing the behaviour of Solnhofen limestone and Carrara marble. The overall
decrease in wave speed is nearly always greater for Solnhofen limestone at a
given set of conditions than it is for Carrara marble. For example, at
$T=20^{\circ}$C, $P=600$ MPa and $\varepsilon=7.5$%, the wave-speed change is
$-5$% in Carrara marble (Figure 2b) but is $-11$% in Solnhofen limestone
(Figure 2a). The only exception to this grain-size dependence is at
$T=400^{\circ}$C and $P=600$ MPa. At these conditions, the final relative
wave-speed change at $\varepsilon=7.5$% is $+2$% in Solnhofen limestone
(Figure 2g) and -2% in Carrara marble (Figure 2 h) dashed dark blue curve).

###  3.2 Wave-speed changes during decompression

Figure 6: Relative change in P-wave speed referenced to the final wave speed
($V_{\mathrm{final}}$) measured during the unloading phase of each experiment
and after sample cooling. Stars denote the wave speed measured at atmospheric
pressure after sample recovery. The grey curve represents measurements of wave
speed in a fused silica blank over the same pressure range.

Wave speed always decreases significantly during decompression (Figure 6). In
Solnhofen limestone, the P-wave speed remains approximately constant during
initial decompression down to around $400$ MPa. Further decompression leads to
a decrease in wave speed, which drops substantially at pressures below $200$
MPa. In samples deformed at room temperature, the wave speed of the recovered
material is as low as 50% of the wave speed measured after deformation but
prior to decompression. This drop becomes less marked in samples deformed at
high temperature, with changes on the order of 25–35% at $T=200^{\circ}$C, and
15–30% at $T=400^{\circ}$C. The behaviour is generally similar in Carrara
marble, with some notable quantitative differences. The wave speed starts
decreasing immediately as pressure in decreased, even in tests conducted at
$P=600$ MPa (Figure 6b,d). The characteristic pressure below which P-wave
speed drops most markedly is on the order of $100$ MPa. The effect of
deformation temperature is similar to that observed in Solnhofen limestone,
with elevated temperature during deformation promoting a more limited
reduction in wave speed during decompression.

###  3.3 Microstructures

Figure 7: Span of microstructural regimes identified in deformed samples.

####  3.3.1 Brittle features

Table 5: Crack densities, $S_{v}$, in selected samples as the volumetric
number of cracks determined from stereological measurements following
Underwood1970

.  Sample Lithology Pressure Temperature $S_{v}$ (mm2/mm3) Run0075 Carrara 200
19.8 18.73 Run0078 Carrara 405 19.8 7.31 Run0084 Carrara 615 19.8 2.28 Run0086
Carrara 236 197 7.60 Run0090 Carrara 213 390 2.05 Run0093 Carrara 599 192 7.82
Run0094 Carrara 594 401 1.30 Run0098 Carrara 769 20.1 4.60 Run0104 Wombeyan
206 19.3 4.02 Run0106 Wombeyan 595 20.2 0.33 Run0110 Wombeyan 231 207 1.51
Run0119 Wombeyan 609 187 0.49 Run0120 Wombeyan 408 19.2 1.60 Run0121 Wombeyan
428 190 0.35 Run0124 Wombeyan 233 408 0.47 Run0126 Wombeyan 574 416 0.36

Figure 8: Brittle microstructures observed in samples after testing. a) Crush
zones, b) geometrically controlled cracks, c) inter-twin cracks and stair-step
cracks, d) grain-boundary cracks and low aspect-ratio cracks, e) stair-step
cracks with orientations controlled by twin boundaries and f) cracks nucleated
from twins.

Systematic observations of cracks within deformed samples reveals three
distinct deformation regimes according to deformation conditions (figure 7).
The low-pressure, low-temperature regime and the high-pressure, low-
temperature regime are characterised by cracks, whereas in the high-
temperature regime cracks are absent.

In the low temperature regimes where cracks are observed, cracks interact with
twins. Dense arrays of cracks confined to individual twin lamellae are
commonly observed (Figure 8c). In addition, some intragranular cracks contain
steps and change orientation across individual twin lamellae (’stair-step’,
Figure 8e). In other instances, microcracks are observed to nucleate from the
tips of twins, and do not reach grain boundaries (Figure 8f).

Low-pressure, low-temperature regime

The low-pressure, low-temperature regime is limited to Carrara marble
experiments conducted at 200 MPa and 20 and 200∘C only. In this regime mode-I
intragranular microcracks are a common feature, and result in the highest
crack densities. These microcracks are typically confined to single grains and
are of low aspect ratio (e.g., Figure 8b,d). In places, the microcracks are
concentrated in discrete regions as crush zones (Figure 8a). In these regions,
cracks often span a few grains and result in a locally elevated crack density.

Cracks are also observed to relate to the geometry of grains and grain
boundaries. Some cracks nucleate at geometric irregularities along grain
boundaries. For example, steps in grain boundaries are often associated with
short tensile cracks that propagate a short distance into the grain interior
(Figure 8d). Smaller grains can also act as indenter grains, and cracks are
nucleated to accommodate the indenter grain shape (Figure 8b). Often, cracks
relating to geometric incompatibilities can be wholly contained within a
single grain (Figure 8b and d).

High-pressure, low-temperature regime

The high-pressure, low-temperature regime spans samples deformed above $P=200$
MPa and at $T=20^{\circ}$ and 200 ∘C for Carrara marble, and all samples
deformed at 20∘C and 200∘C for Wombeyan marble (Figure 7). This regime is
characterised by a reduced crack density ($S_{v}<$8 mm2/mm3) with respect to
the low-pressure, low-temperature regime. Intragranular cracks are not
completely suppressed (Figure 8d), e) and f)). However, thorough observations
revealed no intergranular cracking and crush zones. Another important change
compared to the low-pressure, low-temperature regime is the abundant
observation of open grain boundaries (Figure 8c and e).

Within the high-pressure, low-temperature regime, the crack density decreases
with increasing pressure for Carrara marble and Wombeyan marble. For example
in Carrara marble at 20∘C, $S_{v}=$ 7.31 mm2/mm3 at $P=$ 400 MPa reducing to
$S_{v}=$ 2.28 mm2/mm3 at $P=$ 600 MPa (Table 5, Run0078 and Run0084).
Furthermore, for a given set of conditions in the high-pressure, low-
temperature regime, crack density is lower in the coarser grained Wombeyan
marble.

####  3.3.2 Crystal-plastic microstructures

Figure 9: a) High twin density generated at high-stress, low-temperature
conditions. b) High-density twins bending around an apparent deformation band.
Twins are also nucleated by twins in neighbouring grains. c) Long-range
undulose extinction traversing twin sets. The undulose extinction appears to
be generated by the interaction of neighbouring grains. d) Multiply-twinned
grains and high twin density. e) Corrugation of crystal lattice identified by
wavy undulation. f) Multiple curved twin sets and twinned twin sets. g)
Multiply-twinned grain in Wombeyan marble. Twins are thicker than at lower-
temperature conditions and also appear more patchy. h) Twins formed at high
temperature in Carrara marble. Twins are thicker and the central grain shows
evidence of undulose extinction.

Crystal-plastic microstructural features are dominated by deformation twins
that are present in samples deformed at all conditions (Figure 9). Nearly all
grains are twinned on at least one plane, grains that contain two twin sets
are also common, and occasionally grains contain three twin sets. In
particular, the density of twins, i.e., the number of lamellae per unit
length, is high in samples deformed at low temperature (Figure 9 a-d) and
decreases with increasing temperature (Figure 9g,h). Twins are often curved,
especially in the vicinity of grain boundaries (Figure 9f) or around geometric
irregularities (Figure 9 b). Twins also often appear to have nucleated from
twins within neighbouring grains (figure 9b,d) or from geometric
irregularities at grain boundaries (figure 9f).

On a larger scale than twin lamellae, undulose extinction is widespread
(Figure 9). In Wombeyan marble, the intensity of the undulose extinction
increases when approaching grain boundaries and grain-boundary irregularities
(Figure 9c). In Wombeyan marble, the undulose extinction observed in some
instances suggests corrugation of the crystal lattice (Figure 9e). In Carrara
marble, grains also display undulose extinction, although it is not as common
as in Wombeyan marble. Again, in Carrara marble undulose extinction is
associated to geometric irregularities (Figure 9h).

####  3.3.3 SEM observations

Figure 10: Forescattered-electron orientation-contrast images Prior . (1996)
of deformation twins in (a and b) Solnhofen limestone, (c) Wombeyan marble and
(d) Carrara marble.

Forescattered electron images were used to image twins at high spatial
resolution. These images reveal very thin deformation twins, especially in
Solnhofen limestone, which in some places are on the order of 100 nm in
thickness (Figure 10a,b). Grains of calcite in Solnhofen limestone also
exhibit lower twin incidence than the coarser-grained samples of Wombeyan
marble and Carrara marble (Figure 10c,d). Twins in Solnhofen limestone also
propagate across grain boundaries, and are not bent on the grain scale,
although they do sometimes taper in proximity to grain boundaries. High-
resolution electron images of Wombeyan marble reveal multiple scales of twins,
with micrometre-scale twins contained within thicker twins on the order of
tens of micrometres in thickness (Figure 10c).

Figure 11: EBSD maps of samples deformed at a pressure of 600 MPa and
temperature of 200∘C. (a) Wombeyan marble, (b) Carrara marble, and (c)
Solnhofen limestone. Grains are coloured according to the inverse pole figure.
Twin boundaries are not shown for clarity. White areas were not indexed and
are mostly due to poor quality diffraction patterns along twin boundaries.

Overview EBSD maps demonstrate that most grains have significant internal
misorientation. Lattice curvature is particularly obvious in Wombeyan marble
(Figure 11a) and in Carrara marble (Figure 11b) and in some grains of
Solnhofen limestone (Figure 11c). Regions of lattice curvature are often
related to the geometry of grain boundaries and in places curvature increases
in the vicinity of grain boundaries. Other grains exhibit lattice rotation in
the vicinity of twin planes, indicated by stripes in the inverse pole figure
map (e.g., Figure 11b).

Figure 12: Map of grain reference orientation deviation (GROD) obtained from
EBSD maps of Run0093 ($P$ = 600 MPa, $T$ = 200∘C). Insets (b), (c) and (d) are
small-area EBSD maps showing the local GROD within grains with varying
interaction between lattice distortion and twins. Strong interactions between
twins and lattice curvature are evident in (c), intermediate interaction is
present in (b), and weak interaction is apparent in (d). Insets b) and d) also
exhibit significant lattice curvature in the vicinity of grain boundaries.

Further evidence of internal lattice distortion is revealed by maps of the
grain reference orientation deviation (GROD), which is the misorientation of
each point with respect to the mean orientation of the grain (Figure 12). An
overview map of the GROD in Carrara marble Run0093, deformed at a pressure of
$600$ MPa and temperature of $200^{\circ}$C demonstrates that individual
grains have variable internal structure (Figure 12a). Some grains exhibit
striped patterns of variable GROD that follow the local twin orientation
(Figure 12c). In other grains, the GROD pattern is largely unaffected by twins
and instead exhibits lattice distortion over larger length scales approaching
the grain size (Figure 12d). In some cases, there is a mixed interaction,
where the GROD weakly follows the twin orientation but is also affected by the
grain geometry (Figure 12b).

Figure 13: HR-EBSD maps of (a) the grain in Figure 12b and (b) the grain in
Figure 12c. Each of these subfigures presents maps of the band contrast in the
diffraction patterns, which reveals the locations of twins, GND density, and
heterogeneity in stress ($\sigma_{ij}$). (c) Normal probability plot of
$\sigma_{12}$ in each grain. Straight lines indicate a normal distribution.
(d) Restricted second moment ($\nu_{2}$) versus ln$\sigma_{12}$. Straight
lines indicate that the probability distribution of the stress exhibits the
form $P(\sigma)\propto|\sigma|^{-3}$ expected of a population of dislocations.

HR-EBSD maps (Figure 13) reveal the distributions of GNDs and intragranular
stress heterogeneity in the endmember grains in Figure 12. The grain in Figure
12b, which exhibited the least impact of the twins on the distribution of
GROD, also exhibits negligible impact of the twins on GND density (Figure
13a). Within the grain interior, GND density is generally at or below the
background noise level of approximately $10^{13.5}$ m-2 arising from noise in
the rotation measurements. Apparent GND densities above this noise level are
highly localised in the vicinity of twin boundaries and potentially result
from reduced pattern quality indicated by the band-contrast map. However, GND
densities are significantly elevated to the order of $10^{14}$ m-2 adjacent to
a grain boundary along the right edge of the map. Comparable distributions are
evident in the maps of stress heterogeneity with stresses in the grain
interior being relatively homogeneous and stresses near the grain boundary
being elevated to several hundred megapascals. Different distributions are
apparent in HR-EBSD maps (Figure 13b) from within the grain in Figure 12c,
which exhibited the greatest impact of the twins on the distribution of GROD.
In this grain, zones of elevated GND density and stress extend a few
micrometres from the twin boundaries and beyond the zones of reduced pattern
quality represented in the band-contrast map. Within these zones, GND
densities approach $10^{14}$ m-2 and shear stresses on the order of several
hundred megapascals are common.

The probability distributions of the stress heterogeneity provide further
information on the cause of the stresses. The normal probability plot (Figure
13c) exhibits similar probability distributions for both grains, with stresses
that are significantly greater than those measured on the undeformed Si
standard. Below stress magnitudes of approximately 300–500 MPa, the
distributions fall on a straight line indicating these stresses are normally
distributed. However, at greater stress magnitudues, the distributions depart
from straight lines. These high-stress tails are typical of materials,
including Cu and olivine, deformed by dislocation-mediated mechanisms Jiang .
(2013); Wallis . (2021, 2022). The plot of the restricted second moment
$\nu_{2}$ versus ln($\sigma_{12}$) provides a further test of whether these
high-magnitude stresses are the stress fields of dislocations (Figure 13d). On
this plot, the distributions from the maps from each grain both fall on
straight lines at high stresses, indicating that the distributions have the
$P(\sigma)\propto|\sigma|^{-3}$ form that is characteristic of the stress
field of a population of dislocations Groma Bakó (1998); Wilkinson . (2014);
Kalácska . (2017); Wallis . (2021). These characteristics indicate that the
stresses are, at least in part, the stress fields of dislocations.

Figure 14: The final differential stress at the end of deformation as a
function of twin density for this study is shown in black symbols. Additional
data from previous studies are shown in grey. CM RR 1990, TM RR 1990 and SL RR
1990 are data from Carrara marble, Taiwan marble, and Solnhofen limestone by
Rowe1990, CM RWK 2022 is data from Carrara marble data by Rutter2022, and CM
Ry 2013 is data from Carrara marble by Rybacki2013. The reference line
indicates the gradient of a relationship in which stress is dependant on the
square root of twin density.

In addition to the EBSD mapping, based on the forescatter imaging, we were
able to measure twin spacing and correct their thickness in the same manner as
¡¿[Figure 14]Rutter2022. We report values of twin density, $\rho_{\mathrm{t}}$
(mm-1), which is computed as the inverse of the measured average twin spacing
(Figure 14b). Our data agree with previously reported measurements of stress
versus twin density. Twin density is lowest in Wombeyan marble deformed at a
pressure of 600 MPa and temperature of 200∘C with a value of 130 mm-1,
equivalent to a mean spacing of 7.5 $\mu$m. The highest density is obtained
for Solnhofen limestone deformed at 600 MPa and 200∘C, with a value of about
1500 mm-1, equivalent to a mean twin spacing of 0.7 $\mu$m.

##  4 Discussion

The mechanical data reveal that deformation of calcite rocks across the range
of conditions that we tested is ductile. Almost all experiments demonstrate
strain-hardening behaviour but the precise characteristics of the hardening
vary with pressure and temperature. The deformation of calcite rocks in this
study is accommodated by cracking, twinning and dislocation motion, depending
on experimental conditions. The main question we seek to answer is what
controls the rheological behaviour of calcite rocks in the semi-brittle
regime? Determination of the rheological behaviour of calcite rocks requires
knowledge of the approximate partitioning of strain between each active
deformation mechanism. This can be answered by considering the
microstructural, wave-speed and mechanical data.

###  4.1 Brittle deformation

Figure 15: Hardening modulus computed at 5% strain plotted against post-mortem
crack densities. a) Carrara marble data from room-temperature experiments
(circles this study, stars Fredrich1989) and at $200^{\circ}$C (triangles). b)
Wombeyan marble from experiments at room temperature (circles) and
$200^{\circ}$C (triangles).

The evolution of wave speed combined with post-mortem measurements of crack
density characterise the degree of microcracking occurring during deformation.
At all temperatures at which strength is pressure sensitive, increasing
pressure results in increasing hardening rates, smaller wave-speed decreases
and a reduction in post-mortem crack density (Figure 15 and Table 5). Similar
observations of decreasing post-mortem crack density with increasing pressure
were made by Fredrich1989 in Carrara marble at room temperature. Measurements
of volumetric strain during the deformation of Carrara marble and Solnhofen
limestone at room temperature made by Edmond1972 demonstrate that dilatancy is
suppressed at high pressure. Both of these studies also report increased
hardening rates with increasing pressure. Decreasing strain accommodation by
cracking is therefore systematically associated with increasing hardening
rates (Figure 15).

With increasing temperature, pressure-independent strength is reached at a
lower pressure. In conjunction with this, deformation at elevated pressure is
systematically related to smaller drops in wave speed and increased hardening
rates. Crack density is also lower for a given pressure at higher temperature.
Measurements of reduced pore volume during deformation of Carrara marble at
high temperatures by Fischer1989 support this observation. Taken together,
these observations identify that cracking is suppressed by temperature
increases.

Another interesting observation is the large wave-speed decrease during
decompression. This feature suggests that significant sample damage is accrued
during decompression. Samples with the largest wave-speed decrease are those
deformed at high-pressure, low-temperature conditions, which correspond to a
low intragranular fracture density (table 5). These samples are characterised
by open grain boundaries (Figure 8b, c, e and g), suggesting that grain-
boundary opening is the cause of the observed wave-speed decrease. Similar
observations were made by Edmond1972, who reported increases in volumetric
strain during decompression in deformed Solnhofen limestone and Carrara
marble, with the magnitude of volumetric strain change increasing with the
final stress level. This phenomenon is likely to result from deformation
caused by strain incompatibility between grains Ashby (1970). During
deformation at high pressure, individual grains deform both plastically and
elastically to accommodate the imposed strain in the sample, which results in
heterogeneous internal stresses. Upon removal of the confining pressure, these
internal stresses are no longer in equilibrium with the applied external
forces and are likely to generate interface cracks between grains to
accommodate the strain incompatibilities accrued during deformation.

###  4.2 Grain-size sensitive behaviour

Our results demonstrate that yield stress and the hardening modulus are
functions of grain size. This result is consistent with the works of
Olsson1974 and Fredrich1990 and other published results at both low
temperatures <¿[Figure 4 and Table
4]Paterson1958,Heard1960,Mogi1964,Donath1971a,Edmond1972,Rutter1974,Fredrich1989,Rybacki2021
and high temperatures Renner . (2002).

Based on experiments at room temperature and moderate pressure, Fredrich1990
discussed grain-size strengthening in the model framework of Horii1986.
Horii1986 solved a wing-crack model coupled to a ’plastic zone’, and their
results predicted that coarser-grained materials should be more ’brittle’,
i.e., the ratio of differential stress to confining pressure at the brittle-
ductile transition should decrease with increasing grain size. The results of
Fredrich1990 showed that this ratio was independent of grain size, which they
explained by considering that the plastic yield stress scaled as $D^{-1/2}$.
This hypothesis is supported by our observations. More generally, yield stress
is observed to scale inversely with grain size in metals Cordero . (2016) and
also in olivine Hansen . (2019).

Grain-size strengthening is a common phenomenon in metals and has received
significant attention in the metallurgy literature (see Cordero2016 and Li2016
for reviews). Strengthening with decreasing grain size is usually termed the
Hall-Petch effect Hall (1951); Petch (1953), and is typically of the form

$\sigma=\sigma_{0}+KD^{-m},$ (1)

where $\sigma_{0}$ is the intrinsic resistance of a lattice to dislocation
motion, $K$ is a material-dependent Hall-Petch coefficient and $m$ is a
dimensionless exponent. In the original observations and theoretical arguments
of Hall1951 and Petch1953, the exponent $m=0.5$. Values of $m=0.2–1$ have been
reported in metals Dunstan Bushby (2014); Cordero . (2016); Y. Li . (2016).

Variable |  $\sigma_{0}$ (MPa) |  $K$ (MPa m-0.5)
---|---|---
$T=20~{}^{\circ}$C
$\sigma_{y}$ | 155 | 0.40
$\sigma_{2.5}$ | 244 | 0.73
$\sigma_{5}$ | 274 | 0.87
$T=200~{}^{\circ}$C
$\sigma_{y}$ | 83 | 0.51
$\sigma_{2.5}$ | 149 | 0.72
$\sigma_{5}$ | 185 | 0.79
$T=400~{}^{\circ}$C
$\sigma_{y}$ | 68 | 0.52
$\sigma_{2.5}$ | 91 | 0.86
$\sigma_{5}$ | 108 | 0.96
Table 6: Results of fitting collated mechanical data with the Hall-Petch
relation, with $m=0.5$ (Equation 1).

We fitted equation 1 to our data using a least-squares regression and setting
$m=0.5$ (Table 6), similar to Renner2002. Although our results suggest $m$ in
the range 0.3–0.6 (Figure 4), fitting the exponent significantly modifies
$\sigma_{0}$ and $K$ and makes comparison among our datasets challenging.
Values of the apparent lattice resistance, $\sigma_{0}$, and the Hall-Petch
coefficient, $K$, increase with strain at all temperatures. Values of the
lattice resistance are largest at $T=20^{\circ}$C, with $\sigma_{0}=155–274$
MPa, and are smaller at $T=200~{}^{\circ}$C with $\sigma_{0}=83–185$ MPa,
further reducing at $400~{}^{\circ}$C to $\sigma_{0}=68-108$ MPa. The values
for the lattice resistance are consistent with measurements from single-
crystal experiments by DeBresser1991 and Turner1954 at the same temperatures.
Values of the Hall-Petch coefficient are largely unaffected by temperature
changes. For the reported values of $\sigma_{\mathrm{y}}$, $K=0.4–0.52$ MPa
m-0.5, at larger strain for values of $\sigma_{2.5}$ the Hall-Petch
coefficient increases to $K=0.72–0.86$ MPa m-0.5 with further increases at
large strain of $K=0.79–0.96$ MPa m-0.5 for $\sigma_{5}$ values. When $K$ is
normalised by the product of the shear modulus $G$ and Burgers vector $b$,
with $G$ = 35 GPa and $b$ = 0.74 nm, it falls into the range 0.42–1, which is
consistent with values typically obtained for BCC and HCP metals Cordero .
(2016). In summary, our data combined with literature sources reveal several
key features of the Hall-Petch effect in calcite rocks, 1) the Hall-Petch
effect is amplified by strain, being weakest at yield and strong after plastic
strain, 2) the apparent lattice resistance increases with plastic strain and
decreases with temperature, and 3) the Hall-Petch coefficient is independent
of temperature.

Extensive reviews by Cordero2016 and Li2016 summarise proposed physical models
of the Hall-Petch effect. Li2016 identify four categories of models: (1) the
dislocation pile-up model, in which grain boundaries act as obstacles that
cause pile-ups until the stress in front of the pile-up is sufficient to
induce yielding of neighbouring grains Hall (1951); Petch (1953); (2) the
grain-boundary ledge model, in which grain- and subgrain-boundary
irregularities emit forest dislocations that act as obstacles <see¿[]Li1963;
(3) the plastic-strain model, in which the rate of increase in dislocation
density with plastic strain is inversely proportional to grain size Conrad .
(1967); (4) the elastic-anisotropy model, in which interactions among
elastically anisotropic grains require the introduction of GNDs, and smaller
grains have relatively larger strain gradients and greater GND densities
Meyers Ashworth (1982). All of these models, except that of Meyers1982, arrive
at an expression similar to Equation 1. The coefficient $K$ always includes
the shear modulus, $G$, and the Burgers vector, $b$, as well as other
geometric constants, and $m=0.5$. However, fitting exercises have shown that
for a wide range of metals, $m\neq 0.5$ and may be better fit by $m=-1$ or by
the relationship $\ln d/d$ Dunstan Bushby (2014); Y. Li . (2016).

A grain-size dependence of the yield stress suggests that the plastic-strain
model 3 is not appropriate for calcite as a finite amount of macroscopic
plastic strain is required to generate the Hall-Petch effect Conrad . (1967);
Ashby (1970). As pointed out by Hansen2019, models 1, 2 and 4 also require
finite plastic strain, although this may be sufficiently localised for the
macroscopic behaviour still appear elastic prior to macroscopic yielding Maaß
Derlet (2018). The temperature dependence of the yield stress suggests that
the models must include short-range dislocation interactions, as long-range
interactions are elastic and therefore largely temperature insensitive Hansen
. (2019).

In model 1, the Hall-Petch effect is attributed to pile-ups of dislocations at
grain boundaries, and their role in promoting yield of neighbouring grains
Hall (1951); Petch (1953). In this case, larger grains can support longer
pile-ups, which generate more intense stress concentrations that promote yield
of neighbouring grains. If this model is relevant, we might expect to see
pile-ups of dislocations at grain boundaries and localised strain transfer
across grain boundaries. QuintanillaTerminel2016 use a micro-grid to estimate
microscale strains in deformed Carrara marble and document the occurrence of
strain transfer across grain boundaries, although this appears to be in broad
zones. Further observations using transmission electron microscopy and HR-EBSD
are needed to characterise dislocation structures in the vicinity of grain
boundaries in calcite to test the relevance of this model.

Model 2 considers that dislocations are emitted from grain-boundary ledges
<see¿[]Li1963. As materials with finer grain sizes have greater grain-boundary
areas, there are more potential sites for generation of dislocations,
resulting in a higher dislocation density and therefore stress required for
macroscopic plastic deformation. In our samples it is difficult to assess the
grain-boundary ledge model. Some evidence may be that undulose extinction is
often controlled by the geometry of grain boundaries (Figure 9), indicating
that dislocation activity is influenced by grain boundaries, although it does
not identify whether grain boundaries act as dislocations sources.

The elastic-anistropy model of Meyers1982 relies on incompatibilities in
elastic strain between neighbouring grains. When stress is applied, GNDs form
to accommodate the elastic mismatch between grains. Our microstructural
observations and wave-speed data may be compatible with this model. Open
grains boundaries (Figure 8 c, e and g) and large wave-speed decreases during
decompression (Figure 6) suggest significant relaxation of internal stress
during decompression and, in turn, this internal stress may result from
elastic mismatch among grains and associated GND formation. EBSD observations
also reveal significant lattice curvature (Figure 12) and increases in GND
density in the vicinty of grain boundaries (Figure 13). Furthermore,
microscale observations of strain in deformed calcite rocks by Spiers1979 and
QuintanillaTerminel2016 reveal local heterogeneity in finite strain near to
grain boundaries. In particular, QuintanillaTerminel2016 identify strain
heterogeneity on a scale similar to the grain size. Wallis2018 observed an
increase in misorientation in the vicinity of grain boundaries in naturally
deformed calcite rocks, consistent with the presence of plastic strain
gradients imparted in response to strain incompatibility among neighbouring
grains Meyers Ashworth (1982).

In summary, our observations suggest that either model 1, 2 or 3 maybe
applicable to calcite. More systematic microstructural observations are
required to discriminate among these models. As a generality, all models
predict that the strength is sensitive to the mean free path of dislocations,
which is controlled by grain size. Therefore, Hall-Petch models are closely
related to Kocks-Mecking-Estrin (KME) single state-variable models of
strength, which incorporate the role of dislocation mean free path Kocks
(1966, 1976); Mecking Kocks (1981). To explore the KME model we must consider
the role of twins, which is discussed in the next section.

###  4.3 Is TWIP compatible with the deformation of calcite rocks?

The activity of mechanical twinning in calcite is closely related to the
magnitude of differential stress Jamison Spang (1976); Spiers (1979); Rowe
Rutter (1990). In experiments, Rowe1990 demonstrated that the spacing of twins
decreases with increasing stress, independent of temperature and grain size.
Twins are observed in all our deformed samples, and twin density increases
with differential stress consistently with the observations of
Rowe1990,Rybacki2013,Rutter2022. Rybacki2013 also demonstrated that stress was
proportional to the square root of twin density at low stresses ($<250$ MPa).
However, at high stress, the value of stress saturates with respect to twin
density and the square root dependence breaks down <¿[and Figure 14]Rowe1990.
Transmission Electron Microscope observations of calcite deformation twins
indicate that twins often interact with dislocations Barber Wenk (1979);
Fredrich . (1989); Rybacki . (2013). Given these observations, Rybacki2021
argued that twin spacing linearly decreases as dislocation density increases,
as flow stress is proportional to the square root of dislocation density
Taylor (1934). These observations suggest that twin spacing either is a
consequence of, or controls, the strength of calcite rocks.

Rybacki2021 argued that twin spacing directly controls the strain hardening
rate, and by extension the strength, by drawing analogy to twinning induced
plasticity steels <TWIP,¿[]DeCooman2018. Models of TWIP originate from high
manganese steels, which exhibit high hardening rates (approx. 3% of the shear
modulus) with respect to other steels (approx. 0.05 % of the shear modulus),
as a consequence of mechanical twinning. The mechanism of TWIP originates from
the abrupt changes in crystallographic orientation at twin boundaries, which
can act as barriers to dislocation motion. Progressive twinning leads to
dynamic refinement of the microstructure De Cooman . (2018), in which finer
twin spacing reduces the mean free path of dislocations ($\lambda$) and causes
a dynamic Hall-Petch effect.

In phenomological models of TWIP in the metallurgical literature, strain
hardening is attributed to a dynamic Hall-Petch effect resulting from
progressive twinning Bouaziz . (2008). The model of Bouaziz2008 is formulated
by first considering the Taylor equation that relates shear flow stress
($\tau$) to the total dislocation density ($\rho$),

$\tau=\tau_{0}+\tau_{b}+\alpha\mu b\sqrt{\rho},$ (2)

where $\tau_{0}$ is the initial strength of a polycrystal, $\tau_{b}$ is the
back stress, which may arise from long-range dislocation interactions and
stress fields around twins, $\alpha$ a constant close to unity, $b$ the
Burgers vector and $\mu$ the shear modulus. In their formulation, the final
term represents isotropic hardening due to short-range dislocation
interactions. In calcite, the Taylor equation has been demonstrated to apply
to calcite rocks deformed at temperatures of 550–700∘C JHP. De Bresser (1996),
although the relative contributions of kinematic hardening due to long-range
dislocation interactions that generate back stress and isotropic hardening due
to short-range dislocation interactions have not been separated.

To obtain the evolution of stress with strain, the Taylor relation is combined
with a modified Kocks-Mecking-Estrin equation Kocks (1966, 1976); Mecking
Kocks (1981) to describe the change of $\rho$ with strain Bouaziz . (2008):

$\frac{d\rho}{d\varepsilon}=\frac{1}{b\lambda}-f\rho=\frac{1}{b}\left(\frac{1}{D}+\frac{1}{D_{\mathrm{t}}}+k\sqrt{\rho}\right)-f\rho,$
(3)

where $f$ is a rate- and temperature-dependent dynamic-recovery coefficient,
$k$ is a constant that characterises dislocation storage due to dislocation
interactions, and $D_{t}$ is the twin spacing. In this model, changes in the
dislocation mean free path are sensitive to the total dislocation density,
grain size and twin spacing. An additional term, given by the product of the
recovery factor and the dislocation density, $f\rho$, is subtracted to account
for dynamic recovery processes. Rybacki2021 argued that this model could
potentially capture the rheological behaviour of calcite polycrystals.

The dynamic Hall-Petch model of Bouaziz2008 provides a physical basis for the
dependence of stress on the square root of twin density at low stress.
Rybacki2021 also argued that the high hardening rates (3–5 % of the shear
modulus, Figure 5) observed in calcite rocks are also consistent with TWIP.
However, it should be noted that high hardening rates (compared to typical
expectations in metals of 0.5-1% of the shear modulus) are not unique to
calcite rocks as olivine, which does not twin, exhibits hardening rates up to
5–10 % of the shear modulus when deformed by low-temperature plasticity Hansen
. (2019); Druiventak . (2011).

The TWIP model also suggests that the hardening rate should be dominated by
twin spacing as $D_{\mathrm{t}}$ is always at least 1–2 orders of magnitude
smaller than the grain size $D$ (Fig. 14). Taking the case of Carrara marble
at 600 MPa, 200 ∘C with $D=$ 100 µm and $D_{\mathrm{t}}=$ 5 µm, we might
expect strain hardening of $450\alpha\mu$ (neglecting recovery and forest
hardening). At the same conditions for Wombeyan marble, $D=1$ mm and
$D_{\mathrm{t}}=$ 7.5 µm, so that strain hardening should be of
$360\alpha\mu$. The ratio of hardening rates would therefore be 1.25. The
actual hardening rates observed in our experiments are in a ratio of 2
(considering $H_{5}=$ 1.6 and 0.8 GPa for Carrara marble and Wombeyan marble
respectively), which suggests that twins may have a smaller impact on
hardening than anticipated from Equation 3.

####  4.3.1 Twins as potential barriers to dislocations

Our main observation is that of grain size strengthening (Figure 4), and it is
difficult to assess the exact role of twins in the hardening behaviour from
our microscopic data alone. The TWIP model is founded on the notion that twins
produce further hardening by either retarding or stopping dislocation motion.
In order to test the potential validity of the TWIP model in calcite, in this
Section we assess the respective efficiency of grain boundaries and twins at
impeding dislocations.

Microstructural observations suggest that the interaction between dislocations
and twins boundaries varies between grains. Some grains contain lattice
curvature that is clearly affected by twin boundaries (e.g., Figure 12c)
whereas in other grains the lattice curvature appears to be affected by twin
boundaries (e.g., Figure 9c and Figure 12d). The trapping of dislocations
therefore appears to depend on grain orientation and related twin orientation.

We can assess the effectiveness of twin boundaries and grain boundaries as
barriers to dislocations by considering slip-transmission coefficients. The
simplest form of this analysis is purely geometric and considers only the
slip-plane orientation and direction of the Burgers vector of the incoming and
outgoing slip systems Luster Morris (1995),

$m^{\prime}=(n_{\mathrm{A}}\cdot n_{\mathrm{B}})(b_{\mathrm{A}}\cdot
b_{\mathrm{B}})=\cos(\phi)\cos(\kappa)$ (4)

where $n$ denotes the unit normal vector of the slip plane and $b$ the unit
Burgers vector, and the subscripts $\mathrm{A}$ and $\mathrm{B}$ denote the
incoming and outgoing slip systems. A value of zero for $m^{\prime}$ indicates
an impenetrable barrier to dislocations and a value of one indicates a
transparent boundary.

Figure 16: Computed values of $M^{\prime}$ for a) grain boundaries and b) twin
boundaries.

Extensive work by DeBresser1990,DeBresser1993,DeBresser1997 demonstrated that
$r$10-14¡-2021¿± and $f$-1012¡2-201¿± are the dominant slip systems in calcite
at low temperatures. The critical resolved shear stresses of these slip
systems is about an order of magnitude greater than that of $e$ twinning at
room temperature. Despite their strength, we consider them to be active in our
samples since our stress level is typically significantly above this level and
we observe significant intragranular misorientation. We computed slip
transmission across twin and grain boundaries using a random fabric in MTEX.
Maximum values of $m^{\prime}$ across twin boundaries (Table 4) demonstrate
that $m^{\prime}$ is always greater than 0.4, and in some cases twins are
transparent with a value of 1 (e.g., $r_{2}^{+}$ to $r_{3}^{+}$ across an
$e_{1}$ twin). Given also the large number of available slip systems, this
analysis suggests that twin boundaries may impede dislocation motion less than
randomly oriented grain boundaries.

$S_{\mathrm{A}}$ | maximum $m_{e1}^{\prime}$ | $S_{\mathrm{B}}$ | maximum $m_{e2}^{\prime}$ | $S_{\mathrm{B}}$ | maximum $m_{e3}^{\prime}$ | $S_{\mathrm{B}}$
---|---|---|---|---|---|---
$r_{1}^{-}$ | 0.617 | $r_{1}^{-}$ | 0.414 |  $f_{1}$¡-220-1¿- | 0.414 |  $f_{2}$¡20-2-1¿-
$r_{2}^{-}$ | 0.414 |  $f_{2}$¡20-2-1¿- | 0.617 | $r_{2}^{-}$ | 0.414 |  $f_{2}$¡0-22-1¿-
$r_{3}^{-}$ | 0.414 |  $f_{3}$¡20-2-1¿- | 0.414 |  $f_{3}$¡-220-1¿- | 0.617 | $r_{3}^{-}$
$r_{1}^{+}$ | 0.630 | $r_{1}^{+}$ | 1.000 | $r_{3}^{+}$ | 1.000 | $r_{2}^{+}$
$r_{2}^{+}$ | 1.000 | $r_{3}^{+}$ | 0.630 | $r_{2}^{+}$ | 1.000 | $r_{1}^{+}$
$r_{3}^{+}$ | 1.000 | $r_{2}^{+}$ | 1.000 | $r_{1}^{+}$ | 0.630 | $r_{3}^{+}$
$f_{1}$¡-220-1¿- | 0.541 | $r_{1}^{-}$ | 0.718 |  $f_{1}$¡-220-1¿- | 0.596 |  $f_{1}$¡0-22-1¿-
$f_{1}$¡2-201¿+ | 0.541 | $r_{1}^{-}$ | 0.718 |  $f_{1}$¡-220-1¿- | 0.596 |  $f_{1}$¡0-22-1¿-
$f_{2}$¡0-22-1¿- | 0.596 |  $f_{2}$¡20-2-1¿- | 0.541 | $r_{2}^{-}$ | 0.718 |  $f_{2}$¡0-22-1¿-
$f_{2}$¡02-21¿+ | 0.596 |  $f_{2}$¡20-2-1¿- | 0.541 | $r_{2}^{-}$ | 0.718 |  $f_{2}$¡0-22-1¿-
$f_{3}$¡20-2-1¿- | 0.718 |  $f_{3}$¡20-2-1¿- | 0.596 |  $f_{3}$¡-220-1¿- | 0.541 | $r_{3}^{-}$
$f_{3}$¡-2021¿+ | 0.718 |  $f_{3}$¡20-2-1¿- | 0.596 |  $f_{3}$¡-220-1¿- | 0.541 | $r_{3}^{-}$
$f_{1}$¡0-22-1¿- | 0.541 | $r_{1}^{-}$ | 0.596 |  $f_{1}$¡-220-1¿- | 0.717 |  $f_{1}$¡0-22-1¿-
$f_{1}$¡02-21¿+ | 0.541 | $r_{1}^{-}$ | 0.596 |  $f_{1}$¡-220-1¿- | 0.717 |  $f_{1}$¡0-22-1¿-
$f_{2}$¡20-2-1¿- | 0.596 |  $f_{2}$¡20-2-1¿- | 0.541 | $r_{2}^{-}$ | 0.596 |  $f_{2}$¡0-22-1¿-
$f_{2}$¡-2021¿+ | 0.596 |  $f_{2}$¡20-2-1¿- | 0.541 | $r_{2}^{-}$ | 0.596 |  $f_{2}$¡0-22-1¿-
$f_{3}$¡-220-1¿- | 0.718 |  $f_{3}$¡20-2-1¿- | 0.718 |  $f_{3}$¡-220-1¿- | 0.541 | $r_{3}^{-}$
$f_{3}$¡2-201¿+ | 0.718 |  $f_{3}$¡20-2-1¿- | 0.718 |  $f_{3}$¡-220-1¿- | 0.541 | $r_{3}^{-}$
Table 7: Slip-transmission analysis of twin boundaries. Tabulated results for
the maximum value of $m^{\prime}$ (Equation 4) between slip system
$S_{\mathrm{A}}$ and $S_{\mathrm{B}}$. The $m^{\prime}$ subscript denotes the
twinning system considered: e.g., $m_{e1}^{\prime}$ represents slip transfer
across an $e1$ twin.

To further compare the effect of twin and grain boundaries on dislocation
motion, we can also consider the effects of twin-boundary orientation. This
analysis can be performed with a geometric criterion that quantifies the
degree of misalignment between the line intersections of incoming and outgoing
slip planes with the twin plane, $\mathbf{l}_{\mathrm{A}}$ and
$\mathbf{l}_{\mathrm{B}}$, respectively, and between their slip directions,
$\mathbf{d}_{\mathrm{A}}$ and $\mathbf{d}_{\mathrm{B}}$, respectively. The
following scalar quantity is maximal when the two slip systems on either side
of the boundary are aligned and slip can be transmitted easily across the
boundary Shen . (1986); Bayerschen . (2016):

$\hat{M}=(\mathbf{l}_{\mathrm{A}}\cdot\mathbf{l}_{\mathrm{B}})(\mathbf{d}_{\mathrm{A}}\cdot\mathbf{d}_{\mathrm{B}}).$
(5)

The line intersections $\mathbf{l}$ can be obtained from
$\mathbf{l}=(\mathbf{n}\times\mathbf{n}_{\mathrm{\Gamma}})/|\mathbf{n}_{A}\times\mathbf{n}_{\mathrm{\Gamma}}|$,
where $\mathbf{n}_{\mathrm{\Gamma}}$ is the twin boundary plane normal.

We used the $\hat{M}$ criterion to compare the efficiency of slip transmission
across twin boundaries to slip transmission across grain boundaries. A random
fabric was generated using MTEX, and we computed $\hat{M}$ between random
grain pairs by assuming that the activated slip system (either $r^{\pm}$ or
$f^{\pm}$ slip) was that with highest Schmid factor in each grain. For each
grain, we also computed $\hat{M}$ across a twin boundary hosted in the initial
grain, the activated twin system was assumed to be that with the highest
Schmid factor in each grain. These calculations suggest an average value for
slip transmission of $\hat{M}=0.27$ across grain boundaries, which is
considerably smaller than the average value of $\hat{M}=0.54$ for slip
transfer across twin boundaries.

Figure 17: Map of $\hat{M}$ values for slip transmission across grain
boundaries (coloured according to value) and twin boundaries (grains coloured
according to value) computed from EBSD data obtained for Run0093.

To further test this result, we computed the expected $m^{\prime}$ values for
our EBSD data across the activated twin systems within each grain (Figure 17).
The results demonstrate that grains with low values of $m^{\prime}$ (Figure
17, grain c) exhibit dislocation substructures that are affected by twinning,
that is segmentation of lattice distortion between twins (Figure 12c). In
contrast, grains with large values of $m^{\prime}$ (Figure 17, grain d)
exhibit gradients in lattice distortion over length scales approaching the
grain size, but which are not strongly affected at smaller scales by twin
boundaries. HR-EBSD maps taken from the interior of these grains supports this
conclusion, large residual stresses and GND densities are observed in the
vicinity of twin boundaries in grains for which $m^{\prime}$ is small (Figure
13b and c). Lower residual stresses and GND densities are present in the
vicinity of twin boundaries in grains for which $m^{\prime}$ is low (Figure
13a and Figure 17b). The efficacy of twin boundaries as barriers to
dislocation motion is therefore dependent on the local orientation of
individual grains, but is on average weaker than a grain boundary taken at
random.

The relative ease with which dislocations can transmit across twin boundaries
suggests that Equation 3 requires refinement. We suggest that weights should
be applied to the contributions of twin boundaries and grain boundaries, such
that equation 3 becomes,

$\frac{d\rho}{d\varepsilon}=\frac{1}{b\lambda}-f\rho=\frac{1}{b}\left(\frac{k_{D}}{D}+\frac{k_{\mathrm{t}}}{D_{\mathrm{t}}}+k\sqrt{\rho}\right)-f\rho,$
(6)

in which $k_{\mathrm{D}}$ and $k_{\mathrm{t}}$ are weights to account for the
relative efficacy of grain boundaries and twin boundaries. We expect that
$k_{\mathrm{t}}\ll k_{\mathrm{D}}$. Further microstructural measurements, such
as the evolution of twin density with strain, are required to determine the
value of these weighting factors.

###  4.4 Towards a model of semi-brittle flow in calcite rocks

Our observations combined with previous results suggest several key
characteristics that should be captured by a model of semi-brittle flow in
calcite-rich rocks at $T<400^{\circ}$C: (1) non-linearly increasing strength
and hardening with increasing pressure, (2) decreasing strength with
increasing temperature, (3) increasing strength with decreasing grain size,
(4) strength that is nearly insensitive to strain rate. Rybacki2021 reviewed
proposed models of semi-brittle flow, identifying difficulties in combining
brittle and plastic flow into a simple unified model.

Through the semi-brittle flow regime, the strain contribution of brittle and
crystal-plastic processes changes with depth. At low pressures and
temperatures, at which brittle processes dominate, the macroscopic behaviour
of rocks is described by frictional failure Byerlee Brace (1966); Brace
Kohlstedt (1980). Microphysical models of brittle deformation are typically
based the wing-crack model <e.g.,¿[]Nemat-Nasser1982,Ashby1990, in which
brittle damage is accounted for by the propagation of Mode I wing cracks. In
this regime, strength is pressure sensitive, dependent on grain size and is to
first order strain rate-insensitive. There is no strong temperature
dependence, and plasticity is not considered.

There are only a small number of microphysical models in existence accounting
for coupled brittle-plastic deformation. Horii1986 modified and solved the
problem of a wing crack coupled a ’plastic zone’ by considering a dislocation
pile-up ahead of the shear crack. Their model is sensitive to pressure, grain
size and also temperature as the plastic yield strength in the plastic zone
can vary with temperature. However, their model predicts that materials with
coarser grain size are more brittle, contradicting observations from
experimental data Fredrich . (1990).

More recently, Nicolas2017 derived a model incorporating the propagation of
wing cracks, plastic pore collapse and nucleation of new cracks due to
dislocation pile-ups. The model reproduces several important characteristics
of the deformation of porous limestones. As pointed out by Rybacki2021, this
model does not consider dynamic recovery or twinning, which are important
deformation processes in calcite-rich rocks. The large number of free
parameters make this model challenging to test and may limit its general
application.

An alternative approach to introduce feedbacks between cracking and plastic
flow may be by use of a modified Kocks-Mecking equation. The anticorrelation
between strain hardening and crack density indicated by our results (Figure
15) suggests that cracking acts to reduce stress and hardening. The role of
cracks could be multiple. One possibility is that strength remains dictated by
dislocation density, and correctly predicted by the Taylor equation 2, in
which case a decreasing flow stress would imply a reduction of dislocation
density and thus that and cracks could act as dislocation sinks. This
possibility is compatible with the idea that tensile cracks correspond to free
surfaces within the material, and dislocations intersecting those free
surfaces would form steps and disappear from the crystals. One other
possibility is that cracks relax internal stress and strain incompatibilities
between grains, i.e., act as ”geometrically necessary” structures. A third
option is that deformation at low pressure, at which cracks are pervasive, is
not controlled by dislocation motion but dominated by elastically-accommodated
intergranular slip, and tensile cracks relax the associated internal stresses.

The origin of microcracks during semi-brittle deformation of calcite is
potentially coupled to intracrystalline plasticity. As discussed extensively
by Nicolas2017, cracks can be nucleated due to stress concentrations at the
head of dislocation pile-ups <e.g.,¿[]Stroh1954,Olsson1976,Wong1990 and may
therefore be dislocation sinks, i.e., microcracks could contribute to
dislocation escape from the deformed crystals. Where slip transfer is
inefficient Olsson Peng (1976) and also where geometry results in a high
density of GNDs (e.g., Figure 8d) stresses are high, which can lead to the
nucleation of cracks.

An approach based on the Kocks-Mecking model has the advantage that twinning
and grain size effects can be incorporated, whilst introducing a confining-
pressure dependence due to the propagation of Mode I cracks. At this stage
however, there are insufficient data on dislocation-density evolution during
semi-brittle deformation to make sensible progress beyond the qualitative
statements listed above. Thus, detailed modelling attempts are beyond the
scope of the present work.

##  5 Conclusions

We performed a series of triaxial deformation experiments using three calcite
rocks of variable grain size: Solnhofen limestone, Carrara marble and Wombeyan
marble. Our experimental results demonstrate that the strength and hardening
rates of calcite rocks deformed in the semi-brittle regime are inversely
dependent on grain size.

Microstructural observations using visible-light and electron microscopy
demonstrate that strain is accommodated by cracking, twinning and dislocation
glide. Deformation tests were accompanied by in-situ measurements of P-wave
speed, which generally decreases with strain. Wave speed decreases more at
room temperature than at $200~{}^{\circ}$C and $400~{}^{\circ}$C, which is
consistent with microcracking being more prevalent at low temperature.

Quantitative microstructural observations reveal that microcrack density is
inversely proportional to hardening rate. While the exact role of cracks in
the overall stress-strain behaviour remains unclear at this stage, we propose
the hypothesis that tensile microcracks cause weakening by either relaxing
internal stresses (accommodating strain incompatilibities and reducing the
need for geometrically necessary dislocations), or offering free surfaces
where dislocations can escape individual grains, or a combination thereof.

Furthermore, significant decreases in wave speed are observed upon removal of
confining pressure, indicating the accumulation of sample damage.
Decompression induced wave-speed decreases are greatest in experiments
performed at high pressure (600 MPa) and low temperature (room temperature) in
conjunction with the highest hardening rates. Microstructural observations
from samples deformed at these conditions are characterised by open grain
boundaries, suggesting that wave-speed decreases during decompression
originate from the release of stored elastic strain.

Electron microscopy shows that twin density is high, consistent with previous
studies at similar conditions. Twin spacing is always at least one order of
magnitude smaller than grain size. The spatial distributions of intragranular
misorientation suggest that twin boundaries do not always act as significant
barriers to dislocation motion and slip-transfer computations indicate that
twins are statistically weaker barriers than grain boundaries. Therefore,
grain size exerts a first-order control on strength and strain hardening,
whereas the spacing of twin boundaries may exert a second-order control on
these properties.

Taken together, our results show that semi-brittle flow in calcite is
controlled by grain-size dependent processes that lead to significant
hardening. This behaviour is qualitatively consistent with rheological models
that include dislocation density and twin spacing as key state variables. The
role of cracking in the decrease of strain hardening at low pressure and
temperature requires the addition of a quantity describing crack density (that
should include information on crack spacing, length, and orientation
distribution) as a new state variable, to address fully the stress-strain
behaviour of rocks in the semi-brittle regime.

## Open Research Section

Processed experimental data (stress, strain and velocity change) is available
from Zenodo (https://doi.org/10.5281/zenodo.7347236).

###### Acknowledgements.

Extensive discussions with Erik Rybacki and Brian Evans, who shared some of
their (then) unpublished data, helped to shape this work. Emmanuel David
contributed to early technical developments on the Murrell apparatus.
Technical support from John Bowles and Neil Hughes is greatly appreciated. Ian
Jackson kindly provided the Wombeyan marble. Sarah Incel facilitated thin
section preparation. Sheng Fan helped running SEM sessions. Discussions with
Thomas Breithaupt, Jörg Renner and Chris Spiers contributed to our
understanding of plastic deformation in calcite. This project has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement no
804685/“RockDEaF” to N.B.) and from the UK Natural Environment Research
Council (Grant Agreement No. NE/M016471/1 to N.B.). DW acknowledges support
from a UK Research and Innovation Future Leaders Fellowship [grant number
MR/V021788/1].

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