Insight into the microphysics of antigorite deformation from spherical nanoindentation
Lars N. Hansen, Emmanuel C. David, Nicolas Brantut, David Wallis
IInsight into the microphysics of antigoritedeformation from spherical nanoindentation
Lars N. Hansen , Emmanuel C. David , Nicolas Brantut , David Wallis Department of Earth Sciences, University of Oxford, Oxford, UK. Department of Earth Sciences, University College London, London, UK. Department of Earth Sciences, Utrecht University, Utrecht, The Netherlands.
Abstract
The mechanical behavior of antigorite strongly influences the strengthand deformation of the subduction interface. Although there is microstruc-tural evidence elucidating the nature of brittle deformation at low pres-sures, there is often conflicting evidence regarding the potential for plasticdeformation in the ductile regime at higher pressures. Here, we present aseries of spherical nanoindentation experiments on aggregates of naturalantigorite. These experiments effectively investigate the single-crystal me-chanical behavior because the volume of deformed material is significantlysmaller than the grain size. Individual indents reveal elastic loading fol-lowed by yield and strain hardening. The magnitude of the yield stress isa function of crystal orientation, with lower values associated with indentsparallel to the basal plane. Unloading paths reveal more strain recoverythan expected for purely elastic unloading. The magnitude of inelasticstrain recovery is highest for indents parallel to the basal plane. We alsoimposed indents with cyclical loading paths, and observed strain energydissipation during unloading-loading cycles conducted up to a fixed maxi-mum indentation load and depth. The magnitude of this dissipated strainenergy was highest for indents parallel to the basal plane. Subsequentscanning electron microscopy revealed surface impressions accommodatedby shear cracks and a general lack of lattice misorientation around indents,indicating the absence of dislocations. Based on these observations, wesuggest that antigorite deformation at high pressures is dominated bysliding on shear cracks. We develop a microphysical model that is ableto quantitatively explain the Young’s modulus and dissipated strain en-ergy data during cyclic loading experiments, based on either frictional orcohesive sliding of an array of cracks contained in the basal plane.
Antigorite is one of the dominant hydrous phases in oceanic lithosphere asso-ciated with subduction zones, and its mechanical behaviour plays a key role in1 a r X i v : . [ phy s i c s . g e o - ph ] N ov ontrolling the strength of the subduction interface (e.g., Reynard 2013). Ex-perimental investigations of the rheology of antigorite have revealed a numberof unique characteristics. First, in the brittle regime at confining pressures lessthan 400 MPa and at room temperature, antigorite aggregates experience littleto no pre-failure dilatancy or stress-induced anisotropy in seismic-wave velocities( Escart´ın et al. , 1997;
David et al. , 2018), in sharp contrast to other low-porositycrystalline rocks (
Paterson and Wong , 2005). Second, the ductile regime is ap-parently limited to a high pressure, low temperature domain, with a transitionback to brittle, unstable behaviour as temperature increases towards the dehy-dration temperature of antigorite (
Chernak and Hirth , 2010;
Proctor and Hirth ,2016). Under typical subduction zone conditions, at confining pressures of sev-eral gigapascals and temperatures of ∼ ◦ C, experimental observations areinconclusive regarding the dominant rheological behavior.
Hilairet et al. (2007)and
Amiguet et al. (2012) report power-law creep behaviour consistent with dis-location creep, whereas the results from
Proctor and Hirth (2016) indicate veryhigh stress exponents that are more consistent with exponential creep and plas-ticity. In all experiments conducted under elevated pressures and temperatures,the tendency towards strain localisation appears to complicate the interpreta-tion of macroscopic stress-strain behavior. According to experimental data from
Chernak and Hirth (2010) and
Auzende et al. (2015), it remains unclear whetherit is even possible for antigorite to deform in a fully crystal-plastic regime.Some insight can be gained into the microphysics of antigorite deformationfrom the microstructures produced during deformation. Field observations ofdeformed, antigorite-rich rocks exhumed from subduction-zone environmentstypically reveal strong foliations (e.g.,
Hermann et al. , 2000;
Padr`on-Navartaet al. , 2012). Foliated antigorite also often exhibits a strong crystallographicpreferred orientation (CPO) with (001) mostly parallel to the foliation, whichhas been interpreted as a marker of flow by dislocation creep, although specificslip systems are still debated (
Padr`on-Navarta et al. , 2012). Microstructuralobservations of both experimentally and naturally deformed antigorite tend toindicate that the crystallographic structure of antigorite, with a corrugation ofthe (001) plane in the [100] direction, might prevent dislocation glide in thebasal plane (
Auzende et al. , 2015). In addition, recent deformation experimentsconducted on antigorite single crystals with in situ electron microscopy demon-strate that cleavage opening, delamination, and fracture might be the dominantintracrystalline deformation processes (
Corder et al. , 2018). These experimentaldata are not necessarily in contradiction with the observation of strong CPOin naturally deformed antigorite, considering that a CPO might originate fromcleavage along basal planes and associated grain rotation. Overall, the deforma-tion mechanisms of antigorite remain unconstrained, and only indirect evidencefor the operation of dislocation creep has been obtained. Cleavage and de-lamination along the basal plane has been widely reported in experimentallydeformed samples, but it remains unclear whether dislocation activity couldbecome dominant under geological strain rates.To gain further insight into the intragranular deformation mechanisms ofantigorite, we conducted a series of nanoindentation experiments. This de-2ormation technique spontaneously generates confining pressure and has beenused to study low-temperature, crystal-plastic deformation mechanisms in rock-forming minerals (
Evans and Goetze , 1979;
Basu et al. , 2009;
Kumamoto et al. ,2017). Because of the extremely small scale of these deformation experiments,this technique is well suited to investigate the inelastic deformation of antigoritesingle crystals. We investigate elastic loading, yield, and static internal frictionas a function of crystallographic orientation and then discuss the potential mi-crophysical processes operating during deformation of antigorite.
Mechanical characterization was carried out on a natural antigorite serpentinite.Serpentinite blocks were acquired from the Rochester quarry of Vermont VerdeAntique. Our material is sourced from a block from which similar materialwas characterized by
David et al. (2018). Material from a similar origin andlocation has been characterized in previous work (
Reinen et al. , 1994;
Escart`ınet al. , 1997;
Escart´ın et al. , 1997;
Chernak and Hirth , 2010). This serpentinite isprimarily composed of antigorite ( > × × David et al. , 2018, 2019).The section was cut normal to the antigorite foliation.The section was ground and polished to yield a surface that was as flat andsmooth as possible. The sample section was first bonded onto an aluminumcylinder using a thermoplastic cement (Crystalbond TM µ m. Nanoindentation was carried out using an MTS NanoIndenter XP equipped withcontinuous stiffness measurement (CSM). Indentation tests were performed witha conospherical diamond tip with a nominal tip radius of 10 µ m. Two sets ofindents were created on two different areas of the sample section, each with adifferent methodology.In the first area, we imposed an array of 8 × µ m grid.An initial series of indentation was performed to 100 nm depth, immediatelyfollowed by an additional series in the same locations as the previous indentsbut to 500 nm depth. The initial 100-nm indents were performed to allow easierestimation of the sample modulus, as detailed below, in the same locations asinelastic deformation was induced during the 500-nm indents. Indentation wascontrolled at a constant indentation strain rate of 0.05 s , where strain rate3s defined as the loading rate divided by the load. Once the maximum depthwas reached, the indenter was immediately unloaded at the same rate until aload of 1.5 mN was reached. At this point, the indenter load was held constantand the indenter position was monitored to assess any thermal drift associatedwith temperature changes inside the indenter housing. Throughout each indent,we recorded the indenter load, displacement, and contact stiffness (via CSM),although contact stiffness was not continuously measured during unloading.In the second area, we imposed two arrays of 9 × µ mgrid. For this data set, six loading cycles were performed at each grid point.On the first cycle, the load was increased until a load of 5 mN was reached, atwhich point the indenter was unloaded to 1.5 mN. The loading was then repeatedfive more times following the same procedure but progressively increasing themaximum load to 9, 19, 38, 75, and 150 mN on each successive cycle. Load-ing and unloading were controlled at constant rate of 1.5 mN/s, and a holdwas performed on the final unload to assess thermal drift as described above.Throughout each set of cycles, we recorded the indenter load and displacementbut did not continuously record the contact stiffness. Spherical nanoindentation has been a popular characterization technique fora wide variety of materials. As opposed to indentation with sharp tips (e.g.,Berkovich), spherical indentation benefits from (1) an initial contact that ispurely elastic, (2) analytical solutions for the stress and strain distributionsduring elastic loading, and (3) an easily identifiable transition between elasticand plastic deformation (e.g.,
Basu et al. , 2006;
Field and Swain , 1993;
Angkerand Swain , 2006).The basic configuration of spherical indentation is described in Figure 1.The mechanics of a spherical contact were originally derived by
Hertz (1882)and are typically presented as P = 43 E eff R / h e , (1)where P is the load on the contact, E eff is the effective modulus of the contact, R eff is the effective radius of curvature of the indenter, and h e is the elasticportion of the indentation depth. Assuming the contact is totally elastic, h e isequal to the total indentation depth, h t . The projected area of contact (Figure1) is defined by the contact radius, a = (cid:112) R eff h e , (2)and the contact stiffness, S = dP/dh e , is therefore S = 2 aE eff . (3)In general, h e , is only directly measurable if the indentation is known (or as-sumed) to be entirely elastic, which is typically the case for unloading segments.4SM works by superimposing a small, high-frequency oscillation on top of theprimary loading, which effectively consists of many elastic unloading segmentsand allows S = dP/dh e to be continually measured.Much effort has been put into using the above relationships to produce stress-strain curves from indentation data ( Herbert et al. , 2001;
Bushby , 2001;
Pathakand Kalidindi , 2015). We follow the method reviewed by
Pathak and Kalidindi (2015), in which the indentation stress is defined as the load over the contactarea, σ = Pπa , (4)and the indentation strain is defined as compression of an idealized, cylindricalzone of radius a and height 3 πa/ (cid:15) = 43 π h e a . (5)These definitions are designed to ensure that the initial elastic segments of theresultant stress-strain curves are in agreement with the elastic modulus of thesample.In our single indents in the first area, we record data for P , h t , and duringloading, S via the CSM. Key unknowns are therefore R eff and E eff . The radiusof the indenter tip, R i , is determined through calibration indents on elasticstandards with known modulus (e.g., fused silica) and is related to the effectiveradius by R − = R − + R − , where R s is the radius of curvature of the samplesurface (normally taken to be infinity). With known R eff , we then find thevalue of E eff that best fits a segment of our data shortly after contact thatwe assume is fully elastic. E eff is related to the sample modulus by E − =(1 − ν ) /E i + (1 − ν ) /E s , where E i is the elastic modulus of the diamondindenter, and E s is the elastic modulus of the sample. E s is essentially theYoung’s modulus of the sample in the direction of loading, although the elasticanisotropy tends to be underestimated in highly anisotropic materials (e.g., Kumamoto et al. , 2017). With R eff and E eff known, equations (3), (4), and (5)can be used to generate stress-strain curves.As will be described below, the unloading portions of stress-strain curves areuseful in interpreting our results on antigorite. We did not collect CSM dataduring unloading (as per standard operating methods), which means we do nothave continuous measurements of S to estimate a during unloading. However, acan still be estimated from equation (2) as long as R eff is known. Unfortunately,if inelastic deformation has occurred, R s will no longer be infinite due to theresidual impression that has formed. Therefore, we estimate R eff at the endof loading by fitting equation (1) to an initial segment of the unloading dataand assuming the sample modulus is unchanged. We then use this new value of R eff and equation (2) to estimate the contact radius, stress, and strain duringunloading.For cyclical indentation experiments in the second area, we do not have anyCSM data and therefore cannot estimate the contact area, stress, or strain formost of the deformation path. However, we can measure S at the beginning5 amplesurface total indentationdepth, h Idealized zone of primary deformation contact radius, a diamondsphere Figure 1: Schematic diagram of spherical indentation. The sample surface(black) is deflected by a spherical indenter tip (red). The gray region depictsthe cylindrical region assumed to be the primary region undergoing deformationand used to calculate stress and strain.of each unloading cycle and obtain an estimate of a through equation (3), andtherefore get a single measurement of stress and strain for each cycle. Weestimate the elastic modulus of the sample in the same manner as describedabove. In addition, we are interested in the energy budget during each loadingcycle. The total energy input into the system is simply the integral of the load-displacement curve. We specifically look for differences between the energyrecovered during unloading and the energy input on subsequent reloading tothe same load and depth. We calculate the energy difference for this portionof each loading cycle and normalize that value by the volume of the idealizedcylinder of deformation (Figure 1) represented by the contact radius measuredduring the initial unloading. Subsequent to indentation, we characterized the microstructure in the vicinityof indentation arrays using scanning electron microscopy (SEM) and electronbackscatter diffraction (EBSD). Regions of interest were mapped with eitheran FEI Quanta 650 FEG E-SEM housed at the University of Oxford or with aPhilips XL-30 housed at the University of Utrecht, both equipped with an Ox-ford Instruments Nordlys-Nano EBSD camera and AZtec 3.3 acquisition soft-ware. EBSD data were acquired with an accelerating voltage of 30 kV andcurrents on the order of 10 nA. Diffraction patterns were obtained with either2x2 or 4x4 binning of pixels of the EBSD detector. Diffraction patterns wereindexed by comparison to a match unit based on a crystal structure modified6rom
Capitani and Mellini (2006). The α angle is very sensitive to compositionand was therefore adjusted by modifying the length of the a lattice parameterto 80 based on preliminary tests to achieve optimal indexing.The electron beamwas rastered across the sample using step sizes of 0.2–0.4 µ m.To index antigorite, we used custom-built match units derived from thediffraction analyses of Capitani and Mellini (2006). Rates of successful indexingwere approximately 75%. In post-processing, isolated individual pixels with noorientation relationship to surrounding pixels were removed. Pixels that werenot indexed were filled with the average orientation of neighboring pixels ifthey had 5 or more indexed nearest neighbors. The resulting EBSD maps arepresented in Figure 2, with the numbering of individual indents annotated forreference.Individual indents were also imaged with SEM. We collected secondary-electron images with the sample inclined 70 ◦ relative to the electron beam, as istypically used for EBSD mapping. Images are presented with a tilt correctionto account for foreshortening. Electron imaging in this configuration increasesshadowing around topographic features to emphasize surface characteristics. Examples of mechanical data from three indents from the arrays of single indentsare presented in Figure 3. Data from initial, shallow indents are presented inred. Load is presented as a function of the total indentation depth (top row ofFigure 3), and shallow indents demonstrate that loading and unloading pathsare identical, indicating purely elastic behavior. These load-depth data arepresented as stresses and strains in the bottom row of Figure 3. Red curves arelinear in these plots, again indicative of linear elastic behavior, and are parallelto black dashed lines, which indicate the best-fit elastic moduli. Values of themeasured elastic moduli are discussed in section 3.3.Data from the second array of deeper indents are presented in blue. Loadand depth data exhibit different loading and unloading paths with residual in-dentation on the order of 100 nm, indicating appreciable inelastic deformation.Stress-strain curves demonstrate that the loading path departs from linear elas-ticity at a distinct yield point. Values of the measured yield stresses are dis-cussed in section 3.3. Most loading curves exhibit strain hardening after yield.Load-depth curves also exhibit multiple, near-instantaneous bursts of displace-ment, often referred to as “pop-ins”. Several larger pop-ins are indicated withblack arrows in Figure 3. Pop-ins typically only occur after yield, although theydo occasionally coincide with yield. In stress-strain curves, pop-ins appear asbursts of strain at constant stress, immediately followed by a stress drop alonga path matching the elastic modulus.A notable feature of these indents is the upward curvature of the load-displacement curves during unloading. Although some upward curvature is7 μ m s i ng l e i nden t scyc li c a l i nden t s band contrast IPF coloringband contrast IPF coloring IPF colorscalerelative to normal to section μ m Figure 2: EBSD maps of indent locations. The top row presents maps of thefirst area in which single indents were placed. The bottom row presents mapsof the second area in which cyclical indents were placed. Band contrast mapsare presented with white labels to indicate the indent numbering scheme. Addi-tional maps are presented colored according to an inverse pole figure (IPF) forthe direction normal to the sample surface (i.e., the direction of indentation).Colored maps are transparently overlain on top of band contrast maps. Blacklines indicate calculated grain boundaries.8
Strain
Strain
Strain S t r e ss ( G P a ) Load ( N ) Indentation depth (nm×100) Indentation depth (nm×100) Indentation depth (nm×100) single pop inat yield pop inafter yield e l a s t i c un l oad i ng . % r e c o v e r y e l a s t i c un l oad i ng . % r e c o v e r y pop inafter yield pop inafter yield e l a s t i c un l oad i ng . % r e c o v e r y be s t - fi t e l a s t i c m odu l u s be s t - fi t e l a s t i c m odu l u s Figure 3: Mechanical data from single indents in first area of interest. Threeindent locations are presented with increasing degrees of inelastic strain recoveryduring unloading. The indentation direction relative to the crystal orientaitionis given in Figure 7. Red curves correspond to totally elastic indents to 100nm depth. Blue curves correspond to indents to 500 nm depth. In the toprow, black dashed lines indicate predicted elastic unloading. In the bottom row,the black dashed lines indicate the best-fit elastic modulus during loading. Avariety of pop-in events are indicated by black arrows.expected due to the non-linear nature of Hertzian contacts (Equation 1), theobserved curvature is often more pronounced than expected. As described inSection 2.2.2, we fit an elastic unloading curve to the initial segment of unload-ing data (see black dashed lines in the top row of Figure 3). The predictedindentation depth at which the load is zero represents an estimate of the in-elastic indentation depth, and the predicted amount of recovered displacementrepresents the elastic indentation depth, he, at the end of loading. In some cases,the extrapolated unloading curve reasonably matches the unloading data, forwhich the unloading portion of the stress-strain curves is a straight line witha slope matching the elastic modulus (Figure 3, indent .2 Cyclical indents
Examples of mechanical data from two cyclical indents are presented in Figure4. These data sets exhibit similar characteristics to those observed in singleindents. The first two or three cycles exhibit reversible load-displacement paths,indicating purely elastic behavior. Cycles to larger load amplitudes exhibitresidual displacements, indicating the onset of inelastic behavior. These higher-amplitude cycles are also often characterized by pop-ins and significant plastic-strain recovery during unloading.We further characterize the mechanical behavior of cyclical indents by an-alyzing the energy budget throughout the series of loading cycles. Progressiveloading cycles input increasingly larger amounts of energy. For smaller ampli-tude cycles, nearly all the energy is recovered on unloading, again characteristicof elastic deformation. For larger amplitude cycles, only a fraction of the inputenergy is recovered during unloading. Notably, the unloading path is differentfrom the subsequent reloading path up until the load equals the maximum loadof the previous cycle. At this point, the load-displacement curves for unloadingand subsequent reloading coincide again. We calculate the difference betweenthe amount of energy that is recovered on unloading and the amount of energyinput on the subsequent reloading. The data used for this calculation are pre-sented in red in Figure 4. We normalize this dissipated strain energy by thevolume of deforming material under the indent inferred from the calculated con-tact radius (blue squares in Figure 4). For cycles that are near totally elastic,the energy difference is approximately 10 kJ/m , which represents the smallestmagnitudes resolvable by this technique. Progressively larger amplitude cy-cles result in increasing differences in dissipated strain energy, with maximumobserved values near 10 kJ/m . To aid our interpretation of the micromechanical behavior of antigorite, wefurther consider the mechanical data in the context of the crystallographic ori-entation at the location of each indent. As an initial assessment, we collectedsecondary-electron images of several residual indents to examine the surface ex-pression of antigorite deformation, as illustrated in Figure 5. For imaging, wechose several of the deepest indents, including an indent used for calibratingthe indent location, since these indents exhibit the most visible features. In allcases, residual indents are characterized by significant crack formation along theantigorite basal plane. Cracks appear to be primarily mode II with shear offsetsvisible at the sample surface.Additional images of individual indents mapped with EBSD are presentedin Figure 6. Forescatter images again reveal the presence of shear cracks atthe surface and parallel to the basal plane. Band-contrast maps reveal thatdiffraction patterns were degraded within the indent, presumably associatedwith surface damage due to the indentation process. Some additional degrada-10
Load ( N ) Time (s) E ne r g y i npu t ( kJ ) -11 Load ( N ) Time (s) E ne r g y i npu t ( kJ ) -11 cyclical Indentation depth ( μ m) Indentation depth ( μ m) D i ss i pa t ed s t r a i n ene r g y ( kJ / m ) cycle 6 cycle 6cycle 6 cycle 65 555 4 444 3 332 221 1 D i ss i pa t ed s t r a i n ene r g y ( kJ / m ) Figure 4: Mechanical data from cyclical indents in second area of interest. Theindentation direction relative to the crystal orientatiton is given in Figure 8.Segments highlighted in red indicate unloading and reloading paths that shouldbe identical if the deformation is totally elastic. Two indent locations are pre-sented, one with a small difference between loading and unloading paths (left)and one with a large difference between loading and unloading paths (right).The top row presents load as a function of total indentation depth. The bottomrow presents the total energy input as a function of time (black curves). Bluesquares indicate the difference between the energy recovered during unloadingand the energy input on subsequent reloading. An example of this differenceis indicated by the blue dashed lines. Individual cycles are numbered. Blackarrows indicate larger pop-ins. 11 μ m15 μ m 15 μ m g r a i n b o u n d a r y cyclical Figure 5: Secondary-electron images of individual indents. The sample is tiltedat 70 ◦ to electron beam and tilt correction is applied. The maximum depthof indentation is given for each image. Two images are from cyclical loadingindents, and their numbers correspond to numbering in Figure 2. The third isfrom a calibration indent that was taken to a much greater depth than indentsin the two primary regions of interest.tion is associated with shear cracks and scratches. Importantly, there is littleto no degradation surrounding the indent, contrary to the expectation if signifi-cant crystal-plastic deformation had occurred. Similarly, there is no measurabledistortion of the crystal lattice outside the residual impression, as revealed bythe maps of local misorientation. The magnitude of observed misorientations ison the order of the noise level for this type of measurement in antigorite.We further characterize the role of crystal orientation in the indentation be-havior of antigorite by plotting mechanical data in the crystal reference frame.A series of inverse pole figures (IPFs) are presented for single indents (Figure7) and cyclical indents (Figure 8). For single indents, we investigate the mea-sured elastic modulus, the yield stress, the flow stress at 10% strain, and themagnitude of inelastic-strain recovery. We compare the measured elastic mod-ulus to the Young’s modulus measured by Bezacier et al. (2010) using Brillouinspectroscopy. Although
Marquardt et al. (2015) suggested that
Bezacier et al. (2010) may have mistakenly switched the stiffnesses along the [100] and [010]axes, this difference has little effect on our analysis, and we use the originallyreport values. Our measured elastic moduli range from 74 to 132 GPa, andrelative to the crystal orientation, these values are generally intermediate to theextremes of the previously published elasticity tensor. This reduced anisotropyin our data is characteristic of spherical indentation (
Kumamoto et al. , 2017),which induces a variety of out of plane stresses that result in strains in othercrystallographic directions. However, some anisotropy is still evident in theother measured parameters presented in Figure 7. The lowest yield stresses andlowest stresses at 10% strain tend to be associated with indents nearly paral-lel to the basal plane. Furthermore, the highest magnitudes of inelastic-strainrecovery also tend to be associated with indents nearly parallel to the basalplane.Cyclical indents also exhibit a similar dependence of mechanical behavioron crystallographic orientation. Figure 8 presents a series of IPFs with Young’s12 ore-scattered electron Local misorientation C y c li c a l C y c li c a l Band contract μ m10 μ m Figure 6: EBSD data local to the residual impressions of two cyclical indents.Images are presented from fore-scatter detectors, band contrast of EBSD pat-terns, and local misorientation. White dashed lines represent the approximateextent of the residual impression. Local-misorientation maps reveal relativelylittle signal, with the highest values occurring inside the residual impression andassociated with surface damage of the sample.13 oung's modulus min = 78 GPamax = 208 GPa maxmin
Yield stress min = 2.7 GPamax = 8.9 GPa
Stress at 10% strain min = 3.5 GPamax = 11.4 GPa
Plastic-strain recovery min = -4% GPamax = 23% GPa [100][001][010]
Indent locations
Figure 7: Mechanical behavior of antigorite from single indents as a functionof crystal orientation. Indentation directions are plotted in the crystal refer-ence frame using inverse pole figures (IPFs). Data points are colored accordingto the measured value of modulus, yield stress, stress at 10% strain, or themagnitude of plastic-strain recovery. Indent locations reference the numberingscheme presented in Figure 2. The background contouring for Young’s modulusis calculated using the elastic constants of
Bezacier et al. (2010).moduli and magnitudes of dissipated strain energy observed in cyclical indents.Similar to single indents, our measured values of Young’s modulus are generallyintermediate to the extreme values from previously published results for single-crystal antigorite. The exception, however, is the set of indents near parallelto [001], which is an orientation not sampled by single indents. This crystal-lographic direction is predicted to be the most compliant, and indents in thisdirection do tend to have the lowest values in our data set, matching publishedmagnitudes. Furthermore, the magnitude of dissipated strain energy tends tobe highest for indents parallel to the basal plane, and lowest for indents normalto the basal plane. This pattern is perhaps most distinct for intermediate am-plitude cycles (i.e., maximum loads of 19 or 38 mN). At the highest amplitudes,most indents exhibit magnitudes of dissipation near the maximum observed,although indents near perpendicular to the basal plane still exhibit the lowestvalues.
Our results provide insight on the mechanisms of deformation in antigorite. Thekey observations include: (1) shear cracks in Figure 5 appear to accommodatedeformation at the sample surface; (2) shear cracks are parallel to the basalplane, which is the dominant cleavage plane in antigorite; (3) the lack of crys-14 axmin
Indent locations [100][001][010]
38 mN max load 75 mN max load 5 mN max load 9 mN max load 19 mN max load Young's modulus min = 78 GPamax = 208 GPa
Dissipated strain energy (log scale) min = 10 kJ/m max = 10 kJ/m Figure 8: Mechanical behavior of antigorite from cyclical indents as a functionof crystal orientation. Indentation directions are plotted in the crystal referenceframe using inverse pole figures (IPFs). Data points are colored according to themeasured value of modulus and the magnitude of dissipated inelastic strain en-ergy. Magnitudes of dissipated energy are presented on a separate IPF for eachloading cycle, and the maximum load in that cycle is noted. Indent locationsreference the numbering scheme presented in Figure 2. The background con-touring for Young’s modulus is calculated using the elastic constants of
Bezacieret al. (2010). 15al distortion surrounding residual indents suggests a paucity of crystal-plasticdeformation; and (4) the IPFs in Figure 7 reveal that yield occurs most easilyfor indents parallel to the basal plane, while it is difficult to initiate yield inindents normal to the basal plane. Taken together, these observations suggestthat the basal plane in antigorite is weak and prone to shear microcracking, andthat slip along the basal plane is likely the dominant deformation process inour experiments. As demonstrated in Section 4.2, deformation by sliding shearcracks along basal planes is also compatible with the inelastic strain recoveryobserved in single indents (Figure 7) and the strain-energy dissipation observedin cyclical indents (Figure 8).Slip along basal planes in antigorite can occur in two, possibly nonexclusive,ways: (i) dislocation glide, and (ii) shear fracturing. By contrast with disloca-tion glide, shear fracturing implies bond breakage, delamination, and frictionalslip along (001) planes, with irreversible loss of cohesion between the slippedportions of the crystal. Our SEM observations indicate that shear crackingdid occur during indentation, but are not sufficient to rule out the activityof dislocations entirely. Previous work investigating the potential for plastic-ity in antigorite has suggested that dislocation glide would dominantly be on(001) or on conjugate planes that result in apparent slip on (001) (e.g.,
Amiguetet al. , 2014), and dislocation interactions are an often cited mechanism for thebuildup of backstresses and associated inelastic strain recovery. In addition,even if dislocation glide occurs during unloading, cracks can also form due tostress associated with dislocation interactions (
Kumamoto et al. , 2017). Thus,observations 1 and 2 listed above could potentially explained by dislocationactivity. However, dislocation activity during indentation tends to result ina halo of geometrically necessary dislocations and associated lattice distortionsurrounding residual indents (see Figures 4 and 5 in
Wallis et al. (2019)). Incontrast, Figure 6 reveals ano resolvable lattice misorientation or changes indiffraction-band contrast around indents, suggesting there is little to no dislo-cation accumulation. In addition, dislocation glide in antigorite is also assumedto be predominantly in the [100] direction, and although we observe weak basalplanes, we do not observe any directional dependence of the yield stress forindents within the basal plane.Based on the lack of evidence for dislocation activity, we suggest that ourresults are most consistent with deformation during unloading being accommo-dated primarily by sliding on shear cracks. Cracks are often observed aroundindents on brittle materials, but those are generally related to decompressionduring unloading. Cracks associated with unloading should be roughly parallelto the surface and normal to the primary tensile stresses. In contrast, the shearcracks we observed are normal to the surface and optimally oriented for shearduring loading (
Swain and Hagan , 1976).The observed pop-ins at or subsequent to yield provide some additional infor-mation on the defects available for inelastic deformation. As loading progressesduring spherical indentation, not only does the nominal stress increase, but sodoes the volume of stressed material. Because the stresses at which pop-insoccur are often stochastic in nature, pop-ins are commonly interpreted to re-16ect the point at which the stress field reaches an available defect source (e.g.,
Kumamoto et al. , 2017). Therefore, we suggest that the defects from whichshear cracks nucleate in our samples are of low enough density that they are notimmediately sampled. The idealized volume of deformation (Figure 1) at thepoint of initial pop-in is typically on the order of 1 µ m beneath our indents,suggesting that the initial flaw density is approximately 1 µ m − . Cyclical loading experiments provide estimates of the Young’s modulus and dis-sipated strain energy per cycle, both as a function of crystal orientation andthe stress amplitude (Figure 8). To further test our interpretation that inelas-ticity and anisotropy in antigorite is primarily caused by sliding motion in thebasal plane, we develop a simple two-dimensional analytical model that calcu-lates Young’s modulus, inelastic strain, and dissipated energy as a function ofstress. This “crack sliding” model is based on the previously derived formalismof
Kachanov (1982),
Nemat-Nasser and Obata (1988) and
Basista and Gross (1998) without crack growth. This formulation is a direct extension of the modelof
David et al. (2012) to triaxial stress and cyclic loading. However, here weconsider that all crack faces are initially in contact, an assumption that seemsreasonable for sliding along the basal plane in antigorite grains.The stress conditions during spherical indentation are best described bythose of proportional loading, as can be verified by examining the full Hertziansolutions for the stress field inside the loaded body (e.g.,
Ming and Haiying ,2016). Considering the area of primary deformation given in Figure 1, we findthat the average lateral stress r is proportional to the average axial stress σ , i.e., σ r = kσ , where the constant k is numerically found to be approximately equalto the Poisson’s ratio of the material. Taking ν = 0 .
26 for isotropic antigoriteat room temperature (
Bezacier et al. , 2010), k = 0 . A , of primary deformation (gray regionin Figure 1) to contain an array of N cracks, each of length 2 c with its normaloriented at an angle φ to the applied stress. For simplicity of analysis and avail-ability of analytical solutions, we assume that crack interactions are negligibleand that cracks are embedded into an isotropic solid characterized by a singlevalue of the Young’s modulus, E , and of Poisson’s ratio, ν .The essence of the model is that, for crack sliding to be initiated, the appliedshear stress must overcome a certain shear strength. Two different cases areconsidered: (1) “frictional” crack sliding, in which the resolved shear stresson a crack must exceed a normal stress-dependent, Coulomb-type frictionalresistance and (2) “cohesive” crack sliding, in which the resolved shear stresson a crack must exceed a constant yield stress. The cohesion term can arisefrom several possible physical mechanisms. One possibility is that the cohesioncorresponds to the Peierls stress for moving dislocations. However, cohesion17esulting from bond breakage across a fracture surface is more consistent withthe interpretations presented in Section 4.1. In both frictional and cohesivecases, as sliding proceeds, elastic energy is stored in the material surroundingthe cracks. This stored energy leads to the observed strain hardening afteryield. During unloading, this stored elastic energy provides a restoring forcethat promotes backsliding, which is initiated if the sum of the applied shearstress and the elastic restoring force overcomes the sliding shear strength in thereverse direction (e.g., Nemat-Nasser and Obata , 1988). The activation of slidingand backsliding at different applied stresses during loading and unloading resultsin dissipation of strain energy and produces hysteresis in stress-strain curves.Details of the model and derivations are given in the Appendix. We focushere on two key model outputs: the stress-dependent, effective Young’s modulus E and the dissipated energy per unit volume W during unloading from a maxi-mum stress σ ∗ and subsequent reloading to the same stress. For frictional cracksliding (case 1), the effective Young’s modulus once crack sliding is initiated isgiven by 1 E = 1 E [1 + π Γ sin(2 φ ) M L ] , (6)and the strain energy dissipated per cycle is expressed as W = σ ∗ E (cid:20) π Γ sin(2 φ ) M L (cid:18) − M L M U (cid:19)(cid:21) (7)where σ ∗ is the maximum stress, Γ is the crack density defined as Γ = N c /A . M L and M U are geometrical factors for loading and unloading, respectively,given by M L = (1 − k ) cos φ sin φ − µ [cos φ + k sin φ ] M U = (1 − k ) cos φ sin φ + µ [cos φ + k sin φ ] , (8)where k is the constant of proportional loading given above and µ is the fric-tion coefficient. For the case of cohesive sliding (case 2), the effective Young’smodulus once crack sliding is initiated is given by1 E = 1 E [1 + π Γ sin(2 φ ) M ] , (9)and the energy dissipated per cycle is expressed as W = 2 π Γ sin(2 φ ) τ y E (cid:18) σ ∗ − τ y M (cid:19) (10)if there is backsliding ( σ ∗ ≥ τ y /M ), where τ y is the constant sliding “yieldstress”, and M is a geometrical factor given by M = (1 − k ) cos φ sin φ. (11)We invert for model parameters by comparison to our experimental obser-vations of Young’s modulus and dissipated strain energy energy for five selected18ndents (Figure 9). We assume that all cracks of interest lie in the basal plane(i.e., the crack normal is parallel to [001]). The crack sliding model predictsthat sliding is optimal for cracks approximately at an angle of about 60 ◦ to theloading direction in the frictional sliding case, and an angle of 45 ◦ in the cohe-sive sliding case. We specifically investigate indents 27, 77, 85, and 86 ( φ = ◦ )and indent 80 ( φ = 52 ◦ ), and select cyclical loading data prior to the occurenceof any pop-ins events (if observed). For all indents, the Young’s modulus of theuncracked solid is taken to be E = 97 GPa, the isotropic Young’s modulus forantigorite at room temperature ( Bezacier et al. , 2010) We only need to invertfor two adjustable parameters in each of the models. For the frictional cracksliding model (case 1), the friction coefficient, µ , is imposed to be the same forall indents, and the crack density, Γ, varies among indent locations. For thecohesive crack sliding model (case 2), the strength, τ y , is imposed to be thesame for all indents, while the crack density, Γ, varies among indent locations.For the two selected crystal orientations, both models are able to jointly fitthe modulus deficit relative to the uncracked solid, and the stress-dependent dis-sipation of energy (Figure 9). For the frictional crack sliding model, the best fitis obtained with µ ≈ . τ y ≈ .
15 GPaand Γ in the range 0.1 to 0.4. The reasonable fits obtained with both modelsprovides further validation that sliding on shear cracks is a plausible mechanismresponsible for inelastic deformation in antigorite. However, a significant differ-ence between the two models is that the dissipated energy W is quadratic inapplied stress σ ∗ for the frictional crack sliding model, whereas W is linear inapplied stress σ ∗ for the cohesive crack sliding model (at high stress). A qualita-tive evaluation of Figure 9 suggests the quadratic nature of the frictional slidingcrack model is a better representation of the observations, although with theavailable data, we are currently unable to quantitatively discriminate betweenthe two hypotheses. A variety of deformation mechanisms have been previously proposed to operatein antigorite. Evidence for dislocation-dominated deformation is largely indi-rect, and most investigators infer that dislocations are the primary means ofdeformation based on observed CPOs (
Katayama et al. , 2009; van de Moort`eleet al. , 2010;
Padr`on-Navarta et al. , 2012;
Hirauchi and Katayama , 2013). In-terpretations of CPOs suggest (001) is the dominant glide plane and [100] is thedominant shear direction. However, microscopic evidence for dislocation activ-ity is inconclusive. Because of the modulated, wave-like nature of the layeringin the crystal structure (
Zussman , 1954), dislocations may be difficult to gen-erate in the antigorite. The unit cell dimension along [100] is extremely large(35–43 ˚ A ), which suggests that line energies for dislocations with [100] Burgersvectors would be high. Otten (1993) observed stacking defects in these modula-tions, often referred to as modulation dislocations, but it is unclear whether or19 W , d i ss i pa t ed s t r a i n ene r g y pe r cyc l e , kJ / m frictional modelcohesive modelStress, GPa010002000300040005000 0 0.5 1 1.5 2 2.5 3 Figure 9: Dissipated strain energy per cycle as a function of the maximum stressper cycle for five cyclical indents. Open squares are observations from indents,and full and dashed curves are best fits to observations using the frictional slidingand the cohesive sliding crack models, respectively. Colors and correspondingnumbers indicate the specific indent. Parameters values used in the fittingprocedure are discussed in the text.not these can be carriers of significant plastic deformation.
Amiguet et al. (2014)observed lattice distortion and kink bands in transmission electron microscope(TEM) images of experimentally deformed antigorite, which they interpreted asa result of dislocation activity. In their analysis, slip was interpreted to appearmacroscopically as having occurred on (001), but they suggest that this appar-ent slip plane was the result of simultaneous slip on (101) and (10¯1).
Auzendeet al. (2015) also observed structures in TEM images of naturally and experi-mentally deformed antigorite that they inferred to be built from dislocations.However, based on a range of microstructural observations, they argued thatdislocation activity is severely limited in favor of brittle processes. Similarly,recent in situ experiments by
Corder et al. (2018) did not reveal any dislocationactivity during deformation, instead observing delamination and fracture.Because of the apparent difficulty in activating dislocation slip systems, brit-tle processes dominate the deformation of antigorite under a wide range ofconditions. These brittle processes include cleavage opening near kinks, de-lamination, and shear microcracking. The link between kinking and cleavageopening in antigorite is supported by the correlation of microstructurally ob-served kinks and macroscopically observed brittle deformation (
Nicolas et al. ,1973;
Chernak and Hirth , 2010;
Auzende et al. , 2015;
Proctor and Hirth , 2016).However, recent deformation experiments conducted on aggregates of antigoriteat elevated pressure and room temperature reveal that kinking is localized tothe damage zone near fractures that formed during brittle failure (
David et al. ,20018). In fact, prior to macroscopic failure, the mechanical behaviour of antig-orite aggregates is marked by (1) mostly nondilatant deformation prior to failurein the brittle regime (
Escart´ın et al. , 1997;
David et al. , 2018), (2) significantshear dissipation and the absence of volumetric dissipation during cyclic load-ing at stresses below the brittle failure strength (
David et al. , 2018), and (3)the ubiquitous presence of shear microcracks oriented along the basal (cleavage)planes of antigorite in the ductile (semi-brittle) regime, preferentially orientatedat around 45 ◦ from the maximum compressive stress ( Escart´ın et al. , 1997).Taken together, these observations support the hypothesis that intragranularshear microcracking, equivalent to shear delamination along the cleavage plane,is a key deformation mechanism in antigorite, at least in the low-temperatureregime. Although shear delamination is analogous to, and can be produced by,dislocation glide in the basal plane (e.g.,
Stroh , 1954), delamination need not beaccommodated by dislocations in antigorite, where intracrystalline slip is likelyaccommodated by spatially extended defects like shear cracks.Our results from indentation experiments are consistent with this hypoth-esis, and the operation of shear cracks can explain both the deformation andmicrostructural data. Furthermore, since all of our experiments are confinedto small regions contained within single crystals, it is clear that shear crackingcan occur entirely within grains, suggesting it is possible for shear cracking inmacroscopic triaxial experiments to be dominantly intragranular.
In nature, for instance within subducting oceanic lithosphere, antigorite defor-mation occurs at significantly lower strain rates than in laboratory conditions,and in the presence of aqueous fluids. One key limitation of observations fromlaboratory deformation experiments is that the relatively fast strain rates (andlower temperature) imposed experimentally might limit the mobility of disloca-tions and thus favor shear cracks, whereas dislocation motion could possibly bedominant at much lower strain rates. The relevance of indentation data and ofthe shear-cracking mechanism is discussed here by comparison with experimen-tal data obtained on other silicate minerals, and by analyzing the compatibilityof shear crack-driven deformation with observed CPOs in naturally deformedantigorite.Results from indentation tests conducted on quartz (e.g.,
Masuda et al. ,2000), olivine (e.g.,
Evans and Goetze , 1979;
Kumamoto et al. , 2017), and mica(e.g.,
Basu et al. , 2009) have all demonstrated the activity of one or more dis-location slip systems, even at room temperature, and the rheological behaviourof these minerals determined by indentation tests is consistent with dislocation-dominated deformation mechanisms. Therefore, the self-confining and grain-scale nature of nanoindentation tests typically limits the occurrence of tensilemicrocracks and instead favors intracrystalline flow mechanisms, even in strongsilicate minerals. In contrast, our nanoindents in antigorite do not exhibit de-tectable dislocations, and our data appear to be well explained by an intragran-ular shear cracking mechanism. These observations are strong indicators that21ntigorite does not have any easily activated dislocation glide systems.While crystal-scale observations of naturally deformed antigorite reveal indi-rect signs of dislocation activity in the basal plane (stacking disorder, as reportedby
Auzende et al. (2015)), they also indicate recrystallisation due to dissolution-precipitation processes, which suggests that antigorite typically does not deformby pure climb-limited dislocation creep, even under natural deformation condi-tions. Since dissolution-precipitation processes require the presence of aqueousfluids and therefore the existence of some minimal porosity in the rock, evidencefor such processes in naturally deformed antigorite supports the hypothesis thatsome degree of brittle deformation occurs under natural conditions.Field observations of CPOs in naturally deformed antigorite-rich rocks havecommonly been interpreted as the signature of dislocation creep (e.g., van deMoort`ele et al. , 2010;
Padr`on-Navarta et al. , 2012), with strong implications interms of the rheology of the material. However, intragranular shear crackingby delamination of the basal planes is not necessarily inconsistent with thedevelopment of CPOs, notably by the formation and progressive rotation ofantigorite blades by sliding along (001). Such a mechanism for CPO formationhas recently been suggested for calcite deformed in the brittle regime (
Demurtaset al. , 2019).At this stage, the mechanism of shear cracking inferred from laboratorydeformation experiments at relatively fast strain rates cannot be completelyruled out at geological strain rates, even if CPOs are observed. It remains tobe determined how shear cracking interacts with other deformation processesactivated at lower strain rates and in the presence of fluids, but we emphasisethat intragranular delamination and sliding is likely a significant deformationprocess under natural conditions.
We explored the micromechanics of antigorite using instrumented nanoindenta-tion. Spherical indentation was performed on natural samples of antigorite intwo separate arrays. In the first, a single loading cycle was performed at eachindent location. In the second, multiple loading cycles were performed at eachlocation, with each cycle to a greater maximum load than the previous. Singleindents revealed initial elastic loading, a distinct yield point, and strain harden-ing. During unloading, more strain is recovered than predicted for purely elasticloading. Similarly, cyclical indents recover more strain energy than expected forpurely elastic unloading, which was confirmed by examining the difference inenergy during unloading and subsequent reloading. This range of mechanicalbehavior was also observed to be dependent on crystallographic orientation, withlower yield stresses and increased amounts of strain and strain-energy recoveryfor indents parallel to the antigorite basal plane.We interpret these mechanical data to reflect sliding on shear cracks along thebasal plane. This interpretation is supported by microstructural observationsof shear cracks in and surrounding residual indents at the sample surface. We22urther argue against the activity of dislocations because there is no measurablelattice distortion associated with dislocation accumulation around indents, andthere is no apparent preference for sliding direction in the basal plane thatmight be associated with a Burgers vector. Based on this interpretation, wedevelop a new microphysical model for an isotropic rock containing an arrayof sliding cracks that predicts the effective Young’s modulus and dissipationof strain energy as functions of the maximum stress. The model is able tosuccessfully explain both the modulus deficit and the dissipated strain energymeasured on many indents, with reasonable values of crack density and eitherfriction coefficient (frictional sliding case) or cohesive strength (cohesive slidingcase).
Appendix: calculation of Young’s modulus anddissipated strain energy for a rock containing anarray of sliding cracks, under proportional load-ing
The “effective sliding stress”, τ eff , driving frictional sliding on loading a crackoriented at angle φ to the applied stress is the difference between the resolvedshear stress on the crack, τ , and the frictional resisting stress, τ f = µσ n : τ eff = τ − µσ n , (12)where σ n is the resolved normal stress and µ is the friction coefficient. Underproportional loading ( σ r = kσ , see Section 4.2), the projection of the appliedstress onto a given crack gives τ eff = ( σ − kσ ) cos φ sin φ − µ [ σ cos φ + kσ sin φ ] = σM L (13)where M L is a function of the crack orientation expressed as M L = (1 − k ) cos φ sin φ − µ [cos φ + k sin φ ] . (14)Under the convention that positive stresses are compressive, the condition forsliding on the crack is τ eff >
0. It is easily demonstrated that an increment ofinelastic strain due to an array of sliding cracks is proportional to an incrementof the effective sliding stress (
Nemat-Nasser and Obata , 1988), d(cid:15) i = π Γ sin (2 φ ) E dτ eff = π Γ sin (2 φ ) M L dσ (15)where Γ is the two-dimensional crack density and E is the Young’s modulusof the solid. The effective Young’s modulus, E , is obtained by recalling thatan increment of total strain is the sum of the elastic strain increment and the23nelastic strain increment expressed above. Therefore, for an array of frictionalsliding cracks during loading,1 E = 1 E (cid:104) π Γ sin(2 φ ) M L (cid:105) . (16)During unloading, the effective stress driving backsliding is the difference be-tween the effective sliding stress τ ∗ eff at the maximum stress (which is the restor-ing force accumulated during loading), and the joint action of the frictionalresisting stress and the resolved applied shear stress, which both act againstbacksliding ( Nemat-Nasser and Obata , 1988). τ eff is written as τ eff = τ ∗ eff − ( τ + τ f ) = τ ∗ − τ ∗ f − τ − τ f , (17)and by projecting the applied stress onto a given crack, τ eff can then be expressedas τ eff = ( σ ∗ − σ )(1 − k ) cos φ sin φ − µ ( σ ∗ + σ )(cos φ + k sin φ ) , (18)where the condition for backsliding to occur is, similarly to loading, τ eff > E = E if σ ≥ σ ∗ ( M L /M U ) (19)if there is no backsliding, or by1 E = 1 E (cid:104) π Γ sin(2 φ ) M U (cid:105) (20)if there is backsliding. M U is again a function of the crack orientation expressedas M U = (1 − k ) cos φ sin φ + µ [cos φ + k sin φ ] , (21)and σ ∗ ( M L /M U ) is identified as the “backsliding yield stress” for a given crackorientation.By using the relations given above for the effective Young’s modulus duringloading and unloading, integration of the stress-strain relations allows derivationof the dissipated energy per cycle, W , as a function of the maximum stress σ ∗ , W = σ ∗ E (cid:104) π Γ sin(2 φ ) M L (cid:16) − M L M U (cid:17)(cid:105) . (22) For the purely cohesive crack case, crack sliding during loading occurs if theresolved shear stress simply exceeds a stress-independent cohesive resistance or“crack yield stress”, denoted by τ y . The “effective sliding stress” driving slidingis then simply written as τ eff = τ − τ y . (23)24ollowing the same conventions as applied for the frictional sliding case above, τ eff under proportional loading is expressed as τ eff = σ (1 − k ) cos φ sin φ − τ y = M σ − τ y , (24)where M = (1 − k ) cos φ sin φ .The condition for sliding is again that τ eff >
0. Using similar considerationsas described above for the frictional sliding case, the effective Young’s modulusfor loading of an array of cohesive sliding cracks is1 E = 1 E (cid:104) π Γ sin(2 φ ) M (cid:105) for σ > τ y /M (25)if there is sliding or E = E (26)if there is no sliding, where τ y /M is the “yield stress” for a given crack orienta-tion.During unloading, by analogy to the frictional sliding case, the “effectivebacksliding stress” on a given cohesive crack is τ eff = τ ∗ eff − ( τ + τ y ) = τ ∗ − τ − τ y (27)which yields τ eff = ( σ ∗ − σ )(1 − k ) cos φ sin φ − τ y . (28)The condition for backsliding is again that τ eff >
0. The effective Young’smodulus for unloading is then E = E for σ ≥ σ ∗ − τ y /M (29)if there is no backsliding or1 E = 1 E (cid:104) π Γ sin(2 φ ) M (cid:105) (30)if there is backsliding, where σ ∗ − τ y /M is again identified as the “backslidingyield stress” for a given crack orientation.As described above, the dissipated energy per cycle for the cohesive slidingcase is given by W = 2 π Γ sin(2 φ ) τ y E (cid:16) σ ∗ − τ y M (cid:17) (31)if there is backsliding during unloading, which corresponds to σ ∗ ≥ τ y /M . Acknowledgements
We are grateful to Kathryn Kumamoto and Christopher Thom for useful dis-cussions of nanoindentation. Luiz Morales provided useful insight into EBSDindexing of antigorite. This work was supported by the Natural EnvironmentResearch Council through grant NE/M016471/1 to L. Hansen and N. Brantut,and by the European Research Council under the European Unions Horizon2020 research and innovation programme (project RockDEaF, grant agreement eferences
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