High-temperature shear zone formation in Carrara marble: The effect of loading conditions

Abstract Rock deformation at depths in the Earth's crust is often localized in high temperature shear zones occurring at different scales in a variety of lithologies. The presence of material heterogeneities is known to trigger shear zone development, but the mechanisms controlling initiation and evolution of localization are not fully understood. To investigate the effect of loading conditions on shear zone nucleation along heterogeneities, we performed torsion experiments under constant twist rate (CTR) and constant torque (CT) conditions in a Paterson-type deformation apparatus. The sample assemblage consisted of cylindrical Carrara marble specimens containing a thin plate of Solnhofen limestone perpendicular to the cylinder's longitudinal axis. Under experimental conditions (900 °C, 400 MPa confining pressure), samples were plastically deformed and limestone is about 9 times weaker than marble, acting as a weak inclusion in a strong matrix. CTR experiments were performed at maximum bulk shear strain rates of ~2 ∗ 10−4 s−1, yielding peak shear stresses of ~20 MPa. CT tests were conducted at shear stresses of ~20 MPa resulting in bulk shear strain rates of 1–4 ∗ 10−4 s−1. Experiments were terminated at maximum bulk shear strains of ~0.3 and 1.0. Strain was localized within the Carrara marble in front of the inclusion in an area of strongly deformed grains and intense grain size reduction. Locally, evidences for coexisting brittle deformation are also observed regardless of the imposed loading conditions. The local shear strain at the inclusion tip is up to 30 times higher than the strain in the adjacent host rock, rapidly dropping to 5 times higher at larger distance from the inclusion. At both bulk strains, the evolution of microstructural and textural parameters is independent of loading conditions. Our results suggest that loading conditions do not significantly affect material heterogeneity-induced strain localization during its nucleation and transient stages.


Introduction
Localization of deformation in the deep crust and mantle is a key mechanism involved in the formation of tectonic plates and mountain belts on our planet (Tackley, 2000;Schubert et al., 2001;Bercovici and Karato, 2002;Bercovici, 2003;Regenauer-Lieb & Yuen, 2003; Regenauer-Lieb and Yuen, 2004). Therefore, the knowledge of how deformation is 5 accommodated at plate boundaries and orogenic belts requires the understanding of physical processes that govern localized deformation in the ductile regime and its persistence over geological times.
A multitude of mechanisms have been proposed to be responsible for nucleation of localized deformation in the middle to lower crust, under the generally accepted premise that a 10 dynamic (positive) feedback mechanism is required to induce thermo-mechanical perturbances in otherwise homogeneously deforming mediums (Bercovici, 1996(Bercovici, , 1998 Calcite and natural carbonate materials have been extensively studied in the ductile regime both experimentally (e.g. Schmid et al., 1980;Rutter, 1994, 1995, Pieri et al., 2001a, 2001bBarnhoorn et al., 2004) and in the field (Schmid et al., 1981;40 Bestmann et al., 2000;Rogowitz et al., 2014Rogowitz et al., , 2016, so that the calibration of mechanical, microstructural and textural data for calcite at different deformation conditions is well established. In this contribution, we experimentally investigate the effects of different loading conditions (constant stress versus constant strain rate) on the nucleation and evolution of 45 heterogeneity-induced high temperature shear zones in a carbonate system.

Experimental setup
The torsion experiments presented here were carried out on mono-mineralic calcite aggregates consisting of Carrara marble with elongated Solnhofen limestone inclusions. A thin (0.75 mm) circle segment of Solnhofen limestone with an arc length of about 11.8 mm 50 was inserted in a saw-cut slot in a hollow cylinder (10 mm height, 15 mm outer diameter, 6.1 mm inner diameter) of Carrara marble. Ceramic glue was used to fill up the possible gaps in the slot. Two alumina spacers (for protection of the pistons in the deformation apparatus) and a solid gold cylinder inserted in the inner borehole of the sample completed the setup (Fig.   1). 55 Fig. 1: Elements of the sample assembly. In the small inset, the setup prior to insertion in the copper jacket; indicated is the direction of the applied torsion.
The hollow cylinder configuration was preferred as it guarantees a relatively homogeneous distribution of shear stress within the sample (Paterson and Olgaard, 2000). Carrara marble 60 is a largely used material in experimental rock deformation due to its exceptional purity (> 99% CaCO3) and low to no initial porosity (Rutter, 1995 Twinned crystals are present, but the twins are extremely thin (< 5 µm), straight and not pervasive. Solnhofen limestone is an extremely fine grained (~ 5 µm average grain size) almost pure calcite rock (more than 97 wt% calcite, Rutter, 1972). As a consequence of a large difference in initial grain size, Solnhofen limestone is up to 9 times weaker than Carrara 70 Marble at the imposed experimental conditions of 900 °C temperature and 400 MPa confining pressure (e.g. Rybacki et al., 2014), inducing a strong viscosity contrast between the two materials in our experimental setup.
Experiments were performed in a Paterson-type gas deformation apparatus equipped with a torsion actuator (Paterson and Olgaard 2000). Prior to experimental runs, the samples were 75 inserted into copper jackets of ~ 0.2 mm thickness (to isolate them from the gas confining pressure medium) the strength of which is accounted for in the evaluation of the mechanical data. The samples were first pressurized to 400 MPa, followed by heating at a rate of ~ 30 °C/min up to the desired temperature of 900 °C (with accuracy of ± 2 °C along the sample axis). After test termination, load was maintained constant during cooling (at equal rate to the 80 heating phase) to preserve deformation microstructures and to reduce the amount of static recovery occurring within the samples. Crystallographic orientation mapping was performed systematically in all studied samples using two different step sizes (10 µm and 3.5 µm). The coarser step size map was used to 110 map the orientations of the whole sample and for the construction of mean grain size profiles across the thin sections (top to bottom; orange rectangles in Fig. 2a-d). The fine step size mapping was limited to the area in front of the limestone inclusion, where deformation localizes ( Fig. 3a-d); these maps were used for the study of shape descriptors (section 4.4) and their variations within regions in and outside of the localized zone in front of the inclusion 115 (red rectangles in Fig. 3a-d). of 0.1, meaning that data points with CI < 0.1 are reassigned the orientation and CI of the neighbour data point with the highest CI within individual grains.
To identify very small (dynamically recrystallized) grains and their spatial distribution, a minimum of 5 indexed points per grain criterion (to reduce the loss of small grains by interpolation and smoothing) was used in the finer stepping size maps within a region of 130 interest ( Fig. 3a-d, black rectangles). Unless otherwise specified, grain sizes are expressed as equivalent diameter (diameter of a circle of equivalent area to the grain).
Calculations on the EBSD data were performed using the MTEX 4.3.2 toolbox for Matlab (Hielscher and Schaeben, 2008;Bachmann et al., 2010). Orientation distribution function and pole figure contouring were calculated using a Gaussian half-width of 10° and a 135 maximum harmonic expansion factor of 32.
Grain size evolution across the samples (see section 4.4) was investigated using the coarser EBSD maps (10 µm step size). A moving, partially overlapping (one third of the vertical size) window of size 1.85x0.5 mm  was used to extract data along a profile parallel to the longitudinal axis of the sample cylinder located directly in front of the limestone inclusion 140 in the Carrara marble. Grains transected by the boundary of the moving window were excluded from the calculations. For each window, we estimated the average grain size using the RMS (root mean square) value of the distribution.
To characterize grain shape evolution within the area covered by the finer step size maps (section 4.4), two shape descriptors were considered. First, the inverse aspect ratio 145 (according to the definition of the MTEX toolbox, Hielscher and Schaeben, 2008: = , and being the width and length of the particle, respectively) and secondly a variation of the classical circularity shape factor, defined as follows: shape descriptors assume values between 1 and 0, where the former represents a circle (maximum IAR) with smooth surfaces (maximum circularity) and the latter is characteristic of an infinitely non-circular shape (minimum IAR) with infinitely rough surfaces (minimum circularity).
For the study of shape descriptors a grain size filtering was applied to the datasets in order to 155 remove small grains produced by dynamic recrystallization, which are expected to approach the ideal circular shape at the time of formation. The size filter was set to 20 µm, which is the average grain size of dynamically recrystallized grains determined by optical microscopy.
Six transmission electron microscopy foils (0.15 µm in thickness) from a single thin section (sample CTR03) were prepared using the focused ion beam technique (e.g. Wirth, 2005) 160 and subsequently inspected with transmission electron microscopy (TEM) on a FEI Tecnai G2 F20 X-Twin TEM for the calculation of dislocation densities (see section 4.5).

Mechanical data
We present the results from four experiments, two of which were run at constant twist rate 165 and two at constant torque conditions (CTR and CT, respectively, equivalent to constant strain rate and constant stress) in the torsion setup of a Paterson-type gas deformation apparatus. The final bulk strains reached in the experiments are γ ~ 0.3 and γ ~ 1, respectively, for both loading configurations. Experimental conditions for the four samples presented here are shown in Table 1. 170 up to a factor of about 4 at γ = 1 (Fig. 4b). The slightly higher minimum strain rate of sample 185 CT03 compared to CT1 is related to the higher initially applied stress (Tab. 1).

Strain localization 190
In order to quantify the local distribution of shear strain within the samples, passive strain markers were applied to the copper jackets prior to the experimental runs (except for sample CTR03). A grid of evenly spaced straight lines, parallel to the cylinder axis, was carved in the copper jacket ( Fig. 5; denser spacing is corresponding to the area where the inclusion is located) and recovered after the experiments. The locally imposed shear strain is estimated 195 from the angular deflection of the originally straight lines. After deformation, the distribution of those lines is clearly heterogeneous and reflects the partitioning of strain between the limestone inclusion and the Carrara marble. In particular, a substantial difference in shear strain can be observed between the region directly in front of the Solnhofen inclusion and the surrounding matrix, indicative of ongoing strain localization within the Carrara marble itself, related to the presence of a material heterogeneity. Local shear strains are calculated for this area of incipient localized deformation (named hereafter the process zone) and for the less deformed matrix at some distance from the inclusion tip. A strongly localized shear strain in front of the inclusion is observed (up to a factor ~ 12 compared to the imposed bulk strain), rapidly decaying to background strain level 210 with distance (~ 10 mm) from the inclusion (Fig. 6a). The degree of strain localization (i.e. the ratio between local to bulk shear strain) is not substantially different between the two loading conditions of constant twist rate or constant torque (cf., samples CTR1 and CT1, respectively), especially when one considers the large uncertainty in the calculated local strains when the angular deflection is large and the measurements become less accurate (a 215 3° variation can lead to a difference in of 10).
In Figure 6b the local strain measured in the process zone is normalized to that of the surrounding, less deformed matrix, plotted against distance from the inclusion tip. The localization of deformation in the process zone with respect to the adjacent matrix is higher by a factor ~2-3 than with respect to the bulk strain (Fig. 6a). Clearly, the degree of 220 localization is further developed at higher total bulk strain (samples CTR1 and CT1, as opposed to sample CT03). Again, no substantial difference is noticed between constant stress and constant strain rate samples.

Microstructures 225
In composite micrographs from optical microscopy images we typically observe an area of intense grain size reduction and highly deformed grains in the Carrara marble matrix, close to the Solnhofen limestone inclusion (Fig. 7a-b and f-g). In contrast, the surrounding matrix region remains almost undeformed, and equant grains display similar characteristics to the undeformed Carrara marble (Fig. 7k-m). Note that the bulk shear strain indicated on the 230 micrographs refers to the maximum measured shear strain. Approximate local shear strain within a section of the sample cylinder cut at a radius is about 30% lower (Tab. 1), determined by: ( In all micrographs the Solnhofen limestone inclusion is located on the right side, showing at 235 higher strain stronger distortion from the original undeformed rectangular shape ( Fig. 1). Scanning electron microscope (SEM) images combined with optical close-ups reveal cracks preferentially located at grain boundaries and small voids ahead of the inclusion tip. Small, mostly tensile cracks oriented parallel to the direction of σ1 are visible in the low strain constant twist rate sample ( Fig. 8a-b). In the equivalent strain constant torque sample, a 270 single shear fracture propagating from the tip of the weak limestone inclusion is formed within the host marble (Fig 8c-d). Many grain boundaries in the process zone surrounding the inclusion tip are decorated by strings of pores. It is conceivable that these indicate crack closure and healing during the tests. The displacement associated with the shear fracture is accommodated by the weakest 285 phase (the limestone), as can be observed in the relative movement of the ceramic glue at the contact between the marble and the inclusion. At higher bulk strain ( Fig. 8e-h), a long single shear fracture is observed in the constant torque sample (Fig. 8g-h). In constant twist rate conditions (Fig. 8e-f) a few small incipient intracrystalline cracks developed at the very tip of the soft inclusion. No substantial offset can be discerned along these small cracks. A 290 large, extended fracture oriented consistently to the direction of σ1 is seen in this sample, potentially enhanced during unloading given its extension, developing beyond the microstructurally defined process zone.

Grain size and grain shape evolution
The microstructures of the investigated samples vary strongly on the sample scale as a 295 result of the strain partitioning between limestone inclusion and host Carrara marble, and within the marble matrix due to the local stress concentration at the interface between the two materials. To investigate the distribution of these heterogeneities within samples and between specimens deformed at different conditions we collected EBSD data and performed analysis of grain size distribution and characterized the grain shape evolution (for details on 300 the methods applied, see section 3). Based on optical observations measuring some tens of recrystallized grains per thin section, average grain size of dynamically recrystallized grains was estimated to 20 ± 4 µm.   Table 2 for a transcription of the values). For both low and high bulk shear strains the overall grain size distributions of constant torque and constant twist rate samples within the whole thin section area are largely overlapping with rather similar median values ( Fig. 10a and e). In all samples, the average grain sizes are 330 significantly reduced in the process zones compared to adjacent domains and the starting material. Reduction of average grain size outside the process zone is more pronounced at high strain (cf. median lines in Fig. 10h compared to Fig. 10d) as a consequence of the increased contribution of dynamically recrystallized grains.  For low strain samples (Fig. 10b-d) the strongest grain size reduction occurs within the process zone (Fig. 10c), where the applied loading conditions appear to play no role in the resulting distribution. Upper and lower domains ( Fig. 10b and d) display some subtle 345 differences in evolution between constant twist rate (CTR03) and constant torque (CT03) samples. For constant twist rate samples, grain size distributions are similar above and below the process zone. Samples deformed at constant torque show a generally lower average grain sizes in the domain below the process zone (Fig. 10d). In general, however, grain size evolutions are very similar in samples deformed at different loading conditions. At 350 high strain, the asymmetry in grain size evolution between the upper and lower domain is preserved (also seen in Fig. 9), but the difference between loading conditions is reduced ( Fig. 10f and h). Note, however, that the overall grain size evolution (Fig. 9) and distribution ( Fig. 10a and e) ultimately appear not to be influenced by the imposed boundary conditions.

360
The spatial distribution of dynamically recrystallized grains across the combined three domains (stippled black rectangle in Figure 3a-d) using 3.5 µm step size maps is plotted in Fig. 11. Bivariate histograms are constructed by defining a 25x25 grid colour coded based on how many recrystallized grains (equivalent diameter < 20 µm) are present within each grid square. The frequency of recrystallized grains is largest ahead of the inclusion and increases 365 with increasing bulk strain. No major differences can be observed between the different loading conditions. This is in good accordance with the observation that the area fraction of grains with equivalent diameter smaller than 20 µm is similar in samples deformed at both imposed boundary conditions (Fig. 10a and e).

Grain shape evolution 370
We determined the average grain shape within the aforementioned domains to analyze the degree of plastic deformation of the matrix material (see section 3 for details). In Figure 12al, normalized circularity and inverse aspect ratio data are plotted within the three area domains for the two considered bulk strains ( Fig. 12a-   data, especially at low strains ( Fig. 12d-f). Within the general trend of increased ellipticity (decreased inverse aspect ratio) with respect to the starting material distribution, small differences are present in the process zone of high strain samples (Fig. 12k).

Local stress concentration at the tip of the inclusions
The observations carried out on macrostructural and microstructural data indicate that strain 395 is locally concentrated within the Carrara marble in front of the weak Solnhofen inclusion.
This suggests that the distribution of stress within the samples is also strongly heterogeneous. Stress appears to be locally enhanced in regions around the inclusion tips that experience the highest amounts of dynamic recrystallization and intracrystalline deformation. Several paleopiezometer techniques may be used to estimate local stresses 400 between 500 and 700°C and from Rutter, 1995, who used triaxial compressive and extensional configurations at temperatures between 500 and 1000 °C. The general relationship between stress and recrystallized grain size is given by: where is the equivalent stress, is the recrystallized grain size, and and are two 420 constants. Note that the conversion from shear to equivalent stress is defined as follows  Dislocation densities at increasing distance from the tip of the weak Solnhofen inclusion were estimated using transmission electron microscopy (TEM) of sample CTR03 (low bulk strain constant twist rate). Six foils of 0.15 µm thickness were prepared from areas within relict deformed grains at incremental distances of ~ 500 µm (Fig. 13a). The foils were examined in 440 where is the number of intersections, the length of the transect line and the (constant) thickness of the TEM foil.
The piezometer was calibrated by De Bresser, 1996 on both single crystals and 450 polycrystalline calcite deformed between 550-800 °C, yielding: where is the equivalent stress in MPa and is the dislocation density in m -2 .
Resulting stresses show significant error bars (as a consequence of the uncertainty in the measured dislocation densities), but decrease non-linearly with distance (Fig. 13b). With 455 respect to the applied bulk equivalent stress of 34.8 MPa, the resulting stress concentration at the tip of the inclusion is approximately a factor 2, in accordance with the results from the recrystallized grain size piezometry. Fig. 13 a and b: a) Micrograph of sample CTR03 with approximate locations of FIB foils for TEM analysis, b) local 460 stress with distance from the inclusion tip as calculated using dislocation density (red) and recrystallized grain size (orange) piezometry.

Crystallographic preferred orientation
For the calculation of pole figures, maps produced with a 10 microns step size were used. A rectangular area of the thin sections covering the process zone was analyzed for all samples. 465 Irrespective of applied loading conditions, a strong CPO developed in the deformed samples with increasing strain. Samples deformed at constant torque and constant twist rate show comparable textural evolution (Fig. 14b-e) and pole figures that differ significantly from the starting material (Fig. 14a).
In CTR03 (low strain constant twist rate sample, Fig. 14b  In constant torque samples (Fig. 14d-e Poles of {01-12} are arranged in four symmetrical maxima as described for the constant twist rate sample. A strong preferred orientation is observed for {01-18} poles as well, forming 500 girdles around the normal to the shear plane.

Weakening mechanisms
Plastic strain localization requires the development of an instability in the system undergoing deformation (Poirier, 1980;Hobbs et al., 1990). It is generally assumed that potentially 505 coexisting weakening mechanisms (recrystallization-induced grain size reduction, CPO formation, reaction softening, shear heating) may lead to local strength perturbations and ultimately to strain localization (for a review, see Fossen and Cavalcante, 2017).
At the given experimental conditions, Carrara marble is expected to deform in the dislocation creep regime (e. g. Schmid et al., 1987; Pieri et al., 2001a, b; Rybacki et al., 2014). This 510 assumption is confirmed by the observed presence of a strong crystallographic preferred orientation even at early increments of bulk shear strain (Fig. 14). Moreover, the increased mean aspect ratio of relict grains with respect to the undeformed starting material (Fig. 15ab) associated with the development of a SPO within the process zone of the investigated samples ( Fig. 15c-d) are indicative of intracrystalline deformation.  Fig. 3a-d; (b) comparison of mean aspect ratios of relict (eq diam. >20 µm) and recrystallized (eq. diam < 20 µm) grains within the process zone of the samples; (c, d) shape preferred orientation of relict grains (equivalent diameter > 20 µm) across the vertical profiles defined in Fig. 3a-d. Intense, strain dependent grain size reduction by dynamic recrystallization is observed to 520 develop dominantly in the process zone (Fig. 11), induced by the concentration of stresses and shear strain around the inclusion tip (Fig. 13). Some contribution of grain size sensitive diffusion creep is expected to be active in fine-grained regions in Carrara marble, as shown for the same experimental conditions and for recrystallized grain sizes in the order of 10-15 µm (ref. Fig. 13 in Rybacki et al., 2014). As discussed by the latter authors, the measured 525 stress concentration at the tip of the inclusion may induce switching into the dislocation creep regime, suggesting that the weakening-controlling deformation mechanism in Carrara marble at the examined experimental conditions is grain size insensitive. This is supported by our observations indicating formation of cracks at the inclusion tip.
The textural data (Fig. 14) allows identifying possible slip systems activated in the process 530 zone of the samples, based on the classification of slip systems operating in calcite deformed at high temperature (e.g. De Bresser and Spiers, 1993). A general trend of switching main slip systems with increasing strain is observed in both constant twist rate and constant torque samples; while basal slip (notice the strong alignment of c-poles in Fig. 14c, e) is prevalent in both the high bulk strain samples, together with some remnants of slip along the rhomb r 535 plane in the a-direction {10-14} <20-21>, the low strain samples show a less strongly developed texture in which slip along the rhomb r{10-14} and f{01-12} planes in the a<20-21> direction prevails (Fig. 14b, d). Strengths of the texture within the process zone, as quantified by the calculated j-index (Bunge, 1982) (Fig. 8). As the process zone propagates into the Carrara marble matrix, cracking is overprinted by high-temperature creep of the finegrained recrystallized matrix assisted by crack healing. It is conceivable that some of the 555 cracking is also obliterated during unloading and slow cooling of the samples at the end of the experimental runs.
In CTR samples (Fig. 8a-b and e-f), microcracks and voids display a somewhat different distribution and orientation with respect to constant torque samples. At low strain, a set of small (50-100 µm) cracks often oriented parallel to the direction of σ1 is found (Fig. 8a-b). 560 These are mostly open tensile cracks and in some cases associated with small, dynamically recrystallized material. At higher bulk shear strains (Fig. 8e-f), together with a long, interconnected fracture parallel to σ1, some intracrystalline microcracks occur in parallel to the maximum shear direction. In both constant torque experiments (Fig. 8c-d and g-h), a single fracture forms in plane with the shear propagation direction from the tip of the 565 inclusion, consistent with the far field direction of maximum shear stress. It is, in all cases, difficult to assess with certainty whether any displacement is occurring along these fractures, as the presence of fine recrystallized material related to the ongoing plastic deformation overprints any possible passive marker in the microstructure. The preservation of such a fracture in both low and high bulk strain sample suggests its formation in the early stages of 570 deformation and its further exploitation with increasing strain. Brittle deformation is, in all cases investigated, likely confined to small domains (where local stresses and strain rates are highest) and small intervals of strain, associated with pervasive high-temperature creep of calcite.

Stress distribution and deformation transients 575
The heterogeneous stress distribution produced in the matrix due to the presence of an inclusion is clearly expressed in the resulting microstructures. The stress enhancement in the marble matrix in the process zone in front of the inclusion tip is substantial and has been quantified to a factor of 2-3 with respect to the applied bulk stress (Tab. 3, Fig. 13), although it should be borne in mind that the paleopiezometers applied here were calibrated for steady 580 state conditions not achieved in our tests (Rutter, 1995;De Bresser, 1996;Barnhoorn et al., 2004). Note that the amount of stress concentration surrounding a material heterogeneity depends on the effective viscosity contrast between inclusion and matrix and on coupling of the two materials (Kenkmann and Dresen, 1998). At given thermodynamic conditions of our tests, the initial viscosity contrast between Carrara marble and Solnhofen limestone is 585 expected to be a factor ~ 10 (Rybacki et al., 2014). As suggested by the local shear strain, stress and grain size distributions found in the process zone, an exponential decay is observed with distance from the inclusion towards the matrix.
The time-dependent strain localisation pattern can be additionally investigated using numerical forward models (Döhmann et al., subm. to Journ. Geophys. Res., and Fig. 16). Here we employ 2D Cartesian models with periodic boundary conditions that have been benchmarked to experimental mechanical data. The gradient in flow stress reconstructed for sample CTR03 (Fig. 13) by means of dislocation density piezometry is in general accordance with results from numerical models (Fig. 16). In their study, Döhmann et al. (subm.) found a 600 rapid stress drop, down to roughly far-field levels, within 2-3 mm from the inclusion tip.
Numerical modeling was carried out by means of the geodynamic modeling software SLIM3D (Semi-Lagrangian Implicit Model for 3 Dimensions, Popov and Sobolev, 2008), which was originally intended for the study of lithospheric-scale processes (Brune, 2016) but has been applied to laboratory scale localization models as well (Cyprych et al., 2016). For 605 the applied thermodynamically coupled conservation equations, see Popov and Sobolev, 2008. Experimentally derived (Schmid et al., 1980 andRybacki et al., 2014) flow laws were used to model deformation of Carrara marble and Solnhofen limestone, and a straindependent viscous softening mechanism was implemented (Brune et al., 2014). Modelderived profiles of the second invariant of stress along the inclusion and process zone at the 610 cylinders' outer surface show local stress concentration at the inclusion tip decaying towards the matrix (Fig. 16). Stress concentration is significantly smaller than observed in the deformed samples but the general trend agrees with experimental results. For example, a nonlinear stress decay is observed with increasing distance from the inclusion towards the matrix irrespective of loading conditions. The weakening induced in the marble by the applied 615 high stresses reduces its viscosity and leads to a progressive stress relaxation along the process zone with increasing bulk strain (compare Fig. 16a and c for constant twist rate and 16b and d for constant torque experiments). Note that a stress peak at the inclusion tip is preserved in all cases and regardless of total strain (Table 3 and Fig. 16), as the viscosity contrast between the inclusion and the matrix surrounding it is still high. Fig. 16 also shows a 620 quantitative comparison of stress distribution between constant stress and constant strain rate experiments, highlighting the absence of a substantial difference in the shape and magnitude of the area of enhanced stress.
The paleowattmeter introduced by Austin and Evans, (2007) allows relating grain size of dynamically recrystallized material to mechanical work rather than to flow stress alone. We 625 applied the suggested scaling relationship (eq. 8, 9 in Austin and Evans, 2007) to our data.
Using the measured grain sizes listed in Table 3 and the values of local strain rates derived from the strain markers (see section 4.2), the predicted concentration in stresses with respect to the far field equivalent stress is in the range 3-5, in good agreement with the paleopiezometric estimates (factor 2-4, depending on calibration, Table 3). The approach 630 may also be used to predict the average recrystallized grain size for given differential stresses and strain rates. Resulting recrystallized grain sizes vary between 10-12 µm (far field stress) and 8-10 µm (up to a factor 4 stress concentration, the upper bound derived from paleopiezometry), in good accordance to what measured optically in the samples (Table 3).
Taking into account the uncertainty of measured grain sizes and strain rates, we conclude 635 that our experiments do not allow to resolve if the wattmeter yields more reliable results than the piezometer. It should be borne in mind, however, that our quantitative estimates are conducted on somehow transient microstructures, and the applicability of piezometric and wattmetric techniques to such conditions should be regarded as limited.

Amount and geometry of strain localization 640
Strain localization as indicated by strain markers increased as the process zone propagated into the Carrara matrix (Fig. 6a, b). However, localization remained unaffected by the different loading conditions. Within the process zone at low bulk strains Carrara marble is strongly twinned: thick, often tapered or bent twins are abundant, as are multiple twin sets within single crystals. Deformation twinning in calcite has been extensively studied in the 645 past (e.g. Barber and Wenk, 1979;Wenk, 1985), and twin morphology and intensity  6b). Once weakening is completed with progressive strain, partitioning of shear strain into the 660 localized shear zone saturates at a constant shear strain ratio between shear zone and bulk sample (Fig.6a). The slope in Fig. 6a defines the critical shear strain c required to complete weakening at a critical length of the process zone. Process zone length and shear strain gradient depend on viscosity contrast between strong host rock and weak shear zone and the weakening mechanism(s). 665

Comparison to previous experimental work
High temperature experimental deformation has been conducted extensively on rock-forming minerals to reproduce the processes occurring in natural shear zones. Notably, although mechanical weakening of the deforming materials was described in all cases, localization of deformation at the sample scale was only observed in a small number 675 of these studies, where it appeared to be favoured, e.g. by high initial strength contrast between phases (Holyoke and Tullis, 2006), by a switch in deformation mechanism in only one of the deforming phases, producing locally heterogeneous phase distribution (Barnhoorn et al., 2005) or in the case of imposed constant load (torque) boundary conditions (Hansen et al., 2012). This latter is in good agreement with what was theoretically 680 predicted for the torsion geometry by several authors (Fressengeas and Molinari, 1987;Leroy and Molinari, 1992;Paterson, 2007) who, by means of linearized perturbation analysis, prescribe strain localization to be dependent on the applied boundary conditions: a small enough perturbation of one of the material properties is not expected to produce localization in a constant displacement rate setting even if strain weakening is observed, as 685 opposed to a constant load setup in which localization is always favoured. However, it is important to point out that, as Fressengeas and Molinari (1987) also clarify in their contribution, linear perturbation analysis is carried out with the assumption of small deviations from homogeneity of the material properties: for larger perturbations the field equations cannot be linearized and the analytical solutions are much more complex. We 690 argue that, in our experimental setting, the initial departure from a homogeneous stress distribution that is imposed to the system by the presence of a strong viscosity contrast is too large for the linear approximation to be valid. As a consequence, it can be inferred that, for bi-or multi-phase materials with sufficient viscosity contrast (as is often the case in nature), the expected influence of boundary conditions on localization is absent or minor. 695

Implications for natural shear zones
Our study shows that, in the presence of a material heterogeneity in an otherwise homogeneous medium, localized shear zones form regardless of the imposed loading conditions. Recent theoretical studies accompanied by integration of existing field data (Whipple Mountains core complex, southeast California; Platt and Behr, 2011a and Platt 700 and Behr, 2011c), however, propose a theory for the development of viscous shear zone in the middle to lower crust in a stress-controlled environment. The authors concluded that the yield stress of the undeformed host rock controls the flow stress in the deforming shear zone.
Consequently, a constant velocity boundary condition is always converted into a constant stress one. However, the theory is based on the assumption of steady-state deformation of a 705 homogeneous crustal material. This premise seems to apply to a limited number of tectonic situations, but may not apply to channel flow (Beaumont et al., 2004) or the dynamic feedback between the brittle upper crust and the semibrittle to ductile lower part of the crust during syn-and inter-seismic periods (Trepmann and Stöckhert, 2003). In their review paper on shear zones in the mantle, Vauchez et al. (2012) also point out how the yield stress of a rock is strain rate dependent, where the strain rate itself is a function of shear zone width, so that the constant stress hypothesis is, in fact, intrinsically ambiguous.
Numerous field and experimental studies conducted in the past decades have identified a number of mechanisms that are believed to trigger strain localization at the crustal scale, which can, in general terms, be summarized as the inherited presence of a rheological or 715 structural heterogeneity, or the mentioned interaction with the seismogenic crust (for a review, see Vauchez et al., 2012). If the question still stands, of whether the initiation of shear zones is primarily dependent on the imposed loading conditions or on the activity of one of the aforementioned mechanisms, our results suggest that in the presence of, for example, an inherited rheological heterogeneity, shear zone formation is controlled by this 720 latter while the applied boundary conditions appear to be of secondary importance.

Conclusions
We conducted high temperature torsion experiments to investigate the effect of loading conditions (constant twist rate or constant torque) on the initial and transient stages of strain localization in marble containing a weak material heterogeneity. The inclusion induced stress 725 concentration halos in the stronger surrounding matrix resulted in strain partitioning into localized shear bands propagating into the marble with ongoing bulk deformation.
Progressive localization is associated with strain weakening accommodated by dynamic recrystallization, CPO development and plastic deformation of relict grains within a process zone which is markedly different from the surrounding, relatively intact matrix. High 730 temperature creep of marble is the dominant deformation mechanism at the applied experimental conditions, although evidences for coexisting brittle deformation are found regardless of loading conditions and total strain. The geometry, microstructural and textural features and evolution of the process zone are qualitatively and quantitatively comparable in constant torque and constant twist rate experiments. 735 Overall, our results suggest that the loading conditions do not significantly affect strain localization induced by the presence of a material heterogeneity during nucleation and transient evolution stages.
It should be kept in mind that the experimental setup poses some intrinsic limitations on the number of variables that can be investigated simultaneously. The effects that some of the 740 latter (as the applied confining pressure, temperature or the presence of a second phase in the deforming matrix) might have on the weakening mechanisms and ultimately on the processes favouring strain localization are manifold and would require further investigation and a multidisciplinary approach (with experimental, field based and model based studies).